Multiscale theory of composites and random media

306 Pages·2019·29.917 MB·English

by  Xu, Xi Frank

Multiscale Theory
of Composites and
Random Media
http://taylorandfrancis.com
Multiscale Theory
of Composites and
Random Media
Xi Frank Xu
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB®
software or related products does not constitute endorsement or sponsorship by The MathWorks of a
particular pedagogical approach or particular use of the MATLAB® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2019 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-4822-5624-6 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize
to copyright holders if permission to publish in this form has not been obtained. If any copyright material
has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, trans-
mitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including photocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright
.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the
CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.
Library of Congress Cataloging‑in‑Publication Data
Names: Xu, Xi Frank, author.
Title: Multiscale theory of composites and random media / Xi Frank Xu.
Description: Boca Raton : Taylor & Francis, a CRC title, part of the Taylor & Francis
imprint, a member of the Taylor & Francis Group, the academic division of T&F
Informa, plc, [2019] | Includes bibliographical references and index. |
Identifiers: LCCN 2018015084 (print) | LCCN 2018030719 (ebook) | ISBN 9780429894381
(Adobe PDF) | ISBN 9780429894374 (ePub) | ISBN 9780429894367 (Mobipocket) |
ISBN 9781482256246 (hardback : acid-free paper) | ISBN 9780429470653 (ebook)
Subjects: LCSH: Composite materials--Mathematical models. | Multiscale modeling.
Classification: LCC TA418.9.C6 (ebook) | LCC TA418.9.C6 X824 2019 (print) |
DDC 620.1/18015118--dc23
LC record available at https://lccn.loc.gov/2018015084
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
To Xin and Hannah
v
http://taylorandfrancis.com
Contents
List of symbols and acronyms xi
Preface xvii
Author xix
1 Introduction: Emerging scale-coupling mechanics 1
1.1 Phenomenological methodology
vs. Micro-macro methodology 1
1.2 Multiscale methodology 5
1.3 Scale-coupling mechanics 6
1.3.1 Existing models on scale-coupling effects 6
1.3.2 Nonlocal formulation of scale-coupling mechanics 7
1.3.3 Strain gradient formulation
of scale-coupling mechanics 10
1.3.4 Representative volume element 10
2 Random morphology and correlation functions 13
2.1 Gaussian random fields 14
2.1.1 Preliminaries 14
2.1.2 Random variable representation
of Gaussian random fields 17
2.2 Non-Gaussian random fields: The translation model 20
2.2.1 Definition of the translation model 20
2.2.2 Algorithm to find the underlying Gaussian image 24
2.2.3 Limitations of the translation model 25
2.3 Non-Gaussian random fields: The correlation model 29
2.3.1 The short-range-correlation model 29
2.3.2 Sampling algorithm 31
vii
viii Contents
2.3.3 Simulation of two-phase media 33
2.3.4 Classification of random morphologies 38
Part I
analytical homogenization of scale separation problems 43
3 Green-function-based variational principles 45
3.1 Odd-order variational principle 46
3.1.1 Decomposition of an RVE problem 46
3.1.2 Principle of minimum potential energy 50
3.1.3 Principle of minimum complementary energy 51
3.2 Even-order Hashin-Shtrikman principle 54
3.3 Odd- and even-order variational principles on conductivity 56
4 Nth-order variational bounds 59
4.1 First-order Voigt-Reuss bounds 60
4.1.1 Elastic moduli 60
4.1.2 Conductivity 61
4.2 Second-order Hashin-Shtrikman bounds 61
4.2.1 General results 61
4.2.2 HS bounds for phases characterized
with the isotropic elastic moduli 65
4.2.3 HS bounds in 2D elasticity 70
4.2.4 HS bounds of the effective conductivity 75
4.3 Third-order bounds 76
4.3.1 Third-order bounds of the effective bulk modulus 76
4.3.2 Third-order bounds of the effective shear modulus 84
4.3.3 Third-order bounds in 2D elasticity 91
4.3.4 Third-order bounds of the effective conductivity 96
4.4 Fourth-order bounds 96
4.4.1 Derivation 96
4.4.2 Attainability of ψ 101
4.4.3 Explicit expression of ψ 104
4.5 Fluid–antifluid annihilation 106
5 Ellipsoidal bound 111
5.1 Formulation of the ellipsoidal bound 111
5.1.1 Morphological model 111
5.1.2 Derivation of the ellipsoidal bound 114
Contents ix
5.2 Asymptotic results of the ellipsoidal bound 118
5.2.1 Morphological pattern of isotropic mixture 119
5.2.2 Randomly dispersed spheres 123
5.2.3 Randomly dispersed needle-like rods 124
5.2.4 Randomly dispersed disk-like platelets 129
5.2.5 Ellipsoidal bound of anisotropic mixture 132
5.3 Effective elastic moduli of cracked media 135
5.3.1 Recovery of Griffith criterion 137
5.3.2 Effective elastic moduli of a dilutely cracked solid 141
5.3.3 Effective elastic moduli of a non-dilutely
cracked solid 146
5.3.4 Effective elastic moduli of a solid containing voids 153
6 Prediction of percolation threshold 157
6.1 Percolation threshold of complete dispersion 158
6.1.1 Ellipsoidal bound 158
6.1.2 Percolation threshold of a 3D composite 158
6.1.3 Percolation threshold of a 2D composite 164
6.1.4 Percolation threshold of fillers with various
aspect ratios 165
6.1.5 Concluding remark 168
6.2 Percolation threshold of incomplete dispersion 170
6.2.1 Ellipsoidal bound accounting for clustering effect 170
6.2.2 Asymptotic results 175
Part II
Computational analysis of scale-coupling problems 179
7 Green-function-based variational principles
for scale-coupling problems 181
7.1 Decomposition of a boundary value problem 182
7.1.1 Principle of superposition 182
7.1.2 Stress polarization 185
7.2 Variational principles for a finite body random composite 187
7.2.1 Odd-order variational principle 187
7.2.2 Even-order Hashin-Shtrikman variational principle 192
7.3 Minimum size of representative volume element 193
7.3.1 Series representation of the stress polarization 193
7.3.2 First-order trial function 194
7.3.3 Minimum RVE size in 3D 196
7.3.4 Minimum RVE size in 2D 202