Microstructural Randomness and Scaling in Mechanics of Materials (Modern Mechanics and Mathematics)

P1: Binod July 13, 2007 21:36 C4174 C4174˙C000 Microstructural Randomness and Scaling in Mechanics of Materials P...

Author: Martin Ostoja-Starzewski

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Microstructural Randomness and Scaling in Mechanics of Materials

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Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-417-0 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable eﬀorts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, 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 Ostoja-Starzewski, Martin. Microstructural randomness and scaling in mechanics of materials / Martin Ostoja-Starzewski. p. cm. -- (Modern mechanics and mathematics) Includes bibliographical references and index. ISBN-13: 978-1-58488-417-0 (alk. paper) ISBN-10: 1-58488-417-7 (alk. paper) 1. Strength of materials. I. Title. II. Series. TA405.O88 2007 620.1’12--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Dedication

To Pauline and Michael

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Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

1

Basic Random Media Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Probability Measure of Geometric Objects . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Definitions of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Probabilities on Countable and Euclidean Sample Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2.1 is a Countable Set: = {ω1 , ω2 , . . .} . . . . . . . . . . . . 4 1.1.2.2 is a 1D Euclidean Space: = R. . . . . . . . . . . . . . . .7 1.1.2.3 is a 2D Euclidean Space: = R2 . . . . . . . . . . . . . . . 8 1.1.3 Random Points, Lines, and Planes. . . . . . . . . . . . . . . . . . . . . . .11 1.1.3.1 Random Lines in Two Dimensions . . . . . . . . . . . . . . 11 1.1.3.2 Planes in Three Dimensions . . . . . . . . . . . . . . . . . . . . 14 1.1.3.3 Straight Lines in Three Dimensions . . . . . . . . . . . . . 14 1.2 Basic Point Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.2 Example—Model of a Fiber Structure of Paper . . . . . . . . . . 15 1.2.3 Generalization to Many Types of Outcomes . . . . . . . . . . . . . 17 1.2.4 Binomial and Multinomial Point Fields . . . . . . . . . . . . . . . . . 17 1.2.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.4.2 Simulation of a Binomial Point Field with n Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.4.3 Generalization to a Multinomial Point Field . . . . 20 1.2.5 Bernoulli Lattice Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.6 Poisson Point Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.6.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.6.2 Simulation of a Poisson Point Field . . . . . . . . . . . . . 22 1.2.6.3 Inhomogeneous Poisson Point Field . . . . . . . . . . . . 22 1.2.6.4 Inhibition and Hard-Core Processes . . . . . . . . . . . . 23 1.3 Directional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.2 Circular Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Random Fibers, Random Line Fields, Tessellations . . . . . . . . . . . . . 27 1.4.1 Poisson Random Lines in Plane . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.2 Finite Fiber Field in Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 vii

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viii Random Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.3.2 Planar (Poisson-)Voronoi Tessellations and Delaunay Triangulations . . . . . . . . . . . . . . . . . . . 31 1.4.3.3 Modifications of Voronoi Tessellations . . . . . . . . . . 31 1.4.3.4 Random Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Basic Concepts and Definitions of Random Microstructures . . . . 35 1.5.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 1.5.1.1 Germ-Grain and Boolean Models . . . . . . . . . . . . . . . 35 1.5.1.2 Flocs as Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5.2 Toward Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . 38 1.4.3

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Random Processes and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1 Elements of One-Dimensional Random Fields. . . . . . . . . . . . . . . . . .45 2.1.1 Scalar Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.1.2 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1.2 Vector Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2 Mechanics Problems on One-Dimensional Random Fields . . . . . . 55 2.2.1 Propagation of Surface Waves along Random Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2.2 Fracture of Brittle Microbeams . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.2.1 Randomness of Microbeams . . . . . . . . . . . . . . . . . . . . 57 2.2.2.2 Strain Energy Release Rate in Random Microbeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 2.2.2.3 Stochastic Crack Stability . . . . . . . . . . . . . . . . . . . . . . . 61 2.3 Elements of Two- and Three-Dimensional Random Fields . . . . . . 62 2.3.1 Scalar and Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 2.3.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.3.1.3 Properties of the Correlation Function Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.3.2 Random Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.3.2.1 Second-Rank Tensor Fields . . . . . . . . . . . . . . . . . . . . . 69 2.3.2.2 Fourth-Rank Tensor Fields . . . . . . . . . . . . . . . . . . . . . 71 2.4 Mechanics Problems on Two- and Three-Dimensional Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.4.1 Mean Field Equations of Random Materials . . . . . . . . . . . . . 72 2.4.2 Mean Field Equations of Turbulent Media . . . . . . . . . . . . . . . 73 2.5 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.5.1 Basic Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 2.5.2 Computation of (2.146) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.5.3 Conditions for (2.146) to Hold . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.5.4 Existence of the Limit in (2.146) . . . . . . . . . . . . . . . . . . . . . . . . . 76

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The Maximum Entropy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6.1 Cracks in Plates with Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6.2 Disorder and Information Entropy . . . . . . . . . . . . . . . . . . . . . . 80

Planar Lattice Models: Periodic Topologies and Elastostatics . . . . . . . 87 3.1 One-Dimensional Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.1 Simple Lattice and Elastic Strings . . . . . . . . . . . . . . . . . . . . . . . 87 3.1.2 Micropolar Lattice and Elastic Beams . . . . . . . . . . . . . . . . . . . 88 3.2 Planar Lattices: Classical Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2.1 Basic Idea of a Spring Network Representation . . . . . . . . . . 91 3.2.2 Antiplane Elasticity on Square Lattice . . . . . . . . . . . . . . . . . . . 93 3.2.3 In-Plane Elasticity: Triangular Lattice with Central Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.2.4 In-Plane Elasticity: Triangular Lattice with Central and Angular Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2.5 Triple Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Applications in Mechanics of Composites . . . . . . . . . . . . . . . . . . . . . . 99 3.3.1 Representation by a Fine Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3.2 Solutions of Linear Algebraic Problems . . . . . . . . . . . . . . . . 102 3.3.3 Example: Simulation of a Polycrystal . . . . . . . . . . . . . . . . . . 105 3.4 Planar Lattices: Nonclassical Continua . . . . . . . . . . . . . . . . . . . . . . . . 107 3.4.1 Triangular Lattice of Bernoulli–Euler Beams . . . . . . . . . . . 107 3.4.2 Triangular Lattice of Timoshenko Beams . . . . . . . . . . . . . . . 110 3.4.3 From Stubby Beams to a Perforated Plate Model . . . . . . . 112 3.4.4 Hexagonal Lattice of Bernoulli–Euler Beams . . . . . . . . . . . 114 3.4.5 Square Lattice of Bernoulli–Euler Beams . . . . . . . . . . . . . . . 115 3.4.6 Nonlocal and Gradient Elasticity on a Lattice with Central Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.4.6.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 117 3.4.6.2 Local Continuum Model . . . . . . . . . . . . . . . . . . . . . . 118 3.4.6.3 Nonlocal Continuum Model . . . . . . . . . . . . . . . . . . .118 3.4.6.4 Strain-Gradient Continuum Model . . . . . . . . . . . . 119 3.4.7 Plate-Bending Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.5 Extension-Twist Coupling in a Helix . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.5.1 Constitutive Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.5.2 Harmonic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.5.2.1 Elastic Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.5.2.2 Thermoelastic Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.5.3 Viscoelastic Helices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.5.3.1 Viscoelasticity with Integer-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.5.3.2 Viscoelasticity with Fractional-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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Lattice Models: Rigidity, Randomness, Dynamics, and Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1 Rigidity of Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1.1 Structural Topology and Rigidity Percolation . . . . . . . . . . 133 4.1.2 Application to Cellulose Fiber Networks . . . . . . . . . . . . . . . 137 4.1.2.1 Rigidity of a Graph of Poisson Line Field Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.1.2.2 Loss of Rigidity in a Fiber-Beam Network . . . . . 139 4.2 Spring Network Models for Disordered Topologies . . . . . . . . . . . 141 4.2.1 Granular-Type Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2.1.1 Load Transfer Mechanisms . . . . . . . . . . . . . . . . . . . . 141 4.2.1.2 Graph Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.1.3 Periodic Graphs with Disorder . . . . . . . . . . . . . . . . 145 4.2.2 Solutions of Truss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.3 Mesoscale Elasticity of Paper . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.3.1 Dilemma of Special In-Plane Orthotropy of Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.3.2 Explanation via Random Fiber Network . . . . . . . 153 4.2.4 Damage Patterns and Maps of Disordered Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3 Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3.1.2 Leapfrog Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162 4.3.2.1 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.4 Michell Trusses: Optimal Use of Material . . . . . . . . . . . . . . . . . . . . . 165 4.4.1 Study via Hyperbolic System . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.4.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.4.1.2 Example of an Optimal Layout . . . . . . . . . . . . . . . . 167 4.4.2 Study via Elliptic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5

Two- Versus Three-Dimensional Classical Elasticity . . . . . . . . . . . . . . 171 5.1 Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.1.1 Isotropic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.1.1.1 Three-Dimensional Elasticity . . . . . . . . . . . . . . . . . . 171 5.1.1.2 Two-Dimensional Elasticity . . . . . . . . . . . . . . . . . . . 172 5.1.2 Plane Elasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.1.3 Special Planar Orthotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.2 The CLM Result and Stress Invariance . . . . . . . . . . . . . . . . . . . . . . . . 177 5.2.1 Isotropic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177 5.2.1.1 Basic Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.2.1.2 Two-Phase Composites and Dundurs Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2.2 Anisotropic Materials and the Null-Lagrangian . . . . . . . . 180

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6

5.2.3 Multiply Connected Materials . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.2.4 Applications to Composites. . . . . . . . . . . . . . . . . . . . . . . . . . . .183 5.2.4.1 Effective Moduli of Composites . . . . . . . . . . . . . . . 183 5.2.4.2 Plates with Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.2.5 Extension of Stress Invariance to Presence of Eigenstrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.2.5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.2.5.2 Planar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Two- versus Three-Dimensional Micropolar Elasticity . . . . . . . . . . . . 191 6.1 Micropolar Elastic Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.1.1 Force Transfer and Degrees of Freedom . . . . . . . . . . . . . . . . 191 6.1.2 Equations of Motion and Constitutive Equations . . . . . . . 194 6.1.3 Isotropic Micropolar Materials . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.1.4 Virtual Work Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.1.5 Hamilton’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199 6.1.6 Reciprocity Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.1.7 Elements of Micropolar Elastodynamics . . . . . . . . . . . . . . . 202 6.1.7.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.1.7.2 Plane Monochromatic Waves . . . . . . . . . . . . . . . . . . 204 6.1.8 Noncentrosymmetric Micropolar Elasticity. . . . . . . . . . . . .204 6.2 Classical vis-`a-vis Nonclassical (Elasticity) Models . . . . . . . . . . . . . 205 6.2.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.2.2 The Ensemble Average of a Random Local Medium is Nonlocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3 Planar Cosserat Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.3.1 First Planar Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.3.2 Characteristic Lengths in Isotropic and Orthotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.3.3 Restricted Continuum vis-`a-vis the Micropolar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.4 The CLM Result and Stress-Invariance . . . . . . . . . . . . . . . . . . . . . . . . 216 6.4.1 Isotropic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .216 6.4.2 Anisotropic Materials and the Null-Lagrangian . . . . . . . . 218 6.4.3 Multiply Connected Materials . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.4.4 Applications to Composites. . . . . . . . . . . . . . . . . . . . . . . . . . . .221 6.4.4.1 Two-Phase Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.4.4.2 Effective Moduli of Composites . . . . . . . . . . . . . . . 222 6.4.5 Extensions of Stress Invariance to Presence of Eigenstrains and Eigencurvatures . . . . . . . . . . . . . . . . . . . . . . 223 6.4.5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.4.5.2 Inhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . 225 6.5 Effective Micropolar Moduli and Characteristic Lengths of Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

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xii 6.5.1

From a Heterogeneous Cauchy to a Homogeneous Cosserat Continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228 6.5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7

Mesoscale Bounds for Linear Elastic Microstructures . . . . . . . . . . . . . 237 7.1 Micro-, Meso-, and Macroscales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237 7.1.1 Separation of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.1.3 The RVE Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.2 Volume Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.2.1 A Paradigm of Boundary Conditions Effect . . . . . . . . . . . . 241 7.2.2 The Hill Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.2.2.1 Mechanical versus Energy Definitions . . . . . . . . . 244 7.2.2.2 Order Relations Dictated by Three Types of Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.2.3 Apparent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.3 Spatial Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.3.1 Stationarity of Spatial Statistics . . . . . . . . . . . . . . . . . . . . . . . . 249 7.3.2 Ergodicity of Spatial Statistics. . . . . . . . . . . . . . . . . . . . . . . . . .251 7.4 Hierarchies of Mesoscale Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.4.1 Response under Displacement Boundary Condition . . . . 251 7.4.2 Response under Traction Boundary Condition . . . . . . . . . 254 7.4.3 Scale-Dependent Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.4.4 Homogenization Theory Viewpoint . . . . . . . . . . . . . . . . . . . . 256 7.4.5 Apparent Moduli in In-Plane Elasticity . . . . . . . . . . . . . . . . 257 7.4.5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 257 7.5 Examples of Hierarchies of Mesoscale Bounds . . . . . . . . . . . . . . . . 258 7.5.1 Random Chessboards and Bernoulli Lattices . . . . . . . . . . . 258 7.5.2 Disk-Matrix Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.5.3 Functionally Graded Materials. . . . . . . . . . . . . . . . . . . . . . . . .263 7.5.4 Effective and Apparent Moduli of Multicracked Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.5.4.1 Scale-Dependent Hierarchies of Bounds: Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.5.4.2 Cross-Correlations of the Mesoscale Moduli with the Crack Density Tensor . . . . . . . . 268 7.6 Moduli of the Trabecular Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

8

Random Field Models and Stochastic Finite Elements . . . . . . . . . . . . 273 8.1 Mesoscale Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.1.1 From Discrete to Continuum Random Fields . . . . . . . . . . . 273 8.1.2 Scale Dependence via Beta Distribution . . . . . . . . . . . . . . . . 275 8.1.3 Mesoscopic Continuum Physics Due to Muschik . . . . . . . 276 8.2 Second-Order Properties of Mesoscale Random Fields . . . . . . . . 278 8.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

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xiii 8.2.2 Universal Properties of Mesoscale Bounds . . . . . . . . . . . . . 279 8.2.3 Correlation Structure of Mesoscale Random Fields . . . . . 282 8.3 Does There Exist a Locally Isotropic, Smooth Elastic Material?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284 8.3.1 Correlation Theory Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . 284 8.3.2 Micromechanics Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 8.3.3 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .286 8.4 Stochastic Finite Elements for Elastic Media . . . . . . . . . . . . . . . . . . . 286 8.4.1 Bounds on Global Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.4.2 Example: Torsion of a Duplex-Steel Bar . . . . . . . . . . . . . . . . 288 8.4.3 An Overview of Phenomenological SFE Studies . . . . . . . . 290 8.5 Method of Slip-Lines for Inhomogeneous Plastic Media . . . . . . . 293 8.5.1 Finite Difference Spacing vis-`a-vis Grain Size . . . . . . . . . . . 293 8.5.2 Sensitivity of Boundary Value Problems to Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.5.2.1 Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.5.2.2 Limit Analysis of a Pipe under Internal Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 8.6 Michell Trusses in the Presence of Random Microstructure . . . . 301 8.6.1 Truss-Like Continuum vis-`a-vis Random Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.6.2 Solution via the Hyperbolic System . . . . . . . . . . . . . . . . . . . . 303 8.6.3 Solution via the Elliptic System . . . . . . . . . . . . . . . . . . . . . . . . 305

9

Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.1 Physically Nonlinear Elastic Microstructures . . . . . . . . . . . . . . . . . . 311 9.1.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .311 9.1.2 Power-Law Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.1.3 Random Formation vis-`a-vis Inelastic Response of Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.2 Finite Elasticity of Random Composites . . . . . . . . . . . . . . . . . . . . . . . 318 9.2.1 Averaging Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.2.2 Variational Principles and Mesoscale Bounds . . . . . . . . . . 319 9.3 Elastic-Plastic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 9.3.1 Variational Principles and Mesoscale Bounds . . . . . . . . . . 322 9.3.2 Matrix-Inclusion Composites . . . . . . . . . . . . . . . . . . . . . . . . . . 324 9.3.2.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 9.3.2.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 9.3.3 Geodesic Properties of Shear-Band Patterns . . . . . . . . . . . . 329 9.4 Rigid-Perfectly Plastic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . 333 9.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 9.4.2 Bounding on Mesoscales via Kinematic and Traction Boundary Conditions . . . . . . . . . . . . . . . . . . . . 334 9.4.3 Random Chessboard of Huber–von Mises Phases . . . . . . 336

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xiv 9.5 Viscoelastic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9.6 Stokes Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 9.7 Thermoelastic Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.7.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.7.2 The Nonlinear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 9.8 Scaling and Stochastic Evolution in Damage Phenomena . . . . . . 354 9.9 Comparison of Scaling Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

10 Mesoscale Response in Thermomechanics of Random Media . . . . 359 10.1

10.2

10.3

10.4

10.5

10.6

From Statistical Mechanics to Continuum Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 10.1.1 Dissipation Function of the RVE . . . . . . . . . . . . . . . . . . . . . 359 10.1.2 Departure from the Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Extensions of the Hill Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10.2.1 The Hill Condition in Thermomechanics . . . . . . . . . . . . . 364 10.2.2 Homogenization in Dynamic Response . . . . . . . . . . . . . . 366 Legendre Transformations in (Thermo)Elasticity . . . . . . . . . . . . . 367 10.3.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 10.3.2 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Thermodynamic Orthogonality on the Mesoscale . . . . . . . . . . . . 370 10.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 10.4.2 Homogeneous Dissipation Functions . . . . . . . . . . . . . . . . 371 10.4.3 Quasi-Homogeneous Dissipation Functions . . . . . . . . . 374 Complex versus Compound Processes: The Scaling Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.5.2 Micropolar Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Toward Continuum Mechanics of Fractal Media . . . . . . . . . . . . . 380

11 Waves and Wavefronts in Random Media. . . . . . . . . . . . . . . . . . . . . . . . .385

Basic Methods in Stochastic Wave Propagation . . . . . . . . . . . . . . .386 11.1.1 The Long Wavelength Case . . . . . . . . . . . . . . . . . . . . . . . . . . 386 11.1.1.1 Elementary Considerations . . . . . . . . . . . . . . . . 386 11.1.1.2 Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . .387 11.1.2 The Short Wavelength Case: Ray Method . . . . . . . . . . . . 390 11.1.2.1 Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . 390 11.1.2.2 Markov Character of Rays . . . . . . . . . . . . . . . . . 392 11.1.3 The Short Wavelength Case: Rytov Method . . . . . . . . . . 393 11.2 Toward Spectral Finite Elements for Random Media . . . . . . . . . 395 11.2.1 Spectral Finite Element for Waves in Rods . . . . . . . . . . . 395 11.2.1.1 Deterministic Case . . . . . . . . . . . . . . . . . . . . . . . . . 395 11.2.1.2 Random Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

11.1

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xv Spectral Finite Element for Flexural Waves . . . . . . . . . . . 398 11.2.2.1 Deterministic Case . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.2.2.2 Random Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 11.2.3 Observations and Related Work . . . . . . . . . . . . . . . . . . . . . 402 11.3 Waves in Random 1D Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 11.3.1 Motion in an Imperfectly Periodic Composite . . . . . . . . 403 11.3.1.1 Random Evolutions. . . . . . . . . . . . . . . . . . . . . . . .403 11.3.1.2 Effects of Imperfections on Floquet Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 11.3.2 Waves in Randomly Segmented Elastic Bars . . . . . . . . . 407 11.4 Transient Waves in Heterogeneous Nonlinear Media . . . . . . . . . 408 11.4.1 A Class of Models of Random Media . . . . . . . . . . . . . . . . 408 11.4.2 Pulse Propagation in a Linear Elastic Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 11.4.3 Pulse Propagation in Nonlinear Microstructures . . . . . 414 11.4.3.1 Bilinear Elastic Microstructures . . . . . . . . . . . . 414 11.4.3.2 Nonlinear Elastic Microstructures . . . . . . . . . . 417 11.4.3.3 Hysteretic Microstructures . . . . . . . . . . . . . . . . . 419 11.5 Acceleration Wavefronts in Nonlinear Media . . . . . . . . . . . . . . . . 422 11.5.1 Microscale Heterogeneity versus Wavefront Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 11.5.1.1 Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . 422 11.5.1.2 Mesoscale Response . . . . . . . . . . . . . . . . . . . . . . . 425 11.5.2 Wavefront Dynamics in Random Microstructures . . . . 427 11.5.2.1 Model with One White Noise . . . . . . . . . . . . . . 427 11.5.2.2 Model with Two Correlated Gaussian Noises . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 11.5.2.3 Model with Four Correlated Noises . . . . . . . . 432 11.2.2

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .435 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

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Preface

0.1

Randomness versus Determinism

Partial differential equations form basic models of the mechanics of materials. Thus, typically, we have a deterministic field equation of the form Lu = f,

(0.1)

where L is a differential operator, f is a source or forcing function, and u is a solution field. This needs to be accompanied by appropriate boundary and initial conditions. [Throughout this book we shall interchangeably use the symbolic (u) and the subscript (ui... ) notations for tensors, as the need arises; also an overdot will mean d/dt.] There are three basic ways in which the randomness, indicated by the dependence on an elementary event ω ∈ (a sample space), can be introduced into (0.1): 1. Randomness of the operator: L (ω) u = f.

(0.2)

2. Randomness of the forcing function: Lu = f (ω).

(0.3)

3. Randomness of the boundary and/or initial conditions. Of course, several combinations of these three basic cases are also possible. However, in this book we shall focus primarily on the first case, which is naturally dictated by the presence of imperfect, disordered—i.e., random, in the ensemble sense—material microstructures. In this case, the coefficients of L (ω), such as the elastic moduli C (≡Cijkl ), form a tensor-valued random field, and the stochastic equation (0.1) governs the response of a random medium (or random material) B. The latter is taken as a set of all the realizations B (ω) parametrized by sample events ω of the space B = {B (ω); ω ∈ }.

(0.4)

xvii

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xviii Each of the realizations follows deterministic laws of classical mechanics in that it is a specific heterogeneous material sample; probability is introduced to deal with the set (0.4). Case (1), (2), (3), or any combination thereof, constitutes the stochastic mechanics. When speaking of the material spatial randomness, the key issue is the interpretation of some noise-like observation (measurement/signal), taken over a certain space and/or time domain, as a realization of a random phenomenon. This is, in essence, an epistemological issue: is the physical world, and the particular phenomenon at hand, a deterministic or a random event? In principle, any given physical phenomenon—such as the spatial variability of, say, elastic moduli on a millimeter scale of the particular sheet of paper on which this text is typed—is unique, and therefore, deterministic, albeit nonuniform (disordered). We make an extension to the space of many sheets (i.e., ensemble) to deal with fluctuations in a statistical manner. Thus, in the view of this author—and consistent with the prevailing thinking in stochastic mechanics—the random field of moduli is chosen so as to reflect our inability to obtain a precise mathematical model of spatial composition of the material in each and every realization. Our focus is on what physicists call “quenched” randomness. Alternatively, the choice to work with a random field (or just a set of random variables) may reflect our preference to discard a huge amount of information on space- or time-dependent fluctuations, and work, instead, with a statistical model. However, we consider each and every realization, no matter how complex, to behave according to the laws of deterministic mechanics. It is the objective of stochastic mechanics to determine this behavior: be it ω-by-ω or in terms of a distribution over . Thus, a material’s evolution is stochastic in the sense that there is an ensemble of deterministic specimens. This approach is in line with the foundations of both probability theory and classical statistical mechanics (Primas, 1999), as well as with the principle of determinism (or causality) of continuum mechanics, according to which knowledge of the history of the material up to time t yields knowledge of the state of stresses at t (Truesdell and Noll, 1965).

0.2

Randomness and Scales in Mechanics of Materials

Material spatial randomness forces one to reexamine various basic concepts and results of solid mechanics. Figure 0.1 shows three levels: (a) the microscale d, (b) the mesoscale L, and (c) the macroscale L macro . The reason for introducing the intermediate level (b)—note that meso, just as micro and macro, comes from Greek—is that we cannot claim a priori that the domain sizes involved in (a) are sufficient to homogenize the actual microstructures depicted there and replace them by uniform continua in (c), such as typically involved in

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xix (a)

(b)

(c)

FIGURE 0.1 (a) A Boolean model of a microlayered material; (b) a mesoscale continuum approximation, modeling smoothly inhomogeneous medium by placing everywhere a mesoscale window in the microstructure of (a); (c) a macroscopic body.

conventional solid mechanics. This is the essence of the so-called separation of scales d L L macro . (0.5) d< The first inequality allows one to postulate the existence of a representative volume element (RVE) of continuum mechanics. The second one covers the range of length scales where conventional continuum mechanics applies— this is the domain of spatial dependence of stress, strain and displacement fields one is interested in when solving a boundary value problem. This is

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xx where (0.1), with L being a deterministic operator, applies; the classical Navier equation is an example. When the RVE does not apply, one is faced with fluctuations, and equation (0.2) applies, perhaps with the mesoscale randomness represented as smooth noise about the average stiffness tensor in Figure 0.1(b). Note: Problems lacking separation of scales occur in many situations, ranging from micro/nano-electromechanical systems (MEMS/NEMS) to geophysics. Note: In fluid mechanics one uses the Knudsen number, a dimensionless number defined as the ratio of the molecular mean free path length λ to a representative physical length scale L (e.g., the radius of a body in a fluid) K n = λ/L. Generally, when K n is on the order of one, the deterministic continuum model of fluid mechanics should be replaced by a statistical approach. Suppose we want to determine the ensemble average response u. By inverting (0.2), ensemble averaging and inverting it back again, we find that u is governed by

L−1

−1

u = f.

(0.6)

This begs two questions. 1. How do we determine the operator L−1 −1 ? 2. Can one replace L−1 −1 by L? That is, what is the difference between an average solution of the stochastic problem governed by (0.6) and a deterministic problem obtained by straightforward averaging of (0.7)? This replacement, often implied by deterministic continuum mechanics models, corresponds to L u = f.

(0.7)

This is the basic question of stochastic mechanics for, if there is no difference in a given problem, then one may safely work with its deterministic counterpart. However, theoretical and applied mechanics are ripe with examples to the contrary. Buckling and random vibration are perhaps the most classical instances where fluctuations cause significant changes from the situations governed by homogeneous properties and “nice” inputs equal to the averages of random ones. Various other questions arise here: 3. How does one pass from the microscale d to the mesoscale L, so as to remove the mesoscale fluctuations shown as the grayscale variability in (b)? In other words, unless we reach the RVE, we are faced with a continuum-type random field.

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xxi 4. Supposing we deal with a periodic or disordered lattice-type microstructure instead of the piecewise-constant continuum in (a), what homogeneous continuum models are then dictated? 5. If we are below the RVE scale, how is the continuum random field in (b) defined? Is a locally isotropic random field admissible? 6. What universal statements can be made about the finite size–scaling and convergence of statistical volume element (SVE) domains, such as the one in Figure 0.1(b), to the RVE? 7. How can one proceed with the homogenization in the case of materials having high, or very high, mismatch in phase properties? Note that in those situations the classical homogenization studies, such as the Voigt–Reuss, Hashin–Shtrikman bounds and their improveeff ments, yield very wide bounds on Cijkl . Another dilemma arises when the material to be studied is of an inhomogeneous type, without clearly distinguishable phases, such as shown in Figure 0.1(c). Paper on millimeter to centimeter scales is a perfect example of this for, even though it is well known that paper is made of a cellulose fiber network, the latter displays random grayscale fluctuations directly seen with the naked eye. However, as we move the sheet far away from the eye, the fluctuations vanish and the sheet begins to look homogeneous. Here then we have another motivation to study mesoscale effects. 8. What is the possibility of solving arbitrary boundary value problems of random fields of elastic and inelastic properties governed by (0.2)? 9. What techniques and what results apply when one deals with the inertia term in (0.2)? 10. Can one set up continuum-type field equations for a fractal microstructure where there is no separation of scales whatsoever? 11. How can we analyze wave dynamics in random elastic and/or inelastic media?

0.3

Outline of Contents

Chapter 1 begins with simple aspects of discrete random processes and random geometry and moves on to construction of stochastic models of real microstructures—such as composites, polycrystals, granular, cellular, or fibrous media. By contrast, Chapter 2 focuses on random processes and fields having continuous realizations. These two chapters are written as introductions and guides for nonspecialists who come from deterministic continuum mechanics and need to acquire or recall some tools useful in stochastic mechanics. A deliberate decision was made to only briefly present probability,

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xxii random variables and processes, very classical topics that are expertly and extensively covered in many other books. Our focus is on a basic introduction to random geometry, as well as random processes and fields, because these subjects are scattered in very different places in the literature. Several problems are given in these chapters so as to develop basic “probabilistic” skills in mechanics students and recall some elementary concepts from statistical physics. An analogous foundational role—albeit on the mechanics side—is played by Chapters 3 and 4, where we outline planar lattice (spring) networks: first periodic, then disordered. These constructs have their roots in condensed matter physics and structural mechanics, and over the past few decades provided computational mechanics models in composite materials and granular media. A brief presentation of one-dimensional (1D) models is also made. Spring network models are naturally suited for materials with discrete topology, but they also offer a powerful tool for the representation of piecewise-constant continuous media. They work best in two dimensions (2D), and their continuum counterpart is the planar elasticity discussed in Chapter 5, where we discuss primarily the so-called stress-invariance. Chapter 6 continues this subject in the setting of micropolar elasticity; it also covers several other topics not yet collected in book form, including the passage from a microstructure to an effective micropolar continuum. Chapters 3 through 6 also contain problems, so that the book can be used as a text in a graduate course. Indeed, it is with this combination of several different stochastic and mechanics methods that one can approach problems involving microstructural randomness and scaling. We thus move to Chapters 7 through 11, which present various applications reflecting our own interests. However, it is felt that, in spite of this personal tinge and scope in the presentation, the problems are representative of some important research directions in (micro)mechanics of random media. In particular, Chapter 7 elaborates the finite-size scaling to a classical representative volume element (RVE) and the coupled dependence of moduli on scale and boundary conditions in the sense of Hill (1963). The focus is on linear elastic microstructures. A basic role is played here by a mesoscale window, which may also be called a statistical volume element (SVE). In Chapter 8 we use this approach as a basis for mechanics problems lacking the separation of scales. Thus, we first consider the determination of mesoscale random continuum fields, which, in turn, provide a stepping stone to micromechanically based stochastic finite elements (SFE), slip-lines, and optimal trusses. Here, perhaps, we have a stochastic version of multiscale computational mechanics. Chapter 9 continues the scaling issues of Chapter 7 in more challenging areas: nonlinear elastic and inelastic materials, including a generalization of thermodynamics with internal variables to random media. It is shown again that the SVE rather than the conventional RVE plays an important role here. This leads to a stochastic formulation of thermomechanics with internal variables in Chapter 10. Although the book focuses on nonfractals, Chapter 10’s last section discusses continuum-type equations of fractal media.

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xxiii Chapter 11 focuses on selected topics in wave propagation in random media. Among others, we look at wavefronts, which, given the microscale material randomness, are seen as windows of finite thickness rather than classical discontinuity surfaces—yet another paradigm for the SVE concept. We also report some research on homogenization problems for wave propagation, namely: SFE in the frequency domain, waves in randomly perturbed imperfect periodic structures, and waves in granular media with nonlinear elastic interactions. In fact, it may be a mixed blessing of stochastic mechanics that, depending on the problem, a special subfield of stochastic mathematics is usually needed. Overall, depending on the reader’s interests, the book may require background in several fields: 1. Probability and random variables roughly at the level of Part 1 of Papoulis (1984). 2. Standard continuum (thermo)mechanics (e.g., Ziegler, 1983). 3. Computational methods (finite differences and elements) at the first graduate course level. The book is an outgrowth of graduate courses on micromechanics, mechanics of random media, and probabilistic mechanics that I taught at several universities over the past two decades. The invitation by D.Y. Gao and R.W. Ogden to contribute to their book series has been an honor and I hope that the book meets their expectations. The book would not have materialized in its final form and time frame without the continuing support of Bob Stern and Theresa Delforn at Taylor & Francis. It is a pleasure to acknowledge past and present coworkers—many of them my students—who, through cooperation with me, contributed to this book. My understanding of various randomness and scaling issues in mechanics of materials has been enriched by discussions with many people, including E. Altus, A. Beaudoin, M.J. Beran, J. Carmeliet, A. Chudnovsky, I. Elishakoff, J. Engelbrecht, S. Forest, J.D. Goddard, M. Grigoriu, R.J. Hill, C. Huet, J.T. Jenkins, D. Jeulin, B. Kunin, G.A. Maugin, J.J. McCoy, D.L. McDowell, M. Mi´cunovi´c, V. Mizel, V.F. Nesterenko, M. Pfuff, S.L. Phoenix, A. Pineau, M.-J. Pindera, K. Sab, J. Schicker, S. Schmauder, J. Schulte, K. Sobczyk, D.C. Stahl, A. ´ ech, Swi ˛ S. Torquato, J. Trebicki, ˛ N.J. Triantafyllidis, P. Trovalusci, C.L. Tucker, III, O. Vinogradov, J.R. Willis, A. Zaoui, and A. Zubelewicz. Finally, my special warm thanks go to Iwona Jasiuk, my wife, who supported the idea of writing this book from the outset, while our joint research and her input (including joint work on Chapter 5) substantially helped improve it. My research—and therefore, indirectly, this book—has been made possible by various funding sources over the past decade or so. Principally, these include the AES, AFOSR, CFI, COREM, NSERC, NSF, ONR, USACE, USDA, and I am very grateful for their support.

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xxiv

Bibliography Cristescu, N.D., Craciun, E.-M., and Soos, ´ E. (2003), Mechanics of Elastic Composites, Chapman & Hall/CRC Press, Boca Raton, FL. Hill, R. (1963), Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids 11, 357–372. Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York. Primas, H. (1999), Basic elements and problems of probability theory, J. Sci. Explor. 13(4), 579–613. Truesdell, C. and Noll, W. (1965), The Non-Linear Field Theories of Mechanics, Handbuch der Physik, Vol. III/3, Springer-Verlag, Berlin. Ziegler, H. (1983), An Introduction to Thermomechanics, North-Holland, Amsterdam.

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About the Author

Martin Ostoja-Starzewski is a Professor of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign. He did his undergraduate studies at the Cracow University of Technology, followed by Master’s and Ph.D. degrees at McGill University, all in mechanical engineering. He has been a visiting scientist at Cornell University, the Institute for Mechanics and Materials at UCSD, Ecole des Mines de Paris, Ecole Polytechnique (France), GKSS Research Centre (Germany), and the University of Stuttgart. He has published over 100 journal papers in mechanics, materials science, applied mathematics/physics and geophysics, as well as over 80 conference proceedings papers and book chapters. He is on the editorial boards of Journal of Thermal Stresses, Probabilistic Engineering Mechanics, ASME Journal of Applied Mechanics, Actual Problems of Aviation and Aerospace Systems and International Journal of Damage Mechanics. He is a Fellow of ASME and W.I.F., an Associate Fellow of AIAA, and a Member of ISIMM. He has (co-)edited 10 books/journal special issues and (co-)organized many symposia and conferences. He chairs the PACAM Committee at the American Academy of Mechanics. He is an avid sailor and alpine skier.

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1 Basic Random Media Models

Originating, as legend has it, with the Buffon needle problem (which after two centuries has lost little of its elegance and appeal), geometric probabilities have run into difficulties culminating in the paradoxes of Bertrand which have threatened the fledgling field with banishment from the home of Mathematics. M. Kac, 1976 In his foreword to Santalo’s ´ treatise (1976), partly quoted as a motto to this chapter, Mark Kac pointed out a need for a measure-theoretic formulation of probability theory in general, and random geometry in particular. We shall therefore initially devote some space to one of Bertrand’s paradoxes, and thus motivate the formulation of random geometry as it was accomplished in the first half of the twentieth century. This will set the stage for a review of some basic notions of classical probability theory of engineering and applied science curricula, as well as for an introduction to the simplest random geometric models of disordered microstructures. The review is not complete— for instance, the conditional probability and description of microstructures by joint probability distributions are not treated. The focus of this chapter is on a review of basic concepts of random processes and fields for discrete systems.

1.1

Probability Measure of Geometric Objects

1.1.1 Definitions of Probability The classical definition of probability of an event A—historically, the oldest—is due to de Moivre (1718): P( A) =

NA , N

(1.1)

where NA is the number of all outcomes favorable to A, and N is the number of all possible outcomes. In a simple example of a toss of a fair coin, A = {head}, so that we have NA = 1 and N = 2, and, hence, P( A) = 1/2. However, the 1

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significance of NA, and N is not always clear, as another example shows: roll two dice and consider P( A), where A = {sum of the numbers that equals 7}. Then, basically, we have three ways to proceed: (a) Consider eleven possible sums: 2, 3, . . . , 12. This leads to NA = 1 so that P ( A) = 1/11, which, of course, is a wrong way of reasoning. (b) Consider all possible outcomes of pairs of numbers not distinguishing between the first and the second die: {1, 1}, {1, 2},. . ., {6, 6}. This leads to NA = 3 and N = 21, so that P ( A) = 3/21, which is again wrong. (c) Count all the pairs distinguishing between the first and the second: ⎤ ⎡ 11 . . . 16 ⎥ ⎢ (1.2) = ⎣. . . . . . . . .⎦, 61

...

66

where by we denote the space of all elementary events. Now, NA = 6 and N = 36, so that P( A) = 6/36, which is a correct answer. This is the type of problem that motivated Laplace (1814) to come up with an improved classical definition of probability: “The probability of an event equals the ratio of its favorable outcomes to the total number of outcomes, provided that all outcomes are equally likely.” There are several drawbacks to this definition: 1. It contains a logical error: equally likely meaning equally possible defines a concept in terms of the same concept. 2. It cannot deal with an unfair (loaded) die. The classical definitions are based on the principle of insufficient reason, according to which, in the absence of prior knowledge, we must assume that all the elementary events have equal probabilities. 3. It cannot deal with an infinite set of possible outcomes, as is the case, for example, with Bertrand’s paradox below. One way to deal with the obstacle of an infinite number of outcomes was offered by a relative frequency definition (von Mises, 1931): P( A) = lim

n→∞

nA , n

(1.3)

where n A is the number of occurrences, and n is the number of trials. Let us note, however, that the numbers n A and n might be large but not infinite in any physical experiment. Therefore, this definition can only be accepted as a hypothesis, and not as an experimentally obtainable number. Bertrand’s paradox arises from the following problem: given a circle of radius r , we wish to determine the probability p that the length l of a randomly selected cord AB is longer than the side of an inscribed equilateral triangle.

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(a)

3

(b)

(c)

FIGURE 1.1 Three different solutions of Bertrand’s problem.

There are, in fact, three possible answers: 1. The “random radius” method: Let be a fixed diameter of the circle, having a uniform distribution along it. We draw cords normal to it and find p = 1/2, Figure 1.1(a). 2. The “random endpoints” method: Let be an angle at one vertex of the triangle, having a uniform distribution. Considering all the cords through that vertex, we find p = (π/3)/π = 1/3, Figure 1.1(b). 3. The “random midpoint” method: Let be a circle of radius r inscribed in the triangle and the cord’s center have a uniform distribution over the circle. Then the probability that the cord’s center lies in any region of area A is A/πr 2 . If the cord’s center falls outside a circle of radius r/2, then it is too small. Hence, p = (πr 2 /4)/πr 2 = 1/4, Figure 1.1(c). Clearly, each of the three solutions is correct in its own right as each one of them solves a different problem. The problems differ in the specification of the outcomes of an experiment and the meaning of the term possible (or favorable). The first notion refers to the sample space of elementary events , and the second one to a probability measure P (or, equivalently, a probability distribution) defined on . The basic question to ask is: What is the space in this problem? The correct framework in which to set up and P is offered by an axiomatic definition of probability, with which we shall later briefly return to Bertrand’s paradox. The axiomatic definition (Kolmogorov, 1933) requires that, for a space of elementary events , there is identified a so-called σ − algebra F of subsets of . The system {, F} is called a measurable space (Rudin, 1974). Then, on F one can define a real-valued set function P, called a probability measure, which satisfies three axioms: Axiom I. For every A ∈ F, P is non-negative: P( A) ≥ 0.

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Microstructural Randomness and Scaling in Mechanics of Materials Axiom II. The probability of a certain event equals 1: P() = 1.

Axiom III. For any sequence of mutually disjoint events A1 , A2 , . . . (Ai ∩ Aj = ∅ for i, j = 1, 2, . . .; i = j), there holds the equality ∞ P(∪i=1 Ai ) =

∞

P( Ai ),

i=1

and we say that P is a countably additive function. The sample space , together with F and P defined on it, is called a probability space and denoted by {, F, P}. Here are the basic properties of P (Problem 1): 1. The probability of an impossible event is zero: P( ) = 0. Note that the inverse statement is not true, i.e., P( A) = 0 does not imply A = . 2. P is an additive function: system of pairwise disjoint events for any n n

A1 , A2 , . . . , An , P Ai = P( Ai ). i=1

i=1

3. Probability of an event Aopposite to Ais given as: P( A) = 1− P( A). 4. For any two events A and B, P( A ∪ B) = P( A) + P( B) − P( A ∩ B). 5. If A ⊂ B, then P( A) ≤ P( B). 6. For every A ∈ F, P( A) ≤ 1.

7. For any sequence of events of A1 ⊂ A2 ⊂ . . . such that A = Ai , i

P( A) = limn→∞ P( An ).

We now discuss basic cases of and the corresponding ways of setting up of probability P. 1.1.2 Probabilities on Countable and Euclidean Sample Spaces

1.1.2.1 Ω is a Countable Set: Ω = ω1 , ω2 , . . . We define P on all the elementary events {ωi } in the following way: P({ωi }) = pi , where pi ≥ 0

pi = 1.

(1.4)

(1.5)

i=1

It follows from this and the system of axioms I to III that, if A ⊂ and A = {ω1 , ω2 , . . .}, then P ( A) = P ({ω1 , ω2 , . . .}) = P ({ω1 } ∪ {ω2 } ∪ . . .) = P ({ω1 }) + P ({ω2 }) + . . . = p1 + p2 + . . . .

(1.6)

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The function P defined by (1.6) satisfies axioms I and II because of (1.5), and axiom III due to the fact that the sum of a series of non-negative numbers does not change under arbitrary grouping or change in ordering of the terms in the series. A particular case of the countable space is that of a countably finite set = {ω1 , ω2 , . . . , ω N }, where P ({ω1 }) = P ({ω2 }) = . . . P ({ω N }) =

1 , N

(1.7)

where N is the number of all possible elementary events, and n is the number of the elementary events that favor A. In view of (1.7) and the three axioms, if A ⊂ and A = {ω1 , ωn , . . . , ωn }, then P ( A) = P ({ω1 , ω2 , . . . , ω2 }) = P ({ω1 } ∪ {ω2 } ∪ . . . ∪ {ωn }) = P ({ω1 }) + P ({ω2 }) + · · · P ({ωn }) = p1 + p2 + · · · pn =

n · N

(1.8)

In (1.8) we recognize Laplace’s definition of probability, and its usefulness when is countably finite and (1.7) holds. Example 1: Each of m elements may be assigned to one of n different sets (m < n). Compute the probability that no two elements get assigned to the same set, assuming each of the assignments is equally probable. The solution depends on first observing that there are nm different assignments. Thus, each one is given a probability 1/mn . The sought probability equals [m (m − 1) (m − 2) . . . (m − n + 1)] /mn . Example 2: A box contains seven white and three black balls. We take two balls at random from the box, i.e., with (1.7) satisfied. Find the probability of the event A = {both balls are black}. The solution follows by first noting that has N = 10 = 45 elements, 2 so that each one has the probability 1/45. The event A has NA = 32 = 3 elements. Thus, P( A) = NA/N = 1/15. This example is a special case of the so-called Fermi–Dirac statistics. Classical statistical mechanics offers a number of combinatorial problems of this type, and we collect three basic ones here for the sake of reference. Note the difference between a permutation (P), which is an ordering of a set of objects, with regard to order, and a combination (C), which is an ordering without regard to order. Besides the aspect of ordering, there is also the aspect of “repetitions” versus “no repetitions.” 1. N > M: Number of permutations of N objects taken M at a time without repetitions (sequence important): N PM = N!/( N − M)! The special case of this is: 2. N = M: Number of permutations of N objects taken N at a time without repetitions (sequence important): N PN = N! 3. N = M: Number of permutations of N objects taken N at a time with repetitions (sequence important): N N . Generalizing this to N ≥ or < M, we get N M .

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The latter, which actually shows up in the Maxwell–Boltzmann statistics, is very relevant in the mechanics of microstructures: it is the so-called random chessboard (or checkerboard). Consider a two-phase material in two dimensions (2D): each square cell of M sites of a square lattice L × L = M is occupied, independently of what happens in other cells, with probability p1 and p2 by phases 1 and 2, respectively. Figure 1.2(a) shows one possible elementary event ω, i.e., an assignment of two phases to the lattice. It follows that is made of all such events. Clearly, || = 2 L×L . Given the construction process,

(a)

(b) FIGURE 1.2 (a) One event (or realization) of the random, two-phase chessboard on an L × L lattice, with L = 11. (b) A 16-phase mosaic.

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each ω occurs with probability 1/2 L×L . Figure 1.2(b) shows a realization of a 16-phase material taken out of an accordingly much larger space. 1.1.2.2 Ω is a 1D Euclidean Space: Ω = R Two basic questions arise here: does there exist a function P satisfying the set of axioms I to III, and if it does, how can it be constructed? We proceed by introducing an auxiliary, real-valued function F on R having the following properties: 1o F is nondecreasing. 2o limx→−∞ F (x) = 0, limx→∞ F (x) = 1. 3o F is continuous from the left. As these simple examples show, there exist many such functions: 0 for x ≤ 0, 1 1 F (x) = F (x) = arctan x + π , π 2 1 for x > 0; x 0 for x ≤ 0, F (x) = f (u)du, (1.9) F (x) = F (x) = 1 − e −x for x > 0; −∞ ∞ where f ≥ 0 and −∞ f (u)du = 1. For the function (1.9)4 , the condition 1o implies that, for a < b,

b

F (b) − F (a ) =

f (x) dx ≥ 0,

(1.10)

a

that is, an integral over an arbitrary interval (a , b) is non-negative. Thus, a function P can be constructed from F by taking, for each a < b, P (a , b) = F (b) − F (a ).

(1.11)

This defines P (a probability measure) on the family of all sets of the form [a , b). One can show that P can be extended in a unique way onto S, the Borel σ -field on R, so as to satisfy axioms I to III (e.g., Prohorov and Rozanov, 1969). The function F is called a probability distribution, and the function f is called a probability density. A classical result is that, if P is a probability measure defined on R, then F (x) = P ((−∞, x)) possesses properties 1o to 3o (Problem 2). Basically, there are two possibilities: either to begin with a function P and construct F , or to begin with a function F and construct P. Example 3: Consider this probability distribution:

F (x) =

⎧ 0 ⎪ ⎪ ⎨x − a b−a ⎪ ⎪ ⎩ 1

for

x ≤ a,

for a < x ≤ b, for

b < x.

(1.12)

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This function can be shown to have properties 1o to 3o . Now, let a < c < d < b, so that P (c, d) = F (d) − F (c) =

d −a c−a d −c − = , b−a b−a b−a

(1.13)

which demonstrates that the probability of an interval [c, d) depends on its length, but not on the location of the points c and d in [a , b). It is noteworthy that the function (1.12) can be written as ⎧ 0 for x ≤ a , ⎪ ⎪ ⎪ ⎨ 1 x du for a < x ≤ b, F (x) = (1.14) ⎪ b−a a ⎪ ⎪ ⎩ 1 for b < x. In other words, for x ∈ [a , b), F is an integral of a constant, and such a probability distribution is called uniform (or rectangular). Note: The deterministic case fixed at one value x0 is modeled by a causal distribution F (x) = H (x − x0 ), with H being the Heaviside function. Dirac delta plays the role of the corresponding causal probability density. 1.1.2.3 Ω is a 2D Euclidean Space: Ω = R2 Let F be defined on R2 and have the following properties: 1o F is nondecreasing with respect to each one of its arguments (while the other is fixed). 2o ∀x lim y→−∞ F (x, y) = 0, ∀y limx→−∞ F (x, y) = 0, limx→∞, y→∞ F (x, y) = 1. o 3 F is continuous from the left with respect to each argument (while the other is fixed). o 4 ∀x1 ≤ x2 and ∀y1 ≤ y2 , F (x2 , y2 ) − F (x1 , y2 ) − F (x2 , y1 ) +F (x1 , y1 ) ≥ 0. Similarly as before, an important example of F involves (Problem 3) x y F (x, y) = f (u, v)dudv, (1.15) −∞

−∞

where f is the probability density satisfying ∞ ∞ f ≥ 0, f (x, y)dxdy = 1. −∞

−∞

(1.16)

Now, for every rectangle {(x, y) : x1 < x < x2 , y1 < y < y2 }, P is defined by P({(x, y) : x1 < x < x2 , y1 < y < y2 }) = F (x2 , y2 ) − F (x1 , y2 ) x2 y2 −F (x2 , y1 ) + F (x1 , y1 ) = f (x, y)dxdy ≥ 0. x1

y1

(1.17)

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9

Again, one can show that P can be extended in a unique way onto S, the Borel σ -field on R2 , so as to satisfy all the axioms I to III. Furthermore, it is also possible to prove that every P defined on R2 determines a probability distribution F (x, y) = P({(x , y ) : x < x, y < y})

(1.18)

having properties 1o to 4o . Although the proof of properties 1o to 3o is analogous to the one-dimensional case, it is instructive to demonstrate here the property 4o . To this end, we introduce four sets (Figure 1.3): J 1 = {(x, y) : x < x1 , y < y1 } J 2 = {(x, y) : x1 ≤ x < x2 , y < y1 } J 3 = {(x, y) : x < x1 , y1 ≤ y < y2 ,}

(1.19)

J 4 = {(x, y) : x1 ≤ x < x2 , y1 ≤ y < y2 ,}. The sets J 1 , J 2 , J 3 , and J 4 are disjoint, so that F (x2 , y2 ) − F (x1 , y2 ) − F (x2 , y1 ) + F (x1 , y1 ) = P( J 1 ∪ J 2 ∪ J 3 ∪ J 4 ) − P( J 1 ∪ J 2 ) − P( J 1 ∪ J 3 ) + P( J 1 ) = P( J 1 ) + P( J 2 ) + P( J 3 ) + P( J 4 ) − P( J 1 ) − P( J 2 ) − P( J 1 ) − P( J 3 ) + P( J 1 ) = P( J 4 ) ≥ 0. (1.20) Thus, there is a unique correspondence between the functions P and F with a precision down to within sets of probability measure zero. The above formulation may be extended to = Rn , with F having the same properties as previously. However, the notation becomes quite cumbersome and this is not written here explicitly. Note that the case ⊂ Rn may be treated as a special case of = Rn by taking P (Rn − ) = 0. y

y

y2 J3

b

J4

y1

A x2

x1 J2

J1

x

k

k (a)

b

x

(b)

FIGURE 1.3 (a) Domains J 1 , J 2 , J 3 , and J 4 . (b) Domain A of Example 5, bounded by two lines x − y = ±k.

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(1.21)

It is left an exercise to show that F has the properties 1o to 4o above. Now, let a < a < b < b and c < c < d < d, so that

P( (x, y) : a ≤ x ≤ b , c ≤ y ≤ d ) = F (b , d ) − F (b , c ) − F (a , d ) + F (a , c ) b d b c = f (x, y)d xd y − f (x, y)d xd y −∞

− =

−∞

a

−∞ b

a

−∞

d

−∞ d

c

f (x, y)dxdy +

f (x, y)d xd y =

a

−∞

−∞

c

−∞

f (x, y)d xd y

(b − a )(d − c ) . (b − a )(d − c)

(1.22)

This probability distribution is called uniform in a rectangle ((x, y) : a ≤ x ≤ b, c ≤ y ≤ d). It is interesting to note that the uniform distributions of this example as well as that of (1.12) can also be introduced in another way: Consider to be a subset of Rn , and let μ be a Lebesgue measure in that space, such that 0 < μ () < ∞ and A ⊂ is a Borel set. Then we can introduce a probability P in the following fashion: P ( A) =

μ ( A) . μ ()

(1.23)

Clearly, the distributions encountered in Examples 3 and 4 are special cases of (1.23). It is left as Problem 4 to check that P defined by (1.23) satisfies the set of axioms I to III. F corresponding to it is called a uniform distribution, while P itself is sometimes called a geometric probability. Example 5: Let a = c = 0 and b = d in Example 4. Determine P ( A) for A = {(x, y) : |x − y| < k} with k ∈ (0, b). The solution follows by first observing that = {(x, y) : 0 ≤ x ≤ b, 0 ≤ y ≤ b}. With μ() = b 2 and μ( A) = b 2 − (b − k) 2 , P( A) is readily calculated from (1.23). This example may be used to deal with the following application: Given that two impulses arrive at random instants at a receiver over a period of time T, compute the probability that the difference in their arrival times is smaller than k < T. Another Example is given in Problem 15. In problems of this type the term “random” connotes a uniform probability density in a certain domain. In the case of arbitrary sets of a positive, bounded Lebesgue measure, this implies that the probability distribution is constructed from the formula (1.23).

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1.1.3 Random Points, Lines, and Planes 1.1.3.1 Random Lines in Two Dimensions Let us consider a set A of geometric objects in a 2D Euclidean space, and require that its properties remain invariant under the group of transformations appropriate to such space, that is, all translations, rotations and reflections. Now, all translations are obtained by transforming the coordinates by addition of constants, while all rotations and reflections are effected by an orthogonal transformation. Thus, the coordinates xk are mapped into new coordinates xk by xk = xk(0) + a ki xi ,

(1.24)

where [a ki ] is an orthogonal matrix, xk(0) is the translation, and the set A is mapped into A . If we now denote by E the set of parameter points of A, then E will be a new set corresponding to A . If we now introduce a probability measure on the σ -algebra of all Es, then it is natural to impose here a requirement of invariance of this measure under (a group of) all the transformations (1.24). For example, considering the points in the 2D Euclidean space, the set E of these points must be contained in some region R of finite extent (i.e., of bounded measure), like the inside of a cube or sphere. Then, by μ ( R) we denote the Lebesgue measure of R, and by μ ( E) the Lebesgue measure of E. It follows that the probability of a point lying in E is P ( E) =

μ ( E) . μ ( R)

(1.25)

In the case of a random number of points, the simplest model is to consider the number of these points in E to be a Poisson random variable with mean λμ ( E) = M ( E), where λ is a constant; a formal definition of a random variable is given in Chapter 2. If the number of such points is n, the probability that they lie, respectively, in sets E 1 , E 2 , . . . , E n contained in E, given n, is μ ( E1) μ ( E2) · · · μ ( En) . [μ ( E)]n

(1.26)

Another example concerns straight lines in a 2D Euclidean space. Let each line be defined by an equation of the form ux + vy + 1 = 0,

(1.27)

so that u and v play the role of the line’s coordinates, and the parameter space of all possible events is the (u, v) plane, with exclusion of the point (0, 0); this presents no special problem because the associated Lebesgue measure is zero. The transformation (1.24) takes the form X = a + x1 cos α − x2 sin α

Y = b + x1 sin α + x2 cos α,

(1.28)

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with a and b being arbitrary constants and α ∈ [0, 2π ). The new line is specified by U X + VY + 1 = 0,

(1.29)

with u=

U cos α + V sin α aU + bV + 1

v=

−U sin α + V cos α . aU + bV + 1

(1.30)

Notice that aU + bV + 1 would be zero for lines passing through the origin, and so, again, such lines are being excluded. We now look for a measure M ( E) on the space , defined by an integral of the form M ( E) = F (u, v) dudv, (1.31)

E

such that it is equal to M E under all the transformations (1.18). That is, we want F (u, v) dudv = M ( E) = M E = F (U, V) dUd V. (1.32) E

E

Observe that from F (U, V) = F (u, v) J we obtain the condition where the Jacobian is evaluated as J = (aU + bV + 1) −3 = (u2 + v2 ) 3/2 (U 2 + V 2 ) −3/2 ,

(1.33)

which shows that F (U, V) is to be taken proportional to (u2 + v2 ) −3/2 , and dudv M ( E) = (1.34) 3/2 . 2 E u + v2 Note that (1) for this technique to work, we require that the Jacobian be of the form J = φ (u, v) /φ (U, V) and (2) with the measure M ( E), we can now establish the probability P ( E) of a single line, under the condition that its coordinates are not (0, 0). In the foregoing we began with a standard description of a straight line in the form y = a x + b. However, we note that a line is also completely specified by the distance p from the origin of the (x, y)-system and the angle θ that the foot of the perpendicular to the line makes with the positive axis x, Figure 1.4. The ranges of these two parameters are 0 ≤ p < ∞,

0 ≤ θ ≤ 2π.

(1.35)

In these polar coordinates ( p, θ) one can then represent a line by a so-called Hesse normal form: x cos θ + y sin θ = p.

(1.36)

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13 y

2π p

θ x

θ 0

P

D/2 (a)

(b) FIGURE 1.4 (a) Two steps in generation of a Poisson line within a circle of diameter D. (b) A realization of a Poisson line field of 100 lines in a unit square.

With reference to Figure 1.4(a), we see that lines parametrized by p and θ, and falling within a circle of diameter D, are represented as points in the strip S = (0, D/2] × [0, 2π). Under the ( p, θ) parametrization, and the substitution u=−

cos θ p

v=−

sin θ p

(1.37)

the differential element in the (u, v)-system transforms into one in the ( p, θ)system:

dudv 3/2 → dpdθ. u2 + v2

(1.38)

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Note that this formulation corresponds to the first solution of Bertrand’s problem of Section 1.1. Poincar´e (1912) has shown that dpdθ is the only differential element that remains invariant under the group of all translations and rotations. See also Jaynes (1973). Note: In continuum physics this property results in the invariance of material response under superposed rigid body motions with one observer present, which is unnecessarily restrictive, while the correct path is to require the invariance of material response under the change of observer (Murdoch, 2003). One may continue in this fashion to determine probability measures for straight lines uniformly distributed in 3D space, planes in 3D space, etc. (Santalo, ´ 1976). Indeed, to quote Kendall and Moran (1963): “Once the probability measure of a geometrical set is constituted, the solution of particular problems can proceed without the appearance of paradoxes or of difficulties touching on axiomatization.” 1.1.3.2 Planes in Three Dimensions A plane is represented by the equation ux + vy + wz = 1,

(1.39)

which, similar to the procedure given above, is better represented in terms of the length of the perpendicular to the plane from the origin, together with the polar coordinates of these perpendiculars. The corresponding equation is x sin θ cos φ + y sin θ sin φ + z cos θ = p,

(1.40)

where 0 ≤ θ < π , 0 ≤ φ < π, and 0 < p < ∞. 1.1.3.3 Straight Lines in Three Dimensions This involves a parametrization by four numbers (a , b, p, q ) because a line is conveniently represented by two equations x = a z + p, y = bz + q .

1.2

(1.41)

Basic Point Fields

1.2.1 Bernoulli Trials Let us begin with a few basic facts from the probability theory. A Bernoulli trial is an experiment resulting in a success (event ω1 ) or failure (event ω2 ) with probability p or q , respectively. Here the sample space = {ω1 , ω2 } with P ({ω1 }) = p and P ({ω2 }) = q . A binomial distribution is a series of n Bernoulli trials, all of them independent and identically distributed, in which

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the probability of k successes is given by P{N ( A) = k} =

n k n−k p q k

k = 0, 1, 2, . . . , n.

(1.42)

There are two important asymptotic theorems for this binomial distribution: one due to de Moivre and Laplace, and another due to Poisson. The first of these states that if p 1 and n is so large that np npq 1, then n k n−k 1 2 p q √ e (k−np) /2npq , k 2πnpq

k = 0, 1, 2, . . . , n,

(1.43)

for k being around np. For x = k, the right-hand side in the above equals P

x − np √ npq

1 2 √ e (x−np) /npq , 2π

(1.44)

which is the Gaussian density function. The Poisson theorem applies in the situation where np is of the order of 1, with k being on the order of np, equal, say, to a ; it states that n k n−k ak p q e −a k k!

as

n→∞

p → 0.

(1.45)

1.2.2 Example—Model of a Fiber Structure of Paper It is well known that paper is made of a multitude of cellulose fibers arranged in a disordered fashion, Figure 1.5(a). We now make an assumption very close to reality, namely, that each fiber lies horizontally—i.e., in a plane of the sheet of paper—and there are some n = 7 to 10 layers of fibers, as is the case in a typical paper; the fibers are of a rectangular cross-section and uniform in thickness, Figure 1.5(b). It follows that the process of sampling of voids and fibers by a vertical line is a sequence of n (number of layers) Bernoulli trials, with each trial being a success (finding a void) or failure (finding a fiber) with respective probabilities: P (void) = p,

P (fiber) = q = 1 − p.

(1.46)

As an example, note that the event corresponding to the vertical line X in Figure 1.5(b) is {1110101}. Thus, there are no bonds between the third, fifth, and seventh layer from the top. A quantity of fundamental interest in paper physics is the relative bonded area (RBA), which is defined as the fraction of the total area of fibers, in plan view projection, that is bonded (via hydrogen bonding) to other fibers. A mat of fibers does not form paper for RBA = 0, and, clearly, elasticity and strength of paper strongly depend on a positive-valued RBA. Soszynski ´ (1995)

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(a)

τ 7τ

X X = 1110101 (b) FIGURE 1.5 (a) A perspective view of a sheet of paper; (b) section through an idealized sheet with n = 7, and a sample outcome X ={1110101}. (From Soszynski, ´ 1995, with permission.)

has shown that, for a sheet idealized as a system of layers of fibers, RBA is given by RBA =

n−1 2 q . n

(1.47)

With reference to Figure 1.5(b) we note: (i) The distribution of fibers across the sheet is binomial and formula (1.42) applies. (ii) The probability of the occurrence of bond (fiber–fiber) sites on n − 1 interlayer planes, for an n-layer paper, is k−1 or n−(k−1) k−1 n − (k − 1) k − i Pn (k) = q k p n−k n − 1 i − 1 i i=1 k = 2, . . . , n, (1.48) where i is the so-called bond index; i = 1, . . . , k − 1 or i = 1, . . . , n − (k − 1).

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(iii) The probability of finding all bond sites on all the interlayer surfaces is (Problem 7): n

Pn (k) = q 2

(1.49)

k=2

which, noting that one bond belongs to two neighboring layers, leads to (1.48). 1.2.3 Generalization to Many Types of Outcomes Consider a partition [A1 , A2 , . . . , Am ] of , in a single trial, into m subsets such that P ( Ai ) = pi and p1 + p2 + · · · + pn = 1. Repeat the experiment n times (i.e., the sample space becomes an n-fold Cartesian product n ) and denote by pn (k1 , k2 , . . . , km ) the probability of the event {A1 occurs k1 times, A2 occurs k2 times, . . . , Am occurs km times}, where k1 + k2 + · · · + km = n. Then pn (k1 , k2 , . . . , km ) =

n! km p k 1 p k 2 . . . pm . k1 !k2 ! . . . km ! 1 2

(1.50)

Next, suppose that each npi tends to some constant a i , as n → ∞ and pi → 0 for all i ≤ m − 1. In this case k

pn (k1 , k2 , . . . , km )

a m−1 a 1k1 −a 1 a 2k2 −a 2 e e . . . m−1 e −a m−1 k1 ! k2 ! km−1 !

as

n → ∞.

(1.51)

1.2.4 Binomial and Multinomial Point Fields 1.2.4.1 Basics Let us consider a rectangular window W in the R2 plane. Next, we generate n points in this window under two conditions: (i) The positions of n points (x1 , . . . , xn ) are stochastically independent in the sense that P (x1 ∈ B1 , . . . , xn ∈ Bn ) = P (x1 ∈ B1 ) . . . P (xn ∈ Bn ),

(1.52)

where each Bi is an arbitrary Borel set in W. (In general, we can consider W to be an arbitrary compact set.) (ii) Each of the points (x1 , . . . , xn ) is uniformly distributed in W, i.e., for any Borel set B ⊂ W, we have p B = P (xi ∈ B) =

μ ( B) , μ (W)

(1.53)

where μ () is the area of a given set (i.e., its Lebesgue measure). We observe two basic things from the above. First, the probability that x

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Thus, the number of points in B, N ( B), is called a binomial point process or a binomial point field. We prefer the latter terminology given the fact that things happen spatially rather than temporally, while the word process usually implies some time dependence. From the consideration of the size of our window W, μ (W), and the number of assumed points, n, we immediately obtain the mean number of points per unit area (or areal point density): λ=

n , μ (W)

(1.55)

and the mean number of points per set B, N ( B) = λμ ( B).

(1.56)

From this one can get the so-called emptiness probability P{N ( B) = 0} =

[μ (W) − μ ( B)]n , μ (W) n

(1.57)

or the probability of finding no points in B. 1.2.4.2 Simulation of a Binomial Point Field with n Points There are three basic methods. Method 1: If W is a unit square [0, 1]2 , proceed by generating x = (x1 , x2 ) with both xi s being uniform random numbers in [0, 1], and repeating it n times. Method 2: If W is a square [0, a ]2 , proceed n times by generating x = (x1 , x2 ) with both xi s being uniform random numbers in [0, 1], and scaling each time by a to get xi , i = 1, 2. Method 3: If W is an arbitrarily shaped region, there exist three possibilities: 1. Place W in [0, a ]2 and proceed as in the process in method 2 above. However, accept points falling into W only, and generate up to n such points—see Figure 1.6(a), with W being a unit square. 2. Divide W into a disjoint union of k small squares, which together approximate W. Then, pick each square (i.e., decide whether there is a random point in it) according to its size relative to A(W), and generate its coordinates according to a uniform distribution over that square.

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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 (a)

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 (b)

0.6

0.7

0.8

0.9

1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

FIGURE 1.6 Samples of (a) the binomial point process and (b) the sequential inhibition process, each with 100 points.

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Microstructural Randomness and Scaling in Mechanics of Materials 3. Providing W has adequate symmetry, apply a transformation of coordinates. For example, for a circle of radius R, a random point is described by x = ( p, θ). In order to have a probability distribution 2 F R (r ) = p{r ≤ R} proportional to the area (i.e., to r√ ) we need to use a uniform number u from [0, R], to generate ρ = u. The angle θ is uniform in [0, π].

1.2.4.3 Generalization to a Multinomial Point Field The binomial point field can be generalized to a multinomial one by considering a partition of the window W into a union of m pairwise-disjoint subsets B1 , B2 , . . . , Bm with p Bi = P{μ ( Bi )} =

μ ( Bi ) . μ (W)

(1.58)

It follows that the probability of finding k1 points in B1 , k2 points in B2 , . . . , and km points in Bm , in a sequence of n trials, is Pn {N ( B1 ) = k1 , N ( B2 ) = k2 , . . . , N ( Bm ) = km } n! p k1 p k2 . . . p kBmm . = k1 !k2 ! · · · + km ! B1 B2

(1.59)

This field may be used to model a composite with m different types of inclusions. 1.2.5 Bernoulli Lattice Process Consider a Cartesian lattice of spacing a in R2 , that is L a = {x = (m1 a , m2 a )},

(1.60)

where m1 and m2 are integers. A Bernoulli lattice process p,a on L a is a random subset of the lattice where each point of L a is contained in p,a with probability p independently of all the other points. If the random variable p,a ( B) is the number of points in B, then it is binomially distributed with parameters p and n (the number of lattice points that belong to B). Also, p,a ( B1 ), p,a ( B2 ), . . . , p,a ( Bk ) are independent if B1 , B2 , . . . , Bk are pairwise disjoint, and we have n P{ p,a ( B) = k} = p k (1 − p) n−k k = 0, 1, 2, . . . , n, (1.61) k Simulation of a Bernoulli lattice process: For each point x ∈ L a , generate a random variable zk , uniformly distributed in [0, 1], and accept this point if zx < p. Then, p,a is the union of all such points. The asymptotic limit of a Bernoulli lattice process: It is easy to see that the mean of p,a is (1.62) p,a = np,

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where, as before, n is the number of points that are in B. This leads to a concept of intensity λ = p/a 2 .

(1.63)

If we consider B to be a unit square divided into m × m spacings with each spacing being of the size a = 1/m, we get λ = p/a 2 = pm2 . We now let m2 → ∞ and p → 0 while we keep λ constant, so that, analogously to Section 1.2.1, the random variable p,a ( B) has a distribution that is asymptotically Poisson of mean λμ ( B) P{ p,a ( B) = k}

[λμ ( B)]k −λμ( B) e k!

as

m2 → ∞

p → 0.

(1.64)

1.2.6 Poisson Point Field 1.2.6.1 Basics The homogeneous planar Poisson field (or process), , is the cornerstone of the theory of spatial point fields (or processes). It represents the simplest possible stochastic mechanism for the generation of spatial random point patterns. This process is defined by two postulates: P1: For any λ > 0, and any finite planar region A, N ( A) is a Poisson distribution with mean λ |A|. P2: Given N ( A) = n, the n events in A form an independent random sample from the uniform distribution on A. Note: P1 implies that the intensity λ does not vary over the plane. P2 implies that there are no interactions among the events, that is, no inhibition or encouragement of events in the neighborhood of x given an event at x. The third important property is given in Problem 9. The intensity λ (x) of a spatial point process is defined by λ (x) = lim

|dx|→0

N (dx) , |dx|

(1.65)

where denotes the average. Note that the parameter λ of P1 is the intensity of the process, independent of x. The emptiness probability becomes P{N ( B) = 0} = e −λ|B| .

(1.66)

Furthermore, in analogy to the formula (1.59) for the multinomial field Pn {N ( B1 ) = k1 , N ( B2 ) = k2 , . . . , N ( Bm ) = km } =

[λ |B1 |]k1 −λ|B1 | [λ |B2 |]k2 −λ|B2 | [λ |Bm |]km −λ|Bm | e e e ... . k1 ! k2 ! km !

(1.67)

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Variance of the Poisson point field follows from the variance of the Poisson random variable Var {N ( B)} = λ |B|.

(1.68)

It follows from P1 that, if B1 and B2 are disjoint, N ( B1 ) N ( B1 ) = λ |B1 | λ |B1 |.

(1.69)

It then follows from Problem 10 that, for any two sets B1 and B2 , such that B1 ∩ B2 = D, N ( B1 ) N ( B2 ) = λ |B1 | λ |B2 | − λ |D|.

(1.70)

Now, consider two disjoint, infinitesimal disks B1 and B2 , at a distance r , and of areas d F1 and d F2 , respectively. Then, because the disks are very small, N ( B1 ) N ( B1 ) P ( N ( B1 ) = 1, N ( B2 ) = 1).

(1.71)

On the other hand, PN ( B1 ) = 1, N ( B2 ) = 1 = λd F1 λd F2 = ρ (2) (r ) d F1 d F2 ,

(1.72)

where we have introduced the second-order intensity function ρ (2) (r ) to characterize arbitrary (not necessarily Poisson) point fields. Of course, in the Poisson field case, ρ (2) (r ) = λ2 , and this is the ideal reference case. On this basis, one then defines a pair correlation function for point fields g (r ) ≡ ρ (2) (r ) /λ2 ,

(1.73)

Further concepts, such as a “pair correlation function” and “radial distribution function” and a “k-function” are used to deal with point fields (Math´eron, 1975; Stoyan et al., 1995). 1.2.6.2 Simulation of a Poisson Point Field If a Poisson point process is stationary or homogeneous (i.e., invariant with respect to arbitrary shifts in space), one can consider its restriction to a compact set W of the plane under the condition that (W) = n. This conditioning yields a new point process that is actually the binomial point process in W with n points, simulated according to Section 1.2.5. 1.2.6.3 Inhomogeneous Poisson Point Field This process is defined by two postulates: P1: N ( A) has a Poisson distribution with mean

A λ (x) dx.

P2: Given N ( A) = n, the n events in A form an independent random sample from the distribution on A with the probability density function proportional to λ (x).

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This type of process is very useful for simulating functionally graded materials; see Problem 8. As for a homogeneous Poisson point field, E ( N ( B)) = λ |B|, while N ( B) = 1 = λ (x) |B|, if B is an infinitesimal disk of area |B| = d F. 1.2.6.4 Inhibition and Hard-Core Processes The so-called car parking problem is a point of reference for more complex, analogous problems in higher dimensions. It involves a sequential coverage of an interval (0, x) with x > 1 by intervals of unit length without overlap, until no additional unit length interval can be accommodated. Let N denote the number of all such intervals placed in (0, x). It was shown by R´enyi (1959) that the ratio of the average of N to x tends to 0.74759 . . . as x → ∞, whereas the ratio of the variance of the occupied portion of the interval to its length is 0.03567. Various extensions of this problem have been considered in the probability and statistical physics literature. They involve parking of intervals (or cars) of random length in one and more dimensions. Of primary interest to us is the latter case, which is also called a random space-filling problem, and we list several references. The case of 2D was considered by Palasti (1960) and Gani (1972). The case of 3D was studied in the setting of liquids by Bernal (1960), Bernal et al. (1962), and Scott (1962). Typical inhibition (or hard-core) processes are: 1. Thinning: A Poisson point field of a given intensity λ is thinned by the deletion of all pairs of events a distance less than λ apart. 2. Sequential inhibition process: Throw Poisson points on a plane and keep only those which fall no closer than dmin to any previous ones. Figure 1.6(b) shows a realization of this process with 100 points at dmin = 0.0785, together with disks centered at these points. The volume fraction of the disks is 0.4, their radius being 0.0357. Note that the classical “car-parking” problem is a simple inhibition in 1D. 3. Modified sequential inhibition process: take dmin to be a random variable for each Poisson point. All these processes lead to a development of regularity of point patterns.

1.3

Directional Data

1.3.1 Basic Concepts Many materials involve features whose angular directions—such as fibers or cracks in a matrix, crystal orientations—are of interest. Because these are often of a random character, a basic calculus of angular random variables is needed. First of all, the mean requires a definition modified from that of

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conventional linear variables. Thus, suppose we have N measurements of the angle θ: {θn ; n = 1, . . . , N}. Then the mean angle θ is determined as a solution of the equation tan θ =

S , C

(1.74)

where 1 sin θn , N n=1

1 cos θn . N n=1

N

S=

N

C=

(1.75)

Clearly, S and C play the role of center of gravity of all the data in a polar diagram of coordinates {(cos θn , sin θn ); n = 1, . . . , N}, the distance from the origin being R =

2

2

S + C . We have N

sin(θn − θ) = 0,

(1.76)

n=1

which shows that the deviation about the mean vanishes; this is analogous to the linear case: N

(xn − x) = 0.

(1.77)

n=1

In fact, a shift of the reference zero direction by some angle α in the former case results in θ = (θ − α)mod 2π,

(1.78)

while a shift of origin by a in the latter case gives x = x − a .

(1.79)

These equations show the distinct property of angular measurements as opposed to the linear ones, and this is what motivates the modified concepts below. A full presentation is given in Mardia (1972). The distribution function of θ taking values in (0, 2π ] is defined by F (θ) = P{0 < t ≤ θ}, but this definition is extended to the entire real line, as follows: F (θ + 2π) − F (θ) = 1,

−∞ < θ < ∞.

(1.80)

If F is absolutely continuous, it has a probability density function f such that α

β

f (θ)dθ = F (β) − F (α),

−∞ < α < β < ∞,

(1.81)

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and f has these properties: 1o f (θ) ≥ 0, 2o f (θ + 2π ) = f (θ), 2π 3o 0 f (θ)dθ = 1.

−∞ < θ < ∞, −∞ < θ < ∞,

The Fourier series of f is defined by f (θ) =

∞ 1 φ p e −i pθ , 2π p=−∞

φp =

2π

e i pθ f (θ)dθ,

(1.82)

0

which, in applications, is often written as ⎡ ⎤ ∞ 1 ⎣ f (θ) = (α p cos pθ + β p sin pθ) ⎦ , 1+2 2π p=1 so that the distribution function is ⎡ ⎤ ∞ 1 ⎣ F (θ) = {(α p sin pθ + β p (1 − cos pθ)}/ p ⎦ . θ +2 2π p=1

(1.83)

(1.84)

In view of the considerations at the outset of this section, the mean μθ of a continuous random variable is defined by sin(θ − μθ ) = 0,

(1.85)

and the circular variance is defined as V0 = 1 − cos(θ − μθ ) .

(1.86)

It takes values in [0, 1], unlike [0, ∞] in the linear case. 1.3.2 Circular Models In this section we collect some most often occurring models. (i) Point distribution: P{θ = μ0 } = 1; this may be thought of as an analog of a causal distribution in the linear case. Its Fourier expansion for μ0 = 0 is ∞ 1 f (θ) = cos kθ . (1.87) 1+2 2π k=1 Here V0 = 0, and, in fact, it can be shown that if V0 = 0, then one deals with a point distribution.

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r = 0, 1, . . . , m − 1,

pr ≥ 0,

pr = 1.

r

(1.88) When all pr = 1/m, we get a discrete uniform distribution on the circle. (iii) Uniform distribution: f (θ) =

1 , 2π

0 < θ ≤ 2π.

(1.89)

This and the next cases pertain to a continuous random variable θ. (iv) Cardioid distribution: f (θ) =

1 [1 + 2ρ cos(θ − θ0 )], 2π

0 < θ ≤ 2π,

|ρ| < 1/2. (1.90)

(v) von Mises distribution: 1 e κ cos(θ−μ0 ) , 0 < θ ≤ 2π, 0 ≤ μ0 < 2π, κ > 0 2π I0 (κ) (1.91) where I0 (κ) is the modified Bessel function of the first kind and order zero. μ0 is the mean direction, while κ is the concentration parameter; the higher κ is, the stronger is the concentration of θ about μ0 . See Problem 9. (vi) Wrapped distribution: it arises when we wrap a certain distribution F (x) of a linear random variable x on the circumference of a circle of unit radius. The resulting angular variable xw is f (θ; μ0 , κ) =

xw = x mod 2π,

(1.92)

having the distribution function Fw (θ) =

∞

[F (θ + 2π k) − F (2π k)],

0 < θ ≤ 2π,

(1.93)

k=−∞

and, if x possess a density function, then the density function of xw is f w (θ) =

∞

f (θ + 2π k).

(1.94)

k=−∞

Two important examples of wrapped distributions are given below.

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(vii) Wrapped Gaussian (normal) distribution: f w (θ) =

" ! ∞ 1 (θ + 2π k) 2 √ , exp − 2σ 2 σ 2π k=−∞

0 < θ ≤ 2π,

(1.95)

which follows from the preceding equation by wrapping x from N(0, σ ). (viii) Wrapped Cauchy distribution: f w (θ) =

1 1 − ρ2 , 2π 1 + ρ 2 − 2ρ cos θ

0 < θ ≤ 2π,

0 ≤ ρ < 1, (1.96)

which is seen to reduce to the uniform one for ρ → 0, and to the one fully concentrated at θ = 0 for ρ → 1. Note that V0 , in contrast to the linear case, is finite: it equals 1 − ρ.

1.4

Random Fibers, Random Line Fields, Tessellations

1.4.1 Poisson Random Lines in Plane It was shown in Section 1.1.1 that straight lines are best represented in the ( p, θ)-system, Figure 1.4(a). It follows that under the ( p, θ) parametrization, the set G of all lines g in the (x, y)-plane is equivalent to the semiinfinite strip S of (1.35). Thus, each point of this strip corresponds to a line, and so, each subset of S to a subset of G. Now, let G b( O,r ) be the set of all lines that intersect the disk b ( O, r ), centered at O and of radius r . Clearly, the corresponding subset of S is Sb( O,r ) = {( p, θ) : 0 < p ≤ r, 0 ≤ θ < 2π } .

(1.97)

Now, in order to simulate a line from G b( O,r ) , we use two independent, uniform random variates u and v from [0, 1] and generate p = r u,

θ = 2π v.

(1.98)

Next, consider a point field in R2 . Those points that lie in S form a subpoint field. Each of these points lies in the (x, y)-plane, and the set of these lines is called a line field. Now, consider a Poisson point field of intensity λ in R2 . The points lying in S correspond to a random set of lines, called a Poisson line field, , with parameter ρ = 2λ. An example is shown in Figure 1.4(b). Basic properties of the Poisson line field are (Miles, 1964): 1o The intersection points with a fixed line form a linear Poisson point process with intensity ρ = 2λ. The intersecting angles α are independent random variables and have a probability density (sin α) /2, α ∈ [0, 2π).

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Microstructural Randomness and Scaling in Mechanics of Materials 2o The number of lines that intersect a plane set K has a Poisson distribution with parameter λμ (SK ). 3o The mean total length L A of a line segment of the field in a region of area 1 is π λ, which is known in random geometry as a line density. 4o The probability that more than two lines intersect at a point is exactly 0. o 5 The Poisson line field leads to a so-called Poisson mosaic of Poisson polygons, the average number of edges in a polygon being four. With A denoting the area of a polygon, its first, second, and third moments are A =

π λ2

π2 A2 = 4 2λ

4π 7 . A3 = 7λ6

(1.99)

6o ρ = 2λ is the mean number of lines intersected by a test line segment of unit length. o 7 The probability density of diameters D of circles inscribed into Poisson polygons follows a negative exponential: f ( D) = Ce −λD ,

(1.100)

∞ where C is determined from the normalization condition 0 f ( D) d D = 1, while λ is the coverage parameter (= number of lines per diameter, 2r , of the Poisson line field). Note: Upon letting the Poisson polygons of the Poisson mosaic be occupied at random by either one of two phases (black or white), one can generate a two-phase composite, Figure 1.7(a). The property 7o is relevant in paper technologies, where one studies the retention of spherical particles by sieving as they flow across a planar mat of theoretically infinite, straight fibers, having the geometry of the Poisson line field. Making a correction for the finite width of fibers, d f , amounts to shifting of the density (1.100) to the left by d f . However, trying to determine the network’s ability to retain the particles leads one to consider the relative area of the polygon-inscribed circles, rather than the number of these circles alone. This leads to an area-based probability density having a Gamma function form f A ( D) =

λ3 D2 −λD e , 2

(1.101)

from which we obtain the probability of retention P {retention} = F A ( D) = 0

D

λ2 D2 e −λD . f D d D = 1 − 1 + λD + 2 (1.102)

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(a)

(b)

FIGURE 1.7 (a) A two-phase mosaic generated from a Poisson line field with 100 lines. (b) A finite fiber field.

In effect, this may be replaced by F A in the function of a new dimensionless parameter ϕ = λD, so that ϕ2 P {retention} = F A (ϕ) = 1 − 1 + v + e −ϕ . (1.103) 2 Note: One may consider line fields generated by non-Poissonian point fields in S. A particular case, again motivated by machine-made papers which

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display preferential orientation of fibers, is a generalization of the Poisson line field to an anisotropic model. Here one takes θ as a non-uniform random variable given by the Fourier series model (1.83) with all β p = 0. Interestingly, whereas the formula (1.100) of property 7o was long known to hold for an anisotropic system (Miles, 1964), the area-based formula (1.103) carries over to that case as well (Castro and Ostoja-Starzewski, 2000). Note: Dodson and Sampson (1996) derived (1.101) differently. 1.4.2 Finite Fiber Field in Plane We now consider random fields of finite length fibers (i.e., segments) in plane, Figure 1.7(b). The simulation proceeds as follows: 1. Generate n Poisson points in a square [0, L]2 . 2. Generate an angle ϕ with respect to the x1 -axis, for each Poisson point, according to a specific probability distribution, again allowing for anisotropic patterns. 3. Generate a random fiber of length oriented at angle ϕ, for each Poisson point, according to a specific probability distribution P (l). The centers of the fibers are fixed at their respective Poisson points. Note: There is no continuous passage from this process to the line field. In other words, by extending the fiber lengths l to infinity, one cannot obtain the random (Poisson) line field. Note: One can generate fibers in clusters, see the section on germ-grain models below. 1.4.3 Random Tessellations 1.4.3.1 Basic Concepts The Poisson line field above has been our first example of a tessellation. Basically, a system of polygons in a plane is said to be a tessellation if the constituent polygons are pairwise disjoint and the union of their closures fills the plane. Formally, a polygon is a bounded, convex, open, nonempty set in R2 . The polygons are called the cells of the tessellation, the vertices are called the nodes of a tessellation, and the sides of polygons are called the edges of a tessellation. These concepts may be generalized to 3D, and one can speak of a spatial tessellation as a division of R3 into pairwise disjoint polyhedra. A polyhedron involves a set of vertices, V, a set of edges, E, and a set of faces (i.e. cells), F . Here we recall the Euler formula for finite polyhedra in R3 : |V| − |E| + |F | = 2,

(1.104)

where || stands for the number of elements in a given set. Now, a planar tessellation can be interpreted as the surface of an infinite polyhedron, whereby (1.104) is used to show that, in the limit of an infinitely large polyhedron, λ E = λV + λ F, where λV stands for the intensity of V and so on.

(1.105)

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1.4.3.2 Planar (Poisson-)Voronoi Tessellations and Delaunay Triangulations

Take a realization of points x j ; j = 1, 2, . . . from a Poisson point field in R2 . Typically one considers in a unit square, so that the set of Poisson points is finite. Then, define a cell of the tessellation to be a domain D j of all the points y in R2 closer to a given Poisson point x j than to all other Poisson points xk ( j = k): # # # #

D j = y ∈ R 2 : # y = x j # ≤ # y = xk # , j = k .

(1.106)

Almost all y in R2 have a unique nearest Poisson point, that is, except the edges of cells that are equidistant to the two neighboring Poisson points. The domains Di are called Voronoi (or Dirichlet) cells, and the set of all D j s forms a Poisson–Voronoi tessellation, alternatively called a Poisson–Dirichlet mosaic. Another way of viewing this tessellation is to look at the growth of (circular) disks at a uniform rate, all starting at the same time, until they meet, which happens along the bisector of a line joining two neighboring Poisson points. These latter lines form a Delaunay triangulation. There is a one-to-one relation between the edge set of the Voronoi tessellations and the edge set of the Delaunay triangulation, and another one-to-one relation between the vertex set of Voronoi and the triangles of Delaunay. The average number of Delaunay edges, whether the network is based on a Poisson point field or not, incident onto a vertex is six (Problem 11). Two Voronoi tessellations and their Delaunay triangulations corresponding, respectively, to the point patterns of Figures. 1.6(a) and 1.6(b) are shown in Figure 1.8. Of course, the tessellation of Figure 1.8(b) is not Poisson–Voronoi. Note: While the Voronoi tessellation (and its modifications) is a very popular model of polycrystals, the Delaunay triangulation is useful in mechanics of granular media (Goddard, 2001). Note: A Voronoi tessellation of a 3D space may be carried out similarly to the 2D construct here, and these two models are very popular in mechanics and physics of random media. However, one needs to remember here that a planar intersection of a spatial Poisson–Voronoi tessellation is not a planar Poisson–Voronoi tessellation. It is an open question whether there exists a planar point process whose Voronoi tessellation is identical in distribution to this intersection.

1.4.3.3 Modifications of Voronoi Tessellations There are various ways in which the Voronoi tessellation may be modified, e.g.: 1. Start growing the disks from the Poisson point process as soon as these points fall onto the plane (i.e., sequentially), so as to get a Johnson–Mehl model. 2. Introduce random rates of growth for each disk.

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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

(a)

1 0.9 0.8 0.7

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

(b)

FIGURE 1.8 (a) Voronoi tessellation and its Delaunay triangulation, generated from the 100 points shown in Figure 1.6(a); Voronoi tessellation, Delaunay triangulation, and disks centered at the points of Figure 1.6(b).

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3. Grow the disks by employing a general metric 1/ p n |yi − xi | p d p y, x = 1 ≤ p < ∞,

(1.107)

i

as opposed to the conventional Euclidean d2 metric involved in the Voronoi tessellation described above (Okabe et al., 1992). Figure 1.9 shows such two networks, one for the d1 (Manhattan) and another for the d∞ (sup) metric, both grown from the Poissonian point pattern of Figure 1.6(a) in 2D.

(a)

(b)

FIGURE 1.9 Voronoi tessellations, obtained from the point pattern shown in Figure 1.6(a) in the d1 (a) and d∞ (b) metric, respectively.

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1.4.3.4 Random Crack Model The so-called random crack model is a purely geometric construction designed to simulate some crack patterns encountered in real systems. For one ω ∈ , it is described by the following sequence of steps: 1. Drop N points at random in R2 . 2. Assign an angle sampled from some angular distribution, such as the uniform one (1.89). 3. From each and every point, starting at time t = 0, grow a line segment in both directions at a uniform rate—such as shown in 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

(a) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4 (a)

FIGURE 1.10 The intermediate (a) and the final (b) stages in the generation of the random crack model.

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Figure 1.10(a)—until it meets another line or the window boundary, and then stop. 4. As a result, obtain a tessellation of the plane such as shown in Figure 1.10(b). Note: This model may be modified to periodic boundary conditions. Note: Many models presented here can be generalized to 3D. For instance, by taking random planes according to a parametrization mentioned at the end of Section 1.1, we generate a spatial tessellation made up of (convex) Poisson polyhedra.

1.5

Basic Concepts and Definitions of Random Microstructures

1.5.1 General It should be clear by now that various probabilistic models introduced in previous sections may be used to represent real materials. They are particular examples of random media. Now, a random medium, denoted by B, is defined as B = {B (ω) ; ω ∈ },

(1.108)

where each B (ω) is a particular, spatially heterogeneous realization, following laws of deterministic mechanics; recall this book’s Preface. Equivalently, we also use the terms random microstructure, random material, and random composite. Basically, these are media whose properties vary randomly from point to point, and, therefore, their evolution as an ensemble is stochastic. In the preceding sections we have introduced several random media models, and in the following we discuss several more types. Before we proceed, we state that the media we are interested in are not fractal. That is, their mass (the basic physical property) scales with volume M ( R) = k R D

(1.109)

where M is the mass of fractal medium, R is a box size (or a sphere radius), and D is the spatial dimension of a given problem (i.e., either 1, 2, or 3). The scaling (1.109) is understood “on average”: due to spatial randomness of the material, there are fluctuations of mass, but they vanish upon ensemble averaging. We relax the assumption (1.109) in Section 10.6 of Chapter 10. 1.5.1.1 Germ-Grain and Boolean Models (Flocs of fibers as grains). A germ-grain model is made of grains planted at germs, whose centers are given by some point field in some Euclidean space. A Boolean model is a special case of the germ-grain model in the sense that is the spatially homogeneous Poisson point field. The model is introduced constructively: we start with = {φ1 , φ2 , . . .}, the points φi being germs.

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Next, on each φi we place a grain i , such that 1 , 2 , . . . form a sequence of independent, identically distributed random compact sets in the x, y, z-space, that are independent of the process. The Boolean model is the union of the grains i , translated by the φi in = ∪φi ∈ (i + φi ).

(1.110)

As an example, we now develop a germ-grain model of a fibrous structure of paper, constructed so as to obtain fiber flocculation, commonly seen as nonuniform grayscale effects in a typical sheet of paper held against light. This model is based on a hard-core point process = {φ1 , φ2 , . . .}, where the grains i are flocs of fibers, whereby each fiber’s center is at a position r = (r x , r y ) relative to the floc’s center φi . We focus on realizations of in “windows” of size L x × L y × t in the x, y, z coordinate system. L x , L y , and t are the two in-plane dimensions and the (much smaller) z-thickness of paper, respectively. Upon defining a dimensionless mesoscale parameter δ = L/ l ,

(1.111)

where L = L x = L y and is the average fiber length, we focus on a finitesize random medium Bδ = {Bδ (ω); ω ∈ }. By using a random variable r0 governed by the one-parameter triangular probability density function ! " b2 2 p(r0 ) = r0 + b r0 ∈ 0, , (1.112) 2 b with b being a floc parameter, we generate coordinates for the location of a fiber center relative to the floc center: r x = (1 + a 1 )r0 cos θ

r y = r0 sin θ.

(1.113)

Here θ is the fiber’s in-plane orientation angle controlled via (1.90) with θ0 = 0 and a 1 being a free parameter. The model (1.112–1.113) is chosen so as to obtain clustering in a finite disk of radius 2/b if a 1 = 0, or in an ellipse if a 1 > 0. In the latter case, the flocs are stretched in the machine direction (x) according to the degree of a preferred orientation of fibers so as to reflect a typical structure of machine-made paper. Note that as b increases, fibers are clustered into tight flocs of radius tending to zero as b → ∞, and, as b decreases, they are scattered. This is apparent in (a) and (b) of Figure 1.11 (left column). (We return to the middle and right columns in Section 4.3 of Chapter 4.) We define the density d of the network as the total fiber length per unit inplane area. The coverage (average number of fibers per point) and sheet basis weight (weight per unit area) are obviously directly proportional to d. For a chosen d, a number of fibers to be assigned to the test volume is computed, and then each fiber is assigned at random, uniformly, to any of the germs. The fiber centers within each floc are not generated in a common z-plane; the z-coordinate is sampled from a uniform depth density.

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By keeping d and t independent, we can simulate papers with the same coverage but different degrees of compaction, corresponding to different degrees of pressing during papermaking. The degree of compaction is measured by the relative bonded area RBA defined as RBA =

Abonds , Aprojected

(1.114)

where Abonds is the total area of all bond parallelograms, and Aprojected is the total projected area of all fibers (Deng and Dodson, 1994). This methodology reflects our understanding that the fiber network model is likely to be realistic only for papers that have relatively low RBA, so their primary load-carrying mechanism is the transfer of forces and moments along fiber axes between nodes, rather than a more complex interaction of plate- or solid-like fiber segments. Mechanics of thus constructed fiber networks is studied in various sections of Chapters 4, 7, and 9. 1.5.1.2 Flocs as Continua If one looks at Figure 1.11(a) from afar, it tends to look like a random continuum of Figure 1.1(b) in the Preface. Thus, motivated by the Boolean model of fiber network introduced above, and aiming at the effects of floc-type

(a) Floc parameter b = 2.0

(b) Floc parameter b = 0.4

FIGURE 1.11 Highly (top) and weakly (bottom) flocculated networks in undeformed state (left column) and deformed state amplified for demonstration (center column). Figures of the right column show differences between the true node displacements and those of the uniform strain assumption.

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δ = 10

δ = 10 1 Layer

13 Layers

FIGURE 1.12 Generation of the Boolean model of paper formation through random placement of disks, with coverage at a point ranging from 1 up to 13 discs (layers). Also, zooming in with a mesoscale from δ = 10 to δ = 4 is shown.

formation, we now introduce a Boolean model of a random quasi-continuous paper material. We again focus on realizations B(ω) in 3D windows L x ×L y ×t. In this model each grain i is an elliptical disk of unit thickness t, representing a floc of fibers. This, of course, is a simplification from the preceding section where we dealt with fibers scattered randomly around the grain’s center. After generating a number of disks in the window, we have n disks (layers), each of the same thickness t, stacked at each x,y-location of the plane. Thus, we obtain a sheet of piece-wise variable thickness, n×t, per unit area, which gives us a distribution of basis weight. Figure 1.12 illustrates an ω realization of this simulation process at δ = 10, and a zooming in with δ = 4. If the disks were ellipses and their major axes were to coincide with the x-axis, there would be a biased orientation in the MD direction, just like in a machine-made paper with plane orthotropy. With circular disks the simulation represents a laboratory handsheet with an in-plane isotropy. A somewhat different paper formation can be obtained by taking as a hard-core process, that is, one in which the minimum distance between any two i s is non-zero. This prevents flocs from being arbitrarily close to each other (i.e., near overlap) as in being a Poisson point process, and then turns from a Boolean into a germ-grain model. 1.5.2 Toward Mathematical Morphology The germ-grain models are the cornerstone of the mathematical morphology. Four examples of microstructures generated by that field of mathematics are given in Figure 1.13; see Jeulin (1997). In the following we only introduce a few basic concepts. The covariogram K ( X, h) is a measure (Mes) of the intersection

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of the set X (such as a surface in 2D, or a volume in 3D) with the set X obtained by translation of X by −h: X−h . Thus, (1.115) K ( X, h) = Mes ( X ∩ X−h ) = k (x) k (x + h) d x,

(a)

(b) FIGURE 1.13 (a) Sequential alternate random function (modeling mica, or pack ice, with gray scale); (b) Boolean random function (modeling biological tissue); (c) dead leaves random tessellation of Poisson polygons (e.g., modeling layered material); (d) Boolean model of Poisson polygons (modeling tungsten-carbide [black] and cobalt [white]).

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(c)

(d)

FIGURE 1.13 (Continued).

where

k (x) =

1 if 0 if

x ∈ X, x∈ / X.

(1.116)

The covariogram has the following properties: 1o For h = 0: K ( X, 0) = Mes ( X) .

(1.117)

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2o For a bounded set X: K ( X, ∞) = 0

K ( X, h) = 0 for h > A,

(1.118)

where A is the largest distance between two points in the direction of h. 3o The integral of the covariogram is K ( X, h) dh = [Mes ( X)]2 . Rn

(1.119)

Note: The probabilistic version of the covariogram for a stationary set X is the covariance function C ( X, h): C ( X, h) = P (x ∈ X, x + h ∈ X) ,

(1.120)

which has the property C ( X, h) = C ( X, −h) ≤ C ( X, 0) .

(1.121)

Whether the set X is in 2D or in 3D, we have C ( X, 0) = V ( X),

(1.122)

which is the areal or volume fraction of X in R2 or R3 , respectively. It follows that lim C ( X, h) = [V ( X)]2 ,

h→∞

(1.123)

which says that the covariance asymptotes to the volume fraction squared. The function C ( X, h) is indicative of the connectivity of the set X. If the limit above is reached at some finite h, say, h c (called range of the covariance), the points of the structure beyond h c are uncorrelated. The covariance and its range can be estimated from images by means of the covariogram. This is shown here in terms of the examples of microstructures from the food industry, due to Kanit et al. (2003). In the case of Figure 1.14(a), the range is about 37, and in the case of 1.14(b) it is 19. Also note that, with the horizontal and vertical covariances being very close, both microstructures are isotropic. Following the presentation of Kanit et al. (2003), we introduce the notion of an integral range, which specifies how well the parameters obtained for a domain of finite size statistically represent the random microstructure as a whole. That range is defined (Matheron, 1975) as 1 An = [C( X, h) − C( X, 0) 2 ]dh. (1.124) C( X, 0) − C( X, 0) 2 Rn If this is applied to the volume fraction VV∗ =

Mes( X ∩ V) , Mes(V)

(1.125)

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0.45 Horizontal covariance Vertical covariance Asymptotic value Covariance range

0.4 0.35 0.3 0.25 0.2 0.15 0.1

0

20

40

(a)

60 80 100 Distance (μm)

120

140

160

0.45 Horizontal covariance Vertical covariance Asymptotic value Covariance range

0.4 0.35 0.3 0.25 0.2 0.15 0.1

0

(b)

20

40

60 80 100 Distance (μm)

120

140

160

FIGURE 1.14 Two microstructures (a) and (b) with the same volume fraction but different morphologies. Also shown are the corresponding horizontal and vertical covariances, the asymptotic values, and covariance ranges. (From Kanit et al., 2003. With permission.)

one finds its variance, for a variance, for a microstructure of covariance C( X, h), to be 1 2 ∗ σ P (VV ) = 2 [C( X, x − y) − P 2 ]d xd y. (1.126) V V V For a large specimen (V An ) σ P2 (VV ) =

P(1 − P) An , V

(1.127)

where P = C( X,0) in equation (1.122). Having the variance, one can determine the confidence interval of the average volume fraction P : P ± 2σ P (VV ).

Problems 1. Prove the properties of P given in Section 1.1.1. 2. Prove that, if P is a probability measure defined on R, then F (x) = P {(−∞, x)} possesses properties 1o to 3o of Section 1.1.2.2.

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3. Show that F defined by (1.14) satisfies properties 1o to 4o of Section 1.1.2.3. 4. Prove that P defined by (1.23) satisfies axioms I to III of probability theory. 5. Solve Buffon’s needle problem: given a board ruled with parallel lines of uniform spacing a , what is the probability of a needle of length l ≤ a to cross one of the lines? 6. Solve Buffon’s needle problem by a Monte Carlo method on a computer. Note that this yields an estimate of the number π . 7. Prove the formula (1.49). 8. Justify the expression for the emptiness probability of a binomial field. 9. Show that, for small values of κ, the von Mises distribution (1.91) reduces to the cardioid distribution (1.90). 10. Show that P1 and P2 of Section 1.2.6 imply that, for any two disjoint regions A and B, N ( A) and N ( B) are independent. 11. For a Poisson–Voronoi tessellation of intensity λ, determine the intensities of sets V (vertices) and F (cells). 12. Write computer programs to generate the modifications of Voronoi tessellations listed in Section 1.4.3.3. 13. Write a project on the generation of arbitrary random variables from uniform random variables using the inverse function method and the acceptance-rejection method; consult any book on Monte Carlo methods. 14. Consider a random variable having a Weibull distribution. (a) Simulate random variables from this distribution via the inverse function method. (b) How does the scheme of (a) compare to a generation via the acceptance-rejection method? 15. Two impulses arrive at random instants at a receiver during the time interval [0, b]. Find the probability that at least one impulse arrives before the instant k < b. 16. Prove that P (A|B) defined as P( A∩ B)/P( B) satisfies Axioms I-III of Kolmogorov’s definition of a probability. 17. (a) Consider a lottery of 100 tickets in which 5 tickets are good. Find the probability that exactly one ticket is good using the concept of conditional probability. Hint: consider the events Ak = {k-th ticket wins}, k = 1, . . . 3, and note that P(C1 ∩ C2 ∩ C3 ) = P(C3 |C1 ∩ C2 ) P(C1 ∩ C2 ) = P(C3 |C1 ∩ C2 ) P(C2 |C1 ) P(C1 ). (b) Now, solve that problem employing the concept of combinations.

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Microstructural Randomness and Scaling in Mechanics of Materials 18. Consider an inhomogeneous Poisson point process of Section 1.2.6.3. Generate with a computer a partial realization of this process on A being a unit square, with N( A) = 100 and λ( X) = e xp(−2x1 − x2 ). The figure should show that the intensity gradient in x1 is more pronounced than that in x2 19. Retrace the arguments of Jaynes (1970) to justify the first random radius method as providing the correct solution of Bertrand’s problem. 20. Consider a unit vector n aligned with a random fiber lying in the (x1 , x2 )-plane, and having an orientation θ with respect to x1 . Then, with the density function f (θ) of (1.81), one can introduce second and fourth rank tensors 2π ai j = f (θ)ni n j dθ 0

a i jkl =

2π

f (θ)ni n j nk nl dθ. 0

Next can define traceless orientation tensors 1 b i j = a i j − δi j 2 1 b i jkl = a i jkl − (δi j a kl + δik a jl + δil a jk + δ jk a il + δ jl a ik + δkl a i j ) 6 1 + (δi j δkl + δik δ jl + δil δ jk ). 24 and similarly for higher orders. Show that if tensor basis functions of n are 1 gi j (n) = ni n j − δi j 2 gi jkl (n) = ni n j nk nl 1 − (δi j nk nl + δik n j nl + δil n j nk + δ jk ni nl + δ jl ni nk + δkl ni n j ) 6 1 + (δi j δkl + δik δ jl + δil δ jk ), 24 and similarly for higher orders, then the Fourier series representation of the density f (n) is f (n) =

1 8 2 + b i j gi j (n) + b i jkl gi jkl (n) + · · · 2π π π

See Onat and Leckie (1998) for the 3D case and Advani and Tucker (1987) for a discussion of many other theoretical and applied aspects of that model in 2D and 3D.

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2 Random Processes and Fields

The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient. B.B. Mandelbrot, 1983 With reference to the introduction to this book, the methods of Chapter 1 pertain more to discrete-type microstructures shown in Figure 1.1(a) than continuous ones of Figure 1.1(b) of the Preface. Therefore, in this chapter, we move to random processes and fields having continuous realizations. The material is laid out as a general guide for students of mechanics and materials, who have no particular background in stochastics. Depending on the application, one may have to go well beyond a particular section of random processes or random fields that are sketched here. The objective of our presentation is not rigor, but communicating the gist of multifarious concepts and models of stochastics such as stationarity, ergodicity, and entropy. To that end, some example problems in stochastic mechanics involving random processes and fields are discussed in Sections 2.2 and 2.4.

2.1

Elements of One-Dimensional Random Fields

2.1.1 Scalar Random Fields 2.1.1.1 Basic Concepts A (scalar) random variable is a function Z assigning to an elementary event ω ∈ (an outcome in a sample space) a number z on a real line, that is, Z : → R,

Z(ω) = z.

(2.1)

In some applications it is preferable to define autocorrelation Z as a complex function from the outset. Note: The need for the enigmatic ω may sometimes be discarded by taking Z as a so-called directly given random variable (Prohorov and Rozanov, 1969). 45

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This choice would mean that = R and the function (2.1) would be simply Z(z) = z. Note: The term random variable is a misnomer, as it is a function in the first place. The function F Z (z) = P{Z ≤ z}

(2.2)

is called the probability distribution of Z. Wherever F Z (z) is differentiable, we may define the probability density of Z as f Z (z) =

d F Z (z) . dz

(2.3)

Example 1: Measurements of random paper stiffness. The Young modulus E of 1 ×1 specimens, separated by 1.5 , is measured at n (= 500) points along a paper web, resulting thereby in a string E i , i ∈ I = {1, 2, . . ., n}. We index this sequence of numbers by ω1 , so that we deal with a string E i (ω1 ) = E(ω1 , i) shown in Figure 2.1. For another web, indexed by ω2 , we obtain another string of numbers E i (ω2 ) = E(ω2 , i), i ∈ I , and so on.

FIGURE 2.1 A sample realization of random process E (Young’s modulus), its autocorrelation, and spectrum.

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Guided by this example of Z = E, we are led to say that the set {Z(ω, i); ω ∈ , i ∈ I } of all realizations Z(ω, i) is a (scalar) random field parameterized by the position i. Clearly, we have two options: to keep a discrete parametrization by i, or to consider any string as being parameterized by a continuous spatial coordinate x. In the latter case, the set I is replaced by an appropriate subset X ⊂ R. We say that Z is a 1D random field (alternatively called a random (or stochastic) process in 1D) if it assigns to an elementary event a realization (or trajectory) over X, that is, Z : × X → R,

Z(ω, x) = z.

(2.4)

Note: There exist four possible interpretations of Z: 1. A set, or ensemble, of all functions Z(ω, x): ω and x are variable 2. A single deterministic function (a realization) Z(ω): ω is fixed but x is variable 3. A random variable Z(x): ω is variable but x is fixed 4. A deterministic number z: ω and x are fixed Note: Conventional literature on random/stochastic processes typically introduces a parametrization by the time (t) rather than space (x) coordinate. We prefer x due to our interest in spatially random microstructures, and hence the term “random field.” A dependence on time is more prominent in the chapter on waves (Chapter 11). With reference to (2.1), at each point x we may introduce a function F Z (z; x) ≡ F1 (z; x) = P{Z(x) ≤ e},

(2.5)

called a first-order (or one-point) probability distribution of the process Z; recall the four interpretations above and note that F Z (z; x) is a function of two variables. At points where F Z (z; x) is differentiable with respect to z, we define the corresponding first-order probability density of Z: f Z (z; x) ≡ f 1 (z; x) =

d F1 (z; x) . dz

(2.6)

Given two points x1 and x2 , we consider two random variables Z(x1 ) and Z(x2 ). Their joint second-order (or two-point) probability distribution is F2 (z1 , z2 ; x1 , x2 ) = P{Z(x1 ) ≤ z1 , Z(x2 ) ≤ z2 },

(2.7)

with the corresponding density, if it exists, defined by f 2 (z1 , z2 ; x1 , x2 ) =

∂2 F2 (z1 , z2 ; x1 , x2 ) ∂z1 ∂z2

.

(2.8)

Proceeding in this fashion we may introduce an nth-order (or n-point) probability distribution of Z: Fn (z1 , . . ., zn ; x1 , . . ., xn ) = P{Z(x1 ) ≤ z1 , . . ., Z(xn ) ≤ zn }.

(2.9)

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Note that it is a function of 2n variables. The corresponding nth-order probability density is f n (z1 , . . ., zn ; x1 , . . ., xn ) =

∂n Fn (z1 , . . ., zn ; x1 , . . ., xn ) ∂z1 ...∂zn

.

(2.10)

A question naturally arises here: given a family of functions Fn , can we find a stochastic process such that Fn s are its nth-order probability distributions? The answer is provided by two Kolmogorov conditions: if Fn s are a family of distributions dependent on n, such that for any n, any xi ∈ X, and any permutation i 1 , . . ., i n of numbers 1, . . ., n, the following conditions hold: 1. Symmetry (i.e., invariance with respect to the permutation) Fn (zi1 , . . ., zin ; xi1 , . . ., xin ) = Fn (z1 , . . ., zn ; x1 , . . ., xn );

(2.11)

2. Consistency ∀m < n Fn (z1 , . . ., zm , ∞, . . ., ∞; x1 , . . ., xn ) = Fn (z1 , . . ., zm ; x1 , . . ., xm ). (2.12) Note: Providing the differentiability conditions hold, the above can also be written in terms of the density function (Problem 1). The usefulness of the Kolmogorov conditions is illustrated by this simple example. Example 2: Consider the case n = 2, and 2 −1 z1 z1 z2 z22 1 f (z1 , z2 ; x1 , x2 ) = exp − 2ρ + 2 , σ1 σ2 2 1 − ρ2 σ12 σ2 2π σ1 σ2 1 − ρ 2 (2.13) where σ1 = x1 , σ2 = x2 − x1 , 0 < x1 < x2 , zi ∈ R1 , i = 1, 2. Then the marginal densities are 2

z 1 f (z1 , z2 ; x1 , x2 ) dz2 = √ exp − 12 σ1 x1 2π R1

z22 1 √ exp − (2.14) f (z1 , z2 ; x1 , x2 ) dz1 = 2 (x2 − x1 ) 2 (x2 − x1 ) 2π R1 = ϕ (x1 , x2 , z2 ) . Clearly, ϕ (x1 , x2 , z2 ) defined by (2.14) depends on x1 as well as on x2 , so that the function f (z1 , z2 ; x1 , x2 ) does not satisfy the Kolmogorov conditions, and hence it cannot represent a second-order density function f 2 (z1 , z2 ; x1 , x2 ) of a stochastic process Z (x). The mean (or average) of the process is

Z(x) = zd F1 (z; x) = zf 1 (z; x)dz, (2.15) R

R

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49

which introduces the symbol for the operation of ensemble averaging; it is a function of x. Note that this is ensemble averaging because the integration runs over the entire space . Sometimes, it is convenient to introduce a special symbol, such as µ (≡ µ(x)), for the mean. The autocorrelation of the process is defined by R(x1 , x2 ) ≡ RZ (x1 , x2 ) = Z(x1 ) Z(x2 ) = z1 z2 f 2 (z1 , z2 ; x1 , x2 )dz1 dz2 , R

(2.16) and the autocovariance of the process is the covariance of two random variables Z(x1 ) and Z(x2 ) C(x1 , x2 ) ≡ C Z (x1 , x2 ) = [Z(x1 ) − µ(x1 )][Z(x2 ) − µ(x2 )].

(2.17)

It is easily shown that C Z (x1 , x2 ) = RZ (x1 , x2 ) − µ(x1 )µ(x2 ). By setting x1 = x2 = x, the variance of Z follows as a special case of the above (σ Z(x) is the standard deviation of Z at x) 2 σ Z(x) = C Z (x, x) = RZ (x) − µ2 (x) = Z(x) 2 − Z(x)2 .

(2.18)

Note: The covariance is symmetric C(x1 , x2 ) = C(x2 , x1 ); when the process is complex-valued, it is Hermitian. Furthermore, by the Cauchy–Schwartz inequality, the squared modulus of C(x1 , x2 ) never exceeds the product of the variances σ 2 (x1 ) and σ 2 (x2 ) C 2 (x1 , x2 ) ≤ σ 2 (x1 )σ 2 (x2 ) = C(x1 , x1 )C(x2 , x2 ).

(2.19)

Here is a very simple stochastic kinematics example in space-time (t, x), whereby we briefly switch to a parametrization by t in place of x: Example 3: A point is at ( O, P) at time t0 = 0 in the (t, x)-plane, and then moves with a velocity Q on the straight line. ( P, Q) is a random vector Z. At time t the point is at (t, X (ω, t)) where X (ω, t) = P (ω) + t Q (ω). Realizations of the X (ω, t) process are rays x (t) = p + q t for t ≥ 0; p and q are fixed. The mean and the autocorrelation are found to be (Problem 2)

X(t) = P + t QRX (t1 , t2 ) = P 2 + P Q(t1 + t2 ) + Q2 t1 t2 (2.20) Example 4: With P and Q as above, we form a differential equation for t>0 P

dX + QX = 0 dt

(2.21)

with the initial condition X (t) = 2H (t), H (t) being the Heaviside function. Obviously, the solution is X (t) = 2 exp (−q t/ p) H (t) and this stochastic process consists of a family of exponentials. Example 5: Consider a random process Z (x) = Y sin π x, where FY ( y) is given and x is a deterministic parameter x ∈ [1/4, 1/2]. Then, the first-order

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distribution function is

F Z (z) = P {Y sin π x ≤ z} = P Y ≤

z z = FY . sin π x sin π x

(2.22)

Example 6: Consider sin x Y (ω, x) = 2x

if ω = tails . if ω = heads

(2.23)

Clearly, the process Yx consists of two very regular curves. Nevertheless, it is a stochastic process. These simple examples show that one may have a stochastic process E to be essentially a function of one, two, or more random variables. If they are known for one or two values of the parameter t, they are completely determined for any t. As stochastic processes they are trivial and uninteresting. We are rather interested in processes where randomness is “richer” and “extends into infinity.” A modest step in that direction may be taken by describing a stochastic process, as sometimes encountered in physical applications, in terms of an analytic formula as a function of an independent parameter (such as x) and containing a set of known random variables α (ω) = {α1 (ω) , . . ., αn (ω) , } in the form E (ω, x) = h [x, α (ω)] where the deterministic function h is given. We thus arrive at a classical example of a process specified as Y (ω, x) = A(ω) cos kx + B (ω) sin kx,

(2.24)

where k > 0, and A, B are independent random variables with standard Gaussian (or normal) densities N (0, σ ). It is easily shown that this process is, equivalently, given by Y (ω, x) = C (ω) [cos kx + (ω)] ,

(2.25)

and its mean and autocorrelation are (Problem 4)

Y (x) = 0 RX (x1 , x2 ) = σ 2 cos k (x2 − x1 ) .

(2.26)

This process is a stepping-stone to a more general concept Y (x, ω) =

n

[Ai (ω) cos ki x + Bi (ω) sin ki x] ,

(2.27)

i=1

where ki > 0, and Ai , Bi , for i = 1, . . ., n, are independent random variables with N (0, 1). This example leads to a somewhat richer process, a so-called Rice noise Y (ω, x) =

n

Ai (ω) cos [υi (ω) x + i (ω)] ,

(2.28)

i=1

where Ai s, Bi s, υi s and i s are random variables with known statistics (Problem 4). Typical uses of the Rice noise are random noise currents in physical

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devices and random surfaces (Sobczyk, 1985; Ostoja-Starzewski, 1987) as discussed in the next section. It was shown earlier that, in principle, we describe random processes by their nth-order distribution functions. At this point it should be noted that the knowledge of all such functions does not give the full information about things like continuity, differentiability, or integrability of the realizations of stochastic processes. From the Rice noise we go to a random Fourier series Y (ω, x) =

∞

Vm (ω) e imgx ,

(2.29)

m=1

where the coefficients V1 , V2 , ... are mutually uncorrelated, zero-mean random variables, such that Wm2 if m = n

Vm = 0 Vm Vn = . (2.30) 0 if m =

n An important condition (Papoulis, 1965) states that, in order for Y (ω, x) to (ω, x) adequately (i.e., to minimize the meanrepresent a certain function Y square error) for every x ∈ [−X, X], g should satisfy 4 sin2

gX 1. 4

(2.31)

For Y (ω, x) to represent Y˜ (ω, x) adequately in the interval [−X, X] (i.e., ˜ 2 ), g should satisfy to minimize the mean-square error |Y − Y| 4 sin2

gX 1. 4

(2.32)

Another class of random processes is the one whose values e 1 , . . ., e n , at the respective positions x1 , . . ., xn , are independent random variables. Then f n (e 1 , . . ., e n ; x1 , . . ., xn ) =

n

f 1 (e i ; xi ).

(2.33)

i=1

Problems 5 and 6 focus on two examples of this kind: the Bernoulli process and the binomial process. 2.1.1.2 Homogeneity Homogeneity (or stationarity) of a random process is an important property; there are two kinds. First, a process is called strict-sense stationary (SSS) if all n-order distributions Fn are invariant with respect to arbitrary shifts x , and for any choice of xi s: Fn (z1 , . . ., zn ; x1 , . . ., xn ) = Fn (z1 , . . ., zn ; x1 + x , . . ., xn + x ).

(2.34)

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Next, a process is called wide-sense stationary (WSS) if its mean is constant and its finite-valued autocorrelation depends only on x = x2 − x1 :

Z(x) = µ,

Z(x1 ) Z(x1 + x) = RZ (x) < ∞.

(2.35)

Note: WSS is a much weaker (less restrictive) property than SSS. The latter implies WSS whenever Fn yields a finite second moment. Note: If a random process can be entirely specified in terms of its first and second moments, then it is SSS providing it is WSS. A well-known example is the Gaussian process. Note: Roughly speaking, random processes can be classified as stationary and evolutionary—each requiring specialized techniques. Given our concern with random media, we focus here on random processes of stationary type; evolutionary processes are more common in the case of parametrization by time (also see Chapter 11). It is convenient to employ a normalized autocovariance, or correlation coefficient, which is defined as ρ(x1 , x2 ) ≡ ρ Z (x1 , x2 ) =

C Z (x1 , x2 ) . σ Z1 σ Z2

(2.36)

σ Zi denotes the standard deviation of Z at xi (Problem 8). Two examples of WSS autocovariances (ρ(x1 , x2 ) = ρ(x1 − x2 )) are the Gaussian curve: ρ(x) = exp[−x 2 /2l 2 ];

(2.37)

and the exponential curve: ρ(x) = exp[−x/l].

(2.38)

More information on models of ρ(x) is given in the section on random fields below. One defines the spectral density s(γ ) of ρ(x) as its Fourier transform: ∞ ∞ 1 s(γ ) = ρ(x)e −iγ x d x, ρ(x) = s(γ )e iγ x dγ (2.39) 2π −∞ −∞ Note: For a real-valued random process this gives ∞ ρ(x) = s(γ ) cos(γ x)dγ . −∞

(2.40)

By Bochner’s theorem: every non-negative definite function has a nonnegative Fourier transform, i.e., s(γ ) ≥ 0. As a simple application of this, consider a correlation function with ρ0 , |x| < xc ρ(x) = . (2.41) 0, else

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Random Processes and Fields Now, since s(γ ) =

1 2π

∞

−∞

ρ(x)e −iγ x d x =

53

ρ0 2π

xc

−xc

cos(γ x)d x =

ρ0 sin(γ xc ), πγ

(2.42)

which may take on negative values, model (2.41) is inadmissible. For a WSS process—in analogy to random processes parameterized by time—one can define a correlation length (or correlation radius) as follows: ∞ ∞ 1 lc = 2 R(x)d x = ρ(x)d x. (2.43) σ 0 0 For the Gaussian autocovariance we find lc = l(π/2) −1/2 , whereas for the exponential one lc = l. Note: It is possible for the integral above to diverge, as is, for instance the case with ρ(x) = [1 + x 2 /l 2 ]−a , a < 1/2. Note: For a WSS process one can introduce a Fourier transform of its autocovariance—it is called a spectral density (or power spectrum). In practice, autocovariances and spectra are often estimated from single realizations of random processes (Figures 2.1 and 2.2); this procedure is based on the ergodic assumption (Section 2.5).

FIGURE 2.2 A sample realization of random process σmax , its autocorrelation, and spectrum.

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An important operation that one may want to conduct on a random process is that of local averaging: given any realization Z(ω) of random process Z, consider a new process with realization ZL (ω), defined for each point x by ZL (ω, x) =

1 2L

x+L

Z(ω, x )d x .

(2.44)

x−L

Note: Upon local averaging, the autocorrelations (and hence spectral densities) are changed in function of L. 2.1.2 Vector Random Processes Example: Returning to our measurements of paper properties, we now report strength (σmax ) for the same 500 specimens placed on the paper web, Fig. 2.2. This figure also shows the autocorrelation and spectral density of σmax based on this realization. Note: The use of σ for stress/strength and standard deviation is the wellknown clash of notation between solid mechanics and probability theory. Clearly, we now need to generalize the concept of a scalar random variable (2.1) to a vector random variable:

E

σmax

: → R,

E(ω) = e,

σmax (ω) = s,

(2.45)

whereby E and σmax take values e and s, respectively. Given the parametrization of our measurements by position (either discrete i or continous x), the set of all realizations forms a two-component vector random process. In general, we say that a vector random process (or vector stochastic process) is a vector-valued function:

E

σmax

: × X → R2 ,

E(ω, x) = e,

σmax (ω, x) = s.

(2.46)

In general, we shall simply write Z for an n-component vector random variable, or a processs (or a field later on), taking values z so that previous relation leads to Z : × X → Rn , Z(ω, x) = z. As before, the complete specification of a vector random process is given in terms of all the n-point probability distributions of Z: Fn (z1 , . . ., zn ; x1 , . . ., xn ) = P{Z(x1 ) ≤ z1 , . . ., Z(xn ) ≤ zn },

(2.47)

which again are subject to the Kolmogorov conditions. But, such a description is very difficult to achieve in practice. Therefore, attention focuses on WSS fields.

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Taking the random variables Zi and Z j at points x1 and x2 , respectively, we call the function Cij (x1 , x2 ) = [Zi (x1 ) − Zi (x1 )][Z j (x2 ) − Z j (x2 ) ] (2.48) joint covariance of the process. We may next generalize the normalized autocovariance, or correlation coefficient, as follows: ρij (x1 , x2 ) =

Cij (x1 , x2 ) , σi σ j

(2.49)

where σi denotes the standard deviation of Zi (x1 ). If i = j, we have autocovariance, otherwise crosscovariance, of Z. Putting x1 = x2 = x, we obtain a covariance matrix, and a correlation coefficient, at x ρij (x) =

Cij (x) . σi σ j

(2.50)

Example: In terms of the paper properties, the covariance between E and σmax components of a vector process give basic information about the relation of the variability of E to the variability of σmax , both pointwise and globally. In Ostoja-Starzewski (2001) we reported measurements on two more quantities besides stiffness (E) and strength (σmax ): the strain-to-failure (εmax ) and tensile energy absorption (TEA)—all taken in the same sequence of 500 1“ × 1” specimens. We therefore deal with a vector process ZT = [E, σmax , εmax , TEA].

2.2

Mechanics Problems on One-Dimensional Random Fields

In continuum mechanics, when we are faced with fluctuating fields and their averages, we often take a field Z to be a superposition of the constant mean

Z and the zero-mean fluctuation Z (x): Z(ω, x) = Z + Z (ω, x)

Z = 0

∀ω, x.

(2.51)

Next, we typically assume that the operations of ensemble averaging and differentiation with respect to space (as well as time) commute ∂ ∂ ∂ ∂

Z Z = Z = Z . (2.52) ∂x ∂x ∂t ∂t These concepts are directly generalized to vector random processes and random fields in Sections 2.3 and 2.4 below. Note: In general, formulas (2.52) are not true. This is especially the case when time differentiation of extensive quantities is involved; see Section 10.1 of Chapter 10.

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2.2.1 Propagation of Surface Waves along Random Boundaries The effect of random boundary roughness on the propagation of Rayleightype surface waves can relatively easily be analyzed under the assumption of the wavelength being much larger than the roughness scale of the boundary (Brekhovskikh, 1959; Sobczyk, 1985). When that roughness is expressed in terms of the correlation length lc , and independent of y, we can write for a random surface profile ζx of a half-space {−∞ ≤ x, y ≤ ∞; z ≥ ζ (x, y)} lc 1 λ

dζ 1. dx

(2.53)

That is, the roughness is assumed to be only x-dependent, so that, for the unit vector n = nx , n y , nz normal to the surface we have nx = −

dζ /S 1 dx

nz = 1/S 1,

(2.54)

1/2 where S = 1 − ( ∂ζ /∂x) 2 . Next, ζx is taken as a zero-mean WSS process modeled by the random Fourier series. For a reference, the unperturbed, harmonic (e −iγ t ) Rayleigh wave, propagating in the direction x, is represented by two elastodynamic potentials: φ = e i px−a z

ψ = e i px−bz ,

(2.55)

where a 2 = p 2 − k L2

k L2 =

b =p −

k T2

2

2

k T2

ρω2 λ + 2µ

ρω2 = µ

(2.56)

In principle, the traction boundary should be stated in a ran conditions dom (primed) coordinate system x , y , z , associated locally with every point of the boundary z = ζ (ω, x). However, in view of (2.54), one can approximate such conditions by those at the mean surface z = 0 through 1 σzx − σzx = σzx = 0

1 σzz − σzz = σzz = 0,

(2.57)

where each field, say, σzx , is represented as a superposition of the scattered 1 field (σzx ) and the perturbed field (σzx ). The latter one is calculated by using the transformation of stresses from the unprimed to the primed coordinate system, taking the first expansion of the unprimed stresses , σij (x, ζ ) = σij (x, 0) + ∂z z=0 ∂σij

(2.58)

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and rejecting the higher-order terms σzx

ζ − σ 0 dζ = xx ∂z z=0 dx 0 ∂σzx

σzz

ζ, =− ∂z z=0 0 ∂σzz

(2.59)

0 0 where σzx , σzz correspond to the unperturbed wave of (2.53). In order to find the scattered fields, one introduces a system of plane waves {φn , ψn ; n = 1, 2, ...} where

φn = e i pn x−ηn z ψn = e i pn x−νn z ,

(2.60)

the wavenumbers pn being set equal to p + ng for each n, so that we have ηn = k L − pn2 and νn = k T − pn2 . Thus, we find 1 σzx =

∞ n=1

1(n) σzx =

∞

µ −2An pn ηn + Bn νn2 − ηn2 e i pn x ,

(2.61)

n=1

1 and a similar formula is obtained for σzz (Problem 9). The coefficients of scattered fields are then computed from (2.57), and hence, the random boundary effects on the Rayleigh wave can be determined. Although the methodology outlined here can readily be extended to waves at solid–fluid and solid–solid interfaces (Ostoja–Starzewski, 1987), it is seen that a number of other problems in elasticity, involving weakly random boundaries, can also be analyzed by the perturbation method.

2.2.2 Fracture of Brittle Microbeams 2.2.2.1 Randomness of Microbeams Linear elastic fracture mechanics involves two material properties: the material stiffness tensor C and the surface energy γ [the same symbol as that used for frequency in the previous section). In a 1D situation, such as encountered in a slender microbeam, C is represented by a scalar (e.g., Young’s modulus E), and we have a pair [E, γ ]. Both of these are conventionally taken to be constant, but, given the presence of a randomly microheterogeneous material structure, E and γ are random along the beam’s span x, and the beam is then described by a vector random process [E, γ ]x (Ostoja-Starzewski, 2004). Assume each component of that process to satisfy (2.52). The need to consider randomness of E arises when the representative volume element (RVE) of continuum mechanics cannot be safely applied to the actual beam. Such a case is shown in Figure 2.3(a), where the microbeam is so thin that its lateral dimension L (i.e., the very one defining its Young’s modulus) is comparable to the crystal size d. The “comparable” aspect signifies the problem of scaling from a statistical volume element (SVE) to RVE; see Chapters 7 to 10 for 2D and 3D problems.

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L

P

d

u a (a)

(b)

FIGURE 2.3 (a) Fracturing of a microbeam of thickness L off a substrate, where a statistical volume element imposed by the random microstructure characterized by a scale d is shown. (b) Potential energy ( 1/E) (thick line) and its scatter shown by a parabolic wedge (thin lines), summed with the surface energy = 2a γ (thick line) and its scatter shown by a straight wedge (thin lines), results in ( 1/E) + (thick line) and having scatter shown by a wider parabolic wedge (thin lines). Dashed region indicates the range of a random critical crack length a c ( E (ω)).

2.2.2.2 Strain Energy Release Rate in Random Microbeams First, according to Griffith’s (1921) theory of elastic-brittle solids, the strain energy release rate G is given by G=

∂W ∂A

−

∂U ∂A

= 2γ

(2.62)

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where A is the crack surface area formed, W is the work performed by the applied loads, U is the elastic strain energy, and γ is the energy required to form a unit of new material surface (e.g., Gdoutos, 1993). If we consider the case of dead-loading, the force is not random, but the kinematic variable is. Now, only the second term in (2.62) remains, and, assuming a Euler–Bernoulli beam, the strain energy is

a

U(a ) =

M2 d x, 2I E

0

(2.63)

where a is the crack length, M is the bending moment, I is the beam’s moment of inertia, and E is the elastic modulus. Henceforth, we simply work with a = A/B, where B is the constant beam (and crack) width. In view of Clapeyron’s theorem, the strain energy release rate is G = ∂U/B ∂a . Evidently, U is a random integral

a

U(a , E (ω)) =

M2 d x , 2I E(ω, x)

0

(2.64)

which, upon ensemble averaging, leads to an average energy

U(a , E) = 0

a

M2 d x 2I [ E + E (ω, x)]

.

(2.65)

In the conventional formulation of deterministic fracture mechanics, random microscale heterogeneities E (x, ω) are disregarded, and (2.64) is evaluated by simply replacing the denominator by E, so that

a

U(a , E) = 0

M2 d x . 2I E

(2.66)

We observe that this procedure corresponds to replacing the operator L−1 −1 by L as discussed in the preface. It amounts to postulating a priori that the response of an idealized homogeneous material is equal to that of a random one on average. Therefore, we are interested in making a statement about

U(a , E) versus U(a , E), and about G( E) versus G( E). First, note that, since the random process E is positive-valued almost surely (i.e., with probability one), Jensen’s inequality (Rudin, 1974) yields a relation between harmonic and arithmetic averages of the random variable E (ω) 1 ≤

E

1 E

.

(2.67)

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whereby the x-dependence is immaterial in view of the assumed wide-sense stationarity of E. With (2.65) and (2.66), this implies that U(a , E) = 0

a

M2 d x ≤ 2I E

a

0

M2 2I

1 E

dx =

a

0

M2 d x 2I E(ω, x)

= U(a , E) , (2.68)

since the conditions required by Fubini’s theorem (Rudin, 1974) are met. Now, if we define the strain energy release rate G(a , E) in a hypothetical material specified by E, and the strain energy release rate G(a , E) properly ensemble averaged in the random material {E(ω, x); ω ∈ , x ∈ [0, a ]} G(a , E) =

∂U(a , E)

B ∂a

G(a , E) =

∂ U(a , E)

B ∂a

,

(2.69)

and note that the side condition is the same in both cases U(a , E) |a =0 = 0,

U(a , E) |a =0 = 0,

(2.70)

we obtain G(a , E) ≤ G(a , E) .

(2.71)

This provides a formula for the ensemble average G under dead-load conditions using deterministic fracture mechanics for Euler–Bernoulli beams made of random materials. That is, G computed under the assumption that the random material is directly replaced by a homogeneous material (E(x, ω) = E), is lower than G computed with E taken explicitly as a spatially varying material property. Clearly, G(a , E) is the correct quantity to be used under dead loading, and this result may be generalized to Timoshenko beams (Problem 11). Note: With the beam thickness L increasing, the mesoscale L/d grows, so −1 that E → 0. Thus, E −1 → E, and (2.71) turns into an equality, whereby the deterministic fracture mechanics is recovered. Under the fixed-grip conditions, the displacement is constant (i.e., nonrandom), and the load is random. Take this load to be the force P applied at the tip. Now, only the first term in (2.62) remains, so that G=−

∂U e (a )

B ∂a

=−

u ∂P . 2B ∂a

(2.72)

For a cantilever beam problem, G can be computed by a direct ensemble averaging of E under fixed-grip loading. In the case of mixed loading conditions, both the load and the displacement vary during crack growth, and there is no explicit relation between the crack driving force and the change in elastic strain energy. However, we can bound G under mixed loading (G mi xed ) by the Gs computed under dead-load (G P )

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and fixed-grip (G u ) conditions, from above and below, respectively: G u ≤ G mi xed ≤ G P .

(2.73)

2.2.2.3 Stochastic Crack Stability Recalling the fracture criterion (2.62), we observe that cracking along the x axis is governed by an interplay of two random fields (parametrized by x): the elastic property E and the surface energy density γ . In view of the scaling arguments concerning the SVE versus the RVE given above, the first one is a function of the beam thickness L, but the second one is not. Thus, for statistically stationary and ergodic materials, the randomness of E decreases to zero as L/d → ∞, but the randomness of γ remains constant. To sum up, cracking of microbeams is more sensitive to the material randomness of elastic moduli than cracking of, say, large plates. Crack stability in any particular microbeam (ω ∈ ), in a general loading situation, is governed by the condition of the same form as that in deterministic fracture mechanics  < 0 : unstable equilibrium 2 ∂ ( (ω) + (ω))  (2.74) · = 0 : neutral equilibrium  ∂a 2  > 0 : stable equilibrium Here, both the total potential energy (ω) and the surface energy (ω) are random. Now, under dead-load conditions, the correctly averaged (shown by a solid line) is bounded from above by the deterministic estimated by a straightforward averaging of E: ( 1/E) = ≤ ( E) .

(2.75)

The above follows again from (2.67). Typically, the energy goes like −a 3 . Thus, in Figure 2.3(b), we use two parabolas to indicate a wedge of scatter associated with the mean ( 1/E) = .

(2.76)

Next, the surface energy (ω) = 2a [ γ + γ (ω)] for any ω ∈ . Using two straight lines, we indicate a wedge of scatter about = 2a γ . Consequently, the scatter about the mean of (ω) + (ω) is larger than that of (ω) or (ω) alone and, at the maximum of their sum, we have a stochastic (rather than a deterministic) competition between both contributions. Evidently, according to (2.74), the critical crack length a c becomes a random variable a c ( E (ω)), whose range is shown by a dashed region in Figure 2.3(b). In view of (2.75), there is an inequality between the average a c properly calculated from ( 1/E): ∂2 [ ( 1/E) + ] ∂a 2

= 0 ⇒ a c ( 1/E) = a c ( E) ,

(2.77)

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and the deterministic a c simplistically calculated from ( E): ∂2 [ ( E) + ] ∂a 2

= 0 ⇒ a c ( E) .

(2.78)

The said inequality is a c ( 1/E) ≤ a c ( E) .

(2.79)

Note that the equality a c ( 1/E) = a c ( E) in (2.77) follows from (2.76). Finally, Figure 2.3(b) shows that small random fluctuations in E and γ (i.e., the scatter about the maximum of ( 1/E) + ) lead to relatively much stronger (!) fluctuations in a c . For other aspects of mechanics of random microbeams, see for example Altus (2001); Altus and Givli (2003); Beran (1998); Givli and Altus (2003).

2.3

Elements of Two- and Three-Dimensional Random Fields

2.3.1 Scalar and Vector Fields 2.3.1.1 Basic Concepts Let us first return to the problem of spatial variability of paper properties, this time reporting measurements in two directions x1 and x2 . In fact, Figure 2.4 gives stiffness (E), strength (σmax ), strain-to-failure (εmax ), and tensile energy absorption (TEA)—all in a 25 × 8 array of 7“ × 1” (≡ L 1 × L 2 ) specimens. Clearly, we have a field realization Z(ω)—with an identification of vector components [Z1 , Z2 , Z3 , Z4 ] ≡ [E, σmax , εmax ,TEA]—over a 2D domain in R2 . This motivates us to introduce scalar- and vector-valued random fields. In the first case we say that, say, Z is a random scalar field in D dimensions if it assigns to an elementary event a realization over X ⊂ R D , that is, Z : × X → R,

Z(ω, x) = z,

(2.80)

where x ∈ X. In the second case we say that Z is an n-component random vector field in D parameter dimensions if it assigns to an elementary event a

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1800 1750 1700 1650 1600

13.5 13.0 12.5 12.0 11.5

1.6 1.4 1.2 1.0

0.11 0.10 0.09 0.08 0.07 0.06 FIGURE 2.4 A map (from the top) of stiffness, strength, strain-to-failure, and tensile energy absorption for a 25 × 8 array of 7” × 1” specimens (After DiMillo and Ostoja-Starzewski (1998); paper web provided by Champion Corp.)

realization over X, that is, Z : × X → Rn ,

Z(ω, x) = z.

(2.81)

Note: There exist four possible interpretations of E and Z (an ensemble, a field realization, a random variable at specific position, and a value) just as with random processes. Note: One might say that the random processes parametrized by spatial coordinates, treated in the preceding section, are a special case of random

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fields; the term “process” is reserved by some authors for parametrization by time, and “field” for parametrization by space. Note: As before, the complete specification of a random vector (or scalar) field is given in terms of all the n-point probability distributions of Z—that is Fn (z1 , . . ., zn ; x1 , . . ., xn ) = P{Z(x1 ) ≤ z1 , . . ., Z(xn ) ≤ zn }.

(2.82)

A classical theorem states that, for any complex number λr (r = 1, . . ., m) and any x1 , x2 ∈ Rn the correlation function ρ(x1 , x2 ) has the following properties (the overbar indicates a complex conjugate): 1. ρ(x1 , x1 ) ≥ 0 2. ρ(x1 , x2 ) = ρ(x2 , x1 ) 3. |ρ(x1 , x2 )|2 ≤ ρ(x1 , x1 )ρ(x2 , x2 ) m 4. ρ(x1 , x2 )λ j λ¯ k ≥ 0 j,k=1

Analogous theorems can be set up for the cross-correlation and multiple correlation functions. This theorem can also be stated for WSS random fields— whereby (1) simplifies to ρ (0) ≥ 0—and multiply correlated random fields. We now focus attention on WSS random fields, that is, when

Z(x) = µ, Z(x1 ) Z(x1 + x) = RZ (x) < ∞.

(2.83)

The concepts of correlation (and covariance) functions are very simply generalized by substituting x for x. Thus, taking the random variables Zi and Z j at points x1 and x2 , respectively, we call the function Cij (x1 , x2 ) = [Zi (x1 ) − Zi (x1 )][Z j (x2 ) − Z j (x2 ) ] (2.84) joint covariance of the process. We next generalize the normalized covariance (2.20), or correlation coefficient, as follows: ρij (x1 , x2 ) =

Cij (x1 , x2 ) , σi (x1 )σ j (x2 )

(2.85)

which, for a WSS field, simplifies to ρij (x) =

Cij (x) . σi (x)σ j (x)

(2.86)

ρij (0) =

Cij (0) . σi (0)σ j (0)

(2.87)

Its one-point special case is

A special class of so-called isotropic random fields needs now to be defined. They occur when the autocorrelation (or autocovariance) depends only on

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the magnitude, but not direction, of the vector x = x1 − x2 connecting two points x1 and x2 : √ (2.88) ρ(x) = ρ(x), x ≡ |x| = xi xi . Note: An analog of isotropy property for random fields in 1D is: ρ(x) = ρ(−x), x = x1 − x2 . Note: For WSS isotropic random fields—just as for random processes— one can define a correlation length (or radius) by the same definition as in the previous section. Two rather wide classes of correlation function for isotropic scalar fields are ρ(x) = exp[−Ax α ],

A > 0,

0 < α ≤ 2;

(2.89)

ρ(x) = [1 + Ax α ]−1 ,

A > 0,

0 < α ≤ 2.

(2.90)

Note: For an isotropic field on R D , ρ (x) ≥ −1/D. Note: A valid isotropic ρ(x) in R D2 is always a valid isotropic ρ (x) in R D1 , where D2 > D1 . However, the converse is not true as this example illustrates: the “tent” function σ 2 (1 − |x| /a ), 0 ≤ |x| < a ρ(x) = (2.91) |x| > a 0, is valid in R1 , but not in R2 . Using simpler models, one may construct new, more complex correlation functions. The way to proceed is first to note a few facts from probability theory: 1. A convex combination of probability distributions is a probability distribution; the same holds for density functions. 2. A convex combination of correlation functions is a correlation function. 3. A finite product of correlation functions is a correlation function. On this basis one obtains ρ(x) =

ρ(x) = exp[−

exp[−Ax α ] , 1 + Bx α

r

Ax αs ],

A, B > 0,

As > 0,

0 < α, β ≤ 2;

0 < αs ≤ 2,

s = 1, . . ., r ;

(2.92)

(2.93)

s=1

ρ(x) = [−

r (1 + Bs x βs )ls ], s=1

Bs > 0,

0 < βs ≤ 2,

ls = 1, 2, ...; (2.94)

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ρ(x) =

(cosh Bx α ) s , 1 + Ax α

A + B(2l − s) > 0, 0 < α ≤ 2, s = 1, . . ., r ; (2.95)

A + B(2l − s) > 0,

0 < α ≤ 2,

s = 1, . . ., r. (2.96)

See Szczepankiewicz (1985) for rigorous derivations of the above, as well as more advanced models. Very recently, a new class of so-called Dagum correlation functions has been developed (Porcu et al., 2006) −ε ρ(x) = 1 − 1 + λx −θ , θ < (7 − ε) (1 + 5ε) , ε < 7. (2.97) Here ε and θ act as smoothing parameters and λ is a scale parameter. 2.3.1.2 Example In the problem of paper properties stated above, ρij (x) is estimated invoking both stationarity and ergodicity. With the identification E, σmax , εmax , TEA     Z1 E Z  ε   2   max  (2.98)  ≡ ,  Z3   εmax  Z4

TEA

we compute the following (symmetric) matrix:  ρ E, E ρ sym  E,σmax ρσmax ,σmax ρij (0) ≡  ρ E,εmax ρσmax ,εmax ρεmax ,εmax ρ E,TEA ρσmax ,εmax ρεmax ,TEA 



1 0.43  = 0.10

1 0.56

sym 1

0.14

0.90

0.69

     ρTEA,TEA (2.99)

  .  1

From this we note: 1. Cross-correlations between E and inelastic parameters σmax , εmax , TEA are weak, although we note that ρ E,σmax is greater than ρ E,εmax or ρ E,TEA ; 2. Three cross-correlations between σmax , εmax , and TEA are about the same.

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Next, we present ρij for the nearest neighbors separated by L 1 = 7 and L 2 = 1 , in the x1 and x2 directions, respectively   0.24 0.21 0.27 sym    ρij (L 1 ) =  , 0.00 0.10 0.21  0.14 0.23 0.15 0.22 (2.100)   0.33  0.18 0.13 sym   ρij (L 2 ) =  .  0.09 0.02 0.26 0.05

0.04

0.04

0.04

2.3.1.3 Properties of the Correlation Function Anisotropy Evidently, the field at hand is not isotropic. To account for preferential directions, one generally proceeds as follows: introduce a transformation T(x) = [T1 (x), . . ., T1 (x)],

x = (x1 , . . ., xD )

(2.101)

of R D into R D . If the Jacobian of T is different from zero, there exists an inverse transformation T −1 . Let |x| denote a norm in R D and ||y|| a norm in R D after T. If a WSS field Z has a property that |x| = ||y|| implies ρ(x) = ρ(y),

(2.102)

then Z is called a WSS quasi-isotropic random field. Of particular interest are fields with ellipsoidal structure. Let b ij be the matrix of a positive-definite quadratic form ||y||2 =

n

b ij yi y j ,

(2.103)

i, j=1

whenever y =

0. Consider ρ(||y||). Then the Jacobian of T is the determinant of matrix b ij . For example, the Gaussian autocovariance of a random process on R1 can now be generalized as ρ(y) = exp[−

n

b ij yi y j ].

(2.104)

i, j=1

Another model of a quasi-isotropic field on R D is T(x) = [a 1 x1 , . . .,a D xD ],

all a i > 0.

(2.105)

Note: One needs to distinguish the anisotropy in terms of the correlation function from the anisotropy of realizations.

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Note: Models of the correlation function for isotropic random fields (2.88) carry over to quasi-isotropic fields provided that we replace x by the norm ||T −1 (y)||. For example, (2.89) leads to ρ(x) = exp[−A||T −1 (y)||α ],

A > 0,

0 < α ≤ 2.

(2.106)

Separable structure: An important property that a random field may possess is that of a fully separable ρ(x) = ρ(x1 )ρ(x2 )...ρ(xD ),

(2.107)

or a partially separable structure, such as in ρ(x) = ρ(x1 , x2 )...ρ(xD ).

(2.108)

A common model in fluid mechanics is the one involving a separation between space coordinates and time: ρ(x, t) = ρ(x)ρ(t). Local averaging: This operation, already introduced in (2.44), may be generalized to random fields: given any realization Z(ω, x) of random field Z, consider a new field with a realization ZL (ω, x), defined for each point x, by (e.g., Vanmarcke, 1983) 1 ZL (ω, x) = 2 Z(ω, x )dx , (2.109) L DL where DL is a square-shaped neighborhood NL centered at x. If applied to a stiffness C (respectively, compliance S) tensor field, this would yield a Voigttype (Reuss-type) estimate/bound of the stiffness (compliance) for the domain NL . Clearly, the local averaging is a simple operation, but it may yield very misleading estimates of actual material properties. Note: DL in (2.108) plays the role of a mesoscale with respect to the actual microscale where the random fluctuations of C reside. See Chapter 8 for micromechanically based mesoscale random fields that smooth the microstructures, and the stochastic finite elements based thereon. 2.3.2 Random Tensor Fields Although the previous section treated a random vector field, the vector there was not really considered as a tensor of rank one—it was simply an ordered array of scalars. This is why a simple generalization of stationarity and isotropy with respect to each component of Z was possible. Proceeding in the same way for an actual tensor is not logical because a rotation of the coordinate system is tantamount to a transformation of its components. Let us then consider the vector z = Z(ω, x) to be an actual first-rank tensor, that is, an object transforming according to the rule z = Q · z,

(2.110)

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where Q is a matrix giving x = Q · x upon rotation. Of course, for an nth-rank tensor we have z = Q...Q · z,

(2.111)

where Q appears n times. This suggests defining a random tensor field of first-rank to be wide-sense stationary and isotropic whenever the mean Z(x) and the correlation ρik (x) do not change when subjected to arbitrary shifts and when transforming by rotation in (2.110) according to j j (2.112) Z = Q · Z , ρi (x ) = Q · ρi (x), whereby x is transformed into x according to (2.109). Note: This definition may be extended to strict-sense stationarity (SSS) by requiring the above property to hold for all n-order distributions Fn , not just for the first and second moments. In (2.112) we employ the correlation coefficient

[i (x1 ) − i (x1 )][ j (x2 ) − j (x2 ) ] j ρi (x1 , x2 ) = , (2.113) σi (x1 )σ j (x2 ) j

which is the same thing as (2.85), but the notation ρi is superior to ρij when dealing with tensor fields below. The WSS property means that, for any pair (x1 , x2 ), j

j

ρi (x1 , x2 ) = ρi (x),

x = |x1 − x2 | .

(2.114)

Next, the isotropy says j

j

x ≡ |x| =

ρi (x) = ρi (x),

√

xi xi .

(2.115)

j

Robertson (1940) showed that ρi (x) in 3D admits this representation (see also Lomakin, 1965, 1970; Sobczyk and Kirkner, 2001): j

ρi (x) = K 1 (x) xi xk + K 2 (x) δik

(2.116)

wherein the K i s are real-valued functions of x = |x|: 1 1 K 1 = ρ11 = ρ22

1 K 1 + K 2 = ρ33

(2.117)

j

and ni = xi /x. Here ρij1 stands for the correlation ρi (x) between Cij (x1 ) and Ckl (x2 ) in a coordinate system centered at x1 and directed to x2 . 2.3.2.1 Second-Rank Tensor Fields In general, we say that Z is a random tensor field in D dimensions if it assigns to an elementary event ω ∈ a realization over X ⊂ R D , that is, Z : × X → Rn ,

Z(ω, x) = z

(2.118)

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where x ∈ X. In general, for D = 2, n = 4, while for D = 3, n = 9. Of course, the concept of a random tensor field may be applied to just about any tensor field encountered in continuum mechanics. The normalized correlation function of, say, a second-rank tensor field is

[Zij (x1 ) − Zij (x1 ) ][Zkl (x2 ) − Zkl (x2 )] kl ρij (x1 , x2 ) = , (2.119) σij (x1 )σkl (x2 ) where σij (x1 ) and σkl (x2 ) are the standard deviations of the pair [zij (x1 ) , zkl (x2 )] at respective points. It follows from the tensor property of z that ρijkl is a fourth-rank tensor. As a special case, take Z as an antiplane stiffness tensor C (≡ Cij ) of a hyperelastic material the conductivity-type tensor. Then, the well-known symmetry holding at ∀x ∈ X of every realization C (ω) of the random field Cij = C ji

(2.120)

ρijkl = ρ klji = ρijlk .

(2.121)

implies these symmetries of ρijkl

Now, recall after Robertson (1940), Lomakin (1965, 1970), and Sobczyk and Kirkner (2001) that ρijkl (x) in 3D admits this representation: ρijkl (x) = K 4 (x) δij δkl + K 6 (x) δik δ jl + δil δ jk + [K 5 (x) − K 6 (x)] n j nk δil + ni nl δ jk + ni nk δ jl + n j nl δik + [K 3 (x) − K 4 (x)] ni n j δkl + nk nl δij + [K 1 (x) + K 2 (x) − 2K 3 (x) − 4K 5 (x)] ni n j nk nl ,

(2.122)

wherein the K i s are 1 1 1 K 1 = ρ1111 K 2 = ρ2222 K 3 = ρ1122 1 1 1 K 4 = ρ2233 K 5 = ρ1212 K 6 = ρ2323 ,

(2.123)

they satisfy the relation K 4 + 2K 6 − K 2 = 0,

(2.124)

1 and ni = xi /x. Here ρijkl stands for the correlation between Cij (x1 ) and Ckl (x2 ) in a coordinate system centered at x1 and directed to x2 . In Chapter 8, we return to this result in the context of those tensor fields for which each realization C (ω) is locally isotropic, i.e., ∀x ∈ X. Interestingly, while a second-rank tensor in 3D generally has nine independent components, the assumption of isotropy of its realizations reduces the number of parameters of its correlation function to six.

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2.3.2.2 Fourth-Rank Tensor Fields We say that Z is a random tensor field in D dimensions if it assigns to an elementary event ω ∈ a realization of a fourth-rank tensor z = zijkl over X ⊂ R D , that is, Z : × X → Rn ,

Z(ω, x) = z.

(2.125)

In general, for D = 2, n = 16, while for D = 3, n = 81. For example, in classical (nonmicropolar) hyperelasticity, where Z stands for the stiffness tensor, we generally have n = 21. As an example, if we were able to measure not just the uniaxial stiffness (E) but the entire stiffness tensor (C, or Cijkl ) at each and every point x of a realization of the random medium, we would deal with a random field of stiffness tensor. We focus on Z being WSS. Then, the normalized correlation function of such a fourth-rank tensor field is

[Zijkl (x1 ) − Zijkl (x1 ) ][Z pr st (x2 ) − Z pr st (x2 ) ] prst ρijkl (x1 , x2 ) = , (2.126) σijkl (x1 )σ pr st (x2 ) where σijkl (x1 ) and σ pr st (x2 ) are the standard deviations of the pair [Zijkl (x1 ), Z pr st (x2 )] at respective points. It follows from the tensor property of Z, that prst ρijkl is an eighth-rank tensor. Analogously, for a random fourth-rank tensor field Z we have

[Zijkl (x1 ) − Zijkl (x1 ) ][Z pr st (x2 ) − Z pr st (x2 ) ] prst ρijkl (x1 , x2 ) = , (2.127) σijkl (x1 )σ pr st (x2 ) where σijkl (x1 ) and σ pr st (x2 ) are the standard deviations of the pair [zijkl (x1 ) , z pr st (x2 )] at respective points. It follows from the tensor property of Z, that prst ρijkl is an eighth-rank tensor. prst

Since Z is WSS, ρijkl has the property prst

prst

ρijkl (x1 , x2 ) = ρijkl (x),

(2.128)

where x = x2 − x1 . In the case of isotropy of the correlation function, we have prst

prst

ρijkl (x) = ρijkl (x).

(2.129)

Now suppose that Z stands for a random stiffness tensor C (≡ Cijkl ) in a linear hyperelastic material. In that case, these well-known symmetries holding at ∀x ∈ X of every realization C (ω) Cijkl = Cjikl = Cijlk = Cklij

(2.130)

prst

imply these symmetries of ρijkl prst

prst

prst

prst

rpst

prts

stpr

ρijkl = ρjikl = ρijlk = ρklij = ρijkl = ρijkl = ρijkl .

(2.131)

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Now, let K i (x), i = 1, . . ., 15, be fifteen scalar functions of x. Note from prst Lomakin (1965, 1970) that ρijkl (x) of a WSS and isotropic random field in 3D admits this representation prst

ρijkl (x) = F12 + K 13 (x) δij δkl δ pr δst +K 14 (x) δij δkl δ ps δr t + δij δkl δ pt δr s + δik δ jl δ pr δst + δil δ jk δ pr δst +K 15 (x) δij δ jl δ ps δr t + δik δ jl δ pt δr s + δil δ jk δ ps δr t + δil δ jk δ pt δr s wherein F12 is a linear function of all K i (x), i = 1, . . ., 12, each term involving combinations of the vector x and the Kronecker deltas in pairs from i, j, k, l and p, r, s, t. The concepts of separable structure and local averaging introduced earlier apply, in appropriately generalized forms, to random fourth-rank tensor fields.

2.4

Mechanics Problems on Two- and Three-Dimensional Random Fields

In this section we discuss two problems of random media: (1) mechanics of materials with the randomness inherent in its constitutive law, and (2) turbulence in a fluid with constant Newtonian viscosity, that is, formation of random velocity and stress fields, due to the inherent fluid instability. 2.4.1 Mean Field Equations of Random Materials As noted in the preface, in mechanics of materials with spatial randomness, the field is governed by the equation Lu = f, where L is typically a linear differential operator, f is the source or forcing function, and u is the solution field. Using the decompositions u = u + u

L = L + L ,

(2.132)

and first averaging the original equation and then subtracting the result from it, we obtain an equation governing u

L u + ( I − P) L u = −L u .

(2.133)

Here P is a so-called projection, basically signifying an averaging operation. Solving (2.133) for u , we arrive at an equation governing ( L − M) u = f,

(2.134)

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& −1 ' M = L L + ( I − P) L L

(2.135)

stands for the so-called mass operator; −1 indicates an inversion. Following Fishman and McCoy (1981), a more concrete view of this procedure is given in terms of(the heat conduction problem in a medium de) scribed by a random field K ij (ω, x) ; ω ∈ , under the action of a source term f (x) and the boundary temperature field T 0 (x), both slowly varying on the macroscale. The heat flux is denoted by q i . Thus, for each realization B (ω) ∈ B, we have q i,i = f (x)

x ∈B

q i = K ij (ω, x) T, j

x ∈B

(2.136)

T (x) = T 0 (x)

x ∈ ∂B, Note that (2.136)2 dictates K ij (ω, x) = K ij + K ij (ω, x), so that, upon averaging, we obtain this set of equations:

q i ,i = f (x) x ∈B ' &

q i = K ij T, j + K ij (ω, x) T,j

T (x) = T 0 (x)

x ∈B

x ∈ ∂B.

From the equations governing the fluctuations q i ,i = 0 x ∈B q i = K ij T,j + ( I − P) K ij (ω, x) T,j + K ij (ω, x) T,j T (x) = 0

(2.137)

x ∈B

(2.138)

x ∈ ∂B,

we derive an analog of (2.133) + * K ij T, ji + ( I − P) K ij (ω, x) T,j = −K ij (ω, x) T,j .

(2.139)

Hence, the constitutive equation for the average fields is ∗

q i (x) = K ij T, j + K ij x, x T, j x dx . ,i

(2.140)

,i

B

Note: That the effective constitutive response (2.140) has a nonlocal character; this result carries over to random elastic and inelastic materials (Beran and McCoy, 1970a; Eimer, 1971; McCoy, 1991); Problem 14. 2.4.2 Mean Field Equations of Turbulent Media Mechanics of random materials focuses primarily on problems driven by random fields of constitutive properties, whereas studies of turbulence deal

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with spatiotemporal random velocity fields due to inherent instabilities of fluids. In the latter subject area, we recall the continuity equation, the linear momentum theorem, and the energy conservation law of an incompressible, viscous fluid. vi,i = 0 ρ v˙ i = σi j, j ρ u˙ = σij dij − q i,i ,

(2.141)

where vi is the velocity, ρ is the mass density, dij is the deformation rate tensor, u is the internal energy, and q i is the heat flux. Now, take the velocity vi to be the average superposed with the fluctuation according to (2.51). Assuming the properties (2.52) to hold for all the fields involved, we then find ∗

vi,i = 0 ρ ˙vi = σi∗j, j ρ u˙ ∗ = σij∗ dij − q i,i ,

(2.142)

σij∗ = σij − ρ vi vj

(2.143)

where

is the so-called Reynolds stress (also denoted Rij ), u∗ = u +

1 vv 2 i i

(2.144)

is the sum of the average internal energy and the kinetic energy of random (microscale) fluctuations, while q i∗

= qi −

&

σij vi

'

+ρ

1 u + vj vj 2

vi

(2.145)

is the effective heat flux appearing as the sum of the original heat flux with (1) the rate of work of the fluctuating stresses on the surface of an elementary volume and (2) the stochastic energy convection through the boundary. Note: The average field vi satisfies the same continuity equation as vi . The divergence of the Reynolds stress may be interpreted as the force density on the fluid due to turbulent fluctuations. The Reynolds stress also appears when analyzing the Euler or Navier–Stokes equations (Problem 15). Note: While, formally, vi in the above equations is the ensemble average, in practice, this average is sometimes also thought of as a spatial average over some length scale, or a temporal average. Accordingly, the fluctuation v is then interpreted not as a statistical one, but a spatial or temporal one. In the case of a temporal average, one works with a separation of scales: the time scale of variation of v is much larger than that of v . The equivalence between such averages in statistical turbulence is an open problem, but is justified in the more established field of equilibrium statistical mechanics by the ergodic theorem, as discussd in the next section.

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Ergodicity

2.5.1 Basic Considerations It is clear by now that random fields (or processes) are sets of realizations over some space (respectively, time) domain. Of primary importance is a possibility of determination of probabilistic characteristics (e.g., moments, distributions) of a random process or field in terms of a set of values {z(x1 ), . . . , z(xn )} observed over just one realization Z(ω, x), x ∈ R—as is often the case in practice. At first sight, such an approach appears impossible because it seems that not much statistics (if any) may be obtained from a single realization. Nevertheless, one intuitive explanation of such an approach lies in a possibility of treating the realization Z(ω) at hand as “typical” (in a certain sense, as discussed below) of the whole field Z, recall Figures 2.1 and 2.2 of this chapter. Another explanation involves treating the realization observed in an interval (x0 , xn ) as a set of separate realizations observed in the intervals (x0 , x1 ), (x1 , x2 ), . . ., (xn−1 , xn ). In the engineering literature this kind of supposition is called an ergodic property or ergodicity; in mathematics these terms have a narrower meaning (see below). Now, ergodicity in the mean (or, the process is mean-ergodic) means that any realization Z(ω), ω ∈ is sufficient to get the ensemble average

Z(x) at any x from its spatial average Z(ω) for any ω ∈ taken over a sufficiently large interval: 1 Z(ω) ≡ lim L→∞ 2L

L

−L

Z(ω, x)d x =

Z(ω, x)d P(ω) ≡ Z(x). (2.146)

Several issues arise here. 2.5.2 Computation of (2.146) The left-hand side of (2.141) may be evaluated only with some accuracy, both because of the finite scale discretization of measurements and the impossibility of carrying out the limit L → ∞. In practice, Z(ω) must be replaced, respectively, by a spatial average (generally, volume average) from a finite number of sampling points N (taken over one realization ω) 1 Z(ω, xn ). N n=1 N

Z(ω) ≡

(2.147)

Additionally, also in a real application, Z(x) must be computed from the ensemble average over a finite number M of realizations ω (taken at a chosen

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sampling point x) 1 Z(ωm , x). M m=1 M

Z(x) ≡

(2.148)

2.5.3 Conditions for (2.146) to Hold Assuming the limit in (2.146) exists, its value Z(ω), in general, depends on ω. Under what conditions does it equal the constant µ? The answers are provided by the so-called ergodic theorems. The first one of these states that the process is mean-ergodic iff its autocovariance is such that L L 1 lim C(x1 , x2 )d x1 d x2 = 0. (2.149) L→∞ 4L 2 −L −L This is proved by noting that the variance σ L2 of the random variable SL (ω) = ,L (2L) −1 −L Z(ω, x)d x of the process Z is L L 1 σ L2 = |SL − µ L |2 = C(x1 , x2 )d x1 d x2 (2.150) 4L 2 −L −L where µ L is the mean of SL (ω). Note: A process may be ergodic without being stationary. Upon introducing the WSS property of Z, one readily finds that a process is mean-ergodic if its autocovariance C (x) = R (x) − µ2 is such that (Problem 16): 2L |x| 1 lim C(x) 1 − d x = 0. (2.151) L→∞ 2L −2L 2L ,∞ One gets sufficiency here if −∞ C(x)d x < ∞. There are also other kinds of ergodicity: correlation-ergodic, distribution-ergodic (Papoulis, 1984), for example. The former of these is expressed by L 1 RZ (ω, x) ≡ lim Z(ω, x1 + x) Z(ω, x1 )d x1 L→∞ 2L −L (2.152) = Z(ω, x1 + x) Z(ω, x1 )d P(ω) ≡ Z(ω, x1 + x) Z(ω, x1 ) ,

which, in fact, was the basis for computation of autocorrelations in Figures 2.1 and 2.2. 2.5.4 Existence of the Limit in (2.146) Under what conditions does the limit Z(ω) in (2.146) exist? This is actually known as the so-called ergodic problem in the mathematics literature. This problem of ergodicity has its roots in statistical mechanics, where one is interested in estimating system properties from a single trajectory over a relatively

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very long period of time. Because a trajectory occurs in a phase space, this leads us to the concept of an ergodic flow in the phase space, that is, a flow for which the integral with respect to a time parameter t converges to a random variable Z(ω) 1 lim Z(ω, t)d x = Z(ω). (2.153) L→∞ L More generally, for a random process Z indexed by t, one can define a flow in phase (or state) space of Z as a transformation mapping this space onto itself, whereby any event A is transformed into some other one, Aτ , by means of an operator T(τ ): A → Aτ = T(τ ) A.

(2.154)

As a result, a function g( A) transforms as g( A) → g( Aτ ) = g[T(τ ) A].

(2.155)

In classical statistical mechanics one focuses on Hamiltonian flows (i.e., those where the total energy of the system is conserved), which are measurepreserving in the sense that P( A) = P( Aτ ).

(2.156)

In (2.154) A and Aτ stand for an initial set and a set after the transformation (2.155). The measure is a mathematical term, whose counterpart in physics is the density ρ (q n , pn , t) in the phase space of coordinates q i and conjugate momenta pn , n = 1, . . ., N (the number of degrees of freedom); we return to ρ in Chapter 10. Note that a Hamiltonian system is measure-preserving because of Liouville’s theorem expressed in terms of the convective derivative being zero (Problem 17): dρ ∂ρ ∂ρ ∂ρ = + q˙ n + p˙ n = 0, dt ∂t ∂q n ∂ pn

(2.157)

where the summation convention is implied. Recall here Hamilton’s equations q˙ n =

∂H ∂ pn

p˙ n = −

∂H ∂q n

.

(2.158)

Let us now consider three basic types of measure-preserving flows in phase space, as depicted in Figure 2.5. In each case we let the set A modeling the state of the system evolve while keeping another set B (akin to a control window) fixed. In the case (a), the set A moves in a periodic fashion through the phase space. It visits just a fraction of it without ever entering other regions; a harmonic oscillator is a very simple example: H (q , p) =

p2

2m

1 + kq 2 2

(2.159)

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(a)

(b)

(c)

FIGURE 2.5 Three types of flow in phase space: (a) periodic, nonergodic; (b) ergodic, nonmixing; (c) ergodic, mixing. (After Balescu [1975]). In each case, a square-shaped set B is shown.

In the case (b), we see the set Aonly slowly being altered during its motion, while it sweeps the entire space if observed for a sufficiently long (infinite) time—this is an ergodic flow. It was shown by G.D. Birkhoff in 1931 that, in this flow, every invariant set has the measure 0 or 1 (meaning it comprises almost whole space), or, intuitively, no trajectory can be confined to a finite portion of phase space because it has to wander (ergodos in Greek) through all of it. Another way to express this is that the trajectory of the process is not sensitive to initial conditions. Finally, in the third case (c), the set A not only sweeps the entire space, but its shape is being altered during its motion so as to fill the entire space through a multitude of growing branches, subbranches, and so on—this is a mixing flow. Indeed, Figure 2.5(c) shows three stages: an initial one at time zero when A is just a small blob, a second one when A has already moved and diffused somewhat, and a third stage when A begins to diffuse even more. As time τ goes on, since the flow is mixing, Aτ begins to intersect B, and tends to a “uniformly mixed situation” where the volume fraction of A in B will be the same as the initial volume fraction of A in the whole space , that is, lim P( Aτ ∩ B) = P( A) P( B).

τ →∞

(2.160)

It is easy to show that mixing implies ergodicity: If A is a measurepreserving set, then Aτ = A, and hence Aτ ∩ A = A, so that P( Aτ ∩ A) = P( A).

(2.161)

On the other hand, because the flow is mixing, then by (2.160) above, setting B = A, we find lim P( Aτ ∩ A) = [P( A)]2 .

τ →∞

(2.162)

Comparing (2.160) with (2.162), we find P( A) = [P( A)]2 ,

(2.163)

which is possible only when P( A) equals 0 or 1. Clearly, there may exist ergodic flows that are not mixing, Figure 2.5(b). Note: Evolution of a mixing flow: the set A is painted black so that, following J.W. Gibbs, we can think of ink mixing with water. Some people may prefer to think in terms of mixing cocktails.

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The condition (2.160) is sometimes called strong mixing. A weaker case is the so-called weak mixing described by 1 τ lim (2.164) [P( Aτ ∩ B) − P( A) P( B)] dτ = 0. τ →∞ τ 0 An even weaker case is specified by the condition 1 τ lim [P( Aτ ∩ B)] dτ = P( A) P( B). τ →∞ τ 0

(2.165)

Note: (2.160), (2.164), and (2.165) are various extensions of the conventional independence property [P( A ∩ B) = P( A) P( B)] when the time evolution is involved. Motivated by this simple account of time-dependent flows (processes), we now consider a transformation of the state space with probability distribution P in the context of a directly given random field E, defined on R D , D being the number of spatial dimensions. In analogy to (2.154), consider a family of shift transformations ( j)

g( E) → g( E ξ ) = g[Tξ E]

(2.166)

Tξ g(x1 , . . ., x j , . . ., xD ) = g(x1 , . . ., x j + ξ, . . ., xD ).

(2.167)

where ( j)

( j)

Here g(x) = ω ∈ , each of which takes a set S ⊂ into a set Sξ composed of the functions of S shifted by ξ at their jth parameter x j . If the random field is strictly homogeneous, then these transformations are measure preserving in the sense that ( j)

P(S) = P(Sξ ).

(2.168) ( j)

A set S is called invariant if, for every j and ξ , the sets S and Sξ differ at most by a set of P-measure zero. We say that the strictly homogeneous random field is ergodic if every invariant set has either probability zero or one. See Adler (1981) for a rigorous account of this subject.

2.6

The Maximum Entropy Method

2.6.1 Cracks in Plates with Holes A recent study (Al-Ostaz and Jasiuk, 1997) investigated fracture response in tension of both elastic-brittle (epoxy) and ductile (aluminum) plates having some thirty holes punched in a specific disordered pattern in them. In the case of each material, several macroscopically identical specimens were tested under the same conditions, and each displayed a different crack pattern. The situation is depicted in Figure 2.6 for the elastic-brittle case.

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FIGURE 2.6 Schematic plot of final crack patterns superposed from seven epoxy specimens (in various colors) under the same uniaxial vertical tensile loading conditions, obtained experimentally. (After AlOstaz and Jasiuk [1997].)

The typical experiment involved a number of nominally identical plates (8.25 cm × 33.02 cm), each of a uniform thickness 0.38 cm, and made of epoxy (PSM-5). Each plate had the same nonperiodic, nonuniform distribution of thirty-one randomly distributed circular holes of the same size, an inch in diameter. The locations of the holes’ centers were generated according to a hard-core (sequential-inhibition) point field with exclusion. Each plate was subjected to a uniaxial tensile loading in the y-direction at a constant displacement rate of ˜0.03 cm/s initially, and then decreased to ˜0.0017 cm/s in order to capture more details of the fracture process. The nonuniqueness of the experimental results may be understood by noting, on purely combinatorial grounds, that there exist a large number of geometrically acceptable (plausible) crack paths cutting the specimen across and this number grows rapidly as we consider systems with more holes. Whether we assume a maximum strain energy or a maximum principal stress energy criterion, the respective values associated with all these paths do not differ much from one another. Thus, minute material and geometric imperfections are likely to decide which crack path will actually take place in a particular specimen that is nominally (on the macroscopic scale) the same as the others. The material imperfections arise from the intrinsic nature of materials that are all heterogeneous at a microscopic level. The geometric imperfections include roughness of the holes’ surfaces and microscopic damage (microcracks and other surface flaws) due to drilling, which may give rise to singular stress fields. 2.6.2 Disorder and Information Entropy The phenomenon described above can be (and was) tackled by numerical methods involving finite elements and spring networks (Al-Ostaz and Jasiuk, 1997), resulting in distinct crack paths and distinct overall load-displacement

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diagrams. However, there arose a basic question: which crack path should actually be used as guidance in fine-tuning the computational mechanics model? In principle, by adjusting the elastic parameters and failure criteria, as well as by introducing noise in these parameters, one could reproduce all the crack paths—and this is the methodology of the right column in Figure 2.7. The best-known variational principles of mechanics are those based on minimization of energy. They are the basis on which modern continuum mechanics, having its roots in classical dynamics, is built. These principles are deterministic. Let us consider the minimum potential energy principle in the setting of mechanics of random media: for a given body B(ω) ∈ B min [U (ω) − W]

(2.169)

where U (ω) is the strain energy and W is the work of body forces and surface tractions. In light of the basic considerations in the preface to this book, the above could be further generalized by taking W to be random as well, but this is not crucial here. As is well known, (2.169) forms the starting point for, say, a finite element method (FEM), which leads to a method of quantitatively analyzing each and every specimen of the ensemble separately. The FEM is set up from a system MACROSCALE maximum entropy method (MEM) W

estimate P(Wfailure)

ensemble {B(ω);ω ∈ Ω} (ω) = work-to-failure failure n

times (Monte Carlo) for ensemble ΩMG ⊂Ω

MG

(i) get f–u curves (ii) estimate possible crack paths ∀B(ω) estimate fracture condition: surface energy ℑ

solve for W

(ω) from failure min [U(ω) – W]|constraints

maxH|

constrainsts

postulate failure criterion P(ℑ)

estimate P(ℑ)

minimum potential energy method MICROSCALE FIGURE 2.7 Philosophies of two variational principles in mechanics of random media, and their roles in establishing a connection between microscale and macroscale responses. The left column illustrates a maximum entropy approach where the microscale probabilities are determined from the macroscale statements and observations, which represent constraints. The right column illustrates a (usually) much more familiar variational method of deterministic mechanics extended to a representative subset of heterogeneous specimens of a random medium.

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of equations governing the entire domain of the body B(ω) partitioned into a number of elements. Once the solution is found, one has to repeat the entire numerical analysis from the beginning to establish a solution for another specimen of , see Figure 2.7. Thus, in order to investigate a specific effect of microscale randomness, one needs to carry out a finite, and necessarily limited, number of studies (n MC ) on a subset MC ⊂ . Of course, if an explicit analytical solution is available, one can proceed directly to the stochastic stage, and find, say, probability distributions of macroscopic response in terms of the microscale noise. Unfortunately, such nice solutions are rather rare and a bridge from microscale to macroscale may be too cumbersome or unwieldy. However, Figure 2.7 shows that there is another way of building this bridge: rather than to proceed from micro to macro, one can go the opposite way. This, in fact, is accomplished by a so-called maximum entropy method (MEM): for a statistical ensemble (on which a random variable X = {X(ω); ω ∈ } has been defined), maximize the possible disorder as expressed by the information entropy ∞

− h ( X) = ln p (x) = − p (x) ln p (x) d x (2.170) −∞

subject to the condition that the expected values µ = gi (x) of n known functions gi (x) of X are given. In the case of a discrete-type random variable (taking values i = 1, . . ., I ), h ( X) = − ln p (x) = −

I

pi (x) ln pi (x) ,

(2.171)

i=1

one may show that the probability density of X is given by p (x) = Aexp [−λ1 g1 (x) − · · · − λ I g I (x)] .

(2.172)

Note: The entropy definition (2.170) represents the measure of uncertainty, or lack of information, for a continuous valued random variable. Note: The conditions involved in finding the solution p (x) represent our limited knowledge of possible constraints. If all the constraints were known (and introduced into the problem), we would get exactly the same solution as by conducting a deterministic mechanics study ω-by-ω according to the right column of Figure 2.7—there is no philosophical conflict between both methods. Usually, however, the knowledge of all the constraints is not there, or their introduction into the analysis is prohibitively complex. Consequently, we view the MEM as an approach complementary to the one based on conventional variational principles. The MEM outlined in the left column of Figure 2.7 is ideally suited to study the microscale material randomness, as expressed by the surface energy density fluctuations. We proceed in the following steps:

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1. Fracture occurs at the expense of dissipated energy E(ω), and the fracture path has a length L(ω). The two quantities can be computed from Figure 2.6. 2. Calculate surface energy per unit length γ (ω) = E(ω)/L(ω). Let us note here that γ (and hence its probability density p(γ )) is now assessed on the mesoscale l, which is the mean hole spacing, rather than on, say, a molecular scale. 3. Given the results for N specimens, we get {γ (ωn ); ω1 , . . ., ω N }, each of probability P{ω1 } = ... = P{ω N } =

1 . N

(2.173)

4. Compute moments γ , γ 2 , γ 3 , ... to get, according to (2.172), p(γ ) = Aexp −λ1 γ − λ2 γ 2 − λ3 γ 3 − ...

(2.174)

subject to

∞

exp −λ1 γ − λ2 γ 2 − λ3 γ 3 − · · · dγ = 1

∞

γ exp −λ1 γ − λ2 γ 2 − λ3 γ 3 − · · · dγ = γ

A 0

A

(2.175)

0

A

∞

γ 2 exp −λ1 γ − λ2 γ 2 − λ3 γ 3 − · · · dγ = γ 2 .

0

5. This basic model can be improved to account for: Orientation of cracks with respect to the macroscopic loading direction Local crack (hole–hole) interactions Further crack (hole–hole) interactions, etc. 6. With p(γ ) one can predict the probability of occurrence of any other crack path. Note: The MEM provides a closure method for many nonlinear problems of stochastic mechanics, for example, fragmentation under dynamic impact (Englman et al., 1987), turbulence (Frisch, 1995), random vibration (Sobczyk and Trebicki, 1990, 1993), improved bounds on the effective response of random materials (Kreher and Pompe, 1989), or contact forces in granular media (Goddard, 2004). In Chapter 4 we return to the plate fragmentation experiment with the dynamics taken into account.

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Problems 1. Express the Kolmogorov consistency condition in terms of the probability density function. 2. Suppose that P and Q in Example 3 are jointly Gaussian with the density N (µ1 , µ2 , σ1 , σ2 , ρ). Compute the probability distribution of X (ω, t). 3. Show the equivalence of (2.24) with (2.25), and prove (2.26). Assuming P and Q are jointly normal random variables with the density N (µ1 , µ2 , σ1 , σ2 , ρ), find the first-order probability density of X (t, ω); consult Problem 8 below. 4. Show that, in the case of a Rice noise, we have n n µY (x) = 0, VarY (x) = i=1 σi2 , RY (x1 , x2 ) = i=1 σi2 cos λi (x2 − x1 ) . 5. Make a literature search to define the Bernoulli process and discuss its properties. 6. Make a literature search to define the binomial process and discuss its properties. 7. Prove that |ρ(x1 , x2 )| ≤ 1. 8. If we drop the dependence of f in (2.13) on x1 and x2 , then f (z1 , z2 ) is the joint probability density of zero-mean normal random variables z1 and z2 , commonly denoted N (µ1 , µ2 , σ1 , σ2 , ρ), with µ1 = µ2 = 0 here. Show that ρ is their correlation coefficient. 9. For a Rayleigh wave propagating along a weakly random boundary 1 (Section 2.2) derive σzz and σzz . 10. One of the consequences of Jensen’s inequality is that, for any realvalued function defined over the probability space, exp f d P ≤ e f d P.

(a) Derive from this the inequality between the geometric and arithmetic averages of n positive numbers y1 y2 . . . yn , ( y1 y2 . . . yn ) 1/n ≤

1 ( y1 + y2 + · · · + yn ) . n

Hint: Consider to be a finite set with ( P{ωi }) = p for all i. (b) Derive the inequality between the harmonic and geometric averages −1 1 1 1 1 + + ··· + ≤ ( y1 y2 . . . yn ) 1/n n y1 y2 yn

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11. Generalize (2.71) to Timoshenko beams. 12. State the Kolmogorov consistency conditions for a vector random field. 13. Consider pi to be a potential random vector field (p = ∇ S), where S is a scalar random field with a zero mean and correlation function C (S) (x), x = x2 − x1 . Show that C (S) (x) is related to the correlation ( p) function Cij (x) of p through ∂2 C (S) ∂ x1i ∂ x2 j

( p) = −Cij (x)

where ∂x1i (∂x2 j ) denotes a differentiation with respect to the ith component of x1 ( jth component of x2 ). 14. Following Eimer (1971), discuss the effective constitutive equation approximations of a two-phase, random material (with one phase elastic and another viscous) as Kelvin–Voigt or Maxwell models. 15. Obtain the Reynolds stress by analyzing either the Navier–Stokes or Euler equations in the same fashion as in Section 2.4. 16. Prove the statement of equation (2.151). 17. Prove Liouville’s equation (2.157) by taking the continuity equation for ρ in the phase space, and noting the divergence-free charac˙ p˙ in phase space (recall Hamilton’s ter of the “velocity field” q, equations). 18. Show that the Euler–Lagrange equations remain valid for gyroscopic systems in which the gyroscopic forces G k are derivable from a potential V according to ∂V d ∂V Gk = V q k, q˙ k = f (q k ) q˙ k , − dt ∂q˙ k ∂q k and the Lagrangian is taken in the form L = T − V − V instead of the classical L = T − V. 19. Give an example of a WSS process that is not ergodic. 20. Give an example of an ergodic flow that is not mixing. 21. Consider a collection of particles having a stationary distribution in a conservative field, whose potential is V (x). Using the MEM, find the probability density f (x) of positions x, under the assumption that the mean of V with respect to f is known. 22. Prove properties 1–4 of ρ(X1 , X2 ) listed in Section 2.3.1.1 23. Consider a collection of particles having a stationary distribution in a conservative field, whose potential is V(x). Using the MEM, it is a simple matter to find the probability density P(x) of positions x, under the assumption that the function g(x) = V(x) and the mean

V of V with respect to p are known.

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Microstructural Randomness and Scaling in Mechanics of Materials 24. Verify (2.134) and (2.135). 25. Check whether the random process of equation (2.24) is meanergodic. 26. The one degree-of-freedom oscillator under random loading provides a basic model for equation (0.3) of the Preface. Considering the equation of motion derive the oscillator’s stationary response under a random forcing whose spectral density is (i.e., a white noise). Hint: transform the problem to the frequency domain.

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3 Planar Lattice Models: Periodic Topologies and Elastostatics

Elasticity, thanks to solid state physics, has been reduced from primordial and precise to secondary, approximate and derived. J.A. Wheeler, 1994

Spring network (or lattice) models are based, in principle, on the atomic lattice models of materials. These models work best when the material may naturally be represented by a system of discrete units interacting via springs, or, more generally, rheological elements. It is not surprising that spatial trusses and frameworks have been the primary material systems thus modeled. Spring networks can also be used to model continuum systems by a lattice much coarser than the true atomic one. In engineering mechanics that idea dates back, at least, to Hrennikoff (1941), if not to Maxwell (1869), in a special setting of optimal trusses. This obviates the need to work with an enormously large number of degrees of freedom that would be required in a true lattice model, and allows a very modest number of nodes per single heterogeneity (e.g., inclusion in a composite, or grain in a polycrystal). As a result, spring networks are a relative of the much more widespread finite element method. In this chapter we focus on basic concepts and applications of spring networks, in particular to antiplane elasticity, planar classical elasticity, and planar nonclassical elasticity. The chapter ends with an additional section on the mechanics of a helix, a 1D nonclassical continuum.

3.1

One-Dimensional Lattices

3.1.1 Simple Lattice and Elastic Strings Let us begin with a lattice-based derivation of a wave equation for a 1D chain of particles; see also Askar (1985). The particles (parametrized by i), each of mass m, interact via nearest-neighbor linear elastic interactions, Figure 3.1. 87

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i–1

i

i+1

FIGURE 3.1 A 1 D chain of particles of lattice spacing s, connected by axial springs (thin lines).

For the potential and kinetic energies we have: U=

1 1 Fi (ui+1 − ui ) = K (ui+1 − ui ) 2 2 i 2 i

T=

1 mu˙ i2 , 2 i

(3.1)

where Fi = K (ui+1 − ui ) is the axial force at i, and K is the spring constant between i and i + 1. Using the Euler–Lagrange equations for the Lagrangian L = T − U, we arrive at the dynamical equations K (ui−1 − 2ui + ui+1 ) = mu¨ i ,

(3.2)

which describe a system of coupled oscillators. By taking the Taylor expansion up to the second derivative for the displacement ui±1 ≡ u (xi ± s), 1 ui±1 ∼ = u|xi ± u, x xi s + u, xx xi s 2 , 2

(3.3)

we find from (3.2) an approximating continuum mechanics model, that is, the basic wave equation E Au, xx = ρ A¨u ,

(3.4)

where (A being the cross-sectional area of the rod) E=

Ks A

ρ=

m . As

(3.5)

Of course, (3.4) can also be obtained from Hamilton’s principle for the Lagrangian L expressed in terms of continuum-like quantities, by first introducing (3.3) in (3.1)1 with terms up to the first derivative, U=

AE 2

d

u, x

2

dx

T=

0

Aρ 2

d

˙ 2 dx . ( u)

(3.6)

0

3.1.2 Micropolar Lattice and Elastic Beams We now generalize the preceding lattice model to describe transverse motions of a 1D chain of dumbbell particles (rigid bars) pin-connected by centralforce (axial) springs, Figure 3.2. The particles (again parametrized by i) each have mass m and moment of inertia J . We need to consider two degrees of

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i–1

i (a)

89

i+1 (b)

FIGURE 3.2 (a) A 1D chain of dumbbell particles (vertical rigid bars) of X-braced girder geometry, pinconnected by axial springs (thin lines); (b) the shear and curvature modes of a single bay.

freedom per particle i: total transverse displacement wi and rotation ϕi . The constitutive laws for a single bay (between i and i + 1) are i = K ⊥ [wi+1 − wi − sϕi+1 ] F

Mi = − Kˆ (ϕi+1 − ϕi ) .

(3.7)

Here F˜ i is the shear force and Mi the bending moment at i, whereby the term sϕi+1 is subtracted in (3.7)1 so as to deal with shear only. For this 1D chain of particles, we write down potential and kinetic energies U=

1 K ⊥ [wi+1 − wi − sϕi+1 ]2 + Kˆ (ϕi+1 − ϕi ) 2 2 i

T=

1 mw ˙ i2 + J ϕ˙ i2 . 2 i (3.8)

The first term in (3.8)1 accounts for shear deformations, and the second one for bending. Using the Euler–Lagrange equations for the Lagrangian L = T − U, we obtain a system of equations K ⊥ [wi+1 − 2wi + wi−1 − s(ϕi+1 − ϕi )] = mw ¨i Kˆ [ϕi+1 − 2ϕι + ϕi−1 ] + K ⊥ [wi+1 − wi−1 − sϕi ] s = J ϕ¨ i .

(3.9)

By introducing the Taylor expansions for wi and ϕi in (3.9)1 with terms up to the second derivative, and taking the limit s → 0, we find G A[w, x − ϕ], x = ρ Aw ¨ E I ϕ, xx + G A[w, x − ϕ] = ρ I ϕ, ¨

(3.10)

where (A and I , respectively, being the area and second moment of the crosssection of this beam-like lattice): G=

K⊥s A

ρ=

m As

I =

JA m

E=

Kˆ sm . JA

(3.11)

Relations (3.10) are recognized as the equations of a Timoshenko beam. Evidently, this is a 1D micropolar continuum with two degrees of freedom: displacement w and rotation ϕ.

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As in Section 3.1.1, (3.10) could alternatively be obtained by first introducing the Taylor series with terms up to the first derivative into (3.8)1 , to first get U=

1 2

d

2 2 G A w , x + E I ϕ, x dx

T=

0

1 2

d

˙ 2 dx , A˙ρ ( w) ˙ 2 + Iρ ( ϕ)

0

(3.12) and then, by employing the Hamilton principle. A question arises here: Can other, more complex (micro)structures, especially those made of little beams connected by rigid joints, of a general beamlike geometry—such as shown in Figure 3.3—be sufficiently well described by this beam model? The general answer is: No (Noor and Nemeth, 1980). The basic procedure, however, recommended by those authors is basically the same as that outlined here: y

x

h L

L

L

(a) Vierendeel girder. y

x (b) Pratt girder. y

x (c) X-braced girder. y

x

h L

L

(d) Warren girder. FIGURE 3.3 Planar lattices and their repeating elements. (From Noor and Nemeth, 1980. With permission.)

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1. An equivalent micropolar beam model is set up from the postulate of the same strain and kinetic energies stored in the original lattice when both are deformed identically. 2. A typical repeating element is identified and the energies for this element are expressed in terms of the nodal displacements, joint rotations, as well as the geometric and material properties of the individual members; 3. A passage to an effective continuum is carried out via a Taylor expansion, whereby it turns out that higher-order terms show up in the governing continuum equations, depending on the actual microgeometry of the rods making up the structure; see also Triantafyllidis and Bardenhagen (1993). It is appropriate to note here that beam-like lattices can also be analyzed by a cell transfer matrix approach—the eigenvalues of this matrix are the decay rates relevant in Saint-Venant’s principle for these discrete, rather than continuum, systems (Stephen and Wang, 1996). The associated eigenvectors and principal vectors lead then to equivalent continuum-beam properties. Finally, we note that continuum approximations of plate-like structures were also investigated (Noor, 1988). In that review, among the problems requiring new investigations was listed the effect of microstructural material randomness—we address this topic, in the context of beam vibrations, in Chapter 11. That chapter also outlines the transfer matrix approach in dynamic problems of periodic, possibly disordered, structures.

3.2

Planar Lattices: Classical Continua

3.2.1 Basic Idea of a Spring Network Representation As already demonstrated in the setting of 1D models, the basic idea in setting up 2D and 3D spring network models is based on the equivalence of energies stored in the unit cell (Figure 3.4) of a given network. In the case of static problems, to which we will restrict the discussion henceforth, for a cell of volume V we therefore have Ucell = Ucontinuum .

(3.13)

The unit cell is a periodically repeating part of the network. Two aspects should be noted here: (1) the choice of the unit cell may be nonunique, see Figure 3.4(a); and (2) the inner structure of the unit cell is not necessarily “nicely” ordered—it may be of a disordered microgeometry, with an understanding that it repeats itself in space such as the periodic Poisson–Delaunay network (Chapter 4).

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(a)

x2

α2 α3

α4

x1

α1

(b)

(c) FIGURE 3.4 Three periodic planar lattices: honeycomb, square, and triangular. In each case a possible (not necessarily unique) periodic unit cell is shown.

The energies of the cell and its equivalent continuum, respectively, are Ucell =

b

b 1 Eb = (F · u) (b) 2 b

N

Ucontinuum

1 = 2

σ : εd V.

(3.14)

V

The superscript b in (3.14)1 stands for the bth spring (bond), and Nb for the total number of bonds. Our discussion is set in the 2D setting so that, by a volume we actually mean an area of unit thickness. In the sequel we restrict ourselves to linear elastic springs and spatially linear displacement fields u

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(i.e., uniform strain fields ε), and (3.14) will become |b|

Ucell =

1 (ku · u) (b) 2 b

Ucontinuum =

V ε : C : ε. 2

(3.15)

In (3.15) u is a generalized spring displacement and k its corresponding spring constant. The next step, depending on the particular geometry of the unit element and on the particular model of interactions, will involve making a connection between u and ε, and then deriving C from (3.13). The corresponding procedures and resulting formulas are given below for several elasticity problems set in the square and triangular network geometries. 3.2.2 Antiplane Elasticity on Square Lattice Of all the elasticity problems, the antiplane is the simplest one on which to illustrate the spring network idea. In the continuum setting we thus have the constitutive law σi = Cij ε j

i, j = 1, 2,

(3.16)

where σ = (σ1 , σ2 ) ≡ (σ31 , σ32 ), ε = (ε1 , ε2 ) ≡ (ε31 , ε32 ) and Cij ≡ C3i3 j . Upon the substitution of (3.16) into the equilibrium equation σi,i = 0,

(3.17)

Cij u, j ,i = 0.

(3.18)

we obtain

Henceforth, we are interested in approximations of locally homogeneous media, so that this governing equation becomes Cij u,i j = 0.

(3.19)

In the special case of an isotropic medium, (3.19) simplifies to the Laplace equation Cu,ii = 0.

(3.20)

We now discretize the material with a square lattice network, Figure 3.4(b), whereby each node has one degree of freedom (antiplane displacement u), and the nearest neighbor nodes are connected by springs of constant k. It follows that the strain energy of the unit cell of such a lattice is 1 (b) (b) k l l εi ε j . 2 b=1 i j 4

U=

(3.21)

In the above we employed the uniform strain ε = (ε1 , ε2 ) . Also, l(b) = is the vector of half-length of bond b. In view of (3.13), the stiffness

(l1(b) , l2(b) )

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tensor is obtained as Cij =

4 k (b) (b) l l V b=1 i j

i, j = 1, 2 ,

(3.22)

where V = 4 if all the bonds are of unit length (|l(b) | = 1). This leads to a relation between the bond spring constant k and the Cij tensor C11 = C22 =

k 2

C12 = C21 = 0.

(3.23)

In order to model an orthotropic medium, different bonds are applied in the x1 and x2 directions: k (1) and k (2) . The strain energy of the unit cell is now 1 (b) (b) (b) k li l j εi ε j , U= 2 b=1 4

(3.24)

so that the stiffness tensor is Cij =

4 1 (b) (b) (b) k li l j , V b=1

(3.25)

which leads to relations C11 =

k (1) 2

C22 =

k (2) 2

C12 = C21 = 0.

(3.26)

If one wants to model an anisotropic medium (i.e., with C12 = 0), one may either choose to rotate its principal axes to coincide with those of the square lattice and use the network model just described, or introduce diagonal bonds. In the latter case, the unit cell energy is given by the formula (3.24) with Nb = 8. The expressions for Cij s are C11 =

k (1) + k (5) 2

C22 =

k (2) + k (6) 2

C12 = C21 = k (5) − k (6) .

(3.27)

It will become clear in the next section how this model can be modified to a triangular spring network geometry. 3.2.3 In-Plane Elasticity: Triangular Lattice with Central Interactions In the planar continuum setting (see Chapter 5), Hooke’s law σij = Cijkm εkm

i, j, k, m = 1, 2,

(3.28)

upon substitution into the balance law σij,j = 0,

(3.29)

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results in a planar Navier equation for the displacement ui μui, j j + κu j, ji = 0.

(3.30)

In (3.30) μ is defined by σ12 = με12 , which makes it the same as the classical 3D shear modulus. On the other hand, κ is the (planar) 2D bulk modulus, that is defined by σii = κεii . As in the foregoing section, we are interested in approximations of locally homogeneous media. Consider the regular triangular network of Figure 3.4(c) with central force interactions, which are described, for each bond b, by Fi = ij(b) u j

where

ij(b) u j = α (b) ni(b) n(b) j .

(3.31)

Similar to the case of antiplane elasticity, α (b) is the spring constant of halflengths of such central (normal) interactions—that is, of those parts of the springs that fall within the given unit cell, Figure 3.5(a). The unit vectors n(b) at respective angles of the first three α-springs are θ (1) = 00

n(1) 1 =1

θ (2) = 600

n(2) 1 =

θ (3) = 1200

n(3) 1 =−

n(1) 2 =0 √

1 2

3 2 √ 3 . = 2

n(2) 2 = 1 2

n(3) 2

(3.32)

The other three springs (b = 4, 5, 6) must, by the requirement of symmetry with respect to the center of the unit cell, have the same properties as b = 1, 2, 3, respectively. All the α-springs are of length √ l, that is, the spacing of the triangular mesh is s = 2l. The cell area is V = 2 3l 2 . Every node has two degrees of freedom, and it follows that the strain energy of a unit hexagonal cell of this lattice, under conditions of uniform x2 α(3) 3

β(2)

α(2) 2

β(3)

β(1)

β(4)

β(6)

α(4)4

θ(b + 1) 1 α(1)

5

β(5)

α(5)

6

b

b+1

φ θ(b)

α(6) (a)

x1

(b)

FIGURE 3.5 Unit cell of a triangular lattice model; α (1) , . . ., α (6) are the normal spring constants; β (1) , . . ., β (6) are the angular spring constants; in the isotropic Kirkwood model α (b) = α (b+3) and β (b) = β (b+3) , b = 1, 2, 3. (b) Details of the angular spring model.

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strain ε = (ε11 , ε22 , ε12 ), is U=

6 l 2 (b) (b) (b) (b) (b) α ni n j nk nm εij εkm , 2 b=1

(3.33)

so that, again by (3.13), the stiffness tensor becomes Cijkm =

6 l 2 (b) (b) (b) (b) (b) α ni n j nk nm . V b=1

(3.34)

In particular, taking all α (b) the same, we see that 9 C1111 = C2222 = √ α 8 3

3 C1122 = C2211 = √ α 8 3

3 C1212 = √ α, (3.35) 8 3

so that there is only one independent elastic modulus, and the modeled continuum is isotropic. It is important to note here that the isotropy follows from the triangular lattice having an axis of symmetry of the sixth order. This, combined with the fact that (3.34) satisfies the conditions of Cauchy symmetry (Love, 1934) with respect to the permutations of all the four indices (which is the last equality here) Cijkm = Ci jmk = C jikm = Ckmi j = Cik jm , implies that Cijkm is of the form Cijkm = λ δij δkm + δik δ jm + δim δ jk .

(3.36)

(3.37)

In view of (3.35), we obtain the classical Lam´e constants 3 λ = μ = √ α. 8 3

(3.38)

The above is a paradigm from the crystal lattice theory that the Cauchy symmetry occurs when: 1. The interaction forces between the atoms (or molecules) of a crystal are of a central force type. 2. Each atom (or molecule) is a center of symmetry. 3. The interaction potential in a crystal can be approximated by a harmonic one. Note: The Cauchy symmetry reduces the number of independent constants in general 3D anisotropy from 21 to 15. The first case has been called the multi-constant theory, and the second one the rari-constant theory. Basically, there is a decomposition of the stiffness tensor into two irreducible parts with 15 and 6 independent components, respectively; see Hehl and Itin (2002) for a group-theoretical study of these issues.

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Note: One might try to model anisotropy by considering three different α’s in (3.33 and 3.34), but such an approach would be limited given the fact that only three of those can be varied: one needs to have six parameters in order to freely adjust any planar anisotropy that involves six independent Cijkm values. This can be achieved by introducing the additional angular springs, as discussed below. In fact, angular springs are also the device to vary the Poisson ratio. 3.2.4 In-Plane Elasticity: Triangular Lattice with Central and Angular Interactions We continue with the triangular network, and introduce angular springs acting between the contiguous bonds incident onto the same node. These are assigned spring constants β (b) , and, again by the argument of symmetry with respect to the center of the unit cell, only three of those can be independent. Thus, we arrive at six spring constants: {α (1) , α (2) , α (3) , β (1) , β (2) , β (3) }. With reference to Figure 3.5(b), let θ (b) be the (infinitesimal) angle change of the bth spring orientation from the undeformed position. Noting that n × n = lθ, we obtain θ (b) = e ki j ε j p ni n p ,

(3.39)

where e ki j is the Levi–Civita permutation tensor. The angle change between two contiguous α springs (b and b + 1) is measured by φ = θ (b+1) − θ (b) , so that the energy stored in the spring β (b) is E (b) =

2 1 (b) 1 (b) (b) β |φ|2 = β (b) εki j ε j p ni(b+1) n(b+1) − n n . i p p 2 2

(3.40)

By superposing the energies of all the angular bonds with the energy (3.33), the elastic moduli are derived as (Grah et al., 1996): Cijkm =

6 6 l 2 (b) (b) (b) (b) (b) 1 (b) (b) (b) (b) β + β (b−1) δik n(b) α ni n j nk nm + p n j n p nm V b=1 V b=1

(b) (b) (b) (b) (b+1) (b+1) (b) − β (b) + β (b−1) ni(b) n(b) n p nm j nk nm − β δik n p n j (b) (b+1) (b+1) (b) + β (b) ni(b) n(b+1) n(b+1) nm − β (b) δik n(b) nm j k p n j np (b) (b+1) , + β (b) ni(b+1) n(b) j nk nm

(3.41)

where b = 0 is the same as b = 6. This provides the basis for a spring network representation of an anisotropic material; it also forms a generalization of the so-called Kirkwood model (Kirkwood, 1939) of an isotropic material. The latter is obtained by assigning

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the same α to all the normal and the same β to all the angular springs: Cijkm

6 6 α (b) (b) (b) (b) β (b) = √ ni n j nk nm + √ {2δik n(b) j nm 2 2 3 b=1 2 3l b=1 (b) (b) (b) (b+1) (b+1) (b) (b) − 2ni(b) n(b) n p nm + ni(b) n(b+1) n(b+1) nm j nk nm − δik n p n j j k (b) (b+1) (b+1) (b) (b+1) − δik n(b) nm + ni(b+1) n(b) }. j nk nm p n j np

(3.42)

In accordance with the above, C1111 C1122 C1212

19 2 9 α+ 2 β = C2222 4 l 4 19 1 3 α− 2 β = C2211 = √ l 4 2 3 4 19 1 3 α+ 2 β . = √ l 4 2 3 4 1 = √ 2 3

(3.43)

Condition C1212 = (C1111 − C1122 )/2 is satisfied, so that there are only two independent elastic moduli. From equation (3.43), the α and β constants are related to the planar bulk and shear moduli by 1 κ= √ 2 3

3 α 2

1 μ= √ 2 3

19 3 α− 2 β . 4 l 4

(3.44)

It is noted here that the angular springs have no effect on κ, that is, the presence of angular springs does not affect the dilatational response. The formula for planar Poisson’s ratio (Chapter 5) gives ν=

κ −μ C1111 − 2C1212 1 − 3β/l 2 α = . = κ +μ C1111 3 + 3β/l 2 α

(3.45)

From (3.45) there follows the full range of Poisson’s ratio that can be covered with this model. It has two limiting cases ν=

1 3

ν = −1

if if

β → 0, α β → ∞, α

α − model (3.46) β − model.

For the subrange of Poisson’s ratio between −1/3 and 1/3, one may also use a Keating model (1966), which employs a different calculation of the energy stored in angular bonds.

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A

C

B γ

B

C

β α

A y

A

B (a)

x

C (b)

FIGURE 3.6 (a) A triple honeycomb lattice made of three different spring types α, β, and γ belonging, respectively; to three sublattices A, B, and C; (b) a 42 × 42 unit cell of a triangular lattice of hexagonal pixels, with 11 pixel diameter circular inclusions centered on pixels and randomly placed with periodic boundary conditions. (From Snyder et al. (1992). With permission.)

3.2.5 Triple Honeycomb Lattice It is recalled from Section 3.2.4 that 1/3 is the highest Poisson’s ratio of centralforce triangular lattices with one spring constant. An interesting model permitting higher values, from 1/3 up to 1, was introduced by Day et al. (1992) and Snyder et al. (1992). The model sets up three honeycomb lattices, having spring constants α, β, and γ , respectively, overlapping in such a way that they form a single triangular lattice, Figure 3.6. The planar bulk and shear moduli of a single phase are 1 κ = √ (α + β + γ ) 12

μ=

27 16

1 1 1 + + α β γ

−1

.

(3.47)

In the case of two (or more) phases, a spring that crosses a boundary between any two phases 1 and 2 is assigned a spring constant according to a series rule α = [(2α1 ) −1 + (2α2 ) −1 ], where αi , i = 1, 2, (i.e., α, β, or γ ), is a spring constant of the respective phase. Note: Although this chapter is focused on planar lattice models of elastic solids, there also exist extensions of the lattice approach to 3D and inelastic materials, e.g., Buxton et al. (2001).

3.3

Applications in Mechanics of Composites

3.3.1 Representation by a Fine Mesh As discussed in Section 3.2.2, one may employ the square mesh of Figure 3.4(b) in the x1 , x2 -plane for discretization of an antiplane elasticity problem. Indeed, this approach may be applied to model multiphase composites treated as

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Rigid disks Stiﬀ disks

1.0

Stiﬀ ellipses

Stiﬀ needles

Soft ellipses

Soft needles

Homogeneous medium Soft disks

0.0

Holes

Cracks

(a)

(b) C (i)/C (m)

Rigid inclusions

Stiﬀ pixels 1.0

Stiﬀ ellipses

Stiﬀ needles

Soft ellipses

Soft needles

Homogeneous medium Soft pixels

0.0

Holes

Cracks

(c)

FIGURE 3.7 (a) Parameter plane: aspect ratio of inclusions and the contrast; (b) spring network as a basis for resolution of round disks, ellipses, pixels, and needles in the parameter plane; (c) another interpretation of the parameter plane: from pixels to needles.

planar, piecewise-constant continua, providing a lattice or mesh (very) much finer than a single inclusion is involved, Figure 3.7(b). How much finer it should actually be needs to be assessed on a reference problem. The governing equations for the displacement field u ≡ u3 are u (i, j) [kr + kl + ku + kd ] − u (i + 1, j) kr − u(i − 1, j)kl −u(i, j + 1)ku − u(i, j − 1)kd = f (i, j),

(3.48)

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where f (i, j) is the body force (or source) at node (i, j), while i and j are the coordinates of mesh points, and kr (right), kl (left), ku (up), and kd (down) are defined from the series spring model kr =

1 1 + C(i, j) C(i + 1, j)

−1

−1 1 1 + C(i, j) C(i − 1, j) −1 1 1 ku = + C(i, j) C(i, j + 1) −1 1 1 + kd = . C(i, j) C(i, j − 1) kl =

(3.49)

In (3.49), C(i, j) is the material property at node (i, j). This type of a discretization is equivalent to a finite difference method that would be derived by considering the expansions 2 2 ∂u(i, j) ∂ u(i, j) s + u (i ± 1, j) = u (i. j) ± s ∂ x1 i, j 2! ∂x12 2 2 ∂u(i, j) ∂ u(i, j) s + u (i, j ± 1) = u (i. j) ± s ∂ x2 i, j 2! ∂ x22

i, j

(3.50)

i, j

in the governing equation (recall [3.20]) ∂2 u ∂2 u C + 2 = 0. ∂x12 ∂ x2

(3.51)

However, in the case of in-plane elasticity problems, the spring network approach is not identical to the finite difference method, because the node– node connections of a spring network do really have a meaning of springs, whereas the finite difference connections do not. In the case of a composite made of two locally isotropic phases: matrix (m) and inclusions (i), antiplane Hooke’s law is σi = Cij ε j

i, j = 1, 2 Cij = C (m) δij

or C (i) δij .

(3.52)

The above leads to a so-called contrast (or mismatch) C (i) /C (m) . It is clear that by increasing the contrast we can approximately model materials with rigid inclusions. Similarly, by decreasing the contrast, we go to very soft inclusions and nearly reach a system with holes. Although the disk is the most basic inclusion shape when dealing with composites, a departure from this is of interest. Thus, another basic parameter specifying the composite is the aspect ratio of ellipses a /b, where a respectively,

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(b) is the ellipse’s major (minor) semi-axis. By varying the aspect ratio from 1 up through higher values we can model systems having disk-type, ellipsetype, through needle-type inclusions. We are thus led to the concept of a parameter plane shown in Figure 3.7(a). Resolution of several different types of inclusions by the spring network is shown in Figure 3.7(b). That is, we can model disks, ellipses, needles, etc. Admittedly, this type of modeling is approximate so that a somewhat different interpretation of a parameter plane is given in Figure 3.7(c). It is seen that disks may most simply be modeled as single pixels or more accurately as finite regions; in the latter case arbitrary anisotropies can be modeled. The former case allows one to deal with very large scale systems, while the latter allows a much better resolution of local stress/strain fields within and around inclusions. By decreasing the spring network mesh size, an increasingly better accuracy can be achieved. Depending on the shape functions employed in finite element models, somewhat more accurate results may be obtained, but this comes at a higher price of costly and cumbersome remeshing for each and every new configuration B (ω) from the ensemble B, which is required in statistical (Monte Carlo) studies. It is noteworthy that, in contradistinction to the finite element method, no need for remeshing and constructing of a stiffness matrix exists in our spring network method: spring constants are very easily assigned throughout the mesh, and the conjugate gradient method finds the solution of the equilibrium displacement field u (i, j). In that manner, a system having 106 million degrees of freedom (1000 × 1000 nodes) can readily be handled on a computer with 90 MB of random access memory. For 2000 × 2000 nodes one requires some 360 MB, and so on, because of the linear scaling of memory requirements with the number of degrees of freedom. The quality of approximation of ellipses and needle-type cracks/inclusions can be varied according to the number of nodes chosen to represent such objects. Local fields cannot be perfectly resolved, but the solution by the spring network is sufficient to rapidly establish the elastic moduli of a number of different B (ω) realizations from the random medium B, and the corresponding statistics with a sufficient accuracy. Spring networks are used in later chapters to study scaling laws of various planar composites. Note: Interestingly, the computational method for determining effective moduli of composite materials with circular inclusions due to Bird and Steele (1992) would be very well suited for analysis of this type of stationarity and isotropy.

3.3.2 Solutions of Linear Algebraic Problems The steady-state conductivity and elastostatics problems on spring networks always lead to linear algebraic systems A · x = b,

(3.53)

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because they simply are elliptic problems in discretized forms. There are, in principle, two methods to set up and solve the governing equations. One of them is exactly the same as that conventionally used in the finite element methods—involving the global stiffness matrix accompanied by the connectivity of all the nodes—and will therefore not be elaborated here. The second one comes, just like the spring networks themselves, from the condensed matter physics. It is the so-called conjugate gradient method, which involves the energy of the system as a functional F(x) =

1 x·A·x−b·x 2

(3.54)

of all the relevant degrees of freedom x, and the gradient of this energy ∇ F(x) = A · x − b

(3.55)

with respect to all these degrees. Once written in an explicit form as two subroutines, the program is connected with any of the widely available solvers (e.g., Press et al., 1992). Note that F(x) is minimized when equation (3.55) equals zero, which is then equivalent to (3.53). Of course, one may also employ other algebraic solvers. It is noteworthy that the entire task of mesh generation—such as typically required by the finite element methods—is absent. The energy and energy gradient subroutines are written once and for all for the given mesh of, say, Figure 3.8. The assignment of all the local spring stiffnesses—according to

FIGURE 3.8 A functionally graded matrix-inclusion composite with 47.2% volume fraction of black phase is partitioned into subdomains, corresponding to a 64-processor parallel computer.

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any chosen lattice model of Section 3.2—is done very rapidly in the first stage of the program. These stiffnesses are stored in the common block (in case of a Fortran program) and are readily accessible to the conjugate gradient subroutines that are activated in the second, and main, stage of the program. Once the energy minimum is reached to within any specified accuracy, this energy is used to compute the overall, effective moduli of a given domain of the lattice based on the postulate of the energy equivalence. Here we list several exact relations that may be used in testing the resulting computer programs, homogenization methods, or effective medium theories; see also Garboczi (1998). 1. Suppose we have solutions of two elasticity problems on a certain domain B, with boundary ∂ B, corresponding to the displacement (d) and traction (t) boundary value problems, respectively. Then we can check whether the Betti–Maxwell reciprocity theorem (t) (d) ui ti ds = ui(d) ti(t) ds (3.56) ∂B

∂B

is satisfied numerically within some acceptable accuracy. 2. Perfect series and parallel systems are well known to result in the arithmetic and harmonic averages, or the so-called Voigt (C V ) and Reuss (C R ) bounds f1 f 2 −1 V R + , (3.57) C = f 1 C1 + f 2 C2 C = C1 C2 where f 1 and f 2 are the volume fractions of phases and 1 and 2, respectively. 3. The case of small contrast in properties allows an expansion of, say, effective conductivity to second order in the difference (C1 − C2 ) as follows (Torquato, 1997): C eff = C1 + f 2 (C2 − C1 ) − f 1 f 2

(C2 − C1 ) 1 + O(C2 − C1 ) 3 + · · ·, C1 D (3.58)

where D is the dimensionality of the space. 4. There are many exact relations in the 2D conductivity (Milton, 2002). Perhaps the most well-known one, due to Keller (1964) (also Mendelson, 1975) says that, for a two-phase isotropic system in 2D, C eff (C1 , C2 )C eff (C2 , C1 ) = C1 C2 ,

(3.59)

where C eff (C1 , C2 ) is the effective conductivity of a given system, while C eff (C2 , C1 ) is the effective conductivity with the phases 1 and 2 interchanged.

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5. The so-called CLM result and stress-invariance discussed in Chapters 5 and 6 may be employed for planar elastic (classical as well as micropolar) composites with either twice-differentiable or piecewise-constant properties. In essence, the effective compliance tensor of the transformed material is given by that of the original material plus the same transformation (shift) in bulk and shear compliances as that for the individual phases; see Section 5.2.4 in Chapter 5. When dealing with very large systems, the spring network method is limited by the available computer memory size. This, for example, is the case with a functionally graded material. A composite of that type is shown in Figure 3.8, where the disk–matrix interphase is taken as a finite thickness zone of two randomly mixed phases of the disk (2) and the matrix (1) material. Both phases are locally homogeneous and isotropic—they are described by two constant isotropic conductivities C1 and C2 . We see here three different length-scales: the fine structure of the interphase region, the size and spacing of inclusions, and the macroscopic dimension of the composite. For this type of problem we can also use parallel computing. Thus, in Figure 3.8 we show a partition of the entire simulated domain of a functionally graded composite into 64 = 8 × 8 subdomains, each of which represents a spring network that is assigned to a separate processor of a parallel computer. Thus, the boundary value is solved by using 64 processors operating in tandem. The computational effort is limited by the speed of a single processor (which goes down with the subdomain size) and the communication between the processors (which simultaneously goes up); the latter leads to so-called bottlenecks. Determining an optimal partition is therefore an important task and remains a major challenge today as we move to computers having thousands and tens of thousands of processors. There are, in principle, two ways to execute such a parallel scheme: either to write one’s own software, or to adapt an existing solver running on a given parallel computer.

3.3.3 Example: Simulation of a Polycrystal The generalization of the Kirkwood spring network model outlined in Section 3.2.4 to an anisotropic case was motivated by a need to study micromechanics of a planar polycrystalline aluminum specimen (Grah et al., 1996). The basic strategy is as follows. First, an image of crystal domains (i.e., grains), such as the one showed in Figure 3.9, needs to be scanned and mapped onto a triangular mesh. Next, every bond is assigned its stiffness depending on the domain it falls in. And finally, the mechanics problem of the resulting spring network is solved computationally. In order to assign spring stiffnesses to any node of the spring network mesh, the 3D stiffness tensor Ci jkm for each crystal must be found according

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FIGURE 3.9 Scanned image of a very thin polycrystal aluminum sheet. All the grain boundaries are orthogonal to the plane of the sheet. (From Grah et al., 1996. With permission.)

to its transformation (rotation) matrix a ij (i, j = 1, 2, 3); the latter is provided from the Kikuchi surface electron backscattering technique. Thus, to set up the spring network model, we start from the stiffness matrix of an (anisotropic) aluminum crystal which is given as

Cαβ

⎡ 10, 82 6, 13 ⎢ 6, 13 10, 82 ⎢ ⎢ ⎢ 6, 13 6, 13 =⎢ ⎢ 0 0 ⎢ ⎢ ⎣ 0 0 0 0

6, 13 6, 13 10, 82 0 0 0

⎤ 0 0 0 0 0 0 ⎥ ⎥ ⎥ 0 0 0 ⎥ ⎥ 104 MPa . 2, 85 0 0 ⎥ ⎥ ⎥ 0 2, 85 0 ⎦ 0 0 2, 85

(3.60)

Its corresponding fourth-rank stiffness tensor Cijkm is then set up taking into account three symmetries Cijkm = Ci jmk = C jikm = Ckmi j . We next use a transformation formula for a 4th-rank tensor

Cnpqr = a ni a j p a kp a mr Cijkm

n, p, q , r = 1, 2, 3

(3.61)

to set up the in-plane (2D) part of Cnpqr at every mesh node. This in-plane

part, consisting of C1111 , C2222 , C1112 , C2212 , C1212 is then mapped into the six

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spring constants α1 , α2 , α3 , β1 , β2 , β3 according to ⎡

C1111

⎤

⎡

⎢ ⎥ ⎢ ⎢ C2222 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ C1122 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥=⎢ ⎢ C1112 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢C ⎥ ⎢ ⎣ 1212 ⎦ ⎢ ⎣

C2212

√1 3

1√ 16 3

1√ 16 3

0

9√ 16 3 √ 3 16 1 16 √ 3 16 3 16

9√ 16 3 √ 3 16 1 − 16 √ 3 16 3 − 16

0 0 0 0

√

3 4 √ 3 4

− 4√9 3 3 4 9 √

4 3 − 34

√ √ √

√

3 3

3 2

0 0 0

⎤

⎡ ⎤ 3 α1 4 √ ⎥⎢ ⎥ 3 ⎥⎢α ⎥ 4 ⎥⎢ 2⎥ √ ⎥⎢ ⎥ α3 ⎥ ⎥ − 43 ⎥ ⎢ ⎥. ⎥⎢ ⎢ ⎥ 3 ⎥ ⎢ β1 ⎥ −4 ⎥ ⎢ ⎥ ⎥ 9 ⎥ ⎢ β2 ⎥ √ ⎣ ⎦ 4 3 ⎦ 3 β3 4

(3.62)

While there is no ambiguity concerning the spring constant of any a-bond that entirely belonged to any given crystal domain, special care has to be taken of the bonds that straddle the boundaries of contiguous crystals. The effective stiffnesses are assigned according to a series rule: a = ( 12 α1 + 12 α2 ) −1 . Assignment of the b-springs presents no such ambiguities. A general finite element mapping procedure for defining spring network representations in 2D and 3D for isotropic and anisotropic solids has recently been developed by Gusev (2004).

3.4

Planar Lattices: Nonclassical Continua

3.4.1 Triangular Lattice of Bernoulli–Euler Beams In the solid state physics literature the Kirkwood and Keating models are sometimes referred to as the beam-bending models. This is a misnomer because there is no account taken in these models of the actual presence of moments and curvature change of spring bonds connecting the neighboring nodes. True beam bending was fully and rigorously considered by Wo´zniak (1970) and his coworkers, and, considering a limited access to that book, in this section we give a very brief account of the triangular lattice case. For more modern developments see Trovalusci and Masiani (1999). We focus on the deformations of a typical beam, its bending into a curved arch allowing the definition of its curvature, and a cut in a free body diagram specifying the normal force F , the shear force F˜ , and the bending moment M, see Figure 3.10. It follows that, in 2D, the force field within the beam network is described by fields of force-stresses σkl and moment-stresses mk , so that we have a micropolar medium, see also Chapter 6. The kinematics of the network of beams is now described by three functions: u1 (x)

u2 (x)

ϕ(x),

(3.63)

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~ F (f)

F (f) M (f) n( f ) ~

ϕ (a)

(b)

FIGURE 3.10 The lattice geometry (a); curvature and internal loads in a single beam element (b).

which coincide with the actual displacements (u1 , u2 ) and rotations ϕ at the network nodes. Within each triangular pore, these functions may be assumed to be linear, and hence, the local strain γkl , and curvature, κi , fields are related to u1 , u2 , and ϕ by γkl = ul,k + e lk ϕ

κi = ϕ,i ,

(3.64)

where e lk is the Ricci symbol. It follows from geometric considerations that γ (b) ≡ nk(b) nl(b) γkl

(3.65)

is the average axial strain, with s (b) γ (b) being its average axial length change. Similarly, γ˜ (b) ≡ nk(b) n˜ l(b) γkl = nk(b) n˜ l(b) u(l, k) − ϕ

(3.66)

is the difference between the rotation angle of the beam chord and the rotation angle of its end node. Thus, the difference between the rotation angles of its ends is κ (b) ≡ nk(b) κk .

(3.67)

The elementary beam theory suggests that the force-displacement and moment-rotation response laws of each bond (Figure 3.10) are given as 12E (b) I (b) (b) γ˜ M(b) = E (b) I (b) κ (b) , (3.68) s2 1 where A(b) = wh is the beam cross-sectional area, I (b) = 12 w 3 h is its centroidal moment of inertia with respect to an axis normal to the plane of the network, F (b) = E (b) A(h) γ (b)

F˜ (b) =

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and E (b) is the Young modulus of the beam’s material. All the beams are of length s ≡ s (b) , which is the spacing of the mesh. Turning now to the continuum picture, the strain energy of the micropolar continuum is expressed as (note Chapter 6): Ucontinuum =

V V (1) γij Cijkm γkm + κi Cij(2) κ j , 2 2

(3.69)

(2) Here Cij(2) stands for the tensor Ci3 j3 . From the above we find

(1) Cijkm =

6

(b) (b) ˜ (b) (b) (b) ni(b) nk(b) n(b) n R + n n R j j m m

Cij(2) =

b=1

6

(b) ni(b) n(b) j S ,

b=1

(3.70) where R(b) =

2E (b) A(b) √ s (b) 3

24E (b) I (b) ˜ (b) = √ R 3 s (b) 3

S˜ (b) =

2E (b) I (b) √ . s (b) 3

If we assume all the beams to be the same (R(b) = R, etc.), we obtain (1) (1) (1) (1) ˜ ˜ C1111 = C2222 = 38 3R + R = C2121 = 38 R + 3 R C1212 (1) (1) (1) (1) ˜ ˜ C1122 = C2211 = 38 R − R = C2112 = 38 R − R C1221

(3.71)

(3.72)

(2) (2) ˜ = C22 = 32 S, C11

with all the other components of the stiffness tensors being zero. That is, we have (1) Cijkm = δij δkm + δik δ jm + δim δ jk Cij(2) = δij ,

(3.73)

in which ==

3 ˜ R− R 8

=

3 ˜ R + 3R 8

=

3 ˜ S. 2

(3.74)

We note from (3.58)1 that this beam lattice is an isotropic continuum. The micropolar model (refer to Chapter 6) is conveniently expressed in terms of four compliances A=

1 + + 2

S=

2 +

P=

2 −

M=

2 . 3

(3.75)

The effective bulk and shear moduli are now identified on the basis of (3.73 and 3.74) as κ=

3 3 ˜ , R μ= R+ R 4 8

(3.76)

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which are seen to reduce to the formulas of Section 3.2.3 in the special case of flexural rigidity being absent. Furthermore, the effective Young modulus and Poisson ratio are E = 3R

˜ I + R/R ˜ 3 + R/R

ν=

˜ 1 − R/R . ˜ 3 + R/R

(3.77)

It is noteworthy that the introduction of beam-type effects has a similar influence on E and ν as the introduction of the angular β-interactions in the ˜ Kirkwood model. However, noting that R/R = (w/s) 2 , where w is the beam width, we see that, in view of the slenderness assumption of the beam elements (“neither too thin, nor too thick”), this model admits ν between ∼ 1/3 and ∼ 0.2. With reference to Section 6.1 of Chapter 6, interpreting this lattice in the plane stress formulation does not change ν, whereas adopting the plane strain formulation results in ν between ∼ 0.25 and ∼ 0.16. Finally, the stiffness tensors (3.58) can be inverted to get the compliance tensors (1) Sijkm =

1

δij δkm ( A − S) + δik δ jm (S + P) + δim δ jk (S − P) 4

Sij(2) =

δij , (3.78)

so that, given the definition of the micropolar characteristic length (Chapter 6), l=

s S+ P = 4M 4

1 + 3 (w/s) 2 , 1 + (w/s) 2

(3.79)

where we used the basic facts relating the beam’s cross-sectional area to its moment of inertia. 3.4.2 Triangular Lattice of Timoshenko Beams In the foregoing section we began with the model of Bernoulli–Euler beams, which implies slender connections between the lattice nodes. It is well known that the situation of stubby connections is describable more adequately by Timoshenko beams. The boundary value problem that needs to be solved is that of a beam fixed at both ends and subjected to a shear displacement at one end. That is, with the boundary conditions at the beam’s left end where v is displacement and θ rotation). ν(0) = 0 θ(0) = 0

ν (0) − θ(0) = 0,

(3.80)

and right end ν(s) = γ˜ (b) s

θ(s) = 0

ν (s) − θ(s) = 0,

(3.81)

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it is readily determined from the beam’s governing equations E I θ

+ G A(ν − θ) = 0

G A(ν

− θ) = 0,

(3.82)

that a following relation holds between the shear force F˜ (b) and the displacement s γ˜ (b) F˜ (b) =

12E (b) I (b) (b) s γ˜ . s 3 (1 + β)

(3.83)

Here β=

12E I (b) E w 2 = (b) 2 GA s G s

(3.84)

is the dimensionless ratio of bending to shear stiffness, with A(b) = ta being the beam’s cross-sectional area, and I (b) = ta w 3 /12 its centroidal moment of inertia. Two limiting cases are noteworthy: β → 0, high shear stiffness and, hence, less deflection owing to shear; the Bernoulli–Euler slender beam is recovered. β > 1, low shear stiffness and, hence, deflection owing to shear dominates over that due to the Young modulus E; this is the general case of the Timoshenko beam. Observing that (3.83) replaces (3.68) 2 , we now proceed to derive the effective moduli so that (3.71) is replaced by R(b) =

2E (b) A(b) √ s (b) 3

24E (b) I (b) 1 ˜ (b) = √ R 3 s (b) 31+β

S(b) =

2E (b) I (b) √ , s (b) 3

(3.85)

whereby we note

w 2 1 ˜ R = . R s 1+β

(3.86)

Following the same steps as in the previous section, we see that the effective continuum moduli are given by (3.76 and 3.77) as before. It is now possible to express them in terms of the beam aspect ratio and β. Thus, for E eff (normalized by the beam’s modulus E (b) ) and ν eff we find √ w 1 + (w/s) 2 / (1 + β) E eff = 2 3 , ta E (b) s 3 + (w/s) 2 / (1 + β) ν

eff

1 − (w/s) 2 / (1 + β) = . 3 + (w/s) 2 / (1 + β)

(3.87)

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Considering the geometry of the hexagonal unit cell of the lattice, we can write the above in terms of the porosity p (i.e., the pore’s volume fraction): 2 √

1+ 1 1− 1− p E eff 3(1+β) = 2 1− 1− p 2 , √ 1 ta E (b) 1− 1− p 3 + 3(1+β) (3.88) 2 √ 3 − 1 − 1 − p 3/ (1 + β) eff ν = . 2 √ 9 + 1 − 1 − p 3/ (1 + β) The above are plotted in Figure 3.11(b) and (c) for the special case E (b) = 9G . We see that the consideration of lattice connections as stubby (Timoshenko) beams has a minor softening effect on E eff relative to the BernoulliEuler beam model. This may be explained by noting that the admission of the beam’s angle of rotation as an independent degree of freedom amounts to G (b) being finite, rather than infinite, in the Timoshenko beam model. The Poisson ratio falls off nonlinearly from 1/3 with p increasing in both models. Also here, the admission of finite shear modulus is weak. Also plotted in Figure 3.11(b) and (c) are the results for the central-force lattice of Section 3.2.3, the perforated plate model introduced below, and the Cox model discussed in Chapter 4. (b)

3.4.3 From Stubby Beams to a Perforated Plate Model As the porosity p goes beyond 50%, the beam’s aspect ratio w/s increases so high that one can no longer model the connections between the lattice nodes as beams. Thus, a basic question arises: can any simple explicit model be derived for this low porosity range? One avenue is offered by a perforated plate model. In the limit of p → 1, we have a plate with a regular distribution of triangular-shaped pores, Figure 3.11(a). This is a so-called dilute limit of a locally isotropic material with holes (in either periodic or disordered arrangements). Following Jasiuk et al. (1994) and Jasiuk (1995), the respective formulas are E eff = 1 − α (1 − p) ta E (b)

ν eff = ν (b) − α ν (b) − ν0 (1 − p) .

(3.89)

The coefficients α = 4.2019 and ν0 = 0.2312 have been computed in the above references, and, in fact, analogous coefficients are also available there for plates with other than triangular holes (squares, pentagons, etc.). It is noteworthy that: 1. Both formulas are uncoupled from one another. 2. (3.89)1 models the high porosity range much better than the beam lattice model; E eff is modeled by an upper envelope of all the curves in Figure 3.11(b), that is, curves 3 and 5. 3. (3.89)2 depends on the Poisson ratio ν (b) of the plate material; the latter can be specified only in the Timoshenko beam model.

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(a) 1 4 3

0.8 1

2

0.6 E eﬀ E (b) 0.4

4

0.2

0 0

0.2

0.4

0.6

0.8

1

p (b) 0.5 1,4

0.4

2

3

0.3 v eﬀ 5

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

p (c) FIGURE 3.11 (a) A decrease in pore sizes (left to right): from large holes (slender beams), through a lattice of stubby beams, to a plate perforated with small holes; shown at porosities p = 10%, 50%, and 90%. (b) Effective Young’s modulus E eff , normalized by the beam’s Young’s modulus, as a function of p for: (1) the central-force lattice, (2) the Timoshenko beams lattice, (3) the Bernoulli–Euler beams lattice, (4) the Cox model, and (5) the effective medium theory for a perforated plate. (c) Effective Poisson’s ratio ν eff as a function of p, models (1–5) shown.

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One more question remains in connection with Figure 3.11: What happens when the p values are too high for a beam lattice model to hold and too low for the dilute model to be truly dilute? Or, can anything be done to smooth out the transition between the two curves 3 and 5 at P f around 0.8? One could try here a usual device of micromechanics: an effective medium theory in any one of its guises: differential scheme, self-consistent, etc. (Jasiuk, 1995). However, for the sake of clarity of Figures 3.11(b) and (c), we do not plot these. Summing up, it is seen from Figure 3.11 that, as p grows, beam bending tends to increase the effective Young modulus E eff . In other words, bending effects increase as connections become wider. On the other hand, as they become slender, one can work with segments carrying axial forces only. Thus, beam effects gain in influence as the pore’s volume fraction increases, and lead to an increase of the effective Young modulus relative to the central-force model. Two more things may be said about the beam network model. First, Timoshenko beams, although more sophisticated than Bernoulli–Euler beams, remain, in principle, 1D objects, of micropolar type in fact. Therefore, what they yield is about as far as one gets with a beam model. A better approach would have to consider beam segments as little plates, that is, 2D objects. Finally, lattice nodes that are taken as rigid objects in this model could more realistically be modeled by considering their deformability. 3.4.4 Hexagonal Lattice of Bernoulli–Euler Beams Elasticity of a hexagonal (honeycomb) lattice of beams may be handled in a manner similar to that of the triangular lattice. However, the analysis is somewhat more complicated, due in part to the setup of a unit cell, whose choice is nonunique. Wang and Stronge (1999) analyzed a honeycomb in plane strain, which in a cross section is equivalent to a hexagonal beam lattice; see also Warren and Byskov (2002). As before, the continuum kinematics is described by three functions (3.63). The strain energy of the micropolar continuum is of the general form (3.69), and the stiffness tensors possess the isotropies (3.73). In particular, the non-zero components are √

√ (b) 3 3E (b) w 1 + 3 (w/s) 2 3E w 3 + (w/s) 2 (1) = C1212 = 2 6s 6s 3 1 + (w/s) 1 + (w/s) 2 √ (b) √ (b) 3 3E w 1 − (w/s) 2 3E w 1 − (w/s) 2 (1) (1) C1122 = C = 1221 2 6s 6s 3 1 + (w/s) 1 + (w/s) 2 √ (b) 3 3E w (2) . C11 = 36s

(1) C1111

(3.90)

The effective bulk and shear moduli are now identified as E (b) w κ= √ 2 3s

μ= √

E (b) w , 3s 1 + s 2 /w 2

(3.91)

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which then lead to 4E (b) w E=√ 3s 3 + s 2 /w 2

ν=

1 − (w/s) 2 . 1 + 3 (w/s) 2

(3.92)

Again, considering the slenderness constraints of beam elements (“neither too thin, nor too thick”), this model admits ν between (just under) 1 and ∼ 0.85. With reference to Section 6.1 of Chapter 6, interpreting this lattice in the plane stress formulation does not change ν, whereas adopting the plane strain formulation results in ν between ˜0.5 and ˜0.45. From (3.90) we determine the compliances A=

√ 2 3s E (b) w

P=

√ 2 3s 3 E (b) w 3

√

S=

3s E (b) w

1+

s2 w2

M=

√ 8 3s , E (b) w 3

(3.93)

so that the micropolar characteristic length defined in (Chapter 6) is given by l=

s S+ P = √ 4M 4 2

(w/s) 2 + 3.

(3.94)

This result compares very well with experiments of Mora and Waas (2000). 3.4.5 Square Lattice of Bernoulli–Euler Beams Following the same procedure as in Section 3.4.1, we now analyze a micropolar model of a square lattice network, Figure 3.12. Thus, assuming Bernoulli–

FIGURE 3.12 A square beam lattice.

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Euler beams, by analogy to (3.70), we have (1) Cijkm =

4

(b) (b) ˜ (b) (b) (b) ni(b) nk(b) n(b) n R + n n R j j m m

Cij(2) =

b=1

4

(b) ni(b) n(b) j S , (3.95)

b=1

where R(b) =

E (b) A(b) s (b)

˜ (b) = R

12E (b) I (b) 3 s (b)

S(b) =

E (b) I (b) . s (b)

(3.96)

When all the beams are identical, this leads to (1) (1) (1) (1) (2) (2) ˜ C11 C1111 = C2222 = R C1212 = C2121 =R = C22 = S,

(3.97)

with all the other components of the stiffness tensors being zero. Clearly, this beam lattice results in a special case of an orthotropic continuum. With reference to Chapter 6, we now have two micropolar characteristic lengths S S I s l1 = =r ≡ , (3.98) l2 = = √ ˜R R A 2 3 where the radius of gyration r is easily recognized. For beams of a rectangular √ cross-section, the second one of these becomes l2 = w/2 3. In the foregoing derivation, lattice nodes were taken as rigid objects. As Wo´zniak (1970) showed, this model may be generalized to a situation of deformable nodes, in which case we have ! 4 4 (1) (b) (b) (b) (b) (bb⊥) (b) (b) (b) (b) ˜ (b) Cijkm = ni n j nk nm R + ni n j nk nm R b=1

Cij(2) =

b⊥=1

4

(b) ni(b) n(b) j S ,

(3.99)

b=1

where (bb⊥)

R

=

d 1 − ν( I ) ν( I I )

24E (b) I (b) ˜ (b) = √ R 3 s (b) 3

E˜ ( I ) ν( I ) E˜ ( I ) ν( I I ) E˜ ( I I ) E˜ ( I I ) S(b) =

2E (b) I (b) √ . s (b) 3

!

(3.100)

Recently, an extension of such micropolar models—with an eye for wave propagation and vibration phenomena—has been carried out through the introduction of internal variables (Wo´zniak, 1997; Cielecka et al., 1998). Such

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models, in contradistinction to the more classical homogenization methods, do more correctly account for the internal microstructure. 3.4.6 Nonlocal and Gradient Elasticity on a Lattice with Central Interactions 3.4.6.1 General Considerations Let us now focus our attention on a lattice made of two central force structures, with structure I (a regular triangular network with short-range interactions) and structure II (three regular triangular networks with long-range interactions). These structures are superposed in a way shown in Figure 3.13, so that a typical node communicates with its six nearest neighbors via structure I, and with its six second neighbors via structure II. Generalizing the development of Section 3.2.3, the central-force interaction in the spring connecting the nodes r and r is related to the displacements u(r) and u r of these nodes by

Fi r, r = rr ij u j r, r ,

(3.101)

where

rr rr ij = α ri r j

u j r, r = u j r − u j (r)

ri = ri − ri . (3.102)

Similar to the case of antiplane elasticity, α rr is the spring constant of a halflength of a given node–node interaction. However, assuming the structures I and II to be made of two types of springs, we simply have two kinds of spring constants: α I and α I I ; s I = 2l is the lattice spacing of structure I, while √ II I s = s 3 is that of structure II.

Vo =

3 2 l 2 I II

FIGURE 3.13 Two structures, I and II, resulting in a lattice with local (nearest neighbor) and nonlocal (second neighbor) interactions. Note that structure II consists of three triangular networks having separate sets of nodes, and that all these nodes coincide with the nodes of structure I. (From Holnicki-Szulc and Rogula (1979a. With permission.)

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A question now arises: what continuum model should be set up to approximate this discrete system? Following Holnicki-Szulc and Rogula (1979a), three types of continuum models will now be formulated: local, nonlocal, and strain-gradient. 3.4.6.2 Local Continuum Model Proceeding as in Section 3.2.3 under conditions of uniform strain, and √postulating the equivalence of strain energy in a unit cell of volume V = 2 3l due to all the spring constants E=

1 r,r

1 Fi r, r ui r, r = ij ui r, r u j r, r (3.103) 2 r,r

2 r,r

to equal the strain energy of an effective continuum 1 E= 2

εij Cijkm εkm d V

(3.104)

V

we determine an effective, local-type stiffness tensor II Cijkm = C Ijkm + Cijkm ,

(3.105)

where, recalling (3.34), I (b) I (b) I (b) 2 I Cijkm = √ αI ni n j nk nmI (b) 3 b=1,2,3 I I (b) I I (b) I I (b) 6 II Cijkm = √ αI I ni n j nk nmI I (b) . 3 b=1,2,3

(3.106)

The niI (b) and niI I (b) are given by (3.32). The above tensors are of the form (3.37), so that 3 λI = μI = √ α I 8 3

3 λI I = μI I = √ α I I . 8 3

(3.107)

3.4.6.3 Nonlocal Continuum Model As pointed out in Chapter 6, a non-local model should result in stresses at a point dependent upon the deformation within the range of interactions associated with the point. As a result, the more inhomogeneous is the strain field, the closer is the nonlocal model to grasping the actual strain state of I II the lattice. First, we distribute the values of tensors Cijkm and Cijkm at point r uniformly over the regions of interactions of structures I and II (Figure 3.13),

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and form a new tensor Cijkm r, r such that I II Cijkm r, r = Cijkm r, r + Cijkm r, r

r, r =

I Cijkm h I r, r

II Cijkm h I I r, r

(3.108)

r, r = . AI AI I 2 2 Here AI = π s I , AI I = π s I I are the areas, and h I r, r and h I I r, r

are the characteristic functions of the regions of interactions in the neighborhood of r. Note: Stress in a nonlocal elastic continuum may be determined as a gradient of energy functional in the space of strains, but this stress lacks the clear meaning of a Cauchy stress in a local continuum. The model of nonlocal interactions is valid only in an infinite body, and its justification has to be found in an appropriate limit of the lattice dynamics of an infinite crystal (Maugin, 2002). However, there is a problem at the boundary: constitutive properties in the boundary layer of a homogeneous nonlocal continuum are necessarily different from those in the interior, which contradicts the assumption of homogeneity. I Cijkm

II Cijkm

3.4.6.4 Strain-Gradient Continuum Model Again with reference to Chapter 6, a strain-gradient model is similar to the nonlocal model in that it resolves the local inhomogeneity of deformation within the range of interactions associated with a continuum point. One begins here with a series expansion of the relative displacement field involving two terms—linear and quadratic—that is, 1 u j r, r = εijr r j − r j + γirjk r j − r j rk − rk , 2

(3.109)

where εijr = u(i, j) (r)

γirjk = ε(i, j,k) (r)

(3.110)

are gradients of the first and second orders, respectively, of the displacement field. In view of (3.103), the elastic energy of the structure (3.104) is now expressed as 1 r

1 r,r r

E= εik rk − rk + γikm rk − rk rm − rm 2 r,r ij 2 (3.111)

1 r

r

+ ε jm rm − rm + γ jmn rm − rm rn − rm . 2 Observing the continuum form of energy 1 E continuum = εij Cijkm εkl + γi jk Cijklmn γlmn d V, 2 V

(3.112)

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one can identify I II Cijkl r, r = Cijkl r, r + Cijkl r, r I II Cijklmn r, r = Cijklmn r, r + Cijklmn r, r ,

(3.113)

where 2E I AI I (b) I (b) I (b) I (b) I Cijkl r, r = √ ni n j nk nl 3s I b=1,2,3 I Cijklmn

2E I AI s I I (b) I (b) I (b) I (b) I (b) I (b) √ r, r = ni n j nk nl nm nn , 3 b=1,2,3

(3.114)

II I with completely analogous formulas holding for Cijkl r, r and Cijklmn r, r . Holnicki-Szulc and Rogula (1979b) present further considerations of these models, especially in connection with the setup of boundary value problems, the modeling of surface energy accounting for the heterogeneity of material properties in the boundary layer of the microstructure, and the determination of internal forces. The subject of higher-order gradient theories has been receiving a lot of attention over the past two decades (e.g., Bardenhagen and Triantafyllidis, 1994). We return to it in the context of composite materials in Chapter 9. Various mathematical aspects of lattice, truss, and frame systems are studied by Martinsson and Babuˇska (2006, 2007); see also Pshenichnov (1993), and Cioranescu and Saint Jean Paulin (1999). 3.4.7 Plate-Bending Response We can apply the same approach as that outlined so far for the in-plane problems, to the determination of effective plate-bending response of a periodic beam network. We sketch the basic ideas in terms of a triangular lattice, within the assumptions of a Kirchhoff (thin) plate model. To this end we must consider out-of-plane deformations of a triangular geometry lattice, Figure 3.14. The kinematics is therefore described by three functions, one out-of-plane displacement and two rotations (with respect to the x1 , and x2 axes): u ( x˜ )

w1 ( x˜ )

w2 ( x˜ ) ,

(3.115)

which coincide with the actual displacement (u) and rotations (w1 , w2 ) at the lattice vertices. Within each triangular pore these functions may be assumed to be linear. It follows then that the strain and curvature fields are related to u, ν1 , ν2 by κkl = νl,k

γk = u,k + εkl νl .

(3.116)

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P(b)

M(b)

121

~ M (b)

FIGURE 3.14 A perspective view of a triangular lattice, showing the relevant internal loads in a beam crosssection.

With reference to Figure 3.14, for a single beam the mechanical (forcedisplacement and moment-rotation) response laws are given as M(b) = C (b) κ (b)

˜ (b) = E (b) I (b) κ˜ (b) M

P( f ) =

12E (b) I (b) (b) γ˜ , s (b)

(3.117)

where A(b) = ta is a cross-sectional area of the beam; I (b) = ta3 w/12 is a centroidal moment of inertia of the cross-sectional area of the beam with respect to an axis normal to the plane of the lattice; s (b) = 2l (b) is the full length of each beam. The beam’s parameters are as follows: torsional stiffness C (b) ; Young’s modulus E (b) ; in-plane twisting moment M(b) ; out-of-plane bending ˜ (b) ; shear force P (b) ; shear deformation γ˜ (b) ; curvature κ (b) . moment M The strain energy of the unit cell is Ucontinuum =

V V eff κij Cijkl κkl + γi Aij γ j , 2 2

(3.118)

which is consistent with Hooke’s law mkl = Cijkl κkl

pk = Akl γl .

(3.119)

Here mkl is the tensor of moment-stresses, pk is the vector of shear tractions.

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Proceeding in a fashion analogous to the in-plane problems, Wo´zniak (1970) found Cijkl =

6

(b) (b) (b) ˜ (b) ni(b) nk(b) n(b) + n(b) j nl S j nl S

Aij =

b=1

6

(b) ni(b) n(b) j R , (3.120)

b=1

where S(b) =

C (b) s (b)

S˜ (b) =

E (b) I (b) s (b)

ˆ (b) = R

12E (b) I (b) 2 . s˜ (b) s (b)

(3.121)

In the case of a triangular lattice made of identical beams (E (b) = E, etc.) we find Cijkl = δij δkl + δik δ jl Y + δil δ jk

Aij = δij B,

(3.122)

in which == 2C S= √ s 3

3 S − S˜ 8 2E I S˜ = √ s 3

Y= ˆ = R

3 S + 3 S˜ 8 24E I √ . s3 3

B=

3 ˆ R 2

(3.123)

The same type of derivation may be conducted for a lattice of either rectangular or hexagonal geometry.

3.5

Extension-Twist Coupling in a Helix

3.5.1 Constitutive Properties The Timoshenko beam appears several times throughout this book. Interestingly, while it involves a shear force and a moment normal to the beam’s axis, both mutually orthogonal, in a helix the axial force and the moment are parallel to the main axis. The mechanics of this periodic system presents many interesting phenomena involving at least two scales, and leaves open challenges. The pitch of the helix provides one unit cell. Consider a wire rope made of helically shaped strands (helices), wound along the x-axis (Costello, 1997), Figure 3.15. The helical geometry of strands gives rise—already in the case of a single strand—to a coupling between axial and twisting effects. More specifically, if we replace this helical system by a rod-like element along x, the axial force F parallel to x is directly related not only to the axial strain u, x (= ε) but also to the rotational strain ϕ, x (angle of twist per unit length). Similarly, the torque M acting along x is a function of

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123 M

FIGURE 3.15 A wire-rope made of several strands, each of the same helix angle.

ϕ, x and ε. Thus, the effective constitutive equation of the helix is ⎧ ⎫ F ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ C11 AE = ⎪ ⎪ M ⎪ C21 ⎪ ⎩ ⎭ E R3

C12 C22

!)

u, x ϕ, x

* .

(3.124)

Here A and R = R1 + 2R2 are the effective cross-sectional area and radius of the rod-like element, where R1 is the radius of the center straight strand and R2 is the radius of m helical strands winding around it and thus forming a wire rope. E is the Young modulus of the strands’ materials, while its Poisson ratio ν appears in the expressions for Cij below. Here, from a requirement of a positive definite strain energy density, we obtain two conditions on four constitutive coefficients Cij : C11 > 0 C22 > 0 C12 = C21

C11 C22 − C12 C21 > 0.

(3.125)

In the language of continuum mechanics, the wire rope is a 1D micropolar medium of a noncentrosymmetric type. Basic equations of such continua—also called a hemitropic, antisymmetric, or chiral material—are given in Chapter 6. Costello (1997) analyzed the elastostatics of a bundle of m helical strands (each of radius R2 ) winding around the center straight strand (of radius R1 ), jointly forming a wire rope. The following assumptions were involved: 1. The strands are uniformly spaced along the perimeter of a circle of radius r2 = R1 + R2 , thus forming a ring in a plane perpendicular to the bundle’s axis without touching each other. 2. Each strand’s equilibrium configuration is a helix of constant radius r2 = R1 + R2 and constant helix angle α2 (the subscript 2 refers to the deformed configuration). 3. Strands are linear elastic (with an axial modulus and Poisson’s ratio) and undergo very small strains. 4. Friction and contact deformations are neglected. By reworking Costello’s analysis, the coefficients Cij have been derived explicitly (Shahsavari and Ostoja-Starzewski, 2005b), here we reproduce only

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the first one: C11 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

R12 mR22 + R12 + mR22 R12 + mR22 r2 + r2 tan2 α2 + ν R2

R22 1 − 2 sin2 α2 sin α2 cos2 α2 2 r2 tan α2 − ν R1 sin α2 + 4r2

⎤

⎥ ⎥ ⎥, ⎥ ν 2 R22 sin α2 cos4 α2 R1 + r2 tan2 α2 ⎦ R22 sin3 α2 cos2 α2 (1 + ν) + + 2r2 4 (1 + ν) r2

(3.126)

In Figure 3.16 the strain energies involved in these Cij coefficients are plotted as functions of α. These energies are based on σ 2 /2E and τ 2 /2G as strain energy densities for axial and torsional deformations, and are normalized upon dividing by E. We see that: 1. Strain energy plots involved in C11 and C22 are qualitatively similar to analogous plots of C11 and C22 themselves. As expected, with α approaching π/2, the energy in C11 tends to a maximum. However, the maximum energy in C22 occurs at value of α different from π/2.

× 10–3

0.014

1.2

C21 Energy

C11 Energy

0.012 0.01 0.008 0.006 0.004 0

0

0.6 0.4

0.2

0.4

0.6

0.8 α

1

1.2

1.4

00

1.6

× 10–3

1

6

C22 Energy

0.8 0.6 0.4 0.2 0

1 0.8

0.2

0.002

C21 Energy

July 12, 2007

0

0.2

0.4

0.6

0.8 α

1

1.2

1.4

1.6

0.4

0.6

0.8 α

1

1.2

1.4

1.6

× 10–4

5 4 3 2 1

0.2

0.4

0.6

0.8 α

1

1.2

1.4

1.6

0

0

0.2

FIGURE 3.16 Strain energies associated with Cij ’s as functions of α, at a helical strand’s Poisson ratio ν = 0.25.

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125

2. The strain energy plots involved in coupling terms C12 and C21 are qualitatively similar, but not exactly the same. They are quite close for α > ˜0.5 rad, with the discrepancies stronger below that value. The maximum difference between those strain energies is about 20%. The dependencies on α (with α = 0 rad signifying the case of strands fully aligned with the wire axis) are shown in Figure 3.16. In the first place we note that C12 = C21 , and this points to an open challenge of providing a better derivation of Cij ’s. Of course, in light of the Betti–Maxwell reciprocity theorem applied to the helix on the macro level one should have C12 = C21 . Now, the key assumptions C21 are: the product of higher-order terms resulting from the that cause C12 = strain of a single helical strand is neglected; the changes in the curvature and twist per unit length are linearized; small changes in helix angle (from α1 to α2 ) have been assumed, which then allowed some trigonometric functions to be simplified accordingly. It is not clear which assumption should be relaxed so as to arrive at C12 C21 without dealing with yet more complicated derivations and formulas. Interestingly, an analogous challenge exists in experimental mechanics: to measure C12 and C21 such that they are equal. The phenomenological equations (3.124) generally apply to more complex physical systems than a wire rope, for example a wood fiber made of helically wound fibrils, or a continuum shell. Indeed, it was shown, in the context of structural mechanics (Blouin and Cardou, 1989), that either assumption would lead to a few percent difference (at most ∼ 11%) for any of the Cij coefficients. It remains to be seen, however, what those differences would be for a shell made of a large number of thin cellulose fibrils winding along the axis of a cellulose fiber rather than a few thick wires such as shown in Figure 4.10(d) of Chapter 4. Note that a helical inclusion in an elastic space was studied by Slepyan et al. (2000).

3.5.2 Harmonic Waves 3.5.2.1 Elastic Helix Equations (3.124), together with the equations of motion, led Samras et al. (1974) to derive a system of two coupled wave equations governing the axialtwisting response of a fiber .. C11 u, xx + C12 ϕ, xx = ρ u

.. C21 u, xx + C22 ϕ, xx = J ϕ ,

(3.127)

where ρ is the mass density and J is the mass polar moment of inertia. They considered a monochromatic wave propagation along the fiber u (x, t) = Ue ik(x−ct)

ϕ (x, t) = e ik(x−ct) ,

(3.128)

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and arrived at a dispersion relation whose analysis indicated two possible wave speeds: c 1,2 =

2 (C11 C22 − C12 C21 ) 1/2 .

(C11 J + C22 ρ) ± (C11 J − C22 ρ) 2 + 4ρ J C12 C21

(3.129)

From this it followed, by inspection, that c 1 < c 2 , and, in fact, there may be an order of magnitude difference between both wave speeds. Actual numbers depend, of course, on the choice of a theory employed for the derivation of all Cij s or on the pertinent experiments. Given (3.129), Samras et al. (1974) found that the axial vibrations of the helix are described by two types of waves— slow and fast—each of which consists of forward and backward traveling pulses u (x, t) = U1 e ik(x−c1 t) + U2 e ik(x+c1 t) + U3 e ik(x−c2 t) + U4 e ik(x+c2 t) ϕ (x, t) = 1 e ik(x−c1 t) + 2 e ik(x+c1 t) + 3 e ik(x−c2 t) + 4 e ik(x+c2 t) .

(3.130)

Next, by considering the ratio of axial to torsional amplitudes U/ , they concluded that the waves that are primarily axial in nature (U/ > 1) propagate at speeds c 1 , whereas the waves that are primarily torsional in nature (U/ < 1) propagate at speeds c 2 . Clearly, by assuming C12 = C21 = 0 one immediately arrives at two uncoupled wave equations for purely axial and √ √ torsional waves, respectively, i.e., c 1,2 = C11 /ρ , C22 /J . The above analysis is the stepping stone for deriving the spectral finite element of a helical element. This element is specified via a 4 × 4 spectral +i ) stiffness matrix connecting kinematic quantities (extensions + ui and twists ϕ +i and moments M + i ), all in the frequency and dynamic quantities (forces F space, at both ends (i = 1, 2) of the element of length L: ⎧ ⎫ ⎧+ ⎫ F1 ⎪ + u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨+ ⎨F ⎬ +2 ⎬ u 2 +] = [K . (3.131) +1 ⎪ ⎪ ⎪ +1 ⎪ ϕ M ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ ⎭ +2 ϕ +2 M + ] and a Explicit formulas for all the elements of the spectral stiffness matrix [ K study of their dependencies on frequency and the coupling coefficient C12 = C21 was given by Shahsavari and Ostoja-Starzewski (2005a). 3.5.2.2 Thermoelastic Helix Recently, we have generalized the elastodynamic helix model to account for coupled thermoelastodynamic effects (Ostoja-Starzewski, 2003). Introducing the thermal expansion effect (−α Eθ) into (3.124)1 above, we find =

M =

A1 u, x + A2 ϕ, x − α Eθ A3 u, x + A4 ϕ, x

Q =

−AK θ, x ,

T

(3.132)

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127

where the third equation governs the heat conduction (Q = Aq is the flux in the whole cross-section) according to the basic model of Fourier; K is the thermal conductivity along the strand’s axis. This leads to three coupled equations governing the triplet (u, ϕ, θ) .. ρ A u = A1 u, x + A2 ϕ, x − α Eθ .. ρ J ϕ = A3 u, x + A4 ϕ, x

(3.133)

ρc v θ˙ = K θ, xx − α E T0 u˙ , x . To Upon taking time- and space-harmonic wave forms, we get a system of three algebraic equations, which results in a characteristic equation for the roots c in terms of k and all the material parameters Aρc 2 − AE −A3 cα E T 0 ρc v

−A2 J ρc 2 − A4 0

−i AEα/k 0 =0. K c −ik ρc v

(3.134)

Evidently, the wave motion is not only dispersive, as in the mechanical model above, but also damped. Equation (3.134) leads to a fifth-order algebraic equation for the roots c, which cannot be solved explicitly, and a numerical rootsolving method has been employed to assess the trends in change of c in function of wave number k, in the presence of weak thermal effects expressed by the dimensionless thermoelastic coupling constant (ε) and the thermal diffusivity at constant deformation (kv ) ε=

α 2 E T0 ρc v

kv =

K . ρc v

(3.135)

Considering the values pertaining to an oceanographic steel cable studied by Samras et al. (1974), we have found that, with ε increasing from zero up, while keeping kv = 0, both c 1 and c 2 increase linearly; with ε in metals taking values up to 0.1, those speeds may easily go up by a few percent. On the other hand, increasing kv from zero up, while keeping ε = 0, has no effect on c a and c t . As is well known, it is more correct in coupled thermoelastodynamics to use a wave-type equation for heat propagation. Thus, employing the Maxwell– Cattaneo model, we must replace (3.133)3 by α E T0 ρc v .. .. u˙ , x + τ u = θ, xx − θ˙ + τ θ , ,x K K

(3.136)

where τ is a time-like parameter. We now have a sixth-order algebraic equation for the roots c, but the ensuing numerical analysis reveals that they are only weakly affected by τ .

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3.5.3 Viscoelastic Helices 3.5.3.1 Viscoelasticity with Integer-Order Derivatives Recall from Section 3.5.1 the explicit formulas of Cij in terms of parameters of the geometry and elastic properties of a helical strand (or a bundle of strands). Although these formulas are not perfect (recall C12 = C21 ), one may begin to examine viscoelastic helices, that is, helical strands made of viscoelastic materials. We consider such materials to be non-micropolar and differential type, that is, .. .. T + P1 T˙ + P2 T + · · · = Q0 + Q1 ˙ + Q2 + · · · .

(3.137)

Here T and stand for the Cauchy stress and infinitesimal strain, respectively. We now inquire whether a helical strand made of, say, a Kelvin material will result in an effective Kelvin type or more complex response on the helix level (Shahsavari and Ostoja-Starzewski, 2005b). We proceed by using the correspondence principle of viscoelasticity. That is, we replace the elastic modulus E and elastic Poisson ratio ν by the Laplace transforms of appropriate viscoelastic relaxation functions and viscoelastic Poisson’s ratio, multiplied by the transform parameter s: E → s E (s)

ν → sν (s) .

(3.138)

Here we recall that in most viscoelastic materials, Poisson’s ratio is not a constant, but rather a function of time or frequency (Hilton, 2001; Tschoegl et al., 2002). We also need the relations of E (s) and ν (s) to the transformed bulk and shear moduli κ (s) and μ (s) E (s) =

9κ (s) μ (s) 3κ (s) + μ (s)

ν (s) =

3κ (s) − 2μ (s) . 2s [3κ (s) + μ (s)]

(3.139)

Now, assume that the helical strand material is described by two Kelvin models, one for dilatational and another for distortional response: Tm = Q0 m + Q1 ˙m

T = q 0 + q 1 ˙ ,

(3.140)

where m and denote the isotropic and deviatoric parts of stress and strain tensors, respectively. It follows that 3 q 0 Q0 + (q 0 Q1 + Q0 q 1 ) s + q 1 Q1 s 2 s E (s) = 2Q0 + q 0 + (2Q1 + q 1 ) s Q0 − q 0 + ( Q1 − q 1 ) s sν (s) = . 2Q0 + q 0 + (2Q1 + q 1 ) s

(3.141)

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By using the above equations together with (3.124), we arrive at relations linking the Cauchy stress σ = F /A and the couple-stress μ = M/A, along with their time derivatives, with axial and rotational strains (u, x and ϕ, x ) and their derivatives .. ... .. D0 σ + D1 σ˙ + D2 σ = E 0 u, x + E 1 u˙ , x + E 2 u +E 3 u +B0 ϕ, x ,x

,x

.. ... +B1 ϕ˙ , x + B2 ϕ +B3 ϕ ,x

D0 μ

+

D1 μ ˙

+

D2

,x

.. .. ... μ = E 0 u, x + E 1 u˙ , x + E 2 u +E 3 u +B0 ϕ, x ,x

+B1 ϕ˙ , x

+

B2

,x

.. ... ϕ +B3 ϕ .

,x

,x

(3.142)

Here all the constant coefficients Di , E i , Bi , Di , E i , and Bi , i = 1, . . ., 3, are functions of the strand’s geometry and the Kelvin model parameters. Observe that the constitutive equations of the helix (3.142) are qualitatively different from those of the Kelvin material itself, i.e., (3.140). Moreover, one can show that only in the singular case of ν = 0 do the second- and thirdorder derivatives vanish, and then the helix becomes one of a Kelvin type. Analogous results are obtained upon assuming either Maxwell or Zener models of the strand. This indicates the general trend: the macro-level viscoelastic response of the helix is different in type and more complex than that of the viscoelastic material at the micro-level (the strand material itself). Consequently, direct viscoelastic generalizations of effective constitutive equations of helices, not based on analyses such as those presented here, are likely to be invalid. 3.5.3.2 Viscoelasticity with Fractional-Order Derivatives The above results are amplified in the case of helices governed by viscoelasticity with fractional derivatives. Although the advantages of such a formulation of rheological relations are well known (e.g., Bagley and Torvik, 1983), the thermomechanics should be closely connected with the fractal geometry of the material (Carpinteri and Mainardi, 1997), a subject that still requires development, see Section 10.6 of Chapter 10. Let us start with a generalization of (3.140) Tm = Q0 m + Q1 Dα m

T = q 0 + q 1 Dβ ,

(3.143)

with time derivatives of order α and β meant in the sense of Caputo (1967), e.g., t d d α f (t) 1 Dα f (t) = f (τ ) dτ. (3.144) = (t − τ ) −α α dt (1 − α) 0 dt Proceeding in the same manner as earlier for the model involving integerorder derivatives, we find Pσ = Qu, x + Sϕ, x

P μ = Q u, x + S ϕ, x ,

(3.145)

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where P = c 0 + c 1 Dα + c 2 Dβ + c 3 Dα+β + c 4 D2α + c 5 D2β , Q = h 0 + h 1 Dα + h 2 Dβ + h 3 Dα+β + h 4 D2α + h 5 D2β +h 6 Dα+2β + h 7 D2α+β + h 8 D3β , α

β

α+2β

+ l7 D

α+β

S = l0 + l1 D + l2 D + l3 D +l6 D

2α+β

(3.146)

+ l4 D

2α

+ l5 D

2β

+ l8 D . 3β

Here value for c, h, and l are constant coefficients depending on the geometry and the constitutive parameters of the helical strand. The operators P ,Q , and S also comprise five, eight, and eight terms, respectively, analogously to the three operators in (3.146). When the fractional-order derivative reverts to the integer-order one—that is, upon setting α = β = 1 in (3.140)—the results reduce to (3.142). Thus, the fractional-order model leads to even higher complexity of the helix than the conventional one. The same trend persists when more realistic models (i.e., with more parameters) are adopted (OstojaStarzewski and Shahsavari, 2007). The foregoing observation relating to uniaxial helices also provides guidance for admissible vis-`a-vis inadmissible models of 3D chiral (i.e., helically structured) materials. The constitutive relation of a linear elastic chiral material in 3D involves two coupled equations linking the stress and couple-stress tensors with the strain and torsion-curvature tensors—see equation (6.80) in Chapter 6. In view of the above results, one cannot arbitrarily postulate viscoelastic generalizations of such equations in terms of differential tensorial equations.

Problems 1. The dispersion relation, generally written as ω = ω(q ), expresses a relation between the frequency ω and the wave number q . Verify that the dispersion relation for the 1D chain of particles in Figure 3.1 is ω = ωmax | sin qs/2|ωmax = 2 k/m. Draw a plot to compare this dispersion relation with the one corresponding to the equation (3.4). the range qs [−π, π ] is the so-called 1st Brillouin zone. Finally, determine the wave speed c and the group velocity c g of this 1D lattice, and consider their maximum possible values. 2. Consider a simple orthorhombic lattice with all the bonds having thermal conductivities K i ,where i is the bond direction xi .Derive the effective thermal conductivity of that lattice.

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3. Outline the derivation of (2.35). 4. Generalize the derivation of effective moduli of a hexagonal lattice (Section 3.4.4) to the case of Timoshenko beams. 5. Generalize the derivation of effective moduli of a square lattice (Section 3.4.5) to the case of Timoshenko beams. 6. Determine the effective moduli in plate-bending response of a square lattice with beams of (a) Bernoulli–Euler and (b) Timoshenko type. 7. Demonstrate that, in the case of vibrations of an elastic helix, the waves that are primarily axial in nature propagate at speeds c 1 = √ C11 /ρ, while the waves √ that are primarily torsional in nature propagate at speeds c 2 = C22 /J . 8. Derive the spectral finite element of an elastic rod. 9. Derive the spectral finite element of an elastic helix. 10. Study the literature and write a report on the fractional derivative of Caputo. What is its main advantages over the fractional derivative of Riemann-Liouville?

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4 Lattice Models: Rigidity, Randomness, Dynamics, and Optimality

There is a crack in everything: that’s how the light gets in. Leonard Cohen, 1992 This chapter takes lattice models to more general settings. There are four themes: rigidity, randomness, dynamics, and optimality. The first two of these involve the introduction of spatial randomness into a lattice as a departure from an originally periodic geometry of, say, the central-force triangular network of Chapter 3. One path is through a random depletion of bonds, which leads to a total loss of stiffness, or rigidity, of the lattice. Another way of creating a random lattice is through a random (instead of a regular) network of, say, a Poisson–Delaunay variety. The third theme considered here is a generalization from statics to dynamics, with nodes acting as quasi-particles—here we have a coarse scale cousin to molecular dynamics, and, at the same time, an alternative to finite element methods. Finally, the fourth topic is that of optimal use of material for given loading and support conditions, where a special case of central-force lattices arises, Michell trusses being the basic paradigm.

4.1

Rigidity of Networks

4.1.1 Structural Topology and Rigidity Percolation When considering a central force (or truss) network, a question of fundamental importance is whether such a structure is a sufficiently constrained system or not. In other words: is it an intrinsically rigid body? This is the subject matter of a field called structural topology. In the following we provide its basic concepts. Any central force network is a set of edges (or bars), and vertices (or nodes acting as frictionless pivots). We immediately have an edge set E, and a vertex set V, so that the network is represented by a graph G(V, E). An edge is an unordered pair of two vertices. Structural rigidity can be based either on statics or on kinematics, and, as we shall see below, they are, in a certain sense, equivalent; see Table 4.1. 133

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A Comparison of Various Terminologies from Structural Mechanics, Structural Topology, and Physics Field

Terminology

Terminology

Structural mechanics Structural topology statical approach Structural topology kinematical approach Condensed matter physics

Minimally constrained truss Isostatic structure

Mechanism Nonisostatic structure

Infinitesimally rigid structure

Nonrigid structure

Rigid network

Floppy network

The statical approach involves, in the first place, the concept of an equilibrium load. A system of forces assigned to the nodes of a network is said to be an equilibrium load if and only if (iff ) the sum of the assigned vectors is a zero vector, and the total moment of those vectors about any one point is zero. A network resolves an equilibrium load iff there is an assignment of tensions and compressions to all the bars of E, such that the sum at each node is equal and opposite to its assigned load. A structure is said to be statically rigid iff it resolves all equilibrium loads. The kinematic approach involves the concept of an infinitesimal motion, which is an assignment of velocities to all the nodes of V, such that the difference of velocities assigned to the ends of any bar is perpendicular to the bar itself. This means that the motion does not result in any extension or compression of the bar. Every connected plane structure has at least three degrees of freedom (two translations and one rotation), and this is called a rigid motion. A structure is said to be infinitesimally rigid if and only if all its infinitesimal motions are rigid motions. These statical and kinematical pictures are connected by a theorem due to Crapo and Whiteley (1989): “A structure is statically rigid iff it is infinitesimally rigid.” Next, a structure is said to be isostatic iff it is minimally rigid, that is, when it is infinitesimally rigid but the removal of any bar introduces some infinitesimal motion. Clearly, in an isostatic structure all the bars are necessary to maintain the overall rigidity. In statics this is called a statically determinate structure, as opposed to the indeterminate ones that have more than a minimally sufficient number of bars for the global rigidity. It is a well-known result that, in 2D, a determinate structure of |V| nodes has edges numbering |E| = 2|V| − 3,

(4.1)

where || denotes the number of elements in a given set. As an example, let us consider an incomplete triangular lattice shown in Figure 4.1. Although it satisfies (4.1) it is not at all clear whether it is isostatic. This example shows that |E| = 2|V| − 3 is only a necessary but not a sufficient condition for rigidity. The latter is provided by this theorem (e.g., Laman, 1970; Asimov and Roth, 1978): “A planar graph structure is isostatic if and only if it has 2|V| − 3 bars, and, for every m, 2 ≤ m ≤ |V|, no subset of m nodes has more than 2m − 3 bars connecting it.” This, effectively, allows

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135

FIGURE 4.1 A triangular lattice with 71 edges and 37 vertices; it is generically rigid.

one to check whether the edges of the graph are not distributed spatially in a uniform manner. If they are crowded locally, than the odds are that the structure is not isostatic. The isostatic concept so far discussed falls in the category of generic rigidity, where only the topological information on a graph’s connectivity comes into the picture. However, one may also deal with unexpected infinitesimal motions when, say, two edges incident onto the same vertex lie on a straight line. The applicable techniques for analyzing such geometric problems are reviewed in Guyon et al. (1990). For a review of problems in 3D see Guest (2000). When dealing with very large systems—such as encountered in condensed matter physics—we need to ask the question: what critical fraction, pr , of edges of E needs to be kept so as to render the structure isostatic? We note that we would have |E | = pr |E| new edges of thus modified, or depleted, set |E |. It follows immediately from |E | = 2|V| − 3 that we would have pr = 2/3. This value is a simple estimate of the so-called rigidity percolation, a concept also useful in biophysics (Shechao Feng et al., 1985; Boal, 1993; Hansen et al., 1996). As shown in these references, the actual critical point occurs at a somewhat different value than 2/3; theoretical methods involved include effective medium theories and spring network computations. The latter of these will be demonstrated later on the example of Delaunay networks. Finally, it is important to keep in mind that the rigidity percolation typically occurs above the connectivity percolation, that is, pr > pc . For example, pc = 1/3 in a triangular central-force network, while pr = 2/3. This is demonstrated in Figure 4.2(a) in terms of the planar bulk and shear moduli in function of the volume fraction C2 of phase 2, which is 10−5 times softer than phase 1 of the undepleted network, thus simulating zero stiffness bonds. Note that C2 = 1−C1 , and C1 ≡ p. That figure also shows the bulk and shear moduli of a

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Truss shear modulus µ1/E1 Beam shear modulus µ2/E1 Truss bulk modulus κ1/E1 Beam bulk modulus κ2/E1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 C2

0.6

0.7

0.8

0.9

1

(a) 1

Truss shear modulus µ1/E1 Beam shear modulus µ2/E1 Truss bulk modulus κ1/E1 Beam bulk modulus κ2/E1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 C2

0.6

0.7

0.8

0.9

1

(b) FIGURE 4.2 Dependence of bulk and shear moduli of triangular truss (central-force) and beam networks, obtained by computational mechanics on 50 × 50 lattices at contrasts (a) 10−5 and (b) 10−2 .

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triangular beam network (recall Section 3.4 of Chapter 3) at the same contrast of phases. Clearly, rigid bonding and beam bending are essential in providing the rigidity of the network right down to the point of connectivity percolation at C2 = 2/3. The effect of loss of rigidity becomes less dramatic with a decreasing contrast, see Figure 4.2(b). A much more extensive treatment of rigidity percolation is in Sahimi (2003). 4.1.2 Application to Cellulose Fiber Networks 4.1.2.1 Rigidity of a Graph of Poisson Line Field Geometry The planar Poisson line field was introduced in Chapter 1. If one reads the basic assumptions of the classical article in paper physics/mechanics (Cox, 1952), one arrives at the conclusion that the geometry of a cellulose fiber network assumed therein must be that of the Poisson line field. Note here that: •

A homogeneous field of infinite lines cannot be obtained from a random field of straight and finite segments by extending those segments to infinity.

•

The connectivity of the field of finite fiber segments—which clearly is a more realistic representation of paper—is lower that that of the Poisson line field.

•

The Cox model also assumed the network of not interacting (!) infinite lines/fibers to be held by a frame so as to make it solid-like.

Clearly, the last of these assumptions is not acceptable, but one way to perhaps save the Cox model is first to note that under affine motions (ui = εij0 x j , εij0 = const), straight lines transform into straight lines, and their original (Lagrangian) points of intersection in plane are preserved. We now identify the line segments (between any two consecutive intersections) to be edges of E, and pivots to be vertices of V. Let us recall that the triple-fiber intersections occur with probability zero for isotropic and anisotropic distributions of lines. Thus, we typically have vertices of connectivity 4, that is, V4 . Now, with reference to Figure 4.3(a), which shows a typical realization of the Poisson line field, we see that there are two types of edges: those in direct contact with the square-shaped window, and those entirely in the interior. Clearly, the square window is needed to prevent these boundary layer bonds from dangling, and this immediately renders the entire network a mechanism. However, one may argue that the boundary layer of dangling bonds is very thin relative to the whole field, and ask the question concerning the isostatic condition for the graph G(V, E) representing the interior network of edges not directly in contact with the square window boundary; these are shown in bold in Figure 4.3(a). Here we observe that, while the V4 vertices occur in the interior of this graph, its boundary involves V2 and V3 vertices. Now, since there two vertices to every edge, we can calculate the total number of edges

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(a)

(b)

(c)

(d)

FIGURE 4.3 Samples of (a) a planar Poisson line field and (b) a finite fiber field (recall Section 1.4 of Chapter 1), with a 1 = 1 and all other a i = 0. Test windows of size L × L are considered. (c) Deformation of a network of (b), with 195 fibers with originally straight fibers, with fiber bending present, subjected to axial strain ε11 = 1%. (d) The same network, with fiber bending almost absent, subjected to axial strain ε11 = 1%. All displacements in (c) and (d) are magnified by a factor 8 for clarity. Figure (d) shows large, mechanism-type motions of the network including those of some fibers which spring outside the original domain of the network.

in the bold drawn graph G(V, E) according to |E| = |V2 | + 32 |V2 | + 2|V4 |.

(4.2)

Evidently, since V = V2 ∪ V3 ∪ V4 , the total number of all the vertices is |V| = |V2 | + |V3 | + |V4 |

(4.3)

so that |E| < 2|V| − 3 and the system is not isostatic, it is underconstrained (i.e., a mechanism). Given this observation, the Poisson line field of axial force fiber segments (the so-called Cox model, 1952) is not a valid model of paper, or any other solid material for that matter. The fact that the Cox model does

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give finite values for elastic moduli including the shear modulus is easily explained by the presence of fully stretched fibers, spanning the entire test window. The situation is analogous to a graph of a square lattice topology, which, even though it is an obvious mechanism, will give finite axial moduli in two directions, if fully stretched and subjected to kinematic boundary conditions. In real networks fibers have finite length, so their ends are loose. When fiber ends are removed to eliminate their obvious mechanism motions, the number of vertices in sets V2 and V3 , increases. Consequently, (4.1) is even further from being satisfied. In order to deal with finite fiber effects, Cox and others modified the basic model by a so-called shear-lag theory. However, the latter assumes single-fiber segments to carry axial and shear forces only, which (see Section 4.2), is not a valid model of a solid element: fiber bending should also be included. Paper exhibits finite stiffnesses in 2D as well as in 3D. In the latter case, condition (4.1) is replaced by an even more stringent one as more constraints are needed when dealing with the additional degrees of freedom (Asimov and Roth, 1978). 4.1.2.2 Loss of Rigidity in a Fiber-Beam Network Besides the foregoing structural topology considerations, there is another fact that casts doubt on any fiber network model in which fiber segments are joined by pivots. Namely, any two cellulosic fibers have a finite contact area of hydrogen bonding (Page et al., 1961), which would be sheared by hinge-type connections. While it is very difficult to assess experimentally to what extent this region is deformable, our model will treat it as somewhat deformable in the sense that bonds are rigid but have no dimension, and fiber segments are treated as extensible beams from node to node of the graph G(V, E) (OstojaStarzewski et al., 1999). This modeling of mechanics of fiber networks is similar to that of a cementcoated wood strands composite (Stahl and Cramer, 1998), as well as to the one used for highly porous materials (Chung et al., 1996), and is based on the following assumptions and steps: 1. Generate a system of finite-length straight fibers, such as shown in Figure 4.3(b) according to specific geometric characteristics: distribution of fiber lengths and widths, distribution of angular orientations (θ) of fiber chords, etc. The fibers are laid in three dimensions on top of one another with a possible, slight non-zero out-of-plane angle. θ ∈ [0, π ], that is, the angle a fiber makes with respect to the x-axis is modeled by the Fourier series-type probability density function of Chapter 1. 2. Fibers are homogenous, but each fiber may have different dimensions and mechanical properties, all sampled from any prescribed statistical distribution.

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Microstructural Randomness and Scaling in Mechanics of Materials 3. Each fiber is a series of linear elastic 3D extensible Timoshenko beam elements. Each of these is described by a stiffness matrix written here in an abbreviated form setup in a corotational coordinate system (Cook et al., 1989):  EA      0 0 0 0 0 l F  L          GJ    0    0 0 0 0      l T  θx            2 2    l +3g l −6g  Ma      a 0 0 0 0   θ 4a 2a y y   = 2 2 l +3h l −6h   a  (4.4)  a  0 0 0  Mz   θz    0  4b 2b          2 2   b l −6g l +3g  My   θ yh     0     0 0 0      2a 4a  b  b   2 2 l +3g Mz θz l −6h 0 0 0 0 2b 4a where g = 12

E Iy GA

h = 12

E Iz GA

a=

E Iy l(12g + l 2 )

b=

E Iz . l(12h + l 2 ) (4.5)

Here F and T are the axial force and the twisting moment, while Mya , Mza , Myb , and Mzb are the bending moments around the y and z axes at the a and b ends, respectively. Also, L, θx , θ ya , θza , θ yb , and θzb denote axial elongation, angle of twist, and four angles of rotation. Finally, l, A, J , Ix , and I y are, respectively, the length, cross-sectional area, cross-sectional polar moment of inertia, and the moments of inertia with respect to the x- and y-axes. E and G are the Young modulus and shear modulus of a fiber-beam. 4. After identifying all the fiber–fiber intersection/contact points a connectivity matrix is set up. 5. Equilibrium is found under the boundary condition ui = εij0 x j . 6. All six effective, in-plane stiffness coefficients are determined from the postulate of equivalence of strain energy stored in a squareshaped window of finite thickness with the strain energy of an equivalent continuum. The undeformed network, shown in Figure 4.3(b) in its top view, has the following parameters: window size: 4 × 4 × 0.1 mm, a 1 = 1, and other coefficients in equation (4.4) are zero; fiber length: 2 mm; fiber width: 0.04 mm; fiber height: 0.015 mm. As a result of a Boolean process of fiber placement (Chapter 1), we obtain: 195 fibers with an average of 4.8 bonds per fiber, the whole system having 859 nodes with six degrees of freedom per node. The state of deformation corresponding to axial strain ε11 = 8% is shown in Figure 4.3(c). The analyzed strain is actually 1%, but displacements are magnified for clarity. Compare this deformed network to that in Figure 4.3(d), which shows the same network of fibers subjected to the same strain but with

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the ratio of fiber flexural stiffness to fiber axial stiffness reduced by a factor of 10−4 . Note the following: 1. The sharp kinks we see in both figures are only the artifact of simple computer graphics—the micromechanical model assumes fibers deform into differentiable curves. Magnification creates the appearance of large displacements—actually, an infinitesimal displacement assumption is used in the computational mechanics program. 2. The kinks are far more pronounced when fibers have low flexural stiffness. Portions of the network where connected fibers do not form triangular pores can generate significant forces in response to deformation when fibers have high flexural stiffness, but they cannot do so when fibers rely almost entirely on axial stiffness. These portions of the network are not stable in the sense of loss of generic rigidity discussed earlier. 3. We do not study this rigid-floppy transition by turning, in an ad hoc fashion, all the connections into pivots. Rather, with the model taking into account all the displacements and rotations of nodes, we can study it as a continuous function of fiber slenderness; see also Kuznetsov (1991). Note that this aspect is impossible to investigate with models based on central-force potentials for single fiber segments (e.g., Kellomaki et al., 1996). Our model also fills a gap pointed out in Raisanen et al. (1997) consisting of a need to set up finite element models of 3D disordered fiber networks, yet avoids their simplistic mapping into electrical resistor networks (i.e., second-rank tensor problems) of the same topology. The fiber network model has also found an application in studies of mesoscale stiffness and instability (splitting) of fibril networks found in a bone’s lamellar structure (Jasiuk and Ostoja-Starzewski, 2004), see Figure 7.15 of Chapter 7.

4.2

Spring Network Models for Disordered Topologies

4.2.1 Granular-Type Media 4.2.1.1 Load Transfer Mechanisms The spring network models are most natural when applied to systems that have the same topology as the underlying lattice. One such example has been discussed above: a cellulose fiber network. Another one is offered by a granular medium. Here, the principal method of computational mechanics analyses, dating back to Cundall and Strack (1979), is the so-called discrete element (DE) model.

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V2

2

V1

1

V3

3

(a)

(b)

V2

V2

kn

V1

ka

V1

kn

ka

V3

(d)

V3 (c)

FIGURE 4.4 (a) A cluster of three grains (1, 2, 3) showing the three lines of interactions; (b) a discrete element model showing the normal force, the shear force and the moment exerted by grains 2 and 3 onto grain 1; (c) a most general model showing the same grain-grain interactions as before but augmented by an internal, angular spring constant k a ; a simplified model showing only normal (k n ) and angular (k a ) effects.

Let us employ a graph representation of the planar granular medium: a graph G(V, E), whereby vertices of the set V signify grain centers and edges of the set E represent the existing grain–grain interactions, Figure 4.4(a). We fix an r − θ polar coordinate system at a grain center. There are several types of the DE model that one may consider: •

Central interactions: the total energy is a sum total of central interactions of all the edges U = U cental ,

(4.6)

and this model is a generalization of the basic model of Section 3.2.3 in Chapter 3.

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Central and angular interactions: the total energy is modified to U = U cental + U angular ,

(4.7)

and this model is a generalization of what we saw in Section 3.2.4 in Chapter 3. Continuum mechanics tells us that εθ θ equals r −1 ∂uθ /∂θ + ur /r . This shows that the angular changes φ/φ between two adjacent edges V1 − V2 and V1 − V3 in Figure 4.4 correspond to r −1 ∂uθ /∂θ in the r − θ polar coordinate system fixed at a grain center. This term does not show up in two other expressions for εrr and εr θ . However, ur /r is due to a radial displacement, and so φ/φ does not exactly equal εθ θ , which leads us to call it a “θ θ-type strain.” We now adopt the Kirkwood model (recall Chapter 3) to account for φ/φ in addition to the normal grain–grain interactions, and introduce angular springs of constant k a acting between the edges V1 − V2 and V1 − V3 incident onto the node V1 , Figure 4.4(d); the edges remain straight throughout deformation. •

Central, shear, and bending interactions: the total energy is U = U cental + U shear + U moment ,

(4.8)

and this model is a generalization of what we saw in Sections 3.4.1 to 3.4.2 of Chapter 3. This is a typical DE model, which, of course, may be termed a “locally inhomogeneous micropolar continuum,” with inhomogeneity varying on the scale of grains; see the section below. •

Central, shear, bending, and angular interactions: the total energy is U = U cental + U shear + U moment + U angular .

(4.9)

One may argue that the three-point interaction should be introduced in the DE models so as to better represent the micromechanics, and to make, in accordance with Figure 4.4(c), the strain energy stored in a single Voronoi cell equal to (4.9). However, there exist successful DE models that account for normal and shear forces only (Bathurst and Rothenburg, 1988a,b); this neglect of the contact moment is justified by the fact that only small numerical errors are thus caused in problems of interest in granular materials. In the case of a regular, triangular array of disks, this model is equivalent to a classical Born model of crystal lattices, which is known to lack the rotational invariance (Jagota and Bennison, 1994). 4.2.1.2 Graph Models Let us pursue the planar graph representation of granular media in some more detail (Satake, 1976, 1978). First, we list in Table 4.2 a correspondence between a system of round grains and its graph model. Besides the vertex (V) and edge (E) sets introduced earlier, we also have a loop set L. With this geometric

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Granular Structure and Graph Terminology Assembly of grains

Graph

Index

No. of elements

Grain Contacting point Void (in 2D)

Vertex Edge Loop

v e l

|V| |E| |L|

Note: After Satake (1978).

reference, we can then set up an assignment of load quantities—forces on the left and corresponding kinematic measures on the right—in Table 4.3. The connectivity of the graph is described by the incidence matrix Dve . Let us write down a total of 3|V| scalar equilibrium equations, each one with respect to a typical grain of radius r v and volume V v e Bv F ve v +V D = 0, (4.10) Me Nv where

0 D = v ve e . r D n × Dve Dve

ve

(4.11)

Here ne is the unit vector of edge e in the nonoriented graph. The operator Dve (and its dual Dev mapping from vertices into edges) also plays a key role in the kinematics of all the edges: uv ue ev = −D = 0, (4.12) we wv where

ev = D

Dev 0

−ne × Devr v . Dev

(4.13)

The kinematics is subject to 3|L| compatibility constraints written for all the loops, where we make a reference to Satake’s work. TABLE 4.3

Load Versus Kinematic Quantities Load quantity

Notation

Number of elements

Notation

Body force Body couple Contact force Contact couple

Bv

|V| |V| |E| |E|

uv

Nv Fe Me

Note: After Satake (1978).

wv ue we

Kinematic quantity Grain displacement Grain rotation Relative displacement Relative rotation

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The above should be augmented by 3|E| constitutive equations connecting the contact force Fe and contact moment Me with the relative displacement ue and relative rotation we . Given three global equilibrium conditions, we have a total of 3(|V| − 1 + |E| + |L|) = 6|E|

(4.14)

equations. Taking note of the Euler formula |V| − |E| + |L| = 1 in 2D (as opposed to that in 3D in Section 1.4 of Chapter 1), we see that this budget of equations agrees with the total of 6|E| unknowns: Fe , Me , ue , and we , all defined on edges of the set E. Finally, we note a formal analogy of (4.10) and (4.12) to the equilibrium and strain-displacement equations of Cosserat continua (Chapter 6) σ b γ u div + =0 = grad . (4.15) µ m κ w The similarity of compatibility relations for graph and continuum descriptions has also been shown by Satake (1976, 1978). See Goddard (2006) for an in-depth analysis of these issues. Applications of graph models to cell biomechanics have been explored in Hansen et al. (1996, 1997). While this section focuses on classical continuum modeling, multifield models for granular-type materials are analyzed in Trovalusci and Masiani (2005). 4.2.1.3 Periodic Graphs with Disorder Randomness may be introduced into the periodic networks in various ways. Figure 4.5 displays two basic possibilities: substitutional disorder and topological disorder. The first of these connotes a variability in properties per vertex (or node), and the second consists in a departure from the periodic topology. There is also a third case, of much more interest in solid-state physics: geometric disorder, which involves the variability in the geometry of a network’s structure—like uneven lengths of various bonds—but preserving a

(a)

(b)

FIGURE 4.5 Substitutional (a) versus topological disorder (b) of a hard-core Delaunay network.

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topological periodic structure (Ziman, 1979). The topological disorder is typically caused by an incompatibility of crystal-like domains in a granular material. For example, the material may consist of equisized disks, which are organized into regular, periodic arrays, but the fact that they happen to be differently oriented in space causes an irregular structure and network of domain boundaries. As observed earlier in connection with the DE model carrying torques, the topological disorder leads to a locally inhomogeneous polar, or micropolar, continuum—depending on the type of vertex–vertex interactions—with inhomogeneity varying on the scale of grains. Such a continuum model contains a lot of information, but, in the first place, one wants to establish the eff effective, in the macroscopic sense, first-order moduli Ceff (≡ Cijkl ) of the material. These are obtained from the so-called periodic boundary conditions on an L × L square B: ui (x + L) = ui (x) + εij0 L j

ti (x) = −ti (x + L)

∀x ∈ ∂ B.

(4.16)

Here εij0 is the macroscopic strain, ti is the traction on the boundary ∂ B of B, and L = Lei , with ei being the unit base vector. The periodicity means that the network topology is modified so as to repeat itself with some periodicity L in x1 and x2 directions, whereby L is usually taken much larger than the typical vertex-vertex spacing (or edge length). Now, the periodic conditions (4.16) require that a periodic network be set up, and this, in turn requires a periodic Poisson point field on the L × L square; recall Chapter 1. Topologically, our square turns into a torus, but, as Figure 4.6 shows, there are also three other possibilities. B

A

A

Torus

Cylinder

B

A

A

A

(a) B

A (c)

A

A

A

Klein bottle

Möbius band

June 1, 2007

A

B (b)

A

A (d)

FIGURE 4.6 Mapping of edges of a a square-shaped domain resulting in (a) a cylinder, (b) a Mobius ¨ band, (c) a torus and (d) a Klein bottle. The torus corresponds to periodic boundary conditions.

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FIGURE 4.7 A periodic Poisson–Delaunay network with 200 vertices.

A typical realization B(ω) of a periodic Poisson–Delaunay network, numbering 200 vertices, is shown in Figure 4.7. The set B ={B(ω); ω ∈ } forms the random medium; a single ω indicates one realization of the Poisson point field and a chosen assignment of spring constants. In actual simulations only a minute subset of the entire sample space can be investigated, but by the standard Monte Carlo and ergodicity arguments, this subset is representative of the whole system. Thus, already the response of a single network much larger than the grain size is sufficient to gain a good estimate of Ceff . The ensemble average of that tensor is isotropic for a microstructure of spacehomogeneous and isotropic statistics, but, with the number of vertices large, even one realization of the network should be close to isotropic. Using the formula for the strain energy of a 2D elastic continuum of volume V = L 2: V 1 0 0 0 0 0 0 U= (4.17) κεii εjj + 2 εij εij − εii εjj , 2 2 leads, with reference to Chapter 5, to planar bulk and shear moduli: κ=

V

2U(1) 0 ε11

+

0 2 ε22

µ=

2U(2) 0 2 . V ε12

(4.18)

0 0 Here U(1) and U(2) denote energies under dilatational (ε11 = ε22 ) and shear 0 (ε12 ) strains, respectively. The moduli computed this way were compared to various models in Ostoja-Starzewski et al. (1995):

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A symmetric self-consistent approximation for elliptical disks with perfectly bonded interfaces, which treats the medium as a mixture of N phases without distinguishing any one as a matrix.

•

An asymmetric self-consistent approximation for elliptical disks with perfectly bonded interfaces, which treats the medium as a mixture of N − 1 phases embedded in a matrix.

•

An asymmetric self-consistent approximation for circular disks with perfectly bonded interfaces.

•

An asymmetric self-consistent approximation for circular disks with springy interfaces.

•

Voigt and Reuss bounds for circular disks with springy interfaces.

Comparisons to such continuum models are possible providing one interprets a vertex and all the edges incident onto it as a “spider,” which allows interpretation of that object as a continuum-type inclusion. As a result we have a “spider-inclusion” analogy of Figure 4.8.

4.2.2 Solutions of Truss Models A specialized method of determination of apparent moduli applies in the case of truss-type microstructures. Thus, let us focus on the response of a window, arbitrarily larger than the average edge length, subjected to displacement boundary conditions, Figure 4.9 (Ostoja-Starzewski and Wang, 1989). First, a window is cut out of the network as indicated by a square with dashed lines in Figure 4.9(a), and treated as an independent body (b)—in this case, given the assumption of central forces in all the Delaunay edges, a planar truss. At this stage, all the boundary points (on four sides) of the window are pinned, and this is where the displacement condition is applied—see equation (4.23) below. For example, a state of uniaxial extension and shear is depicted in Figure 4.9(c). The actual mechanical response of this planar truss is now solved by a structural mechanics method.

FIGURE 4.8 A periodic Poisson–Delaunay network with 200 vertices, showing a “spider” of edges incident onto a vertex in the figure on the right.

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(a)

(b)

149

(c)

FIGURE 4.9 A finite window (b) cut out of a Delaunay network (a), and subjected to uniform displacement boundary conditions in (c).

We introduce a vector notation, so that the effective mechanical response of the window is described by    0   C11 C12 C13    σ 11     ε11    0 0 σ 22 ≡ {σ } = [C]{ε } ≡  C21 C22 C23  ε22 .        0  σ 12 C31 C32 C33 ε12

(4.19)

The structural mechanics of the truss is governed by [K ]{u} = {F },

(4.20)

where [K ], {u}, and {F } are the stiffness matrix of the system and the displacement and force vectors at the nodes, respectively. Noting that the set of global equilibrium equations governing all the elements may be partitioned into those corresponding to the degrees of freedom at the window boundaries (b) and the ones in the interior (i), we have K (ii) K (ib) u(i) F (i) [K ] = , {u} = , {F } = . (4.21) u(b) F (b) K (bi) K (bb) The net force on any interior node (and, thus, any interior degree of freedom) must be zero, F (b) = 0, so that the static condensation gives [K ]{u(b) } = {F (b) }

[K ] = [K (bb) ] − [K (bi) ][K (ii) ]−1 [K (ib) ].

(4.22)

For any boundary node the displacement can be written as

u1(b) u2(b)

= [A]{ε} ≡

 0   ε11  x1 0 x2 /2  0  ε22 , 0 y2 x1 /2    0  ε12

(4.23)

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while for an interior node we have [u(i) ] = [K (ii) ]−1 [K (ib) ][A]{ε 0 }.

(4.24)

Now, noting that the strain energy of the window is U=

1 (b) T (b) 1 {u } {F } = {u(b) }T [K ]{u(b) }, 2 2

(4.25)

we obtain, by equivalence with the continuum model, the effective moduli [C] =

1 [A]T [K ][A], V

(4.26)

where the strain-displacement matrix [ A] is obtained through the assembly of the nodal matrices for all the boundary nodes. In the case of traction boundary conditions {F (b) } = [H]{σ },

(4.27)

a force-based homogenization technique has been proposed (Huyse and Maes, 2001). Here [H] is a matrix that depends on the actual boundary’s microgeometry. The structural mechanics of the truss is now rewritten as f (ii) f (ib) [ f ]{F } = {u} [ f ] = , (4.28) f (bi) f (bb) where [ f ] is the flexibility matrix and the strain energy of the window is U=

1 (b) T (b) 1 {u } {F } = {F (b) }T [ f (b) ]{F (b) }. 2 2

(4.29)

From the above equations we obtain, by equivalence of the discrete with the continuum model, the apparent compliances [S] =

1 [H]T [ f (bb) ][H], V

(4.30)

where [ f (bb) ] is obtained through an inversion. 4.2.3 Mesoscale Elasticity of Paper 4.2.3.1 Dilemma of Special In-Plane Orthotropy of Paper Paper is one of the most challenging engineered materials. The difficulties in understanding it are due to its complex multiscale structure—this necessitates random fields of Chapters 1 and/or 2, random network models of this chapter, (non)classical elasticity of Chapters 5 and 6, as well as micromechanics models discussed here and also later. Things get still more complicated when the inelastic behavior is considered, see Chapter 9. For the sake of reference, because we are dealing with the machine-made (i.e., oriented) paper, for the linear elastic tensile range, an orthotropic model

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is adopted (e.g., Uesaka et al., 1979): σ11 =

E1 E 2 ν12 ε11 + ε22 (1 − ν12 ν21 ) (1 − ν12 ν21 )

σ22 =

E 1 ν21 E2 ε11 + ε22 (1 − ν12 ν21 ) (1 − ν12 ν21 )

τ12 = Gγ12 = 2Gε12

(4.31)

where the engineering constants are: E 1 = Young’s modulus in x1 -direction (MD) E 2 = Young’s modulus in x2 -direction (CD) −ε22 ν12 = is the Poisson ratio for strain in x2 -direction (CD) when ε11 paper is stressed in x1 -direction (MD) only −ε11 ν21 = is the Poisson ratio for strain in x1 -direction (MD) when ε22 paper is stressed in x2 -direction (CD) only G = shear modulus in x1 x2 -plane Here MD (x1 ) and CD (x2 ) are the so-called machine- and cross-directions, respectively. To ensure that the stiffness matrix is orthotropic, we require E 1 ν21 = E 2 ν12 .

(4.32)

The shear modulus G is very difficult to measure by quasi-static experiments, but ultrasonic experiments on many paper materials indicate that the following relation is approximately satisfied (Horio and Onogi, 1951; Campbell, 1961): G=

E1 E2 . E 2 + E 1 + E 1 ν21 + E 2 ν12

(4.33)

Thus, paper has a special in-plane orthotropy property where only three elastic constants are independent. This “special orthotropy” means that the shear modulus is (approximately) invariant with respect to rotations of the coordinate system (Horio and Onogi, 1951; Campbell, 1961). Another way to express this invariance, fully equivalent to (4.33) above, is in terms of the in-plane compliances 4S1212 = S1111 + S2222 − S1122 .

(4.34)

Note here that “approximately” reflects the fact that the compliance S is an apparent (i.e., mesoscale) property, typically measured on scales of centimeters, and being actually random since we are below the RVE. Note: Given the multiscale structure of paper shown in Figure 4.10, there is no clear length scale at which homogenization to a perfect RVE can be

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ZD

(a) MD

CD

(b)

(c)

P

S3

S2

S1

(d)

35A

100A

(e)

CH2OH H H OH

O H

O

H H

OH

H1

OH

OH H

H

CH2OH H

O

O

H CH2OH

H

O

H OH

H

H

OH

O

H

H

OH

OH H

H

H

O H

O

O

CH2OH

(f ) FIGURE 4.10 A hierarchy of scales in paper: (a) roll of paper on a paper machine (up to 10 m wide, thousands of kilometers long) with a possible presence of streaks; (b) paper sheet (scale of centimeters); (c) random fiber network (scale of millimeters); (d) cellulose fiber with its layers P, S1, S2, S3 (all made of fibrils) and the lumen (microns); (e) fibrils; (f) molecular chains.

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carried out; see the discussion of RVE in Chapters 7 to 10. Although the constitutive law above applies to length scales of centimeters, specimens display some scatter there. Going to scales of tens of centimeters, one begins to see streaks, while on the scales of meters there is a strong dependence on the cross-direction of the paper machine. There are also complex fluctuations in the machine direction, which do not qualify as a WSS random field. Staying on the scale of centimeters, the question becomes: Why does the relationship (4.34), in fact hardly found in other materials, occur in paper? 4.2.3.2 Explanation via Random Fiber Network The above question is addressed by Ostoja-Starzewski and Stahl (2000) with the help of models introduced earlier. First, we employ the germ-grain model of Chapter 1, constructed so as to obtain nonuniform fiber flocculation, commonly seen as spatially nonuniform grayscale effects in a typical sheet of paper held against light. Next, we take the computational mechanics model of fiber networks of Chapter 3. In brief, fibers are placed in 3D with possible non-zero angles to control out-of-plane orientation of the fiber axis and the “roll” of the fiber about its own axis. Each fiber, depending on its contacts with other fibers, is a series of linear elastic three-dimensional Timoshenko beam elements e, also with torsional response included, between bonds with other fibers. In the analysis of each and every body Bδ (ω), having solved for displacements and rotations of all the bonds under uniform kinematic boundary conditions ui = εij0 x j , we establish the network’s effective stiffness tensor Cijkm (ω) from a postulate of equivalence of the total strain energy of all the networks’ elements e, Utot , with that of an effective continuum (V = L x L y t) Utot =

V εij Cijkm (ω)εkm = [Ueaxial + Ueshear + Uemoment + Uetorsion ]. 2 e∈E

(4.35)

It may be argued, by virtue of a qualitative analogy to in-plane conductivity of networks of identical geometry, that the kinematic boundary condition allows a much faster asymptotic scaling to the RVE than the uniform traction boundary condition ti = σji0 n j . The point is that fiber networks are somewhat analogous to plates with holes, reported in Section 3.4.3 of Chapter 3, recall Figure 3.11 there. Let us now go back to Figure 1.11 of Chapter 1. It displays two networks differing only by the degree of flocculation. That is, parameter b equals 2.0 in (a) and 0.4 in (b), so that we have a highly versus a weakly flocculated 0 network. Their responses under overall ε11 -strain are shown in the middle figures, while the figures on the right show the difference between the resulting displacement of each node and what the displacements would be if the strain field in the interior of the network had been uniform. In the latter case, the figures would consist simply of dots; thus, the lines represent deviation from uniformity. Let us first note that the deviation from uniform displacements is apparent in both networks, but it is certainly higher in the network with b = 2.0 (strong flocculation) than with b = 0.4. (weak flocculation).

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This is so because fibers belonging to a given floc move together, and this motion, being different from the affine one, resembles a swirl. Now, it turns out that the special orthotropy relation (4.34) is satisfied for a network with b between the values employed here. Various parameters of the network may be altered, but some degree of flocculation—neither too much nor too little—is always necessary. [In our original paper the factor “4” in equation (4.34) was inadvertently missed, but this had no effect on the results since we worked with G in computational mechanics.] To sum up, this analysis shows that random network geometry involving two scales— (1) random fiber placement within a floc with random angular orientation and (2) flocculation modeled by the germ-grain process—together with a network model possessing generic rigidity (thanks to beam-type fibers) offers an explanation of the peculiar property of many papers. 4.2.4 Damage Patterns and Maps of Disordered Composites As mentioned in Chapter 3, the lattice method can also be used to simulate damage of heterogeneous materials. This works particularly well in the case of elastic-brittle failure of composites, where one uses a mesh (much) finer than the typical size of the microstructure. In principle, one needs to determine which lattice spacing ensures mesh-independence, or nearly so. Such a study has been conducted for a thin aluminum polycrystalline sheet discussed in Section 3.3 of Chapter 3. Following Ostoja-Starzewski et al. (1997) and Alzebdeh et al. (1998), we now focus on two-phase composites in antiplane shear, under periodic boundary conditions and (necessarily) periodic geometries. Now, since both phases (inclusion i and matrix m) are isotropic and elastic-brittle (Figure 4.11a), the

σ C

Stiﬀ weak

i

C i/C m

Stiﬀ strong

Cm εicr /εm cr εicr (a)

εm cr

ε

Stiﬀ weak

Soft strong (b)

FIGURE 4.11 (a) Elastic-brittle stress-strain curves for matrix and inclusion phases; (b) sketch of the damage plane.

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10.0

C i/ C m

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1.0

0.1

0.1

1.0 i

10.0

m

ε/ε

FIGURE 4.12 Crack patterns in the damage plane on a scale 4.5 times larger than that of the inclusion diameter. The center figure of a homogeneous body is not shown as it corresponds to all the bonds failing simultaneously.

composite can be characterized by two dimensionless parameters i i εcr /εcr

C i /C m ,

(4.36)

i where εcr is the strain-to-failure of either phase, and C its stiffness. This leads to the concept of a damage plane (Figure 4.11b), where we see various combinations of strengths and stiffnesses. While the response in the first and third quarters of damage plane is quite intuitive, this is not so for the second and fourth quarters. In those two quarters there is a competition of either high stiffness with low strength of the inclusions with the reverse properties of the matrix, or the opposite of that. The damage plane is useful for displaying effective damage patterns of any particular geometric realization of the random

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156

σ/σcr

m i

1.0

ε/εm cr

σ/σm cr

ε/εm cr

m

ε/εm cr

C /C

m

m

σ/σm cr

ε/εcr

σ/σm cr

m

ε/εcr

σ/σcr

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0.1 m

m

ε/εcr

ε/εcr

0.1

1.0 i

ε /ε

m

ε/εcr

10.0

m

FIGURE 4.13 Damage maps of statistics of constitutive responses for twenty realizations of the random composite, such as that in Figure 4.12.

composite while varying its physical properties (Figure 4.12) as well as other characteristics, say, statistics of response in the ensemble sense (Figure 4.13). A number of other issues are studied in the referenced papers: •

Stress and strain concentrations

Finite size scaling of response • Function fitting of statistics (where it turns out that the beta probability distribution offers a more universal fit than either Weibull or Gumbel) •

•

Effects of disorder versus periodicity

Also, see Ostoja-Starzewski and Lee (1996) for a similar study under in-plane loading; computer movies of evolving damage are at http://www.mechse. uiuc.edu/research/martinos.

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157

Particle Models

4.3.1 Governing Equations 4.3.1.1 Basic Concepts Particle models are a generalization of lattice models to include dynamic effects, and can also be viewed as an offshoot of molecular dynamics (MD). The latter field has developed over the past few decades in parallel with the growth of computers and computational techniques. Its objective has been to simulate many interacting atoms or molecules in order to derive macroscopic properties of liquid or solid materials (Greenspan, 1997, 2002; Hockney and Eastwood, 1999). The governing Hamiltonian differential equations of motion need to be integrated over long time intervals so as to extract the relevant statistical information about the system from the computed trajectories. Techniques of that type have been adapted over the past two decades to simulate materials at larger length scales, whereby the role of a particle is played by a larger-than-molecular piece of material, a so-called particle or quasi-particle. The need to reduce the number of degrees of freedom in complex systems has also driven models of galaxies as systems of quasi-particles, each representing lumps of large numbers of stars. In all these so-called particle models (PM), the material is discretized into particles arranged in a periodic lattice, just like in the spring network models studied in earlier sections, yet interacting through nonlinear potentials, and accounting for inertial effects, that is, full dynamics. As shown in Figure 4.14, the lattice may be in 2D or in 3D. Note that, by comparison with finite elements (FE), which indeed also involve a quite artificial spatial partitioning, PMs are naturally suited to involve interparticle potentials of the same functional form as the interatomic potentials, providing one uses the same type of lattice. The PM can therefore take advantage of the same numerical techniques as those of MD, and rather easily deal with various highly complex motions. Thus, the key issue is how to pass from a given molecular potential in MD to an interparticle potential in PM. In the case when the molecular interactions are not well known, the PM may still turn out to be superior relative to the FE. Among others, this indeed is the case with comminution of minerals where scales up to meters are involved (Wang and Ostoja-Starzewski, 2006; Wang et al., 2006). In MD, the motion of a system of atoms or molecules is governed by classical molecular potentials and Newtonian mechanics. As an example, let us consider copper. Following Greenspan (1997), its 6–12 Lennard-Jones potential is φ(r ) = −

1.398068 −10 1.55104 −8 10 + 10 erg. r6 r 12

(4.37)

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(a) Z

Y X –4 0 0 –6

–4

2 –2

0 Y (cm)

2

4

6

m)

–2 X (c

Z (cm)

June 1, 2007

4

(b) FIGURE 4.14 Particle models and intermediate stages of fracture in (a) 2D and (b) 3D. (After Wang et al. [2006]).

˚ It follows that the interaction force between two Here r is measured in A. copper atoms is F (r ) = −

dφ(r ) 8.388408 −2 18.61248 =− 10 + dyn. dr r7 r 13

(4.38)

˚ and φ then attains the minimum: In (4.38) F (r ) = 0 occurs at r0 = 2.46A, φ(r0 ) = 3.15045 · 10−13 erg. Let us next recall Ashby and Jones’s (1980) simple method to evaluate Young’s modulus E of the material from φ(r ): compute E=

S0 r0

where

S0 =

d 2 φ(r ) |r . dr 2 0

(4.39)

With this method, we obtain Young’s modulus of copper as 152.942 GPa, a number that closely matches the physical property of copper and copper alloys valued at 120∼150 GPa. Ashby and Jones also defined the continuumtype tensile stress σ (r ) = NF (r ),

(4.40)

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where N is the number of bonds/unit area, equal to 1/r02 . Tensile strength, σTS , results when d F (r )/dr = 0, that is, at rd = 2.73A˚ (bond damage spacing), and yields σTS = NF (rd ) = 462.84 MN/m2 .

(4.41)

This value is quite consistent with data for the actual copper and copper-based alloys reported at 250∼1000 MPa. In the PM, the interaction force is also considered only between nearestneighbor (quasi-)particles and assumed to be of the same form as in MD φ(r ) = −

G H + q. p r r

(4.42)

Here G, H, p, and q , all positive constants, are yet to be determined, and this will be done below. Inequality q > p must hold so as to obtain the repulsive effect that is necessarily (much) stronger than the attractive one. Three examples of interaction force for three pairs of p and q are displayed in Figure 4.15(a). The dependence of Young’s modulus for a wide range of p and q is shown in Figure 4.15(b). The conventional approach in PM, just as in MD, is to take the equation of motion for each particle Pi of the system as Gi d 2 ri Hi rji mi 2 = α − p + q i= j, (4.43) dt rij rij rij j where mi is the mass of Pi and rji is the vector from P j to Pi ; summation is taken over all the neighbors of Pi . Also, α is a normalizing constant obtained by requiring that the force between two particles must be small in the presence of gravity: Gi Hi α − p + q < 0.001 · 980mi . (4.44) D D Here D is the distance of local interaction (1.7r0 cm in this particular example), where r0 is the equilibrium spacing of the particle structure. The reason for introducing the parameter α by Greenspan (1997) was to define the interaction force between two particles as local in the presence of gravity. However, since setting α according to (4.44) would result in a “pseudo-dynamic” solution, we set α = 1. According to equation (4.42), different ( p, q ) pairs result in different continuum-type material properties, such as Young’s modulus E. Clearly, changing r0 and volume of the simulated material V (= A × B × C) will additionally influence Young’s modulus. Therefore, in general, we have some functional dependence E = E( p, q , r0 , V).

(4.45)

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Interaction force (KN)

10

0

–10 (p, q) = 3, 5 (p, q) = 5, 10 (p, q) = 7, 14

–20

–30

–40

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Equilibrium position r0 (cm)

0.9

1

(a) 15 14 13 12 11 10 9 q

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1

2

3

4

5

6

7 8 p

9

10 11 12 13 14

Young’s modulus (GPa)

14

37 59

82 104 127 149 172 195 217 240 262 285 307 330 (b)

FIGURE 4.15 (a) The interaction force for pairs of ( p, q ) exponents, at r0 = 0.2 cm. (b) The variability of Young’s modulus in the ( p, q )-plane.

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In Wang and Ostoja-Starzewski (2005) we formulated four conditions to determine continuum-level Young’s modulus and tensile strength, while maintaining the conservation of mass and energy of the particle system and satisfying the interaction laws between all the particles in the PM model for a given MD model. 4.3.1.2 Leapfrog Method Just like in MD, there are two commonly used numerical schemes in particle modeling: the completely conservative method and the leapfrog method. The first scheme is exact in that it perfectly conserves energy, linear, and angular momentum, but requires a very costly solution of a large algebraic problem. The second scheme is approximate. Since in most problems one needs large numbers of particles to adequately represent a simulated body, the completely conservative method is unwieldy and, therefore, usually abandoned in favor of the leapfrog method (Hockney and Eastwood, 1999). That method is derived by considering Taylor expansions of positions ri,k+1 and ri,k of the particle Pi (i = 1, 2, . . . , N) at times tk = kt and tk+1 = (k +1)t, respectively, about time tk+1/2 = (k + 1/2)t (with t being the time step): ri,k = ri,k+1/2 − ri,k+1 = ri,k+1/2 +

t t 2 t 3 vi,k+1/2 + ai,k+1/2 − a˙ i,k+1/2 + O(t 4 ) 2 8 48 t t 2 t 3 vi,k+1/2 + ai,k+1/2 + a˙ i,k+1/2 + O(t 4 ). 2 8 48

(4.46)

Here vi and ai denote velocity and acceleration. Upon addition and subtraction of these we get the new position and velocity ri,k+1 = 2ri,k+1/2 − ri,k +

t 2 ai,k + O(t 4 ) 4

vi,k+1/2 = (ri,k+1 − ri,k )/t + O(t 2 ),

(4.47)

which shows that the position calculation is two orders of magnitude more accurate than the velocity calculation. However, the error in computation of velocity accumulates only as fast as that in position because it is really being calculated from positions. It is easy to see that the leapfrog method is more accurate than the conventional Euler integration based on vi,k+1 = vi,k + (t)ai,k and ri,k+1 = ri,k + (t)vi,k . Often the leapfrog formulas relating position ri , velocity vi , and acceleration ai for all the particles Pi (i = 1, 2, . . . , N) are written as t ai,0 (starter formula) 2 = vi,k−1/2 + (t)ai,k k = 0, 1, 2, . . .

vi,1/2 = vi,0 + vi,k+1/2

ri,k+1 = ri,k + (t)vi,k+1/2

k = 0, 1, 2, . . . .

(4.48)

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Clearly, the name of the method comes from taking velocities at intermediate time steps relative to positions and accelerations; it is also known as a Verlet algorithm. It can be shown that the global (cumulative) error in position going from ri,k to ri,k+n (i.e., over T = nt) of Pi is error(ri,k+n − ri,k ) = O(t 2 ).

(4.49)

By the argument following (4.46) above, the global error in velocity is also O(t 2 ). Stability is concerned with the propagation of errors. Even if the truncation and round-off errors are very small, a scheme would be of little value if the effects of small errors were to grow rapidly with time. Thus, instability arises from the nonphysical solution of the discretized equations. If the discrete equations have solutions that grow much more rapidly than the correct solution of the differential equations, then even a very small round-off error is certain to eventually seed that solution and render the numerical results meaningless. By the root locus method for an atomistic unit of time, the safe time-step used in the leapfrog method meeting this requirement is 1/2 1 dF t 2 = . (4.50) m dr max We see that, as r → 0, d F /dr → ∞, which results in t → 0. Because this may well cause problems in computation, we introduce the smallest distance between two particles according to these conditions: 1. For a stretching problem of a plate/beam, take (dF/dr) max dF/ dr|r =r0 , which with (4.50) dictates t 10−7 − 10−6 s. 2. For an impact problem, one often needs to set up a minimum distance limiting the spacing between two nearest particles, e.g., rmin = 0.1r0 . It is easy to see from Figure 4.15(a) that, in this case, this suitable time increment is greatly reduced because of a rapid increase in . This leads to t 10−8 s. Following the MD methodology (Napier-Munn et al., 1999), we can also √ set up a criterion for convergence: t < 2 m/k, where m is the smallest mass to be considered, k is the same stiffness as S0 in (4.39)2 . An examination of these two criteria shows there is not much quantitative difference between them in the case of elastic or elastic-brittle, but not plastic, materials. 4.3.2 Examples The maximum entropy formalism of Section 2.6 in Chapter 2 is much more suited to deal with quasi-static rather than dynamic fracture. The dynamic character of fracture in these experiments, combined with the presence of

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multiple incipient spots, was also a big challenge in several computational mechanics models reviewed in Al-Ostaz and Jasiuk (1997) employing commercial finite element programs, as well as in an independent study using a meshless element program (Belytschko et al., 1995). Upon trying various failure criteria, several numerical methods, and even being forced to initialize the cracking process by a subjective choice in the meshless model, the modelers (including this author) have run into uncertainty as to which modeling aspect is more critical, and whether there is a way to clarify it. The recent study by Ostoja-Starzewski and Wang (2006) was motivated by this outstanding challenge, and offered a way to test the PM vis-`a-vis experiments. In Section 2.6 of Chapter 2 we considered crack patterns in epoxy plates perforated by holes. Two analyses—one based on the minimum potential energy formulation and one based on the maximum entropy method—relied on the assumption of quasi-static response. Strictly speaking, although the loading was static, the fragmentation event was dynamic. Clearly, the preparation of mineral specimens—involving measurement of highly heterogeneous and multiphase microstructures—for a direct comparison with the model prediction is very hard. Thus, we can apply the model to the experimentally tested plate with 31 holes and follow this strategy: 1. Decrease the lattice spacing until we attain mesh-independent crack patterns. 2. Find out whether the lattice of (1) will also result in the most dominant crack pattern of Figure 4.16. Indeed, the crack patterns “stabilize” as we refine the mesh. 3. Assuming the answer to (2) is positive, introduce weak perturbations in the material properties—either stiffness or strength—to determine which one of these has a stronger effect on the deviation away from the dominant crack pattern, that is, on the scatter in Figure 4.16. 4.3.2.1 Other Models The PM is but one of the variations on the theme of MD. Here are some other possibilities: •

Molecular statics (MS)—by disregarding the inertia forces, it involves a static solution of the system of atoms (Chen, 1995; Vinogradov, 2006). Although the MD allows simulations of large systems with a constraint to very short time scales (transient phenomena of the order of nanoseconds), the MS allow large (macroscopic type) time scales albeit with a limitation by the size of a (nonlinear) algebraic system one is able to solve and a restriction to 0 K.

•

Derivation of a continuum model from a microscopic model based on the assumption that the displacements on the macroscopic level are the same as those on the molecular level (Blanc et al., 2002).

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FIGURE 4.16 Final crack patterns for four mesh configurations at ever finer lattice spacings: (a) r0 = 0.1 cm; (b) r0 = 0.05 cm; (c) r0 = 0.02 cm; (d) r0 = 0.01 cm. (From Ostoja-Starzewski and Wang, 2006. With permission.)

•

Introduction of a finite extension and spin for continuum-type particles (Yserentant, 1997).

Direct incorporation of interatomic potentials into a continuum analysis on the atomic scale (Zhang et al., 2002). • Computational mechanics of granular media (Hermann and Luding, 1998). •

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165

Michell Trusses: Optimal Use of Material

4.4.1 Study via Hyperbolic System 4.4.1.1 Governing Equations The classical problem of an optimal truss concerns a minimum-weight design of a planar truss T that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range σ0 ≤ σ ≤ σ0 (Michell, 1904). This forms the basis for a problem of layout of a truss whose locally orthogonal members are of a rigidperfectly plastic material with tensile and compressive stresses ±σ0 , (Save`e and Prager, 1985; Rozvany et al., 1995, and references therein). The solution to this problem is provided by a so-called Michell truss-like continuum, whose members form a dense structure having the geometry of an orthogonal net of characteristics. That is, as the mesh spacing becomes infinitesimally fine, the volume (and hence the weight) of the material reaches a minimum. Allowable stresses in all the truss members are in the range σ0 ≤ σ ≤ σ0 . Given a modulus E, we have a range for strains −k ≤ ε ≤ k, where k = σ0 /E. As pointed out by Rozvany et al. (1995), equal permissible stresses in tension and compression are necessary for the Michell (1904) criteria to hold. With length li and cross-sectional area Ai of bar i, the design variables are the yield forces Yi = σ0 Ai , and the design objective is the minimization of the cost =

Yi li ,

(4.51)

i

which is proportional to the total volume of the bars V = i Ai li . As an example let us now consider a problem of optimal layout of a truss set up on a rigid circular foundation F , which can support a force P acting at a point A, Figure 4.17. The solution is provided by a so-called field of type T (Save and Prager, 1985) for which the principal strains have equal absolute values k, but carry opposite signs, and the principal lines are logarithmic spirals. Indeed, plots (a)–(d) of this figure show a sequence of four ever-finer trusses providing supports under the same global conditions (the foundation F and the force P). These trusses are based on, respectively, 2n + 1 (n = 2, 3, 4, and 5) boundary nodes. With n growing, their geometries tend to an optimal truss-like continuum whose principal strain/stress directions are mutually orthogonal characteristics of a quasi-linear hyperbolic system—this is discussed below. This convergence of trusses (of volume V ∗ = V(n), n finite) to the optimal truss-like continuum (of volume V = V(n)|n→∞ ) can be quantified by the efficiency E f f = V/V ∗ . We indicate Eff in all four cases of Figure 4.17.

(4.52)

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P

A

F

A

F

(a) Eﬀ = 0.939

(b) Eﬀ = 0.965 P

P

A

F

A

F

(c) Eﬀ = 0.981

(d) Eﬀ = 0.991

FIGURE 4.17 Successive approximations to the Michell truss, all governed by (3.6) for a homogeneous material, according to meshes based on, respectively, 2n + 1(n = 2, 3, 4 and 5) boundary points on the rigid foundation F.

Our presentation of the governing equations follows Hegemier and Prager (1969). First, we let u and v be the displacement components with respect to the x, y coordinates in the plane of the truss. Then

εx =

∂u ∂x

εy =

∂v ∂y

γ =

∂v ∂x

+

∂u ∂y

/2

W=

∂v ∂x

−

∂u ∂y

/2

(4.53)

are the strain components and the rotation. If by θ we denote the angle between the negative y-direction and arbitrarily assigned positive direction along the line with the unit extension k, then εx = −k cos 2θ

ε y = k cos 2θ

γ = −k sin 2θ.

(4.54)

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It now follows from the above that ∂u ∂v ∂x

∂x

∂u

= −k cos 2θ

∂y

= k(2w − sin 2θ)

= −k(2w + sin 2θ) ∂v ∂y

= k cos 2θ,

(4.55)

where w = W/k. Eliminating u and v from (4.55) by cross-differentiation, we obtain k k

∂w ∂x ∂w ∂y

+ k cos 2θ + k sin 2θ

∂θ ∂x ∂θ ∂x

+ k sin 2θ − k cos 2θ

∂θ ∂y ∂θ ∂y

= 0, (4.56) = 0.

and, setting θ = π/2, we find equations d (w − θ) = 0 ds1

d (w + θ) = 0, ds2

(4.57)

which hold along two characteristics s1 and s2 , at angles specified, respectively, by θ = α + π/4

θ = α + 3π/4.

(4.58)

α is defined as the angle formed by the positive direction along the foundation F with the positive x-direction. On this boundary, w = −1. w = −1.

(4.59)

4.4.1.2 Example of an Optimal Layout In practice, as illustrated by the example of Figure 4.17, the stochastic quasilinear hyperbolic system governing the field is solved by finite differences. At the typical point Q of the foundation F , let the positive direction along this boundary form the angle α with the positive x-direction. As the rigid foundation is inextensible, its tangent and normal at Q bisect the right angles formed by the principal axes of strain at Q. As u vanishes along F , cos α ∂∂ux + sin α ∂∂uy = 0 at Q. From this, along with (4.55) and (4.58) there follows w = ∓1

(4.60)

along F ; in Figures 4.1 and 4.3 the upper sign in the above is appropriate. Thus, we have an inverse Cauchy problem: “find the net of characteristics supporting the given load P at point A, that emanates from the foundation F with conditions (4.58) and (4.59) specified on it.” Figures 4.17(a)–(d) display four deterministic solutions, all governed by (4.57) according to meshes based on, respectively, 2n +1 (n = 2, 3, 4 and 5) boundary points. In the limit n → ∞, we arrive at a truss-like continuum with Eff = 1.

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In case one wants to make (i.e., manufacture) the Michell truss from a metal plate—such as a polycrystal—one encounters the effect of random material microstructure. In effect, the denser the truss—or the finer the mesh spacing— the more significant is the effect of microstructural fluctuations on the plastic limit. This leads to a question we address in Chapter 8: Can one truly reach the limit of truss-like continuum as the mesh is refined ad infinitum? 4.4.2 Study via Elliptic System The foregoing approach applied when one assumed a fixed load and searches for a structure of optimal (i.e., minimal) weight. Another approach, much more in line with the conventional methodology of shape optimization of engineering structures (Bendsøe and Kikuchi, 1988), is to seek the shape of a structure with minimum compliance (maximum stiffness), that possesses a prescribed weight. The minimum compliance problem for a planar (respectively, spatial) body B of volume V in R2 (R3 ) subjected to body forces f and tractions t takes the form: min L(u)

(4.61)

Cijkl ∈ Uad subject to a C (u, v) = L(v), all v ∈ U, design constraints, where a C (u, v) =

ε(u)C(x)ε(v)dx

(4.62)

B

is the energy bilinear form, and

L(u) =

fvdx+ B

∂ Bt

tvdx

(4.63)

is the load linear form. That is, we seek the optimal choice of stiffness tensor C in some given set of admissible tensors Uad . C are generally fields over R2 , so that Uad ∈ (L ∞ (V)) 6 , corresponding to the six independent elements of inplane stiffness tensor. By the “design constraints” we understand constraints on stresses, strains, displacements, etc., while sizing constraints, volume constraints, etc., are accounted for in the choice of Uad . Finally, U is the space of kinematically admissible displacement fields. In the case of optimal shape design, elements C(≡ Cijkl ) of Uad take on the form Cijkl (x) = χ (x)C ijkl ,

(4.64)

where C ijkl is the constant stiffness tensor for the material employed for the construction of the mechanical element, and χ (x) is the indicator function. The discretized formulation of the topology optimization problem can then

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be stated as follows: min f (ρ) s.t. V = ρ j νi ≤ V ∗

(4.65)

η ≤ ρi ≤ 1, i = 1, . . . , N, where f represents the objective function, ρi and νi are element densities and volumes, respectively, V ∗ is the target volume, N is the total number of elements and η is the small number that prevents stiffness matrix from being ill-conditioned. The common objective function is the weighted sum of compliances under all load cases. Note that the problem in (4.65) is a relaxation formulation of the topology problem, where the density should only take value 0 or 1. To force the design to be close to a 0/1 solution, a penalty is introduced to reduce the efficiency of intermediate density elements, namely, by a following power law formulation Ki∗ (ρ) = ρi Ki p

(4.66)

where Ki∗ and Ki represent the penalized and the real stiffness matrix of the ith element, respectively, and p is the penalization factor that is bigger than 1. For a survey of historical development and a summary of theory and techniques of topology optimization, see Rozvany et al. (1995), while for the penalty formulation see Allaire and Kohn (1993). In Chapter 8 we return to Michell trusses in the setting of materials possessing microstructural randomness.

Problems 1. The extension of the condition (4.1) in 3D is the so-called Maxwell’s rule: |E| = 6|V| − 6. Justify this equation. 2. Verify that the network of Figure 4.1 is generically rigid. 3. Using the graph representation, formulate the virtual work principle in granular media relating the work of forces and moments of grain-grain interactions with the work of forces and moments acting at the boundary of the granular medium. 4. In the case of iron, the interatomic potential is φ(r ) = D e −2a (r −r0 ) − 2e −a (r −r0 ) ˚ −1 and where the binding energy D = 0.4174eV, a = 1.3885( A) ϒ0 = 2.845 A˚ (Milstein, 1971). Compute E, c and ωmax . To see the

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1 i

1 K (ui+1 − ui ) 2 + C(ui+1 − ui ) 3 . 2 3

Derive the resulting dynamical equation generalizing (3.2). Then, using a Taylor expansion with terms up to the fourth derivative, obtain the continuum equation s2 E u, xx + u, xxxx + a u, x u, xx = ρ u. ¨ 12 Identify E, α and ρ. 6. Consider two particles (of mass mi , i = 1, 2 located at ri , respectively, separated by a distance rij = rji , and interacting via a centralforce potential φ (rij ). In the completely conservative method of particle modeling (PM) one replaces their Newtonian equations of motion (no summation convention and no tensor index notation) mi

d 2 ri ∂φ ri − r j =− , 2 dt ∂ri j ri j

i, j = 1, 2

by two first order difference equations ri,n+1 − ri,n Vi,n+1 − Vi,n = t 2 φ(ri j,n+1 ) − φ(ri j,n ) ri,n+1 + ri,n − r j,n+1 − r j,n vi,n+1 − vi,n mi =− , i, j =r1, 2 − ri j,n t ri j,n+1 + ri j,n i j,n+1 Where the first subscript specifies the particle number while the second subscript indicates the time t or t+t. These equations can be solved by the Newton’s method. Show that the numerical solution conserves the total energy, the linear and the angular momentum. 7. Consider the system of three particles interacting via central forces. Write their Netwonian equations of motion, and then the corresponding first order difference equations. 8. Show that the numerical method discussed in Problem 5 is invariant with respect to rotations of the frame of coordinates. 9. Verify (4.56), and derive a more general version of the equations governing w and θ for the case of k being smooth in x and y.

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5 Two- Versus Three-Dimensional Classical Elasticity∗

Doing physics in Flatland requires no apologies: many areas of theoretical and condensed matter physics have benefitted greatly from such studies. U. Frisch, 1995 Classical (linear) elasticity is a very old subject, and its planar/2D cases (plane stress and plane strain) equally so. However, the possibility of having the same stress field in 2D materials whose (generally spatially inhomogeneous) elastic moduli are different but satisfy certain relations is a relatively new result. This reduced parameter dependence has consequences for the effective moduli of composite materials, including the special case of a plate perforated by holes (especially up to the percolation point), and lends itself to extensions, such as the presence of body force fields or thermal stresses. These issues, including a short section on poroelasticity, are reviewed in this chapter.

5.1

Basic Relations

5.1.1 Isotropic Relations 5.1.1.1 Three-Dimensional Elasticity The constitutive relations for a linear elastic isotropic 3D material are ε11 =

1 [σ11 − ν3D (σ22 + σ33 )] E 3D

ε12 =

1 + ν3D σ12 E 3D

(5.1)

together with cyclic permutations 1 → 2 → 3. Here E 3D and ν3D stand for conventional 3D Young’s modulus and Poisson’s ratio, and this convention is followed with respect to other quantities. (5.1) is, of course, equivalent to εi j = ∗ Chapter

1 i, j, k = 1, 2, 3 (1 + ν3D ) σi j − ν3D σkk δi j E 3D

(5.2)

written jointly with I. Jasiuk.

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or σi j = λ3D εkk δi j + 2µ3D εi j ,

i, j, k = 1, 2, 3

(5.3)

where the classical relations involving the Lam´e constants and bulk modulus are well known: µ3D =

E 3D 2 (1 + ν3D )

λ3D =

2ν3D µ3D 1 − 2ν3D

κ3D =

E 3D 2 = λ3D + µ3D . 3 (1 − 2ν3D ) 3 (5.4)

5.1.1.2 Two-Dimensional Elasticity In the 2D elasticity there is no x3 direction and thus ε11 , ε22 , and ε12 are the only strains and σ11 , σ22 , and σ12 are the only stresses. Also, there is just one compatibility equation ε22,11 + ε11,22 = 2ε12,12 . For the Hooke law, we have 1 ε11 = [σ11 − ν2D σ22 ] E 2D

ε12 =

(5.5) 1 + ν2D σ12 , E 2D

(5.6)

with cyclic permutation 1 → 2, where the subscript 2D indicates planar (or area) material properties. The above is equivalent to 1 (5.7) εi j = i, j, k = 1, 2 . (1 + ν2D ) σi j − ν2D σkk δi j E 2D Now, in analogy to (5.3) above, we can write σi j = λ2D εkk δi j + 2µ2D εi j ,

i, j, k = 1, 2

(5.8)

and work out expressions for the planar Young modulus and planar Poisson ratio: E 2D = 4µ2D

λ2D + µ2D λ2D + 2µ2D

ν2D =

λ2D . λ2D + 2µ2D

(5.9)

In analogy to the 3D case (e.g., Ziegler, 1983), we can work out the basic inequalities that hold between these planar moduli. First, consider three tests: 1. Uniaxial stress occurring in a narrow strip subjected to tension or compression in, say, x1 direction and defined by the condition that of all three stress components only a single normal stress σ11 is nonzero. The equation (5.6) implies then that ε11 =

1 σ11 E 2D

ε22 =

−ν2D σ11 E 2D

ε12 = 0.

(5.10)

2. Hydrostatic stress (= − p) which leads to ε11 + ε22 = εkk =

−1 p, κ2D

(5.11)

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where we identify the planar bulk modulus κ2D κ2D =

E 2D = λ2D + µ2D . 2 (1 − ν2D )

(5.12)

Of course, in 3D the shear modulus is accompanied by a factor 2/3, as given in equation (5.4). 3. Simple shear stress defined by the condition σ12 = σ21 and σ11 = σ22 = 0, which yields ε12 + ε21 = γ12 =

1 σ12 , µ2D

(5.13)

where the planar shear modulus is given by the same formula in 2D as in 3D µ2D =

E 2D . 2 (1 + ν2D )

(5.14)

Stability of the material—that is, positivity of E 2D , κ2D and µ2D —imposes three inequalities which follow from the above tests: λ2D + 2µ2D > 0

λ2D + µ2D > 0

µ2D > 0.

(5.15)

If µ2D obeys the third of these inequalities, the first one is a consequence of the second and can be dropped. In Figure 5.1 we show the region in the (λ2D ,µ2D )-plane where the two remaining inequalities are satisfied. µ 2D

µ2D = –λ 2D Q

q

λ 2D 0 FIGURE 5.1 Region where the inequalities (5.15) are satisfied.

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On its entire left-hand boundary, Poisson’s ratio given by (5.9)2 assumes the value −1. Since ∂ν2D ∂λ2D

=2

µ2D [λ2D + 2µ2D ]2

> 0,

(5.16)

ν2D increases monotonically on any vector q parallel to the λ2D axis, tending toward 1 with increasing distance from the point Q, Figure 5.1. Note from (5.9)2 that ν2D is seen to range from −1 through +1, in contradistinction to ν3D , which is bounded by −1 and 1/2. 5.1.2 Plane Elasticity Models There are two classes of plane elasticity problems that can be solved by using a reduced form of 3D formulation: plane strain (such as a long cylinder) and plane stress (such as a thin plate). Plane strain. In that case, one requires u3 = 0 in (5.1) along with the independence of all the fields with respect to the x3 direction, so that ε33 = 0, which leads to the following equations: ε31 = ε32 = 0, but σ33 = ε11 =

1 2 2 1 − ν3D σ11 − ν3D + ν3D σ22 E 3D

ε12 =

1 + ν3D σ12 E 3D

(5.17)

again with the cyclic permutation 1 → 2. A comparison with (5.6) readily shows that the relations between the 2D (planar) and the 3D are as follows: 2 1 1 − ν3D = E 2D E 3D

2 ν2D ν3D + ν3D = E 2D E 3D

1 + ν2D 1 + ν3D = . E 2D E 3D

(5.18)

This is a mapping of two constants onto two constants, so that only two relations of the above three are independent. Of particular interest is the relation between the plane strain Poisson ratio and the 3D Poisson ratio ν2D =

ν3D . 1 − ν3D

(5.19)

Now, an inspection of (5.17)3 immediately reveals that the 2D shear modulus has the same form as the 3D one, while applying the concept of bulk modulus to relations (5.2), we infer the plane strain κ2D , which has the form differing from 3D elasticity (5.4)3 . Thus, as already shown in (5.12) and (5.14), κ2D =

E 2D 2 (1 − ν2D )

µ2D =

E 2D . 2 (1 + ν2D )

(5.20)

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Two other very useful relations linking these two-dimensional moduli E 2D , ν2D , κ2D , and µ2D can readily be inferred 4 1 1 = + E 2D κ2D µ2D

ν2D =

κ2D − µ2D . κ2D + µ2D

(5.21)

Plane stress. In that case, one requires σ33 = σ31 = σ32 = 0 (but ε33 = 0) along with the independence of all the fields with respect to the x3 direction, which leads to ε11 =

1 [σ11 − ν3D σ22 ] E 3D

ε12 =

1 + ν3D σ12 , E 3D

(5.22)

with cyclic permutation 1 → 2. A comparison with (5.6) readily shows that the following relationships between these plane-stress and 3D moduli hold ν2D ν3D = E 2D E 3D

1 1 = E 2D E 3D

1 + ν2D 1 + ν3D = . E 2D E 3D

(5.23)

Again the third of these relations is redundant, but, collecting all, we see that E 2D = E 3D

ν2D = ν3D

µ2D =

E 2D 2 (1 + ν2D )

κ2D =

E 2D . (5.24) 2 (1 − ν2D )

Note that (5.9) holds again. A detailed discussion of all these relationships is given in Thorpe and Jasiuk (1992); we also reproduce their Table 1 here (Table 5.1). Unified treatment. Further down, it will be useful to deal with the Hooke law in this form: 4εi j = 2Sσi j + ( A − S) σkk δi j ,

i, j, k = 1, 2

(5.25)

TABLE 5.1

Elastic Constants in Plane Strain and Plane Stress Expressed in Terms of Conventional 3D Moduli Compliance Bulk modulus Shear modulus Young’s modulus Poisson’s ratio

Plane Strain

Plane Stress

κ2D = κ3D + µ3D /3 µ2D = µ3D

κ2D =

2 E 2D = E 3D / 1 − ν3D ν2D = ν3D / (1 − ν3D )

9κ3D µ3D 3κ3D +4µ3D

µ2D = µ3D

E 2D = E 3D ν2D = ν3D

Note: From Thorpe and Jasiuk, 1992. With permission.

2D Relations κ2D = µ2D =

E 2D 2(1−ν2D ) E 2D 2(1+ν2D )

E 2D ν2D

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involving two planar compliances: bulk compliance A and shear compliance S: A=

1 −1 = κ2D 2µ2D

S=

1 , µ2D

(5.26)

where = (3 − ν2D ) / (1 + ν2D ) is the Kolosov constant for plane strain as well as plane stress. Note also that in 3D: plane strain: = 3 − 4ν3D plane stress: =

3 − ν3D . 1 + ν3D

(5.27)

Finally, a uniaxial compliance C C=

1 +1 = E 2D 8µ2D

(5.28)

will also be useful. Observe from (5.21)1 that A + S = 4C.

(5.29)

5.1.3 Special Planar Orthotropies The constitutive relations for a linear elastic planar material may be written in a polar form (Jones, 1975; Vannucci, 2002; Vannucci and Verchery, 2001): C1111 = T0 + 2T1 + R0 cos 4 0 + 4R1 cos 2 1 C1112 = R0 sin 4 0 + 2R1 sin 2 1 C1122 = −T0 + 2T1 − R0 cos 4 0 C1212 = T0 − R0 cos 4 0

(5.30)

C2212 = −R0 sin 4 0 + 2R1 sin 2 1 C2222 = T0 + 2T1 + R0 cos 4 0 − 4R1 cos 2 1 Here T0 , T1 , R0 and R1 are scalars, while 0 and 1 are angles, and these six quantities are related to the six values for Cijkl through 8T0 = C1111 − 2C1122 + 4C1212 + C2222 8T1 = C1111 + 2C1122 + C2222 8R0 e 4i 0 = C1111 − 2C1122 − 4C1212 + C2222 + 4i (C1112 − C2212 )

(5.31)

8R1 e 2i 1 = C1111 − C2222 + 2i (C1112 + C2212 ) The six in-plane compliances Sijkl may be expressed through equations of the same form as (5.30), except that one has to replace T0 through 1 by, say,

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lower-case letters t0 through φ1 ; then a form fully analogous to (5.31) holds for expressing t0 through φ1 in terms of the Sijkl values. The polar form has several interesting features: 1. In a coordinate system rotated by θ around the x3 axis the Cijkl s are expressed by the equations (5.30) with 0 and 1 replaced by 0 −θ and 1 − θ, respectively. 2. The five quantities T0 , T1 , R0 , R1 , and 0 − 1 are invariant with respect to the orientation θ, and are thus invariant. 3. Each component is a Fourier expansion with at most three terms, so that the period equals π or 2π . 4. Orthotropy (C1112 = C2212 = 0) occurs for 0 − 1 being a multiple of π/4, and then the number of independent parameters is only four: T0 , T1 , (−1) K R0 , R1 , with K being either 0 or 1. 5. For an orthotropic material, the usual thermodynamic restrictions result in: T0 > R0

T1 T0 + (−1) K R0 > 2R12

(5.32)

6. One special type of orthotropy occurs when R0 = 0, and this corresponds to the absence of the 4θ harmonic and the shear modulus being invariant to rotations. However, this does not imply r0 = 0, so that the shear compliance is dependent on θ, and the Sijkl tensor still depends on both harmonics. 7. Another special type of orthotropy occurs when R1 = 0; this corresponds to the absence of the 2θ harmonic and implies r1 = 0 for the Sijkl tensor. 8. Isotropy occurs when R0 = R1 = 0, and then, obviously, there are just two independent invariants. 9. In the case of Cauchy symmetry (Cijkl = Cik jl ), isotropy reduces to the stiffness being described by just one elastic constant; recall Chapter 3.

5.2

The CLM Result and Stress Invariance

5.2.1 Isotropic Materials 5.2.1.1 Basic Result Consider a planar elastic solid occupying a simply connected domain B in (x1 , x2 )-plane, in static equilibrium σi j, j (x) = 0

or ∇ · σ (x) = 0.

(5.33)

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Elastic moduli are isotropic and assumed to be twice-differentiable. The solid is subjected to traction boundary conditions σ ji (x) n j = ti(n) (x)

or σ (x) · n = t(n) (x)

∀x ∈ ∂ B,

in such a way that the global equilibrium is satisfied (n) t (x) d S = 0 x × t(n) (x) d S = 0. ∂B

∂B

(5.34)

(5.35)

Now, substituting (5.25) into the compatibility condition (5.5), and using (5.33), we obtain, after some manipulations, ∇ 2[

A+ S (σ11 + σ22 )] − S,11 σ11 + 2S,12 σ12 + S,22 σ22 = 0. 2

(5.36)

The following question may now be asked: “Supposing that A and S are and changed to some A S, then under what restrictions would the original stress field (σ11 , σ22 , σ12 ) remain unchanged?” Denoting by a hat the quantities pertaining to the new material, the so-called invariance of the stress field is written as σ (x) = σ (x)

(5.37)

An examination of (5.36) implies that we must have + A S = m ( A + S)

S,11 = mS,11

S,22 = mS,22

S,12 = mS,12

(5.38)

= mC C

(5.39)

where m is an arbitrary scalar. Note that this means =m A + a + bx + cy A

S = mS − a − bx − cy

where the third equality comes from (5.29). The constants m, a , b, and c are subject to restrictions dictating that the compliances be non-negative. The result that the stress field is unchanged (invariant) under such a shift of compliances is called the CLM stress invariance or transformation after the authors (Cherkaev et al., 1992); see also Milton (2002). However, their original result was that =A + c A

S= S−c

= C, C

(5.40)

while the shift linear in x and y (5.39) is due to Dundurs and Markenscoff (1993). Note that in the original paper of Cherkaev et al. (1992) 1 1 1 = + κ2D

κ2D

1 1 1 = − , 2D µ2D

µ

(5.41)

where constant = 1/c and m = 1. The CLM result holds also for materials with two or more distinct phases (e.g., matrix-inclusion composites) with perfectly bonded or slipping interfaces, as well as for anisotropic materials (see below). Several extensions and

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generalizations of the CLM result are given in this chapter and Chapter 6. Note that a pair of materials satisfying (5.39–5.41) is called equivalent materials. In effect, one obtains a so-called reduced parameter dependence, which is important in parametric studies of composites, both experimental and theoretical. It can be used as a check for numerical and analytical calculations, it reduces the number of output parameters, and facilitates the presentation of results, leading to savings in time and space resources. 5.2.1.2 Two-Phase Composites and Dundurs Constants When dealing with a two-phase, planar composite material (with phases (1) (2) (2) 1 and 2), we have four material constants, for example, κ2D , µ(1) 2D , κ2D , µ2D . (1) (1) (2) (2) This, of course, is equivalent to A , S , A , and S , which undergo the CLM shift A(1) = A(1) + c, S(1) = S(1) − c, S(2) = S(2) − c, A(2) = A(2) + c,

(5.42)

and the stress field remains the same under traction boundary conditions. This implies a reduction in the number of independent constants by one. That result is directly related to the result of Dundurs (1967, 1969). In general, we may expect the stress field to depend on the magnitude of the external loading, expressed by σ0 , and on three dimensionless parameters of material constants: (1) (2) (1) (2) (2) σ (x) = σ0 g x; κ2D (5.43) /κ2D , µ(1) 2D /κ2D , µ2D /κ2D , where g is some second-rank tensor function. Dundurs (1967, 1969) has also shown that, for plane problems, only two dimensionless parameters, α12 =

C (1) − C (2) 1/E (1) − 1/E (2) = (1) (2) C +C 1/E (1) + 1/E (2)

β12 =

A(1) − A(2) 1/κ (1) − 1/κ (2) = , (1) (2) 4 C +C 4 1/E (1) + 1/E (2)

(5.44)

are needed; these are called Dundurs constants. Thus, (5.43) may be replaced by σ (x) = σ0 g (x;α12 , β12 ) .

(5.45)

The choice of α12 and β12 is not unique, and other combinations of elastic constants are possible. Note that α12 and β12 are invariant under the CLM shift. Dundurs’ result holds for two-phase materials with perfectly bonded and slipping interfaces. A generalization to composites with many phases was done by Neumeister (1992).

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5.2.2 Anisotropic Materials and the Null-Lagrangian Note that the 2D compliance Sijkl of an isotropic material is a following function of two constants κ ≡ κ2D and µ ≡ µ2D (here we drop 2D for simplicity of notation): I Sijkl (κ, µ) =

1 1 δi j δkl + δik δ jl + δil δ jk − δi j δkl , 4κ 4µ

(5.46)

its associated stiffness tensor is

−1 I I Cijkl = κδi j δkl + µ δik δ jl + δil δ jk − δi j δkl . (5.47) (κ, µ) = Sijkl (κ, µ) I Equivalently, noting (5.26), Sijkl (κ, µ) can also be given in terms of A and S as I Sijkl ( A, S) =

A S δi j δkl + δik δ jl + δil δ jk − δi j δkl . 4 4

(5.48)

If we let κ = and µ = − in (5.48), it follows that the shift tensor is I Sijkl ( , − ) =

1 1 1 δi j δkl − Rijkl , δik δ jl + δil δ jk = 2 4 2

(5.49)

1 1 δi j δkl − δik δ jl + δil δ jk − δi j δkl 2 2

(5.50)

where Rijkl =

represents the rotation by a right angle of a symmetric second-rank tensor. In the case of an anisotropic material, the starting point is again provided by the compatibility relation (5.5). Proceeding in the same manner as when deriving and analyzing (5.36), we conclude that the stress field remains unchanged when the material constants are modified from Sijkl to I Sijkl = Sijkl + Sijkl ( , − ). Thus, we see from (5.47) that the shift (5.46) in compliances R(1) ijkl σkl is a right-angle rotation of σkl . The local equilibrium equation also holds ∇ · σ (x) = 0,

(5.51)

and this may be written in terms of the Airy stress function ∇ 4 φ = 0.

(5.52)

σ = R · (∇∇φ) · R,

(5.53)

Here the stress is expressed by

in which

0 Rik = −1

1 0

∇∇ =

∂2 /∂x12

∂2 /∂ x1 ∂x2

∂2 /∂x1 ∂ x2

∂2 /∂ x22

.

(5.54)

Thus, Rik is a two-dimensional analog of the Levi–Civita permutation tensor.

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The foregoing development allows us to present the “shift-result” in terms of the null-Lagrangian (Cherkaev et al., 1992). It follows from (5.50) that the strain energy density 1 W Sijkl = σi j Sijkl σkl 2

(5.55)

may now be written as I W Sijkl = Rijkl φ,kl Sijmn Rmnpq φ, pq .

(5.56)

First, let us note that the a minimization of (5.53) via the Euler–Lagrange equations of (5.52) results in compatibility equations in terms of φ for a general anisotropy. On the other hand, the energy density of the shift in compliance becomes I I W Sijkl Rmnpq φ, pq . (5.57) ( , − ) = Rijkl φ,kl Sijmn Now, observing that Rkli j Rijmn Rmnpq = Rklpq

Rijkl Rijmn = δkm δln ,

(5.58)

2

1 I φ,11 φ,22 − φ,12 W Sijkl . ( , − ) = 2

(5.59)

we find

The energy (5.56) can also be written as the divergence of a vector field vk such that 1 1 φ,1 φ,22 − φ,2 φ,12 vk = φ,l Rklpq φ, pq = . (5.60) 2 2 φ,2 φ,11 − φ,1 φ,12 It follows now that vk,k = 0, or that the Euler–Lagrange equations for I I ( , − )) are satisfied identically, justifying the name of W(Sijkl ( , − )): W(Sijkl null-Lagrangian. For a study of symmetries of 2D stiffness tensors see He and Zheng (1996). 5.2.3 Multiply Connected Materials In the case of multiply connected materials, the compatibility condition (5.4) must be accompanied by the so-called Ces`aro integrals. These are three line integrals on a closed contour surrounding any one cavity

3 =

dω3

D1 =

du1

D2 =

du2

(5.61)

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where ω3 = u(2,1) . Following Mindlin and Salvadori (1950), this implies

∂ε12 ∂ε11 ∂ε22 ∂ε12 3 = d x1 + d x2 − − ∂ x1 ∂ x2 ∂ x1 ∂x2

∂ε11 ∂ε11 D1 + y0 3 = − x1 d x1 + d x2 ∂x1 ∂ x2

∂ε11 ∂ε22 ∂ε12 − x2 d x2 d x1 − −2 ∂ x1 ∂ x2 ∂ x2

∂ε22 ∂ε22 D2 − x0 3 = − x2 d x1 + d x2 ∂ x1 ∂ x2

∂ε22 ∂ε12 ∂ε11 − x1 d x1 . (5.62) d x2 + 2 − ∂ x1 ∂ x1 ∂ x2 Here D1 and D2 are dislocation vectors and 3 is a disclination, while (x0 , y0 ) are the coordinates of the starting point on the contour. Using (5.25), we express the above equations in terms of stresses

∂ ∂S ∂S 43 = ds − 2 t2 ds, (5.63) [( A + S) (σ11 + σ22 )] ds − 2 t1 ∂n ∂ x1 ∂ x2

4 ( D1 + y0 3 ) =

x2

+2

∂ ∂n

x1

−2

∂ ∂s

St2 ds − 2

4 ( D2 − x0 3 ) = −

− x1

∂ ∂n

+ x2

St1 ds + 2

( A + S) (σ11 + σ22 ) ds

x2 ∂

∂S ∂x1

t1 +

∂S ∂x2

t2 ds,

∂s

x1

(5.64)

( A + S) (σ11 + σ22 ) ds

∂S ∂x1

t1 +

∂S ∂x2

t2 ds,

(5.65)

where n and s denote the outer unit normal and arc length of the hole boundary. Let us now consider the situation of a continuous displacement field: the dislocations are not allowed, and the left-hand sides in the above equations equal zero. A transformation of the elastic compliances according to (5.39–5.41) should then leave the stress state invariant, which implies that the following equations in tractions hold:

t1 ds = t2 ds = 0 (5.66) (x1 t2 − x2 t1 ) ds = 0. These relations, also called Michell conditions, mean that the net forces and couples applied to each and every cavity must be self-equilibrated. The above results are due to Dundurs and Markenscoff (1993) for linear transformations

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[Equation (5.39)]; those authors also give an analysis of discontinuous compliances, bonded versus slipping interfaces, and intrusions. The constant transformation (5.40) requires only the satisfaction of the first two equations in (5.66)—this is in agreement with Michell (1899), who showed that the stress field in a material with holes, satisfying (5.66)1,2 , is independent of elastic constants. This result has been used extensively in photoelasticity. The effect of body forces b on the reduction of material parameters in multiply connected bodies has been further analyzed by Markenscoff and Jasiuk (1998). In particular, the Michell conditions (of zero traction on the hole boundary) have been generalized to account for the presence of b. For two-phase, multiply connected composites with either perfectly bonded or slipping interfaces, the requirement of reducibility is the continuity of the normal component of the body force across the interface, in addition to the condition that body forces need to be divergence-free:

b n ds = 0 + xb (5.67) ds = 0 ty + yb n ds = 0. (tx n) 5.2.4 Applications to Composites 5.2.4.1 Effective Moduli of Composites As mentioned in the preface, determination of the effective properties of composites is one of the primary applications of the CLM result. It is thus of interest to investigate the shift in macroscopically effective compliance tensors (1)e f f Sijkl connecting the volume averaged (with an overbar) stress and strain tensors eff

εi j = Sijkl σ kl .

(5.68)

As before, there is a restriction to materials with smooth (class C 2 ) properties. Now, stresses σ11 , σ22 , σ12 are the same in two equivalent materials (described by Sijkl (x) and εi j satisfy the Sijkl (x)), so that the strain fields εi j and relation I I εi j = Sijkl σkl = Sijkl σ kl + Sijkl ( , − ) σkl = εi j + Sijkl ( , − ) σkl .

(5.69)

I Carrying out volume averaging of (5.68), and noting that Sijkl ( , − ) is independent of position x, we find eff

eff

I εi j = Sijkl σkl = Sijkl σ kl + Sijkl ( , − ) σ kl ,

(5.70)

which shows that the effective compliance tensor of the transformed material (with hat) is given by that of the first material (without hat) plus the shift given by (5.49), that is, eff eff I Sijkl = Sijkl + Sijkl ( , − ) .

(5.71)

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When the original material is isotropic, these results are expressed by saying that, if the shift of the material properties is governed by (5.41), then the shift of the effective compliances is governed by 1 1 1 = eff +

κ κeff

1 1 1 = eff − eff

µ µ

eff = E eff , E

(5.72)

with the same . Note: This conclusion holds for simply connected inhomogeneous materials with twice-differentiable properties mentioned at the beginning of this section, as well as for composite materials (e.g., matrix-inclusion composites) with two or more discrete phases, with either perfectly bonded or sliding interfaces (Thorpe and Jasiuk, 1992; Dundurs and Markenscoff, 1993; Jun and Jasiuk, 1993; Moran and Gosz, 1994). This exact result applies to any geometry of the composite and is independent of the method used (computational or analytical) to evaluate these elastic constants. Thus, it can serve as a very useful check of such calculations. It should be noted that, strictly speaking, this result is valid under traction boundary conditions, but if the domain is large enough to qualify as the RVE (see Chapter 7), then it will also hold under displacement or mixed-orthogonal boundary conditions, that is, three basic loadings satisfied by the Hill condition, because the derived effective moduli are independent of the boundary conditions. 5.2.4.2 Plates with Holes Thorpe and Jasiuk (1992) have shown that the effective Young modulus E eff of a plate containing holes is independent of the Poisson ratio ν of the matrix material, where E is the Young modulus of the matrix, and the effective Poisson ratio ν eff flows, as the volume fraction of holes increases, toward a fixed point that is reached at percolation of holes (i.e., when E eff = 0): = E eff /E eff / E ν eff − ν eff = ν − ν E eff /E, (5.73) E This is illustrated in Figure 5.2 for a material with randomly placed circular holes (Poisson point field of Chapter 1), in which the effective Poisson ratio ν eff flows toward the fixed point of value 1/3 as the volume fraction of holes increases and reaches the percolation point at the volume fraction 2/3. The fixed point and the percolation point depend on the particular microgeometries and the approximations employed (e.g., different effective medium theories) to find the effective moduli. These results were shown to hold by various effective medium theories (Jun and Jasiuk, 1993), and were first observed numerically by Day et al. (1992). 5.2.5 Extension of Stress Invariance to Presence of Eigenstrains 5.2.5.1 Basic Concepts This section briefly reviews the results of Jasiuk and Boccara (2002). Thus, we begin with the total strain εi j being the sum of the elastic strain e i j and the

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Young’s modulus E*/E1

1.0

185

(a)

0.8 0.6 0.4 0.2 1.0 (b) 0.5

Poisson’s ratio v*

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0

–0.5

–1.0

0.4

0.6

0.8

1.0

q FIGURE 5.2 (a) Young’s modulus E eff /E for a sheet containing randomly distributed circular holes for various values of the planar Poisson ratio ν of the matrix material, in function of the fraction of material remaining q = 1− f , where f is porosity. (b) Flow of the effective planar Poisson ratio ν eff towards 2/3 for a wide range of matrix materials. (From Thorpe and Jasiuk, 1992. With permission.)

eigenstrain εi∗j εi j = e i j + εi∗j

where e i j = Sijkl σkl .

(5.74)

In elasticity with eigenstrains the material is assumed to be free from any external forces and surface constraints (Mura, 1987). If these conditions of free surface are not satisfied, the stress fields can be obtained by a superposition of the stress of a free body and the stress obtained from the solution of a given boundary value problem with non-zero external forces or boundary conditions. Following the approach of Mura (1987), by substituting (5.74) into (5.33) and assuming a homogeneous material, we have ∗ Ci jkl εkl, j = Ci jkl εkl, j,

(5.75)

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and by substituting (5.74) into (5.34) with ti(n) (x) = 0, we obtain Ci jkl εkl n j = Ci jkl εkl∗ n j .

(5.76)

Note that in the absence of eigenstrains (εkl∗ = 0), the left-hand side of (5.75) corresponds to σi j, j , and the left-hand side of (5.76) to σi j n j . Thus, (5.75) is in ∗ the form σi j, j = −Xi where Xi = −Ci jkl γkl, j and (5.76) is in the form σi j n j = ti ∗ where ti = Ci jkl εkl n j . Therefore, the contribution of the eigenstrain εi∗j to the equations of equilibrium (5.75) is mathematically equivalent to a body force, and in the boundary conditions (5.76) is similar to a surface force. We next focus on the planar elasticity with eigenstrains, assuming isotropy in elastic properties. In addition we admit non-zero tractions ti(n) (x) to make the formulation more general. This will not change our conclusions on the reduced parameter dependence. Note that, for the special case of uncoupled thermoelasticity, the eigenstrains εi∗j are defined as εi∗j = αi j T

αi j = 0

if i = j

i, j = 1, 2, 3,

(5.77)

where αi j is the thermal expansion coefficient and T is temperature change. We will refer to this special case in examples. 5.2.5.2 Planar Case In the case of an isotropic 2D material, the constitutive law is generalized from (5.25) to ∗ 4εi j = 2Sσi j + ( A − S) σkk δi j + 4εi∗j + 4ηε33 δi j ,

i, j = 1, 2,

(5.78)

where is the Kolosov constant defined in (5.27). With the assumption of both compliances and eigenstrains being smooth functions of position, the compatibility condition becomes ∗ ∗ ∇ 2 A+S − 2ε22,11 (σ11 + σ22 ) − S, 11 σ11 − S, 22 σ22 − 2S, 12 σ12 = −2ε11,22 2 ∗ ∗ ∗ ∗ + 4ε12,12 − 2∇ 2 ηε33 − 4η, 1 ε33,1 − 4η, 2 ε33,2 − 2η∇ 2 ε33 ,

(5.79)

where, given (5.26–5.27), η3D = ν3D = (1 − A/S) /2 for plane strain, and η = 0 for plane stress. Equation (5.79) remains unchanged under the linear shift (5.39), and thus there is a reduced parameter dependence, in these particular situations: •

For the plane stress case and m = 1, no extra condition is needed

•

For the plane stress case and m = 1, need to satisfy ∗ ∗ ∗ ∗ ∇ 2 ηε33 + 2η,1 ε33,1 + 2η,2 ε33,2 + η∇ 2 ε33 = 0.

•

(5.80)

For the plane stress case and m = 1, need to satisfy ∗ ∗ ∗ ε22,11 − 2ε12,12 + 2ε11,22 = 0.

(5.81)

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For the case of plane strain and m = 1, need to satisfy ∗ ∗ ∗ ∗ ∗ ε11,22 + ε22,11 − 2ε12,12 + ∇ 2 ηε33 + 2η,1 ε33,1 ∗ ∗ +2η,2 ε33,2 + η∇ 2 ε33 = 0.

(5.82)

Besides two special cases: [(1) a homogeneous material with uniform eigenstrains subject to zero traction boundary conditions, and (2) a homogeneous material with nonuniform eigenstrains subject to traction boundary conditions, Jasiuk and Boccara (2002) have also studied the situation of a twophase material with perfectly bonded interfaces. Here each case is governed by equation (5.79) with all the material and field quantities being indexed by ∗( p) ( p) p = 1, 2 (for either phase p), that is, Ap , Sp , η p , σi j , and εi j . The perfect interfaces S12 , with n and s being unit and tangential directions in the plane, are modeled via the continuity of normal and tangential tractions (1) (2) (1) (2) σnn = σnn σns = σns ,

(5.83)

and the continuity of changes in curvature and stretch strains (Dundurs, 1990) (1) (1) K (1) = K (2) εss = εss .

(5.84)

Note: the latter two equations replace the conventional conditions (2) involving the continuity of normal and tangential displacements u(1) n = un , (1) (1) us = us . The linear shift pertaining to either the phase p = 1 or p = 2 takes on the form Ap = m Ap + a + bx + cy

Sp = mSp − a − bx − cy.

(5.85)

Observing that only the boundary conditions (5.84) depend on the compliances A1 , ..., S2 , it was found that, in general, shifts (5.85) are possible in both plane strain and plane stress for either m = 1 or m = 1, providing specific, additional conditions hold. Moreover, these results carry over to both simply and multiply connected two-phase materials, whereby for the multiply connected materials with perfectly bonded interfaces, the Ces`aro integrals do not need to be considered (Markenscoff, 1996).

5.3

Poroelasticity

This theory was introduced to incorporate the change of volume fraction as an additional degree of freedom (Goodman and Cowin, 1972; Cowin and Nunziato, 1983). The constitutive equations in 3D are: σi j = λεkk δi j + 2µεi j + βφδi j

h k = αφ,k

g = −ξ φ − βεrr ,

(5.86)

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where new quantities, relative to the classical elasticity, are the equilibrated stress vector h k , the intrinsic equilibrated body force g, the change in volume fraction from the reference volume fraction φ. The Cauchy stress tensor σi j is symmetric here and the strain-displacement relation is just as in the classical case (εi, j = u(i, j) ). The equilibrium equations are σi j, j = 0

h i,i + g = 0.

(5.87)

Clearly, α, β, and ξ in (5.86) are the new material constants besides the conventional λ and µ, and, at this point, one assumes the internal energy density to be a positive quadratic form, thus implying the inequalities ξ (3λ + 2µ) − 3β 2 > 0

µ>0

α>0

ξ > 0.

(5.88)

Recently, De Cicco and Guarracino (2004) have formulated plane elastostatics of such a material in terms of stress functions. Thus, we have two functions φ and ψ of class C 6 and C 4 , respectively, such that the definitions σ12 = β12

σ22 = φ,11

σ12 = −φ,12

g = −∇ 2 ψ

h 1 = ψ,1

h 2 = ψ,2 .

(5.89)

result in (5.87) being satisfied identically. Clearly, φ plays the same role as the familiar Airy stress function of classical elastostatics. It has been shown that the following four equations result from the above relations: 2µu1,1 = φ,22 − ∇ 2 (vφ + ξ ψ) 2µu(1,2) = −φ,12

2µu2,2 = φ,11 − ∇ 2 (vφ + ξ ψ) ψ=

1 2 ∇ G

[2 (λ + µ) ψ − ηφ] ,

(5.90)

where G = 2ξ (λ + µ) v = λξ − β 2 /G ξ = 2µβ/G.

(5.91)

Note a certain similarity of (5.90) to (6.108) of Chapter 6. From this, assuming the function P1 = ∇ 2 [(1 − v) φ − ξ ψ]

(5.92)

to be harmonic, one can show that φ and ψ satisfy the equations ∇ 4 ∇ 2 − p2 φ = 0

∇ 2 ∇ 2 − p 2 ψ = 0,

(5.93)

which again bear some resemblance to (6.109) of Chapter 6. The coefficient p 2 = ξ (λ + 2µ) −

β α (λ + 2µ)

(5.94)

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appearing in equations (5.93) has dimension of inverse of length squared, so that there is also a characteristic length in Cowin’s theory: l=

1 . p

(5.95)

De Cicco and Guarracino (2004) also established a representation of functions φ and ψ in terms of a pair of complex analytic functions and a real function that satisfies a homogeneous Helmholtz equation. It now remains to check whether there is stress invariance in this theory, and, if so, of what kind.

Problems 1. Prove properties 4. and 5. of Section 5.1.3. 2. What does the Cauchy symmetry imply for the polar representation of a general anisotropic planar material? 3. Dundurs parameters may be written as α12 =

(χ1 + 1) − (χ2 + 1) (χ1 + 1) + (χ2 + 1)

β12 =

(χ1 − 1) − (χ2 − 1) (χ1 + 1) + (χ2 + 1)

Where = µ2 /µ1 and χ1 , χ2 are two Kolosov constants. Show that they are invariant under the CLM shift, or that they agree with (5.44). 4. Consider a unidirectional fiber-reinforced composite with a dilute concentration of fibers. The fibers are in the shape of circular cylinders; their volume fraction is c 2 . The effective shear and bulk moduli are given as follows 1 1 2(1/µ1 + 1/κ1 ) 1 1 = + c2 − eff µ1 µ2 µ1 1/µ1 + 1/µ2 + 2/κ2 µ 1 1 1 1 2(1/κ1 + 1/µ1 ) = + c2 − κ1 κ2 κ1 1/κ2 + 1/µ1 κ eff Show that these effective moduli satisfy the CLM shift. 5. Consider the following boundary value problems involving a linearly elastic plate (made of an isotropic, homogeneous material) with a hole: (a) the plate subjected to a remote uniaxial tension; (b) the hole loaded by two concentrated and opposite forces on its boundary;

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Microstructural Randomness and Scaling in Mechanics of Materials (c) the hole loaded by one concentrated force; (d) the hole loaded by a uniformly distributed torque on its boundary; (e) the plate subjected to a displacement boundary condition ui (x) = εi0j x j . Determine in which of these problems is the stress field independent of the elastic constants. 6. Solve the problem of a large sheet with a circular disk. The sheet and the disk are linearly elastic, isotropic and homogeneous. They are made of distinct elastic constants and have different thermal expansion coefficients. Show that the stress field invariant under the CLM shift. Hint: the problem is axisymmetric. 7. Demonstrate that the stress invariance does not hold in anti-plane elasticity. 8. Verify equation (5.36).

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6 Two- versus Three-Dimensional Micropolar Elasticity

Of course the continuum theory can yield, in principle, less information about a material than does that of a correct, detailed molecular model. C.A. Truesdell, 1966 It was noted in Chapter 3 that lattices of beams should be modeled by micropolar, rather than classical, elastic continua. This chapter outlines the basic theory of such continuum models—both unrestricted and restricted (couple-stress) ones. In particular, we first provide a formulation of basic equations in 3D, and then, in analogy to Chapter 3, we focus on planar micropolar elasticity. Special attention is given to a generalization of the CLM result, and its consequences. Furthermore, we discuss the problem of homogenization of a heterogeneous Cauchy-type composite by a homogeneous micropolar-type material. If conducted properly, one may then reduce the number of degrees of freedom involved in, say, a finite element method, although the method would have to account for a micropolar nature of the approximating body. This also provides more physical insight into the so-called characteristic length, usually an enigmatic concept appearing (and vanishing) in papers on micropolar theories. Although several monographs have been written on the subject of micropolar media, most of the topics discussed in this chapter have never been collected in a book form.

6.1

Micropolar Elastic Continua

6.1.1 Force Transfer and Degrees of Freedom Every course on solid mechanics starts out with an introduction of the Cauchy stress concept. This first involves identification of a finite surface area A (= L 2 )—either in the interior of the body or on its external surface—defined by an outer unit normal n, and a force F acting on A, Figure 6.1(a). Next, 191

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∆M ∆F n ∆A

(a)

(b)

FIGURE 6.1 (a) Force F and couple M acting on an internal (or external) surface area A (= L 2 ) in a continuum; A is the area of any face of a cubic element of side L. (b) A porous medium in 2D, viewed as a beam lattice, with each beam carrying a force and a couple locally. A unit cell of size L is indicated with dashed lines.

one considers the ratio of F to A, and takes the limit F(n) = t(n) . A→0 A lim

(6.1)

It is a basic postulate of conventional solid mechanics that such a limit is well defined, that is, that it is finite except the singularity points in the body, such as crack tips. In the third step, following Cauchy himself, one introduces his force-stress tensor τ as a linear mapping from n into t(n) t(n) = τ · n.

(6.2)

However, any consideration of a finite area A should involve a surface couple M accompanying F. Thus, in analogy to (6.1), we must consider M(n) = m(n) , A→0 A lim

(6.3)

and, following Voigt (1887) and the brothers Eug`ene and Fran¸cois Cosserat (1909), should introduce a couple-stress tensor µ as a linear mapping from n into m(n) m(n) = µ · n.

(6.4)

Both t(n) and m(n) are shown acting on a face ABC of an arbitrary orientation in Figure 6.2. Note: The explicit consideration of µ makes τ nonsymmetric in general, and that is the reason for using τ instead of the conventional Cauchy stress σ .

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193

x3

t(n) m(n)

t(1) C n

m(1)

x2 m(2) B t(2)

O A m(3) t(3)

x1

FIGURE 6.2 Force traction and moment traction acting on face ABC with outer unit normal n of an infinitesimal tetrahedron OABC.

If the microstructure is disregarded, we are dealing with an idealized, homogeneous continuum in which M(n) must vanish in the limit L → 0. To see this, take n to be aligned with n1 , and consider shear stresses τ12 and τ13 . The torque caused by them, proportional to L 3 (τ12 − τ13 ), must disappear as L → 0, because the cube’s volume scales as L 3 . Otherwise, we would be left with a non-zero angular acceleration of a continuum point. This, in fact, is the case with classical/conventional solid mechanics of Cauchy-type continua. One then only has displacement u at a point, and assumes that m(n) = 0. But, if the material intrinsically carries couples, we cannot disregard M(n) . Such a situation occurs when the material has a discrete-type microstructure, such as a beam-lattice shown in Figure 6.1(b), which simply precludes one from taking A → 0. Here one needs to take A equal to the area of the elementary cell’s cross-section, and the moment traction m(n) in (6.3) is defined at A finite. For this model of force distribution in a continuous body to be fully consistent with kinematics, each continuum point is endowed with six degrees of freedom of a rigid body: three displacements ui (i = 1, . . ., 3) (or u) and three rotations ϕi (i = 1, . . ., 3) (or ϕ), which are, in general, independent functions of position and time. In particular, this implies that ϕ is not the same as the macrorotation given by the gradient of u: ϕi =

1 e ijk uk, j 2

Here, as before, e ijk is the Levi–Civita permutation tensor.

(6.5)

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When the inequality above is replaced by an equality, the continuum is kinematically constrained—this will be discussed at various points in this chapter. We now give some key facts in the vein of Nowacki, (1986). 6.1.2 Equations of Motion and Constitutive Equations The equations of motion and constitutive equations of linear elasticity may be derived from the energy conservation principle. This principle, for an adiabatic thermodynamic process and an anisotropic body, has the following form: d (6.6) (U + K ) = ( Xi vi + Yi wi ) dV + (ti vi + mi wi ) dS, dt V S where

U=

ud V V

K =

kd V V

k=

1 ρvi vi + Iij wi w j 2

vi = u˙ i

wi = ϕ˙ i . (6.7)

In the above k is the the kinetic energy density, and u the internal energy, both referred to a unit volume, while ρ denotes mass density and Iij is the rotational inertia tensor. Note that the left-hand side of (6.6) represents the rate of change of kinetic and internal energies, while the right-hand side is the power of body forces and moments and surface forces and moments. Let us now assume that the energy balance is invariant with respect to rigid body motions when Xi , ti , Yi , and mi are kept fixed. Considering a translational motion first, we substitute (with b i being an arbitrary constant vector) vi → vi + b i ,

v˙ i → v˙ i

into (6.6) to get u˙ + ρ (vi + b i ) v˙ i + Iij wi w ˙ j dV = [Xi (vi + b i ) + Yi wi ] dV V V + [ti (vi + b i ) + mi wi ] dS.

(6.8)

(6.9)

S

Subtracting equation (6.6) from (6.9), we obtain bi ( Xi − ρ v˙ i ) dV + b i ti dS = 0, V

(6.10)

S

which, noting (6.2) and the Green–Gauss theorem, becomes bi Xi + τji,j − ρ v˙ i dV = 0. V

(6.11)

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Since this has to hold for an arbitrary volume V, we obtain a local form of the conservation of linear momentum τji,j + Xi = ρ v˙ i .

(6.12)

With the above we can now simplify (6.6) to the form u˙ + Iij wi w j dV = τji vi + Yi wi dV + [ti (vi + b i ) + mi wi ] dS, V

V

S

(6.13) which, noting (6.3) and the Green–Gauss theorem, becomes τji vi, j + µji wi, j dV + µji,j + Yi − Iij w j wi dS. udV ˙ = V

V

(6.14)

S

From this, the local form of conservation of energy may now be written as u˙ = τji vi, j + µji wi, j + µji,j + Yi − Iij w j wi . (6.15) Let us now postulate the energy balance to be invariant with respect to rigid body rotations (with ωk being an arbitrary constant vector): vi, j → vi, j − e ijk ωk

wi, j → wi, j .

(6.16)

Assuming u, Yi , τij , µij , and Iij to be unchanged, and proceeding in a fashion similar as before, we arrive at a local form of the conservation of angular momentum .. e ijk τ jk + µji,j + Yi = Iij ϕ j . This equation brings about a simplification of the energy balance u˙ = τji vi, j − e k ji wk + µji wi, j .

(6.17)

(6.18)

It is now convenient to introduce two tensors describing the deformation of the body—strain γji and torsion κij —as follows: γji = ui, j − e k ji ϕk

κji = ϕi, j .

(6.19)

Evidently, γji and κij are generally nonsymmetric; κij is also called curvature tensor, or torsion-curvature tensor. Just like in a classical continuum, we need compatibility equations, and these are γli,h − γhi,l − e khi κik + e kli κhk = 0

κli,h = κhi,l .

(6.20)

Given (6.19), we can write the energy balance (6.18) as u˙ = τij γ˙ij + µij κ˙ ij .

(6.21)

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Taking u to be a state function of γij and κij , we have u˙ =

∂u ∂γij

∂u γ˙˙ ij + κ˙ ij , ∂κij

(6.22)

whereby we also assume τij and µij not to be explicitly dependent on the temporal derivatives of γij and κij . A comparison of (6.21) with (6.22) then leads to τij =

∂u ∂γij

µij =

∂u ∂κij

.

(6.23)

Clearly, τij , γij and µij , κij are conjugate pairs. Assuming a micropolar material of linear elastic type, its energy density is given by a scalar product u=

1 1 (1) (2) γij Cijkl γkl + κij Cijkl κkl , 2 2

(6.24)

so that Hooke’s law is (1) τij = Cijkl γkl

(2) µij = Cijkl κkl ,

(6.25)

(1) (2) Here Cijkl and Cijkl are two micropolar stiffness tensors. Note that, due to the existence of u, we have the basic symmetry of both stiffness (and hence, compliance) tensors (1) (1) Cijkl = Cklij

(2) (2) Cijkl = Cklij ,

(6.26)

but not the two other symmetries since τij , γij and µij , κij are, in general, nonsymmetric. Indeed, this is the reason for calling this theory an asymmetric elasticity by Nowacki (1970, 1986). The inverse of (6.25) is written as (1) γij = Sijkl τkl

(2) κij = Sijkl µkl .

(6.27)

Note: In this chapter we employ τ and γ to distinguish them, respectively, from the symmetric stress σ and symmetric strain ε of classical continuum theory. This convention will become very useful in homogenization of a heterogeneous Cauchy-type composite by a homogeneous micropolar-type material, Section 6.4. 6.1.3 Isotropic Micropolar Materials Focusing henceforth on the centrosymmetric case, for an isotropic material, (1) (2) Iij = δij I , while tensors Cijkl and Cijkl of (6.25) become (1) Cijkl = (µ − α) δ jk δil + (µ + α) δjl δik + λδij δkl (2) Cijkl = (γ − ε) δ jk δil + (γ + ε) δjl δik + βδij δkl ,

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where λ and µ are the Lam´e constants of classical elasticity, while α, γ , ε, and β are the micropolar constants. The free energy density is given by a scalar product u=

µ+α µ−α λ γji γji + γji γij + γkk γnn 2 2 2 γ +ε γ −ε β κji κji + κji κij + κkk κnn , + 2 2 2

(6.28)

With (6.28), we can write (6.25) in two equivalent forms: τji = (µ + α) γji + (µ − α) γij + λδij γkk

τij = 2µγ(i j) + 2αγ[i j] + λδij γkk

µji = (γ + ε) κji + (γ − ε) κij + βδij κkk

µij = 2γ κ(i j) + 2εκ[i j] + βδij κkk . (6.29)

The round and square brackets indicate symmetric and antisymmetric parts of the tensors, respectively. Of use also will be the inverse forms of this constitutive law, namely, γij = 2µ τ(i j) + 2α τ[i j] + γ δij τkk

κij = 2γ µ(i j) + 2ε µ[i j] + β δij µkk ,

(6.30)

in which 2µ =

1 2µ

2α =

−λ λ = 6µK

1 2α

2γ =

−β β = 6γ

1 2γ

2ε =

2 κ =λ+ µ 3

1 2ε

2 = β + γ. 3

(6.31)

Here we recognize the familiar bulk modulus κ, and its mathematically analogous micropolar quantity . Clearly, there are six material constants: α, β, γ , ε, µ, λ. Considering the fact that u in (6.28) is a positive definite quadratic form, one can show that the following inequalities should hold: 3λ + 2µ > 0 µ>0 µ+α >0 γ +ε >0

3β + 2γ > 0 γ > 0 α>0 ε > 0.

(6.32)

Our constitutive tensors (6.26) may alternatively be expressed in the notation of Eringen (1966, 1999) (1) Cijkl = µ E δ jk δil + (µ E + α E ) δjl δik + λ E δij δkl

(6.33)

(2) Cijkl = β E δ jk δil + γ E δjl δik + α E δij δkl ,

where, using the subscript E to denote quantities in Eringen’s notation, we have µE = µ − α

κ E = 2α

λE = λ

γE = γ + ε

β E = γ − ε.

(6.34)

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We end this section by noting that, just like in the classical elasticity, we can express a micropolar field problem in displacements and rotations, or in stresses and couple-stresses. In the first case, six such equations (for ui and ϕi , i = 1, . . ., 3) are obtained by substituting (6.30) into the equilibrium equations (6.12) and (6.17), and using (6.19): .. (µ + α) ui, j j + (λ + µ − α) u j, ji + 2αe ijk ϕk + Xi = ρ ui .. (γ + ε) ϕi, j j − 4αϕi + (β + γ − ε) ϕ j, ji + 2αe ijk uk + Yi = I ϕ .

(6.35)

This generalization of the Navier equations of classical elasticity is to be supplemented by the kinematic boundary conditions on ∂ Bk and traction conditions on ∂ Bt , where ∂ Bk ∪ ∂ Bt = ∂ B, that is, τij (x, t) n j = ti (x, t)

µij (x, t) n j = mi (x, t)

x ∈ ∂ Bt

t>0

ui (x, t) = f i (x, t)

ui (x, t) = gi (x, t)

x ∈ ∂ Bk

t > 0.

(6.36)

Here ti , mi , f i , gi are prescribed functions. The field equations in stresses are a generalization of the Beltrami–Michell equations. Here we make a reference to a study of Sch¨afer (1967), who generalized the functions of Morrey and Maxwell and Kessel. We will return to this topic in the 2D setting in Section 6.3 below. As pointed out in Section 6.1.1, ϕ is an independent kinematic quantity. However, a special model assuming the equality ϕi =

1 e ijk uk, j 2

(6.37)

is sometimes used, and this is the same definition as in classical elasticity. It is called a pseudo-continuum, a restricted, a couple-stress, or a Koiter-Mindlin model, Koiter (1963); see also Truesdell and Toupin (1960), Grioli (1960), Toupin (1962), Mindlin and Tiersten (1962), and Mindlin (1963). In view of (6.37), the strains γij become symmetric and are (classically) defined as γij = u(i, j) .

(6.38)

A well-known case of such materials is the Bernoulli–Euler beam—indeed, a 1D continuum—in which the cross-section is restricted to rotate according to the gradient of the transverse displacement. 6.1.4 Virtual Work Principle Let us consider fields of virtual displacement δui and rotation δϕi —both of them continuous, consistent with boundary conditions, and infinitesimal. If we multiply equation (6.12) by δui , and equation (6.17) by δϕi , and add the

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.. .. [( Xi − ρ ui )δui + (Yi − I ϕ i )δϕi ]dV V + [τji,j δui + (e ijk τ jk + µji,j )δϕi ]dV = 0.

199

(6.39)

V

Upon transformation of the second of the above integrals, we obtain .. .. ( Xi − ρ ui )δui dV + (Yi − I ϕ i )δϕi dV V

V

+

τji δγji + µji δκji dV,

[ti δui + mi δϕi ] dS = V

(6.40)

V

which expresses an equality between the virtual work of external and internal forces; the latter are conjugate to fields of virtual strain δγji and rotation δκji . Upon introduction of (6.29) into (6.39), we set up the variational principle .. .. [( Xi − ρ ui )δui + (Yi − I ϕ i )δϕi ]dV V

+

[ti δui + (mi δϕi ]d S = δW

(6.41)

S

where

λ W= µγ(i j) γ(i j) + αγ[i j] γ[i j] + γkk γmm 2 V β + γ κ(i j) κ(i j) + εκ[i j] κ[i j] + κkk κmm dV. 2

(6.42)

This principle may be used to derive the energy conservation principle by comparing the functions u and ϕ at a point x and time t with those quantities at x and time t + dt. Thus, introducing δui = vi dt; δϕi = wi dt; vi = u˙ i ; wi = ϕ˙ i into (6.41), we obtain d ( K + W) = ( Xi vi + Yi wi ) d V + (ti vi + mi wi ) d S (6.43) dt V S This is the starting point for the proof of uniqueness of solutions—the procedure is analogous to that in classical elasticity. 6.1.5 Hamilton’s Principle Consider now a micropolar elastic body undergoing some motion between times t = t1 and t = t2 . We now compare the actual displacements u (x, t) and rotations ϕ (x, t) with the virtual u (x, t) + δu and ϕ (x, t) + δϕ, whereby the latter are chosen so as to satisfy the conditions δu (x, t1 ) = δu (x, t2 ) = 0

δϕ (x, t1 ) = δϕ (x, t2 ) = 0.

(6.44)

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The virtual work principle is now written as .. .. δL − (ρ ui δui + I ϕ i δϕi )dV = δW.

(6.45)

V

where

δL =

( Xi δui + Yi δϕi ) dV + V

(ti δui + mi δwi ) dS.

(6.46)

S

Integrating (6.45) over the interval t1 ≤ t ≤ t2 , t2 t2 t2 .. .. δ Wdt = δLdt − dt (ρ ui δui + I ϕ i δϕi )dV, t1

t1

t1

(6.47)

V

while introducing the variation of kinetic energy, ∂ ∂ .. .. δK = ρ ( u˙ i δui ) dV − ρ ui δui dV + I ( ϕ˙ i δϕi ) dV − I ϕ i δϕi dV, ∂ t ∂ t V V V V (6.48) and integrating it also over t1 ≤ t ≤ t2 , and taking note of (6.44), we find t2 t2 t2 .. .. ϕ i δϕi dV. ui δui dV − I δ K dt = −ρ dt dt (6.49) t1

t1

V

t1

V

In view of (6.47) and (6.49), we finally obtain Hamilton’s principle generalized to a micropolar medium t2 t2 δ Ldt. (6.50) (W − K ) dt = δ t1

t1

Variation and integration on the right-hand side of (6.50) commute when the external forces are conservative and derivable from a potential V. In that case, ∂V ∂V ∂V ∂V δL = − (6.51) δui + δϕi = −δ ui + ϕi , ∂ui ∂ϕi ∂ui ∂ϕi and (6.50) becomes δ

t2

( − K ) dt = 0

=W+V

(6.52)

t1

in which denotes the total potential energy. 6.1.6 Reciprocity Relation Consider an isotropic body of volume V and bounding surface ∂ B loaded by a system {X, Y, t, m}, which then results in {u, ϕ}. The initial conditions are

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homogeneous: ui (x, 0) = 0

u˙ i (x, 0) = 0

ϕi (x, 0) = 0 ϕ˙ i (x, 0) = 0 (6.53)

Also, consider

another loading system X , Y , t , m acting on the same body, resulting in u , ϕ , both causes and effects being now denoted by primes. This is subject to analogous initial conditions as in the first case. We now apply Laplace transformation to the constitutive equations (6.29) to get τ ji = (µ + α) γ ji + (µ − α) γ ij + λδij γ kk

ji = (γ + ε) κ ji + (γ − ε) κ ij + βδij κ µ kk , where

τ ji (x, p) =

∞

τji (x, t) e − pt dt, etc.

(6.54)

(6.55)

0

Proceeding in a similar fashion with the primed quantities τ ji (x, p), etc., a statement entirely similar to (6.54) above is obtained. It is easy to verify that the following is true:

κ ji ;

ji κ ji = τ ji γ ji + µ τ ji γ ji + µ ji and this, upon a volume integration, becomes

κ ji dV.

ji κ ji dV = τ ji γ ji + µ τ ji γ ji + µ ji V

(6.56)

(6.57)

V

Next, carry out the Laplace transformation on the equations of motion (6.12) and (6.17) corresponding to the first loading system to get 2

i τ ji,j + Xi = p ρ u

2 i . e ijk τ jk + µ ji,j + Yi = p I ϕ

(6.58)

Noting a completely analogous relation corresponding to the second loading system, (6.57) can be converted to 2 2

i ϕ i dS t i ui + m Xi ui + Yi ϕi − p ρ u i − p I ϕ i dV + V

= V

Xi u i + Y i ϕ i dV +

S

S

ϕ i dS. ti u i + m i

(6.59)

Upon carrying out the inverse Laplace transformation, we arrive at the theorem of reciprocity of work of causes and effects in both loading systems (Sandru, 1966): Xi ∗ ui + Yi ∗ ϕi dV + ti ∗ ui + mi ∗ ϕi dS V

S

= V

Xi ∗ ui + Yi ∗ ϕi dV +

S

ti ∗ ui + mi ∗ ϕi dS,

where ∗ denotes a convolution operation between Xi and ui .

(6.60)

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This is one of the most interesting theorems of micropolar elasticity. Its generality offers a possibility of integration of the equations of elastodynamics using Green’s function. 6.1.7 Elements of Micropolar Elastodynamics 6.1.7.1 Basic Equations We begin by noting that equations (6.35) can also be written in a vector form 2u

+ (λ + µ − α) grad divu + 2α rotϕ + X = 0

4ϕ

+ (β + γ − ε) grad divϕ + 2α rotu + Y = 0,

(6.61)

where 2

= (µ + α) ∇ 2 − ρ ∂2t

4

= (γ + ε) ∇ 2 − 4α + I ∂2t ,

(6.62)

are the d’Alembert and Klein–Gordon operators, respectively, and ∂2t indicates the second derivative with respect to time. The physics represented by this coupled system of hyperbolic differential equations can be understood by operating either with divergence or rotation upon it. In the first case, we find 1 divu

+ divX = 0

3 divϕ

+ divY = 0,

(6.63)

where again we introduced two partial differential operators 1

= (λ + 2µ) ∇ 2 − ρ ∂2t

3

= (β + 2γ ) ∇ 2 − 4α + I ∂2t .

(6.64)

On the other hand, upon carrying out a rotation on (6.61), we find 3 r otu

+ 2α rot rotϕ + r otX = 0

3 r otϕ

+ 2α rot rotu + r otY = 0.

(6.65)

If we now operate with 1 4 on (6.61)1 and employ (6.63)1 and (6.65)1 , and similarly we operate with 3 4 on (6.62)2 and employ (6.63)2 and (6.65)2 , we shall find 2 2 u = − 1 4 − grad div X + 2α rot 1 Y 1 2 4 + 4α ∇ (6.66) 2 2 ϕ = − 2 3 − grad div Y + 2α rot 1 X, 3 2 4 + 4α ∇ where = (λ + µ − α)

4

− 4α 2

= (β + γ − ε)

2

− 4α 2 .

(6.67)

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Now, let us introduce a representation for displacements and rotations, employing two vector functions F and G: u= ϕ=

1

2

− grad div F − 2α rot 3 G 3 − grad div G − 2α rot 1 F.

4

(6.68)

With the substitution of (6.68) into (6.66) we find two equations governing F and G: 1 3

2 2

+ 4α 2 ∇ 2 F + X = 0 2 2 G + Y = 0. 4 + 4α ∇ 4

(6.69)

More insight into what is represented by the equations of motion (6.61) may be gained by using Helmholtz’s theorem for u and ϕ: u = grad + r ot ,

div = 0

ϕ = grad + r ot H,

div H = 0.

(6.70)

If we also introduce the same type of decomposition for body forces and body couples, that is, X = ρ grad ϑ + r ot χ , Y = I grad σ + r ot η ,

div χ = 0 div η = 0.

(6.71)

we find that (6.61) reduces to four wave equations 1 2

+ ρϑ = 0

+ 2α rotH + ρχ = 0

3 4H

+ Iσ = 0

+ 2α rot + I η = 0.

(6.72)

The first of these equations represents a longitudinal wave motion, identical with what is known from classical elastodynamics. The second equation represents a longitudinal wave of (micro)rotation. The third and fourth equations represent propagation of two transverse waves—in displacements and in rotations—which may be cast in the form

2 2

+ 4α 2 ∇ 2 = 2α I rot η − ρ 2 2 H = 2αρ rot χ − I 4 + 4α ∇ 4

These waves were investigated by Ignaczak (1970).

4χ 2 η.

(6.73)

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6.1.7.2 Plane Monochromatic Waves Consider a plane monochromatic wave propagating in an infinite body, and harmonic in time. Let the wavefront at time t = 0 be in the plane p = xi ni , where n is the unit vector normal to the plane. The displacements and rotations are now of the form u j = Aj exp [−ik (t − nk xk )]

ϕ j = B j exp [−ik (t − nk xk )] ,

(6.74)

where k = ω/ = 2π/l is the phase velocity, ω is the angular velocity, and l is the wave length. Introducing (6.74) into (6.35), we arrive at a system of algebraic equations

2αi µ + α − ρ2 Aj + (λ + µ − α) n j xk Ak + e jkl nl Bk = 0, k (6.75) 4α 2αi γ + ε + 2 − I 2 B j + (β + γ − ε) n j xk Bk + e jkl nl Ak = 0. k k

Setting this system’s determinant to zero leads to 4α 2 2 λ + 2µ − ρ β + 2γ + 2 − I k 4α 4α 2 µ + α − ρ2 γ + ε + 2 − I 2 − 2 = 0, k k

(6.76)

from which we determine phase velocities of various plane waves. The first term in (6.76) yields =

λ + 2µ ρ

1/2 (6.77)

The second term in (6.76) yields dispersive (i.e., ω-dependent) wave propagation −1/2 ω2 = 3 1 − 02 , ω

3 =

β + 2γ I

1/2 ,

ω02 =

4α , I

(6.78)

which has physical meaning only for ω > ω0 , because this condition ensures real values of . The third term in (6.76) yields a quartic equation 4α 2 4α µ + α − ρ2 γ + ε + 2 − I 2 − 2 = 0. (6.79) k k 6.1.8 Noncentrosymmetric Micropolar Elasticity Recall that the constitutive equations relate force stresses with strains on one hand, and moment stresses with curvatures on the other. In general, however, there is a possibility of a direct coupling between force-type and couple-type

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effects in the constitutive model, whereby τij would also be a function of κkl , and µij a function of γkl . A simple 1D case of chirality is the helix already discussed in Section 3.5 of Chapter 3. Thus, generalizing (6.25), we write (1) (3) τij = Cijkl γkl + Cijkl κkl , (4) (2) µij = Cijkl γkl + Cijkl κkl .

(6.80)

This is called either non-centrosymmetry or hemitropy (Aero and Kuvshinskii 1964, 1969; Nowacki, 1986), or chirality (Lakes and Benedict, 1982; Lakes, 2001). Note that all the developments of this chapter preceding this equation pertained to centrosymmetric materials. Note that, by the argument of reciprocity, (3) (4) Cijkl = Cijkl .

(6.81)

As Lakes points out, because the coordinate changes in the centrosymmetric material result in the transformation matrix a im = −δjm ,

(6.82)

(1) (1) (1) (1) = a im a jn a ko a lp Cmnop = (−1) 4 Cijkl = Cijkl . Cijkl

(6.83)

(1) the stiffness tensor Cijkl satisfies

6.2

Classical vis-`a-vis Nonclassical (Elasticity) Models

6.2.1 A Brief History Following its birth, the theory of the Cosserat brothers (1896, 1909) remained dormant for half a century, apparently the only exceptions being the works of Somigliana (1910) and Sudria (1935); see also Ball and James (2002). This hibernation was likely due to the theory’s generality (as a nonlinear theory with finite motions and inelastic interactions) and its presentation as a unified theory incorporating mechanics, optics, magnetism and electrodynamics. The dynamic growth of continuum mechanics and thermodynamics (e.g., Ericksen and Truesdell, 1958; Truesdell and Toupin, 1960) begun in the fifties and sixties brought the work of the Cosserat brothers back into focus. Fundamentals of a general linear Cosserat continuum were given by Gunther ¨ (1958), who discussed in detail the 1-, 2-, and 3D Cosserat models, as well as their significance in the dislocation theory, and Sch¨afer (1962), who focused on the planar case. From that period one should also mention several other works. Thus, Grioli (1960) established the constitutive relations for finite deformations of perfectly elastic solids. Aero and Kuvshinskii (1960) independently derived the equilibrium equations and constitutive relations for anisotropic solids in the linearized theory. Mindlin and Tiersten (1962) established the

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boundary conditions; see also Kroner ¨ (1963), Koiter (1963), and Eringen (1968). An expression of the growing interest in Cosserat theory was soon found in symposia (e.g., Kroner ¨ 1968) and monographs on the subject (e.g., Nowacki, 1970, 1986; Stojanovic, 1970; Brulin and Hsieh, 1982). Building on the shoulders of the Cosserats, and to account for increasing levels of complexity, other, more general theories accounting for higher-order interactions such as monopolar, multipolar, and strain-gradient were introduced (see e.g., Green and Rivlin, 1964; Toupin, 1962, 1964; Jaunzemis, 1967; Tiersten and Bleustein, 1974). There are also “micropolar,” “microstretch,” and most generally “micromorphic” continua (Eringen, 1999, 2001; Mariano, 2001). To clarify the key concepts here, following Goddard (2006), let us consider a series expansion of the velocity (or infinitesimal displacement) field v v (x) = v0 + L1 · r + L2 · r2 + · · ·,

(6.84)

where r = x − x 0 , Ln =

1 n!

(∇ ⊗ v) 0T ,

(6.85)

and rn denotes the n-fold symmetric tensor product ⊗n r. This allows an expansion for the global stress-power density in a simple continuum 1 w ˙ = σ : LdV = w ˙ n, (6.86) V V n with w ˙ n = σn : Ln , σn :=

V

σ : rn dV,

(6.87)

where L = (∇ ⊗ v) T is is the first velocity gradient, its dual being the Cauchy stress σ . Furthermore, while we easily see that the higher-order kinematic quantities Ln are conjugate to the stress moments σn , there are two ways to interpret the Ln : 1. Multipolar: the Ln are identical with the higher gradients of the velocity field. This viewpoint was advanced by Green and Rivlin (1964a,b) and Mindlin (1963). 2. Micromorphic: the Ln are intrinsic particulate fields (i.e., pertaining to generally deformable particles making up the macro-continuum), which require their own constitutive equations. This approach dates back to the Cosserat brothers, and was then further pursued by Eringen (1999) leading to microstretch and micromorphic theories. The microscopic treatment dictates that, in a multipolar continuum, the stress moments should satisfy a hierarchy of balances T ∇ · σn+1 + σn = Gn+1 for n = 0, 1, . . .,

(6.88)

where σ0 := 0, σ1 := σ , and the Gs represent extrinsic body moments accumulation of intrinsic multipolar momenta, which effectively vanish in the

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207

(6.89)

In a certain sense, all of these theories can be considered as simpler cases of “nonlocal continuum theories” (Eringen and Hanson, 2002), which, according to these authors, “are concerned with material bodies whose behavior at any interior point depends on the state of all other points in the body—rather than only on an effective field resulting from these points—in addition to its own state and the state of some calculable external field.” Focusing henceforth on micropolar theories, we would like to note their extensions beyond purely elastic material behaviors. An extension pertaining to thermoelasticity was already given by Nowacki (1966) and Tauchert et al. (1968); see Dhaliwal and Singh (1987) for a review. A micropolar generalization of viscoelasticity, with a focus on waves, was presented by Maugin (1974). Beginning with Green and Naghdi (1965), Mi¸sicu (1964), and Sawczuk (1967), there has also been research on (elastic-)plastic continua with microstructure, e.g., Fleck et al. (1994) and Hutchinson (2000). This has then led to strain-gradient models (Aifantis, 1987; Zbib and Aifantis, 1989; Fleck and Hutchinson, 1997; Pamin, 2005). Extensive research has also been done on micropolar fluid mechanics (e.g., Cowin, 1974; Eringen, 2001). Although the nonclassical theories have become very advanced mathematically and explained effects that could not be brought out by classical theories, they usually lacked the input of physically based constitutive coefficients. Besides the beam lattices discussed in Chapters 3 and 4, progress has been made on that front for composite materials; see Section 6.5 below. Mindlin (1963) found that stress concentrations in the presence of holes are lowered in Cosserat-type versus those in Cauchy-type solids. This was followed by studies due to Neuber (1966) Kaloni and Ariman (1967), Cowin (1970a,b) and Itou (1973). On the other hand, an increase of stress concentrations in the vicinity of rigid inclusions was established by Hartranft and Sih (1965) and Weitsman (1964). Micropolar effects also allow a better analysis of localization present in failure of solids then that possible in classical continue (William, 1995). The case of holes motivated one of the earliest experimental studies of couple-stress effects by Schijve (1966), who actually found that effect to be insignificant. However, given the fact that he used aluminum sheets—a macroscopically homogeneous material without, say, reinforcing inclusions—his investigation pertained to couple-stress effects due to the atomic lattice of aluminum. This is not surprising in view of the fact that couple-stress effects vanish on scales much larger than the microscale. Indeed, the situation is much different in, say, a lattice of beams (which may be interpreted as a material with large holes), if one looks at dependent fields on scales comparable to the lattice spacing; recall Chapter 3. Many studies of wave propagation in the context of harmonic disturbances have also been conducted. First, in addition to classical dilatational and shear waves in an unbounded medium, there also exist rotational waves.

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Next, it turns out that only the dilatational waves propagate non-dispersively (Nowacki, 1986; Eringen, 1999). In general, this is indicative of various new dispersion effects in other wave problems, which are not present in classical continua. In some cases of Cosserat continua, entirely new phenomena arise such as, for instance, that a layer on top of an elastic half-plane is not necessary for the propagation of Love waves—in the classical case, a layer is necessary. Many results on periodic and aperiodic waves were collected by Nowacki (1986), see also Eringen (1999). The recent monograph by Dyszlewicz (2004) on micropolar elasticity collects many new results, including the general methods of integration of basic equations (Galerkin, Green–Lam´e, and Papkovitch–Neuber type), formulations of problems (displacement-rotation and pure stress problems of elastodynamics), as well as solutions to various boundary value problems (stationary 2D and 3D problems for a half-space, singular solutions to 2D and 3D elastodynamics and the thermoelastodynamics problems for an infinite space). Several workers, in the 1960s, derived micropolar models explicitly from the microstructure. The work of theoreticians started from lattice-type models enriched with flexural—in addition to central–interactions (e.g., Askar, 1985; Banks and Sokolowski, 1968; Wo´zniak, 1970; Baˇzant and Christensen, 1972; Holnicki-Szulc and Rogula, 1979a,b; Bardenhagen and Triantafyllidis, 1994). From the outset, these models adopted Cosserat-type continua in analyses of large engineering structures such as perforated plates and shells, or latticed roofs. There, the presence of beam-type connections automatically led to micropolar interactions and defined the constitutive coefficients. In principle, such models have their origin in atomic lattice theories (e.g., Berglund, 1982); see Friesecke and James (2000) for the latest work in that direction. Several workers (e.g., Perkins and Thompson, 1973; Gauthier and Jahsman, 1975; Yang and Lakes, 1982; Lakes, 1983, 1986) have provided experimental evidence of micropolar effects in porous materials such as foams and bones. In particular, Lakes (1995) was able to infer micropolar constants from his experiments, both for centrosymmetric and chiral materials. Another interesting application in the context of biomechanics was due to Shahinpoor (1978). It is also to be noted that composite materials may naturally lead to Cosserat models where the nonclassical material constants can directly be calculated from the microstructure; this was done in 1D by Herrmann and Achenbach (1968). But, a similar task in 2D and 3D has only been undertaken recently, and this is described in Section 6.5. In more recent years, progress has been made on derivation of effective (homogeneous) Cosserat models for heterogeneous composite materials of either Cauchy or Cosserat type. We point out in Chapter 4 that a central-force lattice (truss of two-force members) is an example of the former material, while a lattice of beams is an example of the latter one. All the studies in the area of stress singularities due to cracks were preceded by Muki and Sternberg (1965), who studied stress concentrations caused

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by concentrated surface loads or discontinuously distributed surface shear tractions. Next, Sternberg and Muki (1967) and Bogy and Sternberg (1967) studied the implications of the couple-stress theory on unbounded concentrations of stress and on locally infinite deformation gradients. Basically, it was found that, depending on a given situation, where the classical elasticity would predict infinite (singular) stresses, the couple-stress theory may give either finite stresses or weaker singularities, or have an opposite tendency (see also Cowin, 1969; Atkinson and Leppington, 1977). This involves a proper generalization of conservation integrals, which has recently been given in the setting of couple-stress elasticity (Lubarda and Markenscoff, 2000). Recently, Griffith’s fracture theory has been generalized to rectilinear and fractal cracks in micropolar solids (Yavari et al., 2002). In particular, two cases of the Griffith criterion were considered, depending on whether the effects of stresses and couple-stresses are coupled or uncoupled, the key finding being that both cases give equal orders of stress and couple-stress singularities, which is the same result as that in a classical continuum. Also, the effect of fractality of fracture surfaces on the powers of stress and couple-stress singularity was studied. 6.2.2 The Ensemble Average of a Random Local Medium is Nonlocal As pointed out in Chapter 2, the formal solution for the average of a field problem governed by a linear random (and local) operator on the domain B is a deterministic nonlocal operator. This was illustrated in terms of a Fouriertype heat conduction problem, a result that immediately carries over to antiplane elasticity. Moving to a general setting of linear elastostatics on the random field of a fourth-rank stiffness tensor Cijkl = {Cijkl (ω, x) ; ω ∈ , x ∈R2 }, the equations governing the average fields in a Cauchy-type continuum are (Beran and McCoy, 1970a) σij (x) , j = 0 σij (x) = ijkl x, x εkl x dx (6.90) B

εkl =

uk,l + ul,k /2.

(Here we employ σij and εkl to denote symmetric stress and strain tensors.) In (6.90)2 ijkl x, x is an infinite sum of integrodifferential operators, involving moments of all orders of the random field Cijkl (6.91) ijkl x, x = Cijkl + Dijkl x δ x − x + E ijkl x, x , where Dijkl x and E ijkl x, x are functions of the statistical properties of Cijkl and the free-space Green’s function of the nonstatistical problem. Addition of a deterministic body force field fi does not change the results. When the fluctuations in Cijkl are small, Dijkl x and E ijkl x, x may be evaluated explicitly, and this was done by Beran and McCoy (1970) in the special case of the

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realizations Cijkl (ω) being locally isotropic, that is, expressed in terms of a vector random field of two Lam´e coefficients {[λ (ω, x) , µ (ω, x)] ; ω ∈ , x ∈R2 }; recall Section 2.3 of Chapter 2. Next, considering this random field to be statistically homogeneous and mean-ergodic, one may disregard the contributions of this operator for |x−x | > lc (the correlation length). Thus, since only the neighborhood within the distance lc of x has a significant input into the integral (6.85)2 , one may expand εkl (x ) in a power series about x: εkl x = εkl (x) + xm − xm εkl (x) ,m xm − xm xn − xn εkl (x) ,mn . . . + (6.92) 2 so as to obtain σij (x) = ijkl x, x dx εkl (x) B + ijkl x, x xm − xm dx εkl (x) ,m + · · · . (6.93) B

This, in turn, can be rewritten as a sum of local, plus first gradient, plus higher gradient strain effects: ∗ ∗ ∗ εkl + Dijklm εkl ,m + E ijklmn εkl ,mn + · · · . σij = Cijkl (6.94) ∗ Thus, B ijkl x, x dx in (6.93) is recognized as the effective stiffness Cijkl ; indeed the stiffness of a single realization B (ω) of the random material B. If one is given the ensemble B of B (ω), then one may determine the microstruc∗ ∗ tural statistics, and hence the higher-order approximations Dijklm , E ijklmn , and so on.

6.3

Planar Cosserat Elasticity

6.3.1 First Planar Problem There are, in general, two planar problems of Cosserat elasticity (Nowacki, 1986): 1. The so-called first planar problem with u = (u1 , u2 , 0) and ϕ = (0, 0, ϕ3 ), which is a generalization of the classical planar elasticity. 2. The so-called second planar problem u = (0, 0, u3 ) and ϕ = (ϕ1 , ϕ2 , 0), which is a generalization of the classical antiplane elasticity. Focusing on the first planar problem in static setting, from (6.12) and (6.7), the equilibrium equations become τ11,1 + τ21,2 = 0 τ12,1 + τ22,2 = 0

τ12 − τ21 + µ13,1 + µ23,2 = 0,

(6.95)

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while the kinematic relations (6.19) are γ11 = u1,1

γ22 = u2,2

γ12 = u2,1 − ϕ3

κ13 = ϕ3,1

κ23 = ϕ3,2 ,

γ21 = u1,2 + ϕ3

(6.96)

and these satisfy the compatibility equations γ21,1 − γ11,2 = κ13

γ22,1 − γ12,2 = κ23

κ23,1 = κ13,2 .

(6.97)

In the isotropic planar Cosserat medium, compliances of (6.27) become (1) Sijkl =

1 (S + P)δik δjl + (S − P)δil δ jk + ( A − S)δij δkl 4

(2) Si3k3 = δik M,

(6.98)

where A, S, P, and M are four independent planar Cosserat constants, defined in Ostoja-Starzewski and Jasiuk (1995): A=

1 1 = κ λ+µ

S=

1 µ

P=

1 α

M=

1 . γ +ε

(6.99)

Note that A and S define planar bulk and shear compliances of classical elasticity (Dundurs and Markenscoff, 1993), while P and M are two additional Cosserat constants; in the couple-stress elasticity P = 0. The restriction that the strain energy be nonnegative implies the following inequalities: 0 ≤ A≤ S 0 ≤ P

0 ≤ M.

(6.100)

In the case of orthotropy for plane Cosserat elasticity, constitutive equations (6.27) become (1) (1) γ11 = S1111 τ11 + S1122 τ22

(1) (1) γ22 = S2211 τ11 + S2222 τ22

(1) (1) γ12 = S1212 τ12 + S1221 τ21

(1) (1) γ21 = S2112 τ12 + S2121 τ21

(2) κ13 = S1313 µ13

(2) κ23 = S2323 µ23 .

(6.101)

In the above, given (6.26), we have (1) (1) S1122 = S2211

(1) (1) S1221 = S2112 .

(6.102)

Because for the couple-stress formulation γ12 = γ21 (recall equation 6.39), we must have (1) (1) S1212 = S2112

(1) (1) S1221 = S2121 .

(6.103)

This, combined with (6.102)2 above implies (1) (1) (1) (1) = S2112 = S1221 = S2121 , S1212

(6.104)

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so that the constitutive relations (6.101) take on a simpler form (1) (1) γ11 = S1111 τ11 + S1122 τ22

(1) (1) γ22 = S1122 τ11 + S2222 τ22

(1) γ12 = γ21 = S1212 (τ12 + τ21 ) (2) κ13 = S1313 µ13

(6.105)

(2) κ23 = S2323 µ23 .

Finally, for the special type of orthotropy (symmetric, referred to in Section 6.5 below) we have two additional simplifications (1) (1) S1111 = S2222

(2) (2) S1313 = S2323 .

(6.106)

Thus, the constitutive law for such an orthotropic and symmetric planar (1) couple-stress model involves four independent compliance components: S1111 , (1) (1) (2) S1122 , S1212 , and S1313 . 6.3.2 Characteristic Lengths in Isotropic and Orthotropic Media In the early 1960s when the Cosserat models began to undergo a revival following half a century of dormancy after invention by the Cosserats, several people realized that, contrary to classical elasticity, an intrinsic length scale was involved in the governing equations. It was denoted l, and called a characteristic length. Let us now see how this l can be arrived at. Following Mindlin (1963) and Sch¨afer (1962), we employ a stress function formulation, which for the planar Cosserat (as well as the couple-stress) elasticity involves two stress functions, φ and ψ τ11 = φ,22 − ψ,12

τ22 = φ,11 + ψ,12

τ12 = −φ,12 − ψ,22

τ21 = −φ,12 + ψ,11

µ13 = ψ,1

µ23 = ψ,2 .

(6.107)

Note that φ is the Airy stress function known from the classical elastostatics. Recall also that, for the isotropic planar Cosserat elasticity, the compatibility conditions in terms of φ and ψ are given by (e.g., Nowacki, 1986) P+S 2 P+S 2 A+ S 2 A+ S 2 ψ− ψ− =− = ∇ ψ ∇ φ,2 ∇ ψ ∇ φ,1 . 4M 4M 4M 4M ,1 ,2 (6.108) These are the Cauchy-Riemann conditions for the functions A+S ∇ 2 φ and 4M P+S 2 [ψ − 4M ∇ ψ], so that we actually have two harmonic functions P+S 2 ∇ 2∇ 2φ = 0 ∇ 2 ψ − ∇ ψ = 0. (6.109) 4M

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The coefficient ( P + S)/4M appearing above has the dimension of length squared and has thus led to a definition of characteristic length l via l2 =

S(1) P+S ≡ 1212 . (2) 4M S1313

(6.110)

Note: In the couple-stress theory P = 0 in equations (6.107–6.109). For the orthotropic Cosserat elasticity case, the compatibility conditions (6.97) result in (1) (1) (1) (1) [S1111 − S1221 − S1122 ]ψ,122 + S2121 ψ,111 (1) (1) (1) (1) (2) −[S1122 + S2121 + S1221 ]φ,112 − S1111 φ,222 = S1313 ψ,1 (1) (1) (1) (1) [S2222 − S1221 − S1122 ]ψ,112 + S1212 ψ,222 (1) (1) (1) (1) (2) +[S1122 + S1212 + S1221 ]φ,122 + S2222 φ,111 = S2323 ψ,2 ,

(6.111)

which suggest the following definitions of four characteristic lengths (Bouyge et al., 2002): (1) S − S(1) − S(1) 1221 1122 l1 = 1111 (2) S1313

(1) S l2 = 2121 (2) S1313

(1) (1) (1) S − S1221 − S1122 l3 = 2222 (2) S2323

(1) S l4 = 1212 . (2) S2323

(6.112)

In the special case of plane isotropic Cosserat elasticity, the following relations hold (1) (1) (1) (1) (1) (1) (1) (1) S1111 − S1221 − S1122 = S2121 = S2222 − S1221 − S1122 = S1212 ,

(6.113)

and equations (6.112) reduce to a single characteristic length defined in (6.110). For the planar orthotropic couple-stress case, the compatibility equations (6.113) yield (1) (1) (1) (1) [S1111 − S1212 − S1122 ]ψ,122 + S1212 ψ,111 (1) (1) (1) (2) − [S1122 + 2S1212 ]φ,112 − S1111 φ,222 = S1313 ψ,1 (1) (1) (1) (1) [S2222 − S1212 − S1122 ]ψ,112 + S1212 ψ,222 (1) (1) (1) (2) + [S1122 + 2S1212 ]φ,122 + S2222 φ,111 = S2323 ψ,2 .

(6.114)

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Thus, the four characteristic lengths are (1) S − S(1) − S(1) 1212 1122 l1 = 1111 (2) S1313 (1) (1) (1) S − S1212 − S1122 l3 = 2222 (2) S2323

(1) S l2 = 1212 (2) S1313 (1) S l4 = 1212 , (2) S2323

(6.115)

whereupon equations (6.114) can be rewritten as l12 ψ,122 + l22 ψ,111 − l32 ψ,112 + l42 ψ,222 +

(1) (1) + 2S1212 S1122 (2) S1313 (1) (1) + 2S1212 S1122 (2) S2323

φ,112 − φ,122 +

(1) S1111 (2) S1313 (1) S2222 (2) S2323

φ,222 = ψ,1 φ,111 = ψ,2 .

(6.116)

For the special case of plane orthotropic couple-stress case with symmetry, in view of (6.106), there are only two characteristic lengths (1) S − S(1) − S(1) 1212 1122 l1 = 1111 (2) S1313

(1) S l2 = 1212 . (2) S1313

(6.117)

6.3.3 Restricted Continuum vis-a-vis ` the Micropolar Model It is now recalled that a restricted continuum (couple-stress) Cosserat model, with the limitation (6.38), was introduced in the past as a simplified, and somewhat restricted, version of the general micropolar case. First, let us note that, with the definition (6.117), the counterpart of equations (6.108) is [ψ − l 2 ∇ 2 ψ],1 = −2 (1 − ν) l 2 ∇ 2 φ,2

[ψ − l 2 ∇ 2 ψ],2 = −2 (1 − ν) l 2 ∇ 2 φ,1 , (6.118)

and (6.109) holds with ( P + S) /4M replaced by S/4M. The length l appearing in (6.118) is l2 =

2 (1 + ν) B S ≡ . 4M E

(6.119)

where B is the modulus of curvature and µ, E, and ν are the shear modulus, Young’s modulus, and Poisson’s ratio of classical elasticity, respectively. To illustrate the distinction between both models we focus now on the problem of a hole in an infinite body in plane strain under uniaxial tension p. Mindlin (1963) found that in a couple-stress material the maximum stress σm

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(or stress concentration factor) σm = p

3+ F 1+ F

(6.120)

where, in our notation, F =

4 (S + A) /S 4 + (a /l) + 2 (a /l) K 0 (a /l) /K 1 (a /l) 2

(6.121)

where l is defined by (6.119) and a is the hole radius; also, K 0 and K 1 are the modified Bessel functions of the second kind of orders zero and one, respectively. On the other hand, somewhat later Kaloni and Ariman (1967) solved the same problem for a micropolar elastic body with the result F =

4 (S + A) / (S + P) 4 + (a /l) 2 + 2 (a /l) K 0 (a /l) /K 1 (a /l)

(6.122)

where l is defined by (6.110). Finally, we would like to point out that if we set M=

1 4B

P = 0,

(6.123)

in (6.121) and other pertinent micropolar formulas, we recover Mindlin’s couple-stress result. Also, when the details of the microstructure become much smaller than the hole radius, i.e. l → 0, then F → 0, and σm → 3 p, which recovers the classical elasticity result. To see a continuous transition from the classical elasticity to both Cosserat models, it is convenient at this stage to bring in, after Cowin (1969, 1970a, b), a nondimensional constant α S N= = 0 ≤ N ≤ 1. (6.124) α+µ S+ P Then, following Cowin, F is given by F =

8 (1 − ν) N2 4 + ( NL) 2 + 2NL KK 01 (( NL) NL)

(6.125)

whereby, in our notation, A+ S 1−ν = 2S

a L= =a l

4M . S

(6.126)

Indeed, for this example, the case N = 0 leads to the classical elasticity solution, while N = 1 gives the limiting case of couple-stress theory (6.121). The micropolar unrestricted theory follows for any value of N between 0 and 1.

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TABLE 6.1

A Comparison of Various Notations for Micropolar Compliances in the First Planar Problem Compliance

Our Notation

Nowacki

Eringen

Mindlin

Shear compliance 1/G

S

1 µ

1 µ E +κ E /2

2(1+ν) E

Plane strain bulk compliance

A

1 λ+µ

1 λ E +µ E +κ E /2

1−2ν G

Bulk compliance

P

1 α

2 κE

0

Bending or curvature compliance

M

1 γ +ε

1 γE

1 4B

Characteristic length (square of) l 2

P+S 4M

(µ+α)(γ +ε) 4µα

γ E (µ E +κ E ) κ E (2µ E +κ E )

B G

Note: Mindlin’s column refers to the restricted model.

However, in general, the limit N = 0 needs to be used with caution (Lakes, 1985) because the zero value of N does not automatically imply that all the micropolar stiffnesses vanish (i.e., the compliances go to infinity), and, in fact, the microstructural degrees of freedom may remain. Finally, for the sake of reference we provide in Table 6.1 a comparison between several different notations employed in the works referenced here. A similar table linking the notation of other references is included in Cowin (1970a).

6.4

The CLM Result and Stress-Invariance

6.4.1 Isotropic Materials We now allow the micropolar solid to be inhomogeneous by taking all the material coefficients in (6.99) to be class C 2 functions of x, and assume it is simply connected (i.e., no holes are present). Now, note that the compatibility equations (6.97) of Section 6.3 can be written as γ22,11 + γ11,22 = (γ12 + γ21 ) ,12 γ12,22 − γ21,11 = (γ22 − γ11 ) 12 − κ13,1 − κ23,2

(6.127)

κ23,1 = κ13,2 . Substituting (6.98) of Section 6.3 into the compatibility condition (6.127)1 , and using (6.95)1 and (6.95)2 of Section 6.3, we obtain, after rather lengthy manipulations, ∇ 2[

A+ S (τ11 + τ22 )] − [S,1 τ11 ],1 − [S,2 τ22 ],2 − [S,1 τ12 ],2 − [S,2 τ21 ],1 = 0. 2 (6.128)

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Proceeding similarly with respect to the compatibility condition (6.127)2 , we obtain, again after much algebra, P+S (τ12 − τ21 )] + [S,1 τ11 ],2 − [S,2 τ22 ],1 − [S,1 τ12 ],1 − [S,2 τ21 ],2 2 + 2 [Mµ13 ],1 + 2 [Mµ23 ],2 = 0. (6.129)

∇ 2[

The third compatibility condition (6.127)3 yields [Mµ23 ],1 − [Mµ13 ],2 = 0.

(6.130)

Let us note here that (6.128) simplifies to equation (5.36) of Chapter 5 in the case of classical elasticity where τ12 = τ21 . As before, we ask the question: P, then “Supposing that A, P, S, and M are changed to some A, S and M, under what restrictions would the original stress field τ11 , τ22 , τ12 , τ21 , µ13 , µ23 remain unchanged?” An examination of (6.95) implies that we must have + A S = m ( A + S)

S,1 = mS,1

S,2 = mS,2

S,11 = mS,11

S,22 = mS,22

S,12 = mS,12 ,

(6.131)

where m is an arbitrary scalar. Next, note that (6.131) leads to =m A + b A

S = mS − b,

(6.132)

where b is an arbitrary constant restricted by the requirement that the compliances be non-negative. By a similar reasoning, the compatibility equation (6.127)2 implies = nP + c P

S = nS − c

= nM, M

(6.133)

where c is an arbitrary constant restricted by the requirement that the compliances be non-negative. Finally, the compatibility condition (6.127)3 involves a micropolar compliance M and its first derivatives, and so we have the following conditions: = nM M

,1 = nM,1 M

,2 = nM,2 . M

(6.134)

Considering all the above results, it is seen that they are consistent providing m = n and b = c. It thus follows that the stress will be invariant if the following shifts in material compliances are taken: =m A + c A

= mP + c P

S = mS − c

= nM. M

(6.135)

In the terminology of the CLM result (Chapter 5), both materials are equivalent. Clearly, equations (6.135) represent a constant shift in three out of four material parameters, and this is a weaker shift than the linear one: (5.39) in Chapter 5. Note: The second planar problem does not admit a shift.

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6.4.2 Anisotropic Materials and the Null-Lagrangian In the case of an anisotropic material the starting point is provided by relations (6.27). A substitution into the compatibility relations (6.127) and a subsequent inspection leads us to conclude that the stress field remains (1) (2) unchanged when the material constants are modified from Sijkl and Sijkl to I I (1) (1) (1) (2) (2) (1) S = S + S (, −, ) and S = S , providing S (, −, ) is the ijkl

ijkl

ijkl

ijkl

ijkl

ijkl

shift defined by (6.135) for isotropic materials, and

(2) Sijkl

undergoes no shift.

(1) Sijkl

Let us first note that the compliance of an isotropic material is a following function of three constants κ, µ, and α: (1) I Sijkl (κ, µ, α) =

1 4

1 1 + µ α

δik δjl +

1 1 − µ α

δil δ jk +

1 1 − κ µ

δij δkl . (6.136)

Equivalently, noting (6.99), this can also be given in terms of A, P, S as (1) I Sijkl ( A, P, S) =

1 (S + P) δik δjl + (S − P) δil δ jk + ( A − S) δij δkl . 4

(6.137)

If we let κ = , µ = −, and α = in (6.136), it follows that the shift tensor is given as (1) I Sijkl (, −, ) =

1 δij δkl − δil δ jk . 2

(6.138)

Let us now compare this with the CLM shift tensor of classical elasticity (5.49) with the rotation (5.50) of Chapter 5. This leads to a question: “What is (1) I the meaning of Sijkl (, −, )?” The answer is obtained from a consideration of a new rotation tensor defined as R(1) ijkl = δij δkl − δil δ jk

(1) I Sijkl (, −, ) =

1 (1) R , 2 ijkl

(6.139)

showing that R(1) ijkl σkl is also a right-angle rotation of τkl . The foregoing development allows us to present the “shift-result” in terms of the null-Lagrangian. First, let us recall from the theory of micropolar elasticity (Nowacki, 1986) that two stress functions φ and ψ can be introduced such that [recall (6.107)] τij = R(1) ijkl φ,kl − Rik ψ,k j

µi3 = ψ,i ,

(6.140)

where Rik was specified in (5.54) of Chapter 5. The strain energy density (1) (2) (1) (2) , Sijkl ) = τij Sijkl σkl + µi3 Si3k3 µk3 W(Sijkl

(6.141)

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may now be written as (1) (2) (1) (1) W Sijkl , Sijkl φ − R ψ Rmnpq φ, pq − Rmp ψ, pm = R(1) Sijmn ,kl ik ,k j ijkl (2) + µi3 Si3k3 µk3 .

(6.142)

First, let us note that the minimization of (6.142) via the Euler–Lagrange equations of (6.141) results in the compatibility equations in terms of φ and ψ for a general anisotropy. On the other hand, the energy density of the shift in compliance becomes (1) I W Sijkl (, −, ) , Oijkl (1) I (1) = R(1) Sijmn Rmnpq φ, pq − Rmp ψ, pm , (6.143) φ − R ψ ,kl ik ,k j ijkl where Oijkl is a null tensor. Now, observing that (1) (1) (1) R(1) klij Rijmn Rmnpq = Rklpq

(1) R(1) ijkl Rijmn = δkm δln ,

(6.144)

we find 2 1 (1) I W Sijkl φ,11 φ,22 − φ,12 (, −, ) , Oijkl = 2 + ψ,12 φ,22 − φ,11 + φ,12 ψ,22 − ψ,11 .

(6.145)

It is interesting to note here that: 1. The first term in the square brackets is the same as that in the classical elasticity [our (5.59) in Chapter 5, or equation (35) of Cherkaev et al. (1992)]. 2. The second and third terms represent the coupled contribution of φ and ψ potentials. 3. The energy (6.145) can also be written as the divergence of a vector field v,k such that 1 φ,l R(1) klpq φ, pq − φ,l Rkr ψ,rl − φ,i j Rik ψ, j 2 + ψ, j R jk Rnr − Rnk Rjr ψ,rn φ,1 φ,22 − φ,2 φ,12 − φ,1 ψ,12 − φ,2 ψ,22 + φ,21 ψ,1 + φ,22 ψ,2 1 = , 2 φ,2 φ,11 − φ,1 φ,12 + φ,1 ψ,11 + φ,2 ψ,12 − φ,11 ψ,1 − φ,12 ψ,2 (6.146)

vk =

where, again, the first term in each of the square brackets can be recognized as that of classical elasticity. It follows now that vk,k = 0,

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Microstructural Randomness and Scaling in Mechanics of Materials (1) I or that the Euler–Lagrange equations for W(Sijkl (, −, ), Oijkl ) are satisfied identically, implying that it is a null-Lagrangian.

6.4.3 Multiply Connected Materials The requirement that B be simply connected ensures that the stress field σ is single valued. Generalizing the results of Section 5.2.3 of Chapter 5, we have 3 = =

∂ϕ3

d x1 +

∂ x1

∂ε21 ∂x1

−

x2

−

x1

∂ x1 ∂x1

∂ x2

d x1 −

∂ε22

∂ε22

∂ε11

d x1 +

∂x2

d x2 +

−

d x2 ,

∂ x2

(6.147)

∂ε22

∂ε22

∂ε12

d x2

∂ x1

d x1 +

∂ε22 ∂x1

∂ x2

∂ε11

∂ε11

x2

d x1 +

∂ x1

D2 − x0 3 = −

d x2

∂ x2

x1

∂ x2

∂ε11

−

D1 + y0 3 = −

∂ϕ3

−

∂ε12 ∂x2

−

∂ε21

d x2 , (6.148)

∂x2

d x2

∂ε12 ∂x1

+

∂ε21 ∂x1

−

∂ε11

∂ x2

d x1 , (6.149)

where D1 and D2 are dislocation vectors and 3 is a disclination. (Nowacki, 1986; Takeuti, 1973). If, using (6.98), we express the above equations in terms of stresses, they take on the following forms: 3 =

∂ ∂n

[( A + S) (σ11 + σ22 )] ds = 2

+

(S + P)

∂ ∂s

t1

(σ21 − σ12 ) ds +

4 ( D1 + y0 3 ) =

x2

=2

∂ ∂n

− x1

St2 ds − 2

∂x1

ds − 2

(σ12 − σ21 )

∂

∂s

x2

x2 (σ21 − σ12 )

∂ ∂s

t2

∂S ∂ x2

ds

(S − P) ds, (6.150)

( A + S) (σ11 + σ22 ) ds

−2

∂S

∂S ∂ x1

∂S ∂s

t1 +

ds,

∂S ∂ x2

t2 ds (6.151)

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x1

=2

∂ ∂n

+ x2

∂

x1

+2

∂s

St1 ds + 2

221

x1 (σ21 − σ12 )

( A + S) (σ11 + σ22 ) ds

∂S ∂ x1

∂S ∂s

t1 +

∂S ∂ x2

t2 ds

ds,

(6.152)

where n and s denote the outer unit normal and arc length of the hole boundary. With reference to Section 6.3.3, for the problem of an infinite plate with hole, we have these conclusions: 1. For classical elasticity, σm as well as the entire stress field are independent of elastic constants, say A and S (or µ and ν). 2. For a pseudo-continuum which has three constants— A, S, and M (or µ, ν and B)—the stress field depends on two combinations of these constants, such as ( A + S) /S (or ν) and l 2 = S/4M, and thus no shift is possible here. 3. For an unrestricted continuum, which has four constants, A, S, P, and M, the dependence is on two independent combinations of the elastic constants ( A + S) / ( P + S) and l 2 = (S + P) /4M, which, in light of (6.134), allow a shift. Note: Setting P = 0, we get the pseudo-continuum. In this case S + P becomes S, and S by itself is not invariant under shift. 6.4.4 Applications to Composites 6.4.4.1 Two-Phase Materials We continue to generalize the results of Chapter 5. Thus, when the planar body is made up of two or more phases, we must also consider the interface boundary conditions. Assuming perfect bounding between micropolar phases (1 and 2), they have the following (classical) form in the curvilinear coordinate system (n, s, x3 ): (1) (2) τnn = τnn

(1) (2) τns = τns

(2) µ(1) n3 = µn3

(2) u(1) n = un

(2) u(1) s = us

ϕ3(1) = ϕ3(2) .

(6.153)

Alternately, using the boundary conditions proposed by Dundurs (1996), we have (1) (2) τnn = τnn

(1) (2) τns = τns

κn(1) = κn(2)

γss(1) = γss(2)

(2) µ(1) n3 = µn3 (1)

∂ϕ3

∂s

(6.154)

(2)

=

∂ϕ3

∂s

,

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where κn is the change in curvature of boundary curve κn =

∂ ∂s

(γns + γsn ) −

∂γss ∂n

− κγnn ,

(6.155)

and γss is the stretch strain. The advantage of using this second set of boundary conditions is that they can be expressed in terns of stresses, and thus the dependence of the solution on the micropolar constants can be seen more easily. Using the constitutive relations (6.98), which remain of the same form in (n, s, x3 ), and the equilibrium equation ∂τnn ∂n

+

1 ∂ (τns + τsn ) + κ (τss − τnn ) = 0, 2 ∂s

(6.156)

the boundary condition (6.154)4 , which implies the continuity of a change in curvature, becomes, in view of (6.155) and again (6.98), ∂τ (1) ∂ ∂τ (2) ( A2 + S2 ) τss(2) − ( A1 + S1 ) τss(1) + sn ( A1 + S1 ) − sn ( A2 + S2 ) ∂n ∂n ∂s ∂s ∂

(1) + 2τsn (1) + τnn

∂ S1 ∂s

∂ ∂n

(2) − 2τsn

∂ S2 ∂s

(1) + 2τns

∂ ∂s

(S1 − S2 ) − 2 (S1 − S2 )

[( A2 − A1 ) − (S2 − S1 ) + 2 ( A2 − A1 ) κ] = 0.

(1) ∂τns

∂s

(6.157)

Now, taking note of (6.99), the continuity of stretch strain (6.154)5 implies (1) ( A2 − A1 ) − (S2 − S1 ) = 0. (6.158) ( A2 + S2 ) τss(2) − ( A1 + S1 ) τss(1) + τnn

Finally, noting ∂ϕ3 /∂s = κs3 , we observe that (6.154)6 implies (2) M1 µ(1) s3 − M2 µs3 = 0.

(6.159)

Note that these boundary conditions are invariant under the shift (6.135). Thus, if the multiphase material is simply connected (i.e., contains intrusions), the governing equations in terms of stresses are (6.95) and (6.128–6.130) for each phase, and these are invariant under traction loading and boundary conditions (6.154), or, equivalently, (6.154)1−3 and (6.157–6.159). However, if the material is multiply connected, we also need Ces`aro integrals that involve the continuity of displacements. 6.4.4.2 Effective Moduli of Composites We now move to the determination of effective properties of composites, which in the case of classical elasticity turned out to nicely involve the CLM result. We thus examine the shift in macroscopically effective compliance (1)eff (2)eff tensors Sijkl and Sijkl connecting the volume averaged stress and strain tensors (1) ε ij = Sijkl σ kl

(2) κ ij = Sijkl µkl ,

(6.160)

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wherein denotes a volume average. We proceed here in a fashion analogous to that of Cherkaev et al. (1992) with a restriction to materials with continuously varying (class C 2 ) properties. Now, stresses τ11 , τ22 , τ12 , τ21 , µ13 , µ23 are the same in two equivalent (1) (2) (1) (2) materials (Sijkl (x), Sijkl (x)) and ( Sijkl (x), Sijkl (x)), so that the strain fields γij and γ ij satisfy the relation (1) (1) (1) I (1) I γ ij = Sijkl τkl = Sijkl τkl + Sijkl (, −, ) τkl = γij + Sijkl (, −, ) τkl .

(6.161) (1) I Volume averaging (6.161), and noting that Sijkl (, −, ) is independent of position, we find (1) I Si jkl τ kl = Si jkl τ¯kl + Sijkl (, −, ) τ¯kl . γ ij = (1)eff

(1)eff

(6.162)

which shows that the effective compliance tensor of the second material (with hat) is given by that of the first material (without hat) plus the shift given by (6.138) (1)eff (1)eff (1) I Sijkl = Sijkl + Sijkl (, −, ) .

(6.163)

We conclude (by inspection) that there is no shift in the second effective compliance tensor and (2)eff (2)eff Sijkl = Sijkl .

(6.164)

As mentioned at the beginning of this section, this conclusion holds for simply connected inhomogeneous media with twice-differentiable properties. 6.4.5 Extensions of Stress Invariance to Presence of Eigenstrains and Eigencurvatures 6.4.5.1 Basic Concepts Here we extend the results of Section 5.2.3, Chapter 5. First note that one can generalize the elasticity with eigenstrains to Cosserat elasticity. Thus, in analogy to (5.74), the total strain γij is the sum of the elastic strain gij and the eigenstrain γij∗ , γij = gij + γij∗

(6.165)

while the total (generally nonsymmetric) curvature κij is the sum of the elastic curvature kij and the eigencurvature κij∗ κij = kij + κij∗

(6.166)

The eigenstrain γij∗ and eigencurvature κij∗ are inelastic strains and curvatures, respectively. The total strain γij and total curvature κij must satisfy

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compatibility equations γli,n − γni,l − kni κlk + kli κnk = 0 From (6.165 and 6.166) we have (1) (1) τij = Cijkl γkl − γkl∗ gkl = Cijkl

κli,n = κni,l .

(6.167)

(2) (2) µij = Cijkl κkl − κkl∗ . (6.168) kkl = Cijkl

The inverses of (6.168) are (1) gij = Sijkl τkl

(2) kij = Sijkl µkl .

(6.169)

Note that all the quantities may depend on the spatial position x (≡ xi ). We extend the assumption that the material is free from any external forces and surface constraints to Cosserat elasticity with eigenstrains and eigencurvatures. If these conditions of free surface are not satisfied, the force-stress and couple-stress fields can be obtained by a superposition of the force-stress and couple-stress of a free body and the stress obtained from the solution of a given boundary value problem with non zero external forces or boundary conditions. The force-stresses and couple-stresses must satisfy the equations of equilibrium [assume no body and inertia forces in (6.12) and (6.17)] in B τji,j = 0

ijk τ jk + µji,j = 0 i, j = 1, 2, 3,

(6.170)

and zero force-traction or couple-traction free boundary conditions on ∂B τji n j = 0

µji n j = 0.

(6.171)

In analogy to Mura’s approach in classical elasticity, by substituting (6.168) into (6.170) and assuming a homogeneous material, we have (1) (1) ∗ (2) (1) ∗ ∗ Cjikl γkl, j = Cjikl γkl, j C (2) jimn κmn, j = C jimn κmn, j − ijk C jkmn γmn − γmn , (6.172) and by substituting (6.168) into (6.171) we obtain (1) (1) ∗ Cjikl γkl n j = Cjikl γkl n j

(2) (2) ∗ Cjikl κkl n j = Cjikl κkl n j .

(6.173)

Note that in the absence of eigenstrains (γ ∗ = 0), the left-hand side of (6.172)1 corresponds to τji,j , and the left-hand side of (6.173)1 to τji n j . Thus, ∗ (6.172)1 , is in the form τji,j = −Xi where Xi = −Cjikl γkl, j and (6.173)1 is in the ∗ form τji n j = ti where ti = Cjikl γkl n j . Therefore, the contribution of eigenstrain γ ∗ to the equations of equilibrium (6.170) is mathematically equivalent to a body force, and in the boundary conditions (6.171)1 is similar to a surface force. Next, in the absence of eigencurvatures (κ ∗ = 0) the left-hand side of (6.172)2 corresponds to ijk τjk + µji,j , whereas the left-hand side of (6.173)2 to µji n j . Thus, (6.172)2 is in the form ijk τjk + µji,j = −Yi where the couple-body

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∗ ∗ force Yi = −Cijkl γkl, j and (6.173)2 is in the form τji n j = ti where ti = C jikl γkl n j . ∗ Therefore, the contribution of eigenstrain γij to the equations of equilibrium (6.170)2 is mathematically equivalent to a body force, while their contribution to the boundary conditions (6.171)2 is similar to a surface force. In the next sections we focus on the planar elasticity with eigenstrains, assuming isotropy in elastic properties. In addition, we relax the boundary condition (6.171) and admit non-zero tractions to make the formulation more general. This will not change our conclusions on the reduced parameter dependence. Note that the special case of uncoupled micropolar thermoelasticity with eigenstrains εij∗ is defined as

γij∗ = αij T

αij = 0

if

i= j

i, j = 1, 2, 3,

(6.174)

where αij is a thermal expansion coefficient and T is temperature change. We will refer to this special case in examples. We demonstrated the reduced parameter dependence in the in-plane stress fields in the problems governed by plane Cosserat elasticity with eigenstrains and eigencurvatures, if eigenstrains and curvatures satisfy certain conditions. Note that no conditions are needed for the plane stress case for the form of shift with m = 1. These results can be applied for two-phase materials to linear planar uncoupled micropolar thermoelasticity, where eigenstrains are uniform and represent the product of the thermal expansion coefficient and temperature change. The analysis can also be extended to multiphase materials and inhomogeneous multiply connected materials. 6.4.5.2 Inhomogeneous Materials Here we focus on 2D boundary value problems involving applied force- and couple-tractions ti = τji n j

mi = µji n j on ∂B.

(6.175)

It is also useful to write the constitutive laws (recall Section 6.3.1) γ11 = γ22 = γ12 = γ21 = κ13 =

S A+ S ∗ ∗ + ηγ33 (τ11 + τ22 ) − τ22 + γ11 4 2 S A+ S ∗ ∗ + ηγ33 (σ11 + σ22 ) − τ11 + γ22 4 2 S P ∗ (τ12 + τ21 ) + (τ12 − τ21 ) + γ12 4 4 P S ∗ (τ12 + τ21 ) − (τ12 − τ21 ) + γ21 4 4 ∗ ∗ Mµ13 + κ13 κ23 = Mµ23 + κ23 ,

(6.176)

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where 1 µ

S=

1 α

P=

M=

1 , γ +ε

(6.177)

and for plane strain η=

λ , 3K − λ

(6.178)

µ (3λ + 2µ) , λ + 2µ

η = 0.

(6.179)

1 1 , = κplane-strain λ+µ

A= while for plane stress A=

1 κplane-stress

=

If we express λ in terms of Poisson’s ratio v λ=

2vµ , 1 − 2v

(6.180)

then, for plane strain A=

1 κplane-strain

1 − 2v , = µ

1 η=v= 2

A 1− , S

(6.181)

while, for plane stress A=

1 κplane-stress

=

1−v , (1 + v) µ

η = 0.

(6.182)

Now, with the assumption of both compliances and eigenstrains being smooth functions of position, it follows from the substitution of (6.176) into (6.127)1 , and in light of (6.171)1 , that the first compatibility condition in (6.127) is 1 2 1 1 1 ∇ [( A + S) (τ11 + τ22 )] − [S,1 τ11 ],1 − [S,2 τ22 ],2 − [S,1 τ12 ],2 4 2 2 2 1 ∗ ∗ ∗ ∗ ∗ − [S,2 τ21 ],1 = −γ11,22 − γ22,11 + γ12 + γ21 − ∇ 2 ηγ33 ,12 2 ∗ ∗ ∗ −2η,1 γ33,1 − 2η,2 γ33,2 − η∇ 2 γ33 .

(6.183)

Similarly, the compatibility condition (6.127)2 yields 1 2 1 1 1 ∇ [( P + S)] + (S,1 τ11 ) ,2 − (S,2 τ22 ) ,1 − (S,1 τ12 ) ,1 4 2 2 2 1 ∗ ∗ ∗ ∗ + (S,2 τ21 ) ,2 = −γ11,12 + γ22,12 − γ12,22 + γ21,11 2 ∗ ∗ − ( Mµ13 ) ,1 − κ13,1 − ( Mµ23 ) ,2 − 2κ23,2 .

(6.184)

Finally, from the compatibility condition (6.127)3 we have ∗ ∗ = ( Mµ13 ) ,2 + κ13,2 . ( Mµ23 ) ,1 + κ23,1

(6.185)

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Next, following Ostoja-Starzewski and Jasiuk (1995), we seek the conditions for invariance in planar force-stresses and couple-stresses with respect to shift in compliances. We first state the conditions for no eigenstrains and eigencurvatures. Equations (6.183–6.185) remain invariant if the compliances A, P and S undergo a shift =m A + c A

= nP + c P

S = mS − c

= mM. M

(6.186)

Under such a linear shift the force-stress does not change in the absence of eigenstrains and eigencurvatures. Next, we investigate what conditions are needed to be satisfied in the presence of eigenstrains and eigencurvatures. In this analysis, in addition to the plane stress and plane strain cases, which lead to different results, the distinction is made between the cases when m = 1 and m= 1. For the plane stress case and m = 1, (6.183–6.185) remain unchanged under the linear shift (6.186), i.e., the planar stress components remain unchanged, and thus there is a reduced parameter dependence. For the plane stress case and m = 1, (6.183–6.185) give the following conditions on eigenstrains and eigencurvatures: ∗ ∗ ∗ ∗ γ11,22 + γ22,11 − γ12 + γ21 =0 ,12 ∗ ∗ ∗ ∗ ∗ ∗ γ11,12 − γ22,12 + γ12,22 + γ21,11 + κ13,1 + κ23,2 =0 ∗ κ23,2

=

(6.187)

∗ κ13,1 .

For the plane strain case and m = 1, (6.183) remains invariant under the shift (6.186) when ∗ ∗ ∗ ∗ ∇ 2 ηγ33 + 2η,1 γ33,1 + 2ηγ33,2 + η∇ 2 ε33 = 0.

(6.188)

For the special case of uniform eigenstrains, the condition above is satisfied provided that ∗ γ33 =0

∇ 2 η = 0,

or

(6.189)

while for the case of a homogeneous material the condition (6.188) is satisfied if ∗ ∇ 2 γ33 =0

or

η = 0.

(6.190)

For the case of plane strain and m = 1, the following conditions need to be satisfied ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ11,22 + γ22,11 − (γ12 + γ21 ) ,12 + ∇ 2 ηγ33 + 2η,1 γ33,1 − 2η,2 γ33,2 + η∇ 2 γ33 =0 ∗ ∗ ∗ ∗ ∗ ∗ (γ11 − γ22 ) ,12 + γ12,22 + γ21,11 + κ13,1 + κ23,2 =0 ∗ ∗ κ23,1 = κ13,2 .

(6.191)

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6.5

Effective Micropolar Moduli and Characteristic Lengths of Composites

6.5.1 From a Heterogeneous Cauchy to a Homogeneous Cosserat Continuum Any of the lattices considered in Chapter 3 can be viewed as a planar, twophase composite material: one phase (1) is the solid that makes up the beams and another (2), quite trivially, is the vacuum in the pores. Clearly then, the mismatch of elastic properties, that is the ratio of moduli E (1) /E (2) , is infinite (Ostoja-Starzewski et al., 1999). Starting from this consideration, one may now consider “nontrivial” two-phase composites made of two kinds of solids having a finite mismatch, and generalizing the previously established method (i.e., that for lattices) for passage from heterogeneous Cauchy to a homogeneous Cosserat continuum; Fig. 6.3. This passage is done according to the following equality: 1 2

εij Cijkl εkl dV = V

V 0 (1) 0 0 (2) 0 Ci3k3 κk3 , γ C γ + κi3 2 ij ijkl kl

i, j, k, l = 1, 2,

(6.192)

where the left-hand side is the total elastic strain energy stored in the unit cell of the matrix-inclusion composite (a function of Cauchy strain fields εij ), while the right-hand side is the energy of a Cosserat continuum (a function of 0 volume-average strains εij0 and curvatures κi3 of the unit cell). V is the volume of the unit cell B L . Cijkl is the elastic stiffness of the composites’ constituents, (1) (2) while Cijkl and Ci3k3 are the sought (effective) micropolar stiffnesses. The key issue concerns the choice of loading on the periodic unit cell. Following Forest (1989, 1999) and Forest and Sab (1998), the appropriate periodic boundary conditions in terms of displacements are generally nonlinear: ui (x) = Ai + Bij x j + Cijk x j xk + Dijkl x j xk xl

(a)

(b)

(6.193)

(c)

FIGURE 6.3 (a) A periodic, globally orthotropic, matrix-inclusion composite, of period L, with inclusions of diameter d arranged in a square array; (b) a periodic unit cell with soft inclusions at corners; (c) a periodic unit cell with a stiff inclusion at the center.

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where vi takes an equal value on corresponding points of the boundary. The field (6.193) satisfies the Hill condition generalized to the Cosserat material εij σij = γij τij + µij κij

with τij = σij0

µij =

2 e ikl xk σl j . 3

(6.194)

In the notation and language of homogenization theory (e.g., Bensoussan et al., 1978), the Cauchy material in the unit cell is described by a fast coordinate x, while the Cosserat material is described by the slow coordinate y. Accordingly, the displacement, stress and strain fields in the Cauchy continuum are u σ

ε,

(6.195)

while the fields of displacement, rotation, force-stress, couple-stress, strain, and rotation in the Cosserat continuum are U

M

E

(6.196)

K.

We now rewrite (6.194) as εij σij = ij E ij + Mij K ij

with

ij = σij0

Mij =

2 e ikl xk σl j . 3

(6.197)

In the general setting, we have a micromorphic continuum with U (displacement) and χ (microdeformation) fields, and these are determined from a minimization problem (6.198) (U, χ ) = min |u (x) − U − χ · (x − X) |2 The solution is U (X) = u (x)

χ (x) = u⊗ (x − X) · A−1 ,

(6.199)

with A = (x − X) ⊗ (x − X) and respective gradients being U ⊗ ∇X = u ⊗ ∇ x

χ ⊗ ∇x = (u ⊗ x) ⊗ ∇x · A−1 − U ⊗ A−T .

(6.200)

Although none of our boundary conditions were of periodic type, the situation changes when an unrestricted model is used. Indeed, such a derivation has been done in Forest and Sab (1998) by extending the homogenization method (e.g., Sanchez-Palencia and Zaoui, 1987). The loading on ∂ B L in 2D is then effected by boundary conditions involving polynomials of the general form u1 (x) = B11 x1 + B12 x2 − C23 x22 + 2C13 x1 x2 + D12 x23 − 3x12 x2 u2 (x) = B12 x1 + B22 x2 − C13 x12 + 2C23 x1 x2 − D12 x13 − 3x1 x22 . (6.201) Upon a comparison of this with equations (6.193–6.196), we observe that the derivation of the restricted model involves a second-order polynomial, while that of the unrestricted one requires a third-order polynomial. Work has also

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been done on homogenization of a heterogeneous Cosserat-type continuum by a homogeneous one (Forest, 1999; Forest et al., 1999); see also Forest et al. (2000, 2001). Most recently, a related homogenization procedure was outlined by Onck (2002) for the derivation of the micropolar model. In particular, a loading via a skew-symmetric part of the strain tensor, applied in terms of boundary rotations, was proposed to grasp the effect of the difference between the micro-rotation ϕi and the macrorotation e ijk uk, j /2; recall the restricted model’s condition (6.37), where that difference is set to zero. Note: A major outstanding challenge is to develop boundary conditions for the homogeneous Cosserat from a heterogeneous Cauchy continuum. This should be done through a scheme somewhat analogous to that developed so far for the effective micropolar moduli. 6.5.2 Applications A few years ago we computed effective micropolar moduli for planar matrixinclusion composites arranged in periodic arrays: triangular (Bouyge et al., 2001) and square (Bouyge et al., 2002), Figure 6.4, using a finite element method. Several different boundary conditions—ranging from displacementtype to traction-type, and various combinations thereof—were used. For (1) example, using displacement boundary conditions, we determine Cijkl from three tests: 1. Uniaxial extension: u1 (x) = 0

u2 (x) = ε22 x2

∀x ∈ ∂ B

(6.202)

(1) gives C2222 (= 2U cell /V when we set ε22 = 1). For our composite, (1) (1) C1111 = C2222 due to the symmetry of the square arrangement. 2. Biaxial extension:

u1 (x) = ε11 x1 yields

(1) 2C1111

+

(1) C1122

u2 (x) = ε22 x2

∀x ∈ ∂ B

(6.203)

(= 2U cell /V when we set ε11 = ε22 = 1).

3. Shear test: u1 (x) = ε12 x2 yields

(1) C1212

u2 (x) = 0

∀x ∈ ∂ B

(6.204)

(= 2U cell /V when we set ε12 = 1).

(2) , we conduct the fourth test, the bending 4. Finally, to determine Ci3k3 test:

u1 (x) = −κ13 x1 x2

u2 (x) = κ13

x12 2

∀x ∈ ∂ B

(6.205)

(2) (2) (2) gives C1313 . Note that in our study C1313 = C2323 due to the symmetry of square arrangement.

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231

u1 (x) = 0, u2 (x) = y22x2

u1 (x) = y11x1, u2 (x) = y22x2

u1 (x) = y12 x2, u2 (x) = 0

u1 (x) = –x1x2κ13, u2 (x) =

x12 κ 2 13

FIGURE 6.4 (1) (1) (1) (2) Tests for the determination of constants C2222 , C1122 , C1212 , and C1313 of a periodic composite with circular inclusions in a square arrangement under displacement boundary conditions (Bouyge et al., 2002). Left (right) column corresponds to the inclusion at the corner (center). Inclusions can be seen from the mesh pattern.

The resulting deformation modes for the above four tests under displacement boundary conditions are shown in Figure 6.4. Two distinct situations are considered here depending on whether the inclusion is softer or stiffer than the matrix. In the first case, the inclusion is located at the corner, whereas in the second it is located at the center. Typical results for effective moduli are (1) shown in Figure 6.5 in terms of C1212 for a wide range of Poisson’s ratio of the matrix. In the special case of no mismatch in the properties we recover a homogeneous medium of Cauchy type, whereby the composite microstructure

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dd dp tt

0.3 0.2 0.1

0.0001

0.001

0.01

0 0.1 10 Mismatch

100

1000

10000

FIGURE 6.5 (1) The effective moduli C1212 , normalized by E m , from three types of boundary conditions— displacement (dd) displacement-periodic (dp), and traction (tt)—plotted as functions of the stiffness ratio E i /E m for the case of Poisson’s ratios ν m = ν i = 0.3, at inclusions volume fraction of 18.4%. (From Bouyge et al., 2002. With permisssion.)

disappears and no Cosserat continuum is to be set up. Note that when the inclusion is softer, as well as stiffer, than the matrix, the micropolar model provides a better representation of the mechanics of the composite than the classical model. Indeed, this was brought out by the experiments of Mora and Waas (2000) on honeycombs with either porous or very stiff inclusions. In the case of traction boundary conditions, we use

1 2

σij Sijkl σkl dV = V

V 0 (1) 0 0 (2) Si3k3 µ0k3 , i, j, k, l = 1, 2, τij Sijkl τkl + µi3 2

(6.206)

where on the left we have the total complementary strain energy in the unit cell (a function of Cauchy stresses σij ), and on the right we have the complementary strain energy of a couple-stress continuum (a function of volume-average 0 stresses σij0 and couple-stresses µi3 of the unit cell). Here Sijkl (inverse of Cijkl ) is (1) (2) and Si3k3 are the sought effective, the microscale elastic compliance, while Sijkl micropolar compliances. Summing up, for the restricted (or Koiter) model of the composite, the micropolar moduli are bounded from above and below, respectively, by displacement and traction boundary conditions. In fact, as these bounds are wide, we recommend three mixed types of loadings to get tighter results. On the other hand, the characteristic lengths are highly insensitive to the

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Two- versus Three-Dimensional Micropolar Elasticity L

4

Cosserat

ℓ/L = 2/100

04

–6.0

E-00

E-00

.0E -00 4 .0E -0 04

04

–4

4

0E0

.0E

04

-00

4

-00

0E –8.

4

–5.0

–4.

4

-00

.0E

–9

–7

00 4

-00

.0E

04 -0

Cauchy

E-0

.0 E -0

-0

–7 .0E -

0E

–7.0

3

–6

04

0E -0

–1

4

–7 .

–6

.0E –6

Cosserat

04

–8 .

0 -0

.0E

00

4

-00

.0E

3 -00 .0E 1 4 – -00 .0E –9 004 0E–8.

E-

–8

-0 04

u2 .0

.0E

u2 –8

04

E-0

.0 –9

u2

3 E-00 –1.0 - 00 4 –9.0E

003 0E04 –1. E-004 -0 .0 E –9.0 –8

3

-00

.0E

–1

L

u2

–6

u2

–7

233

L

u2

–7 .0E – 6 -004 .0 E00 4

4

Cauchy

Cosserat Cauchy

ℓ/L = 2/10

ℓ/L = 2/1

FIGURE 6.6 Contour lines of vertical components of displacement in a panel loaded as shown, for various scale ratios l/L, where l is the brick length and L is the macroscopic load print. (From Trovalusci and Masiani, 2003. With permission).

mismatch in moduli, especially in the case of stiff inclusions, and this must be contrasted with the sensitivity of moduli. An interesting comparison of boundary value problems set up on structures made of Cauchy vis-`a-vis Cosserat materials was conducted by Trovalusci and Masiani (2003). Their research is motivated by the mechanics of block-type masonry structures, whose stability (and, therefore, safety) is of primary concern in places rich with ancient architecture like Italy and Greece. Figures 6.6 and 6.7 show comparisons of symmetric boundary value problems

L

L

06 E-0 2.00 -006 1.00E

06 E-0 2.00 06 1.00E-0

06

W12

(skw H)12 06

E-0

2.00

06

–1.0E-0

-006 1.00E

007

E–5.0

Cauchy

Cosserat

–1.0E

7 -00 .0E

ℓ/L = 2/10

6

00 0E-

2.0

Cauchy

–5

06 –1.0E-0 6 E-00 –2.0

06

Cosserat

-0

06

–1.0E-0

-0

06 –1.0E-0 006 0E–2.

ℓ/L = 2/100

.0E

06

06 -0 6 0E -00 0E 2.0

–2

–1.0E-0

6

00 E-

.0

–2

(skw H)12

Cosserat Cauchy

1.0

L

(skw H)12 W12

00 E

W12

1.

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-006

6 00 E006 0E2.0

0 1.0

ℓ/L = 2/1

FIGURE 6.7 Contour lines of components of microrotation (Cosserat model) and macrorotation (Cauchy model) in a panel loaded as shown, for various scale ratios l/L, where l is the brick length and L is the macroscopic load print.

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(a)

t t

d

d (b) FIGURE 6.8 (a) Microstructure of trabecular bone at the so-called mesoscale level of Figure 7.15 in Chapter 7; (b) Unit cell of an idealized periodic model following Gibson and Ashby (1988). (From Yoo and Jasiuk, 2006. With permission).

of a plate-type structure, modeled in either of two fashions. It is clear that, as the ratio l/L increases, the discrepancies between both models tend to increase. Another example of determination of micropolar moduli has recently been developed in studies of trabecular bone (Yoo and Jasiuk, 2006), see Figure 6.8. It appears to be a first-ever prediction and evaluation of apparent couplestress moduli for a 3D periodic orthotropic material.

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Problems 1. Derive the local equations of motion of a micropolar continuum from the global equations of conservation of linear and angular momenta. 2. Generalize the Mohr circle concept and analysis to plane stress micropolar elasticity. 3. Prove the inequalities (6.32). Hint: use the Sylvester theorem. 4. Determine the convolution operation involved in the equation (6.60). 5. With reference to equations (6.72), demonstrate that , and H are dispersive waves. 6. Examine the implications of equation (6.79). 7. With reference to equation (6.80), consider an isotropic, hemitropic medium. Develop the corresponding form of the free energy function and obtain restrictions on all the elastic constants, more general than those of (6.32). Hint: introduce three new elastic constants. 8. Extend Kirchhoff’s uniqueness proof from the setting of linear elasticity to linear micropolar elasticity. 9. Outline a theory of micropolar media for finite motions and strains. 10. Formulate the Clausius-Duhem inequality for micropolar elasticdissipative solids, and then outline a formulation of thermomechanics with internal variables. Consult other sources as necessary.

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7 Mesoscale Bounds for Linear Elastic Microstructures

Three statisticians go deer hunting with bows and arrows. They spot a big buck and take aim. One shoots and his arrow flies off three metres to the right. The second shoots and his arrow flies off three metres to the left. The third statistician jumps up and down yelling; We got him! We got him! The problem of effective properties of material microstructures has received considerable, and ever-growing, attention over the past thirty years. By effective (or overall, macroscopic, global) is meant the response assuming existence of a representative volume element (RVE). The RVE in the case of disorder (i.e., lack of microstructural periodicity), implies that there must be some scale larger than the microscale (e.g., single heterogeneity size) to ensure a homogenization limit. Overall, most studies of effective properties simply assume that the RVE is attained and do not specify its size—scarce prescriptions of solid mechanics vaguely state that domains roughly 10 to 100 times larger than the heterogeneity should be taken. In the late 1980s work began on the determination of RVE in the sense of Hill (1963) and on continuum random fields serving as input into stochastic finite element methods. This then led to bounds that explicitly involve the size of a mesoscale domain—this domain also being called a statistical volume element (SVE)—relative to the microscale and the type of boundary conditions applied to this domain. In general, the trend to pass from the SVE to RVE depends on various factors, and displays certain tendencies. This chapter discusses that issue for linear elastic materials, thus setting the stage for nonlinear and/or inelastic materials as well as for continuum random field models and stochastic boundary value problems, topics dealt with in the subsequent chapters.

7.1

Micro-, Meso-, and Macroscales

7.1.1 Separation of Scales Continuum mechanics hinges on the concept of a representative volume element (RVE) playing the role of a mathematical point of a continuum field 237

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approximating the true material microstructure. The RVE is very clearly defined in two situations only: (1) unit cell in a periodic microstructure, and (2) volume containing a very large (mathematically infinite) set of microscale elements (e.g., grains), possessing statistically homogeneous and ergodic properties. The approach via the unit cell is, strictly speaking, restricted to materials displaying periodic geometries. When we consider case (2) we intuitively think of a medium with a microstructure so fine we cannot see it— naturally then we envisage a homogeneous deterministic continuum in its place. This situation, as suggested by Figure 7.1 of the preface, and called the separation of scales d< d

L L macro ,

(7.1)

introduces three scales: 1. The microscale d, such as the average size of grain (or inclusion, crystal, etc.) in a given microstructure; we initially assume the microstructures to be characterized by just one size d. 2. The mesoscale L, size of the RVE (if so justified — see below). 3. The macroscale L macro , macroscopic body size. In equation (7.1) on the left we do admit two options, because the inequality d < L may be sufficient for microstructures with weak geometric disorder and weak mismatch in properties; otherwise a much stronger statement d L applies. Note also that the first inequality in (7.1)1 could even be a weak one because we may be considering a microstructure with a nearly periodic geometry and small mismatch in the properties of the phases. As opposed to the periodic homogenization which relies on a periodic window directly taken as the RVE (Figure 7.1[a]), the homogenization in random media is more complicated. In any case, the issue of central concern is the trend—either rapid, moderate, or slow—of mesoscale constitutive response, with L/d increasing, to the situation postulated by Hill (1963): “a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are macroscopically uniform.” In essence, (a) is a statement about the material’s statistics, while (b) is a pronouncement on the independence of effective constitutive response with respect to the boundary conditions. Both of these are issues of mesoscale L of the domain of random microstructure over which smoothing (or homogenization) is being done relative to the microscale d and macroscale L macro . These considerations, however, are not rigorous, because neither spatial statistics nor mechanics (or physics) definitions of properties have yet been introduced.

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239

(a)

(b) FIGURE 7.1 (a) A disordered microstructure of a periodic composite with a periodic window of size L; (b) one realization of a random composite B L/d of size L.

7.1.2 Basic Concepts A note on determinism. In principle, any realization B(ω) of the composite B = {B(ω); ω ∈ }, while spatially disordered (i.e., heterogeneous), follows deterministic laws of mechanics. The most preferred approach, dictated by stochastic mechanics, would be to first ascertain what happens to each and

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every B(ω) of B, starting from a certain random microstructure model, and then pass to ensemble setting, by taking the averages or higher moments as the need arises. In many situations, however, this may generate enormous amounts of perhaps not very useful information. Ensemble versus volume averaging. We reserve the overbar for spatial (volume-type) averages, and for ensemble averages. That is, if we have a random (n-component, real valued) field defined over some probability space {, F , P} − F being a σ -field and P a probability measure—over some domain X in R D of volume V : × X → Rn ,

(7.2)

the said averages are 1 (ω) ≡ V

(ω, x)dV

(x) ≡

v

(ω, x)dP.

(7.3)

We assume the conditions necessary for the fulfillment of commutativity of both operations to be satisfied (i.e., requirements of Fubini’s theorem), so that = . (7.4) The existence of the integral (7.3)2 is assumed in accordance with the ergodic theorem, while the interchangeability of both operations in (7.4) is the subject of so-called ergodicity, or ergodic property, discussed in Chapter 2 and shortly recalled in Section 7.3 below. 7.1.3 The RVE Postulate The random material onmesoscale, such as shown in Figure 7.1(b), is denoted B L/d = B L/d (ω); ω ∈ with B L/d (ω) being one realization. Properties on mesoscale are also described by an adjective apparent (Huet, 1995), as opposed to effective. The latter term pertains to the limit L/d → ∞ as it connotes the passage to the RVE, while any finite mesoscale involves statistical scatter and, therefore, describes some statistical volume element (SVE). Note that the separation of scales is also known in the solid mechanics literature as the MMM principle (Hashin, 1983). In the following, it will be convenient to describe the mesoscale by a nondimensional parameter δ = L/d

(7.5)

in the range [0, ∞), so that B L/d , a mesodomain, will be written Bδ , etc. Thus, δ = 0 signifies the pointwise description of the material, while δ → ∞ is the RVE limit. The setting is one of quasi-static loading, so that the body is governed locally by the equilibrium equation σij,j = 0,

(7.6)

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with σij being the Cauchy stress, body forces being disregarded. The case of dynamic loading is considered in Chapter 9. For a mesoscale body Bδ (ω) of volume Vδ , such as the microstructure shown in Figure 7.1(b), we define volume average stress and strain 1 1 σ (ω, x)dV ε(ω, x)dV. (7.7) σ δ (ω) = ε δ (ω) = Vδ Vδ Assuming we deal with a linear elastic microstructure, the problem is to pass from the random field of stiffness with fluctuations on the microscale σ = C(ω, x) : ε,

ω ∈ ,

x∈B

(7.8)

to some effective Hooke’s law σ δ = Ceff : εδ ,

(7.9)

whereby (a) the dependence on ω (i.e., randomness) would be removed, (b) the dependence on x (i.e. spatial fluctuations) of strain and stress fields would also vanish, and (c) the independence of response with respect to boundary conditions would be attained. It is intuitively expected that δ needs to be large, but exactly how large it should be, is the key problem. Suppose we are tackling a boundary value problem of a body on macroscopic length scales. In general situations, we have to use computational mechanics—such as, say, a finite element meshing of the body—and this approach conventionally assumes that every single finite element is at least as large as the RVE, although this is rarely verified. Thus, there arises a need to know the rate of approach of SVE to RVE in function of δ, mismatch and microgeometry of the phases, etc., and, whether that rate is too slow in a given problem. If the latter is actually the case, one then needs to set up a stochastic finite element (SFE) scheme to account for the microstructure-borne noise on the mesoscale of any given finite element, see Chapter 8. Let us note here that the typical recipes of solid mechanics (e.g., Lemaitre and Chaboche, 1994) say that δ should be about 10 to 100 for the RVE to apply, but the analyses reported below show that δ depends on the type of problem studied.

7.2

Volume Averaging

7.2.1 A Paradigm of Boundary Conditions Effect Clearly, the attainment of the RVE is a function of the scale δ as well as the mismatch in properties of inclusions versus matrix. To illustrate this point, let us consider boundary distributions of displacement u3 and stress traction t3 in two boundary value problems of the mesodomain Bδ (ω) of the matrixinclusion specimen of Figure 7.2 in antiplane elasticity. The material is piecewise uniform with perfectly bonded, isotropic phases, so that the governing

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FIGURE 7.2 Antiplane elastic responses of a matrix-inclusion composite, with nominal 35% volume fraction of inclusions, at decreasing contrasts: (a) C (i) /C (m) = 1, (b) C (i) /C (m) = 0.2, (c) C (i) /C (m) = 0.05, (d) C (i) /C (m) = 0.02. For (b–d), the first figure shows response under Dirichlet boundary conditions, while the second shows response under Neumann boundary conditions with σ 0 equal to the volume average σ of stress computed in the Dirichlet problem.

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equation is C ( p) ∇ 2 u = 0

p = m, i.

(7.10)

Here m and i denote matrix and inclusion, respectively, and C3i3 j = C ( p) δij (δij is the Kronecker delta) is the phase stiffness. Two boundary value problems are considered: one of Dirichlet type: 0 u3 (x) = ε3i xi

∀x ∈ ∂ Bδ ,

(7.11)

where the strain ε30 j is prescribed, and the other of Neumann type: t3 (x) = σ30j n j

∀x ∈ ∂ Bδ ,

(7.12)

where the stress σ30j is prescribed; ∂ Bδ denotes the boundary of Bδ . Figure 7.2(a) treats the situation of no mismatch in the material properties: C (i) /C (m) = 1. and so we can interpret it as either a uniform displacement field 0 0 0 on the boundary ∂ Bδ under ε 0 = (ε31 , ε32 ), with ε32 = 0, resulting in a uniform 0 0 stress field on ∂ Bδ , or a uniform stress field on ∂ Bδ under σ 0 = (σ31 , σ32 ), with 0 σ32 = 0, resulting in a uniform displacement field on ∂ Bδ . Evidently, both problems are perfectly interchangeable because the microstructure is trivially homogeneous. This then is the situation of the RVE. Both boundary value problems become much more interesting when C (i) /C (m) =

1. In Figures 7.2 (b–d) we decrease the mismatch by first setting it to 0.2, then 0.05, and finally 0.02. In each case, we first solve the Dirichlet 0 problem under ε 0 = (ε31 , 0), and find t(x). Next, we compute the volume average t of t(x) on ∂ Bδ , and set t0 = t to run the Neumann problem. We keep ε0 identical in all four cases (a–d). Let us now define an “apparent stiffness” Cd in the displacement controlled (d ) problem (7.11) via equation σ = Cd : ε 0 ,

(7.13)

and “apparent compliance” St in the Neumann (t ) problem (7.12) via equation ε = St : σ 0 .

(7.14)

The latter allows us to define “apparent stiffness” in the traction controlled −1 problem as Ct = St . The following points are noteworthy: 1. The volume average displacement of the resulting u3 (x) distribution in the problem (7.11) differs from that in the problem (7.12). 2. The “apparent stiffness” in one boundary value problem is different from that in the other one; this should not be surprising given the preceding observation.

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A Collection of Diverse Physical Problems Governed by the Laplace Equation Physical Subject

u(= u3 )

ε(= ε3i )

C(= C3i3j )

σ(= σ3i )

Antiplane elasticity

Displacement

Strain

Elastic moduli

Stress

Thermal conductivity

Temperature

Thermal gradient

Thermal conductivity

Heat flux

Torsion

Stress function

Strain

Shear moduli

Stress

Electrical conduction

Potential

Intensity

Electrical conductivity

Current density

Electrostatics

Potential

Intensity

Permittivity

Electric induction

Magnetostatics

Potential

Intensity

Magnetic permeability

Magnetic induction

Diffusion

Concentration

Gradient

Diffusivity

Flux

3. The degree to which Cd is different from (St ) −1 may be regarded as an indication of the departure from the effective moduli Ceff in separation of scales; as a measure of their closeness to Ceff one might use the deviation of the product Cd : St from unity. 4. Although the governing partial differential equation is linear (and we could even replace (7.10) by (Cij u3, j ) ,i = 0), the resulting property is nonlinear as a function of actual realization ω, scale δ, mismatch C (i) /C (m) , and type of boundary conditions (i.e., Dirichlet or Neumann); see Markov (2000) for a related discussion. For the sake of completeness, in Table 7.1 we collect various classical analogies of problems locally governed by the Laplace equation C T,ii = 0 (or by (Cij T, j ) ,i = 0) in two dimensions; see also (Hashin, 1983). Note here, with reference to equations (10.39) and (10.40) of Chapter 10, that the thermal dissipation th (which is analogous to the strain energy) is a scalar product of q and ∇T, divided by the absolute temperature T. However, when looking for response in a steady-state and for small temperature changes (which is where the linear conductivity applies), one can approximately treat T as a constant (effectively, a volume-averaged quantity). 7.2.2 The Hill Condition 7.2.2.1 Mechanical versus Energy Definitions Let us consider a body Bδ (ω) with a given microstructure, in which, as a result of some boundary conditions and in the absence of body and inertia

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forces, there are stress and strain fields σ and ε. If we represent them as a superposition of the means (σ and ε) with the zero-mean fluctuations (σ and ε ) σ (ω, x) = σ + σ (ω, x)

ε(ω, x) = ε + ε (ω, x),

we find for the volume average of the energy density over Bδ (ω) 1 1 1 1 U≡ σ (ω, x) : ε(ω, x)dV = σ : ε = σ : ε + σ : ε . 2V Bδ (ω) 2 2 2

(7.15)

(7.16)

Thus, we see that for the average of a scalar product of stress and strain fields to equal the product of their averages σ : ε = σ : ε,

(7.17)

σ : ε = 0.

(7.18)

we need

Relation (7.17) is called the Hill condition in the (conventional) volume average form (Hill, 1963; see also Kroner, ¨ 1972, 1986; Huet, 1982, 1990; Sab, 1991). Some authors (e.g., Stolz, 1986) call it the Hill–Mandel macrohomogeneity condition, after J. Mandel (1966). Writing (7.18) in the index notation, we have 1 σij εij = (σij − σij )(εij − εij )dV V V 1 = {[(σij − σij )(ui − ui )], j − (σij,j − σij,j )(ui − ui )}dV V V 1 = [(σij − σij )(ui − ui )]n j dS V ∂V 1 = [(ti − σij n j )(ui − εij x j )]dS. (7.19) V ∂V Now, for an unbounded space domain (δ → ∞), the fluctuations are negligible, but for a finite mesoscale, we find the necessary and sufficient condition for (7.17) to hold σ : ε = σ : ε ⇐⇒ (t − σ · n) · (u − ε · x)dS = 0. (7.20) ∂ Bδ

This is satisfied by three different types of boundary conditions on the mesoscale: uniform displacement (also called kinematic, essential, or Dirichlet) boundary condition (d) u(x) = ε0 · x ∀x ∈ ∂ Bδ ;

(7.21)

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uniform traction (also called static, natural, or Neumann) boundary condition (t) t(x) = σ 0 · n ∀x ∈ ∂ Bδ ;

(7.22)

uniform displacement-traction (also called orthogonal-mixed) boundary condition (dt)

t(x) − σ 0 · n · u(x) − ε 0 · x = 0 ∀x ∈ ∂ Bδ .

(7.23)

Here we employ ε0 and σ 0 to denote constant tensors, prescribed a priori. and note, from the strain average and stress average theorems: ε 0 = ε and σ0 = σ. Each of these boundary conditions results in a different mesoscale (or apparent) stiffness, or compliance tensor; Huet uses the term apparent. Either of these terms is used to make a distinction from the macroscale (or effective, global, overall) properties that are typically denoted by eff or ∗ , see also NematNasser and Hori (1993). For a given realization Bδ (ω) of the random medium Bδ , taken as a linear elastic body (σ = C(ω, x) : ε), on some mesoscale δ, condition (7.21) yields an apparent random stiffness tensor Cdδ (ω)—sometimes denoted Ceδ (ω)—with the constitutive law σ = Cdδ (ω) : ε 0 .

(7.24)

Similarly, the boundary condition (7.22) results in an apparent random compliance tensor Stδ (ω)—sometimes denoted Snδ (ω)—with the constitutive law being stated as ε = Stδ (ω) : σ 0 .

(7.25)

The third type of boundary condition, (7.23), involves a combination of (7.21) and (7.22); it results in a stiffness tensor Cdt δ (ω). In fact, this condition may best represent actual experimental setups; the other two are nearly impossible to realize physically. For example, (7.23) may signify displacement boundary conditions on two parallel sides, and traction-free boundary conditions on the remaining two parallel sides. Or, it may signify pure shear-type loading through boundary conditions (see Figure 7.3): 0 0 ε11 = −ε22

0 σ12 = 0.

(7.26)

7.2.2.2 Order Relations Dictated by Three Types of Loading Note from the discussion above that the use of (7.17) assures the equivalence of the properties from the mechanical standpoint—that is, via apparent Hooke’s law (7.21) or (7.22)—with the properties from the energy standpoint.

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FIGURE 7.3 Possible loading under orthogonal-mixed boundary condition (7.23).

Both approaches are equivalent for a homogeneous material—that is, the RVE—but not necessarily so for a heterogeneous one, Bδ (ω), of size finite relative to the microscale heterogeneity (δ < ∞). In fact, the following relation ordering the Neumann and Dirichlet apparent moduli holds: [Stδ (ω)]−1 ≤ Cdδ (ω). A proof of the above in the framework of functional analysis has been given by Suquet (1986), and another one in the framework of general convex analysis applied to the nonlinear case by Willis and Talbot (1989), which was followed by Huet (1990). Later on, Hazanov and Huet (1994) extended it to the following: d [Stδ (ω)]−1 ≤ Cdt δ (ω) ≤ Cδ (ω),

(7.27)

that is, the modulus of Bδ (ω) obtained under mixed dt-conditions (7.21) always lies between the moduli obtained under the t-conditions (7.22) and the d-conditions (7.23). Other consequences of (7.27)—especially, in the context of orthotropic materials—were discussed by Hazanov and Amieur (1995). Another important result, also due to Huet (1990), is that

Stδ (ω)

−1

≤ Ceff (ω) ≤ Cdδ (ω) .

(7.28)

That is, the effective modulus Ceff (ω) always lies between the harmonic average of moduli obtained under the Neumann boundary conditions on the ensemble Bδ and the arithmetic average of moduli obtained under the Dirichlet conditions on the same ensemble. Hill (1967) and Mandel (1966) gave a qualitative estimate of the error between Stδ (ω) and Cdδ (ω) Stδ (ω) : Cdδ (ω) = 1 + O (1/δ) 3

(7.29)

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Many quantitative estimates of δ-dependence—or what may be called scaling—were computed for many different materials by Huet and coworkers, this author and coworkers; see also Terada et al. (2000), Kanit et al. (2003), Vinogradov (2001). Note: Other definitions of the RVE than that employed here, without the concept of SVE, have also been considered (e.g., Stroeven et al., 2004). Recently, Sab and Nedjar (2005) introduced a periodization of random media, which supports the computational mechanics results of Gusev (1997) involving an assumption of some finite scale periodicity in random microstructures and confirmed that analysis. 7.2.3 Apparent Properties In general, if we consider a body Bδ (ω) of volume V subjected to a volume average strain by the boundary condition (7.21): ui (x) = εij0 x j

∀x ∈ ∂ Bδ ,

(7.30)

given the continuity of displacements throughout, we have the average strain theorem: 1 1 ui, j + u j,i dV = ui n j + u j ni dS ε ij = 2V V 2V S 0 1 = εik xk n j + ε 0jk xk ni dS = εij0 . (7.31) 2V S d d The apparent stiffness Ckli j (≡ C ) of Bδ (ω) made of a linear elastic microstructure may now be defined by d 0 σ ij = Cijkl εkl ,

(7.32)

where σ is the volume average stress. Alternatively, we may consider the volume average energy density in Bδ (ω) 1 1 1 U= σij εij dV = σij ui, j dV = σij ui n j dS 2V V 2V V 2V S 1 1 1 d 0 = σij εij0 dV = σ ij εij0 = εkl0 Ckli (7.33) j εij , 2V V 2 2 d where the equilibrium σij,j = 0 was used. Thus, the apparent stiffness Ckli j may be defined either from the mean stress σ or from the mean energy density U if the boundary condition (7.30) is imposed; see also Willis (1981) and NematNasser and Hori (1993). On the other hand, considering the traction boundary condition (7.22), that is,

ti (x) = σij0 n j (x)

∀x ∈ ∂ Bδ ,

(7.34)

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249

(7.35)

We can now define the apparent compliance St from this relation t ε ij = Sijkl σij0 ,

or from the volume average energy density in Bδ (ω): 1 1 1 U= σij εij dV = σij ui, j dV = σij ui n j dS 2V V 2V ∂V 2V ∂V 1 1 1 t 0 = σ 0 ui, j dS = σij0 ε ij = σkl0 Skli j σij . 2V V ij 2 2

(7.36)

(7.37)

t This shows that the apparent compliance Skli j may be defined either from ε or from U if the boundary condition (7.34) is imposed.

7.3

Spatial Randomness

7.3.1 Stationarity of Spatial Statistics In the following we revisit the key concepts of spatial homogeneity (stationarity) and ergodicity from the standpoint of what is required by Hill’s definition of RVE. With reference to Chapter 2, we shall do this in terms of a material property (or a vector of properties) Z entering the heterogeneous medium model as a random field over the D-dimensional physical space Z : × R D → R.

(7.38)

In the case of an r -phase material microstructure, Z is itself described by a random indicator function χr of all the phases. All that follows can then be generalized to a vector or tensor random field Z, as may be the case with a linear elastic microstructure represented by a random stiffness field C. Now, let us recall two well-known classes of spatial homogeneity: Strict-sense stationary (SSS) random fields. This requires that all n-order probability distributions Fn are invariant with respect to arbitrary shifts x , and for any n and any choice of xi , they satisfy Fn (z1 , . . ., zn ; x1 , . . ., xn ) = Fn (z1 , . . ., zn ; x1 + x , . . ., xn + x ).

(7.39)

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Wide-sense (or weak-sense) stationary (WSS) random fields. The ensemble mean is constant and its finite-valued covariance depends only on the shift h from x to x + h Z(x) ≡ µ,

[Z(x) − Z(x)] [Z(x + h) − Z(x + h)] ≡ K Z (h) < ∞. (7.40)

Clearly, SSS implies WSS whenever the first-order distribution F1 yields a finite second moment. It is through the covariance function K Z (h) that we introduce the concept of correlation distance lc , which, in turn, gives an indication of decay of correlations between two different points. If this decay is such that Z(x) and Z(x + h) become asymptotically uncorrelated according to lim|h|→∞ K Z (h) → 0, then, using lc we would rewrite the separation of scales (7.1) as lc < L L macro . (7.41) lc In other words, the RVE of size L = V 1/D (D = 1, . . ., 3) should be at least larger than lc , and the material could be taken as homogeneous beyond L. This concept can also be applied to the SSS fields. There also exist two more general classes of spatially homogeneous fields: (1) intrinsically stationary (locally homogeneous) random fields, and (2) quasi-stationary random fields. These types of random fields become relevant when, in general, there is no hope of establishing even the WSS property. In those situations, strictly speaking, there is no certainty of having the RVE in the sense employed in this chapter, and the SVE is needed. 7.3.2 Ergodicity of Spatial Statistics Basic considerations. Following Chapter 2, let us recall that the ergodic property means that the spatial average over any realization ω equals the ensemble average at any specific point x: Z(ω) = Z(x).

(7.42)

Also as noted earlier, this is a property that: 1. Must be checked a priori for a given material microstructure. 2. Is hard to verify because the measurements are necessarily finite in number both in as well as in B. 3. Is practically impossible to verify because the measurements are supposed to be carried out in a pointwise sense, on domains of infinite spatial extent. The framework we develop below applies to materials satisfying (7.42). Ergodic response. Any given boundary value problem for an elastic body B = {B(ω); ω ∈ } is defined by the equilibrium equation (7.8), Hooke’s law,

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compatibility conditions, and some boundary conditions. Now, subject B(ω) to a uniform boundary condition such as (7.15) or (7.20) and find the ensemble of solutions {ε(ω, x), σ (ω, x), u(ω, x); ω ∈ , x ∈ V},

(7.43)

which would represent a complete solution of the stochastic mechanics problem. Then, at any point x, we may define effective moduli Ceff (x) as σ (x) = C(x) : ε(x) = Ceff (x) : ε(x).

(7.44)

Note: 1. For any given boundary conditions, Ceff is, in general, a function of x. 2. Ceff also generally depends on the boundary conditions applied. 3. Assuming B is ergodic in the sense that the stiffness field C is ergodic, the explicit dependence on the location in (7.44) vanishes and we can write ε(x) = ε

or

σ (x) = σ ,

(7.45)

providing (7.29), respectively (7.30), is applied and the displacement and stress fields are continuous; consequently, σ = Ceff : ε.

(7.46)

4. Providing the mesodomain tends to macroscale δ → ∞, write the Hill condition (7.17) as ε : σ = ε : σ . Of course, by analogy to (7.18), we have ε : σ = 0,

(7.47)

(7.48)

which means that stresses and strains are statistically uncorrelated.

7.4

Hierarchies of Mesoscale Bounds

7.4.1 Response under Displacement Boundary Condition We take the 2D square-shaped mesodomain (a mesoscale window) of a composite body Bδ (ω) of Figure 7.4 to be described everywhere by the local stress-strain relations σ = C(ω, x) : ε. Suppose that this body is now evenly partitioned into four square-shaped bodies Bδs (ω), s = 1, . . ., 4. We define two types of uniform displacement boundary conditions, in terms of a prescribed

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FIGURE 7.4 Partition of the window of Figure 7.1(b) into four subwindows.

constant strain ε 0 , over the mesodomain with any given microstructure: unrestricted u(x) = ε 0 · x ∀x ∈ ∂ Bδ ,

(7.49)

and: restricted ur (x) = ε 0 · x ∀x ∈ ∂ Bδs

s = 1, . . ., 4.

(7.50)

The superscript r in equation (7.50) indicates a “restriction.” That is, (7.49) is given on the external boundary of the mesodomain, whereas (7.50) is given on the boundaries of each of the four submesodomains, Figure 7.4. Let us note, by the strain averaging theorem (assuming perfect bonding throughout the material), that the volume average strain is the same in each sub-mesodomain and also equals that in the large mesodomain ε0 = ε = εs .

(7.51)

Let σ , ε be any kinematically admissible fields: they satisfy everywhere the local stress-strain relations σ = C(ω, x) : ε and the displacement boundary condition (7.49), with εij = u(i, j) but σ is not necessarily in equilibrium. Now, there is a minimum potential energy principle for the fields σ , ε in Bδ (ω) (e.g., Hill, 1950) ∂ Bδt

dS − t·u

1 2

σ : εdV ≤ Bδ

∂ Bδt

t · udS −

1 2

σ : εdV. Bδ

(7.52)

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For the displacement boundary condition, ∂ Bδu (ω) = ∂ Bδ (ω) and ∂ Bδt (ω) = ∅, so that 1 1 σ : εdV = U (ω, ε) ≤ U ω, ε = σ : εdV, (7.53) 2 Bδ 2 Bδ or σ : ε ≤ σ : ε.

(7.54)

However, the Hill condition (7.17), combined with the fact that ε 0 = ε, allows us to write σ : ε ≤ σ : ε. (7.55) εr under the restricted condition (7.50) is an admisBecause the solution σ r , sible distribution under unrestricted conditions (7.49) (but not vice versa), we have

σ : ε ≤ σ r : εr .

(7.56)

This in turn implies a weak inequality between the mesoscale stiffness tensors obtained under unrestricted (Cdδ (ω)) and restricted (Cdr δ (ω)) conditions 1 d C (ω) 4 s=1 δs 4

Cdδ (ω) ≤ Cdr δ (ω) =

∀δ = δ/2

(7.57)

where it follows from the relation 1 d C (ω). 4 s=1 δs 4

0 σ = Cdr δ (ω) : ε

Cdδ (ω) =

(7.58)

That is, the effective stiffness of a partitioned domain subjected to (7.50) involves respective stiffnesses of four subdomains. Now, in view of the tacitly assumed statistical homogeneity and ergodicity d of the ensemble d material, d averaging of (7.57) allows us to replace Cδ (ω) by dr Cδ , and Cδ (ω) by Cδ , so that d d Cδ ≤ Cδ (7.59) ∀δ = δ/2. By applying this inequality to ever larger mesodomains ad infinitum we get a hierarchy of bounds on Cd∞ from above d (7.60) C∞ ≤ . . . ≤ Cdδ ≤ Cdδ ≤ . . . ≤ Cd1 ≡ CV ∀δ = δ/2. In fact, we have

eff Cd∞ = Ceff ∞ =C

(7.61)

for the macroscopically effective response (δ → ∞) because, by the ergodicity argument, it must be deterministic. On the upper end, the hierarchy stops at

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the scale of a single heterogeneity. Now, since the single heterogeneity—like an inclusion or a crystal—is homogeneous, the uniform strain is true strain, so that ensemble averaging gives the Voigt bound CV . 7.4.2 Response under Traction Boundary Condition The above suggests an analogous procedure in proving a hierarchy bounding Ceff (≡ (Seff ) −1 ) from below. The proof is analogous, providing we first replace the displacement by traction boundary conditions (in terms of a prescribed constant stress σ 0 ), again of two types: unrestricted t(x) = σ 0 · n ∀x ∈ ∂ Bδ ,

(7.62)

and: restricted tr (x) = σ 0 · n ∀x ∈ ∂ Bδs

s = 1, . . ., 4.

(7.63)

The superscript r indicates a “restriction” as before. Next, by the stress averaging theorem, the volume average stress is the same in each sub-mesodomain and also equals that in the mesodomain σ 0 = σ = σ s.

(7.64)

The key inequality is obtained from a minimum complementary energy principle for statically admissible fields σ , ε in Bδ (ω) 1 2

1 σ : εdV − t · udS ≤ u 2 Bδ ∂ Bδ

σ : εdV − Bδ

∂ Bδu

t · udS,

(7.65)

and noting that ∂ Bδt (ω) = ∂ Bδ (ω) and Bδu (ω) = ∅, so that σ : ε ≤ σ : ε. Using the Hill condition leads then to Stδ (ω) ≤ Str δ (ω),

(7.66)

and, upon ensemble averaging, to t t Sδ ≤ Sδ

∀δ = δ/2.

(7.67)

By applying this inequality ad infinitum we get toever-larger mesodomains a hierarchy of bounds on St∞ from above (i.e., on Cd∞ from below)

St∞ ≤ . . . ≤ Stδ ≤ Stδ ≤ . . . ≤ St1 ≡ S R

∀δ = δ/2,

(7.68)

where

eff St∞ = Seff = (Ceff ) −1 . ∞ =S

(7.69)

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7.4.3 Scale-Dependent Hierarchy Combined hierarchy. Combining relations (7.60) with (7.68), we arrive at a scale-dependent hierarchy of bounds on the macroscopically effective moduli t −1 d −1 t −1 d d S1 ≤ . . . ≤ Stδ ≤ Sδ ≤ . . . ≤ Ceff ∞ ≤ . . . ≤ C δ ≤ Cδ ≤ . . . ≤ C 1 ∀δ = δ/2.

(7.70)

These hierarchies were essentially first derived by Huet (1990); see also (Huet, 1991, 1997). A more rigorous proof using techniques of homogenization and probability theories was given by Sab (1992), see Section 7.4.4 below. The decrease of the upper (displacement-controlled) bound with increasing scale appears to have first been demonstrated on planar random networks of Delaunay topology by Ostoja-Starzewski and Wang (1989); as mentioned in Chapter 4, the uniform traction conditions, however, could not be applied in a unique way to such a disordered discrete system. Spatial statistics aspects. Considering that the hierarchy (7.70) is stated in terms of the averages, it suffices to choose the setting of material properties specified via wide-sense stationary random fields. Furthermore, note that in the case of C being an isotropic random field, Ceff should become an isotropic tensor involving two Lam´e constants λeff and µeff . On the other hand, for an orthotropic random field C, Ceff should become an orthotropic tensor; a twodimensional example in the setting of machine-made paper is discussed in Chapter 9. Extension to noncommensurate partitions. The hierarchy (7.70) has been shown to hold for commensurate partitions, i.e., δ = δ/2. We may, however, extend these inequalities to an arbitrary pair of mesoscales δ < δ—not just for commensurate ones involving partitions in which δ = nδ , n being a natural number. It will suffice to focus on the Dirichlet bounds, because then the Neumann bounds follow by an analogous argument. To this end, consider two separate cases of the hierarchy (7.60) for commensurate partitions: one at an arbitrary δ1 and another at δ2 , whereby δ1 < δ2 < 2δ1 . Thus, in the first case, we have a sequence of inequalities · · · ≤ Cd4δ1 ≤ Cd2δ1 ≤ Cdδ1 · · · , (7.71) while in the second case, we have another sequence · · · ≤ Cd4δ2 ≤ Cd2δ2 ≤ Cdδ2 · · · ·

(7.72)

We prove by contradiction: assume that inequalities (7.71) and (7.72) are not consistent with each other in the following way · · · > Cd4δ2 < Cd4δ1 > Cd2δ2 < Cd2δ1 > Cdδ2 < Cdδ1 > · · · (7.73) This, however, would imply that the microstructure is characterized by a Dirichlet bound which, while displaying an overall decreasing behavior

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with δ, also fluctuates with a period increasing like nδ. One can certainly construct a random, ergodic microstructure with such a scale-dependent fluctuation. This would, however, contradict the assumption of statistical homogeneity of the material properties which is always taken in the definition of the RVE. Summarizing, we conclude that the hierarchy (7.70) holds for noncommensurate mesoscale sequences. Other than (7.73) types of inconsistencies between (7.71) and (7.72) may also be considered, but then one is led to similar contradictory conclusions as above. 7.4.4 Homogenization Theory Viewpoint We assume the composite—such as that of Figure 7.2—to be made of a finite number of r phases each of which is linear elastic and elliptic: 0 < e : C : e < ∞, ∀e =

0. Next, following Sab (1992), let F ( B, ) represent a real functional of the open-bounded domain B of volume V and random field with the following five properties: 1. F is a property of the medium invariant with respect to any translation in the material domain. 2. F or any partition of the domain B into n disjoint subdomains, F satisfies a subadditivity property F ( B, ) ≤

n

Vi F ( Bi , ) V i=1

n B = ∪i=1 Bi .

(7.74)

3. F is a measurable mapping with respect to the sample space of outcomes ω. 4. is a statistically homogeneous, ergodic random field (recall Chapter 2). 5. F is uniformly bounded in B and ω in the sense that there exists a real b, such that |F ( B, )| ≤ 0

∀B, ω.

(7.75)

Let us now take B to be a square-shaped domain, with side of length L, and which contains some microstructure of characteristic microscale d, see Figure 7.2. With the conditions 1–5 satisfied, we can adopt the result that there exists a nonrandom (i.e., deterministic) constant F hom such that, for all ω with probability one, we will have lim |F ( B, )| = F hom

L/d→∞

∀B, ω,

(7.76)

with the bound inf( F ( B, )) = F hom L/d

∀B, ω.

(7.77)

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This limit is understood in the sense of the homogenization theory: x → F (x) = F (x/ ) ≡ F (y), where x and y are the so-called slow (macroscopic) and fast (microscopic) variables, respectively, and is a small parameter, reciprocal of our δ = L/d. The limit δ → ∞ should be taken here in such a way that L is kept finite so as to keep the energy finite. If F represents the volume average elastic energy density (or complementary) energy density, then stands for the stiffness C (respectively, compliance S) tensor of the domain B, and from (7.77) one has sup S−1 = (Shom ) −1 = Chom = inf S, L/d

L/d

(7.78)

where (Shom ) −1 = Chom is the macroscopic (effective) stiffness tensor Ceff in the sense of Hill (1963). However, (7.78) does not assert that the averages S−1 and C are monotonic functions of L/d (= δ). Finally we note that, for the RVE Bδ→∞ (ω), in the notation of homogenization theory, we have the macrostress and macrostrain (ω) = σ δ→∞ (ω)

E(ω) = ε δ→∞ (ω).

(7.79)

7.4.5 Apparent Moduli in In-Plane Elasticity 7.4.5.1 General Considerations When considering the apparent constitutive law of a planar elastic material σij = Cijkl (ω, x)εkl

i, j, k, l = 1, 2,

(7.80)

we must, in general, deal with an arbitrary anisotropy. Thus, to determine six d t unknown values for Cijkl (or Sijkl ) for Bδ (ω) we need six tests, Figure 7.5. When d seeking Cijkl s, each test is run by applying the affine displacements on ∂ B so that the strain energy density is (V is the volume of Bδ (ω)) 0 2 0 2 0 2 V V V C1111 ε11 σ ij εij0 = εij0 Cijkl εkl0 = + C2222 ε22 + 4C1212 ε12 2 2 2 0 0 0 0 0 0 . (7.81) + 2ε11 C1122 ε22 + 4ε22 C2212 ε12 + 4ε12 C1211 ε11

Uδ =

In each and every test, separately, the energy is found by computational mechanics, and this is set equal to a corresponding special form of (7.81).

FIGURE 7.5 Six tests: #1, #2, . . ., #6 from left to right, to determine the six unknowns of the in-plane stiffness tensor Cijkl .

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For example, in test #1, Uδ =

0 2 V C1111 ε11 , 2

(7.82)

from which we infer C1111 . Similarly, from the test #2 we find C2222 , while test #3 gives C1211 . We are then in a position to find C1122 from Uδ =

0 2 0 2 V 0 0 [C1111 ε11 + C2222 ε22 + ε11 C1122 ε22 ], 2

(7.83)

and then C2212 as well as C1211 in analogous fashion. In practice, one can proceed by truly carrying out six tests, or by carrying out only the three tests #1–#3 of Figure 7.5, and then combining their results through a superposition. In any case, the solution necessarily involves some computational mechanics method for discretization of the composite, such as spring networks, finite elements, or boundary elements—recall Chapters 3 to 5. Indeed, in Chapter 4 we discussed such a method for truss systems. In the case of traction boundary conditions, determination of the apparent compliances follows the same type of approach and one works with the complementary energy, that is, 0 2 0 2 0 2 V V V Uδ∗ = σij0 ε ij = σij0 Sijkl σkl0 S1111 σ11 + S2222 σ22 + 4S1212 σ12 2 2 2 0 0 0 0 0 0 0 0 0 . (7.84) + 2σ11 S1122 σ22 + 4σ22 S2212 σ12 + 4σ12 S1211 σ11 An extension of this methodology to three dimensions is straightforward. Examples of bounds obtained in this way are given in the aforementioned works of Huet and his coworkers (Amieur et al., 1995). A numerical method for materials having a linear elastic truss microstructure is discussed in Section 4.2 of Chapter 4.

7.5

Examples of Hierarchies of Mesoscale Bounds

7.5.1 Random Chessboards and Bernoulli Lattices In this section we focus on antiplane elasticity of random microstructures, which, as recalled in Table 7.1, also gives information on various other physical problems of material systems having the same morphology. With reference to Chapter 1, let us now consider the Bernoulli lattice process p,a on a Cartesian lattice of spacing a with each point of this lattice being of type 1 (or 2) with probability p (respectively, q = 1 − p) independently of all the other points. Evidently, p and q define the volume fractions of both types of phases (1 and 2). Clearly, the local stress and strain concentrations cannot be resolved, but the statistics of such a simple system gives an indication of the

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statistics of random media because this, perhaps, is the simplest setup in which to investigate the scale and volume fraction dependence of the ensemble average estimates based on the essential (e) and natural (n) boundary conditions. The notation e and n is equivalent to d and t, respectively. After Ostoja-Starzewski and Schulte (1996), in Figures 7.6 through 7.8 we give Ceδ −1 and Snδ at δ = 4, 10, and 20 (for the highest contrast) and their comparison with Hashin-Shtrikman) upper and lower bounds CuH = C2 + f 1 [1/(C1 − C2 ) + f 2 /2C2 ]−1 ClH = C1 + f 2 [1/(C2 − C1 ) + f 1 /2C1 ]−1 .

(7.85)

Observe that CuH and ClH are outside the mesodomains of size 4. In other words, relatively very small mesodomains can give tighter mesoscale bounds than those of Hashin. Note that the problem of δ-dependence, especially in the setting of such binary systems, is akin to the so-called finite-size scaling in statistical physics, but the attention in that area has always been focused on the phase transition problems (Cardy, 1988). The approach to such a transition at about 2/3 volume fraction of the soft phase is shown in Figure 7.8— at contrast 1000. However, in contradistinction to the terminology of phase transitions, we now have a different tool to describe the scale dependence. The particular case of p = q = 0.5 has been studied in Ostoja-Starzewski and Schulte (1996), and it was found that Ceδ = exp[−δ −m ] Snδ = exp[δ −n ],

(7.86)

where m and n are functions of the contrast α m = 3.8α 0.14

n = 2.4α 0.59 .

(7.87)

These results were obtained from computations over a range of scales 1 through 1000. While the smallest scale can be calculated explicitly as the Voigt and Reuss bounds, the largest involved a lattice of 1000 × 1000 nodes, that is, having 106 degrees of freedom. The parameter space of contrast and volume fraction is vast, and therefore only select cases can be run numerically at these large scales. But, (7.86 and 7.87) give an idea of the functional forms of scaling laws for other volume fractions. Now, the Bernoulli lattice at volume fraction below, say, 30% can also be interpreted as a very crude model of a disk matrix composite—again with one degree of freedom per disk. Given the fact that a more realistic spring network model requires several (at least five) lattice spacings per disk, a lattice of some 5000×5000 nodes (25·106 degrees of freedom) would have to be run. Thus, the above scaling laws provide the best available indication of finite-size scaling of both bounds, Ceδ and Snδ , of disk-matrix composites.

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4 C eδ

δ

S nδ

–1

3

2

1

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FIGURE 7.6 −1 Bounds on tr (Ceff ) for a random two-phase lattice at contrast 10, showing tr Ceδ and tr Snδ at δ = 4 and 10; also shown, by dashed lines, are Hashin upper and lower bounds CuH and ClH .

50 4 10

40 e

Cδ

δ

S nδ

–1

30

20

10

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FIGURE 7.7 −1 Bounds on tr (Ceff ) for a random two-phase lattice at contrast 100, showing tr Ceδ and tr Snδ at δ = 4 and 10; also shown, by dashed lines, are Hashin bounds CuH and ClH .

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500 4 10

400

20 300

C eδ

δ

n –1

Sδ

200

100

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FIGURE 7.8 Bounds on tr (Ceff ) for a random two-phase lattice at contrast 1000, showing tr Ceδ and tr Snδ −1 at δ = 4, 10 and 20; also shown, by dashed lines, are Hashin bounds CuH and ClH .

7.5.2 Disk-Matrix Composites The hierarchy of bounds (7.70) is now illustrated on the example of a diskmatrix composite at volume fraction 20%. These microstructures have been generated via the planar Poisson point process for disk centers, with sequential inhibition rule (recall Chapter 1) that prevents any two Poisson points from coming closer than 110% of diameter, so as to avoid the numerically and analytically difficult problem of very narrow necks between disks. We consider two examples of this composite: one with relatively soft and another with relatively hard inclusions. In the first case, we have a soft matrix (C (m) = 1) and inclusions C (i) = 102 in Figure 7.9(a), and C (i) = 104 in Figure 7.9(b). Clearly, as the contrast in the composite increases, the bounds take larger mesodomains to converge. Basically, in order to attain the equality Snδ : Ceδ = 1 within, say, 10%, one has to take mesodomains that are some ten and fifty times larger than a single inclusion, for these two contrasts, respectively. In Figure 7.10 we show results for the opposite case: soft inclusions in a hard matrix, with the same volume fraction 20%. Again, we consider two cases of contrast—C (i) /C (m) = 10−2 and 10−4 —while keeping the matrix at C (m) = 1. The first one is shown in Figure 7.10(a), the second one in Figure 7.10(b), and both were obtained with exactly the same spring network resolution as above. As before, an increase in the contrast in the composite has the effect of slowing down the convergence of Snδ : Ceδ to unity with δ increasing, but, by comparison with Figure 7.9, this convergence is relatively much slower (!) for

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0

10

20

–1

S nδ

30

40

50

30

40

50

δ

(a)

300 200 e

Cδ

100 0

10

0

Seδ

–1

20

δ

(b)

FIGURE 7.9 A hierarchy of scale-dependent bounds on tr (Ceff ) of the disk-matrix composite at contrasts 102 (a), and 104 (b) (After Ostoja-Starzewski, 1998). 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

C eδ n –1

Sδ

0

10

20

30

40

50

30

40

50

δ

(a) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

e

Cδ

n –1

Sδ 0

10

20

δ

(b)

FIGURE 7.10 A hierarchy of scale-dependent bounds on tr (Ceff ) for the disk-matrix composite at contrasts 10−2 (a), and 10−4 (b). (After Ostoja-Starzewski, 1998).

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extreme contrasts. It follows that one needs to go to very large scales in order to homogenize such a composite material. This is the principal difference from the case of high contrasts, and is indicative of all the material systems with soft inclusions of whatever shape. Note: The tendency for both bounds to converge onto Ceff increases as the contrast C (i) /C (m) tends to unity. Thus, for low contrasts, the RVE size may be taken of the order of just a few inclusion diameters. Note: In all the systems one has the option of using the orthogonal-mixed boundary condition, which results in an intermediate mesoscale response, having a weaker scale dependence than Ceδ and Snδ . The trade-off is this: no bounding property, but the tendency to asymptote to, and attain Ceff , is most rapid. This property will be displayed in an elastoplastic composite in a later chapter. The above results were explored in more depth by Jiang et al. (2001). They studied responses under several different boundary conditions: uniform displacement, uniform traction, periodic, and mixed ones (a combination of any of the first three) to evaluate the mesoscale moduli.

7.5.3 Functionally Graded Materials Interfaces in composite materials influence their local fields and effective properties. Theoretical studies in this area represent the interface as either a 2D bounding surface, or as a 3D region of certain microstructure, called an interphase (e.g., Drzal, 1990), Figure 7.11(a). The inhomogeneity of the interphase may be due, for example, to the chemical reaction(s) or diffusion. Composites with inhomogeneous interphases have been studied by various researchers, whereby the interphase region was assumed to be isotropic with one property, such as Young’s modulus, varying linearly or nonlinearly and Poisson’s ratio taken as constant. Materials with interphases—also called functionally graded materials (FGM)—present a fundamental challenge to the mechanician not only because of their spatially varying nature, but also due to the need to account for the spatially graded random microstructure. In Ostoja-Starzewski et al. (1996a) we represented the interphase as a zone of two randomly interpenetrating phases with radially dependent statistics, assuming a linear distribution of the indicator function P{χ(black) = 1} = A1r + A2 .

(7.88)

To make things specific, with reference to Figure 7.11(a), we admitted two models of the microstructure: a fine-grained model with topology of a random chessboard, and a coarser-grained model with a geometry of a Voronoi tessellation, whose cells are occupied at random by either one of two phases. The analysis of in-plane conductivity Cij (r ) was then based on a mesoscale window larger than the scale of heterogeneity in the FGM. To define apparent

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Matrix Fiber Interphase

a0 b0 r

(a) FIGURE 7.11(a) Sketch of the fiber-interphase matrix system, and the mesoscale window shown in a random chessboard model (top right) and a two-phase Voronoi model (bottom right).

properties of this window, we employed two boundary value problems: essential and natural. As a result, local (radially dependent) properties of the interphase turned out to be orthotropic and nonlinear in r ; they could be approximated by polynomials of the form Cij (r ) = k0 + k1r + k2r 2 + k3r 3 + · · · ·

(7.89)

These functions (in the r, θ-coordinate system) represent ensemble aver −1 for a given window scale δ, Figure 7.11(b). The temperages Ceδ and Snδ ature field T(r, θ) in the interphase zone resulting from the field equation Crr

∂2 T ∂r 2

+

1 1 ∂Cr θ ∂T Cr θ ∂2 T ∂T ∂C θ θ ∂2 T + Crr + +2 + 2 =0 ∂r r ∂r r ∂r ∂θ r ∂r ∂θ r ∂θ 2 (7.90)

∂Crr

was then used as input to the effective medium theory allowing prediction of the overall transverse conductivity of the composite. At this latter stage we used the “composite cylinders assemblage” model to account for the interactions of the fibers. Let us note that the mesoscale δ may be set up quite arbitrarily over a range of values, but, in contrast to other problems discussed earlier, the finite size of the interphase precludes a passage to infinity. We thus have two

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L C(f )

C(m)

0

a

b0

a0

b

r

(b) FIGURE 7.11(b) A typical bi-phasic distribution along the radial coordinate r, where a 0 ≤ r ≤ b 0 , displays the heterogeneity of the interphase region. C ( f ) is the fiber phase, while C (m) is the matrix (e) (n) (n) phase. Four curves, top to bottom, are Cθ(e) θ , Crr , C θ θ a nd Crr . Also, a mesoscale L = 2(a 0 − a ) = 2 (b 0 – b) is indicated.

fields bounding the global response—a precursor of two random fields to be introduced in the next chapter.

7.5.4 Effective and Apparent Moduli of Multicracked Solids 7.5.4.1 Scale-Dependent Hierarchies of Bounds: Numerical Results Let us again make use of one of the random media models developed in Chapter 1. This time, as we are interested in planar fields of randomly placed needles, we consider a random fiber field generated from the Poisson point field— Figure 7.12(a) presents its typical realization B(ω); (Ostoja-Starzewski, 1999a). The field density chosen here is ∼ 0.015. As in the previous section, the material has two locally isotropic phases—matrix (m) and inclusions (i)—and we keep C (m) = 1 and vary C (i) . Hierarchies of bounds are shown in Figure 7.12(b). Before we proceed further, we note that the work on effective moduli of materials with microcracks dates back to Vakulenko and Kachanov (1971) and Budiansky and O’Connell (1976), while a comprehensive review was given by Kachanov (1993). An experimental confirmation of the results of an effective medium (Mori–Tanaka) theory for low crack densities, typically employed by Kachanov and others, has been given in Carvalho and Labuz (1996). The effective medium theory, however, cannot say anything about the finite-size scaling of (mesoscale) moduli and their statistics; nor is it reliable for higher crack densities. The importance of mesoscale bounds is now increasingly being recognized in wave propagation studies in geophysics (Saenger et al., 2006).

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Microstructural Randomness and Scaling in Mechanics of Materials trC/2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (a)

e

Cδ choles S nδ

0

1

2

3

4

5

6

–1

7

8

9 10

δ= L d

(b)

FIGURE 7.12 A field of long (1 × 100) needles in a 1000 × 1000 square, generated from a Poisson point field of density λ = 10−4 , i.e., there are 100 needles; (b) effect of increasing window scale on the convergence of bounds (7.70) for soft needles C (i) = 10−4 ; the effective stiffness Choles 0.65 is computed by a mean field method of Garboczi et al. (1991).

Note also that these problems—especially in connection with finite-size scaling—can be treated by the lattice (spring-network) method of Chapter 3 bearing in mind two approximations: needles have a finite thickness, and the contrast is finite although it can be made very close to zero. In Figure 7.13 we display the scaling of Ceδ and Snδ −1 for needle systems at contrasts 10−2 and 10−4 at two mesoscales: δ = 10 and δ = 50. Of principal interest here is the same type of slow approach to the RVE as that in soft diskmatrix systems noted earlier in Figure 7.10. Now, in the studies of effective moduli of heterogeneous materials, the resulting Ceff is typically presented versus the volume fraction x of one of the phases. For our system of short needles, this is shown in terms of Choles against x = nL 2e f f almost all the way to the percolation point at ∼ 5.9; Choles is computed by the physicist’s mean field method (Garboczi et al., 1991) for needle-shaped holes of any aspect ratio and with arbitrarily strong interactions. To sum up, this figure displays (1) a very slow approach of Ceδ and n −1 Sδ to the RVE (i.e., curve Choles ), and (2) a discrepancy between Choles and the Dirichlet as well as the Neumann bounds. Note that Choles corresponds to an effective (macroscopic) response of a very large random system, which is typically computed under periodic boundary conditions. For the system at hand, there is an interesting result concerning the statistics of second invariants of Ceδ (ω) and Snδ (ω). Namely, within a few percent, their coefficients of variation are constant for a given type of boundary conditions (either Dirichlet or Neumann), are independent of the changing mesoscale δ, and independent of the contrast in the material,

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(a) trC/2 1 0.9 (C eδ) δ = 10 0.8 e (C δ) δ = 50 0.7 0.6 0.5 0.4 0.3 n 0.2 (S δ )–1 δ = 50 choles n –1 (S ) 0.1 δ = 10 δ x 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (b)

FIGURE 7.13 (a) A 1000 × 1000 window of 2000 randomly placed 10 × 1 needles with an isotropic distribution; a subwindow of size (δ = 50) is indicated. (b) Normalized overall moduli Ceδ and Snδ −1 , at δ = 10 and δ = 50, and the effective stiffness Choles for a random field of short (1 × 10) needles (such as that of Figure 7.12), as functions of the volume fraction x. Data were computed only at discrete intervals x = 1.21, 2.42, 3.63, and 4.84.

except, of course the singular and trivial case of C (m) /C (i) = 1. We return to this property, also with reference to other microstructures, in Section 8.2 of Chapter 8. 7.5.4.2 Cross-Correlations of the Mesoscale Moduli with the Crack Density Tensor The aforementioned studies of effective moduli (Ceff ) of materials with cracks of elliptical (or ellipsoidal) pores rely on a crack density tensor (Kachanov, 1993), which may also be called a fabric tensor, βij =

π 2 (a ni n j + b 2 mi m j ) (k) . A k

(7.91)

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m b

a n

(a)

(b)

FIGURE 7.14 (a) Randomly located ellipses, with overlap permitted: (b) basis for the fabric tensor.

Here a and b are the semi-axes of an ellipse, Figure 7.14. We focus on the e tensors correlation of the fabric tensor βij with the mesoscale response C δ and n Sδ . This is done via the cross-covariance function K βij Ckl = βij Ckl − βij Ckl , where Ckl stands for either Ceδ or Snδ ; we suppress the δ parameter for simplicity of notation. It is more convenient to use the correlation coefficient βij Ckl − βij Ckl ρβij Ckl = . (7.92) σβij σCkl Although our computations, given the vast extent of the parameter space, have so far been restricted, some interesting trends could be observed. For example, for isotropic systems of moderately soft needles (contrast 0.1) of aspect ratio either 10 or 20, at window sizes either δ = 5 or 10, the following fabric-property cross-correlations hold:  ρβij Ckle

=

 ρβ11 C e > 0 12  ρβ11 C22e > 0 

ρβij Skln

=

ρβ11 C11e > 0

ρβ11 S11n < 0

 ρβ11 S12n < 0  ρβ11 S22n < 0

ρβ12 C12e < 0



ρβ22 C22e > 0

ρβ12 C22e < 0

 ρβ22 C12e > 0 , ρβ22 C22e > 0

ρβ12 S12n > 0

ρβ22 S22n < 0

ρβ12 C12e < 0

ρβ12 S12n > 0 ρβ12 S22n > 0



(7.93)

 ρβ22 S12n < 0.  ρβ22 S22n < 0

This implies that for finite windows the correlation of βij with Ckle tends to be opposite in sign to that of βij with Skln ρβij Ckle = −ρβij Skln .

(7.94)

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Now, Ceδ and (Snδ ) −1 must converge to the macroscopic, deterministic Ceff = limδ→∞ Ceδ , so that the only way that (7.94) may hold is for βij to become uncorrelated with Ceδ and Snδ in the infinite scale limit—that is, ρβij Ckle and ρβij Skln should tend to 0. This implies that the second-order moments of geometry of a material with cracks, such as the conventional fabric tensor βij , may be insufficient in describing macroscopic structure-property relations. This in turn indicates that one should consider higher moments of βij and other geometric measures of the crack network connectivity. This conclusion is supported by a study of the cross-correlation structure of systems with anisotropy: a 1 = 1 in the Fourier series angular distribution of the long axis of the needles, and higher a i s equal zero, while all the other parameters are the same as before. This leads to fabric-property crosscorrelations with these inequality signs:

ρβij Ckle

ρβij Skln

=

=

 ρβ11 C11e < 0  ρβ11 C e < 0 12  ρβ11 C22e > 0

ρβ12 C12e > 0 ρβ12 C12e > 0 ρβ12 C22e < 0

 ρβ11 S11n > 0 ρβ12 S12n < 0  ρβ11 Sn > 0 ρβ12 Sn < 0 12 12  ρβ11 S22n < 0 ρβ12 S22n > 0

ρβ22 C22e < 0



 ρβ22 C12e < 0 , ρβ22 C22e > 0 ρβ22 S22n > 0



(7.95)

 ρβ22 S12n > 0 . ρβ22 S22n < 0

Note that, although the inequality signs in (7.95) are not the same as those in (7.93), they satisfy (7.94).

7.6

Moduli of the Trabecular Bone

The trabecular bone has a multiscale, random 3D structure, Figure 7.15. Evidently, there are at least five length scales present. Focusing on the mesoscale of that structure, Wang and Jasiuk (2006) have recently considered a model of a trabecular bone as having a 2D periodic prismatic structure and they predicted computationally the apparent elastic moduli of such a model. They studied a periodic structure in order to avoid the complexity of accounting for the actual random structure of trabecular bone (by using a micro-CT based finite elements) and to avoid the need to obtain many realizations to obtain the ensemble average of response. In order to explore the effect of scale and boundary conditions, they varied the size of the “window” from one unit cell to larger window sizes (multiple unit cells) and

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X 100

Mesoscale (a)

Trabecular network

X 1,000

Microscale (b)

Single trabeculae

X 10,000 Submicroscale (c)

Single lamella

X 100,000 Nanoscale (d)

Collagen fibers apatite crystals

FIGURE 7.15 Hierarchical structure of trabecular bone, showing (a) mesostructure (0.5–10 cm) of trabecular network; (b) microstructure (10–500 µm) single trabeculae; (c) submicrostructure; and (d) nanoscale. (After Jasiuk, 2005.)

applied four types of boundary conditions: uniform displacement, traction, mixed and periodic. The results calculated using periodic boundary conditions give effective response while the remaining three boundary conditions give apparent moduli. The apparent moduli calculated using the uniform displacement boundary conditions bound effective moduli from above while the moduli obtained using the uniform traction boundary condition bound the effective moduli from below. The larger was the window, the closer were the bounds.

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In the analysis the trabecular bone is modeled as a two-phase material consisting of bone tissue (hard phase) forming a trabecular network and soft tissue (soft phase) present in pores. Both phases are assumed linear elastic and isotropic. Bone tissues are assigned Young’s modulus of 13.0 GPa and Poisson’s ratio of 0.3 while Young’s modulus of soft tissue is given as 1.3 kPa (giving moduli mismatch of 106) with the same Poisson’s ratio of 0.3. The bone tissue’s volume fraction is estimated at 20%.

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8 Random Field Models and Stochastic Finite Elements

Il n’y a pas de probabilit´e en soi; il n’y a que des modeles probabilistes.∗ G. Matheron, 1978 Chapter 7 began with a discussion of the separation of scales, and then focused on the problem of material properties below the RVE. With reference to equation (7.1) there, our concern was with the left part of that inequality: d < L or d L. The issue we focus on in this chapter is the right part of (7.1): L L ma cr o . That is, if the RVE on scale L is not justified, only a random continuum can be used—recall Figure 1(b) of the Preface. As a result, we need to establish some methods to deal with solution of macroscopic boundary value problems having the mesoscale SVE as input. Such problems are necessarily stochastic, and this leads us to a formulation of random fields of material properties from the SVE information, and their input into numerical methods leading then to so-called stochastic finite element (SFE) and stochastic finite difference methods.

8.1

Mesoscale Random Fields

8.1.1 From Discrete to Continuum Random Fields As discussed earlier, a random medium is a set of deterministic media: B = {B(ω); ω ∈ }. Suppose we deal with the antiplane elasticity of a matrixinclusion composite (B(ω) = Bm ∪ Bi ) with locally isotropic phases of properties C (m) (matrix) and C (i) (inclusion), respectively. The most complete description of this two-phase microstructure is given in terms of an indicator (or characteristic) function 1 i f x ∈Bm χm (ω, x) = or χm : × R2 → {0, 1}. (8.1) 0 i f x ∈Bi ∗ Probability

in itself does not exist; there are only probabilistic models.

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Ceδ 

Cm

Snδ

–1 

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x FIGURE 8.1 The setup of random fields: from a piecewise-constant realization of a composite to two approximating continua at a finite mesoscale.

Of course, the indicator function of C (≡ Cii /2) gives the local property at any point C(ω, x) = χm (ω, x)C (m) + [1 − χm (ω, x)] C (i) .

(8.2)

The indicator function of either phase is a random field with discrete-valued realizations with a continuous parameter x: {χm (ω, x); ω ∈ , x ∈R2 }. Material properties are given in terms of a random field C : × R2 → {IC (i) , IC (m) }

or

{C(ω, x); ω ∈ , x ∈R2 },

(8.3)

Clearly, this random field is also discrete-valued with a continuous parameter, as illustrated in Figure 8.1 in terms of one realization along x. In view of the considerations leading to the derivation of the δ-dependent hierarchy of bounds (7.70) in Chapter 7, at any point x in the material and at any mesoscale δ, two estimates of effective properties may be introduced: Ceδ and Cnδ ≡ (Snδ ) −1 . Consequently, at any mesoscale δ there are two approximating tensor-valued random fields Ceδ : × R2 → R3

Snδ : × R2 → R3 .

(8.4)

These two mesoscale random fields are continuous-valued with continuous parameter x ∈ R2 . The random composite material B is now described, in an approximate way, by two sets of realizations: {Ceδ (ω, x); ω ∈ , x ∈R2 } and {Snδ (ω, x); ω ∈ , x ∈R2 }, see Figure 8.1. These approximations, as they directly depend on the choice of δ, provide two alternate inputs to the field equation governing the global response on the smoothing mesoscale δ [Ci j (ω, x)T, j ]i = 0.

(8.5)

Let us also recall from Chapter 7 that, besides Cd and St we can define the mesoscale response via uniform orthogonal-mixed boundary conditions. As a result, we thus have three different approximating random fields B d = {B(ω); ω ∈ }, B t = {B(ω); ω ∈ }, B dt = {B(ω); ω ∈ },

(8.6)

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with Cdδ and Stδ tensors bounding Cdt δ . If we generalize this line of thought to a more general microstructure, we return to Figure 1 of the Preface and conclude that there is no unique way of setting up a random field with continuous realizations in (b). 8.1.2 Scale Dependence via Beta Distribution Assume the statistics of χm to be homogeneous and isotropic. Recalling our definition of the mesoscale parameter δ, we see that the equation (8.2) is a pointwise limit δ → 0, described by a probability density p[C(x)] = f (m) δ C(x) − C (m) + f (i) δ C(x) − C (i) . (8.7) The Dirac deltas on the right are weighted by the volume fractions f (m) = χm and f (i) = 1 − χm of phases m and i, respectively. Suppose now that we sample the local properties not in this pointwise limit but, rather, with some finite mesoscale δ1 , that is smaller than the inclusion size: 0 < δ1 < 1. Figure 8.2 shows that if we take such a finite size window, it can fall into either of two phases, or on the boundary of inclusions. The former possibility of the pointwise limit, considered in the preceding paragraph, corresponds to a Lebesgue measure zero and thus we have simply had the equation above. This discrete distribution is now replaced by a continuous one such as shown by the curve p1 . Note that the probability mass is distributed continuously between C (m) and C (i) , but not outside this finite range. As δ grows, we see a redistribution, or flow, of the probability mass away from the end points of the interval [C (m) , C (i) ] toward some region indicated by the curve p2 . When finally δ → ∞, p[C(x)] tends to the causal distribution centered at C eff ≡ C∞ —the graph p∞ . These considerations indicate that of all the classical probability densities, beta is the most convenient one to describe this scale effect while keeping all admissible values within a finite range. It is given by p C, a , b, C (m) , C (i) =

C a −1 (1 − C) b−1 f or C (m) < C < C (i) , [C (i) − C (m) ]B(a , b)

(8.8)

where B[a , b] =

(a + b) (a )(b)

for

C (m) < C < C (i) ,

(8.9)

with being the gamma function. In Figure 8.3 we assess its statistical character by displaying probability e densities of tr (Ce ) (a) and C12,max (b) at δ = 10 under essential boundary condition. The first of these plots confirms the beta character suggested in Figure 8.2, while the second one indicates that the radius of the Mohr circle is strongly positively skewed. When comparing Figure 8.3 with corresponding plots for Sn , we observed that densities of traces are similar; this conclusion does not carry over to skewnesses of Ce and Sn (Ostoja-Starzewski, 1998).

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p(Cδ )

P2

Cm

P•

C eﬀ

P1

Ci

C iδ

FIGURE 8.2 Sampling of the mesoscale property (trace of apparent tensor Cδ ) of a disk-matrix composite via windows of different sizes. The beta distribution gives a practical approximation for the entire range of window sizes, showing four cases: the pointwise limit of equation (8.7); the scale δ1 and fit p1 ; the scale δ2 and fit p2 ; and the scale δ∞ and the causal distribution p∞ . (After Ostoja-Starzewski, 1998.)

8.1.3 Mesoscopic Continuum Physics Due to Muschik The line of studies by Blenk et al. (1991), Papenfuβ and Muschik (1998), and Muschik et al. (2000) introduces the concept of a statistical element as a mesoscopic distribution function (MDF) f (m, x, t) generated by the different values of the mesoscopic variables in a volume element f (m, x, t) ≡ f (.)

(.) ≡ (m, x, t) ∈ M × R D × Rt .

(8.10)

the distriThe MDF is defined on the mesoscopic space M×R D ×Rt describing bution of m in a volume element at (x, t), normalized by f (m, x, t) dM = 1 as it should. The macroscopic properties are obtained via averaging with respect to the distribution f (m, x, t).

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(a)

(b) FIGURE 8.3 Two planar random microstructures: (a) four-phase Poisson–Voronoi mosaics; (b) superposition of a matrix-disk composite with a matrix-needle composite.

Thus, when trying to integrate Muschik’s mesoscopic continuum physics approach with ours, we first recognize M to be analogous to the random field on mesoscale L. In terms of an elastic microstructure problem, f (m, x, t) is analogous to a probability distribution of the mesoscale stiffness. On one hand, this analogy necessarily forces one to choose L in mesoscopic continuum physics, and on the other, it shows that the mesoscale continuum thermodynamics approximation is nonunique for we have three possible choices

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stemming from three uniform boundary conditions: either displacement, traction, or orthogonal-mixed. This appears to be a serious dilemma and perhaps no single choice is universally good for all the applications, although Muschik’s model works well for liquid crystals, for which it was developed.

8.2

Second-Order Properties of Mesoscale Random Fields

8.2.1 Governing Equations First, consider the equilibrium equation of antiplane statics σ j, j = f,

(8.11)

where f i represents the negative of the body force b i per unit volume. Take both fields as superpositions of their averages and zero-mean random fluctuations σ j (x) = σ j + σ j (x) f (x) = f + f (x), (8.12) where f i represents the negative of the body force b i per unit volume. Multiplying σ j, j at x1 by σ j, j at x2 , and then ensemble averaging, yields (recall Problem 13 of Chapter 2) j

∂2 Ci (x1 , x2 ) ∂ x1i ∂x2 j

=

∂2 σi (x1 ) σ j (x2 ) ∂ x1i ∂ x2 j

= f (x1 ) f (x2 ) = F (x1 , x2 ). (8.13)

Here we introduced Ci as the correlation function of the stress field σ , and F (x1 , x2 ) as the correlation function of the body force field f . The same approach may be used in the general case of statics governed by j

σi j, j = f i ,

(8.14)

to obtain the relation ∂2 Ciklj (x1 , x2 ) ∂x1 j ∂ x2l

= Fik (x1 , x2 ),

(8.15)

where Ciklj = σi j (x1 )σkl (x2 ) and Fik (x1 , x2 ) = f i (x1 ) f k (x2 ). One can now proceed to statics of a micropolar body with random forcestress, couple-stress, and body force fields (but zero body moment fields) τji (x) = τji + τji (x) µji (x) = µji + µ ji (x) f (x) = f + f (x). (8.16)

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Recalling from Chapter 6 the governing equations (6.12) and (6.17), we first derive ∂2 Tjilk (x1 , x2 ) ∂ x1 j ∂x2l

= Fik (x1 , x2 ),

(8.17)

where Tjilk = τji (x1 ) τlk (x2 ) and Fik = f i (x1 ) f k (x2 ) is the same as above. Next, from the angular momentum balance we obtain qp

∂2 Mji (x1 , x2 ) ∂x1 j ∂ x2k

= e iab e pcd Tabcd (x1 , x2 ),

(8.18)

qp

where Mji = µ ji (x1 ) µq p (x2 ) is the correlation function of the couple-stress field µ . Differentiating (8.18) with respect to xa at x1 , and with respect to xc at x2 , we find qp

∂4 Mji (x1 , x2 ) ∂ x1 j ∂x2k ∂x1a ∂ x2c

= e iab e pcd Fbd (x1 , x2 ).

(8.19)

Other relations of that kind remain to be developed. 8.2.2 Universal Properties of Mesoscale Bounds An interesting statistical property of radius R of Ci j has been found to hold for four types of microstructures (Ostoja-Starzewski, 2000): 1. Matrix-needle composites with stiff needles 2. Multiphase Poisson–Voronoi mosaics, Figure 8.4(a), 3. Matrix-disk composites with circular or elliptical disks, 4. Superpositions of the latter with matrix-disk composites, Figure 8.4(b). In essence, we deal here with random two-dimensional microstructures of spatially homogeneous, isotropic, and ergodic statistics, that are generated from planar Poisson point fields. Thus, besides the Poisson–Voronoi mosaics, we are considering here Boolean models (recall Chapter 1); hard-core point processes are excluded, and so we do see partial overlaps of inclusions in Figure 8.4(b). In the first case, the material of each Voronoi cell is sampled at random from either two, three, or four types of phases; p = 1, . . ., 4, depending on the actual choice of a p-phase random microstructure. The sampling is done sequentially, independent of the states of other cells of the mosaic. An example of a mosaic with four phases present is shown in Figure 8.4(a). In the case of Boolean models, we generate inclusions sampled at random from any one of two (or three, or four) types of phases; the matrix is phase p = 1, and inclusions are or p = 2, 3, or 4. Also here, the sampling is done sequentially—one inclusion after another—independent of the states of other cells of the composite. Each phase is locally homogeneous and isotropic, and

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1

0.5

0.5

0

0

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–0.5

–1 0

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10

15

20 20

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0

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10

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10

(d)

(a)

–1 0

20 20

15

5

10 15 20 20 15 (c)

10 5

0

–1 0

20 20 (f )

0

FIGURE 8.4 Graphs of the correlation coefficient ρi jkl (r) ≡ ρCi j Ckl (r) of the components of Ceδ : (a) ρ1111 (r); (b) ρ1212 (r); (c) ρ1112 (r); (d) ρ1122 (r); (e) ρ1212 (r) under uniform strain; (f) ρ1111 (r) under uniform strain. r1 (respectively, r2 ) axis goes to the right (left).

it is characterized by its volume fraction f ( p) and conductivity C ( p) . Thus, the contrast for a phase p = 1 is α ( p) = C ( p) /C (1) . For each of the mesoscale second-rank tensors—conductivity Ceδ (essential) and resistivity Snδ (natural), respectively—for any specific configuration

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281

(8.20)

Thus, in the ensemble sense, for any scale δ and any type of boundary conditions (e or n), we have two random invariants: {Rδe (ω); ω ∈ } or {Rδn (ω); ω ∈ }. We next consider the coefficient of variation of each of these invariants CV eδ =

σ ( Rδe ) , µ( Rδe )

CV nδ =

σ ( Rδn ) . µ( Rδn )

(8.21)

In the above, µ stands for the ensemble average and σ for the standard deviation of the given invariant. We have carried out a range of numerical experiments on microstructures of Voronoi mosaic and Boolean type to determine CV eδ and CV nδ , usually employing a very fine spring network (Chapter 3) for the resolution of the microstructure and solution of both types of boundary value problems. It has turned out that, whatever the point in the parameter space, the coefficients of variation of both invariants (i.e., CV eδ and CV nδ ), at any δ > 1, equal about 0.55 ± 0.1 irrespective of: 1. The window size δ. 2. The boundary conditions applied to the window (uniform Dirichlet or uniform Neumann). 3. The contrasts α ( p) ( p = 2, . . ., 4), and the shape of the inclusion. 4. The volume fraction f ( p) of any phase p = 1, . . ., 4, providing its conductivity is not 0 or ∞. This result indicates a universal nature of CV eδ and CV nδ for planar random media generated from Poisson point patterns. The fluctuations of up to ±0.1 around 0.55 appear to be due to the finite number of realizations of the random microstructure (generated by a Monte Carlo method) in any given parameter case. An exact mathematical analysis and proof of the constancy of these coefficients of variation does not appear possible at the present stage of theories of random media. However, we offer some observations that may prove vital to such a proof in the future: •

The Poisson point process does not possess any intrinsic length scale, which fact seems consistent with CV eδ and CV nδ being independent of the window size δ.

•

If our microstructures are generated from hard-core point processes (i.e., non-Poisson point fields), then CV eδ and CV nδ are usually lower than 0.55 for window sizes on the order of several grains (δ 5), and then rise and stabilize around 0.55 at higher δ (Jiang et al., 2001a).

•

Although there are no explicit formulas for the conductivity or resistivity tensors for heterogeneous domains of finite size, we can

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The second invariant of the conductivity tensor Ceδ (as well as the resistivity tensor Snδ ) of the material possessing isotropic statistics goes to zero as the window size δ → ∞. Thus, the mean µ( Rδe ) and standard deviation σ ( Rδe ) of this invariant also go to zero as δ → ∞. In view of (8.21), the constancy of CV eδ (and CV nδ ) with δ implies that the mean and standard deviation remain in the same ratio as they both go to zero.

Finally, the third invariant C(3) also appears to have similar universal properties. In fact, observing that, in planar problems (i, j = 1, 2), 2 C(3) ≡ Ci j C jk Cki = C(1) [3C(2) − C(1) ]/2,

(8.22)

C(3) should decrease with δ due to a decrease of C(1) with δ. In effect, (8.22) allows determination of any statistic of C(3) from those of C(1) and C(2) . 8.2.3 Correlation Structure of Mesoscale Random Fields For a composite having SSS statistics of its properties—such as its indicator function (8.1)—the mesoscale random field C is stationary in the same sense too, that is, ρCi j Ckl (x1 , x2 ) = ρCi j Ckl (r)

r = x1 − x2 .

(8.23)

A natural question arises here: For a composite having also an isotropic statistics of its properties, is the C field isotropic in terms of its correlation function? This isotropy property is expressed by ρCi j Ckl (r) = ρCi j Ckl (|r|),

(8.24)

and is, of course, different from the isotropy of its realizations. In OstojaStarzewski (1993a,b, 1994) we addressed this issue through Monte Carlo computations of antiplane mechanics of two systems: binomial fields on square lattices and disk-matrix composites. For each and every realization B(ω) ∈ B of the given random material, the study necessarily involved finding Ci j and Ckl for two windows: one placed at the origin (r = 0) and another at some arbitrary position r = (r1 , r2 ). Resulting plots for the Ceδ tensor, at δ = 10, for the binomial field at a nominal 50% volume fraction of either phase, at contrast 10, are shown in Figure 8.5; surface fluctuations are due to a finite number of B(ω)s. The computations were discontinued for windows separated by several disks because then the apparent moduli became independent. This observation shows that any long-range correlation function

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FIGURE 8.5 A two-phase material with a Voronoi mosaic microgeometry of a total 104,858 black and white cells, at volume fraction 50% each.

models—for example of exponential type—need to be substantiated by a longrange “microstructural memory” such as, say, reinforcing rods. Other observations with regard to Figure 8.5 are:

1. The autocovariance ρC11 C11 is not isotropic as there is a stronger correlation in x1 than in x2 . 2. The autocovariance ρC12 C12 is isotropic. 3. The crosscovariance ρC11 C12 is practically zero. 4. The crosscovariance ρC11 C22 attains, at the origin, the maximum value of ∼ 0.75 rather than 1.0 as might intuitively be expected. 5. Practically identical plots of ρCi j Ckl are obtained under the assumption of uniform strain, as evidenced by Figure 8.4 (e) and (f). 6. Practically identical plots of ρCi j Ckl are obtained for the Snδ tensor, with uniform strain in (5) being replaced by uniform stress.

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8.3

Does There Exist a Locally Isotropic, Smooth Elastic Material?

8.3.1 Correlation Theory Viewpoint The need to include lower than the RVE length scale phenomena in elasticity, combined with their nondeterministic character—as discussed at the beginning of this chapter—has motivated a number of researchers to adopt random stiffness fields, assuming invertibility of the constitutive law ε = S(ω, x): σ

σ = C(ω, x): ε

S(ω, x) = C−1 (ω, x),

(8.25)

with σ and ε being uniform fields applied to a rather hypothetical and unspecified RVE (or SVE) of a random medium. When dealing with fourthrank tensors of elasticity, typically a locally isotropic form ε=

1 [(1 + ν(ω, x))σi j − ν(ω, x)δi j σkk ] ω ∈ E(ω, x)

x ∈ B ⊂ RD

(8.26)

is adopted by simply postulating E and ν (often just E) to be random fields— usually of Gaussian type—with differentiable realizations. To simplify the setting somewhat, let us consider random fields of secondrank tensors Ci j (ω, x); ω ∈ ; x ∈ B , such as those of Section 2.3 in Chapter 2. Assuming the isotropy of realizations, we have Ci j (x) = C (x) δi j or C (x) = C (x) I .

(8.27)

However, in view of the developments of Chapter 7, a smooth stiffness tensor field C (x) is really a mesoscale continuum approximation Ci j (x, δ) = C (x) δi j or C (x, δ) = C (x, δ) I ,

(8.28)

with δ expressing the ratio of L to d (heterogeneity size) of the underlying microstructure, while δi j stands for the Kronecker delta. Thus, we should have 1 1 = ρ2323 = 0, ρ1212

(8.29)

with the obvious symmetries (2.121) of Chapter 2 present, so that (2.123) of that chapter implies K 5 = K 6 = 0.

(8.30)

Clearly, the 2D setting of antiplane elasticity is a special case of this, and we come to conclude that, in particular, ρ1212 (x) = 0; it must be because C12 = 0 everywhere. At this point we refer to our computational mechanics studies in Section 8.2.3 above. We see that ρi jkl is strongly dependent on the particular pair [Ci j , Ckl ] as well as on the direction x. Here, we make correlation-type (C) observations:

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C1: As expected, ρ1111 transforms into ρ2222 upon the rotation of x = (x1 , 0) into x = (0, x2 ). C2: Against expectation, ρ1212 = 0, which indeed can be understood from physical considerations alone, especially at x = 0, without any recourse to numerics. This provides an argument via reductio ad absurdum against the admissibility of (8.27) and, by extension of the same arguments to higher-rank tensors, of (8.26). C3: Only ρ1112 and ρ2212 turn out to be null. 8.3.2 Micromechanics Viewpoint From the standpoint of the Hill condition, we have three loadings to choose from when setting up a random field with continuous realizations, and this leads to nonuniqueness of response on mesoscales. Now, in place of the disordered microstructure of Figure 8.1(a) of the Preface, let us consider a special case of a smooth elastic continuum (8.27) without any microstructure. Assume the dependence of C (x) on x1 -direction only. Henceforth, by Cδ(i j) we denote the components of the mesoscale stiffness Cδ . We make analysis-type (A) observations: A1: The mesoscale response Cδ(11) of Cδ of the L ×L window is calculated exactly under the assumption of a uniform stress σ1 (x) = σ¯ 1 , ∀x ∈ B L , because for this loading we have a smooth “microstructure” of a series-type. A2: The mesoscale response Cδ(22) of the L × L window is calculated exactly under the assumption of a uniform strain ε2 (x) = ε¯ 2 , ∀x ∈ B L , because for this loading we have a smooth “microstructure” of a parallel type. A3: Loadings dictated by A1 and A2 jointly correspond to the special case of the orthogonal-mixed boundary condition. Then, assuming any smooth function C (x) in (8.27), Cδ(i j) on the left of equation (8.11) can be evaluated via integration of C (x) over the mesoscale domain B L . This would involve a calculation of compliance Sδ(11) and of stiffness Cδ(22) , which, given the fact that the axes x1 and x are oriented along the principal directions, would allow one to determine all the elements of Cδ(i j) . Next, one can take the limit lim Cδ(i j) (x) = C (x) δi j , δ→0

(8.31)

to recover the original smooth continuum. Suppose now that a heterogeneous material microstructure with statistics depending on x is introduced. We then need a mesoscale continuum at δ > 0, and, in view of A1–3 above, Cmix δ (x, ω) must be determined. That tensor is anisotropic! At this point we recall (7.27) of Chapter 7: d [Stδ (ω)]−1 ≤ Cdt δ (ω) ≤ Cδ (ω),

(8.32)

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We now make micromechanics-type (M) observations: M1: Each one of the three tensors in (8.32) is anisotropic. M2: As the mesoscale δ → ∞, the tensors Cdδ (ω, x1 ), Cdt δ (ω, x1 ) and [Snδ (ω, x1 )]−1 converge to Ceff , but this is far above the scale of all the fluctuations, that is, beyond Figure 1(b) of the Preface. In principle, this is assured by the separation of scales limit (assured in turn by the ergodic and SSS properties of the microstructure) where the homogeneous continuum applies—a situation of no interest to us because fluctuations of Figure 1(b) arise below that limit. N −1 M3: The ensemble average tensors CδD (x1 ), Cmix are δ (x1 ) and Sδ (x1 ) orthotropic; a computed example was given in a study of functionally graded materials in Section 7.5.3 of Chapter 7.

8.3.3 Closure The observations C1–3, A1–3 and M1–3 of Sections 8.3.1 and 8.3.2 show that a locally isotropic, smooth elastic continuum is untenable, unless one is willing to disregard the underlying microstructure. Here we may quote Truesdell and Noll (1965): “Continuum mechanics presumes nothing regarding the structure of matter.” That is, continuum theories are not only phenomenological, but some may well have no physical content. This can happen with deterministic as well as stochastic models. Evidently, generalizing the notion of a uniform isotropic elastic continuum to an inhomogeneous, smooth and isotropic one does not appear to violate any principles of continuous media. However, when the micromechanics is brought into the analysis of antiplane elastic response—indeed, one of the simplest models in continuum mechanics—we arrive at contradictions above, and these extend to higher-rank tensors and/or inelastic responses.

8.4

Stochastic Finite Elements for Elastic Media

8.4.1 Bounds on Global Response The mesoscale random fields introduced in Sections 8.1 and 8.2 provide inputs to two finite element schemes, based on minimum potential and complementary energy principles, respectively, for bounding the global response (Ostoja-Starzewski, 1993). As an example of such an approach, a (scalar) problem of torsion of a bar made of a two-phase microstructure has been analyzed (Ostoja-Starzewski and Wang, 1999; Ostoja-Starzewski, 1999b). Assuming a deterministic body force f , its governing field equation is [Ci j (ω, x)φ, j ],i + f = 0,

x ∈ B(ω)φ = 0,

x ∈ ∂ B(ω).

(8.33)

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Here φ is a stress function and C(ω, x) (≡ Ci j (ω, x)) corresponds to one particular realization B(ω) (of volume V) of the random medium B, so that the problem is entirely deterministic; recall (8.5). The lower and upper bounds on global response of B(ω) are obtained, respectively, from two, energy principles: the minimum potential energy principle

1 1 inf η T Cδ (ω)ηd V − f φd V, (8.34) 2 V φ∈H01 (V) 2 V and the minimum complementary energy principle,

1 inf ξ T Sδ (ω)ξ d V ∀ξ ∈ H = {ξ ∈ (L(V) 2 |∇ · ξ + f = 0}. φ∈H01 (V) 2 V

(8.35)

Here Cδ (ω) and Sδ (ω) are the stiffness and compliance tensor fields on mesoscale δ (set by the size L of the finite element relative to the grain size d), while η and ξ stand for ∇φ and Cδ (ω)η, respectively. We thus have two algebraic problems [K(ω)] {} = { f }

ω ∈ ,

(8.36)

where [K(ω)] is the global stiffness matrix, and [L(ω)] {ϕ} = {}

ω ∈ ,

(8.37)

where [L(ω)] is the global flexibility matrix. Here and ϕ are the respective vector solutions; see the first reference above for all the details. The essence of this setup is that these two energy principles ensure a monotonic convergence of the lower and upper bounds of the energy norm from below and above, respectively, in terms of the energy norm

1 φ E = η T C(ω)ηd V ω ∈ . (8.38) 2 V provided (1) we have a homogeneous material and (2) the mesh resolution δ → 0 (e.g., Brezzi and Fortin, 1991). This is the classical limit of infinitesimal finite elements solving a deterministic continuum problem without identifying any microstructure. The situation, however, is not that straightforward in the case of a heterogeneous material. Namely, because the effective stiffness tensor on mesoscales is nonunique, and (8.34) is set up in displacements, Cδ (ω) determined from the displacement boundary condition, should enter this principle; the global stiffness matrix in (8.36) is then built, element by element, from Cδ (ω)s. On the other hand, given the fact that (8.35) is set up in stresses, the apparent compliances Sδ (ω) from natural boundary condition should be used as input to the minimum complementary energy formulation; the global flexibility matrix is then built, also element by element, from Sδ (ω)s. The tendency of global FE methods to converge with δ decreasing—as pointed out following (8.38)—is now hindered by the fact that the mesoscale

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(i.e. apparent) responses Cδ (ω) and Sδ (ω) tend to diverge as δ decreases. Evidently, we have a competition of two opposing scaling trends. Nevertheless, both FE methods provide bounds on the global, macroscopic response for a given choice of δ, and, in fact, a possibility of an optimal mesoscale emerges. To sum up, we are dealing here with the situation lacking separation of scales alluded to in the Preface.

8.4.2 Example: Torsion of a Duplex-Steel Bar The ideas discussed above are illustrated in the context of two-dimensional, two-phase microstructures of linear elastic materials governed locally by (8.33). The microstructural geometry is specified by a Poisson–Voronoi mosaic. Each cell of the mosaic is being occupied by either phase 1 or 2, according to a probability equal to the global volume fraction, which is chosen at 50% in Figure 8.5. Because the Voronoi cells are six-connected on average, we have a percolating system—strictly speaking, bipercolating—which clearly lacks any periodicity, and so, a periodic unit cell cannot be setup. The isotropy of both phases’ stiffness tensors leads to a contrast α = C (2) /C (1) . When α = 1 the material is homogeneous, otherwise it is heterogeneous. Note that the two-phase Poisson–Voronoi mosaic microstructure chosen here may be applied to model a range of different materials—examples are offered by duplex steels for a finite α (e.g., Werner et al., 1994), or porous materials for the extreme cases of α = 0 or ∞. In Figure 8.6(a) we show the case of a homogeneous material (α = 1): as discussed in connection with equation (8.38), the finer the mesh—that is, the smaller the mesoscale—the closer are both estimates of the global response. However, in the case of a heterogeneous material (α = 1), there exists an opposing trend according to the hierarchy of mesoscale bounds, Figure 8.7. Thus we observe a competition of two opposing trends: 1. The global responses, computed by (8.36) and (8.37), tend to converge as δ decreases. 2. The mesoscale responses, serving as input to (1), computed respectively from the essential and natural boundary conditions, tend to diverge as δ decreases. The results of this competition are shown in terms of the energy norm (8.38), in function of the increasing finite element size, for three contrasts α = 10, 100, 1000 in Figures 8.6(b), (c) and (d), respectively. Note that in the case (b) of relatively weak contrast (α = 10) an optimal finite element mesh size, or mesoscale δopt , can clearly be seen—-it gives the closest upper and lower bounds. As the contrast increases—-cases (c) and (d)—the bounds diverge further away from one another and only the crudest meshing of the entire domain provides a relatively useful estimate of the global response. Our methodology employing mesoscale finite elements is checked by a comparison to the response of the same material computed by a finite element

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8000 a = 10

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2.15 2.05

E

||φ||

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||φ||

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Computational micromechanics solution

6000

Energy formulation

Energy formulation

1.75

5000

20 40 60 Number of elements per side (a)

20 40 60 Number of elements per side (b) 5500

5000 a = 103 Complementary energy formulation

E

3000 2000 Computational micromechanics solution

1000

a = 102

4500

||φ||

E

4000

0

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Complementary energy formulation

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Computational micromechanics solution

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Energy formulation

0

Complementary energy formulation

7000 Complementary energy formulation

1.85

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0

20 40 60 Number of elements per side (d)

FIGURE 8.6 Behavior of the energy norm (8.38) with respect to a sequence of self-accommodating finite element meshes, in terms of the increasing finite element resolution, for: (a) a homogeneous material domain contrast = 1, and (b) a heterogeneous domain of Figure 8.5 for contrast α = 10, (c) the same domain for α = 100, and (d) the same domain for α = 1000. In (a–d) computational micromechanics solutions taking account of the entire microstructure of Figure 8.5 are also shown.

mesh much finer than any single grain—it presents an absolute and best available, albeit very costly, reference solution. Thus, in all four cases of Figure 8.6 this “computational micromechanics” solution directly taking into account the entire microstructure of 104,858 black and white cells is also shown. It is seen that the micromechanics solutions always fall between the bounds based on the micromechanics moduli. The mesoscale window, or the SVE, is identified as a “mesoscale finite element” of the global finite element mesh. With the demonstration of the method for a single realization B(ω) of random microstructure, it is a rather simple matter to generalize it to an ensemble response. Thus, instead of computing the mesoscale moduli Cd and St from Dirichlet and Neumann boundary value problems, one could generate them, ω by ω and for any specific δ, based on the statistics such as those presented in Chapter 7 and in the preceding sections. The rest of the procedure would then be identical.

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1.9 〈 C eδ 〉

1.8 1.7 1.6

〈 Snd 〉

–1

1.5 1.4

FIGURE 8.7

0

10

20

30

40 δ

50

60

70

80

−1

for the two-phase microstructure of Figure 8.5 at Hierarchies of bounds Ceδ (ω) and Snδ (ω) volume fraction and contrast α = 10; five data sets are shown.

Finally we must note that, so far, this formulation is suited to handle problems with Dirichlet-type boundary conditions on the macroscale. In order to deal with Neumann-type (or, generally, mixed) conditions on the macroscale, a mixed variational formulation is necessary. 8.4.3 An Overview of Phenomenological SFE Studies The foregoing development is different from the conventional SFE, which basically proceed as follows: (1) assume a random field of constitutive coefficients, (2) use it as input into the global FE scheme, usually based on the minimum potential energy principle, and (3) derive the global response either for the first two moments or for the ensemble in the Monte Carlo sense (Benaroya and Rehak, 1988; Brenner, 1991; Ditlevsen, 1996; Liu et al., 1995; Kaminski, ´ 2002); see also Elishakoff and Yongjian (2003). In the following we review basic tenets of the SFE. With reference to the preface, given a deterministic field equation in mechanics Lu = f,

(8.39)

randomness may enter through either the operator (i.e., material properties), the forcing function (temporal in nature), or the boundary and/or initial conditions. However, the choice of randomness of forcing f in time is fundamentally different from the randomness of the field operator L in physical space.

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The point is that f(ω, t) in (8.39) implicitly involves some local averaging in the time domain

t+t/2 1 f t (ω, t) = f (ω, t )dt , (8.40) t t−t/2 which is needed to smear out fluctuations in, say, wind forcing on a structure, over time scales too short to have any influence on the oscillator. Commonly the subscript t in f on the left-hand side is suppressed, and we simply write f (ω, t). On the other hand, the local averaging in physical space is not consistent with the concepts of micromechanics, and, as shown in Chapter 7, should be replaced by stochastic homogenization, which, by offering three optional boundary conditions, leads to a nonuniqueness of continuum approximation. Now, if local averaging is applied to a stiffness (respectively, compliance) tensor field, it yields a Voigt-type (Reuss-type) estimate of stiffness (compliance) for some spatial domain of the microstructure. The need for development of finite element methods taking into account the uncertainty in structural material parameters has been recognized since the late 1970s (Contreras, 1980); one also needs to mention here stochastic boundary element methods (Liu et al., 1995; Saigal, 1995). As a result of the observation that many engineering structures are described by spatially random material properties, several theoretical methodologies were developed in the civil engineering literature. Most of these studies—broadly called stochastic (probabilistic) finite elements (SFE)—are based on a direct generalization of Hooke’s law to random fields such as in (8.25) and (8.26). It has to be noted, however, that the effort and merit in SFE has been on the development of efficient numerical methods for solution of boundary value problems, rather than on development of a connection to the material microstructure. Moreover, most cases have been restricted to the case of weak fluctuations in material properties, whereby the stiffness matrix is expressed as the sum of the mean and noise ω ∈ . (8.41) [K (ω)] = [K ] + K (ω) Perturbation method. This approach consists in a replacement of the random system by a (theoretically infinite) number of identical deterministic systems each of which depends on the solution for the lower order equations. Thus, to second order, for the static problem—[K (ω)] {U} = { f }—the solution is expressed as the sum {U} = {U0 } + {U1 } + 2 {U2 },

(8.42)

where 1. This leads to a system of equations {U0 } = [K ]−1 { f } {U1 } = − [K ]−1 K (ω) {U0 } {U2 } = − [K ]−1 K (ω) {U1 } .

(8.43)

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Neumann series method. This method (Shinozuka and Yamazaki, 1988), is based on a Neumann series for the inverse of the random operator [K (ω)], which takes the following form: [K (ω)] = ( I − P(ω) + P 2 (ω) − P 3 (ω) + · · ·) [K ]−1 P(ω) = [K ]−1 K (ω) .

(8.44)

The approach was introduced as an avenue for a speedier way of solving the stochastic problem by a Monte Carlo simulation. To that end also a Cholesky decomposition of [K (ω)] is implemented. Weighted integral method. In contradistinction to the above two methods, this one focuses on the determination of the random stiffness matrix [K (ω)]. The idea, in the setting of an elastic plate problem (e.g., Deodatis and Graham, 1997), is to start with a locally isotropic random field of, say, Young’s modulus and assign it to all the finite elements according to i i E(ω, x) = i E + 1 +i f (ω, x) f = 0. (8.45) Next, the stiffness of each element of i V is calculated as

i K (ω) = [i B]T [C(ω)] [i B]dx = i K 0 +i X0 (ω) i K 0 (ω) ,

(8.46)

iV

where [i K 0 ] and [i K 0 (ω)] are deterministic matrices, and i X0 (ω) is a random variable given as

i i X0 (ω) = f (ω, x)dx. (8.47) i

A

From a micromechanics standpoint, this approach gives a Voigt-type estimate for the effective stiffness of the ith finite element. Similarly, if applied to the compliance, it would yield a Reuss-type estimate of flexibility. Spectral method. It is well known that, in a representation of a random function by a Fourier series, the coefficients of the expansion become, in general, correlated. In order to retain the uncorrelatedness while obtaining the desired orthogonality of random coefficients, a Karhunen–Lo´eve expansion (e.g., Papoulis, 1984; Yaglom, 1957) is introduced. This idea has been employed by Ghanem and Spanos (1991) to represent the spatial variability of random field of Young’s modulus such as in (8.45). This method claims not to be limited to weak fluctuations and to avoid the inconsistencies between various other methods involved in the inversion of the random stiffness matrix [K (ω)]. Also, it is designed to do away with the problem of dealing with a large number of random variates resulting from a pointwise representation of the random field E(ω, x). Conclusions. There is: (1) a necessity of a correct link to micromechanics in setting up of the continuum random fields and of the random stiffness matrix [K (ω)]; and (2) a need for a careful interpretation of the variational principles as a basis for SFE.

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Thus, although an assumption such as (8.25 and 8.26) is generally incorrect, it follows that the classical SFE methodologies are amenable to modifications to incorporate the micromechanical input of the type described above. More precisely, once continuum random fields of a given material are found, an existing approach—for example, a Neumann series—may be applied to determine the upper and lower bounds on response according to the stochastic variational formulation given in Section 8.4.1. For applied mathematics trends in multiscale methods see e.g., E et al. (2005) and Gloria (2006b).

8.5

Method of Slip-Lines for Inhomogeneous Plastic Media

` 8.5.1 Finite Difference Spacing vis-a-vis Grain Size The breakdown of separation of scales—represented by equation (7.1) of Chapter 7—can also be encountered in mechanics of plastic materials. Let us now suppose that we deal with a specific boundary value problem of a rigid perfectly plastic material of Huber–von Mises type by the method characteristics. This means we want to determine the planar stress field (σ11 , σ22 , σ12 ) from the equilibrium equations combined with the yield condition (e.g., Hill, 1950; Kachanov, 1971) σ11,1 + σ12,2 = 0 σ22,2 + σ12,2 = 0

2 = 4k 2 . (σ11 − σ22 ) 2 + 4σ12

(8.48)

As is well known, solution proceeds by setting up a system of two equations that hold along two mutually orthogonal families of characteristics, and then employs a finite difference method. If the separation of scales holds, k can be taken as constant throughout the domain of material. This implies that the crystal size in a material effectively (on the scale of RVE) described by (8.48)3 is infinitesimal relative to the spacing of the finite difference mesh. However, if we have the situation depicted in Figure 8.8, we ought to consider k in (8.48)3 as a smoothing, mesoscale random field kδ , in the (x1 , x2 )plane where δ is set by the ratio of the window size L to the average crystal size d. Following Ostoja-Starzewski and Ilies (1996), the random plastic medium Bδ is now given by the set {Bδ (ω); ω ∈ } = {kδ (ω, x); ω ∈ , x ∈ D} with

kδ (ω, x) = kδ + kδ (ω, x) kδ (ω, x) = 0, (8.49) where kδ (ω, x) is zero-mean noise. To make things simple and tractable, we assume—just like in phenomenological SFE studies of Section 8.3.3 above—a high signal-to-noise ratio

k (ω, x) kδ . (8.50) δ If we now introduce a deterministic, homogeneous plastic medium Bdet defined by kdet = kδ , the basic question that arises is this: What is the

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β

α N

1

2

FIGURE 8.8 Mesoscale windows (or SVEs) involved in finding apparent plasticities k at 1, 2, and N. Evidently, the separation of scales would be recovered with crystal size becoming infinitesimal relative to the spacing of characteristics.

difference, if any, between the solution of Bdet and the ensemble average solution of Bδ ? We do know that in the limit both solutions should coincide, but, to answer the question we first have to modify the derivation of characteristics. As is usual in the theory of slip-lines (here we follow Szczepinski, ´ 1979), two functions p and ϕ are now introduced σ11 = p + kδ cos 2ϕ

σ22 = p − kδ cos 2ϕ

σ12 = kδ sin 2ϕ.

(8.51)

Upon substitution of (8.48) into (8.48)1,2 , and setting ϕ = − p/4 on differentiation, we get ∂p ∂x1

+ 2kδ

∂ϕ ∂ x1

=

∂kδ

∂p

∂x2

∂ x2

− 2kδ

∂ϕ ∂ x2

=

∂kδ ∂x1

,

(8.52)

where the rectangular axes are now along the local slip-line directions. Equations (8.52) become independent of the orientation of the axes if ∂/∂x1 and ∂/∂x2 are replaced by the tangential derivatives ∂/∂s1 and ∂/∂s1 along the s1 and s2 characteristics, respectively. Thus, we find d p + 2kδ dϕ =

∂kδ ∂s2

s1

d p − 2kδ dϕ =

∂kδ ∂s1

s2 .

(8.53)

These relations represent a system of two quasi-linear hyperbolic equations driven by the random terms involving kδ , both on the right- and left-hand sides. Clearly, the randomness vanishes as kδ tends to a constant. The basic setup is completed by corresponding characteristic directions which are, in

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fact, specified by the same relations as in the homogeneous medium case s1 :

d x2 = tan(ϕ + π/4) d x1

s2 :

d x2 = tan(ϕ − π/4). d x1

(8.54)

Equations (8.53) and (8.54) form the basis for determination of the Hencky– Prandtl network of slip-lines in a given boundary value problem. In cases of Cauchy and characteristic problems studied below this relies on the method of finite-differences for finding {x1 , x2 , p, ϕ} N at a new point N given the data {x1 , x2 , p, ϕ}i at the two preceding points i = 1, 2. It follows that, due to the randomness in k at 1 and k at 2 and k N , as well as the possible randomness in the initial data { p, ϕ}1 and { p, ϕ}2 two characteristics of the deterministic problem are replaced here by two cones (or wedges) of forward dependence, which, in the ensemble sense, contain all the characteristics of the stochastic problem emanating from points 1 and 2, Figure 8.8. The new values { p, ϕ} N are found explicitly by finite differencing from (8.53), while {x1 , x2 } N are found either by backward or forward differencing from (8.54). The latter choice itself follows either the scheme recommended by Hill (1950), yN − yi = (xN − xi ) tan [(ϕi + ϕ N )/2 ± π/4]

i = 1, 2,

(8.55)

or that by Szczepinski ¯ (1979), yN − yi =

1 (xN − xi ) [tan(ϕi ± π/4) + tan(ϕ N ± π/4)] 2

i = 1, 2. (8.56)

8.5.2 Sensitivity of Boundary Value Problems to Randomness 8.5.2.1 Cauchy Problem The Cauchy boundary value problem we consider is defined as one in which the normal stress and the shear stress are specified on a line AB, whereby AB intersects only once each of the characteristics. Henceforth, we adopt AB to lie along the x1 -axis over the interval 0.0 to 9.0 so that the normal stress is σ22 and the shear stress is σ12 , while the domain of influence lies in the first quadrant of the (x1 , x2 )-plane, Figure 8.9. The particular case we give in that figure involves an inhomogeneous boundary condition x1 , (8.57) 10 where we take n = 0, 1, . . ., 9 and increments x = 1.0. The sensitivity of these boundary value problems to the medium’s randomness is studied through a comparison of three settings: σ22 = −1.5

σ12 = −0.6 +

1. Deterministic case (medium Bdet )—zero noise: kδ = 0

2. Random case—very small noise of 0.5% about the mean: kδ ∈ [−0.0025, 0.0025] 3. Random case—small noise of 5% about the mean: kδ ∈ [−0.025, 0.025]

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(a)

x

y

(b)

x

y

(c)

x

FIGURE 8.9 The Cauchy problem with boundary data (8.57) for: (a) deterministic homogeneous medium— case of zero noise; (b) random medium—case of very small noise; (c) random medium—case of small noise.

Here kδ = 1 and kδ is a uniform random variate in a given interval. Forward dependence cones of Hencky–Prandtl nets in the boundary value problem (8.57) in all three cases are shown in Figure 8.9(a), (b) and (c), respectively. The latter two were obtained from 100 simulated realizations of B(ω). Studies of such Cauchy as well as characteristic (Goursat) problems lead to these conclusions regarding the effect of noise kδ in the yield limit kδ : •

There is practically no difference between the ensemble average net of slip-lines of the stochastic problem (i.e., for a random medium B) and the net of the corresponding deterministic problem (i.e., for a homogeneous medium Bdet ) for very small noise; however, this

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difference increases with the growing inhomogeneity in the boundary data. •

The scatter about this average has a coefficient of variation which tends to quickly increase over that of kδ on the length scales L ma cr o of the boundary value problem; it increases with the growing inhomogeneity in the boundary data such as that modeled by (8.57).

•

For noise growing above 5% there is an increasing change in the average, and very amplified scatter, in the random fields of stress as well as velocity.

•

There is practically (to within four significant figures) no difference in resulting slip-line nets between the explicit and implicit integration methods; here, the explicit formulas were used first as a predictor and then, without any numerical convergence problems, the implicit formulas were employed as a corrector.

•

We have also made a comparison of the uniform versus truncated Gaussian randomness of kδ . As equivalence criterion between both kinds of probability distributions we took the same standard deviation σ for the case of very small noise (σkδ = 14.437 · 10−5 ), and the case of small noise (σkδ = 144.37 · 10−5 ). With this setup all our Gaussian-based simulations resulted in practically the same slipline networks as the ones shown here.

•

To sum up, we conclude that only in the case of very small noise may one safely replace the average solution of a stochastic problem by the solution of a deterministic problem with ke f f = kdet = kδ , that is Be f f = Bdet .

All these results were obtained under the assumption of independence of random variables kδ at all the points of the finite difference nets. By introducing the correlatedness between neighboring windows, the scatter in slip-lines tends to decrease, but only for δ < 10 (i.e., very small mesoscale windows) is this a significant effect. 8.5.2.2 Limit Analysis of a Pipe under Internal Loading As a further application of this method we consider limit analysis of a cylindrical tube loaded by a uniform traction on its internal boundary. The solution to this classical problem (e.g., Kachanov, 1971) in the case of a homogeneous material is based on the solution of a Cauchy problem for an axisymmetric stress field (σrr , σθ θ , σr θ ) in the (r, θ) coordinate system in a plate with circular hole, under a pressure boundary condition σrr = − p < 0

σr θ = 0

at

r = a.

(8.58)

With the yield condition σθ θ − σrr = 2k the stress field is determined explicitly by the formula r (8.59) σθ = σr + 2k. σrr = − p + 2k ln a

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The slip-line field is now given by a system of two orthogonal, logarithmic spirals θ + ln

r =α a

θ − ln

r = β. a

(8.60)

It follows from (8.59)1 that, in case of a tube of external radius b, the condition σrr = 0 at r = b gives the limit pressure b p = 2k ln . a

(8.61)

Having this formula we may regard p as given and ask for the external radius b as a function of p . To make the example concrete, we set k = 1.5, p = 3.0 and a = 1, which then results in b( p ) = e p /2k = 2.723. In the case of a tube made of an inhomogeneous material, however, the net of slip-lines becomes distorted from such perfect pattern and the above “nice” relations no longer apply. The finite differencing outlined above has now to be used to determine the slip-lines as well as the stress field in any particular realization of B. Now, the evolution of σr along any slip-line emanating from r = a is a certain random walk in the body domain D starting from the value − p there and stopping at 0 at r = b. Thus, the condition σrr = 0 plays the key role in the definition of an excursion set of the random field σrr (r, θ) = {σrr (ω); ω ∈ } (e.g., Adler, 1981) A0 (σrr , D) = {(r, θ) ∈ D|σrr (r, θ) ≥ 0}.

(8.62)

This then leads to a set of level crossings ∂ A0 (σr , D) = {(r, θ) ∈ D|σrr (r, θ) = 0}

(8.63)

where ∂ A0 (σrr , D) is a set of closed contours of the plastic zone, which in the case of a homogeneous material Bdet under the pressure boundary condition would be a circle of radius b( p ) = 2.723. Thus, in any given random material the set of level crossings is a random set in plane, which, assuming spatial homogeneity and isotropy of field {kδ (ω, r, θ); ω ∈ , (r, θ) ∈ D}, is a circle containing all the possibilities. Now, let b max and b min be a maximum and minimum distance for a given realization, respectively, from the origin to the contour. Since b max determines the minimal amount of material needed for a tube to withstand the internal pressure p , we ask: Is b max ( p ∗ ) smaller, equal to, or larger than b( p ∗ ) of the homogeneous medium problem? In Figure 8.10(a) we plot patterns of slip-lines—actually as wedges of forward dependence—under condition (8.67) corresponding to 400 realizations of B with kδ = 1.5 and kδ ∈ [−0.025, 0.025]. The set of level crossings is shown as a ring containing all 400 piecewise-constant, noncircular closed curves. Clearly, b max ( p ∗ ) is always larger than b( p ∗ ), whereby b max increases

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(a)

(b)

FIGURE 8.10 Slip-line patterns in a randomly inhomogeneous material with kδ ∈ [−0.025, 0.025], under: (a) the boundary condition (8.58), and (b) the boundary condition (8.64). In each case 400 realizations of B(ω) are used.

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as the noise level in k increases. Thus, the principal conclusion is that the presence of material inhomogeneities requires a thicker tube than what is predicted by the homogeneous medium theory. Let us also note that the higher the pressure p, the greater is the external radius b, and hence, the greater is the spread of the forward evolution cones implying an increase in the scatter of b max and b min . Interestingly, both random variables b max and b min are symmetrically distributed about the deterministic radius b( p ∗ ) of the homogeneous medium problem. The same qualitative conclusions carry over to the case of a combined pressure and shear boundary condition σrr = − p < 0

σr θ = q = 0

at

r = a,

(8.64)

The slip-lines are still spirals, albeit no longer of logarithmic type. The stress field is now given by (Kachanov, 1971)  σrr = − p ± k 2 ln a − r

r 2

−

q k

+

1−

q k

+

a

r 2 a

1+

+

q k

q 2 r 4 q 2 − + 1− , a k k

σθ θ − σrr = ±

k2 − q 2

a 4 r

.

q k

(8.65)

(8.66)

The general character of spirals is displayed in Figure 8.10(b), which shows the random slip-line network, also for 400 realizations, corresponding to q ∗ = 1.3, and all other parameters the same as before. These data result in the external radius b( p ∗ , q ∗ ) = 3.028. We note here that the addition of shear traction has a strongly amplifying effect on the scatter of dependent field quantities, and most notably on the spread of slip-lines. Note: While this section focused on metal plasticity, similar concepts have recently been developed for Mohr-Coulomb plasticity of granular materials (Kamrin and Bazant, 2007). In it a stochastic flow rule has been introduced to model fluidization (stick-slip) transition along non-deterministic slip-lines. Note: Another related study focused on the plastic collapse (under a fixed load) of an elastic/perfectly plastic medium with the yield strength taken as Gaussian random field (Ku and Nordgren, 2001). Theorems of limit analysis and methods of reliability theory were used to develop algorithms for the computation of upper and lower bounds on the probability of plastic collapse. Significantly, three-dimensional results were found to differ from those of the plane problem.

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301

Michell Trusses in the Presence of Random Microstructure

8.6.1 Truss-Like Continuum vis-a-vis ` Random Microstructure We now return to the issue concerning the actual making of a Michell truss-like continuum introduced in Section 4.4.1 of Chapter 4, but when the material to be used for the manufacture has a random microstructure, such as in a polycrystal plate. First, consider the situation in which we are going to use the same realization B(ω) to manufacture the trusses of sequence shown in Figure 8.11, and continuing for ever higher discretizations (n increasing). As illustrated in that figure, the finer the truss, the shorter and narrower are the truss members. Associated with this there must be a growing dependence of the yield limit k on the chosen mesh spacing L. Although we employ our nondimensional parameter δ = L/d,where d is the typical size of microscale imperfections (e.g., grain in a polycrystal), we keep in mind that now L may strongly vary in space. For the truss problem introduced, L is smallest near the rigid foundation’s boundary and largest at the point A of application of force P. From the standpoint of plasticity of random heterogeneous media, we note two effects: (1) the scatter in k grows as δ decreases, and (2) the statistical average k may change as δ decreases. Here we shall assume that the mean of k is constant while its standard deviation, σk , is inversely proportional to δ k = const

σk (δ) ∼ 1δ .

(8.67)

This suggests an impossibility to physically attain the limit of a truss-like continuum—a deterministic continuum with an infinitesimally fine spacing of truss connections—due to competition of two opposing effects (OstojaStarzewski, 2001b): (1) noise in k decreases to zero as the truss spacing L grows; and (2) classical deterministic solutions tend to hold as the spacing L decreases. Similar to the plastic media problem of Section 8.5, k is a random field, parametrized by location x = (x1 , x2 ) in the truss plane. Relations (8.67) describe the apparent plastic limit, as a result of local smoothing of the random microstructure (one realization B(ω) of which is shown in Figure 8.11), of a given truss member according to the given mesh spacing L = δd. This leads us again to the concept of a random medium: a set B = {B(ω); ω ∈ } = {k(ω); ω ∈ }, where is a sample space of inhomogeneous continua locally smoothed according to (8.67). We can write k(ω) = k + k (ω)

k (ω) = 0,

where k is the zero-mean noise in k. In the following we assume:

(8.68)

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(a)

(b) FIGURE 8.11 Manufacturing of a truss from one mosaic of a polycrystal using either a coarse (a) or a twice finer (b) refinement of the mesh. It is understood that the material in the interior of squares is removed, thus leaving an orthogonal grid of bars. Clearly, the scatter of the effective plastic limit of a bar on the scale of a single cell (mesoscale) increases as we go from (a) to (b). And, simultaneously, the thickness of the bars decreases with the mesh refinement. The dash-dot lines are the axes of bars of the truss (i.e., characteristics of the hyperbolic system).

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1. The truss spacings L of interest to us are greater than the grain size d, so that k may be treated as a field of independent random variables when entering the finite difference formulation generalized to inhomogeneous media (see below). 2. The underlying material microstructure is space-homogeneous and ergodic. 8.6.2 Solution Via the Hyperbolic System The governing equations need now to be modified from the homogeneous medium case given in Section 4.4 of Chapter 4, following Hegemier and Prager (1969). The modification consists in recognizing k not to be constant, so that, with all the prior definitions, we obtain k k

∂w ∂x1 ∂w ∂x2

+ k cos 2θ + k sin 2θ

∂θ ∂ x1 ∂θ ∂ x1

+ k sin 2θ − k cos 2θ

∂θ ∂x2 ∂θ ∂x2

=− =

2w + sin 2θ ∂k cos 2θ ∂k + 2 ∂ x1 2 ∂x2

cos 2θ ∂k 2w − sin 2θ ∂k + . (8.69) 2 ∂x1 2 ∂x2

Setting θ = π/2, we obtain ∂w ∂x

−

∂θ ∂x

=−

w ∂k 1 ∂k − k ∂x 2k ∂ y

∂w ∂y

+

∂θ ∂y

=−

1 ∂k w ∂k − . 2k ∂x k ∂y

(8.70)

If k = const, we recover the equations of a Michell truss made of a homogeneous material d (w − θ) = 0 ds1

d (w + θ) = 0, ds2

(8.71)

which hold along two characteristics s1 and s2 , at angles specified, respectively, by θ = α + π/4

θ = α + 3π/4.

(8.72)

α is defined as the angle formed by the positive direction along the foundation F with the positive x-direction. On this boundary w = −1. If k is a random field, the characteristics’ directions are still given by (8.72) and w = −1 on the foundation F , but the evolution of w and θ along the characteristics is governed by (8.70). Now, if the derivatives ∂/∂x and ∂/∂ y are replaced by the tangential derivatives ∂/∂s1 and ∂/∂s2 along the s1 and s2 characteristics, equations (8.70) become independent of the orientation of the axes, and result in dw − dθ = −

w 1 ∂k ds1 dk − k 2k ∂s2

dw − dθ = −

w 1 ∂k ds2 . dk − k 2k ∂s1

(8.73)

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Summarizing, with k being a random function of x and y, we have a stochastic quasi-linear hyperbolic system governing the field. Given the presence of randomness, we have a stochastic inverse Cauchy problem, a set of deterministic inverse Cauchy problems, each one corresponding to one realization k(ω) of random field k: “find the net of characteristics supporting the given load P at point A and emanating from the foundation F with conditions (8.72) and (8.73) specified on it.” A solution to any one of these Cauchy problems—denoted by a truss T(ω)—represents a departure from that of the deterministic homogeneous medium Bdet . Because all nets meet at A, we deal with an ensemble T = {T(ω); ω ∈ } of nets of backward dependence wedges, which contains all T(ω)s, Figure 8.12(a)–(d). In these figures we show the wedges in black and the solutions of Bdet in white; the latter are repeated from Figure 4.17(a)–(d) of Chapter 4 for a reference. The net of characteristics for every realization k(ω), according to meshes based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary points, was determined from equations (8.72 and 8.73) by finite differencing. We recall here that an analogous scatter of Hencky–Prandtl nets of characteristics, also due to the resolution by finite mesoscales, was observed in plasticity problems in Section 8.4 above. Given our interest there in direct (not inverse) Cauchy problems, we dealt accordingly with forward (rather than backward) dependence wedges. In Figure 8.12(d), where truss spacings on the foundation arc F are about 10% of the radius, we take the noise k to be a zero-mean uniform random variable having standard deviation 0.015; the mean of k is k = 1.0. All noises k away from F in Figure 8.12(d), as well as in Figures 8.12(a–c), are scaled according to (8.67)2 , that is, in inverse proportion to the local truss spacing L. For the small noise-to-signal ratio adopted here, taking Gaussian as opposed to uniform noise, while keeping the same variance, made hardly any difference on the final results. The efficiencies Eff of four random truss systems are given in Figure 8.13— they are defined by generalizing the concept used for the deterministic truss Eff = V/ V ∗ . (8.74) Here V(ω) is the volume of T(ω) from T , and V ∗ = V(n) (n finite) is the ensemble average volume. We observe at once that values for Eff are lower than those obtained under the deterministic, homogeneous medium assumption involved in Section 4.4 of Chapter 4. Most importantly, however, a finer random truss than that shown in Figure 8.12(d) is not possible (!) because the noise is too strong for the net of characteristics to continue in a stable manner up to the point A—the characteristics just tend to intersect prior to A. This suggests that, should a larger number than 25 + 1 boundary nodes on F be desired, a different truss topology—one of a disordered type—rather than that of an orthogonal net might be needed to solve the problem. This shows that the random microstructure prevents attainment of the classically predicted optimal shape using the homogeneous continuum. Additionally, the efficien-

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305 P

A

A

F

F

(b) Eﬀ = 0.935

(a) Eﬀ = 0.93 P

P

A

F (c) Eﬀ = 0.942

A

F (d) Eﬀ = 0.95

FIGURE 8.12 Shown in black are successive sets of trusses, all governed by (8.54), according to meshes based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary points on the rigid foundation F . Shown in white are the trusses of Bdet , the same as those of Figure 4.17(a–d) in Chapter 4.

cies in Figure 8.12 are lower than those of the corresponding Figure 4.17 in Chapter 4, although that effect might be different—perhaps even opposite— depending on the choice of scaling law such as the one assumed in (8.67). 8.6.3 Solution via the Elliptic System The elliptic system approach has already been outlined for a deterministic material in Section 4.4.2 of Chapter 4. Recall that the shape optimization problem is set up in the space of Ua d of admissible tensor fields of C(x). Any such field is now taken as a particular realization C(ω, x) of the random body {B(ω); ω ∈ }, and we look at optimization in the ensemble sense. Since even

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A F

(a)

(b) FIGURE 8.13 (a) Michell truss problem: design space and boundary conditions. (b) Random distribution of the Young modulus on the 27 × 27 mesh.

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307

in the deterministic problem a computational solution needs to be employed, we can only think of employing a Monte Carlo method in the stochastic case at hand. The formulation outlined above is also applied to the classical Michell problem of finding an optimal structure within the unit square domain, supported by a circular foundation boundary F placed within and subjected to a loading condition (force down) at A, Figure 8.13(a). This domain is partitioned into a 27 × 27 mesh of square-shaped finite elements, each one with a linear elastic and isotropic stiffness tensor C(ω, x), but of random magnitude; the Poisson ratio was ν = 0.3. That is, the stiffness tensor is taken as a superposition of a mean C(ω, x) and of a zero-mean random fluctuating part

C (ω, x):

C(ω, x) = C + C (ω, x)

C (ω, x) = 0,

(8.75)

Here the overbar indicates the volume average mean. Tensor C (ω, x) is generated, without any dependence on x (i.e., no spatial “memory”), by multiplying each element’s mean part C by a random number r sampled from a uniform distribution [−1, 1] and multiplied by a constant scale factor s

C (ω) = rs C

(8.76)

As an example, for s = 0.3, the Young modulus is varied from 0.7 to 1.3; its sample realization is shown in Figure 8.13(b). The scale factor was introduced to account approximately for smoothing and finite-size scaling of a heterogeneous material microstructure with variability on a smaller scale. This device, of course, is an oversimplification for we already know that the mesoscale stiffness of a heterogeneous material should be derived from one of three boundary value problems. Figure 8.14 shows the results of the optimization problem at radius of foundation equal to 0.1 of the square domain size. In (a) we display the homogeneous medium case (a), and then consecutive realizations of the random field with increasing scale factor. In particular, figures of the sequence (b) through (h) correspond, respectively, to the multipliers s = 0.1, 0.15, 0.2, 0.25, 0.3, 0.6, 0.9. In all these figures immediately evident is the breakdown of global symmetry of the truss structure due to the spatial nonuniformity of material—a feature that could not be attained via the hyperbolic system. On the other hand, we note that the CPU time involved in generation of any single truss realization of Figure 8.12 is just a few seconds, as opposed to about four hours on an identical computer for, say, Figure 8.14(b). Our other principal conclusions (Liszka and Ostoja-Starzewski, 2003) are: •

Our method assumes that the actual stochastic realization is known a priori during the optimization process. This does not correspond to a typical engineering process, but rather represents an “organic” type optimization, like the growth of veins in leaves on a tree.

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(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

FIGURE 8.14 Topology optimization method applied to the mesh of square-shaped finite elements of Fig. 8.13(b) with uniform properties (a) and then increasing random properties: from (b) through (h).

•

In general, introduction of stochastic properties reduces the quality of the results, but in some cases stochastic material properties may improve the compliance of optimal design because the optimization is performed on a given random distribution, so that the design process has an opportunity to choose “stiffer” cells and discard those with weaker material.

•

Our study is a preparation to the robust design itself. In particular, we propose and plan this course of action: In the course of an optimization procedure, the choice of a goal function in terms of local variabilities will represent an intermediate step.

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As the particular function form is decisive with respect to the choice of an optimal shape, the sensitivities need to be modified so as to account for a characteristic of spatial material randomness and a statistical goal function. Save and Prager (1985) said: “Although a truss-like continuum is not practical, it uses the smallest possible volume V of structural material for the considered behavioral constraint and thus furnishes a useful basis for computing the efficiencies of practical structures.” Similarly, the present study uses the Michell truss-like continuum to provide a better guidance on optimal structures in the presence of practically unavoidable microscale material randomness. One can argue that Michell-type truss-like continua with infinitesimal spacing are rather theoretical concepts in conventional engineering structures, and relatively few members can already ensure a high efficiency as indicated by the trend of Eff in Figure 8.14. This, however, is not necessarily the case with (very) small-scale systems, such as encountered in nanotechnology and biostructures. It is here that the practically unavoidable microscale noise may significantly alter predictions of conventional continuum mechanics in that it may prevent the realizability of optimal, deterministic truss-like continua.

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9 Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures

Law 0: “Your friend is my friend” Law I: “You cannot win” Law II: “You cannot break even” Law III: “You cannot even get out of the game” It should be clear from the derivation of mesoscale bounds for linear elastic materials that three properties are required for their derivation: (1) the statistical homogeneity and ergodicity; (2) the Hill condition leading to admissible boundary conditions; and (3) an extremum (or variational) principle. It follows that the mesoscale bounds can be generalized to various other types of material behavior. More specifically, in this chapter we give hierarchies of mesoscale bounds for nonlinear elasticity, elastoplasticity, rigidplasticity, elastic-brittle damage, viscoelasticity, flow in porous media and thermoelasticity.

9.1

Physically Nonlinear Elastic Microstructures

9.1.1 General Consider physically nonlinear elastic materials in the range of infinitesimal strains, described by the constitutive law σ = σ (ε) =

∂w(ε) ∂ε

ε = ε(σ ) =

∂w ∗ (σ ) ∂σ

,

(9.1)

where the energy densities are related by w ∗ = σ : ε − w; w is a statistically homogeneous and ergodic field. As in Chapter 7, we assume the absence of body and inertia forces. The Hill condition, and its implication for the type of admissible boundary conditions, is σ : dε = σ : dε ⇐⇒

∂ Bδ

(t − σ · n) · (du − dε · x)dS = 0,

(9.2) 311

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where σ = σ 0 and ε = ε 0 . This results in the same three types of uniform boundary conditions on the mesoscale as (7.21–7.23) in the linear elastic case of Chapter 7. For a given realization Bδ (ω) of the random medium Bδ on some mesoscale δ, the uniform kinematic boundary condition yields an apparent constitutive law σ = σ (ε 0 ).

(9.3)

Similarly, the uniform traction condition results in an apparent constitutive law σ = σ (ε) =

∂wδd (ε 0 ) ∂ε 0

ε = ε(σ 0 ) =

∂wδt∗ (σ 0 ) ∂σ 0

.

(9.4)

Next, we have the minimum potential energy principle 1 (ε) ≤ 2

w( ε)dV − Bδ

∂ Bδt

t · u dS,

(9.5)

and the minimum complementary energy principle 1 (σ ) ≤ 2

w( σ )dV − Bδ

∂ Bδu

t · u dS.

(9.6)

Here tildes indicate admissible fields. From this, using the same partition concept as in Chapter 7, the apparent constitutive responses are shown to be related by wδd (ε0 , ω) ≤ wδd (ε 0 , ω)

δ = ∀δ/2,

(9.7)

wδt∗ (ε 0 , ω) ≤ wδt∗ (ε0 , ω)

δ = ∀δ/2.

(9.8)

and

By passing to the ensemble, we get a hierarchy of bounds (Jiang et al., 2001b) from above d w∞ ≤ . . . ≤ wδd ≤ wδd ≤ . . . ≤ w1d

∀δ = δ/2,

(9.9)

and from below t∗ w∞ ≤ . . . ≤ wδt∗ ≤ wδt∗ ≤ . . . ≤ w1t∗

∀δ = δ/2.

(9.10)

Note: wδt∗ = σ : ε 0 − wδd because the mesoscale response depends on the type of loading.

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 313 9.1.2 Power-Law Materials Consider a random medium of the specific constitutive law (Kroner, ¨ 1994) σ = σ (ε) =

N

Cn : ε n =

∂w(ε)

n=1

∂ε

ε = ε(σ ) =

N

Sn : σ n =

∂w ∗ (σ )

n=1

∂σ

,

(9.11)

While the above are in mechanical form, the energetic laws are w(ε) =

N

Cn :

n=1

εn n+1

w ∗ (σ ) =

N

σn . n+1

Sn :

n=1

(9.12)

This leads to apparent responses under uniform displacement and traction boundary conditions σ =

N

Cdnδ (ω) : (ε 0 ) n =

∂wδd (ε 0 ) ∂ε 0

n=1

ε=

N

Stnδ (ω) : (σ 0 ) n =

∂wδt∗ (σ 0 )

n=1

∂σ 0

.

(9.13)

σn . n+1

(9.14)

and, respectively, the energetic forms w(ε) =

N n=1

Cn :

εn n+1

w ∗ (σ ) =

N

Sn :

n=1

The apparent constitutive laws for Bδ (ω) are next shown to be related by a partition theorem (Hazanov, 1998, 1999) (Stnδ (ω)) −1 ≤ Cnδ (ω) ≤ Cdnδ (ω), eff

(9.15)

and, in view of the statistical homogeneity and ergodicity of the material, we have hierarchies of bounds from above Cdn∞ ≤ . . . ≤ Cdnδ ≤ Cdnδ ≤ . . . ≤ Cdn1 (9.16) ∀δ = δ/2, and from below Stn∞ ≤ . . . ≤ Stnδ ≤ Stnδ ≤ . . . ≤ Stn1

∀δ = δ/2.

(9.17)

It is known that, under proportional monotonic loading, strain-hardening elastoplastic composites may be treated in the framework of deformation theory of plasticity, which is formally equivalent to physically nonlinear, smalldeformation elasticity, such as dealt with here. In (Jiang et al., 2001b) we have assumed this equivalence to also hold for apparent elastoplastic response, and have thus obtained energy bounds on random elastoplastic composites.

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In that reference we have also proposed an approach via tangent moduli. Finally, the effect of imperfect interfaces on the hierarchies of bounds has been investigated by Hazanov (1999). Hashin–Shtrikman-type and more advanced bounds, without scaling laws, for the effective energy function of more general type than power-law were investigated by various authors, for example, Gibiansky and Torquato (1998), Ponte Castaneda ˇ and Suquet (1998), Talbot (2000), and Torquato (2002). 9.1.3 Random Formation vis-a-vis ` Inelastic Response of Paper The linear elastic response of paper from the standpoint of its random fiber network structure was discussed in Chapter 4. Going to higher strains in tensile loading—be it uniaxial or biaxial in-plane loading—paper becomes inelastic without any clear elastic limit, or a yield point. (Compressive loading has to be treated separately because of an obvious tendency to buckle.) All the experiments show that paper’s inelasticity (i.e., plastic deformation) occurs beyond the linear range as a plastic-hardening response, and the end of this range is commonly taken as the yield point. Since here we are interested in a paper’s response in that regime in loading only, we generalize the continuum model given in Chapter 4 to a physically nonlinear elastic model (Castro and Ostoja-Starzewski, 2003):   tanh (bε11 )      Q11 Q12 0  σ11   tanh b ν21 ε 1     22  (9.18) ν12 ,  σ22  =  Q12 Q22 0     b   σ12 0 0 G tanh (bε11 ) γ12 ε11 where, evaluating at zero strain (ε = 0), we have Q11 =

(1, 0) (0) E1 dσ11 , = dε11 (1 − ν12 ν21 )

Q12 =

(0, 1) (0) E 1 ν21 dσ11 = Q21 , = dε22 (1 − ν12 ν21 )

Q22 =

(0, 1) (0) E2 dσ22 . = dε22 (1 − ν12 ν21 )

(9.19)

Two experiments on a typical specimen and their fitting by the above formulas are shown in Figure 9.1. Thus, MD and CD indicate the resulting responses under applied strains (ε11 , ε22 ), equaling either (1, 0) or (0, 1). Clearly, for very small strains (below ≈ 0.1%), we effectively have a linear elastic model. Note that the MD(1,0) curve goes up to 1.5% strain, and then drops due to yield, damage and failure occurring at smaller strains than those for the CD(0,1). The thing to note is that, for the initial slopes, we have: Q12 = Q21 .

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Q11 tanh (b e)

40 Stress (MP a)

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MD (1,0)

35 30

CD (0,1)

25 20 15

Q11

10 5 0 0.0%

MD (0,1)

Q22

CD (1,0)

Q12 0.5%

1.0%

1.5%

2.0%

2.5%

Strain e FIGURE 9.1 Stress-strain curves obtained from testing 145-mm ×145-mm specimens. Loading cases (1, 0) 0 0 , ε22 applied on the biaxial tester. and (0, 1) indicate the strains ε11

Curves such as those shown in Figure 9.1 actually display scatter from specimen to specimen, and so, it is of interest to address the effect of paper’s formation on scaling of its mechanical properties. To model formation we employ the Boolean model of Chapter 1, wherein d represents the paper’s fiber floc size, which varies typically between 3 and 6 mm depending on grade and furnish (i.e., a particular fiber suspension going into the paper manufacture). The paper sheet is now discretized by a very fine mesh of rectangular finite elements with material properties of equation (9.19) assigned to each element proportionally to its basis weight (weight per unit area). Arbitrary in-plane, biaxial stress-strain tests on such nonhomogeneous material samples can now be run with the finite element model. In the following, we apply uniform displacement and traction boundary conditions to realizations of the Boolean model having isotropic statistics and flocs of size d = 5 mm, introduced in Figure 1.12 of Chapter 1. Thus, we work with two square-shaped mesoscale domains 20 mm × 20 mm and 50 mm × 50 mm at δ = 4 and δ = 10, respectively. We resolve the Boolean microstructure by a fine mesh finite element model, where each finite element is 1 mm × 1 mm, which means that each B4 (ω) and B10 (ω)—sampled, respectively, from the ensembles B4 = {B4 (ω); ω ∈ } and B10 = {B10 (ω); ω ∈ }—is modeled through meshes of 20 × 20 and 50 × 50 elements, respectively. Figure 9.2 shows mesh deformations of B10 (ω) from the previous figure under uniform displacement and traction boundary conditions (7.21) and (7.22) in pure shear loading. In both cases the nonuniform deformations of the mesh are clearly visible, whereby loading by the uniform displacement condition results in a more uniform deformation than that under the uniform traction. Of course, should the material be perfectly homogeneous, all the mesh lines would be parallel and would deform affinely under both kinematic and traction boundary conditions.

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δ = 10

Traction b. c.

Number of Layers Displacement b. c. FIGURE 9.2 Material’s response under pure shear loading (uniform displacement or traction boundary conditions) at mesoscale δ = 10.

This line of thought is pursued in Figure 9.3, which shows a comparison of shear stress fields under both types of boundary conditions, and for both 0 mesoscales considered. The loading is in MD: uniaxial extension (ε11 = 0.5%), 0 and uniaxial tension (σ11 = 0.5%). Due to the material inhomogeneity, there is a non-zero σ12 field, and the difference between both stress fields is stronger at δ = 4 than at 10. Indeed, we note a clear tendency at δ = 10 to form orthogonal shear bands running at angles ±45o to the horizontal. This suggests also that the patterns of strain become very alike at this mesoscale, which occurs well below the δ → ∞ limit of the “perfect” RVE. With numerical simulations carried out in the Monte Carlo mode involving ten (at δ = 10) and twenty (at δ = 4) realizations, we obtain ensemble averaged shear stress-strain curves of Figure 9.4. They demonstrate scaling trends of potential and complementary energies, as well as those on tangent moduli for this model paper material. On that basis we establish that the RVE in the sense of Hill is approximately reached on scales about ten times larger than the floc size.

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δ′ = 4

1.00 + 00 3.00E + 00 5.00E + 00 7.00E + 00 9.00E + 00 1.10E + 01 1.30E + 01

δ = 10

1.00 + 00 3.00E + 00 5.00E + 00 7.00E + 00 9.00E + 00 1.10E + 01 1.30E + 01

Displacement b.c.

Traction b.c.

FIGURE 9.3 Comparison of shear stress fields according to applied uniform boundary conditions (either displacement or traction) and the mesoscale δ, computed for two formation fields of Figure 1.12 of Chapter 1.

100,000 δ′ = 4 displ.

90,000 Average shear stress (Pa)

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80,000

δ = 10 displ.

70,000

δ = 10 traction

60,000

δ′ = 4 traction

50,000 40,000 30,000 20,000 10,000 0 0

0.002

0.004 0.006 Average shear strain

0.008

0.01

FIGURE 9.4 Stress-strain curves of shear response resulting from uniform displacement (respectively, traction) boundary conditions on mesoscales δ = 4 and 10.

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The inelastic response of paper under both monotonic and cyclic uniaxial tension has also been studied via an elastic-plastic fiber network model, effectively a generalization of a linear elastic model reported in Chapter 7 (Bronkhorst, 2003).

9.2

Finite Elasticity of Random Composites

9.2.1 Averaging Theorems The key assumption of the finite hyperelasticity theory is the existence of a strain energy function ψ per unit volume of an undeformed body, which depends on the deformation of the object and its material properties. Here we restrict ourselves to the reference configuration, so that the equation of state of the material takes the form: Pij =

∂ψ ∂ Fij

,

(9.20)

where Pij is the first Piola–Kirchhoff stress tensor and Fij is the deformation gradient tensor. With reference to Hill (1972), Nemat-Nasser (1999), Lohnert ¨ and Wriggers (2003), and Costanzo et al. (2004, 2005), the average strain and stress theorems of the infinitesimal strain case (Chapter 7) generalize to 1 Fij ≡ Fij (X)dV = Fij0 , (9.21) V0 Bδ 1 Pij ≡ V0

Bδ

Pij (X)dV = Pij0 .

(9.22)

Indeed, outof several possible pairs for finite motions (Macvean, 1968), the pair Pij , Fij is dictated by the Hill condition for finite motions Pij Fij = Pij Fij .

(9.23)

Thus, we now have three types of boundary conditions: 1. Uniform kinematic

ui (X) = Fij0 − δij x j

∀X ∈ ∂ Bδ ,

(9.24)

where Fij0 is prescribed. 2. Uniform traction ti (X) = Pij0 n j where Pij0 is prescribed.

∀X ∈ ∂ Bδ ,

(9.25)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 319 3. Uniform orthogonal-mixed ti (X) − Pij0 n j ui (X) − Fij0 − δij x j

∀X ∈ ∂ Bδ ,

(9.26)

where the averaging theorems (9.21–9.22) have been used. In the sequel, following Khisaeva (2006) and Khisaeva and Ostoja-Starzewski (2006a,b), we use these conditions in setting up mesoscale bounds for random microstructures in finite elasticity. 9.2.2 Variational Principles and Mesoscale Bounds Consider the functional

( u) =

ψ( ui,k )dV − Bδ

∂

Bδt

ui dS, ti0

(9.27)

is an admissible displacement field such that u = u on the portion where u u of the boundary ∂ Bδ where displacement is prescribed, and ti0 is the specified boundary traction on the remaining part of ∂ Bδ . This is the finite elasticity counterpart of the principle of minimum potential energy, in that the functional ( u) assumes a local minimum for the actual solution u if ∂2 ψ di,k d p,q dV > 0 (9.28) Bδ ∂ui,k ∂u p,q for all non-zero di such that di = 0 on ∂ Bδu (Lee and Shield, 1980). Under the uniform displacement boundary condition (9.24) the functional (9.27) reduces to ( u) = ψ( ui,k )dV. (9.29) Bδ

At this point we again consider an alternative loading of the domain Bδ (ω) ∈ B by restricted boundary conditions—in the same sense as (7.50) vis-`a-vis (7.49) of Chapter 7. Thus, if the inequality (9.28) holds, the energy stored in the body under the restricted boundary condition ( r (F0 , ω)) is related to that under the unrestricted one ( (F0 , ω)) as

(F0 , ω) ≤ r (F0 , ω).

(9.30)

Upon passing to the ensemble, we get a hierarchy of bounds on the energy density of the RVE ( ∞ (F0 )) from above

∞ (F0 ) ≤ . . . ≤ δ (F0 ) ≤ δ (F0 ) ≤ . . . ≤ 1 (F0 ) ∀δ = δ/2. (9.31) We now turn to the derivation of a reciprocal expression for the lower bounds. It is well known that, in nonlinear elasticity, the strain-energy function can be nonconvex and therefore noninvertible, that is, strain cannot be

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expressed in terms of stress. As a result, the principle of minimum complementary energy cannot be used without certain restrictions placed on the strain energy function. In order to avoid these difficulties, the following functional was proposed by Lee and Shield (1980): Q( u) =

∂ψ ∂ uik

Bδ

uik − ψ dV −

∂

∂ψ Bδt

∂ uik

ui dS, nk

(9.32)

where uik is a “trial” function, which satisfies the following conditions: ∂

∂ xk

∂ψ

= 0 in Bδ ,

ui,k ∂

∂ψ

ui,k ∂

nk = ti0 on ∂ Bδt

(9.33)

It was shown that the functional Q is stationary for uik = uik , where ui , the actual solution of a problem, assumes a local minimum if

∂2 ψ Bδ

∂ui,k ∂u p,q

dik dpq dV > 0

(9.34)

for all non-zero dik satisfying the following conditions: ∂ ∂ xk

∂2 ψ ∂ui,k ∂u p,q

dpq

= 0 in Bδ ,

∂2 ψ ∂ui,k ∂u p,q

dpq nk = 0 on ∂ Bδt .

(9.35)

Now, under the uniform traction boundary condition (9.25), the functional Q reduces to Q( u) = Bδ

∂ψ

uik ∂

uik − ψ dV.

(9.36)

Thus, the following inequality between responses under restricted and unrestricted traction boundary conditions holds:

∗ (P0 , ω) ≤ ∗r (P0 , ω).

(9.37)

where ∗ (P0 , ω) = Bδ (uik ∂ψ/∂uik − ψ) dV. From this, upon ensemble averaging, we can derive a scale-dependent hierarchy of lower bounds on the effective property Q∞ (P0 ) ∗

∞ (P0 ) ≤ . . . ≤ δ∗ (P0 ) ≤ δ∗ (P0 ) ≤ . . . ≤ 1∗ (P0 )

∀δ = δ/2. (9.38)

The bounds (9.31) and (9.38) are illustrated on the example of a planar matrix-inclusion composite in Figure 9.5, and the corresponding stress-strain

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FIGURE 9.5 The finite element mesh of a composite in (a) undeformed and (b) deformed (traction boundary conditions) configurations.

curves are shown in Figure 9.6. In particular, we model the matrix by a strain energy function of Ogden form:

(m) =

N 2µi i=1

αi2

N 2i 1 el α α α J −1 λ1 i + λ2 i + λ3 i − 3 + Di i=1

(9.39)

with µ1 = 4.095 · 105 Nm α1 = 1.3 µ2 = 0.03 · 105 Nm α2 = 5.0 D1 = 4.733 · 10−8 Nm, µ3 = 0.01 · 105 Nm α3 = −2.0 and the inclusions by a strain energy function of a neo-Hookean form

(i) = C10 I 1 − 3 +

2 1 el J −1 , D1

(9.40)

with C10 = 2.062 · 106 Nm D1 = 4.733 · 10−8 Nm. The mismatch between both phases is set at µ(i) 0 µ(m) 0

= 10

with µ(m) 0 =

N

µi

µ(i) 0 = 2C 10 .

(9.41)

i=1

It is clear that the approach to RVE is rapid, and it is attained at δ = 16 with very good accuracy. For other studies of homogenization in finite elasticity see e.g., Gloria (2006a) and He et al. (2006).

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5.E + 06 5.E + 06 4.E + 06 4.E + 06 3.E + 06 Hard phase Soft phase KUBC δ = 16 KUBC δ = 8 KUBC δ = 4 KUBC δ = 2 KUBC δ = 1 SUBC δ = 16 SUBC δ = 8 SUBC δ = 4 SUBC δ = 2 SUBC δ = 1 MIXED δ = 16 MIXED δ = 8 MIXED δ = 4 MIXED δ = 2 MIXED δ = 1 Voigt bound Reuss bound

3.E + 06 2.E + 06 2.E + 06 1.E + 06 6.E + 05 0.E + 00 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

FIGURE 9.6 Stress-strain curves of a planar matrix-inclusion composite made of neo-Hookean inclusions (m) in an Ogden matrix with µ(i) = 10 under traction and kinematic boundary conditions at 0 /µ0 δ = 1, . . ., 16; also shown are the individual responses of both phases.

9.3

Elastic-Plastic Microstructures

9.3.1 Variational Principles and Mesoscale Bounds Let us consider a multiphase elastic-plastic-hardening material with perfect bonding between the phases p = 1, . . ., ptot . Each realization B (ω) of B is described by an associated flow rule (e.g., Hill, 1950)

dεij = dεij = dε =

dσij 2G p dσij 2G p

+λ

∂f ∂σij

d fp

whenever

1 − 2ν p dσ 2G p (1 + ν p )

whenever

fp = cp

and d f ≥ 0

fp < cp everywhere

(9.42) (dε = dεii /3

dσ = dσii /3).

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 323 Here G p (shear modulus), ν p (Poisson’s ratio), and c p (yield limit) form a vector, each component of which (described by its indicator function χ p ) gives rise to a scalar random field, such as G (ω, x) =

ptot

G p χ p (ω, x)

∀ω ∈ .

(9.43)

p=1

The entire body B = {B (ω) ; ω ∈ } is described by a random vector field = {G, ν, c}. On the microscale we have tangent moduli, CδT (ω) or SδT (ω), of the body Bδ (ω), which connect stress increments with strain increments applied to it dσ = CδT (ω) : dε

dε = SδT (ω) : dσ.

(9.44)

Consequently, the Hill condition, and its implication for the type of admissible boundary conditions, is dσ :dε = dσ : dε ⇐⇒ (dt − dσ · n) · (du − dε · x)d S = 0, (9.45) ∂ Bδ

where dσ = dσ 0 and dε = dε 0 . Thus, we have du(x) = d ε¯ · x

∀x ∂ Bδ

(9.46)

dt(x) = d σ¯ · n

∀x ∂ Bδ

(9.47)

and

Next, we recall two extremum principles (e.g., Hill, 1950): one for kinematically admissible fields 1 1 dS − dt · d u d σ : d εdV ≤ dt · dudS − dσ : dεdV, 2 Bδ 2 Bδ ∂ Bδt ∂ Bδt (9.48) and another for statically admissible fields 1 1 dσ : dεdV − dt · dud S ≤ d σ : d εdV − dt · dudS. 2 Bδ 2 Bδ ∂ Bδu ∂ Bδu (9.49) From these, noting that ∂ Bδt = ∅ under kinematic boundary conditions and ∂ Bδu = ∅ under traction boundary conditions, we find Td CTd δ (ω) ≤ Cδ (ω),

Tt STt δ (ω) ≤ Sδ (ω),

∀δ = δ/2.

(9.50)

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Upon ensemble averaging, and applying this to ever-larger windows ad infinitum we get a hierarchy of bounds on the macroscopically effective T T −1 tangent modulus C∞ = (S∞ )

STt 1

−1 Tt −1 T ≤ . . . ≤ STt ≤ Sδ ≤ . . . ≤ C∞ δ Td Td ≤ C ≤ . . . ≤ C ≤ . . . ≤ CTd 1 δ δ

−1

∀δ = δ/2,

(9.51)

−1 where STt and CTd 1 1 are recognized as the Sachs (1928) and Taylor (1938) bounds, respectively. Also, see Suquet (1997) and Ponte Castaneda ˜ and Suquet (1998) for comprehensive reviews of effective (RVE level) properties of nonlinear composites. For elastic-perfectly plastic materials, similar results have been obtained by He (2001) using a mathematically more advanced analysis involving gauge functions.

9.3.2 Matrix-Inclusion Composites 9.3.2.1 Case 1 Jiang et al. (2001) studied a composite with an elastic-hardening plastic matrix phase and linear elastic inclusions. The latter were of the same modulus as that of the matrix in the elastic range. The study confirms the bounding character of responses computed under (9.46) and (9.47), respectively. In particular, patterns of plastic shear bands under these two boundary conditions are displayed in Figure 9.7, and the corresponding hierarchy of bounds, computed here on two scales, is shown in Figure 9.8. Also note: 1. At a smaller scale (δ = 6), the response under uniform kinematic (9.46) is much more uniform than that under uniform stress boundary conditions (9.47). 2. At a larger scale (δ = 20), the discrepancy due to different types of loading is much smaller, which fact shows a tendency to homogenize with mesoscale tending to ∞. 3. The macroscopic response is quite well approximated by the uniform stress (but not the uniform strain) assumption. This situation would be reversed for an elastic-plastic matrix with soft, rather than hard, inclusions, and helps explain why Taylor bound works well for polycrystals with dislocations. An application of this type of approach to elastoviscoplastic composites of Perzyna’s type has been explored by van der Sluis et al. (2000). 9.3.2.2 Case 2 Li and Ostoja-Starzewski (2006) considered an elastoplastic, matrix-inclusion composite, whose stress-strain response is characterized by a power law

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PEEQ VALUE +0.00E−00 +3.89E−02 +7.78E−02 +1.17E−01 +1.56E−01 +1.94E−01 +2.33E−01 +2.72E−01 +3.11E−01 +3.50E−01 +3.89E−01 +4.28E−01 +4.67E−01 +5.06E−01

PEEQ VALUE +0.00E−00 +1.17E−02 +2.35E−02 +3.52E−02 +4.69E−02 +5.87E−02 +7.04E−02 +8.21E−02 +9.39E−02 +1.06E−01 +1.17E−01 +1.29E−01 +1.41E−01 +1.53E−01

PEEQ VALUE +0.00E−00 +5.18E−02 +1.04E−01 +1.55E−01 +2.07E−01 +2.59E−01 +3.11E−01 +3.63E−01 +4.14E−01 +4.66E−01 +5.18E−01 +5.70E−01 +6.22E−01 +6.73E−01

PEEQ VALUE +0.00E−00 +5.22E−02 +1.04E−01 +1.57E−01 +2.09E−01 +2.61E−01 +3.13E−01 +3.65E−01 +4.17E−01 +4.70E−01 +5.22E−01 +5.74E−01 +6.26E−01 +6.78E−01

FIGURE 9.7 Two samples (realizations Bδ (ω)) of a random matrix-inclusion composite at δ = 6 (left column, top row) and δ = 20 (right column, top row); corresponding von Mises strain patterns shown under traction (middle row) and displacement boundary conditions (bottom row). (After Jiang et al. (2001b.)

(Dowling, 1993)

 ε  if ε ≤ ε0 ,    ε0 σ = N  σ0 ε   else.  ε0

(9.52)

The material parameters are given in Table 9.1. The Huber–von Mises yield criterion, with an associated flow rule, is assumed for each phase. Shear

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f

0.8 (Average stress)/σ0

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f

T

f

0.4 f m T S

0.2

S

m

m S

0.6

m

kinematic b.c., δ = 6 traction b.c., δ = 6 kinematic b.c., δ = 20 traction b.c., δ = 20 ﬁber matrix Taylor bound Sachs bound mixed b.c., δ = 6 mixed b.c., δ = 20

0 0

3 2 (Average strain)/ε0

1

4

5

FIGURE 9.8 Stress-strain curves of a random matrix-inclusion composite, shown in Figure 9.12, with a linear elastic inclusion (i) phase and a power-law hardening matrix (m) phase: ensemble averages of eight sample windows at mesoscale δ = 6, and four sample windows at mesoscale δ = 20, under displacement, mixed and traction boundary conditions. (After Jiang et al. (2001b.)

loading is applied through either a uniform kinematic or traction boundary condition with ε0 = 0.04 or σ0 = 1.7 × 102 MPa. Due to the heterogeneity of the material, stress distribution is nonuniform under uniform loading, which results in the local stress reaching the yield stress level somewhere in the material domain, even when the volume average stress is (much) lower than the yield stress of material. Thus, it would not be reasonable to define the yield condition of a sample as the stress level when the first yield occurs in the specimen. We therefore adopt the model of Dvorak and Bahei-Ei-Din (1987), who considered the overall yield of a sample to indicate magnitudes of the overall stress σ , which causes the local volume average stress to satisfy the yield condition in any phase. Thus, for Huber– von Mises materials, the mesoscale yield condition of the composite in the space of overall stresses is defined as Fδ () = inf{ ∈ R3×3 |∃σ (x) , σ = , f p (K p σ ) = c p , ∀x ∈ Bδ , p = 1.2}, (9.53) where K p is the mechanical stress concentration factor in the form of a 6 × 6 matrix, with σ treated as a (6 × 1) vector. The computational application of TABLE 9.1

Material Properties of the Matrix-Inclusion Composite Material Properties Soft phase Hard phase

ε0

σ0 [MPa]

N

0.001036 0.001425

75 295

0.25 0.15

E [GPa] 72.4 207

ν 0.33 0.32

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 327 (9.53) involves an equality between the local stress of the soft phase and the yield stress of that phase within a 3.5% error. Patterns of the von Mises plastic strain, as well as those of the ensemble average stress-strain responses, are similar to those of Case 1. Of particular interest are the yield surface and flow rule on mesoscales. At this point, let us introduce two different loading programs: displacement increment control and traction increment control. In the traction increment control T = · n,

(9.54)

where T is the increment of traction, and is the increment of volume average stress. In the displacement increment control U = E · x,

(9.55)

where U is the increment of displacement, E is the increment of volume average strain. For a specific load (E 11 /E 22 = const or 11 /22 = const) in both loading programs, we fix the ratio of increment (E 11 /E 22 = const or 11 /22 = const) and continue to increase it until the volume average stress satisfies equation (9.53); we then get one specific yield point. The mesoscale responses are collected in Figure 9.9, while the loading paths are shown in Figure 9.10(a). Actually, for each sample we apply 17 different loading paths corresponding to 17 different ratios of loading so as to obtain 17 yield points, whose ratios (11 /22 ) would vary from −1 to 1 (albeit only approximately so for the displacement control). These 17 points cover quite densely one quarter of the yield surface, which, by symmetry arguments, is representative of the entire surface. The loading paths are always linear for the traction increment control, but not so for the displacement increment control because there are some spots becoming plastic due to the local stress concentrations, even when the volume average stress is still lower than the overall yield stress. (For a random chessboard composite not shown here this nonlinearity is even stronger.) Interestingly, the plastic strain rate is not always normal to the yield surface and the shape of the yield surface is not perfectly elliptical. Figure 9.10(b) shows the ensemble average yield surface on different mesoscales under two different loading controls. Clearly, with the increasing mesoscale δ, the yield surface bounds under displacement and traction boundary conditions become tighter. We find a departure of the mesoscale flow rule from normality— under both uniform kinematic and traction boundary conditions. That departure is the strongest when the in-plane ensemble averaged principal stresses are in the ratio of about 0.3 ∼ 0.9. Given the limitation of available computers, we cannot establish the expected trend to recover normality as the mesoscale domain (i.e., SVE) grows and tends to the macroscale (RVE). Our understanding of normality in plasticity follows the thermomechanicsbased argument of Ziegler (1983), who points out the much more fundamental ˙ role played by the thermodynamic orthogonality in the space of velocities, E, and notes that only when the dissipation function depends on velocities

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3.5

× 108

Ensemble average stress-strain response for matrix-inclusion composite

3

2.5 Von mises stress (Pa)

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2

1.5 Soft phase Hard phase

1

Sachs bound Taylor bound Displacement B.C., δ = 6

0.5

Traction B.C., δ = 6 Displacement B.C., δ = 24

0

Traction B.C., δ = 24

0

0.005

0.01

0.015 0.02 0.025 Average equivalent strain

0.03

0.035

0.04

FIGURE 9.9 Ensemble average stress-strain responses for different mesoscales δ under various boundary conditions for the random matrix-inclusion composite. Also shown are the responses of both constituent phases, as well as the Sachs and Taylor bounds.

alone in its arguments, the normality carries over to the space of dissipative stresses. In the situation when depends also on other quantities, say, stresses or internal variables, the normality gets violated. Now, in the case of a heterogeneous material, is also a function of the particular microstructure of B (ω), which may roughly be represented by an internal variable α. The latter must be chosen so that, in the case of homogeneity, it becomes null and the dependence of on α vanishes. Perhaps, the simplest candidate for α may be the ratio of yield limits of both phases or the volume fraction of inclusions. Another viewpoint on the loss of normality is offered by making a reference to the classical result of nonlinear homogenization where the existence of a plasticity potential at the micro-level implies the existence of a macro-potential from which the effective constitutive equations are derived; the macro-potential is the mean value of the local ones (Suquet, 1997), and so, the normality is preserved by a scale transition. In our study, according to (9.53), the macro-yielding takes place as soon as the local plastic flow begins

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8

6

4

Stress Σ 22(Pa)

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2

0

−2 Traction B.C. Displacement B.C. Matrix Inclusion Plastic strain rate Plastic strain rate

−4

0.0013

2.5013

5.0013 Stress Σ 11(Pa)

7.5013

10.0013 × 107

(a) FIGURE 9.10 (a) Loading paths and yield surfaces for one sample of a matrix-inclusion composite at δ = 6 (After Li and Ostoja-Starzowski, 2006).

for the first time at some point of the heterogeneous material. While such a macro-yield criterion is not very useful in practical applications, in the case of a more realistic (tolerant) yield criterion, the loss of normality would also persist under scale transition. 9.3.3 Geodesic Properties of Shear-Band Patterns It is clear from the preceding section that the shear bands display geometric shapes that conform to the actual spatial distribution of the material

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3.5

× 108

3

2.5

2

1.5 Stress Σ 22(Pa)

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1

0.5

0

−0.5

Displacement B.C., δ = 6 Traction B.C., δ = 6 Displacement B.C., δ = 12

−1

Traction B.C., δ = 12 Matrix

−1.5

Inclusion

0.5

1

1.5

2 Stress Σ 11(Pa)

2.5

3

3.5 × 108

(b) FIGURE 9.10 (b) A hierarchy of ensemble averaged mesoscale responsed (yield loci and flows) at δ = 6 and δ = 12 (After Li and Osteja-Starzewski, 2006).

microstructure. With the latter being spatially irregular, the shear bands are irregular too: they take paths of lowest plastic resistance while avoiding the obstacles of high yield limit. Conceptually, therefore, they may well behave as geodesics—curves of shortest path joining two specific points in space; see also Fermat’s principle, discussed in Chapter 11. In accordance with this

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(a)

(b)

FIGURE 9.11 (a) A matrix-inclusion composite studied in this paper; (b) shear bands (bright points) obtained by computational micromechanics.

hypothesis, the geodesics should join the opposite faces of a specimen Bδ (ω), such as that in Figure 9.11(a), to which shear loading is applied. The basic concepts of geodesics for mechanics of materials, especially in the context of fracture mechanics of random media, were given by Jeulin (1993). Taking the material model of Case 1, Figure 9.11(b) displays patterns of equivalent plastic strain, found by computational micromechanics due 0 0 0 0 to uniform kinematic loading on ∂ B: ε11 = ε22 = 0, ε12 = ε21 .While in a homogeneous material perfectly straight (horizontal and vertical) shear bands would form, in the heterogeneous material of Figure 9.11(a) a distortion of these bands, due to the presence of elastic inclusions, occurs. We postulate that these distorted shear bands follow the shortest paths in the matrix, and proceed to estimate, by geodesic propagations, shortest paths on the twodimensional configuration of the composite of domain Bδ (ω). As potential shear bands, we therefore consider two families of shortest paths obtained by geodesic propagations in two orthogonal directions that avoid (black) inclusions. For a horizontal propagation as source S and destination D we first take the left and right faces, and then invert their roles, obtaining the set G-hor. For a vertical propagation as source S and destination D we first take the top and bottom faces, and then invert their roles, obtaining the set G-ver. We consider propagations obtained on hexagonal lattices, so that we deal with the so-called hexagonal geodesic distances. Final results, following Jeulin and Ostoja-Starzewski (2000), are synthesized as follows: 1. Addition of G-hor with G-ver to obtain the set G-add, Figure 9.12(a). 2. Supremum of G-hor with G-ver to obtain the set G-sup, Figure 9.12(b).

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(a)

(b)

FIGURE 9.12 (a) Shortest paths (in dark) that avoid (black) inclusions obtained by (a) addition or (b) supremum of horizontal with vertical geodesics. Note that the darkest (brightest) points here correspond to the brightest (respectively, darkest) ones in (b) of the previous figure.

The grayscale at any given point in Figures 9.12(a) and (b) indicates the (appropriately normalized) distance it takes to connect the opposite edges of the square-shaped domain Bδ (ω): the darker the point, the shorter is this distance. Given two fields such as {g(i); i = 1, . . ., I } (geodesic) and {e(i); i = 1, . . ., I } (equivalent strain), both defined on a square lattice of the same size (256 × 256), so that I = 2562 , we compare them by using the normalized covariance (or crosscorrelation coefficient) ρge : equation (2.36) of Chapter 2. We compute under the ergodic assumption (2.152)

ρge = !

1 I 1 I I

i=1

I i=1

g 2 (i) −

g(i)e(i) − I 1 I

i=1

I 1 I

g(i)

i=1

g(i)

I 1

2 " ! I 1 I

i=1

I

i=1

e 2 (i) −

e(i) I 1 I

i=1

e(i)

2 "

.

(9.56) Basically, working with a single realization forces us to invoke the ergodic assumption. It turns out that the crosscorrelation of G-sup (Figure 9.12[b]) with the true solution, e, obtained by finite elements (Figure 9.11[b]) is only ∼ 0.2, and a lower number (∼ 0.1) is obtained for the G-add geodesic propagation (Figure 9.12(a)). The advantage of this purely geometric method is an extremely fast computation of the pattern of plastic deformation as opposed to a full computational mechanics approach. This may then offer a very rapid way

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(a)

(b)

FIGURE 9.13 The von Mises strain distribution obtained by a computational mechanics simulation (a) versus the geodesic pattern (b).

of a purely geometric assessment of zones of plastic flow in disordered heterogeneous materials, although much more research needs to be done to assess the appropriate ranges of parameters. One situation where the geodesics may be very useful occurs in northern latitudes, where a highly heterogeneous ice field is covering a large body of water. Given only an image—with the gray scale roughly indicative of the ice thickness, and hence of its strength—an ice breaker wants to find the shortest path, yet one of least resistance, to go from point A to B. This methodology is also applied to the composite material Case 2 and Figure 9.13 shows a typical result with ρge = 0.85. Mapping out the ranges of composite material parameters for which geodesics work as well as here (or even better) and developing a solid theoretical basis is presently an open problem.

9.4

Rigid-Perfectly Plastic Microstructures

9.4.1 Background The random rigid-plastic material B = {B(ω); ω ∈ }, is defined by stating that, for any grain (i.e., on microscale),

dij = λ

∂ F σij , ω ∂σij

,

σij dij = σij λ

∂ F σij , ω ∂σij

≥ 0.

(9.57)

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Thus, in each grain the admissible stress states lies within the set P1 (ω) = {σ |F (σ, ω) ≤ 0} ≡ {σ |σeq ≤ σY }, where σeq is the equivalent stress; σY is the yield stress. Next, if on any mesoscale δ = L/d, we define volume averages σ δ = B1δ σ (x) dV dδ = B1δ d (x) dV ,

(9.58)

(9.59)

then, on the macroscale, Bδ→∞ (ω), in the notation of homogenization theory (Suquet, 1993): = σ δ→∞ (ω)

D = dδ→∞ (ω) .

(9.60)

For the RVE (δ → ∞), the yield (or extremal) surface of the composite B∞ delimits the set P hom = { ∈ R3×3 |∃σ (x) with σ = , divσ = 0, F (σ, ω) ≤ 0, ∀x ∈ B}, (9.61) and the elementary Taylor and Sachs bounds on P hom are expressed via a hierarchy of inclusion relations {|eq ≤ inf σY (x)}Sachs ⊂ P hom ⊂ {|eq ≤ σY }Taylor . x∈Bδ

(9.62)

On any finite mesoscale—a scale below RVE—there holds some form of an apparent yield surface Fδ (σ , ω), bounding some set Pδ (ω) = {σ δ |∃σ (x) , divσ = 0, F (σ, ω) ≤ 0, ∀x ∈ Bδ }.

(9.63)

In contradistinction to P hom , Pδ (ω) depends on the configuration ω and the mesoscale δ, and these are two issues we investigate in the following. 9.4.2 Bounding on Mesoscales via Kinematic and Traction Boundary Conditions Let us start by recalling the upper bound theorem allowing for discontinuities in the velocity field, according to Kachanov (1971), but in a form more suited for our purposes. First, we consider an arbitrary kinematically admissible velocity field v∗ of the body Bδ . If σij∗ is the stress field associated with vi∗ by (9.57)1 and also satisfying (9.57)2 , the basic energy balance equation can now be written as # # ti vi∗ dS = σij dij∗ dV + τY #[v∗ ]# dS, (9.64) ∗ ∂ Bδ

∗

Bδ

S[v ]

∗

M Sm[v ] is the set of internal surfaces of discontinuity in v∗ . where S[v ] = ∪m=1 Recalling the inequality (e.g., Hill, 1950)

σij dij∗ ≤ σij∗ dij∗ ,

(9.65)

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∂ Bδ

ti vi∗ dS

≤ Bδ

σij∗ dij∗ dV

+ τY

# ∗# #[v ]# dS.

(9.66)

S[v]∗

Upon subjecting the body Bδ (ω) on its entire boundary ∂ Bδ = ∂ Bδv to a uniform kinematic boundary condition (dij is a constant and equals the volume average of dij ) vi (x) = dij x j

∀x ∈ ∂ Bδ ,

(9.67)

noting (9.64), we write this upper-bound theorem as ∗ ∗ σij dij dV + |τY | ti vi dS ≤ σij dij dV + |τY | [v] dS = ∂ Bδ

S[v]

Bδ

Bδ

∗ S[v ]

# ∗# #[v ]# dS. (9.68)

This states that the upper bound on the actual external forces at which plastic deformations begin may be found by assuming an arbitrary kinematically admissible deformation mechanism of the body under consideration. Next, as vi∗ let us take a “restricted” uniform kinematic boundary condition vi∗ (x) = dij x j

S ∀x ∈ ∪s=1 ∂ Bδs

s = 1, . . ., S,

(9.69)

which acts on all the boundaries of the partition. In the context of Figure 9.6(a), (9.69) applies to the external square-shaped boundary ∂ Bδ as well as to the internal cross. Because the solution under the condition (9.69) on the partition S ∪s=1 Bδs is a kinematically admissible distribution under the condition (9.67), but not vice versa, from (9.68) we have # ∗# ∗ ∗ #[v ]# dS. σij dij dV + |τY | σij dij dV + |τY | [v] dS ≤ ∗ Bδ

S ∪s=1 Bδ

S[v]

s

S[v ]

(9.70) This says that the actual external forces at which plastic deformations begin in Bδ under (9.67) are bounded from above by forces at which plastic S Bδs under (9.69). If by Pδv and Pδvs we deformations begin in the partition ∪s=1 denote domains in stress space bounded by volume average yield stresses S σ δY and σ δ Y corresponding to Bδ and ∪s=1 Bδs , respectively, we can write Pδv (ω) ⊆ Pδvs (ω).

(9.71)

These two domains are bounded by the surfaces Fδv (σ , ω) and Fδvs (σ , ω), respectively. We can next take ensemble averages of σ δY and σ δ Y so as to $v and define mesoscale yield surfaces Fδv and Fδv , respectively. Thus, if P v v δ v $ denote two domains bounded, respectively, by F and F , we can P δ δ δ write $δv ⊆ P $v P δ

∀δ = δ/2.

(9.72)

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Here we simply write δ in place of δs , because the information about a particular part s of the partition is lost. By induction, we obtain a hierarchy of inclusions v $δv ⊆ P $v ⊆ . . . ⊆ P $v P∞ ⊆ ... ⊆ P δ 1

∀δ = δ/2.

(9.73)

v where P∞ is the yield locus P hom of RVE, for which the statistical average may $v is the domain bounded by the average be dropped. On the other hand, P 1 yield locus of a single grain, or the Taylor (1938) bound of (9.62), which results from prescribing a uniform deformation rate field everywhere. Although an analytical derivation of a lower hierarchy of bounds under uniform traction boundary conditions is not available, our computational results lead us to conjecture that this hierarchy of inclusions holds: t P∞ ⊇ ... ⊇ $ Pδt ⊇ $ Pδt ⊇ . . . ⊇ $ P1t

∀δ = δ/2,

(9.74)

$t denote a domain in stress space bounded by ensemble where $ Pδt and P δ averaged yield surfaces Fδt and Fδt . 9.4.3 Random Chessboard of Huber–von Mises Phases We consider a two-phase (r = 1, 2) material with the microgeometry of a random chessboard in two dimensions (x1 and x2 ), see Figures 9.6 and 9.7. This material is a system of phase 1 (light) and phase 2 (dark) square-shaped grains d × d. More precisely, the microgeometry of phase 1 is taken as a Bernoulli lattice process p,d ( p is a site probability) on a Cartesian lattice of spacing d in R2 Ld = {x = (m1 d, m2 d)} m1 , m2 ∈ N.

(9.75)

Any configuration B(ω) of this piecewise-constant material is described by an indicator function χ (ω, x) taking values 1 and 0 according as x falls in phase r = 1 and outside (i.e., phase 2), respectively. We take grains to be incompressible, rigid-plastic, isotropic of the Huber– von Mises type (HM), that is, having a yield condition F (σij , ω) = σ(2) − kr2 (ω) = 0,

r = 1, 2.

(9.76)

Here σ(2) is the second invariant of the stress deviator σij . The above shows that randomness appears at the level of yield stress kr (ω) of a homogeneous grain; prime denotes a deviator. As is well known, in this case, the flow rule becomes: dij = λσij whenever F (σij , ω) = 0 and d F ≥ 0. Given k1 and k2 of both phases, we also introduce a mismatch (or contrast) α = k2 /k1 . We set α and the nominal volume fraction of light and dark phases to 50%, so that we have 1 ,d . On finite scales (δ < ∞) there are, of course, local 2 fluctuations from one Bδ (ω) to another, and this is reflected in Figures 9.14. More specifically, we consider two domain sizes δ = 10 and 50. The response of any given specimen Bδ (ω), generated according to the Bernoulli lattice

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Hard phase

Soft phase

FIGURE 9.14 Two samples [realizations Bδ (ω)] of a random chessboard composite at δ = 10 (left column, top row) and δ = 50 (right column, top row); corresponding von Mises strain patterns shown under traction (middle row) and kinematic boundary conditions (bottom row). (After Ostoja-Starzewski (2005b.)

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process, and subjected to in-plane loadings, is obtained by a computational mechanics method described below. We study mechanical responses of 50 specimens at mesoscale δ = 10, and 20 specimens at δ = 50. These numbers reflect the fact that the larger the scale, the smaller is the scatter, and hence, the number of realizations required. In plane stress the yield condition (9.76) now simplifies to 2 2 2 Fr (σ ) = σ11 − σ11 σ22 + σ22 + 3σ12 − 3kr2 ,

(9.77)

and the flow rule becomes d11 d22 d12 = = . 2σ11 − σ22 2σ22 − σ11 6σ12

(9.78)

Graphics in Figure 9.14 display plastic responses—in terms of the von Mises strain—of the Huber–von Mises two-phase chessboards at contrast 5, on mesoscales δ = 10 (a) and 50 (b). The loading is again of shear type: in the left column for the kinematic boundary conditions, and in right column for the traction boundary conditions. As before, in each case, bottom pictures show the resulting plastic zones superposed onto the original mesh. The inhomogeneity of strain fields, especially under the kinematic condition, is evident. However, as the scale δ increases from 10 to 50, there is a clear tendency for responses under both types of loading to attain spatial uniformity and similarity. It is interesting to compare the plastic responses of Figure 9.14 for the twophase 10 × 10 chessboard: there is a strong difference between the kinematiccontrolled responses, but not between the traction-controlled, as we switch from the plane deformation rate to the plane stress. This comparison may be continued for the two-phase 50 × 50 chessboard by considering apparent plastic responses (always shown in terms of the von Mises strain). Also note the formation of shear band features under the kinematic condition, which was brought out at 0.01 of the maximum von Mises strain. Figure 9.15 gives an ensemble view of apparent yield surfaces in the (σ1 , σ2 )-plane on mesoscales δ = 10 and 50, whereby, respectively, 50 and 20 realizations were employed. Hard and soft phase yield loci at contrast 5, having the classical HM elliptical shape, are shown for reference. Also drawn is the ensemble average kinematic-controlled yield locus and two most extreme ones for scale δ = 50; their symmetry with respect to the origin of (σ1 , σ2 )-plane was noted. Two Huber–von Mises ellipses are drawn in broken lines for reference so as to display the departure of these apparent (mesoscale) responses from the HM response. This departure is made more visible in Figure 9.16 representing a close-up view of a part of Figure 9.15, focusing on the quarter of apparent yield surfaces ranging from pure shear up to uniaxial (ex)tension, on mesoscales δ = 10 and 50. In general, we are led to a model that could describe the ensemble of yield surfaces Fδ = {Fδ (ω); ω ∈ } ,

(9.79)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 339 6.0 σ2(10^8 Pa) 5.0 4.0 Hard phase

3.0

Average

2.0

Soft phase

1.0

−6.0

−5.0

−4.0

−3.0

−2.0

−1.0

0.0 0.0 −1.0

1.0

2.0

3.0

4.0

5.0

6.0

σ1(10^8 Pa)

−2.0 −3.0 −4.0 −5.0 −6.0 FIGURE 9.15 An ensemble view of apparent yield surfaces under kinematic conditions in plane stress on mesoscales δ = 10 and 50. Hard and soft phase yield loci, as well as an average kinematiccontrolled locus and two most extreme ones on δ = 50, are shown.

where Fδ is of an elliptical (finite thickness) shell shape. The shell converges to an infinitesimal thickness in the δ → ∞ limit, but bifurcates into two distinct HM ellipses in the δ → 1 limit. Figure 9.16 also displays the d vectors, ensemble averaged for each one of four yield surfaces here (at δ = 10 and 50 under both kinematic and traction conditions). Noteworthy is the departure of apparent (mesoscale) flow rule, for each case, from the normality. Observations: Neither in plane deformation rate, nor in plane stress, is the flow rule normal for any given realization of the random medium. • In plane deformation rate, the flow rule is normal upon ensemble averaging, but not so in plane stress. •

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6.0

5.0

4.0

Average 10 × 10 Traction b.c. Average 50 × 50 Traction b.c.

3.0

Average 50 × 50 Kinematic b.c. 2.0 Average 10 × 10 Kinematic b.c. 1.0

0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

−1.0

−2.0 FIGURE 9.16 A close-up from Figure 9.15 of the ensemble view of apparent yield surfaces in plane stress, focusing on the first quarter in the space of principal stresses, on mesoscales δ = 10 and 50. The d vectors, shown for ensemble average yield surfaces at δ = 10 and 50 under kinematic and traction conditions, display the departure from normality. (After Ostoja-Starzewski (2005b.)

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In plane deformation rate the yield surface, for any δ and ω of random medium, consists of two parallel lines. In plane stress, the yield surface is ellipse-shaped, but, for both boundary conditions (kinematic and traction), of a smaller aspect ratio than that of the HM material. This suggests a convenient form of Fδ (σ ) for homogeneous transversely isotropic materials (Mellor, 1982) 2 Fδ (σ ) = (σ1 + σ2 ) 2 + (1 + 2Rδ ) (σ1 − σ2 ) 2 − 2 (1 + Rδ ) σδY ,

•

•

9.5

(9.80)

where σδY is the yield stress in uniaxial tension and Rδ 0.7. It is not yet clear what scale is necessary to approximately attain the classical HM material for the RVE. All we can say is that much larger domains need to be simulated, and these involve much more computing power. Scatter of yield loci is decreasing with mesoscale increasing, under either type of boundary conditions, be it plane strain or plane stress.

Viscoelastic Microstructures

Here we give a very brief account of research carried out on heterogeneous viscoelastic materials by Huet (1995, 1997, 1999a), First, the Hill condition involves strain rates σ : ε˙ = σ : ε˙ .

(9.81)

Translated to the mesoscale, it implies that σ : ε˙ = σ : ε˙ 0 or σ 0 : ε˙ ,

(9.82)

depending on whether the strain rate (˙ε0 ) or stress (σ 0 ) is prescribed. On the microscale (i.e., locally) the material is governed by a formula involving the relaxation modulus tensor (r) t σ (t) = r t − t : dε t dt , (9.83) 0

or a dual one involving the creep compliance tensor (f) t ε (t) = f t − t : dσ t dt .

(9.84)

0

On the mesoscale, under the kinematic boundary condition, the material domain Bδ (ω) is governed by a formula involving the mesoscale relaxation

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modulus tensor (rδ )

σ (t) = 0

t

rδ t − t : dε 0 x, t dt ,

(9.85)

and, under the traction boundary condition, a similar one involving the mesoscale creep compliance tensor (fδ ) t ε (t) = fδ t − t : dσ 0 x, t dt . (9.86) 0

On the macroscale the two tensors become dual, and then we have reff (t) and feff (t). Huet has shown that reff (t) is bounded by the hierarchy reff (t) ≤ . . . ≤ rδ (t) ≤ rδ (t) ≤ . . . ≤ r1 (t)

∀δ = δ/2, ∀t ≥ 0,

(9.87)

∀δ = δ/2, ∀t ≥ 0

(9.88)

and feff (t) is bounded by the hierarchy feff (t) ≤ . . . ≤ fδ (t) ≤ fδ (t) ≤ . . . ≤ f1 (t)

A number of related results on that subject are in Huet’s papers referenced above.

9.6

Stokes Flow in Porous Media

Under consideration is a steady, single-phase fluid flow in a random porous medium with explicit account of a spatially disordered microstructure. In phenomenological, deterministic continuum mechanics, such a flow is described by Darcy’s law. With the slow flow in a porous medium also considered as incompressible and viscous, Darcy’s law is governed by the equations (Dullien, 1979) 1 U = − K·∇p µ

∇ · U = g (x)

(9.89)

where U is the Darcy velocity (volume averaged) velocity, ∇ p is the applied pressure gradient driving the flow, µ is the fluid viscosity, and K is the permeability, a second-rank tensor that depends on the microstructure of the porous medium. Equation (9.89)2 is the continuity equation and g (x) is the source/sink term. In the general studies on property bounds, the variational principles usually provide the basic starting point (Prager, 1961; Torquato, 2002). Now, Darcy’s law (9.89) is not a local constitutive relation, since the slow flow in the voids is governed by the Stokes equation and the continuity equation µ∇ 2 v = ∇ p

∇ ·v=0

(9.90)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 343 In any chosen element dV of a porous medium the energy dissipation can be expressed in two different quadratic forms: one in the space of pressure gradient, and another in the space of Darcy velocity K (9.91) φ (∇ p) = V1 V µij p, j p, j dV φ ∗ (U) = V1 V µRij U j U j dV The Hill condition for Darcy’s law is ∇ p·U = ∇ p · U,

(9.92)

which leads to three possible loadings: 1. A uniform Dirichlet (or essential) boundary condition p(x) = p¯ ,i xi

∀x ∈ ∂ Bδ

(9.93)

2. A uniform Neumann (or natural) boundary condition ¯ ,i ni Ui (x)ni = U

∀x ∈ ∂ Bδ

3. A uniform orthogonal-mixed boundary condition & % ¯ ,i ni = 0 ∀x ∈ ∂ Bδ [ p(x) − p¯ ,i xi ] Ui (x)ni − U

(9.94)

(9.95)

At this point, on the mesoscale, we introduce apparent dissipation functionals φδ (∇ p) =

K ij p, j p, j µ

φδ∗ (U) = µRij U j U j ,

(9.96)

which, respectively, are homogeneous functions of degree 2 in the volume average pressure gradient ∇ p (≡ p,i ) and the volume average Darcy velocity U (≡ Ui ). Thus, we have (Ziegler, 1983) ¯i = U

1 ∂φδ 2 ∂ p,i

p¯ ,i =

1 ∂φδ∗ 2 ∂Ui

(9.97)

With this we have a full analogy between the Stokesian flow in porous media and the antiplane elasticity in a matrix of finite stiffness, containing rigid (infinitely stiff) inclusions. Employing the minimum potential energy principle with the essential (e ) condition (9.93), we find a scale-dependent hierarchy of upper bounds on Ke∞ e K∞ ≤ . . . ≤ Keδ ≤ Keδ ≤ . . . ≤ Ke1 ∀δ = δ/2 (9.98) where

eff Ke∞ = Keff ∞ =K

(9.99) is the effective (RVE level) permeability. Note that Ke1 pertains to the smallest scale where Darcy’s law still applies; by analogy to elasticity of random media, it may be called a Voigt bound.

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Similarly, from the minimum complementary energy principle together with the (n ) natural (9.94), we obtain a hierarchy of upper bounds condition on the resistance Rn∞ n R∞ ≤ . . . ≤ Rnδ ≤ Rnδ ≤ . . . ≤ Rn1 (9.100) ∀δ = δ/2 where

−1 Rn∞ = Keff .

(9.101)

Collecting these inequalities, we have a two-sided hierarchy

Rn1

−1

−1 n −1 ≤ . . . ≤ Rnδ ≤ Rδ ≤ . . . ≤ Keff ≤ . . . ≤ Keδ ≤ Keδ ≤ . . . ≤ Ke1 (9.102) ∀δ = δ/2.

Figure 9.17 illustrates the comparison between both kinds of boundary conditions in terms of the pressure fields. We can clearly see in (b) that the pressure distributes nonlinearly on side boundaries in order to maintain the flow within the top and bottom boundary according to the requirment of Neumann boundary condition. At the same time, the velocity field is nonuniform in (a). As the mesoscale increase, all the fluctuations becomes so negligible that the results of the Neumann boundary value problem begin to coincide with those of the Dirichlet problem (except for the obvious nonuniqueness of the Neumann problem). Figure 9.18(a) displays the simulation results of the random medium at 60% porosity in terms of the ensemble average half-traces of the permeability and resistance tensors. The bounding character of these tensors and their convergence to the RVE (effective medium) with the increasing mesoscale are clearly seen. In fact, at δ = 80, the ensemble averages of both tensors are approximately equal within 0.4%. Another viewpoint of that convergence to RVE may be offered in terms of the departure of one half of the ensemble averaged scalar product of Keδ and Rnδ from unity, because, at δ → ∞, we must have 1 e K · Rnδ = 1. 2 δ

(9.103)

The statistical isotropy of the underlying Poisson point field with exclusion dictates the isotropy of not only the effective (RVE level) properties but also the isotropy of Keδ and Rnδ . Thus, we can work with d = Keδ,1 j · Rnδ,1 j − 1

(9.104)

to characterize the above-mentioned departure from the RVE properties; here 1 j are the indices of particular components of the Keδ and Rnδ tensors. The said departure is shown in Figure 9.18(b), both for the preceding case of porosity, as well as for the entire range: 50% through 80%. Clearly, the scaling

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Pressure on boundary

(a)

(b) FIGURE 9.17 Pressure fields in a realization of the random porous medium on mesoscale δ = 4 under uniform Dirichlet (a) and Neumann (b) boundary conditions.

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( –1) (×10–9m2)

45

40

Dirichlet B.C.

35

Neumann B.C.

30

25

Round disks: 40%

20 0

20

40

60

80

100

δ(L/d) (a) 2.4 Round disk model, porosity: 80%

2.2

70% KijeRjk n(I = k = 1, or 2)

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2

60% 50%

1.8 1.6 1.4 1.2 1 0

20

40

60

80

100

 (L/d) (b) FIGURE 9.18 (a) Effect of increasing mesoscale on the convergence of the permeability/resistance tensor hierarchy. (b) Scaling of Rn11 Ke11 to unity. (After Du & Ostoja-Starzewski, 2006b).

effect of permeability in porous media depends more strongly on the size of micro-channels than on the disk diameter d itself. The rate of convergence to RVE decreases with the porosity increasing. For example, at δ = 32, the RVE size can be considered to be attained at porosity 50%. However, for a higher porosity (e.g., 80%) the RVE size is not achieved even at δ = 80.

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9.7

Thermoelastic Microstructures

9.7.1 Linear Case Let us now consider a linear thermoelastic composite material (Rosen and Hashin, 1970; Christensen, 1979). Each specimen B(ω) ∈ B is described locally (within each phase) by either εij = Sijkl (ω, x)σkl + αij (ω, x)θ

or

σij = Cijkl (ω, x)εkl + ij (ω, x)θ, (9.105)

−1 where Sijkl (ω, x) = Cijkl (ω, x), θ is the temperature rise, and

ij (ω, x) = −Cijkl (ω, x)αkl (ω, x)

(9.106)

are the thermal stress coefficients computed from the thermal expansion coefficients αkl . We also recall from thermoelasticity that the stress and entropy are derivable from the Helmholtz free energy density as ∂A ∂A (9.107) σij = S=− ∂εij T ∂T εij where A=

1 1 θ2 εij Cijkl εkl + ij εij θ − c v , 2 2 T0

(9.108)

for a small temperature change θ = T/T0 , and c v is the specific heat under constant volume. Noting the Legendre transformation, the potential energy is defined as U= Ad V − Fi ui dV − ti ui dS, (9.109) V

S

St

where St is the part of boundary S with traction prescribed on it, and Fi is the body force. On the other hand, we have ∂G ∂G εij = − S=− (9.110) ∂σij T ∂T σij where the Gibbs free energy is G=

1 1 θ2 σij Sijkl σkl − αij σij θ − c p , 2 2 T0

(9.111)

with c p being the specific heat under constant traction. Again by the Legendre transformation, the complementary energy is defined as ∗ U = Gd V + ti ui dS, (9.112) V

Su

where Su is the part of S with displacement prescribed on it.

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Of interest is the derivation of scale-dependent bounds on α eff ≡ α∞ as well as c v and c p , using the already available bounds on Ceff ≡ C∞ (recall Chapter 7). Note here that eff

eff

eff

ij = −Cijkl αkl ,

eff

eff

eff

eff c eff p − c v = T0 C ijkl αij αkl .

(9.113)

Of course, the energy relations for a mesoscale above have to be reconsidered in light of having specific applied loading. Thus, it has been shown that the scale-dependent random functional of potential energy under displacement controlled boundary conditions is 1 1 θ2 U εij , θ, ω = εij0 Cijkl,δ (ω) εkl0 + i j,δ (ω) εij0 θ − c v,δ (ω) , 2 2 T0 where i j,δ (ω) = ij +

1 V

V

Dijkl kl dV

c v,δ (ω) = c v +

T0 V

V

E ij ij dV.

(9.114)

(9.115)

In (9.115) is the local fluctuation, whereas the tensor Dijkl relates the applied strain to the local elastic strain fluctuation e ij , and the tensor E ij relates the temperature change to the thermal strain fluctuation e ijth e ij (x) = Dijkl (x) εkl0

e ijth (x) = E ij (x) θ.

(9.116)

It is shown in the works referenced above that (macroscopically) effective eff thermal expansion coefficients αkl can be derived in terms of the effective eff eff −1 moduli Cijkl = (Sijkl ) and the information on the distribution of individual phases. That is, in the case of two phases, 1 and 2, eff

eff

(2) (2) αij = (αkl(1) − αkl(2) ) Pklmn (Smni j − Smni j ) + αij ,

(9.117)

where eff Pklmn (Smnr s − Smnr s ) = Iklr s =

1 (δkr δls + δks δlr ). 2

(9.118)

Alternatively, as pointed out in the aforementioned works, bounds on α eff can be obtained by using bounds on Ceff , and such a result was produced employing the Hashin–Shtrikman bounds. A study of the scaling from the SVE to the RVE was recently reported by Du and Ostoja-Starzewski (2006b). First, the Hill condition is extended so as to develop the equivalence between the energetic and mechanical formulations of constitutive laws of thermoelastic random heterogeneous materials at arbitrary mesoscale δ. Let us note there that, while the potential energy of a homogeneous material is UP =

1 eff 1 θ2 Cmni j εij εkl + ij εij θ − c v , 2 2 T0

(9.119)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 349 where c v is the specific heat at constant volume and T0 is the reference temperature, the potential energy of a heterogeneous material in a mesoscale domain Bδ is 1 1 θ2 eff UP = Cmni j εij εkl + ij εij θ − c v , (9.120) 2 Vδ 2 T0 Indeed, both energy forms (potential and complementary) at the SVE level can be cast in the same form as those of the homogeneous material, that is, (9.108) and (9.111). Next, taking an ensemble averages and noting the scale dependent hierarchy (7.70) of Chapter 7, leads to two scale-dependent hierarchies for the isotropic part αδt of αitj,δ : (i) α (1) ≥ α (2) ≥ 0 and κ (1) > κ (2) : α ∗ ≥ . . . ≥ αδt ≥ αδt ≥ . . . ≥ α1t ≡ α R

∀δ = δ/2.

(9.121)

(ii) α (1) ≥ α (2) ≥ 0 and κ (1) < κ (2) : α ∗ ≤ . . . ≤ αδt ≤ αδt ≤ . . . ≤ α1t ≡ α V

∀δ = δ/2.

(9.122)

where α R is the Reuss-type bound on α ∗ . Furthermore d (2) Cnni j − δij κ ijd = (1) − (2) − (2) δij . κ (1) − κ (2)

(9.123)

with the help of (7.70) we derive two hierarchical relations for the isotropic part δd of idj,δ : (i) 0 ≥ (1) ≥ (2) ≥ 0 and κ (1) > κ (2) : ∗ ≤ . . . ≤ δd ≤ δd ≤ . . . ≤ 1d ≡ V

∀δ = δ/2.

(9.124)

(ii) 0 ≥ (1) ≥ (2) ≥ 0 and κ (1) < κ (2) : ∗ ≥ . . . ≥ δd ≥ δd ≥ . . . ≥ 1d ≡ V

∀δ = δ/2.

(9.125)

where V is the Voigt-type bound on ∗ . In view of (9.113)2 , this provides a two-sided bounding hierarchy on α∞ , or, equivalently, on ∞ . Both inequalities are obtained under the uniform traction and displacement boundary conditions, respectively. Figures 9.19 and 9.20 display simulation results under these two loadings. Note: Due to the presence of a nonquadratic term in energy formulas, the mesoscale bounds on the thermal expansion are more complicated than those on the stiffness tensor and heat capacity. In general, upper and lower bounds correspond to loading of mesoscale domains by essential and natural

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(a)

(b) FIGURE 9.19 Numerical results at (a) δ = 2 and (b) δ = 32 under the displacement boundary condition (9.44) at εij0 ; disks do not touch.

boundary conditions. Depending on the property mismatches, the upper and lower bounds can be provided either by essential boundary conditions or natural boundary conditions. Suppose we deal with a two-phase composite material with locally isotropic phases. To fully characterize it, we need three mismatches between

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 351

(a)

(b) FIGURE 9.20 Numerical results at (a) δ = 2 and (b) δ = 32 under the traction boundary condition (9.50) at σij0 ; disks do not touch.

both phases: E (i) /E (i) µ(i) /µ(i) α (i) /α (i) .

(9.126)

Figure 9.21 shows the hierarchies of bounds for a composite with 40% volume fraction of inclusions.

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i Ei = 3, vi = 1, αi = 0.5, cp = 2. m m m m c E v α p

2.8681

ctv〉

〉

cδv × 10–6 (J/m3K)

2.8682

〉

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cdv〉

2.868

2.8679

0

8

16

24

32

40

δ(L/d) FIGURE 9.21 Scaling of specific heat capacities from SVE toward RVE at 40% volume fraction of inclusions.

Finally, there exist similar hierarchies on the effective specific heat coefficient: one at constant volume d d d d c v∗ ≡ c v,∞ ≥ c v,δ ≥ . . . ≥ c v,1 ≥ . . . ≥ c v,δ ≡ c vV

∀δ = δ/2, (9.127)

and another at constant pressure c ∗p ≡ c tp,∞ ≤ . . . ≤ c tp,δ ≤ c tp,δ ≤ . . . ≤ c tp,1 ≡ c Vp

∀δ = δ/2. (9.128)

9.7.2 The Nonlinear Case We refer back to Section 9.2. By generalizing the derivation of the hierarchy (9.31) to thermoelastostatics, we obtain

∞ (F0 , T0 ) ≤ . . . ≤ δ (F0 , T0 ) ≤ δ (F0 , T0 ) ≤ . . . ≤ 1 (F0 , T0 ) ∀δ = δ/2,

(9.129)

where T0 is the prescribed temperature. Similarly, (9.38) is replaced by G ∞ (P0 , T0 ) ≥ . . . ≥ G δ (P0 , T0 ) ≥ G δ (P0 , T0 ) ≥ . . . ≥ G 1 (P0 , T0 ) ∀δ = δ/2,

(9.130)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 353 where G(P0 , T0 , ω) = (V0 ) −1 Bδ g X, P0 , T0 , ω dV. Utilizing the Legendre transformation relating Gibbs (g) and Helmholtz (ψ) energy densities ∂ψ ∂ψ g ,T =ψ− xi, j , (9.131) ∂ui, j ∂ui, j and applying stress-free boundary conditions (i.e., a free expansion) P0 = 0, the hierarchy (9.130) transforms to ∗

∞ (P0 ) ≤ . . . ≤ δ∗ (P0 ) ≤ δ∗ (P0 ) ≤ . . . ≤ 1∗ (P0 ) ∀δ = δ/2, (9.132) which provides a lower bound on the effective Helmholtz free energy. Note that under stress-free boundary conditions, the elastic part of equals zero for a homogeneous body. In contrast, for a heterogeneous material, the elastic contribution increases with δ. For a numerical example, we consider the simplest form of the potential ψ given by a neo-Hookean strain-energy function (Dhont, 2004) ψ=

& 1% µ (T) λ˜ 21 + λ˜ 22 + λ˜ 23 − 3 + κ (T) ( J M − 1) 2 + T˜ (T) λ˜ 2 = J −1/3 λa . 2 (9.133)

Here µ (T) and κ (T) are temperature-dependent initial shear and initial bulk moduli. The quantity T˜ (T) is the purely thermal contribution to the free energy and, because it does not change with scale, can be ignored in the hierarchies above. The Jacobian is decomposed into purely mechanical (J M ) and purely thermal (J T ) parts according to J = J M J T , with J T = (1 + αT) 3 . In general, the free energy of the composite can be written as % & ψ = 12 µ (T) λ˜ 21 + λ˜ 22 + λ˜ 23 − 3 + κ (T) ( J M − 1) 2 + T˜ (T) λ˜ 2 = J −1/3 λa . (9.134) where ψ

' 1 F0 , T0 , ω = µ (T) ( λ˜ 01 ) 2 + ( λ˜ 02 ) 2 + ( λ˜ 03 ) 2 − 3 2 2 ( 0 J + κ (T) −1 + T˜ (T) . JT

(9.135)

and ψ X, F0 , T0 , ω is a local fluctuation of ψ. Various comparisons of thermal expansion/stress coefficients have been carried out by Khisaeva (2006); see also Khisaeva and Ostoja-Starzewski (2007). As an example, 9.22, we show the scaling trend of the ther in Figure mal stress coefficient tr ij δ on the temperature change T for one particular composite. Note the smooth transition of response into that predicted by the linear theory as T decreases.

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0.42

0.40 Γ/Γmax

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0.38

0.36

0.34

0.32 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 δ

FIGURE 9.22 Dependence of the thermal stress coefficient on the temperature change T for the rubber– polystyrene composite.

9.8

Scaling and Stochastic Evolution in Damage Phenomena

Consider a material whose elasticity is coupled to damage state, as described by the constitutive equation (Lemaitre and Chaboche, 1994) σij = (1 − D)Cijkl εkl .

(9.136)

Here Cijkl is isotropic, and must be coupled with a law of isotropic damage, that is, ∗ ˙ = ∂ , D ∂Y

(9.137)

with Y = −∂ /∂ε, being the (Helmholtz) free energy. This formulation is set within the TIV framework, see Chapter 10. In particular, the scalar D evolves with the elastic strain ε = εii , which is taken as a time-like parameter, according to ∂D ∂ε

=

) ∗ (ε/ε0 ) s 0

when

ε = εD

and dε = dε D > 0,

when

ε < εD

and dε < 0.

(9.138)

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 355 Integration from the initial conditions D = ε D = 0 up to the total damage, D = 1, gives D = (ε/ε0 ) s

∗

% ∗ &s ∗ +1 ε R = (1 + s ∗ )εsD

+1

% & ∗ σ = 1 − (ε/ε R ) s +1 Eε,

(9.139)

where σ = σii . This formulation is understood as the effective law for the RVE, that is, eff

Cijkl = Cijkl |δ→∞

Deff = D|δ→∞

eff = |δ→∞

eff = |δ→∞ ,

(9.140)

as well as a guide for adopting the form of apparent responses on mesoscales. Thus, assuming that the same types of formulas hold for any mesoscale δ, we have an apparent response for any specimen Bδ (ω) σ = (1 − Dδd )Cdδ (ω) : ε 0

(9.141)

under uniform displacement boundary condition. The notation Dδd expresses the fact that the material damage is dependent on the mesoscale δ and the type of boundary conditions applied (d ). In fact, although we could formally write another apparent response ε = (1Dδd ) −1 Stδ (ω) : σ 0 , we shall not do so because the damage process under the traction boundary condition (t) would be unstable. It is now possible to obtain scale-dependent bounds on Dδd through a procedure analogous to that for linear elastic materials, providing one assumes a WSS microstructure. One then obtains a hierarchy of bounds d and ergodic on D∞ ≡ Deff from above (Ostoja-Starzewski, 2002b):

Dδd ≤ Dδd ≤ . . . ≤ D∞d

∀δ = δ/2.

(9.142)

These inequalities are consistent with the much more phenomenological Weibull model of scaling of brittle solids saying that the larger is the specimen the more likely it is to fail. Next of interest is the formulation of a stochastic model of evolution of Dδd with ε to replace (9.138)1 . Said differently, we need a stochastic process Dδd = {Dδd (ω, ε); ω ∈ , ε ∈ [0, ε R ]}. Assuming, for simplicity of discussion, just as in Lemaitre and Chaboche (1994), that s ∗ = 2, we may consider this setup: d Dδd (ω, ε) = Dδd (ω, ε) + 3ε 2 [1 + rδ (ω)]dt,

(9.143)

where rδ (ω) is a zero-mean random variable taking values from [−a δ , a δ ], 1/δ = a δ < 1. This process has the following properties: 1. Its sample realizations display scatter ω-by-ω for δ < ∞, that is, for finite body sizes. 2. It becomes deterministic as the body size goes to infinity in the RVE limit (δ → ∞).

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Microstructural Randomness and Scaling in Mechanics of Materials 3. Its sample realizations are weakly monotonically increasing functions of ε. 4. Its sample realizations are continuous. 5. The scale effect inequality (9.142) is satisfied, providing we take ε R as a function of δ with a property ε R (δ) < ε R (δ )

∀δ = δ/2.

(9.144)

Let us observe, however, that, given the presence of a random microstructure, mesoscale damage should be considered as a sequence of microscopic events—shown as impulses in Figure 9.23(a)—thus rendering the apparent damage process Dδd one with discontinuous sample paths, having increments d Dδd occurring at discrete time instants, Figure 9.23(c). To satisfy this requirement one should, in place of the above, take a Markov jump process whose range is a subset [0, 1] of real line (i.e., where Dδd takes values). This process would be specified by an evolution propagator, or, more precisely, by a nextjump probability density function defined as follows: p(ε , Dδd | ε, Dδd )dε d Dδd = probability that, given the process is in state Dδd at time ε, its next jump will occur between times ε + ε and ε + ε + dε , and will carry the process to some state between Dδd + Dδd and Dδd + Dδd + d Dδd . Figure 9.23(b) shows one realization Cδd (ω, ε); ω ∈ , ε ∈ [0, ε R ], of the apparent, mesoscale stiffness, corresponding to the realization Dδd (ω, ε); ω ∈ , ε ∈ [0, ε R ], of Figure 9.23(c). In Figure 9.23(a) we see the resulting constitutive response σδ (ω, ε); ω ∈ , ε ∈ [0, ε R ]. Calibration of this model (just as the simpler one above)—that is, the specification of p(ε , Dδd | ε, Dδd )dε d Dδd —may be conducted by either laboratory or computer experiments such as those discussed in Chapter 4. Note that in the macroscopic picture (δ → ∞) the zigzag character and randomness of an effective stress-strain response vanish. However, many studies in mechanics/physics of fracture of random media (e.g., Herrmann and Roux, 1990), indicate that the homogenization with δ → ∞ is generally very slow, and hence that the assumption of WSS and ergodic random fields may be too strong for many applications. Extension of the above model from isotropic to (much more realistic) anisotropic damage will require tensor, rather than scalar, Markov processes. This will lead to a somewhat greater mathematical complexity, which may be balanced by choosing the first model of this section rather than the latter. These issues are quite secondary. Our goal has been to outline a stochastic continuum damage mechanics that (1) is based on, and consistent with, micromechanics of random media as well as the classical thermomechanics formalism, and (2) reduces to the classical continuum damage mechanics in the infinite volume limit. Over the past two decades numerous studies of damage in both classical and non-classical as well as deterministic and stochastic settings have been conducted, e.g., Carmeliet and de Borst (1995), Pamin (2006), Bazant (2007).

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Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures 357

C = E (1-D)

ε/εR (a)

ε/εR (b)

D

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ε/εR (c)

FIGURE 9.23 Constitutive behavior of a material with elasticity coupled to damage where ε/ε R plays the role of a controllable, time-like parameter of the stochastic process. (a) Stress-strain response of a single specimen Bδ from, B having a zigzag realization; (b) deterioration of stiffness Cδd (ω, ε); (c) evolution of the damage variable. Curves shown in (a–c) indicate the scatter in stress, stiffness, and damage at finite scale δ. Assuming spatial ergodicity, this scatter would vanish in the limit δ → ∞, whereby unique response curves of continuum damage mechanics would be recovered.

9.9

Comparison of Scaling Trends

We now recall question 6 of the Preface, which may be rephrased as: on what mesoscales is the RVE attained with the same accuracy for various types of random microstructures? Said differently, given a specific mesoscale, what are

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Mismatch and Discrepancy Values on Mesoscale δ = 16 Physical Subject

Mismatch µ(i)

Linear elasticity

= 10

µ(m)

D [%] κ (i) κ (m)

=1

Linear thermoelasticity

µ(i) = 10 µ(m)

κ (i) = 10 κ (m)

Plasticity

h (i) = 10 h (m)

E (i) =1 E (m)

µ(i) 0

κ0(i)

Finite elasticity

µ(m) 0

Flow in porous media

= 10

(i)

tr K

tr K(i)

κ0(m)

2.28 α (i) 1 = 10 α (m)

=1

=∞

5.51 2.29 5.86 27

the discrepancies between the bounds obtained from kinematic and traction boundary conditions for various types of random microstructures? Following a recent study (Ostoja-Starzewski et al., 2007), a comparison of such results is given in terms of a discrepancy D (Table 9.2). The latter is defined as D=

Rδe − Rδn , Rδe + Rδn /2

(9.145)

where, depending on the case, Rδe and Rδn are the responses under kinematic and traction loadings, respectively. In each case, the same planar random matrix-inclusion composite is used, where the centers of inclusions are generated via a Poisson random field with inhibition. We close this chapter with these observations: •

Scaling from SVE to RVE slows down when: 1. We go from elastic to inelastic microstructures 2. The mismatch in properties grows 3. We go from 2D or 3D setting 4. The microscale geometries exhibit nonuniform effects (from disks to ellipses, spatial clustering of inclusions, etc.)

•

Scaling from SVE to RVE in linear elastic microstructures: 1. Mesoscale moduli of stiff matrix with soft inclusions converge (much) more slowly to RVE than those of a soft matrix with stiff inclusions 2. Convergence to RVE is slowest in antiplane, faster in inplane, and fastest in 3D elasticity.

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10 Mesoscale Response in Thermomechanics of Random Media

. . . the dissipation function is the key to irreversible thermodynamics. H. Ziegler, 1983 The presence of dissipative phenomena in mechanics of materials necessitates a formulation of continuum mechanics consistent with principles of thermodynamics, leading to so-called thermomechanics or continuum thermodynamics. As elaborated by Maugin (1999), there are now four classical continuum thermodynamics theories: thermodynamics of irreversible processes (TIP); thermodynamics with internal variables (TIV); rational thermodynamics (RT); extended (rational) thermodynamics (ET). All of these are deterministic, homogeneous continuum theories without clear account of the underlying random compositions of materials—that is, they a priori postulate the existence of the RVE. Strictly speaking, some statistical treatments were carried out as a bridge from micro to macro levels for select variants of the above theories—for example, by Ziegler (1963, 1970) for TIV (see below), or by Muschik et al. (2000) for TIP (see Chapter 8)—but such studies were only concerned with providing foundations from the standpoint of statistical physics directly to the level of the RVE, without making clear what the size of the RVE actually should be. Given the widespread use of TIV in mechanics of materials (e.g., Lemaitre and Chaboche, 1990), we now set out to generalize it so as to provide a link to random microstructures.

10.1

From Statistical Mechanics to Continuum Thermodynamics

10.1.1 Dissipation Function of the RVE In this section, following Ziegler (1962, 1963, 1972), and consistent with Gibbs (1902), we consider a mechanical system with a very large number of N particles (i.e., with 6N degrees of freedom) with generalized coordinates q n (n = 1, . . ., 3N), pn (n = 1, . . ., 3N), and a k (k = 1, . . ., K ). The q n ’s and pn ’s are the so-called microcoordinates, while a n ’s are the so-called macrocoordinates. 359

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The first of these, as they refer to atoms or molecules in an RVE, are too numerous and miniscule in magnitude to be directly measured, as opposed to the latter, which are perceptible to a macroscopic observer and, therefore, used to described the system’s macrostate. Thus, the a n s are the strain components of the RVE and possibly the internal parameters necessary to study, say, an elastic-plastic material. Ziegler begins with a motion of the microsystem described by Hamilton’s equations [recall (2.158) of Chapter 2], where the Hamiltonian H (q n , pn , a k ) represents the total energy of the system. In view of equations (2.158) of Chapter 2, the time rate of H is ˙ = ∂ H a˙ k , H ∂a k

(10.1)

which means that H is modified through the macrocoordinates alone. On a continuum mechanics/thermodynamics level we have a deterministic system, modeling the RVE, which is described by the a k s, which vary slowly compared with the q n s. The evolution of H in the phase space {(q n , pn ) ; n = 1, . . ., N} follows Hamilton’s equations (2.158) of Chapter 2. As mentioned there, the discrete distribution, and indeed the motion, of phase points is described by a so-called canonical density ρ (q n , pn , t). The latter is henceforth treated as a distribution over a continuum since N 1. Thus, we have a continuum-like, incompressible phase fluid, for which there must hold a conservation of phase points, as expressed by the Liouville theorem dρ = 0. dt

(10.2)

Here we may recall the continuity equation for a real fluid of density ρ in continuum mechanics 0=

dρ dρ + ρvi,i = , dt dt

(10.3)

where the second equality holds if the fluid is incompressible. Equation (10.3) allows us to write an integral form df d ρ dV = ρ f dV , (10.4) dt dt where f is an arbitrary function. Also in a real continuum, the material rate of change of internal energy u per unit volume due to heat flux q alone is du + q i,i = 0. dt

(10.5)

Returning to the phase space, the role of dV is played by dq 1 , . . ., dq N , d p1 , . . ., dp N , and the integration over that space means the ensemble averaging relative to the density ρ, as shown in (10.11) below. Also note the

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361

ρdV = 1.

(10.6)

Ziegler assumes the particle dynamics to be holonomic, scleronomic, and admits gyroscopic forces derivable from a potential (Problem 18 of Chapter 2). The latter feature allows one to admit, say, Coriolis or Lorentz forces occurring, respectively, in rotating coordinate systems or magnetic fields. Ziegler also identifies degrees of freedom of microsystem ν, and notes that ρ (q n , pn , t) supplies a canonical probability w (q n , pn , t) of finding the phase point of a given microsystem in the unit volume centered at (q n , pn ) at the time t. In general, they are proportional w (q n , pn , t) =

hν ρ (q n , pn , t), N

(10.7)

where h is the unit of action—such as Planck’s constant in quantum mechanics. According to a postulate of Gibbs, the surfaces H = const and ρ = const coincide in a state of statistical equilibrium, that is, all phases of the microsystem characterized by the same value of H are equally probable. Thus, ρ is a function of the energy H of the system ρ = ρ ( H),

(10.8)

and, as shown by Gibbs, the simplest function suitable for ρ is ρ (ω) = exp

− H (ω) , kT

(10.9)

where is the free energy and k is the Boltzmann constant. Hereinafter, we explicitly indicate the dependence of ρ on the actual outcome ω, and hence the random character of H. It now follows that the so-called Gibbs’ index of probability (or Gibbs phase) is η (ω) = ln ρ (ω) =

− H (ω) , kT

(10.10)

and, consistent with Section 2.6 of Chapter 2, the information entropy h is again the ensemble average of − ln p h = − η = − ρ ln ρdV. (10.11) While from (10.11) we have −

d d η = h, dt dt

(10.12)

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from (10.10)1 we have dη/dt ≡ η˙ = ρ/ρ, ˙ which, in view of (10.6), gives d η ˙ = ρ˙ dV = ρ dV = 0. (10.13) dt Comparing this with (10.12), we see that the time differentiation does not generally commute with the ensemble averaging: d d η = η . (10.14) dt dt This type of result carries over to the Hamiltonian. First, from (1.10)2 we see H (ω) = − Tkη (ω),

(10.15)

which, upon ensemble averaging, gives H = + Tkh = + T S,

(10.16)

where the second equality follows from the relation between the information and thermodynamic entropies kh = S.

(10.17)

From this, upon time differentiation, we obtain d ˙ ˙ + T˙ S + T S. H = dt

(10.18)

On the other hand, subjecting (10.15) first to time differentiation, and then ensemble averaging, we get ˙ = ˙ + T˙ S, H (10.19) where (10.11), (10.13), and (10.17) have been used. Thus, d d d H + T S. H = dt dt dt

(10.20)

Comparing (10.20) with (10.4), we see that there is no continuity in the phase space for nonisentropic processes. In other words, if it were not for the pres˙ continuity would hold for H. To reflect the fact that the Liouville ence of S, theorem is not satisfied, Ziegler replaces (10.2) by a transport equation dρ ∂a n ∂b n + + = 0, dt ∂q n ∂ pn

(10.21)

where the vector pair (a, b), called flux, represents the change of density ρ due to the creation/annihilation of the phase fluid, called flux. Note that (10.21) combines two types of transport exemplified by (10.3)1 and (10.5).

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363

Now, upon recognizing in the average Hamiltonian the RVE’s internal energy H = U,

(10.22)

where both sides are now dependent on S and the vector a of a k s but not q i s and pi s, we recall the key relations of (deterministic) continuum thermomechanics (Ziegler, 1983): the energy conservation law, ˙ = L + Q∗ = A · a˙ + Q∗ , U

(10.23)

where L is the power of forces A corresponding to kinematic parameters a, and Q∗ is the heat supply per unit time; and the second law of thermodynamics, (10.24) S˙ = S˙ (r ) + S˙ (i) S˙ (r ) = Q∗ /T S˙ (i) = A(d) · a˙ /T ≥ 0, where we have a decomposition of entropy rates S˙ (r ) (reversible) and S˙ (i) (irreversible), as well as a decomposition of forces into quasi-conservative and dissipative parts: A = A(q ) + A(d) .

(10.25)

These forces are determined from the free energy and the dissipation function ( a˙ ) = T S˙ (i) .

(10.26)

as (q ) Ak

=

∂ ∂a k

A(d) k

=ν

∂ ∂a˙ k

ν=

∂ ∂a˙ k

a˙ k

−1 ,

(10.27)

wherein (10.27)2 expresses Ziegler’s thermodynamic orthogonality. It leads to the principle of maximal dissipation rate: Provided the dissipative force A(d) is prescribed, the actual velocity a˙ maximizes the dissipation rate L (d) = A(d) · a˙ subject to the side condition ( a˙ ) = A(d) · a˙ = L (d) ≥ 0.

(10.28)

Ziegler (1970) proves this orthogonality on the basis of statistical mechanics (Ziegler, 1970): Consider variations of a˙ by δ a˙ during a small time interval. For variations δ a˙ that do not affect the various sides in (10.28), the behavior of the macrosystem during that time interval is the same. It follows that not only a single velocity a˙ is compatible with the prescribed force A(d) , but also all those varied velocities a˙ + δ a˙ for which the dissipation function, and hence also the scalar product with A(d) remain unchanged. However, this is possible only if δ a˙ is infinitesimal and A(d) is orthogonal to the tangential plane passing through the end point of a˙ . Note: The application of the maximum entropy method (Chapter 2), subject to the condition (10.22), results in the exponential form of density (10.9).

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10.1.2 Departure from the Second Law of Thermodynamics The second law of thermodynamics applies to macroscopic systems, not microscopic ones. In the 1990s the so-called fluctuation theorem (FT) was developed; the proof was given in Evans and Searles (1994). It is a statement about the probability distribution of the irreversible entropy production, t , averaged over a time interval t. The FT states that, in systems not in thermodynamic equilibrium over a time interval t, the ratio of the probability that t = A and the probability that it takes the opposite value, −A, is e At :

P t = A

= e At . P t = −A

(10.29)

That is, as the time or system size increases (because t is extensive), the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. When the FT is applied to macroscopic systems, the second law is recovered. The FT has consequences in nanomechanics and biophysics. For example, very small-scale machines will spend part of their time actually running in “reverse,” that is, in a way opposite to that for which they were designed. Note: With the help of the FT, the MEM has recently been shown to provide a more fundamental basis for Ziegler’s orthogonality principle (Dewar, 2005).

10.2

Extensions of the Hill Condition

10.2.1 The Hill Condition in Thermomechanics In order to extend the continuum thermomechanics to random media at mesoscale, that is, below the RVE but above the atomic/molecular level of the previous section, we first need to establish other expressions of the Hill condition. As is well known, the field forms of five basic laws of (deterministic) continuum thermomechanics—the conservation of mass, conservation of linear momentum, conservation of angular momentum, conservation of energy, and second law of thermodynamics (entropy production inequality)—are ∂ρ ∂t

+ div (ρv) = 0

divσ + b − ρ v˙ = 0 σT = σ ρ u˙ − σ : d = r − divq r q ρ s˙ − − div = ρ s˙i ≥ 0. T T

(10.30)

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Here ρ is the mass density, v the velocity, σ the Cauchy stress, d the deformation rate, u the internal energy, r the heat source, q the heat flux, T the temperature, s the entropy, and s˙i the irreversible part of entropy production. Noting these rules involving the averaging operation, in addition to relations (2.52) of Chapter 2, pertaining to field (intensive) quantities V Z = V Z + V Z a 0 + a i Zi = a 0 + a i Zi ; a i nonrandom dZ ∂ Z + v · grad Z + v · gradZ , = dt ∂t

(10.31)

Huet (1982) developed Hill-type conditions for these laws: ∂ ρ ∂t

+ div(ρ v) = −div ρ v

div σ + ρ b − ρ v· = ρ v · gradv + ρ v˙ − ρ b σ T = σ

ρ u· − σ : d − r + div q = − ρ u − ρ v · grad u + σ : d 1 1 ρ s· − r + div q − ρ si · T T 1 ρ =− v · grad s − ρ s˙ + r T 1 −div r + ρ v · grad si + ρ s˙i T · ρ si ≥ ρ v · grad si − ρ s˙i (10.32) Here · indicates a time derivative acting on . These six equations lead to the so-called assimilation conditions, which ensure homogenization at the effective continuum level div(ρ v ) = 0 ρ v · gradv + ρ v˙ = ρ b ρ v · grad u + ρ u = σ : d ρ v · grad si + ρ s˙i = 0 1 1 ρ v · grad s + ρ s˙ = r − div b . T T

(10.33)

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10.2.2 Homogenization in Dynamic Response Moving from quasi-statics to dynamics, one has to admit inertial effects. The Hill condition of earlier chapters needs to be modified. Starting from the local equations of motion σij,j = ρ u¨ i ,

(10.34)

Wang (1997) found ε˙ i j σi j − ε˙ i j σi j + υi σi j, j − ε˙ i j σki , k x j = 1 υi (X) − vi,, j x j nk (X) σki (X) − σki0 d S Vδ ∂ Bδ

(10.35)

When dealing with strain rates and velocities instead of strains and displacements, the left hand side in the above may also be written as ε˙ i j σi j − ε˙ i j σi j + 1 d ρ (υi υi ) − ε˙ i j ρ υ˙ i x j . 2 dt In the case (ρ u¨ i = 0) and when either one of three uniform boundary conditions of Chapter 7 is prescribed, the Hill condition εi j σi j = εi j σi j is satisfied. However, in the dynamic case, when velocities prescribed on ∂ Bδ are affine, the right hand side of (10.35) equals zero, and that equation reduces to ε˙ i j σi j = ε˙ i j σi j + υi σi j, j − ε˙ i j σki , k x j 1 d = ε˙ i j σi j + ρ (υi υi ) − ε˙ i j ρυi x j . 2 dt

(10.36)

On the other hand, in the dynamic case with uniform tractions prescribed, one finds a simpler relation only the rate of strain energy computed involving from volume averages ε˙ i j σi j , the volume average of rate of strain energy, and the rate of kinetic energy ε˙ i j σi j = ε˙ i j σi j + υi σi j, j = ε˙ i j σi j +

1 d ρ (υi υi ) 2 dt

(10.37)

Clearly, we see a generalization of the conventional Hill condition. Z.-P. Wang (1997) has developed dynamic plasticity models of porous materials on the RVE level with this formulation. A still outstanding challenge is the development of such models to randomly heterogeneous materials on the SVE (mesoscale) level, with explicit inclusion of scale effects. It is also, of interest to note the effective (RVE level) equation of motion, derived by Wang and Sun (2002): σi j, j + Fi = ρ u¨ i ,

(10.38)

where Fi may be regarded as an effective body force resulting from the micro-inertia.

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367

Legendre Transformations in (Thermo)Elasticity

10.3.1 Elasticity On the microscale we have ∗ (σ ) + (ε) = σ : ε,

(10.39)

where (ε) and ∗ (σ ) are also functions of position x and realization ω ∈ . Passing to a mesoscale δ > 0, given the randomness of response of Bδ (ω), there are two Legendre transformations linking the potential ((ε0 )) and complementary ( ∗ (σ 0 )) energies, depending on whether strain (ε 0 ) or stress (σ 0 ) is controlled via uniform displacement or traction boundary conditions: δ∗ (σ (ω)) + δ ε0 = σ (ω) : ε 0 δ∗ σ 0 + δ (ε (ω)) = σ 0 : ε (ω). (10.40) Upon ensemble averaging on the SVE level, we obtain ∗ 0 δ∗ (σ ) + δ ε0 = σ : ε 0 δ σ + δ (ε) = σ 0 : ε. (10.41) Thus, in the first case here, σ is the ensemble average outcome and it becomes the argument of ∗ . In the second case, ε is the ensemble average outcome and it becomes the argument of . As the mesoscale increases indefinitely (δ ≡ L/d → ∞)—or, in other words, as the SVE turns into the RVE—the relations (10.41)1 and (10.41)2 should coincide and turn into the classical statement of a deterministic continuum theory: ∗eff σ 0 + eff ε 0 = σ : ε 0 (10.42) whereby the distinction between ε0 and ε, as well as between σ 0 and σ , vanishes, as we are dealing with the RVE situation. Accordingly, we have dropped the subscript δ on ∗eff and eff in the above. 10.3.2 Thermoelasticity When the material is thermoelastic, the single Legendre transformation is generalized to a quartet of partial Legendre transformations linking internal energy, enthalpy, Gibbs and Helmholtz energies (Sewell, 1987, Collins and Houlsby, 1997; Houlsby and Puzrin, 2000). To start with, in analogy to (10.39), consider the case of a homogeneous continuum—effectively the RVE level. Thus, depending on what we take as the reference, or controllable, loading case, we have either (ε0 ) or ∗ (σ 0 ) playing the role of X(xi , αi ) in Figure 10.1; x(≡ xi ) is ε0 or σ 0 , and we have to add α(≡ αi ) or β(≡ βi ) as temperature or entropy.

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Microstructural Randomness and Scaling in Mechanics of Materials X(xi, αi) ∂X , β = ∂X yi = i ∂xi ∂αi X(xi, α) + Y(yi, αi) = xi yi

X(xi, α) + Y(xi, βi) = αiβi

Y( yi, αi)

V(xi, βi)

∂Y , β = –∂Y xi = i ∂yi ∂αi

∂V , α = ∂V i ∂xi ∂βi

yi = –

Y(yi, αi) + W(yi, βi) = –αiβi

V(xi, βi) + W(yi, βi) = –xi yi W(yi, βi)

∂W ∂W x i = – , αi = – ∂yi ∂βi (a) X(xi, αi) ∂ X, β =∂ X yi = i ∂αi ∂xi X(xi, αi) + Y( yi , αi ) = xi yi

X(xi, αi) + V(xi, βi ) = α i βi

Y( yi , αi) xi =

V(xi, βi )

∂Y , β = –∂Y i ∂ yi ∂αi

yi = –∂V , αi = ∂V ∂xi ∂ βi

Y( yi , αi) + W( yi , βi ) = –α i βi

V(xi, βi ) + W( yi , βi ) = –xi yi

W( yi , βi ) xi = –∂W , αi = – ∂W ∂ yi ∂ βi (b) FIGURE 10.1 (a) A quartet of deterministic partial Legendre transformations for pairs xi ↔ yi and αi ↔ βi for the functional X (xi , αi ). (After Sewell (1987). (b) A quartet of ensemble averaged, partial Legendre transformations for pairs xi ↔ yi and αi ↔ βi , when the pair (xi , αi ) is controllable and the functional X (xi , αi ) is random.

For a homogeneous continuum the complementary energy under a prescribed traction boundary condition and the potential energy under a prescribed displacement boundary condition are the negative of each other: U P = −U C .

(10.43)

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We also have this classical Legendre transformation linking the Helmholtz and Gibbs energies: A(ε, θ) = G(σ, θ) + σ : ε.

(10.44)

Here we simply write A, G, σ , and ε. This, of course, is one pair out of all four possible Legendre transformations in a quartet linking the internal energy, Helmholtz free energy, enthalpy, and Gibbs energy when the temperature (θ) and entropy (s) are kept as passive variables. In a random medium (i.e., for the SVE Bδ (ω) ∈ Bδ ), the quartet must be reinterpreted carefully according as either uniform displacement or traction boundary conditions are applied. In the case of the first one of these loadings, we have the Helmholtz energy Aδ (εij0 , θ, ω) =

1 1 θ2 0 Cijkl,δ (ω)εij0 εkl + ij,δ (ω)εij0 θ − c v,δ (ω) , 2 2 T0

(10.45)

and, in the case of traction loading, we have the Gibbs energy 1 1 θ2 G δ (σij0 , θ, ω) = − Sijkl,δ (ω)σij0 σkl0 − αij,δ (ω)σij θ − c p,δ (ω) . 2 2 T0

(10.46)

Upon ensemble averaging, (10.45) and (10.46) become, respectively, Aδ (εij0 , θ) =

1 1 θ2 0 Cijkl,δ εij0 εkl + ij,δ εij0 θ − c v,δ , 2 2 T0

(10.47)

and 1 1 θ2 G δ (σij0 , θ) = − Sijkl,δ σij0 σkl0 − αij,δ σij0 θ − c p,δ . 2 2 T0

(10.48)

Clearly, under displacement boundary conditions (εij0 controlled), the volume average stress is random (i.e., σ ij (ω)), so that G δ (σ ij (ω), θ) = Aδ (εij0 , θ, ω) − σ ij (ω)εij0 ,

(10.49)

and hence, the ensemble average Gibbs energy on mesoscale δ should be calculated from Aδ (εij0 , θ) according to G δ (σ ij , θ) = Aδ (εij0 , θ) − σ ij εij0 ,

(10.50)

rather than as G δ (σ ij , θ). Similarly, under traction boundary conditions (σij0 controlled), the volume average strain is random (i.e., ε ij (ω)), so that Aδ (ε ij (ω), θ) = G δ (σij0 , θ, ω) + σij0 ε ij (ω),

(10.51)

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and hence, the ensemble average Helmholtz energy on mesoscale δ should be calculated from G δ (σij0 , θ) according to Aδ (εij , θ) = G δ (σij0 , θ) + σij0 ε ij ,

(10.52)

rather than as Aδ (ε ij , θ). The above relations are used in computation of bounds for thermoelastic microstructures discussed in the previous chapter. When the mesoscale SVE reaches the RVE, the dependence on the type of boundary conditions vanishes, and we recover the classical relation (10.44) for a homogeneous material. Note: When the orthogonal-mixed boundary condition is employed, we obtain an intermediate response—very likely with much weaker scale effects than the other two.

10.4

Thermodynamic Orthogonality on the Mesoscale

10.4.1 General The version of thermomechanics we adopt here belongs to the category of thermodynamics with internal variables (TIV) originated by Ziegler (1957); see also Germain et al. (1983), Ziegler and Wehrli (1987), Maugin (1999). As is well known, the RVE response in TIV is described by the free energy and the dissipation function , both of which are scalar products =

1 (q ) e σ :ε 2

= th + intr = T S˙ (i) ≥ 0,

(10.53)

where the Clausius–Duhem inequality expresses the second law of thermodynamics with a summation of th (the thermal dissipation) and intr (the ˙ intrinsic dissipation); S˙ (i) is the irreversible part of entropy production rate S. The said scalar products are th = −q · ∇T/T

intr = Y · a˙ = σ (d) : d + β (d) : α. ˙

(10.54)

In the above σ (q ) is the quasi-conservative stress, σ (d) is the dissipative stress, β (d) is the internal dissipative stress, ε e is the elastic strain, d is the deformation rate, α ˙ is the rate of internal parameters α, and q is the heat flux. In view of (10.54), we may simply view as the scalar product = Y · a˙ ,

(10.55)

where Y is the dissipative force vector and a˙ is the velocity vector (rate of a). If we want to generalize the formulas relating the dissipative force with the velocity via functions δ and ∗δ , we must recognize that the situation will be analogous to conservative processes on mesoscale, where we encountered two types of the functional , and two types of ∗ , depending on the prescribed loading. An ensemble representation of dissipation surfaces

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δ a˙ ,ω in the velocity space and its dual ∗δ Y, ω in the force space is shown Figure 10.2; see also Ostoja-Starzewski (2002c). First, we have a generalization of the extremum principles of deterministic thermomechanics (Ziegler, 1983) to the mesoscale random medium Bδ = {Bδ (ω); ω ∈ }. Velocity control: a˙ is prescribed and Y(ω) follows from the ensemble of random dissipation surfaces δ ( a˙ , ω) according to Yi (ω) = λ(ω)δ ( a˙ , ω)/∂a˙ i ; Figure 10.2(a). The principle of maximal dissipation rate for Bδ reads: Provided the dissipative force Y is prescribed, the actual velocity a˙ maximizes the dissipation ˙ subject to the side condition rate L (d) δ = Y · a δ ( a˙ ) = Y · a˙ = L (d) (10.56) δ >0 The principle of least dissipative force for Bδ reads: Provided the value δ ( a˙ ) of the dissipation function and the direction n of the dissipative force Y are prescribed, the actual velocity a˙ minimizes the magnitude of Y subject to the side condition (10.56). Force control: Y is prescribed and a˙ (ω) follows from the ensemble of random dissipation surfaces ∗δ (Y, ω) according to a˙ i (ω) = µ(ω)∗δ (Y, ω)/∂Yi ; Figure 10.2(c). The principle of maximal dissipation rate reads now: Provided the dissipative force Y is prescribed, the actual velocity ˙a maximizes the dissipation rate a subject to the side condition L (d) δ = Y · ˙ δ a˙ = Y · a˙ = L (d) (10.57) δ >0 The principle of least dissipative force for a random medium Bδ reads: Provided the value δ (˙a) of the dissipation function and the direction n of the dissipative force Y are prescribed, the actual velocity ˙a minimizes the magnitude of Y subject to the side condition (10.57). 10.4.2 Homogeneous Dissipation Functions We now work with mesoscale dissipation functions for each body Bδ (ω) ∈ Bδ , such that: 1. The function depends on a˙ alone, and is star-shaped, convex, and homogeneous of degree r ∂δ a˙ , ω (10.58) a˙ i = r δ a˙ , ω ; ∂a˙ i 2. The function ∗ is star-shaped, convex and homogeneous of degree s = 1−r ∂∗δ Y, ω Yi = s∗δ Y, ω . (10.59) ∂Y i

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Φδ∗ ( Yδ )

∆Yδ Φδ(aδ)

aδ (b) Y = Y∞

Φ∞∗ (Y )

a = a∞

Φ∞(a)

(c) Yδ

Φδ∗(Yδ)

aδ

Φδ( aδ )

∆ aδ FIGURE 10.2 Thermodynamic orthogonality in: (a) spaces of velocities a˙ δ and ensemble average forces Yδ on mesoscale δ, with Yδ showing the scatter in Yδ ; (b) spaces of velocities a˙ ≡ a˙ ∞ and forces Y ≡ Y∞ on the RVE level, where the scatter is absent; (c) spaces of ensemble average velocities ˙aδ and forces Yδ on mesoscale δ, with ˙aδ showing the scatter in a˙ δ .

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Note that δ ( a˙ , ω) and ∗δ (Y, ω) are almost surely not inverses of one another because perfectly homogeneous domains of material carry probability zero in the space. The joint dependence of material response on the mesoscale δ and on the choice of independent variable (i.e., velocity a˙ or dissipative force Y) leads to two Legendre transformations for any Bδ (ω) ∈ Bδ : 1. Case of a˙ being an independent variable ∗δ Y, ω + δ a˙ = Y (ω) · a˙

(10.60)

2. Case of Y being an independent variable ∗δ Y + δ a˙ , ω = Y · a˙ (ω) .

(10.61)

Again, depending on how we take the ensemble averages (Ostoja-Starzewski, 1990), we find ∗δ Y + δ a˙ = Y · a˙ (10.62) and

∗δ Y + δ a˙ = Y · a˙ .

(10.63)

In the δ → ∞ limit, (10.60–10.63) become ∗eff

δ

eff Y + δ a˙ = Y · a˙ .

(10.64)

It is of interest to note here that the conventional Onsager–Casimir reciprocity relations, that is, those that apply to Figure 10.2(b), need to be reconsidered depending on whether we work in the space of a˙ or Y for finitesized bodies in Figures 10.2(a) and (c), respectively. Thus, in the first case we actually have two choices: when we are either on the surface δ ( a˙ ) of Figure 10.2(a) ∂ Yi ∂ Yj = (10.65) ∂a˙ j ∂a˙ i or on the surface δ (˙a) of Figure 10.2(c) ¯i ∂Y

∂Y = j . ∂ a˙ j ∂ a˙ i

(10.66)

When working in the space of Y, we also have two choices: when we are on the surface ∗δ (Y) of Figure 10.2(a), we have ∂a˙ i ∂Y j

=

∂a˙ j ∂Yi

(10.67)

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while on the surface ∗δ (Y) of Figure 10.2(c), we have ∂a˙ i ∂Y j

=

∂a˙ j ∂Y i

.

(10.68)

10.4.3 Quasi-Homogeneous Dissipation Functions A wide class of dissipative processes is described by dissipation functions ( a˙ , ω) of quasi-homogeneous type (Ziegler, 1963). Following the general framework given in Ostoja-Starzewski (1990), we now consider the apparent behavior to be described by dissipation functions of also quasi-homogeneous type on mesoscale, so that δ ( a˙ , ω) pertains to a finite-sized body Bδ (ω): a˙ i

∂δ ( a˙ , ω) ∂a˙ i

= f (δ ( a˙ , ω)),

(10.69)

where f is arbitrary. This, of course, implies that the mesoscale dissipation functions ∂∗δ (Y, ω) in the space of dissipative forces, Y, are quasi-homogeneous too, that is, Yi

∂∗δ (Y, ω) ∂Y i

= g(∗δ (Y, ω)).

(10.70)

Given the nonuniqueness of the mesoscale response, these two functions are not perfectly dual of each other. Clearly, we have two alternatives: 1. Assume velocity a˙ to be prescribed (controllable) for the body Bδ (ω), the result being Y. 2. Assume Y to be prescribed (controllable) for the body Bδ (ω), the result being a˙ . In the first case, on account of (10.70), for any Bδ (ω) we have Yi (ω) =

δ ( a˙ , ω) ∂δ ( a˙ , ω) ∂a˙ i f (δ ( a˙ , ω))

If for every Bδ (ω) we define a function φδ ( a˙ , ω) from δ ( a˙ , ω) by δ φδ ( a˙ , ω) = dδ f (δ )

(10.71)

(10.72)

and let the additional constant in (10.72) be fixed by setting φδ (δ ( a˙ , ω) = 0) = 0, upon ensemble averaging, we obtain ∂ φδ ( a˙ , ω) ∂φδ ( a˙ , ω) = Yi = . (10.73) ∂a˙ i ∂a˙ i Turning now to the space of dissipative forces, we may proceed in an analogous fashion. That is, we may either consider a random dissipation

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function ∗δ (Y, ω) in the space of controllable forces resulting in a random a˙ , or a deterministic ∗δ (Y) in the space of ensemble averaged forces Y such that ∂∗δ ( Y ) a˙ i = ν (10.74) ∂ Yi Relevant to our analysis leading to (10.74) is the latter situation. On account of (10.70) the connection between a˙ and Y reduces to ∗δ ( Y ) ∂∗δ ( Y ) ∂∗δ ( Y ) a˙ i = =µ (10.75) ∂ Yi g(∗δ ( Y )) ∂ Yi where

µ=

∗δ

−1 . ∂ Yi

∂∗δ ( Y )

If we now define a function ψδ (Y) from ∗δ (Y) by ∗δ d∗δ ψδ ( Y ) = g(∗δ ) and let ψδ (∗δ (Y) = 0) = 0, we can write, instead of (10.75) ∂ ψδ ( Y ) a˙ i = ∂ Yi

(10.76)

(10.77)

(10.78)

whereby φδ ( a˙ = 0) = 0

ψδ ( Y ) = 0) = 0.

(10.79)

We will now consider two curves: C in velocity space and its image C in force space. Curve C connects the origin O with a point P with coordinates a˙ , while C connects the origin O with the image P of P having coordinates Y. Thus, we have (in index notation) a˙ · d Y = (10.80) Y · d a˙ + d Y · a˙ = Y · a˙ . C

C

C

In light of (10.73), (10.78) and (10.79), this leads to a Legendre transformation corresponding to case (1) φ( a˙ ) + ψδ ( Y ) = Y · a˙ = ∗δ ( Y ). (10.81) An analogous analysis for case (2) results in a very similar Legendre transformation (duality between the results in the velocity space and those in the force space) φδ ( a˙ ) + ψδ (Y) = Y · a˙ = δ ( a˙ ) (10.82)

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where Yi = and

∂ a˙ i

∂φδ ( a˙ )

(10.83)

∂ ψδ (Y) a˙ i = . ∂Y i

(10.84)

The functions φδ (˙a) and ψδ (Y) in the above are defined by δ φδ ( a˙ ) = dδ δ ≡ δ ( a˙ ) f (δ )

(10.85)

and, for every Bδ (ω), ψδ (Y, ω) =

∗δ d∗δ ∗δ ≡ ∗δ (Y, ω). g(∗δ )

(10.86)

In both cases above we note the feature already made transparent earlier: when the ensemble average acts on the argument of a given functional, it acts on the dual functional itself.

10.5

Complex versus Compound Processes: The Scaling Viewpoint

10.5.1 General Considerations According to Ziegler (1983) the dissipative processes appearing in the Clausius–Duhem inequality may be classified as either elementary, compound, or complex. The first of these is characterized by a single coherent set of velocities a˙ , and governed by the thermodynamic orthogonality. The compound case (exemplified by the well-known split into mechanical and thermal processes) is the situation where each of the constituent, elementary processes (s = 1, . . ., S) is governed separately by its own thermodynamic orthogonalities Yk(d s)

=

∂(s) ν (s) (s) ∂a˙ i

ν

(s)

=

(s)

∂(s)

a˙ (s) (s) i ∂a˙ i

−1 =

S

(s) .

(10.87)

s=1

Here the last equality gives the dissipation function of the entire process in terms of those of elementary processes. On the other hand, the complex case is the situation where all the constituent, elementary processes are governed jointly by a single

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thermodynamic orthogonality Yk(d

s)

=ν

∂ (s)

∂a˙ i

ν=

∂

a˙ (s) (s) i

∂a˙ i

−1 ,

(10.88)

whereby there is no splitting of into dissipation functions of elementary processes. Suppose we now consider an unrestricted micropolar medium with displacement u and rotation ϕ being independent degrees of freedom of each continuum particle. (The case of a restricted [or couple-stress] medium is not interesting because, as discussed in Chapter 6, ϕ would then be given in terms of u.) The dissipation is involved in the processes described by velocities u˙ as well as ϕ. ˙ On a continuum level, we have two sets of conjugate variables:         velocity gradient l = vi, j dissipative force stress τ (d) . ←→  curvature rate κ   dissipative couple-stress µ(d)  ˙ = ϕ˙ i, j

(10.89) Thus, the question to ask is this: How do we decide whether it is a compound or a complex process? For guidance, given the ubiquity of helical structures in biomaterials, let us now consider a model chiral material made of helices (h) placed randomly in a matrix (m), Figure 10.3. Furthermore, both the helix and matrix material are of Cauchy type. There are three sources of dissipation: inelasticity of helices,

FIGURE 10.3 A realization B (ω) of a chiral material made of helices randomly placed in a matrix.

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inelasticity of the matrix and imperfect interfaces between helices and matrix. Observe: •

On the level of the material making up each helix or matrix: (symmetric) deformation rate d and symmetric Cauchy stress σ (d) are linking for either phase (s = h or m).

• (d p)

σij

=ν

∂( p)

=

∂dij

(h)

d

ν = ( p) (h)

+

(m)

∂( p) ∂dij

−1 dij

d(h) .

(10.90)

This a compound process. •

On the level of a single helix (as if it were isolated from the matrix) there is a coupling of axial with torsional dissipative actions just like that of an elastic helix (along its local axis x). Thus, we have a complex process: F (d) ∂(h) =ν AE ∂u˙ , x

M(d) ∂(h) = ν , E R3 ∂ϕ˙ , x

(10.91)

where (h) denotes the dissipation function of the helix. Note that the constitutive equations in the viscoelastic case are in the category of a complex process, Section 3.5.3 of Chapter 3. •

On the microscale (δ 1) through mesoscale: τij(d) = ν

∂δ ∂vi, j

µ(d) ij = ν

∂δ ∂ϕ˙ i, j

,

(10.92)

where, in principle, mesoscale bounds in the sense of the preceding section should be used. This would involve δ ( a˙ , ω) and ∗δ (Y, ω) for B(ω) ∈ B. •

On the macroscale (where δ is sufficiently large to reach RVE) the micropolar effects become negligible, so that equation (10.92) is replaced be a relation between the dissipative Cauchy stress σ (d) and the deformation rate d: −1 ∂ δ ∂ (d) (10.93) σij = ν ν= dij , ∂dij ∂dij where is responsible for the dissipation on all the lower scales.

We conclude: 1. The length scale is a factor deciding between the compound or complex nature of the dissipative process. 2. Even if the helical geometry of inclusions were removed, and we simply dealt with, say, classical ellipsoidal-shaped inclusions, by

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the argument of Section 6.5 of Chapter 6 extended to inelastic media, there would be micropolar effects on mesoscales from δ 1 and up. The corresponding dissipative process would be complex, but its micropolar effects would vanish in passage to very large scales δ. 10.5.2 Micropolar Plasticity As another example, let us focus on two possible extensions of the J 2 flow theory to micropolar materials (Muhlhaus, ¨ 1995; Forest and Sievert, 2003). Following these authors, start from the total strain rate γ˙ij as the sum of the elastic strain rate g˙ ij and the plastic strain rate γ˙ij∗ , with an analogous split of the total curvature rate κ˙ ij into the elastic (k˙ ij ) and plastic (κ˙ ij∗ ) parts κ˙ ij = k˙ ij + κ˙ ij∗ ,

γ˙ij = g˙ ij + γ˙ij∗

(10.94)

On one hand, one may formulate an extension of the Huber–von Mises criterion via a single equation f (σij , µij ) = J 2 (σ ji , µ ji ) − Y J 2 (σij , µij ) = a 1 sij sij + a 2 sij s ji + b 1 µij µ ji + b 2 µij µ ji ,

(10.95)

where sij is the stress deviator of σij , while Y, a 1 , a 2 , b 1 and b 2 are material parameters. The plastic rates follow this flow rule γ˙ij∗ = λ˙ σ

∂ f (σij , µij ) ∂σij

κ˙ ij∗ = λ˙ µ

∂ f (σij ,µij ) . ∂µij

(10.96)

On the other hand, one may set up a multicriterion model f σ (σij ) = J 2 (σ ji ) − Yσ ! J 2 (σ ji ) = a 1 sij sij + a 2 sij s ji

f µ (µij ) = J 2 (µij ) − Yµ J 2 (µ ji ) =

!

(10.97)

b 1 µij µ ji + b 2 µij µ ji ,

whereby the flow rule is γ˙ij∗ = λ˙ σ

∂ f σ (σij ) ∂σij

κ˙ ij∗ = λ˙ µ

∂ f µ (µij ) ∂µij

.

(10.98)

In view of the remarks immediately preceding this section, the model (10.95 and 10.96) is a complex process, and as such is better justified than (10.97 and 10.98), which is a compound process.

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Toward Continuum Mechanics of Fractal Media

This is the only place in the book where we admit fractal geometries of materials. While much has been said and written over the past three decades about mathematics and physics of fractals, field equations in the vein of continuum mechanics have been lacking. However, the recent work of Tarasov (2005a,b) allows progress in that direction. We start by relaxing the assumption (1.109) of Chapter 1: the properties of fractal media like mass m obey a power law relation m ( R) = k R D

D < 3,

(10.99)

where R is a box size (or a sphere radius), D is a fractal dimension of mass, and k is a proportionality constant. It follows that the power law (10.99) describes the scaling of mass with R. Note: In this section we depart from the convention used in the book of D indicating the physical dimension. That is, d shall denote the latter, while D is now reserved for the fractal dimension. Focusing on porous media, the power law relation (10.99) is rewritten as D R m D ( R) = m0 , (10.100) Rp where R p is the average radius of a pore, and m0 is the mass at R p = R; this is a reference case. At this point, the conventional equation giving mass in a 3D region W of volume V m (W) = ρ (r) d 3 r (10.101) W

has to be generalized m3d (W) =

23−D (3/2) ( D/2)

ρ (r) |r − r0 | D−3 d 3 r,

(10.102)

W

where µ D is the measure in the 3D space. Assuming the fractal medium to be spatially homogeneous ρ (r) = ρ0 = const, the equation (10.102) is replaced using a fractional integral 23−D (3/2) |R| D−3 d 3 r, m3d (W) = ρ0 ( D/2) W

(10.103)

(10.104)

where R = r − r0 . That is, the fractal medium with a noninteger mass dimension D is described using a fractional integral of order D. This allows an interpretation of the fractal (intrinsically discontinuous) medium

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as a continuum. In particular, the next step is Tarasov’s reformulation of the Green-Gauss Theorem Aυκ nk dSd = c 3−1 ( D, R) div(c 2 (d, R) Av)dV D, (10.105) ∂w

W

Where A is an arbitrary function, v is the velocity, and dSd = c 2 (d, R)dS2

dV D = c 3 ( D, R)dV 3

(10.106)

On account of (2.9), the " left hand side in (2.8) is a fractional integral, equal to a conventional integral ∂W c 2 (d, R) AvdS2 . Similarly, the"right hand side in (2.8) is a fractional integral, equal to a conventional integral W div(c 2 (d, R) Av)dV 3 . See also (2.14) below. The above formulation allows the derivation of fractional-type balance equations of fractal media: •

the fractional equation of continuity: d ρ = −ρ∇kD υk , dt D

(10.107)

•

the fractional equation of balance of density of momentum: d ρ υk = ρ f k + ∇lD σkl , (10.108) dt D

•

the fractional equation of balance of density of energy: d ρ u = c( D, d, R)σkl υk,l − ∇kD q k . dt D

(10.109)

In the above σkl is the Cauchy stress (symmetric according to the balance of angular momentum, employed just like in non-fractal media), and the following operators (or, generalized derivatives) are used ∂

∇kD A = c 3 ( D, R) [c 2 (d, R) A] ≡ c 3 ( D, R)∇k [c 2 (d, R) A] , ∂xk d ∂A ∂A + c ( D, d, R) υk A= , (10.110) dt D ∂t ∂xk where 2 D−d−1 ( D/2) (3/2)(d/2) 22−d c 2 (d, R) = |R| D−3 (d/2)

c( D, d, R) = |R|d+1−D

c 3 ( D, R) = |R|2−d

23−D (3/2) ( D/2)

c( D, d, R) = c 3−1 ( D, R)c 2 (d, R).

(10.111)

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Henceforth, for simplicity of notation, we write c, c 2 , and c 3 . Note that, in a non-fractal medium (D = 3, d = 2) c (D, d, R) = 1, whereby one recovers conventional forms of local relations of continuum mechanics. Of further use will also be the fact that the relation ∇kD AB = A∇kD B + B∇kD A does not hold, and should instead be replaced by ∇kD AB = A∇kD B + c B∇k A.

(10.112)

Proceeding in this vein, the Clausius-Duhem inequality for fractal media takes the form d d d T, k q k (d) 0 ≤ Tρ , , s (i) = σi(d) u +β αi j − c( D, d, R) (i j) j ij dt D dt D dt D T (10.113) where lim

D→3

d dt

u(i , j) = υ(i j) ≡ di j ,

(10.114)

D

which is just the deformation rate. Also, for small strains di j is (d/dt) D with an analogous limit lim

D→3

d dt

εi j,

εi j

≡ ε˙ i j .

(10.115)

D

It is most interesting that generalized derivatives appear only for the time rates of external and internal strains but do not arise in the third term in (10.113) except for the coefficient c( D, d, R). One can now identify the term Tρ (d/dt) D s (i) as the dissipation function φ in three velocity-like arguments { [(d/dt) D u(i ], j), (d/dt) D αi j, q k } and a number of relations of Ziegler’s theory, including the thermodynamic orthogonality, carry over to fractals. We can also generalize the Hill condition to quasi-static loadings, and on account of previous relations, we have c σ : d = σ : d ⇔ c σ : d = 0.

(10.116)

Adapting the same route as in Section 7.2.2.1, we find the necessary and sufficient condition for (10.116)2 to be (t − σ · n) · (v − d · x) c 2 d A2 = 0. (10.117) ∂W

which dictates possible loadings on the boundary of a fractal body. Observe that Ziegler’s thermomechanics is a formalism very suitable for generalization to random media precisely because it allows scale-dependent homogenization in the vein of Hill condition where (i) either the applied strain

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or the applied stress are on an equal footing, and (ii) the energy or power of dissipation is the key criterion for equivalence between a heterogeneous structure and a smoothing continuum. By contrast, the rational thermomechanics of Truesdell does not jibe with Hill’s condition because there the stress is taken as a primary quantity while the energy as a secondary one (Ball & James, 2002).

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11 Waves and Wavefronts in Random Media

To deal with thermodynamics is to look for trouble. G.A. Maugin, 1999 Heeding the above warning, we now switch to a field where thermodynamics is not of central importance: waves in random media, also called stochastic wave propagation. That research activity, fast growing since the middle of the twentieth century, has primarily been motivated by various problems arising in acoustics, atmospheric physics, geophysics, and composite materials, see the reviews and books of Chernov (1960), Frisch (1968), Dence and Spence (1970), Uscinski (1977), Sobczyk (1985, 1986), Rytov et al. (1987), Papanicolaou (1998). Mathematical problems in all these applications have typically been set up as ordinary of partial differential equations on random fields with either discrete or continuous realizations. A key characteristic of random fields has been the correlation length, and most studies have focused on the most tractable situations of wavelengths being either much smaller or much larger than the typical size of heterogeneity; recall Section 2.2 in Chapter 2. Generally speaking, in stochastic wave propagation we must have three length scales: (1) the typical propagation distances L macro ; (2) the typical wavelength λ or wavefront thickness L; and (3) the typical size of inhomogeneity d. Most studies have focused on linear elastic waves, and in this chapter we only give a very brief introduction to these topics, which are expertly covered in the classical references listed above, and then consider less conventional problems. Thus, in Section 11.1 we first discuss two basic cases of wavelengths λ being either much longer or much shorter than the heterogeneity, and in the latter case we reconsider random geometric acoustics when, in contradistinction to the common assumption, the elastic medium is not necessarily locally isotropic. In Section 11.2 we introduce the concept of stochastic spectral finite elements, which provides a setting for analyses of steady-state vibrations in random structural elements, without any assumptions on separation of scales between L macro , λ, and d, but requires a numerical solution for obtaining quantitative results. Section 11.3, while still set in the harmonic regime, focuses on a surprisingly unconventional behavior of disordered one-dimensional (1D) composites; the presentation outlines a little-known stochastic homogenization technique (random evolutions) that may also be useful in other

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mechanics problems. Section 11.4 is devoted to transient waves in piecewise constant random media with linear or nonlinear elastic, and then elasticdissipative, responses. It is shown through a generalization of the method of characteristics to random media setting, that even weak material randomness may strongly affect conventional solutions of homogeneous media. Finally, Section 11.5 examines the evolution of acceleration wavefronts whose thickness is not infinitesimal as conventionally done in the singular surface wave analysis (e.g., Chen, 1976), but finite. This aspect forces us to replace the classical RVE of deterministic continuum mechanics by the mesoscale SVE, leading us to uncover various characteristics of blow-up of acceleration wavefronts in random media.

11.1

Basic Methods in Stochastic Wave Propagation

11.1.1 The Long Wavelength Case 11.1.1.1 Elementary Considerations The starting point in classical analyses of wave propagation in random media is offered by the wave equation for a scalar field u in a domain X ∇ 2u =

1 c 2 (ω,

∂2 u

x) ∂t 2

,

ω ∈ ,

x ∈ X.

(11.1)

Here c(ω, x) is a random field, that is, an ensemble {c(ω, x), ω ∈ , x ∈ X }. Formally speaking, we have a triple (, F,P), where is the space of elementary events, F is its σ -algebra, and P is the probability measure defined on it. We sometimes write ω explicitly to show the random character of a given quantity like c(ω, x), or else, we suppress it for clarity of notation. Assuming harmonic time dependence (e iγ t ), we obtain the scalar stochastic Helmholtz equation ∇ 2 u + k02 n2 (ω, x)u = 0,

ω ∈ ,

x ∈ X,

(11.2)

where, writing k02 n2 (ω, x) for k(ω, x), we introduce a random wave number to deal with the spatial randomness of the medium. Thus, k0 = γ /c 0 is the wave number of a reference homogeneous medium where c 0 is its phase velocity, and n(x, ω) is a random index of refraction. Hereinafter, we employ γ for the frequency, rather than the conventional ω, which has been reserved to denote an outcome ω (i.e., a random medium’s realization) from the sample space . Equation (11.2) is a valid ansatz whenever the time variation in the refractive properties of the medium is much slower than the wave propagation itself; thus, for example, swirling as rapid as the wave motion violates the monochromaticity assumption. The random field {n(ω, x), ω ∈ , x ∈ X } is determined from experimental measurements. At this point, it is convenient to

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consider the model n2 (ω, x) = 1 + µ(ω, x)

µ(ω, x) = 0

µ = O (1) ,

(11.3)

so that all the randomness is present in the zero-mean random field µ. The key role is played by a correlation function of µ, which, given (11.3)2 , is Cµ (x, x ) = µ(x)µ(x ).

(11.4)

Usually, µ is taken as a wide-sense stationary random field Cµ (x, x ) = Cµ (x − x ) < ∞,

∀x, x ,

(11.5)

possessing an ergodic property almost surely in and X µ (x) = µ (ω).

(11.6)

Rytov et al. (1987) also discuss more general random field models such as, say, those with stationary increments. A special class of so-called isotropic random fields occurs when ρ(x) depends only on the magnitude, but not direction, of the vector x √ Cµ (x) = Cµ (r ), r = |x| = xi xi . (11.7) A very common model for the correlation coefficient is the Gaussian form (a special case of (2.89)) Cµ (r ) = µ2 exp[−r 2 /a 2 ], (11.8) where a is the so-called correlation radius. Determination of the random field {µ(ω, x), ω ∈ , x ∈ X } is made on the basis of experimental measurements. As Chernov (1960) shows, this function may present a very good fit, Cµ (r ) = µ2 exp[−r/a ], (11.9) but one must bear in mind that it corresponds to random fields with discontinuous, rather than continuous, realizations. Indeed, Hudson (1968) took precisely this form to model scattering in a granular/cellular structure of a polycrystal. 11.1.1.2 Series Expansion Returning to the stochastic Helmholtz equation, one takes 1, and considers its solutions in the form of an expansion with respect to the powers of u(ω, x) = u0 (x) + u1 (ω, x) + 2 u2 (ω, x) + · · · ·

(11.10)

Note that the Russian school (Chernov, 1960; Rytov et al., 1987) takes n(ω, x) = 1 + µ(ω, x) in place of (11.3)1 .

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Upon substituting (11.3)1 and (11.10) into (11.1) and equating terms of the same order in , one obtains a system of recurrence equations for u0 , u1 , u2 , and so on ∇ 2 u0 + k02 u0 = f (x) ∇ 2 u1 + k02 u1 = −k02 µ(ω, x)u0 (x)

(11.11)

∇ 2 u2 + k02 u2 = −k02 µ(ω, x)u1 (x). It is seen that the solution u0 to a homogeneous medium problem serves as forcing to the first correction u1 , which then drives the third equation in (11.11) governing u2 , and so on. Thus, the perturbation approach reflects a multiple scattering nature of stochastic wave propagation. Using Green’s function for a free space Helmholtz equation of the homogeneous medium exp[ik0 |x − x |] G 0 x, x = , −4π |x − x |

(11.12)

the solutions to equations (11.11) may be calculated from u0 (x) = G 0 (x, x ) f x dx u1 (x) = −k02 u2 (x) =

k04

G 0 x, x µ ω, x u0 x dx G 0 x, x G 0 x , x µ ω, x µ ω, x u0 x dx dx . (11.13)

Five aspects are important with respect to (11.13): 1. The above solution is nonlinear in the boundary conditions. 2. The ensemble average of u1 (x) is zero. 3. Solution ui (x) represents the perturbation of the wave field original ui−1 (x) caused by the inhomogeneity field µ ω, x , i = 1, 2, . . .. This characterizes stochastic wave propagation as a successive multiple scattering. 4. The above solution may be obtained using an integral equation formulation. First, write the solution to (11.2) as 2 u1 (x) = u0 (x) − k0 G 0 (x, x )µ(ω, x )u x dx (11.14) and iterate to obtain the perturbation expansion. To find the first iteration of (11.14) we write the value of the field at x = x u1 x = u0 x − k02 G 0 (x , x )µ(ω, x )u x dx (11.15)

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and substitute this into the right-hand side of (11.14) so as to get 2 u1 (x) = u0 (x) − k0 G 0 (x, x )µ(ω, x )u x dx +k04

G 0 (x, x )µ(ω, x )u x dx

G 0 (x , x )µ(ω, x )u x dx . (11.16)

To find the second iteration we write the value of the field at x = x and substitute the result into the right-hand side of (11.16). This leads to an infinite perturbation (or Neumann) series, for the integral equation (11.14), the first three terms of which are given in (11.13). Following Frisch (1968), it can be shown that this series’ convergence requires Mk02 D2 /2 < 1,

(11.17)

where D is the diameter of the scattering region, and |µ (ω)| < M almost surely. 5. A particular case of (11.13) occurs if one considers single scattering only by assuming u (ω, x) = u0 (x) + u1 (ω, x) ,

(11.18)

rather than (11.10)—this is called (first) Born approximation. It is popular in applications as it simplifies the analysis, but it imposes strong restrictions on the validity of this perturbation approach: (a) The noise-to-signal ratio must be low: 1 in (11.3). (b) The size of inhomogeneity must be much smaller than the wavelength: d λ. Given the particular form (11.8) of the correlation function, this would be equivalent to stating that the correlation length needs to be much smaller than the wavelength: d λ. In fact, in that case, under the assumption that the dimensions of the scattering domain V are much larger than the correlation radius a , the intensity of the scattered field is computed as

I |u1 ( P)|

2

k04 a 3 V µ2 exp − k02 a 2 sin2 θ/2 . = √ 16 π R02

(11.19)

Here R0 is the distance from the scattering region to the observation point P, and θ is the scattering angle—the angle between the wave vector ki of the incident wave and the wave vector ks of the scattered field. The above result may be used to obtain various physically interesting parameters. For example, considering that the scattered energy in a region of size L = V 1/3 must be much smaller than the total energy, we obtain the

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condition

µ2 k 0 L

1

k0 a 1 − e −k0 a

2 2

.

(11.20)

Because k0 a = 2πa /λ, this shows that the correlation radius a must go down as the domain size L goes up, for the Born approximation to be valid. 11.1.2 The Short Wavelength Case: Ray Method 11.1.2.1 Fermat’s Principle It is well known that the case of short wavelengths falls into the realm of geometric acoustics (or optics) in which all the disturbances propagate according to Fermat’s principle

t1

dt = min .

(11.21)

t0

Here t0 and t1 denote the initial and final instants on a path from points P0 (x) to P1 (x). Assuming the medium is pointwise (i.e., locally) isotropic, we have c = c(x), and thence follows a description by a field of circular indicatrices of Figure 11.1(a). Using the Euler–Lagrange equations, in a d-dimensional setting, one obtains the well-known equations of ray dynamics (e.g., Hudson, 1980) dxi = cyi ds

dyi ∂ = ds ∂xi

1 i = 1, . . ., d, c

(11.22)

where s is the arc length along the ray x(s), propagating in direction y(s) at a local speed c(x). Associated with (11.22) there is an eikonal equation: (11.31) below. All of the above involves the assumption of a locally isotropic, inhomogeneous medium, possibly of a spatially random character, and so c(x) should be interpreted as c(ω, x). However, a spatial gradient of smooth elastic moduli suggests that a constitutive response in one principal direction is very likely different from that in another. Therefore, turning to a medium of anisotropic type, we should have c(x, x˙ ), and, corresponding to it, a pointwise description by a field of nonspherical indicatrices of Figure 11.1(b); see also Nye (1957). Here x˙ i = ∂xi /∂l, l being a parameter of the path. Indeed, in taking c(x, x˙ ) we are motivated here by the discussion of Huygens’ principle in Arnold (1978). The equations governing the dynamics of a ray—that is, its position x and direction y (= c −1 dx/ds)—are now found by generalizing the variational procedure leading from (11.21) to (11.22) (Ostoja-Starzewski, 2001c). Thus, for

t1

t0

dt =

t1

t0

ds = c

t1

t0

σ dl = min, c

(11.23)

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p

y

(a)

y

p

(b) FIGURE 11.1 Indicatrix envelopes in two realizations of a random medium: (a) locally isotropic; (b) locally anisotropic; see also Arnold (1978). In both cases, the direction of the wavefronts motion p and the direction of the ray y are shown.

we identify the Lagrangian as L=

σ 1 = [x˙ i x˙ i ]1/2 i = 1, . . ., d, c(x, x˙ ) c(x, x˙ )

(11.24)

where the ray x(s) propagates at a local speed c(x, x˙ ). From the Euler-Lagrange equations, with s (or l) playing the role of time, we now find a dynamical system of rays for an anisotropic medium dxi = cyi ds

d ∂ σ ∂c ( yi − 2 )= ds c ∂x˙ i ∂ xi

1 i = 1, . . ., d c

(11.25)

in place of (11.22) in the isotropic case. Let us now consider the SH wave motion of a generally anisotropic medium of local elastic property Cij (x) ≡ Cikjm (x); i, j = 1, 2; k, m = 3. Clearly, the relevant governing equation is then (Cij u, j ) ,i = ρ u. ¨

(11.26)

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For isotropy (Cij (x) ≡ C√ 0 (x)δij ), (11.26) reduces to the classical wave equation ¨ 2 , with c(x) = [C0 (x)] /ρ being the local wave speed. u,ii = u/c Let us now assume a trial solution of the form u(x, t) = A(x) exp [i[κ0 S(x) − ωt]] ,

(11.27)

where κ0 = ω/c 0 = 2π/λ0 is the wave number in the reference homogeneous medium (c(x) = c 0 = const), and A(x) and S(x) are to be determined. The latter two quantities are supposed real as there is no dissipation. Upon substituting (11.27) into (11.26), we find equations governing, respectively, the real part Cij A,ij − κ02 ACij S,i S, j − A,i Cij,j = −Aρω2 ,

(11.28)

and the imaginary part 2Cij A,i S, j + ACij S,ij + ACij,i S,i = 0.

(11.29)

As a first approximation for a slowly varying medium, we may ignore the second derivatives of A as well as the products of first derivatives of Cij with those of A in (11.28), obtaining the eikonal equation for an anisotropic medium κ02 Cij S,i S, j = ρω2 ,

(11.30)

which, for Cij (x) ≡ C0 (x)δij , reduces to the classical eikonal equation for an isotropic medium S,i S,i = n2

or

|∇ S|2 = n2 .

(11.31)

Here n(x) ≡ c 0 /c(x) is the refractive index. Similarly, ignoring the products of first derivatives of Cij with those of S in (11.29), we find 2Cij A,i S, j + ACij S,ij = 0,

(11.32)

which, for Cij (x) ≡ C0 (x)δij , reduces to the classical form (e.g., Elmore and Heald, 1969) 2A, j S, j + AS,ii = 0

or

2∇ A · ∇ S + A∇ 2 S = 0.

(11.33)

A stepping-stone to extending this analysis to general elastodynamics is ´ in Epstein and Sniatycki (1992). 11.1.2.2 Markov Character of Rays Already in the simplest case when c is a locally isotropic random field—that is, when we have equation (11.22)—there arose various analytical problems and approximations. First, Kharanen (1953) and Chernov (1960) treated the problem in a “dishonest” way; that is, they assumed that the direction of the ray y(s) was Markov in s. This assumption enabled them to formulate a Fokker–Planck equation in order to find the probability density of the rays

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p(y; s). On the other hand, Keller (1962) employed a perturbation expansion to obtain results valid only for short ray paths; this was an honest method in that no unjustified probabilistic assumptions were made a priori. An intuitive justification of the Markov property can be provided by an analogy of evolution of y(s) to the evolution of the velocity of a Brownian particle that has suffered many independent collisions. Thus, the dishonest methods are good for long ray paths only. A solution method valid for all ray paths has been developed by Frisch (1968), who, using a first-order perturbation, cast the system (11.22) for a locally isotropic medium, in the form of a stochastic Liouville equation for the rays, which led to a kinetic equation for random geometric optics. Motivated by the limitation of the Markovian assumption, P´erez et al. (2004) introduced a fractional Brownian motion model so as to account for memory along the ray’s paths. Another line of approach to the problem of evolution of stochastic rays in a random medium has been pursued by Brandstatter and Schoenberg (1975), whose simplified model was dx = c (x) edt + g (x)

e = cy,

(11.34)

where g(x) is a random noise with zero mean, finite covariance and all moments higher than the second being on the order O( t), e is the unit vector. The effort of mathematical analysis, in the framework of dynamic programming, was on finding a policy e (x) such that (11.21) obtains in the ensemble average sense. Here we observe that, substituting dt/ds = 1/c, (11.22) becomes dx = c 2 (x) ydt

dy = c (x) ∇ydt.

(11.35)

Clearly, (11.35)1 —or, equivalently, dx = c (x) edt—stands in stark contrast to (11.25), and we conclude that the approach of Kharanen (1953) consisted in assuming the Markov character for the y variable according to the dynamics of (11.35)2 without taking account of (11.35)1 , while the approach of Brandstatter and Schoenberg (1975) consisted in replacing the random system (11.35) by an equation, with an additive, rather than multiplicative, noise for the position of the mean ray. Our equation (11.25) shows that the Markov character of {x, y}s is preserved in a locally anisotropic medium, but an explicit solution for the statistics of rays will be more challenging than in the isotropic case. In fact, such a solution via Itoˆ calculus appears unwieldy, and a recourse to a numerical solution of the stochastic dynamical system is necessary. 11.1.3 The Short Wavelength Case: Rytov Method While the Born method is based on the expansion of the wave field u with respect to a small parameter , the Rytov method relies on the expansion of ln u. First, the solution to the stochastic Helmholtz equation without forcing (11.2), with (11.3) as before, is represented as u (ω, x) = A0 exp [−i (ω, x)].

(11.36)

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Here A0 is the wave amplitude in the reference homogeneous medium. Substituting (11.36) into (11.2), we obtain a nonlinear equation (∇ ) 2 + i∇ 2 = k02 n2 (ω, x).

(11.37)

If we look for its solution in the form of an expansion

(ω, x) = 0 (x) + 1 (ω, x) + 2 2 (ω, x) + · · ·,

(11.38)

again under the assumption (11.3), we obtain a recurrent sequence of equations for 0 , 1 , 2 , and so on. The simplest situation results if we restrict ourselves to the first two terms of (11.38), (ω, x) = 0 (x) + 1 (ω, x), in which case we arrive at (∇ 0 ) 2 + i∇ 2 0 = k02 2 (∇ 0 ) (∇ 1 ) + i∇ 2 1 = k02 µ (ω, x) − (∇ 1 ) 2 .

(11.39)

Equation (11.39)1 has a plane-wave solution 0 = k0 x (x ≡ x1 ), so that (11.39)2 becomes 2k0

∂ 1 ∂x

+ i∇ 2 1 = k02 µ (ω, x) − (∇ 1 ) 2 .

(11.40)

Using an auxiliary function φ1 (x), defined via 1 (x) = φ1 (x) e −ik0 x , we find an inhomogeneous Helmholtz equation ∇ 2 φ1 + k02 φ1 = −2ik02 µ (ω, x) e −ik0 x .

(11.41)

whose solution is φ1 =

ik02 2π

1 ik0 ( R+ξ ) µ (ξ, η, ζ ) dξ dηdζ. e R

(11.42)

where R = [(x − ξ ) 2 + ( y − η) 2 + (z − ζ ) 2 ]1/2 is the distance between the observation point at r and the scattering element at (ξ, η, ζ ). This results in

1 (ω, r ) =

ik02 2π

1 ik0 [R−(x−ξ )] e µ (ξ, η, ζ ) dξ dηdζ, R

(11.43)

which shows the advantage of the Rytov method: the random wave field

1 (ω, r ) is expressed as a linear transformation of a given random field µ (ω, r ), and hence, the moments of (ω, r ) = 0 (r ) + 1 (ω, r ) can be computed by averaging over and integrating over the physical space. The Rytov method and the older method of geometric optics apply to situations in which the size of inhomogeneity is much larger than the wavelength: d λ. However, since the geometric optics is also limited by a condition on the length L of the path propagated by the ray, that is, d λL, the Rytov method offers a more powerful avenue. These and other related issues are discussed at length and in depth in the references listed in this

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section. See also Sections 4–6 in a recent review of methods for stochastic Helmholtz equation (Beran, 2001). In particular, he discusses single and multiple scattering problems, and the wave transmission problem in a 1D random medium with back-scattering fully accounted for. Two reviews of elastodynamics of random media, especially from the standpoint of variational principles, are given in Willis (1997, 2001).

11.2

Toward Spectral Finite Elements for Random Media

11.2.1 Spectral Finite Element for Waves in Rods 11.2.1.1 Deterministic Case The classical approach via stochastic Helmholtz equation (11.2) grasps, through the random field of refraction index, the spatial variability in the mass density but not the variability in elastic moduli or geometric parameters. This is immediately seen by considering the equation governing axial motions in a rod with a space-dependent mass density ρ, elastic modulus E, and cross-sectional area A, namely,

∂ ∂ ∂2 A(x, ω) E(x, ω) u(x, t) = ρ(x, ω) A(x, ω) 2 u(x, t), ω ∈ . (11.44) ∂x ∂x ∂t In order to remove the aforementioned restrictions—at least in this 1D model—and to analyze the relative effects of spatial randomness of mass density, elastic properties, as well as cross-sectional geometric properties, we may consider stationary responses of rods in longitudinal vibrations, and of Timoshenko beams in flexural vibrations. This naturally leads to a spectral approach, in which, given the randomness of a rod or a beam, we seek a stochastic spectral finite element (Ostoja-Starzewski and Woods, 2003), which presents a generalization of a spectral finite element (e.g., Doyle, 1997). In principle, techniques such as those presented in Section 11.1 could be employed to tackle equation (11.44). However, the analysis would have to be restricted to the situation where the separation of scales would at least approximately be satisfied, that is, where either λ d or λ d holds. For the sake of reference, let us first recall basic concepts of deterministic spectral finite elements through a paradigm of a rod made of a homogeneous material. The elastodynamic equation governing the axial response of a rod (assuming zero external forcing) is well known: ∂2 u ∂x2

=

1 ∂2 u . c a2 ∂t 2

(11.45)

√ Here c a = E/ρ denotes the phase velocity of axial waves; E being the elastic axial modulus and ρ the mass density. Assuming u(x, t) = u(x)e iγ t , the Helmholtz equation corresponding to (11.45) set up over the domain X of

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∧ F1

∧

u1

u2

∧

F2

FIGURE 11.2 Finite element for a rod in longitudinal motion.

size L, is d 2 u dx2

u = 0, + k2

u(0) = u1 ,

u(L) = u2 ,

(11.46)

where k = γ /c a . Then, the spectral matrix expresses a connection between 1 , F 2 }, at both the kinematic and dynamic quantities, that is, { u1 , u2 } with { F ends of the rod (1 and 2) in Figure 11.2, where the hat signifies the quantities in the frequency space. Considering solutions in the form u(ξ ) = Asin kξ + 1 = − F (0) and F 2 = F (L), we readily B sin k(L − ξ ), with the definitions F find the following spectral matrix 1 F k cot k L −k csc k L u1 = AE . (11.47) 2 −k csc k L k cot k L u2 F This representation demonstrates the purely real nature of the spectral matrix. In Figure 11.3 we plot the k11 -component of this matrix as a continuous black line, with the peaks of k cot k L representing the resonant frequencies of the system with A = 10−4 m, E = 27.4 GPa and ρ = 2, 400 kg/m3 . These values correspond to a rod made of concrete. We are now interested in the change from this “crisp” functional form, and the associated scatter as we go to the random rod. 11.2.1.2 Random Case The frequency space version of the stochastic equation (11.44), again with Dirichlet boundary conditions, is

d d u A(x, ω) E(x, ω) + ρ(x, ω) A(x, ω) u(x) = 0, ω ∈ dx dx u(0) = u1 ,

u(L) = u2 .

(11.48)

There are many ways to simulate imperfect microstructures (e.g., Jeulin and Ostoja-Starzewski, 2001), and some definite choices have to be made in the case of rods, which themselves are simplified 1D models of 3D random bodies. Focusing on a random field model of a band-limited type, rather than on trying to approximate some “nice” function, we therefore assume the mass density, elastic modulus, and cross-sectional area to be described by random Fourier series (Chapter 2) with a typical/average characteristic size of inhomogeneity d, which is either smaller, comparable to, or larger than the wavelength. The third length scale entering the problem, but kept constant,

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5

(a)

4

Log10 κ11

3 2 1

0 0.0E + 00 1.0E + 04 2.0E + 04 3.0E + 04 4.0E + 04 5.0E + 04 6.0E + 04 7.0E + 04 –1 –2 –3 Frequency (Hz) 5

(b)

4

Log10 κ11

3 2 1

0 0.0E + 00 1.0E + 04 2.0E + 04 3.0E + 04 4.0E + 04 5.0E + 04 6.0E + 04 7.0E + 04 –1 –2 –3 Frequency (Hz) 5

(c)

4 3 Log10 κ11

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FIGURE 11.3 Rod vibrations in the case of random modulus E showing k11 (black line) for the reference homogeneous medium and < k11 > (gray line) for the random case with: (a) g = 0.1, (b) g = 1.0, (c) g = 10.0.

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is the rod length. Thus, for A we have 10 (i) (i) A(x, ω) = A0 1 + ε A a A (ω) cos igx + b A (ω) sin igx ,

(11.49)

i=1

with completely analogous models for ρ(x, ω) (having coefficients a ρ(i) , b ρ(i) ) (i) = 1, . . . , 10, and E(x, ω) (having coefficients a (i) E , b E ). Here, for i (i) (i) a A (ω), . . . , b E (ω) come from a uniform distribution on [−1/2, 1/2] and ω ∈ . The spectral finite element is now given by a relation 1 u1 k11 (ω) k12 (ω) F = , ∀ω ∈ , (11.50) 2 k21 (ω) k22 (ω) u2 F but, since we deal with a differential equation with inhomogeneous coefficients (11.49) in (11.48), the kij s cannot be determined explicitly. In the ensemble setting, in view of the spatial homogeneity of the random field (11.49), the averages satisfy k11 = k22 . A numerical method has been developed to obtain the kij s, and a sample of results for k11 , corresponding to a random E for g = 0.1, 1 and 10, respectively, is shown in Figure 11.3. The deterministic case already discussed above is shown as a crisp, black line, while the random case is shown as a gray thicker line, possibly overlapping the first one. Thus, whenever we only see the gray line, there is no difference between the deterministic and the mean of the stochastic problem. Here we see a strong departure from the reference case at and around the resonant frequencies. Note that the “scatter interval” increases with the increasing frequency. The effects decrease with increasing g but, regardless of the value of g, the effects are most noticeable at higher frequencies. Very similar results are obtained for random mass density ρ, and for ρ and E randomized together. On the other hand, the effect of random cross-section A is quite different—basically, it has influence only at g ≥ 1.0 and, in particular, for higher frequencies. 11.2.2 Spectral Finite Element for Flexural Waves 11.2.2.1 Deterministic Case The dynamics of a Timoshenko beam with spatially inhomogeneous properties is governed by two coupled equations: ∂ ∂ ∂x

EI

∂φ ∂x

∂x

+ G Aκ

G Aκ

∂v ∂x

∂v ∂x

−φ

=

−φ

+ ρ I ω2 φ =

∂2 v ∂t 2 ∂2 φ ∂t 2

, .

(11.51)

Clearly, there are two kinds of wave motion in such a beam: flexural and rotational; it is a 1D micropolar continuum indeed. The spatial inhomogeneity

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is modeled again by random Fourier series, although it now involves five (rather than three) independent parameters appearing in the governing equations: mass density ρ, elastic modulus E, Poisson’s ratio ν, area Aand moment of inertia I of the cross-section. It appears more physical, however, to work with the cross-sectional height h and cross-sectional width w instead of the latter two. Thus, the beam is described by a five component random field [ρ, E, ν, h, w]. Specializing to the spatially homogeneous case, the frequency space equations governing the transverse deflection v(x, t) = v(x)e iγ t and the transverse shearing deformation w (as measured by the difference ∂v/∂x − φ, with iγ t φ(x, t) = φ(x)e ) of a Timoshenko beam, assuming zero external forcing, are: d 2 v dφ G Aκ − + ρ Aγ 2 v=0 2 dx dx

d v d 2φ = 0. − φ + ρ I γ 2φ E I 2 + G Aκ (11.52) dx dx Here G is the shear modulus, Ais the cross-sectional area, κ is the shape factor of the cross-section, ρ is the mass density, E is the elastic modulus, and I is the cross-sectional area moment of inertia. The spectral stiffness matrix expresses now a connection between the kine1 , M 1, V 2 , M 2 }, 1 , 2 } with {V matic and the dynamical quantities, that is, { v1 , φ v2 , φ at both ends of the beam, Figure 11.4. In the derivation of the spectral stiffness matrix for the Timoshenko beam equations (11.52) one employs the boundary conditions v(0) = v1 ,

1 φ(0) =φ

v(L) = v2 ,

2 φ(L) =φ

(11.53)

set up over a domain of length L, and considers a solution of the form v(x) = B1 Rt cos k1 x − B2 Rt sin k1 x + C1 Rh cosh k2 x + C2 Rh sinh k2 x φ(x) = B1 sin k1 x + B2 cos k1 x + C1 sinh k2 x + C2 cosh k2 x. ∧

V1

∧

v1

∧

φ1 ∧

M1

FIGURE 11.4 Finite element for a beam in flexural motion.

∧

v2

∧

V2 ∧

φ2 ∧

M2

(11.54)

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Here Rt =

G Aκk1 ρ Aγ 2 − G Aκk12

and

Rh =

G Aκk2 ρ Aγ 2 + G Aκk22

(11.55)

are the so-called amplitude ratios and the boundary conditions of (11.53) specify the constants B1 , B2 , C1 , and C2 . See Doyle (1997) for a complete derivation. The k11 -component of the resulting spectral stiffness matrix is plotted in three plots of Figure 11.5 in black, as the reference case; the same material values as before are employed, and the peaks indicate the resonant frequencies. 11.2.2.2 Random Case From (11.52) we easily find the frequency space equations of an inhomogeneous Timoshenko beam

d Ehwκ d v −φ + ρhwγ 2 v = 0 dx 2(1 + ν) dx

d Eh3 w d φ ρh3 w 2 dv Ehwκ −φ + γ φ = 0. (11.56) + dx 12 dx 2(1 + ν) dx 12 We again employ a random field of a band-limited type, and model the mass density ρ, elastic modulus E, cross-sectional height h, cross-sectional width w, and Poisson’s ratio ν in the same way as in (11.49). The spectral stiffness matrix      1 V v1 k11 k12 k13 k14        M1   k21 k22 k23 k24   φ1     (11.57)   = v  2  V2   k31 k32 k33 k34   2 k41 k42 k43 k44 2 φ M is computed by the numerical method outlined before. Figure 11.5 shows three cases of k11 corresponding to the random E for g = 0.1, 1, and 10, respectively, wherein the deterministic case is shown as a crisp, black line, while the random case is shown as a gray thicker line, possibly overlapping the first one. Evidently, random E has a significant impact on the averaged solution for all but the lowest frequencies. Moreover, for g = 0.01, after no more than 10 kHz, the averaged solution k11 resembles random noise. In fact, this disordered behavior is a result of shifts in the resonant frequencies of the solution for various realizations. The conclusion, then, is that for these low values of g, we can rely on a homogenized solution in only the lowest frequency ranges. The situation improves as we go to higher g values for ρ, E, and h. Already at g = 1.0, there is some agreement between the deterministic and the mean at lower frequencies. As we go to g = 10.0 and 100.0, we tend to have an excellent agreement.

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1.00E + 01

(a)

9.00E + 00 8.00E + 00 Log10 κ11

7.00E + 00 6.00E + 00 5.00E + 00 4.00E + 00 3.00E + 00 2.00E + 00 1.00E + 00 0.00E + 00 0.0E + 00 1.0E + 01 2.0E + 01 3.0E + 01 4.0E + 01 5.0E + 01 6.0E + 01 7.0E + 01 Frequency (Hz) 1.00E + 01

(b)

9.00E + 00 8.00E + 00 Log10 κ11

7.00E + 00 6.00E + 00 5.00E + 00 4.00E + 00 3.00E + 00 2.00E + 00 1.00E + 00 0.00E + 00 0.0E + 00 1.0E + 01 2.0E + 01 3.0E + 01 4.0E + 01 5.0E + 01 6.0E + 01 7.0E + 01 Frequency (Hz) 1.20E + 01

(c)

1.00E + 01 Log10 κ11

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FIGURE 11.5 Timoshenko beam vibrations in the case of random modulus E showing k11 (black line) for the reference homogeneous medium and mean < k11 > (gray line) for the random case with: (a) g = 0.1, (b) g = 1.0, (c) g = 10.0.

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Interestingly the effect of varying w or ν for any value of g has almost no effect on the averaged solution at any of the frequency levels we studied. To sum up, the diffusion of resonances away from those of homogeneous rod and beam will always occur. In particular, in the case of rods, the effects of random mass density and elastic modulus—but not of cross-sectional area— are strong. In the case of beams, the effects of random mass density, elastic modulus and beam’s height—but not of Poisson’s ratio and beam’s width— are strong. Another new aspect is the very high level of second, third and fourth moments of response for a much weaker level of noise in the material. In all the results reported here the introduction of dissipation in the material has a tendency of removing the singularities of mean response at resonant frequencies. 11.2.3 Observations and Related Work In the deterministic case, by connecting all the elements according to the spatial geometry, a global stiffness matrix of a given 2D or 3D structure is constructed and a global response due to a specified impulse is studied—first by going over all the frequencies and then by transforming to the time domain, which is conveniently done by the fast Fourier transform (FFT), Doyle (1997). This procedure still needs to be generalized to the stochastic case—the main problem being that we do not have explicit forms of spectral finite elements for random field properties. Note here that the classical static stiffness matrix is actually obtained from the spectral stiffness matrix in a zero-frequency limit. Other related work on elastodynamics of structures described by random fields has been reported, among others, in Adhikari and Manohar (1999, 2000). Gupta and Manohar (2002), Manohar and Adhikari (1998). Their approach follows the stochastic finite element method. This method is a close relative of the stochastic finite difference method (Kaminski, ´ 2002). The frequency shift in the dispersion relation for waves on random strings has been studied by Howe (1971)—albeit in the long (λ d) and short (λ d) wavelength limits only. Those results could not be verified by our numerical simulations, and the dispersion at those special as well as the general case of arbitrary wavelengths still remains an open issue. The 1D wave motion in the case of variable E and ρ was treated analytically, using a Liouville transformation, in Belyaev and Ziegler (1994), but the results could only be obtained in the limits λ d, or λ d. Finally, there is also a method of analysis based on the Kramers–Kronig relations (Beltzer, 1989). This is based on the primitive causality condition, which expresses the observation that the output cannot precede the input. It enables one to evaluate the dispersion curves for the entire frequency interval (0 ≤ γ ≤ ∞) and yields a bound relating the static and dynamic frequency responses.

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403

Waves in Random 1D Composites

11.3.1 Motion in an Imperfectly Periodic Composite 11.3.1.1 Random Evolutions In many problems of stochastic mechanics even a weak material disorder may have a significant impact on the effective material response. In this section we report on such an occurrence in a relatively classical setting: harmonic wave motion in an imperfectly periodic composite. The classical subject of mechanics of periodic composites has seen a great deal of research, but here we consider a composite with geometric randomness in a nominally periodic layered structure. We follow B´ecus (1978, 1979), who himself employed the method of random evolutions (Hersch, 1974). Let us consider a dynamical system evolving in any one of several modes and whose switching from one mode to another is governed by a certain random mechanism described through a stochastic process. In general, the dynamical system is represented by a function f (an element in a Banach space B), and each mode of operation is specified by a linear operator TAω (t) on B with an infinitesimal generator Aω . These operators have a semigroup property, that is, TAω (t1 + t2 ) = TAω (t1 ) TAω (t2 ).

(11.58)

Here the sample space = {1, 2, . . ., n} is a countable index set, and we assume the switching process to be a Markov chain Z(t) with transition probabilities pij and an infinitesimal matrix Q = [q ij ] = [ pij (0)]. Also, let τ j (z) and N(t, z) denote the time of the jth jump and the number of jumps up to time t, respectively, for the sample path z(t) of Z(t). Now, a product M (t) = TZ(0) (τ1 (z)) TZ(τ1 (z)) (τ2 (z) − τ1 (z)) . . . TZ(τ N (t,z)) (t − τ N (z)) (11.59) defines a random evolution on B n . It can then be proved that: 1. The ensemble average system evolution (indicated by a tilde) is given by a semigroup of expectation (ensemble average) operators (t) f = M (t) f Z(t) f ∈ B n , T (11.60) where subscript i indicates the state in which the process began. 2. The Cauchy problem ∂ ui ∂t

ui + = Ai

n

ui q ij

i = 1, 2, . . ., n u (0) = f

j=1

(t) f . for the unknown vector u (t) is solved by u (t) = T

(11.61)

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There is also another result from modern probability theory that has usefulness in wave motion in composite materials because it generalizes the law of large numbers to the products of random matrices (Berger, 1984). To this end, consider the sequence of the products of real random matrices P K (ω) =

K

(11.62)

M j, K (ω).

j=1

It is assumed that, for K → ∞, the matrices M j, K (ω) can be represented by M j, K (ω) = Id +

1 B j, K (ω) + R j ( K , ω), K

(11.63)

where B j, K (ω) for j = 1, 2, . . ., K are independent, identically distributed random matrices, integrable with respect to the probability measure P and |R j ( K , ω)| = o( K −1 ) for large K . Under these conditions the law of large numbers takes place and lim P K (ω) = exp B j, K (ω) (11.64) K →∞

in the sense of convergence in distribution of all the vectors obtained by multiplication of the random matrix by an arbitrary deterministic vector. 11.3.1.2 Effects of Imperfections on Floquet Waves 11.3.1.2.1 Floquet wave Let us consider a plane wave propagating in direction x in a periodic composite, of period p, made up of two linear elastic alternating layers 1 and 2. The equations governing stress σ and displacement u in each layer ∂σ ∂x

= ρ (x)

∂2 u ∂t 2

σ = E (x)

∂u ∂x

(11.65)

led, in the case of a harmonic motion (u(x, t) = e iγ t u(x)), to an ordinary differential equation

d d u E (x) + ρ (x) γ 2 u = 0. (11.66) dx dx Given the periodic structure ρ (x + p) = ρ (x)

E (x + p) = E (x),

(11.67)

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according to Floquet’s theorem (e.g., Stoker, 1950) there exist solutions of the form u (x) = e iq x v (x).

(11.68)

where q is the Floquet wave number and v is a periodic function v(x + p) = v(x). The periodicity over one period [x0 , x0 + p] of displacements and stresses leads then to a statement of quasi-periodic boundary conditions d u d u (x0 + p) = (x0 ) e iq p . dx dx

u (x0 + p) = u (x0 ) e iq p

(11.69)

The solution of the Sturm–Liouville problem (11.66) and (11.69) is of the form u (x, t) = e i[q x+γ t] v (x)

(11.70)

and v (x) is called a Floquet wave. 11.3.1.2.2

Transfer matrix approach

Going to the frequency domain, one can rewrite (11.65) as a first-order system (Ziegler, 1976): ∂X ∂x

where

u X= σ

= A(x) X,

(11.71)

0 1/E (x) A= . −ρ (x) γ 2 0

(11.72)

Now, in each homogeneous layer I , E, and ρ are constant, and so the evolution over this layer is X (x + h) = TA (h) X (x),

(11.73)

were TA is a transfer matrix TA (h) = e

Ah

−1 sinh kh cosh kh Ek = , −1 − Ek sinh kh cosh kh

(11.74)

√ where k = γ ρ/E is the wave number in I . We see immediately that TA has the property (11.58), and, since A is piecewise constant, there is a switching from one layer to another. The global evolution is given by an equation entirely analogous to (11.59), namely, X (x0 + p) = TAn (ln ) TAn−1 (ln−1 ) . . . TA1 (l1 ) X (x0 ), where li is the length of the ith interval, in which A = Ai .

(11.75)

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(11.76)

for the Floquet wave X (x) = ξ (x) e iq x and ξ (n + p) = ξ (x), and, from (11.75), arrive at a matrix eigenvalue problem iq p 1 0 TAn (ln ) TAn−1 (ln−1 ) ...TA1 (l1 ) X (x0 ) − e ξ = 0. (11.77) 0 1 An eigenvalue e iq p with q real (complex) corresponds to a frequency γ in a passing (stopping) band, where q = mπ/ p (m = 1, 2, ...) is a Floquet wave number. 11.3.1.2.3 Floquet waves in a random composite We are now ready to consider a situation more general than that of a periodic composite: there are two layers made of materials 1 and 2, having deterministic properties (ρ, E) 1 and (ρ, E) 2 , respectively, but randomly varying thicknesses α1 and α2 ; the average thicknesses are a 1 and a 2 . Clearly, within the general framework of Section 11.3.1, the sample space has two elements {1, 2}, and the infinitesimal generators Aγ are 0 −E 1−1 0 −E 2−1 A1 = A2 = . (11.78) −ρ1 γ 2 0 −ρ2 γ 2 0 Furthermore, the switching process Z (t) is taken as a generalized telegraph process with an infinitesimal matrix −1 −a 1 a 1−1 Q= , (11.79) a 2−1 −a 2−1 so that the switching from layer 1(2) to 2(1) takes place over distances distributed according to a Poisson process with intensity a 1−1 (a 2−1 ). 1 and X 2 denoting the average solution to (11.71), an application of With X (11.61) and (11.77) results in     j j X X d     (11.80)   = Oj  d X j  , j = 1, 2, j dx d X dx dx where D j can be determined explicitly from all the parameters of the problems. This then leads to an eigenvalue problem, whose solution indicates that there is only one frequency of an average Floquet wave when (ρ1 − ρ2 )( E 1−1 − E 2−1 ) < 0. On the other hand, assuming a perfectly periodic composite without structural imperfections, one finds a whole spectrum of frequencies of Floquet waves. A diffusion approximation for this system

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was developed in B´ecus (1979), and this may become especially useful when there are more than two layers. 11.3.2 Waves in Randomly Segmented Elastic Bars Various studies of wave transmission and reflection phenomena, with a focus on the spectral response, have been conducted (e.g., Kotulski, 1994). He considered a more general model composite made of 2K segments (i.e., twolayered again) with random lengths li (ω) , i = 1, 2, . . ., 2K , whereby also the material properties of the segments (mass density, elastic modulus and cross-sectional area) are random vectors ( j = 1, 2, . . ., 2K ): ρ2 j−1 (ω) , E 2 j−1 (ω) , A2 j−1 (ω) ρ2 j (ω) , E 2 j (ω) , A2 j (ω) . (11.81) The lengths of the segments are assumed to satisfy a special relation:

l2 j−1 (ω) l2 j (ω) l2 j−1 (ω) , l2 j (ω) = , 2K 2K

j = 1, 2, . . ., K

(11.82)

are independent, identically distributed random variables with the means l2 j (ω) = L 2 . l2 j−1 (ω) = L 1 (11.83) With this setup, he proceeded to study the asymptotic behavior of the randomized equation for the amplitudes of the waves by the law of large numbers for the products of random matrices stated in Section 11.3.1. The transfer matrix M j, K (γ , ω) is not reproduced here explicitly, but once again we note that our notation of ω for elementary event and γ for frequency is the reverse of that used by Kotulski. It has turned out that the randomization of the bar results in a much slower homogenization of response, in function of the increasing K , than in the deterministic case (i.e., one where all the properties are perfectly periodic). Analogous, but more complex, effects of coupled thermoelastic wave propagation in such composite media were studied in Kotulski and Pretczy ˛ nski ´ (1994). As a starting point they took the well-known equations of thermoelasticity in 1D ∂2 u ∂t 2

ρc ε

= (λ + 2µ)

∂ϑ ∂t

=β

∂ ϑ 2

∂x2

∂2 u ∂x2

− (3λ + 2µ) α

− T0 (3λ + 2µ) α

∂ϑ ∂x

2

∂ u ∂t ∂ x

,

(11.84)

where λ and µ are the Lam´e constants, T0 is the reference temperature, c ε is the specific heat at constant strain, β is the heat conductivity coefficient, and α is the thermal expansion coefficient. Passing to the frequency domain, they

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set up a first-order system of the type (11.71), where   u    σ   X=  ϑ  ϕ



0

  −ργ 2 A=   iγ α(3λ+2µ)T0  β

1 (λ+2µ)

α(3λ+2µ) (λ+2µ)

0

0

0

0

0

iγρc ε

0

0



 0 . 1 β

(11.85)

0

Here, each layer was described by two random vectors ( j = 1, 2, . . ., 2K ): ρ2 j−1 (ω), λ2 j−1 (ω), µ2 j−1 (ω), α2 j−1 (ω), β2 j−1 (ω), c ε,2 j−1 (ω), l2 j−1 (ω) , ρ2 j (ω), λ2 j (ω), µ2 j (ω), α2 j (ω), β2 j (ω), c ε,2 j (ω), l2 j (ω) (11.86) and the ensuing analysis was analogous to that outlined above. In the next section we will disregard the backscattered waves, as our interest will be in nonlinear (in-)elastic microstructures where the forward transmitted pulse is of primary importance. The backscattering due to mismatch at differing layers, however, plays an important role in geophysical problems and results, for example, in an apparent slowing down of the main pulse carried forward (Asch et al., 1991). See also Foias and Frazho (1990), Kennett (1981, 1983), and Papanicolaou (1998). The latter paper reviews a number of aspects important in geophysical applications: coherent versus incoherent fields, localization and transport.

11.4

Transient Waves in Heterogeneous Nonlinear Media

11.4.1 A Class of Models of Random Media In this section we consider the random medium B = {B(ω); ω ∈ } to be made of realizations B(ω) (i.e., specimens) that are one-dimensional and semiinfinite in the physical domain: X ∈ X = [0, ∞]. (With strains being infinitesimal, no distinction is required between the material and spatial coordinates— but here we use X instead of x of the previous sections, because x will be employed for a value of the random variable in the diffusion model to be developed below.) Material properties vary in a discontinuous fashion from grain to grain, but remain constant within each grain. In the stochastic terminology this is a chain process. Let us introduce the following classification of microstructures. Linear elastic media: length l, mass density ρ, and elastic modulus E are random, Figure 11.6(a). • Bilinear elastic media: length l, mass density ρ, and two elastic moduli E 0 , E 1 are random. The stress level σ ∗ separating both linear •

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409 σ

(a)

(b)

σ∗

<E1 >

<E> <E0 > ε

ε σ

σ

<E0 > (c)

(d) <E> <E1 >

FIGURE 11.6 Constitutive laws: (a) linear elastic; (b) soft bilinear elastic; (c) soft nonlinear elastic; (d) linearhysteretic. In each case, randomness of modulus, or moduli, is indicated.

elastic ranges is assumed deterministic. The stress-strain law of each grain is σ = E 0 (ω) ε if |σ | < σ ∗ , σ = σ ∗ + E 1 (ω) (ε − ε0 ) if |σ | ≥ σ ∗ . (11.87) Thus, B is described by a vector random process parameterized by X, that is: {l, E 0 , E 1 } X . Three general types of this bilinear elastic granular medium may be considered: 1. All the grains are of a soft characteristic, Figure 11.6(b) 2. All the grains are of a hard characteristic 3. Both types of grains are present •

Nonlinear elastic granular media: length l, mass density ρ, and elastic modulus E are random; see Figure 11.6(c). The stochastic stress-strain law is σ = E (ω) ε n ,

(11.88)

where either n > 1, or < 1. Three general types of this model analogous to those of the bilinear model may be considered here. •

Linear-hysteretic granular media: length l, mass density ρ, and two elastic moduli E 0 , E 1 are random; see Figure 11.6(d). The stress-strain curve is a straight line on initial loading; its slope

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(11.89)

If the material is reloaded, it follows that the second line backs up, and then continues along the initial loading line. In all these models material properties are generally assumed to have a Markov property in X, while the noise-to-signal ratio of these properties is assumed small. 11.4.2 Pulse Propagation in a Linear Elastic Microstructure In this section we describe a method of analysis of transient waves propagating in a linear elastic microstructure, which forms the basis for study of nonlinear models. We start by considering a space-time graph of a disturbance propagating in a semi-infinite sequence of linear elastic grains. By a disturbance we understand any single point of the pulse f (t) applied at the free face X = 0, see Figure 11.7. This pulse results in a wavefront moving into the material domain. Our strategy is to find the rules of evolution of the disturbance, and use these to construct the wavefront at any later stage. Now, disturbance propagation in every grain occurs as a Riemann wave. We follow the forward propagating disturbance only, that is, we do not keep track of the waves backscattered from all the grain boundaries. Due to a (random) variation of the properties of the grains, there are two effects: 1. The amplitude ζ of the disturbance undergoes a change with passage from grain to grain; ζ denotes either stress or velocity. This change of ζ is described by the transmission coefficient T (it) defined by ζ (t) = T (it) ζ (i) ,

(11.90)

in which i and t denote the incident and transmitted quantities, respectively. In case of an ideal (nonslip) grain boundary model, and with ζ standing for stress, T (it) is given by the well-known formula T (it) =

2χ (it) , 1 + χ (it)

with the relative impedance being ! χ (it) =

ρ (t) E (t) . ρ (i) E (i)

(11.91)

(11.92)

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t (X,t)

c–(t0)

–1

t0

τ(X,ω)

t(X,ω)

c+(0) Sample path ω Fastest path Slowest path ξ(t,ω)

f(t)

X(t,ω)

X

FIGURE 11.7 Space–time graph of pulse propagation. Shown are the backward causality cone C − (t0 ) and the forward causality cone C + (t0 ), the latter being bounded by the slowest and fastest paths, and

contains two paths at c −1

−1

and c, as well as a sample (random) path c (ω, X).

2. The disturbance propagates forward with a phase velocity varying randomly from grain to grain. Thus every point at (X = 0, t = t0 ) gives rise to a set of all possible characteristics contained within a so-called forward causality cone C + (t0 ), where each single characteristic corresponds to a disturbance propagating in a single specimen B(ω) of B, where t0 = 0. For the assumed piecewise-constant random medium model these characteristics are continuous piecewiselinear. The random walk is therefore a natural stochastic model for their forward evolution, and it may be described by either t ( X, ω) if X is chosen as an independent parameter, or by X (t, ω) if t plays that role. However, in order to grasp the scatter in the arrival times we may introduce a so-called dispersion time τ ( X, ω) = t ( X, ω) − t ( X),

(11.93)

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Microstructural Randomness and Scaling in Mechanics of Materials where the mean (ensemble average) characteristic is defined by t ( X) = c −1 X, (11.94) while, in order to describe the scatter in distances covered in a given time t, we may introduce a so-called dispersion distance ξ (t, ω) = X (t, ω) − X (t),

(11.95)

where the mean characteristic is defined by X (t) = c t.

(11.96)

In view of the above points, we have a following sequence of implications: Assume (l, ρ, E) X is a Markov process • Then (l, ρ, E) , χ (it) , c X is a Markov process • Then (l, ρ, E) , T (it) , c X is a Markov process • Then [(l, ρ, E) , ζ, τ ] X is a Markov process •

•

Then [(l, ρ, E) , ζ, ξ ]t is a Markov process.

It follows that a complete description of the disturbance evolution— hereinafter denoted by W, but not implying the Wiener process per se—is obtained through either one of two vector processes (taking values w in the state space W) WX = [(l, ρ, E) , ζ, τ ] X

or

Wt = [(l, ρ, E) , ζ, ξ ]t ,

(11.97)

depending on whether a parametrization with respect to X or t is preferred. An important and natural property of this approach is that the transition probability function of WX , or Wt , is derivable from the microstructure’s statistics. In case these statistics are space-homogeneous, the transition function of WX is space-homogeneous too, while that of Wt is time-homogeneous. Now, Markov processes having time-homogeneous transition functions satisfy naturally (without transformation of the state space) the semigroup property M (t1 + t2 ) = M (t1 ) M (t1 ) where

M (0) = I,

(11.98)

M (t) [g (w1 )] =

W

g (w2 ) P (t, w1 , w2 ) dw2 .

(11.99)

The above represents a stochastic form of Huygens’ minor principle for the disturbance evolution (Ostoja-Starzewski, 1989). Markov processes Wx and Wt that model propagating disturbances may conveniently be approximated by diffusion processes. In the following we

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discuss the forward Fokker-Planck (FP) approximation, which is expressed by ∂p ∂t

=−

∂ 1 ∂2 Bij (w) p , [Ai (w) p] + ∂wi 2 i, j ∂wi ∂w j i

(11.100)

in which p is the conditional probability or the probability density of W at time t. The dependence on time of the drift and diffusion coefficients is present if the transition function of the (l, ρ, E) X process is not homogeneous in X. In the special case of a complete mutual independence of the properties of the grains (δ-correlatedness on scale l) we find for p(t, z, x) for the [ζ, ξ ] vector at time t ∂p ∂t

=−

1 ∂2 1 ∂2 ∂ ∂2 p + p + B B Aζ p − Aξ p + Bζ ξ p . ζ ζ ξ ξ 2 2 ∂z ∂x 2 ∂z 2 ∂x ∂z∂x (11.101) ∂

Thus, z and x denote values of the random variables ζ and ξ , while the drift and diffusion coefficients have forms Aζ = Az,

Aξ = 0,

Bζ ζ = Bz , 2

Bξ ξ = D,

Bζ ξ = E z,

(11.102)

which involve constants A, B, D and E—these can be computed from random fluctuations of the material. A glance at (11.101) reveals that the wave process WX is multiplicative in its amplitude ζ , a property characteristic of all wave phenomena studied hereinafter. The forms of drift and diffusion coefficients in (11.102) permit a transfor mation of the Wt = (ζ, ξ ) t process into a vector Wiener process ζ , ξ t , as follows: (11.103) (ζ, ξ ) → ζ , ξ , ζ (ζ ) = ln (ζ ). Accordingly, the new drift and diffusion coefficients become Aζ = 0, Bζ ζ = B,

Aξ = 0, Bξ ξ = D,

Bζ ξ = E,

(11.104)

so that the FP equation governing the probability density p = p (t, x, y) is ∂ p ∂t

=

∂2 p D ∂2 p B ∂2 p + + E , 2 ∂ y2 2 ∂x2 ∂ y∂ x

(11.105)

in which y is a value in the range of the random variable ζ . Thus, the solution to the above, subject to the initial condition p (0, x, y) = δ ( y − y0 ) δ (x),

y0 = ln (z0 ),

(11.106)

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has a bivariate Gaussian form with a constant mean and a covariance matrix proportional to B E . (11.107) E D The probability density in the original variables is then found as p (z, y) = e −y p ( y, x).

(11.108)

It is possible to find from (11.101) the evolution of first and second moments (covariances) ζ (t) = z0 e At , ξ (t) = 0, ζ (t) = (z0 ) 2 e 2At e Bt − 1 , ξ (t) = Dt,

ζ (t) ξ (t) = E z0 te At .

(11.109)

It follows that every point t0 on the t−axis is the origin of a forward causality cone C + (t0 ) centered about the mean characteristic. The latter is defined either by c or c −1 −1 depending, respectively, on whether time or position is chosen as a controlling parameter. Thus, the entire space-time is covered by the C + cones rather than by forward characteristics of the homogeneous deterministic linear elastic medium problem. The ensemble average amplitude at an arbitrary point (x, t = t0 + xc −1 ) in space-time may by calculated by considering all the characteristics within the backward causality cone C − (t0 ) and conducting an integration of the initial pulse f (t) over its base at x = 0, see Figure 11.7. 11.4.3 Pulse Propagation in Nonlinear Microstructures 11.4.3.1 Bilinear Elastic Microstructures Let us now extend the preceding analysis to pulse propagation in a microstructure of soft bilinear elastic grains (Ostoja-Starzewski, 1991a). First, we discuss the rules of disturbance propagation, and then proceed to infer from this the response to an initial pulse at the front end x = 0 f (t) = at,

a = const.

(11.110)

Recalling Figure 11.6(b), we observe: (1) if σ (t0 ) < σ ∗ , a disturbance will propagate in the medium as if it was a linear elastic medium; (2) if σ (t0 ) > σ ∗ , the propagation velocity is initially slow—c 1 corresponds to E 1 —and upon σ reaching σ ∗ , due to attenuation, it becomes fast—c 0 corresponds to E 0 . Strictly speaking, there is a possibility of the propagation becoming slow in one or several grains due to random fluctuations, but the loss of energy at the wavefront soon takes over and the propagation is fast. The location of the ensemble average point O = ( X , t ) of transition from fast to slow propagation is readily found using the results of Section 11.4.2. Thus, the

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mean amplitude behavior is described by (11.109)1 , so that σ ∗ = σ0 e At ,

(11.111)

yields the time t . Thin lines in Figure 11.8(a) show the outlines of scatter about the mean forward characteristic, that is, of the forward cone C + (t0 ), where " √ sξ (t) = ξ (t) = t D (11.112) denotes the standard deviation of t . At the point O we begin the second forward cone with the mean characteristic corresponding to c 0 (or, in case of parametrization by x, to the harmonic average of c 0 ). Clearly, there is a scatter about point O , whose range is characterized by two times t1 and t2 . The first one is the earliest time of reaching the stress level σ ∗ and is calculated

t t'2 t'1

t

t' 0' t'2 t'1

0'

t'

X

X (a)

(b)

t

t <∑(t)>

<φ(t)>

t∗

t∗

X

f (t) σ∗ (c)

X

f (t) σ∗ (d)

FIGURE 11.8 Space-time graphs of disturbances propagating in (a) soft and (b) hard bilinear elastic media; (c) linear pulse and the acceleration wavefront (t) propagating in the soft bilinear medium; (d) linear pulse and the shock wavefront (t) propagating in the hard bilinear medium.

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" from the condition z0 e At1 − 3 ξ(t1 ) = σ ∗ . On the other hand, t2 is the latest possible of reaching the stress level σ ∗ , and is to be found from " time At2 z0 e + 3 ξ(t2 ) = σ ∗ . These two times define the outlines of the second forward cone, that are shown in thin lines again. We are now ready to analyze wavefront propagation due to the pulse (11.110), see Figure 11.8(c). Thus, if by t ∗ we denote the time for which σ equals σ ∗ , it becomes apparent that all the disturbances originating at times t0 < t ∗ will propagate fast, while those that originate after t ∗ will propagate initially slowly and later—due to a decrease of their amplitudes down to σ ∗ —will propagate fast. In the ensemble average sense, this switch to fast propagation occurs at

∗ σ 1 σ∗ ∗ t = t0 + ln . (11.113) , t0 ≥ A a t0 a We observe that ∂σ/∂t is discontinuous at a point of switch from slow to fast propagation. Thus, the loci of all the points (c 1 (t − t0 ), t ) in the space-time graph, denoted by X = (t ), represent the ensemble average acceleration wavefront. In the case of a microstructure made of grains with hard bilinear elastic response, we observe a disturbance displaying behavior reverse to that discussed in the preceding section providing σ (t0 ) > σ ∗ . That is, the propagation velocity is initially fast—c 1 corresponds to E 1 —and upon σ reaching σ ∗ due to attenuation, it becomes fast—c 0 corresponds to E 0 ; see Figure 11.8(b). The location of the ensemble average point O = ( X , t ) of transition from fast to slow propagation, and the range of scatter described by t1 and t2 may be found with the same type of formulas as (11.111–11.113). Turning our attention to wavefront propagation due to the initial pulse (11.110), we first recall a solution to a corresponding deterministic homogeneous medium problem (Wlodarczyk, 1972), which corresponds to the case of no randomness in the properties of the grains. It is well known that a shock wave (t) (made of two parts 1 (t) and 2 (t)) will form. More specifically, •

1 (t) is an intersection √ of fast and slow characteristics and is straight with velocity c 0 = E/ρ.

•

2 (t) corresponds to fast characteristics propagating into an undisturbed region and it curves progressively to become faster.

These results serve as a reference basis in the case of the same loading of a random microstructure. In fact, each of the straight characteristics originating at t0 is to be replaced by a forward causality cone C + (t0 ). Consequently, the shocks 1 (t) and 2 (t) are Markov random processes evolving within a region shown by two broken lines in Figure 11.8(d). They may be approximated by diffusion processes, with the transition functions—and hence, the drift and diffusion coefficients—being derived from the rules of evolution in the deterministic problem. The idea is to consider the rates—and their conditional moments—of change of (t) and (t + t) over an interval t

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corresponding to the passage of through one grain. This then provides a framework for analysis of random variation of shock strength [ζ ] = ζ + − ζ + at the wavefront, where + and − refer to the quantities ahead of and behind

(t). The vector W = (ζ + , [ζ ], ) t evolves as a Markov process, and the drift and diffusion coefficients can be derived from the rules of evolution in the deterministic problem. 11.4.3.2 Nonlinear Elastic Microstructures It is well known that, in the absence of body forces and for small strains, the equations of motion and continuity in a homogeneous continuum result in a system of two first-order quasi-linear hyperbolic partial differential equations ∂σ ∂X

=ρ

∂v

∂v

∂t

∂X

where

! c (σ ) =

=

1 ∂σ , ρc 2 (σ ) ∂t

1 ∂σ ρ ∂ε

(11.114)

(11.115)

is the propagation velocity. This is the situation of physically nonlinear elastic grains shown in Figure 11.6(c). Following Ostoja-Starzewski (1995a), if ρ and c are random fields in X, (11.114) is stochastic; the explicit dependence of σ , v, ρ, and c on ω is suppressed here for clarity of presentation. In accordance with our basic formulation, we develop the rules of disturbance evolution first, and then use them to study response due to the initial pulse (11.110). As before, there are two choices: parametrization with respect to X—formula (11.94)1 —or with respect to t—formula (11.96)2 . We realize that τ is now dependent on ζ in case of the first choice, or analogously, ξ is dependent on ζ in case of the second choice; ζ itself is being driven by the (l, ρ, E) x vector process. The change of ζ is described by the transmission coefficient T (it) of (11.90). Using an ideal (nonslip) grain boundary model for two nonlinear elastic grains in contact, we derive the following relation governing T (it) : # $2n/(1+n) −1 (it) (1+n)/2n T (it) + χ (it) −1 −1=0 T 1/2 (t) (i) 1/2n . E /E χ (it) = ρ (t) /ρ (i)

(11.116)

This reduces to (11.91 and 11.92) in the special linear elastic case of n = 1. The fact that (11.116)1 is implicit does not pose a problem in the diffusion formulation that follows. However, before developing it, we note the dependence of phase velocity on the stress amplitude c (σ ) = (nE/ρ) 1/2 [σ/E](n−1)/2n .

(11.117)

The diffusion model of a propagating disturbance is formulated now for the Wx process in the particular case of space-homogeneous statistics;

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parametrization with respect to x is used in order to avoid the dependence on ζ of the passage time t referred to a single grain, where x = l is used. Thus, we work with Wx = [σ, τ ]x for which (11.101) holds, and the drift and diffusion coefficients are Aσ = Aσ, Bσ σ = Bσ 2 ,

Aτ = Cσ (1−n)/2n , Bτ τ = Dσ (1−n)/2n ,

Bσ τ = Eσ (1+n)/2n .

(11.118)

All the constants A through E may be computed, most conveniently, through a Monte Carlo procedure. We observe that: •

The functional forms of drift and diffusion coefficients of the σ process are the same as in the linear (or bilinear) elastic case, that is, Aσ is linear while Bσ σ is quadratic in σ .

•

The drift and diffusion coefficients of the τ -process are nonlinear in σ and hence the process Wx is nonlinearly multiplicative in σ ; so that an analytical solution of (11.101) with (11.118) is unwieldy.

Let us again consider the response of a semi-infinite body, this time made of soft grains of Figure 11.6(c), due to the pulse (11.110). Figure 11.9(a) shows the graph of the homogeneous medium, and (b) gives the graph of the random medium. It is seen in the first case that as the stress increases at X = 0, the propagation velocity of Riemann waves is successively smaller, and as the pulse is carried away from the front end it is being “washed out” in time. In case (b) this phenomenon is modified by the curving and diffusion of characteristics within their forward evolution cones, thereby reflecting the accompanying attenuation of stress. Finally, we consider the response of a material with hard grains to the same pulse (11.110). As expected, the effect of an increasing pulse will be opposite to that observed above: instead of washing out we have a compression of the pulse resulting in a so-called loading shock wave (Nowacki, 1978). Thus, Figure 11.10(a) illustrates the classical homogeneous medium response. Solution by characteristics is continued here until the slower ones are overtaken by the faster ones. Using ten initial characteristics we obtain an envelope of the shock wave propagating into an undisturbed body. In the random medium case (Figure 11.10(b)) we see a qualitative modification of this phenomenon due to a replacement of all straight characteristics by the forward evolution cones. Their curving up leads to a delay in the arrival of the shock, which actually has a progressively weaker strength than that of the homogeneous problem, due to the stress attenuation. In both cases presented in Figure 11.10 the computation started at a very small (non-zero) value of f (t) in order to avoid the situation of a so-called “sonic vacuum,” which calls for a zero propagation speed at zero stress in a medium with Hertzian contacts (Nesterenko, 2001).

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t

(a)

x

t

(b)

x FIGURE 11.9 Washing out of a pulse in a medium of soft bilinear elastic grains due to a linear forcing: (a) homogeneous material, (b) random material. Note the curving down of characteristics in case (b).

11.4.3.3 Hysteretic Microstructures The same method as formulated above can be employed to study pulse propagation in a hysteretic microstructure described by Figure 11.6(d). The free face is now subjected to a square pulse

f (t) =

− p0 0

for otherwise

0 ≤ t ≤ t1

.

(11.119)

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(a)

x t

(b)

x FIGURE 11.10 Formation of a loading shock wave in a medium of hard bilinear elastic grains due to a linear forcing: (a) homogeneous material, (b) random material. Note the curving up of characteristics.

A problem with this initial condition for a deterministic, homogeneous medium (Salvadori et al., 1960) forms the reference basis for solution of a stochastic problem. In the space-time of Figure 11.11(a) we see several regions: I, II, III, IV, V, . . .. Thus, region I is that of an undisturbed body σ I = 0, ε I = 0, v I = 0, while region II corresponds to a material in which σ I I = − p0 , ε I I = σ I I /E 0 , v I I = −c 0 ε I I and the entire x, t-plane can be analyzed in this fashion. Turning now to a random medium problem we first observe that each of the lines in Figure 11.11(a) representing the discontinuity waves (i.e., shocks) can be considered as the mean path providing reference for stochastic processes ξ and ζ . Thus, for example, the leading shock is a reference for a family of characteristics X (0, ω, c 0 ) = X (t, ω, c 0 ) |t=0

∀ω ∈ .

(11.120)

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V

IV

III

0'

t1 II

I

p(t) –σ

x

0 (a) t C

F G B H t1

A

0' D

E

p(t) –σ

x

0 (b)

FIGURE 11.11 (a) Space–time graph of response of a deterministic, linear-hysteretic medium to a square pulse. (b) Intersection of forward dependence cones in a random medium case showing strong scatter about O .

Similarly, the line bounding the region II from above is a family of characteristics X (t1 , ω, c 1 ) = X (t, ω, c 1 ) |t=t1

∀ω ∈ ,

(11.121)

and so forth. The cones corresponding to the ξ processes along the two above-mentioned mean characteristics (paths) are shown in Figure 11.11(b). Clearly, the point of intersection will be diffused about the reference point (x = c 0 t2 , t2 ).

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Following an analysis of characteristics in the space–time, it is seen that, in case of dependence between the random variables E 0 and E 1 , the scatter in the point of intersection of two characteristics—as measured by the distance between points A and C—increases for the ratio E 0 /E 1 decreasing to 1. This indicates that even a weak randomness in the medium’s properties may alter certain aspects of its response in a significant way! On the other hand, in the case of dependence defined by E 0 (ω) = E 0 if and only if E 1 (ω) = E 1 ,the scatter in the location of intersection point of two characteristics—as measured by the distance between points B and D—would be much weaker. The foregoing analysis of the intersection of characteristics carries over to any intersection point, as well as to all such points of other related problems. We also note that extensions to 2D and 3D problems are possible and, in fact, very natural for problems with cylindrical or spherical symmetry (of both loading and material’s statistics) where analyses would be conducted in a space–time having radius in place of X. Finally, we observe a possibility of treatment of a nonlinear elastic laminated composite (Chen and Gurtin, 1973), made of alternating two-phase layers, with each phase being described by three variables: mass density, tangent modulus and second-order modulus. The elastic nonlinearity leads to a wave amplification, while the layer-to-layer mismatch of properties has an opposite effect. A study of this type of competition, albeit in the random continuum setting, is reported in the next section.

11.5

Acceleration Wavefronts in Nonlinear Media

11.5.1 Microscale Heterogeneity versus Wavefront Thickness 11.5.1.1 Basic Considerations It is a general finding of continuum mechanics of nonlinear elastic/dissipative media that acceleration waves, that is, moving singular surfaces with a jump in particle acceleration, are governed by a Bernoulli equation (Coleman and Gurtin, 1965) dα = −µα + βα 2 . dx

(11.122)

Here x denotes position (as in all the previous sections except 11.4.), α is the jump in particle acceleration, and the coefficients µ and β represent, respectively, two effects: dissipation and elastic nonlinearity. The interesting aspect of acceleration waves uncovered through this equation is that, due to the competition between these two effects, there is a possibility of blow-up, and hence, of shock formation in a finite distance x∞ , providing the initial amplitude α0 exceeds a critical amplitude αc . x∞ is also called distance to blow-up or

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distance to form a shock. It is easy to establish that, for a homogeneous medium, αc =

µ , β

x∞ = −

µ 1 ln 1 − . µ βα0

(11.123)

Various other cases of deterministic spatial dependence of µ and β on x were investigated in the wake of the aforementioned reference; see also Chen (1973), McCarthy (1975), and Menon et al. (1983). It is interesting that (11.122) was also derived independently from the continuum thermodynamics with internal variables (Bland, 1969); see also Engelbrecht (1997). The third setting in which such a problem appears is rational extended thermodynamics (e.g., Wilmanski, ´ 1998), a theory which, basically speaking, sets up all the field equations from a hyperbolic systems standpoint. Now, all these studies were set in the context of deterministic mechanics (and thermodynamics), as expressed by the fundamental requirement of separation of scales d L L macro .

(11.124)

Here d is the characteristic scale of the microstructure, L is the RVE size and L macro is the macroscopic body size. This says that, in the case of wavefront propagation, d must be infinitesimal relative to the wavefront thickness L. Using a fine-grained mosaic, Figure 11.12(a) suggests this with the understanding that truly infinitesimal grains cannot really be shown. Evidently, the RVE limit implied in Figure 11.12(a) corresponds to the classical concept of the wavefront, treated as a singular surface, as shown in Figure 11.13. Let us therefore recall that a jump in f (x, t) in the classical case is defined and denoted by [[ f ]] = f 2 − f 1 ,

(11.125)

where f 1 and f 2 are, respectively, the quantities immediately ahead of and behind the wavefront. It is well known from continuum mechanics that, when f is continuous, we have the first-order kinematical and dynamical compatibility conditions

∂f ∂t

= −c

∂f ∂x

,

σij p j = −ρc

∂ui ∂t

.

(11.126)

Given the limit d λ, the tractions and displacements on either side are uniform because we effectively deal with a classical continuum. This means that the constitutive law of the RVE, in order to assure that mechanically defined response should be identical to the energetically defined response, satisfies the Hill condition. When d L, assuming spatially homogeneous and ergodic statistics of material properties, we have a separation of scales and all three conditions result in the same (i.e. unique) constitutive response. For example, if also isotropy applies, then a linear elastic law σij = λδij εkk + 2µεij

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f |t = 0

t ron vef a W 0

L

p (a)

x

x0 + L

x0

L

p

d

L

p

p

(b)

(c)

FIGURE 11.12 Propagation of a wavefront f (x, t) in space–time. The wavefront is a zone of finite thickness L (between x0 and x0 + L at time t = 0) propagating in the direction p, in a microstructure of characteristic grain size d. Three cases are distinguished: (a) L d, which shows the trend to a classical (deterministic) continuum limit, in which fluctuations die out to zero; (b) L finite relative to d, where spatial fluctuations render the wavefront a statistical mesoscale element; (c) L d, which leads to a piecewise-constant evolution.

1

p

2

FIGURE 11.13 A singular surface propagating from region 2 to 1.

D

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holds for B L/d , and we simply have σij = λδij [[uk,k ]] + µ ui, j + u j,i ,

(11.127)

because jumps in displacements imply jumps in components of σij across the wavefront surface. However, when d is not infinitesimal relative to L, we face a nonuniqueness of constitutive responses of B L/d , and an accompanying statistical scatter. The latter is due to the fact that microstructure changes as the wavefront mesodomain travels across it. The mesodomain B L/d is the statistical volume element (SVE). Thus, if the mosaic of Figure 11.12(b) is linear elastic everywhere, we have a random response law (almost surely anisotropic) σij = Ci jkl (ω, x, λ/d)εij,

(11.128)

where Ci jkl (ω, x, λ/d) is a random stiffness tensor field. In place of (11.131), we should then have σij = Ci jkl (ω, x, λ/d) ui, j + u j,i /2.

(11.129)

Clearly, the SVE B L/d is set up on a mesoscale L relative to d, and the wavefront’s evolution is stochastically affected by the random mesoscale fluctuations of the microstructure, Figure 11.12(b). To this end, one must consider the wavefront’s modulation according to the Bernoulli equation (11.122), but now with material coefficients µ and β taken as random processes in x, that is, jointly forming a vector random process [µ, β]x . Finally, there is also a third possibility, shown in Figure 11.12(c), where the wavefront thickness L is much smaller than the grain size d; the grain signifies a layer. In that case, the RVE assumption pertains to the microstructure much finer than d, not shown here, and the evolution involves transmissions and reflections at consecutive boundaries, resulting in a jump process for the forward propagating wavefront. This is a special case of what has been discussed in Section 11.4. 11.5.1.2 Mesoscale Response Focusing henceforth on the case of Figure 11.12(b), we deal with a stochastic Bernoulli equation driven by [µ, β]x , a process having continuous realizations. The question that arises is how to set up (or specify) such a process. At this point, we recall the explicit formulas for the dissipation coefficient, the nonlinear amplification coefficient and the velocity of acceleration wave

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are (Coleman and Gurtin, 1965, and other references above) ! 0 G 0 G0 E µ=− , c= , β=− . 2G 0 2G 0 c ρR

(11.130)

Here G 0 is called the instantaneous modulus, G 0 is the coefficient responsible 0 is called the instantaneous second-order tangent modulus for dissipation, E and ρ R is the mass density in the reference state. Thus, in the random medium case, we really have a dynamical system 0 % ρR dα G E = 0 α− α2, dx 2G 0 2G 0 G 0

(11.131)

' & 0, ρR . driven by a four-component random process x = G 0 , G 0 , E x

Clearly, it would be most desirable to specify x , according to the mesoscale L relative to d, rather than via [µ, β]x . However, mesoscale properties for the wavefront, except for the mass density where a straightforward volume averaging is valid, would require a combination of mathematical morphology (for generation of realizations of random geometries) with computational mechanics of nonlinear elastic/dissipative microstructures (for boundary value problems according to either one of (11.128) through (11.130), the actual loading in Figure 11.12(b) being, of course, unknown. While this procedure has been shown to provide mesoscale bounds for various linear and nonlinear elastic as well as some inelastic materials (see previous chapters), here we would also need to compute the second-moments as well as spatial crosscorrelations of x . Assuming we go ahead with this, we would then be faced with a differential equation (11.131) driven by x , for which a quite complicated parametric study of various dependencies between the four component processes would still need to be carried out. Therefore, in our studies of (11.122) to date, we have considered the three most fundamental cases of the [µ, β]x process: full positive, zero and full negative cross-correlation of µ with β. Note that this approach gives bounds on the stochastic problem at hand (!) in that any particular situation of the four-component vector process from the said 4 × 4 matrix must fall within our bounds, yet our analysis is much more tractable. Now, for small mismatches in microscale material parameters or for a wavefront’s thickness L rather large relative to d, we can definitely argue that any micromechanically based mesoscale model would lead to µ and β being two random processes with small noise-to-signal ratios. This, in fact, has been our starting point in the previous analyses of the subject (Ostoja-Starzewski, 1993c, 1995c; Ostoja-Starzewski and Trebicki, ˛ 1999, 2003), its most fundamental feature being a stochastic rather than a deterministic competition between the elastic nonlinearity and dissipation in (11.122), and the resulting random character of µ and β. Hence, the question we have been asking: how different are the averages αc and x∞ for the random medium from the values given by (11.123) in which the random noises in [µ, β]x are neglected?

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11.5.2 Wavefront Dynamics in Random Microstructures 11.5.2.1 Model with One White Noise The simplest way to introduce randomness into the Bernoulli equation (11.122) is to consider µ and β to take them as constants perturbed by the same zeromean white noise process ξ (x) with two strengths S1 and S2 , respectively, µ (x) = µ + S1 ξ (x) S1 , S2 ≥ 0

β (x) = β + S2 ξ (x) S1 + S2 = S,

,

(11.132)

where µ S1

β S2 .

(11.133)

Introducing (11.132) into (11.122) we obtain a stochastic differential equation for α dα = − µ α + β α 2 + S2 α 2 − S1 α ξ (x) dx

α (x0 ) = α0 ,

(11.134)

where the initial condition is deterministic. Now, given the fact that perturbations entering (11.134) are of a parametric type, we set up a Stratonovich equation in the sense that ξ (x) is treated as a Stratonovich-type differential d W(S) (x) of the Wiener process (e.g., Schuss, 1980) dα = − µ α + β α 2 + S2 α 2 − S1 α d W(S) (x) dx

α (x0 ) = α0 . (11.135)

The Itoˆ equation equivalent to (11.135) is (α) dx dα = A(α) dx + B where A(α) =

α (x0 ) = α0 ,

(11.136)

1 2 3 S − µ α + β − S1 S2 α 2 + S22 α 3 2 1 2

(α) = S2 α 2 − S1 α B

(11.137)

are, in fact, the drift and diffusion coefficients of the diffusion Markov process ζ . As our interest is in determining the blow-up (or escape) of α to ∞, it is more convenient to study the decay of the inverse (or reciprocal) amplitude ζ = 1/α to zero, Figure 11.14. This way the problem of blow-up in α is converted to the classical problem in evolutionary random processes: crossing the boundary at ζ = 0. The Itoˆ equation for ζ is dζ = [b 1 ζ + b 2 ] dx + (S1 ζ − S2 ) d W (x)

ζ (x0 ) =

1 . α0

(11.138)

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Acceleration wave

3

2

1

0 (a) 2.0

1.5 Inverse amplitude

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1.0 z0 = zc 0.5

0 2

4

6

x (b) FIGURE 11.14 Simulation of ten exemplary evolutions of an acceleration wavefront α (a) and its inverse ζ = 1/α (b) originating from the critical amplitude of a reference homogeneous deterministic medium αc(det) = µ / β as functions of distance x in a random medium described by one white noise. Observe that either a growth to ∞ or a decay to 0 occur. Parameters: µ = 1, β = 1, S1 = 0.2, and S2 = 0.35. After Ostoja-Starzewski and Trebicki (1995), with permission.

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where b 1 = µ +

1 2 S 2 1

b 2 = − β −

1 S1 S2 . 2

(11.139)

In a similar fashion one can set up formulas for the moments of ζ . Of primary interest is the equation governing the first moment ζ ζ = b 1 ζ + b 2

ζ (x0 ) =

1 . α0

(11.140)

From this, noting from (11.123) that the critical amplitude αc(det) of the reference deterministic homogeneous medium is αc(det) =

µ , β

(11.141)

we find the following relationships between the average critical amplitude of the random medium, αc , and αc(det) αc < αc(det)

for

β S2 > , µ S1

αc > αc(det)

for

β S2 < . µ S1

(11.142)

11.5.2.2 Model with Two Correlated Gaussian Noises A richer, and more realistic model can be constructed when the random processes µ and β are taken as two separate processes. In the following we sketch this for Gaussian processes, whereby we note that (1) in reality (and strictly speaking) µ and β cannot be Gaussian, but (2), for weak randomness levels, non-Gaussian noises result in effectively the same results for probability distributions of dynamical systems as the Gaussian ones. Also, working with µ and β having coefficients of variation not more than a few percent, we deal with random perturbations having the skewness and flatness parameters very close to zero, that is, just about the same as for the Gaussian processes themselves. With two processes we can study all three cases—fully positive, zero and fully negative cross-correlation between µ and β—which cover the full range of all the possibilities between µ and β. We generalize the original Bernoulli equation (11.122) by introducing randomness in µ and β according to dα = −(µ + ξ1 (x))α + (β + ξ2 (x))α 2, dx

(11.143)

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where ξ1 (x) and ξ2 (x) are two zero-mean stationary Gaussian noises having a correlation matrix g11 g12 R= (11.144) , g12 = g21 = g, g11 g22 ≥ g 2 . g21 g22 Here gij (i, j = 1, 2) are intensities of both noises. A small level of randomness in µ and β now implies µ

√

√

β

g11

g22 .

(11.145)

In general, the noises ξ1 (x, ω) and ξ2 (x, ω) are being interpreted here as stochastic processes equivalent, respectively, to two real processes X1 (x, ω) and X2 (x, ω) with sufficiently small correlation radii, that perturb the material parameters µ and β. The conditions for introducing the equivalent noises ξ1 (x, ω) and ξ2 (x, ω), as well as their relation to the correlation functions X1 (x, ω) and X2 (x, ω), were discussed at length in Ostoja-Starzewski and Trebicki ˛ (1999). Interpreting (11.143) in the Stratonovich sense again, we arrive at the equivalent Itoˆ equation dα = A(α) + B (α) d W (x)

α (x0 ) = α0 ,

(11.146)

where

A(α) =

1 3 g11 − µ α + β − g α 2 + g22 α 3 2 2

1 B (α) = g11 α 2 − gα 3 + g22 α 4 . 2

(11.147)

Next, the transformation of variables ζ = 1/α leads to an Itoˆ equation for the inverse amplitude process ζ dζ = A(ζ ) +

√

B(ζ )d W

ζ (x0 ) =

1 , α0

(11.148)

where the drift A(z) and diffusion B(z) coefficients of the Markov process ζ are 1 1 A(ζ ) = (µ + g11 )ζ − (β + g) 2 2

2 B(z) = (g11 ζ − 2gζ + g22 ). (11.149)

This leads to the average critical amplitude of the random medium αc =

µ + 12 S12 β + 12 S1 S2

.

(11.150)

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We note here: 1. In general, the case of full positive correlation among the noises √ (g = g11 g22 ) corresponds to the situation of equivalence, in a probabilistic sense, of processes ξ1 (x, ω) and ξ2 (x, ω) in the governing system (11.149), and hence in (11.146), thus leading to a weaker randomness (i.e., closest to a homogeneous medium) than in cases of √ zero (g = 0) and of full negative cross-correlation (g = − g11 g22 ), see Figure 11.15. 2. In the case of g = 0, this model reduces to that with two independent white noises µ (x) = µ + ξ1 (x)

β (x) = β + ξ2 (x) .

(11.151)

3. In the case of g → S1 S2 , g11 → S12 , and g22 → S22 , the model reduces to that given in the preceding subsection. Figures 11.14 and 11.15 also show that x∞ is a random variable, whose scatter is strongly sensitive to even weak perturbations in the material. In the language of stochastic processes, the problem of finding x∞ is a nonstationary stochastic evolution problem with absorbing boundary. Although it is governed by a linear differential equation, it generally does not have an explicit analytical solution. In Ostoja-Starzewski and Trebicki ˛ (2003) the method of 1.0 g11 = 0.001 g22 = 0.001 0.8 Trajectories of inverse amplitude

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Full positive correlation

0.6

Full negative correlation

0.4

0.2

Deterministic

Zero correlation

0.0 0.0

0.4

0.8

1.2

1.6

2.0

FIGURE 11.15 Effect of cross-correlations between the noises ξ1 and ξ2 on the evolution of a single trajectory of the inverse amplitude process. After Ostoja-Starzewski and Trebicki (2002), with permission.

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maximum entropy was employed to find an approximate solution for the density pζ (z, x) of ζ (x). 11.5.2.3 Model with Four Correlated Noises All the above studies were based on some choice of the wavefront thickness L relative to the microscale d, and an implicit assumption that the ratio L/d would not change in the course of evolution. It is more realistic, however, to admit a change of L as a function of the wavefront amplitude α, reflecting the fact that, as α grows, the wavefront thickness L decreases—because at α → ∞ we have the shock or caustic formation—showing that the wave becomes more and more sensitive to random microstructural details of the material. Motivated by this, Ostoja-Starzewski and Trebicki (2006) proposed a generalization of the original stochastic Bernoulli equation (11.122) by introduc 0 , ρ R (and simultaneously in processes ing randomness in processes G 0 , G 0 , E µ and β, which are functions of G 0 , G 0 , E 0 , ρ R ) in function of the amplitude α, in the following way: dα = −µ (U (x, ω)) α + β(U (x, ω))α 2 dx " dU = −a (α)Ud x + σU (α) 2a (α)d Wx (x, ω) dσU = C 1 α m1 dx da = C 2 α m2 , dx

(11.152)

with conditions σU (x0 ) = 1, a (x0 ) = a 0 , C1 , C2, m1 , m2, > 0.

(11.153)

Here U stands for the Ornstein–Uhlenbeck (O-U) process. Thus, as α grows, so does the standard deviation σU of the driving random process O-U, and its correlation length 1/a decreases. The same will also occur 0, ρR to the standard deviations and correlation lengths of processes G 0 , G 0 , E which, are linear functions of the O-U process. It is again more convenient to express the entire dynamics in terms of the 1 inverse amplitude ζ = so that, we have α dζ = µ (U (x, ω)) ζ − β(U (x, ω)) dx " dU = −a (ζ ) dx + σ (ζ ) a (ζ )d Wx (x, ω) dσU = C1 ζ −m1 dx

(11.154)

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da = C2 ζ −m2 , dx

(11.155)

with σ (x0 ) = σ0 = 1,

a (ζ0 ) = a 0 .

(11.156)

Our principal conclusions stemming from this model may be summarized as follows: 1. The introduction of coupling of four random fields of material properties to the wavefront amplitude process causes growth of the variance of x∞ , and, therefore, an even higher probability of blow-up. 2. Overall, the coupling of acceleration wavefront dynamics to material randomness has a dominant effect irrespective of whether there 0 , and ρ R , is a zero or non-zero cross-correlation among G 0 , G 0 , E and that is an interesting point on which we end this book.

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Author Index A Achenbach, J.D., 208 Adhikari, S., 402 Adler, R.J., 79 Advani, S.G., 44 Aero, E.L., 205 Aifantis, E.C., 207 Al-Ostaz, A., 79–80, 154, 163 Alava M.J., 141 Allaire, G., 169 Altus, E., 62 Alzebdeh, K., 105, 154 Amieur, M., 247, 258 Ariman, T., 207, 215 Arnold, V.I., 390 Asch, M., 408 Asimow, L., 134, 139 Askar, A., 87, 208 Askes, H., 248 Atkinson, C., 209

Bernal, J.D., 23 Bird, M.D., 102 Blanc, X., 163 Bland, D.R., 423 Blenk, S., 276 Bleustein, J.L., 206 Blouin, F., 125 Boal, D.H., 135 Boccara, S., 184, 187 Bogy, D.B., 209 Boots, B., 33 Bouyge, F., 230 Brandstatter, J.J., 393 Brekhovskikh, L.M., 56 Brenner, C.E., 290 Brezzi, F., 287 Bronkhorst, C.A., 318 Budiansky, B., 265 Buxton, G.A., 99 Byskov, E., 114

C B Babuska, I., 120 Bagley, R.L., 129 Bahei-Ei-Din, Y.A., 326 Ball, J.M., 205, 383 Banks, C.B., 208 Bardenhagen, S., 91, 120, 208 Bathurst, R.J., 143 Bazant, M.Z., 300 Bazant, Z.P., 208, 356 Beltzer, A.I., 402 Belyaev, A.K., 402 Belytschko, T., 163 Benaroya, H., 290 Bendsoe, M., 165, 168–169 Benedict, R., 205 Bennison, S.J., 143 Bensoussan, A., 229 Beran, M.J., 62, 73, 209, 395 Berger, M.A., 404 Berglund, K., 208

Campbell, J.G., 151 Caputo, M., 129 Cardou, A., 125 Carmeliet, J., 356 Carvalho, F.C.S., 265 Castro, J., 30, 314 Chaboche, J.-L., 241, 354–355, 359 Chen, P.J., 163, 386, 422–423 Cherkaev, A.V., 178, 181, 219, 223 Chernov, L.A., 385, 387, 392 Chien, S., 135, 145 Christensen, M., 208 Christensen, R.M., 347 Chung, J.W., 139 Cielecka, I., 116 Cioranescu, D., 120 Claus, W.D., 207 Cohen, L., 133 Coleman, B.D., 422, 426 Collins, I.F., 367 Contreras, H., 291

459

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18:26

460

C4174

C4174˙C013

Microstructural Randomness and Scaling in Mechanics of Materials

Cook, R.D., 140 Cosserat, E., 205 Cosserat, F., 205 Costanzo, F., 318 Costello, G.A., 122–123 Cowin, S.C., 187, 207, 209, 215–216 Cox, H.L., 137 Cramer, S.M., 139 Crapo. H., 134 Cundall, P.A., 141

D Day, A.R., 99, 184 de Borst, R., 356 De Cicco, S., 188–189 Dence, D., 385 Deng, M., 37 Deodatis, G., 292 Dhaliwal, R.S., 207 Ditlevsen, O., 290 Dodson, C.T.J., 30, 37 Dowling, N.E., 325 Doyle, J.F., 395, 400, 402 Drzal, L.T., 263 Du, X., 348 Dullien, F.A.L., 342 Dundurs, J., 179, 182, 184, 211, 221 Dvorak, G.J., 326 Dyszlewicz, J., 208

E Eastwood, J.W., 157, 161 Ehrentraut, H., 276, 359 Eimer, C., 73, 85 Elishakoff, I., 290 Engelbrecht, J., 423 Englman, R., 83 Ericksen, J.L., 205 Eringen, A.C., 206–208 Evans, D.J., 364

F Feng, S., 135 Fishman, L., 73 Fleck, N.A., 207 Foias, C., 408 Forest, S., 41, 228–230, 248, 379 Fortin, M., 287 Frazho, A.E., 408

Friesecke, G., 208 Frisch, U., 83, 171, 385, 389, 393

G Gani, J., 23 Garboczi, E., 99, 104, 266 Gauthier, R.D., 208 Gdoutos, E.E., 59 Germain, P., 370 Ghanem, R.G., 292 Gibbs, J.W., 359, 367 Gibiansky, L., 314 Givli, S., 62 Gloria, A., 293, 321 Goddard, J.D., 31, 83, 145, 206 Gosz, M., 184 Grah, M., 105 Graham, L., 292 Gray, G.L., 318 Green, A.E., 206–207 Greenspan, D., 157, 159 Griffith, A.A., 58 Grioli, G., 198, 205 Guarracino, F., 188–189 Guest, S.D., 135 Gupta, S., 402 Gurtin, M.E., 422, 426 Gusev, A.A., 107, 248

H Hansen, J.C., 135, 145 Hanson, G.W., 207 Hartranft, R.J., 207 Hashin, Z., 240, 244, 347 Hazanov, S., 247, 258, 314 He, Q.-C., 181, 321 He, R., 324 Hegemier, G.A., 166, 303 Hehl, F.W., 96 Herrmann, G., 208 Herrmann, H.J., 164, 356 Hersch, R., 403–404 Hill, R., 237–238, 245, 247, 252, 293, 295, 318, 322, 334 Hilton, H.H., 128 Hockney, R.W., 157, 161 Holnicki-Szulc, J., 118, 120, 208 Hori, M., 246, 248 Horio, M., 151 Houlsby, G.T., 367 Howe, M.S., 402 Hrennikoff, A., 87 Huang, Y., 164

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Author Index

461

Hudson, J.A., 390 Huet, C., 240, 245, 247, 255, 341 Hutchinson, J.W., 207 Huyse, L., 150

Kuvshinskii, E.V., 205 Kuznetsov, E.N., 141

I

Labuz, J.F., 265 Lakes, R., 205, 208, 216 Laplace, P.S., 2 Le Quang, H., 321 LeBris, C., 163 Leckie, F.A., 44 Lee, J.D., 156 Lee, S.J., 319 Lemaitre, J., 241, 354–355, 359 Leppington, F.G., 209 Li, W., 324 Liszka, T., 307 Lohnert, ¨ S., 318 Lomakin, V.A., 69–70 Love, A.E.H., 96 Lu, Y.Y., 163 Lubarda, V.A., 209 Luding, S., 164 Lurie, K.A., 178, 181, 219, 223

Ilies, H., 293 Itin, Y., 96 Itou, S., 207

J Jagota, A., 143 Jahsman, W.E., 208 James, R.D., 205, 208, 383 Jasiuk, I., 79–80, 112, 114, 141, 163, 175, 183–184, 187, 211, 227, 230, 269 Jaunzemis, W., 206 Jaynes, E.T., 14, 44 Jeulin, D., 38, 331, 396 Jiang, M., 263, 281, 313, 324 Jones, R.M., 176 Jun, S., 184

K Kac, M., 1 Kachanov, L.M., 265, 293, 297, 334 Kachanov, M., 265, 267 Kaloni, P.N., 207, 215 Kaminski, M., 290, 402 Kamrin, K., 300 Kanit, T., 41, 248 Keating, P.N., 98 Keller, J.B., 104, 393 Kendall, M.G., 14 Kennett, B.L.N., 408 Kharanen, V.Y., 392 Khisaeva, Z.F., 319, 353 Kikuchi, N., 168 Kirkner, D., 69–70 Kirkwood, J.G., 97 Knauss, W.G., 128 Kohler, W., 408 Kohn, R.V., 169 Koiter, W., 198, 206 Kotulski, Z., 407 Kravtsov, Y.A., 385, 387 Kreher, W., 83 Kroner, ¨ E., 206, 245 Krylov, V.I., 125 Ku, A.P.-D., 300

L

M Maes, M.A., 150 Malkus, M.E., 140 Mandel, J., 245, 247 Mandelbrot, B.B., 45 Manohar, C.S., 402 Mardia, K.V., 24 Mariano, P.M., 206 Markenscoff, X., 182–184, 187, 209, 211 Markov, K., 244 Martinsson, P.G., 120 Masiani, R., 107, 145 Masson, J., 23 Mateau, J., 66 Matheron, G., 41, 273 Maugin, G.A., 119, 207, 359, 370, 385 Maxwell, J.C., 87 McCarthy, M.F., 423 McCoy, J.J., 73, 209 Mellor, P.B., 341 Mendelson, K.S., 104 Menon, V.V., 423 Michell, A.G.M., 165 Michell, J.M., 183 Miles, R.E., 27, 30 Milton, G.W., 104, 178 Mindlin, R.D., 182, 198, 205–207, 212, 214 Misicu, M., 207 Mora, R., 115

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18:26

462

C4174

C4174˙C013

Microstructural Randomness and Scaling in Mechanics of Materials

Moran, B., 184 Morrell, S., 162 Muki, R., 208–209 Muller, G.M., 207 Mura, T., 185 Murdoch, A.I., 14 Muschik, W., 276, 359

Press, W.H., 103 Pretczynski, Z., 407 Prohorov, Y.V., 7, 45 Pshenichnov, G.I., 120 Puzrin, A.M., 367

N

Quoc Son, N., 370

Naghdi, P.M., 207 Napier-Munn, T.J., 162 Nemat-Nasser, S., 246, 248, 318 Nemeth, M.P., 90 Nesterenko, V., 418 Neuber, H., 207 Neumeister, J.M., 179 Noll, W., 286 Noor, A.K., 90–91 Nordgren R.P., 300 Nowacki, W., 194, 196, 205–208, 210, 212, 218, 220, 418 Nunziato, J.W., 187 Nye, J.F., 390

R

O O’Connell, R.J., 265 Okabe, A., 33 Onat, E.T., 44 Onck, P.R., 230 Onogi, S., 151 Ostoja-Starzewski, M., 30, 51, 57, 123, 126, 128, 130, 139, 141, 147–148, 153–154, 156–157, 161, 163, 211, 227–228, 255, 259, 263, 265, 275, 279, 286, 293, 307, 314, 319, 324, 331, 348, 353, 355, 358, 371, 373–374, 390, 395–396, 412, 414, 417, 426, 430–432

P Page, D.H., 139 Palasti, I., 23 Pamin, J., 207, 356 Papanicolaou, G.C., 229, 385, 408 Papenfuβ, C., 276 Papoulis, A., 292 Perkins, R.W., 208 Pinsky, M., 404 Pompe, W., 83 Ponte Castaneda, ˜ P., 314, 324 Porcu, E., 66 Prager, S., 342 Prager, W., 165–166, 303, 309

Q

Raisanen V.I., 141 Rehak, M., 290 Rivier, N., 83 Rivlin, R.S., 206 Robertson, H.P., 69–70 Rogula, D., 118, 120, 208 Roos, A., 139 Rosen, B.W., 347 Roth, B., 134, 139 Rothenburg, L., 143 Roux, S., 356 Rozanov, Y.A., 7, 45 Rozvany, G.I.N., 165, 169 Rudin, W., 3, 59–60 Rytov, S.M., 385, 387

S Sab, K., 228–229 Sab, V., 245, 248, 255–256 Sachs, G., 324 Sahimi, M., 137 Saigal, S., 291 Saint Jean Paulin, J., 120 Salvadori, M.G., 182, 420 Sampson, W.W., 30 Samras, R.K., 125–127 Sanchez-Palencia, E., 229 Santalo, ´ L.A., 1, 14 Sarkani, S., 209 Satake, M., 143, 145 Save, M., 165, 309 Sawczuk, A., 207 Schijve, J., 207 Schoenberg, M., 393 Schreurs, P.J.G., 324 Schulte, J., 259 Scott, D.G., 23 Searles, D.J., 364 Sewell, M.J., 367 Shahinpoor, M., 208 Shahsavari, H., 123, 126, 128, 130 Sharma, V.D., 423

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C4174˙C013

Author Index Shield, R.T., 319 Shinozuka, M., 292 Siegmund, T., 288 Sievert, R., 379 Sih, G.C., 207 Singh, A., 207 Skalak, R., 420 Skop, R.A., 125–127 Slepyan, L.I., 125 Snyder, K.A., 99, 184 Sobczyk, K., 51, 56, 69–70, 83, 385 Sokolowski, M., 208 Somigliana, C., 205 Soszynski, R., 15 Spanos, P.D., 292 Spence, J.E., 385 Stahl, D.C., 139, 153 Steele, C.R., 102 Stephen, N.G., 91 Sternberg, E., 208–209 Stojanovic, R., 206 Stoker, J.J., 405 Strack, O.D.L., 141 Stroeven, M., 248 Stronge, W.J., 114 Sudria, J., 205 Sun, C.T., 366 Suquet, P.M., 247, 314, 324, 328 Szczepankiewicz, E., 66 Szczepinski, W., 294

T Takeuti, Y., 220 Talbot, D.R.S., 314 Tarasov, V.E., 380 Tauchert, T.R., 207 Taylor, G.I., 324, 336 Terada, K., 248 Teukolsky, S.A., 103 Thomson, D., 208 Thorpe, M.F., 135, 175, 184, 266 Tiersten, H.F., 198, 205–206 Torquato, S., 314, 342 Torvik, P.J., 129 Toupin, R.A., 198, 205–206 Trebicki, J., 83, 426, 430–432 Triantafyllidis, N., 91, 120, 208 Trovalusci, P., 107, 145 Truesdell, C., 191, 198, 205, 286 Tschoegl, N.W., 128 Tucker, C.L., 44 Tydeman, P.A., 139

463 U Uscinski, B.J., 385

V Vakulenko, A.A., 265 van der Sluis, O., 324 Vannucci, P., 176 Verchery, G., 176 Vinogradov, O., 163, 248 von Mises, R., 2

W Waas, A.M., 115 Wang, C., 148, 255, 269 Wang, G., 157, 161, 163 Wang, P.J., 91 Wang, X., 114, 286 Wang, Z.-P., 366 Warren, W.E., 114 Wehrli, Ch., 370 Weitsman, Y., 207 Werner, E., 288 Wheeler, J.A., 87 Whiteley, W., 134 William, K., 207 Willis, J.R., 247–248, 395 Wilmanski, K., 423 Wlodarczyk, E., 416 Woods, A.N., 395 Wozniak, C., 107, 116, 208 Wriggers, P., 318

Y Yaglom, A.M., 292 Yamazaki, F., 292 Yang, J.F.C., 208 Yavari, A., 209 Yongjian, R., 290 Yserentant, H., 164

Z Zaoui, A., 229 Zbib, H., 207 Zhang, P., 164 Zheng, Q.-S., 181 Ziegler, H., 172, 327, 343, 359, 363, 370–371, 374, 376, 402 Ziman, J.M., 146

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C4174˙C013

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19:27

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C4174˙C014

Subject Index A

C

Acceleration wave, 422, 425, 428 Acceleration wavefront, 386, 415–416, 422, 428, 433 Angular distribution, 34, 269 Anisotropic material, 180–181, 218–220 Antiplane elasticity, 93–94 square lattice, 93–94 Asymptotic theorem, 15 Average/averaging theorem, 246, 252, 254, 318–319

Cardioid distribution, 26, 43 Cauchy-Schwartz inequality, 49 Cauchy continuum, 229–230 Cauchy distribution, 27 Cauchy stress, 119, 128–129, 188, 191–192, 206, 232, 241, 365, 378, 381 Causal distribution, 25, 275 Central interactions, elasticity, lattice with, 117–120 Characteristic length, 110, 115–116, 189, 212–214, 216, 228, 232 Classical continuum, 91, 145, 195–196, 208–209, 356, 359, 423 Classical elasticity, 171–190, 197–199, 205–210, 217–219 Clausius-Duhem inequality, 235, 382 CLM, 105, 177–179, 183, 189–191, 216–227 Complex process, 377–379 vs. compound, 376–379 Compound process, 376, 378–379 Connectivity percolation, 135, 137 Conservation principle, 194, 199 Constants Dundurs, 179 Lame´e constants, 96, 172, 197, 255, 408 multi-constant theory, 96 rari-constant theory, 97 Continuum Cauchy, 229–230 classical, 91, 145, 195–196, 208–209, 356, 359, 423 Cosserat, 145, 205, 208, 228–229, 232 fractal media mechanics, 380–383 homogeneous cosserat, 228–230 local model, 118 mesoscopic physics, 276–278 micromorphic, 206, 229 micropolar, 89, 109, 114, 143, 146, 235, 398 multipolar, 206 nonlocal, 118–119, 207 nonlocal model, 118–119

B Beam Bernoulli–Euler, 107–110, 114–117 elastic, 88–91 fiber network rigidity, 139–141 microbeams, 57–62 stubby, 112–114 Timoshenko, 60, 85, 89, 110–114, 122, 131, 140, 153, 395, 398–401 Bernoulli lattice, 20–21, 258–261, 336 process, 84 trial, 14–15, 18 Bernoulli–Euler beam, 107–110, 114–117 Beta distribution, 275–276 scale dependence, 275–276 Bilinear elastic microstructure, 414–417 Binomial distribution, 14–15, 18 Bone material, 141, 208, 234, 269–271 Boundary condition displacement response, 251–254 effect paradigm, 241–244 kinematic, 139, 153, 198, 312, 322–323, 335, 337–338, 341 mixed-orthogonal, 184 traction, 315–317, 334–336 uniform, 251, 278, 312, 317, 366

465

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19:27

466

C4174

C4174˙C014

Microstructural Randomness and Scaling in Mechanics of Materials

pseudo-continuum, 221 random, 37, 237, 273, 292–293, 422 random fields, 273–275 restricted, 214–216 strain-gradient, 119 strain-gradient model, 119–120 thermodynamics, 359–364 truss-like, 301–303 Correspondence (or viscoelasticity) principle, 128 Cosserat continuum, 145, 205, 208, 228–229, 232 Couple-stress, 129–130, 191–192, 198, 207, 209, 211–215, 224, 227, 229, 232, 278–279, 377

D Damage, 80, 154–156, 159, 311, 314, 354–357 map, 156 patterns, 154–155 stochastic (random) evolution, 354 Diffusion process, 412, 416 Directional data, 23 Discrete distribution, 275, 360 Disk-matrix composite material, 259, 261–262, 282 Displacement response boundary condition, 251–254 Dissipation function, 371, 374, 377 homogeneous, 371–374 quasi-homogeneous, 374–376 Distribution angular, 34, 269 beta, 275–276 scale dependence, 275–276 binomial, 14–15, 18 cardioid, 26, 43 Cauchy, 27 causal, 25, 275 discrete, 275, 360 lattice, 26 mesoscopic, 276 point, 25 Poisson, 21–22 probability, 1, 3, 7–10, 20, 30, 48, 65, 82, 84, 156, 277, 297, 364, 429 radial, 22 stationary, 85 von Mises, 26 Weibull, 43 wrapped, 26

Dundurs constant, 179 Duplex-steel material, 288 Dynamic response, Hill(-Mandel) condition, 366

E Eigencurvatures, stress invariance, 223–227 Eigenstrains, stress invariance, 184–187, 223–227 Elastic beam, 88–91 Elastic beams, 88–91 Elastic helix, 125, 131, 378 Elastic media, stochastic finite elements, 286–293 Elastic-plastic microstructure, 322–333 Elastic strings, simple lattice, 87–88 Elasticity, 367 antiplane, 93–94 classical, 171–190, 197–199, 205–210, 217–219 finite, 318–322 gradient, 117–120 in-plane, 94–99, 257–258 mesoscale, 150–154 micropolar, 191–231, 233, 235 noncentrosymmetric micropolar, 204–205 nonlocal, 117–120 paper, 150–154 planar Cosserat, 210–216 plane models, 174–176 three-dimensional, 171–172 two-dimensional, 172–174 Entropy, 45, 79–82, 162–163, 347, 363–365, 367, 369–370, 432 maximum, 79, 81, 162–163, 363, 432 production rate, 370 Ergodic/ergodicity, 45, 53, 61, 66, 74–76, 78–79, 85, 147, 238, 240, 249–251, 253, 256, 279, 286, 303, 311, 313, 332, 355–357, 387, 423 Ergodic theorem, 74, 76, 240 Euler–Bernoulli beams, 107–110, 114–117 Extremum or variational principle, 81–82, 292, 311, 319, 322–323, 342, 371, 395

F Fermat’s principle, 330, 390–392 Fiber network rigidity beam, 139–141 Finite elasticity, 318–322 Hill(-Mandel) condition, 318

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Subject Index

467

Finite elements spectral, 126, 131, 395, 398, 402 for flexural waves, 398–402 for random media, 395–402 for waves in rods, 395–398 stochastic for elastic media, 286–293 random field models and, 273–309 Flexural wave, 398–402 spectral finite elements, 398–402 Floquet wave, 404–407 Fractal, 35, 129, 209, 380–382 mechanics of, 380 Functionally graded material, 23, 105, 263–265, 286

I

G

Jensen’s inequality, 59, 84

Gaussian process, 52, 429 Geodesic, 329, 331–333 Gradient elasticity, 117–120 Green-Gauss theorem, 381

K

In-plane elasticity, 94–99, 257–258 Inelastic microstructure, 311–358 Inequality Cauchy-Schwartz, 49 Clausius-Duhem, 235, 382 Jensen’s, 59, 84 Inhibition process, 19, 23 Inhomogeneous material, 225–227 Insufficient reason, principle of, 2 Internal variables, thermomechanics, 235 Inverse amplitude process, 430–431 Isotropic material, 177–179, 216–217 Isotropic micropolar material, 196–198

J

Kinematic boundary condition, 139, 153, 198, 312, 322–323, 335, 337–338, 341

L H Hamilton’s principle, 88, 90, 199–200 Hard-core, inhibition process, 23 Harmonic wave, 125–127, 403 Heat conduction, 73, 127, 209 Helix, 87, 122–123, 125–126, 128–131, 205, 377–378 elastic, 125, 131, 378 thermoelastic, 126 Hexagonal lattice, Bernoulli–Euler beams, 114–115 Hill(-Mandel) condition, 184, 229, 245, 251–253, 285, 311, 323, 341, 343, 348, 364–365, 382–383, 423 dynamic response, 366 in dynamic response, 366 in finite elasticity, 318 in thermomechanics, 364 Homogeneous Cosserat continuum, 228–230 Homogeneous dissipation function, 371–374 Homogenization, 104, 117, 150–151, 191, 196, 229–230, 237–238, 255–257, 291, 321, 328, 334, 356, 365–366, 382, 385, 407 Honeycomb lattice, 99, 114 Huygens’ principle, 390 Hysteretic microstructure, 419–422

Lame´e constant, 96, 172, 197, 255, 408 Lattice Bernoulli, 20–21, 258–261, 336 with central interactions, elasticity, 117–120 distribution, 26 hexagonal, Bernoulli–Euler beams, 114–115 honeycomb, 99, 114 micropolar, 88 elastic beams, 88–91 models, 133–170 one-dimensional, 87–91 periodic, 157 planar, 87, 89–99, 101, 103, 105, 107–123, 125, 127, 129, 131 planar models, 87–131 simple, 87 elastic strings, 87–88 square, 93–94, 115, 131, 139, 282, 332 antiplane elasticity, 93–94 Bernoulli–Euler beams, 115–117 triangular, 94–99, 107, 110, 114, 120–122, 134–135 Bernoulli–Euler beams, 107–110 Timoshenko beams, 110–112 triple honeycomb, 99 Legendre transformation, 347, 353, 367–369, 373, 375

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Linear elastic microstructure, 237–271, 410–414 Liouville theorem, 360, 362 Locally isotropic, smooth elastic material, 284–286

M Markov process, 356, 412, 417, 427, 430 Material anisotropic, 180–181, 218–220 bone, 141, 208, 234, 269–271 disk-matrix composite, 259, 261–262, 282 duplex-steel, 288 fractal, 35, 129, 209, 380–382 mechanics of, 380 functionally graded, 23, 105, 263–265, 286 isotropic, 177–179, 216–217 isotropic micropolar, 196–198 locally isotropic, smooth elastic, 284–286 matrix-inclusion composite, 103, 178, 184, 228, 242, 320, 322, 324–326, 328–329, 331, 358 multiply connected, 181–183, 220–221 null-Lagrangian, 180–181 optimal use, 165–169 paper, 15–17, 150–154, 314–318 plastic, 162, 165, 168, 207, 293, 298, 300–302, 314, 324, 327–333, 335, 338, 379 power-law, 313–314 random, mean field equations, 72–73 random chessboard, 258, 263–264, 327, 336–337 two-phase, 221–222 Mathematical morphology, 38, 426 Matrix-inclusion composite, 103, 178, 184, 228, 242, 320, 322, 324–326, 328–329, 331, 358 Maximum entropy, 79, 81, 162–163, 363, 432 Mechanics of fractal, 380 Mesoscale, 425–426 bounding, kinematic, 334–336 bounds hierarchies, 251–269 linear elastic microstructures, 237–271 nonlinear, 311–358 universal properties, 279–282 variational principles, 319–324 crack density tensor, 267–269 elasticity, paper, 150–154

hierarchies of, 251, 258, 311, 313, 315, 317, 319, 321, 323, 325, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 353, 355, 357 property, 151, 276, 426 random fields, 273–278 correlation structure, 282–283 second-order properties, 278–283 random media, thermomechanic response, 359–383 thermodynamic orthogonality, 370–376 universal properties of, 279 Mesoscale elasticity, 150–154 Mesoscopic distribution, 276 Mesoscopic physics, 276–278 Microbeams, 57–62 Micromorphic continuum, 206, 229 Micropolar continuum, 89, 109, 114, 143, 146, 235, 398 Micropolar elasticity, 191–231, 233, 235 Micropolar lattice, 88 elastic beams, 88–91 Microstructure bilinear elastic, 414–417 elastic-plastic, 322–333 hysteretic, 419–422 inelastic, 311–358 linear elastic, 237–271, 410–414 nonlinear, 311–358, 414–422 nonlinear elastic, 311–318, 417–419 random, 35–44, 301–309, 427–433 rigid-perfectly plastic, 333–341 thermoelastic, 347–354 viscoelastic, 341–342 Mixed-orthogonal boundary condition, 184 MMM principle, 240 Multi-constant theory, 96 Multiply connected material, 181–183, 220–221 Multipolar continuum, 206

N Noncentrosymmetric micropolar elasticity, 204–205 Nonlinear elastic microstructure, 311–318, 417–419 Nonlinear media wavefront, 422–433 Nonlinear microstructure, 311–358, 414–422 Nonlocal continuum, 118–119, 207 Nonlocal elasticity, 117–120 Null-Lagrangian material, 180–181 Numerical solutions, 170, 385, 393

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Subject Index O One-dimensional composite wave, 403–408 One-dimensional lattice, 87–91 Operator, 59, 72–73, 144, 209–210, 290, 292 random, 292 Optimal, 87, 105, 133, 165, 167–168, 288, 304, 307–309 structure, 307, 309 truss-like continuum, 165 use material, 165–169 Orthogonality, thermodynamic, 327, 363, 370, 372, 376–377, 382

P Paper, 15–17, 150–154, 314–318 elasticity, 150–154 fiber structure, 15–17 in-plane orthotropy, 150–153 mesoscale elasticity, 150–154 random formation, 314–318 Particle model, 157–158 Partition theorem, 313 Percolation, 133, 135, 137, 171, 184, 266, 282 connectivity, 135, 137 rigidity, 133, 135, 137 Periodic lattice, 157 Physics, mesoscopic, 276–278 Planar Cosserat elasticity, 210–216 Planar lattice, 87, 89–99, 101, 103, 105, 107–123, 125, 127, 129, 131 Plane elasticity, 174 Plane models elasticity, 174–176 Plane monochromatic wave, 204 Plane wave, 57, 204, 404 Plastic material, 162, 165, 168, 207, 293, 298, 300–302, 314, 324, 327–333, 335, 338, 379 Point distribution, 25 Point process, 18–19, 21–22, 27, 31, 36, 38, 44, 261, 279, 281 Poisson distribution, 21–22 inhomogeneous, point field, 22–23 line field geometry, 137–139 point field, 21–23, 27, 31, 35, 146–147, 184, 265–266, 279, 344 simulation of, 22 process, 406 random lines in plane, 27–30 Voronoi tessellations, 31 Power-law material, 313–314 Principle

469 conservation, 194, 199 correspondence (or viscoelasticity), 128 extremum or variational principle, 81–82, 292, 311, 319, 322–323, 342, 371, 395 Fermat’s, 330, 390–392 Hamilton’s, 88, 90, 199–200 Huygens’, 390 of insufficient reason, 2 MMM, 240 Saint-Venant’s, 91 variational, mesoscale bounds, 319–324 virtual work principle, 169, 198–200 Probability definitions of, 1–4 distribution, 1, 3, 7–10, 20, 30, 48, 65, 82, 84, 156, 277, 297, 364, 429 measure, 1, 7, 9, 11, 14, 42, 386 geometric objects, 1–14 Process Bernoulli, 20–21, 84 complex, 377–379 complex vs. compound, 376–379 compound, 376, 378–379 diffusion, 412, 416 Gaussian, 52, 429 hard-core, inhibition, 23 inhibition, 19, 23 inverse amplitude, 430–431 Markov, 356, 412, 417, 427, 430 point, 18–19, 21–22, 27, 31, 36, 38, 44, 261, 279, 281 random, 45–86, 425–427 stochastic, 47–51, 54, 355, 357, 403, 420, 430–431 vector random, 54–55 Wiener, 413, 427 Production rate entropy, 370 Pseudo-continuum, 221

Q Quasi-homogeneous dissipation function, 374–376

R Radial distribution, 22 Random boundaries, 56–57 chessboard, 258–261, 336–341 composite, 406–407 crack model, 34–35 evolutions, 403–404

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fiber network, 153–154 fibers, 27–35 field (See Random field) line fields, 27–35 lines, 11–14 local medium, 209–210 materials, 72–73 mean field equations material, 72–73 media, 359–383, 385–433 class of models, 408–410 mesoscale response, 359–383 models, 1–44 spectral finite elements, 395–402 wavefronts, 385–433 waves, 385–433 media models, 1–44 microbeams, 58–61 microstructure, 35–44, 301–309, 427–433 one-dimensional composites, 403–408 points, 11–14 Poisson lines, 27–30 process, 45–86, 425–427 processes, 45–86 tessellations, 30–35, 39 vector processes, 54–55 Random chessboard material, 258, 263–264, 327, 336–337 Random continuum, 37, 237, 273, 292–293, 422 Random field continuum, 237, 273, 292–293 discrete to continuum, 273–275 mesoscale, 273–283 models, 273–309 one-dimensional, 45–55 scalar, 45–54, 85, 323 three-dimensional, 62–72 two-dimensional, 62–72 vector, 85, 210 Random media spectral finite elements, 395–402 thermomechanics, 359–383 wave, 385–433 wavefront, 385–433 Random microstructure, 35–44, 301–309, 416, 427–433 Random process, 45–86, 425–427 Rari-constant theory, 97 Restricted continuum, 214–216 Reynolds stress, 74, 85 Rigid-perfectly plastic microstructure, 333–341 Rigidity percolation, 133, 135, 137 Rods, wave, 395–398 RVE (Representative Volume Element)

dissipation function, 359–363 postulate, 240–241 size, 263, 346, 423

S Saint-Venant’s principle, 91 Scaling complex vs. compound processes, 376–379 in damage phenomena, 354–357 laws, 102, 259, 314 trend, 353 trends, 357–358 Segmented elastic bars, wave, 407–408 Separation of scales, 74, 237, 240, 244, 250, 273, 286, 288, 293–294, 385, 395, 423 SFE (Stochastic Finite Elements) elastic media, 286–293 phenomenological studies, 290–293 phenomenological study, 68, 237, 241, 273–309 random field models and, 273–309 Shear band, 316, 324, 329–331, 338 Shock wavefront, 415 Simple lattice, 87–88 elastic strings, 87–88 Slip-line, 293–300 Spectral finite elements, 126, 131, 395, 398, 402 for flexural waves, 398–402 for random media, 395–402 Square lattice, 93–94, 115, 131, 139, 282, 332 Bernoulli–Euler beams, 115–117 Stationary distribution, 85 Stochastic finite elements for elastic media, 286–293 random field models and, 273–309 Stochastic process, 47–51, 54, 355, 357, 403, 420, 430–431 Stochastic propagation wave, 386–395 Stochastic (random) evolution, 354, 385, 403–404, 431 in damage phenomena, 354 Strain-gradient continuum, 119–120 Stress Cauchy, 119, 128–129, 188, 191–192, 206, 232, 241, 365, 378, 381 couple-stress, 129–130, 191–192, 198, 207, 209, 211–215, 224, 227, 229, 232, 278–279, 377 Reynolds, 74, 85 Stress invariance, 105, 216 CLM result, 177–187, 216–227

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Subject Index eigencurvatures, extensions to presence of, 223–227 eigenstrains, extension to presence of, 184–187, 223–227 Stubby beam, 112–114 Surface wave, 56, 386 propagation, 56–57

T Tessellations, random, 30–35, 39 Theorem asymptotic, 15 average/averaging, 246, 252, 254, 318–319 ergodic, 74, 76, 240 Green-Gauss, 381 Liouville, 360, 362 partition, 313 upper bound, 334 Thermodynamic orthogonality, 327, 363, 370, 372, 376–377, 382 Thermodynamics continuum, 359–364 Thermoelastic helix, 126 Thermoelastic microstructure, 347–354 Thermomechanics Hill(-Mandel) condition, 364–365 with internal variables, 235 random media, 359–383 Thickness, wavefront, 422, 432 Three-dimensional elasticity, 171–172 Timoshenko beam, 60, 85, 89, 110–114, 122, 131, 140, 153, 395, 398–401 Topology, 133–134, 139, 141, 145–146, 168–169, 255, 263, 304, 308 structural, 133–134, 139 Traction boundary condition, 315–317, 334–336 Transformation, Legendre, 347, 353, 367–369, 373, 375 Transient wave, 386, 408–422 Trial, Bernoulli, 14–15, 18 Triangular lattice, 94–99, 107, 110, 114, 120–122, 134–135 Bernoulli–Euler beams, 107–110 Timoshenko beams, 110–112 Triple honeycomb lattice, 99 Truss-like continuum, 165, 301–303

471 Two-dimensional elasticity, 172–174 Two-phase material, 221–222

U Uniform boundary condition, 251, 278, 312, 317, 366 Upper bound theorem, 334

V Variational, mesoscale bounds principle, 319–324 Variational principle, 81–82, 292, 311, 319, 322–323, 342, 371, 395 Vector random process, 54–55 Virtual work principle, 169, 198, 200 Viscoelastic microstructure, 341–342 Viscoelasticity principle, 128 von Mises distribution, 26

W Wave acceleration, 422, 425, 428 flexural, 398–402 Floquet, 404–407 harmonic, 125–127, 403 one-dimensional composites, 403–408 plane, 57, 204, 404 plane monochromatic, 204 random media, 385–433 rods, 395–398 segmented elastic bars, 407–408 stochastic propagation, 386–395 surface, 56, 386 surface propagation, 56–57 transient, 386, 408–422 Wavefront acceleration, 386, 415–416, 422, 428, 433 nonlinear media, 422–433 random media, 385–433 random microstructures, 427–433 shock, 415 thickness vs. heterogeneity, 422–426 Weibull distribution, 43 Wiener process, 413, 427 Wrapped distribution, 26

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