Advanced Engineering Materials and Modeling 978-1-119-24246-8

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Advanced Engineering Materials and Modeling

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Advanced Materials Series he Advanced Materials Series provides recent advancements of the fascinating ield of advanced materials science and technology, particularly in the area of structure, synthesis and processing, characterization, advanced-state properties, and applications. he volumes will cover theoretical and experimental approaches of molecular device materials, biomimetic materials, hybrid-type composite materials, functionalized polymers, supramolecular systems, information- and energy-transfer materials, biobased and biodegradable or environmental friendly materials. Each volume will be devoted to one broad subject and the multidisciplinary aspects will be drawn out in full. Series Editor: Ashutosh Tiwari Biosensors and Bioelectronics Centre Linköping University SE-581 83 Linköping Sweden E-mail: [email protected] Managing Editors: Sachin Mishra and Sophie hompson Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Advanced Engineering Materials and Modeling

Edited by

Ashutosh Tiwari, N. Arul Murugan and Rajeev Ahuja

Copyright © 2016 by Scrivener Publishing LLC. All rights reserved. Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey, and Scrivener Publishing LLC, Beverly, Massachusetts. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best eforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciically disclaim any implied warranties of merchantability or itness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. he advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. For more information about Scrivener products please visit www.scrivenerpublishing.com. Cover design by Russell Richardson Library of Congress Cataloging-in-Publication Data: ISBN 978-1-119-24246-8

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents Preface

xiii

Part 1 Engineering of Materials, Characterizations, and Applications 1 Mechanical Behavior and Resistance of Structural Glass Beams in Lateral–Torsional Buckling (LTB) with Adhesive Joints Chiara Bedon and Jan Belis 1.1 Introduction 1.2 Overview on Structural Glass Applications in Buildings 1.3 Glass Beams in LTB 1.3.1 Susceptibility of Glass Structural Elements to Buckling Phenomena 1.3.2 Mechanical and Geometrical Inluencing Parameters in Structural Glass Beams 1.3.3 Mechanical Joints 1.3.4 Adhesive Joints 1.4 heoretical Background for Structural Members in LTB 1.4.1 General LTB Method for Laterally Unrestrained (LU) Members 1.4.2 LTB Method for Laterally Unrestrained (LU) Glass Beams 1.4.2.1 Equivalent hickness Methods for Laminated Glass Beams 1.4.3 Laterally Restrained (LR) Beams in LTB 1.4.3.1 Extended Literature Review on LR Beams 1.4.3.2 Closed-form Formulation for LR Beams in LTB 1.4.3.3 LR Glass Beams Under Positive Bending Moment My

3 4 5 5 5 8 9 10 14 14 17 18 23 23 24 28 v

vi

Contents 1.5 Finite-element Numerical Modeling 1.5.1 FE Solving Approach and Parametric Study 1.5.1.1 Linear Eigenvalue Buckling Analyses (lba) 1.5.1.2 Incremental Nonlinear Analyses (inl) 1.6 LTB Design Recommendations 1.6.1 LR Beams Under Positive Bending Moment My 1.6.2 Further Extension and Developments of the Current Outcomes 1.7 Conclusions References

2 Room Temperature Mechanosynthesis of Nanocrystalline Metal Carbides and heir Microstructure Characterization S.K. Pradhan and H. Dutta 2.1 Introduction 2.1.1 Application 2.1.2 Diferent Methods for Preparation of Metal Carbide 2.1.3 Mechanical Alloying 2.1.4 Planetary Ball Mill 2.1.5 he Merits and Demerits of Planetary Ball Mill 2.1.6 Review of Works on Metal Carbides by Other Authors 2.1.7 Signiicance of the Study 2.1.8 Objectives of the Study 2.2 Experimental 2.3 heoretical Consideration 2.3.1 Microstructure Evaluation by X-ray Difraction 2.3.2 General Features of Structure 2.4 Results and Discussions 2.4.1 XRD Pattern Analysis 2.4.2 Variation of Mol Fraction 2.4.3 Phase Formation Mechanism 2.4.4 Is Ball-milled Prepared Metal Carbide Contains Contamination? 2.4.5 Variation of Particle Size 2.4.6 Variation of Strain 2.4.7 High-Resolution Transmission Electron Microscopy Study 2.4.8 Comparison Study between Binary and Ternary Ti-based Metal Carbides

31 32 32 35 38 38 39 42 44 49 50 50 50 51 51 52 53 54 55 56 58 58 60 60 60 65 69 71 72 74 76 76

Contents 2.5 Conclusion Acknowledgment References 3 Toward a Novel SMA-reinforced Laminated Glass Panel Chiara Bedon and Filipe Amarante dos Santos 3.1 Introduction 3.2 Glass in Buildings 3.2.1 Actual Reinforcement Techniques for Structural Glass Applications 3.3 Structural Engineering Applications of Shape-Memory Alloys (SMAs) 3.4 he Novel SMA-Reinforced Laminated Glass Panel Concept 3.4.1 Design Concept 3.4.2 Exploratory Finite-Element (FE) Numerical Study 3.4.2.1 General FE Model Assembly Approach and Solving Method 3.4.2.2 Mechanical Characterization of Materials 3.5 Discussion of Parametric FE Results 3.5.1 Roof Glass Panel (M1) 3.5.1.1 Short-term Loads and Temperature Variations 3.5.1.2 First-cracking Coniguration 3.5.2 Point-supported Façade Panel (M2) 3.5.2.1 Short-term Loads and Temperature Variations 3.6 Conclusions References 4 Sustainable Sugarcane Bagasse Cellulose for Papermaking Noé Aguilar-Rivera 4.1 Pulp and Paper Industry 4.2 Sugar Industry 4.3 Sugarcane Bagasse 4.4 Advantageous Utilizations of SCB 4.5 Applications of SCB Wastes 4.6 Problematic of Nonwood Fibers in Papermaking 4.7 SCB as Raw Material for Pulp and Paper 4.8 Digestion 4.9 Bleaching

vii 80 80 80 87 87 89 92 93 94 94 96 96 98 101 101 102 106 109 111 114 117 121 122 123 124 129 130 131 134 135 135

viii

Contents 4.10 Properties of Bagasse Pulps 4.10.1 Pulp Strength 4.10.2 Pulp Properties 4.10.3 Washing Technology 4.10.4 Paper Machine Operation 4.11 Objectives 4.12 Old Corrugated Container Pulps 4.13 Synergistic Deligniication SCB–OCC 4.14 Elemental Chlorine-Free Bleaching of SCB Pulps 4.15 Conclusions References

5 Bio-inspired Composites: Using Nature to Tackle Composite Limitations F. Libonati 5.1 Introduction 5.2 Bio-inspiration: Bone as Biomimetic Model 5.3 Case Studies Using Biomimetic Approach 5.3.1 Fiber-reinforced Bone-inspired Composites 5.3.2 Fiber-reinforced Bone-inspired Composites with CNTs 5.3.3 Bone-inspired Composites via 3D Printing 5.4 Methods 5.4.1 Composite Lamination 5.4.2 Additive Manufacturing 5.4.3 Computational Modeling 5.5 Conclusions References

Part 2

136 137 137 138 138 138 139 141 150 156 158 165 166 169 172 172 176 177 179 180 181 182 183 185

Computational Modeling of Materials

6 Calculation on the Ground State Quantum Potentials for the ZnSxSe1-x (0 < x < 1) G.H.E Alshabeeb and A.K. Arof 6.1 Introduction 6.2 Ground State in D-Dimensional Coniguration Space for ZnSxSe1-x Zincblende Structure 6.3 Ground States in the Case of Momentum Space 6.4 Results and Discussion

193 193 194 196 199

Contents 6.5 Conclusions Acknowledgment References 7 Application of First Principles heory to the Design of Advanced Titanium Alloys Y. Song, J. H. Dai, and R. Yang 7.1 Introduction 7.2 Basic Concepts of First Principles 7.3 heoretical Models of Alloy Design 7.3.1 he Hume-Rothery heory 7.3.2 Discrete Variational Method and d-Orbital Method 7.3.2.1 Discrete Variational Method 7.3.2.2 d-Electrons Alloy heory 7.4 Applications 7.4.1 Phase Stability 7.4.1.1 Binary Alloy 7.4.1.2 Multicomponent Alloys 7.4.2 Elastic Properties 7.4.3 Examples 7.4.3.1 Gum Metal 7.4.3.2 Ti2448 (Ti–24Nb–4Zr–8Sn) 7.5 Conclusions Acknowledgment References 8 Digital Orchid: Creating Realistic Materials Itikhar B. Abbasov 8.1 Introduction 8.2 Concept Development 8.3 hree-dimensional Modeling of Decorative Light Fixture 8.4 Materials Creating and Editing 8.5 Conclusion References 9 Transformation Optics-based Computational Materials for Stochastic Electromagnetics Ozlem Ozgun and Mustafa Kuzuoglu 9.1 Introduction 9.2 heory of Transformation Optics

ix 201 201 201 203 203 204 207 207 212 212 214 215 215 215 218 219 222 222 223 226 226 226 229 230 230 231 232 239 240 241 242 245

x

Contents 9.3

Scattering from Rough Sea Surfaces 9.3.1 Numerical Validation and Monte Carlo Simulations 9.4 Scattering from Obstacles with Rough Surfaces or Shape Deformations 9.4.1 Numerical Validation and Monte Carlo Simulations 9.4.2 Combining Perturbation heory and Transformation Optics for Weakly Perturbed Surfaces 9.5 Scattering from Randomly Positioned Array of Obstacles 9.5.1 Separate Transformation Media 9.5.1.1 Numerical Validation & Monte Carlo Simulations 9.5.2 A Single Transformation Medium 9.5.2.1 Numerical Validation & Monte Carlo Simulations 9.5.3 Recurring Scaling and Translation Transformations 9.5.3.1 Numerical Validation & Monte Carlo Simulations 9.6 Propagation in a Waveguide with Rough or Randomly Varying Surface 9.6.1 Numerical Validation and Monte Carlo Simulations 9.7 Conclusion References 10 Superluminal Photons Tunneling through Brain Microtubules Modeled as Metamaterials and Quantum Computation Luigi Maxmilian Caligiuri and Takaaki Musha 10.1 Introduction 10.2 QED Coherence in Water: A Brief Overview 10.3 “Electronic” QED Coherence in Brain Microtubules 10.4 Evanescent Field of Coherent Photons and heir Superluminal Tunneling through MTs 10.5 Coupling between Nearby MTs and their Superluminal Interaction through the Exchange of Virtual Superradiant Photons 10.6 Discussion

248 252 254 259 260 264 265 267 269 271 272 274 274 279 283 284 287 288 291 297 301 308 312

Contents xi 10.7 Brain Microtubules as “Natural” Metamaterials and the Ampliication of Evanescent Tunneling Wave Amplitude 10.8 Quantum Computation by Means of Superluminal Photons 10.9 Conclusions References 11 Advanced Fundamental-solution-based Computational Methods for hermal Analysis of Heterogeneous Materials Hui Wang and Qing-Hua Qin 11.1 Introduction 11.2 Basic Formulation of MFS 11.2.1 Standard MFS 11.2.2 Modiied MFS 11.2.2.1 RBF Interpolation for the Particular Solution 11.2.2.2 MFS for the Homogeneous Solution 11.2.2.3 Complete Solution 11.3 Basic Formulation of HFS-FEM 11.3.1 Problem Statement 11.3.2 Implementation of the HFS-FEM 11.3.4 Recovery of Rigid-body Motion 11.4 Applications in Functionally Graded Materials 11.4.1 Basic Equations in Functionally Graded Materials 11.4.2 MFS for Functionally Graded Materials 11.4.3 HFS-FEM for Functionally Graded Materials 11.5 Applications in Composite Materials 11.5.1 Basic Equations of Composite Materials 11.5.2 MFS for Composite Materials 11.5.2.1 MFS for the Matrix Domain 11.5.2.2 MFS for the Fiber Domain 11.5.2.3 Complete Linear Equation System 11.5.3 HFS-FEM for Composite Materials 11.5.3.1 Special Fundamental Solutions 11.5.3.2 Special n-Sided Fiber/Matrix Elements 11.6 Conclusions

315 321 325 326 331 332 334 334 336 337 338 339 340 340 342 345 345 345 346 349 353 353 356 356 356 357 358 358 359 361

xii

Contents Acknowledgments Conlict of Interest References

12 Understanding the SET/RESET Characteristics of Forming Free TiOx/TiO2–x Resistive-Switching Bilayer Structures through Experiments and Modeling P. Bousoulas and D. Tsoukalas 12.1 Introduction 12.2 Experimental Methodology 12.3 Bipolar Switching Model 12.3.1 Resistive-Switching Performance 12.3.2 Resistive-Switching Model 12.4 RESET Simulations 12.4.1 I–V Response 12.4.2 Inluence of TE on the CFs Broken Region 12.5 SET Simulations 12.6 Simulation of Time-dependent SET/RESET Processes 12.7 Conclusions Acknowledgments References

362 362 362

369 370 372 376 376 379 385 385 389 394 397 399 400 400

13 Advanced Materials and hree-dimensional Computer-aided Surgical Worklow in Cranio-maxillofacial Reconstruction 407 Luis Miguel Gonzalez-Perez, Borja Gonzalez-Perez-Somarriba Gabriel Centeno, Carpóforo Vallellano and Juan Jose Egea-Guerrero 13.1 Introduction 408 13.2 Methodology 409 13.3 Findings 414 13.4 Discussion 423 References 432 14 Displaced Multiwavelets and Splitting Algorithms Boris M. Shumilov 14.1 An Algorithm with Splitting of Wavelet Transformation of Splines of the First Degree 14.1.1 “Lazy” Wavelets 14.1.2 Examples of Wavelet Decomposition of a Signal of Length 8

435

439 440 443

Contents xiii 14.1.3 “Orthonormal” Wavelets 14.1.4 An Example of Function of Harten 14.2 An Algorithm for Constructing Orthogonal to Polynomials Multiwavelet Bases 14.2.1 Creation of System of Basic Multiwavelets of Any Odd Degree on a Closed Interval 14.2.2 Creation of the Block of Filters 14.2.3 Example of Orthogonal to Polynomials Multiwavelet Bases 14.2.4 he Discussion of Approximation on a Closed Interval 14.3 he Tridiagonal Block Matrix Algorithm 14.3.1 Inverse of the Block of Filters 14.3.2 Example of the Hermite Quintic Spline Function Supported on [−1, 1] 14.3.3 Example of the Hermite Septimus Spline Function Supported on [−1, 1] 14.3.4 Numerical Example of Approximation of Polynomial Function 14.3.5 Numerical Example with Two Ruptures of the First Kind and a Corner 14.4 Problem of Optimization of Wavelet Transformation of Hermite Splines of Any Odd Degree 14.4.1 An Algorithm with Splitting for Wavelet Transformation of Hermite Splines of Fith Degree 14.4.2 Examples 14.5 Application to Data Processing of Laser Scanning of Roads 14.5.1 Calculation of Derivatives on Samples 14.5.2 Example of Wavelet Compression of One Track of Data of Laser Scanning 14.5.3 Modeling of Surfaces 14.5.4 Functions of a Package of Applied Programs for Modeling of Routes and Surfaces of Highways 14.6 Conclusions References Index

446 450 452 452 455 457 459 460 460 461 463 466 467 471 474 481 486 486 486 486 488 490 490 495

Preface he engineering of materials with advanced features is driving the research towards the design of innovative high-performance materials. New materials oten deliver the best solutions for structural applications, precisely contributing to the inest combination of mechanical properties and low weight. Furthermore, these materials mimic the principles of nature, leading to a new class of structural materials which include biomimetic composites, natural hierarchical materials and smart materials. Meanwhile, computational modeling approaches are valuable tools which are complementary to experimental techniques and provide signiicant information at the microscopic level and explain the properties of materials and their existence itself. he modeling further provides useful insight to propose possible strategies to design and fabricate materials with novel and improved properties. Depending upon the pragmatic computational models of choice, approaches vary for the prediction of the structure- and element-based approaches to fabricate materials with properties of interest. his book brings together the engineering materials and modeling approaches generally used in structural materials science. Research topics on materials engineering, characterization, applications and their computational modeling are covered in this book. In general, computational modeling approaches are routinely used as cost-efective and complementary tools to get information about the materials at the microscopic level and to explain their electronic and magnetic properties and the way they respond to external parameters like temperature and pressure. In addition, modeling provides useful insight into the construct of design principles and strategies to fabricate materials with novel and improved properties. he use of modeling together with experimental validation opens up the possibility for designing extremely useful materials that are relevant for various industries and healthcare sectors. his book has been designed in such a way as to cover aspects of both the use of experimental and computational approaches for materials engineering and fabrication. Chapters 1 through 6 are devoted to experimental characterization of materials and some of their applications relevant to the paper xv

xvi

Preface

industry and healthcare sectors. Chapters 7 through 13 are devoted to computational materials modeling and their fabrication using atomisticand inite-element-based approaches. Speciically discussed in Chapters 7 and 8 are irst-principles-based modeling approaches to predict the structure and electronic properties of extended systems. he remaining chapters contribute with theoretical approaches to understanding hybrid materials and stochastic electromagnets and to modeling complex processes like tunneling of superluminal photons. he book is written for readers from diverse backgrounds across chemistry, physics, materials science and engineering, medical science, pharmacy, environmental technology, biotechnology, and biomedical engineering. It ofers a comprehensive view of cutting-edge research on materials engineering and modeling. We acknowledge the contributors and publisher for their prompt response in order that this book could be published in a timely manner. Editors Ashutosh Tiwari, PhD, DSc N. Arul Murugan, PhD Rajeev Ahuja, PhD 10 June 2016

Part 1 ENGINEERING OF MATERIALS, CHARACTERIZATIONS, AND APPLICATIONS

Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (1–48) © 2016 Scrivener Publishing LLC

1 Mechanical Behavior and Resistance of Structural Glass Beams in Lateral–Torsional Buckling (LTB) with Adhesive Joints Chiara Bedon1* and Jan Belis2 1

University of Trieste, Department of Engineering and Architecture, Trieste, Italy 2 Ghent University, Department of Structural Engineering, Laboratory for Research on Structural Models – LMO, Ghent, Belgium

Abstract Glass is largely used in practice as an innovative structural material in the form of beams or plate elements able to carry loads. Compared to traditional construction materials, the major inluencing parameter in the design of structural glass elements – in addition to their high architectural and aesthetic impacts – is given by the well-known brittle behavior and limited tensile resistance of glass. In this chapter, careful attention is paid to the lateral–torsional buckling (LTB) response of glass beams laterally restrained by continuous adhesive joints, as in the case of glass façades or roofs. Closed-form solutions and inite-element numerical approaches are recalled for the estimation of their Euler’s critical buckling moment under various loading conditions. Nonlinear buckling analyses are then critically discussed by taking into account a multitude of mechanical and geometrical aspects. Design recommendations for laterally restrained glass beams in LTB are inally presented. Keywords: Lateral–torsional buckling (LTB), glass beams, analytical models, inite-element modeling, structural adhesive joints, composite sections, incremental buckling analysis, imperfections, buckling design methods, buckling curve

*Corresponding author: [email protected] Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (3–48) © 2016 Scrivener Publishing LLC

3

4

Advanced Engineering Materials and Modeling

1.1 Introduction Glass is largely used in practice as an innovative structural material, e.g. in the form of beams or plate elements able to carry loads. Oten, structural glass components are used in structures in combination with other materials, such as timber [1–6] or composites [1, 7–9]. However, especially in façades, roofs, and building envelopes, the use of glass panels combined with steel frames, aluminum bracing systems, or cable nets represents one of the major conigurations, for which a wide set of case studies and technological possibilities are available [1, 2, 10–15]. Compared to traditional construction materials, the major inluencing parameter in the design of structural glass elements – in addition to their high architectural and aesthetic impact – is given by the well-known brittle behavior and limited tensile resistance of glass. he use of thermoplastic interlayers alternated to two (or more) glass sheets in the form of laminated glass (LG) elements – despite the high sensitivity of the bonding foils to the efects of temperature and load-duration – represents the typical solution for buildings, automotive applications, etc. due to the intrinsic ductility and post-breakage resistance. In those cases, the typical conigurations for structural glass assemblies are oten derived – and properly modiied, to account for the brittle behavior of glass – from practice of traditional construction materials (e.g. steel structures and sandwich structures). he connections used in such LG assemblies are traditionally properly designed and well-calibrated mechanical connections (e.g. steel fasteners and bolted joints) able to ofer a certain structural interaction among multiple glass components. However, due to continuous scientiic (material) improvements, technological innovations and architectural demands, recent design trends are oten oriented towards the minimization of mechanical joints and toward the development of frameless glazing systems, in which glass to glass interaction is provided by chemical connections such as sealant joints or adhesives only. his is the case for beams, such as glass elements used in practice as stifeners for façade or roof panels, where the coupling between them is oten provided by continuous adhesive joints. From a structural point of view, the efect of such joints can be compared to a partially rigid shear connection, and consequently its mechanical efectiveness should be properly taken into account. Bolted point ixings or continuous adhesive joints currently represent the two most used typologies of connections and can both be employed in glass façades or roofs, e.g. to provide the mechanical interaction between the glass beams and the supported glass roof panels.

Mechanical Behavior and Resistance

5

While in the irst case the bolted connectors and their related efects can oten be rationally described in the form of ininitely rigid intermediate restraints, the coniguration of glass beams with continuous adhesive joints requires appropriate studies and related analytical methods. Adhesive joints are in fact characterized by moderate shear stifness, and consequently they act as a continuous, lexible joint between the beams and the connected panels. Adhesives of common use in practice are also characterized by moderate shear/tensile resistance; hence, an appropriate design approach should be taken into account for them, regardless of possible LTB phenomena. his chapter, in this context, aims to present an extended review of glass beams in LTB, including a discussion of the main inluencing parameters, mechanical properties, geometrical aspects, available analytical methods, and inite-element (FE) approaches. A detailed discussion of the LTB mechanical response of glass beams, laterally unrestrained or restrained by means of continuous adhesive joints, will then be proposed.

1.2 Overview on Structural Glass Applications in Buildings Structural glass applications are mainly associated, in current practice, to aesthetic, architectural or thermal, and acoustic requirements. Glass is, in fact, synonymous of transparency and lightness, hence inds primarily application in building envelopes, roofs, canopies, etc. and solutions in which transparency is mandatory. Major structural glass assemblies – oten of complex geometry – are obtained by appropriate conjunct use of glass elements with metal frameworks and substructures (Figure 1.1). Structural conigurations combining glass elements with timber components (Figure 1.2) also represent a solution of large interest for designers and engineers, especially in those applications aiming to strong energy eiciency [24].

1.3

Glass Beams in LTB

1.3.1 Susceptibility of Glass Structural Elements to Buckling Phenomena he exposure of structural components in general to signiicant compression, shear, bending, or a combination of them is the irst cause of buckling

6

Advanced Engineering Materials and Modeling

(a)

(c)

(b)

(d)

Figure 1.1 Example of structural glass applications in buildings, in conjunction with metal frameworks and substructures. Pictures taken from (a) [16], (b) [17], (c) [18], and (d) [19].

failure mechanisms (Figure 1.3). As far as these structural elements are slender and/or afected by several inluencing parameters, such as initial geometrical imperfections, eccentricities, and residual stresses, the susceptibility to buckling phenomena increases and represents an important issue to be properly predicted and prevented. his is the case of both isotropic and orthotropic plates, beams, columns, but also laminates and composites in general. he presence of rather unconventional materials, in particular, represents one of the major inluencing parameters to be properly assessed, especially in the presence of materials whose mechanical properties can be afected by time/temperature-dependent degradation. In structural glass beams, a multitude of efects strictly related to mechanical properties, geometrical features, initial imperfections, etc., should be properly taken into account to prevent possible LTB failure mechanisms.

Mechanical Behavior and Resistance

(a)

7

(b)

(c)

(d)

Figure 1.2 Example of structural glass applications in buildings, in conjunction with timber components and assemblies. Pictures taken from (a) [20], (b) [21], (c) [22], and (d) [23].

Plate buckling

Column buckling N

a

F L0

w

v

w

LTB buckling σx L0

x

x y z

h

x

y z

Figure 1.3 Buckling phenomena in columns, beams, and plates.

y z

b

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Advanced Engineering Materials and Modeling

1.3.2 Mechanical and Geometrical Inluencing Parameters in Structural Glass Beams Structural glass beams ind primary applications in façades and roofs in the form of stifeners. here, both mechanical and adhesive joints can be used to provide a certain structural interaction between the glass beams and the supported panels (see Sections 1.3.3 and 1.3.4). Compared to beams composed of traditional construction materials, such as steel, the out-of-plane bending response of glass ins is characterized by speciic mechanical and geometrical aspects that should be properly taken into account when assessing their expected structural response. First, glass is a material characterized by a relatively small modulus of elasticity E compared to steel, see Table 1.1 and Figure 1.4), and by a typical brittle elastic tensile behavior with limited characteristic strength (Figure 1.4b). Table 1.1 Soda lime silica glass properties [25]. Symbol Density Young’s modulus

E

Soda lime silica glass

kg/m3

2500

MPa

70,000 0.23

–

Poisson’s ratio Coeicient of thermal expansion

σ

Unit

9 × 10–6

K–1

t

σ [MPa]

Steel

FT

120 Glass

HS

70

AN

45 Concrete 0 (a)

ε (b)

0

1

2

3

ε [‰]

Figure 1.4 Mechanical properties of monolithic glass for structural applications. (a) Qualitative comparison of glass tensile behavior vs. traditional construction materials, such as steel and concrete; (b) tensile constitutive law of glass, depending on the adopted pre-stressing technique (FT = fully tempered, HS = heat-strengthened, AN = annealed).

Mechanical Behavior and Resistance (b)

(c)

(d)

h

(a)

9

t

t1

tint

t1

tint

t1

tint

t2

Figure 1.5 Typical cross sections of common use in structural glass applications (edge chamfers are neglected). (a) Monolithic cross section or (b), (c), and (d) laminated cross sections.

Although thermal or chemical pre-stressing processes can increase the reference characteristic tensile strength of annealed glass (AN) by a factor of about two (for heat-strengthened glass, HT) or even three (in the case of fully tempered glass, FT), the occurrence of both local or global failure mechanisms due to the tensile peaks should be properly prevented. Careful consideration should be given to glass especially in the vicinity of supports and point ixings, as well as to LG cross sections (Figure 1.5), representing the majority of structural glass applications but being typically characterized by the presence of two (or more) glass layers and one (or more) intermediate foils able to act in the form of a lexible shear connection only between them. Common interlayers are, in fact, composed of PVB [26, 27], SG [28], or Ethylene vinyl acetate (EVA) [29] components, e.g. by thermoplastic ilms whose shear stifness Gint strictly depends on several conditions (e.g. time loading, temperature (Figure 1.6)). Also in the case of cross sections composed of multiple glass layers (e.g. Figure 1.5), glass beams are moreover characterized by relatively high slenderness ratios, e.g. large h/t ratios with long spans L.

1.3.3 Mechanical Joints Glass elements can be used in constructions in diferent ways, including a variety of metal point ixings and connectors able to provide a certain restraint and interaction between multiple structural components.

10 Advanced Engineering Materials and Modeling 100

G [N/mm2] Gräf et al. [24] Van Duster et al. [25]

10 T = 10 °C T = 20 °C

1

T = 30 °C

T = 50 °C

t [s/]

0.1 1

10

100

1000

10000

100000

1000000 10000000

Figure 1.6 Mechanical constitutive law of common interlayers for structural glass applications. Data provided for PVB ilms [26, 27] in the form of shear modulus Gint as a function of load duration, for diferent temperatures.

Common options in structural glass applications are in fact represented by metal clamp ixings, drilled ixings, and auxiliary metal connectors (Figure 1.7). Characterized by strong aesthetic impact, mechanical point supports generally provide a fully rigid restraint to the joined glass panels, hence allowing to minimize the presence of metal frameworks and substructures. On the other hand, speciic design rules and requirements must be satisied, to avoid possible local failure mechanisms in glass, etc.

1.3.4 Adhesive Joints An important aspect that should be properly taken into account in the design of structural glass assemblies in general, but in particular for the LTB calculation of glass beams, is represented by the presence of adhesive joints at the interface between the beams themselves and the supported elements. Some examples are provided in Figure 1.8a and b, while Figure 1.8c and d presents a schematic view of a typical adhesive joint, with the corresponding analytical model. As shown in Figure 1.8c, the typical joint consists in fact of small, continuous layers of adhesive and special setting blocks, which act as spacers during application and curing of the adhesive. In addition, they provide an appropriate joint stifness

Mechanical Behavior and Resistance

(a)

(b)

(c)

(d)

11

Figure 1.7 Examples of mechanical joints in structural glass applications. Pictures derived from (a) [30], (b) [31], (c), and (d) [32].

in the direction of the applied external loads. In this hypothesis, under the action of loads applied on the glass panels (e.g. distributed pressures due to live loads on the roof), the adhesive joint behaves as continuous, ininitely rigid link in the z-direction, while the same joint acts as a lexible shear connection toward possible out-of-plane phenomena (e.g. y-direction). In general, when designing an adhesive connection for structural glass applications, several aspects should be properly taken into account. he strength of a given adhesive joint – compared to mechanical fasteners – is in fact strictly related to a multitude of inluencing parameters, such as the joint geometrical properties (e.g. shape, thickness), its mechanical properties (e.g. the type of adhesive), the duration of loading (e.g. due to possible degradation of the reference mechanical properties), and further environmental parameters including temperature, moisture, UV light, time curing, adhesion, etc. For analytical calculations or reined FE studies and analyses related to the LTB response of laterally restrained glass beams, as shown in Sections 1.4 and 1.5, the actual mechanical

12 Advanced Engineering Materials and Modeling

(a)

(b)

Glass panel

h

ad

z

Adhesive joint Setting block

ky kθ

ZM

Glass beam

zx y (c)

tint

h

G

y

L0

hadh

t tint

θ Wadh

t

t (d)

Figure 1.8 Typical examples of structural glass applications with adhesive joints. Pictures taken from (a) [33] and (b) [34]. (c and d) Schematic view of a typical adhesive joint for application in a glass roof. (c) Overview and (d) transversal cross section of the reference analytical model.

properties of common adhesive joints represent a key input parameter. In Table 1.2, some nominal mechanical properties are proposed for various adhesives of common use for glass-to-glass, glass-to-steel, and glassto-timber connections. Alternatively, the simplest way to determine the constitutive law of a given adhesive type for structural applications in glass beams and ins takes the form of a pure shear test. Shear tests, for example, were performed at Ghent University [39] on a series of adhesive specimens composed of Dow Corning 895 (DC 895) [37], a one-component sealant largely used in practice for glass structures (Figure 1.9).

Mechanical Behavior and Resistance

13

Table 1.2 Mechanical properties of common adhesive types for structural glass applications. Adhesive type Silicone sealant

Ultimate tensile strength (MPa)

Young’s modulus (MPa)

Elongation at rupture (%)

Reference

1.20

0.70

460

[35]

2.20

0.90

450

[36]

1.17

0.95

525

[37]

n.a.

1.05

400

[38]

n.a.: not available.

Figure 1.9 Progressive shear failure of a structural silicone specimen [39].

Displacement-controlled shear tests were carried out at 23 °C, with a constant speed deformation of 5 mm/min as recommended by ETAG 002 [40]. Small sealant samples (total length ladh = 100 mm) with diferent square section size (wadh = 6 and 15 mm, in accordance with [37]) were tested, in accordance with Figure 1.8. An almost linear elastic behavior was noticed, up to failure, with ky = 0.184 N/mm2 the average elastic stifness per unit of length. All the shear tests generally highlighted in fact an almost stable behavior for the specimens, attaining large displacements before failure, with an ultimate elongation u ≈ (du – wadh)/wadh equal

14 Advanced Engineering Materials and Modeling to ≈416% and ≈406% for series A and B, respectively, and du denoting the maximum displacement at failure. he obtained average ultimate elongation u,avg resulted well comparable to nominal values of structural sealants available in commerce (e.g. Table 1.2). In terms of ultimate shear/tensile stress u,avg, this parameter was also derived from experimental measurements as the average ratio between the failure load Fu of each specimens and the corresponding resisting cross-sectional area Aadh, hence resulting in u,avg = 0.94 N/mm2. Again, the so-calculated strength was found to be in rather good agreement with the nominal ultimate tensile resistance of common structural sealants and adhesives for glass applications (e.g. Table 1.2).

1.4 heoretical Background for Structural Members in LTB 1.4.1 General LTB Method for Laterally Unrestrained (LU) Members In practice, the possible buckling failure of structural members composed of traditional construction materials (e.g. steel, timber or concrete beams, columns, panels) is usually checked by means of standardized design methods. In them, normalized buckling curves properly calibrated are used to express the efective design resistance of a given structural element, as a function of Euler’s theoretical buckling load and a multitude of inluencing parameters, strictly related to material properties, geometrical imperfections, residual stresses, defects, eccentricities, etc. For safe design purposes, Euler’s buckling load is in fact conventionally reduced from the ideal value, by means of well-calibrated imperfection factors and buckling reduction coeicients. he structural stability represents an essential phase of design for several structural typologies and for steel structures in particular so that the irst analytical approaches for a standardized buckling veriication have been implemented for steel members irst. For clarity of presentation, let us consider the monolithic beam depicted in Figure 1.10, with the given nominal dimensions and coordinate system. he beam is simply supported by fork bearings on its span L0 along the x-axis, while (zy) denotes the plane of its general cross section. In EN-1993-1-1: 2005 [41], the capacity of a structural member with regard to buckling and instability has been irst expressed in the form of a buckling reduction factor LT, being a normalized factor

15

Mechanical Behavior and Resistance z’

z

My

My

x L0

x’

My Deformed z z’

x 2 L 0/

z

z’

v

G

y’

y

y

θ

y

Undeformed z

h

(a)

G’ t

G u G’

(c)

θ

(b)

Figure 1.10 LTB of a laterally unrestrained monolithic beam under constant bending moment My. (a) Elevation, (b) axonometry, and (c) transversal cross section.

strictly related to the member’s susceptibility to buckling phenomena and to the so-called slenderness parameter λ LT . While λ LT is typically expressed as the inverse square root of Euler’s critical moment M cr( E ) , e.g. in the form:

λ LT =

M pl Mcr( E )

=

Wzσ yk Mcr( E )

,

(1.1)

with Mpl denoting the plastic moment, Wz = ht3/6 the elastic resistant modulus, and the yielding stress of steel. yk he buckling reduction factor LT representative of the capacity of the member with regard to stability is generally calculated as:

χ LT =

1 Φ LT + Φ

2 LT

−λ

2 LT

,

χ LT ≤ 1.0 ,

(1.2)

16 Advanced Engineering Materials and Modeling where

(

(

)

2

)

Φ LT = 0.5 ⋅ 1 + α imp ⋅ λ LT − α 0 + λ LT , 0

= 0.2

(1.3) (1.4)

and the imperfection factor imp depends on the cross-sectional properties, the steel grade and the buckling case (e.g. weak or strong axis) under consideration. In accordance with this formulation, the LTB veriication of a given member can be considered satisied when the maximum applied design moment My,Ed does not exceed the buckling design resistance Mb,Rd:

M y , Ed ≤ Mb , Rd = χ LT

Wz f yk

γ M1

,

(1.5)

with M1 the partial safety factor for steel. he major advantage of Eqs. (1.1–1.5) is represented by the general validity of the method, once the non-dimensional slenderness ratio λ LT of a given cross-section (e.g. the geometrical and mechanical properties) are known. At the same time, the loading and boundary condition is implicitly taken into account in the form of correction factors able to modify the general expression for Euler’s critical moment value:

Mcr( E ) = Mcr =

π 2 EI z C1 (kz L0 )2

kz kw

2

I w (kz L0 )2 GIt ⋅ + + (C2 zG )2 − C2 zG , (1.6) 2 Iz π EI z

where kz is the efective length factors for lateral bending, kw is the efective length factor for warping (=1, unless special provisions are provided), zG is the distance between the point of application of the load and the middle axis, C1 and C2 are coeicients depending on the loading and end restraint condition (Table 1.3), Iz is the second moment of area of the section, about the minor axis, It is the torsion constant, Iw is the warping constant.

17

Mechanical Behavior and Resistance Table 1.3 Loading/restraint coeicients for the calculation of Euler’s critical moment of simply supported, fork-end restrained beams [41]. End restraint

Moment distribution

kz

C1

C2

Free rotation about the weak axis

Parabolic

1.0

1.12

0.45

Triangular

1.0

1.35

0.59

Complete restraint against rotation

Parabolic

0.5

0.97

0.36

Triangular

0.5

1.05

0.48

1.4.2 LTB Method for Laterally Unrestrained (LU) Glass Beams In the past years, several experimental, analytical, and numerical studies have been dedicated to the assessment of the LTB response in LU glass beams. A buckling design approach strictly related to Eq. (1.5) was, for example, proposed for LU glass beams in LTB [42–44], by assuming in (Eq. 1.6) the characteristic tensile resistance of glass fyk ≡ Rk and M = 1.4 as a partial safety factor. he advantage of this approach is given to its general formulation, but namely represents the extension of earlier consistent studies on the topic (e.g. [45, 46]). In the hypotheses of a rectangular cross section for a fork-end restrained glass beam, Eq. (1.6) leads in fact to

Mcr( E ) = Mcr =

π EI z GIt , L0

(1.7)

where E ≡ Eg and G ≡ Gg in Eq. (1.7) represent Young’s and shear moduli of glass, respectively, while Iz = ht3/12 signiies the moment of inertia about the minor z-axis and It ≈ ht3/3 (for h/t > 6) is the torsional moment of inertia. Extended comparison proposed in [45] and [46] highlighted the good correlation between analytical critical load predictions derived from Eq. (1.7) and detailed FE models, both for monolithic and laminated cross sections belonging to beams in LTB with various geometrical and mechanical aspects. In the latter case, viscoelastic FE calculations were assessed toward equivalent thickness approaches (e.g. Section 1.4.2.1) applied to LG beams under well-deined load-time and temperature conditions. he main advantage deriving from the application of equivalent thickness-based methods to LG elements is given by the assumption of fully monolithic glass sections with equivalent bending and torsional stifnesses, hence resulting in simpliied but rational and practical design methods, especially for buckling purposes.

18 Advanced Engineering Materials and Modeling In [43, 44], based on classical Euler’s buckling moment deinitions (e.g. Eq. 1.7) and the standardized method proposed by the Eurocode 3 for steel structures (Eq. 1.5), calibration of the imperfection factors deining χLT was then carried out on the base of LTB experimental data available in literature for monolithic and LG beams, as well as extended FE and analytical calculations. Figure 1.11b presents the result of this calibration, for the so-called “Eurocode-based” design buckling curves for glass beams in LTB, compared to previous studies (Figure 1.11a).

1.4.2.1 Equivalent hickness Methods for Laminated Glass Beams When applying equivalent thickness methods to LG sections in LTB, two main aspects should be properly taken into account, namely the appropriate estimation of both the equivalent bending stifness and torsional stifness required in Eq. (1.7). Several formulations are available in literature for this purpose. In [48], for example, extended assessment of some of these existing analytical models based on the equivalent thickness concept and primarily intended for the calculation of the critical LTB moment in three-layered sandwich beams was discussed and further extended for the LTB analysis of LU LG beams, ater an appropriate validation toward FE viscoelastic and experimental data. In the following sections, some of these formulations are recalled for LG beams. Analytical models are proposed for symmetric cross sections composed of two glass layers only (e.g. Figure 1.5, case b). 1.4.2.1.1 Method I For the analysis of the LTB behavior of LG members, Luible [45] irst applied the analytical formulations originally developed for sandwich structural elements to glass sections. he mentioned analytical approach, in particular, is based on the concepts of equivalent bending stifness EIz,ef and equivalent torsional stifness GIt, where EIz,ef is calculated depending on the speciic loading condition (constant bending moment My, distributed load q, concentrated load F at mid-span), while GIt depends on the geometrical/mechanical properties of the cross section only. he expression proposed for EIz,ef is given as a function of the slenderness of the beam (t1, h, L0), the elastic stifness of glass (E), the thickness of the interlayer (tint), and its mechanical properties (Gint). Based on [45],

Mechanical Behavior and Resistance 1.4 FE, M cost. [45] (θ0= L0/270 h rad)

1.2

FE, q dist. [45] (θ0= L0/270 h rad) TEST, F conc. [45] (monolithic) TEST, F conc. [45] (laminated) TEST, F conc. [47]

1.0

χLT

0.8 0.6 0.4 Euler

0.2

Buckling curve (αimp= 0.26, α0= 0.20) for θ0= L0/270 h rad [45]

0.0 0.0

0.5

1.0

(a)

_ λLT

1.5

2.0

2.5

1.4 Buckling curve (αimp= 0.26, α0= 0.20), for θ0= L0/270 h rad [45]

1.2

“EC-based curve” (αimp= 0.45, α0= 0.20), for θ0= L0/200 h rad [44] FE, M cost. [44] (monolithic, θ0= L0/200 h rad)

1.0

χLT

0.8 0.6 0.4 0.2 0.0 0.0 (b)

0.5

1.0

_ λLT

1.5

2.0

2.5

Figure 1.11 Calibration of design buckling curves for LU beams in LTB, by assuming diferent geometrical imperfection amplitudes.

19

20 Advanced Engineering Materials and Modeling according to the laminated cross section of Figure 1.4, case (b), and considering the beam subjected to a constant bending moment My, as proposed in Figure 1.10, EIz,ef is in fact deined as follows:

EI z ,eff =

E (2 I1 + I s ) , 2 12 48 sinh λ /2 − 1+ α L λ 2 α L λ 3 sinh λ

(1.8)

with

I s = h (2t1z12 )

(1.9)

I1 = h t13 /12

(1.10)

αL = β=

Gint

λ=

2 I1 , Is

t int EI s , 2 h (2z1 ) L20 1 + αL , αLβ

z1 = 0.5 (t1 + t int )

(1.11)

(1.12)

(1.13) (1.14)

Based on the same approach, the equivalent torsional stifness GIt was also derived from the classical theory of sandwich elements (Figure 1.11 [45]). For a symmetric two-layer LG beam, in particular, GIt can be calculated as follows:

GIt = G (2 It ,1 + It ,comp ) = G(It ,abs + It ,comp ) ,

(1.15)

where the expression for It,1 and It,comp are listed in Table 1.4. Speciically, Eq. (1.15) takes into account the efective torsional contribution It,comp due to the adopted interlayer. Stamm and Witte originally derived the expressions, partly collected in Table 1.4 [45], for the estimation of this torsional stifness term, typically occurring in a faced “sot” core within a lat sandwich panel subjected to a torsional moment MT.

Mechanical Behavior and Resistance Glass

Shear stresses due to the interlayer

21

Interlayer Glass

Shear stresses due to torsion of the glass layers MT h

t1

tint

t2

Figure 1.12 Qualitative torsional behavior of a LG beam in accordance with the analytical model recalled in [45].

Table 1.4 he Stamm–Witte equivalent parameters for the calculation of the torsional stifness term in layered cross sections [45]. Symbol

Deinition

It,1

ht13 /3

It,comp

It,LT

λLT

I s , LT ⋅ 1 −

tanh(hλLT /2)

4 ⋅ (t1 + t int )2 ⋅

hλLT /2 t12 h 2t1

Gint 2 G t1t int

heir model basically applies to layered elements in which the cross section is uniform along the total length L0. Largely used for the analysis of sandwich elements and recalled in several handbooks [49–51], Eq. (1.15) has been applied successfully to LG elements. 1.4.2.1.2 Method II An alternative analytical model for the lateral–torsional buckling (LTB) veriication of LG beams has been assessed in [48]. In that case, the

22 Advanced Engineering Materials and Modeling theoretical model was based on the Wölfel–Bennison expression for the equivalent thickness teq, e.g. on the concept of an equivalent, monolithic lexural stifness EI z ,eff = ht eq3 /12 with

t eq = 3 2t13 + 12Γb I s ,WB ,

(1.16)

and

0 ≤ Γb =

1 ≤1 2 2 E t1 t int 1+π 2t1 Gint L20

(1.17)

the shear transfer coeicient representative of the shear transfer contribution of the adopted interlayer, where Is,WB in Eq. (1.17) is equal to Is/h (Eq. 1.9). Due to the shear transfer coeicient b, the efective stifness of the interlayer can be rationally taken into account within a range conventionally comprised between an “abs” layered limit (e.g. Gint → 0) and “full” monolithic limit (e.g. Gint → ∞). Analytical calculations highlighted that based on Eq. (1.17) the lexural stifness EIz,ef = f(teq) exactly coincides, for the boundary and loading conditions considered in this contribution, with calculations provided by exact analytical models (e.g. derived for example from Newmark’s theory of beams with partially rigid interaction [42]). To be used for LTB purposes, the “Method II” further requires the calculation of the torsional stifness term GIt, that also in this case is calculated based on Eq. (1.15). 1.4.2.1.3 Other Available Formulations he so-called “Method I” and “Method II” represent two analytical approaches of large use for structural glass applications. Other formulations – with almost the same efects – are anyway available in the literature. Based on [52], for example, the torsional stifness of laminated cross sections is calculated as a multiple of the “abs” torsional stifness corresponding to a null shear stifness of the interlayers (Gint → 0), e.g. by introducing a parameter f ≥ 1 so that

GIt = (GIt )abs ⋅ f is the equivalent torsional stifness of the laminated member, where

(1.18)

Mechanical Behavior and Resistance

f =

Gint 2 2 2 ) h (4t1 + 6t1t int + 3t int G G t12 6t1t int + int h2 G

6t13t int +

23

(1.19)

in the case of a symmetric laminated cross section as given in Figure 1.5, case (b), while speciic expressions are provided for “f ” as far as the cross section is unsymmetrical or not, and composed of two or three glass foils, respectively (e.g. Figure 1.5, cases a and c).

1.4.3 Laterally Restrained (LR) Beams in LTB Difering from the reference LU theoretical coniguration depicted in Figure 1.10a, the typical glass beam supporting a roof or façade panel is usually connected in practice to the adjacent construction elements by means of continuous structural silicone sealant joints acting as linear shear lexible connections between the LU glass beam and the supported plates (see also Figure 1.8). In these hypotheses, it is expected that the lateral restraint provided by the sealant joints could improve the LTB resistance of the LU reference beams (Figure 1.10). However, at the same time, the actual strengthening and stifening contribution deriving from sealant joints on the LTB response of LU glass beams must be irst properly assessed. his latter aspect represents in fact a crucial diference between structural glass applications and traditional steel–concrete or timber–concrete composite sections, where almost fully rigid connections at the beam-to-roof interfaces (namely consisting in steel stud connectors) generally ensure the occurrence of possible LTB phenomena.

1.4.3.1

Extended Literature Review on LR Beams

LTB of structural beams with lateral restraints has been widely investigated and assessed in the past years. However, careful consideration has been primarily focused on the LTB response of steel members whose behavior is not directly comparable to structural glass beams and ins. In [53], research studies have been in fact dedicated to the typical LTB response of doubly symmetric steel I beams, with careful attention for possible distortional buckling phenomena in the steel webs. Khelil and Larue proposed in [54–56] a simple analytical model for the assessment of the critical buckling moment in steel I sections with LR tensioned langes,

24 Advanced Engineering Materials and Modeling highlighting that the presence of rigid continuous lateral restraints in steel I beams under LTB can have a weak inluence, compared to their unrestrained Euler’s critical buckling moment. he same authors implemented also a further analytical approach for the LTB assessment of I beams continuously restrained along a lange by accounting for the buckling resistance of an equivalent, isolated “T” proile. he latter approach, due to its basic assumptions, typically resulted in conservative analytical predictions for the LTB resistance of rigidly LR steel I beams. Conversely, the main advantage of this method was given by the availability of the Appendix values of practical use for designers. he LTB behavior of thin-walled cold-formed steel channel members partially restrained by steel sheeting has been assessed, under various boundary conditions, by Chu et al. [57], based on an energy-based analytical model. Bruins [58] numerically investigated the LTB response of steel I section proiles under various loading conditions (e.g. distributed load q, mid-span concentrated force F, constant bending moment My) and laterally restrained by single, elastic, discrete connectors, highlighting through parametric FE numerical studies and earlier experiments that partial elastic restraints can have signiicant inluence on the overall LTB response. he efects deriving from initial geometrical curvatures with diferent shape were also emphasized by means of FE simulations, while simple equations were proposed as strength design method for “rigid” discrete lateral restraints. Further assessment of the main structural efects deriving from discrete rigid supports on the buckling behavior of steel beams and braced columns can be found also in [59–62].

1.4.3.2 Closed-form Formulation for LR Beams in LTB As far as a certain lateral restraint is provided to a given structural member in LTB, the most eicient tool for design purposes is represented by closedform solutions of practical use. Oten, however, these analytical models are diicult to obtain. As also highlighted in [54–56], when continuous elastic lateral restraints are introduced to prevent LTB of beams, the analytical description of the corresponding structural phenomenon becomes rather complex, and closed-form, practical expressions, and simpliied analytical models can be derived only for simple loading/boundary conditions, thus requiring the use of sophisticated FE numerical models and computationally expensive simulations. A further diiculty in elastic buckling calculations is given, when applying the analytical model of Larue et al. [54] for fully rigid laterally restrained steel members to glass beams with continuous sealant joints, by

Mechanical Behavior and Resistance

25

yq q

zq z

z’ k

ky

zM

G

θ

y’ y

Undeformed coniguration Deformed coniguration

Figure 1.13 Reference transversal cross section for the analytical model of laterally restrained beam presented in [54].

the intrinsic mechanical properties of the sealant joints themselves, namely characterized by relatively small shear stifnesses ky, as well as by the typically high slenderness ratios of glass beams and ins. In this work, based on Figure 1.13, the LTB behavior of LR monolithic glass beams is irst investigated by means of the analytical approach originally proposed in [54] for the prediction of the elastic LTB moment of doubly symmetrical steel I beams with fully rigid and continuous lateral restraints (e.g. ky = ∞). With reference to the loading condition depicted in Figure 1.10a and to the schematic cross-sections provided in Figure 1.13, the elastic LTB behavior of the LR beam subjected to a constant, positive bending moment My can in fact be described by [54] 2

4

EI z

v x4

kyv

4

EI

x

( xMy ) x2

ky zM

2 x 4

GI t

x

(1.23a)

0

x 2

x 2

x (z q q z

k )

v My x2

k y z M (v z M

x

) 0,

(1.23b)

where ky represents the translational (shear) rigidity of the continuous elastic restraint, per unit of length, along the y-axis; k is the rotational rigidity of the continuous elastic restraint, per unit of length, about the x-axis; My is the applied bending moment;

26 Advanced Engineering Materials and Modeling qz represents a possible transversal distributed load; zM is the distance between the continuous lateral restraint and the middle x-axis; zq is the distance between the point of introduction of qz load and the x-axis; v represents the vertical delection of the beam, in the z-direction; is the rotation of the cross-section about the longitudinal x-axis of the x beam, while Iz, It, Iw are deined according to Eq. (1.6). For the full mathematical and matrix formulation of the reference analytical model, the reader is referred to [54]. he main assumptions of the second order diferential system given by Eqs. (1.23a) and (1.23b) are that: t
the geometrical and mechanical properties of the beam are kept constant along the buckling length L0, t
the resisting cross section remains lat and undistorted ater bending of the beam, t
the beam is fork supported at the ends, while the continuous lateral joints introduce translational (ky) and rotational (k ) restraints only. Based on the deinitions given above, the critical buckling moment M cr( E, R) for the simply supported, fork-end restrained beam of Figure 1.10a subjected to a constant, positive bending moment My is given by: M

(E) cr , R

= M cr , R = z M k y +

EI z

nR L0

L0 nR

2

2

+ ky

L0 nR

2

EI

nR L0

2

L + GI t + (z k y + k ) 0 nR 2 M

2

,

(1.24) where nR ≥ 1 is an integer able to minimize the critical load given by Eq. (1.24). he main efect of this assumptions is that for a rectangular cross section composed of glass (e.g. Figure 1.10b), I = 0. A second aspect is strictly related to the torsional contribution of adhesive joints, that at irst instance could be taken equal to k = 0. As a result, the LR critical buckling moment can be calculated as follows:

27

Mechanical Behavior and Resistance

M

(E) cr , R

L0 nRπ

= Mcr ,R = z M k y ±

EI z

nRπ L0

2

+ ky

2

(1.25)

L0 nRπ

2 2 M

GIt + z k y

L0 nRπ

2

.

In accordance with Eq. (1.25), for a given t × h × L0 monolithic glass beam under a positive, constant bending moment My, it can be also concluded that: t
in presence of a lexible adhesive joint (0 < ky < ∞)

Mcr( E, R) = RM ⋅ Mcr( E ) ,

(1.26)

t
while as far as the adhesive joint is “weak” (ky = 0), the critical buckling moment is given by

Mcr( E, R) = Mcr( E ) ,

(1.27)

where in both the cases Mcr( E ) is expressed by Eq. (1.7). he ampliication factor RM = f (ky, b, t, L0, zM, nR) > 1 of Eq. (1.26) is representative of the efects deriving from a multitude of aspects, e.g. the joint shear rigidity ky, the beam aspect ratio, its elastic bending and torsional stifnesses, as well as the position of the applied restraints (zM) or the number nR of half-sine waves able to minimize, based on Eq. (1.25), the expected critical buckling moment Mcr( E, R) . When zM = h/2 (Figure 1.3b) and h/t > 6, for example, Eqs. (1.25) and (1.26) leads to

RM =

6hL30 k y + (3h2 L20 k y + 4(nRπ )2 Ght 3 ) (12L40 k y + (nRπ )4 Eht 3 ) 2nR 2π 3ht 3 EG

,

(1.28) and in the speciic case of glass (E = 70 GPa and G ≈ 0.41E [25]) to

RM ≈

6 hL30 k y + (3h2 L20 k y + 16nR2 Eht 3 ) (12L40 k y + 97.4nR4 Eht 3 ) 39.5 nR2 ht 3 E

.

(1.29)

28 Advanced Engineering Materials and Modeling

1.4.3.3 LR Glass Beams Under Positive Bending Moment My Eqs. (1.25) and (1.29) – although not fully exhaustive for the description of the expected LTB behavior – can provide a irst assessment of the efects due to continuous, adhesive joints acting as lexible lateral restraints along the top edge of a given glass beam in LTB, both in terms of expected magnifying factors RM as well as corresponding critical buckling shapes. Some qualitative analytical calculations derived from Eqs. (1.25) and (1.29) are in fact collected in Figure 1.14 to investigate the sensitivity of the LTB theoretical resistance of glass beams with several properties, when modifying some of the input parameters. In doing so, various geometrical (beam thickness t, height h, buckling length L0) and mechanical (joint shear stifness ky) parameters were properly modiied within a suficiently wide range of practical interest for structural glass applications (1000 mm ≤ L0 ≤ 6000 mm, 6 mm ≤ t ≤ 25 mm, 100 mm ≤ b ≤ 350 mm, with 0 < ky < ∞ the shear joint stifness). Several RM,i magnifying coeicients given by Eq. (1.29) were irst calculated – for each beam/adhesive joint stifness – as a function of a speciic number of critical half-sine waves nR, with 1 ≤ nR ≤ 20. he actual critical coeicient RM was then detected as the minimum of the so-calculated RM,i values. In general, the actual “critical” RM coeicient representative of the lowest critical buckling moment should in fact be properly estimated, by 29

L0 =3000mm, h = 300mm t = 10mm

25

nR=1

RM

21

2

17 3

13 10

9 5 1 10–4

10–2

100 ky [N/mm2]

102

Figure 1.14 Analytical estimation of the non-dimensional RM coeicient (Eq. 1.29) for monolithic glass beams under positive bending moment. Efects of the number of halfsine waves nR on the expected strengthening contribution of continuous lateral restraints.

Mechanical Behavior and Resistance

29

considering not only the assigned mechanical/geometrical properties of a given LR beam, but also the geometrical coniguration (e.g. number of half-sine waves nR) leading to LTB collapse. In this context, the major diference between the LTB response and theoretical resistance of a LU or LR glass beam is strictly related to nR. While for LU beams the lowest buckling load is associated to a single half-sine shape (e.g. nR = 1), this is not the case of LR beams. Some practical examples are proposed in Figure 1.14 for a selected set of geometrical conigurations. In Figure 1.14a, in particular, the lower envelope of the RM,i values calculated as described above is represented by the black thick line. As shown, it is clear that for a given beam geometry, as far as the adhesive shear stifness ky increases, the corresponding fundamental buckling shape modiies (nR > 1). Consequently, calculations derived from Eq. (1.25) with a reference single sine-shaped coniguration (e.g. nR = 1) would strongly underestimate the actual LTB resistance and overall response of the examined LR beams. Once assessed the correlation between half-sine waves and adhesive joints, further analytical calculations were carried out by changing the beams geometrical properties (e.g. in the form of slenderness ratios and torsional stifness). In doing so, the nominal dimensions only (e.g. thickness, width, and length) were taken into account for all the examined glass members. As a result, based on product tolerances for structural glass elements [25] and also highlighted in past research studies related to buckling phenomena in glass beams and plates (e.g. [45, 46, 63], it is expected that the qualitative comparisons discussed in this contribution would be further emphasized by the discrepancy between nominal and actual dimensions. he beneits deriving from the shear stifness ky of the adopted lateral restraints typically resulted, for example, rather sensitive to the beam slenderness ratio. Comparative calculations are proposed in Figure 1.15a for 10-mm-thick and 300-mm-high beams (with L0 = 1000, 2000, and 3000 mm their buckling length) or in Figure 1.15b for 300-mm-high and 3000-mm-long beams (with nominal thickness t comprised between 6 and 19 mm, respectively). Variation in the beam height h and h/L0 ratio, conversely, was generally detected as a minor inluencing parameter for the RM analytical predictions obtained by means of Eq. (1.29). Examples are proposed in Figure 1.15c for 10-mm-thick beams having diferent buckling length (L0 = 1000 and 3000 mm) and height (b = 100 and 300 mm). While for the long beams it can be seen that RM ampliication factors higher than ≈10 are almost non-sensitive to their h/L0 ratio, for the short

30 Advanced Engineering Materials and Modeling 29

29

h = 300mm, t =10mm

25

L0=3000mm

17

RM

RM

2000

13

9

9 1000

8 10 12 15

17

13

5

19

5 1

1 10–4

10–2

100

102

ky [N/mm2]

(a) 29

21

10–4

10–2

100

102

ky [N/mm2]

(b) 29

t =10mm L0=3000, h=100 L0=3000, h=300 L0=1000, h=100 L0=1000, h=300

25

25

ky = 0.6136N/mm2

L0 =1000mm L0=5000mm

21

B

17

17 RM

RM

t = 6mm

21

21

13

13

9

9 A

5

5

1 10–4

(c)

L0 =3000mm, h =300mm

25

10–2

100 ky [N/mm2]

ky=0.184N/mm2 ky=0.6136N/mm2 ky=0.184N/mm2

1

102

6 7 8 9 10 11 12 13 14 15 16 17 18 19

(d)

t [mm]

Figure 1.15 Analytical estimation of the non-dimensional RM coeicient (Eq. 1.29) for monolithic glass beams under positive bending moment. Efects of (a) beam length L0, (b) thickness t, (c) height h, and (d) qualitative response of LR monolithic glass beams with average adhesive joint stifnesses (0.184 N/mm2 ≤ ky ≤ 0.6136 N/mm2 [35–39], with kq = 0 and h = 100 mm).

eams (L0 = 1000 mm) the obtained RM values are primarily dependent on the ky shear stifness of the adopted lateral restraints, as far as the critical number of half-sine waves is nR < 4 (e.g. point A of Figure 1.15c, for the selected beam geometries). As a result, for a given ky value (with™), the beneit deriving from adhesive joints is higher (e.g. higher RM coeicient) as far as the h/L0 ratio is lower. Further increase in the shear stifness ky, otherwise, would result in a further increase of the number of half-sine waves able to minimize Eq. (1.28) in Figure 1.15c, as well as in the higher involvement of the beam torsional stifness GIt. his efect can be qualitatively seen in Figure 1.15c, point B, where for a given ky value > ≈ 50 N/mm2, a

Mechanical Behavior and Resistance

31

lower RM coeicient is obtained for the more slender beam (h = 100 mm, L0 = 1000 mm), while the maximum increase of the LTB reference LR resistance is found – due to a combination of ky, GIt contributions – for the higher beam (h = 300 mm, L0 = 1000 mm). Maximum structural beneits deriving from application of continuous adhesive joints in glass beams in LTB is also emphasized in Figure 1.14d, in the form of a magnifying factor RM proposed as a function of various thicknesses t for beams with b = 100 mm and L0 = 1000 mm or L0 = 5000 mm, respectively. Compared to the other plots, the main characteristic of Figure 1.15d is that the collected RM values are derived from Eq. (1.28) as the minimum envelope of the analytical estimations obtained for each beam geometry (t, b, L0) and joint stifness values ky of common use in structural glass applications.

1.5 Finite-element Numerical Modeling Analytical calculations partly discussed in Section 1.4 highlighted, although in terms of critical buckling moment and fundamental buckling shape only, that the presence of adhesive joints with small dimensions and moderate shear stifness (compared to the beams themselves) can have important efects on the LTB response of glass beams and ins. Careful consideration should be consequently paid in their LTB design and veriication, to take into account a multitude of additional inluencing parameters and their consequences on the actual LTB resistance, compared to theoretical values. In this sense, a practical tool in current practice for an accurate structural analysis of glass beams in LTB is certainly represented by the FE numerical approach. While FE models can describe in detail a combination of several geometrical and mechanical aspects – when properly assessed toward experimental and analytical data – in the speciic case of LR glass beams, careful consideration should be focused on the glass mechanical response (e.g. brittle tensile behavior) and laminates (e.g. via equivalent thickness methods), on the characterization of the adhesive joints (e.g. stifness contribution and possible collapse), on the efects of initial geometrical imperfections (shape and form) as well as on possible stress peaks in the vicinity of the supports (e.g. possible crushing mechanisms). In this section, the LTB response of glass beams laterally restrained by means of adhesive joints is further assessed by means of eigenvalue and incremental buckling FE numerical simulations. While several studies have been dedicated to the assessment of the LTB structural behavior

32 Advanced Engineering Materials and Modeling of laterally unrestrained (LU) glass beams, as well as to the calibration of potential standardized LTB design methods (e.g. see Section 1.3), the extension of the same simpliied analytical methods should be properly checked.

1.5.1 FE Solving Approach and Parametric Study 1.5.1.1 Linear Eigenvalue Buckling Analyses (lba) FE models were developed with the computer sotware ABAQUS/ Standard [64], to verify irst the accuracy of Eq. (1.25). Speciically, eigenvalue buckling (lba) analyses were carried out on a wide range of beam geometrical properties, with continuous lateral restraints characterized by a suiciently extended set of shear stifnesses ky. For each FE model, the irst 30 eigenvalues and eigen shapes were numerically predicted. hroughout this FE study, two main aspects were in fact assessed. he critical buckling moment Mcr,R of each beam subjected to a constant, positive bending moment My was numerically predicted and compared to the corresponding analytical estimation (Eq. 1.25). At the same time, the correspondence between the FE numerical and analytical critical numbers of half-sine waves nR was also properly assessed. In doing so, the typical FE model consisted of S4R 4-node, quadrilateral, stress/displacement shell elements with reduced integration and largestrain formulation (type S4R of ABAQUS element library), see the detailed view of Figure 1.16. To ensure the accuracy of FE results, a reined and regular mesh pattern was used, with lmesh the characteristic size of quadrilateral shell elements typically comprised between 3 and 15 mm, depending on the h × L0 dimensions of the studied beams. he positive bending moments My acting on each simply supported, fork-end restrained beam in LTB were introduced in the FE models in the form of point bending moments applied at the middle node of the end sections. Similarly, the simply supports and fork-end boundaries were described in the form of translational and rotational restraints for the same beam cross-sectional nodes. For lba purposes only, glass was described in the form of a fully isotropic, indeinitely linear elastic material (E = 70 GPa, = 0.23 [25]). Careful attention was paid to the description of the adhesive joints applied along the top edge of the beams, e.g. to save the modeling and computational cost of parametric FE simulations but at the same time to preserve the accuracy of the same FE models. A series of indeinitely linear elastic

Mechanical Behavior and Resistance Ky

33

Spring local axis

x’

z’ y’ Fork-support Simply support

Middle axis X Z

h

Y My L0 t Imesh

Figure 1.16 FE numerical model for a laterally restrained glass beam with continuous adhesive joints, under the action of a constant positive bending moment My (ABAQUS/ Standard).

springs (“axial” connectors available in the ABAQUS library) directly connected to the ground and characterized each one in terms of elastic stifness Ky, was in fact introduced along the top edge of each beam (zM = h/2, Figure 1.16), according to the adopted mesh pattern. he elastic stifness Ky of each spring, being dependent on the reference length lmesh, was estimated as Ky = ky × lmesh (with the exception of the axial springs at the corners of each beam, where the value Ky = ky × 0.5 lmesh was taken into account). he same approach was carried out for several beam geometries as well as for various adhesive joint stifnesses, with 10–4 N/mm2 ≤ ky ≤ 104 N/mm2. In terms of analytical and FE numerically predicted ampliication factors RM, a rather close agreement was generally found between the socollected data, at least for a wide range of geometrical and mechanical conigurations of practical interest for structural glass applications with adhesive joints. Some examples are proposed in Figure 1.17, where data are associated to several joint shear stifness ky. An average percentage discrepancy RM equal to ≈ ±0.5% was found between the collected data, for joint shear stifnesses per unit of-length ky of practical interest for continuous sealant restraints (e.g. ≈0.184 N/mm2 ≤ ky ≤ ≈0.6136 N/mm2), with:

RM

= 100

(RM )Analytical − (RM )ABAQUS (RM )ABAQUS

(1.30)

34 Advanced Engineering Materials and Modeling 4 0

RM

[%]

–4 –8

Avg.Exp.

–12 –16 –20 10–4

10–2

100

102

ky [N/mm2]

Figure 1.17 Comparison between analytical and FE magnifying coeicients RM (Eq. 1.30) calculated for various glass beam geometries, by changing the lateral restraint stifness ky.

As expected, FE lba analyses also conirmed that the presence of lexible adhesive joints provides a substantial modiication of their reference buckling shape (e.g. critical nR value), thus a variation of their global LTB response. In this sense, a detailed incremental buckling investigation should be performed by taking into account the actual critical buckling shape for each geometrical coniguration, with appropriate amplitude. As a irst estimation, in absence of more detailed experimental measurements and analyses on LR beams, this maximum amplitude could be taken equal to u0,max = L0/400, as experimentally obtained in [65]. Parametric studies also highlighted an almost exact correlation between the analytical and numerical number of half-sine waves nR associated to comparative data collected in Figure 1.17, especially in presence of lateral restraints able to provide a buckling strength increase up to ≈10 times the unrestrained beams (e.g. 1 ≤ RM ≤ 10 with 1 ≤ nR ≤ 8–10 depending on the beam geometry). Some examples are proposed in Figure 1.18 for two conigurations with identical geometrical properties but diferent joint stifnesses. A minor lack of correlation between analytical and numerical critical numbers of half-sine waves nR was found, through the full FE study, only in the presence of very stif joints typically associated to a large number of half-sine waves (e.g. 10 N/mm2 ≤ ky ≤ 102 N/mm2 and ≈ 10 ≤ nR ≤ ≈ 18, in this investigation), but not of primary interest for structural glass and adhesive applications.

Mechanical Behavior and Resistance

1.5.1.2

35

Incremental Nonlinear Analyses (inl)

Following the preliminary lba assessments, additional geometrical nonlinear, static incremental simulations were successively carried out on the same geometrical/mechanical conigurations. In each FE analysis, the shape of the initial geometrical imperfection was derived from the buckling mode belonging to the lowest lba eigenvalue (e.g. Figure 1.18). he maximum amplitude of the so-scaled fundamental buckling shapes were assumed – in the absence of more detailed experimental measurements and investigations for LR glass beams in LTB – equal to 1/400 of the total span L0, as this approach is generally accepted for the analysis of structural glass members and for the calibration of design buckling curves [65]. he mechanical characterization of the adhesive joints was also properly modiied, compared to lba studies. At this stage, a brittle elastic constitutive law was in fact taken into account, so that a possible progressive failure mechanism in the adhesive layers could be properly simulated. his mechanical calibration was carried out by taking into account, for the equivalent axial springs depicted in Figure 1.16, the ultimate shear stress and elongation values derived from shear experiments performed on small adhesive specimens (Section 1.3.4). Some examples of the parametric FE results derived from this further inl study are proposed in Figures 1.19 and 1.20 for a L0 = 3000 mm × h = 300 mm × t = 10 mm beam subjected to positive constant bending moment My. In Figure 1.19, speciically, the maximum envelope of out-ofplane displacements umax is proposed as a function of the RM ampliication factor for the reference LU beam, as well as for the same beam geometry laterally restrained by means of continuous adhesive joints (LR). For both the LU and LR beams, the maximum amplitude of the initial geometrical

x

z

(a)

x

y

nR = 1 (ky= 0)

z y

(b)

nR = 4 (ky= 0.814 N/mm2)

Figure 1.18 Critical buckling shapes of (a) LU glass beams, compared to (b) LR glass beams. L0 = 3000 mm, h = 300 mm, t = 10 mm; ABAQUS/Standard, white-to-black contour plot.

36 Advanced Engineering Materials and Modeling 8 7 Mcr,R(E)

6

RM

5 4 3 2 1

5.70Mcr(E) My

A FT , & , HS , AN

B

L0 = 3000mm, h = 300mm, t = 10mm ky = 0.184N/mm2 Mcr(E)

u0,max = L0/400 LU, nR = 1 LR, nR = 4

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (umax - u0, max) [m]

Figure 1.19 LTB response of a monolithic glass beam laterally unrestrained (LU, nR = 1) or restrained (LR, nR = 4). Efects of continuous lateral restraints (ABAQUS-inl).

imperfection is set equal to u0,max = L0/400, being the corresponding lba buckling shapes obtained performed on both the FE models (with nR = 1 and nR = 4 for the LU and LR beams, respectively). As shown, preliminary neglecting possible cracking mechanisms in glass, the LU beam would ideally carry on a maximum bending moment asymptotically tending toward the theoretical critical buckling moment M cr( E ) given by Eq. (1.27) (e.g. RM → 1 for the LU beam). he laterally restrained beam (LR), otherwise, would be theoretically able to ofer (when neglecting possible cracking in glass and damage in the adhesive joint) a signiicantly higher buckling resistance, e.g. up to ≈5.7 times the LU geometry, almost comparable to the corresponding theoretical critical buckling moment (e.g. M cr( E, R) ≅ 5.70 M cr( E ) ). In the same igure, it is also possible to notice that for the LR beam both possible failure mechanisms occurring in glass or in the adhesive joint would result in marked decrease of its ideal LTB resistance. By assuming in the same beam an indeinitely linear elastic mechanical behavior for glass, for example, the LTB failure mechanism would be governed by the progressive collapse of few axial connectors, representative of the adhesive joint, hence typically resulting in an ultimate failure load signiicantly lower than the theoretical Mcr( E, R) value (point A of Figure 1.19, RM = 4.22). Due to the progressive failure of these equivalent axial connectors along the beam buckling length L0, the post-cracked LTB response would also be characterized by an unsymmetrical deformed coniguration (e.g. path AB of Figure 1.19).

Mechanical Behavior and Resistance 1.0

My

0.8

RM = 3.78 (FT)

0.6 0.4

RM = 2.81 (HS)

x/L0 = 0

0.2 Rσ

37

RM = 1.94 (AN)

x/L0 = 1

0.0 –0.2 –0.4 –0.6 –0.8 –1.0

L0 = 3000 mm, h = 300 mm, t = 10 mm ky = 0.184 N/mm2, nR = 4, u0,max = L0/400 0.0

0.1

0.2

0.3

0.4

0.5

RM = 4.22 (joint) 0.6

0.7

0.8

0.9

1.0

x / L0

Figure 1.20 Stress ratio R evolution in the silicone joint, as a function of the applied bending moment (ABAQUS-inl).

his latter efect can be noticed in Figure 1.20, where the R stress coeicient – denoting the ratio between the measured stress max in each axial connector and the corresponding ultimate resistance u – is proposed for the adhesive joint belonging to the LR beam of Figure 1.19. he sodeined R values are shown, along the beam buckling length x/L0 (with 0 ≤ x ≤ L0), as a function of the applied bending moment (e.g. the RM loading conigurations derived from Figure 1.19). As shown, due to the assumed geometrical coniguration for the examined LR beam (nR = 4), the damage in the axial connectors (e.g. R = ± 1) irst occurs where the beam undergoes the maximum out-of-plane delections. In the same igure, it is also possible to notice – according to Figure 1.19 – that the ultimate LTB resistance of the examined LR beam would be clearly afected by the limited tensile resistance of glass. Depending on the type of glass and the corresponding characteristic tensile strength Rk, the LTB collapse would occur due to premature glass failure (with AN, HS, FT in Figures 1.19 and 1.20 denoting the attainment of the tensile resistance for AN, HS, and FT glass types, respectively). It is interesting to notice, in this context, that almost the same buckling collapse mechanism was found for all the beam geometries taken into account in this parametric investigation, and the failure of the adhesive joints, accordingly, typically occurred for higher bending loads only. However, a detailed LTB investigation for a general beam and adhesive joint geometrical/mechanical coniguration should necessarily take into

38 Advanced Engineering Materials and Modeling account both these possible collapse mechanisms, since strictly related to a combination of several inluencing parameters.

1.6

LTB Design Recommendations

1.6.1 LR Beams Under Positive Bending Moment My In conclusion, to preliminary assess the possible extension of the LTB design curve proposed, for example, in [44] for LU beams, further extended FE investigations were carried out, as partly discussed in Section 1.5. In doing so, for all the examined LR beams, the failure condition was identiied as the irst coniguration associated to i. glass tensile failure, ii. collapse of the adhesive joint, thus, the “failure” bending moments Mu* = min {(i), (ii)} were separately collected for each FE simulation. Some of the so-collected numerical predictions are proposed in* Figure 1.21, where normalized FE results are * expressed in the form ( χ LT , λ LT ) and compared to the design LTB curve proposed in [44] (thus with imp = 0.45, 0 = 0.20), being

χ

Mu* = , Wzσ Rk

(1.31)

1.6

1.2

u0,max = L0/400 LTB design curve [44] (αimp= 0.45, α0= 0.20) ABAQUS-inl, My

1.0 0.8

1.2 LR

1.0

0.6

u0,max = L0/400 LTB design curve [44] (αimp= 0.45, α0= 0.20) ABAQUS-inl, My

1.4

χLT

χLT

* LT

0.8 0.6

0.4

0.4 0.2

0.0

0.0 0.0

(a)

LU

0.2 0.5

1.0

_1.5 λLT

2.0

2.5

3.0

0.0

(b)

0.5

1.0

_1.5 λLT

2.0

2.5

3.0

Figure 1.21 Design buckling curve for glass beams in LTB under constant bending moment My. Calculation example for (a) LR glass beams and (b) comparison for a same beam geometry laterally unrestrained (LU) or continuously restrained (LR) by means of adhesive joints (ky = 0.184 N/mm2).

Mechanical Behavior and Resistance *

λ LT = Wzσ Rk / Mcr ,R ( E ) ,

39

(1.32)

with M cr , R ( E ) the minimum critical buckling moment obtained from Eq. (1.25). Monolithic glass beams with various geometrical properties (buckling length L0 = 1000–5000 mm, with step increment of 500 mm between each series of beams; height b = 100, 200, and 300 mm; nominal thickness t = 6, 8, 10, 12, 15, and 19 mm) and glass types (AN, HS, and FT) – opportunely combined with each other – were analyzed. Each beam, subjected to a constant, positive bending moment MEd ≡ My, was assumed afected by an initial geometrical imperfection of maximum amplitude u0,max = L0/400, obtained as the scaled critical buckling shape for each coniguration (ABAQUS-lba). Continuous silicone joints were also characterized as discussed in Section 1.5, and shear stifness values ky were assumed comprised between ky = 0.184 N/mm2 (average experimental value, Section 1.3) and ky = 0.6136 N/mm2 (maximum nominal value derived from [35–39]). As shown, compared to LU beams, the primary efect of additional continuous lateral restraints, due to increased stifness and overall resistance, typically results in a decrease of the normalized slenderness ratio λ LT (Eq. 1.32) and an increase in the maximum load carrying capacity, hence providing an increase of the buckling coeicient LT. he normalized FE predictions collected in Figure 1.21 are in fact characterized by limited slenderness ratios λ LT (in the order ≈ 0.2 ≤ λ LT ≤ ≈ 1.2) due to the adopted adhesive joints. he same LU beam geometries, otherwise, would have higher slenderness values, typically up to λ LT ≈ 2–2.5. he maximum structural beneits deriving from continuous adhesive joints, in this context, were found for beams with small h/L0 ratios. However, an interesting structural eiciency was generally obtained for all the examined beams. In conclusion, based on extended assessment and validation of methods discussed in this paper, it is expected that the LTB veriication of glass beams subjected to constant bending moments My and laterally restrained by means of continuous structural silicone joints could be performed by means of Eq. (1.5), with LT given by Eq. (1.31) and λ LT given by Eq. (1.32).

1.6.2 Further Extension and Developments of the Current Outcomes In the previous sections, it was shown how the presence of continuous adhesive joints in glass beams can afect the typical LTB response, compared

40 Advanced Engineering Materials and Modeling to a fully unrestrained beams, and how this strengthening and stifening contribution can be estimated by means of analytical and FE methods, as well as taken into account for LTB design purposes. In doing so, simpliications have been introduced at the level of the lateral restraints, e.g. by fully neglecting for example the rotational stifening contribution of these lexible joints, as well as any possible interaction between the so-restrained glass beams and the supported panels/façade plates. It is expected, in particular, that the presence of a roof or façade panel connected to the examined beams could be further increase the expected LTB ultimate strength. On the other hand, experimental prototypes would be necessarily required to properly assess this efect, as well as the full discussion of comparisons provided in the previous sections. In this chapter, to provide some solid background for further extended investigations, the presence of roof or façade plates was preliminary assessed in the form of FE models able to take into account – based on various modeling assumptions – the inertial properties of the so assembled, composite “T-section” (Figure 1.21a). An example of this further efort is provided in Figure 1.22, where a schematic representation is ofered for possible FE approaches to be taken into account for the investigated system. Figure 1.22b, speciically, refers to a simpliied but rather eicient FE model of a glass beam inclusive of the shear stifening contribution of the adhesive joints (ky ≠ 0). In it, both the torsional contribution (k = 0) and the additional stifness term due to the supported plate are fully neglected. he advantage of this approach, otherwise, is represented by the applicability of the method to 3D solid as well as 2D shell elements also, hence resulting, in the latter case, in a computationally eicient FE model type. In Figure 1.22c, a geometrically reined FE model type is shown. In it, the glass beam, the adhesive joint and the supported glass plate are fully described in the form of 3D solid elements and the respective material properties. Figure 1.22d, inally, represents a top limit coniguration for the expected LTB performances for the examined systems, in the form of a FE model type including the glass beam, the shear stifening contribution of the adhesive joint (ky) and the torsional efect due to the supported glass plate. he torsional term per unit of length k should be calculated as follows:

kθ = G ⋅ ITp .

lref z M2

,

(1.33)

Mechanical Behavior and Resistance

41

Glass plate

Adhesive joint z bp tp

zM h y

G

(a)

Glass beam t z

z

z

Ky

Ky K0

G

(b)

“M1”

G

y

(c)

“M2”

y

G

(d)

y

“M3”

Figure 1.22 Comparison of diferent FE modeling approaches (transversal cross sections). (a) Reference geometrical coniguration; (b) “M1” simpliied model of laterally restrained glass beam with shear stifness contribution only (kq = 0); (c) “M2” geometrically reined model, including the glass beam, the adhesive joint, and the supported glass plate; and (d) “M3” approximate model, including the glass beam and the shear/torsional stifening terms.

with G the shear modulus of the glass plate, zM the distance between the middle axis of the beam and the adhesive joint (see Figure 1.22d), ITp the torsional moment of inertia of the supported plate, and lref the distance between each equivalent connector representative of the adhesive connection. Some preliminary comparisons are collected in Figure 1.23 for several geometrical conigurations. FE numerical data derived from lba simulations are compared by changing the glass beam thickness for a “T-section” assembly characterized by L0 = 3000 mm, h = 300 mm for the beam geometry, tp = 30 mm, bp = 1000 mm for the roof plate, and average experimental values for the adhesive joints (see Section 1.3.4).

42 Advanced Engineering Materials and Modeling X106 300 250

(E) Mcr, R [Nmm]

200 LR “M3”

LR “M2”

150 100

LR “M1”

50 LU

10

15

20 t [mm]

25

30

Figure 1.23 Comparative lba calculations carried out on fully LU glass beams and LR beams described according with Figure 1.22.

FE data are obtained, for several beam thicknesses t, by means of FE models discretized as proposed in Figure 1.22, and compared in the form of Euler’s critical loads, as a function of the beam thickness t, as obtained for the LU beam and the LR geometries (“M1”, “M2”, and “M3” approaches). As shown, the resistance of the fully LU beams can be clearly distinguished from the LR conigurations, as expected. However, a strong modiication in terms of theoretical resistance of the same geometries is also found for the “M1”, “M2”, and “M3” methods. While the “M1” approach is able to take into account the shear stifening contribution due to adhesive joints, the “M3” method, conversely, fully overestimates the contribution of the connected roof plate (e.g. due to the assumption of a fully rigid torsional restraint between the beam and the plate). he “M2” approach, inally, provides almost an intermediate response to “M1” and “M3” cases, due to the reined description of the adhesive joint with both the nominal shear and torsional stifening contribution.

1.7 Conclusions In this chapter, results of a recent research activity on the LTB behavior of beam glass elements with continuous lateral restraints have been discussed.

Mechanical Behavior and Resistance

43

Depending on combinations of silicone joint stifnesses and beam geometrical properties, analytical and FE numerical calculations highlighted that their critical buckling moment can be strongly increased, especially in the case of slender beams. However, Euler’s critical moment yields poor information only on the actual LTB ultimate resistance of laterally restrained (LR) glass beams, and detailed incremental analyses should be generally carried out to properly assess the efects of multiple mechanical and geometrical aspects (e.g. beam-to-joint stifness ratio, failure mechanisms in glass or joints, initial geometrical imperfections, and loading condition). For this purpose, reined incremental nonlinear FE analyses were performed on a large number of glass beams, to properly assess their global LTB response up to failure. Speciically, parametric analyses were carried out to take into account the efects of possible geometrical imperfections – both in terms of maximum amplitude and reference shape – as well as the premature buckling failure deriving from glass cracking in tension, or the occurring of possible failure mechanisms in the structural silicone joints. Simulations generally conirmed the appreciable eiciency of structural silicone joints, compared to laterally unrestrained (LU) beams, and highlighted – although in the presence of continuous lateral supports – that their ultimate buckling resistance is strictly related to failure of glass in tension. Based on earlier contributions of literature, inally, a design LTB curve recently calibrated for the veriication of LU glass beams has been recalled and applied to LR beams. As shown, in this latter case, a rather good agreement was found, hence suggesting its possible extension for the design and veriication of the studied structural typology. Certainly, further improvements of the current method could be derived from extended experimental validations, as well as reined FE investigations able to account for the structural interaction between the studied glass beams and the supported glass roof/plates. In this latter case, in particular, it was shown that FE methods could provide further valid support for LTB studies. However, the basic FE assumptions (e.g. torsional stifening contribution and presence of the supported glass roof plates) could strongly afect the expected FE predictions. In the case of LG elements composed by two or more glass plies interacting together by means of shear deformable interlayers, moreover, the full direct applicability of the current LR buckling design approach should be further assessed, with particular attention for the accuracy of simpliied equivalent thickness methods in the presence of LR beams. In any case, it is expected that the proposed review and comparisons could provide useful background and suitable tools for practical applications.

44 Advanced Engineering Materials and Modeling

References 1. Martens, K., Caspeele, R., and Belis, J., Development of composite glass beams – a review. Engineering Structures, 101, 1–15, 2015. doi: 10.1016/j .engstruct.2015.07.006 2. Martens, K., Caspeele, R., and Belis, J., Development of reinforced and posttensioned glass beams: review of experimental research. Journal of Structural Engineering (ASCE), in press, 2015, doi: 10.1061/(ASCE) ST.1943-541X.0001453. 3. Blyberg, L., Timber/Glass Adhesive Bonds for Structural Applications. Linnaeus University, Report 10, 2011, ISBN: 978-91-86983-06-2 4. Blyberg, L., Lang, M., Lundstedt, K., Schander, M., Serrano, E., Silfverhielm, M., and Stalhandske, C., Glass, timber and adhesive joints – innovative load bearing building components. Construction and Building Materials, 55, 470–478, 2014. 5. Fadai, A., and Winter, W., Application of timber-glass composite (TGC) structures for building construction. Proceedings of Challenging Glass 4 & COST Action TU0905 Final Conference, 235–242, 2014, doi: 10.1201/ b16499-36 6. Eriksson, J., Ludvigsson, M., Dorn, M., Enquistand, B., and Serrano, E., Load bearing timber glass composites – a WoodWisdom-Net project for innovative building system. Proceedings of COST Action TU0905 Mid-term Conference on Structural Glass, 269–276, 2013, doi: 10.1201/b14563-38 7. Neto, P., Alfaiate, J., Valarinho, L., Correia, J.R., Branco, F,A., and Vinagre, J., Glass beams reinforced with GFRP laminates: Experimental tests and numerical modelling using a discrete strong discontinuity approach. Engineering Structures, 99, 253–263, 2015. 8. Speranzini, E., and Agnetti, S., Flexural performance of hybrid beams made of glass and pultruded GFRP. Construction and Building Materials, 94, 249–262, 2015. 9. Speranzini, E., and Agnetti, S., Strengthening of glass beams with steel reinforced polymer (SRP). Composites Part B: Engineering, 67, 280–289, 2014. 10. Weller, B., and Engelmann, M., 9m Glass bridge with post-tensioned reinforcement a Glasstec 2014 [in German]. Stahlbau, 84(1), 455–464, 2015. 11. Bedon, C., and Louter, C., Exploratory numerical analysis of SG-laminated reinforced glass beams. Engineering Structures, 75, 457–468, 2014. 12. Louter, C., Belis, J.; Frederic, V., and Lebet, J.P., Structural response of SG-laminated reinforced glass beams; experimental investigations on the efects of glass type, reinforcement percentage and beam size. Engineering Structures, 36, 292–301, 2011. 13. Nielsen, J.H., and Olesen, J.F., Post-crack capacity of mechanically reinforced glass beams (MRGB). Fracture Mechanics of Concrete and Concrete

Mechanical Behavior and Resistance

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

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Structures:Recent advances in Fracture Mechanics of Concrete,Oh, B.H., et al., (Ed.), ISBN 978-89-5708-180-8, 2010. Jordão, S., Pinho, M., Neves L.C., Martin, J.P., Santiago, A., Behaviour of laminated glass beams reinforced with pre-stressed cables. Steel Construction, 7(3), 2014. Silvansky, M., heoretical veriication of the reinforced glass beams. Procedia Engineering, 40, 417–422, 2012. www.bdonline.co.uk www.wagner-biro.com www.openbuildings.com www.vangviet.com http://www.decoist.com/2014-06-06/victorian-house-renovationmelbourne/ http://www.storyphotography.com.au https://www.canalengineering.co.uk/news-article.asp?NID=87 www.aeccafe.com Leskovar, V.Z., Premrov, M., Energy-Eicient Timber-Glass Houses – Green Energy and Technology. London: Springer-Verlag, 2013. EN 572-1. Glass in buildings: basic soda lima silicate glass products – Part 1: deinitions and general physical and mechanical properties. CEN, 2004. Gräf, H., Schuler, C., Albrecht, G., and Bucak, Ö., he inluence of various support conditions on the structural behaviour of laminated glass. Proceedings of Glass Processing Days, 408–411, 2003. Duser, A.V., Jagota, A., and Bennison, S.J., Analysis of glass/polyvinyil butyral laminates subjected to uniform pressure. Journal of Engineering Mechanics, 125, 435–442, 1999. Callewaert, D., Belis, J., Delincé. D., and Van Impe, R., Experimental stifness characterisation of glass/ionomer laminates for structural applications. Construction and Building Materials, 37, 685–692, 2012. EVALAYER , http://interlayersolutions.com/eva-layer/ PS Glass Fittings, www.psglassitting.com IHK Building in Munich (DE), http://www.slideserve.com/zuri/reinforcedglass-beams-lecture-for-verre-2006 www.enclos.com www.arcspace.com. Oices for the Castilla Leòn Government in Zamora (photo: Javier Callejas) www.theartobuilding.co.uk Bostik V-70 – Technical Data Sheet. Sikasil SG-20 – Technical Data Sheet. DowCorning – Technical Data Sheet. Henkel Pattex SL 690 solyplast – Technical Data Sheet. Bedon, C., Belis, J., and Amadio, C., Structural assessment and lateraltorsional buckling design of glass beams restrained by continuous sealant joints. Engineering Structures, 102, 214–229, 2015.

46 Advanced Engineering Materials and Modeling 40. EOTA. ETAG 002. Guideline for European technical approval for structural sealant glazing systems (SSGS) – Part 1: supported and unsupported systems, 1999. 41. EN 1993-1-1: 2005. Eurocode 3 – design of steel structures – Part 1-1: general rules and rules for buildings, CEN. 42. Amadio, C., and Bedon, C., Buckling of laminated glass elements in out-ofplane bending. Engineering Structures, 32, 3780–3788, 2010. 43. Amadio, C., and Bedon, C., A buckling veriication approach for monolithic and laminated glass elements under combined in-plane compression and bending. Engineering Structures 53, 220–229, 2013. 44. Bedon, C., and Amadio, C., Design buckling curves for glass columns and beams. Proceedings of the Institution of Civil Engineers – Structures and Buildings, 168(7), 514–526, 2015. 45. Luible, A., Stabilität von Tragelementen aus Glas. Dissertation, EPFL Lausanne, hése 3014, 2004. 46. Belis, J., Kipsterkte van monolithische en gelamineerde glazen liggers. Ghent: Ghent University, 2005. 47. Rosati, G., Orlando, M., and and Piscitelli, L,R., Flexural-torsional buckling tests on laminated glass beams. Proceedings of XXVII ATIV Conference, Parma, Italy, 137–143, 2012. 48. Bedon, C., Belis, J., and Luible, A, Assessment of existing analytical models for the lateral torsional buckling analysis of PVB and SG laminated glass beams via viscoelastic simulations and experiments. Engineering Structures, 60, 52–67, 2014. 49. Zenkert, D., he Handbook of Sandwich Construction. United Kingdom: Engineering Materials Advisory Service Ltd, 1997. 50. Davies, J.M., Lightweight Sandwich Construction. United Kingdom: Blackwell Science Ltd, 2001. 51. Plantema, F., Sandwich Construction – he Bending and Buckling of Sandwich Beams, Plates and Shells. John Wiley & Sons, New York, 1966. 52. Scarpino, P., Berechnungsverfahren zur Bestimmung einer äquivalent Torsionssteiigkeit von Trägern in Sandwichbauweise. Lehrstuhl für Stahlbau RWTH Aachen: Diplomarbeit, 2002. 53. Kalkan, I., and Buyukkaragoz, A., A numerical and analytical study on distortional buckling of doubly symmetric steel I-beams. Journal of Constructional Steel Research, 70, 289–297, 2012. 54. Larue, B., Khelil, A., and Gueury, M., Elastic lexural-torsional buckling of steel beams with rigid and continuous lateral restraints. Journal of Constructional Steel Research, 63, 692–708, 2006. 55. Larue, B., Khelil, A., and Gueury, M., Evaluation of the lateral-torsional buckling of an I beam section continuously restrained along a lange by studying the buckling of an isolated equivalent proile. hin-Walled Structures, 45, 77–95, 2007.

Mechanical Behavior and Resistance

47

56. Larue, B., Khelil, A., and Gueury, M., Elastic lexural-torsional buckling of steel beams with rigid and continuous lateral restraints. Journal of Constructional Steel Research, 63, 692–708, 2006. 57. Chu, X.-T., Kettle, R., and Li, L.-Y., Lateral-torsional buckling analysis of partial-laterally restrained thin walled channel-section beams. Journal of Constructional Steel Research, 60, 1159–1175, 2004. 58. Bruins, R.H.J., Lateral-torsional buckling of laterally restrained steel beams. Report No. A-2007-7, Technische Universiteit Eindhoven, 2007. 59. Bradford, M,A., Inelastic buckling of I-beams with continuous elastic tension lange restraint. Journal of Constructional Steel Research, 48, 63–77, 1998. 60. Bradford, M.A., Strength of compact steel beams with partial restraint. Journal of Constructional Steel Research, 53, 183–200, 2000. 61. Nguyen. C.T., Joo, H.S., Moon, J., and Lee, H.E., Flexural-torsional buckling strength of I-girders with discrete torsional braces under various loading conditions. Engineering Structures, 36, 337–350, 2012. 62. Zhang, L., and Tong, G.S., Lateral buckling of eccentrically braced RHS columns. hin-Walled Structures, 49, 1452–1459, 2011. 63. Belis, J., Bedon, C., Louter, C., Amadio, C., and Impe, R,V., Experimental and analytical assessment of lateral torsional buckling of laminated glass beams. Engineering Structures, 51, 295–305, 2013, 64. Simulia, ABAQUS v.6.9 Computer Sotware and Online Documentation, Dassault Systèmes, 2009. 65. Belis, J., Mocibob, D., Luible, A., and Vandebroek, M., On the size and shape of initial out-of-plane curvatures in structural glass components. Construction and Building Materials, 25(5), 2700–2713, 2011.

2 Room Temperature Mechanosynthesis of Nanocrystalline Metal Carbides and heir Microstructure Characterization S.K. Pradhan1* and H. Dutta2 1

Department of Physics, he University of Burdwan, West Bengal, India 2 Department of Physics, Vivekananda College, West Bengal, India

Abstract he need of improved materials in high-temperature structural applications has stimulated research into the mechanical behavior of a number of materials including the refractory hard metals. Mechanical alloying process is a very useful method for the production of high-melting-point materials like metal carbides, which additionally fabricate nanocrystalline structure with improved properties. his chapter presents the microstructure characterization of nanocrystalline binary/ternary metal carbides synthesized at room temperature by mechanical milling of diferent metals and graphite powders under inert atmosphere at room temperature. Microstructure of the prepared materials is characterized in terms of lattice imperfections by analyzing the X-ray powder difraction data employing Rietveld’s structure and microstructure reinement method. he high-resolution electron microscopy study of inal ball-milled sample in each case gives direct supportive evidence of structural and microstructural evaluation by X-ray difraction pattern analysis. Finally, a comparative study of microstructural changes between binary and ternary metal carbides has completed our investigation about inding out an easy cost-efective method to prepare metal carbides. Keywords: Nanocarbides, mechanical alloying, microstructure, Rietveld’s analysis, electron microscopy

*Corresponding author: [email protected]; [email protected] Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (49–86) © 2016 Scrivener Publishing LLC

49

50 Advanced Engineering Materials and Modeling

2.1 Introduction 2.1.1 Application Nowadays, the interest in metal–carbon systems has substantially been increased due to the possibility of obtaining nanocrystalline metal particles encapsulated by crystalline or amorphous carbon ilm [1]. hese metal carbides possess some excellent properties similar to refractory materials, like high melting temperatures, high hardness, high chemical stability, and high Young’s modulus at elevated temperatures and are being used in abrasives, cutting tools, grinding wheels, and coated cutting tips [2–6]. In addition, they also exhibit some metallic properties like high-temperature ductility, good thermal shock resistance, etc. [7]. hey also have excellent thermal and electronic conductivities as well as superconductivity [8, 9]. Beside these, they are also now known as catalysts for ammonia synthesis and decomposition, hydrogenolysis, isomerization, methanation, and hydroprocessing. Such unique properties help this kind of materials to possess a wide variety of potential applications in high-tech ields, such as electrodes and high-temperature structural materials [7].

2.1.2 Diferent Methods for Preparation of Metal Carbide Diferent technologies have been developed to prepare metal carbides. hese methods can be grouped into two diferent categories: (i) top-down and (ii) bottom-up methods as given below: i. Top-down method: mechanical alloying (MA). ii. Bottom-up method: hot isostatic pressing (HIP), combustion synthesis (CS), spark plasma sintering (SPS), hot pressing (HP), and self-propagation high-temperature synthesis (SHS). All the above-mentioned methods, except for MA, require rigorous conditions such as high temperature, high pressure, and long time and also have the disadvantage of synthesis of materials with large particle size, and further extensive processing is needed to crush them into nanocrystalline powders or for thin-ilm deposition by physical vapor deposition, sputtering, or laser ablation technique due to their very high melting temperature.

Room Temperature Mechanosynthesis 51

2.1.3 Mechanical Alloying High-energy ball milling also known as MA is one of the most promising solid-state synthesis routes to prepare nanocrystalline alloy/compound powders by milling the elemental powders in a suitable ball mill. In the dry milling process, fracturing and cold welding are dominant processes by which micron-order precursor elemental powders gradually transform to nanocrystalline powders through diferent stages of simultaneous chemical reactions. his technique has been extensively explored for many metallic, ceramics, metallic–ceramic, and ceramic–ceramic systems [10–18]. Diferent types of ball mills such as attrition mill, vibratory ball mills, and planetary ball mills are being used for diferent purposes. he milling media (balls and bowls) are generally made of diferent materials, and a few of the common materials are chrome–steel, alumina, zirconia, agate, and tungsten carbide.

2.1.4 Planetary Ball Mill he laboratory scale ball mill “pulverisette 5” commonly known as P-5 (Fritsch, GmbH, Germany) can be used universally for high-speed grinding of solid or liquid inorganic or organic samples for synthesis of nanocrystalline materials both by wet and dry milling (Figure 2.1a). It is also being used to mix and homogenize dry samples, emulsions, or pastes. All metal carbide nanocrystalline powders reported in this review work are synthesized by mechanical alloying (dry milling) the stoichiometric mixture of elemental powders under Ar atmosphere in a P-5 ball mill using chrome–steel milling media.

Rotation of the grinding bowl

Horizontal section

(a)

Movement of the supporting disc

Centrifugal force

(b)

Figure 2.1 (a) Planetary ball mill. (b) Grinding mechanism of planetary ball mill.

52 Advanced Engineering Materials and Modeling he material to be ground is crushed and torn apart by grinding balls in two or four grinding bowls. he centrifugal forces created by the rotation of the grinding bowls around their own axis and the rotating supporting disc are applied to the grinding bowl charge of material and grinding balls. Since the directions of rotation of grinding bowls and supporting disc are opposed (Figure 2.1b), the centrifugal forces are alternately synchronized and opposite. hus, friction results from the grinding balls and the material being ground alternately rolling on the inner wall of the bowl, and impact results when they are lited and thrown across the bowl to strike the opposite wall. he impact is intensiied by the grinding balls striking one another. he impact energy of the grinding balls in the normal direction attains values up to 40 times higher than gravitational acceleration. Lossfree combination is guaranteed by a hermetic seal between grinding bowls and lid. he mechanics of this mill are characterized by the rotation speed of the disk, Ω; that of the container relative to the disk, ω; the mass, m; size and the number of balls; the radius of the disk, R; and the radius of the container, r. Gafet [19] has shown that depending on the relative values of ω/Ω and r/R, two extreme regimes may be achieved: (i) the ball rolls on the inner surfaces of the container, or (ii) it escapes and impacts an opposite portion of the surface. For both cases, the energy transferred per unit area scales with mΩ2 and the frequency of the occurrence of the impacts scales with ω. he power induced to the powder sample therefore scales as P ∝ ml2Ω2ω, where l2 is a characteristic area of the order R2 or rR for the rolling or impact regime, respectively.

2.1.5 he Merits and Demerits of Planetary Ball Mill Merits: 1. his is a one-step method. 2. his top-down method has the advantage of obtaining nano or amorphous materials. 3. A room temperature preparation of the material is possible. 4. Alloying and complete solid solubility of materials can be done. 5. Solid-state amorphization of materials can be achieved. 6. he cation distribution of the prepared ferrite materials may by unusual. 7. he nanosized particles can be achieved in a short duration of time.

Room Temperature Mechanosynthesis 53 8. Energy of the milling media can be controlled by a large number of parameters such as ball-to-powder mass ratio (BPMR), rpm, duration of milling, and size of balls/bowls.

Demerits: 1. 2. 3. 4.

Contamination from grinding media. Stickiness of material during dry grinding. Excessive heating of material to be ground. Combustible liquids with boiling point 1/2), and preferred orientation parameter. he proile of X-ray powder pattern can be well described by a convolution equation

Yc (2 ) = [B∗(I∗A)](2 ) + bkg,

(2.1)

where ∗ is the convolution symbol and ‘bkg’ is a polynomial function of degree four for reproducing the background intensity. he experimental proiles were itted with the most suitable pseudo-Voigt (pV) analytical function [52, 65] because it takes individual care for both the particle size and strain broadening of the experimental proiles. For both the K 1 and K 2 proiles, the line broadening function B(2 ) and the symmetric part of instrumental function S(2 ) may be represented by the pseudo-Voigt function [52, 65]:

pV (x ) = ∑ Int [ηC(x ) + (1 − η )G( x )]

(2.2)

α1α 2

where the Cauchyian component, C(x) = (1 + x2)–1 and the Gaussian component, G(x) = exp [–(ln2)x2] with x = (2 – 2 0)/HWHM (HWHM = half width at half maxima of the X-ray peaks) and HWHM = ½ FWHM = (Utan2 + Vtan + W)1/2 , where U, V, and W are coeicients of the quadratic polynomial, is the gaussianity of X-ray proile, 0 is the Bragg angle of K 1 peak, and Int is the scale factor. Due to the anisotropy in particle size and microstrain values, proiles with diferent Miller indices are broadened in diferent manner and this efect creates problems frequently in Rietveld’s structure reinements. To consider their inluence in the proile shapes, in the adopted method, the tensors

Room Temperature Mechanosynthesis 59 similar to the temperature factors, for particle sizes and microstrains in different crystallographic directions have been used as follows:

∑ D h h ∑δ h h 2 ij i

D(h1 , h2 , h3 ) =

j

ij i

ij

1/2

,

j

(2.3)

ij

1/2

ε2

1/2

∑ε

(h1 , h2 , h3 ) =1/2

2 ij

ij

hi h j

∑δ h h ij i

j

,

(2.4)

ij

where ij = 0, if Dij = 0 or < 2>ij = 0, representing for (2.3) and (2.4) and = 1, in the other cases [54]. ij he powder difraction patterns were simulated providing all necessary structural information and some starting values of microstructural parameters of the individual phases with the help of the Rietveld sotware, the MAUD 2.26 [48]. Initially, the positions of the peaks were corrected by successive reinements of systematic errors taking into account the zero-shit error and sample displacement error. Considering the integrated intensity of the peaks as a function of structural parameters only, the Marquardt least squares procedures were adopted for minimization the diference between the observed and simulated powder difraction patterns and the minimization was carried out by using the reliability index parameter, Rwp (weighted residual error) and RB (Bragg factor) deined as follows:

Rwp =

∑ w (I i

1/2

− Ic )

∑w I

2

o

2

i o

i

,

(2.5)

i

and

RB = 100 ∑ I o − I c

∑I .

(2.6)

o

he goodness of it (GoF) is established by comparing Rwp with the expected error, Rexp:

Rexp =

(N − P ) ∑ w I

2 i o

1/2

,

(2.7)

i

where Io and Ic are the experimental and calculated intensities, respectively; wi (= 1/Io) and N are the weight and the number of experimental observations; and P is the number of itting parameters. his leads to the value of GoF [52, 54, 66]:

GoF = Rwp/Rexp

(2.8)

60 Advanced Engineering Materials and Modeling he process of successive proile reinements modulates diferent structural and microstructural parameters of the simulated pattern to it the experimental difraction pattern. Proile reinement continues until convergence is reached in each case, with the value of the quality factor (GoF) approaching 1.

2.3.2 General Features of Structure We have reported synthesis of four binary and three ternary metal carbide systems in this chapter. All the Ti-based carbides are isostructural, and the corresponding XRD patterns are simulated in accordance with the structural information incorporated in TiC JCPDF File (Table 2.2). Lattice types and JCPDF File No. of all other binary metal carbides and the elemental phases present in the unmilled homogeneous metal carbide mixture powder are also tabulated in Table 2.2a and b, respectively. Some of the atomic models are represented in Figure 2.2a and b for a better understanding of the structure of metal carbides.

2.4 Results and Discussions 2.4.1 XRD Pattern Analysis X-ray powder difraction patterns of some selected metal–graphite mixtures before milling and ater milling at diferent durations at room temperature are shown in Figure 2.3. he XRD powder patterns of unmilled homogeneous powder mixture for all metal carbide samples show only the relections of precursor elemental phases. Figure 2.3 clearly illustrates that XRD pattern of unmilled sample comprises of quite sharp peaks and highangle relections are well resolved into CuKα1-α2 components. It indicates that all elemental powders are composed of quite large particles which are almost free from lattice strain. In general, XRD patterns of unmilled sample (Figure 2.3) show that relections of graphite (hexagonal) powder are extremely oriented along . his kind of preferred orientation leads to a major problem in determining the average particle size as well as the quantitative estimation of phases in a multiphase material. Within a short milling time, all graphite relections in metal carbide composites disappear and the relections of elemental metal phases become signiicantly broad without any shit in peak positions. he absence of graphite relection in the XRD patterns of longer milling time samples may be attributed to the following reasons: (i) the carbon atoms

Ti0.9Al0.1C Ti0.9Ni0.1C

Ti0.9W0.1C

SiC

Ni3C

Fe3C

# 32-1383

# 32-1383

# 32-1383

# 32-1383

# 29-1129

# 06-0697

# 35-0772

Cubic; Fm3m Cubic; Fm3m Cubic; Fm3m Cubic; Fm3m Cubic; F43m Rhombohedral; Orthorhombic; R3c Pnma

TiC

Si Cubic; Fd3m:1 # 27-1402

Hcp; P63/ mmc

# 44-1294

Lattice type; space group

JCPDF File

Phase present

α-Ti

# 04-0850

Cubic; Fm3m

Ni

# 41-1487

Cubic; Im3m

α-Fe

# 44-1294

Cubic; Fm3m

Al

# 04-0806

Cubic; Im3m

W

# 41-1487

Hexagonal; P63/mmc

C (Graphite)

Table 2.2(b) Structural information of elemental powders used for synthesis of nanocrystalline metals carbides by mechanical alloying.

JCPDF ile

Lattice type; space group

Metal carbide

Table 2.2(a) Structural information of metal carbides with respective ICSD iles.

Room Temperature Mechanosynthesis 61

62 Advanced Engineering Materials and Modeling C

W

W C W

C

Ti

Ni

W

C

Ni Ti

Ti C

(a)

C Ti C

W C

Ti C

C W

W C

(b)

Figure 2.2 Atomic model of (a) rhombohedral Ni3C phase and (b) TiWC (unit cell).

occupy interstitial positions in metal lattice of major phase Ti/Ni/Si/Fe, (ii) thin graphite layer stick into the inter-grain boundaries of major phase grains, or (iii) amorphization of graphite layers (amorphous carbon). (i) Ti-based binary/ternary metal carbide: With the progress of milling, texturing of -Ti phase increases along . -Ti particles are more likely embedded on the graphite layer and also aligned along the same direction due to structural similarity. he reason of texturing may be due to transitional bonding between some -Ti and C atoms caused by the lubricating nature of graphite. As a result, graphite particles stick to the -Ti surface and try to pull (arrange) the -Ti particles along its own orientation [20]. Relections of all metals nearly disappear in the XRD pattern at the early stage of milling. he TiC and TiMC (fcc) powder formed ater 35–55 min of milling (Table 2.3) via mechanically induced self-propagating reaction (MISPR). Relections in the XRD patterns appeared to be as from an annealed standard material with clearly resolved Cu K 1 − 2 doublets even at lower scattering angle. It clearly indicates that metal carbides are composed of large crystallites without any lattice imperfection. Full formation of stoichiometric TiMC takes more time than that of TiC as shown in Table 2.3. To prepare nanocrystalline TiMC powder, the as-prepared TiMC powder with a trace amount of unreacted Ti and M powders ater formation time of milling was milled for longer durations. (ii) SiC: Ater 3 h of milling of Si and graphite powders, graphite relections disappeared completely and Si relections broaden signiicantly owing to the

Room Temperature Mechanosynthesis 63

(331)

(420)

(400)

(311)

(222)

(220)

3000

6h

2500

1h 2h

2000

45 m 35 m

1500

34 m 30 m 15 m

1000 500

5m 0h

--C

(0 02 ) (1 00 (0 ) 02 (1 ) 01 ) (1 02 (0 ) 04 ) (1 10 ) (1 03 ) (1 12 (2 ) 01 )

0 –500

--TiC --Ti

20

40

60

(a)

80

100

120

1500

(222)

Ti-Ni-C Ti Ni C

(220)

(111)

4000

(200)

2θ (degree)

(311)

Intensity (arb.unit)

3500

(200)

(111)

4000

Intensity (arb.unit)

12 h 8h 1000 4h 55 min 500 45 min 0h (002)

0 20

30

40

(b)

60

70

80

(223)

(306)

(119)

--Ni3C --Ni --C

(3 00 )

(116)

(1 13 )

(110) (006)

100000

Intensity (arb.unit)

50 2θ (degree)

8h 5h

80000

3h 2h 1.75 h

60000

1.5 h 1h

40000

30 m 15 m

20000

30 (c)

40

50

60

70

80

90

(222)

(311)

(220)

(200)

(002)

0

(111)

0h

100

2θ (degree)

Figure 2.3 X-ray powder difraction patterns of unmilled and ball-milled stoichiometric mixture of elemental metals and graphite powders milled for diferent duration under argon medium representing for (a) TiC, (b) TiNiC, and (c) Ni3C.

64 Advanced Engineering Materials and Modeling Table 2.3 Formation time of diferent metal carbides by mechanical alloying the stoichiometric powder mixture of elemental metal powders with graphite under Ar. Metal carbide

TiC

Formation time 35 min Stoichiometric phase formation time Final milling time

Ti0.9Al0.1C Ti0.9Ni0.1C Ti0.9W0.1C SiC Ni3C Fe3C 48 min

55 min

50 min

5 h 1.5 h

3h

35 min

3h

8h

8h

15 h

5h

5h

6h

10 h

12 h

8h

15 h

8h

8h

continuous reduction in particle size and accumulation of lattice strain with increasing milling time. A part of crystalline Si transforms to amorphous Si with a very low particle size (~6 nm) and high lattice strain. Ater 5 h of milling, the amount of amorphous Si increases gradually at the expense of crystalline Si phase with increasing milling time. At the same time, SiC phase is noticed to form and its amount increases continuously with increasing milling time at the expense of both the amorphous Si and graphite phases. Ater 10 h of milling, nanocrystalline Si phase disappears completely and it totally converts into amorphous state. Full formation of cubic SiC phase is observed just ater 15 h of milling. (iii) Ni3C: Ater 30 min of milling of Ni and graphite powders, only Ni relections are noticed to present in the XRD pattern with signiicant peak broadening (Figure 2.3c). It suggests that within this short duration of milling Ni–C solid solution alloy is formed. Further milling up to 1.5 h results in a signiicant amount of peak shit, peak asymmetry, and peak broadening in all Ni–C relections. It reveals that the ball-milled powders are deformed plastically due to high-energy impact. Ater 1.5 h of milling, there is an indication of formation of Ni3C phase (visualize clearly in XRD pattern of 2 h milled sample) with the appearance of the strongest (113) relection around 2 ~45°. In the course of milling up to 5 h, the amount of Ni3C phase increases continuously in the expense of Ni-C solid solution phase. (iv) Fe3C: Ater just 30 min of milling of Fe and graphite powders, graphite relections disappear completely and -Fe relections broaden signiicantly owing to

Room Temperature Mechanosynthesis 65 the continuous reduction in particle size and accumulation of lattice strain, with increasing milling time. Ater 2 h of milling, Fe3C phase is noticed to form from Fe–C solid solution and its amount increases continuously with increasing milling time at the expense of both the -Fe and carbon phases. In the present study, Rietveld’s structure and microstructure reinement method [48–56] has been employed for accurate estimation of phase contents and microstructure parameters of individual phases found in metal–graphite mixture in the course of milling. For each prepared composite, the experimental patterns (I0) are itted with the theoretically simulated patterns (Ic) and some of the selected patterns are shown in Figure 2.4. Fitting residual (I0–Ic) plotted at the bottom of respective itted patterns gives evidence of accuracy in itting (Figure 2.4). Almost linear plots of residue except at the peak positions ensure that all the relections of all phases have been itted very well. he GoF values (~1.0) for ittings of all experimental data also reveal that the simulated powder patterns are properly reined to it the experimental powder patterns. XRD patterns of metal carbide powder prepared at higher milling time appear with considerable amount of peak broadening and peak shiting [41–46]. Peak broadening of metal carbide relections increases continuously in the course of milling, which is prominent in relatively higher milling time ater full formation of metal carbide phase. his peak broadening is itted very well by considering both the efect of small particle size and lattice strain, which are reasoned due to cold working on metal carbide lattice during ball milling.

2.4.2 Variation of Mol Fraction Change in mol fraction of diferent phases for metal carbides with increasing milling time is explained in details in this section. Figure 2.5 depicts the nature of variation of mol fraction of diferent phases in diferent metal carbide powders with increasing milling time. Rietveld’s analysis of unmilled sample in each case difers from the original experimental molar ratio and this change in composition arises due to highly oriented graphite layers along . Actual representations of mol fraction values are shown by hollow symbols and dotted lines in the plots for TiNiC (Figure 2.5c). Phase content of graphite reduces to zero within a short time of milling. From the plots of mol fraction, an enhancement in mol fraction of major metal phases is clearly noticed without any major change in other structural or microstructural parameters within an initial stage of milling. his apparent enhancement may be attributed to the formation of amorphous carbon and then ater gradual difusion of all C-atoms into Ti/Ni/Fe/Si lattice.

66 Advanced Engineering Materials and Modeling 4500 o I o Ic

4000 3500

48 min

3000 Io-Ic

Intensity (a.u)

2500 2000

1h

1500

Io-Ic

1000

3h

500 10 h 0

Io-Ic Io-Ic Ti-Al-C

–500

Al Ti

–1000 20

30

40

(a)

50

60

70

80

(311)

(222)

TiWC (220)

2000

(200)

(111)

2θ (degree)

8h 1500

Intensity (arb. units)

2h Io-Ic 50 min

1000

Io-Ic 45 min 500

Io-Ic Pure

0

Io-Ic TiWC W Ti C

-500 20 (b)

30

40

50

60

70

80

2θ (degree)

Figure 2.4 Typical Rietveld’s output of X-ray powder difraction patterns of ball-milled. (a) TiAlC. (b) TiWC powder mixtures. Experimental data points are shown as hollow circles, while reined simulated patterns are shown as continuous lines. he diference between the experimental data (I0) and the itted simulated pattern (Ic) is shown as a continuous line (I0 – Ic) under each difraction pattern.

Room Temperature Mechanosynthesis 67 1.0 0.8

0.75

Mol fraction

Mol fraction

1.00

C Ti TiC

0.50 0.25

Ti-Al-C Al Ti C

0.6 0.4 0.2

0.00 0

20

40

(a)

60

80

0.0

100 120 140 358 360

0

1.0

Mol fraction

Mol fraction

Ti Ni Ti-Ni-C C

0.6 0.4 0.2

8

10

0.8 W Ti C TiWC

0.6

0.0 0

2

4

6 8 Milling time(h)

10

12

Graphite Si Si- Amorphous SiC

0.8

(d)

0.4

3 4 5 6 Milling time(h)

7

8

0.0

0.0 6 8 10 Milling time (h)

12

14

16

Ni Ni3C C

0.4 0.2

4

2

0.6

0.2

2

1

0.8

0.6

0

0

1.0

Mol fraction

1.0

Mol fraction

6

0.4

0.0

(e)

4

Milling time(h)

1.0

0.8

(c)

2

(b)

Milling time (min)

(f)

0

2

4 6 Milling time (h)

8

1.0

Mol fraction

0.8 0.6

α-Fe Fe3C

0.4

Graphite

0.2 0.0

(g)

0

2

4 6 Milling time (h)

8

Figure 2.5 Variation of phase content (mol fraction) of diferent phases in unmilled and ball-milled mixture of metals and graphite at diferent milling time during the formation of (a) TiC, (b) TiAlC, (c) TiNiC, (d) TiWC, (e) SiC, (f) Ni3C, and (g) Fe3C.

68 Advanced Engineering Materials and Modeling (i) Ti-based binary/ternary metal carbide: Mol fraction of graphite phase reduces to nil within 30–48 min of MA the binary/ternary metal–graphite mixture powder. he new phase TiC or TiMC is noticed to form ater a short time of milling with a considerable high mol fraction (>0.8) value with complete inclusion of carbon atoms into -Ti lattice. A critical observation of mol fraction plots reveals that in ternary composite, phase content of TiMC increases signiicantly with the solid solution of M phase in the course of milling. Further milling ater formation time results in full formation of TiC or TiMC phase (formation of single-phase material contributing phase has mol fraction 1.0) through MISPR via an instant combustion reaction. (ii) SiC: Mol fraction of graphite phase reduces to zero within 3 h of milling, but that of the crystalline Si phase increases from 0.42 to 0.61 and ~0.39 mol fraction transforms to amorphous Si powder. Within 3 h of milling, all carbon atoms (graphite) difuses into Si matrix and the enhancement in Si phase content leads to the total contribution in XRD pattern. Ater 5 h of milling, SiC phase starts to form and its content gradually increases with increasing milling time until full formation of SiC is observed ater 15 h of milling. (iii) Ni3C: For Ni3C compound, a Ni–C solid solution is formed within 15 min of milling followed by difusion of amorphous carbon in Ni lattice. he Rietveld analysis approximates the whole contribution in the XRD pattern is only from Ni (total mol fraction = 1). Ater 1.5 h of milling, Ni–C solid solution transforms partly to Ni3C phase and a core–shell structure is formed. Within 3 h of milling, mol fractions of both Ni and Ni3C phases become almost equal in proportion. It reveals the fact that the volumes of the Ni–C core and Ni3C shell turn equal in the course of milling through growth of more and more Ni3C shells on continuously fractured surface of Ni–C cores (Figure 2.6). Further milling up to 5 h results in full formation of Ni3C phase through fracture and re-welding mechanism of these core–shell particles and up to 8 h of milling there is no signiicant change in content of stoichiometric Ni3C phase. (iv) Fe3C: In Fe3C compound, due to the absence of graphite phase at the early stage of milling, the total contribution in XRD pattern is considered solely for

Room Temperature Mechanosynthesis 69

Graphite

Ni

Ni-C core

Ni3C

Figure 2.6 Schematic diagram of development of core–shell structure during Ni3C formation in the course of ball milling of stoichiometric mixture of Ni and graphite powders.

-Fe phase (mol fraction = 1). Ater 2 h of milling, Fe3C phase starts to form. Full formation of Fe3C is observed ater 5 h of milling.

2.4.3 Phase Formation Mechanism XRD pattern analysis by Rietveld’s method explores nature of variation of phase abundances in metal carbides with increasing milling time and reveals the formation mechanism of diferent metal carbides as given below. (i) Formation of Ti-based binary/ternary metal carbide: In MA technique, two kinds of reaction mechanism have been commonly accepted; one is a gradual reaction through elemental difusion, and the other is a self-sustaining reaction ignited ater a certain period of activation time. he MISPR was usually observed in highly exothermic systems such as Ti–B [67], Ni–Al [68], and Mo–Si [69]. In this study, synthesis of all binary and ternary Ti-based metal carbides using a planetary ball mill also has been followed by MISPR [70]. In case of TiC and TiMC–metal composites, the absence of the graphite relections ater short duration of milling, broadening of -Ti relections without any peak shit, and texturing of -Ti particles along graphite layers reveal the fact that prior to the combustion reaction the possibility of amorphization of graphite powder at the initial stage of milling can be ruled out. Instead, there may be a transitional bonding state between some of C and -Ti atoms, preferably located on the nanocrystalline -Ti grain boundaries [20, 71, 72]. Whereas at early stage of milling, formation of -Ti–C interstitial solid solution is only accountable for initiation of binary TiC phase formation. However, ternary Ti-based metal carbide formation is more complex. he formation of the ternary Ti–M–C phase can be interpreted as a result of formation

70 Advanced Engineering Materials and Modeling of two types of solid solutions: (i) interstitial solid solutions of -Ti–C followed by the amorphization of graphite phase, and (ii) substitutional solid solution of Ti–M–C due to accumulation of any of M atom in the Ti–C matrix in the course of milling. It can be seen that the binary TiC and ternary Ti–M–C–metal composites proceed toward full formation ater 35–55 min of milling time period (Table 2.3) through MISPR. he most probable reason for MISPR may be due to the fact that high-energy ball milling results in rapid decrease in particle size and thereby rapid accumulation of defects that lowers the activation energy for the reaction and brings the powder to a critical pre-combustion condition. he reaction can then be ignited easily by the energy of the colliding milling media. he high-shock pressure experienced by powder trapped between colliding milling balls acts as the ignition of the reaction [73]. (ii) Formation of SiC: he phase transformation kinetics of formation of nanocrystalline SiC via MA can be described in a three-step process. Step I: pure graphite + pure silicon = amorphous Si + nanocrystalline Si + thin graphite layer embedded at grain boundary of Si (beyond the detection limit of X-ray). Step II: nanocrystalline Si amorphous Si Step III: amorphous Si + thin graphite layer = nanocrystalline SiC (iii) Formation of Ni3C: he phase transformation kinetics studied by the Rietveld method reveals that the graphite powder becomes amorphous at the early stage of milling and then difuses into Ni matrix and a Ni–C solid solution is formed within 1 h of milling. he high-resolution electron microscopy reveals that the Ni3C phase is initiated as a thin shell on Ni–C core within 2 h of milling and becomes stoichiometric in composition within 5 h of milling. (iv) Formation of Fe3C: At the early stage of milling, carbon atoms could not entered into the -Fe matrix and thin graphite layers are started to distribute around the -Fe grain boundaries. Ater 1 h of milling, carbon atoms started to difuse through the boundaries of nano -Fe grain and an interstitial Fe–C solid

Room Temperature Mechanosynthesis 71 solution has been formed with a trace amount of carbon. he orthorhombic Fe3C phase is formed ater 2 h of milling and remains stable up to 8 h of milling.

2.4.4

Is Ball-milled Prepared Metal Carbide Contains Contamination?

As we have mentioned earlier in Section 2.1.5, the main disadvantage of using planetary ball mil for metal carbide preparation is the probability of appearance of contamination in the as-milled powder material particularly in powder sample milled for a long duration. hough in the XRD pattern of any of the prepared metal carbides does not show trace of contamination, to make sure we have analyzed the Mössbauer spectra of Fe3C for case study to investigate the Fe contamination from the chrome–steel milling media. Figure 2.7 shows the Mössbauer spectra of 3, 5, and 8 h ball-milled Fe3C powder. he spectrum of 3 h milled sample reveals the presence of -Fe (elemental phase in this case) in the sample apart from the stoichiometric Fe3C phase. Only the Fe3C phase has been detected in the samples milled for relatively higher milling time 5 and 8 h with high energy; no -Fe contamination (from chrome–steel milling media) is detected in the corresponding Mössbauer spectra, which agrees well with the results obtained from Rietveld’s analysis.

Fe3C

8h

(c) Intensity (a.u.)

Fe3C 5h (b) Fe3C

α-Fe 3h

(a) 0

50

100 150 Channel. no.

200

250

Figure 2.7 he Mössbauer spectra of -Fe and graphite powder (3:1 mol) mixture ball milled for (a) 3 h, (b) 5 h, and (c) 8 h duration under argon atmosphere.

72 Advanced Engineering Materials and Modeling

2.4.5

Variation of Particle Size

Nature of variation of particle (coherently difracting domain) size of elemental phases and synthesized metal carbide phases with increasing milling time as obtained from Rietveld’s analysis are shown in Figure 2.8. Particle size of all the phases present in the unmilled sample or formed in the ball-milled powder mixture in the course of milling is considered as isotropic. In the present study, the lowest particle size of each metal carbide achieved during MA is given in Table 2.4. (i) Ti-based binary/ternary metal carbide: he particle size of -Ti phase decreases rapidly to ~50 nm due to high-energy impact within an early stage of milling. For ternary TiMC within 1 h of milling, particle size of M phase reduces sharply from starting value via fracturing and ater then reduces slowly due to cold working on M lattice until it disappears completely. Due to the high-temperature reaction of MISPR, particles 350

W Ti TiWC

250

60

200 150 100

40 30 20 10

50 0

0 0

1

2

(a)

3

4

5

6

7

8

0

4

6

8

10

12

14

16

Milling time (h)

150

70

125

Ni

60

Particle size (nm)

Particle size (nm)

2

(b)

Milling time (h) 80

50 40 30 20

0

Fe Fe3C

100 75 50 25

Ni3C

10

0 –25

0

(c)

Si SiC

50

Particle size (nm)

Particle size (nm)

300

1

2

3

4

5

Milling time (h)

6

7

8

0

(d)

2

4

6

8

Milling time (h)

Figure 2.8 Variation of particle size of diferent phases in unmilled and ball-milled mixture of metals and graphite at diferent milling time during the formation of (a) TiWC, (b) SiC, (c) Ni3C, and (d) Fe3C.

10 10 5 0.4316

13

4.85

0.4318

Particle size (nm)

R.M.S strain ×103

Lattice parameter (nm)

Ti0.9Al0.1C

6

TiC

Milling time (h)

Metal carbide

0.4324

4.85

7

8

Ti0.9Ni0.1C

0.4299

6.37

11

8

Ti0.9W0.1C

0.4356

28.24

5

15

SiC

4.5499(a) 12.9483(c)

5.74

9

5

Ni3C

0.5092(a) 0.6784(b) 0.4521(c)

8.22

5

5

Fe3C

Table 2.4 Microstructure parameters of diferent metal carbides with the lowest particle size formed in stoichiometric composition by mechanical alloying the elemental metal and graphite powder mixtures under Ar.

Room Temperature Mechanosynthesis 73

74 Advanced Engineering Materials and Modeling of TiC and TiMC phases grow with a relatively large size ater formation. It is evident from the observation that reduced particle size plays a vital role in inclusion of M atoms in -Ti–C matrix. Particle size of all Ti-based metal carbides reduces gradually with increasing milling time toward a saturation value (~10 nm) ater a long duration of milling. (ii) SiC: he particle size of Si decreases very sharply from ~586 to ~36 nm within 3 h of milling and then decreases linearly up to ~5 nm within 7 h of milling. However, size of SiC particles remained almost invariable in the course of milling. Average particle size of SiC phase obtained from Rietveld’s analysis is ~5 nm. (iii) Ni3C: In the course of milling, particle size of unmilled Ni powder decreases from ~98 to ~3 nm within 3 h of milling. In contrary to Ni phase, the isotropic particle size of Ni3C shell formed ater 1.5 h of milling with ~1.5 nm size and increases slowly to ~9 nm ater 5 h of milling. (iv) Fe3C: he particle size of -Fe decreases very sharply from ~114 to ~20 nm within 30 min of milling and then decreases slowly up to 3 h of milling to ~3 nm. Ater new phase formation, no signiicant change in isotropic particle size of ball-milled Fe3C powder is noticed. It reveals that the gradual peak broadening of ball-milled Fe3C relections is not inluenced by the small particle size.

2.4.6

Variation of Strain

he rms strain generated in elementary metals and synthesized metal carbide phases are obtained from Rietveld’s analysis, and their variations with increasing milling time are shown in Figure 2.9. With increasing milling time, rms lattice strain of all elementary metal phases, in general, increases rapidly which indicates that ball milling introduces high strain in the produced powder material. High lattice strain in metal matrix plays a vital role in peak broadening of corresponding relections. (i) Ti-based binary/ternary metal carbide: he TiC/TiMC phase is formed in a short time of milling with nearly zero lattice strain value. he strain value of Ti-based carbide phase increases

Room Temperature Mechanosynthesis 75 0.045

35

25 20 15 10

Si SiC

0.035 0.030

R.M.S. Strain

R.M.S strain x103

0.040

Ti-Al-C Al Ti

30

0.025 0.020 0.015 0.010

5

0.005

0

0.000 0

2

(a)

4

6

8

10

0

(b)

Milling time (h)

2

4

6 8 10 Milling time (h)

12

14

16

12 Fe Fe 3C

10

Ni3C Ni

R.M.S strain x103

R.M.S strain x103

30

20

10

8 6 4 2 0

0

–2 0

(c)

1

2

3

4

5

Milling time (h)

6

7

0

8

(d)

2

4

6

8

Milling time (h)

Figure 2.9 Variation of rms strain of diferent phases in unmilled and ball-milled mixture of metals and graphite at diferent milling time during the formation of (a) TiAlC, (b) SiC, (c) Ni3C, and (d) Fe3C.

signiicantly at the early stage of milling time, i.e. more strain is accumulated in TiC/TiMC matrix during the time of full formation of single TiC/TiMC phase. his nature also shows that lattice strain is a crucial parameter for peak broadening in TiC/TiMC relections in relatively lower milling time. Aterward, strain value variation of TiC/TiMC shows a steady nature. Ater full formation of binary or ternary Ti-based metal carbide phase, strain value of the same phase remains almost unchanged until the end of milling. (ii) SiC: SiC phase is initiated with a small amount of lattice strain as that of Si. Ater the formation, the SiC particles are subjected to high-energy impact and these particles accumulate lattice strain with increasing milling time. (iii) Ni3C: Initially, a sharp increase in lattice strain of Ni matrix seems to be due to the inclusion of carbon atoms in Ni matrix. Aterward, a slow rate of

76 Advanced Engineering Materials and Modeling increment suggests that the most of the carbon atoms are incorporated within 3 h of milling. he Ni3C shell formed with high density of lattice strain. Release of lattice strain in Ni3C lattice ater 2 h of milling clearly signiies the formation of Ni3C phase with proper composition in course of milling. (iv) Fe3C: Fe3C phase is initiated almost without any lattice strain. It suggests that the Fe3C is formed instantly via the re-welding mechanism of nanocrystalline -Fe and thin graphite layers. Fe3C particles accumulate lattice strain on the way to complete formation with increasing milling time.

2.4.7 High-Resolution Transmission Electron Microscopy Study HRTEM study of some of the nanocrystalline metal carbide systems is presented in this section. he HRTEM image of 8 h Ti–W–C ball-milled sample and histogram of particle size distribution obtained from HRTEM image analysis are shown in Figure 2.10a and b, respectively. It is evident from Figure 2.10a that the metal carbide particles are almost isotropic (spherical) in shape with a corresponding size distribution ~9–10 nm (Figure 2.10b), which is of same order of magnitude as found from XRD data analysis of Ti–W–C by Rietveld’s method. he indexed selected area electron difraction (SAED) pattern of 10 h Ti–Al–C ball-milled sample presented in Figure 2.10c clearly reveals the presence of only cubic Ti-based metal carbide phase in the inal ball-milled sample. Figure 2.10d represents HRTEM image of 12 h ball-milled Ti–Ni–C powder in which nanoparticles metal carbide are clearly noticed (marked by white rings). Interplanar spacing of the relecting planes in nanocrystalline metal carbide particles is calculated from Figure 1.10e and f for inal ball-milled powder mixture of Ti–Al–C and Ti–W–C, respectively, and the values 0.251 and 0.242 nm corresponding to the (111) plane of both the system conirms the full formation of cubic metal carbide phase in inal ball-milled sample.

2.4.8 Comparison Study between Binary and Ternary Ti-based Metal Carbides Till now, we have prepared Ti-based binary TiC and ternary Ti–Al–C, Ti–Ni–C, and Ti–W–C metal carbides by MA. A comparative study between the binary and ternary Ti-based metal carbides ball-milled at

Room Temperature Mechanosynthesis 77 (b) 30 Frequency

(a)

20

10

0

0

5 10 15 20 Particle size (nm)

(d)

(c) (111) (200) (220) (311)

(111)

(222) (400) (331)

(200)

2 1/ nm

5 nm

(e)

(f) ~2.42 Å (111)

(111)

0.251nm 2 nm

~8.7 nm

Figure 2.10 HRTEM (a) transmission micrograph, (b) histogram plot of 8 h ball-milled homogeneous mixture of TiWC powders, (c) SAED pattern of 10 h TiAlC ball-milled sample, (d) HRTEM image of 12 h ball-milled TiNiC nanoparticle, (e) micrograph containing (111) planes in a TiAlC nanocrystalline particle, and (f) (111) planes in TiWC nanocrystalline particle.

diferent time period can be enlightened from the results obtained from the Rietveld analysis of XRD data. Figure 2.11a illustrates the variations of Ti-based carbide phase formation with and without the presence of any of the solute Al/Ni/W with progress of milling time. Nature of phase content variation evident that the stoichiometric mixture of -Ti and graphite powders under mechanical milling at room temperature and argon atmosphere produced binary stoichiometric TiC ater 35 min of milling. Critical observation of Figure 2.11a points out that at the early stage of milling, formation

78 Advanced Engineering Materials and Modeling 0.434

Mol fraction

0.98

Lattice parameter (nm)

1.00 TiC TiWC TiAlC

0.96 0.94

TiNiC

0.92 0.90

TiC TiWC

0.433 TiNiC 0.432 0.431 0.430 TiAlC

0.88 0.429 0

2

4

(a)

6

8

10

0

12

TiC TiWC

3

4

400

50 40 30 20 TiAlC

10

6

7

8

9 10 11 12 13

TiAlC

5

60

5

TiC TiWC

6

strainx103

Particle size (nm)

2

Milling time (h) 7

450

TiNiC

4 3 2 1

TiNiC

0

0 0

(c)

1

(b)

Milling time (h)

2

4

6

8

Milling time (h)

10

12

1

(d)

2

3

4

5 6 7 8 9 10 11 12 Milling time (h)

Figure 2.11 Variation of microstructural parameters of binary and ternary Ti-based metal carbides for (a) mol fraction, (b) lattice parameters, (c) particle size, and (d) rms strain.

of metal carbide phase is retarded due to introduction of solute metal M atom in Ti–C metal matrix. Ternary Ti-based metal carbides needed more milling time compared to TiC for formation as well as becoming complete stoichiometric in composition (Table 2.3, Figure 2.11a). Among them, Ti0.9Al0.1C forms and also turns into stoichiometric powder in quickest milling time (48 min and 3 h, respectively). Ti0.9W0.1C and Ti0.9Ni0.1C started to form ater 50 and 55 min of milling time, respectively, and become completely stoichiometric ater same milling time (8 h). However, at the time of formation phase, contents of both Ti0.9Ni0.1C and then Ti0.9Al0.1C are less than that of Ti0.9W0.1C in their respective powder mixtures. It is clear that on the way of preparation of Ti–M–C, W atom and then Ni atom difuse slowly in Ti–C metal matrix compared to Al atom. Figure 2.11b depicts the nature of change in lattice parameter values for all cubic Ti-based metal carbides. It is evident from the illustration that at the time of formation, lattice parameter of ternary phases does not difer much from that of binary Ti–C phase. Lattice parameter of TiC remains nearly constant during milling process, but inclusion of other metal atom

Room Temperature Mechanosynthesis 79 in α-Ti-C matrix in the process of ternary phase formation leads to diferent nature of variation as plotted in Figure 2.11b. Obviously, expansion or contraction of ternary Ti-based metal carbide lattice depends upon comparative atomic radius of solute metal atom and that of Ti atom. Also at relatively higher milling time cold working on ball-milled powder mixture may results in lattice expansion in some cases. A comparison study between binary and ternary Ti–C system on the basis of particle size variation with increasing milling time can be described from the analysis of Figure 2.11c. Nature of variation of particle size clearly shows that TiC phase formed at 35 min of milling with larger particle size (~448 nm). Within 6 h of milling, rapid decrease in TiC particle size (~13 nm) indicates that TiC phase was formed through MISPR. All ternary Ti-based metal carbides under consideration formed via MISPR with a relatively small particle size than that of TiC at the time of formation. Compared to the variation of particle size of binary metal carbide phase, same plots for ternary phases show a rather slow variation at the early stage of milling. It is interesting to note that to attend the same order of particle size value ternary phases needed more time than binary phase. Ater 2 h of milling, particle size values for all phases difer in the increasing order of particle sizes, i.e., TiC < Ti0.9Al0.1C < Ti0.9Ni0.1C < Ti0.9W0.1C (Figure 2.11c). Evidently, in case of ternary metal carbide formation, inclusion of Al/Ni/W atoms in -Ti–C matrix plays a vital role in rate of reduction of particle size. Observation of particle size plots (Figure 2.11c) also reveals that all metal carbide phases have same particle size value ater 10 h of milling (~10 nm). Figure 2.11d shows that rms lattice strain value of all binary and ternary Ti–C increases nonlinearly with increasing milling time and a sudden enhancement in strain value is noticed ater 2 h of milling. In the course of milling, both particle size and rms strain values of TiC, Ti0.Al0.1C, and Ti0.Ni0.1C tend to saturate at relatively higher milling time. However, rms lattice strain value of Ti0.W0.1C increases signiicantly till its particle size saturates ater 8 h of milling. It is to be noted that Ti0.W0.1C phase acquires a higher rms strain value in comparison to nearly same strain value of other binary or ternary Ti-based metal carbides. From the plots of lattice strain variation it is evident that in particular, difusion of W atom in TiC metal matrix leads to rapid accumulation of enormous amount of stored energy which is manifested in rapid increase in lattice strain value of Ti0.9W0.1C ater 8 h of ball milling. Rapid increase in rms strain value compared to slow reduction of particle size value in general leads to the conclusion that peak broadening of metal carbide phase is basically controlled by the high degree of lattice strain induced by ball milling.

80 Advanced Engineering Materials and Modeling

2.5 Conclusion In this chapter, we have discussed about preparation and microstructure characterization of nanocrystalline binary/ternary metal carbides synthesized at room temperature by mechanical milling of diferent metals and graphite powders under inert atmosphere at room temperature. he stoichiometric nanocrystalline metal carbide phases are produced in a cost-efective way within a very short duration by ball-milling process. Structure and microstructure characterization and formation mechanism of these synthesized nanocarbides are made by using XRD data employing Rietveld’s analysis. Formation of single-phase nanocrystalline metal carbide is also conirmed from HRTEM study. Average particle size of nanocrystalline metal carbide powder obtained from Rietveld’s analysis is very close to that obtained from HRTEM observation.

Acknowledgment he authors wish to thank to the University Grants Commission (UGC), India for granting CAS-I programme under the thrust area “Condensed Matter Physics including Laser applications” to the Department of Physics. S.K.P. also thankful to UGC for granting the Major Research Project [F. No. 41-845/2012(SR)] under the inancial assistance of which the work has been carried out.

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Room Temperature Mechanosynthesis 81 6. Koc, R.C., Meng, C., Swit, G.A., Sintering properties of submicron TiC powders from carbon coated titania precursor. J. Mater. Sci., 35, 3131, 2000. 7. Toth, L.E., Margrave, J.L., Transition Metal Carbides and Nitrides, Academic Press, New York and London, 1971. 8. Schwarz, K., Band structure and chemical bonding in transition metal carbides and nitrides. Crit. Rev. Solid State Mater. Sci., 13, 211, 1987. 9. Gubanov, V.A., Ivanoskii, A.L., and Zhukov, V.P., Electronic Structure of Refractory Transition Metal Carbides and Nitrides. Cambridge University Press, Cambridge, 1994. 10. Ren, R., Yang, Z., and Shaw L.L., Polymorphic transformation and powder characteristics of TiO2 during high energy milling. J. Mat. Sci., 35, 6015, 2000. 11. Begin-Colin, S., Girot, T., Le Caer, G., and Mocellin, A., Kinetics and mechanisms of phase transformations induced by ball-milling in anatase TiO2. J. Solid State Chem., 149, 41, 2000. 12. Dutta, H., Sahu, P., Pradhan, S.K., and De, M., Microstructure characterization of polymorphic transformed ball milled anatase TiO2 by Rietveld method. Mater. Chem. Phys., 77, 153, 2002. 13. Patra, S., and Pradhan, S.K., Microstructure and optical characterization of CdTe quantum dots synthesized in a record minimum time. J. Appl. Phys., 108, 083515, 2010. 14. Bhaskar, U.K., Bid, S., and Pradhan, S.K., Microstructure evaluation of nanostructured Ti0.9Al0.1N prepared by reactive ball milling. J. Alloys Compd, 509, 620, 2011. 15. Patra, S., Satpati, B., and Pradhan, S.K., Quickest single-step mechanosynthesis of CdS quantum dots and their microstructure characterization. J. Nanosc. Nanotechonol., 11, 1, 2011, 16. Enayati, M.H., Seyed- Salehi, M., and Sonboli, A., Development of Fe3C, SiC and Al4C3 compounds during mechanical alloying. J. Mater. Sci., 42, 5911, 2007. 17. Calka, A., Mechanical alloying: technology and properties of prepared materials. Key Eng., 81–83, 17, 1993. 18. Suryanarayana, C., Mechanical alloying and milling. Prog. Mater Sci., 46, 1, 2001. 19. Abdellaoui, M., and Gafet, E., he physics of mechanical alloying in a planetary ball mill: mathematical treatment. Acta Metall Mater., 43, 1087, 1995 20. Wu, N.Q., Lin, S., Wu, J.M., and Li, Z.Z., Mechanosynthesis mechanism of TiC powders. J. Mater. Sci. Tech., 14, 287, 1998. 21. Kudaka, K., Kiyokata, I., Sasaki, T., Mechanochemical synthesis of Titanium Carbide, Diboride and Nitride. J. Ceram. Soc. Jpn, 107, 1019, 1999. 22. Xinkun, Z., Baochang, C., Quishi, L., Xiuquin, Z., Tieli, C., and Yunsheng, S., Synthesis of nanocrystalline TiC powder by mechanical alloying. Mater. Sci. Eng. C, 16, 103, 2001. 23. Deidda, C., Doppiu, S., Monagheddu, M., and Cocco, G., A direct view of the self combustion behaviour of TiC system under milling. J. Metastable Nanocryst. Mater., 15–16, 215, 2003.

82 Advanced Engineering Materials and Modeling 24. Zhang, B., and Li, Z.Q, Synthesis of vanadium carbide by mechanical alloying. J. Alloy Compd, 392, 183, 2005. 25. Kim D.-Joo, and Choi, D.-Jin, Microhardness and surface roughness of silicon carbide by chemical vapour deposition. J. Mater. Sc. Lett., 16, 286, 1997. 26. Vandamme, N.S., Que L., and Topoleski, L.D.T., Carbide surface coating of Co-Cr-Mo implant alloys by a microwave plasma-assisted reaction. J. Mater. Sci., 34, 3525, 1999. 27. Factor, M., and Roman, I., Microhardness as a simple means of estimating relative wear resistance of carbide thermal spray coatings: Part 1. characterization of cemented carbide coatings. J. hermal Spray Tech., 11, 468, 2002. 28. Tang, B., Liu, H., Wang, L., Wang, X., Gan, K., Yu, Y., Wang, Y., Sun T., and Wang, S., Fabrication of titanium carbide ilm on bearing steel by plasma Immersion ion implantation and deposition. J. Surf. Coat. Tech., 186, 320, 2004. 29. Jiang, G., Zhuang H., and Li, W., Field-activated, pressure-assisted combustion synthesis of tungsten carbide-nickel composites. Mater. Lett., 58, 2855, 2004. 30. Khairaldien, W., and Khalil, A., Bayoumi, Production of aluminum-silicon carbide composites using powder metallurgy at sintering temperatures above the aluminum melting point. J. Test. Eval., 35, 13, 2007. 31. Xia, Z.P., Shen, Y.Q., Shen, J.J., and Li, Z.Q., Mechanosynthesis of molybdenum carbides by ball milling at room temperature. J. Alloy Compd, 453, 185, 2008. 32. Xiang, J.Y., Liu, S.C., Hu, W.T., Zhang, Y., Chen, C.K., Wang, P., He, J.L., Yu, D.L., Xu, B., Lu, Y.F., Tian Y.J, and Liu, Z.Y., Mechanochemically activated synthesis of zirconium carbide nanoparticles at room temperature: a simple route to prepare nanoparticles of transition metal carbides. J. Eur. Cer. Soc., 31, 1491, 2011. 33. Fan, Q.C., Chai, H.F., and Jin, Z.H., Role of iron addition in the combustion synthesis of TiC–Fe cermet. J. Mater. Sci., 32, 4319, 1997. 34. Lee, W.C., and Chung, S.L., Ignition phenomena and reaction mechanisms of the self-propagating high-temperature synthesis reaction in the Ti-C-Al system. J. Am. Ceram. Soc., 80, 53, 1997. 35. Shon, I.J., and Munir, Z.A., Synthesis of TiC, TiC‐Cu Composites, and TiC‐ Cu Functionally Graded Materials by Electrothermal Combustion. J. Am. Ceram. Soc., 81, 3243, 1998. 36. Mishra, S.K., Das, S.K., Ray, A.K., and Ramchandrarao, P., Efect of nickel on sintering of self-propagating high-temperature synthesis produced titanium carbide, J. Mater. Res., 14, 3594, 1999. 37. Brinkman, H. Zupanic, J. Duszczyk, F., and Katgerman, J.L., Production of Al-Ti-C grain reiner alloys by reactive synthesis of elemental powders: Part I. Reactive synthesis and characterization of alloys. J. Mater. Res., 12, 2620, 2000.

Room Temperature Mechanosynthesis 83 38. Zhang, X.H., Han, J.C., Du, S.Y., and Wood, J.V, Microstructure and mechanical properties of TiC-Ni functionally graded materials by simultaneous combustion synthesis and compaction, J. Mater. Sci., 35, 1925, 2000. 39. Zhang, Y.M., Han, J.C., Zhang, X.H., He, X.D., Li, Z.Q., and Du, S.Y., Rapid prototyping and combustion synthesis of TiC/Ni functionally gradient materials. Mater. Sci. Eng. A, 229, 218, 2001. 40. Han, J.C., Zhang, X.H., and Wood, J.V., In-situ combustion synthesis and densiication of TiC-xNi cermets. Mater. Sci. Eng. A, 280, 328, 2000. 41. Ghosh, B., and Pradhan, S.K., Microstructure characterization of nanocrystalline TiC synthesized by mechanical alloying, Mater. Chem. Phys., 120, 537, 2010. 42. Ghosh, B., and Pradhan, S.K., Microstructural characterization of nanocrystalline SiC synthesized by high-energy ball-milling. J. Alloy Compd, 486, 480, 2009. 43. Ghosh, B., and Pradhan, S.K., Microstructure characterization of nanocrystalline Fe3C synthesized by high-energy ball milling. J. Alloy Compd, 477, 127, 2009. 44. Ghosh, B., Dutta, H., and Pradhan, S.K., Microstructure characterization of nanocrystalline Ni3C synthesized by high-energy ball milling. J. Alloy Compd, 479, 193, 2009. 45. Dutta, H., Sen, A., Bhattacharjee, J., and Pradhan, S.K., Preparation of ternary Ti0.9Ni0.1C cermets by mechanical alloying: Microstructure characterization by Rietveld method and electron microscopy. J. Alloy Compds, 493, 666, 2010. 46. Dutta, H., Sen, A., and Pradhan, S.K., Microstructure characterization of ballmill prepared ternary Ti0.9Al0.1C by X-ray difraction and electron microscopy. J. Alloy Compd, 501, 198, 2010. 47. Bandyopadhyay, S., Dutta, H., and Pradhan, S.K., XRD and HRTEM characterization of mechanosynthesized Ti0.9W0.1C cermet. J. Alloy Compd, 581, 710, 2013. 48. Lutterotti, L., MAUD version2.26, http://www.ing.unitn.it/~luttero/maud 49. Rietveld, H.M., Line proiles of neutron powder-difraction peaks for structure reinement. Acta Cryst. 22, 151,1967. 50. Rietveld, H.M., A proile reinement method for nuclear and magnetic structures. J. Appl. Cryst., 2, 65, 1969. 51. Wiles, D.B., and Young, R.A., A new computer program for Rietveld analysis of X-ray powder difraction patterns. J. Appl. Cryst., 14, 149, 1981. 52. Young, R.A., and Wiles, D.B., Proile shape functions in Rietveld reinements. J. Appl. Cryst., 15, 430, 1982. 53. Young, R.A., (Ed.), he Rietveld Method. Oxford University Press/IUCr, 1996. 54. Lutterotti, L., Scardi, P., and Maistrelli, P., LSI-a computer program for simultaneous reinement of material structure and microstructure. J. Appl. Cryst., 25, 459, 1992.

84 Advanced Engineering Materials and Modeling 55. Dutta, H. Manik, S.K., and Pradhan, S.K., Phase transformation kinetics and microstructure characterization of high- energy ball-milled ZrO2-10 mol% TiO2 by Rietveld analysis. J. Appl. Cryst., 36, 260, 2003. 56. Warren, B.E., X-ray Difraction. Addison-Wesley, Reading, 1969. 57. Toraya, H., Weighting scheme for the minimization function in Rietveld reinement. J. Appl. Cryst., 31, 333, 1998. 58. Mitchell, R.H., Chakhmouradian, A.R., A structural study of the perovskite series Na1/2+xLa1/2−3xh2xTiO3. Solid State Chem., 138, 307, 1998. 59. Yartys, V.A., Fjellvag, H., Hauback, B.C., Riabov, A.B., and Sorby, M.H., Neutron difraction studies of Zr-containing intermetallic hydrides with ordered hydrogen sublattice: III. Orthorhombic Zr3FeDx(x=1.3, 2.5 and 5.0) with partially illed Re3B-type structure. J. Alloy Compd, 287, 189, 1999. 60. Jorgensen, J.D., Hu, Z., Teslic, S., Argyriou, D.N., and Short, V., Pressureinduced cubic-to-orthorhombic phase transition in ZrW2O8. Phy. Rev. B, 59, 215, 1999. 61. Yamazaki, S., and Toraya, H., Rietveld reinement of site-occupancy parameters of Mg2-xMnxSiO4 using a new weight function in least-squares itting. J. Appl. Cryst., 32, 51, 1999. 62. Dutta, H., Lee Y.-C., and Pradhan, S.K., Microstructure characterization and polymorphic transformation kinetic study of ball milled nanocrystalline a-TiO2 – 20 mol% m-ZrO2 mixture by x-ray difraction and electron microscopy. Physica E, 36, 17, 2007. 63. Toraya, H., Estimation of statistical uncertainties in quantitative phase analysis using the Rietveld method and the whole-powder-pattern decomposition method. J. Appl. Cryst., 33, 1324, 2000. 64. Lala, S., Satpati, B., Kar, T., and Pradhan, S.K., Structural and microstructural characterizations of nanocrystalline hydroxyapatite synthesized by mechanical alloying. Mat. Sci. Eng. C, 33, 2891, 2013. 65. Enzo, S., Fagherazzi, G., Benedetti, A., and Polizzi, S., A proile-itting procedure for analysis of broadened X-ray difraction peaks. I. Methodology. J. Appl. Cryst. 21, 536, 1988. 66. Guizard, C., Cygankiewicz, N., Larbot, A., and Cot, L., Sol-gel transition in zirconia systems using physical and chemical processes. J. Non Cryst. Solids, 82, 86, 1986. 67. Park, Y.H., Hashimoto, H., Abe, T., and Watanabe, R., Mechanical alloying process of metal-B (M ≡ Ti, Zr) powder mixture. Mater. Sci. Eng. A, 1291, 181, 1994. 68. Atzmon, M., In situ thermal observation of explosive compound-formation reaction during mechanical alloying. Phys. Rev. Lett., 64, 487, 1990. 69. Takacs, L., Soika, V., and Bal´aˇz, P., he efect of mechanical activation on highly exothermic powder mixtures. Solid State Ionics, 641, 141, 2001. 70. Ye, L.L., Liu, Z.G., Quan, M.X., and Hu, Z.Q., Diferent reaction mechanisms during mechanical alloying Ti50C50 and Ti33B67. J. Appl. Phys., 80, 1910, 1996.

Room Temperature Mechanosynthesis 85 71. Loshe, B.H., Calka, A., and Wexler, D., Efect of starting composition on the synthesis of nanocrystalline TiC during milling of titanium and carbon. J. Alloy Compd, 394, 148, 2005. 72. Loshe, B.H., Calka, A., and Wexler, D., Synthesis of TiC by controlled ball milling of titanium and carbon. J. Mater. Sci., 42, 669, 2007. 73. Lee, W.H., Reucrot, P.J., Byun, C.S., and Kim, D.K., Efect of Si powder reining on the self-propagating high temperature synthesis reaction of titanium silicide induced by mechanical alloying. J. Mater. Sci. Lett., 20, 1647, 2001.

3 Toward a Novel SMA-reinforced Laminated Glass Panel Chiara Bedon1* and Filipe Amarante dos Santos2 1

Department of Engineering and Architecture, University of Trieste, Trieste, Italy 2 CEris, ICIST, Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa, Quinta da Torre, Caparica, Portugal

Abstract Shape-memory alloys (SMAs) are a class of metal materials that exhibit two outstanding properties, namely the superelastic and the shape-memory efects. Taking advantage of these intrinsic properties, several applications have been proposed over the past years in robotic, automotive, and biomedical engineering in the form of SMA actuator wires and plates replacing conventional pneumatic or hydraulic systems. In this chapter, a novel design concept of SMA-reinforced laminated glass panels is proposed, and the feasibility of an adaptive embedded reinforcement system built up of martensitic SMA wires is explored. Glass panels are oten used as cladding walls in façades or roofs in buildings to cover large surfaces with typically high size-to-thickness ratios. Major restrictions in their design are thus represented by prevention of glass failure and large delections. It is expected, based on the current investigation, that useful design recommendations could be derived for further reinement of this novel concept. Keywords: Laminated glass (LG), shape-memory alloys (SMAs), glass panels, reinforcement techniques, pretension, inite-element modeling, experimental characterization, temperature variations

3.1 Introduction In this chapter, shape-memory alloy (SMA) wires are used as eicient reinforcing system for traditional laminated glass (LG) panels. Due to mainly *Corresponding author: [email protected] Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (87–120) © 2016 Scrivener Publishing LLC

87

88 Advanced Engineering Materials and Modeling architectural and lightening demands, LG panels are typically used as cladding walls and load-carrying elements in façades, enclosures, or roofs in buildings (see, for example, Figure 3.1 and [5–7]). In order to cover large surfaces, these panels are oten characterized by usually high size-to-thickness ratios. he intrinsic characteristic of LG panels is that in them two or more glass sheets – with pure brittle tensile behavior and limited characteristic resistance [8] – are generally bonded together by means of high loadtime and temperature sensitive, thin interlayers with viscoelastic mechanical response (e.g. [9–12]). As a result, major limitations in their design – when subjected to permanent, accidental (e.g. crowd), and dynamic (e.g. wind) pressures – are represented by prevention of glass failure (e.g. local peaks of tensile stresses in the vicinity of the connection systems or at the mid-span cross section) and limitation of maximum delections. he avoidance of failure propagation and redundancy plays also an importat role. In this context, the main objective of the present research activity is to explore the concept of an adaptive embedded reinforcement system for LG panels built up of martensitic SMA wires, highlighting the possible structural beneits and overall efects of the proposed SMA system. In addition to providing a higher stifness and resistance to the traditional LG panels, minimizing the probability of glass failure, the novel hybrid system aims to optimize the glass panels themselves. hese primary efects are

(a) (b)

(c)

(d)

Figure 3.1 Examples of glass applications in façades: (a) [1], (b) [2], (c) [3], and (d) [4].

Toward a Novel SMA-reinforced Laminated Glass Panel 89 achieved by controlling the maximum delections and stresses occurring in the glass panes, when subjected to higher temperatures, through the automatic thermal activation of the embedded SMA wires. Difering from traditional reinforced structural glass applications, where FRP sheets or steel tendons and pre-stressed cables are used to improve the irst-cracking and post-cracked response of glass beams or panels [13–22], the current enhancement technique is characterized by an adaptive reinforcing system that helps to mitigate the degradation of the structural performances in LG assemblies under high temperatures. he eiciency of this novel approach is assessed in this chapter by means of parametric inite-element (FE) investigations [23] carried out on some selected conigurations of practical interest. It is expected, based on the current exploratory study, that useful design recommendations and implementations could be used for further reinement of this novel structural glass concept.

3.2

Glass in Buildings

Structural glass beams and plates ind primary applications in façades and roofs, in the form of stifeners and structural components able to carry on loads. here, the beams act as stifeners and intermediate supports for roof and façade panels, based on mechanical or adhesive joints able to provide a certain structural interaction between them [24]. In the case of glass panels, a wide range of possible geometrical conigurations and restraint solutions can be found in practice, involving the use of aluminum or steel frameworks and cable nets (Figure 3.2a), timber frames (Figure 3.2b and [27–30]), adhesives, or metal point ixings only (Figure 3.2c and d). Compared to structural components composed of traditional construction materials like steel, timber, or concrete, glass elements in buildings are characterized by speciic mechanical and geometrical aspects that should be properly taken into account when assessing their expected response. First, glass is a material characterized by a relatively small modulus of elasticity Eg, compared to steel (e.g. in the order of 70 GPa, see Table 3.1 and Figure 3.3a) and by a typical brittle elastic tensile behavior characterized by a limited strength (Figure 3.3b). Although thermal or chemical prestressing processes can increase the nominal characteristic tensile strength of annealed glass (AN, 45 MPa [8]) up to two times (for heat-strengthened glass, HT) or up to three times (in the case of fully tempered (FT) glass) the reference value, the occurrence of both local on global failure mechanisms due to tensile stresses exceeding the corresponding limit value should be properly prevented.

90 Advanced Engineering Materials and Modeling

(a)

(b)

(c)

(d)

Figure 3.2 Examples of structural glass assemblies: (a) [25], (b) [26], (c) [31], and (d) [32].

Table 3.1 Soda lime silica glass properties [8]. Symbol Density Young’s modulus Poisson’s ratio Coeicient of thermal expansion

Soda lime silica glass

Unit (kg/m )

2500

(MPa)

70,000

3

g

Eg g

(−)

t,g

(K ) –1

0.23 9 × 10–6

On the other hand, the relatively high nominal compressive strength of glass (with a theoretical value in the order of 1000 MPa [8]) makes structural glass components – with the exception of applications characterized by particular boundary conditions – less sensitive to crushing phenomena. Careful consideration should be also given to the appropriate analysis and design of LG cross sections (Figure 3.4), representing the majority of structural glass applications. hese conigurations are in fact characterized by the presence of two (or more) glass layers and one (or more) intermediate

Toward a Novel SMA-reinforced Laminated Glass Panel 91 σ

Steel

σ[MPa] FT

120 Glass

HS

70

AN

45 Concrete ε

0 (a)

0 (b)

1

2

ε [‰]

3

Figure 3.3 Mechanical properties of structural glass. (a) Qualitative comparison between glass tensile behavior versus traditional construction materials like steel and concrete and (b) tensile constitutive law of glass, depending on the adopted pre-stressing technique (with AN = annealed glass; HS = heat strengthened; FT = fully tempered). (b)

(c)

(d)

h

(a)

t

t1

tint

t1

tint

t1

tint

t2

Figure 3.4 Typical cross sections of common use in structural glass applications. Examples referred to the speciic case of beam sections with edge chamfers fully neglected: (a) monolithic or (b–d) LG elements.

foil that able to provide a lexible shear connection only between them. he typical interlayers are in fact composed of Polyvinyl butyral (PVB) [8, 9], SentryGlas (SG) [10], or ethylene-vinyl acetate (EVA) [11] components, in which the shear stifness Gint strictly depends on several conditions [e.g. time loading, temperature (Figure 3.5)]. Also in the case of cross sections composed of multiple glass layers (e.g. Figure 3.4, cases b–d), glass beams and panels are moreover characterized by relatively high slenderness ratios, e.g. small thickness t compared to their overall dimensions. As a result, maximum deformations due to

92 Advanced Engineering Materials and Modeling 100

Gint [N/mm2] Gräf et al. Van Duster et al.

10 T = 10 °C T = 20 °C

1

T = 30 °C

T = 50 °C

t [s/]

0.1 1

10

100

1000

10000

100000

1000000 10000000

Figure 3.5 Mechanical constitutive law of common PVB interlayers for structural glass applications [8, 9] in the form of shear modulus Gint as a function of time, for several reference temperatures T.

ordinary quasi static loads should be properly limited in accordance with current standards for the design of structural glass elements [33–35].

3.2.1

Actual Reinforcement Techniques for Structural Glass Applications

In the past decades, the progress in technology of glass and the so-obtained LG elements represented for long time the “conventional” cross section for glazing applications, since able to provide a certain amount of additional plasticity to typical brittle systems, as well as to guarantee them a certain amount of post-cracked deformations and energy absorption under impact. he continuous request of a combination of multiple aspects, like structural optimization and performance enhancement, weight minimization, post-failure redundancy, and enhanced ductility, together with the ongoing production technology and materials science improvements, leads progressively toward the so-called hybrid glass components. Based on the actual design standards and recommendations for glass elements, a certain level of redundancy and post-failure resistance should be generally guaranteed. For this purpose, several studies have been carried out in order to improve the typical structural performance of traditional glass assemblies (in the pre-cracked and post-breakage phases, or in the latter one only)

Toward a Novel SMA-reinforced Laminated Glass Panel 93 by the application to LG elements of additional external reinforcements [36]. For an exhaustive review of the current developments and available techniques for the reinforcement of structural glass elements, the reader is encouraged to refer to [13, 14].

3.3

Structural Engineering Applications of Shape-Memory Alloys (SMAs)

SMAs are a class of metal materials that exhibit two outstanding properties, namely the superelastic and the shape-memory efects (SMEs). Superelasticity (SE) is associated to the capacity of these materials to recover from high imposed strains (up to 8%) with no residual deformations, while developing hysteresis. he SME allows the material to return to its original shape, upon permanent deformation, through a heat cycle. hese properties are due to the ability of SMAs to develop martensitic transformations, which are solid state, difusion-less transformations between high-energy and low-energy crystallographic phases, e.g. (i) the austenitic and (ii) the martensitic phases, respectively. Taking advantage of these two properties, several applications of SMAs have been proposed over the past years. In the ields of robotic, automotive, machinery, and biomedical solutions, for example, SMAs usually appear in the form of actuator wires and plates, where they replace conventional pneumatic or hydraulic systems and devices [33, 38]. A considerable amount of progress has been also recently achieved in the development of applications for civil engineering structures that take advantage of SMAs properties, and a wide range of solutions are reported in the literature, e.g. in the form of seismic links able to enhance the resistance of cultural heritage structures [32], bracing systems [39–42], base isolation systems [43–45], bridge hinge restraining systems [46–49], and structural connections [50, 51]. Several adaptive-passive vibration control devices using SMA actuators have been also investigated. he control of the shape and natural frequencies of composite beams, reinforced with NiTi SMAs, has been reported by Baz and co-workers [52, 53]. here, the authors showed that the vibration modes of the activated NiTi-reinforced composite beams can be readily shited to higher frequency bands, compared to those of the inactivated or un-reinforced beams. Williams et al. [54, 55] assessed the control of a steel beam by using a SMA adaptive tuned vibration absorber, in which the elastic modulus of the SMA element was controlled by heating. Besides the frequency shit capabilities related to the temperature-induced phase transformation associated with the SME,

94 Advanced Engineering Materials and Modeling vibration absorbers based on these materials can also be designed to take advantage of the hysteretic behavior due to SE. Related numerical and experimental studies, exploiting both the SME and the SE efects, can be found for example in [56–58]. A large-scale civil engineering application is discussed by Li et al. [59], where the behavior of two smart concrete beams with embedded SMA bundles has been investigated. he mentioned beams, integrated into a smart bridge in a freeway, were manufactured and properly analyzed, concluding that SMAs can be successfully used in civil engineering structures. NiTi actuators, based on the SME, present outstanding performance also in actuation stress and reliability [60], with a high corrosion resistance [61].

3.4 he Novel SMA-Reinforced Laminated Glass Panel Concept 3.4.1

Design Concept

he novel design concept investigated in this chapter consists in a reinforcement system composed of martensitic NiTi SMA wires. hese wires are included within the interlayer thickness, e.g. in the form of a reinforcing SMA net fully embedded in the typical cross section of a traditional LG panel (Figure 3.6). In order to provide a fully structural interaction between the SMA wires and the LG panels, each wire is supposed rigidly connected at its ends by means of small metal setting blocks. Glass

Shell (S4R) Solid (C3D83) Beam (B31)

Interlayer SMA wire

B

B

L

L iSMA

iSMA SMA

SMA

(a)

t tint

(b)

t tint

Figure 3.6 Overview of the (a) SMA-reinforced laminated panel concept and (b) the corresponding FE modeling approach

Toward a Novel SMA-reinforced Laminated Glass Panel 95 he key aspect of this possible reinforcing technique is given by the interaction between a traditional LG panel and a net of SMA wires able to provide enhanced stifness and resistance, especially in higher-temperature loading scenarios, when the laminated panel shits from the monolithic toward the layered behavior (see Figure 3.7). As far as the temperature T increases, the shear stifness Gint ofered by the interlayer foils progressively tends toward zero (e.g. Figure 3.5) and to the so-called layered limit for the LG element characterized by an almost lack of shear bonding between the glass panes. In this context, the proposed SMA wires exploit at the best their potentiality. While at ordinary temperatures these SMA wires provide a limited stifness contribution only (e.g. in the form of inactive embedded cables), any increase in temperature allows the activation of the SMA wires, with a consequent pretension efect in the glass panes. It is thus expected, once the wires are activated, a positive efect consisting in both enhanced stifness (e.g. limited deformations for the LG panel) and resistance (e.g. limited tensile stresses in glass, due to pretension of the SMA wires). In doing so, an implicit advantage is that SMA wires can take the form of cables with very limited cross-sectional dimensions, e.g. with nominal diameters SMA in the order of ≈1 mm or multiples. No external wires nor additional bracing systems are thus required compared to other reinforcing and pre-stressing techniques. Since SMA wires can be structurally efective although with small cross-sectional dimensions, as discussed in the following sections, it is expected that the proposed design Monolithic limit Glass t tint

Interlayer SMA wire

Layered limit t tint

Figure 3.7 Qualitative efect of temperature variations in terms of distribution of stresses in the glass panels and stresses in glass due to the activation of the SMA wires (transversal cross section).

96 Advanced Engineering Materials and Modeling concept could result extremely advantageous especially in “ultra-thin” and “light-weight” glass applications, as the current architectural design trends are emphasizing [62]. On the other hand, a possible limitation of such design concept is that the SMA wires are embedded in the traditional LG section. As a result, the diameter SMA of each wire has to be smaller than the thickness of the interlayer, e.g. tint = 0.76, 1.52, or 4.52 mm, as in the case of glass panels designed to resist impacts and exceptional loads. In this latter case, as a result, the optimal solution would necessarily result from a combination of structural requests as well as aesthetic requirements (e.g. minimization of SMA wires and maximization of transparency).

3.4.2 Exploratory Finite-Element (FE) Numerical Study As a preliminary investigative study, the potentiality of SMA wires embedded in traditional LG panels was investigated by means of FE numerical models implemented in ABAQUS/Standard [23]. Exploratory FE simulations were carried out on several geometrical conigurations of practical interest for structural glass applications in buildings, as partly discussed in the following sections, so that rational assessment of their potentiality could be obtained. It is expected, based on the obtained outcomes, that the current results could be successively extended and validated by means of full-scale experiments, in the prospective of prototyping the so-called “SMA-reinforced LG panels”, as well as in view of a possible implementation of design rules and recommendations of practical use.

3.4.2.1 General FE Model Assembly Approach and Solving Method Nonlinear static incremental simulations (inls) were carried out on several conigurations. Various inluencing parameters (e.g. glass panel dimensions, boundary conditions, loads, LG cross-sectional properties, as well as geometrical properties of the embedded SMA wires) were taken into account. For clarity of presentation, two main case studies and applications of SMA embedded reinforcements are discussed in detail in this chapter, see Table 3.2. For both the M1 and M2 cases, as well as for further geometrical conigurations not included in this contribution, the parametric study was carried out by means of FE models with speciic features, but typically assembled with almost an identical method. he typical FE model consisted in fact of 3D solid elements for the interlayer ilms, 2D monolithic shell elements for the glass panels, and

Toward a Novel SMA-reinforced Laminated Glass Panel 97 Table 3.2 Reference geometrical conigurations for the SMA-reinforced LG panels object of the parametric FE investigations. Case study Cross section

Boundaries

Loads

Typical application

M1

Doubly Long edges Roof q = 1 kN/m2 symmetric, simply (wind pressure, three-layer LG supported; 3 s) section short edges fully (e.g. Figure 3.4b) unrestrained

M2

Doubly Point-supported q = 1 kN/m2 Façade panel symmetric, glass panel (wind pressure, three-layer LG 3 s) section (e.g. Figure 3.4b)

1D beam elements for the embedded SMA wires (Figure 3.6b). A key role was assigned to several types of FE interactions in order to reproduce the expected mechanical performance of the examined SMA-reinforced LG panels. A rigid “tie” connection was in fact irst introduced at both the interfaces between the interlayer and glass elements. In this manner, all the possible relative displacements and rotations among the corresponding mesh nodes belonging to the middle solid elements and to the adjacent external shell elements were fully neglected. At the same time, each FE analysis was carried out in the form of three separate steps so that the efects of temperature variations in the SMA wires (e.g. initial pre-stress) as well as the transmission of these efects to the adjacent LG panels, could be properly take into account. hese steps were detected and deined respectively as follows: t
Phase I: application of the temperature variation in the SMA wires. he “initial condition” option available in the ABAQUS/Standard library [23] for predeined conditions was used. At this stage, each SMA wire was assumed to be simply supported at the ends and rigidly connected to the adjacent LG elements, via localized “tie” fully rigid connectors. he initial principal stress in them due to temperature variations was calculated, for each simulation, based on the available experimental stress–temperature relationship (see Figure 3.10), by taking into account a speciic operating temperature T.

98 Advanced Engineering Materials and Modeling t
Phase II: application of the structural interaction between the SMA wires and the adjacent LG panel, via an “embedded” constraint able to provide a full coupling between the SMA wires and the surrounding PVB ilm. At the beginning of Phase II, the end restraints of each SMA wire were fully released, while the desired boundary condition (e.g. continuous simply supports, point supports) was assigned to the examined LG panels. t
Phase III: loading phase. Once reproduced the desired efects due to temperature variations T, each SMA-reinforced LG panel was subjected to a linear increasing, uniform pressure q acting on the surface of glass.

3.4.2.2 Mechanical Characterization of Materials he structural performance of SMA-reinforced glass panels was investigated with careful consideration for two major aspects: (i) the pre-cracked response and (ii) the irst-glass cracking coniguration. he (iii) post-cracked response was neglected, at this current stage of investigation. It is expected, however, that further reined assessment of the SMA-reinforcement technique for LG panels could emphasize additional advantages deriving from this novel design concept. While the performance of structural glass elements in general is strictly related to the limited tensile strength of glass, it is expected in fact that the embedded SMA wires could provide structural eiciency both in the pre-cracked stage (e.g. increased elastic stifness and resistance of the composite cross section) and in the post-cracked phase (e.g. additional ductility and residual resistance, compared to the traditional LG section, due to the presence of the SMA net). For this purpose, the mechanical properties of the basic materials were properly calibrated. 3.4.2.2.1 Glass and Interlayer Glass was described in ABAQUS/Standard in the form of a linear elastic material, with Eg = 70 GPa the nominal elastic modulus and g = 0.23 its Poisson’s ratio [8]. he post-cracked mechanical behavior of glass was then described by means of the “concrete-damaged plasticity” material model available in the ABAQUS/Standard library [23]. A crucial phase was represented by the reined calibration of its key input parameters so that the expected tensile brittle behavior of glass could be properly reproduced in accordance with earlier research studies (e.g. [19]). At the same time, a compressive ultimate characteristic strength equal to 1000 MPa was conventionally taken into account, e.g. coinciding with the nominal theoretical compressive resistance of glass (Section 3.2).

Toward a Novel SMA-reinforced Laminated Glass Panel 99 he thermoplastic PVB ilm bonding together the glass panes was described in the form of an equivalent, isotropic, elasto-plastic material with int = 0.49 Poisson’s ratio. he yielding stress of PVB was conventionally set equal to y,int = 11 MPa, in accordance with the FE approach validate in [64]. At the same time, hardening efects in its plastic response were fully neglected, with int,u = 400% the elongation at failure. Concerning the elastic modulus Eint, its value was properly calculated by taking into account a speciic time loading and temperature condition based on master curves available in the literature (see Figure 3.5). 3.4.2.2.2 SMA Wires Regarding the SMA wires, their mechanical calibration was based on experimental measurements. SMA NiTi actuator wires, straight, oxidefree, martensitic wire samples, with a 0.51 mm diameter circular cross section were used, obtained from Dynalloy, Inc. In order to characterize the phase transformation temperatures of the NiTi wire specimens, a diferential scanning calorimetry (DSC) test was performed, using a SETARAMDSC92 thermal analyser (Figure 3.8). he temperature program comprised a thermal cycle where the sample, tested as-received, was heated up to 130  °C, held at this temperature for 6 min, and then cooled to –20 °C, with heating and cooling rates of 7.5 °C/min. Prior to the DSC experiment, the sample was submitted to a chemical etching (10 vol.% HF + 45 vol.% HNO3 + 45 vol.% H2O) in order to remove the oxide and the layer formed by the cutting operation. According to the obtained results, the transformation temperatures associated to the start and end of transformation between the martensite and austenite phases, during heating, were found to be around 40 °C and

0.2 0.15

DSC[mW/mg]

0.1 0.05 0 –0.05 –0.1 –0.15 –0.2 –0.25 –20

T[°C] 0

20

40

60

Figure 3.8 Diferential scanning calorimetry (DSC) test.

80

100

120

100 Advanced Engineering Materials and Modeling 80 °C. his range of transition temperatures seem to be particularly suited for the proposed adaptive approach, since the shear stifness degradation of PVB ilms due to temperature increase occurs in almost the same range (e.g. Figure 3.5). In order to characterize the full relation between force and temperature in the NiTi wires, during a thermal cycle, a NiTi wire specimen with a total length of 715 mm was placed within a rigid frame, which was equipped with a miniature load cell (Figure 3.9). he load cell enabled the force readings in the NiTi wire. Heating of the NiTi wire actuator was done via Joule heating, which is resistively heating the actuator using electric current. For this efect, a Sorensen programmable DC power supply (PPS), model XHR 40-25, was used. he temperature of the NiTi wire was monitored with the same T-type thermocouple (Copper-Constantan), with a temperature reading range from –40 °C to 100 °C, connected to a NI SCXI-1112 8 Channel hermocouple Ampliier. he general platform for the data acquisition and control was a NI PXI-1052. During the experiment, the temperature program comprised a thermal cycle where the NiTi wire was heated up to 110 °C and then cooled to 20 °C, with heating and cooling rates of 20 °C/min. According to the obtained stress–temperature relationship (see Figure 3.10), it is possible to notice that within the expected temperature range for the proposed adaptive LG panel (up to T ≈ 50–60 °C, for façade and roof panels exposed to sunshine), the relation between

Clamp Frame Sma wire Thermocouple PPS Load cell

Electrical wires

Figure 3.9 Overview of the experimental prototype.

Toward a Novel SMA-reinforced Laminated Glass Panel 101 700 600

Stress (MPa)

500 400 300 200 100 0 –100 20

T[°C] 30

40

50

60

70

80

90

100

110

Figure 3.10 Experimental stress–temperature relationship for the NiTi wire.

temperature and stress in the NiTi wire is almost linear, with measured stresses up to about 120 MPa. As expected, maximum stress higher than 500 MPa could also be obtained for the same NiTi wire, but for very high temperatures only, typically requiring an activation system (T ≈ 100 °C). By limiting the stress levels of the SMA wires to ≈120 MPa, moreover, it is rationally expected that the SMA net could guarantee an exceptional performance of the system, regarding stability with cycling. For FE investigation purposes, the SMA wires were described by means of an equivalent, elasto-plastic material with ESMA Young’s modulus and = 0.3 Poisson’s ratio. Based on test data, ESMA was assumed equal to SMA 40 GPa for temperatures up to 50 °C and equal to 50 GPa for T = 60 °C, respectively. Although the FE study was carried out in a range of temperatures within 30 °C and 60 °C, the elasto-plastic constitutive law was used to ensure the attainment of maximum principal stresses in the SMA wires exceeding the value of ≈500 MPa (Figure 3.10). In doing so, for each reference temperature T, the initial pre-stress in the SMA wires was also derived from the experimental data, based on the stress–temperature relationship of Figure 3.10.

3.5 Discussion of Parametric FE Results 3.5.1

Roof Glass Panel (M1)

he typical “M1” LG panel was considered simply supported along its longest edges, with short edges fully unrestrained, being this loading and boundary coniguration well representative of a glass plate for a roof.

102 Advanced Engineering Materials and Modeling he irst analyzed, reference LG assembly consisted in a 1000 mm wide × 3000 mm high panel composed of two 5-mm-thick glass sheets and a middle PVB foil, 4.52 mm in thickness. he unreinforced panel (“M1-0”, in the following) was subjected to a uniform pressure q = 1 kN/m2 (e.g. representative of an ordinary wind load), with a conventional load duration of tL = 3 s. In doing so, diferent temperature scenarios were also taken into account (with T = 30 °C, 40 °C, 50 °C, and 60 °C, respectively). Several geometrical properties of SMA wires were successively taken into account for the so-assembled “M1” panel. In Sections 3.5.1.1 and 3.5.1.2, some selected SMA-reinforced conigurations are identiied as “M1-i” cases.

3.5.1.1 Short-term Loads and Temperature Variations A qualitative assessment of SMA wires efects on the pre-cracked response was irst performed in terms of maximum delections but also distribution and entity of maximum principal stresses in the full assembly components (e.g. the glass panels, the middle interlayer, and the SMA wires). Aim of this exploratory FE investigation phase was in fact a preliminary assessment of the structural eiciency of the SMA net compared to the traditional M1-0 LG panel, as well as a irst optimization in terms of SMA wires size (φSMA) and interspacing (iSMA). Table 3.3 provides an overview of the irst set of all the examined conigurations, as derived from the M1-0 reference case. As shown in the following sections, rather “conventional” temperature variations in the SMA-reinforced LG panels can have a typical double efect. he irst of them is strictly related to the mechanical performance of the middle PVB ilm, being characterized by a strongly temperature-dependent Table 3.3 Reference geometrical conigurations for the M1 glass panel. SMA net

Panel geometry

Loading condition

φSMA

iSMA

B

L

T

q

(mm)

(mm)

(mm)

(mm)

(°C)

(kN/m2)

1000

3000

30, 40, 50, 60

1 (3 s)

No SMA

M1-0

–

–

SMA wires

M1-1

4

650

M1-2

450

M1-3

250

Toward a Novel SMA-reinforced Laminated Glass Panel 103 shear modulus Gint. As far as the reference temperature T increases, speciically, a drastic decrease of Gint is expected (e.g. Figure 3.5), with Gint progressively tending to zero and leading to almost a total lack of any shear connection between the external glass faces. On the other hand, an increase in the reference temperature T is also associated to a certain pretension level in the SMA wires (e.g. Figure 3.10). Due to the presence of rigid restraints between each wire end and the adjacent glass plates, in this sense, a transmission of compressive stresses from the SMA wires to the glass panes is expected as a consequence of moderate temperature variations. Since the SMA wires are fully embedded in the thermoplastic interlayer, however, a combination of both the efects discussed above is expected to be the main inluencing parameter for the typical response of the tested panels. In this sense, reined FE methods certainly represent a rational and practical tool for accurate studies and optimization processes. In Figure 3.11, 3.12, and Table 3.4, in particular, the main FE results are proposed for the investigated LG panel in the form of maximum stresses and deformations in each assembly component. In Table 3.4, the Rw and Rσ percentage ratios are calculated as follows:

Ri

100

( pi

p0 ) p0

(3.1)

where “p” denotes the given response parameter calculated for the “i” coniguration with SMA wires, while the suix “0” is associated to the unreinforced LG panel. In the same table, wq denotes the maximum delection occurring at the panel center, while q and int,q represent the maximum tensile stresses in glass and PVB respectively (maximum envelopes). In terms of overall lexibility under ordinary, short-term wind pressures, the SMA wires highlighted for all the examined conigurations an almost linear correlation between the interspacing of wires (e.g. iSMA) and the corresponding decrease of maximum delections wq (Figure 3.11a). A non-linear dependency was found, conversely, between the amount of SMA wires and the maximum tensile stresses q in glass, by varying the reference temperature T. his efect can be clearly seen from Table 3.4 as well as from Figure 3.11b, where higher temperatures are associated (in presence of SMA wires) to limited tensile stresses in glass only. A similar result can be rationally justiied by the positive efect deriving from the activation of the SMA net, e.g. by a higher Young’s modulus for them (ESMA = 50 GPa, at T = 60 °C), a higher initial pre-stress in the SMA wires

104 Advanced Engineering Materials and Modeling 3

2

1

0

wq (mm)

6 5 4 3 30°

40°

50°

60°

2 250

450

650

3

2

1

850 2000 iSMA (mm)

(a)

2500

3000

0

14

σq (MPa)

30°

40°

50°

60°

12 10 8 6 250

450

650

850 2000 iSMA (mm)

(b)

2500

3000

16 iSMA (mm) 250

σ0 (MPa)

12

450

650

8 4 0 30

(c)

40

50

60

T (°C)

Figure 3.11 Efect of temperature variations on the structural response of a SMAreinforced glass panel under wind pressures (ABAQUS/Standard) in terms of (a) maximum deformations at the panel center wq, (b) maximum tensile stresses in glass , and (c) maximum compressive stress 0 in glass, due to temperature variation only. q

Toward a Novel SMA-reinforced Laminated Glass Panel 105

S, Min. in-plane principal envelope (min) (Avg: 75%) –6.138e+05 –2.139e+06 –3.665e+06 –5.191e+06 –6.717e+06 –8.242e+06 –9.768e+06 –1.129e+07 –1.282e+07 –1.434e+07 –1.587e+07 –1.740e+07 –1.892e+07

Figure 3.12 Typical distribution of compressive stresses 0 in glass (in [Pa]; ABAQUS/ Standard, minimum envelope), due to the initial pre-stress of the SMA wires, as a consequence of temperature variations (iSMA = 250 mm, SMA = 4 mm).

Table 3.4 Comparative FE results for the M1 subjected to short-term wind loads, under the efects of temperature variations. Glass

30 °C

40 °C

M1

M1

Interlayer

wq

Rw

(mm)

(%)

(MPa)

(%)

(MPa)

0

3.30

–

7.41

–

0.036

1

2.81

–15.0

7.09

–4.3

0.056

2

2.64

–19.8

6.89

–7.1

0.042

3

2.57

–22.0

6.81

–8.1

0.041

0

3.87

–

8.23

–

0.031

1

3.15

–18.2

7.68

–6.7

0.058

2

2.88

–25.2

7.27

–11.7

0.039

3

2.86

–25.9

7.26

–16.7

0.037

q

R

int,q

(Continued)

106 Advanced Engineering Materials and Modeling Table 3.4 Cont Glass

50 °C

60 °C

M1

M1

Interlayer

wq

Rw

(mm)

(%)

(MPa)

(%)

(MPa)

0

4.56

–

10.2

–

0.024

1

3.57

–21.5

8.50

–16.6

0.061

2

3.29

–27.6

7.99

–21.6

0.038

3

3.17

–30.2

7.81

–23.4

0.030

0

5.85

–

13.73

–

0.011

1

4.25

–27.4

9.71

–29.3

0.035

2

3.84

–34.4

8.63

–35.4

0.031

3

3.66

–37.5

8.49

–38.2

0.025

q

R

int,q

(e.g. Figure 3.10) and hence a signiicant amount of initial compressive stresses 0 in glass (see Figure 3.11c). Figure 3.12 represents the typical distribution of these initial compressive stresses 0 (minimum envelope) deriving from the pretension of the SMA wires, as a consequence of the temperature increase. As shown, as expected, the maximum peaks of stresses were generally found to be located in the vicinity of the wires’ ends. Toward the center of the panel, conversely, a partial transmission only of compressive stresses was found for all the examined conigurations, due to the presence of a shear lexible PVB bonding foil unable to bond together the glass plates. Compared to the M1 unreinforced panel, an almost identical deformed shape was generally obtained for all the examined conigurations (see Figure 3.13a). Conversely, an interesting efect was found in terms of distribution of maximum tensile stresses in glass. Some examples are collected in Figure 3.13b–f.

3.5.1.2 First-cracking Coniguration A second aspect of primary importance for the overall structural response of the examined panels is represented by the so-called “irst-cracking coniguration”, namely associated to the attainment on glass of maximum tensile stresses max at least equal to the corresponding characteristic tensile resistance Rk.

Toward a Novel SMA-reinforced Laminated Glass Panel 107

U. U3

S. Max. principal envelope (max) (avg: 75%)

+5.848e-03 +5.356e-03 +4.864e-03 +4.373e-03 +3.881e-03 +3.389e-03 +2.389e-03 +2.406e-03 +1.914e-03 +1.423e-03 +9.310e-04 +4.394e-04 –5.228e-05

(a)

+1.114e+07 +1.021e+07 +9.285e+06 +8.357e+06 +7.428e+06 +6.500e+06 +5.571e+06 +4.643e+06 +3.714e+06 +2.786e+06 +1.857e+06 +9.285e+05 +0.000e+00

M1-0

S. Max. in-plane principal envelope (max) (avg: 75%)

+6.812e+06 +6.249e+06 +5.686e+06 +5.123e+06 +4.560e+06 +3.997e+06 +3.434e+06 +2.871e+06 +2.308e+06 +1.745e+06 +1.182e+06 +6.185e+05 +5.547e+04

M1-2

S. Max. in-plane principal envelope (max) (avg: 75%)

(d)

M1-3

S. Max. in-plane principal envelope (max) (avg: 75%)

+7.728e+06 +7.025e+06 +6.322e+06 +5.618e+06 +4.915e+06 +4.212e+06 +3.508e+06 +2.805e+06 +2.102e+06 +1.398e+06 +6.952e+05 –8.162e+03 –7.115e+05

(e)

M1-0

S. Max. in-plane principal envelope (max) (avg: 75%)

+6.889e+06 +6.329e+06 +5.768e+06 +5.208e+06 +4.647e+06 +4.087e+06 +3.526e+06 +2.966e+06 +2.405e+06 +1.845e+06 +1.285e+06 +7.240e+05 +1.636e+05

(c)

(b)

+6.240e+06 +5.672e+06 +5.103e+06 +4.535e+06 +3.966e+06 +3.398e+06 +2.829e+06 +2.261e+06 +1.692e+06 +1.124e+06 +5.554e+05 –1.315e+04 –5.817e+05

M1-2

(f)

M1-3

Figure 3.13 Efect of temperature variations in the structural response of a SMAreinforced glass panel (ABAQUS/Standard). (a and b) maximum delection and tensile stresses in glass (T = 30 °C), (c and d) maximum tensile stresses in glass for the SMA conigurations 2 and 3 (T = 30 °C), and (e and f) maximum tensile stresses in glass, for the SMA conigurations 2 and 3 (T = 60 °C). Values expressed in [m] and [Pa].

108 Advanced Engineering Materials and Modeling For this purpose, the same FE models described in Section 3.6.1.1 were further analyzed and subjected to a linearly increasing, distributed pressure q. In each simulation, the maximum envelope of tensile stresses in glass was continuously monitored so that the analyses could be stopped at the irst occurrence of maximum tensile stresses max exceeding Rk. In doing so, the examined LG panel was assumed composed of HS glass, with Rk = 70 MPa [8]. he corresponding pressure q = q1 was separately collected for each M1 coniguration, together with the main output parameters of practical interest. Some of these results are proposed in Table 3.5, where F1 and K1 represent, respectively, the reaction force at the panel support and the corresponding ratio F1 versus maximum delection w1 at the panel center, at the irst cracking of glass. Also in this case, the typical distribution of maximum principal stresses in glass and in the PVB ilm at the attainment of irst cracking resulted partly afected by the geometrical features of the SMA net, e.g. in agreement with Figure 3.14. Again, the maximum structural beneits deriving from the embedded SMA wires were emphasized at high temperatures. Figure 3.15 presents a comparison of stifness increment for the M1-0 and the i-reinforced panels, in the form of K1,i/K1,0 ratio versus the wires interspacing iSMA, as obtained at T = 30 °C and T = 60 °C, respectively. Table 3.5 Comparative FE results for the M1 panel at the irst-cracking coniguration. Reaction force at the support

30 °C

60 °C

M1

M1

Panel stifness

F1

RF

K1

Rk

(N)

(%)

(N/mm)

(%)

0

14719

–

457.7

–

1

14822

0.7

537.2

17.4

2

15356

4.3

568.1

24.4

3

15446

4.9

584.2

27.7

0

7931

–

258.8

–

1

9497

19.7

352.3

36.1

2

10922

37.7

391.6

51.3

3

11972

50.9

411.0

58.8

Toward a Novel SMA-reinforced Laminated Glass Panel 109

S. Max. principal envelope (max) (avg: 75%) +7.048e+07 +6.464e+07 +5.880e+07 +5.296e+07 +4.711e+07 +4.127e+07 +3.543e+07 +2.959e+07 +2.375e+07 +1.790e+07 +1.206e+07 +6.221e+06 +3.796e+05

S. Max. principal (avg: 75%) +2.607e+05 +2.327e+05 +2.047e+05 +1.768e+05 +1.488e+05 +1.208e+05 +9.282e+04 +6.484e+04 +3.686e+04 +8.885e+03 –1.909e+04 –4.707e+04 –7.505e+04

(b)

(a)

Figure 3.14 Distribution of maximum tensile stresses (a) in glass and (b) in the PVB ilm (maximum envelope, in [Pa]) for the M1-3 panel, at irst cracking (ABAQUS/Standard). 3

2

1

0

1.6

K1,i / K1,0 (–)

30°

60°

1.4

1.2

1.0 250

450

650

850

2000 iSMA (mm)

2500

3000

Figure 3.15 Stifness variations for the M1 panels at the attainment of irst cracking in glass at T = 30 °C and 60 °C, respectively (ABAQUS/Standard).

3.5.2 Point-supported Façade Panel (M2) he M2 case study represents a point-supported façade panel composed of two t = 10-mm-thick HS glass layers ( Rk = 70 MPa [8]) and a middle tint = 4.52-mm-thick PVB ilm, with global nominal dimensions B = 2500 mm and L = 1500 mm, in accordance with Figure 3.16, and d = 104 mm. he panel is subjected to the action of self-weight in the vertical direction as well as to major live loads taking the form of wind pressures q = 1 kN/m2 (with tL = 3 s the conventional load duration) acting in the direction perpendicular to the surface of glass (Figure 3.16). Mechanical point supports

110 Advanced Engineering Materials and Modeling d

t

Hole

HS

d

PVB L

B

tint Ø 59

M 14 x 1,5

(a)

109 67 64

(b)

Figure 3.16 M2 case study. (a) Reference geometrical properties for the conventional unreinforced LG panel and (b) example of a rotule connector (axonometry and transversal cross section, nominal dimensions in [mm]).

Table 3.6 Reference geometrical conigurations for the M2 glass panel.

No SMA SMA wires

M1-0 M1-1 M1-2 M1-3 M1-4

SMA net iSMA SMA (mm) (mm) – – 4 450 350 250 150

Panel geometry B L (mm) (mm) 2500 1500

Loading condition T q (°C) (kN/m2) 30, 40, 1 50, 60 (3 s)

are realized in the form of articulated steel rotules, e.g. point ixings able to provide fully rigid translational restraints but enabling, at the same time, possible rotations. Difering from the M1 case study, the point-supported panel is characterized by maximum stresses in the vicinity of the point restraints and at mid-span, as well as by high deformability, due to the absence of continuous supports along the edges. Table 3.6 summarizes the main input data for some of the examined M2-reinforced conigurations.

Toward a Novel SMA-reinforced Laminated Glass Panel 111

3.5.2.1 Short-term Loads and Temperature Variations he irst parametric study was carried out on several SMA geometries (see Table 3.6), by progressively increasing the reference temperature T. he major results are collected in Table 3.7, where the “R” percentage ratios are calculated in accordance with Eq. (3.1).

Table 3.7 Comparative FE results for the M2 panel subjected short-term wind loads under the efects of temperature variations. Glass

30°

40°

50°

60°

Interlayer

wq,M (mm)

Rw,M (%)

wq,E (mm)

Rw,E (%)

0

5.93

–

0.661

1

5.87

–1.01

2

5.86

3

(MPa)

R (%)

(MPa)

–

8.08

–

0.009

0.605

–8.47

8.13

0.61

0.010

–1.18

0.598

–9.53

8.14

0.74

0.011

5.84

–1.52

0.587

–11.20

8.15

0.99

0.011

4

5.82

–1.85

0.586

–11.37

8.15

0.99

0.012

0

6.06

–

0.657

–

8.18

–

0.008

1

6.01

–0.83

0.606

–7.76

8.22

0.48

0.009

2

5.99

–1.16

0.603

–8.21

8.16

–0.25

0.009

3

5.98

–1.29

0.600

–8.68

8.08

–1.22

0.024

4

5.96

–1.65

0.595

–9.44

7.95

–2.81

0.021

0

6.19

–

0.652

–

8.28

–

0.005

1

6.15

–0.64

0.615

–5.67

8.17

–1.32

0.014

2

6.13

–0.97

0.606

–7.05

8.09

–2.29

0.016

3

6.12

–1.13

0.604

–7.36

8.06

–2.66

0.023

4

6.09

–1.62

0.594

–8.90

7.81

–5.68

0.023

0

6.37

–

0.644

–

8.42

–

0.002

1

6.32

–0.78

0.609

–5.43

7.87

–6.53

0.028

2

6.31

–0.94

0.608

–5.59

7.57

–10.09

0.028

3

6.29

–1.32

0.607

–5.75

7.47

–11.28

0.028

4

6.26

–1.73

0.607

–5.75

6.57

–21.97

0.031

q

int,q

112 Advanced Engineering Materials and Modeling

wq,M wq,E

Glass 1 (g1)

Y Z

X

Glass 2 (g2)

Figure 3.17 Qualitative distribution of out-of-plane delections in the M2 panel due to the applied wind pressure (vectorial representation, ABAQUS/Standard).

In accordance with Figure 3.17, the maximum delections due to the applied wind pressure were monitored both at the panel center (wq,M) and along the unrestrained edges (wq,E) due to the presence of point supports only. As shown in Table 3.7 and Figure 3.18, the presence of SMA wires can strongly improve the overall response of the examined panel at high temperatures. Compared to the M1 case, however, the speciic geometry and boundary conditions of the M2 panel manifests higher beneits in terms of stress control in the glass panels, rather than decrease of maximum delections. An interesting modiication in terms of distribution of maximum tensile stresses in glass as well as in the PVB foils was in fact generally noticed, due to the interaction occurring between the traditional laminated panel and the embedded wires, once the SMA net is activated. his efect can be clearly seen from some selected conigurations proposed in Figure 3.18, where the position of SMA wires within the interlayer ilm can be clearly distinguished, as well as from Figures 3.19–3.21. As a primary efect of pre-stressing of SMA wires, a partial introduction of compressive stresses in the glass panes was found, as expected. As in the case of the M1 panel, however, the presence of a shear lexible middle PVB foil typically resulted in a moderate increase of compressive stresses in the vicinity of the wires’ ends only. Some examples of compressive stresses

Toward a Novel SMA-reinforced Laminated Glass Panel 113 M2-3 (iSMA= 250mm. φSMA= 4mm)

M2-0 (unreinforced panel) U. U3

+6.614e–04 +1.120e–04 –4.374e–04 –9.868e–04 –1.536e–03 –2.086e–03 –2.635e–03 –3.184e–03 –3.734e–03 –4.283e–03 –4.833e–03 –5.382e–03 –5.932e–03

(a) S. Max. in-plane principal envelope (max) (avg: 75%) +5.413e+06 +4.929e+06 +4.445e+06 +3.962e+06 +3.478e+06 +2.994e+06 +2.510e+06 +2.027e+06 +1.543e+06 +1.059e+06 +5.756e+05 +9.194e+04 –3.918e+05

U. U3

+5.969e–04 +6.016e–05 –4.766e–04 –1.013e–03 –1.550e–03 –2.087e–03 –2.624e–03 –3.160e–03 –3.697e–03 –4.234e–03 –4.771e–03 –5.308e–03 –5.844e–03

(b) S. Max. in-plane principal envelope (max) (avg: 75%) +4.718e+06 +4.297e+06 +3.875e+06 +3.453e+06 +3.032e+06 +2.610e+06 +2.819e+06 +1.767e+06 +1.345e+06 +9.237e+05 +5.021e+05 +8.049e+04 –3.411e+05

(c)

(d)

S. Max. in-plane principal envelope (max) (avg: 75%) +8.071e+06 +7.506e+06 +6.941e+06 +6.375e+06 +5.810e+06 +5.244e+06 +4.679e+06 +4.113e+06 +3.548e+06 +2.983e+06 +2.417e+06 +1.852e+06 +1.286e+06

S. Max. in-plane principal envelope (max) (avg: 75%) +8.151e+06 +7.573e+06 +6.996e+06 +6.419e+06 +5.842e+06 +5.246e+06 +4.687e+06 +4.110e+06 +3.533e+06 +2.955e+06 +2.378e+06 +1.801e+06 +1.224e+06

(e)

(f)

S. Max. principal (avg: 75%) +9.495e+03 +4.110e+03 +1.275e+03 –6.660e+03 –1.204e+04 –1.743e+04 –2.281e+04 –2.820e+04 –3.358e+04 –3.897e+04 –4.435e+04 –4.974e+04 –5.512e+04

(g)

S. Max. principal (avg: 75%) +1.165e+04 +6.064e+03 +4.806e+02 –5.102e+03 –1.069e+04 –1.627e+04 –2.185e+04 –2.743e+04 –3.302e+04 –3.386e+04 –4.418e+04 –4.977e+04 –5.535e+04

(h)

Figure 3.18 Comparison of (a and b) maximum delections, (c and d) maximum principal stresses in glass (g1), (e and f) maximum principal stresses in glass (g2), and (g and h) maximum stresses in the PVB foil (ABAQUS/Standard), under the action of wind (q = 1 kN/m2, tL = 3 s, T = 30 °C). Values expressed in [m] and in [Pa].

attained in the glass panes as a consequence of temperature variations and SMA wires activation is shown in Figure 3.20. Almost negligible efects were noticed in terms of initial compressive stresses in glass due to the SMA wires activations, by varying the interspacing iSMA of the wires themselves (Figure 3.21d). A general beneicial contribution – especially in terms of overall maximum stresses in glass – was in any case provided by the activation of the SMA net (Figure 3.21). In terms of PVB, although the attainment of local peak of stresses in the region of PVB immediately adjacent to the SMA wires (especially the wires ends), almost negligible maximum stresses were generally found in the same PVB ilms, e.g. in the order of ≈0.5 MPa, hence conirming a fully linear elastic behavior of the PVB foils and the total lack of possible damage phenomena.

114 Advanced Engineering Materials and Modeling M2-3 (iSMA = 250 mm. φSMA = 4 mm)

M2-0 (unreinforced panel) U. U3

+6.443e–04 +5.953e–05 –5.253e–04 –1.110e–03 –1.695e–03 –2.280e–03 –2.864e–03 –3.449e–03 –4.034e–03 –4.619e–03 –5.204e–03 –5.789e–03 –6.373e–03

(a) S. Max. in-plane principal envelope (max) (avg: 75%) +4.921e+06 +4.490e+06 +4.059e+06 +3.628e+06 +3.197e+06 +2.766e+06 +2.335e+06 +1.904e+06 +1.473e+06 +1.041e+06 +6.103e+05 +1.792e+05 –2.520e+05

(c) S. Max. in-plane principal envelope (max) (avg: 75%) +8.427e+06 +7.854e+06 +7.282e+06 +6.709e+06 +6.137e+06 +5.564e+06 +4.992e+06 +4.419e+06 +3.846e+06 +3.274e+06 +2.701e+06 +2.129e+06 +1.556e+06

(e) S. Max. principal (avg: 75%) +1.349e+03 –6.690e+02 –2.687e+03 –4.705e+03 –6.723e+03 –8.742e+03 –1.076e+04 –1.278e+04 –1.480e+04 –1.681e+04 –1.883e+04 –2.085e+04 –2.287e+04

(g)

U. U3

+6.075e–04 +3.274e–05 –5.420e–04 –1.117e–03 –1.691e–03 –2.266e–03 –2.841e–03 –3.416e–03 –3.990e–03 –4.565e–03 –5.140e–03 –5.714e–03 –6.289e–03

(b) S. Max. in-plane principal envelope (max) (avg: 75%) +3.929e+06 +3.562e+06 +3.195e+06 +2.828e+06 +2.460e+06 +2.093e+06 +1.726e+06 +1.359e+06 +9.915e+05 +6.243e+05 +2.571e+05 –1.101e+05 –4.773e+05

(d) S. Max. in-plane principal envelope (max) (avg: 75%) +7.471e+06 +6.915e+06 +6.359e+06 +5.803e+06 +5.247e+06 +4.691e+06 +4.135e+06 +3.579e+06 +3.023e+06 +2.467e+06 +1.911e+06 +1.355e+06 +7.955e+05

(f) S. Max. principal (avg: 75%) +2.826e+04 +2.407e+04 +1.988e+04 +1.570e+04 +1.151e+04 +7.325e+03 +3.139e+03 –1.047e+03 –5.233e+03 –9.419e+03 –1.361e+04 –1.779e+04 –2.198e+04

(h)

Figure 3.19 Comparison of (a and b) maximum delections, (c and d) maximum principal stresses in glass (g1), (e and f) maximum principal stresses in glass (g2), and (g and h) maximum stresses in the PVB foil (ABAQUS/Standard), under the action of wind (q = 1 kN/m2, tL = 3 s, T = 60 °C). Values expressed in [m] and in [Pa].

3.6 Conclusions In this contribution, a novel design concept of SMAs-reinforced LG panels has been preliminary assessed by means of FE investigations and proposed as possible innovative technique for the enhancement and optimization of traditional LG elements. LG panels are in fact widely used in common practice, in the form of façade, roof, or inter-storey elements. here, in order to cover large surfaces, LG units have typically high size-to-thickness ratios. As a result, major restrictions in their design can be represented by prevention of glass tensile failure and the attainment of large delections. In them, however, the typical resisting cross section is obtained by bonding two (or more) glass plates and one (or

Toward a Novel SMA-reinforced Laminated Glass Panel 115

S. Max. in-plane principal envelope (max) (avg: 75%) –4.056e+04 –3.498e+05 –6.590e+05 –9.683e+05 –1.278e+06 –1.587e+06 –1.896e+06 –2.205e+06 –2.514e+06 –2.824e+06 –3.133e+06 –3.442e+06 –3.751e+06

iSMA = 450 mm (M2-1)

S. Max. in-plane principal envelope (max) (avg: 75%) –7.162e+04 –3.731e+05 –6.746e+05 –9.761e+05 –1.278e+06 –1.579e+06 –1.881e+06 –2.182e+06 –2.484e+06 –2.785e+06 –3.087e+06 –3.388e+06 –3.690e+06

iSMA = 350 mm (M2-2)

S. Max. principal envelope (min) (avg: 75%) –7.162e+04 –3.731e+05 –6.746e+05 –9.761e+05 –1.278e+06 –1.579e+06 –1.881e+06 –2.182e+06 –2.484e+06 –2.785e+06 –3.087e+06 –3.388e+06 –3.690e+06

S. Max. in-plane principal envelope (max) (avg: 75%) –5.285e+05 –7.960e+05 –1.064e+05 –1.331e+05 –1.599e+06 –1.866e+06 –2.134e+06 –2.401e+06 –2.669e+06 –2.936e+06 –3.204e+06 –3.471e+06 –3.739e+06

iSMA = 250 mm (M2-3)

iSMA = 150 mm (M2-4)

Figure 3.20 Distribution of compressive stresses 0 in glass, due to the pre-stress of the SMA wires, as a consequence of temperature increase (minimum envelope, in [Pa]), T = 60 °C ( SMA = 4 mm).

more) intermediate interlayers with oten mechanical properties rather sensitive to temperatures, loading time, humidity, etc. As a result, careful consideration should be generally paid for their appropriate design and veriication. In this context, the feasibility of an adaptive embedded reinforcement system built up of martensitic SMA wires has been explored, by taking into account some conigurations of practical interest for structural glass applications. SMAs are in fact a class of metal materials that exhibit two outstanding properties, namely the superelastic (SE) and the SMEs. Taking advantage of these intrinsic properties, several applications have been

116 Advanced Engineering Materials and Modeling 3

2

1

0 0.67

6.3

0.65 wq,E [mm]

wq,M [mm]

4 6.5

6.0 5.8 30°

40°

50°

250

350

(a) 3

2

1200

150

250

350

0

0.61 30°

(b)

1

1

40°

50°

60°

0.57 450 550 iSMA [mm]

1200

1500

0

8.5

4.0

8.0

3.0

7.5 7.0

iSMA [mm] 250 450

2.0 1.0

30°

40°

50°

60°

6.5 150 (c)

2

0.63

1500

σ0 [MPa]

σq [MPa]

4

450 550 iSMA [mm]

3

0.59

60°

5.5 150

4

250

350

450 iSMA [mm]

550

1200

0.0

1500

30 (d)

40

T [°C]

50

60

Figure 3.21 Efect of temperature variations in the structural response of a SMAreinforced glass panel under wind pressures (ABAQUS/Standard) in terms of (a) maximum deformations at the panel center wq,M and (b) unrestrained edge wq,E, (c) maximum tensile stresses in glass q, and (d) maximum compressive stress 0 in glass, due to temperature variation only.

proposed over the past years in robotic, automotive, and biomedical engineering in the form of SMA actuator wires and plates replacing conventional pneumatic or hydraulic systems. Several applications can be found also in civil engineering applications. Parametric FE studies generally highlighted – for the discussed cases – that an embedded SMA reinforcement can have positive structural efects on traditional LG panels. As far as these LG elements are subjected to ordinary temperatures, the SMA wires provide a moderate stifness contribution only, e.g. in the form of fully embedded cables. Conversely, when the temperature increases, the intrinsic transformation phase of the SMA wires manifests in the form of an embedded pre-tensioned net able to provide a certain level of additional stifness and resistance to the same LG elements. Certainly, extended experimental studies are currently required for a full development and optimization of this novel design concept. However, based on the promising results, it is expected that the current outcomes could represent a rational and useful background for the development of further reinements and design recommendations.

Toward a Novel SMA-reinforced Laminated Glass Panel 117

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4 Sustainable Sugarcane Bagasse Cellulose for Papermaking Noé Aguilar-Rivera Facultad de Ciencias Biológicas y Agropecuarias, Universidad Veracruzana, Córdoba, Veracruz, México

Abstract Sugarcane bagasse (SCB) is a raw material for pulp, paper, and wood composites manufacturing. SCB pulp has a greater lexibility in a wide range of paper grades, but the main constraints are as follows: slow drainage, low opacity, morphological heterogeneity, short iber, high ines content and pith, poor wet strength, high ash content, and low runnability on the paper machine and washing system, and may require changes to the forming section of the paper machine and by use of drainage aids and krat sotwood pulp (up to 20–30%) as a major component of mixture for pulps for the best performance. herefore, the objective of this paper was to evaluate the incorporation of recycled old corrugated container (OCC) pulps, since cooking process (conventional alkaline deligniication) of SCB (sugarcane bagasse), as long iber source to replace sotwood chemical pulp for highquality paper SCB production. OCC was collected, pulped, and cleaned. Technical Association of the Pulp and Paper Industry standard was used to study the SCB chemical pulping with OCC in 10, 20, and 30% and to evaluate physical properties. he pulps were bleached using an elemental chlorine-free sequence. SCB– OCC soda pulp was found to be very efective because they have higher tensile, tear, burst strength, and lower drainability and porosity resistance ater reining. Keywords: Sugarcane bagasse, OCC, alkaline deligniication, ECF bleaching, drainability

Corresponding author: [email protected] Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (121–164) © 2016 Scrivener Publishing LLC

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4.1 Pulp and Paper Industry Papermaking started from nonwood materials in China almost 2,000 years ago, when paper itself was invented. Pulp and paper (P&P) are manufactured from raw materials containing cellulose ibers, generally wood, recycled paper, and agricultural residues as bagasse. In developing countries, about 60% of cellulose ibers originate from nonwood raw materials such as bagasse, cereal straw, bamboo, reeds, esparto grass, jute, lax, and sisal. P&P mills are highly complex and integrate many diferent process areas including raw material preparation and handling, pulping, pulp washing and screening, chemical recovery, bleaching, stock preparation, and papermaking to convert wood and nonwood pulps and others materials to the inal product. Processing options and the type of iber processed are oten determined by the inal product. he pulp for papermaking may be produced from virgin iber by chemical or mechanical means or may be produced by the repulping of paper for recycling. Wood is the main original raw material. Paper for recycling accounts for about 50% of the ibers used, but in a few cases bagasse, straw, hemp, grass, cotton, and other cellulose-bearing material can be used. Paper production is basically a two-step process in which a ibrous raw material is irst converted into pulp, and then the pulp is converted into paper [1, 2]. P&P industry is considered a large user and producer of biomass-based energy and materials the P&P industry is facing important reforms in both environmental performances and production processes, in order to satisfy stringent environmental regulations, to maintain their proitability, and to overcome the declining and competitive markets [3]. It is evident that nonwood ibers will play a vital role in the world P&P industry and lignocellulosic bioreineries and sustainability [4, 5]. Several nonwood ibers could be made available and even greater quantities grown. One of the major advantages of nonwood ibers is that they can be pulped economically on a small scale with conventional technology, requiring relatively low investment. It has been proved that by properly selecting the appropriate mixture of nonwood ibers and the pulping process, many grades of paper and paperboard can be produced. In fact, some grades are already being produced with 100% nonwood ibers. hough the use of nonwood iber is the option for developing countries, one should not be surprised if these materials also become valuable in other countries. When we consider the tightening pulpwood supply situation and the resultant increased cost of pulpwood in the major producing areas, increased impetus will undoubtedly be given to the use of sugarcane

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bagasse (SCB) and the huge quantities of other nonwood ibrous raw materials that are available all over the world. heir utilization is mainly a question of economic and technological feasibility. Although nonwood plant ibers are used now to only a limited extent in North America and Europe, they represent one of the major sources of ibrous raw material for many of the developing countries of the world. heir importance promises to become greater to many of the developed countries, as well. Among the many nonwood ibers utilized for pulp manufacture, SCB stands out, more than any other, as being one which promises to become a major ibrous raw material for the world’s P&P industry. his material is readily available and easily accessible in a great many countries of the world, and is especially abundant in some of the wood-poor countries. SCB has a distinct advantage over the other nonwood ibers in that it involves no great problem of collection. he costs of collecting, crushing, and cleaning the material are borne by the sugar extraction process, and the SCB when delivered to the pulp mill, is in excellent condition for further processing. It has been found that properly stored SCB will keep in good condition for extended periods of time. Furthermore, with the development of highly eicient bulk storage methods for SCB, and the elimination of the necessity for high cost baling and stacking, there has been a tremendous decrease in the overall cost of this raw material delivered to the pulp mill. SCB iber dimensions are fairly similar to those of hardwoods ibers such as eucalyptus. SCB pulps are now used in practically all grades of paper, including newsprint, bag, wrapping, printing, writing, toilet tissue, toweling, glassine, corrugating medium, linerboard, bleached boards, and coating base stock. It is possible that even will be produced with a high content of SCB and dissolving pulp for chemicals. From an overall standpoint, bleached SCB pulp can be used on an equal basis as a replacement for bleached hardwood pulp without any decrease in quality of the inal product [6–8]. he state of knowledge relative to the production and use of almost all grades of SCB pulp, paper, paperboard, and cardboard is already well developed to a high technological level—and this knowledge is being put to use in large-scale commercial operations.

4.2 Sugar Industry Sugarcane (Saccharum spp.) is a perennial monocotyledonous plant belonging to the Gramineae (Poaceae) family that has received considerable recent

124 Advanced Engineering Materials and Modeling attention due to its potential for biofuel production. Sugarcane has been cultivated for nearly 500 years for sugar production. he plant originates from Asia, but it is well adapted to most tropical and subtropical climates where it has become one of the most important bioenergy crops. Its C4 photosynthesis makes it one of the most eicient species in carbon conversion and one of the most productive among all cultivated crops. Sugarcane is the main feedstock for sugar production, accounting for nearly twothirds of the world’s production and is also the main raw material for the production of ethanol [9]. he valorization of sugarcane by-products as SCB by biological, chemical, or integrated routes with conventional approaches is a key technology that contributes to the development of sustainable processes and the generation of value-added products.

4.3 Sugarcane Bagasse SCB or by-product of the sugar industry is the ibrous residue obtained ater crushing and extraction of the sugarcane juice by either tandem milling or difusion in the sugar production process is one of the largest agro industrial residues in the world. he possibilities of the utilization of bagasse are various and diverse. It is an attractive feedstock to produce second-generation bioethanol and other value-added products in the context of the lignocellulosic bioreinery. SCB is composed of two parts iber and pith (Figure 4.1) SCB is primarily composed of lignin (20–30%), cellulose (40–45%), and hemicelluloses (30–35%), an amorphous polymer usually composed of xylose, arabinose, galactose, glucose, and mannose. Because of its lower ash content, 1.9% bagasse ofers numerous advantages compared with other nonwood residues. he most common use for SCB is the energy and steam production by combustion. One ton of sugarcane generates 280 kg of SCB (135–140 kg dry weight), which corresponds to about 35% of the total cane weight. he high moisture content of SCB, typically 40–50%, is detrimental to its use as a fuel. his product represents a great morphological heterogeneity. About 50% of this residue is used in distillery plants and sugar mills as a source of energy; the remainder is stockpiled. herefore, because of the importance of SCB as an industrial waste, there is great interest in developing methods for the biological production of fuel and chemicals that ofer economic, environmental, and strategic advantages.

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Figure 4.1 Components of SCB (a) integral, (b) iber, (c) pith, and (d) depithed.

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Figure 4.2 Countries producer of sugar from sugarcane (Saccharum spp.).

In the approximately 100 sugarcane producing countries, there is a potential to make better use of the SCB. Subjected to improved energy eiciency sugar producers could supply energy either as co-generated electricity, or as fuel ethanol through cellulose hydrolysis followed by fermentation considering that about 92% of SCB is burned in the industry for process heat generation, if the remaining 8% could be used for other productions [10] (Figure 4.2).

126 Advanced Engineering Materials and Modeling SCB can be used either for energy production or for non-energy applications, and currently, there is much research on the uses of SCB; it can be used as paper-making material and for cellulose derivatives (methylcellulose, cellulose acetate, and microcrystalline cellulose); it can also be utilized as a source of raw material in the chemical industry to produce xylose, xylitol, and furfural and their derivatives as bio oil. As it is rich in fermentable sugars (but not directly available), it has aroused the interest of researchers for the production of ethanol with chemical pre-treatments such as enzymatic or hydrolysis. In addition, SCB can be used in various other ways: as a constituent of animal feed, dietary iber, to produce useful materials for the construction industry and furniture manufacturing, such as plywood, particleboard, and other engineered wood products; and as an adsorbent for environmental remediation to remove oil by-products from contaminated water. Nowadays, there are many companies around the world producing paper products (e.g., tableware) made of SCB. hese products have many advantages over plastic goods: they are 100% biodegradable, nontoxic, harmless, recyclable, microwaveable, and resistant to diferent temperatures (hot or cold). It can substitute for virgin paper and cellulose in many applications, saving the countless trees used to produce these products [11–19]. Mushroom cultivation on SCB is also practiced in some countries [20]. Transformation of SCB into addedvalue products, emphasizing on fuel ethanol production, is discussed by Cardona [21].

Pulp & paper

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α cellulose

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Figure 4.3 Products from SCB.

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here has been increasing interest in cellulose-based materials due to the abundance, renewable, and eco-friendly nature of cellulose [22]. Hence, the peculiar characteristics of SB, particularly the large amount of carbohydrates in the form of hemicelluloses and cellulose, have encouraged research on bioconversion processes of this material for the production of bioproducts. In fact, some authors do not consider SCB a by-product or residue of the sugar and alcohol industry, but instead as a high value coproduct, which can be used as a raw material for the production of biofuels and bioproducts [23]. In order to expand the use of SCB, it is essential that information on iber characteristics and the factors that afect the performance of that iber be available. Speciic knowledge of SCB material is useful to predict its behavior during transformation processes. Physical properties involved in the heat and mass transfer, such as density and porosity, play an important role in the design, the estimation of other properties, the characterization of materials, and the prediction of heat transfer operations during processing and handling. Rodriguez-Ramirez [24] reported a revision of the published results and industrial experience about the viability of satisfying the energy needs of a sugar factory with half of its residual SCB. hen, the remainder can be used in more than 40 applications. In the future, with many options for the construction of an energetic matrix, SCB could either be used for burning to generate bioelectricity, P&P, or for the production of cellulosic ethanol or chemicals, and the decision will be the market, emphasizing the most proitable option. With future technological improvements of boilers and processes, it will be possible to supply the same quantity of energy with less SCB and use the surplus for the bioreinery production of ethanol and other added value chemicals [25]. Due to SCB composition, it is considered as an ideal ingredient to be applied and utilized as reinforcement iber in composite materials for the purposes of creating new materials that possess distinct physical and chemical properties. he structural components are distributed in a lamellar structure. Fractionation of SCB into its main components and their characterization is essential to enable this renewable feedstock for conversion to valueadded products according to the bioreinery concept [26]. he composition of SCB can vary among other factors, climatic conditions, and the soil properties where sugarcane was grown. However, studies carried out with diferent varieties of SCB show that its composition does not difer signiicantly regarding the main chemical components. In

128 Advanced Engineering Materials and Modeling general, the same trend is observed when the diferent physical fractions of bagasse are compared [27, 28]. he iber biometry characteristics of SCB were reported by several papers [29–31]. According to [32] have been identiied two important types of ibrous residue occurring in SCB: (a) the tough, hard-walled, cylindrical cells of the rind and vascular tissues (true iber); and (b) the sot, thin-walled, irregularly shaped parenchymatous cells of the inner stalk tissue (pith). he vessel segments, also associated with the vascular bundles, are oten, but not always, considered as a pith fraction. the true iber and the pith have almost the same chemical composition, but their structures difer widely. he true ibers have a fairly high ratio of length to diameter (approximately 70), and a relatively high coeicient of expansion and contraction upon wetting and subsequent drying. his results in close bonding of one iber with another and accounts for the strength, cohesiveness, and ability to become matted when subjected to pulping processes. he pith cells are of irregular size and shape, with a length-to-diameter ratio of about 5. hey are characterized by their absorption properties. hey do not adhere and so tend to weaken any pulp in which they are incorporated and, further, prevent its rapid drying. However, they can absorb many times their mass of liquid (Figure 4.4). Each SCB particle is the lignocellulosic skeleton of aggregated sugarcane cells. Particle size, shape, and internal structure depend on the represented sugarcane tissues as well as on the pattern of cane disintegration during crushing for juice extraction. SCB particles are commonly classiied as pith, iber, and rind. Pith fraction is derived from parenchyma; it is made of iner particles with irregular shape and near unitary aspect ratios. he so-called ibers are not true ibers; they are rather fragments of ibro vascular bundles, with hundreds of microns in diameter and high aspect ratio. Rind particles make the coarser fraction of SCB; they are derived from the outermost layers of the culm and are aggregates of epidermal cells, ibro vascular bundles, and associated parenchyma. Several mean morphological properties of SCB fractions have been determined, including skeletal and envelope densities, cell dimensions, porosity, speciic surface areas, and particle size distributions. In addition, local SCB morphologies have been imaged by optical and electron (scanning and transmission) microscopies [33–36]. he application of X-ray microtomography (XMT) to raw and treated SCB. Sugarcane anatomy, crushing damages, treatment efects, and particle porosity were discussed based on 3D images to obtain the irst threedimensional images from SCB [37].

Sugarcane cross-cutting and radial

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Microscopy sugarcane Bagasse (SCB)

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Fiber bundle

Figure 4.4 Sugarcane anatomy.

A review of literature by [38–44] indicates that SCB is favored in the manufacturing of high-quality green products given its low production cost. his is mainly attributed to the abundant availability of raw materials from the sugar processing plants and its low pre-treatment costs.

4.4 Advantageous Utilizations of SCB SCB wastes are chosen as an ideal raw material in manufacturing new products because of its low fabricating costs and high-quality green end material. It is ideal due to the fact that it is easily obtainable given the extensive sugarcane cultivation making its supply constant and stable. he associated costs of extraction, chemical modiications, and/or other pre-treatments

130 Advanced Engineering Materials and Modeling of SCB in the transformation process to ready-to-be-used materials are potentially reduced as the complex processes are simpliied by the mere usage of SCB. When appropriate modiications and manufacturing procedures are applied, SCB displays improved mechanical properties such as tensile strength, lexural strength, lexural modulus, hardness, and impact strength. SCB is also found to be easily treated and modiied with chemicals besides blending well with other materials to form new types of composite materials. It also satisies the greening requirements by being biodegradable, recyclable, and reusable.

4.5 Applications of SCB Wastes SCB have been applied in the following instances: i. Cellulose, lignin, and hemicelluloses production; ii. Mixed with tapioca starch and glycerol to produce composite materials; iii. Mixed with gelatin, starch, and agar to produce tableware packaging material; iv. SCB ash and sugarcane straw ash (SCSA) can partially replace cement and act as pozzolanic additive in manufacturing of concrete and ash block; v. SCB ash mixed with Arabic gum and water to produce ceramic and refractory products; and vi. Both sugarcane and their mixture with hardwood are used with phenol formaldehyde resin and wax to manufacture composite board. SCB can act as efective reinforcement iber in the manufacture of polymeric composites. It may also be applied and utilized for composite materials manufacturing and applications in various forms, such as cellulose iber, lignin extracted, pith, sugarcane bagasse ash (SCBA), SCSA, and more. Additionally, SCB produces good reactions when mixed with other additives and chemicals, which produces materials with improved and desired properties and sometimes creating new characteristics. Chemical modiications of SCB wastes are vital and can efectively improve the matrix– iber adhesion in the composites thus enhancing those desired mechanical properties and functions on the materials manufactured [45, 46].

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4.6 Problematic of Nonwood Fibers in Papermaking In papermaking, it is very oten that more than one type (or grade) of pulps are used to develop paper sheet properties necessary for both machine runnability and requirement of the end users. In many paper grades, long ibers from sotwood bleached krat pulp are oten used in combination with short iber from hardwood bleached krat pulp. In this combination, the long iber component provides the strength (especially tear resistance) to the paper sheet, while the short iber helps to improve the paper functional properties, such as sheet smoothness and formation. It is generally accepted that ibers such as straw or SCB are not likely to be used on a 100% basis for paper manufacturing. Particularly, their poor drainability, permeability, and low tearing strength make it necessary to use furnish mixtures containing a certain amount of long ibers from wood. he key element for sustainable utilization of the nonwood ibers as SCB is to understand their special qualities and how they afect the technical aspects involved. Low bulk density, short iber, and high content of ines are the most important features. Other disadvantages include transportation and storage problems, comparatively high silica content, and very quick degradation (high losses). hese disadvantages have prevented the emergence of nonwood plant ibers as a source of cost competitive pulp for both printing/writing and cellulose products particularly in regions of the world where wood supplies are adequate. he use of nonwood ibers, however, is common in wood-limited countries [47]. he advantages, disadvantages, and related problems to nonwood pulping and papermaking as SCB are discussed [48]. Hunter [49] reported that the P&P industry is a global industry producing mainly commodity products in large-scale facilities using wood as the primary iber raw material. It is a diverse industry that produces a wide range of pulp, paper, and paperboard products to meet speciic end user requirements where about 89% of all papermaking is done using wood ibers leaving only about 11% produced using a variety of nonwood ibers. When assessing nonwood plant ibers, it is also important to understand that there is a huge diversity of plant materials available that could be used to produce P&P. It is also important to understand that this diversity of plant iber raw materials ofers a wide range of iber characteristics and that paper makers can use speciic nonwood plant ibers to impart desirable properties to their inished products.

132 Advanced Engineering Materials and Modeling Combinations of common and specialty nonwood pulps will permit the production of virtually any grade of paper to meet any quality requirements demanded in the global market. Adding possible combinations that include wood pulp, nonwood pulp as SCB, and recycled wastepaper pulp increases the possibilities for developing paper with speciic sheet properties designed to meet speciic customers’ needs. he current uses of nonwood pulps include virtually every grade of paper produced including, but not limited to: t
t
t
t
t
t


Printing and writing papers Linerboard Corrugating medium Newsprint Tissue Specialty papers

Specialty papers such as currency, cigarette papers, tea bags, and dielectric paper may be made from a furnish of 100% nonwood specialty pulps, but sotwood krat is added to provide the strength requirements to the paper. In some cases, wastepaper pulp is blended in the furnish. he nonwood portion can vary from 50 to 90% and even up to 100% depending on the grade and required quality. he possible combinations are endless and can be adjusted to meet market requirements. Furthermore, nonwood pulps can be used as an additive to wood-based papers for a variety of reasons such as the following: t
To provide the papers with certain speciic desired properties—i.e. production of ultra-lightweight papers or papers with increased opacity or better bulk, etc. t
To ofset higher wood costs. t
To provide an incremental increase in mill capacity in a region where woods resources are inite. he issues surrounding the production of P&P using agricultural residues and/or iber crops are many and include both technical and economic matters. A vast body of knowledge has already been developed on the use of nonwood plant ibers. Many of the technical questions raised below have been addressed many times in many countries throughout the world by engineering consultants and equipment suppliers with expertise in the use of nonwood plant ibers.

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Sadawarte [50] concluded that the main constraints for the widespread use of nonwood and annual plants in the P&P industry are as follows: 1. Length/diameter ratios of agricultural residue ibers are very high compared to those of wood ibers. his adversely efects the physical properties of paper. 2. he high pentosan content of some nonwood pulps reduces the power consumption in reining due to easier swelling of ibers. 3. When long iber pulps are used along with agricultural residue pulps, separate reining of the pulps is preferable before inal reining to attain homogeneity. 4. he separate reining of long iber and wastepaper pulp requires the use of instrumentation and controls to maintain the furnish composition. Any variation in the percentages of short ibers and broke pulp results in hours of production loss due to breakages, particularly when bagasse and straws are the major portion of the furnish. 5. Slow drainage, a limit on the dryness at the press, poor wet strength, a tendency to blister and cockle due to rapid drying, and high shrinkage are problems associated with the use of nonwood pulps in papermaking. 6. Paper machines need a wire 20–30% longer than on a wood-iber machine to provide a drainage area large enough to maintain the required stock consistency at the headbox. Foils, vacuum foils, and additional suction boxes with large open areas on covers are other features. Monoilament and multiple-ilament synthetic fabrics are typical. 7. Even with an improved press coniguration, web moisture content will be higher than with wood pulps. Web dryness improves when nonwood iber is only 35–50% of the total furnish. In bleached printing and writing papers, the moisture content of the paper web is 62–65% when nonwood iber forms 75–80% of the furnish. 8. he web should be supported in the press section to avoid breaks, as it is likely to break at the slightest change in tension. Retention of additives such as modiied starches would help reduce the problem of ines and luf. 9. hese papers need an extended drying surface area at high temperatures, and there is high cross-directional shrinkage,

134 Advanced Engineering Materials and Modeling although camber rolls at critical points reduce the shrinkage. Dry fabrics or mesh dryers in the section following the irst dryer group provide greater permeability, and evaporated water can be easily removed. Pocket ventilation and high-velocity hoods can also improve drying. 10. he presence of water in iber walls is typical of these ibers. his sets a limit to the dryness that can be achieved by increased bonding at the press and results in crushing of the sheet and marking of the felt. his can be overcome to a large extent by improved press design. A smoothing press ater a wet press is essential. 11. he use of high-porosity open felts helps to achieve further dryness. Fines bound with size, and iller particles, tend to clog up the felt pores and reduce water absorbency. Regular cleaning of felts is necessary.

4.7 SCB as Raw Material for Pulp and Paper One of the principal advantages of SCB as a raw material for P&P as compared with other agricultural residues and nonwood is its easy collection; however, the economic situation of P&P SCB plants is directly related to both the sugar industry and the luctuations in prices of P&P products. herefore, there are a number of issues relating to their use, which must be considered by P&P industry from SCB: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Raw material supply Harvesting, transportation, and storage Raw material preparation Pulping technology Washing technology Bleaching technology Silica and chemical recovery Furnishes for paper grades Paper machine operation Capital costs Operating costs

Many pulping and papermaking processes have been studied to SCB; most of these processes are based on optimized methods to the wood P&P, but

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the application to produce brown grades such as krat sacks and liner board is generally avoided. he key to successful SCB pulping and papermaking is thorough depithing because the pith content of SCB (30–35%) has a negative inluence on the quality of pulp for paper and boards, creating serious technological problems such as undesirable properties of the inal product, increased costs for handling and storage, increased chemical consumption, and negative efects on the environment. Undoubtedly, for P&P production, the removal of pith or the ine fraction from SCB is more necessary than in the case of other technologies. It is not possible to produce a high-quality paper with a poorly depithed bagasse. Rainey [51, 52] reported a review of the depithing process; Lois [53] about storage SCB; and Aguilar [54] related to the efect of storing of SCB on physical properties from cellulose for paper.

4.8 Digestion he soda process (NaOH) has been found to be the best pulping process to date for SCB. Either continuous or batch cooking can be employed for pulping. In general, SCB needs rather moderate cooking conditions for satisfactory results as compared with wood in respect of chemical application, cooking temperature, and cooking time. Because of the bulkiness of SCB, a much higher liquor ratio as compared with that employed in case of wood pulping has to be employed to insure complete immersion, which is a necessary condition for uniform cooking. A liquor ratio of 7/1 is usually considered suitable for SCB pulping. he moisture content of SCB entering the digester should be as low as is feasible, since unduly moist bagasse interferes with penetration of cooking liquor to such an extent as to prevent satisfactory cooking. In addition to commercially established technologies as krat or soda process, there are a many new “emerging” pulping and deligniication technologies, which may be applied to SCB [55, 56].

4.9

Bleaching

he cooked SCB is irst washed, screened, and centrifugally classiied to a satisfactory degree of cleanliness and uniformity of iber dimensions and then subjected to bleaching if bleached pulp is to be produced. While SCB pulp can be more easily bleached than wood pulp, irrespective of the

136 Advanced Engineering Materials and Modeling cooking process employed, it is always a sound practice to employ multistage bleaching, especially for the bleaching of sulfate or soda bagasse pulp, for both technical and economic reasons. A conventional four-stage bleaching process consisting of chlorination (C), alkali extraction (E), and two hypochlorite (H) oxidation stages or chlorine dioxide (D). Pulp bleaching by chlorine dioxide oxidation has also the advantage of yielding a pulp of higher viscosity, and hence better strength characteristics to produce the best grade of bleached SCB pulp ever produced, which will be comparable in all respects with superior-grade bleached hardwood. Pulps from depithed bagasse can be bleached to 80% brightness by a conventional Chlorination, Alkaline Extraction and Sodium Hypo-chlorite (CEH sequence). he consumption of chemicals is low due to the easy bleachability of the pulp [57].

4.10 Properties of Bagasse Pulps SCB pulps have properties normally considered similar, or a little inferior, to those of hardwood pulps. Some issues speciic to bagasse according to Rainey and Clark [58] are as follows. he seasonal nature of cane harvesting means that SCB may be stored for months in order to provide a continuous supply to the pulp mill. Some 30–40% of the incoming iber is pith that must be removed prior to downstream processing or product quality is much reduced. SCB pulp has slow water drainage characteristics, particularly if depithing is not eicient, with reduced pulp mat permeability, which oten requires paper machines to be run more slowly. SCB pulp oten has high dirt counts and low brightness resulting from colored impurities (e.g. carbon from cane burning). Fiber damage during cane preparation (hammer milling) and crushing leads to lower pulp strength properties, requiring increased addition of (imported) bleached sotwood chemical pulps to the paper machine furnish to provide runnability and product quality. SCB ibers average 1.4 mm in length by 0.02 mm in width. he lengthto-width ratio of bagasse iber is thus of the same order as that of some wood ibers and nonwood plant ibers. his is a deinite advantage of SCB pulp over hardwood pulps, the ibers of which have about the same average length as SCB ibers but greater width, meaning smaller lengthto-width ratio.

Sustainable Sugarcane Bagasse Cellulose for Papermaking

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SCB iber is very easily hydrated, which characteristic is other desirable feature of SCB pulp in respect of power requirement for strength development. However, easy hydration gives rise to high sheet density as well as unduly low sheet opacity of the inal paper made from SCB pulp. herefore, due to these particular characteristics, SCB pulp possesses good bursting strength (mullen), fair tensile strength, but low tear. SCB pulp therefore has to be mixed with longer and less easily hydrated iber(s) for the manufacture of ine papers in spite of its numerous advantages. SCB has been used in many grades where local factors limit the availability of alternative pulps, but grades in which SCB has proved particularly suitable include: printings and writings, tissue grades, corrugating medium, and newsprint; but in most cases, SCB pulp is most competitive when a moderate proportion of another pulp is added, usually to give better runnability. It is possible to add small quantities (up to 20–30%) of SCB pulp to primarily wood pulp-based papers without impairing paper properties or paper machine runnability. his provides wood-based mills, which are hardwood deicient but located within a region with available SCB resources with the option of adding-on a straw pulping line to supplement their iber requirements [59].

4.10.1 Pulp Strength he pulps produced are not generally of high strength. hey are similar or slightly deicient to hardwood pulp and generally a long iber component must be added where high tear strength or high machine speed are required. he SCB pulp in the furnish is chemical pulp. he long iber pulp typically would be krat or sulite chemical pulp (or a mixture of the two) made from sotwoods, and bleached, semi-bleached or unbleached depending on the type of paper or paperboard.

4.10.2 Pulp Properties SCB pulps are generally smooth and sot. However, chemical pulps have poor opacity. It does not produce ‘universally applicable pulps’ under open market conditions. herefore, it is important to consider the inal use very carefully when planning a mill [60]. In relation at pulp processing, the SCB pulps have two speciic problematic related to the characteristics of the iber.

138 Advanced Engineering Materials and Modeling

4.10.3 Washing Technology Pulps from SCB are slower draining than wood pulps, thus requiring larger brown stock washer areas than wood pulps, and the same applies to bleach plant washers and the same applies to bleach plant washers. Errors in washer sizing in nonwood pulp mills are common when the mills are designed by persons who are more familiar with the design of wood pulp mills that with the requirements of SCB pulp. In the wood pulp industry, recent developments have included a new generation of press washing systems commercially operating in a number of the larger wood pulp mills. hese new compact press washers ofer many advantages; however, they have not been commercially tested on bagasse pulps, because there may be problems with applying press washing to bagasse pulps because the ibers are more fragile than wood ibers and may be damaged during press washing. Given the many potential advantages of press washing, further research and development should be carried out on the impact of press washing on SCB pulps.

4.10.4

Paper Machine Operation

he design of paper machines will difer for furnishes with a high nonwood content. For example, a longer wet end with more drainage elements will be required for papers with a high cereal straw or SCB content. Press loading will be lower to avoid sheet crushing, and the dryer section will require more sections to account for the higher shrinkage of the mainly nonwood sheet. Also, machine speeds for high nonwood content sheets typically are lower than for wood pulp papers. However, if the percentage of nonwood pulp in the sheet is in the order of 10 to 30% with the balance being wood pulp and/or recycled pulp, there should be little or no efect on the design and operation of the paper machine. Accordingly, other long iber supply pulp as old corrugated containers (OCC), for increment strength, permeability, and drainability, may be used instead of sotwood krat or sulite pulp, thus producing a 100% nonwood paper.

4.11 Objectives SCB has become the most important raw material within the P&P sector in developing countries. However, SCB pulps have several constraints for successful performance in papermaking, derived from its anatomic/morphological characteristics as lignocellulosic nonwood, damage to the iber during crushing on sugar mills, the wet and dry depithing process, and

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139

the microbiological degradation during storage. he resulting pulp has low draining capability, high ines content, poor wet strength, high ash content, and low runnability and permeability, because the ibers are not as long and they do not bond as well as sotwood pulp and may require technological changes in the washing system and formation at the Fourdrinier paper machine, use of drainage aids and mainly by adding unbleached or bleached krat sotwood pulp (up to 20–30%) as a major component for SCB pulps on paper and linerboard production. he issues of this paper were (1) to evaluate the incorporation of recycled OCC pulps krat liner, since cooking process (conventional alkaline deligniication) of SCB, as long iber source to replace unbleached sotwood chemical pulp for the production of high-quality paper with SCB pulps and (2) to determine the impact of quality of storaged SCB for 12 months on mechanical and drainage pulp properties compared with those of corresponding bleached and unbleached pulp made from mills SCB (fresh SCB) and with OCC pulp in synergistic deligniication with fresh SCB (Figure 4.5).

4.12 Old Corrugated Container Pulps Recycling paper (secondary iber) for diferent recycled pulp grades has been seen as the solution for the lack of forest products for P&P production. However, on secondary iber, both reversible and irreversible changes as horniication, loss in lexibility, swelling properties, conformability and lower strength, and bonding between ibers take place in progressive and repeated recycling, which results in a weaker, lower grade of paper. OCC consisting of corrugated container (54%), double-liner krat corrugated cutting (DLK) (30%), and boxboard cutting (13%) are the most signiicant category of waste papers or post-consumer waste for recycling based to protect contents inside box from compression forces during packing, storage, and distribution. OCCs are made of unbleached sotwood chemical long ibers that also have a low lignin content of about 10–15% and a high cellulose content of 50%, and recycled cardboard of diferent fraction by length, where the short fraction is used as the corrugated medium, while the stronger long fraction is used for the liner. However, most OCC has unique characteristics: (a) high amount of ines and short iber length; (b) low water retention value; (c) poor drainage properties (below 200 mL CSF); (d) high amount of contaminants such as inks, stickies, plastics, inorganic particles and polymeric and nonpolymeric materials; and (e) high dissolved solid (mainly primary and secondary ines or material passing a 200 mesh screen). hese characteristics

140 Advanced Engineering Materials and Modeling Sugarcane Bagasse SCB

Depithing

Pith

Old Corrugated Containers (OCC)

Repulping

Chemical and morphological composition

SCB Fiber Chemical and morphological composition

OCC Pulp

Mixed fibers

White liquor (NaOH)

Black liquor

Optical, mechanical and drainability properties

Bleached SCB Unbleached SCB from pulp industry

ECF Bleaching

Synergistic delignification SCB-OCC

SCB-OCC Pulps

Mechanical and drainability properties

Figure 4.5 Experimental.

could be derived from two reasons. First OCC contains various diferent kinds of pulps such as coated waste and MOW, which usually cause problems with the control of paper machine parameters as runnability and resulting paper qualities. he second is a severe horniication due to the increased number of recycling. herefore, the simple separation of ines with whole or slit-type fractionators is not believed to give the desired separation of the ines and contaminants from the OCC furnish. he loss in tensile or burst strength of chemical pulps from OCC is about 30% ater a few times recycling. Tear strength, by contrast, has a gain, which is about 30%. hese strength changes are caused by the loss in iber-to-iber bonds [61]. Many attempts have been made and designed to improve and/or modify the pulp characteristics and to increase the suitability of recycled ibers for papermaking. hese include various techniques of beating and chemical

Sustainable Sugarcane Bagasse Cellulose for Papermaking

141

treatment of recycled ibers. here are four possible ways to recover the loss of bonding of recycled ibers: (1) beating and reining for internal ibrillation but reining does not completely reverse horniication, (2) chemical treatment, (3) blending with virgin ibers, (4) iber fractionation, and (5) chemical treatment. Each method has its own advantages and disadvantages [62, 63] Traditional methods to increase the strength of OCC recycled paper involve adding dry strength agents and reining. Fibers are also shortened during reining, and more ines are produced, which leads to drainage problems and afects the productivity of the paper machine; therefore, current recycling processes have a limited capability to produce high-quality iber from the available recycle grades and require high-cost upgraded raw material [64, 65] hus, in many cases, it is necessary to upgrade OCC pulps before reusing them, removing contaminants, brightening the pulp, and improving the papermaking properties of the ibers. Upgraded pulps can then be used for the production of higher value-added products such as folding boxboard, and even writing and printing papers [66, 67]. It was observed that strength properties of recycled paper are improved ater the treatment of oxygen deligniication resulting from lignin, extractives, and chemical removal with the treatments [68–71]. Deligniication and bleaching are keys to upgrading of OCC iber potential for papermaking alone or mixtures with wood and nonwood iber as bagasse, straw, bamboo or kenaf, and other pulps grades [72–75].

4.13 Synergistic Deligniication SCB–OCC Fresh bagasse obtained from the harvest period in a sugar mill from Jalisco, Mexico was depithing to 87/13 ratio iber/pith, which the cooking trials with furnishes SCB–OCC: 100% SCB, 90/10% SCB–OCC, 80/20% SCB–OCC, and 70/30% SCB–OCC were carried out in a batch stainless steel rotary digester, electrically heated with temperature control system, and maximum pressure of 8.5 kg cm–2. he white liquor used was prepared from sodium hydroxide (NaOH) at 13% as Na2O based on O.D. weight of pulp with liquor to iber ratio 10:1, reaction time 30 min, and temperature 170 °C. At the end of pulping (lignin removal), the pulps were washed, disintegrated in a laboratory pulp mixer, and screened.1 1 Additionally, unbleached and bleached SCB soda pulp (100% OLD SCB) prepared in a local Pulp and Paper Mill located in Veracruz Mexico, under conventional industrial conditions: depithing, storing, pulping to conventional soda-based process and bleaching by CEH sequence (chlorination in acidic, alkaline extraction and alkaline hypochlorite bleach) was evaluated.

142 Advanced Engineering Materials and Modeling Table 4.1 he TAPPI test for P&P properties. Testing Kappa number of pulp

T 236 cm-85

Fiber Length of Pulp by Classiication (Bauer-McNett)

T-233-cm-82

Forming hand sheets for physical test of pulp

T-205-om-88

Standard conditioning and testing atmospheres for paper, board, pulp hand sheets, and related products

T-402-om-93

Basis weight or grammage

T410-om-98

Tensile strength

T494 om-96

Bursting strength of paper

T-403-om-91

Internal tearing resistance of paper

T-414-om-04

Forming hand sheets for relectance testing of pulp

T-218-om-91

Brightness of pulp, paper, and paperboard

TAPPI 525-om-98

Opacity

T-519-om-96

CED viscosity

T-230-om-89

Brightness reversion

T-um-200

Folding endurance

T-423

Pulp beating

T-248-wd-97

Reining level

SCAN-m3:65

Drainage time

T-205-sp-95

Porosity, air permeability (Gurley method)

T-460-om-96

he papermaking potential, pulp properties, and yield of SCB–OCC furnishes and industrial SCB pulp (100% OLD–SCB) were determined according to the Technical Association of the Pulp and Paper Industry (TAPPI) test (Table 4.1). Pulps were reined at a consistency of 10% to obtain a reining curve, later forming hand sheets, respectively, based on 60 g/m2 for physical tests of strength and drainability properties (Tables 4.2–4.9 and Figures 4.6–4.13). It was found that the blending of OCC pulp with SCB iber, since the deligniication stage, gave well-balanced pulp in strength and drainability properties because the alkali treatment of deligniication, increased tear, tensile and

Sustainable Sugarcane Bagasse Cellulose for Papermaking

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Table 4.2 Pulp-reining degree (Schopper–Riegler °S.R.). Reining (min) 0 4 8 12 16 20

100% 100% FRESH 90/10% 80/20% SCB OLD–SCB SCB–OCC SCB–OCC 17 14 14 13.5 24 20 21 21 32 26 27 29 48 32 34 33 61 38 36 42 66 51 42 46

70/30% SCB–OCC 13.5 19 25 30 31 47

Table 4.3 Freeness (mL C.S.F.) Reining (min) 0 4 8 12 16 20

100% OLD–SCB 653 521 401 228 129 97

100% FRESH SCB

90/10% 80/20% SCB–OCC SCB–OCC 721 733 574 574 472 443 375 388 351 285 285 247

70/30% SCB–OCC 733 612 504 428 415 237

100% 100% FRESH 90/10% 80/20% SCB OLD–SCB SCB–OCC SCB–OCC 3672 5616 4462 4610 4243 7763 8240 7618 6643 8939 8744 8280 6546 9145 8825 9036 6324 9110 8986 9244 7687 9256 9302 8885

70/30% SCB–OCC 4365 7303 8198 8348 8783 8421

721 593 488 401 328 203

Table 4.4 Breaking length (m). Reining (min) 0 4 8 12 16 20

Table 4.5 Burst index (kPam2/g). Reining (min) 0 4 8 12 16 20

100% OLD–SCB 2.53 3.47 4.72 4.94 5.43 5.61

100% FRESH SCB 3.59 5.56 6.77 7.22 7.32 7.58

90/10% SCB–OCC 2.92 5.75 6.07 6.15 7.21 6.75

80/20% SCB–OCC 3.14 5.74 6.35 6.65 7.64 6.80

70/30% SCB–OCC 3.13 5.68 6.25 6.33 6.53 6.48

144 Advanced Engineering Materials and Modeling Table 4.6 Tear index (mNm2/g). Reining (min) 0 4 8 12 16 20

100% OLD–SCB

100% FRESH SCB

90/10% SCB–OCC

80/20% SCB–OCC

70/30% SCB–OCC

9.15 8.87 8.51 8.31 7.85 6.54

7.85 10.06 11.05 9.80 9.10 8.28

16.83 11.12 10.46 9.82 9.75 9.70

21.45 12.09 10.78 10.29 9.20 9.15

28.67 13.72 12.76 11.76 11.11 11.11

Table 4.7 Folding endurance (double folds). Reining (min) 0 4 8 12 16 20

100% 100% 90/10% 80/20% FRESH SCB OLD–SCB SCB–OCC SCB–OCC 30 213 171 158 84 156 172 206 220

362 521 688 722 869

428 567 756 624 580

688 871 973 657 525

70/30% SCB–OCC 125 618 708 729 857 556

Table 4.8 Porosity (s/100 cm3). Reining (min) 0 4 8 12 16 20

100% 100% 90/10% 80/20% FRESH SCB OLD–SCB SCB–OCC SCB–OCC 5 6 2.0 2.0 26 38 34.6 18.7 53 82 37.5 30.7 79 138 59.1 38.7 135 239 92.7 78.4 410 343 93.3 75.8

70/30% SCB–OCC 2.0 9.8 14.0 17.5 32.5 19.5

Table 4.9 Drainage time of pulp (s). Reining (min) 0 4 8 12 16 20

100% 100% 90/10% 80/20% OLD–SCB FRESH SCB SCB–OCC SCB–OCC 4.75 4.55 4.25 4.27 5.93 7.70 9.07 12.08 16.01

5.76 6.90 8.05 8.92 11.39

5.74 6.46 7.28 8.69 8.19

5.31 6.26 6.94 8.29 8.65

70/30% SCB–OCC 4.29 5.12 5.50 5.86 6.34 6.33

Pulp-reining degree (Schopper–Riegler °SR)

Sustainable Sugarcane Bagasse Cellulose for Papermaking 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

0

4

8 12 Reining time (Minutes)

100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

16

20

90/10 SCB-OCC

Figure 4.6 Pulp-reining degree (Schopper–Riegler °S.R.).

800 700 Freeness (mL)

600 500 400 300 200 100 0

0

4

8 12 Reining time (Minutes)

100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

Figure 4.7 Freeness (mL C.S.F.).

16

20

90/10 SCB-OCC

145

146 Advanced Engineering Materials and Modeling 10000

Breaking Length (m)

9000 8000 7000 6000 5000 4000 3000 2000 1000 0

0

4

8

12

16

20

Reining time (Minutes) 100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

90/10 SCB-OCC

Figure 4.8 Breaking length (m).

8.00

Burst index (kPam2/g)

7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

0

4

8 12 Reining time (Minutes)

100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

Figure 4.9 Burst index (kPam2/g).

16

20

90/10 SCB-OCC

Sustainable Sugarcane Bagasse Cellulose for Papermaking 30.00 27.00 Tear Index(mNm2/g)

24.00 21.00 18.00 15.00 12.00 9.00 6.00 3.00 0.00

0

4

8

12

16

20

Reining time (Minutes) 100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

90/10 SCB-OCC

Folding endurance (Double folds)

Figure 4.10 Tear index (mNm2/g).

900 750 600 450 300 150 0

0

4

8

12

16

Reining time (Minutes) 100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

Figure 4.11 Folding endurance (double folds).

90/10 SCB-OCC

20

147

148 Advanced Engineering Materials and Modeling 450 Gurley Porosity (sec/100cm3)

400 350 300 250 200 150 100 50 0

0

4

8 12 Reining time (Minutes)

100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

16

20

90/10 SCB-OCC

Drainage time of pulp (Sec)

Figure 4.12 Porosity (s/100 cm3).

16.50 16.00 15.50 15.00 14.50 14.00 13.50 13.00 12.50 12.00 11.50 11.00 10.50 10.00 9.50 9.00 8.50 8.00 7.50 7.00 6.50 6.00 5.50 5.00 4.50 4.00

0

4

8

12

16

Reining time (Minutes) 100 % OLD SCB

100 % FRESH SCB

80/20 SCB-OCC

70/30 SCB-OCC

Figure 4.13 Drainage time of pulp (s).

90/10 SCB-OCC

20

Sustainable Sugarcane Bagasse Cellulose for Papermaking

149

burst strength at diferent levels of reining in pulps. Besides, the drainage time decreased considerably and porosity of pulps was increased, because the inter-iber bonding and long iber proportion were increased with more lexible OCC pulps and chemical pulping provided relatively cleaner surfaces, which is consistent with the removal of wax, pectin, lignin, and hemicelluloses he highest tensile strength, burst, and folding endurance were likely obtained by 80/20, and tear, drainage time of pulp, and porosity by 70/30 and 90/10 were similar to FRESH SCB. In general, SCB serves as a bonding component between long ibers from OCC in the mixture, because strength properties are mainly inluenced by iber length and extend of inter-iber bonding as a result of deligniication process of both ibers, also that elimination of supericial layer was able to increase the contact area because the ibrils became more exposed. his will be very important for pulp washers, runnability in paper machine, press room, and converting machine. Besides, the SCB pulps were separated in the Bauer-McNett classiier into ive iber length fractions, and the respective mass percentages are represented in Table 4.10. For SCB–OCC pulps, the largest fraction corresponds to the longest ibers (30 and 50 mesh), but the ines (200 mesh) are the third largest fraction. he positive increase in the longest ibers (30 and 50 mesh) will have a synergistic efect in strength and drainability properties. hey will expand the raw SCB–OCC pulp in diferent types of paper with several characteristics and end use comparable to wood pulp. Nowadays, cellulose from SCB can be applied in many products, for example, composites, chemical derivatives, and others. In relation to the OLD–SCB pulp, the higher reining values (low freeness and greater S.R.) in a shorter time, are due to the breaking of cells, higher ines generation, and lower proportion of long ibers that FRESH SCB and SCB–OCC bagasse pulp due to the processes of degradation by depithing and storage. In SCB–OCC pulp, the reining tended toward lexibility, swelling and no cutting and ibrillation instead OLD–SCB pulp to Table 4.10 Pulp classiication Mesh 30 50 100

100% OLD–SCB 0.5 26.9 30.6

100% FRESH SCB 18.9 34.6 26.1

90/10% SCB–OCC 27.7 22.2 26.5

80/20% SCB–OCC 29.1 24.9 20.9

70/30% SCB–OCC 32 27.2 11.8

200 1, the q-Gaussian fade away at r = and q − 1) ( 1 . he probability density ρ(r) = |ψ (r) |2 related equal zero for r > (q − 1) to the wave function maximizes Tsallis’ power-law entropic

Sq =

(

1 1− ∫ q −1

q

dr

)

(6.7)

196 Advanced Engineering Materials and Modeling under the restriction of the expectation and normalization value of r2. Using Eq. (6) into Eq. (6.1), the potential in the ground state can be written as

V=

− D + r 2 ( D(q − 1) + 3 − 2q ) 2

(

1 − (q − 1) r 2

)

2

− Eg 0 +

1 2

(6.8)

When q ≤ 1 the potential V is inite for all r ∈ D. Besides, when q > 1, the potential function is singular when r takes the value

r* =

1 (q − 1)

(6.9)

Physically, when q > 1, Eq. (6.8) has an ‘‘ininite wall’’ at r = r* and the quantum particle is restricted on the region r ≤ r*. herefore, the wave function Eq. (6.6) fade away at r = r*, and equal to zero when r ≥ r*. his gives an example of the Tsallis cut-of condition [10]. When q → 1, the q-Gaussian wave function Eq. (6.6) turn into standard Gaussian and the potential given in Eq. (6.8) decreases to the D-dimensional isotropic harmonic oscillator potential

V (r ) = −

D 1 + 2 2

2 2

r − Eg 0 +

1 2

(6.10)

his estimation becomes accurate in the limit q→ 1.

6.3 Ground States in the Case of Momentum Space In this space we consider the q-Gaussian wave function 1 2 2(q −1)

( p) = C(1 − (q − 1) p )

,

(6.11)

D

where p2 = ∑ pi2 , and q and β are positive factors, and C is normalization i =1

constant. We consider q-Gaussians in momentum space with q < 1. Now, applying the Fourier transform of the Eq. (6.11)

1 (r ) = 2

∞

∫

−∞

( p) eipr dp

(6.12)

197

Calculation on the Ground State Quantum Potentials then we can write the wave function as

(I −v (r ) − Iv (r )) =

v

1 r v (r ) = Γ(v ) 2

sin(v )

v

1 r Γ(v ) 2

K v (r ) ,

(6.13)

where Kν is known as the modiied Bessel function of the 2nd kind with ∞

(−1)m 1 J v (r ) = ∑ r m= 0 m ! Γ (m + v + 1) 2

2m+ v

(6.14)

and

J − v (r ) = (−1)v J v (r )

(6.15)

with r = |r|. he derivation rule for the function ψν (r) is given as

1 ∂ r ∂r

v (r )

=−

1 2(v − 1)

v −1 (r )

(6.16)

where

∂ K v (r ) ∂r

=−

1 K (r ) + K v +1 (r ) . 2 v −1

(

)

(6.17)

Using Eq. (6.13) into Eq. (6.1) we get

−

1 2r

D −1

∂ D −1 ∂ v (r ) r + V (r ) ∂r ∂r

v (r )

1 = (− Eg 0 − ) 2

v (r )

(6.18)

this gives −

1 2r

D −1

(D − 1)r D −2

∂

v (r )

∂r

+ r D −1

∂2

v (r ) 2

∂r

+ V (r )

v (r )

1 = ( − Eg 0 − ) 2

v (r )

(6.19) hence

1 (D − 1) ∂ v (r ) ∂2 v (r ) + − + V (r ) r ∂r 2 ∂r2

v (r )

1 = (− Eg 0 − ) 2

v (r )

(6.20)

198 Advanced Engineering Materials and Modeling therefore −

1 (D − 1) 2 −2(v − 1)

v −1 (r ) −

1 2(v − 1)

v −1 (r ) +

1 r2 4(v − 1)(v − 2)

+ V (r )

v (r )

v −2 (r )

1 = (− Eg 0 − ) 2

v (r )

(6.21)

Consequently

−

D 1 2 −2(v − 1)

v −1 (r ) +

1 r2 4(v − 1)(v − 2)

+ V (r )

v (r )

v −2 (r )

1 − Eg 0 − ) = (− 2

v (r )

(6.22)

Now, the Bessel function Kν follows the diference equation

rK v (r ) = rK v −2 (r ) + 2(v − 1)K v −1 (r )

(6.23)

r v K v (r ) = r 2r v −2 K v −2 (r ) + 2(v − 1)r v −1K v −1 (r )

(6.24)

hence

and v (r )

= r2

1 4(v − 1)(v − 2)

v − 2 (r ) +

v −1 (r )

(6.25)

Now, we can rewrite Eq. (6.22) as

−

D 1 2 −2(v − 1)

v −1 (r ) −

v −1 (r ) +

v −1 (r ) +

1 r2 4(v − 1)(v − 2)

+ V (r )

v (r )

v −2 (r )

1 = (− Eg 0 − ) 2

v (r )

(6.26) then

−

1 2

−D −1 2(v − 1)

v −1 (r ) +

v (r )

+ V (r )

v (r )

1 = (− Eg 0 − ) 2

v (r )

(6.27)

199

Calculation on the Ground State Quantum Potentials herefore, ψν (r) is an eigenfunction of the potential

V (r ) = −

v −1 (r )

D 1 +1 2 2(v − 1)

with eigenvalue equal to − Eg 0 +

v (r )

− Eg 0

(6.28)

1 . 2

6.4 Results and Discussion For v =1/2, the potential given in Eq. (6.28) reads

Vv (r )

(2(v − 1) + D ) r

1+

1 (1 − 2v ) + ....... − Eg 0 2r

(6.29)

In Table 6.1 some of potential results from diferent half-integer values of ν are listed. hese potentials are depicted in Figure 6.1 for two sulphur concentrations (x = 0.12 and 0.9). Table 6.1 he potential function Vν(r) corresponding to the values of the parameter q, as a function of the space dimension D, for various half-integer values of the parameter ν. v

q

Vv

1/2

D + 2 Eg 0 + 1 D + 2 Eg 0 + 2

1 − − Eg 0 r

3/2

D + 2 Eg 0 + 3 D + 2 Eg 0 + 4

−

1 1+ D − Eg 0 2 1+ r

5/2

D + 2 Eg 0 + 5 D + 2 Eg 0 + 6

−

1 2

D + 2 Eg 0 + 7 D + 2 Eg 0 + 8

−

2 1 (5 + D ) r + 3r + 3 − Eg 0 2 r 3 + 6r 2 + 15r + 15

7/2

(3 + D )(r + 1) r 2 + 3r + 3

(

− Eg 0

)

For a ixed value of d, we get q→ 1 when D → ∞. When the space dimensions → ∞, q-Gaussian describes ground state in the momentum space access a standard Gaussian.

200 Advanced Engineering Materials and Modeling r –2.3

0

2.5

5

7.5

10 12.5 15

17.5 20 22.5 2.5

–2.6

= 3/2, x = 0.12, D = 1

–2.9

= 3/2, x = 0.12, D = 3

–3.2 V (r)

= 3/2, x = 0.12, D = 1

= 3/2, x = 0.9, D = 3

–3.5

= 5/2, x = 0.12, D = 1

–3.8

= 5/2, x = 0.9, D = 1

–4.1

= 5/2, x = 0.12, D = 3

–4.4

= 5/2, x = 0.9, D = 3

–4.7

Figure 6.1 he potential functions Vν appearing in Table 6.1, corresponding to ν equal to 3/2 and 5/2 for x = 0.12 and x = 0.9 with D = 1 and 3. 2.7 2.6

V(r)

2.5 2.4 2.3 2.2 2.1 2 2.8

2.9

3

3.1 Eg0

3.2

3.3

3.4

Figure 6.2 D-dimensional isotropic harmonic oscillator potential V(r) as a function of Eg0 for ZnSxSe1-x.

When ν = 1/2+ d, with d integer, the parameter q characterizing the q-Gaussian is represented as

q=

D + 2 Eg 0 + 2d + 1 . D + 2 Eg 0 + 2d + 2

Vv(r) decreases with increase in D for constant v and x and increase with increases in x for constant v and D. Figure 6.2 shows the decreasing of D-dimensional isotropic potential, V(r) when the unperturbed energy band gap on the ZnSxSe1-x is increasing. Since Eg0 increase with sulphur content [17, 18] it can be inferred that V(r) decreases with increasing of sulphur concentration.

Calculation on the Ground State Quantum Potentials

201

6.5 Conclusions We have been speciied the potential functions in D-dimensional coniguration space momentum space with the ground states of the q-Gaussian form for ZnSxSe1-x. he q-Gaussian ground states in coniguration space, we speciied the D-dimensional isotropic harmonic oscillator potential as a speciic case related to q → 1. Furthermore, we obtained a family of potentials for ZnSxSe1-x which can help to understand this potential in the ground state when taking into account the ground states having the form of a q-Gaussian in momentum space.

Acknowledgements his work was supported by funds from University of Malaya PPP grant Project No.PS331/2014A.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

C.Y. Yeh, S.H. Wei, and A. Zunger, Phys. Rev. B, 50, 2715, 1994. M. Van Schlfgaarde, A. Sher, and A. B. Chen, J. Cryst. Growth, 178, 8, 1997. L. Sugiura, J. Appl. Phys. 8, 11633, 1997. A. Garcia and M. L. Cohen, Phys. Rev. B, 47, 4215, 1993. Chin-Yu Yeh, Z. W. Lu, S. Froyen, and A. Zunger, Phys. Rev. B, 45, (1992) 12130; 46 (1992) 10086. J. C. Phillips, Bonds and Bands in Semiconductors, Academic, (New York) 1973. J. St. John and A. N. Bloch, Phys. Rev. Lett., 33, 1095, 1974. J. R. Chelikowsky and J. C. Phillips, Phys. Rev. B, 17, 2453, 1978. M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy: Interdisciplinary Applications, (Oxford, Oxford University Press) 2004. C. Tsallis, Introduction to Nonextensive Statistical Mechanics, (New York, Approaching a Complex World Springer) 2009. J. Naudts, Generalized hermostatistics, (New York, Springer) 2011. A. Nagy, E. Romera, Phys. Lett. A 373, 844, 2009. A.R. Plastino, M. Casas, A. Plastino, A. Puente, Phys. Rev. 52, 2601, 1995. A.R. Plastino, A. Plastino, Phys. Lett. A 181, 446, 1993. A.M.C. Souza, Phys. A 390, 2686, 2011. haller B., Advanced Visual Quantum Mechanics, ( Verlag, Springer) (2004). Ghassan H.E. Al-Shabeeb and A.K. Arof, Materials Research Innovations, 15, S2, 132–136, 2011. Ghassan H.E. Al-Shabeeb and A.K. Arof, Eur. Phys. J. Plus, 128, 153, 2013.

7 Application of First Principles heory to the Design of Advanced Titanium Alloys Y. Song1*, J. H. Dai1, and R. Yang2 1

School of Materials Science and Engineering, Harbin Institute of Technology at Weihai, China 2 Institute of Metal Research, Chinese Academy of Sciences, Shenyang, China

Abstract First principles calculations have become a powerful tool to explore the physical and chemical properties of materials. It provides fundamental information in exploring and designing advanced materials as modern technologies. It has been successfully applied to predict new materials such as cubic BN, superconductors, electrode, and so on and also used to investigate material phenomena such as the mechanical properties, phase transformations, difusion, and so on. Titaniumbased alloys have shown promising applications as a class of high-temperature structural materials due to their low density, high speciic strength, and good high-temperature creep resistance. hey have been applied as new-generation biomedical materials with relatively low modulus and non-toxicity. his chapter will focus on the application of irst principles theory for developing new titaniumbased alloys. he methods to evaluate the mechanical properties, especially the elastic, and phase stability from irst principles total energy calculations will be introduced. Applications of irst principles to the design of new titanium alloys such as the high-strength titanium alloys and low-modulus titanium alloys will be discussed in detail combining with experimental researches. Keywords: Alloy design, titanium alloy, irst principles

7.1 Introduction Materials design is a process to determine optimal combination of material chemistry, processing routes, and parameters to robustly meet speciic *Corresponding author: [email protected] Ashutosh Tiwari, N. Arul Murugan, and Rajeev Ahuja (eds.) Advanced Engineering Materials and Modeling, (203–228) © 2016 Scrivener Publishing LLC

203

204 Advanced Engineering Materials and Modeling performance requirements such as mechanical, chemical, and thermal properties. Recent advances in computer calculation techniques and hardware technology have accelerated theoretical approaches to develop new materials. Calculations based on density functional theory (DFT) can be used to predict physical behavior that originates from the nature of atomic bonding, providing a basic insight into the electronic tendencies as a function of compositions and a guideline to future alloy design approaches in contrast to commonly applied in the ield of metallurgical alloy design, reduce the huge eforts commonly associated with experiments to screen alloy, and reveal shortcuts to advanced alloy, as opposed to metallurgical trial-and-error methods. he DFT provides a scale-jumping modeling approach to directly predict intrinsic properties, such as structure and elasticity, of materials barely relating to mesoscale methods. It can be applied to predict certain aspects of the plastic behavior of materials, such as ideal strengths or dislocation behaviors of structural materials. However, history of applying DFT to predicting properties of a material is relatively short. One fundamental step for materials design is to predict the stable structures of alloys from their constituents. he traditional approach to this problem is to extract trends from available experimental data and apply them to uncharacterized systems. hese rules usually depend on simple parameters, e.g., atomic number, atomic radius, electronegativity, ionization energy, and melting temperature or enthalpy. Several well-known methods include the Hume-Rothery rules [1], the Miedema formation enthalpy method [2], the Pettifor maps [3], the molecular orbital method [4, 5], and more recently the high-throughput calculation approach [6]. Nowadays, DFT has become a powerful tool in materials research and design.

7.2 Basic Concepts of First Principles In general, the atomistic principles underlying the structure and functional behavior of materials are astonishingly simple: (a) for most purposes, atomic nuclei can be treated as classical particles with a given mass and positive charge; (b) electrons are particles of half spin, thus obeying the Pauli exclusion principle, and their kinetic behaviors are described by quantum mechanics; and (c) the only relevant interactions are of an electrodynamics nature, in particular, attractions and repulsions governed by Coulomb’s law. Based on these fundamental principles, which are oten called the irst principles, it is conceptually possible to explain and predict the wonderful richness of most physical and all chemical properties of

Application of First Principles Theory to the Design 205 matter such as the structure and stability of crystalline phase. he concept that the mechanical properties of matter are being determined by the electronic structure is essential to modern materials science. he electronic structure of matter can be obtained by solving the Schrödinger equation. Due to the nonlocal exchange and correlation interaction of electrons, it is diicult to obtain an exact solution of the Schrödinger equation for multi-particles systems. An efective tool to solve the Schrödinger equation is the density functional approach. DFT is primarily a theory of the electronic structure of atoms, molecules, and solids in their ground states (GSs), in which the electronic density distribution (r) plays the central role in determination of the GS and other properties of a system of electrons in an external ield. A series of approximations have been made to solve the Schrödinger equation for systems of interest in condensed matter physics or in materials science, while the suicient accuracy is kept to provide a reliable description of studied systems and processes. One of the widely used treatments is the Born–Oppenheimer approximation (also called adiabatic approximation) relying on the fact that the mass of the nuclei is three to four orders of magnitudes greater than the mass of electrons. Consequently, any change in positions of nuclei is almost immediately (adiabatically) followed by electrons. hus, to a good approximation, the nuclei are considered as ixed in static conigurations and the electrons are treated in the ield of the stationary nuclei. his allows us, in this approximation, to separate the wavefunction into a product of nuclear and electronic terms. Here, the nuclei may be considered as classical particles, and their positions can be taken as parameters that appear in the potential of the electron part of the Schrödinger equation. Electron states in a solid may be described by the non-relativistic many-electron Schrödinger equation:

H

E

(7.1)

with H being the electronic Hamiltonian with the nuclei at the positions, Ψ being the many-electron wave function, and E being the energy of the system. Eq. (7.1) is an N-electron problem. Virtually, all the approximations to the N-particle Hamiltonian have been aimed at constructing an accurate Hamiltonian for a single electron and to approximate the true manyelectron GS wave functions in terms of one-electron wave functions. With two theorems proposed by Hohenberg and Kohn, it can be reduced to the single-electron problem. he irst theorem is that the single-electron density (r) as a variable is suicient to describe the GS of a system

206 Advanced Engineering Materials and Modeling with interacting electrons. According to this theorem, the ground-state single-particle density (r) implicitly determines (to within a trivial constant) the external potential Vext(r) acting on the electron system. here is a one-to-one relationship between the external potential Vext(r) and the (nondegenerate) GS wave function Ψ, and there is a one-to-one relationship between Ψ and the GS density (r) of an N-electron system,

(r )

N dr2

(r , r2 ,

drN

rN ) (r , rn ,

rN )

(7.2)

Knowledge of the density then determines the external potential to within a constant so that all terms in the Hamiltonian are known. All ground-state characteristics of the system in general and the total groundstate energy in particular may, therefore, be considered as a function of the single-particle density (r). he second theorem is the variational principle that the total energy of the N-electron system, E[ ], is minimized by the ground-state electron density:

EGS

min E[ ,Vext ] (r )

(7.3)

Eq. (7.3) is key to the atomic-scale understanding of electronic, structural, and dynamic properties of matter. Furthermore, the derivative of the total energy with respect to the nuclear position of an atom gives the force acting on that atom, i.e.:

F

R

E( (r ), {R })

(7.4)

Within the Kohn–Sham (KS) scheme [11], the complicated manybody problems are simpliied to a non-interacting system. he KS scheme attempts to calculate the energy functional, which depends on the electron density. For an efective one-electron theory, the electrons move in an efective potential that is constructed to generate the ground-state density for these electrons,

E[ ρ (r )] = T[ ρ (r )] + ∫ ρ (r )v(r )d 3r +

1 ρ (r )ρ (r ) 3 3 e2 d rd r + Exc [ ρ (r )] ∫∫ 2 |r − r |

(7.5)

Here, T[ (r)] is the kinetic of one-particle electrons, the second term of the right-hand side contains interactions between the electrons and an external

Application of First Principles Theory to the Design 207 potential, and the third term is the classical Hartree interaction. he last term, the exchange-correlation energy, as functions of the electron density (r), is deined as the diference between the true (unknown) energy functional and the three irst terms on the right-hand side of Eq. (7.5). he KS equation with an efective potential Vef for the ith particle is given by

1 − ∇2 + Veff ψ i = Eiψ i 2 Veff (r ) = v(r ) + ∫ e 2 xc

(r ) =

ρ (r ) 3 dr + |r − r |

(7.6)

xc

(r )

dExc [ ρ (r )] d ρ (r )

N

ρ (r ) = ∑ fi |ψ i (r )|2

(7.7) (7.8) (7.9)

i =1

where xc is the so-called exchange-correlation potential. Eq. (7.8) is formally exact in the sense that it does not contain any approximation to the complete many-body interaction. However, the exchange-correlation energy is not known. To solve the KS equation (7.6), various approximations for the exchangecorrelation energy and the expansions of the wave function on a basis are made. Various methods used in the electronic structure calculations may be distinguished according to the choice of the basis functions and the form of the efective potential. In the most general case, the efective potential is constructed from all electrons, and no additional approximation is made to its shape. From the viewpoint of practical calculation, this is not a very convenient means. So, some approximations were made on the efective potential, such as the jellium model, pseudopotential, muin-tin approximation, and so on. he better we choose basis functions, the smaller number of them is needed for a description of the one-electron wave functions.

7.3 7.3.1

heoretical Models of Alloy Design he Hume-Rothery heory

he foundation stone for the application of the electron theory to alloy design was laid by William Hume-Rothery in 1926, based on studies conducted on certain alloys of the group IB metals, i.e., Cu, Ag, and Au

208 Advanced Engineering Materials and Modeling alloys [7]. Study on the stability of Cu–Zn alloys showed that for e/a = 1.5 the structure of the compounds is body-centered cubic, but changes to the γ-brass type for e/a = 1.61; with further increase in the e/a ratio to 1.75, the hexagonal close-packed structure becomes stable [7]. hus, for this kind of compounds, the electron concentration determines the structure, which is called the Hume-Rothery electron compounds. Hume-Rothery listed four factors that are important in relation to the electronic structure of alloys: (a) the diference between the electro negativities of two metals, (b) a tendency for elements near the end of short periods and B subgroups to complete their octet of electrons, and a similar tendency to ill the d shell in later transition elements, (c) orbital-type restrictions in structures with certain types of hybrid bonding, and (d) the formation of deinite crystal structures at characteristic e/a ratios [8], and proposed that there existed a correspondence between the crystal structures and the average number of valence electrons per atom, i.e. e/a ratio, of these alloys. He pointed out the importance of e/a ratio in controlling the phase stability and phase boundaries in binary alloys [9]. Extensive researches show the relationships between e/a and physical properties of materials, such as the stability of solid solutions, the formation of intermetallics, the stacking fault energy, and the elastic moduli. he extent and diversity of the relationship of e/a ratio to physical properties suggest that it may be a useful parameter for optimization of physical properties. Application of the Hume-Rothery theory to the prediction of solid solubility and phase stability of metals and binary alloys has gained great successes. Hume-Rothery and co-workers clariied the factors limiting the solid solubility of solutes, the electrochemical nature, the diference in the atomic size and the valence number, and evaluated their inluence on the solid solubility of various alloying elements in copper and silver alloys. Results illustrate that when atomic size misit and electrochemical diference are small, the e/a ratio exerts a limited inluence on the extent of the solid solubility. For the vacancy containing phase, Tiwari et al. [10] proposed an expression linking the formation energy of vacancy (Ev) and the e/a ratio using the standard relation between the Fermi energy and the e/a ratio,

Ev = K 2[e /a]5/3 L−2

(7.10)

where L is the lattice parameter and K2 is another constant. hey further extended this relationship to activation energy for difusion (Q) because for a group of solids having identical physical and chemical properties, the ratio between Ev and Q is a constant. Figure 7.1 shows that the linearity for

Application of First Principles Theory to the Design 209

QsolventL2 (10–14 Jmol–1 m2)

3.2

3.0

AgSb

2.8

AgTI

AgCd

2.6

AgIn 2.4 1.0

1.1

1.2 (e/a)5/3

1.3

1.4

Figure 7.1 Activation energy for self-difusion of solvent versus the e/a ratio for silver alloy system (from Ref. [10]).

individual alloy systems is maintained. However, these lines are not superimposed implying that the e/a ratio is not the only factor inluencing Q in an alloy. Many attempts have been made to apply the electron theory approach to achieve a better understanding of phase transformation in alloys and compounds. One such area of application is in the martensitic transformation in shape memory alloys. Researchers have studied the inluence of the electron concentration in the Cu–Al–Mn alloy systems on the ultimate martensitic structure of the alloys [11]. In the studied Cu–A1–Mn alloys, which had manganese contents in the range of 4.4–61.7%, a relationship between the martensite phase structure and the electron concentration in the alloy was observed. he martensite structure was found to depend on the electron concentration (e/a); a greater value of e/a than 1.45 yielded the 2H structure, while for values less than 1.45 an 18R structure is stable. In Cu–Zn–A1, Cu–A1–Ni, and Cu–A1–Be alloys the entropy diferences, S, caused by the martensitic transformation increase with e/a leading to a relationship [12]:

S = (0.345 ± 0.012)e/a – 0.337[kB] with kB being the Boltzmann constant.

(7.11)

210 Advanced Engineering Materials and Modeling While the role of e/a ratio in phase stability is well catalogued, it is not oten recognized that this ratio can afect the mechanical behavior. he increasing of the yield stress of the alloy and the strengthening produced by the solute additions are designated as solid solution hardening. he tensile behavior of the copper based zinc, gallium, germanium, and arsenic alloys with approximately identical grain size showed similar true stress strain curves having approximate e/a ratio of 1.087 [13]. Experimental evidence for the e/a ratio as controlling parameter in plastic deformation of dilute copper-based alloys, which have nearly identical grain size as well as lattice parameter to eliminate any possible inluence of these factors in comparison with plastic yielding behavior, can be seen in Figure 7.2 [14]. Similar to the linear relationship between the yield stress and the e/a ratio in copper-based alloys as shown in Figure 7.2, the e/a ratio shows a fairly linear relationship with the bulk modulus. Cho studied the correlation between the bulk modulus and the electron concentration, the number of valence electrons per volume, of pure metals as shown in Figure 7.3 [15]. he inluence of e/a on the elastic constants in Zr–Nb–Mo–Re alloy systems has been examined in detail [16]. In these alloy systems, alloying is merely to change the density of the negative charge and the shape of band. In this case, when the elastic constants are plotted against e/a, there are no singularities and the relationship is properly represented by a straight line (Figure 7.4). he singularity in the e/a versus elastic

Yield stress (N/m2X10–4)

3500 3000

Ge

2500

Al

2000

Ga

1500 Zn

1000 500 0

1.10

1.15

1.20

1.25

(e/a)

Figure 7.2 Variation of yield stress values for copper-based alloys of equal lattice parameters via electron concentration (from Ref. [14] ater Hibbard Jr.).

Application of First Principles Theory to the Design 211 20

Cr

Bulk modulus (B X 10–9) g/cm–3

Ni V

15

Au

Ti

10

Cu

Be Ag

Zn

Al

BCC FCC HCP Others

Ga Sn In Pb Mg Sc Cd

05

0

Co

Na Ca Cs Sr Li K Ba Ka

Tl

Hg

10

20 30 40 Electron concentration (ne X 10–22) cm–3

50

Figure 7.3 Relationship between bulk modulus and electron concentration of pure metals (from Ref. [15]).

2.8 2.6 2.4

1011N/m2

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8

4

5 6 Electrons / Atom

Figure 7.4 Relationship between elastic constant C11 and electron concentration of Zr–Nb–Mo–Re alloy systems (from Ref. [16]).

212 Advanced Engineering Materials and Modeling constant plot is an indication of a signiicant alteration in the electronic structure. More researches were carried out. Ogwu and Davies have described the relationships between electronic structure and (a) dislocation motion and ductility in metals, intermetallics, semiconductors, and ceramics; (b) formation of martensite in shape memory alloys; (c) hardness of carbides and borides; (d) work of adhesion between metals and ceramics; (e) transformation kinetics in steel; (f) transition metal oxide sintering additives on the densiication of ceramics; and (g) propagation of cavities in superplastic ceramics [17].

7.3.2

Discrete Variational Method and d-Orbital Method

7.3.2.1 Discrete Variational Method he discrete variational method (DVM) is a full numerical self-consistent ield method based on the DFT [18]. he one-electron wave functions of a molecule or an atomic cluster are expressed by atomic orbitals with the numerical type (or other types) obtained from the DFT. he multi-dimensional Diophantus numerical integral method has been introduced into the atomic or molecular systems, and thus the integration of the Hamiltonian and overlap matrix elements can be calculated by a summation over sample points in real space. his is the characteristics of DVM, which distinctively enhances the calculation speed. Similar to the usual DFT, the key problem in calculating the electronic structure of a multi-electron system is to solve the KS equation (7.6):

1 − ∇2 + Veff (r ) ψ i (r ) = ε iψ i (r ) 2

(7.12)

where the one-electron wave function i of a molecule or an atomic cluster is expanded with the basis of a linear combination of atomic orbitals (LCAO) basis j(r),

ψ i = ∑ Cij χ (r )

(7.13)

j

where Cij is the coeicients. he multi-dimension Diophantus numerical integral method is applied to Eq. (7.12) and gives

(H − ε S)C = 0

(7.14)

Application of First Principles Theory to the Design 213 where H, S, and C are the Hamiltonian, overlap, and coeicient matrix, respectively:

H ij = ∫ χi∗ H χ j dr

(7.15)

Sij = ∫ χi∗ χ j dr

(7.16)

he coeicients Cij and the eigenvectors i can be obtained by solving Eq. (7.14) with the self-consistent ield method. Application of the Rayleigh–Ritz variational procedure on a discrete grid of sample points then gives the secular matrix equation (7.14), and the matrix elements were evaluated by using the DVM. A major weakness of variational methods is the necessity of adequate basis function completeness. Since the quality of the basis and the representation of the molecular potential are linked, it is necessary to explore options available for basis generation. he choice of the basis set was mainly specialized to the following cases: (i) he minimal basis set contains the occupied atomic wave functions, which are obtained either from a self-consistent atomic calculation or from a spherical projection of the molecular potential obtained by a repulsive well to bound the n, l state. It is obvious from this construction that the electronic structure can be interpreted easily in terms of these functions, in terms of LCAO or partial wave functions. (ii) It has been known for some time that the basis chosen as above is not accurate enough for most purposes to describe covalent bonds. herefore, the virtual atomic states have been usually used to improve basis convergence. It should be noted that virtual s and p states are almost proportional to the corresponding covalence functions typically up to ~1/2 nearest neighbor distances. A sizable occupation of difuse virtual states is oten found due to large overlaps with states centered on neighboring atoms. hus, one has to be careful about assuming that variational freedom can be enhanced simply by adding further s, p, and d states. (iii) An alternative way of extending a minimal basis is to add a further valence orbital set obtained from a diferent spherical potential, again localized to nuclear sites. A irst way of constructing this basis is to introduce an artiicial change

214 Advanced Engineering Materials and Modeling in the original atomic potential at larger distances. he basis functions obtained this way may be approximately equivalent to the muin-tin-orbitals scheme using energy derivative functions to obtain some further variational freedom.

7.3.2.2 d-Electrons Alloy heory In early 1990s, Morinaga and colleagues proposed a method for theoretical alloy design based on the basis of molecular orbital calculation of electronic structure (DMV cluster method) [4, 5]. he aim of this method is to calculate quantum parameters that take into consideration the alloying element efect for predicting the stable phase of alloys. Two parameters, the bond order (Bo) and the d-orbital level (Md), are proposed. he former corresponds to the overlap population between the host atom and the alloying element M, which is a measure of the covalent bond strength between them. he latter is the mean energy of virtual d-orbitals centered upon the alloying element, which correlates well with the electronegativity and the metallic radius of elements. Each element has a speciic Bo value and a speciic Md value based on calculations using the DV-Xα molecular orbital method as illustrated in Table 7.1 [19]. For multicomponent alloys, the bond order and the d-orbital level are evaluated as the average values weighted with the atomic fraction of each alloying element (xi) [20]: n

Bo = ∑ xi Boi

(7.17)

i =1 n

Md = ∑ xi Mdi

(7.18)

i =1

where n is the number of alloying elements in the alloy. By combining Coulomb’s law and the d electrons alloy theory, the interatomic bonding force of titanium alloys can be estimated by [19]

Fb ∝

Z eff Bo Md 2

(7.19)

where Z eff is the average efective nuclei charge of alloy. Bonding force is related not only to atom types and the distances between atoms, but also to crystal structure. From Table 7.1, it can be seen that the

Application of First Principles Theory to the Design 215 Table 7.1 List of Bo and Md values for various alloying elements in hcp-Ti and bcc-Ti clusters (from Ref. [19]). Bo (hcp-Ti)

Bo (bcc-Ti)

Md (eV)

Ti

3.513

2.790

2.447

V

3.482

2.805

1.872

Cr

3.485

2.779

1.478

Mn

3.462

2.723

1.194

Fe

3.428

2.651

0.969

Co

3.368

2.529

0.807

Ni

3.280

2.412

0.724

Cu

3.049

2.114

0.567

Zr

3.696

3.086

2.934

Nb

3.767

3.099

2.424

Mo

3.759

3.063

1.961

Hf

3.664

3.110

2.975

Ta

3.720

3.144

2.531

W

3.677

3.125

2.072

Al

3.297

3.426

2.200

Si

3.254

3.561

2.200

Sn

2.782

2.283

2.100

Element

Bo values in the hcp-Ti cluster are larger than those in the bcc-Ti cluster, and the trend for Md values is similar to that for Bo values. Applying these parameters to the Ti–Nb–Zr alloys, good agreement between the bonding force and Young’s modulus validates the bonding force indicating that it can be used as a useful parameter for designing low Young’s modulus alloys in the Ti–Nb–Zr alloy system [19].

7.4 Applications 7.4.1 Phase Stability 7.4.1.1 Binary Alloy he stability of Ti–M alloys (M = transition element) can be evaluated from the ratio of valence electrons and the number of atoms (e/a) [21], the Bo–Md diagram [22], and the total energy maps.

216 Advanced Engineering Materials and Modeling For an alloy, the average values of Bo and Md are deined simply by taking the compositional average of each parameter, denoted Bo and Md, respectively. So, the alloy position moves on the Bo–Md diagram as the alloy composition changes. he characteristic of each alloying element, M, is described by using the alloying vector as shown in Figure 7.5. he alloying vectors shown in Figure 7.5 are directed upward and letward for the β-stabilizing isomorphous elements (e.g. Nb, Ta) and the eutectoid elements (e.g. Cr, Fe), respectively. he vectors of the -stabilizing elements (e.g. Al, Sn) are directed downward. However, the neutral elements (e.g. Zr, Hf) take a completely diferent vector direction from other elements that are upward and rightward in the diagram. It was shown that a small addition of Zr to -type Ti alloys shits the / + (+ ) phase boundary to the region where the content of the -stabilizing elements is poorer on the Bo–Md diagram [22]. Also, the composition of the least stable -phase alloy correlates in some ways with the emergence of many unique properties, such as non-linear superelasticity and very low work hardening [23]. It has been shown that -phase in Ti–M alloys is stabilized if the e/a ratio is 4.2 or more [24]. In Ti–Nb alloy system, the e/a ≥ 4.2 starts from Ti–22Nb onward. hus, a minimum niobium content of ~22–25 at.% is expected to stabilize the -phase of Ti–Nb. he -phase stability of Ti–Nb system with 25 at.% Nb content can also be understood using bond order

3.25 W Nb

Mo

3.00 2.75 Bo

Mn

V

Cr

Ti

Fe 2.50

Ta

Si

Co

Zr Hf neutral

stabilizer

Al

Ni

Sn

2.25 Cu 2.00

0.5

1.0

1.5

2.0

2.5

3.0

Md (eV)

Figure 7.5 Alloying vectors, starting from pure Ti and ending at the position of Ti-10 at.% M of elements in titanium illustrated on the Bo–Md diagram (re-plotted using data from Ref. [22]).

Application of First Principles Theory to the Design 217 (Bo) and d-orbital energy levels (Md) [22]. he stability of the β-phase increases with the increase of the Nb content in the Ti–Nb alloy with signiicant increase of Bo but a narrow variation of Md as shown in Figure 7.6 [25]. In this diagram, the single -phase region is clearly separated from the + phase region. he / + phase boundary is shown in Figure 7.6, which is close to the slip/twin boundary. he boundary for Ms = RT (room temperature) is also presented by using a dotted curve, below which the -phase coexists with the - and/or -phase in the alloy at RT. In addition, the boundary for Mf = RT is given by a solid curve, below which the martensite phase exists predominantly in the alloy at RT. he calculations of formation energy of alloying elements in -Ti with the supercell approach show that most of the transition elements are good stabilizers as alloying formation energies are negative except the V and Zr [26]. It was also illustrated that the phase stability of alloys is monotonously depends on the concentration of alloying element. For the phase stabilizers, the phase stability is increased with the concentration of alloying element. While for the non-stabilizers, the stability is reduced with the increasing of concentration of alloying element. 2.96 2.94 2.92 2.90

BO

2.88 2.86 2.84 2.82 2.80 2.78 2.35

2.40

2.45

2.50

2.55

2.60

Md

Figure 7.6 Extended Bo–Md diagram taken from Ref. [22] superimposed with calculated binary Ti–x(at.%)Nb (x = 6.25, 12.5, 18.75, 25, 31.25, 37.5, and 50) (from Ref. [25]).

218 Advanced Engineering Materials and Modeling

7.4.1.2 Multicomponent Alloys he application of the Bo–Md diagram to the multicomponent titanium alloys also releases useful information for alloy design particularly for the low-modulus bio-titanium alloys. For example, Ti–Fe–Ta and Ti–Fe–Ta–Zr alloys and low-cost Ti–Fe–Nb–Zr alloys (TFNZ) with Young’s modulus of 75 GPa and an ultimate tensile strength (UTS) of 1169 MPa were developed for bio-implant applications with the aid of the Bo–Md diagram. he Bo and Md values of the single -phase and non-single -phase Ti–Nb–Zr ternary alloys [27, 28] were irst calculated and shown in Figure 7.7. It was found that the boundary between the single -phase and the non-single -phase is almost a straight line (the solid straight line in Figure 7.7). Following these results, new low Young’s modulus Ti–Nb–Zr alloys with the Bo and Md values along the metastable β-phase boundary were designed [19]. Zirconium has been considered to have a neutral efect on the phase stability (Figure 7.5). Zr works as a weak -stabilizing element in Ti–Nb–Zr [27] and Ti–Cr–Zr [29] alloys, as compared to the so-called α stabilizing elements (i.e., O, Al, and Sn). For examples, Ti–5Zr–xNb–10Ta–0.23O alloys and Ti–5Zr–30Nb–yTa–0.23O alloys are shown in Figure 7.8 [30]. It is evident from this igure that the β-phase becomes more stable with increasing of Ta or Nb contents and the formation of the ω-phase is suppressed in these alloys. Furthermore, the combined use of O and Zr shits

3.15 Pure Nb

3.10 3.05

Bo

3.00 2.95

6 metastable β-phase boundary without Zr 4

2.90

5 Ti50-y/2 Zr50-y/2 Nby(y = 0,4,8,10,12)

2 3 Ti-30Zr-xNb(x = 12,14,16) 1 Ti-24Nb Ti-22Nb-xZr(x-2,4,6,8) 2 Ti-20Nb-12Zr 1 (77Ti-23Zr) 3 Ti-17Nb-21Zr 100-z Zrz(z = 0,3,6,9,12) 4 Ti-11Nb-38Zr 5 Ti-6Nb-53Zr Pure Ti 6 Ti-70Zr

2.85 2.80 2.75

Pure Zr metastable β-phase boundary of Ti-Nb-Zr alloy system

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

Md (eV)

Figure 7.7 he Bo–Md diagram of the -phase Ti–Nb–Zr alloys. he solid and open symbols are for single and non-single phases, respectively (re-plotted from Ref. [19]).

Application of First Principles Theory to the Design 219 2.91 x = 40 (80) y = 20 (75)

2.90

Bo

2.89

x = 35 (78)

2.88

x = 30

2.87 2.86

y = 15 (70)

β y = 10 (66.9)

y = 5 (72) β+ω

x = 25 (86) y = 0 (85.2)

2.85 2.455

x = 20 (100)

2.460

Ti- xNb-5Zr-10Ta-0.23O T i-30Nb-5Zr-yTa-0.23O Shifted boundary Norminal boundary

2.465

2.470

Md (eV)

Figure 7.8 Phase stability changes with the Nb and Ta contents in Ti–Nb–5Zr–Ta–0.23O alloys. By co-alloying of O and Zr, / + phase boundary moves from the nominal one (solid curve) to the shited one (dotted curve). Numbers in parentheses are the value of Young’s modulus in unit of GPa (re-poltted from Ref. [30]).

the / + phase boundary to the part of further lower content of the β-stabilizing elements (i.e., Ta and Nb), as indicated by a dashed curve. he Bo–Md diagram is useful for treating the phase stability problem of Ti-based alloys. As shown in Figure 7.6, in the lower part of this diagram, the Bo value is less than 2.84. he region of the -type alloys is extended over the high Bo range in this diagram. he -type alloys are known to be deformed by either the slip or the twin mechanism, depending largely on the phase stability of alloys [4]. he plastic deformation mode changes from the twin to slip mechanism with the increase of stability of the -phase. In response to this change, the region in the Bo–Md diagram is separated into either the slip or the twin dominant sub-region.

7.4.2 Elastic Properties It is well known that Young’s modulus and other mechanical properties change with the type of the phases in the alloy [31]. hus, the Bo–Md diagram has been used for the design of Ti-based alloys. Studies illustrated that the -type Ti-based alloys deform by either slip or twin mechanism [32]. he stress-induced martensitic transformation also takes place in some alloys upon external stress [32, 33]. he occurrence of these phenomena depends on the -phase stability and hence will be controlled by alloying.

220 Advanced Engineering Materials and Modeling Also, it is known that the slip/twin boundary is close to the / + + ( boundary [22]. At this boundary, the elastic anisotropy factor is rather high since the value of the elastic shear modulus, C = (C11 − C12)/2, is diminishing with the alloy approaching to this boundary. his value even approaches zero when the e/a value is about 4.24 [34]. he stability of the high-temperature bcc phase with respect to hcp increases with increasing of e/a, but is interrupted by the formation of metastable and phases above a certain e/a value corresponding to the peak in Young’s modulus. If these metastable phases can be suppressed, a minimum in Young’s modulus might be obtained (the broken line) as schematically illustrated in Figure 7.9 [35]. he elastic properties of Ti–M alloys have been estimated by calculating the formation energy of Ti–M alloys with DFT methods within the frameworks of DVM. he formation energy of a hypothetical Ti–M alloy with structure based on a certain parent structure and with a unit cell volume V is

EMλ (V ) = Ebλ (M ,V , x ) − [(1 − x )Ebλ (Ti,VTi ) + xEbλ (M ,VM )] (7.20) where x is the atomic concentration of alloying atom M and Eb(M, VM) is the binding energy per atom of the M with the unit cell of VM. To estimate the formation energy, an assumption was made that the alloying atom is resolved from the metal M at equilibrium and solute into the matrix Ti at the equilibrium. In general, the binding energy per atom depends on the concentration of the alloying addition if the interaction between alloying atoms cannot be ignored. In the dilute limit, the interaction

Young’s modulus

β + α” + ω

4.0

α”

β+ω

4.1

4.2

β

4.3

4.4

4.5

Electron/atom (e/a) ratio

Figure 7.9 Schematic variation of Young’s modulus with e/a ratio in binary Ti–TM system (from Ref. [35]).

Application of First Principles Theory to the Design 221 between alloying atoms is weak, and the bulk modulus of a Ti–M alloy is estimated by

B =V

d2 ( EMλ (V )) dV 2 V =V

(7.21)

0

he results of the relative bulk modulus of Ti–M alloys to the pure -Ti calculated with eq. (7.21) are illustrated in Figure 7.10. he above treatment can be extended to multi-element titanium alloys in the dilute limit. Similar to the case of binary titanium alloys, the average binding energy per atom of an alloy consisting of m alloying elements, Ebλ , is evaluated by m

Ebλ (V ) = Ebλ (Ti,V ) + ∑ x M Ebλ (M ,V )

(7.22)

M =1

where xM is the atomic concentration of the Mth alloying element. he bulk modulus and theoretical strength of a multi-element alloy are estimated by

B =V

d2 (Ebλ (V )) dV 2 V =V

(7.23)

0

V Cr Fe Ni Cu Zr Nb Mo Ta Ai Sn -1.3

1.2

1.2

1.1

1.1

1.0

1.0

0.9

0.9

0.8

0.8

0.7

Relative binding energy of Ti-M alloys to -Ti

Relative bulk modulus of Ti-M alloys to -Ti

Ti 1.3

0.7 Ti

V Cr Fe Ni Cu Zr Nb Mo Ta Ai Sn -Alloying element

Figure 7.10 Relative binding energy and bulk modulus of Ti–M alloys to the -Ti.

222 Advanced Engineering Materials and Modeling 125 Calculated Bulk Modulus (GPa)

124

β-type

123

Ti-13V-11Cr-3Al

122 Ti-15V-3Cr-3Al-3Sn

121 120 119

Ti-3Al-8V-6Cr-4Mo-4Zr (Beta C)

118 117 116 115

Ti-15Mo

Ti-15Mo-3Al-3Nb Ti-11.5Mo-6Zr-4.5Sn Ti-12Mo-6Zr-2Fe

60

65

70 75 80 85 90 95 100 105 110 Experimental Tensile Modulus (GPa)

Figure 7.11 Comparison between the calculated bulk modulus and experimental tensile modulus for some practical titanium alloys.

he bulk modulus and theoretical strength of some practical titanium alloys were calculated with these equations. A comparison between the theoretical bulk modulus and the experimental tensile modulus is made in Figure 7.11. Alloys in Figure 7.11 are either - or near -type and the experimental data are taken from Ref. [36] for Ti–11.5Mo–6Zr–4.5Sn and Ti–3Al– 8V–6Cr–4Mo–4Zr, and from [37] for others. In general, the microstructure of the -type titanium alloys depends on heat treatment. For example, Ti–15Mo alloy retains a ine grain structure if rapidly quenched ater solution annealing at 800 °C [38]. he Ti–12Mo–6Zr–2Fe alloy has a single -phase structure ater solution annealing [38]. For comparison at similar conditions, only experimental data obtained from annealed specimens are included in Figure 7.11. It can be seen from Figure 7.11 that a linear relationship exists between the theoretical bulk modulus and the experimental tensile modulus. his shows that the present model correctly describes the bonding between atoms and the bulk modulus of these alloys and can be considered a viable method for the estimation of the bulk modulus of practical alloys.

7.4.3 Examples 7.4.3.1

Gum Metal

Based on the irst principles DV-Xα calculation of the electronic structures of binary Ti alloys, Satio et al. designed a new Ti alloy following three

Application of First Principles Theory to the Design 223 electronic parameters: (i) the compositional average valence electron number, e/a = 4.24; (ii) the bond order Bo = 2.87; and (iii) the d electron-orbital energy level Md = 2.45 eV [39]. Various alloy composition combinations of Group IVa and Va elements and oxygen, to meet these criteria, such as Ti-12Ta-9Nb-3V-6Zr-O and Ti–23Nb–0.7Ta–2Zr–O [molar percent (mol %)], wherein each alloy has a simple body-centered cubic (bcc) crystal structure. hese alloys exhibit multiple “super” properties and dramatical changes in physical properties ater plastic deformation at room temperature, and simultaneously ofer super elasticity, super strength, super cold workability, and Invar and Elinvar properties. he properties emerge only when all three parameters are satisied simultaneously. In order to exhibit these properties, each alloy system requires substantial cold working and a certain amount of oxygen restricting in 0.7–3.0 mol %. he Ti-based binary, ternary, and quaternary alloys with Nb, Ta, Zr, Hf, V, W, and Mo additions, and also oxygen were prepared by a powder process owning a variety of compositions around e/a = 4.24 to examine the efect of the additives on the mechanical strength and the elastic properties. he maximum concentration of oxygen should be