IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials: Proceedings of the IUTAM Symposium held in Kazimierz ... (Solid Mechanics and Its Applications, 135) 1402045654, 9781402045653

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IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials

SOLID MECHANICS AND ITS APPLICATIONS Volume 135 Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials Proceedings of the IUTAM Symposium held in Kazimierz Dolny, Poland, 23--27 May 2005 Edited by

TOMASZ SADOWSKI Lublin University of Technology, Lublin, Poland

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4565-4 (HB) 978-1-4020-4565-3 (HB) 1-4020-4566-2 (e-book) 978-1-4020-4566-0 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

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All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

CONTENTS

Preface

ix

Keynote Lecture 1: A Computational Damage Micromodel for Laminate Composites P. Ladevèze, G. Lubineau, D. Violeau, D. Marsal........................................................ 1 Keynote Lecture 2: Numerical Methods for Debonding in Composite Materials: A Comparison of Approaches R. de Borst……………………………………………………………………………………..13 Keynote Lecture 3: Atomic-Continuum Transition at Interfaces of Silicon and Carbon Nanocomposite Materials R. Pyrz………………………………………………………………………………………….23 Multiscale Modelling for Damaged Viscoelastic Particulate Composites A. Dragon, C. Nadot-Martin, A. Fanget ..................................................................... 33 A Shakedown Approach to the Problem of Damage of Fibre-Reinforced Composites D. Weichert, A. Hachemi........................................................ .................................... 41 Impact Behavior of Cellular Solids and their Sandwich Panels H. Zhao, I. Nasri , Y. Girard ....................................................................................... 49 Influence of Delamination on the Predicition of Impact Damage in Composites A. Johnson, N. Pentecôte, H. Körber........................................................................... 57 Debonding or Breakage of Short Fibres in a Metal Matrix Composite V. Tvergaard......................................................... ...................................................... 67 A Microscale Model of Elastic and Damage Longitudinal Shear Behaviour of Highly Concentrated Long Fiber Composites S. Lenci.........................................................................................................................77 Analysis of Metal Matrix Composites Damage under Transverse Loading E. Oleszkiewicz, T. àodygowski...................................................................................89

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Contents

On the Out-of-Plane Interactions between Ply Damage and Interface Damage in Laminates D. Marsal, P. Ladevèze, G. Lubineau........................................................................ 97 The Elastic Modulus and the Thermal Expansion Coefficient of Particulate Composites Using a Dodecahedric Multivariant Model E. Sideridis, G. A. Papadopoulos, V. N. Kytopoulos, T. Sadowski………………….. 105 Prediction of Crack Deflection and Kinking in Ceramic Laminates D. Leguillon, O. Cherti Tazi, E. Martin.....................................................................113 Dynamic Buckling of Thin-Walled Composite Plates T. Kubiak....................................................................................................................123 Numerical and Experimental Models of the Fracture in the Multi -Layered Composites M. Jaroniek.................................................................................................................131 Multiscale Method for Optimal Design of Composite Structures Incorporating Sensors F. Collombet, M. Mulle, Y-H. Grunevald, R. Zitoune ...............................................141 Numerical Multiscale Modelling of Elasto-Plastic Behaviour of Superconducting Strand B. A. Schrefler, D. P. Boso, M. Lefik ........................................................................151 Anisotropic Failure of the Biological Multi-composite Wood: A Micromechanical Approach K. Hofstetter, Ch. Hellmich, H. A. Mang...................................................................159 Antiplane Crack in a Pre-Stressed Fiber Reinforced Elastic Material E.-M. Craciun…………………………………………………………………………........ 1. 67 Characterization and Practical Application of a Multiscale Failure Criterion for Composite Structures F. Laurin, N. Carrère, J. F. Maire, D. Perreux………………………………………….177

Contents

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Stress Field Singularities for Reinforcing Fibre with a Single Lateral Crack G. Mieczkowski, K .L. Molski………………………………………………………………185 Numerical Modelling of Mechanical Response of a Two-Phase Composite E. Postek, T. Sadowski, S. Hardy………………………………………………………….193 Modelling of Delamination Damage in Scaled Quasi-Isotropic Specimens S. R. Hallett, W. G. Jiang, M. R. Wisnom, B. Khan……………………………………...201 Damage in Patrimonial Masonry Structures: The Case of the O-L Cathedral in Tournai (Belgium) L. Van Parys, D. Lamblin, G. Guerlement, T. Descamps………………………………209 The Macroscopic Strength of Perforated Steel Disks at Maximum Elastic and Limit State S. Datoussaïd, D. Lamblin, G. Guerlement, W. Kakol………………………………….217 Importance of Surface/Interface Effect to Properties of Materials at Nano-Scale J. Wang, B. L. Karihaloo, H. L. Duan, Z. P. Huang…………………………………...…227 On a Deformation of Polycrystalline Structures L. Berka, N. Ganev, P. Jenþuš, P. Lukáš ………………………………………………...235 A Multiscale Damage Model for Composite Laminate Based on Numerical and Experimental Complementary Tests C. Huchette, D. Lévêque, N. Carrère……………………………………………………..241 Influence of Strength Heterogeneity Factor on Crack Shape in Laminar Rock-Like Materials J. Podgórski, J. Jonak…………………………………………………………………….…249 Crack Propagation in Composites with Ceramic Matrix K. Konopka……………………………… …………………………………………………….. 255 Experimental Investigations and Modelling of Porous Ceramics S. Samborski, T. Sadowski………………………………………………………………….263 Numerical Analysis of Stress Distributions in Adhesive Joints J. Kuczmaszewski, M. Wáodarczyk………………………………………………………...271

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Contents

APPENDIX: The Scientific Programme....................................................................279

The following papers were presented during Symposium but are not published in this volume: Materials Modeling from Atomistics Macro Behaviour S. Schmauder A Synergistic Multiscale Modelling Approach in Damage Mechanics of Composite Materials R. Talreja A Nonlocal Plasticity and Damage Model for Size Effect in Metal Matrix Composites G. Z. Voyiadjis, R.. K. Abu Al-Rub Designing, Testing and Manufacturing of Composite Aviation Products at “PZL-ĝwidnik” S.A. P. Chojnacki, M. Greguáa, M. PaĔko, K. Siedlecki. Advances and Trends in Composite Material Testing G. Socha Rheology and Fracture of Composite Materials S. Elsoufiev Strength Criteria for Composites – Current State and Future Trends H. Altenbach Multiscale Model for Upscaling of Strength Properties of Bituminous Composites J. Füssl, R. Lackner, J. Eberhardsteiner Multi-Scale Testing for Simple Micromechanical Models of Concrete J. G. M. van Mier and P. Trtik

PREFACE

The IUTAM Symposium on “Multiscale Modelling of Damage and Fracture Processes in Composite Materials” was held in Kazimierz Dolny, Poland , 23 -27 May 2005. The Symposium was attended by 48 persons from 15 countries. During 5 day meeting, 4 keynote lectures and 39 invited lectures were presented. This volume constitutes the Proceedings of the IUTAM Symposium. The main aim of the Symposium was to discuss the basic principles of damage growth and fracture processes in different types of composites: ceramic, polymer and metal matrix composites, cement and bituminous composites and wood. Nowadays, it is widely recognized that important macroscopic properties like the macroscopic stiffness and strength, are governed by processes that occur at one to several scales below the level of observation starting from nanoscale. Understanding how these processes influence the reduction of stiffness and strength is essential for the analysis of existing and the design of improved composite materials. The study of how these various length scales can be linked together or taken into account simultaneously is particular attractive for composite materials, since they have a well-defined structure at the nano, micro and meso-levels. The well-defined microstructural level can be associated with small particles or fibres, while the individual laminae can be indentified at the mesoscopic level. Moreover, the advances in multiscale modelling of damage and fracture processes to the description of the complete constitutive behaviour in composites which do not have a very well-defined microstructure, e.g. cementitious, bitumous composites and wood was analysed. In particular, the fracture mechanics and optimization techniques for the design of polymer composite laminates against the delamination type of failure was discussed. Computational modelling of laminated composites at different scales: microscopic mesoscopic and macroscopic with application of suitable plate/shell elements for thin composites was presented. With regard to ceramic matrix composites (CMC) the damage and fracture processes were described in three scales. The important problem of damage process of interfaces surrounding particles, grains or fibres in composites was analysed for different properties of the components of composites and in different scales. This Symposium clearly showed growing interests in the development of multiscale modelling approaches to model the macroscopic behaviour of different types of composite materials. The Proceedings present the current state of advances in multiscale modelling of composites and provide a reasonable knowledge to understanding the importance of the local phenomena and the spatial composite components distribution on the non-linear character of the overall response.

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Preface

The International Scientific Committee responsible for the Symposium comprised the following: R. de Borst (The Netherlands), B. Karihaloo (UK), P. Ladeveze (France), G. Maier (Italy), Z. Mróz (Poland), R. Pyrz (Denmark), T. Sadowski (Poland, Chair), J. Salençon (France, IUTAM representative), G. Voyiadjis (USA) The Editor wish to thank the Bureau of IUTAM, the International Scientific Committee, session chairmen and the local organising committee. Thanks to authors and referees who have made this volume publication possible. Special thanks to Mrs J.Sadowska and Mrs A.Bukowska for help during organisation and preparation of the Symposium. Thank you for financial support to the International Union of Theoretical and Applied Mechanics, Kluwer Academic Publishers, Precious Metals Mint, International Scientific Network for Advanced Materials and Structures, Polish Aviation Works Świdnik S.A., MTS System GmbH – Polish Representative ELHYS Ltd, Lublin University of Technology, Marshall of the Lubelskie Voivodeship, Lubella Pasta Company.

Tomasz Sadowski Lublin, November 2005

A COMPUTATIONAL DAMAGE MICROMODEL FOR LAMINATE COMPOSITES

P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal LMT-Cachan (E.N.S. de Cachan / Université Paris 6 / C.N.R.S.) 61 Avenue du Président Wilson 94235 Cachan Cedex France

Abstract:

A major challenge in the design of composites is to calculate the intensities of the damage mechanisms at any point of a composite structure subjected to complex loading and at any time until final fracture as a result of strain and damage localization. Such final fracture mechanisms always involve delamination and most of the time lead to delamination macrocracks. The huge number of tests carried out on stratified composites in the aerospace industry shows the low level of confidence in models. A significant improvement in this situation, i.e. a drastic reduction in the number of industrial tests, could be achieved if one could create a real synergy among the approaches on different scales which, today, are followed quite independently of one another in the case of laminated composites. Numerous theoretical and experimental works carried out in micromechanics introduce microscale models. An intermediate scale called mesoscale enables one to take into account the mechanisms of damage easily. However, there are only few links between the two scales. The questions discussed here are how to bridge the micro and mesomechanics of laminates and its impact to the micro and meso computational damage modellings for delamination prediction.

Key words:

laminates, micromechanics, cracking, damage

1.

INTRODUCTION

The analysis of composite structures subjected to complex loading may require models to predict the intensities of the damage mechanisms and their evolution until final fracture. Even in the case of laminated composites (which are the most studied and, therefore, the best understood), the 1 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 1–12. © 2006 Springer. Printed in the Netherlands.

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P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

prediction of damage evolution up to and including final fracture remains a major challenge in modern mechanics of composite materials and structures. Existing approaches are based either on refined concepts on the microscale (which thus, provide precise information on specific mechanisms, but whose formalisms are not convenient for structural analysis), or on formalisms suitable for calculations (which, controversially, provide only very coarse and insufficient information on the micro nature of the degradations). The Damage Mesomodel for Laminates developed since twenty years at Cachan is one of those computational approaches. This model provides global measures of the degradation and no information on the microstate of degradation. Therefore, recent works have focused on the development of bridges between the micro- and mesomechanics, in order to create a more robust damage mesomodel, providing more detailed information on the actual origin of the degradations. It is proved that there exist intrinsic operators which link the solutions on the two scales for any state of degradation. This legitimizes the description of these mechanisms on the mesoscale. On this basis, one can define a refined mesomodel using micro considerations. Another multiscale approach, which is rather simple, is called the "computational damage micromodel". Based on a semi-discrete and probabilistic description, this model is directly linked to the microscale, and is able to take into account several mechanisms observed at the microscopic level. However, this model leads to prohibitive calculation costs with the use of current industrial codes. The adaptation of a multiscale computational strategy is absolutely essential. Several points are mentioned in order to show the capabilities and the limitations of the different models.

2.

MICRO AND MESO COMPUTATIONAL DAMAGE

2.1 The micromechanics of laminates Up to now, there have been numerous theoretical and experimental works on the micromechanics of laminates (e.g. [1], [2], [3], [4]); the micromechanics approach provides a relatively good understanding of damage mechanisms. However, these micromechanical models are lacking in some respects: in particular, they are far from being complete for the prediction of localization and final fracture. In order to establish micro-meso relations, five degradation mechanisms are clearly identified on the microscale : diffuse damage in both the ply and the interface, transverse microcracking, local delamination and fiber

A Computational Damage Micromodel for Laminate Composites

3

breakage (see figure 1). The diffuse damage is associated mainly with fibermatrix debonding, a fundamental mechanism in order to explain the behaviour of the ply and often unused by micromechanical analysis. Transverse cracks are assumed to span the entire thickness of the ply. At the tip of the transverse microcracks, interface degradations start.

Figure 1. The mechanisms of degradation on the microscale. 1: transverse microcracking, 2: local delamination, 3: diffuse damage, 4: diffuse delamination.

In most practical cases, damage is initiated by diffuse damage. Then, the accumulation of fiber-matrix debonding instances followed by their coalescence leads to transverse microcracking. The competition between transverse microcracking and diffuse delamination ends with the saturation of transverse microcracking and is relayed by the development of local delamination, which can lead to macroscopic interlaminar debonding.

2.2

The damage mesomechanics of laminates

One of the computational approaches which can be applied to laminated composites is based on what we call a "damage mesomodel for laminates (D.M.L) " (see [5], [6], and also [7] for alternative approaches). In this approach, one assumes that the behavior of any laminated composite for any loading and any stacking sequence can be modeled using two elementary constituents which are continuous media: the ply and the interface (see figure 2). The ply takes into account diffuse damage, transverse microcracking, local delamination and fiber breakage. The interface is a surface entity representing the thin layer of matrix which exists between two adjacent plies characterized by their relative orientation. Its shear stiffness is equal to the shear modulus of the matrix divided by its thickness which is taken, by assumption, as the thickness of an elementary ply divided by 10. The interface provides for the transfer of strains and stresses between plies. The level of each mesoconstituent is quantified by damage indicators, whose evolution is governed by damage forces through an assumed damage law. An important point of the model is that the state of damage is assumed to

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P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

remain constant throughout the thickness of a single layer (of course, it can vary from one layer of the laminate to the next). In other word, the main hypothesis of this computational model is that the damage evolution law is intrinsic to a ply and does not depend on the stacking sequence in which the ply is placed. In this classical version of the D.M.L, the indicators of damage are only global measures of the degradation which provide no information on the micro state of degradation. Several examples have been treated using the D.M.L (see [8]).

Figure 2. The mesoconstituents of the damage mesomodel for laminates.

3.

TOWARD A BRIDGE BETWEEN MICRO AND MESOMECHANICS

3.1

Micro-meso relationships

The central question we aim to discuss here is this: how can one bridge the micro and mesomechanics of damage? The belief that such a complete bridge could exist is not shared by all people working in micromechanics. A first attempt at building such a bridge was made in ([9], [10]) for plane macrostresses. The mesomodel was found to be fully compatible with the microdamage mechanisms. The micro-meso relations introduce quantities or relations which we call "approximately ply-material", which are intrinsically related to the characteristics and, therefore, independent of the characteristics of the cracked ply of the other plies. Recently, additional work has extended this approach to out-of-plane stresses (see [11]). This more complex situation involves non-local mesomodels, as there are interactions between the interface’s damage and the microcracking mechanisms of the adjacent plies. The method of investigation is now entering what is called a "virtual testing" stage, in which numerous numerical experiments using the

A Computational Damage Micromodel for Laminate Composites

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micromodel and involving various possible stacking sequences, thicknesses,... are performed. The link between the micro and meso scales requires the resolution of two basic problems. The first problem, which concerns the homogenized single layer, is an extension of the former 2D-ply problem to non-plane stresses. The second one, which concerns the homogenized interface, is a new 3D problem. A fundamental link between the micro and meso scales exists for both problems : two mesoquantities (the plane part of the mesostrains and the out-of-plane part of the mesostress) can be interpreted as mean values of the corresponding microquantities. The objective is to build a continuum damage mechanics model which is quasi equivalent, from an energy standpoint, to the damaged laminate micromodel. Consequently, the potential energy stored in any part of a complete structure must be the same on the microscale and on the mesoscale, as illustrated in figure 3. The micromodel is characterized by periodic micro patterns, at least locally, which is consistent with most practical situations. The level of microcracking is quantified by a cracking rate U defined by U HL ,

U  [0 007] . The local delamination is described at each transverse crack R tip by a local delamination ratio W H , W  [0 004] . From experiments, the material can be considered as fully damaged for higher level of degradation. The evolution of these microvariables is governed by energy release rates in the frameworks of fracture mechanics or finite fracture mechanics.

Figure 3. Energy equivalence between the micro and meso interpretations of damage.

The equivalent mesomodel is considered to be completely achieved if two basic situations of equivalence are established, one for the ply and one for the interface. The determination of the potential energy of the domain on the microscale involves the calculation of the stress field at any point of the

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P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

domain. The homogenization must be carried out for any stacking sequence defined by the thicknesses and orientations of the plies, but also for any kind of microdamage state. As previously stated, the traditional basic problem defines only the state of damage of a specified ply S. This procedure gives excellent results for inplane stresses because the energy is really confined to the ply S. In the case of out-of-plane stresses, however, the damage state of ply S induces some non-negligible energy in adjacent plies S’- and S’+. Consequently, the damage state of the ply being considered depends on the state of damage of the adjacent plies. To handle these considerations, one defines an extended ply problem. It is a 3D problem with periodic conditions which can be approximated by two 2D problems. Prior to carrying out any identification between the micro- and mesooperators, one must verify the consistency of the homogenized operator with the hypotheses of the mesomodel. Numerical simulations made for a given set of interface parameters show that, to within 5 %, the identified damage indicators do not depend on the stacking sequence.

3.2

An enhanced damage mesomodel for laminates

The standard mesomodel has been extensively used for the resolution of impact and quasi-static engineering problems. We are now developing an enhanced version in order to improve the prediction of delamination. This development is an application of the bridge which has just been presented. The interfaces’s damage mesomodel is made non local by coupling it with the microcracking of the adjacent plies. Another improvement consists in using the micro-meso relations to describe damage in terms of micromechanics. This mesomodel needs to go through the identification procedure again. A further development would be to work with the true, complete mesomodel, which is non local both for the interface and for the ply.

4.

THE COMPUTATIONAL DAMAGE MICROMODEL LAMINATES - BASIC ASPECTS

Let us start with the initial state in which residual stresses occur. These can be calculated from the simulation of the process, but a more pragmatic and standard approach is simply to apply a uniform negative temperature variation, and calculate the corresponding residual stresses under elastic assumptions.

A Computational Damage Micromodel for Laminate Composites

7

On the microscale, the structure is described as an assembly of layers, made of the fiber-matrix material, and interfaces, which can be cracks. The proposed computational micromodel for laminates is hybrid. The surfaces of the cracks are described using a discrete model by introducing "minimum cracked surfaces". This belongs in what is called "finite fracture mechanics" [12]. The minimum cracked surfaces related to transverse microcracking are square surfaces, parallel to the fiber, whose dimension is the thickness of the ply. This choice of the lateral dimension derives from the fact that for values equal to or greater than the ply thickness the tunneling energy release is nearly constant. For minimum cracked delamination surfaces localized at the interfaces, the corresponding areas are about 0.1h x 0.1h. The results should vary only slightly depending on this choice.

Figure 4. The computational damage micromodel: minimum cracking surfaces.

For the rest, diffuse damage and plasticity are described through continuum mechanics models. Consequently, the fiber-matrix material which constitutes elementary homogeneous volumes between cracking surfaces follows the classical damage mesomodel for laminates: diffuse damage [13] and, if necessary, (visco) plasticity. Thus, the continuous part of the model is completely classical and will not be detailed further in this paper. From now on, we will focus on the description of the discrete part for transverse microcracking and local delamination.

5.

THE DAMAGE MICROMODEL FOR MICROCRACKING

5.1

Some key points of micromechanics

Key point 1: modelling of initiation / propagation - the thickness effect

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P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

Most basic papers on the subject are not recent (e.g. [14], [15], [16]). For tension tests of stacking sequences built with 0° and 90° plies, two main observations have been made: first, the behaviour of thick 90° plies is different from that of thin 90° plies: in thick plies, the transverse microcracks always run throughout the width of the specimen, whereas in thin plies they may stop near the edges. The second observation is related to the thickness effects (see Figure 5): the transition thickness h is about twice that of the elementary ply.

Figure 5. Failure strain as a function of the number of 90 ° plies.

The theoretical explanation of the existence of transverse microcracks was given a long time ago and is well-known. Let us consider a flaw in the form of a penny-shaped crack which can propagate either in the longitudinal or in the transverse direction. It has been proven that the transverse energy release rate is much higher than the longitudinal energy release rate and, therefore, that the flaw propagates in the transverse direction, which is the thickness of the layer [1] (Figure 6).

Figure 6. Tunneling energy release rate as a function of the lateral dimension.

Another point is that there is a difference between the initiation of a crack or a minimum cracking surface, and the propagation of an existing crack. The criteria should be different: the latter is an energy release rate criterion and depends on the thickness; the former is a stress criterion which does not depend on the thickness. Consequently, one can tale a new look at the curve

A Computational Damage Micromodel for Laminate Composites

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of Figure 5: the right part for thick ply is, in fact, related to the initiation criterion; initiated cracks propagate immediately. The left part is associated with the propagation criterion, in this case for thin plies. Key point 2: microcracking as a stochastic phenomenon Several probabilistic models have already been proposed [17] [18] …. For high cracking densities, heuristic coefficients were introduced in order to characterize the imperfect periodicity (e.g. [19]). This was necessary in order to obtain reasonable agreement with experiments.

Figure 7. Max., mean and min. values of the microcracking rate vs. the longitudinal strain.

Along with [20], we support the idea that the process is stochastic, but quasi-independent of the probabilistic law. We thus introduce simply uncertainty on the critical energy release rate {Gc } . It is assumed to vary near a mean value: {Gc }  {Gcmean }'' , where ' is a small parameter. One then prescribes a uniform probability density over the domain:

{M _ maxM {G ( M )}  {G ( M )} d '}

(1)

Figure 7 shows several samples for different values of ' ; the "mean" curve appears to be insensitive to the probabilistic parameter ' and to the samples, and is a quasi-deterministic curve, which is quite different from the curve related to a periodic pattern.

5.2

The model

Let us consider a minimum surface which can crack and let

{Y } {Y I  Y II  Y III } be the corresponding discrete damage force. The failure criterion is expressed as:

P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

10

1

) ({Y }{Yc })

ª§ Y I ·D § Y II ·D § Y III ·D º D «¨ I ¸  ¨ II ¸  ¨ III ¸ » t 1 «¬© Yc ¹ © Yc ¹ © Yc ¹ »¼

(2)

For initiation, i.e. in the case of a minimum surface which is not connected to other, already cracked, surfaces, one introduces: x the damage force {Y }init

min{{Gh }  r {Gh }} , where {G} is the discrete

energy release rate for the Modes I, II, III, h the thickness of the ply, h the transition thickness and r a material coefficient; x the critical damage force {Yc }init

{Gc } h

, where h is transition thickness

and {Gc } the discrete critical energy release rate. In summary, for initiation, one has:

{G} {G} {Gc } ) (min{  } ) t1 h h h

(3)

In the case of propagation, i.e. when the minimum surface being studied is connected to existing cracking surfaces, one introduces: x the damage force {Y } {G} , where {G} is the discrete energy release rate; x the critical damage force {Yc } {Gc } . Thus, the propagation criterion is:

) ({G} {Gc }) t 1

(4)

Presently, we use the same critical value for Modes II and III. Moreover, for Mode I, an additional criterion can be introduced by separating traction from compression. When an elementary surface is cracked, unilateral contact conditions with friction apply. The critical values {Gc } are stochastic fields for which we assume that the correlation length is not greater than the thickness h. Then, after discretization, they can be replaced by independent stochastic variables for which a modified normal law is introduced. Remark: The energy release rate related to microcracking could be calculated simply by using the tunneling value. For delamination, we again introduce minimum surfaces for cracking. We also use a criterion which connects:

A Computational Damage Micromodel for Laminate Composites

11

x the damage force {Gdel } (the finite energy release rate for Modes I, II, III) and x the critical force {G cc} . This criterion is: )c({Gdel } {G cc}) t 1 (5) Remark: These criterions are chosen in agreement with the keypoints previously defined, and enable us to take into account major experimental observations. Two criteria can also be proposed to introduce fiber breakage.

6.

CONCLUSION

The computational micromodel developed in this paper seems to be a very promising approach, because it enables one to take into account all classical micro- and macroinformation. Further research is necessary, especially in computational mechanics, to develop this model into a true engineering tool.

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[2] [3]

[4]

[5]

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[8]

N. Laws and G.J. Dvorak, Progressive transverse cracking in composite laminates, Journal of Composites Materials, 1988, 22, 900916 Z. Hashin Analysis of cracked laminates: a variational approach, Mechanical of Materials, 1985, 4 ,121-136. J.A. Nairn, The strain energy release rate of composite microcracking: a variational approach, Journal of Composites Materials, 1989, 23, 1106-1129 J. M. Berthelot, Transverse cracking and delamination in cross-ply glass-fiber and carbon-fiber reinforced plastic laminates: static and fatigue loading, Appl. Mech. Rev., 2003, 56, 1, 1-37 P. Ladevèze, On damage mechanics of composites, C.Bathias and D.Menkès, Proceedings of JNC5 ,667-683, 1986, Paris, Pluralis Publication, (in french) P. Ladevèze and O. Allix and J.F. Deu and D. Lévèque, A mesomodel for localisation and damage computation in laminates, Computer Methods in Applied Mechanics and Engineering, 2000, 183, 105-122 R. Talreja, Transverse cracking and stiffness reduction in compositelaminates, Journal of Composites Materials, 1985, 19, 355375 O. Allix and D. Guédra-Degeorges and S. Guinard and A. Vinet, Analyse de la tenue aux impacts à faible vitesse et à faible énergie des stratifiés composites par la mécanique de l’endommagement, Mec. Inc., 2000, 1, 27-35

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P. Ladevèze, G. Lubineau, D. Violeau and D. Marsal

[9]

P. Ladevèze and G. Lubineau, On a damage mesomodel for laminates: micro-meso relationships, possibilities and limits., Composite Science and Technology, 2001, 61, 15, 2149-2158 P. Ladevèze and G. Lubineau, On a damage mesomodel for laminates: micromechanics basis and improvement, Mechanics of Materials, 2003, 35, 8, 763-775 P. Ladevèze and G. Lubineau and D. Marsal, Towards a bridge between the micro- and the mesomechanics of delamination for laminated composites, Composites Science and Technology, 2005 Z. Hashin, Analysis of stiffness reduction of cracked cross-ply laminates, Engineering Fracture Mechanisms, 1986, 25, 771-778 P. Ladevèze and E. Le Dantec, Damage modeling of the entary ply for laminated composites, Composite Science and Technology, 1992, 43, 3 , 257-267 A. Parvisi and J.E. Bailey, On multiple transverse cracking in glassfiber epoxy cross-ply laminates, Journal of Material Sciences, 1978, 13, 2131-2136 F.W. Crossman and A.S.D Wang, The dependence of transverse cracking and delamination on ply thickness in graphite/epoxy laminates, K.L. Reifsnider, Damage in composite materials, 118-139, American society for Testing and Materials, 1982, ASTM-STP 775 L. Boniface and P.A. Smith and S.L. Ogin and M.G. Bader, Observations on tranverse ply crack growth in a [0  902 ]s CFRP laminate under monotonic and cyclic loading, Proceedings of the 6th International Conference on Composite Materials, 1987, 3, 156-165 A.S.D Wang and P.C. Chou and S. Lei, A stochastic model for the growth of matrix cracks in composite laminates, Journal of Composite Materials, 1984, 18, 239-254 H. Fukunaga and T.W. Chou and P.W.M. Peters and K. Schulte, Probabilistic failure strength analyses of graphite/epoxy cross-ply laminates, Journal of Composite Materials, 1984, 18, 339-356 J. Nairn, Matrix microcracking in composites, Taljera-Manson, Polymer Matrix Composites, Ch.13, 2000 P. Ladevèze, Multiscale computational damage modelling of laminate composites, Course CISM, 2005, to appear

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NUMERICAL METHODS FOR DEBONDING IN COMPOSITE MATERIALS A Comparison of Approaches R. de Borst Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, NL-2600 GB Delft, The Netherlands and I.N.S.A. de Lyon, UMR-CNRS 5514, F-69621 Villeurbanne, France [email protected]

Abstract:

An overview is given of discretization methods that are capable of simulating debonding in composite materials and structures: conventional interface elements, finite element methods that exploit the partition–of–unity property of shape functions, and discontinuous Galerkin methods. Their interrelations are discussed.

Key words:

Composites, delamination, debonding, interface elements, partition-of-unity, discontinuous Galerkin method

1.

INTRODUCTION

A major failure mode in composite structures is debonding, either between two structural components, or between different layers within a structural part. Conventionally, special interface elements methods are placed a priori between the continuum finite elements to capture debonding at locations where they are expected to emerge (Allix and Ladeveze 1992, Schellekens and de Borst 1994). More recently, discretization methods have been proposed, which are more flexible than standard finite element methods, while having the potential to capture propagating debonding cracks in a robust, efficient and accurate manner. Examples are finite element methods that exploit the partition–of– unity property of finite element shape functions, and discontinuous Galerkin methods. This contribution will review these methods and point out the similarities and differences.

13 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 13–22. © 2006 Springer. Printed in the Netherlands.

R. de Borst

14

2.

INTERFACE ELEMENTS

The classical way to represent delamination and debonding is to introduce zero–thickness interface elements between two neighbouring (solid) finite elements. The governing kinematic quantities in interfaces are relative displacements: vn , vs , vt for the normal and the two sliding modes, respectively. When collecting these relative displacements in a relative displacement vector v, they can be related to the displacements at the upper (+) and lower sides (−) of the + − + − + interface, u− n , un , us , us , ut , ut , by v = Lu with uT =

+ (u− n , ..........., ut )



(1)

and L an operator matrix:

 −1 +1 0 0 0 0 0 −1 +1 0 0  LT =  0 0 0 0 0 −1 +1

(2)

The displacements contained in the array u are interpolated in a standard manner, as   u = Ha , H = diag h h h h h h (3) with h an 1 × N matrix containing the interpolation polynomials, and a the array that contains the nodal displacements of the element. The relation between nodal displacements and relative displacements for interface elements is derived from eqs (1) and (3) as: v = Bi a

(4)

with Bi = LH the relative displacement–nodal displacement matrix for the interface element. For analyses of fracture propagation that exploit interface elements, cohesive– zone models are used almost exclusively. In this class of fracture models, a discrete relation is adopted between the interface tractions t i and the relative displacements v: ti = ti (v, κ), with κ a history parameter. After linearization, necessary to use a tangential stiffness matrix in an incremental–iterative solution procedure, one obtains: t˙ i = Tv˙

(5)

with T the material tangent stiffness matrix of the discrete traction–separation law. A key element is the presence of a work of separation or fracture energy, Gc , which governs crack growth and enters the interface constitutive relation in addition to the tensile strength f t . It is defined as the work needed to create a unit area of fully developed crack:  ∞ Gc = σdvn (6) vn =0

Numerical Methods for Debonding in Composite Materials ε 11

r3

% ε 1.0 u

15

∆ +

0.75 + ∆

r2 r1

θ

0.5

+ ∆

0.25

h 0.0

b

ε 11

n=1 n=2 n=3 24 16 8 number of plies

Figure 1. Left: Uniaxially loaded laminated strip. Right: Computed and experimentally determined values for the ultimate strain
u as a function of the number of plies (Schellekens and de Borst 1994). Results are shown for laminates consisting of eight plies (n = 1), sixteen plies (n = 2) and twenty-four plies (n = 3). The triangles, which denote the numerical results, are well within the band of experimental results. The dashed line represents the inverse dependence of the ultimate strain on the laminate thickness.

with σ the stress across the fracture process zone. Evidently, cohesive–zone models as defined above are equipped with an internal length scale, since the quotient Gc /E, with E a stiffness modulus for the surrounding continuum, has the dimension of length. Conventional interface elements have to be inserted in the finite element mesh at the beginning of the computation, and therefore, a finite stiffness must be assigned in the pre–cracking phase with at least the diagonal elements being non-zero. Prior to crack initiation, the stiffness matrix in the interface element therefore reads: T = diag [dn ds dt ], with dn the stiffness normal to the interface and ds and dt the tangential stiffnesses. With the material tangent stiffness matrix T, the element tangent stiffness matrix can be derived in a straightforward fashion, starting from the weak form of the equilibrium equations, as:  K= Γi

BT i TBi dΓ

(7)

where the integration domain extends over the surface of the interface Γ i . For comparison with methods that will be discussed in the remainder of this paper, we expand the stiffness matrix in the pre–cracking phase as (cf. Schellekens and de Borst (1994)):  0 Kn 0 Ks 0  K= 0 0 0 Kt 

(8)

R. de Borst

16 with the submatrices Kπ , π = n, s, t defined as:  hT h −hT h Kπ = dπ hT h −hT h

(9)

with dπ the (dummy) stiffnesses in the interface prior to crack initiation. An example where the potential of cohesive–zone models can be exploited fully using conventional discrete interface elements, is the analysis of delamination in layered composite materials (Allix and Ladeveze 1992, Schellekens and de Borst 1994). Since the propagation of delaminations is then restricted to the interfaces between the plies, inserting interface elements at these locations permits an exact simulation of the failure mode. Figure 1 shows an example of a uniaxially loaded laminate. Experimental and numerical results (which were obtained before the tests were carried out) show an excellent agreement, Figure 1, which gives the ultimate strain of the sample for different numbers of plies in the laminate (Schellekens and de Borst 1994). A clear thickness (size) effect is obtained as a direct consequence of the inclusion of the fracture energy in the model. Since conventional interface elements have to be inserted in the finite element mesh at the beginning of the computation, a (finite) stiffness must be assigned also in the pre–cracking phase with at least the diagonal elements being non–zero. This causes undesired elastic deformations which can largely be suppressed by choosing a high value for the stiffness d n . This, however, can lead to spurious traction oscillations in the pre–cracking phase for high stiffness values (Schellekens and de Borst 1994), which can cause erroneous crack patterns. When analysing dynamic fracture, spurious wave reflections can occur as a result of the introduction of such artificially high stiffness values prior to the onset of delamination. Moreover, the necessity to align the mesh with the potential planes of delamination, restricts the modelling capabilities.

3.

PARTITION-OF-UNITY CONCEPT

A unifying approach to discretization methods for the analysis of propagation of delamination and debonding is enabled by the partition–of–unity concept (Babuska and Melenk 1997). A

collection of functions φ ı , associated with nodes ı, form a partition of unity if nı=1 φı (x) = 1 with n the number of discrete nodal points. For a set of functions φ ı that satisfy this property, a field u can be interpolated as follows:   n m   u(x) = φı (x) a ¯ı + ψ (x)˜ aı  (10) ı=1

=1

with a ¯ı the ‘regular’ nodal degrees–of–freedom, ψ (x) enhanced basis terms, and a ˜ı the additional degrees–of–freedom at node ı which represent the am-

17

Numerical Methods for Debonding in Composite Materials

plitudes of the th enhanced basis term ψ  (x). In conventional finite element notation we can thus interpolate a displacement field as: ˜ a) u = H(¯ a + H˜

(11)

˜ the enhanced basis terms where H contains the standard shape functions, H and ¯ a and ˜ a collect the conventional and the additional nodal degrees–of– freedom, respectively. A displacement field that contains a single discontinuity ˜ = HΓ I, with can be represented by choosing (Belytschko and Black 1999): H i HΓi the Heaviside function. Substitution into eq. (11) gives H˜ a u =  H¯ a +HΓi  ¯ u

(12)

˜ u

Identifying the continuous fields u ¯ = H¯ a and u ˜ = H˜ a we observe that eq. (12) exactly describes a displacement field that is crossed by a single discontinuity, but is otherwise continuous. Accordingly, the partition–of–unity property of finite element shape functions can be used in a straightforward fashion to incorporate discontinuities in a manner that preserves their discontinuous character. The partition–of–unity property of finite element shape functions is therefore a powerful method to introduce discontinuities in continuum finite elements. Using the interpolation of eq. (12) the relative dis˜ |x∈Γi . When using a placement at the discontinuity Γi is obtained as: v = u cohesive–zone model, the tractions at the discontinuity can directly be derived from the discrete interface relation. We take the balance of momentum ∇ · σ + ρg = 0

(13)

as point of departure and multiply this identity by test functions w w , taking them from the same space as the trial functions for u, w = w¯ + HΓi w˜ . Applying the divergence theorem and requiring that this identity holds for arbitrary w w ¯¯ and w ˜ yields the following set of coupled equations:    sym ∇ w ¯ : σ dΩ = w ¯ · ρgdΩ + w ¯ · tdΓ (14a) Ω

 Ω+

∇

sym

w ˜ : σ dΩ +

 Γi

Ω

w ˜ · ti dΓ =

Γ

 Ω+

w ˜ · ρgdΩ +

 Γ

HΓi w ˜ · tdΓ (14b)

where in the volume integrals the Heaviside function has been eliminated by a change of the integration domain from Ω to Ω + . Interpolating the trial and the test functions in the same space, u ¯ = H¯ a , u ˜ = H˜ a w ¯ = Hw ¯ , w ˜ = Hw ˜

(15)

18

R. de Borst

¯ and and requiring that the resulting equations must hold for any admissible w ˜ we obtain the discrete format: w,    BTσ dΩ = ρBT gdΩ + HT tdΓ (16) Ω



T

Ω+

Ω



B σ dΩ +

T

Γi

H ti dΓ =

Γ



T

Ω+

ρB gdΩ +

 Γ

HΓi HT tdΓ

(17)

where B = ∇H. After linearization, the following matrix–vector equation is obtained:   ext    fa¯ − fa¯int d¯ a Ka¯a¯ Ka¯a˜ = (18) d˜ a Ka¯a˜ Ka˜a˜ fa˜ext − fa˜int with fa¯int , fa˜int given by the left–hand sides of eqs (14a)–(14b), fa¯ext , fa˜ext given by the right–hand sides of eqs (14a)–(14b) and   T Ka¯a¯ = B DBdΩ , Ka¯a˜ = BT DBdΩ (19) Ω

Ka˜a˜ =

 Ω+

BT DBdΩ +

Ω+



HT THdΓ

(20)

Γi

If the material tangential stiffness matrices of the bulk and the interface, D and T respectively, are symmetric, the total tangential stiffness matrix remains symmetric. When the discontinuity coincides with a side of the element, the traditional interface element formulation is retrieved. For this, we expand the term in K a˜a˜ which relates to the discontinuity as    0 Kn 0 Ks 0  HT THdΓ =  0 (21) Γi 0 0 Kt with Kπ = dπ hT h (Simone 2004), which closely resembles eqs (8) – (9). ¯ and a ˜ as primary variable a on Defining the sum of the nodal displacements a ¯ at the - side and rearranging then the + side of the interface and setting a = a leads to the standard interface formulation. However, even though formally the matrices can coincide for the partition–of–unity based method and for the conventional interface formulation, the former does not share the disadvantages of oscillating traction profiles and spurious wave reflections prior to the onset of decohesion, simply because the partition–of–unity concept permits the placement of cohesive surfaces in the mesh only at onset of decohesion, thereby by–passing the whole problem of having to assign a high dummy stiffness to the interface prior to crack initiation.

Numerical Methods for Debonding in Composite Materials

Figure 2.

19

Double cantilever beam with initial delamination under compression.

4

Perfect bond Debonding (dense mesh) Debonding (coarse mesh)

3 2 1 0

1

2

3

4 u (mm)

5

6

7

8

Figure 3. Left: Load–displacement curves for delamination–buckling test. Right: Deformation of coarse mesh after buckling and delamination growth (true scale), after de Borst and Remmers (2005).

To exemplify the possibilities of this approach we consider the double cantilever beam of Figure 2 with an initial delamination length a = 10 mm. This case, in which failure is a consequence of a combination of delamination growth and structural instability, has been analysed using conventional interface elements in Allix and Corigliano (1999). The beam is subjected to an axial compressive force 2P , while two small perturbing forces P 0 are applied to trigger the buckling mode. Two finite element discretizations have been employed, a fine mesh with three elements over the thickness and 250 elements along the length of the beam, and a coarse mesh with only one (!) element over the thickness and 100 elements along the length. Figure 3 shows that the calculation with the coarse mesh approaches the results for the fine mesh closely. Steady–state delamination growth starts around a lateral displacement u = 4 mm. From this point onwards, delamination growth interacts with geometrical instability. Figure 3 (right) shows the deformed beam for the coarse mesh at a tip displacement u = 6 mm. Note that the displacements are plotted at true

20

R. de Borst

scale, but that the difference in displacement between the upper and lower parts of the beam is for the major part due to the delamination and that the strains remain small. The excellent results obtained in this example for the coarse discretization have motivated the development of a layered plate/shell element in which delaminations can occur inside the element between each of the layers (Remmers et al. 2003).

4.

DISCONTINUOUS GALERKIN METHODS

Discontinuous Galerkin methods have classically been employed for the computation of fluid flow. More recently, attention has been given to their potential use in solid mechanics, and especially for problems involving debonding (Mergheim et al. 2004), or for constitutive models that incorporate spatial gradients (Wells et al. 2004). In the latter case, the fact that discontinous Galerkin methods do not require interelement continuity a priori, by–passes the requirement of C 1 –continuity on the damage or plastic multiplier field which plague the implementation of many gradient models in a continuous Galerkin finite element method. In the former case, use of a discontinuous Galerkin formalism can be an alternative way to avoid traction oscillations in the pre– cracking phase. For a discussion on the application of spatially discontinuous Galerkin to fracture it suffices by dividing the domain in two subdomains, Ω − and Ω+ , separated by an interface Γi . In a standard manner, the balance of momentum is multiplied by a test function w w , and after application of the divergence theorem, we obtain:      − − ∇symw : σ dΩ− w + ·t+ dΓ− w ·t dΓ = w ·ρgdΩ+ w ·tdΓ i i Ω/Γi

Γi

Γi

Ω/Γi

Γ

(22) where the surface (line) integral on the external boundary Γ has been explicitly separated from that on the interface Γ i . Prior to crack initiation, continuity of displacements and tractions must be enforced along Γ i , at least in an approximate sense: − u+ − u− = 0 , t+ (23) i + ti = 0 + − − + − with t+ i = nΓi · σ and ti = nΓi · σ . Assuming small displacement gradi− ents, we can set nΓi = n+ Γi = −nΓi , so that the expressions for the interface + + − tractions reduce to ti = nΓi · σ and t− i = −nΓi · σ . A classical procedure to enforce conditions (23) is to use Lagrange multi− pliers. Then, λ = t+ i = −ti along Γi , and eq. (22) transforms into:     sym + − w −w w )·λ λ dΓ = ∇ w : σ dΩ− (w w ·ρgdΩ+ w ·tdΓ (24) Ω/Γi

Γi

Ω/Γi

Γ

21

Numerical Methods for Debonding in Composite Materials augmented with:

 Γi

z · (u+ − u− )dΓ = 0

(25)

z being the test function for the Lagrange multiplier field λλ. After discretization, eqs (24) and (25) result in a set of algebraic equations that are of a standard mixed format and therefore give rise to difficulties when using solvers without pivoting. For this reason, alternative expressions are often sought, in + which λ is directly expressed in terms of the interface tractions t − i and ti . One such possibility is to enforce the second identity of eqs (23) pointwise, so that: λ = −ti and one obtains:     w + − w − ) · ti dΓ = ∇symw : σ dΩ + (w w · ρgdΩ + w · tdΓ Ω/Γi

Γi

Ω/Γi

Γ

(26) With aid of relation (4) between the relative displacements v = u + − u− and the nodal displacements at both sides of the interface Γ i , and the interface traction–relative displacement relation (5), the second term on the left–hand side can be elaborated in a discrete format as:    + − T T w − w ) · ti dΓ = w (w Bi TBi dΓ a (27) Γi

Γi

which, not surprisingly, has exactly the same format as obtained for a conventional interface element. Another possibility for the replacement of λ by an explicit function of the tractions is to take the average of the stresses at both sides of the interface: σ + + σ − ). The surface integrals for the interface in eq. (24) can λ = −21 nΓi · (σ now be reworked as:   1 + + − σ + + σ − )dΓ w − w − ) · nΓi · (σ w − w ) · λ dΓ = (28) (w −(w Γi Γi 2 To ensure a proper conditioning of the discretized equations, one has to add eq. (25), so that the modified form of eq. (22) finally becomes:   1 + sym σ + + σ − )dΓ− w − w − ) · nΓi · (σ ∇ w : σ dΩ − −(w Ω/Γi Γi 2  1 α −(∇symw + + ∇symw − ) : D · nΓi · (u+ − u− )dΓ = (29) Γi 2   w · ρgdΩ + w · tdΓ Ω/Γi

Γ



w + − w −) · To ensure symmetry, α = 1, but then a diffusionlike term, Γi τ (w (u+ − u− )dΓ has to be added to ensure numerical stability. The numerical

R. de Borst

22

parameter τ = O(|k|/h), with |k| a suitable norm of the diffusionlike matrix that results from elaborating this term and h a measure of the grid density. For the unsymmetric choice α = −1, addition of a diffusionlike term may not be necessary (Baumann and Oden 1999).

5.

CONCLUDING REMARKS

Discretization methods for delamination and debonding have been discussed. In particular, conventional interface elements, partition–of–unity based approaches, and discontinuous Galerkin methods were treated.

REFERENCES Allix, O. and Ladeveze P. (1992). Interlaminar interface modelling for the prediction of delamination. Composite Structures, 22: 235–242. Allix, O. and Corigliano, A. (1999). Geometrical and interfacial non-linearities in the analysis of delamination in composites. International Journal of Solids and Structures, 36: 2189–2216. Babuska, T. and Melenk, J.M. (1997). The partition of unity method. International Journal for Numerical Methods in Engineering, 40: 727–758. Baumann, C.E. and Oden, J.T. (1999). A discontinuous hp finite element method for the Euler and Navier–Stokes problems. International Journal for Numerical Methods in Fluids, 31:, 79–95. Belytschko, T. and Black T. (1999). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45: 601–620. de Borst, R. and Remmers J.J.C. (2005). Computational Modelling of Delamination. Composites Science and Technology, in press. Mergheim, J., Kuhl, E. and Steinmann, P. (2004). A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Communications in Numerical Methods in Engineering, 20:, 511–519. Remmers, J.J.C., Wells G.N. and de Borst R. (2003). A solid–like shell element allowing for arbitrary delaminations. International Journal for Numerical Methods in Engineering, 58: 2013–2040. Schellekens, J.C.J. and de Borst, R. (1992). On the numerical integration of interface elements. International Journal for Numerical Methods in Engineering, 36: 43–66. Schellekens, J.C.J. and de Borst, R. (1994). Free edge delamination in carbon-epoxy laminates: a novel numerical/experimental approach. Composite Structures, 28: 357–373. Simone, A. (2004). Partition of unity–based discontinuous elements for interface phenomena: computational issues. Communications in Numerical Methods in Engineering, 20: 465–478. Wells, G.N., Garikipati, K. and Molari, L. (2004). A discontinuous Galerkin formulation for a strain gradient–dependent damage model. Computer Methods in Applied Mechanics and Engineering, 193:, 3633–3645.

ATOMIC-CONTINUUM TRANSITION AT INTERFACES OF SILICON AND CARBON NANOCOMPOSITE MATERIALS

Ryszard Pyrz Institute of Mechanical Engineering, Aalborg University, Pontoppidanstræde 101, 9220 Aalborg East, Denmark

Abstract:

The mechanical performance of composite materials is critically controlled by the interfacial characteristics of the reinforcing phase and the matrix material. Her we report a study on the interfacial properties of a silicon nanowirepolypropylene nanocomposite system through molecular dynamics simulations. Carbon nanotube polypropylene nanocomposite serves as a reference system for comparison. Results of a silicon nanowire pullout simulation suggest that the interfacial shear stress transfer of this novel system is comparable with corresponding interfacial shear stress of carbon nanotube system. A new atomic strain concept is formulated that allows calculation of continuum quantities directly within a discrete atomic (molecular) system. The concept is based on the Voronoi tessellation of the molecular system and calculation of atomic site strains, which connects continuum variables and atomic quantities when the later are averaged over a sufficiently large volume treated as a point of the continuum body.

Key words:

silicon nanowires; carbon nanotubes; molecular dynamics; interfacial properties; atomic strain; Voronoi tessellation.

1.

INRODUCTION

Nanocomposites are a new class of composites that are particle-filled polymers for which at least one dimension of the dispersed particles is in the nanometer range. Three types of nanocomposites can be distinguished depending on dimensions of the dispersed phase are in nanometer scale. 23 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 23–32. © 2006 Springer. Printed in the Netherlands.

24

R. Pyrz

When the three dimensions are in the order of nanometers, we are dealing with isodimensional nanoparticles such as spherical silica and aluminum nanoparticles or semiconductor nanoclusters. When two dimensions are in the nanometer scale and the third is larger, forming an elongated structure, we speak about nanotubes and nanowires. Finally, the third type of nanocomposites is characterized by only one dimension in the nanometer range. In this case the filler is present in the form of sheets of a few nanometer thickness and up to one micrometer long. This family of nanocomposites, known as polymer layered crystal nanocomposites, is almost exclusively obtained by the intercalation of the polymer inside the galleries of layered host crystals. Polymer-layered crystal nanocomposites are now commercially available and have been throughout investigated in a large amount of publications during last ten years 1-3. Nanocomposites with elongated structural fillers have recently attracted many investigations. A focus is exclusively on nanocomposites with carbon nanotubes (CNT) due to carbon nanotubes unique properties including mechanical, thermal, optical and electrical 4-7. However, after nearly a decade of research, their potential as reinforcement for polymers has not been utilized to a full extent not mentioning their high price. Therefore, seeking an alternative seems to be well motivated. Inorganic nanotubes and nanowires have been the focus of extensive research due to their technological importance in the field of nanotechnology. New forms of stable inorganic nanowire-structures have been explored 8-14. Particularly, silicon nano-structures are required to sustain the current silicon-based technology and as a by-product it might bring benefits to the field of structural nanocomposites. The knowledge and technology of silicon nanowires is well advanced 15-20. The fabrication of silicon nanowires with diameters ranged from 1.3 to 190 nm, growing rate up to the order of 8 micrometer per minute and length of a few micrometers is now well established. A number of micromechanical investigations have been performed to predict behaviour of composite interfaces, showing that the detailed behaviour of the material at these interfaces frequently dominates the behaviour of the composite as a whole. In that respect nanocomposite materials are not an exception. On the contrary, since the surface area of nanowires and nanotubes is by a few orders of magnitude larger that corresponding surface area of classical fibers at the same volume fraction, an interpretation of interfacial interaction in nanocomposites becomes a critical issue. The interfacial interaction is an extremely complex process due to continuous evolution of interfacial zones during deformation and poses severe interpretation problems going from atomic to continuum descriptions. In the present work we report a study of interfacial properties of silicon

Atomic-Continuum Transition at Interfaces of Silicon

25

nanowire and CNT embedded in polypropylene matrix. These nanoinclusions have been placed in the same cavity of a polymeric network and subject to energy minimization in order to obtain a stable reference configuration. Nano-inclusion pullout simulations were performed for both systems. The pullout process consisted of a number of predetermined deformation steps, where a fully embedded nano-inclusion was being “pulled” out from the polypropylene matrix. The interfacial shear stress has been calculated from the energy difference between the fully embedded nano-inclusion and the complete pullout configuration, which is equal to the work required for nano-inclusion pullout. However, the Cauchy stress calculated in this manner is not well defined at atomic level. Therefore, a new atomic strain concept is formulated that allows calculation of continuum quantities directly within a discrete atomic system. The concept is based on the Voronoi tessellation of the molecular system and calculation of atomic site strains.

2.

CALCULATION OF STABLE STRUCTURES

Marsen and Sattler16 have produced silicon nanowires using magnetron sputtering technique showing that on the nanometer size scale silicon nanowires may differ significantly from the crystalline bulk. Without silicon oxide shielding there is no preferential growth direction associated with the diamond-type bulk silicon lattice. Therefore the structure of nanowire has to have the anisotropy axis intrinsically build into molecular configuration as illustrated in Fig. 1. The wire unit cell consists of two hexagons (light gray),

Figure 1. Silicon nanowire cage structure.

R. Pyrz

26

in the center of which lies the wire axis, and twelve pentagons. In order to find stable cage structure the quantum mechanical density functional method has been used. The method is included in commercially available molecular orbital package MOPAC200018. The resulting cage structure illustrated in Figure 1 exhibits hexagonal bond length 2.262 nm and pentagonal bond length 2.256 nm. The density functional method is very demanding and the CPU time spent to create stable cage structure was 9.3x104 sec. This is a prohibitive task for larger molecular structures and AM1 semi-empirical quantum mechanical method22 has been used to find mechanical properties of silicon nanowires. Figure 2 illustrates deformation stages of the silicon wire as it is subject to axial tension and Figure 3 shows changes of potential

Figure 2. Snapshots of Si wire deformation during axial tension.

Figure 3. Potential energy of Si wire during deformation.

Atomic-Continuum Transition at Interfaces of Silicon

27

energy during deformation. Under large axial deformation the phase transition of the nanowire structure is taking place and that point of deformation is taken as an ultimate strength of the nanowire. The Young modulus has been calculated from the initial part of the energy curve yielding the value of 133 GPa. This is significantly lower than CNT modulus, which is reported to be in the range 0.7-1.4 TPa. Nevertheless, it is expected that the full potential of such high modulus of elasticity can never be addressed in the context of interfacial properties of polymer nanocomposites due to intrinsic stiffness and strength limitations of the polymer itself. The final structures of the silicon nanowire and the CNT are shown in Figs. 4 and 5. The single wall carbon nanotube structure has been created in

Figure 4. The structure of silicon nanowire.

Figure 5. The structure of carbon nanotube.

order to compare interfacial properties of these two nanostructures. The length and the diameter of silicon nanowire and CNT have been sought as close as possible to avoid size effects in a subsequent pullout analysis.

3.

INTERFACIAL LOAD TRANSFER

Single silicon nanowire- and CNT-polypropylene models have been constructed. Nano-inclusions have been placed in the same cavity of a polymeric network and subject to energy minimization in order to obtain a reference configuration. The energy minimization was performed employing molecular mechanics MM+ code from HyperChem©. Nano-inclusion pullout simulations were performed for both systems. The pullout process consisted of a number of predetermined deformation steps, where a fully embedded nano-inclusion was being “pulled” out from the polypropylene matrix. Snapshots of the pullout process for both systems are shown in Fig. 6.

R. Pyrz

28

Figure 6. Pullout snapshots of CNT system (a) and Si wire system (b).

Figure 7. Potential energy (a) and interaction energy (b).

Atomic-Continuum Transition at Interfaces of Silicon

29

After each step, the energy minimization has been performed and an energy of the nano-inclusion has been recorded, Fig. 7a. The silicon nanowire clearly shows the “stick and slip” behaviour , which results from the nanowire cage structure. Contrary to the silicon nanowire, the CNT pullout takes place in a smooth manner with higher energy level as compared to the silicon counterpart. It means that the nanoinclusion/polymer interaction is stronger for this system. Interaction energy i.e. the energy difference between the fully embedded nano-inclusion and the complete pullout configuration is equal to the work required for nanoinclusion pullout yielding

Eemb  E0

³

L 0

S ( L  x ) W i dx

(1)

where IJi denotes interfacial shear stress, and S and L are the circumference and the length of nano-inclusions, respectively. The energy difference is shown in Fig. 7b. The shear interfacial stress transfer for silicon nanowire and CNT estimated from this expression is correspondingly 15.98 MPa and 24.91 MPa, which indicates that the silicon nanowire may serve as a feasible replacement for CNT reinforcement in polymer based nanocomposites despite large difference in their stiffness and strength properties.

4.

ATOMIC STRAIN

An important issue is the development of definitions for continuum quantities that are calculable within a molecular structure. The most frequently used form for the stress at atomic level is based upon the Clausius virial theorem, which determines the stress field applied to the surface of a fixed volume containing interacting particles (atoms). It has been shown that the virial stress cannot be directly related to the classical Cauchy stress that appears in Eq. (1), and different modifications have been proposed 23,24. It seems that the relationship between local displacements of atoms and the strain tensor is not ambiguous as the concept of atomic stress. Although different strain measures can be formulated all of them rely on the coordinates of atoms. Given a set of atom coordinates the structure of the molecular system can be analyzed by means of the Voronoi tessellation, which divides space into regions centered on these atoms 25, 26. The Voronoi polygon (polyhedron in 3D) around central atom is composed of a set of sub-

R. Pyrz

30

Figure 8. Atomic strain tensor in 2D: Voronoi tessellation (a); interaction cell between neighboring atoms (b); deformed configuration (c); interaction cell after deformation (d).

Figure 9. Voronoi tessellations for Si nanowire in reference configuration (a) and after deformation (b).

polygons whose number is determined by a number of neighbors to the central atom. During motion, the Voronoi polygon associated with the atom changes its shape. The interaction cell defined for each pair consisted of a central atom and its neighbors is a unique region for which it can be assumed that it is influenced only by these atoms. For illustrative purposed the geometrical construction of the atomic strain is shown in Fig. 8 in two dimensional case. A detailed description of the atomic strain concept is presented elsewhere27. The following relation defines interaction cell strain

İijc

1 c c hi u j Vc

(2)

where Vc is an area (volume in 3D) of the interaction cell and ujc are components of the displacement vector of a central point that bisects the

Atomic-Continuum Transition at Interfaces of Silicon

31

distance between neighboring atoms. Vector h is collinear with a line connecting central atom and its neighbor. Its length equals the length of polygonal edge (area of the polyhedron face in 3D), which is common for the central atom and its neighbor. A total strain, which can be identified with the continuum strain, is calculated by volume averaging over the total volume of all interaction cells within representative volume element 27. Figure 9 shows silicon nanowire in its reference state and deformed state along with corresponding Voronoi tessellations. The axial strain calculated from boundary conditions is o.0234 whereas the strain calculated according to Eq. (2) reads 0.0193. During deformation the Voronoi polyhedra change their size and shape. However, as long as the polyhedra do not change their topological properties i. e. number of vertices, edges and faces is constant, we may assume that it correspond to elastic reversible deformation. The change of topology would indicate irreversible deformation.

5.

CONCLUSION

A novel silicon nanowire-polypropylene nanocomposite has been investigated with respect to interfacial characteristics using molecular simulations. The adhesion energy and the interfacial shear stress has been determined indicating sufficient bonding quality as compared to CNT system. An extended simulation study should consider influence of nanowire geometry such as aspect ratio and diameter on interfacial stress distribution, both of which affect stress transfer behavior. Furthermore, an immobilization of polymer chains in a vicinity of nano-inclusions should be investigated in order to determine an effective domain of reinforcement of the polymeric matrix. All these analyses would be facilitated by the use of atomic strain concept, which allows analyzing the molecular structure atom by atom.

REFERENCES 1. 2. 3. 4. 5.

P.C. LeBaron, Z. Wang and T.J. Pinnavaia, Polymer-layered silicate nanocomposites. an overview, Appl.Clay Sci. 15, 11-29 (1999). M. Alexandre and P. Dubois, Polymer-layered silicate nanocomposites: preparation, properties and uses of a new class of materials, Mat. Sci. Eng. R 28, 1-63 (2000). D. Schmidt, D. Shah and E.P. Giannelis, New advances in polymer/layered silicate nanocomposites, Current Opinion in Solid State & Mat. Sci. 6, 205-212 (2002). D. Srivastava, C. Wei and K. Cho, Nanomechanics of carbon nanotubes and composites, Appl. Mech. Rev. 56(2), 215-230 (2003). H. Rafii-Tabar, Computational modelling of thermo-machanical and transport properties of carbon nanotubes, Phys. Rep. 390, 235-452 (2004).

R. Pyrz

32 6. 7. 8. 9. 10. 11. 12.

13. 14.

15.

16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27.

R. Andrews and M.C. Weisenberger, Carbon nanotube polymer composites, Current Opinion in Solid State & Mat. Sci. 8, 31-37 (2004). E.T. Thostenson, C. Li and T-W. Chou, Nanocomposites in context, Comp. Sci. Techn. 65, 491-516 (2005). O. Gülseren, F. Ercolessi and E. Tosatti, Noncrystalline structures of ultrathin unsupported nanowires, Phys. Rev. Lett. 80(17), 3775-3778 (1998). X. Duan, J. Wang and Ch.M. Lieber, Synthesis and optical properties of gallium arsenide nanowires, Appl. Phys. Lett. 76(9), 1116-1119 (2000). O. Shenderova, D. Brenner and R.S. Ruoff, Would diamond nanorods be stronger than fullerene nanotubes? Nanolett. 3(6), 805-809 (2003). J.W. Kang and H.J. Hwang, Molecular dynamics simulations of ultra-thin Cu nanowires, Comp. Mat. Sci. 27, 305-312 (2003). X. Wang, Ch.J. Summers and Z.L. Wang, Large-scale hexagonal-patterned growth of aligned ZnO nanorods for nano-optoelectronics and nanosensor arrays, Nanoletters 4(3), 423-426 (2004). L.W. Yin, Y. Bando, Y.C. Zhu and M.S. Li, Controlled carbon nanotube sheathing on ultrafine InP nanowires, Appl. Phys. Lett. 84(26), 5314-5316 (2004). B. Akdim, R. Pachter, X. Duan and W. Wade Adams, Comparative theoretical study of single-wall carbon and boron-nitride nanotubes, Phys. Rev. B 67(24), 245404-1/8 (2003). Y.F. Zhang, Y.H. Tang, N. Wang, C.S. Lee, I. Bello and S.T. Lee, One-dimensional growth mechanism of crystalline silicon nanowires, J.Cryst.Growth 197, 136-140 (1999). B. Marsen and K. Sattler, fullerene-structured nanowires of silicon, Phys. Rev. B 60(16), 11 593-11 600 (1999). R.Q. Zhang, S.T. Lee, C-K. Law, W-K. Li and B.K. Teo, silicon nanotubes. Why not? Chem. Phys. Lett. 364, 251-258 (2002). J.J.P. Steward, MOPAC 2000 Manual (Fujitsu Limited, 1999). Y. Zhao and B.I. Yakobson, what is the ground-state structure of the thinnest Si nanowires? Phys. Rev. Lett. 91(3), 035501-1/4 (2003). M. Menon, D. Srivastava, I. Ponomareva and L.A. Chernozatonskii, nanomechanics of silicon nanowires, Phys. Rev. B 70, 125313-1/6 (2004). Y. Wu, Y. Cui, l. Huynh, C.J. Barrelet, D.C. Bell and C.M. Lieber, Controlled growth and structures of molecular-scale silicon nanowires, Nanolett. 4(3), 433-436 (2004). M.J.S. Dewar and C. Jie, AM1 calculations for compounds containing silicon, Organometallics 6, 1486.1490 (1987). R.J. Hardy, Formulas for determining local properties in molecular dynamics simulations: Shock waves, J. Chem. Phys. 76(1), 622-628 (1982). M. Zhou, A new look at the atomic level virial stress: on continuum-molecular system equivalence, Proc. R. Soc.Lond. A 459, 2347-2392 (2003). R. Pyrz, in: Comprehensive Composite Materials, Vol. 1, edited by A. Kelly and C. Zweben (Elsevier, Oxford, 2000), pp. 465-478. B. Bochenek and R. Pyrz, Reconstruction of random microstructures – a stochastic optimization problem, Comp. Mat. Sci. 31, 93-112 (2004). R. Pyrz and B. Bochenek, Atomic strain tensor for molecular systems, in preparation.

MULTISCALE MODELLING FOR DAMAGED VISCOELASTIC PARTICULATE COMPOSITES André Dragon1, Carole Nadot-Martin1 and Alain Fanget2 1

Laboratoire de Mécanique et de Physique des Matériaux (UMR CNRS n°6617), Ecole Nationale Supérieure de Mécanique et d'Aérotechnique, BP 40109, 86961 FuturoscopeChasseneuil cedex, France; 2Centre d'Etudes de Gramat, 46500 Gramat, France

Abstract:

1.

The problem of micromechanically based constitutive description of composite time-dependent materials for which damage consists in grain-matrix debonding is treated starting from the homogenization framework proposed by Christoffersen1. The localization relations and the homogenized stress are established and discussed for a fixed state of damage (i.e. for a given actual number of open and closed cracks) and using the hypothesis of no sliding on closed crack lips. The formulation is completed by a second stage leading to a thermodynamically consistent and simplified formulation of the model.

INTRODUCTION

This paper deals with a two step scale transition, for modelling the anisotropic damage behaviour of viscoelastic particulate composites, starting from the methodology initially proposed by Christoffersen1 for elastic bonded granulates. By means of a specific geometrical and kinematical description giving much attention to the granular character, this methodology offers an advantage of accounting - in a direct manner – for morphology and internal organization of a large class of particulate composites. The recent extension of the method (Nadot-Martin et al.2), for composites involving a viscoelastic matrix, has confirmed its efficiency since it allows to account for genuine viscoelastic interactions between constituents and for their consequence, the “long range memory” effect. The present contribution attempts to extend further the technique in presence of damage by grain-matrix debonding. Sect. 2 recapitulates the main ingredients and results of the scale transition with special attention paid to coupling between damage and viscoelasticity. In Sect. 3, a complementary 33 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 33–40. © 2006 Springer. Printed in the Netherlands.

A. Dragon, C. Nadot-Martin and A. Fanget

34

localization-homogenization procedure appears crucial to ensure both the thermodynamic consistence and further applicability of the model final formulation.

2.

VISCOELASTIC SCALE TRANSITION IN PRESENCE OF DAMAGE

2.1

Micro-mechanical analysis and behaviour

Fig.1 (a) shows a close-up schematic for grains separated by matrix layers according to the scheme initially proposed by Christoffersen1 for sound, i.e. undamaged, particulate composites. The grains are considered as polyhedral; any two of them are interconnected by a thin material layer of a given uniform thickness. The grain-layer interfaces are characterized by their orientation ( n D for the Dth layer). Some granulometry is accounted for through the vectors linking grain centroids ( d D for the Dth layer). The damage by grain/matrix debonding is incorporated in the form of material discontinuities (cracks) located at grain/layer interfaces (see Fig. 1 b). .

. .

dD

h

D

n

. D

Layer D

Debonded interface

dD

hD

c

nD

D

I2

.

D

Layer D

Debonded interface

I D1

Figure 1. (a) Two neighbouring grains with an interconnecting material layer according to Christoffersen1. (b) A layer with cracks at its boundaries.

The grains are assumed isotropic and linear elastic. The matrix occupying each elementary layer D is considered as viscoelastic and isotropic according to a thermodynamically consistent internal variable representation (NadotMartin et al.2). The dissipative process related to viscoelastic relaxation is accounted for via the symmetric, strain-like, tensorial internal variable J The free energy per unit volume and correspondingly the total stress, are decomposed into two terms, a reversible one, function of the total strain H, and a viscous one, function of J. The reversible and viscous stresses are obtained by partial derivation of the free energy respectively with respect to H and J. The evolution of J which can be interpreted as ‘delayed elastic’ strain is given by a law employing, for simplicity, a single relaxation time X

Multiscale Modelling for Damaged Viscoelastic Particulate Composites 35 w İ, Ȗ J 

1 J X

1 1 H : L( e) " : H  J : L( v ) : J , V 2 2 H , J t

0 0

d (v)

V ( r )  V ( v)

ı (v) : İ  Ȗ

L(e)" : İ  L(v) : Ȗ

1 Ȗ : L(v) : Ȗ t 0 ȣ

(1)

L(e )" and L( v ) are respectively the fourth-order tensors of elastic and viscous

stiffness for the matrix.

2.2

Local problem approach and overall stress

The material discontinuities and relative displacement jumps are incorporated in a compatible way with the Christoffersen’s original kinematical description. The latter is defined by four assumptions for the local displacement field that are recalled below (see Nadot-Martin et al.2 for relative analysis). The kinematics of grain centroid is characterized by the global displacement gradient F . The grains are supposed homogeneously deformed and the corresponding displacement gradient f 0 assumed to be common to all members of the statistically representative volume element (SRVE). Each layer is subject to a homogeneous displacement gradient, proper to the layer D and noted f D . Local disturbances at grain edges and corners are neglected in regard to thinness of the layers. In addition to imply the displacement jump linearity, the previous assumptions impose only two possible configurations for a layer D: either its two boundaries are cohesive, or they are both debonded. The second configuration is described with two mean displacement discontinuity vectors of opposite signs. By taking into account relative conditions of displacement jump, the displacement gradient f D for a debonded layer D (see Fig. 1 b), is given by: f ijĮ





f ij0  Fik  f ik0 d Įk n Įj /h Į  f ijĮD

b iĮ

I1Į

 b iĮ

I Į2



1 ĮD Į f ij c j 2

(2)

with h D the thickness of the layer and c D the vector connecting the centres of two opposite interfaces. For a layer D whose both interfaces are cohesive f D is obtained by using the continuity of displacements on the grain/layer interfaces (Christoffersen1): f ijĮ





f ij0  Fik  f ik0 d Įk n Įj /h Į

(3)

The supplementary term f DD in Eq. (2) represents the specific contribution of two microcracks located at the boundaries of the debonded layer

A. Dragon, C. Nadot-Martin and A. Fanget

36

considered. In view of Eqs. (2)-(3), strain and rotation in any layer depend on its geometrical features. Such a dependence allows to account for microstructure effects on deformation mechanisms of the matrix. Due to the assumption neglecting interlayer zones, the transmission through grains-andlayers assembly is strongly involving grains (see f 0 in Eqs. 2-3). By considering the SRVE loaded on its boundaries by uniform tractions represented by a given macroscopic stress and using the generalized Hill lemma, one may prove that: Ȉ ij

ı ij

V

1 V

¦ t iĮ d Įj

, t iĮ

ı Įki n Įk A Į

(4)

Į

V represents the volume of grains and layers excluding interlayer zones, A D is the projected area of the Dth layer and t D represents the total force transmitted through the interfacial layer, ı D being the average stress in this

latter. The second expression of 6 is specific to the Christoffersen-type approach: stresses are seen from a granular viewpoint as forces transmitted from grain to grain by layers acting as contacts zones. The following consists in searching f 0 as such a way that the real stress field satisfies Eq. (4). The grains have identical moduli noted L0. The mechanical properties of the matrix ( L(e)" , L(V), X) are the same for all layers. Consequently, as H D Sym. f D is uniform over the Dth layer, the corresponding relaxation J(see Eq. 1) is also uniform, it is noted J D . Furthermore, the average stress ı D in Eq. (4) becomes a uniform field for the Dth layer and t D is the mean force transmitted through its first interface I1Į . For a debonded layer D, two cases are considered. When the cracks located at its boundaries are open then t D 0 . When they are closed, it is supposed that no sliding is allowed so that t D is integrally transmitted. The solution is then: fij0

Id - Bc 1

1

: Ac



ijkl Flk

­ 1 (v) ° 1 Lmukl®  Bcijuv °¯ V

½°

1

¦ȆĮvmc ȖĮlkcAĮchĮc  įvm V ¦ȖȕlkAȕ hȕ ¾° Įc

­ ½ 1 ȕ ȕ° 1 (e)" ° 1 f f İȕD Lmukl® ¦Ȇfvm İfD  Bcijuv lk A h  įvm ¦ lk A h ¾ V ȕ °¿ °¯ V f

ȕ

¿ (5)

with H ȕD Sym. f ȕD , H fD Sym. f fD , ȆijĮ įij  diĮ nĮj /hD and where A c , B c , degraded by the presence of damage, are defined by:

Multiscale Modelling for Damaged Viscoelastic Particulate Composites 37 e) L(ijkl

A cijkl

V

e)" į im D im  L(mjkl

e) L(ijkl

A ijkl



V

e)"  L(ijkl



e)" e)" į im  D im  L(mjnl A ijkl  L(mjkl T imkn  D imkn 1 1 dȕ nȕ Aȕ D ijkl d ȕ n ȕ d ȕ n ȕ A ȕ /h ȕ V ȕ i j k l V ȕ i j 1 d iĮ n Įj d Įk n Įl A Į /h Į V Į

B cijkl

¦

D ij

¦

(6)

¦

T ijkl

Superscripts D, Dc , E and f denote summations over respectively all layers, layers either cohesive or with closed cracks, layers with open cracks only and layers with closed cracks only. The form of Eq. (5) represents a remarkable decomposition into a reversible term, a viscous one f 0(v) function of internal variables JD for D=1,…,N - with N being the total number of layers inside the SRVE - and a damage-induced one f 0(d) involving the full set H kD H fD ‰ H ȕD . These three contributions depend on the damage

^ ` ^ ` ^ `

state through the tensors D and D (see A c , B c ). The same can be done for f D with Eqs. (2)-(3). The local strain field is then given by:

­İDD for y debonded layer D İy C y, D, D : E  İ(v)y  İ(d)y  ® ¯0 elsewhere 0 ­°Cijkl D, D Id  Id : Bc1 : Ac ijkl for y  grains Cijkl y, D, D ® Į Į 1 °¯Cijkl D, D Idijkl  Idijuv Bc : Ac vmkl Ȇmu for y  layer Į, Į (7) ­°İij0(v) ȖD , D, D Idijkl flk0(v) for y  grains (v) İij y ® Į(v) D 0(v) Į °¯İij Ȗ , D, D Idijuv fvm Ȇmu for y  layerD, D ­°İij0(d) İkD , D, D Idijkl flk0(d) for y  grains İij(d)y ® Į(d) kD 0(d) Į °¯İij İ , D, D Idijuv fvm Ȇmu for y  layerD, D









^ ^ ^ ^

with Id ijkl



^ `

^ `

 A : Bc1 : Acº : E  Ȉ(v) ȖD , D, D  Ȉ(d) İkD , D, D »¼ 1 ȖD , D, D A : f 0(v) ȖD , D D  L(v) : ¦ȖĮ AĮ hĮ V Į kD 0(d) kD (e)" 1 İ , D, D A : f İ , D D  L : ¦İkD Ak h V k V

^ ` ^ `

Ȉ(d)





1 (į ik į jl  į il į jk ) . The overall stress is derived from Eq. (4): 2

Ȉ ª L(e) «¬ Ȉ(v)

   ` ` ` `

^ ` ^ `

(8)

A. Dragon, C. Nadot-Martin and A. Fanget

38

^ `

The strain field depends through İ ( v ) on the full set of relaxations JD . D

J representing the Dth layer's memory, such a dependence clearly indicates

viscoelastic interactions between the matrix layers and the grains in the SRVE. This dependence results from the term f 0(v) which, by means of Eqs. (2)-(3) appears in the expression of f D and therefore in that of the strain field. The form of f 0(v) (second term in Eq. 5) indicates strong complexity of local viscoelastic interactions. Secondly, the strain field depends on damage through the tensors D and D but also through the term H (d ) involving the H fD ‰ H ȕD related to the effect of any kind of cracks (open full set H kD

^ ` ^ ` ^ `

or closed). The latter dependence results from the term f 0(d ) (third term in Eq. 5). In particular, for a debonded layer D one may distinguish two kinds of contribution of damage on its strain: a “local” one, H DD , related to the cracks located at its own boundaries and a “non local” one, H D (d ) , involving the effect of the others cracks inside the SRVE. The overall stress is split into a reversible and a viscous part, both degraded by D and D and completed by a damage-induced stress Ȉ(d) . The first terms of Ȉ(v) and Ȉ(d) , (i.e. A : f 0(v) and A : f 0(d) in Eq. 8), correspond respectively to the macroscopic consequence of viscoelastic interactions and damage “nonlocal” effects. Damage “non-local” effects lead also to the quadratic dependence of the total stress on D. It is already seen that A c , B c and therefore the reversible moduli are only degraded by the open cracks via D and D . This is due to the assumption of no sliding on closed crack lips. These two tensors are natural candidates for macroscopic damage variables. Being tensorial by nature, they allow to account for damage induced anisotropy. Moreover, the moduli may be recovered with crack closure showing that the model is potentially capable of describing unilateral effects.

3.

COMPLEMENTARY LOCALIZATION ANALYSIS

^ `

The homogenized stress (Eq. 8) conveys a full set H kD in addition to the damage

^ `



^ ` ^İ `, D, D

^ `

Ȉ(d)2

^ `

^ `



^ `



that

^ `

Ȉ(d)1 İED , D, D  Ȉ(d)2 İfD , D, D 1 ED A : f 0(d)1  L(e)" : İED AE hE V E 1 İfD , D, D A : f 0(d)2  L(e)" : İfD Af h V f

Ȉ(d) İkD , D, D 6(d)1

Remarking in Eq. (5) D and D. f 0( d )1 İ ED , D, D  f 0( d ) 2 İ fD , D, D , Ȉ(d) becomes:

tensors

f 0( d ) İ kD , D, D

¦

¦

(9)

Multiscale Modelling for Damaged Viscoelastic Particulate Composites 39

^ `

In Eq. (9), H fD

acquire the status of macroscopic internal variables

accounting for the distortion due to the blockage of closed cracks and 6 (d)2 appears as the corresponding residual stress. At the microscopic level, H ED represents, for a layer E the “local” contribution on its strain of the open cracks located at its own boundaries. It seems natural to think that the crack opening depends on the total strain E and therefore H ED also, so that each H ED cannot a priori be considered as a macroscopic variable independent of E. This is confirmed when noting that reversible moduli in Eq. (8) have not all the symmetries required for effective moduli tensor suggesting that Ȉ(d)1 must depend, through H ED , on E. Moreover, Eq. (8) contains the whole set

^

^ `

`

D

J , D 1 N . Considering the possible large number of layers in the SRVE, this is not really the most convenient form for engineering applications. So, it is assumed that a single second-order symmetric strainlike tensorial variable * can be substituted for the set J D to describe the relaxation state of the composite. The purpose consists then in establishing two kinds of relations, the first one: J D J D * for D=1,…,N as it was done by Nadot-Martin et al.2 for the sound material but here should be done in the presence of damage and, simultaneously, the second: İ ED İ ED E, ī for each layer E with open cracks at its boundaries. These relations are not a priori postulated so that the advanced strategy can be viewed as a complementary “localization” analysis involving two aims. To reach these two aims simultaneously, thermodynamic framework is used as a guide. The solution is given by considering D and D as parameters related to the damage configuration considered. In this way, the expression of the overall viscoelastic dissipation with respect to E and * leads to search the above

^ `

mentioned relations in a way that Ȉ

ı

w w V



w w wī

V

. The problem

J D * and the decomposition of

is solved by postulating the linearity of J D İ ED

wE

V

İ ED E, ī in two parts respectively linear in E and * and independent

^ `

of H fD . After some manipulations, it follows: ­ ȕD °İ ij ® °Ȗȕij ¯ ȖijĮc



@

1 Idijmu dȕv n ȕm /h ȕ Bc1 : Ac  Kc uvkl E  ī lk  rijȕD 2  layer E - Idijvu Bc1 : Ac  Kc uvkl  Idijkl īlk  ȖȕRes ij -

>  > - Id Ȇ Bc : Ac  Kc ijmu

Įc vm

1

uvkl

@

 Idijkl īlk  ȖijĮcRes for others layers

(10)

A. Dragon, C. Nadot-Martin and A. Fanget

40

where K c is a structural tensor depending on A , A c and B c (Eq. 6) and on their equivalents in which L( v ) replaces L(e)" . In view of Eq. (10), the strain induced in a layer by the open cracks at its interfaces is controlled by the state variables E and *, D and D the damage parameters, but also by the geometrical features of the layer E considered. The constant r ED represents a residual strain induced in this layer by a residual opening of the cracks at its boundaries when E * 0 . The constants J ERes and J DcRes correspond to non perfectly relaxed states of the layers when the composite is relaxed ( * 0 ). When there is no open crack, i.e. for only closed cracks or for the sound material, one may observe that the expression of J D for any layer corresponds, as expected, to that obtained for the sound material by NadotMartin et al.2. With Eq. (10), one may formulate the whole model, giving local and global responses of the composite, in terms of state variables E, * H fD and damage parameters D and D . The evolution law of * is obtained by averaging the local rule (Eq. 1) over all layers. The homogenized degraded elastic and viscous moduli have then all the symmetries available.

^ `

4.

CONCLUSION

A non-classical homogenization method that constitutes an extension of the Christoffersen-type approach for both viscoelasticity and damage by grain/matrix debonding is presented. It leads to the natural emergence of two macroscopic damage tensorial variables, involving granular aspects, in order to describe moduli degradation, induced anisotropy and unilateral effects. These variables - in addition to the textural tensor T (Eq. 6) related to morphology and internal organisation of constituents - allow to account, in a general 3D context, for coupling of primary anisotropy with the damageinduced one. A complementary localization-homogenization is advanced in order to express the damage-induced strains related to open cracks as functions of macroscopic state variables and simultaneously to replace the set of relaxation variables by a single internal variable.

REFERENCES 1. J. Christoffersen, Bonded granulates, J. Mech. Phys. Solids 31, 55-83 1983. 2. C. Nadot-Martin, H. Trumel, A. Dragon, Morphology-based homogenization method for viscoelastic particulate composites. Part I: Viscoelasticity sole, Eur. J. Mech. A/Solids 22, 89-106 (2003).

A SHAKEDOWN APPROACH TO THE PROBLEM OF DAMAGE OF FIBERREINFORCED COMPOSITES

D. Weichert, A. Hachemi Institute of General Mechanics, RWTH-Aachen, 52056 Aachen, Germany

Abstract: Failure of fiber-reinforced composites under variable repeated loads is investigated by means of shakedown theory and using a two-scale approach: On the microscopic scale, shakedown analysis is carried out in order to study the influence of each component (matrix, fiber and interface) on the behavior of the composite under variable repeated loads. Then, using the concept of averaging techniques overall properties of the composite can be predicted under the assumption of periodicity on the macroscopic scale. The irreversible material features taken into account in this paper are plasticity, material damage related to the matrix material and debonding effects.

Key words:

1.

Shakedown Analysis, Direct Methods, Composites, Plasticity, Failure

INTRODUCTION

Shakedown analysis (SA) is a powerful method to predict if under variable loads in a structure failure occurs or not. Its characteristic feature is that the information, if failure will occur or not, is obtained directly, without solving an evolutionary problem. The reader interested in the fundamentals of SA and, more general, “Direct Methods” referred to1-5.

41 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 41–48. © 2006 Springer. Printed in the Netherlands.

D. Weichert and A. Hachemi

42

A rather new area of application is the study of structured materials, basically composites with at least one ductile phase. These materials can be considered as mechanical structures by themselves and therefore treated with the same methods as mechanical structures taken in the usual sense. The present paper, in continuation of earlier work6,7, focuses on dissipative effects in the interface between matrix material (here considered as metaltype) and reinforcing fibers. The process of debonding is modeled within the framework of interface damage mechanics, where the displacement discontinuities during progressive decohesion are related to constitutive equations extended by an anisotropic damage model8,9. The work is based on a two-scale approach: On the macroscopic scale, the global response of the composite is analysed. On the microscopic scale, the influence of each component (matrix, fiber and interface) on the behavior of the composite is investigated. The methods used are averaging technique, in particular the homogenisation technique10, combined with shakedown analysis, applied to a representative volume element on the micro-level. This allows, under the assumption of periodicity of the composite, to link the results of shakedown analysis on the micro-level to the overall material properties on the macrolevel.

2.

GENERAL ASSUMPTIONS AND RELATIONS

We consider the problem of ductile (metal-type) matrix materials reinforced by straight fibers, justifying in the case of transverse loading the hypothesis of plane state of strains. Thus, the problem is reduced to a twodimensional case with periodicity in the fiber section plane. The macroscopic behavior of this inhomogeneous material is observed on the scale x and the mesoscopic behavior on the scale y. The representative element (RVE) of total surface Ω presented herein includes the fiber phase, the matrix phase occupying the domains Ω + and Ω −, respectively and the cohesive zone represented by the band-shaped domain Ω δ of width δ 1, a time-independent field of ° (r) , a time-independent vector of traction forces periodic residual stresses § (r) ¨° at the interface and an admissible domain P of macroscopic states of stress 

[

\

− − P =  ∃§(s), ∃¨(s) , §(s)(y) ∈ F(y), ¨(s)(y) ∈ G(y) ,

(25)

then, the fiber-reinforced composite material shakes down to the given (s) domain of loading. Safe state of stresses §(s) and traction forces ¨ are defined, respectively, by ° (r) , ¨ = α ¨(c) + ¨° (r) §(s) = α §(c) + § (s)

(26)

such that − F(y) = {§(s)  f (§(s), y) < 0, ∀y ∈ Ω − ∪ Ω +}

(27)

The Problem of Damage of Fiber − Reinforced Composites

47

and

\

[

− (s) (s) G(y) = ¨ g(¨ , d, y) ≤ 0, ∀y ∈ Ω δ .

(28)

° (r) are respectively stress field and traction force in a Here, §(c), ¨ and § purely elastic RVE under the same boundary conditions as for the RVE of the original problem satisfying the according equilibrium- and periodicity conditions. For details of the proof of the theorem we refer to13. (c)

5.

NUMERICAL EXAMPLES

By means of FEM it is possible to calculate safe loading domains for composites using this theorem. Here, we present for illustration the typical graph of safe domains for a simple unit cell under bi-axial loading. For details on this example we refer to13,15. Other calculations can be found in15. 0,8 (a) Limit analysis

E22

(b) Shakedown

0,6

E11 E22 0 E22

0,4 (a) (b)

3D-analysis

0,2

2D-analysis

0 0

0,2

0,4

0,6

0,8

E11 0 E11

Fig 1. Fiber-reinforced composite and admissible domains.

ACKNOWLEDGEMENT The authors gratefully acknowledge the support provided for this research by the Deutsche Forschungsgemeinschaft (DFG grant no. WE 736/22).

D. Weichert and A. Hachemi

48

REFERENCES 1. 2.

3. 4. 5.

6. 7.

8.

9.

10. 11. 12. 13. 14. 15.

Melan, E.: Zur Plastizität des räumlichen Kontinuums. Ing. Arch. 9 (1938) 116-126 Koiter, W.T.: General theorems for elastic-plastic solids. In: Sneddon, I.N., Hill, R. (eds.), Progress in Solid Mechanics, North-Holland, Amsterdam, pp. 165-221 (1960) König, J.A.: Shakedown of elastic-plastic structures. Elsevier: Amsterdam 1987 Kleiber, M.; König, J.A.: Inelastic solids and structures (A. Sawczuk memorial volume). Pineridge Press, Swansea 1990 Maier, G.; Pastor, J.; Ponter, A.R.S.; Weichert, D.: Direct Methods of Limit and Shakedown Analysis. In: De Borst, R. ; Mang, H. A. (eds.), Numerical and computational methods, Chapter 12, Vol. 3. In: Milne, I.; Ritchie, R. O.; Karihaloo, B. (eds.), Comprehensive structural integrity, Elsevier-Pergamon, Amsterdam 2003 Weichert, D.; Hachemi, A.; Schwabe, F.: Application of shakedown analysis to the plastic design of composites. Arch. Appl. Mech. 69 (1999) 623-633 Hachemi, A.; Weichert, D.; On the problem of interfacial damage in fibrereinforced composites under variable loads. Mech. Res. Comm. 32 (2005) 1523 Carrère, N.; Kruch, S.; Vassel, A.; Chaboche, J.-L.; Damage mechanisms in unidirectional SiC/Ti composites under transverse creep loading: Experiments and modeling. Int. J. Damage Mech. 11, (2002) 41-63 Döbert, C.; Mahnken, R.; Stein, E.: Numerical simulation of interface debonding with a combined damage/friction constitutive model. Comput. Mech. 25 (2000) 456-467 Suquet, P.: Analyse limite et homogénéisation. C. R. Acad. Sci. 206 (1983) 1355-1358 Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11 (1963) 357-372 Ju, J.W.: On energy-based coupled elastoplastic damage theories: Constitutive modelling and computational aspects. Int. J. Solids Struct. 25 (1989) 803-833 Hachemi, A.; Mouhtamid, S.; Weichert, D.: Progress in shakedown analysis with application to composites, in print in Arch. Appl. Mech. Bazant, P.Z; Prat, P.C.: Microplane model for brittle plastic material. J. Eng. Mech. 114 (1988) 1672-1702 Schwabe, F.: Einspieluntersuchungen von Verbundwerkstoffen mit periodischer Mikrostruktur. Mechanik Berichte, 2, Institute of General Mechanics, RWTH-Aachen, 2000.

IMPACT BEHAVIOR OF CELLULAR SOLIDS AND THEIR SANDWICH PANELS

Han Zhao, Ibrahim Nasri1 and Yannick Girard2 1

Laboratoire de Mécanique et Technologie, ENS-Cachan/CNRS/University Paris 6, 61, Avenue du président Wilson, 94235 Cachan cedex, France 2EADS-CCR Suresnes, 12 bis rue Pasteur, 92152 Suresnes cedex, France

Abstract:

This paper presents a study of the strength enhancement under impact loading of metallic cellular materials as well as sandwich panels with cellular core. It begins with a review of likely causes responsible for the strength enhancement of cellular materials. A testing method using 60mm diameter Nylon Hopkinson pressure bars is used to investigate the rate sensitivity of various metallic cellular materials. In order to identify the factor responsible for the strength enhancement of those materials, a multiscale analysis is performed on a model structure which is a square tube made of rate insensitive materials. At the macroscopic level, significant enhancement is experimentally observed under impact loading, whereas the crushing mode is nearly the same under both static and impact loading. Numeric simulations and theoretical models give a satisfactory explanation of the role of the lateral inertia. An inversed perforation test on sandwich panels with an instrumented pressure bar is also presented. Such a new testing setup provides piercing force time history measurement, generally inaccessible. Testing results show a notable enhancement of piercing forces, even though the skin aluminum plates and the foam cores are nearly rate insensitive.

Key words:

Cellular material, sandwich panels, impact loading, SHPB.

1.

INTRODUCTION

Metallic based cellular materials such as honeycomb, foam, hollow spheres (especially sandwich panels with cellular solid core) are promising 49 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 49–56. © 2006 Springer. Printed in the Netherlands.

50

H. Zhao, I. Nasri and Y. Girard

structural materials which can be used in lightweight structures, impact energy absorption, acoustical wave attenuation, etc. A large number of experimental, numerical and analytical studies on the behaviour of cellular materials have been reported in the open literatures (Wierzbicki, 1983; Gibson & Ashby, 1988; Klintworth & Stronge, 1988; Reid & Peng, 1997; Deshpande & Fleck, 2000a; Banhart, 2001; Hanssen et al., 2002, Zhao, 2004). However, most of previous works concerns studies under quasi-static loading. There is still a lack of reliable experimental data under impact loading, which makes it difficult to understand the behaviour of cellular solids and their sandwich panel under impact. This paper presents at first an experimental study on the behaviour of cellular materials under impact loading. Various factors that may be potentially responsible for the strength enhancement of metallic foam are discussed as well as a testing method using large diameter Nylon pressure bars in a SHPB set-up and experimental results are presented in Section 2. A multiscale analysis on a model-structure (square tubes) in Section 3 shows that the inertia effect is the main reason of measured macroscopic rate sensitivity of the studied cellular materials. In the last section, a new inversed perforation impact test on the sandwich panel is presented, which provides an accurate measurement of piercing force time history. A notable enhancement of piercing force under impact is observed.

2.

POTENTIAL CAUSES FOR RATE SENSITIVITY, NYLON SHPB, SOME EXPERIMENTAL RESULTS

There exist some known causes for the rate sensitivity of the cellular materials: namely, the rate sensitivity of the base material, the pressure of the air entrapped in the cell, the shock enhancement, wavelength changes, etc. All these potential sources may contribute to the macroscopic rate sensitivity in metallic cellular materials. Actually, it has been shown in micromechanics analyses that the ratio of cellular material strength over that of the cell wall material depends on a power function of the relative density (Gibson and Ashby, 1988). From such an analysis, the rate sensitivity of the base material induces the macroscopic rate sensitivity of a cellular material (Zhao, 2004). The strain rate sensitivity of a cellular material may be due also to the existence of a gas/fluid phase (Gibson and Ashby, 1988; Deshpande and Fleck, 2000b). Taking an open cell foam as an example, when the foam is loaded slowly, the gas filled in the foam can move out without resistance. Under impact, the gas entrapped in cells may have no time to move out. They will then take part in the whole resistance of cellular materials by the pressure increase due to the volume change. The quicker the loading, the more the air is entrapped. At high impact speed (>50m/s), Reid and Peng (1997) reported the strength enhancement by the formation of

Impact Behavior of Cellular Solids and their Sandwich Panels

51

shock wave. If the constitutive relation is concave, a single shock wave front can be formed. However, such kind of concave constitutive function is unusual for solid materials under uniaxial stress condition. For cellular materials, the densification part of the stress-strain curve is a concave function and the shock wave may be formed under usual one-dimensional stress conditions. Another possible factor that may cause strength enhancement is the micro-inertia effect. It was shown that the buckling of a column under impact compression occurs at a delayed time because of lateral inertia, so that the apparent critical buckling force is higher than the quasi-static one (Calladine and English,1984). It is observed for Balsa wood that the folding wave-length has become smaller (Vural and Ravichandran, 2003) under impact loading and this can also explain a higher crushing strength. The above analyses show that the rate sensitivity of cellular materials comes from various sources. It is therefore difficult to predict the rate sensitivity by a simple theoretical model. Experimental studies are definitely the main investigating means. To perform tests at high strain rates, the most widely used method is the spilt Hopkinson pressure bar test (Hopkinson, 1914; Kolsky, 1949). A summary of the development of the SHPB technique can be found in (Zhao and Gary, 1996, 1997). However, from an experimental point of view, impact tests on cellular materials using SHPB have two major difficulties. One is the large scatter due to the small ratio between the specimen size and the cell size. To overcome this difficulty, a large diameter pressure bar is necessary to host a larger specimen. Another is the weak signal due to the weak strength of cellular materials, which leads to a low signal-noise ratio. Here, soft nylon pressure bars are used to solve these problems. However, soft Nylon bars are viscoelastic materials and the wave dispersion increases greatly with the diameter of the bars. Kolsky's original SHPB analysis is based on the basic assumptions that the wave propagation in the bars can be described by a onedimensional wave propagation theory. However, the correction of wave dispersion due to a viscoelastic behaviour coupled with a geometrical effect is necessary for the large diameter Nylon bar. Such a correction is not equivalent to a simple additive correction of a nominal elastic geometrical dispersion and a one-dimensional viscous dispersion and attenuation. As the attenuation coefficient is also affected by the geometric effect, it has been proved that such an approach gives less accurate results (Zhao and Gary, 1995). The 60mm diameter Nylon Hopkinson pressure bar allows for an accurate measurement, because a reasonable large specimen can be used and the signal/noise ratio is improved by soft Nylon bars. It is applied to measure the rate sensitivity of various cellular materials. The dynamic out-of-plan behaviours of 5056 and 5052 aluminium honeycombs were studied, which involved three different cell sizes and wall

H. Zhao, I. Nasri and Y. Girard

52

thicknesses. Figure 1a shows a typical progressive buckling pattern observed for all the tested honeycombs. Figure 1 presents typical mean stress (crushing force divided by cross section) – displacement curves. The bold part is under impact loading, which is followed by the quasi-static test. It shows that the enhancement under impact loading does exist. IFAM AA6061 foam samples are obtained from Aluminium powder blowing process. The final specimens are cylinders 45 mm in diameter and 60 mm long with closed outer skins. The density of the specimen is about 620kg/m3.. It shows an enhancement of about 15% for the mean flow stress. In contrast, 45 mm x 45mm x 55mm Cymat foam samples are cut from foam plates of about 200mm thickness manufactured by the gas injection process with an average density is about 250kg/m3; There is nearly no notable rate sensitivity. Actually, the failure modes are different. Figure 3 indicates that the main crushing mode is progressive folding of IFAM foam and figure 4 shows the failure mode by multiple cracks of the cell walls of rather brittle cymat foam.

Figure 1. A typical mean stressdisplacement curve.

3.

Figure 2. Folding of cell wall of IFAM foam.

Figure 3. Fractures of Cymat foam.

MULTISCALE ANALYSIS OF INERTIA EFFECTS IN A SUCCESIVE FOLDING PROCESS

In order to determine the factor responsible for strength enhancement under impact in a successive folding process, we studied a model-structure: commercially available brass square tube (after suitable partial annealing). The observed crushing mode of such tubes is also successive folding of walls. In addition, the behaviour of brass is known to be rate insensitive for a large range of strain rates. Experimental studies of brass square tubes are presented to confirm that strength enhancement under impact can occur without changes in the folding wavelength. Material characterization tests under static and dynamic loading up to 2500/s were performed to ensure that there is no rate sensitivity for brass in this range of strain rates (Fig.4). Static crushing tests were performed on heat-treated tubes with a universal testing machine. For dynamic loading, the direct impact configuration (Hauser, 1966) with a large scale SHPB system (diameter 80

Impact Behavior of Cellular Solids and their Sandwich Panels

53

mm, input bar of 6m and output bar 4m). A typical comparison between static and dynamic tests is shown in figure 5. We can see that not only the initial peak load but also the successive peak load were measured.

Figure 4. Rate sensitivity of base material.

Figure 5. strength enhancement of tubes.

In the crushing tests presented above, only force-displacement recordings were made. There are no simple methods to know what was happening in the tube, especially under impact loading. An alternative is to simulate the crushing of the tubes numerically. The explicit version of the commercial code LS-DYNA was used to perform the simulations under impact loading, and the implicit version was adopted for static loading. Figure 6 provides a comparison between the force-time histories of the rigid wall under both static and dynamic loading. The simulated crushing tests also reproduce the strength enhancement under dynamic loading not only for the initial peak load but also for the successive peak load, which is another proof of the value of simulations. A careful examination of the sequence of events in successive crushing reveals that the folding cycles are composed of two stages. At the start, crush is obtained by bending in the middle of the flat plates (the two trapezoids around nodes B or B’ in figure 7) and there exist small areas around the four corner lines (the two adjacent triangles around node A in figure 7) which remain vertical (no significant lateral displacement). In the simulated images of a deformed tube (left part of figure 7), the corner line between the two triangles around node A is straight whereas the flat plate is already bent. Subsequently, the second stage begins with the buckling of the corner line areas as shown on the right of figure 8. The buckling of these edge zones (the two adjacent triangles around node A in figure 7) corresponds to a decrease of the global crushing load. The simulation provides the displacement in two perpendicular lateral directions at node A at the corner with the corresponding global crushing force. It shows that the time when the corner node begins to move laterally correspond to the time when the global crushing force begins to decrease. It reveals that in the regions close to the edge, elements are more compressed (higher strain) under dynamic loading because of inertia, This is similar to what was found for a straight column. As the base material (brass) strain hardens, a higher strain is

54

H. Zhao, I. Nasri and Y. Girard

reached under dynamic loading before buckling leading to higher crushing forces. Such a simulation is validated by post-mortem microhardness measurement and a theoretical column box model is also developed, which gives an excellent agreement on the peak force level (Zhao & Abdennadher, 2004).

4.

INVERSED IMPACT PERFORATION TEST OF SANDWICH PANELS

The main deficiency of classical penetration/perforation tests is the lack of force record during the perforation process (Radin and Goldsmith, 1988, HooFatt and Park, 2000). For this purpose, an instrumented long rod can be used at the place of projectile as a Hopkinson pressure bar (Hopkinson, 1914). However, it was very difficult to launch a long rod at a uniform speed without friction during the test (A length of several meter is necessary because the measuring duration is determined by the length over wave speed in the rod). An alternative is to launch the target sandwich panel to strike the perforating long rod. Therefore, the proposed inversed perforation testing setup used a gas gun with a 70mm inner diameter barrel and a 16mm diameter and 6m-long rod with a semi-spherical nose at one end. The rod is instrumented by strain gauges aimed at accurate force measurement during perforation process. The target sandwich panel is launched with the aid of a hollow bullet projectile which is made from an aluminium tube with a welded bottom plate at one end. Two Teflon rings are screwed on the tube to reduce the friction between projectile and barrel of the gas gun. The circular sandwich plate is mounted between the open end of the aluminium tube and an aluminium collar. The fixture is realised by six uniformly distributed bolts slightly tightened. Figure 8a gives schematic drawing and figure 8b provides an illustration this bullet projectile with a mounted sandwich plate. Such a fixture is close to a clumped condition. Another technical point is to stop the launched sandwich panel mounted on the hollow projectile after complete perforation. The adopted solution is to use aluminium honeycomb bumper to absorb the residual velocity of the sandwich and projectile. Figure 9 shows the

Impact Behavior of Cellular Solids and their Sandwich Panels

55

aluminium honeycomb bumper and semi-spherical nose end of the pressure bar.

Figure 8. a) details of projectile and clamping ring; b) Projectile

Figure 9. perforation setup.

Figure 10. Quasi-static and dynamic curves.

The tube form bullet-projectile together with fixture has a weight of 720 g. The gas gun can launch such a mass to a speed up to 46m/s. Perforation tests were performed by firing the mounted target sandwich plate to strike the spherical nosed end of the pressure bar. With the aforementioned inversed perforation testing setup, the piercing strain impulse can be record by the strain gauges cemented in the pressure bars. The piercing force and velocity time history are calculated by the following equation. F( t ) Sb E b H( t ) v ( t ) C b H( t )

Where Sb, Eb, Cb are respectively the cross sectional area,

the elastic modulus and elastic wave speed of the pressure bar. The velocity of the launched sandwich plate before the strike is also available and is measured with two optical barriers. With this initial impact velocity, it is possible to estimate the piercing displacement. Indeed, the sandwich plate mounted on the projectile is decelerated by the piercing force measured by pressure bar. In this way, a corresponding force-displacement diagram can be drawn, which make possible a quantitative comparison between the static and impact piercing behaviour. These impact piercing results are compared with quasi static tests. A notable enhancement of peak load under impact loading is found (Figure 10). Such enhancement is also a structural effect because both the skin plates and the core foam (cymat foam) are rate insensitive.

56

H. Zhao, I. Nasri and Y. Girard

REFERENCES Banhart, J., Manufacture, characterisation and application of cellular metals and metal foams. Progress in Material Science 46 (2001), 559-632. Calladine, C.R., English, R.W, 1984. Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure. International Journal of Mechanical Science 26 (1984), 689-701. Deshpande, V.S, Fleck, N.A, High strain rate compression behaviour of aluminium alloy foams. International Journal of Impact Engineering 24 (2000), 277-298. Gibson, L.J, Ashby, M.F, 1988. Cellular Solids. Pergamon Press. Hauser F. E, Techniques for measuring stress-strain relations at high strain rates. Experimental Mechanics 6 (1966), 395-402. Hanssen, A.G, Hopperstad, O. S., Langseth, M, Ilstad, H., Validation of constitutive models applicable to aluminium foams, International Journal of Mechanical Sciences 44(2002), 359-406. HooFatt M.S, Park K.S., Perforation of sandwich plates by projectiles. Composites: Part A: applied science and manufacturing 31(2000), 889-899. Hopkinson, B, A method of measuring the pressure in the deformation of high explosives or by the impact of bullets. Phil. Trans. Roy. Soc., A213 (1914), 437-452. Klintworth, J.W., Stronge,. W.J, Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs. International Journal of Mechanical Science 30 (1988), 273-292. Kolsky, H., An investigation of the mechanical properties of materials at very high rates of loading. Proceeding of Physical Society B62 (1949), 676-700. Radin, J. Goldsmith W., Normal projectile penetration and perforation of layered targets. International Journal of Impact Engineering 17(1988), 229-59. Reid, S.R., Peng, C., Dynamic uniaxial crushing of wood. International Journal of Impact Engineering 19 (1997), 531-570. Vural M., Ravichandran G., Dynamic response and energy dissipation characteristics of balsa wood: experiment and analysis, International Journal of Solids and Structures 40 (2003), 2147-2170. Zhao, H., Gary, G., A three dimensional analytical solution of longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. Application to experimental techniques. Journal of Mechanics and Physics of Solids 43 (1995), 1335-1348. Zhao, H., Gary, G., On the use of SHPB techniques to determine the dynamic behaviour of materials in the range of small strains. International Journal of Solids and Structures 33 (1996), 3363-3375. Zhao, H., Gary, G., A new method for the separation of waves. Application to the SHPB technique for an unlimited measuring duration. Journal of Mechanics and Physics of Solids 45 (1997), 1185-1202. Zhao, H., Gary, G., Crushing behaviour of aluminium honeycombs under impact loading. International Journal of Impact Engineering 21 (1998), 827-836. Zhao, H., 2004, Cellular materials under impact loading, IFTR-AMAS Edition, Warsaw, Poland. ISSN 1642-0578. Zhao, H. and Abdennadher, S., On the strength enhancement under impact loading of square tubes made from rate insensitive metals, International Journal of Solids and Structures 41(2004) 6677-6697. Zhao, H Nasri, I. and Abdennadher S., An experimental study on the behaviour under impact loading of metallic cellular materials, International Journal of Mechanical Science 47 (2005) 757-774.

INFLUENCE OF DELAMINATION ON THE PREDICTION OF IMPACT DAMAGE IN COMPOSITES Alastair Johnson, Nathalie Pentecôte, Hannes Körber German Aerospace Center (DLR), Institute of Structures and Design, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany

Abstract:

This paper describes recent progress on materials modelling and numerical simulation of fibre reinforced composite panel structures subjected to transverse impact loads. A model of impact damage is presented which includes the contribution from delamination by introducing cohesive interfaces in the laminate model, which may degrade or fail when a delamination failure energy criterion is reached. A continuum damage mechanics model is applied to model the delamination failure at these interfaces which has been implemented in a commercial explicit finite element (FE) code. A comparison of contact load pulses and failure modes from numerical simulations and drop tower impact tests shows good agreement for the prediction of delamination damage and penetration under a range of impact conditions. Results show the importance of including delamination damage in the impact models.

Key words:

Polymer composites, impact simulation

1.

impact behaviour, damage mechanics, delamination,

INTRODUCTION

To reduce development and certification costs for composite aircraft structures, efficient computational methods are required by the industry to predict structural integrity and failure under dynamic loads, such as crash and impact. By using meso-scale models based on continuum damage mechanics (CDM), proposed by Ladevèze1 and Allix2, it is possible to define materials models for FE codes at the structural macro level which embody the salient micromechanics failure behaviour. CDM provides a framework within which in-ply and delamination failures may be modelled. In previous 57 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 57–66. © 2006 Springer. Printed in the Netherlands.

A. Johnson, N. Pentecôte and H. Körber

58

work3 a delamination model was obtained by applying the CDM framework to the ply interface with fracture mechanics concepts introduced to relate the total energy absorbed in the damaging process to the interfacial fracture energy4. The delamination model has been implemented into a commercial explicit FE crash and impact code PAM-CRASH™ 5 using stacked shell elements and contact interfaces, which may separate when the interface fracture energy condition is reached, to model delamination failure. The paper concentrates on code validation of the delamination interfaces by comparing in some detail test data from a series of drop tower impact tests on UD cross-ply composite plates with a steel impactor at low velocities in the range 2.52 – 4.44 m/s. FE simulation results are presented, which simulate numerically the observed impact failure modes and failure progression. It is shown that when simplified layered shell models are used with no delamination failure the contact impact forces are significantly overestimated. Applying the stacked shell model with delamination interfaces gives good correlation with impact test data, since delaminations near the impactor reduce local shell bending stiffness and hence peak contact forces. These results give experience with the stacked shells and delamination model, which has now been used in prediction of damage under gas gun impacts from soft bodies (synthetic birds) and foreign object damage (FOD).

GIC

V33m unload/ reload

Stress

u3O

Displacement

u3m

Figure 1. Idealised mode I interface stress-displacement function V33 - u3.

2.

COMPOSITES FAILURE MODELLING

2.1

Delamination model

Delamination failures occur in composite structures under impact due to local contact forces in critical regions of load introduction and at free edges. They are caused by the low, resin dominated, through-thickness shear and

Influence of Delamination on the Prediction of Impact Damage

59

tensile properties found in laminated structures. In composites delamination models2, the thin solid interface is modelled as a sheet of zero thickness, across which there is continuity of surface tractions but jumps in displacements. The equations of the model are given here for the case of mode I tensile failure at an interface. Let V33 be the tensile stress applied at the interface, u3 the displacement across the interface, and k3 the tensile stiffness. Following Crisfield4 an elastic damaging interface stressdisplacement model is assumed: V33 = k3 (1- d3) u3,

d3 = c1 (1 - u30 / u3 ) , o fr u30 d u3 d u3m

(1)

with tensile damage parameter d3, and c1 = u3m / (u3m - u30 ). It can be verified that with this particular choice of damage function d3, the stressdisplacement function has the triangular form shown in Fig. 1, and u30 , u3m correspond to the displacement at the peak stress V33m and at ultimate failure. The damage evolution constants are defined in terms of V33m and GIC, the critical fracture energy under mode I interface fracture by u30 = V33m / k3 and u3m = 2GIC / V33m. From these expressions it can be shown that the area under the curve in Fig. 1 is equal to the fracture energy GIC. This interface model therefore represents an initially elastic interface, which is progressively degraded after reaching a maximum tensile failure stress V33m so that the mode I fracture energy is fully absorbed at separation. For mode I interply failure the interface energy GI , defined as u3

GI

³V

33

du 3

(2)

0

is monitored and, if this is found to exceed the critical fracture energy value GIC, then the crack is advanced. For mode II interface shear fracture a similar damage interface law to (1) is assumed, with equivalent set of damage constants, u130 , u13m and critical fracture energy GIIC.. In general there will be some form of mixed mode delamination failure involving both shear and tensile failure. This is incorporated in the model by assuming a mixed mode failure condition, which for mode I/mode II coupling could be represented by the interface failure envelope:

§ GI ¨¨ © G IC

n

· § G II ¸¸  ¨¨ ¹ © G IIC

· ¸¸ ¹

n

eD d 1 (3)

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A. Johnson, N. Pentecôte and H. Körber

where GI and GII are the monitored interface strain energy in modes 1 and 2 respectively, GIC and GIIC are the corresponding critical fracture energies and the constant n is chosen to fit the mixed mode fracture test data. Typically n is found to be between 1 and 2. Failure at the interface is imposed by degrading stresses when eD < 1 using (1) and the corresponding shear relation. When eD = 1 there is delamination and the interface separates.

2.2

Composites ply and laminate model

The composite laminate is modelled by layered shell elements or stacked shells with a contact interface which may fail by delamination. The shells are composed of composite plies which are modelled as a homogeneous orthotropic elastic damaging material whose properties may be degraded on loading by microcracking prior to ultimate failure. This paper uses the PAMCRASHTM 5 FE code, which contains several materials models and special elements for laminated composite materials, including a recently implemented CDM mesoscale model for UD and fabric plies. The simulation results presented here are based on the ‘bi-phase’ ply model for UD plies in which it is supposed that damage evolution is dependent on strain invariants. The simplifying assumption is made that the composite ply has a single damage function d acting on the stiffness constants which is assumed to be a function of the second strain invariant HII, or the effective shear strain. Because of this shear strain dependence the damage parameter degrades the matrix dominated shear properties more strongly than the fibre properties. In addition the code allows different damage functions in fibre tension and compression. Uniaxial stress-strain curves are assumed to have a damage evolution equation bilinear in strain, as indicated schematically in Fig. 2, in which there are two constants d1 and du determined from ply test data. In typical uniaxial stress-strain curves Hi is the strain at the onset of initial damage, H1 the strain at peak failure stress, and Hu is a limiting strain above which the stress is assumed to take a constant value Vu. The constant d1 is related to the departure from linearity at the first 'knee' in the stress-strain curves, and is thus small in tension, whilst the constant du determines a residual stress value Vu. In the layered shell element the stiffness properties of the UD plies are degraded in this way as the shear strain invariant increases. Eventually a damaged shell element is eliminated from the computation when the shear strain invariant reaches a pre-defined critical value. Explicit FE methods have proved successful for the analysis of dynamic, non-linear problems, particularly where contact plays an important role. The commercial explicit code PAM-CRASHTM 5 uses a bilinear 4-node quadrilateral isoparametric shell element with uniform reduced integration in

Influence of Delamination on the Prediction of Impact Damage

61

bending and shear. Hour glass control is applied to compensate for the under-integration. A central difference explicit integration scheme is used in time with geometrical nonlinearities accounted for in an updated Lagrangian scheme with co-rotational description. A Mindlin-Reissner shell formulation is used with a layered shell description to model a composite ply, a sublaminate or the complete laminate. The layered shells contain one integration point per ply, so that at least 4 plies are required in a layered shell for the correct bending stiffness.

Figure 2. Schematic fracturing damage function and corresponding stress-strain curve5.

The delamination model of § 2.1 is implemented in the PAM-CRASHTM code, with the laminate modelled as a stack of shell elements. Each ply or sublaminate ply group is represented by a set of layered shell elements and the individual sublaminate shells are connected together using a contact interface with an interface traction-displacement law. The interface is a contact constraint not an interface element in which a penalty force procedure is used to compute contact forces between adjacent shells. Contact may be broken when the interface energy dissipated reaches the mixed mode delamination energy criteria (3). This ‘stacked shell’ approach is an efficient way of modelling delamination, with the advantage that the critical integration timestep is relatively large since it depends on the area size of the shell elements not on the interply thickness. Full details of the implementation of the delamination model as a tied interface with failure between stacked shell elements is given in the paper by Greve and Pickett6. The paper shows validation studies in which mixed mode delamination tests were simulated by stacked shell elements with measured GIC and GIIC data for carbon/epoxy, giving very good agreement with test data.

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3.

VALIDATION OF IMPACT MODELS

3.1

Low velocity impact tests on composite plates

In order to provide test data for validation of the impact models with UD laminates where there is interaction between UD ply and delamination failures, a set of low velocity impact tests were carried out on cross-ply UD carbon/epoxy plates within the EU HICAS project7. During the test the force pulse and the vertical displacement at the impact head are measured. The plates tested had dimensions 300 mm x 300 mm. The composite was UD carbon/epoxy with cross-ply lay-up [0/90]20 giving a nominal plate thickness of 5 mm. A square steel frame with an inner diameter of 250 mm x 250 mm was used as support frame. A steel sphere with a diameter of 50 mm was used as impactor head, which was attached to the mass and guides giving a total impactor mass of 21 kg. Three tests were carried out with different normal impact velocities: 2.52 m/s, 3.11 m/s and 4.44 m/s giving impact energy levels in the range 67 - 207 J so that extensive plate damage was observed with failure modes ranging from delamination to complete penetration. Plate CF0901 was impacted at 4.44 m/s which led to complete fracture of the plate and impactor penetration. At 3.11 m/s, the impactor rebounded in Plate CF0902 which caused rear face cracking (Fig. 3a) with extensive delamination. The impactor also rebounded in Plate CF0903 at 2.52 m/s causing minor rear face damage with less delamination (Fig. 3b).

Figure 3. Rear face damage - a) V0 = 3.11 m/s

b) V0 = 2.52 m/s.

Influence of Delamination on the Prediction of Impact Damage

63

Ultrasonic C-scans were carried out to determine the extent of delamination damage and Fig. 4 compares the size of the delaminated regions for the two rebound tests, showing a greater delamination area after impact at higher velocity. The results obtained are thus an excellent source for validation of the impact simulations since distinct failure modes are observed.

Figure 4. C-scan data showing delamination a) V0 = 3.11 m/s

3.2

b) V0 = 2.52 m/s.

FE simulation of composite plate impact damage

For the simulation a stacked shell laminate model is used with 40 plies. In order to reduce CPU time the laminate has been simplified by using eight sub-laminates of layered shells connected by seven delamination interfaces. The composite plate is simply supported over the frame and impacted at its centre with the sphere, which is modelled here as a rigid impactor. Numerical results are presented for the three impact test cases. A half-plate is modelled with 14,400 layered shell elements. Ply material parameters were determined from the HICAS7 material tests programme, with the delamination data for GIC and GIIC reported by Greve and Pickett6. Fig. 5 shows the simulated deformation plot for the impact velocity of 4.44 m/s, with contours of the ply damage parameter. The simulation predicts well the complete penetration of the plate. Nevertheless, the observed fracture of the plate into 4 pieces was not simulated numerically. The second simulation refers to the lowest impact velocity of 2.52 m/s, which led in the test to a rebound failure mode. Fig. 6 shows the simulated damage plots after the impact event. At this lower impact velocity the impactor rebounds and

64

A. Johnson, N. Pentecôte and H. Körber

the plate has some fibre cracking. The rebound was successfully modelled in the FE simulation. The size of delamination predicted in Fig. 7 is also found to be in good agreement with C-scan tests in Fig. 4b. However, the characteristic shape of the delamination contour was not reproduced since crack propagation inside the plies is not available in the ply model.

Figure 5. FE-model of plate CF0901 - Damage after 15 ms.

Figure 6. FE-model of plate CF0903 - Damage after rebound at t = 20 ms.

Figure 7. FE-model of Plate CF0903 - Delamination interface damaged in the centre area near symmetry after rebound at t = 20 ms.

Influence of Delamination on the Prediction of Impact Damage

65

Figs. 5 to 7 demonstrate that PAM-CRASH with the delamination model can successfully predict the failure modes in impact, by penetration or rebound, with convincing contour plots of ply damage and delamination. However to validate the code, quantitative comparisons of impact loads with measured test data are required. The measured load pulse at V0 = 3.11 m/s is plotted in Fig. 8 and compared with simulations using the ‘degenerated’ biphase model with a single layered shell (without delamination), and with 8 sublaminate stacked shells (with delamination).

Figure 8. Comparison of the contact forces for CF0902 between test and simulations including results from a FE 1-plate and FE 8-plate model.

The single layered shell model over-predicts the measured failure loads by a factor of three after delamination occurs. In contrast, the simulation with the 8-sublaminates including delamination model shows a significant decrease of the loads after the peak, which is much closer to the test data. The simulation demonstrates that delamination in the plate reduces the plate bending stiffness near the impactor which leads to lower loads so that the model without delamination gives poor agreement with test data. The results for the delamination model are encouraging and show that delamination effects are significant in plate impact and that these can be successfully modelled with the improved code. Modifications to the delamination failure initiation stresses and interface fracture energies in the model leads to further changes in the peak loads, showing that the impact simulations are sensitive to the interface parameters, which are not well documented for these materials. More experience and relevant test data are now needed to further improve the simulation results.

A. Johnson, N. Pentecôte and H. Körber

66

4.

CONCLUDING REMARKS

The paper discusses a materials failure model for UD composites laminates which includes both interply delamination and intraply damage which has been implemented in the dynamic FE code PAM-CRASHTM. Detailed code validations were successfully carried out on the application of the model to predict impact damage to cross-ply composite plates under low velocity impact. The FE simulations gave good predictions of failure modes, observed damage and measured impactor load-time pulses. In particular it demonstrated the importance of delamination damage during impact. Along with similar studies on fabric reinforced composite plates3, the results indicate the general validity of the mesoscale damage mechanics and delamination models, and the associated code developments. There are a number of issues to be solved before the methods may be applied with confidence to predict impact damage in composite aircraft structures. These include the number of delamination interfaces in the model, which introduces a mesh dependency. The models include ply and delamination failures which are not directly coupled, although micromechanics studies suggest this coupling to be important in UD laminates. In future work it is suggested that attention is given to including this effect by initiating interface failures in the stacked shells according to neighbouring ply damage parameters.

REFERENCES 1. Ladevèze P. Inelastic strains and damage. Chapt. 4 in Damage Mechanics of Composite Materials, R. Talreja (ed), Composite Materials Series, Vol 9, Elsevier, 1994. 2. Allix O, Ladevèze P. Interlaminar interface modelling for the prediction of delamination. Composites Structures, 22, 235-242, 1992. 3. Johnson AF, Pickett AK, Rozycki P. Computational methods for predicting impact damage in composite structures. Comp Sci and Tech, 61, 2183 –2192, 2001. 4. Crisfeld MA, Mi Y, Davies GAO, Hellweg HB. Finite Element Methods and the Progressive Failure Modelling of Composite Structures. Computational Plasticity – Fundamentals and Applications’, ed Owen DRJ et al., CIMNE Barcelona, 239-254, 1997. 5. PAM-CRASH™ FE Code, Engineering Systems International, F-94578 Rungis Cedex, France. 6. Greve L, Pickett AK. Delamination testing and modelling for composite crash simulation. Comp Sci Tech, in press. 7. HICAS: High Velocity Impact of Composite Aircraft Structures, CEC DG XII BRITEEURAM Project BE 96-4238, 1998 - 2000.

DEBONDING OR BREAKAGE OF SHORT FIBRES IN A METAL MATRIX COMPOSITE

Viggo Tvergaard Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

Abstract:

For aluminium reinforced by aligned, short SiC fibres, a cell-model is used to analyse a rather short fibre surrounded by smaller particulates. Competition between failure by fibre fracture or by fibre-matrix decohesion is considered for the longer fibre, which is surrounded by an array of smaller particulates that do not fail. In the numerical, axisymmetric model, a cohesive zone model is used to represent debonding, while fibre fracture is represented by a critical value of the average tensile stress on a cross-section.

Key words:

Plasticity, Fracture, Composite materials, Cohesive zone.

1.

INTRODUCTION

Micromechanical studies for metal matrix composites, by numerical cellmodel analyses, are an important tool for obtaining a parametric understanding of the effect of various material parameters, as such analyses allow for an accurate representation of the fibre shape, etc., thus leading to realistic stress and strain fields around the fibres (Needleman et al. [1]). Effects of fibre clustering or of different fibre sizes are important and this can be rather easily included in a planar model of a discontinuously reinforced metal, but planar models have the disadvantage that fibre geometries are not realistically represented. Full three-dimensional cell model analyses can represent realistic material geometries, but are computationally very heavy when damage evolution towards final failure is to be modelled. 67 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 67–76. © 2006 Springer. Printed in the Netherlands.

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68

Axisymmetric cell-models that allow for rather realistic fibre geometries have been used recently (Tvergaard [2, 3]) to represent some effects of clustering and different fibre sizes, and these models are further explored in the present study. Experimental studies of failure in metals reinforced by short brittle fibres (McDanels [4], Divecha et al. [5]) have shown that the reinforcement increases the stiffness and tensile strength, but also leads to poor ductility and low fracture toughness. Furthermore, observations of matching pairs of fracture surfaces for Al-SiC composites have demonstrated the occurrence of failure by debonding as well as by brittle fracture of elongated particles aligned with the tensile direction (Mummery and Derby [6], Zok et al. [7]). Several cell-model analyses have considered a characteristic material volume containing only a single fibre, e.g. see Nutt and Needleman [8], Tvergaard [9,10] and Finot et al. [11]. Three dimensional cell model analyses have been used to study ductile failure in the metal matrix between brittle fibres [12], also for cells containing different size fibres [13]. The axisymmetric models considering a number of different size fibres inside a cell are a generalization of the staggered fibre model in [9,10]. As in [3] the cell-model analysed contains a rather short fibre surrounded by smaller particulates. Competition between failure by fibre fracture or by fibre-matrix decohesion is considered for the longer fibre, while the particulates do not fail. In the present study an array of particulates is placed in two different ways relative to the longer fibre, and the difference in the failure evolution is analysed.

2.

PROBLEM FORMULATION

The metal matrix composites with different size brittle fibres are here studied by a numerical analysis of an axisymmetric unit cell model, as in Tvergaard [2,3]. On the curved side of the circular cylindrical cell for transversely staggered fibres, equilibrium and compatibility with the neighbouring cells is represented in an approximate manner. A neighbouring cell is identical to that analysed, but is rotated 180$ so that it points in the opposite direction (see Fig. 1). Compatibility and equilibrium in the axial direction are directly specified in terms of the axial edge displacements and nominal tractions. In the radial direction compatibility is represented by the requirement that the total cross-sectional area (consisting of an equal number of cross-sections of the two types of neighbouring cells considered) is independent of the axial coordinate. When U1 denotes the increase of the length of the cell, and U 2 is the radius increase at the centre of the cell (at x1 " c / 2 ) , the average

Debonding or Breakage of Short Fibres logarithmic

strains

in

the

H1 "n(1  U1 / " c ) and H 2

axial

69

and

transverse

directions

are

"n(1  U 2 / rc ) , respectively. The average

nominal stresses ¦ij are computed as the appropriate area averages of the microscopic nominal stress components on the surface (considering both the cell analysed and one of the neighbouring cells of opposite kind). The same set of boundary conditions also applies to a unit cell with several fibres along the axis, as long as the neighbouring cell is identical to that analysed, but is rotated 180$ so that it points in the opposite direction. Such two neighbouring cells are shown in Fig. 1, for a case where the cell contains two additional particulates. In the present studies the cells contain four additional particulates, with the length, radius and position specified by " i , ri and ai , for particulate No. i . The value of the total fibre volume fraction is given by

f

4 § 2 · 2 2  r " ¨ f f ¦ (ri " i ) ¸ / rc " c i 1 © ¹

(1)

The matrix material is taken to be described by J 2 flow theory with isotropic hardening. A convected coordinate, Lagrangian formulation of the field equations is used, in which gij and Gij are metric tensors in the reference configuration and the current configuration, respectively, with determinants g and G , and Kij ½( Gij gij ) is the Lagrangian strain

Figure 1. Axisymmetric unit cell containing three different short fibres. The neighbouring cell, rotated 180q , is indicated by dashed lines.

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tensor. The contravariant components W ij of the Kirchhoff stress tensor on the current base vectors are related to the components of the Cauchy stress G / g V ij . The finite strain generalization of J 2 tensor V ij by W ij flow theory gives an incremental stress-strain relationship of the form Wij Lijk "Kk " , where Lijk " is the tensor of instantaneous moduli. The uniaxial stress-strain behaviour is represented in terms of standard power hardening. Here, E is Young’s modulus, V y is the uniaxial yield stress, and n is the strain hardening exponent. Both fibre breakage and the possibility of fibre-matrix debonding are modeled for the fibre with length " f and radius rf . For convenience this fibre is modelled as rigid, while the other four fibres are modelled as elastic. It has been found for perfectly bonded SiC whisker-reinforced aluminium that predictions for elastic or rigid fibres differ rather little when plastic yielding has occurred, since the fibre elastic modulus is much higher than that of aluminium ( E f | 5.7 E A1 ) . Fibre breakage is taken to occur when the average tensile stress in the middle of the fibre (at x1 0 ) reaches a critical value V F (see also Tvergaard [10]). When the peak fibre stress reaches the critical value, the two fibre halves are allowed to separate, and the axial force on the fibre cross-section, V F S rf2 , is stepped down to zero, while the axial displacement of the rigid half-fibre grows positive. The debonding behaviour for the fibre with length " f and radius rf is specified in terms of a cohesive zone model proposed by Tvergaard [9], where the interface constitutive relations give the dependence of the normal and tangential tractions Tn and Tt on the corresponding components un and ut of the displacement difference across the interface. In purely normal separation (ut { 0) the maximum traction is V max , and total separation occurs at un G n , while in purely tangential separation (un { 0) the maximum traction is DV max , and total separation occurs at ut G t . Approximate solutions of the principle of virtual work are obtained by a linear incremental method, using a finite element approximation of the displacement fields. In the computations a fixed ratio is enforced, U V 2 / V 1 , of the average true stresses in the transverse and axial directions.

3.

RESULTS

Two different material geometries are considered here, as illustrated by the initial, undeformed meshes in Fig. 2. Both material geometries have " f / rf 5 , rf / rc 0.538 and " c / rc 18.001 , the total fibre volume fraction is f 0.13 , and the larger fibre accounts for one third of the fibre

Debonding or Breakage of Short Fibres

71

Figure 2. Initial meshes used, with particulates shown as hatched regions. (a) Material I. (b) Material II.

volume fraction. In addition, all the smaller particulates have the radii ri rf and the aspect ratio " i / ri 2.5 , so that the length of each particulate is one quarter the length of the larger fibre continuing across the symmetry plane at x1 0 . In material I , the locations of the four particulates in the unit cell are specified by a1 / " c 0.538 , a2 / " c 0.658 , a3 / " c 0.778 and a4 / " c 0.897 . In material II , the locations are a1 / " c 0.194 , a2 / " c 0.314 , a3 / " c 0.434 and a4 / " c 0.553 . The material parameters used here are chosen to represent the particular aluminium alloy 2124-SiC whisher-reinforced composite investigated by Christman et al. [14], that has also been modeled in previous studies of failure in MMC’s (Tvergaard [10]). For this material it was found that the uniaxial stress-strain curve of the matrix is reasonably well approximated by the power law with V y / E 0.005 and n 7.66 (i.e. V y 0.3GPa and E 60GPa ) , and the value of Poisson’s ratio is taken to be Q 0.3 ,

Figure 3. Stress-strain curves predicted for V F /Vy 12 , Vmax /V y . n 5.

5, n

7.66 and U

0

72

V. Tvergaard

while the value of Poisson’s ratio for the fibers is taken to be Q f 0.12 . Furthermore, in the debonding model the parameter values G n G t 0.02 rf and D 1 are used, while different values of the peak stress V max are considered. In the representation of failure by fibre breakage different values are considered for the critical value V F of the average tensile stress in the middle of the longest fibre. In Fig. 3 the critical stress values are taken to be V max / V y 5 and V F / V y 12 , the strain hardening exponent for the metal matrix is n 7.66 , and the figure shows computed stress-strain curves for overall uniaxial tension ( U V 2 / V 1 0 ) . For material II failure at the longest fibre occurs first by fibre breakage, which leads to a sudden reduction of the overall stress level, and subsequently fibre pull-out develops by debonding of the fibre-matrix interface near the fibre crack. Fig. 3 shows that at a later stage also debonding at the fibre end occurs for material II. For material I, failure initiates later than found for material II, and here debonding at the fibre end occurs first, while fibre fracture does not occur in the range analysed. Considering Fig. 2a and the staggered fibre array (Fig. 1), with the neighbouring unit cell rotated 180 degrees, it is seen that for material I the arrays of particulates in the neighbouring cells are adjacent to the longer fibre, so that they provide a shielding effect that reduces the stress levels around the longer fibre. For the same volume fractions material II has a different distribution of of the particulates, which does not provide such shielding. Thus, the differences between the curves in Fig. 3 illustrate an effect of fibre clustering, which is here obtained for realistic geometries of the fibres and particulates by using an axisymmetric model.

Figure 4. Stress-strain curves predicted for V / V 11 , V max/V y 5 , n F y

7.66 and U

0.

Debonding or Breakage of Short Fibres

73

Figure 5. Stress-strain curves predicted for V F / V y

n

7.66 and U

10 ,

V max / V y

5,

0.

In Figs. 4 and 5 the same computations have been repeated with the slight differences, that V F / V y 11 and V F / V y 10 , respectively. In both cases failure occurs earlier for material II than for material I, as in Fig. 3, due to the shielding effect of particulate clusteres in material I. Also, for both materials in Figs. 4 and 5 first failure occurs by fibre fracture, different from material I in Fig. 3, which shows the sensitivity to the slight change in the value of V F / V y . Once first failure has switched, the stress levels are reduced, so that material I in Fig. 4 does not show debonding at the fibre

Figure 6.

and either n

Stress-strain curves predicted for V F / V y

5 or n

10 .

12 , V max / V y

5,

U

0

74

V. Tvergaard

ends. For material II in Fig. 4, where the fibre fracture occurs earlier than in Fig. 3, shows a corresponding delay in the later occurrence of debonding at the fibre end, and in Fig. 5 this later debonding is not found at all in the range analysed. Fig. 6 considers the effect of a lower or a higher strain hardening behaviour, for materials otherwise identical to those considered in Fig. 3. Both for n = 5 and for n = 10 the shielding effect results in later onset of failure in material I than in material II, as in the previous figures. For n = 5 , where the stress levels are higher, both materials fail first by fibre fracture, and material II shows subsequent failure by debonding at the fibre ends. For n = 10 , where the stress levels are lower, both materials fail by debonding at the fibre ends, and no fibre fracture is found in the range analysed. The effect of an increased stress triaxiality is investigated in Fig. 7 by taking U V 2 / V 1 0.5 , with the material parameters identical to those in Figs. 3 and 6. The increased stress triaxiality gives higher stress levels, as has been found in earlier investigations if metal matrix composites, and this results in earlier fibre breakage than found in Fig. 3. After breakage, the average tensile stress in material I grows higher again, but then suddenly decays, as a matrix void around the fibre crack rapidly develops towards coalescence with neighbouring voids (see Fig. 8c). For material II, debonding at the fibre ends is predicted after fibre fracture, but although a void grows around the fibre crack (Fig. 8d), the stage of final coalescence has not been reached in the range analysed.

Figure 7. Stress-strain curves predicted for V F / V y and for increased stress triaxiality, U 0.5.

12 , Vmax / V y 5 , n

7.66 ,

Debonding or Breakage of Short Fibres

75

Four different examples of deformed unit cells are shown in Fig. 8, with contours of the maximum principal logarithmic strain drawn into the matrix material regions. The first figure, for material I, illustrates the situation at the end of the computation in Fig. 4, where fracture at the centre of the longer fibre has occurred and large shear strains have developed along the fibre sides while tangential debonding and subsequent fibre pull-out takes place. Fig. 8b shows material II at the end of the computation in Fig. 4, where first the longer fibre has broken in the middle and subsequently debonding has just occurred at the ends of this fibre. The two last figures show materials I and II, respectively, at the ends of the computations in Fig. 7, where an applied transverse stress gives an increased stress triaxiality. It is clearly seen, as discussed in relation to Fig. 7, that in Fig. 8c the void around the fibre breakage has grown large, approaching coalescence, whereas in Fig. 8d the corresponding void has grown much less, as the growth rate is slowed down by the occurrence of debonding at the fibre ends.

Figure 8. Contours of maximum principal logarithmic strain, for n 7.66 and V max / V y 5 . (a) Material I, V F / V y 11 , U 0 at H 1 0.143 . (b) Material II, V F / V y 11 , U 0 at H 1 0.104 . (c) Material I, V F / V y 12 , U 0.5 at H 1 0.083 . (d) Material II, V F / V y 12 , U 0.5 at H 1 0.086 .

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REFERENCES 1 A. Needleman, S.R. Nutt, S. Suresh, V. Tvergaard, in: Fundamentals of Metal Matrix Composites, edited by S. Suresh, A. Mortensen and A. Needleman (ButterworthHeinemann, Boston, MA, 1993), pp. 233-250. 2 V. Tvergaard, Debonding of short fibres among particulates in a metal matrix composite, Int. J. Solids Structures 40, 6957-6967 (2003). 3 V. Tvergaard, Breakage and debonding of short brittle fibres among particulates in a metal matrix, Materials Science and Engineerring A369, 192-200 (2004). 4 D.L. McDanels, Analysis of stress-strain, fracture, and ductility behaviour of aluminum matric composites containing discontinuous silicon carbide reinforcement, Metall. Trans. A 16, 1105-1115 (1985). 5 A.P. Divecha, S.G. Fishman, S.D. Karmarker, Silicon carbide reinforced aluminum-a formable composite, J. Met. 33, 12-17 (1981). 6 P. Mummery, B. Derby, The influence of microstructure on the fracture behaviour of particulate metal matrix composites, Mater. Sci. Engng. A135, 221-224 (1991). 7 F. Zok, J.D. Embury, M.F. Ashby, O. Richmond, in: Mechanical and Physical Behaviour of Metallic and Ceramic Composites, edited by S.I. Andersen et al. (Risø National Laboratory, Denmark, 1988), pp. 517-526. 8 S.R. Nutt, A. Needleman, Void nucleation at fibre ends in Al-Sic composites, Scripta Metallurgica 21, 705-710 (1987). 9 V. Tvergaard, Effect of fibre debonding in a whisker-reinforced netal, Mater. Sci. Eng. A125, 203-213 (1990). 10 V. Tvergaard, Model studies of fibre breakage and debonding in a metal reinforced by short fibres, J. Mech. Phys. Solids 41, 1309-1326 (1993). 11 M. Finot, Y.-L. Shen, A. Needleman, S. Suresh, Micromechanical modeling of reinforcement fracture in particulate-reinforced metal-matrix composites, Metall. Mater. Trans. 25A, 2403 (1994). 12 V. Tvergaard, Effects of ductile matrix failure in three dimensional analysis of metal matrix composites, Acta Mater. 46, 3637-3648 (1998). 13 V. Tvergaard, Three-dimensional analyses of ductile failure in metal reinforced by staggered fibres, Modelling Simul. Mater. Sci. Eng. 9, 143-155 (2001). 14 Christman, T., Needleman, A., Suresh, S., An experimental and numerical study of deformation in metal-ceramic composites, Acta Metall. 37, 3029-3050 (1989).

A MICROSCALE MODEL OF ELASTIC AND DAMAGE LONGITUDINAL SHEAR BEHAVIOR OF HIGHLY CONCENTRATED LONG FIBER COMPOSITES Stefano Lenci Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica della Marche, via Brecce Biance, 60131 Ancona, Italy, [email protected]

Abstract:

The elastic and damage longitudinal shear behavior of highly concentrated long fiber composites are studied by a micro-mechanical model. Attention is focused on the effects of circular cracks along the fiber-matrix interfaces on the overall properties of composites. The fracture is initiated by the touching of adjacent fibers, and develop according to the Griffith energy criterion. The longitudinal shear moduli Px, Py and Pxy, which are important in the case of non-uniform tractions applied to the fibers, are investigated, and their dependence on the damage (the fiber-matrix cracks) are determined. The statistical properties of the elastic behavior are detected by Monte Carlo simulations. The crack propagation is then investigated, and various evolution curves are presented. The elastic and the ultimate limits are finally determined and discussed in detail.

Key words:

Micromechanics; damage; composites; fracture; elastic behaviour

1.

INTRODUCTION

In unidirectional long-fiber composites a very high fiber volume fraction is usually employed to improve the mechanical performances, specially the longitudinal stiffness and strength1. Due to the high concentration, the fibers frequently come into contact with each other. At the touching point, a circumferential fiber-matrix crack is initiated2. This mechanism causes a damage on a microscopic scale, which represents one important limitation to the macroscopic performances of these composites, and stands as a serious drawback in their practical utilization. 77 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 77–87. © 2006 Springer. Printed in the Netherlands.

78

S. Lenci

Not all the elastic constants are equally influenced by the previous phenomenon. In fact, the longitudinal modulus Ez=Czzzz (z is the axis in the fibers direction, C is the tensor of elasticity, Cijhk their components - so that Vij=6hkCijhkJhk) is only scarcely modified by the considered cracks, while being strongly influenced by other damage micro-phenomena such as fiber breaking. The most influenced constants, on the other hand, are the in-plane moduli Ex=Cxxxx, Ey=Cyyyy and µxy=Cxyxy, which are of minor interest because for a proper use these materials should have only small loads normal to the fibers; and the longitudinal shear moduli µx=µxx=Cxzxz, µy=µyy=Cyzyz and µxy=Cxzyz, which are very important because they determine the elastic behaviour in the case of non-uniform traction on the fibers. Indeed, this is the mechanism through which the load is transferred to the fibers, so that they represent the key point in the actual load bearing capacity of these materials. Based on the previous considerations, a micromechanical model for the computation of µx, µy and µxy was previously developed by the author3. It relies on the assumptions (i) that each fiber touches all its six adjacent ones (maximum packing), and (ii) that the fibers are rigid, which allows to describe quite accurately the behavior of real highly concentrated fiberreinforced composites. These approximations, on the other hand, are balanced by the fact that the problem has been solved analytically, by means of the complex variable method applied to the underlying anti-plane elastic problem3. The previous work3 was mainly aimed at developing the model, and at obtaining the closed form solution. Only a preliminary, although systematic, analysis of the model behavior was performed, and some mechanical properties were emphasized by means of examples. They show a very rich response scenario of the model, and call for a deeper investigation of its characteristics, which constitutes the object of this paper. After summarizing the model and the elastic solution (Sect. 2), a statistical analysis is performed by a Monte Carlo simulation based on a random distribution of fiber-matrix cracks (Sect. 3). This is motivated by the observation that in practice the exact distribution of initial cracks is unknown and, usually, random. Thus, it is hoped that the results of this section will help in the interpretation of experimental data. The second important investigated issue (Sect. 4) is the failure limit, while in Ref. 3 only the elastic limits were considered. It is obtained by the systematic construction of the curves of evolution of the damage (crack propagation), and provides information on the ultimate load bearing capacity of the composite, and on its dependence on the load direction. Furthermore, all of the six sub-unit cells are here considered, while in Ref. 3 only one is

Elastic and Damage Behavior of Long Fiber Composites

79

dealt with for the sake of simplicity. Thus, more general results are herein obtained, which add to and extend the previous ones3.

2.

THE MICROMECHANICAL MODEL

The considered mechanical model has been developed and illustrated in Ref. 3, which is referred to for further details. Here only the aspects important for the present work are summarized. The representative volume element (RVE) is an hexagonal unit cell (Fig. 1a), which repeats periodically, and which is constituted by six triangular sub-unit cells differently oriented with respect to the central fiber (Fig. 1b).

Figure 1. (a) The hexagonal unit cell and (b) the circular triangular sub-unit cell.

As the fibers are rigid, only the matrix in the circular triangle of Fig. 1b deforms elastically. The corresponding anti-plane elastic problem taking into account fiber-matrix cracks in neighborhoods of the fibers touching points Į, ȕ and Ȗ is shown in Fig. 1b. It has been solved in closed form by the complex variable method in Ref. 3 by means of an appropriate conformal transform. The elastic moduli corresponding to a given crack configuration are computed by assuming that the elastic energy E stored in the unit cell equals to that of the homogenized (anisotropic) material. They can be expressed in the form µij=µm[ Pˆ ij(23)+ Pˆ ij(34)+ Pˆ ij(45)+ Pˆ ij(56)+ Pˆ ij(67)+ Pˆ ij(72)]/6 (µm is the shear modulus of the matrix), which highlights the contribution of each subunit cell. In Ref. 3 only the effects of Pˆ ij(23) are considered in the examples. To study the crack propagation, i.e. the damage growth, the classical linear elastic fracture mechanics is employed. The energy release rate is given by4 Gk=í˜E/˜sk, where sk=Rijk is the length of the k-th fissure, and can be expressed in the form

S. Lenci

80 wPˆ y wPˆ x P ª Gk= m «J xz  J yz F 12 R «¬ wM k wM k

2

º § wPˆ xy · ¸¸ . » , Ȥ= sign¨¨ © wM k ¹ »¼

(1)

According to the Griffith5 criterion, the k-th crack does not propagate if

J xz

wPˆ x wM k

 J yz F

wPˆ y wM k

 $4U 2U 3  '4U 2  (*$  %' )U 3U 4 @¾

(10)

where: $ >(U 1  U 2 )(1  Q 2 )  2U 1 (1  2Q 2 )@E1  2U 2 (1  2Q 1 ) E 2 % 3(U 1  U 2 )(1  Q 2 ) E1 *

E 2U 2 >(1  U 4 )(1  Q 3 )  2(U 1  U 2 )(1  2Q 3 )@  E 3U 3 >U 1 (1  Q 2 )  2(U 1  U 2 )(1  2Q 2 )@

'

3E 3U 3U 1 (1  Q 2 )

=

3E 2 U 2 (1  U 4 )(1  Q 3 )

+

E 3U 3 >(1  Q 4 )  2(1  U 4 )(1  2Q 4 )@  E 4U 4 >(U 1  U 2 )(1  Q 3 )  2(1  U 4 )(1  2Q 3 )@

4 3E 4U 4 (U 1  U 2 )(1  Q 3 )

4.

EXPERIMENTAL WORK

The specimens used in this work have as matrix a system based on a diglycidil ether of bisphenol A resin (Epikote 828) as prepolymer, with an epoxy equivalent 185-192, a molecular weight between 370 and 384 and a viscocity of 15000 cP at 25oC. As curing agent, 8% triethylenetetramine hardener per weight of the epoxy resin was employed. The test pieces were machined from each casting. The density was measured and compared with theoretical values given by: Uc

U1U 1  U 2U 2  U 3U 3  U 4U 4

(11)

The Elastic Modulus and the Thermal Expansion Coefficient

111

In order to measure the fracture stress, fracture strain and the elastic modulus of the particulate composite and the matrix material the tensile experiments were carried out with an Instron-type testing machine at the room temperature. Four specimens of each material were tested at a rate of extension of 0.2 mm/sec. The specimens were of dogbone type with dimensions at the measuring area 50x10-3x20x10-3x9x10-3 m3 and of a total length of 150x10-3 m. In order to obtain the stress-strain diagrams for each material strain gauges (KYOWA type, gauge factor k=1.99) were located on each specimen to measure the strains. In order to measure the thermal expansion coefficient of the material square shaped samples 5x10-3 m and 2-3x10-3 m thick were cut from each volume fraction and tested on a Dupont 990 thermomechanical analyzer. The test piece for each material was loaded at ambient temperature and subsequently the temperature was measured at a constant rate. A minimum of three samples per volume fraction were tested during the thermal experiments at a heating rate of Hr=10 oC/min. Each value on the diagram is the mean value of the obtained results. The properties of the constituent materials which where used during the theoretical calculations are given in Table 1. Table 1. Properties of the materials used

Parameter Elastic Modulus, E (N/m2) Poisson ratio, Ȟ Density, ȡ (gr/cm3) Thermal expansion coefficient,

5.

D (oC-1)

Iron 210x109 0.29 7.80 15x10-6

Epoxy resin 3.5x109 0.36 1.19 65.26x10-6

CONCLUSIONS

1. The distribution of the neghbouring inclusions influences the thermomechanical constants of the composite. 2. The proposed model presents restrictions concerning the filler content and it is valid for medium volume fractions (Uf=0.20 y 0.35) given that their maximum values are Uf=0.50 y 0.60. 3. Different theoretical values were obtained from the various combinations of the proposed model. However, only the values derived from the first variation are in good agreement with experimental results whereas those derived from the third variation show large discrepancies especially for higher filler volume fractions. 4. Among the other theoretical expressions the Counto model [15] yields values which show the best agreement with those derived from the proposed model as well as the experimental results for the elastic

112

E. Sideridis, G.A. Papadopoulos, V.N. Kytopoulos and T. Sadowski modulus and Tummala-Friedberg [24] and expressions for the thermal expansion coefficient.

Fahmy-Ragai

[21]

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

L.E. Nielsen, Mechanical Properties of Polymers and Composites, 2, Marcel Dekker Inc., New York, 1974. E.H. Kerner, Proc. Phys. Soc., B69, 808 (1956). Z. Hashin and J. Shtrikman, J.Mech. Phys. Solids, 11, 127 (1963). Z. Hojo, W. Toyoshima, M. Tamura and N. Kawamura, Polym. Eng. Sci., 14, 604 (1974). H. Alter, J. Appl. Polym. Sci., 9, 1525 (1966). W.M. Baldin, Acta Mech., 6, 141 (1958). S.K. Bhattacharya, S. Basu and S.K. De, J. Mat. Sci., 13, 2109 (1978). G. Landon, G. Lewis and G.F. Boden, J. Mat. Sci., 12, 1605 (1977). Y. Benveniste, Mech. Mater., 4, 197 (1985). Z. Hashin, Mech. Mater., 8, (1990). A. Einstein, Über die von Molekularkinetischen theorie der Warme geförderten Bewegung von in Ruhenden, (1911). Flussigkeiten suspendten Teilchen, Ann Physic, 17,549 (1905). Eine neue Bestimmung der Molekuldimensionen, Ann Physic, 19, 289 (1906). Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Molekuldimensionen, 34, 591 (1911). B. Paul, Trans. Miner. Inst. Mech. Eng., 36, 218 (1960). G. Guth, J.Appl. Phys., 16, 20 (1945). H.M. Smallwood, J. Appl. Phys., 15, 758 (1944). R. Counto, Mag. Concr. Res., 16, 129 (1964). K. Takahashi, M. Ikenda, K. Harakawa and K. Tanaka, J. Pol. Phys. Ed., 16, 415 (1978). R.A. Schapery, J. Comp. Mat., 2, 380 (1968). W.D. Kingery, J. Amer. Ceram. Soc., 40, 351 (1957). G. Arthur and J.A. Coulson, J. Nucl. Mater., 13, 242 (1964). J.P. Thomas, General Dynamics, Fort Wporth, Tex., AD 287-826 (1960). A.A. Fahmy and A.I. Ragai, J. Appl. Phys., 41, 5108 (1970). T.T. Wang and T.K. Kwei, J. Pol. Sci., A-2, 7, 889 (1969). A. Malliaris and D.T. Turner, J. Appl. Phys., 42, 614 (1971). R.R. Tummala and A.L. Friedberg, J. Appl. Phys., 11(13), 5104 (1970). L. Nicolais, Pol. Eng. Sci., 11, 194 (1971). M. Schrager, J. Appl. Polym. Sci., 22, 2379 (1978).

PREDICTION OF CRACK DEFLECTION AND KINKING IN CERAMIC LAMINATES D. Leguillon 1*, O. Cherti Tazi 1, E. Martin 2 1

LMM, CNRS UMR7607, Université P. et M. Curie, 4 place Jussieu, 75005 Paris, France; 2 LCTS, CNRS UMR 5801, Université Bordeaux 1, 3 allée de la Boétie, 33600 Pessac, France. Abstract:

The arrangement of ceramic layers in laminated structures is an interesting way to enhance the flaw tolerance of brittle ceramic materials. The interfaces are expected to deflect cracks, increasing the fracture energy of the laminate compared to a monolithic material and thus raising the toughness. The crack deflection has been studied in a previous paper, the analysis focuses now on the crack kinking out of the interface. The criterion for kinking prediction relies on a two-scale analysis taking into account the laminated structure of the material. It can be written in terms of a single material parameter: the volume fraction of pores. Crack deflection is promoted only for very high values of the porosity. Predicted values agree satisfactorily with experiments. This porosity is generally high enough to ensure that the crack remains within the interface after deflection without kinking out. Then, the final ruin of the structure can occur only by successive onsets of new cracks.

Keywords:

fracture; toughness and toughening; ceramic laminates; crack kinking, porous interlayers.

1.

INTRODUCTION

The arrangement of ceramic layers in laminated structures is an interesting way to enhance the flaw tolerance of brittle ceramic materials.1-6 The interfaces are expected to deflect cracks, increasing the fracture energy of the laminate compared to a monolithic material and thus raising the toughness. Next the deflected cracks must remain within the interface and not kink out in order to delay the final ruin of the structure. Laminates are elaborated by alternating dense and porous layers (≈20 layers) of the same material, SiC or B4 C, in order obtain a good 113 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 113–122. © 2006 Springer. Printed in the Netherlands.

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D. Leguillon, O.C. Tazi and E. Martin

chemical compatibility between the laminas and almost no thermal residual stresses. Porosity, in the porous layers, is achieved by organic particles which are burned out during the elaboration process. In this context, the goal is to predict the volume fraction of pores, in the porous layer, required first to cause crack deflection and next to avoid kinking out. The first part of the analysis has been proposed in a recent paper 7 and the present analysis focuses on the second mechanism. The criterion derives from an energy balance and can be written in terms of two relevant material parameters: the dense and porous Young’s modulus and toughness ratios. A unique function depending on the volume fraction of pores can be used to express the two above mentioned ratios.7 Assuming a cubic lattice of spherical voids, the parameters of the porous ceramic depend linearly on the porosity and vanish at percolation of pores ( V = π / 6 ≈ 0 .52 ): 6V (1) E p = H (V ) Ed ; G cp = H (V )Gdc with H (V ) = 1 − π where Gcp and Gdc and Ed and E p are the toughness and Young's modulus respectively of the porous and the dense ceramics. The Poisson's ratio ν is kept unchanged (ν = 0. 16 , see Leguillon et al. 4 for an analysis of the minor role of the Poisson's ratio). As a consequence, the criteria can be rewritten in term of a single parameter: the volume fraction of pores V .

Figure 1. Different failure scenarios for the 3 -point bending specimen: (a) initial notched specimen, (b) crack growth through the layers, (c) deflected crack, (d) interface crack growth, (e) crack kinking out of the interface, (f) new crack onset along (or close to) the symmetry axis of the specimen.

Prediction of Crack Deflection and Kinking

115

The analyses of crack deflection and kinking by interfaces are generally based on two models due to He and Hutchinson. 8,9 Both are carried out in an unbounded domain made of two elastic materials filling two half spaces. In the first model, the primary crack lies in one material and impinges on the interface and the competition between deflection and penetration is studied. In the second model, the primary crack is along the interface and the ability of the crack to kink out of the interface is examined. The two criteria involve the toughness of the materials and of the interface. Nevertheless the laminated environment of the crack tip is ignored in these approaches. In the present approach, the laminated structure is taken into account through a two-scale analysis as explained in the next section and different scenarios of failure are examined as shown in figure 1. For the cases corresponding to figures 1b and 1c, it has been shown in a recent paper by Leguillon et al.4 first that the deflection cannot occur when the primary crack impinges on an interface between a dense and a porous layer, as expected; and second that deflection can occur at the porous dense interface only for high porosities (V > 0 .4 ) as observed in the experiments. 2.

THE MATCHED ASYMPTOTICS

The model is based on a two-scale analysis, the small parameter being the layers thickness e . At the macro scale the laminated microstructure is ignored, as a first approximation, the whole laminate is treated as a homogeneous material. It is homogenized using a rule of mixture for simplicity, since more sophisticated homogenization processes do not bring significant differences in the final results.10 The Poisson's ratio ν is unchanged and the homogenized Young's modulus is denoted Eh . There is an extensive horizontal deflected crack (figure 1d) and the two symmetric crack tips undergo the classical singular fields: U 0 ( x1 , x2 ) = U 0 (0, 0) + k1 r u1 (θ ) + k2 r u 2 (θ ) + ... (2) The constant term Ct is present for consistency reasons but plays no role, x1 and x 2 stand for the Cartesian coordinates and r and θ for the polar ones. The coefficients k j ( j = 1,2 ) are the modes 1 and 2

stress intensity factor and u j (θ ) denotes the corresponding angular shape function.

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Considering now a small horizontal crack extension l or a small vertical kink, the perturbed solution is expressed as a correction brought to the initial term: U l(x1 ,x2 ) = U 0(x1,x2 ) + small correction (3) The small correction is assumed to vanish as l → 0 . The micro scale is obtained by stretching the domain around the primary crack tip by 1 / e , e being the layer thickness e ≈ ( 100µm). Considering the limit e → 0 , the problem is now settled in an unbounded domain, so-called inner domain. In order to have tractable computations, this domain is artificially bounded at a large distance r ∞ >> 1 , where r ∞ is the radius of the artificial boundary and 1 the dimensionless stretched thickness of the la yers. Moreover, only few (3) dense and porous layers are kept in the vicinity of the primary crack tip, the remaining part being replaced by the homogenized material. The validity of this simplification is discussed in Leguillon et al. 4

Figure 2. Schematic view of the i nner (unbounded) stretched domain, only 3 layers are kept around the crack tip in the model, the remaining layers are replaced by the homogenized material.

Using the change of variable (stretching) yi = x i / e ( ρ = r / e ), U l can be expanded as (near field): l

l

U ( x1 , x 2 ) = U ( ey1 , ey 2 ) = Ct + k 1 e W 1 ( y1 , y 2 , µ ) + k 2 e W 2 ( y1 , y 2 , µ ) + ... (4)

117

Prediction of Crack Deflection and Kinking

where µ = l / e is the dimensionless crack extension length. Equation (4) is valid either for l = 0 (no crack extension) or l ≠ 0 . A far field mode mix parameter m is classically defined by: k m= 2 (5) k1

The functions W j are solutions to the following system:

− ∇ y .σ j  σj   j  σ .n  

=0 = C : ∇ yW

j

(6)

= 0 along the crack faces j W behaves like ρ u j (θ ) at infinity

The first equation is the balance of momentum (equilibrium). The symbol nabla ∇ y holds for derivatives with respect to y1 and y 2 . The second equation is the constitutive law, C is the elastic operator, it takes different values in the dense and porous layers and in the homogenized remaining part. The third equation expresses that the crack faces are free of traction. Finally, the last one is the matching condition with the mode j term involved in the far field (see eq. (2)).

3.

THE CRITERION FOR KINKING OUT

The leading term of the change in potential energy between the two states (prior to and following a crack extension) writes:10 δP = [k12 A11 ( µ ) + k1 k 2 ( A12 (µ ) + A21 ( µ )) + k 22 A22 (µ )] e d + ... (7) where d stands for the specimen depth (plane elasticity). The functions Aij depend on the Young’s modulus ratio E p / Ed (section 1). They are numerically derived from the displacement fields W j using a contour integral:10,11 j j Aij (µ ) = ψ W ( y1 , y2 , µ ) − W ( y1 , y2 ,0), ρ u i (θ ) (8) with 1 ψ (U ,V ) = ∫ (σ (U ).n.V − σ (V ).n.U ) dS 2 Γ Γ is any contour in the inner domain surrounding and located far from the crack tip and its extension, n is its normal pointing toward the crack tip. The integral ψ in eq. (8) is path independent for any U and V fulfilling equilibrium equations like (61 ) and (63 ).

(

)

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118

A necessary condition for the crack growth is a consequence of an energy balance: δP ≥ Gc l d ⇒ k12 A(µ , m) ≥ Gc (9) µ with A( µ, m) = A11 ( µ ) + m[ A12 (µ ) + A21 (µ )] + m 2 A22 ( µ) Here l d is the newly created crack surface and G c is the toughness in the direction of fracture. This expression must be considered twice, once for the kink (index kink in the following) and once for the growth along the interface (index grow). Moreover, the coefficients Agrow ij turn out to be independent of the crack increment and the coupling terms vanish: ~ Agrow (m) = Agrow11 + m2 Agrow 22 = Agrow(1 + m 2 ) (10)

~ 1− ν 2 with Agrow11 = Agrow 22 = Agrow = Eh Then, kinking is promoted if the above inequality (9) holds true for the kink but is wrong for the straight growth, it leads to: Akink ( µkink , m) Gc ≥ c d1 (11) µkink Agrow( m) G p (m) It is assumed here that the interface toughness is that of the porous material. Indeed, if the interface was stronger then the crack would grow within the porous medium at a short distance from the interface. Such a choice is also suggested in Fujita et al. 12 Moreover, the

emphasize is put on the dependence of this toughness Gcp on the mode mix parameter m . A phenomenological formulation due to Hutchinson and Suo13 can be adopted:  6V  Gcp (m) = Gcp 1 (1 + ξ m2 ) = Gdc1 1 − (1 + ξ m 2 ) (12) π   The toughness Gpc1 is the pure opening mode 1 toughness of the porous material. On the other hand we still use the opening mode 1 toughness Gdc1 for the failure of the dense layer in eq. (11). Now we make the following reasonable additional assumption: if the crack kinks in the next layer then it breaks it completely, l kink = e ⇒ µ kink = 1 . Finally the criterion takes the simplified form: g (m ) =

Akink (1, m) Gc ≥ c d 1 = K (V , m) = Agrow( m) G p (m)

1  6V  (1 + ξm2 )1 −  π  

(13)

Prediction of Crack Deflection and Kinking

119

The extreme values of the adjustable parameter ξ = 0 and ξ = 1 correspond either to an interface toughness insensitive to the mode mix or to failure insensitive to the mode 2. In the last case, the pure mode 2 fracture is inhibited. Figure 3 illustrates the criterion in eq. (13). The left side of the figure ( m < 0 ) is meaningless since it is associated with the kinked crack closure. For a low porosity ( V = 0 .21 ), it is clear on figure 3 (top) that kinking can occur for m ≈ 1 , whatever the mode mix dependence of the interface toughness. For a medium value of the porosity ( m = 0 .35 ), figure 3 (middle) shows that, choosing ξ = 0 leads to predict that no

Figure 3. Variations of the functions g and K in eq. (13) with the mode mix parameter m for 3 different porosities V. The function K eq. (13) is represented by the dotted and dashed lines corresponding respectively to ξ = 0 and ξ = 1 .

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120

kinking can occur while for ξ = 1 kinking can still take place for a mode mix parameter m ≈ 1 . Obviously ξ must be adjusted from experiments, in order to have a reliable prediction. For a large value of the porosity ( V = 0 .50 ), a high mode mix parameter, at least m ≈ 3 , is required to promote kinking out (figure 3 (bottom)). The case V = 0 (i.e. the homogeneous case) allows recovering the results of He and Hutchinson. 9

4.

THE ONSET OF A NEW CRACK

Two mechanisms are opposed to the crack growth along the interface: first, the energy release rate decreases with the crack length for a fixed applied load (leading to a stable crack growth); second, the mod mix parameter increases with the crack length (table 1) and so does the toughness as a consequence of eq. (12). Then, a continuous crack growth occurs only if the applied load increases enough to surmount them and to fulfil the Griffith criterion (see eqs. (9) and (10)): ~ G = k12 Agrow 1 + m 2 ≥ G cp ( m) (14)

(

)

Results are illustrated in table 1. A specimen with an aspect ratio 20 is considered. The deflected crack is located along the middle layer of the specimen and its length varies from 4% to 90% of the overall specimen length. The mode mix parameter m changes from 0.8 to 1.9. Within the same range, the applied load must be multiplied by a coefficient varying between 7 and 11 (depending on the extreme values of ξ from 0 to 1, see eq. (12)) to keep the crack growing, independently of the porosity in the porous layers. Table 1. Variation of the mode mix parameter m with the ratio l / L : crack length to overall specimen length. The two bottom lines show the required increase in the applied load ∆f to keep the interface crack growing from l / L = 0.04 to l / L = 0.90 .

l/L

0.04

0.10

0.25

0.50

0.75

0.90

m

0.81

0.82

0.85

0.94

1.18

1.90

∆f , ξ = 0 1.00

1.06

1.25

1.82

3.38

6.82

∆f , ξ = 1

1.07

1.28

1.95

4.07

11.38

1.00

Prediction of Crack Deflection and Kinking

121

The far field tension on the free surface of the specimen increases in the same proportion. Moreover, as a consequence of the homogenization process, the corresponding tension in the dense layer t d (near field) is: 1 (15) td = t 3V 1π This can give raise to a new crack onset (figure 1f) if the tension in the neighbouring dense layer reaches the critical strength of the dense material. The porosity intensifies this phenomenon as shown by eq. (15). The middle point has been selected for symmetry reasons, nevertheless, it is clear that the tension t keeps constant along the free edge on a wide domain (depending on the deflected crack length) and that the new crack can start at any point depending on some randomly distributed micro flaws. The energy condition for the onset of a new crack across the neighbouring dense layer writes: G = t 2 D e ≥ Gdc1 (16) The coefficient plays a similar role to A in eq. (9). There is no stress concentration and the asymptotic field to take into account is the uniform tension parallel to the free edge ( λ = 1 ). This condition competes with the crack growth along the interface eq. (14). Moreover a numerical computation shows that the relation between k1 and t turns out to be almost constant within the analysed crack length range: t = τ k1 (17) Then the condition that promotes a new crack onset derives from eqs. (14) and (16):  6 V  1 + ξ m2 F e 1 − (18) ≥1  π  1 + m2  With m = 1 .5 and the previously defined layer thickness e ≈100µm, ~ the constant F (depending on Agrow , D and τ ) being computed once for all, the above inequality predicts that, for ξ = 0 the crack remains in the interface if V ≥ 0 .35 and for ξ ≥ 0.5 if V ≥ 0 .48 . For ξ = 1 , a new crack appears whatever the porosity. Nevertheless, it is clear that the layer thickness e (equal to the crack length) plays an important role in eq. (18), there is a size effect.

D. Leguillon, O.C. Tazi and E. Martin

122

5.

CONCLUSION

The main conclusion to draw is that deflection is very difficult to promote by porous layers obtained by the addition of spherical pore forming particles. This prediction correlates well with experimental results: Reynaud et al.4 did not observe any extensive deflection in SiC below V ≈ 40 % and Tariolle et al. 5 found a significant deflection for a rather high value V ≈ 46 % in B4 C. Once an extensive deflected crack exists, three mechanisms can take place: the crack goes on growing along the interface, the crack kinks out of the interface in the next dense layer or the deflected crack stops and a new crack appear along the symmetry axis of the 3-point bending specimen. It is found that these two latter are inhibited provided the interface toughness is not too much sensitive to the mode mix parameter of loading. REFERENCES 1. K.S. Blanks, A. Kristofferson, E. Carlström, W.J. Clegg, Crack deflection in ceramic laminates using porous interlayers, J. Eur. Ceram. Soc., 1998, 18, 19451951. 2. W.J. Clegg, K.S. Blanks, J.B. Davis, F. Lanckmans, Porous interfaces as crack deflecting interlayers in ceramic laminates, Key Engng. Mater., 1997, 132-136, 1866-1869. 3. J.B. Davis, A. Kristoffersson, E. Carlström, W.J. Clegg, Fabrication and crack deflection in ceramic laminates with porous interlayers, J. Am. Ceram. Soc., 2000, 83(10), 2369-2374. 4. C. Reynaud, Thévenot, T. Chartier, J.L. Besson, Mechanical properties and mechanical behaviour of SiC dense-porous laminates, J. Eur. Ceram. Soc., 2004, in press, available on line. 5. S. Tariolle, C. Reynaud, F. Thévenot, T. Chartier, J.L. Besson, Preparation and mechanical properties of SiC-SiC and B4C- B4C laminates, J. Solid State Chemistry, 2004, 177, 487-492. 6. J. Ma, H. Wang, L. Weng, G.E.B. Tan, Effect of porous interlayers on crack deflection in ceramic laminates, J. Eur. Ceram. Soc., 2004, 24, 825-831. 7. Leguillon D., Tariolle S., Martin E., Chartier T., Besson J.L., Prediction of crack deflection in porous/dense ceramic laminates, J. Eur. Ceram. Soc., 2005, in press, available on line. 8. M.Y. He, J.W. Hutchinson, Crack deflection at an interface between dissimilar elastic materials, Int. J. Solids Structures, 1989, 25(9), 1053-1067. 9. M.Y. He, J.W. Hutchinson, Kinking of a crack out of an interface, J. Appl. Mech., 1989, 111, 270- 278. 10. O. Cherti Tazi, Comportement à la rupture d'un assemblage formé de matériaux fragiles, PhD thesis, university P. and M. Curie, Paris, France, 2005. 11. D. Leguillon, E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity, Masson, Paris, John Wiley, New-York, 1987. 12. H. Fujita, G. Jefferson, R.M. McMeeking, F.W. Zok, Mullite/Alumina mixtures for use as porous matrices in oxide fiber composites, J. Am. Ceram. Soc, 2004, 87(2), 261-267. 13. Hutchinson J.W., Suo Z., Mixed mode cracking in layered materials, Advances in Applied mechanics, 1992, 29, 63-191.

DYNAMIC BUCKLING OF THIN -WALLED COMPOSITE PLATES

Tomasz Kubiak Technical University of Lodz, Department of Strength of Materials and Structures, Stefanowskiego 1/15, 90-924 Lodz, Poland

Abstract:

The paper deals with a dynamic response of thin rectangular plate subjected to in-plane pulse loading of different triangular and rectangular shape. The investigated plates are made of unidirectional fibre composites, which are modelled as orthotropic material with principal directions of orthotropy parallel to the plate edges. The structures are assumed to be simply supported at the loaded ends and five different boundary conditions at non-loaded edges. Dynamic response of thin plates are calculated using analytical-numerical method and finite element method for comparison.

Key words:

dynamic buckling; thin plate; in-plane pulse loading; unidirectional fibre composite; orthotropic material

1.

INTRODUCTION

Dynamic stability of thin-walled structures has been discussed in many works since the 1960’s. In the majority of studies numerous simplifications that allow in practice for an effective analysis of stability of the thin-walled structure are assumed. Mathematical models tend to aim at higher precision and closer approximation of real structures, which enables one to analyse more and more exactly the phenomena occurring during and after the loss of dynamic stability. The analysis of dynamic stability of plates under in-plane pulse loading can be divided into three categories depending on pulse duration and magnitude of its amplitude. For pulses of high intensity the impact phenomenon is observed and for pulses of low intensity the problem 123 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 123–130. © 2006 Springer. Printed in the Netherlands.

T. Kubiak

124

becomes quasi-static. The dynamic pulse buckling occurs when the loading process is of intermediate amplitude and the pulse duration is close to the period of fundamental natural transverse vibrations (in range of milliseconds). In such case the effects of dumping are neglected. In word literature one can find many criteria allowing for determining dynamic critical or failure loads. One of the simplest is the criterion proposed by Volmir1 – the dynamic critical load corresponds to the amplitude of pulse force (of constant duration) at which the maximal plate deflection is equal to some constant value k (k=one plate thickness). The next one is Budiansky-Hutchinson2 that states: dynamic stability loss occurs when the maximal plate deflection grows rapidly with the small variation of the load amplitude. Both mentioned above criteria of dynamic stability are chosen to determine and compare the critical value of dynamic buckling. The paper deals with the dynamic response of thin plates subjected to inplane pulse loading of different shapes. The problem is investigated on the basis of asymptotic analytical-numerical method and finite element method.

2.

PROBLEM FORMULATION

The square and rectangular plates are simply supported at loaded ends and five different boundary conditions (both simply supported – further denoted as ss, both clamped – cc, one edge simply supported and the second clamped sc, one edge clamped and the second free – ce, one edge simply supported and the second free – se) at non-loaded edges are considered. The plates are made of unidirectional fibre composite. Composite material is modeled as orthotropic with principal directions of orthotropy parallel to the plate edges. In order to determine homogenised orthotropic material properties, the equations3 (Eq.1) based on theory of mixture are used. The analysed plate is subjected to compression pulse load. The duration time of pulse load is equal to period of natural vibration Tp. In numerical calculations three triangular (further denoted as T1 – Fig.1a,T2– Fig.1b,T3– Fig.1c), rectangular (R) and sinusoidal (S) shapes of pulse loading are considered. The material all the plates are made of is subject to Hooke’s law. It is assumed that the loaded edges remain straight and parallel during loading. Young's modulus and the Poisson's ratios occurring in equations have to satisfy the Betty-Maxwell’s theorem.

Dynamic Buckling of Thin – Walled Composite Plates Ex

E m 1  f  E f f ;

Ey

Em

Q yx

Qm 1 f  Qf f ;

G



Gm

 

@

Em 1 f  Ef f ; Em 1 f 1 f  Ef f 1 f

Gm

>





f 1  f  G >1  f 1  f @ G f  G 1  f

125

(1)

f

m

f

where: Ex, Ey –Young’s modulus of elasticity in longitudinal x and transverse y direction for composite material; G – shear modulus for composite material; Qyx – Poisson ratio for composite material; Em, Ef – Young’s modulus of elasticity for isotropic matrix (subscript m) and for isotropic fibre (subscript f); Gm, Gf – shear modulus for isotropic matrix and for isotropic fibre; Qm, Qf – Poisson’s ratio for isotropic matrix and for isotropic fibre; f – fibre volume fraction.

Figure 1. Shapes of pulse loading.

To describe the middle surface strains a complete strain tensor for thin plates has been assumed4,5: Hx

u , x  12 ( w ,2x  u ,2x  v ,2x ),

Hy

v , y  12 ( w ,2y  u ,2y  v ,2y ),

2H xy

J xy

(2)

u , y  v,x  w ,x w , y  u ,x u , y  v,x v, y ,

where: u, v, w - displacements parallel to the respective axes x, y, z of the local Cartesian system of co-ordinates, whose plane xy coincides with the middle surface of the plate before its buckling (Fig. 2).

T. Kubiak

126

Figure 2. Dimensions of the plate and the assumed local co-ordinate system.

The differential equations of equilibrium obtained from Hamilton’s Principle for a single plate can be written as:

  N u  N u  N u  N v  N v  N v  N v w  N  N w  N w  N w 

 Uhu , tt  N x , x  N xy, y  N y u , y  Uhv , tt  N xy, x  N y, y



 Uhw , tt  N x , x  N xy, y

x

,x

,y

x

,x ,x

xy

,x ,y

xy

,y ,x

,x ,x

y

,y ,y

xy

,x ,y

xy

,y ,x

y, y

 2 N xy w , xy  M x , xx  2M xy, xy  M y, yy

xy , x

,y

x

, xx

y

0 0

(3)

, yy

0

Let us obtain the equations of motion of a compressed plate assuming that the natural modes of vibration coincide with the buckling modes. Let O be a load factor, and U - the linear buckling mode with the minimal critical load factor values Ocr. We assume the following expansion of dynamic displacements field (Koiter’s type expansion for the buckling problem)5-8: U { (u , v, w )

OU 0  [( t ) U1  [ 2 ( t ) U 2  ...

(4)

where: [ – amplitude of the buckling mode (normalised with the equality condition between the maximum deflection and the thickness of the plate h); U 0 – the prebuckling static displacement field; U1 – the first order displacement field; U 2 – the second order displacement field. For plate containing geometric imperfections u (only linear initial imperfections determined by the shape of buckling modes), where U [ U then, similarly to the Koiter’s theory6 for the buckling problem, the potential energy can be written in the form7: P

§ O 1 1 a 0 O2  [ 2 ( t )a 1 ¨¨1  O 2 2 cr ©

· 1 O 1 ¸¸  a 111[ 3 ( t )  a 1111[ 4 ( t )  a 1[*[( t ) O 3 4 cr ¹

(5)

where coefficients a0, a1, a111, a1111 are determined in literature7,8 and [* is the amplitude of the imperfection in the form of the buckling mode.

Dynamic Buckling of Thin – Walled Composite Plates

127

Moreover, it was assumed that initial velocity of displacement equals zero [*, t 0 . Neglecting the inertia forces associated with second order inertia terms related to buckling, the kinetic energy with the account of expansion (Eq.5) and conditions of orthogonality for U1 and U2 is as follows 7: h 2

T

Ulb u ,2t  v ,2t  w ,2t dxdydz 2 ³0 ³0 ³





h 2

1 m[ ,2t 2

(6)

Then the Lagrange’s equations are: § O 1 [ , tt ( t )  ¨¨1  O Z02 cr ©

· O ¸¸[( t )  b111[ 2 ( t )  b1111[ 3 ( t )  [* O cr ¹

0;

(7)

where: Z02

a1 ; b111 m

a 111 ; b1111 a1

a 1111 ; Tp a1

2S Z0

(8)

The initial conditions8 are: [(t=0) = 0 and [ , t ( t 0) 0

(9)

The paper takes into account buckling modes for the minimal critical load and the fundamental vibration modes. For most cases buckling modes and vibrations mode are the same so the solution of eigenvalue problem is searched for various values of m-th harmonic8. For the free vibration we set O=0. The system of the ordinary differential equilibrium equations (Eq.3) for the first and the second order approximation is solved by the modified numerical transition matrix method in which the state vector of the final edge is derived from the state vector of the initial edge by numerical integration of the differential equations along the circumferential direction using the Runge-Kutta formulae by means of the Godunov orthogonalization method. Solution of this system in form of trigonometric series was presented in paper by Kubiak5. The equation of motion (Eq.7) is solved by the numerical Runge-Kutta method (with step size control and density output). The proposed method of solution allows one to take into account an influence of initial imperfections on the free vibration frequency and of critical loads in an easy way9.

128

3.

T. Kubiak

RESULT OF CALCULATION

Numerical calculations presented in this paper were conducted using own software based on equation presented in paragraph 2 (result denoted as A-N). To check the correctness of the obtained results the same cases were calculated using software based on finite element method – ANSYS 9.0 (results denoted as FEM). In dynamic analysis small imperfection ([*=0.01) was assumed. All presented examples of the calculation were conducted for epoxyglass fibre composite3 (Table 1.) square plate (a = b) with width to thickness ratio equals b/h = 100 and for the boundary conditions mentioned above. Different material properties depend on volume fibre fraction f=0.2 to 0.8 was analysed. Table 1. Material properties for fibre and matrix Density Young’s modulus U [kg/m3] E [GPa] Epoxy 1246 3.5 Glass 2450 71

Kirchoff’s modulus G [GPa] 1.25 30

Poisson ratio Q>@ 0.33 0.22

The buckling force Ncr [N] obtained for square plate using both methods are presented in Table 2. According to Budiansky-Hutchinson criterion (B-H) the curves representing maximal dimensionless plate deflection ([=w/h) as a function of impulse amplitude force to critical static buckling force (Nx0/Ncr) for different analysed cases, were prepared. These curves are shown in Figs. 3-6. Fig. 3 presents the results obtained from analytical-numerical method for plate clamped on non-loaded edges, for volume fibre fraction equals f=0.5 (cc05) and for all analysed impulses. In figure 4 the results obtained from analytical-numerical method (A-N) and finite element method (FEM) for triangular (T1) and rectangular (R) shape impulses are presented. The curves in Fig.4 present the results for a plate simply supported on all edges and volume fibre fraction equals to f=0.5 (ss05). The obtained results are consistent. The results obtained for all analysed volume fibre fraction for plate clamped on non-loaded edges (cc) and subjected to triangular shape impulse (T3) are presented in Fig.5. The curves almost overlie it means that for all volume fibre fraction the same DLF’s is obtained but the dynamic critical buckling load are not the same, because this value depends on DLF’s and corresponding critical static buckling load. Figure 6 shows boundary condition influence on dynamic response for case denoted as T2_02 – triangular shape impulse and volume fibre fraction equals to 0.2.

Dynamic Buckling of Thin – Walled Composite Plates Table 2. Critical buckling forces for square plates  f = 0.2 Boundary condition Vcr [MPa] A-N Vcr [MPa] FEM 2.8 ss 2.8 5.1 5.0 cc 3.6 3.6 sc 1.6 1.6 se 1.7 1.7 ce

Figure 5. Dynamic curves for cases T3cc.

129

f = 0.7 Vcr [MPa] A-N 8.1 14.4 10.5 4.8 5.4

 Vcr [MPa] FEM 8.2 14.6 10.6 4.8 5.1

Figure 6. Dynamic curves for cases T2_02.

The Budiansky-Hutchinson and Volmir criteria were compared for square plate with fibre volume fraction equals 0.8, for all analysed boundary condition and impulses. Obtained DLF’s (critical value of the ratio of impulse amplitude to critical buckling load Nx0/Ncr) are presented in Table 3. Table 3. DLF’s for fibre volume fraction equals f=0.8 T1 T2 T3 B-H V B-H V B-H V 3.1y3.3 3.1 2.4y2.6 2.3 2.4y2.6 2.1 ss 2.2 2.5y2.7 2.1 3.2y3.4 3.0 2.4y2.6 cc cs 3.2y3.4 3.1 2.4y2.6 2.3 2.4y2.6 2.1 se 2.6 2.0y2.2 2.2 2.8y3.0 3.6 2.0y2.2 ce 2.8y3.0 3.6 2.0y2.2 2.6 2.0y2.2 2.2

R B-H 1.4y1.6 1.4y1.6 1.4y1.6 1.2y1.4 1.2y1.4

V 1.4 1.4 1.4 1.6 1.7

S B-H 3.0y3.2 3.0y3.2 3.0y3.2 3.0y3.2 3.0y3.2

V 2.9 2.9 2.9 3.0 3.1

130

4.

T. Kubiak

CONCLUSION

The fibre composite materials are very strong for instance the ultimate compression strength for the analysed composite with volume fibre fraction f=0.6 in longitudinal direction is equal to 600 MPa. If this value is compared with critical buckling stress (Table 2) it is seen that the static and dynamic buckling critical stresses are many times less than the ultimate stress. It means that buckling analysis is very important for thin composite plates. The results obtained for different volume fibre fraction almost overlie (for all analysed cases, example in Fig.5). It means that volume fibre fraction does not have any influence on the DLF value. It should be remembered that DLF is only a ratio. To obtain the dynamic critical load the DLF should be multiply by corresponding critical static buckling load Ncr, and then the influence fiber volume fraction on critical dynamic buckling load appears. The most dangerous impulse shape is the rectangular one (Fig.3). It gives the smallest value of DLF’s for all analysed cases and the higher out of plane deflection. According to presented results (Table 3) it can be said that for some cases the Budiansky-Hutchison criterion is more strict than Volmir criterion. The big differences in obtained DLF’s using two mentioned above criteria appear for the plates with one edge free, subjected to triangular shape impulses (T1 and T2). The results obtained using two mentioned above method are consistent (Fig.4). It means that proposed analytical -numerical method gives proper results.

REFERENCES 1. S.A. Volmir, Nonlinear dynamics of plates and shells, /in Russian/ Science, Moscow, 1972 2. J.W. Hutchinson, B. Budiansky, Dynamic buckling estimates, AIAA Journal, 4-3, 525-530, (1966) 3. A. Kelly (ed.), Concise Encyclopedia of Composite Materials, Pergamon Press, (1989) 4. Z. Kolakowski, K. Kowal-Michalska (eds.), Selected problems of instabilities in composite structures, A Series of Monographs, Technical University of Lodz, Poland, (1999) 5. T. Kubiak, Postbuckling behaviour of thin-walled girders with orthotropy varying widthwise. Int. J. Solids Structures, 38, 4839-4855, (2001) 6. W.T. Koiter, Elastic Stability and Post-Buckling Behaviour. Proc. Symposium on Nonlinear Problems, Univ. of Wisconsin Press, Wisconsin, USA, (1963) 7. B. Budiansky, Theory of buɫkling and post-buckling behaviour of elastic structures. Advances in Applied Mechanics, 14, Acad. Press, 1-65, (1974) 8. A. Schokker, S. Sridharan, A. Kasagi, Dynamic buckling of composite shells. Computers & Structures, Vol. 59, No. 1, 43-55, (1996) 9. I. Elishakoff , V. Birman, J. Singer, Small vibrations of an imperfection in the vicinity of a nonlinear static state. J. Sound Vibr. 114, 57-63, (1987)

NUMERICAL AND EXPERIMENTAL MODELS OF THE FRACTURE IN THE MULTI-LAYERED COMPOSITES

Mieczysł aw Jaroniek 1 1

Department of Materials and Structures Strength, Technical University of àódĨ, POLAND Abstract: Advanced mechanical and structural applications require accurate assessment of the damage state of materials during the fabrications as well as during the service. Due to the complex nature of the internal structure of the material, composites including the layered composite often fail in a variety of modes. The failure modes very often are influenced by the local material properties that may develop in time under heat and pressure, local defect distribution, process induced residual stress, and other factors. Consider a laminate composite in plane stress conditions, multi-layered beam bonded to planes having shear modulus G i and Poissons ratio QI respectively, subjected to bending. The behaviour of the cracks depends on the cracks configuration, size, orientation, material properties, and loading characteristic. The fracture mechanics problem will be attacked using the photoelastic visualisation of the fracture events in a model structure. The proposed experimental method will developed fracture mechanics tools for a layered composite fracture problem.

1.

INTRODUCTION

The development of the failure criterion for a particular application is also very important for the predictions of the crack path and critical loads. Recently, there has been a successful attempt to formulate problems of multiple cracks without any limitation. This attempt was concluded with

131 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 131–139. © 2006 Springer. Printed in the Netherlands.

132

M. Jaroniek

the series of papers summarising the undertaken research for isotropic [2], an isotropic [4] and non-homogeneous class of problems [5] and [4]. Crack propagation in multi-layered composites of finite thickness is especially challenging and open field for investigation. Some results have been recently reported in [5]. The numerical calculations were carried out using the finite element programs ANSYS 5.4 and 5.6 [8]. Two different methods were used: solid modeling and direct generation. 2.

MATERIAL PROPERTIES

Material properties exert an influence on the stress distribution and concentration, damage process and load carrying capacity of elements. In the case of elastic-plastic materials, a region of plastic strains originates in most heavily loaded cross-sections. In order to visualise the state of strains and stresses, some tests have been performed on the samples made of an ”araldite”-type optically active epoxy resin (Ep-53), modified with softening agents in such a way that an elastic material has been obtained. Properties of the components of experimental model are given in table 1. Table 1. Mechanical properties of the experimental model components

3.

Layer

Young's Modulus Ei [MPa]

1 2 3 4

3450.0 1705.0 821.0 683.0

Poisson’s ratio Qi [1] 0.35 0.36 0.38 0.40

Photoelastic constants Photoelastic constants in in terms of stresses terms of strain kV [MPa/fr.] fH [-/fr

1.68 1.18 0.855 0.819

6.572 ˜10-4 9.412 ˜10-4 14.31 ˜10-4 16.79 ˜10-4

EXPERIMENTAL RESULTS

The stress distribution in was determined using two methods: Shear Stress Difference Procedure (SDP – evaluation a complete stress state by means the isochromatics and the angles of the isoclines along the cuts) [3]. Method of the characteristics (the stress distribution were determined using the isochromatics only and the equations of equilibrium [8]. In a general case [7], the Cartesian components of stress: Vx , Vy and Wxy in the neighbourhood of the crack tip are:

Numerical and Experimental Models of the Fracture

Vx

1 2Sr

Vy

[ K I cos 42 (1  sin 42 sin 324 )  K II sin 42 ( 2  cos 42 cos 324 )]  V ox

1 2Sr

W xy

133

[ K I cos 42 (1  sin 42 sin 324 )  K II sin 42 cos 42 cos 324 ]

1 2Sr

4 2

4 2

[ K I sin cos cos

34 2

4 2

4 2

)  K II cos (1  sin sin

24 2

( 1)

)]

From which: (V 1  V 2 ) 2

1 ( K I sin 4  2 K II cos 4) 2  ( K II sin 4) 2  2Sr

>

@

(2)

V 4 2 ox sin >K I sin 4(1  2 cos 4)  K II (1  2 cos 2 4  cos 4)@  V ox2 2 2Sr

a)

Figure 1. a) Four-layer beam with cracks. Photoelastic model under four point bending, the isochromatic patterns ( V1  V 2 ) distribution b). Initial loading (P=20.0 N). c) P=50.0 N - tension of layers 2, 3 and 4.

M. Jaroniek

134

By inserting the values kV˜mi =V1-V2 into (2) we obtain the isochromatics curves in polar coordinates (r, 4). For each isochromatic loop the position of maximum angle 4m corresponds to the maximum radius of the rm. This principle can also be used in the mixed mode analysis [11] by employing information from two loops in the near field of the crack, if the far field stress component - Vox(4) = const. Differentiating Eqn (2) with respect to 4, setting 4=4m and r=rm and using Eqn (wW m/ w4m=0) gives: g ( K I , K II , V ox ) 2 

1 K I2 sin 24  4 K I K II cos 24  3K II2 sin 24 2Sr

>

@

V ox 4 sin ^ >K I (cos 4  2 cos 24)  K II (2 sin 24  sin 4)@ 2 2Srr

1 4 cos >K i (sin 4  sin 24)  K II (2  cos 24  cos 4)@ 2 2

f ( K I , K II ,V ox )

V 1  V 2 2  (k

V

˜ m) 2

`

0

2

and

g ( K I , K II , V ox )

w[(V 1  V 2 ) ] w4 m

0

(3)

Substituting the radii rm and the angles 4m from these two loops into a pair of equations of the form given in eqn (3) gives two independent relations dependent on the parameters KI, KII and Vox. The third equation is obtained by using eqn (2). The three equations obtained in this way have the form

g i ( K I , K II , V ox ) 0 g j ( K I , K II , V ox ) 0

(4)

f k ( K I , K II , V ox ) 0 In order to determine KI, KII and Vox it is sufficient to select two arbitrary points ri, 4i and apply the Newton-Raphson method to the solution of three simultaneous non-linear equations (4). The values KC according to mixed mode of the fracture were obtained from

KC

K I2  K II2

(5)

Example of the numerical results obtained from (4): m=12.5, r1 =0.6mm 41 =1.484 r2 =10.45mm 42 =1.416 (4) (4) (4) KI = 0.14 MPa—m, KII = 1.05 MPa—m Vox = 0.039 Mpa, KC = 1.05 MPa—m

135

Numerical and Experimental Models of the Fracture

By inserting the values ri, 4i in three selected arbitrary points into (2) we obtain three non-linear equations (i = 1, 2, 3)

f i ( K I , K II , V ox )

0

(6)

and apply the Newton-Raphson method to the solution we have KI, KII and Vox. Example of the numerical results (shown in Fig. 3) obtained from (6): for m1 =12.5 r1 =0.72mm 41 =1.484 m2 =8.0, r2 =1.15mm 42 =1.37, 5 m 3 =5.5mm r3= 1.85 43 = 1.315 KI

(4)

= 0.702 MPa—m, KII

A

B

A

B

(4)

= 1.043 MPa—m Vox = 0.152 Mpa, KC

(4)

= 1.257 MPa— m

Figure 2. The isochromatic patterns ( V1  V 2 ) distribution according to the propagation of the crack obtained experimentally. BS

P=540N

A P=540N

-17.64MPa

-4.29 MPa

Vx 2.78 MPa 2.57 „

-3.45MPa

Vx

m=4.55 3.72MPa

m=3.5

-3.54MPa

2.95 MPa

Mg=29700Nmm

Mg=27000Nm

2.95MPa m=2.95

5.45 MPa

5.51 MPa

5.77MPa m=4.1

3.98MPa m=6.25 5.5 MPa

41.1 MPa 0.5 mm

-15.41MPa

-5.88 MPa e=4.5mm

BS

A

Figure 3. Distribution of stresses V x in cros sections A-A and B-B 0.5mm with respect to crack obtained experimentally.

M. Jaroniek

136

4. NUMERICAL DETERMINATION OF STRESS DISTRIBUTION The distribution of stresses and displacements has been calculated using the finite element method (FEM) [9]. Finite element calculations were performed in order to verify the experimentally observed the isochromatic distribution observe during cracks propagation. The geometry and materials of models were chosen to correspond to the actual specimens used in the experiments. The numerical calculations were carried out using the finite element program ANSYS 5.4 and by applying the substructure technique. For comparison the numerical (from FEM) and experimental sochromatic fringes ( V1  V 2 ), distribution was shown in Fig. 3. A finite element mesh of the model (used for numerical simulation) are presented in Fig.5 and the stresses Vx are shown in Figs. 6 and 7.

Figure 4. A finite element mesh of the model (for numerical simulation).

The strain energy release rate GC equal in this case to the Rice J-integral: wu J ³ ( 12 V ij H ij dx 2  Ti n i ds )] (7) wx1 s or from numerical calculation using the finite element method: J

­ ª

°1 1 ¦ ®° 2 «« E V j

¯ ¬

i

2 yi



 V xi2 

½ ªW xyi V xi QV yi ˜ n2i  W xyin1i  V yin2i 'vi º»°¾ ˜ 'Si » ˜ n1i  « 'xi ¼ °¿ 2G ¼» ¬ Ei

2 º W xyi

(8)

Numerical and Experimental Models of the Fracture

137

Figure 5. Numerical determination of stress distribution (Ansys 5.4). Distribution of the stresses Vx along the crack.

The values KC according to the fracture in the 4-layer were determined from

K C(i )

E i ˜ GC

(9)

Figure 6. Numerical determination of stress distribution (Ansys 5.4). Distribution of the stresses Vx (cracks length a=6.0 mm ).

M. Jaroniek

138

V ,. Figure 7. Numerical results. (Ansys 5.4). Distribution of the stresses x cracks length 1. a=6.0mm and 2. a=9.0mm, thickness of layers h=10mm. T able 2. Experimental and numerical results. Critical values K (1)IC according to the (2) propagation of the crack and K IC

5.

Crack length. a [mm]

Critical force Pcr [N]

6.0 9.0 9.8

265.0 205.0 185.0

Experimental

results

[MPa—m]

KI

(4)

1.177 0.702 0.14

KII

(4)

0.8793 1.043 1.05

KC

Numerical res. [MPa—m]

(4)

1.419 1.257 1.05

Vox [MPa] 2.58 0.152 0.039

GC

(4)

KC

(4) n

[MN/m] MPa—m 3.08 1.45 2.39 1.28 1.97 1.16

CONCLUSIONS

Photoelasticity was shown to be promising in stress analysis of beams with various number and orientation of cracks. It is possible to fabricate a model using various photoelastic materials to model multi layered structure. Finite element calculations (FEM) were performed in order to verify the experimentally observed branching phenomenon and the isochromatic distribution observed during cracks propagation. The agreement between the finite element method predicted isochromaticsfringe patterns distribution and those determined photoelasticaly was found to be within 3y5 percent.

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REFERENCES 1. Cherepanov G. P., (1979): Mechanics of Brittle Fracture. Mc Graw Hill, New York 2. Cook T. S. and Erdogan F. (1972) In:, Int. Journ. of Eng. Science, Vol. 10, 677-697. 3. Frocht M. M., (1960) Photoelasticity, John Wiley, New York 4. Gupta A. G., A (1973): In: International Journal of Solids and Structures, No36 (1845-1864)

5. Hilton P. D. and Sin G. C., (1971), In: International Journal of Solids and Structures, No 7, 913 6. Neimitz A.,( 1998 ) Mechanics of fracture. (in Polish) P. W. N. Warsaw, 7. Sanford R. J. and Dally J. (1979) In: Eng. Fract. Mech. Vol. 2, 621-633 8. SzczepiĔski W. In: Archiwes of Applied Mechanics 5 (13) 1961. 9. User’s Guide ANSYS (1999): 5.4, 5.6, Ansys, Inc., Huston, USA 10. Zienkiewicz O. C., The Finite Element Method in Engineering Science. Mc Graw - Hill, London, New York 1971

MULTISCALE METHOD FOR OPTIMAL DESIGN OF COMPOSITE STRUCTURES INCORPORATING SENSORS

Francis Collombet*, Matthieu Mulle*†, Yves-Henri Grunevald†, and Rédouane Zitoune* * LGMT-PRO²COM, IUT P. Sabatier, 133c av. de Rangueil, BP 67701, F31077 Toulouse cedex 4, e-mail: [email protected]

† DDL Consultants, Pas [email protected]

de

Pouyen,

F83330

Le

Beausset,

e-mail:

Abstract:

This paper deals with the challenge related to the improvement of the dialogue "calculation - test" in the design of composite structures by taking into account the influence of the manufacturing conditions of the composite material within the structure in the frame of a multiscale method. The incidence of discrepancies on the meso and the macro scales between the measurements and the calculation are evaluated on the variability of the input data used for numerical calculation (concerning the elastic characteristics of the composite material), thanks to the scientific added value of in core measurements using optical fibers with Bragg gratings.

Key words:

Laminated composite structures, Optical fibers with Bragg gratings, Multiscale approach for designing

1.

MULTISCALE APPROACH

For the composite, the intrinsic material data does not exist. Usually, reference material data are the basis of the pyramid of tests in aeronautics. It leads to a well-known issue: the composite material in coupons manufactured with the same raw materials is different from the composite material in the structure. Such a situation comes from the variability due to manufacturing conditions; operators, raw materials, and design 141 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 141–149. © 2006 Springer. Printed in the Netherlands.

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singularities… Regarding design singularities, material is different from those in the current zone. In addition, the singular zones are not reproducible from the manufacturing point of view. This paper is a contribution to discuss the scientific added value of in situ instrumentation in the frame of a "multiscale" approach. "Multiscale" material requests "multiscale" data. In such a way, the "addressable" scales are macro (or structural scale) and meso (ply scale). Microscale is rather an "analyzable" scale thanks to the knowledge of manufacturing phase of the structure. This study is placed within an original framework where the number of composite structures is important (48) in order to carry out an identification of properties of the material within the structure itself, without resorting to the use of the specimens known as "elementary" which compose classically in aeronautics the base of pyramid of tests (obviously costly and long).

2.

TECHNOLOGICAL DEMONSTRATORS

What are the objectives to improve calculation of composite parts to achieve an efficient "Virtual Testing approach"? To start with, we need input data representative of the effective material properties within the structure. Achieving this objective implies to address the material on different scales of the composite part. The technological demonstrators address composite material under conditions representative of the real structure with onboard drop offs, free edges, variations of thickness … Our approach deals with a comparison with experimental measurements (on the scales respectively, mesoscopic thanks to longitudinal deformation provided by the sensors and macroscopic thanks to the resulting force applied to the demonstrator) with information from a numerical model. A family of 48 technological demonstrators of "beam" type (T.B.D.) is obtained by water jet cutting of "mother" plates (cf. figure 1c). The "mother" plates are manufactured using prepreg of UD M21/35%/268/T700GC (provided by Hexcel Composites, cf. figure 1a) 300 mm wide and are polymerized in autoclave (cf. figure 1b).

(a) During laying up in the (b) After curing in the (c) During water jet white room autoclave cutting Figure 1. From the "mother" plate to the technological demonstrator beam types.

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The T.B.D. 30 mm wide consist in a central zone of 28 plies, surrounded by two current zones of 20 plies (cf. figure 2a and 2b). The stacking sequence, respectively percentages of plies with 0°, +/- 45°, 90°, is selected to correspond to those qualified in fatigue for aeronautical structures: 50/40/10 for the current zones, with an addition of 25/50/25 in the central zone. The T.B.D. are subjected to three-point bending tests on 250 mm span (as a skin of wing plane). 14 demonstrators have an embedded instrumentation composed of optical fibers with Bragg gratings (FBG) (cf. figure 2c).

Figure 2. Technological beam demonstrators with: (a) dimensions, (b) stacking sequence with the location of optical fibers with Bragg gratings and (c) examples.

The M21 matrix is a third generation epoxy prepreg from Hexcel Company. This matrix has very interesting properties as far as properties under impact due to a thermoplastic solid phase are concerned. This solid phase (nodules) is used to improve tenacity and resistance of cracks propagation. For the curing phase of the epoxy matrix, the temperature is about 180 °C which corresponds to 50 °C over the temperature of glass transition of the thermoplastic. It means that the thermoplastic nodules could grow. During the cooling phase down to the room temperature, it is impossible for the nodules to recover their previous diameter due to interaction between the two chemical networks (epoxy and thermoplastic).

3.

EMBEDDED INSTRUMENTATION

The placement of the sensors is studied at the early moment of the design phase of the demonstrators. There is an obvious interest in placing optical fibers parallel to carbon fibers and at an X-coordinate direction assuring the

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integrity of the sensor until the rupture of the TBD. The limit in longitudinal deformation of a FBG is about 1.10-2. Under these conditions, it is relevant to place optical fibers in one of the principal directions of the stresses taking into account the operating mode of the structure. With technological beam demonstrators subjected to a three-point bending test (case of the skin of a plane wing), optical fibers are integrated in 0° plies. The strategy of instrumentation depends on the use of the structure in study. As shown on figure 2b, the choice of the plies N° 3 and N° 26 is natural. These plies are far away from the median plan of the demonstrators. Moreover, they are voluntarily placed between two plies whose direction of fibers corresponds to + or - 45°. Thus the position of the FBG in the thickness of the mother plate is controlled by limiting their migration during the curing phase. The dimensions of optical fibers without coating are defined through a diameter equal to 125 µm whereas the average thickness of the ply is 264 µm. Their position is controlled in a range of + or - 70 µm. FBG's are exploited as sensors in many industrial fields like geology, naval, aeronautics, and railway (Grunevald, Collombet et al., 2003) or wind energy. They have already been described in various papers since the last decade. We just remind that they are sensitive to temperature (T), strain (H), and to hydrostatic pressure (P). FBG's deliver information related to a variation of the Bragg wavelength (Esquer et al., 2002). With respect to the study, we have found that the pressure and temperature influences were not significant compared to the other factor. Thus, they are considered to be negligible.

4. 4.1

DATA FROM T. B. DEMONSTRATORS Manufacturing influence on the meso-scale

In aeronautics, thousands of tests are performed with normalized thin specimens to determine all the mechanical characteristics on qualified staking sequences such as (50/40/10), (40/40/10), (25/50/25) or (10/80/10) (with or without holes and/or impacts). After that, these data are used for all stress analyses but in some cases such as thick structures or complex parts with singularities this leads to unsatisfactory numerical results, in comparison with those obtained in the current zones. The T.B.D. are manufactured in autoclave with respect to the same parameters than those used for the normalized specimens (compaction, curing cycle …) so we are supposed to get an average 260 µm thickness per ply and a 58 % fiber content (normalized values on thin specimens). The thicknesses of the 48 demonstrators are measured (cf. figure 2a) on 6 points along the X axis. Thanks to these 288 data, we define 288 mean thicknesses per ply. We

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obtain a minimum value of about 0.258 mm, a maximum value of about 0.270 mm, a general mean thickness of 0.264 mm (+ 1.5 % in comparison with the theoretical value), a standard deviation (SD) of 0.002 mm and a 3 sigma tolerance (3ST) of ± 0.007 mm (± 2.7 %). With such a set of data, it is possible to calculate an average fiber content of 57.02 %, a minimum value of 56.61 %, a maximum value of 57.52 %, a SD of 0.25 % and 3ST of about ± 0.74 %. At first sight, these results seem to confirm the suitability of this manufactured material as regards to the aeronautical standards. A 1.5 % variation concerning the thickness leads to a 4.6 % variation for inertia and stiffness. A 1 % reduction of the fiber content involves a 1.7 % reduction concerning longitudinal Young modulus on a 0° ply (cf. table 1) and, in a first approximation, a reduction of - 1.3 % (in central zone) and - 1.7 % (in lateral zones) concerning the general longitudinal Young modulus. So at the end, it is credible to get a stiffness variation ranging from + 2.8 % to + 3.2 %. Table 1. Mechanical characteristics of the reference UD and "rectified" data.

Reference data Rectified data

Vf % 58 57

E11t GPa 141.9 139.5

E11c GPa 114.3 112.5

E22t = E33t GPa 8.4 8.2

E22c = E33c GPa 10.5 10.3

G12 = G13 GPa 4.6 4.5

ȡ kg/m3 1564 1559

The "rectified" properties are obtained thanks to the mixing law (cf. table 1). So we notice that with the same process parameters, we have a first variation on meso scale concerning the mechanical properties of the composite material in the structure. Our ignorance of the initial state of the composite material does not simply consist in terms of residual cure stresses but in more elementary ways in terms of basic parameters i.e. real ply thicknesses and respective fiber content.

4.2

Manufacturing influence on the micro-scale

We have also measured the mass and the volume of 28 demonstrators which were not instrumented. On this population of demonstrators, we obtain a minimum of density of about 1509.8 kg/m3, a maximum of about 1530.4 kg/m3 and an average value of 1522.1 kg/m3. The SD is 5.5 kg/m3 and the 3ST corresponds to ± 16.5 kg/m3. These results are obtained with respect to three products and the 3ST is very small (less than ± 1 % which is smaller than the 3ST on the fiber and matrix density) so we can maintain that the process is under control. However, the results are not in accordance with the rectified value of 1559 kg/m3 (- 2.4 %) previously obtained.

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NUMERICAL MODEL FOR "STRESS ANALYSESTESTS" MULTISCALE COMPARISON

Analyzing the material properties within a structure means facing the numerical problems of the stress analysis. Figure 3 presents some details of the numerical representation of the technological beam demonstrators in a configuration a three-point bending test. We use three-dimensional laminated finite element of type 11 with degree 1 and 33 degrees of freedom (24 + 9 incompatible) each, available in the library of the SAMCEF software. The model involves 1040 elements representing 6264 degrees of freedom. The loading is carried out using conditions in imposed displacements. Each element contains 10 plies for the current or side zone and 8 plies in the element of the patch zone (cf. figure 3a). The zone which is rich in resin in the drop offs was not represented (cf. figure 3b). Indeed, it is not possible to use the composite volumic element to represent a composite prism. This type of element exists in the SAMCEF software but it is not used because it cannot be placed in the direction of the layer sequence. The size of the elements is 3 mm according to X, 6 mm according to Y. According to Z, are respectively considered 2.6 mm on the flat zones and 2.1 mm in the drop offs. There is thus a ratio of 3 between the dimensions of the volumic elements (cf. figure 3c).

Figure 3. Details (a), (b) and (c) of the volumic mesh of the T.B.D.

6.

THREE-POINT BENDING TESTS

Figure 4. The "load versus displacement" curves up to rupture of the T.B.D.

From 0 to 18 mm, the “load versus displacement” curves are almost linear (cf. figure 4). After 19 mm, audible cracks are heard and an inflexion on the curve is noticed. The rupture appears from 23 mm displacement. It leads to test the TBD with a range of an imposed displacement up to 18 mm.

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We have to carry out "in situ" strain measurement up to 23 mm displacement. On figure 4, the curve of the demonstrator noted TBD-0209 presents a slope slightly different from the others. However this demonstrator is placed at the periphery of the "mother" plate (cf. figure 1) which can imply particular manufacturing conditions and thus properties. Anyway, a structure cannot be like another. From the "rectified" material Characteristics (cf. table 1), we calculate the longitudinal strain (H11) in plies 3 and 26 in the centre of all finite elements placed in the central line of the mesh (cf. figure 5). Estimating the length of a grating equal to 10 mm along the X axis allows to Figure 5. "H11 versus X-coordinate" curves of T.B.D. in establish that it is not plies 3 and 26 for 8 mm displacement. possible to get a variation of information below 31.10-6 per mm of displacement or 250.10-6 for 8 mm. At the X coordinate -90 mm, the measurement at the sensor will never exceed the value of maximum strain of fiber equal to 10-2 until the rupture of the demonstrator.

7.

IMPROVEMENT OF THE "STRESS ANALYSESTESTS" DIALOGUE VIA DATA ADJUSTMENT

Considering data of Table 1, the difference between calculation and average measurement is about 5 % on a mesoscopic scale (H11) as well as on a macroscopic scale (resultant force) (cf. figure 6). It is interesting to note that this 5 % variation is a classical satisfying value in terms of "stress analysis-test" correlation. But what does happen to variations on the elastic characteristics when trying to reduce the 5 % variation at the same time on a mesoscopic scale (H11) and a macroscopic scale (resultant force)? By numerical iterations (cf. figure 7), three values of mechanical characteristics are "adjusted", respectively the Young modulus in traction in the longitudinal direction (E11t goes down from 139.5 to 125.0 GPa i.e. a 10 % variation), the Young modulus in compression in the longitudinal direction (E11c goes down from 112.5 to 118 GPa i.e. a 5 % variation) and its shear modulus (G12 goes down from 4.5 to 3.7 GPa i.e. a 18 % variation). The variations between the calculation and the average values of tests are reduced to 1 % for all scales.

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Figure 6. "Stress analyses-tests" in 3pt. bending test with "rectified" properties.

Figure 7. "Stress analyses-tests" in 3pt. bending test with "adjusted" properties.

In the case of complex numerical models, the improvement of the description of the boundary conditions is not sufficient to improve significantly the "stress analyses-tests" correlation starting from the only data reference material, even though fiber content is corrected using "rectified" properties. The same results are obtained with material properties "adjusted" on a given level of modeling (mesh finesse, description of the design singularities and representation of the boundary conditions).

8.

CONCLUSION

In the studied demonstrator representative of aeronautical structures, the "adjusted" elastic properties are different from those "rectified" using tests carried out on elementary specimens. However the numerical calculations performed with the "rectified reference values" could give satisfaction at first sight: variation of 5 % between the calculations and the average of measurements respectively, on H11 (on a mesoscopic scale) and on the resultant force (on a macroscopic scale). The multi-scale methodology led to a significant adjustment (20 % for 14 T.B.D. instrumented out of 48) of the composite material elastic properties in the structure. Gathering information thanks to the sensors allows a return of experiment during the whole life cycle of the structures. The proposed multi-scale methodology is ready to be applied for most industrial composite structural designs and is a contribution to the "Virtual Testing approach" requested in aeronautics to improve the "stress analysis-test" dialogue in order to decrease cost.

ACKNOWLEDGEMENTS The authors thank DDL Consultants Company for financing a PhD thesis, Hexcel Composites Company for the HexPly® pre-preg provision

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and DGA (French Ministry of Defense) for its financial support through the upstream program AMERICO conducted by ONERA (French National Office for Aeronautical Studies).

REFERENCES Grunevald, Y.H., Collombet, F., et al., 2003, Global approach and multi-scale instrumentation of composite structures in the field of transportation, 8th Japan Int. SAMPE, Tokyo, Japan. Esquer, P.M., et al., 10-12 July 2002, Optical fibre sensors in Airbus Spain, SHM2002 - First European Workshop on Structural Health Monitoring, Cachan, France.

NUMERICAL MULTISCALE MODELLING OF ELASTO-PLASTIC BEHAVIOR OF SUPERCONDUCTING STRAND

Bernhard A. Schrefler1 , Daniela P. Boso1, Marek Lefik2 1 Department of Structural and Transportation Engineering Università degli Studi di Padova, via F. Marzolo 9, 35131 Padova, Italy; 2 Department of Mechanics of Materials, Technical University of àódz, Al.Politechniki 6, 93590 Lodz, Poland

Abstract:

In the paper we present the analysis of multi-scale composite "from bottom to top of the hierarchy". At each step of the analysis we update the values of material properties of homogeneous components and compute new effective material parameters. Two different numerical techniques of homogenization in non-linear range, basing on numerical experiments, are shown.

Key words:

Homogenization, Hierarchical composite, Superconducting strands, Thermomechanical coupling.

1.

INTRODUCTION

Many natural and man-made materials exhibit an internal structure at more than one length scale. Superconducting cable can be regarded as very good examples of hierarchical structures where lower structural levels influence the global behaviour. According to the current design, the superconducting (SC) alloy is formed into fine filaments, which are twisted together and embedded in a low-resistivity matrix of normal metal to make the elementary strand. The strands, in turn, are aligned and twisted together in triplets and quadruplets to form a complex structure of the cable. In the paper we focus our attention on four initial substructures. A scale separation between each of the two successive levels is very sharp. These levels are enumerated here in increasing order of their characteristic dimension: a) the super-conducting alloy; b) the filament; c) the strand; d) the bundle of

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strands. The whole structure of the super-conducting coil is more complicated and few subsequent steps upstairs of the hierarchy can be distinguished.

Figure 1. Multi-hierarchical composite and a corresponding spatial discretisation with finite element meshes of three initial levels of the structural hierarchy. The Figure represents the cable of the VAC type.

Because of the scale separation between structural levels a mesh fine enough for the micro level would result in a huge number of elements and unknowns at the macro level. Such a model would be numerically difficult to manage and, first of all, not necessary. The common solution for the analysis of such a structure is the estimation of global mechanical and thermal characteristics. The number of structural levels does not complicate very much the analysis when the global characteristics remains constant and thus can be computed once for ever at the beginning of the analysis. In this case the standard homogenization procedure can be simply repeated as many times as many hierarchical levels are distinguished. The problem arises when the global material characteristics changes during the loading process, depending on yield state, on the temperature or on other factors. In the case of superconductor, the dependence of the Young modulus and the Poisson ratio on the temperature is sharp, even within the elastic range of stress. The yielding superposes further the influence of the stress level on the value of these characteristics of the components.

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The purpose of this paper is to provide a short review of numerical tools we use in the analysis of the hierarchical composite. In the elastic regime we compute, at each structural level, the effective thermal and mechanical characteristics. This is done by an analysis of the micro-cell of periodicity, according to the classical theory of homogenization. We use our own methods of numerical definition of yield surface for a fictitious homogenized body in the space of effective stresses, published earlier in Pellegrino et al.(1999) and by Lefik and Schrefler (2003). We show also how an artificial neural network can be alternatively applied to this end. In the paper, particular steps of the analysis are exemplified with different types of superconducting cable. The interested reader is referred to Boso et al. (2005) for details of these various microstructures, treated always with a similar algorithmic framework.

2.

ALGORITHM OF THE ANALYSIS

The basic algorithm contains three main steps: multi-level homogenization, FE solution of boundary value problem (BVP) defined at the macro level over the homogenized body and finally, unsmearing. Homogenization consists in defining coupled constitutive relationships between mean strains and mean stress, average temperature, mean heat flux and mean temperature gradient. Solution of the BVP for the whole composite results with average thermal and mechanical fields that are starting point for unsmearing. This last step attributes to the mean values of thermal and mechanical quantities - the repartitions of stress and temperature localized to the meso and micro cell. This allows us to check yielding conditions and true values of temperature dependent materials properties at the level of homogeneous components. After this, with possibly changed values of the micro-data, we come back to the homogenization step and complete the main loop of the analysis. Homogenization starts at the micro level. Input data are the geometry of the cell of periodicity and material properties at current temperature and current stress level. Output is the matrix of effective material properties of the homogenized material at the current level of hierarchy. The composite material plays the role of a component at the next structural level. The homogenization is repeated once more at the higher structural level. While the material is elastic we apply the classical asymptotic approach as described by Sanchez-Palencia (1980). According to it, the so called homogenization functions are obtained as a solution of BVP with periodic boundary conditions formulated over a repetitive cell of periodicity, loaded

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with mean unit strains. The vector of homogenization functions is useful at the stage of unsmearing. It allows us to attribute a field of localized stress and temperature over the cell of periodicity at the lower structural level for each values of the mean strains and mean temperature gradient in any integration point of the element of FE mesh drawn for the FE solution at the current structural level. A special purpose FE code has been developed to perform homogenization and unsmearing automatically, through all structural levels of the composite. When the material behavior becomes not elastic, the superposition principle is no more valid; the asymptotic homogenization cannot be applied and is replaced by the method of numerical experiment. We solve, namely, a BVP for kinematically loaded cell of periodicity. For a given mean strains state E*, computed in the integration point of the current FE mesh, the following formulae are applied:

ui

Eij* x j

6 ij

1 Y

³V

ij

n j xi ds

G x  wY

(1)

wY

Figure 2. Two exemplary numerical experiments. The microstructure represented in the Figure is that of LMI cable. On the right, the scheme and the deformed finite element mesh for the mean uniform extension along axis x1 are shown. The picture on the left hand side shows the numerical experiment of shear of the representative cell.

Simple algebraic manipulations allow us to compute the current stiffness at the integration point, at the current temperature and in the current yield conditions. Two examples of many numerical experiments prescribed by (1) are shown in Figure 2. The right is a simple stretching in the direction x2, the left is for testing the shear stiffness. Although the examples in Figure 2 are solved with commercial FE code, we have developed our own special

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purpose FE program that computes the numerical experiments trough all levels of hierarchy. We start the numerical analysis of the composite at the micro level with estimation of effective mechanical and thermal coefficients of the immediate, next structural level. We perform then the analysis level by level up to top of the hierarchy. The technique of numerical experiment is fundamental in collecting the constitutive information but this knowledge about the materials behavior is to be elaborated and used in the context of FE algorithm. In the sequel of the paper we present synthetic information about two different methods of processing these data: numerical construction of yield surface and approximation of the nonlinear constitutive law with Artificial Neural Network (ANN).

3.

NUMERICAL CONSTRUCTION OF YIELD SURFACE IN THE SPACE OF STRESSES

A global strain tensor E* is imposed to the cell and it is monotonically increased to generate a kinematic loading path. This means that, numerically, a large number of equal kinematic steps are applied to the unit cell. Parameter D0 is chosen in such a way that the elastic frontier is reached and then the displacement is proportionally increased with steps all equal to 1/10 of the first one. The homogenized stress tensor 6 is then computed, by means of Eq. (12) for each step of the load history. The sequence of steps characterized by a fixed E*, generates a sequence of points in the stress space. Therefore we have one point, in the stress space, for each load step. These points are called interpolation points: here the behavior of the homogenized material is known. ­ ° ° ® ° Eij °¯

microscopic constitutive laws divV

1 Y

0 (micro equilibrium) D · § H ij dY ¨D 0  m 0 ¸ Eij* ; m 0,1,..mmax 10 ¹ © Y

(2)

³

Repeating the procedure for several different given tensors E* we know the behavior of the homogenized material in a discrete number of points and for a discrete variety of load situations. At this point we introduce a simplifying hypothesis: we assume that all the interpolation points, on different loading paths, characterized by the same number of steps, are on the same plastic surface, i.e. they are labeled by the same value of an internal variable. In this manner by connecting points relative to the corresponding

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steps of different loading paths it is possible to construct a series of plastic surfaces generated by the numerical experiments (see Fig. 3). This surface can be elaborated numerically in order to obtain elements needed for FE procedure, as it is described by Pellegrino et al. (1999). The very intuitive illustration of this post-processing is given in Fig. 3.

Figure 3. numerically defined yield surface (left) and a zoom illustrating the post-processing of the collected data for (right), for a detailed description we refer to Pellegrino et al. (1999).

4.

APPROXIMATION OF NONLINEAR CONSTITUTIVE LAW WITH ARTIFICIAL NEURAL NETWORK

Neural network can be considered as a collection of simple processing units that are mutually interconnected with variable weights. This system of units is organized to transform an input signal into an output signal. Both input and output signals are suitably defined to possess a needed physical interpretation. In our case this is a sequence of corresponding increments of average stresses and strains (for the homogenized model):

'Ȉ

ANN @^Ȉ 'E`

(3)

The sign “@” denotes an action of the neural operator on its arguments at the input layer. The weights of interconnections are modified by an iterative procedure to force the desired output signal as a response to a given input pattern. This process is called “training” and is continued until the error between the neural network output and the desired (known) output is minimized for a whole set of pairs: given input - known output. Nonsymbolic model is constructed as follows: neural network is trained first to reflect correctly the set of observed, experimental data (in our case, the input

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- output pairs come from numerical experiments described above in this paper). Then the networks automatic generalization capability (interpolation between some data sets) enables us to predict the material behavior i.e. to produce the graph stress-strain for an arbitrary sequence of stress or strain values. The networks simulation can be checked against the numerical (or real, experimental) results at this step. If the network’s prediction is satisfactory, the model is ready, if not, the new experimental data can be added to the existing training set and the network should be taught again. According to this approach the sufficiently trained neural network replaces the symbolic, mathematical description of the problem. Usually the neural network with one or two hidden layers is used.

Figure 4. Scheme of the ANN for approximation the constitutive data and the approximation of strongly non-linear behavior of the bundle of strands of VAC type superconductor (labeled with d in Fig. 1). We refer to Lefik and Schrefler (2002) for details.

An incremental (or rate) form of the constitutive law is preferable for the neural modeling. In this representation of the material behavior a stress increment 'V V at the output of the ANN is obtained as a response of the ANN on the input vector presented at the input layer. The input vector (for path dependent behavior) contains strains increment 'HH and current stress tensors Vcomponents (see Fig. 4 for this scheme). ANN can also approximate the functional dependence of elements of the effective constitutive tensor Dijkl on the constitutive tensors Di of each of n components of the composite and on scalar parameters ci characterising the geometry of the microstructure: eff Dijkl

^

ANN @ D (1) .. D ( n ) c k

`

(4)

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Independent variables are now the mechanical properties of the components and some parameters invented to describe its geometrical repartition in the cell. The functions to be approximated, namely Deff, are known from examples furnished by direct application of the homogenisation procedure for the exemplary cells.

5.

CONCLUSIONS

The global analysis consists in solving the homogenized problem. The solution is then localized, level by level, in order to discover the true state of stress at the micro level. At the micro level the modification of the physical properties of the materials are introduced according to their state of yielding. The new effective properties are then computed and the next loop of the analysis restarts. In the range of non-elastic behavior, we perform the homogenization that simply records the numerically modeled behavior of the structure. We avoid thus any “a priori” hypothesis concerning the shape of yield surface in the space of average stress.

ACKNOWLEDGEMENTS Support for this work was partially provided by PRIN 2004094015_002 “Thermo-hydraulic-mechanical and Electro-mechanical Modelling of ITER Superconducting Magnets” and by “KMM-NoE - Knowledge-based multicomponent materials for durable and safe performance - Network of Excellence”. This support is gratefully acknowledged.

REFERENCES C. Pellegrino, U. Galvanetto, B.A. Schrefler, 1999, Numerical homogenisation of periodic composite materials with non linear material components, Int. J. Numerical Methods Engineering, 46:1609-1637. M. Lefik, B.A. Schrefler, 2003, Artificial neural network as an incremental non-linear constitutive model for a finite element code. Computer Methods in Applied Mechanics and Engineering, 192/28-30: 3265-3283. D. Boso, M. Lefik, B.A. Schrefler, 2005, A multilevel homogenized model for superconducting strand termomechanics, Cryogenics, 45: 259-271 E. Sanchez-Palencia, 1980, Non-Homogeneous Media and Vibration Theory, Springer, Berlin. M. Lefik, B.A. Schrefler, 2002, One-dimensional model of cable-in-conduit superconductors under cyclic loading using artificial neural networks, Fusion Engineering and Design, 60(2): 105-117.

ANISOTROPIC FAILURE OF THE BIOLOGICAL MULTI-COMPOSITE WOOD: A MICROMECHANICAL APPROACH

K. Hofstetter, CH. Hellmich, H.A. Mang Vienna University of Technology (TU Wien), Institute for Mechanics of Materials and Structures, Karlsplatz 13/202, 1040 Vienna, Austria

Abstract:

Biological materials are characterized by an astonishing variability and diversity. Their hierarchical organizations are often well suited and seemingly optimized to fulfill specific mechanical functions. Still, once a (hierarchical) composite material has been adopted within a class of living organisms, its fundamental building principles, morphologies, or universal patterns of architectural organization) remain largely unchanged during biological evolution. Hence, entire material classes of biological materials exhibit common (universal) principles of (micro)mechanical design. In the theoretical framework of continuum micromechanics, such a building principle was recently expressed in quantitative terms, allowing for a prognosis of tissuespecific (inhomogeneous and anisotropic) elasticity properties of wood from tissue-specific volume fractions of (amorphous and crystalline) cellulose, hemicellulose, lignin, and water, as well as of lumen and vessel pores, based on universal elastic properties of (amorphous and crystalline) cellulose, hemicellulose, lignin, and water. We here extend these investigations to tissuespecific anisotropic strength properties. Macroscopic material strength is governed by strain peaks in the material microstructure, which can be suitably characterized by quadratic strain averages over material phases, being effective for material phase failure. Macroscopic stress states estimated from local shear failure of lignin agree very well with corresponding strength experiments. This expresses the paramount role of lignin as strengthdetermining component in wood.

Key words:

wood, lignin, anisotropic strength properties, continuum micromechanics, experimental validation

159 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 159–166. © 2006 Springer. Printed in the Netherlands.

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1.

INTRODUCTION

Biological materials are characterized by an astonishing variability and diversity. Their hierarchical organizations are often well suited and seemingly optimized to fulfill specific mechanical functions. This has motivated much research in the fields of bionics and biomimetics. The aforementioned optimization is primarily driven by selection during the evolution process. However, it is of great importance to notice that selection is realized at the level of the individual plant or animal (and not at a material level). Therefore, material optimization in the strictest sense of the word does not take place. Rather, once a (hierarchical) composite material has been adopted within a class of living organisms, its fundamental building principles (morphologies or universal patterns of architectural organization1) remain largely unchanged during biological evolution. Hence, entire material classes of biological materials exhibit common (universal) principles of (micro)mechanical design. In the theoretical framework of continuum micromechanics2,3, Hofstetter et al.4 recently expressed such a building principle in quantitative terms, allowing for a prognosis of tissue-specific (inhomogeneous and anisotropic) elasticity properties of wood from tissuespecific volume fractions of (amorphous and crystalline) cellulose, hemicellulose, lignin, and water, as well as of lumen and vessel pores, based on universal elastic properties of (amorphous and crystalline) cellulose, hemicellulose, lignin, and water. We here extend these investigations to tissue-specific anisotropic strength properties.

2.

FUNDAMENTALS OF CONTINUUM MICROMECHANICS

In continuum micromechanics2,3, a material is understood as a microheterogeneous body filling a representative volume element (RVE) with characteristic length l, l  d , d standing for the characteristic length of inhomogeneities within the RVE. The 'homogenized' mechanical behavior of the material, i.e. the relation between homogeneous deformations acting on the boundary of the RVE and resulting (average) stresses, can then be estimated from the mechanical behavior of different homogeneous phases (representing the inhomogeneities within the RVE), their dosages within the RVE, their characteristic shapes, and their interactions. Based on matrix-inclusion problems5,6, an estimate for the 'homogenized' stiffness of a material reads as3

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161 1

^hom

¦

f r ^ r :  r with  r

r

1 1 ½ 0 0 0 ª    º : ­®¦ f s ª    :  ^ s  ^0 º¼ ¾ , r :  ^ r  ^ ¼ ¬ ¬ ¯ s ¿

(1) where ^ r and f r denote the elastic stiffness and the volume fraction of phase r, respectively,  r is the concentration tensor of phase r, and  is the fourth-order unity tensor. The two sums are taken over all phases of the heterogeneous material in the RVE. The fourth-order tensor r0 accounts for the characteristic shape of phase r in a matrix with stiffness ^0 . The choice of this stiffness describes the interactions between the phases. For ^0 coinciding with one of the phase stiffnesses (Mori-Tanaka scheme), a composite material is represented (contiguous matrix with inclusions); for ^0 ^ hom (self-consistent scheme), a dispersed arrangement of the phases is considered (typical for polycrystals). If a single phase exhibits a heterogeneous microstructure itself, its mechanical behavior can be estimated by introduction of RVEs within this phase, with dimensions l2  d , comprising again smaller phases with characteristic length d 2  l2 , and so on, leading to a multistep homogenization scheme. Macroscopic ('homogenized') strength properties are derived from local phase failure, e.g. of the von Mises type: ‚( H r ,v , H r ,d ) =0, e.g. ‚( H r ,d ) = (2µ r H r ,d )2- sr2 = 0,

(2)

with µ r and sr as shear modulus and strength of phase r. H r ,v and H r ,d are the quadratic strain averages (second-order estimates) over phase domain Vr of the volumetric and the deviatoric part of a (microscopic) strain field H , H v tr H and H d H  1 3 H v 1 , where 1 is the second-order unity tensor7. For isotropic materials built up by isotropic phases, H r ,v and H r ,d can be related to macroscopic strains E prevailing at the boundary of the RVE, through derivatives of the potential energy stored in the RVE with respect to the bulk modulus kr and the shear modulus P r of (isotropic) phase r7,8: f r H r2,v 4 f r H r2,d

w ^ hom :E w kr

wk hom wP hom Ed : Ed ,  tr E 2  2 w kr w kr

w ^ hom E: :E w Pr

wk hom wP hom Ed : Ed .  tr E 2  2 w Pr w Pr

E:

(3)

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K. Hofstetter, Ch. Hellmich, H.A. Mang

Transversely isotropic material and phase behavior is approximated by bulk and shear moduli krapp and P rapp , related to ('average microscopic') strain,

Hr

 r : E , and ('average microscopic') stress, V r krapp {

^r

V r ,m V r ,d , P rapp { , H r ,v 2 H r ,d

(4)

with mean stress V r ,m

1/ 3trV r , volumetric strain H r ,v

deviatoric stress V r , d

V r  V r , m 1 , V r ,d

deviatoric strain H r ,d

3.

: H r , in phase r 9:

H r  1/ 3H r ,v 1 , H r ,d

trH r , equivalent

1/ 2 V r ,d : V r ,d and equivalent

 H r ,d : H r ,d .

MICROMECHANICAL MODEL FOR WOOD

The components of the wood cell wall, namely amorphous cellulose, crystalline cellulose, hemicellulose, lignin, and water lumped together with extractives, exhibit universal properties inherent to all wood species (Tab. 1). Their mechanical interactions are considered through four homogenization steps4 (Fig. 1): 1. Within an RVE of polymer network with 8-20 nm characteristic length, hemicellulose, lignin, and water are intimately mixed, occupying volume fractions fhemcel , flig , and fH 2Oext 1  fhemcel  flig . The disorder and the intense mixing of the chemical components in the network motivate the use of a self-consistent scheme with inclusions of spherical shape. The microscopic strength of lignin can be described adequately by a von Mises strength criterion. 2. Within an RVE of cell wall material with 0.5-1 µm characteristic length, cylindrical fiber-like aggregates of amorphous cellulose and of crystalline cellulose, exhibiting typical diameters of 20-100 nm, are embedded in the polymer network of the previous homogenization step. Respective volume fractions are f amocel , f crycel , and f polynet 1  f amocel  f crycel . The fibrils are helically wound around the lumens within the cell wall with an average inclination angle to the cell axis (microfibril angle T ) of 20←10. The behavior of such a composite material is suitably estimated by a Mori-Tanaka scheme. 3. Within an RVE of softwood with 100-150 µm characteristic length, cylindrical pores with characteristic diameters of 20-40 µm and volume fraction fˆlum , representing the cell lumens, are embedded in a contiguous (transversely isotropic) matrix built up by the cell wall material of homo-

Anisotropic Failure of the Biological Multi-Composite Wood

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Figure 1. Four-step homogenization scheme.

genization step 2. Again a Mori-Tanaka scheme is used for estimation of the stiffness of the composite material 'softwood'. 4. Within an RVE of hardwood with 2-4 mm characteristic length, cylindrical pores with characteristic diameters of 400-500 µm and a volume fraction f ves , representing vessels, are embedded in the contiguous matrix built up by the softwood-type porous material of homogenization step 3. Stiffness estimates for the composite material 'hardwood' are again obtained by application of a Mori-Tanaka scheme. The effective strains in the phases are characterized by second-order estimates at all levels of the homogenization scheme, allowing for assignment of an effective lignin shear strain to a macroscopic strain state E through application of Eqs. (3) and (4) at each homogenization step. The macroscopic strain states related to H lig , d fulfilling the microscopic strength criterion (see Eq. 2) define an estimate for a (tissue-dependent, anisotropic) macroscopic strength criterion. In tension perpendicular to the lumen axis, interface failure between the polymer network and the cellulose fibers precedes the (shear) failure of lignin. Thus, the effective lignin strain in the corresponding limit state is estimated for de-activated stiffness of cellulose fibers ( ^ amocel , ^  ). Also the ultrastructural water in the cell wall does not crycel contribute to the stiffness of wood up to tensile failure, because of its incapacity to carry related tensile stresses ( ^ H 2O  ).

K. Hofstetter, Ch. Hellmich, H.A. Mang

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4.

VALIDATION OF THE MICROMECHANICAL MODEL

The micromechanical model is validated for stiffness and strength properties based on two independent sets of experimental data11,4. Stiffness values ^ SW and ^ HW predicted by the micromechanical model, on the basis of universal phase stiffness properties of Tab. 1 [experimental set I], for tissue and sample-dependent volume fractions of the universal constituents as well as of lumen and vessel pores [experimental set IIa4, on the basis of a degree of crystallinity of cellulose of 0.6612], are compared to corresponding experimentally determined stiffness values [experimental set IIb4]; across a variety of softwood and hardwood species (Fig. 2). Stiffness predictions and experiments agree very well, both longitudinally (in the direction of the lumen axis) and transversely (perpendicular to the lumen axis), see Fig. 2. Validation of strength predictions requires extension of experimental set I by the shear strength of lignin, which is the only universal strength property of wood in the micromechanical model. Lignin is a complex phenylpropanoid polymer14, the (tensile) strength of which does not fall below 35 MPa15. This corresponds to a shear strength of 35/¥3=20.2 MPa (Tab. 1). As for strength, experimental set IIb contains tensile and compressive strengths, st and sc, of a variety of wood species, loaded either parallel or perpendicular to the lumen axis16-27. The good agreement between modelpredicted strength values and corresponding experimental results (Fig. 3) expresses the central role of lignin as strength-governing component in wood. As regards the tensile strength parallel to the lumen axis, the model Table 1. Experimental set I: 'Universal' (tissue-independent) phase properties (see [13] for experimental sources) Phase Amorph. cell. Hemicellulose Lignin Water+extract.

Crystall. cell.

Material behavior isotropic isotropic isotropic isotropic

trans. isotropic

Bulk modulus Shear modulus k [GPa] µ [GPa] kamocel=5.56 µamocel=1.85 khemcel=8.89 µhemcel=2.96 µlig =2.30 klig =5.00 µH2oext=0 kH2oext=2.30 Stiffness tensor components cijkl [GPa] ccrycel,1122=0 ccrycel,1111=34.86 ccrycel,3333=167.79 ccrycel,2233=0

Shear strength s [MPa]

slig=20.2

ccrycel,1313=5.81

Anisotropic Failure of the Biological Multi-Composite Wood

165

Figure 2. Comparison of predicted and experimental elastic constants.

provides a lower limit for the macroscopic strength of wood. After the failure of lignin, the cellulose fibers may keep carrying the load as separated strands. Their strength may remarkably improve the macroscopic tensile strength.

Figure 3. Comparison of predicted and experimental strength values.

Given its impressive predictive capabilities, the micromechanical model for wood presented herein is expected to support optimization of the wood drying process as well as analyses of wood structures.

REFERENCES 1. S.J. Gould and R.C. Lewontin, The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptionist programme, Proceedings of the Royal Society London Series B 205(1161), 581-598, 1979. 2. P. Suquet, editor, Continuum Mircomechanics (Springer Verlag, Wien, New York, 1997). 3. A. Zaoui, Continuum micromechanics: Survey, ASCE Journal of Engineering Mechanics, 128(8), 808-816, 2002. 4. K. Hofstetter, Ch. Hellmich, and J. Eberhardsteiner, Development and experimental validation of a continuum micromechanics model for the elasticity of wood, European Journal of Mechanics A/Solids, accepted for publication, 2005.

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5. J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proceedings of the Royal Society London Series A 241, 376-396, 1957. 6. N. Laws, The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material, Journal of Elasticity 7(1), 91-97, 1977. 7. L. Dormieux, A. Molinari, and D. Kondo, Micromechanical approach to the behavior of poroelastic materials, Journal of Mechanics and Physics of Solids 50, 2203-2231, 2002. 8. W. Kreher, Residual stresses and stored elastic energy of composites and polycrystals, Journal of Mechanics and Physics of Solids 38(1), 115-128, 1990. 9. W. Kreher and A. Molinari, Residual stresses in polycrystals as influenced by grain shape and texture, Journal of Mechanics and Physics of Solids 41(12), 1955-1977, 1993. 10. K. Hofstetter, Ch. Hellmich, and J. Eberhardsteiner, Poro-microelasticity of wood : the relevance of the microfibril angle, Proceedings of the 3rd Biot Conference on Poromechanics, Oklahoma, USA, 2005. 11. Ch. Hellmich, J.-F. Barthélémy, and L. Dormieux, Mineral-collagen interactions in elasticity of bone ultrastructure - a continuum micromechanics approach, European Journal of Mechanics A/Solids 23, 783-810, 2004. 12. D. Fengel and G. Wegener, Wood. Chemistry, Ultrastructure, Reactions (Verlag Kessel, Remangen, 2003). 13. K. Hofstetter, Ch. Hellmich, and J. Eberhardsteiner, A continuum micromechanics approach to wood elasticity, Proceedings of the 16th International Conference on Computer Methods in Mechanics, Czestochowa, Poland, 2005. 14. G. Sengupta and P. Palit, Characterization of a lignified secondary phloem fibre-deficient mutant of jute (Corchorus capsularis), Annals of Botany 93, 211-220, 2004. 15. eFunda (April 10, 2005); http://www.efunda.com/materials/polymers/properties/polymer datasheet.cfm ?MajorID=phenolic&MinorID=3. 16. F.-W. Bröker, Technologische Untersuchungen an Pinus radiata-Importhölzern aus Chile, Holz als Roh- und Werkstoff 38, 345-350, 1980. 17. F.-W. Bröker and S. Olsen, Orientierende holztechnologische Untersuchungen an Pinus radiata D. Don aus Neuseeland, Holz als Roh- und Werkstoff 46, 335–339, 1988. 18. J.R. Goodman and J. Bodig, Orthotropic strength of wood in compression, Wood Science 4(2), 83–94, 1971. 19. K. Hemmer, Versagensarten des Holzes der Weißtanne (Abies alba) unter mehrachsiger Beanspruchung (Doctoral Thesis, Universität Karlsruhe, 1984). 20. W. Kokocinski and J. Raczkowski, Einfluß der Reibung zwischen Prüfkörper und Druckplatten auf die Druckfestigkeit parallel zur Faser, Holz als Roh- und Werkstoff 36, 241–246, 1978. 21. F. Kollmann, Technologie des Holzes und der Holzwerkstoffe (Springer Verlag, Berlin, Heidelberg, New York, second edition, 1982). 22. D.E. Kretschmann and D.W. Green, Modeling moisture content-mechanical property relationships for clear southern pine, Wood and Fiber Science 28(3), 320–337, 1996. 23. H. Kuehne, H. Fischer, J. Vodoz, and T. Wagner, Über den Einfluß von Wassergehalt, Faserstellung und Jahrringstellung auf die Festigkeit und die Verformbarkeit schweizerischen Fichten-, Tannen-, Lärchen-, Rotbuchen- und Eichenholzes (Technical report No. 183, Eidgenössische Materialprüfung, Versuchsanstalt Ind., Zürich, 1955). 24. K. Möhler and P. Beyersdorfer, Festigkeitsuntersuchungen an einheimischem Douglasienholz als Bauholz, Holz als Roh- und Werkstoff 45, 49–58, 1987. 25. L. Vorreiter, Verbreitung und Verwertung der Schwarzkiefer, Holz als Roh- und Werkstoff 7/8(1), 42, 1944/45. 26. G.B. Walford, The Mechanical Properties of New Zealand-Grown Radiata Pine for Export to Australia (Bulletin No. 93, Forest Research Institute, 1985). 27. Wood Handbook (Forest Products Society, USA, 1999).

ANTIPLANE CRACK IN A PRE-STRESSED FIBER REINFORCED ELASTIC MATERIAL

Eduard-Marius Craciun Ovidius University of Constanta, Faculty of Mathematics and Informatics, Bldv. Mamaia nr.124, Constanta, cp. 900529, ROMANIA, e-mail: [email protected].

Abstract:

The behavior of a fiber-reinforced elastic material subject to initial stresses in antiplane state is considered in the paper. We determine the incremental values of the shear stresses producing crack propagation and we determine the crack propagation direction in a pre-stressed orthotropic or isotropic elastic material. The problem of two collinear cracks interaction it is also studied in the paper.

Key words:

antiplane cracks, Guz’s representation formula, Sih’s criterion, Griffith-Irwin fracture theory.

1.

INTRODUCTION

In this paper we study the incremental fields of cracked fiber-reinforced elastic materials subjected to initial stresses in antiplane state. The problem of representation of the elastic fields for anisotropic materials was studied by Lekhnitsky [1], by Sih and Leibowitz [2] and for cracked pre-stressed materials by Guz [3]. For a Mode III crack contained in a pre-stressed orthotropic or isotropic elastic material we determine the critical values of incremental shear stresses which produce crack propagation and the direction of crack propagation. We obtain that for a pre-stressed isotropic material the critical value of incremental shear stress producing crack propagation predicted by Sih’s theory is always smaller as that given by classical Griffith’s and Irwin’s theory.

167 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 167–176. © 2006 Springer. Printed in the Netherlands.

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168

Also we study the behavior of two collinear and unequal cracks in a prestressed, orthotropic, linear elastic material representing a fiber reinforced elastic composite. Using Guz’s representation theorem of incremental fields (see [3]) we solve our mathematical problem using the theory introduced to solve the Riemann-Hilbert problem (see [4], [5]) and assuming that the initial stresses have a constant value. We determine the critical values of incremental shear stresses which end to crack propagation and we study the interaction of the cracks.

2.

REPRESENTATION OF THE INCREMENTAL FIELDS IN MODE III

The incremental field of a pre-stressed, orthotropic, linear elastic material representing a fiber reinforced-elastic composite is presented in this Section following Guz (see [3]). In the absence of initial applied stresses the following representation reduce to those given by Lekhnitsky (see [1]). We consider a pre-stressed orthotropic linear elastic material. We take as coordinate planes the symmetry planes of the orthotropic material. We assume an unbounded material containing a right crack of length 2a situated in x1 x3 - plane. We suppose that the initial applied stress V 0 acts in the direction of the x1 - axis. We assume that V 0 is sufficiently small such that it produces infinitesimal initial strains. We assume that the upper and the lower faces of the crack are loaded by symmetrically distributed incremental shear stresses having constant value W ! 0 and acting in the direction of the x3 -axis. Finally, we suppose that the initial deformed equilibrium configuration of the body is locally stable. Under the assumed condition, the incremental equilibrium state of the material is an antiplane state relative to the x1 x3 - plane, and the third fracture mode occurs. We have the following conditions:

u1

u2

0, u3

u3 ( x1 , x2 ), T31

Z1331u3,1 , T 23 Z2332u3,2 , (2.1) ___

___

where ui (i

1,3) and respectivelly T i j , (i, j 1,3) are the components

of the incremental displacement and respectivelly of___the nominal stress. The instantaneous elasticities Z klmn (k , l , m, n 1,3) involved in (2.1) can be expressed through the elastic coefficients of the material and through the initial applied stress V 0 acting in the direction of the fibers, by the following relations:

Z1331 C55  V 0 , Z2332

C44 ,

C55

G13 ,

C44

G23 , (2.2)

Antiplane Crack in a Pre-stressed Fiber Reinforced Elastic Material

169

where G13 and G23 are the shear moduli of considered material. We have the following representation due to Guz (see [3]) of the incremental fields:

u 3 2

1

Z 2332

Re

) 3 ( z3 )

P3

, T13

2 Re P 3< 3 ( z 3 ) and T 23

2 Re < 3 ( z 3 ), (2.3)

where

< 3 ( z3 )

d ) 3  z3 , z3 dz 3

x1  P 3 x 2 and P 3

iq with q

Z1331 .(2.4) Z 2332

The boundary conditions on the two faces of the crack for  a  x1  a are: (2.5) T 23  x1 T 23 x1 , 0 T 23  x1 T 23 x1 , 0 W  x1 .









We also suppose that at large distances from the crack the body is not perturbed. Thus,

lim ^u3  x1 , x2 ,TD 3  x1 , x2 ` 0, lim ^) 3  z3 , < 3  z3 ` 0, r r of

x12  x22 .

z3 of

Eqs. (2.5) expressed with the limit values of the potential < 3  z3 become: 



< 3  x1  < 3  x1 W  x1 , < 3  x1  < 3  x1 W  x1 . (2.6) Using the Eqs. (2.6), we obtain a Riemann-Hilbert problem which is solved using Plemelj function and Cauchy integrals (see [5]). For a constant shear stress , W  x1 W ! 0 we have following representation of the complex potentials:

· z3 W§ W < 3  z3  ¨  1¸ , ) 3  z3  ¸ 2 ¨ z32  a 2 2 © ¹

3.





z32  a 2  z3 . (2.7)

SIH’S FRACTURE CRITERION We denote by w the involved incremental strain energy density; we have

2w T13u3,1  T 23u3,2 .

(3.1)

It follows from (2.1) that 1 1 u3, 1  2Z 2332 Re P31< 3 ( z3 ), u3, 2  2Z 2332 Re < 3 ( z3 ). (3.2)

Thus we get

E.-M. Craciun

170 2

1 w 2Z 2332 < 3 ( z3 ) .

(3.3)

The expression (2.7) of the complex potential shows that near a crack tip, the incremental strain energy density has a singular part as well as a regular part. To obtain the singular part w, we denote by r and M the radial distance from the considered crack tip and the angle between the radial direction and the line ahead the crack. Starting with the expression (2.7) of the complex potential and using (3.3), after long manipulations, we conclude that near the considered crack tip the incremental strain energy density has the following structure:

w(r , M )

1 S (M )  regular part, r

(3.4)

where

S (M )

K 2a s (M ), s (M ) 4Z2332

1 2

cos M  q 2 sin 2 M

.

(3.5)

Extending the validity of Sih’s energetical fracture criterion (see [6]) for orthotropic or isotropic pre-stressed materials, we assume that: H1: Crack propagation starts in a radial direction Mc along which the incremental strain energy density S (M ) is a minimum. H2: The critical intensity

Sc Smin S (Mc )

(3.6)

governs the onset of crack propagation and represents a material constant, independent of crack geometry, loading and initial applied stresses. The assumed independence of Sc of the initial applied stress is similar to Guz’s assumption (see [3]) concerning the independence of Griffith’s specific strain energy J on the initial applied stresses. This generalization of Sih’s fracture criterion requires no a priori assumption concerning the direction in which the energy is released by the separating crack surfaces. At the same time, according to (3.5) and (3.6) the critical incremental shear stress Kc for which crack propagation starts in the critical direction Mc can be obtained using the equation (3.7) aK c2 4Z2332 Sc / s (Mc ) . In the above equation, Sih’s new material parameter Sc replaces Griffith’s specific strain energy J in the new theory of brittle fracture. Once Sc is known, (3.7) can be used to obtain Kc. As it was shown by Sih (see [6]) for isotropic materials without initial stresses, S c can be expressed through J . In [7] it was shown that Sih’s relation is true even for a pre-stressed

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171

isotropic material if the initial applied stress produces only infinitesimal strains and are also given conditions under which Sc can be expressed through J for orthotropic materials. From (3.3) we get s min = s(0) = 1 if q < 1 and s min = s ( rS / 2 ) = q 1 if q > 1. (3.8) Hence the critical angle Mc along which crack propagation will start is given by the equations (3.9) Mc = 0 if q < 1 and Mc = rS / 2 if q > 1.

4.

PRE-STRESSED ISOTROPIC MATERIAL

For a pre-stressed isotropic material the instantaneous elasticities are given by the equations Z1331 G  V 0 , Z 2332 G, (4.1)   where G represents the shear modulus. We denote by Tc and Tc the critical shear stresses given by Sih's theory for q < 1 and for q > 1, respectively. From (3.7), (3.8) and (4.1) we get

 aTc2 4GSc and Mc 0 if V 0 < 0

(4.2)

 V aTc2 4G 1  0 Sc and Mc r S / 2 if V 0 > 0. G

(4.3)

and

Hence the crack will propagate along its line if the initial applied load is a compression, and the crack will propagate perpendicular to its line if the initial applied load is a traction. At the same time, from (4.2) and (4.3) we can see that the critical incremental shear stress corresponding to an initially compressed material is smaller as that which corresponds to a material which is initially loaded in traction. This is obviously a plausible result of Sih’s theory. According to (3.8), Griffith’s and Irwin’s classical theory for any admissible value of the initial applied stress V 0 gives the following value of the critical  angle Mc and of the critical incremental shear stress W c :

V  aW c2 4G 1  0 J and Mc 0 , for any V 0 . G 

(4.4)

As in Sih’s theory, W c is decreasing if the initial applied load is a compression, and it is increasing if the initial applied load is a traction. But

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E.-M. Craciun

according to this classical theory, the crack will propagate always along its line. As we have mentioned the above results are true only if the initial applied stress V 0 produces only infinitesimal strains. As was shown in [7] in this case in the first fracture mode the critical angle is always zero, even if the material is initially deformed. Consequently, Sih’s relation [6] connecting the material parameters J and Sc is true and we have (4.5) Sc J 1  2Q / ª¬S 1 Q º¼ , where v is Poisson's ratio. For realistic values of v, from (4.2) and (4.3) we can conclude that the critical incremental shear stress given by Sih’s theory is smaller than that given by the classical theory, even if the initial applied load is a compression, i.e.,   (4.6) Tc  W c . Obviously, the above predictions of Sih’s new theory must be confirmed or infirmed by fracture experiments concerning third mode crack propagation in pre-stressed isotropic materials. Ending this section we observe that if V 0 = 0, i.e. q = 1, from (3.2) we get s( M ) = 1, for any M . Hence, in this case, Sih’s fracture criterion can say nothing about the critical angle M c along which crack propagation takes place. Obviously, the same situation appears if G 13 = G 23 and the considered anisotropic material is not submitted to an initial applied load.

5.

TWO COLLINEAR UNEQUAL CRACKS

We consider an unbounded composite containing two colinear cracks situated in the same plane, parallel with the reinforcing fibers and with the initial applied stress V 0 . The plane containing the cracks is taken as x1 x3 -plane, the x 2 - axis being perpendicular to the crack faces. We suppose that given incremental antiplane shear stresses IJ = IJ( x1 ) act on the upper and lower faces of the cracks. The produced incremental state is an anti-plane state relative to x1 x 2 - plane. Hence we can use the complex representation described in the previous section to study our problem corresponding to the Mode III in classical fracture mechanics. We denote by L the cut corresponding to the segments (  a, l 2 a ) and ( l1a, a ). Here a > 0 is a given positive constant and l1 and l 2 are given positive numbers satisfying the restriction 0 < l1 , l 2 < 1. In this case we have the same boundary and large distance conditions as in Section 2, but for x1    a, l 2 a ‰  l1a, a .

Antiplane Crack in a Pre-stressed Fiber Reinforced Elastic Material

173

Assuming W ( x1 ) W const. ! 0 , using the Plemelj’s function theory and imposing the uniformity of the incremental displacement field, we get the following equation for the complex potential < 3 ( z 3 ) :

§ · z32  Maz3  a 2 N  1¸ , < 3 ( z 3 ) = W ¨ ¨ ( z  a 2 )( z  l a )( z  l a ) ¸ 3 3 2 3 1 © ¹

(5.1)

where

M

R2 S0  S 2 R0 , N R0 S1  S0 R1

M (l1 , l2 )

N (l1 , l2 )

R2 S1  S 2 R1 R0 S1  S0 R1

(5.2)

and 1

Rn

R(t )

tn ³l R(t ) dt , 1

1

Sn

tn ³ S (t ) dt , l2

(1  t 2 )(t  l2 )(t  l1 ) , S (t )

n 1, 2,3,

(5.3)

(1  t 2 )(t  l2 )(t  l1 ) . (5.4)

In order to evaluate the energy release rate corresponding to a crack tip we have to know the asymptotic values of the complex potential < 3 ( z 3 ) and of the incremental displacement u3 and incremental shear stress T 23 in a small neighborhood of the crack tip. We start to obtain these asymptotic values near the crack tip a of the crack ( l1a, a ) by taking x1 a  r cos M , x2 r sin M , (5.5) where the polar coordinates r and M denote the radial distance from the crack tip and the angle between the radial line and the line ahead of the crack, respectively. Now, from (5.1) we obtain the asymptotic values of the complex potentials in a small neighborhood of the tip a,

< 3 ( z3 )

§ a  m1W ¨ ¨ 2r ©

1 · ¸ , ) 3 ( z3 ) F 3 (M ) ¸¹

 m1W





2ar F 3 (M ) , (5.6)

where

m1

m1 (l1, l 2 )

l12  l1M  N (1  l12 )(l1  l 2 )

and F 3 (M )

cos M  P3 sin M . (5.7)

Through (2.3)1,3 and (5.6) we obtain the asymptotic values of the incremental displacement u3 and of the incremental shear stress T 23 and according to Irwin’s formula for the energy release rate G (a ) corresponding to the propagation of the tip a of the crack ( l1a, a ) we get the following value: G (a ) = S aW 2 m12 q 2 / 2 , (5.8)

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In a similar way we get the corresponding energy release rates of the other tips: G (l1a ) S aW 2 m22 q 2 / 2, G (l2 a ) S aW 2 m32 q 2 / 2, G ( a ) S aW 2 m42 q 2 / 2. (5.9)

6.

CRITICAL VALUES OF ANTIPLANE SHEAR STRESSES CRACKS INTERACTION

In order to obtain the critical values of the applied incremental stress for which one of the tips starts to propagate, we use Griffith’s energetical criterion. According to Griffith’s criterion (see for instance [7]) the tip a starts to propagate if the following condition is fulfilled G (a ) = 2J . (6.1) Let IJ(a) be the critical value of the applied incremental stress for which the tip a starts to propagate, than from (5.8) and (6.1) we get

W ( a ) 2q

J (1  l2 )(1  l1 ) , S a 1 M  N

(6.2)

where J 0 is the constant positive number introduced by (5.3). The other values corresponding to the propagation of the other tips are consequently

W (l1a) q W (l2 a)

2 2J (1  l1 )(l1  l2 ) , S a l12  l1M  N

2 2J (1  l2 )(l1  l2 ) q , W (a) S a l22  l2 M  N

2q

(6.3)

J (1  l2 )(1  l1 ) . S a 1 M  N

To study the interaction of the crack we introduce now the following dimensionless quantities:

F1 (l1, l 2 )

W (a) F

W (l 2 a) F 3 (l1, l 2 ) F F 4 (l1, l 2 )

W (a) F

2(1  l1 )(1  l 2 ) W (l1a ) , F 2 (l1, l 2 ) 1 M  N F

(1  l1 2 )(l1  l 2 ) , l12  l1M  N

(1  l1 2 )(l1  l 2 ) , l 22  l 2 M  N 2(1  l 2 )(1  l1 ) , with F 1  M  N

q

2J . S a0

(6.4)

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175

The analysis of the interaction of cracks requires the knowledge of the order between the four incremental stresses, for different values of the dimensionless parameters l1 and l 2 . The necessary order relation will be obviously known if the values of the dimensionless quantities F 1 , F 2 , F 3 and F 4 are known. The tip which will propagate first is that which corresponds to the minimum value of F j for j 1, 4 . Numerical analysis for the equal and unequal cracks led us to the following : - if the distance between the equal cracks is much smaller than their length, there exists a strong interaction between the cracks and they tend to unify; - if the distance between the equal cracks is much greater than their length, there is a weak interaction between the cracks, - the tips of the longer crack start to propagate simultaneously and the interaction between cracks is relatively weak when the distance between them is relatively large compared with the length of the shorter crack. - that if the distance between the cracks is relatively small compared with the length of the short crack, the interaction between the cracks is strong. Therefore, the inner tips start to propagate first.

7.

FINAL REMARKS

Using the new fracture theory we determine the direction of crack propagation as well as the critical incremental shear stress assuming the existence of the third fracture mode. Also, in the case of a pre-stressed isotropic material we compare the results obtained with those given by the classical Griffith’s and Irwin’s theory. Finally we can conclude that the critical incremental shear stress producing crack propagation and predicted by Sih’s theory is always smaller as that given by the classical theory. Moreover, according to the new theory, the crack will propagate along its line if the initial applied load is a compression and perpendicular to its line if the initial applied load is a traction. For the case of two collinear cracks we find that if the lengths of the cracks are much greater than the distance between them, the interaction between the cracks is strong and the cracks tend to unify and if the lengths of the cracks are much smaller than the distance between them, the interaction between cracks is weak. In case of equal cracks all the tips start to propagate simultaneously and in the case of unequal cracks the tips of the longer crack start to propagate nearly in the same time. Using the same Muskhelisvili technique (see [5]) of complex potentials and following Bigoni et all. (see [8], [9]) many other topics of importance in

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fracture mechanics, such as the effects of pre-stressed cracks in incompressible solids or effects of cracks bounded with linear interface in a pre-stressed elastic composite remain to be investigated.

ACKNOWLEDGEMENTS The author gratefully aknowledge to Professor Tomasz Sadowski and to IUTAM for the support at the Symposium.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

S.G. Lekhnitski, Theory of elasticities of anisotropic body, (Holden Day, San Francisco, 1963). G.C. Sih and H. Leibowitz, Mathematical theories of brittle fracture. in Fracture An Advanced Treatise Mathematical Fundamentals, edited by H. Leibowitz, (Academic Press, New York, 1968). A.N. Guz, Mechanics of Brittle Fracture of Pre-stressed Materials, (Visha Schola, Kiev, 1983). N.D. Cristescu, E.M. Craciun and E. Soós, Mechanics of Elastic Composites, (Chapman & Hall/CRC Press, Boca Raton, 2003). N.I. Muskhelisvili, Some Basic Problems of the Mathematical Theory of Elasticity, (Noordhoff Ltd., Groningen, 1953). G.C. Sih, A special theory of crack propagation. In: G.C. Sih (Ed-), Mechanics of Fracture,Vol. 1, pp. XXI-XLV, Noordhoff, Leyden, 1973. E.M. Craciun and E. Soos, Sih’s generalised fracture criterion for prestressed orthotropic and isotropic materials, Rev. Roum. Sci Techn.Mec.Appl., 44, 533-545, (1999). E.Radi, D. Bigoni and E Capuani, Effects of pre-stress on crack tip fields in elastic, incompressible solids, Int. J. Solids Structures, 39, 3971-3996, (2002). D. Bigoni, S.K. Serkov, A.B. Movchan and M. Valentini, Asymptotic models of dilute composites with imperfectly bonded inclusions Int J. Solids Structures, 35, 3239-3258, (1998).

CHARACTERIZATION AND PRACTICAL APPLICATION OF A MULTISCALE FAILURE CRITERION FOR COMPOSITE STRUCTURES

F. Laurin, N. Carrère, J.-F. Maire and D. Perreux 1

ONERA/DMSE, BP72, 92322 Chatillon Cedex, France ; 2LMARC, 25030 Besançon, France.

Abstract:

This paper presents a multiscale progressive failure criterion for composite materials which permits to predict the failure of laminates from elementary unidirectional ply data (behaviour and strength). This kind of approach is predictive for different stacking sequences and takes into account the effects of plies failure on macroscopic behaviour. The present model allows to obtain good predictions of the stress/strain curves and failure envelopes of multilayered composites under complex tensile loadings. Nevertheless, for laminates subjected to compressive loadings, structural phenomena as buckling occur and lead to premature laminate failure. In order to predict accurately the laminate failure under compressive loadings, it is necessary to consider the laminate as a structure and to develop methods which are able to describe the competition between material (plies failure) and structural (buckling) aspects. Finally, we present a comparison with experimental data from literature on tubes under axial compressive loadings.

Key words: Multiscale, failure, progressive, degradation, criterion, structure, buckling.

1.

INTRODUCTION

Laminate composite materials are widely used in aerospace and automobile industries because of their very attractive ratio weight / rigidity / strength. Prediction of laminate composite failure is a key point for the design of engineering structures, there is still a lack of confidence 177 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 177–184. © 2006 Springer. Printed in the Netherlands.

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into existing failure criteria important safety factors. To address this issue, a recent effort has been by Hinton and Soden1, called the “World Wide Failure Exercise” (WWFE), to assess the state-ofthe-art of prediction capabilities for laminate behaviour and failure. The main conclusions of this exercise were: (a) “best” approaches predicted correctly the final failure for classical laminates under complex tensile loading, but (b) the predictions of macroscopic behaviour and final failure for [±ș]s laminates are in poor agreement with test data and (c) all the material failure approaches overestimate the failure of laminates subjected to compressive loading even for classical quasi-isotropic [0°/±45°/90°]s laminates. The aim of this study is to propose a failure approach which can (i) predict failure behaviour for various stacking sequences under a large range of loading conditions from the properties (behaviour and strength) of the unidirectional ply, but also (ii) take into account the effect of buckling on laminates final failure under compressive loadings. The multiscale progressive failure approach is presented in section 2 and compared with test data from literature for multilayered laminates under tensile loadings. The method, which permits to model the buckling in a finite element (FE) code, is briefly described in section 3. Finally, the competition between material (plies failure) and structural (buckling) aspects is discussed and the results of FE calculations on composite tubes under axial compressive loadings are compared with experimental data.

2.

MULTISCALE FAILURE APPROACH

We have chosen to develop a mesoscopic failure approach in order to be predictive for different stacking sequences and take into account the effects of plies failure on macroscopic behaviour. The principle of the presented multiscale progressive failure approach can be shared into four sub problems: (i) choice of the mesoscopic behaviour, (ii) definition of a mesoscopic failure criterion, (iii) introduction of the degradation model of the failed ply in a laminate, and (iv) finally definition of the final laminate failure. In the following, these four considerations are discussed in details. We proposed a failure criterion which is based on Hashin’s2 assumptions. Two failure modes are considered: a fibre failure mode (FF) and an interfibre failure mode (IFF). For each failure mode, we distinguish the failure in tension and in compression, because the failure mechanisms are very different.

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The interfibre failure (IFF) mode distinguishes the tension failure (f2+) and the compression failure (f2-) by modelling each one by a failure criterion. ­ ° f 2 °° ® °  ° f2 ¯°

2

· § § V 22 · W 12 ¸¸ ¸¸  ¨¨ ¨¨ © Sc 1  p V 22 ¹ © Yt ¹ 2

§ V 22 · § · W12 ¨¨ ¸¸  ¨¨ ¸¸   Y S 1 p V 22 ¹ © c ¹ © c

2

1  d f 2

if V 22 t 0

(1) 2

1  d f 2

if V 22  0

where Yt, Yc, Sc are respectively the transverse tension and compression strengths and the in plane shear strength. The two original points of this interfibre failure criterion are: (i) a better description of the strength of the UD ply under shear and transverse loadings (thanks to the introduction of the coefficient p which reinforced or decreased the shear strength Sc in function of transverse loading) and (ii) the use of a degradation variable df, based on classical damage model, to represent the degradation of the interfibre strengths due to premature single fibres failure (statistical effect on fibre strength). The fibre failure (FF) mode distinguishes the tension failure (f1+) and the compression failure (f1-) by modelling each one by a failure criterion. f1

§ V 11 · ¨ ~ ¸ ¨ X d ¸ © t 2 ¹

f1

§ V 11 · ¸¸ ¨¨ © Xc ¹

2

~ 1 with X t

X t e  h1r d 2  X tyarn (1  e  h1r d 2 ) if V 11 t 0

(2)

2

1 if V 11  0

where Xt, Xc are respectively the longitudinal tension and compression strengths of the UD ply and Xtyarn is the strength of the yarn fibres. Several authors3 have demonstrated experimentally that the state of degradation of the matrix has a strong influence on the longitudinal tension strength of the UD ply. So, we have introduced a coupling between the efficient ~ longitudinal tension strength X t and the interfibre degradation of the matrix d2. The h1r coefficient represents the effect of the matrix degradation on the longitudinal tensile strength, which is connected to the fibre strength distribution. Fig. 1 presents failure envelopes of UD plies for different materials, the model results are in good agreement with experimental data4 mainly due to the improvements introduced into the interfibre failure mode.

F. Laurin, N. Carrère, J.-F. Maire and D. Perreux

180

Figure 1. Failure envelopes of UD ply a) in plane stress (ı22,IJ12) for two different materials (T300/LY556 and Eglass/LY556) and b) in plane stress (ı11,ı22) for Eglass/MY750.

In order to predict correctly a ply failure, it is necessary to have a good estimate of the mesoscopic stresses; hence, the choice of the behaviour at the ply scale is essential. The thermo-viscoelastic behaviour which is used to simulate the behaviour of polymer matrix composites is presented in Eq. (3).

V

C 0 : (H T  H th  H ve ) with H th

D 'T

(3)

where ı is the stress, C0 the elastic stiffness tensor, İT the total strain, İth the thermal strain (in order to take into account the thermal residual stresses which could have a strong influence on the prevision of the first ply failure) and İve the viscous strain, expressed in the material axis. Besides, the nonlinear aspect of the behaviour under shear loading is addressed. The viscoelastic strain İve is calculated using a nonlinear viscoelastic spectral behaviour. The main idea of this approach consists in decomposing the viscosity in a sum of elementary viscous mechanisms.

H ve

g V ¦[i i

with [i

1

Wi

P g V S i

R

: V  [i



(4)

where SR is the viscous compliance tensor. Each elementary viscous mechanism (ȟi) is defined by its relaxation time (IJi) and its weight (µi). These two quantities define the temporal spectrum which is supposed to have a Gaussian form. The non linear function g(ı) permits to describe the non linear aspect of viscosity in creep tests. This kind of behaviour is able to describe creep and relaxation tests and to take also into account the influence

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of the loading rate on the macroscopic behaviour and final failure contrary to non linear elastic behaviour widely used. The progressive degradation model is based on damage models which were developed at Onera. After a first failure of a ply in a laminate (failure ~ criterion value is higher than 1), the effective elastic compliance S of the failed ply is increased with two terms as it is written in Eq. (5):

~ S

0

S  d1H1  d 2 H 2

­ ° FF : d1 D ° ® ° ° IFF : d 2 E ¯

f1r  1 f 2r



d1 t 0 

1

(5) d2 t 0

Where S0 is the elastic (initial) compliance, d1H1 and d2H2 are tensors that represent respectively the effect of fibre failure and interfibre failure on the compliance of the broken ply. For each degradation model, we distinguish the kinetics of the degradation with the scalar variables di and the effect of failure on mesoscopic behaviour with effect tensors Hi±. The evolution of the degradation variable is positive in order to insure the Clausius-Duhem inequality. This condition permits also to describe the realistic behaviour even under unloading. The effect tensors are determined with a micromechanical approach5 which is based on failure mechanics. Consequently, no additional test is necessary for identification of the effect tensors. The definition of laminate failure depends on the industrial application. Only tubes are addressed in the WWFE, so obviously fibre failures (tension and compression) and transverse compression failure (leading to explosive wedge effect) are considered as ultimate failure for laminates. An important loss of macroscopic rigidity is also considered as catastrophic for the laminates (especially for [±ș]s laminates). Fig. 2 presents a) macroscopic behaviour of a laminate [±45°]s in Eglass/MY750 subjected to a loading ratio of Ȉxx:Ȉyy =-1:1, where the viscosity of the matrix has a strong effect on the macroscopic behaviour and final failure; and b) a failure envelope (Ȉxx, Ȉyy) of a [90°/±30°]s laminate in Eglass/LY556.

The predictions of the failure approach are in good agreement with experimental data6 for tensile loadings, nevertheless this material model overestimates dramatically the strength of the laminate subjected to compressive loadings.

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F. Laurin, N. Carrère, J.-F. Maire and D. Perreux

Figure 2. a) Macroscopic behaviour for a [±45°]s laminate in Eglass/MY750 (Ȉxx:Ȉyy=-1:1) and b) failure envelope (Ȉxx,Ȉyy) of a [90°/±30°]s laminate in Eglass/LY556.

3.

FAILURE UNDER COMPRESSIVE LOADINGS

One of the main conclusions of the WWFE is that all the failure approaches overestimates the laminate strength under compressive loadings because the structural buckling is neglected. It is necessary to consider the laminate as a structure in order to predict accurately the final failure for compressive loadings. Buckling is classically defined as a loss of stability initiated by some imperfections (material, geometry). The prevision of the buckling and postbuckling, with FE modelling, is still a very difficult issue. Methods found in literature could be divided into two classes: (i) numerical methods, difficult to implement and use in a practical way, but determine all the possible post buckling modes and (ii) methods which try to model the imperfections thus leading only to the most realistic buckling mode. The fabrication process of composites with polymer matrix induces a spatial variability of mechanical properties in the structure (due to heterogenous fibres distribution for instance). This method is very simple to use in a FE code; at each Gauss point of the structure, different material properties are affected. This spatial material variability creates local imperfections which lead to the buckling of the structure. The validation

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183

with analytical solutions on plates and the demonstration of the predictive capabilities of this method were presented elsewhere7. Soden et al8 demonstrated experimentally by axial compression on [±55°]s laminate tubes in Eglass/MY750 with different thicknesses, that there is a competition between material and structural aspects (see Fig. 3).

Figure 3. Competition between material and structural aspects on [±55]s composite tubes.

On one hand, for thick tubes (e>3mm), final failure is due to ply failure, which is a function of the strength of the unidirectional ply. On the other hand, for thin tubes (e5), shearing failure mechanism is observed (Fig. 6).

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J. Podgórski and J. Jonak

Figure 6. Results of the simulation of crack propagation in laminar material for hf = 5.

4.

CONCLUSIONS

The Finite Elements Method analysis showed explicitly that in the case of laminar brittle materials the mechanics of the crack development process depends strictly on the value of strength heterogeneity in the material layers. Greater layer strength asymmetry causes the material in its “weaker” layer to be more easily destroyed. As a result of this the whole laminar material is delaminated faster. Three different cracking mechanism has been found by numerical analysis as result of change the heterogeneity value of laminar material. For some value (hf=4 in tested FEA model) the delamination has been observed.

ACKNOWLEDGMENTS This paper has been prepared within the framework of a grant of the Polish Scientific Research Committee No 5 T12A 015 23 and 8 T12A 064 21.

REFERENCES 1. Podgórski J., General Failure Criterion for Isotropic Media. Journal of Engineering Mechanics ASCE, 111 2, 188-201 (1985). 2. Podgórski J., Influence Exerted by Strength Criterion on Direction of Crack Propagation in the Elastic- Brittle Material. Journal of Mining Science 38 (4); 374-380, (2002) JulyAugust, Kluwer Academic/Plenum Publishers. 3. Podgórski J., Jonak J., Jaremek P. The Strength Asymmetry Effect in Laminar Rock-Like Materials on Crack Propagation, MPES Proceedings, Wrocáaw 1.09-3.09. 2004

CRACK PROPAGATION IN COMPOSITES WITH CERAMIC MATRIX Katarzyna Konopka Warsaw University of Technology, Faculty of Materials Science and Engineering, Warszawa 02-507 ul. Woáoska 141

Abstract:

The purpose of the present study was to describe the change of the crack profile and as a consequence its length due to the metal particles introduced into the ceramic matrix. The results are based on the experimental observations of the crack propagation in ceramicmetal composites (CMCs). Changes of crack profile by the metal particles affect the fracture toughness of the composite. In the method of measuring the fracture toughness which is based on the Vickers indentation technique the changes of the crack length by the metal particles lead to increase the fracture toughness. The real crack length measured according to the crack profile deflection by the metal particles is longer than that measured as a straight line and can be divided into two parts. One part is the length of the crack which is moving through the ceramic matrix. The second part is equal to the length of the crack profile that results from the interaction between the metal particles and the crack. Knowing the path of crack propagation it is possible to predict the fracture of the composites and to design composites with the desired fracture toughness.

Key words: ceramics, composites, fracture toughness

255 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 255–262. © 2006 Springer. Printed in the Netherlands.

256

1.

K. Konopka

INTRODUCTION

In the last few years applications of ceramic matrix composites has significantly increased due to their properties. Ceramic-metal composites (CMCs) have been developed to overcome the brittleness of the ceramics. In these materials, the resistance to brittle fracture can be increased as a result of the action of the mechanisms (Wachtman 1996): 1) bridging the cracks by the fibers, particles or grains of a plastic phase, 2) deflecting the fracture path as a result of a deviation or rotation of the fracture plane with respect to the grains and the regions that contain the other phase, 3) branching the cracks into two or more parallel cracks, 4) shearing the crack tip by forming micro-cracks and plastic deformation zones within the crack surroundings, There is a wide spectrum of CMCs depending on the chemical composition of the matrix and the reinforcement. Non-oxide CMCs have been the most studied. A given ceramic matrix can be reinforced with either a discontinuous reinforcement, such as particles, whiskers or chopped fibers, or with continuous fibers. In the microstructure of ceramic-metal composites, the reinforcing phase may be uniformly distributed throughout the ceramic matrix. This is the case in the composites fabricated by traditional methods, i.e., powder metallurgy and casting (Trasty and Yeomans 1997; Schicker et al. 1999). It should also be mentioned about the in-situ technique, increasingly used, in which the reinforcing phase is formed due to a chemical reaction between the individual components of the composite (Tjong and Ma 2000; Chen et al.1999). Another technique, consisting of infiltrating a porous ceramic material by a liquid metal results in the metallic phase being continuously distributed within the ceramic matrix (Prielipp et al.1995). It is also possible to produce a ceramic-metal composite with a gradient-type distribution of the metallic phase, i.e., with the volumetric share of the metal varying with increasing distance from the outer surface of the composite. In all of these types of composites the metal introduced into ceramic matrix influences the mechanical properties of composites. There are relationships between the size of the metal particles and the fracture toughness (Sigl et al. 1988; Ashby et al. 1989). The purpose of the present study was to describe the crack propagation in CMCs composites. In particular, emphasis was laid on the change of the profile and length of the crack as a result of metal particles being introduced into the ceramic matrix. Changes of the crack profile affect the fracture toughness of the composite. Knowing the path of crack propagation it is

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Crack Propagation in Composites with Ceramic Matrix

possible to predict the failure of the composites and to design composites with the desired fracture toughness. The results are based on experimental observations of the crack propagation in ceramic-metal composites. It should be underlined that in the presented work only geometrical changes of the crack path induced by metal particles were considered. The process of the plastic deformation of the metal particles during crack propagation and its effect on the profile and length of the crack was not considered. However, it is a very efficient mechanism of the dispersion of the energy of propagating cracks and thereby increases the CMCs composites fracture toughness.

2.

RESULTS

Our earliest work devoted to this effect and the modeling of the influence of the metal particles on the fracture toughness of ceramic matrix composites showed that metal particles change the length of the crack and in consequence affect the fracture toughness (Konopka et al. 2003). One method of measuring the fracture toughness is based on the Vickers indentation technique. One of the equations used in this method is (Niihara et al. 1982): KIC = 0.018 (E/HV)0.4 Hv a 0.5 (L/a) – 0.5

(1)

where: E - Young modulus, HV-hardness, L –crack length, a - one half of the mean indent half-diagonal length. This method was developed for brittle materials mostly ceramics. However, some attempts are made in using it for calculating the fracture toughness of other materials such as composites with a ceramic matrix (Tanaka 1996). In this method the fracture toughness of the material is calculated by measuring the length of the cracks propagated from the corner of the indent. It is obvious that the way of measuring the length of the crack is important, especially in composites with a brittle ceramic matrix and a ductile phase incorporated in it. One of the most commonly used methods is the measurement of the distance along the straight line from the corner of the indent where the crack starts to the end of the crack (Cahn et al. 1994) as it is shown in Fig.1. Since, in the composites, the crack profile can be deflected by the ductile phase this measurement underestimates the true length of the crack. As a consequence, the value of KIC obtained by the Vickers method differs from the value calculated when the real crack length is measured. The value of KIC calculated from Eq. 1 for the real crack length is lower. This is reason why the measurement of the crack profile is so important, especially

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K. Konopka

in composites where metal particles due to the interactions with crack change the crack path. Generally in CMCs the interactions possible to occur between the crack and a spherical metal particle (diameter 2r) are: 1. the crack passes through the particle, the crack length is equal to 2r , 2. the crack deflects on either side of the particle which gives the length of the crack equal to ʌr, 3. the crack moves beneath the metal particle along the particle/matrix interface and the length of the crack is ʌr, 4. the crack surrounds the metal particle and the crack length is 2ʌr

Figure 1. Schematic view of an indentation crack and the measurement of its length in the straight line from the corner of the indenter .

The first of the possible interactions between metal particles and crack gives the minimum real length of the crack, the second and the third equal in their effect give an increase of the length of the crack profile, whereas, the last gives the biggest change of the crack profile. Moreover, the interactions of type 2, 3 and 4 lead to metal particles being pulled out from the material. The total length of the crack profile in a composite (being the result of an interaction between the metal particles and crack) is: for interaction 1 L1 = M 2r for interaction 2 L2 = N ʌr

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259

for interaction 3 L3 = O ʌr for interaction 4 L4 = P 2 ʌr where: M, N, O, P – are the number of particles which interact with the crack. These changes of the length of the crack profile and the process of pulling out the metal particles lead to dispersion of the fracture energy and a significantly change the fracture toughness. For example, in a composite of Al2O3-with 20 volume % of Mo for the case when the crack surrounds the metal particles the fracture toughness increases by about 22% compared with the value of KIC in Al2O3 (Konopka et al. 2003). The examples of these interactions between a crack and the metal particles in a composite Al2O3-Mo are shown in the figures below (Fig 2, 3, 4). The real crack length (L) measured according to the crack profile deflection by the metal particles will be longer than that measured as a straight line and can be divided into two parts. One part is the length of the crack which is moving through the ceramic matrix (LCM). The second part is equal to the length of the crack profile that results from the interaction between the metal particles and the crack. It should be underlined that in composites very often all the interactions possible between the metal particles and a crack are observed (Fig. 2) so the value of L can be calculated as: (2) L = LCM + L1 + L2 + L3 + L4 (3) L = LCM + r(2M + Nʌ + Oʌ + 2Pʌ)

Figure 2. Crack propagation in an Al2O3-Mo composite – crack deflection and bridging by the Mo particles light- grey area – Mo particles.

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K. Konopka

10 µm

Figure 3. Crack propagation in an Al2O3-Mo composite – crack deflection by the particles.

100 µm

Figure 4. Hole left by the Ni particles pulled out from the Al2O3 matrix.

The length of the crack LCM in the ceramic matrix is not a straight line either. In ceramic materials, intergranular crack propagation is mostly observed (Fig. 5). This means that the length of the crack which propagates through the ceramic matrix is proportional to half of the circumference of the ceramic grain (assuming spherical shape of ceramic grain, diameter 2R). The real length of crack in CMCs can be calculated as: L =S ʌ R + r(2M + Nʌ + Oʌ + 2Pʌ) (4) where: S – is the number of ceramic grains

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Figure 5. Crack propagation in A2O3 ceramic , SEM image.

This reasoning permits predicting the real path of the crack in the composites and thereby designing the composites toughening by the metal particles. The description of the crack propagation in CMCs show that the profile of the crack is controlled by the processes that take place between a metal particle and the crack. The real length of the crack depends on the size and shape of the metallic phase. The size of the ceramic grains is also important. It is also evident that the volume fraction and the metal particles distribution in the ceramic matrix influences the crack propagation. Given a defined volume fraction of metal particles in the matrix and the size of these particles, a different level of uniformity of their distribution will result in a different number of metal particles encountered by the propagating crack. For example, in an Al2O3-Mo composite with a non-uniform distribution of the Mo particles the fracture toughness was lower (6.86 MPa m 0.5) than in a sample with uniform arrangement of Mo particles (8.02 MPa m 0.5) (Konopka et al. 2003). In designing composites of various kinds, produced by various methods and controlling their microstructure it is important to find correlations between the factors described above. Especially in functionally graded composites where the metal particle arrangement changes together with the length of the sample. Typically, in regions where there are no metal particles, the crack propagates as in brittle ceramics, whereas when the crack enters to the region with the metal particles it changes its profile.

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CONCLUSIONS

The presented results are the preliminary work concerning the influence of metal particles, their size, shape and distribution on the fracture toughness of the composites with ceramic matrix. Knowing the path of crack propagation it is possible to predict the fracture of the composites and to design CMCs with the desired fracture toughness. In the Vickers indentation technique for the estimation of the KIC value the length of the crack is used . The length of the crack is changed by the metal particles and as a consequences the fracture toughness is change. Further investigations of describing the crack propagation in ceramic-metal composites are in progress.

ACKNOWLEDGMENT This research has been in part financially supported by the State Committee for Scientific Research (project PBZ-KBN-100/T08/2003).

REFERENCES Ashby M.F., Blunt F.J., Bannister M., 1989, Flow characteristic of highly constrained metal wires, Acta Metal. 37(7): 1847-1857 Cahn R.W., Haasen P., Kramer E.J.. ,1994, Materials science and technology, structure and properties of ceramics, vol.11, Weinheim, New York Chen G, Sun G, Zhu Z., 1999, Study on reaction-processed Al-Cu/ Į Al2O3(p) composites, Mater. Sci. Eng , A265: 197-201 Konopka K, Maj M., Kurzydáowski K.J., 2003, Studies of the effect of metal particles on the fracture toughness of ceramic matrix composites, Materials Characterization 51: 335-340 Konopka K., Matysiak H., Olszyna A., 2000, Homogeneity of the microstructure of Al2O3Mo composite depending on the mixing time of the powders, Proc. of Sixth International Conference Stereology and Image Analysis in Materials Science STERMAT, 20-23 September, Kraków, POLAND, 203-208 Niihara K.A., Morena R., Hasselman D.P.H., 1982, Evaluation of KIC of brittle solids by the indentation method with low crack-to-indent ratios, J. Mater. Sci. Letters : 13-16 Prielipp H., Knechtel M., Claussen N., Streiffer S.K., Müllejans H., Rühle M., Rödel J., 1995, Strength and fracture toughness of aluminium/alumina composites with interpenetrating networks, Mater. Sci. Eng. A197: 19-30 (1995) Schicker S., Erny T., Garcia D.E., Janssen R. and Claussen N., 1999, Microstructure and mechanical properties of Al-assisted sintered Fe/Al2O3 cermets, J. Eur. Ceram. Soc. 19: 2455-2463 Sigl L.S. , Mataga P.A., Delgleish B.J. , Mc Meeking R.M., Evans A.G., 1988, On the toughness of brittle materials reinforced with a ductile phase, Acta Metall. 36 (4): 945953 Tanaka M., 1996, Fracture toughness and crack morphology in indentation fracture of brittle materials , J. Mater. Sci. 31:749-755 Tjong S.C., Ma Z.Y., 2000, Microstructural and mechanical characteristics of in situ metal matrix composites, Mater. Sci. Eng. 29: 49-113 Trasty P.A., Yeomans J.A., 1997, The toughnening of alumina with iron: effect of iron distribution fracture toughness, J. Eur. Ceram. Soc. 17: 495-504 J.B. Wachtman: „Mechanical properties of ceramics”, John Wiley and Sons New York, 1996

EXPERIMENTAL INVESTIGATIONS AND MODELLING OF POROUS CERAMICS

S. Samborski1, T. Sadowski2 1 Faculty of Mechanical Engineering, Lublin University of Technology, 36 Nadbystrzycka St., Lublin 20-618, Poland; 2Faculty of Building and Architecture, Lublin University of Technology,40 Nadbystrzycka St., Lublin 20-618, Poland

Abstract:

This article describes a new research method of the mechanical features estimation and damage evolution assessment for porous ceramics in compression. The method couples micromechanical and phenomenological modelling with experiment. It requires determination of the initial material structure by SEM, quasi-static tests of ceramic samples subjected to compressive loading with unloading and multiple reloading. As a result, a detailed analysis of material constants and damage assessment is possible. At micromechanical level of modelling sets of distributed pores and cracks are described by application of Representative Surface Element with averaging procedure, what yields a macroscopic material behaviour. The phenomenological approach bases on macroscopic material damage variable (the second order tensor) with some criteria of cracks nucleation, their propagation and final failure of the material sample. Experiments were conducted on ceramic samples of alumina and magnesia with various porosity fraction (0 to 30%). The main advantages of the proposed method: examination of brittle materials in the process of controlled deformation, estimation of the global material stiffness tensor by deformation analysis and current damage anisotropy assessment.

Key words:

porous ceramics, crack growth, damage evolution

1.

INTRODUCTION

Presently ceramics is getting a wider and wider field of application. As multiphase materials, with a gas in pores as a second phase, it is applied for 263 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 263–270. © 2006 Springer. Printed in the Netherlands.

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filtration purposes, sound or heat insulation, furnace lining in steel fabrication and others (Davidge , 1979; Evans et al., 1970; Pampuch, 1988). Mechanical strength of ceramic materials is strongly determined by their initial structure, which may be characterised by grain dimensions, grain boundaries’ properties, an existence of pores, cracks etc. Under compression (loading and unloading) behaviour of ceramics is strongly related to damage growth, that considerably influences macroscopic mechanical characteristics of the material. The need of better understanding of the nature of these processes results from the growing field of practical applications of ceramic materials in various branches of technology. The proposed method presented here is based on micromechanical modelling considering the real internal structure of porous ceramics. The examination of ceramics in a process of controlled deformation and it’s theoretical description will allow a new way of determination of elastic characteristics change of brittle materials. Moreover, understanding and modelling of damage growth mechanisms will become possible. The main advantages of the new method are: research of porous ceramics under compressive loading-unloading-reloading process, theoretical description of material deformation taking into account the anisotropy of microcracks propagation, introduction of phenomenological approach with application of tensorial damage parameter, coupling of two levels of modelling (micromechanical and phenomenological) in order to give physical interpretation of macroscopic damage parameter. Research of damage processes of ceramic materials is one of the most upto-date problems of polycrystalline material behaviour. However, there is still a lack of more accurate experimental data of ceramic materials under compression (Munz and Fett, 1999). Furthermore, there was not considered the process of gradual degradation of elastic characteristics following the development of material deformation. Up to now the researchers were only determining final strength related to current material damage state. Thus, understanding of the interdependence between structure parameters and global material response is still unsatisfactory. In our work, theoretical explanation of phenomena proceeding inside material during compressive loading is being described on the basis of “wing-crack” model. The idea of Moss and Gupta (1982) was developed by Horii and Nemat-Nasser (1983), Ashby and Hallam (1986), as well as Nemat-Nasser and Obata (1988). There was also a comprehensive work on wing-cracks propagation in ceramics by Sadowski (1994), where internal structure parameters were considered. Unfortunately, there was no porosity taken into account. The basic formulae describing effective elastic constants, as Young’s modulus and Poisson’s ratio for porous materials with cracks gave Kachanov (1993). Sammis and Ashby (1986) found earlier, that pores

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may act as crack sources. Hardy and Green (1995) stated, that mechanical features (Young’s modulus, fracture toughness) of porous alumina (with porosity from 20 to 40 percent) depend on degree of densification and sintering. This indicates, that microstructure analysis is essential. The influence of various parameters on the dependence of mechanical characteristics on porosity under tension discussed Rice (1993, 1998). On the other hand, Lu et al. (1999) on the basis of pure experimental data introduce theoretical models defining macromechanical features. Those are, however, purely experimental relations without physical microanalysis of damage growth in porous materials. Proper analysis of brittle fracture in compression must take into account full understanding of cracking mechanisms and finding universal fracture criterion. We must understand mechanical qualities of ceramics in relation to their microstructures. Thus, it is necessary to elaborate more sophisticated experimental techniques and improve theoretical models towards their better compatibility with the real materials.

2.

MODELLING

An idea of theoretical description of internal structure changes in polycrystalline porous ceramics was proposed by the authors of this paper in several articles (Sadowski et al., 2000, 2002, 2003a, 2003b; Samborski and Sadowski, 2003, 2005). We propose to use for the purpose of the ceramics behaviour description the following models: x micromechanical, based on averaging procedures of existing defect arbitrary distributed inside material, x phenomenological, in which the changes of structure are described by internal parameters. Macroscopic constitutive equation describing all defects propagation inside ceramic material is the following:

H ij

Sijkl  p, Z nm , p V kl

(1)

where: H ij , V kl are deformation and stress tensors, respectively. Sijkl is the compliance tensor of rank four, dependent on a few internal parameters: porosity p , damage tensor of rank two Z nm and plastic effects p . Being aware, that deformation in ceramics is very small, we make an assumption about the global deformation tensor ( H ij ). It can be decomposed as follows: purely elastic strains H ije , as the material matrix response, strains connected

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with pore occurrence H ijpo , H ijcr describing crack growth and H ijpl standing for plastic effects. Thus, we may write:

H ij

H ije  H ijpo  H ijpl  H ijcr

(2)

In general case of loading of the material damage development induces anisotropic behaviour of the ceramics, which can be described by a tensorial damage parameter of the following form:

Zij

I1G ij  I2V ij  I3V imV mj

(3)

where I1 , I2 , I3 are scalar functions of the stress tensor V ij . The last two terms of the above-written equation specify anisotropy. Particular forms of the constitutive equations for analysed porous ceramics are currently being found.

3.

TESTING PROCEDURE

Experiments are carried out as uniaxial compression tests of alumina (Al2O3) and magnesia (MgO) samples. Various porosity fraction enables an estimation of this parameter on the global material response to applied load (initial compliance tensor). Microscopic analysis of material structure before compression tests and after failure give essential data for micromechanical modelling. There are three types of tests: x uniaxial compression of cylindrical specimens subjected to loading-unloading and subsequent reloading. The analysis of the permanent deformation after each unloading cycle enables estimation of damage level and current material anisotropy caused by cracking; x three-point bending of notched beams in order to get critical value of the stress intensity factor (KIc); x microscopic observations of material structure changes concerning: initial porosity estimation, grain diameter and grain boundary texture as well as fracture surface observation leading to determination of the way of cracking (intergranullar or intragranullar).

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3.1

Damage description by repeated loading

As we mentioned above, one of the most important experiments within the framework of our method is uniaxial testing with unloading and subsequent reloading. A scheme of the process is presented in Fig. 1. Remember, that there is the possibility of plotting analogous graph for transversal direction. F

EL

n

Fn 3

F3 2

F2

0

EL

H L 2

H L1

H

Figure 1.

EU

1

F1

p 1 L

H

p 2 L

H Lp 3

HL

HL n

H L3

He

H Lp n 'HLn

A scheme of uniaxial repeated loading.

According to this, we can calculate damage parameters for two n directions: longitudinal ( DL ): n n EUL §¨ p, V L ·¸ n § n ·   © ¹ (4) DL ¨ p, V L ¸ 1  ©

¹

EL ( p )

n and transversal ( D  ): T

n n DT §¨ p, V L ·¸ 1  ©

¹

n n EUT §¨ p, V L ·¸ © ¹ ET  p

(5)

Here EL and ET are initial Young’s moduli for two directions. They are  n and E  n stand for observed to be a function of material porosity (p). EUL UT unloading moduli at each n-th stage of unloading. (see Fig. 1). Thus, it is possible to introduce anisotropy of damage using the two damage parameters.

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EXPERIMENTAL RESULTS

From a wide range of experimental data we chose two plots reflecting material deterioration of the material with a certain loading history (cf. Fig. 1). At Fig. 2 one can see the comparison of initial (ET) and unloading (EUT) elastic moduli for alumina as a function of initial porosity (p). The unloading Young’s modulus was measured just before the sample rupture. E, GPa

400 350 300 250 200 150 100 50 0

‘ ET U˜˜˜ EUT

0

0,1

0,2

0,3

0,4 p

Figure 2. Comparison of Young’s moduli for porous alumina.

0,60

D

U···· DL, …— DT

0,40

0,20

0,00 0

0,1

0,2

0,3

0,4 p

Figure 3. Damage parameters vs. porosity for alumina.

Note, that it’s values are lower than ET, what reflects a growth of material structure deterioration. Figure 3 shows the difference between the two damage parameters: DL and DT at the very moment of rupture. It is clear from Fig. 3, that porosity makes ceramic material more damage tolerant.

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The fact, that transversal damage is higher than this for axial direction is physically reflected by axial splitting of samples during loading, as presented in Fig. 4.

Figure 4. Ceramic sample after splitting.

5.

CONCLUSIONS

A new method of the brittle materials behaviour description requires both experimental and theoretical approaches. The proposed method will be a convenient research tool for damage evolution specification during variable loading of the material considering most essential phenomena developing inside material structure. The consistency of experimental macroscopic test with theoretical approach makes the proposed method applicable for description of all brittle or quasi-brittle engineering materials with internal structure.

ACKNOWLEDGEMENT This work was funded by Polish State Committee for Scientific Research within the years 2004-2006 as a research project No. 3 T08D 027 26.

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REFERENCES Ashby M.F. and Hallam S. D., 1986, The Failure of Brittle Solids Containing Small Cracks under Compressive Stress States, Acta Metall. 34(3): 497-510. Davidge R.W., 1979, Mechanical Behaviour of Ceramics, Cambridge University Press; London. Evans A.G., Gilling D. and Davidge R. W., 1970, The Temperature-Dependence of the Strenght of Polycrystalline MgO, J. Mat. Sc. 5: 187-197. Hardy D. and Green D. J., 1995, Mechanical Properties of a Partially Sintered Alumina, J. Eur. Ceram. Soc. 15: 769-775. Horii H. and Nemat-Nasser S., 1983, Overall Moduli of Solids with Microcracks: LoadInduced Anisotropy, J. Mech. Phys. Sol. 31(2): 155-171. Kachanov M., 1993, On the Effective Moduli of Solids with Cavities and Cracks, Int. J. Fract. 59: R17-R21. Lu G., Lu G. Q. and Xiao Z. M., 1999, Mechanical Properties of Porous Materials, J. Porous Mat. 6: 359-368. Munz D., Fett T., 1999, Ceramics. Mechanical Properties, Failure Behaviour, Materials Selection, Springer, Berlin. Moss W.C. and Gupta Y. M., 1982, A Constitutive Model Describing Dilatancy and Cracking in Brittle Rocks, J. Geoph. Res. 87(B4): 2985-2998. Nemat-Nasser S. and Obata M., 1988, A microcrack model of dilatancy in brittle materials, J. Appl. Mech. 55: 24-35. Pampuch R., 1988, Ceramic Materials. An Outline of the Inorganic-Non-metallic Materials Science, Polish National Scientific Publishing House, Warsaw. Rice R. W., 1993, Evaluating Porosity Parameters for Porosity-Property Relations, J. Am. Ceram. Soc. 76(7): 1801-1808. Rice R. W., 1998, Porosity of Ceramics, Marcel Dekker, Inc., New York. Sadowski T., 1994, Modelling of Semi-Brittle MgO Ceramic Behaviour under Compression, Mech. Mat. 18: 1-16. Sadowski T.and Samborski S., 2000, Damage Process of Initially Porous Polycrystalline Ceramics, Conference Proceedings of Brittle Matrix Composites 6: 566-575. Sadowski T., Samborski S. and Mróz Z., 2002, Gradual degradation of initially porous polycrystalline ceramics subjected to quasi-static tension, Kluwer Academic Publishers, 401-405. Sadowski T. and Samborski S., 2003a, Prediction of the mechanical behaviour of porous ceramics using mesomechanical modeling, Comp. Mat. Sc. 28: 512-517. Sadowski T. and Samborski S., 2003b, Modeling of Porous Ceramic Response to Compressive Loading, J. Am. Ceram. Soc., 86(12): 2218-2221. Samborski S. and Sadowski T., 2003, On the different behaviour of porous ceramic polycrystalline materials under tension and compression stress state, Comp. Fluid. Sol. Mech.: 615-618. Samborski S. and Sadowski T., 2005, On the method of damage assessment in porous ceramics, Conference proceedings of 11th Conference on Fracture, Turin. Sammis C. G.and Ashby M. F., 1986, The Failure of Brittle Porous Solids under Compressive Stress States, Acta Metall., 34(3): 511-526.

NUMERICAL ANALYSIS OF STRESS DISTRIBUTIONS IN ADHESIVE JOINTS

Józef Kuczmaszewski and Maciej Wáodarczyk LUBLIN UNIVERSITY OF TECHNOLOGY, Departament of Production Engineering, Nadbystrzycka 36str., 20-618 Lublin, Poland

Abstract:

The paper discusses an effect of finite-element lattice type selection on stress distribution in adhesive joints of metallic elements. Results of modeling by tetra- and hexagonal lattices have been compared. Methods for joining various types of finite-element lattices in the vicinity of areas where local stress concentrations occur and their effect on the accuracy of calculation results have also been presented. Numerical analysis has been performed for two different types of glue models: a linear and a non-linear one, taking into account various patterns of glue-joint division i.e. into 2,3,and 4 layers. The Abaqus software has been applied as a numerical tool. The obtained results present a comparison of various methods for modeling and calculation of adhesive joints and their effect on stress distributions especially in the areas of strong concentrations that determine strength of the whole joint.

Key words:

adhesive joints, stress distributions, non-linear glue model

1.

INTRODUCTION

Finite Elements Method (FEM) is the most popular computing instrument to support engineering works. Nowadays, this method is used to test all kinds of physical processes related to construction mechanics, with cracking mechanics in particular. In the presented paper the FEM has been applied to analyze adhesive joints. Structure of such joints makes it very difficult to determine their stress state by means of direct methods. The FEM can also be applied to analyze the effect of heat and the adhesive joint heterogeneity on the life and creep strength of such joints as well as to perform a dynamic analysis. 271 T. Sadowski (ed.), IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials, 271–278. © 2006 Springer. Printed in the Netherlands.

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In order to obtain correct calculation results it is necessary to elaborate a joint model whose parameters are the closest possible to a real object. It is also important to define proper boundary conditions. In the light of the performed investigations on elaborating a FEM computational model it seems to be equally important to select an adequate finite-element lattice that is dense enough in the areas of expected local stress concentrations. The paper presents those research fragments that concern the FEM application to analyze stress and strain in adhesive joints. The testing has been performed for various glue models as well as for various models of finite-element lattices. Results obtained for linear and non-linear glue models, tetra- and hexagonal lattices as well as for various divisions in glue layers have been compared.

2.

ELABORATION OF A ADHESIVE JOINT MODEL

The performed analysis has applied a number of computational models that differ from each other in the FEM lattice parameters, in the lattice division and in the density of local stress concentrations. A FEM computational model of a lap joint has been used as a basic model. (Fig.1) .

Figure 1. A basic model for FEM calculations .

The model has been made as an assembly of two plates, of the following parameters: length: 100 [mm], width: 25 [mm], thickness: 1,5 [mm], separated by a glue layer of the following parameters: length: 12,5 [mm], width: 25 [mm], and thickness: 0,15 [mm]. Aluminum of the Young’s modulus of 70 000 [MPa] and the Poisson ratio of 0.34 has been applied as a lap material.

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Two glue characteristics - a linear and a non-linear one - have been accepted for the calculations. The linear glue material – epoxy resin E5 hardened by poliaminonide C– has been developed based on the following assumptions: the Young’s modulus of 2500 [MPa] and the Poisson ratio of 0.34. The non-linear glue characteristic has been assumed to be a multi-linear characteristic developed on the basis of an experimentally obtained curve - Fig.2. Non-linear glue

60 55 54

50 46

į [MPa]

40

30

20

10

0

0

0

0,02

0,02652

0,0311

İ

Figure 2. Non-linear glue characteristic [8].

In the basic model the lap element has been partitioned into a few zones that are characterized by different structures of the applied finite-element lattices. The glue body has been divided into layers as shown in Fig.3.

Figure 3. Division of a adhesive joint into 3 layers.

Boundary conditions and external load have been defined within the model global coordinate system that is described by 1,2,3 axes (Fig.1) as follows: x the upper plate edge has been fixed by no possibility of node displacement in any direction (1,2,3) and this way the model fixing has been modeled (by no rotational freedom degrees in solid element nodes),

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x the free end of a adhesive joint has no displacement possibility in the 2 and 3 directions within the global coordinate system, External load of the joint lap model is a static load of the value of 3125 [N] that is applied to the lower lap free end in the “-1”direction.

3. ANALYSIS OF STRESS IN A ADHESIVE JOINT The numerical analysis has been performed in successive discrete “time” moments with a specified load-increment step that corresponds to them and eventually yields a total load level for the construction. As a result the strain process course is obtained in successive steps of load increment. Printouts of the adhesive joint strain state and reduced stress distributions (according to the Huber-Mises hypothesis) presented in the form of colored contour maps illustrate the analysis results. Fig.4 presents the reduced stress distribution pattern of the basic model in the form of colorful contour maps.

Figure 4. Distribution of reduced stress in adhesive joints according to the Huber-Mises hypothesis.

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4.

COMPARISON OF LINEAR AND NON-LINEAR GLUE CHARACTERISTIC MODELS

Among FEM analyses of adhesive joints those that are performed at simplifying assumptions regarding characteristics of materials that occur in the model over the whole loading range are dominant and they are considered to be ideally linearly elastic. This assumption is correct only to the moment when some limit of stress gets exceeded under the working load. From that moment on the glue characteristic changes, which is followed by an increase of the material strain at an insignificant stress change. Regarding the above, for comparative purposes an analysis of stress distributions in a adhesive joint has been performed taking into consideration linear and nonlinear properties of the glue material. Fig.5 presents the analyses results obtained for linear and nonlinear glue models in the form of diagrams that illustrate stress reduced along the length of a adhesive joint. linear glue model

non-linear glue model

75

stress, Mises [MPa]

65 55 45 35 25 15 5 0

1

2

3

4

5

6

7

8

9

10

11

12

13

length joint [mm]

Figure 5. Reduced stress distribution of in a glue layer at the load of 3125 [N].

It follows from the comparison of the stress distributions that their courses are similar with differences in the initial zone values. For the linear material a rapid growth of medium reduced stress values by nearly 61% can be observed at the length of ca 0.6 mm away from the lap edge. For the nonlinear glue model, that increase is of only 37%. The stress differences get even more conspicuous in the surface glue layers, close to the lap ends. In the linear model stresses grow at the segment of 0.6 mm away from the values from 52 to 92 MPa and for the non-linear glue, over the same length for the values from 49 to 54 MPa.

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It follows from the performed numerical analyses that when the non-linear glue material properties are ignored an error of the 30% order in the estimation of medium reduced stress values at the lap ends occurs and when the maximum reduced stress values at the adhesive joint edge layers are concerned the error value reaches 41%. The values are considerable and their effect on the forecasting of a adhesive joint strength is significant By comparing stresses in the lap materials it can be found that in both models they remain at a similar level because the difference in the maximal reduced stress values does not exceed 10%. Fig. 6 presents stress levels in the plates.

Figure 6. Reduced stress layers in the lap: a) non-linear, b) linear.

5.

THE EFFECT OF A GLUE-LAYER DIVISION PATTERN ON THE ACCURACY OF RESULTS

The FEM lattice generation process requires some optimization on account of the number of obtained elements, which translates into a number of freedom degrees of the computational model and most of all into the computation time. For that reason it seems to be purposeful to start investigations into the "economical" significance of the glue division into layers. In the tested models the glue body has been divided into 2,3 and 4 layers. The obtained results show that joints where the glue body is divided into 3 layers exhibit the best characteristic. Smaller number of layers brings about lower accuracy of stress distribution results. Any increase of accuracy has not been observed at the non-linear glue division into 4 layers, which is shown in Fig.7. Stress distributions along the lap length in adhesive joints at the 1 mm distance away from the edge have been analyzed, which is shown in Fig.8.

277

Numerical Analysis of Stress Distributions in Adhesive Joints The glue division on 3 layers The glue division on 4 layers

55

50

45

stress, Misesa [MPa]

40

35

30

25

20

15

10

5 0

1

2

3

4

5 6 7 8 length joint [mm]

9

10

11

12

Figure 7. Medium reduced-stress distribution in glue at its division into 3 and 4 layers.

56

56

54

54

52

52 50

48

layer 1 layer 2

46

layer 3 44

layer 4 mean value

42

stress, Mises [MPa]

stress, Mises [MPa]

50

48

layer 1 46

layer 2 layer 3

44

mean value

42

40

40

38

38

36

36 34

34 0

0,1

0,2

0,3

0,4

0,5

0,6

length joint [mm]

0,7

0,8

0,9

1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

length joint [mm]

Figure 8. Reduced stress distribution at various glue divisions.

It follows from the diagrams that the highest stress increase can be observed at the distance of ca 0.1 mm. For the other lap side the runs are similar. The boundary effect is visible in the stress analysis. Within classical analytical considerations stresses at the lap ends approach infinity. It follows from the accepted boundary conditions for the solved differential equations. The FEM yields solutions that are close to real conditions.

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CONCLUSIONS

The following conclusions can be drawn on the basis of the performed analysis: 1. Reduced stress distributions that have been obtained by the FEM according to the Huber-Mises hypothesis along the lap length of a adhesive joint resemble analytical estimation of the stress state. 2. When the non-linear glue material characteristic is ignored the maximal reduced-stress values at the lap ends can get overstated, which can yield errors in the load-capacity estimation for such joints. 3. Adequate selection of elements and their concentration in areas where stress concentration is expected as well as a correct location of zones that connect various lattice types guarantee linear distribution of the obtained results. 4. Division of the glue body into more than 3 layers is of no significant effect on the stress distribution over the lap length in a adhesive joint.

REFERENCES 1. 2. 3. 4. 5. 6.

7.

8.

9.

ABAQUS: Getting started with ABAQUS, Version 6.4, Abaqus Inc. 2003. ABAQUS: ABAQUS/STANDARD User Manual Volume III, Version 6.1, Abaqus Inc. 2000 ABAQUS: Introduction to ABAQUS/CAE, Version 6.4, Abaqus Inc 2003. ABAQUS: Introduction to ABAQUS/CAE Workshop, Version 6.4, Abaqus Inc 2003 Cagle, Ch.V.; Glue and gluing. WNT, Warsaw 1977 Godzimirski, J.; Kotlarz, W.; The choose of finite elements model to calculations of adhesive joints with use the packet of programme the System WAT-KM. Biul. WAT 494(1993) 10 121-127 Godzimirski, J.; Shear strength of constructional adhesive joints WNT, Warsaw 2002 Godzimirski, J.; Tkaczuk, S.; The possibility of using the FEM to strength calculation of adhesive joints. The Technology and the automation of assembly nr 2/2001 43–46 Kuczmaszewski, J.; The constructional and technological bases of the estimation of metals’ adhesive joints strength. Academic Publishing Houses of Lublin University of Technology, Lublin (1995)

APPENDIX APPENDIX: The Scientific Programme Monday 23.05.2005 Keynote Lecture 1 P. Ladevèze, G. Lubineau, D. Violeau, D. Marsal - A computational damage micromodel for laminate composites. Session I S. Schmauder – Materials modeling from atomistics macro behaviour. A. Dragon, C. Nadot-Martin, A. Fanget - Multiscale modelling for damaged viscoelastic particulate composites. Session II R. Talreja - A synergistic multiscale modelling approach in damage mechanics of composite materials. G.Z. Voyiadjis, R.K. Abu Al-Rub - A nonlocal plasticity and damage model for size effect in metal matrix composites. E. Sideridis, G.A. Papadopoulos, V.N. Kytopoulos, T. Sadowski – The elastic modulus and the thermal expansion coefficient of particulate composites using a dodecahedric multivariant model. Session III A. Hachemi, D. Weichert - A shakedown approach to the problem of damage of fibre-reinforced composites. H. Zhao, I. Elnasri – Perforation of sandwich panels with cellular solid core under impact loading. A. Johnson, N. Pentcôte,– Influence of delamination on the predicition of impact damage in composites. Session IV V. Tvergaard - Debonding or breakage of short fibres in a metal matrix composite. S. Lenci – A microscale model of elastic and damage longitudinal shear behaviour of highly concentrated long fiber composites. E. Oleszkiewicz, T. Łodygowski - Analysis of metal matrix composites damage under transverse loading.

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D. Marsal, P. Ladevèze, G. Lubineau - On the out-of-plane interactions between ply damage and interface damage in laminates. Tuesday 24 Keynote Lecture 2 P. Chojnacki, M. Greguła, M. Pańko, K. Siedlecki – Designing, testing and manufacturing of composite aviation products at “PZL-Świdnik” S.A. Session V G. Socha - Advances and trends in composite material testing. S. Elsoufiev – Rheology and fracture of composite materials. H. Altenbach - Strength criteria for composites – current state and future trends. Session VI D. Leguillon, O. Cherti Tazi, E. Martin - Prediction of crack deflection and kinking in ceramic laminates. T. Kubiak – Dynamic buckling of thin-walled composite plater. M. Jaroniek - Numerical and experimental models of the fracture in the multi layered composites. Session VII F. Collombet, M. Mulle, Y-H. Grunevald – Multiscale method for optimal design of composite structures incorporating sensors. B.A. Schrefler, M. Lefik, D.P. Boso – Numerical multiscale modelling of elasto-plastic behaviour of superconducting strand. K. Hofstetter, Ch. Hellmich, H.A. Mang – Mechanical properties of wood investigated by means of continuum micromechanics. Session VIII E.M. Craciun - Antiplane crack in a pre-stressed fiber reinforce elastic material. J. Füssl, R. Lackner, J. Eberhardsteiner – Multiscale model for upscaling of strength properties of bituminous composites. F. Laurin, N. Carrere, J.F Maire, D. Perreux - Characterization and practical application of a multiscale failure criterion for composite structures.

Appendix

281

Wednesday 25.05.2005 Session IX AMAS M. Białas, Z. Mróz - Crack and delamination patterns in thin layers under monotonic and cyclic temperature loading. G. Mieczkowski, K.L. Molski - Stress field singularities for reinforcing fibre with single lateral crack. E. Postek, T. Sadowski, S. Hardy – Mechanical Response of a Two-phase Composite

Thursday 26.05.2005 Keynote Lecture 3 R. de Borst – Numerical methods for debonding in composite materials: A comparison of approaches. Session X S.R. Hallett, W.G. Jiang, M.R. Wisnom – Modelling of delamination damage in scaled quasi-isotropic specimens. L. Van Parys, D. Lamblin, G. Guerlement, T. Descamps – Damage in patrimonial masonry structures: the case of the O-L cathedral in Tournai (Belgium). S. Datoussaïd, D. Lamblin, G. Guerlement, W. Kakol – Macroscopic strength of perforated steel plates at maximum elastic and limit state. Session XI J. Wang, B.L. Karihaloo – Importance of surface/interface effect to properties of materials at nano-scale. J.G.M. van Mier and P. Trtik - Multi-scale testing for simple micromechanical models of concrete. L. Berka – On a deformation of polycrystalline structures. C. Huchette, D. Leveque, N. Carrere - A multiscale damage model for composite laminate based on numerical and experimental complementary tests. J. Podgórski, J. Jonak – Influence of strength heterogeneity factor on crack shape in laminar rock-like materials.

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Friday 27.05.2005 Keynote Lecture 4 R. Pyrz - Atomic-continuum transition at interfaces of silicon and carbon nanocomposite materials. Session XII AMAS K. Konopka - Crack propagation in composites with ceramic matrix. S. Samborski, T. Sadowski – Experimental investigations and modelling of porous ceramics. J. Kuczmaszewski, M. Włodarczyk – Numerical analysis stress distributions in adhesive joints.

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

R.T. Haftka, Z. G¨urdal and M.P. Kamat: Elements of Structural Optimization. 2nd rev.ed., 1990 ISBN 0-7923-0608-2 J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990 ISBN 0-7923-0712-7 P. Karasudhi: Foundations of Solid Mechanics. 1991 ISBN 0-7923-0772-0 Not published Not published. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-2 O.O. Ochoa and J.N. Reddy: Finite Element Analysis of Composite Laminates. ISBN 0-7923-1125-6 M.H. Aliabadi and D.P. Rooke: Numerical Fracture Mechanics. ISBN 0-7923-1175-2 J. Angeles and C.S. L´opez-Caj´un: Optimization of Cam Mechanisms. 1991 ISBN 0-7923-1355-0 D.E. Grierson, A. Franchi and P. Riva (eds.): Progress in Structural Engineering. 1991 ISBN 0-7923-1396-8 R.T. Haftka and Z. G¨urdal: Elements of Structural Optimization. 3rd rev. and exp. ed. 1992 ISBN 0-7923-1504-9; Pb 0-7923-1505-7 J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-1610-X H.S. Tzou and G.L. Anderson (eds.): Intelligent Structural Systems. 1992 ISBN 0-7923-1920-6 E.E. Gdoutos: Fracture Mechanics. An Introduction. 1993 ISBN 0-7923-1932-X J.P. Ward: Solid Mechanics. An Introduction. 1992 ISBN 0-7923-1949-4 M. Farshad: Design and Analysis of Shell Structures. 1992 ISBN 0-7923-1950-8 H.S. Tzou and T. Fukuda (eds.): Precision Sensors, Actuators and Systems. 1992 ISBN 0-7923-2015-8 J.R. Vinson: The Behavior of Shells Composed of Isotropic and Composite Materials. 1993 ISBN 0-7923-2113-8 H.S. Tzou: Piezoelectric Shells. Distributed Sensing and Control of Continua. 1993 ISBN 0-7923-2186-3 W. Schiehlen (ed.): Advanced Multibody System Dynamics. Simulation and Software Tools. 1993 ISBN 0-7923-2192-8 C.-W. Lee: Vibration Analysis of Rotors. 1993 ISBN 0-7923-2300-9 D.R. Smith: An Introduction to Continuum Mechanics. 1993 ISBN 0-7923-2454-4 G.M.L. Gladwell: Inverse Problems in Scattering. An Introduction. 1993 ISBN 0-7923-2478-1

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 24. 25. 26. 27. 28. 29. 30. 31. 32.

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G. Prathap: The Finite Element Method in Structural Mechanics. 1993 ISBN 0-7923-2492-7 J. Herskovits (ed.): Advances in Structural Optimization. 1995 ISBN 0-7923-2510-9 M.A. Gonz´alez-Palacios and J. Angeles: Cam Synthesis. 1993 ISBN 0-7923-2536-2 W.S. Hall: The Boundary Element Method. 1993 ISBN 0-7923-2580-X J. Angeles, G. Hommel and P. Kov´acs (eds.): Computational Kinematics. 1993 ISBN 0-7923-2585-0 A. Curnier: Computational Methods in Solid Mechanics. 1994 ISBN 0-7923-2761-6 D.A. Hills and D. Nowell: Mechanics of Fretting Fatigue. 1994 ISBN 0-7923-2866-3 B. Tabarrok and F.P.J. Rimrott: Variational Methods and Complementary Formulations in Dynamics. 1994 ISBN 0-7923-2923-6 E.H. Dowell (ed.), E.F. Crawley, H.C. Curtiss Jr., D.A. Peters, R. H. Scanlan and F. Sisto: A Modern Course in Aeroelasticity. Third Revised and Enlarged Edition. 1995 ISBN 0-7923-2788-8; Pb: 0-7923-2789-6 A. Preumont: Random Vibration and Spectral Analysis. 1994 ISBN 0-7923-3036-6 J.N. Reddy (ed.): Mechanics of Composite Materials. Selected works of Nicholas J. Pagano. 1994 ISBN 0-7923-3041-2 A.P.S. Selvadurai (ed.): Mechanics of Poroelastic Media. 1996 ISBN 0-7923-3329-2 Z. Mr´oz, D. Weichert, S. Dorosz (eds.): Inelastic Behaviour of Structures under Variable Loads. 1995 ISBN 0-7923-3397-7 R. Pyrz (ed.): IUTAM Symposium on Microstructure-Property Interactions in Composite Materials. Proceedings of the IUTAM Symposium held in Aalborg, Denmark. 1995 ISBN 0-7923-3427-2 M.I. Friswell and J.E. Mottershead: Finite Element Model Updating in Structural Dynamics. 1995 ISBN 0-7923-3431-0 D.F. Parker and A.H. England (eds.): IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics. Proceedings of the IUTAM Symposium held in Nottingham, U.K. 1995 ISBN 0-7923-3594-5 J.-P. Merlet and B. Ravani (eds.): Computational Kinematics ’95. 1995 ISBN 0-7923-3673-9 L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Mechanics and Inverse Problems. 1996 ISBN 0-7923-3849-9 J. Menˇcik: Mechanics of Components with Treated or Coated Surfaces. 1996 ISBN 0-7923-3700-X D. Bestle and W. Schiehlen (eds.): IUTAM Symposium on Optimization of Mechanical Systems. Proceedings of the IUTAM Symposium held in Stuttgart, Germany. 1996 ISBN 0-7923-3830-8 D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky: Solution of Crack Problems. The Distributed Dislocation Technique. 1996 ISBN 0-7923-3848-0 V.A. Squire, R.J. Hosking, A.D. Kerr and P.J. Langhorne: Moving Loads on Ice Plates. 1996 ISBN 0-7923-3953-3 A. Pineau and A. Zaoui (eds.): IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Proceedings of the IUTAM Symposium held in S`evres, Paris, France. 1996 ISBN 0-7923-4188-0 A. Naess and S. Krenk (eds.): IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics. Proceedings of the IUTAM Symposium held in Trondheim, Norway. 1996 ISBN 0-7923-4193-7 D. Ie¸san and A. Scalia: Thermoelastic Deformations. 1996 ISBN 0-7923-4230-5

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J.R. Willis (ed.): IUTAM Symposium on Nonlinear Analysis of Fracture. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4378-6 A. Preumont: Vibration Control of Active Structures. An Introduction. 1997 ISBN 0-7923-4392-1 G.P. Cherepanov: Methods of Fracture Mechanics: Solid Matter Physics. 1997 ISBN 0-7923-4408-1 D.H. van Campen (ed.): IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems. Proceedings of the IUTAM Symposium held in Eindhoven, The Netherlands. 1997 ISBN 0-7923-4429-4 N.A. Fleck and A.C.F. Cocks (eds.): IUTAM Symposium on Mechanics of Granular and Porous Materials. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4553-3 J. Roorda and N.K. Srivastava (eds.): Trends in Structural Mechanics. Theory, Practice, Education. 1997 ISBN 0-7923-4603-3 Yu.A. Mitropolskii and N. Van Dao: Applied Asymptotic Methods in Nonlinear Oscillations. 1997 ISBN 0-7923-4605-X C. Guedes Soares (ed.): Probabilistic Methods for Structural Design. 1997 ISBN 0-7923-4670-X D. Fran¸cois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume I: Elasticity and Plasticity. 1998 ISBN 0-7923-4894-X D. Fran¸cois, A. Pineau and A. Zaoui: Mechanical Behaviour of Materials. Volume II: Viscoplasticity, Damage, Fracture and Contact Mechanics. 1998 ISBN 0-7923-4895-8 L.T. Tenek and J. Argyris: Finite Element Analysis for Composite Structures. 1998 ISBN 0-7923-4899-0 Y.A. Bahei-El-Din and G.J. Dvorak (eds.): IUTAM Symposium on Transformation Problems in Composite and Active Materials. Proceedings of the IUTAM Symposium held in Cairo, Egypt. 1998 ISBN 0-7923-5122-3 I.G. Goryacheva: Contact Mechanics in Tribology. 1998 ISBN 0-7923-5257-2 O.T. Bruhns and E. Stein (eds.): IUTAM Symposium on Micro- and Macrostructural Aspects of Thermoplasticity. Proceedings of the IUTAM Symposium held in Bochum, Germany. 1999 ISBN 0-7923-5265-3 F.C. Moon: IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics. Proceedings of the IUTAM Symposium held in Ithaca, NY, USA. 1998 ISBN 0-7923-5276-9 R. Wang: IUTAM Symposium on Rheology of Bodies with Defects. Proceedings of the IUTAM Symposium held in Beijing, China. 1999 ISBN 0-7923-5297-1 Yu.I. Dimitrienko: Thermomechanics of Composites under High Temperatures. 1999 ISBN 0-7923-4899-0 P. Argoul, M. Fr´emond and Q.S. Nguyen (eds.): IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics. Proceedings of the IUTAM Symposium held in Paris, France. 1999 ISBN 0-7923-5450-8 F.J. Fahy and W.G. Price (eds.): IUTAM Symposium on Statistical Energy Analysis. Proceedings of the IUTAM Symposium held in Southampton, U.K. 1999 ISBN 0-7923-5457-5 H.A. Mang and F.G. Rammerstorfer (eds.): IUTAM Symposium on Discretization Methods in Structural Mechanics. Proceedings of the IUTAM Symposium held in Vienna, Austria. 1999 ISBN 0-7923-5591-1

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P. Pedersen and M.P. Bendsøe (eds.): IUTAM Symposium on Synthesis in Bio Solid Mechanics. Proceedings of the IUTAM Symposium held in Copenhagen, Denmark. 1999 ISBN 0-7923-5615-2 S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. 1999 ISBN 0-7923-5681-0 A. Carpinteri: Nonlinear Crack Models for Nonmetallic Materials. 1999 ISBN 0-7923-5750-7 F. Pfeifer (ed.): IUTAM Symposium on Unilateral Multibody Contacts. Proceedings of the IUTAM Symposium held in Munich, Germany. 1999 ISBN 0-7923-6030-3 E. Lavendelis and M. Zakrzhevsky (eds.): IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems. Proceedings of the IUTAM/IFToMM Symposium held in Riga, Latvia. 2000 ISBN 0-7923-6106-7 J.-P. Merlet: Parallel Robots. 2000 ISBN 0-7923-6308-6 J.T. Pindera: Techniques of Tomographic Isodyne Stress Analysis. 2000 ISBN 0-7923-6388-4 G.A. Maugin, R. Drouot and F. Sidoroff (eds.): Continuum Thermomechanics. The Art and Science of Modelling Material Behaviour. 2000 ISBN 0-7923-6407-4 N. Van Dao and E.J. Kreuzer (eds.): IUTAM Symposium on Recent Developments in Non-linear Oscillations of Mechanical Systems. 2000 ISBN 0-7923-6470-8 S.D. Akbarov and A.N. Guz: Mechanics of Curved Composites. 2000 ISBN 0-7923-6477-5 M.B. Rubin: Cosserat Theories: Shells, Rods and Points. 2000 ISBN 0-7923-6489-9 S. Pellegrino and S.D. Guest (eds.): IUTAM-IASS Symposium on Deployable Structures: Theory and Applications. Proceedings of the IUTAM-IASS Symposium held in Cambridge, U.K., 6–9 September 1998. 2000 ISBN 0-7923-6516-X A.D. Rosato and D.L. Blackmore (eds.): IUTAM Symposium on Segregation in Granular Flows. Proceedings of the IUTAM Symposium held in Cape May, NJ, U.S.A., June 5–10, 1999. 2000 ISBN 0-7923-6547-X A. Lagarde (ed.): IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics. Proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France, August 31–September 4, 1998. 2000 ISBN 0-7923-6604-2 D. Weichert and G. Maier (eds.): Inelastic Analysis of Structures under Variable Loads. Theory and Engineering Applications. 2000 ISBN 0-7923-6645-X T.-J. Chuang and J.W. Rudnicki (eds.): Multiscale Deformation and Fracture in Materials and Structures. The James R. Rice 60th Anniversary Volume. 2001 ISBN 0-7923-6718-9 S. Narayanan and R.N. Iyengar (eds.): IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics. Proceedings of the IUTAM Symposium held in Madras, Chennai, India, 4–8 January 1999 ISBN 0-7923-6733-2 S. Murakami and N. Ohno (eds.): IUTAM Symposium on Creep in Structures. Proceedings of the IUTAM Symposium held in Nagoya, Japan, 3-7 April 2000. 2001 ISBN 0-7923-6737-5 W. Ehlers (ed.): IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials. Proceedings of the IUTAM Symposium held at the University of Stuttgart, Germany, September 5-10, 1999. 2001 ISBN 0-7923-6766-9 D. Durban, D. Givoli and J.G. Simmonds (eds.): Advances in the Mechanis of Plates and Shells The Avinoam Libai Anniversary Volume. 2001 ISBN 0-7923-6785-5 U. Gabbert and H.-S. Tzou (eds.): IUTAM Symposium on Smart Structures and Structonic Systems. Proceedings of the IUTAM Symposium held in Magdeburg, Germany, 26–29 September 2000. 2001 ISBN 0-7923-6968-8

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Y. Ivanov, V. Cheshkov and M. Natova: Polymer Composite Materials – Interface Phenomena & Processes. 2001 ISBN 0-7923-7008-2 R.C. McPhedran, L.C. Botten and N.A. Nicorovici (eds.): IUTAM Symposium on Mechanical and Electromagnetic Waves in Structured Media. Proceedings of the IUTAM Symposium held in Sydney, NSW, Australia, 18-22 Januari 1999. 2001 ISBN 0-7923-7038-4 D.A. Sotiropoulos (ed.): IUTAM Symposium on Mechanical Waves for Composite Structures Characterization. Proceedings of the IUTAM Symposium held in Chania, Crete, Greece, June 14-17, 2000. 2001 ISBN 0-7923-7164-X V.M. Alexandrov and D.A. Pozharskii: Three-Dimensional Contact Problems. 2001 ISBN 0-7923-7165-8 J.P. Dempsey and H.H. Shen (eds.): IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics. Proceedings of the IUTAM Symposium held in Fairbanks, Alaska, U.S.A., 13-16 June 2000. 2001 ISBN 1-4020-0171-1 U. Kirsch: Design-Oriented Analysis of Structures. A Unified Approach. 2002 ISBN 1-4020-0443-5 A. Preumont: Vibration Control of Active Structures. An Introduction (2nd Edition). 2002 ISBN 1-4020-0496-6 B.L. Karihaloo (ed.): IUTAM Symposium on Analytical and Computational Fracture Mechanics of Non-Homogeneous Materials. Proceedings of the IUTAM Symposium held in Cardiff, U.K., 18-22 June 2001. 2002 ISBN 1-4020-0510-5 S.M. Han and H. Benaroya: Nonlinear and Stochastic Dynamics of Compliant Offshore Structures. 2002 ISBN 1-4020-0573-3 A.M. Linkov: Boundary Integral Equations in Elasticity Theory. 2002 ISBN 1-4020-0574-1 L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell: Functional Analysis. Applications in Mechanics and Inverse Problems (2nd Edition). 2002 ISBN 1-4020-0667-5; Pb: 1-4020-0756-6 Q.P. Sun (ed.): IUTAM Symposium on Mechanics of Martensitic Phase Transformation in Solids. Proceedings of the IUTAM Symposium held in Hong Kong, China, 11-15 June 2001. 2002 ISBN 1-4020-0741-8 M.L. Munjal (ed.): IUTAM Symposium on Designing for Quietness. Proceedings of the IUTAM Symposium held in Bangkok, India, 12-14 December 2000. 2002 ISBN 1-4020-0765-5 J.A.C. Martins and M.D.P. Monteiro Marques (eds.): Contact Mechanics. Proceedings of the 3rd Contact Mechanics International Symposium, Praia da Consola¸ca˜ o, Peniche, Portugal, 17-21 June 2001. 2002 ISBN 1-4020-0811-2 H.R. Drew and S. Pellegrino (eds.): New Approaches to Structural Mechanics, Shells and Biological Structures. 2002 ISBN 1-4020-0862-7 J.R. Vinson and R.L. Sierakowski: The Behavior of Structures Composed of Composite Materials. Second Edition. 2002 ISBN 1-4020-0904-6 Not yet published. J.R. Barber: Elasticity. Second Edition. 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6 C. Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August 2001. 2003 ISBN 1-4020-1170-9

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 109. P. St˚ahle and K.G. Sundin (eds.): IUTAM Symposium on Field Analyses for Determination of Material Parameters – Experimental and Numerical Aspects. Proceedings of the IUTAM Symposium held in Abisko National Park, Kiruna, Sweden, July 31 – August 4, 2000. 2003 ISBN 1-4020-1283-7 110. N. Sri Namachchivaya and Y.K. Lin (eds.): IUTAM Symposium on Nonlinear Stochastic Dynamics. Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 – 30 August, 2000. 2003 ISBN 1-4020-1471-6 111. H. Sobieckzky (ed.): IUTAM Symposium Transsonicum IV. Proceedings of the IUTAM Symposium held in G¨ottingen, Germany, 2–6 September 2002, 2003 ISBN 1-4020-1608-5 112. J.-C. Samin and P. Fisette: Symbolic Modeling of Multibody Systems. 2003 ISBN 1-4020-1629-8 113. A.B. Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics. Proceedings of the IUTAM Symposium held in Liverpool, United Kingdom, 8-11 July 2002. 2003 ISBN 1-4020-1780-4 114. S. Ahzi, M. Cherkaoui, M.A. Khaleel, H.M. Zbib, M.A. Zikry and B. LaMatina (eds.): IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials. Proceedings of the IUTAM Symposium held in Marrakech, Morocco, 20-25 October 2002. 2004 ISBN 1-4020-1861-4 115. H. Kitagawa and Y. Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of Fracture Process and Materials Strength. Proceedings of the IUTAM Symposium held in Osaka, Japan, 6-11 July 2003. Volume in celebration of Professor Kitagawa’s retirement. 2004 ISBN 1-4020-2037-6 116. E.H. Dowell, R.L. Clark, D. Cox, H.C. Curtiss, Jr., K.C. Hall, D.A. Peters, R.H. Scanlan, E. Simiu, F. Sisto and D. Tang: A Modern Course in Aeroelasticity. 4th Edition, 2004 ISBN 1-4020-2039-2 117. T. Burczy´nski and A. Osyczka (eds.): IUTAM Symposium on Evolutionary Methods in Mechanics. Proceedings of the IUTAM Symposium held in Cracow, Poland, 24-27 September 2002. 2004 ISBN 1-4020-2266-2 118. D. Ie¸san: Thermoelastic Models of Continua. 2004 ISBN 1-4020-2309-X 119. G.M.L. Gladwell: Inverse Problems in Vibration. Second Edition. 2004 ISBN 1-4020-2670-6 120. J.R. Vinson: Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction. 2005 ISBN 1-4020-3110-6 121. Forthcoming 122. G. Rega and F. Vestroni (eds.): IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics. Proceedings of the IUTAM Symposium held in Rome, Italy, 8–13 June 2003. 2005 ISBN 1-4020-3267-6 123. E.E. Gdoutos: Fracture Mechanics. An Introduction. 2nd edition. 2005 ISBN 1-4020-3267-6 124. M.D. Gilchrist (ed.): IUTAM Symposium on Impact Biomechanics from Fundamental Insights to Applications. 2005 ISBN 1-4020-3795-3 125. J.M. Huyghe, P.A.C. Raats and S. C. Cowin (eds.): IUTAM Symposium on Physicochemical and Electromechanical Interactions in Porous Media. 2005 ISBN 1-4020-3864-X 126. H. Ding and W. Chen: Elasticity of Transversely Isotropic Materials. 2005ISBN 1-4020-4033-4 127. W. Yang (ed): IUTAM Symposium on Mechanics and Reliability of Actuating Materials. Proceedings of the IUTAM Symposium held in Beijing, China, 1–3 September 2004. 2005 ISBN 1-4020-4131-6 128. J.-P. Merlet: Parallel Robots. 2006 ISBN 1-4020-4132-2

Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L. Gladwell 129. G.E.A. Meier and K.R. Sreenivasan (eds.): IUTAM Symposium on One Hundred Years of Boundary Layer Research. Proceedings of the IUTAM Symposium held at DLR-G¨ottingen, Germany, August 12–14, 2004. 2006 ISBN 1-4020-4149-7 130. H. Ulbrich and W. G¨unthner (eds.): IUTAM Symposium on Vibration Control of Nonlinear Mechanisms and Structures. 2006 ISBN 1-4020-4160-8 131. L. Librescu and O. Song: Thin-Walled Composite Beams. Theory and Application. 2006 ISBN 1-4020-3457-1 132. G. Ben-Dor, A. Dubinsky and T. Elperin: Applied High-Speed Plate Penetration Dynamics. 2006 ISBN 1-4020-3452-0 133. X. Markenscoff and A. Gupta (eds.): Collected Works of J. D. Eshelby. Mechanics and Defects and Heterogeneities. 2006 ISBN 1-4020-4416-X 134. R.W. Snidle and H.P. Evans (eds.): IUTAM Symposium on Elastohydrodynamics and Microelastohydrodynamics. Proceedings of the IUTAM Symposium held in Cardiff, UK, 1–3 September, 2004. 2006 ISBN 1-4020-4532-8 135. T. Sadowski (ed.): IUTAM Symposium on Multiscale Modelling of Damage and Fracture Processes in Composite Materials. Proceedings of the IUTAM Symposium held in Kazimierz Dolny, Poland, 23–27 May 2005. 2006 ISBN 1-4020-4565-4

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