Computational Modelling of Concrete Structures [1 ed.] 0415584795, 9780415584791, 9780203848333

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COMPUTATIONAL MODELLING OF CONCRETE STRUCTURES

PROCEEDINGS OF EURO-C 2010, ROHRMOOS/SCHLADMING, AUSTRIA, 15–18 MARCH 2010

Computational Modelling of Concrete Structures Editors

Nenad Bi´cani´c Department of Civil Engineering, University of Glasgow, Scotland, UK

René de Borst Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands

Herbert Mang Institute for Mechanics of Materials and Structures, Vienna University of Technology, Vienna, Austria

Günther Meschke Institute for Structural Mechanics, Department of Civil Engineering, Ruhr University, Bochum, Germany

CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2010 Taylor & Francis Group, London, UK Typeset by Vikatan Publishing Solutions (P) Ltd., Chennai, India Printed and bound in Great Britain by Antony Rowe (a CPI Group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: CRC Press/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com - www.taylorandfrancis.co.uk - www.balkema.nl ISBN: 978-0-415-58479-1 (Hbk) ISBN: 978-0-203-84833-3 (eBook)

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Table of contents

Foreword

XI

Keynote lectures Modeling of concrete creep and hygrothermal deformations, and computation of their structural effects Z.P. Bažant & Q. Yu

3

Alternate approaches to simulating the performance of ductile fiber-reinforced cement-based materials in structural applications S.L. Billington

15

Recent developments on computational modeling of material failure in plain and reinforced concrete structures A.E. Huespe, J. Oliver, G. Díaz & P.J. Sánchez

31

Concrete under various loadings, way to model in a same framework: Damage, fracture and compaction J. Mazars, F. Dufour, C. Giry, A. Rouquand & C. Pontiroli

45

Upscaling quasi-brittle strength of cement-based materials: A continuum micromechanics approach B. Pichler & C. Hellmich

59

C-Crete: From atoms to concrete structures F.-J. Ulm, R.J.-M. Pellenq & M. Vandamme

69

Constitutive and multi-scale modelling Pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix H. Aljewifi, B. Fiorio & J.L. Gallias

77

How to enforce non-negative energy dissipation in microplane and other constitutive models for softening damage, plasticity and friction Z.P. Bažant, J.-Y. Wu, F.C. Caner & G. Cusatis

87

A multiscale approach for nonlinear hysteretic damage behaviour of quasi-brittle disordered materials J. Carmeliet, S. Mertens & P. Moonen

93

Modeling of reinforced cementitious composites using the microplane damage model in combination with the stochastic cracking theory R. Chudoba, A. Scholzen, R. Rypl & J. Hegger

101

A statistical model for reinforced concrete bond prediction Z. Dahou, Z.M. Sbartaï, A. Castel & F. Ghomari

111

Introduction of an internal time in nonlocal integral theories R. Desmorat & F. Gatuingt

121

Elastoplastic constitutive model for concretes of arbitrary strength properties G. Etse & P. Folino

129

Properties of concrete: A two step homogenization approach E. Gal & R. Kryvoruk

137

V

Stress state influence on nonlocal interactions in damage modelling C. Giry, F. Dufour, J. Mazars & P. Kotronis A model for the uniaxial tensile behaviour of Textile Reinforced Concrete with a stochastic description of the concrete material properties J. Hartig & U. Häußler-Combe Multi-axial modeling of plain concrete structures based on an anisotropic damage formulation M. Kitzig & U. Häußler-Combe Determination of cement paste mechanical properties: Comparison between micromechanical and ultrasound results S. Maalej, Z. Lafhaj & M. Bouassida The simulation of microcracking and micro-contact in a constitutive model for concrete I.C. Mihai & A.D. Jefferson Simulations of dynamic failure in plain and reinforced concrete with regularized plasticity and damage models J. Pamin, A. Winnicki & A. Wosatko

145

153 163

173 179

187

Micromechanical approach to viscoelastic properties of fiber reinforced concrete V.F. Pasa Dutra, S. Maghous & A.C. Filho

197

Boundary and evolving boundary effects in non local damage models G. Pijaudier-Cabot & F. Dufour

207

Homogenization-based model for reinforced concrete E. Rumanus & G. Meschke

217

Development of constitutive model of shear stress transfer on concrete crack surface considering shear stress softening Y. Takase, T. Ikeda & T. Wada

225

Microplane approach for modeling of concrete under low confinement N.V. Tue & J. Li

233

Meso- and macroscopic models for fiber-reinforced concrete S.M. Vrech, G. Etse, G. Meschke, A. Caggiano & E. Martinelli

241

Gradient damage model with volumetric-deviatoric split A. Wosatko & J. Pamin

251

Advances in numerical methods Continuous and discontinuous modeling of cracks in concrete elements J. Bobi´nski & J. Tejchman

263

Topological search of the crack path from a damage-type mechanical computation M. Bottoni & F. Dufour

271

On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches J. Cervenka & V. Cervenka

281

Lattice Discrete Particle Model for Fiber reinforced concrete (LDPM-F) with application to the numerical simulation of armoring systems G. Cusatis, E.A. Schauffert, D. Pelessone, J.L. O’Daniel, P. Marangi, M. Stacchini & M. Savoia

291

Nonlocal damage based failure models, extraction of crack opening and transition to fracture F. Dufour, G. Pijaudier-Cabot & G. Legrain Convergence aspects of the eXtended Finite Element Method applied to linear elastic fracture mechanics W. Fleming & D. Kuhl

VI

301

309

Applicability of XFEM for the representation of crack bridge state in planar composite elements J. Jeˇrábek, R. Chudoba & J. Hegger

319

Localization properties of damage models M. Jirásek & M. Horák

327

Numerical multiscale solution strategy for fracturing of concrete Ł. Kaczmarczyk, C.J. Pearce & N. Bi´cani´c

337

A 3D lattice model to describe fracture process in fibrous concrete J. Kozicki & J. Tejchman

347

Limit analysis of 3D reinforced concrete frames K.P. Larsen, P.N. Poulsen & L.O. Nielsen

355

The role of domain decomposition techniques for the study of heterogeneous quasi-brittle materials O. Lloberas Valls, D.J. Rixen, A. Simone & L.J. Sluys Modelling cohesive crack growth applying XFEM with crack geometry parameters J.F. Mougaard, P.N. Poulsen & L.O. Nielsen

363 373

Strong discontinuities, mixed finite element formulations and localized strain injection, in fracture modeling of quasi-brittle materials J. Oliver, I.F. Dias & A.E. Huespe

381

Model for the analysis of structural concrete elements under plane stress conditions: Finite element implementation M. Pimentel & J. Figueiras

391

A partition of unity finite element method for fibre reinforced concrete F.K.F. Radtke, A. Simone & L.J. Sluys

401

A discrete cracking model for sequentially linear analysis A.V. van de Graaf, M.A.N. Hendriks & J.G. Rots

409

Relations between structure size, mesh density, and elemental strength of lattice models M. Voˇrechovský & J. Eliáš

419

Time-dependent and multi physics phenomena Prediction of the permeability of damaged concrete using a combined lattice beam-crack network approach M. Abreu, J. Carmeliet & J.V. Lemos

431

Modelling the CaO hydration in expansive concrete B. Chiaia, A.P. Fantilli, G. Ferro & G. Ventura

441

A two-scale approach for fluid flow in fracturing porous media R. de Borst, J. Réthoré & M.-A. Abellan

451

A probabilistic approach for modelling long-term behaviour and creep failure of a concrete structure subjected to calcium leaching T. de Larrard, F. Benboudjema, J.B. Colliat, J.M. Torrenti & F. Deleruyelle

461

A coupled transport-crystallization FE model for porous media H. Derluyn, R.M. Espinosa-Marzal, P. Moonen & J. Carmeliet

471

A numerical model for early age concrete behavior G. Di Luzio, L. Cedolin & G. Cusatis

481

Hygro-mechanical model for concrete specimens at the meso-level: Application to drying shrinkage A.E. Idiart, C.M. López & I. Carol

VII

487

Comparison of approaches for simulating moisture content changes in concrete A.D. Jefferson & P. Lyons Modeling of Chloride and CO2 transport in intact and cracked concrete in the context of corrosion predictions of RC structures M. Kemper, J.J. Timothy, J. Kruschwitz & G. Meschke

497

503

Thermal activation of basic creep for HPC in the range 20°C–80°C W. Ladaoui, T. Vidal, A. Sellier & X. Bourbon

513

Modelling of the THM behaviour of concrete at the macroscopic and mesoscopic scale T.T.H. Le, H. Boussa & F. Meftah

519

Application of enhanced elasto-plastic damage models to concrete under quasi-static and dynamic cyclic loading I. Marzec & J. Tejchman

529

Propagation of cracks and damage in non ageing linear viscoelastic media Nguyen Sy Tuan, L. Dormieux, Y. Le Pape & J. Sanahuja

537

Micromechanical modelling of concrete V.P. Nguyen, M. Stroeven & L.J. Sluys

547

3D finite element analysis of concrete under impact load J. Ožbolt, V. Travaš & I. Kožar

553

Investigation into the form of the load-induced thermal strain model C.J. Robson, C.T. Davie & P.D. Gosling

563

Development of service life model CHLODIF++ I. Stipanovi´c Oslakovi´c, D. Bjegovi´c, D. Mikuli´c & V. Krsti´c

573

Modelling of concrete structures Some remarks concerning the shear failure in prestressed RC beams B. Belletti & C. Damoni

581

Verification of experimental tests on roller bearings by means of numerical simulations S. Blail & J. Kollegger

591

Numerical crack modelling of tied concrete columns C. Bosco, S. Invernizzi & G. Gagliardi

595

Numerical study of a massive reinforced concrete structure at early age: Prediction of the cracking risk of a massive wall L. Buffo-Lacarrière & A. Sellier

603

Numerical modelling of failure mechanisms and redistribution effects in steel fibre reinforced concrete slabs L. Gödde & P. Mark

611

Transverse rebar affecting crack behaviors of R.C. members subjected to bending D. Han, M. Keuser & L. Ruediger

623

A numerical method for RC-boxgirders under combined shear bending and torsion U. Häußler-Combe

629

Computation of optimal concrete reinforcement in three dimensions P.C.J. Hoogenboom & A. de Boer

639

Sequentially linear modelling of local snap-back in extremely brittle structures S. Invernizzi, D. Trovato, M.A.N. Hendriks & A.V. van de Graaf

647

Simulation of masonry beams retrofitted with engineered cementitious composites M.A. Kyriakides, M.A.N. Hendriks & S.L. Billington

655

VIII

Setting and loading process simulation of Push-in anchor for concrete Y.-J. Li, N. Chilakunda & B. Winkler

665

Nonlinear FE modelling of shear behaviour in RC beam retrofitted with CFRP Y.T. Obaidat, O. Dahlblom & S. Heyden

671

Material optimization for textile reinforced concrete applying a damage formulation E. Ramm & J. Kato

679

A multifibre approach to describe the ultimate behaviour of corroded reinforced concrete structures B. Richard, F. Ragueneau & Ch. Crémona

689

Simulation of shear load behavior of fifty year old post-tensioned concrete bridge girders A. Schweighofer, M. Vill, H. Hengl & J. Kollegger

699

Multi-scale modelling of concrete beams subjected to three-point bending Ł. Skar˙zy´nski & J. Tejchman

703

Experimental and numerical analysis of reinforced concrete corbels strengthened with fiber reinforced polymers R.A. Souza

711

Computational modeling of the behaviour up to failure of innovative prebended steel-VHPC beams for railway bridges S. Staquet & F. Toutlemonde

719

Numerical investigations of size effects in notched and un-notched concrete beams under bending E. Syroka, J. Bobi´nski, J. Górski & J. Tejchman

729

FE modeling and fiber modeling for RC column failing in shear after flexural yielding K. Tajima, N. Shirai, E. Ozaki & K. Imai

737

Numerical simulation and experimental testing of a new bridge strengthening method W. Traeger, J. Berger & J. Kollegger

749

Failure studies on masonry infill walls: Experimental and computational observations K. Willam, B. Blackard & C. Citto

757

Numerical study on mixed-mode fracture in LRC beams R.C. Yu, L. Saucedo & G. Ruiz

767

Hazards, risk and safety (fire, blast, seismicity) Numerical investigation of damage and spalling in concrete exposed to fire C.T. Davie & H.L. Zhang

775

Numerical simulation of slender structures with integrated dampers P. Egger & J. Kollegger

785

Textile reinforced concrete sandwich panels: Bending tests and numerical analyses J. Finzel & U. Häußler-Combe

789

Structural behavior of tunnels under fire loading including spalling and load induced thermal strains T. Ring, M. Zeiml & R. Lackner

797

Numerical assessment of the failure mode of RC columns subjected to fire S. Sere˛ga

805

Ultimate load analysis of a reactor safety containment structure B. Valentini, H. Lehar & G. Hofstetter

815

Author index

821

IX

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Foreword

The long standing EURO-C conference series returned again this spring to the mountains of Austria, with the conference held in Schladming, Salzburgerland from the 15th to 18th March 2010. These proceedings comprise over 90 invited and contributed papers presented at the conference, which keeps its main focus and its traditional format in line with earlier conferences in the series (Innsbruck 1994, Badgastein 1998, St Johann im Pongau 2003, Mayrhofen 2006). The EURO-C series has emerged as a joint undertaking following an increase of research in computational mechanics of concrete generated by the ICC 1984 conference held in Split, Croatia, the SCI-C 1990 conference in Zell am See, Austria and the two Concrete Mechanics Colloquia, held in Delft, The Netherlands in 1981 and 1987. Once again we express our sincere gratitude to the members of the International Advisory Panel (Zdenek Bažant, Sarah Billington, Alberto Carpinteri, Tony Jefferson, Milan Jirásek, Karen Lundgren, Koichi Maekawa, Jacky Mazars, Javier Oliver, Jerzy Pamin, Gilles Pijaudier-Cabot, Marco di Prisco, Ekkehard Ramm, Jan Rots, Tadahiko Tanabe, Franz-Josef Ulm and Kaspar Willam) for their invaluable support and considerable effort in reviewing around 130 abstracts. EURO-C conference series keeps its tradition of a very rigorous reviewing process, thereby ensuring the high quality of presented papers. In order to reflect the current advances in computational modelling of concrete and concrete structures, conference papers are grouped into five distinct, yet strongly related, sections – (A) Constitutive and Multiscale Modelling of Concrete (B) Advances in Computational Modelling, (C) Time Dependent and Multiphysics Problems (D) Modelling of Concrete Structures and (E) Hazard, Risk and Safety. We hope that the EURO-C 2010 Conference proceedings will continue to provide a valuable referential source for an up-to-date information and scientific debate on the research advances in computational modelling of concrete and concrete structures, as well as on their application and relevance to the structural engineering practice. Nenad Bi´cani´c René de Borst Herbert Mang Günther Meschke Glasgow, Eindhoven, Vienna, Bochum January 2010

XI

Keynote lectures

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modeling of concrete creep and hygrothermal deformations, and computation of their structural effects Zdenˇek P. Bažant & Qiang Yu Northwestern University, Evanston, IL, USA

ABSTRACT: This study discusses a physically based formulation of material model for concrete creep and shrinkage, and presents an effective computational approach for structural analysis of creep and shrinkage effects. As an instructive case study, excessive deflections of a prestressed box girder bridge of world-record span, which was built in Palau in 1977 and collapsed after remedial prestressing in 1996, is investigated. A new version of the step-by-step computational algorithm, based on the continuous retardation spectra of compliance curves for different ages at loading, is implemented as a driver program for repeated use of ABAQUS for three-dimensional analysis. The excessive creep deflections are studied by finite element analysis, and their causes are identified. They reveal a need to improve the current standard recommendations of engineering societies. A limited improvement can be achieved by statistical analysis of the existing database from worldwide testing. However, a major improvement will require adopting a model based on the theory of physical processes of creep and shrinkage in the nano-porous structure of cement gel. A model based to a large extent on such a theory is model B3. Its basic features are reviewed. 2 EXCESSIVE DEFLECTIONS AND COLLAPSE OF KB BRIDGE IN PALAU

1 INTRODUCTION Portland cement concrete is a rather unusual porous material, characterized by both capillary porosity and sub-capillary nano-porosity. The physical and chemical processes in the nano-pores (or gel pores) are believed to be the cause of complex creep properties (Bažant 2001), very different from those of other viscoelastic materials. The nano-porosity of cement gel has intriguing consequences whose mathematical formulation material model B3 will be reviewed. This model rests on the microprestress-solidification theory (Bažant & Prasannan 1989a,b, Bažant et al. 1997, Bažant et al. 1997), which is a theory that has achieved a unified description of the known creep properties of concrete, including the long-term aging after the hydration process has terminated, the Pickett effect, and the transitional thermal creep. The purpose of this paper is to summarize (based on a report by Bažant et al. 2007) the lessons from excessive long-time deflections of a record-setting segmentally erected prestressed concrete box girder, discuss the importance of selecting the correct material model for creep and shrinkage, critically comment on the statistical validation of the material model, and review the theoretical physical basis of a realistic material model.

The Koror-Babeldaob (KB) Bridge (Yee 1979, Mc-Donald et al. 2003, Burgoyne & Scantlebury 2006) was built in 1977 in the Republic of Palau, situated in the tropical western Pacific. It connected the islands of Koror and Babeldaob, the former containing the airport and the latter the country capital. As shown in Figure 1a, the main span consisted of two symmetric simultaneously erected cantilevers connected at mid-span by a horizontally sliding hinge. Each cantilever consisted of 25 cast-in-place segments of depths varying from 14.17 m (46.5 ft.) to 3.66 m (12 ft.). The segmental erection of the box girder took about 6–7 months. At the time of completion, the main span of 241 m (790 ft.) set the world record for a prestressed concrete box girder. In design, the long-term deflection of the bridge was expected to remain in the tolerable range with the final mid-span sag ranging from 0.46 to 0.58 m (18.2 to 23 in.). In the early years, the deflections were benign but then accelerated unexpectedly. After 18 years, the deflection increase measured since the installation of the mid-span hinge that joined the opposite segmentally erected cantilevers reached 1.39 m (54.6 in.) and kept growing (Fig. 1b). If compared to the design camber, an additional creep deflection of 0.22 m (9 in.)

3

analyses, which can be most effectively carried out by a commercial general purpose finite element program. The program ABAQUS was selected. Similar to other general purpose programs, ABAQUS is not designed to handle concrete creep. Therefore, a user subroutines that implements the creep integration in time and calls ABAQUS in each time step has been formulated. Another material subroutine was developed to describe the constitutive model for concrete creep and shrinkage. If all the equations of the constitutive model, converted to a rate form, are put in a proper incremental form, an incremental elastic problem with eigenstrains (or inelastic strains) is obtained for each subsequent time step. The incremental elastic moduli matrices are generally anisotropic and are different for each integration point of each finite element of each time step. So are the eigenstrains, which are non-isotropic. The incremental elastic relations can be obtained from the differential equation of Kelvin chain model according to the exponential algorithm which is unconditionally stable (Jirasék & Bažant 2002). When the non-aging Kelvin chain model is applied to model B3, one and the same relaxation spectrum, determined in advance, can be used for all the time steps (Bažant & Xi 1995). Since the cantilevers are symmetric, only one half of the bridge is modelled and simulated. A computational model with 5036 8-node hexahedral elements for concrete and 6764 bar elements for prestressed and unprestressed steel bars is built in ABAQUS; see Figure 2. 316 tendons (Dywidag alloy bars), which are densely packed in up to 4 layers within the top slab, run through the pier. The jacking force of each tendon is about 0.60 MN (135 kips), and the total jacking force at the pier is about 190 MN (42660 kips). The segmental construction sequence, the moving of the formwork traveller and the process of pre-stressing are all reproduced in the simulation by utilizing the functions provided by ABAQUS. There is no material model in the ABAQUS material library to capture the characteristics of the creep

Figure 1. (a) KB Bridge after construction; (b) sag at the mid-span; (c) collapse of KB Bridge.

was accumulated earlier during segmental erection, and so the total deflection was 1.61 m (63.6 in.). The serviceability of the bridge was severely impaired by the excessive deflection, and so a retrofit was carried out in 1996. The mid-span hinge was removed and external prestress within the box was added. Unfortunately, three months later the bridge collapsed (with 2 fatalities and many injuries); see Figure 1c. After legal litigation in which the deflections and collapse remained unexplained, all the data were sealed in perpetuity. However, on November 6, 2007, the 3rd Structural Engineers’ World Congress in Bangalore endorsed a resolution (proposed by the first writer, with the support of many experts) which called, on the grounds of engineering ethics, for the release of all the technical data necessary for analyzing major structural collapses, including the bridge in Palau. The resolution was circulated to major engineering societies. Two months later, the Attorney General of Palau gave his permission to release the technical data. This made it possible to analyze the deflections. 3 NUMERICAL MODELING OF KB BRIDGE Since structural creep analysis can be broken down to a series of many incremental elastic finite element

Figure 2. 3-dimensional model of KB Bridge built in ABAQUS.

4

compliance functions J (t, t ′ ) of the creep and shrinkage prediction models used in practice (t = duration of a unit uniaxial stress and t ′ = age when this sustained stress is applied). Therefore, the appropriate material model has been developed and supplied to ABAQUS. Given the fact that no severe cracking damage was found in the box girders and the observed cracks were sparsely distributed, the concrete of KB Bridge has been assumed as isotropic, characterized by a time-independent Poisson’s ratio ν = 0.21 (JICA 1990). As a compromise between simplicity and accuracy, the creep in concrete is generally considered to follow aging linear viscoelasticity, which implies the principle of superposition in time. The direct application of this principle gives the stress-strain relation in the form of a history integral. However, for the sake of efficiency in large-scale computer analysis, it is advantageous to avoid computation of history integrals. This is made possible by converting the compliance function to an equivalent rate-type form, which has here been based on the Kelvin chain model (Fig. 3). In model B3, which is based on the solidification theory, the creep is defined for a non-aging constituent of growing volume (≈ cement gel) (Bažant & Baweja 1995, Bažant & Baweja 2000), and in that case the conversion of the compliance function to a rate-type creep law is particularly easy. This can be done according to the solidification theory, in which the aging is taken into account by means of volume growth of the solidifying component, and by a gradual increase with age of the flow term viscosity. This makes it possible to use a non-aging compliance function for the solidifying component, for which one can uniquely determine a continuous retardation spectrum by a simple explicit formula (Widders’s formula) for inversion of Laplace transform (Bažant & Xi 1995). The parameters of the Kelvin chain model are in this case constant (i.e., non-aging) and are simply obtained as a discrete representation of the continuous spectrum. For example, if J (t, t ′ ) = φC(t − t ′ ) where φ is a non-aging factor, the continuous retardation spectrum can be

expressed as: Aµ = φ(t ′ )L(τµ ) ln 10(log τµ ) (−kτµ )k (k) C (kτµ ) k→∞ (k − 1)!

L(τµ ) = − lim

(1) (2)

where τµ is µth retardation time, k is a positive integer, and C (k) represents the kth order derivative of function C (K ≤ 3 is usually sufficient). In Figure 3, the non-aging spectrum of the basic creep of model B3 is plotted. Note the spectral value Aµ does not diminish as τµ increases. The reason is that the basic creep according to model B3 is unbounded. For empirical models, such as those of ACI (2008), CEB (1990) (or ‘fip’ 1999), JSCE (1991) and GL (Gardner 2000, Gardner & Lockman 2001), the creep analysis is slightly more complicated since the aging is not separated from the compliance function. Therefore, compliance curves that change with the age at loading must be used. Such a situation was handled in the 1970s by considering the retardation (or relaxation) spectrum to be age dependent, and the age dependence of Kelvin chain elastic moduli was identified from tests data by simultaneous fitting of creep data for various ages at loading. However, the identification problem turned out to be ill-conditioned and the resulting moduli as functions of age non-unique. As a new simpler approach, one exploits the fact that, during a short time step, compliance function may be considered as age-independent. The continuous retardation spectrum can thus be obtained easily from Eqs. (1) and (2) corresponding to the loading age at each time step, but the spectrum is different for each different age. This continuous retardation spectrum is then approximated by a set of discrete spectral values Aµ (µ = 1, 2, 3, . . . ), one set for each time step. These spectral values are then used in the individual time steps of the exponential algorithm based on Kelvin chain. No continuous function for Aµ (t) need be identified and used. A surface describing the compliance function of ACI for different ages is plotted in Figure 3, with the x-axis representing the retardation time, τµ , and the y-axis showing age t ′ . The disappearance of Aµ at large retardation time τµ is due to the fact that the ACI compliance function is (incorrectly) assumed to be bounded. After obtaining the Kelvin chain moduli, the exponential algorithm, which enables more data and more efficient creep computation, is implemented; see the flowchart in Figure 4. In this algorithm, the stress is assumed to vary linearly in each time step. The initial time steps since the bridge closing at mid-span were 0.1, 1, 10 and 100 days. After that, the time step was kept constant at 100 days up to 19 years.

Figure 3. Kelvin chain model and spectrums of non-aging and aging models.

5

Figure 5. Brooks. Figure 4.

B3 model compared with 10-year creep data by

Flowchart of algorithm for creep calculation.

4 COMPARISON OF PREDICTIONS BASED ON DIFFERENT MODELS For the concrete creep and shrinkage properties, models B3, GL, ACI, CEB (identical to ‘fib’ 1999) and JSCE are considered and predictions compared. Model B3, in contrast to the others, does not necessarily give a unique prediction because, in addition to concrete design strength, it involves several input parameters depending on the composition of concrete mix, on which there exists no information. These parameters can be set to their default values, but they can also be varied over their plausible range in order to ascertain the range of realistic predictions. The findings are as follows:

4.

1. For model B3, reasonable values of input parameters can be found to match all the measured deflections (as well as the few existing creep tests of duration longer than 10 years; Brooks 1984,2005), while for other models the maximum deflections cannot be approached and the recorded shape of laboratory creep curves cannot be reproduced; see Figure 5. 2. The 18-year deflections calculated by threedimensional finite elements according the ACI, CEB, JSCE and GL models (with consideration of differential shrinkage and drying creep) are about 66%, 62%, 46% and 53% less than the observed values. On the other hand, the observed deflections are closely matched by calculation on the basis of model B3, provided that model B3 is calibrated to match the 10-year creep tests of Brooks. If the default parameters are used in model B3, its predicted deflection is 43% less than the observed value; see Fig. 6. 3. The bridge in Palau is unique in that the pre-stress loss in grouted tendons was measured by stress

5.

6.

7.

6

relief tests. Three sections of each of three tendons were bared, gages installed, tendons cut and the stress determined from the shortening of the tendon. The result was an average prestress loss of 50%, much larger than the prestress loss of 22% assumed in design. Comparisons of the predictions of various models with the measured prestress loss are shown in Figure 7. The deflection is highly sensitive to prestress loss because it represents a small difference of two large numbers (deflections due to self-weight, and to prestress). According to the ACI, CEB, JSCE and GL models, the prestress loss obtained by the same finite element code is, respectively, about 56%, 52%, 42% and 46% smaller than the measured prestress loss, respectively. Contrarily, the prestress loss predicted by model B3 is very close to what measured in the strain relief tests. Model B3 is in agreement with the measurements if three-dimensional finite elements with step-bystep time integration are used to calculate both the deflections and the prestress losses, and if the differences in shrinkage and drying creep properties caused by differences in slab thickness, temperature and damage are taken into account; Figures 6, 7. The shear lag cannot be neglected. What makes its effect large is that it increases the downward deflection due to self-weight much more than the upward deflection due to prestress. The traditional beam-type analysis, in which the creep and shrinkage properties are assumed to be uniform throughout each cross section, gives grossly incorrect predictions for deflections and prestress loss. These differences can be captured closely by model B3, but poorly or not at all by other models. The compliance function must be determined separately for each slab in the cross section, depending on its thickness and humidity exposure (as well as temperature).

Figure 6. Deflections obtained by B3, ACI, JSCE, CEB and GL models compared with measured. Left: linear scale; right: logarithmical scale.

Figure 7. Prestress loss obtained by B3, ACI, JSCE, CEB and GL models compared with the strain relief tests. Left: linear scale; right: logarithmical scale.

calibration by a larger database, with a rational statistical calibration procedure compensating for the database bias for short times and for small specimen sizes; c) Adjustment of the model to the shape of creep curves observed in individual longtime tests (5 to 30 years), which is obscured when the database is considered only as a whole; and d) rational physical basis of the model. 2. Three-dimensional analysis of deflections and prestress loss, which is much more realistic than the beam bending analysis, especially because it can capture different effects of shear lag on the downward deflection due to self-weight and on the upward deflection due to prestress. 3. Realistic representation of nonuniform shrinkage and drying creep properties in the cross sections, caused by the effect of different wall and slab thicknesses on the shrinkage and drying creep

8. The results agree with a recent statistical study of Bažant & Li (2009a), which shows that model B3 gives smaller errors in comparison to a comprehensive database than do the ACI, CEB and GL models (whose errors can be of the order of 100%). 9. Creep and shrinkage are notorious for their relatively high random scatter. Therefore, the design should be based on the 95% confidence limits. These limits can be calculated by Latin hypercube sampling of the input parameters analysis. The reasons why model B3 performs better than the other models can be summarized as follows: 1. A significantly higher long-time creep in model B3, compared to the ACI, CEB and GL models. There are four causes of that: a) theoretical advances on nano-porous materials during the last three decades, incorporated in model B3; b) Model

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where q1 , q2 , q3 , q4 and n are primary material parameters to be determined from concrete composition, if known; t0 is the age at the start of drying; ζ = t − t ′ = stress duration; C(t, t ′ ) is the compliance of basic creep (independent of moisture loss); and Cd (t, t ′ , t0 ) is the compliance function of additional creep caused by the drying process (or moisture content change). In model B3, the drying creep and shrinkage strain ǫsh can be written as:

rates and half-times, as well as by the differences in permeability due to temperature differences (Bažant & Kaplan 1996) and cracking (Bažant et al. 1987). 4. A larger number of input parameters in model B3, which include the water-cement ratio and aggregate-cement ratio. If these parameters are not specified, their variation allows exploring a greater range of responses, compared to the ACI, CEB and GL models. These models are inflexible because of missing these input parameters, and thus provide a unique response for a given design strength of concrete.

 ′ Cd (t, t ′ , t0 ) = q5 (e−8H (t−t0 ) − e−8H (t −t0 ) ) ǫsh (t, t0 ) = −ǫsh∞ (1 −

Similar conclusions have been obtained by studying the observations in some other large-span prestressed box girders on which sufficient data could be accessed, particularly the Tsukiyono Bridge, Koshirazu Bridge, Konaru Bridge and Urado Bridge in Japan, and the Dˇecín Bridge in Czech Republic. Together with those bridges, the experience with the KB Bridge in Palau reveals that the current design practice based on empirical creep and shrinkage models and beam-type analysis may lead to dangerous underestimates of the longterm deflections of prestressed large box girders and prestress losses. Improvements in the currently used creep and shrinkage prediction models are, therefore, required. The improvement needs to based on better statistical evaluation of the existing data. However, the existing data are insufficient by far. Consequently, a physical justification based on the processes in the nano-porous microstructure is vital.

= −ǫsh∞ (1 −

− t0 )  (t − t0 )/τsh

h3e ) tanh

(5)

(6)

where q5 is material constant; he is environmental relative humidity; ǫsh∞ is ultimate shrinkage strain determined by the concrete property; τsh = shrinkage halftime = kt (ks D)2 , where Ks is the cross section shape factor, D is the effective cross section thickness, kt is a parameter determined by diffusivity (or permeability); and H (ˆt ) = 1 − (1 − he )S(ˆt ). There is a mathematical reason—the self similarity in time—for which the unbounded individual physical processes involved in creep and shrinkage should be described by power laws, and those that approach a finite bound by decaying exponentials. Eqs. (3) to (6) give a simplified model for practice which approximately describes the mean behavior of a cross section of a slab of thickness D when the bending moment is negligible, i.e., for essentially central in-plane loading. For a point-wise constitutive model, which can be applied only when the thickness of each slab is subdivided into many finite elements, the solidificationmicroprestress theory can be characterized by the nonlinear first-order differential equation for the microprestress in the nano-porous structure:

5 MODEL BASED ON PHYSICALLY BASED THEORY AND STATISTICS Among the existing models, model B3 is the only one with some theoretical foundation—the solidification theory and the theory of microprestress buildup and relaxation in the nano-porous structure of cement gel. The theoretical basis include the following phenomena: 1) process without a characteristic time and asymptotic limit; 2) solidification process; 3) microprestress relaxation; 4) activation energies of creep and hydration; 5) diffusion of water; 6) arguments based on capillarity, surface tension, and free and hindered adsorbed water; 7) cracking or damage; and 8) rate of chemical processes causing autogeneous volume change. After certain idealizations and simplifying hypotheses, consideration of the phenomena led to a compliance function of the form: J (t, t ′ ) = q1 + C(t, t ′ ) + Cd (t, t ′ , t0 ) q  n−1 q4 ˙ t ′ ) = 2 + q3 nζ + C(t, tm 1 + ζn t

h3e )S(t

˙ s˙ + as2 = c|T lnh|

(7)

The microprestress s initially produced by the disjoining pressures in nano-pores and by volume mismatch of various constituents relaxes with time at a decaying rate over many years. If there is drying or wetting, or a change of temperature T , the fields of pore humidity h (and temperature) must be calculated for each time step from the nonlinear diffusion equation of drying, and the humidity and temperature change at each integration point of each finite element causes a change in the right-hand side, which produces a buildup of microprestress, and thus an acceleration of creep. ˙ at each material The free shrinkage rate ǫ˙sh = ksh h, point, is proportional to the local pore humidity rate. This is much simpler than the approximate expression in Eqs. (5) and (6) for the overall average shrinkage of

(3) (4)

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the entire cross section, which includes the approximate effect of the self-equilibrated nonuniform shrinkage stresses and of the cracking that they produce. Although the form of the constitutive law is physically based, the model B3 parameters have to be obtained by statistical calibration from test data, same as the parameters of purely empirical models. Therefore, statistical study on a large database, is important. Can the best creep and shrinkage model be identified purely by a standard statistical regression of the existing database? The answer is—no (unless a statistically perfect database were available, which is not the case). In the last several decades, numerous tests have been conducted around the world to investigate the concrete creep and shrinkage. In 1978, the first comprehensive database, consisting of about 400 creep tests and approximately 300 shrinkage tests, was compiled at Northwestern University (Bažant & Panula 1978). A slight expansion based on this database led to what known as RILEM database (M¨uller & Hilsdorf 1990, M¨uller 1993, M¨uuller et al. 1999), which was widely used to calibrate various creep and shrinkage models. Now a significantly expanded database, named the NU-ITI Database, is assembled at Northwestern, comprising 621 creep tests and 490 shrinkage tests (Bažant & Li 2009b). Although there are thousands of test points in this expanded database, the statistical comparison is useless since the scatter is huge and very little difference can be seen among different models if the entire database is treated as a statistical population (ensemble statistics). The reasons are three: 1) The data points are not sampled uniformly in terms of the loading or drying duration, age of loading, start of drying, and cross section thickness; 2) The scatter caused by various concrete compositions is enormous; and 3) The trends of creep and shrinkage evident from individual tests are obfuscated. Most laboratory test data on creep and shrinkage have durations ≤5 years, while the data for 10 to 30 years are very scant (micrometer in size), is one source in the delay of shrinkage and of the part of creep due to drying. But normal structures are so thick that the rate of shrinkage and drying creep of a structure is controlled by diffusion of moisture through structure. This diffusion is highly nonlinear, with the permeability and diffusivity

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decreasing about 20-times as the h decreases from 95% to 65%. Despite this nonlinearity, the drying (weight loss) as well as the shrinkage and drying creep at constant environmental humidity exhibit three basic properties of diffusion processes (Bažant & Xi 1994): a) The drying half-times scale as a square of structure size D (or cross section thickness) (Fig. 8b, c); b) They begin as a square-root function of time (Fig. 8b); and c) They approach the final equilibrium state as a decaying exponential of a power of time (Fig. 8c). Asymptotic matching of properties b) and c) for shrinkage leads to the tanh-function of the duration (Fig. 8c), normalized by half-time τsh which is proportional to D2 . A similar function should apply to drying creep compliance J . Since concretes have thousands of different compositions, it is important to conduct, before design, shorttime tests of creep and shrinkage, so as to update the parameters of creep prediction equations (Fig. 8c, d). For creep, which has no upper bound, the update is simple: It suffices to scale the entire creep curve up or down (Fig. 8d). Not so for shrinkage. There is a trap in that the half-time value and the final asymptotic shrinkage value cannot be predicted from short-time shrinkage measurements alone. An important idea in 1995, which still has not penetrated the practice, is that these values can be easily predicted (by a certain linear regression) if the loss of weight (due to loss of pore water) is simultaneously recorded, and if the total water loss upon heating of the shrinkage specimen at the end of the short-time tests (typically of 1 month duration) is determined. In this manner one can avoid wrong extrapolations shown at lower left and lower right of Fig. 8d. A convenient property of concrete creep is that, in absence of microcracking, it is linear (within the range of service stresses allowed in structure), i.e., follows the Volterra principle of superposition in time. However there is major complication—the aging. It causes that the creep properties vary significantly with time (at a decaying rate), because of two phenomena: a) the chemical hydration of cement, which lasts at normal temperature for about a year; and b) the gradual relaxation of microprestress in the nano-structure (possibly combined with polymerization of C-S-H), which continues for decades. The use of microprestress appears indispensable as a unifying concept that captures simultaneously the drying creep (Pickettt effect), the transitional thermal creep and the long-time aging. However, it is not yet clear what is its precise physical mechanism in the C-S-H. Researches at Northwestern University (by H. Jennings) and at M.I.T. (by F.-J. Ulm) may clarify this fundamental question. The aging causes some thermodynamic restrictions on the formulation of the constitutive equation, which have often been ignored. The easiest way to formulate

a thermodynamically legitimate constitute law with aging is to take into account the fact that the newly produced hydrates are deposited on the surface of capillary voids in an unstressed state. Thus aging may be considered as a consequence of the growth of volume fraction v(t) of a certain solidifying constituent (approximately the solid cement gel), which itself is considered to have non-aging properties (Fig. 8e). This represents the essence of the solidification theory, which simplifies the analysis of aging creep in structures. The absence of aging for the solidifying constituent makes it possible to co-opt the theory of viscoelasticity of polymers, including the rate-type approximation of the non-aging compliance function by a Kelvin (or Maxwell) chain model (Fig. 8e) with an easily obtainable retardation spectrum. This spectral approach avoids the use of history integrals. Also, it is amenable to introducing the effects of variable pore humidity h, temperature T (Bažant et al. 2004) and the softening (Bažant 1995) due to distributed cracking (which cannot be introduced into a Volterra history integral). Although the Kelvin chain model corresponds to a discrete retardation spectrum, it is better to start with a continuous spectrum. Such a spectrum can be identified from the compliance curve by means of Widder’s formula, known from the viscoelasticity of polymers. Discretization of this formula at discrete times, which are best chosen to be spaced by decades in the log-time scale, yields then the Kelvin chain moduli. A direct least-square determination of the Kelvin chain moduli is possible and was used in the 1970s, but it suffers from ill-conditioning and near non-uniqueness. The continuous retardation spectrum is unique, and thus eliminates the problem with ill-conditioning. Besides, its use is computationally more efficient, as demonstrated by the analysis of the KB Bridge in Palau. The effect of temperature rise on creep and shrinkage is twofold: a) Acceleration of creep, which can be described as a decrease of viscosities of Kelvin units (Fig. 8e) according to the Arrhenius equation, characterized by an activation energy of creep; and b) at the same time an increase in the rate of aging, characterized again by another Arrhenius equation with a different activation energy that governs the rate of hydration. The latter reduces creep. Thus the temperature effect on creep is a competition of accelerating and decelerating influences. For shrinkage it is similar. Temperature increase accelerates diffusion but accelerated hardening hinders shrinkage. One consequence of ignoring the thermodynamic restrictions due to chemical aging is the condition of non-divergence of the compliance function J (t, t ′ ), defined as the strain at time t due to a unit stress applied at age t ′ ; see Fig. 8e. If this condition is ignored, the principle of superposition can produce, after unloading, a recovery reversal (Fig. 8f), and, the relaxation

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Figure 8.

Basic physical phenomena of creep and shrinkage in concrete.

such compliance functions are still used in the design codes of ACI, JSCE and CEB (or ‘fib’). The effect of aging on the conventional elastic modulus E (which is really the inverse of compliance function for the duration of about 15 min.) can be greatly simplified by noting that the compliance

curves at constant strain can cross into the opposite sign. These phenomena violate the second law of thermodynamics if realistic Kelvin or Maxwell chain models with aging are considered (violation, though, need not occur for some contrived rheologic models that give unrealistic compliance curves). Nevertheless,

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Therefore, design measures that mitigate deflections should be sought. To have a better estimate of long-term deformation in bridge design, the following is necessary: a. Use a realistic creep and shrinkage model (such as B3 model built on solidification theory); b. update the model by tests of creep, shrinkage and water loss; c. analyze the structure three-dimensionally, in time steps; and d. calculate the 95% confidence limits instead of mean response.

Figure 9. Creep curves for different ages at loading and the asymptotic modulus E0 .

Practical measures can also be undertaken to mitigate the deflections, for example: a. no mid-span hinge; b. choice of a concrete with low creep, especially a low long-time creep; c. use of a higher level of prestress, even so high that an upward deflection is predicted; d. giving concrete more time to gain strength before prestressing; e. using stiffer girders of low slenderness; and f. installing some empty ducts into which additional tendons can be placed later.

curves for different ages approximately intersect at one point when plotted versus the 0.1-power of stress duration (Fig. 9). The intersection point gives the socalled asymptotic modulus E0 , which corresponds to a fictitious extrapolation of creep curve to the load duration of about 1 ns. With this concept, the compliance function automatically includes the dependence of the conventional elastic modulus E(t) on age t (Fig. 9).

Two more points deserve mention: In continuous girders, the deflections are particularly sensitive to tendon layout and can be reduced by the right layout. But a layout benefiting the stress state can at the same time be harmful from the deflection viewpoint. Second, although the absence of a mid-span hinge reduces deflections, excessive deflections may occur, as documented by the Dˇecˇ ín in Bridge.

7 CONCLUSIONS If the advances in research of creep and shrinkage in last three decades were incorporated into the practice, and if the models and methods available today were used at the time of design, the observed deflections of the KB Bridge in Palau would have been expected. This would have forced a radical change of design of this bridge, precluded the fatal retrofit, and thus prevented the final catastrophe. Numerical simulation of the KB Bridge deflections shows that the main causes of error in design are as follows:

ACKNOWLEDGEMENT Financial support from the U.S. Department of Transportation through Grant 0740-357-A222 from the Infrastructure Technology Institute of Northwestern University is gratefully appreciated. Thanks are due to Dr. Khaled Shawwaf of DSI, Inc., Bolingbrook, Illinois, for providing valuable information on the analysis, design and investigations of the bridge in Palau.

1. incorrect compliance and shrinkage functions which are empirically based and rely on a database overwhelmingly dominated by short-term tests (≤3 years duration) and small size tests; 2. neglect of differential shrinkage and drying creep compliance due to a) different thicknesses of cross section walls; b) different temperatures; and c) cracking; 3. lack of three-dimensional analysis. The simplified one- or two-dimensional model leads to inaccurate calculation of shear lags in the slabs and webs; 4. absence of a statistical estimate of the 95% confidence limits; 5. inaccurate time integration and neglect of temperature and cracking; 6. lack of updating of the material parameters by short-time tests. 7. errors in top-bottom differences (a 5% error in prestress can cause a 50% error in the total deflection).

REFERENCES ACI Committee 209. 2008. Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete ACI Guide 209.2R-08, Farmington Hills. Bažant, Z.P. 1995. Creep and damage in concrete Materials Science of Concrete IV, J. Shalny and S. Mindess, Eds., Am. Ceramic. Soc., Westerville, OH, 355–389. Bažant, Z.P. 2000. Criteria for rational prediction of creep and shrinkage of concrete. Adam Neville Symposium: Creep and Shrinkage—Structural Design Effects, ACI SP-194,

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A. Al-Manaseer, ed., Am. Concrete Institute, Farmington Hills, Michigan, 237–260. Bažant, Z.P. 2001. Creep of concrete. Encyclopedia of Materials: Science and Technology, K.H.J. Buschow et al., eds., Elsevier, Amsterdam, Vol. 2C, 1797–1800. Bažant, Z.P. and Baweja, S. 1995. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. Materials and Structures 28, 357–367. Bažant, Z.P. and Baweja, S. 2000. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. Adam Neville Symposium: Creep and Shrinkage—Structural Design Effects, ACI SP-194, A. Al-Manaseer, ed., 1–83. (update of RILEM Recommendation published in Materials and Structures Vol. 28, 1995, 357–365, 415–430, and 488–495). Bažant, Z.P., Cusatis, G. and Cedolin, L. 2004. Temperature effect on concrete creep modeled by microprestresssolidification theory. J. of Eng the Mechanics ASCE 130 (6) 691–699. Bažant, Z.P., Hauggaard, A.B. and Baweja, S. 1997. Microprestress-solidification theory for concrete creep. II. Algorithm and verification, J. of Engrg. Mech. ASCE 123(11), 1195–1201. Bažant, Z.P., Hauggaard, A.B., Baweja, S. and Ulm, F.-J. 1997. Microprestress-solidification theory for concrete creep. I. Aging and drying effects, J. of Engrg. Mech. ASCE 123(11), 1188–1194. Bažant, Z.P. and Kaplan, M.F. 1996. Concrete at High Temperatures: Material Properties and Mathematical Models, Long-man (Addison-Wesley), London (2nd printing Pearson Education, Edinburgh, 2002). Bažant, Z.P. and Li, G.-H. 2009a. Unbiased statistical comparison of creep and shrinkage prediction models. ACI Materials Journal, 106(6), 610–621. Bažant, Z.P. and Li, G.-H. 2009b. Comprehensive database on concrete creepa and shrinkage. ACI Materials Journal, 106(6), 635–638. Bažant, Z.P., Li, G.-H. and Yu, Q. 2008. Prediction of creep and shrinkage and their effects in concrete structures: critical appraisal. Proc., 8th International Conference on Concrete Creep and Shrinkage (CONCREEP-8), IseShima, Japan, T. Tanabe et al. eds., CRC Press/Balkema, 1275–1289. Bažant, Z.P. and Panula, L. 1978. Practical prediction of time-dependent deformations of concrete. Part I: shrinkage. Part II: creep. Materials and Structures, 11(65), 307–328. Bažant, Z.P. and Prasannan, S. 1989a. Solidification theory for concrete creep: I. Formulation. Journal of Engineering Mechanics ASCE 115(8), 1691–1703. Bažant, Z.P., and Prasannan, S. 1989b. Solidification theory for concrete creep: II. Verification and application Journal of Engineering Mechanics ASCE, 115(8), pp. 1704–1725. Bažant, Z.P., Sener, ¸ S. and Kim, J.-K. 1987. Effect of cracking on drying permeability and diffusivity of concrete. ACI Materials Journal, 84, 351–357. Bažant, Z.P. and Xi, Y. 1994. Drying creep of concrete: Constitutive model and new experiments separating its mechanisms. Materials and Structures, 27, 3–14. Bažant, Z.P. and Xi, Y. 1995. Continuous retardation spectrum for solidification theory of concrete creep J. of Engrg. Mech. ASCE 121(2), 281–288.

Bažant, Z.P., Yu, Q. and Li, G.-H., Klein, G., and Kˇrístek, V. (2007). ‘‘Explanation of excessive long-time deflections of collapsed record-span box girder bridge in Palau’’, Prelim. Structural Engrg. Report 08-09/A222e, Infrastructure Technology Institute (ITI), Northwestern University. Brooks, J.J. 1984. Accuracy of estimating long-term strains in concrete. Magazine of Concrete Research, 36(128), 131– 145. Brooks, J.J. 2005. 30-year creep and shrinkage of concrete. Magazine of Concrete Research, 57(9), 545–556. Burgoyne, C. and Scantlebury, R. 2006. Why did Palau bridge collapse? The Structural Engineer, 30–37. CEB-FIP Model Code 1990. Model Code for Concrete Structures. Thomas Telford Services Ltd., London, Great Britain; also published by Comité euro-international du béton (CEB), Bulletins d’Information No. 213 and 214, Lausanne, Switzerland. Deryagin, B.V. 1955. The definition and magnitude of disjoining pressure and its role in the statics and dynamics of thin fluid films, Kolloid Zh. 17, 205–214. FIB 1999. Structural Concrete: Textbook on Behaviour, Design and Performance, Updated Knowledge of the CEB/FIP Model Code 1990. Bulletin No. 2, Federation internationale du béton (FIB), Lausanne, Vol. 1, 35–52. Gardner, N.J. 2000. Design provisions of shrinkage and creep of concrete Adam Neville Symposium: Creep and Shrinkage- Structural Design Effect ACI SP-194, A. AlManaseer, eds., 101–104. Gardner, N.J. and Lockman, M.J. 2001. Design provisions for drying shrinkage and creep of normal strength ACI Materials Journal 98(2), Mar.–Apr., 159–167. Japan International Cooperation Agency (JICA). 1990. Present Condition Survey of the Koror-Babelthuap Bridge, Feburary, 42 pages. Jirásek, M. and Bažant, Z.P. 2002. Inelastic analysis of structures, John Wiley & Sons, London and New York. JSCE 1991. Standard Specification for Design and Construction of Concrete Structure, Japan Society of Civil Engineers (JSCE), in Japanese. McDonald, B., Saraf, V. and Ross, B. 2003. A spectacular collapse: The Koro-Babeldaob (Palau) balanced cantilever pre-stressed, post-tensioned bridge The Indian Constrete Journal Vol. 77, No.3, March 2003, 955–962. M¨uller, H.S. 1993. Considerations on the development of a database on creep and shrinkage tests. Creep and Shrinkage of Concrete, Proceedings of the 5th International RILEM Symposium, Barcelona, Spain, Bažant Z.P. and Carol I., eds., E&F Spon, London, UK, 859–872. M¨uller, H.S., Bažant, Z.P. and Kuttner, C.H. 1999. Database on creep and shrinkage tests. RILEM Subcommittee 5 Report, RILEM TC 107-CSP, 81 pages. M¨uller, H.S. and Hilsdorf, H.K. 1990. Evaluation of the TimeDependent Behaviour of Concrete: Summary Report on the Work of the General Task Force Group No. 199, Comité Euro-Internationale du Béton, Lausanne, Switzerland, 201 pages. Pickett, C. 1942. The effect of change in moisture-content on the creep of concrete under sustained load. ACI J. 38, 333–356. Yee, A.A. 1979. Record span box girder bridge connects Pacific Islands Concrete International 1 (June), 22–25.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Alternate approaches to simulating the performance of ductile fiber-reinforced cement-based materials in structural applications S.L. Billington Stanford University, CA, USA

ABSTRACT: Structural-scale applications of ductile, fiber-reinforced cement-based materials are being evaluated by researchers and implemented in practice more and more. These materials exhibit significant ductility in direct tension and generally do not spall in compression. Little guidance exists in codes or specifications for designing with these materials. Validated simulation approaches are called for to understand how these materials can best be applied and will perform in structural applications. This paper presents several approaches to simulating plain and steel-reinforced ductile, fiber-reinforced cement-based materials in both new design and retrofit applications, with an emphasis on the ability of simple approaches, using basic material properties, to predict performance. Specifically, the use of nonlinear finite element analysis, phenomenological spring modeling, and sequentially linear finite element analysis is presented. Methods of simulating ductile cement-based composites in structures and structural components range from sequentially linear finite element analysis (Billington, 2009) to nonlinear finite element analysis for both two-dimensional (Lee & Billington, 2008, Suwada & Fukuyama, 2006, Hung & El-Tawil, 2010) and three-dimensional simulations (Sirijaroonchai & El-Tawil, 2007) to multi-phase and multi-scale methods (Chuang & Ulm, 2002, Kabele, 2007) to phenomenological spring models for cyclic loading (Lignos et al., 2009). In this paper, several modeling approaches for two ductile fiber-reinforced cement-based materials are evaluated by modeling a variety of experiments conducted on components and structural configurations

1 INTRODUCTION Many researchers and practitioners are investigating structural-scale applications of ductile, fiberreinforced cement-based materials. These materials exhibit fine, multiple cracking in direct tension, reaching strains of 0.5 to 3 percent and generally exhibit little to no spalling in compression. The materials are often referred to as high performance fiber-reinforced concrete (HPFRC) or high performance, fiber-reinforced, cement-based composites (HPFRCC) if they contain no coarse aggregate. One class of HPFRCC being heavily researched is engineered cementitious composites (ECC). The term high performance is used when the fiber volume fraction is relatively low (2% or less) and the tensile properties are either high strength (e.g. increase in tensile strength after first cracking) or high ductility (e.g. >3% strain while maintaining full tensile carrying capacity) (Figure 1). Applications in service and under investigation range from bridge decks, to energy-dissipating hinge regions in seismicresistant frames, to repair and retrofit of structures. Little guidance exists in codes or specifications for designing with these materials. Therefore validated simulation approaches are called for to understand how these materials can best be applied and will perform in structural applications. This paper presents several example approaches to simulating ductile fiberreinforced cement-based materials in both new design and retrofit applications in particular for resistance to seismic loads, with an emphasis on the ability of simple approaches, using basic material properties, to predict performance.

Figure 1. Schematic of uniaxial tensile responses possible from ductile fiber-reinforced cement-based composites.

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using these materials. The two materials focused on here are ECC (Li, 2003) and a self-compacting HPFRC (SC-HPFRC) developed by Liao et al., (2006). Nonlinear finite element analysis, phenomenological spring modeling, and sequentially linear finite element analysis are evaluated. An additional example of modeling ECC as applied to masonry beams is given in Kyriakides et al. (2010).

2 NONLINEAR FINITE ELEMENT ANALYSIS TO IDENTIFY OPTIMAL ECC PROPERTIES FOR DUCTILE BRIDGE PIERS Figure 2.

2.1 Background and motivation Ductile fiber-reinforced cement-based materials have been investigated for use in regions of high moment and shear (often referred to as hinge regions, i.e. where plastic hinges will likely form under overloads) in particular for bridge supports. Both experimental (Rouse, 2004, Saiidi and Wang, 2006) and analytical (Lee & Billington, 2008) research has been conducted, including the use of nonlinear finite element analysis to interpret an unexpected experimental failure mode (Lee & Billington, 2010). Here, an example application of nonlinear finite element analysis to evaluate and propose alternate material properties to improve structural performance of bridge columns is presented. In this case, ECC was used in the hinge regions of precast segmental bridge columns that were reinforced with vertical, unbonded post-tensioning. While the performance of the columns with ECC hinge regions was superior to that of a column using all traditional concrete segments, it was believed that more ductility should be possible in a column with ECC hinge regions than was observed experimentally. As its name suggests, ECC can be engineered to achieve a variety of tensile properties (Li and Leung, 1992). Therefore a study was carried out to evaluate the impact of alternate ECC properties on structural performance to identify what ECC properties lead to improved structural performance, which was defined as increased ductility under cyclic lateral loading. The original experiment modeled is briefly reviewed here and further detail on both the experiments and simulations are given in Rouse (2004) and Lee and Billington (2008). The experiment simulated was one of a pair of large-scale unbonded post-tensioned bridge columns tested together (Figure 2a). The column specimen had a height of 3.7 m and a 460-mm-square cross-section. The specimen consisted of four precast segments each 1.067 m in length, with the two end segments embedded into a cap and a foundation block. Reinforcement details of the column segments can be seen in Figure 2b. The precast segments, which had been match cast, were joined using a flowable epoxy mortar,

Test set-up and bridge column segment details.

and had no continuous, bonded reinforcing across the segmental joints. The segments were post-tensioned together with six 15.2-mm-diameter low-relaxation strands stressed to 690 MPa. Two columns were tested simultaneously to achieve double curvature in order to represent the behavior of a column in a multiple-column bent configuration subjected to lateral loading. Specimens were oriented horizontally (longitudinal axis parallel to the floor) with their foundation blocks connected to a steel reaction frame and cap blocks connected to one another to provide rotational restraint. The specimens were subjected to quasi-static cyclic lateral loads while under a constant axial load of 720 kN (applied with a hydraulic actuator), representing dead load from a bridge superstructure. The hinge segments were precast with ECC that demonstrated a compressive strength of roughly 40 MPa and a uniaxial tensile response as shown schematically in Figure 3 as ‘‘ECC’’ (based on experimental testing in Rouse, 2004). Lightweight concrete was used for the remaining segments (including cap and foundation blocks) with a nominal compressive strength of 55 MPa. The reinforcement in the column segments was detailed to meet only shear and shrinkage requirements. 2.2 Finite element modeling A plane stress model was used in the analyses, as outof-plane stresses were assumed to be negligible for the testing configuration. Each column was modeled individually. The finite element model for a single column is shown in Figure 4. The concrete and ECC were modeled using nine-noded, quadrilateral isoparametric plane stress elements with a 3 by 3 Gauss integration scheme. All longitudinal and transverse bonded, mild steel reinforcing bars were modeled with threenoded embedded reinforcement elements, which were assumed to have perfect bond with the surrounding

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tendons, was applied to the truss elements. Cracking was not modeled explicitly between the segmental joints (e.g. with interface elements) because it was neither observed in this region experimentally nor expected (the epoxy mortar joining the segments a was stronger in tension than the segment materials). The foundation block was modeled as fixed by providing pin supports at all nodes along the bottom of the model. The top nodes of the specimen (at the top of the cap block) were modeled as being rotationally fixed to represent the fixity provided by the connection to the cap of the other specimen. Because rotational degrees of freedom do not exist for the quadrilateral elements, rigid three-noded beam elements were added along the top of the cap and their rotational degrees of freedom were constrained to provide the appropriate fixity. Point loads with a total magnitude equal to the magnitude of the applied axial load were applied to the top nodes of the cap block. Lateral loading was applied through applied displacements at two control nodes in the cap. Geometric nonlinearity was included in the analyses in addition to material nonlinearity as described next. The concrete elements in the footing and the cap were modeled as linear elastic, as non-linear behavior (manifested by cracking or crushing) in those regions was neither expected nor observed during testing. The elastic modulus for the concrete in these and all other concrete segments was assumed to be 24.8 GPa based on the American Concrete Institute (ACI) 318-05 (2005) equation for lightweight concrete using measured compressive strength data. A smeared, rotating crack model with a total strain formulation was used (Feenstra et al., 1998) for the concrete in tension. The tensile modulus was assumed equivalent to the compressive modulus (24.8 GPa). Post-cracking behavior was defined to be fracture energy-based with linear tensile softening. The tensile strength was assumed to be 3.45 MPa (ACI, 2005) and the tensile fracture energy was assumed to be 0.4 N-mm/mm2 (CEB-FIP, 1990). A bilinear model was used for the bonded, mild steel reinforcement with yield strength of 460 MPa, elastic modulus of 200 GPa, and post-yield (hardening) slope of 2 percent of the elastic modulus. The post-tensioning tendons were modeled as linear elastic, as designed and as observed during testing, with an elastic modulus of 186 GPa. The ECC was modeled with a total strain-based, rotating crack model developed by Han et al. (2003) using the loading and unloading parameters proposed therein. This model is based on the observed responses from a series of reversed cyclic tests on uniaxially loaded ECC specimens and was validated against cyclically loaded components. The envelope curves are shown in Figure 5. The unloading and reloading paths of the constitutive model were developed to capture the unique cyclic behavior of ECC’s. The tensile

Figure 3. Tensile properties of tested ECC (ECC) and three alternate ECC tensile property designs (Models 1–3).

Figure 4. Finite element model of bridge column with concrete and ECC segments joined with unbonded posttensioning.

plane stress elements. The bonded longitudinal reinforcing bars in the column segments did not have adequate development length to reach their yield strength in all areas. Where reinforcing bars did not have adequate development, the yield stress of the elements comprising the reinforcing bar were reduced in a stepwise fashion from 100 percent of the yield stress at the point of full development to zero at the end of the bar in order to approximate the linear reduction in bond stress (Lee & Billington, 2010). The unbonded post-tensioning tendons were modeled with two-noded truss elements that were constrained at their end nodes to the concrete element nodes at the anchorage locations. This connectivity allowed the strains to be distributed evenly along the length of the post-tensioned tendons. An initial stress of 690 MPa, equal to the experimental prestress in the

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Figure 5. Envelope curves for ECC constitutive model in tension (left) and compression (right).

stress and strain values for the envelope curve of the model were based on uniaxial tensile tests on the ECC (Kesner et al., 2003). The peak compressive stress was based on cylinder compression tests and the elastic modulus and peak compressive strain were estimated based on tests performed on ECCs with similar mix proportions and similar compressive strengths. To determine the effect of the ECC tensile parameters on the global response, the model was also analyzed for three additional tensile stress-strain responses (Models 1–3), shown in Figure 3. The three models represent behavior that can reasonably be expected from an ECC mix. 2.3 Simulation results A cyclic analysis was performed and the loaddisplacement response is shown in Figure 6. The envelope of the simulated cyclic response generally follows the experiment well. During the lower drift cycles, low residual displacements (displacement at zero load) are seen in both the experiment and the simulation. Additionally, up to the 3 percent drift cycle, the simulation captures similar peak strength and displacement as the experiment and the observed gradual softening to 3 percent drift is also simulated. It is noted that the downward slope to increasing drifts (e.g. from 2 to 3 percent drift) is indicative of P-delta effects due to the gravity load on the column and is less prominent in the simulation, although geometric nonlinearity is included in the simulations. At the largest drift cycle (to 3 percent drift) the simulated response no longer models the experimental response well particularly in unloading. The simulated response displays the expected flag-shaped (origin-oriented) response while the experimental response follows a much less stiff unloading path. The less stiff experimental unloading is caused by the change in boundary conditions in the experiment relative to the simulation (i.e. the lower stiffness of deformation in single-curvature once one of the two specimens tested simultaneously begins to form a plastic hinge, versus double-curvature at the beginning of the test, Figure 2a). The wider hysteresis

Figure 6. Cyclic response of experiment and simulation using measured ECC material properties (top), and regions of crack and compressive strain localization after peak load (bottom).

loops are attributed to the spread of ECC crushing not captured in the simulation and additional effects that are not modeled such as possible slip of bonded steel reinforcement after cracks localize. The column reaches its peak lateral load soon after crack localization occurs in the column near the joint between the end segments and the adjacent concrete segments (as shown in Figure 6, bottom). Although there is longitudinal, bonded mild steel reinforcement at this section, it is unable to contribute significantly to the moment capacity because it cannot reach its yield strength due to its development length (recall that the mild steel reinforcement is not continuous across the segmental joints). The ECC is therefore relied on primarily to resist tensile stresses. After the ECC begins to soften in tension (crack localization), the crack progresses toward the compression region, reducing the size of the compressive zone and causing the ECC in compression to quickly reach its peak strain and begin to soften. At this point the column begins to lose its

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lateral load resistance. The fact that there is no steel yielding leads to a relatively brittle and therefore undesirable failure. The formation of the localized crack near the joint and the subsequent compressive failure of the ECC were both observed in the experiment. The column with ECC as simulated above had a low drift capacity and had a relatively brittle failure mode (both experimentally and in the simulation), both of which are unfavorable for structures in seismic regions. As ECC is an engineered material, its tensile response can be tailored to meet the requirements of the engineer. Therefore, a parameter study was performed to determine the effect of the ECC tensile parameters on the load-drift behavior of the columns. The goal of this study was to evaluate possible benefits of engineering new ECC mix designs, specifically with respect to increasing drift capacity and introducing greater ductility to the column prior to failure. Three alternate sets of tensile properties for ECC were evaluated as described above and shown in Figure 3. Particular attention is given here to the results of Model 3. The load-drift response of the column using Model 1 led to a 2 percent increase in the peak load and 36 percent increase in the drift at peak load. The localization of cracking in this case occurred in the concrete segments above the ECC segments rather than in the ECC segments themselves due to the cracking strength of the ECC exceeding the concrete cracking strength. The column fails when the compressive strength of the ECC is reached and begins to soften shortly after at the segmental joint between the ECC and concrete segments near where the crack localization occurs, just as observed in the original simulation. Model 2 led to a lightly higher increase in strength and ductility (details in Lee and Billington, 2008) but the critical section remained in the joint region where the mild steel reinforcement was unable to contribute significantly to the moment capacity due to its development length. Using ECC with properties of Models 1 and 2 would provide only minimal improvements to the overall behavior of the column relative to what was tested. Model 3 provided an increase in the strain hardening capacity of the ECC in tension while limiting the peak strength so as not to exceed the tensile strength of the concrete. The resulting load-drift response for Model 3 is compared in Figure 7 with the experimental response and the simulated response using the original ECC model. The response of the column with ECC Model 3 shows a distinct difference from the ECC used to model the experiment. By 3 percent drift, the column is continuing to carry additional load. In this case, the maximum tensile strains occur at the ends of the columns (column-base interface), and localization of cracking does not occur near the segment joints between the ECC and concrete. Cracking and strain hardening of the ECC occur throughout most

Figure 7. Cyclic response of experiment, original simulation (‘‘Baseline’’ using model ECC) and Model 3.

of the hinge segment, with the most tensile straining occurring at the base of the column. Therefore, the mild steel is able to contribute significantly to the moment capacity (and ductility) because in this region it has adequate development length to yield. 2.4 Further discussion The finite element model was found to capture the response of the tested column well, with respect to both global behavior (lateral load versus displacement response) as well as local, damage (development of damage and mode of failure). The mode of failure of the column at a relatively low drift was captured well. The parameter study of the ECC tensile behavior revealed that the best method of improving the performance of the columns is to increase the ductility of the ECC without increasing the ultimate tensile strength. Increasing the ultimate tensile strength of the ECC to a value greater than that of the tensile strength of the concrete is predicted to lead to crack localization above the base segment (the plastic hinging region). High ECC tensile strength would lead to compressive failure near the segmental joint between the ECC and adjacent concrete segment. When increasing the tensile ductility while keeping the peak tensile stress below the tensile strength of the concrete, the tensile benefits of the ECC are better utilized. Cracking is predicted to occur throughout the ECC segment and a crack will eventually localize at the base of the column. In addition, the failure location is shifted away from sections where mild steel reinforcement is not able to fully yield (i.e. near the joints) to sections where it can contribute more significantly to moment capacity and improve ductility (i.e. at the ends of the column). Experimental validation of

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the proposed ECC material properties for improving column performance is needed and it is recommended that additional experimental projects on realistic structural applications be conducted to evaluate the ability of achieving a variety of structural performance characteristics through altering ECC material properties alone.

The panels to be modeled were single panels representing one half of the double-height panels shown in Figure 8, fixed at the base and laterally loaded at the top of a single panel (Figure 9a), representing the mid-point of the double height panels, or, the point of inflection for the panel system under lateral load. The reinforcing details of the two panels modeled are shown in Figure 9b–c. All reinforcement was modeled as embedded, which implies perfect bond to the concrete and/or SC-HPFRC material. Perfect bond was assumed to be representative of the deformed bars used but not the welded wire fabric (WWF) mesh reinforcement, which was made up of smooth wire. All modeling was 2-dimensional using 8-noded plane stress elements. The geometry for each panel was meshed with a user-defined mesh discretization size of ∼25 mm squares. The boundary condition at the base was fixed against vertical and horizontal deflection. The SC-HPFRC and concrete materials were considered to be isotropic. In compression, all materials were modeled as linear elastic as no compressive failure was expected (nor observed) during testing. The concrete panel was assumed to be brittle in tension (tensile strength of 4.1 MPa), while the SC-HPFRC was idealized as multi-linear with hardening and then softening in tension (similar to the ECC in Section 2). For both materials, a total strain fixed crack model was used in tension (Feenstra et al., 1998) with a constant shear retention factor of 0.2 and secant unloading and reloading. The multi-linear tension model for the SC-HPFRC was selected based on an inverse analysis of the performance of unreinforced SC-HPFRC modulus-of-rupture beams (Billington & Olsen, 2009). The steel for the connections was modeled as linear elastic, as designed (E = 200 GPa). The composite

3 NONLINEAR FINITE ELEMENT ANALYSIS TO SIMULATE HPFRC INFILL PANEL RESPONSE AND PREDICT STRUCTURAL PERFORMANCE 3.1 2D plane stress analysis of infill panels in cyclic flexure and shear An infill panel system using a self-compacting HPFRC (SC-HPFRC) has been proposed to retrofit fracturecritical steel frames for seismic loading (Ol-sen & Billington, 2009) (Figure 8). The infill panels are precast, and intended to be both easily installed and rapidly replaced after an earthquake, if damaged. Panels were tested and proved to be robust under repeated cyclic load. With the SC-HPFRC a ductile, flexure failure was observed rather than a more brittle shear failure as seen in an equivalent panel cast with traditional concrete. In order to best design the infill panels for various retrofit configurations, two-dimensional nonlinear finite element analysis is proposed to evaluate strength, ductility and failure modes of various infill panel designs. Modeling strategies are under investigation and one example is presented here, in comparison to the modeling of a traditional concrete panel. Only measured material properties are used in the initial analyses as the objective of the investigation is to evaluate the ability of the simulation approach to be predictive of behavior.

Figure 8. Ductile fiber-reinforced concrete infill panel system proposed for seismic retrofitting of steel frame structures.

Figure 9. (a) Single panel test set-up, and details of the (b) SC-HPFRC Panel and (c) Concrete panel modeled here.

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connection area where the panel base was sandwiched with the U-channel with pre-stressed bolts and grout was also modeled as linear elastic (E = 69 GPa) since no damage was expected in this region during testing. The modulus of elasticity in the connection region was increased based upon a strain compatibility analysis of the different materials and thicknesses used in the connection. Finally, the reinforcing steel was modeled with the Von Mises elas-to-hardening yield plasticity assumption using measured material properties from uniaxial tensile tests (9.5 mm-dia. bars: E = 200 GPa, fy = 483 MPa, fu = 656 MPa at a strain of 0.2, and WWF: E = 200 GPa, fy = 690 MPa, fu = 828 MPa at a strain of 0.05). Tensile fracture was not modeled. The controlled cyclic displacement pattern that was used in the experiments was also used for the cyclic simulations, except drift cycles were not repeated in the simulations as they were in the experiments. The displacement step size was chosen to avoid numerical divergence of the solution while minimizing computation time. A Newton-Raphson iteration scheme was used with a force norm for the convergence criteria with a tolerance of 0.07%. If convergence could not be reached, the simulations were not allowed to continue. A line search algorithm was also implemented to aid in convergence. Figure 10 compares the hysteretic response of the SC-HPFRC panel experiment with the finite element analysis. Strength is well captured by the simulation. However the stiffness, strength degradation and hysteretic energy dissipation are not well captured with this simplified approach to modeling. The higher simulated stiffness is attributed primarily to connection flexibility and connection slip, which were not modeled using the fixed base assumption. The inability to capture post-peak strength degradation is attributed to the WWF reinforcement being modeled as having perfect bond to the SC-HPFRC. The contribution to shear strength provided by the

WWF, particularly the horizontal bars likely did not occur in the experiment because the WWF could not be fully developed as discussed above. Further evaluation of the WWF is given at the end of this sub-section. Finally, the level of pinching seen in the experiments is not well captured and is attributed to the idealized perfect bond assumption (by using embedded reinforcement) and the use of secant unloading rather than the more accurate ECC-type unloading/reloading proposed by Han et al. (2003). As large cracks form, some level of slip would be expected leading to a more pinched response. Figure 11a shows the principle tensile strain contours at +1% drift (∼9 mm, used as a reference point to compare failure locations and failure modes between the simulations and experiments) with the maximum contour representing 0.002 strain, which is where tensile softening is expected from the multi-linear tensile approximation for the SC-HPFRC. The damage is located near the base and matches the observed experimental results (Figure 11b). The average compressive strain in the compression region was observed to be around 0.001, which is in the elastic range of the material. Figure 12 shows the concrete panel simulation vs. experimental hysteretic response. The simulated panel capacity is accurate in the negative drift direction, and slightly underestimated in the positive direction. The simulated initial stiffness is very close to that seen in the experiments. The concrete panel simulation also uses a brittle tension model instead of the SC-HPFRC multi-linear tension model. The simulation was unable to converge after the peak was reached. Figure 13 shows the principal tensile strains at peak load and at 1% drift (∼9 mm). While there is evidence

Figure 10. Cyclic response of the SC-HPFRC panel experiment and initial simulation.

Figure 11. (a) Simulated principal tensile strains at 1% drift, and (b) final failure of SC-HPFRC panel experiment.

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ECC under cyclic loading with its unique loading and unloading response. Regarding the tensile response of the SC-HPFRC material, performing uniaxial tensile tests to characterize cast material is difficult to standardize and simpler methods are needed to estimate this necessary material response for finite element simulations of structural configurations. As an example of the impact such changes in assumptions can make, Figure 14 shows a monotonic pushover analysis for the SC-HPFRC panel where the full WWF is modeled and where only the vertical WWF is modeled, both compared to the cyclic experimental response. In these simulations, a stronger multi-linear tensile curve for the SC-HPFRC accounts for the difference in strength relative to the simulation shown in Figure 10. However the significant difference is seen in the post-peak behavior when no horizontal WWF steel is included. Finally, for a more rapid evaluation of strength and ductility for various designs, sequentially linear analysis (discussed in Section 4) is currently being investigated for these panel designs.

Figure 12. Cyclic response of the concrete panel experiment and simulation.

3.2 Seismic analysis of a frame structure with HPFRC infill panels modeled as rotational springs Large-scale hybrid (pseudo-dynamic) testing of the infill panels described in Section 3.1 installed in a two-story steel frame is scheduled for early 2010. The testing involves a physical subassembly of a one bay sub-frame that includes the SC-HPFRC panels, and a numerical subassembly to represent the remainder of the moment resisting frame and modeled in the Open System for Earthquake Engineering Simulation (OpenSees, 2009) platform. Two horizontal translational degrees of freedom at each one of the floor levels of the experimental subassembly are to be controlled during the hybrid test.

Figure 13. (a) Simulated principal tensile strains at 1% drift, and (b) final failure of the concrete panel experiment.

of cracking damage throughout the panel, most damage is concentrated at mid-height and is at a 45-degree angle, indicative of shear cracking and failure, which is expected in the concrete material in this geometry. The location of damage at mid-height is likely influenced by the 9.5 mm-diameter dowel reinforcing bar cutoff at this height (Figure 9c). Both the location and orientation of the most severe damage corresponds to that seen in the experiment. Several improvements to this simplified modeling approach are currently being investigated such as implementing springs at the base to better represent the flexibility in that connection, reducing the bond (or area) for the WWF, which cannot fully develop, evaluating the sensitivity of the panel response to variations in the multi-linear tension model where tension hardening, yielding and tension softening show significant differences in component post-peak behavior, and adopting the Han et al. (2003) material model for

Figure 14. Evaluation of SC-HPFRC panel response with and without horizontal WWF reinforcement.

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Prior to performing the experiment, it is necessary to perform a ‘‘virtual test’’ wherein simulations represent both the portion of the structure to be modeled analytically during testing and the portion to be modeled experimentally. This modeling work is described here. The prototype structure that is used for seismic retrofitting with the SC-HPFRC panels is a two story 3-bay office building with perimeter steel moment resisting frames designed based on 1980s U.S. seismic provisions. The building does not meet the retrofit objectives based on current accepted guidelines in the U.S. A 2/3-scale model of the east west (EW) moment resisting frame (predominant period of 0.75 seconds) of the building is to be retrofit with five SC-HPFRC double panels per story installed in the first bay. The 2/3-scale frame of the prototype frame is designed with a W10 × 45 exterior and interior columns and W14 × 26 and W10 × 30 first and second floor beams, respectively. These sizes are based on similitude laws for strength and stiffness based on Mon-carz and Krawinkler (1981). The predominant period of the retrofitted scale frame is 0.39 sec. The bf /2tf and hf /tw ratios of the selected scaled sections are almost the same with the ones of the prototype steel momentresisting frame, i.e. the deterioration parameters of the components of the scaled frame represent reasonably well those of the prototype frame. The geometry of the physical subassembly is shown schematically in Figure 15 and details are given in Lignos & Billington (2010). Connection details of the panels are described in detail in Olsen and Billington, (2009) and Lignos et al. (2009). Although the prototype frame was designed based on pre-Northridge seismic provisions, the four steel moment connections in the physical subassembly are designed as standard welded, unreinforced flange-bolted web connections per FEMA-350 (2000) provisions to avoid any control instability of actuators during testing due to beam-column joint fracture. Fracture is simulated in the numerical portion of the hybrid model though recognizing the possibility of having fracture(s) at design level earthquake events.

During hybrid testing the two horizontal translational degrees of freedom are the control quantities and the physical subassembly is connected with a link that is designed to behave elastically. Two 975-kN dynamic actuators will impose the computed displacements and also measure the force and displacement quantities from the physical subassembly. The 2-dimensional test frame is modeled in OpenSees with elastic beam-column elements that have concentrated plasticity springs at their ends. The hysteretic response of these springs is bilinear. The springs simulate component deterioration based on the modified Ibarra-Krawinkler model (Lignos and Krawinkler, 2009). Deterioration parameters of the components are determined from relationships for deterioration modeling that were derived from a recently developed database of steel components for deterioration modeling (Lignos and Krawinkler, 2007, 2009). The modified deterioration model is able to simulate brittle fracture with an ultimate deformation parameter θ u that is set to be 2% for one end of the first floor exterior beam, recognizing the possibility of having a brittle fracture at an early inelastic cycle as reported in FEMA-351 (2000). The analytical model developed to capture the hysteretic response of the SC-HPFRC infill panels is shown in Figure 16. Two rigid links are connected together with a hinge connection in the middle that allows vertical movement of one panel with respect to the other. Each infill panel at its fixed end has a concentrated plasticity spring that utilizes the modified Ibarra-Krawinkler deterioration model with peakoriented hysteretic behavior. Experimental data provided by Hanson and Billington (2009) is used to calibrate the infill panel model. The calibrated moment vs. rotation diagram of the SC-HPFRC double panel is shown in Figure 16c. P-Delta effects are simulated numerically with a leaning column that does not contribute to the lateral stiffness of the building. Two percent Rayleigh damping is assigned to the bare steel moment frame. For the retrofitted steel moment resisting frame, 3% Rayleigh damping is assumed in order to consider the effect of the SC-HPFRC panels on viscous damping of the building. Two testing phases are scheduled wherein the same steel moment frame is used but the SC-HPFRC infill panels are replaced between the two phases. In both phases the ground motion records are scaled appropriately to represent levels of intensity that are of particular interest for the engineering profession. The two testing phases are summarized in Table 1. Phase I is concerned with seismic performance of the retrofitted 2-story frame during two subsequent design level earthquakes (DLE). Phase II involves experimental testing with a maximum considered earthquake (MCE) based on the scaled component of the Petrolia record from the Cape Mendocino earthquake

Figure 15. Coupled simulation model of two-story steel frame with SC-HPFRC infill panel retrofit in one bay.

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in 1992. After the end of this event the frame is subjected to the unscaled component of the JR Takatori record from the 1995 Kobe earthquake in Japan. Simulation results for the two testing phases are presented here based on a new simulation method that couples two or more displacement-based structural finite element analysis programs together through a generic adapter element approach, which is implemented in the Open-source Framework for Experimental Setup and Control (OpenFresco) (Takahashi and Fenves, 2006; Schellenberg et al., 2007). The numerical subassembly of the 2-story steel moment-resisting frame shown in Figure 15 is analyzed in OpenSees and is connected with a generic Super-Element within OpenSees that represents the physical subassembly tested in the laboratory (referred to as the Master Program). The physical subassembly itself is modeled in OpenSees but as a separate input file (referred to as the Slave Program). The master program imposes boundary conditions on all of the subassemblies. The 2-node adapter element connects to the interface nodes of the physical subassembly in the slave program and is responsible for imposing trial displacements on such subassembly. Details of the couple simulation theory, adapter elements and implementation are given in Schellenberg et al. (2007, 2008). During DLE-I and DLE-II motions of Phase I (Table 1) the retrofitted frame does not exceed a maximum story drift ratio (SDR) of roughly 1.5% compared to 2.5% of the bare frame ground motions. Figure 17 illustrates the drift histories of the bare versus retrofitted frame for DLE-I level of intensity. This result indicates that during the DLE events no fracture occurs in any of the steel moment frame connections. At the end of both DLE-I and II levels of Phase I the residual drift ratios of the retrofitted frame are almost zero compared to a prediction of roughly 0.6% in the bare frame at both stories. After replacing the SC-HPFRC infill panels with new ones (Phase II) and subjecting the retrofitted frame to the MCE level ground motion (105% of the unscaled Petrolia record, Table 1) peak story drift ratios of the retrofitted frame stay below 2.5% at the first story. Due to fracture of the connection at the end location of the first floor beam of the bare frame, the story drift ratio increases in the bare frame are about 30% larger compared to the retrofitted ones indicating that the proposed retrofit system is effective for retrofitting existing steel moment frames with fracture-critical beam-column connections. Similarly, residual story drifts of the retrofitted frame are reduced by roughly 50% compared to those of the bare frame (Figure 18). Based on the ‘‘virtual’’ hybrid simulation for the two scheduled testing phases it was found that: (1) peak story drift ratios of the retrofitted frame are expected to be reduced by roughly 30% compared to the peak story drift ratios of the bare frame when they are both subjected to a design level and maximum considered

Figure 16. Modeling of double-height SC-HPFRC infill panels.

Table 1. Level of intensity Phase I Service level Design level Design level

Hybrid testing experimental program. Gr. motion Earthquake Notation intensity record SLE

30%

DLE-I

70%

DLE-II

100%

Phase II Service SLE level Maximum MCE considered level Collapse CLE level

30% 105% 100%

Petrolia (Cape Mendocino, 1992) Petrolia (Cape Mendocino, 1992) Canoga Park (Northridge, 1994) Petrolia (Cape Mendocino, 1992) Petrolia (Cape Mendocino, 1992) JR Takatori (Kobe, 1995)

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4 SEQUENTIALLY LINEAR ANALYSIS OF ECC IN FLEXURE For large-scale structural analyses, sequentially linear finite element analysis has been proposed as an alternative to nonlinear finite element analysis (Rots & Invernizzi, 2004). To remain a simple modeling approach, sequential linear analysis is most useful for monotonic loading. Applied to structures using ductile cement-based materials, sequential linear analysis has the potential to simplify the evaluation of alternate designs of large-scale applications, such as the new designs proposed in Section 2 or the retrofit system presented in Section 3, where a monotonic pushover analysis could be employed for initial design evaluations in terms of panel design details and locations within a steel frame. An advantage of using sequential linear analysis would be that detailed behavior (e.g. local damage, failure modes) of the ductile cementbased materials could be obtained relatively easily (e.g. no convergence issues) using basic material properties as input. The evaluation of alternate tensile properties, as demonstrated in Section 2, or alternate reinforcing details as performed experimentally for the infill panels in Section 3, could be performed through simulation to identify the best material and component designs for desired structural-scale behavior. Sequentially linear finite element analysis has been evaluated for modeling traditional fiber-reinforced concrete, which shows tensile-softening behavior after cracking (Belletti et al., 2008) and more recently tensile hardening materials such as ECC (Billington, 2009). Here, a comparative analysis of small-scale slender beams made of ECC and tested in four-point bending is performed using nonlinear analysis with smeared cracking and using sequentially linear analysis. The beams to be modeled are shown in Figure 19 and displacement was measured at mid-span (details given in Billington, 2009). For the nonlinear analyses, two-dimensional, 8-noded, plane stress elements of constant thickness and a 3 × 3 Gaussian integration scheme were used to model the beam with 4 elements along the height and 48 along the length. The ECC was modeled as linear elastic in compression as no softening in compression was expected nor observed in the experiments. Young’s modulus was measured from an average of 4 cylinders tested in compression to be E = 11,032 MPa. A total strain-based fixed crack model (Feenstra et al., 1998)

Figure 17. Comparison of story drift ratio histories between the bare and retrofitted 2-story frame during design level earthquake I of testing Phase I.

Figure 18. Comparison of story drift ratio histories between bare and retrofitted 2-story frame during maximum considered level earthquake of testing Phase II.

event. The implication is that brittle fracture of beamto-column moment connections does not occur during design level events and is delayed during maximum considered earthquake events, and (2) residual deformations of the retrofitted frame are expected to be zero for a design level event and reduced by roughly 50% during a maximum considered event. Hybrid simulation testing of the retrofitted 2-story steel moment frame is under way at the Network for Earthquake Engineering Simulation (NEES) experimental facility at The University of California at Berkeley in order to confirm and improve pre-test analytical simulations and validate if the proposed SC-HPFRC infill panel retrofit system and modeling approach can be used to design retrofits of existing steel moment frames.

Figure 19. ECC flexural beam test set-up.

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energy dissipation using the sawtooth model, whereby the triangle above the baseline curve is ‘‘extra’’ energy dissipated by loading along E2 up to the upperbound curve, and the triangle below the curve is the ‘‘missing’’ energy dissipation upon reloading along E3, again to the upperbound curve. These two triangles are roughly equal, thus allowing for the preservation of fracture energy dissipation with reasonable accuracy. Again, a shear retention factor of 0.05 was adopted and Poisson’s ratio was taken as 0.15. The linear analyses were run in load control as explained above and were repeated for 2000 steps. Figure 21 compares the load-displacement response of the nonlinear analysis, the sequentially linear analysis and the three experiments on the ECC beams. The nonlinear analysis failed to converge in the second step after the peak load was reached, which coincided with the first integration point to reach the softening branch of the ECC constitutive model. The sequential linear analysis captured the likely snap-back experienced during failure (although snapback was not attempted to be measured experimentally). In sequentially linear analysis, only one critical point is identified in each analysis and as a result, an asymmetric failure mode can easily be captured, which is representative of experimental results. However, the nonlinear analysis will typically predict symmetric failure modes if no prior imperfections are introduced (Figure 22). The colors (shading) in Figure 22 represent different regions in the ECC tensile strain response as labeled in the figure. Both analysis methods show that at this displacement, the ECC has experienced significant multiple cracking but that failure has localized in one or a pair of cracks. Upon localization of the failure cracks and unloading of the beam, the regions of the beam that had been tensile-hardening also unloaded in both analyses (top portion of beam, between loading points) as observed in experiments. In Figure 23, the mesh sensitivity of the sequentially linear analysis is observed when comparing the original analysis (mesh of Figure 20) with an analysis

was used with a trilinear tensile stress-strain response assumed based on previous tensile tests of ECC dogbone specimens and defined by the following four stress-strain points: 1) (0,0), 2) (3.1 MPa, 0.000281), 3) (4.1 MPa, 0.025), and 4) (0, 0.05). A shear retention factor of 0.05 was used for the cracking and Poisson’s ratio was taken as 0.15. While the shear retention factor for concrete is often taken to be 0.2 to represent, for instance, aggregate interlock and dowel action of reinforcement across crack surfaces, a lower value of 0.05 is adopted here because the materials contain no steel reinforcement or coarse aggregate. This value 0.05 for shear retention did not cause ill-conditioning of the stiffness matrix. The analyses were run in displacement control with a step size of 0.05 mm for 100 steps followed by 100 steps of 0.025 mm, all applied at the loading point. The nonlinear analyses used a regular NewtonRaphson iteration scheme, including line estimation, with a maximum of 30 iterations. Both a displacement and force tolerance of 1% were used for the convergence criteria. For the sequential linear analyses, the same finite element model for the nonlinear analysis was used as described above, with the exception of using a 2 × 2 integration scheme rather than 3 × 3 in the elements. The ECC was modeled as linear elastic in compression as no softening in compression was expected nor observed in the experiments. Young’s modulus was measured from an average of 4 cylinders tested in compression to be E = 11,032 MPa. In tension, the same multi-linear tensile stress-strain curve from the nonlinear analysis was adopted for the baseline curve. To preserve fracture energy dissipation, a sawtooth model using an upper and lower bound curve, relative to the baseline curve, was adopted as shown in Figure 20. The upper and lower bound curves for the tension hardening and tension softening regions have the same stiffness as the baseline curve and represent a roughly 15% increase and decrease in stress at first cracking and peak cracking strength. Twenty stiffness reductions (similar to damage states) resulted to represent the full stress strain response. The grey triangles in Figure 20 highlight the preservation of fracture

Figure 21. Load-displacement response of experiments, and nonlinear and sequentially linear analyses.

Figure 20. Saw-tooth modeling of ECC in tension.

26

The sequential linear analysis is able to capture likely snapback behavior after crack localization. In the nonlinear analysis, a more advanced solution procedure would be required to capture the snap back. Mesh dependence was observed in the sequential linear analysis for the ECC beams, as the adopted crack model was not regularized for mesh size. A regularization procedure for tensile hardening-softening materials may be required but further study is warranted. 5 CONCLUSIONS Numerous simulation methods exist for modeling ductile fiber-reinforced cement-based materials. For structural applications, it is desirable to have simple modeling methods that can be used for rapid evaluation of alternate component designs and structural configurations using these ductile cement-based materials. Three different modeling approaches have been reviewed here including simulations of structural applications for new design as well as seismic retrofitting of existing structures. Using measured material properties and standard assumptions, nonlinear finite element analysis was demonstrated to capture well the cyclic response of precast segmental bridge piers using unbonded posttensioning and ECC in hinging regions, a complex combination of materials and components. It was demonstrated that this modeling approach can be used to identify optimal ECC properties to achieve a variety of structural responses, including a more robust, ductile response as would be desired for seismic design. A similar approach using nonlinear finite element analysis to capture the response of SC-HPFRC as well as traditional concrete infill panels to retrofit steel frames was observed to predict strength and failure modes well. However in the absence of detailed information on bond-slip characteristics of smooth and deformed steel reinforcement in SC-HPFRC materials, softening and hysteretic energy dissipation is not well predicted. Furthermore it was observed that panel simulation response post-peak is particularly sensitive to the presence of the smooth welded wire fabric reinforcement, whose contribution is not clear given the limited of bond and development possible. When experimental data is available, phenomenological rotational spring models can be adopted and an example application of such models to predict the seismic performance of a retrofitted frame was demonstrated here to limit beam-column joint fracture in a fracturecritical steel frame and reduce residual displacements after a seismic event. Finally, it was demonstrated that sequential linear analysis could be used to predict the tensile-hardeningsoftening performance of ECC in flexure. In comparison with a nonlinear analysis using a total-strain based

Figure 22. Principle strain contours of beams at failure (roughly zero load).

Figure 23. Load-displacement response of ECC beams from sequential linear analysis and two mesh sizes.

performed on a beam with twice as many elements in each direction. Adopting the same saw-tooth softening model as for the original mesh, one would expect to see a more brittle response using the fine mesh, which is observed in Figure 21. A regularization procedure for tensile hardening-softening materials may be needed but further study, particularly for ECC materials reinforced with conventional reinforcement is warranted. In summary, both the nonlinear finite element analysis with a smeared, fixed crack model based on total strain, and the sequential linear finite element analysis are able to capture the load-displacement response of deflection hardening ECC beams well using a tensile hardening-softening crack model. In both analyses, similar extents of multiple cracking were observed and softening of the ECC began at the same displacement (and load). A symmetrical pair of localized cracks formed in the nonlinear analysis due to the symmetry of the beam, loading and test set-up. With the sequential linear analysis, an asymmetrical failure mode, as expected experimentally is easily captured.

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fixed crack model, both analyses predict deflectionhardening response well. Unlike the nonlinear analysis, the sequential linear analysis is able to predict an asymmetric failure mode (as expected experimentally) as well as snap-back behavior with little computational effort. Applying sequential linear analysis to large-scale structural evaluations of ductile cementbased materials could prove to be an easy-to-adopt approach for designers and researchers to study both local and global behavior of a wide variety of material and structural designs.

Modelling of Concrete Structures, Proceedings of EURO-C 1998, de Borst, Bicanic, Mang & Meschke (eds), Balkema, Rotterdam, 13–22. Han, T.S., Feenstra, P.H. and Billington, S.L. (2003). Simulation of Highly Ductile Cement-Based Composites. ACI Structural Journal, 100(6): 749–757. Hanson, J.V. and Billington, S.L. (2009). Cyclic testing of a ductile fiber-reinforced concrete infill panel system for seismic retrofitting of steel frames, Report TR. 173, John Blume Earthquake Engineering Center, Stanford, CA. Hung, C-C. and El-Tawil, S. (2009). ‘‘Cyclic Model for High Performance Fiber Reinforced Cementitious Composite Structures,’’ Proc. ATC-SEI conference, Improving the Seismic Performance of Existing Buildings and Other Structures, December 9–11, 2009 San Francisco, CA. Kabele, P. (2007). Multi-scale framework for modeling of fracture in high performance fiber reinforced cementitious composites, Engineering Fracture Mechanics, 74: 194–209. Kesner, K.E., Billington, S.L. and Douglas, K.S. (2003). Cyclic Response of Highly Ductile Fiber-Reinforced Cement-Based Composites. ACI Materials J., 100(5): 381–390. Kyriakides, M.A., Hendriks, M.A.N. and Billington, S.L. (2010). Simulation of Masonry Beams Retrofitted with Engineered Cementitious Composites, Computational Modeling of Concrete Structures, Proc. EURO-C 2010, de Borst, Bicanic, Mang & Meschke (eds), March 15–18, Rohrmoos, Austria. Lee, W.K. and Billington, S.L. (2008). ‘‘Simulation of Self-Centering Fiber-Reinforced Concrete Columns,’’ Proceedings of ICE, Engineering and Computational Mechanics, 161(2): 77–84. Lee, W.K. and Billington, S.L. (2010). Simulation and Performance-Based Earthquake Engineering Assessment of Self-Centering Post-Tensioned Concrete Bridge Systems, Research Report, Pacific Earthquake Engineering Research (PEER) Center, to appear. Li, V.C. (2003). On Engineered Cementitious Comopsites (ECC)—A Review of the Material and Its Applications, J. Advanced Concrete Technology, 1(3): 215–230. Li, V.C. and Leung, C. (1992). Steady-State and Multiple Cracking of Short Random Fiber Composites, Journal of Engineering Mechanics, 118(11): 2246–2264. Liao, W.-C., Chao, S.-H., Park, S.-Y. and Naaman, A.E. (2006). ‘‘Self-Consolidating High Performance Fiber Reinforced Concrete (SCHPFRC)— Preliminary Investigation,’’ Report UMCEE 06-02, Dept. of Civil & Env. Engineering, Univ. of Michigan, Ann Arbor, MI, 76 pp. Lignos, D.G. and Billington, S.L. (2010). Hybrid Testing of a Retrofitted Steel Moment Resisting Frame with High Performance Fiber Reinforced Concrete Infill Panels, 9th US National & 10th Canadian Conference on Earthquake Engineering, Toronto, Canada, July. Lignos, D.G., Hunt, C.M., Krebs, A.D. and Billington, S.L. (2009). Comparison of retrofitting techniques for existing steel moment resisting frames, Proceedings ATC/SEI Conference, San Francisco, CA, December. Lignos, D.G. and Krawinkler, H. (2007). A database in support of modeling of component deterioration for collapse prediction of steel frame structures, ASCE Structures Congress, Long Beach CA, SEI.

ACKNOWLEDGEMENTS Financial support provided by the National Science Foundation (CMS-0530383), by the Pacific Earthquake Engineering Research (PEER) Center, and the Faculty of Civil Engineering at the Technical University of Delft, The Netherlands is gratefully acknowledged. The modeling research presented herein was carried out by the author as well as graduate research assistants Dr. Won Lee, Cole Olsen and post-doctoral scholar Dr. Dimitrios Lignos. The opinions expressed in this paper do not reflect those of the financial sponsors. REFERENCES American Concrete Institute. (2005). Building Code Requirements for Structural Concrete and Commentary. ACI 318-05 and ACI 318R-05, Farmington Hills, Michigan, 2005. Belletti, B., Hendriks, M.A.N. and Rots, J.G. (2008). Finite Element Modelling of FRC Structures —Pitfalls and How to Avoid Them, 7th Int’l RILEM Symp. on Fibre Reinforced Concrete: Design and Applications, BEFIB-2008, Chennai, India, September. Billington, S.L. (2009). ‘‘Evaluation of Sequentially Linear Finite Element Analysis to Simulate Nonlinear Behavior in Mortar and Engineered Cementitious Composites in Flexure,’’ ACI Special Publication 265-12, November. CEB-FIP. (1990). Bulletin d’information, CEB, Lausanne, Switzerland. Chuang, E. and Ulm, F.-J. (2002). Two-phase composite model for high performance cementitious composites, J. Engineering Mechanics, 128(12): 1314–1323. Federal Emergency Management Agency—FEMA 350 (2000). Recommended seismic design criteria for new steel moment frame buildings, Washington, DC, July. Federal Emergency Management Agency—FEMA 351 (2000). Recommended seismic evaluation and upgrade criteria for existing welded steel moment-frame buildings, Washington, DC: Federal Emergency Management Agency, July. Feenstra, P.H., Rots, J.G., Arnesen, A., Teigen, J.G. and Hoiseth, K.V. (1998). A 3D constitutive model for concrete based on a con-rotational concept. Computational

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Saiidi, M.S. and Wang, H. (2006). Exploratory Study of Seismic Response of Concrete Columns with Shape Memory Alloys Reinforcement, ACI Structural J., 103(3): 435–442. Schellenberg, A.H. (2008). ‘‘Advanced implementation of hybrid simulation’’, PhD Dissertation, Civil and Environmental Engineering, University of California, Berkeley. Schellenberg, A., Mahin, S. and Fenves, G. (2007). Software framework for hybrid simulation of large structural systems, Proceedings, Structures Congress, ASCE, Long Beach, CA, United States. Sirijaroonchai, K. and El-Tawil, S. (2007). Three Dimensional Plasticity Model for High Performance Fiber Reinforced Cement Composites, Proc. HPFRCC-5, H.W. Reinhardt and A.E. Naaman eds., Mainz, Germany, July,. Suwada, H. and Fukuyama, H. (2006). Nonlinear Finite Element Analysis on Shear Failure of Structural Elements using HPFRCC, J. Advanced Concrete Technology, 4(1): 45–57. Takahashi, Y. and Fenves, G. (2006). Software framework for distributed experimental computational simulation of structural systems, Earthquake Eng. and Structural Dynamics, 35(3): 267–291.

Lignos, D.G. and Krawinkler, H. (2009). Sidesway collapse of deteriorating structural systems under seismic excitations, Report TR 172, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Moncarz, P.D. and Krawinkler, H. (1981). Theory and application of experimental model analysis in earthquake engineering, Report No. 50, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Olsen, C. and Billington, S.L. (2009). Evaluation of precast, high-performance fiber-reinforced concrete infill panels for seismic retrofit of steel frame building: Phase 1-cyclic testing of single panel components, Technical Report No. TR 158, John A Blume Earthquake Engineering Center, Stan-ford University, Stanford CA. OpenSees. (2009). ‘‘Open System for Earthquake Engineering Simulation,’’ Pacific Earthquake Engineering Research Center (PEER), (http://opensees.berkeley.edu). Rots, J.G. and Invernizzi, S. (2004). Regularized sequentially linear saw-tooth softening model, International Journal for Numerical and Analytical Methods in Geomechanics, 28: 821–856. Rouse, J.M. (2004). Behavior of Bridge Piers with Ductile Fiber Reinforced Hinge Regions and Vertical Unbonded Post-Tensioning, PhD thesis, Cornell Univ., New York.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Recent developments on computational modeling of material failure in plain and reinforced concrete structures A.E. Huespe, J. Oliver & G. Díaz Technical University of Catalonia, Campus Nord UPC, Barcelona, Spain

P.J. Sánchez INTEC-UNL-CONICET, Santa Fe, Argentina

ABSTRACT: New developments for the computational simulation of plain and reinforced concrete structures are presented. Two models based on different length scales are proposed: one, at the macroscopic level, considers a homogenized material. Another, at the mesoscale level, describes specifically each mechanical component of the reinforced concrete members. Numerical applications of both approaches addressed to different problems are presented. Particular attention is given to the plain concrete simulation in confined regimes in order to capture the high stress triaxiality compressive states, which are typical of reinforced structures. The plain concrete is described via a new phenomenological model which can be characterized by a low number of parameters. plane concrete constitutive model which is capable of capturing the wide range of failure modes observed in these structures. For example, the (distributed or concentrated) crack patterns displayed in tensile stress zones of RC structural members and the crushing effect produced in zones of high stress confinement. In order to take into account this characteristic response, in subsection 2.2 we present a new phenomenological constitutive relation for concrete that is well adapted for capturing this salient behavior. Its regularization is performed via the Continuum StrongDiscontinuity Approach (CSDA) Oliver, (2000) summarized in section 3. Numerical applications of this model are presented in section 4. The mesoscopic scale approach is adopted for simulating corroded reinforced concrete structures. In these structures, the effect of the slip fiber-concrete mechanism plays a crucial role in the structural strength. We show in section 5 that, although the corrosion problem is essentially a 3D phenomenon, it is amenable to be analyzed as a 2D problem via a two stages procedure. Finally, in section 6, some concluding remarks close this work.

1 INTRODUCTION Reinforced concrete (RC), constituted by concrete with long fibers (reinforcements) oriented in different directions embedded in it, can be analyzed following two different conceptual models implying different length scales: i. a macroscopic scale model describing the response of the composite material via a homogenized constitutive model. In this case, the success of the model relies on the homogenization procedure, which becomes the key issue in this type of conceptual approach; ii. a mesoscopic scale model describing the response of every composite material constituent (matrix, interface and reinforcement) as an independent subsystem that is mechanically interacting with the neighbor constituents. The numerical simulation of mesoscopic models requires demanding computational costs. In this contribution, we present applications of both conceptual models addressed to analyze RC structures. Following the macroscopic scale approach, it is possible to develop a rather simple homogenization procedure based on the mixture theory, which provides a computational model that captures the most salient phenomena governing the failure of reinforced concrete structures. In this case, an important feature is that it only requires a reduced computational effort. This procedure is described in section 2. In any case, whatever is the adopted conceptual approach, it is necessary to include a regularized

2 MACROSCOPIC APPROACH BASED ON A HOMOGENEIZED RC MODEL We summarize in this section a macroscopic model based on the mixture theory, which is adopted as a procedure for homogenization of the RC material response. This model was initially proposed by Linero, (2006), see also Oliver et al., (2008b), FOR additional details.

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2.1 Constitutive model for the composite Reinforced concrete is assumed to be a composite material constituted of a matrix (concrete) and long fibers (steel rebars) arranged in different directions, as shown in Figure 1. According to the basic hypothesis of the mixture theory, the composite material is a continuum where each infinitesimal volume is occupied by all the constituents in terms of a volumetric fraction given by the factor k i ≤ 1 (for the i-th constituent). Assuming a parallel layout, all constituents are subjected to the same composite deformation ε. The composite stress, σ, is obtained by summing up the stress of each constituent, weighted according to its corresponding volumetric fraction k i , as shown in Figure 2. Thus, the matrix strain, εm , coincides with the composite strain, ε: εm = ε

(1) Figure 2. Constitutive model for the composite using a mixf f ture theory:  m , σ , τ are the constitutive material laws for the matrix, the fiber uniaxial effect and dowel action constituents, respectively.

Considering a fiber f in the direction r, a local orthogonal reference frame (r, s), can be associated with it, as shown in Figure 1. Then, the extensional strain, εf , of the fiber f is equal to the component εrr of the composite strain field in that direction, that is: εf = r · ε · r

rf (f = 1, 2, . . ., nf ) can be obtained using the following weighted sum of each contribution:

(2) σ = k m σm (εm ) +

In order to take into account the dowel action, the fiber shear strains, γ f , are obtained as the shear components of the composite strain field. The fiber shear f component εrs is given by: εfrs =

γf =r·ε·s 2

nf  f =1

k f σ f (εf )(r f ⊗ r f )

+ 2τ f (γf )(rf ⊗ sf )sym

(4)

where k m and k f , are the volumetric fraction of the matrix and fiber f , respectively, σm is the matrix stress tensor, σf is the fiber axial stress, and τf is the fiber shear stress component. In eq. (4) it is assumed that the normal and tangential stress components of the fibers are related to the corresponding strains by means of specific constitutive equations in a completely decoupled behavior of the matrix response. The incremental form of the composite constitutive equation can be written as:

(3)

The stresses of a composite having nf fibers (or fiber bundles) oriented in different directions

σ˙ = Dtg : ε˙ Dtg = k m Dm tg + +

nf  f =1

f 4Gtg (rf

f

k f [Etg (rf ⊗ rf ) ⊗ (rf ⊗ rf )

⊗ sf )sym ⊗ (rf ⊗ sf )sym ]; f

Figure 1.

(5) f

m m f f where: Dm tg = ∂σ /∂ε ; Etg = ∂σ /∂ε ; Gtg = f f ∂σ /∂ε are the tangent moduli for the involved constitutive relations of concrete and fiber respectively.

Reinforced concrete as a composite material.

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2.2 Constitutive model for concrete

Table 1. Constitutive damage-plastic model for the concrete constituent.

Concrete shows very different responses for either tensile or compressive stress regimes. In each case, the most striking difference is observed in the failure mode. Under tensile stresses, the concrete displays a much lower strength than in compressive states. Also, it shows a higher brittleness in tensile stress conditions. In fact, formation of cracks is expected only if tensile states are observed. While in compressive states, the concrete behaves like a plastic material, sometimes displaying failure mechanisms like shear bands. Additionally, in reinforced concrete structures, concrete is generally subjected to high confinement stress states due to stirrups, a condition that plays a very important role in the structural strength. Therefore, it is advisable to use a concrete constitutive relation having the ability to capture the phenomenology observed under both tensile and compressive stress conditions. Previous concrete models addressed to capture this phenomenology where reported by Feenstra & de ˇ Borst, (1996) and Cervenka & Papanikolaou, (2008). The present constitutive model for the concrete matrix is described by a phenomenological damageelasto-plastic law, with the following main features:

Stress strain relationship: q σ = (1 − d)C : (ε − εp ) = σ¯ r Brittle-Ductile behavior: If (σm ≥ 0) or (σm = 0) & Tr( C : ε˙ ≥ 0) then Damage model, only possible evolution of variable d(˙εp = 0) ft ft r|t=0 = r0 = √ ; fc = ; η E √ −1 g(σ , q) = τσ − q; τσ = σ : C : σ;

r˙ = λ;

q˙ = H (r)˙r ; λ ≥ 0;

H < 0;

g ≤ 0;

qo = ro

λg = 0;

else if (σm < 0) or (σm = 0) & Tr(C: ε˙ < 0) then Elasto-Plastic model, only possible evolution of variable ε˙ p (d˙ = 0) rˆ (e, θ ) ψ(σ, ξ ) = Sβ + ξ(α)σm − km ; 2 ξ(α) ∈ [ξo , ξmax ]; ξ > 0; 1 S2 + χ ξ G σm − kmG ; 2 G ]; ξ G ∈ [ξoG , ξmax ∂G ∂G ; α˙ = −γ G ε˙ p = γ ∂σ ∂ξ γ ≥ 0; ψ ≤ 0; γ ψ = 0 G(σ, ξ G ) =

i. the fracture phenomenon, typical of tensile stress states, are described by an isotropic damage evolution law, identical to the model presented in Oliver, (2000), which is regularized by the CSDA methodology. The damage variable evolution is only admissible in tensile states that are identified by the mean stress, σm , being positive (σm ≥ 0). In this case, the material softening, caused by the damage evolution, induces material instability and strain localization, this being the precursor mechanism for the crack formation; ii. in compressive stress regimes, (σm ≤ 0), the concrete behaves like a plastic material that follows closely the Willam’s elastoplastic model, see Willam & Warnke, (1974). There is no damage evolution and, furthermore, the plastic model is equipped with a hardening modulus in order to induce a stable material response.

end

ii. λ and γ are standard damage and plastic multipliers; iii. ft and fc are the uniaxial tensile and compressive ultimate stress, respectively, and η = ft /fc is a material parameter governing the ratio between them (fc ≫ ft ). iv. (r, α) is the set of scalar, strain-like, internal variables, r being related with the damage model and α with the lastic model, respectively. v. (q, ξ ) is the set of scalar stress-like internal variables; from the thermodynamic point of view there is a duality relation between q and r. vi. The standard damage variable is defined by: d = 1 − q/r. Damage and plastic strain evolutions are mutually exclusive. Decision of which inelastic mechanism evolves depends only on the sign of σm . vii. In tensile regimes, while the material is not completely degraded, it is verified that: 0 < q/r ≤ 1. Thus, the damage criterion is given by the function: g(σ, q) = 0. Geometrically, the surface g = 0, as defined in Table 1, is an ellipsoid in the Haigh-Westergaard’s space. The intersection between this surface and the octahedral plane is

In Table 1, the governing basic equations of the damage-plastic model are presented. The notation and some important observations are the following: i. σ, ε and εp are, respectively, the Cauchy stress, the total strain tensor and the plastic component of the strain tensor; σ¯ is the effective stress, associated with the damage model, being defined by the relation: σ¯ = C : (ε − εp ). The mean stress is: σm = tr(σ)/3 and the deviatoric stress tensor: S = dev(σ). With the symbol C we refer to the fourth order isotropic elastic Hooke’s tensor. E is the Young’s modulus;

33

√ a circle whose radius is: R = S = ( 2µ)q. During damage evolution, q → 0, and therefore the radius R tends to zero, or equivalently, the circle shrinks to a point. viii. The plastic yield surface is given by ψ(σ , ξ ) = 0. This surface intersects the octahedral plane as the circle of radius: R = S = (2km )1/β , since rˆ = 1 whenever σm = 0 (see next point). Thus, defining the parameter km through the simple restriction  2µq = (2km )1/β (6)

Note that rˆ depends on the eccentricity function e(σm ) which, in the present work, is empirically defined in terms of the mean stress value as follows: e(σm ) = eˆ + [1 − eˆ ] exp(3σm ); e(σm = 0) = 1;

  √ 1 3 3J3 θ = arccos ; 3/2 3 2J2 J3 = det(S);

rˆ (e = 1, θ) = 1

J2 =

1 (S : S) 2

(9)

x. The plastic-flow evolution is non-associative. In order to define this law, the potential function G(σ, ξ G ) is included. For determining the plastic potential surface G(σ, ξ G ) = 0 the parameter kmG (which is immaterial) could be computed from the equation:  2µq = (2kmG )1/2 (10)

− 1)2

θ + (2e ; 2(1 − e2 ) cos θ  +(2e − 1) 4(1 − e2 ) cos2 θ + 5e2 − 4e

rˆ = 

(8)

where eˆ is the eccentricity coefficient. Following to Kang, (1999), this parameter must satisfies: eˆ ∈ [0.5, 1]. In this contribution we have adopted eˆ = 0.54 Also, rˆ depends on the Lode’s angle θ defined by:

we force the continuity of both surfaces g(σ, q) = 0 and ψ(σ, ξ ) = 0 onto the octahedral plane for σm = 0, for any parameter q and, therefore, for any value of the damage variable d. Characterization of ξ is given in the next section. ix. The function rˆ (e, θ ), due to Willam & Warnke, (1974) (see also Kang, (1999)), defines the roundness of the plastic yield surface octahedral planes, see Figure 3-b, and it is given by: 4(1 − e2 ) cos2

e(σm → −∞) = eˆ

(7)

Characterization of ξ G is given in the next section. The scalar factor χ accounts for the degree of dilatancy introduced in the model. xi. From the discussed expressions it is clear that the evolution of damage modifies the plastic yield surface ψ = 0 as well as the potential surface G = 0, but it does not modify the set of plastic internal variables {ε p , α}. On the other hand, and since the hardening of the elasto-plastic model affects only the pressure dependent term, the evolution of plasticity have not any influence on the damage constitutive relation, neither in the value of their internal variables {q, r} nor in the evolution of its limit surface g = 0. 2.2.1 Parameter characterization for the concrete model A salient feature of this model is the relatively small number of parameters necessary to characterize the material. Similarly to the isotropic damage model presented in Oliver, (2000), the required damage model parameters, are the standard elastic parameters: Young’s modulus E and Poisson’s ratio ν; the tensile strength ft , and the fracture energy Gf which identifies the intrinsic softening parameter H¯ as will be explained in the next section. Furthermore, the plastic model is characterized by the (pseudo) cohesion km which is fully defined

Figure 3. Reinforced concrete model; a) damage and plastic yield surfaces in the meridian plane; b) cross-sections of the plastic yield surface with octahedral planes.

34

through the variable q, as indicated in eq. (6). The parameter β adjusts the failure surface profile in the plane (σm S), see Figure 3-a. By using several experimental works of concrete failure in confinement stress regimes, see for example the literature cited in Kang, (1999), a good fitting is obtained with β = 1.48. In order to preserve the material stability, the hardening function ξ(α) is a monotonous increasing function with an asymptotic limit: ξmax = lim α→∞ ξ(α) as shown in Figure 4. ξmax can be characterized through a uniaxial compression test with an ultimate compressive strength given by fc , while ξo , which determines the initial yield condition, is identically characterized using a uniaxial compressive strength of value: fc /3. With this material model is possible to reproduce reasonably well a number of classical and well known uniaxial, biaxial and triaxial concrete tests. For example, Figure 5 displays the classical biaxial experimental results reported by Kupfer in 1973. We remark

the close description of the model, particularly, in the second and fourth quadrants. 2.3 Constitutive model for the steel fiber Steel fibers are regarded as one-dimensional elements embedded in the matrix. They can contribute to the composite mechanical behavior introducing axial or shear strength and stiffness. The axial contribution of each fiber bundle depends on its mechanical properties and the matrix fiber bond/slip behavior. The combination of both mechanisms is modeled by the slipping-fiber model described below. 2.3.1 Bond-slip effect The fiber axial contribution can be modeled through one-dimensional constitutive relations, relating extensional strains with normal stresses. The assumed compatibility between matrix and fiber strains allows capturing the slip effect due to the bond degradation by means of a specific strain component associated with the slip. Thus, the fiber extensional strain, ε f , given by eq. (2), can be assumed as a composition of two parts: one due to fiber mechanical deformation, ε d , and the other related to the equivalent relaxation due to the bond-slip in the matrix-fiber interface, εi : εf = ε d + ε i

(11)

Assuming a serial composition between fiber and interface, as illustrated in Figure 6-a, the normal stress of the slipping-fiber model, σ f , is equal to each component of stress:

Figure 4. Reinforced concrete model; description of the function ξ(α) governing the plastic hardening.

σf = σd = σi

(12)

The stress associated to the fiber elongation, as well as the one associated with the matrix-fiber slip effect, can be related to the corresponding strain component by means of a uniaxial linearly elastic/perfectly plastic constitutive model as shown in Figure 6-b with material parameters defined from the composition of both models: Ef =

1/E d

1 ; + 1/E i

i σyf = min[σyd , σadh ]

(13)

in which E d and σyd are the Young’s modulus and yield stress of the steel, respectively, E i is the interd face elastic modulus and σadh is the interface bond limit stress, which is an upper bound for the fiber—matrix f interface adherence: σy .

Figure 5. Reinforced concrete model; profile of the failure criteria (ξ = ξmax ) under biaxial conditions. Results with the proposed model are plotted in solid line.

35

f

τ˙ f = Gtg γ˙ f

(14) f

where the shear tangent modulus Gtg is given by: ⎧ f ⎪ ⎨G f Gtg = Gf H f τ ⎪ ⎩ f G + Hf τ

(elastic/unloading) (15) (loading)

The constitutive elastic shear modulus, G f , is given f by: G f = E f /2 and the yield shear stress, τy , is given f f √ by τy = σy / 3 . The fiber shear hardening/softening modulus is commonly assumed as H f τ = 0. 3 CONSTITUTIVE REGULARIZATION VIA CONTINUUM-STRONG DISCONTINUITY APPROACH (CSDA) Introducing strain softening (H (r) < 0) in the damage model, forces the consideration of some type of constitutive regularization technique in order to preserve the mathematical and physical consistency of the (local) Boundary Value Problem. For this purposes, the so called Continuum Strong Discontinuity Approach (CSDA) was adopted in the present work. The basic foundations and theoretical concepts about the CSDA can be found in many previous contributions, see for example Oliver et al., (2002) and references cited therein. Here, we only summarize its main features. As a consequence of material softening, strain localization modes and (probably) macro discontinuities (like fractures or cracks) can develop in the concrete under tensile regimes. This particular phenomenology is considered in the CSDA by means of an enhanced kinematical representation which introduces additional discontinuous modes in the displacement field. Such kinematics (commonly know as Strong Discontinuity Kinematics) can be formally described in terms of the nomenclature displayed in Figure 7.

Figure 6. Slipping-fiber model: the composition (+) of both elements (d) and (i) must be understood as a serial mechanical system, in the sense that, deformations are additive (ε. f = ε d + εi ) and the stresses are common (σ f = σ d = σ i ).

2.3.2 Parameter characterization of the bond— slip model: slipping-fiber model The parameters required for the bond/slip model characterization can be calibrated from the pull-out test, in which a bar embedded into a concrete core is subjected to a force applied at its free end. Additional details can be seen in Linero, (2006). 2.3.3 Constitutive model of rebars in shear mode (shear-resistant fiber). Dowel action model In a reinforced concrete member, when cracks open in mode II, the internal locking between particles (aggregate interlock) withstands some shear forces across the crack interfaces. The steel bars also introduce an important contribution to the shear strengthening effect, known as dowel action. This phenomenon has been widely studied and several authors have included this effect into their numerical simulations, see Belletti et al., (2001; Kollegger & Mehlhorn, (1990; Pietruszczak & Winnicki, (2003). Based on the previous considerations, the dowel action is modeled by means of a one-dimensional shear stress-strain elasto-plastic constitutive model, similar to the previously mentioned one for the axial stressstrain. In this case, the 1D constitutive model relates the fiber shear stress, τ f , associated to the local coordinate system (r, s) (see Figure 1), with the corresponding fiber shear stain, γ f , by means of:

3.1 Strong discontinuity kinematics and its regularization Let us consider a body  displaying a discontinuous displacement field across the surface S with normal n and splitting the body  into two disjoints parts, + and − . An admissible displacement field, u(x), exhibiting discontinuities can be described by: u(x, t) = u(x, t) + HS (x) · [[u]] (x, t)

(16)

where u(x) is a smooth field, HS is the Heaviside/step function shifted to S. The second term captures the displacement jump, given by [[u]], at the discontinuity interface S.

36

path. Additional details of this particular aspect can be seen in Oliver, (2000) and Oliver et al., (2002).

4 NUMERICAL EXAMPLES WITH THE MACROSCOPIC MODEL 4.1 Reinforced concrete beam test

Figure 7.

In order to assess the macroscopic RC approach of the previous sections, we simulate one of the classical beam cases that were tested by Vecchio & Shim, (2004) and previously by Bresler & Scordelis, (1963) more than 40 years ago. Vecchio et al. observes that the failure mode of this series of beams is highly influenced by crushing of concrete below the loading plate. Also, they have observed that triaxial confinement effect in that zone is important and must be taken into account in the numerical approach. We simulate the case B3, in the reference work, using a 3D numerical FE model. Details of the beam dimensions, as well as the distribution and quantity of the reinforcement bars and stirrups, are depicted in Figure 8. In this case, these authors reported that the observed failure mode corresponds to flexurecompression induced by crushing of the concrete in the compressive zone. The material parameters for the numerical model are taken from Table 2, which closely correspond with that values reported in the reference work. In Figure 9 we depict the load vs. mid-point vertical displacement curve and compare the numerical solution with the experimental results. The simulation excellently matches the initial beam stiffness and reasonably captures the ultimate loading states.

Strong discontinuity kinematics.

The strain field, kinematicaly compatible with the discontinuous displacement field u(x), is given by: sym ε = ∇ sym u = ∇ sym S n ⊗ [[u]])

 u + (δ

  regular

singular

µS = ∇ sym u + (n ⊗ [[u]])sym k

(17)

where the emerging surface Dirac’s delta function, δ s , has been regularized in (17) in terms of a, very small, regularization parameter k and a collocation function µs (x) on the discontinuity interface S. 3.2 Constitutive model regularization A key point in the (CSDA) is the fact that bounded stress states σ S should appear at the discontinuity interface S, even when singular strains (εS ) are present, see equation (17). This is achieved via a regularization of the continuum strain softening modulus which is defined in a distributional sense: H = δS−1 H

(18)

where H¯ is an intrinsic softening modulus that can be computed from the classical parameters used in Fracture Mechanics of quasi-brittle materials. An adequate definition of the function: H¯ = f (fc ; Gf ; E); makes the dissipative effects, captured by the continuum constitutive relation, consistent with the specific fracture energy Gf of the material. The re-interpretation given in expression (18) assures the compatibility between a singular kinematics and the continuum damage material model depicted in Table 1. Besides, when the strong discontinuity kinematics, equations (16) and (17), is consistently introduced in this continuum setting, a classical cohesive-type constitutive law (traction vs. displacement jump) is reproduced at the fracture interface as a projection of the original continuum constitutive model onto the strong discontinuity

Figure 8. Beam case ‘‘B3’’ Vecchio et al. (2004) experimental test. a) main geometrical dimensions; b) cross section; c) reinforcement distribution; d) cross section FE discretization.

37

Table 2.

Material parameters of the concrete beam test.

Concrete Poisson’s ratio Young’s modulus Compresive strength Tensile strength Fracture energy Steel Young’s modulus Yield stress Ultimate stress Dowel action stiffness Yield shear stress No slip effect was assumed

ν = 0.18 E = 26,075 [MPa] fc = 34.3 [MPa] ft = 2.65 [MPa] Gf = 0.063 [N/mm] E = 200 [GPa] σy = 436 [MPa] σu = 700 [MPa] G f = E/2 √ f τy = σy / 3 f (E f = E; σy= σy )

Figure 10. Beam case ‘‘B3’’ Vecchio et al. (2004): a) iso-displacement lines in the central zone; b) experimental result displaying the crack pattern distribution; c) numerical distribution of the damage variable.

Figure 11. Beam case ‘‘B3’’ Vecchio et al. (2004): Contour fill of the scalar variable (ltrack) displaying the crack pattern distribution captured by the numerical strategy.

the numerically tracked cracks. They include all those cracks that appeared and were tracked at some stage of the loading process but not necessarily active at the end of analysis. Figure 11 displays a very large number (more than 100) of these cracks. This shows the ability of the numerical model to handle a large number of evolving three-dimensional cracks, and, thus, its qualification for reproducing the crack distribution in heavily reinforced concrete structures (characterized by a large number of cracks).

Figure 9. Beam case ‘‘B3’’ ,Vecchio et al. (2004). Load vs. mid-span vertical displacement.

Description of the simulated crack pattern in concrete is shown in Figure 10-a. This figure displays, through the x-isodisplacement lines, four well developed vertical cracks (macro cracks) in the center zone of the beam. Figure 10-c presents the beam damage distribution obtained with the numerical model at the end of analysis. This result agrees with the experimental crack pattern displayed in Figure 10-b showing a rather distributed crack pattern. We have to remark that the crushing zone below the upper load, although considered in the model, is not represented in this damage variable distribution. Figure 11 confirms the results of Figure 10-c. In this figure we plot the scalar field (ltrack) defining

4.2 L-shaped panel A series of RC L-shaped structural member was tested by Winkler et al., (2001). In the present subsection, we show the numerical solution obtained with the macroscopic model of the panel corresponding to the series D reported in that paper. We compare our results with the experimental test. Dimensions of the specimen, loading condition and the reinforcement distribution are displayed in

38

Figure 12. L-shaped panel: a) geometric properties and boundary conditions (specimen width = 100 mm, the load P is located at 20 mm away from the free vertical face). Dimensions in mm; b) 3D FE model; c) FE grid accounting for the homogenized composite material model. Remaining elements are modeled as plane concrete. Table 3.

Figure 13. L-shaped panel. Plot of the Force P vs. the vertical displacement of the load application point. Experimental and numerical solutions.

Material parameters.

Concrete Poisson’s ratio Young’s modulus Compresive strength Tensile strength Fracture energy Steel Young’s modulus Yield stress Ultimate stress

ν = 0.18 E = 26,075 [MPa] fc = 29.45 [MPa] ft = 2.65 [MPa] Gf = 0.063 [N/mm] E = 179.0 [GPa] σy = 526 [MPa] σu = 584 [MPa] Figure 14. L-shaped panel. Comparison of the crack pattern distribution: experiments vs. numerical solution. Cracks in the numerical solutions are depicted through the contours of iso-displacement lines.

Figure 12-a. As can be seen there, a welded orthogonal reinforcement grid at 45◦ has been used in this specimen series. Steel bars of 6 mm diameter were used in the reinforcement grid having a grid spacing of 50 mm. The material parameters are specified in Table 3. We remark the large scattering of the concrete tensile strength, depending on the test technique, reported in the reference work. Figure 13 plots the load P vs. the vertical displacement of the load application point, computed with the numerical model and compared with the experimental results. It can be observed a rather good approximation in the initial stages, for vertical displacement less than 1.5 mm. After that point, the numerical solution shows a spurious drop not observed in the panel, which exhibits a ductile behavior. Figure 14 compares the crack pattern distribution of the numerical results with the experimental observation. As it can be seen in this case, results compare qualitatively well with the experiment.

5 MESOSCOPIC MODEL: AN APPLICATION TO CORRODED RC BEAMS 5.1 Problem description The corrosion phenomenon observed in reinforced concrete structures, largely limits the service life of these structures. In the present example we show an application of the proposed model to simulate the mechanical consequences of it: typically the loss of the structural load carrying capacity due to the reinforcement corrosion. The mechanism for the steel corrosion to cause a loss of the structural strength is the expansion of the

39

corroded rebar, which induces cracking in the concrete cover, loss of steel-concrete bond, as well as the net area reduction of the steel fiber. The effects of the mentioned mechanisms on the structural load carrying capacity can be analyzed as a function of the reinforcement corrosion degree. Therefore, the model makes possible to determine the influence and sensitivity of this key variable, the reinforcement corrosion level, in the structural deterioration problem. The proposed numerical strategy can be applied to beams, columns, slabs, etc., through two successive and coupled 2-D meso and macroscopic mechanical analyzes, as follows: i. For a number of structural cross sections of the beam we simulate, at mesoscopic level, the reinforcement expansion due to the volume increase of the steel bars as a consequence of the corrosion products accumulation (see next subsection 5.1.1). Then, the damage distribution and crack patterns in the concrete bulk and cover are evaluated, and the corresponding concrete loss of stiffness in the structural member is calculated. ii. A second macroscopic longitudinal analysis, now considering the results of the previous analysis in terms of the initial damage distribution and the rebars net section, evaluates the mechanical response of the structural member subjected to an external loading system. This evaluation, presented in subsection 5.1.2, determines the global response and the macroscopic mechanisms of structural failure. iii. An intermediate evaluation is performed in order to couple both analyses. Results of the stage i) are projected onto the structural analysis ii) as it will be explained in subsection 5.1.3.

Figure 15. Corroded beam problem: cross section analysis of the structural member (expansion mode).

interface. Each of them is characterized by a different constitutive response, and FE technology, that takes into account the main mechanisms involved in the corrosion process. A steel-concrete interface model is considered in order to capture the possible friction and slipping between both constituents once the concrete fractures. A contact linear triangular element adopted in the present model has been taken from Oliver et al., (2008a), see Figure 16-a, where additional details about its formulation can be obtained. In Figure 17, we display two different numerical solutions related to the expansion mechanism of steel bars for a predefined corrosion attack depth, and the degradation induced in concrete cover at the cross section level. From a qualitative point of view, it can be observed that the proposed (mesoscopic plane strain) numerical model captures physically admissible failure mechanisms. The introduction at the interface of friction/contact finite elements becomes crucial to obtain consistent crack patterns that match very well the semi analytical predictions published in the literature.

5.1.1 Cross section analysis of the structural member (expansion mode) Let us consider the cross section of an arbitrary RC structural member, as displayed in Figure 15-(b), whose reinforcement bars are experiencing a corrosion process, not necessarily identical in all of them. The products derived from the steel bar corrosion, such as ferric oxide rust, reduce the net steel area and accumulate, causing volumetric expansion of the bars (see Figure 15-(a)), what induces a high hoop tensile stress state in the surrounding concrete. As a consequence the cover concrete undergoes damage and a degradation process which results in two typical fracture patterns: (i) inclined cracks and (ii) delamination cracks, as observed in Figure 15-(c). Clearly, these induced cracks can increase the rate of corrosion process in the structural member. The two-dimensional plane strain mesoscopic model, idealized in Figure 15-(b), considers three different domains of analysis: (i) the concrete matrix, (ii) the steel reinforcement bars and (iii) the steel-concrete

5.1.2 Macroscopic model to simulate the structural load carrying capacity (flexural mode) The model of subsection 5.1.1 provides qualitative information related to the concrete degradation mechanisms due to the steel expansion. Nevertheless, it does not give additional information about the mechanical

40

behavior of a deteriorated RC structure subjected to external loads. Therefore, a 2-D macroscopic homogenized composite model, as the one described in section 2, is used to evaluate the residual load carrying capacity of the corroded RC member. The idealized scheme of the discrete model is shown in Figure 18. We have modeled the plain concrete by means of the model presented in section 2.2 which is regularized by means of the CSDA approach of section 3. 5.1.3 Coupling strategy Figure 19 shows an idealized scheme of the adopted strategy to couple the models presented above in sections 5.1.1 and 5.1.2. We transfer, from the domain of the generic geometric cross section, to the structural member domain, the average value of the damage variable ‘‘d’’ across horizontal slices in the cross section geometry. This projection is consistent because both analyses use the same continuum isotropic damage model for simulating the concrete domain. Thus, the final degradation state of concrete, induced by the rebar volumetric deformation process, is considered to be the initial damage condition for the subsequent longitudinal structural analysis. This means that we are assuming that the two models are coupled in only one direction, neglecting the structural load effects on the concrete damage evaluation determined in the cross section analysis.

Figure 16. Corroded beam problem. Contact finite element at the interface: a) representative scheme; b) typical contact element; c) scheme of the constitutive law for the contact normal stress σηη ; d) scheme of the constitutive law for the friction shear stress σηt .

5.1.4 Numerical results of the corroded structural member The structural analysis of the RC beam with the model of sections 5.1.1–5.1.3. provides structural responses taht change with the rebar corrosion attack depth. Sánchez et al., (2009) have used this approach to compare the numerical solution of several beam series tests performed by Rodriguez et al., (1995). From this study, in Figure 20 we reproduce the structural response of an identical beam subjected to different

Figure 17. Analysis of the rebar corrosion problem for two different reinforcement bar distribution. Plane strain expansion analysis: a) iso-displacement contour lines (pattern of cracks); b) scaled deformed configuration; c) Damage contour fill.

Figure 18. Corroded beam problem. Macroscopic FE model to simulate the structural load carrying capacity of a structural member (flexural mode).

41

Figure 21. Corroded beam problem Numerical results of the macroscopic FE model: a) contour fills of the damage variable displaying the crack pattern distribution captured by the model; b) iso-displacement contour lines displaying the active crack pattern at the end of analysis.

corrosion levels. There, it can be observed the sensitivity of the model with the reinforcement corrosion attack. The same structural analysis provides the damage distribution across the structural member, as shown in Figure 21-(a) for one case of corrosion level. Also, Figure 21-(b) shows the iso-displacement contour lines, which display the active crack pattern at the end of analysis and, therefore, the final failure mechanism. Figure 19. Corroded beam problem. Coupling strategy between the cross section analysis and the structural member analysis.

6 CONCLUSIONS Two different approaches for modeling the nonlinear mechanical behavior of RC structural members have been presented: i. a macroscopic material model that considers reinforced concrete as a composite material, with concrete and rebars as constituents, and ii. a mesoscopic model that takes into account the mechanical response of every constituent of the RC member in a separate way and includes, explicitly, their interaction effects. The specific ingredients of the constitutive models for plane concrete and rebars, and the framework for the material failure simulations have also been described. Within the resulting setting, a number of threedimensional simulations have been run, for assessment purposes, in heavily reinforced linear structural members (beams and panels). In most of cases they have displayed acceptable agreements between the physical reality, observed from experiments, and the simulation results. Ongoing research is done to assess if the proposed methodology can be extended to more complex reinforced concrete members as slabs and shells.

Figure 20. Corroded beam problem. Numerical results of the macroscopic FE model to simulate the structural load carrying capacity (load P vs. mid-span vertical displacement plots). The curves correspond to an identical RC beam subjected to a range of corrosion attack depths.

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ACKNOWLEDGMENT

Oliver, J., Huespe, A.E. & Cante, J.C. 2008a. An implicit/ explicit integration schemes to increase computability of non-linear material and contact/friction problems. Comput. Meth. App. Mech. Eng., 197, 1865–1889. Oliver, J., Huespe, A.E., Pulido, M.D.G. & Chaves, E. 2002. From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics, 69(2), 113–136. Oliver, J., Linero, D.L., Huespe, A.E.& Manzoli, O. 2008b. Two-dimensional modeling of mateiral failure in reinforced concrete by means of a continuum strong discontinuity approach. Comp. Meth. Appl. Mech. in Eng., 197, 332–348. Pietruszczak, S. & Winnicki, A. 2003. Constitutive model for concrete with embedded sets of reinforcement. Journal of Engineering Mechanics—ASCE, 129(7), 725–738. Rodriguez, J., Ortega, L. & Casal, J. (1995). Load carrying capacity of concrete structures with corroded reinforcement. Proc. of the 4th Int. Conf. on Structure Faults and Repair.: Engineering Tech. Press., Edinburgh, U.K. Sánchez, P.J., Huespe, A.E., Oliver, J. & Toro, S. 2009. Mesoscopic model to simulate the mechanical behavior of reinforced concrete members affected by corrosion. Int. J. Solids & Struct., published ‘‘on-line’’, doi: 10.1016/j.ijsolstr.2009.10.023. Vecchio, F.J. & Shim, W. 2004. Experimental and analytical reexamination of classical concrete beam tests. J. of Struct. Eng., ASCE, 130(3), 460–469. Willam, K. & Warnke, E. (1974). Constitutive model for triaxial behaviour of concrete. Proc. of the Concrete Struct. Subjected to triaxial stresses: Institution Assoc. for Bridges and Struct. Eng.,19, Section III., Zurich. Winkler, B., Hofstetter, G. & Niederwanger, G. 2001. Experimental verification of a constitutive model for cracking. Proc. Instn. Mech. Engrs., 215(Part L), 75–86.

Financial support from the Spanish Ministry of Science and Innovation through grant BIA2008-00411 is gratefully acknowledged. REFERENCES Belletti, B., Cerioni, R. & Iori, I. 2001. Physical approach for reinforced-concrete (PARC) membrane elements. Journal of Structural Engineering. ASCE, 127(12), 1412–1426. Bresler, B. & Scordelis, A.C. 1963. Shear strength of reinforced concrete beams. J. Am. Concr. Inst., 60(1), 51–72. ˇ Cervenka, J. & Papanikolaou, V.K. 2008. Three dimensional combined fracture—plastic material model for concrete. Int. J. of Plast., 24(12), 2192–2220. Feenstra, P.H. & de Borst, R. 1996. A composite plasticity model for concrete. Int. J. os Solids & Struct., 33(5), 777–730. Kang, H.D. & Willam, K.J. 1999. Localization characteristics of triaxial concrete model. ASCE, JEM, 941–950. Kollegger, J. & Mehlhorn, G. 1990. Material model for the analysis of reinforced concrete surface structures. Computational Mechanics, 6, 341–357. Linero, D.L. (2006). A model of material failure for reinforced concrete via Continuum Strong Discontinuity Approach and mixing theory.PhD. Thesis, Technical University of Catalonia (UPC), Barcelona. Oliver, J. 2000. On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. International Journal of Solids and Structures, 37(48–50), 7207–7229.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Concrete under various loadings, way to model in a same framework: Damage, fracture and compaction J. Mazars, F. Dufour & C. Giry Lab. 3S-R, Institut Polytechnique de Grenoble, France

A. Rouquand & C. Pontiroli DGA/C.E.G. Gramat, France

ABSTRACT: The causes of the non linear behavior of concrete until failure are numerous and complex, particularly for non monotonic and rapid loadings. We present hereafter a strategy based on damage mechanics or a coupling of damage model and plasticity model including several effects: development and closure of cracks, damping, compaction, strain rate effects,...The idea being to describe with the same tools a wide variety of problems: static loading including the evaluation of crack opening, young concrete behaviour, dynamic loading at low and high velocity . . . The models are of explicit form, that makes possible their implementation into explicit numerical scheme well adapted to the treatment of fast dynamic problems. In this context the F.E. ‘‘Abaqus explicit’’ code is used for the PRM model which couples damage and plasticity. It has been successfully applied during the past few years to model the response of reinforced concrete structures subjected to severe loadings. In this paper the main model concepts are presented and some examples of numerical simulations are given (with some related to impact) and compared to experimental results in order to demonstrate the efficiency of the concepts used. 1 INTRODUCTION It has been shown (Mazars, 1986) that basically three different damage modes have to be considered (see Figure 1): A/Situation dominated by mode I related to local extension (εi > 0), B/Situation dominated by mode II (or/and 3) without any local extension, C/Situation related to the application of a strong hydrostatic pressure which leads to consolidation (pore collapse in the cement matrix). To simulate the behavior of concrete, plasticity (Ottosen 1979), damage models (Mazars 1986, Mazars et al. 1989, La Borderie et al. 1994, Jirasek 2004, Gatuingt et al. 2008) or fracture based approabreak ches (Bazant et al. 1996) are used. They are adapted to simulate the situation ‘‘A’’, often present in classical reinforced concrete structures. For severe loadings, related to natural or technological hazards (accidental or intentional actions) two further aspects must be considered: the dynamic nature of the loading and locally, high confinement pressure. However very few models are able to simulate both phenomena. Figure 1 presents the different situations listed before related to the type of loading. To model the behaviors which arise, the strategy is the coupling

A

B

With extensions in at least one direction

Without any extension

Under hydrostatic pressure

C

Mode I

Mode II

Pore collapse

With instability

With ductility

With compaction

Damage Model local extension state pilots threshold and damage evolutions

Plasticity Model Von Mises stress is used for plastic threshold and plastic flow

Compaction is created after a threshold through the evolution of the bulk modulus

Figure 1. Local damage modes in relation with the type of loading (Mazars 1986) and the kind of models used.

of a damage model and a plasticity model including compaction effects. This paper presents how the models have been set up, the experiments requisite to identify the material

45

parameters and series of applications at the material level and on reinforced concrete structures for which experimental results testify their effectiveness.

is expressed in terms of the principal extensions. An equivalent strain is defined as:  3 εi 2+ (4) ε eq = i=1

2 MODELLING CONCEPTS

where •+ is the McCauley bracket and εi are the principal strains. Accounting for isotropic hardening κ(D) the yield criterion of damage follows:

Constitutive laws for concrete are based on the principles of damage mechanics following the usual approach (Lemaitre et al. 1990). After choosing the state variables and the expression of free energy, derivations give the state laws that lead to the constitutive equations. Two different models are presented hereafter, the first one adapted to monotonic loadings having one scalar damage variable, and the second one adapted to cyclic loadings having two scalar damage variables and including crack closure and permanent strain effects.

f (ε, D) = εeq − κ(D)

Two evolution laws for damage are considered for tension and compression (index i refers either to traction or compression): Di = 1 −

εd0 (1 − Ai ) − Ai exp(−Bi (εeq − εd0 )) (6) εeq

εd0 is the initial damage threshold; Ai and Bi are material parameters. The resulting damage to be introduced in the constitutive equation is a combination of those two scalar damage variables using the following weighting coefficients αt and αc [9]:

2.1 Mazars damage model Concrete—like most of the geomaterials and ceramics—is perceived like brittle in tension and more ductile under compression loading. During experimental tests, a network of microscopic cracks nucleates perpendicularly to the direction of extension, which coalesces until complete rupture. Whereas under uniaxial tension a single crack propagates, under compression and due to the presence of heterogeneities in materials (aggregate surrounded by a cement matrix) tensile transverse strains generate a self-equilibrated stress field orthogonal to the loading direction. A pure mode I (extension) is thus considered to describe the behavior in compression. The influence of microcracking due to external loads is introduced via a single scalar damage variable D ranging from 0 (undamaged material) to 1 (completely damaged material). The free energy ψ for this model takes the following form: 1 ρψ = ε : (D) : ε (1) 2 (D) is the Hooke elasticity tensor depending on the actual value of D through the form (D) = 0 (1 − D), 0 being the elasticity tensor for the virgin material. From the state equations, σ = ρ∂ψ/∂ε, the constitutive state law for a scalar damage model coupled to elasticity leads to:    1 σ = (1 − D) KTr(ε)I + 2G ε − Tr(ε)I (2) 3 or, ε = 1/E(1 − D)[(1 + ν)σ − νTr(σ )I]

(5)

β

D = αt Dt + αcβ Dc

(7)

We denote σ + and σ − (σ = σ + +σ − ) the tensors in which appear only the positive and the negative principal stresses, respectively, and ε t , εc the strain tensors defined as: ε t = −1 : σ + and εc = −1 : σ −

(8)

(D) is a fourth-order symmetric tensor interpreted as the secant stiffness matrix and it is a function of damage. The weights αt and αc are defined by the following expressions: αt =

3  1

Hi

3  εti (εti + εci ) εci (εti + εci ) αc = Hi (9) 2 2 εeq εeq 1

αt and αc define the contribution of each type of damage. αt (respectively αc ) ranges from 0 (pure 3D compression state—respectively traction state) to 1 (pure 3D traction state—respectively compression state). Hi = 1 if εi = εci + εti ≥ 0, otherwise Hi = 0. From equation (9) it can be verified that for uniaxial tension αt = 1, αc = 0, D = Dt and vice versa for compression. β is a shear factor, generally equal to 1.06. Responses under compression and tension of this model are presented Figure 2. For the general case of a loaded structure, D is calculated from the local maximum value of εeq reached during the loading. Then, whatever is the following loading path (compression or traction) the local Young’s modulus is equal to E(1-D) which cannot allow to

(3)

K and G are the bulk modulus and the shear modulus respectively, E and ν are the Young’s modulus and the Poisson’s ratio respectively. I denotes the second order identity tensor. In order to introduce the nonsymmetric behavior of concrete, the failure criterion

46

2.2 Applications Such a model is useful for different kind of applications for monotonous loading. It allows to describe global and local behaviour including damage contours and can be completed in order to simulate thermomechanical effects and to evaluate crack opening. 2.2.1 Damage field and crack opening The sample presented Figure 3 is notched (2 notches at different levels) and loaded with a uniaxial traction until the propagation of cracks. Calculation is performed using the non local damage model presented before and a post treatment is used to deduce the crack opening from the strain field. The procedure used is from (Dufour et al. 2008), it consists: – to fit the line where ε¯ eq is maximum which gives the crack path (2 cracks are found in the present case, which is consistent with the experiment, cf. figure 3). – to consider short segments along the crack path and to get a perpendicular profile in the middle of each segment; the strain tensor is projected on these profiles in order to get the local evolution for εeq on each profile. If on the same path, a crack is considered instead of a damage field, displacement is locally discontinue and the corresponding strain tensor projected on the same profiles as before, is a Dirac function the intensity of which is directly related to the local crack opening; this Dirac function regularized using the non local expression for εeq (equation 10) gives a curve which can be compared to the one coming from the non local damage modelling. The crack opening is obtained assuming same maximum value for

Figure 2. Traction and compression response of the Mazars damage model.

simulate the well-known crack- closure effect (unilateral effect) which appears when the stress goes from traction to compression. This model is used in a non local version to avoid mesh dependency with the following concepts (Pijaudier-Cabot et al. 1987): – An average value for εeq is used at point M(x): ε¯ eq (x) =



− s)εeq (s)dv

φ(x − s)dv

φ(x



(10)

– is the characteristic volume on which ε¯ eq is calculated and φ is the weighting function:  

x − s 2 φ(x − s) = exp − 2 lc

(11)

It has been shown that when is a circle (plane problem) with a constant size during the whole loading process some artifacts appear (close to an edge or close to a crack). A new proposition is done by (Giry et al. 2010) in which, during the loading,

becomes elliptic, the orientation and the size of which depending on the local stress field.

Figure 3. Notched specimen in traction—On the left ‘‘non local damage’’ - On the right the crack opening deduced from post treatment; bottom, zoom on the experimental crack (Shi et al. 2000).

47

in the French design code giving the strength at day j: fcj = 0.685 fc28 log(j + 1). For the general case of concrete structures, some others phenomena must be taken into account such as dilation and shrinkage or creep and relaxation. This is done through the use of a partition of the strain tensor:

both non local ε¯ eq curves. This has been performed on the notched sample presented in Figure 3 and the results (crack path and opening) are consistent with the experimental data (Shi et al. 2000). 2.2.2 Prevision of the behaviour of young concrete For young concrete, hydration of the cement paste generates a progressive increase of mechanical properties. This can be taken into account through the use of a new state variable: maturity M , a function of temperature T and the actual time t: M=

K(T ) 1 + K(T )

ε = εe (E, D, T ) + ε d (T , M ) + εc (E, M ) e

ε is the total strain, ε is the elastic strain, ε is the dilation strain and εc the creep strain. Details can be found in (Mazars et al. 1998). 2.3 PRM damage model for cyclic loading

(12)

Based on the previous model (Mazars, 1986) and from works of Pontiroli (1995) and Rouquand (1995 & 2005), this model, named ‘‘PRM model’’, simulates the cyclic behavior of concrete for low confinement (type A in figure 1). The main additionnal aspect is crack-closure effects and permanent strains. As for the previous model, the PRM model distinguishes the behavior under traction and the behavior under compression, but Dt and Dc have, each one, the status of a state variable individually instead of D (issue from the coupling of Dt and Dc –equation (7)) in the Mazars model. Between these two loading states a transition zone is defined by (σft , εft ), where σft and εft are the crack closure stress and the crack closure strain respectively. The main constitutive equations of the PRM model for a uniaxial loading are the following.

exp(− Tb )

K(T ) = k derives from the Arrhenius law, T is the temperature at time t, t0 corresponds to the beginning of the hardening phase, k and b are material parameters. M evolves between 0 at t = t0 and 1 at tf (infinite time), when hydration is over. The Young’s modulus and the Poisson’s ratio evolve with maturity. Simple laws have been proposed (Mazars et al. 1998): E(M ) = Ef M

and ν(M ) = 0.5(1 − M ) + νf M (13)

Ef and νf are respectively the Young’s modulus and the Poisson’s ratio at complete maturation (M = 1). Loading of young concrete generates damage. This can be forecast using the following coupling: E(M , D) = Ef M (1 − D)

(15) d

for traction:

(14)

(σ − σft ) = E(1 − Dt ) · (ε − εft )

Then, introducing this expression in equation (3), one can study the effects of time, temperature and loading on the material. In that way, the serie of curves Figure 4 gives the evolution of the compression behaviour from 0.5 to 28 days of a concrete cured at 20◦ C. These results are consistent with results reported

(16)

for compression: (σ − σft ) = E(1 − Dc ) · (ε − εft )

(17)

Before any damage in compression εft = εft0 and σft = σft0 (material parameters), afterwards σft is directly link to Dc as follows: σft = σft0 (1 − Dc )2

(18)

E is the initial Young’s modulus; Dt = 1 −

εd0t (1 − At ) − At exp(−Bt (εeq − εd0t )) εeq (19)

(εfc , σfc ) defines a focal point from which the damage compression Young’s modulus E(1 − Dc ) is deduced (see figure 5). Then: Dc = Figure 4. Mazars model, coupling of Damage and Maturity: prevision of the evolution of the compression behaviour with time (temperature of curing is T = 20◦ C; stress and strain are considered positive for compression).

DcM 1 − εfc /εm

(20)

with DcM = f (εeq , ε0c , Ac , Bc ) as in equation (6), εm is the maximum compression strain reached during the loading path.

48

Remarks:

σ d = σ − σ ft ; ε d = ε − εft where σ ft and εft are the crack closure stress and strain tensors respectively, used to manage crack-closure effects. D remains a scalar and is issued from the combination law of the two damage modes Dt and Dc seen before (equation 22). αt evolves between 0 and 1 and the actual value depends on the sign of Tr(σ − σ ft ). This formulation is an explicit one. It has been implemented into ‘‘Abaqus explicit’’ and is used mainly for dynamic structural simulations. For more details see (Mazars 1986, Rouquand & Pontiroli 1995, Rouquand 2005).

– for both equations (19), (20), the equivalent strain is used with an expression adapted to the presence of permanent  strains: 2 εeq = i xi + and, depending on the sign of the stress in the direction i, xi = ει in compression and xi = (ε − εft )i in traction. – the thresholds in traction ε0t and compression ε0c can be different Figure 5 illustrates the corresponding response for a uniaxial cyclic loading. One can observe that if we use the combination damage rule as in equation (7), the behaviour can be described by the classical equation: σd = E0 (1 − D)εd

2.3.1 Strain rate effects—Internal friction, damping It is well known that concrete is strain rate dependent, particularly under tensile loading. This effect is d d accounted for using dynamic thresholds (ε0t and ε0c ) instead of static one’s (ε0t and ε0c ) through the use of a dynamic increase factor R = ε0d /ε0 . Its value for a compressive dynamic loading takes the following form:

(21)

with σd = σ − σft , εd = ε − εft and D = αt Dt + (1 − αt )Dc

(22)

αt is the activation factor. In the uniaxial case: αt = 1 if σd > 0 and

αt = 0

if σd < 0

Rc = min(1.0 + ac ε˙ bc , 2.50) (23)

and for a dynamic traction loading:

In the framework of isotropic damage evolutions (Dt and Dc are scalars), the general 3D constitutive equations are set up on the scheme used for 1D situation, which leads to:

Rt = min[max(1.0 + at ε˙ bt , 0.9˙ε0.46 ), 10.0]

(26)

ac , bc and at , bt are material coefficients identified from experimental results. For a high strain rate, the traction dynamic increase factor is supposed to follow an empirical formula (0.9˙ε0.46 ) that well agrees with the experimental data obtained by Brara & Klepaczko 1999 on Hopkinson bar tests on ordinary micro concrete (necessary due to the size of the tested specimen). Looking forward new results it is assumed that the same trend can be used for the concrete considered in the applications hereafter. Figure 6 illustrates the evolution of the compressive (dashed line) and tensile (continuous line) dynamic increase factors versus the strain rate.

   1 σ d = (1 − D) KTr(εd )I + 2G ε d − Tr(ε d )I 3 (24)

Figure 5.

(25)

Uniaxial response of the PRM damage model.

Figure 6.

49

PRM damage model: strain rate effects.

For cyclic loading, as the one encountered during an earthquake loading, friction stresses induce significant dissipated energy during unloading and reloading cycles. To account for this important phenomenon, an additional damping stress is introduced in the model: σ − σ ft = (σ − σ ft )damage + σ damping

confined cyclic loading (Rouquand 2005). For very high dynamic loads leading to a higher pressure level, an elasto-plastic model is more appropriate. For example, the impact of a projectile striking a concrete plate at 300m/s induces local pressures near the projectile nozzle of several hundred MPa, which induces a combination of type B and C modes seen Figure 1. The previous damage models can neither simulate the pore collapse phenomena nor the shear plastic strain occurring at this pressure level. To overcome these limitations, the elasto-plastic model proposed by (Krieg 1978), for geomaterials has been chosen to simulate this kind of problem. From this simple elasto-plastic model an improvement has been introduced in order to take into account the non linear elastic behavior encountered during an unloading and reloading cycle under a high pressure level. Another recent improvement has been done to take into account the water content effects, introducing an effective stress theory as described by C. Mariotti et al. (2003). This effect induces changes on the pressure volume curve and on the shear plastic stress limit. Finally, both models have been coupled to simulate the combination of the three types of situation (A, B, C) seen Figure 1.

(27)

The damping stress generates a hysteretic loop during the unloading and the reloading cycle. This stress is calculated from the damping ratio ξ , classically defined as the ratio between the area under the closed loop and the area under the linear elastic-damage stress curve, which gives for the uniaxial case: ξ=

Ah E(1 − D)(εmax − εft )2

(28)

Ah is the loop area under the stress strain curve, E(1 − D) is the current material stiffness. εmax is the maximum strain before unloading; εft is the closure strain that defines the transition point seen before. From (28) the damping stresses are computed in such a way that the damping ratio ξ is related to the damage D according to the relation: ξ = (β1 + β2 D)

2.4.1 The modified Krieg plastic model The Krieg model can be applied to describe the behavior of a dry material. It is based on a classical elastic purely plastic description using a parabolic deviatoric plastic threshold including a cut-off linked to the porosity of the material. We will see hereafter that this cut-off is also linked to the water content η figure 8 and § 2.4.2 for the relation between qmax and η):

(29)

β1 is a damping ratio for an undamaged and perfectly elastic material. β1 + β2 is the damping ratio for a fully damaged material. Usually β1 can be chosen equal to 0.02 and β2 can be chosen equal to 0.05. Figure 7 shows, for cyclic traction-compression loading, the strain-stress curve including damping stresses.

q = min (q0 =

2.4 Damage and plasticity. PRM coupled model

a0 + a1 P + a2 P 2 , qmax )

(30)

with q = (3/2σ d : σ d )1/2 (Von Mises stress), the stress tensor being σ = −P I + σ d , P is the confinement pressure and σ d the deviatoric stress tensor. The improvement made here concerns the spheric part of the behavior which is non linear and pressure

The previous damage model is very efficient to simulate the behavior of concrete for unconfined or low

Figure 8. Krieg modified model: shear yield threshold for dry and partially satured materials (qmax decreases when the water content η increases).

Figure 7. PRM damage model: cyclic loading including damping stresses.

50

dependent. This non linearity is more pronounced when the pore collapse phenomenon has progressed a lot. Figure 9 shows a typical P − εv curve used in this model (εv = Tr [ε], is the volumic strain). For pressure values lower than P1 , the behavior is linear and elastic. For a pressure greater than P1 , the pore collapse mechanism becomes effective. During the loading process the pressure-volume response follows a curve identified from experiments. During the unloading the behavior is elastic and non linear. The bulk modulus becomes pressure dependent. It is equal to Kmax at the first unloading point and decreases to Kmin when the tensile pressure cut-off Pmin is reached (this value is generally negative, which means that traction is necessary to recover the initial volume). This pressure cut-off becomes smaller and smaller as the maximum pressure Pmax increases. When Pmax reaches Pcons , Kmax becomes equal to Kgrain and Kmin is equal to K0grain . So the non linearity becomes more and more important as the pore collapse phenomenon has taken place a lot. When Pmax reaches Pcons , all the voids are crashed in the material which means that the pore collapse phenomenon is achieved. At this pressure level the material is consolidated and the behavior is purely elastic and non linear.

Figure 10. Water content effect on pressure volume relationship.

partially saturated material, the relation between pressure and volume is given by the response of the dry material until all the voids (part of the pores without water) are removed from the medium (at pressure Pvps ). Thereafter, the dashed curve gives the response of the solid and water mixture. The intersection of the dotted curve with the horizontal axis gives the porosity of the dry material. The intersection of the dashed curve with the horizontal axis gives the ‘‘free porosity’’ εvps of the partially saturated material. Consequently, when the material becomes drier and drier, the dashed curve moves to the right. To define the behaviour of this solid and water mixture (dashed curve) the pressure is assumed to increase in the same way in both phases (solid and liquid). So an iterative procedure as to be run in order to find the relative volume changes of each phase (see Rouquand 2005). Schematically the behaviour remains similar to the behaviour of a dry material until all the voids are removed. Thereafter the pressure difference between the two phases remains constant. At this point, the effective stress concept can be introduced:

2.4.2 Improvement for partially saturated materials Concrete and geologic media contain an open porous network. Internal water can move through the porous media from one void to another. To understand more easily the water effect, the material structure can be studied as a mixture of a solid medium with a void partially filled with water. For high dynamic loads, the time scale is very low (few milliseconds or less) so the water has no time to move inside the material, then undrained conditions can be considered. Figure 10 shows the generic pressure—volumic strain response of a partially saturated material. For a dry material, the response follows the solid curve. When the pressure is sufficient to collapse all the voids, the response is given by the dotted curve. In case of a

Peff = Pdry (εv ) if Pmax < Pvps

Peff = Pdry (εv ) − Pw (εv − εvps ) if Pmax > Pvps

(31)

Pw comes from the state equation of Mie-Gruneisen (Jonhson 1968) which describes the liquid behaviour under dynamic pressure. The solid phase behaviour is the non linear elastic model seen before (Figure 9). Water content η(ratio of the water volume on the pores volume)has also an effect on the shear behaviour. In the Krieg model, the plastic shear strength q (computed as the Von Mises stress) is pressure dependent (Figure 8). As the pressure increases, the shear yield stress increases too. This effect is the consequence of the porous structure of the material. During the pore collapse phenomenon the void ratio decreases, the contact area of the solid grains inside the material matrix increases so the pressure increases and

Figure 9. Krieg modified model: pressure—volumic strain behavior (P and εv are considered positive for compression).

51

the shear forces inducing sliding motions between the solid grains also increases. When all the voids have collapsed, the shear strength remains constant and becomes pressure independent because of the contact area which cannot increase any more. The material becomes ‘‘homogeneous’’ and the shear strength reaches a limit that is material dependent. Then the proposition is to relate shear strength to the effective pressure. For a dry material, the effective pressure is always equal to the total pressure. But for a partially saturated material, the effective pressure is the total pressure like in dry material until all the voids are removed (equation (31)). After consolidation the interstitial (or the effective) pressure does not increase any more because the pressure in the solid phase and in the water increases together and consequently the shear strength remains constant. As the water content increases, the pressure level Pvps at the consolidation point decreases and then the shear strength q also decreases. Figure 8 illustrates the effect of the effective pressure concept. The solid line gives the shear strength versus the pressure for a dry material. For a partially saturated one, the shear strength follows the solid line until the pressure Pvps is reached. Afterwards the shear strength does not increase and it follows the dashed horizontal line. Then the shear yield limit can be described by:   2 q = min q0 = a0 + a1 Peff + a2 Peff ,

allows to perform triaxial tests on cylindrical samples ( = 7 cm, h = 14 cm) with a confining pressure possible up to 1 GPa. Figure 11 shows a serie of tests on dry concrete used to calibrate the model. The hydrostatic loading part of five triaxial tests is used to fit the P − εv curve of the model and, on the shear yield threshold is fitted from a series of triaxial tests performed at increasing confinement (Gabet et al. 2008—Vu, Malécot et al. 2009). It is assumed that the threshold is reached when the evolution of the volume variation is reversed (dεv /dσ > 0). 2.5.2 Ability of the model to simulate various loading situations Figures 12 (experiment) and 13 (model) show the static response obtained on a cylindrical specimen for triaxial tests with increasing lateral pressure. The response of the model is in good agreement. It exhibits the activation of the damage part for low confinement, the plasticity part for high confinement (visible when there is a plateau), a combination of

 qmax (Pvps)

(32)

The PRM damage model has been coupled with the modified Krieg model. The coupling procedure ensures a perfect continuity between the two model responses. The damage model is activated if the maximum pressure is too low to start the pore collapse phenomenon or if the shear stress is too low to reach the shear yield stress. If not, the plastic model is the one activated and it drives the evolutions until the extensions sufficiently increase to lead to a damage failure. This model has been implemented in Abaqus explicit (Hibbitt et al. 2000) and to avoid mesh dependency the Hillerborg regularization method is used (Hillerborg 1976). This method insures constant fracture energy (Gf ) whatever is the size of finite elements. 2.5 Response of the model at the material level 2.5.1 Identification of the model In order to identify and to validate the model, specific tests have been performed at 3S-R Grenoble. As for the Mazars model the material parameters for the PRM damage model, are calibrated from uniaxial tests (traction and compression) and from existing data base for strain rate effects and damping. The identification of the modified Krieg model needs very specific tests. The GIGA machine at 3S-R

Figure 11. Tri-axial tests performed on the GIGA machine (3S-R)—Above, calibration of the P − εv response of the model—At the bottom, identification of the shear yield threshold (P, ε are considered >0 for compression, PRP means proportional test and TRX means triaxial test).

52

Beam elements, 2D plane stress elements and 3D solid elements are used to model the reinforced concrete beam. A single element is used in the depth direction with the 3D model. Taking advantage of the symmetry, only one half of the beam is modelled. The reinforcement material model is the classical Johnson & Cook (1983) plasticity model. The concrete and the steel reinforcement are supposed to be perfectly bonded. Figure 15 shows, at the end of the dynamic test, the tensile damage contours on the 3D beam model (upper part of the figure). The lower part shows the corresponding observed crack pattern. The computed cracks are mainly concentrated in the central part of beam like in the experiment.

Figure 12. Experimental response of concrete subjected to triaxial compression for various confinements (Gabet et al. 2008).

450 120 MPa

400

Axial stress (MPa)

350

80 MPa

300 40 MPa

250

20 MPa

200 150

10 MPa

100

5 MPa

50 0

-1

0 MPa

0

1

2

3 4 5 6 Axial strain (%)

7

8

9

10

Figure 13. Response of the PRM coupled model for triaxial compression at various confinements.

both in between and the reactivation of damage when failure occurs. Figure 14. Dynamic three point bending tests: experimental devices.

3 APPLICATION TO REINFORCED CONCRETE STRUCTURES 3.1 Dynamic three points bending test on a reinforced concrete beam Figure 14 shows the experimental device and the beam characteristics (in mm). These tests have been performed by (Agardh, Magnusson & Hanson 1999) in Sweden on a high strength reinforced concrete beam.

Figure 15. Computed and observed crack pattern.

53

Figure 16 shows the evolution in time of the measured force (cross points) compared to the three computed forces resulting from the three different meshes. The beam model gives the lower force. The 2D and 3D models give very similar results.

3.2 Impact of a soft projectile on a plate The problem presented here concerns the crash of a soft missile impacting a reinforced concrete structure. This kind of impact is not a very highly dynamic event but it can generate severe structural damage. Numerical simulations of this problem are not easy since there are strong interactions between the target behaviour and the missile crash behaviour. Correct predictions suppose that the response of the target and of the missile is properly modelled. In order to evaluate the capabilities of the explicit finite element code Abaqus including the PRM coupled model, 3-D numerical simulations of Meppen tests (tests n◦ 12 and n◦ 20 presented below) have been done (Riech, 1984, Rouquand, 2006). A 3-D finite element model with solid brick elements is the most appropriate to reproduce the local and the global complex strain field generated during the impact on the target. The projectile is composed of a thin steel tube which can be efficiently modelled using 3-D shell elements. The target size is 6.5 m by 6.0 m the thickness is 70 cm in test n◦ 12 and only 50 cm in test n◦ 20 (see the scheme for test n◦ 12 and test n◦ 20 on Figure 19 and figure 20 respectively). Figure 17 shows a view of the meshes. About 30 000 3-D solid elements are used to model the target. The reinforcement is introduced using the Abaqus rebar option. This option allows to take into account the stiffness and the mass contribution of the reinforcement in the elementary stiffness matrix and in the elementary mass matrix associated to the 3-D solid elements. The rebar definition requires the definition of the reinforcement constitutive material.

Figure 17. Soft shock: mesh of the projectile and of the target with the location of the points where displacements are measured (tests n◦ 12).

The projectile mesh is composed of about 6500 ‘‘S4R’’ shell elements used for thin or moderately thick structures. This projectile is a generic missile. Its length is 6 m and its diameter is 0.6 m. The thickness of the steel envelop is 7 mm in a first part and 10 mm in the second part. An additional mass is incorporated on the rear part of the projectile to model ballast. The behaviour of the metallic bars is modelled using a standard elastic and plastic model without any strain rate effect which remains low in the reinforcement. The behaviour of the metallic missile material is modelled using the Johnson Cook (1983) elasto-plastic model. Strain rate is now accounted for because during the projectile crash, it can reach about 1000/s (Souroushian et al., 1987, Karagiozova et al. 2002). In the physical tests, the reinforced concrete slab was put on a vertical position at the end of a rail. The projectile is accelerated along the rail and impacts the plate in its middle point. The slab is supported on a very stiff metallic frame of 5.4 m square centred on its rear face. To simulate this support, all the nodes on the rear face located along the frame, are fixed (zero displacement) in the plate normal direction. The missile initial velocity is 241.5 m/s in test n◦ 12 and 197.7 m/s in test n◦ 20. The missile axis and the velocity vector are perpendicular to the reinforced concrete plate. 3.2.1 Test n◦ 12 Figure 18 shows a comparison between measured and computed displacements on three points (w10, w6 and

Figure 16. Measured and computed dynamic load.

54

Figure 18. Soft shock: comparison between measured (dashed line) and computed (solid line) displacements for test n◦ 12.

O

w8—see figure 17). The agreement is good with a difference less than 10%. 3.2.2 Test n◦ 20 Figure 19 shows the comparison between the observed crack pattern and damage areas. In fact for practical reasons the contours are related to maximum tensile strains which are directly linked to damage through the equivalent strain (see § 2.2). This comparison shows that the conical zone with open tensile cracks corresponds more or less to the computed cracked zone (light grey contours). Figure 20 shows the comparison between the computed maximum tensile strains and the observed crack pattern. In this experiment severe damage is obtained in the concrete ahead the projectile nozzle. Because of the reduction of the slab thickness in this test, a concrete plug is now clearly observed in the experiment. This damage mode is the result of large shear strains around the plug. This kind of damage is correctly predicted in the numerical simulation.

Figure 19. Soft shock (test n◦ 12): comparison between computed and observed damages (O: impact location).

3.3 Shock wave propagation in a partially saturated media (tuff specimen) R.P. Swift (1973) details a serie of experimental tests where a spherical LX—0401 explosive charge, embedded in a geologic tuff specimen, detonates (Figure 21). The charge diameter is 2.5 cm. The size of the geologic material is large enough to prevent the comeback of reflecting waves during the first microseconds. For each test, 3 electromagnetic velocity gages are incorporated into the tuff sample to measure the material velocity (numbered from 1 to 3 on Figure 21). Four physical tests have been performed with different water content. The Abaqus explicit finite element code simulates the detonation of the pyrotechnic device and the shock

Figure 20. Soft shock (test n◦ 20): comparison between computed and observed damages (O: impact location).

55

Figure 21. Experimental shock wave tests in a geologic tuff specimen with various saturation ratios.

wave propagation through the tuff material. A single row of 2D axi-symmetric elements is used to model the problem. The total number of the finite elements is 550. The size of each element is 1 mm. The J.W.L. equation of state (Lee et al. 1973) simulates the behaviour of the explosive combustion products. The PRM coupled model with adapted material parameters is used for the tuff. As an example, the upper part of Figure 22 shows a comparison between experimental and numerical velocities at different points in the geologic specimen. Here, we consider a test with a water content of 58.8%. Three sets of curves are given. Each refers to a particular velocity transducer point (see Figure 21). The amplitude of the peak velocity and the arrival time of the shock wave are correctly predicted. The numerical simulation gives a stiffer material response so the arrival time is a little bit overestimated. The lower part of figure 22 shows the same kind of results for a water content of 81.4%. These results clearly show the great influence of the water contents on the shock wave characteristics (amplitude and arrival time) and confirm the ability of the PRM coupled model to simulate this effect.

Figure 22. Material velocity profiles for two water contents (58.8% and 81.4% respectively).

3.4 Impact on a T shape reinforced concrete structure (hard shock) This study is related to the analysis of the vulnerability of concrete structures under intentional actions. More specifically, the effect of a projectile of about 80 kg striking at more than 300m/s a reinforced concrete structure is studied. Such an experiment has been done by E. Buzaud et al. (2003). The 35NCD16 steel projectile has an ogival nozzle. Its diameter is 160 mm and its length is 960 mm. An accelerometer system is placed inside the projectile to measure the axial and lateral accelerations during the tests. Figure 23 shows the test configuration with the T shape concrete target. The size of each reinforced concrete square plate composing the target is 3 m. The thickness of the front part of the concrete structure is 400 mm and the thickness of the rear part is 300 mm. Reinforcement is composed of two steel layers (one on each side of the concrete plate) with 16 mm diameter bars.

Figure 23. Hard shock: test configuration, impact on T structure and the projectile mesh.

Other 10 mm diameter bars link each reinforcement mesh node of the face to face layers. The distance that separates each bar is 100 mm. The distance between the reinforcement layer and the top (or the bottom) plate surface is 50 mm (concrete cover). 3D numerical simulations have been done using the ABAQUS explicit finite element code. The total number of the finite elements is about 530 000 for the entire

56

model. The projectile material (Figure 23) is simulated using an elastic and perfectly plastic model with a plastic yield stress of 1300 Mpa. The reinforcement is also modelled with an isotropic hardening elastoplastic model. The initial yield stress is 600 Mpa and reaches 633 MPa for a failure strain ε = 0.13. The concrete behaviour is simulated with the PRM coupled model and an erosion method, applied to high twisted elements, is used for perforation zones. On Figure 24, the measured deceleration is compared to the computed value. Some differences can be seen but the overall deceleration shape is correctly predicted. Figure 25 shows the tensile damage contours at the end of the numerical simulation (T = 20 ms). The first part of the target is perforated and a rebound on the rear

310

Experiment

Velocity (m/s)

280

PRM coupled model

250 220 190 160 0

2

4

6

8 10 12 14 16 18 20

Time (ms)

Figure 26. Hard shock: measured and computed projectile velocity.

part is noticed. This has been observed experimentally. The projectile velocity is also close to the measured one (Figure 26).

4 CONCLUSION In the framework of damage and plasticity mechanics general constitutive models has been developed for concrete and concrete structures submitted to a large range of loadings (monotonic, cyclic, at high velocity and high confinement). Two models are presented. The simplest one (Mazars model) includes one scalar damage variable, is developed with non local concepts and is proposed with two recent improvements: the evaluation of crack opening and the simulation of young concrete behaviours. The second one (PRM coupled model) includes two damage variable and can simulate a lot of physical mechanisms like crack closure effects, strain rate effects, material damping induced by internal friction, shear plastic strains and compaction of porous media under high pressure. To validate this particular coupling of plasticity and damage, an extensive experimental programme has been performed at 3S-R Grenoble using the GIGA machine which permits high confinement up to 1 GPa (Gabet et al. 2008), and a new program is in progress on the large Hopkinson bar at JRC Ispra to complete the data base under high velocity loading. The new model implemented into the F.E. Abaqus explicit code, has been extensively used and can advantageously simulate a large range of problems going from quasi-static simulations on concrete structures to high dynamic problems related to the effect of low (Mazars et al., 2009) and high velocity impacts. The simulations presented here, compared to experimental results, show the relevance of the modelling used which allows to carry out true numerical

Figure 24. Hard shock: measured and computed projectile decelerations.

Figure 25. Hard shock: tensile damage contours at 20 ms. The projectile has perforated the upper part and penetrated the right part after a rebound.

57

experiments very useful for complex structures and/or extreme loadings.

La Borderie C., Mazars, J., Pijaudier-Cabot G., 1994. ‘‘Damage mechanics model for reinforced concrete structures under cyclic loading’’, A.C.I, 134:147–172, edited by W. Gerstle and Z.P. Bazant. Lee E., Finger M., Collins W., 1973. JWL equation of state coefficients for high explosives, Technical report UCID16189, Laurence Livermore National Laboratory, Livermore CA, USA. Lemaitre, J. and Chaboche, J.L., 1990. Mechanics of solids material. Cambridge University Press. Mariotti, C., Perlat, J.P., Guerin, J.M., 2003. ‘‘A numerical approach for partially saturated geomaterials under shock’’, International Journal of Impact Engineering 28 717–741. Mazars, J., 1986. ‘‘A description of micro and macro scale damage of concrete structures,’’ Engineering Fracture Mechanics, 25(5/6), pp. 729–737. Mazars, J., Pijaudier-Cabot, G., 1989. ‘‘Continuum damage theory—application to concrete,’’ Journal of Engineering Mechanics. 115(2), pp. 345–365. Mazars J., Bournazel J.P., 1998, Modelling of Damage Processes due to Volumic Variations for Maturing and Matured Concrete in Concrete: from Material to Structure, Edit. JP Bournazel & Y. Malier, RILEM publications, Paris. Mazars J., Rouquand A., Pontiroli C., Berthet-Rambaud P., Malécot Y., 2009, ‘‘Damage tools to model severe loading effects on reinforced concrete structures’’, Proc. ACI fall convention, New Orleans, USA. Ottosen, N.S., 1979. ‘‘Constitutive model for short time loading of concrete’’, Journal of Engineering Mechanics, ASCE, 105, pp. 127–141. Pijaudier-Cabot, G. and Bažant, Z.P., 1987. ‘‘Nonlocaldamage theory’’, Journal of Engineering Mechanics, 113, 1512–1533. Pontiroli, C., 1995. ‘‘Comportement au souffle des structures en béton armé, analyse expérimentale et modélisation,’’ PhD thesis ENS Cachan—France. Riech, H., Rüdiger, E., 1984. ‘‘Results on MEPPEN TESTS II/11 to II/22)’’, Technischer Bericht 1500 408 (RS 467). Rouquand, A., Pontiroli, C., 1995. ‘‘Some considerations on explicit damage models,’’ Proc.FRAMCOS-2, Ed.F.H. Wittmann, AEDIFICATIO Publish., Freiburg. Rouquand, A., 2005. ‘‘Presentation d’un modèle de comportement des géomatériaux, applications au calcul de structures,’’ C.E.G., report T2005-00021/CEG/NC. Shi, C., van Dam, A.G., van Mier, J., Sluys, B., 2000. Crack Interaction in Concrete, in Materials for Buildings and Structures, Editor F.H. Wittmann, Publisher J. Wiley & Sons. Soroushian, P., Choi, K.B., 1987. ‘‘Steel mechanical properties at different strain rates,’’ Journal of structural engineering, 4, pp. 663–205. Swift, R.P., 1973. Dynamic response of earth media to spherical stress waves, Final report, Physics International Compagny. Vu, X.H., Malecot, Y., Daudeville, L., Buzaud, E., 2009. ‘‘Experimental analysis of concrete behavior under high confinement: Effect of the saturation ratio,’’ International Journal of Solids and Structures, 46, 1105–1120.

ACKNOWLEDGEMENT The authors thank the French research network VOR, the ANR programme Mefisto and the French ministry of defense (DGA) which participated in this research financing numbers of tests and calculations performed. REFERENCES Agardh, L., Magnusson, J., Hansson, H., 1999. ‘‘High strength concrete beams subjected to impact loading, an experimental study’’, FOA Defence Research Establishment report, FOA-R-99-01187-311—SE. Bazant, Z.P., 1994. ‘‘Nonlocal damage theory based on micromechanic of crack interaction’’, Journal of Engineering Mechanics ASCE 120, pp. 593–617. Brara, A., 1999. «Etude expérimentale de la traction dynamique du béton par écaillage», PhD thesis Metz university—France. Dufour, F., Pijaudier-Cabot, G., Choinska, M., Huerta, A., 2008. ‘‘Extraction of a crack opening from a continuous approach using regularized damage models’’, Computers and Concrete, 5(4), 375–388. Gabet, T., Malecot, Y., Daudeville, L., 2008. ‘‘Triaxial behavior of concrete under high stresses: Influence of the loading path on compaction and limit states,’’ Cement and Concrete Research, 38(3), 403–412. Gatuingt, F., Desmorat, R., Chambart, M., Combescure, D., Guilbaud, D., 2008. ‘‘Anisotropic 3D delay-damage model to simulate concrete structures’’. Revue Européenne de Mécanique Numérique. 17, pp. 740–760. Giry, C., Dufour, F., Mazars, J., Kotronis, P., 2010. ‘‘Stress state influence on nonlocal intractions in damage modelling, Euro-c 2010’’, Rohrmoos-Shladming, Austria. Hibbit, Karlssonn & Sorensen Inc., 2000. Abaqus manuals, version 6.4. Hillerborg, A. et al., 1976. ‘‘Analysis of crack formation and growth in concrete beams of fracture mechanics and finite elements’’, Cement and Concrete Research, 6, pp. 773–782. Jirásek, M., 2004. ‘‘Non-local damage mechanics with application to concrete’’, Revue française de génie civil, 8, pp. 683–707. Johnson, G.R., Cook, W.H., 1983. ‘‘A constitutive model and data for letals subjected to large strains, high strain rates and high temperatures’’. Proc. 7th International Symposium on Ballistics. pp. 541–547. Johnson, James N., 1968. ‘‘Single-particle Model of a Solid: the Mie-Grüneisen Equation’’, American Journal of Physics, 36(10), pp. 917–919. Karagiozova, D., Jones, N., 2002. ‘‘Stress wave effects on the dynamic axial buckling of cylindrical shells under impact,’’ in Structures under shock and impact VII, Jones N., edit. CA Brebia. Krieg, R.D., 1978. ‘‘A simple constitutive description for soils ans crushable foams,’’ Sandia National Laboratories report, SC-DR-72–0833, Albuquerque, New Mexico.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Upscaling quasi-brittle strength of cement-based materials: A continuum micromechanics approach Bernhard Pichler & Christian Hellmich Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Vienna, Austria

ABSTRACT: It is well known from experiments that the strength of cement-based materials depends linearly on the degree of hydration, once a critical hydration degree has been surpassed. It is less known about the microstructural material characteristics which drive this dependence, nor about the nature of the hydration degree-strength relationship before the aforementioned critical hydration degree is reached. In order to elucidate the latter issues, we here present a micromechanical explanation for the hydration degree-strength relationships of cement pastes covering a large range of water-to-cement ratios: Therefore, we extend the one step elastobrittle homogenization scheme of Pichler, Hellmich, and Eberhardsteiner (2009) to a more realistic two-step approach. Accordingly, we envision, at a scale of fifteen to twenty microns, a hydrate foam (comprising spherical water and air phases as well as needle-shaped hydrate phases oriented isotropically in all space directions), which, at a higher scale of several hundred microns, acts as a contiguous matrix in which cement grains are embedded as spherical clinker inclusions. It is the mixture- and hydration degree-dependent load transfer of overall, cement paste-related, uniaxial stress states down to deviatoric stress peaks within the hydrate phases triggering local brittle failure, which determines the first nonlinear, and then linear dependence of brittle paste strength on the degree of hydration. When extending this approach to the concrete level, it is expected to support, among others, the performance-based design of shotcrete in the framework of the New Austrian Tunneling Method. Hellmich 2008). Following the related line of elastic limit-based (‘‘brittle’’) strength homogenization, we here extend and validate a micromechanics-based strength model for hydrating cement paste against experimental data by (Taplin 1959) who measured early-age strength evolutions of hydrating cement pastes with water-cement ratios ranging from 0.157 to 0.8. This contribution is structured as follows. After recalling fundamentals of continuum micromechanics and presenting cement paste as a hierarchical twolevel material (Section 2), we describe continuum micromechanics models for upscaling elasticity and compressive strength (Section 3). In Section 4, we identify, from dense hydrate foams with very low porosity, elastic properties which are, on average, representative for all hydrates. On this basis, our new micromechanics models predict elasticity and strength of cement pastes as functions of watercement ratio and degree of hydration (Section 5). Finally, the upscaling scheme for cement paste strength is quantitatively verified through the landmark experiments of Taplin (1959), in Section 6, before we conclude by comparing our brittle upscaling scheme with recently published ductile schemes related to the upscaling of confined hardness measurements on cement pastes and concretes (Section 7).

1 INTRODUCTION Cement paste is the binder for cement-based materials, including cement mortar, concrete, shotcrete, and soilcrete. Therefore, a reliable prediction of mechanical properties of cement paste is paramount for subsequent modeling activities, be they related to material behavior of cement-based composites or to the structural behavior of engineering constructions built up from these materials. Challenging applications even require modeling of the evolution of mechanical properties of hydrating cement-based materials, e.g. drill and blast tunneling according to the principles of the New Austrian Tunneling Method (NATM), where shotcrete tunnel shells are loaded by the inward moving rock, while the material still exhibits rather small maturities and undergoes the chemical hydration process. This provides the motivation for the present contribution which focuses on upscaling elasticity and strength of hydrating cement paste by means of continuum micromechanics. Within cement paste, hydration products establish the links that constitute a network of connected particles. Their non-spherical phase shape as well as their brittle failure behavior are essential for reliable micromechanics-based prediction of the strength evolution of cement-based materials (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and

59

2 FUNDAMENTALS OF CONTINUUM MICROMECHANICS

equilibrium conditions (disregarding volume forces)

2.1 Representative volume elements and separation of scales principle

and linear strain-displacement relations

divσ (x) = 0,

1 (3) (∇ξ + t ∇ξ ), 2 where x denotes the position vector, σ and ε, respectively, stand for the second-order tensors of stresses and strains, C for the fourth-order elastic stiffness tensor, and ξ for the displacement vector. The boundaries ∂ of the RVEs are subjected to linear displacements corresponding to a second-order strain tensor E, i.e. we prescribe so-called Hashin boundary conditions (Hashin 1983), also referred to as uniform strain boundary conditions ε(x) =

In continuum micromechanics (Hill 1963; Suquet 1997; Zaoui 1997; Zaoui 2002), a material is understood as a macro-homogeneous, but micro-heterogeneous body filling a representative volume element (RVE) with characteristic length ℓ. The separation of scales requirement implies (i) ℓ ≫ d, where d is standing for the characteristic length of inhomogeneities within the RVE, and (ii) ℓ ≪ D, where D stands for the characteristic lengths of dimensions or loading of a structure built up by the material defined on the RVE. Notably, ‘‘much smaller (≪)’’ does not necessarily imply more than a factor of 4 to 5 between the characteristic length of the heterogeneities and that of the RVE (Drugan and Willis 1996). In general, the microstructure within one RVE cannot be described in complete detail. Therefore, quasi-homogeneous subdomains with known physical quantities (such as volume fractions or elastic properties) are identified. They are called material phases. Once their mechanical behavior, their dosages within the RVE, their characteristic shapes, and the mode of their interactions are identified, the ‘‘homogenized’’ mechanical behavior of the overall material can be estimated, i.e. the relation between homogeneous deformations acting on the boundary of the RVE and resulting (average) stresses, or the ultimate stresses sustainable by the RVE, respectively. In the framework of multiscale homogenization theory, a material phase, identified at a specific scale of observation ‘‘A’’, exhibits a heterogeneous microstructure on a lower scale of observation ‘‘B’’. The mechanical behavior of this microheterogeneous phase can be estimated by that of an RVE with a characteristic length being smaller than or equal to the characteristic length of the aforementioned phase, i.e. that of inhomogeneities identified on observation scale ‘‘A’’, see, e.g. (Fritsch and Hellmich 2007).

ξ (x) = E · x. 2.3

(4)

Homogenization of elasticity

The geometric compatibility of the microscopic strain field ε(x) with boundary condition (4) implies the following strain average rule   1 ε(x)dV = fp ε p , (5) E=  p 

where p denotes an index running over all phases of the considered RVE, fp stands for the volume fraction of phase p, and ε p for the second-order tensor of average phase strains defined as  1 ε(x) dV . (6) εp = p p

In (6), p denotes the subvolume of the RVE occupied by phase p. Analogously to (5), macroscopic stresses  are set equal to the spatial average of the equilibrated local stresses σ (x) inside the RVE,   1 σ (x) dV = fp σ p , (7) =  p 

with σ p as the second-order tensor of average phase stresses, defined by analogy to (6). Linearity of the field equations (1)–(3) implies a linear strain concentration rule

2.2 Statement of the studied problem For cement paste, we employ two RVEs: The first one relates to the a polycrystalline hydrate foam (with spherical phases representing water and air, and with needle-shaped phases of hydration products exhibiting isotropically distributed orientations), and the second one relates to cement paste (with a spherical phase representing clinker embedded in a continuous hydrate foam matrix), see Figure 1. We are left with presenting the field equations and the boundary conditions. Within the volume  of both RVEs, we consider field equations of linear elasticity, i.e. generalized Hooke’s law accounting for linear elastic material behavior σ (x) = C(x) : ε(x),

(2)

εp = Ap : E,

(8)

with Ap as the fourth-order strain concentration tensor of phase p. Specification of the elastic constitutive law of phase p σ p = C p : εp ,

(9)

for strain concentration rule (8) and insertion of the resulting expression for the phase stresses σ p into the stress average rule (7) delivers a relation between macrostress  and macrostrain E. Comparison of this relation with the macroscopic elastic law

(1)

60

Figure 1. Micromechanical representation of cement paste microstructure through a two-step homogenization scheme: a) polycrystalline RVE of ‘‘hydrate foam’’ built up of water, hydrates, and air; (b) RVE of matrix-inclusion composite ‘‘cement paste’’ where clinker is embedded in hydrate foam matrix.

 = Chom : E allows for identification of the homogenized elasticity tensor as  Chom = fp C p : A p . (10)

Pichler, Scheiner, Hellmich 2008), on bone biomaterials (Fritsch, Dormieux, Hellmich, and Sanahuja 2009), and on gypsum (Sanahuja, Dormieux, Meille, Hellmich, and Fritsch 2009), we here model quasibrittle fracture in the framework of elastic limit analysis. Microscopic phase failure resulting in the failure of the entire RVE is governed by strain peaks rather than by average phase strains defined in (6). Strain levels corresponding to volumetric and deviatoric strain peaks can be estimated by corresponding quadratic strain averages (Dormieux, Molinari, and Kondo 2002). They can be easily derived from the elastic energy stored in the RVE, provided that the phases within the RVE exhibit isotropic elastic behavior

p

Eq. (10) highlights that knowledge of phase strain concentration tensors Ap allows for homogenization (upscaling) of phase stiffnesses to the homogenized elasticity tensor. As a rule, the concentration tensors Ap are not known up to analytical precision. Still, they can be estimated based on classical matrixinclusion problems of Eshelby (Eshelby 1957) and Laws (Laws 1977), for details see Zaoui (Zaoui 2002) and Benveniste (Benveniste 1987) Ap = [I + P0p : (Cp − C0 )]−1 −1  : fq [I + P0q : (Cq − C0 )]−1 .

(11)

Cp = 3kp J + 2µp K,

(13)

q

where Cp denotes the fourth-order isotropic elasticity tensor of phase p, kp and µp , respectively, denote the bulk modulus and the shear modulus of phase p, K stands for the deviatoric part of the fourth-order unity tensor, being defined as K = I−J, with I as the symmetric fourth-order unity tensor with components Iijrs = 1/2 (δir δjs + δis δjr ), and J = 1/3(1 ⊗ 1) stands for the volumetric part of the fourth-order unity tensor, where 1 denotes the second-order unity tensor with components δij (Kronecker delta), δij = 1 for i = j, and δij = 0 otherwise. In the context of modeling the uniaxial compressive strength evolution of cement-based materials, the quadratic phase average of the deviatoric strain field ε dev (x) = ε(x) − [tr ε(x)/3]1 over hydrates with specific orientations is of interest (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008), see also Section 3.4. At this stage, we recall the expression for the quadratic average of the deviatoric strain field over a general phase p, which reads according to

Specification of (10) for (11) delivers the related estimate of the homogenized elastic stiffness tensor as Chom =

 p

:

fp Cp : [I + P0p : (Cp − C0 )]−1

 q

fq [I +

P0p

0

−1

: (Cq − C )]

−1

(12) .

In (11) and (12), P0p denotes the fourth-order Hill tensor, accounting for the characteristic shape of phase p embedded in a matrix with stiffness C0 . Choice of C0 describes the interactions between material phases, see Sections 3.2 and 3.3 for details. 2.4 Homogenization of strength In the line of earlier work on cementitious materials (Pichler, Hellmich, and Eberhardsteiner 2009;

61

(Dormieux, Molinari, and Kondo 2002) as

εpdev =



1 p

=



1 4 fp E



p

:

1 dev (x) 2ε

products, and air: fclin (ξ ) =

: ε dev (x) dV (14)

∂Chom : E. ∂µp

Related deviatoric stress peaks follow simply as

σpdev

=



1 p



p

1 2 s(x)

: s(x) dV = 2µp εpdev

20(1 − ξ ) ≥ 0, 20 + 63(ω/c)

fH2 O (ξ ) =

63[(ω/c) − 0.42ξ ] ≥ 0, 20 + 63(ω/c)

fhyd (ξ ) =

43.15ξ , 20 + 63(ω/c)

fair (ξ ) =

3.31ξ . 20 + 63(ω/c)

In (16), ω/c denotes the water-to-cement mass ratio, and ξ stands for the hydration degree which is defined as the mass of currently formed hydrates over the mass of hydrates formed at completed hydration. Notably, air-filled pores are created since hydration products occupy a smaller volume than the reactants clinker and water.

(15)

where s(x) denotes the field of the deviatoric stress tensor, defined as s(x) = σ (x) − [tr σ (x)/3]1.

3.2 Homogenization step I: hydrate foam

3 CONTINUUM MICROMECHANICS OF CEMENT PASTE

The polycrystal-type mutual interaction of hydration products, water, and air suggests usage of the selfconsistent scheme (Hershey 1954; Kröner 1958; Hill 1965; Budiansky 1965) for determining the homogenized stiffness tensor of the hydrate foam: Chom hf . Accordingly, C0 in (12) is chosen to be equal to the homogenized stiffness of the hydrate foam itself, and the sum over p in (12) now extends over the air and water phases, as well as over an infinite amount of hydrate phases oriented in all space directions indicated by polar angles ϕ and ϑ. Consequently, we arrive at the following implicit tensorial expression for Chom hf :

3.1 Model inputs for cement paste Herein, we collect input data required for specifying the discussed elasticity and strength models for cement paste. Elasticity constants listed in Table 1 comprise intrinsic values for clinker, water, and air taken from the open literature, while properties of ‘‘hydration products’’ represent average elasticity constants of all types of hydrates, including Portlandite, Ettringite, and Calcium Silicate Hydrates (C-S-H) of all densities, see Section 4 for the related identification strategy. The evolution of cement paste-related phase volume fractions during hydration is accounted for by Powers’ hydration model (Powers and Brownyard 1948), providing the relative volumes of clinker, water, hydration

Chom hf =

 p

 −1 hf f˜p Cp : I + Psph : (Cp − Chom hf )

+ f˜hyd Chyd : Table 1. Intrinsic mechanical properties of microstructural constituents of clinker, water, and air. Bulk modulus k [GPa]

Shear modulus µ [GPa]

Source

Clinker

kclin = 116.7

µclin = 53.8

(Acker 2001)

Water (drained RVE) Water (sealed RVE)

kH2 O = 0.0

µH2 O = 0.0

kH2 O = 2.3

µH2 O = 0.0

Hydration products Air

khyd = 18.7

µhyd = 11.8

kair = 0.0

µair = 0.0

Phase

(16)

2ππ 

hf

I + Pcyl (ϕ, ϑ)

0 0

−1 sin ϑdϑdϕ  ) : (Chyd − Chom hf 4π   −1 hf : f˜p I + Psph : (Cp − Chom hf ) p

+ f˜hyd

2π π  0

(Bilaniuk and Wong 1993) Section 4

hf

I + Pcyl (ϕ, ϑ)

0

: (Chyd − Chom hf )

−1 sin ϑdϑdϕ −1 . 4π

(17)

In (17), the hydrate foam-related volume fractions of hydration products, water, and air (f˜hyd , f˜H2 O , and

62

f˜air ) follow from the cement paste-related volume fractions of clinker, water, hydration products, and air (fclin , fH2 O , fhyd , and fair , given in (16)) according to fj f˜j = 1 − fclin

⎧ ⎨hyd, j = H2 O, ⎩air.

almost no pre-peak nonlinearities (Pichler, Hellmich, and Eberhardsteiner 2009), strength of cement paste may be estimated through an elastic limit analysis. In this context, we essentially envision that each hydrate behaves linear elastically as long as microscopic deviatoric stress peaks remain below a specific critical value:

(18)

dev dev ≤ σhyd,crit max σhyd,ϕ,ϑ

For detailed explanations regarding the numerical evaluation of (17), as well as for computation of cp the Hill tensors for spherical material phases, Psph , cp and for cylindrical phases, Pcyl , respectively, see (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008).

ϕ,ϑ

(22)

Eq. (22), together with (20) and (21) constitutes a micromechanics-based brittle failure criterion for cement pastes, considering through derivation of the homogenized stiffness tensor (19), the effect of their constituents’ elastic properties, as well as of their microstructures and compositions. In words, this criterion reads as follows: If, due to (compressive uniaxial) dev macroscopic load increase, this critical value σhyd,crit is reached in the most heavily stressed region of the hydrate phase, the elastic limit on the microscale is reached, which, in turn, corresponds to the macroscopic elastic limit of cement paste, associated with failure of the material (under macroscopic uniaxial compression). An algorithm for evaluation of (20), i.e. for computation of deviatoric stress peaks in needle-shaped hydrates within a two-step homogenization approach, is presented in (Pichler, Scheiner, Hellmich 2008).

3.3 Homogenization step II: cement paste The interaction of the spherical clinker phase embedded in a continuous matrix built up by the hydrate foam suggests usage of the Mori-Tanaka scheme (Benveniste 1987; Mori and Tanaka 1973; Zaoui 2002) for determining the homogenized stiffness tensor of 0 cement paste: Chom cp . Accordingly, C in 12 is chosen to be equal to the homogenized stiffness of the hydrate foam, resulting in an explicit expression for Chom cp : hom Chom cp = {(1 − fclin )Chf hf

−1 + fclin Cclin : [I + Psph : (Cclin − Chom hf )] } hf

4 ELASTICITY OF HYDRATION PRODUCTS

: {(1 − fclin ) I + fclin [I + Psph −1 −1 : (Cclin − Chom hf )] } .

3.4

(19)

For the sake of simplicity, we introduce only one elasticity tensor representing the average behavior of all different types of hydration products including Portlandite, Ettringite, and Calcium Silicate Hydrates (C-S-H) of all densities. In this context, it would be desirable to produce a material sample consisting of hydration products only, and to perform elasticity tests on such a material. Deplorably, this is not possible. As a remedy, we consider the material coming closest to the aforementioned ‘‘pure hydrate agglomerate’’, i.e. to the situation where, in Figure 1, the volume fractions of water, air, and clinker are close to zero, while the volume fraction of hydrates approaches one. According to Powers’ hydration model, the aforementioned volume fractions are most closely attained in a fully cured (ξ = 1) cement paste with water-cement ratio amounting to 0.42. These fractions would be:

Strength modeling

Deriving the homogenized stiffness of cement paste (19) with respect to the shear modulus of a hydrate phase with a specific orientation (ϕ, ϑ) provides access to the deviatoric stress peak within hydrates of that orientation. According to (14) and (15) the deviatoric stress peak is expressed as a function of the macrostrain Ecp imposed on a RVE of cement paste dev σhyd,ϕ,ϑ =



µ2hyd fhyd,ϕ,ϑ

Ecp :

∂Chom cp ∂µϕ,ϑ

: Ecp

(20)

the macrostrain being itself related to the macrostress cp through the inverse elasticity law according to −1  : cp , Ecp = Chom cp

fclin = fH2 O = 0, (21)

fhyd = 92.9%,

fair = 7.1%. (23)

To get experimental access to the elastic properties of such a ‘‘close-to-a-pure-hydrate-agglomeration’’ cement paste, we consider data from (Helmuth and Turk 1966). These authors tested three different types of cement pastes: one was produced using almost

with Chom cp following from (19). Since macroscopic stress-strain diagrams obtained in uniaxial compression tests on cement pastes exhibit

63

exclusively tricalcium silicate (C3 S), and the remaining two contained Portland cements (two different products). Water-cement ratios ranged from 0.3 to 0.6. Young’s modulus was deduced from measurements of the fundamental flexural and torsional resonance frequencies of specimens that were allowed to cure for six to 14 months (C3 S samples) and for six to 24 months (Portland cement samples), respectively. Interpolating between experimental data by (Helmuth and Turk 1966) implies that the almost pure hydrate agglomeration (fully-cured cement paste with ω/c = 0.42) can be expected to exhibit a macroscopic Young’s modulus of 25 GPa, see circle in Figure 2. When studying the almost pure hydrate agglomeration by means of the proposed micromechanics model, (23) implies that the two-step homogenization scheme illustrated in Figure 1 degenerates to a one-step scheme where the hydrate foam (consisting exclusively of hydration products and air) is equal to cement paste. Hydrate foam-related volume fractions follow from inserting (23) into (18) as f˜H2 O = 0 ,

f˜hyd = 92.9% ,

f˜air = 7.1% .

and Lemarchand 2003), as being representative for all hydration products. Specifying (17) for the volume fractions (24), for vanishing bulk and shear moduli of air, for νhyd = 0.24, and for Ehyd = 29.158 GPa (this is equivalent to the elasticity constants of hydration products listed in Table 1) results in the sought result Ehf (ξ = 1, ω/c = 0.42) = 25 GPa. A first relevance check of the hydrate elasticity in terms of Ehyd and νhyd can be made when predicting the experimental data of Figure 2, referring to cement pastes with a wide range of water-cement ratios: Therefore, the two-step homogenization scheme (17) and (19) needs to be specified for the identified average hydrate elasticity constants, for the intrinsic elastic properties of clinker, for the vanishing effective stiffnesses of the drained fluid phases air and water, see Table 1, as well as for the phase volume fractions (16) and (18) specified for the maximum attainable hydration degree ⎧ ⎨ ω/c for ω/c ≤ 0.42 max ξ = 0.42 (25) ⎩ 1 for ω/c ≥ 0.42

(24)

Model predictions and experiments agree very well (Figure 2).

In the following, we use the elasticity model (17) to identify the average elasticity constants of the hydration products which, together with the vanishing bulk and shear moduli of air, results in a prediction of macroscopic Young’s modulus of the almost pure hydrate agglomeration as Ehf (ξ = 1, ω/c = 0.42) = 25 GPa. Since experimental data by (Helmuth and Turk 1966) provide access to one constant of isotropic elasticity only, we have to make an assumption on the average Poisson’s ratio of the hydration products. Considering that Calcium Silicate Hydrates are commonly the dominating hydration products, we treat their Poisson’s ratio νCSH = 0.24, see (Bernard, Ulm,

5 PERFORMANCE OF ELASTICITY AND STRENGTH MODELS Herein, we provide model-predicted evolutions of elasticity and strength of hydrating cement pastes with water-cement ratios ranging from 0.157 to 0.8. As expected, the hydration degree-induced stiffness increase is the larger the lower the water-cement ratio (Figures 3 and 4), and interestingly, this increase is of convex nature for low water-cement ratios, but of S-shaped nature for higher water-cement ratios, for both bulk and shear modulus evolutions (see Figures 3 and 4). In all model predictions, non-zero stiffnesses are reached already for very early hydration degrees,

Figure 2. Final Young’s modulus of cement paste: experimental results by (Helmuth and Turk 1966), taken from (Sanahuja, Dormieux, and Chanvillard 2007), and outputs of the proposed model.

Figure 3. Model-predicted evolution of bulk modulus of hydrating cement pastes.

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testing, the prisms were broken into two halves resulting in two specimens (2 by 0.5 by 0.5 in): On each of them, two (destructive) uniaxial compression tests in cross direction were carried out. This was possible, since the load plates were of rather small size, such that in each test an effective specimen length of less than three-quarters of an inch was destroyed. Because the crushed part did not always correspond exactly to the area of the load plate, (Taplin 1959) published the actually measured ultimate forces, rather than postprocessed compressive strength values, see the ordinate in Figure 6. After testing, (Taplin 1959) determined the bound water content of the crushed samples, which is a measure for the hydration degree, for a more elaborate theoretical discussion see, e.g. (Byfors 1980). In order to make our model predictions comparable to the experimental results by (Taplin 1959), we have to establish a link between the ordinates of dev Figures 5 and 6, whereby the unknown value of σhyd,crit should, preferably, not influence the result. This provides the motivation to divide the ordinate of Figure 5 by the dimensionless compressive strength predicted for a fully-cured cement paste with ω/c = 0.157, and the ordinate of Figure 6 by the maximum compressive strength measured on samples with ω/c = 0.157, see Figure 7 for the obtained result. This normalization makes sense since the largest compressive strength measured by (Taplin 1959) on a sample with ω/c = 0.157 exhibits ξ = 0.366 (see Figure 6), which is by only 2% smaller than the largest hydration degree possible at this ω/c-value, amounting to max ξ = 0.157/0.42 = 0.374, see (25). Model-predicted strength evolutions agree satisfactory with corresponding independent results by (Taplin 1959), see Figure 7. It appears worth emphasizing that the model predictions illustrated in Figure 7 do not depend on the unknown deviatoric strength of the hydration products, but only on the elasticity model proposed herein, i.e. the elasticity constants listed in Table 1, on Powers’ hydration model-related volume

Figure 4. Model-predicted evolution of shear modulus of hydrating cement pastes.

Figure 5. Model-predicted dimensionless compressive strength evolution of hydrating cement pastes.

and this is due to the prolate (‘‘infinitely long’’) hydrate shape adopted herein. It might be more realistic to introduce ‘‘cigars’’ with an aspect ratio somewhat smaller than infinity, if precise percolation modeling would be the key feature of interest. Herein, however, our focus is on strength, see Figure 5 for the modelpredicted uniaxial compressive strength of cement paste, normalized by the deviatoric hydrate strength. In contrast to the elastic predictions, strength turns out to be an always convex function of the hydration degree, whose slope obviously increases with decreasing water-cement ratio.

6 VALIDATION OF THE STRENGTH MODEL Aiming at a strictly quantitative test of the predictive capabilities of the presented strength model, we now compare model predictions with independent experimental data. (Taplin 1959) produced cement pastes with water-cement ratios ranging from 0.157 to 0.8. Specimens were rectangular prisms measuring 4 by 0.5 by 0.5 in, which were cured at 25◦ C. At the time of

Figure 6. Compressive strength values of hydrating cement pastes: experimental results by (Taplin 1959).

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the deviatoric hydrate strength, which can be identified from classical macroscopic tests on cement pastes. Combining, for instance, the uniaxial compressive strength fcu = 54.1 MPa of a cement paste with a water-cement ratio amounting to 0.50 (Bernard, Ulm, and Germaine 2003), with related model predictions for a fully-cured paste from Figure 5, suggests a deviatoric hydrate strength of magnitude 50 MPa. As to our knowledge, except for our earlier approaches (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008), no random homogenization tools for upscaling strength of cementbased materials have been proposed in the open literature so far. However, there has been intensive activity in strength upscaling in the framework of yield design approaches and nanoindentation testing, see, e.g. (Constantinides and Ulm 2007; Constantinides 2006). Interestingly, the latter approaches suggest hydrate hardness values of some hundreds of MPa, which could be regarded as somehow contradictory to our deviatoric hydrate strength in the tens of MPa range. On a second glance, however, this turns out to be not contradictory at all, but just the outcome of considering one problem (‘‘strength of cement-based materials’’) from two different viewpoints, which, in the end, might be integrated into a consistent overall understanding of the problem. In this context, two threads of argumentation are of interest:

Figure 7. Strength evolution of hydrating cement pastes: experimental results by (Taplin 1959) and predictions of the proposed model.

fractions (16) and (18), as well as on the morphological representation of cement paste including the hierarchical organization, the mode of phase interactions, and the phase shapes illustrated in Figure 1. Nonetheless, the comparison of model predictions with experimental results is of strict quantitative nature: the quadratic correlation coefficient is as high as r 2 = 93%, while the mean prediction error and the standard deviation amount to satisfactory −3% and to 7.3%, respectively. Given the simplicity of the proposed model and the vast range of considered water-cement ratios, we conclude that model validation is accomplished successfully.

1. Nanoindentation tests with indentation depths ranging from 170 ± 66 nm to 201 ± 90 nm (Constantinides and Ulm 2007) can be expected to probe the plastic material hardness at a scale of 700–800 nm (Ulm, Vandamme, Jennings, Vanzo, Bentivegna, Krakowiak, Constantinides, Bobko, and Van Vliet 2010). This is much smaller than 3 to 5 microns, which is the scale at which we introduce the hydrates as a material phase. Hence, the nanoindentation tests might give access to strength at a level well below that where we introduce the deviatoric strength value, and the latter might represent the strength of hydrate agglomerations with weak interfaces in between (which the nanoindentation tests do not ‘‘see’’), rather than the strength of ‘‘individual’’ hydrate phases probed by nanoindentation testing. 2. The hardness obtained from nanoindentation tests with indentation depths ranging from 170 ± 66 nm to 201 ± 90 nm (Constantinides and Ulm 2007) relates to a state of confined plastic (‘‘ductile’’) flowing of the hydrates, while the uniaxial tests considered in the present contribution refer to quasi-brittle failure of unconfined hydrates. In other words, our modeling scheme turns out to be well suited for representing brittle failure of unconfined hydrates, but it may have its limitations for more confined loading states. First computations of

7 DISCUSSION AND CONCLUSIONS For the sake of simplicity, we do not distinguish between different types of hydrates, but identify, from dense hydrate foams with very low porosity, elastic properties which are, on average, representative for all hydration products. On their basis, we then relate overall stresses acting on the cement pastes to higherorder deviatoric stress averages in the hydrates. When the latter reach a critical strength value, the overall stresses refer to the (brittle) macroscopic uniaxial compressive strength of cement pastes. A single deviatoric hydrate strength can explain, via higher-order average upscaling within a two-step random homogenization scheme, the uniaxial compressive strengths of cement pastes, as functions of the hydration degree, over a wide range of water-cement ratios. Since the experimental data were all normalized to a maximum strength attained by a material with a very low water-cement ratio, our upscaling scheme did not even require a single strength parameter to be introduced—this may be regarded as an additional strong indication for the validity of our approach. When absolute, rather than normalized, values are of interest, however, we need to introduce a number for

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ours (to be documented in a later publication) have actually shown that a dependence of hydrate failure on hydrostatic stress states is beneficial to be introduced if non-uniaxial strengths at the macroscopic level are to be represented (and depending on the magnitude of this hydrostatic stress, the relevant deviatoric hydrate strength will increase, possibly even up to the hundreds of MPa regime). From the extension of our model towards a Drucker-Prager type hydrate failure (comprising the current formulation as the special case for zero hydrostatic stress, the relevance of which was shown herein), it is only a small conceptual steps towards a full elastoplastic upscaling of strength of cement-based materials; in particular, since a very first model of that type was already proposed for bone (Fritsch, Hellmich, and Dormieux 2009), and since the ‘‘transformation field analysis by (Dvorak and Benveniste 1992)’’ has been, very recently, extended to general microstructures (including those of Figure 1 representation) with eigenstrains (inclusive of plastic strains), (Pichler and Hellmich 2010). However, the aforementioned ‘‘small conceptual step’’ will trigger extensive algorithmic work to be done—and this will probably open a new chapter in the scientific community devoted to the ‘‘Computational Modeling of Concrete Structures’’.

Constantinides, G. and F.-J. Ulm (2007). The nanogranular nature of CSH. Journal of the Mechanics and Physics of Solids 55(1), 64–90. Dormieux, L., A. Molinari, and D. Kondo (2002). Micromechanical approach to the behavior of poroelastic materials. Journal of Mechanics and Physics of Solids 50(10), 2203–2231. Drugan, W. and J. Willis (1996). A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. Journal of the Mechanics and Physics of Solids 44(4), 497–524. Dvorak, G.J. and Y. Benveniste (1992). On transformation strains and uniform fields in multiphase elastic media. Proceedings of the Royal Society London 437, 291–310. Eshelby, J. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London A 241, 376–396. Reprinted in (Markenscoff and Gupta 2006). Fritsch, A., L. Dormieux, C. Hellmich, and J. Sanahuja (2009). Mechanical behavior of hydroxyapatite biomaterials: An experimentally validated micromechanical model for elasticity and strength. Journal of Biomedical Materials Research—Part A 88(1), 149–161. Fritsch, A. and C. Hellmich (2007). Universal microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity. Journal of Theoretical Biology 244(4), 597–620. Fritsch, A., C. Hellmich, and L. Dormieux (2009). Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength. Journal of Theoretical Biology 260(2), 230–252. Hashin, Z. (1983). Analysis of composite materials—a survey. Journal of Applied Mechanics 50(3), 481–505. Helmuth, R. and D. Turk (1966). Elastic moduli of hardened portland cement and tricalcium silicate pastes: Effect of porosity. Symposium on structure of Portland cement paste and concrete 90, 135–144. Highway Research Board, Washington, DC. Hershey, A. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. Journal of Applied Mechanics (ASME) 21, 226–240. Hill, R. (1963). Elastic properties of reinforced solids. Journal of the Mechanics and Physics of Solids 11(5), 357–372. Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13(4), 213–222. Kröner, E. (1958). Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls [Computation of the elastic constants of a polycrystal based on the constants of the single crystal]. Zeitschrift für Physik A Hadrons and Nuclei 151(4), 504–518. In German. Laws, N. (1977). The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. Journal of Elasticity 7(1), 91–97. Markenscoff, X. and A. Gupta (Eds.) (2006). Collected Works of J.D. Eshelby—The Mechanics of Defects and Inhomogeneities, Volume 133 of Solid Mechanics and Its Applications. Springer.

REFERENCES Acker, P. (2001). Micromechanical analysis of creep and shrinkage mechanisms. In F.-J. Ulm, Z. Bažant, and F. Wittmann (Eds.), Creep, Shrinkage and Durability Mechanics of Concrete and other Quasi-brittle Materials, 6th International Conference CONCREEP@MIT, Amsterdam, pp. 15–26. Elsevier. Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6(2), 147–157. Bernard, O., F.-J. Ulm, and J. Germaine (2003). Volume and deviator creep of calcium-leached cement-based materials. Cement and Concrete Research 33(8), 1127–1136. Bernard, O., F.-J. Ulm, and E. Lemarchand (2003). A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cement and Concrete Research 33(9), 1293–1309. Bilaniuk, N. and G. Wong (1993). Speed of sound in pure water as a function of temperature. The Journal of Acoustical Society of America 93(3), 1609–1612. Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids 13(4), 223–227. Byfors, J. (1980). Plain concrete at early ages. Technical report, Swedish Cement and Concrete Research Institute, Stockholm, Sweden. Constantinides, G. (2006). Invariant properties of calcium silicat hydrates (C-S-H) in cement-based materials: instrumented nanoindentation and microporomechanical modeling. Ph.D. thesis, MIT.

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Sanahuja, J., L. Dormieux, S. Meille, C. Hellmich, and A. Fritsch (2009). Micromechanical explanation of elasticity and strength of gypsum: from elongated anisotropic crystals to isotropic porous polycrystals. Journal of Engineering Mechanics (ASCE). Online in advance of print, doi:10.1061/(ASCE)EM.1943.7889.0000072. Suquet, P. (1997). Continuum Micromechanics, Volume 377 of CISM Courses and Lectures. Springer Verlag, Wien New York. Taplin, J. (1959). A method of following the hydration reaction in portland cement paste. Australian Journal of Applied Science 10(3), 329–345. Ulm, F.-J., M. Vandamme, H. Jennings, J. Vanzo, M. Bentivegna, K. Krakowiak, G. Constantinides, C. Bobko, and K. Van Vliet (2010). Does microstructure matter for statistical nanoindentation techniques? Cement and Concrete Composites 32(1), 92–99. Zaoui, A. (1997). Structural morphology and constitutive behavior of microheterogeneous materials. In P. Suquet (Ed.), Continuum Micromechanics, Vienna, pp. 291–347. Springer. Zaoui, A. (2002). Continuum micromechanics: Survey. Journal of Engineering Mechanics (ASCE) 128(8), 808–816.

Mori, T. and K. Tanaka (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21(5), 571–574. Pichler, B. and C. Hellmich (2010, conditionally accepted for publication). Estimation of influence tensors for eigenstressed multiphase elastic media with non-aligned inclusion phases of arbitrary ellipsoidal shape. Journal of Engineering Mechanics. Pichler, B., C. Hellmich, and J. Eberhardsteiner (2009). Spherical and acicular representation of hydrates in a micromechanical model for cement paste—Prediction of early-age elasticity and strength. Acta Mechanica 203 (3–4), 137–162. Pichler, B., S. Scheiner, and C. Hellmich (2008). From micron-sized needle-shaped hydrates to meter-sized shotcrete tunnel shells: Micromechanical upscaling of stiffness and strength of hydrating shotcrete. Acta Geotechnica 3(4), 273–294. Powers, T. and T. Brownyard (1948). Studies of the physical properties of hardened portland cement paste. Research Laboratories of the Portland Cement Association Bulletin 22, 101–992. Sanahuja, J., L. Dormieux, and G. Chanvillard (2007). Modelling elasticity of a hydrating cement paste. Cement and Concrete Research 37(10), 1427–1439.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

C-Crete: From atoms to concrete structures Franz-Josef Ulm Massachusetts Institute of Technology, Cambridge, USA

Roland J.-M. Pellenq Massachusetts Institute of Technology, Cambridge, USA Centre Interdisciplinaire des Nanosciences de Marseille, CNRS and Marseille Universite, Marseille, France

Matthieu Vandamme Ecole des Ponts Paris Tech, Universite Paris-Est, France

ABSTRACT: We review recent developments of a multiscale bottom-up approach for concrete. The approach starts at the electron and atomic scale to nanoengineer the fundamental building block of concrete; to assess the properties by nanoindentation; and upscale strength, fracture and stiffness properties from nanoscales to macroscales of day-to-day concrete engineering applications. The key to all this is mechanics at the interface of physics and engineering. empirical approaches, we have chosen a bottom-up approach that starts at the electron and atomic scale to nanoengineer the fundamental building block of concrete; to assess the properties by nanoindentation; and upscale strength, fracture and stiffness properties from nanoscales to macroscales of day-today concrete engineering applications. The key to all this is mechanics at the interface of physics and engineering. This paper reviews some recent developments of this bottom-up approach.

1 INTRODUCTION More concrete is produced than any other synthetic material on Earth. The current worldwide cement production stands at 2.3 billion tons, enough to produce more than 20 billion tons or one cubic meter of concrete per capita per year. There is no other material that can replace concrete in the foreseeable future to meet our societies’ legitimate needs for housing, shelter, infrastructure, and so on. But concrete faces an uncertain future, due to a non-negligible ecological footprint that amounts to 5–10% of the worldwide CO2 production. It now appears that mechanics can be the discipline that enables the development of a sustainable green concrete future. We here adopt the perspective originating from Galileo’s Strength of Materials Theory, that weight, and thus CO2 -emission, increases with the volume of the produced material, while strength of structural members increases with the section. Hence, as one increases the strength of a material by a factor of x, one reduces the environmental footprint by 1/x for pure compressive members such as columns and perfect arches and shells, x−2/3 for beams in bending, and x−1/2 for slabs. Similarly, if one adopted a Linear Elastic Fracture model, an increase of the fracture toughness KIc = y KIc0 , would entail a reduction of the environmental footprint by 1/y for columns and y−4/5 for (notched) beams in bending or in torsion. All this hints towards a critical role of mechanics, and in particular Strength of Materials and Fracture Mechanics of concrete, in redesigning concrete materials and structures for the coming of age of global warming. In contrast to the classical top-down

2 A REALISTIC MOLECULAR MODEL FOR CALCIUM-SILICATE-HYDRATES The first step in setting up a bottom-up approach is to address the fundamental unit of concrete material behavior at electron and atomic scale. But, despite decades of studies of calcium-silicate-hydrate (C-S-H), the structurally complex binder phase of concrete, the interplay between chemical composition and density remains essentially unexplored. Together these characteristics of C-S-H define and modulate the physical and mechanical properties of this ‘‘liquid stone’’ gel phase. 2.1 Background Much of our knowledge of C-S-H has been obtained from structural comparisons with crystalline calcium silicate hydrates, based on HFW Taylor’s postulate that real C-S-H was a structurally imperfect layered hybrid of two natural mineral analogs: tobermorite of

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2.2 Molecular properties of C-S-H

14-Å interlayer spacing and jennite. While this suggestion is plausible in morphological terms, this model is incompatible with two basic characteristics of real C-S-H; specifically the calcium-to-silicon ratio (C/S) and the density. Recently, small-angle neutron scattering measurements have fixed the C/S ratio at 1.7 and the density at 2.6 g/cm3 (Allen et al. 2007), values that clearly cannot be obtained from either tobermorite (C/S = 0.83, 2.18 g/cm3 ) or jennite (C/S = 1.5 and 2.27 g/cm3 ); see for instance Shahsavari et al. (2009). From the standpoint of constructing a molecular model of C-S-H, this means that these crystalline minerals are not strict structural analogs. This brought about the development of a realistic molecular of C-S-H, based on a bottom-up atomistic simulation approach that considers only the chemical specificity of the system as the overriding constraint (Pellenq et al. 2009). By allowing for short silica chains distributed as monomers, dimers, and pentamers, this C-S-H archetype of a molecular description of interacting CaO, SiO2 , and H2 O units provides not only realistic values of the C/S ratio and the density computed by grand canonical Monte Carlo simulation of water adsorption at 300 K. The model, displayed in Figure 1, with a chemical composition of (CaO)1.65 (SiO2 ) (H2 O)1.75 , also predicts other essential structural features and fundamental physical properties amenable to experimental validation. This model suggest that the C-S-H gel structure includes both glass-like shortrange order and crystalline features of the mineral tobermorite.

One of the great advantages of having a realistic molecular model of C-S-H is that it is possible to probe the structure mechanically. The first quantity of interest is the elasticity content of the molecular structure, which is given in Table 1 in form of the components of the elasticity tensor. As one may expect from a glassy—layered hybrid, the elasticity exhibitsa high degree of anisotropy. For further applications it will be useful to consider the random polycrystal properties, in form of the Voigt-Reuss-Hill approximation classically used in mineralogy. As shown by Povolo & Bolmaro (1987), both Voigt and Reuss models are built using the invariance of the trace of the 9×9 matrix representing the stiffness and compliance tensors, respectively. This leads to the observation (made by Hill) that the Voigt and Reuss averages only use 9 of the 21 independent elastic constants. Denoting by I1 = Ciijj and I1∗ = Ciijj the traces (or linear invariants) of tensors Ciikl and Cijjl , respectively, the Voigt average is obtained from a comparison of those traces with their corresponding isotropic expressions, leading to: K Voigt =

1 1 I1 ; G Voigt = (3I ∗ − I1 ) 9 30 1

(1)

Applying a similar procedure to the compliance −1 , the Reuss average is obtained: tensor Sijkl = Cijkl K Reuss =

1 15 ; G Reuss = ∗ J1 6J1 − 2J1

(2)

where J1 = Siijj and J1∗ = Sijij are the corresponding traces of the compliance tensors Siikl and Sijjl , respectively. The Reuss-Voigt-Hill polycrystal properties are obtained as the arithmetic mean of Reuss and Voigt bounds, which yields for C-S-H: K C−S−H = 49GPa; G C−S−H = 23GPa

(3)

A second useful information the molecular model can provide is about the strength behavior, by simulating for instance the stress-strain behavior of the Table 1. Elastic tensor of C-S-H determined from molecular dynamic simulations (Voigt notation) [from Pellenq et al. 2009].

Figure 1. Computational molecular model of C-S-H: the blue and white spheres are oxygen and hydrogen atoms of water molecules, respectively; the green and gray spheres are inter and intra-layer calcium ions, respectively; yellow and red sticks are silicon and oxygen atoms in silica tetrahedra (adapted from Pellenq et al. 2009).

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Cij/GPa

1

2

3

4

5

6

1 2 3 4 5 6

93.5

45.4 94.9

26.1 30.0 68.5

0.6 −4.6 −4.3 19.2

−0.1 1.8 −2.7 0.3 16.1

3.46 −3.0 −0.6 1.82 −0.4 31.2

C-S-H model in affine shear deformation (strain controlled) after first relaxing the computational cell using MD at 300 K, under constant NVT ensemble conditions. A series of shear strains in increments of 0.005 is imposed; after each increment the atomic configuration is relaxed and the shear stress determined from the virial expression. Figure 2 displays the shear-stress-strain response of the C-S-H model. Two configurations are herein considered: the ‘‘wet’’ model, with water present particular in the inter-layer space, and the ‘‘dry’’ model, in which all water molecules have been removed. The stressstrain response shows that the present of water in the inter-layer space leads to a localization of deformation into a narrow band defined by a wet interlayer, akin to a fracture process. By contrast, the ‘‘dry’’ model shows a plastic deformation behavior, with irreversible (plastic) deformation upon unloading. This shows that there are some strength reserves at a molecular scale of C-S-H that could enhance strength and (ductile) deformation behavior of C-S-H.

the comparison with nanoindentation experiments. Nanoindentation measurements of C-S-H do not probe the particle properties, but rather the C-S-H gel properties. From a straightforward dimensional analysis, the link between particle properties and microstructure is provided by (Ulm et al. 2007): P = hs × H (µ, η, η0 ) Ac (dP/dh)hmax M =c √ = ms × M (v, η, η, η0 ) Ac

H=

(4)

where H the hardness, the indentation modulus, that are accessible from the applied force, Pand the indentation depth, h, and the projected contact area Ac . In return, hs and ms are particle hardness and plane stress modulus, the latter being defined for an isotropic material by: ms = 4G

3K + G = 65 GPa 3K + 4G

(5)

where we used the values for K and G in Eq. (3) determined from MD simulations. Furthermore, functions H and M in Eq. (4) are dimensionless functions of the friction coefficient µ, the Poisson’s ratio ν, and the packing density η. Linear and nonlinear microporomechanics (Dormieux et al. 2006) provides a convenient way to determine these functions analytically (Constantinides & Ulm, 2007; Cariou et al. 2008). Moreover, combining the elastic properties determined from the C-S-H model with these micromechanics models with no adjustable parameters, one can probe the texture and extent of anisotropic structures within cement paste at micrometer length scales of randomly oriented C-S-H particles. Figure 3 compares the prediction of two micromechanics models along with

2.3 From molecular properties to microtexture The molecular model of C-S-H has been validated against several experimentally accessible properties, including density, extended X-ray absorption fine structure (EXAFS) spectroscopy signals measuring short-range order around Ca atoms, longer range correlations revealed in X-ray diffraction intensity, vibrational density of states measured by infrared spectroscopy, and nanoindentation measurements of elasticity and strength properties (for details, see Pellenq et al. 2009). Here we restrict ourselves to

Figure 3. Nanoindentation data (obtained by Vandamme & Ulm 2009) compared with two micromechanics model. The input for the model are the elastic C-S-H properties (indentation modulus = 65 GPa, Poisson’s ratio = 0.3). [adapted from Pellenq et al. 2009].

Figure 2. Relationship between the shear stress and the shear strain for the C-S-H model with (solid line) and without (dashed line) water molecules (results obtained by A. Kushima and S. Yip; adapted from Pellenq et al. 2009).

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3.2 Implementation for back-analysis of packing density distributions

nano-indentation results; one is a porous bicontinuous matrix approach captured by the so-called MoriTanaka scheme, and the other a granular approach captured by the self-consistent scheme. From this comparison, we observe first that the granular approach better describes the experimental data over the entire domain of C-S-H particle packing fractions. Second, both approaches give acceptable predictions at larger packing fractions. That is, at the micrometer-scale, Mori-Tanaka and self consistent micromechanics approaches, parameterized only with nanoscale derived elasticity constants, indicate that cement paste can be conceptualized as a cohesive granular material rather than a porous bi-continuous matrix.

Consider then a series of N indentation tests on a heterogeneous material. What is measured in these tests are N times (M , H ) values at each indentation point representative of a composite response. Assuming that the solid phase is the same, i.e. C-S-H with now well defined elasticity properties, and that all what changes between indentation points is the packing density, there are two particle unknowns (hs and µ) and N packing densities. Hence, for N ≫ 2, the system is highly over-determined, which makes it possible to assess microstructure and particle properties. One can then proceed with a statistical analysis of the packing density distributions, as shown in Figure 5. The interesting feature which emerges from this analysis is the presence of different C-S-H gel phases in the microstructure; namely a Low-Density (LD)

3 PROBING C-S-H MICROSTRUCTURE BY NANOINDENTATION With the molecular properties of C-S-H in hand it becomes possible to assess the microstructure of cement-based materials at micrometer scale, and ultimately to make the link between microstructure, and meso- and macro-scale properties. The key herein are the dimensionless scaling relations H and M in Eq. (4). 3.1 Does particle shape matter? The determination of functions H and M requires the choice of particle morphology. For perfectly disordered materials, the key quantity to be considered is the percolation threshold, that is the critical packing density below which the composite material has no strength, nor stiffness. This percolation threshold depends on the particle shape (Sanahuja et al. 2007). Clearly, as seen in TEM images of C-S-H, the elementary particle has an aspect ratio. However, as far as the mechanics response is concerned, it turns out that particle shape does not matter as soon as the packing density of the porous material is greater than 60%. This is obviously the case of cement-based materials (as their industrial success is due to their strength performance), and also of most other natural composite materials like bone, compacted clays (shale), etc. One can thus conclude that the effect of particle shape is negligible as far as the mechanical performance is concerned (Ulm & Jennings 2008); see Fig. 4. The negligible effect of the particle shape on the homogenized mechanical properties largely simplifies the micromechanical analysis. It thus suffices, to consider spherical particles and a distinct disordered morphology of the solid phase, similar to a polycrystal, characterized by a solid percolation threshold of η0 = 0.5. These relations can be found, for the indentation modulus in Constantinides & Ulm (2007), and for the indentation hardness in Cariou et al. (2008).

Figure 4. Normalized stiffness vs. packing density scaling relations. The percolation threshold of 0.5 corresponds to a perfectly disordered material composed of spherical particles, while lower percolation thresholds are representative of disordered materials with particle shape (here ellipsoids). [Adapted from Ulm & Jennings 2008].

Figure 5. Probability density plot (PDF) of packing density distribution determined from 400 nanoindentation tests on a w/c = 0.3 cement paste; together with a phase deconvolution obtained by fitting Gaussian distributions to the experimental PDF [adapted from Vandamme & Ulm, 2009].

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phase and High Density (HD) phase as originally proposed by Jennings and co-workers (for a recent review, see Jennings et al. 2007), with mean packing densities close to two limit packing densities of spherical objects, namely the 64% packing density of the random close-packed limit for the LD-phase, and 74% for the ordered face-centered cubic (fcc) or hexagonal close-packed (hcp) packing for the HD-phase. In addition, a third Ultra-High-Density (UHD) phase appears in the packing fraction distribution, particularly for cement pastes with low water-to-cement (w/c) ratios (Vandamme et al. 2010). Indeed, a comprehensive nanoindentation analysis of cement pastes prepared at different w/c ratio shows that LD dominates cement-based materials prepared at high w/c mass ratios; HD and UHD control the microstructure of low w/c ratio materials (Fig. 6). This shows that nanoindentation combined with micromechanicsbased indentation analysis provides a means to probe the microstructure of the C-S-H gel at micrometer scale of cement-based materials.

allow the determination of the packing density distribution. The difference between the two types of test is in a holding phase: following a fast loading (5s), in nanoindentation creep tests a constant load is applied for some 180s; in contrast to microstructure tests in which the dwelling time is 5s. During the 180s dwelling time, the change in indentation depth h(t) = h(t)−h0 is recorded as a function of time. In all tests on C-S-H phases, after correcting for thermal drift effects of the indenter equipment, a least square fitting of h(t) = h(t) − h0 demonstrates an indentation creep compliance of the C-S-H phases that is logarithmic with regard to time (Fig. 7). The indentation creep compliance rate is determined as (Vandamme & Ulm 2006, 2009): ˙ max = 1/(Ct) L˙ = 2aU h/P

(6)

√ where aU = Ac /π is the contact radius at the end of the dwelling phase; while C, which has the same dimension as an elastic modulus, is justly termed contact creep modulus (Vandamme & Ulm 2009). This creep modulus C depends on the packing density as shown in Figure 8. It is on the order of CLD = 120.4 ± 22.6 GPa, CHD = 183.6 ± 30.5 GPa and CUHD = 318.6 ± 32.2 GPa. The value for CLD = 120.4 ± 22.6 GPa comes remarkably close to macroscopic values of the long-term creep of normal strength concrete, while the value for CUHD = 318.6 ± 32.2 GPa is not far off from the long-term creep values of high-strength concrete -with one major difference: a nanoindentation creep test takes 3 minutes, a longterm macroscopic creep test requires several years of force, deformation control under highly controlled environmental conditions (Acker & Ulm 1999). This observation opens new perspectives for creep assessment of concrete materials and structures: By probing micrometer-sized volumes of materials, nanoindentation creep experiments provide quantitative results in a 6 orders of magnitude shorter time

3.3 Nanogranular origin of concrete creep It is now possible to make the link between microstructure and mechanical performance. Besides elasticity and strength, a main concern for concrete is creep. Concrete creep occurs at a rate that deteriorates the durability and truncates the lifespan of concrete structures. One challenge in establishing this link between creep properties and microstructure is that creep of C-S-H needs to be experimentally measured at the scale of the microstructure of the C-S-H gel. We here consider nanoindentation creep tests, in addition to nanoindentation tests of stiffness and elasticity that

Figure 7. During the 180s dwelling time, the change in indentation depth is recorded as a function of time, and fit with a logarithmic function (for details see Vandamme & Ulm 2009).

Figure 6. Volume fraction distributions in the microstructure: (a) volume fractions of the cement paste composite; (b) volume fractions of the hydration phases [adapted from Van-damme etal. 2010].

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As engineers move into the fundamentals of the physics of materials, and physicist move into engineering applications, the boundary between physics and engineering is finally blurred. It is on this basis that we expect that progress on the concrete front will translate into societal benefits. Along this way, we believe that computational and experimental mechanics of concrete will be redefined. REFERENCES Acker P. & Ulm F.-J. 2001. Creep and shrinkage of concrete: physical origins and practical measurements. Nuclear Engineering and Design 203(2–3): 143–158. Allen A.J., Thomas J.J. & Jennings, H.M. 2007. Composition and density of nanoscale calcium-silicate-hydrate in cement. Nature Materials 6(4): 311–316. Cariou S., Ulm F.J. & Dormieux L. (2008). Hardness-packing density scaling relations for cohesive-frictional porous materials. Journal of the Mechanics and Physics of Solids 56: 924–952. Constantinides G. & Ulm F.-J. (2007). The nanogranular nature of C-S-H. Journal of the Mechanics and Physics of Solids 55(1): 64–90. Dormieux L., Kondo D. & Ulm, F.-J. (2006). Microporomechanics. J. Wiley & Sons, Chichester, UK. Jennings H.M., Thomas J.J., Gevrenov J.S., Constantinides G. & Ulm, F-J. 2007. A multi-technique investigation of the nanoporosity of cement paste. Cem. Concr. Res. 37 (3): 329–336. Pellenq R.J-M., Kushima A., Shahsavari R., Van Vliet K.J., Buehler, M.J., Yip S. & Ulm F.-J. 2009. A realistic molecular model of cement hydrates. Proc. Natl. Acad. Science 106(38): 16102–16107. Povolo F. & Bolmaro R.E. 1987. Average elastic-constants and tensor invariants. Phys. Stat. Sol. 99(2) 423–436. Sanahuja J., Dormieux L. & Chanvillard G. 2007. Modelling elasticity of a hydrating cement paste. Cem. Concr. Res. 37: 1427–1439. Shahsavari R, Buehler M.J., Pellenq R.J.-M. & Ulm F.-J. 2009. First-Principles Study of Elastic Constants and Inter-layer Interactions of Complex Hydrated Oxides: Case Study of Tobermorite and Jennite. J. American Ceramic Society 92(10): 2323–2330. Ulm F.J. & Jennings H.M. 2008. Does C-S-H particle shape matter? A discussion of the paper ‘Modelling elasticity of a hydrating cement paste’, by Julien Sanahuja, Luc Dormieux and Gilles Chanvillard. CCR 37 (2007) 1427–1439. Cem. Concr. Res. 38(8–9): 1126–1129. Ulm F.-J., Vandamme M., Bobko C., Ortega J.A., Tai K. & Ortiz C. 2007. Statistical indentation techniques for hydrated nanocomposites: Concrete, bone, and shale. J. Am. Ceram. Soc. 90 (9): 2677–2692. Vandamme M. & Ulm F.-J. 2006. Viscoelastic solutions for conical indentation. Int. J. Solids and Structures 43(10): 3142–3165. Vandamme M. & Ulm F.-J. 2009. Nanogranular origin of concrete creep. Proc. Natl. Acad. Science 106(26): 10552–10557. Vandamme M., Ulm F.-J. & Fonollosa P. 2010. Nanogranular packing of C-S-H at substochiometric conditions. Cem. Concr. Res. 40(1): 14–26.

Figure 8. Creep modulus—packing density relation [adapted from Vandamme & Ulm 2009].

and on samples 6 orders of magnitude smaller in size than classical macroscopic creep tests. This ‘‘lengthtime equivalence’’ (large time scales can be accessed by looking at small length scales) may turn out invaluable for the implementation of sustainable concrete materials whose durability will meet the increasing worldwide demand of construction materials for housing, schools, hospitals, energy and transportation infrastructure, and so on. 4 CONCLUSIONS The premise of the bottom-up approach is to identify new degrees of freedoms for concrete material and structural design, from electrons and atoms to structures. While still in its infancy, the first results obtained with this bottom-up approach reveal some interesting perspectives for enhancing the sustainability of this omnipresent material. With a focus on strength enhancement, we note: • On the atomic scale, strength enhancement may be achieved by fine tuning the chemistry of cementbased materials. Classical cement-based materials exhibit a calcium-to-silicon ratio (C/S) of 1.7, which determines mechanical properties of the molecular structure. A change in chemistry is expected to enhance those properties. • At the scale of the microtexture, a denser packing of elementary C-S-H particles has the premise to enhance both the strength behavior and the durability performance, for instance in terms of concrete creep. • From micro-to-macro of concrete structures, continuum micromechanics provides a powerful framework to translate progress in materials science into day-to-day engineering applications.

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Constitutive and multi-scale modelling

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix H. Aljewifi, B. Fiorio & J.L. Gallias Université de Cergy-Pontoise, Laboratoire de Mécanique et Matériaux du Génie Civil, Cergy-Pontoise cedex, France

ABSTRACT: This paper describes the pull-out behavior of glass multi-filaments yarns embedded in a cementitious matrix. The pull-out behavior is characterized by direct pull-out tests performed on different glass rovings. For these tests, different pre-treatments of the roving are used, that induce different impregnation patterns. In this way, the effect of the impregnation on the pull-out behavior is characterized. The effect of the constitutive parameters of the yarn on the mechanical behavior is also studied through the use of three different glass yarns for the experimental study.

1 INTRODUCTION

yarn consists of several hundreds up to thousands of single filaments. Therefore the fineness of the yarn, in tex (g/km), depends on the number of filaments, filament diameter (range between 10 to 30 µm) and fibre density. Filaments can be gathered in two different ways to produce a yarn: direct roving and assembled roving. Direct roving consists in a mass agglomeration of thousands of single filaments, coated with a sizing (see below). Assembled roving consists in the agglomeration of strands, themselves constituted from hundreds of single filaments. Assembled roving are a three levels structure (filament-strand-yarn), when direct roving presents a two level structure (filament-yarn). In most of the applications, filaments are coated with a sizing which goal is to enhance the interaction with the matrix or to ease the building process of the yarn. The sizing material is a chemical mixture of polyhydroxyphenols, silane, polymer emulsion (PVAC) and additives. The detail of the chemical composition of the sizing is generally unknown, because of the industrial protection imposed by manufacturers. The type of sizing influences the interfacial properties of the cement matrix —filaments bond. Therefore, the mechanical properties of TRC may be influenced by the sizing, as the quality of the adhesion between filaments and matrix influences the composite performances. But the main element that influences the mechanical comportment of yarns reinforced concrete is the specific structure of the yarn itself. The tensile load-strain relationship of the yarn is significantly influenced by the specific organisation of the filaments in the yarn (Chudoba et al. 2006). If all filaments in the yarn were strictly straight and aligned, the stress in each filament of a yarn submitted to a given strain should

1.1 Textile reinforced concrete Textile reinforced concrete (TRC) is a new cementitious composite that appeared in the early 1980’s, when the combination of new yarn types with 3D-production processes leads to an increasing number of new textile applications. Materials used for TRC application are mainly alkali-resistant glass fibres and carbon fibres. The yarn structures include cabled yarns and friction spun yarns which were developed to improve the bonding behaviour. They both have a structure that looks like the one of steel reinforcements with rods. The development of TRC is based on the fundamentals of shortcut glass fibre reinforced concrete. In order to increase the effectiveness of the fibres embedded in the concrete matrix, the fibres are aligned in the direction of the tensile stresses similar to ordinary steel reinforced concrete. Textile reinforced concrete offers many advantages compared to traditional concrete (Hanisch and et al., 2006). The most important reason for a reinforcement of concrete parts with textiles is that the concrete parts can be very thin as there is no risk of corrosion of the reinforcement materials. This allows to reduce the weight of the concrete parts, and therefore to contribute to the reduction of the environmental impact of construction. In addition, the reinforcement is more flexible and therefore the shape of the concrete elements can be varied in a wide range. 1.2 Multi-filaments reinforcement 1.2.1 Yarns characteristics Yarns are multi-filaments reinforcements made of a bundle of elementary fibres, so-called filaments. One

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be homogeneous in the yarn and the stress-strain relationship of the yarn would correspond to those of filaments, i.e. to the linear elastic brittle behaviour of the bulk material, in the case of glass or carbon filaments. In this case, the failure of the yarn should be associated to the simultaneous failure of all constitutive filaments. Real yarns lead to a different behaviour, as filaments are not initially aligned. As an example, Figure 1 shows the tensile behaviour of a glass yarn measured by the way of a direct tensile test. The first stage of the loading is characterized by the progressive stiffening of the mechanical behaviour, induced by the differed tension of the filaments due to their initial mis-alignement. When all filaments are tensed, the load-strain curve is linear until the first filaments begin to fail. This results in the progressive softening of the behaviour until the load reaches its maximum values Pmax . For increasing strain, filaments breaking continue until full failure of all filaments. Residual friction between failed filaments then results in a residual load that progressively reduces to zero for high strain. The main observation is that during yarn elongation, the filaments work almost independently one from the other, and that the yarn behaviour results from the coupling action of the individual behaviour of all yarns.

1mm

Penetration depth

Figure 2. Cross section (left); longitudinal section (right) of multi-filaments yarn embedded in micro concrete.

The inner filaments are not reached by the hydrated cement paste and thus are not directly anchored in the cementitious matrix. Only the friction between these filaments can generate a certain bond resistance when the yarn is pulled out, when differential slips appear between filaments. This phenomenon induces a non-uniform stress and strain distribution in the yarn. Two interactions between the yarn and the matrix can been formed. Ohno et al., 1994 & Langlois, 2004 assume two families of yarns: sleeve filaments with direct contacts with the matrix and core filaments without direct contacts. Sleeve filaments are mechanically anchored to the matrix. Core filaments are submitted to tension due to friction with sleeve filaments. As filaments are not straight and parallel into the yarn, a filament that is located in the center part of the yarn, in a given cross section of the yarn, can be located in the impregnated peripherical part of the yarn in another cross section. This means that the relative importance of the sleeve filaments and core filaments does not only depends on the impregnation process, but depends also on the embedded length of the yarn. The longer this length is, the more numerous the anchored sleeves filaments are. This phenomenon also depends on the fiber/matrix chemical bond strength (Kabele et al., 2006) that conditions the filament debonding and the filament/ filament and filament/matrix slip properties. From this point of view, the sizing may also play an important part in the pull-out behaviour. As explained, the impregnation process greatly influences the bond conditions of each individual filament of the yarn. The resulting bond conditions of the yarn are then also mainly guided by the impregnation process, as well as by the matrix and yarn properties. This specificity of multi-filaments reinforcements, compared to monolithic reinforcements, induces a very specific mechanical behaviour, which cannot be modelized with the same methods as monolithic reinforcements. The aim of this paper is to give some experimental explaination of the mechanisms involved in the yarn/matrix interaction, and to provide the modelization community with experimental data that describe the pull-out behaviour of multi-filaments reinforcement.

1.2.2 Influence of the yarn impregnation on the mechanical behaviour Due to the specific geometry of the yarn and to the penetration of the cementitious matrix inside the yarn during the casting of TRC, the behaviour of impregnated yarn in the cementitious matrix is very dissimilar from the behavior of ordinary steel rods. The penetration of the cementitious matrix is not homogeneous: the cross section of the impregnated yarn shows that the outer filaments are embedded in a hardened cement paste that provides them a good anchorage (figure 2). The penetration depth of the cement paste inside the yarn is not sufficient to ensure a full impregnation of the yarn.

Figure 1. Typical load vs. strain curve observed for multifilaments yarn in direct tension.

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2 EXPERIMENTAL PROCESS 2.1 Objectives of the experimentation The hereafter-presented experimentation focuses on the influence of the fibre impregnation on the pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix, in relation with the constitutive parameters of the yarn. In this experiment, pull-out tests have been performed to characterize the micromechanical behaviour of the multi-filaments yarn/cementitious matrix interface. In these tests, a yarn was embedded into a matrix with various condition of impregnation and loaded in tension until the filaments slip, or break. Parameters of the study were the yarn properties, the embedded length and the matrix impregnation. Yarn properties were controlled through the use of three different types of yarns. The impregnation was varying trough the use of three preparation process applied to the yarn before the casting of the pull-out samples. 2.2 Materials 2.2.1 Glass yarn Three types of glass yarns (named in the following OC1, OC2 and SG2) have been used in this study. OC1 and OC2 yarns are made from E-glass filaments and came from a first manufacturer. SG2 yarn is made from AR-glass and came from a second manufacturer. OC1 and SG2 yarns are assembled rovings. OC2 is a direct roving (see 1.2.1). The main characteristics of these yarns are given in table 1. Figure 3 gives the results of tensile tests performed on the three yarns. These tests were performed on 10 cm long yarns of the three studied type. The five curves in each chart correspond to the same type of yarn and give an idea of the variability classically observed in the case of yarns.

Figure 3.

composition). The compressive and flexural strength were measured according to NF 196-1. Mean values of these properties were 55 MPa and 10.83 MPa respectively.

2.2.2 Cementitious matrix Fine grain (1.25 mm maximum size) CEM I 52.5 mortar was used as cementitious matrix. Mixing proportion was as follow: W/ C = 0.5, S/C = 1.4 and SP/C = 0.0035 (W, C, S and SP are respectively the mass of water, cement, sand and superplasticizer used for the

2.3 Samples 2.3.1 Samples preparation Samples were 3.4 cm diameter cylinder made from the mortar described in the previous part. A straight yarn was positioned along the axis of the cylinder. The height of the cylinder varied from 1 to 25 cm, which results in a variation of the embedded length of the yarn. One of the three hereafter described pre-treatment was applied to each yarn before casting the sample:

Table 1. Characterization of the glass multi-filaments yarns (n: number of strands in assembled rovings).

Yarn

Filament diameter µm

OC1 OC2 SG2

12 17 14

n

Fineness tex (g/km)

Young’s modulus GPa

Glass bulk density kg/m3

39 / 30

2400 2400 2450

57.73 59.38 49.11

2530 2530 2680

Tensile test on yarns.

• Pre-wetting (W): yarn was saturated with water so that the inter-filaments voids are filled with water, which prevent the cement paste to enter the yarn

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during casting. It should be noticed that in this case, capillarity forces induce the agglomeration of the glass filaments which became more parallel than without this treatment. • Drying (D): yarn was air dried at room temperature before casting. Capillarity leads to penetration of water and cement particles in the yarn at the time of casting. As the filaments act as a filter, penetration of cement particles into the yarn was limited. • Cement pre-impregnation (PI): the yarn was manually saturated with a cement slurry before casting. Saturation was obtained by manual action on the yarn placed in a slurry batch. The cement slurry composition corresponds to the micro-concrete matrix cement paste composition.

filaments partially interlocked due to the mechanical action during pre-treatment. Dried yarn (D) shows an intermediate facies, with the main part of the filaments embedded in the matrix, which signifies that the free length of most of the filaments (the distance between two embedded points) is smaller than the length of the specimen (4 cm). Similar observations were made for all other types of yarns. The penetration depths of the cementitious matrix in the yarns were estimated from the SEM observations. An impregnation index iy was calculated from these values as the ratio of he impregnated area of the yarn to the apparent area of its cross section. Values of iy for the different combinations of yarns and pretreatments are given in table 2. Flow tests realized by Aljewifi et al., 2009 give complementary information about the porosity of the impregnated yarn. This test allows to measure the water flow rate along an embedded yarn under a constant pressure gradient of 107.5 kPa/cm. For a given yarn, the measured flow rate decreases when the impregnation becomes more important (see figure 5). Figure 5 also shows that the lowest value of the flow rate corresponds to the PI pre-treated yarns and that this value is roughly independent from the type of yarn. This is the signature of a small waterflow inside the tortuous cement paste porosity of the impregnated yarn.

After 24 h hardening at room conditions (about 50% relative humidity and 20◦ C), samples were removed from mould and placed in 20◦ C water during 24 days. They were then removed from water for one day to allow the air drying of the yarn. The free end of the yarn was glued in between two epoxy plates for fastening in the pull-out device. 2.3.2 Characterization of the yarn impregnation Aljewifi et al., 2010 have studied the impregnation of the yarns obtained from the three pre-treatments applied to the three types of yarns used in this study. Different methods of characterization were used, among them scanning electron microscopy (SEM) observation of longitudinal sections of yarns and flow tests. The main result of these SEM observations is the strong influence of the pre-treatment on the impregnation. As an example, figure 4 gives the three SEM photographs corresponding to each pre-treatment applied to OC1 yarn. The pre-wetting of the yarn (W) leads to straight parallel mainly un-impregnated filaments. Conversely, the pre-impregnation with the cement slurry (PI) induces a full impregnation of the yarn, with

W

D

PI

Table 2. Diameter of the impregnated yarns (mm) and values of the impregnation index iy (%).

OC1 OC2 SG2

W

D

PI

3.52 mm 38% 2.69 mm 24% 2.84 mm 29%

3.26 mm 67% 3.39 mm 19% 3.14 mm 70%

4.19 mm 100% 4.26 mm 100% 3.42 mm 100%

1 mm

1 mm

1

Figure 4. OC1 yarns; longitudinal sections corresponding to W, D and PI yarn pre-treatment.

Figure 5.

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Flow rate vs. impregnation index iy .

along the fibre/matrix interface. Once the debonding process has reached the end of the embedded fibre length, a dynamic mechanism of pull-out is observed. Moreover, a displacement at the end of the free length is also accompanied by a displacement at the embedded end (Naaman et al., 1991). In the case of yarn, the above described behaviour is observed for individual filaments but cannot be directly applied to the whole yarn.

In the case of the lowest impregnation index, the flow rate is higher and influenced by the yarn type. This is explained by a flow of water that takes place inside the inter-filaments porosity of the un-impregnated center part of the yarn. As the water is in direct contact with the filaments in this case, the chemical composition of the filament surface (sizing) influences the flow, which explains the differences observed for the yarns in this situation. As a conclusion of this part, it is shown that there are strong evidences that the yarn impregnation is only partial in most of the case. As a consequence, the pullout behaviour of the yarn will be strongly influenced by this fact and the overall behaviour will diverge from the pull-out behaviour of monolithic reinforcements.

3 RESULTS 3.1 Load/displacement curves 3.1.1 Measurements As the stress is not homogeneous in all filaments of the tested yarns, it was chosen to present the results of the pull-out tests as load/displacement curves instead of stress/strain curves. Load and displacement are measured through the press sensor. In particular, the displacement is measured through the displacement transducer of the test device: it is the relative displacement of the upper grip supporting beam regardless to the lower grip support (figure 6). Therefore, the measured displacement is not directly the extracted length of the yarn but also take into account the strain of the free length of the yarn. As this length stay constant over all the tests, it is considered that the comparison between the behaviours observed in all different cases that have been tested remains pertinent. This assumption is clearly true when working with a given type of yarn. It remains roughly true for comparison between two different types of yarns, as yarns mechanical properties are in the same range. Figures 7 to 9 give the load/displacement curves for each of the three tested yarns. Each figures presents three charts, one for each type of yarn’s pre-treatment. In each chart, the results corresponding to the different measurements made for a given embedded length are presented as a single average behaviour curve. This curve is obtained from three tests performed in the same conditions.

2.4 Pull-out test Pull-out tests are performed on yarns embedded in a cylinder of micro-concrete. As explain in 2.3.1, the free end of the yarn (i.e. the end which is not embedded in the micro-concrete) is glued between two epoxy plates. A universal mechanical device is used to perform the tests (see also figure 6). The micro-concrete cylinder is positioned in a specific basket clamped on the upper grip of the test device. The free end of the yarn is clamped on the lower grip of the tension device. The mortar cylinder is precisely centred so as to ensure that the tension load is applied parallel to the embedded yarn. For all test, the free length LL of the yarn (i.e. the length of the yarn from the micro-concrete cylinder to the epoxy plates, figure 6) has a constant values of 10 cm. Tests are realized at constant speed (1 mm.min−1 ). In this test, a load P is applied to the tip of the yarn embedded in a cementitious matrix over an embedded length Le which was a parameter of the study. In the case of monolithic fibre, a monotonic increase in the load P is accompanied by a displacement from the tip of the fibre and leads to progressive debonding

Fixed member: Gl: lower grip D: displacement transducer

L Gu

D

S

Le Mobile member: Gu: upper grip LL B: sample basket L: load cell

SS

B E

P Gl

Figure 6.

3.1.2 Three stages behaviour Generally, pull-out curves are divided into three stages (Hegger and et al., 2004). In our study, all of the obtained results obey this statement, except may be for the third stage, as explain later. The first stage is determined by the elasticity of the adhesion bond and corresponds to the progressive tension of the filaments that constitute the yarn. Stiffening is generally observed at the beginning of the loading, associated to the delayed tension of the filaments. The slope of the linear ascending portion is not similar for all curves and seems to be an increasing function of the embedded length.

Sample: S: mortar cylinder with embedded yarn E: epoxy plates P: applied load

Experimental setup.

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W pre-treated OC1 yarn

W pre-treated OC2 yarn

D pre-treated OC1 yarn

D pre-treated OC2 yarn

PI pre-treated OC1 yarn

Figure 7. Load vs. displacement curves for OC1 yarn and different embedded lengths in fine concrete.

PI pre-treated OC2 yarn

Figure 8. Load vs. displacement curves for OC2 yarn and different embedded lengths in fine concrete.

In the second stage of the behaviour, the chemical bond between the strands and the matrix break (the smooth surface of the filaments do not ensure a good mechanical anchorage of the strands on short length. The breaking of the strand/matrix bond is accompanied by filaments breaking. This phenomenon induces a smoothing of the mechanical behaviour. When broken filaments are in sufficient number, failure progressively appears as a slow decrease of the load. After the breaking of all the filaments and strands (tensile or adhesion failure), broken filaments are extracted from the matrix. During this stage, frictional stresses appear at the filament/filament and filament/matrix contacts. This results in a residual load which progressively reduces to almost zero, when all filaments become extracted from the matrix. This observation corresponds to the third stage of the behaviour mentioned by Hegger and et al., 2004.

3.1.3 Complementary elements on the residual load The values of the residual frictional load for all types of yarn and embedded lengths show a tendency of the residual load level to increase with the embedded length, which is consistent with the increase of contact points along the filaments. In some case, this tendency is not respected and some lower embedded length gives highest frictional level. This can be attributed to the appearance of local disorganization in the yarn, due to differential slip of the filaments. Evidences of this phenomenon have been seen on longitudinal sections of samples made after the pull-out test (figure 10). By comparison with reference longitudinal sections made before pull-out test, the structure of the loaded yarns has been disorganized. In some location, node looking

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structures (see detail on figure 10) appear and seem to be the consequence of the blocking of a group of filaments in a small cement cluster pulled by another groups of filaments. It should be noted that this third stage of the behaviour is in some cases reduced to almost nothing. This is systematically the case for PI pre-treated yarns, whatever the embedded length was. The explanation is, that in this case, the anchorage length necessary to anchor the whole yarn (i.e. all its filaments) is very short. Most of the filaments’ failures then take place in the vicinity of the surface of the concrete, producing failed filaments with short embedded length. As the embedded length of the failed filaments is short, the remaining frictional load is very low and comes back to zero for small extraction displacement. 3.2 Behaviour law parameters 3.2.1 Maximum pull-out load Pmax Raw data described in 3.1 were used to determine the maximum value Pmax of the pull-out load during the test. Values of Pmax are given in figure 11. As a general trend, it is seen that Pmax is an increasing function

Figure 9. Load vs. displacement curves for SG2 yarn and different embedded lengths in fine concrete.

Figure 10. Longitudinal section of OC1 D yarn after pullout test. Arrow: direction of extraction.

Figure 11. Values of Pmax vs. embedded length.

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of the embedded length for the low values of the embedded length. For higher values of the embedded length, Pmax remains roughly constant but do not reach the tensile strength of the yarn, which indicates that a portion of the filaments remains inactive during the pull-out test. SG2 yarn gives the best efficiency (about 90% of the tensile strength), when OC1 and OC2 give lower efficiencies. Concerning the effect of the pre-treatment, D and PI pre-treatment does not show major differences for the evolution of Pmax : PI pre-treatment gives higher values of Pmax than D pre-treatment, which itself gives higher values than W pre-treatment. In the case of OC1 yarn, values of Pmax are rather independent of the pre-treatment. 3.2.2 Optimum embedded length and efficiency of the yarn. The minimum embedded length Lmin necessary to reach the maximum value of Pmax was determined for each pre-treated yarn. The maximum value Pmax lim of Pmax was determined as the average values of Pmax for embedded length greater than Lmin . The pull-out efficiency epo was calculated as the ratio of Pmax lim to the tensile strength of the yarn. Values of theses parameters are given in table 3. The efficiency is all the more high that the pre-treatment favours the penetration of cement paste in the yarn. The best efficiency is obtained for SG2 yarn. 3.2.3 Stiffnesses κbond and κdebond are respectively the ascending and descending stiffnesses measured for the ascending and descending part of the behaviour curve. Calculation is made for load values in the range 0.5 to 0.8 Pmax . Figure 12 gives the values of these two parameters for the different configurations used in the study. Values lower than those measured in direct tension (right bars on figure 12) reveal a lake of adhesion of

Table 3. yarns.

Figure 12. Values of κbond and κdebond vs. the embedded length.

Optimum embedded length and efficiency of Lmin cm

Pmax lim N

epo %

OC1

W D PI

10 5 3

333 334 354

29 30 34

OC2

W D PI

10 10 5

206 367 484

21 36 48

SG2

W D PI

15 15 3

395 452 748

49 55 81

the sleeve filaments. This is specially the case for OC2 W pre-treated yarn. OC2 and SG1 yarns give κbond values for pull-out tests comparable to those measured in direct tension. κdebond is an increasing function of the embedded length. The observed values for κdebond are dependent of the pre-treatment. In particular, PI pre-treated yarns give high values of κdebond , allowing κdebond to reach the values measured in direct tension. This signifies that in this case the anchorage of the sleeve filaments is very efficient and that this family of filaments behaves as if they were in direct tension.

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3.3 Core and sleeve filaments

the value of the ratio is in some case an over estimated value of the real fraction of core filaments (this fact is particularly true for PI yarns, which explains why in some case the ratio exceeds 100%). Figure 14 gives the values of the ratio mLe /mLL for the different yarns, pre-treatments and embedded lengths. The main trends that should be noted are the decrease of the ratio when the embedded length increases (for OC1, the ratio falls to zero when the embedded length is high enough), the similarity of W and D pre-treated yarns (except for OC2, PI pretreatment leads to much better anchoring of the filaments, which appears as a marked reduction of the mLe /mLL ratio) and the specificity of OC2 yarn, which kept a large amount of core filaments, even for high embedded length.

To evaluate the ratio between sleeve and core filaments, the ratio of the linear mass mLe of the extracted part of the yarn (damaged due to filaments failure) to the linear mass mLL of the undamaged yarn was determined after each pull-out test (figure 13). This ratio gives an idea of the amount of un-anchored core filaments in the loaded yarn. As these filaments could include some small cement particles in between them,

Free length LL: Embedded length Le : Linear mass mLL Linear mass mLe

4 RELATIONSHIP TO THE IMPREGNATION

Figure 13. Determination of the yarn’s linear masses after the pull-out test.

4.1 Optimal embedded length and efficiency

Figure 14. Values of mLe /mLL vs. embedded length.

Figure 15 gives the relationship between the optimal embedded length Lmin , the efficiency and the impregnation index Iy determined from SEM observation (see table 2). The optimal embedded length is a decreasing function of the impregnation index. This is directly connected to the increase of cement paste presence inside the yarn, which reduces the distance between the anchorage points of the filaments. The efficiency tends to increase when the impregnation index increases, which is linked to the increase of the sleeve filaments due to the penetration of the matrix in the yarn. It should be noted that the efficiency never reach 100%. OC1 and OC2 data are very similar, despite of the difference of structure of these two yarns. This may be related to the type of sizing, not design in this case for a use with concrete. Sizing for SG2 yarn is dedicated to concrete applications and favours the load transfer between yarn and matrix. Concerning the effect of the impregnation index on the stiffness, the three yarns, despite of their differences, roughly present the same increasing of κbond lim with the impregnation index (figure 16). This is consistent with the evolution of Lmin : when Lmin becomes lower, the working length of the filaments (i.e. length from the grip to the first anchorage point in concrete) decreases. As a consequence, the displacement is reduced for a given load, which traduces an increase of stiffness. Values of κdebond lim also increase with the impregnation index. They are about ten times highest than κbond lim one for full impregnation. κbond lim and κdebond lim are the values of κbond and κdebond that correspond to Lmin .

85

However, despite of this complexity, the experimental approach presented above allow the main parameters that determined the pull-out behaviour of the yarn to be highlighted. These parameters are the following: the structure of the yarn itself, its sizing, the embedded length, the state of the yarn before the casting of concrete. Most of these parameters influence the way the concrete penetrate the yarn at the time of casting and the resulting impregnation appears as playing a major part in the control of the pull-out behaviour. Impregnation, by influencing the partition between sleeve and core filaments, influences the maximum pull-out load and the efficiency of the yarn. It also determined the anchorage length necessary to reach the maximum efficiency of the yarn. What is now needed is to take into account this complexity in models, so as to evaluate the effect of the different scenarios that can explain the pull-out behaviour. REFERENCES

Figure 15. Lmin and epo vs. impregnation index iy .

Figure 16. index iy .

κbond lim and κdebond lim vs.

Aljewifi, H., Fiorio, B., Gallias, J.L., 2010. Caracterization of the impregnation by a cementitious matrix of five glass multi-filaments yarns. European Journal of Environmental and Civil Engineering, EJECE (in press). Aljewifi, H., Fiorio, B., Gallias, J.L., 2009. Quantitative methods used to characterize the impregnation of a glass multifilament yarn by a cementitious matrix. 4th Colloquium on Textile Reinforced Structures (CTRS4). 3–5 June Dresden: Germany. Chudoba, R., Voˇrechovsky, M., Konrad, M., 2006. Stochas tic modeling of multi-filament yarns. I. Random properties within the cross-section and size effect. International Journal of Solids and Structures 43: 413–434. Hanisch, V., Kolkmann, A., Roye, A., Gries, T., 2006. Yarn and textile structures for concrete reinforcements. FERRO8, Bangkok, Febuary 6th. Hegger, J., Bruckermann, O., Chudoba, R., 2004. A smeared bond-slip relation for multi-filament yarns embedded in fine concrete. 6th International RILEM Symposium on Fibre Reinforced Concretes. Kabele, P., Novák, L., Nemecek, J., Kopecký, L., 2006. effects of chimical exposure on bond between synthetic fiber and cementitious matrix. ICTRC—1st International RILEM Conference on Textile Reinforced Concrete 10: 91–99. Langlois, V., 2004. Etude du comportement mécanique des matériaux cimentaires à renforts synthétiques longs ou continues, PhD, Université de Cergy-Pontoise, France. Naaman, A.E., Namure, G.G., Alwan, J.M., Najm, H.S., 1991. Fiber pullout and bond slip. I: Analytical study. ASCE, Journal of Structural Engineering. 1117(9): 2769–2790. Naaman, A.E., Namure, G.G., Alwan, J.M., Najm, H.S., 1991. Fiber pullout and bond slip. II: Experimental validation. ASCE, Journal of Structural Engineering. 1117(9): 2791–2800. Ohno S., Hannant, D.J., 1994. Modelling the stress-strain response of continuous fiber reinforced cement composites. ACI Materials Journal, vol. 91, pp. 306–312.

impregnation

5 CONCLUSION The results presented in this paper give detailed information on the influence of the pull-out behaviour of multi-filaments yarns. They show all the complexity induced by the non-monolithic structure of the yarn. This complexity is also a consequence of the variability generated by the specific constitution of yarns.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

How to enforce non-negative energy dissipation in microplane and other constitutive models for softening damage, plasticity and friction Zdenˇek P. Bažant Northwestern University, Evanston, Illinois, USA

Jian-Ying Wu State Key Laboratory of Subtropical Building Science, South China University of Technology, China

Ferhun C. Caner UPC, Barcelona, Spain

Gianluca Cusatis RPI, Troy, New York, USA

ABSTRACT: Material constitutive models must be formulated in such a way that the energy dissipation can never become negative during deformation increments within the range of intended applications. However, checking this obvious thermodynamic condition for complex models such as the microplane model (e.g. Bažant 1984, Bažant and Caner 2000) is not a trivial task and is often complicated by incomplete, ambiguous or unrealistic definition of unloading or reloading. Ignoring such incompleteness may result in a misleading appraisal of the performance of the model for monotonic loading. Here an attempt is made to clarify this problem and suggest a simple way of ensuring non-negativity of dissipation. The condition of non-negative increment of energy dissipation density at each continuum point of each loading step in an incremental computation of structural response is formulated in the context of the microplane model. If a negative dissipation is detected, the trial constitutive law is adjusted by a change in the unloading compliance and, if necessary, also by a change of the final stresses in the loading step. This adjustment represents an integral part of the constitutive law and must be considered in calibrating the model by test data. A similar correction is then formulated for tensorial constitutive models. Further it is pointed out that without specifying the unloading behavior, the dissipation inequality cannot be checked, and that by modifying the hypothesis about unloading, negative dissipation increments can be changed to positive. Thus the dissipation inequality is not too important for constitutive models intended only for monotonically applied loads, provided that unloading for the individual microplane strain components either does not occur or occurs only rarely. The dissipation check is very sensitive to the assumption about unloading, and so it makes no sense to get alarmed by a check of the dissipation inequality for constitutive models whose characterization of unloading is known to be simplistic and unrealistic. But for models intended for cyclic loading, this inequality is, of course, an essential criterion of soundness. 1 INTRODUCTION AND DEFINITIONS

where the superior dots denote the derivatives with respect to time t. Two expressions for U may be considered:

In constitutive models intended to describe damage such as distributed microcracking, the elastic stiffness tensor Eijkl as well as is inverse, the compliance tensor Cijkl , is variable (the subscripts refer to Cartesian coordinates xi , i = 1, 2, 3). Under isothermal con˙ is the ditions, the rate of energy dissipation density, D, rate of work of stress tensor σij on the rate of strain tensor, ǫ˙ij , minus the rate of change of the stored strain energy U (e.g., Jirásek and Bažant 2002). Thus we have: ˙ = σij ǫ˙ij − U˙ ≥ 0 D

U =

1 e ǫ Eijkl ǫkle 2 ij

(2)

or, equivalently, U =

1 σij Cijkl σkl 2

(3)

where ǫ e = elastic part of strain tensor. Under isothermal conditions, the former represents the Helmholtz free energy density (or isothermal potential energy),

(1)

87

and the latter the Gibbs free energy density (or the isothermal complementary energy). In some works (e.g., Lubliner 2006), Eq. (1) is enhanced by the term k Qk θk where θk are internal variables and Qk are the thermodynamically associated internal forces. However, such an enhancement is appropriate only if some physically different types of work, e.g., the work of a muscle driven by chemical energy (Lubliner 2006), are present, which is not the case considered here. Alternatively, the internal forces may be considered as a partial or full replacement of the term σij ǫ˙ij ; but this is appropriate only if the plastic work itself is described by a set of internal variables (Rice 1970). Otherwise the internal variables do not belong into Eq. (1).

U

O (a)

2 DISSIPATION IN DAMAGE MODELS The contribution to U from the plane of stress σi j versus strain ǫij is represented by the cross-hatched triangular areas in Fig. 1(a). Generally, the unloading from a damaged state cannot terminate at the origin. An exception is the special case of an isotropic damage model, for which unloading terminates at the initial stress-free state for which the residual stresses σi0j vanish (which corresponds to perfect closing of all microcracks). In this special case Eqs. (2) and (3) may be written as U =

4

C dC

(4)

3

6

7 d

p

(b)

or U =

2

5

1 0

1−ω e 0 e ǫij Eijkl ǫkl 2

D d = 1 2dC 2 Dp = d

1

1 0 σkl σij Cijkl 2(1 − ω)

Figure 1. Areas representing various parts of work or energy dissipation in the one-dimensional case.

(5) where σ , ǫ and C are the stress, strain and compliance. This equation may be rewritten as

0 0 where Eijkl and Cijkl are the fourth-order tensors of the initial elastic moduli and compliances. In real materials, though, plastic frictional deformations always accompany microcracking. The consequence is that the material cannot unload to its initial stress-free state and, after unloading of the material, nonzero residual stresses σij0 or strains ǫij0 are always locked in. Thus U , as given by Eq. (2) or (3), represents the triangular area cross-hatched in Fig. 1(a). Consider now the one-dimensional case, representing the uniaxial loading, or one pair of tensorial components, or one component of the microplane stress or strain vector. The increment of energy dissipation density is, according to Eq. (1)

˙ = σ ǫ˙ − d D dt



1 Cσ 2 2



˙ = 1 σ Cσ ˙ + σ (˙ǫ − C σ˙ − Cσ ˙ ) = 1 σ Cσ ˙ + σ ǫ˙ p D  2 2    ˙d D

˙p D

(7)

˙ . Eq. (7) has a simwhere we have set ǫ˙ p = ǫ˙ −C σ˙ −Cσ ple geometrical interpretation in the one-dimensional stress-strain diagram of Fig. 1(b) depicting an infinitesimal loading increment from point 1 to point 2 for the case of post-peak softening damage: The total ˙ is represented by the energy dissipation increment D area 42634 which is first-order small in terms of strain increment ǫ (cf. Bažant 1996), by subtracting the area of the triangular 2672 (i.e., the change

1 ˙ (6) = σ (ǫ˙ − C σ˙ ) − σ Cσ 2

88

of elastic strain energy, U˙ ) from the area of parallelogram 42734 (i.e., the rate of work of stress, σ ǫ˙ ). The triangular area 1241 is second-order small and thus negligible in comparison. The first term ˙ d , is equal to triangular area 4534 and of Eq. (7), D represents the energy dissipation by damage alone. ˙ p , corresponds to parallelogram The second term, D area 52635 and represents the frictional-plastic energy dissipation. The special case in which the second term vanishes for all load increments (i.e., area 52635 = 0, or length 36 = 0) represents the unloading to the origin, for which there is no plastic-frictional energy dissipation. In concrete, however, the plastic-frictional deformation in the fracture process zone generally dissipates more energy than the microcracking (Bažant 1996).

unknown parameters α and β as follows: D =





(σN δǫN + σ T · δǫT ) d





˙ N + σ T · ǫ˙T )d − U˙ ≥ 0 (σN Dǫ

(10)

This equation represents a summation of the contributions defined by Eq. (7) over all the microplane stress components and all the microplanes. Subscripts i and j label the beginning and end of the loading step in which the strain increments are prescribed; subscripts N and T label the microplane normal and shear components; CN and CT are the normal and shear compliances specified by the microplane constitutive law; and σN and σT represent the normal component and the shear stress vector on each microplane. The use of averages such as 12 (βσN , j + σN ,i ) makes Eq. (6) a central difference approximation. In computations, the integral over the unit hemisphere surface is approximated by a summation based on an optimal Gaussian integration formula. At the end of computation of each small loading step (ti , tj ), one evaluates D from Eq. (8) assuming that α = β = 1. If D, no change is made. But if D is detected, one solves a new value of α from the condition D = 0

(11)

still assuming that β = 1. This means that the unloading compliances are changed from CN and CT are changed to αCN and αCT . However, if the new αCN or new αCT is greater than the initial elastic compliance for one or more microplanes (which is inadmissible), a revised α and a new β must be obtained from the condition that both αCN − CN0 ≥ 0 and αCT − CT0 ≥ 0 for all the microplane while D > 0. The minimum value of β satisfying these inequality conditions should be used, which is achieved by decreasing β in small steps until all the aforementioned inequalities are satsified. The α and β corrections are implemented at each integration point of each finite element at the end of calculation of each loading step. These corrections, which adjust the unloading moduli and the final microplane stresses, must be regarded as part of the microplane constitutive law. Thus the initially assumed constitutive law of the microplane model represents only a trial constitutive law, and the α and β corrections based on Eq. (10) complete the definition of the constitute law. These corrections must, of course, be considered in data fitting and calibration of the microplane constitutive model.

(8)

and the energy dissipation density is ˙ = 3 D 2π



1 (βσN , j + σN ,i )(ǫN ,j − ǫN ,i ) 2

− (αCN , j · βσN2 , j − αCN ,i · βσ 2T ,i )

− (αCT , j · βσ 2T , j − αCT ,i · βσ 2T ,i ) d

The microplane model was conceived as a counterpart of the classical Taylor model (Taylor 1938, Batdorf and Budianski 1949), permitting the softening to be modeled. In this model, the total energy density dissipated, D, is the sum of the energies dissipated on all the microplanes. The contribution to D from each microplane can be positive or negative but the sum (or integral) of all these contributions must be nonnegative. In the fitting of complex multiaxial data for complex loading histories, such as those for concrete, it is often not easy to ensure a priori that the dissipation inequality be always satisfied. In microplane model M1 (Bažant and Oh 1986), in which only the normal and shear components of the stress and strain vectors on the microplanes are considered, the virtual work equation is given by 3 2π



+ (βσ T , j + σ T ,i )(ǫT ,j − ǫT ,i )

3 ENFORCING NON-NEGATIVE DISSIPATION IN MICROPLANE MODEL

δW =

3 2π

(9)

Therefore, it is proposed to make in each small loading step from time ti to time tj the following correction to the a priori assumed constitutive law: If a negative increment of the total D is detected, the compliance increment or the stress increment, or both, are reset so as to be make D non-negative. To this end, one may introduce, for an explicit finite element program,

89

4 ADAPTATION TO MICROPLANE MODEL WITH VOLUMETRIC-DEVIATORIC SPLIT

are also an equilibrium system of forces, but such a postulate is not consistent with the calculation of the dissipated work. ˙ in microplane Therefore, the energy dissipation D models M2, M3 and M4 should be expressed as

In microplane models M2 (Bažant and Prat 1988), M3 and M4 (Bažant et al. 2000), the normal strain and stress on the microplanes are split into the volumetric and deviatoric components. Upon substitution of the relations ǫN = ǫV + ǫD and σN = σV + σD Eq. (10), based on the principle of virtual work, becomes δW =

3 2π 

˙ VD ˙ =W ˙ − U˙ = W ˙ − U˙ +W D  m  ˙m D

˙m = W ˙ m − U˙ ≥ 0. It is easy to ensure that D Therefore, it is only necessary to enforce the condition ˙ VD ≥ 0, i.e., W  ˙ VD = 3 W (σD ǫ˙V ) d = 3σ¯ D ǫ˙V ≥ 0 (18) 2π



(σV δǫV + σD δǫD + σT · δǫT ) d

  δWm  3 σD δǫV d

+ (12) 2π

   δWVD

where σ¯ D is the average deviatoric stress over all microplanes  1 σ¯ D = σD d

(19) 2π

The microplane volumetric and deviatoric stress components are expressed separately in terms of the corresponding microplane strains ǫV and ǫD ; σV = fV (ǫV ),

σD = fD (ǫD )

Condition (18) requires that the sign of the average deviatoric stress σ¯ D be the same as that of ǫV . If ǫV ≥ 0, the average deviatoric stress σ¯ D should be non-negative, i.e., the positive deviatoric stress on the microplanes under deviatoric tension should overall be greater in magnitude than the negative deviatoric stress on those under deviatoric compression. If ǫV < 0, the average deviatoric stress σ¯ D should be negative (this property is necessary to describe the di-latancy exhibited under uniaxial and biaxial compression loadings).

(13)

In microplane model M2, functions fV and fD are formulated as a microplane damage model, whereas in microplane models M3 and M4, these functions are implied by the strain and stress boundaries. In model M4, they are further constrained by the condition: σV = min



1 2π





 σN d , fV (ǫV )

(14)

In M4 (Bažant et al. 2000), only the first term, δWm (Eq. 12), is considered in the virtual work equation, i.e.,  3 (σV δǫV + σD δǫD + σ T · δǫT ) d (15) δW = 2π

5 ENFORCING NON-NEGATIVE DISSIPATION IN TENSORIAL FORM OF CONSTITUTIVE MODEL For a constitutive law in the classical tensorial form, the increment of energy dissipation density loading step (tr , ts ) is given by

The microplane volumetric stress σV is defined by the virtual work equation 3 σkk δmm = 3 2π



σV ǫV d

(17)

1 1 (σ r + σ s ) : ǫ − βσ s : αCs : βσ s 2 2 1 + σ r : Cr : σ r ≥ 0 2

D =

(16)



which leads to σV = σkk /3. This is an equilibrium definition of microplane volumetric stress σV . As one can see from Eqs. (8), (12), (15) and (16), the stress tensor σij , microplane normal stress σN and shear stress vector σ T are an equilibrium system of forces, and can thus be used to calculate the first-order work and energy dissipation, which underlies Eq. (10). One can, of course, introduce a postulate that the stress tensor σij , the microplane volumetric stress σN , the deviatoric stress σD and the shear stress vector σ T ,

(20)

(now σ , ǫ and C are all tensors). At the end of the computation of each load step, the procedure is as follows: • Set first α = β = 1. • Check if D ≥ 0. • If satisfied, go to the next integration point. If not, find α from the condition D = 0, which amounts to adjusting the constitutive law for damage.

90

• But if 21 σ s : αCs : σ s < 12 σ s : C0 : σ s , then reset also β so that, in this condition, ‘. Moreover, regarding the positioning of cracks as a problem of random placement of objects with the length δ along a line the average crack spacing in the final stage III (no more cracks fit in) can be quantified as (Widom 1966) csF = 1.337δ.

Figure 7. Strains between two cracks with nonoverlapping debonding zones.

(6)

Evolution of crack spacing: During the cracking stage II, multiple cracks in the matrix occur. The matrix strength is assumed random with the probability of failure prescribed by the cummulative Weibull distribution

σm m . (7) P(σm ) = 1 − exp − σmu Here, m denotes the Weibull modulus and σmu the scale parameter of the distribution. As the loading level σ¯ increases, new cracks occur along the specimen at random locations. Their average spacing cs is getting finer until the final average crack spacing csF (see Eq. 6) is reached. The hypothesis behind the stochastic cracking model is that in the stage II the crack spacing is inverse proportional to the cummulative probability distribution of the matrix strength: cs =

1 csF . P(σm )

Using Eqs. (7) and (4) the development of crack spacing from the start to the saturated stage is given as 

−1 σ¯ m cs = csF 1 − exp − . σ¯ R

(8)

Figure 8. Strains between two cracks with overlapping debonding zones.

debonding zones the strain profile becomes triangular (Fig. 8). The macroscopic (effective) strain can be obtained using the standard averaging technique: the effective composite strain ε¯ can be obtained as the average strain in the reinforcement εf  within a sufficiently long segment of the specimen. More precisely, due to the periodicity of the strain field the necessary and sufficient averaging volume is equal to the half of the crack spacing at the instantaneous stress level σ¯ .

¯ mu /Em is the referential comHere, the σ¯ R = Eσ posite stress, i.e. the composite stress at average level of the matrix strength. Averaging of strains: With the above aligns at hand it is possible to specify the strain profile between two neighbouring cracks at any level of the loading stress σ¯ . For low stress level, the debonding zones do not ovelap so that the strain profile has a trapezoidal shape with strain equality between the debondinz zones (Fig. 7). For higher stress level with overlapping

107

ε¯ =

2 cs

cs 2

0

εf (x) dx.

For low stress level, with 2δ < cs, the trapezoidal shape of the εf stress profile (Fig. 7) the averaging delivers the effective strain in the form 1 ε¯ = E¯

αδ σ¯ , 1+ cs

(9)

where (1 − ρ) Em . α= ρ Ef Once the debonding zones start to overlap 2δ > cs (Fig. 8) the averaging of fiber strains renders

1 α cs ε¯ = − σ¯ . (10) ρEf 4δ E¯ Remarks: • The strain hardening curve obtained using Eqs. (9) and (10) does not depend on the the frictional stress level τ and radius of the fiber r. • The stochastic cracking model does not reflect the effect of the reinforcement ratio on the matrix strength. • The described mesoscopic resolution of the damage process into cracking of the matrix and debonding is limited to a simple frictional behavior. In spite of the mentioned limitations, the model can be used in its present form as a meso-scopic representation of the failure process that can be integrated within the macroscopic damage model. Moreover, the limitations regarding the effect of the reinforcement ratio on the level of matrix cracking stress and the microscopic effects in the crack bridge can be included in the model by further enhancements. An extension of the stochastic cracking model with the microscopic representation of a crack bridge has been presented in (Konrad, Jerabek, Vorechovsky, and Chudoba 2006). 6 DIRECTION DEPENDENT REINFORCEMENT RATIO The application of the described mesoscopic model has been primarily motivated by the need to reflect the effect of the reinforcement ratio on the strainhardening behavior of the composite. The same model shall be employed for reflecting the directionally dependent response of the composite due to the orientation of the reinforcement. Indeed, by introducing an assumption that the effective reinforcement ratio changes with the angle between the crack plane and the reinforcement orientation the closed form of the strain-hardening curve given by Eqs. (9) and (10) could be applied to extrapolate the response measured in a tensile test with aligned reinforcement to a response with inclined reinforcement. For the present paper a simple projection of the reinforcement ratio on the crack plane is used (see Fig. 9). Full alignement of the yarn with the loading direction is assumed. Further, the effective cross sectional area of the yarn is assumed to remain constant in the crack bridge. For a composite reinforced with uni-axial fiber reinforment (no textile fabrics)

108

Figure 9. plane.

Projection of the reinforcement ratio on the crack

these assumptions lead to the following relationship between the reinforcement ratio and the inclination α of the reinforcement: ρ(α) = ρ0 cos(α),

(11)

with ρ0 representing the reinforcement ratio for α = 0. This mapping accounts for the two extreme cases of full reinforcement ratio ρ(α = 0) = ρ0 and no reinforcement ration ρ(α = π2 ) = 0. The transition between those two extremes is described by cos(α) reflecting an increasing distance between the yarns in the plane parallel to the crack. The assumptions applied for the simple directional mapping of the reinforcement ratio using Eq. (11) do not reflect the local microscopic effects occurring in a crack bridge. In particular, the snubbing, debonding and lateral pressure at the crack edges are not included. As dicussed in (Bruckermann 2007) the complete alignement of the filament with the loading direction holds primarily for the outer filaments. The link can be improved by a more detailed micromechanical resolution of the crack bridge distinguishing individual filament groups. The micromechanical models provide a more realistic description of the crack bridge behavior. They can be linked with the mesoscopic model in terms of the fiber crack bridging stress and the corresponding stress transfer length δ (Konrad, Jerabek, Vorechovsky, and Chudoba 2006). However, even for the simplified projection of the reinforcement ratio to the crack plane the validity of the macroscopic model is significantly extended by

establishing a link to the mesoscopic failure mechanisms, namely the matrix cracking and debonding.

7 CONCLUSIONS Figure 10 gives an overview of the entire modeling approach. The tensile test specimen (1) is used to calibrate the mesoscopic stochastic cracking model (SCM). The mesoscale model together with the direction dependent reinforcement ratio introduced in Sec. 6 can then be used to obtain the response for varying inclinations of the reinforcement with respect to the loading direction (2). This information is the basis for the calibration procedure of the directiondependent damage functions (3) of the microplane damage model (MDM). After the model has been calibrated the multi-axial behavior of a TRC-structure can be simulated as exemplified by a depicted quarter of a thin plate with hinge supports at the corners (4). The red/dark color shows the amount of dissipated energy. Summarizing, the modeling framework combines the mesoscopic model for uniaxial tension reflecting multiple cracking and debonding with the macroscopic damage model. The flexibility of the microplane model is exploited in order to construct a direction dependent damage functions. Further adjustments of the introduced directional dependency of the damage specification for TRC are necessary. In particular the kinematic behavior of the model during the localization process must be adjusted in order to reflect the meso-level damage mechanisms occurring in the tested material. The microscopic aspects of the crack bridge behavior, as well as effects connected with unloading and cyclic loading (Konrad, Chudoba, and Kang 2006) have not been considered in the present paper.

ACKNOWLEDGEMENT The work has been supported by Deutsche Forschungsgemeinschaft (DFG) in the framework of the collaborative research center SFB 532 Textilereinforced concrete, development of a new technology. The support is gratefully acknowledged.

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Figure 10. Steps included in the modeling of the cementitious composite.

109

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A statistical model for reinforced concrete bond prediction Z. Dahou University of Béchar, Department of Civil Engineering, Béchar, Algeria

Z.M. Sbartaï Université Bordeaux, Talence, France

A. Castel Université de Toulouse UPS, INSA, LMDC, Toulouse cedex, France

F. Ghomari University of Tlemcen, Department of Civil Engineering, Tlemcen, Algeria

ABSTRACT: This study proposes a statistical approach, based on artificial neural network model, for modelling the bond between conventional ribbed steel bars and concrete. Then and according to RILEM test configuration (Rilem 1970), the ultimate pull-out load is predicting from the concrete mix constituents and from the steel bar diameter. More than hundred pullout tests were carried out in order to investigate experimentally the bond behaviour between three concrete mixes, with different constituent proportions, and two different diameters of ribbed bars. These experimental results show that the pull-out load is as affected by concrete mixes as strength of concrete and the bar diameter. On the whole, six input data were used for the ANN model and the ultimate pull-out load was the output data. The network was Multi-Layer-Perceptron trained according to a back-propagation algorithm. The agreement between experimental results of other authors and simulation of the proposed model proves a satisfactory accuracy. 1 INTRODUCTION A fundamental property of reinforced concrete is the bond between the reinforcement and the concrete. In anchorage zones, the two important aspects that control the bond are the force transfer mechanism between the reinforcement and the surrounding concrete and the capacity of the concrete to resist reinforcement pull-out. Chemical adhesion, friction and mechanical actions are the mechanisms that assure bond efficiency. The contribution of each of these mechanisms depends on the surface and geometry of the reinforcement. In the case of ribbed bars, the force transfer is mainly governed by the blocking of the ribs in the concrete (Eligehausen et al. 1999). Thus, the bond depends on numerous and various factors, which basically concern the reinforcing bars, the concrete and the stress state in both the reinforcing and the surrounding concrete. Generally, bond strength can be assessed by different test in laboratories. Also in codes of practice, equations are proposed for the bond evaluation. This relations, referred usually to a correlation between the strength of concrete and the bond stress.

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The main goal of this study was to develop a bond model allowing the pull out load prediction from concrete mix constituents, concrete maturity and the ribbed steel bar diameter. So, an experimental program was developed by conducting pull-out tests on ribbed bars with a nominal diameter of 10 mm and 12 mm. Three concretes (two grades 40 and one grade 30) were studied. The specimens were tested at various ages ranging from 1 to 28 days. Based on these experimental data, an Artificial Neural Network (ANN) was used in order to predict the ultimate pull-out load. The basic strategy for implementing a neural based model of material behaviour is to train a neural network on the results of a series of experiments on a material. If the experimental results contain the relevant information about the material behaviour, then the trained neural network will contain sufficient information to qualify as a material model. Such a trained neural network would not only be able to reproduce the experimental results on which it was trained but should also, through its generalization capability, be able to approximate the results of other experiments (Ghaboussi et al. 1991).

2 ARTIFICIAL NEURAL NETWORKS An artificial neural network (ANN) can be represented as a simplified model of the nervous system consisting of a large number of information processing elements. ANN has been successfully applied to a wide range of engineering and scientific applications such as classification, pattern recognition, process control and prediction based on historical data. The fundamental principle of ANN has been reported by several researchers (Dreyfus et al. 2002), (Jodouin 1994), (Bishop, 1995), (Rafiq et al. 2001). The basic element of the method is the artificial neuron called the Perceptron, which is a mathematical model of a biological neuron as presented in (Fig. 1). Artificial neurons connected together form a network. Depending on the way in which the neurons are connected, many types of artificial neural networks are distinguished (Dreyfus et al. 2002), (Jodouin 1994). The Multy-Layer Perceptron (MLP) is the most popular (Yeh 1998), (Dias et al. 2001), (Nehdi et al. 2001), and is adopted in this work. The MLP is composed of an input layer, one or more hidden layers and an output layer. The relationship between the inputs (xi ) and the output (y) can be written as follows:   n  (1) y =F b+ w i xi i=1

where F is an activation or transfer function. Several functions are available but the sigmoid is the must common (Fig. 2). Upper and lower limits of output from this function are generally 1 and 0 respectively. It is therefore recommended to normalise the input and output data before presenting them to the network. The vectors wi and b design weights and bias vectors. They governed the rate of information transmitted between input and output. These network parameters are optimised by the use of a training process. Several training algorithms exist but back-propagation (BP) commonly provides satisfactory results (Rumelhart et al. 1987). BP is a gradient descent algorithm which consists of changing the weights and bias according to the negative of the error function. They are expressed

Figure 2.

Sigmoid function.

by equations 2 and 3. This process must be repeated until the network error is minimized. To enhance the generalization capacity of the network, a crossvalidation method is generally used which consists of testing the network on new data at each iteration (Weigend et al. 1991), (Morgan et al. 1990). These new data are not used in the training processes and are composed of approximately 20% of the database. Evaluation of the network prediction must therefore be based on the testing data prediction. wi+1 = wi − η∇Ei/w bi+1 = bi − η∇Ei/b

(2) (3)

wi+1 is the corrected weight value, wi is the weight value at iteration i, η refers to the learning rate, ∇i is the error gradient computed at iteration i: ∂Ei (4) ∂w Ei designates the Root Mean Square Error (RMSE) and it is defined by:   N 1  |en |2 (5) Ei =  N ∇Ei/w =

n=1

3 EXPERIMENTAL PROCEDURES The standard RILEM pull-out test was used to study the anchoring capacity of the rebar in the concrete. The influence of the diameter of the ribbed bars, the concrete mixes and maturity were studied experimentally. Concrete mix variables were the type of cement (R for fast hydration and N for normal hydration), the water to cement ratio (W/C), the gravel to sand ratio (G/S) and the crushed to rolled gravel ratio (Gc /Gr ). Only standard 52.5 R (MPa) and 52.5 N (MPa) grade Portland cement was used. 3.1 Concrete mixtures

Figure 1.

Three concretes (two grade 40 MPa and one grade 30 MPa) were studied. Table 1 summarizes the mix proportions. The ratio of crushed gravel to rolled

Artificial neuron (Perceptron).

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Composition of the vibrated concretes.

Table 2.

425

375

710 532.5 532.5 185 0.435 1.5 1

755 336 790 187.5 0.5 1.5 2.35

Digital VC30 (kg/m3 )

Silico

Bond length L

VC40b (kg/m3 )

325 811 382 731 195 0.6 1.37 1.91

Total length: 2 x

Cement 52.5 R Cement 52.5 R Sand 0/4 Rolled Gravel 4/10 Crushed Gravel 10/14 Total water W/C G/S Gc/Gr

VC40a (kg/m3 )

5 (no

Table 1.

Fixed Rebar

Plastic

Mechanical properties of the concretes.

Concretes

Compressive strength MPa

Elastic modulus GPa

VC40a VC40b VC30

43.9 43.8 36.8

32.0 34.2 25.1

F Figure 3. tions.

gravel, ranging from 1 to 2.35, was the main difference between the two 40 MPa grade concretes. Also, the water to cement ratio was significantly higher for concrete VC40b. The specimens were removed from their moulds 24 hours after casting and stored for 28 days in a controlled environment (T◦ = 20◦ C, RH% = 60%). The compressive strength and the instantaneous elastic modulus were measured at 28 days on concrete cylinders (diameter = 110 mm, height = 220 mm). Table 2 shows the mechanical properties of the tested concrete.

Pull-out test according to RILEM recommenda-

The bond strength was calculated assuming a uniform distribution of bond stresses along the bond length. It was calculated from the ultimate pull-out load using the relation (6): τu =

Fu π ·φ·L

(6)

where τu is the ultimate bond stress (MPa), φ is the rebar diameter (mm), L is the bond length (mm) and Fu is the ultimate pull-out load (N).

4 TESTS RESULTS AND PARAMETERS OF THE BOND MODEL

3.2 Pull-out tests

4.1 Load-slip curves

3.2.1 Description of the tests specimens The pull-out test was carried out according to Figure 3. The experimental set up was similar to the pull-out test described in the RILEM recommendations (Rilem 1970). The specimens had a concrete cross-section of 10 × 10 cm. There were two possible diameters for the ribbed bars used: 10 mm and 12 mm. All bars had a yield strength of 500 MPa.

In each pull-out test, the slip of the free end of the reinforcement was recorded on a plotting table versus the load applied. For the three concrete, pull-out tests were performed for each ribbed bar diameter (10 mm and 12 mm) at different concrete maturities. The VC40a specimens were tested at 5, 7, 14 and 28 days. For the VC40b, the tests were conducted at 1, 3 and 11 days. Experiments on VC30 concrete were carried out both at early age and at 7, 14 and 28 days. For each configuration (concrete mix and diameter of the bar), four specimens were tested. For all specimens, failure occurred at the interface between the reinforcing bar and the surrounding concrete and was due to the steel pull-out. For example, the curve of the Figure 4 shows the typical load-slip behaviour obtained. It correspond to

3.2.2 Testing The pull-out bond specimens were tested up to bond failure by applying progressively a tensile load to the end of the bar. It was pulled from the concrete cube by controlling the applied force, at rate of 0.1 kN/s. To plot the bond stress—slip relationship, the slip was measured at the free end of the bar by a digital transducer with an accuracy of 0.001 mm (Fig. 3).

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a VC30 concrete specimen, with a 10 mm or a 12 mm ribbed bar, carried out after 14 days of curing. The curves show a different behaviour in relation to the diameter. Due to the simplified uniform bond stress distribution assumed, this difference do not appear clearly for the ultimate bond stress (Fig. 5) given by relation (6). For the whole set of specimens tested, more than 95% showed a standard deviation of less than 3.5 KN (Fig. 6).

4.2 Effect of the parameters of materials In turn for the concrete and the bars, the impact of the individual parameters was studied to establish their influence on the ultimate pull-out load. For both bars tested the Figures 7–9, show the variation of the ultimate pull-out load with respect to the age, for the three concretes mixtures respectively. The results reported correspond to the average ultimate pull-out load obtained from four tests. For all the concrete mixes, the ultimate pull-out load increased from 10 kN to 30 kN for the specimens with the 10 mm

35 HA10

HA12

60

25

HA12

HA10

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25

50 Ultimate Pull-out load (kN)

Pull out load (kN)

30

20 15 10 5 0

40 30 20 10

0

1

2

3

4

5

6

7

8

9

10

Slip (mm)

0 1

Figure 4. Pull-out-load versus slip curves of the two deformed bars (VC30 concrete used and specimens tested at 14 days).

4

7

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13

16

Age (days)

19

28

Figure 7. Average ultimate pull-out load versus age for the VC40a concrete. 60

16 HA10

HA12

HA12

HA10

22

25

50

12

Ultimate Pull-out load (kN)

Bond stress (MPa)

14

10 8 6 4

40 30 20 10

2 0 0

1

2

3

4

5

6

7

8

9

0

10

1

Slip (mm)

Figure 5. Bond stress versus slip curves of the two deformed bars (VC30 concrete used and specimens tested at 14 days).

4

7

10

13

16

Age (days)

19

28

Figure 8. Average ultimate pull-out load versus age for the VC40b concrete. 60 HA12

30%

HA10

50 Ultimate Pull-out load (kN)

25%

Frequency

20% 15% 10%

40 30 20 10

5%

0

0% 0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

1

5,5

4

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10

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16

Age (days)

19

22

25

28

Standard Deviation [KN]

Figure 6.

Frequency distribution of the standard deviation.

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Figure 9. Average ultimate pull-out load versus age for the VC30 concrete.

diameter bars and from 15 kN to 50 kN for the specimens with the 12 mm diameter bars. The ultimate pullout load measured on the VC30 concrete remained lower than those obtained on the VC40 specimens. So, the influence of the ribbed bar diameter and the concrete compressive strength (VC40 and VC30) was clearly apparent. Also, the ultimate pull-out load increased significantly with the concrete maturity. After 1 day and 3 days, grade 40 MPa concretes, cast with fast hydration cement type R, reached about 50% and 75% respectively (average value obtained on the 10 mm and 12 mm diameter steel bars) of the bond strength measured after 28 days. For the grade 30 MPa concrete, cast with normal hydration cements type N, only about 35% and 55% (average value obtained on the 10 mm or 12 mm diameter steel bars) of the bond strength measured after 28 days was reached after 1 day and 3 days respectively. It is well known that concrete constituents, such as the type of cement, the water-cement ratio and the granularity, influence the strength of concrete, which is a key factor for enhancement of the bond with ribbed steel bars. The water to cement ratio is obviously a key parameter influencing the bond strength but results obtained on both VC40 show that the ratio of crushed gravels to rolled gravels is also significant. VC40a and VC40b have water to cement ratios equal to 0.435 and 0.5 respectively (Table 1) and, according to Figures 7 and 8, show almost the same bond strength. This is due to the crushed to rolled gravel ratio, which is significantly higher for the VC40b. Therefore, both grade 40 MPa concretes had the same compressive strength after 28 days (Table 2) in spite of their different watercement ratios. Only grade 52.5 MPa cement was used in this experimental program, but two types were studied: N (Normal hydration) and R (Fast hydration). According to the experimental results, this factor influences the compressive strength of the concrete at early age and, thus, the bond strength. Finally, as the gravel interaction with the steel ribs significantly increases the bond strength, the characteristics of the gravels (sizes and types) affect both the compressive strength (Guinea et al. 2002) and the bond strength and will therefore be taken into account in the development for the statistical ANN bond model.

Instead of considering the concrete compressive strength as input data to the ANN model, the water to cement ratio (W/C), gravel to sand ratio (G/S), and crushed to rolled gravel ratio (Gc/Gr) were used. Moreover, the type of cement (R: fast hydration or N: normal hydration) was added because it influences the concrete strength at early age. The age of the concrete and the ribbed steel bar diameter were directly used as ANN input factors. Thus, a total of six factors were considered as input data of the ANN bond model.

5 DEVELOPMENT OF THE ANN PULL-OUT MODEL The experimental set of data was divided into two sets: a training set and a testing set. The first one represented 80% of the whole database and the reset was reserved for the testing process. MATLAB software was used for the numerical implementation of the ANN and the training algorithm. The network structure: six inputs, one hidden layer of 10 neurons and one output were adopted. The structure of the network is shown in Figure 10. The variations of training RMSE and testing RMSE are plotted with respect to the number of epochs in Figure 11.The training process in this figure was limited when the optimum error was reached. It’s corresponding to a minimum error of the testing data of about 1.2 × 10−3 achieved after 2000 epoch. The performance of the model was assessed mainly by analyzing the prediction ability related to data set.

[W]

bH

W/C

1

[Z]

G/S

2 b0

Gc /Gr

3 Fu

Cement

…

9

4.3 Summary and ANN input parameters Experimental results show that the concrete compressive strength, the concrete maturity and the diameter of the bar tested have a significant effect on the ultimate pull-out load. Thus, the input data of the proposed ANN-bond model must be selected to take into account the influence of these factors on the pull-out load, chosen as the output of the ANN model.

115

10 age INPUT

HIDDEN LAYER

OUTPUT LAYER

Figure 10. Structure of the network bond model.

The correlation coefficient (R) is equal to about 0.98 for the training set and to about 0.97 for the testing set. Moreover, 90% of the training patterns were located near the perfect prediction line with an absolute accuracy range of ±3 kN. For the testing set, 81% of the patterns showed the same accuracy. Only 5% of the tested patterns presented absolute errors exceeding ±5 kN. 6 VALIDATION OF THE ANN MODEL ON EXPERIMENTAL RESULTS FROM THE LITERATURE Experimental results obtained by other researchers (Castel et al. 2006), (Daoud et al. 2002), (Koning et al. 2001), (Zhu et al. 2004), (Xiao et al. 2007) were considered in this section in order to test the validity and the accuracy of the proposed ANN bond model. Table 3 gives the specifications of concrete and reinforced for each research. As some experimental factors did not fit with our experimental range used for the development of the ANN bond model, the following strategy was applied:

Figure 11. Model network training and testing RMSE variation versus the number of epochs. 52

R² = 0.97

ANN Pull-out load (kN)

48 44 40 36 32 28 24 20

– In the case of concrete mixes incorporating only crushed or only rolled gravels, the crushed to rolled gravel ratio could not be taken to be equal to infinity or zero. So the bond model was used with this ratio fixed at the smallest value (Gc/Gr = 1) for mixes with rolled gravel and the largest value (Gc /Gr = 2.35) for concrete with crushed gravel. – For the application of the proposed model to concrete mixes with lower grades of cement, we assumed that this factor had the same effect on the bond strength as on the compressive strength of concrete.

16 12 8 4 0 0

4

8

12

16

20

24

28

32

36

40

44

48

52

Experimental Pull-out load (kN)

ANN Pull-out load (kN)

Figure 12. ANN-predicted versus measured ultimate pullout load—Training data set. 52 48 44 40 36 32 28 24 20 16 12 8 4 0

R² = 0.94

The Bolomey law (7) (Dreux et al. 1998), assumes a linear relationship between the concrete compressive strength (fc ) and the cement grade (FCE ) by taking the cement to water ratio (C/W) and the coefficient (G), Table 3. A detail of Concretes mixes and ribbed bars for referenced research.

0

4

8

12

16

20

24

28

32

36

40

44

48

Authors

52

Cement W/C G/S

φ Bond Gc /Gr [mm] length

Experimental Pull-out load (kN)

Castel et al.

Figure 13. ANN-predicted versus measured ultimate pullout load—Testing data set.

Predicted and measured values are presented and compared in Figure 12 for the training set and Figure 13 for the testing set. They show that predicted and measured ultimate pull-out loads are well correlated.

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32.5 R 52.5 R Daoud et al. 52.5 R Konig et al. 42.5 R Zhu et al. 42.5 N 42.5 N Xiao et al. 32.5 R

0.5 0.43 0.57 0.55 0.68 0.68 0.43

1.51 1.49 1.93 2.17 1.15 1.15 2.24

2.58 1 C* R* C* C* C*

12 12 16 10 12 20 10

5φ 5φ 5φ 5φ 10φ 6φ 5φ

* The letters C or R specifies that only crushed or rolled gravel was used.

Daoud et al. Konig et al. Zhu et al. Xiao et al.

(*)

(**)

Variation [MPa]

12.70 22.47 15.11 20.17 10.24 7.89 17.38

20.07 22.11 13.34 26.05 10.67 7.97 28.31

12.04 / / 20.84 8.54 6.38 16.99

0.66 0.36 1.77 0.67 1.70 1.51 0.39

* Corresponding to calculated values. ** Corresponding to equivalent values.

Table 4 shows ultimate bond stress measured experimentally by different researchers and those estimated by the proposed model. The comparison shows that the model is able to predict the bond strength with acceptable accuracy. Although the coefficients characterizing the granularity were, in the majority of the concrete mixes reported in table 3, outside our experimental margin, the model reproduced the experimental ultimate bond stress with a maximal variation of 2 MPa. 7 PARAMETRIC STUDY OF THE ANN BOND MODEL

45 40 35 30 25 20 15 10 5 0 60

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (a)

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (b)

50 40 30 20 10 0

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (c)

Figure 14. Comparison of the effect of water-to-cement ratio on value of the pull-out load for two ribbed bars. (a) Concrete maturity 3 days (b) Concrete maturity 7 days (c) Concrete maturity 28 days.

45 N cement

Ultimate Pull-out Load (kN)

which characterizes the quality of gravel, as constant. In our case, G was equal to 0.45 and, for a fixed cement to water ratio, grade 42.5 MPa and grade 32.5 MPa cement led to a reduction in compressive strength of about 20% and 40% respectively in comparison to a concrete cast with grade 52.5 MPa cement. Therefore, a 20% or 40% reduction was applied to the calculated ANN pull-out load when the cement grade used was 42.5 or 32.5 MPa respectively. The pull-out load obtained is labelled equivalent pull-out load.  C − 0.5 (7) fc = G · FCE W

Ultimate Pull-out Load (kN)

Castel et al.

Experimental τu [MPa]

Ultimate Pull-out Load (kN)

Authors

Estimated τu [MPa]

40 35 30 25 20 15 10 5 0

Ultimate Pull-out Load (kN)

Table 4. Evaluation of the bond model for predicting ultimate bond stress.

R cement

40

35

30

25

20

The ANN bond model can be used to simulate the effects of concrete mix on the ultimate pull out load. The simulation results are shown in Figures 14 to 17. Firstly, figures 14a, b, and c compare the ultimate pull-out load for both diameters for different value of the water to cement ratio. The ultimate pull-out load, and so the bond strength, increase significantly with the decrease of the water to cement ratio. At 28 days, for a water-cement ratio increasing from 0.435 to 0.6, the decrease in ultimate pull-out load is 30% for the 10 mm diameter bar and about 35% for the 12 mm diameter bar. So, the water to cement ratio has the

117

0

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10

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14

Age (days)

Figure 15. Effect of type of cement (N or R) on ultimate pull-out load.

same effect on both the bond strength and compressive strength. Figure 15 shows the variation in ultimate pullout load versus the type of 52.5 MPa cement. Fast hydration cement (R) leads to a significant increase in the bond at early age compared to normal hydration

The concrete mixes used did not contain any chemical or mineral admixtures. The experimental results provided a database for implementing a Neural Network model for the ultimate pull-out prediction. The following conclusions can be drawn from the present study:

60

Ultimate Pull out Load [KN]

3 days

7 days

14 days

28 days

50 40 30 20

– The results show that artificial neural networks can be implemented to model the experimental relationship between the ultimate pull-out load and parameters such as water-cement ratio, gravel-sand ratio, crushed-rolled gravel ratio, type of cement, diameter of bar and concrete maturity. The bond strength can then be deduced according to the RILEM method. – The statistical model, trained, tested and validated according to the large laboratory database, shows good accuracy in the ultimate pull-out load prediction and a high capacity for generalization. – The ANN model was based only on experimental results with bar diameters of 10 mm or 12 mm, the prediction of bond strengths measured by different researchers on specimens with larger diameter bars was also accurate.

10 0 1,36

1,38

1,4

1,42

1,44

1,46

1,48

1,5

1,52

Gravel-sand ratio

Figure 16. Ultimate pull-out load versus gravel to sand ratio. 60 1

1,5

2

2,35

Ultimate pull out load (kN)

50 40 30 20 10

Further experiments including other input concrete parameters could extend the field of application of the proposed ANN bond model. The influence of the cement grade could be investigated experimentally, which may enhance the accuracy of the prediction. Concrete mixes including water reducers or plasticizers could be studied. A wider variation range of the experimental input parameters, such as water to cement ratio or gravel to sand ratio, could be also useful.

0 3

7

14

28

Age (days)

Figure 17. Effect of crushed to rolled gravel ratio on ultimate pull-out load.

cement (N). This accelerated hydration effect on bond strength disappears with increasing concrete maturity. For concrete mixtures with gravel to sand ratios increasing from 1.37 to 1.5, the ultimate pull-out load calculated increases of about 17% for any concrete maturity (Fig. 16). An increase in crushed gravel proportion also leads to a significant improvement of the ultimate pull-out load (Fig. 17). After 28 days, an increase in the Gc /Gr ratio from 1 to 2.35 leads to an increase in the ultimate pull-out load of about 18%. This result is in accordance with other studies (Appa et al. 2002) which have shown that concrete shear strength, and thus bond strength with ribbed steel bars, increases as the roughness of the aggregate increases. 8 CONCLUSIONS The bond between concrete and steel bars in reinforced concrete structures is governed by numerous factors. Experimental work has been performed to study how the ultimate pull-out load, according to the RILEM recommendation, is affected by the concrete mix proportions, the concrete maturity and the diameters of the ribbed bars. Two grades of concrete strength, of about 30 and 40 MPa, were considered.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Introduction of an internal time in nonlocal integral theories Rodrigue Desmorat & Fabrice Gatuingt LMT-Cachan, ENS Cachan/CNRS/Université Paris 6/PRES Universud Paris, Cachan, France

ABSTRACT: Nonlocal damage models are now commonly used. Their ability to make finite element computations with softening laws robust and mesh independent is well established. There are nevertheless still a few open questions as the identification of the so-called internal length lc , as its loading or its boundary independency. One focuses in the present note on the boundary conditions problem in nonlocal integral approaches and on the feature that points separated by a crack or a hole should not interact as they do in Pijaudier-Cabot and Bazant initial nonlocal theory. Instead of defining an internal length one proposes to make the nonlocal weight function as a function of the information time propagation of an elastic wave normalized by an internal time τc . 1 INTRODUCTION The effect and the formulation of boundary conditions— such as free edges, notches and initial cracks— remain an open question for nonlocal models. The main drawback of the classical nonlocal integral theory (Pijaudier-Cabot & Bazant (1987)) consists in the nonphysical interaction, through the nonlocal averaging process, of points across a crack or a hole. The definition of natural boundary conditions of vanishing strain normal derivative at a free edge is still under discussion for gradient formulations (Aifantis (1987); Peerlings et al (1996)). The continuous nucleation of a crack of zero thickness is not so simple as the thickness of a localization band is more or less proportional to the internal length introduced. Local behavior along free edges—i.e. with a vanishing internal length—has been obtained by some authors (Pijaudier-Cabot et al (2007); Krayani et al (2009)). The consideration of an internal length evolving with damage (Geers et al (1998); Pijaudier-Cabot et al (2004)) seems a way to properly bridge Damage Mechanics and Fracture Mechanics as the internal length may then vanish for large values of damage. One attempts here to propose a solution—bringing also questions—to these main difficulties. The idea is to keep the nonlocal averaging process but to quantify the distance between points as an effective distance, i.e. as a distance function for instance of the geometry and the matter encountered between interacting points. One proposes to define such an effective distance with respect to a dynamic process: how information or wave propagates between interacting points. This can be made through the introduction of an internal time τc , constant, instead of a internal length ℓc , measured as evolving. Dynamics is important to define a link between a characteristic time and a characteristic length, either

when viscosity is introduced (Needleman (1988); Allix & Deü (1997)) or when the physical defects obscuration phenomenon encountered in high speed dynamics and multi-fragmentation is taken into account (Denoual C. & Hild (2000)). Some authors even introduce the classical nonlocal theory by comparing the characteristic wavelength of the deformation field to an intrinsic length of the material (Jirasek (2003)), still a dynamics vocabulary. Wave propagation will give us information on interacting/non interacting points for the definition of the nonlocal averaging. One does not intend here to solve the problem in the general case and focuses only on the nonlocal integral theories. 2 NONLOCAL THEORY WITH INTERNAL TIME Softening constitutive equations classically lead to spurious dissipation modes and to mesh dependency. The need of the definition—and the introduction—of an internal length in the models is now established. But in which form? In a gradient form? In an integral form? From an internal viscosity (delay-damage) combined then with dynamics? The main idea of such regularizations is to average the variable—i.e. the thermodynamics force, denoted next V in the general case or Y or ǫˆ for damage—responsible for the strain localization. The procedure to define a nonlocal variable V nl from its local expression V introduces a characteristic length lc considered as a material parameter.

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2.1 Nonlocal integral theories—Boundary effect The classical nonlocal theory (Pijaudier-Cabot & Bazant (1987)) uses the integral

   x − s 1 V(s)dVs V (x) = ψ Vr  lc    x − s dVs Vr (x) = ψ lc  nl

This illustrates the need to deal with such a boudary condition effect but also to make equivalent a real crack and a highly damaged zone.

(1) (2)

2.2 Time instead of distance

over the whole domain  in order to define a nonlocal quantity V nl —built from the local variable V—to be used in thermodynamics state or evolution laws. In Eq. (1) ψ is the nonlocal weight function, positive and decreasing with respect to the distance x − s between the considered point x and all the structure points s. The normalizing factor is denoted Vr (x) and lc is an internal or characteristic length. Two classical weight functions are (Bazant & Jirasek (2002)). ψ(ξ ) = e

− 12 ξ 2



or

ψ(ξ ) = 1 − ξ

 2 2

In order to solve the problem, one proposes in this work to keep the nonlocal integral framework and to consider the information time propagation τxs between points x and s instead of the classical distance x − s. A nonlocal theory with internal time is then simply defined by replacing Eq. (1) by:    1 τxs V(s)dVs ψ Vr  τc    τxs Vr (x) = dVs ψ τc 

V nl (x) =

(3)

The expressions (1) do then define the same contribution to V nl of points x and s

(4) (5)

with τxs the information propagation time (Desmorat et al (2009)) taken next as the time for a wave to propagate from point x to point s and τc a material parameter. As the wave time propagation τsx from point s to point x is identical to τxs the nonlocal weight function thus built is symmetric. Such a nonlocal averaging process may apply to physical laws of different nature. Concerning continuum mechanics and elasticity, plasticity and/or damage, the variables V are often equivalent strains, accumulated plastic strains or strain energy densities.

– across a crack than any points x and s separated by the same distance x − s (for instance path x1 –x6 versus path x2 –x5 of figure 1); this flaw has been pointed out and studied recently by Jirasek & Marfia (2006) and Pijaudier-Cabot et al (2007). A numerical averaging adaptation neglecting the communication between some integration points can be found in (de Vree et al (1993)); – across a damaged zone: for example the interaction between points x3 –x4 is not affected by the zone at a damage level D0 , i.e. at a Young’s E(1 − D0 ) much lower than the Young’s modulus of the virgin material; much less studies address this point.

2.3 Example: nonlocal damage models

Specimen (a) x1

x2

x3

x1

x2

x3

x6

x5

x4

x6

x5

x4

Specimen (c) x1

x2

x3

D=D0 x6 Figure 1.

x5

For example, local isotropic damage theories for quasibrittle materials define

Specimen (b)

˜ with – the damage as a loss of stiffness D = 1 − E/E ˜E (resp. E) the damaged (resp. initial) Young modulus, equation also rewritten in 3D as the elasticity law coupled with isotropic damage σ = E (1 − D) : ǫ

(6)

with σ , ǫ and E respectively the stress, the strain and the Hooke tensors; – the damage evolution as a function of a local variable V either equal to the thermodynamics force Y = 12 ǫ : E : ǫ associated with damage (Marigo model) √ or to an equivalent strain as Mazars strain ǫˆ = ǫ+ : ǫ+ , D = g(V)

x4

(7)

with g a nonlinear function and ·+ the positive part of a tensor (in terms of principal values).

Notched and damaged specimens.

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The non local damage law is simply written D = g(V nl )

– Induced damage anisotropy governed by the positive extensions, (8)

˙ ǫ 2+ D = λ˜

instead of Eq. (7) with the nonlocal averaging process (4). For concrete, the microcracks due to tension are mainly orthogonal to the loading direction, when the microcracks due to compression are mainly parallel to the loading direction. The damage state has then to be represented by a tensorial variable D either a fourth rank tensor or a second rank tensor (Murakami & Ohno (1978); Cordebois & Sidoroff (1982); Murakami (1988); Lemaitre & Desmorat (2005); Lemaitre et al (2009)). The use of a second order damage tensor is more convenient for practical applications (as well as for the material parameters identification) and this is the choice which has been made. The damage anisotropy induced by either tension or compression is simply modeled by the consideration of damage evolution laws ensuring a damage rate proportional to the positive part of the strain tensor, i.e. a damage governed by the principal extensions (Mazars et al (1990)). The full set of constitutive equations for the local anisotropic damage model reads (Desmorat (2004); Desmorat et al (2007); Souid et al (2009))

ν 1+ν σ˜ − tr σ˜ 1 E E

There are 5 material parameters introduced: E, ν for elasticity, κ0 as damage threshold and A and a as damage parameters. The model is simply made nonlocal, either from the classical integral theory or from the new integral nonlocal with internal time theory, by replacing Mazars equivalent strain ǫˆ by its nonlocal form in the damage criterion function, becoming f = ǫˆ nl − κ(tr D )

instead of Eq. (11) with the nonlocal averaging process (4),    τxs 1 ǫˆ nl (x) = ψ (15) ǫ(s)dV ˆ s Vr (x)  τc

In a plain and uncracked medium the internal length and internal time concepts are equivalent as

(9)

x − s = c τxs

(16)

with c the information celerity taken as the wave speed. Then, if the internal time is related to the internal length as lc = c τc the weight functions are equal,     x − s τxs ψ =ψ (17) lc τc

(10)

where(.)D stands for the deviatoric part and . for the positive part of a scalar. – Damage criterion, f = ǫˆ − κ(tr D )

(14)

2.4 Effective or ‘‘dynamic’’ distance—Vanishing internal length

with E the Young modulus and ν the Poisson ratio. – Effective stress,  D σ˜ = (11 − D )−1/2σ D (11 − D )−1/2

1 tr σ  + − −tr σ  1 3 1 − tr D

(13)

In the rate independent formulation, the damage multiplier λ˙ is determined from the consistency condition f = 0, f˙ = 0.

– Elasticity, ǫ=

ǫ˜ = E −1 : σ

(11)

so that the condition f < 0 → elastic loading or unloading, f = 0, f˙ = 0 → damage growth, where √ ǫˆ = ǫǫ + : ǫǫ + is Mazars equivalent strain built from the positive part ǫǫ + of the strain tensor and where  κ

tr D 0 (12) + arctan κ(tr D ) = a · tan aA a

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In a homogeneously damaged medium at D = D0 , the wave speed is proportional to the square root of the damaged Young’s modulus and depends on the damage level as

c˜ = c 1 − D0 (18) One has in this case

τxs =

x − s c˜

(19)

and x − s x − s x − s τxs = = √ > τc c˜ τc lc lc 1 − D0

(20)

which shows that the effective or ‘‘dynamic’’ distance x − seff

x − s = √ 1 − D0

and on upper damaged zone sides. When the damage D0 becomes large (0.99 in the figure), notched and damaged specimens are found equivalent.

(21)

3.1 Connectivity matrices

between two points increases, as expected, with damage. Eq. (20) defines in an equivalent manner an effective evolving internal length

˜lc = lc 1 − D0 (22)

which tends to zero when D0 tends to unity in accordance with Pijaudier-Cabot et al. results (PijaudierCabot et al (2007)) of a material behavior becoming local on free edges (at least in the direction normal to the edge). In 1D and in the non uniform case, the effective distance is gained as the integral over the path [s, x] as x | s (1 − D(x′ ))−1/2 dx′ |, in 3D as c τxs . 3 EQUIVALENCE BETWEEN A CRACK AND A DAMAGED ZONE

X2

X3

0.1 m

X4

X6

X5

X4

y z x

X6

1.414

0

1

1.414

1

1

0

1

1.414

1.414

1

0

1

1

1.414

1

0

1.414

2.236

2

1

1

⎤

⎥ 1.414⎥ ⎥ 2.236⎥ ⎥ ⎥ 2 ⎥ ⎥ ⎥ 1 ⎦ 0

(23)

The classical connectivity matrix C is the same (and is symmetric) for the 3 structures (a), (b) and (c). It does not take into account the presence of notches nor the occurence of damage as C16 = C25 = C34 = l/lc . In order to determine the connectivity matrix with the new internal time formulation, one proceeds as follows:

X2

X1

0.5 m

X1

2.236

For the concrete square plate (E = 35000 MPa, ν = 0.2, ρ = 2400 kg/m3 ) and notched specimen one ends up to:

X3

X5 0.5 m

2

Specimen (c)

Specimen (b)

0.5 m

X6

1

– an impulse force according to the out of plane z-axis (close to a Dirac) is applied at each point x = xp in a dynamic finite element analysis of a 3D model of a plate (thickness of 0.1 m, 720000 nodes and 360000 elements, free boundary conditions), – the times τxs at which the first pic of the z-acceleration information arrives at point s = xq are recorded. Divided by τc they are put in the form of connectivity matrices C τ , – the expressions for τxs /τc are synthesized by use of the relation lc = c τc in the form C τ = llc · A and compared to table 1 results for the different specimens.

Specimen (a)

X2

0 ⎢ ⎢ 1 ⎢ l ⎢ ⎢ 2 C= ·⎢ lc ⎢2.236 ⎢ ⎢ ⎣1.414 ⎡

1

In order to illustrate the formulation ability, consider the specimens of figure 2: (a) a square plate, (b) a notched specimen and (c) a specimen with a damaged zone at D = D0 . Vertical z-acceleration fields are also plotted at the same instant for the 3 specimens. It is clearly noticed that the information initiated as an impulse at x = x1 takes longer to reach point x6 for both notched and damaged specimens (the wave generated in the example has to turn around the notch). Note the waves reflexion on upper notch

X1

To quantify the approach, consider six points xp in these specimens. The distances between these points, used in the classical nonlocal theory, are calculated in table 1 for the plate (a) (with l = 0.5 m). The connectivity table√ is rewritten √ in a matrix form as a connectivity matrix ( 2 and 5 replaced by their numerical values to make easier further comparisons),

X5

X3

⎡

X4

C (a) τ

Damage zone at D = D0

Figure 2. Square plate, notched and damaged specimens. Geometry and pictures of wave propagation at the same instant.

124

0

⎢ 1 ⎢ ⎢ l ⎢ ⎢2.008 = ·⎢ lc ⎢2.233 ⎢ ⎢ ⎣1.420 1

1

2.008

2.233

1.420

0

1

1.420

1 1.420

1

0

1

1.420

1

0

1

1

1.420

1

0

1.420

2.233

2.008

1

1

⎤

1.420⎥ ⎥ ⎥ 2.233⎥ ⎥ ⎥ 2.008⎥ ⎥ ⎥ 1 ⎦ 0

(24)

3.2 Straight 1D wave propagation

Table 1. Connectivity table built from the classical distances x − s = xp − xq . xp − xq  lc

x1

x2

x3

x1

0

l lc

2l lc l lc

l lc 2l l √c

x2 x3

5l lc √ 2l lc l lc

x4 x5 x6

⎡

⎢ ⎢ l ⎢ ⎢ (b) Cτ = ·⎢ lc ⎢ ⎢ ⎣

0 l lc √ 2l lc l lc √ 2l lc

0 l lc √ 2l lc √ 5l lc

x4 √

x5 √ 2l lc l lc √ 2l lc l lc

5l

l √c 2l lc l lc 0 l lc 2l lc

0 l lc

For a better understanding, focus on times τ16 , τ25 , τ34 wave propagation in specimen (c) from x1 to x6 , from x2 to x5 , from x3 to x4 . The corresponding distances are equal in the 3 cases (equal to l). Due to the presence of the notch the time τ16 is of course larger than the time τ25 correspnding to a straight path with no notch (and τ25 /τc = l/lc ). A wave propagating along the path x1 − x6 in specimen (c) slows down to the celerity c˜ when meeting the damaged zone. If one only consider the straight path x1 − x6 for the wave propagation, one has

x6 l lc √ 2l lc √ 5l lc 2l lc l lc

0

1

2.008

2.233

1.420

1.494

0

1

1.420

1

1.420

2.008

1

0

1

1.420

2.233

2.233

1.420

1

0

1

2.008

1.420

1

1.420

1

0

1

1.494

1.420

2.233

2.008

1

0

(27)

√

e 1 − 1 − D0 l τ16 1+ · √ = τc lc l 1 − D0

(28)

or

0

1

√ e 1 − 1 − D0 l + · √ c c 1 − D0

τ16 =

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(25)

where the notch presence is taken into account (boxed terms), leading for the notched specimen to Cτ(b)16 = (b) (a) /τc = 1.494 l/lc intead of Cτ(a)16 = τ16 /τc = l/lc τ16 for the plate with no notch. For the specimen with the damaged zone at D = D0 = 0.99, the connectivity matrix corresponding to the nonlocal internal time analysis reads

where e is the thickness of the damaged zone at D = D0 . The figure 3 shows the normalized increase of time (c) with respect to the damage value D0 and τ34 = τ34 for different ratios e/l. The ratio (τ34 /τc )/(l/lc ) in case (c) is equal (resp. close) to unity for a zero (resp. small) damage, the nonlocal theory with internal time recovering then the classical nonlocal theory. The very large increase obtained for large values of the damage, enhanced by a large damaged zone thickness, proves that both a real crack and a highly damaged zone are equivalent in the proposed nonlocal framework. This property is emphasized when the ratio of the weight functions ψ(τ34 /τc )/ψ(l/lc ) is drawn (figures 4 to 6) for l/lc = 5, 1 and 0.2 and for the Gaussian weight 1 2 function ψ(ξ ) = e− 2 ξ : the nonlocal spatial interaction between points across a damaged zone strongly diminishes with damage increase, the larger the points 10

⎢ ⎢ l ⎢ = ·⎢ lc ⎢ ⎢ ⎣

0

1

2.008

2.233

1.420

1.500

1

0

1

1.420

1

1.420

2.008

1

0

1

1.420

2.233

2.233

1.420

1

0

1

2.008

1.420

1

1.420

1

0

1

1.500

1.420

2.233

2.008

1

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(26)

e/l=0,1 e/l=0,5 e/l=1

8 (tau/tau_c)/(l/l_c)

(c) Cτ

⎡

6

4

2

Again ony the terms 16 and 61 of the connectivity matrices are changed and found close to the value 1.494 l/lc obtained with reals notches: the damaged zone behaves as a notch, damage and notch being both taken into account by the proposed nonlocal with internal time analysis.

125

0 0

0,2

0,4

0,6 Damage

Figure 3.

Ratio

τ34 l τc / lc

vs damage D0 .

0,8

1

1,2

distance l with respect to the internal length lc , the stronger the effect.

1,2 e/l=0,1

e/l=0,5

e/l=1

Ratio of weigth functions

1

4 BAR WITH A DAMAGED ZONE

0,8

As an example emphasizing how a damaged zone is taken into account, consider a bar in tension at a stress level σ . The length of the bar is 2L, a damaged zone at large D = D0 corresponds to x ∈ [−l, l]. The non local analyses perform the integrals (1) over the whole bar,

0,6

0,4

0,2

0 0

0,2

0,4

0,6

0,8

1

ǫ nl (x) =

1,2

Damage

Figure 4.

Ratio



2

τ exp − 34 2

exp

τc 2 − l2 lc





L

Vr (x) =

ψ · ǫ(s)ds

−L



L

ψ ds (29)

−L

with as strain ǫ(x) = σ/E if |x| > l, ǫ(x) = σ/ E(1 − D0 ) else and where ψ = ψ( |x−s| lc ) for the



 vs damage D0 for l/lc = 5.

30

1,2 e/l=0,1

e/l=0,5

e/l=1

a)

1

25

0,8 20

0,6

Vr

Ratio of weigth functions

1 Vr

15

0,4

Vr0 Vrnew 10

0,2

5

0 0

0,2

0,4

0,6

0,8

1

1,2

Damage 0

Figure 5.

Ratio

exp

τ2 − 34 τc2 2 − l2 lc



exp



-60



-20

0

20

40

60

x

 vs damage D0 for l/lc = 1.

100

b) 80

1,2 e/l=0,1

e/l=0,5

e/l=1

1 60

nl0 nlnew local

nl /

0,8

.

Ratio of weigth functions

-40

40

0,6

0,4

20

0,2 0 -60

0 0

0,2

0,4

0,6

0,8

1

-40

-20

0

20

40

60

x

1,2

Damage

Figure 6.

Ratio



τ2

exp − 34 τ2 exp

c 2 − l2 lc



Figure 7. a) Comparison of normalizing factors Vr (x); b) Comparison of normalized nonlocal strains (lengths in cm, black: nonlocal with internal time, grey: classical nonlocal, dot: local strain).



 vs D0 for l/lc = 0.2.

126

classical nonlocal analysis, ψ = ψ( ττxsc ) for the new nonlocal analysis with internal time. The Gaussian 1 2 weight function ψ(ξ ) = e− 2 ξ is considered. The normalizing factors Vr (x) are compared for both analyses in figure 7a where D0 = 0.99, l = 2.5 cm, L = 50 cm and where the characteristic length is taken as lc = 10 cm (twice the size 2l of the damaged zone). The classical normalization does not ‘‘see’’ the damaged zone and averages across it when the new nonlocal with internal time approach behaves for the undamaged domains almost as for two independent bars, as expected. The nonlocal strains obtained with both approaches are compared in figure 7b. In this particular piecewise constant strain field, the formulation with internal time gives as expected a nonlocal strain field closer to the local strain field than the classical nonlocal approach for which too much importance is given to points over the damaged zone when performing the integral (29). Note that in case of structural failure, strain localization leads to non homogeneous fields. The proposed nonlocal averaging then fully acts and makes the solution regular.

between a localized zone and a crack will not be obtained. To gain this last feature, two possibilities are: – to consider an elasticity coupled with damage even if the plasticity part of the model remain not affected by damage, – to define τxs from plastic waves propagation (no need of damage then).

5 CONCLUSION A new nonlocal integral formulation is proposed. An internal time is introduced leading to the equivalent definition of an effective or ‘‘dynamic’’ distance and of an evolving internal length. The important feature is that the distance bewteen points is not the argument of the averaging weight function anymore. It is replaced by the information or wave time propagation between these points. The nonlocal weight function build is then symmetric, even in non homogeneous bodies. Pre-computations of wave propagation in the considered structure allow to build the corresponding nonlocal connectivity matrix, with of course the open question of the wave type. The cracks and notches presence—and if necessary their closure—are naturally taken into account within the wave propagation study and PijaudierCabot and Bazant nonlocal theory is recovered far from the boundaries. The proposed approach makes equivalent a crack and a highly damaged zone, as points across a notch have a small contribution in the nonlocal averaging. Important point, no assumption on the medium isotropy is made. The proposed nonlocal framework includes anisotropy, either initial or induced. Last, when no damage is considered as in plasticity with negative hardening models, the internal time concept may still be used, for example by making nonlocal—through Eq. (4)—the accumulated plastic strain. The presence of existing notches and cracks will be naturally taken into account if propagation of elastic waves defines the time τxs but the equivalence

127

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Elastoplastic constitutive model for concretes of arbitrary strength properties Guillermo Etse & Paula Folino Faculty of Engineering, University of Buenos Aires, Argentina

ABSTRACT: The increasing use of high strength concretes in civil constructions and structures is demanding the development of reliable constitutive laws for accurate structural analysis by means of computational methods. One of the main difficulties in this regard is the strong variation of the relevant mechanical features of concretes with the intrinsic quality of this material. This variation is not only relevant when comparing normal (NSC) with high strength concretes (HSC), but also within the wide spectrum of high strength concretes. Experimental evidence demonstrates that besides maximum strength, one of the mechanical properties of concrete most affected by the variation of material quality is the ductility and, therefore, the transition point from ductile to brittle failure modes. Most of the constitutive laws for concrete in the literature are only valid for NSC. Very few of them can be also used for HSC with the same level of accuracy as in case of NSC. But the most significant limitation is that up to date very few proposals can be found, related to concrete models covering the wide spectrum from NSC to very HSC. In this work, a performance dependent constitutive model for concretes of arbitrary strengths is presented. The model is based on the flow theory of plasticity and takes into account fracture energy concepts for the formulation of the softening law. The maximum strength surface of the model as well as hardening/softening evolution laws are described in terms of the so-called concrete performance parameter. This parameter is introduced to evaluate concrete quality in terms of uniaxial strength and of the water/binder ratio, as proposed by the authors [Folino, Etse & Will 2009]. The pre and post-peak behavior of concrete are described by means of non-uniform hardening and isotropic softening formulations. Volumetric dilatancy of concretes of arbitrary strengths is described by means of a volumetric non-associated flow rule that is also expressed in terms of the performance parameter. The proposed model is being implemented in finite element tool for failure analysis of concrete elements of different qualities. The most relevant features of model formulation as well as of the considerations made to take into account the influence of concrete quality in the response behavior of this quasi-brittle material in pre and post-peak regimes are described in this work. 1 INTRODUCTION Several authors have extensively discussed the fundamental differences between mechanical response behavior of NSC and HSC. Among others, we refer here to the works by Imran & Pantazopoulou 1996 and Xie et al. 1995. A very important fact that follows from the experimental evidence is that the improvement of concrete mechanical properties, particularly the ductility in pre and post regimes, but also the reduction in of the volumetric dilatancy during monotonic compressive loading does not linearly vary with the increase of the uniaxial compressive strength ( fc′ ). Actually, this variation is highly complex and nonlinear. The same can be said regarding the variation of the mechanical response sensitivity with respect to the acting confinement. By far, this sensitivity is higher in case of NSC. Contrarily, there are other mechanical features that decrease (in a non-linear form as well) for increasing fc′ or concrete strength properties, such as the fracture energy release in Mode I and II

type of failure. Finally, another relevant aspect to be considered is that the variation of the uniaxial tensile strength ( ft′ ) does not linearly depend on the variation of fc′ . This means that in fact, maximum strength surface of concrete cannot isotropicaly vary when going from NSC to HSC. This relevant property was taken into account by the authors, see Folino et al. 2009, for the formulation of the performance dependent failure criterion for concrete. In this work a model for concrete of arbitrary strength and performance is presented. The model is based on the flow theory of plasticity. A fundamental feature of the proposed formulation is that it encompasses in one single equation the yield surfaces in pre and post-peak regimes for all stress stages and all types of considered concrete qualities. In hardening zone the yield surfaces have cap-cone form with C1 -continuity. This allows a simpler numerical implementation of the elostoplastic model in finite element codes. Cone portion of yield condition agrees with the maximum strength surface, while the cap portion is based on an

129

The evolution of the state variables t κ˙ is defined by the hardening/softening laws. In the next sections, the failure criterion, hardening and softening formulations of the proposed material model for concretes of arbitrary performance are described.

elliptic formulation that perpendicularly intersects the hydrostatic axis. Thus, concrete non-linear response under pure hydrostatic loading can also be realistically simulated with this model. Under increasing plastic loading in the pre-peak regime the cone-cap intersection moves to the confinement direction. In the proposed formulation the ductility of the hardening mechanical response depends on both concrete quality and current confining pressure. To account for mesh regularization during post peak regime softening formulation of the model is based on fracture energy concepts following original proposals by Etse & Willam 1994. The presented model incorporates the influence of the performance dependent parameter in the evaluation of the fracture energy-based characteristic length both in mode I and II type of fracture. In this way the ductility in post-peak regime depends on the material quality, on current confining pressure and on the fracture mode. To reduce excessive volumetric dilatancy of concrete when loading in the low confining region the proposed formulation includes a non-associative formulation of the plastic flow based on volumetric variation of the yield condition. The volumetric nonassociativity depends on both the confining pressure and the concrete quality by means of the performance parameter.

3 PERFORMANCE DEPENDENT FAILURE CRITERION FOR CONCRETE The performance dependent failure criterion (PDFC) for concretes of arbitrary strength considered in this model was proposed by the authors (see Folino, Etse & Will 2009) on the basis of the so-called performance parameter. It covers the entire spectrum of concrete qualities from NSC to HSC. The PDFC is defined in terms of the Haigh Westergaard stress coordinates ξ and ρ which are functions of the first and second invariant to the stress and deviatoric tensor, respectively, and of the Lode angle θ . The last one is function of the second and third invariants of the deviatoric stress tensor. According to this criterion, concrete failure occurs when actual 2nd Haigh Westergaard stress coordinate ρ reaches the shear strength ρ ∗ F=

2 THEORETICAL FRAMEWORK The proposed constitutive formulation is based on smeared crack approach and the elasto-plastic incremental flow theory. Only infinitesimal strains are considered. Elasticplastic coupling is neglected, accepting the additive Prandtl-Reuss decomposition of the infinitesimal strain rate tensor into its elastic and plastic parts described as p

ε˙ ij = ε˙ ije + ε˙ ij

σ˙ ij = Eijkl ε˙ kl

(2)

ρ = ρc∗ /r

(5)

being r the ellipticity factor r=

4(1 − e2 ) cos2 θ + (2e − 1)2  − 1) 4(1 − e2 ) cos2 θ + 5e2 − 4e

2(1 − e2 ) cos θ + (2e

( 6)

and e = ρt∗ /ρc∗ the so-called eccentricity. The failure surface is represented by (7) F = A r 2 ρ ∗ 2 + Bc rρ ∗ + C ξ¯ − 1 = 0

In the above equation σ˙ ij is the Cauchy stress rate tensor, Eijkl the fourth order elasticity tensor depending on the material Young’s modulus Ec and Poisson’s ratio υ. Inelastic material response is governed, in general, by the following non associated flow rule where mij =

∂Q ∂σij

(4)

In the deviatoric plane, the C1 -continuity elliptic interpolation of Willam & Warnke (1974) between second stress coordinate in compressive ρc∗ and tensile ρt∗ meridians is followed

(1)

The elastic constitutive response is defined by the generalized Hooke law

p ˙ ij ε˙ ij = λm

ρ −1=0 ρ∗

(3)

Being Q (σ ; κ) = 0 the plastic potential. Plastic parameter follows from the well-known consistency condition.

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Whereby ρ ∗ = ρ ∗ /fc′ and ξ¯ = ξ/fc′ are the normalized first and second stress coordinates. For the compressive and tensile meridians the failure surface reduces, respectively, to Fπ/3 = Aρc∗ 2 + Bc ρc∗ + C ξ¯ − 1 = 0 F0 =

Aρt∗ 2

+

Bt ρt∗

+ C ξ¯ − 1 = 0

(8) (9)

Coefficients A, Bc , Bt and C in Eqs. (7) to (9) are defined by means of explicit expressions, see Folino, Etse & Will (2009), of four material parameters, as follow:

– uniaxial compressive strength fc′ , – uniaxial tensile strength ft′ (through uniaxial strength ratio αt = ft′ /fc′ ), – biaxial compressive strength fb′ (through the biaxial compressive strength ratio αb = fb′ /fc′ ), – parameter m: tangent to the compressive meridian on the peak stress’s shear component corresponding to the uniaxial compression test. Considering the difficulties involved in determining material parameters αt , αc and m, internal calibration functions were proposed by the authors in terms of the performance parameter βP βp = 0.001 fc′ /(W /B)

(10)

being (W /B) the water/binder ratio, considered as a fundamental property of concrete mix controlling the material performance. Consequently, βp is directly related to the porosity of the mortar. When W /B is unknown, a range of possible βp can be defined, limited by upper and lower bounds proposed by Folino et al. (2009). Thus, once fc′ and βp are defined, the whole set of remaining material parameters αt , αb and m can be evaluated using calibration functions and, subsequently, coefficients A, Bc , Bt and C from the explicit expressions by Folino, Etse & Will (2009). As a result, the performance dependent maximum strength criterion is fully defined. Figure 1 illustrate maximum strength meridians predicted by the PEDFC for different concrete qualities. Figures 2 and 3 depicts comparisons of experimental results in terms of peak stresses with the proposed failure criterion. In Figures 4 and 5 validations of triaxial and biaxial tests results are presented. As can be observed good agreements between experimental results and the proposed failure criterion are obtained in all cases.

Figure 1. Compressive and tensile meridians predicted by the PDFC for different concrete qualities—Normalized plot.

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Figure 2. Triaxial tests validation, for two concretes of different qualities: fc′ = 65 and 120 MPa—Normalized plots.

Figure 3. Biaxial tests validation, for two concretes of different quality: fc′ = 43 and 97 MPa—Normalized plots.

of C 1 -continuity. Cone portion of yield surfaces in hardening agrees with the failure criterion F defined by Equation (7) while cap sector in meridian plane is defined by ellipse portions that perpendicularly intersect the hydrostatic axis. fh can be mathematically expressed as  F =0 if 2 fh =  ξ¯ − ξ¯cen /a2 + r 2 ρ¯ 2 /b2 − 1 = 0 if

Figure 4. Loading surfaces in hardening fc′ MPa—Normalized plot.

=

20

ξ¯ ≥ ξ¯1 (k) ξ¯ < ξ¯1 (k) (11)

whereby the hardening parameter k controls the evolution of coefficients a, b and ξ¯cen defining the ellipse’s size and location in the meridian plane. Haigh Westergaard second stress coordinate of the cap-cone intersection point P1 in current position of hardening yield locus is defined in terms of k as  (12) ρ¯p1 = k 2/3 Haigh Westergaard first stress coordinate of P1 depends on concrete quality as well. During hardening process under increasing confining pressure, P1 evolutes along the maximum strength surface defining starting point of current cap. The evolution of the hardening parameter k from its initial lower bound ko has no defined limit. Figures 4 and 5 illustrate normalized plots of loading surfaces in hardening regime corresponding to two different concrete qualities. 4.2 Initial loading surface in cap regime The initial cap requires definition of the lower bound ko of the hardening parameter and of k = kel corresponding to the initiation of plastic process in uniaxial compression test. Initial cap in compressive meridian is an ellipse centered in the ξ axis. Coordinates of this ellipse centre are ξ¯ = ξ¯cen o and ρ = 0. For evaluation of ξ¯cen o and the two ellipse radiuses ao , bo the following three boundary conditions are introduced, see Fig. 6:

Figure 5. Loading surfaces in hardening fc′ MPa—Normalized plot.

= 120

4 HARDENING FORMULATION FOR PRE-PEAK REGIME In this section the hardening formulation of the performance dependent constitutive model for concrete is described. 4.1 Yield surfaces in hardening Meridian traces of the yield criterion in hardening regime fh are defined by cap-cone loading surfaces

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1. Initial ellipse passes through P1 on the compressive meridian of the failure surface with coordinates ξ¯1o and ρ¯1o . 2. C 1 -continuity at P1 . 3. Initial ellipse also passes through P2 corresponding to plastic process initiation of uniaxial compression test. P2 with coordinates ξ¯2o and ρ¯2o lies in the line between the uniaxial compression maximum strength and the stress coordinates origin. From these conditions result explicit expressions for ξ¯cen o , ao and bo , see Folino & Etse (2009). Additional geometrical constrains of ellipse radiuses a2 > 0, b2 > 0 and a2 /b2 > 0 imply, as expected, that ξ¯1 > ξ¯2 . On the other hand, it also implies that

Figure 6.

Figure 7. Hardening level as a function of the level of confinement and of the developed work hardening.

Initial cap ellipse—Normalized plot.

ρ¯1 o min < ρ¯1 o < ρ¯1 o max , defining an admissible range for the second stress coordinate of P1 at initial ellipse. 4.3 Hardening surface evolution Subsequent cap yield surfaces after reaching elastic limit are obtained through the evolution of P1 along the maximum strength surface that, in turn is controlled by the evolution of k. New cap ellipses keep the ratio between major and minor axes invariant with respect to the initial cap. From this condition follow explicit equations for the ellipse parameters ξ¯cen , a and b in current position during hardening regime. 4.4 Hardening evolution law In this proposal, hardening parameter k is defined in terms of the normalized work hardening measure κh by means of an elliptic variation as follow (See Fig. 9)   k = ko + k max − ko κh (2 − κh ) (13)

being k max the maximum possible hardening parameter corresponding to actual work hardening measure κh that, in turn, is compatible with current confinement level. k max is defined as    −Bc + Bc2 − 4A 1 − C ξ¯ 1 max = √ (14) k 2A 2/3 while the evolution law of the work hardening measure is p

κ˙ h = ω˙ ap /Wt

(15)

Thereby are

ω˙ ap = and

p σij ε˙ ji

(16)

p

p Wt

=

εmax 0

p

σij ε˙ ji

p

where εmax is the maximum plastic strain at peak stress under current confinement pressure. A bilinear interpolation formula is proposed for numerical evaluation p p of Wt , see Folino and Etse (2009). Once Wt is known, the evolutions of plastic work and of work hardening measure can be obtained from previous Eqs. (16) and (15), respectively. Then, the hardening parameter can be updated with Eq. (14).

(17)

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5 ISOTROPIC SOFTENING FORMULATION FOR POST-PEAK REGIME In this section, the softening law of the proposed model is summarized. It is based on fracture energy concepts for Mode I and II type of failure. 5.1 Yield surface evolution in softening regime Once the cone is reached, a progressive softening process starts, controlled by the softening parameter cs that varies between 1 and 0. Yield surfaces in softening regime are defined by the expression fs = A r 2 ρ¯ 2 + B r ρ¯ + C ξ¯ − cs = 0

(18)

Figures 8 and 9 show compressive and tensile meridian views of yield surfaces in softening regime for two different concrete qualities. 5.2 Fracture energy-based softening model Similarly to the formulation by Etse & Willam (1994), softening parameter cs defines the degradation of the tensile strength during post-peak regime in terms of fracture parameters and variables  σt −κs ≥0 (19) 1 ≥ cs = ′ = exp ft ur

6 NON-ASSOCIATED FLOW RULE To limit volumetric dilatancy during plastic range in low confinement regime, a restricted form of nonassociativity is considered that only involved the volumetric plastic flow. In other words, plastic potential in hardening/softening regime is based on a volumetric modification of the yield condition in cone and/or cap zone, as follow Plastic potentials in hardening: (22) Qcone = A r 2 ρ ∗ 2 + B rρ ∗ + Cηcone ξ¯ − 1 = 0 2 2 2 2 2 ¯ ¯ Qcap = ηcap [ξ −ξcen ] /[a ] + r ρ¯ /[b ] − 1 = 0 (23) being Figure 8. Meridian views of yield softening—fc′ = 20 MPa—Normalized plot.

surfaces

  ηcone = ηcone βp ,

in

  ηcap = ηcap t κ; βp

(24)

Plastic potential in softening:

Qs, cone = A r 2 ρ ∗ 2 + B rρ ∗ + Cηcone ξ¯ − cs = 0 (25) with   ηs, cone = ηs, cone cs ; βp

(26)

Thereby are ηcone , ηcap and ηs, cone the grade of volumetric non-associativity in terms of the performance parameter and the state parameter corresponding to hardening/softening regime. Explicit expressions of the grades of volumetric non-associativity are given in Folino & Etse (2009). 7 CONCLUSIONS

Figure 9. Meridian views of yield surfaces softening—fc′ = 120 MPa—Normalized plot.

in

Thereby κs is the fracture energy-based softening measure and ur the maximum crack opening displacement. In this formulation it is considered that the evolution lay of κs is defined in terms of

 (20) κ˙ s = κ˙ s δ; ε˙˜ f ; lcmII ; βp

whereby, δ defines the shape of the decay function, βp is the performance parameter and incorporate the influence of concrete quality in the post-peak evolution law, ε˙˜ f is the rate of fracture strain that extracts only tensile components of the rate of plastic strain tensor in principal components. Finally, lcmII is the characteristic length for general mode II type of fracture lcmII = [GfII /GfI ]ht

(21)

with GfII and GfI the fracture energy releases in mode II and mode I type of rupture, respectively, and ht the characteristic length in uniaxial tensile failure mode.

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In this work a constitutive model is proposed to evaluate failure behavior of concrete of arbitrary performance and strength capacity. The material model is formulated within the mathematical framework of the flow theory of plasticity. Maximum strength criterion of the model is sensitive not only to the variation of basic material parameters of concrete such as friction, cohesion and uniaxial compressive/tensile strengths, but, moreover, of the particular concrete quality that is considered. Concrete quality is defined by means of the so-called performance parameter together with uniaxial compressive strength. Constitutive model is completed with the formulation of hardening/softening laws as well as non-associated flow rule. All of them include the performance parameter in their respective equations to incorporate the influence of the material quality in pre and post-peak ductility as well as in the volumetric dilatancy of concrete when loading in the low confinement regime. One particularity of the model is the formulation of yield condition in hardening regime that represents a cap-cone surface with C 1 -continuity. The cone portion coincides with the maximum strength surface while the cap one is describe by an elliptic trace that evolves during work hardening process. Softening

formulation includes fracture mechanics concepts and, consequently, incorporates a characteristic length allowing regularization of post-peak behavior. Both hardening and softening laws are sensitive to the level of confinement in order to reproduce the increase of ductility in pre and post-peak regime with the acting confining pressure.

ACKNOWLEDGEMENTS The authors acknowledge partial financial supports for this work by FONCYT (Argentine agency for research & technology) through Grant PICT1232/6, by CONICET (Argentine council for science & technology) through Grant PIP6201/05 and by University of Buenos Aires through Grant UBACYT2006-2009 Project I813.

REFERENCES Chen, W.F. & Han, D.J. 1988. Plasticity for structural engineers, Springer Verlag. Dvorkin, E., Cuitiño, A. & Gioia, G. 1989. A concrete material model based on non-associated plasticity and fracture. Journal Engineering Computations, Vol. 6, No. 4, 281–294. Etse, G. & Willam, K. 1994. Fracture energy formulation for inelastic behavior of plain concrete. ASCE Journal of Engineering Mechanics, Vol. 120, No. 9, 1983–2011. Folino, P., Etse, G. & Will, A. 2006. Modelación inelástica de hormigones de distintas resistencias basada en el índice de prestación. Mecánica Computacional Vol. XXV, 1915–1925. Folino, P. & Etse, G. 2008. Endurecimiento a través de superficies con capa dependientes del grado de prestación del hormigón, Mecánica Computacional Vol. XXVII, 909–926. Folino, P., Etse, G. & Will, A. 2009. A performance dependent failure criterion for normal and high strength concretes, ASCE Journal of Engineering Mechanics, in press. Folino, P. and Etse, G. 2009. Performance dependent constitutive model for concretes of arbitrary strength. Submitted to Int J. Comp. Methods in Applied Mech & Eng.

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Fossum, A.F. & Fredrich, J.T. 2000. Cap plasticity models and compactive and dilatant pre-failure deformation. Proc. 4th North American Rock Mechanics Symp., A.A. Balkema, Rotterdam, 1169–1176. Grassl, P., Lundgren, K. & Gylltoft, K. 2002. Concrete in compression: a plasticity theory with a novel hardening law. International Journal of Solids and Structures, Vol. 39, 5205–5223. Han, D.J. & Chen, W.F. 1987. Constitutive modeling in analysis of concrete structures. ASCE Journal of Engineering Mechanics, Vol. 113, No. 4, 577-593. Hussein, A. & Marzouk, H. 2000. Behavior of High-Strength Concrete under biaxial stresses. ACI Materials Journal, Vol. 97, No. 1, 27–36. Imran, I. & Pantazopoulou, S.J. 1996. Experimental study of plain concrete under triaxial stress. ACI Materials Journal, Vol. 93, No. 6, 589–601. Kang, H. & Willam, K. 1999. Localization characteristics of triaxial concrete model. ASCE Journal of Engineering Mechanics, Vol. 125, No. 8, 941–950. Lade P. 1977. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. International Journal of Solids and Structures, Vol. 13, 1019–1035. Lu, X. 2005. Uniaxial and triaxial behavior of high strength concrete with and without steel fibers. Phd Thesis, New Jersey Institute of Technology. Ohtani, Y. & Chen, W.F. 1988. Multiple hardening plasticity for concrete materials. ASCE Journal of Engineering Mechanics, Vol. 114, No. 11, 1890–1910. Oller, S. 1988. Un modelo de daño continuo para materials friccionales. Tesis Doctoral UPC, Barcelona, España. Rossi, P., Ulm, F.J. & Hachi, F. 1996. Compressive behavior of concrete: physical mechanisms and modeling. ASCE Journal of Engineering Mechanics, Vol. 122, No. 11, 1038–1043. Sfer, D., Carol, I., Gettu, R. & Etse, G. 2002. Experimental study of the triaxial behavior of Concrete. ASCE Journal of Engineering Mechanics, Vol. 128, No. 2, 156–163. van Geel, E. 1998. Concrete Behavior in multiaxial compression. Doctoral Thesis, Technische Universiteit Eindhoven. van Mier, J.G. 1997. Fracture Processes of Concrete. CRC Press. Willam, K.J. & Warnke, E.P. 1974. Constitutive model for the triaxial behavior of concrete. Proc. Intl. Assoc. Bridge Struct. Engrg., Report 19, Section III, Zurich: 1–30. Xie, J., Elwi, A. & Mac Gregor, J. 1995. ‘‘Mechanical properties of three high-strength concretes containing slica fume’’, ACI Materials Journal, Vol. 92, No. 2, 135–145.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Properties of concrete: A two step homogenization approach E. Gal & R. Kryvoruk Department of Structural Engineering, Ben-Gurion University of The Negev, Beer-Sheva, Israel

ABSTRACT: This paper describes the development of a two step homogenization approach for evaluating the elastic properties of concrete. For that purpose a finite element model of the concrete unit cell is generated. Prior to the generation of the unit cell finite element model the Interface Transition Zones (ITZ) and the aggregates are homogenized using an analytical approach. This approach makes it possible to exclude the ITZs from taking part in the finite element model of the unit cell. This is essential for achieving a practical calculating effort, since the ITZ typical dimensions are three orders of magnitude smaller than the typical aggregate dimensions; it thus requires a much more detailed model to represent explicitly the ITZ shells. For verification and validation purposes three types of concrete were tested, in two of them the aggregate distribution was obtained using sieve analysis while in the third the aggregate distribution was obtained according to the Fuller curve. 1 INTRODUCTION Evaluating the elastic properties of concrete becomes complicated due to the fact that concrete has a variety of microstructures and eventually has no one distinctive microstructure. The variety of micro-structures include: addition of fibers made up from different materials; variation of aggregate size, shape and type; water to cement ratio etc.. The authors suggest that the use of multi-scale analysis evidently is the appropriate way to model the behavior of concrete structures by coupling between the concrete micro-structures and its macroscopic properties needed to analyze the concrete structure (e.g. Markovic & Ibrahimbegovic 2004, Ibrahimbegovic & Markovic 2003, Gitman et al. 2006, 2007, 2008, He et al. 2009, Kouznetsova et al. 2001, 2002, Feyel 2003, Gutierrez 2004, Ghosh et al. 2001, Nadeau 2003, Lee et al. 2009, Pichler et al. 2007, Mang et al. 2003, de Borst et al. 1999, Füssl et al. 2008, Asferg et al. 2007, Oliver 1996, Jirasek 2000, Wells & Sluys 2001, Moes & Belytschko 2002, Dumstorffz & Meschke 2007, de Borst 2002, Meschke & Dumstorff 2007, Simone & Sluys 2004, Wriggers & Moftah 2006, Haffner et al. 2006, Wang et al. 1999, Cusatis & Cedolin 2007, Gal et al. 2008). The method for obtaining the macroscopic behavior of the concrete, based on its microstructure is referred to as the theory of homogenization, by which the heterogeneous material is replaced by an equivalent homogeneous continuum. The method is performed on a statistically representative sample of material, referred to as a material unit cell. Numerous theories have been developed to predict the behavior of composite materials. Starting from the various effective properties obtained by the models of Eshelby (1957),

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Hashin (1962), Mori & Tanaka (1973), self-consistent approaches of Hill (1965) and various mathematical homogenization methods (e.g. Christensen 1979) pioneered by Bensoussan (1978) and Sanchez-Palencia (1980). Unfortunately, most of these analytical models can only give estimates or boundaries for the macroscopic properties, and the simplifying assumptions used, result in, the major differences obtained. Computational procedures for implementing homogenization have been an active area of research starting with the contribution by Guedes & Kikuchi (1990) for linear elasticity problems. Over the past decade major contributions have been made to extend the theory of computational homogenization to nonlinear domains Terada & Kikuchi (1995), Fish et al. (1997), Fish & Shek (1999), Fish & Yu (2001) and improving fidelity and computational efficiency of numerical simulations (Terada & Kikuchi 2001, Matsui et al. 2004, Aboudi 1991, Aboudi et al. 2003, Aboudi 2003, Smit et al. 1998, Miehe & Koch 2002, Kouznetsova et al. 2001, Feyel & Chaboche 2000, Ghosh et al. 1995, 1996, Geers et al. 2001). These developments established the Finite Element Method (FEM) as one of the most efficient numerical methods, whereby the macroscopic responses can be obtained by volumetrically averaging numerical solutions of unit cells (e.g. Zohdi & Wriggers 2001, 2005). It has been established that the presence of the aggregates in the mortar causes a thin layer of the mortar material surrounding each inclusion to be more porous than the other surrounding matrix. This thin layer is named the interfacial transition zone (ITZ) Bentz et al. (1992), (1993), Scrivener & Nemati (1996), Sun et al. (2007). The ITZ between the paste matrix (mortar) and aggregate phases played an important role in the properties of a concrete composite

of the following quadratic equation where the positive square root is the physical answer.    2 G G +C =0 (1) + 2B A G2 G2

He et al. (2009), Scrivener & Gariner (1988), Prokopski & Halbiniak (2000). It has been found by experiment that the elastic moduli of concrete are intimately related to the elastic modulus and volume fraction of the ITZ regions Simeonov & Ahmad (1995), Nilsen & Monteiro (1993). Therefore the concrete microstructure is to include three phases consisting of aggregates, paste matrix and ITZ. The variety of homogenization techniques suggest modeling the concrete microstructure using a spherical aggregate to be surrounded by a concentric layer of ITZ, all embedded in the cement paste matrix. This paper adopts this assumption which enables performing a two step homogenization for evaluating the concrete module of elasticity. As the first step the properties of a spherical aggregate and its concentric ITZ layer were homogenized via the procedure suggested by Gaboczi & Berryman (2001). Then the homogenization procedure suggested by Gal et al. (2008) is applied to obtain the macroscopic properties of the concrete. Finally verification and validation of three types of concrete were performed, in two of them the aggregate distribution was obtained using sieve analysis while in the third the aggregate distribution was obtained according to the Fuller curve.

where G2 and G3 are the shear modulus of the ITZ and aggregate respectively and A = 8z (4 − 5ν2 ) η1 p10/3 − 2 (63zη2 + 2η1 η3 ) p7/3 + 252zη2 p5/3 − 50z(7 − 12ν2 + 8ν22 )η2 p

+ 4 (7 − 10ν2 ) η2 η3

B = − 2z (1 − 5ν2 ) η1 p10/3 + 2 (63zη2 + 2η1 η3 ) p7/3 − 252zη2 p5/3 + 75z (3 − ν2 ) η2 ν2 p

+

3 (15ν2 − 7) η2 η3 2

(2b)

C = 4z (5ν2 − 7) η1 p10/3 − 2 (63zη2 + 2η1 η3 ) p7/3 + 252zη2 p5/3 + 25z(ν22 − 7)η2 p − (7 − 5ν2 ) η2 η3

η1 = z (7 − 10ν2 ) (7 + 5ν3 ) + 105 (ν3 − ν2 )

(2c)

η2 = z (7 + 5ν3 ) + 35 (1 − ν3 )

(2d)

G3 −1 G2 3  r p= r+t

(2e)

2 MACROSCOPIC MATERIAL PROPERTIES OF CONCRETE

η3 = z (8 − 10ν2 ) + 15 (1 − ν2 )

2.1 Homogenization of the aggregate and the ITZ-Step 1

z=

Here, developing the homogenization of the concrete starts from homogenizing the aggregates and the ITZs. This is essential since in the following stage is using the finite element method (FEM) to evaluate the homogenized property of the concrete. Using the FEM requires a detailed model so that each constituent of the concrete has to be represented through using many finite elements. As the differences in typical dimensions of the aggregate (∼2000–19000 µm) and the ITZ (∼10–50 µm) is about three orders of magnitude, creating a 3D finite element model which links, the finite elements representing the ITZs and the finite elements representing the aggregates, probably will be unpractical or/and inaccurate. Therefore we suggest excluding the ITZ from our finite element model by homogenizing it with the aggregate. The homogenization of a spherical particle surrounded by a spherical shell of different elastic properties can be performed as suggested by Christensen (1979), (1990). In this paper we follow the formulation suggested by Christensen (1990) as it has been applied by Gaboczi & Berryman (2001) to homogenize the concrete aggregate with its ITZ. Following Gaboczi & Berryman (2001), the effective shear modulus is obtained from the solution

(2a)

where ν2 and ν3 are the Poisson’s ratio of the ITZ and aggregate respectively, r is the aggregate radius and t is the ITZ thickness. The effective bulk modulus, K, is given by: K = K2 +

p (K3 − K2 )

1 + (1 − p)

2 ) (K3 −K K2 + 43 G2

(3)

where K2 and K3 are the bulk modulus of the ITZ and aggregate respectively. 2.2 Asymptotic theory of homogenization-Step 2 The asymptotic theory of homogenization is based on the following asymptotic expansion of the displacement field: u (x, y) = u0 (x, y) + ξ u1 (x, y) + ξ 2 u2 (x, y)

138

(4)

where x is the macroscopic scale position vector; y = x/ξ are the micro scale position vectors as 0 < ζ ft

where ft is the tensile strength of the respective material. While this is a priori an appropriate approach for the glass fibres, it is known that cementitious matrices like concretes and mortars show certain post-cracking resistance also known as tension softening. However, previous investigations, see (Hartig et al. 2009), showed that considering realistic values of the fracture energy, the stress-strain response of the composite is

Discretization

matrix

Transverse direction:

Longitudinal direction:

j

x

no matrix cracking

no matrix cracking

yarn sleeve direction (j )

(1)

u

core direction (i)

i

i

h cr h rr

longitudinal ng dinal inal direction (x ) r

h rr node

matrix element

reinforcement element

bond element

Deterministic constitutive relations

stress

bond stress

E 1

strain

Figure 2.

t

8

Bond law h rr : 10

max)

bond degradation

6 4 2 0

friction (sres, res) unloading 0.2 0.4 0.6 0.8 1.0 1.2 slip s [10-5 m]

Segment model (schematic) and deterministic constitutive laws.

155

[N/mm²]

[N/mm²]

ƒt

Bond law h cr : 10 (smax,

bond stress

Tensile material law for matrix and reinforcement:

8 6 4 2 0

(smax,

max)

= (sres,

res)

friction unloading

0.2 0.4 0.6 0.8 1.0 1.2 slip s [10-5 m]

only less affected. Since it also further complicates the computations, tension softening of the matrix is neglected here. The bond behaviour between the matrix and the reinforcement as well as between parts of the reinforcement is modelled with bond laws formulated as bond stress-slip (τ -s) relations. As the bond law was also presented in some detail in (Hartig et al. 2008) only important properties will be described here. Corresponding to the two bond zones inside the yarns, two bond laws are used, which only differ concerning the parametrization. In the case of matrix-filament interaction in the fill-in zone, the bond law hcr starts with a steep increase corresponding to the assumption of adhesional load transfer until the bond strength τmax at the slip smax , see also Figure 2. Subsequently, bond degradation with a transition from adhesion to friction is assumed, which is finished at the residual bond stress value τres at the slip sres . Afterwards, purely frictional load transfer is assumed, which is modelled with a constant bond stress equal to τres . The bond law also includes a description of unloading based on the concept of plasticity. The interpolation between the supporting points of the bond law, e. g. (smax , τmax ) is performed with the PCHIP method, see (Fritsch & Carlson 1980). The load transfer in the core of the yarns is assumed to occur by friction. Thus, in the respective bond law hrr the values of τmax and τres are equal, see also Figure 2. The limited tensile strength of matrix and reinforcement as well as the nonlinear bond laws lead to a nonlinear system of equations, which needs to be solved with an incremental iterative solution method. Therefore, the BFGS approach in combination with line search, see for instance (Matthies & Strang 1979), is used. In the case of reaching the tensile strength of a bar element, the Young’s modulus is set to zero. In every load step, only one bar element is allowed to crack and subsequently the system is solved again at the same load level. The load steps are chosen such small that after a crack at least one load step without element failure follows. However, the resulting cracking stresses are always somewhat larger than the tensile strength depending on the load step length, which might influence a statistic evaluation of results of the model. 3.3

properties of both the aggregates and the hardened cement paste. Thus, the material properties of a certain material region, which is assumed to be a representative section of the whole body, are homogenized properties of the material points in this region. As the body is at least initially assumed to be a continuum, it might be assumed that also the material properties show some continuous distribution due to the influence of the closer neighbourhood. This influencing is called correlation in statistics and smooths the spatial distribution of the material parameters. Also between different material properties correlation might exist, which is called cross-correlation. A method to model smooth fluctuations of the material properties is given with random fields. A convenient approach for the simulation of cross-correlated random fields was presented by (Voˇrechovský 2008) and is applied here. As the concrete and the reinforcement are modelled each essentially one-dimensional according to Section 3.1, the approach by (Voˇrechovský 2008) can be substantially simplified. Because the bar and bond elements used in the model have only one integration point, the material properties are constant in one element and, thus, the number nRV of the random variables X of a random field describing one material property is equal to the number of considered elements nel , e. g. the number of bar elements representing the concrete. It is assumed that all random variables have the same underlying distribution function. Depending on the position, two random variables X and X ′ at the coordinates x and x′ are assumed to influence one another more or less, which can be modelled with the autocorrelation coefficient ρ(X, X ′ ) where the approach     |x − x′ | 2 ′ ρ(X, X ) = exp − (3) lcor

Stochastic constitutive relations

All described material parameters show more or less scatter in reality. While on the nanoscopic scale and eventually also on the microscopic scale the material parameters might fluctuate purely randomly, on the mesoscopic and the macroscopic scale some kind of homogenization occurs. For instance Young’s modulus and tensile strength of concrete on the macroscale, which is also the scale that is considered for the concrete in the model, depend on the respective

156

is used and where lcor is the correlation length, which determines the range of influence. Small values of lcor lead to very local influence while large values extend the influence to larger regions. The values of the mutual correlation of all random variables can be assembled in a so-called auto-correlation matrix Cauto , which is real, symmetric, dense and of order nRV . The Karhunen-Loève expansion is applied to characterize the random fields in a domain , which incorporates the solution of the Fredholm integral of the second kind defining an eigenvalue problem  Cauto (X , X ′ )ψ i (X ′ )ddim = λi ψ i (X ) (4) 

with eigenvectors ψ i and eigenvalues λi of Cauto . Equation (4) needs not to be solved explicitly. Cauto rather can be decomposed into independent matrices of eigenvectors  auto and eigenvalues auto via Cauto =  auto auto  Tauto .

(5)

An advantageous property of this decomposition is that the only accumulation point of the eigenvalues is zero, see (Sudret & Der Kiureghian 2000), which can be used to reduce the number of random variables at a given accuracy by means of truncating the series after the nred -th term, see (Voˇrechovský 2008). The eigenvectors ψ i and the eigenvalues λi resulting from Equation (5) are used to expand the random field H H (X ) =

nred √ 

λi ξ i ψ i (X )

(6)

i=1

where ξ is a vector of length i of uncorrelated standard Gaussian random coefficients. In the one-dimensional case, H is a vector containing the realisations of the random field. The previously described procedure is only valid if the underlying distribution function is Gaussian. In the case of non-Gaussian distributions, which are often needed to model e. g. the tensile strength, the procedure is more complicated. At first, the correlation coefficients determined with Equation (3) have to be transformed from the non-Gaussian domain to the Gaussian domain. Therefore, usually the Nataf transformation is applied ρNG (X , X ′ ) =



∞



∞



F −1 ( (X )) − E(X ) D(X ) −∞ −∞  −1  ′ F ( (X )) − E(X ′ ) × D(X ′ ) ′

′

× φ2 (X , X , ρG (X , X ))dXdX

′



(7)

where ρNG (X , X ′ ) and ρG (X , X ′ ) are the correlation coefficients in the non-Gaussian and the Gaussian domain, E(.) and D(.) are expected value and standard deviation of the non-Gaussian variables, (·) is the standard Gaussian cumulative distribution function (CDF), F −1 (·) is the percent point function (PPF) of the non-Gaussian distribution and φ2 is the bivariate standard Gaussian probability density function (PDF). The solution of Equation (7) with respect to ρG (X , X ′ ) is quite complicated as it has to be performed iteratively and it is not always guaranteed to have a solution, see (Vio et al. 2001; Lebrun & Dutfoy 2009). Having solved for ρG (X , X ′ ) the next steps are identical to the case of a Gaussian distribution. Only at the end, the values of H corresponding to Equation (6) have to be transformed back to the (original) non-Gaussian domain by means of (X ) = F −1 [ (H (X ))], H

(8)

which (Grigoriu 1998) calls translation process. As mentioned previously, also between different material properties some correlation, called crosscorrelation in this case, might exist depending on

the underlying physical process, as e. g. for the tensile strength of the concrete and the bond strength between matrix and reinforcement. In the approach by (Voˇrechovský 2008), the cross-correlations between nprop different material properties, which are each modelled as random fields, are introduced with constant factors assembled in the so-called crosscorrelation matrix Ccross of order nprop . Ccross has entries in the range −1 to 1 with entries equal to 1 at the main diagonal and it is symmetric and dense. Similar to Cauto , the correlation coefficients of Ccross have to be transformed from the, in general, underlying non-Gaussian domain to the Gaussian domain using Equation (7). In contrast to the case of Cauto , the underlying distribution functions are not identical. Again, Ccross in the Gaussian domain has to be decomposed into eigenvectors  cross and eigenvalues cross similar to Equation (5). Again, an eigenvalue truncation is possible, but because nprop is usually small the reduction is often negligible. With the entries of Ccross a so-called block crosscorrelation matrix D of order nprop · nRV is virtually established, where only the main diagonal and the main sub-diagonals of the nRV × nRV partial blocks have non-zero entries. The eigenvectors  D and eigenvalues D of D are nRV -multiples of  cross and cross as (Voˇrechovský 2008) showed. This is used to calculate a cross-correlated random vector

χ D =  D D ξ (9)

where ξ is a vector of independent standard Gaussian random variables of order nprop · nRV . χ D can be j splitted into subvectors χ D of order nRV where j = 1, 2, . . . , nprop . Cauto is identical for all nprop fields. Thus, the realisations of each random field H j corresponding to a certain material property can be calculated using Equation (6) with the eigenvectors ψ i , the eigenvalues λi of the (Nataf-corrected) auto-correlation matrix Cauto as well as the respective cross-correlated random j subvector χ D . Finally, the realisations of the random fields have to be transformed back into the (original) non-Gaussian domain via Equation (8) leading j . to random fields H In the following simulations, only the material properties of the concrete are modelled stochastically. Although, this would be also possible for the reinforcement in principle, this exceeds the scope of this paper and has to be devoted to further investigations. For the simulation of the random fields, a correlation length lcor of 2 mm is assumed, which corresponds to two times the maximum grain size of the matrix. This has to be seen as a first approach. A better approximation might be possible e. g. with the approach by (Baxter & Graham 2000). The expected value of the Young’s modulus of the matrix Ec , which is modelled with a logarithmic normal distribution,

157

Between these material properties a relatively strong correlation is assumed while the correlation between fct and τmax is expected to be slightly higher than with Ec . This has to be seen also as a first approach. The realisations of the random fields are computed at the positions of the integration points of the bar elements. As the integration point positions of the bond elements and the respective bar elements do not coincide, an interpolation of the bond strength values is necessary, which is realised by means of the calculation of the mean value of the bond strength at the integration points of the two neighbouring bar elements of the bond element under consideration. Furthermore, it is assumed that the concrete only cracks in a range of 0.3 m in the centre of the model while at the ends in ranges of 0.1 m each corresponding to the clamping zones in the experiments the concrete is not allowed to crack. In contrast, Ec and τmax are modelled stochastically along the whole length of the model. In order to reduce disturbances of the random fields at the boundaries the discretization range is artificially increased for 0.02 m at both ends of the model.

relations is carried out first. The material properties are assumed to be the mean values given in the previous section while the geometrical properties of the model were specified in Section 3.1. In Figure 3, the simulated stress-strain relation is observable where the stress is the calculated reaction force at the concrete’s end nodes divided by the cross-sectional area of the concrete while the strain is the mean strain of the concrete elements at a length of 0.2 m in the centre of the specimen. Despite the deterministic modelling of the material parameters, a relatively good agreement between the simulated and the experimental results is observable, cp. Figure 1. While the linear-elastic part corresponding to the uncracked state essentially coincides in the simulation with the experiments, most differences occur in the state of multiple cracking. The first difference is the stress where the first concrete crack occurs, which is considerably lower in the experiments. Furthermore, the experiments show a relatively smooth increase of the stress-strain relation during the crack development, which is not reproduced by this model. In the simulation, the cracks occur at several discrete stress levels. In the presented case, the last cracks develop between existing cracks at a mean strain of about 0.5%. Concerning the post-cracking state, the simulation overestimates the stiffness compared to the experiments. The calculated stiffness corresponds to the stiffness of the reinforcement, which is not the case in the experiments as mentioned in Section 2. As the ultimate strain in the simulation corresponds to those of the experiments, the ultimate stress is overestimated in the simulation. Scatter of matrix tensile strength

4.2

In the following, the influence of the scatter of the matrix tensile strength on the material behaviour of TRC is investigated. Therefore, only the tensile 30 25 mean stress [N/mm2]

is assumed with 28,500 N/mm2 , according to (Jesse 2004), with a relative standard deviation of 2% based on tests on a similar matrix by (Brockmann 2005). The tensile strength of the matrix fct , which is modelled with a two-parametric Weibull distribution, is assumed with an expected value of 6.7 N/mm2 and a relative standard deviation of 10% corresponding to results of flexural tensile strength tests by (Jesse 2004). The Young’s modulus and the tensile strength of the reinforcement is modelled deterministically with values of 79,950 N/mm2 for the Young’s modulus and 1357 N/mm2 for the tensile strength, see (Abdkader 2004). The parametrisation of the bond laws is chosen corresponding to Figure 2. For the bond strength τmax of the bond law hcr , which has an expected value of 9 N/mm2 , a relative standard deviation of 10% corresponding to the tensile strength is assumed, because experimental results are missing. Furthermore, τmax is also modelled with two-parametric Weibull distribution. The cross-correlation between fct , Ec and τmax is assumed as ⎤ ⎡ (f ) ct 1.0 0.8 0.9 (E ) c 1.0 ⎦ Ccross = ⎣ 0.8 (10) 0.8 (τmax ) 1.0 0.9 0.8

20 15 10

4 RESULTS AND DISCUSSION

5

4.1 Deterministic case

0

In order to determine the influence of the stochastic modelling of the material parameters on the simulated behaviour of the tensile specimen corresponding to Section 2, a simulation with deterministic constitutive

158

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 3. Simulated stress-strain relation with deterministic material laws.

strength is modelled stochastically with expected value and standard deviation according to Section 3.3. Ten simulations with different realisations of the random field for the tensile strength are carried out, see Figure 4. As previously mentioned, limited tensile strength is only applied to the concrete elements in a range of 0.3 m in the centre of the specimen. The simulated stress-strain relations, see Figure 5, show a more realistic course compared to the deterministic calculation. As to be expected, the results of the simulations show scatter because of the stochastic modelling of the tensile strength. The state of multiple cracking is strongly influenced by the varying tensile strengths. At least the first crack can be forecast from the simulated strength distribution. For instance, for the simulation with the black strength distribution curve in Figure 4, the first crack will occur at the minimum at x = 0.1542 m. Subsequently, cracking depends also on the force transmission length between matrix and reinforcement while the local strength minima are only preference locations for 9 8 7 fct [N/mm2]

6 5 4 3 2 1 0

0.1

Figure 4.

0.15

0.2

0.25 x [m]

0.3

0.35

0.4

Realizations of random fields for fct .

cracks. Compared to the deterministic case, the stress at the first matrix crack decreases in all cases. Furthermore, contrary to the deterministic simulation, the stress levels of subsequent cracks increase successively similar to the experiments. However, at least the standard deviation seems to be chosen somewhat too high as the experimental results show a flatter mean slope at the begining of the multiple cracking state. For the post-cracking state, the statements given for the deterministic simulation apply. Additionally, the courses of the stress-strain relations in the post-cracking state show scatter. While this can be explained to some extent with different tension stiffening of the matrix due to different numbers of cracks also the slope slightly varies between the computations. This might be explained with different activations of the reinforcement via bond. 4.3 Scatter of matrix Young’s modulus Further simulations are carried out to investigate the influence of variations of Ec along x. Therefore, again ten simulations were carried out with the stochastic properties according to Section 3.3. The respective realisations of the random fields are shown in Figure 6. Because of the smaller standard deviation of Ec compared to random fields for fct , the fluctuations are considerably lower. The simulated stress-strain relations are shown in Figure 7. In principle, similar effects as for fct should be observable as the matrix stress σc is coupled to Ec via Equation (1). As the standard variation is considerably smaller for Ec and the matrix is also connected to the reinforcement, the impact on the stress-strain relations should be, however, smaller. In Figure 8, the qualitative courses of σc before and after the first crack as well as Ec along x are shown for one of the simulations. The course of σc before the crack is smoother compared to Ec as the stiffness differences

30 35 30

20

25 Ec [103 N/mm2]

mean stress [N/mm2]

25

15

20 15

10

10

5 0

5

0

0.5

1.0 mean strain [%]

1.5

2.0

0

Figure 5. Simulated stress-strain relations with stochastic modelling of fct .

159

0

Figure 6.

0.1

0.2

x [m]

0.3

0.4

Realizations of random fields for Ec .

0.5

are balanced by the activation of the reinforcement via bond. Furthermore, it is observable that the matrix cracks in one of the stiffest elements first. Thus, the stress at the first crack is always smaller in the stressstrain relations compared to deterministic simulation although the effect is subordinated in this parameter combination. The same applies for the subsequent matrix cracking where at the beginning only small differences between the simulations with fluctuating Ec but also to the deterministic case exist. Scatter in the stress-strain relation starts to appear only at the crack development between existing cracks. Furthermore, the stresses where cracks occur successively increase, see Figure 7, in contrast to the deterministic simulation. As in the case of fluctuating fct , variations of the stress-strain courses between the simulations appear in the post-cracking state, which can be explained similar to the case of varying fct . 30

mean stress [N/mm2]

25 20 15

4.4 Scatter of bond strength Another parametric study with stochastically modelled bond strength τmax of bond law hcr between matrix and reinforcement was performed. Again all other material properties were assumed to be deterministic and ten simulations were carried out. The courses of τmax are similar to those of fct , see Figure 4, as only the expected value is assumed differently between both properties. At least in this parameter combination, the stochastic modelling of τmax has only minor influence on the stress-strain relations, which are shown in Figure 9. As the bond laws influence the load transmission length between matrix and reinforcement, only the crack plateaus of the matrix cracks, which develop between existing cracks, show scatter in the stress-strain relations. Furthermore, it is observable that the cracking stresses do not successively increase as in the cases of stochastically modelled fct and Ec but several crack plateaus with almost constant stress develop. This can be explained with the constant value of fct , which does not allow for such stress fluctuations. The scatter in the post-cracking state appears somewhat lower compared to the previous cases and seems to be essentially a result of different extents of tension stiffening due to different numbers of cracks. Simultaneous scatter of matrix tensile strength, matrix Young’s modulus and bond strength

10

4.5

5

In a final parametric study, the three parameters fct , Ec and τmax , which were studied in the previous sections independently of each other, are modelled now simultaneously with cross-correlated random fields with the stochastic properties given in Section 3.3. In this study twenty simulations were carried out, where an exemplary realisation of the cross-correlated random fields for the three stochastically modelled properties

0

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 7. Simulated stress-strain relations with stochastic modelling of Ec .

matrix cracking allowed

30 25 mean stress [N/mm2]

Ec σc (right before first crack) σc (right after first crack)

20 15 10 5

0

0.1

0.2

x [m]

0.3

0.4

0

0.5

Figure 8. Qualitative courses of Ec and σc along x before and after cracking.

160

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 9. Simulated stress-strain relations with stochastic modelling of τmax of bond law hcr .

is observable in Figure 10. The cross-correlations between the different properties are observable. In Figure 11, the stress-strain relations resulting from the simulations are shown. The courses are similar to those of the simulations with varying fct , which seems reasonable because this property appears to have the strongest impact on the stress-strain behaviour with the chosen parameter combination as the previous parametric studies showed. However, the strong cross-correlation between the properties should lead to a smaller variation of the matrix cracking stresses because now two types of preference locations for cracking exist. As shown in the Sections 4.2 and 4.3, both locations with low fct and high Ec are preferred for matrix cracking while in this parameter combination the locations with low fct have more impact. Regarding the post-cracking state, the variation of the slopes increases compared to the previous parametric studies, which needs further investigations to be explained completely. However, the explanations given in Section 4.2 on this issue certainly apply. The influence of the variation of τmax , which was already Ec [103 N/mm 2]

35 30

fct [N/mm 2]

max

[N/mm 2]

25 12 10 8 6 4 9 8 7 6 5 4 0

0.1

0.2

x [m]

0.3

0.4

0.5

Figure 10. Realizations of cross-correlated random fields for fct , Ec and τmax . 30

mean stress [N/mm2]

25

In this contribution, a model for the uniaxial tensile behaviour of TRC with a stochastic constitutive law for the concrete and the bond between concrete and reinforcement was presented. Based on parametric studies on tensile specimens the applicability of the model was shown and first conclusions can be drawn. For instance, it can be concluded that although the variations of the matrix tensile strength has the largest influence on matrix cracking also the other stochastically modelled parameters influence the cracking behaviour of the matrix. This complicates a clear identification of the respective impact if all material parameters are modelled simultaneously in a stochastic manner. Furthermore, the estimation of stochastic material properties is difficult, especially for the tensile strength where only minimum values are accessible by means of experiments but the whole range of occurring values is needed to establish a distribution function. Although is was not studied in this work, the results of the simulations strongly depend on the chosen correlation length where a consistent theory for its estimation seems to be missing. Furthermore, the enhanced model allows for investigations concerning size effects. While the description of the energetic size effect was already possible with the deterministic model, the stochastic modelling of the material properties allows also to incorporate the statistical size effect. However, to complete the model also the material properties of the reinforcement need to be modelled stochastically. While the application of the random fields to the reinforcement is unproblematic, the considerable increase of the system size, which is necessary for a sufficiently fine discretization of the reinforcement up to single filaments, is still challenging.

The authors gratefully acknowledge the financial support of this research from Deutsche Forschungsgemeinschaft DFG (German Research Foundation) within the Sonderforschungsbereich 528 (Collaborative Research Center) ‘‘Textile Reinforcement for Structural Strengthening and Retrofitting’’ at Technische Universität Dresden.

15 10 5 0

5 SUMMARY AND CONCLUSIONS

ACKNOWLEDGEMENTS

20

0

relatively small when it was varied exclusively, seems to be completely masked by the effects of fct and Ec .

0.5

1.0 mean strain [%]

1.5

2.0

REFERENCES

Figure 11. Simulated stress-strain relations with simultaneous stochastic modelling of fct , Ec and τmax of bond law hcr .

161

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Betonbewehrung. Ph. D. thesis, Technische Universität Dresden, Dresden. Baxter, S. & Graham, L. (2000). Characterization of random composites using moving-window technique. Journal of Engineering Mechanics 126(4), 389–397. Brockmann, T. (2005). Mechanical and fracture mechanical properties of fine grained concrete for textile reinforced composites. Ph. D. thesis, RWTH Aachen, Aachen. Chudoba, R.; Voˇrechovský, M. & Konrad, M. (2006). Stochastic modeling of multi-filament yarns. I.Random properties within the cross-section and size effect. International Journal of Solids and Structures 43, 413–434. Curbach, M.; Graf, W.; Jesse, D.; Sickert, J. & Weiland, S. (2007). Segmentbrücke aus textilbewehrtem Beton: Konstruktion, Fertigung, numerische Berechnung. Beton- und Stahlbetonbau 102(6), 342–352. Cuypers, H. & Wastiels, J. (2006). Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading. Materials and Structures 39, 777–786. Daniels, H. (1944). The statistical theory of the strength of bundles of threads. i. Proceedings of the Royal Society of London A183, 405–435. Fritsch, F. & Carlson, R. (1980). Monotone piecewise cubic interpolation. SIAM Journal on Numerical Analysis 17(2), 238–246. Grigoriu, M. (1998). Simulation of stationary non-gaussian translation processes. Journal of Engineering Mechanics 124(2), 121–126. Hartig, J.; Häußler-Combe, U. & Kitzig, M. (2009). Effect of matrix tension softening on the uniaxial tensile behaviour of textile reinforced concrete. In Smojver, I. & Sori´c, J. (eds.), Proceedings of the 6th International Congress of Croatian Society of Mechanics (ICCSM), Dubrovnik, 30 September–2 October 2009, p. 46 (Book of Abstracts and CD of full papers). Zagreb: Croatian Society of Mechanics. Hartig, J.; Häußler-Combe, U. & Schicktanz, K. (2008). Influence of bond properties on the tensile behaviour of textile reinforced concrete. Cement & Concrete Composites 30(10), 898–906.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Multi-axial modeling of plain concrete structures based on an anisotropic damage formulation M. Kitzig & U. Häußler-Combe Institute of Concrete Structures, Technische Universität Dresden, Dresden, Germany

ABSTRACT: In this contribution, a strain-based constitutive law for concrete within the framework of continuum damage mechanics is proposed. The model allows for the multi-axial simulation of predominantly tensile loaded plain concrete. A second-order integrity tensor is chosen as the internal damage variable to consider the phenomenon of load-induced anisotropy. The model is implemented in the finite element method. Hence, a fracture-energy based regularization approach is included to overcome the mesh-dependence of the computational results. This local regularization method is contrasted with a non-local model version, in which the damage variable is obtained based on weighted strain averages over a spatial neighborhood of the point under consideration. Consistent tangent stiffness formulations are derived for both model versions to achieve fast convergence within the incremental-iterative solution procedures. The applicability of the model and the influence of the used regularization technique on the numerically obtained results are demonstrated by means of the simulation of a well documented experiment with plain concrete specimens. 1 INTRODUCTION Concrete is one of the dominating construction materials in various fields of civil engineering. The demand for slender supporting structures in combination with an increase of load-carrying capacity reserves requires the appropriate specification of the complex multiaxial material behavior in constitutive laws. Up to now, such universally valid descriptions only succeed with restriction, especially with regard to practical applications. The material characteristics of concrete are wellknown from a large number of comprehensive experiments. Investigations on concrete for biaxial and triaxial stress states were performed e.g. by (Kupfer & Hilsdorf 1969), (Imran & Pantazopoulou 1996), (Hussein & Marzouk 2000) and (Lee et al. 2004). Major characteristics of the quasi-brittle material are the strength increase under multiaxial compressive loading compared to the uniaxial compressive strength, the pronounced non-linear stress-strain response for tensile and compressive loading due to propagation and growth of micro-defects and a softening behavior after exceeding the limit states of the material. A further important material property is the loadinduced anisotropy, i.e. the phenomenon that the initially isotropic elastic stiffness becomes cumulatively anisotropic for increased loading. Continuum damage mechanics provides a constitutive theory for the macro-scale description of the progressive degradation of material stiffness due to micro-cracking. For a theoretical framework refer

163

e.g. to (Carol et al. 1994). The appropriate choice of the internal damage variable is one of the kernel issues of a damage formulation within this framework. This state variable, introduced to quantify the orientation and density of defects on the micro-scale, can be of scalar-valued, second-order or fourth-order tensor-valued type. The use of scalar internal variables enables the description of isotropic damage, wheras tensor-valued state variables are needed for anisotropic damage formulations. Detailed information can be found e.g. in (Lemaitre 1992), (Krajcinovic 1996) and (Skrzypek & Ganczarski 1999). In this contribution, a strain-based formulation is proposed, which allows for the multi-axial simulation of predominantly tensile loaded plain concrete structures. Based on the concept of effective and nominal stresses and strains as well as the energy equivalence requirement in terms of both quantities, the secondorder integrity tensor is chosen as the internal damage variable, see (Carol et al. 2001a). This choice enables the specification of an orthotropic material behavior. The eigenvalues of the state variable 1 can be understood as damage measures in the directions of the associated eigenvectors. These directions are established under the assumption that the principal axes of the strain tensor coincide with those of the incremental integrity tensor. Hence, load-induced anisotropy can be described. The proposed constitutive law is implemented in the finite element method. The use of material models with softening in numerical calculations leads to localization phenomena, i.e. the concentration of large

strains in narrow bands, and to highly meshsensitive results. Hence, regularization methods are required to overcome the dependence of the results on the used discretization. A popular approach is based on a rescaling of the post-peak branch of the stress-strain curve as a function of the element size, see e.g. (Pietruszczak & Mróz 1981), (Bažant & Oh 1983), (Willam et al. 1986). Advanced techniques to ensure mesh insensitivity of numerical results exist in form of localization limiters. This group includes the Cosserat continuum theory, see e.g. (de Borst 1991), nonlocal models of the integral or gradient type, see e.g. (Pijaudier-Cabot & Bažant 1987) and (M¨uhlhaus & Aifantis 1991), and higher-order gradient approaches, see e.g. (Aifantis 1984). A computationally convenient localization limiting technique is provided by the concept of nonlocal weighted averaging of a proper quantity over a spatial neighborhood of the point under consideration. For the motivation and a survey of nonlocal integral formulations refer e.g. to (Bažant & Jirásek 2002). In this contribution, the constitutive law proposed in Section 2 includes a fracture-energy based regularization approach, which is described in Section 2.3, and a nonlocal version of the integral type presented in Section 3. Consistent tangent stiffness formulations are derived for both the local and the nonlocal model version to achieve fast convergence in incrementaliterative procedures like the Newton-Raphson method in the finite element analysis. The applicability of the constitutive law is demonstrated with the numerical simulation of a well documented experimental setup in Section 4. In this context, the independence of the computationally obtained structural response on the chosen discretization as well as the influence of the chosen regularization method on the determination of the localization zone geometry in the calculation is presented.

that the initially isotropic stiffness becomes cumulatively anisotropic for increased loading. This internal variable is obtained based on the concept of effective and nominal stresses and strains. Within this concept, effective quantities are defined as stresses and strains to which the material between the distributed microdefects is subjected, whereas nominal quantities are the stresses and strains that can be measured externally. The requirement of energy equivalence, i.e. the same elastic energy shall be stored in terms of the effective and in terms of the nominal quantities, leads to the choice of the second-order integrity tensor φ¯ as the internal damage variable, see e.g. (Carol et al. 2001a). According to this choice, E = λφ¯ ⊗ φ¯ + 2µφ¯ ⊗ φ¯

can be written for the secant stiffness tensor. In Equation (2), ⊗ and ⊗ symbolize the dyadic products (A ⊗ B)ijkl = Aij Bkl and (A ⊗ B)ijkl = 12 (Aik Bjl + Ail Bjk ) of second-order tensors A and B. Furthermore, the Lamé constants λ = ν 0 E 0 /[(1 + ν 0 )(1 − 2ν 0 )] and µ = E 0 /[2(1+ν 0 )] are defined in terms of the Young’s modulus E 0 and the Poisson’s ratio ν 0 of the initially isotropic material. 2.2 Damage evolution and limit condition The formulation of the damage evolution starts with the spectral decomposition of the incremental integrity ˙¯ The principal axes of the incremental damtensor φ. age variable are assumed to coincide with those of the (i) strain tensor ǫ and the principal values φ˙¯ are related to damage measures D(i) according to 3 3   ˙ (i) (i) −D d(i) ⊗ d(i) . φ˙¯ = φ˙¯ d(i) ⊗ d(i) = √ (i) 2 1 − D i=1 i=1 (3)

2 ANISOTROPIC DAMAGE CONSTITUTIVE LAW 2.1 Secant relations and damage variable In the following, an anisotropic formulation for concrete within the framework of continuum damage mechanics is proposed. A characteristic equation of elastic degradation is the secant stress-strain relation σ =E:ǫ

(1)

which couples Cauchy stresses σ and elastic strains ǫ via the fourth-order secant stiffness tensor E. Plastic strains are assumed to be negligible since predominant tension is considered. An internal damage variable must be chosen to describe the density and orientation of distributed micro-defects in the material. This variable must be a tensor of order two or higher to consider the load-induced anisotropy, i.e. the phenomenon

(2)

The vectors d(i) in Equation (3) are the eigenvectors of the strain tensor ǫ. The damage measures D(i) are functions of equivalent strain measures κd(i) ⎧ 0 κd(i) ≤ ed0 ⎪ ⎪ ⎞gd ⎛ ⎨ (i) ⎜ κd − ed0 ⎟ D(i) = (4) −⎝ ⎠ ⎪ ⎪ ed ⎩ (i) 1−e κd > ed0

164

with (i) κd

(i) (i)

=β ǫ

,

β

(i)

=

αt ǫ (i) ≥ 0 . −1 ǫ (i) < 0

(5)

This coupling enables the description of damage in three mutually orthogonal material directions. Values for the damage measures D(i) lie within the interval

[0, 1] with D(i) = 1 representing the completely damaged state in direction i. In Equation (5), ǫ (i) are the eigenvalues of the strain tensor, while the damages parameters αt , ed0 , ed and gd in Equations (4) and (5) are material constants, which can be obtained from uniaxial stress-strain relations. Their determination is described in detail in (Häußler-Combe & Hartig 2008). Values for the damage parameters are listed in Table 1 for the particular concrete of the L-shaped panel test described in Section 4. The stress-strain relation for tensile loading in the uniaxial special case is shown in Figure 1. ˙ (i) remain to be defined. The damage multipliers D For this purpose, loading functions Fd(i) (i)

(i)

Fd = β (i) ǫ (i) − κd

(i)

˙ (i) ≥ 0, D

(6)

˙ (i) F (i) = 0. D d

(7)

The corresponding consistency conditions F˙ d(i) = 0 lead to the definition of the damage multipliers Table 1.

(i)

˙ (i) = 0 β (i) (d(i) ⊗ d(i) ) : ǫ˙ − h(i) d D

are introduced which control the evolution of damage and differentiate between damage in tension and compression. Damage can increase only if the current material state reaches the boundary of the elastic domain Fd(i) = 0. This is expressed by the loadingunloading or Kuhn-Tucker conditions Fd(i) ≤ 0,

(i)

∂Fd ∂Fd ∂κd (i) ˙ =0 D : ǫ˙ + (i) ∂ǫ ∂κd ∂D(i)

Material parameters for the L-shaped panel test.

˙ (i) = D

β (i) h(i) d

(d(i) ⊗ d(i) ) : ǫ˙

(8)

(i) (i) with 1/h(i) d = ∂D /∂κd . Substitution of Equation (8) in Equation (3) results in the formulation for the incremental damage variable φ˙¯ 3

(i)

 −β (i) /h d φ˙¯ = (d(i) ⊗ d(i) ⊗ d(i) ⊗ d(i) ) : ǫ˙ . √ (i) 2 1 − D i=1 2.3 Regularization based on fracture energy

The application of the proposed damage law in the tension regime leads to a limited tensile strength followed by a descending branch in the stress-strain relation, see Figure 1. It is well-known that the use of material models with softening for numerical calculations leads to localization phenomena, i.e. the concentration of large strains in narrow bands or localization zones, respectively, and to highly mesh-sensitive results. A fractureenergy based calibration of the softening part of the uniaxial stress-strain curve provides an opportunity to overcome this mesh dependence. For this purpose, the equivalent strains κd(i) are replaced by modified values (i) κd,mod

L-shaped panel Damage parameter ed0 Damage parameter ed Damage parameter gd Damage parameter αt Young’s modulus E 0 Poisson’s ratio ν 0 Strengths fct / fc Fracture energy GF

[-] [-] [-] [-] [MN/m2 ] [-] [MN/m2 ] [Nm/m2 ]

(i) κd,mod = κd,fct + (1 − γ )κd,fct ln

−1.99 · 10−4 3.07 · 10−3 2.00 11.49 25, 850 0.18 2.7 / − 31.0 90

(9)

κd(i) − γ κd,fct (1 − γ )κd,fct

(10)

if and only if the equivalent strain κd,fct corresponding to the concrete tensile strength is exceeded. Otherwise, (i) (i) the relation κd,mod = κd holds. This is demonstrated in Figure 2. The modified equivalent strains shall

Kd,mod 0.005

without regularization = 0.2 = 0.3 = 0.4

0.004 0.003 0.002 0.001

Kd Kd,fct

Figure 1.

Uniaxial tensile stress-strain curves.

Figure 2.

165

0.004

0.006

Modified equivalent strains.

0.008

ensure a continuously differentiable stress function at the position of the maximum stress. This requirement is fulfilled by the approach presented in Equation (10). In this equation, γ is a further model parameter depending on the characteristic length lel,ch of the elements used in the finite element computation. Values for γ can be obtained by means of involvement of the fracture energy GF , i.e. the energy required to induce a tensile crack of unit area. For the determination of γ it is assumed that the limit strain κd,fct is exceeded within the whole width bcal = f (lel,ch ) of the localization zone in the calculation. Then the regularization parameter γ can be derived from the relations for the specific fracture energy gf ,cal , which is the area below the uniaxial stress-strain curve  ǫ GF (i) gf ,cal = = σ (κd,mod , ǫ) dǫ . (11) bcal ǫfct

the accurate formulation of the tangent stiffness tensor Etan to achieve fast convergence. The incremental constitutive relation, which relates increments of the stresses with increments of the strains, is obtained by means of differentiation of Equation (1) with respect to time: ˙ : ǫ = Etan : ǫ˙ . σ˙ = E : ǫ˙ + E (12)

With a presumed width of the localization zone, which varies between one and two times the characteristic element length, Equation (11) can be solved for γ , see Figure 3. For the numerical simulation of the L-shaped √ panel test described in Section 4 a value bcal = 2lel,ch is chosen. Modified uniaxial stressstrain relations are shown in Figure 1. The advantages of the fracture-energy based regularization are the numerical robustness and relatively small effort with respect to the implementation in the finite element method. On the other hand, an essential drawback is the fact that the width of the localization zone must be a priori predefined in dependence on the element length by the choice of bcal . This disadvantage can be eliminated by means of a refined regularization technique based on weighted averaging of a local quantity over a certain neighborhood of the point under consideration, which is presented in Section 3.

˙ : ǫ in Equation (12) needs further elabThe term E oration. The utilization of compact tensor formalism, see e.g. (Rizzi & Carol 2001), and of the symmetries of both the strain and the integrity tensor, leads to the expression ˙ : ǫ = [λ(φ¯˙ ⊗ φ¯ + φ¯ ⊗ φ) ¯˙ E

˙¯ : ǫ + 2µ(φ˙¯ ⊗ φ¯ + φ¯ ⊗ φ)]

= [λ((φ¯ : ǫ)Is + φ¯ ⊗ ǫ)

¯ + (φǫ) ¯ ⊗ I )] : φ˙¯ + 2µ(I ⊗ (φǫ)

˙¯ ˆ : φ, =E

(13)

which in association with Equation (9) can be substituted in Equation (12) to obtain the nonsymmetric fourth-order tangent stiffness tensor Etan ˆ : Etan = E + E

3  −β (i) /h(i) d d(i) ⊗ d(i) √ (i) i=1 2 1 − D

⊗ d(i) ⊗ d(i) , Etan = E + E .

(14)

3 NONLOCAL DAMAGE MODEL 3.1 Nonlocal averaging

2.4 Tangent stiffness The proposed constitutive equations are implemented in a finite element code. Equilibrium iterations are performed by means of the Newton-Raphson method. The use of this incremental-iterative procedure requires

Figure 3.

Mesh-dependent parameter γ .

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The constitutive law proposed in Section 2 is extended to a nonlocal model version. This advanced regularization method introduces an additional material parameter, the characteristic length, which can be understood as a measure of material inhomogeneities and controls the width of the localization zone in numerical calculations. The introduction of nonlocal approaches to model the multiaxial material behavior of concrete is motivated by a number of reasons. First of all, in continuum models the heterogeneous microstructure of concrete is homogenized on a small scale compared to the overall dimensions of the structure. The externally measurable stress, which is averaged over an representative volume element (RVE), depends not only on the strain in one material point, but on the average strain values within this RVE. A further cause is the mutual interaction of microcracks, which in dependence on their orientation and size can shield or amplify the stress intensity of one another. Then, microcrack growth is controlled not only by the stresses and strains

in the particular point of the continuum representing the center of the crack, but by the energy release from a spatial domain around the crack. Finally, a different and not physically motivated reason for the introduction of nonlocal constitutive laws is the fact that damage models with strain-softening behavior lead to localization of damage in a zero thickness or volume zone, which is associated with a mesh-sensitivity in the finite element method. In this regard, nonlocal strain-softening models may serve as localization limiters. For more details regarding the motivations of nonlocality refer e.g. to (Bažant & Jirásek 2002). In this contribution, attention is focused on a nonlocal continuum model of the integral type. In this approach, the local elastic strains ǫ are replaced by their nonlocal counterparts ǫ˜ , which are obtained by weighted averaging over a spatial neighborhood of each point x under consideration  g(x, ξ ) ǫ(ξ ) dξ . (15) ǫ˜ (x) = V

Based on the nonlocal quantity, the internal dam¯ ǫ ) is determined as a function of age variable φ˜¯ = φ(˜ the nonlocal strain tensor, and the secant stress-strain relation Equation (1) can be rewritten as ˜ :ǫ σ˜ = E(˜ǫ ) : ǫ = E

(16)

E˜ = λφ¯˜ ⊗ φ¯˜ + 2µφ¯˜ ⊗ φ¯˜ .

(17)

with

The weight function g in Equation (15) remains to be specified. Without an influence of a boundary of the considered body, g is only a function of the distance r = ||x − ξ || between the point x under consideration and the affecting point ξ in the neighborhood of x. Near a boundary, the nonlocal quantity is obtained by averaging only over the domain part that lies within the body. Hence, the weight function must be scaled to satisfy the normalizing condition  g(x, ξ ) dξ = 1 ∀x ∈ V . (18) V

This can be done by reformulating the weight function in the form g(x, ξ ) = 

V

′

g (r) . g (||x − ζ ||) dζ ′

(19)

The function g ′ can be chosen e.g. as the Gaussian distribution function or as a bell-shaped polynomial approach. In this distribution, a formulation ⎧ ⎪ 0 |r| > R ⎪ ⎨ ′   2 g (r) = (20) r2 ⎪ ⎪ |r| ≤ R ⎩ 1− 2 R

167

Figure 4. Nonlocal damage law: (a) interaction length R, (b) weight function g.

is chosen, see Figure 4(b). The interaction length R determines how far away a point ξ can be located from the point x under consideration to have an influence on the nonlocal quantity. Thus, Equation (20) represents a bounded weight function, which is advantageous with respect to the bandwidth of the stiffness matrix if the nonlocal model is implemented in the finite element method. 3.2 Tangent stiffness In analogy to Section 2, a consistent tangent stiffness formulation shall be derived also for the nonlocal model to achieve fast convergence in incrementaliterative procedures like the Newton-Raphson method. For this purpose, the incremental constitutive relations are derived in tensorial notation similarly to Section 2. Subsequently, these equations are rewritten in a form suited for the implementation in the finite element method. A respective derivation was shown for an isotropic damage law e.g. in (Jirásek & Patzak 2002). Firstly, the incremental integrity tensor based on the nonlocal strain tensor ǫ˜ is obtained in analogy to Section 2.2 if the local quantities D(i) , hd(i) , β (i) and ˜ (i) , h˜ (i) , β˜ (i) d(i) are replaced by their counterparts D d (i) and d˜ determined from ǫ˜ ˙ φ˜¯ =

3  −β˜ (i) /h˜ (i) d  (d˜ (i) ⊗ d˜ (i) ⊗ d˜ (i) ⊗ d˜ (i) ) : ǫ˙˜ . ˜ (i) i=1 2 1 − D (21)

Differentiation of Equation (16) with respect to time leads to the incremental relation ˜ˆ ˙˜¯ ˙˜ : ǫ = E˜ : ǫ˙ + E ˜ : ǫ˙ + E σ˙˜ = E :φ ˜ : ǫ˙˜ . ˜ : ǫ˙ + E = E

(22)

˜ similar to Equations (12), (13) and (14). Eˆ is obtained ˆ if φ¯ is replaced by φ˜¯ in the expression for E. In the following, the tangent stiffness is presented for the proposed anisotropic approach in the form

required for the implementation in the finite element method. For this purpose, the notation for stresses, strains and elastic stiffness is changed from tensor to matrix and vector notation. Starting point is the vector f int of internal nodal forces  BT (x) σ˜ (x) dx (23) f int =

which in association with Equation (22) can be substituted in Equation (27): K=

f int =

 i

wi BTi · σ˜ i

(24)

with the weight factors wi for the numerical integration. In a next step, σ˜ i is replaced by the expression ˜ i · ǫ i as the matrix notation of Equation (16) σ˜ i = E ˜ i and the (6 × 1) with the (6 × 6) elasticity matrix E local strain vector ǫ i at the integration point under consideration. Substitution in the previous equation leads to  ˜ i · ǫi. f int = wi BTi · E (25) i

The tangent stiffness matrix K is obtained as the derivative of the vector f int of internal nodal forces with respect to the vector u of the nodal displacements K=

∂f int . ∂u

=



∂

i



(26)

i

˜ i · ǫi wi BTi · E ∂u

˜ tan wi BTi · E i · Bi .

j

 i

j

˜ ·B wi wj gij BTi · E i j

(29)

with the secant stiffness matrix K sec and the matrix ˜ of the fourth-order tensors E ˜ and ˜ i and E notations E i ˜E, respectively, from Equation (22). The obtained expression for K can immediately be utilized for the implementation in a finite element code. From Equation (29) it can be seen that the tangent stiffness is equal to the secant stiffness if for all integration points the stress-strain relations are either still in the linear elastic range or if unloading occurs according to the Kuhn-Tucker conditions Equation (7). Moreover, it becomes evident that the nonlocal tangent stiffness matrix turns out to be nonsymmetric. One of the reasons is the fact that the relation gij = gji for the weight function g is valid only if the mesh of the finite element model is regular and extends over an infinite domain. Since irregular discretizations with existing boundaries are used in practice, g is not symmetric with respect to the considered integration points i and j. The second reason for the nonsymmetry of K results from the expressions of the fourth-order ˜ˆ tensor E which possesses no major symmetry, but this holds for the respective quantity in the local model version, too. Hence, the nonsymmetry of the nonlocal tangent stiffness matrix means no important drawback compared to the constitutive relations presented in Section 2.

Substitution of Equation (25) in Equation (26) results in

K=

i

   ˜ · ˜ i · Bi + E w g B wi BTi · E i j ij j

= K sec +

V

with the strain-displacement matrix B and the (6 × 1) vector σ˜ of the stress components. Within the finite element method, the integral in Equation (15) is replaced by the sum over the finite number of integration points i of the elements used for the discretization. For reasons of convenience, a quantity with subscript i denotes a scalar, vector or matrix at the integration point i with the coordinates xi in the following, e.g. Bi = B(xi ). Thus, Equation (23) can be rewritten in the form



(27)

Now we replace the integral form Equation (15) by the sum over all integration points j in the considered neighborhood of the integration point i, see Figure 4(a), to obtain  wj gij ǫ j (28) ǫ˜ i = j

168

4 APPLICATION Experimental data are commonly used for the calibration of material models and for the demonstration of their applicability. The L-shaped panel (LSP) test performed by (Winkler 2001) provides appropriate data for the validation of constitutive models under mode-I conditions. The geometry and the boundary conditions of the test specimens are shown in Figure 5, values for the elastic constants, strengths and fracture energy, which are indicated in Table 1, are taken from (Feist et al. 2004). Two discretizations of the concrete specimen are investigated to demonstrate the meshobjectivity of the numerical results and the dependence of the computed crack paths on the mesh alignment. The first mesh consists of 2161 nodes and 2079 fournode quadrilaterals with regularly arranged elements in the region of the expected process zone, see Figure 6. In contrast, 1829 three-node triangular elements and

500 250

220

30

uh

500

250

t = 100

250

F, uv

[mm]

Figure 5. L-shaped panel (LSP): geometry and boundary conditions. Figure 7.

LSP mesh 2: triangular elements.

mesh 1 nonlocal

mesh 1 local

mesh 2 local

exp. scatter

Figure 8.

Figure 6.

LSP mesh 1: quadrilaterals.

LSP: load-displacement uv relation.

mesh 1 nonlocal

968 nodes are used for the second discretization, see Figure 7. Plain stress states are considered in both cases. All nodes along the lower horizontal edge of the vertical leg are fixed. The vertical tensile load F is applied under displacement control. The displacements of the point of load application and of the upper left corner have been measured. First of all, computations were performed for both discretizations using the local fracture-energy based regularization technique presented in Section 2.3. The corresponding force-displacement relations for the point of load application are contrasted with the experimental data in Figures 8 and 9. The maximum test load is well approximated for both meshes, whereas the value for the triangular element mesh exceeds the one obtained with the quadrilateral mesh by 6%. This

169

mesh 2 local mesh 1 local

experiment

Figure 9.

LSP: load-displacement uH relation.

minor difference leads to variances in the descending branches, which nevertheless exhibit similar courses and qualitatively agree with the test data. Generally, the slope of the pre-peak load-displacement curve

5 SUMMARY AND OUTLOOK

(a)

(b)

(c)

(d)

Figure 10. Crack paths: (a) test; (b) mesh 2; (c), (d) mesh 1.

observed in the tests is considerably lower compared to the numerical results. This fact can be explained by the disregard of the elasticity of the support construction in the experiment, see (Kitzig et al. 2009). The involvement of a rigid body rotation of the specimen leads to an improved agreement of the numerical and the test data, which is not demonstrated in the present contribution. The computational crack paths are plotted for mesh 2 in Figure 10(b) and for mesh 1 in Figure 10(c) and are compared to the observed scatter in the experiment in Figure 10(a). For this purpose, the maximum eigenvalues of the tensorial damage variable φ¯ in the last calculated step are presented. It can be clearly seen, that the light inclination of the cracks in the test is adequately reproduced only by the discretization using triangular elements. In contrast, a straight horizontal crack arises for the regular quadrilateral mesh, which reflects the dependence of the path on the element alignment. However, the widths of the damaged bands are in accordance with the a priori predefined value bcal described in Section 2.3. A further computation was performed based on the nonlocal version of the constitutive law presented in Section 3 using the quadrilateral mesh. An interaction length R = 12 mm, see Equation (20), was chosen, which corresponds to 1.5 times the maximum aggregate size of the concrete used in the test. Again, the corresponding force-displacement relation for the point of load application is shown in Figures 8 and 9. The experimental maximum load is well approximated before numerical problems occur. The crack path for the last computed load step represented by the maximum eigenvalue of φ˜¯ is plotted in Figure 10(d). In contrast to the result of the local model version, this path exhibits a clearly observable inclination.

170

An anisotropic damage formulation was proposed, which allows for the simulation of predominantly tensile loaded plain concrete. The choice of the second-order integrity tensor as the internal damage variable allows for the consideration of the loadinduced anisotropy. The material directions were assumed to coincide with the eigenvectors of the strain tensor, while the decoupled damage measures were associated with equivalent damage strains in the respective directions. The proposed model was implemented in the finite element method. Hence, regularization techniques are necessary to overcome the mesh-sensitivity of the numerically obtained results. For this purpose, firstly a local approach was described in Section 2 based on a fracture-energy dependent rescaling of the softening parts of the uniaxial stressstrain relations in the considered material point. Furthermore, a more refined regularization technique was formulated in Section 3 in association with a nonlocal version of the constitutive law including the replacement of the elastic strains by their nonlocal counterparts obtained by means of averaging over a certain neighborhood of the considered point. The major advantage of the latter method is the fact, that the width of the localization zone does not need to be a priori predefined as in the firstly mentioned approach. Indeed, the local regularization technique is more robust and less time consuming with respect to its implementation in numerical calculation procedures. Finally, consistent tangent stiffness formulations were derived for the local as well as for the nonlocal model version to achieve fast convergence in incremental-iterative procedures. The applicability of the proposed damage law was demonstrated in Section 4 by means of the simulation of a well-documented experiment with plain concrete specimens. Moreover, this section also served for the presentation of the characteristics of the implemented regularization techniques. The experimental loads were well approximated by both the local and the nonlocal model version. The width of the numerically obtained localization zone in the case of the energy-based regularization was in good agreement with the value bcal predefined in Section 2.3, whereat the slightly inclined crack paths observed in the tests could only adequately be reproduced using the nonregular discretization. Furthermore, the maximum test load as well as the inclined band of damage were proven to be numerically obtained if the nonlocal regularization technique is applied and the interaction length R is properly chosen in dependence on the aggregate size, which can be understood as a measure of the material inhomogeneity for concrete. Further investigations are related to the extension of the proposed constitutive law by a plastic model part

to enable the simulation of typical concrete properties like dilatancy under compressive loading. Moreover, the mutual interaction of the presently decoupled damage directions shall be considered. While a tensile damage in one direction has only a marginal influence on the damage measures in the remaining directions, this effect is stronger pronounced for compressive damage. In this case and in the case of simultaneously appearing damage in multiple directions in one material point, a mutual dependence is obvious. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of this research from Deutsche Forschungsgemeinschaft DFG (German Research Foundation) within the project ‘‘Multiaxial Concrete Constitutive Laws Based on Anisotropic Damage and Plasticity’’. REFERENCES Aifantis, E.C. (1984). On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology 106, 326–330. Bažant, Z. & Jirásek, M. (2002). Nonlocal integral formulations of plasticity and damage: Survey of progress. Journal of Engineering Mechanics 128, 1119–1149. Bažant, Z. & Oh, B. (1983). Crack band theory for fracture of concrete. Materials and Structures 16, 155–177. Carol, I., Rizzi, E. & Willam, K. (1994). A unified theory of elastic degradation and damage based on a loading surface. International Journal of Solids and Structures 31, 2835–2865. Carol, I., Rizzi, E. & Willam, K. (2001). On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo logarithmic damage tensor rate. International Journal of Solids and Structures 38, 491–518. de Borst, R. (1991). Simulation of strain localization: a reappraisal of the cosserat continuum. Engineering Computations 8, 317–332. Feist, C., Kerber, W., Lehar, H. & Hofstetter, G. (2004). A comparative study of numerical models for concrete cracking. In Neittaanmäki, P., Rossi, T., Korotov, S., Oñate, E., Périaux, J. & Knörzer, D. (Eds.), Proceedings of ECCOMAS 2004, Jyväskylä, Finland, 1–19. Häußler-Combe, U. & Hartig, J. (2008). Formulation and numerical implementation of a constitutive law for concrete with strain-based damage and plasticity. International Journal of Non-Linear Mechanics 43, 399–415.

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Hussein, A. & Marzouk, H. (2000). Behavior of high-strength concrete under biaxial stresses. ACI Structural Journal 97, 27–36. Imran, I. & Pantazopoulou, J. (1996). Experimental study of plain concrete under triaxial stress. ACI Materials Journal 93, 589–601. Jirásek, M. (1998). Nonlocal models for damage and fracture: Comparison of approaches. International Journal of Solids and Structures 35, 4133–4145. Jirásek, M. & Patzak, B. (2002). Consistent tangent stiffness for nonlocal damage models. Computers & Structures 80, 1279–1293. Kitzig, M. & Häußler-Combe, U. (2009). Damage modeling of plain concrete based on an anisotropic constitutive law. In Smojver, I. & Sori´c, J. (Eds.), Proceedings of 6th ICCSM 2009, Dubrovnik, Croatia. Krajcinovic, D. (1996). Damage Mechanics. North-Holland, Elsevier, 1996. Kupfer, H. & Hilsdorf, H.K. (1969). Behavior of concrete under biaxial stresses. ACI Journal 66, 656–666. Lee, S.-K., Song, Y.-C. & Han, S.-H. (2004). Biaxial behavior of plain concrete of nuclear containment building. Nuclear Engineering and Design 227, 143–153. Lemaitre, J. (1992). A Course on Damage Mechanics. Springer, 1992. Mühlhaus, H.B. & Aifantis, E.C. (1991). A variational principle for gradient plasticity. International Journal of Solids and Structures 28, 845–857. Pietruszczak, S. & Mróz, Z. (1981). Finite element analysis of deformation of strain-softening materials. International Journal for Numerical Methods in Engineering 17, 327–334. Pijaudier-Cabot, G. & Bažant, Z.P. (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Rizzi, E. & Carol, I. (2001). A formulation of anisotropic elastic damage using compact tensor formalism. Journal of Elasticity 64, 85–109. Skrzypek, J. & Ganczarski, A. (1999). Modeling of Material Damage and Failure of Structures. Springer, 1999. Willam, K., Bi´cani´c, N. & Sture, S. (1986). Composite fracture model for strain-softening and localised failure of concrete. In Hinton, E. and Owen, D.R.J. (Eds.), Computational Modelling of Reinforced Concrete Structures, 122–153. Pineridge Press, Swansea. Winkler, B. (2001). Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes für Beton. Ph.D. thesis, Universität Innsbruck.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Determination of cement paste mechanical properties: Comparison between micromechanical and ultrasound results S. Maalej & Z. Lafhaj LML UMR CNRS 8107, Ecole Centrale de Lille, Villeneuve d’Ascq, France

M. Bouassida Unité de Recherche Ingénierie Géotechnique, Ecole Nationale d’Ingénieur de Tunis, Tunisia

ABSTRACT: This work presents a comparison between micromechanical and experimental elastic properties of a cement paste material. First, principal micromechanical models found in the literature are described. Second, the ultrasonic and porosity methods used to constitute the experimental database are detailed. The investigated material in this study is a cement paste prepared with three water/cement ratios. Finally, a comparison between micromechanical and experimental obtained results is discussed and analyzed. 1 INTRODUCTION A composite is a combination of two materials or more, which are not soluble in each other. Generally, one component is designed as the matrix (the continuous component) and the other constituents are designed as inclusions. Thus, a cementitious material (cement paste, mortar or concrete) is considered to be a composite material made up of components of different sizes (inclusions) held together by nonporous cement paste (matrix). The determination of effective elastic properties of composite materials is of great interest for many studies mainly geophysical and mechanical fields since they are considered as indicator of stability and durability. Various computational and experimental methods have been investigated to estimate mechanical, hydraulic and acoustic properties of a composite material. The first theoretical development, focused on determining overall macroscopic transport phenomena, was proposed by (Maxwell, 1873). Voigt determined the effective mechanical properties of composite material with a contribution by Reuss (Hill, 1963, Watt et al. 1976). Later, several researches (Berryman & Berge, 1996, Kaczmarek & Goueygou, 2006, NolenHoeksema, 2000, Watt et al. 1976) were interested in developing micromechanical models to estimate the effective properties of composite materials. But, few ones were interested in cementitious materials. To estimate experimentally the effective elastic properties of cementitious porous material, ultrasonic techniques were adopted. These methods are considered as non-destructive promising tools. They are characterized by the elastic property of ultrasonic

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waves and the well-known relationships between pulse velocities and elastic moduli. The composite and its constituents are considered to be isotropic and homogenous. The overall bulk, K hom , and shear, Ghom , moduli are related to the longitudinal and transverse wave velocities (VL and VT ) and overall density, ρ hom , by (Brown, 1997):   4 K hom = ρ hom VL2 − VT2 ; G hom = ρ hom VT2 (1) 3 Micromechanical formulas for effective elastic moduli are used to evaluate some properties which are considered as indicators of the durability of material mainly porosity, permeability or diffusivity. In this paper, micromechanical models relating macroscopic and microscopic properties are described. These models are applied to both dry and fully saturated cement paste material. The latter is modelled as a two-phase porous material. In the first part, four micromechanical models were detailed. In the second part, ultrasonic measurements were carried out to determine the longitudinal and transverse velocities. Finally a comparison between the micromechanical models’ estimates and the experimental results obtained on cement paste will be presented and analyzed. 2 TWO PHASE MICROMECHAICAL MODELS The micromechanical models are based on the knowledge of mechanical properties of the considered composite’s constituents. In this study, micromechanical models are formulated for cement paste material. The latter is perceived as two-phase composite

with multiple inclusions of the same material (dry or saturated pores) within a matrix. The composite is considered to be isotropic. Four micromechanical models are presented below; (K1 , G1 ) and (K0 , G0 ) denote respectively the bulk and shear moduli, of inclusions and cement matrix (at zero porosity). 2.1 Dilute inclusion model The dilute inclusion model was introduced by Eshelby. He gives (Eshelby, 1957) the overall elastic moduli of a material containing a dilute dispersion of ellipsoidal inclusions within a uniform strain. This was accomplished by using a set of imaginary cutting, straining and welding operations. The pores are assumed to be spherical and diluted at small volumetric concentration, in the matrix. The expressions of effective shear and bulk moduli are as follows: K hom = K0 (1 + Ap)−1 ;

G hom = G0 (1 + Bp)−1 (2)

where p is the porosity, A and B are constants depending on phases’ elastic properties such that: A= α=

K1 − K0 ; (K0 − K1 )α − K0 3K0 ; 3K0 + 4G0

β=

B=

G 1 − G0 (G0 − G1 )β − G0

2 3K0 + 6G0 5 3K0 + 4G0

It can be noted that the interaction between the inclusions is neglected, thus expressions above are valid only for low porosities, which represent the inclusion’s volume fraction. 2.2 Hashin-Shtrikman bounds Hashin and Shtrikman (Hashin, 1960, Hashin & Shtrikman, 1963) gave bounds for the effective shear and bulk moduli of multiphase composite using variational principles in elasticity. The inclusions are supposed to be spherical. The effective bulk K hom and shear G hom moduli are comprised between lower hom and upper bounds values denoted respectively (Klow , hom hom hom Kup ) and (Glow , Gup ) such that: hom hom < K hom < Kup ; Klow

hom hom Glow < G hom < Gup (3)

Where: hom Klow = K1 + hom = K0 + Kup

1 K0 − K1 1 K1 − K0

1−p +

3p 3K1 + 4G1

+

3(1 − p) 3K0 + 4G0

p

;

and hom = G1 + Glow hom = G0 + Gup

1 G1 − G0

1−p

;

+

6(K1 + 2G1 )p 5G1 (3K1 + 4G1 )

+

6(K0 + 2G0 )(1 − p) 5G0 (3K0 + 4G0 )

p

It can be observed that for dry material, the lower Hashin bound are equal to zero. This is due to the fact that elastic properties of air, which is considered as the material of the pore phase, are null. The range between Hashin-Shtrikman bounds becomes too wide to be of any practical interest when the difference between the phases’ moduli is very large. Several authors showed that the Hashin-Shtrikman bounds are considerably tighter than the Voigt and Reuss bounds (Kaczmarek & Goueygou, 2006, Watt et al. 1976). 2.3 Kuster-Toksöz model Kuster and Toksöz derived (Kuster & Toksöz 1974a,b) a multiphase model for porous material by considering a process of wave propagation in an inhomogeneous elastic material. In the Kuster-Toksöz model, the effective bulk and shear modulus of the composite material are given by:   K0 K0 + 43 G0 + 34 G0 (K1 − K0 ) α 12 p K hom = (4) K0 + 43 G0 − (K1 − K0 )α 12 p G hom = ζ0 =

G0 (G0 + ζ0 ) + (G1 − G0 ) ζ0 β 12 p ; G0 + ζ0 − (G1 − G0 ) β 12 p G0 9K0 + 8G0 6 K0 + 2G0

(5)

Coefficients α 12 and β 12 depends on the aspect ratio of pores and are function of the elastic moduli of both matrix and inclusion phases (Berryman & Berge, 1996). It can be noted that for porous cement paste with spherical pores (for all saturation states), the KusterToksöz approximations coincide with the upper Hashin bounds. 2.4 Mori-Tanaka model The non-interaction assumption (between the composite phases) considered in Eshelby model is unrealistic. Thus, involving slight modifications in the dilute model, Mori and Tanaka (Mori & Tanaka, 1973) carried out a generalized formulation for the effective moduli of a composite. Mori-Tanaka suggested the following expressions (Benveniste, 1987, Berryman & Berge, 1996): K hom = K0 +

174

1 G0 − G1

(K1 − K0 ) α 12 p (1 − p) + α 12 p

(6)

G hom = G0 +

(G1 − G0 ) β 12 p (1 − p) + β 12 p

(7)

where α 12 and β 12 are expressions that depends on the aspect ratio of pores and the elastic moduli of phases (Berryman & Berge, 1996). These expressions are the same as those presented in Kuster-Toksöz model. It is to notice that for spherical inclusion, and for all saturation degree, the Mori-Tanaka approximations coincide with the upper Hashin bounds. In the particular case of two-phase porous material with spherical pores, the effectives moduli given by Kuster-Toksöz, Mori-Tanaka and upper Hashin bound coincide. 3 EXPERIMENTAL SETUP In order to obtain various porosity values, cement paste samples were prepared with three different water/ cement (w/c) ratios, namely 0.3, 0.4 and 0.5. The cement paste used is made up of cement CPA CEM I 52.5. The samples were either dry or fully saturated. Porosity was measured by the gravity method (AFPC-AFREM 1977), using vacuum saturation. Porosity, p, is then determined using the following formula: p=

Msat − Mdry ρw V

(8)

where ρ w is the unit mass of water, V is the volume of the sample, Mdry and Msat denote, respectively, the weight of the dried and fully saturated sample. Broadband ultrasound spectroscopy (Eggers & Kaatze 1996) was used to obtain ultrasonic parameters of the materials (longitudinal and transverse velocities). Here, this technique is applied in pulsed mode, using a conventional through-transmission setup with a pair of compression and shear wave contact transducers (Lafhaj et al. 2006, Ould Naffa et al. 2002). The initially emitted ultrasonic pulse is wideband, with a central frequency of 500 kHz. The pulse velocity is obtained from: VL,T =

2e t

(9)

where e is the thickness of the sample and t is the time delay between the first and the second received signals. 4 RESULTS AND DISCUSSION Each sample of cement paste was subjected to the ultrasonic tests described previously. For each w/c ratio and water saturation state, the presented result is an average of measurements performed on three samples taken from the same core.

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Table 1. Dynamic cement paste elastic properties at zero porosity. State

Bulk (MPa)

Shear (MPa)

Dry Saturated

28857 28393

13878 15635

In order to evaluate the micromechanical models, elastic moduli of the cement paste with diminishing pore volume is needed. The latter are determined by a linear regression of the experimental data describing relationships between dry and fully saturated modulus and porosity. The zero porosity parameters were computed and obtained data are given in Table 1. Dry pores are considered as voids and thus both bulk and shear moduli are null. On the other hand, the fully saturated samples presented full water pores with null shear modulus and with 2.2 GPa bulk modulus. The bulk and shear moduli of dry and fully saturated cement paste were evaluated using the above micromechanical models based on two-phase formulations and assuming spherical inclusions (dry and fully saturated pores). The obtained results are presented in Figures 1, 2. Figures 1, 2 show the variation of normalized bulk and shear moduli for the two saturation states. The micromechanical and experimental elastic properties decrease when porosity increases. Bulk and shear moduli estimated by the dilute inclusion model, fit the data with a good agreement for low porosities, up to 20% (Fig. 1b). But, in general, dilute inclusion model overestimates the moduli, especially for higher porosity samples. Such result was predicted as this model assumes low inclusion fraction. It can be noted that Hashin-Shtrikman upper bound curve coincides with the Kuster-Toksöz and MoriTanaka models, which assume spherical inclusions. Such result was predicted from the formulation of the three models as noted previously. According to Figures 1, 2, the Hashin-Shtrikman upper bound, the Kuster-Toksöz and Mori-Tanaka models present a general good agreement with the experimental determined bulk and shear moduli. This agreement is observed for all the porosity ranges, even relatively high porosity (around 35%). This is due to the fact that these models take into account the interaction between inclusions. The discrepancy between micromechanical and experiment bulk moduli, remarked in figure 1b, would be caused by the effect of capillary pressure present in the saturated state and not considered when modeling the saturated cement paste. Comparison between Figure 1a and 1b shows that the measured and theoretical dynamic bulk modulus varies significantly between the dry and the fully saturated states. For instance, the measured bulk modulus

Normalized saturated bulk modulus (Knom/K0)

Normalized dry bulk modulus (Knom/K0)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(a)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(b)

Normalized saturated shear modulus (Ghom/G0)

Normalized dry shear modulus (Ghom/G0)

Figure 1. Normalized experimental and computed dynamic bulk modulus vs porosity (a) dry (b) saturated. • Experiments Eshelby _ _ _ Hashin upper bound, Mori-Tanaka, Kuster-Tuksöz. 0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(a)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(b)

Figure 2. Normalized experimental and computed dynamic shear modulus vs porosity (a) dry (b) saturated. • Experiments Eshelby _ _ _ Hashin upper bound, Mori-Tanaka, Kuster-Tuksöz.

increases by 13.3% to 36.82% between the dry and the saturated samples. However, the dynamic shear modulus (Fig. 2) presents a little increase between the dry and saturated states for both measured and theoretical values. These results were predicted by theoretical micromechanical formulations. In fact, for the case of shear modulus, the pores’ shear modulus was considered, for the both saturation cases, to be null. So, the shear modulus was expected to be independent from the saturation state. While, the pores’ bulk modulus was considered to be non-null for the case of saturated cement paste and null for the dry one.

5 CONCLUSION In this paper, the relationship between homogenized elastic parameters (bulk and shear) of a dry and fully

176

saturated cement paste and porosity was investigated. Transverse and longitudinal pulse velocities were used to determine the experimental dynamic elastic parameters. Four models were used to express theoretically the bulk and shear parameters as a function of porosity. The adopted micromechanical models considered cement paste as two phase material: the matrix (cement paste) and spherical inclusions (pores). Comparisons showed that the relationship between elastic moduli and porosity is correctly represented by dilute inclusion model for low porosity and by the three other models for higher porosities. The large scatter observed for the case of dilute inclusion model in high porosity is due to the fact that this model assumes no interaction between inclusions and that is defined for low porosities. The presented work, dealt with the particular case of two-phase composite with spherical inclusion. Considering spheroidal inclusions with different

aspect ratio can influence the elastic moduli predictions. Considering micromechanical models that depend on inclusions’ aspect ratio is the subject of ongoing works. REFERENCES AFPC-AFREM 1977. Concrete durability: determination of apparent density and water accessible porosity: 121–125. Benveniste, Y. 1987. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6: 147–157. Berryman, J.G. & Berge, P.A. 1996. Critique of two explicit schemes for estimating elastic properties of multiphase composites. Mechanics of Materials 22, pp. 149–164. Brown, A.E. 1997. Rational and summary of methods for determining ultrasonic properties of materials. Report at Lawrence Livermore national laboratory, California. Eggers, F. & Kaatze, U. 1996. Broad-band ultrasonic measurement techniques for liquids. Measurement Science & Technology 7: 1–19. Eshelby, J.D. 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 241: 376–396. Hashin, Z. 1960. The elastic moduli of heterogeneous materials. Technical Report 9 Office of Naval Research. Hashin, Z. & Shtrikman, S. 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. and Phys. of Solids 11: 127–140.

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Hill, R. 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11: 357–372. Kaczmarek, M. & Goueygou, M. 2006. Dependence of elastic properties of materials on their porosity: Review of models. Journal of Porous Media 4: 335–355. Kuster, G.T. & Toksöz, M.N. 1974a. Velocity and attenuation of seismc waves in two-phase media: Part I. Theoretical formulations. Geophysics 39: 587–606. Kuster, G.T. & Toksöz M.N. 1974b. Velocity and attenuation of seismic waves in two-phase media: Part II. Experimental results. Geophysics 39: 607–618. Lafhaj, Z., Goueygou, M., Djerbi, A. & Kaczmarek, M. 2006. Correlation between porosity, permeability and ultrasonic parameters of mortar with variable water/cement ratio and water content. Cement and Concrete Research 36(4): 625–633. Maxwell, J.C. 1873. A treatise on electricity and magnetism. Clarendon Press, Oxford. Mori, T. & Tanaka, K. 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21: 571–574. Nolen-Hoeksema, R.C. 2000. Modulus–porosity relations, Gassmann’s equations, and the low-frequency elasticwave response to fluids. Geophysics 65: 1355–1363. Ould Naffa, S., Goueygou, M., Piwakowski, B. & BuyleBodin, F. 2002. Detection of chemical damage in concrete using ultrasound. Ultrasonics 40: 247–251. Watt, J.P., Davies, G.E.R. & O’Connell, J. 1976. The elastic properties of composite materials. Reviews of geophysics and space physics 14(4): 541–563.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

The simulation of microcracking and micro-contact in a constitutive model for concrete I.C. Mihai & A.D. Jefferson Cardiff University School of Engineering, UK

ABSTRACT: The paper describes a recently developed model for concrete which is based on micromechanical solutions. The model employs solutions of a material with a matrix phase, spherical inclusions, penny-shaped cracks and combines these with a rough surface contact sub-model. The components of the two-phase composite are modelled using the Eshelby inclusion solution and its properties are derived using the Mori-Tanaka scheme. This approach allows the model to predict the development of tensile stresses within the mortar phase under uniaxial compression and thus the model is able to simulate compressive splitting cracks in a natural way. A novel aspect of the model is the inclusion of a multi-component rough contact sub-model which aims to represent the surface of a crack, and associated post-cracking contact behaviour, at three scales; namely those appropriate to hardened cement paste, fine aggregate and coarse aggregate. An attempt to derive the parameters for the three components from micrographs of cracked concrete is also described. The inclusion of the embedded contact sub-model allows the model to simulate the dilatant behaviour of concrete subject to compression with reasonable accuracy. Finally, the paper presents a consistent formulation for the constitutive tangent matrix and a number of examples which describe predictions for uniaxial and biaxial compressive and tensile loading paths. 1 INTRODUCTION The attainment of a robust and accurate constitutive model for concrete remains a goal for researchers in the field. A number of macroscopic phenomenological models based on damage, plasticity and plasticdamage theories have been developed over the years (Comi & Perego 2001, Prisco & Mazars 1996, Este & Willam 1994, Feenstra & de Borst 1995, Grassl et al. 2002). Many of these models have been implemented in FE codes and have been used effectively. However, some such models use parameters that are difficult to establish and no one model is able to fully describe and predict the complete range of the complex behaviour of concrete. Considerable work has been undertaken in recent years to develop micromechanical models as an alternative approach in order to overcome the drawbacks experienced with the phenomenological models. Pensée et al. (2002) formulated an anisotropic damage model based on a micromechanical solution for an elastic solid containing penny-shaped micro-cracks and an energy release rate-based damage criterion that includes crack closure effects. Gam-barotta (2004) proposed an anisotropic friction-damage model using the solution of an elastic body containing plane cracks. More recently, Zhu et al. (2008, 2009) developed an anisotropic model based on the classic Eshelby inclusion solution and Ponte-Castaneda and Willis homogenization scheme that takes into account the

179

interaction and spatial distribution of microcracks and unilateral effects. A thermodynamics framework is used to derive the damage evolution law which is then coupled with frictional sliding on closed cracks based on a Coulomb friction criterion. The model presented in this paper is based on the work of Jefferson & Bennett (2007, 2009) and employs micromechanical solutions in an attempt to capture the macroscopic behaviour of concrete by modelling the physical phenomena that take place at microscale. 2 GENERAL PRESENTATION OF THE MODEL The model utilizes an Eshelby-based framework to simulate a two-phase composite comprising a matrix (m) that represents the mortar and spherical inclusions () that represent the coarse aggregate particles. The composite incorporates randomly distributed pennyshaped microcracks with rough surfaces on which stress can be recovered. It is assumed that the microcrack initiation and propagation takes place in the matrix phase. Figure 1 illustrates the basic concepts of the model. The theoretical aspects of the two-phase composite as well as the solutions for microcracking and stress recovery on crack surfaces that regain contact are described in detail in Jefferson & Bennett (2007, 2009). However, for completeness a summary is given below.

Figure 1.

Microcracking and rough contact concepts. Figure 2.

Reference system for a penny-shape microcrack.

2.1 Two-phase composite solution Eshelby based micromechanical solutions and the Mori-Tanaka averaging method for a non-dilute distribution of inclusions (Nemat-Nasser & Hori 1999) are employed in order to derive the equivalent homogeneous elastic tensor Dm . Similarly, a fourth-order tensor Wm  that links the matrix and average stresses is obtained. σ¯ = Dm : (¯ε − ε a )

all directions (integrating over a hemisphere) and considering that the crack density parameter is a function of orientation f = f (θ, ψ): ⎛

⎜1 εa = ⎝ 2π

(1)

σ m = Wm : σ¯

(2)

2π

π 2

⎞

⎟ NT : Cα : N f (θ, ψ) sin(ψ)dψdθ⎠ : σ¯

(4)

Cα contains the elastic compliance terms in Equation 3 and N is the transformation matrix (Jefferson 2003). 29 sample directions corresponding to McClauren integration rule are used in numerical simulations in order to evaluate the integral over a hemisphere.

where Dm = (f D : T + fm Dm ) : (f T + fm I4s )−1 , Wm = Dm : (f D : T + fm Dm )−1 · fm and f = volumetric proportions of the matrix material and of the inclusions, noting that fm + f = 1. T = (I4s + S : A ), A = [(D − Dm ) : S = Dm ]−1 ; (Dm − D ), S = Eshelby tensor for spherical inclusions (Nemat-Nasser & Hori 1999), D and Dm = elasticity tensors for the inclusion and matrix phase respectively and I4s = fourth order identity tensor. σ¯ and ε¯ = average or far-field stress and strain tensors respectively and εa = total additional strain due to penny-shaped microcracks with random orientation.

2.3 Cracking stress and damage evolution Using the Mori-Tanaka averaging scheme for a nondilute distribution of inclusions throughout all stages of damage leads to an over brittle post peak response in compression because if the matrix strength is reduced in a direction lateral to a compressive principal axis, the formulation leads to a reduction in strength in the compressive direction, as explained more fully in Jefferson and Bennett (2009). Hence, a cracking stressed is introduced which ensures that initial cracking depends on the matrix stress whereas the latter stages are governed by the average stress as follows:

2.2 Additional strain due to penny-shaped microcracks First, the additional strains generated by a dilute set of microcracks that have the same orientation, given by the normal vector r = [r s t]T (Fig. 2), are written in terms of a crack density parameter f employing the classical solution of Budiansky and O’Connell (1976):

sα = N :  : σ¯

(5)

where  is a tensor that provides the transition from the matrix to the average stress.  = rWm + 1(−r)I4s ,

⎡ ⎤ ⎤ σ¯ rr εαrr 2) 16(1 − ν ⎢ 4 σ¯ ⎥ ε α = ⎣γαrs ⎦ = f ⎣ 2−ν rs ⎦ = f Cα : s¯ 3E γαrt 4 σ¯ ⎡

2−ν rt

 

ξ −εtm

(3)

Next, the total additional strain is calculated by transforming and summing the contributions from

where r = e− 7εtm is the transition function. ζ = effective local strain parameter and εtm = strain in the matrix at uniaxial damage. Expressing damage in terms of the crack density parameter of Budiansky and O’Connell, f proved to be rather inconvenient and it was found to be more

180

advantageously characterized by a one-dimensional damage parameter ω. For the undamaged state ω = 0 whereas for the fully damaged state ω = 1. The equivalence between f and ω is described by: f Cα =



 ω CL 1−ω

where CL =

1 E



(6)

1 0

0

4 2−ν

0 0

0

0

4 2−ν



. This leads to the relation-

ship between the local stress tensor and the equivalent local strain tensor given by: s = (1 − ω)DL εL

(7)

where DL = C−1 L · εL incorporates the elastic strain and the additional strain due to microcracking εL = εLe + εα . The damage evolution law ω = ω(ζ ) has the same expression in all directions and is an exponential function in terms of the effective local strain parameter: ω =1−

εm e ζ

tm −5 εζ −ε 0 −εtm

(8)

where ε0 = uniaxial strain at full damage. The damage function that governs the effective local strain parameter ζ includes both normal and shear components of the effective local strain vector εL : 2 =0 F = γL2 − q 2 ζ 2 − (q 2 + µ2 )ζ εLrr + µ2 εLrr

(9)

The effective local strain εL can be written in a similar fashion as the cracking stress of Equation 5, considering the transition function as well as the stress and strain transformation matrices: εL = rCL : s∗m + (1 − r)Nε : ε¯

The consequence of the use of an effective or average matrix stress is that microcracks may initiate anywhere in the matrix phase. The authors are currently examining the possibility of obtaining a better estimate for the cracking stress by assuming that microcracks first appear in the interfacial transition zone between aggregate particles and the hardened cement paste. A framework that employs the so called exterior point Eshelby solution is proposed in order to achieve this and the effective matrix stress s∗m in equation (10) can be replaced accordingly by: s∗m (x) = N : Dm : [I4s − SE (x) : (A + S )−1 ] : (f T + fm I4s )−1 : ε¯

(12)

where SE (x) = the exterior point Eshelby tensor and A is a fourth order phase-mismatch tensor. This work is currently under study and will be addressed in detail in a forthcoming journal paper. 2.4 Rough crack contact model—Stress recovery The rough crack contact model implemented at this stage is based on the one used in the Craft concrete model (Jefferson 2003). The motivation behind crack closure model is that contact can be regained with normal and shear movement and consequently stress can be recovered across rough crack surfaces, as indicated by experimental evidence. The main principles are presented in Figures 4 and 5. The asperities of the crack surface are represented as right circular cones. The degree of roughness of the crack surface is characterized by parameter mg which also defines the slope of the interlock contact surface. Mathematically, rough crack contact is introduced by splitting the local stress tensor into two components: an average stress on undamaged material and a

(10)

∗ where sm = effective matrix stress obtained when applying the Mori-Tanaka averaging scheme:

s∗m = N : Dm : (f T + fm I4s )−1 : ε¯

Figure 3.

(11)

Figure 4. Rough crack with elastic region either side showing the angle of contact surface.

Damage surface.

181

relationship is obtained σ¯ = Ds : ε¯ , where    1 NT : Ccα : Ds = I4s + Dm 2π 2π

π 2

N :  sin(ψ)dψdθ

−1

Dm

(17)

3 PRESENT DEVELOPMENTS Figure 5.

Contact regions in relative displacement space.

recovered stress on microcracks in contact: s − su + sr = [(1 − ω)DL : ε L ] + [Hf (εL )ωDL : g] (13) The recovered stress depends on the state of damage—through the damage parameter ω—and on the state of contact—through the embedment strain g. g =  : εL

(14)

where  is a matrix that depends on the state of contact, as follows:  = 0 if crack surfaces are not in contact (open region).  = g if crack surfaces are in contact (interlock region).  = I if are in fully closed region.

The expression of the contact matrix in the interlock region was demonstrated to be: 1 g = 1 + mg2



∂φint ∂ε L



∂φint ∂ε L

T

∂ 2 φint + φint ∂ε 2L

 (15)

 where φint (εL ) = mg εLrr − εL2 rs εL2 rt is the mathematical expression for the interlock contact surface. Hf is a shear contact reduction function that accounts for the fact that the potential for shear transfer reduces with increasing crack opening: Hf = e−ηe , where −εtm ηe = εLrr εrange · εrange defines the strain range on which function stress recovery mechanism is active. Making use of Equations 13 and 14 and removing the elastic compliance, the added compliance with incorporated contact is obtained:   Ccα = [(1 − ω)I2s + Hf ω]−1 − I2s · CL

(16)

Rearranging Equation 1 and making use of Equations 4, 5, 6 and 16 the average stress—average strain

182

Present investigations include exploration of potential improvements of the contact model, and the transition function. Also, the use of the so called exterior point Eshelby solution for improving the initial cracking criterion as well as the derivation of a consistent formulation for the constitutive tangent matrix has been examined. The present paper focuses on the developments of the contact model and on the constitutive tangent matrix formulation. 3.1 Multilevel contact The use of non-destructive and non-invasive techniques has led to progress in understanding the process of microcrack formation, growth and propagation. Several assumptions regarding crack evolution, supported by experimental evidence provided by the aforementioned techniques, are exerted in this model. The interfacial transition zone between hardened cement paste and aggregate particles, fine and coarse, is generally considered to be the weakest link in normal concrete (van Mier 1997) and therefore microcracks are presumed to form initially at the cement—aggregate interface. With the increase of the macrostrain they link with neighbouring cracks through the hcp following paths that go around, rather than through, fine and coarse aggregate particles (Avram et al. 1981). The process eventually results in macrocrack formation, frequently referred to as damage localization. Following the investigation of micrographs (van Mier 1997, Avram et al. 1981, Mouret et al. 1999, Nichols & Lange 2006, Elaqra et al. 2007) it has been concluded that the roughness of the crack surface varies at different stages of damage. Stress recovery on crack surfaces that regain contact depends upon surface tortuosity through parameter mg . Thus, the contact model has been enlarged in order to capture the variation of the crack surface unevenness. For this, three contact components have been introduced at different scales each of them being linked to material constituents as follows: the first component characterizes microcracks in hcp, the second component is related to the crack surface in the vicinity of fine aggregate particles and the third component characterizes the crack surface around coarse aggregate particles.

int (u, v) = mg u − |v|

(18)

where u = crack opening and v = shear displacement at which contact is regained. The linear function of Jefferson reasonably matches, in a particular relativedisplacement range, the regression analysis-based relation of Walraven and Reinhardt (1981) that in turn fits experimental results with considerable accuracy: τ =−

fcu + [1.8u−0.80 + (0.234u−0.707 − 0.20) · fcu ] · v 30 (19)

where τ = shear stress, fcu = compressive strength. In the multi-component contact model each component is active on a certain relative-displacement range, given in the exponent of the shear contact reduction function. The values of mg from Table 1 generate a sensible match for Walraven and Reinhardt’s function in those specific ranges (Figs. 6, 7). Replacing the single-component with the multicomponent contact model the added compliance given in equation (16) becomes: ⎤ ⎡ −1 3  pk Hfk k − I2s⎦ · CL Ccα = ⎣ (1 − ω)I2s + ω

Fine aggregate mg=0.35

shear strain

0.00016 0.00012 0.00008 0.00004 0 0

0.0002

0.0004

0.0006

0.0008

normal strain

Walraven&Reinhardt

Contact model

Figure 6. Contact surface for fine aggregate component.

Coarse aggregate mg=0.22 0.0004

shear strain

A set of three parameters, namely, proportion p, slope of the interlock surface mg and exponent ηe of the shear transfer reduction function defines each contact component. Following a parametric study and careful examination of micrographs the parameters for the three components were proposed (Table 1): A rigorous study of micrographs indicated that the surfaces of fine micrographs in hcp are, at a microscale, locally approximately smooth (van Mier 1997, Avram et al. 1981). This observation explains a value of mg that tends to infinity for the first contact component. As cracks propagate and link with neighbouring cracks they form sinuous paths that gently bridgeover fine aggregate particles increasing the roughness of the crack surface accordingly (i.e. mg decreases). Around coarse aggregate particles the tortuosity is further increased due to particle debonding. The theoretical development of the contact model is presented by Jefferson (2002) in which the author derives the expression of the interlock surface in the form of a linear function:

0.0003 0.0002 0.0001 0 0

0.001

0.002

0.003

normal strain

Walraven&Reinhardt

Contact model

Figure 7. Contact surface for coarse aggregate component.

3.2 Constitutive tangent matrix The consistent formulation for the constitutive tangent matrix is derived below for the case of a two-phase model incorporating micromechanics based damage evolution and crack closure. The tangent constitutive matrix is subsequently used in a New-ton-Raphson method based algorithm which significantly improves the convergence properties in comparison with those of a direct iteration method. For the aforementioned case the constitutive relationship can be written as:

k=1

(20)

Table 1. Component

mg

εrange

Proportion p

Hcp Fine aggregate particles Coarse aggregate particles

∞ 0.35 0.22

12εtm 25εtm ε0

0.2 0.5 0.3

σ¯ = Ds : ε¯

 −1 d T where Ds = I4s +Dm ni=1 Ni Ccαi Ni i wi · Dm is the secant constitutive matrix and Ccαi is given by Equation 20. Differentiating Equation 21 gives: d σ¯ = Ds : d ε¯ +

183

(21)

∂Ds d ε¯ : ε¯ ∂ ε¯

(22)

For expediency, the parameter r was kept constant within each strain increment. the following  d Using NiT Ccαi Ni i wi , B = notations, A = I4s + Dm ni=1  (1 − ω)I2s + ω 3j=1 pj Hfj j and the expression of the derivative of an inverse

∂Y−1 ∂x

−1 = −Y−1 ∂Y ∂x Y :

∂Ds ∂A ¯ −1 Dm d ε¯ = −A−1 d εA ∂ ε¯ ∂ ε¯

(23)

and:    nd   ∂A ∂Bi NiT −B−1 d ε¯ = Dm d ε¯ · B−1 i · i CL Ni i wi ∂ ε¯ ∂ ε¯ i=1

(24)

For each sample direction i = 1..nd , mined using the chain rule:

∂B ∂ ε¯

is deter-

4 NUMERICAL RESULTS A number of single point strain path simulations were undertaken with the proposed model. Numerical predictions for uniaxial tension, uniaxial compression as well as a biaxial failure envelope are shown below (Figs. 8–12). The values of the material parameters used for the simulations are (Case 1): Em = 32000 N/mm2 , E = 49000 N/mm2 , νm = 0.1, ν = 0.28, f = fm = 0.5, q = 3.67, µ = 1.76, ε0 = 0.003, ftm = 1.5 N/mm2 . Figure 9a indicates that the three component contact model significantly improves the predicted ductility relative to the previous version in which an average degree of the crack surface roughness was assumed at all stages of damage (single component contact model). The model is able to capture the post-peak dilatancy observed in experiments (Figs 10a, b, Case 2).

∂B ∂ω ∂ζ ∂ε L ∂B · d ε¯ = · · d ε¯ ∂ ε¯ ∂ω ∂ζ ∂ε L ∂ ε¯

4

3  ∂B ∂Hf j ∂ε L d ε¯ + · · ∂Hf j ∂ε L ∂ ε¯ 3  ∂B ∂j ∂ε L · · d ε¯ ∂ ∂ε L ∂ ε¯ j j=1

xx-stress (MPa)

j=1

+

(25)

It can be proven that the differential of the embedded strain in Equation 14 is:

Column k (k = 1, 2, . . . 6) of matrix Dadt is hence obtained: (28)

0 0

0.001

0.002

0.003

Figure 8.

Predicted uniaxial tensile response, Case 1.

single component contact Case 1 3-component contact Case 1 40 35 30 25 20 15 10 5 0 0

Equation 22 becomes: d σ¯ = (Ds + Dadt ) : d ε¯

1

xx-strain

∂A−1 ∂A ∂A d ε¯ k = −A−1 · d ε¯ k · y = −A−1 · · y · d ε¯ k ∂ ε¯ k ∂ ε¯ k ∂ ε¯ k (27)

(k) Dadt

2 1.5

(26)

∂ Hence, ∂ε = 0 and the second summation in L Equation 24 vanishes. The partial derivatives can be straightforwardly derived. The consistent tangent matrix is then formed in a ‘‘column by column’’ manner so that dε¯ is extracted element by element as shown below. Denoting y = A−1 · Dm : ε¯ one can obtain:

∂A ·y = −A−1 · ∂ ε¯ k

3 2.5

0.5

xx-stress (compression +ve) MPa

dg =  : dε L

uniaxial tension

3.5

0.0005 0.001

0.0015

0.002 0.0025

0.003

xx-strain (compression +ve)

(29) Figure 9a. Predicted response for uniaxial compressive paths, Case 1.

Consequently, Dt = Ds + Dadt .

184

3-component contact Case1

xx-stress (compression +ve) MPa

single component contact Case 1

40 35 30 25 20 15 10 5 0 -0.0008

-0.0004

0

0.0004

volumetric strain

xx-stress (compression +ve) MPa

Figure 9b. Predicted dilatancy for uniaxial compressive paths, Case 1.

the model. Therefore, the authors acknowledge that the current model is not complete and believe that the incorporation of microplasticity will address its shortcomings. This work is underway and the complete model will be the subject of a forthcoming publication. The introduction of the multi-component contact sub-model generates a biaxial failure envelope (Fig. 11) that favourably matches the experimental data of Kupfer (1969).

40

sx/sy=1/0 Case 2

35

Figure 11. Predicted biaxial failure envelope, Case 1.

30 25 20 15 10 5 0 0

0.001

0.002

0.003

0.004

5 CONCLUSION

xx-strain (compression +ve)

xx-stress (compression +ve) MPa

Figure 10a. Predicted uniaxial compressive response, Case 2.

-0.0005

40 35 30 25

sx/sy=1/0 Case2

20 15 10 5 0

The constitutive model presented in this paper is able to predict many of the key characteristics of the complex behaviour of concrete under uniaxial and biaxial tension and compression. The expanded rough crack contact model enables the model to improve the predicted ductility. On-going work on the model involves the development of a crack criterion based on an exterior point Eshelby solution as well as the introduction of microplasticity on the crack planes.

REFERENCES 0

0.0005

0.001

volumetric strain

Figure 10b. Predicted dilatancy for uniaxial compressive path, Case 2.

However, it proved necessary to modify some of the material parameters in order to obtain such a response. Also, following a parametric study it was observed that a reasonable ductile and dilatant response could not be predicted simultaneously with this version of

185

Avram, C. Facaroiu, I. Filimon, I. Mirsu, O. & Tertea, I. 1981. Concrete strength and strains. Developments in civil engineering. Elsevier. Budiansky, B. & O’Connell, R.J. 1976. Elastic moduli of a cracked solid. International journal of solids and structures. 42: 81–97. Comi, C. & Perego, U. 2001. Fracture energy based bidissipative damage model for concrete. International Journal of Solids and Structures. 38: 6427–6454. di Prisco, & M. Mazars, J. 1996. Crush-crack: a non-local damage model for concrete. Mechanics of Cohesive Frictional Materials.; 1: 321–347.

Elaqra, H. Godin, N. Peix, G. R’Mili, M. & Fantozzi, G. 2007. Damage evolution analysis in mortar during compressive loading using acoustic emission and X-ray tomography: effects of the sand/cement ratio. Cement and concrete research. 37: 703–713. Este, G. & Willam, K. 1994. Fracture energy formulation for inelastic behaviour of plain concrete. Journal of Engineering Mechanics ASCE. 120(9): 1983–2011. Feenstra, P.H. & de Borst, R. 1995. A plasticity model and algorithm for mode-I cracking in concrete. International Journal for Numerical Methods in Engineering. 38: 2509–2529. Grassl, P. Lundgren, K.G. & Gylltoft, K. 2002. Concrete in compression: a plasticity theory with a novel hardening law. International Journal of Solids and Structures; 39: 5205–5223. Gambarotta, L. 2004. Friction-damage coupled model for brittle materials. Engineering fracture mechanics. 71: 824–836. Jefferson, A.D. 2002. Constitutive modelling of aggregate interlock in concrete. International journal for numerical and analytical methods in geomechanics. 26: 515–535. Jefferson, A.D. 2003. Craft – a plastic–damage–contact model for concrete. I. Model theory and thermodynamic Considerations. International journal of solids and structures. 40: 5973–5999. Jefferson, A.D. & Bennett, T. 2007. Micro-mechanical damage and rough crack closure in cementitious composite materials. International journal for numerical and analytical methods in geomechanics. 31(2): 133–146. Jefferson, A.D. & Bennett, T. 2009. A model for cementitious composite materials based on micro-mechanical solutions and damage-contact theory. Computers and structures. In press.

Kupfer, H.B. Hilsdorf, H.K. & Ruch, H. 1969. Behaviour of concrete under biaxial stresses. Journal of ACI. 66(8): 656–666. Mouret, M. Bascoul, A. & Escadeillas, G. 1999. Microstructural features of concrete in relation to initial temperature—SEM and ESEM characterization. Cement and concrete research. 29: 369–375. Nemat-Nasser, S. & Hori, M. 1999. Micromechanics: Overall properties of heterogeneous materials. North-Holland. Nichols, A.B. & Lange, D.A. 2006. 3D surface image analysis for fracture modelling of cement-based materials. Cement and concrete research. 36: 1098–1107. Pensee, V. Kondo, D. & Dormieux, L. 2002. Micromechanical analysis of anisotropic damage in brittle materials. Journal of engineering mechanics ASCE. 128(8): 889–897. van Mier, J.G.M. 1997. Fracture processes of concrete. CRC Press. Walraven, J.C. & Reinhardt, H.W. 1981. Theory and Experiments on the Mechanical Behaviour of Cracks in Plain and Reinforced Concrete Subjected to Shear Loading. Heron. 26(1A). Zhu, Q.Z. Kondo, D. & Shao, J.F. 2008. Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: Role of homogenization scheme. International journal of solids and structures. 45: 1358–1405. Zhu, Q.Z. Kondo, D. & Shao, J.F. 2009. Homogenizationbased analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks. International journal for numerical and analytical methods in geomechanics. 33: 749–772.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Simulations of dynamic failure in plain and reinforced concrete with regularized plasticity and damage models Jerzy Pamin, Andrzej Winnicki & Adam Wosatko Faculty of Civil Engineering, Cracow University of Technology, Poland

ABSTRACT: The paper presents an overview of two enhanced constitutive models: Burzy´nski-Hoffman viscoplastic consistency model and scalar gradient damage model. They are used for the simulation of wave propagation in two-dimensional benchmark configurations: Brazilian split test and direct tension test. The influence of loading rate on the results is examined. 1 INTRODUCTION The paper is focused on a comparison of the performance of two regularized continuum models in the simulations of selected benchmarks of dynamic response. These models are: Burzy´nski-Hoffman viscoplastic consistency model (Winnicki et al. 2001) and scalar damage model enhanced with higher order strain gradients (Peerlings et al. 1996). For the latter model a coupling to plasticity can also be considered (de Borst et al. 1999). The numerical modelling of dynamic processes using continuum models equipped with localization limiters was for the first time fully covered in (Sluys 1992). Only for nonlocal or rate-dependent models a proper formation of standing localization wave can be reproduced and pathological mesh sensitivity of discrete solutions can be avoided. In the paper the employed models are briefly reviewed. The results of two standard wave propagation tests on plain and reinforced concrete are presented: Brazilian split of a cylinder and direct tension of a bar. Two- and three-dimensional finite element models are built, but the paper is limited to 2D configurations. Linear kinematic relations are assumed. The constitutive models have been implemented in the FEAP package (Taylor 2001). The standard Newmark scheme combined with the consistently linearized Newton procedure is used. The issue of failure prediction is discussed whereby the influence of loading rate is considered. The physical significance and regularization effect of the gradient-enhancement and viscosity are assessed.

idea proposed by (Wang 1997) for metals, named ‘‘viscoplastic consistency model’’. A similar approach was proposed by (Kleiber 1998) and later developed by (Heeres et al. 2002). In the considered model the yield function can expand or shrink depending on the current viscoplastic strain rate. The stress state remains on the yield surface and the consistency condition is invoked. There is no additional equation defining a viscoplastic multiplier and this approach is very close to the classical rate independent plasticity. Therefore, the established numerical algorithms like the closest point projection or the cutting plane algorithm (Simo and Hughes 1997) can easily be adapted. The Burzy´nski-Hoffman yield surface in its isotropic form is selected, since it was successfully employed in the analysis of concrete structures (Bi´cani´c et al. 1994; Pearce 1993). This yield surface has the form: f vp = 3J2σ + I1σ (fc − ft ) − fc ft = 0

(1)

where J2σ and I1σ are the usual stress invariants, fc and ft are the uniaxial compressive and tensile strengths, respectively. Two internal variables κc and κt are postulated which are both functions of the equivalent viscoplastic strain. They separately describe the material hardening/ softening in compression and tension, respectively. Moreover, two more internal variables ηc and ηt are assumed to determine the increase/decrease of compressive and tensile strengths due to the actual equivalent viscoplastic strain rate. Thus, the respective strengths are:

2 ENHANCED MATERIAL MODELS 2.1 Overview of Burzy´nski-Hoffman viscoplasticity

fc = fc (κc , ηc ) and ft = ft (κt , ηt )

The first employed model for concrete has been developed within the viscoplasticity theory and follows the

The rates of the internal variables depend on the current stress and the rates of internal variables

187

(2)

The dependence of fc on κc and ηc is formulated in a general way as:

κ and η: κ˙ c = gc (σ )κ˙

and

κ˙ t = gt (σ )κ˙

(3)

η˙ c = gc (σ )η˙

and

η˙ t = gt (σ )η˙

(4)

fc = fc′ Hc (κc )Sc (ηc )

where fc′ is the initial compressive strength. Functions Hc and Sc can for instance be specified as piecewise linear. Similarly, the actual tensile strength is computed as:

In the above equations gc and gt are scalar functions accounting for independent processes of damage in compression and tension. In turn, κ˙ is defined as an equivalent viscoplastic strain rate assuming work hardening, while η˙ depends on the time derivative of the viscoplastic strain rate: 1 κ˙ = − σ : ǫ˙ vp σ˜

and

η˙ =

1 σ : ǫ¨ vp σ˜

ft = ft′ Ht (κt )St (ηt )

1

ǫ˙ = ǫ˙ e + ǫ˙ vp

n : σ˙ +

and Hooke’s law is valid for the elastic part: σ˙ = E : ǫ˙

(7)

The viscoplastic flow is defined similarly to the classical associated plasticity: ˙ ǫ˙ vp = λn,

n=

∂f vp ∂σ

∂f vp ∂f vp ∂f vp ∂f vp κ˙ c + η˙ c κ˙ t + η˙ t = 0 ∂κc ∂ηc ∂κt ∂ηt (11)

Using Equations (3), (4), (5), (8) derivatives κ˙ c , κ˙ t , η˙ c , η˙ t can be expressed as:

(6)

e

(10)

Example functions, based on experimental data from (Kupfer 1973) and (Suaris and Shah 1985), are presented in (Winnicki 2007). In order to establish the viscoplastic multiplier λ˙ the consistency equation is used:

(5)

where σ˜ = (3J2σ ) 2 . It should be noted that in the course of loading the yield surface can change its shape due to the separate hardening/softening processes of the compressive and tensile strengths, but the surface remains convex at all times. It depends not only on the internal parameter κ, but also on the additional one η. Since the total value of η depends on the first derivatives of viscoplastic strains, the yield surface is rate dependent, i.e. it expands for higher and shrinks for lower viscoplastic strain rates. As a result, this model can correctly predict basic viscoplastic phenomena like creep and relaxation. The strain rate is decomposed into its elastic and viscoplastic parts:

(9)

(8)

˙ κ˙ t = gt g λ, ˙ η˙ c = gc g λ, ¨ η˙ t = gt g λ¨ κ˙ c = gc g λ, where g is equal to: g=

1 σ : n σ˜

188

(13)

Using the above relationships the consistency equa¨ tion can be written in a form depending on λ˙ and λ: n : σ˙ − hλ˙ − sλ¨ = 0

(14)

where h is the classical generalized plastic modulus and s is the generalized viscoplastic modulus. Due to the last term the consistency equation is no longer an algebraic equation for the viscoplastic multiplier, but a differential equation of the first order, to be solved for an appropriate initial condition. The generalized plastic and viscoplastic moduli are computed as: h = ac Sc hc + at St ht ,

The functions gc and gt from Equations (3)–(4) are selected according to the experimental evidence taking into account the influence of the damage process in compression on the concrete strength in tension and, conversely, the influence of the damage process in tension on the concrete strength in compression. In (Winnicki 2007) two options are considered: (1) gc = gt = 1 (damage is assumed to be an isotropic phenomenon) and (2) gc + gt = 1 (with extreme cases of dominant compressive stress gc = 1, gt = 0 and dominant tensile stress gc = 0, gt = 1).

(12)

s = ac sc Hc + at st Ht

(15)

where: hc =

dHc dHt dSc dSt , ht = , sc = , st = dκc dκt dηc dηt

(16)

In turn, coefficients ac and at are: ac = fc′ (ft − tr σ )gc g,

at = ft′ (fc + tr σ )gt g

(17)

When functions Sc and St are constant, their derivatives vanish and Equation (14) reduces to the form known from the classical rate independent plasticity.

It is discussed in (Winnicki 2007) how the material model parameters can be linked with fracture energies Gf t and Gfc which are the actual material properties (fib 1999; van Mier 1984; Vonk 1992) via the width of the localization zone. For the case St = Sc = 1, i.e. when the presented model is not a localisation limiter, the width of the localisation zone usually coincides with one row of finite elements. However, when viscous effects are active the width of localisation zone becomes larger and can only be estimated a posteriori.

The equivalent strain measure ǫ˜ can be defined in different ways. In this paper we employ the modified von Mises definition involving the first and second strain invariants, I1ǫ and J2ǫ , respectively, (de Vree et al. 1995) and depending on the ratio of compressive and f′ tensile strengths k = fc′ : t    (k − 1)I1ǫ 1 (k − 1)I1ǫ 2 12kJ2ǫ + ǫ˜ = + 2k(1 − 2ν) 2k 1 − 2ν (1 + ν)2 (21)

2.2 Summary of gradient damage model The simplest model of continuum damage mechanics which can describe elastic stiffness degradation in quasi-brittle materials is scalar damage. This means that one damage measure ω, which grows monotonically from 0 to 1 (Kachanov 1958), is a function of damage history parameter κ d and depends on the deformation of a body. For a material without any defect (no damage) the parameter ω equals to 0. For a material with a complete loss of stiffness (complete damage) we have ω = 1. Considering the damage evolution we distinguish the actual body with strains ǫ and stresses σ and its fictitious undamaged counterpart with stresses σˆ and strains ǫˆ . The fictitious counterpart represents the undamaged ‘‘skeleton’’ of the body, and the stresses σˆ acting on it are called effective. We adopt the postulate that the strains ǫ observed in the actual body and in its undamaged representation ǫˆ are equal (Lemaitre 1971; Simo and Ju 1987). Then, the real stress tensor σ and its effective counterpart σˆ are related by the above-mentioned parameter ω: σ = (1 − ω)σˆ

(18)

where: σˆ = E : ǫ e

(19)

here E is the elastic stiffness operator. The elastic strain tensor ǫ e is equal to the strain tensor ǫ when the standard elasto-damage model is considered, but the model can easily be coupled with a plastic behaviour of the undamaged ‘‘skeleton’’ (de Borst et al. 1999), and then ǫe = ǫ − ǫp. The strain equivalence is related with a loading function f d , also called damage activation function, defined in the strain space: f d (ǫ, κ d ) = ǫ˜ (ǫ) − κ d = 0

(20)

Here ǫ˜ is an equivalent strain measure. During the damage evolution the history parameter κ d is equal to the largest value of ǫ˜ reached in the loading history and obeys the standard loading/unloading conditions.

189

Although this definition introduces the sensitivity of the model to the sign of strains and allows for damage not only under tension but also under (biaxial) compression, the interaction between tensile and compres-sive action in concrete is not as well represented as in the Hoffman model. In order to improve the description, one would need to use an isotropic version of the damage model with two damage parameters degrading the tensile and compressive stiffness separately, see for instance (Mazars and Pijaudier-Cabot 1989; Comi 2001). It is assumed that κ d grows from damage threshold κo and damage ω asymptotically increases but never reaches 1. We define the damage growth function according to (Mazars and Pijaudier-Cabot 1989):  κo  d ω(κ d ) = 1 − d 1 − α + αe−η(κ −κo ) (22) κ The respective parameters η and α are responsible for material ductility and residual stress, respectively. The former parameter is thus related with concrete fracture energy Gf . The latter one prevents the complete loss of material stiffness and leads to a more stable numerical response. If unloading is considered, irreversible strains are usually observed in concrete. In this case the motivation for coupling the damage model with plasticity is substantial. A combination of a plasticity theory formulated in the effective stress space with the above damage theory formulated in the strain space is described for instance in (de Borst et al. 1999). In addition, the constitutive relations can incorporate a crack-closing projection operator which is important for cyclic loading and extensive stress redistributions, see e.g. (Pamin et al. 2003). Following (Peerlings et al. 1996), the damage evolution in the gradient-enhanced model is governed by the following damage loading function: f d (ǫ, κ d ) = ǫ¯ (˜ǫ (ǫ)) − κ d = 0

(23)

where the averaged (nonlocal) strain measure ǫ¯ satisfies the Helmholtz equation: ǫ¯ − c∇ 2 ǫ¯ = ǫ˜ .

(24)

The parameter c > 0 has a unit of length squared and is related to an internal length scale. It is assumed here to be constant, although, with some modifications in the formulation, it can be made a function of ǫ˜ or ǫ¯ (Geers 1997), which might be physically relevant.

Analogically, at integration points we decompose secondary fields, for example the stress:

2.3 FE implementation

σ (i+1) = σ (i) + σ

In this subsection Voigt’s notation is used. We consider a certain domain B, occupied by the material body, with boundary ∂B. The weak form of motion equations is the virtual work equation:   δuT ρ udV ¨ + δǫ T σ dV

Hence the constitutive relation is written in its incremental version:

B

B

=



B

δuT bdV +



δuT tdS

B

B

and

ǫ¯ = hT e

(27)

where N and h contain suitable shape functions. From the above interpolations the secondary fields ǫ and ∇ ǫ¯ are computed: ǫ = B a and ∇ ǫ¯ = gT e

(28)

where B = L N, gT = ∇hT , L is a differential operator matrix. After discretization, applied also for variations δu and δ ǫ¯ , Equations (25) and (26) must hold for any admissible δa and δe. The boundary value problem is linearized, hence at nodal points the increments of the primary fields from iteration i to iteration i + 1 are introduced according

190

(29)

(30)

σ = (1 − ωi )E ǫ − ω σˆ i

(31)

The increment of damage ω depends on the increment of averaged strain ¯ǫ :

ω =

In the ensuing two-field formulation averaged strain measure ǫ¯ is discretized in addition to displacements u. These primary fields are interpolated in the following way: u=Na

ai+1 = ai + a and ei+1 = ei + e

(25)

∂B

where the superscript T is the transpose symbol, ρ is the material density, b is the body force vector and t is the traction vector. The finite element algorithm for the Burzy´nskiHoffman viscoplastic model is similar to the classical rate-independent plasticity, i.e. linearization is performed and the weak form of incremental motion equations is discretized (Winnicki 2007). For the gradient damage model a two-field FE model is required. The weak form of Equation (24) is derived as follows. The variation of the averaged strain measure δ ǫ, ¯ Green’s formula and the natural boundary condition (∇ ǫ¯ )T ν = 0 are introduced to obtain:    δ ǫ¯ ǫ¯ dV + (∇δ ǫ¯ )T c∇ ǫ¯ dV = δ ǫ¯ ǫ˜ dV (26) B

to the following decomposition:



∂ω ∂κ d

i 

∂κ d ∂ ǫ¯

i

¯ǫ = G i hT e

(32)

In the Helmholtz equation the increment of equivalent strain measure ˜ǫ is computed from the interpolated displacement increment a:

˜ǫ =



dǫ˜ dǫ

i

ǫ = [sT ]i B a

(33)

Finally, we rewrite Equations (25) and (26) in a matrix form, so that the gradient damage formulation can be written as the coupled matrix problem (Peerlings et al. 1996):   Maa 0 a¨ 0 0 e¨    K Kae a f − fint + aa = ext (34) fǫ − f e Kea Kee e In the absence of damage growth Ka¯ǫ = 0 and the motion equations are uncoupled from the averaging equation. It is noted that no structural damping is included in the analysis, however both employed models involve energy dissipation, so they are capable of representing material damping. 3 SIMULATIONS FOR PLAIN CONCRETE—BRAZILIAN SPLIT TEST This section presents the results of simulations of the Brazilian split test on concrete specimen using the described regularized models. A comparison of simulations of the same configuration under static loading is presented in (Wosatko et al. 2009). The geometry of the Brazilian tests is based on papers (Feenstra 1993; Winnicki et al. 2001; Ruiz et al. 2000). Plane strain

conditions are analyzed. Due to double symmetry only a quarter of the domain is considered. If not stated otherwise, the load is applied through a stiff elastic platen. The platen is perfectly connected with the specimen. The fundamental material data for concrete are: Young’s modulus Ec = 37700 MPa, Poisson’s ratio νc = 0.15, density ρ = 2405 kg/m3 . For the platen Es = 10 · Ec and the other parameters are the same as for concrete.

3.1 Viscoplasticity simulations

2

2

1.5

1.5

1

St

Sc

In this section the cylinder radius is equal to 40 mm according to (Feenstra 1993; Winnicki et al. 2001). Standard eight-noded finite elements are used. The adopted material parameters are: initial tensile strength ft′ = 3 MPa, initial compressive strength fc′ = 30 MPa. Isotropic version of the model κt = κc is assumed. Linearly decreasing softening moduli are adopted with Hc decreasing from 1 to 0 for κc = 0.015 and Ht decreasing from 1 to 0 for κt = 0.001. In Figure 1 the employed piecewise linear functions Sc and St are shown. The load is assumed to grow linearly with time. Figure 2 presents the structural response for three different loading rates: (1) ‘‘ slow’’ dynamic process (vertical displacement of the platen equal to 0.2 mm is reached in 300 s), (2) ‘‘moderate’’ (300e-4 s) and (3) ‘‘fast’’ dynamic process (300e-5 s). The higher strain rates result in the larger maximum loads and

0.5 0

corresponding displacements. Also the post critical behaviour is less steep for the higher strain rate. Figure 3 presents contour plots for the total strain values (horizontal component ǫ11 , vertical component ǫ22 and internal variables κt = κc at the end of the loading process for the smallest strain rate (slow process) and the highest strain rate (fast process). The horizontal strain component is highly localised whereas the vertical component exhibits a rather distributed pattern. Damage is localised in the narrow vertical zone in the middle of the specimen forming a wedge at the top under the platen. For the higher strain rates the width of the localisation zone becomes larger which proves the viscoplastic regularization is active. A comparison for the fast loading between the dynamic test computed in the quasi static manner (i.e. without inertia effects using the standard Newton-Raphson procedure) and the full dynamic procedure using the Newmark algorithm for integration in time domain shows that inertia effects for the considered range of strain rates are negligible (Winnicki 2007).

(a) Slow, ∈11

(b) Slow, ∈22

(c) Fast, ∈11

(d) Fast, ∈22

1 0.5

0

0.2

0.4

0.6

0.8

1

0

0

(a) Function Sc

Figure 1.

0.2

0.4

0.6

0.8

1

(b) Function St

Material functions for viscoplasticity.

0.8 0.7

P/2 [kN]

0.6 0.5 0.4 0.3 fast rate medium rate slow rate

0.2 0.1 0

0

0.02

0.04

0.06

0.08 [mm]

0.1

0.12

0.14

Figure 2. Load-displacement plots for three different load rates (8-noded elements).

191

Figure 3. Contour plots of strains and internal variables for different loading rates.

p pi

ti Figure 4.

td

vertical displacement

[mm]

0

t

Coarse Medium Fine

0

Loading history for gradient damage simulations.

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(a) Displacement in time. 1000 0

192

Coarse Medium Fine

0

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(b) Velocity in time. 1e+09

acceleration a[ mm ] s2

Coarse

5e+08 0

Medium Fine

0

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(c) Acceleration in time.

Figure 5. History diagrams for gradient simulations—three different mesh densities.

damage

0 vertical displacement ν [mm]

The input data adopted in this section are similar to (Ruiz et al. 2000). The radius of the cylinder is 25.4 mm. The material data adopted for the scalar gradient damage model are as follows: internal length l = 4 mm, exponential softening with threshold κo = 0.0001195 (tensile strength ft′ = 4.53 MPa), ductility parameter η = 600 and to prevent the complete loss of material stiffness α = 0.98. The modified von Mises definition of damage loading function with parameter κ = 14.8 (compressive strength fc′ = 67 MPa) is selected. A linear-constant type of the pulse loading is adopted as shown in Figure 4, where for instant ti = 48 µs the traction intensity pi = 175 MPa is reached, then remains constant till time td = 100 µs and drops to 0. The elements with quadratic interpolation of the displacements, linear interpolation of the averaged strain measure and 2 × 2 Gauss integration are employed for three mesh refinements. In Figure 5 the displacement, velocity and acceleration in time are plotted in history diagrams for a point at the top of the platen. We notice differences for the final stage of deformation history, i.e. when damage is advanced and the specimen is close to failure. If we compare results for specimens with or without platen (Wosatko 2008) the character of diagrams and crack patterns is quite similar, so next aspect of computations is performed for the specimen without platen. Displacement history diagrams in Figure 6 are obtained for cases with different times when the load changes from linear to constant. The shorter time ti is the more rapidly failure follows. Hence, the largest loading rate when ti = 40 µs induces the most sudden response and displacements go to infinity very fast. In Figure 7 quite similar damage patterns are observed for each mesh. The plots are depicted for time t = 110 µs, so 10 µs after the load is removed. The splitting process in the central vertical zone in each case is present, while the damage pattern is a continuum representation of crack distribution. The split zones are observed in each contour plot of Figure 8, however for the case with ti = 40 µs damage is visible also at the top where the load is apllied.

velocity V [

mm ] s

3.2 Gradient damage simulations

ti = 56 µs ti = 48 µs ti = 40 µs

0

0.00005

0.0001 t [s]

0.00015

0.0002

Figure 6. Different loading rates—displacement history diagrams for gradient damage simulations.

4 SIMULATIONS FOR REINFORCED CONCRETE–DIRECT TENSION TEST The aim of this test is a two-dimensional analysis of tensile wave propagation in a bar not only made from plain concrete, like in (Pamin 2005), but also

(a) Coarse mesh.

Figure 7.

(b) Case ti = 48 µs, t = 90 µs.

(c) Case ti = 56 µs, t = 210 µs.

Damage patterns for different loading rates.

A

Figure 9.

(c) Fine mesh.

Damage patterns for three different meshes in instant t = 110 µs.

(a) Case ti = 40 µs, t = 65 µs. Figure 8.

(b) Medium mesh.

Dynamic direct tension test.

strengthened with reinforcement (Wosatko et al. 2006; Wosatko 2008). As shown in Figure 9, we consider a bar supported at the symmetry axes and loaded with a time-dependent normal traction at both ends. The reinforcement is located along the axis of the bar. The length of the bar is L = 250 mm, the width B = 60 mm. Figure 9 also shows the employed discretization, where the central zone is refined because of the expected localization. Eight-noded two-field gradient damage elements for concrete and elastic truss finite elements for the reinforcement are introduced. Plane stress with thickness t = 50 mm and full bond between steel and concrete are assumed. For the gradient damage model of concrete the material data are as follows: E = 18000 MPa, ν = 0.0,

ρ = 2320 kg/m3 . Exponential damage growth function is used with κo = 0.00188 (tensile strength ft′ = 3.4 MPa), α = 0.99 and η = 500. The modified von Mises definition of the equivalent strain is employed. For the internal length scale parameter 3 values, namely l = 2/4/8 mm, are considered. The steel reinforcement is modelled with E = 200000 MPa, ν = 0.0 and ρ = 7800 kg/m3 and assumed cross section A = 30 mm2 (the reinforcement ratio equals 1%). The time step is 2 × 10−6 s, ti = 3 × 10−5 s and the traction intensity is pi = 2.4 MPa. The diagrams depicted in Figure 10 confront the behaviour of the bar with and without the reinforcement. For plain concrete the elongation tends to infinity due to fracture, while for reinforced concrete the horizontal displacement oscillates around a certain state. Nevertheless, for both responses the standing wave and localization are observed in the centre of the bar, cf. Figures 10(e) and 10(f). Figure 10 shows the internal length study of this test for plain and reinforced concrete. The elongationtime diagrams for plain concrete, which are presented in Figure 10(a), reveal that the smaller the value of

193

l = 2 mm l = 4 mm l = 8 mm

0.15

0.025 Elongation at point A [mm]

Elongation at point A [mm]

0.2

0.1 0.05 0

0

0.02 0.015 0.01 l = 2 mm l = 4 mm l = 8 mm

0.005 0

5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 t [s] (a) Diagrams for plain concrete.

0

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 t [s] (b) Diagrams for reinforced concrete.

(c) Averaged strain, plain, l = 2 mm.

(d) Averaged strain, reinforced, l = 2 mm.

(e) Averaged strain, plain, l = 4 mm.

(f) Averaged strain, reinforced, l = 4 mm.

(g) Averaged strain, plain, l = 8 mm.

(h) Averaged strain, reinforced, l = 8 mm.

Figure 10. Internal length study for plain concrete and reinforced bar.

parameter l is, the faster the elongation grows. Figure 10(b) points out that for reinforced concrete together with the decrease of l the amplitude of vibrations diminishes. For l = 2 mm a much smaller period of the vibrations occurs. We observe a similar upper limit of the elongation for each l, which is related to the reinforcement action. All contour plots in Figure 10, which illustrate the averaged strain distribution, are performed for the time instant t = 0.0006 s (after 300 time steps). Considering the results, for plain concrete and l = 2 mm (Figure 10(c)) two separated zones of localization appear, which can be a numerical effect. For the other cases one central zone emerges, which is consistent with the analytical solution for the strain-softening bar presented in (Baˇzant and Belytschko 1985). The standing decohesion wave is located exactly in the centre. The general rule for gradient-enhanced models that the width of zone expands together with larger l is valid. For plain concrete the localization zones are parallel to the width of the domain and averaged strain contours form even bands, which is expected due to the loading direction. On the right-hand side of Figure 10 the averaged strain distributions for reinforced concrete are shown. Here again with the growth of l the zone of localization enlarges, but a new visible effect is that the reinforcement delays the progress of cracking along

the whole width. Along the bar axis localization does not seem to occur. The most active zones are the farthest from the reinforcement, but similarly to plain concrete the standing wave and damage zone are placed in the central part of the tensile bar. The full bond between concrete and the rebar influences the shape of the damage zones, which do not look very realistic. In reality more than one localized cracks, which gradually develop in the vertical direction next to the reinforcement, are observed. To simulate this behaviour a proper representation of bond slip and, from the numerical point of view, the implementation of interface elements are required as shown in (Sluys and de Borst 1996). 5 FINAL REMARKS Two constitutive models for concrete have been summarized and applied in the simulation of the Brazilian split test. The Burzy´nski-Hoffman viscoplastic consistency model is quite general, since it incorporates separate representations of damage in tension and compression as well as intrinsic loading rate dependence. Since it involves many parameters, an extensive case study is needed and has been to a certain extent provided in (Winnicki 2007). The scalar gradient damage model can be extended to its isotropic version with

194

two independent damage variables, thus enabling a more accurate representation of damage in tension and compression. Nevertheless, the scalar model is capable of representing the splitting effect in the dynamic Brazilian test. It has also been used to predict a standing localization wave in the direct tension test, whereby a comparison of the response of the plain versus reinforced bar has been performed. The gradient damage simulations of the dynamic four-point bending test have been presented in (Wosatko et al. 2006; Wosatko 2008). REFERENCES Bažant, Z.P. and T. Belytschko (1985). Wave propagation in a strain-softening bar: exact solution. ASCE J. Eng. Mech. 111(3), 381–389. Bi´cani´c, N., C.J. Pearce, and D.R.J. Owen (1994). Failure predictions of concrete like materials using softening Hoffman plasticity model. In H. Mang, N. Bi´cani´c and R. de Borst (Eds.), Computational modelling of concrete structures, Euro-C 1994, Volume 1, Innsbruck, Austria, pp. 185–198. Pineridge Press. Comi, C. (2001). A non-local model with tension and compression damage mechanisms. Eur. J. Mech. A/Solids 20(1), 1–22. de Borst, R., J. Pamin, and M.G.D. Geers (1999). On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. Eur. J. Mech A/Solids 18(6), 939–962. de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Comput. & Struct. 55(4), 581–588. Feenstra, P.H. (1993). Computational aspects of biaxial stress in plain and reinforced concrete. Ph.D. dissertation, Delft University of Technology, Delft. fib (1999). Structural Concrete. The Textbook on Behaviour, Design and Performance. Updated knowledge of the CEB/FIP Model Code 1990, Volume 1, Bulletin No. 1. fib. Geers, M.G.D. (1997). Experimental analysis and computational modelling of damage and fracture. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Heeres, O., A. Suiker, and R. de Borst (2002). A comparison between the Perzyna viscoplastic model and the Consistency viscoplastic model. Eur. J. Mech. A/Solids 21, 1–12. Kachanov, L.M. (1958). Time of rupture process under creep conditions. Izd. Akad. Nauk SSSR, Otd. Tekh. Nauk 8, 26–31. (in Russian). Kleiber, M. (1998). Plasticity problems. In M. Kleiber (Ed.), Handbook of Computational Mechanics. Survey and Comparison of Contemporary Methods, Berlin, Germany, pp. 201–252. Springer Verlag. Kupfer, H. (1973). Das Verhalten des Betons unter mehrachsiger Kurzzeitbelastung under besonderer Berücksichtigung der zweiachsigen Beanspruchung. Number 229. Berlin: Deutscher Ausschuss fur Stahlbeton. Lemaitre, J. (1971). Evaluation of dissipation and damage in metals. In Proc. I.CM., Volume 1, Kyoto, Japan. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. ASCEJ. Eng. Mech. 115, 345–365.

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Pamin, J. (2005). Gradient plasticity and damage models: a short comparison. Computational Materials Science 32, 472–479. Pamin, J., A. Wosatko, and A. Winnicki (2003). Two- and three-dimensional gradient damage-plasticity simulations of cracking in concrete. In N. Bi´cani´c et al. (Eds.), Proc. EURO-C 2003 Int. Conf. Computational Modelling of Concrete Structures, Rotterdam/Brookfield, pp. 325–334. A.A. Balkema. Pearce, C.J. (1993). Computational aspects of the softening Hoffman plasticity model for quasi-brittle solids. M.Sc., University Of Wales, College Swansea. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Meth. Engng 39, 3391–3403. Ruiz, G., M. Ortiz, and A. Pandolfi (2000). Threedimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders. Int. J. Numer. Meth. Engng 48, 963–994. Simo, J. and T. Hughes (1997). Computational Inelasticity. New York, Berlin: Springer Verlag. Simo, J.C. and J.W. Ju (1987). Strain- and stress-based continuum damage models–I. Formulation, II. Computational aspects. Int. J. Solids Struct. 23(7), 821–869. Sluys, L.J. (1992). Wave propagation, localization and dispersion in softening solids. Ph.D. dissertation, Delft University of Technology, Delft. Sluys, L.J. and R. de Borst (1996). Failure in plain and reinforced concrete–an analysis of crack width and crack spacing. Int. J. Solids Struct. 33, 3257–3276. Suaris, W. and S.P. Shah (1985). Constitutive model for dynamic loading of concrete. J. of Struct Engng ASCE 111(3), 563–576. Taylor, R.L. (2001). FEAP—A Finite Element Analysis Program, Version 7.4, User manual. Technical report, University of California at Berkeley, Berkeley. van Mier, J.G.M. (1984). Strain-softening of concrete under multiaxial loading conditions. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Vonk, R.A. (1992). Softening of concrete loaded in compression. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Wang, W.M. (1997). Stationary andpropagative instabilities in metals—a computational point of view. Ph.D. dissertation, Delft University of Technology, Delft. Winnicki, A. (2007). Viscoplastic and internal discontinuity models in analysis of structural concrete. Series Civil Engineering, Monograph 349, Cracow University of Technology, Cracow. Winnicki, A., C.J. Pearce, and N. Bi´cani´c (2001). Viscoplastic Hoffman consistency model for concrete. Comput. & Struct. 79, 7–19. Wosatko, A. (2008). Finite-element analysis of cracking in concrete using gradient damage-plasticity. Ph.D. dissertation, Cracow University of Technology, Cracow. Wosatko, A., J. Pamin, and A. Winnicki (2006). Gradient damage in simulations of behaviour of RC bars and beams under static and impact loading. Archives of Civil Engineering 52(1), 455–477. Wosatko, A., A. Winnicki, and J. Pamin (2009). Numerical analysis of brazilian split test on concrete cylinder. Computers & Concrete. Submitted for publication.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Micromechanical approach to viscoelastic properties of fiber reinforced concrete V.F. Pasa Dutra, S. Maghous & A. Campos Filho Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil

ABSTRACT: This paper studies some aspects of fiber reinforced concrete (FRC) within a micromechanical framework. As a starting point, the overall elastic properties are determined by modeling the fibers as flat prolate spheroids and implementing a Mori-Tanaka homogenization scheme. In a second step, the formulation of the non-aging viscoelastic behavior is carried out by making use of results previously obtained in elastic homogenization and the Elastic-Viscoelastic Correspondence Principle. Adopting a Zener model for the behavior of the concrete matrix the homogenized creep compliance functions are derived analytically. The validity of the model is established by means of comparison with available experiment measurements of creep strain in steel fiber reinforced concrete under compressive load and analytical models formulated within a one-dimensional setting. 1 INTRODUCTION Fiber reinforced concrete (FRC) is a composite material formed by the association of a cement matrix and embedded fibers. Several studies have shown that the addition of steel fibers causes improvement of the mechanical properties of concrete such as ductility, post-cracking behavior, and resistance to static and dynamic actions. In the perspective of a wide use in structural engineering, a comprehensive formulation of the corresponding behavior is necessary. The micromechanics-based approach aims at establishing a connection between the macroscopic behavior of FRC and the characteristics of its constituents. A considerable advantage of the micromechanical approach lies in the fact that the closed-form expression for the homogenized properties makes it possible to easily analyze the impact of the reinforcement characteristics at the macroscopic level. This study focuses on elastic and viscoelastic behavior of the steel FRC by means of a micromechanical approach. Reasoning on the representative elementary volume (REV) of such a composite, the overall elastic characteristics are determined from the knowledge of the elastic properties of its individual constituents. Modeling the fibers as prolate inclusions embedded within the concrete matrix, the Mori-Tanaka scheme, which is based on the results established by Eshelby (1957), is used to derive estimates of the elastic coefficients. It was found that the micromechanical predictions for the overall stiffness proved to be considerably close to the experimental data, as well as to the finite element solutions obtained from numerical analysis of a REV

197

of steel FRC (modeled as a randomly heterogeneous medium). The formulation of the non-aging viscoelastic behavior is then carried out by making use of results previously obtained in elastic homogenization and the Elastic-Viscoelastic Correspondence Principle.

2 MACROSCOPIC ELASTIC BEHAVIOR OF FIBER REINFORCED CONCRETE 2.1 Estimates of the elastic properties of FRC This section deals with the evaluation of the elastic behavior of FRC through homogenization approach. The main interest of this approach lies on the possibility to use the obtained effective behavior to perform computations at the scale of the structure, considering the homogenized structure instead of the initial heterogeneous one. A central concept of the homogenization procedure is the existence of a representative elementary volume, which must comply with the scale separation condition (Zaoui 2002). The effective behavior of the composite is obtained from the response of its REV to a mechanical loading analyzed in the framework of a boundary value problem (Suquet 1987). In the framework of linear elastic homogenization of random heterogeneous material, considering the imposition of homogeneous strain boundary conditions, the macroscopic constitutive equation reads  = C hom : E =

∼

=

(1)

where E represents in fact the macroscopic strain and

with

it is the volume average of the microscopic strain field over the whole REV,  is the macroscopic stress ten-

[1 + P m,α : (c f − c m )]−1 fib

=

=

∼

sor and it is taken by definition as the volume average of the microscopic stress field over the whole REV and C hom denotes the overall homogenized (macroscopic)

=

∼

elasticity tensor: C hom = ∼

n  r=1

2π π

[1 + P m,α (ϕ, θ) : (c f − c m )]−1

×

(2)

∼

∼

∼

∼

∼

∼

∼

conditions E 0 at infinity. =

The Mori-Tanaka scheme (Mori & Tanaka 1973), which is based on the results established by Eshelby, considers a composite with a matrix phase m clearly identified surrounding all the others. This homogenization scheme adopts this matrix as the reference medium in the Eshelby inhomogeneity problem: c 0 = ∼

and takes into account the interaction between the

phases in an indirect way. More precisely, this interaction is expressed by means of the value of the reference strain E 0 which is equal to the average strain in the

(4)

∼

∼

is the corresponding volume average over the domain occupied by phase r. As emphasized by the Equation 2, the determination of the overall elasticity tensor requires being able to compute estimates of the average of strain concentration tensor over each phase. This is usually achieved through an appropriate homogenization scheme integrating some information on the microstructure morphology. From the viewpoint of Eshelby-based theory (Eshelby 1957), the average strain concentration tensor is estimated from the elastic solution of an ellipsoidal inhomogeneity embedded within an infinite medium (reference medium) with stiffness c 0 subjected to uniform strain boundary

sin θ dθ dϕ 4π

stiffness tensor of fibers, respectively. The fibers are represented by prolate spheroids which differ in orientation (θ , ϕ). The shape and orientation of the fibers are taken into account by the Hill tensor P m,α for pro-

the stiffness tensor of phase r. The fourth-order tensor A is the so-called strain concentration tensor and Ar

∼

late spheroid with aspect ratio α (Mura 1987). fib is the volume average over the domain occupied by the fibers and is obtained through integration over all the orientation of space, as it can be seen in Equation 4. The assumption of elastic isotropy for both concrete matrix and fibers, and the random distribution of the fibers imply the elastic isotropy of FRC at the macroscopic level. The homogeneity assumed for the concrete matrix can result of a preliminary homogenization process, which accounts for possible microcracks. The homogenized elasticity tensor C hom,MT ∼

can therefore be completely defined by means of the homogenized bulk modulus K hom and shear modulus G hom , respectively Equations 5 and 6. These coefficients are function of the elastic modulus of concrete matrix (k m , g m ) and fibers (k f , g f ), as well on the volume fraction f and aspect ratio α of fibers. For simplicity, we shall restrict the subsequent analysis to the situation of fibers with infinite value for the aspect ratio (α → ∞), that is the case where the diameter parallel to the axis of revolution is considerably larger than the diameter perpendicular to the axis of revolution.

=

K hom =

matrix: E 0 = ε m . As a result of the latter consider=

∼

∼

where f and c f denotes the volume fraction and the

in which n is the number of different phases of the REV, fr = the volume fraction of phase r and c =

cm ∼

∼

ϕ=0 θ=0

fr c r : Ar ∼

∼

((3g m + g f )[(1 − f )k m + fk f ] + 3k m k f ) ((3g m + g f ) + 3(fk m + (1 − f )k f )) (5)

=

ations, the homogenized stiffness tensor of the MoriTanaka scheme for a uniform orientation distribution of fibers is ∼

[1 + P ∼

m,α

: (c f − c m )]−1 fib } :

∼

(c f − c m )]−1 fib ∼

5 

di

i=1

n1 = 9fk m k m (g f )3 ,

∼

(1 − f )1 + f : [1 + P ∼

ni



∼

∼

∼

=

6  i=1

C hom,MT = {(1 − f )c m + f c f : ∼

G

hom

∼

∼

m,α

:

n2 = 3g m (g f )2 (7fg f (k m + k f ) + 15k m k f (1 + f ) (3)

198

+ 5g f k m ),

(6)

n3 = (g m )2 g f (35(g f )2 (35 + 37f ) m f

f

1.5%}. Figures 1 and 2 respectively display the MoriTanaka estimates of E hom and ν hom , together with the experimental results. The Ashour et al. experiments (2000) correspond to steel fibers with steel fibers with an aspect ratio of α = 75 and f ∈ {0.5%, 1%}. The comparison of the micromechanical estimates of the homogenized Young modulus with the experimental results are shown in Figure 3. As it appears from the above figures, the MoriTanaka estimates prove to be slightly lower than the measured values provided in the mentioned experimental works (Thomas & Ramaswamy 2007, Ashour et al. 2000). A good agreement is obtained and the consistency with experimental data can be termed as reasonable. The discrepancy observed between the micromechanical predictions and the experimental results remains lower than 10%.

f

+ 15k g (5 + 3f ) + 3k g (35 + 17f ) + 10k m k f (9 − f )),

n4 = (g m )3 ((g f )2 (145 + 39f ) + 21k m g f (5 − f ) + 3k f g f (40 − 19f ) + 45k m k f (1 − f )),

n5 = (g m )4 (45k m (1 − f ) + 15k f (1 − f ) + g f (125 − 61f )),

n6 = 15(g m )5 (1 − f ),

d1 = 3k m (g f )2 (5g f (1 − f ) + 3k f (5 − 4f )),

d2 = g m g f (35(g f )2 (1 − f ) + 3k m g f (25 − 13f ) + 21k f g f (5 − 4f ) + 90k m k f )

d3 = (g m )2 ((g f )2 (145 − 73f ) + 15k m g f (7 + f ) + 12k f g f (10 + 3f ) + 9k m k f (5 + 4f ),

d4 = (g m )3 (9k m (5 + 4f ) + 3k f (5 + 16f ) + g f (125 + 59f )),

d5 = (g m )4 (15 + 49f ).

2.2 Comparison with experimental results The micromechanical predictions of FRC elastic properties using a results Mori-Tanaka scheme are compared herein to available experimental results performed on steel fiber reinforced concrete. The values of Young modulus and Poisson ratio of the composite components as provided in the experimental studies are listed in Table 1. In Thomas & Ramaswamy’s (2007) experiments, the aspect ratio of the used steel fibers was α = 55 and several volume fractions were tested f ∈ {0.5%, 1%, Table 1.

Figure 1. Young modulus: Mori-Tanaka estimate and experimentally measured value (Thomas & Ramaswamy 2007).

Material properties of steel fiber and concrete.

Material

Young modulus Poisson E (MPa) ratio ν

Thomas & Ramaswamy (2007) Steel Fiber Concrete 35 MPa (C35) Concrete 65 MPa (C65) Concrete 85 MPa (C85)

210,000 28,700 37,500 41,700

0.300 0.182 0.201 0.210

Ashour et al. (2000) Steel Fiber 210,000 Normal Strength Concrete NSC 24,612 Medium Strength Concrete MSC 35,443 High Strength Concrete HSC 38,423

0.300 0.200 0.200 0.200

Figure 2. Poisson ratio: Mori-Tanaka estimate and experimentally measured value (Thomas & Ramaswamy 2007).

199

2.3 Finite element numerical solutions In order to asses to accuracy of the Mori-Tanaka estimates, the concentration problem (Eq. 1) is now solved numerically through the implementation of a finite element (FE) method. For sake of simplicity, a cubic REV of side l is considered. Owing to the macroscopic isotropy of FRC, the restriction of the analysis to a loading mode defined by uniaxial compression is sufficient to obtain the overall elastic properties. Hence, a macroscopic stress  = −l e1 ⊗ e1 with l > 0 is applied to the REV. It is recalled that the scale separation condition ensures the equivalence between homogeneous boundary strain and stress conditions on the REV (Hill 1967). Twenty nodded quadratic hexahedral elements were used for the concrete discretization of the concrete matrix geometry. As regards the reinforcement components, the fibers are randomly generated and embedded within the concrete matrix finite elements.

Figure 3. Young modulus: Mori-Tanaka estimate and experimentally measured value (Ashour et al. 2000). Table 2.

It is recalled that perfect bonding is assumed between fibers and concrete matrix. We herein refer to the socalled ‘‘embedded model’’ (Elwi & Hrudey 1989) in which each fiber has the same kinematics than the coincident points of the embedding concrete matrix finite element. The spatial distribution and orientation of fibers are randomly generated by means of a specific procedure using the intrinsic function RAN of Fortran programming language. Once the finite element displacement solution is obtained, the macroscopic strain could also be computed. It is expected that E = −El e1 ⊗ e 1 + Et (e 2 ⊗ e 2 + e 3 ⊗ e 3 ), =

with El , Et ≥ 0

(7)

The homogenized elastic parameters are then obtained by E hom = l /El and ν hom = Et /El . In the objective to qualitatively address the scale effect, two values for the REV side l = 20, 40 cm as well as two values for the fiber length d = 1, 3 cm have been considered in the numerical study. The values adopted for the aspect ratio and volume fraction of fibers range between 10–100 and 0.5%–5%, respectively. The numerical results for each of the above configuration defined by the set {l, d, α, f } are given in Tables 2 and 3, where the Mori-Tanaka predictions are also reported. It is noted that E hom and ν hom stand for the numerical estimates of the effective Young modulus and Poisson ratio obtained from the FE simulations. As it clearly appears from the above results, the micromechanical predictions are very close to the FE numerical solutions. This emphasizes the consistency of the Mori-Tanaka scheme to estimate the elastic properties of FRC. In addition even though the condition d to the average strain < ǫ >:

locality at the macro scale in the presence of strong gradients which does not vanish upon upscaling. Non locality vanishes only in three cases: (i) when the distribution of micro-craking is uniform and the solid is subjected to a homogeneous remote state of stress, local quantities are equal to non local ones; (ii) at the boundary of a solid, non locality should vanish in the direction normal to the boundary; (iii) upon a macrocrack formation, non locality should also vanish on the macro-crack faces (in their normal direction).

= L :< ǫ > −  T : ∇ 2 (< ǫ >)

4 NON LOCAL DAMAGE MODEL

where the average strain < ǫ > is defined as:  Ŵ(x, ξ )dξ < ǫ >= ǫ0 +

(5)

We are going to examine now how such results can be incorporated in a continuum formulation and extended to the case of evolving boundaries, macro-crack surfaces described by a zone in which damage is equal to 1 according to the continuum model (Mazars and Pijaudier-Cabot 1996). Although the constitutive relations are discussed in a full 3D format, we will restrict applications of the modified non local model to the one-dimensional case which is simple but not entirely representative of full boundary effects (no tangential effects on the boundary). 2D and 3D generalisation are left for further developments.

(6)



Function Ŵ(x, ξ ) is a measure of the local interactions derived with the help of Green’s functions and ǫ0 is the strain in the homogeneous comparison solid (equal to the strain in the matrix if inclusions are very dilute). The above equation is very similar to Eq. (2), except that it is expressed in term of strain rather than stress. Since the material is elastic, the difference is not really fundamental. In the approach by Drugan (Drugan and Willis 1996), there is an evaluation of the size of the representative volume element (RVE) in the homogenised material. The size of the RVE is such that the second order gradient appearing in Eq. (5) becomes negligible. In other words, the constitutive equations relating the average stress to the average strain become local. This definition of the RVE was also used by Delaplace and co-workers (Delaplace et al. 1996) in order to compute a correlation length in a lattice model. According to Drugan and Willis, the size of the RVE for a material containing voids with a volume fraction less than 0.33 is about two times the diameter of the void. As pointed out in the same paper, this size of RVE is much less than what is commonly expected in heterogeneous materials. A ratio of five to ten times the size of the voids or cracks is usually used for defining the size of the RVE needed in order to average in a statistical sense the fluctuations of the stress or strain. Hence, the operator relating the stress to the elastic strain may be defined locally as non local second order gradient terms are negligible but at the same size of RVE, averages of strains and stresses may not be statistically representative. A non local strain may be required in order to capture damage growth, although the constitutive relations may remain defined locally since higher order terms are negligible. Pijaudier-Cabot and Dufour (Pijaudier-Cabot and Dufour 2010) have discussed this issue and arrived to the conclusion that crack interaction at the microstructural scale results into non

4.1 Isotropic (scalar) damage model The classical stress-strain relation for this type of model reads: σij = (1 − D)Cijkl εkl

(7)

where σij and εkl are the components of the stress and strain tensors, respectively (i, j, k, l ∈ [1, 3]) and Cijkl are the components of the fourth-order elastic stiffness tensor. The damage variable D represents a measure of material degradation which grows from zero (undamage material with the virgin stiffness) to one (at complete loss of integrity). The material is isotropic, with E and ν the initial Young’s modulus and Poisson’s ratio respectively. For the purpose of defining damage growth, a scalar equivalent strain εeq is introduced, which quantifies the local deformation state in the material in terms of its effect on damage. In this contribution, Mazars’ definition of the equivalent strain is used (Mazars 1984):

εeq

  3  =  (εi + )2

(8)

i=1

where εi + are the positive principal strains. Damage growth is governed by the loading function: g(ε, k) = εeq (ε) − k

210

(9)

k equals the damage threshold εD0 initially, and during the damage process it is the largest ever reached value of εeq . The evolution of damage is governed by the Kuhn-Tucker loading-unloading condition: g(ε, k) ≤ 0,

k˙ ≥ 0,

˙ kg(ε, k) = 0

(10)

The damage variable D is determined as a linear combination of two damage variables Dt and Dc , that represent tensile damage and compressive damage respectively, by the help of two coefficients αt and αc which depend on the type of stress state (Mazars 1984): D = αt Dt + αc Dc

(11)

Dt,c = 1 −

(12)

1 − At,c At,c + εeq exp(Bt,c (εeq − εD0 ))

with respect to its arguments x and ξ . This lack of symmetry leads to the non-symmetry of the tangent operator (Bažant and Pijaudier-Cabot 1988; PijaudierCabot 1995; Jirasek and Patzák 2002). A symmetric non local formulation exists also (Borino et al. 2003).

Standard values of the model parameters in the damage have been given in Ref. (Mazars 1984). 4.2 Original non local formulation In the integral-type non local damage models, the local equivalent strain is replaced by its weighted average: 

(x, ξ )εeq (ξ )dξ (13) ε¯ eq (x) = 

with  the volume of the structure and (x, ξ ) the weight function. It is required that the non local operator does not alter the uniform field, which means that the weight function must satisfy the condition: 

(x, ξ )dξ = 1 ∀x ∈  (14)

4.3 Modified non local formulation The goals of the modified non local formulation are twofold: (1) at the boundary of the solid, non locality should vanish; (2) when a crack has formed—i.e. when a band of damage close to 1 has been created, the constitutive relations should again be local in this band. By comparison to the modifications discussed in (Krayani et al. 2009) which dealt with a boundary effect only, here we try to account also for boundary effects occurring when a new free surface has formed in the process of cracking. We are going to rely on a simple argument: in the process of cracking, the quantity of information that can be transmitted between two neighbors depends on the state of damage. The argument is rather similar to that of Desmorat and Gatuingt (Desmorat and Gatuingt 2007). Some internal time is introduced in the constitutive relation which induces a decrease of the internal length upon damage growth. Here, the analogy with the propagation of wave is used but the weight function is regarded as an attenuation function in the transmission of information from one point to its neighbor. Let us denote as s the distance appearing in Eq. (16), defined as x − ξ . This distance is now modified according to the following incremental relation: du =



For this reason, the weight function is recast in the following form (Pijaudier-Cabot and Bažant 1987):

0 (x − ξ ) r (x) 

0 (x − ξ )dξ with r (x) =

(x, ξ ) =

(15)



where r (x) is a representative volume and 0 (x − ξ ) is the basic non local weight function which may be taken as a polynomial bell-shaped function (Bažant and Jirasek 2002), or here as a Gauss distribution function:  4x − ξ 2

0 (x − ξ ) = exp − (16) lc2 lc is the internal length of the non local continuum. Preserving the uniform field in the vicinity of the boundary makes the averaging in Eq. (15) not symmetric

211

x − dξ  ds = γ (ξ ) γ (ξ )

(17)

where γ (ξ ) is a function of the state of damage at point of coordinate ξ . In the computation of the average centered at point x, we use now the distance u instead of the distance s:  ds u= (18)  γ (ξ ) In this remapping of the neighborhood around point of coordinate x, each point of coordinate ξ is now defined by the new coordinate u. In a spherical coordinate system centered at point x, the distance is defined by the above equation and the two angles are kept the same, they are invariant through the mapping function. The weighted average in Eq. (13) becomes: ε¯ eq (x) =





(u)εeq (x + u)du

(19)

γ (ξ ) should be equal to 1 when the material is not damaged at point ξ , and it decreases when damage

grows. As a consequence, the distance between two close neighbors will be increased as damage grows and in the weighted average process, the non local effect of point ξ will decrease. Function γ (ξ ) remains to be defined completely. We use for this an analogy with the attenuation during the propagation of waves. In the non local average defined in Eq. (13) the weight function (x−ξ ) is seen as the attenuation applied to a wave of amplitude εeq generated at point ξ and propagating toward point x.

(x−ξ )·εeq is the interaction of point ξ on point x. It is attenuated as the information is propagated in the solid. We may now consider that the wave speed depends on the state of damage: the time needed for the interaction effect to cover distance s is going to increase if the material on the path of propagation of the wave is getting damaged. the wave √ √In a one-dimensional setting, speed c = E/ρ becomes cd = (1 − D) · E/ρ where cd is the velocity of the damaged material. This increase of time for the information to be transmitted in the damaged medium between the two points ξ and x is converted into a fictitious increase of distance between these two points. The time needed for the interaction to propagate between the two points is kept the same, but since we consider now that the interaction is propagated in an undamaged medium the distance between the points is increased. When this distance is increased, the attenuation is increased accordingly. Let us consider Eq. (17) and divide both terms by the wave speed c: du ds ds = = c γ (ξ ) ∗ c cd

(20)

This equation states that the time needed for an information to propagate over a distance du in an undamaged medium is the same as the time needed for an information to propagate over the length ds in the damaged medium. Upon damage, the distance between these two points is increased, which is equivalent to a slower wave propagation and consequently, it is more attenuated than in the case where damage is not observed. From this equation we have: γ (ξ ) =

cd c

(21)

This qualitative reasoning implies that damage enters in the function γ . The difficulty is that the formulation becomes implicit. In order to keep the simplicity of the approach, we substitute to the non local value of damage the local expression and we take: γ (ξ ) = (1 − F(εeq (ξ )))

1 2

arbitrary. In fact, it fulfills the requirements stated above: when the material is not damaged, this function is equal to 1 and it becomes equal to zero if the material is totally damaged locally. At this stage, the incremental value of the modified distance between two points becomes infinite and the non local interaction in between them vanishes. The exponent 1/2 has been chosen by analogy to wave propagation. This new definition of the non local averaging needs also to fulfill the conditions on the free boundary demonstrated in the previous section. This can be performed by considering that on a free boundary local damage is set equal to 1. According to Eq. (17), a close neighbor to a point located on the boundary of the solid is located at a distance du which becomes infinite. Consequently, the non local effect of this neighbor to the non local average centered at the boundary is equal to zero.

5 COMPARISON BETWEEN THE ORIGINAL AND MODIFIED MODELS In order to compare the original and the modified non local formulation, we are going to consider two simple one-dimensional tests. 5.1 Dynamic failure of a bar The first test is quite classical (see e.g. (PijaudierCabot and Bažant 1987)). A bar of length 2L, in which two constant strain waves converge toward its center is considered. The amplitude of the wave is 0.7 times the deformation at the peak load in tension. When the two waves meet at the center, the strain amplitude is doubled, the material enters the softening regime suddenly and failure occurs in the middle of the bar. The bar length is taken equal to 30 cm. The parameters used in this example are: the mass per unit volume ρ = 1 kg/cm3 , the Young’s modulus E = 1 N/cm2 and the velocity boundary condition v = 0.7 cm/s applied at the two bar ends. The other model parameters are At = 1, Bt = 2, εD0 = 1 and the internal length lc is 4 cm (there is no damage in compression). A fixed mesh of 99 constant strain elements is used. Time integration is performed according to an explicit, central difference scheme. The time step is t = 0.3s. Figures 5 present the evolution with time of the profiles of damage and non local strain. The two waves meet at the center of the bar at time t = 15s.

(22)

where F(εeq ) is the function defined in Eq. (12). F(εeq ) is the local value of damage computed at the considered point. This definition of function γ is rather

212

X U(-L, t) = -vt

U(L, t) = vt L

Figure 4.

Principle of the one dimensional computation.

Figure 6. Evolution of the damage profile with time over the bar according to the original damage model.

Figure 5. Evolution of the damage (top) and non local strain (bottom) profiles with time over the bar.

Note that damage develops over a band of finite—non zero–width. Compared to computations with the original damage model shown in Fig. 6, the profile of damage is almost triangular instead of being almost rectangular, forming a band of damage equal to 1 according to the original formulation. This difference is due to the modification of the weight function as damage develops. This is illustrated in Figs. 7. The weight function centered in the middle element where complete failure is expected to occur (e.g. where damage is equal to 1) shrinks progressively as damage develops. When damage is equal to 1 in this element, it is a Dirac delta function and the material response becomes local. When damage is equal to 1, it is as if the bar would be cut into two pieces. At this point, the material response is local and this is in agreement with micro-mechanics of crack interaction as demonstrated in this paper. In neighboring elements, however, the weight function evolves differently. We have computed the weight function nearby the element in which failure occurs, two elements farther on the left. Figure (7) shows that the weight is cut at the center of the bar. Information coming from material points located behind the point at which failure occurs is screened by damage. This is a difference with the original non local damage formulation in which non local interactions (weights) is transmitted even across a macro-crack. We have also

213

Figure 7. Evolution of weight function with damage: weight centered in the middle element (top), weight centered near the middle element (bottom).

checked in Fig. (8) that the distribution of damage is not subject to spurious mesh dependency. For the finite element meshes used which are already quite refined, the largest element size is smaller than the internal length used in the computations, the profiles of damage are almost the same. Convergence of the damage profiles with respect to mesh refinement means also that the energy dissipated at failure is a constant. It is the sum of the energy dissipated due to damage at each material point in the damage band.

Figure 8. meshes.

Evolution of the damage profiles for different

Figure 9. Evolution of the maximum strain il the middle element with the finite element size. l0 = l/2

The damage profiles are not dependent on the finite element size. The issue is now whether the strain distribution should be also independent of the finite element size or not. Before complete failure, this result is expected. Without this property, the damage band which is a function of the maximum non local strain recorded in each finite element over the history of loading would not be expected to be mesh independent. When damage is equal to one in the middle element, convergence of the strain with respect to the size of this element may be questioned. Indeed, in the finite element located in the middle of the mesh, where complete failure occurs, the stress is vanishing and the relative displacement at the extremity of the finite element should be equal to the crack opening. According to the finite element discretisation however, there is no displacement and strain discontinuity and the crack opening is smeared over the element. Let us denote as h the length of this finite element, the crack opening [u] is: [u] = ǫ × h

(23)

If the finite element calculation has converged with respect to mesh refinement, upon complete failure the crack opening should be independent of the element size and therefore that the strain in the element located in the middle of the mesh evolves as a function of h−1 . We have checked this in Fig. (9). According to the modified non local model, the maximum strain is indeed a power law of the finite element size. Exponent −1 is recovered. At the same time, the crack opening displacement is measured. We obtain [u] = 4,81 cm. Interestingly, the same crack opening displacement, computed according to the technique proposed by Dufour and co-workers (Dufour et al. 2008) is rather close [u] = 5,07 cm. For this calculation, we have chosen a constant weight function for the computation of the estimate of the crack opening. It is important to remark that the original non local formulation is far from exhibiting the same property. The

214

tension t < t0 u = ct t > = t0, u = 0 l = t0 E r

compression

Figure 10. Principle of the dynamic tension test.

maximum strain is constant and therefore, in this simple application it may not provide the displacement discontinuity properly. 5.2 Spalling test The second example is the spalling test proposed by (Krayani et al. 2009). This example has been designed so that the location of localised failure may occur very close to one extremity of the bar, involving modified non local interactions due to the extremity of the bar and to the occurrence of failure at the same time. The geometry and applied load is shown in Fig. 10. A square compression signal is generated in the bar. Upon reflection at the end of the bar, the compression signal turns into a tensile one. This signal is added to the incoming compression. If the absolute amplitude of the compression signal is greater than the tensile strength, failure is initiated at a distance from the boundary equal to half the signal length. Depending on the duration of the compression signal, it is possible to initiate failure in the material at any location, near the boundary or far from it. The bar length is 20 cm. The parameters used in this example are: the mass per unit volume ρ = 1 kg/cm3 , the Young’s modulus E = 1 N/cm2 and the velocity boundary condition c = −1.5 cm/s applied at the left bar end. The other model parameters are the same as in the previous computations. Time integration is performed according to an explicit, central difference scheme. The signal length is calculated as l = t0 υ and

its amplitude is c/υ where υ is

E ρ.

The time step is

t = 0.2s and the signal length is 4s so that failure is initiated at a distance from the boundary equal to half the internal length. Figure (11) shows the evolution of the damage profiles according to the original non local model. We observe that maximum damage is reached exactly at the bar end. This result is in agreement with those obtained in (Krayani et al. 2009). On the contrary, the location of maximum damage is approximately located at 2 cm from the bar end according to the modified formulation (Fig. 12). The modified computation of the non local average devised in this paper provides results that are close to those proposed by Krayani and co-workers who discussed boundary effects, but not the case of cracks viewed as emerging boundaries. The modified formulation is capable of describing spalling failure with a spall of finite thickness, same as the modified formulation in (Krayani et al. 2009). There is, however, a difference between the two models that needs to be further investigated: the distribution of damage in Figure (12) decreases slowly in between the location of the maximum of damage and the finite element located at the bar end, and then it jumps to zero in the last finite element. In the formulation by

Figure 11. Evolution of the damage profiles during spalling failure according to the original formulation.

Krayani and co-workers, the non local strain becomes local as it reaches the boundary and the neighborhood over which the average is computed is decreased gradually as the point at which the average is computed is getting close to this boundary. This produces a smooth decreasing damage profile going towards the boundary after the peak damage located inside the bar.

6 CONCLUSIONS Boundary effects are difficult issues in non local models. In either integral or gradient formulations, boundary conditions are rather arbitrary. We have recalled first that on a free boundary non local interactions should vanish. In the course of failure, when a macrocrack is formed, new boundary surfaces are appearing. This should also be taken into account in non local formulation, with the same requirements as for an initial boundary in the solid. We have presented a prototype damage model that accounts for the progressive shielding effect induced by a crack appearing in the material. This is achieved by a remapping of the non local averaging, in the same spirit as in (Krayani et al. 2009), but with a different mathematical formulation. One dimensional finite element calculations show that the modified non local model describes failure with a finite non zero fracture energy, that the damage profiles are triangular and not rectangular as observed in the original formulation, that the model is capable of approaching a discontinuous formulation at complete failure, whereas the original formulation is not capable of reaching this limit case, and that the model is capable of capturing spalling failure more properly that the original one. Grassl and Jirasek (Grassl and Jirasek 2006) showed that boundary effects had an influence on the energy of fracture. The variation of fracture energy, measured experimentally and derived from finite element computations, may help at the validation of the present model, e.g. with the help of structural size effect tests.

ACKNOWLEDGEMENTS Financial support from ERC advanced grant Failflow (27769) to the first author is gratefully acknowledged.

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Figure 12. Evolution of the damage profiles during spalling failure according to the modified formulation.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Homogenization-based model for reinforced concrete Erkan Rumanus & Günther Meschke Institute for Structural Mechanics, Ruhr University Bochum, Germany

ABSTRACT: In the paper, a constitutive model for reinforced concrete considering cracking and the relevant interactions between steel and concrete, such as bond slip and dowel action, is presented. Without explicit discretization of the reinforcing bars, reinforced concrete is modeled as a composite material by adopting concepts of continuum micromechanics to describe the macroscopic mechanical response. The formulation is based on the Mori-Tanaka homogenization scheme assuming a three-phase composite material formed by a continuous matrix (concrete) and two sets of straight rebars (reinforcement). For each phase, the nonlinear preand post-peak behavior is described separately, while considering the interactions between the reinforcement bars and the concrete. Concrete cracking is modeled by means of a fracture-energy-based elasto-plastic damage model. A classical J2 -plasticity model describes the elasto-plastic response of the steel reinforcement. Interfacial debonding is taken into account by adopting a two-phase serial system consisting of the steel bar and of the bond surface. The redistributed stresses due to crack formation at the steel-concrete interface are governed by slip strains. The related macroscopic bond-slip vs. bond-stress relationship is determined from classical pull-out tests. The residual shear stiffness of cracked reinforced concrete (dowel action) is derived from homogenization of the shear force vs. displacement relation obtained from the ‘‘beam on elastic foundation’’ theory considering each set of steel bar as a beam embedded in the surrounding concrete. The proposed model is implemented in the finite element code Msc.Marc and validated by means of selected experimental investigations on reinforced concrete panels and beams. 1 INTRODUCTORY REMARKS The efficiency of the load-carrying characteristics of reinforced concrete (RC) is attributed to the compensation of the brittle behavior of concrete in tension by the ductile material properties of steel. The combination of these two materials generates a more ductile composite material and a more distributed mode of cracking with reduced crack widths. Both factors play a predominant role with regards to the serviceability and the ultimate load-carrying capacity of reinforced concrete structures. Since the material behavior of the composite differs completely from the material properties of the constituents, besides the constitutive laws for steel and concrete, also the corresponding interactions have to be considered adequately. Even though the interactions between steel and concrete are acting at the scale of the steel bars, their influence on the structural level is relevant and must be considered in a computational model for reinforced concrete structures. In the first part of the paper, the proposed nonlinear constitutive laws for concrete and steel are presented separately. Subsequently, the main steelconcrete interactions—bond slip and the dowel action—and the respective models for their numerical representation are summarized. Adopting concepts of continuum micromechanics, the composite material consisting of concrete and steel is formulated and

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the homogenized constitutive relations for reinforced concrete considering the steel-concrete interactions in cracked concrete are established in the following sections. The performance of the proposed reinforced concrete model is finally demonstrated in the last section of this paper by means of re-analyzes of selected experimental investigations performed on RC panels and beams.

2 CONSTITUTIVE MODELS 2.1 Concrete matrix material The constitutive model for (plain) cracked concrete is originally formulated within the framework of poromechanics according to the Biot-Coussy theory (Coussy 2004). Using an elasto-plastic damage model, this multiphase concept allows consideration of shrinkage induced damage and its evolution of damage and long-term creep deformations resulting from combined external and thermo-hygral loading conditions (Meschke and Grasberger 2003; Grasberger and Meschke 2004). Since the focus of the present study lies on the mechanical behavior of concrete, the time-dependent state variables such as temperature, moisture and long-term creep are not included in the formulation of the model in this paper.

The linearized strain tensor εm of the concrete is assumed to be small and can therefore be additively decomposed into elastic strains εem and irrecoverable p strains εm ε m = εem + εpm .

(1)

In the following, the index m refers to the matrix (concrete). To characterize the behavior of concrete in tension and in compression, a multi-surface fracture energy-based damage-plasticity theory is adopted Meschke et al. 1998. Degradation mechanisms and inelastic deformations are controlled by four threshold functions fk defining a region of admissible stress states in the space of the matrix stresses σ m E = {(σ m , qk )| fk (σ m , qk (αk )) ≤ 0,

In the following, the subscript s refers to the steel reinforcement. The secant stress-strain relationship is obtained as σ s = Cs0 : ε es = Cs0 : (ε s − εps ),

where Cs0 denotes the isotropic tensor of elasticity. The admissible stress field σ s within the steel reinforcement is controlled by a J2 –plasticity model (Simo and Hughes 1998)  fs (S, α) = S  − 2/3[σy + Kαs ] ≤ 0.

k = 1, . . ., 4}. (2)

Cracking of concrete is accounted for by means of a fracture energy based Rankine criterion, employing three failure surfaces perpendicular to the axes of principal stresses fR,A (σ m , qR ) = σA − qR (αR ) ≤ 0,

A = 1, 2, 3

(4)

with Cm0 denoting the undamaged elasticity tensor and ψ the isotropic integrity parameter capturing the damage state of the concrete. The evolution of the irrecovp erable strains εm associated with the mismatch of the crack faces is defined in an associative format. The evolution of the integrity ψ is governed by (ψ˙−1 ) =

3  k=1

γ˙k

ε˙ dm C0 ε˙ dm ε˙ dm : σ

,

(5)

where εdm are the damage strains representing the effect of the stiffness degradation. Details of the formulation and the numerical implementation within a finite element code are contained in (Meschke et al. 1998). 2.2 Steel reinforcement The tensor of total strains ε s of the steel rebar is decomp posed into an elastic ε es and a plastic part ε s ε s = ε es + εps .

(8)

In 8 s denotes the deviatoric stress tensor, K a constant isotropic hardening plastic modulus and αs the isotropic hardening parameter governed by α˙ s =

 2/3γs ,

(9)

with γs as the consistency parameter. The evolution of p the irreversible plastic strains εs follows an associative flow rule.

(3)

with qR (αR ) = −∂U/∂αR denoting the softening parameter and the index A refers to the principal direction. The ductile behavior of concrete subjected to compressive loading is described by a hardening/ softening Drucker-Prager plasticity model. Based on a secant formulation, the stress field within the matrix σ m can be calculated by the elastic strains εem given in equation (1) σ m = ψCm0 : εem = Cm : (ε m − εpm )

(7)

(6)

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3 CONCRETE-STEEL INTERACTIONS In this section the two relevant mechanical interactions between concrete and steel, bond slip and dowel action, are briefly discussed and the adopted models are presented. While bond slip is concerned with the transmission of forces in the longitudinal direction of the steel bar, dowel mechanisms become relevant when shear forces across a crack are transmitted by the reinforcing bar. 3.1 Bond slip As long as the interface between the concrete and the embedded rebars is undamaged, a full (perfect) bond can be assumed between the constituents which is manifested by strain compatibility. By exceeding the tensile strength of the concrete, however, relative displacements between the steel bar and the surrounding concrete occur, inducing local bond stresses in the vicinity of the cracks. Since the focus of the proposed model lies on structural analyses, a macroscopic approach is adopted for the modeling of bond behavior. Accordingly, the assumption of strain compatibility between steel and concrete in the bar direction is maintained and therefore no additional degree of freedoms need to be introduced. Furthermore, a macroscopic description of the bond behavior can easily be formulated within the framework of homogenization concepts. Hence, without focusing on local bond mechanisms, the proposed bond model is based on redistribution of steel stresses due to bond damage (Luccioni et al. 2005;

Linero 2006; Manzoli et al. 2008). Adopting a twophase serial system consisting of the steel bar and of the bond surface, the strain tensor of the steel given in equation (6) is extended by slip strains εi εsi = εs + εi = εse + εsp + εi .

(10)

Note that in the local coordinate system of the steel bar, the strain tensor εi consists only of the component in the bar direction. The modified steel law including slip strains has the format σ s = Cs0 : ε es = Cs0 : (ε si − ε ps − ε i ).

(11)

Adopting suggestions made in (Linero 2006; Manzoli et al. 2008) a relatively simple elasto-plastic bond slip model is adopted (Figure 1). Note, that consequently the decreasing branch of the shear stress— slip law observed in experiments is neglected. The parameters (Ei , τmax ) may be obtained from classical pull-out tests where bond damage origins from sliding of the rebar along the steel-concrete interface as Ei =

P l , u As

τmax =

Pmax ≤ σy As

(12)

with As denoting the cross section of the embedded bar. 3.2 Dowel action The transmission of global shear stresses across a crack is governed—in absence of aggregate interlock—by dowel mechanisms of the reinforcing bar which undergoes additional shear and bending deformations in the vicinity of the crack. The dowel action can have a considerable influence on the structural stiffness, in particular when shearing or bending loading conditions are taken into account. Besides the global loading conditions the direction of the crack αc has a significant influence on the residual shear stiffness of a cracked reinforced element. In Figure 2a the dowel

Figure 2. a) Dowel action mechanisms. b) ‘‘Beam on elastic foundation’’—theory (He and Kwan 2001).

action is illustrated schematically. The cracked reinforced concrete element loaded by shear strains γ ∗ , has a residual (dowel) stiffness G D which considerably depends on the crack direction αc . In the proposed model the residual shear stiffness of cracked reinforced concrete is derived from the relation between shear stresses and strains obtained from the ‘‘Beam on elastic foundation’’—theory (He and Kwan 2001; El-Ariss 2006) in which the foundation modulus kf depends on the surrounding concrete (see Figure 2b). Without considering the influence of the crack direction αc , the related differential equation based on the Timoshenko beam—theory is first solved analytically. Afterwards, by means of a cracked reinforced element with an arbitrary crack direction αc the equilibrium state is formulated, leading to the effective dowel stiffness G D (αc ) (He and Kwan 2001; El-Ariss 2006). In this work, the dowel action is incorporated in the constitutive law of the reinforcement. Consequently, the shear modulus of steel Gs is replaced by the effective dowel modulus G D (αc ). Since the proposed dowel action model is formulated within a continuum mechanics model concept, it is consistent with the adopted continuum approaches for concrete cracking and the reinforcement.

4 CONTINUUM MICROMECHANICS 4.1 Reinforced concrete as a homogenized composite

Figure 1. Adopted macroscopic bond slip law derived from pull-out experiments.

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Reinforced concrete is represented as a three-phase composite material consisting of a continuous matrix formed by the concrete and by two sets of straight rebars representing the steel reinforcement and forming the reinforcement layer. In this Section, all parameters associated with concrete are subscripted with ‘‘m’’ (matrix) and the steel sets with ‘‘1’’ for the first set and ‘‘2’’ for the second set, respectively. The direction of the rebars and the geometry of the

cross section may be arbitrary within the (2–3)—plane. Figure 3 contains an illustration of the composite material ‘‘reinforced concrete’’. Such a configuration is typical for reinforced shell-like and plate structures as well as beam structures. Besides steel bars also fibers made of steel or of other materials (glass, polymers) and textile reinforced concrete may be covered by the proposed homogenization-type approach. In (Richter 2005), a similar micromechanical framework has been employed for the modeling of concrete reinforced by textile materials. In order to cover a broad class of reinforced composite materials the proposed micromechanically oriented material model for RC presented in the following section is formulated in a rather general format. 4.2 The representative volume element (RVE)

Figure 4.

A widely used approach in continuum micromechanics is based on the consideration of a representative volume element (RVE) representing an arbitrary material point of a structure. Thereby, the complex morphology of the microstructure is captured in a simplified manner by the RVE in order to estimate the related effective (macroscopic) response by means of an averaging procedure. The RVE can therefore be regarded as a homogeneous counterpart to a heterogeneous material with effective material properties. To confirm the representative character of the RVE, its size l has to be large enough in order to ensure a statistical distribution of the constituents with a characteristic size d and at the same time it has to be essentially smaller than a length of the structure L d ≪ l ≪ L.

(13)

In Figure 4 the assumed RVE is depicted schematically. The considered microstructure is governed by two straight steel rebars with the tensors C1 , C2 which are embedded in a continuous matrix (concrete) with the tensor Cm . It should be emphasized that in the present work the material tensors C1 and C2 consist

Representative volume element (RVE).

only of the stiffnesses in the longitudinal direction of the rebar provided by the Young’s modulus of steel Es and by the shear stiffnesses given either by the shear modulus of steel Gs or by the effective dowel stiffness G D (αc ). C1 and C2 have to transformed to the global coordinate system. Depending on the related angles α1 , α2 and volume fractions of each rebar, the expected effective mechanical response of the RVE is therefore in general anisotropic or transversal isotropic. Hill’s condition requires the equality of the energy on the micro and macro level independently of the constitutive law. This condition is a priori fulfilled by homogeneous strain boundary conditions applied by prescribing linear displacements at the boundary of the RVE (Zohdi and Wriggers 2005) u(x) = ε ∗ · x,

x ǫ∂V ,

(14)

where ε∗ defines the macroscopic (constant) strain tensor. 4.3 Micro-macro mapping In order to evaluate the homogenized values of the strains and stresses, the local strain and stress fields within the RVE are averaged over the total volume V   1 V = ε(x)dV = ci i (15) V v i   1 V = σ (x)dV = ci i . (16) V v i

Figure 3. Illustration of reinforced concrete as a three-phase composite material.

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Since the local averaged field values are assumed to be constant within each phase (σ i =i and εi =i ), they can be summed up according to the volume fraction ci = Vi /V , whereby Vi is the total volume of the phase i within the RVE. Hence,

the volume of the RVE of the considered three-phase composite is assumed to be filled completely by all phases i.e. c1 + c2 + cm = 1. According to the average strain theorem, for any perfectly bonded heterogeneous body the averaged strains V given by equation (15) can be identified as the macroscopic strain tensor ε ∗ applied on the RVE i.e. V = ε∗ , which is independent of the considered constitutive laws (Zohdi and Wriggers 2005). For a three-phase composite, a reformulation of equation (15) and (16) leads to ∗

ε =v = c1 ε1 + c2 ε2 + cm ε m

(17)

describing the homogeneous macroscopic strains ε∗ applied on the composite material and to σ ∗ = v = c1 σ 1 + c2 σ 2 + cm σ m

(18)

for the macroscopic stresses σ ∗ representing the composite stress field. The related local stress tensors σ i (εi ) are calculated according to the constitutive laws which are already given in equations (4) and (7). The unknown local strain fields ε i have to estimated by means of the forth-order localization (concentration) tensor Ai which relates the homogenized macroscopic strains ε∗ to the local strains within each phase εi = Ai : ε ∗ ,

i = 1, 2, m.

(19)

The tensor Ai of each phase accounts for the morphology of the microstructure by considering the elasticity, the volume fraction, the aspect ratio, the orientation and the shape of each constituent. Since the distribution of the stiffness within a reinforced concrete structure is discontinuous, the strains of the matrix may differ from the strains of the reinforcement even when full bonding between reinforcement and matrix is assumed. Therefore the concentration tensor of the matrix Am usually differs from that of the reinforcement. It should be emphasized that Ai relates micro and macro quantities and depends on the chosen micromechanical model. If a three-phase composite is considered only two concentration tensors have to be known. The third one can be determined from the average value V = c1 A1 + c2 A2 + cm Am = 11,

(20)

with 11 denoting the forth-order unit tensor. Due to the different orientation and shape of the inhomogeneities, which is captured by Ai , the mechanical response of the related homogenized stiffness tensor C ∗ is in general anisotropic even if all constituents are isotropic. As long as all constituents are in the elastic regime, the mechanical constitutive relation for a composite

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material is defined by σ ∗ = C ∗ : ε∗ ,

(21)

where C ∗ can be derived from the localization tensors of each phase  C ∗ =V = ci Ci : Ai , i = 1, 2, m. (22) i

4.4 Three-phase Mori-Tanaka approach An appropriate homogenization scheme to derive the effective mechanical response of a RVE is provided by the widely used Mori-Tanaka approach (Mori and Tanaka 1973). This micromechanical model ensures the continuity of the matrix phase and accounts for mechanical interactions between the inclusions in an average manner. According to this homogenization scheme, the reference material playing the predominant morphological role of the composite is the continuous matrix. The inclusions and their states of strain and stress are directly affected by the matrix material. Within this approach, which is also denoted as effective field theory, for a two-phase composite (c2 = 0) the limit cases are covered. If, for example, no inclusions are considered (c1 = 0), the macroscopic stiffness is given by the matrix phase C ∗ = Cm and if no matrix phase is considered (cm = 0) the macroscopic stiffness is solely manifested by the inclusion material C ∗ = C1 . In the following, the Mori-Tanaka equations for a non-linear three-phase composite are summarized. The relation between the strains of the phases ε i and the applied macroscopic strains ε∗ on the boundary of the RVE is formulated in the general format : ε∗ , ε i = AMT i

i = 1, 2, m.

(23)

In order to identify the fourth-order concentration of each phase, the related assumptions tensor AMT i of the Mori-Tanaka approach have to be taken into account. As mentioned before, the average strains of the inclusions (ε 1 , ε2 ) are defined by the average strains of the matrix εm ε 1 = T1 : ε m

and ε 2 = T2 : ε m .

(24)

Based on Eshelby’s equivalent inclusion approach, the fourth-order tensors T1 and T2 of each inclusion can be estimated by reformulating the inclusion inhomogeneity problem as a homogeneous problem with eigenstrains (Eshelby 1957). The solution for a single elastic inhomogeneity with an ellipsoidal shape perfectly bonded to a surrounding homogeneous matrix is given by Tj = [11 + S j : (Cm−1 : Cj − 11)]−1

j = 1, 2

(25)

For an ellipsoidal geometry of the inclusions the related forth-order Eshelby tensor Sj is solely dependent on the aspect ratio of the inclusion and on Poisson’s ratio νm of the surrounding isotropic matrix Cm . Since a cylindrical shape can be regarded as an ellipsoidal geometry with a particular aspect ratio, the solution of the Eshelby tensor for the considered straight rebars can be computed explicitly (Eshelby 1957). In Figure 5, the cylindrical inhomogeneity representing a single rebar is illustrated. Depending on the aspect ratio s = a2 /a1 the cross section of the rebar may have an elliptical or circular shape. In this Figure, over-bars are used to characterize the local coordinate system. Note that for each set of rebars with the orientation αj ( j = 1, 2) within the RVE, the related Eshelby tensor Sj has to be transformed from the local to the global coordinate system. The tensor T given by equation (25) correlates exactly to the localization tensor resulting from the dilute approach, where no interactions between inclusions are considered (Benveniste 1987, Gross and Seelig 2006). The application of the dilute approach is, however, limited to composites with very small volume fractions of the inclusions. For the present three-phase composite the concenintroduced in equation (25) related tration tensors AMT i to each phase can be identified from combining equation (17) for the homogenized macroscopic strains ε∗ with the microscopic strains ε i given by equation (24) as −1 AMT + cm T1−1 ]−1 1 = [c1 11 + c2 T2 : T1

(26)

−1 + c2 11 + cm T2−1 ]−1 AMT 2 = [c1 T1 : T2

(27)

−1 AMT m = [c1 T1 + c2 T2 + cm 11] .

(28)

is a function of T1 and T2 , it is obviSince AMT i ous that the strains in each phase are affected by the

other constituents, which allows for the consideration of micromechanical interactions within the MoriTanaka strategy. 4.5 Influence of damage and plasticity As soon as the mechanical response of the matrix becomes inelastic, the fourth-order tensor Tj given in equation (25) and the related concentration tensor AMT have to be defined according to the actual damage i state. Hence, in the post-cracking regime the tensors Tj are calculated with the degenerated secant stiffness of the matrix phase Cm = ψCm0 where ψ is the remaining integrity (see equation (4)). The secant stiffnesses of the steel reinforcement C1 and C2 , however, remain unchanged even in the post-yielding regime. Since the Mori-Tanaka homogenization approach is based on the existence and dominance of the matrix phase, for ψ → 0 the calculated composite shear stiffness tends to zero. Hence, the dowel stiffness would be reduced to zero. In the longitudinal direction of the rebar however, for ψ → 0 the composite stiffness is represented correctly by the axial steel stiffness. In order to reproduce the correct residual shear stiffness (dowel action) for a completely cracked matrix material (ψ = 0), the shear components of the Eshelby tensor S are scaled with ψ 2 . This modification allows to reproduce the correct limits regarding the macroscopic shear behavior of a cracked reinforced concrete element. In the post-cracking range of the matrix or in the yielding regime of the rebars, the homogenized tangent operator of the composite C ∗,tan needs to be computed in the context of the solution of the linearized equilibrium equations. Hence, depending on the damage or yielding state, the tangent stiffness Citan for each phase is first derived from the adopted constitutive laws given in Sections 2.1 and 2.2, respectively, Citan = dσ i /dε i

i = 1, 2, m.

(29)

Finally, the homogenized non-linear tangent operator C ∗,tan which is used for formulating the stiffness matrix of the composite can be calculated dσ ∗ dσ 1 dσ 2 dσ m = c1 ∗ + c2 ∗ + cm ∗ dε∗ dε dε dε dσ 1 dε1 dσ 2 dε2 dσ m dε m = c1 : + c2 : + cm : dε1 dε ∗ dε 2 dε∗ dε m dε∗

C ∗,tan =

tan MT tan MT = c1 C1tan : AMT 1 + c2 C2 : A2 + cm Cm : Am

= Cmtan + c1 (C1tan − Cmtan ) : AMT 1 Figure 5.

+ c2 (C2tan − Cmtan ) : AMT 2 .

Representation of an inhomogeneity.

222

(30)

As mentioned before, in the non-linear regime also is affected by the damthe concentration tensor AMT i age or yielding state, which is manifested by Ti given in equation (25). As long as all constituents are within the elastic range, equation (30) coincides with equation (22) and the classical micromechanical laws of elastic composites are restored.

5 VALIDATION OF THE MODEL 5.1 3-point-bending tests The performance of the proposed reinforced concrete model is illustrated by reanalyzes of 3-point-bending tests by means of the Finite-Element-Method using 1116 volume-elements. Two identical concrete beams of the dimensions 1350 · 300 · 50 mm3 (length · height · depth) with different steel ratios of ρs = 0.065% and ρs = 0.13% were loaded by prescribed displacements. The structures were reinforced with ripped steel bars of 2φ2.5 mm and 4φ2.5 mm, respectively. The experimental procedure, boundary conditions and the related material parameters are given in (Ruiz et al. (1998)). For the concrete and for the steel reinforcement the proposed elasto-plastic damage model described in Section 2.1 and the elasto-plastic material model (section 2.2), respectively, are used. The bond properties are estimated by the performed pull-out experiments documented in (Ruiz et al. (1998)). Figure 6 shows the numerical results vs. the measured data. For each beam three simulations are performed. The load-displacement curve obtained for the same beam analyzed as a plain concrete structure is included in dark grey. The two numerical results depicted in black correspond to the reinforced structure, whereby the solid curve is obtained by assuming a small notch in the center of the beam (with predamage). A comparison of the experimental data (light grey) with the numerical simulations shows good agreement for both reinforced concrete beams.

Figure 6. Re-analyzes of Three-Point-Bending tests on RC beams (Ruiz et al. 1998). Steel ratio: ρs = 0.065% (left) and ρs = 0.13% (right).

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5.2 Shear RC panel In order to investigate the influence of the bond quality and dowel action on the structural load carrying capacity, a shear loaded reinforced panel with the dimensions 890 · 890 · 70 mm3 (length · height · depth) is reanalyzed numerically. The panel is reinforced biaxially with the steel ratio ρy = ρz = 1.79% with two layers of reinforcement in thickness direction. The experimental set-up, the boundary conditions, the measuring technique and the governing material parameters are documented in (Collins et al. 1985; Vecchio and Collins 1986). In the documentation of the experimental investigation, the considered panel is denoted as PV27. For the numerical simulation, some missing material parameter are estimated by the measured compressive strength according to (Vecchio and Collins 1986; CEB-FIP 1990). Since no information concerning the bond quality and dowel action are documented in the experimental study, these properties are assumed according to lower and upper limits in the numerical analyses performed with the proposed reinforced concrete model including the presented bond slip and dowel action mechanisms. For the finiteelement simulation, 1024 volume-elements are used whereby only one element layer is chosen in the thickness direction. Since the reinforcement is distributed homogeneously within the panel, the material properties of reinforced concrete have been assigned to all finite elements. Elements containing plain concrete elements are not used in this analysis. The shear-stresses τ versus the equivalent shear-strains γ are shown in Figure 7. The structural shear-strains are computed from the prescribed displacements at the edges of the panel, and the corresponding shear-stresses are obtained by averaging the local shear-stresses also calculated at the edges of the panel. The solid lines in black and dark gray color are obtained by assuming the same (low) bond quality. The dashed line corresponds to the dark gray curve (with dowel action), assuming, however, an improved

Figure 7. Re-analysis of a shear test on RC panels (Collins et al. 1985; Vecchio and Collins 1986).

bond quality. The numerical result in which the dowel mechanisms are completely disregarded (black curve) underestimates the structural response considerably. If the residual shear stiffness of the cracked reinforced concrete is taken into account (solid dark gray curve), a relatively good agreement between the experimental and numerical results is obtained with regards to the structural stiffness. To capture the structural load carrying capacity accurately, however, an improved bond behavior has to be adopted. In this analysis the maximum load carrying capacity of the panel is caused by exhausting the ultimate bond stresses and by concrete crushing. Yielding of the reinforcement was not observed. 6 CONCLUDING REMARKS Based on a continuum micromechanics oriented concept, a constitutive model for reinforced concrete including bond slip and dowel action mechanisms is presented in this paper. For each constituent, the nonlinear pre- and post-peak behavior is described separately. The proposed homogenization approach allows considering steel-concrete interactions without an explicit discretization of the reinforcing bars. As was shown by selected validation examples on a structural level, the main mechanisms governing the structural behavior of RC structures are well captured by the proposed model. The concrete model presented in Section 2.1 is originally formulated within a poromechanics framework. This more general model framework allows to employ the proposed model for durability-oriented numerical analyses of reinforced concrete structures considering also moisture and heat transport. This becomes relevant for the numerical analysis of corrosion induced damage in RC structures which is the topic of future research. REFERENCES Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6, 147–157. CEB-FIP (1990). Model Code 1990, Bulletin d’Information. Lausanne, Switzerland: CEB. Collins, M., Vecchio, F. and Mehlhorn, G. (1985). An international competition to predict the response of reinforced concrete panels. Canadian Journal of Civil Engineering 12, 624–644.

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Coussy, O. (2004). Poromechanics. Chichester, England: Wiley. El-Ariss, B. (2006). Shear mechanism in cracked concrete. International Journal of Applied Mathematics and Mechanics 2(3), 24–31. Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London Series A, 241, 376–396. Grasberger, S. and Meschke, G. (2004). Thermohygromechanical degradation of concrete: From coupled 3D material modelling to durability-oriented multifield structural analyses. Materials and Structures 37, 244–256. Gross, D. and Seelig, T. (2006). Bruchmechanik—Mit einer Einführung in die Mikromechanik. Springer. He, X. and Kwan, A. (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Computers and Structures 79, 595–604. Linero, D. (2006). A model of material failure for reinforced concrete via continuum strong discontinuity approach and mixing theory. Ph. D. thesis, Universitat Politècnica de Catalunya, Barcelona. Luccioni, B., López, D. and Danesi, R. (2005). Bond-slip in reinforced concrete elements. Journal of Structural Engineering 131(11), 1690–1698. Manzoli, O., Oliver, J., Huespe, A. and Diaz, G. (2008). A mixture theory based method for three dimensional modeling of reinforced concrete members with embedded crack finite elements. Computers and Concrete 5(4), 401–416. Meschke, G. and Grasberger, S. (2003). Numerical modeling of coupled hygromechanical degradation of cementitious materials. Journal of Engineering Mechanics (ASCE) 129(4), 383–392. Meschke, G., Lackner, R. and Mang, H. (1998). An anisotropic elastoplastic-damage model for plain concrete. International Journal for Numerical Methods in Engineering 42, 703–727. Mori, T. and Tanaka, K. (1973). Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574. Richter, M. (2005). Entwicklung Mechanischer Modelle zur analytischen Beschreibung der Materialeigenschaften von textilbewährtem Feinbeton. Ph.D. thesis, TU Dresden, Germany. Ruiz, G., Elices, M. and Planas, J. (1998). Experimental study of fracture of lightly reinforced concrete beams. Materials and Structures 31, 683–691. Simo, J. and Hughes, T. (1998). Computational inelasticity. Berlin: Springer. Vecchio, F. and Collins, M. (1986). The modified compression-field theory for reinforced concrete elements subjected to shear. ACI Material Journal, March– April, 219–231. Zohdi, T.I. and Wriggers, P. (2005). Introduction to Computational Micromechanics. Springer.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Development of constitutive model of shear stress transfer on concrete crack surface considering shear stress softening Y. Takase & T. Ikeda Research Institute of Technology, Tobishima Corporation, Chiba, Japan

T. Wada Hokkaido Polytechnic College, Hokkaido, Japan

ABSTRACT: In a previous paper by the authors, local stress softening phenomenon found on surface of crack subjected to under shear loading was observed by constant contact ratio tests for the first time. Therefore, the authors constructed a nouveau constitutive model that enables to estimate for shear stress transfer on concrete crack surface. The proposed model was based on certain physical theory. Details of the proposed model, which consides stress softening based on the experimental results are presented in this paper. Moreover, the relidity of this model is comfirmed by comparing with experimental results. 1 INTRODUCTION Studies for shear stress transfer mechanism on concrete crack surface has begun since the 1960s, and until now a multitude of shear transfer models with various interpretive concepts has been proposed. Current studies indicate that ‘‘shear stress transfer models of physical contact type’’ possess higher reliability and possibilities than other types of models. In order to construct a model of this type, experimental data with a high degree of accuracy is indispensable. Accurate verification of constitutive model can only be possible with the following requirements: a shape measurement method, which is able to measure concavo-convex shape on local crack surface accurately; an analytical method and also a shear loading experiment equipment which is accurately controllable on the amount of physical local contact between crack surfaces. It is seemed that there are no studies that fulfill the above requirements. Under the previously mentioned study background, the authors conducted study by taking following steps: 1) Shape measurement and analysis by light-projection method; 2) Development of highly accurate shear loading equipment based on the PID (Proportional, Integral, Deferential) control theory; 3) Experimental verification of shear stress transfer mechanism based on the contact theory. With the findings in the previous studies by the authors, completed with the work carred out in the study fulfill the requirements mentioned above, a nouveau constitutive model was developed. The validity of the model was then verified by comparing the experimental results with those previously obtained.

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2 MODELING OF CONTACT STRESS-CONTACT DISPLACEMENT CURVE WITH STRESS SOFTENING Figure 1 shows mechanism of shear stress transfer and, Figure 2 shows the conceptual depiction of contact stress operating on local crack surface hypothecating that effect of friction stress is ignorable. In the shear stress transfer model based on the contact theory, shear stress transfer is acquired by integration of contact stress occurring on a local surface with regards to the whole crack surface. As a for re-result, the contact stress-contact displacement curve for a local surface is found to be vital and inegrectable in the shear stress transfer model. That is the reason that the authors conducted constant contact ratio tests in a previous study to get to know the behavior of contact stress of local surface. The results will be discussed in the later section.

Vertical Stress

Shear Disp.

Shear Stress

Crack Width

Figure 1.

Mechanism of shear transfer on crack surface.

Shear displacement Crack width

cn

Contact stress Incline angle

Contact displacement

Figure 2. Conceptual depiction of contact stress operating on local crack surface hypothecating that friction stress is ignorable.

With the results, it became clear that the contact stress curve in a post-peak area is in a nonlinear trend accompanied by stress softening. Based on these experimental results, the authors try formulated a mathematical model for σcn –ωθ curve which considers stress softening. In the final part of this paper, discussion are on implement this model using the contact theory to construct the proposed model for shear stress transfer on concrete crack surface. 2.1 Envelope model of contact stress-displacement curve

√

e·

  ω2 σcnc · EXP − θ2 · ωθ ωθc 2ωθc

K0: Initial contact Unloading

Reloading K0

K0

Contact displacement (mm)

Figure 3. Conceptual behavior shape of curve model proposed in this paper.

Contact displacement is given by the equation (2) which is used in the same previous models of physical contact type by the shear displacement and the crack width. ωθ = δ · sin θ − ω · cos θ(ωθ ≥ 0)

(2)

2.2 Impact evaluation of cyclic history

Figure 3 shows conceptual behavior shape of curve model proposed in this study adopted in. In the contact ratio constant tests by the authors, it is difficult to evaluate σcn –ωθ curve on arbitrary incline angle on local surface for the lack of data at this point, because incline angle of contacting local surface is confined to a certain degree during the shear loading. However it is possible to know approximate behavior of σcn –ωθ curve from ‘‘the average contact stress—average contact displacement’’ curve analyzed by the crack shape model of experimental results. For the above reason, applying the experimental results to develop σcn –ωθ model needs next two steps. First, using examples from ‘‘average contact stress—average contact displacement’’ curve for modeling it. Next introduction the model in contact theory, simulating experimental curve of constant contact ratio, if confirmed by the possibility of simulated the experimental curves, the formulation model is judged as contact stress—contact displacement curve on local surface. In addition, previous contact stress models are mostly elastic—plastic models, so it is clearly different from the experimental results of contact ratio constant tests by the authors. And so the actual nonlinearity and the softening phenomenon of post-peak areas are duplicated by exponent function in this model. Then the coordinate of peak value has shown by the peak contact stress and the contact displacement. σcn =

Contact stress (MPa)

cn

(1)

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Behavior of the model with cyclic shear history is preferable to con figure modeling based on the experimental results. However, it was difficult to figure out qualitative nature on cyclic behavior using highprecision controlled loading system of the authors. In the proposal model, as shown in Figure 2, incremental contact stress is written in the form of the equation (3) supposing stress process under unloading and reloading is able to duplicate by the initial contact stiffness.  √ dσ cn = e · σ cnc ωθc · dωθ (ωθq ≤ ωθ ≤ ωθp ) (3)

In the equation above ωθp represent maximum contact displacement history on the local surface. ωθq is the contact displacement when the contact stress zero, drawing a tilted straight line of initial contact stiffness from stress value of contact displacement. 2.3 Maximum contact displacement on local surface occurs in adjacent spaces and the displacement

Figure 4 shows stress field by usual contact theory. The Figure 4 enables to understand that only detached local surfaces have contact with each other. However Figure 5 and Figure 6 show contact concept on local surface and the model noticing ‘‘steep-sided surface’’ and ‘‘mildly-sided surface’’ in the crack surface. From the results of contact ratio constant tests, as the contact ratio increases the peak value of contact stress decreases, stress softening curve of post-peak becomes shallow curve, and the displacement on peak

ωθc = 2.5 · cos3 θ

Contact area cn

Figure 4.

3 ADOPTATION OF CONTACT STRESS-DISPLACEMENT MODEL TO CONTACT THEORY

Stress field by usual contact theory.

Steep-sided surface Aggregate

Contact Area

Figure 5.

Adequacy of the contact model will be verified by comparison with the experimental results later.

Crack width Shear displacement Contact displacement Contact stress

cn

Due to modeling shear stress transfer stress on concrete crack surface, the process applying the contact behavior model, which was developed in previous chapter with the shape model on crack surface, to contact theory will be described in this chapter.

Contact stress circumferentially exerted in actual phenomenon

Contact stress evaluated in previous models

3.1 Equilibrium equation of stress based on contact theory

Contact concept on steep-sided surface.

In the previous contact theory, it is assumed that the sum of contact stress balances with shear stress transfer stress occurring throughout crack surfaces. The proposal model adopts the same assumption. In order to set up shear stress and normal stress based on the assumption, formulated contact stress into shear element and vertical element, then each component are multiplied by the crack shape function and integrated. The formulation is shown in equation (6) and (7).

contact stress exerted in Contact stress evaluated actual phenomenon Mildly-sided in previous models surface Aggregate

Contact Area

Figure 6.

Contact area evaluated in contiguous steep-sided surface

Contact concept on mildly-sided surface.

tends to increase. These results show that contact phenomenon occurs around local areas and the breakdown progression quickens on a steep-sided local surface with larger contact quantity. On the other hand, the breakdown progression slows on a mildlysided local surface with small contact displacement. Above-mentioned two types of contact condition were reflected into σcn –ωθ curve model. In order to apply contact condition on crack surface to modeling, formulation of model that maximum contact stress vary with incline angle was attempted. However critical contact angle of the contact ratio constant tests were based on only three experimental data in the 8π /36 to 10π/36 range. For this reason modeling beyond the range just had to be estimated from the experimental results. In the proposal model, the equation (4) shows the maximum contact stress using incline angle to estimate the experimental results of above mentioned three data. σcnc = 480 sin3 |θ|

(5)

(4)

Also the equation (5) shows the contact displacement under the maximum stress with fracture mechanism changing, as the incline angle on local surface is getting moderate.

227

τ=

π/2

kcn · σcn · sin θ · (θ)dθ

(6)

π/2

kcn · σcn · cos θ · (θ)dθ

(7)

−π/2

σ =

−π/2

The kcn in the equation above is the experimental coefficient, which is the physical quantity to compute stress, imponderable only with the simple 2D contact, on local crack surface in the proposal model. And this kcn shown in the equation (8) is compatible with previous experimental results, however, further analyses will be a future agenda. kcn = 15/8

(8)

3.2 Simulation of crack surface shape Figure 7 is a chart that compares 2D incline density function model with the process of measurement result. There are many possible elements in shape property on concrete crack surface. Constructing high accuracy model needs formulation with various physical parameters. Nevertheless the proposal model is a simplified

Table 1.

1.5

List of specimens.

(a) Constant contact ratio series.

Average incline density 1.0

Proposal equation

0.5

Specimen

Cont. ratio

Critical cont. angle (rad.)

Cont. area (mm2 )

RCN0157 RCN0271 RCN0439

0.0157 0.0271 0.0439

10/36π 9/36π 8/36π

118 203 309

Cont. = contact.

(b) Constant crack width series.

0.0 0 θ (rad.)

δ( mm)

Figure 7. Chart that compares 2D incline density function model with the process of measurement results.

one because the development is the initial step. In the proposal model, 2D incline density function model, which is regarded as significant influence for shear stress transfer in published papers, is used. Approximation of 2D incline density function model is shown in the equation (9) and (10). (θ) =

 3 a0 + a1 · |θ |n · cos3 θ 2

n + 1  π −1 , a0 = 2n 2

a1 = −

ω0 (mm) +1 −1

+2

−2

+3 −3

W03δ06O3 W05δ08O3 W10δ09O3 W01δ05R1 W03δ05R1 W05δ08R1 W10δ08R1 W03δ06R2 W05δ09R2 W10δ09R2

0.3 0.5 1.0 0.1 0.3 0.5 1.0 0.3 0.5 1.0

0.45 0.6 0.7 – – – – 0.6 0.9 0.9

– – – – – – – −0.6 −0.9 −0.9

0.6 0.8 0.9 – – – – – – –

0.3 0.4 0.5 0.5 0.5 0.8 0.8 0.4 0.6 0.6

– – – −0.5 −0.5 −0.8 −0.8 −0.4 −0.6 −0.6

– – – – – – – – – –

(9) Table 2.

n + 1  π −(n+1) 2n

Specimen

2

Concrete mix. Mix (kg/m3 )

(10)

Size of agg.

W/C

Water

Cem.

Agg.

Adm.

In the above equation, set n to 0.6. Undoubtedly the approximation has high adequacy of the measurement result of shapes on crack surface in Figure 7.

20 mm

42%

173

412

1, 718

4.491

4 ADEQUACY VALIDATION OF EXPERIMENTAL VALUE OF SHEAR STRESS TRANSFER MODEL

Table 3.

In this chapter, the adequacy will be verified by comparing the experimental results of the proposal model with the previous model. 4.1 Active parameter of specimen in comparison Table 1 shows active parameter list of specimen, Table 2 shows concrete mix, and each material property are listed in Table 3. Adequacy of σcn –ω model on a local surface will be verified among specimens of constant contact ratio series shown in Table 1(a). As their specimen named, RCN code and contact ratio values are used. And Table 1(b) shows specimen of three types of constant crack width series with different loading history. Using

228

Cem. = cement, Agg. = aggregate, Adm. = admixture.

Material properties of concrete.

Comp. strength

Tensile strength

Young’s modulus

40.1 N/mm2

2.61 N/mm2

23.69 N/mm2

those specimens, the comparison between the proposal model and previous models will be noted. Name of specimen is composed with a combination of decided crack width, maximum shear displacement, and loading cycle. 4.2 Verifying of contact stress σcn -contact displacement ωθ in the proposal model There is no previous study verifying adequacy or compatibility of contact stress—contact displacement on

a local surface. As far as the authors know, above mentioned data of constant contact ratio test conducted by the authors is the only experimental data of the behavior of contact stress σcn –ωθ displacement on local surface. In this chapter, implementing σcn –ωθ model previously formulated on local crack surface in contact theory, and the adequacy of the model will be verified by comparing experimental value with the answer of numerical calculation under the contact ratio constant condition. Focus on the shear stress-shear displacement curves shown in each center figures in three different contact ratio specimens in Figure 8 (a), (b) and (c), maximum shear stress value and the tendency of experimental curve, like softening behavior on a post-peak, have been reproduced roughly by the proposal model. About shear stress—normal stress curve in figures on the right side, experimental value of stress gradient and peak value in the proposal model shows overall tendency. Seeing the comparison result, the σcn –ωθ model in Chapter Three can be identified as a predictable model for contact stress-contact displacement behavior on an actual local surface. 4.3 Compatibility validation of the proposal model and LI model on cyclic loading Using three types of experimental results in Table 1(b), the compatibility of the proposal model and Li model,

1.6

(mm)

RCN0157 fc = 40MPa 1.2 0.8 0.4 0

4

(MPa)

4

Starting point con =

4.3.1 In case of the cyclic shear history of one way three cycle Figure 9(a), (b) and (c) show the experimental results of specimen W03d06O3, W05d08O3, and W10d09O3, setting each crack width to 0.3 mm, 0.5 mm, and 1.0 mm, with one way cyclic shear pressing and analysis values of each models. In those figures, figures on the left show crack width–shear displacement curves, center figures show shear stress—shear displacement curves, and figures on the right show shear stress normal stress curves. What follows is comparison between experimental values, the proposal model, and Li model in all of those figures. At first the result of W03d06O3 specimen, target crack width to 0.3 mm, is shown in Figure 9(a). In the center, shear stress τ –δ shear displacement curve, the proposal model is almost correspond with the experimental value on the maximum value of shear stress and shear stiffness, but somewhat higher curve. However, in Li model, the proposal model is almost coincident with the experimental value on the maximum stress value, but both loading and unloading rigidity were neither reproducible nor describable. Looking at the shear stress-normal stress curve on the right side, the proposal model estimates normal stress rather higher, but the tendency is almost coincident with

(MPa) Model

lim

Target line 2 ch1 ch2 ch3 ch4 tan lim=1.192 (mm) 1 2 0

which is most frequently used among published papers, most reliable and expansible among previously analyzed models, will be verified in this chapter.

1

0.3

Exp.

Exp. (mm) 2 0

(MPa) 4

2

(mm)

RCN0271 fc = 40MPa 1.2 0.8 0.4 0

6

(MPa)

-0.1 0 6

Starting point

0.9

Exp.

Model 0.0271 1

3

(MPa) 6

0.1

(b) RCN0271 specimen 1.6

(mm)

RCN0439 1.2 fc = 40MPa 0.8 0.4 0

6

(MPa)

6

Target line 3 ch1 ch2 ch3 ch4 tan lim=0.839 (mm) 1 2 0

0

1.4

Model 3

1.0

Starting point lim con =

0.0439 1

(mm) 2 0

(MPa) Proposal Li-Maekawa Exp.

(MPa) -1 2

0

(MPa)

(mm) ch1 ch2 ch3 fc = 40MPa ch4 1

(mm) 1

0

(MPa)

12

Proposal Li-Maekawa Exp.

Proposal Li-Maekawa Exp.

(MPa) -1 2

0

(b) W05d08O3 specimen

(MPa) Exp.

Model Exp.

12

(mm) 1

0

12

W05 08O3 0 = 0.5mm

0.5

Model (mm) 2 0

(MPa) Proposal Li-Maekawa Exp.

(a) W03d06O3 specimen (mm)

3

lim con =

(mm) ch1 ch2 ch3 fc = 40MPa ch4 1

(MPa)

Exp.

Target line 3 ch1 ch2 ch3 ch4 tan lim=1.000 (mm) 1 2 0

12

W03 06O3 0 = 0.3mm

2 Model

(a) RCN0157 specimen 1.6

(mm)

0.7

0.0157

3

(MPa) 6

0.60

(mm)

12

W10 09O3 0 = 1.0mm (mm) ch1 ch2 ch3 fc = 40MPa ch4 1

0

(MPa) Proposal Li-Maekawa Exp.

12

(mm) 1 0

(MPa) Proposal Li-Maekawa Exp.

(MPa) -1 2

(c) W10d08O3 specimen

(c) RCN0439 specimen

Figure 8. Comparisons between experimental results and models of contact ratio constant.

229

Figure 9. Comparison between experimental results and models of one way three cycles.

the experimental value. Previous models are almost coincident with the experimental value. Specimen W05d08O3 target crack width to 0.5 mm in Figure 9(b) is the next topic. In the shear stress-shear displacement curve in the middle, the same with the specimen W05d08O3 setting crack width to 0.3 mm in the same figure, magnitude of maximum shear stress is nearly described. However, it is impossible to duplicate a concave upward carve, which is shown in the experiment, neither in the proposal model nor in Li model. Nonetheless, the curve of unloading time in the proposal model is mostly pursuing experimental values and it is more compatible than Li model. In the shear stress-normal stress curve on the right side of the same figure, the proposal model is almost coincident with experimental results but in Li model the normal stress is computed rather low. And finally, specimen W10d09O3 target crack width to 1.0 mm in the same figure will be verified. It is visible in shear stress—shear displacement curve in the middle of the same figure that the proposal model is estimated smaller than the experimental value. On the other hand, the stress value of Li model is near the experimental value. The adaptation of differences, as described in Section 3 Paragraph 3, including the experimental results of other loading type, target crack width to 1.0 mm, will be conciderd again at the end of this paper. Aside from the stress discussion, large and small, the proposal model describes stress gradient well as the comparison between both models about shear stress-normal stress curve on the right side of the same figure. Adaptation of the proposal model has been cleared. Compatibility comparison of the proposal model and previous Li model have been mentioned above. It will be later described that the results of verification of specimen series, other RV-1 and RV-2, differ little from the consideration in this paragraph. In the validation, starting from the next paragraph, Li model will be mentioned for reference, but not thorough detailed consideration. 4.3.2 In case of shear history of reversed one cycle Figure 10(a), (b), (c) and (d) show that comparison results of specimen W01d05R1, W03d05R1, W05d08R1, and W10d08R1 target crack width to 0.1 mm, 0.3 mm, 0.5 mm, and 1.0 mm under the shear pressing of reversed one cycle. In those figures, analysis curves of the experimental results, the proposal model, and Li model were shown together. First of all, the result of specimen setting and kept the crack width to 0.1 mm in Figure 9(a) is observed. The specimen target crack width to 0.1 mm is the only one among three loading types, therefore Li model will be noted here. The proposal model almost coincide the maximum stress value of the experimental value rather than shear stress-shear displacement curve in

(mm)

0.5

P roposal Li-Maekawa Exp.

0 =0.1mm

0.1

(mm)

(MPa) 15 Proposal Exp.

(MPa)

15

W01 05R1 fc = 40MPa

0

(mm)

0

(MPa)

ch1 ch2 ch3 ch4 -0.3 -1

1 -15-1

0

0

1

-15

LiMaekawa 0

-1 5

(a) W01d05R1 specimen (mm)

0.7

(MPa)

12

W03 05R1 fc = 40MPa

(MPa)

12

Proposal

Proposal

Li-Maekawa Exp.

0 =0.3mm

0.3

(mm)

0

(mm)

Exp.

0

(MPa)

ch1 ch2 ch3 ch4 -0.1 -1

0

1

-12 -1

0

1

-12

LiMaekawa 0

-1 2

(b) W03d05R1 specimen 0.9

(mm)

12

W05 08R1 fc = 40MPa 0 =0.5mm

(MPa)

(MPa)

12

Proposal Li-Maekawa

Proposal

Exp. 0.5

(mm)

0

(mm)

Exp.

0

(MPa)

ch1 ch2 ch3 ch4 0.1 -1

0

1

-12 -1

0

1

-12

LiMaekawa 0

-1 2

(c) W05d08R1 specimen 1.4

(mm)

12

W10 08R1 fc = 40MPa 0 =1.0mm

1.0

(mm)

(MPa)

12

Proposal Li-Maekawa Exp.

0

Exp. Proposal (mm)

0

ch1 ch2 ch3 ch4 0.6 -1

(MPa)

(MPa) LiMaekawa

0

1

-12 -1

0

1 -12 0

-1 2

(d) W10d08R1 specimen

Figure 10. Comparison results of reversed one cycle between models and experimental results.

the middle of the same figure, but whole hysteresis behavior has a bigger tendency than the experimental value. However, shear displacement of Li model yields with shear displacement as 0.2 mm and it has completely different behavior from the experimental curve. Also looking at the τ –σ curve on the right side of Figure 9(a), the proposal model estimates normal stress rather higher but basically compatible with the experimental values. Next, focus on the central figure of Figure 10(b) target crack width to 0.3 mm, the proposal model is able to pursue the τ –δ curve with high accuracy. About the τ –σ relation on the right side of the same figure, it is clear that the proposal model almost correspond with the experimental value. Furthermore, the specimen setting crack width to 0.5 mm in Figure 10(c) will be verified. Looking at the maximum value of the shear stress in the middle

230

figure and the normal stress in the right figure, it is distinctive in the experimental value that the stress in the negative loading is bigger than the one in the positive loading. It is impossible to evaluate the phenomenon from the theoretical assumption of the proposal model, it is clear that the experimental values are basically traceable in both figures; shear stress-shear displacement curve in the middle of same figure and shear stress—normal stress curve on the right side of the same figure. Finally, the maximum stress value of the proposal model is smaller than the experimental value in middle figure in Figure 10(d), target crack width to 1.0 mm, which has same tendency with the specimen in Figure 9(c). And about the shear stress-normal stress curve in the same figure on the right side, stress value of the proposal model is underestimated but stress ratio of normal and shear is properly described.

4.3.3 In case of cyclic shear history of reversed two cycles Figure 11(a), (b) and (c) show comparison between experimental results of specimen W03d06R2, W05d09R2, and W10d09R2, target crack width to 0.3 mm, 0.5 mm, and 1.0 mm under the shear loading of reversed two cycle, and two models.

0.7

ω (mm)

12

W03δ 06R2 fc = 40MPa

τ (MPa)

12

Proposal

τ (MPa) Exp.

Li-Maekawa

ω 0 =0.3mm

Exp. 0.3

(mm)

δ 0

(mm)

Proposal

δ 0

0

1

-12 -1

0

1

-12

5 CONCLUSION

σ

(MPa)

ch1 ch2 ch3 ch4 -0.1 -1

In this study, a unique contact stress model, which adapts the softening phenomenon to the contact stress of local concrete crack surface, was developed. Adapting this model to the previous theory create a 2D shear stress transfer model. In addition, by comparing the proposed model with the representative previous model, in terms of comparison of some experimental data; the compatibility of the model has verified. Summaries of this study are shown as follows:

LiMaekawa 0

-1 2

(a) W03d06R2 specimen 0.9

ω (mm)

12

W05δ 09R2 fc = 40MPa

τ (MPa)

12

Proposal

τ (MPa) Proposal

Li-Maekawa Exp.

ω 0 =0.5mm 0.5

(mm)

δ 0

(mm)

Exp.

δ 0

0

1

-12 -1

0

1

-12

σ

(MPa)

ch1 ch2 ch3 ch4 0.1 -1

LiMaekawa 0

-1 2

(b) W05d08R2 specimen 1.4

ω (mm)

12

W10δ 09R2 fc = 40MPa ω0 =1.0mm

1.0

τ (MPa)

12

Proposal

τ (MPa) Exp.

Li-Maekawa Exp.

(mm)

δ 0

(mm)

δ 0

ch1 ch2 ch3 ch4 0.6-1

First, focus on the specimen in Figure 11(a) setting crack width to 0.3 mm, it appears that the experimental results are totally accurate by the proposal model. Though there is a difference of stress values between the proposal model and the experimental curve during the negative loading in the shear stress-shear displacement curve. The proposal model is able to pursue the experimental value more than the shear stressnormal stress curve on the right side of the same figure. Figure 11(b) target crack width to 0.5 mm is the next topic. In the first cycle, positive and negative, the proposal model has lower initial stiffness than the shear stress—shear displacement curve in the middle of the same figure, and the transition of stress is smaller than the experimental value. In the second cycle, stiffness, maximum stress value, and the experimental value of unloading curve are described with a high degree of accuracy. In the shear stress-normal stress curve on the right side of Fig. 11, stress value of the proposal model is slightly smaller but it seems that the compatibility is higher than Li model. Then looking into the specimen in Figure 11(c) setting crack width to 1.0 mm, the maximum stress value of positive loading is slightly smaller in the shear stress-shear displacement curve in the middle, but it is clear that the experimental curve is captured by the proposal model in negative loading. In the shear stress-normal stress curve on the right side of Figure 11(c), it shows that the proposal model is able to pursue experimental values with high accuracy.

Proposal σ (MPa) LiMaekawa

0

1

-12 -1

0

1 -12 0

-1 2

(c) W10d08R2 specimen

Figure 11. Comparison results of reversed two cycles between given models and experimental results.

231

1. The proposal analysis curve, which adapts σcn –ωθ model of local surface to contact theory, is compatible with the curve of contact ratio constant tests. The proposal model has sufficient validity as contact stress σcn –ωθ displacement model on local concrete crack surface. 2. In τ –δ curve of specimen with each crack width, the proposal model is not only compatible with stiffness under loading and peak value, but also enable unloading stiffness and reloading curve to reproduce with high accuracy. 3. In τ –σ curve of each specimen, the proposal model, which almost coincides with stress gradient of the experimental curve, show rather high compatibility.

The authors will continue the experiment to study shear stress transfer mechanism for different concrete compressive strengths to develop extended models based on this proposed model.

Van Mier, J.G.M., Nooru-Mohamed, M.B., & Timmers, G. 1991. An experimental study of shear fracture and aggregate interlock in cement based composites, Heron, Vol. 36, No. 4. Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1996. Development of Mesurement of the Concrete Crack Surface by Laser Beam and Proposal of 2-Demensional Analytical Method of the Measured Image A basic study on shape properties of the concrete crack surface Part. 1-, Journal of Struct. Constr. Eng., AIJ, No. 490: 179–188. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1998. Dimensional Shape Analysis of the Concrete Crack Surfaces Introduced by Various Kinds of Stress—A basic study on shape properties of the concrete crack surface Part. 2-, Journal of Struct. Constr. Eng., AIJ, No. 504: 81–86. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1999. Proposal of 3-dimensional analytical method for the concrete crack surface image measured by laser beam—A basic study on shape properties of the concrete crack surface Part. 3-, Journal of Struct. Constr. Eng., AIJ, No. 524: 111–118. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 2000. Dimensional shape analyses of the concrete crack surfaces introduced by various kinds of stress—A basic study on shape properties of the concrete crack surface Part. 4-, Journal of Struct. Constr. Eng., AIJ, No. 534: 103–111. (in Japanese) Walraven, J.C. 1981. Fundamental Analysis of Aggregate Interlock, ASCE, Vol. 107, No. ST11: 2245–2270.

ACKNOWLEDGEMENTS The authors would like to express our gratitude to the graduates of 2005 and 2006 in Wada laboratory of Hokkaido Polytechnic College, for their generous effort in tests conducted for this study.

REFERENCES Bazant, Z.P. & Gambarova, P. 1980. Rough Cracks in Reinforced Concrete, Journal of Structure Division, ASCE, Vol. 106, No. 4: 819–842. Bujadham, B. 1991. The Universal Model for Transfer across Crack in Concrete, Department of Civil Engineering, The Graduate School of The University of Tokyo. Fenwick, R.C., & Paulay, T. 1968. Mechanism of Shear Resistance of Concrete Beams, Journal of Structure Division, ASCE, Vol. 94, No. 10: 2325–2350. Feenstra, P.H., Borst, R., & Rots, J.G. 1991. Numerical Study on Crack Dilatancy Part 1 Models and Stability Analysis, Journal of Engineering Mechanics, ASCE, Vol. 117, No. 4: 733–753. Li, B., Maekawa, K., & Okamura, H. 1989. Contact Density Model for Stress Transfer across Cracks in Concrete. Journal of the Faculty of Eng. The University of Tokyo(B), Vol. 40, No. 1: 9–52. Michael N. Fardis, & Oral, B. 1979. Shear Transfer Model for Reinforced Concrete, ASCE, Vol. 105, No. EM 2: 255–275. Millard, S.G. & Johnson, R.P. 1984. Shear Transfer across Cracks in Reinforced Concrete due to Aggregate Interlock and Dowel Action, M. of Concrete Research, Vol. 36, No. 126: 123–137. Sato, R., Wada, T., & Ueda, M. 2001. Fast Fourier One-dimensional Analysisi of Concrete Crack Surface, Proceedings of Fracture Mechanism of Concrete and Concrete Structures, Framcos-4: 423–430. Sato, R., Wada, T., & Ueda, M. 2003. Study on Shape Properties of Concrete Crack Surface in Freaquency Domain, Proceedings, Computation modelling of concrete structures, 315–324. Shinohara, Y., Kawamichi, K., & Ishitobi, S. 2001. Shear Behavior in Precracked Concrete under Cyclic Loading at Constant Crack Width, Journal of Struct. Constr. Eng., AIJ, No. 548: 101–106. (in Japanese) Takase, Y., Ueda, M., & Wada, T. 2007. Proposal of Optimal Experimental Method for Evaluation of the Concrete Shear Transfer Mechanism, Proceedings of Fracture Mechanism of Concrete and Concrete Structures, Framcos6: 333–339. Catania, Italy. Takase, Y., Wada, T., & Ueda M. 2007. Verification of Shear Transfer Mechanism Based on Contact Theory—Study on shear transfer mechanism on concrete crack surface Part 1-, Journal of Struct. Constr. Eng., AIJ, No. 622: 155–162. (in Japanese)

APPENDIX The following symbols are used in this paper: a0 = a1 = δ = dσcn = dωθ = n = θ = θlim = σ = σcn = σcnc = τ = (θ) = ω = ωθ = ωθc =

232

ωθp ωθq

Modulus of incline density function Modulus of incline density function Shear displacement Increment of contact stress Increment of contact displacement Multiplier of incline density function Contact angle Critical contact angle Normal stress Contact stress Maximum contact stress Shear stress Incline density Crack width Contact displacement Contact displacement of maximum contact stress = Maximum contact displacement = Contact displacement when the contact stress zero, drawing a tilted straight line of initial contact stiffness from stress value of maximum contact displacement

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Microplane approach for modeling of concrete under low confinement Nguyen Viet Tue & Jiabin Li Leipzig University, Leipzig, Germany

ABSTRACT: This paper mainly deals with the computational modeling of concrete subjected to low confinement. To this end, a new concrete material model is developed on the basis of the well known microplane theory. The new model, called model M4L, follows the same approach as a previous microplane model M4 but with enhanced constitutive formulation (stress-strain boundaries). The main contributions include a novel concept of confinement-adjusted effective microplane moduli, two confining pressure dependent microplane deviatoric stress boundaries and a new shear stress boundary. Numerical experiments indicate that model M4L is able to fit a wide range of test data in a realistic way. Finally, the model is implemented into the finite element software (SOFiSTiK) and used to model the behavior of tied high-strength concrete (HSC) and normal-strength concrete (NSC) column as well as a sandwich joint between HSC column and NSC slab subjected to axial compression. Numerical results indicate that the simulated response agree well with the experimental observations. for concrete under low confinement, both from the material and structural level.

1 INTRODUCTION The failure of concrete under uniaxial compression is often characterized by volume dilitancy. However, the tendency for internal cracking and volume increase is reduced when lateral confining pressure (σL ) exists. As a result, the ultimate strength, deformation capacity and the post peak ductility of concrete are greatly enhanced. In conventional reinforced concrete structures, the confining pressure level is relatively low (smaller than 30% of the uniaxial compressive strength fc ). Reliable numerical modeling of confined concrete structures and members requires a concrete material model that can accurately represent the responses of concrete under low confinement. Various theories have been used to describe the behavior of concrete subjected to low confining pressure, such as the elasticity, plasticity, damage, coupled damage-plasticity as well as endochronic and microplane theory. Each theory has its own advantages and disadvantages. Amongst of them, the microplane theory is a promising approach. The appealing aspects of microplane theory are the conceptual simplicity and the three dimensional formulation. In this paper, a new material model that can realistically capture the behavior of concrete under low confinement is developed within the microplane framework. The new model is called microplane model M4L and can be regarded as an enhanced version of the model M4 developed by Prof. Bažant and his coworkers (Bažant et al. 2000, Caner and Bažant 2000). The new model follows the same approach as model M4 but with improved constitutive formulation. Numerical experiments reveal that the model M4L can accurately reproduce a wide range of experimental observations

233

2 2.1

MICROPLANE THEORY Theoretical framework

The essential idea of the microplane modeling approach is to characterize the constitutive law in terms of stress and strains vectors on various planes in the material, called microplanes. A microplane is an arbitrary plane which cuts through the material at a point and is defined by the orientation of its normal unit vector n = (n1 , n2 , n3 ), which represents one point in the spatial unit sphere. In microplane models for concrete, the microplanes might be imagined as the damage planes or weak planes in the micro-structure, such as the inter particle contact planes, interfaces between mortar and aggregates or planes of microcracks, as shown in Figure 1. In order to generalize a stable post peak softening damage, a kinematic constraint is usually used in modeling the nonlinear behavior of quasi-brittle materials. That is, the strain vectors on microplanes are assumed to be the projection of the macroscopic strain tensor. Given the strain vectors on various microplanes, the stress vectors of work conjugacy for various microplanes can be determined through the predefined stress-strain relationships for the microplanes. Once the microscopic stress components for each microplane are known, the macroscopic stress tensor can be obtained though the principle of virtual work (Bažant et al. 1996) or the principle of energy equivalence (Carol et al. 2001). The flow path of the microplane model is given in Figure 2.

strain is imposed, ensuring energy dissipation on the microplane. After the stress components on each microplane are known, the macroscopic stress tensor σij is derived by accumulating the contributions of each microplane as: 3 2π

    δij σD Nij − 3   + σM Mij + σL Lij d

σij = σv δij +

Figure 1.

(2)

A full explicit computational algorithm is adopted in the microplane model M4, which is very helpful for achieving numerical efficiency.

Imagined microplanes in concrete.

2.3 Numerical accuracy of microplane model M4

Flow path of microplane modeling approach.

2.2 Microplane model M4 The microplane model M4 was developed by Prof. Bažant and his coworkers in 2000 (Bažant et al. 2000; Caner and Bažant 2000). This model adopted a volumetric-deviatoric-tangential (V-D-T) split for describing the strain components on each microplane. A concept of stress-strain boundary is used to simulate the inelastic behavior of concrete. There are altogether 6 stress-strain boundaries in microplane model M4. They include: – A normal stress boundary (σNb ); – Two deviatoric stress boundaries for compression and tension (σDb− , σDb+ ); – A shear stress boundary (σTb ); and – Two volumetric stress boundaries for compression and tension (σVb− , σVb+ ). Within the boundaries, the behavior of the microplane is incrementally elastic, i.e., σv = Ev εv ,

 σM = ET εM ,

 σD = ED  εD , and

σL = ET εL

– The model leads to an increase of the peak stress with the increase of the confining pressure, which is consistent with the experimental observations;

90 Model M4 Test

75 in MPa

Figure 2.

Numerical experiments indicate that the microplane model M4 is very powerful and versatile in capturing many complex inelastic behavior of concrete under various stress states (Caner and Bažant 2000), such as tension and compression softening, cracking and damage, volume dilitancy and strength dependence on high confining pressure. The model has been adopted by some popular finite element packages such as ATENA and OOFEM, as a standard concrete model. However, it was found that the microplane model M4 still exhibits some undesired responses (Tue et al. 2008). For instance, the model predictions for concrete under low confinement are quite inadequate. This can be seen from Figure 3, which shows the model M4 reproductions and the test data of Imran (Imran 1994) for concrete subjected to low confinement. A comparison between the model response and the test data indicates the following noticeable aspects:

60 45 8.4

30

(1)

15

where, EV , ED and ET are the microplane volumetric, deviatoric and tangential moduli, respectively. The exceeding of the stress-strain boundary value is never allowed. If the stress exceeds the boundary, a vertical drop to the boundary at the current

0

234

0.0 0

5

1.05 10

2.1

15 in ‰

4.2

20

25

Figure 3. A comparison of model M4 simulations and test data for concrete under low confinement.

0.8

Model M4 Ottosen model Test

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

Figure 4. A comparison of model M4 simulations and test data for the Poisson’s ratio of concrete in uniaxial compression.

– The magnitudes for the peak stress and strain at peak stress from model M4 are considerably lower than that in the test; – The model reproductions for the softening branch are also less satisfactory. As can be seen, the softening branches for different confining pressures are almost parallel to each other indicating that the lateral confinement has nearly no influence on the post peak ductility. Furthermore, model M4 gives also insufficient prediction for the lateral response of concrete under uniaxial compression, as shown in Figure 4. It can be seen that model M4 gives too much lateral expansion at an early stage. This will lead to an overestimation of the stiffness in case that the concrete expansion is an important issue, e.g. passive confinement. The above two weaknesses must be eliminated before the model can be used to accurately predict the behavior of confined concrete members in nonlinear analysis. The aim of this paper is to remedy these insufficiencies through formulating a new microplane model, called model M4L.

strain dependent (except for the shear stress boundary), the damage induced by strain increasing can be captured. However, for concrete subjected to multiaxial compression, experimental observations indicate that the slope of the stress-strain curve is influence by both the strain and the stress states. Although it is usually reasonable to assume that the initial tangential elastic modulus under multiaxial compression is independent on the confining pressure (Attard and Setunge 1996). However, if the initial hydrostatic pressure is taken into account, namely, if the stress-strain curve is plotted to begin from the confining pressure, a reduction of the tangential elastic modulus here denoted as E∗ with the increase of the confining pressure can be observed (Dahl 1992, Sfer et al. 2002). Figure 5 presents the test data of Sfer et al. (2002). It can be clearly seen from Fig. 5 that the tangential elastic modulus E ∗ decreases with the increase of the confinement. This reduction can be approximately expressed in a bilinear manner. However, the combination of constant EV , ED and ET and the stress-strain boundaries is insufficient in reproducing this behavior since the stress-strain boundaries are insufficient for reflecting the influence of the stress states. In order to improve the model performance for concrete under low confinement, the elastic description of the microplane is improved in the new model through re-determining the microplane moduli EV , ED and ET according to the stress states. A simple way is to make the microplane elastic moduli dependent on the confining pressure. The following index is chosen for representing the confining pressure in the model M4L. θ=

−σ1  ≥0 fc

where, σ1 is the first principle stress; fc is the uniaxial compressive strength of concrete, and x =

3 MICROPLANE MODEL M4L: FORMULATION AND VALIDATION

(3)

x + |x| 2 1.2

3.1 Confinement-adjusted effective microplane elastic moduli

1.0

235

0.8 E*/E

In model M4, the initial values for the microplane elastic moduli EV , ED and ET are always used in representing the elastic behavior of the microplanes. Through the use of the pre-defined stress-strain boundaries, the decrease of the toughness of the stress-strain curve, i.e., the damage to the elastic modulus with increased of stress (or strain) level, can be correctly predicted which can be seen from the uniaxial compression or tension simulations in Caner and Bažant (2000). This is because most of the stress-strain boundaries are

0.6 0.4 0.2 0.0 0.0

0.4

0.8

1.2 L

1.6

2.0

/f c

Figure 5. Influence of confining pressure on the elastic modulus of concrete (Sfer et al. 2002).

1.0

ED( )/ED

0.8 0.6 0.4 0.2 0.0 0.0

0.2 =

0.4 ( 1, f c )

0.6

0.8

0.6

0.8

(a) ED 1.0

ET( )/ET

0.8 0.6 0.4 0.2 0.0 0.0

0.2 =

0.4 ( 1, f c )

(b) E T Figure 6. Variation of microplane elastic moduli with lateral confining pressure.

In model M4L, the following microplane elastic moduli are used in describing the elastic behavior of the microplane: EV is kept constant while ED and ET are made to be varied with θ , as shown in Figure 6.This idea is on the basis of experimental observations and the work of Phillips and Zienkiewicz (1976). The detailed equation for the ED and ET can be found in Li (2009). Finally, the elastic behavior for the microplane in model M4L is calculated as: σv = Ev εv ,

 σD = ED (θ) εD ,

 σM = ET (θ)εM ,

and

σL = ET (θ)εL (4)

3.2 Stress-strain boundaries Following the same approach as in model M4, the inelastic behavior of concrete is also characterized through the concept of stress-strain boundary in the new model M4L. The normal stress boundary σNb simulates the tensile fracture of concrete while the volumetric boundaries

236

σVb− , σVb+ simulate the behaviour of concrete under hydrostatic loadings. They have only slightly influence on the model responses for concrete under low confinement. These boundaries remain in model M4L. However, the deviatoric stress boundaries σDb− , σDb+ and the shear stress boundary σTb must be enhanced for better simulating the responses of concrete subjected to low confining pressure and the lateral responses under uniaxial compression. The failure of concrete under unconfined compression or low pressure is a result of the formulation and propagation of splitting tension cracks parallel to the loading direction. This means that the cohesion plays a dominant role. Beyond the peak, with increasing deformation, the cohesion continues to decrease until it vanishes, resulting in strain softening. The influence of the confining pressure can be well reflected in the deviator space. As found by Li (2009), the microplane deviatoric strain εD comes from the macroscopic deviatoric strain tensor and is oriented in the normal direction of the microplane, which represents the interparticle cohesion in the material. In model M4, two deviatoric boundaries including a compressive and a tensile one are used to simulate the compressive softening behavior of concrete. The compressive deviatoric boundary controls the axial strain of concrete under unconfined and low confinement while the tensile deviatoric boundary controls the lateral behavior, the volume change as well as the transverse opening of the axial distributed cracks under compression. However, numerical simulations show that the two boundaries in model M4 fail to capture the behavior of lightly confined concrete in a realistic way. With the new procedure for determining the elastic behavior of the microplane with confinement-adjusted effective microplane elastic moduli, the increased deformation capacity in multiaxial compression can be obtained, however, the enhancement in the ultimate strength and especially the post peak ductility is still less satisfactory. This means that the deviatoric stress boundaries in model M4 has to be improved. In model M4L, two new deviatoric stress boundaries are developed, shown in Figure 7. The mathematic expressions are given in Li (2009). The distinct characteristics of the new boundaries are both of the boundaries are scaled according to θ . The two boundaries, together with the new procedure for describing the microplane elastic behavior, is very efficient in capturing the improvement of the ultimate strength, the deformation capacity and the post peak ductility for concrete under low confinement. This can be clearly seen in the examples given later. The shear stress boundary σTb in model M4 given in Eq. (5) has a nonlinear form and describes the frictional interaction between the normal stress σN and shear stress σT . When the normal stress σN is not

Decrease of

( 1, fc)

3.0 in MPa

-10.0

4.0

-20.0 -30.0

1.0

-40.0 -12.0 -9.0 -6.0 -3.0 0.0 3.0 6.0 in ‰ D (a) Compressive deviatoric stress boundary

0.0 -1.2

Figure 8.

in MPa b+ D

6.0 3.0

ET k1 c11 1 + c12 εv 

0.3

0.6

σN0 in model M4L.

3.3 Numerical validation

very large, the shear boundary is an approximate linear function. This boundary simulates the friction and slip.

σN0 =

-0.3 0.0 in ‰

12.0

Deviatoric stress boundaries in model M4L.

σTb = FT (−σN , εv ) =

-0.6

is given in Li (2009). The newly formulated σN0 makes it possible for a realistic prediction for the Poisson’s ratio of concrete subjected to uniaxial compression, as shown later. A similar numerical algorithm as model M4 is adopted in model M4L. For more details, see Li (2009).

9.0

Decrease of ( 1, fc) 0.0 -6.0 -3.0 0.0 3.0 6.0 9.0 in ‰ D (b) Tensile deviatoric stress boundary

-0.9

V

12.0

Figure 7.

2.0

0 N

bD

in MPa

0.0

ET k1 k2 c10 −σN + σN0  ET k1 k2 + c10 −σN + σN0 

(5a) (5b)

However, the shear stress boundary in model M4 is found to be too low in the elastic regime. As a result, the microplane shear stresses σM and σL are determined from the boundary value. This shifts the shear stress components into inelastic regime too early. Hence, the Poisson’s ratio under uniaxial compression begins to increase when the stress level is still very low, as shown in Fig. 4. A very direct remedy is to make the shear boundary higher through enlarging σN0 . As found by Di Luzio (2007) an increase of the parameter from its reference value 0.2 to 0.95 is very beneficial for producing realistic Poisson’s ratio of concrete under uniaxial compression. However, such a way makes the shear stress boundary rather high in the inelastic regime and shifts the peak strain under uniaxial compression too far away. In model M4L, a new formulation of σN0 is developed, as illustrated in Figure 8. The detailed expression

237

To validate the accuracy of the model M4L, a comparison of the model responses and the test data in the literature is carried out. The used model parameters can be found in Li (2009). Figure 9 (a) shows the simulation of model M4L and the test data of Imran (1994) for concrete under low confinement. It is obvious that the simulated response is very close to the experimental observations. The increase of ultimate strength, deformation capacity as well as the post peak ductility with increasing the confining pressure can be successfully represented by model M4L. This mainly attributes to the newly developed procedure for evaluating the microplane elastic behavior and the novel deviatoric stress boundaries. Figure 9 (b) illustrates the model simulations for the low confined compression data obtained by Richart et al. (1928). One can see that the model reproduces the test data fairy well. Figures 9 (c) and 9 (d) show a comparison between the simulated responses and the test data of Sfer et al. (2002) and Candappa et al. (2001), respectively. It is evident that the model M4L reproduces the both of the test data quite accurate. A comparison of the model predicted Poisson’s ratio under uniaxial compression and the measured test data by Ma (2009) is presented in Figure 9 (e). It is clear that model M4L realistically captures the variation of the Poisson’s ratio with the relative stress. The mode predicted Poisson’s ratio under different lateral pressures is shown in Figure 9 (f). It can be seen from the figure that the simulated response is quite consistent with the experimental observations of Dahl (1992).

75

50

75 9.0

4.2 2.1

30

1.05 0.0

15

15 in ‰ (a) Stress-strain response (Imran 1994) 120 0

5

10

20

25

90 in MPa

3.79

30 20

1.24

10

Model M4L Test

0

60

5.38 in MPa

45

0

40

8.4 in MPa

in MPa

60

10

20

30 40 in ‰ (b) Stress-strain response (Richart et al. 1928) 1.0 Model M4L Test 0.8

50

0.6 60

1.5 0.0

0

0

Model M4L Test

5

10 in ‰ (c) Stress-strain response (Sfer et al. 2002) 0.8 0.0 1.05 0.6 2.1 4.2 8.4 0.4

15

20

0.4 0.2

0.2

30 0 -20 -15 -10 -5

Model M4L Test

0 5 10 15 20 25 in ‰ (d) Stress-strain response (Candappa et al. 2001)

Figure 9.

30 15

Model M4L Test 0

4.5

45

0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

(e) Poisson’s ratio in uniaxial compression (Ma 2009)

0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

(f) Poisson’s ratio under low confinement

Comparisons of model M4L simulations and test data for concrete under low confinement.

The above numerical simulations imply that model M4L is adequate for representing the behavior of concrete under low confinement.

4 STRUCTURAL APPLICATION EXAMPLE To further evaluate the performance of model M4L, the model is implemented in the finite element software SOFiSTiK and used to model a common problem in high rise buildings—the transmission of high-strength concrete (HSC) column loads through normal-strength concrete (NSC) slabs. Due to the different lateral deformation properties of HSC and NSC, the slab is subjected to the confinement from the column. As a result, the bearing capacity of the joint is higher than that of column with the same concrete. A corner joint is considered in this paper as this kind of joints is more critical owing to the absence of a surrounding slab. The test specimen by Lee et al. (2007) is simulated with model M4L. The test consists of a sandwich specimen and two reference columns, shown in Figure 10. 4.1 Specimen details All the specimens have a cross section of 250 × 250 mm2 and a height of 750 mm. In the sandwich specimen, the NSC joint layer has a thickness of 150 mm. The layout of the longitudinal and the transverse reinforcements as well as the measured properties of concrete and reinforcement for each specimen are given in (Lee et al. 2007).

238

Figure 10. Specimen details (Lee et al. 2007).

4.2 Finite element model The finite element model for each specimen is illustrated in Figure 11. The concrete was modeled by Hexahedral elements with 8 integration points. Truss elements were used for modeling the reinforcement. Each model consists of a sum of 4608 concrete elements and 384 reinforcement elements. The total number of the nodes is 5577. The model parameters are calibrated according to the uniaxial compression behavior and the knowledge about the variation of the Poisson’s ratio for different concretes in uniaxial compression. The parameters used in the calculation are given in Li (2009). The predicted Poisson’s ratio under uniaxial compression for both concretes of the sandwich specimen is illustrated in Figure 12.

(a) HSC column (a) HSC column

(b) NSC column

6000 HSC

5000

1.2

P in kN

NSC HSC

1.0 0.8 0.6 0.4

4000 NSC

3000 2000

Predicted Test

1000

0.2

0 0

0.2

0.4

0.6 /f c

(c) Sandwich column

Figure 13. Deformed mesh at failure.

(c) Sandwich column

Figure 11. Finite element mesh.

0.0 0.0

(b) NSC column

0.8

1.0

1

2

3

4

5

in ‰

1.2

Figure 14. Predicted and observed load-deformation relation for HSC and HSC columns.

Figure 12. Model predicted Poisson’s ratio for concrete.

6000

4.3 Numerical results

239

P in kN

5000

The simulated responses for each specimen are illustrated in the following. Figure 13 shows the deformed FE-mesh at failure. The model is proven to be able to correctly reproduce the failure mode. For the sandwich column, the predicted failure by the numerical model is due to the crushing of the joint under compression, as observed in the test. Figure 14 illustrates a comparison of the predicted load-deformation behavior of each specimen. One can see from the figure that for both the HSC and the NSC columns, the calculated results are in good agreement with the test data. This implies that model M4L is adequate for simulating the behavior of confined concrete columns under axial compression. The predicted response for the sandwich column is also shown in Figure 15. It can be seen that the numerical model successfully simulates the strength and deformation enhancement of the joint due to the confinement of the top and bottom HSC column. However, the predicted peak load and strain at failure are a little higher than that of the test. A better simulation might be achieved through optimizing the parameters for the concretes both in the top and bottom columns and slabs. In this calculation, the model parameters for the concretes are calibrated only from

4000 3000 2000 Predicted Test

1000 0 0

1

2

3

4

5

in ‰

Figure 15. Predicted and observed load-deformation relation for sandwich columns.

uniaxial compression test data in the axial direction, this might be insufficient.

5 CONCLUSIONS In this paper, a new material model, called model M4L for representing the behavior of concrete under low confinement is developed. The model is formulated on the basis of microplane theory and follows the same approach as a previous microplane model M4. Numerical results indicate that the model M4L can accurately

describe the enhanced ultimate strength, deformation capacity as well as the post peak ductility of concrete subjected to low confinement. REFERENCES Attard, M.M. & Setunge, J.A.S.D. 1996. Stress strain relationship of confined and unconfined concrete. ACI Materials Journal, 93(5), 432–442. Bažant, Z.P., Caner, F.C., Carol, I., Akers, S.A. & Adley, M.D. 2000. Microplane mode M4 for concrete. Part I: Formulation with work-conjugate deviatoric. Journal of Engineering Mechanics, ASCE, 126(9), 944–953. Bažant, Z.P., Xiang, Y. & Prat, C. 1996. Microplane mode M4 for concrete. I: Stress-strain boundaries and finite strain. Journal of Engineering Mechanics, ASCE, 126(9), 944–953. Candappa, D.C., Sanjayan, J.G. & Setunge, S. 2001. Complete triaxial stress-strain curves of high-strength concrete. Journal of Materials in Civil Engineering, ASCE, 13(3), 209–215. Caner, F.C. & Bažant, Z.P. 2000. Microplane mode M4 for concrete. Part II: Algorithm and calibration. Journal of Engineering Mechanics, ASCE, 126(9), 954–961. Carol, I., Jirásek, M. & Bažant, Z.P. 2001. A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. International Journal of Solids and Structures, 38(17), 2921–2931.

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Dahl, K. 1992. A Failure Criterion for Normal- and Highstrength Concrete. Technical University of Denamrk. Di Luzio, G. 2007. A symmetric over nonlocal microplane model M4 for fracture in concrete. International Journal of Solids and Structures 44(13), 4418–4441. Imran, I. 1994. Application of Nonassociated Plasticity in Modelling the Mechanical Response Concrete. PhD thesis, University of Toronto. Lee, J.-K., Yoon, Y.-S., Cook, W. & Mitchell, D. 2007. Benefits of using puddle HSC with fibers in slabs to transmit HSC column loads. Journal of Structural Engineering, ASCE, 133(12), 1843–1847. Li, J. 2009. Development of a New Material Model for Concrete on the Basis of Microplane Theory. PhD Thesis, Leipzig University. Ma, J. 2009. Production and Performance of Ultra-highperformance Concrete. PhD Thesis, Leipzig University. Richart, F.E., Brandtzaeg, A. & Brown, R.L. 1928. A Study of the Failure of Concrete under Combined Compressive Stresses. Bulletin No. 185, University of Illinois. Sfer, D., Carol, I., Gettu, R. & Este, G. 2002. Study of concrete under triaxial compression. Journal of Engineering Mechanic, ASCE, 128(2), 156–163. Tue, N.V., Li, J., Caner, F.C. & Püschel, T. 2008. A new Microplane constitutive model for concrete. The Eleventh East Asia-Pacific Conference on Structural Engineering and Construction (EASEC-11), pp. 548–549.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Meso- and macroscopic models for fiber-reinforced concrete Sonia M. Vrech & Guillermo Etse CONICET, Universities of Tucuman and Buenos Aires, Argentina

Günther Meschke Institute for Structural Mechanics, Ruhr University Bochum, Bochum, Germany

Antonio Caggiano & Enzo Martinelli Department of Civil Engineering, University of Salerno, Fisciano (SA), Italy

ABSTRACT: Fiber reinforced concrete is analyzed and modeled at two different levels of observation. On the one hand, a macroscopic formulation based on the non-linear microplane theory is presented. Following approaches recently proposed in Pietruszczak & Winnicki (2003) and Manzoli et al. (2008), the mixture theory is used to describe the coupled action between concrete and the fiber reinforcement. The parabolic DruckerPrager maximum strength criterion is considered at the microplane level. Post-peak behavior is formulated in terms of the fracture energy release under mode I and/or II failure modes. On the other hand, a mesoscopic model of fiber reinforced concrete (FRC) is also presented which is based on three constituents: aggregate, mortar and aggregate-mortar interfaces. Aggregates are considered to be elastic while cracks are represented in a discrete format by means of interface elements. The presence of steel fibers is considered within the framework of the mixture theory. Consequently, mortar-mortar interfaces account for both fiber-mortar debonding and dowel effects according to the fiber volume content. After describing the constitutive models the paper focuses on numerical analysis of FRC failure behavior including re-analyzes of the experimental tests of Hassanzadeh (1990). The capabilities and shortcomings of both approaches for FRC failure analyses are evaluated. 1 INTRODUCTION Fundamental deficiencies of cement-based materials like concrete and mortar such as low tensile strength and brittleness can be mitigated by adding steel fibers into the matrix. Fibers play a major role in the postcracking behavior of fiber reinforced mortar composites (FRMC) by bridging the cracks and providing resistance to the crack opening process. Actually, FRMC may achieve quasi-ductile response exhibiting strain-hardening response with multiple cracks and relatively large energy absorption prior to fracture localization. In this case the composite takes the name of high performance steel fiber reinforced mortar composite (HPFRMC). Regarding structural behavior of concrete members, the addition of fibers leads also to significant improvements in the ductility in pre- and post-peak regimes as well as in the tensile peak stress. This is a consequence of the increase of dissipation attributed to the action of the fibers bridging opening microcracks and the reduction of volumetric expansion in the low confinement regime. Other relevant advantages attributed to FRMC is the reduced water permeability.

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Different approaches on different scales have been proposed for the modeling of FRMCs. They can be broadly categorized as follows: • Micro-scale models (more correctly denoted as meso-scale models): models which describe the interaction among the phases of the composite material, i.e. fiber, matrix, fine and coarse aggregates, and interfacial zones between them on the scale of the individual fibers. • Macro-scale models: the fibers and the matrix inside the FRMC at this scale of observation are indistinguishable. In this context FRMC is considered as a homogeneous material. Among others, we refer to the macro-scale models for FRMC by Hu et al. (2003) who proposed a single smooth biaxial failure surface for steel fiber reinforced concrete (SFRC), the proposal by Seow & Swaddiwudhipong (2005) based on a five parameter failure criterion for FRC with straight and hooked fibers and that of Minelli & Vecchio (2006) based on a modification to the compression field theory. Other relevant works are those by Guttema (2003), Pfyl (2003), etc. • Structural-scale models: these models capture the essence of the material behavior at the structural

level, for example, cross-sectional moment versus curvature behavior or panel shear force versus lateral displacements. Semi-analytical models, for the flexure behavior of fiber-reinforced concrete (FRC) materials, based on the equilibrium of forces in the critical cracked section, have been proposed by Zhang & Stang (1997). Stang & Olesen (1998) present a design approach for fiber reinforced concrete structures. Lee & Barr (2003) characterize the complete load/stress-deformation curve, under various loading conditions, using one continuous four-exponential function. • Multi-scale models: the performance of these models are based on coupling ingredients of different scale models: micro, meso, macro and structural models (e.g. Kabele (2002)). The present work deals with failure analysis of FRC material at the macro and mesoscopic level of observation. Both approaches use the composite theory as a basis for the simulation of the interaction between matrix and steel fibers. The main aim of this research is the evaluation of fundamental properties at the mesoscopic level controlling the mechanical response behavior of FRMC during monotonic loading beyond the elastic range. To this end, macroscopic constitutive formulation which incorporates information on the fibre-mortar-interaction at the meso-level is used. For the 2D mesoscopic analysis in this work a new methodology is followed based on considering FRMC as a three phase material: aggregate, mortar and the interfaces between the other two constituents. The non-linear behavior of steel fiber reinforced mortar is captured by means of zero-thickness joint elements. To this end the original interface model by Carol et al. (1997) is reformulated to include the interaction between steel fibers and mortar based on the composite theory by Manzoli et al. (2008). The macroscopic approach is based on the microplane theory combined with the flow theory of plasticity and the parabolic Drucker-Prager maximum strength criterion. In this case, the composite theory accounts for the interaction between steel fibers and concrete (including the effect of aggregates at the macroscopic level). The numerical analysis presented in this work include predictions of both approaches of FRC failure behavior in direct shear and uniaxial traction. Also the stress paths of experimental tests are considered. The results demonstrate the potentials of the mesoscopic approach in this work to evaluate relevant aspects of FRC such as the influence of the aggregate maximum size, of the ratio between this size and the fiber length, and of the failure mechanism of this complex material. On the other hand, the microplane based theory for macroscopic evaluations of FRC failure behavior allows the consideration of arbitrary and

multiple directions for steel fibers and, eventually, of non-homogeneous materials. The next steps of this research consider the inclusion of strain gradients in the constitutive formulation of the microplanes to capture the size effect of FRC and, particularly, the influence of the ratio between maximum aggregate size and the fiber length.

2 CONSTITUTIVE MESO-MECHANICAL MODEL Fracture analysis of FRMC on the meso-level follows the approach proposed by Vonk (1992) and then also used by Lopez et al. (2008a,b). While continuum mortar elements are assumed as linear elastic, the non-linear dissipative behavior of plain and fiber reinforced mortar is fully localized in the interface elements. According to the basic hypothesis of the mixture theory, the composite is considered as a continuum in which each infinitesimal volume is ideally occupied simultaneously by all constituents. In the plane of the interface, all constituents are subjected to the same strain fields and the corresponding composite stresses are given by the weighted (in terms of the volume fraction) sum of the constituent stresses. Being u the interface relative displacement vector, axial displacement of the fiber (in n direction) is given by uN = u · n, while in the transversal direction, uT = u · nT , being the unit vector perpendicular to n, see Figure (1). Consequently, the axial and angular fiber strains are given by εN = uN /lf and γT = uT /lf respectively, being lf the length of the fiber. Similarly to the flow theory of plasticity, the interface constitutive model is formulated in rate form. According to the composite theory, the rate of the stress vector ˙t = [σ˙ , τ˙ ]t in the interface plane is calculated as the sum of each constituent weighted by the volumetric fraction, k

242

Figure 1. fiber.

Schematic configuration of joint crossed by one

m t˙ = k m t˙ + k f σ˙ f (˙εN ) nt + k f τ˙f (γ˙T ) (nT )t

(1)

The superscripts m and f refer to mortar matrix and fibers, respectively, while t means the transposition operation for tensors. The incremental stress-displacement relation of the proposed composite joint model can be expressed in compact form as t˙ = E ep · u˙

rate form is defined by u˙ = u˙ el + u˙ cr

(4)

u˙ el = C −1 t˙

(5)

t˙ = C (u˙ − u˙ cr )

(6)

(2)

where the constitutive tangent matrix E ep is ep

ep

E ep = k m C ep + k f

Ef

lf

nt n + k f ep

Gf lf

(nT )t nT

(3) ep

With C ep = ∂t/∂u, Ef = ∂σf /∂εN and Gf = ∂τf /∂γT as tangent operators defined in the following sections. General aspects of the constitutive formulation, which models the discontinuity behavior at the interface in presence of steel fibers, are the following: • Fracture energy-based joint constitutive law: the constitutive model is formulated in terms of normal and shear stresses on the interface plane, and corresponding relative displacements. The failure criterion (F = 0) of the interface model is described by the three-parameter hyperbola by Carol et al. (1997). Main features of the elastoplastic interface model formulation are summarized in the Section 2.1. • Fiber bond-slip effects are considered as a combination of the uniaxial elasto-plastic model, for fibers, and a uniaxial debonding dissipative model for the interface mortar-fibers, resulting in a global constitutive model for the constituent slipping fiber. • Dowel action of reinforcement short fibers crossing cracks in mortar is modeled in a smeared manner. ‘‘Beam on elastic foundation theory’’ is utilized to derive the dowel force-displacement law, which is expressed in terms of dowel stress and strain in order to be compatible with the previously indicated constitutive laws. Both the bond-slip axial model of the fiber and the dowel action of short fiber reinforcements crossing cracks in mortar/concrete are similarly considered in both meso- and macroscopic approaches. These models are detailed in Section 4. 2.1

Fracture-based interface constitutive model

In this section, the interface model, originally proposed by Gens et al. (1988), is summarized. The elasto-plastic formulation of the interface model in

243

where u˙ t = [˙u, v˙ ] is the rate vector of relative displacements, decomposed into the elastic and crack opening components u˙ el and u˙ cr , respectively. C defines a fully uncoupled normal/tangential elastic stiffness at the interface C=

  kn 0 0 kt

(7)

The yield-loading condition of the interface constitutive model is defined as F = τ 2 − (c − σ tanφ)2 + (c + χ tanφ)2

(8)

withτ and σ as the interface stress components. The tensile strength χ (vertex of the hyperbola), the shear strength c (cohesion strength) and the internal friction angle φ are model parameters. In Eq. (8), two limit situations can be distinguished: (a) cracking under pure tension, with zero shear stress (Mode I), when the yield surface is reached along the horizontal axis, and (b) cracking under shear and very high compression, when the yield surface is reached in its asymptotic region, where the hyperbola approaches a MohrCoulomb criterion. The last one is called ’’asymptotic Mode II (or Mode IIa)’’. The evolution of fracture process is driven by the cracking parameters χ and c, which depends on the energy release during interface degradation Wcr . Details are given in Carol et al. (1997). 3 MACROSCOPIC MODEL BASED ON MICROPLANE THEORY For the mesoscopic analysis of FRC failure behavior a fracture energy-based elastoplastic microplane model was developed. Instead of the existing spherical microplane models, see a. o. Beghini et al. (2007), Carol et al. (2001), Kuhl et al. (2001), the proposed constitutive theory considers a 2D stress and strain fields using disk microplanes according to the proposal by Park & Kim (2003). As a result, a reduced number of microplanes is required. According to this approach, the fiber-reinforced concrete is idealized as a disk of unit radius and constant

thickness b, which agrees with that of the analyzed material patch. The following assumptions are considered:

stress K in terms on the strain-like internal variable κ, and the friction and cohesion parameters α and β, respectively, that are defined as functions of the uniaxial compression and tensile strengths, ft′ and fc′ , respectively, according to

• Macroscopic stresses are uniform in the disk and are equilibrated by the surface tractions on the microplanes. • Microscopic strains in normal (ε) and tangential (γr ) directions to each microplane with normal direction n, are obtained from macroscopic strains ε (kinematic constrains) ε = ni nj εij γr =

1 [ni δjr + nj δir − 2ni nj nr ]εij 2

∂[ρ0 ψ0mic ] , ∂ε

τr =

∂[ρ0 ψ0mic ] ∂γr

with ρ0 denoting the material density. The macroscopic free-energy potential per unit mass of material in isothermal conditions, ψ0mac (ε, κ), with κ a set of thermodynamically consistent internal variables, result ψ0mac

1 = bπ



ψ0mic (tε , κ)dV

κ˙ =

(10)

(11)

 ′  f c − ft ′ , fc′

β=

3 fc′

∂F = λ˙ ∂K

(15)

A non-associated flow is adopted in order to avoid the excessive inelastic dilatancy. The plastic potential is based on a volumetric modification of the yield condition in the compressive regime. Then, the gradient tensor m of to the plastic potential can be obtained by a modification of the gradient tensor nσ to the yield surface as nσ m=  1−

(12)

V

with V as the disk volume. 3.1 Microplane constitutive laws



→ if p > 0 p pdil



nσ

→ if 0 < p < pdil

(13)

in terms of the pressure p, the second invariant of the deviatoric stress tensor J2 , the dissipative plastic

244

(16)

where pdil is a model parameter representing the value at which the dilatancy vanishes, see Figure (2). In the post-peak regime the evolution of the dissipative stress, due to micro-fracture process at the microplane level, is defined through the homogenization process of the fracture energy released during crack formation with the plastic dissipation of an equivalent continuum, similarly to the fracture energybased plasticity model by Willam et al. (1985) and

The microplane constitutive law is based on the mixture theory by Truesdell & Toupin (1960) and, similarly to the interface model formulation used in the analysis at the mesoscopic level of observation, on the hypothesis of FRC to be represented by a composite model (Oliver et al. (2008)). Then, the rate of stress vector t˙ at each microplane is obtained by Eq. (1) where superscript m refers now to the concrete matrix variables. The constitutive model of the concrete matrix is based on the parabolic Drucker-Prager strength criterion. The yield condition in hardening/softening regime is defined by the unified equation F(t, k) = βJ2 + αp − (ft′ − K(κ))

(14)

The evolution of the internal variable is defined in terms of the plastic parameter rate λ˙ as

(9)

• Normal and tangential microscopic stresses t = [σ , τr ]t are obtained from the microscopic free energy potential σ =

α=3

Figure 2.

Macroscopic constitutive model.

Etse & Willam (1994), as



K˙ = ft′ 1 − exp −5

I ht Gf ε˙ fr ur GfIIa

The yield criterion, in tension as well as in compression, is represented by the following expression

Ff = |σf | − (σy, f + Qf )

(17)

where σy, f is the elastic limit. The evolution, in the post-elastic regime of the 1D surface is driven by the stress-like internal variable Qf , given in incremental form as

with the equivalent fracture strain ε˙ fr = mI  κ˙

(18)

˙ f = λ˙ f H f Q

where ht represents the characteristic length associated with the active fracture process and, more specifically, the distance or separation between microcracks. Moreover, ur represents the maximum crack opening displacement in mode I type of failure. GfI and GfIIa are the fracture energies in modes I and II of failure, respectively. The McAuley brackets in Eq. (18) indicate that only tensile principal plastic strains contribute to the fracture strain during fracture process. In the special case of uniaxial tension state, the evolution of the dissipative stress can be obtained with the simplified expression    ht K˙ = ft′ 1 − exp −5 ε˙ fr ur

(23)

(24)

with ε˙ p// = λ˙ f ∂Ff /∂σf = λ˙ f sign[σf ] representing the plastic flow law, λ˙ f is the non-negative plastic multiplier and H f is the hardening/softening modulus. The incremental stress-strain relationship is ep

σ˙ f = Ef ε˙

(25) ep

where the elasto-plastic tangent modulus Ef takes the two following distinct following values, see Figure (3)  ep Ef = E f → Elastic response (26) ep Ef = Ef E /H1 f +1 → Elasto-plastic regime f

(19)

4 FIBER-MORTAR/CONCRETE INTERACTION

It is assumed that the fiber strain ε is decomposed in two additive parts, one due to the intrinsic fiber uniaxial deformation εs and another one associated with the interface debonding εd (27)

In this section the models for the interaction between steel fibers and mortar (mesoscopic model), as well as steel fibers and concrete (macroscopic model), considering the bond-slip and dowel effects, are presented.

ε = εs + ε d

4.1 Bond-slip axial model of the fiber

1/Ef = 1/Es + 1/Ed

The uniaxial behavior of the steel fiber is approcimated by means of a simple 1D elasto-plastic model. The following set of equations is considered

where Es and Ed are the steel Young’s modulus and an equivalent elastic modulus of matrix-fiber interface, respectively. Two limit situations can be recognized:

ε˙ = ε˙ el + ε˙ p ε˙ el =

σ˙ f Ef

σ˙ f = Ef (˙ε − ε˙ p )

Assuming a serial model constituted by the fiber and the fiber-mortar joint the corresponding total deformability 1/Ef is given by

(20) (21) (22)

where the rate of the axial fiber strain ε˙ is decomposed into a elastic part and a plastic component, ε˙ el and ε˙ p , respectively. Ef represents an equivalent uniaxial elastic modulus including the uniaxial response of the steel and the bond-slip effect of the short reinforcement. σ˙ f is the rate of bond-slip axial stress of the steel fiber.

245

Figure 3.

Uniaxial bond-slip model for the fiber.

(28)

• Ed → 0: the stiffness of the complete structure becomes null and the effect of the fibers vanishes. • Ed → ∞: represents the case of a perfect adherence between matrix-fibers. In this context, and to complete the bond-slip axial constitutive models presented by the Eqs. (20) to (26), the following material parameter are defined σy, f = min[σy,s , σy,d ] f

H =



Hs Hd

If σy,s < σy,d otherwise

following values ⎧ ⎨Gfep = Gf ⎩Gfep = Gf

(29)

(30)

γ˙ = γ˙ el + γ˙ pl γ˙ el =

τ˙f Gf

τ˙f = Gf (γ˙ − γ˙ pl )

(31) (32) (33)

being γ˙ the rate of the shear fiber strain, which is decomposed into a elastic and a plastic part, γ˙ el and γ˙ pl , respectively. Gf represents the shear modulus, while τ˙f is the rate of dowel shear stress of the interaction between fiber-matrix. The model is completed with:

→ plastic regime

Vd = Es Is λ3 

(35)

(36)

in which Is = πdf4 /64 is the moment of inertia of the fiber (df diameter of the fiber) and λ parameter representing the relative stiffness between the fiber and the foundation, defined as  kc df λ= 4 (37) 4Es Is

Dowel action of reinforcement short fibers crossing cracks in mortar matrix.

The dowel effect of fibers crossing cracks in mortar, is taken account in the joint model by means of a 1D shear stress-strain elasto-plastic constitutive model, similar to the previously mentioned one for the axial stressstrain. In this case, the following equations are utilized

Gf /H dow + 1

H dow is the hardening/softening modulus of the uniaxial dowel model, commonly assumed as H dow = 0. Dowel effect can be analyzed treating each reinforcement fiber as beam on elastic foundation (Winkler theory) to deal with the interaction between the fibers and the surrounding mortar (He and Kwan, 2001, Rumanus and Meschke, 2010). The fiber can be treated as a semi-infinite beam on the elastic foundation, loaded with a concentrated load at one extreme representing the dowel resultant Vd . The analytical solutions of the beam on elastic foundation in Figure (4) results in the following forcedisplacement relationship Vd − 

in which σy,s and σy,d are the material yield stress and the equivalent interface elastic limit, respectively; while the super-indices s and d refer to steel and debonding, respectively. The parameters Ed , σy,d and H d required for the bond/slip model characterization can be calibrated from a simple pull-out test (Oliver et al. 2008). 4.2

→ elastic response 1

Thereby is kc the foundation modulus of the surrounding mortar that governs the dowel stiffness. Experimental data, available for RC specimens (Dei Poli et al. 1992), show that the same coefficient take values ranging from 75 to 450 N/mm3 . Other tests (Soroushian et al. 1987) show that the coefficient kc increases as the strength of surrounding mortar increases and when the volume fraction of the reinforcement increases. An equivalent shear elastic modulus can be calculated as Vd = Es Is λ3  = Gf

 Af Lf

⇒

Gf = Es Is λ3

Lf Af (38)

where As = πdf2 /4 is the cross area section of the bar.

• yielding criterion, similar to Eq. (23); • hardening/softening law, similar to Eq. (24). The incremental shear stress-strain relation, can be written as ep

τ˙f = Gf γ˙

(34)

where, in a similar manner as in the bond model, ep the tangent shear modulus Gf takes the two distinct

246

Figure 4.

Dowel effect based on Winkler beam theory.

At the limit stage, local crushing of the surrounding mortar and/or yielding of the dowel bar occurs. Based on experimental results for RC specimens, the following equation has been proposed by Dulacska (1972) for the dowel force at the limit stage    (39) Vdu = kdow df2  fc′ σy,s 

In (39) kdow is a non-dimensional coefficient (kdow = 1.27, for RC-structures), while fc′ is the compressive strength of the concrete or the surrounding mortar. Finally, the equivalent shear yield stress, τy, f results as Vdu τy, f = (40) Af 5 NUMERICAL ANALYSIS

In this section numerical analyses are performed with both the mesoscopic and macroscopic models and considering failure behavior of FRMC and FRC, respectively. 5.1 Mesoscopic evaluation of FRMC failure Behavior

Firstly, a uniaxial tensile test is performed by imposing homogeneous vertical displacements to all four nodes of the upper quadrilateral element. The results in terms of vertical nominal stress vs. vertical displacements are shown in Figure (6) when the test is performed with different amount of fibers crossing the interface (df = constant = 0.8 mm), and in Figure (7) where the specific case of 5 fibers are considered but with six different values of df . The results in Figures (6) and (7) indicate that the proposed model is able to reproduce the sensitivity of FRMC regarding peak load and post-peak ductility to both the amount of fibers and the diameter of the fibers (when equal number of fibers are considered). Moreover, the sensitivity of the post-peak ductility is significantly more important than that of the peak load with respect to fiber amount and diameter. Post-peak responses in both figures show increasing reloading effects with the increment in the amount and diameter of the fibers. This is due to the increasing composite (inhomogeneity) effect that results by enlarging the amount or diameter of the fibers. Figures (8) and (9) show the model performance when compared against the experimental results by Hassanzadeh (1990) (included with dotted lines). These tests are characterized by imposing combined normal and shear relative displacements to

To evaluate the predictive capabilities of the proposed non-linear dissipative interface model for FRMC the element patches shown in Figure (5) are considered. Thereby, one interface element is placed between 2D plane stress isoparametric four node elements subjected to the indicated boundary conditions in terms of impeded displacements. As indicated in the same Figure (5) six different cases are considered with one, two, three, five, seven and nine steel fibers crossing the interface element (all of them having the same diameter df ). Figure 6. Normal stress vs. vertical displacement performed with different amount of fibers with df = constant = 0.8 mm.

Figure 5. Interface configuration with: (a) one, (b) two, (c) three, (d) five, (e) seven and (f) nine steel fibers (df = 0:8 mm).

247

Figure 7. Normal stress vs. vertical displacement performed with six different values of df (nf = constant = 5).

Figure 8. Single crack crossed by fibers: numerical tests with θ = 30◦ , df = constant = 0.8 mm and different number of fibers: (a) normal stress vs. relative displacement and (b) shear stress vs. relative displacement.

the peak strength is reached. From that point, normal and shear relative displacements are applied simultaneously in a fixed proportion characterized by a constant value of the relation tanθ = u/v, with u and v the normal and tangential relative interface displacements, respectively. These tests were re-analyzed with θ = 30◦ and θ = 60◦ , both for df = 0.8 mm. Model parameters value used in these numerical analysis are: kN = 200 MPa/m , kT = 200 MPa/m, tanφ = 0.9, χ0 = 2.8 MPa, c0 = 7.0 MPa, GfI = 0.1 N/mm, σdil = 30 MPa, αχ = 0 and αc = 0, for the interface, while ν = 0.2 and Em = 25000 MPa, were used for the elastic modulus and the Poisson’s ratio, respectively, for the mortar. Results in Figure (8a) show, as expected, that the tangential and normal displacement control in the second part of the Hassanzadeh test for θ = 30◦ is responsible for a stronger softening of the normal tensile stress than in case of the pure tension test. The fiber content affects mainly the last portion of the curves in the reloading zone. With other words, the strong post-peak decrease (connected with severe cracking) in the first portion of the softening regime of this test practically suppresses the fiber contribution to the ductility (to compare the results with that corresponding to plain mortar). In contrast, results in terms of shear stress vs. tangential relative displacement in Figure (8b) show relevant fiber contribution in the preand post-peak regimes as well as in the maximum strength. When comparing these curves with results in Figures (9a) and (9b) we observe, as expected, less severe softening both in normal and shear stress components. Moreover, only in case of plain mortar or low content of fibers the softening branch leads to zero in the compressive regime of the normal stress. With three fibers (or more then three) the softening regime fully remains in the tensile regime indicating a significant increase of the ductility as compared to plain mortar case. In conclusion, the proposed interface model for mesoscopic analyses of FRMC failure behavior seems to provide realistic predictions of peak stresses, ductility and post-peak behavior of this material when different fiber directions and fiber contents are considered crossing a single crack.

Figure 9. Single crack crossed by fibers: numerical tests with θ = 60◦ , df = constant = 0.8 mm and different number of fibers: (a) normal stress vs. relative displacement and (b) shear stress vs. relative displacement.

a developing crack in a prismatic concrete specimen of 0.07 × 0.07 m2 square cross section with a 0.015 m deep notch. During first part of the numerical test, a pure tension stress state is imposed until

248

5.2 Macroscopic evaluation of failure behavior of FRC To evaluate the capabilities of the proposed macroscopic model based on microplane theory to simulate failure behavior of FRC, preliminary numerical studies considering a unidirectional and an isotropic distribution of fibers, are performed. A single element problem in plane strain conditions and subjected to a homogeneous stress/strain state is considered. The concrete matrix is characterized by

Figure 10. Uniaxial tensile test for microplane model. Fibers oriented in loading direction.

Figure 12. Uniaxial compression test for microplane plasticity. Fibers oriented in loading direction.

Figure 13. Uniaxial compression test for microplane plasticity. Isotropic distribution of fibers. Figure 11. Uniaxial tensile test for microplane model. Isotropic distribution of fibers.

the following material parameters E = 19000 MPa, ν = 0.2, fc′ = 22 MPa, ft′ = 3.0 MPa, ht = 108 mm, ur = 0.127 mm, GfIIa /GfI = 10, pil /fc′ = 1.2. Fiber material parameters are those of Subsection 5.1. Firstly, a uniaxial tensile test is performed. Results for fibers oriented in the loading direction and with isotropic distribution of fibers are illustrated in Figures (10) and (11). Different volume contents Vf of the fibers are considered including the extreme case of plain concrete (Vf = 0).The elastic stiffness increases in case of bias fiber as compared to the plain concrete case and with the increment of Vf . In both cases of fiber orientations a slight increase of the peak tensile strength and a re-stiffening effect in post-peak regime are observed. Figures (12) and (13) show the results obtained in the uniaxial compression test when both uniaxially oriented fibers and an isotropic distribution of

249

fiber directions are considered. In case of bias fiber, as expected, the stiffness in the pre-peak regime and the overall dissipated energy during post-peak regime increase with Vf . 6 CONCLUSIONS Mesoscopic and macroscopic models for fiber reinforced cement composite materials were presented. The mesoscopic model takes into account a three phase mesostructure composed by elastic aggregates, mortar and mortar-aggregate interfaces. This model also includes mortar-mortar interfaces to simulate the dissipative response behavior of this constituent. The macroscopic model is formulated within the theoretical framework of microplane theory. Both the interface model and the microplane model for meso- and macroscopic analyses, respectively, are based on flow rule of plasticity, mixture theory by

Truesdell & Toupin (1960) and composite model by Oliver et al. (2008). The interactions between steel fibers and mortar/concrete associated with debonding and dowel effects are considered in both models. Preliminary numerical studies presented in this paper demonstrate (partially on a conceptual level) their capabilities to reproduce the most relevant aspects of failure behavior of steel fiber reinforced concretes under tensile, shear and compressive stresses.

ACKNOWLEDGEMENTS The first two authors acknowledge financial support for this work by FONCYT (Argentine agency for research & technology) through Grant PICT1232/6, and by CONICET (Argentine council for science & technology) through Grant PIP6201/05.

REFERENCES Beghini, A., Bazant, Z.P., Zhou, Y. Gouirand, O. and Caner, F.C. 2007. Microplane Model M5f for Multiaxial Behavior and Fracture of Fiber-Reinforced Concrete. Journal of Engineering Mechanics. 133: 66–75. Carol, I., Prat, P.C. and Lopez, C.M. 1997. Normal/shear cracking model: Applications to discrete crack analysis. Journal of Engineering Mechanics. 123(8): 765–773. Carol, I., Jirasek, M. and Bazant, Z. 2001. A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. International Journal of Solids and Structures. 38: 2933– 2952. Dei Poli, S., Di Prisco, M. and Gambarova, P.G. 1992. Shear response, deformations, and subgrade stiffness of a dowel bar embedded in concrete. ACI Struct. J. 89(6): 665–75. Dulacska, H. 1972. Dowel action of reinforcement crossing cracks in concrete. ACI J. 69(12): 754–7. Etse, G. and Willam, K. 1994. A fracture energy-based constitutive theory for inelastic behavior of plain concrete. J. Engineering Mechanics, ASCE. 120: 1983–2011. Gens, A., Carol, I. and Alonso, E. 1988. An interface element formulation for the analysis of soil-reinforcement interaction. Comput. Geotechnics. 7: 133–151. Guttema, T.B. 2003 Ein Beitrag zur realitatsnahen Modellierung und Analyse von stahlfaserverstarkten Stahlbeton und Stahlbetonflachentragwerken. PhD Thesis. Univ. Kassel. Hassanzadeh, M. 1990. Determination of fracture zone properties in mixed mode I and II. Engineering Fracture Mechanics. 35 (4/5): 845–853. He, X. and A. Kwan (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Computers and Structures 79: 595–604. Hu, X.D., Daz, R. and Dux, P. 2003. Biaxial failure model for fiber reinforced concrete. Journal of materials in civil engineering. 15(6): 609–615. Kabele P. 2002. Equivalent continuum model of multiple cracking. Engineering Mechanics. 9(1/2): 75–90. Kuhl, E., Steinmann, P. and Carol, I. 2001 A thermodynamically consistent approach to microplane theory. Part II.

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Disipation and inelastic constitutive modeling. International Journal of Solids and Structures. 38: 2921–2931. Lee, M.K. and Barr, B.I.G. 2003. A fourexponential model to describe the behavior of fibre reinforced concrete. Materials and Structures. 37(7): 464–471. Lopez, C.M., Carol, I. and Aguado, A. 2008a. Mesostructural study of concrete fracture using interface elements. I: numerical model and tensile behavior. Materials and Structures. 41: 583–599. Lopez, C.M., Carol, I. and Aguado, A. 2008b. Mesostructural study of concrete fracture using interface elements. II: compression, biaxial and Brazilian test. Materials and Structures. 41: 601–620. Manzoli, O.L., Oliver, J., Huespe, A.E. and Diaz, G. 2008. A mixture theory based method for three-dimensional modeling of reinforced concrete members with embedded crack finite elements. Computers and Concrete. 5(4): 401–416. Minelli, F. and Vecchio, F.J. 2006. Compression Field modeling of fiber-reinforced concrete members under shear loading. ACI Structural Journal. 106(2): 244–252. Oliver, J., Linero, D.L., Huespe, A.E. and Manzoli, O.L. 2008. Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach. Comput. Methods Appl. Mech. Engrg. 197: 332–348. Park, H. and Kim, H. 2003. Microplane model for reinforcedconcrete planar members in tension-compression. Journal of Structural Engineering. 129: 337–345. Pfyl, T. 2003 Tragverhalten von Stahlfaserbeton. PhD. Thesis. Eidgenssischen Technischen Hochshule Zrich. Switzerland. Pietruszczak, S. and Winnicki, A. 2003. Constitutive Model for Concrete with Embedded Sets of Reinforcement. Journal of Engineering Mechanics. 129(7): 725–738. Rumanus, E. and Meschke, G. 2010. Homogenization-based model for reinforced concrete. Computational Modeling of Concrete Structures (EURO-C 2010), in print. Seow, P.E.C. and Swaddiwudhipong, S. 2005. Failure Surface for Concrete under Multiaxial Load—A Unified Approach. Journal of Material in Civil Engineering. 17(2): 219– 228. Soroushian, P., Obaseki, K., Rojas, M.C. 1987. Bearing strength and stiffness of concrete under reinforcing bars. ACI Mater. J. 84(3): 179–84. Stang, H. and Olesen, J.F. 1998. On the interpretation of bending tests on FRC materials. Fracture Mechanics of Concrete Structures. 1: 511–520. Truesdell, C. and Toupin, R. 1960. The classical field theories. Handbuch der Physik, Springer Verlag, III/I, Belin. Vonk, R. 1992. Softening of concrete loaded in compression. Ph.D. thesis, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, the Netherlands. Willam, K., Hurbult, B. and Sture, S. 1985. Experimental and constitutive aspects of concrete failure. In US-Japan Seminar on F.E. Anal. of R.C. Struct. ASCE-Special Public. 226–254. Zhang, J. and Stang, H. 1998. Applications of Stress Crack Width Relationship in Predicting the Flexural Behavior of Fibre—Reinforced Concrete. Cement and Concrete Research. 28(3): 439–452.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Gradient damage model with volumetric-deviatoric split Adam Wosatko & Jerzy Pamin Faculty of Civil Engineering, Cracow University of Technology, Poland

ABSTRACT: The decomposition of the isotropic damage model according to a volumetric-deviatoric split involves two damage parameters and in the case of gradient enhancement can produce a two- or three-field formulation depending on the adopted strain measure. The paper discusses the theory of isotropic gradient damage starting from the constitutive equations and finishing with proper matrix systems. The theory is verified by means of one-element test, however a more advanced simulation of the splitting effect in the Brazilian test is also performed. 1 INTRODUCTION

The constitutive equation becomes:

Departing from scalar damage, but remaining within the isotropic description, it is possible to generalize the damage model with two strain measures and/or two damage parameters (Ju 1990). One way of such modification can be the decomposition of damage into two parts related to tensile and compressive actions, cf. for instance (Mazars and Pijaudier-Cabot 1989; Comi 2001). However, the proposal discussed in this paper results from a volumetric-deviatoric split given for example in (Lubliner et al. 1989; Comi and Perego 2001). The first damage parameter influences the bulk modulus and the second one reduces the shear modulus. In the simplest approach one damage history parameter and one damage loading function are assumed, while two different damage growth functions are distinguished. Then, in a gradient-enhanced continuum model, one Helmholtz averaging equation is still the basis of a formulation as for the scalar gradient damage (Peerlings et al. 1996). Total separation leads to a model with two history parameters, corresponding to the volumetric and deviatoric strain measures. In this proposal two damage loading functions are introduced and two different averaging equations are applied. The theory is described in Sections 2 and 3. Computational verification of both approaches is performed in Sections 4 and 5.

σ = E KG ǫ

2 CONSTITUTIVE MODEL

and the variables on the right-hand side are: θ = IIT ǫ

The continuum damage formulation satisfies the isotropy condition if two damage parameters ωK and ωG for the volumetric and deviatoric parts respectively are considered, cf. (Lubliner et al. 1989; Ju 1990; Comi and Perego 2001; Carol et al. 2002).

(1)

where: E KG = (1 − ωK )KT + 2(1 − ωG )GQ

(2)

σ is the stress tensor and ǫ is the strain tensor, both in a vector form. In the above relation the damage parameter ωK degrades bulk modulus K and the parameter ωG reduces the deviatoric stiffness represented by shear modulus G. The following relations are introduced: 1 Q = Q0 − T 3 1 Q dev = I − T 3

(3) (4)

T where in  three dimensions:  = [1, 1, 1, 0, 0, 0] and 1 1 1 Q0 = diag 1, 1, 1, 2 , 2 , 2 . Note that the strain and stress vectors are split into volumetric and deviatoric parts:

1 θ + ǫ dev 3 σ = p + ξ ǫ=

(5) (6)

– dilatation, ǫdev = Qǫ –deviatoricstrain, p = 13 T σ – pressure and ξ = Q dev σ – deviatoric stress. The stress rate is obtained differentiating Equation (1):

251

σ˙ = E KG ǫ˙ − ω˙ K KT ǫ − 2ω˙ G GQǫ

(7)

The basic issue is how the damage and the equivalent strain which governs it should be measured. Two approaches can be proposed. 2.1 One strain measure The simplest case is to assume κ d = κkd = κGd so that one damage loading function: f d (ǫ, κ d ) = ǫ˜ (ǫ) − κ d = 0

ω˙ G =

(8)

ωK = ωK (κ d )

3 ISOTROPIC GRADIENT DAMAGE This section concerns the derivation of isotropic damage with a gradient enhancement. The gradientenhanced model according to (Peerlings et al. 1996; Geers 1997; Pamin 2004) is nonlocal and guarantees producing mesh-independent results. During a failure process, from the onset of localization and until the total loss of the stiffness, the governing system of equations remains elliptic and regularization allows one to avoid a spurious mesh sensitivity. The equations of the boundary value problem (BVP) are almost the same as for scalar damage, but the tangent stress-strain relation is defined in Equation (7).

(9)

ωG = ωG (κ d )

(10)

This option is further called ‘‘one strain measure’’. If one history parameter κ d governs the damage evolution, the rates of damage parameters ωK and ωG during loading are respectively:

ω˙ G =

dωK dκ d d ǫ˜ ǫ˙ dκ d d ǫ˜ dǫ dωG dκ d d ǫ˜ ǫ˙ dκ d d ǫ˜ dǫ

3.1 Gradient enhancement for one strain measure

(11)

If ‘‘one strain measure’’ approach is assumed, the Helmholtz equation:

(12)

ǫ¯ − c∇ 2 ǫ¯ = ǫ˜

2.2 Two strain measures A more general concept involves a total separation of damage multipliers applied to the bulk and shear modulus, hence it is called ‘‘two strain measures’’. Now two damage loading functions are employed— separately for the volumetric strains (dilatation): fKd (θ, κKd ) = θ˜ (θ ) − κKd = 0

f d (ǫ, κ d ) = ǫ¯ (˜ǫ (ǫ)) − κ d = 0

(13)

= ǫ˜dev (ǫ dev ) −

=0

ω˙ K = (14)

The aim of distinguishing two history parameters κKd and κGd is that damage then increases independently for the volumetric and deviatoric strains: ω˙ K =

dωK dκKd d θ˜ ǫ˙ dκKd d θ˜ dǫ

(18)

Hence the damage rates are computed as:

and for the deviatoric (shear) strains: κGd

(17)

is still the basis of the two-field formulation like for scalar gradient damage (Peerlings et al. 1996). The above relation involves the second gradients of the averaged strain ǫ¯ . The parameter c > 0 has a unit of length squared and it is connected with the internal length scale l of a material. The relation c = 12 l 2 is derived for instance in (Askes et al. 2000). Instead of the local equivalent strain ǫ¯ the averaged strain ǫ¯ now governs the damage progress:

and during unloading both ω˙ K and ω˙ G are equal to 0.

fGd (ǫ dev , κGd )

(16)

During unloading it can be assumed that either ω˙ K or ω˙ G are equal to 0. A similar approach with total separation of damage parameters is described in (Carol et al. 2002), where two damage variables influence two loading functions and that model is called ‘‘bi-dissipative isotropic model’’.

satisfying the Kuhn-Tucker conditions is applied with the equivalent strain measure ǫ˜ and the damage history parameter κ d which grows from the initial value κ o . This means that function ǫ˜ (ǫ) can be defined like in the scalar damage model. However, two different damage growth functions are distinguished:

ω˙ K =

dωG dκGd d ǫ˜dev ǫ˙ dκGd d ǫ˜dev dǫ

ω˙ K =

(15)

252

∂ωK ∂κ d ˙ ǫ¯ = GK1 ǫ˙¯ ∂κ d ∂ ǫ¯

(19)

∂ωG ∂κ d ˙ ǫ¯ = GG1 ǫ˙¯ ∂κ d ∂ ǫ¯

(20)

The weak form of equilibrium equations can be written as follows:    δǫ T σ dV = δuT bdV + δuT tdS (21) B

B

∂B

where u is the displacement field, b is the body force vector, t is the traction vector. A weak form of Equation (17) is derived using Green’s formula and the non-standard boundary condition (∇ ǫ¯ )T ν = 0:    T δ ǫ¯ ǫ¯ dV + (∇δ ǫ¯ ) c∇ ǫ¯ dV = δ ǫ¯ ǫ˜ dV (22) B

B

B

Independent interpolations of displacements u and averaged strain measure ǫ¯ are employed in the semidiscrete linear system and a two-field formulation ensues. The primary fields are interpolated in this way: u=Na

ǫ¯ = hT e

and

(23)

where N and h contain suitable shape functions. From the above interpolations the secondary fields can be computed in the following way: ǫ = Ba

T

and ∇ ǫ¯ = g e

(24)

where B = LN and g T = ∇hT . The discretized equations must hold for any admissible δa and δe, therefore:    BT σ dV = N T bdV + N T tdS (25) B



B

∂B

B

(hhT + cgg T )edV =



h˜ǫ dV

(26)

and et+ t = et + e

(27)

Analogically, at the integration points we employ decomposition of ǫ, σ and ǫ˜ . The equilibrium equations then become:  BT (σ t + σ )dV b

=



B

T t+ t

N b

dV +



T t+ t

N t

dS

(28)

∂B

and averaging equation is derived as:   T T t (hh + cgg )(e + δe)dV = h(˜ǫ t + ˜ǫ )dV B

t hT e

ωK = GK1

KG t+ t t K KG aa a + K ae e = f ext − f int

K KG aa =



BT E tKG Bd V 

B



f t+ t = ext f tint =

(34)

B

K KG ae = −

B



t t BT [GK1 KT + 2GG1 GQ]ǫ t hT dV

(35) N T bt+ t dV +



N T t t+ t dS

(36)

∂B

BT σ t dV

(37)

B

In Equation (29) the increment of equivalent strain measure ˜ǫ is computed from the interpolated displacement increment a: t

d ǫ˜

˜ǫ = dǫ 

ǫ = [sT ]B a

(38)

and Equation (29) can be formulated as follows: K ea a + K ae e = f tǫ − f te

(39)

The matrices and vectors in Equation (39) are similar to the scalar damage formulation, cf. (Peerlings et al. 1996): K ea = −



h[sT ]t BdV

(40)

K ee = −



(hh + cgg T )dV

(41)



B

B

h˜ǫ t dV

(42)

B

f te = K ee et

(29)

253

(33)

where:

B

t t

σ = E tKG B a − [GK1 KT + 2GG1 GQ]ǫ t hT e (30)

(32)

and the derivatives are determined at instant t. We rewrite Equation (28) in a matrix form:

f tǫ =

The incremental constitutive relation is derived starting from Equation (7):

(31)

t

ωG = GG1 hT e

B

The BVP is linearized, hence at nodal points the increments of the primary fields from time instant t to time instant t + t are introduced in such a way: at+ t = at + a

where the damage increments have been calculated as:

(43)

Eventually, the following system of equations is used: 

K KG aa K ea

K KG ae K ee



  t+ t 

a f ext − f tint =

e f tǫ − f te

(44)

3.2 Gradient enhancement for two strain measures In this option two damage loading functions for two averaged strain measures are introduced: fKd (θ, κKd )

= θ¯ (θ˜ (θ )) −

κKd

=0

(45)

fGd (ǫ dev , κGd ) = ǫ¯dev (˜ǫdev (ǫ dev )) − κGd = 0

(46)

The damage rates depend on two different fields: ∂ωK ∂κKd ˙ θ¯ = GK2 θ˙¯ ∂κKd ∂ θ¯

(47)

∂ωG ∂κGd ω˙ G = ǫ˙¯ dev = GG2 ǫ˙¯ dev ∂κGd ∂ ǫ¯dev

(48)

ω˙ K =

θ¯ − cK ∇ 2 θ¯ = θ¯

(49)

ǫ¯dev − cG ∇ ǫ¯dev = ǫ˜dev

(50)

2

and in result a three-field formulation is obtained. Note that in this approach we can adopt two different values of internal lengths, namely cK for the volumetric part and cG for the deviatoric part. The weak form of equilibrium equations is identical to the previous approach. Using Green’s formula and the non-standard boundary conditions (∇ θ¯ )T v = 0 and (∇ ǫ¯dev )T v = 0 for Equations (49) and (50) the weak forms of the above averaging equations are obtained:    ¯ ˜ δ θ¯ θ¯ dV + (∇δ θ¯ )T cK ∇ θdV = δ θ¯ θdV (51) B

B

B

δ ǫ¯dev ǫ¯dev dV + =



B

B



T

B

(∇δ ǫ¯dev ) cG ∇ ǫ¯dev dV

δ ǫ¯dev ǫ¯dev dV

(52)

u = N a,

T

ǫ¯dev = h d

(53)

It is noted that the same shape functions are used here for the volumetric and deviatoric averaged strain, but in general they can be different. The discretized equilibrium is governed by Equation (25). In addition we obtain two discretized averaging equations:   (hhT + cK gg T )o dV = hθ˜ dV (54) B



h˜ǫdev dV

(55)

B

which are valid for any δo and δd. The BVP is linearized and corresponding increments of the primary fields at nodal points and the secondary fields at integration points from time instant t to time instant t + t are written. The equilibrium equation (28) holds, but instead of Equation (29) we obtain:   ˜ (hhT + cK gg T )(ot + o)dV = h(θ˜ t + θ)dV B

(56)



B

(hhT + cG gg T )(d t + d)dV =



t h(˜ǫdev + ˜ǫdev )dV

B

(57)

As previously, the incremental form of constitutive relation is derived from Equation (7), but now the interpolation of three fields is considered: t

σ = E tKG B a − GK2 KT ǫ t hT o t GQǫ t hT d + 2GG2

(58)

Here the damage increments are functions of different discretized fields in such a way: t

ωK = GK2 hT o

(59)

ωG =

(60)

t GG2 hT d

We can formulate Equation (28) using matrix notation: t+ t t K KG aa a + K ao o + K ad d = f ext − f int

(61)

where:

Three primary fields are distinguished, so now we employ interpolations of displacements u, volumetric averaged strain measure θ¯ and deviatoric averaged strain measure ǫ¯dev : θ¯ = hT o and

B

(hhT + cG gg T )d dV =

B

Therefore, two different averaging equations are introduced:





B

254

K ao = −



K ad = −2

B



t Bt GK2 Kt ǫ t ht dV

(62)

t Bt GG2 KGQεt ht dV

(63)

B

The increments of equivalent strain measures θ˜ and ˜ǫdev , which occur in Equations (56) and (57), can be calculated on the basis of primary fields as follows:

θ˜ =



d θ˜ dǫ

t

ǫ = [sKT ]t B a

(64)

d ǫ˜dev dǫ



˜ǫdev =

t

T t

ǫ = [sG ] B a

(65)

Hence, Equations (56) and (57) take the matrix form: K oa a + K oo o = f tvol − f to K da a + K dd d =

f tdev

−

f td

The matrices and vectors denote:  K oa = − h[sKT ]t B dV

(66)

the degradation of bulk modulus K and shear modulus G. This permits one to control damage evolution governed by dominating volumetric or shear failure. All calculated cases are set in Table 1. There are presented symbols used to distinguish a given case, and corresponding damage growth data. During the damage evolution κ d can grow in different ways. In this paper linear softening law (Peerlings et al. 1996):

(67) ω(κ d ) =

f tvol =



B



B

(hhT + cK gg T ) dV

hθ˜ t dV

f to = K oo ot  T t K da = − h[sG ] B dV

and exponential softening law (Mazars and PijaudierCabot 1989; Peerlings et al. 1998):

(69)

ω(κ d ) = 1 −

(70)

are employed. Using Equation (77) for the description of damage evolution firstly κ d exceeds threshold κo and then grows to ultimate value κu which corresponds to total damage. The parameters η and α in Equation (78) are responsible for the rate of softening and residual stress which in one dimension tends to (1 − α)Eκo . In our computations in the case of linear softening, we assume equal or different ultimate values κu for volumetric and deviatoric degradation. For exponential softening ductility parameter η is varied. Hence lower index i = K, G is applied in order to distinguish the analyzed tests. The cases with identical data for both damage growth functions can be treated as reference, because these in fact are computations using scalar damage. The same parameter α = 0.98

(71) (72)

B

K dd = − f tdev =





B

(hhT + CG gg T ) dV

t h˜ǫdev dV

(73) (74)

B

f td = K dd d t

(77)

(68)

B

K ee = −

κu κ d − κo κ d κu − κo

(75)

Finally, the matrix system of equations for the threefield formulation has the form: ⎤⎡ ⎤ ⎡ ⎡ KG t ⎤ K aa K ao K ad f t+ t

a ext − f int ⎥⎣ ⎦ ⎣ t ⎢ t 0 ⎦ o = f vol − f o ⎦ (76) ⎣ K oa K oo

d f tdev − f td K da 0 K dd

 κo  d 1 − α + αe−η(κ −κo ) d κ

(78)

Table 1. Tension in one direction—all cases computed using one strain measure. Linear softening Ultimate value of history parameter

4 ONE-ELEMENT BENCHMARK TEST One three-dimensional finite element (FE) with eight nodes is subjected to static tension in one direction. Such a simple test permits one to observe the numerical response of the isotropic damage model at the point level. Better understanding is necessary to perform more advanced analyses. The following material data are introduced: Young’s modulus E = 20000 MPa, Poisson’s ratio ν = 0.20.

Symbol of case

Volumetric

Deviatoric

lin lin, K lin, G lin, K&G

κuK κuK κuK κuK

κuG κuG κuG κuG

••• ..... –––

One damage threshold κo (equal here to 0.0001) is used in the loading function (18). The simple modification of the scalar theory makes it possible to adopt two different damage growth functions, separately for

255

= 0.002 = 0.002 = 0.003 = 0.003

Exponential softening Ductility parameter Symbol of case

4.1 Computations for one strain measure

= 0.002 = 0.003 = 0.002 = 0.003

exp exp, K exp, G exp, K&G

Volumetric ---–––

ηK ηK ηG ηK

= 1000 = 750 = 1000 = 750

Deviatoric ηK ηG ηG ηG

= 1000 = 1000 = 750 = 750

in all cases with exponential softening is assigned. It is also possible to adopt two different damage laws during the failure process, cf. (Wosatko 2008). The normalized elastic energy release rate (Ju 1989) is applied in computations as the loading function, however Mazars definition (Mazars 1984) and modified von Mises definition (de Vree et al. 1995) for one three-dimensional FE subjected to tension in one direction are all equivalent. The diagrams of strain ǫ11 versus ωK and ωG in the final phase of damage are shown in Figure 1 for the chosen cases. These diagrams are helpful in understanding how different damage growth functions can influence the results. Firstly, in Figure 2 cases with linear softening are depicted. As it was expected, after the peak, diagrams for lin, K and lin, G start to descend between diagrams 1 0.95 K

0.9 0.85 0.0005

G

0.001

11

0.0015

for lin and lin, K&G. However, the nonlinear character of softening is observed although the damage growth function related to linear softening is assumed for cases lin, K and lin, G. Moreover, in case lin, G, where the deviatoric part has larger κuG arc-length control must be applied during computations because a snapback occurs. It suffices to change the ultimate value κu for either the volumetric or deviatoric part in order to obtain nonlinear response. In Figure 3 we can see stress-strain diagrams for damage growth functions related to exponential softening. The interpretation is easy since each softening branch has an exponential character. For cases exp, K and exp, G with different ductility parameters the diagrams run between the extreme cases exp and exp, K&G. Analogically to the cases with linear functions a certain tendency can be noted. If the fracture energy for the deviatoric part is increased (compare cases exp and exp, G), a larger difference in response is noticed than when fracture energy grows for the volumetric part. It is doubtful to consider fracture energy only for a chosen part of the stiffness, however in order to simplify the explanations this concept is used. Before the next figure is described the following definition of Poisson’s ratio depending on the stiffness

0.002

(a)Case lin,K.

2

1

lin,K&G

σ11 [MPa]

K

0.9 0.85 0.0005

lin,G

1.5

0.95

G

0.001

11

0.0015

0.002

lin,K lin

1

0.5

(b)Case lin,G. 1

0

0

0.0005

K

0.9 0.85 0.0005

0.001

0.0015

0.0025

0.003

Figure 2. Influence of ultimate κui (i = K, G) in linear softening for one strain measure.

G

0.001

11

0.0015

0.002

2 exp,K &G

(c)Case exp,K. 1

exp,G

σ11 [MPa]

1.5

0.95 K

0.9 0.85 0.0005

0.002

11

0.95

11

0.0015

exp

1 0.5

G

0.001

exp,K

0.002 0

(d)Case exp,G.

0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 11

Figure 1. Final variation of damage parameters in damagestrain diagrams for chosen cases—one strain measure.

256

Figure 3. Influence of ductility parameter ηi (i = K, G) in exponential softening for one strain measure.

degradation is introduced: νω =

3(1 − ωK )K − 2(1 − ωG )G 2[3(1 − ωK )K + (1 − ωG )G]

For linear softening (77) ultimate value κu = 0.002. If exponential softening (78) is considered, then η = 750 and α = 0.98. However, for two history parameters (κKd for volumetric part and κGd for deviatoric part) two damage thresholds are employed. For two averaged strain measures we introduce two equivalent strain definitions. The simplest proposal is:

(79)

To distinguish the Poisson’s ratio which is given as an elastic material parameter from the one defined in Equation (79) the subscript ω is additionally applied. Hence, parameter νω is computed during the damage process. As it is shown in Figure 4, if only damage growth functions are predefined for a particular case then this parameter changes. Furthermore, in the case of linear softening the value of νω drastically tends to a lower or upper limit. These limits can be perceived as controversial results and they are not physically motivated. Such extreme behaviour in simulation and as a consequence of nonlinear relation between ǫ11 and σ11 seems to be undiserable. A complete degradation for the volumetric part in case lin, G gives finally νω equal to –1. On the other hand the zeroed shear stiffness in case lin, K leads to νω = 0.5 like for incompressible materials, cf. (Carol et al. 2002). Concrete is rather a material where microcracks under tension decrease the Poisson’s effect during the failure process (Ju 1990). The problem is that non-physical values below zero appear. If exponential softening is used (see Figure 4, case exp, G) a smooth drop to zero is observed, but this is not always so. Naturally the starting value of νω and also the configuration of the considered test decide on whether νω becomes negative. For quasibrittle materials like concrete generally it is expected that Poisson’s ratio tends to 0 during the damage evolution (Carol et al. 2002). Summarizing, case exp, G seems to be physically correct.

θ˜ = 0.5(I1ǫ + |I1ǫ |)  ǫ˜dev = 3J2ǫ

(80) (81)

where I1ǫ and J2ǫ are strain invariants. Analogically to the previous computations, the considered cases with symbols and corresponding data are given in Table 2. The thresholds for case t are chosen in such manner that they are equivalent to threshold κo = 0.0001 for scalar damage. For case t, K&G the choice of values is connected with a higher threshold κo = 0.00015 for scalar damage. Figure 5 shows stress-strain diagrams for the cases with linear softening. It is visible that the lower value of damage threshold decides about the onset of softening. Moreover, for case t, G only the deviatoric part of stiffness undergoes degradation, while the volumetric part remains elastic. For the opposite case t, K the deviatoric stiffness remains without changes and the

4.2 Computations for two strain measures In this subsection we continue the analysis for one three-dimensional FE, but the proposal of two strain measures in the formulation is verified. The same damage growth data is assumed to focus our attention.

Table 2. Tension in one direction—cases computed using two strain measures. Damage threshold Symbol of case

Volumetric

Deviatoric

t t, G t, K t, K&G

κoK κoK κoK κoK

κoG κoG κoG κoG

= 0.00006 = 0.00009 = 0.00006 = 0.00009

= 0.00012 = 0.00012 = 0.00018 = 0.00018

3 2.5

0.2

2

σ11 [MPa]

0.5 0.4

0

exp,K &G

1

0.5

exp,G lin,G

0.0005 0.001

t,

1.5

lin,K exp,K

0

t t,G

0

0.0015 0.002 0.0025 0.003 0.0035

0

0.0005

0.001

0.0015

0.002

11

11

Figure 4. Sensitivity of Poisson’s ratio νω to adopted damage growth functions for one strain measure.

257

Figure 5. Influence of damage threshold κoi (i = K, G) in linear softening for two strain measures.

3 t, &G

[MPa]

2.5

t

2

t,G t,

11

1.5 1 0.5 0

0

0.0005

0.001 0.0015 0.002 0.0025 0.003

0.0035

11

Figure 6. Influence of damage threshold κoi (i = K, G) in exponential softening for two strain measures.

Figure 7. Table 3.

volumetric part is reduced. In spite of this behaviour a linear character of softening is preserved. Analogical diagrams for exponential softening are presented in Figure 6. Like previously, the lower value of damage threshold activates the failure process. It is observed that damage of either volumetric or deviatoric part causes a change of Poisson’s ratio νω , so for the volumetric degradation (here case t, K) νω still decreases below zero. 5 SIMULATION OF BRAZILIAN TEST The splitting test is used to establish the concrete tensile strength, because the compression between the loading platens induces the perpendicular tensile force action in the centre of the specimen. This phenomenon and the snapback response in the Brazilian test is not simple to reproduce in numerical computations, because damage can localize directly under the platen. In fact, different mechanical models have been verified numerically using this test, starting from plasticity theories like in (Chen and Chang 1978). Due to a double symmetry and in order to reduce the computation time only a quarter of the domain (with radius equal to 40 mm) is considered. The general geometry data are based on (Winnicki et al. 2001), but plane strain conditions are assumed. The load is applied to the specimen indirectly via a stiff platen (width – 5 mm, height – 2.5 mm). The platen is perfectly connected with the specimen. In these computations only one mesh shown in Figure 7 is employed and attention is focused on the details of isotropic version of the gradient damage model. However, mesh insensitivity for gradient damage is widely discussed in (Wosatko 2008). The load acts downwards at the top of the platen. The material data are presented in Tables 3 and 4. It is obvious that in the gradient model not only the internal length parameter decides about the results of the test, but also other parameters, for instance the choice of the damage growth function. Here four

Applied mesh for Brazilian test. Brazilian test—material model data.

Specimen

damaging

Young’s modulus: Poisson’s ratio: Equivalent strain measure: Fracture energy: Internal length scale: Threshold:

Ec = 37700 MPa ν = 0.15 modified von Mises, k = 10 Gf = 0.075 N/mm l = 6 mm, i.e. c = 18.0 κo = 7.9576 × 10−5

Platen

elastic

Young’s modulus: Poisson’s ratio:

Es = 10 · Ec ν = 0.15

Table 4. Brazilian test—cases computed using one strain measure. Symbol of case

ηK

ηG

Damage growth

exp exp, K exp, G exp, K&G

1200 600 1200 600

1200 1200 600 600

more intensive ωK < ωG ωK > ωG less intensive

---–––

options of exponential softening are analyzed, where different combinations of values of ductility parameter η decide whether the damage process is more or less brittle. The parameter α equals 0.99 for each case. Because of a possible snapback response the test is computed using the arc length method. Load-displacement diagrams in Figure 8 are plotted for the four considered cases. It is noticed that for cases exp,K and exp the softening paths are mono-tonically decreasing and without snapback. The same value of parameter ηG = 1200 governs the solution. On the other hand, for cases exp,G and exp,K&G the snapback response is retrieved. This means that deviatoric damage is more important in the stiffness degradation

258

P/2[kN]

0.45 0.4

exp, &G

0.35 0.3 0.25 0.2

exp,G

A

exp, exp

0.15 0.1 0.05 0

B

0

0.05

0.1

0.15

0.2

0.25

0.3

(a)

K,

exp,G, point A.

(b)

K,

exp,G, point B.

(c)

K,

exp,K, point A.

(d)

K,

exp,K, point B.

(e)

G,

exp,G, point A.

(f )

G,

exp,G, point B.

(g)

G,

exp,K, point A.

(f )

G,

exp,K, point B.

[mm]

Figure 8. Brazilian test—influence of ductility parameter ηi (i = K, G), load-displacement diagrams.

(a) Case exp,G, point A.

(c) Case exp,K, point A.

Figure 9.

(b) Case exp,G, point B.

(d) Case exp,K, point B.

Brazilian test—contour plots of averaged strain ǫ¯ .

and decides about the proper behaviour in the Brazilian test. Contour plots in Figures 9–10 are presented for the peak and the final state (in Figure 8 the respective points A and B are marked). The splitting is observed only for the cases which correspond to the larger value of fracture energy Gf for the deviatoric part, i.e. the ductility ηG = 600. It is confirmed by means of Figure 9, where the distributions of averaged strain are plotted for the two stages—points A and B. Therefore the interaction between the compressive loading and the tensile response seems to be transferred via the deviatoric characteristics in the model. Figure 10 shows the damage patterns for cases exp,G and exp, K. According

Figure 10. Damage patterns in Brazilian test.

to the assumptions included in Table 4, the domination of damage paramter ωK in case exp, G and inversely κG —in case exp, K is visible. The splitting effect observed for case exp, G is expected in the Brazilian test and the coincidence with the decrease of νω during the process (not shown in this paper) is understandable for concrete. However, the problem with negative values is still present, but the aim in our computations is to observe how sensitive the model is

259

to the change of parameters, even if one strain measure approach is used. 6 CONCLUSIONS Two proposals of gradient enhancement for isotropic damage with the volumetric-deviatoric split are formulated and examined. The option with one strain measure assumes one damage loading function, one averaging equation and two different damage growth functions. Then two-field finite element formulation as for the scalar gradient damage is obtained. A more general approach introduces two damage loading functions, two different averaging equations and a threefield formulation is derived. One of the features of the isotropic models is that evolving Poisson’s ratio is simulated, which is characteristic for degrading quasibrittle materials. It is questionable whether negative values of this ratio should be admitted in the simulated process. In the proposed models this is not excluded, hence only by means of appropriate values of input parameters negative Poisson’s ratio can be avoided. However, this problem can be removed if the model is implemented with restrictions similar to the ones presented in (Ganczarski and Barwacz 2004). REFERENCES Askes, H., J. Pamin, and R. de Borst (2000). Dispersion analysis and element-free Galerkin solutions of secondand fourth-order gradient-enhanced damage models. Int. J. Numer. Meth. Engng 49, 811–832. Carol, I., E. Rizzi, and K. Willam (2002). An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate. Eur. J. Mech. A/Solids 21(5), 747–772. Chen, W.F. and T.Y.P. Chang (1978). Plasticity solutions for concrete splitting tests. ASCE J. Eng. M Div. 104(EM3), 691–704. Comi, C. (2001). A non-local model with tension and compression damage mechanisms. Eur. J. Meek A/Solids 20(1), 1–22.

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Comi, C. and U. Perego (2001). Numerical aspects of nonlocal damage analyses. Revue europenne des lmentsfinis 10(2-3-4), 227–242. de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Comput. & Struct. 55(4), 581–588. Ganczarski, A. and L. Barwacz (2004). Notes on damage effect tensors of two-scalar variables. Int. J. Damage Mechanics 13, 287–295. Geers, M.G.D. (1997). Experimental analysis and computational modelling of damage and fracture. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Ju, J.W. (1989). On energy-based coupled elasto-plastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct. 25(1), 803–833. Ju, J.W. (1990). Isotropic and anisotropic damage variables in continuum damage mechanics. ASCE J. Eng. M 116(12), 2764–2770. Lubliner, J., J. Oliver, S. Oller, and E. Ofiate (1989). A plastic-damage model for concrete. Int. J. Solids Struct. 25(3), 299–326. Mazars, J. (1984). Application de la mecanique de l’edommagement au comportement non lineaire et a la rupture du beton de structure. Ph.D. dissertation, Universite Paris 6, Paris. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. ASCE J. Eng. Mech. 115, 345–365. Pamin, J. (2004). Gradient-enhanced continuum models: formulation, discretization and applications. Series Civil Engineering, Monograph 301, Cracow University of Technology, Cracow. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Meth. Engng 39, 3391–3403. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and M.G.D. Geers (1998). Gradient-enhanced damage modelling of concrete fracture. Mech. Cohes.-frict. Mater. 3, 323–342. Winnicki, A., C.J. Pearce, and N. Bicanic (2001). Viscoplastic Hoffman consistency model for concrete. Comput. & Struct. 79, 7–19. Wosatko, A. (2008). Finite-element analysis of cracking in concrete using gradient damage-plasticity. Ph.D. dissertation, Cracow University of Technology, Cracow.

Advances in numerical methods

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Continuous and discontinuous modeling of cracks in concrete elements J. Bobi´nski & J. Tejchman Gdansk University of Technology, Gdansk, Poland

ABSTRACT: The paper presents results of numerical simulations of concrete elements using two different approaches to model cracks. First, the cracks were modeled in a smeared way by defining an elasto-plastic, a continuum damage or a smeared crack model. In elasto-plasticity, a Rankine criterion was used. The degradation of the stiffness in the second model was described as a scalar variable of a equivalent strain measure. A rotating crack model and a multi-fixed orthogonal crack model were also used. To ensure mesh-independent results, all these models were enriched by a characteristic length of a micro-structure using a non-local theory. Second, the cracks were simulated as discontinuities with the aid of cohesive elements. Two benchmark problems of concrete elements under mixed mode conditions (simultaneous occurrence of the failure mode I and II) were examined: a Nooru-Mohamed test (1992) and Schlangen (1993) test. The obtained numerical results were compared with the corresponding experimental ones. 1 INTRODUCTION Concrete belongs to quasi-brittle materials which exhibit a gradual decrease of strength with increasing strain after the peak. It is a highly heterogeneous and non-linear material. A choice of the constitutive model, the knowledge about its limitations and drawbacks is essential to obtain results that agree with experimental outcomes. This problem is especially important when the formation and evolution of cracks is considered. There are two main approaches to simulate cracks in concrete within continuum mechanics. The first one simulates cracks in a smeared sense, i.e. it assumes a finite width of cracks by using elasto-plasticity, damage mechanics or a smeared crack approach. These continuum models have to be enriched in a softening regime by a characteristic length of micro-structure to properly capture strain localization, to restore the well-posedness of a boundary value problem and to obtain objective numerical solutions. The alternative options using e.g. cohesive (interface) elements analyze cracks as zero width discontinuities while the material remains continuous in the remaining region. The more advanced numerical techniques utilize a strong discontinuity approach or XFEM. The concrete can be also modeled as a set of discrete bonded grains by DEM. The behaviour of concrete always follows a continuous-discontinuous failure description. At the beginning of failure, a localized fracture zone of diffused micro-cracks with a defined thickness is created and next a discrete crack is formed with increasing strain. Thus, a realistic constitutive concrete model should reflect a finite width of the fracture zone at the

263

beginning and its discrete nature at the end of the cracking process. The aim of our paper is to present FE results of concrete specimens under quasi-static conditions obtained with different continuum constitutive laws (Marzec et al. 2007, Majewski et al. 2008, Bobi´nski & Tejchman 2008). The continuum models (elastoplastic, damage mechanics, smeared crack) were enriched by a integral non-local theory. In the case of an elasto-plastic model, a Rankine criterion in a tensile regime was adopted. Within damage mechanics, different definitions of the strain measure were used (by Rankine and modified von Mises). In the case of smeared cracks, a rotational and fixed crack model were applied. Alternatively, cohesive elements were used to simulate a crack formation. In interface elements, a damage constitutive relationship between the traction vector and separation vector (both quantities with normal and shear terms) was assumed. All models were implemented into the commercial FE-code Abaqus/Standard. Two benchmark problems with curved cracks in concrete were carefully analyzed: a Nooru-Mohamed (1992) test and a Schlangen (1993) test. The FE results were compared with corresponding laboratory outcomes. Advantages and disadvantages of constitutive models were outlined. 2 CONSTITUTIVE MODELS 2.1 Elasto-plasticity A Rankine criterion was used with a yield function f defined as: f = max {σ1 , σ2 , σ3 } − σt (κ),

(1)

where σ1 , σ2 and σ3 = the principal stresses, σt = the tensile yield stress and κ = the hardening/softening parameter (equal to the maximum principal plastic p strain ε1 ). An associated flow rule was assumed. To define the softening under tension, a curve proposed by Hordijk (1991) was chosen: σt (κ) = ft ((1 + A1 κ 3 ) exp(−A2 κ) − A3 κ),

way in a tension-compression regime. The coefficient c reflects the influence of the principal compression stress. With c = 0, Equation (6) is recovered. In the text, Equation (8) will be called a ‘modified Rankine’ definition. Alternatively, a modified von Mises definition in terms of strains was used:   k −1 ε 2 12k 1 k −1 Jε I1 + I1 + ε˜ = 2k(1 − 2ν) 2k 1 − 2ν (1 + ν)2 2 (9)

(2)

where A1 , A2 and A3 are defined as A1 =

c1 , κu

A2 =

c2 , κu

A3 =

 1  1 + c13 exp (−c2 ) κu (3)

where I1ε and J2ε = the first invariant of the strain tensor and second invariant of the deviatoric strain tensor, respectively:

with κu = the ultimate value of the softening parameter κ (material constants c1 = 3.0 and c2 = 6.93).

I1ε = ε11 + ε22 + ε33 ,

ε J21 =

2.2 Damage mechanics Damage of the material was described by a scalar parameter D growing from zero (undamaged) to one (total damage). The damage variable D acts as a stiffness reduction factor: e εkl , σij = (1 − D) Cijkl

(5)

κ(t) = max{˜ε(τ )}. τ ≤t

During loading, the parameter κ grew, and during unloading and reloading, it remained constant. To define the equivalent strain measure ε˜ , a so-called Rankine definition was used:

D =1−

max{σi } E

The total strains εij were decomposed into elasticεije and inelastic strains εijcr (coupled with the existence of cracks): εij = εije + εijcr .

eff

e εkle . σij = Cijkl

ε˜ =

eff σ1

eff

−

eff c−σ2

E eff



(8)

with σ1 > σ2 and a non-negative coefficient c. This formulation is equivalent to Equation (6) in a tension—tension regime, but it behaves in a different

(13)

Between stresses and cracked strains, the following relation held (in a local coordinate system):

(7)

Due to some numerical problems, Eq. (6) was modified as:

(12)

The stresses were related to the elastic strains by the following traditional relationship:

where E = the modulus of elasticity and σi = the principal values of the effective stress tensor defined as: eff

(11)

2.3 Smeared crack approach

(6)

e σij = Cijkl εkl .

κ0 (1 − α + αe−β(κ−κ0 ) ), κ

wherein κ0 = the initial value of the damage threshold parameter κ, and α and β = the material parameters.

eff

ε˜ =

(10)

The parameter k in Eq. (9) denotes the ratio between the compressive and tensile strength of the material. To describe the evolution of the damage variable D, an exponential softening law was used for all definitions of ε˜ :

(4)

e where Cijkl = the elastic stiffness matrix and εkl = the total strain tensor. The growth of the variable D was controlled by a damage threshold parameter κ which was defined as the maximum of the equivalent strain measure ε˜ reached during the entire load history:

1 1 εij εij − (I1ε )2 . 2 6

cr cr σij = Cijkl εkl

(14)

cr with the secant cracked matrix Cijkl (defined only for open cracks). To simplify the calculations, the matrix cr was defined as a diagonal one. A crack was creCijkl ated when the maximum tensile stress σ1 exceeded the tensile strength ft . To define material softening in a normal direction under tension, a Hordijk (1992) curve was adopted again. In this model, the parameter

264

κu in Eqs. (2)–(3) was replaced by an ultimate value of the cracked strain in tension εnu and the parameter κ by εmax , respectively. The quantity εmax was the largest cracked strain εicr in a i-direction obtained during the entire loading history. By combining Eqs. (12)–(14), the following relationship between stresses and total strains was derived (in a local coordinate system): s σij = Cijkl εkl

separation δi vector (both quantities with normal and shear terms) was assumed: ti = (1 − D) E0 Iij δj

with the penalty (dummy) stiffness E0 and unit tensor Iij . To take into account both the normal and shear terms in the separation vector, an effective displacement was introduced (Camacho & Ortiz 1996):

(15)

s with the secant stiffness matrix Cijkl defined as: s e e e cr −1 e = Cijkl − Cijrs (Crstu + Crstu ) Ctukl . Cijkl

δeff = (16)

s Cijij

δn 2 + ηδs2

(20)

   ft σt (δmax ) = ft exp −β δmax − , E0

(17)

(18)

δmax = max δ(τ ),

(22)

where β = the model parameter. 3 NON-LOCAL THEORY As a regularization technique, a non-local theory was used. It is based on a spatial averaging of tensor or scalar state variables in a certain neighborhood of a given point. In plasticity, softening parameters κ were treated non-locally according to the Brinkgreve’s (1994) proposal: κ(x) ¯ = (1 − m)κ(x) + m



α0 (x − ξ )dκ(ξ )dξ  , α0 (x − ξ )dξ (23)

where κ¯ = the non-local softening parameter, m = the coefficient greater than one, x = the coordinates of the considered (actual) point, ξ = the coordinates of the surrounding points and α0 = the weighting function. As the weighting function α0 , the Gauss distribution was assumed:

where εsu = the ultimate value of the cracked strain in shear (usually εsu = εnu ) and p = the material parameter. 2.4 Cohesive elements Cracks were simulated as the discontinuities in a bulk continuum using cohesive elements. These elements were defined at the edges between standard finite elements to nucleate cracks and propagate them following the deformation process. They governed the separation of crack flanks in accordance with irreversible cohesive laws. The bulk elements were modeled as linear elastic ones. In the interface elements, a simple damage constitutive relationship between the traction vector ti and

(21)

with: τ ≤t

The second formulation allowed one a creation of three mutually orthogonal cracks in 3D-problems (2 cracks in plane simulations, respectively). The orientation of the crack was described by its primary inclination at the onset, i.e. the crack could not rotate during loading. A shear modulus G was reduced by a shear retention factor β which was constant (usually β = 0.2) or it decreased with increasing strains across the crack. Here, the power law proposed by Rots and Blaauwendraad (1989) was chosen:   εcr p , β = 1− i εsu



with a coefficient η. To describe softening after the cracking, an exponential law was assumed (Camacho & Ortiz 1996):

After cracking, the isotropic elastic stiffness matrix was replaced by the orthotropic one. Two formulations were investigated: a rotating crack model and a multi-fixed orthogonal crack model. In the first approach, only can crack was created which could rotate during deformation. To keep the principal axis of total strains and stresses aligned, the secant stiffness coefficient was calculated according to: σii − σjj = . 2(εii − εjj )

(19)

r 2 1 α0 (r) = √ e−( l ) , l π

(24)

where r = x − ξ  = the distance between points x and ξ , and l = the characteristic length of microstructure. It should be noted that the averaging was restricted only to a small area around each material point; the influence of points at the distance of r = 3 × l is only of 0.01%.

265

In a damage model, a local definition of the equivalent strain measure ε˜ from Eqs.(6) or (9) was replaced in Eq. (5) by its non-local counterpart ε¯ :  α0 (x − ξ )˜ε(ξ )dξ ε¯ (x) =  . (25) α0 (x − ξ )dξ

In a smeared crack approach, a constitutive law in the form: s (¯εkl )εkl σij = Cijkl

(26)

was adopted following the proposal by Jirásek and Zimmerman (1998). The symbol ε¯ kl denotes the nonlocal total strain tensor defined as (independently for all tensor components):  α0 (x − ξ ) εkl (ξ ) dξ  ε¯ kl (x) = . (27) α0 (x − ξ ) dξ

s was calculated from Thus, the secant matrix Cijkk the non-local strain tensor.

4 NOORU-MOHAMED TEST 4.1 Problem A double-edge notched (DEN) specimen under various different loading paths of combined shear and tension was analyzed (Nooru-Mohamed (1992)). The dimensions of the largest specimen and boundary conditions are presented in Fig. 1. The length and height of the element was 200 mm. The thickness was 50 mm. Two notches with dimensions of 25 × 5 mm2 were placed in the middle of the vertical edges. The loading

was prescribed by rigid steel frames glued to concrete. During one of the loading paths (number 4), a shear force Ps was applied until it reached a specified value, while the horizontal edges were free. At the second stage, the shear force remained constant and vertical tensile displacement was prescribed. In the experiment, two curved cracks with an inclination depending on the shear force (for a small value of Ps —almost horizontal, for a large value of Ps —highly curved) were obtained. The following elastic material parameters were chosen in the FE-analyses: E = 32.8 GPa and v = 0.2. A FE-mesh was composed of 12600 3-node triangle finite elements. 4.2 FE-results within elasto-plasticity The tensile strength was assumed as ft = 2.6 MPa and parameter κu = 0.033. The characteristic length was equal to l = 1 mm and the non-local parameter was m = 2. Figure 2 presents the obtained FE results with the shear force Ps = 5 kN (path 4a). A very good agreement was achieved with respect to both the forcedisplacement curve and strain localization, although the calculated maximum force P was too large. The FE-results for the path 4b (Ps = 10 kN) are shown in Fig. 3. The force-displacement curve is satisfactorily reproduced. Two curved localized zones

Figure 1. Geometry of DEN specimen (Nooru-Mohamed (1992)) (dimensions in mm).

266

Figure 2. The force-displacement curves and the contour map of non-local parameter κ¯ for shear force Ps = 5 kN within elasto-plasticity (l = 1 mm, κu = 0.033).

Figure 3. The force-displacement curves and contour map of non-local parameter κ¯ for shear force Ps = 10 kN within elasto-plasticity (l = 1 mm, κu = 0.033).

Figure 4. The force-displacement curves and contour map of parameter D for shear force Ps = 10 kN within damage mechanics (Rankine definition, Eq. 6).

were numerically obtained again, but they were too flat as compared to the experiment (wherein they were more curved and the distance between them was larger). 4.3 FE-results within damage mechanics First, the Rankine definition of the equivalent strain measure ε˜ was used, Eq. (6). The following material parameters were assumed: κ0 = 7 × 10−5 , α = 0.92, β = 100 and l = 0.5 mm. Figure 4 presents results at the shear force Ps = 10 kN. The force-displacement curve overestimated the maximum value, and its slope after the peak was too high. One curved localized zone was numerically obtained instead of two experimental ones. To improve the behaviour of the damage model, the modified Rankine definition (Eq. (8)) was also used. The coefficient c was taken as 0.15. The pattern of localized zones and force-displacement diagram reflect the experimental results much better than those with a standard Rankine definition (Fig. 5). Afterwards, the modified von Mises definition was used with the same set of parameters as for the Rankine definition. The coefficient k in Eq. (9) was equal to 10. Both the force-displacement curve and pattern of localized zones are in good agreement with the experiment (Fig. 6). 4.4 FE-results with smeared crack model The characteristic length was equal to l = 2 mm. The tensile strength was taken as ft = 2.2 MPa and the

267

Figure 5. The force-displacement curves and contour map of parameter D for shear force Ps = 10 kN within damage mechanics (modified Rankine definition, Eq. 8).

ultimate normal crack strain ε nu = 0.02. For fixed crack models, the shear retention parameters were assumed as: εsu = 0.02 and p = 8. Figure 7 shows results obtained with a multi-orthogonal fixed crack model. The force-displacement curve was reproduced quite well. One obtained two straight localized zones similarly as in the experiment.

Figure 6. The force-displacement curves and the contour map of the parameter D for shear force Ps = 10 kN for damage model with modified von Mises definition.

Figure 7. The force-displacement curves and the contour map of the εmax for shear force Ps = 5 kN for multi-fixed orthogonal crack model.

The calculations with a rotating crack model were also carried out, but they were not successful. Serious numerical problems with convergence took place shortly after the peak. The obtained localization pattern

268

Figure 8. The force-displacement curves and deformed specimen for shear force Ps = 5 kN using cohesive elements.

Figure 9. The force-displacement curves and deformed specimen for shear force Ps = 10 kN using cohesive elements.

was similar to that obtained with a damage model using a Rankine definition of the equivalent strain measure (one horizontal localized zone or two almost horizontal localized zones were created).

4.5 FE-results with cohesive elements The following parameters were assumed: ft = 2.2 MPa, η = 0.0 and β = 30000. Figures 8 and 9 show the results obtained for a shear force Ps equal to 5 kN and 10 kN, respectively. In both cases, a very good agreement was obtained between experimental and numerical crack patterns and force-displacement curves.

5 SCHLANGEN TEST 5.1 Problem A single-edge notched (SEN) concrete beam under four-point shear loading (anti-symmetric loading) was analyzed (Schlangen, 1993), Fig. 10. The length and height of the beam were equal to 440 mm and 100 mm, respectively. The depth of the notch was equal to 20 mm. The thickness was 5 mm. In the experiments, a curved crack starting from the lower-right part of the notch towards a point to the right of the lower right support was obtained. A FE-mesh consisted of 6556 3-node triangle finite elements. The Young modulus was taken as E = 35 GPa and the Poisson ratio as ν = 0.2. The deformation was induced by linearly increasing the distance δ2 (due to the snap-back behaviour of vertical displacements at the points where the forces were applied) (Fig. 10). Therefore, a so-called indirect displacement control method was implemented into the program Abaqus Standard. It required the definition of the second independent mesh (the same as the basic one) to collect the needed data and to calculate the loading factor (Bobi´nski et al. 2009).

Figure 11. The force-displacement curves and contour map of non-local parameter κ¯ in central part of beam within elastoplasticity (l = 1 mm, κu = 0.04).

5.2 FE-results within elasto-plasticity Figure 11 demonstrates results obtained with the ela-sto-plastic model (ft = 3 MPa, κu = 0.040 and l = 1 mm). The localized zone was curved and its shape matched well the experiment. A satisfactory agreement between a numerical and experimental force-displacement diagram was also achieved.

Figure 12. The force-displacement curves and contour map of damage parameter D in central part of beam within damage mechanics (with Rankine equivalent strain definition, Eq. 6).

5.3 FE-results within damage mechanics

Figure 10. Geometry of SEN specimen (Schlangen, 1993) (dimensions in mm).

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The numerical calculations were performed with the Rankine definition (Eq. (6)) of the equivalent strain measure. The following material parameters were assumed: l = 1 mm, κ0 = 8.5 × 10−5 , α = 0.92 and

shown that the influence of the material description in the tensile-compression regime has to be taken into account. A fixed crack model was not able to reproduce curved cracks. The worst results were obtained with a rotating crack model. In turn, the approach with cohesive elements provided the best approximation of experiments. ACKNOWLEDGEMENTS The FE-calculations were performed at the Academic Computer Centre in Gdansk TASK. REFERENCES

Figure 13. The force-displacement curves and contour map of damage parameter D in central part of beam within damage mechanics (with modified von Mises equivalent strain definition).

β = 150. Both the force displacement-diagram and strain localization differ significantly from the experimental outcome (Fig. 12). To obtain a better agreement with the experiment, again the modified von Mises definition (Eq. (9)) of the equivalent strain measure was used with the following parameters: l = 1 mm, κ0 = 8.0 × 10−5 , α = 0.92 and β = 150. Although, the slope of the calculated force-displacement curve was too sharp, the shape of the localized zone was properly reproduced (Fig. 13). 6 CONCLUSIONS The results of our FE simulations within continuum mechanics of the concrete behaviour under mixed mode conditions have shown that a proper choice of a constitutive law is a very important issue. The models show a different capability to capture the crack phenomenon. In general, the elasto-plastic model with the Rankine failure criterion was the most effective among continuous models. The use-fulness of an isotropic damage model depended on the definition of the equivalent strain measure. Our simulations have

Bobi´nski, J. & Tejchman, J. 2008. Quasi-static crack propagation in concrete with cohesive elements under mixedmode conditions. In B.A. Schrefler, U. Perego (eds.), Proc. 8th World Congress on Computational Mechanics WCCM 2008, Venice. Bobi´nski, J., Tejchman, J. & Gorski, J. 2009. Notched concrete beams under bending–calculations of size effects within stochastic elasto-plasticity with non-local softening. Archives of Mechanics, 61(3–4): 1–25. Brinkgreve, R.B.J. 1994. Geomaterial Models and Numerical Analysis of Softening. PhD Thesis, Delft University of Technology. Camacho, G.T. & Ortiz, M. 1996. Computational modelling of impact damage in brittle materials. Int J Sol Struct, 33(20–22): 2899–2938. Hordijk, D.A. 1991. Local approach to fatigue in concrete. PhD Thesis, Delft University of Technology. Jirásek, M. & Zimmermann, T. 1998. Rotating crack model with transition to scalar damage. J Eng Mech-ASCE, 124(3): 277–284. Majewski, T., Bobi´nski, J. & Tejchman, J. 2008. FE-analysis of failure behaviour of reinforced concrete columns under eccentric compression. Engineering Structures, 30(2): 300–317. Marzec, I., Bobi´nski, J. & Tejchman, J. 2007. Simulations of crack spacing in reinforced concrete beams using elasticplasticity and damage with non-local softening. Computers and Concrete, 4(5): 377–403. Nooru-Mohamed, M.B. 1992. Mixed mode fracture of concrete: an experimental approach. PhD Thesis, Delft University of Technology. Rots, J.G. & Blaauwendraad, J. 1989. Crack models for concrete, discrete or smeared? Fixed, Multi-directional or rotating?. Heron, 34(1): 1–59. Schlangen, H.E.J. 1993. Experimental and numerical analysis of fracture processes in concrete. PhD Thesis, Delft University of Technology, 1993.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Topological search of the crack path from a damage-type mechanical computation Marina Bottoni Électricité de France, Research and Developement, Clamart, France

Frédéric Dufour Grenoble Institute of Technology, Grenoble, France

ABSTRACT: In this paper, we present a method to track the crack path from a continuum finite element simulation using non linear material models. The work belongs to a larger project, aiming at extracting the crack opening from the results of a computation performed by means of a regularized damage model, where a damaged zone represents the crack in a diffuse manner. In the developed algorithm, we look at the distribution of the history variable rather than the internal variable, since it is not limited to 1 and thus its profile is sharper. Then, we find the crack path by means of a step-by-step procedure. At each step, a new point is found as the location where the history variable is maximum at a given direction and distance. The algorithm makes use of some parameters, whose range of use is discussed and calibrated. Finally, we applied the method to two examples and the crack path is automatically found along the ridge of the studied variable. 1 INTRODUCTION A correct evaluation of crack opening is important in many concrete structural applications. For some type of structure, crack opening estimation is required in relation with permeability issues, in order to limit gas or liquids releases. Examples are confinement vessels and cooling towers of nuclear power plants, dams and silos. In other cases, where concrete is exposed to environmental agents, such as for bridges, the notion of crack opening is necessary to assess durability in the sense of penetration of external chemical aggressive agents or simply for aesthetic reasons. Another approach to assess structural durability is the use of a matching law which relates material permeability with damage variable (Pijaudier-Cabot et al. 2009). Even in this method a simple analytical relation between crack opening and damage has been used and can be improved. Finite elements and continuum damage models are an useful and widespread method to compute concrete structures in the nonlinear range. They are often employed together with nonlocal (Pijaudier-Cabot and Bažant 1987) or gradient enhanced models (Peerlings et al. 1998), since conventional continuum damage descriptions suffer from ill-posedness beyond a certain level of damage. Some other computational methods seem to be more suited to compute crack opening, such as finite elements with re-meshing of the crack (Askes and Sluys 2000) and methods incorporating discontinuities, such as the X-FEM (Belytschko

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and Black 1999; Moës et al. 1999). However, the application of the first one is limited by excessive computational time, whereas the use of the second one is limited to crack propagation only. Another method consists in combining the continuum formulation for the bulk response and a discrete cohesive model to represent the crack formation, see for example (Simo and Oliver 1994; Oliver 1996). However, this approach has intrinsic difficulties by establishing the transition point between the two kind of modelisations, in terms of stresses, damage or strain equivalence (Comi et al. 2007; Simone et al. 2003). Our approach consists in the extraction of the opening directly from a damage computation, as detailed by Dufour and co-workers (Dufour et al. 2008). This procedure is able to predict crack opening, if the crack path is a priori known. The aim of this paper is to obtain the line representing the crack path in automatic way. We restrict ourself to 2D domains with a single fracture process zone FPZ. Coherently with damage mechanics, a crack is represented by a damage distribution. The challenge is to develop a procedure which extract a 1D geometry, i.e. the crack path, from a 2D space, i.e. the numerical FPZ. However, damage as internal variable is limited to the value of 1, resulting in an undifferentiated zone where it takes the same value. Hence, we assumed the history variable (maximum value taken by the equivalent strain during loading) as representative of the crack, since this quantity has no upper limit. We propose here a procedure to extract this discrete

loses mesh-objectivity. In literature, there are different regularization techniques to overcome this problem. In finite elements with C 0 continuity, regularization can be performed, for example, by nonlocal models or implicit gradient models. In nonlocal models (Pijaudier-Cabot and Bažant 1987; Jirásek and Bažant 2002), a non local variable g(ε) is computed by averaging the local g(ε) with a weighting function  defined on a neighbourhood of the point of coordinates x, so that:   (ξ − x) g(ε(ξ )) dξ (4) g(ε(x)) = V  V  (ξ − x) dξ

information from the distribution of the history variable. The main idea is to identify the crack path with the ‘‘ridge’’ of this distribution on the process zone. One of the advantages of our algorithm, is that it is applied as a post-processing and not incrementally at each time-step of the simulation, since informations about the loading history are incorporated in the history variable. In this paper, we remind firstly basic notions of regularized damage models. Then, we explain the algorithm in details. In the third part, we apply it to an artificial result in order to study the sensitivity to parameters. Finally, we test it to a more complex artificial result and to one mechanical simulation.

For , a number of functions can be employed; often, a gaussian function is used:     2ξ − x 2 (ξ − x) = exp − (5) lc

2 NON LOCAL DAMAGE APPROACH 2.1 Damage models In damage models, damage growth is controlled by a loading function f , which can be expressed in terms of stresses or strains. In the following, this function is given in terms of strains, the most common choice to be implemented in finite element codes (strain driven problem). Supposing for simplicity to have a single scalar variable for damage (isotropic damage), the loading function can be written as: f = f (g(ε) − Y )

where lc is a characteristic length. In implicit gradient models (Peerlings et al. 1998; Geers et al. 2000), the regularized variable is related to the local quantity by the following differential relationship and associated boundary conditions: g(ε(x)) − c2 ∇ 2 g(ε(x)) = g(ε(x)) ∇g(ε) · n = 0

(1)

˙ ≥ 0, D

˙ =0 fD

(2)

The choice of function g(ε) is related to the material characteristics. For concrete, usually damage is related to positive principal strains, see among others (Mazars and Pijaudier-Cabot 1989; Desmorat et al. 2007; Papa and Taliercio 1996). The constitutive law can be written as: σ = (1 − D) Eε

(6b)

where n is the vector normal to the boundary and c is the characteristic length of the gradient model. For implementation in a finite element code, Equation 6a is then restated in a weak form. In the nonlocal/gradient enhanced formulation of damage models, the constitutive relationship given by Equation 3 remains unaltered, whereas the evolution of damage depend on regularized quantities: D = D(Y ).

where g(ε) is a function of strains and Y is the history variable, which starts from an initial value Y0 and is, for each time instant, equal to the maximum value taken by g(ε) during the loading history. An evolution law for damage, D = D(Y ), must also be defined. The damage growth is controlled by the Kuhn-Tucker conditions: f ≤ 0,

(6a)

(3)

being E the elastic stiffness tensor. D varies from 0 for a sound material to 1 upon the total loss of integrity. 2.2 Regularization techniques for damage models For many materials, such as concrete or rock, damage models introduce strain softening and consequently the ill-posedness of the continuum. After the domain discretization due to finite elements, the solution

3 THE NUMERICAL PROCEDURE 3.1 Damage and history variable: an example We present in this section an example of the damage and history variable fields to show the difference in their distribution. Fields are issued from a 2D-mechanical simulation of a brazilian test using Mazars’ damage model (Mazars and Pijaudier-Cabot 1989), performed with the finite element code Code_Aster (www.codeaster.org). In Figure 1 it is shown the obtained distribution of damage. The field is very flat in the FPZ since damage is saturated to 1 by definition. For this reason, the extraction of the crack path from this quantity is not possible. On the contrary, the history variable has a much more pronounced ridge, as shown in Figure 2, easier to follow during our procedure.

272

D

interpolation functions are used for displacements and linear interpolation functions for regularized strains. Hence, regularized strains are defined on the apex of elements; all quantities derived from regularized strains, such as the equivalent strain and the history variable Y are then also known on the same nodes and have the same degree of approximation on the finite element. Nonlocal integral models can be applied to finite elements with all possible approximations, but are usually applied to linear elements due to the large size of non-zero components in the connectivity matrix. In the damaged zone, the search consists in a stepby-step procedure, according to which a new point is found at each step. Except for the first two points (initialisation), a starting point and a search direction must always be fixed. At each step, the following actions must be performed (see also the scheme of Figure 3):

1 0

0 40

15

30

x Figure 1.

20

10

30

y

0 45

Damage map for a brazilian test.

Figure 2. Y map for a brazilian test.

3.2 The algorithm A precise definition of the ridge cannot be given; however, this notion lies on some intuitive assumptions. To begin with, there is somehow the idea that the ridge contains the maxima of the field; still, it cannot be easily defined as the geometric one-dimensional locus containing the field maxima, because the field varies strongly along the crack path, from the crack notch to the point where the crack is first activated. The second implicit assumption is that the damaged zone from which the crack is extracted, is mainly distributed in one direction; to that is related the idea of a directional search. Thirdly, it can be stated that the tangent to the crack is continuous; this hypothesis is not verified in true concrete due to heterogeneities, but acceptable in the frame of continuum mechanics. An important clarification is about the discretization of the history variable, its level of approximation depending on the degree of the interpolation polynoms of the adopted finite elements. The examples shown in this contribution (see Section 5) are based on a linear interpolation of the field Y , which is the most common for both regularisation techniques. In fact, equations describing the gradient enhanced method are usually implemented in finite element codes by means of a mixed formulation; quadratic

273

1. the starting point is the point found in the previous step; 2. the search direction is determined by the last two points found; 3. the next point is first estimated by moving from the starting point in the search direction, at a distance a (‘‘step length’’); actually, this choice for the search direction is related to the assumption of continuous tangent; 4. a line orthogonal to the search direction is traced; 5. the field Y is projected onto this line, by means of the element shape functions; 6. the profile Y (s) obtained by projecting the field is regularized in order to find a curve without discontinuities in the first derivative; in fact, since the history variable field is stepwise linear due to the discretization, the projected field on the orthogonal line also shares the same property. Then, the new point of the ridge is the point where Y (s) is maximum. In this regard, it is easy to understand why the

Figure 3.

Scheme of the topological search procedure.

regularisation on the orthogonal profile is important. In fact, since Y (s) is stepwise linear, the maxima of the field would be necessarily at an intersection between the orthogonal line and an element edge; the precision in finding the good ridge would be then strongly dependent on mesh refinement. We will see in Section 4 how the precision of the procedure can be improved with respect to mesh refinement. To obtain a smooth curve, a convolution product is used, analogously to the regularization operated by the nonlocal integral model, so to obtain the quantity Y(s) as a function of the curvilinear abscissa s on the orthogonal profile : Y (s) =



l

(|ς − s|) Y (ς) dς  l (|ς − s|) dς

Search direction

a

3 (P1 ) 1

Orthogonal profile

r=a Circular profile

(7)

where l is the orthogonal line and a Gaussian function is employed as weight function :     2|ς − s| 2 (ς − s) = exp − lreg

2 (P2)

Figure 5.

Initialisation of the topological search.

(8)

An example of the projected field Y (s) and its regularization Y (s) is shown in Figure 4. As far as the initialisation procedure is concerned, the scheme is shown in Figure 5. The procedure starts from the point where the field Y is maximum. Due to the discretization, this point is located on a node. Hence, an iteration is required in order to set its position again. Moreover, if this node does not belong to the geometry boundaries, a search must take place in two opposite directions. In the following list, we detail the actions to find the first two points: 1. the point with the maximum value of Y is found on the damaged zone; due to the linear interpolation of the field, the maximum is always on a node;

Figure 4. Projected fields Y (s) and Y (s) on the orthogonal profile, obtained by regularization through the convolution product.

274

Figure 6. Projected fields Y (s) and Y (s) on the circle obtained by regularization through the convolution product.

2. a circle is drawn, with radius equal to the step length a; 3. the field Y is projected onto this circle, by means of the element shape functions; 4. the projected Y (s) field is regularized in order to find a curve without discontinuities in the first derivative. A convolution product is used, analogously to the typical step (see Equations 7 and 8), by integration on the circle. An example of the projected field and its regularization are shown in Figure 6; 5. the second ridge point (P2 ) is the point where Y (s) is maximum; 6. two possible search directions are determined by −−→ −−→ the points P1 and P2 , i.e. P1 P2 and P2 P1 . After the two first ridge points are found, the search is managed as detailed for the typical step.

The procedure is then stopped in one direction when the identified ridge value is smaller than a given value defined a priori, which is another parameter of the algorithm. However, damage models for brittle or quasi brittle materials already contain a damage threshold in terms of an equivalent strain, below which damage cannot take place. In the present work, the same value is employed as a threshold for the search algorithm, beyond which the crack is not propagated, i.e. no damage has occured. A higher value can eventually be set, in case it is decided that for the corresponding value of damage there is no real crack, but only diffuse microcracks. Furthermore, if the crack path is looked for computing the crack opening then the limit value will be the one below which the opening is computed with an associated error larger than a given limit according to Dufour and co-workers (Dufour et al. 2008). Moreover, there are other situations requiring the search to stop in one direction, that is, when the crack path reaches the material boundary. Obviously, the procedure is arrested when all points on the orthogonal profile lay outside the material, but this condition is very strict and seldom verified; hence, in the current implementation, the procedure is simply stopped if the prediction point is located outside the material.

size Lx = 52 and Ly = 32 (the units are not important). Three squared meshes with different refinements have been used. The three meshes are here denoted as coarse, medium and fine; corresponding element sizes are  = 2,  = 1.333 and  = 0.8. The crack path is given by the following equation:

4 CALIBRATION OF PARAMETERS

y = 10 sin ([0.045 (x + 10)]2 ) + 15

whereas on the orthogonal plane, the field is determined by an inferiorly limited parabola, with the minimum value equal to Y min = 5 · 10−7 : Y =



10−3 (−t 2 + 10) if Y > Y min Y min if Y ≤ Y min

(10)

t is the distance between any point belonging to the line orthogonal and the crack path in the xy plane. The intersections of the parabola with the plane z = Y min determine the numerical fracture process zone width, in this case equal to about 6.32. Hence, there are about 3 elements in the process zone in the case of the coarse mesh, 7 for the fine mesh. The function representing the theoretical crack path (Equation 9) is represented in Figure 7. The curve has remarkable curvatures at points A and B, that is 0.127 in A and 0.382 in B; these values correspond to radii of the osculating circles equal to 7.86 and 2.62 in A and B

On the whole, the procedure requires the calibration of three parameters: • the search step a; • the length of the orthogonal profile lort ; • the regularisation length lreg ;

(9)

30 A

25

y

20 15 10 5

4.1 Definition of the testing field In order to test the numerical procedure, we have built on a given mesh a numerical field representing the history variable. The aim is to compare the crack path predicted by the algorithm with a theoretical curve, while controlling the parameters defining the field. In order to build the field, first a mesh must be determined on the geometric domain xy. Then, two functions are needed, the first one defining the crack path in the xy−plane, and the second one the values of the field on all nodes of the geometric domain. The second function is defined on all points of the crack path, on the straight line orthogonal to the crack path itself; it has a maximum on the crack path and is always positive, since it represents the history variable. For the calibration of the procedure parameters, we set a constant width for the process zone, and a constant value for the history variable on the ridge. The field is defined on a rectangle in the plane xy, having

275

0

B 0

10

20

30 x

40

50

60

Figure 7. Theoretical ridge for the testing of the procedure and parameters calibration.

Figure 8. Numerical field used for parameters calibration; the color of each element facet is associated to average value of the field on the element nodes.

respectively. We tested the algorithm also for high curvatures, even for values which are hardly found in a real crack. 4.2 Calibration Search step. At first, the influence of the search step a is studied. In the following simulations, the values of all other parameters are constant: lort = 8, lreg = 0.18lort . In Figure 9, the obtained ridge points are shown together with the theoretical curve for the coarse mesh (element size equal to 2) and for three values of the search step. The algorithm provides points with good correspondence with the initial curve for a large range of a. For big values of the search step, the algorithm find it difficult to follow the real path near point B. Looking closely to the curves of Figure 9, an error can be observed near extremities; the numerical crack path ridge deviates from the theoretical one. This loss of accuracy is due to the fact that there the field Y is projected only onto the part of the orthogonal profile belonging to the geometric domain; the resulting orthogonal profile can be much smaller and asymetrical with respect to the prediction point, thus boundary problems can arise, in analogy of what happens with nonlocal formulations (Peerlings et al. 2001). However, this boundary error remains relatively small and confined in a very small region. Furthermore, looking at a recent work (Krayani et al. 2009), one can conclude that for nonlocal damage simulation cracks are always othogonal to free boundaries. Thus, the symmetry of the orthogonal profile will be maintained. To compare quantitatively the results, the average residue defined in Equation 11 is used as an error indicator: Np 1  · |yi,theo − yi,num | Q= Np i

with Np being the number of points identified by the numerical procedure, yi,num their ordinates on the xy plane, and yi,theo the theoretical ordinates at the same abscissa. In Figure 10, the error is depicted as a function of the search step a and for the three meshes. The points near the crack path extremities are not taken into account, in order to exclude the boundary effect. The error remains nearly constant for steps going from 0.4 to 2.6; then, from 2.6 the error increases rapidly. However, if the geometric domain of the field is reduced to x ≤ 24, so excluding the area with the strong curvature near point B, the error remains of the same order of magnitude until 4, the last tested value of the search step. We can conclude that the step length must be smaller than the radius of the osculating circle (equal to 2.62 in B) in order to well approximate the curve. A final remark is about the behaviour of the algorithm for different meshes. In fact, the error depends strongly on the mesh refinement, as pointed out again on Figure 10. On the contrary, the value of the step from which the algorithm cannot follow the theoretical path is the same for the three meshes since it depends on the geometrical properties of the path (curvature).

(11)

Figure 9. Crack paths identified by the developed algorithm compared with the theoretical path: results for different values of the search step, coarse mesh.

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Length of the orthogonal profile. Figure 11 shows the error for different values of the orthogonal profile lort . The other parameters are constant and have the following values: a = 1 and r = 0.28. As seen for the calibration of the step length, the error depends on the refinement. For lort bigger than a certain value, which depends on mesh refinement, the error is minimum for that refinement and more or less constant; this value is about 4−5 times the finite element size. It is anyway recommended to adopt a value larger than the FPZ width. Regularization length. Figure 12 shows the error defined by Equation 11 for different values of the

Figure 10. Error for different values of the search step: results for the three meshes.

near extremities are excluded. As remarked by the calibration of the search step and of the length of the orthogonal profile, the error depends on mesh refinement. A loss of accuracy can be observed for all three meshes when r is bigger than 0.52; the lower limit of the optimal range depends slightly on the refinement. This fact is easily explained by noticing that for lreg → 0, the regularized curve Y (s) coincides with the original, stepwise linear curve Y (s), so that the points of the crack path are on the element edges due to the linear discretization of the field. In Figure 13, the obtained crack paths for three values of r are shown. For larger values of r, the curvature at point B is caught less precisely. Figure 11. Error for different values of the length of the orthogonal profile: results for the three meshes.

5 EXAMPLES 5.1 Artificial result We first tried our procedure onto an artificial result, obtained as described in Section 4.1. The crack path is the fourth order polynom running through points P1 = [10, 0], P2 = [35, 20] P3 = [60, 0], P4 = [85, 20], P5 = [110, 0]. We modulated the field along the crack path assuming for the parabola y = a1 x2 + a2 on the orthogonal profile two coefficients a1 ,a2 , varying with x with equation: a1 · 10−6 = −60 cos (0.1x) + 100 a2 · 10

Figure 12. Error for different values of the regularisation length: results for the three meshes.

−6

= 500 cos (0.3x) + 1000

(12a) (12b)

The obtained field is shown in Figure 14. The identified crack path is shown in Figure 15; for the search procedure, we used the following parameters: a = 2, lort = 20, lreg = 0.18lort . The procedure follows the ridge with accuracy also when the fracture process zone width and the value of the field on the ridge have strong variations along the crack path. 5.2 Double-notched specimen in tension In this section, the procedure is applied to the results of a F.E. mechanical simulation, performed with the

Figure 13. Crack paths identified by the developed algorithm compared with the theoretical path: results for different values of the regularization length, medium mesh.

regularization length lreg and for the three meshes. The error is represented as a function of the ratio between the regularization length and the length of the orthogonal profile, r = lreg /lort . The other parameters are constant and are: a = 1 and lort = 8. The points

277

Figure 14. Artificial results for testing of the procedure.

means of a non local integral model. For the presented FE-simulation, we made use of the following parameters: Young modulus E = 31000 MPa, Poisson ratio ν = 0.2, damage threshold εd0 = 0.000097, At = 0.95, Bt = 9000, Ac = 1.25, Bc = 1000; the internal lenght of the nonlocal model is lc = 0.0216 m. The distribution of damage on the specimen at the end of the computation is shown in Figure 17. A single damaged zone is observable in the center of the specimen. Nevertheless, the simulation predicts two distinct cracks developing from the notches. These two cracks are still recognizable from two distinct zones in the history variable map (see Figure 17). Since the algorithm applies to one crack at a time, the specimen has been divided into two separated zones by tracing an

Figure 15. Crack path for the manifactured results.

thickness = 10 mm

2 mm 120 mm

5 mm

40 mm 10 mm

10 mm

60 mm

(a)

(b)

Figure 16. (a) geometry and (b) boundary conditions of the mechanical test.

software Cast3M (www.cast3m.cea.fr). Geometry and boundary conditions are shown in Figures 16a, b. For the analysis of the double-notched specimen, the Mazars damage law has been employed. According to this law, the damage (scalar) variable D is a combination of two components: Dt which is damage due to tension and Dc which is damage due to compression: D = αt Dt + αc Dc

(13)

where αt and αc depend on the strain tensor. Damage in tension is: Dt = 1 −

εd0 (1 − At ) At − Y exp (Bt (Y − εd0 ))

Figure 17. Scalar map of damage on the double-notched specimen in tension.

(14)

where εd0 is the damage threshold (Y0 in Section 2.1), At , Bt are other two material parameters. Damage in compression Dt is calculated in the same way by using material parameters Ac , Bc instead of At , Bt . For this law, function g(ε) defined in Section 2.1 is an equivalent strain εeq , equal to:

εeq



3

 =  ε 2

i +

(15)

i=1

εi + denotes the positive part of the principal strain εi . Strain regularization has been performed by

278

Figure 18. Scalar map of the Y field and crack path for the double-notched specimen in tension: zoom on the central part of the specimen.

horizontal line at the center of the specimen. Then, the procedure has been run onto the two zones in order to find the two cracks. In Figure 17, the points of the cracks path identified by our algorithm are plotted. For the search, we used the following parameters: a = 0.001 m, lort = 0.015 m, lreg = 0.1lort . 6 CONCLUSIONS In this paper, we describe a method to track the crack path from a finite element computation making use of a regularized damage model. The main advantage is to use continuum models and to apply the tracking algorithm as a post-processing, so without making the simulation computationally more expensive. The point is to perform a topological search, so to find the ‘‘ridge’’ of the history variable field, which is less flat than the damage field. The notion of crack direction is essential for the search; moreover, the regularization of the history variable field decreases appreciably the error on the crack path. The method makes use of some parameters, such as the search step, the regularization length and the length of the orthogonal profile. The method performs well, working for a wide range of the parameters. More into details, the search step must be smaller of the minimum radius of the osculating circle; the length of the orthogonal profile must be bigger than about 5 times the finite element size and of the fracture process zone width; the regularization length has an optimal range depending slightly on mesh refinement and ranged between about 0.2 and 0.52. The total error on the crack path depends on mesh refinement too. We have also tested the procedure on an artificial result, which shows a strong curvature of the crack path and a modulation on the ridge value and of the numerical FPZ width. The other application is on the results of a mechanical simulation, performed with an integral nonlocal model on a double-notched specimen loaded in tension. In both applications, the algorithm has managed well to find the crack path, even with a rather coarse mesh. REFERENCES Askes, H. and L.J. Sluys (2000). Remeshing strategies for adaptative ale analysis of strain localisation. European Journal of Mechanics A/Solids 19(3), 447–467. Belytschko, T. and T. Black (1999). Elastic crack growth in finite elements with minimal remeshing. Int. J. Num. Methods Eng. 45, 601–620.

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Comi, C., S. Mariani, and U. Perego (2007). An extended fe strategy for transition from continuum damage to mode I cohesive crack propagation. International Journal for Numerical And Analytical Methods in Geomechanics 31, 213–238. Desmorat, R., F. Gatuingt, and F. Ragueneau (2007). Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials. Engineering Fracture Mechanics 74(10), 1539–1560. Dufour, F., G. Pijaudier-Cabot, M. Choinska, and A. Huerta (2008). Extraction of a crack opening from a continuous approach using regularized damage models. Computers and Concrete 5(4), 375–388. Geers, M., R. Peerlings, W. Brekelmans, and R. de Borst (2000). Phenomenological nonlocal approaches based on implicit gradient-enhanced damage. Acta Mecanica 144, 1–15. Jir´asek, M. and Z.P. Bažant (2002). Non local integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149. Krayani, A., G. Pijaudier-Cabot, and F. Dufour (2009). Boundary effect on weight function in non-local damage model. 76, 2217–2231. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—Application to concrete. J. Eng. Mech. Div. 115(2), 345–365. Moës, N., J. Dolbow, and T. Belitschko (1999). A finite element method for crack growth without remeshing. Int. J. Num. Methods Eng. 46, 131–150. Oliver, J. (1996). Modelling strong discontinuities in solids mechanics via strain softening discontinuities equations. Part 1: Fundamentals. Int. J. Numeric. Methods Eng. 39(21), 3575–3600. Papa, E. and A. Taliercio (1996). Anisotropic damage model for the multiaxial static and fatigue behaviour of plain concrete. Engineering Fracture Mechanics 55(2), 163–179. Peerlings, R., R. de Borst, W. Brekelmans, and M. Geers (1998). Gradient-enhanced damage modelling of concrete fracture. Mechanics of Cohesive-Frictional Materials 3, 323–342. Peerlings, R., M. Geers, R. de Borst, and W. Brekelmans (2001). A critical comparison of nonlocal and gradientenhanced softening continua. IJSS 38, 7723–7746. Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. ASCE Journal of Engineering Mechanics 113(2), 1512–1533. Pijaudier-Cabot, G., F. Dufour, and M. Choinska (2009). Permeability due to the increase of damage in concrete: from diffuse to localised damage distributions. 135(9), 1022–1028. http://dx.doi.org/10.1061/(ASCE)EM.1943 7889.0000016. Simo, J. and J. Oliver (1994). A new approach to the analysis and simulation of strong discontinuities, pp. 25–39. E&FN Spon. Simone, A., H. Askes, R. Peerlings, and L. Sluys (2003). Interpolation requirements for implicit gradientenhanced continuum damage models. Commun. Numer. Meth. Engng 19, 563–572.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches J. Cervenka & V. Cervenka Cervenka Consulting Ltd., Prague, Czech Republic

ABSTRACT: Numerical modeling of shear failure of reinforced concrete beams with or without shear reinforcement still remains a challenging task even after several decades of active research. The paper concentrates on the shear failure analysis of large beams without and with shear reinforcement. As an example, it uses the experiments performed by Collins and Yoshida (2006). The paper discusses the main issues affecting the reliability of shear strength predictions and will evaluate the effectiveness of the discrete and smeared crack approaches in resolving the numerical problems in shear failure modeling of strain softening materials. 1 INTRODUCTION Numerical models for brittle materials such as concrete were for the first time introduced already in the 70’s by landmark works of Ngo & Scordelis (1967), Rashid (1968) and Cervenka V. & Gerstle (1971). Many material models for concrete and reinforced concrete were developed in 70’s, 80’s and 90’s such as for instance the models by Suidan & Schnobrich (1973), Lin & Scordelis (1975), De Borst (1986), Rots (1989), Pramono & Willam (1989), Etse (1992) or Lee & Fenves (1998). These models were based on the finite element method. A concrete material model was formulated as a special constitutive model that is used at each integration point for the evaluation of internal forces. It was soon realized that material models with strain softening, if not formulated properly, exhibit severe mesh dependency (De Borst & Rots 1989), and tend to zero energy dissipation if the element size is reduced (Bažant 1976). This was attributed to the local nature of the constitutive material description, which results in the loss of hyperbolicity of the governing differential equation in the softening region (Belytschko et al. 1986). This deficiency means that mathematically a solution can be found, but its uniqueness cannot be guaranteed. In numerical analysis this results in mesh sensitivity and/or numerical instabilities such as convergence problems. The crack band approach was proposed by Bažant and Oh (1983) to remedy the convergence towards ˇ zero energy dissipation. It was shown by Cervenka V. (1995) that proper formulation of the crack band size can severely reduce also the mesh bias of these smeared crack approaches. A more rigorous solution of the ill-posed nature of the strain softening problem is the introduction of higher-order continuum models: such as non-local

damage model by Bažant & Pijaudier-Cabot (1987), gradient plasticity model by de Borst & Muhlhaus (1992) or gradient damage model by de Borst et al. (1996). The non-local models introduce additional material parameters related to an internal material length scale, which is however difficult to derive from existing material tests. Currently these models are mathematically rigorous, but appear to be too fundamental for practical applications. Another solution for the strain softening problem is the discrete crack model, where the discontinuities arising from strain localization are directly included into the numerical model. This model was first introduced with automatic remeshing and crack propagation by Saouma & Ingraffea (1981). In the classical form of the discrete crack approach a crack is simulated as a cohesive interface, which is inserted into the finite element model whenever a certain criterion for crack initiation or propagation is satisfied. This means that whenever a crack is initiated or an existing crack propagates remeshing is necessary. In recent years, a whole new class of methods has emerged based on enhanced finite element formulations or various mesh free methods. A comprehensive treatise on these approaches is for instance provided in de Borst et al. (2003) or Jirásek (2003). The mesh free method was proposed by Belytschko et al. (1994). This method has a strong potential for solving crack propagation problems, but it contains still many open issues such as large computational demand, difficulties in 3D implementation and the need to use a background mesh for the numerical integration. The enhanced finite element formulations such as EED (elements with embedded discontinuities) or X-FEM (extended finite elements) are nicely classified by Jirásek (2003). These models attempt to bridge the gap between the discrete and smeared models by enhancing the finite element formulation to

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better capture the discontinuity arising from the strain localization. Their basic idea is however very close to the classical discrete approach, where the ‘‘traditional’’ remeshing is replaced by enhancing the finite element formulation in the elements where cracking occurs. Similarly to the classical discrete crack approach it is often necessary to trace the crack propagation through the model, guarantee its continuity across element boundaries, solve crack branching, crack intersection etc. The paper discusses the main issues affecting the reliability of shear strength predictions. Second objective is to evaluate the effectiveness of the discrete approach in resolving the ill-posed nature of the underlying mathematical problem; since it is often believed that introduction of discontinuities into the numerical formulation can resolve this issue. The reliable prediction of shear strength in real structures requires a rather complex material formulation, which should capture at least the most important features of concrete behavior, such as: compressive crushing, compressive softening, shear response of cracked concrete and reinforcement yielding. The paper evaluates the importance of these factors on the reliability of shear strength predictions. 2 FRACTURE-PLASTIC MATERIAL MODEL The smeared crack analyses presented in this paper ˇ were performed with program ATENA (Cervenka et al. 2009) using the combined fracture-plastic model ˇ of Cervenka & Pappanikolaou (2008). The material model formulation assumes small strains, and is based on the strain decomposition into p f elastic (εije ), plastic (εij ) and fracture (εij ) components. The stress development can be then described by the following rate equations describing the progressive degradation (concrete cracking) and plastic yielding (concrete crushing): p

f

σ˙ ij = Dijkl · (˙εkl − ε˙ kl − ε˙ kl )

to define analogous quantities for the fracturing model, i.e. λ˙ f is the inelastic fracturing multiplier respectively and g f is the potential defining the direction of inelastic fracturing strains in the fracturing model. The consistency conditions can be than used to evaluate the change of the plastic and fracturing multipliers.

The constitutive equations of the both models can be summarized as follows: Flow rule governs the evolution of plastic and fracturing strains: p

p

f

f

Fracture model: ε˙ ij = λ˙ f · mij ,

p

mij = f

∂g p ∂σij

mij =

∂g f ∂σij

nij =

f f˙ f = nij · σ˙ ij + H f · λ˙ f = 0,

nij =

p

f

∂f p ∂σij

(4)

∂f f ∂σij

(5)

H p and H f is hardening modulus for plastic model and fracturing model respectively. This represents a system of two equations for the two unknown multiplier rates λ˙ p and λ˙ f , and is analogous to the problem of multisurface plasticity (Simo et al. 1988). The details of ˇ the model implementation can be found in Cervenka ˇ et al. 1998 and Cervenka & Pappanikolaou (2008). The model is using Rankine criterion for tensile fracture with exponential softening of Hordijk (1991) (see Figure 1). The compressive behavior is modeled by the plasticity model, which is using the three parameter surface of Menentrey & Willam (1995) (see Figure 2) and hardening softening is defined according to the laws p described in Figure 3 where εeq is the equivalent plastic strain. The softening in tension and compression is adjusted using a crack band approach of Bažant & Oh (1983). The crack band Lt as well as crush band size Lc are adjusted with regard to the crack orientaˇ tion approach proposed by Cervenka V. et al. (1995). This method is described in Figure 4 and in (6).

(1)

Plastic model: ε˙ ij = λ˙ p · mij ,

p f˙ p = nij · σ˙ ij + H p · λ˙ p = 0,

Lt′ = γ Lt

and

Lc′ = γ Lc

γ = 1 + (γmax − 1)

θ , 45

θ ∈ 0; 45

The basic idea is to adjust the crack band size depending on the crack orientation with respect to the element edges. This reflects the fact that a crack cannot localize into a single element if the crack direction is not aligned with the element edges.

(2) (3)

where λ˙ p is the plastic multiplier rate and g p is the plastic potential function. Following the unified theory of elastic degradation of Carol et al. (1994) it is possible

282

(6)

Figure 1.

Tensile softening (Hordijk 1991).

2.1 Special features of reinforced concrete When it comes to nonlinear analysis of reinforced concrete, i.e. when reinforcement is to be considered, it becomes important to consider additional special issues related to the reinforcement and the composite reinforced concrete material. The most important phenomena are:

Figure 2. Three paramater (Menetrey & Willam 1995).

criterion

for

concrete

a. shear strength and stiffness of cracked concrete, i.e. aggregate interlock b. compressive strength reduction due to crack opening in perpendicular direction c. reinforcement yielding d. tension stiffening e. dowel action and bending stiffness of the reinforcement e. bond failure between concrete and reinforcement. In the used constitutive model, the items (a) and (b) are considered according to the modified compression field theory of Collins (Bentz et al. 2006). In this theory, the compressive strength is reduced using the formula σc = rc fc′ rc =

(7)

1 , 0.8 + 170 ε1

rclim ≤ rc ≤ 1.0

(8)

where ε1 is the tensile strain in the crack. In ATENA the largest maximal fracturing strain is used for ε1 and the compressive strength reduction is limited by rclim . In this work rclim = 0. The shear strength of crack concrete is also assumed according to the modified compression field theory MCFT (Bentz et al. 2006) as:

Figure 3.

σij ≤

Hardening and softening in compression.

 0.18 fc′

0.31 +

24 w ag +16

;

i = j

(9)

where fc′ is the compressive strength in MPa, ag is the maximum aggregate size in mm and w is the maximum crack width in mm at the given location. The modified compression field theory does not give any formula for the shear stiffness, but it is an important parameter, which significantly affects the reinforced concrete response. In the present formulation the crack shear stiffness Ktcr is calculated directly from crack normal stiffness using a scaling factor sF . Ktcr = sF Kncr

(10)

where Kncr comes directly from the tensile softening law in Figure 1 as: Figure 4. Crack band size adjustment based on crack direction orientation.

283

 Kncr = ft (wt ) wt

(11)

This appears to be a very natural assumption as this makes the shear stiffness dependent on the crack opening displacement. Reinforcement is modeled using the embedded approach with truss elements, and a multi-linear stressstrain law is used to capture reinforcement yielding. Tension stiffening can be activated in the present ˇ model, but was not used. It was shown by Cervenka & Margoldová (1995) that if sufficiently fine mesh is used the tension stiffening effect can be very well captured by an appropriate cracking model. The dowel action and reinforcement bending stiffness is not considered in the present model. The analyzed beams are only slightly reinforced; therefore these effects cannot play a major role. Reinforcement bond failure can play an important role in the analyzed problems. A bond modeling was discussed by authors in a separate paper Jendele & ˇ Cervenka (2006). It was shown that a bond model can strongly improve the results if large finite elements are used in heavily reinforced structures. The problems presented in this paper are only lightly reinforced and the largest element size is 200 mm. It was therefore decided not to use the bond model to limit the number of investigated parameters. 3 DISCRETE CRACK MODEL The analyses calculated with the discrete crack model in this paper are using a simple approach, where a crack is modeled as a zero thickness interface with Mohr-Coulomb type of criterion with tension cut-off (see Figure 5). τ ≤ c − σ φ;

σ ≤ ft

(12)

where c is cohesion and φ is frictional coefficient. The Hordijk’s (1991) law is used for tensile softening. The cohesion softening is also modeled by the same law but the displacement values in the softening diagram for cohesion are 10 times increased. This approximately corresponds to the assumption that the shear response should be more ductile then the tensile one, and that

Figure 5. elements.

Mohr-Coulomb criterion for discrete crack

284

the mode II fracture energy GIIF is about 10 times larger then the fracture energy for mode I. GFII ∼ 10 GFI

(13)

4 SHEAR FAILURE IN PLAIN CONCRETE In the first example, the discrete and smeared crack models are compared on a typical shear problem without reinforcement (see Figure 6). This is the well known Iosipescu’s shear beam. The geometry corresponds to the SEN beams tested by Schlangen (1993). The tested beams have dimensions 440 × 100 × 100 mm. They were cast from concrete with modulus of elasticity E = 35 GPa, Poisson’s ratio ν = 0.15, tensile strength ft = 2.8 MPa and the specific fracture energy GF = 70 N/m. This test setup was originally proposed by Iosipescu (1967) for shear tests of metals. This test was later used by Bažant & Pfeifer (1986) for shear testing of concrete. It was discovered by Ingraffea & Panthaki (1985) that the crack propagation in this kind of test is mainly dominated by mode I, i.e. tensile cracking. Since then it has become a typical test problem for crack propagation analysis, because it is a common believe that smeared crack models cannot predict the behavior correctly and some kind of enhanced formulation is necessary. The load displacement diagrams are compared in Figure 7. The figure shows a single discrete crack analysis and several smeared crack results. The discrete crack analysis has been performed previously ˇ by Cervenka J. (1994). The peak load in the discrete crack analysis is captured very well as well as the overall shape of the response. In post-peak the response is slightly lower. This could be probably improved by increasing the shear properties of the cohesive interface model. The crack path was determined (see Figure 8) by the direction of maximal principal stresses at the crack tip. The discrete crack

Figure 6. Geometry of the modified Iosipescu’s beam (Schlangen 1993), the dimensions are in mm.

bias, and an incorrect vertical crack is reported, which ends to left of the bottom plate. This improved behavior of the current model can be attributed to the crack band size formulation (6). The smeared crack analyses labeled with ‘‘Std’’ indicate analyses using the model described in Section 2, but the special features for reinforced concrete are not activated. The shear factor sF is set to the low value of 20, which means that the shear stiffness on the crack surface is almost identical to the normal one. The main findings from this study can be summarized as follows: Figure 7. Comparison of load-displacement diagrams for the Iosipescu’s beam.

Figure 8.

Crack pattern by discrete analysis.

a. The results confirm that the crack propagation is mainly in mode one. b. The shear properties of the crack do not influence the results significantly. c. The cracked area is quite localized so no numerical problems occur in the smeared crack analysis d. The peak load is predicted correctly by all models. e. The coarse finite element models show lower peak values, which is quite common situation in the crack band model. f. The discrete model predicts more accurately the curved shape of the crack path. g. The crack path predicted by the smeared model is acceptable for practical applications. 5 SHEAR FAILURE IN LARGE RC BEAMS

Figure 9. Crack pattern by smeared analysis. (a) coarse mesh 10 mm, (b) fine mesh 2 mm, (c) experiment (Schlangen 1993).

model can nicely capture the curved shape of the crack path. On the contrary, the crack path curvature is even slightly overestimated. This is caused by extending the crack by a certain non-infinitesimal length a at each propagation. Because of that the crack extension is overestimated, and the crack needs to curve strongly to return to the correct path. Figure 1 shows also the results from several smeared crack analyses. All smeared crack analyses showed the crack path depicted in Figure 9, i.e. a more or less straight crack path towards the right side of the bottom loading plate. So the curved crack path is not obtained, but the crack ends at the right side of the loading plate. For instance the smeared crack results reported by Schlangen (1993) are strongly affected by the mesh

In the next example large beams tested at the University of Toronto by Collins and Yoshida (2000) were investigated numerically. Two beams from the experimental program of Yoshida are considered: Beam YB2000/0 with bending reinforcement and no shear reinforcement and beam YB2000/4 with vertical reinforcement by 8 T-headed bars. The beams are schematically depicted in Figures 10 and 11. The longitudinal reinforcement in both beams is identical. The reinforcing ratio of bottom reinforcement of 6×M30 bars is 0.0074. The ratio of vertical reinforcement of T-headed bars T#4, spacing 0.59 m is 0.00071. The beams are only lightly reinforced. The shear span ratio a/d = 2.86 indicates a shear critical geometry.

285

Figure 10. Beam YB2000/0 dimensions and reinforcement.

is considered in the analysis but not included in the monitored load P. 5.1 Discussion on best-fit results

Figure 11. Beam YB2000/4 dimensions and reinforcement.

The material properties denoted as ‘‘Std’’ and listed in Table 1 correspond to a standard material setup. It is approximately identical with the standard EC2 concrete class C30/37. The used set of material parameters can be recognized as mean properties of this concrete class.

Table 1. Concrete material properties of RC beams, this material set is denoted as ‘‘Std’’, i.e. ‘‘Standard’’.

fctm = fck + 8 = 30 + 8 = 38∼37 MPa

Concrete property

Value

Elastic modulus Ec [MPa] Compressive strength fc [MPa] Tensile strength ft [MPa] Specific fracture energy Gf [N/m] Poisson ratio µ [−] Plastic strain at fc (peak) εcp [−] Plastic end displacement wd [mm] Shear factor sF MCFT fc reduction MCFT aggregate interlock

34 000 37 2.8 80 0.2 0.001 0.5 20 none none

Table 2.

This was the initial set used for the analyses. This set of parameters is very similar to the one used in Section 4. This set of parameters does not include any of the special provisions for reinforced concrete analysis from Section 2.1. It provided very good results for the Iosipescu’s beam (see Figure 7) and also for the case of beam YB2000/0 (see Figure 10). The loaddisplacement diagrams for this beam are compared in Figure 22. However for the beam with stirrups (Figure 11), the peak load was greatly underestimated. These results are reported in Figure 22 under the label ‘‘FP-Std’’. The peak load is underestimated by almost 50%. The input parameters had to be modified in order to obtain a good agreement. This best-fit response is shown in Figure 13, and the adjusted parameters are listed in Table 3.

Reinforcement properties of RC beams.

Steel property

Value

Elastic modulus Es MPa Yield stress fy MPa Max. stress fs,max MPa Limit strain εlim

200 000 470 680 0.11

(14)

The experimental study Yoshida (2000) offered for concrete property only a compressive strength at the date of testing, which was obtained from cylinder tests. In the tests, slightly different properties were found in two specimens. However, in this study it was decided to use identical concrete properties in both specimens in order to keep the effect of different shear reinforcing not influenced by other parameters. The assumed set of parameters for concrete and reinforcement is shown in Tables 1 and 2. The parameters reported in this table are referred to a ‘‘Standard’’ or ‘‘Std’’. In some analysis certain parameters are modified to evaluate their influence on the results. The finite element analysis was done for a symmetrical half of the beam in plane stress representation. Quadrilateral 4-node isoparametric elements, sizes 50–200 mm, were used for concrete and embedded truss elements for bars. The total load P = 2 V acting in the top centre of the beam is considered as the global resistance. Like in experiment, self weight

286

Figure 12. Beam YB200/0 L-D diagram comparison. Analysis is based on ‘‘Std’’ properties and mesh 200 mm.

Figure 13. Beam YB2000/4 L-D diagram comparison. Analysis is based on modified ‘‘Std’’ properties, sF = 300, εcp = 0.002, wd = 50 mm.

Table 3. Adjusted parameters for best fit for beam YB2000/4. Concrete property

Value

Plastic strain at fc (peak): εcp [−] Plastic end displacement: wd [mm] Shear factor: sF

0.002 50 300

Figure 15. Beam YB2000/4 crack pattern comparison.

Figure 14. Beam YB2000/0 crack pattern comparison.

Figure 16. Beam YB2000/4 concrete crushing and yielding of stirrups in numerical analysis.

From the adjusted parameters it is clear that the deficiencies of the initial parameter set were: – brittle response in compression – low shear stiffness of the cracked material. The beam YB2000/0 is failing due to a diagonal cracking. The diagonal cracks can fully open and therefore no significant shear stress can be transferred across the cracks. This failure pattern is nicely documented in Figure 14, which also shows a good agreement between the calculated and observed crack patterns. This should be contrasted by the behavior of beam YB2000/4. This is a beam with shear reinforcement. The reinforcement limits the crack opening so the crack cannot open so much, and significant shear is transferred across each crack. If the shear stiffness is underestimated, a premature failure is calculated. Figure 16 shows the calculated failure mode for this beam. The final failure is due to concrete crushing near the top loading plate and stirrups yielding. Also the bottom bending reinforcement is yielding at this point. However, to obtain a ductile response as in the experiment, it is necessary to increase the ductility of the concrete in compression; otherwise the concrete near the top loading plate fails by a brittle compression failure. In both examples, it is rather difficult to obtain a stable solution in the post-peak. This can be attributed to the following facts: – It is a large beam with lot of elastic energy, which needs to be released.

287

– Large areas of the model are cracked, and there exist multiple similar solutions, which of these cracks should close and which to open. The second point exactly corresponds to the deficiency of the smeared crack models reported in the introduction to this paper. The mathematical problem of the strain-softening material becomes ill-posed and the uniqueness of the solution is not guaranteed. 5.2 Discrete crack analyses Now, it will be interesting to explore if an application of discrete crack model can help to resolve this issue of non-uniqueness and numerical stability. Both beams are analyzed using a discrete crack model with cohesive zero thickness elements as described in Section 3. These elements are placed along the expected crack paths. It should be noted that in this study no automatic remeshing and crack propagation is used. It is not necessary since the objective is to verify if the addition of discrete discontinuities into the model can help to resolve the localization problem of the strain softening material. During the localization process some of the initially created cracks need to close while some should open. The results from the discrete crack analyses are summarized in Figure 17 for the beam YB2000/0, i.e. the beam with no shear reinforcement. In this figure several discrete analysis are shown with different number of inserted cohesive cracks. The number of assumed cracks ranges from 1 to 11. The first crack,

Figure 17. Beam YB2000/0 results including discrete analyses.

Figure 18. Beam YB2000/4 results including discrete analyses.

Figure 19. Discrete models for beam YB2000/0 with 1 and 11 discrete cracks.

which quite naturally comes to our mind, is the diagonal shear crack (Figure 19 top), which corresponds to the final failure mode of this beam as it was shown in the previous Section 5.1 in Figure 14. The diagonal crack is not the first crack that appears in reality. The previous analyses showed that the cracking is first initiated in the middle of the beam as bending cracks that later on spread through the whole bottom part of the beam. When these bending cracks are not included

288

Figure 20. Discrete model for beam YB2000/4 with 22 discrete cracks.

in the discrete model an extremely stiff response is obtained as shown in Figure 17. In order to correct the pre-peak stiffness, new models were created with multiple bending cracks in the middle of the beam. One such model is shown in Figure 19 (bottom) with altogether 11 discrete cracks. It is interesting to note that as the number of discrete cracks in the model is increasing the stiffness of the pre-peak is improving as well. Figure 17 also shows that once the number of discrete cracks increases to 11 it becomes very difficult to obtain a stable post-peak solution. Analogical results were obtained for the beam YB2000/4. In this case, a model with cca 22 discrete cracks was used (Figure 20). The load displacement diagram is shown in Figure 18. It is clear that even 22 discrete cracks are not enough to capture the reduction of stiffness due to the diffused crack pattern in the pre-peak regime. The results show that similarly to the smeared model in Figure 13, it is necessary to modify the shear properties of the crack model to obtain at least a correct peak load. Standard discrete parameters underestimated the peak load by more then 50%. It was necessary to increase the cohesion to 4.2 MPa and friction coefficient to 0.35 to obtain good match of the peak load (Standard values were c = 2.8 MPa and φ = 0.3). It was also very difficult to obtain a stable solution once the peak load was reached. Many of the discrete cracks are opened, many similar solutions exist. For the numerical solver it is difficult to determine which of them should close and which should continue to open and localize the failure. 5.3 Effect of special reinforced concrete features Various special issues related to the constitutive modeling of reinforced concrete were introduced in Section 2.1. It will be interesting to examine their effect on the numerical solution. Some effects were already discussed in Section 5.1 and 5.2 and additional results are shown in Figures 21 and 22.

Figure 21. Beam YB200/0 effect MCFT and ductility.

Figure 22. Beam YB2000/4 effect of MCFT and ductility.

Figure 21 clearly shows that these special features play only a minor role when no shear reinforcement is present. This is also confirmed by the results of the Iosipescu’s beam in Section 4. Totally different situation is in the case of beam YB2000/4, i.e. the one with shear reinforcement. Although the beam is only lightly reinforced, its strength is determined by reinforcement yielding as shown in Figure 16. The results also show that the shear properties of the crack concrete, i.e. both the shear strength as well as the shear stiffness should be considered properly. In this case the shear stiffness of the cracked concrete was a major factor. The MCFT (Bentz et al. 2006) parameters such as the aggregate interlock and the reduction of compressive strength due to cracking did not play a major role. As already pointed out in Table 3 it was the concrete compressive ductility and the shear stiffness of cracked concrete that proved to be critical for good prediction of the beam behavior. 6 CONCLUSIONS Paper discusses various aspects of numerical predictions of shear strength of plain and reinforced concrete structures. One of the objectives is to verify whether

289

the introduction of strong displacement discontinuities into the numerical solution can be used as a remedy for the known problem of softening materials, i.e. the ill-posed nature of the mathematical solution, which results in a non-unique solution. In plain concrete the discrete crack model definitely improves the crack path predictions; however a good smeared crack model can provide almost identical results. This is especially true if the randomness and heterogeneity of the concrete material is taken into account. In reality, the crack path will always differ in all tests, so minor deviations from the exact path should be tolerated. In reinforced concrete, the discrete crack model is applicable only if large number of discontinuities is introduced into the model. This may be difficult to accomplish with the classical form of the model with remeshing, but can be handled by its modern variants such as X-FEM. With increasing number of discontinuities, i.e. cracks, it is apparent that the same problem of solution non-uniqueness will appear. This shows that the enhanced finite element method cannot be used as a remedy to this problem of softening materials. The only proper solution would be a nonlocal approach or a full dynamic analysis with rate dependent formulation. The reinforced concrete beam shows that shear properties of the crack concrete are critical for good predictions, although the current level of knowledge is quite limited in this area. The aggregate interlock as well as the fc reduction proposed by Modified Compression Field theory of Bentz et al. 2006 did not play an important role for the shear strength of the analyzed beams.

ACKNOWLEDGEMENTS This research was partially supported by research grants from Czech Grant Agency no. 103/07/1660. and 103/08/1527. The financial support is greatly appreciated.

REFERENCES Bažant, Z.P. 1976. Instability, Ductility and Size Effect in Strain Softening Concrete, J. Engrg. Mech., ASCE, Vol. 102, No. 2, pp. 331–344. Bažant, Z.P., and Oh, B.H., 1983. Crack band theory for fracture of concrete. Materials and Structures, RILEM 16 (3), 155–177. Bažant, Z.P., and Pfeifer, P.A. 1986. Shear Fracture Tests of Concrete. Materiaux at Constructions, Vol. 19, No. 110, pp. 111–121. Bažant, Z.P., and Pijaudier-Cabot, G. 1987. Nonlocal continuum damage, localization instability and convergence. Journal of Applied Mechanics, ASME 55 (2), 287–293.

Belytschko, T., Bažant, Z.P., Hyun, Y.W., and Chang, T.P. 1986. Strain Softening Materials and Finite-element Solutions, Computers and Structures, Vol. 23, No. 2, pp. 163–180. Belytschko, T., Lu, Y.Y., and Gu, L. 1994. Element-free Galerkin methods, Int. J. Num. Meth. Eng., Vol. 37, pp. 229–256. Bentz, E.C., Vecchio, F.J., and Collins, M.P. 2006. Simplified Modified Compression Field Theory for Calculating Shear Strength of Reinforced Concrete Elements. ACI Material Journal, Jul/Aug 2006. Carol, I., Rizzi, E., and Willam, K., 1994. A unified theory of elastic degradation and damage based on a loading surface. International Journal of Solids and Structures 31 (30), 2835–2865. ˇ Cervenka, J. 1994. Discrete Crack Modeling in Concrete Structures, PhD. Thesis, University of Colorado at Boulder. ˇ ˇ Cervenka, J., Jendele, L., and Cervenka, V. 2009. ATENA Program documentation. Cervenka Consulting, www.cer venka.cz. ˇ Cervenka, J., and Pappanikolaou, V. 2008. Three dimensional combined fracture-plastic material model for concrete. Int. J. of Plasticity, Vol. 24, 12, 2008, ISSN 0749-6419, pp. 2192–2220. ˇ Cervenka, V., and Gerstle, K., 1971. Inelastic analysis of reinforced concrete panels. Part I: Theory. Publication I.A.B.S.E. 31 (11), 32–45. ˇ Cervenka, V., and Margoldová, J. 1995, Tension Stiffening Effect in Smeared Crack Model, Engineering Mechanics, Stain Sture (Eds), Proc. 10th Conf., Boulder, Colorado, pp. 655–658. ˇ Cervenka, V., Pukl, R., Ozbold, J., and Eligehausen, R. 1995. Mesh Sensitivity Effects in Smeared Finite Element nalysis of Concrete Fracture, Proc. Fracture Mechanics of Concrete Structure II, Vol. II, ISBN 3-905088-12-6, pp. 1387–1396. de Borst, R. 1986. Non-linear analysis of frictional materials. PhD Thesis, Delft University of Technology, The Netherlands. de Borst, R., and Rots, J.G. 1989. Occurence of Spurious Mechanisms in Computations of Strain-Softening Solids, Eng. Computations, Vol. 6, pp. 272–280. de Borst, R., and Mühlhaus, H.B. 1992. Gradient dependant plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering 35 (3), 521–539. de Borst, R., Benallal, A., and Heeres, O.M. 1996. A gradientenhanced damage approach to fracture. J. de Physique IV, C6, pp. 491–502. de Borst, R., Remmers, J.J.C., Needleman, A., and Abellan, M.A. 2003. Discrete vs smeared crack models for concrete fracture: bridging the gap, Computational Modelling of Concrete Structures, Bicanic et al. (eds), ISBN 90 5809 536 3, pp. 3–17. Etse, G. 1992. Theoretische und numerische untersuchung zum diffusen und lokalisierten versagen in beton. PhD Thesis, University of Karlsruhe.

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Hordijk, D.A. 1991. Local approach to fatigue of concrete. PhD Thesis, Delft University of Technology, The Netherlands. Ingraffea, A.R., and Panthaki, M.J. 1985. Analysis of shear fracture tests of concrete beams, Finite Element Analysis of Reinforced Concrete Structures, Eds. Meyer, C., and Okamura, H., ASCE, N. York, pp. 151–173. Iosipescu, N. 1967. New Accurate Procedure for Single Shear Testing of Metals, Journal of Materials, Vol. 2, pp. 537–566. ˇ Jendele, L., and Cervenka, J. 2006. Modelling Bar Reinforcement with Finite Bond, Computers and Structures, 84, 1780–1791. Jirásek, M. 2003. Models and algorithms for localized failure, Computational Modelling of Concrete Structures, Bicanic et al. (eds), ISBN 90 5809 536 3, pp. 19–31. Lee, J., and Fenves, G.L. 1998. Plastic-damage model for cyclic loading of concrete structures. Journal of Engineering Mechanics, ASCE 124 (8), 892–900. Lin, C.S., and Scordelis, A. 1975. Nonlinear Analysis of RC Shells of General Form, ASCE, J. of Struct. Eng., Vol. 101, No. 3, pp. 152–163. Menétrey, P., and Willam, K.J. 1995. Triaxial failure criterion for concrete and its generalization. ACI Structural Journal 92 (3), 311–318. Ngo, D., and Scordelis, A.C. 1967. Finite element analysis of reinforced concrete beams, J. Amer. Concr. Inst. 64, pp. 152–163. Pramono, E., and Willam, K.J. 1989. Fracture energy-based plasticity formulation of plain concrete. Journal of Engineering Mechanics, ASCE 115 (6), 1183–1204. Rashid, Y.R. 1968. Analysis of prestressed concrete pressure vessels. Nuclear Engineering and Design 7 (4), 334–344. Rots, J.G., and Blaauwendraad, J. 1989. Crack models for concrete: Discrete or smeared? Fixed, multi-directional or rotating? Heron 34 (1). Saouma, V.E., and Ingraffea, A.R. 1981. Fracture Mechanics Analysis of Discrete Cracking, Proc. IABSE Coll. in Advanced Mechanics of Reinforced Concrete, Delft, June 1981, pp. 393–416. Simo, J.C., Kennedy, J.G., and Govindjee, S., 1988. Nonsmooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. International Journal for Numerical Methods in Engineering 26 (10), 2161–2185. Schlangen, E. 1993. Experimental and Numerical Analysis of Fracture Processes in Concrete, Ph.D. dissertation, Delf University of Technology. Suidan, M., and Schnobrich, W.C. 1973. Finite Element Analysis of Reinforced Concrete, ASCE, J. of Struct. Div., Vol. 99, No. ST10, pp. 2108–2121. Yoshida, Y. 2000. Shear Reinforcement for Large Lightly Reinforced Concrete Members, MS Thesis, Prof. Collins, Univ. of Toronto, Canada.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Lattice Discrete Particle Model for Fiber reinforced concrete (LDPM-F) with application to the numerical simulation of armoring systems G. Cusatis & E.A. Schauffert Rensselaer Polytechnic Institute, Troy, NY, USA

D. Pelessone ES3, Solana Beach, CA, USA

J.L. O’Daniel Engineer Research and Development Center, Vicksburg, MS, USA

P. Marangi, M. Stacchini & M. Savoia University of Bologna, Bologna, Italy

ABSTRACT: In this paper, the Lattice Discrete Particle Model (LDPM) is extended to include the effect of dispersed fibers with the objective of simulating the behavior of fiber reinforced concrete for armoring system applications. Within the LDPM framework, the effect of dispersed fibers is taken into account through the following procedure. 1) Fibers are randomly placed in the volume of interest according to the given fiber volume ratio and fiber geometry. 2) The number and orientation of fibers crossing each facet are computed along with the fiber embedment length on each side of the facet. 3) At the facet level, fibers and plain concrete are assumed to be coupled in parallel. 4) The contribution of each fiber to the facet response is formulated on the basis of a micromechanical model of fiber-matrix interaction. The developed model, named LDPM-F, is validated by carrying out numerical simulations of direct tension and three-point bending tests on fiber reinforced concrete mixes characterized by various fiber volume fractions. Finally, LDPM-F is applied to the analysis of the penetration resistance of fiber reinforced slabs. 1 THE LATTICE DISCRETE PARTICLE MODEL Since the mid-eighties, many mesoscale models for concrete have appeared in the literature. The main advantage of these models over classical constitutive models for concrete is their ability to simulate material heterogeneity and its effect on damage evolution and fracture. Noteworthy examples of mesoscale models include: Roelfstra et al. (1985); Wittmann et al. (1988); Bažant et al. (1990); Schlangen & VanMier (1992); Bolander & Saito (1998); Bolander et al. (1999); Bolander et al. (2000); Carol et al. (2001); Lilliu & Van Mier (2003); Cusatis et al. (2003a,b); Cusatis et al. (2006); and Cusatis & Cedolin (2006). In this paper, recent results obtained at Rensselaer Polytechnic Institute with the Lattice Discrete Particle Model (LDPM) are presented and discussed. LDPM simulates concrete mesostructure by taking into account only the coarse aggregate pieces, typically with characteristic size greater than 5 mm. The mesostructure is constructed through the following steps. 1) The coarse aggregate pieces, whose shapes are assumed to be spherical, are introduced into the

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concrete volume by a try-and-reject random procedure. 2) Zero-radius aggregate pieces (nodes) are randomly distributed over the external surfaces. 3) A three-dimensional domain tessellation, based on the Delaunay tetrahedralization of the generated aggregate centers, creates a system of cells interacting through triangular facets, which can be represented in a two-dimensional sketch by straight line segments (Fig. 1). A vectorial constitutive law governing the behavior of the model is imposed at the centroid of the projection of each single facet (contact point) onto a plane orthogonal to the straight line connecting the particle centers (edges of the tetrahedralization). The projections are used instead of the facets themselves to ensure that the shear interaction between adjacent particles does not depend on the shear orientation. The straight lines connecting the particle centers define the lattice system. Rigid body kinematics describes the displacement field along the lattice struts and the displacement jump, [uC ], at the contact point. The strain vector is defined as the displacement jump at the contact point divided

way similar to simple damage models (see derivation in Cusatis et al. 2003a): σN = σ

εN ; ε

σM = σ

αεM ; ε

σL = σ

αεL ε

(4)

The effective stress, σ , is incrementally elastic (σ˙ = E0 ε˙ ), and must satisfy the inequality 0 ≤ σ ≤ σbt (ε, ω). The strain dependent boundary, σbt (ε, ω), can be expressed as:   εmax − ε0 (ω) (5) σbt = σ0 (ω) exp −H0 (ω) σ0 (ω) in which the brackets • are used in Macaulay sense: x = max{x, 0}. The internal variable ω is defined as follows (Cusatis et al. 2003a): √ εN σN α tan ω = √ = σT αεT Figure 1. a) Mesostructure tessellation. b) Threedimensional discrete particle. c) Definition of nodal degrees of freedom and contact facets in two-dimensions.

by the inter-particle distance, L. The components of the strain vector in a local system of reference, characterized by the unit vectors n, l, and m, are the normal and shear strains: εN =

nT [uC ] ; L

εL =

lT [uC ] ; L

εM =

mT [uC ] (1) L

The unit vector n is orthogonal to the projected facet and the unit vectors l and m are mutually orthogonal and lie in the projected facet. The elastic behavior is described by assuming that the normal and shear stresses are proportional to the corresponding strains:

(6)

and it characterizes the coupling between normal and shear strains (or stresses). The σbt boundary evolves exponentially as a function of the maximum effective strain, which is a history-dependent vari-

able defined as εmax = εN2 ,max + αεT2 ,max , where εN ,max (t) = max[εN (τ )], and εT ,max (t) = max[εT (τ )] τ vd :    β(v − vd ) v − vd 1+ P(v) = P0 1 − L df

(20)

The critical slippage, vd , represents the extent of slippage leading to full debonding, and is defined as: vd =

2τ0 L2 + Ef d f



8Gd L2 Ef df

1/2

(21)

In Eqs. 19 to 21: P0 = πLdf τ0 ; Ef is the fiber Young’s Modulus; τ0 is the frictional bond strength; Gd is the debonding fracture energy associated with ‘‘tunnel-type’’ cracking over the embedment length as the fiber debonds from the matrix; β is a dimensionless material parameter characterizing the extent to which the frictional pullout of the fiber will be slip-hardening (low values, near zero or negative, will cause the frictional pullout to be a softening process); and L is the current embedment length, defined as L = Lsf − sf and L = Llf − sf for the short and long embedment lengths, respectively. The previous equations are valid only if the fiber force does not cause fiber rupture. This can be checked by making sure that the fiber force magnitude is less than the rupture force, Puf : Pf ≤ Puf = ′ 0.25πdf2 σuf e−krup ϕf , where krup is a material parameter and σuf is the ultimate tensile strength of the fiber associated with the theoretical situation where the fiber’s bridging segment and its embedded segments are collinear. For the definition of the spalling length, sf , various models have been proposed in the literature, including Cailleux et al. (2005) and Leung & Li (1992). Herein, the formula proposed by Yang et al. (2008) is adopted under the assumption that the shear components of the fiber force do not cause spalling: sf =

PfN sin(ϕ/2) ksp σt df cos2 (ϕ/2)

Since the spalling length, sf , depends on the fiber force, and the force versus slippage relationship is nonlinear, the problem of computing the fiber force for a given crack opening is highly nonlinear and needs to be solved iteratively.

3 SIMULATION OF EXPERIMENTAL DATA In this section, the Lattice Discrete Particle Model for Fiber Reinforced Concrete (LDPM-F), formulated above, is validated by comparison with experimental data relevant to direct tension tests and three-point bending tests. 3.1 Direct tension tests Simulation of the fiber effect on the tensile fracturing behavior of concrete is shown in Fig. 3. The experimental data are relevant to experiments reported by Li et al. (1998). In this experimental investigation, rectangular specimens were subjected to direct tension. The tests were controlled through displacement measurements over a measured length of 120 mm to ensure stability in the post-peak softening regime. The simulated fibers are Dramix steel fibers with hooked ends characterized by a diameter of 0.5 mm and a length of 30.0 mm. Figure 3a shows experimental and numerical stress versus displacement curves for four different fiber volume fractions (Vf ): 0% (plain concrete), 2%, 3%, and 6%. LDPM-F is able to predict the increased strength

(22)

where ϕ = arccos(nfT wf )/wf is the orientation of the crack-bridging segment in the absence of spalling (Fig. 2d), and ksp is a dimensionless material parameter.

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Figure 3.

Direct tension tests of FRC specimens.

and ductility due to the effect of fibers. The behavior gradually transitions from softening for plain concrete and low Vf , to hardening for high Vf . LDPM-F numerical results are further investigated in Fig. 3b, where contours of the mesoscale crack opening at the end of the simulations are reported for three fiber volume fractions. For plain concrete, the crack pattern is characterized by one localized crack that propagates from one side of the specimen towards the other. As fracture propagates, material outside the crack unloads as the overall load applied to the specimen tends to zero. For the 2% Vf , there is still one main crack propagation, but the entire specimen features diffuse cracking and no unloading occurs. Absence of unloading outside the main crack is due to the fact that even though the overall behavior is softening, the stress versus displacement curve shows a non-zero residual stress associated with the fiber crack bridging effect. Finally, for the 6% Vf , the crack pattern is characterized by several branched cracks whose propagation is arrested by the effect of the fibers. No unloading occurs outside the main cracks since the overall behavior is strain-hardening and, up to a displacement of 0.5 mm (average nominal strain of 0.5 mm/120 mm ≈ 0.42%), no reduction of the load carrying capacity can be observed. 3.2

Three-point bending tests

In this section, LDPM-F is validated by simulating three-point bending tests (Buratti et al. 2008) carried out on prismatic notched specimens of plain concrete and FRC with two different Vf values: 0.26% and 0.45%. Figure 4 shows the specimen geometry and

Figure 4. Specimen geometry and discretization for the three-point bending tests.

the adopted numerical model, in which only the central part (where damage is expected to occur due to the presence of the notch) is modeled through LDPM-F, while the two lateral parts are modeled with standard elastic finite elements. All the specimens have an out-of-plane thickness of 150 mm. The LDPM-F model parameters were calibrated by fitting the load versus CMOD (crack mouth opening displacement) curves relevant to plain concrete and 0.26% Vf as reported in Figs. 5a and b. In Fig. 5a, both the numerical and experimental curves represent the average of the response of two specimens. In Fig. 5b, the numerical and experimental response of seven specimens is reported along with the average curves (solid curve and circles, respectively). The seven different numerical simulations were obtained by seven different random fiber distributions. The agreement between the numerical results and the experimental data is very good. Figure 5c shows the same comparison for the 0.45% Vf . Again the response of seven different specimens and the average curves are reported. In this case the numerical results produce a post-peak load carrying capacity higher than the experiments. For a CMOD of 0.8 mm the difference between the average numerical and experimental curves is about 45%. In order to clarify this apparent discrepancy, it is useful to compare the actual number of fibers bridging the crack in the experiments and the in numerical simulations. During the experimental campaign, postmortem evaluation was performed and the number of fibers on the crack surface was counted for all specimens of both FRC mixes. In Buratti et al. (2008), the authors report the number of fibers in each third of the ligament length (i.e. the reduced cross-section width associated with the notch, as shown in Fig. 6a). The same information was extracted from the numerical simulations. In Fig. 6b, for CMOD = 0.8, the load increment due to the effect of fibers, calculated as 100(1 − P/P0 ), where P is the load for the FRC specimens and P0 is the average load for the plain concrete specimens, is plotted versus the number of fibers in the lower two-thirds of the ligament length, i.e. the two-thirds closest to the notch. The choice of plotting the number of fibers in the lower two-thirds of the ligament length was motivated by analyzing the crack pattern in the numerical simulations. Figure 6a shows gray-scale isosurfaces from a typical mesoscale crack opening. Each isosurface corresponds to a different crack opening value. The minimum value that was plotted (very light gray) corresponds to a crack opening magnitude of 2vd (see Sec. 2.2). This corresponds to a situation at which a straight fiber, orthogonal to the crack surface, would experience full debonding (onset of the frictional pullout phase). At this level of crack opening, fibers can be considered fully active. As one can see from Fig. 6a, the zone characterized by fully active fibers extends to

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Figure 6. a) Crack opening isosurfaces; b) load increment due to fiber versus number of active fibers.

Figure 5. Load versus CMOD curves: a) Vf b) Vf = 0.26%; c) Vf = 0.45%.

= 0%;

approximately two-thirds of the ligament length above the notch. The plot shows clearly a linear relationship between the load increment and the number of fibers detected, and that both the experimental data and the numerical results have a similar trend. There is, however, an

297

inconsistency in the number of detected fibers for the experimental specimens. For the numerical specimens, the range of detected fibers increases from (22–39) to (47–70) as the reported fiber volume fraction nearly doubles from Vf = 0.26% to Vf = 0.45%. For the experimental specimens, the increase in number of detected fibers, from (17–39) to (17–50), is not consistent with the near doubling of Vf . For the higher Vf , the numerically simulated specimens feature more active fibers than the experimental ones. This explains the discrepancy in the curves shown in Fig. 5c, while confirming the validity of the fiber-matrix interaction strategy adopted in this study. The difference between the number of active fibers within the experimental and numerical crack ligaments is due to the fact that in the numerical model a uniform fiber distribution was obtained, while such uniformity apparently was not obtained in the experimental specimens with higher Vf . Based on the fact that fiber dispersion can be significant in terms of load carrying capacity, LDPM-F should be extended to include non-unform fiber distribution, and, in addition,

experimental data should provide as much information as possible on the actual fiber distribution obtained during specimen casting. 4 PROJECTILE PENETRATION OF FRC SLABS 4.1 Effect of fibers on penetration resistance In this section, simulations of steel projectile impacts into FRC slabs at various impact velocities are presented. The numerical simulations were performed by using the same material parameters used in Sec. 3.2. The simulated slabs were 508 mm squares with two different thicknesses, t = 50.8 mm and 101.6 mm. Impact velocities up to 450 m/sec and three different fiber volume fractions (Vf = 0%, 2%, and 3%) were considered. Figure 7 reports the projectile residual velocity, Vr , as a function of the striking velocity, Vs , for the different simulations. As one can see, the ballistic limit (the highest strike velocity associated with zero residual velocity) increases for increasing Vf . The FRC slabs with 2% Vf have ballistic limits that are approximately 175% and 80% higher than the plain concrete, for t = 50.8 mm and 101.6 mm, respectively. The 3% Vf simulations show, however, that an additional increase in fiber content does not necessarily correspond to a significant additional increase of the ballistic limit. Finally, the effect of the fibers becomes less and less significant for high striking velocities. This is due to the fact that at very high striking velocity, the penetration phenomenon is governed more by the mass of the system and the confined compressive resistance, rather than by the tensile fracturing behavior, which is significantly influenced by the presence of the fibers.

design of lightweight and low-cost armoring systems. Within this effort, experimental characterization of a very high strength (fc′ = 157 MPa) FRC mix was performed both quasi-statically (Akers, et al. 1998) and dynamically. The dynamic experiments consisted of impacts from a Fragment Simulating Penetrator (FSP) into FRC panels. Parameters of the tests, including FSP velocity and target panel thickness, were chosen to generate extreme deformation and failure of the

4.2 Design of armoring systems The Engineer Research Development Center (ERDC) is currently performing research towards the innovative

Figure 7.

Figure 8. Numerical simulations of penetration into FRC slabs performed at ERDC.

Projectile penetration of concrete slabs.

298

material, including generation of surface craters and perforation through the target. Of concern were the damage generated by the FSP to the target plates and the residual velocity when the projectile exited the targets. The adopted fibers were steel of 25.4 mm length and bent ends, and were randomly distributed during the casting phase. Penetration experiments were conducted to measure the FRC’s resistance to ballistic projectile penetration. Panels tested were 304.8 mm squares of 25.4, 50.8, and 76.2 mm thickness. Some preliminary results relevant to the LDPM-F simulations of these tests are reported herein. LDPM-F was used to model the target panels. The FSP was modeled with hexahedral finite elements. A typical LDPM-F mesh for a 50.8 mm thick panel contained approximately 20,000 fibers, 115,000 LDPM particles, and 670,000 LDPM tetrahedral elements. Figures 8a, b, and c show snapshots of the penetration event characterized by the formation of entrance and exit craters. Figure 8d shows a comparison between the post-penetration damage observed during the experiments (left) and that predicted by the simulation (right).

5 CLOSING REMARKS In this paper the Lattice Discrete Particle Model was extended to include the effect of randomly dispersed fibers. Fiber-matrix interaction was modeled by using an earlier formulated micromechanical theory providing the fiber crack bridging force as a function of the crack opening. This law was coupled to the LDPM constitutive behavior at the facet level. The contribution of each individual fiber is taken into account by detecting the intersection between the fibers and the LDPM facets. The formulated model, named LDPM-F, is able to reproduce the fiber toughening mechanisms and, once calibrated, is able to predict the macroscopic fracturing properties as a function of different fiber volume fractions. Finally, LDPM-F was used to investigate the effect of fibers on the penetration resistance of FRC slabs. Preliminary numerical results show that fibers influence significantly the penetration response for striking velocities close to the ballistic limit, whereas their effect tends to become less significant as striking velocity increases.

ACKNOWLEDGMENT This effort was sponsored in part by the US Army Engineer Research and Development Center (ERDC). Permission to publish was granted by the Director, Geotechnical and Structures Laboratory, ERDC. The work of the first two authors was also supported under

DTRA grant HDTRA1-09-1-0029 to Rensselaer Polytechnic Institute. This financial support is gratefully acknowledged. REFERENCES Akers, S.A., Green, M.L. and Reed, P.A., 1998. Laboratory characterization of very high-strength fiber-reinforced concrete. US Army Corps of Engineers, Waterways Experiment Station, TR, SL-98-10. Bažant, Z.P. and Oh, B.H., 1983. Crack band theory for fracture of concrete. Mater. Structures, RILEM, 16(93): 155–177. Bažant, Z.P., Tabarra, M.R., Kazemi, T. and PijaudierCabot, G., 1990. Random particle model for fracture of aggregate or fiber composites. J. Eng. Mech., 116(8): 1686–1705. Bolander, J.E. and Saito, S., 1998. Fracture analysis using spring network with random geometry. Engng. Fracture Mech., 61(5–6): 569–591. Bolander, J.E., Yoshitake, K. and Thomure, J., 1999. Stress analysis using elastically uniform rigid-body-spring networks. J. Struct. Mech. Earthquake Engng., (JSCE), 633 (I-49): 25–32. Bolander, J.E., Hong, G.S. and Yoshitake, K., 2000. Structural concrete analysis using rigid-body-spring networks. J. Comp. Aided Civil and Infrastructure Engng., 15: 120–133. Buratti, N., Mazzotti, C., Savoia, M. and Thooft, H., 2008. Study of the behavior of concrete reinforced through steel and polymeric fibers (in Italian). XVII Congress C.T.E., 5–8 November, 2008. Roma, Italia. Cailleux, E., Cutard, T. and Bernhart, G., 2005. Pullout of steel fibers from a refractory castable: experiment and modeling. Mechanics of Materials, 37: 427–445. Carol, I., López, C.M. and Roa, O., 2001. Micromechanical analysis of quasi-brittle materials using fracture-based interface elements. Internat. J. Numer. Methods Engrg., 52: 193–215. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2006. Confinementshear lattice model for fracture propagation in concrete. Comput. Methods Appl. Mech. Engrg., 195: 7154–7171. Cusatis, G. and Cedolin, L., 2006. Two-scale analysis of concrete fracturing behavior. Engng. Fracture Mech., 74: 3–17. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2003a. Confinement-shear lattice model for concrete damage in tension and compression: I. Theory. J. of Engrg. Mech. (ASCE), 129(12): 1439–1448. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2003b. Confinement-shear lattice model for concrete damage in tension and compression: II. Computation and validation. J. of Engrg. Mech. (ASCE), 129(12): 1449–1458. Leung, C.K.Y. and Li, V.C., 1992. Effect of fiber inclination on crack bridging stress in brittle fiber reinforced brittle fiber matrix composites. J. Mech. Phys. Solids, 40(6): 1333–1362. Li, Z., Li, F., Chang, T.-Y.P. and Mai, Y.-W., 1998. Uniaxial tensile behavior of concrete reinforced with randomly distributed short fibers. ACI Material Journal, 95(5): 564–574. Lilliu, G. and VanMier, J.G.M., 2003. 3D lattice type fracture model for concrete. Engng. Fracture Mech., 70: 927–941.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Nonlocal damage based failure models, extraction of crack opening and transition to fracture Frédéric Dufour 3S-R Lab. (Soils, Solids, Structures—Risks), Grenoble Institute of Technology, France

Gilles Pijaudier-Cabot Laboratoire des Fluides Complexes, Univ. de Pau et des Pays de l’Adour, France

Grégory Legrain GeM Research Institute, Ecole Centrale de Nantes, France

ABSTRACT: Damage models are capable to represent initiation and somehow crack propagation in a continuum framework. Thus crack openings are not explicitly described. However for concrete structures durability analysis, crack opening through transfer properties is a key issue. Therefore, in this contribution we present a new approach that is able from a continuum modelling to locate a crack from internal variable field and then to estimate crack opening along its path. Results compared to experimental measures for a three point bending test are in a good agreement with an error lower than 10% for a widely opened crack (40 µm). 1 INTRODUCTION For many concrete structures, crack opening is a key parameter needed in order to estimate durability. Cracks are preferential paths along which fluids or corrosive chemical species may penetrate inside concrete structural elements. For structures such as confinement vessels, reservoirs or nuclear waste disposals for instance, tightness to gas or liquids is a major serviceability criterion that is governed by Darcy’s relation in which permeability of the material is involved. Hence, the prediction of the durability of structural components requires models that describe failure, crack locations and crack openings in the present example too when damage has localised. Enhanced continuum and integral damage models are capable of representing diffuse damage, crack initiation and possibly crack propagation (Pijaudier-Cabot and Bažant 1987; Peerlings et al. 1996). They regard cracking as an ultimate consequence of a gradual loss of material integrity. These models, however, do not predict crack opening as they rely on a continuum approach to fracture. Fictitious crack models are based on an explicit description of the discontinuity within the material (e.g. cohesive crack model model (Hillerborg et al. 1976)). They relate the crack opening to the stress level and they are based on the linear elastic (or plastic) fracture mechanics. Cohesive crack models needs proper algorithms for crack propagation, and more importantly they are not capable of describing crack initiation.

301

Ideally, the prediction of durability that involves inception of failure, crack location, propagation and crack opening would require to merge the continuum damage approach and the discrete crack approach into a single, consistent, computational model bridging the continuous and discrete approaches. Bridges between damage and fracture have been devised in the literature (see e.g. (Mazars and Pijaudier-Cabot 1996; Planas et al. 1993)). They rely on the equivalence between the dissipation of energy due to damage and the energy dissipated in order to propagate a crack. Given the energy dissipated in the damage process, the equivalent crack length is computed, knowing the fracture energy. Generally, the entire energy that is dissipated in the fracture process zone is ‘‘converted’’ into a crack length (Mazars and Pijaudier-Cabot 1996). Some part of this energy may be dissipated in the process zone outside from the crack and it follows that the crack length and opening are probably overestimated. The strong discontinuity approach initiated by (Simo et al. 1993) and widely used over the last decade (e.g. (Oliver et al. 2002; Larsson et al. 1998)) offers the possibility of merging in the same formulation a continuous damage model for the bulk response and a cohesive model for the discontinuous part of the kinematics. It is certainly a combination of continuum—discrete modelling that is sound from a theoretical point of view and appealing from the point of view of the physics of fracture. The issue in combining the continuum based model for crack initiation and then a discrete crack model for propagation is, however, the threshold upon which one switches from one

analysis to the other. Usually, it is considered that the discontinuity appears when damage, stresses or strain energy reach a certain threshold fixed beforehand, which remains arbitrary (Comi et al. 2007; Simone et al. 2003). Besides as damage and fracture models do not rely on the same material description and thus on the same internal variables, jumps in time are observed on variables of interest (strain and stress) at the switch time. As we will see further, one of the outcome of the present paper is to provide an indicator on the basis of which the appearance of a discontinuity during a damage process can be defined, with an indication of accuracy. Instead of trying to combine continuum and discrete models in computational analyses, it would be attractive to derive from the continuum approach an estimate of crack opening, without considering the explicit description of a discontinuous displacement field in the computational model. This derivation could be based on some post-processing of the distribution of strain and damage in the considered structure. The main purpose of this paper is to present such an estimate of crack opening derived from a continuum model description. First, we recall the continuum approach that will be considered: the (integral) nonlocal damage model. Nonlocal models are known to possess shortcomings such as spurious boundary effects on fracture propagation (Jirasek et al. 2004; Pijaudier-Cabot et al. 2009) or incorrect initiation of damage at a crack tip (Simone et al. 2004). Still these defects in the model formulation do not alter their ability to capture a fully localised (mode I) crack. The location of the crack in the computational domain and the estimate of its opening are discussed in the second part in which we propose an improvment of an existing approach (Dufour et al. 2008). Finally, we compare our numerical procedure with experimental results obtained on a 3 point bending test on plain concrete beam in terms of crack location and opening.

2 NUMERICAL MODELLING 2.1 Nonlocal damage approach The scalar isotropic damage model (Mazars and Pijaudier-Cabot 1989) will be used in the finite element computations for representing the progressive failure. This constitutive relation exhibits strain softening. Thus a regularization technique shall be considered in order to avoid mesh dependency and ill-posedness of the governing equations of equilibrium. In this model the tensorial stress σ —strain ε relationship is expressed as follows: σ = (1 − D)C : ε

(1)

302

where D is the damage scalar variable and C is the elastic stiffness tensor of the sound material. Damage is a combination of two components: Dt and Dc which are damages due to tension and compression based loads respectively: D = αt Dt + αc Dc

(2)

αt and αc depend on both strain and stress tensors. Damage evolution laws for both traction and compression components read:   YD0 1 − At,c At,c Dt,c = 1 − (3) − B Y¯ −YD0 )] ¯ [ ( t,c Y e where At , Ac , Bt , Bc and YD0 are model parameters and Y¯ is defined by:   (4) Y¯ = max Y¯ , ε¯ eq

with Y¯ = YD0 initially. The nonlocal equivalent strain ε¯ eq (Pijaudier-Cabot and Bažant 1987) is defined as a weighted average of the local equivalent strain εeq :  φ(x − s)εeq (s)ds (5) ε¯ eq (x) =    φ(x − s)ds

Several weight functions exist in the literature, we choose the most used, i.e. the Gaussian function:     2x − s 2 (6) φ (x − s) = exp − lc where lc is the internal length of the model. Finally the local equivalent strain is defined according to Mazars criterion:

3

ε =  ε 2 (7) i +

eq

i=1

+ denotes the positive part of the principal strain εi .

2.2 Location of a crack In (Dufour et al. 2008), an approach was proposed to extract an equivalent crack opening from a nonlocal damage computation. However, the crack position was supposed a-priori known and the computational domain was reduced to 1D. In order to apply this approach in a more general context (2D and 3D with unknown crack position), it is necessary to be able to locate an idealized crack from the nonlocal computation. Some approaches have already been proposed in the field of damage/fracture transition in order to update the crack position during the propagation. In (Comi et al. 2007), the authors proposed to fit a fourthorder polynomial on the damage field, then to propagate the crack in a direction that is perpendicular to the maximum curvature of the polynomial at the crack-tip. The main drawback is that when the damage profile does not exhibit a clear peak but a region with a small

curvature, the fitting may be obtained with a degraded accuracy. Moreover, the accuracy of the fitting may not be sufficient at the crack-tip, which should lead to extra difficulties for the estimation of the crack direction. In (Mariani and Perego 2003), the authors proposed a similar procedure, but working on the stress-field in a half disc centered at the crack-tip. Finally, the crack is introduced perpendicular to the fitted maximal principal stress. This approach allows to work on a ‘‘sharper’’ mechanical field, but the influence of the degree of the polynomial fitting was not discussed. However, the authors reported that a third order polynomial fit was not sufficient, and that in the proposed examples a fourth order one provided consistent results. Alternatively, averaging approaches have been proposed in (Mediavilla et al. 2006) and (Wells et al. 2002). In the later, the directions pointing from the tip to the most damaged points at various radius are averaged on a V-shaped window. A last approach was proposed by (Oliver and Huespe 2004a), called ‘‘Global tracking algorithm’’. This approach was first used within the strong discontinuity approach (SDA) (Oliver and Huespe 2004b) to evaluate the crack propagation direction. The resolution of a heat-conduction like problem leads to a scalar function whose iso-values represent all the possible directions of propagation. The selection of the iso-value emanating from the crack-tip makes possible its propagation. The approach has been modified (Feist and Hofstetter 2007) in order to restrict the heat-conduction problem on a subset of elements already or potentially crossed by the crack. The approach was also applied in the context of the extended Finite Element Method (Moes et al. 1999) by (Dumstorff and Meschke 2007), and compared with various crack branching criteria. Here, we propose to use this approach in order not to propagate the crack since we use a continuum modeling but to locate it from mechanical variables at hand.

T(x)

Crack

Figure 1. Global tracking algorithm: envelopes of the vector field T(x), possible crack path and real crack.

This condition can be formulated as the following linear boundary value problem (Oliver and Huespe 2004a): div(K Grad θ) = 0

in ()

(−K Grad θ) · n = 0 θ = θd

on ∂q 

(9)

on ∂θ 

where () is the domain occupied by the solid, n is the unit vector normal to ∂q , θd is a prescribed value for the Dirichlet boundary condition and K is a second order tensor defined as: K (x) = T (x) ⊗ T (x)

2.2.1 Global tracking algorithm According to this approach, the evaluation of the propagation direction is obtained as a separate problem (linked to the mechanical one). The crack is assumed to be located along a surface (or a line in 2D) which is tangent to a vector field T (x) (with unit norm). The construction of the envelopes of T (x) supplies all the possible discontinuity lines at time t (see Figure 1). The envelopes of T (x) are described by a function θ(x) whose level contours (θ (x) = constant) define all the possible discontinuity lines, as described in Figure 1. The gradient of this function must be normal to T (x) in each point: ∂θ T (x) · Grad θ = =0 ∂ T

Possible crack path

(8)

303

(10)

The θ field can be assimilated as a temperature field, −K Grad θ as a heat flux, and K as an anisotropic conductivity tensor. If the Dirichlet boundary conditions are compatible with Equation (8), then a solution satisfying: ∂θ =0 (11) ∂ T is solution of the boundary value problempresented in Equation 10. In order to overcome the singularity of the problem (K is rank one), the conductivity tensor is modified as (Oliver and Huespe 2004a):

θ(x)  = constant ;

K (x) = T (x) ⊗ T (x) + ǫI

(12)

where ǫ is a small isotropic algorithmic conductivity, and I is the second order identity tensor. Once the problem is solved, the crack can be propagated along the path defined by the iso-value of θ that passes at the crack-tip. 2.2.2 Location of the crack using the global tracking algorithm To apply this approach to the problem at hand, two main ingredients have to be adapted: (1) the definition of the T field, and (2) the location of one point of the crack. We make here the hypothesis that the idealized crack is perpendicular to the principal direction

associated to the maximum principal strain εmax which represents the opening direction in mode I dominated loading. The T field is thus taken perpendicular to the principal direction associated to εmax . The knowledge of this field in the body makes possible to solve the boundary value problem and obtain the θ field. The last operation consists in selecting the right isovalue. We make here a last hypothesis by considering that the crack passes at the Gauss point where εmax is maximal on the body. One could have chosen the most damage Gauss point, but this choice proved to be a bad one since the damage profile is very flat, leading to crack paths that depends on the loading level. The algorithm that summarizes the process is presented in Figure 2. In practice, the thermal-like problem is not solved on the full structure, but only on the damaged zone. This allows to speed-up the process and decrease the computer requirement of the procedure. 2.3 Reduction to a 1D problem At the end of the tracking process, a mesh of the crack is built using the iso-temperature defining the crack. The second step consists now in evaluating the opening across this idealized crack. In this contribution, it is proposed to re-use the approach that was presented in (Dufour et al. 2008). Once the crack is meshed, it is possible to apply the 1D approach on lines perpendicular to the elements (segments in 2D, triangles in 3D) defining the crack surface. In 2D, for example, a set of lines is generated from the middle of each segment of the crack (see Figure 3). In 3D this set of lines would emanate from the center of the elements that are used to discretize the crack surface.

Once these profiles are defined, the component of the local strain field along the 1D profile εN = N ·ε· N is first computed (see Figure 4). Then, this axial strain field is projected on the 1D profile as an input for the 1D crack opening procedure.

2.4 Estimation of a crack opening We summarize in this part the key idea developped by (Dufour et al. 2008) to estimate the crack opening in a 1D structure that we use along perpendicular profiles to the idealized crack. If we assume a bar upon failure, the displacement field is a step with a jump {U } at the crack location x0 . The derivation of the displacement field gives a Dirac function for the local strain and a nonlocal strain with an amplitude of {U } and the same shape than the averaging function ψ used in the convolution product. Remark: For any regularized damage models ψ can be defined independently of the mechanical model. However, since with the nonlocal model used in the present work we already have defined a weighting function φ, we keep it, thus ψ = φ (see Equation (6)). With this procedure a nonlocal measure of strain is analytically obtained assuming a strong discontinuity kinematical field upon failure. The key point of (Dufour et al. 2008) is to compare this function with the nonlocal strain obtained by the FE mechanical computations. Several possibilities do exist in order to compare two profiles to each other. In the original paper, only the strong link were developped, i.e. the crack opening is computed so that both profiles are equal at their maximum x = x0 : ε¯ sd (x0 ) = ε¯ eq (x0 ) ⇒ ⇒

Figure 2.

Algorithm for the location of the idealized crack.

Figure 3. (red).

1D profiles (blue) generated from the crack mesh

304

{U }ψ(0) = ε¯ eq (x0 ) ψ(x 0 − s)ds Ŵ  ε¯ eq (x0 ) Ŵ ψ(x0 − s)ds {U } = ψ(0) 

(13)

Figure 4. 1D profiles (blue) generated from the crack mesh (red).

Furthermore, the distance between both profiles gives an indicator of the quality of the solution obtained by the FE computation using nonlocal damage model with respect to an analytical strong discontinuity approach:

  ε¯ sd − ε¯ eq  dŴ Ŵ (14) εI (x) = ε¯ eq dŴ

(c)

(b)

(a)

Ŵ

Thus, it is not an error on the crack opening itself but on the capacity of nonlocal damage to reproduce local kinematic field across the crack as in the strong discontinuity approach. A new profile comparison technique, named weak form, is proposed in the present work by equating the integral of both profiles, i.e.:

ε¯ eq dŴ = ε¯ sd dŴ (15) Ŵ

Ŵ

Thus it gives a different value for the crack opening. Since this error measure gives only a quality estimation of the model, we have performed experimental test in order to estimate by comparison the error on the crack opening itself.

(d)

Figure 5. (a) Initial picture in undeformed state, (b) picture during the crack propagation, (c) horizontal displacement field and its 3D view (d). 3

(1)

2.5 Comparison vs experimental results

305

200

(2) (3) (4)

2 1,5

(5)

(9) (6)

1

(7) (8)

0,5

-25

Opening [micron]

160

Top of the beam

In order to quantify our approach against experimental measurements, three point bending test were performed on a notched beam. The beam dimensions are 400 mm span, 100 mm high, 50 mm thick and the notch is 20 mm high. The test is driven by the CMOD measure at the notch mouth. In order to measure the crack length and opening, we use a Digital Image Correlation technique (see Figure 5). For practical reasons, the picture frame is limited to 55 mm high from the notch tip. The crack is assumed to be vertical and thus the crack opening is estimated as the horizontal displacement jump in the horizontal direction. In order to get the crack opening evolution along the crack, 30 horizontal profiles are drawn and the displacement jumps are estimated along those profiles. For particular values of CMOD 20 (corresponding to peak load), 30, 40, 50, 60, 80, 100, 150 and 200 microns, a linear curve is fitted through the 30 measurements (see Figure 6). The fitted solid lines are prolongated in dashed line up to the CMOD measure at the notch mouth. For the numerical simulation of this test, we use the nonlocal version of Mazars’ damage model described in 2.1. Model parameters are fitted (before any crack opening estimation) on the experimental global response, i.e. force vs CMOD. Comparison between experimental and numerical curves is shown in Figure 7.

Force (kN)

2,5

0 0

50

100

150

(8)

200

COD (micron)

120 (7) (6)

80

(5) (4) (3) (2)

40

(1)

0

0

25

Y [mm] Picture frame

Figure 6. process.

(9)

50

75

Notch

Crack opening at different stage of the loading

A good fit is obtained for material parameters summarized in Table 1. For a given CMOD, the crack shape is compared between experimental measurements and numerical estimation both using the strong and the weak link approaches. A relative error is computed between experimental crack opening and its numerical counterpart. Just after the peak (CMOD = 50 microns), the two numerical approaches are quite similar (see Figure 8.a) and slightly underestimate the measured crack opening. However for large CMOD (200 microns) the strong approach yield a large error (see Figure 8.b and d) and the weak approach always

properties for a structure are naturally dominated by large crack openings. The numerical approach systematically underestimates the experimental crack opening, at least for a 3 point bending test. Although it is not on the safe side for an engineering use, it can be clearly explain from crack propagation considerations and by recalling that experimental crack opening are measured on the surface whereas the numerical one is performed on a 2D plane stress simulation:

Figure 7. Experimental and numerical Force vs CMOD responses.

Table 1.

Parameter fitting using nonlocal damage model.

E [GPa]

At

ν

30

0.2

0.9

Bt

YD0

lc [mm]

4 000

410−5

8

50

200

Opening [micron]

Opening [micron]

Expe Num strong Num weak

40 30 20

120 80

10

40

0 -25

Expe Num strong Num weak

160

0

0

25

50

75

-25

0

25

Y [mm]

75

(b)

0.6

0.4

Strong approach Weak approach

0.5

0.3 0.4

Error [-]

Error [-]

50

Y [mm]

(a)

0.3

Strong approach Weak approach 0.2

0.2 0.1 0.1 0

0 0

10

20

30

Y [mm]

(c)

40

50

60

0

10

20

30

40

50

60

Y [mm]

(d)

Figure 8. Comparison betweem strong and weak approaches vs experimental crack opening for COD = 50 microns (a) and 200 microns (b). Corresponding errors between numerical and experimental crack openings for COD = 50 microns (c) and 200 microns (d).

provides a better estimation of the measured crack opening. The strong approach relies only on the regularized equivalent strain at one given point that may be affected by boundary effect for instance (Pijaudier-Cabot et al. 2009) and is thus more sensitive to numerical perturbations. The larger the crack opening, the better the estimation. This is a rather important result since the transfer

306

• The stress state is close to a plane stress condition at the beam free surfaces whereas it is close to a plane strain condition in the bulk of the beam that reduces the crack propagation velocity due to confinement. • Due to casting process the material contains less aggregate close to the boundaries and is thus weaker in the sense that aggregates are obstacles for cracking. For these two reasons, on the surface the crack is more developped in length and opening than in the core of the beam. It is clearly proved if one looks carefully at the experimental measurements of the crack opening for CMOD of 200 microns (see Figure 8-b). The extension of the plot gives a zero opening above the top of the beam, i.e. the neutral axis is out of the beam. For a bending test it means that the applied load is null. However in Fig 7 one can see that for CMOD of 200 microns the bearing capacity is not yet zero and thus outside the surface the crack has not yet propagated to the beam limits. Besides due to 2D assumption the numerical modelling gives an average crack geometry between the surface and the core of the beam. In contrary, the experimental measurements are made on the surface where the crack length and opening are the largest. Furthermore, part of the inaccuracy in the crack opening estimation is due to the spreading of strain profile that occurs in nonlocal damage models. At the end the strain profile width is related to the internal length of the model (Giry et al. 2010).

3 CONCLUSIONS In this contribution we have presented a complete procedure used in a post-treatment analysis to get crack location, crack opening and an estimation of the error done for tensile failure. The tracking is perform solving a conduction-like FE problem based on mechanical variables. The crack opening is estimated by nonlocal strain profile comparisons with those analytically obtained from the strong discontinuity approach. Results are in good agreement with crack opening measured on a 3 point bending test by DIC technique.

REFERENCES Comi, C., S. Mariani, and U. Perego (2007). An extended fe strategy for transition from continuum damage to mode i cohesive crack propagation. International Journal for Numerical and Analytical Methods in Geomechanics 31(2), 213–238. Dufour, F., G. Pijaudier-Cabot, M. Choinska, and A. Huerta (2008). Extraction of a crack opening from a continuous approach using regularized damage models. Computers and Concrete 5(4), 375–388. Dumstorff, P. and G. Meschke (2007). Crack propagation criteria in the framework of x-fem structural analyses. International Journal for Numerical and Analytical Methods in Geomechanics 31(2), 239–259. Feist, C. and G. Hofstetter (2007). Three-dimensional fracture simulations based on the sda. International Journal for Numerical and Analytical Methods in Geomechanics 31(2), 189–212. Giry, C., F. Dufour, J. Mazars, and P. Kotronis (2010). Stress state influence on nonlocal interactions in damage modelling. In Computational Modelling of Concrete Structures. Rohrmoos-Shladming, Austria. Euro-C. Hillerborg, A., M. Modeer, and P. E. Pertersson (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6, 773–782. Jirasek, M., S. Rolshoven, and P. Grassl (2004). Size effect on fracture energy induced by non-locality. International Journal for Numerical and Analytical Methods in Geomechanics 28(7–8), 653–670. Larsson, R., P. Steinman, and K. Runesson (1998). Finite element embedded localization band for finite strain plasticity based on a regularized strong discontinuity. Mechanics of Cohesive-Frictional Materials 4, 171–194. Mariani, S. and U. Perego (2003). Extended finite element method for quasi-brittle fracture. International Journal for Numerical Methods in Engineering 58(1), 103–128. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. Journal of Engineering Mechanics 115(2), 345–365. Mazars, J. and G. Pijaudier-Cabot (1996). From damage to fracture mechanics and conversely: a combined approach. International Journal of Solids and Structures 33, 3327–3342. Mediavilla, J., R.H.J. Peerlings, and M.G.D. Geers (2006). A robust and consistent remeshing-transfer operator for ductile fracture simulations. Computers and Structures 84(8–9), 604–623.

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Moes, N., J. Dolbow, and T. Belytschko (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150. Oliver, J. and A.E. Huespe (2004a). Continuum approach to material failure in strong discontinuity settings. Computer Methods in Applied Mechanics and Engineering 193(30–32), 3195–3220. Oliver, J. and A.E. Huespe (2004b). Theoretical and computational issues in modelling material failure in strong discontinuity scenarios. Computer Methods in Applied Mechanics and Engineering 193(27–29), 2987–3014. Oliver, J., A.E. Huespe, M.D.G. Pulido, and E.W.V. Chaves (2002). From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics 69, 113–136. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39, 937–953. Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Pijaudier-Cabot, G., A. Krayani, and F. Dufour (2009). Boundary effect on weight function in nonlocal damage model. Engineering Fracture Mechanics 76(14), 2217–2231. Planas, J., M. Elices, and G.V. Guinea (1993). Cohesive cracks versus nonlocal models: Closing the gap. International Journal of Fracture 63, 173–187. Simo, J.C., J. Oliver, and F. Armero (1993). An analysis of strong discontinuities induced by strain-softening in rateindependent inelastic solids. Computational Mechanics 12, 277–296. Simone, A., H. Askes, and L.J. Sluys (2004). Incorrect initiation and propagation of failure in non-local and gradient-enhanced media. International Journal of Solids and Structures 41, 351–363. Simone, A., G.N. Wells, and L.J. Sluys (2003). From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Computer Methods in Applied Mechanics and Engineering 192(41–42), 4581–4607. Wells, G., L.J. Sluys, and R. de Borst (2002). Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. International Journal for Numerical Methods in Engineering 53(5), 1235–1256.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Convergence aspects of the eXtended Finite Element Method applied to linear elastic fracture mechanics Wagner Fleming Department of Civil Engineering, Universidad Católica del Norte, Chile

Detlef Kuhl Institute of Mechanics and Dynamics, Universität Kassel, Germany

ABSTRACT: In the framework of linear elastic fracture mechanics, the non-smoothness of the solution influences the accuracy of the classical p-finite element solution and consequently the optimal convergence rate, associated with smooth solutions, is not achieved. For example, in the two-dimensional case, the convergence rate for the energy norm is limited to be of order O(h1/2 ), where h is the mesh size, being independent of the order of the finite element basis (4; 12). Considering that the determination of the crack propagation direction is a crucial step in fracture modelling and that an inaccurate calculation of this parameter can lead to locking in the solution, other techniques are needed in order to improve the accuracy of the p-finite element solution. In this paper, the extended finite element method (10) is exploited in order to improve the convergence rate of the classical finite element method for problems involving cracks in brittle materials. The extended finite element method allows to model a discontinuity in the displacement field by means of the enrichment of classical finite element basis with a generalized Heaviside function and specific near-tip functions that consider the singularity of the stresses at the crack tip within the framework of the partition of unity (1). Different near-tip enrichment techniques like the classical approach enriching only the nodes of the element containing the crack tip, the enrichment of a domain of fixed size around the crack tip, the use of a cut-off function and the gathering of the degrees of freedom are considered. The influence of all these approaches on the convergence rates of the following mechanical quantities of interest is studied: • • • •

L2 -norm over the entire domain and over a subdomain not containing the crack tip. Energy norm over the entire domain and over a subdomain not containing the crack tip. Stress intensity factors. T -stress.

This paper complements the works (3; 5; 7) in order to give a better understanding of the performance of the extended finite element in fracture modelling. cific near-tip functions, which take into account the discontinuity in the displacement field and the local behaviour of the solution near the crack tip. The properties of the XFEM have been analyzed in (3; 5; 7), and some variations in order to improve the method have been proposed:

1 INTRODUCTION The modelling of crack propagation using the finite element method has several difficulties, the mesh needs to match the crack and due to this, remeshing is needed as the crack propagates. Furthermore, in linear elasticity, the finite element solution over regular meshes can not capture the singularity of the stress field properly and special elements are needed at the crack tip, see for example the quarter-point elements (2). In recent years, several techniques have been proposed in order to modell the fracture process (6). One if this techniques is the extended finite element method XFEM, introduced by Moës and co-workers (10), which offers the possibility of modelling crack growth without remeshing. The idea behind this method is to enrich the classical finite element approximation with a generalized Heaviside step function and spe-

• The enrichment with near-tip functions of a domain of fixed size around the crack tip. • The gathering of the degrees of freedom associated to the near-tip functions. • The use of a cut-off function. In this paper, the effect of these variations on the errors in the L2 -norm, energy norm, stress intensity factors (SIFs) and T -stress is studied for different Lagrange finite elements of polynomial order p = 1, 2 and 3.

309

The paper is structured as follows: The strong and weak form of the model problem are formulated in Section 2. In Section 3, the finite element method is briefly described, together with the classical extended finite element method and its above-mentioned variations. The convergence properties of the different methods are analyzed in Section 4. Finally, concluding remarks are given in Section 5. 2 FORMULATION OF THE PROBLEM In this section, classical results of linear elasticity are summarized in order to describe the problem in its strong form and weak form.

where σ is the Cauchy stress tensor, ε is the linear strain tensor, b the vector force per unit volume and C the elasticity tensor. Equations (1) and (2) define the strong form of the problem. 2.2 Weak form This section is devoted to obtain the weak form of the problem. For this purpose a test function δu is considered, which is zero on the boundary Ŵu and has at least square-integrable derivatives (δu ∈ H 1() ), that is δu = 0 ∀X ∈ Ŵu .

Multiplying (1) by the test function δu and integrating over the domain  leads to   δu · Div (σ ) dV + δu · b dV = 0. (4)

2.1 Governing equations A cracked body with open domain  ∈ R3 and piecewise smooth boundary Ŵ is considered. The boundary of the domain is the junction of the disjointed parts Ŵu , Ŵσ , Ŵd+ and Ŵd− , being n its outward unit normal vector. Prescribed displacements u∗ are imposed on Dirichlet boundary Ŵu , while prescribed stresses t ∗ are imposed on Neumann boundary Ŵσ . Ŵd+ and Ŵd− represent the crack surfaces, which are assumed to be traction-free, see Figure 1. The problem is to find the displacement u, satisfying the governing equations Div (σ ) + b = 0, σ = C : ε, in   1 ∇u + ∇ T u , ε= 2 together with the boundary conditions



δu · Div(σ ) = Div (δu · σ ) − ∇(δu) : σ ,







δu · (σ · n) dS

+



Ŵ

δu · b dV .

(6)



∇(δu) : σ = ∇ s (δu) : σ = δε : σ . 

n

δε : σ dV =



δu · (σ · n) dS

+



Ŵ

(8)

Separating the boundary integral in the domains Ŵu , Ŵσ , Ŵd+ and Ŵd− , and using the boundary conditions (2) together with (3) leads to   δu · (σ · n) dS = δu · t ∗ dS. (9)

d

Ŵ

t* Cracked body and boundary conditions.

δu · b dV .



t

+ d

(7)

Replacing this expression in (6) follows

(2)

e1

Figure 1.

∇(δu) : σ dV =

Defining now δε = ∇ s (δu) and considering the symmetry of the stress tensor, it is easy to see that

Ω

e3

(5)

replacing this in (4) and using the Gauss’s theorem, the previous equation can be re-written as



e2



Considering now that

(1)

u(X ) = u∗ (X ) ∀ X ∈ Ŵu , σ (X ) · n = t ∗ (X ) ∀ X ∈ Ŵσ , σ (X ) · n = 0 ∀ X ∈ Ŵd+ , σ (X ) · n = 0 ∀ X ∈ Ŵd− ,

u

(3)





310

Ŵσ

Finally, substituting (9) into (8) yields   δε : σ dV = δu · t ∗ dS + δu · b dV . Ŵσ



(10)

Here K is the stiffness matrix, F the vector of external forces and u, abusing of the notation, the vector grouping the nodal degrees of freedom in the form

T (17) u = u11 u21 u12 u22 . . . u1NN u2NN .

This is the weak form of the equilibrium equation. No assumption related to the material behaviour is implicit. The weak form of the problem can now be formulated as follows: Given b, C, u∗ and t ∗ . Find u(X ) ∈ U such that ∀ δu(X ) ∈ U0 , equation (10) is satisfied. Here, U and U0 are the spaces defined by U = {u(X ) ∈ H 1() | u(X ) = u∗ (X ) on Ŵu }, U0 = {δu(X ) ∈ H 1() | δu(X ) = 0 on Ŵu }.

(11) (12)

3 CRACK MODELLING WITH XFEM This Section describes the different numerical methods analyzed in this paper. The analysis is restricted to two-dimensional problems. 3.1 FEM In order to fix the notation, the classical FEM approach is resumed as follows. First, the domain  is partitioned into NE only at their boundaries overlapping elements e with 1 ≤ e ≤ NE. The element vertices are called nodes and the total number of nodes in the domain is NN . Associated to every node X i , i ∈ I = {1, 2, . . . , NN }, there is a shape function φ i , which takes the value 1 at this node and 0 at all the other nodes. Formally  1 if i = j, i j φ (X ) = δij = i, j ∈ I . (13) 0 if i  = j.

3.2 XFEM approach The principal idea behind XFEM is to enrich the classical finite element approximations (14) and (15), with local information about the solution u in order to capture its local features. Thus, in the context of Linear Elastic Fracture Mechanics considered in this work, the facts that the displacement field is discontinuous on the crack and that the asymptotic behaviour of the solution in a neighborhood of the crack tip is known (see Appendix A), can be incorporated into the finite element approximation. The discontinuity across the crack is treated using a generalized Heaviside step function while the asymptotic behaviour is incorporated by means of near-tip functions. With this new formulation, the method allows to model crack growth without remeshing. Consider a Lagrange finite element basis of order p, B = {φ i (X )}i∈I , where φ i is the shape function associated to the node X i . On the same mesh a Lagrange basis of order 1, B1 = {ψ i (X )}i∈IPr can be constructed, where IPr is the set of primary nodes of the mesh. With this definitions, a first version of the XFEM approximation for a boundary crack (see Figure 2) can be written as   uh (X ) = φ i (X )ui + φ i (X )H (X )bi i∈I

+

Thus, the support wi of the shape function φ i is given by the union of the element sub-domains connected to the node X i . In this paper, only Lagrange shape functions are considered. Now, the basic idea of the finite element method is to approximate the displacement  uand the test function δu using the same basis B = φ i i∈I , that is, u ≈ uh (X ) =



ui φ i (X ),

δu ≈ δuh (X ) =

δui φ i (X ),

 i∈IT

(14)

j=1 k=1

⎞

j i,j Fk (X )ck ⎠,

(18)

The Heaviside function takes the value 1 on one side of the crack, and the value 0 on the other side. j The near tip functions Fk are given by

(15)

i∈I

j

2j−1 2

j

2j−1 2

j

2j−1 2

j

2j−1 2

F1 (r, θ) = r

ui

where are the unknown nodal degrees of freedom and δui are arbitrary vectors. Inserting this approximations in the weak form (10), a system of linear equations, called the structural equation, is obtained

F3 (r, θ) = r

(16)

F4 (r, θ) = r

Ku = F.

ψ i (X ) ⎝

p 4  

where IH is the set of nodes enriched with the Heaviside function and IT the set of nodes enriched with the near-tip functions   (19) IT = i ∈ IPr | X t ∈ wi ,   / wi . (20) IH = i ∈ I | wi ∩ Ŵd  = ∅ ∧ X t ∈

i∈I



i∈IH

⎛

F2 (r, θ) = r

311

(2j − 1)θ , 2 (2j − 1)θ cos , 2 (2j − 5)θ sin , 2 (2j − 5)θ cos . 2 sin

(21)

At this point some remarks about the XFEM approximation (18) are worth pointing out in order to achieve an optimal convergence rate (the convergence rate of the FEM for smooth solutions): First, the shape functions multiplying the discontinuous function H (X ) need to be of the same order as the finite element basis B. This had been shown in (7) comparing the p-FEM and the XFEM approximations for the displacement jump. If p-FEM is considered, the displacement jump is approximated by a p-order polynomial. On the other hand, in the XFEM approach, if an element completely cut by the crack is considered, the displacement jump is given only by the discontinuous part of the displacement approximation, which is associated with the Heaviside function. Thus, if for example only a linear basis is considered for the Heaviside enrichment, the displacement jump would be approximated linearly, which is not compatible with the p-FEM approximations and will lead to a nonoptimal convergency rate. Another way to see this is comparing the DOFs of both approximations. Second, p sets of 4 near-tip functions (F-functions) need to be added to the displacement approximation. This functions span the singular part of the displacement field (see the asymptotic expansion in Appendix 5), related to the fractional potencies of r, which is responsible for the well known lack of convergence in the classical FEM approach. Furthermore, it is neither neccesary nor desirable to multiply the F-functions by the high order functions φ i and the order-1 ψ functions are the right choice. This is due to the fact that it is only neccesary to be able to reproduce the F-functions, which can be achieved simply combining the products ψF. This reduces the number of extra degrees of freedom and minimizes the convergence problems in the partially enriched elements (blending elements).

Figure 2. Modified near-tip enrichment: All nodes within a radius amax are enriched by the near-tip functions.

  IH = i | wi ∩ Ŵd  = ∅ ∧ X t ∈ / wi ,

where X t is the position of the crack tip, X i the position of the node i and wi its support. It will be shown in Section 4, at least numerically, that if amax is independent of the mesh size h, this enrichment strategy leads to the optimal convergence rate. Nevertheless, a consequence of this strategy is that many extra degrees of freedom are now added to the classical FEM approach, and due to this, the conditioning of the stiffness matrix deteriorates with respect to classical FEM stiffness matrix. In order to reduce this effect, a specialized preconditioner is proposed in (3), while in (7) the gathering of the extra degrees of freedom related to the near-tip enrichment is considered. The second approach is studied in the next section. 3.4

3.3 Near-tip enrichment on a fixed area As the mesh size goes to zero, the influence zone of the near-tip enrichment vanishes and the optimal convergence rate is not achieved. Due to this, a modification of the enrichment strategy is considered, see also (3; 7). The idea is to enrich all the nodes within a radius amax by the near-tip functions. In this zone, the singular behaviour of the stress field is assumed to dominate the solution. Thus, a node with coordinates X i is enriched by the near-tip functions if its distance to the crack tip X t is less than amax . Figure 2 shows this enrichment strategy for a boundary crack. Within this framework, there are nodes with both enrichments, Heaviside and near-tip. The sets IH and IT can be formally defined as follows IT = i | X t ∈ w i ∨ X t − X i ≤ amax , 



(23)

DOF gathering

In order to reduce the number of extra degrees of freedom and, consequently, to improve the conditioning of the stiffness matrix, the gathering of the degrees of freedom associated with the near-tip enrichment is considered. Inserting the gathering condition i,j

j

ck = d k

(22)

312

i ∈ IT , k = 1, 2, 3, 4

and j = 1, 2, . . . , p,

(24)

in the XFEM approximation (18), leads to uh (X ) =

 i∈I

φ i (X )ui +



φ i (X )H (X )bi

i∈IH

p 4   j j Fk (X )d k , + χ h (X ) j=1 k=1

(25)

with χ h (X ) =



ψ i (X ).

(26)

i∈IT

where r is the distance of a point X to the crack tip, see Figure 3. This approach leads to the optimal convergence rate, provided that χ is smooth enough such p  j j that u − χ j=1 4k=1 d k Fk ∈ H k+1 for some conj

Compared with the XFEM approach without gathering, which adds 8p degrees of freedom for each node enriched with near-tip functions, the DOF gathering technique reduces the number of extra degrees of freedom to only 8p for each crack tip. Note that χ h (X ) is identical to one on those elements with all their nodes enriched with the near-tip functions, vanishes in the standard elements and varies continuously from zero to one in the blending elements (elements with only some enriched nodes). The fact that the zone where χ h falls from 1 to 0 vanishes as h → 0, leads to a lost of 1/2 in the convergence rate in the energy norm (over the entire domain), independently of the order of the polynomial approximation, see (7).

stants dk , see (12). Functions of class C k , with k = 1, 2, 3, lying in H k+1 can be defined as a function of x = (r − r1 )/(r2 − r1 ) as follows

χ|[0 1] = 1 − 3x2 + 2x3

⇒

χ |[0 1] = 1 − 10x3 + 15x 4 − 6x5

χ ∈ C1 ⇒

χ ∈ C2

χ |[0 1] = 1 − 35x4 + 84x 5 − 70x6 + 20x7 ⇒ χ ∈ C3 (28)

4 ANALYSIS In this section, five methods are compared:

3.5 Cut-off function

• The finite element method (FEM), Section 3.1. • The classical extended finite element method, where only the nodes whose support contains the crack tip are enriched with near-tip functions (XFEM), Section 3.2. • XFEM with near-tip enrichment on a fixed area (XFEM-FA), Section 3.3. • XFEM considering the gathering of the degrees of freedom associated to the near-tip enrichment (XFEM-FA-Gath), Section 3.4. • XFEM using a cut-off function (XFEM-Cutoff), Section 3.5.

In this section the use of a cut-off function varying continuously from 1 to 0 in a zone independent of the mesh size h is studied. This idea is actually very old and was considered in the 70’s by Strang & Fix in the framework of the classical FEM (12), and more recently by Chahine and co-workers (5) in the framework of the extended finite element method. Within this approach, the XFEM approximation is similar to (25) but with χ h replaced by a mesh independent cut-off fuction χ given by ⎧ 1 if r < r1 ⎪ ⎨ χ (r) = smooth if r1 < r < r2 ⎪ ⎩ 0 if r2 < r

r2

2

r1 1

d

r 1 , r 2 constant! Figure 3.

Enrichment with a cut-off function.

For each method, the convergence rate r, in terms of the mesh size h, is studied for the following parameters: (27)

• L2 -norm over the entire domain and over a subdomain not containing the crack tip. • Energy norm over the entire domain and over a subdomain not containing the crack tip. • Stress intensity factors. • T -stress.

The tests are carried out considering Lagrange finite elements of order p = 1, 2 and 3. The domain  is the unit square [0, 1] × [0, 1] (in meters) and the crack is defined by the segment going from (0, 0.5) to (0.5, 0.5). Different regular meshes are obtained by dividing the unit square. Divisions going from 5 to 75 (in both directions) are considered, see Figure 4 for an example mesh. The mesh size h is choosen as the side length of the elements. In the case of the methods XFEM-FA and XFEMFA-Gath a value of amax = 0.1 m is choosen, while for the XFEM-Cutoff the values r1 = 0.1 m and r2 = 0.4 m are considered.

313

The L2 -norm, energy norm and stress intensity factors are calculated for the solution obtained by imposing the asymptotic displacement field (30) with KI = KII = 1 N/mm3/2 as Dirichlet boundary condition, while for the T -stress the values KI = KII = 10 N/mm3/2 and T = 1 N/mm2 are used. Considering that the exact solution of the problem does not involve terms of order r 3/2 or higher, only one set of near-tip functions is added in the XFEM approximation.

Since the Heaviside enrichment is active at the Dirichlet boundary Ŵu , the known nodal degrees of freedom at the boundary are calculated by minimizing 

u∗ − uh L2 (Ŵu ) = u∗ − uh 2 dŴ. (29) Ŵu

The subdomain considered for the L2 -norm and energy norm is the square [0.6, 0.8] × [0.6, 0.8] in the case of the XFEM methods, while for the FEM the subdomain is given by [0.625, 0.875]×[0.625, 0.875]. In the case of the stress intensity factors and T -stress, they are extracted from the numerical solution using the domain interaction integral J (1,2) and appropiate auxiliary solutions, as described in Appendix B. The domain over which J (1,2) is evaluated, is choosen as composed of all the elements with a node within a circle of radius 0.2 m around the crack tip. Thus, this domain depends only weakly on the mesh size. In all cases, the stiffnes matrix of the element containing the crack tip was obtained using the integration technique proposed in (7), where the integration points of the sub-triangles with vertex on the crack tip are obtained from the unit square via mapping. The results of the convergence test are summarized in Table 1, while Figure 5 shows the convergence curves for the stress intensity factor KII . From this results some remarks are worth mentioning:

u = u * (KI = 1, KII = 1)

L

• The finite element method shows a convergence rate equal to 1 for all the studied parameters, with exception of the energy norm in the entire domain, where a poor convergence rate r = 1/2 is achieved. This

L Figure 4. Table 1.

Example mesh used for the analysis.

Convergence rate r in terms of the mesh size h (side length of the elements). L2 -norm

L2 -norm-sub

Energy norm

Energy norm-sub

KII

T -stress

FEM p=1 p=2 p=3

1.0 1.0 1.0

1.0 1.0 1.0

0.5 0.5 0.5

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0

XFEM p=1 p=2 p=3

0.9 1.4 1.5

0.9 1.0 1.0

0.5 0.5 0.5

1.0 2.0 2.9

1.0 1.0 1.0

0.9 1.0 1.0

XFEM-FA p=1 p=2 p=3

2.4 3.3 4.2

2.4 3.1 3.9

1.2 2.2 3.1

1.0 2.0 3.0

2.5 4.2 −

2.5 4.2 −

XFEM-FA-Gath p=1 p=2 p=3

0.9 2.6 3.6

0.9 3.0 3.8

0.5 1.5 2.6

1.0 2.0 3.0

1.0 3.0 4.8

0.9 3.0 4.6

XFEM-Cutoff p=1 p=2 p=3

1.7 3.2 3.9

1.7 3.0 4.0

0.9 1.9 2.9

0.8 2.0 3.0

1.8 3.8 5.7

1.8 3.9 5.6

314

10

02

10

04

10

06

FEM p=1 (r=1.0) FEM p=2 (r=1.0)

K II 10

1

FEM p=3 (r=1.0)

08

XFEM p=2 (r=1.0)

XFEM p=1 (r=1.0)

XFEM p=3 (r=1.0) XFEM FA p=1 (r=2.5) XFEM FA p=2 (r=4.2)

10

10

10

12

10 Figure 5.

2

h/L

10

1

Convergence analysis for the stress intensity factor KII and the different enrichment strategies.

results are independent of the order of the finite element basis p. The classical XFEM presents the same behaviour, the only surprising result was a convergence rate r ≈ p for the energy norm in a subdomain not containing the crack tip. This result needs further investigation. • The methods XFEM-FA and XFEM-Cutoff show a similar behaviour, with convergence rates r ≈ p + 1 in the L2 -norm and r ≈ p in the energy norm. In the case of the stress intensity factors and T -stress, a convergence rate r ≈ 2p is achieved. For the XFEM-FA with p = 3 it is difficult to recognize a convergency rate in the case of the stress intensity factors and T -stress, probably due to the fact that the accuracy of the method is comparable with the machine precision and that the stiffnes matrix becomes very sensitive due to its poor conditioning. Actually, the condition number for p = 3 is of order O(h−8.7 ), while the XFEM-Cutoff shows an order O(h−3.0 ). • The method XFEM-FA-Gath shows a lost of convergence with respect the optimal convergence rates, but the convergence rates increase with the order of

315

the finite element basis. In the energy norm over the entire domain, the method shows a lost of half an order (independent of p). The same occurs for the L2 -norm over the entire domain with exception of the case p = 1, where one order was lost. In the case of the SIFs and T -stress, one order was lost in the convergence rate (again independent of p), with respect to the XFEM-FA and XFEM-Cutoff methods. This lost of convergence is attributed to the blending elements, see (7).

5 CONCLUDING REMARKS In this paper, a numerical analysis of the extended finite element method and some of its variations was carried out. The considered variations were: • The enrichment of a domain of fixed size around the crack tip. • The gathering of the extra degrees of freedom associated to the near-tip enrichment. • The use of a cut-off function.

The main point of analysis was the convergence rate for quantities of special interest as the L2 -norm, energy norm, SIFs and T -stress. Even though the enrichment of a fixed domain presents the best results from the convergence point of view, the fact that many extra degrees of freedom need to be added, presents an important disadvantage for this method. In comparison, the extended finite element method with a cut-off function achieves similar convergence rates without adding many extra degrees of freedom. The performance of the XFEM with the gathering technique is also worth mentioning, the method shows an improvement over the classical XFEM and since the dof gathering is nothing more than a static condensation, this technique can be easily included in a commercial software. Finally, in the context of quasi-brittle materials, when a polynomial cohesive law is considered, the asymptotic fields have been given in ?, showing that the first term of the displacement is proportional to r 3/2 . Thus, the solution u is smoother than in the case of brittle materials and better convergence rates are expected in the case of the FEM and the classical XFEM.

APPENDICES

f21n (r, θ) =

    nθ r n/2 n κ − − (−1)n sin √ 2 2 2G 2π +

f22n (r, θ) =

r n/2 √ 2G 2π



 n   n sin −2 θ 2 2

    n nθ − κ − + (−1)n cos 2 2  n   n − cos −2 θ 2 2

(31)

and κ is the Kolosov’s constant defined as ⎧ ⎨3 − 4ν κ = 3−ν ⎩ 1+ν

Plane strain, Plane stress.

(32)

The displacements corresponding to n = 0 are rigid body translations, representing the displacement of the crack tip. In (30), KI1 and KII1 are the stress intensity factors, denoted normally by KI and KII , respectively. Furthermore KI2 is related to the T-Stress by T = √ 4KI2 / 2π.

A asymptotic displacement field The asymptotic solution for the displacement field is given in polar coordinates relatively to the crack tip by (see for example (9))  ∞   f11n (r, θ ) f12n (r, θ ) = f21n (r, θ ) f22n (r, θ ) u2 (r, θ) n=0



u1 (r, θ)



×

 KIn , KIIn



(30)

where f11n , f12n , f21n & f22n are given by f11n (r, θ) =

    r n/2 n nθ κ + + (−1)n cos √ 2 2 2G 2π

B.1 Extraction of the SIFs Consider a structure containing a crack and suppose that by some means, for example a numerical method, the displacements, strains and stresses are known, at least approximately. Then, the question is how to extract from this solution the stress intensity factors. In this appendix, the attention is concentrated on the extraction of the SIFs using the domain interaction integral following (8; 11). The main idea is to consider two admissible states (u(1) , ε(1) , σ (1) ) and (u(2) , ε (2) , σ (2) ), and their associated stress intensity factors (KI(1) , KII(1) ) and (KI(2) , KII(2) ), respectively. Consider the well known expresion for the J -integral for linear elastic fracture J =

 n   n − cos −2 θ 2 2 f12n (r, θ) =

B SIFs AND T-STRESS EXTRACTION

KI2 KII2 + , E∗ E∗

(33)

where

    nθ r n/2 n κ + − (−1)n sin √ 2 2 2G 2π

∗

E =

 n   n −2 θ − sin 2 2

⎧ ⎨ ⎩

316

E 1 − ν2

Plane strain,

E

Plane stress.

(34)

(2)

J (1+2) =

(KI(1) + KI(2) )2 + (KII(1) + KII(2) )2 E∗

= J (1) + J (2) +

(1)

KI

2 (K (1) K (2) + KII(1) KII(2) ). E∗ I I (35)

J =−

P : ∇q d,

=

E ∗ (1,Mode I) J . 2

(42)

In a similar way, the stress intensity factor KII(1) can be extracted from the numerical solution, considering the auxiliary state as the Mode II of fracture (KII(2) = 1 and KI(2) = 0), which leads to

On the other hand, for a traction-free crack, the J -integral can be written in a domain form as 

(2)

asymptotic solution for Mode I (KI = 1 and KII = (1) 0), the stress intensity factor KI can be obtained from

For the superposition of these two states, the J integral calculated from (33) for in-plane loading, takes the form

KII(1) =

(36)

E ∗ (1,ModeII) . J 2

(43)



B.2 Extraction of the T -stress

where P is the Eshelby tensor P = ω1 − ∇ T u · σ ,

The extraction of the T -stress is very similar to that of the SIFs. In this case, the auxiliary solution can be taken as the leading term of the solution associated to a concentrated load F per unit thickness, acting at the crack tip in the e1 direction, see Figure 6. The stress field of this auxiliary solution, satisfying Div(σ ) = 0, is given by

(37)

and q(X ) is an auxiliary vector field, which vanishes on a prescribed contour Ŵ1 and takes the value e1 on the crack tip, formally q=

 0

e1

∀ X ∈ Ŵ1 ,

if X = 0.

(38) σrr = −

Here, the coordinate system is defined locally, with the basis vector e1 tangent to the crack at its tip, as shown in Figure 6. Furthermore, in order for (36) to be valid, q needs to be tangent to the crack (see (11)), that is q·n=0

on

Ŵd .

σrθ = σθ θ = 0,

(44)

or equivalently,

σ11 = −

F cos3 θ , πr

σ22 = −

F cos θ sin2 θ πr

(39)

Considering again two states and evaluating (36) for the sum of these states gives J (1+2) = J (1) + J (2) + J (1,2) ,

F cos θ , πr

σ12 = −

F cos2 θ sin θ , πr (45)

(40)

where J (1,2) is the domain interaction integral given by J (1,2) = −



(σ (1) : ε(2) 1 − ∇ T u(2) · σ (1)



− ∇ T u(1) · σ (2) ) : ∇q d.

(41)

Equalling now (40) and (35), and considering the state 1 as the numeric solution and state 2 as an auxiliary solution, for example, the leading term of the

317

Figure 6. Concentrated load per unit thickness acting at the crack tip.

and the displacement field derivatives are −F (κ cos θ + cos 3θ ), 8π Gr F ((2 − κ) cos θ + cos 3θ ), u2,2 = 8π Gr −F ((2 + κ) sin θ + sin 3θ ), u1,2 = 8π Gr F (κ sin θ − sin 3θ ). u2,1 = 8π Gr u1,1 =

(46)

The J -integral vanishes for this auxiliary solution. Now, if the J -integral is calculated for the superposition of the auxiliary solution and the general asymptotic solution (30), including the r −1/2 singularity, this yields J (1+2) = J (1) + J (2) +

FT , E∗

(47)

where the last term can be identified as the domain interaction integral and then, after choosing F = 1, the T -stress can be calculated by T = E ∗ J (1,Aux) ,

(48)

where J (1,Aux) is the domain interaction integral defined in (41). REFERENCES [1] I. Babuška and J.M. Melenk. The partition of unity method. International Journal for Numerical Methods in Engineering, 40, 725–758, 1997. [2] R.S. Barsoum. Application of quadratic isoparametric finite elements in linear fracture mechanics. International Journal of Fracture, 10, 603–605, 1974. [3] E. Bechet, H. Minnebo, N. Moës and B. Burgardt. Improved implementation and robustness study of the X-FEM for stress analysis around cracks. International Journal for Numerical Methods in Engineering, 64, 1033–1056, 2005.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Applicability of XFEM for the representation of crack bridge state in planar composite elements J. Jeˇrábek RWTH Aachen University, Aachen, Germany

R. Chudoba & J. Hegger Institute of Structural Concrete, RWTH Aachen University, Aachen, Germany

ABSTRACT: This paper reviews an enriched finite element representation of crack bridges suitable for simulating the complex damage processes in textile reinforced concrete. The heterogeneity of both the matrix and the reinforcement occurs at similar length scales of the material structure. Consequently, an improved accuracy of approximation at the hot-spots of damage is required in order to capture the relevant damage mechanisms. The explicit representation of the matrix crack is thus inevitable. Besides that, the quality of the strain and stress fields along the discontinuity is of special importance. This paper presents numerical studies focused on the reflection of the peak values in the crack bridge for selected elementary configurations. The studied approach combines the XFEM approximation of matrix displacement field with higher-order approximation of the reinforcement field. After showing the difficiencies of this approach, the possibilities of a consistent enrichment of both fields are briefly discussed. 1 INTRODUCTION Textile reinforced concrete (TRC) is a composite material combining the advantages of fiber reinforced concrete and steel reinforced concrete. Textile reinforcement made of glass, carbon or aramid is embedded as fabrics in a cementitious matrix. The heterogeneity of both the matrix and the reinforcement occurring on similar length scales of the material structure demands for an improved accuracy to capture the relevant damage mechanisms. In the present paper, we focus on the strategy applicable for the simulation of coarse crack pattern emerging in structural details and in shear zones. In these zones a complex stress state develops leading to localization of matrix damage in a few dominant cracks (Fig. 1). With further loading the bond and yarn damage develops particularly at the crack bridges. Therefore, an accurate assessment of the ultimate failure is only possible with an improved kinematics reflecting the discontinuities in the displacement fields. We remark that in zones with fine and regular crack pattern developing under uniaxial loading (Fig. 2) the smeared approach to the damage modeling is appropriate as documented e.g. in (Scholzen, Chudoba, and Hegger, 2008) presenting a smeared model with initial and damage-induced anisotropy. The possibility to introduce discrete crack in order to better reflect the stress state in the crack bridge is studied using 2D implementation. The constructed displacement approximation is intended to serve as a

319

basis for improved coupled meso-micro scale simulation of planar boundary value problems with coarse crack pattern. The explicitly obtained values of crack opening and crack sliding provide the necessary input for the micro-scale representation of critical crack bridges. For the related work on this issue we refer to (Konrad, Chudoba, and Jeˇrábek 2007; Konrad, Jeˇrábek, Voˇrechovský, and Chudoba 2006). We emphasize that this work is primarily focused on the comparison and verification of the formulated enrichment approaches. An improved local kinematics in the vicinity of the crack bridge can be achieved with the help of the

Figure 1.

Shear zone with few dominant cracks.

Figure 2.

TRC tensile specimens with fine crack patterns.

extended finite element method (XFEM) providing an efficient and elegant tool for introducing discontinuities and material interfaces into an originally smooth discretization. The discontinuities are included in a standard finite element framework by augmenting the set of displacement approximation functions with additional discontinuous fields through the partition of unity method (Melenk and Babuška, 1996). The major advantage of this method is that the enrichment is independent of the mesh and, thus, no or only minimum remeshing is required. XFEM has its roots in the work of (Belystchko and Black, 1999) who used the local partition of unity enrichment of finite elements. They enriched the nodes around a crack tip with the near-tip, linear elastic fracture mechanics (LEFM) solution. An enrichment for the nodes the support of which is fully cut by the crack introduced by (Moës, Dolbow, and Belytschko 1999). By enriching these nodes with a Heaviside function the treatment of cracks in the interior of elements became possible. In the following years the method was further enhanced and applied to numerous problem dealing with discontinuities. XFEM for three-dimensional crack modeling was first used by (Sukumar, Moës, Moran, and Belystchko 2000). An algorithm that couples the XFEM with the level set method (Osher and Sethian 1988) by applying a discontinuous function in modeling two-dimensional linear elastic crack-tip displacement fields was presented in (Stolarska, Chopp, Moës, and Belytschko 2001). Above mentioned methods were unified and further extended in (Belytschko, Moës, Usui, and Parimi 2001). A new crack tip elements and their application to cohesive cracks introduced in (Belytschko, Zi, Xu, and Chessa 2003; Zi and Belytschko 2003; Dumstorff and Meschke 2007). 2 VARIATIONAL FORMULATION OF TWO-FIELD BOUNDARY VALUE PROBLEM

 with the boundary Ŵ. The domain  is decomposed according to the layout of the material components: m for the cementitious matrix, f for the fibers or reinforcement and mf for the intersection of these subdomains (see Fig. 3). Further, we introduce the zones of potential debonding e ⊆ mf where the displacement fields are explicitly resolved in the numerical representation. The boundaries of the subdomains are denoted by Ŵm and Ŵf and are further distinguished into essential (Ŵum , Ŵuf ) and natural boundary conditions (Ŵtm , Ŵtf ). The stress and strain components are aggregated in vectors σ (·) = {σ(·)xx , σ(·)yy , σ(·)xy }T and ε (·) = {ε(·)xx , ε(·)yy , ε(·)xy }T , respectively. The index (·) stands for the two material phases [m, f ]. The surface friction is denoted as τ mf = {τx , τy }T For the sake of simplicity the derivations shall be demonstrated on one-dimensional debonding problem. The local equilibrium of both layers with an implicitly assumed unit thickness is given as (see Fig. 3 right demonstrating the uni-axial stress state) ∂ T σ m − τ mf = 0,

∂ T σ f + τ mf = 0

where ∂ T denotes the differential operator with the following distribution of partial derivatives along x and z   · (·), y (·), x ∂T = . · (·), y (·), x The corresponding essential and natural boundary conditions are specified as um = u¯ m uf = u¯ f

on Ŵum on Ŵuf

and and

nσ m = t¯m nσ f = t¯f

on Ŵtm

on Ŵtf

(2)

where n represents the boundary operator to project the stress tensor into surface tractions (n1 , n2 are the components of the normal surface vector)   · n2 n . n= 1 · n2 n1 The weak form of the boundary value problem given in Eqs. (1) and (2) can be constructed using the variation fields vm and vf   v m , ∂ T σ m − τ m  + (vm , um − u¯ m )Ŵum     + vm , nσ m + t¯m Ŵt + vf , ∂ T σ f + τ m  m   + (uf , uf − u¯ m )Ŵu + uf , nσ f + t¯f Ŵt = 0 f

The variational framework is established for a twolayered, two-dimensional body occupying the domain

(1)

f

(3)

where (u, v)V denotes the integration of the product of the terms u, v over V . The constitutive laws for matrix and fibers and the corresponding kinematic relations

320

Figure 3.

Left: Two-field boundary value problem; Right: Infinitesimal segment of two-phase composite.

have the form σ (·) = D(·) ε(·) ,

(4)

ε(·) = ∂ u(·) · · ·

Substituting Eqs. (4) into Eq. (3) and applying the integration by parts and assuming that essential boundary conditions are implicitly satisfied by the chosen approximation we obtain the reduced weak formulation as  T  ∂ vm , Dm ∂um  + (v m , τ mf )     − vm , t¯m Ŵ + ∂ T v f , Df ∂uf  t   − (v f , τ mf ) − v f , t¯f Ŵ = 0.

(5)

t

In the next two sections the approximation for the sought displacement fields um , uf leading to a discrete form of Eq. (5) shall be formulated and tested.

3 GLOBAL TWO-FIELD APPROXIMATION The reinforcement displacement field is approximated using standard shape functions uf (x) = N d f =



Ni (x)df , i

(6)

i∈I

where I denotes the set of all nodes, Ni is the standard finite element shape function at node i and df , i the corresponding degree of freedom. The displacement field of the matrix is additionally enriched to introduce a crack at xξ applying the XFEM method (see Fig. 4) um (x) = N d m =

 i∈I

Ni (x)d¯ m, i +



Figure 4. XFEM enrichments N s (xξ ) by example of four node element bilinear shape functions and a single crack ξ at position xξ = 0.5.

with J denoting the set of nodes with a crack cutting its support, d¯ m and dˆ m representing the standard and additional degrees of freedom, respectively and with j introducing the XFEM jump function for the enrichment of node j: j (x) = sign( (x)) − sign( j ),

j ∈ J.

(8)

Here, represents the level set function implicitly specifying the position xξ of the crack as (xξ ) = 0. By substituting the displacement approximations (6) and (7) into the variational formulation given in Eq. (5) and applying the kinematics to both the trial and the test functions (u(·),x = N,x d (·) = Bd (·) ) the discrete weak form of the boundary value problem is obtained as     δd Tm BT , Em Am B  d m − N T , τmf       − N T , ¯tm Ŵ + δd Tf BT , Ef Af B  d f

Nj (x)j (x)dˆ m, j

t

j∈J

(7)

321

     + N T , τmf  − N T , ¯tf Ŵt = 0.

(9)

Here, also the fact was exploited that the variations δd f and δd m are constant within the spatial integrals. The above equation must be fulfilled for arbitrary variations of δd f and δd m resulting in two sets of possibly nonlinear equilibrium equations R(d) =



   Rm (d m , d f ) 0 = Rf (d m , d f ) 0

Inserting Eqs. (14) and (15) into Eq. (13) yields the algorithmic stiffness matrix at the iteration step (k)   (k−1) = BT Em Am B dx + N T Dτ(k−1) N dx K mm 

(10)

with the residuals Rm and Rf defined as

      Rf = BT , Ef Af B  d f + N T , τmf  − N T , ¯tf Ŵ . t (11) In case of nonlinear material behavior assumed either for the matrix, reinforcement or bond, Eq. (10) must be prepared for iterative solution strategies by means of linearization i.e. by Taylor expansion neglecting quadratic and higher order terms. The expansion up to the linear term reads

R(d

(k)

) ≈ R(d

(k−1)

∂RT (d) d (k) . )+ ∂d d (k−1)

(12)

By substituting for R(d) in Eq. (10) and denoting the derivatives of residuals K (ij) = ∂R(i) /∂d (j) with i, j ∈ [m, f ] we can write down the standard iterative expression for achieving equilibrium in the form

(k−1) K mm (k−1)

K fm

(k−1) K mf (k−1)

K ff



d (k) m

d f (k)

=−

Rm(k−1) Rf(k−1)

. (13)

For the sake of simplicity, throughout this paper the material behavior of matrix and reinforcement is assumed linear elastic. A generally nonlinear bond law can be introduced for τmf (s) with the slip field defined as s = um − uf . The derivatives of the constitutive law with respect to the degrees of freedom then read ∂τmf ∂τmf ∂s ∂(−um + uf ) = = Dτ (s) ∂d m ∂s ∂d m ∂d m

(k−1) =− K mf

N T Dτ(k−1) N dx

(k−1) K fm =−



N T Dτ(k−1) N dx

K ff(k−1) =

      Rm = BT , Em Am B  d m − N T ,τmf  − N T, ¯tm Ŵt

(14)











BT Ef Af B dx +







N T Dτ(k−1) N dx. (16)

The applicability of this approach is shown on the example of a two layer 2D composite rectangle (3 × 2 meters) with uniform displacement loading of the reinforcement layer uf in x-direction (ur = 0.3 m) at the right end. The bond law is assumed linear in the form τmf = Dτ s with constant stiffness Dτ . The matrix layer um is cut by a single crack with varied orientation. The values of the material parameters are provided in Tab. 1. The example was computed using of n = 30 × 20 elements for both the matrix and the reinforcement fields with the total of 2688 degrees of freedom (including degrees of freedom for the XFEM contribution in the cracked matrix element). The obtained stress/strain state fields (Fig. 5) shall now be examined in a more detail with respect to their • convergence to the exact solution for perpedicular crack with macroscopically uniform stress field, and • ability to reproduce the kink in the reinforcement at the crack bridge. The convergence to an exact solution has been studied for the case of the matrix crack perpendicular (90◦ ) to the uniform tensile loading (σxx = 1 MPa). At the left hand side, both fields were fixed (um = 0, uf = 0). The strain/stress in y-direction is constant and the stress level of the composite in x direction is equal to one (Fig. 6). At the crack bridge σmexact = 0 and σfexact = 1. Evidently, the linear element with constant strain in the x-direction cannot capture the peak stress/strain in the reinforcement. Such a pure reflection of strains in the crack bridge, the hot spot of fiber damage, calls for further improvements of the approximation. The

= −Dτ (s)N Table 1.

and, analogically ∂τmf = Dτ (s)N . ∂d f

Parameters of the running example.

Em [Pa]

νm [−]

Ef [Pa]

νf [−]

Dτ [Pa/m]

α [◦ ]

2.0

0.0

1.0

0.0

10.0

30, 45, 60, 90

(15)

322

Figure 5. Tensile specimen with inclined matrix crack discretized by 30 × 20 bilinear elements with varying crack angle (measured relatively to the loading direction); a) 90◦ ; b) 60◦ ; c) 45◦ ; d) 30◦ .

cracked

Figure 7. Maximal value of the stress in the reiforcement in the crack bridge to number of degrees of freedom in the direction of loading.

simplest option is the uniform h- and p-refinement. Figure 7 shows the convergence to an exact value of fiber stress for an increasing number of DOFs. Even for higher-order elements very slow convergence can be observed. The fact, that the kink in the reinforcement strain field is ignored leads to an underestimation of the peak stress. The fundamental problem lies in the inability of the shape function to reflect the shape of exact strain profile in the cracked element. The situation is qualitatively sketched in Figure 8. The profile of the reinforcement strain is compared

with the shape function derivatives of the linear, quadratic and cubic approximation. For linear elastic bond, the profile of the exact solution consists of two convex functions centered at the crack. Obviously, the linear and quadratic approximation can only reflect the average constant strain within the element. For a cubic element, the continuous approximation leads to concave reinforcement strain in the crack element. This contradiction to the convexity of the exact solution results in the underestimation of the peak strain. As the applied error criterion is local, it would be more appropriate to compare the number of DOFs for

Figure 6. Displacement and strain profile of specimen (a).

90◦

323

non-uniform refinement with higher-order elements applied only in the in the crack-bridging elements. However, this would require kinematic constraints between elements of different order and even then the

Figure 8. Qualitative comparison between the exact solution and linear, quadratic and cubic approximation of σf .

Figure 9. Comparison of the quality of the approximation of the stress peak in the reinforcement field.

mismatch between the convexity of the exact solution and concavity of the shape functions would remain. This mismatch has the consequence, that isolated inclusion of the displacement jump into the matrix without corresponding kink in the strain field cannot reproduce a stress free crack in the matrix. For a generally inclined crack shown in Figure 5a, b and c, additional characteristics of the strain field at the crack bridge can be recognized. For linear elements, the low order of strains results in saw-tooth profile of the fiber stress along the crack bridge depicted in detail in Figure 9 (left). The consequence is the same as for the perpendicular crack: underestimated peak stress in the reinforcement and spurious stress transfer across the matrix crack. The depicted ‘‘hangovers’’ in the stress field can be reduced using the element with higher polynomial approximation. In the profile shown in Figure 9 (right) the serendipity bicubic elemets were used. Their polynomial basis is able to form the peak and visibly decreeses the strain incompatibilities at the element edges. Still, as for the perpendicular crack, the shape of the strain profile with a kink cannot be accurately reproduced using higherorder elements. The advantage of the described approach is that it can be implemented into finite element software with XFEM support without the need to modify the core components. Figure 10 shows an example of a rectangle with multiple matrix crack subjected to tensile loading. The cubic serendipity elements have been used showing an acceptable quality of the peak

Figure 10. Tensile specimen with inclined matrix cracks discretized by 30 × 20 serendipity bicubic elements; a) ideaqlization and discretization; b) strain field in the reinforcement; c) slip filed; d) strain field in the matrix.

324

reflection. The example shows also the associated slip field and the matrix stresses. However, for the purpose of local improvement of the field approximation in the crack bridge, this approach seems to be a numerical overkill. Moreover, the present study has shown that an isolated inclusion of a crack into the matrix without a kink reinforcement strain field leads to an inconsistency in the strain fields and very slow convergence to an exact local value. More focused improvements with a consistent enrichment of the fiber displacement field is desirable. 3.1 CONSISTENT ENRICHMENT OF THE REINFORCEMENT The displacement jump in the matrix um induces a kink in the gradient of the reinforcement displacement field uf . This holds for any type of the bond law between the matrix and reinforcement. Therefore, a consistent enrichment could be achieved by introducing a weak discontinuity in uf . For this purpose, the abs-enrichment and its extensions could be applied (Moës, Choirec, Cartaud, and Remacle 2003; Fries 2007). In these methods, the weak discontinuity is introduced only approximately. Even though the sufficient convergence rate in the global sense (L2 -norm) is achievable, the required quality of the strain field along the discontinuity is not generally guaranteed. An alternative approach to achieving the consistent fields has been suggested by the authors (Chudoba, Jeˇrábek, and Peiffer 2009). The performed construction of the enrichment can be summarized in two steps: • use the same XFEM approximation for the common part of uf and um , introducing a crack in both fields, • add an additional jump term with oposite signs to both the matrix and the reinforcement fields, • constrain the reinforcement field to recover the displacement continuity.

      +N˜ d˜ m um 1  N d + ξ N d + = uf 1 −N˜ d˜ f

The major advantage of such an approach is the possibility to center the enrichment in the crack and to keep the original discretization in the regions without cracks. Thus, the matrix and reinforcement fields are only distinguished in the zones with debonding. 4 CONCLUSIONS The purpose of the present study was to assess the feasibility of the numerical approximation for twofield problems with XFEM discontinuity introduced in one field. The reproduction of the field values in the discontinuity has been used as a criterion for qualitative evaluation of the approximation. The isolated inclusion of the jump in one field leads to an underestimated value of the bridging stresses in the reinforcement phase. This issue is critical, since the failure of the reinforcement initiates the ultimate failure of the structure. The work motivates further improvements of the crack bridge enrichments. One possible approach has been presented recetly by the authors in (Chudoba, Jeˇrábek, and Peiffer 2009). ACKNOWLEDGEMENT

The approximation has the following structure: 

Even though the formulation of the kinematic constraint for 2D problems is possible, it leads to a rather complex data structure required for the implementation. The complexity arises from the need to distinguish the DOFs at the edges cut by the crack. Further, for higher order elements the number and positioing of the kinematic constraints must reflect the order of the shape functions. • In the latter case, the variational formulation of the equilibrium condition is augmented with terms corresponding to the crack bridging forces. The most robust approach of this class seems to be the Nitsche’s method (Hansbo 2005).

(17)

The shared part of the displacement (first term) introduces a jump in both fields. The same local approximation term containing a jump N˜ is added to both fields with an oposite sign. As a consequence, the approximation of both fields remains of the same order. In order to recover the continuity of uf in Eq. (17), either the kinematic or the static constraint can be applied: • The recovery of the reinforcement continuity using a kinematic constraint for uniaxial case has been presented in (Chudoba, Jeˇrábek, and Peiffer 2009).

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The work has been supported by Deutsche Forschungsgemeinschaft (DFG) in the framework of the collaborative research center SFB 532 Textile-reinforced concrete, development of a new technology. The support is gratefully acknowledged. REFERENCES Belytschko, T. and T. Black (1999). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45, 601–620. Belytschko, T., N. Möes, S. Usui, and C. Parimi (2001). Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering 50, 993–1013.

Belytschko, T., G. Zi, J. Xu, and J. Chessa (2003). The extended finite element method for arbitrary discontinuities. In Proccedings of Computational Mechanics— Theory and Practice, Barcelona, Spain. CIMNE. Chudoba, R., J. Jeˇrábek, and F. Peiffer (2009). Crack-centered enrichment for debonding in two-phase composite applied to textile reinforced concrete. International Journal for Multiscale Computational Engineering 7(4), 309–328. Dumstorff, P. and G. Meschke (2007). Crack propagation criteria in the framework of x-fembased structural analyses. Int. J. Numer. Anal. Meth. Geomech. 31, 239–259. Fries, T.P. (2007). A corrected xfem approximation without problems in blending elements. Int. J. Numer. Meth. Engng. 75, 503–532. Hansbo, P. (2005). Nitsches method for interface problems in computational mechanics. GAMM-Mitt. 28, 183–206. Konrad, M., R. Chudoba, and J. Jeˇrábek (2007). Influence of the heterogeneity at the microlevel on the multi-cracking performance of textile reinforced concrete. In SEMC 2007: The Third International Conference on Structural Engineering, Mechanics and Computation. Konrad, M., J. Jeˇrábek, M. Voˇrechovský, and R. Chudoba (2006). Evaluation of mean performance of cracks bridged by multifilament yarns. In EURO-C 2006: Computational Modelling of Concrete Structures, pp. 873–880. Taylor and Francis Group, London. Melenk, J. and I. Babuška (1996). The partition of unity finite element method basic theory and applications. Moës, N., M. Choirec, P. Cartaud, and J.F. Remacle (2003). A computational approach to handle complex microstructure geometries. Comp. Meth. Appl. Mech. Engng 192, 3163–3177.

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Moës, N., J. Dolbow, and T. Belytschko (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46. Osher, S. and J. Sethian (1988). Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 12–49. Scholzen, A., R. Chudoba, and J. Hegger (2008). Numerical simulation of textile reinforced concrete using a microplane-type model with initial anisotropy. In B. Topping and M. Papadrakakis (Eds.), Ninth International Conference on Computational Structures Technology. Stolarska, M., D. Chopp, N. Moës, and T. Belytschko (2001). Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering 51(8), 943–960. Sukumar, N., N. Moës, B. Moran, and T. Belytschko (2000). Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering 48(11), 1549–1570. Zi, G. and T. Belytschko (2003). New crack-tip elements for XFEM and applications to cohesive cracks. International Journal for Numerical Methods in Engineering 57, 2221–2240.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Localization properties of damage models Milan Jirásek & Martin Horák Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic

ABSTRACT: The classical localization condition based on vanishing determinant of the acoustic tensor is explored in detail for several types of damage models. It is shown that the localization properties of isotropic models are not in full agreement with intuitive expectations. More satisfactory results are found for the rotating crack model, considered as a special type of an anisotropic damage model. a discontinuity in certain components of the strain rate. Combination of the traction continuity conditions, rate form of the constitutive equations and Maxwell compatibility conditions leads to the classical criterion for the loss of ellipticity based on singularity of the acoustic tensor. The fundamental question is under which conditions the inelastic strain increments can localize in one or more narrow bands separated from the remaining part of the body by weak discontinuity surfaces. Across such surfaces, the displacement field remains continuous but the strain field can have a jump. At the onset of localization, the current strains are still continuous and the jump appears only in the strain rates. Classical localization analysis was inspired by the early works of Hadamard (1903) and Hill (1958) and developed, among others, for plasticity by Rice (1975) and Runesson (1991) and for damage by Rizzi et al. (1996). Under certain assumptions, the necessary condition for the formation of a weak discontinuity is the existence of a unit vector n for which the tensor

1 INTRODUCTION Realistic description of the mechanical behavior of quasibrittle materials such as concrete requires constitutive laws with softening. From the physical point of view, softening can be attributed to the propagation and coalescence of defects, e.g. voids and cracks. It is well known that softening may lead to localization of inelastic strain into narrow process zones. For traditional models formulated within the classical framework of continuum mechanics, such zones have an arbitrarily small thickness, and failure can occur at extremely low energy dissipation, which is not realistic. The mathematical model becomes ill-posed and the numerical solutions suffer by pathological sensitivity to the discretization parameter, e.g. to the size of finite elements. It is therefore important to clearly understand the conditions under which localization may occur, and to limit the application of traditional continuum damage mechanics to the range of material states that do not allow for localization of damage into arbitrarily thin bands. Beyond this range, special enhancements acting as localization limiters are necessary. This paper analyzes the localization properties of

Q =n·D·n

• a family of simple isotropic damage models with one damage variable and with different definitions of the damage-driving equivalent strain (Mazars, Rankine, modified Mises, energy release rate), • an isotropic damage model for concrete with two damage variables characterizing separately tensile and compressive damage, • a rotating crack model, which can be interpreted as an anisotropic damage model. 2 INCIPIENT WEAK DISCONTINUITIES From the mathematical point of view, the onset of localization can be characterized as the appearance of

327

(1)

becomes singular. Here, D denotes the tangent material stiffness tensor. Vector n for which the localization condition det Q = 0

(2)

is satisfied represents the normal to a potential discontinuity surface. The normalized nontrivial solution m of the homogeneous equation Q·m=0

(3)

is the so-called polarization vector, which serves as an indicator of the discontinuity mode. The angle between vectors m and n characterizes the failure mode, ranging from tensile splitting with m = n (Fig. 1b) to shear sliding with m perpendicular to

(a)

(b)

n

nm

(c) n m

Figure 1. (a) Body with a localization band, (b) tensile splitting (mode I), (c) shear sliding (mode II).

n (Fig. 1c) and further to compaction with m = −n. From the mathematical point of view, singularity of the localization tensor1 Q indicates the loss of ellipticity. The localization tensor defined in (1) depends on the tangent stiffness tensor D and on the unit normal to the discontinuity surface, n. With certain exceptions (e.g. models with multiple loading conditions or incrementally nonlinear models), the tangent stiffness can be considered as dependent on the current state only and thus known. The vector n, however, is not given in advance. Therefore, localization analysis consists in searching for a unit vector n for which the localization tensor becomes singular. If such a vector does not exist, the strain field must remain continuous. Singularity of the localization tensor for a certain vector n indicates that a strain jump can develop across a surface with normal n. One should bear in mind that this condition for the appearance of a weak discontinuity is only necessary but not sufficient because the localization analysis presented here is purely local, restricted to the level of a material point and its infinitely small neighborhood. Whether the discontinuity surface indeed develops in a finite body depends on the state of the surrounding material and on the boundary conditions. Nevertheless, analysis of the localization tensor is widely used as a powerful indicator of potential discontinuous failure modes. In the artificial but illustrative case of a body under uniform stress, the localization condition (2) can be satisfied at all points simultaneously. The local discontinuities potentially appearing at individual points can then easily merge into global discontinuity planes. For instance, one can imagine solutions with a band (layer) under uniform strain rate, enclosed by two parallel discontinuity planes that separate the band from the remaining part of the body, in which the strain

1 The localization tensor Q is sometimes referred to as the acoustic tensor, because if the tangent stiffness tensor D is taken as the elastic stiffness tensor De , the eigenvalues of the corresponding acoustic tensor Qe = n · De · n divided by the mass density are squares of the speeds of elastic waves propagating in the direction of n. The corresponding eigenvalues are polarization vectors that determine the type of waves (longitudinal, transversal, mixed).

328

rate is also uniform but different from the strain rate inside the band; see Fig. 1a. If the polarization vector m is aligned with the normal vector n, the difference between the strain rates inside and outside the band corresponds to stretching of the band in the normal direction; see Fig. 1b. This discontinuous mode is a precursor to splitting failure and is denoted as mode I. On the other hand, if the polarization vector m is perpendicular to vector n, i.e., parallel with the discontinuity planes, the failure occurs by shear slip and this is referred to as mode II; see Fig. 1c. For general vectors m, failure is of a mixed type, and the angle between m and n characterizes the failure mode. Under compressive loading, compaction failure with a negative scalar product m · n can arise. 3 SIMPLE ISOTROPIC DAMAGE MODELS 3.1 Basic Equations In this section, we consider a family of simple isotropic damage models with one scalar damage variable ω, driven by the equivalent strain. The basic equations consist of the stress-strain law σ = (1 − ω)De : ε

(4)

damage law ω = g(κ)

(5)

and loading-unloading conditions f (ε, κ) ≤ 0,

κ˙ ≥ 0,

f (ε, κ) κ˙ = 0

(6)

in which g is the damage evolution function, f (ε, κ) = εeq (ε) − κ

(7)

is the damage loading function, εeq is a scalar measure of the strain level called the equivalent strain, and κ is an internal variable that corresponds to the maximum level of equivalent strain ever reached in the previous history of the material. The choice of the specific expression for the equivalent strain directly affects the shape of the elastic domain in the strain space and, as will be shown in Section 3.4, also the localization properties of the model. From the rate form of the basic equations it is easy to derive the (elastic-damaged) tangent stiffness tensor Ded = (1 − ω)De − g ′ σ¯ ⊗ η

(8)

Here, (1 − ω)De is the unloading (secant) stiffness, g ′ = dg/dκ is the derivative of the damage function, σ¯ = De : ε is the effective stress, and η = ∂εeq /∂ε is a second-order tensor obtained by differentiation of

the expression for equivalent strain with respect to the strain tensor. 3.2 Localization condition The specific form of the necessary condition for the onset of localization (incipient weak discontinuity) depends on the particular choice of the equivalent strain definition, but its general form can be elaborated for the entire family of simple isotropic damage models (4)–(7). As shown in detail e.g. by Jirásek (2007), it is fully sufficient to restrict attention to the case when the tangent stiffness is the same on both sides of the discontinuity, which leads to the localization condition det Qed = 0

(9)

where Qed = n · Ded · n is the elasto-damage localization tensor. Substituting from (8), we obtain Q ed = (1 − ω)Qe − g ′ (n · σ¯ ) ⊗ (η · n)

(10)

the simplest case of a one-dimensional damage model. All tensors become scalars, the elastic stiffness tensor De is replaced by Young’s modulus E, the equivalent strain εeq is the strain εeq itself (we consider monotonic tensile loading), and tensor η is replaced by the scalar η = dεeq /dε = 1. The unit normal vector n is also replaced by the scalar n = 1, and so there is no difference between the localization tensor and the tangent stiffness. Realizing that the effective stress under uniaxial loading is σ¯ = Eε and substituting into (8) we get the tangent modulus Eed = (1 − ω)E − g ′ Eε = (1 − ω − g ′ ε)E

The localization condition det Qed = 0 is now written as Eed = 0, which means that the loss of ellipticity occurs when the peak of the stress-strain curve is reached. This is of course the result that would be expected intuitively. The value of g ′ corresponding to vanishing tangent stiffness Eed is

where Qe = n · De · n is the elastic acoustic tensor. Due to the special structure of the localization tensor (10), it is easy to show that condition (9) is satisfied if and only if ′

g n·η·

Q −1 e

· σ¯ · n = 1 − ω

′ gcrit =

1−ω ¯ · n) maxn=1 (n · η · Q −1 e ·σ

′ gcrit =

1−ω ε

(14)

This is exactly what we obtain from the general formula (12) by substituting 1 for η and n, E for Qe and Eε for σ¯ .

(11)

¯ · n depends on the elasThe product n · η · Q −1 e ·σ tic constants, on the current state of the material and on the assumed direction of discontinuity plane. For given η, De and σ¯ , it is a fourth-order polynomial of the components of the unit vector n. Variable g ′ (the derivative of the damage function with respect to the equivalent strain) indicates how ‘‘dramatically’’ damage evolves. If g ′ is sufficiently small, the left-hand side of (11) is less than the right-hand side for all unit vectors n and the strain must remain continuous. The minimum value of g ′ for which (11) can hold is

(13)

3.4 Multi-dimensional case Formula (13) has been derived under uniaxial loading but it turns out that the tangent modulus Eed is directly related to the the slope of the stress-strain curve even under multiaxial stress, provided that the loading is proportional. Suppose that the strain evolution is described by

(12)

′ , the localization tensor Q ed is regular If g ′ < gcrit for all possible directions n, and strain discontinuities ′ , the localization tensor Qed are excluded. If g ′ < gcrit is singular for that particular direction n which maximizes n · η · Q −1 ¯ · n, and a weak discontinuity e · σ across a plane perpendicular to that direction can start ′ , there exist infinitely evolving. Finally, if g ′ < gcrit many directions n for which Qed is singular, and the discontinuity can evolve even ‘‘more easily’’.

3.3 One-dimensional case To get more insight into the meaning of the critical value of g ′ , let us reduce the results derived so far to

329

ε = µεref

(15)

where µ is a scalar multiplier that parameterizes the loading process and εref is the reference value of the strain tensor, which specifies the type of loading. Assuming that the expression for equivalent strain is positively homogeneous of degree 1 (which is true for all the equivalent strain definitions presented in Section 3.5), the corresponding evolution of stress under proportional loading with monotonically increasing parameter µ is expressed from (4)–(7) as σ = [1 − g(µκref )]µσ¯ ref

(16)

where κref = εeq (εref ) is the equivalent strain at the reference state and σ¯ ref = De : εref is the effective stress at the reference state. Denoting Dν = De /E = elastic stiffness tensor constructed for a unit value of Young’s

modulus, we can rewrite (16) and its rate form as

terms of the principal components gives the fourthorder polynomial

σ = Eu Dν : ε

(17)

σ˙ = Eed Dν : ε˙

(18)

⎡ ⎤ 3 3 3 ηI n2I σ¯ I n2I ⎥ 1 ⎢ 2 I =1 I =1 ϕ(n1 , n2 , n3 ) = ⎣ ηI σ¯ I nI − ⎦ G 2(1 − ν) I =1

where

(25)

Eu = [1 − g(µκref )]E

(19)

is the unloading modulus and Eed = [1 − g(µκref ) − µκref g ′ (µκref )]E

(20)

is the tangent modulus. Obviously, (20) is a generalized form of (13). At the onset of localization, g ′ is ′ equal to gcrit evaluated from (12) with ω = g(µκref ). The corresponding ratio between the tangent and unloading moduli is 

Eed Eu



=1−

crit

κref maxn=1 ϕ(n)

(21)

where ¯ ref · n ϕ(n) = n · η · Q−1 e (n) · σ

(22)

is an auxiliary function of the unit vector n. The critical stiffness ratio (21) is much easier to interpret than the critical value of g ′ . For a specific version of simple damage model (with a given expression for the equivalent strain) and given value of Poisson’s ratio, the critical stiffness ratio depends only on the type of loading, characterized by the reference effective stress σ¯ ref . Evaluation of εref = D−1 : σ¯ ref , κref = εeq (εref ) and η(εref ) e is straightforward. The only demanding step is the maximization of function ϕ. In the general three-dimensional setting, the inverse of the elastic acoustic tensor is given by Q−1 e =

1 G

  n⊗n δ− 2(1 − ν)

in which ηI and σ¯ I , I = 1, 2, 3, are the principal values of η and σ¯ , and NI , I = 1, 2, 3, are the components of n with respect to the principal coordinate system. This polynomial has to be maximized under the constraint n21 + n22 + n23 = 1. Renaming n21 as N1 , n22 as N2 and n23 as N3 , we convert the objective function f into a quadratic function of arguments N1 , N2 and N3 , and the constraint into a linear one, but additional constraints N1 ≥ 0, N2 ≥ 0 and N3 ≥ 0 must be imposed. The resulting optimization problem has a strictly concave objective function and a convex admissible domain, and so the maximum exists and is unique. It can be obtained by methods of quadratic programming. Two-dimensional localization analysis under plane stress conditions can be based on the general formulae (12) or (21), but all tensors must be interpreted as two-dimensional ones, with indices running from 1 to 2. Furthermore, the elastic acoustic tensor must be derived from the elastic stiffness tensor for plane stress, and its inverse is given by

(23)

where G is the elastic shear modulus and δ is the unit second-order tensor (Kronecker delta). The function ϕ to be maximized with respect to n is thus

Q−1 e

1 = G

  1+ν n⊗n δ− 2

(26)

It is also important to realize that even if the problem is analyzed under plane stress conditions, the evaluation of equivalent strain must take into account the outof-plane normal strain ε3 , which is in general nonzero and can be expressed from the condition of zero outof-plane normal stress as ε3 (ε1 , ε2 ) = −

ν (ε1 + ε2 ) 1−ν

(27)

where ε1 and ε2 are the in-plane principal strains. If εeq (ε1 , ε2 , ε3 ) is the original expression for equivalent strain, in plane-stress calculations it is replaced by ∗ εeq (ε1 , ε2 ) = εeq (ε1 , ε2 , ε3 (ε1 , ε2 ))

(28)

and its derivatives are evaluated as   1 (n · η · n)(n · σ¯ ref · n) n · η · σ¯ ref · n − ϕ(n) = G 2(1 − ν)

η1∗ =

(24) The principal directions of tensors σ¯ ref = De : εref and η = ∂εeq /∂ε are the same, and (24) rewritten in

330

η2∗ =

∗ ∂εeq

∂ε1 ∗ ∂εeq

∂ε2

=

∂εeq ∂εeq ∂ε3 η3 ν + = η1 − ∂ε1 ∂ε3 ∂ε1 1−ν

(29)

=

∂εeq ∂εeq ∂ε3 η3 ν + = η2 − ∂ε2 ∂ε3 ∂ε2 1−ν

(30)

Principal values η1∗ and η2∗ are then substituted into (25) instead of η1 and η2 , and the sums are taken over I = 1, 2. After all the described modifications, we obtain a polynomial  1 ∗ ϕ (n1 , n2 ) = η∗ σ¯ 1 n21 + η2∗ σ¯ 2 n22 (1 − ω)G 1  1+ν ∗ 2 (η1 n1 + η2∗ n22 )(σ¯ 1 n21 + σ¯ 2 n22 ) − 2

(31)

which has to be maximized under the normalizing constraint n21 + n22 = 1. Renaming n21 as N1 and n22 as N2 = 1−N1 , we convert the objective function ϕ ∗ into a quadratic function of one single argument N1 and we automatically satisfy the normalizing constraint, but additional inequality constraints 0 ≤ N1 ≤ 1 must be imposed. The coefficient multiplying the quadratic term N12 is a positive constant times −(η1∗ − η2∗ ) (σ¯ 1 − σ¯ 2 ). Without any loss of generality, we can order the principal stresses such that σ¯ 1 ≥ σ¯ 2 . The equality sign corresponds to the special cases of equibiaxial tension or equibiaxial compression. In all other cases, the function to be maximized is strictly concave, provided that η1 > η2 (which is indeed verified for all the expressions for equivalent strain presented in the next section). The value of N1 at which the first derivative of ϕ ∗ with respect to N1 vanishes is N1∗

η∗ σ¯ 1 + νη2∗ σ¯ 2 − (1 + ν)(η1∗ σ¯ 2 + η2∗ σ¯ 1 )/2 = 1 (1 + ν)(η1∗ − η2∗ )(σ¯ 1 − σ¯ 2 )

(32)

If N1∗ is between 0 and 1, the unit normal to the potential discontinuity curve has coordinates n1,crit =

± N1∗ and n2,crit = ± 1 − N1∗ . Formally, four solutions are obtained, but only two of them represent physically different directions, which are symmetrically placed with respect to the principal axes. If formula (32) gives N1∗ ≥ 1, then the unit normal has components n1,crit = 1 and n2,crit = 0 and the discontinuity is perpendicular to the major principal axis. If formula (32) gives N1∗ ≤ 0, then the unit normal has components n1,crit = 0 and n2,crit = 1 and the discontinuity is perpendicular to the minor principal axis. 3.5 Specific expressions for equivalent strain Localization properties will be scrutinized for models incorporating the following formulae for equivalent strain: 1. Mazars (1984) defined the equivalent strain as  3 εeq = ε  = εI 2 (33) I =1

331

where εI , I = 1, 2, 3, are the principal strains, and the brackets . . . denote the positive part. The corresponding tensor η is obtained as η=

∂εeq ε = ∂ε εeq

(34)

2. For the Rankine-type definition of equivalent strain σ¯ 1 E

εeq =

(35)

we get η=

1 ∂ σ¯ 1 ∂ σ¯ 1 : = (p1 ⊗ p1 ) E ∂ σ¯ ∂ε E   ν 1 : De = δ + p 1 ⊗ p1 1 + ν 1 − 2ν

(36)

where σ¯ 1 is the maximum principal effective stress and p1 is the unit vector in the corresponding principal direction. 3. de Vree, Brekelmans, and van Gils (1995) introduced the modified von Mises expression εeq =

1 (k − 1)I1ε + 2k(1 − 2ν) 2k



12kJ2ε (k − 1)2 2 I + (1 − 2ν)2 1ε (1 + ν)2

(37) in which I1ε = δ : ε

(38)

is the first strain invariant (trace of the strain tensor), J2ε =

1 e:e 2

(39)

is the second deviatoric strain invariant, e = ε − (I1ε /3)δ is the deviatoric part of strain, and k is a model parameter controling the ratio between the uniaxial compressive strength fc and uniaxial tensile strength ft . The corresponding tensor η is given by (k−1)2

6k

I δ + (1+ν)2 e k −1 (1−2ν)2 1ε δ+  η= 2k(1 − 2ν) (k−1)2 2 12kJ2ε 2k (1−2ν)2 I1ε + (1+ν) 2

(40)

4. Formulations based on the energy release rate can be presented in the format considered here if the

equivalent strain is defined as the scaled energy norm, (41)

which leads to η=

1 De : ε = Eεeq εeq



I1ε δ e + 3(1 − 2ν) 1 + ν



critical stiffness ratio

ε : De : ε E

3.6 Localization characteristics To get an overall idea about the influence of the definition of equivalent strain on localization properties, the localization condition has been evaluated for proportional loading under plane stress conditions with different ratios of the in-plane principal stresses σ1 and σ2 . The type of stress is characterized by the ratio σ2 /σ1 or, more conveniently, by the stress angle ζ chosen such that tan ζ = σ2 /σ1 . This can be achieved by setting the principal values of the reference effective stress to σ¯ ref ,1 = E cos ζ and σ¯ ref ,2 = E sin ζ . To get σ1 ≥ σ2 , values of ζ are considered in the range from −135◦ (equibiaxial compression) to 45◦ (equibiaxial tension). Values ζ = −90◦ , −45◦ and 0◦ correspond respectively to uniaxial compresion, shear and uniaxial tension. The principal strains at the reference state are given by

simple Mazars Rankine modified Mises Mises energy

0.5

0

-0.5

-1 -135

(42)

-90

90

εref ,2 = sin ζ − ν cos ζ

(44)

εref ,3 = −ν(cos ζ + sin ζ )

(45)

Depending on the specific version of the model, the equivalent strain at the reference state κref = εeq (εref ) an the tensor η(εref ) are evaluated according to the appropriate formulae (33)–(42). For each value of stress angle ζ , the critical stiffness ratio (21) is then calculated. The direction of the potential discontinuity plane is characterized by the critical angle, defined as the angle between the normal vector n and the direction p1 of maximum (in-plane) principal stress. The mode of discontinuity is characterized by the polarization angle, defined as the angle between the normal n and the polarization vector m. The results are graphically presented in Figs. 2–4 for Poisson’s ratio ν = 0.18. For the modified Mises definition of equivalent strain, the ratio between the compressive and tensile strength is set to k = 10. For comparison, the (non-modified) Mises definition with k = 1 is also considered. Fig. 2 shows the dependence of the critical ratio Eed /Eu on the stress angle ζ . It is interesting to note that the critical stiffness remains nonpositive only for the energy-based equivalent strain, which is the only

332

45

simple Mazars Rankine modified Mises Mises energy

75 60 45 30 15 0 -135

-90

-45

0

45

stress angle [deg]

Figure 3. Simple isotropic damage models, plane stress analysis: dependence of the critical angle on the stress angle ζ .

polarization angle [deg]

(43)

0

Figure 2. Simple isotropic damage models, plane stress analysis: dependence of the critical stiffness ratio Eed /Eu on the stress angle ζ .

180

εref ,1 = cos ζ − ν sin ζ

-45 stress angle [deg]

critical angle [deg]

εeq =



1

simple Mazars Rankine modified Mises Mises energy

135

90

45

0 -135

-90

-45 stress angle [deg]

0

45

Figure 4. Simple isotropic damage models, plane stress analysis: dependence of the polarization angle on the stress angle ζ .

approach that leads to a symmetric tangent stiffness. For the classical Mises definition based on the second deviatoric invariant (i.e., on the part of elastic energy stored in the change of shape), the critical modulus can become slightly positive. The other definitions of equivalent strain lead for certain stress combinations to relatively large positive values of critical modulus. This means that if the damage law is formulated such that the stress-strain curve has a hardening part, localization can occur already before the peak. This is the case for stress states close to shear or uniaxial compression, but for the modified Mises model even for uniaxial tension. On the other hand, for stress states

introduced two damage parameters, ωt and ωc , that are computed from the same equivalent strain (33) using two different damage functions, gt and gc . Function gt is identified from the uniaxial tensile test and gc from the compression test. The damage parameter entering the constitutive equation (4) is ω = ωt under tension and ω = ωc under compression. Under general stress, the value of ω is obtained as a linear combination

(a)

(b)

Figure 5. Results of finite element simulations on skewed meshes showing numerically computed localized strain for simple damage model with (a) Mazars and (b) Rankine definition of equivalent strain.

close to biaxial tension or biaxial compression the critical modulus is negative and localization can occur only if the softening is sufficiently ‘‘steep’’. Fig. 3 shows the dependence of the critical angle on the stress angle ζ . Here it is perhaps surprizing that, under uniaxial tension, all formulations (with the exception of the modified Mises model) predict a discontinuity plane that is not perpendicular to the loading direction. The deviation is considerable not only for the energy-based and classical Mises formulations, which are not really appropriate for quasibrittle materials, but also for the Rankine and Mazars formulations. This is confirmed by the results of a finite element simulation of the uniaxial tensile test in Fig. 5. Skewed meshes containing layers of elements of different inclinations are used, in order to allow for the formation of a localized damage band in an ‘‘arbitrary’’ direction. It appears that, on such meshes, no imperfection is needed to trigger the bifurcation from a uniform state. The numerical solution tends to spontaneously localize into one layer of elements with the most favorable orientation. This orientation is very close to the theoretically predicted one (indicated in the figure by the inclined line). Fig. 4 shows the dependence of the polarization angle on the stress angle ζ . Again, under uniaxial tension all formulations with the exception of the modified Mises model predict a nonzero polarization angle, i.e., a mixed-mode discontinuity rather than pure tensile splitting.

(46)

ω = αt ωt + αc ωc

where the coefficients αt and αc take into account the nature of the stress state. In the original model, Mazars (1984) used αt = α and αc = 1 − α where α=

3 εtI εI 2 εeq I =1

(47)

is a dimensionless factor that depends on the principal values εtI , I = 1, 2, 3, of the part of strain εt = D−1 e : De : ε

(48)

that corresponds to the positive part of stress. In more recent implementations of Mazars model, an additional improvement is achieved by setting αt = α β ,

αc = (1 − α)β

(49)

where the exponent β = 1.06 slows down the evolution of damage under shear loading (i.e., when principal stresses do not have the same sign). Note that if all principal stresses are nonnegative we have αt = 1, αc = 0, and ω = ωt , and if all principal stresses are nonpositive we have αt = 0, αc = 1, and ω = ωc . These are the ‘‘pure tensile’’ and ‘‘pure compressive’’ stress states. For intermediate stress states, the value of ω is between ωt and ωc , depending on the relative magnitudes of tensile and compressive stresses.

4 MAZARS MODEL

0 -5 stress [MPa]

2.5 2 1.5 1 0.5 0

4.1 Basic equations

(b)

(a) 3 stress [MPa]

For concrete and other materials with a high ratio of compressive to tensile strength, the simple models presented in the preceding section can provide realistic results only if the failure has a predominantly tensile character. If the model should be used in rather general loading situations, certain modifications must be introduced.

333

-15 -20 -25

0

0.1

0.2

0.3

strain [1/1000]

A popular damage model specifically designed for concrete was proposed by Mazars (1984, 1986). He

-10

0.4

-30

-3 -2.5 -2 -1.5 -1 -0.5

0

strain [1/1000]

Figure 6. Stress-strain curves for Mazars damage model constructed for (a) uniaxial tension, (b) uniaxial compression.

Functions characterizing the evolution of damage were proposed by Mazars (1984) in the form ε0 − At e−Bt (κ−ε0 ) κ ε0 gc (κ) = 1 − (1 − Ac ) − Ac e−Bc (κ−ε0 ) κ

gt (κ) = 1 − (1 − At )

(50)

Switch factors HIε or HIσ = H (σI ) turn on and off certain terms depending on the signs of principal strains or stresses. Substituting (55)–(56) into (54) and using (47) and the relation ε˙ eq = η : ε˙ leads to

(51)

where ε0 is the equivalent strain at the elastic limit and At , Bt , Ac , and Bc are material parameters related to the shape of the uniaxial stress-strain diagrams. Formulae (50)–(51) are valid above the initial damage threshold, i.e., for κ > ε0 , while for κ ≤ ε0 both gc and gt vanish and the response is purely elastic. To ensure a continuous variation of slope of the compressive stress-strain curve, it is necessary to satisfy the condition Ac Bc ε0 = Ac − 1, which reduces the number of independent parameters to four. A sample set of parameters used by Saouridis (1988) is ε0 = 10−4 , At = 0.81, Bt = 10450, Ac = 1.34, and Bc = 2537.

ξ=

= kα α˙ + (αt gt′ + αc gc′ )η : ε˙

ω˙ = k ′ ξ : ε˙ + g ′ η : ε˙

ωt − ωc ωt − ωc

(52)

for β > 1 for β = 1

  3 ε˙ tI εI + εtI ε˙I 2εtI εI ˙εeq α˙ = − 2 3 εeq εeq I =1

kα , εeq

g ′ = (αt gt′ + αc gc′ − 2αk ′ )

(53)

(54)

Ded = (1 − ω)De − σ¯ ⊗ (k ′ ξ + g ′ η)

(61)

ε˙I = HIε PI : ε˙

(55)

where PI = pI ⊗pI and HIε = H (εI ), with H denoting the Heaviside function, and also that ε˙ tI = ξ I : ε˙

(62)

is again a rank-1 correction of the elastic stiffness tensor, but the term g ′ η from (8) is replaced by k ′ ξ + g ′ η, and g ′ is not the derivative of a given damage function g but depends on the derivatives of functions gt and gc and on the current state according to (61). The corresponding localization tensor Q ed = n · Ded · n is singular if and only if

Due to material isotropy, the strain, stress and effective stress tensors as well as the tensor εt have the same principal directions pI , I = 1, 2, 3. Using the spectral representation, it is possible to show that

(56)

n · (k ′ ξ + g ′ η) · Q−1 ¯ ·n=1−ω e ·σ

(63)

For a material with no damage, the left-hand side vanishes (for all directions n) and the right-hand side is equal to 1. Due to damage, the left-hand side increases (at least for some directions n) and the right-hand side decreases. Therefore, the onset of localization can be characterized by the condition min L(n) = 0

n=1

(64)

where

where ν ξI = 1 − 2ν

(60)

where k′ =

where η is given by (34), and kα =

(59)

The resulting tangent stiffness tensor

ω˙ = α˙ t ωt + αt ω˙ t + α˙ c ωc + αc ω˙ c

β(1 − α)β−1

3 1 (εI ξ I + HIε εtI P I ) εeq I =1

Finally, using (58), expression (52) for the damage rate can be rewritten as

Differentiating (46)–(49) with respect to time we obtain

βα β−1

(58)

where

4.2 Localization condition



ξ − 2αη : ε˙ εeq

α˙ =



HIσ

+ HIσ P I −

3 ν σ − H 1 + ν J =1 J

ν 1+ν

3

HJσ P J



¯ ·n L(n) = 1 − ω − n · (k ′ ξ + g ′ η) · Q −1 e ·σ

I

(57)

J =1

334

(65)

is the localization indicator. Of course, instead of L one could use the determinant of the localization tensor, but the expression for L is simpler and, in some cases, equation (64) can be solved analytically.

4.3 Localization characteristics Under monotonic proportional loading, scalars α, αt , αc and tensors ξ and η remain fixed, while σ¯ grows proportionally to the load multiplier µ and scalars ω, g ′ and k ′ evolve in a nonlinear fashion, depending on the selected form of damage functions gt and gc . The minimum value of the localization indicator becomes a function of µ, and the critical value of µ for which the localization condition (64) is satisfied can be found numerically. This has been done for plane stress conditions, with the stress type characterized by the stress angle ζ related to the ratio of the in-plane principal stresses, as explained in Section 3.6. The dependence of the critical stiffness ratio, critical angle and polarization angle on the stress angle ζ is plotted in Figs. 7–8. The results are only slightly affected by parameter β and are similar to the results obtained for the simple model with Mazars definition of equivalent strain. A peculiar feature is that the dependence of the localization properties on the stress angle exhibits a discontinuity at ζ = −90◦ , i.e., at uniaxial compression. This is caused by the jump change of the switch factor H2σ , which depends on the sign of principal stress σ2 . 1

5 ROTATING CRACK MODEL Many different anisotropic damage formulations have been proposed in the literature, and it is not possible to cover all of them or to derive generally valid statements. For illustration, we will look at a simple rotating crack formulation, which can be interpreted as a special type of anisotropic damage. Rotating or fixed crack models are usually presented in the engineering notation. The physical motivation and detailed derivation can be found e.g. in Jirásek and Zimmermann (1998). For the purpose of localization analysis it is useful to rewrite the basic equations in the tensorial notation. Based on the additive split of strain into the elastic part and the inelastic part due to cracking, the stress-strain law can be written as

initial, beta=1.06 beta=1.00 critical, beta=1.06 beta=1.00

0.5 stiffness ratio

Fig. 7 shows not only the critical stiffness ratio at the onset of localization but also the initial stiffness ratio at the onset of damage propagation. Under uniaxial or biaxial compression, the stress-strain curve is smooth and the tangent stiffness at the onset of damage is equal to the elastic stiffness, which gives initial stiffness ratio equal to 1, while for uniaxial and biaxial tension softening starts right at the elastic limit and the initial stiffness ratio is negative (and in a certain range is below the critical stiffness ratio). For stress angles ζ between −100◦ and 19◦ , the critical modulus is positive and localization can occur already on the hardening branch of the stress-strain diagram. The largest critical stiffness ratio Eed /Eu = 0.468 is found for stress angle ζ = −86.5◦ , i.e., for principal stress ratio σ2 : σ1 = −1 : 0.061.

0

-0.5

-1 -135

-90

-45

0

45

σ = De : (ε − εc )

stress angle [deg]

Figure 7. Mazars model, plane stress analysis: dependence of the initial and critical stiffness ratio Eed /Eu on the stress angle ζ .

critical and polarization angle [deg]

180

135

For the rotating crack model, the inelastic (cracking) strain εc is fully described by a single variable c , which is the only nonzero component of crackεnn ing strain in the local coordinate system aligned with the crack (it is the normal strain in the direction perpendicular to the crack). The cracking strain tensor is expressed as

polarization angle, beta=1.06 beta=1.00 critical angle, beta=1.06 beta=1.00

90

c c εc = εnn p1 ⊗ p1 = εnn P1

45

0 -135

-90

-45 stress angle [deg]

0

45

Figure 8. Mazars model, plane stress analysis: dependence of the critical angle and polarization angle on the stress angle ζ .

(66)

(67)

where p1 is the unit vector in the direction of maximum principal strain (assumed to be normal to the crack), and P 1 = p1 ⊗ p1 . The cracking law relates the cracking strain ε c to the normal component of the traction transmitted by the (cohesive) crack,

335

σnn = p1 · σ · p1 = P 1 : σ

(68)

For our purpose, it is sufficient to consider the cracking law in the rate form c σ˙ nn = Dc ε˙ nn

(69)

where Dc is the tangent crack stiffness. By eliminating the cracking strain from the rate form of (66)–(67), it is possible to derive the tangent stiffness tensor c ˜ N− Dec = De − εnn

D e : P1 ⊗ P 1 : D e e Dc + D1111

(70)

where N˜ is a certain fourth-order tensor. The specific expression for N˜ is not important because in our localization analysis we will focus attention on the state c just after crack initiation, when the cracking strain εnn c ˜ is zero and the term εnn N can be deleted from (70). The stiffness tensor then becomes a rank-1 modification of the elastic stiffness tensor, and the conditions for a vanishing determinant of the localization tensor can be established using the same techniques as in the localization analysis of isotropic damage models. The resulting necessary condition for an incipient weak discontinuity is e (n · De : P 1 ) · Q−1 e · (n · De : P 1 ) = Dc + D1111

(71)

where as usual, n is a unit vector normal to the discontinuity and Qe = n · De · n is the elastic acoustic tensor. The critical value of the softening modulus is e Dc,crit = max ϕ(n) − D1111 n=1

(72)

where ϕ(n) = (n · De : P 1 ) · Q −1 e · (n · De : P 1 )

(73)

e and D1111 = P 1 : De : P 1 is an elastic constant. For isotropic elasticity in a three-dimensional setting, we obtain e = D1111

E(1 − ν) (1 + ν)(1 − 2ν)

Eν ϕ(n) = 1 − ν2



(74)

ν2 + 2(p1 · n)2 + (p1 · n)4 1 − 2ν



The scalar product p1 · n represents the cosine of the angle between the discontinuity plane and the crack, and it can vary between 0 and 1. It is easy to show that the maximum of ϕ(n) is attained for n = p1 , e and the maximum turns out to be equal to D1111 , which means that the critical modulus is Dc,crit = 0.

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The actual softening modulus Dc used by the rotating crack model is always negative and thus smaller than the critical one. The weak discontinuity can appear right at the onset of cracking and is perfectly aligned with the crack, independently of the Poisson ratio and of the stress state. It is also possible to show that the discontinuity type corresponds to pure mode I, which is consistent with the idea of an opening crack. This confirms that the model is appropriate for quasibrittle materials such as concrete. Of course, the crack is initiated only if the maximum principal stress reaches the tensile strength, and so compressive failure cannot be captured. ACKNOWLEDGMENT Financial support of the Czech Science Foundation ˇ under project GACR 106/08/1508 is gratefully acknowledged. REFERENCES de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Computers and Structures 55, 581–588. Hadamard, J. (1903). Leçons sur la propagation des ondes. Paris: Librairie Scientifique A. Hermann et Fils. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids 6, 236–249. Jirásek, M. (2007). Mathematical analysis of strain localization. Revue Européenne de Génie Civil 11, 977–991. Jirásek, M. and T. Zimmermann (1998). Analysis of rotating crack model. Journal of Engineering Mechanics, ASCE 124, 842–851. Mazars, J. (1984). Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure. Thèse de Doctorat d’Etat, Université Paris VI., France. Mazars, J. (1986). A description of micro and macroscale damage of concrete structures. International Journal of Fracture 25, 729–737. Ottosen, N. and K. Runesson (1991). Properties of discontinuous bifurcation solutions in elastoplasticity. International Journal of Solids and Structures 27, 401–421. Rizzi, E., I. Carol, and K.Willam (1996). Localization analysis of elastic degradation with application to scalar damage. Journal of Engineering Mechanics, ASCE 121, 541–554. Rudnicky, J.W. and J.R. Rice (1975). Conditions for the localization of deformation in pressuresensitive dilatant materials. Journal of the Mechanics and Physics of Solids 23, 371–394. Saouridis, C. (1988). Identification et numérisation objectives des comportements adoucissants: Une approche multiéchelle de l’endommagement du béton. Ph. D. thesis, Université Paris VI.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical multiscale solution strategy for fracturing of concrete Łukasz Kaczmarczyk, Chris J. Pearce & Nenad Bi´cani´c Department of Civil Engineering, University of Glasgow, Glasgow, UK

ABSTRACT: This paper presents a numerical multiscale modelling strategy for simulating fracturing of concrete where the fine-scale heterogeneities are fully resolved. The fine-scale is modelled using a hybrid-Trefftz stress formulation for modelling propagating cohesive cracks. The very large system of algebraic equations that emerges from detailed resolution of the fine-scale structure requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. This paper presents a two-grid strategy for construction of the preconditioner that utilizes scale transition techniques derived for computational homogenization and represents an adaptation and extension of the work of Miehe and Bayreuther (IJNME, 2007). For the coarse scale, this paper investigates both classical C 0 -continuous displacement-based finite elements as well as C 1 -continuous elements. The preconditioned GMRES Krylov iterative solver with dynamic convergence tolerance is integrated with a constrained Newton method with local arc-length control and line searches. The convergence properties and performance of the parallel implementation of the proposed solution strategy is illustrated on a numerical examples. 1 INTRODUCTION Multiscale analysis aims to predict the macroscopic constitutive behaviour of materials with heterogeneous microstructures. Such techniques not only determine macroscopic ‘‘effective’’ continuum material properties but also provide understanding of the relationship between microstructural phenomena and the overall macroscopic behaviour. Computational approaches (Moulinec and Suquet 1998, Miehe and Koch 2002, Kouznetsoca et al. 2002) typically utilize nested multilevel finite element analyses with discretisation at both the microscale and macroscale—so-called computational homogenization. A fundamental restriction of these techniques is a clear separation of scales, such that the characteristic length of a representative volume element (RVE) is sufficiently small compared to the macrostructural characteristic length. Clear separation of scales permits the assumption of uniformity of the macroscopic strain field across the microstructure, as adopted in first-order homogenization schemes. In cases where the existence of an RVE necessitates a less well defined separation of scales, the assumption of uniform strains may be inappropriate in some situations, e.g. strain localization, boundary layers, etc. Second-order schemes have been proposed to overcome such short-comings (Kouznetsova et al. 2002, Feyel 2003, Kaczmarczyk et al. 2008), whereby the macroscopic material behaviour is described using a higher-order continuum theory (e.g. strain gradient, Cosserat, micropolar). In such cases the material response at a macroscopic point also depends on the response in the neighbourhood of

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that point, thereby introducing a material length scale into the macroscopic constitutive model, and enables geometrical size effects to be captured. Fracturing leads to an evolving microstructure which makes it impossible to define a priori the size of the RVE. Once strains start to localise or fractures coalesce, material instability occurs, scale separation is no longer possible, the RVE becomes undefined and it is not possible to use scale-transition homogenization techniques. Various strategies for such situations have been presented in the literature, including Belytschko et al. (Belytschko et al. 2008), Gitman (Gitman et al. 2007, Gitman et al. 2008) and Markovic and Ibrahimbegovic (Markovic and Ibrahimbegovic 2006). Miehe and Bayreuther (Miehe and Bayreuther 2007) presented unifying computational procedures for the analysis of heterogeneous materials in the extremes of scale separation and in particular a multi-grid solution strategy (referred to as numerical multiscale) for situations without scale separation. This approach was inspired by the formulations, and in particular the scale transition techniques, of computational homogenization. Here Miehe and Bayreuther’s numerical multiscale solution strategy (Miehe and Bayreuther 2007) is extended for the case of fracturing in concrete at the level of observation below the macroscale (so-called meso-level (1–10 cm)), identifying individual aggregates embedded in a matrix, with a weak interfacial transition zone. An efficient two-grid (fine and coarse mesh) preconditioner for Krylov iterative solvers is constructed and derivation of the homogenizationbased scale transition operator is fully described.

stiffness matrix is very small and computationally efficient to solve. The bulk response of the material is described by stress and displacement fields which are approximated by means of a HTS approximation. The stress field within an element is approximated directly as:

The analysis of fracturing heterogeneous materials requires a robust solution strategy for tracing the unstable equilibrium path. Thus the preconditioned iterative solver is embedded in a constrained Newton method with local arc-length control and line searches. In this paper, the cohesive crack methodology is utilized together with Hybrid-Treffz stress elements (Kaczmarczyk and Pearce 2009) for the fine mesh that fully resolves the heterogeneous mesostructure. Next, the overall solution strategy for fracturing heterogeneous materials is discussed, before describing in detail the proposed two-grid preconditioner and in particular the construction of the homogenization-based scale transition operators. Finally performance of the proposed model is demonstrated with a numerical example and investigates the use of both C 0 -continuous and C 1 -continuous elements for the coarse mesh.

σ = Sv v

where Sv is a matrix of field approximation functions and v is the unknown vector of generalised stress degrees of freedom. In Equation (1), the stress approximation field is chosen so as to automatically satisfy equilibrium: LT Sv = 0.

(2)

An additional and independent approximation of displacements uŴ on the traction boundary Ŵσ of the element is introduced:

2 HYBRID-TREFFTZ STRESS ELEMENTS The analysis of fracturing heterogeneous materials necessitates full resolution of the fine-scale structure, that evolves during mechanical loading. This requires a robust model for cohesive cracking, where multiple cracks, crack branching and crack coalescence are the norm and where different constitutive models are required for the various phases of matrix, inclusions and interface. A hybrid-Trefftz stress (HTS) formulation is adopted, the detailed description of which can be found in (Kaczmarczyk and Pearce 2009) for the extension to heterogeneous quasi-brittle materials. The current formulation is restricted to 2D but is fully extendable to 3D. Displacement discontinuities are restricted to element interfaces and the material response within each element is assumed to be hyperelastic. Such an approach is deemed realistic (Tijssens 2001), and can be justified by the observation of fracturing phenomena, for which localization occurs and material unloads in the vicinity of the crack. Unstructured fine-scale meshes are adopted in order to reduce the influence of the mesh on fracture propagation. Furthermore, for heterogeneous materials, the finite element mesh size is significantly constrained by the size and spacing of the inclusions. It has been shown that for the type of problem considered here, the results are mesh objective (Kaczmarczyk and Pearce 2009). The HTS finite element formulation is characterised by the approximation of stresses within the domain of the element and by the fact that the stiffness can be expressed via a boundary, rather than domain, integral. Thus, compared to their classical FEM counterpart, HTS elements exhibit faster convergence of the stress fields. Furthermore, the displacements are approximated on element boundaries and the displacement basis is defined independently on each element interface. Consequently, the overall bandwidth of the

(1)

uŴ = UŴ q.

(3)

The primary unknowns are the stress degrees of freedom v within the element and the displacement degrees of freedom on the element boundary q. However, the stress degrees of freedom can be eliminated conveniently from the global system of equations by application of static condensation, leaving only the displacement degrees of freedom to be determined. Following solution of the displacement degrees of freedom, the stress degrees of freedom can be recovered on an element by element basis, a process which lends itself to parallelization. The cohesive crack model on the element interfaces adopted here assumes that all inelastic deformation in the vicinity of a crack is concentrated onto a line and expressed in terms of tractions and displacements (Hillerborg et al. 1976). In order not to over-complicate the formulation, a straightforward material model for cohesive failure in two-dimensions is adopted. The model response depends on three material parameters: tensile strength ft , fracture energy Gf and α which assigns different weights to the shear and normal opening displacements. As such, a discontinuity (i.e. displacement jump) κ is introduced as a history variable when the effective traction ˜t on an element face exceeds the tensile strength of the material; the subsequent traction transferred across the interface is dependent on the magnitude of the displacement jump. A bilinear softening law is adopted here. Full details of the HTS element formulation for modelling cohesive cracking in heterogeneous materials are described in (Kaczmarczyk and Pearce 2009). The current paper focuses on the solution strategy for this class of problem.

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3.2

3 SOLUTION STRATEGY The analysis of fracturing heterogeneous materials represents a significant computational challenge. First, tracing the unstable equilibrium path, including overcoming critical stability points, requires a robust solution scheme. Second, the full resolution of the heterogeneous fine-scale structure results in a large system of algebraic equations that needs to be solved efficiently. This section will initially discuss Newton’s method with local arc-length control and line searches for tracing the nonlinear response before going on to present a numerical multi-scale preconditioner for Krylov solvers that is specifically designed for this application.

δ T = [g1 , g2 , . . . , gN ]

3.1 Constrained Newton method The equilibrium equation can be expressed as r(q, λ) = f ext (λ) − f int (q) = 0

(4)

where q are the displacement degrees of freedom, λ is the load parameter, f ext and f int are the external and internal forces, and r is the vector out-of-balance forces. In order to trace the dissipative load-displacement path, an arc-length scheme is adopted. The corresponding constraint equation is written as: r λ = b(q) − l = 0

with

(6)

bT δq = 0

(δ c )T = [g1c , g2c , . . . , gNc ].

(7)

where q is the displacement increment and δq is the iterative change in displacement.

339

(9)

The constraint equation is now written as a function of displacement jumps:

(5)

where K = −∂q r is the tangent stiffness matrix and p = ∂λ f ext is the vector of externally applied reference loads. Since it is assumed that function b(q) remains unchanged during each load step (Alfano and Crisfield 2003), b in Equation (5) can be expressed as b = bT q

(8)

where gi is the displacement jump in the normal direction to the face, at integration point i. Only active cracks are included, i.e. gi > 0 in the previous increment. A critical value, gic , is associated with each component of δ, and is chosen to be the displacement jump associated with the change in slope of the adopted interface bilinear softening curve. The magnitude of this value will vary for different phases of the composite material.

where l represents the increment length for the current load step, the magnitude of which is computed via an automatic incremental procedure described later. The equilibrium and constraint equations together form a nonlinear system of equations that is solved using Newton’s method in combination with an efficient iterative solver for the associated linear system of equations. Linearization of (4) results in: Kδq − pδλ = r

Local arc-length control

The concept of local arc-length control with line searches was presented in (Alfano and Crisfield 2003) for delamination analysis and is adapted here to the analysis of fracturing heterogeneous materials. The basic idea of local arc-length control is to restrict attention to those DOF’s associated with the active process zone and thereby control the dissipative evolution of the structure. The arc-length vector b in (7) is a function of the displacement jumps along element interfaces. The magnitude of the displacement jumps in the direction normal to the interface at the integration points are collected together in δ as:

bT q =

N  i=1

vi sgn(δis−1 )(δi − δis−1 ) = l

(10)

with 1 δc

vi = N i

1 i=1 δic

.

(11)

Although local arc-length control has proven to be generally effective, the procedure is still found to be unstable in some specific situations, typically related to overcoming sharp limit points and non-smooth nonlinearities. Line searches have been shown to provide an effective remedy (Alfano and Crisfield 2003) and is adopted here. 3.3 Global system of equations The linearized problem (6) and Eq. (5) can be expressed together in matrix form as: 

K bT

−p 0



q λ



=



r rλ



(12)

where δ has been omitted from δq and δλ for simplicity. In condensed form, the above equation can be rewritten as:

qrc = (K c )−1 Rr

and

qλc = (K c )−1 Rp

(17)

Solution to this system of equations is found using a Generalized Minimum Residual Method (GMRES) solver. Although the GMRES method is well founded theoretically, it can suffer from slow convergence for problems of the type discussed in this paper. In general, preconditioning can be applied to the system described by Eq. (13) (Balay et al. 2009, Saad 1996) as follows:

where the superscript ‘‘c’’ has been added to indicate the coarse mesh. The construction of the stiffness matrix Kc on the course mesh will be discussed in the next section. The grid transfer operator R, and its counterpart P (introduced shortly), are at the heart of the proposed solution strategy since they account for the fine-scale heterogeneities, including cracks. Detailed derivation of these are also presented in the next Section. Typically, since the size of the coarse mesh system of equations is significantly smaller than that of the fine mesh, the coarse mesh solution can be computed using a direct solver. The solutions computed on the coarse mesh (Eq. (17)) are prolongated back onto the fine mesh to give:

ˆ R−1 )(MR q) (ML−1 KM ˆ = ML−1 fˆ

qrf = Pqrc = P(K c )−1 Rr

ˆ qˆ = fˆ K

(13)

4 PRECONDITIONER

(14)

where ML and MR indicate ‘‘left’’ and ‘‘right’’ preconditioning matrices respectively. In this paper, attention is restricted to left preconditioning, i.e. MR = I & ML = M. An important feature of this solution strategy is that the convergence tolerance is dynamic and ensures that the GMRES algorithm does not converge to an unnecessarily small tolerance while the Newton iterations are still far from equilibrium, thereby representing an important speed-up with respect to direct solvers.

A multi-grid preconditioning strategy utilises coarser meshes for fast smoothing of the long-wave modes of the error. Typically, a hierarchy of coarse meshes would be used, although here we restrict ourselves to just two meshes—fine and coarse. To construct the two-grid preconditioner, it is convenient to consider the variables in Eq. (12) separately. The first equation in (12) can be written as: (15)

where the superscript ‘‘f ’’ has been added to indicate the fine mesh. Rearranging this equation, we obtain an additive decomposition of qf as: qf

f −1

f −1

= (K ) r + λ(K ) p

= qrf + λqλf

and qλf = Pqλc = P(K c )−1 Rp

(19)

Thus, the improved solution for qf can be determined from Eq. (16) once λ has been calculated. Substituting (16) into the second equation of (12) gives:   bT qrf + λqλf = r λ

(20)

which can be rearranged to give λ:

4.1 Two-grid preconditioner

r = K f qf − λp

(18)

(16)

An approximation to qrf and qλf can be obtained by first restricting r and p onto the coarse mesh using a restriction operator R as:

340

  λ = r λ − bT qrf /bT qλf

(21)

The algorithm also includes a relaxation process to remove high frequency errors, which exhibit local variations in the solution, with a straightforward relaxation method such as Gauss-Seidel (Xu 1997). In subsequent sections we focus on the key issue of deriving the coarse-to-fine prolongation operator P, fine-to-coarse restriction operator R and coarse stiffness matrix K c .

5 GRID TRANSFER OPERATORS The two-grid preconditioner utilises a coarse mesh in addition to the fine mesh. A patch of fine mesh elements, which fully resolves the fine-scale heterogeneities, is associated with a single coarse mesh element, see Figure 1. The construction of the homogenization-based coarse mesh stiffness matrix K c is described later. Attention is currently focused on derivation of the prolongation operator P based on a decomposition of the fine-scale displacements

at hand, the fine-to-coarse mesh restriction operator R, is defined as

Nodal displacements & element shape functions qf = Pqc , uf = uc

R ≡ PT .

qf

Geometric centre for Taylor expansion

=

B.C. s & Solve BVP uf + rf = uc

Pave qc ,

5.1 Long-wave prolongation operator P

Elements of different shades depict heterogeneities

The long-wave coarse-to-fine mesh prolongation operator P relates the displacement field in a coarse mesh element to the displacement field in the corresponding patch of fine mesh elements and is computed from interpolation of the displacement field on the coarse mesh. Here two different discretization methods are used, i.e. displacement type finite elements and HTS elements for the coarse and fine mesh problems, respectively. For such problems, the prolongation operator relating coarse mesh nodal degrees of freedom with those on the faces of the corresponding fine mesh patch is given by

qf = Phom qc uf = x

X

   P = A N c xif

Figure 1. Long-wave and short-wave prolongation operators. Top: long-wave operator P. Middle: average component of the short-wave operator  Pave . Bottom: homogeneous component of short-wave operator  Phom .

 + qf qf   long-wave short-wave

(22)

The long-wave contribution of the fine-scale displacements are associated with the homogeneous contribution of the coarse mesh approximation and determined from interpolation of the coarse mesh displacements. The short-wave displacements represent fluctuations due to the fine-scale heterogeneous structure of the patch, e.g. inclusions and deformation induced cracks. Long- and short-wave contributions to the fine mesh approximation of displacements are obtained by prolongation of the coarse mesh displacements qc qf = Pqc

and  qf =  Pqc .

5.2 Short-wave prolongation operator  P

The short-wave component of the prolongation operator,  P, reflects the influence of the heterogeneous nature of the fine mesh patch of elements corresponding to a single coarse-scale element. Following (Miehe and Bayreuther 2007), this contribution can be formulated using the scale transition techniques for computational homogenization. A truncated Taylor series expansion of the displacement vector about the geometric centre of the coarse mesh element, using Voigt notation, yields:

(23)

As a consequence, the prolongation operator P in (??) is decomposed into long-wave P and short-wave fluctuation  P components as: P P = P +

(26)

where matrix function A(. . .) depends on the fine mesh face approximation functions adopted and relates the interpolated coarse mesh nodal displacements with the fine mesh face displacements of the fine mesh patch. In a classical geometric multi-grid strategy, this long-wave prolongation operator associated with the homogeneous part of the deformation represents the only component of the total prolongation operator, i.e. P = P. However, following the approach in (Miehe and Bayreuther 2007), the long-wave prolongation operator is augmented by a short-wave component reflecting the fine-scale heterogeneities.

into long-wave and short-wave contributions by the additive split: qf =

(25)

u(x) = u0 + Xε + w(x)

(27)

(24)

This decomposition is also described in Figure 1. With the coarse-to-fine mesh prolongation operator P

where u0 is the displacement vector at the geometric centre of the coarse mesh element, ε is the strain state at element centre, w(x) is the displacement fluctuation

341

over the fine mesh patch and X is a matrix of relative position coordinates:   1 2x 0 y (28) X= 0 2y x 2

boundary conditions represent a minimal condition that fulfill the Hill-Mandel theorem, e.g. see (Kaczmarczyk et al. 2008, Kouznetsova et al 2002, Miehe and Bayreuther 2007). Given the HTS discretization of displacements (3) and the expression for the displacement fluctuation (29), the constraint equation (35) is expressed as:

Rearranging (27) yields an expression for the displacement fluctuation field within the fine mesh patch:

Cqf − DBc qc = 0

(29)

w(x) = u(x) − Xεε − u0

where C is a constraint matrix given by

Relating the displacement fluctuation to the coarse element degrees of freedom qc yields, after HTS discretization (3), the short-wave component of the prolongation operator is introduced as: w(x) = UŴ Pqc

C=

HUŴT UŴ dŴ,

(37)

∂c

(30) and D as

where UŴ is the matrix of displacement approximation functions on the element faces in the fine mesh patch. Similarly, Pave qc + u0 u(x) = UŴ

(36)

and

Phom qc Xεε = UŴ

D=

(31)

(32)

 Phom P = Pave − 

(33)

 Phom = A(Xi Bc )

(34)

where the short-wave component of the prolongation operator is:

The homogeneous component of this prolongation operator,  Phom , associated with coarse mesh deformations, simply takes the form:

where Xi is evaluated at the fine mesh nodes and the approximation prolongation function A is the same function as used in (26). Bc is the coarse mesh straindisplacement matrix. The fluctuation component of the short-wave prolongation operator,  Pave , is determined by the solution of a boundary value problem (BVP), whereby deformation of the fine mesh patch is enforced, via appropriate boundary conditions, according to a given coarse element average strain. The boundary conditions for the fine mesh patch must satisfy:

w ⊗ n dŴ = 0 (35)

(38)

K f qf + C T λ = 0

(39)

Cqf = DBc qc

(40)

∂c

where n is the normal vector to the coarse element boundary domain ∂c . It can be shown, that such

HUŴT XdŴ

∂c

The terms of matrix H reflects the specific nature of the boundary conditions used, i.e. linear deformation or uniform traction. The influence of the type of boundary condition will be investigated with numerical examples in the next section. Each term in matrix H can be interpreted as an admissible distribution of nodal traction forces on the boundary of the elements patch. Details on the construction of H can be found in (Kaczmarczyk et al. 2008). The solution of the discretized BVP for the finescale patch of elements can be expressed as a constrained quadratic problem. A common method to solve such a problem is to introduce Lagrange multipliers. However, such an approach increases the number of unknowns and alters the character of the system matrix (to an indefinite saddle point problem). Moreover, the numerical solution of Euler’s conditions for the stationary point of the Lagrangian is rather inefficient and therefore not suitable for solving computationally complex multiscale problems where the constrained quadratic problem has to be solved for every integration point. Alternative numerical techniques include the penalty method or Uzawa method (Zienkiewicz and Taylor). However, in order to express the short wave prolongation operator in closed-form, the approach taken in (Kaczmarczyk et al. 2008) is briefly presented, where the work of Ainsworth (Ainsworth 2001) is applied. The fine-scale BVP can be expressed as:

Substitution of (30) and (31) into (29) yields Pqc = UŴ ( UŴ Pave −  Phom )qc

342

element volume. Inserting (47) into the above, the coarse element stress-strain relationship is obtained:

where λ are the Lagrange multipliers. Following (Ainsworth 2001), the following matrices are defined: Q = I − CT (CCT )−1 C,

F = CT (CCT )−1

(41)

σ = Cεε

Matrix Q projects the fine mesh stiffness matrix K f onto a sub-space that is orthogonal to that of the constraints expressed by CT C. Solution for qf is then given as: f

˘ q =K

−1 ˘

(43)

where ˘ = CT C + QT K f Q K

(44)

and f˘ , expressed in terms of the coarse element displacement degrees of freedom, is: f˘ = g˘ Bc qc ,

  g˘ = CT D − QT K f FD

(45)

˘ −1 g˘ Bc with  Pave = K

(46)

Thus, Equations (42) & (45) provide an expression for  Pave as: Pave qc qf = 

The coarse scale element stiffness is computed from averaging of the evolving fine-scale stiffness and represents a coupled-volume approach (Gitman et al. 2007, Gitman et al. 2008). For computation of the coarse element stiffness Kc , it is convenient to rewrite (46) in terms of the coarse element strain ε¯ , rather than degrees of freedom qc , using an alternative prolongation operator as:

w(x) = u(x) − XBc qc − ZGc qc − u0

1 T 1 D λ = − DT FT K f qf V V

(47)

(48)

where the final expression has been arrived at by substitution of λ from Equation (43) and V is the coarse

(51)

where 1 Z= 4

To determine the coarse element stress-strain relationship, the average stress can be expressed in terms of the Lagrange multipliers (Kaczmarczyk et al. 2008) as

σ =

(50)

where η = ∂xx u is a second-order strain measure. Detailed description of second-order computational homogenization can be found in Kaczmarczy et al. 2008 Here we limit ourselves to the salient equations required for extension of the first-order scheme developed in the previous sections. The displacement fluctuation in (50) expressed in terms of the coarse mesh degrees of freedom have the form

5.3 Construction of coarse element stiffness

ave ˘ −1 ˘ Pave q = ε ε with Pε = K g

(49)

1 u(x) = u0 + x · ε + x ⊗ x : η + w(x) 2

Pave to give  P, Recall that  Phom is subtracted from  which is then added to P to yield the prolongation operator P = P + ( Pave −  Phom ).

f

1 T T f ave D F K Pε V

For problems involving softening and fracturing of materials, there is further potential for improvement of the preconditioner. The two-grid preconditioner described above can be described as a first-order approach, adopting only first-order deformation modes in the computation of Pave . A natural extension of this can be achieved by taking into account secondorder strain measures, in the spirit of the second-order homogenization method (Kouznetsova et al. 2002, Kaczmarczyk et al. 2008). In the computation of Pave , such an approach enhances the coarse mesh displacement field approximation space and improves numerical efficiency of the preconditioner, which is manifested by faster convergence of the Newton method with Krylov iterative solver. This will be demonstrated with numerical examples in the next section. A second-order scheme is realized by expanding the fine mesh displacements with a Taylor series, truncated after the second-order term

and λ = −FT K f qf

C=−

5.4 Generalization to second-order continuum

(42)

f

where



2x2 0

0 2y2

2y2 0

0 2x2

xy 0

0 xy



(52)

and Gc is the coarse element second-order equivalent of Bc . The homogeneous part of the prolongation operator has the form  Phom = A(Xi Bc + Zi Gc )

343

(53)

where Xi and Zi are evaluated at the fine mesh nodes.

The boundary conditions for the fine mesh patch, in addition to Equation (35), are given by

n ⊗ x ⊗ w dŴ = 0. (54) ∂c

This equation enforces deformation of the fine mesh patch according to the average coarse mesh second-order strain η. Following discretization of the fine mesh displacements, the boundary conditions (35) and (54) are expressed in a modified constraint equation (compare with (36)): Cqf − DBc qc + EGc qc = 0

(55)

where

HUŴT ZdŴ E=

(56)

∂c

With this enhanced constraint equation (55) the second-order fluctuation part of the short-wave contribution to the coarse-to-fine mesh operator  Pave can be constructed in a similar manner to the first-order scheme. In order to keep the coarse mesh problem as simple and efficient as possible for practical implementation, the formulation is restricted to the antisymmetric part of η . Thus, in 2D, there are two bending modes in addition to the simple first-order deformation modes. 6 NUMERICAL EXAMPLE The fracture of dog-bone concrete specimens subject to tensile loading is analysed in order to demonstrate the performance of the proposed solution strategy for relatively large-scale problems and fracturing heterogeneous materials. Details about geometry, boundary conditions, aggregate size distribution, material model and parameters can be found in (Vliet 2000). In Figure 2, a simplified geometry for the specimens is Mesostructure 0.25D

50mm

D

D

100mm

presented. Three sizes of dog-bone samples are considered, D = 50 mm, D = 100 mm and D = 200 mm. Each sample size has a meso-structure generated with the same aggregates size distribution. Specific details of the aggregate structure can be found in Appendix B of (Vliet 2000) and in (Walraven 1980). The mesostructure for all problems is fully resolved, whereby aggregates, matrix and the interfacial transition zone are discretely modelled. The characteristic size of the fine mesh for all sample sizes are the same and controlled by the aggregate size and spacing. Such an approach has been shown to lead to mesh independent results (Kaczmarczyk and Pearce 2009). The circular shape of aggregates is discretized by a piecewise linear approximation, with the length of the segment equal to 1/10 of the radius of the smallest aggregate. The characteristic size of the coarse mesh is chosen arbitrarily as the distance between aggregate centres. Table 1 shows the number of degrees of freedom for both the fine and coarse meshes. For the aggregate, the elastic modulus is EA = 98 MPa and for the matrix EM = 35 MPa. Poisson’s ratio ν = 0.2 for all phases. The steel loading plates are also modelled as an elastic material with ES = 420 MPa and ν = 0.3. The cohesive crack material parameters for matrix, aggregate and interfacial transition zone have the following values: ftM = 12 MPa, GfM = 190 N/m, ftA = 6 MPa, GfA = 190 N/m, ftITZ = 2.0 MPa and GfITZ = 30 N/m, respectively. For all interfaces α = 1.0 is assumed. For the smallest sample D = 50 mm attention is focused on convergence for four types of grid transfer operators, as in the previous example, i.e. CST/DISP, CST/TRAC, GRADIENT/DISP and GRADIENT/ TRAC, see Figure 3 and Table 1. In all cases the preconditioned GMRES solver was used in combination with local arc-length control and line searches. The analysis using uniform traction boundary conditions to compute the short-wave prolongation operator together with elements (i.e. CST/TRAC) does not converge. The analyses with linear and quadratic displacement boundary conditions (i.e. CST/DISP and GRADIENT/DISP) exhibit good convergence. The C 1 -continuous elements with linear distribution of tractions (GRADIENT/TRAC) shows the fastest convergence. The relative number of iterations and relative cumulative computational time are shown in Table 2. The GRADIENT/TRAC analysis, with 8 processors, is 10% faster than GRADIENT/DISP, whereas the cumulative number of iterations is 40%

Ls = 1.2D

Table 1.

0.2D

200mm

Figure 2. Dog-bone geometry (left), mesostructure for different specimen sizes (right).

344

Number DOF’s for each sample size.

Size

Coarse DOF

Fine DOF

50 mm 100 mm 200 mm

3342/1114 9084 28296

71496 (+114324) 248080 (+390204) 868936 (+1364160)

Nominal Stress [MPa] Number of iterations GRADIENT/DISP GRADIENT/TRAC 2.5 CST/DISP 5k CST/TRAC 2.0

Nominal Stress [MPa]

mag. x103

not converged

6k

4k 3k 2k

1.5

0.1

2.5 2.0

0.5

0

10

20

30

40

Elongation of Ls [ m] Relative Error

1.5

0.0 60

50

1.0 0.5

0.01 0.001

1000

50mm 100mm 200mm

mag. x102

1.0

1k 0

3.0

0.0

1020

1040

1060

resNewton

1080

Number of iterations

resGMRES 1100

0

10

20

30

40

Elongation of Ls [µm]

50

60

1120

Figure 3. Model performance for 50 mm specimen. Nominal stress-displacement response and cumulative number of iterations (top left). Deformed fine mesh at peak and post-peak (top right). Illustration of convergence of GRADIENT/TRAC analysis, demonstrating the influence of the dynamic tolerance adopted for the GMRES algorithm (bottom).

Figure 4. Nominal stress vs displacement plots for D = 50 mm, 100 mm and 200 mm dog bone specimens. Experimental peak stress and standard deviation also show.

50mm

100mm

Table 2. Relative number of iterations and relative cumulative computational time. Analysis

Rel. num. iterations

Rel. time

GRADIENT/TRAC CST/TRAC GRADIENT/DISP CST/DISP

1.0 not converged 1.3 1.4

1.0 – 1.2 1.1

200mm

smaller. However, since the current implementation of the second-order scheme is not optimized, the authors believe that there is potential to significantly improve efficiency. All analyses with displacement boundary conditions (DISP) converged more slowly than for the second-order scheme with traction boundary conditions (GRADIENT/TRAC). The reasons for this is that where a crack crosses the fine mesh patch boundary, displacement fluctuation continuity is not enforced by the traction boundary conditions. Furthermore, the coarse mesh problem gives too stiff a response as a result of the residuals restricted from the fine mesh in the case of displacement boundary conditions. In summary, the type of preconditioner adopted (CST/DISP, CST/TRAC, GRADIENT/DISP or GRADIENT/TRAC) does not affect the response of the structure, it only affects the rate of convergence and consequently the solution time. GRADIENT/TRAC was found to be the most efficient. Finally, for each of the D = 50 mm, D = 100 mm and D = 200 mm specimens, four random mesostructures were generated, resulting in 12 separate analyses. The GRADIENT/TRAC preconditioner was used

Figure 5. Fracture patterns for D = 50 mm, 100 mm and 200 mm dog bone specimens.

345

for all analyses. Figure 4 shows the nominal stressdisplacement response for all 12 of these analyses and it can be seen that there is a degree of scatter in the response for all specimen sizes. This scatter can be directly attributed to the randomness of the fine-scale structure only and is not the result of the preconditioner used. Moreover, these numerical results support the experimental observation of van Vliet (Vliet 2000), that for a range of small sample sizes (i.e. 50–200 mm) no clear size effect for nominal strength is observed (i.e. reducing with size). It can also be seen that the response becomes more brittle with increasing size. Figure 5 shows the crack patterns that developed, illustrating that the shape of the developing crack is clearly influenced by the spatial arrangement of aggregates.

7 CONCLUSIONS This paper presents a modelling strategy for simulating the behaviour of fracturing heterogeneous materials

such as concrete where the fine-scale heterogeneities are fully resolved. The fine-scale is modelled using HTS elements, where cracks are restricted to element interfaces. This represents an efficient framework for modelling propagating cohesive cracking. The very large nonlinear system of algebraic equations that emerges from this fine scale resolution requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. A significant extension to the work of Miehe and Bayreuther (Miehe and Bayreuther 2007) has been proposed that constructs a preconditioner using a two-grid strategy that utilizes the scale transition techniques derived for computational homogenization. Both displacement and traction boundary conditions are considered in construction of the grid transfer operators for both first and second-order schemes. The proposed two-grid strategy has been demonstrated for both elastic and fracturing heterogeneous materials, illustrating both the convergence properties of the proposed scheme and efficiency of the parallel implementation. The use of the preconditioned GMRES Krylov iterative method with dynamic tolerance in combination with a constrained Newton method with local arc-length control and line searches represents a robust and efficient simulation framework.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the UK Engineering and Physical Sciences Research Council (Grant Ref: EP/D500273). The authors also wish to thank Dr. Lee Margetts of The University of Manchester for access to HECToR (UK’s national supercomputing service) through the UK Engineering and Physical Sciences Research Council HECToR capability challenge grant (Grant Ref: EP/F055595/1).

REFERENCES Ainsworth, M. (2001). Essential boundary conditions and multi-point constraints in finite element analysis. Computer Methods in Applied Mechanics and Engineering 190(48), 6323–6339. Alfano, G. and M.A. Crisfield (2003). Solution strategies for the delamination analysis based on a combination of localcontrol arc-length and line searches. International Journal for Numerical Methods in Engineering 58(7), 999–1048. Balay, S., K. Buschelman, W. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B. Smith, and H. Zhang (2009). PETSc Web page. http://www.mcs.anl.gov/petsc. Belytschko, T., S. Loehnert, and J.H. Song (2008). Multiscale aggregating discontinuities: A method for circumventing loss of material stability. International Journal for Numerical Methods in Engineering 73(6), 869–894.

Feyel, F. (2003). A multilevel finite element method (fe2) to describe the response of highly non-linear structures using generalized continua. Comput. Methods. Appl. Mech. Engrg. 192(-), 3233–3244. Gitman, I.M., H. Askes, and L.J. Sluys (2007). Representative volume: Existence and size determination. Engineering Fracture Mechanics 74(16), 2518–2534. Gitman, I.M., H. Askes, and L.J. Sluys (2008). Coupledvolume multi-scale modelling of quasi-brittle material. European Journal of Mechanics A-Solids 27(3), 302–327. Hillerborg, A., M. Moder, and P.E. Petersson (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6(6), 773–781. Kaczmarczyk, L. and C.J. Pearce (2009). A corotational hybrid-trefftz stress formulation for modelling cohesive cracks. Computer Methods in Applied Mechanics and Engineering 198(15–16), 1298–1310. Kaczmarczyk, L., C.J. Pearce, and N. Bicanic (2008). Scale transition and enforcement of rve boundary conditions in second-order computational homogenization. International Journal for Numerical Methods in Engineering 74(3), 506–522. Kouznetsova, V., M.G.D. Geers, and W.A.M. Brekelmans (2002). Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering 54(8), 1235–1260. Markovic, D. and A. Ibrahimbegovic (2006). Complementary energy based fe modelling of coupled elasto-plastic and damage behavior for continuum microstructure computations. Computer Methods in Applied Mechanics and Engineering 195(37–40), 5077–5093. Miehe, C. and C.G. Bayreuther (2007). On multiscale fe analyses of heterogeneous structures: From homogenization to multigrid solvers. International Journal for Numerical Methods in Engineering 71(10), 1135–1180. Miehe, C. and A. Koch (2002). Computational micro-tomacro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4–5), 300–317. Moulinec, H. and P. Suquet (1998). A numerical method for computing the overall response of non-linear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering 157, 69–94. Saad, Y. (1996). Iterative Methods for Sparse Linear Systems. PWS. Tijssens, M.G.A. (2001). On the cohesive surface methodology for fracture of brittle heterogeneous solids. Ph.D. thesis, Technische Universiteit Delft. Vliet, M.V. (2000). Size effect in tensile fracture of concrete and rock. Ph.D. thesis, Technische Universiteit Delft. Walraven, J. (1980). Aggregate interlock: A theoretical and experimental analysis. Ph.D. thesis, Technische Universiteit Delft. Xu, J. (1997). An introduction to multigrid convergence theory. In Iterative methods in scientific computing, pp. 169–241. Springer. Zienkiewicz, O. and R. Taylor.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A 3D lattice model to describe fracture process in fibrous concrete J. Kozicki & J. Tejchman Gda´nsk University of Technology, Gda´nsk-Wrzeszcz, Poland

ABSTRACT: Paper deals with simulations of fracture process in concrete including steel fibres. A discrete geometric linear lattice 3D model was used. Concrete was described at a meso-scale as a five-phase material composed of aggregates, cement matrix, steel fibres and interfacial zones between matrix and aggregate and between matrix and fibres. The lattice elements were stochastically distributed in the form of a mesh using a Delaunay’s construction scheme. The calculations were carried out with notched concrete specimens subjected to uniaxial extension. 1 INTRODUCTION Concrete is still the most widely used construction material since it has the lowest ratio between cost and strength as compared to other available materials. However, it has two undesirable properties, namely: low tensile strength and large brittleness that cause the collapse to occur shortly after the formation of the first crack. Therefore, the application of concrete subjected to impact, earth-quaking and fatigue loading is strongly limited. To improve these two negative properties and to achieve a partial substitute of conventional reinforcement, the addition of short discontinuous randomly oriented fibres (steel, glass, synthetic and natural) can be practiced among others. Steel fibres are the most used in concrete applications due to economy, manufacture facilities, reinforcing effects and resistance to the environment aggressiveness. By addition of steel fibres, the following properties of plain concrete: tensile splitting strength, flexural strength, first cracking strength, toughness (energy absorption capacity), stiffness, durability, impact resistance, fatigue and wear strength increase, and deflection, crack width, shrinkage and creep are reduced (Shah 1971, Bentur & Mindess 1990, Balaguru & Shah 1992, Zollo 1997). In turn, compressive strength can slightly increase (Mohammadi et al. 2008) or slightly decrease (Altun et al. 2007). The addition of steel fibres aids in converting the brittle characteristics to a ductile one. Fibres limit the formation and growth of cracks by providing pinching forces at crack tips. They bear some stress that occurs in cement matrix themselves and transfer the other portion of stress at stable cement matrix portions. Real effects of fibre addition can be observed as a result of the bridging stress offered by the fibres after the peak load. The fibre reinforced concrete specimens develop first a pattern of fine distributed cracks instead of directly failing in one localized crack. Fibre-reinforced concrete has found many applications in tunnel linings, wall

347

cladding, bridge desks, pavements, slabs on grounds, factory (industrial) floors and slabs, dams, pipes, fire protection coatings, spray concretes (Balaguru and Shah 1992, Krstulovic-Opara et al. 1995, Falkner & Henke 2000, Schnüttgen & Teusch 2001, Walraven & Grünewald 2002). It can be also used as a method for repair, rehabilitation, strengthening and retrofitting of existing concrete structures (Li et al. 2000). The degree of improvement of fibrous concrete depends upon many different factors such as: size, shape, aspect ratio, volume fraction, orientation and surface characteristics of fibres, ratio between the fibre length and maximum aggregate size, and volume ratio between long and short fibres and concrete class. The fibre orientation depends on the specimen size and flow direction of the fresh concrete. On the other hand, fibres hinder the flowability of fresh concrete decreases workability by increasing the pore volume, resulting in strength decrease (what is visible during compression). The most suitable volume values for concrete mixes are between 0.5% and 2.5% by volume of concrete. The aspect ratios of steel fibers used in concrete mix are varied between 50 and 100. A probability of heterogeneous distribution and flocculation of fibers in concrete mix is increased by increasing aspect ratios. Homogeneous distribution of fibers at mixing and placing is required regardless of the type of fibers. A better workability of concrete mix is obtained as the percentage of shorter fibres increased in the mix in comparison to percentage of longer fibres (Mohammadi et al. 2008). A strengthening effect in concrete increases with the ratio of the fibre length to the size of the coarse aggregate (optimum ratio: 1.5–2.0, Chenkui & Guofan 1995) and by combining two different fibre types: short and long fibers (optimum content: 65% long and 35% short, Mohammadi et al. 2008). A small addition of non-metallic fibres results in good fresh concrete properties and reduced early age cracking (Sivakumar & Santhanam 2007) and an increased impact energy, toughness and

ductility (Komlos et al. 1995). The toughness indices are higher for lightweight concrete than for normal weight concrete and for normal strength concrete than for high strength concrete (Balendran et al. 2002). In spite of positive properties, fibrous concrete did not find such acknowledgment and application as usual concrete. There do not still exit consistent dimensioning rules due to the lack sufficient large-scale static and dynamic experiments taking into account the effect of the fibre orientation. There is a general lack of confidence in the design particularly under bending in spite of existing tools on different scale. An analytical micro-scale approach has been proposed in (Lim et al. 1987). On the meso scale, a truss model has been used to study the material behavior (Li et al. 2006, Bolander & Saito 1997, van Hauvert & van Mier 1998). In turn, on the macro scale, constitutive models have been developed that can be used in finite element computations (Al-Taan & Ezzadeen 1995, Kooiman et al. 2000). Recently, a numerical continuum approach was proposed by Radtke et al. (2008) wherein the existing continuum approach to model concrete failure was combined with a discrete representation of fibres by adding extra nodal forces at fibre ends measured during the pull-out of a fibre from a matrix specimen. In turn, a semi-analytical method was proposed by Jones et al. (2008) to predict the flexural behaviour of steel fibers using a stress-block approach. Kabele (2007) has used a multi-scale framework for modeling of fracture in high performance fiber reinforced cementitious composites. The intention of this paper is to describe the fracture process at the meso-scale in fibrous concrete using a geometrical linear lattice type model (Kozicki & Tejchman 2007, Kozicki & Tejchman 2008). In the model, individual steel fibres were explicitly modeled. Three dimensional calculations were performed with concrete considered as five-phase material (aggregate, cement matrix, steel fibres and interfacial transition zones between both cement matrix and aggregate and cement matrix and steel fibres). Attention was paid to the effect of the amount of steel fibres and their orientation on the material behavior during uniaxial tension and uniaxial compression. Our lattice model was successfully used to model the fracture process in two-dimensional and three-dimensional concrete specimens subject to uniaxial tension, uniaxial compression, three-point bending and shear-extension test (Kozicki & Tejchman 2007, 2008). The effect of aggregate density, mean aggregate size and specimen size was realistically captured.

(Schlangen et al. 1997, van Mier et al. 1995, van Mier & van Vliet 2003) in that it consists of rods with flexible nodes and longitudinal deformability, rotating in the form of a rigid body rotation (Fig. 1). Thus, shearing, bending and torsion are represented by a change of the angle between rod elements connected by angular springs. This quasi-static model is of a kinematic type. The calculations of element displacements are carried out on the basis of the consideration of successive geometrical changes of rods due to translation, rotation and normal and bending deformation. Thus, the global stiffness matrix is not built and the calculation method has a purely explicit character. In spite of necessity of the application of small displacement increments (what is the inherent property of explicit numerical procedures), the computation time is significantly reduced as compared to implicit solutions. In our model, the quasi-brittle material was discretized in the form of a 3D tetrahedral grid including lines. The distribution of elements was assumed to be completely random using a Delaunay’s construction scheme. First, a tetrahedral grid of nodes was created in the material with the side dimensions equal to g. Then each node was randomly displaced by a vector of the random magnitude smaller than s. The nodes randomized in this way were connected with each other. Thus, each edge in the Delaunay mesh formed a lattice. The model needs 2 parameters to randomly distribute elements in the lattice. In the calculations, we assumed mainly the parameters g and s as g = 2 and s = 0.6 g. The elements possessed longitudinal stiffness described by the parameter kl (controlling the changes of the element length), bending stiffness described by the parameter kb (controlling the changes of the angle between elements) and torsional stiffness described by the parameter kt (controlling torsion between two elements in 3D calculations). The parameter kr was assumed kr = kt . The displacement of the center of each rod element (Fig. 2) was calculated as the average displacement of its two end nodes from the previous iteration step  i X

=

A  i X

+ BiX 2

(1)

wherein AX and BX are the displacement of the end nodes A and B in the rod element i, respectively.

2 LATTICE MODEL Our linear lattice model (Kozicki & Tejchman 2007, 2008) differs from a classical lattice beam model composed of beams connected by non-flexible nodes

348

Figure 1. Rods connected by angular springs (rods do not bend) (Kozicki & Tejchman 2007).

Figure 2.

General scheme to calculate displacements of elements in the 2D lattice (Kozicki & Tejchman 2007).

The displacement vector of each element node was obtained by averaging the displacements of the end of elements belonging to this node caused by translation, rotation, normal and bending deformation (Fig. 1):

j

X =

 iW  + iR  jn sum i +



1    i i d init (i D i kl + iB i kb + iT i kt ) ,  1 i i dinit (i kl + i kb + i kt )

Figure 3. Lattice composed of rods to model five phases of concrete: fibres, aggregate, cement matrix and interfaces between aggregate and cement matrix and between fibres and cement matrix.

(2)

 = wherein: i X = resultant node displacement, i W  = node displacement due to the rod translation, i R  = node node displacement due to the rod rotation, i D displacement due to a change of the rod length  = node displacement due to (induced by kl ), i B a change of the angle between rods (induced by kb ),  = node displacement due to torsional spring i T between two neighboring rods (induced by kt ), i = successive rod number connected with node j, j = node number, j nsum = number of rods belonging to node j and i dinit = initial length of rod i. The node displacements were calculated successively during each calculation step beginning first from elements along boundaries subject to prescribed displacements. By applying Eq. 2, the equilibrium of strains was obtained in each node (what required always about 10 iterations). The resultant force F in a selected specimen’s cross-area A was determined with the aid of corresponding normal strain ε, shear strain γ, stiffness parameters kl , kl , kt , modulus of elasticity E, shear modulus G and cross-section area A:

F =A

 (kl εE + kb γ G),

(3)

where the sum is extended over all elements that cross a selected specimen’s cross-section. For the bending stiffness parameter kb = 0 (Eq. 2), the elements behave as simple bars. The element was removed from the lattice if the local critical tensile strain εmin was exceeded. All presented numerical calculations were strain-controlled. The strain increment in a single calculation step expressed as scrit = gεmin /l (where l—displacement of specimen edge, εmin —local critical tensile strain, g—average rod length) should be larger than 500. It means that minimum 500 iterations were required to remove a single rod. In the calculations, different properties were prescribed to lattice elements to simulate the behaviour of aggregate, cement matrix, bond between aggregate and cement matrix, fibres and bond between fibres and cement matrix (Fig. 3, Tab. 1). The aggregate volume percentage was 60% in 3D, and the mean aggregate diameter was d50 = 12 mm (Fig. 4). The aggregate distribution was generated following the method given

349

Table 1.

Material parameters used in calculations (five-phase material, p = kb /kl ). Young’s modulus E [GPa]

p = kb /kl tension

p = kb /kl compression

kl

Tensile strain εmin [%]

Cement matrix

20

0.6

0.2

0.01

0.2

Aggregate

60

0.6

0.2

0.03

0.133

Aggregate interface Fibrous interface

14 20 14 14 14 14 14 14 14 14

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.007 0.01 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007

0.05 0.2 0.025 0.05 0.1 0.2 0.5 1 10 90

160

0.6

0.2

0.08

90

Steel fibres

Figure 5. Approximation of the grading curve with discrete number of aggregate sizes (Cusatis et al. 2003). Figure 4. Aggregate sieve curve (for two different specimens), d50 = 12 (Kozicki & Tejchman 2007).

by Cusatis et al. (2003) and Eckardt anf Könke (2006). First, a grading curve was chosen (Fig. 4). Next, the certain amounts of particles with defined diameters were generated according to curve in Fig. 5. Finally, the spheres describing aggregates were randomly placed in the specimen preserving the particle density and a certain mutual minimum distance (van Mier et al. 1995)

Dp > 1.1

D1 + D 2 , 2

(4)

where D is the distance between two neighboring particle centers and D1 and D2 are the diameters of these two particles. In 3D calculations, the minimum element length was 0.6 mm, and the maximum one was

about 4 mm (2 mm on average). The size of 3D concrete specimen was 5 × 5 × 5 cm3 . The fibre length was 20 mm. The amount of elements was 200 000. Tab. 1 includes the material parameters assumed for analyses. The material stiffness parameters kl and kb for aggregate, cement matrix and bond between matrix and aggregate were determined empirically (Kozicki & Tejchman 2007, 2008) to match experimental results at the macro-scale with numerical ones on the basis of a uniaxial tension and compression test for concrete (van Mier et al. 1995). The calculations resulted in the following values of the material parameters kl and εmin : kl = 0.01 and εmin = 0.2% in the cement matrix, kl = 0.03 and εmin = 0.133% in the aggregate and kl = 0.007 and εmin = 0.05% in the bond zones (Kozicki et al. 2007, Kozicki et al. 2008). The ratios between the parameters kl and εmin for different phases were assumed in a similar way as the ratios between the moduli of elasticity and tensile strengths for each concrete phase (van Mier et al.

350

1995). The weakest phase was bond between aggregate and cement matrix. A different value of the parameter kb was used in compression and in tension (Kozicki & Tejchman 2008). For the sake of simplicity, one assumed that same ratio of p = kb /kl for all phases (0.6 in tension and 0.2 in compression) (Tab. 1). The chosen ratio of p resulted in the following Poisson’s ratios: 0.22 (uniaxial compression) and 0.07 (uniaxial tension) (Kozicki & Tejchman 2008). In the case of the fibrous interface, different cases were taken into account (Tab. 1) by changing the parameter εmin (the values of E and p were similar as for the aggregate interface): A. no interface between fibre and cement matrix (properties the same as for cement matrix), B. strength of fibrous interface smaller than strength of aggregate interface (local critical tensile strain εmin = 0.025%), C. strength of fibrous interface equal to strength of aggregate interface (εmin = 0.05%), D. strength of fibrous interface stronger than strength of aggregate interface (εmin = 0.1%, 0.2%, 0.5%, 1.0%, 10% and 90%).

Figure 7. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 0.1%, c) fibrous interface with εmin = 0.2%, d) fibrous interface with εmin = 0.5% (σ22 —vertical normal stress, ε22 —vertical normal strain).

The material parameters for steel fibres were: E = 160 GPa, p = 0.6 (tension), p = 0.2 (compression), kl = 0.08 and εmin = 90% (Tab. 1). 3 SIMULATIONS OF UNIAXIAL TENSION The results of calculated stress-strain curves during uniaxial tension with a notched cubical 5 × 5 × 5 cm3 concrete specimen with smooth horizontal edges are shown in Figs. 6–11 for different properties of the

Figure 8. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 1%, c) fibrous interface with εmin = 10%, d) fibrous interface with εmin = 90% (σ22 —vertical normal stress, ε22 —vertical normal strain).

Figure 6. Calculated stress–strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) without fibrous interface, c) fibrous interface with εmin = 0.025%, d) fibrous interface with εmin = 0.05% (σ22 —vertical normal stress, ε22 —vertical normal strain).

Figure 9. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) without fibrous interface, c) fibrous interface with εmin = 0.025%, d) fibrous interface with εmin = 0.05% (σ22 —vertical normal stress, ε22 —vertical normal strain).

351

Figure 10. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 0.1%, c) fibrous interface with εmin = 0.2%, d) fibrous interface with εmin = 0.5% (σ22 —vertical normal stress, ε22 —vertical normal strain). Figure 12. Distribution of aggregate in concrete specimens 5 × 5 × 5 cm3 subjected to uniaxial extension (with 2.25% fibre volume).

Figure 11. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 1%, c) fibrous interface with εmin = 10%, d) fibrous interface with εmin = 90% (σ22 —vertical normal stress, ε22 —vertical normal strain).

fibrous interface (Tab. 1). The distribution of aggregate and steel fibres in the concrete specimen is shown in Figs. 12 and 13. In turn, Fig. 14 demonstrates the deformed concrete specimens at failure. The results show that both concrete strength and ductility increase with increasing amount of steel fibres if the fibrous interface is significantly stronger than the aggregate interface, i.e. εmin ≥ 0.2% (both strength and ductility increase with increasing εmin ≥ 0.2%). Due to a high particle density of 60%, percolation of bond zones occurs early in the loading history. Since the interface between cement matrix and aggregate is the weakest component of the system, the material becomes initially weak there, cracks

Figure 13. Distribution of steel fibres in concrete specimens 5 × 5 × 5 cm3 subjected to uniaxial extension (with 2.25% fibre volume).

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are created along aggregate and the pre-peak nonlinearity does not appear. The cracks propagate from the notch in the mid-region of the specimen. Since the amount of aggregates is relatively large, the cracks cannot propagate in long lines. Instead of this, several discontinuous macro-cracks propagate in a tortuous manner. Between fibres, the cracks overlap and form

branches. The crack propagation is clearly disturbed by the presence of steel fibres which delay their development. If the fibrous interface is weaker than the aggregate interface, both material strength and ductility decrease due to the fact that several straight cracks are only created along fibres which act as imperfections promoting cracks (Kozicki & Tejchman 2009).

4 CONCLUSIONS The following conclusions can be drawn from numerical results for fibrous concrete described as five-phase material at meso-scale using a linear lattice model where individual steel fibres were explicitly modeled. The lattice model in spite of its simplicity is capable to simulate fracture process. The obtained results of stress-strain curves for fibrous concrete during uniaxial tension qualitatively compare with experimental results. By using an elastic-purely brittle local fracture law at the particle level of the material, global softening behavior is obtained. The advantage of our quasi-static lattice model lied is its explicit character. Thus, a large amount of elements could be taken into account when using parallelized computers. Both strength and ductility of concrete specimens increase with increasing amount of fibres during uniaxial extension, if strength of the fibrous interface is significantly higher than strength of interfacial zones between aggregate and cement matrix. In this case, the crack propagation is delayed by the presence of steel fibres. The calculations with a lattice model will be continued. Further calibration studies will be performed by taking into account the real micro-structure of fibrous concrete specimens. The possibility of a crack closure will be considered. In addition, inertial forces will be taken into account during dynamic calculations.

REFERENCES

Figure 14. Crack distribution at failure in concrete specimens 5 × 5 × 5 cm3 with 2.25% fibre volume subjected to uniaxial extension including fibrous interface with: a) εmin = 0.1%, b) εmin = 10%, c) εmin = 90%.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Limit analysis of 3D reinforced concrete frames Kasper P. Larsen Ramboll Denmark DTU Byg, Technical University of Denmark

Peter N. Poulsen & Leif O. Nielsen DTU Byg, Technical University of Denmark

ABSTRACT: In this paper we present a new finite element framework for lower bound limit analysis of reinforced concrete beams subjected to loadings in three dimensions. The method circumvents the need for a complex section force based yield criteria by creating a discrete representation of the internal stress state in the beam, and employ the yield criteria on the stress state level. The stress state is decomposed into concrete and reinforcement stresses and separate yield criteria are applied to each stress component. Simple upper- and lower bounds are applied to the reinforcement stresses and the classical Mohr-Coulomb criteria are applied to the concrete stresses. The exact Mohr-Coulomb criteria are implemented using Semidefinite Programming and an approximation using Second-Order Cone Programming is developed for improved performance. The element is verified by comparing the numerical results with analytical solutions. 1 INTRODUCTION Structural analysis of reinforced concrete structures applying limit state analysis always results in an optimization problem. In order to solve the optimization problem the yield criteria needs to be established for the given structural component. Originally, yield criteria have often been sought on a sectional level, e.g. for beams (Damkilde and Hoyer 1993), plates (Krabbenhoft and Damkilde 2002), stringers (Damkilde, olsen, and Poulsen 1994) and disks (Poulsen and Damkilde 2000). In most of these yield criteria approximations have been applied in order to simplify the formulation. Another way of establishing the load bearing capacity of a structural component would be to divide the section into smaller parts/zones for which the yield criteria is well-known and then get the capacity by summing up the different contributions. In a beam the yield criteria could be established on a sectional level as the N-M-diagram can be found by analyzing typical situations for a beam. For a 2D beam this approach is applicable but for a 3D beam this turns out to be very troublesome. Instead the capacity for a 3D beam can be formulated by dividing the crosssection into zones. This approach has been applied in (Niebling, Vinther, and Larsen 2007) where a reinforced concrete 3D-beam was established using the zone model, though the cross section modeling capabilities was limited to rectangular beams with simple reinforcement. Recent advances in optimization algorithms and computational power has made it possible to solve medium to large scale problems with complex yield criteria within reasonable time. The Finite

355

Element Method (FEM) is the most common method for numerical analysis of structures. In this paper we present an element which expands on the work done by (Niebling, Vinther, and Larsen 2007) by allowing engineers to model a wider range of cross section types. Similar to the element presented in (Niebling, Vinther, and Larsen 2007), a discrete formulation of the internal stress state is used, and the yield criteria are applied at the stress level instead of at the section force level. The classic Mohr-Coulomb failure criteria are applied to the concrete while simple upper- and lower bounds are applied to the reinforcement. We will here implement the exact Mohr-Coulomb criteria using Semidefinite Programming (SDP) as described in (Larsen, Poulsen and Nielsen 2009). We will also develop an approximate method based on second-order cones (SOCP) which solves faster and more efficiently. The lower bound limit analysis problem is formulated as minimize −λ subject to −Rλ + H σ = R0 σ T ∈ Ct

(1.1)

where λ is the scalar load factor sought optimized (by minimizing the negative load factor we will find the maximum load capacity). The vector σ contains the n stress parameters used to define the stress state in within the structure. The matrix H defines the equilibrium equations and the continuity conditions, the vectors R and R0 are the variable and static load vectors respectively, and the Ct is a convex cone.

2 ELEMENT MODEL Here we consider a geometric linear beam element with local coordinate axes as shown on Figure 1. The element length is measured along the local x-axis and the width and height is measured along the y- and z-axis respectively. Here, the x direction will also be referred to as the longitudinal direction and the y- and z-direction will be referred to as the transverse directions. The element is capable of carrying loads in the longitudinal and transverse directions as well as torsional moments as illustrated on Figure 1. The element is intended to be compatible with a shell element, such that full 3D structures can be modeled. The external load intensities are coupled to the internal section forces through the following equilibrium equations. py + Vy,x = 0

(2.1)

pz + Vz,x = 0

(2.2)

tx + Mx,x = 0

(2.3)

The bending moments are coupled to the external loads through the shear forces by two additional equilibrium equations Mz,x + Vy = 0

(2.4)

My,x − Vz = 0

(2.5)

Finally, a set of static boundary conditions are required to get at complete description of the internal section forces. These are applied at the beam ends, see Figure 1, and defined as QA = {−NxA , −VyA , −VzA , −MxA , −MyA , −MzA }

(2.6)

QB = {NxB , VyB , VzB , MxB , MyB , MzB }

(2.7)

2.1 Zone model To obtain a lower bound solution, the element must be in a safe state i.e. the yield criterion must be obeyed at all points within the element. To circumvent the need for a yield criterion based on section forces, we

Figure 1.

3D Beam element model.

apply the material constraints on a stress state level. In (Niebling, Vinther, and Larsen 2007), a zone model was used to discretize the internal stress state in the beam element. Here, we will adopt this zone model and extend it to work with an arbitrary number of zones, hereby enable engineers to model and analyze more complex cross sections. The zones must be aligned with the local coordinate axis and the reinforcement is smoothed over the zone area. We will here assume that the resulting transverse normal stresses, i.e. the sum of transverse reinforcement and concrete stresses, are equal to zero. The transverse normal stresses in the concrete and reinforcement does not have to be zero, as long as the total transverse stresses are. Additionally, only the transverse shear stresses, τxy and τxz , are considered in the cross section. The stress state at a point within the beam is then defined by ⎤ ⎡ σx τxy τxz 0⎦ (2.8) σ = ⎣τxy 0 0 τxz 0 Besides the section force equilibrium equations described in the previous section, the stress state within a zone must also fulfil the following equilibrium equation σx,x + τxy,y + τxz,z + fx = 0

where fx is the volume load in the longitudinal beam direction. The normal stresses, σx , are chosen constant in the yz-plane and the shear stresses, τxy and τxz , varies linearly in the y- and z-direction respectively. A fourth degree variation of the normal stresses and a third degree variation of the shear stresses in the longitudinal direction requires 5 normal stress variables and 4 shear stress variables along each side of the zone, see Figure 2. The stress state in a zone can be parameterized by a total of 21 stress variables. To obtain a statically admissible stress field within the element, traction continuity between zones must be ensured. Because the zones are axis-aligned and transverse normal stresses are disregarded, this is simply done by ensuring shear stress continuity along shared zone faces. Because the internal stress state is discretized by the zones, the 6 section forces required by the equilibrium

Figure 2.

356

(2.9)

Sketch of zone model.

equations and the static boundary conditions can be computed from a set of summations e.g. Mx =

n  1 (2) (1) (2) (2) Ai ((τxz + τxz )yi + (τxy + τxy )zi ) (2.10) 2 i=1

where n is the number of zones, Ai is the cross section area of zone i and yi and zi are the geometric center point coordinates of the zone cross section. 3 YIELD CRITERION The equilibrium equations defined in the previous section ensures that the obtained solution is statically admissible. Because a lower bound solution requires the structure to be in a safe state, a set of material constraints must be applied as well. The stress state given in Eq. (2.8) is decomposed into a set of concrete- and reinforcement stresses as σ = σc + As σs where ⎡

σcx σc = ⎣τcxy τcxz

τcxy σcy 0

The separation strength, fA , will often be equal to the uni-axial tensile strength, ft . k is a parameter determined from the frictional angle, φ. Here, k = 4 are used which is equivalent to a frictional angle of approximately 37◦ . The principal stresses are sorted as σ1 ≥ σ2 ≥ σ3 . The yield criteria on the reinforcement and the concrete will be applied in a number of control points along the length of the beam element. The SDP formulation of the Mohr-Coulomb criteria is, see (Larsen, Poulsen, and Nielsen 2009) for further details: For a given stress state σc in the concrete, the yield criteria can be written as two linear matrix inequalities and two linear inequalities σc + kαI  0 σc − β I  0 β ≤ fA

(3.1) β≤ ⎤ τcxz 0 ⎦ σcz

are the concrete stresses, ⎡ ⎤ σsx 0 0 σs = ⎣ 0 σsy 0 ⎦ 0 0 σsz

(3.2)

(3.8)

fc −α k

where α and β are scalar variables. 3.1 Approximation to Mohr-Coulomb using SOCP

(3.3)

are the reinforcement stresses and As is then a diagonal matrix containing reinforcement area per unit area perpendicular to the local element axes ⎡ ⎤ Asx 0 0 As = ⎣ 0 Asy 0 ⎦ (3.4) 0 0 Asz Since only normal stresses are considered in the reinforcement, simple upper- and lower bounds are applied to these −fYc ≤ σs ≤ fYt

(3.5)

kσ1 − σ3 ≤ fc

(3.6)

σ1 ≤ fA

(3.7)

where fYc and fYt are the compression and tensile strength of the rebars respectively. The Mohr-Coulomb failure criterion, which consists of a sliding- and a separation criteria as shown in Eq. (3.6) and Eq. (3.7), will be applied to the concrete stresses.

In Eq. (3.7), fA is the separation strength and fc is the uni-axial compression strength of the concrete.

357

Despite the attractive properties of the SDP formulation of the Mohr-Coulomb criteria given in the previous section, performance considerations in practical applications calls for a yield criterion based on second-order cones. The major challenge here is to model the tri-axial effects which occurs in areas with reinforcement in all three directions. Initially, we will make the assumption that the separation strength of the concrete material is equal to zero. Because of the low separation strength found in concrete, this assumption is often made in practical engineering. All tensile stresses must therefore be carried by the reinforcement and transverse reinforcement must be present in zones that carry shear stresses. Figure 3 shows a rectangular beam reinforced with longitudinal reinforcement bars and stirrups. The beam has been discretized into 9 zones as marked with the dashed lines. The figure also illustrates the three possible zone types used when modeling cross sections. The zone types differ by the transverse reinforcement present in the zone. The first type has no transverse reinforcement and because the concrete is not able to carry any tensile stresses, only a uni-axial stress state is possible in these zones. The second zone type has transverse reinforcement in one of the transverse directions making it similar to a reinforced concrete disk with a biaxial stress state. In the third zone type, reinforcement is present in both of the transverse directions making a tri-axial stress state possible. The zone types

t 2

3

h

where

t

1

2

3

t

i =

Asi fYt Aci fc

for i = {x, y, z}

(3.12)

This type of constraint can be handled by some solvers (these are known as Quadratic Constrained Programming (QCP) problems) but it is often recommended to use cone constraints because they solve more efficiently, (MOSEK). Eq. (3.11) can be translated into a rotated quadratic cone as shown below

2

2x1 x2 ≥ x32 + x42 2

3

3

1 x1 = − σcx y z 2

t

w

x2 = 1 x3 = y τcxy x4 = z τcxz

Figure 3. Rectangular beam example with longitudinal rebars and stirrups.

are marked by the numbers 1–3 on Figure 3. Separate yield criteria are applied in each of the three zone types. 3.1.1 Triaxial stress state Here, we will propose an approximation to the full Mohr-Coulomb using second order cone constraints only. The approximation will, to some extent, be able to utilize the tri-axial stress state in the concrete. The approximation is based on the observation that zones subjected to tri-axial stress states often will have the same amount of reinforcement in the transverse directions. Since zones with reinforcement in both direction often will be corner zones, see Figure 3, in which a stirrup wraps around the longitudinal reinforcement, this assumption seems reasonable. The principal stresses are determined by the eigenvalues of the concrete stress tensor σc ⎡ ⎤ σcx − λ τcxy τcxz ⎣ τcxy σcy − λ 0 ⎦=0⇒ (3.9) τcxz 0 σcz − λ (σcx − λ)(σcy − λ)(σcz − λ)

2 (σ − λ) − τ 2 (σ − λ) = 0 − τcxy cz cxz cy

For the sliding criterion, Eq. (3.6), we will assume that the transverse normal stresses are of equal magnitude i.e., σcy = σcz . If the transverse normal stresses are not equal, we apply the largest (compression is negative) of the transverse normal stresses in both directions. It should be noted that if the difference between the reinforcement in each direction becomes large, the criterion will be too restrictive. Therefore, a simplified criterion must be applied in the biaxial stress state. First, we will assume that σcz is the largest of the transverse normal stresses. If we substitute σcy with σcz in Eq. (3.10) we get (σcx − λ)(σcz − λ)2 2 2 − τcxy (σcz − λ) − τcxz (σcz − λ) = 0

It is evident that one eigenvalue must be given by



τcxy fc

2

+ y



τcxz fc

2

≤0

(3.14)

λ1 = σcz

(3.10)

In (Andreasen 1985) the separation criterion, Eq. (3.7), for a reinforced solid is utilizing Eq. (3.1) and Eq. (3.10) approximated on quadratic form as   σcx − x − y z fc + z

(3.13)

(3.11)

358

The remaining two eigenvalues are computed from

σcx + σcz λ2 = λ3 2 ±

 

1 (σcx − σcz ) 2

2

2 + τ2 + τcxy cxz

(3.15)

Since we do not know the order of the three eigenvalues we must check every combination hereof.



From Eq. (3.15) we see that λ2 ≥ λ3 which means that it is sufficient to check three combinations of the eigenvalues. In order to formulate the yield criterion on a second-order cone form, we introduce the auxiliary variable 1 1 =

1 (σcx − σcz ) 2

    σcx + σcz 1 fc − (k − 1) k +1 2

(3.16)



σcx + σcz 2 2 + τ2 + 1 + τcxy cxz 2



≤ fc ⇒ (3.17)

σ + σcy 2 + τ 2 ≤ f − k · σ + cx 12 + τcxy (3.18) c cz cxz 2 The above inequality is a quadratic cone constraint. Case II: k · λ2 − λ1 ≤ fc Here, λ1 is assumed to be the smallest eigenvalue making λ2 the largest. Inserting this into Eq. (3.6) we get k·



 σcx + σcz 2 2 2 + 1 + τcxy + τcxz − σcz ≤ fc ⇒ 2 (3.19)

f σ σ + σcz 2 + τ 2 ≤ c + cz − cx 12 + τcxy cxz k k 2

k·



σcx + σcz + 2

−





12

σcx + σcz − 2



σcx + σcz 2 +





+

2 τcxy

+

2 τcxz

(3.24)



2 + τ2 ≤ α 22 + τcxy 2 cxz

(3.25)

1 (σcx − σcz ) 2

(3.26)

2 =

 1 σcx − σcy 2

(3.20)

(3.28)

(3.29)

α1 ≤

    1 σcx + σcz fc − (k − 1) k +1 2

(3.30)

σcx + σcy 2

(3.31)

α2 ≤

σcy σcx + σcy fc + − k k 2

(3.32)

α2 ≤

    σcx + σcy 1 fc − (k − 1) k +1 2

(3.33)

≤ fc ⇒ (3.21)

(k − 1)

2 + τ 2 (k + 1) ≤ f ⇒ 12 + τcxy c cxz

σcx + σcz 2

fc σcz σcx + σcz + − k k 2

α2 ≤ fc − k · σcy +



(3.27)

α1 ≤



2 + τ2 12 + τcxy cxz

2 + τ2 ≤ α 12 + τcxy 1 cxz

1 =

which yields another quadratic cone constraint.





α1 ≤ fc − k · σcz +

Case III: k · λ2 − λ3 ≤ fc The last combination is when λ1 lies between λ2 and λ3 which gives the following constraints when inserted into Eq. (3.6)

(3.23)

The above conditions are used when σcz is the largest of the transverse normal stresses. The case when σcy is the largest normal stress is handled by interchanging the z-indicies with y in the above constraints adding an additional three cones and another auxiluary variable 2 to the set of inequality constraints. We can reduce the number of cones by introducing two more auxiluary variables, α1 and α2 . The complete yield criterion can be written as a combination of two cones, two linear equality constraints and six linear inequality constraints as shown in Eq. (3.24)–(3.33).

Case I: k · λ1 − λ3 ≤ fc Here we assume that λ1 is the largest of the three eigenvalues. λ3 must therefore be the smallest and when inserted into Eq. (3.6) we get k · σcz −

2 + τ2 ≤ 12 + τcxy cxz

4 NUMERICAL TESTS (3.22)

This section shows some numerical examples of the beam element presented in this paper. In the first

359

example a single element is subjected to some basic load cases and the results are compared to analytical solutions for verification. The second examples utilizes the flexibility of the zone model to analyze a beam with a more complex cross section.

reinforcement varies along the length as described in the figure. Because a single element must have the same shear reinforcement throughout, 5 elements are required to model the beam. The cross section is modelled using 37 zones as illustrated on Figure 4 and the material parameters are noted on the figure. Initially, the uni-axial bending capacity of the cross section with respect to the y-axis is found to be My = −2073 kN/m. When a constant line load is applied along the geometric center line, the load bearing capacity is found to be pz = 119 kN/m. It should be noted that the maximum bending moment is only 689 kN/m The load bearing capacity is therefore governed by the shear strength of the beam. As part of a renovation project in Copenhagen, Ramboll conducted a series of tests on beams similar to the one shown on Figure 4. Five tests were done and the capacity was here found to be pz = 136 kN/m. The material strength given on Figure 4 are design strength provided by the

4.1 Basic load cases For the basic element tests, a simple rectangular beam modeled using the 9 zones shown on Figure 3 is used. The width and height of the beam is 250 mm and 600 mm respectively and the thickness, t, of the outer zones is 76 mm. The reinforcement diameters are 20 mm and 6 mm for the longitudinal and the stirrups respectively. The material parameters for concrete and reinforcement are as shown below fYt = 550 MPa fYc = 0 fc = 25 MPa ft = 0

k=4

All examples are analyzed using both the exact and the approximate Mohr-Coulomb model. The tests results are presented in Table 1. 4.2 Analysis of inverse T-Beam The rectangular beam analyzed in the previous section is similar to the one presented in (Niebling, Vinther, and Larsen 2007). The major advantage of the element presented here is its ability to model more complex cross sections. This example demonstrates how the flexible zone model can be used to determine the load bearing capacity of the inverse T-beam shown on Figure 4. The beam is 6.8 m long and the shear

Figure 4. Cross section of inverse T-beam with zone division model.

Table 1. Results obtained from the exact, and approximate method compared to an analytical solution. Forces are in kN, moments in kNm and line loads in kN/m. Static model

SDP SOCP

Analytical

Comments

Fx

691

691

691

Fx

4626

4626

4500

My

272

272

273

The lower capacity found by the numerical method is caused by the zone meshing.

22

22

21

The maximum moment is identical to the numerical moment capacity found above.

47

47

51

The lower capacity found by the numerical method is caused by the combined shear stresses in the corner zones.

pz

Mx

360

The numerical and analytical solutions are identical because no approximations are made to the tensile strength of the reinforcement. The higher capacity found by the numerical method is caused by the tri-axial effects found in the corner zones.

manufacturer which are often lower than the actual strength. It is therefore expected that the capacities found from tests are higher than the one found by the numerical model.

and analyzed and the numerical results were compared to those obtained from physical tests. The load bearing capacity found by the numerical model was only 13% less than those found from the tests which is considered very good when taking the the uncertainties of the material properties into account.

5 CONCLUSION We have presented a method for determining the load bearing capacity of 3D beams subjected to both bending- and torsional moments as well as normal and shear forces. The method circumvent the need for a complex yield criteria based on section forces by representing the internal stress state by a discrete set of zones. The material constraints are then applied directly at the stress-state level. The Mohr-Coulomb failure criteria is used to constrain the stress state in the concrete. Two different methods are employed to solve the non-linear problem posed by this criteria. The first method is Semidefinite Programming which makes it possible to implement the exact Mohr-Coulomb criteria. It does suffer from some performance issues on the current solvers. A method using only second-order cones was therefore developed to reduce computation times. This method makes good approximations to the exact Mohr-Coulomb and is sufficient for most analysis cases. The numerical model was tested using a range of basic load cases. The results were then compared with analytical results and it was found that the correlation was very good. An inverse T-beam was also modelled

REFERENCES Andreasen, Bent Steen Nielsen, M.P. (1985). Armering af beton i det tredimensionale tilfælde. Bygningsstatiske Meddelelser 56(2–3). Damkilde, L. and O. Hoyer (1993). An efficient implementation of limit state calculations based on lower-bound solutions. Computers and Structures 49(6), 953–962. Damkilde, L., J.F. Olsen, and P.N. Poulsen (1994). A program for limit state analysis of plane, reinforced concrete plates by the stringer method. Bygningsstatiske Meddelelser 65(1), 1–26. Krabbenhoft, K. and L. Damkilde (2002). Lower bound limit analysis of slabs with nonlinear yield criteria. Computers and Structures 80(27–30), 2043–2057. Larsen, K.P., P.N. Poulsen, and L.O. Nielsen (2009). Limit analysis of solid reinforced concrete structures. In The First International Conference on Computational Technologies in Concrete Structures. MOSEK. The mosek optimization toolbox for matlab manual. Niebling, J., A. Vinther, and K.P. Larsen (2007). Numerisk modellering af plastiske betonkonstruktioner. Poulsen, P.N. and L. Damkilde (2000). Limit state analysis of reinforced concrete plates subjected to in-plane forces.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

The role of domain decomposition techniques for the study of heterogeneous quasi-brittle materials O. Lloberas Valls, D.J. Rixen, A. Simone & L.J. Sluys Delft University of Technology, Delft, The Netherlands

ABSTRACT: We focus on the analysis of fracture in quasi-brittle materials by exploiting the potential of domain decomposition techniques. More specifically, we restrict our attention to the FETI (Finite Element Tearing and Interconnecting) method which is used as a solver in our non-linear solution scheme. We develop criteria to anticipate the linear/non-linear character of different regions in the structure according to a damage constitutive model. A first application of our scheme focuses on the efficiency increase of a monoscale analysis by simplifying the computations in those areas that remain linear. The second application treats the problem in a multiscale fashion where the resolution of the non-linear domains is increased in order to describe failure phenomena with a higher degree of accuracy. Both applications represent a significant improvement from a computational standpoint when the main non-linear regions are small compared to the size of the whole specimen. This is often the case for brittle and quasi-brittle materials where strain localization is expected to occur upon failure. 1 INTRODUCTION The study of cracking and failure phenomena is of utmost importance in the design of many engineering materials. Damage nucleation and growth can eventually cause the collapse of an existing structure. For this reason, the accurate modeling of these phenomena has been a topic of ongoing research in the last decades. It is therefore crucial to model, with great care, those regions in the material that can potentially show failure and strain localization. In this study we develop strategies to focus the computational effort preferentially at those regions of the structure that undergo the main non-linear processes. A domain decomposition method is adopted to split the discretized structure in several domains and, based on certain criteria, allow for a different treatment between linear and non-linear regions. Some preliminaries of the selected domain decomposition technique are introduced in Section 2. The criteria to predict the linear and non-linear constitutive behaviour of the material is described in Section 3. Two applications of the present framework are elaborated in Sections 4 and 5 including an illustrative example for each of them. They basically consist on: (i) an improvement of the solver performance by simplifying a number of operations at the elastic domains; (ii) a zoom-in at the areas where damage growth is taking place.

equations. Consequently, different strategies based on a divide and conquer approach are well suited for these complex systems. One of the most popular choices is the use of domain decomposition techniques. These methods can be used as parallel efficient solvers for the partitioned systems of equations arising from the mechanical analysis of a decomposed structure. In the following we introduce the basic principles of the FETI (Finite Element Tearing and Interconnecting) technique introduced by Farhat and Roux (1991) which constitutes the starting point of the present framework. Consider a body which is divided in two domains (Fig. 1). Continuity of the solution field between domains is enforced by the interface constraint u(1) = u(2)

at ŴI .

(1)

The variational forms of the equilibrium problem arising from each of the domains in combination with the continuity condition (Eq. 1) results in a hybrid variational principle. Finite element discretization using a standard Galerkin procedure transforms the hybrid

(2)

(1)

2 THE FETI METHOD I

The computation of structures with a high degree of resolution leads to the solution of large systems of

363

Figure 1.

Decomposition in two domains.

The operator F I represents the flexibility of the interface. In absence of the rbm the matrix F I sets the relation between the forces λ and the displacements d at the interface. The operator G I is built of the rigid body modes of each domain restricted onto the interface and e is the part of the applied force that is out of balance with respect to the subdomain rigid body modes. The FETI method can be regarded as a solver for a decomposed structure which is formed by a blend of direct solvers for the independent local problems (Eq. 3) and iterative solvers for the coupled interface problem (Eq. 5). A short overview of primal and dual domain decomposition methods is presented by Rixen (2001). The FETI method, and in particular its dual-primal version has become one of the most used parallel solution techniques.

variational principle in the set of equations ⎡

K (1) ⎣ 0 B(1)

0 K (2) B(2)

⎤ ⎡ ⎤ T ⎤⎡ B(1) f (1) u(1) T (2) (1) ⎦, B(2) ⎦ ⎣ u ⎦ = ⎣ f λ 0 0

(2)

where K (s) and B(s) represent the stiffness and signed Boolean matrices for domain (s) . The solution field is composed by the domain displacements u(s) and the connecting forces λ. The solution of Equation 2 for a given set of λ forces and a particular configuration of the rigid body modes (rbm) reads T

+

u(s) = K (s) (f (s) − B(s) λ) − R(s) α (s) .

(3)

(s) +

is introduced in The generalized inverse K order to handle local singularities induced by floating domains (i.e. domains which are not fixed by any boundary condition once they are disconnected from its neighboring domains). Conversely, the matrix + K (s) is the inverse of K (s) if it is non-singular (i.e. the domain is isostatically or hyper-statically constrained). All floating domains exhibit rbm R(s) spanning the null space of K (s) . The amplitudes α (s) of the rbm are determined by imposing self-equilibrium between the prescribed external forces f (s) and the tying forces T B(s) λ between domains: T

T

R(s) (f (s) − B(s) λ) = 0.

3 LINEAR/NON-LINEAR STRAIN PREDICTORS The formulation of the domain strain indicators is obviously linked to the considered constitutive relation. In the following we restrict to a particular damage model for the modeling of softening materials. All inertia effects are neglected in this study (i.e. quasistatic loading) and we consider small deformations and rotations. However, it should be stressed that similar predictors can also be formulated on the basis of different inelastic laws.

(4)

The tying forces λ and the amplitudes α of the rbm are determined by the solution of the interface problem by substituting Equation 3 in the compatibility condition (Eq. 1) and taking into account the orthogonality condition in Equation 4:      FI GI d λ , (5) = T e α GI 0

3.1 Constitutive model and implementation in the FETI framework Failure mechanisms are simulated in this study considering a continuous degradation of the elastic material moduli via the introduction of a damage variable as described by Lemaître and Chaboche (1994). The total stress σ and strain ǫ tensors are related through the isotropic elasticity-based damage constitutive law

with

FI =

d=

Ns 

B(s) K

(s)+

B

(s)T

σ = (1 − ω)De ε,

+

B(s) K (s) f (s) ,

f (˜ε, κ) = ε˜ − κ,

s=1

GI =



B(1) R(1) . . . B(s) R(s)

⎤ α (1) ⎢ . ⎥ α = ⎣ .. ⎦ ⎡

α (s)

(7)

The range of the scalar variable ω represents the transition from a virgin material (ω = 0) with intact elastic moduli De into a fully damaged one (ω = 1). Damage growth is controlled by the damage loading function

,

s=1

Ns 

ω ∈ [0, 1].



,

⎤ T R(1) f (1) .. ⎢ ⎥ and e = ⎣ . ⎦. ⎡

(6)

T

R(s) f (s)

364

(8)

where κ is a history dependent parameter which reflects the loading history. The equivalent strain ε˜ corresponds to an invariant of the strain tensor. In the following we adopt a specific definition of the equivalent strain introduced by Mazars and Pijaudier-Cabot (1989) in which only the tensile strains are considered relevant. The initial strain κ0 dictates the initiation of damage in a particular point. The evolution of the

deformation history parameter κ is governed by the Kuhn-Tucker relations f ≤ 0,

κ˙ ≥ 0,

f κ˙ = 0.

(9)

An exponential damage evolution law ω(κ) is considered in this study where critical damage (ω = 1) can only be reached asymptotically at infinite strain values. Without a regularization strategy strains would localize into a narrow band of infinitesimal width while its value would approach infinity. This would cause the problem to become ill-posed. A way to circumvent this drawback is the introduction of a non-local strain quantity as argued by Bazant and Pijaudier-Cabot (1988). In this study we consider a differential version of the non-local damage model introduced by Peerlings et al. (1996) as the Gradient Enhanced Damage model. The governing equations result in a coupled system between the equilibrium equation and a diffusion equation. In the absence of body forces the governing equations read ∇ ·σ =0 ε˜ (x) = ε˜ nl (x) − c∇ 2 ε˜ nl (x),

(10)

preconditioners can improve the convergence as indicated by Rixen and Farhat (1999). 3.2 Domain strain indicators The goal of an appropriate indicator is to predict sufficiently early (i.e. at least one step ahead of the most recent computed information) the linear or non-linear behaviour of a domain (or part of it). Moreover, the information on which the predictor is based should not be expensive. A convenient strategy seems to be the use of solution field data so it is not needed to descend to the Gauss point level. 1In the current constitutive model the non-local equivalent strain ε˜ nl , available at the solution field, and its history can be used to estimate a strain prediction for the coming steps. This procedure can be performed domain-wise (i.e for each domain (s) ) with the evaluation of the corresponding domain loading function f (s) :

p,(s) 

c,(s)  p,(s) f (s) ǫ˜eq − κ (s) ǫ˜eq = ǫ˜eq , (12)

with

p,(s) c,(s) c,(s) = ǫ˜eq + δ ǫ˜eq , c,(s) c ǫ˜eq = max(˜ǫeq,i ), ∈ (s) ,

c,(s) 

c,(s) (s)  ,κ . = max ǫ˜eq κ (s) ǫ˜eq

ǫ˜eq

ε˜ nl (x) being the non-local equivalent strain and ε˜ (x) 2 the local equivalent strain. Moreover c = l2 and l is the internal length scale of the gradient enhancement. This parameter represents the internal length scale needed to regularize the localization of deformation and is related to the width of the localization band. The variational formulation of the governing equations results in the hybrid variational form, in which the inter-domain continuity condition is accounted for. After discretization using FE and linearization, a set of equations analogous to the one shown in Equation 2 (partition in two domains) is recovered. The field of connecting forces λ now reads   λx , (11) λ= λε˜nl being λx the field of spatial connecting forces and λε˜nl the connector that glues the non-local equivalent strain dof arising from the diffusion equation. The FE stiffness resulting from the implementation of the coupled governing equations turns out to be non-symmetric and, consequently, the flexibility operator F I of the interface problem (Eq. 5) becomes non-symmetric as well. If an iterative solver is chosen for the solution of the interface problem, the Bi-Conjugate Stabilized Gradient (Bi-CGSTAB) or Generalized Minimal Residual (GMRES) can be appropriate candidates as explained by Barret et al. (1993). For the case of a highly heterogeneous interface problem (e.g. when considering heterogeneous materials or growth of damage in a structure) special

365

(13)

The superscripts c and p denote current and predicted values, respectively. The subscript i indicates the ith nodal point of the corresponding domain (s). The growth of f (s) is controlled by the Kuhn-Tuker loading-unloading conditions f (s) ≤ 0,

κ˙ (s) ≥ 0,

f (s) κ˙ (s) = 0.

(14)

The construction of a safe strain increment for each c,(s) is of crucial importance to build a cordomain δ ǫ˜eq rect strain prediction. In Equation 15, two different c,(s) are described choices for the strain increment δ ǫ˜eq considering a constant load stepping. If the external load increments are not constant the strain increment should be scaled to the future load step increment. The first option (∗) is a more optimal choice whilst the latter (∗∗) provides always a safer guess. This is because the first choice computes the maximum of all nodal increments. The second choice computes the difference between minimum and maximum strains at step t and t − 1 of different nodal points. The proposed strain increments read    c,t c,(s) c,t−1  (∗)δ ǫ˜eq = max ˜ǫeq,i − ǫ˜eq,i ,

(s) (s)  c,(s) (∗∗)δ ǫ˜eq = max δ ǫ˜1 , δ ǫ˜2 , 

c,t 

c,t−1  (15)  δ ǫ˜1(s) = max ǫ˜eq,i − min ǫ˜eq,i  , ∈ (s) , 

c,t−1 

c,t   δ ǫ˜2(s) = min ǫ˜eq,i − max ǫ˜eq,i  , ∈ (s) .

The non-linear behaviour of domain (s) is deterp,(s) mined when the predicted strains ǫ˜eq reach the initial (s) domain strain κ0 . In the absence of non-linearities over the whole specimen the transition to a non-linear regime can be predicted at (s) using strain increments which are based on past information. It is stressed that deformations and rotations are small and, for this reason, no geometrical non-linearities can occur. Nevertheless, when damage growth is initiated in a certain region of the specimen the strain history at a particular linear domain (s) might not be sufficient to construct a safe strain increment. In this scenario, it is expected p,(s) that the predicted strains ǫ˜eq for step step t are found c,(s) to be smaller than the current strains at time t ǫ˜eq . In this situation, it might be dangerous to assume a linear behaviour at domain (s) at time t (see Section 4) and a correction should be done in the previous step (i.e. the linear enhancements described in Section 4 can not be applied) in order to avoid deviating from the true solution path. Considering the previous domain strain indicators it is possible to predict the linear character of the forthcoming steps using the criteria shown in the following flowchart (Fig. 2). Three different situations are possible: linear loading (switch = 0), non-linear loading (switch = 1) and linear unloading (switch = −1). The value assigned to the switch is exploited in the next sections in order to trigger the enhancements to the FETI framework.

(s)

c (s) eq

+


0 [uyd (ζi )] =   3 1 2  1 h l  d 2 for y < 0 [v2 ] ζ2 + 2 ζ1 1 − l (35) defines outside the crack ud = [uyd ]      d − ζ3 + 12 ζ1 1 − hl [v2 ] for y > 0 d      N = for y < 0 [v2d ] ζ2 + 12 ζ1 1 − hl vd = [v2d ]

(36)

3.2 Strains Outside the crack the strains are defined as usual. Here ǫx = 0, so the strain vector reduces to     uy,y ǫy = (37) ǫ= γxy uy,x From the continuous displacements in Equation 26 are obtained

h = ζ1 l

∂ (ζ2 − ζ3 )v2c ∂y

c ǫyc = uy,y =

  h 1 ζ2′ = ζ2 + ζ1 1 − 2 l   h 1 ζ3′ = ζ3 + ζ1 1 − 2 l

1 In the crack ζ2 = ζ3 = 1−ζ 2 . Then Equation 28 gives      h d h 1 d v2 = − 1 − ζ1 v2 un = −2 ζ3 + ζ1 1 − 2 l l (32)

(30)

With Equation 30 and 27 the discontinuous displacements can be expressed in area coordinates from triangle 123     − ζ + 1 ζ 1 − hl v2d for y > 0  uyd (ζi ) =  3 1 2 1 ζ2 + 2 ζ1 1 − hl v2d for y < 0 (31)

377

−x3 + x1 −x1 + x2 − 2A 2A

=



=

v2c 2h c v = 2 b 2 12 h2b

c = γxyc = uy,x

=



v2c

(38)

∂ (ζ2 − ζ3 )v2c ∂x 

y 1 − y2 y3 − y1 − 2A 2A



v2c = 0

(39)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ d B =  ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Writing Equation 38 and 39 on the form d



ǫyc γxyc



=

1 b





0

determines  c  ǫy ǫc = γxyc

v2c

(40)



Bc =

1 b

0



(41)

From the discontinuous displacements in Equation 35 is obtained in the upper half sub triangle d = ǫyd = uy,y

∂ ∂y



d = γxyd = uy,x

h v2d 2 12 h2b

=

v2d 2b

(42)

    ∂ 1 h − ζ3 + ζ1 1 − v2d ∂x 2 l 

= −

1 h y1 − y2 − 1− 2A 2 l 



vd hb v2d = 2 = 1 2l l2 2 h2b

y 2 − y3 2A

ǫyd γxyd



for y < 0

(47)

(48)



The crack parameter l is found in Equations 36, 47 and 49. The non-zero crack growth derivatives occur in Equations 9, 15 and 16, and they give   ∂NdT (x, y, l) 1 h = − ζ1 2 ∂l 2 l

v2d

⎧  ⎪ 0 − 2l12 ⎨ ∂B (x, y, l) =   ⎪ ∂l ⎩ 0 12 2l dT

(43)

(44)

(49)

Crack growth derivatives

3.3

3.4

(50)

for y > 0 (51) for y < 0

(52)

Crack growth matrices

The element load matrix from crack growth is defined in Equation 20, inserting Equation 50 gives

i.e. v2d 2b

γxyd = −

v2d 2l

(45)

ra =

=

Writing Equation 42 and 43 on the form

ǫyd γxyd



   h ǫ cr = [un ] Bcr = − 1 − ζ1 l

ǫyd (−y) = ǫyd (y)



1 2b −1 2l

   h [un ] = − 1 − ζ1 [v2d ] l

  ∂BcrT (x, y, l) h = −ζ1 2 ∂l l

ǫyd =

for y > 0

In the crack the strains are given by Equation 16, in this case only un is non-zero. Writing the equation on the form

Symmetry about the x-axis gives in the lower half sub triangle γxyd (−y) = γxyd (y)



defines

    h 1 − ζ3 + ζ1 1 − v2d 2 l

    −x1 + x2 1 h −x2 + x3 d = − − 1− v2 2A 2 l 2A =

ǫ =



1 2b 1 2l

⎧ ⎪ ⎪ ⎪ ⎪  ⎪ ⎨ =  ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 2b 1 2l



1 2b −1 2l



v2d

for y > 0

tipel−cr

1 h − ζ1 2 fy = 2 l subtri

(46) v2d

=−

for y < 0

∂NdT (x, y, a) f ∂a

bfy ≡ ra 6

1 1 − ζ1′ fy 2 l subtri

(53)

The internal element nodal force matrix from crack growth is defined in Equation 19, and gives inserting

which determines

378

the results from Equation 51 and 52, when symmetry along the x-axis is utilized qa =

tipel−cr

∂BdT (x, y, a) σ ∂a

=2



+

−ζ1

cr

uhs

cr

=

∂BcrT (x, y, a) σ ∂a

+

0 − 2l12





(54)

σy τxy

h σy ≡ q a l2

1 bl bh bl 1 h−l = σyc 2b + σyc+d 2 = σyc + σyd 2 A 2 A A A (56)

  E c bl bh 1 + σyd = v2 h + v2d l = ft (57) = σyc A A A 2

The incremental form of Equation 57 is 1 1 hdv2c + ldv2d + v2d dl = 0 2 2

(58)

3.6 FEM equations Extending the FEM equations 21 with 58 gives k cd k dd 1 2l

k cd = k dc =

⎤⎡ c ⎤ ⎡ c ⎤ 0 dv2 dr qa − r a ⎦ ⎣ dv2d ⎦ = ⎣ dr d ⎦ 1 d 0 dl 2 v2



1 b

0



E 0

BcT DT Bd



triangle

1 b

0

 1 

0 G

b

0



E 0

0 G

=

Eh b



1 2b ± 2l1

(60)



=

El 2b

(61)

(55)

where the stress superscript indicates the contributing field. Using the constitutive condition σy = Eǫy , where E is Young’s modulus outside the crack. Equation 56 gives together with Equation 38,42 and 23

k cc ⎣ k dc h

triangle

BcT DT Bc

triangle

The nominal stresses in the area immediately in front of the crack-tip is determined with an area weighting of the triangle and sub triangle stresses at T. Because of symmetry the nominal shear stress τxy vanish, while

⎡

triangle

=

3.5 Crack tip conditions

σynom



where uhs means the upper half sub triangle.

σynom

k cc =

(59)

where the matrix elements k cc , k cd , k dc , k dd are obtained as follows

379

k dd =

BdT DT Bd +

=



subtriangle

subtriangle

+

1 2b

±

1 2l

cr



E 0

cr BcrT Dcr T B

0 G

1 2b ± 2l1





   

  h h [Dn ] − 1 − ζ1 − 1 − ζ1 l l cr

  b l 1 1 E +G + Dn l = 4 b l 3

(62)

Equation 59 represent the extended system of equilibrium including the crack length as a variable. 4 CONCLUSION In the present work crack geometry parameters has been included as direct variables in a XFEM formulation. By including these crack geometry parameters directly in the formulation and introducing the incremental relations the propagation of the crack can be handled more directly and with fewer iterations. The crack parameters are introduced through the principle of virtual work, and appear as direct variables in the equations. A set of conditions concerning crack growth has been set up in order to determine the crack parameters. An example with a partly cracked CST element has been considered. In this example a weighting of the stresses has been applied. The example shows the setup the crack growth matrices. In the present work the scope has been to clarify the principles of introducing crack parameters as variables, and the example with the partly cracked CST element allowed for calculations by hand. A natural extension of this work is to implement the LST element, where the performance can be tested.

REFERENCES Belytschko, T. and T. Black (2003). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering (45), 601–620. Hillerborg, A., M. Moder, and P.-E. Peterson (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res (6), 773–782. Meschke, G. and P. Dumstorff (2007). Energy-based modeling of cohesive and cohesionless cracks via x-fem. Computer Methods in Applied Mechanics and Engineering 196(21–24), 2338–2357.

380

Mougaard, J.F., P.N. Poulsen and L.O. Nielsen (2009a). Modelling concrete structures applying xfem with a mixed mode constitutive model. Submitted to FraMCoS7. Mougaard, J.F., P.N. Poulsen and L.O. Nielsen (2009b). A partly and fully cracked xfem element based on higher order polonomial shape functions for modeling cohesive fracture. Submitted to: International Journal for Numerical Methods in Engineering. Nielsen, L.O., J.F. Mougaard, J.S. Jacobsen and P.N. Poulsen (2009). A mixed mode model for fracture in concrete. Submitted to FraMCoS7.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Strong discontinuities, mixed finite element formulations and localized strain injection, in fracture modeling of quasi-brittle materials J. Oliver & Ivo F. Dias Technical University of Catalonia, Campus Nord UPC, Edifici C-1, Barcelona, Spain

A.E. Huespe INTEC-UNL-CONICET, Santa Fe, Argentina

ABSTRACT: The presented work explores a combination of strain localization methods and strong discontinuity approaches to remove the flaws (stress-locking and mesh bias dependence) of the classical strain localization methods, and reduce the sophistication of the strong-discontinuity based methods. The concept of strong discontinuity is initially substituted by that of weak discontinuity capturing the displacement jump through its distribution (smearing) in a finite length: the oriented with of a finite element through which the fracture passes in a rather diffuse manner. In other words, the classical strain localization concept is brought to the approach, no strong discontinuity enrichment is made and standard constitutive models are used in a classical strain localization setting. The standard mesh refinement recovers the strong discontinuity concept by taking the element width to zero. However, in order to remove the spurious mesh orientation dependence, in a second stage specific localization modes are injected, via mixed finite elements formulations, to the path of elements that are going to capture the cracks. These modes, as well as their injection time, are characterized from the information provided by the SDA. Then, the obtained approach enjoys the benefits of the strong discontinuities one, at a complexity similar to the classical, and simpler, localization methods. 1 MOTIVATION A review of the available approaches used for numerical modeling of material failure in quasi-brittle materials, allows roughly classifying them as: a. Strain localization methods: They use standard finite elements and classical continuum stressstrain constitutive models equipped with strain softening (Borst, Sluys et al. 1993). It is well known that they suffer from classical spurious mesh size and mesh-bias dependences. Mesh size dependence can be easily overcome by using appropriate regularizations of the softening modulus values and resorting to the fracture energy concept. However, the spurious mesh-bias dependence cannot be easily removed. Only non-trivial, and sometimes unphysical, modifications of the constitutive model by introducing an internal characteristic length (viscous, non-local or gradient regularizations) and requiring very fine meshes to capture fracture line/surfaces have proved to remove it. However, their use for engineering purposes, in modeling fracture of real 3D concrete and reinforced concrete structures, seems limited due to their inherent high computational cost (Oliver and Huespe 2004). b. A subsequent sophistication, stemming from the 90’s of the past century, brought the so-called

381

strong discontinuity approach (SDA) (Simo, Oliver et al. 1993). There, a real discontinuity in the displacement field (strong discontinuity) was aimed at being modeled and captured via an appropriate mechanical theory (the strong discontinuity theory) and different families of standard finite elements enriched by discontinuous deformation modes, namely: finite elements with embeddeddiscontinuities (elemental-based enrichment) or X-fem methods (Belytschko, Moes et al. 2001) (nodal-based enrichment) (Oliver, Huespe et al. 2006). At the cost of resorting to some additional sophisticate tools, like multiple crack tracking algorithms, specific integration algorithms etc., the method proves to be efficient in providing results free from the mesh-size and orientation dependences, and showed able to account for the propagation of multiple cracks in two and even three-dimensional settings at a relative low computational cost (Oliver and Huespe 2004). However, the sophistication of the required tools seems to place a limitation on their incorporation to commercial simulation codes and to their use for real-life industrial purposes. The presented work explores a combination of both approaches to remove the flaws of the classical strain localization methods and reduce the sophistication of

the strong-discontinuity based methods. The concept of strong discontinuity is initially substituted by that of weak discontinuity capturing the displacement jump through its distribution (smearing) in a finite length: the oriented with of a finite element through which the fracture passes in a rather diffuse manner. In other words, the classical strain localization concept is brought to the approach, no strong discontinuity enrichment is made and standard constitutive models are used in a classical strain localization setting. The standard mesh refinement recovers the strong discontinuity concept by taking the element width to zero. However, in order to remove the spurious mesh orientation dependence, in a second stage specific localization modes are injected, via mixed finite elements formulations, to the path of elements that are going to capture the cracks. These modes, as well as their injection time, are characterized from the information provided by the SDA. Then, the obtained approach enjoys the benefits of the strong discontinuities one, at a complexity similar to the classical, and simpler, localization methods.

Bh , containing S, and µS stands for the h-regularized Dirac’s delta function whose support is Bh . In this sense, the strain field can be approximated as:  ε¯ (x, t) ε(x, t) ≈ 1 S h ([[u]] ⊗ n) (x, t)

x ∈ B/Bh x ∈ Bh

On the other hand, strain localization can be understood as a concentration and intensification of the strain in a geometrical band, Bloc ⊂ B, of width ℓ, containing the path of a strong discontinuity. The corresponding strain field reads:  ε¯ (x, t) ε(x, t) ≡ εloc (x, t)

x ∈ B/Bloc x ∈ Bloc

2 STRAIN LOCALIZATION AND WEAK DISCONTINUITIES Let us consider a strong discontinuity in the displacement field, u(x, t), occurring at a discontinuity line, S, of unit normal n, in a body B (see Figure 1):

β = (1/ℓ) [[u]] (x, t)

(1)

where u¯ stands for the regular (continuous) displacement field, [[u]] is the displacement jump and HS is the step (Heaviside) function on S. The regularization of the corresponding infinitesimal strain, ε, (weak discontinuity kinematics) reads: ε ≡ ∇ s u = ε¯ +

µS ([[u]] ⊗ n)S h

(2)

where ε¯ is the regular strain, h is the (very small, thus tending to zero) width of the regularization band,

Figure 1.

(4)

Comparison of equations (3) and (4), and the fact that they aim at capturing the same physical phenomenon (a propagating discontinuity) suggest that a physically meaningful localized strain field should take the format:  ε¯ (x, t) ε(x, t) ≡ εloc (x, t) = (β ⊗ n)S

u = u¯ + HS [[u]]

(3)

x ∈ B/Bloc x ∈ Bloc

(5)

where β is a field proportional to the jump, obtained by comparison of equations (3) and (4), in terms of the inverse of the localization bandwidth ℓ. Equation (5) is the basis of the strain injection concept in the proposed finite element approach to strain localization and the term εloc (x, t) = (β ⊗ n)S will be termed, from now on, the strain localization mode. 2.1 Mesh size and mesh bias dependency and the strain localization mode On the light of equation (5) one could think that the ability of a given finite element to appropriately capture strain localization depends on the measure that the localization mode can be reproduced and propagated trough the finite element mesh. Regarding this ability there are two issues to be considered here: a. The dependence of the strain on the localization bandwidth ℓ. In fact, this gives raise to the classical mesh-size dependence issue. Different kinds of remedies provide a solution for this problem: e.g. by resorting to concepts of the Strong Discontinuity Approach (SDA) the regularization of the softening modulus, H = ℓHint , in terms of the intrinsic softening modulus Hint which is considered a material

Strong discontinuity in a body

382

property, computed, (for isotropic continuum damage models) as: Hint =

σy2

(6)

2EGf ℓ

in terms of the uniaxial peak stress, σy , the Young modulus E and the Fracture Energy Gf , overcomes such a dependence also for strain localization approaches. b. The dependence of the resulting strain localization on the orientation of the finite element mesh (mesh-bias dependence): i.e. the obtained localization path tends to propagate spuriously parallel to the sides of the elements, thus dramatically changing with the finite element mesh orientation in structured meshes. In the context of strain localization methods, using standard finite elements, this flaw can only be overcome by changing the assumed physical properties of the material: i.e.: the constitutive model. Introduction of rate dependence (viscous models), non-locality or gradient dependency, partially solves the problem at the cost of using tinny elements at the localization zone, thus substantially affecting the computational cost. In this work, we consider the injection, in the finite element mesh, of specific strain localization modes as a possible remedy for the spurious mesh-bias dependency.

solution, n  = 0 and, in general, two solutions (n ≡ n, n ≡ m) for the normal to the weak discontinuity. Let us also consider the constitutive model equipped with strain softening, and the strain-like internal variable, α (where, as usual, α˙ = 0 for elastic/unloading cases and α˙ > 0 for loading cases) and the corresponding stress-like internal variable, q(α) defining the hardening/softening law (see Figure 2). In terms of this, the injection zone is determined as the geometrical locus of the in-loading bifurcated points (see Figure 3), i.e.: ˙ t) > 0} Binj (t) := {x ∈ B | t ≥ tB (x); α(x,

(8)

3.2 Injection of an elemental constant strain mode As soon as a given finite element (whose bifurcation time is characterized, for numerical implementation purposes, by that of its central point) belongs to the injection domain Binj , it is equipped with a strain field whose ability to capture and propagate localization modes is superior to that of the standard underlying element. Thinking of two-dimensional problems, we consider here the four-noded bilinear quadrilateral as the underlying element. Then, a first injection of an elemental constant strain mode is performed via a mixed u − ε formulation restricted to Binj . The

3 THE INJECTION PROCEDURE The proposed method lies on the following steps: a) determination of the injection domain, b) determination of the localization domain and c) the injection of specific deformation modes. They will be described in next sections. Figure 2.

Hardening softening law and injection threshold.

Figure 3.

Injection and localization zones.

3.1 Determination of the injection domain Again, we borrow from the SDA the concept of the inception of a weak discontinuity indicated by the discontinuous bifurcation analysis. According to this, a weak discontinuity onsets as the following condition takes place, for the first time, at the bifurcation time tB , at a given material point x: det Q(x, tB ) = 0;

Q(n(x), tB ) = n · C(x, tB ) · n (7)

where Q is the so called localization tensor, computed in terms of the tangent constitutive operator C (fulfilling σ˙ = C : ε˙ ), which stems from the chosen constitutive model. Equation (7) provides the bifurcation time tB (the first time that it produces a nontrivial

383

corresponding variational equations, in rate form, read as follows:  ˙ s u)dB ∇ s η : (∇ ˙ B\Binj

+ 

 

B(e)

e∈Binj

(e) ∇ s η : (ε˙¯ )dB − Gext = 0

(e)

B(e)

µ(e) : {∇ s u˙ − ε˙¯ }dB = 0 ∀e ∈ Binj

(9a)

(9b)

where η stands for the virtual admissible displacements, (•) for the function returning the stresses in the constitutive model, ε¯ (e) the element-wise constant injected strain field and µ(e) the corresponding element-wise constant weighting function. Solution of equation (9)-(b) is then trivially found as: (e) (e) ε¯˙ = ∇ s u˙

(10)

(e)

where (•) stands for the mean value of (•) in the element (e). Substitution of equation (10) into (9)-(a) yields:  ˙ s u)dB ∇ s η : (∇ ˙

where Fint and Fext stand, respectively, for the internal and external forces, A is the assembling operator, B(e) is the standard deformation matrix, σ is the stress vector, K is the tangent stiffness matrix and D is the tangent constitutive matrix (Voigt’s notation is considered everywhere in the equations). As it will be shown in the examples, the injection of that constant strain mode at the injection domain exhibits remarkable properties as for the correct propagation of the strain localization in initial stages. However, as deep localization takes place, in some cases it still exhibits a limited capability to replicate the strain localization mode, εloc , of equation (5), and some degree of stress-locking can appear. Therefore, an additional injection mode is required.

3.3

Recalling the bifurcation time, tB , already defined as the first time that equation (7) is fulfilled, the corresponding value of the stress-like internal variable is defined as: qbif (x) = q(x, tbif ) : ∀x ∈ Binj )

+

e∈Binj

(e)

B(e)

˙ s u˙ )dB − Gext = 0 ∇ s η : (∇

qloc (x) = γ qbif (x) : γ ∈ [0, 1]

(11)

Fint − Fext = 0  Fint = A e∈B\Binj

+ A

e∈Binj

A

e∈B\Binj



+ A

e∈Binj

B(e)



(e)

B(e)

B(e)



˙ s u˙ (e) )dB B(e)T · σ(∇

B(e)T · D · B(e) dB B

B(e)

(e)T

·D·B

(e)

˙ t) > 0} Bloc (t) := {x ∈ Binj |q(x, t) ≤ qloc (x) : α(x,

(15)

where parameter γ in equation 14 defines the degree of softening of points in Bloc ⊂ Binj . Then Bloc is the locus of bifurcated in-loading points, which have experienced a sufficient degree of softening determined by γ . For practically uses it is taken γ ∈ [0.95 − 0.99].



˙ s u˙ )dB B(e)T · σ(∇



3.4 Injection of an elemental localization mode Similar to what has been done in section 3.2, a specific localization mode is injected in all the elements whose central point belongs to Bloc .



dB

(14)

Consequently, the localization zone is defined as (see Figure 3):

which justifies the use if the term injection for the used procedure. In fact, equation (11) corresponds to the virtual work principle where the strain field, ∇ s u˙ (e) , is injected in the constitutive equation in those elements belonging to the injection zone Binj . After the corresponding discretization procedure, equation 3.3 yields the following equations in matrix form:

K=

(13)

Then, a new variable, controlling the softening depth, is defined at each point as:

B\Binj

 

Determination of the localization domain



(12)

(e) (x, t) = (β (e) ⊗ n(e) )S εloc

384

(16)

such that:

The corresponding variational equations read: 

B\Binj

+

+ 

B(e)

Ploc : (•) = (•) − (•)tt (t ⊗ t)

˙ s u)dB ∇ s η : (∇ ˙ 



e∈Binj \Bloc

 

e∈Bloc

(•)tt = (t ⊗ t) : (•) = t · (•) · t

(e)

B(e)

∇ s η : (ε˙¯ )dB

(17a)

and Ploc : (•) ∈ Eloc . After the corresponding discretization procedure, equation (18) yields the following equations in matrix form:

∇ s η : (˙εloc )dB − Gext = 0 (e)

B(e)

Fint − Fext = 0

(e) µ(e) : {∇ s u˙ − ε˙¯ }dB = 0 ∀e ∈ Binj \Bloc

Fint =

(17b) 

B(e)

(e) δβ (e) ⊗ n(e) : {∇ s u˙ − ε˙ loc }dB = 0 ∀e ∈ Bloc

A

e∈B\Binj

+

(17c)



A

e∈Binj \Bloc

+ A

Solving equations (17)-(b)-(c) and inserting them into equation (17)-(a) leads to: 

(21)

e∈Bloc



B(e)



B(e)

˙ ε(e) )dB B(e)T · σ(˙

B(e)



T (e) ˙ ε˙¯ )dB B(e) · σ(

˙ ε(e) B(e)T · σ(˙ loc )dB





(22)

where, in matrix notation, s

B\Binj

+

+

˙ s u)dB ∇ η : (∇ ˙ 

e∈Binj \Bloc

 

e∈Bloc



B(e)

∇ s u˙ (e) ≡ ε˙ (e) → B(e) · u˙ (e) (e)

B(e)

˙ s u˙ )dB ∇ s η : (∇

s

˙ (e) : ∇ η : (P loc

− Gext = 0

∇ s u˙

(e)

(e) (e) ¯ (e) (e) ·u˙ = B¯ loc · u˙ (e) ≡ ε˙ (e) loc → πloc · B  

(e) B¯ loc

(23)

(18) ¯ (e) is the B(e) being the standard elemental B-matrix, B (constant) B-matrix evaluated at the center of the element and B(e) loc is the localization B-matrix computed (e) in terms of the matrix version, πloc , of the localization (e) projector Ploc in equation (19):

(19)

where I stands for the symmetric fourth order unit tensor and t(e) is the tangent unit vector orthogonal to n(e) (see Figure 1) computed at the center of the element. It is trivially shown that Ploc : (•) is the orthogonal projection of a symmetric second order tensor (•) onto the strain localization sub-space εloc defined as: εloc := {εloc = (β ⊗ n)s }

(e) ¯ (e) · u˙ (e) ≡ ε˙¯ → B

Ploc : ∇ s u˙

(e) ∇ s u˙ )dB

(e) is the fourth order localization projector where Ploc defined as:

(e) Ploc = I − t(e) ⊗ t(e) ⊗ t(e) ⊗ t(e)

(e)

(e) ¯ (e) B(e) loc = πloc · B ;

(20)

385

a(e)

⎧ ⎫ ⎨(t1 )2 ⎬ = (t2 )2 ⎩2t t ⎭ 1 2

(e) πloc = 1 − a(e) ⊗ a(e)

(24)

where 1 is the second order unit tensor and t1 and t2 are the components of the tangent vector t ≡ [t1 , t2 ]T .

4.2 Injection of an element wise constant strain mode in an isotropic continuum damage model: double cantilever beam (DCB) with diagonal loads

As for the stiffness matrix, stemming from equations (22) and (24), it reads:

K=

A

e∈B\Binj

+



B(e)

A

e∈Binj \Bloc

+ A

e∈Bloc

B(e)T · D · B(e) dB





B(e)

B B(e)

(e)T

·D·B



(e)

B(e)T · D · B(e) loc dB

dB

Figure 4 shows the geometric description and the spatial and temporal loading conditions of the problem (Bazant and Cedolin 1980). The diagonal compression forces, F2 , are initially introduced together with the wedge loads, F1 , increasing along the time, until reaching 3.78 [KN]. Then, the diagonal loads remain constant while the wedge loads increase. The material parameters are, Young’s modulus: E = 30500 [MPa], Poisson’s ratio: ν = 0.2, fracture energy: Gf = 100 [N/m], ultimate tensile strength: σu = 3 [MPa] and specimen thickness: 50.8 [mm]. The reported experimental crack path follows a straight line (inclined α = 19◦ with the vertical axis) Therefore, it is rather simple to construct structured meshes such that the discontinuity path intersects the elements in arbitrary directions. Those meshes will





(25)

where the non-symmetric character of the stiffness matrix in the last term of equation (25) can be noticed.

4 REPRESENTATIVE NUMERICAL SIMULATIONS In order to assess the effectiveness of the proposed methodology in capturing strain localization processes, a set of two-dimensional examples have been considered. 4.1

Table 2.

Elasto-plastic model.

Free energy:

φ(εe , α) = 21 ε e : Ce : εe + φ p (α) ε = εe + εp

Internal variables

α˙ = λ˙ ; ˙ ; ε˙ p = λξ

Constitutive models

As representative of a broad set of models, two families of constitutive models for computational material failure have been tackled: isotropic damage models and elastoplastic models (summarized in Table 1 and 2). The reader is referred to (Oliver, Huespe et al. 2006) for an specific description of both models. Table 1.

Constitutive equation Damage/ yield function Loading-unloading condition

Free energy: Internal variables

ψ(ε, α) = (1 − d)ψo ; ψo = : εd(α) = 1 − q(α)/α √ α˙ = λ˙ ; αt=0 = αo = σu / E q α

Constitutive equation

σ = (1 − d)C : ε =

Damage/yield function

g(ε, α) ≡τε (ε) − α τε (ε) ≡ σ¯ : (Ce )−1 · σ¯

Loadingunloading condition Stress-like internal variable evolution Constitutive tangent operator

λ˙ ≥ 0;

g ≤ 0;

C : ε =

q α

Ce

:

σ = Ce : εe = Ce : (ε − ε p ) g(σ , q) = (σ ) − q λ˙ ≥ 0;

g ≤ 0;

λ˙ g = 0

Stress-like internal variable evolution

q˙ = −H α; ˙ q|t=0 = σu ;

Constitutive tangent operator

σ˙ = Ctan  : ε˙u C ≡ Ce tan C = Cl ≡ Ce −

Isotropic continuum damage model. 1 2ε

αt=0 = 0 ξ = ∂σ (σ )

q≥0 q|t=∞ = 0

Ce :ξ ⊗Ce :ξ ξ :Ce :ξ +H

σ¯ ;

σ¯

(a)

(b)

˙ =0 λg

q˙ = H α; ˙ q ≥ 0√ q|t=0 = αo = σu / E ;

q|t=∞ = 0

σ˙ =  C : ε˙ ; Cu ≡ (1 − d)Ce = αq Ce C= α Cl ≡ αq Ce − q−H σ¯ ⊗ σ¯ α3

Figure 4. Double cantilever beam with diagonal loads: (a) geometrical data and (b) loading data.

386

strongly challenge the standard localization formulations based on quadrilateral elements, which perform particularly well when the propagation direction is parallel to the element sides but exhibit mesh-bias dependence otherwise. Figure 5 shows the results (in plane stress), in terms of the localization pattern, obtained with one of those meshes for two cases: (a) using the standard bi-linear quadrilateral element and, (b) injecting in the domain Binj an element-wise constant strain mode, following the procedure indicated in section 3.2. There, it

is displayed the good match of the obtained localization pattern in case (b) with the reported experimental crack. In Figure 6 the evolution of the injection domain, Binj is displayed in four representative time steps. There, it can be observed a bulb-shaped domain, at the tip of the advancing localization band, where the material initially bifurcates and remains in inloading state, so that the elemental-wise constant deformation mode is injected. Soon later, most of the bifurcated elements behind the bulb unload (i.e.: α˙ = 0) so they leave the Binj domain, according to equation (8), excepting for an inclined band, behind the bulb and encompassing one element size, which remains in inelastic loading and defines the localization band. It is remarkable the sharp definition of this localization band, characterizing a good resolution of the weak discontinuity in a one-element-width band, propagating independently of the mesh bias (unlike for the standard quad element) and following the reported experimental crack path. No further type of injection was required in this example.

(a) 19º

4.3 Injection of constant strain and localization modes in a J2 plasticity model: strip in homogeneous uniaxial tensile stress state Figure 7 sketches the uniaxial tensile test in a plane strain state. The material properties are the following: Young’s modulus: E = 120 [Kpa], Poisson’s ratio: ν = 0.49, fracture energy: Gf = 22.72 [N/m], and ultimate tensile strength: σu = 1 [Kpa]. In the context of the strong discontinuity approach the solution of the problem consists of a straight shear band inclined 45◦ (Oliver and Huespe 2004).

(b) 19º

Figure 5. Double cantilever beam with diagonal loads: localization patterns (iso-displacement contours) with different approaches a) standard quadrilateral element b) Injection of an elemental constant strain mode.

387

t=1

t=2

t=3

t=4

Figure 6. Double cantilever beam with diagonal loads: evolution of the injection domain (shaded zone) , Binj , along different times of the analysis.

(a)

P

(b)

0.5m

P

Figure 7. Strip under uniaxial tensile stress state. Oriented mesh of quadrilaterals and theoretical discontinuity path.

In order to trigger the discontinuity, the initially homogeneous problem is slightly perturbed in terms of the tensile strength, reduced in a 10%, at two elements placed at the upper edge of the strip. The theoretical solution in terms of the P − δ curve is also displayed in Figure 10 for the linear softening case. In order to check the ability to circumvent the mesh bias dependence, a structured mesh of quadrilaterals, with an orientation of 65◦ is considered (see Figure 7) and checked in front of a number of formulations. Three of alternatives are then considered: I. The standard (irreducible) formulation II. Injection of an element wise constant strain mode in Binj ≡ Bloc (as explained in section 3.2) III. Injection of an element wise constant strain mode in Binj and, then, a localization mode in Bloc ⊂ Binj (as explained in section 3,4).

(c)

Figure 8. Strip under uniaxial tensile stress state: localization patterns with the different approaches: (a) I-standard element, (b) II-injection of a constant strain mode and (c) III-injection of a localization mode.

(a)

In Figure 8 the obtained localization patterns, in terms of the contours of the total displacement field, are shown. In addition, in Figure 9 the corresponding localization domains, for the alternatives II and III, are shown. Finally in Figure 10 the load-displacement curves are presented. There, the following facts can be noticed: • The very poor resolution, both in terms of the propagation and the sharpness of the localization pattern for the standard formulation in alternative I • The good resolution of the constant strain injected deformation modes (alternative II above) as for the propagated localization pattern. However, it exhibits still some stress-locking, manifested in terms of a little diffuse localization pattern, which encompasses more than one element. • The clear improvement, in terms of the localization pattern (now much sharper) introduced by the injection of the localization mode (alternative III above). • As for the quantitative results, although alternative II translates into a significant improvement with

388

(b)

Figure 9. Strip under uniaxial tensile stress state: Localization domain with the different approaches: (a) II-injection of a constant strain mode and (b) III-injection of a localization mode.

Research on the extension of these techniques to other target elements and formulation is currently undertaken.

Theoretical Option III Option II Option I

0.7 0.6

P(kN)

0.5 0.4

ACKNOWLEDGEMENT

0.3

Financial support from the Spanish Ministry of Science and Innovation through grant BIA2008-00411 is gratefully acknowledged.

0.2 0.1 0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

REFERENCES

(m)

Figure 10. Strip under uniaxial tensile stress state: Load deflection curves for the different approaches.

respect to standard formulations (alternative I), the injection of a localization mode in alternative III, results in a softer action-response curve, this stating the practical cancelling of the stress-locking effects for this case. 5 FINAL REMARKS In the preceding sections an exploratory work, about the effects of some new techniques in the ability of standard quadrilateral elements to capture mesh size and mesh-bias indifferent strain localization, has been presented. The presented preliminary results show that the combination of some concepts borrowed from the Strong Discontinuity Approach (SDA) and the injection of specific deformation modes, via mixed formulations, translate into a large increase of the strain localization properties of that target element. The result is a new technique that enjoys the benefits of both the SDA, in terms of mesh-size and mesh bias indifference, and the simplicity of the classical strain localization methods. These benefits have been obtained with two representative constitutive models (continuum damage and elasto-plasticity), which is encouraging in terms of the broadness of the application of the new technique.

389

Bazant, Z. and L. Cedolin 1980. ‘‘Fracture mechanics of reinforced concrete.’’ Journal of the Engineering Mechanics Division ASCE: 1287–1305. Belytschko, T., N. Moes, et al. 2001. ‘‘Arbitrary discontinuities in finite elements.’’ International Journal for Numerical Methods in Engineering 50(4): 993–1013. Borst, R.d., L.J. Sluys, et al. 1993. ‘‘Fundamental issues in finite element analyses of localization of deformation.’’ Engineering Computations 10: 99–121. Oliver, J. and A. Huespe 2004. ‘‘Continuum approach to material failure in strong discontinuity settings.’’ Computer Methods in Applied Mechanics and Engineering 193: 3195–3220. Oliver, J. and A.E. Huespe 2004. ‘‘Continuum approach to material failure in strong discontinuity settings.’’ Computer Methods in Applied Mechanics and Engineering 193(30–32): 3195–3220. Oliver, J. and A.E. Huespe 2004. ‘‘Theoretical and computational issues in modelling material failure in strong discontinuity scenarios.’’ Computer Methods in Applied Mechanics and Engineering 193(27–29): 2987–3014. Oliver, J., A.E. Huespe, et al. 2006. ‘‘Stability and robustness issues in numerical modeling of material failure in the strong discontinuity approach.’’ Comput. Methods Appl. Mech. Engng. 195: 7093–7114. Oliver, J., A.E. Huespe, et al. 2006. ‘‘A comparative study on finite elements with embedded discontinuities: E-FEM vs. X-FEM.’’ Computer Methods in Applied Mechanics and Engineering 195: 4732–4752. Simo, J., J. Oliver, et al. 1993. ‘‘An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids.’’ Computational Mechanics 12: 277–296.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Model for the analysis of structural concrete elements under plane stress conditions: Finite element implementation M. Pimentel & J. Figueiras Laboratory for the Concrete Technology and Structural Behaviour (LABEST), Faculty of Engineering of the University of Porto, Portugal

ABSTRACT: The implementation in a finite element code of a recently developed material model for the analysis of reinforced concrete (RC) cracked membranes is presented. The model aims for the analysis of large scale structural elements that can be considered an assembly of membrane elements, such as bridge girders, shear walls, transfer beams or containment structures. The code was implemented as a user-supplied sub-routine in the general purpose finite element software DIANA 9.3. The equilibrium equations of the cracked membrane element are established directly at the cracks eliminating the need to resort to averaged stress-strain relations. The average stress fields are obtained as a by-product of the local behaviour at the cracks and of the bond stress transfer mechanisms between the reinforcement and concrete. Special attention is devoted to the issues related with the implementation of the material model in a finite element code. A total strain based formulation was developed allowing the direct calculation of the stresses given a trial average strain tensor, without having to resort to inner loops at the constitutive level. The material model, previously calibrated at the element level with the results of monotonic loading tests on RC panels under in-plane shear and axial stresses, is now generalized to accommodate all possible stress trajectories, such as loading reversals, biaxial compression and the existence of two orthogonal cracks. The path dependent behaviour of the model is illustrated and some validation examples with experimental results from full scale tests with common shear critical structural elements are presented. 1 INTRODUCTION In this work a macroscopic scale treatment of the reinforced concrete (RC) behaviour is adopted. The accurate reproduction of RC behaviour at the membrane level is taken as a departure point for the analysis of large scale RC structures via the finite element method. In the past 30 years a strong investment has been made on the research of suitable constitutive laws for reinforced concrete membranes under in-plane shear and axial stresses. Given the present state of knowledge, a phenomenological approach is adopted instead of the existing theoretical frameworks that can also be used for modelling concrete behaviour, such as the plasticity theory, continuum damage theory, microplane theory, etc. The direct implementation of such a phenomenological approach usually leads to less elegant theoretical formulations than the ones based on the above mentioned theories. Nonetheless, it has the advantage of being rooted on strong experimental evidence and on mature physical models specially developed for RC. The behaviour of the basic cracked membrane element is discussed in references (Pimentel 2010; Pimentel et al. 2010), where the detailed formulation and extensive validation of the post-cracking concrete relations is presented. One of the most striking aspects

391

of structural concrete behaviour is that, after cracking, the slip between concrete and reinforcing bars leads to a highly irregular stress field. This aspect is commonly dealt with by treating reinforced concrete as a new material with its own stress-strain characteristics, which are valid only in spatially average terms (Hsu & Zhu 2002; Maekawa et al. 2003; Vecchio 2000; Vecchio & Collins 1986). The relation between these averaged constitutive laws and the mechanical phenomena taking place at the cracks—which are known to govern the behaviour of cracked concrete elements—is not always straightforward. Therefore, it is not possible to establish a link between the usual averaged constitutive relations and the significant research effort that has been made throughout the last decades on RC mechanics topics like bond, aggregate interlock, concrete tensile fracture, etc. The main advantage of the proposed model is that the equilibrium equations of the cracked membrane element are established directly at the cracks eliminating the need to resort to averaged stress-strain relations. Henceforth, the response the cracked membrane element is computed considering the individual contributions of the involved mechanical phenomena in a transparent manner. The basic membrane element is assumed to contain several cracks. Therefore, the strain field calculated

from the nodal displacements must be understood as a spatially-averaged strain field. The strain profile along the reinforcing bars is recovered from the averaged strain field assuming and stepped rigid bond shear stress-slip law between the rebars and the surrounding concrete according to the Tension Chord Model (TCM) (Marti et al. 1998). With the TCM the steel stresses at the cracks can be calculated allowing the establishment of the equilibrium in terms of stresses at the cracks. Moreover, the bond stress transfer mechanisms can be accounted for in a rational manner both in the pre- and post- yielding regimes, and the crack spacing and crack widths can be calculated from first principles instead of relying in empirical based expressions. In the following the generalization of the proposed model allowing its implementation in a finite element code is described. This comprised the formulation of the uncracked concrete nonlinear behaviour, the establishment of suitable loading/unloading conditions, the existence of two orthogonal cracks in the same integration point, and the treatment of the strain localization issues. 2 TOTAL STRAIN BASED MODEL 2.1 Uncracked concrete For uncracked concrete, an efficient elasticity based isotropic formulation is adopted, requiring only one state variable for tracing the damage evolution and establishing the loading/unloading/reloading conditions. 2.1.1 Equivalent stress The equivalent stress S is here defined as a scalar measure of the applied stress level under biaxial plane stress conditions. For S = 0 concrete is totally unloaded while for S = 1 the biaxial failure envelope has been reached. As proposed by Maekawa & Okamura (1983), the equivalent stress can be defined as a function of the mean and deviatoric invariant components of the stress tensor, σm and τd :  2  2 a b σ + τ (1) S= m d fc′ fc′ √ √ 2 2 |σ1 − σ2 | (2) σm = (σ1 + σ2 ) ; τd = 2 2 In the above expressions, fc′ is the standard cylinder uniaxial compressive strength and the two parameters a = 0.61 and b = 1.27 were determined from experimental data for the uniaxial compression and equal biaxial compression stress trajectories. In Figure 1 the resulting failure envelope (S = 1) is presented in the σ1 − σ2 plane and it is compared to the Kupfer &

1.5

1.0 2/

Kupfer and Gerstle

.52:

f c'

1

Present criteria: S = 1

0.5 1

Crack detection surface

0.0 0.0

0.5

1.0 1/

1.5

f c'

Figure 1. Graphical representation of the proposed failure criteria in the σ1 − σ2 stress plane.

Gerstle (1973) failure criteria, which was adopted by the CEB in the Model Code 1990 (CEB 1993). The crack detection envelope represented in the figure is used for detecting if cracking has occurred. When this envelope is violated, the cracked concrete subroutines are invoked, see section 2.2. 2.1.2 Equivalent total strain The equivalent total strain E is a scalar measure of the current biaxial total strain level. Similar to the equivalent stress, for E = 0 concrete is unloaded while for E = 1 the failure envelope is reached. The equivalent total strain is also defined as a function of the two invariants of the total strain tensor (Maekawa & Okamura 1983), εm and γd : E=



εm =

√

c εm ε0′

2

+



2 (ε1 + ε2 ); 2

d γd ε0′

2

γd =

(3)

√ 2 |ε1 − ε2 | 2

(4)

The two constants c = 0.62 and d = 1.10 were derived such as to fit the peak strains obtained in the uniaxial compression and equal biaxial compression tests and ε0′ is peak strain in a standard uniaxial compression test. 2.1.3 Stress-strain relations and loading/unloading conditions In a monotonic loading process it is assumed that the equivalent stress S can be univocally obtained from the equivalent total strain E through the following nonlinear relationship:

S=

392

⎧ k E − E2 ⎪ ⎪ ⎪ ⎨ 1 + (k − 2) E , ⎪ ⎪ ⎪ ⎩

1+

η 2

E≤1

1

, E > 1 1 − 2E + E 2

(5)

For the ascending branch, E ≤ 1, the Sargin’s law was adopted (fib 1999), while for the descending (post-peak) branch a modified version of the function proposed by Krätzig & Pölling (2004) was defined, ensuring a smooth transition between the two branches. The parameter k determines the shape of the curve in the ascending branch. With k = 2, a parabolic relationship is obtained. In general k = Eci · ε0′ /fc′ , with Eci being the initial tangential modulus of elasticity in a uniaxial compressive test. The parameter η controls the shape of the curve in the descending branch. In order to guarantee objectivity of the results with respect to the mesh size, the post-peak response is regularized according to the dimension of the strain localization band, hC , which is related to the finite element size. It is proposed to use 1 η= 2



π f ′ ε′ c 0 ′ GC hC − fc′ ε0 2

2

a region in the total strain space where no damage evolution occurs—the reversible process domain. This region expands as the maximum total strain increases and can never contract. The resulting algorithm is very simple as can be summarized as follows: 1. calculate the current equivalent total strain E t+t from current the total strain tensor ε t+t according to expression (3); 2. calculate the current maximum equivalent total t+t strain and the equivalent strain ratio: Emax = t , E t+t ), r = E t+t/E t+t , respectively; max(Emax max t+t 3. calculate the equivalent stress Smax corresponding t+t to Emax according to expression (5); 4. calculate the current equivalent stress S t+t = t+t r Smax .

(6)

where GC is the compressive fracture energy. Unless otherwise stated, GC is estimated using the Compressive Damage Zone (CDZ) model proposed by Markeset & Hillerborg (1995) considering k = 3, r = 1.25 mm, wc = 0.6 mm and Ld = 2.5 · e, being e the element thickness (k, r, wc and Ld are the CDZ model parameters): GC = 1500 GF e + 0.0003fc′

(7)

In Figure 2, the resulting uniaxial stress-strain curves for an element with length/thickness of 0.25 × 0.10 m2 are presented for 5 different uniaxial concrete strengths. The fracture energy GF , the Young modulus Eci and the peak uniaxial strain ε0′ were calculated according to the expressions given in (fib 1999). In the present model, a loading step always leads to damage evolution while in an unloading/reloading step damage remains constant. The variable that is used to monitor damage evolution is the maximum equivalent total strain, Emax , which by definition is a non-decreasing variable. A given Emax value bounds 1

100 f ' c = 20 40 60 (Mpa) 80 100

S

-

0

2

0

0

E

3

0

-

0.006 2

Figure 2. Graphical representation of the proposed stressstrain curve in the uniaxial compression case for 5 different concrete strengths: (a) Equivalent stress vs. equivalent strain; (b) Resulting uniaxial stress-strain curves.

393

2.1.4 Constitutive and iteration stiffness matrices So far, only a scalar relationship between the equivalent stress and equivalent total strain was defined. To complete the formulation, a directional relationship between the total strain and stress tensors is required. In the present model, before cracking, concrete is treated as an isotropic material. At higher loading levels this is a simplifying assumption, as shown by Maekawa & Okamura (1983). Nonetheless, it greatly simplifies the model while the error that is introduced is small and has little significance on the overall response of a RC member under biaxial stress conditions. Taking into account the definition of equivalent stress given by (1), the secant constitutive matrix relating concrete stresses with the total strains is derived in the form of a nonlinear elasticity matrix: ⎡ ⎧ ⎫ 1 ⎨ σcx ⎬ Ec,sec ⎢ν σcy = ⎣ ⎩τ ⎭ 1 − ν 2 0 cxy Ec,sec =

fc′  ε0  a

fc′

ν 1 0

⎤⎧ ⎫ 0 ⎨ε ⎬ 0 ⎥ εx ⎦ 1−ν ⎩ y⎭ γxy 2

S 2  εm + fb′ 1−ν c

γd 1+ν

2

(8)

(9)

where Ec,sec is the secant Young modulus and ν is the Poisson ratio for which a constant value ν = 0.2 may be taken. The secant stiffness matrix of (8) is used to calculate stresses for the given strain state. The tangent stiffness matrix Dc is used for construction of an element stiffness matrix for the iterative solution at the structural level. This matrix can readily be obtained in the global coordinate system replacing in (8) Ec,sec by the tangential modulus Ec,tan , which is the slope of the equivalent stress-strain curve at a given total strain: Ec,tan = dS/dE. In the present implementation, whenever the slope of the curve is less than the minimum value (Ec,tan )min the value of the tangent modulus is set equal to the minimum. This occurs in the softening range and near the compressive peak.

2.1.5 Examples In Figure 3 the results from the analytical model are compared to a classical set of experimental data (Kupfer & Gerstle 1973) with three compressive biaxial stress trajectories (see also Figure 1). The agreement is quite good in view of the simplifications considered in the model, such as the isotropy assumption and the constant value for the Poisson coefficient. In Figure 4 the model response for the case of non proportional loading under biaxial compression stress states is presented. In a first stage, a uniaxial compression loading was applied till the equivalent stress reached S = 0.7, after which followed an unloading stage. Then a biaxial compression loading path was applied following the stress trajectory σ1 :σ2 = −.52:−1 till the equivalent stress reached S = 0.85. After unloading, a biaxial compression loading path was finally applied following the stress trajectory σ1 :σ2 = −1:−1. The stress trajectories are

1.5 0.52/ 1

1.0

1: 1

' 2 / fc

1:

2=

0: 1

0.5

0.0 1.0

0.0 1

1.0

/ '0

2

2.0

/ '0

Figure 3. Comparison of analytical model (k = 2.375, ν = 0.2) with experimental data from Kupfer & Gerstle (1973). 1.5

represented in Figure 4a and the corresponding strain trajectories in Figure 4b. 2.2 Cracked concrete After cracking, concrete is treated as an orthotropic material and a local coordinate system is introduced (n-t coordinates) where the constitutive laws are established. The total strains in the global coordinate system are transformed into the local coordinate system through the usual strain transformation relationship. Once cracking occurs, two uniaxial stress fields are assumed along the two orthogonal n-t directions and the Poisson effect in the cracked concrete is neglected. The crack direction is determined by the principal tensile direction of the concrete stress tensor at impending cracking, θr . The crack angle θr remains fixed and is kept in memory as a state variable. 2.2.1 Compression model parallel to the cracks direction The compressive stress parallel to the crack direction is determined from the total strain along the corresponding local using the base curve (5) already adopted in the uncracked stage. However, it is well known (Belarbi 1991; Vecchio & Collins 1986) that the compressive strength of cracked concrete is reduced (or softened, as usually termed in RC literature) with increasing tensile strains in the orthogonal direction. Therefore, the compressive strength must be replaced by the effective compressive strength fc,ef = ζf · ζe · fc′ . Also the peak uniaxial strain is reduced to ε0,ef = ζf · ζe · ε0′ . The softening coefficients are formulated as (ε⊥ is the strain in the direction normal to the compressive stress being evaluated):

1.5

1

1.0 2/

f c'

E=1.0 2/

0.52: 1 0.5

ςf = (fc0 /fc′ ) 3;

1.0

S=1.0

0: 1

1:

2=

ςe =

'0

1: 1 0.85

1 Cm (1.08 + 81ε⊥ )

(10)

0.5 0.568

0.7

0.411

0.0

with fc0 = 30 MPa. For details refer to (Pimentel 2010; Pimentel et al. 2010). For each local axis (n- or t-), the equivalent stress and the equivalent strain are given by

0.0

0.0

0.5

1.0 1/

1.5

0.0

0.5

f c'

1.0

(a)

1.5

'0

1/

(b) 1.5

E = 0.411 S = 0.7

1.0

2: 1

S=−

1.0 E = 0.568 S = 0.85

S

' 2 / fc

0.5

2: 2

0.5

0.0

0.0 0.0

0.5

E

(c)

1.0

1.5

0.0

1.0

0.5 1

/ '0 ;

2

1.5

/ '0

(d)

Figure 4. Model behaviour under non-proportional biaxial compression loading paths: (a) Load paths in the principal stress space and (b) corresponding principal strain trajectories; (c) equivalent stress-equivalent strain relationship; (d) minimum compressive stress ratio versus the two principal strain ratios.

394

σi fc,ef

;

E=−

εi ε0,ef

;

i = n or t

(11)

2.2.2 Tension model normal to the cracks direction At the cracks, the normal stresses are due to crack bridging stresses and to crack dilatancy stresses arising from the shear transfer mechanics. For very small crack widths the former govern while the later are important for fully developed cracks. The dilatancy stresses are dealt by the crack shear transfer model. The crack bridging stresses are calculated with the

expression proposed by Hordijk (1992):   3  wr wr wr σi e−6.93 wc − 0.027384 = 1 + 27 fct wc wc GF wc = 5.14 , fct

wr = hεi ,

i = n or t

(12) (13)

where h is the crack band width, which is related to the finite elements size. 2.2.3 Coupled tension-compression model The direct combination of the compression and tension models results in a uniaxial path-dependent model. The unloading/reloading is performed secant to the origin as already illustrated for the uncracked state. For reproducing this type of behaviour two state variables are required for each local coordinate system direction: the maximum and minimum total strain experienced in the past history. In the present implementation, instead of the minimum compressive strain, the state variable governing the compressive behaviour is the maximum equivalent total strain in the past history, given by expression (11)2 . The loading/unloading determination is made independently for the n- and t-direction. Once concrete is cracked, the biaxial confinement effects cannot be recovered. 2.2.4 Shear transfer model in the local coordinate system 2.2.4.1 One active crack in the coordinate system The last stage of the tensile fracture process corresponds to the formation of a macroscopic crack that cannot transmit normal tensile stresses. Shear transfer across these cracks cannot be simply formulated as a relation between shear stress and shear displacement, but is a more complex mechanism, in which shear stress, shear displacement, normal stress and crack width are involved. The formulation of the equilibrium conditions directly at the cracks, allows the adoption of previously developed theoretical models for reproducing this complex behaviour, instead of adopting some oversimplified formulations, such as the ones based on shear retention factors. A closed form solution based on the crack density model (Li et al. 1989), which was posteriorly adapted by Pimentel (Pimentel 2010; Pimentel et al. 2010), is adopted for calculating the crack shear and crack dilatancy stresses from the normalized shear displacement β = δr,t /δr,n . If a regular array of cracks is formed then it is possible to show that βi = γnt /εi,max , where γnt is the current shear strain in the local n-t coordinate system and εi,max is the maximum normal strain along the local axis perpendicular to the crack. The state variable εi,max (see 2.2.2) is used

395

instead of the current normal strain εi in the determination of the normalized shear strain, βi . This was required for ensuring: (1) the stability in the algorithm in the unloading/reloading regimes; and (2) a smooth transition from loading to unloading states. For loading/unloading states in the positive side of the normalized shear strain, β > 0, the shear and dilatancy stresses are given by: βi2 1 + βi2   |βi | π σdil,i = −τLIM ,i − cot−1 |βi | − 2 1 + βi2

τLIM ,i = 3.83fc′1/3 g εi,max , srmθ τagg,i = τLIM

g εi,max , srmθ =

(14)

0.5 ≤1 0.31 + 200 srmθ εi,max

In the expressions above srmθ is an estimate of the crack spacing according to the TCM (Pimentel 2010; Pimentel et al. 2010). In the negative side of the normalized shear strain, β < 0, a minus sign must be placed in the τagg expression. For simplicity of the resulting algorithm, especially when two cracks arise in the same integration point, loading and unloading are performed following the same curve. This equivalent to disregarding the energy dissipation occurring at the crack lips during shear loading reversals. 2.2.4.2

Equilibrium and compatibility for two orthogonal cracks Material models for reinforced concrete must deal with situations were at least two cracks arise in the same integration point. In the present formulation, postcracking stresses are evaluated in the local orthotropy axes, which are, by definition, orthogonal. Therefore, only two orthogonal cracks are allowed. In an RC element containing two cracks the total shear strain γnt must be distributed by each crack (Figure 5) so that the corresponding shear and dilatancy stresses can be calculated. Equilibrium requires that the shear stress transmitted along both cracks is the same. Therefore,

(a) Compatibility

(b) Equilibrium

(c) Problem to be solved

Figure 5. Equilibrium and compatibility for two-way orthogonal cracks.

2.3 Reinforcing steel

the problem is governed by the two following conditions: τagg = τagg (β1 ) = τagg (β2 );

The reinforcement stresses at the cracks are calculated from the average strains, as described elsewhere (Pimentel 2010; Pimentel et al. 2010). The algorithm starts by calculating the maximum diagonal crack spacing srm0 given the crack angle θr , the reinforcement ratios, the assumed rigid plastic bond shear-stress slip law, the concrete stresses at the cracks—given by (16)—, and the fact that maximum concrete tensile stresses at the centre between the cracks cannot exceed the concrete tensile strength fct . Figure 6 provides polar representations of the solution. The steel stresses at the cracks are calculated in the form (Pimentel 2010; Pimentel et al. 2010):

γnt = γnt,1 + γnt,2 (15)

′ . with β1 = γnt,1 /εn,max and β2 = γnt,2 /εt,max It can be shown that the determination of the shear strains fulfilling the equilibrium and compatibility conditions (15) can be reduced to the problem of finding the roots of a polynomial of fourth degree. Since only one real root exists in the interval, γnt,1 ∈ [0, γnt ], the bisection method, which can be shown to be unconditionally convergent, was implemented for finding the solution. The algorithm starts by checking how many active cracks exist. An active crack is defined as having tensile normal strains, otherwise is considered to be closed or inactive. If two active cracks are detected, a subroutine is called for calculating the corresponding crack shear strains γnt,1 and γnt,2 from the total shear strain γnt . The crack shear stress τagg and the crack dilatancy stresses σdil,1 and σdil,2 are then calculated according to (14).

2.2.5 Assembly of the concrete stress vector In the local n-t coordinate system, the concrete stress vector at the cracks is simply obtained from: σcr,n = σn + σdil,1 σcr,t = σt + σdil,2 τcr,nt = τagg

(16)

2.2.6 Iteration stiffness matrix In the local coordinate system, the stiffness matrix used for the iterative process has the form:

Dc,nt

⎡ D11 =⎣0 0

0 D22 0

⎤ 0 0⎦ D33

(17)



σsr,i = f εi , Es,i , fsy,i , fsu,i , εsu,i , srm,i , τb0,i , τb1,i

with the subscript (·)i indicating the x- or y- reinforcement direction, εi the average strain along the reinforcement i-direction, Es,i the steel Young modulus, fsy,i , fsu,i and εsu,i the steel yielding strength, tensile strength and corresponding rupture strain, respectively, srm,i the component of the crack spacing along the reinforcement i-direction, and τb0,i and τb1,i the plastic bond stresses prior to and after the i-reinforcement yielding. For implementation in a finite element procedure, the model needs to be complemented with suitable loading/unloading/reloading conditions. These can be derived considering rigid-plastic behaviour of the bond shear stress-slip law as depicted in Figure 7. In the unloading stage, it is here assumed that bond shear stress drops to zero. In reality the bond shear stress may reverse sign and, although a rapid degradation of bond is usually seen, negative bond shear stresses are expected in the unloading stage. This may be important for an accurate modelling of the cracks widths in service conditions and should be subject of future generalization of the present model. If reversed bond action is neglected, when the unloaded length is equal to the crack spacing, the

The non-zero members of (17) are determined in a closed form by differentiation of the corresponding stress-strain laws. In the case of two active cracks, the shear stiffness is obtained from: 1 = D33

1 D33,1

1 +

1 D33,2

(19)

x= xy /f ct =

s rmy0

(18)

In theory, the adoption of a tangent stiffness matrix neglecting the effect of the cross terms weakens the convergence characteristics of the algorithm if a fully Newton-Raphson procedure is adopted for the incremental-iterative solution procedure. Nonetheless, this effect was found not to be severe and good convergence rates can still be achieved.

396

3 2 1

3

x=

y

Øx = 2 Øy cr,nt

/f ct = 0

cr,nt

s rm0

s rm0 r

0 0

s rmx0

3

y

Øx = 2 Øy

r

0

/f ct = 0.5

xy /f ct =

3 2 1

s rmx0

Figure 6. Polar representations of the maximum diagonal crack spacing srm0 .

(a)

(b)

Figure 8. Reinforcement stresses under non-proportional loading (Note: εsm = εi ).

(c)

(d)

Figure 7. Formulation of the reinforcement model under non-proportional loading: (a) Bond stress-slip law; (b) Stress and strain distributions between the cracks (pre-yielding stage); (c) Resulting stress-strain laws for the steel; (d) Corresponding average stresses in the concrete (Note: εsm = εi ).

stresses in the reinforcement are equal to σs,min , see Figure 7b. From that point on, the naked bar stressstrain relation is adopted. For simplicity, a linear relationship is assumed until the unloaded length is equal to the crack spacing. It can be shown that the unloading/reloading stiffness in this region is twice the steel young modulus Es if the steel stresses along the tension chord are all above or below the tensile strength fsy . In these cases the following equalities hold: σsr = σsr − σs,min = 2(σsr − σsm )

(20)

σsr = 2 Es ε

(21)

In Figure 7c and d the resulting loading/unloading/ reloading behaviour is illustrated. The determination of the notable points of the unloading/reloading σsr (ε) curve, i.e. the points where the unloading/reloading slope changes 2·Es to Es , is made using the equilibrium conditions of the differential element as discussed in (Pimentel 2010; Pimentel et al. 2010) and the constitutive laws for the reinforcement steel and bond shear stress. For the cases where partial yielding has occurred along the tension chord, i.e. σs,min < fsy < σsr , the equalities (20) and (21) are only good approximations of the real solution. Nonetheless, these were adopted for simplicity of the resulting algorithm.

397

The naked bar stress-strain law is adopted also when the stresses go onto the compressive side. The plastic strains are used as offsets of the steel response, as depicted in Figure 8. 2.3.1 Assembly of the stress vector The composite stress vector in the reinforcement directions is finally assembled as:  σ xy = Tσ (θr )σ nt + ρx σsr,x

ρy σsr,y

0

T

(22)

with Tσ being the stress transformation matrix and ρx and ρy the reinforcement ratios in the x- and ydirections. 3 VALIDATION Following a previously performed validation at element level, which included an extensive set of results obtained from tests on RC panels (Pimentel et al. 2010), here the validation at the structural level of the proposed constitutive model is presented. The validation is made using the results from a set of experimental tests on shear critical structural elements. Four validation examples are presented here. 3.1 Beams VN2 and VN4 The first two beams, VN2 and VN4, belong to an experimental campaign performed by Kaufmann and Marti at the ETH Zürich. The goal of the VN test series was to investigate the behaviour of webs of structural concrete girders with no plastic deformation in the chords and having low shear reinforcement ratios (ρw = 0.335%). The state of stress is similar to

that observed outside the support regions of a continuous girder. For that purpose, a specially developed testing facility was developed allowing the investigation of elements of beams rather than entire girders. The beams were subjected to a constant shear force. The rotation of the beams ends was restricted and the forces at the element ends were controlled such that the axial force remained constant. In beam VN2 the axial force was null while in beam VN4 a compressive force of 1 MN was applied. In the case of beam VN2 collapse was triggered by stirrup rupture after some minor spalling of the web cover concrete. Beam VN4 failed after crushing of the web concrete. For further details refer to test report (Kaufmann & Marti 1996) and to (Pimentel 2010). The beam elements are 5.84 long and 0.78 m high, see Figure 9. The model predicts the same failure modes as observed in the tests. In the case of cracked concrete, the state variable Emax = −εt /ε0,ef is an indicator of whether concrete crushing has or has not occurred. If Emax > 1 the post-peak branch of the stress strain curve has been achieved. In this stage of the curve, signs of concrete cover spalling or concrete delamination are to be expected in real specimens. The contour levels with the values of Emax at failure are presented in Figure 10. Only minor signs of web cover spalling are predicted near the flanges for beam VN2. Similarly to the test, collapse was triggered by rupture of the vertical reinforcement. This can be confirmed from inspection of Figure 11, where the contour levels with

the vertical reinforcement stresses at the cracks are presented. The red colour corresponds to regions where the tensile strength of the stirrups ( fsu = 604 MPa) was exceeded. The cracking patterns at failure are depicted in Figure 12 and the web crack widths, which constitute a direct output of the analysis, are presented in Figure 13. In Figure 14 the calculated and the measured average vertical deformations in a 1.60 m long web segment at the centre of the beams are compared. In the case of beam VN2 the curve is presented till the load step where the steel stresses reached fsu .

Figure 11. Beam VN2: deformed shape (x10) with the σsr,y contour levels at failure.

(a)

(b)

Figure 12. Experimental and calculated cracking patterns at failure: (a) VN2; (b) VN4. 2.5

LS 4 wr [mm]

Exp.

Calculated 0 1.6

LS 3 wr [mm]

Figure 9. Beams of the VN series: internal forces, cross section and adopted finite element mesh.

0 -2

0

2

x [m]

Figure 10. Deformed shape (x10) with the Emax contour levels at failure.

398

Figure 13. Experimental (Kaufmann & Marti 1996) and calculated web crack widths of beam VN2. The load steps LS3 and LS4 are identified in Figure 14.

Peak load

Peak load

F [kN]

LS3

LS5

LS4

LS2

LS3 LS2

Exp.

LS1

Exp.

VN4

Calculated

LS5

Exp.

LS1

Exp.

VN2

LS4

Calculated

0 0 y

3

(x10 )

30 0 y

3

(x10 )

30

Figure 14. Force vs. average vertical strain curves.

3.2 Beams MVN2 and MVN4 The four beams of the series MVN have the same geometry and the same shear reinforcement content as the ones from the series VN (Kaufmann & Marti 1996). However, in contrast to series VN, an additional vertical jack was provided at midspan. The loading procedure simulated the behaviour of an (inverted) intermediate support of a continuous girder, see Figure 15. In this case, bending moments have a significant influence on the behaviour of the girder and usually chords are plastically deformed. This bending-shear interaction can be explained by the decreasing softened compressive strength of the web concrete with increasing longitudinal strains. Here only the analyses to specimens MVN2 and MVN4 are presented. In the first phase of the test, horizontal forces were controlled such that moments at the element ends were zero. In the second phase, after reaching the yield moment at midspan, rotations of both element ends were prevented. This second phase of the test simulated a redistribution of moments from the support region (midspan) into the span (element ends) of a continuous girder. The forces applied at the element ends were controlled such that normal forces remained constant throughout the test and shear forces on either side of the concentrated load were of equal magnitude. Specimens MVN2 and MVN4 were subjected to axial compression of 1.30 MN before being loaded in shear and the axial force was held constant during the test. The specimen MVN4 was designed not to yield at midspan during the entire test and was post-tensioned with a bonded VSL 6-7 cable (Pu = 1.85 MN) tensioned to an initial force of 1.30 MN. All the girders contained the same amount of web reinforcement (ρw = 0.335%, equal to the one adopted in the series VN). When specimen MVN2 failed, half of the girder was shifted over the remaining part of the element along an inclined failure surface made up of existing web shear cracks (see Figure 18a). Also some spalling of the concrete cover in the web was observed. This failure mode is here classified as a web sliding failure. At peak load the model predicts that some regions of the web

399

are crushing, as can be seen in Figure 16a. Nonetheless, collapse only occurred with concrete crushing in the top flange, near the loading platen. This can be confirmed by the incremental deformed shape in the post peak branch of the load-deformation curves, see Figure 16b, which is an indication of the occurrence of a sliding shear failure. The specimen MVN4 exhibited a typical web crushing failure (Figure 18b). Failure was more ductile and the web concrete crushed progressively on both sides of the girder. Altough this failure mode was correctly predicted the failure load is slightly underestimated (Fu,exp /Fu,calc = 1.09). In Figure 17 the calculated and the measured average vertical deformations are compared. The deformations were averaged over two symmetrically located and 0.80 long web segments. In general, the

Figure 15. Beams of the MVN series.

Figure 16. (a) Deformed shape (x10) with the Emax contour levels at failure; (b) Beam MVN2: incremental deformed shape (x1000) after the peak load with the contour levels of the crack shear strains γnt . 2000 Peak load Peak load

F [kN]

su 700

MVN2

Exp. Exp. Calculated

MVN4

0 0

ε y (x103)

20 0

ε y (x103)

Figure 17. Force vs. average vertical strain curves.

20

REFERENCES

(a)

(b) Figure 18. Experimental (Kaufmann & Marti 1996) and calculated cracking patterns at failure: (a) MVN2; (b) MVN4.

agreement between the measurements and the calculations is good. It must be remarked that these are average web strains which are directly related with the shear behaviour. Usually, a good fit between numerical and experimental results is much easier to obtain if flexure related parameters, (midspan displacements, per example) are compared. The observed and calculated cracking patterns at failure are depicted in Figure 18. 4 CONCLUSIONS A new numerical model is presented for the analysis of large scale structural concrete elements. A macroscopic scale treatment of the RC behaviour is adopted and the accurate reproduction of structural concrete behaviour at the membrane level is taken as a departure point for the analysis of large scale RC structures via the finite element method. The model uses a classical kinematic description of the displacement field based on the concept of ‘‘weak discontinuities’’. The developed total strain formulation revealed to be robust and showed good convergence characteristics. The results from the preliminary validation examples showed that the model can successfully reproduce some of typical shear failure models usually observed on RC beams. ACKNOWLEDGMENTS The support by the Portuguese Foundation for Science and Technology (FCT) through the PhD grant SFRH/BD/24540/2005 attributed to the first author is gratefully acknowledged.

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Belarbi, A. (1991). ‘‘Stress-Strain Relationships of Reinforced Concrete in Biaxial Tension-Compression,’’ Doctoral Thesis, University of Houston, Houston. CEB. (1993). CEB-FIP Model Code 1990, Thomas Telford, London. fib. (1999). ‘‘Bulletin n◦ 1: Structural Concrete. Text book on Behaviour, Design and Performance. Vol. 1.’’ fib, Lausanne. Hordijk, D.A. (1992). ‘‘Tensile and tensile fatigue behaviour of concrete: experiments, modelling and analyses.’’ Heron, 37(1), 1–79. Hsu, T.T.C. & Zhu, R.R.H. (2002). ‘‘Softened Membrane Model for Reinforced Concrete Elements in Shear.’’ ACI Structural Journal, 99(4), 460–469. Kaufmann, W. & Marti, P. (1996). ‘‘Versuche an Stahlbetonträgern unter Normal- und Querkraft.’’ Swiss Federal Institute of Technology Zürich, Zürich. Krätzig, W.B. & Pölling, R. (2004). ‘‘An elasto-plastic damage model for reinforced concrete with minimum number of material parameters.’’ Computer & Structures, 82, 1201–1215. Kupfer, H.B. & Gerstle, K.H. (1973). ‘‘Behaviour of Concrete Under Biaxial Stresses.’’ Journal of Engineering Mechanics Division, 99(EM4), 853–866. Li, B., Maekawa, K. & Okamura, H. (1989). ‘‘Contact Density Model for Stress Transfer across Cracks in Concrete.’’ Journal of the Faculty of Engineering of Tokyo, XL(1), 9–52. Maekawa, K. & Okamura, H. (1983). ‘‘The Deformational Behavior and Constitutive Equation of Concrete Using the Elasto-Plastic and Fracture Model.’’ Journal of the Faculty of Engineering of Tokyo, XXXVII(2). Maekawa, K., Pimanmas, A. & Okamura, H. (2003). Nonlinear Mechanics of Reinforced Concrete, Spoon Press, London. Markeset, G. & Hillerborg, A. (1995). ‘‘Softening of concrete in compression—Localization and size effects.’’ Cement and Concrete Research, 25(4), 702–708. Marti, P., Alvarez, M., Kaufmann, W. & Sigrist, V. (1998). ‘‘Tension Chord Model for Structural Concrete.’’ Structural Engineering International, 98(4), 287–298. Pimentel, M. (2010). ‘‘Advanced numerical modelling applied to the safety examination of existing concrete bridges,’’ Doctoral Thesis, Faculty of Engineering of the University of Porto, Porto. Pimentel, M., Brühwiler, E. & Figueiras, J. (2010). ‘‘Extended cracked membrane model for the analysis of RC panels.’’ Engineering Structures, (Submitted for publication). Vecchio, F.J. (2000). ‘‘Disturbed Stress Field Model for Reinforced Concrete: Formulation.’’ Journal of Structural Engineering, 126(9), 1070–1077. Vecchio, F.J. & Collins, M.P. (1986). ‘‘The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear.’’ ACI Journal, 83(2), 219–231.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A partition of unity finite element method for fibre reinforced concrete F.K.F. Radtke, A. Simone & L.J. Sluys Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

ABSTRACT: In this contribution we present a partition of unity based approach to model discrete fibres embedded in a matrix material without discretization of the fibres. For that purpose, the fibres are superimposed on a background mesh. Fibres are incorporated into the model by enriching the displacement field. We develop an element that incorporates fibre material characteristics, geometrical fibre position, bonding between fibre and matrix and matrix material behaviour. For the constitutive behaviour of the matrix simple fracture energy regularised isotropic damage with an exponential softening law is used. The behaviour of the fibre-matrix bond follows a non-linear relation. The performance of the approach and the influence of different numerical and material parameters are investigated. 1 INTRODUCTION Fibres are employed in a number of materials to improve their mechanical behaviour. Notable examples are fibre reinforced concrete and fibre reinforced polymers. With the extensive use of these materials, there is an increasing need for understanding the influence of fibres and for predicting the interaction between the micro-structure of the fibre reinforced material and the mechanical properties of the composite. Finite elements provide a very flexible and easy to use tool for this purpose. Unfortunately, accurate analysis of fibre reinforced materials are commonly unfeasible due to the prohibitive costs of the meshing process. Here, we present an approach based on the partition of unity property of finite elements that allows the inclusion of an arbitrary number of discrete fibres in an element without meshing them.

2 APPROACH Fibres are incorporated into finite elements by employing the partition of unity properties of the finite element method (Duarte et al. 2000) (pufem approach). By means of a suitable enrichment function it is possible to superimpose discrete fibres on a background mesh as depicted in Figure 1. The action of the fibres is represented by the enrichment function χ as depicted in Figure 2. The enrichment function is equal to one at the fibre and zero elsewhere. We make the following main assumptions: 1) the fibre thickness is small; 2) the interface between fibre and matrix is imperfect, thus a relative deformation between matrix and fibre may occur already from the beginning of the simulation; 3) all fibres are fully embedded in the matrix material. Since we treat fibre, matrix and

401

fibre matrix interface independently we have to specify a constitutive relation for each of them. The matrix is modelled using simple fracture energy regularised isotropic damage with an exponential softening law (Jirásek and Patzák 2002). The tangential component of the fibre matrix bond is described using a relation given by (Häußler-Combe and Hartig 2007), shown in Figure 3, which has been modified including secant unloading. The normal component of the bond law is chosen such that a normal opening of the gap between matrix and fibre is prevented. No coupling between normal and tangential component is assumed. The fibre is considered linear elastic.

3 VALIDATION OF THE PUFEM APPROACH IN THE LINEAR ELASTIC RANGE We compare the proposed pufem approach to the shear lag theory as described in Bentur and Mindess (2006) for fibre reinforced concrete to validate the results in the linear elastic range. For that purpose the shear stress along the fibre and the normal stress in the fibre are computed following both approaches. Contrary to the pufem approach the shear lag theory assumes perfect bond between fibre and matrix. It does not account for an interface allowing for slip between fibre and matrix, while this is one of the basic assumptions of the pufem approach (see Section 2). Thus, to compare both approaches it is necessary to consider an almost rigid interface in the pufem approach to minimise slip along the fibre. The influence of the interface stiffness is studied in Section 3.2 in more detail. Furthermore, the analytical solution neglects the influence of the fibre on the overall strain field of the matrix. But as long as this influence is limited, the results of both approaches can be compared.

Figure 1.

matrix

fibres

fibre reinforced concrete

background mesh

discrete fibres

fibres do not coincide with mesh

Discrete fibres distributed in a continuum matrix discretised with a background mesh.

thick fibre

thin fibre

(a)

matrix

F

F

mechanical system

enrichment function

(b) dfib Figure 2.

dfib

0

Enrichment function χ for thick and thin fibres being pulled out from a matrix material.

elastic

debonding

at its left end and pulled at its right end. The test is performed using displacement control with the right end of the sample being pulled by 0.1 mm. This leads to a theoretical strain in the sample of

frictional

(stmax , ttmax )

tt

ε11 = εm =

(stres , ttres ) unloading branch st Figure 3. Relation between the tractions tt and the relative deformation between fibre and matrix st describing the tangential component of the fibre-matrix bond.

3.1 Test setup For the comparison between the pufem approach and the analytical solution given by the shear lag theory a test setup as shown in Figure 4 is used. The fibre is embedded in the centre of a sample which is fixed

0.1 mm = 0.01, 10 mm

which is used as input for the analytical solution. The fibre has a length of 6 mm while the specimen has a length of 10 mm. The distances between fibre and sample end have been chosen such that the influence of boundary effects on the stresses in the fibre are minimised (Goh et al. 2004). The fibre has a diameter of 0.025 mm. Poisson’s ratio of matrix and fibre are chosen to be 0.2. Matrix, fibre and interface are linear elastic. 3.2 Influence of the stiffness of the fibre matrix interface in comparison to the analytical solution In this example the influence of the stiffness of the fibre matrix interface in the pufem approach is studied in comparison to the shear lag theory assuming a rigid

402

1

fibre

2 Figure 4.

6

2

[mm]

Test setup for the comparison between analytical solution and pufem approach. 8

analytical

6 4

/

0

2 0

pufem

Dbt = 10

3

N/mm

2

pufem

Dbt = 10

4

N/mm

2

pufem

Dbt = 10

5

N/mm

2

pufem

Dbt = 10

6

N/mm

2

Dbt = 10

7

N/mm

2

pufem

0

0.1

0.2

0.3

0.4

0.5 x / lf

0.6

0.7

0.8

0.9

1

/

0

0

0.95

1

x / lf

Figure 5. Influence of the stiffness of the fibre matrix interface on the shear stress at the fibre matrix interface in comparison to the analytical solution; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum shear stress in the interface following the shear lag theory.

interface. In the pufem approach the stiffness of the fibre matrix interface is varied from Dbt = 103 N/mm2 to Dbt = 107 N/mm2 . The test setup has been described in the previous section. Young’s modulus of matrix and fibre are chosen to be equal (Ef = Emat = 20000 N/mm2 ). The shear stresses along the fibre are plotted in Figure 5. The x-axis is normalised with respect to the fibre length while the y-axis is normalised with respect to the maximum shear stress from the analytical solution. Regarding the shear stress along the fibre the influence of the interface stiffness and the difference between the different approaches is barely visible. In the detail shown in the lower part of Figure 5 it can be seen that the increase of the stiffness of the fibre matrix interface leads to higher shear stresses and higher gradients at the fibre ends, which seems to agree with experimental findings as described in (Bentur and Mindess 2006).

The normal stresses in the fibre are plotted along the fibre axis in Figure 6. The x-axis is normalised as before. The y-axis is normalised with respect to the maximum normal stress computed with the shear lag theory. The weak interface allows for rather large amounts of slip that lead to a less steep increase of the normal stresses in the fibre. The higher the interface stiffness the more the pufem approach approximates the analytical solution. But naturally the pufem approach always considers some small amount of slip in the interface, while the analytical solution does not. An interesting aspect is that the pufem approach predicts higher stresses in the fibre than the analytical solution does. This is due to the fact that the local strain field around the fibre endpoints is influenced by the fibre. This is not taken into account by the analytical solution but occurs in measurements of real systems (Bentur and Mindess 2006).

403

3.3 Comparison of different ratios between fibre and matrix Young’s modulus

maximum shear stress following from the analytical solution for the case of Ef /Em = 1. For all cases the differences between the pufem approach and the analytical solution are small apart from the endpoints where the same phenomenon can be observed as has been described already in Section 3.2: the pufem approach yields higher values than the shear lag theory, which agrees with experimental results according to Bentur and Mindess (2006). The plots of the normal stresses in the fibre as given in Figure 8 show more clearly the influence of the stiffness variation. The x-axis is normalised with

We compare the response of the sample using different ratios between fibre and matrix stiffness computed with the pufem approach to the response following from the analytical solution. The interface stiffness is set to Dbt = 107 N/mm2 . We vary the ratio between fibre and matrix Young’s modulus Ef /Em from 1 to 8. The shear stress distributions are plotted in Figure 7. The x-axis is normalised with respect to the fibre length while the y-axis is normalised with respect to the

1.2

analytical

1 0.8

pufem

Dbt = 10

3

N/mm

2

pufem

Dbt = 10

4

N/mm

2

Dbt = 10

N/mm

2

0.6

pufem

Dbt = 10

6

N/mm

2

pufem

Dbt = 10

7

N/mm

2

0.4

f

/

0

pufem

5

0.2 0

0

0.2

0.4

0.6

0.8

1

x / lf

Figure 6. Influence of the stiffness of the fibre matrix interface on the normal stress in the fibre in comparison to the analytical solution; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum normal stress in the fibre following the shear lag theory. 15 10

analytical pufem

Ef /Em =2

analytical pufem

Ef /Em =4

analytical pufem

Ef /Em =8

analytical pufem

/

0

5

Ef /Em =1

0

0

0.2

0.4

0.6

0.8

x / lf

1

0

/

0

analytical solution

pufem approach

0 .9 7 5

1

x / lf

Figure 7. Comparison of the shear stress at the fibre interface computed following the shear lag theory and the pufem approach for different ratios between fibre and matrix Young’s modulus; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum shear stress following the shear lag theory for the case of an equal Young’s modulus in fibre and matrix.

404

9

analytical pufem

Ef / Em = 8

8 7

0

5

f

6

4

Ef / Em = 4

3

Ef / Em = 2

2

Ef / Em = 1

1 0

0

0.2

0.4

0.6

0.8

1

1.2

x / lf

Figure 8. Comparison of the normal stress in the fibre computed following the shear lag theory and the pufem approach for different ratios between fibre and matrix Young’s modulus; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum normal stress in the fibre following the shear lag theory for the case of an equal Young’s modulus in fibre and matrix.

respect to the fibre length. The y-axis is normalised using the maximum normal stress in the fibre for a ratio between fibre and matrix stiffness of 1. The results show good agreement between the analytical and the pufem solution. Only for a ratio Ef /Em = 8 the results differ considerably. This is due to the fact that in this case the fibre influences the strain field of the sample not only locally around the fibre endpoints, but also globally by stiffening the sample in the centre part. This basically means that the fibre is pulled less than assumed in the analytical solution and thus the normal stress in the fibre computed with the pufem approach is lower than the one following from the analytical solution. In general, the pufem approach and the shear lag theory compare well if the fibre matrix interface in the pufem approach is chosen sufficiently stiff. Differences are mainly due to the fact that the analytical solution does not take into account the influence of the fibre on the strain field of the sample—neither locally around the fibre endpoints nor globally on the strain field further away from the fibre. The higher shear stresses produced by the pufem approach at the fibre endpoints seem to be in better agreement with reality than the ones coming from the shear lag theory (Bentur and Mindess 2006). 4 EXAMPLES USING NON-LINEAR CONSTITUTIVE EQUATIONS We present a comparison of three different fibre distributions as shown in Figure 9, each sample containing 9 fibres. One specimen contains only parallel, horizontal

fibres while the fibres in distribution 1 and 2 are arbitrarily placed in the sample. The samples are squares of 4 mm length. The system is discretised with 625 bilinear quadrilateral elements. The sample is fixed at its left boundary and pulled at its right boundary. The results are given in Figures 9 and 10. To begin with, we study the linear elastic case. The matrix has a Young’s modulus of 20000 N/mm2 and Poisson’s ratio of 0.2. The fibre is 0.2 mm thick, it has a Young’s modulus of 500000 N/mm2 and a Poisson’s ratio of 0.2. The interface stiffness is set to Dbt = 50000 N/mm2 . Naturally, the sample containing 9 horizontal fibres is stiffer than the other samples. The stiffer response of the second distribution compared to the first distribution can be explained by examining the main orientation of the fibres in the samples with respect to the horizontal loading direction: in distribution 2 more fibres are orientated horizontally than in distribution 1 leading to a higher influence of the fibres. Next, the influence of the nonlinear bond law is examined while the matrix material is kept linear elastic. As shown in Figure 3 the following parameter are chosen: stmax = 10−6 mm, ttmax = 50 N/mm2 , stres = 10−3 mm and ttmax = 2 N/mm2 . The debonding process and the frictional pull-out stage can be observed for all samples. Again, the horizontal fibre sample shows the stiffest reaction. Furthermore, distribution 1 is slightly less ductile than distribution 2. This shows the influence of the orientation of the fibres with respect to the loading direction. Compared to the linear elastic example the influence of the fibres diminishes with the use of the nonlinear bond law. This is due to the shape of the bond law as shown in Figure 3. While

405

linear elastic case 160 no fibres horizontal fibres distribution 1 distribution 2

F [N]

120

80 mechanical system

u

40

0

0

0.004

0.008 u [mm]

0.012

0.016

horizontal fibres

F [N]

120

80 no fibres horizontal fibres distribution 1 distribution 2

40

0

0

0.004

0.008 u [mm]

0.012

distribution 1

0.016

distribution 2

no fibres horizontal fibres distribution 1 distribution 2

F [N]

20

10

0

Figure 9. bond.

0

0.004

0.008 u [mm]

0.012

0.016

Comparison of three fibre distributions employing nonlinear constitutive behaviour for matrix and fibre-matrix

the interface is rather stiff in the beginning, its stiffness decreases rapidly after the fibre has been pulled out of the matrix for a distance larger than a specified peak value stmax . But as soon as the interface weakens, the load carrying capacity of the fibre as part of the composite material reduces. Finally, we allow damage growth in the matrix material. The matrix strength is set to 5 N/mm2 . In the middle of the sample a vertical zone of weak material is assumed to have a strength of 2 N/mm2 . The pure matrix sample responds in a very brittle fashion as can bee seen in the force-displacement plot shown in the bottom of Figure 9. The addition of fibres not only leads to a strong increase in peak strength but also to a more ductile post-peak behaviour. Furthermore, the results clearly show the influence of the discrete fibre distributions. All fibre samples develop

406

the same strength which is mainly determined by the matrix strength outside of the weak zone in the middle of the sample. While the horizontal fibre sample shows the stiffest behaviour in the pre-peak part, fibre distribution 2 behaves clearly more ductile in the post peak part. Distribution 1 is only slightly more ductile than the horizontal fibre sample. Apart from the fact that in the horizontal fibre sample all fibres undergo the debonding process simultaneously while in the arbitrarily distributed fibre samples some fibres are already debonding while others are still in the elastic branch the described behaviour can be explained by considering the damage patterns as depicted in Figure 10. Without fibres damage localises in the weak part in the centre of the sample. Adding horizontal fibres leads to failure at the supports of the sample. Damage localises at the support which is

no fibres

horizontal fibres

distribution 1

distribution 2

ω Figure 10.

0

1

Comparison of damage patterns of three different fibre distributions (damage plots are smoothed).

pulled while only a small amount of damage occurs at the clamped support. Regarding distribution 1 some damage occurs at the weak zone in the middle of the sample although localisation takes place at the right support of the sample. The amount of damage is only slightly higher than in the sample with horizontal fibres. Thus only a minor increase in ductility is visible in Figure 9 between the horizontal fibre sample and distribution 1. In distribution 2 damage occurs at both supports of the sample and parts of the weak zone in the middle of the specimen. This leads to a higher energy distribution and thus to an increased ductility compared to the other samples. For all cases, the purely linear elastic setting, the nonlinear fibre bond, and the nonlinear fibre bond in combination with damage in the concrete matrix,

407

the influence of the discrete fibre distribution on the mechanical response of fibre reinforced concrete and thus the importance to include discrete fibres in the study of fibre reinforced materials has clearly been shown in this example.

5 CONCLUSIONS We have presented an approach based on the partition of unity finite element method in connection with damage mechanics. This approach enables us to study the influence of discrete fibre distributions in a sample on the mechanical properties of the material. The computationally expensive mesh generation process

of each fibre is avoided. Since we treat fibre reinforced concrete as a composite material consisting of concrete matrix, fibres and interfaces between fibres and matrix, the constitutive behaviour of each constituent can be directly taken into account. This enables a detailed study of the mechanical behaviour of fibre reinforced concrete or other fibre reinforced materials. REFERENCES Bentur, A. and S. Mindess (2006). Fibre reinforced cementitious composites (2. ed.). Taylor & Francis.

408

Duarte, C., I. Babuška, and J. Oden (2000). Generalized finite element methods for three-dimensional structural mechanics problems. Compt. Struct. 77, 215–232. Goh, K., R. Asden, K. Mathias, and D. Hukins (2004). Finite–element analysis of the effect of material properties and fibre shape on stresses in an elastic fibre embedded in an elastic matrix in a fibre–composite material. Proc. R. Soc. Lond. A. 460, 2339–2352. Häußler-Combe, U. and J. Hartig (2007). Bond failure mechanisms of textile reinforced concrete (TRC) under uniaxial tensile loading. Cem. Concr. Compos. 29, 279–289. Jirásek, M. and B. Patzák (2002). Consistent tangent stiffness for nonlocal damage models. Compt. Struct. 80, 1279–1293.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A discrete cracking model for sequentially linear analysis A.V. van de Graaf, M.A.N. Hendriks & J.G. Rots Delft University of Technology, Delft, The Netherlands

ABSTRACT: Over the past few years it has been shown that the sequentially linear analysis scheme is able to model fracture in quasi-brittle materials in an effective manner. So far, in nearly all publications a smeared cracking model was adopted. However, some phenomena can be modeled in a better way by using discontinuum interface elements. Consider for instance the case of a pull-out test in which the bond-slip relation plays an important role. The aim of this paper is to start the implementation of this family of finite elements by considering the simplest case, viz. a discrete cracking model. It is demonstrated that results obtained with the new implementation match the results obtained with finite element analyses based on an incremental-iterative scheme. It is also shown that a saw-tooth softening law constructed with the so-called ripple model gives smoother results in terms of load-displacement diagrams than a saw-tooth softening law constructed with a model based on equal energy dissipation per damage increment. 1 INTRODUCTION Numerical simulation of fracture in quasi-brittle materials by means of a finite element analysis based on an incremental-iterative scheme can be a real hassle, in particular for large-scale structures. In these cases it may be hard to obtain a properly converged solution. To address the problem of poor convergence, Rots (2001) introduced the sequentially linear analysis (SLA) technique which was later elaborated by Rots & Invernizzi (2004), DeJong et al. (2008) and DeJong et al. (2009). In nearly all the work that followed, a smeared cracking model was adopted. Rots et al. (2006b) have initiated the development of a sawtooth softening model for interface elements, but was never continued nor extended to fields other than discrete cracking. The aim of this paper is to resume this work by reproducing the discrete cracking model for sequentially linear analysis and to present a new model to set up saw-tooth softening laws based on equal energy dissipation per damage increment. 1.1 Brief review of the sequentially linear analysis procedure Traditionally physically nonlinear finite element analysis is carried out by applying an incremental-iterative scheme. The basic concept of SLA is to replace this scheme by a series of scaled linear analyses. After each linear analysis, the applied loads are to be multiplied by a scalar λ which has to be chosen such that a critical stress state is obtained. This means that in one integration point—the so-called critical integration point—the maximum stress equals the current strength of that point whereas in every other

409

integration point the maximum stress remains smaller than the respective strength. Next, a damage increment is applied to the critical integration point which comes down to an instantaneous reduction in stiffness and strength of that point in the next linear analysis. This implies that the constitutive law needs to be discretized, resulting in a so-called saw-tooth softening law. Summarizing, the outlined procedure amounts to continuously executing the following steps: 1. Perform a linear-elastic analysis with a representative (unit) load. 2. Determine which integration point has the largest maximum stress over tensile strength ratio (i.e. identify the critical integration point). 3. Multiply the representative (unit) load by the inverse of the largest ratio of step 2. In this way a critical stress state is obtained. 4. Update the stiffness and strength properties of the critical integration point in accordance with the adopted saw-tooth softening law.

2 CRACK SIMULATION WITH INTERFACE ELEMENTS: A DISCRETE CRACKING MODEL 2.1 Basic formulation Interface elements can be used in a finite element model to anticipate geometrical discontinuities due to fracture or debonding processes. For these structural interface elements the relative displacements across the interface u are related to the tractions t acting at both sides of the interface. In a two-dimensional

model the relative displacement vector u consists of two components   un , (1) u = ut

tn 3

ft

2.5

mother curve 2 1.5

where un represents a crack opening displacement and ut a crack sliding displacement. Similarly, the traction vector t consists of the components   t t= n , (2) tt where tn and tt are the normal and tangential tractions respectively. The relation between these two vectors can be written compactly as t = Du,

(3)

where D is the constitutive matrix which—in case of discrete cracking without dilatancy—reads   0 k , (4) D= n 0 kt with kn the normal stiffness and kt the shear stiffness. 2.2 Crack initiation and growth criterion In the context of SLA a crack arises or grows if for one and only one integration point the following equation holds ∗

tn (λ) = ft ,

(5)

where ft∗ equals the current tensile strength of the integration point under consideration. For all other integration points the following inequality should hold tn (λ) < ft∗ .

(6)

p1 ×fft t

1 0.5

p2 ×fftt

kn( i )

0 0

0.02

0.04

0.06

un 0.08

0.1

un;ult

Figure 1. Saw-tooth model 1 which is based on a fixed band around the mother curve.

and p2 (see Fig. 1) are determined from an iterative procedure such that the following two conditions are fulfilled. First, the obtained saw-tooth softening law should be invariant with respect to the fracture energy GfI , i.e. GfI − GfI ∗ GfI

= 0,

(7)

where GfI ∗ is the area enclosed by the saw-tooth softening law. Simultaneously, the obtained saw-tooth softening law should be invariant with respect to the ultimate relative displacement un;ult , i.e. ∗ un;ult − un;ult

un;ult

= 0,

(8)

∗ where un;ult is the ultimate relative displacement of the saw-tooth softening law. Note that both conditions (7) and (8) have been normalized to ease the iterative procedure for finding p1 and p2 .

3.2 Model 2: Saw-tooth model based on equal energy dissipation per damage increment

3 SAW-TOOTH DISCRETISATION OF A LINEAR STRAIN-SOFTENING LAW In this section two ways to set up a saw-tooth softening law are presented. The first saw-tooth softening model is a slightly modified version of the ripple model which is based on a fixed band around the mother curve (Rots et al. 2006a). The second saw-tooth model is based on equal energy dissipation per damage increment. 3.1 Model 1: Saw-tooth model based on a fixed band around the mother curve (ripple model) The input parameters for this model are the initial dummy stiffness kn(1) and the desired number of sawteeth N . Then the unknown band width parameters p1

410

Starting point of this new saw-tooth softening model is that the area under the softening curve—which equals the mode I fracture energy GfI —is split in N triangles of equal area (see Fig. 2). In other words, after each damage increment an equal amount of fracture energy GfI is dissipated, assuming secant unloading. Then for every triangle the relative displacement of its rightmost vertex un(i) is calculated. These values are used as input for the saw-tooth softening model. Starting from the ultimate relative displacement a new series of triangles is constructed. The height of every triangle—which effectively comes down to a traction drop—is calculated from the required triangle area and the corresponding relative displacement. Every triangle is

225

tn 3

225 F

2.5

100

2 I f

G =

1.5

G If N

1

t = 100 mm

0.5

E = 30,000 N/mm2

un

0 0

0.02

0.04

0.06

(a)

0.08

50

= 0.2

0.1

5

un(i )

tn

Figure 3. Model setup of the analyzed single-edge-notched beam. All dimensions are in mm.

3 2.5 2

G If * = G If

1.5 1

(a) 0.5

un

0 0

0.02

0.04

0.06

0.08

0.1

(b) (b)

Figure 2. Saw-tooth model 2 which is based on equal energy dissipation per damage increment. Divide the area under the softening curve in N equal parts of GfI and calculate un(i) (i)

for every triangle (a). Then use un as a basis to construct a new series of triangles with an individual area of GfI ∗ (b).

positioned such that it fits exactly on top of the previous triangle (see Fig. 2b).

(c)

(d)

Figure 4. Adopted meshes: medium meshes with linear (a) and quadratic (b) elements and fine meshes with linear (c) and quadratic (d) elements.

4 NUMERICAL SIMULATION OF A THREE-POINT-BEND TEST ON A NOTCHED BEAM 4.1 Beam model and meshes The effectiveness of the presented discrete cracking model is demonstrated by numerical simulation of a three-point-bend test on a single-edge-notched beam (Grassl & Jirásek, 2005). The simply supported beam has a span of 450 mm, a depth of 100 mm and a thickness of 100 mm (see Fig. 3). At midspan a notch of 50 mm deep and 5 mm wide is present. The unreinforced concrete has a Young’s modulus of 30,000 N/mm2 , a Poisson’s ratio of 0.2 and has been modeled with triangular plane stress elements. The anticipated crack has been indicated in Figure 3 with a dashed line and has been modeled with line interface elements. To investigate the objectiveness of the obtained solution with respect to the chosen mesh, four meshes have been applied (see Fig. 4). Variations regard the mesh size

411

(10 interface elements for the medium mesh vs. 20 interface elements for the fine mesh) and the element shape functions (linear versus quadratic). Table 1 shows the characteristics of each mesh. 4.2 Adopted saw-tooth softening laws For the adopted saw-tooth softening laws linear softening was assumed with a tensile strength ft of 2.4 N/mm2 and a mode I fracture energy GfI of 113 J/m2 (see Fig. 5). The ultimate relative displacement un;ult was obtained as (2*GfI )/ft ≈ 0.094 mm. To investigate the effect of the adopted saw-tooth softening law six laws have been set up. Variations regard the adopted saw-tooth model (the models of Sections 3.1 and 3.2) and the number of saw-teeth (25, 50 and 100). See Table 2 for an overview

Table 1.

Characteristics of the adopted meshes.

Label

Mesh refinement

Mesh 1 Mesh 2 Mesh 3 Mesh 4

Medium Medium Fine Fine

tn

Element shape functions

25 teeth kn(1) = 1.0 104 N/mm3

2.5

Linear* Quadratic** Linear* Quadratic**

p1 = 12.3%

2

p2 = 12.0% 1.5

* Numerical integration of the line interface elements was carried out with a two-point Newton-Cotes scheme. ** Numerical integration of the line interface elements was carried out with a three-point Newton-Cotes scheme.

p1 ft

1 0.5

tn

p2 ft

kn(i )

0 0

0.02

0.04

0.06

un 0.08

0.1

(a)

3

tn

ft = 2.4 N/mm 2

ft

Saw-tooth law A

3

2.5

G If = 0.113 Nmm/mm 2 un ;ult

2

Saw-tooth law B

3

50 teeth kn(1) = 1.0 104 N/mm3

0.094 mm 2.5

1.5

p1 = 6.16%

2 1

p2 = 5.91%

G If

1.5

0.5

un

0 0

0.02

0.04

0.06

0.08

1

0.1

0.5

un;ult

Figure 5. For the line interface elements a linear softening relation relation was assumed.

un

0 0

0.02

Characteristics of adopted saw-tooth softening Number of saw-teeth

Saw-tooth law A Saw-tooth law B Saw-tooth law C Saw-tooth law D Saw-tooth law E Saw-tooth law F

Model 1* Model 1 Model 1 Model 2** Model 2 Model 2

25 50 100 25 50 100

0.1

100 teeth kn(1) = 1.0 104 N/mm3

2.5

Saw-tooth model

0.08

Saw-tooth law C

3

Label

0.06

(b)

tn Table 2. laws.

0.04

p1 = 3.13%

2

p2 = 2.88% 1.5 1 0.5

* Based on the ripple model. ** Based on equal energy dissipation per damage increment.

of the characteristics of each saw-tooth softening law. Figures 6 and 7 show the adopted saw-tooth softening laws. Note that the initial dummy stiffness cannot be chosen freely for saw-tooth model 2. Figure 8 shows the cumulative energy dissipation as a function of the relative displacement un for both models. Note that for model 1 the dissipated energy per damage increment increases with un whereas for model 2 it is an equal amount per damage increment. 4.3 Analysis results An overview of all performed analyses is given in Table 3. For comparison reasons, also nonlinear finite

412

un

0 0

0.02

0.04

0.06

0.08

0.1

(c)

Figure 6. Saw-tooth softening laws which have been constructed with the saw-tooth model of Section 3.1 with 25 (a), 50 (b) and 100 (c) saw-teeth. For all saw-tooth laws an initial dummy stiffness of 1.0 × 104 N/mm3 was used.

element analyses (NLFEA) have been carried out using an incremental-iterative scheme. Figure 9 shows the load-displacement curves obtained with a medium mesh consisting of linear elements. The applied saw-tooth softening laws have been based on the ripple model. Obviously, all models are able to capture the course of the nonlinear finite element analysis sufficiently accurate. It is also clear that if more saw-teeth are adopted a smoother structural response

tn

(

Saw-tooth law D

4 3.5 3

G If ) [Nmm/mm 2 ]

0.12

25 teeth kn(1) = 1.023 ×103 N/mm3

0.08

G If = 4.52 × 10 3 Nmm/mm 2

0.06

Saw-tooth law A

0.1

2.5 0.04

2 0.02

G If

1.5

un [mm]

0 0

1

G

I f

(

0.5

un

0 0

0.02

0.025

0.05

0.075

0.1

0.05

0.075

0.1

(a)

0.04

0.06

0.08

G If ) [Nmm/mm 2 ]

0.12

0.1

Saw-tooth law D

0.1

(a)

0.08

tn

0.06

Saw-tooth law E

4 3.5

k

3

(1) n

0.04

50 teeth = 2.071× 103 N/mm3

I f

3

G = 2.26 × 10 Nmm/mm

0.02

2

un [mm]

0 0

2.5

0.025

(b)

2

Figure 8. Cumulative energy dissipation (GfI ) as a function of the relative displacement un for saw-tooth model 1 (a) and saw-tooth model 2 (b).

1.5 1 0.5

un

0 0

0.02

0.04

0.06

0.08

Table 3.

0.1

(b)

tn

Saw-tooth law F

4 3.5

kn(1)

3

100 teeth = 4.167 ×103 N/mm3

G If = 1.13 × 10 3 Nmm/mm 2

2.5 2 1.5 1 0.5

un

0 0

0.02

0.04

0.06

0.08

0.1

(c)

Figure 7. Saw-tooth softening laws which have been constructed with the saw-tooth model of Section 3.2 with 25 (a), 50 (b) and 100 (c) saw-teeth. The initial dummy stiffness depends on the number of saw-teeth applied.

is obtained. Figure 10 shows the results for the same mesh, but for these analyses the applied saw-tooth softening laws have been based on equal energy dissipation per damage increment. From Figure 10a it is clear that the initial dummy stiffness of saw-tooth law D is not large enough compared to the stiffness of the surrounding continuum elements. Also note that the obtained curves are less smooth compared to those of Figure 9 and that a few

413

Overview of all performed analyses.

Label

Adopted mesh

Adopted saw-tooth law

SLA01 SLA02 SLA03 SLA04 SLA05 SLA06 SLA07 SLA08 SLA09 SLA10 SLA11 SLA12 SLA13 SLA14 SLA15 SLA16 SLA17 SLA18 SLA19 SLA20 SLA21 SLA22 SLA23 SLA24

Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 3 Mesh 4

Saw-tooth law A Saw-tooth law A Saw-tooth law B Saw-tooth law B Saw-tooth law C Saw-tooth law C Saw-tooth law A Saw-tooth law A Saw-tooth law B Saw-tooth law B Saw-tooth law C Saw-tooth law C Saw-tooth law D Saw-tooth law D Saw-tooth law E Saw-tooth law E Saw-tooth law F Saw-tooth law F Saw-tooth law D Saw-tooth law D Saw-tooth law E Saw-tooth law E Saw-tooth law F Saw-tooth law F

minor snap-backs are observed as well as a small bump around a displacement of 0.35 mm. In Figures 11 and 12 the load-displacement curves obtained with a medium mesh and quadratic elements are shown for different numbers of saw-teeth. For the

2

2

nonlin NLFEA

nonlin NLFEA sla0 SLA01

sla0 SLA13

1.5

Load [kN]

Load [kN]

1.5

1

1

0.5

0.5

0

0 0

0.25

(a)

0.5

0.75

0

1

0.25

0.5

1

2

2

nonlin NLFEA

nonlin NLFEA sla0 SLA03

sla0 SLA15

1.5

Load [kN]

1.5

Load [kN]

0.75

Displacement [mm]

(a)

Displacement [mm]

1

1

0.5

0.5

0

0

0

0

0.25

0.5

0.75

0.25

(b)

(b)

0.5

1

Displacement [mm]

Displacement [mm] 2

2

nonlin NLFEA

nonlin NLFEA

sla0 SLA17

1.5

sla0 SLA05

Load [kN]

1.5

Load [kN]

0.75

1

1

1

0.5

0.5 0 0

0 0

0.25

0.5

0.75

1

(c)

0.25

0.5

0.75

1

Displacement [mm]

(c)

Figure 9. Load-displacement diagrams for a medium mesh with linear elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

Figure 10. Load-displacement diagrams for a medium mesh with linear elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

curves in Figure 11 the saw-tooth model of Section 3.1 was applied. Just like the curves in Figure 9, the curve obtained with the nonlinear finite element analysis is resembled well. Also here it holds true that when more saw-teeth are applied, a smoother response is obtained. Compared to linear elements, quadratic elements do not yield significantly better results, except for a slight reduction in the band width of the curve. Compared to Figure 10 the load-displacement curves of Figure 12 look better, particularly because the

strange little bump has now disappeared. Nevertheless, the minor snap-backs remain. So for saw-tooth model 2 switching to quadratic elements yields an improvement in response. Figure 13 reveals the effect of a mesh refinement. In that case the band of small jumps and drops—which is characteristic for results obtained with SLA— becomes even smaller. The almost perfect resemblance of the load-displacement curve obtained with nonlinear finite element analysis is remarkable. Note that the the curve in Figure 13a is slightly rougher than the curve in Figure 9b, despite that on average

414

2

2 nonlin NLFEA

sla0 SLA02

sla0 SLA14

1.5

Load [kN]

Load [kN]

1.5

nonlin NLFEA

1

0.5

1

0.5

0

0 0

0.25

0.5

0.75

1

0

Displacement [mm]

(a)

0.25

(a)

2

0.5

sla0 SLA16

1.5

Load [kN]

Load [kN]

nonlin NLFEA

sla0 SLA04

1.5

1

0.5

1

0.5

0

0 0

0.25

0.5

0.75

1

0

Displacement [mm]

(b)

0.25

0.5

0.75

1

Displacement [mm]

(b)

2

2 nonlin NLFEA

nonlin NLFEA

sla0 SLA06

sla0 SLA18

1.5

Load [kN]

1.5

Load [kN]

1

2 nonlin NLFEA

1

0.5

1

0.5

0

0

0

(c))

0.75

Displacement [mm]

0.25

0.5

0.75

1

0

Displacement [mm]

(c)

0.25

0.5

0.75

1

Displacement [mm]

Figure 11. Load-displacement diagrams for a medium mesh with quadratic elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

Figure 12. Load-displacement diagrams for a medium mesh with quadratic elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

the same amount of energy—fracture energy times area per integration point—is dissipated per damage increment. The load-displacement curves in Figure 14 also show a smoothening of the structural response upon mesh refinement. However note that the peak value of the load is still slightly underestimated. Also note that a mesh refinement does not help to get rid of the minor snap-backs that we have seen before.

Figure 15 shows the obtained load-displacement curves for a fine mesh consisting of quadratic elements. As expected this combination gives the best results. In particular for Figure 15c it yields that there is no visible deviation between the curve obtained with SLA and the one obtained with NLFEA. Figure 16 shows that if saw-tooth laws based on the saw-tooth model of Section 3.2 are used the results are improving upon mesh refinement and an increasing number of saw-teeth. Nevertheless, it

415

2

2 nonlin NLFEA

sla0 SLA07

sla0 SLA19

1.5

Load [kN]

Load [kN]

1.5

nonlin NLFEA

1

0.5

1

0.5

0

0 0

0.25

(a)

0.5

0.75

1

0

Displacement [mm] 2

0.75

nonlin NLFEA

nonlin NLFEA

sla0 SLA09

sla0 SLA21

1.5

Load [kN]

Load [kN]

0.5

1

Displacement [mm] 2

1.5

1

0.5

1

0.5

0

0 0

0.25

(b)

0.5

0.75

1

0

Displacement [mm]

0.25

0.5

0.75

1

Displacement [mm]

(b)

2

2 nonlin NLFEA

nonlin NLFEA

sla0 SLA11

sla0 SLA23

1.5

Load [kN]

1.5

Load [kN]

0.25

(a)

1

0.5

1

0.5

0

0 0

0.25

(c)

0.5

0.75

1

0

Displacement [mm]

(c)

Figure 13. Load-displacement diagrams for a fine mesh with linear elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

should be concluded that in all cases saw-tooth softening laws set up with the ripple model give the best results.

5 CONCLUSIONS This paper has confirmed that sequentially linear analysis can be applied successfully to finite element models that contain interface elements with discrete cracking models. A new saw-tooth model has been presented based on equal energy dissipation per damage

416

0.25

0.5

0.75

1

Displacement [mm]

Figure 14. Load-displacement diagrams for a fine mesh with linear elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

increment. Although the quality of the obtained structural response with this new saw-tooth model can be called reasonable, it was shown that the best results (in terms of smooth curves and resemblance of the curve obtained by an incremental-iterative approach) are obtained with the ripple model. Apparently, it is more important to resemble the original softening curve as closely as possible by adopting a constant stress overshoot line—like in the ripple model—than having equal energy dissipations per damage increment. Vice versa, the snap-backs in the global loaddisplacement curves, which are so typical for SLA

2

2 nonlin NLFEA

sla0 SLA08

sla0 SLA20

1.5

Load [kN]

Load [kN]

1.5

nonlin NLFEA

1

1

0.5

0.5

0

0 0

0.25

0.5

0.75

0

1

0.5

nonlin NLFEA

nonlin NLFEA

sla0 SLA10

sla0 SLA22

1.5

Load [kN]

1.5

1

1

0

0 0

0.25

0.5

0.75

0

1

0.25

0.5

0.75

1

Displacement [mm]

(b)

Displacement [mm]

(b)

2

2

nonlin NLFEA

nonlin NLFEA sla0 SLA12

sla0 SLA24

1.5

Load [kN]

1.5

Load o d [kN] k

1

0.5

0.5

1

1

0.5

0.5

0

0 0

(c)

0.75

Displacement [mm] 2

2

Load [kN]

0.25

(a)

Displacement [mm]

(a)

0.25

0.5

0.75 .

0

1

(c)

Displacement [mm] [m ]

Figure 15. Load-displacement diagrams for a fine mesh with quadratic elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

0.25

0.5

0.75

1

Displacement [mm]

Figure 16. Load-displacement diagrams for a fine mesh with quadratic elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

REFERENCES results with relatively rough meshes and/or rough sawtooth diagrams, are more related to the variations of tensile strengths than to variations in dissipated energy per damage increment. Furthermore, for both saw-tooth models it was observed that a mesh refinement and/or an increase in the number of saw-teeth improved the quality of the obtained results.

417

DeJong, M.J, Belletti, B., Hendriks, M.A.N. & Rots, J.G. 2009. Shell elements for sequentially linear analysis: lateral failure of masonry structures. Engineering Structures 31(7): 1382–1392. DeJong, M.J., Hendriks, M.A.N. & Rots, J.G. 2008. Sequentially linear analysis of fracture under non-proportional loading. Engineering Fracture Mechanics 75(18): 5042–5056.

Grassl, P. & Jirásek, M. 2005. Nonlocal damage-plasticity model for failure of plain concrete. Proc. 11th Int. Conf. Frac., Turin, Italy. Rots, J.G. 2001. Sequentially linear continuum model for concrete fracture. In R. de Borst, J. Mazars, G. PijaudierCabot & J.G.M. van Mier (eds), Fracture mechanics of concrete structures: 831–839. Rotterdam: Balkema. Rots, J.G., Belletti, B. & Invernizzi, S. 2006a. On the shape of saw-tooth softening curves for sequentially linear analysis. In G. Meschke, R. de Borst, H. Mang & N. Bi´cani´c (eds), Computational modelling of concrete structures: 431–442. Leiden: Taylor & Francis/Balkema.

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Rots, J.G. & Invernizzi, S. 2004. Regularized sequentially linear saw-tooth softening model. Int. J. Numer. Anal. Meth. Geomech. 28: 821–856. Rots, J.G., Invernizzi, S. & Belletti, B. 2006b. A sequentially linear saw-tooth model for interface elements. In G. Meschke, R. de Borst, H. Mang & N. Bi´cani´c (eds), Computational modelling of concrete structures: 203–212. Leiden: Taylor & Francis/Balkema.

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Relations between structure size, mesh density, and elemental strength of lattice models M. Voˇrechovský & J. Eliáš Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic

ABSTRACT: We study the effect of discretization of lattice models. Two basic cases are examined: (i) homogeneous lattices, where all elements share the same strength and (ii) lattices in which the properties are assigned to the elements according to their correspondence to three phases of concrete, namely matrix, aggregates, and the interfacial transitional zone. These dependencies are studied with both, notched and unnotched beams loaded in three point bending. We report the results for regular discretization and irregular networks obtained via Voronoi tessellation. This is done for two types of models: with and without rotational springs (normal and shear springs are always present). All the springs are ideally brittle, i.e. after reaching the strength criterion; they are irreversibly removed from the structure. The dependence of strength is compared to various size effect formulas, and we show that in the case of homogeneous lattices, the fineness of discretization of the specimens of the same size can mimic variations in the size of lattice models with the same discretization density. In the case of heterogeneity (ii), we report how both the peak force and fracture energy depend on the mesh resolution both for notched and unnotched structure. 1 INTRODUCTION Lattice models are well established tool for fracture modeling and they appear to be very helpful especially thanks to the increasing power of modern computers. In classical lattice models, the material is represented by a set of discrete elements interconnected by springs. The combination of simple constitutive models with a material structure incorporated from the meso-level (Lilliu & van Mier 2003, Bolander et al. 1998) or by randomness of material parameters which somehow mimics this structure (Grassl & Bažant 2009, Alava et al 2008) makes it a powerful tool able to model quasibrittle structural response. It is an alternative to relatively complex constitutive laws applied in classical continuum models. The simplest models are those involving only elasto-brittle springs. This type of model is studied in this contribution. The weak point of using purely brittle springs is strong dependency of the results on a network density. Since the network does not represent any real underlying structure, this dependency is understood as a bias which should be removed. If one insists on keeping the brittleness of elements as we do (no softening of elements is incorporated), the mesh size dependency can be overcome e.g. by scaling the strength of elements according to their lengths and a chosen internal length parameter (Jagota & Bennison 1995) or, as is believed, by incorporating the material inhomogeneities (voids, grains, microcracks) that introduces an internal length as well. In this paper, we extend our recent results (Voˇrechovský & Eliáš 2009) by the effect of rotational

springs acting on facets of Voronoi cells. Correction functions for previously derived scaling formulas are found and compared with the results. We study both homogeneous and heterogeneous lattice models. By homogeneous models, we mean a lattice in which all elements share the same deterministic material strength criterion (and elastic modulus E). Otherwise, there are several ways to represent disorder or heterogeneity of material. This can be achieved e.g. (a) by spatial randomization of the properties of elements or, (b) by attributing element properties depending on their phase which is obtained by projecting a granular structure on the mesh. From here on, heterogeneous models are those obtained by alternative (b), i.e. by projecting the simulated meso-level material structure on the network and changing the properties based on the phase classification. Both types of models can be used with either regular (structured, REN) or irregular (IRN) geometry of the network (or mesh). In this paper, we use both types of discretization. If we speak of structured network (REN), we use unstructured meshes with a regular mesh in a certain small region of interest. In this work, we focus on the effect of varying the network density (or mesh density) on the overall structural response. We consider that varying the network density in homogeneous models corresponds to changes in structural size of the structure modeled. In other words, by changing the network density, we might model different sizes of the specimen. Several papers concerning the effect of network density have been published but, according to authors’ knowledge, two issues have not been studied yet:

419

Figure 1. Specimens with a central notch: nTPBT (relative notch depth α = 1/3). Two types of meshes around a notch are presented: REN and IRN. The Delaunay triangulation corresponding to the dual graph of the Voronoi tessellation is illustrated for a given mesh density. The configuration of uTBPT is identical except for the missing notch.

(i) the effect of size on the strength of the homogeneous model with a random network geometry (IRN) and, (ii) the effect of grain microstructure projected on the specimen with varying network densities (the grain layout properties to be used to classify elements is kept, but the network density—mesh—is varied). These mesh-density effects are studied for both: specimens that fail by crack initiated from a smooth surface and notched specimens. In particular, we have performed numerous simulations with either notched or unnotched three-point-bent specimens (denoted either nTPBT or uTPBT). The geometry of the specimens is illustrated in Figure 1 top. The span S = 3D is equal to three times the specimen depth D. We have found this topic interesting because the irregularity of the network influences the strength unexpectedly. Not all the sources of the observed behavior were identified and analytically analyzed, thus, the contribution predominantly present our (mostly numerical) observations. The first part of the contribution describes briefly the model adopted. The following sections (3–6) are devoted to the observed size effects in homogeneous lattice models with both REN and IRN. The last part presents a short study describing how (whether) the meso-level concrete structure projected onto the model reduces these size (or mesh density) effects.

(1998). The fracture criteria are taken from the same article, i.e. Mohr-Coulomb surface with tension cutoff is adopted. More detailed description can be found in Eliáš (2009). In this paper, we study two different mechanical models that differ in how internal forces (between rigid bodies) are transmitted at the connections of adjacent facets. In the first model type (denoted NS), only normal and shear springs act. In the NSR model type, also rotational springs transferring local bending moments are added. However, only stresses in normal and shear spring contribute to the fracture criteria in both model types. As mentioned above, in the case of homogeneous models, the strength criterion defined by the breaking stress is identical for all springs. Tensile strength of all elements is set to 5 MPa. Also the E-modulus is the same for all springs. Forces carried by springs are influenced by the corresponding cross-sectional area A, spring length l and Poisson ratio ν. The cross-sectional area is calculated from the contact area between the rigid bodies (Figure 1). The springs representing the contact areas operate on the actual eccentricity coming from the discretization. 2.2 Meshing algorithm It has been proven by several authors (e.g. Schlangen & Garboczi 1997, Jirásek & Bažant 1995) that irregular geometry of the network helps to avoid directional preference of crack propagation. Thus, it has been chosen for the present model. The meshing algorithm is based on Voronoi tessellation, which is performed on the set of pseudorandomly placed triangulation nodes within the domain. The only restriction is that their minimal mutual distance equals to a predefined parameter l min . When a notch is to be modeled, it is included by mirroring nodes by the notch line in the notch vicinity, see Figures 1b and d. Voronoi tessellations then creates a straight line and all springs on that line are subsequently removed to model the notch. In order to place the notch tip exactly at the desired coordinate, three points are placed with a prescribed distance from the tip. This procedure guarantees an exact location of the shared vertex—the interface of the three corresponding rigid bodies at the notch tip. 2.3 Deviations from the theory of elasticity

2 BRIEF DESCRIPTION OF THE MODEL 2.1 Mechanics of discrete model Several lattice-type models can be found in literature. Here, the rigid-body-spring network developed by Kawai (1978) is used. In basics, the model is very similar to the one published by Bolander & Saito

420

The stiffness of springs is derived to represent an underlying imaginary isotropic, linearly elastic homogeneous continuum (Kawai 1978). However, the elastic behavior differs from the assumed theory. The effect is clearly described by Schlangen & Garboczi (1996). Simply, the isotropic elastic material should exhibit uniform stress under uniform strain. Voronoi tessellation can satisfy this

criterion for zero Poisson’s ratio ν. However, as showed by Bolander et al. (1999), for nonzero Poisson’s ratios, the stress distribution of a body under remote uniform uniaxial strain is not uniform any more. The greater the deviation from zero ratio ν, the more fluctuation in stress occurs (see error bars in Fig. 2). We also observed that Poison’s ratio severely influences the stress in the surface layer of elements. The average values of stresses in the lowermost elements of uTPBT diverge from the linear stress profile approximately obtained for zero ratio ν (Fig. 2a). Elsewhere, the average stresses roughly correspond to values given by ν = 0. The higher the Poisson’s ratio is and the finer discretization is used, the higher mean value of the stress is received in the boundary layer. This was observed also in a simulation of uniaxial tensile test without friction (i.e. uniform applied stress). Due to

Figure 4. Dependency of peak load on the REN network density (structural size D). Comparison with the size effect formulas (Equations 8 and 15).

Figure 5. Crack patterns at the peak load for various sizes of the unnotched beam with irregular network geometry. Left horizontal lines indicate the average height cf reached by the crack and its standard deviation.

Figure 2. Effect of poisons ratio on stresses σxx . a) the lowermost part of stress profile of uTPBT and b) stress profile close to the notch tip of nTPBT with regular net geometry loaded by force 10 N. Error bars show averages and sample standard deviations computed from 50 realizations.

this surface effect, the nominal strength is decreased, see Fig. 4. There is no significant effect of Poisson’s ratio on nominal strength observed in the case of notched specimens (nTPBT). The stresses in the elements around the notch tip are shown in Figure 2b. The greater is the deviation of Poisson’s ratio from 0, the greater is the variation in the stress. But the averages seem to be independent of value ν selected.

3 SIZE EFFECT SIMULATIONS Figure 3. On derivation of the peak moment in a bent specimen.

421

The size of a concrete specimen typically affects the observed nominal strength. Several sources of this phenomenon are documented (Bažant & Planas 1998),

we name the statistical and deterministic effects. Two main types of the deterministic size effects are distinguished. Structures with preexisting notches (positive geometry exhibiting type II size effect) and structures without any notch or with a small notch with respect to material internal length (negative geometry exhibiting type I size effect). Notched (type II, nTPBT) and unnotched (type I, uTPBT) are used to study this size effect in homogeneous brittle-spring networks. The density of the network is denoted as l min . Since there is no internal length in our constitutive law/model, we can represent varying size by varying network density. The characteristic size (depth) D is kept constant at a reference size D0 = 0.1 m, whereas the network density l min is varied; and we can mimic varying of the intrinsic size D by writing:

D = D0

l0min l min

Let us now deliver a closed-form expression for the observed size effect. Consider the midspan rectangular cross-section BD. The depth is discretized into 2N rigid bodies’ contacts of the same size, see Figure 3. Therefore the stress profile is a piecewise constant function along the depth D and approximates the actual (almost perfectly) linear profile. When the outermost spring reaches the extreme tensile stress f ∞ , the crosssection reaches its maximum bending moment M. Due to the symmetry along the neutral axis we can consider only the lower bottom of the depth (N elements) and calculate the bending moment as a doubled sum of force contributions times the corresponding arm. Each force contribution can be written as (Figure 6): Ti = B ·

(1)

where l0min = 0.02 m is the selected reference mesh density. Since we deal, in fact, with models of the same size, it is not necessary to report the size dependence on nominal strength (nominal stress at peak load). It suffices to report the loading forces F (D). On the other hand, however, the lengths (e.g. crack length) must be recalculated in a similar fashion as we did for D (see Equation 1). Removal of one element of the same size is interpreted as a crack of different lengths in models of various mesh densities. In order to evaluate the effect of network irregularity, all the results are computed for REN and IRN. Since the network in REN models is only regular in the vicinity of notch or midspan, the rest of the specimen (meshed by a lattice of irregular geometry) causes fluctuations of forces acting on the ‘‘crack faces’’. Subsequently, the obtained nominal forces are scattered. This effect is emphasized in unnotched structures, see e.g. Figure 4. In the regular networks, the rupture of the first element (beam or spring) causes the collapse of the whole structure. This holds both in the nTPBT and uTPBT. Therefore, the measured peak loads F p equal the elastic limits F e in the case of REN.

D i − 1/2 ∞ · f , 2N N − 1/2

i = 1, . . . , N

(2)

where B is the bar thickness [m], the second factor is the bin width l min = D/(2N ) [m] and the third factor is the corresponding constant stress in that bin [N/m2 ]. Each such a force has the following arm from the neutral axis:   1 D i− , i = 1, . . . , N (3) ri = 2N 2 where again, the first factor is the bin width. The resisting moment is a double of the sum (i = 1, . . . , N ): M (N ) = 2

N  i=1

Ti ri =

 N  1 2 BD2 f ∞  i − 4N 2 N − 1/2 2 i=1 (4)

4 SIZE EFFECT OF UNNOTCHED STRUCTURE In the case of regular mesh geometry (REN), the crack can only propagate along the axis of symmetry through regularly placed squared elements of exact size l min . Figure 4 shows the maximal load (that is the elastic limit at the same time) depending on the density of the REN net (or, the size of the structure D).

422

Figure 6. Plot of elastic limits and peak loads of beams with irregular network and smooth bottom surface. Average values and standard deviations are computed from 50 realizations for every size.

Calculating the sum yields:   BD2 ∞ 2N + 1 f M (N ) = 6 2N

where f ∞ is the strength limit for infinitely large structures and Db = 2 cf , i.e. double of the thickness of the boundary layer of cracking. If we take r = 1 which is a special case derived by Bažant and Li (1995), and rewrite Equation 9 in forces:

(5)

As N grows to infinity, the bending moment converges to the well-known value: M

∞

=f

∞

2

BD /6

(6)

The external moment equals the support reaction times the half span: M = F/2 · 3D/2. Putting this equal to Equation (5) yields:   2BD ∞ 2N + 1 F= f (7) 9 2N Equation (7) can be transformed into the dependence of peak force on bin width l min = D/(2N ): c

Fmax

c

       2BD ∞ l min l min 1+ = f = F∞ 1 + (8) 9 D D

This equation is plotted in Figure 4 and compared to the computed data. We can introduce a new length constant Db = l min = 20 mm to make it identical with Equation 10 introduced later. What remains to be clarified is the choice of the extreme stress f ∞ . An obvious choice would be the direct tensile strength (fl∞ = 5 MPa) of the model. This is because very large specimens fail at initiation of crack right at the midspan bottom face, which must equal the tensile strength. It would yield the asymptotic force Fl∞ = 11.11 kN. Unfortunately, the stress profile in not perfectly linear in reality. The real stress profile is affected by wall effects (the span of the beam is only 3D) and by the local compressive stress concentration around the point load. The nonzero Poisson’s ratio causes additional deviation from linear stress profile, see Sec. 2.3. As an approximation, we used nonlinear square fitting procedure to determine the two free parameters Db and F ∞ in Equation 8. One can calculate form fitted constant F ∞ the theoretical stress at the bottom layer for infinitely small mesh caused by load 10 N, see Sec. 2.3. This stress σ ∞ is added into Fig. 2 to show consistency with our fits. Is it worth pointing that another way exists to nicely fit the data—to consider the bent specimen being made of a quasibrittle material. One can assume a linear stress profile along the depth except for the damaged zone in the bottom tensile part. If we consider that the boundary layer of cracking has a constant size cf irrespective the specimen size, the following size effect formula can be derived (see pages 41–43 of Bažant 2005) for scaling of nominal strength (modulus of rupture):   rDb 1/r σN = fr (D; f ∞ , Db , r) = f ∞ 1 + (9) D

Fmax

       2BD ∞ Db Db 1+ = f = F∞ 1 + 9 D D    F∞

(10)

when Db = l min , Equations 8 and 10 match. Now, a question appears: What if we consider rotational springs? In each element (or contact area), the spring add a new additional moment Mi , see the last strip in Figure 6 right. These contributions are equal (for i = 1, . . . , 2N ). In each bin there is a pair of forces Ti that represent two triangles (below and above the constant stress σi . Each of these triangles are half of the strip long (= D/(4N)) and the maximum stress difference is σ . The stress σ is one half of the difference between the current strip and the adjacent strip:   f∞ i − 0.5 − i + 1.5 (σi − σi−1 ) = σ = 2 2 N − 1/2 =

f∞ 2N − 1

(11)

The pair of forces Ti representing the two triangles are (for i = 1, . . . , N ): Ti =

1 D BD f∞ BD σ · B · = σ = · 2 4N 8N 2N − 1 8N (12)

Each of these two forces act over the distance of D/(6N ) from the ‘‘neutral’’ state and form an additional moment increase to the total bending moment. The magnitude of each such moment contribution (in each strip-bin) is twice the arm times force Ti : Mi = 2 · Ti ·

423

BD2 D = f∞ 2 6N 24N (2N − 1)

(13)

In total, there are 2N such partial moments over the whole cross-section and therefore the total moment increment is 2N times the contribution Mi : M = 2N · Mi = f∞

1 BD2 6 2N (2N − 1)

(14)

This moment increment is not reflected in the failure condition. Transforming it into the increment of

maximal force gives F =

1 4 M = F∞ 3D 2N (2N − 1)

= F∞

l min l min D D − l min

(15)

Adding this increment to the total force from Equarot = tion 8 yields the upgraded size dependence Fmax Fmax + F. The first term uses Db , while the increment F uses l min . In other words F increases the maximal load, especially for small sizes. If the structure is represented by just one element (N = 1/2), the increment is infinite (an arbitrarily high load is supported by the rotational spring, no tensile forces appear). As the size becomes larger (or discretization finer), the total moment tend to that in Equation 6. The irregularity of the network geometry (IRN) allows the model to choose the ‘‘weakest’’ area to initiate and propagate the crack. That is why the elastic limits are, on average, lower in IRN compared to REN, see Figure 6. The load applied to break the first spring F e in IRN model is, on average, also much lower than the peak forces. The peak forces in IRN models are greater than those of REN. The first crack appears at the weakest spring loaded by high forces: the crack prefers short facets. Qualitatively, however, both force dependencies of IRN are similar to REN and follow the tendency proposed by Equation (8), respectively (15). The deviations for larger specimens (finer mesh densities) are caused by local stress deviation described in Section 2.3. Namely, we mean the stress fluctuations in the lowermost layer caused by Poisson’s ratio. Both the elastic forces and peak forces can drop below this horizontal asymptote; see Figure 9. Instead of one crack, many small cracks are created inside the bottom area of the specimen (Figure 5) and the model allows for redistribution of forces after many such local ruptures. These cracks do not form a continuous line. On average, the thickness of the boundary zone of distributed cracking is >l min . The fact that the zone has approximately the same height for all sizes (Figure 5), supports our claim that the data can be approximated reasonably well by Equations 8 and 10.

at the notch tip leads to collapse of the whole beam. The peak forces of REN (that are the elastic limits at the same time) are plotted against the net density lmin (or size D) in loglog plot. They fall exactly on a line of slope –1/2 (see Figure 7). This result is not new and corresponds to the remedy of size dependency of homogeneous regular lattice models proposed by Jagota & Bennison (1995). Again, rotational springs cause deviation from the theoretical size effect (LEFM power law). The deviation diminishes with increasing size of the structure (refining the mesh). We use the same procedure as in Section 4 to modify the power law. In large structures (or fine meshes) the stress profile behind the notch tip follows the well known formula σxx = √

KI 2π r

,

(16)

One could compute the F increment (correction) assuming this function over the whole stress profile. This would have to be done numerically because expression for Mi is too complicated to be summed analytically. However, this nonlinear stress function is valid only in a short region close to the notch tip. The correction for rotational spring is only necessary when the mesh is very coarse (small structure). There however, the stress profile is almost linear, not singular as in Equation 16. Therefore we finally assume linear stress profile and derive the correction term F for notched specimens caused by additional moments the same way as we did in Equations 11–15. The only difference is that instead of the total beam depth D, we substitute 2/3 D, because 1/3 Dis occupied by the notch. The formula for pair of forces Ti in strip i using σ from Equation 11 reads: Ti =

1 2 DB DBf ∞ σ = . 2 3 4N 12N (2N − 1)

(17)

5 SIZE EFFECT OF NOTCHED STRUCTURE A somewhat different situation appears when the beam fails by crack propagation from sharp notch (such as our nTPBT). In our numerical simulations, the notched specimens have similar features to the uTPBT. In the case of regular geometry (REN), the first rupture of the beam

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Figure 7. Simulations of nTPBT beams with regular mesh REN. Each circle is an average of 50 simulations, error bars are not included as the standard deviation is extremely small. Fitted by LEFM straight line of slope –1/2.

Contribution to the moment is given by multiplying by arm of the pair of forces (similar to Eq. (13)). Mi = 2Ti

2 D D2 Bf ∞ = 3 6N 54N 2 (2N − 1)

(18)

Summing all moment contributions, we receive a moment correction M that can be transformed into the load correction F: M = 2

F =

N  i=1

Mi = 2N · Mi =

D2 Bf ∞ 27N (2N − 1)

4DBf ∞ 4 M = 3D 81N (2N − 1)

(19) Figure 8. Simulations of nTPBT beams with irregular mesh (IRN). Each circle is an average value of 50 simulations.

(20)

Finally, we rewrite the function in terms of l min using N = D/(3 l min ): F =

2(l min )2 4Bf ∞ (l min )2 = F∞ min 9(2D − 3l ) D(2D − 3l min )

(21)

Adding this correction function to the power law obtained for large sizes, we can explain the trend of the data also for small sizes (see Fig. 7). The effect of springs is underestimated because the basic stress profile was assumed to be linear. Network irregularity (IRN), brings a new effect. Since the element placed right above the notch tip is angled and has varying size, the external load F e necessary to break it is affected. One has to break usually more than one element to reach peak force F p , thus F e < F p. The elastic F e limit obeys the LEFM slope of –1/2 and lies very close to the previous fit with regular nete e works (FIRN = 1.92D−0.5 , FREN = 1.94D−0.5 ). This is surprising because two effects working one against the other appear here. (i) The angle of the first element (deviation from the horizontal direction) increases the elastic limit force, i.e. shifts the line upwards. (ii) The average area of the first broken element is lower in IRN than in REN where all broken elements have the area l min × thickness. This leads to downward shift of the size effect line. Apparently, effects of those the upward and downward shifts cancel each other. Looking at the peak force data, a good fit in loglog plot is a straight line of slope –0.424, see Figure 8. The source of the observed deviation from the LEFM slope of –1/2 was found in the crack behavior. Figure 9 shows crack patterns at the peak load for all considered sizes. These crack length are recalculated into the intrinsic magnitudes (Equation 1). Apparently, the larger the specimen, the longer the length at the peak load: the crack initiation from the notch is followed by an increase in peak crack length with size D. This increase can be fitted by a power law with exponent

Figure 9. Crack patterns at the peak load for various sizes of the notched IRN beam (with irregular mesh geometry).

1/2 (see Fig. 10) both for NS and NSR models. The slower declination of the fitted power law in Figure 8 (–0.424) can be attributed to the described growth in crack length with specimen size D.

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6 DISCUSSION Some interesting points have been shown in the two preceding section and we now discuss some of the features in a more detail. It was mentioned previously that the length of a crack at the peak load seems to be increasing in the nTPBT IRN simulations while the average crack length at the peak load is about constant in uTPBT, see Figures 5 and 9.

Let us also mention recent results of Alava et al (2008) who show, using random fuse model, that strength of notched beams of various sizes and notch depths is influenced by the amount of disorder. The strength dependence on notch depth deviates from LEFM power law with an increasing disorder. Specimen strength made of highly disordered material, with a small notch, is driven mainly by the disorder and not only by the stress concentration. 7 REDUCTION OF SPURIOUS SIZE EFFECT The influence of network density has to be understood as a spurious phenomenon, because the mesh is arbitrary, artificial and does not arise from any real material structure. Some authors believe that the network density dependency might be removed by projecting the material inhomogeneities onto the lattice. This introduces the internal length, which decrease this dependency (van Mier & van Vliet 2003). In the following, this expectation is subjected to critical study, which shows limits of such a procedure. 7.1 Incorporating of grain structure Figure 10. Length of the crack at the peak for notched and unnotched TPBT: a) NS model, b) NSR model with rot. springs.

Surprisingly, the increase in the crack length of IRN nTPBT specimens obeys a power law with exponent 1/2 (see the thin angled line in Figure 10ab). In the case of unnotched specimens, the constant peak crack length is dependent on the selected reference network density l0min . This was checked numerically by additional simulations with various lengths l0min (otherwise kept constant here); see the thin horizontal lines in Figure 10a which shows that the average peak crack length roughly lies in the range of 1.1–1.3 l0min . In conclusion, scaling both the network size and the specimen size by the same positive scaling factor yields an identical result as for the original sizes. The peak force depends only on the quality of the stress profile approximation. We have performed a sensitivity analysis to identify the influence of various parameters on the elastic limit load and the peak load. In particular, Spearman nonparametric correlation coefficient was used. The greatest absolute correlation to the peak load was found with the maximum vertical coordinate of the crack—i.e. the thickness of the zone with distributed cracking cf . In the case of notched specimens, the elastic limit force F e is sensitive to the initial crack length as well as to the inclination of the initial crack from vertical direction (corr. coeff. approx. 0.8). Unfortunately, no dominant variable affecting the peak load was identified.

426

The grain structure that is used here is generated by computer algorithm using the Fuller curve (see e.g. Cusatis et al. 2006). Typically, maximal grain diameter dmax is chosen according to the real batch contents, and the minimal dmin according to the network density. The length of network elements should be at least three times smaller than dmin (van Mier 1997), otherwise the particles coalesce in the mesh. Grains smaller than dmin are ignored in the procedure. The larger dmin , the coarser mesoscopic structure is incorporated; yet, the generated coarse grains still correspond to the requested content of coarse grains, i.e. reducing dmin has no effect on coarse grains—it only adds finer aggregates. Grains were projected onto the lattice to attribute springs with the three material phases—aggregate, matrix and ITZ. These are distinguished according to the positions of nodes with respect to the mesostructure (see van Mier et al. 1997). Each phase has a different strength and Young’s modulus. Values from the article by Prado & van Mier (2003) were used. In the following part, we will test the hypotheses that: when finer the mesoscopic structure is considered a lower mesh sensitivity of the model is observed. For this reasons, six different grain contents were generated. The first one corresponds to the homogeneous case without any grains (studied above), the others differ by dmin (Figure 11); whereas maximum grain diameter dmax is kept equal to 32 mm. Mesh varied from density 2.5 mm up to density 0.625 mm. Note that not all the densities can be used for all the grain contents. The finer the grains, the finer

mesh is required. For all possible combinations, 50 realizations of notched and unnotched TPBT were simulated. In order to have an idea about the model behavior, Figure 12 shows average load-deflection diagrams for some of considered densities and mesostructures. Notice the drastic strength decrease when one incorporates even only few grains in the unnotched structure. The crack position can be sampled from many possible positions and thus fracture always propagates along the weak ITZ. Such a steep drop in strength is not observed with notched structures: because the crack propagates from the deep notch independently of positions of grains. The results for the peak loads are shown in Figure 13. The homogeneous model of the notched TPBT follows a straight line of slope 0.424 in the notched case (previously described in Figure 8). Models with a grain contents seem to reduce this dependency. The

best results (almost a horizontal line) are achieved by the most detailed grain contents. In the unnotched case, the presence of grains apparently increases the dependence on network density. The more detailed structure, the less dependency is visible. But still, the dependence of the peak force on the mesh resolution is stronger than in the homogeneous models. The second monitored parameter is the area under load-deflection curves, which has meaning of energy. No substantial reduction in the dependence of this energy on the mesh density (by incorporation of the grain structure) was observed for the notched structure—Figure 14a documents that all the lines

Figure 11. An example of crack patterns observed in a simulation of (a) nTPBT and (b) uTPBT with the model including concrete mesoscopic grain structure of varying fineness.

Figure 13. Dependence of maximum load for (a) notched TPBT and (b) uTBPT on grain structure incorporated.

Figure 12. Average load-deflection diagrams of 50 realizations of (a) nTPBT and (b) uTPBT for some of the considered densities and mesostructures.

Figure 14. Dependence of the area under load-deflection curve for (a) notched TPBT and (b) unnotched TPBT on grain structure incorporated.

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share approximately the same rising slope. The energy dependency is only slightly reduced for the unnotched structure, see Figure 14b. Yet, the line plotted for the finest resolution is far from being horizontal. 8 CONCLUSIONS The effect of discretization of lattice models was studied. The basic cases are examined: (a) homogeneous lattices, where all elements share the same strength and (b) lattices in which the properties are assigned to the elements according to their correspondence to three phases of concrete, namely matrix, aggregates, and the interfacial transitional zone (ITZ). These dependencies are studied with both, notched and un-notched beams loaded in three point bending. We report the results for regular discretization and irregular networks obtained via Voronoi tessellation. The dependence of strength is compared to various size effect formulas and we show that in the case of homogeneous lattices, the fineness of discretization of the specimens of the same size can mimic variations in the size of lattice models with the same discretization. In the case of heterogeneity (b), we report that even though the peak force dependence is influenced by the mesh resolution, and almost disappears in notched structures, a strong dependence of the fracture energy remains. This is important for fracture studies with lattice models. ACKNOWLEDGEMENTS This research was conducted with the financial support of the Ministry of Education, Youth and Sports of the Czech Republic under project No. 1M06005 within activities of the CIVAK research centre, and with the financial support of the Czech Science Foundation, project GACR 103/09/H085. The second author acknowledges the support from the Fulbright commission within the framework of Fulbright-Masaryk fellowship. REFERENCES Alava, M.J., Nukala, P.K.V.V & Zapperi, S. 2008. Fracture size effects from disordered lattice models. International Journal of Fracture 154: 51–59. Bažant, Z.P. (2005). Scaling of Structural Strength; 2nd updated ed., Elsevier Butterworth-Heinemann. Bažant, Z.P. & Li, Z. (1995). Modulus of rupture: size effect due to fracture initiation in boundary layer. J. of Struct. Engrg—ASCE, 121(4): 739–746. Bažant, Z.P. & Planas, J. 1998. Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton and London.

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Bolander, J.E., Hikosaka, H. & He, W.-J. 1998. Fracture in concrete specimens of different scale. Engineering Computations 15: 1094–1116. Bolander, J.E. & Saito, S. 1998. Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics 61: 569–591. Bolander, J.E., Yoshitake, K. & Thomure, J. 1999. Stress analysis using elastically homogeneous rigid-body-spring networks. Journal of Structural Mechanics and Earthquake Engineering 633: 25–32. Cusatis, G., Bažant, Z.P. & Cedolin, L. 2006. Confinementshear lattice CSL model for fracture propagation in concrete. Computer Methods in Applied Mechanics and Engineering 195: 7154–7171. Eliáš, J. 2009. Discrete simulation of fracture processes of disordered materials. Ph.D. thesis, Brno University of Technology, Faculty of Civil Engineering, Brno, Czech Rep. Grassl, P. & Bažant, Z.P. 2009. Random lattice-particle simulation of statistical size effect in quasi-brittle structures failing at crack initiation. Journal of Eng. Mechanics 135: 85–92. Jagota, A. & Bennison, S.J. 1995. Element breaking rules in computational models for brittle materials. Modelling and Simulation in Materials Science and Engrg 3: 485–501. Jirásek, M. & Bažant, Z.P. 1995. Particle model for quasibrittle fracture and application to sea ice. Journal of Engineering Mechanics 121: 1016–1025. Kawai, T. 1978. New discrete models and their application to seismic response analysis of structures. Nuclear Engineering and Design 48: 207–229. Lilliu, G. & van Mier, J.G.M. 2003. 3D lattice type fracture model for concrete, Eng. Fracture Mechanics 70: 927–941. Schlangen, E. & Garboczi, E.J. 1997. Fracture simulations of concrete using lattice models: computational aspects. Engineering Fracture Mechanics 57: 319–332. Schlangen, E. & Garboczi, E.J. 1996. New method for simulating fracture using an elastically uniform random geometry lattice. Int. Journal of Engineering Science 34: 1131–1144. Prado, E.P. & van Mier, J.G.M. 2003. Effect of particle structure on mode I fracture process in concrete. Engineering Fracture Mechanics 70: 1793–1807. van Mier, J.G.M. 1997. Fracture Processes of Concrete: Assessment of Material Parameters for Fracture Models. CRC Press, Boca Raton, Florida. van Mier, J.G.M. & van Vliet, M.R.A. 2003. Influence of microstructure of concrete on size/scale effects in tensile fracture. Engineering Fracture Mechanics 70: 2281–2306. van Mier, J.G.M., Chiaia, B.M. & Vervuurt, A. 1997. Numerical simulation of chaotic and self-organizing damage in brittle disordered materials. Computer Methods in Applied Mechanics and Engineering 142: 189–201. Voˇrechovský, M. & Eliáš, J. 2009. Mesh dependency and related aspects of lattice models. FRaMCoS 7, Proc. of 7th Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures, Jeju, South Korea, May 23–28, 2010, in press.

Time-dependent and multi physics phenomena

Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Prediction of the permeability of damaged concrete using a combined lattice beam-crack network approach M. Abreu Swiss Federal Institute of Technology ETHZ Zürich, ETH-Hönggerberg; Materials Department, Portuguese National Laboratory for Civil Engineering, LNEC-Lisbon

J. Carmeliet Swiss Federal Institute of Technology ETHZ Zürich, ETH-Hönggerberg; Swiss Federal Laboratories for Materials Testing and Research, EMPA-Dübendorf, Building Technologies

J.V. Lemos Concrete Dams Department, Portuguese National Laboratory for Civil Engineering LNEC-Lisbon

ABSTRACT: The article describes a combined damage and permeability 3D model for concrete. The objective of the model is to predict the permeability of damaged concrete. For the mechanical model a lattice of beam elements is used where the damage is modelled by a step-by-step removal of beams. For the calculation of the permeability the removed beams are connected by transport elements with a aperture proportional to the relative displacement between the nodes previously connected by the beans. The numerical simulations are then compared with experimental results from a diffuse tensile cracking and permeability test. Two lattice models are compared with the experimental results: in the first one the concrete is considered as a homogenous material and in the second the aggregates are explicitly modelled. It is shown that when the aggregates are included, the simulation agrees better with the test results for lower strains, while for higher strains the homogeneous model agrees better. 1 INTRODUCTION The accurate knowledge of the concrete permeability is crucial for the assessment of durability of concrete, since liquid transport into the material is a determining key factor. The penetration of liquids may initiate several damage processes, such as frost damage, corrosion, chemical degradation and salt damage, etc. Damage in the form of cracks significantly modifies the permeability of the material and allows a preferential transport along the crack network into the material several orders of magnitude faster compared to the transport in sound concrete (Roels et al. 2006). Therefore, when designing durable concrete constructions, not only the prediction of cracking is needed, but also the estimation of the permeability of cracked concrete is necessary. The permeability of cracks highly depends on the average crack width, the crack tortuosity and connectivity (Carmeliet et al. 2004). Therefore, we present in this paper a model taking into account these important crack aspects. In order to predict the permeability of damaged concrete a combined crack and transport model is presented. The crack model is based on a three dimensional lattice of beam elements where damage is

431

modelled by the consecutive removal one-by-one of beams. Therefore the model can be classified as of a repetitive linear elastic type. The transport elements are formed by connecting the centre of the removed beams and their individual permeability depends on the crack width, taken as the relative displacement between the nodes previously connected by the beans. The global permeability of the cracked concrete is then determined from the obtained crack network considering saturated transport. The model accuracy depends on the properties of the beam elements and the criterion used for their removal, which are obtained by calibration against experiments. 2 LATTICE MODEL The lattice crack model is based on developments at the Delft University and therefore often known as Delft Lattice (Schlangen & Garboczi 1996, 1997) (Lilliu. & van Mier 2000). In this type of model the lattice of beams is composed of one dimensional finite elements with axial, bending and shear stiffness. In a 3D model there are 6 degrees of freedom per node, these are the three displacements and the three rotations (figure 1).

A beam element will connect two nodes, therefore the elemental stiffness matrix has 12×12 dimensions (figure 1) (CALFEM, 1999) were Ke is the stiffness matrix of a beam element, u1...12 are the displacements and rotations at the beam nodes, E is the beam material modulus of elasticity, Iy,z are the beam inertia in each direction, L is the beam length, k1 is the beam axial stiffness defined as EA/L where A is the beam area and k2 is the beam shear stiffness defined as GJ/L where G is the beam shear modulus and J is the polar inertia. To generate the model first a pack of spheres is created. If a regular lattice is used the model nodes will be located on the centre of the spheres, if a irregular lattice is used the nodes are randomly located inside a cube with side dimensions equal to the spheres diameter. The distortion factor z controls how irregular the lattice will be (Lilliu & van Mier, 2000). If z = 0 the lattice is regular and the node is in the centre of the sphere, if z = 1 the distortion is maximal as the node may be located in any position inside the cube. In all the models presented in this article z = 0.9, this means that the nodes may be located anywhere inside the cube up to 0.05x the distance from its borders. After all the nodes are placed, a type of 3D Delaunay tessellation is used to connect them. The objective of the tessellation is to create regular tetrahedrons as close as possible and that adjacent tetrahedrons have adjoining faces. The beam elements are the edges of the tetrahedrons. Figure 2 shows three small examples of one regular and two irregular meshes, where z = 0.5 and 1.0. After the boundary conditions are applied the fundamental equation Ku = F is solved, where K is the lattice global stiffness matrix obtained from assembling all the individual elemental matrices accordingly to

Figure 1. Beam element degrees of freedom and elemental stiffness matrix (CALFEM, 1999).

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their connectivity, u are the displacements and rotations of the nodes and F the forces and moments. At each calculation step, a linear elastic analysis is performed and the forces and displacement are calculated. A beam will be removed when it fails according to a defined failure condition. The beam to be removed is the one closest to failure. The failure condition used in the presented lattice models is a maximum value for the beam positive (tensile) axial force. Negative axial forces (compression), bending moments, torsion and transversal forces are not considered for the failure condition. This failure condition was selected since it is result in a less brittle behaviour for a lattice under a tensile load (Schlangen, 1993) and agrees better with the experimental results. On the next calculation step the stiffness of the beam is removed from the global stiffness matrix. The procedure is then repeated until the lattice breaks completely and therefore in no longer able to transfer the load. Figures 3a and 3b show an example of a uni-axial tensile simulation. The forcedisplacements required to break each beam are plotted sequentially in figure 3b. As it can be seen in this image, frequently the strain necessary to break a element is smaller than the previous one, which creates ‘‘backward steps’’.

Figure 2. Lattice mesh examples for z = 0 (left), z = 0.5 (center) and z = 1.0 (right).

Figure 3a. Example of a lattice simulation of a tensile test on a concrete cylinder with a thinner middle section, the removed beams are highlighted.

Figure 3b. Force-displacement graphic for the lattice shown in figure 3a. Figure 4. Possible locations of the transport element when three beams are removed.

3 TRANSPORT NETWORK MODEL For the transport network model a finite element mesh of triangles with three nodes is used. There is only one degree of freedom per node: the pressure. The pressure varies linearly within the finite element and the flow is constant. The transport triangles are introduced in the lattice of tetrahedrons when the beams on the edge of a tetrahedron are removed. They are created when at least one of the four nodes of the tetrahedron is no longer directly connected to another node belonging to the same tetrahedron. When this happens the tetrahedron is considered split and a transport triangle is created at the splitting surface. The triangle will connect the centre points of the removed beams, which become nodes of the transport model. At least three beams belonging to the same tetrahedron must be removed in order for a transport element (triangle) to be created. Depending on the position of the three removed beams, the triangle location on the tetrahedron will change as shown in figure 4, where the removed beams are highlighted (bold lines). If four beams are removed two cases are considered: 1) if the two remaining beams are adjacent which means that three of the lattice nodes are still connected, then only one transport triangle is created, as shown in figure 4; 2) if the two reaming beams are not adjacent to each other this means that the tetrahedron is now split by a square. As shown in figure 5 there are three possible positions for this square surface. Note that in order to use only triangular finite elements, the square surface is divided in two triangles. If five beams are removed from the tetrahedron several of the situations presented in figure 4 and 5 are combined. There will be two triangles and one square inside the same tetrahedron. The possible combinations are: (using the numbers and letters

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Figure 5. Possible locations of the transport element when four beams are removed.

shown on figures 4 and 5) A+1+2, A+3+4, B+1+4, B+2+3, C+1+3 and C+2+4. The presence of several splitting surfaces inside a tetrahedron means that several crack surfaces are connected on this tetrahedron showing crack splitting or branching. Finally if all the six beams are removed, four triangles and three squares will be present inside the tetrahedron showing a complex pattern of crack connectivity. The aperture of the transport triangle is calculated at each of the three nodes (figure 6). First the relative displacement between the nodes previously connected

Figure 6.

The elemental permeability matrices are assembled in the global permeability matrix and the fundamental equation for the saturated (steady) case K1 P = Q can then be solved; where K1 is the global permeability P is the pressure at the nodes and Q the flow. The pezrmeability for the entire lattice is generally obtained by imposing a unitary pressure gradient between two of the lattices faces, the sum of the flows at one of those borders is then equal to the lattice permeability. In figure 7 we present an example of the calculated flow across a simulated crack surface. The crack in this example is almost planar because a regular and very brittle lattice is used. Nevertheless some irregularities do exist and they divert the flow increasing its velocity around them, as it can be seen in the image.

Aperture of the transport triangle at each node.

4 SIMULATION OF A DIFFUSE CRACKING TEST (P.I.E.D.)

Figure 7. Example of calculated flow inside a simulated crack surface: a) lattice at failure with the removed beams highlighted; b) stress-strain curve; d) flow in the direction of the arrow, perspective and top views.

In this section, the model is used to simulate a diffuse cracking test on concrete and the results are compared to the ones presented in Gérard et al, (1994). In a P.I.E.D. (Pour le Identification de le Endommagement Diffuse) test, the test apparatus allows for a permeability test to be performed on the specimen during different stages of the damage process. As shown in figure 8, in this type of test a concrete cylinder specimen is attached between two steel plates with a cylindrical hole. The diameter of these holes is half the concrete cylinder diameter (55 mm). The holes in the steel plates allow water to cross the sample during the permeability test. The steel plates are submitted to tensile loading, so that the deformation is applied indirectly to the concrete. As the steel plates are loaded the concrete is subjected to quasi uni-axial tensile stresses.

with a beam is calculated. The aperture is assumed to be the relative displacement normal to the triangle surface. The aperture at each of the three triangle nodes is then averaged so that an element with constant aperture is obtained. Knowing the geometry of the transport model defined by the transport triangles and their corresponding aperture and applying a pressure difference as boundary condition the transport problem can be solved. The element permeability for each transport triangle is calculated using the equations for flow thorough parallel plates as presented in equation (1) (Vandersteen, 2002): ⎡ ⎤ k1 0 0 ρ2 (1) K1e = ⎣ 0 k1 0 ⎦ k1 = t 12µ 0 0 k 1

where: Ke1 is the elemental permeability matrix, ρ is the density of the fluid, µ is the dynamic viscosity of the fluid and t is the aperture of the transport triangle.

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Figure 8.

P.I.E.D. test scheme (B. Gérard et al, 1994).

Because the concrete is attached to the steel plates the damage will not be localized in one simple crack, but a diffuse crack pattern is obtained. Figure 9 shows two cracking patterns obtained with the P.I.E.D. test for a normal strength concrete (left) and a high strength concrete (right). In both cases two main cracks are present, but for the normal concrete a diffuse secondary cracking is clearly visible. The secondary cracks probably do not connect with each other and with the main crack, they probably also do not cross the entire thickness of the cylinder therefore not contributing much to the permeability. In figure 10 the lattice model of the P.I.E.D. test is shown. The lattice consists of about 7000 nodes and 50,000 beams. With this number of elements it is possible to simulate the geometry of the P.I.E.D. test, although as shown in the figure the model geometry is still quite rough. The thickness of the steel plates is increased to 8 mm in order to facilitate its modelling. To generate the model a pack of spheres with 5 mm

Figure 9. Diffuse cracking patterns obtained with a P.I.E.D. test for a normal strength concrete (left) and a high strength concrete (right) (B. Gérard et al, 1994).

Figure 10. Lattice model 1 (reference).

diameter is used. This means that the average length of the beams will be close to this value. The beams are cylindrical with a 2.25 mm diameter; this value is used so that the entire lattice behaves as a material with a Poisson coefficient of about 0.2. The beams at the interface between the concrete and the steel plates have the same properties as the beams modelling the concrete. The elastic modulus of the beams modelling the concrete is 60 GPa; this value is used so that the elastic modulus of the entire lattice will be about 20 GPa. The elastic modulus of the beams modelling the steel is 1000x higher than the concrete ones. The limit stress for the beams modelling the concrete is 6 MPa. Note that this stress only limits the beam positive axial force. The 6 MPa value is chosen so that the lattice breaks for tensile stress of about 2 MPa and at a strain of 1.0E-4 m/m. For the steel an infinite value was used for this limit stress in order to guarantee that there will be no damage in the steel plates. Figure 11 compares the damage in the concrete cylinder obtained by the lattice simulation and by a characteristic P.I.E.D. test (we have no information if the results are for a normal or high strength concrete). Damage is defined according to the loss of stiffness of the concrete: damage is zero when the concrete stiffness equals its elastic stiffness, and the damage has a value of one when the concrete stiffness is zero, therefore D = 1 − Ei /Ee where D is the damage, Ei is the concrete stiffness for calculation step i and Ee is the concrete initial or elastic stiffness. As shown in figure 27 the point when damage starts is accurately simulated by the lattice but then the lattice behavior is more brittle, showing a too fast increase of damage in the simulation. In figure 12 the results for the permeability are presented. The permeability is presented in a logarithmical scale as it increases very fast and ranges over several orders of magnitude. The permeability is calculated every 100 calculation steps by solving equation (1) given a pressure gradient of 1.0 Pa between the back and the front of the cylinder. Because of the characteristic lattice ‘‘backward steps’’ (figure 3) the

Figure 11. Damage versus strain for model 1 (reference).

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Figure 12. Permeability versus strain for model 1 (reference).

simulated permeability is not a perfect curve. The simulated results lower limited is 5.0e-13 which is the value obtained for the permeability of the sound concrete. In the simulations the permeability remains low for too long and starts to increase only for a strain of 6.3E-04 m/m, then it quickly rises and for values above 8.5E-4 m/m the simulation and test results agree. Although the permeation tests for the P.I.E.D. experiments are performed in unloaded specimens, the results are plotted for the maximum strain achieved during the loading. Nevertheless the obtained permeability values seem to be comparable to the ones simulated with the lattice model, possibly because: 1) in the P.I.E.D. test the steel plates are yield up to plastic load levels, so they will not be able to recover all their strain; 2) as stated by the authors; small pieces of concrete seem to detach inside the cracks preventing their closing after unloading (B. Gérard et al, 1994). In figure 13 the simulated crack distribution and flow across the cylinder for step calculation 4800 and a strain of 6.8E-4 m/m is plotted. Comparing with the experimental results shown in figure 9 we may observe also two main cracks, one slightly bellow and the other above the centre of the cylinder. As it can be seen in the image in perspective the maximum flow is located in the crack bellow the centre of the cylinder. In figure 14 the simulated crack distribution and flow is again plotted but for calculation step 7300 and for a strain of 1.0E-3 m/m. It is now possible to observe considerable more damage but the main flow is still located in the same crack. In the next lattice larger aggregates are included in the model. For the aggregate positioning a set of 9 images is used. These images are shown in figure 15. The aggregates are idealized as spherical or ellipsoidal particles. The images are shown along the x axis of the model, and the steel plates can be seen on the left and right side of the images. The model length in the x direction is 110 mm (this is also the diameter of the cylinder so the 9 images are positioned at intervals of 12.2 mm. During the material assignment

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Figure 13. Simulated crack distribution and flow across the cylinder for step 4800 and a strain of 6.8E-4 m/m for model 1 (reference).

step the model scan the images and assign properties according to the material present in the image pixel closest to each node. If both nodes of a beam element belong to material A, material properties A are assigned to that beam. If one node of a beam element is associated with material A while the other node is associated with material B, the beam will be considered as belonging to an interface and the properties will be assigned accordingly. For the beams modelling the mortar the same properties were considered them in the previous model, the beams modelling the aggregates are two times more resistant, and the interface beams connecting the aggregate and the mortar are two times less resistant. In figure 16 the resulting lattice model is shown. Some of the aggregates at the edges of the concrete cylinder are highlighted.

Figure 15. Set of 9 images used for the positioning of the aggregates on model 2.

Figure 14. Simulated crack distribution and flow across the cylinder for step 7300 and a strain of 1.0E-3 m/m for model 1 (reference).

The damage versus strain for the lattice model including the aggregates is shown in figure 17. Comparing with figure 11 we can conclude that adding aggregates has almost no influence on the damagestrain behaviour. In figure 18 the permeability is plotted for the lattice model with aggregates. The permeability is calculated every 50 simulation steps. Compared with figure 12, the permeability now starts to increase earlier and agrees with the tests results, but as the strain increases the increase of the permeability is slower. For large strains the permeability is lower than the one estimated by the lattice without aggregates and does not agree so well with the test results. The simulated crack distribution and flow across the cylinder for step 2000 and a strain of 2.7E-4 m/m is presented in figure 19. Comparing with the experimental results shown in figure 9 we note that in this simulation there is only one main crack crossing the specimen. This crack is not planar, it shows a wave

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Figure 16. Lattice model 2, some of the aggregates are highlighted.

Figure 17. Damage versus strain for model 2 (with aggregates).

Figure 18. Permeability versus strain for model 2 (with aggregates).

Figure 20. Simulated crack distribution and flow across the cylinder for step 4000 and a strain of 5.2E-4 m/m and for model 2 (with aggregates).

pattern, probably formed by the presence of the aggregates around which the crack develops. Most of the flow (dark colour) is located in the centre of the crack, which coincide with the middle of the cylinder. In figure 20 the simulated crack distribution and flow for calculation step 4000 and a strain of 5.2E-4 m/m is plotted. At this level of strain there is a lot more damage, but most of this damage consists of secondary isolated cracks that don’t contribute to the transport. The main flow is still on the main crack near the centre of the cylinder as it was for step 2000.

5 CONCLUSIONS AND FUTURE DEVELOPMENTS Figure 19. Simulated crack distribution and flow across the cylinder for step 2000 and a strain of 2.7E-4 m/m and for model 2 (with aggregates).

438

The article describes a combined damage and permeability 3D model for concrete where for the damage model a lattice of beam elements is used and for the

calculation of the saturated permeability a transport network model is used. The numerical simulations are then compared with experimental results from a diffuse tensile cracking and permeability test (P.I.E.D.). Two lattice models are compared with the experimental results: in the first one the concrete is considered as a homogenous material and in the second the aggregates are explicitly modelled. We may conclude that adding aggregates has almost no influence on the damage-strain (mechanical) behaviour of the lattice, but it influences the simulated permeability. When the aggregates are added the permeability agrees better with the experimental results for small strains, but for the large strains the results from the lattice without aggregates agree better with the experimental results. The early increase of the model permeability probably is a direct result of the lower resistance of the beams modelling the mortar-aggregate interface. The too low permeability of the model with aggregates for the high strains is probably a result of the fact that now the crack is less planar, developing around the aggregates, which means that there will be a longer path for the water to cross the model. The simulation results may depend significantly on the aggregates geometry and position, therefore it is advisable to test different aggregates distributions. An accurate prediction of the cracks geometry improves the prediction of its permeability. The lattice of beams crack model is capable of simulating quite realistic crack geometries and therefore it seems a good candidate for a model when coupling between damage and permeability is required. The main drawback of the lattice model is that, because of its simplicity, there are too few control parameters that may be changed to fine-tune the model behaviour. Currently an experimental study is planned where the main aim is to develop a test capable of simultaneous loading to produce damage, measuring the permeability and imaging the cracks in 3D. This test would be used to further calibrate the lattice-transport model.

ACKNOWLEDGMENTS Founding for this work was provided by the Portuguese Foundation for Science and Technology (SFRH/BD/ 30435/2006). REFERENCES Schlangen, E. 1993. Experimental and numerical analysis of fracture processes in concrete. Dissertation. Delft University of Technology. Gérard, B. et al. 1996, Cracking and permeability of concrete under tension. Materials and Structures/Materiaux et Constructions, Vol. 29, April 1996, pp. 141–151. Schlangen, E. & Garboczi, E.J. 1996. New method for simulating fracture using an elastically uniform random geometry lattice. Int. J. Engineering Sci. Vol. 34, No. 10, pp.1131–1144. UK: Elsevier. Schlangen, E. & Garboczi, E.J. 1997. Fracture simulation of concrete using lattice models: Computational aspects. Engineering Fracture Mechanics Vol. 57, No. 2/3, pp.319–332. UK: Elsevier. CALFEM, A finite element toolbox to MATLAB, Version 3.3 Reference book. Division of Structural Mechanics and the Department of Solid Mechanics, Lund University, Sweden, 1999. Lilliu, G. & van Mier, J.G.M. 2000. Simulation of 3D crack propagation with the lattice model. Proceedings Materials Week 2000. Available in 2005-07-08 at: http://www.proceedings.materialsweek.org/proceed/ mw2000_634.pdf Vandersteen, K. 2002. Unsaturated water flow in fractured porous media. Dissertation. Catholic University of Leuven. Carmeliet, J. Delerue, J.-F. Vandersteen, K. Roels, S. 2004. Three-dimensional liquid transport in concrete cracks. Int. J. Num. Anal. Meth. Geomech. 28: p. 671–-687. Roels, A. Moonen, P. de Proft, K. Carmeliet, J. 2006, A coupled discrete-continuum approach to simulate moisture effects on damage processes in porous materials. International Journal for Computational Methods in Applied Mathematics 195(52): 7139–7153.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modelling the CaO hydration in expansive concrete B. Chiaia, A.P. Fantilli, G. Ferro & G. Ventura Politecnico di Torino, Torino, Italy

ABSTRACT: In order to investigate the average expansion of concrete members having the same percentage of free lime and impurities, but not the same grain diameter, a size-effect theory is introduced. More specifically, by drawing an analogy with the structural size-effect, the expansion of lime hydration must show a characteristic length, whose dimension remains constant at all scales. Thus, if a grain of CaO is assumed to be a sphere, only a thin layer on the external surface hydrates and doubles the volume. As the thickness of such layer is constant, independently of lime manufacture, the expansion of CaO is proportionally lower in grains of larger diameter. This phenomenon, confirmed by the results of a 2D finite element model, agrees with some experimental observations. For practical application, a simplified chemo-mechanical model of lime expansion is eventually proposed, to predict the consequences of uncombined lime in concrete structures. 1 INTRODUCTION Most of the burnt lime (CaO) contained in the Portland cement is generally hydrated at the moment of concrete mixing, in accordance with the well known ‘‘slaking’’ reaction: CaO + H2 O ⇒ Ca(OH)2 + 15.5 kcal

(1)

Eq. (1) is a strongly exothermic reaction, in which the formation of the calcium hydroxide has a larger volume (expansive reaction). The ratio of volume change from CaO particle to Ca(OH)2 is 33/17 ∼ =2 (Nagataki & Gomi, 1998). Sometimes, the presence of residual grains of burnt lime, not yet hydrated, can be detected within the concrete members. In such particles, a very slow slaking reaction takes place, and continues for months after concrete casting. Generally, this lime consists of high-density CaO grains, originated by a deadburn process, which hydrate very slowly because of a reduced porosity. If during the hydration of cement the formation of the calcium hydroxide from such CaO grains is incomplete, the slaking process [Eq. (1)] continues also during the concrete hardening. In some cases the expansion by hydration of CaO, similarly to the alkali aggregate reaction, can cause concrete cracking or the so-called pop-out phenomenon (Lee & Lee, 2009). The latter consists of the rising, and the subsequent expulsion, of concrete around the grains closer to the concrete surface. The presence of magnesium oxide (MgO), which derives from dolomitic limestone (MgCO3 ), can intensify the deleterious effects of expansion by hydration, as it shows a slower hydration.

Obviously, if expansion takes place in an unrestrained concrete, it can cause cracking. Conversely, if the expansion is properly restrained, its magnitude reduces and a prestress develops. When the compressive stress is between 0.1 and 0.7 MPa, which is adequate to compensate the tensile stress from restrained drying shrinkage, a sort of shrinkage compensation can be achieved (Metha & Monteiro, 2006). This compensation has been largely used in making crack-free concrete structures, such as pavements and slabs. It must be remarked that the mechanism of shrinkage compensation generated by expansive agents like CaO, differs from that produced by the Surface Reducing Admixture (SRA). In fact, SRA, not considered in the presents work, reduces drying and/or autogenous shrinkage due to the decrease of the surface tension of water within the cement paste (Collepardi et al., 2005). Only the use of CaO-based expansive agents is here investigated. In accordance with Collepardi (2003), in concretes having the same hardening conditions and the same amount of reinforcement, the type of CaO plays a fundamental rule in reducing the deleterious effect of shrinkage. Precisely, although the slaking reaction [Eq. (1)] is always the same, the physical and chemical properties of calcium oxide can vary, as well as its expansion.

2 PROPERTIES OF CALCIUM OXIDE X-ray diffraction reveals that a pure calcium oxide crystallizes in the cubic system depicted in Figure 1. The edges of the cube are about 4.8 Å in length, with calcium atoms located in between (Boynton, 1966). The physical and chemical properties of quicklime, which is the name of CaO when it is obtained by the

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Figure 1. Crystal structure (unit cell) of calcium oxide (Boynton, 1966).

calcination process (i.e., cooking of limestone at a temperature higher than 575◦ C), depend, respectively, on the cooking temperature (the so-called calcination temperature) and on the purity of limestone (CaCO3 ). 2.1 Physical properties of CaO The physical properties of CaO, such as porosity, bulk density, the dimensions of grains, and hydration rate are strongly related to the calcination temperature. Depending on the temperature reached in the kiln where limestone is cooked, hard-burned lime, averageburned lime and soft-burned lime can be obtained (Schiele & Berens, 1976). With respect to the hard-burned lime (resulting from high calcination temperature), soft-burned lime (resulting from low calcination temperature) shows the following properties: – – – – –

Lower grain dimensions. Larger specific surface. Higher porosity (but a lower dimension of pores). Lower bulk density. Higher chemical reactivity.

are generally made by CaO aggregation, and show cracked and corrugated surfaces. In this quicklime, large contact surfaces between grains, caused by a sintering process at high temperature, can be detected. In hard-burned limes, the diameter of grains can be larger than 10 µm, whereas the diameter of pores reaches 20 µm (Schiele & Berens, 1976). Table 1 shows the values of the main physical properties of the soft-burned, average-burned and hardburned limes, having the same unit weight (Schiele & Berens, 1976). The calcination temperature, the bulk density, and the specific surface as well, are strictly connected. In the case of soft-burned limes the specific surface can be three times larger than that measured in hard-burned limes. As a consequence, also the average diameter of grains increases in hard-burned limes. All these observations are extremely important in evaluating the chemical reactivity of lime (and the consequent expansion). It is unanimously accepted that the surface area is the most reliable parameter which can be used to evaluate the property of quicklime hydration: the higher this value, the higher the chemical reactivity. This has been confirmed by various chemical tests (Boynton, 1966). Also fineness, as determined by particle size or grain diameter, is indicative of the degree of reactivity. However, both the specific surface and the mean diameter of grains are related to the calcination temperature. As reported in Figure 2, if calcination temperature increases, but is maintained for the same duration, porosity (Fig. 2a) and specific surface area Ss (Fig. 2b) decreases, while grain diameter size (CaO ) increases (Fig. 2c). Thus, hard-burned limes are generally characterized by moderate to low chemical reactivity (Boynton, 1966; Schiele & Berens, 1976). 2.2 Chemical properties of CaO The hydration or slaking reaction of quicklime [Eq. (1)] is strongly related to the temperature. Up to 100◦ C, the rate of reaction increases. In such situations, as supplementary heat is developed, a further increment of temperature can be generated and, consequently, the chemical reaction is accelerated.

Generally, soft-burned lime is made of little grains, whose maximum diameter varies between 1 µm and 2 µm, and shows a bulk density of about 1.51 g/cm3 . Most of the pores have a diameter comprised between 0.1 µm and 1 µm. If the calcination temperature increases, an agglomeration of the single grains can be observed, although pores increase in diameter (all the observed cavities are in the range 1 ÷10 µm). These effects are even more pronounced in the case of hard-burned limes, which can show a bulk density of 2.44 g/cm3 . Moreover, due to the high calcination temperature, grains

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Table 1. Some of the physical properties measured in the most common limes (Schiele & Berens, 1976). Type of cooking

Unit weight (g/cm3 ) Bulk density (g/cm3 ) Porosity (%) Specific surface (m2 /g)

soft

average

hard

3.35 1.5÷1.8 46÷55 >1.0

3.35 1.8÷2.2 34÷46 0.3÷1.0

3.35 >2.2