Homogenization of Coupled Phenomena in Heterogenous Media

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Author: Jean-Louis Auriault | Claude Boutin | Christian Geindreau

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Homogenization of Coupled Phenomena in Heterogenous Media


Homogenization of Coupled Phenomena in Heterogenous Media

Jean-Louis Auriault Claude Boutin Christian Geindreau

First published in France in 2009 by Hermes Science/Lavoisier entitled: Homogénéisation de phénomènes couplés en milieux hétérogènes volumes 1 et 2 © LAVOISIER 2009 First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2009 The rights of Jean-Louis Auriault, Claude Boutin and Christian Geindreau to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Auriault, J.-L. (Jean-Louis) [Homogénéisation de phénomènes couplés en milieux hétérogènes. English] Homogenization of coupled phenomena in heterogenous media / Jean-Louis Auriault, Claude Boutin, Christian Geindreau. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-161-2 1. Inhomogeneous materials--Mathematical models. 2. Coupled problems (Complex systems) 3. Homogenization (Differential equations) I. Boutin, Claude. II. Geindreau, Christian. III. Title. TA418.9.I53A9513 2009 620.1'1015118--dc22 2009016650 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-161-2 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Contents

Main notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

PART ONE . U PSCALING M ETHODS . . . . . . . . . . . . . . . . . . . . . . .

27

Chapter 1. An Introduction to Upscaling Methods

29

. . . . . . . . . . . . . .

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Heat transfer in a periodic bilaminate composite . . . . . . . . 1.2.1. Transfer parallel to the layers . . . . . . . . . . . . . . . . 1.2.2. Transfer perpendicular to the layers . . . . . . . . . . . . 1.2.3. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Characteristic macroscopic length . . . . . . . . . . . . . 1.3. Bounds on the effective coefficients . . . . . . . . . . . . . . . 1.3.1. Theorem of virtual powers . . . . . . . . . . . . . . . . . . 1.3.2. Minima in the complementary power and potential power 1.3.3. Hill principle . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4. Voigt and Reuss bounds . . . . . . . . . . . . . . . . . . . 1.3.4.1. Upper bound: Voigt . . . . . . . . . . . . . . . . . . 1.3.4.2. Lower bound: Reuss . . . . . . . . . . . . . . . . . 1.3.5. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6. Hashin and Shtrikman’s bounds . . . . . . . . . . . . . . . 1.3.7. Higher-order bounds . . . . . . . . . . . . . . . . . . . . . 1.4. Self-consistent method . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Boundary-value problem . . . . . . . . . . . . . . . . . . . 1.4.2. Self-consistent hypothesis . . . . . . . . . . . . . . . . . . 1.4.3. Self-consistent method with simple inclusions . . . . . .

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. . . . . . . . . . . . . . . . . . . .

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29 30 31 33 35 35 36 36 38 39 40 40 42 44 45 46 46 47 48 49

6

Homogenization of Coupled Phenomena

1.4.3.1. Determination of βα for a homogenous spherical inclusion 1.4.3.2. Self-consistent estimate . . . . . . . . . . . . . . . . . . . . 1.4.3.3. Implicit morphological constraints . . . . . . . . . . . . . 1.4.4. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 52 53

Chapter 2. Heterogenous Medium: Is an Equivalent Macroscopic Description Possible? . . . . . . . . . . . . . . . . . . . . . . . .

55

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Comments on techniques for micro-macro upscaling . . . . . . . . . 2.2.1. Homogenization techniques for separated length scales . . . . 2.2.2. The ideal homogenization method . . . . . . . . . . . . . . . . 2.3. Statistical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Method of multiple scale expansions . . . . . . . . . . . . . . . . . . 2.4.1. Formulation of multiple scale problems . . . . . . . . . . . . . 2.4.1.1. Homogenizability conditions . . . . . . . . . . . . . . . 2.4.1.2. Double spatial variable . . . . . . . . . . . . . . . . . . . 2.4.1.3. Stationarity, asymptotic expansions . . . . . . . . . . . . 2.4.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Parallels between macroscopic models for materials with periodic and random structures . . . . . . . . . . . . . . . . . . 2.4.3.1. Periodic materials . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2. Random materials with a REV . . . . . . . . . . . . . . . 2.4.4. Hill macro-homogenity and separation of scales . . . . . . . . 2.5. Comments on multiple scale methods and statistical methods . . . . 2.5.1. On the periodicity, the stationarity and the concept of the REV 2.5.2. On the absence of, or need for macroscopic prerequisites . . . 2.5.3. On the homogenizability and consistency of the macroscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4. On the treatment of problems with several small parameters . .

. . . . . . . . . . .

55 56 57 59 60 61 61 61 62 64 65

. . . . . . .

68 68 68 69 69 69 70

. .

71 72

Chapter 3. Homogenization by Multiple Scale Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Separation of scales: intuitive approach and experimental visualization 3.2.1. Intuitive approach to the separation of scales . . . . . . . . . . . 3.2.2. Experimental visualization of fields with two length scales . . . 3.2.2.1. Investigation of a flexible net . . . . . . . . . . . . . . . . 3.2.2.2. Photoelastic investigation of a perforated plate . . . . . . . 3.3. One-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Elasto-statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 78 78 81 84 85

Contents

3.3.1.1. Equivalent macroscopic description . . . . . . . . . . . . 3.3.1.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Elasto-dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1. Macroscopic dynamics: Pl = O(ε2 ) . . . . . . . . . . . 3.3.2.2. Steady state: Pl = O(ε3 ) . . . . . . . . . . . . . . . . . 3.3.2.3. Non-homogenizable description: Pl = O(ε) . . . . . . 3.3.3. Comments on the different possible choices for spatial variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Expressing problems within the formalism of multiple scales . . . . 3.4.1. How do we select the correct mathematical formulation based on the problem at hand? . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Need to evaluate the actual scale ratio εr . . . . . . . . . . . . 3.4.3. Evaluation of the actual scale ratio εr . . . . . . . . . . . . . . 3.4.3.1. Homogenous treatment of simple compression . . . . . 3.4.3.2. Point force in an elastic object . . . . . . . . . . . . . . . 3.4.3.3. Propagation of a harmonic plane wave in elastic composites . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3.4. Diffusion wave in heterogenous media . . . . . . . . . . 3.4.3.5. Conclusions to be drawn from the examples . . . . . . . PART TWO . H EAT AND M ASS T RANSFER

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7

86 89 91 92 95 95

. 97 . 100 . . . . .

100 101 102 103 104

. 104 . 105 . 106

. . . . . . . . . . . . . . . . . . . 107

Chapter 4. Heat Transfer in Composite Materials . . . . . . . . . . . . . . . 109 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Heat transfer with perfect contact between constituents . . . . . . 4.2.1. Formulation of the problem . . . . . . . . . . . . . . . . . . . 4.2.2. Thermal conductivities of the same order of magnitude . . . 4.2.2.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.2.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.2.2.3. Example: bilaminate composite . . . . . . . . . . . . . 4.2.3. Weakly conducting phase in a connected matrix: memory effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.3.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.2.3.3. Example: bilaminate composite . . . . . . . . . . . . . 4.2.4. Composites with highly conductive inclusions embedded in a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2. Macroscopic model . . . . . . . . . . . . . . . . . . . . 4.3. Heat transfer with contact resistance between constituents . . . . 4.3.1. Model I – Very weak contact resistance . . . . . . . . . . . .

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109 109 110 113 113 117 119

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121 122 124 125

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126 127 129 130 132

8


4.3.2. 4.3.3. 4.3.4. 4.3.5. 4.3.6. 4.3.7.

Model II – Moderate contact resistance . . . . . . . . . . Model III – High contact resistance . . . . . . . . . . . . . Model IV – Model with two coupled temperature fields . Model V – Model with two decoupled temperature fields Example: bilaminate composite . . . . . . . . . . . . . . . Choice of model . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5. Diffusion/Advection in Porous Media

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133 135 138 140 141 142

. . . . . . . . . . . . . . . 143

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Diffusion-convection on the pore scale and estimates . . . . . . . . 5.3. Diffusion dominates at the macroscopic scale . . . . . . . . . . . . . 5.3.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.1. Boundary value problem for c∗(0) . . . . . . . . . . . . . 5.3.1.2. Boundary value problem for c∗(1) . . . . . . . . . . . . . 5.3.1.3. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.3.2. Macroscopic diffusion model . . . . . . . . . . . . . . . . . . . 5.4. Comparable diffusion and advection on the macroscopic scale . . . 5.4.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1.1. Boundary value problems for c∗(0) and c∗(1) . . . . . . . 5.4.1.2. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.4.2. Macroscopic diffusion-advection model . . . . . . . . . . . . . 5.5. Advection dominant at the macroscopic scale . . . . . . . . . . . . . 5.5.1. Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1.1. Boundary value problem for c∗(0) . . . . . . . . . . . . . 5.5.1.2. Boundary value problem for c∗(1) . . . . . . . . . . . . . 5.5.1.3. Boundary value problem for c∗(2) . . . . . . . . . . . . . 5.5.2. Dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Very strong advection . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Example: Porous medium consisting of a periodic lattice of narrow parallel slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1. Analysis of the flow . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2. Determination of the dispersion coefficient . . . . . . . . . . . 5.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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143 143 146 146 146 147 148 148 149 149 149 149 150 151 151 151 151 153 154 154

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155 156 157 159

Chapter 6. Numerical and Analytical Estimates for the Effective Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2. Effective thermal conductivity for some periodic media . . . . . . . . 162 6.2.1. Media with spherical inclusions, connected or non-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Contents

6.2.1.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . 6.2.1.2. Solution to the boundary value problem over the period 6.2.1.3. Effective thermal conductivity . . . . . . . . . . . . . . . 6.2.2. Fibrous media consisting of parallel fibers . . . . . . . . . . . . 6.2.2.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . 6.2.2.2. Solution to the boundary value problem over the period 6.2.2.3. Effective thermal conductivity . . . . . . . . . . . . . . . 6.3. Study of various self-consistent schemes . . . . . . . . . . . . . . . 6.3.1. Self-consistent scheme for bi-composite inclusions . . . . . . . 6.3.1.1. Granular or cellular media . . . . . . . . . . . . . . . . . 6.3.1.2. Fibrous media . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.3. General remarks on bi-composite models . . . . . . . . . 6.3.2. Self-consistent scheme with multi-composite substructures . . 6.3.2.1. n-composite substructure . . . . . . . . . . . . . . . . . . 6.3.2.2. Treatment of a contact resistance . . . . . . . . . . . . . 6.3.3. Combined self-consistent schemes . . . . . . . . . . . . . . . . 6.3.3.1. Mixed self-consistent schemes . . . . . . . . . . . . . . . 6.3.3.2. Multiple self-consistent schemes . . . . . . . . . . . . . 6.4. Comparison with experimental results for the thermal conductivity of cellular concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. Dry cellular concrete . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2. Damp cellular concrete . . . . . . . . . . . . . . . . . . . . . . .

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9

162 163 163 168 168 169 170 175 175 175 178 179 181 181 183 184 185 185

. 188 . 189 . 190

PART THREE . N EWTONIAN F LUID F LOW T HROUGH R IGID P OROUS M EDIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 7. Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.1. 7.2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state flow of an incompressible Newtonian fluid in a porous medium: Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Comments on macroscopic behavior . . . . . . . . . . . . . . . 7.2.2.1. Physical meaning of the macroscopic quantities . . . . . 7.2.2.2. Structure of the macroscopic law . . . . . . . . . . . . . 7.2.2.3. Study of the underlying problem . . . . . . . . . . . . . 7.2.2.4. Properties of K∗ . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.5. Energetic consistency . . . . . . . . . . . . . . . . . . . . 7.2.3. Non-homogenizable situations . . . . . . . . . . . . . . . . . . 7.2.3.1. Case where QL = O(ε−1 ). . . . . . . . . . . . . . . . .

. 197 . . . . . . . . . .

199 201 203 203 204 205 205 206 206 207

10


7.2.3.2. Case where QL = O(ε−3 ) . . . . . . . . . . . . . . . 7.3. Dynamics of an incompressible fluid in a rigid porous medium . 7.3.1. Local description and estimates . . . . . . . . . . . . . . . . 7.3.2. Macroscopic behavior: generalized Darcy’s law . . . . . . 7.3.3. Discussion of the macroscopic description . . . . . . . . . . 7.3.3.1. Physical meaning of macroscopic quantities . . . . . 7.3.3.2. Energetic consistency . . . . . . . . . . . . . . . . . . 7.3.3.3. The tensors H∗ and Λ∗ are symmetric . . . . . . . . 7.3.3.4. Low-frequency behavior . . . . . . . . . . . . . . . . 7.3.3.5. Additional mass effect . . . . . . . . . . . . . . . . . 7.3.3.6. Transient excitation: Dynamics with memory effects 7.3.3.7. Quasi-periodicity . . . . . . . . . . . . . . . . . . . . 7.3.4. Circular cylindrical pores . . . . . . . . . . . . . . . . . . . 7.4. Appearance of inertial non-linearities . . . . . . . . . . . . . . . 7.4.1. Macroscopic model . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Macroscopically isotropic and homogenous medium . . . . 7.4.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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208 209 209 211 213 213 213 215 215 215 216 216 216 220 221 224 226 226

Chapter 8. Compressible Newtonian Fluid Flow Though a Rigid Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Slow isothermal flow of a highly compressible fluid . . . 8.2.1. Estimates . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Steady-state flow . . . . . . . . . . . . . . . . . . . . 8.2.3. Transient conservation of mass . . . . . . . . . . . . 8.3. Wall slip: Klinkenberg’s law . . . . . . . . . . . . . . . . 8.3.1. Pore scale description and estimates . . . . . . . . . 8.3.2. Klinkenberg’s law . . . . . . . . . . . . . . . . . . . 8.3.3. Small Knudsen numbers . . . . . . . . . . . . . . . . 8.3.4. Properties of the Klinkenberg tensor Hk . . . . . . . 8.3.4.1. Hk is positive . . . . . . . . . . . . . . . . . . 8.3.4.2. Symmetries . . . . . . . . . . . . . . . . . . . 8.4. Acoustics in a rigid porous medium saturated with a gas 8.4.1. Harmonic perturbation of a gas in a porous medium 8.4.2. Analysis of local physics . . . . . . . . . . . . . . . . 8.4.3. Non-dimensionalization and renormalization . . . . 8.4.4. Homogenization . . . . . . . . . . . . . . . . . . . . 8.4.4.1. Pressure and temperature . . . . . . . . . . . . 8.4.4.2. Velocity field . . . . . . . . . . . . . . . . . . .

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229 229 230 231 235 238 238 240 241 243 243 244 245 246 247 249 251 251 252

Contents

11

8.4.4.3. Macroscopic conservation of mass . . . . . . . . . . . . . 252 8.4.5. Biot-Allard model . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Chapter 9. Numerical Estimation of the Permeability of Some Periodic Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Permeability tensor: recap of results from periodic homogenization 9.3. Steady state permeability of fibrous media . . . . . . . . . . . . . . 9.3.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Transverse permeability . . . . . . . . . . . . . . . . . . . . . . 9.3.2.1. Mesh, velocity fields and microscopic pressure fields . . 9.3.2.2. Transverse permeability KT . . . . . . . . . . . . . . . . 9.3.3. Longitudinal permeability . . . . . . . . . . . . . . . . . . . . . 9.3.3.1. Mesh, velocity fields . . . . . . . . . . . . . . . . . . . . 9.3.3.2. Longitudinal permeability KL . . . . . . . . . . . . . . . 9.4. Steady state and dynamic permeability of granular media . . . . . . 9.4.1. Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Steady state permeability . . . . . . . . . . . . . . . . . . . . . 9.4.4. Dynamic permeability . . . . . . . . . . . . . . . . . . . . . . . 9.4.4.1. Effect of frequency . . . . . . . . . . . . . . . . . . . . . 9.4.4.2. Low-frequency approximation . . . . . . . . . . . . . . . 9.4.4.3. High-frequency approximation . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

257 259 259 259 260 261 262 264 264 264 267 267 267 269 269 269 270 272

Chapter 10. Self-consistent Estimates and Bounds for Permeability . . . . 275 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. Notation and glossary . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Intrinsic (or steady state) permeability of granular and fibrous media 10.2.1. Summary of results obtained through periodic homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1.1. Global and local descriptions – energetic consistency . . . 10.2.1.2. Connections between the micro- and macroscopic descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2. Self-consistent estimate of the permeability of granular media . 10.2.2.1. Formulation of the self-consistent problem . . . . . . . . . 10.2.2.2. General expression for the fields in the inclusion . . . . . 10.2.2.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . . 10.2.3. Solution and self-consistent estimates . . . . . . . . . . . . . . . 10.2.3.1. Pressure approach: p field . . . . . . . . . . . . . . . . . 10.2.3.2. Velocity approach: v field . . . . . . . . . . . . . . . . .

275 277 278 279 280 281 281 281 283 285 288 288 289

12


10.2.3.3. Comparison of estimates . . . . . . . . . . . . . . . . . . . 10.2.4. From spherical substructure to granular materials . . . . . . . . . 10.2.4.1. Cubic lattices of spheres . . . . . . . . . . . . . . . . . . . 10.2.4.2. Bounds on the permeability of ordered or disordered granular media . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4.3. Empirical laws . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5. Intrinsic permeability of fibrous media . . . . . . . . . . . . . . . 10.2.5.1. Periodic arrangements of identical cylinders . . . . . . . . 10.2.5.2. Permeability bounds for ideal ordered and disordered fibrous media . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Dynamic permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Summary of homogenization results . . . . . . . . . . . . . . . . 10.3.1.1. Global and local description – Energetic consistency . . . 10.3.1.2. Frequency characteristics of dynamic permeability . . . . 10.3.2. Self-consistent estimates of dynamic permeability . . . . . . . . 10.3.3. Formulation of the problem in the inclusion . . . . . . . . . . . . 10.3.3.1. Expressions for the fields . . . . . . . . . . . . . . . . . . . 10.3.3.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . 10.3.4. Solution and self-consistent estimates . . . . . . . . . . . . . . . 10.3.4.1. P estimate: p field . . . . . . . . . . . . . . . . . . . . . . 10.3.4.2. V estimate: v field . . . . . . . . . . . . . . . . . . . . . 10.3.4.3. Commentary and comparisons with numerical results for periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4.4. Bounds on the dynamic permeability of granular media . 10.3.4.5. Bounds on the real and imaginary parts of K(ω) . . . . . . 10.3.4.6. Bounds on the real and imaginary parts of H(ω) . . . . . . 10.3.4.7. Low-frequency bounds . . . . . . . . . . . . . . . . . . . . 10.3.4.8. High-frequency bounds for tortuosity . . . . . . . . . . . . 10.4. Klinkenberg correction to intrinsic permeability . . . . . . . . . . . . 10.4.1. Local and global descriptions obtained through homogenization 10.4.2. Self-consistent estimates of Klinkenberg permeability . . . . . . 10.5. Thermal permeability – compressibility of a gas in a porous medium 10.5.1. Dynamic compressibility obtained by homogenization . . . . . 10.5.2. Self-consistent estimate of the thermal permeability of granular media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3. Properties of thermal permeability . . . . . . . . . . . . . . . . . 10.5.4. Significance of connectivity of phases . . . . . . . . . . . . . . . 10.5.5. Critical thermal and viscous frequencies . . . . . . . . . . . . . . 10.6. Analogy between the trapping constant and permeability . . . . . . . 10.6.1. Trapping constant . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 291 291 292 296 297 298 298 299 300 300 302 304 304 305 306 307 308 309 310 314 315 316 317 318 318 318 319 322 322 323 324 326 327 328 328

Contents

10.6.1.1. Comparison between the trapping constant and intrinsic permeability . . . . . . . . . . . . . . . . . . . 10.6.1.2. Self-consistent estimate of the trapping constant for granular media . . . . . . . . . . . . . . . . . . . . . . . 10.6.2. Diffusion-trapping in the transient regime . . . . . . . . . . . 10.6.3. Steady-state diffusion-trapping regime in media with a finite absorptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

. . 330 . . 331 . . 332 . . 333 . . 334

PART FOUR . S ATURATED D EFORMABLE P OROUS M EDIA . . . . . . . . . 337 Chapter 11. Quasi-statics of Saturated Deformable Porous Media . . . . . 339 11.1. Empty porous matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1. Local description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2. Equivalent macroscopic behavior . . . . . . . . . . . . . . . . . 11.1.2.1. Boundary-value problem for u∗(0) . . . . . . . . . . . . 11.1.2.2. Boundary-value problem for u∗(1) . . . . . . . . . . . . 11.1.2.3. Boundary-value problem for u∗(2) . . . . . . . . . . . . 11.1.3. Investigation of the equivalent macroscopic behavior . . . . . . 11.1.3.1. Physical meaning of quantities involved in macroscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3.2. Properties of the effective elastic tensor . . . . . . . . . . 11.1.3.3. Energetic consistency . . . . . . . . . . . . . . . . . . . . 11.1.4. Calculation of the effective coefficients . . . . . . . . . . . . . 11.2. Deformable saturated porous medium . . . . . . . . . . . . . . . . . 11.2.1. Local description and estimates . . . . . . . . . . . . . . . . . . 11.2.2. Diphasic macroscopic behavior: Biot model . . . . . . . . . . . 11.2.2.1. Boundary-value problem for u∗(0) . . . . . . . . . . . . 11.2.2.2. Boundary-value problem for p∗(0) and v∗(0) . . . . . . . 11.2.2.3. Boundary-value problem for u∗(1) . . . . . . . . . . . . 11.2.2.4. First compatibility equation . . . . . . . . . . . . . . . . 11.2.2.5. Second compatibility equation . . . . . . . . . . . . . . . 11.2.2.6. Macroscopic description . . . . . . . . . . . . . . . . . . 11.2.3. Properties of the macroscopic diphasic description . . . . . . . 11.2.3.1. Properties of macroscopic quantities and effective coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3.2. The coupling between (11.31) and (11.32) is symmetric, α = γ. . . . . . . . . . . . . . . . . . . . . . 11.2.3.3. The tensor α∗ is symmetric . . . . . . . . . . . . . . . . 11.2.3.4. The coefficient β ∗ is positive, β ∗ > 0 . . . . . . . . . . . 11.2.3.5. Specific cases . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

340 340 342 342 343 344 345

. . . . . . . . . . . . . .

345 346 348 348 349 350 352 352 352 353 354 355 355 355

. 355 . . . .

356 356 357 357

14


11.2.3.6. Homogenious matrix material . . . . . . . . . . . . . 11.2.3.7. Homogenous and isotropic matrix material and macroscopically isotropic matrix . . . . . . . . . . . 11.2.3.8. Diphasic consolidation equations: Biot model . . . . 11.2.3.9. Effective stress . . . . . . . . . . . . . . . . . . . . . . 11.2.3.10. Compressible interstitial fluid . . . . . . . . . . . . . 11.2.4. Monophasic elastic macroscopic behavior: Gassman model 11.2.5. Monophasic viscoelastic macroscopic behavior . . . . . . . 11.2.6. Relationships between the three macroscopic models . . .

. . . 357 . . . . . . .

. . . . . . .

. . . . . . .

358 359 361 361 362 363 365

Chapter 12. Dynamics of Saturated Deformable Porous Media . . . . . . . 367 12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Local description and estimates . . . . . . . . . . . . . . . . . . . . . . 12.3. Diphasic macroscopic behavior: Biot model . . . . . . . . . . . . . . 12.4. Study of diphasic macroscopic behavior . . . . . . . . . . . . . . . . . 12.4.1. Equations for the diphasic dynamics of a saturated deformable porous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2. Rheology and dynamics . . . . . . . . . . . . . . . . . . . . . . . 12.4.3. Additional mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4. Transient motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.5. Small pulsation ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.6. Dispersive waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5. Macroscopic monophasic elastic behavior: Gassman model . . . . . . 12.6. Monophasic viscoelastic macroscopic behavior . . . . . . . . . . . . . 12.7. Choice of macroscopic behavior associated with a given material and disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1. Effects of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1.1. Transition from diphasic behavior to elastic behavior . . . 12.7.1.2. Transition from viscoelastic behavior to elastic behavior . 12.7.2. Effect of rigidity of the porous skeleton . . . . . . . . . . . . . . 12.7.3. Effect of frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.3.1. Low-dispersion P1 and S waves . . . . . . . . . . . . . . . 12.7.3.2. Dispersive P2 wave . . . . . . . . . . . . . . . . . . . . . . 12.7.4. Effect of pore size . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.5. Application example: bituminous concretes . . . . . . . . . . . .

367 368 370 374 374 375 376 376 376 376 377 378 380 382 382 383 384 384 384 385 385 385

Chapter 13. Estimates and Bounds for Effective Poroelastic Coefficients . 389 13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.2. Recap of the results of periodic homogenization . . . . . . . . . . . . 389 13.3. Periodic granular medium . . . . . . . . . . . . . . . . . . . . . . . . . 391

Contents

13.3.1. Microstructure and material . . . . . . . . . . . . . . . . . . . 13.3.2. Effective elastic tensor c . . . . . . . . . . . . . . . . . . . . . 13.3.2.1. Methodology . . . . . . . . . . . . . . . . . . . . . . . 13.3.2.2. Compressibility and shear moduli . . . . . . . . . . . . 13.3.2.3. Degree of anisotropy . . . . . . . . . . . . . . . . . . . 13.3.2.4. Young’s modulus and Poisson’s ratio . . . . . . . . . . 13.3.3. Biot tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. Influence of microstructure: bounds and self-consistent estimates 13.4.1. Voigt and Reuss bounds . . . . . . . . . . . . . . . . . . . . . 13.4.2. Hashin and Shtrikman bounds . . . . . . . . . . . . . . . . . . 13.4.3. Self-consistent estimates . . . . . . . . . . . . . . . . . . . . . 13.4.4. Comparison: numerical results, bounds and self-consistent estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Comparison with experimental data . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

15

391 392 392 394 396 396 398 398 399 399 400

. . 401 . . 403

Chapter 14. Wave Propagation in Isotropic Saturated Poroelastic Media . 407 14.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2. Comments on the parameters . . . . . . . . . . . . . . 14.2.2.1. Elastic coefficients . . . . . . . . . . . . . . . . 14.2.2.2. Dynamic permeability . . . . . . . . . . . . . . 14.2.3. Degrees of freedom and dimensionless parameters . . 14.3. Three modes of propagation in a saturated porous medium 14.3.1. Wave equations . . . . . . . . . . . . . . . . . . . . . . 14.3.2. Elementary wave fields: plane waves . . . . . . . . . . 14.3.2.1. Homogeneous plane waves . . . . . . . . . . . . 14.3.2.2. Inhomogenous plane waves . . . . . . . . . . . 14.3.3. Physical characteristics of the modes . . . . . . . . . . 14.3.3.1. Low frequencies: f fc . . . . . . . . . . . . 14.3.3.2. High frequencies: f fc . . . . . . . . . . . . 14.3.3.3. Full spectrum . . . . . . . . . . . . . . . . . . . 14.4. Transmission at an elastic-poroelastic interface . . . . . . . 14.4.1. Expression for the conditions at the interface . . . . . 14.4.2. Transmission of compression waves . . . . . . . . . . 14.5. Rayleigh waves . . . . . . . . . . . . . . . . . . . . . . . . . 14.6. Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1. Source terms . . . . . . . . . . . . . . . . . . . . . . . 14.6.2. Determination of the fundamental solutions . . . . . . 14.6.3. Fundamental solutions in plane geometry . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

407 408 408 410 410 410 411 412 413 416 416 417 419 419 421 423 423 426 428 430 432 432 433 437

16


14.6.4. Symmetry of the Green’s matrix, and reciprocity theorem 14.6.5. Properties of radiated fields . . . . . . . . . . . . . . . . . 14.6.5.1. Far-field – near-field – quasi-static regime . . . . . 14.6.5.2. Decomposition into elementary waves . . . . . . . 14.6.6. Energy and moment sources: explosion and injection . . 14.7. Integral representation . . . . . . . . . . . . . . . . . . . . . . . 14.8. Dislocations in porous media . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

438 439 441 442 442 445 448

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Main notations

Operators ⊗ : · × , x div, divx grad, gradx curl, curlx a aΩα

tensor product doubly contracted tensor product scalar product vector product Laplacian, Laplacian with respect to the variable x divergence, divergence with respect to the variable x gradient, gradient with respect to the variable x curl, curl with respect to the variable x mean of a over the period Ω mean of a over the domain Ωα

Dimensions, bases, spatial variables Ω Ωα ∂Ω Γ l L ε = l/L ei n X = Xi ei x∗ = X/Lc = x∗i ei y∗ = X/lc = yi∗ ei

period domain occupied by constituent α boundary of the period interface between two constituents microscopic length [m] macroscopic length [m] separation of scales parameter unit vector – orthonormal basis unit vector normal to the interface Γ physical spatial variable [m] macroscopic dimensionless variable [-] microscopic dimensionless variable [-]

18


Dimensionless numbers B Kn Pe Re Rt S

Biot number [-] Knudsen number [-] Péclet number [-] Reynolds number [-] transient Reynolds number [-] Strouhal number [-]

Properties and physical quantities a = aijkl ei ⊗ ej ⊗ ek ⊗ el c cα Cα c = cijkl ei ⊗ ej ⊗ ek ⊗ el D(v) = Dij (v) ei ⊗ ej dif ei ⊗ ej Ddif = Dij dis dis ei ⊗ ej D = Dij mol Dmol = Dij ei ⊗ ej e(u) = eij ei ⊗ ej E h H = Hij ei ⊗ ej H(ω) = Hij (ω) ei ⊗ ej H R , HI H(ω) = K−1 = HI + iHR H(ω) = Hij (ω) ei ⊗ ej k ei ⊗ ej Hk = Hij I K K = Kij ei ⊗ ej k ei ⊗ ej Kk = Kij K(ω) = KR + iKI K(ω) = Kij (ω) ei ⊗ ej K R , KI M p t Tα us = usi ei

elastic tensor [MPa] concentration [-] volume fraction of constituent α [-] heat capacity of constituent α [J/(Kg.K)] effective elastic tensor [MPa] strain rate tensor [s−1 ] effective diffusion tensor [m2 /s] effective dispersion tensor [m2 /s] molecular diffusion tensor [m2 /s] strain tensor [-] Young’s modulus [MPa] inverse of the thermal contact resistance inverse of the intrinsic permeability tensor [m2 ] inverse of the dynamic permeability tensor K [m−2 ] real and imaginary parts of the H tensor inverse of the dynamic permeability of an isotropic medium inverse of the dynamic hydraulic conductivity tensor Λ = ηK(ω) Klinkenberg tensor [m2 ] identity tensor intrinsic permeability of an isotropic medium [m2 ] intrinsic steady state permeability tensor [m2 ] Klinkenberg permeability tensor [m2 ] dynamic permeability of an isotropic medium [m2 ] dynamic permeability tensor [m2 ] real and imaginary parts of the K tensor form factor fluid pressure [Pa] time [s] temperature of constituent α [˚K] solid displacement [m]

Main notations

v = vi ei α δ η λα λeff = λeff ij ei ⊗ ej λ Λv Λ(ω) = Λij (ω) ei ⊗ ej μ ν ρ σ = σij ei ⊗ ej τ0 , τ∞ φ ω

fluid velocity [m/s] Biot tensor [-] boundary layer thickness (thermal, diffusion...) dynamic fluid viscosity [Pa.s] thermal conductivity of constituent α [W/(m.˚K)] effective thermal conductivity tensor [W/(m.˚K)] Lamé coefficient [MPa] viscous length hydraulic conductivity tensor shear modulus [MPa] Poisson’s ratio [-] fluid density [kg/m3 ] Cauchy stress tensor low- and high-frequency tortuosity porosity [-] pulsation (angular frequency) [rad/s]

Subscripts αc α∗ Ql QL

characteristic value of α dimensionless quantity (α = αc α∗ ) dimensionless number Q estimated from the microscopic viewpoint dimensionless number Q estimated from the macroscopic viewpoint

19


Introduction

The result of a fusion of mathematical and physical concepts, homogenization has established itself as a method of overcoming the usual framework based on a description of elementary phenomena in a homogenous medium, to achieve the objective of a global description of coupled phenomena in heterogenous media. This book aims to present the key methodological points and their relevance to engineering science in a pedagogical format. What is the nature of the problem? Even brief observation of natural or industrial materials reveals that they often consist of a combination of different constituents in various structures, and they are therefore heterogenous. For example, take the behavior of civil engineering materials. The descriptions of the properties that they exhibit – and consequently the design rules for structures built using these materials – are, for the most part, issues related to the mechanics of continuous media applied to homogenous media. This theory has been widely proven, and a huge number of constructions designed using these principles can attest to its success, as can the accuracy of modeling performed using this approach. This simple observation leads us to believe that heterogenous materials can, at least subject to certain constraints, be treated similarly. But why and to what extent is this concept useful? Furthermore, although the heterogenous nature of the material may not be obvious for this apparently continuous medium, it is on the other hand clear that its behavior depends on the characteristics of the heterogenities. How then do we proceed if we are to account for the properties of the constituents when defining the behavior of an equivalent continuous homogenous medium? These two points are of great practical importance. On one hand, understanding of the limits of a method is an important safety consideration, and on the other hand determination of the equivalent continuous medium allows us to better understand the

22


parameters that govern its behavior (for a natural material) or to adapt the constituent parts to achieve the desired performance (for an artificial material). The homogenization methods have been developed to answer these questions. They make it possible – under well-specified conditions – to obtain a description of the behavior of heterogenous materials starting from the behavior of the heterogenities. A condition essential for the existence of an equivalent continuum is that the physical mechanism under study should vary on a length scale which is very large compared to the scale of the heterogenities. This requirement for a difference in length scales gives rise to the expressions “upscaling method” and “method of multiple scales”. The term “homogenization” also arises from this, because considering the heterogenities to be of infinitessimal size compared to the effects under study naturally leads us to consider the medium as a homogenous or, more precisely, homogenized continuum. Linking the large-scale observable behavior to microscopic mechanisms is an ageold preoccupation of physicists. One famous example is that of elasticity, where Navier (1821) and Poisson (1829) obtained a single macroscopic isotropic elastic coefficient from a particular molecular model: the two Lamé coefficients are equal. Cauchy (1828) obtained a two-coefficient isotropic elastic model starting from a more sophisticated molecular model. From among these well-known names we also draw attention to the preliminary work of Rayleigh (1892) on the conductivity of media containing impurities present in a parallelopiped lattice, and that of Einstein (1906) on the viscosity of suspensions and sedimentation rates. These attempts remained intermittent until the 1950s when the needs of industrial development demanded a detailed understanding of the behavior of natural materials (the oil industry), manufactured materials (in particular steels and alloys), and the design of new materials (mainly for aeronautics). In order to understand the significance of small scale mechanisms on global behavior, scientific approaches at the time involved phenomenological micromechanical models built on thermodynamic principles. The pioneering works of Biot (1941) and of Hill (1965) took this approach. It was in the 1970s that a new school of thought was born, started by Keller who used a rather different angle to tackle the question of the change of scales. This involved the method of asymptotic expansions at multiple scales. Initially built on an approach which was more mathematical than mechanical, this method was a true conceptual leap forwards in terms of its rigor and formalism. The works of Bensoussan et al. (1978) and Sanchez-Palencia (1980) are still important references on this subject, with similar ideas developed in Russia by Bakhvalov and Panasenko (1989).

Introduction

23

Initially confined to very specialist circles, these methods blossomed considerably in the 1990s. During this period their fields of application were broadly diversified across all traditional engineering fields, and also in life sciences, particularly biomechanics. This undeniable success is thanks to the effectiveness of asymptotic methods for treating complex physics on a microscopic scale, and their ability to include coupling between different phenomena. However, while their use has become almost routine for some research groups, it remains poorly documented at present in a form suitable for engineers and researchers working in related fields. These considerations convinced us there was a demand for a book which would set out a coherent picture of these approaches, and render them accessible to a wider audience than just specialists in this research area. Rather than be exhaustive (which would not be an easy task), we have chosen to pick a few problems where the main points can be presented in a simple manner. The aim is also – using a unified treatment – to illustrate the common thought processes connecting the issues addressed in this book. In keeping with this approach, the bibliography does not attempt to be exhaustive, but shows the reader the seminal works in the field, and the references corresponding to the main steps of the problems we consider. This volume, which has grown out of the Mechanics of Heterogenous Media course taught at the University of Grenoble by J.L. Auriault, is intended to be a basic course in upscaling methods, aimed at advanced students, engineers and graduate students. With this pedagogical aim, we have used a progressive approach to each subject, starting out with traditional problems and then following them with recent developments. We also thought it useful to illustrate the potential applications of the results of homogenization. With this in mind, for each of the themes we treat, the theoretical results are followed by an example of the development through homogenization which provides a concrete example of the advances in a particular field of application. This book is divided into four parts. Part 1 is an introduction to the philosophy of homogenization methods. We discuss methods aimed at periodic and random materials while emphasizing their physical significance and their potential applications to real materials, which are often neither perfectly periodic or perfectly random. The basic examples given in Chapter 1 give an understanding of the fundamental tools underlying both methods. Chapter 2 goes into more detail on the techniques and discusses connections between them and the details which distinguish them. There is a detailed discussion of conditions for their application, delineating the range of validity of these approaches. This overview of methods underlines the power of the asymptotic

24


method at multiple scales for the treatment of complex physics with many coupled effects in materials with simple or heirarchical morphologies. Combining rigorous formalism and intuitive reasoning, Chapter 3 presents the methodology of the multiple scale approach which will be used throughout the rest of the book. The emphasis is on the systematic use of dimensional analysis combined with the separation of length scales. We also detail the means of expressing a practical problem involving real materials in the context of homogenization. This methodological basis is applied in the following sections where we specifically treat the physical mechanisms involved in coupled phenomena. Part 2 presents a first field of application of homogenization. We study the physics of transport by diffusion, convection and advection, phenomena which allow us to apply the basic tools of upscaling methods to engineering problems. Chapter 4 focuses on thermal transfer in heterogenous media. Going beyond the classical model of thermal transfer in a composite, we find a diverse range of macroscopic models depending on the level of contrast in the conductive properties of the constituents and their interfaces. In particular, memory effects arise from the presence of local non-equilibrium of a very weakly conducting phase, and twotemperature models can be developed for quasi-insulating interfaces. The transport of solutes in porous media is examined in Chapter 5. We highlight the different descriptions associated with the local physics of pure diffusion and then with diffusionadvection. This second situation, which is reached at high transport rates, results in a macroscopic dispersion. The range of validity of each of these models is explicitly specified. Chapter 6 makes use of, and extends, these results, focusing on specific materials. The numerical procedure of periodic homogenization is illustrated by determining the coefficients for fibrous and granular materials. By way of comparison, we recall the classical self-consistent analytical estimates. Finally, comparison with experimental results enables us to judge the appropriateness of these models for describing the properties of materials. Part 3 is dedicated to the modeling of Newtonian fluid flows in rigid porous media. Chapter 7 discusses incompressible fluids using multiple-scale asymptotic expansions. It starts with the canonical problem of Darcy’s law (in the regime of steady-state laminar flow). It continues taking into account inertial effects, both in the dynamic linear regime which leads to memory effects through visco-inertial coupling, and in the steady-state advective regime, where the correction due to weak nonlinearities is established. The flow in porous media of compressible fluids such as gases is the subject of Chapter 8. Using the asymptotic method, we treat in succession high pressure steady-state flows, wall slip effects in rarefied gases and, in the dynamic regime, the acoustic description under weak pressure perturbations with thermal coupling. The transfer of theoretical results for homogenization to their numerical formulation is illustrated in Chapter 9. The solution to local problems derived by

Introduction

25

periodic homogenization is given for calculation of the Darcy permeability of granular and fibrous materials. Finally, Chapter 10 returns to the same problems, which are discussed in the context of a self-consistent approach. We use this to establish analytical estimates and bounds for steady state and dynamic permeabilities, thermal effects, wall slip corrections and – by analogy – for the trapping constant. Part 4 focuses on the behavior of deformable saturated porous media. Chapter 11 considers the behavior in the quasi-static regime, first examining that of the empty porous medium (a specific case of an elastic composite) and then that of the saturated medium, introducing the fluid-solid coupling. Depending on the level of contrast between the shear properties of the fluid and the solid, the asymptotic method of multiple scales leads to three distinct behaviors whose properties are discussed. The study of poroelastic behavior is extended to the dynamic regime in Chapter 12. The characteristics of the three possible behaviors – including the Biot biphasic model – are analyzed in detail, particularly properties of the effective coefficients. The range of validity of each of the descriptions is specified. Chapter 13 puts the homogenization results to numerical use in order to carry out a parametric analysis of the elastic and coupling coefficients in the biphasic model. At the same time these results, obtained for cohesive granular media, are compared to traditional selfconsistent estimates and to bounds. In Chapter 14, the homogenized biphasic behavior is used with the aim of describing the propagation of waves in saturated porous media. After specifying the properties of the three propagation modes, the transmission of waves across a poroelastic interface is examined. We also establish the expression for Green’s functions in the context of poroelasticity, the integral formulation, and the fields radiated by abrupt dislocations. To complete our summary of this text, it is worth mentioning certain important subjects which are not treated here (or only discussed briefly). One of these subjects is complex microstructures. In fact, we will only consider media whose local geometry is sufficiently simple that it can be characterized by a typical length scale of the heterogenity, and whose local problems can be formulated in terms of continuous media. This choice means that we omit: – Media whose architecture involves very different characteristic sizes (such as double porosity media). These can give rise to various interacting physical effects on each length scale. These many possible couplings vastly increase the diversity of the possible macroscopic behaviors, with some behaviors only being possible in such media as; – Microstructure whose behavior can be reduced to that of various interacting points within the material (for example the nodes in trellis structures). For these it is preferable to use a locally discrete description, and to move to a continuum description through homogenization. This alternative approach will not be discussed here.

26


A second aspect only outlined is that of the corrections to macroscopic descriptions which have been established to first order. Indeed, for the most part, the results presented here are restricted to the first significant term, and lead to descriptions involving a continuous medium which is materially simple, and descriptions valid in the bulk of the heterogenous medium. There are two corrections which can usefully be applied to these descriptions: – those which appear on the boundary of the medium. They lead to a boundary layer with a thickness of the order of the size of the representative elementary volume. This makes it possible to reconcile local boundary conditions and boundary conditions used at the macroscopic scale; – those which make it possible to treat situations with weak separation of scales, obtained by including higher-order terms within the homogenized descriptions. These correctors to simple continuum models introduce non-local effects whose spatial extent is of the order of the size of the representative elementary volume. Finally, we will not discuss the taking into account of non-linearities. All the cases that we present involve linear effects, or sometimes weakly non-linear ones where the non-linearity can be treated as a perturbation of the linear solution. Whether they have material or geometric origins, non-linearities introduce considerable theoretical difficulties compared to linear situations. While the establishment of criteria through local limit analysis – the rheology of elastic composites with a non-linear power law, or the flow of non-Newtonian power-law fluids in porous media – has been successfully achieved, in general non-linearities present a real challenge to upscaling methods. These three omitted themes – complex microstructures, corrections and nonlinearities – are very rich and interesting, and they deserve further discussion on their own. We hope that this volume will offer a sufficiently clear and solid basis to guide the reader who may wish to explore these fields. This work is the fruit of a long collaboration between its authors. It has of course been supported by the work and suggestions of numerous friends, colleagues and research students, whom we are delighted to thank for the assistance that they have given us, and in particular: P. Adler, I. Andrianov, L. Arnaud, P. Y. Bard, J. F. Bloch, G. Bonnet, L. Borne, L. Dormieux, H. Ene, M. Lefik, T. Levy, J. Lewandowska, C.C. Mei, X. Olny, L. Orgéas, P. Royer, E. Sanchez-Palencia, T. Strzelecki. We extend particular gratitude to P. Adler, whose sound advice and criticism has added a great deal to this work.

PART ONE

Upscaling Methods


Chapter 1

An Introduction to Upscaling Methods

1.1. Introduction In engineering applications, media with small-scale heterogenities are often used in volumes which contain a very large number of these heterogenities. Despite progress in numerical modeling, the small-scale description of such volumes remains difficult (Figure 1.1). The aim of homogenization techniques – also know as upscaling methods – is to model the heterogenous medium as a simpler equivalent continuous medium, whose description is valid at a scale which is very large compared to that of the heterogenities. The aim of this chapter is to introduce the problem and the basics of three common upscaling methods, specifically the technique of homogenization of periodic structures, the determination of bounds to the effective coefficients, and self-consistent estimates. By way of example, we will consider steady-state heat transfer in a heterogenous medium and for this simple physical phenomenon we will discuss: – the explicit calculation of macroscopic conduction coefficients for a periodic medium with a known microstructure. Here we will consider a bilaminate composite, for which analytic solutions can be found (section 1.2); – the determination of limits which bound the macroscopic conduction coefficients for heterogenous media, regardless of the microstructure (section 1.3); – the self-consistent estimate of a macroscopic conduction coefficient, valid for media with random microstructures which obey certain morphological constraints (section 1.4).

30


Figure 1.1. Two examples of heterogenous materials: concrete and polyurethane foam. These photographs illustrate the large number of heterogenities contained within the volumes that are typically used

1.2. Heat transfer in a periodic bilaminate composite Consider the steady-state heat transfer in a periodic bilaminate composite (Figure 1.2). The composite is periodic, with period l in the direction e2 . The volume fraction of medium a is denoted ca , and that of medium b is cb = 1 − ca . The constituents are isotropic and homogenous, with thermal conductivities λa and λb . Γ designates the interfaces. We consider a sample consisting of a large number of periods. The equations describing the temperature field T (x) are: divQ = 0 Q = −λα gradT,

(1.1) λα > 0,

α = a, b

(1.2)


31

e2 Q·n=0

L2 T0

T1

l

e1

(a)

Q·n=0

L1

e2

1 (1 cb )l

a

(b)

cb l

b

e1

Figure 1.2. (a) Macroscopic sample undergoing a transfer parallel to the layers and (b) period of the bilaminate composite

[Q · n]Γ = 0

(1.3)

[T ]Γ = 0

(1.4)

where Q(x) is the heat flux. In the above equations, [φ]Γ represents the jump in φ across the interface Γ. Can we replace this description with a simpler equivalent continuum? 1.2.1. Transfer parallel to the layers Consider a sample (Figure 1.2) subjected at x1 = 0 and x1 = L1 to temperatures T0 and T1 . The boundaries x2 = 0 and x2 = L2 are perfectly insulating: Q · n = 0 Clearly T only depends on x1 , and the same is therefore true of Q. Since: d dT (λα )=0 dx1 dx1

32


the temperature gradient is constant in each phase. And with (1.4), it is constant everywhere. However Q1 = −λα ddxT depends on the material. If we take the average, 1 with in this case: Q1 =

1 L2

L2

Q1 dx2 0

we find that: Q1 = −λα

dT dx1

where . represents the average volume. We will make the following hypothesis (H) which we will use throughout the rest of this chapter: (H): the macroscopic behavior is described by the same formulation as the local behavior, with a Fourier law and a balance equation. In the particular case we are considering here, we have: d Q1 = 0 dx1 Q1 = −λeff 11

(1.5) dT dx1

(1.6)

Hence the macroscopic conductivity λeff 11 is such that: 1 L2

λeff 11 = λα =

L2

λα dx2 0

The length L2 is a number (not necessarily integer) of periods: L2 = N l + l0 ,

l0 < l,

N integer

Thus we have: λeff 11

1 = N l + l0

N l + l0 eff λ11 Nl

=

0

Nl

1 λα dx2 + N l + l0

1 Nl

Nl

λα dx2 + 0

l0

λα dx2 0

1 Nl

l0

λα dx2 0


= =

l0 1 λα dx2 + λα dx2 Nl 0 0 l0 1 int λ11 + λα dx2 Nl 0 1 l

33

l

where λint 11 , a constant independent of L2 , is intrinsic to the material. It is the arithmetic mean of the conduction coefficients of the constituents of the period. λint 11 = ca λa + (1 − ca )λb Defining: ˜ 11 = 1 λ l

l0

0

λα dx2 < λint 11

it follows that: λeff 11 =

Nl Nl ˜ 11 λ λint 11 + N l + l0 (N l + l0 )N

λeff 11 is not intrinsic, and depends on L2 . If N is large, we have: int λeff 11 ≈ λ11 −

1 int l0 ˜ (λ + λ11 ) N 11 l

When N tends to infinty, N ≈

L2 and: l

N →∞

int λeff 11 −→ λ11 int eff int is intrinsic. Since λeff 11 = λ11 [1+O(l/L2 )], the approximation of λ11 by λ11 becomes better when the separation of scales is good. In other words the ratio l/L2 tends to 0. λint 11 is also known as the effective coefficient.

1.2.2. Transfer perpendicular to the layers Consider a sample (Figure 1.3) subject at x2 = 0 and x2 = L2 to temperatures T0 and T1 . The boundaries at x1 = 0 and x1 = L1 are perfectly insulated: Q · n = 0. T and Q now only depend on x2 . We have: d dT (λα )=0 dx2 dx2

34


e2 T1

L2

l

Q·n=0

Q·n=0

e1 L1

T0

Figure 1.3. Macroscopic sample subject to transfer perpendicular to the layers

Q is constant in each material, and with (1.3) it is constant throughout. We therefore have: Q dT =− dx2 λα and by integration: T1 − T0 Q =− L2 L2

L2 0

dx2 λα

For the boundary-value problem we are considering here, the left hand side represents the macroscopic temperature gradient. With the hypothesis (H) made in the previous section, it follows that: −1 (λeff 22 )

1 = L2

L2

0

dx2 1 = λα N l + l0

N l+l0 0

dx2 λα

Thus λeff 22 is not intrinsic and, as in the preceding section, we can write: 1 −1 int −1 1 + O( (λeff ) , ) = (λ ) 22 22 N

−1 (λint = 22 )

1 l

0

l

dx2 λα

λint 22 is the geometric mean of the conduction coefficients within the period: ca 1 − ca 1 = + int λ λb λ22 a λint 22 is intrinsic.

or alternatively

λint 22 =

λa λb ca λb + (1 − ca )λa


35

1.2.3. Comments The above results lead to various observations: – The effective coefficients are only an approximation. The error is O(l/L2 ). When l/L2 tends to zero, λeff tends to the intrinsic value λint . – When the stratified material has a random structure, we get the same kind of result. The period is then replaced by the representative elementary volume. This is a volume containing a sufficiently large number of layers that its properties can be considered as equivalent to that of the macroscopic sample. l represents the thickness of the representative elementary layer. – It would be useful if we could abandon the prerequisite (H) on the macroscopic scale. – Here the geometry of the heterogenities is very simple. Can we extend certain results to more complex geometries? – Our demonstration here makes use of very simple conditions in the macroscopic limit (homogenous). What happens in more general cases? – We have studied the transfer in the steady state regime. What are the effective coefficients in the transient regime? Do they still exist?

1.2.4. Characteristic macroscopic length The two examples above demonstrate the role played by the sample length L1 or L2 . This macroscopic length characterizes the size of the sample. We note that L = Lgeom = L1 , L2 , for geometric L. We can also define a phenomenological macroscopic length Lphen which characterizes the heat flux [BOU 90]: Lphen =

T |gradT |

(1.7)

In the two cases considered here: Lphen = Lgeom

(1.8)

The geometric and phenomenological macroscopic lengths are the same in our case. This is a very special case, though. In many problems: Lphen = Lgeom

(1.9)

and the macroscopic length scale that must be used will be: L = min(Lphen , Lgeom )

(1.10)

36


In particular, this will be the case for transient phenomena where Lphen can be compared to a “wavelength”. We note also that Lphen may vary from one point to another in the sample, thus introducing a separation of scales which is not constant across the sample. We will return in detail to this issue in Chapter 3. 1.3. Bounds on the effective coefficients We now consider arbitrary local geometries, either periodic or random. In contrast with the layered structure where the microstructure was explicitly defined, we assume here that the only information available on the material is the concentration of the constituents and their properties. Naturally, depending on the morphology of the medium, the effective coefficients will take different values: without additional assumptions we cannot therefore offer estimates of the effective coefficients. We can however show with the help of (i) the principle of virtual powers, (ii) minimization of potential and complementary energies and (iii) the Hill principle, that the range of possible values of the effective conditions is bounded. These energetic methods using variational approaches make up a powerful theoretical tool with which to frame the macroscopic properties of the material [WIL 81]. Only requiring the bare minimum of information on the medium (concentrations and coefficients of the constituents), these bounds create a very widely-applicable result. In order to present this approach, we will again study steady-state heat transfer, and we will assume the constituents to be isotropic. 1.3.1. Theorem of virtual powers In the steady-state regime, heat transfer in a heterogenous medium, with a symmetric, positive definite thermal conductivity tensor λ(x) at all points, which is described by: divQ = 0

(1.11)

Q = −λ gradT

(1.12)

with continuity conditions on the temperature and flux on the surface Γ of any discontinuities which may be present: [Q · n]Γ = 0

(1.13)

[T ]Γ = 0

(1.14)

We will consider the boundary-value problem shown in Figure 1.4. A volume V of material is bounded by the surface S = ΓQ ∪ ΓT . On ΓQ we impose a normal flux Qn (x) and on ΓT we impose a temperature T0 (x).


37

Q · n = Qn

n

V

Q T (x)

T0 T Figure 1.4. Macroscopic boundary conditions

We define two virtual fields: – a field of admissible flux Q∗ , satisfying: divQ∗ = 0

within V

Q∗ · n = Qn

over ΓQ

– a field of admissible temperature T , satisfying: T = T0

over ΓT

In general Q∗ = −λ gradT , except if we choose the actual fields Q and T , which are the correct solutions to the boundary-value problem for Q∗ and T . We will now calculate the virtual power developed by the fields T and Q∗ . We have: T divQ∗ dV = 0 V

so that: ∗ div(T Q )dV − Q∗ · gradT dV = 0 V

V

And using the divergence theorem: T Q∗ · ndS − Q∗ · gradT dV = 0 S

V

Finally, decomposing S to reflect the type of boundary condition imposed: Q∗ · gradT dV = T Qn dS + T0 Q∗ · n dS (1.15) V

ΓQ

ΓT

38


The virtual power dissipated equals the exterior virtual inputs. This equality, which forms the theory of virtual power, is satisfied in particular for the solution Q∗ = Q and T = T . 1.3.2. Minima in the complementary power and potential power Consider the following quadratic form: 1 ∗ (Q + λgradT ) · λ−1 (Q∗ + λgradT ) dV Φ(Q∗ , T ) = 2 V Clearly: Φ(Q∗ , T ) 0 and Φ = 0 for Q∗ = Q, and T = T , since Q = −λ gradT . Making use of the properties of λ, expansion of Φ gives: 1 ∗ −1 ∗ 1 Q ·λ Q dV + λ gradT · gradT dV + Q∗ · gradT dV Φ(Q∗ , T ) = V 2 V 2 V The last integral is given by the theory of virtual powers. Φ can then be written in the form: Φ(Q∗ , T ) = Φ1 (Q∗ ) + Φ2 (T ) ∗

Φ1 (Q ) = V

Φ2 (T ) =

V

1 ∗ −1 ∗ Q · λ Q dV + 2

T0 Q∗ · n dS ΓT

1 λ gradT · gradT dV + 2

T Qn dS

ΓQ

The complementary dissipation Φ1 (Q∗ ) and the potential dissipation Φ2 (T ) vary independently with Q∗ and T , and their sum is minimized for Q∗ = Q and T = T . As a result, Φ1 is minimized for Q∗ = Q and Φ2 is minimized T = T . From this we deduce two inequalities: Φ1 (Q) Φ1 (Q∗ )

(1.16)

which states that the minimum in the complementary dissipation is attained for the flux field Q which is the solution to the boundary-value problem, and: Φ2 (T ) Φ2 (T )

(1.17)


39

which states that the minimum in the potential dissipation is attained for the temperature field solution T . In conclusion, since Φ(Q, T ) = Φ1 (Q) + Φ2 (T ) = 0, for any admissible temperature field T and admissible flux field Q∗ , we arrive at the following framework: −Φ2 (T ) −Φ2 (T ) = Φ1 (Q) Φ1 (Q∗ ) 1.3.3. Hill principle The Hill principle makes it possible to establish a link between the microscopic and macroscopic descriptions. It is therefore fundamental to the upscaling methods for which the micro-macro connection does not fall out of the homogenization procedure. This principle is based on two hypotheses: – hypothesis (H) (see section 1.2) which, we recall, stipulates (i) that the global variables are the volume means of the local flux and gradients, and (ii) that the conservation and constitutive equations have the same structure at microscopic and macroscopic scales; – the assumption of energetic consistency – known as the Hill principle – which imposes equality on the energy contained within the medium, whether it is expressed in local variables or using macroscopic variables defined according to (H). In the case of heat transfer, the Hill principle applies to dissipation expressed in terms of the microscopic variables – in other words the thermal conductivity tensor λ, gradT and Q. It is also expressed in terms of the dissipation relating to the macroscopic variables which are, according to (H), the effective thermal conductivity tensor λeff , gradT and Q: gradT · Q dV = gradT · λgradT dV = V gradT · λeff gradT (1.18) V

V

Q · λ−1 Q dV = V Q · λeff−1 Q

= V

where . represents the volume mean: 1 . dV . = V V In the following section, we will show that when the conditions at the boundary of an arbitrary volume V are homogenous, either in terms of temperature gradient or flux, then: V gradT · Q dV = gradT dV · Q dV V

V

V

40


Under these conditions (and these conditions alone), and in the context of (H), the Hill principle becomes the Hill theorem. In addition, we will see many times in subsequent chapters that even if (H) is not always respected – the Darcy law is a blatant example – energetic consistency can be demonstrated through the method of asymptotic expansions, without the requirement of homogenous conditions at the boundary. More precisely, we show that in its present form the Hill principle is just an approximation, which becomes better when the separation of scales is large. 1.3.4. Voigt and Reuss bounds 1.3.4.1. Upper bound: Voigt Consider the case where the medium is subject over its entire boundary S to an imposed temperature T0 , corresponding to a uniform gradient G (Figure 1.5) so that: ΓQ = ∅,

and over

ΓT = S,

T0 = G · x

This load induces a temperature field T in the medium. We can show that this satisfies the Hill hypothesis. This indicates that in any arbitrary heterogenous material, under this type of load, the mean gradient in the medium is identical to that followed by the temperature imposed at the boundary: gradT = G

V

T0 = G · x

T (x)

T Figure 1.5. Temperature imposed over the entire boundary S, corresponding to a uniform gradient G

This results from: V gradT = T ndS = T0 ndS = (G · x)ndS S

S

S

xi ndS = Gi

= Gi S

ei dV = V G V


41

where we have adopted, as we will throughout the rest of this work, the Einstein convention: summation over all repeated indices in a term. Also, applying the theorem of virtual powers, the dissipated power takes the form: gradT · Q dV = T0 Q · n dS = (G · x)Q · n dS = Gi xi Qj nj dS V

S

S

S

and, using the divergence theorem as well as the property of zero divergence of flux: (∂xi Qj ) Gi xi Qj nj dS = Gi dV = Gi δij Qj dV = Gj Qj dV ∂xj S V V V It follows from this that: gradT · Q dV = V gradT · Q V

Assuming (H), and further supposing that λeff is homogenous (necessarily) and eff isotropic (out of convenience), λeff ij = λ δij , the macroscopic description is given by: Q = −λeff gradT = −λeff gradT This finally leads us to the quoted result for this field: λ gradT · gradT dV = V λeff gradT · gradT

(1.19)

V

Consider now the temperature field T with homogenous gradient gradT = gradT . This field is admissible in temperature since over V : T = gradT · x = G · x

and hence

T = T0

over S

Furthermore, the field T satisfies the minimum of the potential dissipation Φ2 (T ) Φ2 (T ), in other words, given that ΓQ = ∅: λ gradT · gradT dV λ gradT · gradT dV V

V

λ gradT · gradT dV = gradT · gradT

= V

λ dV V

Using the energy property (1.19), it follows that: V λeff gradT · gradT gradT · gradT

λ dV V

42


which gives: λeff λ

(1.20)

The arithmetic mean λ is the upper bound of Voigt [VOI 87]. In the case of a composite with two homogenous constituents with volume fractions ca and cb : λeff ca λa + cb λb We recover the value established for transfer parallel to the layers in a stratified sample. In fact, the bound of Voigt [VOI 87] is reached when gradT = gradT = gradT , in other words if the solution field presents a homogenous gradient. 1.3.4.2. Lower bound: Reuss Now consider the case where the medium is subject across its entire boundary S to a uniform imposed flux Q (Figure 1.6), so that: ΓT = ∅,

and over ΓQ = S

Qn = Q · n

Qn = Q · n

n

V T (x)

Q Q

Figure 1.6. Uniform flux imposed over the entire boundary S

This load induces a flux field Q in the medium. We will show that it also satisfies the Hill hypothesis of energetic consistency. First of all we will prove the following remarkable property, which states that in any arbitrary heterogenous medium, under this type of load, the mean flux within the medium is identical to that imposed at the boundary: Q = Q For each flux component: V Qi =

δij Qj dV =

V

Qj V

∂xi dV = ∂xj

V

∂(xi Qj ) − xi divQ dV ∂xj


43

The last term of the integral is zero because Q has zero divergence. The remaining term can be transformed successively using the divergence theorem and the boundary conditions: xi Qj nj dS = xi Qj nj dS = Qj xi nj dS V Qi = S

S

S

div(xi ej )dV = Qj

= Qj V

δij dV = V Qi V

Having achieved this result, we now apply the theorem of virtual work:

gradT · QdV = V

T Q · n dS = S

T Q · n dS = Q · S

T n dS S

gradT dV = V Q · gradT

=Q· V

Invoking (H), and assuming once again that λeff is homogenous (out of necessity) eff and isotropic (for convenience) then λeff ij = λ δij , and the macroscopic description is given by: gradT = −λeff−1 Q from which we deduce the following energetic property: V

Q·Q Q · Q dV = V λ λeff

(1.21)

Now consider the field Q∗ defined by Q∗ = Q. This field is flux admissible since on the one hand Q∗ is uniform, and so has zero divergence, and on the other hand, since Q = Q, we have over S: Q∗ · n = Q · n = Q · n = Qn

(1.22)

The field Q∗ therefore satisfies the minimum of the complementary dissipation: Φ1 (Q) Φ1 (Q∗ ) which, since ΓT = ∅, gives: V

Q·Q dV λ

V

Q∗ · Q∗ dV = Q · Q λ

V

1 dV λ

44


and, using the energetic property (1.21) of Q: Q · Q V Q · Q λeff

V

1 dV λ

Finally: 1 1 λeff V

V

1 dV, λ

or alternatively

λeff λ−1 −1

(1.23)

The geometric mean λ−1 −1 is the lower Reuss bound [REU 29]. In the case of two homogenous constituents, this lower bound can be written as: λeff

λa λb cb λa + ca λb

This expression is identical to that obtained for transport in a stratified medium perpendicular to the layers. Indeed the Reuss bound [REU 29] is reached when Q = Q∗ = Q, in other words if the solution field presents a homogenous flux.

1.3.5. Comments – The same comments still apply as for the bilaminate composite. In particular, the above results only have meaning in the presence of a good separation of length scales. In this sense λeff is only an approximation, and is not therefore intrinsic to the material. – In the case of homogenous constituents with known properties, the Voigt and Reuss bounds only require a knowledge of their volume fractions. – The preceding results were, for convenience, established for the isotropic case. They can be extended without difficulty to the anisotropic case, where they become: ∂T ∂T ∂T (λeff 0, ∀ ij − λij ) ∂xi ∂xj ∂xi which corresponds to the upper bound of Voigt, and: − (λ−1 )ij )Qj 0, Qi (λeff−1 ij

∀Qi

which corresponds to the lower bound of Reuss. – Since the Voigt and Reuss bounds are in fact attained by certain stratified materials, they cannot be improved upon without the addition of further information on the medium. – By way of example, consider a composite with two isotropic constituents with: λa = 10 λb , ca = cb = 0.5. We obtain: 0.182 λa λeff 0.55 λa .


45

– The bounds are clearly looser when the contrast in the properties of the constituents – which can be quantified by the product λλ−1 – is larger. When ca is varied from 0 to 1, we obtain the graph in Figure 1.7, known as Hill’s diagram. 1

e& / a

0.8

0.6

Reuss

HS+ HS-

0.4

Voigt 0.2

0

0

0.2

0.4

0.6

0.8

1

ca Figure 1.7. Hill’s diagram, and bounds of [HAS 63] for a composite with λa /λb = 10

1.3.6. Hashin and Shtrikman’s bounds By advancing the additional hypothesis that the material is isotropic, and using other variational principles, we can obtain improved bounds. In the case of two constituents, Hashin and Shtrikman [HAS 63] derive the following bounds when λa > λb : λb +

ca +

1 λa −λb

cb 3λb

λeff λa +

cb +

1 λb −λa

ca 3λa

(1.24)

In the above example, λa = 10λb , ca = cb = 0.5, and we obtain: 0.28λa λeff 0.47λa Of course, the calculation made by Hashin and Shtrikman leads implicitly to restrictive assumptions on the spatial distribution of the two phases. However it is not possible to explicitly formulate these morphological constraints using this method. Thus, even when the morphology of the microstructure is known, it is not generally possible to make a judgement on the macroscopic isotropy and hence on the validity of the bounds. However, for periodic media, the isotropy of the conduction tensor is assured if the period presents three orthogonal planes of symmetry: in this case Hashin and Shtrikman’s bounds are known to be valid.

46


In addition, Hashin showed that these bounds give effective conduction for materials formed of aggregates of bi-composite homothetic spheres filling space (see Chapter 6). The upper bound is reached when each sphere consists of a core of the less conductive constituent surrounded by a shell of the more conductive material, and the lower bound is reached in the reverse situation. Since the bounding values can be reached, these bounds are optimal for the level of information available (the properties of the constituents, their proportions, macroscopic isotropy). 1.3.7. Higher-order bounds It is possible to define higher-order bounds. Their definition requires other morphological parameters in addition to the volume fraction of each phase. On this subject, the reader is referred for example to [BOR 01; TOR 02; and MIL 02]. 1.4. Self-consistent method The aim of the self-consistent method [BRU 35; BRI 49; HIL 65; BUD 65; HAS 68] is to estimate the effective coefficients using minimal information, in other words the properties and concentrations of the constituents. Naturally, this estimate requires us to introduce an additional hypothesis known as self-consistency. This assumes that each inclusion, or more generally each generic substructure [ZAO 87] “sees” the rest of the medium not in terms of the actual heterogenous material, but as the equivalent homogenous material whose effective properties we are trying to determine. The idea of generic inclusion is to approach the mechanisms at a local scale. The inclusion leads to assumptions on the morphology, but these do not appear in an explicit manner. The simplicity of this approach, and the flexibility in the choice of generic substructures – on which the macroscopic model will depend – make this method widely used in practical terms. We will present the self-consistent approach by drawing on the same physics as in the earlier sections, that is to say steady-state thermal transfer in a composite: divQ = 0 Q = −λ gradT,

(1.25) λ>0

(1.26)

with: [Q · n]Γ = 0

(1.27)

[T ]Γ = 0

(1.28)

over any surfaces Γ of discontinuity.


47

1.4.1. Boundary-value problem We will consider a composite with N homogenous and isotropic constituents α = 1...N (Figure 1.8) occupying a volume V . We will denote as Vα the volume occupied by phase α and its volume fraction as cα = Vα /V . In order to apply the

kn k2 k1

kn k1

k1

k2

k1

kn k1

k2

kn

k1

kn kn

V

k2

T0 = G · x

k2 k1 T

Figure 1.8. Boundary-value problem

self-consistent method we must assume that the medium is subject at its boundary S to homogenous conditions either in terms of temperature gradient or flux (identical to those introduced for the Voigt and Reuss bounds). While the type of boundary condition does not matter, homogenity is essential because then – as previously seen – the solution must respect Hill’s condition of energetic consistency. To give a concrete example, we will assume that the medium is exposed at its boundary S to an imposed temperature T0 corresponding to a uniform gradient G of magnitude G in the direction ej , so that: T0 = G · x. We have shown that the solution T also satisfies: gradT = G (see the Voigt bound). The macroscopic flux Q is given by: 1 Q = V

V

with: QVα =

1 Vα

N N 1 QdV = QdV = ci QVα V Vα α=1 α=1 QdV Vα

Since our boundary-value problem is linear, the solution T is a linear function of the amplitude G = |gradT | of the temperature gradient imposed at the boundary. The same applies to the flux. As a result, the means over each constituent can be expressed in the form: QVα = −aα |gradT |

48


where the vector aα only depends on the microstructure and physical properties. We will adopt the aforementioned hypothesis (H) and will assume the material to be macroscopically homogenous (out of necessity) and isotropic (out of convenience). Thus Q, gradT and ej are collinear and: Q = −λeff gradT so that: −

N

cα aα |gradT | = −λeff |gradT | ej

α=1

Setting: aα · ej = βα and projecting the preceding equality onto ej , we find: λeff =

N

cα βα

α=1

λeff is thus determined once the βα are known. However, a rigorous calculation of these coefficients would need to take into account all the complexities of the interactions between the constituents, and would therefore be very involved. The selfconsistent method relies on a physically acceptable hypothesis which allows us to carry out a simple estimate of its value. 1.4.2. Self-consistent hypothesis As already mentioned, the very shrewd point of the self-consistent method [BRU 35; BRI 49; HIL 65; BUD 65; HAS 68], is to assume that each inclusion (or more generally each generic substructure) “sees” the rest of the medium not as the actual heterogenous material, but as its equivalent homogenous material whose conductivity λeff we are seeking to determine. To put it another way, βα is determined by considering the inclusion α to be embedded in a material of conductivity λeff subject to the same homogenous boundary conditions (Figure 1.9). Thus the βα appear as functions of λeff and λα : βα = βα (λeff , λα ) and λeff is then determined by solving the following equation: λeff =

N α=1

cα βα (λeff , λα )

(1.29)


T0 = G · x

T0 = G · x

V K

49

V

1

…

K

T

n

T

Figure 1.9. Self-consistent scheme with simple inclusions: boundary-value problems for inclusions 1 and n, treated as spheres

For simple geometries of inclusions, and considering the volume V to be infinite, the calculation can then be performed analytically.

1.4.3. Self-consistent method with simple inclusions The original self-consistent method consists of considering homogenous spherical inclusions. Other structures will be considered in Chapter 6. 1.4.3.1. Determination of βα for a homogenous spherical inclusion In order to determine βα , the spherical inclusion represented by constituent α is placed in a volume V of the equivalent medium, which is assumed infinite. The ensemble is subject to a homogenous gradient G at infinity (Figure 1.10). We will work in spherical coordinates (r, θ, ϕ), with the origin at the center of the inclusion defined by r < R, and the θ = 0 axis defined by G. Bearing in mind the symmetries of the problem, the temperature fields in the inclusion α (Tα for r < R) and in the equivalent homogenous medium (T for r > R) take the form: Tα = fα (r)cos(θ),

T = f (r)cos(θ)

and their gradients are given by: gradTα = fα cos(θ)er − gradT = f cos(θ)er −

fα sin(θ)eθ r

f sin(θ)eθ r

Consider the external medium (r > R). The Fourier equation can be written: divQ = div(λgradT ) = λeff ΔT = 0

50


G ez

ez

er

M

r R

Equivalent medium V

ey

O

Constituent V

ex

Figure 1.10. Spherical inclusion representing the constituent α placed in a volume V (assumed infinite) of the equivalent medium

where: λeff 2 f ) − 2f =0 (r r2 which has solutions of the form: f (r) = A

R2 r +B 2 r R

for r > R

Following the same line of reasoning in the inclusion, we have:

fα (r) = a

R2 r +b r2 R

for r < R

Since the temperature must take finite values throughout, a = 0 and as a consequence the flux is uniform within the inclusion. In order to determine this flux, we write the continuity condition on the temperature and the flux normal to the interface (r = R), as well as the condition of homogenous gradient at infinity: T (R) = Tα (R) =⇒ λeff gradT (R) · er = λα gradTα (R) · er

=⇒

gradT (∞) = G

=⇒

A+B =b λα λeff (−2A + B) = b R R B=G


51

Solving this system of equations leads us to: A=G

λeff − λα , λα + 2λeff

b=G

3λeff kα + 2λeff

This means that the flux within the inclusion is: Q = λα

3λeff G λα + 2λeff

And as a result: βα (λeff , λα ) = 3

λeff λα λα + 2λeff

1.4.3.2. Self-consistent estimate Using these values of βα (λeff , λα ), equation (1.29) which defines λeff becomes: N 1 λα cα = 3 α=1 λα + 2λeff

(1.30)

This equation (which can be written in the form of a polynomial of degree N ) only had a positive root of λeff > 0, the second term decreases Nsince, in the domain eff monotonically, from α=1 cα = 1 for λ = 0, to 0 for λeff = ∞. Returning to the example of the bi-composite already considered in the preceding sections, λa = 10 λb , ca = cb = 0.5. λeff is given by: 1 1 = 3 2

10λb λb + 10λb + 2λeff λb + 2λeff

and so: (λeff )2 −

11 λb λeff − 5 λb 2 = 0 4

Rejecting the negative solution to λeff we find: λeff = 4 λb = 0, 4 λa The different results, the bounds of Voigt (V), of Reuss (R), of Hashin and Shtrikman (HS) and the self-consistent estimate (SCE) (which of course falls within the bounds) are summarized in Figure 1.11.

52

Homogenization of Coupled Phenomena 1

a / b = 10

e& / a

0.8

0.6

R

0.4

SCE HS-

HS+

V

0.2

0

0

0.2

0.4

0.6

0.8

1

ca

1

a / b = 100

e& / a

0.8

0.6

0.4

R HS+

0.2

SCE HSV

0

0

0.2

0.4

0.6

0.8

1

ca Figure 1.11. Comparison between self-consistent estimates and the bounds for a bi-composite for two conductivity contrasts

1.4.3.3. Implicit morphological constraints Although the self-consistent method can be applied to general microstructures (here simple inclusions), the morphological constraints which these impose are not explicitly formulated. Because of this, it is not possible to define the class of microstructures to which the self-consistent result can be applied. This is an inherent limitation of the method: a result is obtained, but its conditions of applicability are not specified. To illustrate these implicit constraints, consider the case of media A and B with two constituents. One has finite conductivity λ, the other of concentration c has


53

zero conductivity (medium A) or infinite conductivity (medium B). The self-consistent model gives: λeff A =λ

2 − 3c 2

for c < 2/3,

λeff A = 0 for

λeff B =λ

1 1 − 3c

for c < 1/3,

λeff B =∞

c > 2/3

for c > 1/3

Thus, the self-consistent method with a simple inclusion imposes – without this being specified a priori – a connectivity threshold at a concentration of 1/3: – any constituent with a concentration greater than 1/3 is connected (medium B is infinitely conductive when c > 1/3); – any constituent with a concentration of less than 1/3 is non-connected, but is dispersed within the other phase (medium A is perfectly insulating when the concentration of the conductive phase 1 − c falls below 1/3). Many materials do not respect this constraint. In order to avoid this difficulty, other generic layouts tailored to specific microstructures can be imagined. We mention in particular the bi-composite spheres method of Kerner [KER56b, a] or Hashin [HAS 68] (see Chapter 6) which allows us to ensure the connectivity of one of the phases, whatever the concentration, with the other necessarily being disperse. Particularly well suited to media formed of inclusions within a matrix, this is also one of the few models with which we can associate an exact or approximate explicit morphology.

1.4.4. Comments – The result obtained with the self-consistent model is only meaningful if the material and the temperature field exhibit a good separation of scales. If we calculate the βα over finite volumes V , they depend on the position of the inclusion in the volume. They are not therefore intrinsic to the material, and neither is λeff . The infinite medium and the macroscopic homogenous thermal flux field correspond to an ideal scale separation l/L = 0. In this case the effective conductivity is intrisic. The question is: under what conditions does the result obtained remain valid for inhomogenous macroscopic flux fields or non-steady-state flows? – The bounds presented above, those of Voigt, Reuss and Hashin and Shtrikman, as well as the self-consistent effective conductivity, can be superseded in the general context of a systematic theory of random heterogenous materials [KRÖ 72]. A more and more precise understanding of the material, given by correlation functions of increasing order n, offer tighter and tighter bounds on the effective conductivity. For n = 1, in other words when we only know the volume fractions of the constituents,

54


we obtain the bounds of Voigt and Reuss. For n = 2, for uncorrelated conductivities of neighboring inclusions and an isotropic material, we obtain the bounds of Hashin and Shtrikman. Finally, in the case of perfect disorder (zero correlations at all orders, up to infinite n) we obtain the self-consistent result. – All the models discussed up to now rely on conservation of potential energy, and the absence of any other type of energy. They therefore apply to elliptical problems. As already mentioned, it remains to be seen if the results are applicable to hyperbolic or parabolic problems.

Chapter 2

Heterogenous Medium: Is an Equivalent Macroscopic Description Possible?

2.1. Introduction Heterogenous materials represent an important and expanding field. In the field of mechanics, we can think of composite materials consisting of several solids, of mixtures of fluids, of suspensions of solids in a fluid, or of porous media. In fact all materials are heterogenous on one scale or another, at least if this only occurs when they are considered on an atomic scale. Such materials are often used in volumes consisting of a very large number of heterogenities (Figure 2.1) so that the description of the physical processes involved becomes difficult, or indeed impossible, if we take into account every heterogenity. The idea – a very old and traditional one – is then to determine, where possible, a macroscopically equivalent medium, known also as the homogenized medium. This is a continuous medium which behaves “on average” like the heterogenous material. The description thus obtained must be intrinsic to the material and the perturbation under consideration, and independent of the macroscopic boundary conditions. In what follows, the scale of the heterogenities will be referred to as the microscopic, or local, scale – as opposed to the macroscopic scale, the scale on which the equivalent continuum is defined. The continuum macroscopic description of the heterogenous materials introduces behavior, conservation equations, physical quantities and effective coefficients, in place of the behavior, the conservation equations, physical quantities and coefficients valid on the scale of the heterogenities. The study of the relationship between the local and macroscopic descriptions is of very great interest.

56


1200 μm

Figure 2.1. Microstructures of heterogenous materials: aluminum with aluminum oxide reinforcement (left) and SiC-Titanium composite (right)

The macroscopic description can be obtained either by phenomenological or experimental investigation directly at the macroscopic scale, or by a homogenization technique, in other words by working from a microscopic description to a macroscopic description. It is this second line of attack which we will describe here. It is not possible to give a complete account of all the techniques of homogenization, of which there are many. We will therefore mainly restrict ourselves to presenting two of them: homogenization using multiple scale expansion for small-scale periodic structures (HPS) technique, with the foundation laid by Sanchez-Palencia [SAN 74] and introduced by others [KEL 77; BEN 78; SAN 80], and statistical modeling (SM) developed by Kröner [KRÖ 86], for materials with random structures. These are without doubt the most effective techniques and we will by analogy be able to draw a number of general concepts from them. Section 2.2 makes some general comments on homogenization techniques, as well as the relationships between some of them. SM is then briefly discussed in section 2.3. Similarly, the technique of multiple scale expansions is the subject of section 2.4 followed in section 2.5 by a comparison of the usefulness of these two techniques. 2.2. Comments on techniques for micro-macro upscaling To start with, we observe (and this will be understood throughout the introduction below), that the homogenization of a medium with a high density of heterogenities is only possible if we consider regions containing a large number of these heterogenities. If l is a characteristic dimension of the heterogenities and L is a characteristic dimension of the volume of material or of the phenomenon of interest, the condition of separation of scales can be expressed as: ε = l/L 1

Is an Equivalent Macroscopic Description Possible?

57

Two main classes of material can then be defined, depending on whether these two scales are effectively separated or not. For the first class there is no other intermediate scale. Homogenization is then possible. The materials are then characterized by a translational invariance (or quasiinvariance) at the microscopic scale, in a sense that we will explain later on. There may alternatively be more than two coexisting scales, but only if they are well separated from each other. We then move from the smallest scale to the larger scale by homogenization, but with the possibility of strong interaction between the different length scales depending on the different degrees of separation [AUR 92; AUR 93a]. When there is a continuum of non-separated length scales (the second class), results can be obtained when the structure is length-scale invariant. For example, in the case of a regular lattice of conducting or non-conducting rods organized in a random fashion, the structure of the conducting part of the lattice is length-scale invariant when the probability of a rod being conductive is close to the critical probability which forms the conductive–non-conductive threshold of the lattice. Percolation theory [BRO 57; CLE 83] then makes it possible for such structures to obtain the change in conductivity of the lattice close to this threshold. As we will see, this is also a type of homogenization since an “average” property, here the effective conductivity of the lattice is obtained. Nevertheless, we will in general reserve the term homogenization to micro-macro transition techniques involving structures with separated length scales. In what follows we will only consider this last class of materials.

2.2.1. Homogenization techniques for separated length scales We will start by giving a few examples of heterogenous materials in the context of civil engineering. The characteristic microscopic length scale l is only “microscopic” in comparison to the characteristic macroscopic length scale L. Thus l can take values which are not remotely microscopic compared to the human scale. If l is the size of the pores in a clay, maybe 50 angstroms, a glass fiber-epoxy resin composite would have l ≈ 1 mm, concrete l ≈ 1 cm, reinforced earth or a bank drained with geotextiles l ≈ 50 cm, and finally a pile foundation l ≈ 5 m. While it is not possible to give an exact cutoff, it is generally considered that homogenization becomes effective for L > 10 l or ε < 0.1. As we will see in the examples below, two more classes of material will become evident: materials with periodic structures, and such random structures as possess a representative elementary volume. The former can be studied using the technique of homogenization for periodic structures (HPS), which uses multiple scale expansions and which was developed by Bensoussan et al. [BEN 78] and Sanchez-Palencia [SAN 80]. Nowadays it is applied to many fields of physics. We will revisit the method of multiple scale expansions in section 2.4 where we will present a

58


methodology [AUR 91] which in particular allows us to investigate the conditions of homogenizability of a material subject to a given excitation. It will be used in specific examples in the following chapters. The second class of materials, those with random structures, are the subject of many techniques, and we cannot discuss all of them here. We will refer to Kröner’s method of statistical modeling (SM) [KRÖ 86], the methods developed by Gelhar [GEL 87] or Matheron [MAT 67], selfconsistent methods (see for example [HAS 68] and [ZAO 87]) and, in general terms, the averaging techniques [NIG 81; BED 83; GIL 87; QUI 93; WHI 99; ELH 02]. In section 2.3 we will briefly present the technique introduced by Kröner [KRÖ 86]. The study of materials with random structures presupposes the assumption of stationarity or quasi-stationarity. Even if it is not always stated, this assumption underlies all the techniques and is required for homogenization to be possible. It corresponds to the property of periodicity in periodic structures, and captures the property of translational invariance which is required for separation of length scales, as opposed to the invariance by self-similarity which is used for certain materials with non-separable length scales. Another general characteristic of all the methods is that they use mean values to define macroscopic quantities. These may be introduced in a phenomenological manner however – and this constitutes an assumption on the macroscopic scale which leads to the definition of the macroscopic physical quantities. Alternatively they may be introduced in the micro-macro transition, which guarantees that we will obtain the “correct” macroscopic physical quantity. We note on this subject that the volume averaging theorem, which allows us to take the mean of a system of partial differential equations, and which is often used to determine the macroscopic description, is valid whether or not the problem is homogenizable. It is interesting to note the points in common between the most effective homogenization techniques. On one hand the self-consistent techniques appear to be equivalent to the SM method in the case of perfect disorder [KRÖ 72; KRÖ 86]. For an elastic composite, the (macroscopic) effective elasticity coefficients have an identical formal structure, whether they are obtained by the SM technique for a random composite, or using the HPS method for a periodic structure [KRÖ 80]. For filtration problems, it can easily be seen that a similar result is valid for a porous medium, by comparing the formal structure of the effective permeability coefficient obtained by Matheron [MAT 67], and that obtained by the HPS method. More recently, Bourgeat and Piatnitski [BOU 04] established a connection between homogenization of periodic and random materials in the context of heat transfer. In general terms, though, the larger the separation of scales, the better the results. That gives a clear advantage to the HPS method. Indeed the periodicity introduces a perfect separation of scales, with l defined as the characteristic dimension of the period. Such a precise definition is not possible for materials with a random structure. The results will therefore be more accurate for periodic materials.


59

2.2.2. The ideal homogenization method Starting from a complete description at the microscopic scale, the ideal homogenization method is intended to determine the complete description at the macroscopic scale without additional assumptions. Thus at the microscopic scale we assume that we know: – the conservation equations, – the rheologies and the values of the parameters, – the physical quantities which describe the phenomena of interest, – the geometry. In addition, the process of the micro-macro upscaling must give an equivalent macroscopic description, intrinsic to the material and to the phenomenon of interest, specifically: – the conservation equations, – the rheologies and the effective parameters, – the correct physical quantities which describe the phenomena on the macroscopic scale. In short, the ideal process should be independent of any assumption on the physics of the model on the macroscopic scale in order to ensure the quality of the result. When we consider that the structures of the macroscopic model can be very different to the structures of the corresponding microscopic model, we can understand the importance of this point. One example is filtration in a rigid porous medium: on the scale of the pores, the description for a Newtonian fluid matches the Stokes description with the corresponding rheology. On the macroscopic scale the model is that of Darcy’s law, which is of course also an expression of the fundamental principles of mechanics, but with a very different structure, where the rheology is now intimately combined with the dynamics [AUR 80]. The ideal procedure must also permit localization, that is the determination of the local fields of physical quantities starting from the values of macroscopic physical quantities. It is then clear that the only method which meets these requirements is the HPS method, since its periodicity ensures a complete description of the material. On the other hand, it is never possible to completely describe a random material (even in the limiting case of perfect disorder). Thus the methods associated with random structures cannot be as effective. Since the local description is incomplete, it is necessary to introduce assumptions on the macroscopic scale. The HPS technique also makes it possible to treat regions close to macroscopic boundaries where, perpendicular to these boundaries, the separation of length scales, as well as the stationarity, are violated. The introduction of matching boundary layers then makes it possible to complete the solution to the problem.

60


2.3. Statistical modeling This technique, developed by Kröner [KRÖ 86], is mainly used to obtain information about the effective coefficients of an elastic composite subject to a static load. Its assumptions are: – the medium, in the steady state, has infinite dimensions. The scales are separated: ε = l/L ≈ 0. This assumption underlies the technique, but systematic use of this assumption is not made during the micro-macro transition; – the material has a stable, random structure; – the assumption of ergodicity – the mean of the ensemble is equal to the volume mean – is made; – the Hill principle is applied: σ : e = σ : e, where the . operator represents the volume mean, σ the stress and e the strain. This implies that: – the macroscopic stress is σ, – the macroscopic strain is e, – the material has a macroscopically elastic behavior, – the mean of the local elastic energy density is equal to the macroscopic elastic energy density. As we indicated above, the local description of a material with a random structure is never perfectly known. The point of Kröner’s statistical modeling is to perfectly adapt to this situation: the technique provides upper and lower bounds for the effective conditions, with these bounds becoming tighter as the information available is increased. The technique can be presented starting from the integral formulation of the boundary-value problem for an infinite medium, taking into account all the heterogenities and using the modified Green’s function for the strain. Since this Green’s function is clearly unknown, bearing in mind the complexity of the problem, Kröner [KRÖ 86] introduces the modified Green’s function for the strain, associated with a fictitious homogenous elastic medium. This function is exactly known. Kröner then shows that the true Green’s function can be expressed based on this fictitious Green’s function and on the field c of the elastic tensor. Then, using the following definition for the effective elastic tensor Ceff which follows from the aforementioned assumption: σ = c : e = Ceff : e Ceff is represented as a series expansion with respect to correlation functions of increasing order in c. Then, considering energy, we can show that the truncated series, which only requires the knowledge of correlation functions up to a given order n, corresponds


61

to an upper bound on Ceff , which decreases with n. The same approach with s = c−1 leads in the same way to an increasing series of lower bounds. The least precise bounds are those of Voigt, c, and Reuss, c−1 −1 . They only require a knowledge of the means of c and s. The introduction of second-order correlation functions leads to tighter bounds. If the material is macroscopically isotropic, and the moduli of elasticity of two neighboring grains are uncorrelated, these are the bounds of Hashin and Shtrikman [HAS 63], and so on. The knowledge of all the correlation functions leads in theory to the actual value of the effective coefficient. 2.4. Method of multiple scale expansions The method of multiple scale expansion, whose foundations were laid by SanchezPalencia [SAN 74], was introduced [KEL 77; BEN 78; SAN 80] in order to study problems involving the homogenization of structures with small-scale periodicity. The methodology presented here [AUR 91] makes use of these developments and allows us to answer the question of homogenizability. It also demonstrates the common points between periodic and random materials. When homogenization is possible, the structure of the equivalent macroscopic behaviors is identical in both cases. Only the effective coefficients require different treatments to obtain them. 2.4.1. Formulation of multiple scale problems Before embarking on this subject, we must define the concept of the macroscopically equivalent medium. We would like to obtain an equivalent macroscopic boundary-value problem, in other words relations between the macroscopic quantities (in practice these are average quantities, whose meaning will be specified later on) and the effective parameters. These relations are either constitutive laws or conservation laws. The description is intrinsic to a class of media subject to a given type of disturbance the macroscopic description must be valid for all boundary-value problems belonging to this class. The macroscopic description is continuous, as opposed to the microscopic description which can be discontinuous on a small scale. Here we will only consider media and disturbances which are piecewise continuous on the microscopic scale. It is clear that the macroscopic descriptions obtained are also valid for macroscopic problems which are piecewise continuous, on the condition that each macroscopic region of continuity is itself homogenizable. 2.4.1.1. Homogenizability conditions We will now investigate the homogenizability conditions. It is clear, based on the considerations developed above, that the macroscopic domain must contain a (very) large number of heterogenities and that the size of these and their organization must be such that the representative elementary volume (REV) for the medium is

62


small compared to the macroscopic volume. In order to be representative of the heterogenities’ geometry and the constituents’ properties, the REV must contain a sufficiently large number of heterogenities. Its size does not however tend to exceed 10 heterogenities in each spatial direction [CHE 88; ANG 94; ROL 07]. Thus we can normally compare the characteristic size of the heterogenities to that of the REV: the separation of scales parameter is not significantly modified. The separation of scales is a property which depends as much on the geometry of the medium as on the phenomenon. The REV reduces to the unit cell when the medium is periodic. Let lc be a characteristic length of the REV or period, and Lc a macroscopic length. Lc represents either a characteristic length of the volume of material under consideration, or a macroscopic characteristic length of the phenomenon. Separation of scales requires that: lc =ε1 Lc

(2.1)

We again emphasize that the physical quantities must satisfy this condition of separation of scales. For example, consider a periodic elastic composite satisfying (2.1) from the geometric point of view, but subject to a dynamic excitation of wavelength O(lc ). Now diffraction comes into play. The Lc of the perturbation is O(lc ) and condition (2.1) is not satisfied. We cannot obtain a macroscopic description which satisfies the conditions of homogenization presented above. This example will be revisited in more detail in Chapter 3. Furthermore, periodic and random media behave in very different manners. In the case of periodic media, diffraction introduces forbidden frequency ranges, and if the structure is random then the excitation is confined close to the sources. 2.4.1.2. Double spatial variable Condition (2.1) is taken as a base assumption for all homogenization processes, even if most of them do not make systematic use of it. The two characteristic length scales lc et Lc introduce two dimensionless spatial variables y∗ = X/lc and x∗ = X/Lc where X is the physical spatial variable (an asterisk superscript indicates a dimensionless quantity and the subscript c indicates a characteristic value). Due to the separation of scales, each quantity Φ appears as a function of these two dimensionless variables rather than anything else. (In the literature, it is common to use physical spatial variables, in other words X and Y = X/ε.) The variable x∗ is the macroscopic (or slow) spatial variable and y∗ is the microscopic (or fast) spatial variable. The fast variable y∗ describes the short-range interactions O(lc ), whereas the slow variable x∗ describes the long-range interactions O(Lc ). Two equivalent notations are therefore possible: – the first corresponds to the macroscopic viewpoint, since the analysis is conducted with the spatial variable x∗ : Φ = Φ(x∗ , y∗ ),

y∗ = x∗ /ε,

Φ = Φ(x∗ , ε)

(2.2)


63

– the second corresponds to the microscopic viewpoint, with the analysis conducted with the spatial variable y∗ . Φ = Φ(x∗ , y∗ ),

x∗ = εy∗ ,

Φ = Φ(y∗ , ε)

(2.3)

Let Φ be the mean of Φ. For a random medium, the mean is taken over a representative elementary volume with respect to the variable y∗ , whereas for a periodic medium the volume considered is the unit cell. Separation of scales implies that for Φ (see Figure 2.2): Φ = Φ + (Φ − Φ) where Φ varies with x∗ and (Φ − Φ) varies with y∗ , and: ∂Φ =O ∂y ∗

∂Φ ∂x∗

(2.4)

The symbol O(.) must be interpreted relative to ε: A = O(B)

ε1/2 |A/B| ε−1/2

if

lc

Lc (x , y )

(x , y )

h i h i x = X/Lc

1

y = X/lc

Figure 2.2. Macroscopic and local variation of Φ

1

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The local gradient of Φ is of the same order of magnitude as the macroscopic gradient of Φ. In general terms, lc is known for a given material. Lc is determined a posteriori, after solving the macroscopic boundary-value problem, using [BOU 89b]: ⎞ ⎛ ⎜ Φ ⎟ Lc = O ⎝ ⎠ ∂Φ ∂X

(2.5)

The separation of scales condition is then written as: ⎞ ∂Φ lc ⎟ ⎜ ε= = O ⎝ ∂X ⎠ 1 Lc Φ ⎛

lc

(2.6)

2.4.1.3. Stationarity, asymptotic expansions Figure 2.2 shows that the variation of Φ as a function of x∗ over a distance O(lc ) is small, and is in fact zero in the limit ε → 0. This means that Φ satisfies the property of y∗ -stationarity at the local scale. The y∗ -stationarity of Φ is defined as follows. Let Φ and ΦS be the volume and surface means of Φ, over a representative elementary volume and surface respectively. The choice between a volume or surface mean depends on the physical meaning of Φ. If Φ is for example a density, the volume mean is the appropriate choice. If Φ is a stress, then it is the surface mean which should be considered, etc. Φ is y∗ -stationarity if its local mean is invariant under a local translation, of order lc and whatever the actual structure of a given instance of the random medium (ergodic hypothesis). Consider for example a surface mean. Let Σ∗1 and Σ∗2 be two arbitrary straight, parallel sections of a REV (see Figure 2.3). The stationarity condition can be written: ΦdS ∗ = ΦdS ∗ (2.7) Σ∗ 1

Σ∗ 2

A similar property is satisfied when the microstructure is periodic, with the REV replaced by the unit cell. In this case the property is stronger: Φ is y∗ -periodic. It is now clear that the small parameter ε is the keystone of the homogenization process. This leads us to look at fields for unknown physical quantities in the form of asymptotic expansions at multiple scales [BEN 78; SAN 80] in powers of ε. For a dimensionless quantity Φ∗ , the expansion can be written: Φ∗ (x∗ , y∗ ) = Φ∗(0) (x∗ , y∗ ) + εΦ∗(1) (x∗ , y∗ ) + ε2 Φ∗(2) (x∗ , y∗ ) + · · ·

(2.8)

The equivalent macroscopic behavior is estimated to be first order by the behavior of Φ∗(0) . The stationarity or the periodicity then implies the y∗ -stationarity or the


P 1

y = X/lc

65

P 2

1

Figure 2.3. Representative elementary volume for a randomly-structured material

y∗ -periodicity of Φ∗(i) . In return, this property ensures a good separation of scales, and hence homogenizability: if the unknowns can be found in form (2.8), where Φ∗(i) are y∗ -stationarity or y∗ -periodic, then homogenization is possible. In the converse case, the medium and excitation are not homogenizable. The method is consistent with itself (self-consistent in this sense). 2.4.2. Methodology More precisely, the method to be followed is as described here. We assume the local description to be given, and we are looking for the equivalent macroscopic description. – First of all we choose the macroscopic or microscopic viewpoint. These lead to equivalent results, and the choice depends purely on convenience for the problem being studied. The expansions are then made in form (2.8) with y∗ = x∗ /ε or x∗ = εy∗ respectively. The variable x∗ (y∗ ) is the directional spatial variable and Lc (lc ) is the characteristic length to be used to non-dimensionalize the various quantities which appear in the description. The processes of homogenization corresponding to the two viewpoints are illustrated schematically in Figure 2.4. – We then proceed to the non-dimensionalization of the local description. The dimensionless numbers are evaluated as a function of powers of ε (other types of evaluation are possible, depending on the problem being considered). A dimensionless quantity Φ∗ is said to be O(εp ) if: εp+1/2 |Φ∗ | εp−1/2 The normalization is an important stage during which the physics of the problem is taken into account. It is required before using expansions in powers of ε. After this

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lc

X Lc

1

y

x

1

1

y

1

x

X Lc Figure 2.4. The two possible viewpoints: microscopic (left), normalized with lc and Φ∗ = Φ∗ (ε−1 y∗ , y∗ ) with x∗ = ε−1 y∗ and macroscopic (right), normalized with Lc and Φ∗ = Φ∗ (x∗ , εx∗ ) with y∗ = εx∗

operation, the equations appear in the form: εqp Ψ∗p = 0

(2.9)

p

where the Ψ∗p operators are dimensionless. – Finally we substitute the asymptotic expansions in the form (2.8), where Φ∗(i) are y∗ -periodic and Φ∗(0) is O(1), into the normalized local description. The identification of terms of the same power in ε leads us to solve the various problems thus obtained over the unit cell. This last stage is that described in Bensoussan et al. [BEN 78] for periodic media when the macroscopic viewpoint is adopted. In the case of linear problems with convex energy, it is generally possible to show the existence and uniqueness of the solutions to these successive problems and to obtain a numerical solution to them.


67

In the case of random media things are not the same. Nevertheless, considering the boundary-value problem over a macroscopic volume (Figure 2.1) as well posed (although difficult to solve in practice due to the large number of heterogenities), each representation of the medium possesses one solution, and only one solution. The local stationarity then implies that the surface (or volume mean) of the quantity under consideration is independent of the representation used, which leads to the uniqueness of the macroscopic description. The cornerstone of homogenization is a necessary and sufficient condition, often known as the compatibility condition, for the existence of solutions to each problem which needs to be solved in succession over the period. The equations to solve take the form (they follow from conservation laws!): divy∗ Φ∗(i) = −divx∗ Φ∗(i−1) + W ∗

(2.10)

where the indices x∗ and y ∗ indicate that the derivatives are taken with respect to x∗ and y∗ respectively. This equation represents the local conservation of Φ∗(i) , where −divx∗ Φ∗(i−1) + W ∗ appears as a source term. Since Φ∗(i) are locally periodic or stationary, the source must have a mean of zero: divx∗ Φ∗(i−1) − W ∗ = 0,

or

divx∗ Φ∗(i−1) − W ∗ = 0

(2.11)

This property corresponds to the Fredholm alternative: the operator in y∗ on the left-hand side of of the equation to be solved, given appropriate boundary conditions (periodicity and possibly others depending on the problem at hand), has an eigenvalue of zero, associated with an eigenfunction independent of y∗ . The necessary and sufficient condition for the existence of a solution lies in the orthogonality of the righthand side of the equation to this eigenfunction, which leads to the result. Due to the non-dimensionalization, the first non-zero terms of the asymptotic expansions must be such that the orders of magnitude of the dimensionless numbers estimated during normalization are respected. In other words, the first non-zero terms of Ψ∗p must satisfy (2.9). This will be made clearer in the one-dimensional example considered in the following chapter, in the context of elasticity (Chapter 3). In this case, the compatibility conditions lead either to the equivalent macroscopic description or to non-homogenizability when the first non-zero terms of Ψ∗p do not satisfy (2.9); what happens is that such values will end up modifying the order of magnitude of the dimensionless numbers characterizing the medium and the phenomenon under study. We will return to this argument in a systematic manner in all the physical problems discussed in the following sections. In conclusion, we note that convergence problems for solutions when ε −→ 0 will not be covered in this work. On this subject, the reader may wish to refer to: [SPA 68; GIO 75; BEN 78; MUR 78; TAR 78; MUR 95; ALL 92; ALL 96].

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2.4.3. Parallels between macroscopic models for materials with periodic and random structures Let us recall the characteristics of these two types of material when they exhibit a separation of scales, as well as the phenomena to which they are subjected. We will also take a viewpoint that is distant from any macroscopic boundary. 2.4.3.1. Periodic materials The geometry is periodic. The physical quantities, and the derived quantities, are also periodic with respect to the fast variable y∗ . The dependence of the macroscopic physical quantities with respect to the boundary conditions of a given macroscopic problem are expressed in terms of the slow variable x∗ alone. 2.4.3.2. Random materials with a REV The geometry is random. The physical quantities, and derived quantities, are stationary with respect to the fast variable y∗ . The dependence of the macroscopic physical quantities with respect to the boundary conditions of a given macroscopic problem, for a given realization of the structure, is expressed in terms of the slow variable x∗ alone. For both types of material, we will consider macroscopic samples subjected to phenomena, for which the modeling at the local scale leads to a well-posed boundaryvalue problem (even if in practice the numerical solution is difficult to obtain due to the large number of heterogenities contained within the sample). We can therefore be sure that a unique solution does exist in both cases, for the periodic and the random material. If we assume that the asymptotic expansions are unique, the same then applies for successive terms in the expansions. These successive terms are solutions to the boundary-value problem across the period or the REV, with the same partial derivative equations and boundary conditions where discontinuities occur. The only differences are boundary conditions on the surface of the period or the REV, which specify either periodicity or stationarity. Also, the variable of integration is y∗ , with the slow variable x∗ only playing the role of a parameter here. The macroscopic boundary conditions are not therefore involved in these problems, and the structure of solutions at each order of expansions is the same for both types of material. Finally, the determination of compatibility conditions such as the symmetric properties of the effective coefficients only involves integrals over the boundary of period or REV. Periodicity and stationarity are equivalent from the macroscopic viewpoint. Thus, if we restrict ourselves to looking for the structure of the macroscopic description, random and periodic media are equivalent when homogenization is possible. Another means of comparing media with period and random structures is to consider a rectangular parallelepiped REV of the random medium. We will


69

apply the three symmetries to the three orthogonal faces of this REV so that we can ensure the connectivity of the phases. We thus obtain a period equivalent to the REV in terms of ε. The two media will have the same macroscopic behavior, except the anisotropy behavior which may be affected by the planar symmetries we have introduced.

2.4.4. Hill macro-homogenity and separation of scales Hill [HIL 63; HIL 67] defines the representative volume (RV) as a volume of material with the following two properties: – The RV is on average structurally representative of the whole material; – Macro-homogenity. The RV contains a sufficiently large number of heterogenities that the effective coefficients are independent of the boundary conditions for strain or load applied to the RV. The contribution of the matching boundary layer along the edges of the RV is thus negligible. If d is the characteristic size of a heterogenity and L the characteristic size of the RV, d/L 1. Property a corresponds to the definition of the REV introduced above. When applied to the asymptotic expansion method, property b implies a large number of REV in the RV: in this way the effective coefficients are definitely independent of the boundary conditions on the RV. The RV is a volume which ensures the separation of length scales and L may thus be compared to a macroscopic length Lc .

2.5. Comments on multiple scale methods and statistical methods 2.5.1. On the periodicity, the stationarity and the concept of the REV Both methods deal with materials with separated length scales: they have a small parameter ε = lc /Lc 1. Of course, the lack of systematic use of this property in the statistical methods explains certain weaknesses that it suffers from, which we will mention later. The same formal structure of the effective coefficients of an elastic composite shows that the two methods are similar, and it is reasonable to assume this property extends to all elliptical problems. The translational invariance (periodicity or stationarity) represents a third point in common. Finally, although the result is obtained by very different means, both methods make it possible to study the physical characteristics of the volume means. The solution follows from the assumption of ergodicity in the statistical methods, and from the periodicity or stationarity, and the local solenoidal nature of the quantities, in the case of the multiple scale method (see [AUR 86; AUR 91] for some simple examples). The method of multiple scales however allows the consideration of examples where the volume mean is different to the surface mean [LEV 81; AUR 87a; AUR 89].

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Now we return to selection of the representative elementary volume (REV). While there is no issue for periodic media, because REV is the period, the REV is necessarily approximate for random materials [ROU 88; DRU 96; GUS 97; KAN 03; KAN 06; FOR 06; ROL 07]. For a porous structure the optimal REV for porosity differs from the optimal REV for permeability. A good investigation of this topic involving unsaturated porous media can be found in Rouger [ROU 88]. The works of Cherel et al. [CHE 88] and Kanit et al. [KAN 03; KAN 06] demonstrate that the optimal REV depends on the physical quantity of interest (thermal conductivity, modulus of elasticity, etc.), on the contrast between the different phases and the precision required for the property being investigated. They also show that it is possible to obtain a precise estimate of the effective properties of a material with a random microstructure, starting from its “apparent” properties identified over a number of volumes the size which is much smaller than the REV. This leads to the question of whether we can transfer some of the advantages of the multiple scale method applied to periodic media to the study of random materials, to the calculation of effective coefficients. When homogenization is possible, we have seen that the structure of the macroscopic description is independent of whether the material is periodic or random. So, applying the method of multiple scales to a fictitious periodic material which is “analogous” to the random material, we can obtain effective coefficients which we then need to relate back to those of the random material. Various works [CHE 88; ANG 94; BOU 04; KAN 03; KAN 06] show, for example, that the assumption of periodicity is not an obstacle in estimating the effective properties of a material with a random microstructure. The same issue approached from a different angle has led to estimates for the permeability of periodic and/or random porous media [BOU 00; BOU 08] (see Chapter 10).

2.5.2. On the absence of, or need for macroscopic prerequisites One crucial difference between the methods is the absence (multiple scale methods) or need (statistical methods) for assumptions in order to establish the macroscopic description. These assumptions do not help when it comes to determining “exotic” descriptions, in other words descriptions which do not follow the normal macroscopic phenomenological rules. Assumptions impose a priori restrictions on the macroscopic description and limit the generality of the statitical theory and most other homogenization methods. The method of multiple scales, which does not require any starting assumptions, stands out as the only entirely satisfactory method in this respect: – In the field of elastic composites the work of Duvaut [DUV 76] can be consulted. He shows without any starting assumptions that steady state microscopic and macroscopic behaviors have the same structure to their description (see Chapters 3 and 11).


71

– The study of plastic composites and materials with non-linear behavior is another more complicated example [deB 87; deB 86; PAS 86; SUQ 87; deB 91; SUQ 97; PON 98; BOR 01]. – The study of filtration in a rigid porous structure represents one situation where the structures of the microscopic and macroscopic descriptions are very different (the Navier-Stokes equations and Darcy’s law) even though they represent the expression of fundamental principles of mechanics on both scales [ENE 75; AUR 80]. The case of dynamic perturbations [LEV 79; AUR 80] shows the unusual property that the macroscopic filtration, or generalized Darcy’s law, has “exotic” dynamics: the filtering fluid has a memory of past accelerations (see Chapter 7). – The study of suspensions in a Newtonian liquid [FLE 83; LEV 83b; SAN 85] as well as fluid mixtures [LEV 81; AUR 89; BOU 93] is a clear example, as is the study of any such mixture, of the microscopic causes of mono- or pluriphasic macroscopic behavior. – For porous media (see Part 4), references to the homogenization of various physical problems can be found in Hornung [HOR 97]. More recently, Moyne and Murad [MOY 02; 03; 06] showed that the consideration of electrochemical effects in a saturated porous medium (fluid with dissolved solute) leads to the concept of osmotic stress on the macroscopic scale. 2.5.3. On the homogenizability and consistency of the macroscopic description The question of homogenizability has been considered in Auriault [AUR 91]. The reader is also referred to Auriault and Boutin [AUR 92; 93a; 94] for porous media with double porosity [AUR 93a], and [AUR 95] for Taylor dispersion in porous media, as well as [BOU 90] for the acoustics of bituminous concretes and [BOU 93] for that of Newtonian liquids with high concentrations of gas bubbles. Physical consistency of the results is a very important consideration. The macroscopic description is obtained through conservation equations and rheological laws, with macroscopic quantities whose physical meaning needs to be specified. It is also necessary to ensure energetic consistencies. Thus for an elastic composite under static load, the mean of the local elastic energy density must equal the macroscopic elastic energy density defined using the correct macroscopic physical quantities. The method of multiple scales is well suited to investigating this. As far as the statistical method is concerned, the consideration does not apply, since consistency is included as a prerequisite (at least as far as elastic composites are concerned). Localization is another advantage of the method of multiple scales, as applied to periodic media, whether it is used to obtain local fields in the bulk of the material or to escape the simplistic restriction of an infinite material. Close to macroscopic boundaries, the separation of scales (and the stationarity) is broken perpendicular

72


to these boundaries. The introduction of matching boundary layers means that the method of multiple scales can be applied to boundary-value problems on finite regions and can be used to investigate the problems associated with them [ENE 75; AUR 87; TUR 87; LEV 75; LEV 77; SAN 87]. The method of multiple scales makes it possible to deduce whether the point measurement of a physical quantity Φ is reasonable or not within the heterogenous material. The dimensionless macroscopic behavior is given to first order by that of Φ∗(0) (x∗ , y∗ ) and the whole problem can be reduced to that of determining whether the macroscopic quantity Φ∗(0) equals the quantity Φ∗(0) (x∗ , y∗ ) at all points across the period. Or to put it another way, whether Φ∗(0) (x∗ , y∗ ) is independent of the variable y∗ . This is, for example, the case for a temperature T in heat transfer problems in a composite of materials with conductivities of the same order of magnitude with respect to ε: T ∗(0) as a function of the slow variable x∗ alone, where point measurement is permitted [AUR 83]. In the problem of filtration of a Newtonian liquid in a porous medium, the pressure is such that p∗(0) = p∗(0) (x∗ ) while the velocity (clearly) depends on the fast variable y∗ , v∗(0) = v∗(0) (x∗ , y∗ ). Point measurement of the pressure is permitted, whereas measurement of the velocity must be global (a surface mean!). Such conclusions are also possible for other homogenization methods, and the reasoning relies to a lesser or greater extent on the presence of separated length scales. On this topic the reader is referred to: [HAS 79; WHI 86; NOZ 85]. 2.5.4. On the treatment of problems with several small parameters The systematic use of the small parameter ε offers a decisive advantage to the method of multiple scales. It is not possible to describe the dynamics of an elastic composite using the statistical method [KRÖ 86]. On the other hand, the method of multiple scales rapidly demonstrates that the macroscopic rheology obtained in the steady state regime is useful as a macroscopic dynamic description, at least in most normal situations [AUR 85a]. Other examples involve the possibility of treating problems with several length-scale separations using the method of multiple scales, and problems where, as well as the small homogenization parameter ε, there are other small parameters involving the internal geometry of the period or rheological coefficients of the composite materials. Thus many situations can be studied depending on the respective levels of various small parameters. Double porosity rigid structures can also be added to this category [LEV 88; ROY 92; ROY 94]. Three well-separated length scales are used (Figure 2.5). The first scale is that of the pores, where behavior can be described using Stokes equations. The second scale is that of fissures. On this scale the Stokes equations are valid in the fissures, and the medium equivalent to the micro-porous material on the preceding scale is described by Darcy’s law. Finally, on the macroscopic scale


73

Figure 2.5. Doubly porous material: macroscopic scale of the same, mesoscopic scale of the fractures and microscopic scale of the pores

homogenization leads to a Darcy law. When the matrix is deformable, [AUR 92; 93a; 94a] interaction between the length scales becomes strong, and the macroscopic description depends on the relative values of the separation of scales as well as the contrast in rheological properties of the matrix and the fluid. This makes it possible to confirm or discount [ROY 94] various macroscopic descriptions introduced directly on this length scale through phenomenological studies, such as those of Barrenblatt et al. [BAR 60] or Warren and Root [WAR 63]. Along similar lines, double conductivity media were modeled in [AUR 83]. There again, the study made it possible to confirm or discount the existing models. Finally the case of porous media with strong heterogenities on the macroscopic scale, which introduces a third characteristic length, are homogenized on a larger scale [MEI 89]. Filtration in a standard porous medium (where the geometry is only described in terms of one small parameter) is governed by Darcy’s law. When the matrix consists of particles (or fibers) which are small compared to the period, by adding a small geometric parameter, the description becomes that of Brinkman’s law [LEV 83a]. Along similar lines, a study can be found of the Dirichlet problem for a region containing small holes [CIO 85]. Banks drained using geotextile sheets [AUR 82] introduce, in addition to ε, small parameters characterizing the geometry of the period (ratio of the thicknesses of geotextile and soil) and also the hydraulic properties (ratio

74


of permeabilities of soil and geotextile). The study of reinforced earth [PAS 86], steady state diffusion in a body with small highly-conducting bodies [CAI 83], the behavior of elastic bodies reinforced with thin highly-rigid fibers [CAI 81], or the steady state analysis of thin elastic layers with a small-scale periodic structure [CAI 82], are all also problems involving three small parameters. Finally we mention homogenization of reticulated structures, which makes it possible to replace these structures with an equivalent continuum. This sort of problem, treated as a continuous medium on the local scale, introduces a second small parameter characterizing the geometry of the period [CIO 86; 88] in addition to ε. It can also be treated using a description based on beam elements on the local scale, which leads to the methods of homogenization for discrete structures [CAI 89; BAK 89; BOU 03]. Recent developments in structural mechanics are summarized in Andrianov et al. [AND 04].

Chapter 3

Homogenization by Multiple Scale Asymptotic Expansions

3.1. Introduction Following discussion of the multiple scale method and its formalism, in this chapter we will explain in detail how it can be implemented. We will begin by using basic experiments to show how the concepts presented in the previous chapters apply in reality and how they match up to physical intuition. We will then show how the homogenization process is carried out for a one-dimensional example with an analytical solution. Finally, the last section focuses on the translation of physical problems into the framework of the multiple scale method.

3.2. Separation of scales: intuitive approach and experimental visualization The concept of multiple scales, and its use in homogenization methods, may appear an abstract one that could be taken as a mathematical trick. It is no such thing, because in fact this idea represents an actual physical reality. Here we will try to help the reader grasp this by using an intuitive approach illustrated with simple experimental examples.

3.2.1. Intuitive approach to the separation of scales We have already seen that homogenization involves the search for a given phenomenon in a given heterogenous material, for an equivalent – or “homogenized” – global description, which does not make any explicit reference to local fluctuations.

76


This statement incorporates, as a subtext, the concept of separation of scales, since a global description has no meaning if the phenomenon of interest only varies on a local scale. As indicated in the preceding chapters, it is this crucial concept of separation of length scales which makes it possible to look for a homogenized description. The concept can be described in terms of two requirements: – the first involves the medium, which must be such that we can define a characteristic length l, which is only possible if the material has a representative elementary volume (without a REV, there is no characteristic length!); – the second involves the phenomenon: a quantity associated with it must exhibit a characteristic length L, which is large compared to l.

The graphics in Figure 3.1 (top) give a visual depiction of the separation of scales required for homogenizability: as long as the phenomenon of interest (the runner) has a scale of motion (his stride) which is large compared to the REV of the material (a sand or pebble beach), a global description (of the speed and trajectory of the runner) which ignores local fluctuations (the exact positions of the grains of sand or pebbles) is possible. It can be clearly seen in this example that for the phenomena involved a homogenized description is more efficient – and also more realistic – than a description which incorporates every last detail of reality without removing all but the essential parts of the picture. Outside this framework, in other words without a separation of scales, the search for a macroscopic description is doomed to failure (Figure 3.1 below): on a route consisting of meter-sized rocks, neither the trajectory nor the speed of the runner can be known independently of the distribution of the blocks. This would also be the case for an insect on the pebble beach (despite the fact that it is homogenizable for the runner). This illustrates the fact that homogenizability is a property not intrinsic to the material or the phenomenon, but which depends on the material/phenomenon pair. The role of the periodic or random nature of the microstructure (when the constituents follow the same connectivity conditions) was discussed in Chapter 2. The images in Figure 3.2 (top) illustrate the main conclusions: when the separation of scales is obvious, whatever the organization on the local scale (pebbles arranged periodically or laid out randomly), the mechanisms (the determination of the runner’s trajectory) are the same, and as a result the macroscopic behavior of the material (what the runner experiences) will be qualitatively the same. Here we justify the use of the method of periodic media for treating real aperiodic materials when there is a separation of scales. Conversely, the closer the macroscopic scale gets to the microscopic scale, the more sensitive it becomes to local fluctuations, and consequently the organization of the microstructure. At the limit of the homogenizable domain, Figure 3.2 (bottom – the runner striding across separate blocks), the


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Figure 3.1. The separation of scales is the sine qua non condition for a global description. Here it is only the case in the top picture: the property of homogenizability only has a meaning for the combination of the material and the phenomenon together (illustrations by Jacques Sardat)

phenomena in periodic media (where the runner can jump from block to block) and random media (where the runner falls between blocks that are too far apart) diverge: without the separation of scales, homogenization loses its meaning, and the type of organization within the microstructure becomes critical.

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Figure 3.2. The role of the microstructure layout is more significant when there is not a good separation of scales (illustrations by Jacques Sardat)

3.2.2. Experimental visualization of fields with two length scales Here we will investigate, with the help of basic experiments on two periodic twodimensional media: the manifestation of local and global scale variations, and the (quasi-)periodicity or local periodicity of the fields. 3.2.2.1. Investigation of a flexible net The photos in Figure 3.3 show a net with a diamond mesh (period Ω) fixed at its edges to a square framework consisting of four rigid, articulated rods. If we apply a distortion to the frame, we impose a homogenous distortion to the net: – photo (a) in Figure 3.3 shows the starting position, where the net is undistorted; – photos (b) and (c) in Figure 3.3 show the geometries obtained when a moderate, and then considerable, distortion is applied to the supporting framework. It is clear that the mesh is distorted but that the structure remains periodic. So, for homogenous deformations, the property of periodicity of the initial medium is preserved by the perturbations, even for large deformations.


79

(a)

(b)

(c) Figure 3.3. Visualization of a periodic net (a) of the periodicity under (b) moderate and (c) large deformations

What happens for loadings which lead to inhomogenous distortions? – In photo (a) in Figure 3.4 the net is dragged in the plane by a rigid rake which applies a tension across several units of the mesh. The deformation produced in the net is not homogenous. Nevertheless, a local (quasi-)periodicity (i.e. Ω-periodicity relative to the microscopic variable) is visible. What we mean is that all the meshes adjacent to mesh A have an almost identical geometry. The same is true of mesh B. However, meshes A and B, which are fairly far apart from each other, have a very different geometry. We also point out that the geometries of the deformed meshes are the same as we have already seen under homogenous distortion. This situation shows

80


the two-length-scale variations, where each cell is deformed, but the amplitude of this deformation varies gradually across distances corresponding to many mesh cells. – If instead of being spread out the tension is applied at one point (photograph (b) in Figure 3.4), a new effect appears which is characterized by a violation of the local periodicity on either side of the line of the pull. In these regions where there is a high gradient of deformation, there is no longer a separation of scales because the phenomenon is concentrated on the local scale (which leads to the loss in periodicity): perpendicular to the direction of the pull the problem is not homogenizable.

Figure 3.4. Inhomogenous load: the quasi-periodicity relies on a separation of length scales: (a) a load which respects the separation of scales: the local quasi-periodicity is modulated by large-scale variations; (b) localized loading: the periodicity is lost along the line of the pull

This net makes it possible to directly observe the deformed geometry of the lattice. However, in many homogenization problems the period is considered to be fixed, as is effectively the case of flow in rigid porous media, cases of heat transfer or diffusive solute transport, etc. or where it is an approximation which can be justified by the low level of deformation such as when considering elastic composites, poroelastic behavior, etc. In this case the homogenizability conditions apply to the fields which develop within this periodic geometry. We will consider such a case in the next example.


81

3.2.2.2. Photoelastic investigation of a perforated plate Consider a plexiglass plate drilled with oblong windows distributed in a periodic staggered pattern, which we will subject to small deformations in plane. Through the photoelastic effect we can visualize the deviatoric stresses which develop in the plate under different loads: – when the plate is loaded uniformly in its plane (photograph (a) in Figure 3.5) it is very obvious that the field is periodic, matching up with the periodicity of the plate; – if the loading area on the upper edge is reduced (photograph (b) in Figure 3.5) while maintaining the entire contact surface on the lower edge, the local quasiperiodicity and the global fluctuations can both be seen; – “large-scale” intensity variations are even clearer in photograph (c) in Figure 3.5 where the load is pointlike. It is clear in this case that close to where the force is applied the phenomenon is not homogenizable, but that it becomes so outside a region around the point of loading (which extends for around one period). We will return to this aspect of the problem at the end of this chapter. We also remark that outside the areas of concentrated load, the local distribution of the deviatoric stresses looks the same, but its global evolution depends on the load. If on the other hand the same load is applied in a different orientation relative to the plate (photographs (a) and (b) in Figure 3.6) the local and global distributions of the deviatoric stresses are completely changed (but of course the periodicity is still retained). Hence the anisotropy of the distribution of the perforations has a direct impact on the local, and hence global, strains: this illustrates the fact that the macroscopic description is tightly linked to the microscopic structure. Also, across all the pictures, an edge effect can be seen at the boarder of the plate, which rapidly fades towards the middle of the periodic medium. This rapid decrease can be confirmed in photograph (c) of Figure 3.6 where, under homogenous compression, the periodicity remains obvious when the plate only consists of oneand-a-half periods! The edge effects which result from the loss of periodicity at the boundary can be treated from a theoretical point of view by the introduction of a boundary layer [see for example SAN 87; AUR 87a]. In conclusion, these two examples show how the principles of homogenization have a basis in physical reality. They also show that these principles apply even some way from the ideal separation of scales which the theoretical developments require. Indeed, in situations of inhomogenous loading, the actual scale ratio εr is, at very best, in the order of the inverse of the number of periods contained in the smallest dimension of the experiment, so εr 0.1 for the net and εr 0.3 for the plate. This possibility of extending the field of applicability is also one of the main reasons

82


(a)

(b)

(c) Figure 3.5. Condition of separation of scales, and quasi-periodicity of the fields in a periodic medium. Perforated plate subject to: (a) homogenous compression exerted by pressure across the width of the plate, (b) inhomogenous compression exerted by a pressure from above across a narrower width, (c) inhomogenous compression under point loading of the upper surface


(a)

(b)

(c) Figure 3.6. Role of the microstructure in the distribution of local and global forces: (a) inhomogenous compression parallel and (b) perpendicular to the holes, (c) quasi-periodicity and edge effects under homogenous compression for a plate which only consists of one-and-a-half periods

83

84


why the results of homogenization are so good at describing real situations: results established rigorously in the context of ideal assumptions retain their pertinence for real physical situations corresponding to weakened hypotheses. From a theoretical viewpoint, this observation is analogous to proof that the results converge when the scale ratio approaches zero. To clarify what is meant by this, we will return to these issues, and to the importance of the actual separation of scales in a real-life problem, after we have demonstrated application of the method to a simple example.

3.3. One-dimensional example Now, and in what follows, we will systematically apply the method of multiple scales, following the methodology laid out in Chapter 2, section 2.4.2. In order to present the various stages of the process, we have selected a one-dimensional example which has an analytical solution. Due to its simplicity, this example cannot include all the problems inherent to homogenization techniques. We will encounter them in subsequent chapters during the study of multi-dimensional problems. Here we will consider a one-dimensional elastic Galilean medium with an oedometric modulus E and density ρ, subject to a dynamic perturbation. The medium is periodic, with a small period lc , and we will consider a sample of length Lc lc . The displacement u is governed by the equation of dynamic equilibrium: divX (E gradX u) = ρ

∂2u ∂t2

(3.1)

where divX and gradX are the divergence and gradient operators with respect to the spatial variable X, which are the same here since the problem is one-dimensional. We recall that E(X) is a positive quantity, as is the density ρ. They are both periodic with period lc , and may exhibit discontinuities. Figure 3.7 shows an example of the variation of E.

E

lc Figure 3.7. Periodic variation of E

X


85

Across the discontinuities Γ, the stress σ and the displacement u are continuous: [σ] = [E gradX u]Γ = 0

(3.2)

[u]Γ = 0

(3.3)

In the above equations, [φ]Γ indicates the jump in φ across the interface Γ. We will first consider the steady state problem, where the second member of (3.1) is zero. We will then treat the dynamic case.

3.3.1. Elasto-statics Equation (3.1) now becomes: divX (E gradX u) = 0

(3.4)

Equations (3.2, 3.3, 3.4) do not introduce any dimensionless numbers. We will take the microscopic viewpoint. With: X = lc y ∗ ,

E = Ec E ∗ ,

u = u c u∗

where Ec and uc are characteristic values, we have: σ = E gradX u = σc σ ∗

with σc = Ec uc /lc

Equation (3.4) becomes: divlc y∗ (Ec E ∗ gradlc y∗ uc u∗ ) = 0 Carrying out the same change of variables in (3.2, 3.3), we find after simplification of the terms referring to the same characteristic values: divy∗ (E ∗ grady∗ u∗ ) = 0,

[E ∗ grady∗ u∗ ]Γ∗ = 0,

[u∗ ]Γ∗ = 0

The unkown u∗ must be found in the form of the following expansion: u∗ (x∗ , y ∗ ) = u∗(0) (x∗ , y ∗ ) + εu∗(1) (x∗ , y∗ ) + · · · ,

x∗ = εy ∗

(3.5)

where ε = lc /Lc and u∗(i) are periodic with respect to the local variable y ∗ = X/lc , of period 1. The differential operators are therefore operators with respect to the variable y ∗ , and in (3.5) x∗ = εy ∗ . The equivalent macroscopic description will be valid when the perturbation satisfies the condition of separation of scales (we assume that the condition on the separation of geometric scales is met).

86


3.3.1.1. Equivalent macroscopic description The method involves the introduction of the expansion (3.5) into the dimensionless system and identifying the powers of ε. We note that due to the two spatial variables and the choice of the microscopic viewpoint, the spatial derivative takes the following form: ∂ ∂x∗ ∂ ∂ ∂ + ∗ ∗ = +ε ∗ ∗ ∗ ∂y ∂y ∂x ∂y ∂x The local description becomes: (

∂ ∂ ∂ ∂ ∗ ∗ ∗ + ε ) E (y )( + ε )u =0 ∂y ∗ ∂x∗ ∂y ∗ ∂x∗

with: [E ∗ (y ∗ )(

∂ ∂ + ε ∗ )u∗ ]Γ∗ = 0 ∗ ∂y ∂x

[u∗ ]Γ∗ = 0 across the discontinuities. Substituting the expansion (3.5) into these expressions, we obtain in succession the following results, separating out terms of the same power of ε: First order in ε0 : the system defining u∗(0) is the following: ∂ ∂y ∗

∂u∗(0) E (y ) ∂y ∗ ∗

[E ∗ (y ∗ )

∗

=0

(3.6)

∂u∗(0) ]Γ∗ = 0 ∂y ∗

[u∗(0) ]Γ∗ = 0 where u∗(0) is 1-periodic in y ∗ . By successive integration of (3.6) it follows, making use of the conditions at the discontinuities, that: E ∗ (y ∗ )

∂u∗(0) = σ ∗(0) (x∗ ) ∂y ∗

u∗(0) (x∗ , y ∗ ) = σ ∗(0) (x∗ )

y∗ 0

E ∗−1 (y ∗ )dy ∗ + u∗(0) (x∗ , 0)


87

where the constants of integration σ ∗(0) (the zero-order stress) and u∗(0) (x∗ , 0) are functions of x∗ alone. Also, the periodicity can be expressed as: u∗(0) (x∗ , 1) = u∗(0) (x∗ , 0) which leads us to: 1 E ∗−1 (y ∗ )dy ∗ = 0 σ∗(0) (x∗ ) 0

which means that σ ∗(0) = 0, since E ∗ > 0. Finally: u∗(0) (x∗ , y ∗ ) = u∗(0) (x∗ ) proving that at the dominant order, the displacement is a function of x∗ alone. In other words it does not fluctuate over the course of a period. Second order in ε: the following order gives us: ∂ ∂y ∗

∂u∗(1) du∗(0) + ) E (y )( ∂y∗ dx∗ ∗

[E ∗ (y ∗ )(

∗

=0

∂u∗(1) du∗(0) + )]Γ∗ = 0 ∂y ∗ dx∗

[u∗(1) ]Γ∗ = 0 where u∗(1) is 1-periodic in y ∗ . The general solution to the differential equation can be obtained as before: E ∗ (y ∗ )(

∂u∗(1) du∗(0) + ) = σ ∗(1) (x∗ ) ∗ ∂y dx∗

u∗(1) (x∗ , y ∗ ) = σ ∗(1)

y∗

E ∗−1 (y ∗ )dy ∗ − y ∗

0

du∗(0) + u∗(1) (x∗ , 0) dx∗

where σ ∗(1) (first-order stress) and u∗(1) (x∗ , 0) are functions of x∗ alone. Again the periodicity of the unknown u∗(1) (x∗ , 1) = u∗(1) (x∗ , 0) allows us to determine σ ∗(1) : σ ∗(1) (x∗ )

1

E ∗−1 (y ∗ )dy ∗ −

0

σ ∗(1) (x∗ ) = E ∗−1 −1

∂u∗(0) ∂x∗

du∗(0) =0 dx∗

88


where . represents the mean operator over the period, here:

1

. =

. dy ∗

0

Third order in ε2 : Compatibility condition. At this order we have: ∂ ∂y ∗

∂u∗(2) ∂u∗(1) + ) E (y )( ∂y ∗ ∂x∗ ∗

∗

[E ∗ (y ∗ )(

∂ =− ∗ ∂x

∂u∗(1) du∗(0) + ) E (y )( ∂y ∗ dx∗ ∗

∗

(3.7)

∂u∗(2) ∂u∗(1) + )] = 0 ∗ ∂y ∂x∗

[u∗(2) ] = 0 where u∗(2) is 1-periodic in y ∗ . We do not need to calculate u∗(2) as we did for u∗(0) and u∗(1) , at least if we limit ourselves to studying the first macroscopic order. In fact the differential equation represents the conservation of the periodic quantity: σ ∗(2) = E ∗ (y ∗ )(

∂u∗(2) ∂u∗(1) + ) ∗ ∂y ∂x∗

in the presence of the source term: ∂ − ∗ ∂x

∗

∂u∗(1) du (0) + ) E (y )( ∂y∗ dx∗ ∗

∗

=−

∂σ ∗(1) (x∗ ) ∂x∗

In accordance with the analysis presented in section 2.4.2, equation (3.7) is the exact analog of equation (2.10) with W ∗ = 0. By integrating this conservation equation over the period, we have:

∂σ ∗(1) (x∗ ) ∂σ ∗(2) = − ∗ ∂y ∂x∗

But, due to the periodicity, the left hand side is zero:

∂σ ∗(2) = ∂y ∗

1 0

∂σ ∗(2) ∗ dy = σ ∗(2) (y ∗ = 1) − σ ∗(2) (y ∗ = 0) ∂y ∗

Thus we have established the compatibility condition requiring the source to have a mean of zero (see equation (2.11)):

∂σ ∗(1) =0 ∂x∗


89

so that, swapping the derivation with respect to x∗ and integration with respect to y ∗ , and introducing the expression for σ∗(1) : d dx∗

E

∗−1 −1 du

∗(0)

=0

dx∗

(3.8)

This compatibility equation represents, in dimensionless form and to first order of approximation, the equivalent macroscopic description that we were looking for. With: x∗ =

X , Lc

E∗ =

E , Ec

u∗ =

u , uc

¯ u = u(0) + O(ε)

the stress can be written in dimensional variables: σ = σc σ ∗(1) = (Ec uc /lc )E ∗−1 −1

du∗(0) du = E −1 −1 dx∗ dX

In the same way, the model can be written in dimensional variables: d −1 −1 du ¯ E = O(ε) dX dX ¯ where O(ε) is a term of relative order ε. 3.3.1.2. Comments 3.3.1.2.1. Effective coefficient The structure of the macroscopic description is identical to that of the local description. In particular, the property E ∗ > 0 is preserved because the macroscopic effective elastic coefficient is such that: E eff∗ = E ∗−1 −1

>0

This result is incidentally a classical one, and does not require any particular homogenization technique to prove it (see Chapter 1 where the equivalent thermal problem was treated). We find in this one-dimensional steady state problem that the stress is constant: σ=E

du = constant dX

Taking the mean of σ/E over the period, we find:

1 du σ = σ = E E dX

90


which leads to the result when it is observed that the mean strain is the macroscopic strain. Finally we note that E(y∗ ) tends to E(y ∗ ) when ε tends to zero, weakly in L2 [SAN 80], but that in general terms: E eff = E(y ∗ ) 3.3.1.2.2. Macroscopic physical quantities The dimensionless physical quantities – the displacement u∗ and the stress σ ∗ – are given to first order by: u∗ = u∗(0) (x∗ ) σ ∗ = εσ ∗(1) = εE ∗ (y ∗ )(

du∗(0) ∂u∗(1) du∗(0) + ) = εE ∗−1 −1 ∗ ∗ ∂y dx dx∗

They are independent of the local variable y ∗ and represent macroscopic quantities, without any mean operator. The physical significance of the macroscopic quantities does not therefore pose any problem here because it is identical to those introduced locally. 3.3.1.2.3. Accuracy of the macroscopic description Returning to the displacement u∗ , the dimensionless macroscopic description (3.8) can be written: d dx∗

E ∗−1 −1

du∗ dx∗

= O(ε)

In practice the small parameter ε is non-zero and the equivalent macroscopic description is only approximate. This is the case for any macroscopic description of a heterogenous material. 3.3.1.2.4. Quasi-periodicity: macroscopically heterogenous material The case of quasi-periodicity where the modulus E ∗ is not only a function of y∗ but also of x∗ does not pose any difficulty, as long as the variations are sufficiently slow that a separation of scales is retained. The effective coefficient is still written as E ∗−1 −1 , but now it depends on the variable x∗ . What happens is that x∗ plays the role of a parameter in the process: we recall that the differential systems that must be solved involve the variable y ∗ . This observation can of course be applied to all homogenization problems, thus making it possible to systematically extend the results to slightly macroscopically heterogenous media.


91

Finally, when the material is not strictly periodic, in other words when the period Ω∗ depends on x∗ , (3.8) becomes: d dx∗

|Ω∗ |E ∗−1 −1

du∗ dx∗

= O(ε)

3.3.2. Elasto-dynamics We will now include the inertial term. The local description is then given by the system of equations (3.1, 3.2, 3.3). The change is that this system introduces a dimensionless number denoted P, the ratio of the inertial term to the elastic term: ∂2u | ∂t2 P= |divX (EgradX u)| |ρ

We will again adopt the microscopic viewpoint here, so that the characteristic length for non-dimensionalization is lc . With: X = lc y ∗ ,

E = Ec E ∗ ,

u = u c u∗ ,

ρ = ρc ρ ∗ ,

t = t c t∗

it follows in dimensionless form that: divy∗ (E ∗ grady∗ u∗ ) = Pl ρ∗

∂ 2 u∗ ∂t∗2

(3.9)

[σ ∗ ]Γ∗ = [E ∗ grady∗ u∗ ]Γ∗ = 0

(3.10)

[u∗ ]Γ∗ = 0

(3.11)

with: Pl =

ρc lc2 Ec t2c

Typically the time tc is linked to the period of the wave, or to its pulsation ωc by tc = 1/ωc . The physical significance of the dimensionless number Pl , the value of P using lc as the characteristic length, should be clarified. We can anticipate that the effective elastic modulus E eff , if it exists, is of the order of magnitude of the characteristic modulus Ec . The wave velocity is then: c=O

Ec ρc

92


and the wavelength λ for pulsations of order ωc is: 2π Ec Ec λ=O = O 2πtc ωc ρ c ρc Finally, Pl is the squared product of the wavenumber (2π/λ) and the length of the geometric period: 2 2πlc (3.12) Pl = O λ We will again look for a displacement u∗ of the form: u∗ (x∗ , y ∗ , t∗ ) = u∗(0) (x∗ , y ∗ , t∗ ) + εu∗(1) (x∗ , y ∗ , t∗ ) + · · ·

(3.13)

with x∗ = εy ∗ , where ε = lc /Lc and u∗(i) are periodic with respect to the local variable y ∗ , of period 1. Before beginning any homogenization, we must evaluate Pl as function of powers of ε. Different values of Pl can in fact be imagined, which reveals whether the situation can be homogenized or not. We will begin with the local description which leads to an equivalent macroscopic description of the dynamics. This situation corresponds to a Pl = O(ε2 ). Then we will consider values close to Pl = O(ε3 ) which lead to a macroscopic description which is steady state to first order of approximation, the case investigated in section 3.2, and finally Pl = O(ε) which corresponds to a local description which cannot be homogenized. 3.3.2.1. Macroscopic dynamics: Pl = O(ε2 ) 3.3.2.1.1. Normalization We are looking for the local description corresponding to macroscopic dynamics. It must be homogenizable, and so the geometry and disturbance must exhibit a separation of scales. We will assume that this is the case for the geometry. As far as the perturbation goes, λc /2π is a good candidate to define a characteristic macroscopic length Lc , as we will demonstrate in the following section. The separation of scales then requires that: 2πlc =ε1 λ and with (3.12): 2 2πlc = O(ε2 ), Pl = O λ

so that Pl = ε2 Pl∗ with Pl∗ = O(1)


93

We can reasonably hope that this estimate of Pl represents a homogenizable local description which will lead to a macroscopic description of the dynamics. This is proven below. We observe that the condition of separation of scales, in imposing a wavelength which is large relative to lc , implies as a consequence a frequency ω which must be sufficiently low: ω < ωdif (diffraction becomes significant for frequencies O(ωdif ) such that λ is of the order of lc ). Equation (3.9) becomes: divy∗ (E ∗ grady∗ u∗ ) = ε2 Pl∗ ρ∗

∂ 2 u∗ ∂t∗2

(3.14)

3.3.2.1.2. Homogenization Substituting the expansion (3.13) into the dimensionless equations, it is easy to see that the way the first two problems we solved, for unknowns u∗(0) and u∗(1) , are identical to those obtained for the steady state case. We therefore have: u∗(0) = u∗(0) (x∗ , t∗ ) u∗(1) (x∗ , y ∗ , t∗ ) = σ ∗(1) (x∗ , t∗ )

y∗

0

E ∗−1 (y ∗ )dy ∗ − y ∗

du∗(0) + u∗(1) (x∗ , 0, t∗ ) dx∗

with: σ ∗(1) (x∗ , t∗ ) = E ∗−1 −1

∂u∗(0) ∂x∗

By way of contrast, the next order is modified, with the appearance of the inertial term −ω ∗2 ρ∗ Pl∗ u∗(0) in the source term: ∂ ∂y ∗

E ∗ (y ∗ )(

∂ =− ∗ ∂x [E ∗ (y ∗ )(

∂u∗(2) ∂u∗(1) + ) ∗ ∂y ∂x∗

2 ∗(0) ∂u∗(1) du∗(0) ∗ ∗∂ u + ) + ρ P E (y )( l ∂y ∗ dx∗ ∂t∗2

∗

∗

∂u∗(2) ∂u∗(1) + )]Γ∗ = 0 ∂y ∗ ∂x∗

[u∗(2) ]Γ∗ = 0 Once again we find an equation analogous to equation (2.10), where W ∗ is the inertial term. Setting the mean of the source to zero, this leads us to the compatibility condition which gives the macroscopic description: ∗(0) d ∂ 2 u∗(0) ∗−1 −1 du E = ρ∗ Pl∗ (3.15) ∗ ∗ dx dx ∂t∗2

94


in dimensionless form, with: x∗ =

d dX

X E ω ρ u(0) u , E∗ = , ω∗ = , ρ∗ = , u∗(0) = = + O(ε) Lc Ec ωc ρc uc uc

∂2u du Ec t2c ¯ ρPl∗ 2 + O(ε) E −1 −1 = 2 dX ρ c Lc ∂t

and since: Pl = O

2πlc λ

2

= ε2 Pl∗ = O(ε2 )

we have: PL = O

2πLc λ

2 = O(1)

It follows that: d dX

du ∂ 2u ¯ E −1 −1 = ρ 2 + O(ε) dX ∂t

The return to dimensional variables then occurs without ambiguity. In the next sections, the dimensionless numbers will be taken as equal to their order ε estimate (which is equivalent to taking Pl∗ = 1). 3.3.2.1.3. Comments – The estimate does indeed correspond to a homogenizable situation which leads to a macroscopic description of the dynamics. – The effective elastic modulus to be used in the dynamic regime is the same as that in the steady state regime! – The effective density is the mean volume of the local density. – The dynamic description incorporates the steady state situation as a special case. We just need to set ω ∗ = 0. – The macroscopic description is an approximation of order O(ε). – The considerations in section 3.2 about the physical meaning of the macroscopic quantities still apply here.


95

3.3.2.2. Steady state: Pl = O(ε3 ) The normalization of equation (3.9) is obvious: divy∗ (E ∗ grady∗ u∗ ) = ε3 ρ∗

∂ 2 u∗ ∂t∗2

with the relations at the discontinuities remaining unchanged. It is clear that now, up to third order, the problems to be solved are identical to those obtained in section 3.2 for the steady state case. There is now an equivalent macroscopic description given by (3.8): d dx∗

E

∗−1 −1 du

∗(0)

dx∗

=0

As with the other macroscopic descriptions obtained up to now, this is only an approximation. The investigation of the next order (the fourth problem), gives a second approximation of order ε. As can easily be anticipated, this approximation includes an inertial term. For Pl = O(εp ), p 2, the dynamics appear at the (p−2)th order of approximation. 3.3.2.3. Non-homogenizable description: Pl = O(ε) Again the normalization is clear: divy∗ (E ∗ grady∗ u∗ ) = ερ∗

∂ 2 u∗ ∂t∗2

with the relations at the discontinuities remaining unchanged. But now only the first problem is the same as that obtained above, with: u∗(0) = u∗(0) (x∗ , t∗ ) The dynamics appear in the second problem, which can be written: ∂ ∂y ∗

E ∗ (y ∗ )(

[E ∗ (y)(

∂u∗(1) du∗(0) + ) ∂y∗ dx∗

= ρ∗

∂ 2 u∗(0) ∂t∗2

∂u∗(1) ∂u∗(0) + )]Γ∗ = 0 ∂y ∗ ∂x∗

[u∗(1) ]Γ∗ = 0 where u∗(1) is 1-periodic in y ∗ . The first equation is the conservation of a periodic quantity, and includes the source term ρ∗ ∂ 2 u∗(0) /∂t∗2 . The compatibility condition

96


implies that this term must have a mean of zero (Fredholm alternative): ρ∗

∂ 2 u∗(0) =0 ∂t∗2

so, since ρ∗ > 0: ∂ 2 u∗(0) =0 ∂t∗2

4

3

2

1

c 2 Pl = ( 2$l )

Di&raction Not homogenizable

Static 2$

1

À Lc =

2$

lc À lc

2

1

1

lc

2

c 2 PL = ( 2$L )

Dynamic 2$

= Lc = 1 lc À lc

Figure 3.8. Macroscopic descriptions that may or may not be valid depending on the values of Pl or PL

This result is impossible since ∂ 2 u∗(0) /∂t∗2 is O(1) by construction. The estimate Pl = O(ε) is a non-homogenizable description. It corresponds to: Pl = O

2πl λ

2 = O(ε)

and hence to: lc λ = √ Lc ε The dynamic excitation does not fulfill the condition of separation of scales. To conclude, the different situations are shown in Figure 3.8 as a function of the values of Pl . We observe that the richest macroscopic description, which corresponds to dynamic behavior Pl = O(ε2 ), lies at the limit of the homogenizable situations.


97

3.3.3. Comments on the different possible choices for the spatial variables In order to analyze the previous example we transformed the dimensional spatial variable X into dimensionless spatial variables x∗ = X/Lc and y∗ = X/lc . In addition, the normalization was carried out by adopting the microscopic viewpoint. The problem was then examined in the space of dimensionless variables, with the return to dimensional variables being carried out at the end of the process. In the literature, the change into the variables x∗ and y ∗ is often omitted. The treatment is carried out directly in a system of variables x and y, where in general x refers to the normal unit of length, the meter. Alternatively the normalization is carried out by adopting either the micro- or macroscopic viewpoints. The use of these different approaches, although equivalent, is sometimes a source of confusion. It is for this reason that we will now revisit these different methods. We recall that variables x∗ and y ∗ are particularly well suited to the analysis of problems with a double length scale because, by construction, x∗ is the measure of the distance X when using the distance Lc as unit length, and y∗ is the measure of the same distance X using the distance lc = εLc as the unit length. Thus: – x∗ varies by 1 over the macroscopic length Lc (and hence by ε over lc ); – y ∗ varies by 1 over the microscopic distance lc (and hence by ε−1 over lc ). We also note that as a measure of distance in some systems of units, x and y are both dimensionless variables. We will choose x for metric value X, and will designate c and lc = εL c as the metric values of lengths Lc and lc . Denoting a respectively L meter by “1m ”, we have the following: c 1m = x 1m X = x∗ Lc = x∗ L and: X = y ∗ lc = y ∗ lc 1m = x 1m = yε 1m whence it follows that: x y c = ∗ =L ∗ x y which shows that the variables x and y are homothetic to x∗ and y ∗ . We note that here y is a measurement of X in ε m (for example in millimeters for ε = 10−3 , etc.). From these we deduce that the derivative operator can take the following equivalent forms: 1 ∂ 1 ∂ ∂ = = ∂X Lc ∂x∗ 1m ∂x

98


or: ∂ 1 ∂ 1 ∂ = = ∗ ∂X lc ∂y ε1m ∂y To illustrate this we will return to the preceding problem in the dynamic regime. We will return to the initial equation, written for convenience in the harmonic regime: divX (E gradX u) = ρω2 u which can also be written in terms of the variable y ∗ (microscopic viewpoint): 1 divy∗ (E grady∗ u) = ρω 2 u (εLc )2

(3.16)

or alternatively, in terms of x∗ (macroscopic viewpoint): 1 divx∗ (E gradx∗ u) = ρω 2 u L2c

(3.17)

Physical analysis showed us that the dynamic regime was characterized by: Pl =

lc2 ρc ωc2 = O(ε2 ) Ec

or alternatively

PL =

L2c ρc ωc2 = O(1) Ec

Substituting these expressions into (3.17) and (3.16), and changing to the double variable operators, i.e. for the microscopic viewpoint: ∂ ∂y ∗

becomes

∂ ∂ +ε ∗ ∗ ∂y ∂x

and for the macroscopic viewpoint: ∂ ∂x∗

becomes

∂ ∂ + ε−1 ∗ ∂x∗ ∂y

we obtain the dimensionless formulations established starting with the microscopic viewpoint (already given in the previous section) and the macroscopic viewpoint. We can show that they of course lead to the same equations: (divy∗ + εdivx∗ ) E ∗ (grady∗ + εgradx∗ )u∗ = ε2 ρ∗ ω ∗2 u∗ or: (divx∗ + ε−1 divy∗ ) E ∗ (gradx∗ + ε−1 grady∗ )u∗ = ρ∗ ω ∗2 u∗


99

Also, using the expressions for the derivative operators, we can transform the equations by writing them in terms of the variable y (microscopic viewpoint): 1 divy (E grady u) = ρω 2 u (ε1m )2 or x (macroscopic viewpoint): 1 1m 2

divx (E gradx u) = ρω 2 u

As before, these two equations are normalized in order to describe the dynamic regime. For the mathematical treatment, the unit (1m ) is neutral (since all of the variables and parameters are expressed in the metric system) and so we can abstract ourselves from it. Thus we obtain the formulations in x and y resulting from the microscopic and macroscopic viewpoints: (divy + εdivx ) E(grady + εgradx )u = ε2 ρω 2 u or: (divx + ε−1 divy ) E(gradx + ε−1 grady )u = ρω 2 u which, again, are the same. We observe that the use of variables x and y is inconvenient because we lose the unit variation over the micro- or macroscopic distances. The advantage is we can continue to use the normal system of units (metric), and maintain the dimensional physical parameters throughout the treatment. At the end of the process all that needs to be done is to restore the meter as the unit. In other words, replace the value x with the distance X in order to obtain the dimensional formulation. We also note that the equations in x∗ , y ∗ or x, y are formally identical and lead to an identical treatment. As a final example, consider the quasi-static case corresponding to: Pl =

lc2 ρc ωc2 = O(ε3 ) Ec

or alternatively

PL =

L2c ρc ωc2 = O(ε) Ec

The normalizations are, in terms of the variable y ∗ (microscopic viewpoint): 1 divy∗ (E grady∗ u) = ερω 2 u (εLc )2 and, in terms of the variable x∗ (macroscopic viewpoint): 1 divx∗ (E gradx∗ u) = ερω 2 u L2c

100


Transforming the derivative operators, we find in terms of the variable y (microscopic viewpoint), and after canceling out the meter units: divy (E grady u) = ε3 ρω 2 u and, in terms of the variable x (macroscopic viewpoint): divx (E gradx u) = ερω 2 u After introducing derivative operators for the double variables, we again reach the same conclusions about the equivalence of the different approaches.

3.4. Expressing problems within the formalism of multiple scales The above example shows the general approach to be taken in order to establish various behaviors depending on the assumptions made. However, when a macroscopic description is sought for a real phenomenon within a given material, one of the difficulties is that of expressing the assumptions within the formalism of homogenization, in accordance with the problem under investigation. In the previous example, the question would be the following: if a material (for example a soft rock) has the following characteristic values: lc = 1 mm, Ec = 8 × 109 Pa, ρc = 2 × 103 kg/m3 , and cycles of testing at a frequency of 3 kHz are performed on a lattice of size H = 10 cm, which of the models that we obtained is the appropriate one to use?

3.4.1. How do we select the correct mathematical formulation based on the problem at hand? The macroscopic description will only be valid if the physics at the microscopic scale is described correctly. The physical analysis of the problem is thus a crucial stage that must occur before the process of homogenization. We have seen that dimensional analysis is an extremely useful tool for carrying out this process correctly. The problem is expressed in dimensionless form and, in order to correctly account for the importance of each term, the dimensionless numbers are expressed in powers of ε. This normalization phase is a key point in the process because that is where the physics of the phenomenon is taken into account. We emphasize that normalizing the dimensionless numbers in terms of powers of ε ensures that the various physical effects are accounted for to the same order, independent of the value of ε 1. Thus when a description is normalized it retains the nature of the physics that applies to the situation, but does not contain any reference to the effective value of ε which, although small, is still not zero.


101

Nevertheless, it is rare that the normalization follows naturally from the problem under consideration. In particular, when several small parameters are involved (ratios of properties, characteristic times, etc.), several possibilities are available and one should be chosen which applies to the situation being examined. Examples include bituminous concretes, whose behavior varies strongly with temperature and frequency of the load [BOU 89b; BOU 90] (see Chapter 9), or cement pastes which change from a fluid to solid state when they set [BOU 95]. We will show later that this difficulty can be overcome by analyzing the value taken by the scale ratio εr in the actual problem. This idea is clear in the previous example where the choice of model depends on the value of Pl as a function of ε. For the problem in question, with the numerical values given above, Pl can be estimated objectively: Pl =

ρc lc2 2 103 (10−3 )2 = 10−4 Ec t2c 8 109 (2π 3 103 )−2

However ε is not specified. We should also point out that if we assume ε to be infinitely small, this is equivalent to considering Pl = O(1), which is a situation that cannot be homogenized! Also, considering an arbitrary value of ε to give a scale to Pl is equivalent to making an arbitrary choice in the constitutive model. To avoid this impasse we are therefore forced to come up with a realistic estimate of ε for the problem being considered. 3.4.2. Need to evaluate the actual scale ratio εr The difficulty here is the gulf between: – the mathematical view, where ε = lc /Lc → 0 is infinitely small and the macroscopic description in this limit is infinitely accurate, corresponding to heterogenities which are infinitely small compared to the macroscopic scale, or alternatively to macroscopic dimensions which are infinitely large compared to the heterogenities; – the physical reality where this ideal situation is not reached because the size of the REV is finite (lc = 0) and the macroscopic scale is not infinite (Lc = ∞) so that the actual scale ratio takes a value which is small but non-zero (0 < εr 1). We can reconcile these two viewpoints by evaluating εr . Indeed if εr can be estimated, the dimensionless numbers of the real problem can be evaluated in terms of powers of εr . Thus we can define a normalization which is consistent with the physics of the problem. If, with this normalization, we carry out homogenization, we obtain a macroscopic description in which all the physical mechanisms act with the same strengths as in the actual problem. Because of this, the problem being considered is only an imperfect example (for ε = εr ) of the macroscopic description we have

102


developed, with the discrepancy being smaller when εr is small, in other words when the separation of scales is clear. In this case, the zero-order description matches the actual behavior up to order O(εr ). To summarize, there are two reasons we need to evaluate εr : the correct description of the local physics and the estimation of error in the macroscopic description.

3.4.3. Evaluation of the actual scale ratio εr For a given problem, lc is known for the medium, but the characteristic macroscopic size Lc which, as we have just seen, is crucial for selection of the correct model, is one of the unknowns. The literature is still rather unclear on this issue: this dimension is often associated with the size of the medium under study but, depending on the problem of interest, Lc might alternatively depend on the boundary conditions imposed or on a characteristic dimension of phenomenon such as a wavelength, or a thickness of a viscous layer, etc. In order to evaluate Lc (and εr ), we will follow the approach proposed by [BOU 89b; 89a] which consists of observing that the process of homogenization must necessarily lead to a quantity – in the case considered above, the displacement u(0) – with the following dimensional form: u(0) (X) + εu(1) (X, ε−1 X) + ...

with

O(u(0) ) = O(u(1) )

Turning the problem around, we can say that results of the homogenization will only be applicable to the real situation if this (necessary) condition is satisfied when ε takes the value εr . In other words if the variations in u(0) are effectively negligible (i.e. O(εr )) over the period. If we consider for example the growth of u(0) over a period in the direction X1 , we must therefore necessarily have: |u(0) (X1 + lc ) − u(0) (X1 )| O(εr |u(0) (X1 )|) On the macroscopic scale lc is very small and we can write: (0) ∂u (0) −1 |u | O(εr ) lc . ∂X1 This gives an underestimation of εr . However since εr is a measure of macroscopic accuracy, the optimum value is the smallest one that is permissible, which means we can write: (0)

ε r lc

| ∂u ∂X1 | |u(0) |

or

Lc

|u(0) | (0)

| ∂u ∂X1 |

(3.18)


103

In the general case where the displacement field is three-dimensional, we have: ⎛ εr lc max ⎝

(0)

⎞

∂ui ∂Xj | ⎠ (0) |uj |

|

⎛

Lc

and

⎞

(0) |u | min ⎝ i(0) ⎠ ∂u | ∂Xi j |

Locally, an order of magnitude for εr is thus given by relative variation of the displacement field over a period. This is equivalent to the estimate that would be obtained by dimensional analysis carried out directly on the macroscopic scale: the slower (or faster) the spatial variations in u(0) the larger (or smaller) Lc is, and the “smaller” (larger) εr is (in other words the accuracy is greater (or smaller)). We note that εr depends on the geometry of the field, and because of this it is not generally constant in the material, but can vary depending on load, boundary conditions, etc. Our estimate of (3.18) can answer the questions of accuracy and validity of the zeroorder macroscopic description. We will now give an evaluation of εr in several familiar situations. 3.4.3.1. Homogenous treatment of simple compression The displacement in a sample of height H takes the form (Figure 3.9): u(0) = aX, (0) = a, so that: from which it follows that O(u(0) ) = aH and ∂u ∂X Lc =

|u(0) | (0) | ∂u ∂X |

=

aH =H a

and

εr =

lc H

It follows that an accuracy of order 10% for the constitutive law requires samples with dimensions which are around 10 times larger than the size of the heterogenities. u(0)

u(0)

Lc

Figure 3.9. Estimate of the physical scale ratio εr in the case of simple compression: εr = O(lc /Lc )

104


3.4.3.2. Point force in an elastic object This is a case where the value of εr is not constant in the medium. In fact the displacement field varies as ∼ 1/r2 (Figure 3.10), which gives: Lc

|u(0) | (0) | ∂u | ∂X

=

r 2

and

εr = 2

lc r

P

r

Figure 3.10. Estimate of the physical scale ratio εr for a point source in a porous medium: εr = 2lc /r

where r is the distance to the point force. From this we can deduce that close to the point force is applied, the phenomenon is not homogenizable. Taking into account the effects of the microstructure, it becomes homogenizable beyond a radius R ≈ 10lc . The simple continuum description becomes acceptable at distances greater than 200lc (with an accuracy in the order of a few percent). 3.4.3.3. Propagation of a harmonic plane wave in elastic composites The displacement created by a plane wave in an infinite medium (Figure 3.11) has the form: u(0) (X, t) = |u(0) | exp(2iπ(t/T − X/λ)) and consequently: ∂u(0) = −(2iπ/λ)|u(0) | exp(2iπ(t/T − X/λ)) ∂X


105

whence: Lc =

|u(0) | (0) | ∂u ∂X |

=

λ 2π

and

εr =

2πlc λ

Again we find that the closer we get to the diffraction regime, the poorer the zeroorder description performs, so that we require higher order corrections [BOU 96b; AUR 05a]. For wavelengths shorter that 2πlc , homogenization is no longer applicable.

Lc u(0) r u(0) lc = r L c

Figure 3.11. Estimate of the physical scale ratio εr for wave propagation: εr = 2πlc /λ

3.4.3.4. Diffusion wave in heterogenous media For a harmonic plane wave of thermal diffusion, temperature takes the form: √ θ(0) (X, t) = |θ(0) | exp(2iπ(t/T − X i/δt )) where δ is the wavelength of thermal diffusion: λ δt = ρCω λ is thermal conductivity and ρC is heat capacity. As a result: √ √ ∂θ(0) = −(2 iπ/δt )|θ(0) | exp(2iπ(t/T − X i/δt )) ∂X so that: Lc =

|θ(0) | (0) | ∂θ ∂X |

=

δt 2π

and

εr =

2πlc δt

106


The same applies for diffusive waves as for elastic waves, and it is the wavelength which determines the macroscopic scale. 3.4.3.5. Conclusions to be drawn from the examples Let us return to the example of dynamic measurements of a rock sample. At a frequency of 3 kHz we have the following estimate of the wavelength: Ec 1 8.109 1 λ c = = 0.1m 2π ω ρc ω 2.103 2π3.103 which corresponds here to the height H of the sample. The value of εr is thus (with lc = 1 mm): εr =

2πlc lc = = 10−2 λ H

so that, following the value of Pl estimated above: Pl = 10−4 = ε2r It is thus natural that we should use the model corresponding to Pl = O(ε2 ), in other words the dynamic description. If, for the same material, tests were carried out at f = 300 Hz, we would have: P l =

ρc lc2 2 103 (10−3 )2 = 10−6 Ec t2c 8 109 (2π3 102 )−2

and

λ 1m > H 2π

In this case the macroscopic size is no longer defined by the wavelength but by the dimensions of the sample, and this time we have: εr =

lc = 10−2 H

so that:

P l = 10−6 = ε3r

which leads us to use the model for Pl = O(ε3 ), in other words the quasi-static description. Conversely, tests carried out at 30 kHz give Pl 10−2 , λ/2π √ 1 cm < H, and εr = 10−1 , so that Pl = O( εr ), putting the tests in the dynamic regime, at the limit of what is homogenizable. As for the diffractive regime where homogenization is no longer applicable due to the absence of a separation of scales, this is reached at frequencies where (λ/2π) lc so that εr 1.

PART TWO

Heat and Mass Transfer


Chapter 4

Heat Transfer in Composite Materials

4.1. Introduction The first works which made it possible to determine the effective conductivity through upscaling techniques were those of Rayleigh [RAY 92], for materials with low concentrations of inclusions and with a weak thermal conductivity contrast. The study of thermal transfer in heterogenous media has been the subject of many investigations, an excellent summary of which can be found in the works of Kaviany [KAV 99; 01]. Here we focus on thermal transfer in a composite material with two constituents. In order to obtain an equivalent macroscopic description we will use the method of multiple scale expansions described in Part 1. In order to apply this technique, we will assume the medium to be periodic. (The main results were obtained by [AUR 83; LEW 04; 05; and AUR 94a].) Section 4.2 focuses on thermal transfer with perfect contact between the constituents. A contact resistance will be introduced in section 4.3. The theoretical developments are systematically illustrated in the case of a bilaminate composite.

4.2. Heat transfer with perfect contact between constituents In this section we assume perfect contact between the constituents. Section 4.2.2 covers composites with thermal properties of the same order of magnitude. Section 4.2.3 treats composite media consisting of a matrix with inclusions of low thermal conductivity. Finally, section 4.2.4 considers thermal transfer for composites consisting of highly conductive inclusions embedded in a matrix.

110


4.2.1. Formulation of the problem The composite material is periodic, with a period Ω. Material a occupies the volume Ωa , and material b occupies the volume Ωb , with their interface being Γ (Figure 4.1). We assume that |Ωa |/|Ωb | = O(1); and Ω, Ωa and Ωb have the same characteristic size lc . In the transient regime, the heat transfer equation in each of the constituents α = a, b can be written: divX (λα gradX Tα ) = ρα Cα

∂Tα ∂t

(4.1)

Lc

lc

n

b

Sb Sa

a (a)

(b)

Figure 4.1. Composite medium: (a) macroscopic sample, (b) representative elementary volume (REV) with period Ω

By way of simplification, we assume the two conductivities to be isotropic and constant: λa = λa I and λb = λb I. We will also assume their densities ρα and their heat capacities Cα to be constant. In addition, it is convenient to consider heat transfer in the harmonic regime, with a temperature of the form: Tα (X)eiωt where the temperature Tα is a function of the spatial coordinate X only. When ω = 0, the heat transfer is permanent. Since the problems we will consider are all linear, the time dependence eiωt will be omitted throughout the following text. In general, heat transfer in the composite is governed by the Fourier law, which leads to the following system of equations: divX (λa gradX Ta ) = ρa Ca iωTa divX (λb gradX Tb ) = ρb Cb iωTb

in Ωa in Ωb

(4.2) (4.3)


111

with continuity in temperature and normal flux across Γ: T a − Tb = 0

over Γ

(4.4)

(λa gradX Ta − λb gradX Tb ) · n = 0

over Γ

(4.5)

where n is the outward unit normal to constituent a. We introduce into the system of equations (4.2-4.5), Ta = Tac Ta∗ ,

Tb = Tbc Tb∗ ,

ρa = ρac ρ∗a ,

λa = λac λ∗a ,

λb = λbc λ∗b ,

Ca = Cac Ca∗ ,

ρb = ρbc ρ∗b Cb = Cbc Cb∗

where the quantities with the subscript c and the superscript ∗ are respectively the characteristic and dimensionless magnitudes, and take the macroscopic view X = x∗ Lc . Then the dimensionless microscopic description can be written in the form: divx∗ (λ∗a gradx∗ Ta∗ ) = PL ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

L divx∗ (λ∗b gradx∗ Tb∗ ) = PL C ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (λ∗a gradx∗ Ta∗ − Lλ∗b gradx∗ Tb∗ ) · n = 0

(4.6) (4.7) (4.8)

over Γ∗

(4.9)

The above system involves three dimensionless quantities: PL =

ρac Cac ωc L2c |ρa Ca iωTa | taL = = |divX (λa gradX Ta )| λac tc

L=

λbc λac

C=

ρbc Cbc ρac Cac

which characterize heat transfer in the composite. The quantity PL is the inverse of the Fourier number. It represents the ratio between the characteristic times taL and tc in material a. taL = ρac Cac L2c /λac is the characteristic time associated with conductive transfer phenomena in the sample with characteristic dimension Lc . tc is the characteristic time associated with the harmonic or transient regime. In order to simplify the treatment, we will limit the following analysis to constituents with heat capacities of the same order of magnitude, C = ρbc Cbc /ρac Cac = O(1).

112


We will study harmonic regimes where the effects of macroscopic conduction will equal the thermal inertia, so that PL = O(1). If PL = O(εp ), p > 0, the macroscopic behavior is permanent to first order. If PL = O(εp ), p < 0, the regime is not homogenizable, because there is not enough of a separation in length scales between the wavelength of thermal diffusion and the size of constituents. A similar situation has already been analyzed in section 3.3.2.3 for the example of dynamic behavior of elastic composites. The interesting cases are the following: – Composite materials, without any restriction on connectivity of the constituents, and whose conductivities are of the same order of magnitude L = O(1) (Figure 4.2 (a) and (b), section 4.2.2). In this case, we recover the classical heat transfer equation [SAN 80; AUR 83]. It is simple to show that L = O(ε) is a particular case of the classic example with negligible λb . – Composite materials with high conductivity contrast, with the strongly conducting b phase embedded in a matrix formed by the a phase, in other words L = O(ε2 ) (Figure 4.2 (a) and (b), section 4.2.3). We obtain a non-standard description with memory effects. This means that transfer time in the macroscopic structure is of the same order of magnitude as the transfer time in weakly conducting inclusions [AUR 83]. – Composite materials with high conductivity contrast, with the strongly conducting b phase embedded in a matrix formed by the a phase, in other words L = O(ε−1 ) (Figure 4.2 (b), section 4.2.4). We once again recover a classical heat transfer equation, with the effective conductivity arising from local non-standard effects along the lines of those treated by Levy [LEV 90] for the flow of a Newtonian fluid in a porous medium; and Lewandowska and Auriault [LEW 04] and Lewandowska et al. [LEW 05] for non-linear diffusion problems.

lc

n

lc

b

Sb

n

Sa

b b n

Sa a a

a (a)

(b)

Figure 4.2. Unit cell of a composite material: (a) the two phases a and b are embedded, (b) inclusions buried in the matrix a. Phase a is connected, phase b is non-connected


113

4.2.2. Thermal conductivities of the same order of magnitude We assume PL = O(1) and L = O(1). Media a and b are either connected or non-connected (Figure 4.2). The thermal transfer time in Ωa and Ωb are of the same order of magnitude. The dimensionless description (4.6-4.9) can be written as: divx∗ (λ∗a gradx∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

(4.10)

divx∗ (λ∗b gradx∗ Tb∗ ) = ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

(4.11)

Ta∗ − Tb∗ = 0 over Γ∗

(4.12)

(λ∗a gradx∗ Ta∗ · n − λ∗b gradx∗ Tb∗ ) · n = 0

over Γ∗

(4.13)

We look for solutions to the unknowns Ta∗ and Tb∗ of the form: Ta∗ (x∗ , y∗ ) = Ta∗(0) (x∗ , y∗ , ) + εTa∗(1) (x∗ , y∗ ) + · · · ∗(0)

Tb∗ (x∗ , y∗ ) = Tb

∗(1)

(x∗ , y∗ ) + εTb

(x∗ , y∗ ) + · · ·

∗(i)

where Tα are Ω∗ -periodic in y∗ and with y∗ = ε−1 x∗ . The technique involves the introduction of these expansions into the dimensionless equations (4.10-4.13) and the identification of the powers of ε. We recall that, because of the two spatial variables and adoption of the macroscopic point of view, the spatial derivation takes the following form: gradx∗ −→ gradx∗ + ε−1 grady∗ Also, henceforth all un-subscripted variables defined for Ω∗ will take values subscripted by a in Ωa and by b in Ωb . For example, the conductivity λ∗ will be given as λ∗a in Ω∗a and λ∗b in Ω∗b . 4.2.2.1. Homogenization ∗(0)

4.2.2.1.1. Boundary value problem for Ta

∗(0)

and Tb

Equations (4.10-4.11) of order ε−2 , (4.12) of order ε0 and (4.13) of order ε−1 give ∗(0) ∗(0) the boundary value problem required by the Ω∗ -periodic fields Ta and Tb : divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

divy∗ (λ∗b grady∗ Tb

in Ω∗a

(4.14)

) = 0 in Ω∗b

(4.15)

114

Homogenization of Coupled Phenomena ∗(0)

Ta∗(0) − Tb

over Γ∗

=0

(4.16) ∗(0)

(λ∗a grad∗y∗ Ta∗(0) − λ∗b grad∗y∗ Tb

)·n=0

over Γ∗

(4.17)

This is a system of differential equations in terms of the variables y∗ , where x∗ is only present as a parameter. We note that T ∗(0) is only defined up to an additive constant, which is independent of y∗ . In these terms, the issue in the unit cell is a steady-state conduction problem without source terms. An obvious solution is a constant temperature field. Over a given period: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ , y∗ ) = T ∗(0) (x∗ )

To formalize the solution to this problem, as well as subsequent ones, we introduce the space V of regular functions θ defined over Ω∗ and Ω∗ -periodic, satisfying the condition of having zero mean, a condition which has no bearing on the problem at hand: 1 θ dΩ∗ = 0 (4.18) |Ω∗ | Ω∗ that has been introduced to solve the issue of the indeterminate nature of T ∗(0) . The bilinear, symmetrical positive definitive form: (θ1 , θ2 )V =

Ω∗

λ∗ grady∗ θ1 · grady∗ θ2 dΩ∗

(4.19)

defines a scalar product over V. Multiplying (4.14) and (4.15) by θ ∈ V, integrating by parts over each constituent, we find: λ∗a grady∗ Ta∗(0) · grady∗ θ dΩ∗ − λ∗a grady∗ Ta∗(0) θ · na dS ∗ + Ω∗ a

∂Ω∗ a

∗(0)

Ω∗ b

λ∗b grady∗ Tb

· grady∗ θ dΩ∗ −

∗(0)

∂Ω∗ b

λ∗b grady∗ Tb

θ · nb dS ∗ = 0

where na and nb are the outward normals to Ω∗a and Ω∗b . Decomposing the surface integrals into integrals over the interface Γ∗ and over the cell boundaries Sa∗ = ∂Ω∗a ∩ ∂Ω∗ and Sb∗ = ∂Ω∗b ∩ ∂Ω∗ , we have: ∂Ω∗ a

λ∗a grady∗ Ta∗(0) θ

∗

· na dS +

∂Ω∗ b

∗(0)

λ∗b grady∗ Tb

θ · nb dS ∗ =


Sa∗

Sb∗

λ∗a grady∗ Ta∗(0) θ · na dS ∗ +

∗(0) λ∗b grady∗ Tb θ

∗

· nb dS +

Γ∗

λ∗a grady∗ Ta∗(0) θ · na dS ∗ +

∗(0)

Γ∗

115

λ∗b grady∗ Tb

θ · nb dS ∗

Also, over Γ∗ (Figure 4.2), we have na = −nb = n, and using the condition of flux continuity (4.17), the sum of the two integrals over Γ∗ cancel out. As for the integrals over Sa∗ and Sb∗ , they come to zero because of their periodicity. Thus, the equivalent variational formulation of the initial problem over space V can be written, ∀θ ∈ V: Ω∗ a

λ∗a grady∗ Ta∗(0)

∗

· grady∗ θ dΩ +

Ω∗ b

∗(0)

λ∗b grady∗ Tb

· grady∗ θ dΩ∗ = 0

Alternatively, in a more compact form: ∀θ ∈ V,

(T

∗(0)

, θ)V =

Ω∗

λ∗ grady∗ T ∗(0) · grady∗ θ dΩ∗ = 0

(4.20)

Making use of the Lax-Milgram theorem, this formulation ensures the existence and uniqueness of solution T ∗(0) which belongs to V. Here it is clear that this solution is exactly zero. The solution to the initial problem can be obtained by adding a constant field which is independent of y∗ . After all this, we obtain T ∗(0) in the form: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ , y∗ ) = T ∗(0) (x∗ )

As a result of this, to first order the temperature, independent of y∗ , is therefore constant over the period. 4.2.2.1.2. Boundary value problem for T ∗(1) and the canonical problem P(λ∗a , λ∗b ) Equations (4.10-4.11) of order ε−1 , (4.8) of order ε and (4.9) of order ε0 lead to ∗(1) ∗(1) the following problem over the period Ω∗ for Ta and Tb , which are Ω∗ -periodic: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)

divy∗ (λ∗b (grady∗ Tb ∗(1)

Ta∗(1) − Tb

=0

+ gradx∗ T ∗(0) )) = 0

over Γ∗

in Ω∗b

(4.21) (4.22) (4.23)

116


(λ∗a (grady∗ Ta

+ gradx∗ T ∗(0) ) − ∗(1)

λ∗b (grady∗ Tb

+ gradx∗ T ∗(0) )) · n = 0

over Γ∗

(4.24)

This is a steady-state conduction problem, but different to the preceding problem in that there is a forcing term gradx∗ T ∗(0) . In order to solve this problem, we follow the same approach as before. Multiplying (4.21) and (4.22) by θ ∈ V, and integrating by parts and using the conditions on Γ∗ and the periodicity, the equivalent variational formulation to the initial problem over space V can be written: ∀θ ∈ V

(T ∗(1) , θ)V =

Ω∗

λ∗ grady∗ T ∗(1) · grady∗ θ dΩ∗ =

−

Ω∗

λ∗ gradx∗ T ∗(0) · grady∗ θ dΩ∗

(4.25)

In the same way as before, this formulation ensures the existence and uniqueness of the solution in V. The linearity of the problem makes it possible to decompose the solution into three elementary solutions t∗i (y∗ ) associated with unit macroscopic gradients in the three directions gradx∗ T ∗(0) = ei . In this way, the solution in V can be written t∗i (y∗ )∂T ∗(0) /∂x∗i . The solution to the initial problem can be obtained by adding a constant field T¯ ∗(1) which is independent of y∗ . We arrive at T ∗(1) in the form: T ∗(1) (x∗ , y∗ ) = t∗i (y∗ )

∂T ∗(0) + T¯ ∗(1) (x∗ ) ∂x∗i

The specific solutions t∗i are obtained by solving three distinct boundary value problems over the period, which form the canonical problem P(λ∗a , λ∗b ): ∗ ∂ ∗ ∂tai (λ ( + Iij )) = 0 in Ω∗a a ∂yj∗ ∂yj∗

(4.26)

∗ ∂ ∗ ∂tbi + Iij )) = 0 ∗ (λb ( ∂yj ∂yj∗

(4.27)

t∗ai − tbi = 0 (λ∗a (

in Ω∗b

over Γ∗

∗ ∂t∗bi ∗ ∂tbi + I ) − λ ( + Iij )).nj = 0 ij b ∂yj∗ ∂yj∗

(4.28) over Γ∗

(4.29)


1 |Ω∗ |

Ω∗ a

t∗ai dΩ∗

+

117

Ω∗ b

t∗bi dΩ∗

=0

(4.30)

For convenience, we combine the three solutions into a vector t∗ (y∗ ), writing: T ∗(1) (x∗ , y∗ ) = t∗ (y∗ ) · gradx∗ T ∗(0) + T¯ ∗(1) (x∗ )

(4.31)

and with the help of vector t∗ the three problems can be written in the following compact form: divy∗ (λ∗ (grady∗ t∗ + I)) = 0 in Ω∗ [t∗ ]Γ∗ = 0 over Γ∗ [λ∗ (grady∗ t∗ + I)]Γ∗ · n = 0

over Γ∗

t∗ = 0 where [φ∗ ]Γ∗ represents the step change in φ∗ at the interface Γ∗ and t∗ is defined by equation (4.30). 4.2.2.2. Macroscopic model Equations (4.10-4.11) of order ε0 , (4.12) of order ε2 and (4.13) of order ε lead us ∗(2) to the following boundary valve problem over the period of Ω∗ -periodic fields Ta ∗(2) and Ta : divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = ρ∗a Ca∗ iω ∗ T ∗(0) ∗(2)

divy∗ (λ∗b (grady∗ Tb

∗(1)

divx∗ (λ∗b (grady∗ Tb ∗(2)

Ta∗(2) − Tb

=0

∗(1)

+ gradx∗ Tb

in Ω∗a

(4.32)

in Ω∗b

(4.33)

))+

+ gradx∗ T ∗(0) )) = ρ∗b Cb∗ iω ∗ T ∗(0)

over Γ∗

(λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) )− ∗(2)

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)) · n = 0

over Γ∗

118


The existence of a periodic solution imposes a compatibility condition which can be establised by integrating (4.32) and (4.33) over Ω∗a and Ω∗b . First we note that: Ω∗ a

∗(2)

divy∗ (λ∗a (grady∗ Ta

Γ∗

since: Sa∗

∗(1)

+ gradx∗ Ta

))dΩ∗ =

∗(2)

λ∗a (grady∗ Ta

∗(1)

+ gradx∗ Ta

) · ndS ∗

(4.34)

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · ndS ∗ = 0

due to the periodicity of the fluxes. Similarly, since n is the outward normal to Ω∗a , we have: ∗(2) ∗(1) divy∗ (λ∗b (grady∗ Tb + gradx∗ Tb ))dΩ∗ = Ω∗ b

−

∗(2)

Γ∗

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

) · ndS ∗

(4.35)

As a result, using the continuity of the flux over Γ∗ , the sum of integrals (4.34) and (4.35) is zero. We also note that: grady∗ T ∗(1) + gradx∗ T ∗(0) = (grady∗ t∗ + I)gradx∗ T ∗(0) From this it follows, swapping the integration with respect to y∗ and derivation with respect to x∗ , that: Ω∗

divx∗ (λ∗ (grady∗ T ∗(1) + gradx∗ T ∗(0) ))dΩ∗ =

divx∗

(

Ω∗

λ∗ (grady∗ t∗ + I)dΩ∗ )gradx∗ T ∗(0)

All this leads us to the following macroscopic description: divx∗ (λeff∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eff iω ∗ T ∗(0)

(4.36)

where: λeff∗ ij

1 = ∗ |Ω |

∂t∗ λ∗a (Iij + ai∗ ) dΩ∗ + ∂yj Ω∗ a

∂t∗ λ∗b (Iij + bi∗ ) dΩ∗ ∂yj Ω∗ b

(4.37)


∗

∗ eff

(ρ C )

1 = ρ C = ∗ |Ω | ∗

∗

Ω∗ a

ρ∗a Ca∗

∗

dΩ +

Ω∗ b

119

ρ∗b Cb∗

dΩ

∗

(4.38)

The variational form (4.25) associated with the symmetry and positivity of the scalar product makes it possible to show the symmetry and positivity of the effective thermal condictivity λeff [AUR 83]. For an outline of this proof, which we will not discuss here, the reader is referred to Chapter 7, section 7.3.3 and Chapter 11, section 11.1.3.2, where the same arguments are used to show the symmetry and positivity of the permeability and elasticity tensors. Returning to the dimensional variables, the macroscopic model can be written: divX (λeff gradX T (0) ) = ρC

∂T (0) ∂t

(4.39)

¯ But T = T (0) + O(ε) and as a result, for the physical variable T , the model can be written: divX (λeff gradX T ) = ρC

∂T ¯ + O(ε) ∂t

(4.40)

¯ where O(ε) is a small term, of order ε relative to the other terms in the equality. The order 0 model is accurate up to ε. A result of the separation of length scales is that the presence of transient effects does not have any effect on the effective coefficients. Also, the form of the result that we have established for bi-composites can be applied to all types of composite where properties of the constituents are of the same order. 4.2.2.3. Example: bilaminate composite Using this method we will return to the case of the bilaminate composite that we considered in Chapter 1 (cf. Figure 4.3). The specific solutions t∗i are obtained by solving the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) over the period Ω∗ . The calculation is performed using dimensionless variables. The stratification perpendicular to e2 requires that the solutions t∗i are functions of y2∗ only. If we impose a macroscopic gradient, gradx∗ T ∗(0) = e1 , the problem can be written for t∗a1 and t∗b1 as: dt∗ d (λ∗a a1 )=0 ∗ dy2 dy2∗

in Ω∗a

d dt∗ (λ∗b b1∗ ) = 0 in Ω∗b ∗ dy2 dy2 t∗a1 − t∗b1 = 0

over Γ∗

120


e2 L2 l2

e1

(a)

L1 e2

1 (1 cb )

a

cb (b)

b

e1

Figure 4.3. Bilaminate composite material: (a) macroscopic structure, (b) dimensionless period Ω∗

λ∗a

dt∗a1 dt∗ − λ∗b b1∗ = 0, ∗ dy2 dy2

over Γ∗

The obvious solution to this is: t∗a1 (y2∗ ) = t∗b1 (y2∗ ) = 0 In the same way, with a macroscopic gradient along e3 : t∗a3 (y2∗ ) = t∗b3 (y2∗ ) = 0 If we impose a macroscopic gradient, gradx∗ T ∗(0) = e2 , the problem can be written for t∗a2 and t∗b2 as: dt∗ d (λ∗a a2 + 1) = 0 in Ω∗a ∗ dy2 dy2∗ ∗ d ∗ dtb2 (λ + 1) = 0 b dy2∗ dy2∗

t∗a2 − t∗b2 = 0

over Γ∗

in Ω∗b


λ∗a (

dt∗a2 dt∗ + 1) − λ∗b ( b1∗ + 1) = 0, ∗ dy2 dy2

121

over Γ∗

The solution to this satisfying the condition t∗a2 + t∗b2 = 0 (see equation 4.30) is:

t∗a2 =

cb (λ∗b − λ∗a ) cb λ∗a + (1 − cb )λ∗b

t∗b2 =

(cb − 1)(λ∗a − λ∗b ) ∗ cb (y − ) cb λ∗a + (1 − cb )λ∗b 2 2

y2∗ −

(1 + cb ) 2

in Ω∗a

in Ω∗b

Substituting these expressions into equation (4.82), we obtain the effective conductivity tensor in dimensionless form, which is consistent with the results of Chapter 1: eff eff λeff 12 = λ13 = λ23 = 0 eff λeff 11 = λ33 = cb λb + (1 − cb )λa

λeff 22 =

λb λa cb λa + (1 − cb )λb

4.2.3. Weakly conducting phase in a connected matrix: memory effects We still assume that PL = O(1), but we now consider a strong conductivity contrast: L = (λbc /λac ) = O(ε2 ). In addition, medium a is always connected, but medium b may be connected or non-connected (Figure 4.3). With these assumptions, we have: ρbc Cbc lc2 ρac Cac L2c =O λac λbc This indicates that thermal transfer time in the macroscopic structure is of the same order of magnitude as the transfer time in Ωb . Bearing in mind these estimates, the description in (4.6-4.9) becomes: div∗x∗ (λ∗a grad∗x∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗

in Ω∗a

ε2 div∗x∗ (λ∗b grad∗x∗ Tb∗ ) = ρ∗b Cb∗ iω ∗ Tb∗

in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (λ∗a grad∗x∗ Ta∗ − ε2 λ∗b grad∗x∗ Tb∗ ) · n = 0

(4.41) (4.42) (4.43)

over Γ∗

(4.44)

122


4.2.3.1. Homogenization ∗(0)


∗(0)

and Tb

Substituting the asymptotic expansions into the equations (4.41-4.44), equations (4.41) of order ε−2 and (4.44) of order ε−1 make up the boundary value problem for ∗(0) Ta (Ω∗ -periodic): divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 λ∗a grady∗ Ta∗(0) · n = 0

in Ω∗a

(4.45)

over Γ∗

which has an obvious solution of the form: Ta∗(0) (x∗ , y∗ ) = Ta∗(0) (x∗ ) This is obtained in the same way as in the previous section, with functions in space V now defined over Ω∗a (c.f. equations (4.19) and (4.18)). ∗(0)

The boundary value problem for Tb (4.42) and (4.43) of order ε0 : ∗(0)

divy∗ (λ∗b grady∗ Tb ∗(0)

Tb

= Ta∗(0)

(Ω∗ -periodic) is obtained from equations

∗(0)

) = ρ∗b Cb∗ iω ∗ Tb

in Ω∗b

(4.46)

over Γ∗

∗(0)

∗(0)

where Ta (x∗ ) is known. Setting Tb problem becomes:

∗(0)

= Ta

divy∗ (λ∗b grady∗ W ) = ρ∗b Cb∗ iω ∗ (Ta∗(0) + W ),

(x∗ ) + W , the boundary value in Ω∗b

W = 0 over Γ∗

(4.47) (4.48)

This problem is very different to the preceding case, because it involves both transient effects and a source term resulting from the temperature imposed at the boundary. To solve this problem, we introduce W, the space of complex θ-regular functions, defined over Ω∗b , Ω∗ -periodic, zero over Γ∗ , and with the Hermitian product: (θ1 , θ2 )W =

Ω∗ b

(grady∗ θ1 · grady∗ θ˜2 + θ1 θ˜2 ) dΩ∗


123

where θ˜ is the complex conjugate of θ. Multiplying (4.47) by θ˜ ∈ W, integrating by parts and making use of the condition on Γ∗ as well as the periodicity, we reach the equivalent formulation: ˜ dΩ∗ = ∀θ ∈ W, (λ∗b grady∗ W · grady∗ θ˜ + ρ∗b Cb∗ iω ∗ W θ) Ω∗

(W, θ)W = −

Ω∗

ρ∗b Cb∗ iω ∗ Ta∗(0) θ˜ dΩ∗

(4.49)

This formulation implies the existence and uniqueness of the solution. The last ∗(0) part depends linearly on the forcing term associated with Ta : ∗(0)

W (x∗ , y∗ ) = Tb

(x∗ , y∗ ) − Ta∗(0) (x∗ ) = −τ ∗ (y∗ , ω∗ )Ta∗(0) (x∗ ) ∗(0)

= 1. This is a complex function which where τ ∗ is the specific solution for Ta depends on ω ∗ . It is clear that τ ∗ (ω ∗ = 0) = 0 and τ ∗ (ω ∗ → ∞) = 1. To first order, the temperature is constant in the matrix, but not in the inclusions. ∗(1)


Equations (4.41) of order ε−1 and (4.44) of order ε0 make up the boundary value ∗(1) problem for the Ω∗ -periodic Ta : divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = 0 in Ω∗a λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = 0

(4.50)

over Γ∗

∗(0)

where gradx∗ Ta appears as a forcing term. This system can be studied using the space V associated with Ω∗a . Once again the linearity allows us to write the solution in the form: Ta∗(1) (x∗ , y∗ ) = m∗a (y∗ ) · gradx∗ Ta∗(0) + T¯ ∗(1) (x∗ ) where T¯ ∗(1) is an arbitrary function of x∗ , m∗a (y∗ ) is y∗ -periodic and has a mean of zero in Ω∗a . The specific solutions m∗ai are solutions to three distinct problems associated with the unit macroscopic temperature gradients gradx∗ T ∗(0) = ei : ∗ ∂ ∗ ∂mai ∗ ∗ (λa ( ∗ + Iij )) = 0 in Ωa ∂yj ∂yj

λ∗a (

∂m∗ai + Iij ).nj = 0 ∂yj∗

over Γ∗

(4.51)

(4.52)

124


1 Ω∗

Ω∗ a

m∗ai dΩ∗ = 0

(4.53)

which can be written in the following form: divy∗ (λ∗a (grady∗ m∗a + I)) = 0 in Ω∗a λ∗a (grady∗ m∗a + I) · n = 0

over Γ∗

m∗a = 0 We note that we have recovered the canonical problem P(λ∗a , λ∗b ) (4.26-4.30), where here we have λ∗b = 0, as a result of the assumption we have made about the contrast. 4.2.3.2. Macroscopic model ∗(2)

The boundary value problem for Ω∗ -periodic Ta of order ε0 and (4.9) of order ε:

is obtained from equations (4.6)

divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = ρ∗a Ca∗ iω ∗ Ta∗(0)

(4.54)

with continuity of normal flux across Γ∗ : ∗(0)

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = λ∗b grady∗ Tb

·n

(4.55)

∗(2)

The existence of a periodic solution Ta requires a compatibility condition which we establish by integrating (4.54) over Ω∗a . Using in succession the divergence theorem, conditions (4.55) and (4.46), we find: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) )) dΩ∗ Ω∗ a

=

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n dS ∗

Γ∗

=

∗(0)

λ∗b grady∗ Tb

Γ∗

=−

Ω∗ b

· n dS ∗ = −

∗(0) ρ∗b Cb∗ iω ∗ Tb

∗

dΩ = −

∗(0)

Ω∗ b

Ω∗ b


) dΩ∗

ρ∗b Cb∗ iω ∗ (1 − τ ∗ )Ta∗(0) dΩ∗


125

The volume average of (4.54) can then be written: ∗(0) divx∗ (λeff∗ ) = (ρC)eff∗ iω ∗ Ta∗(0) a gradx∗ Ta

(4.56)

which gives the macroscopic description, with the following effective coefficients: λeff∗ aij

1 = ∗ |Ω |

Ω∗ a

λ∗a (Iij +

(ρC)eff∗ = ρ∗ C ∗ −

1 |Ω∗ |

∂m∗aj ) dΩ∗ ∂yi∗ Ω∗ b

(4.57)

ρ∗b Cb∗ τ ∗ dΩ∗

(4.58)

Equation (4.56) represents the macroscopic behavior of the composite at constant ∗(0) frequency, to first order. In this equation, Ta represents the temperature in medium eff∗ ∗(0) the macroscopic heat flux. We note that λeff∗ is identical a and λa gradx∗ Ta a to that of a composite where material b is a perfect insulator and that the effective specific heat capacity is complex and depends on the frequency. This unusual property results from the local non-equilibrium in the inclusions. This follows from the similar transfer times in the inclusions and in the macroscopic volume, and leads to a memory effect. We note that in the steady-state regime the effective specific heat capacity is the mean of that of the two constituents, and that at high frequency, only the specific heat capacity of material a has an effect on the macroscopic scale. We write macroscopic model (4.56) in dimensional form and in the time domain:

(0)

(0) divX (λeff a gradX Ta ) = ρC

∂Ta ∂t

− ρb Cb

t

−∞

M (t − t )

where the memory function M (t) is the inverse Fourier transform of The properties of M can be found in Auriault [AUR 83].

(0)

∂ 2 Ta ∂t2

dt (4.59)

1 iω|Ω|

τ dΩ. Ωb

4.2.3.3. Example: bilaminate composite Let us return to the composite depicted in Figure 4.3. Medium a (the more conductive) is connected in directions e1 and e3 . The results derived above can therefore be applied to thermal fluxes in these directions. In this specific case, the solution to the boundary-value problem (4.51-4.53) over Ω∗ means that m∗a = 0. As a result, using equation (4.57), the effective dimensional thermal conductivity becomes: eff eff λeff a12 = λa13 = λa23 = 0 eff λeff a11 = λa33 = (1 − cb )λa

126


λeff a22 = 0 The function τ (y2∗ ) can be determined from the following system of equations: d ∗ dτ ) = ρ∗b Cb∗ iω ∗ (1 + τ ), ∗ (λb dy2 dy2∗ τ =0

at

y2∗ = 0 and

in 0 < y2∗ < cb

y2∗ = cb

After integration and return to dimensional variables: 1 |Ω|

τ dΩ = cb Ωb

tanh(i1/2 β) 1− , i1/2 β

β=

ωρb Cb λb

1/2

cb h 2

The function M can be found by inverse Fourier transform: M (t) = 8cb

2 2 ∞ e−(2p+1) π α/4

p=0

(2p +

1)2 π 2

,

α=

4λb t ρb Cb c2b h2

4.2.4. Composites with highly conductive inclusions embedded in a matrix We still assume PL = O(1), and also a high conductivity contrast. Compared to the preceding situation, the more conductive phase is embedded in the matrix, so that L = (λac /λbc ) = O(ε−1 ). Medium a is therefore connected and b is not (Figure 4.4). Here we look at a specific traditional case considered in section 4.2.2. However λb λa and the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) with λ∗b −→ ∞ is singular. The method of asymptotic expansions makes it possible to remove that singularity. A similar treatment can be found in [LEW 04] and [LEW 05] for solute diffusion. With these assumptions, the description (4.6-4.9) becomes: divx∗ (λ∗a gradx∗ Ta∗ ) = ρ∗a Ca∗ iω ∗ Ta∗ divx∗ (λ∗b gradx∗ Tb∗ ) = ερ∗b Cb∗ iω ∗ Tb∗

in Ω∗a in Ω∗b

Ta∗ − Tb∗ = 0 over Γ∗ (ελ∗a gradx∗ Ta∗ − λ∗b gradx∗ Tb∗ ) · n = 0

over Γ∗


e2

127

S a2

l2

n

S a1

b b n a a

e1 l1

Figure 4.4. Period Ω∗ of a composite material consisting of inclusions Ω∗b embedded in a matrix Ω∗a

4.2.4.1. Homogenization ∗(0)

4.2.4.1.1. Boundary value problems for Ta

∗(0)

and Tb

∗(0)

The boundary value problems for Ω∗ -periodic Tb ∗(0)

divy∗ (λ∗b grady∗ Tb ∗(0)

λ∗b grady∗ Tb

are:

) = 0 in Ω∗b

·n=0

over Γ∗

The solution has the form: ∗(0)

Tb

∗(0)

(x∗ , y∗ ) = Tb

(x∗ )

It can be obtained as before using space V associated with Ω∗b (see equations (4.19) ∗(0) and (4.18)). The boundary value problem for Ω∗ -periodic Ta can be written: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)

Ta∗(0) = Tb

(x∗ )

in Ω∗a

over Γ∗

This is a conduction problem without source terms, where the temperature imposed at the boundary of the inclusions is uniform. The obvious solution is: ∗(0)

Ta∗(0) (x∗ , y∗ ) = Tb

(x∗ ) = T ∗(0) (x∗ )

To first order, the temperature is constant across the whole period.

128


4.2.4.1.2. Boundary value problem for Tb ∗(1)

The Ω∗ -periodic temperature Tb ∗(1)

divy∗ (λ∗b (grady∗ Tb ∗(1)

λ∗b (grady∗ Tb

satisfies:

+ gradx∗ T ∗(0) )) = 0

+ gradx∗ T ∗(0) ) · n = 0

in Ω∗b

(4.60)

over Γ∗

where gradx∗ T ∗(0) appears as a forcing term. Multiplying the two terms from (4.60) ∗(1) by (y · gradx∗ T ∗(0) + Tb ), integrating over Ω∗b , then by parts, and using the ∗ condition over Γ , we find: ∗(1) λ∗b (grady∗ Tb + gradx∗ T ∗(0) )2 dΩ∗ = 0 Ω∗ b

∗(1)

The positivity of λ∗b implies that grady∗ Tb ∗(1)

Tb

+ gradx∗ T ∗(0) = 0, and then:

(x∗ , y∗ ) = −y∗ · gradx∗ T ∗(0) + T¯∗(1) (x∗ )

(4.61)

where T¯ ∗(1) is an arbitrary function of x∗ . Thus we obtain an explicit solution for ∗(1) Tb . We note that the gradient is order 0 in temperature: ∗(1)

grady∗ Tb

+ gradx∗ T ∗(0) = 0

as a consequence of our assumption of a high conductivity in the inclusion. ∗(1)

4.2.4.1.3. Boundary value problem for Ta ∗(1)

The Ω∗ -periodic temperature Ta

satisfies:

divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)

Ta∗(1) = Tb

= −y∗ · gradx∗ T ∗(0) + T¯∗(1) (x∗ )

(4.62)

over Γ∗

where gradx∗ T ∗(0) appears as a forcing term. T¯∗(1) (x∗ ) can also be considered as ∗(1) a forcing term, and the obvious solution associated with it is Ta = T¯ ∗(1) (x∗ ). The above boundary-value problem is non-singular and the solution can be written in the form: Ta∗(1) (x∗ , y∗ ) = s∗a (y∗ ) · gradx∗ T ∗(0) + T¯ ∗(1) (x∗ ) The vector s∗a (y∗ ) is the solution to the following boundary-value problem: divy∗ (λ∗a (grady∗ s∗a + I)) = 0 s∗a + y∗ = 0 over Γ∗

in Ω∗a

(4.63) (4.64)


129

4.2.4.2. Macroscopic model ∗(2)

Let us consider the equations satisfied by Ta

∗(3)

and Tb

:

divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = ρ∗a Ca∗ iω ∗ T ∗(0) ∗(3)

∗(2)


+ gradx∗ Tb

∗(2)

divx∗ (λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

in Ω∗a

(4.65)

in Ω∗b

(4.66)

)+ )) = ρ∗b Cb∗ iω ∗ T ∗(0)

with the condition of normal flux continuity over Γ∗ : ∗(3)

λ∗b (grady∗ Tb

∗(2)

+ gradx∗ Tb

)·n=

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n

over Γ∗

(4.67)

We integrate (4.65) and (4.66) over Ω∗a and Ω∗b respectively. After applying the divergence theorem, using (4.67) and dividing by |Ω∗ |, we find: 1 ∗ ∗(1) ∗(0) ∗ λa (grady∗ Ta + gradx∗ T ) dΩ + divx∗ |Ω∗ | Ω∗ a 1 divx∗ |Ω∗ |

Ω∗ b

∗(2) λ∗b (grady∗ Tb

+

∗(1) gradx∗ Tb )

dΩ∗

ρ∗ C ∗ iω ∗ T ∗(0)

=

(4.68) ∗(2)

We cannot make direct use of the macroscopic behavior in (4.68) because Tb is not known. We therefore need to to transform (4.68). The heat fluxes in each constituent can be written: q∗a = λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) ∗(2)

q∗b = λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)

They satisfy: ∗ = qαk

∗ ) ∂(yk∗ qαi ∗ ∂yi

where

q∗α = divy∗ (y∗ ⊗ q∗α )

130


Thus the integrals in (4.68) can be written, after use of the divergence theorem, as: ∗ ∗ ∗ ∗ ∗ qa dΩ = (y ⊗ qa ) · ndS + (y∗ ⊗ q∗a ) · ndS ∗ Ω∗ a

Ω∗ b

Γ∗

q∗b dΩ∗ = −

∂Ω∗

Γ∗

(y∗ ⊗ q∗b ) · ndS ∗

where n is the unit external normal to Ω∗a , and ∂Ω∗ is the contour Ω∗ (Figure 4.4). We emphasize that these expressions make use of the fact that the inclusions are buried in the matrix. Substituting these expressions into (4.68), and making use of the condition of flux continuity over Γ∗ : (q∗a − q∗b ) · n = 0 we find that: divx∗ (λ+eff∗ gradx∗ T ∗(0) ) = ρ∗ C ∗ iω ∗ T ∗(0)

(4.69)

where the effective conductivity λ+eff∗ is defined by: ∂s∗aj 1 ∗ = λ ( + δij )yk∗ ni dS ∗ λ+eff∗ kj |Ω∗ | ∂Ω∗ a ∂yi∗ The macroscopic dimensional model can be written to first order of approximation as: divX (λ+eff gradX T ) = ρC

∂T ¯ + O(ε) ∂t

(4.70)

Thus we recover the classical model of conduction. To first order, the effective conductivity is that which would be obtained by assuming the inclusion infinitely conductive (so that it does not support any temperature gradient and the flux across it is constant). Our treatment gives the solution to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) when λ∗b −→ ∞. 4.3. Heat transfer with contact resistance between constituents In this section we investigate the influence on the equivalent macroscopic model of having imperfect contact between the constituents, which leads to a contact resistance (Figure 4.5) [AUR 94b]. The system describing thermal transfers is the same as introduced above. Over Γ the continuity of temperatures is replaced by a Biot boundary condition (in this section we will revert to explicit notation of the time t): divX (λa gradX Ta ) = ρa Ca

∂Ta , ∂t

in Ωa

(4.71)


131

lc

n

Sb

Sa

b

a

contact resistance

Figure 4.5. Period Ω of a composite material with contact resistance

∂Tb , ∂t

in Ωb

(4.72)

λa gradX Ta · n = −h(Ta − Tb )

over Γ

(4.73)

divX (λb gradX Tb ) = ρb Cb

(λa gradX Ta − λb gradX Tb ) · n = 0 over Γ

(4.74)

We recall that n is the normal unit of Γ exterior to Ωa . The scalar h > 0 measures the contact conductivity. The contact resistance is characterized by 1/h. We introduce into the equations (4.71-4.74): Ta = Tac Ta∗ ,

Tb = Tbc Tb∗ ,

ρa = ρac ρ∗a ,

ρb = ρbc ρ∗b

λa = λac λ∗a ,

λb = λbc λ∗b ,

Ca = Cac Ca∗ ,

Cb = Cbc Cb∗ ,

h = hc h∗

and take the macroscopic viewpoint X = x∗ Lc . Then the microscopic description, in dimensionless form, can be written: divx∗ (λ∗a gradx∗ Ta∗ ) = PL ρa Ca∗

∂Ta∗ , ∂t∗

Ldivx∗ (λ∗b gradx∗ Tb∗ ) = PL Cρ∗b Cb∗

in Ω∗a

∂Tb∗ , ∂t∗

λ∗a gradx∗ Ta∗ · n = −BL h∗ (Ta∗ − Tb∗ )

in Ω∗b

over Γ∗

(λ∗a gradx∗ Ta∗ − Lλ∗b gradx∗ Tb∗ ) · n = 0 over Γ∗

(4.75)

(4.76) (4.77) (4.78)

132


This description involves four dimensionless numbers: ∂Ta | ρac Cac L2c ∂t PL = = |divX (λa gradX Ta )| λac tc |ρa Ca

L=

λbc λac

C=

ρac Cac ρbc Cbc

BL =

|h(Ta − Tb )| h c Lc = |λa gradX Ta · n| λc

For simplicity, we assume the following orders for the usual parameters: PL = O(1), L = O(1) and C = O(1). We still need to estimate the value of the Biot number BL , which depends on the conductivity h. The situations of interest are: BL = O(εp ),

p = −2, −1, 0, 1, 2

They correspond to different macroscopic models which we denote respectively by I, II, III, IV and V. The criteria for choosing between these different models will be discussed in section 4.3.7. We will look for solutions to Ta∗ and Tb∗ of the form: Ta∗ (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , y∗ , t∗ ) + εTa∗(1) (x∗ , y∗ , t∗ ) + · · · ∗(0)

Tb∗ (x∗ , y∗ , t∗ ) = Tb ∗(i)

where Tα

∗(1)

(x∗ , y∗ , t∗ ) + εTb

(x∗ , y∗ , t∗ ) + · · ·

(4.79) (4.80)

are Ω∗ -periodic in y∗ and y∗ = ε−1 x∗ . As a result, we have:

gradx∗ −→ gradx∗ + ε−1 grady∗ 4.3.1. Model I – very weak contact resistance In this first case we have: PL = O(1), L = O(1), C = O(1) and BL = O(ε−2 ). It is easy to see that by substituting (4.79-4.80) into the system (4.75-4.78), the Biot condition (4.77) leads to temperature continuity in the first two orders: ∗(0)

Ta∗(0) − Tb

=0

Heat Transfer in Composite Materials ∗(1)

Ta∗(1) − Tb

133

=0

As a result, the analysis of section 4.2.2 remains appropriate, and model I corresponds to the model defined by (4.36): divx∗ (λIeff∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eff with: eff∗

(ρC)

λIeff∗ ij

1 = ρ C = ∗ |Ω |

1 = ∗ |Ω |

∗

∗

Ω∗ a

ρ∗a Ca∗

∂T ∗(0) ∂t∗

dΩ +

∂t∗ λ∗a (Iij + ai∗ ) dΩ∗ + ∂yj Ω∗ a

∗

Ω∗ b

ρ∗b Cb∗

dΩ

∗

∂t∗ λ∗b (Iij + bi∗ ) dΩ∗ ∂yj Ω∗ b

(4.81) (4.82)

where vectors t∗a and t∗b are solutions to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30). They depend on y∗ , λ∗a , λ∗b and are independent of h∗ . Finally, we can see that the Biot numbers O(εp ), p < −2 lead to the same model. In dimensional variables, this model can be written: divX (λIeff gradX T ) = (ρC)eff

∂T ¯ + O(ε) ∂t

4.3.2. Model II – moderate contact resistance Contact resistance (1/h) is increased by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε−1 ). The first order in temperature, Ω∗ -periodic in y∗ , satisfies: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0, ∗(0)


) = 0,

in Ω∗a

(4.83)

in Ω∗b

(4.84) ∗(0)

λ∗a grady∗ Ta∗(0) · n = −h∗ (Ta∗(0) − Tb λ∗a grady∗ Ta∗ · n = λ∗b grady∗ Tb∗ · n

∗(0)

)

over Γ∗

over Γ∗ ∗(0)

(4.85) (4.86)

respectively, and integrate over We multiply (4.83) and (4.84) by Ta and Tb their respective domains of definition. Integration by parts and use of the divergence theorem for each of these integrals leads with (4.86) to: ∗(0) ∗ ∗(0) 2 ∗ λa (grady∗ Ta ) dΩ + λ∗b (grady∗ Tb )2 dΩ∗ = 0 Ω∗ a

Ω∗ b

134


The thermal conductivities λ∗a and λ∗b are positive, and as a result grady∗ T ∗(0) = 0. We deduce from this that: ∗(0)

Ta∗(0) (x∗ , y∗ , t∗ ) = Tb

(x∗ , y∗ , t∗ ) = T ∗(0) (x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisfies: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) )) = 0 in Ω∗a ∗(1)


+ gradx∗ T ∗(0) )) = 0

(4.87)

in Ω∗b

(4.88) ∗(1)

λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) · n = −h∗ (Ta∗(1) − Tb

) over Γ∗

(4.89)

λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) · n = ∗(1)

λ∗b (grady∗ Tb

+ gradx∗ T ∗(0) ) · n,

over Γ∗

(4.90)

where gradx∗ T ∗(0) appears as a forcing term. The boundary conditions are linear. ∗(1) ∗(1) The unknowns Ta and Tb are linear functions of gradx∗ T ∗(0) , up to an arbitrary ∗ function of x : ∗ ∗(0) + T¯ ∗(1) (x∗ , t∗ ) Ta∗(1) (x∗ , y∗ , t∗ ) = χII∗ a (y ) · gradx∗ T ∗(1)

Tb

∗ ∗(0) (x∗ , y∗ , t∗ ) = χII∗ + T¯ ∗(1) (x∗ , t∗ ) b (y ) · gradx∗ T

(4.91) (4.92)

II∗ ∗ The superscript II refers to the model type. Vectors χII∗ a and χb , Ω -periodic in ∗ ∗ ∗ ∗ y , depend on y , λa , λb and h . They are solutions to: ∗

divy∗ (λ∗a (grady∗ χII∗ a + I) = 0

in Ω∗a

(4.93)

∗ divy∗ (λ∗b (grady∗ χII∗ b + I) = 0 in Ωb

(4.94)

∗ II∗ II∗ ∗ λ∗a (grady∗ χII∗ a + I) · n = −h (χa − χb ) over Γ

(4.95)

∗ II∗ ∗ λ∗a (grady∗ χII∗ a + I) · n = λb (grady ∗ χb + I) · n over Γ

(4.96)

II∗ χII∗ a + χb = 0

(4.97)

When h → ∞, this boundary condition is equivalent to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30). Finally, the temperature T ∗(2) , Ω∗ -periodic


135

in y∗ , satisfies: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ T ∗(0) ) = ρ∗a Ca∗ ∗(2)


∗(1)


∗(1)

+ gradx∗ Tb

∂T ∗(0) ∂t∗

in Ω∗a

(4.98)

∂T ∗(0) ∂t∗

in Ω∗b

(4.99)

))+

+ gradx∗ T ∗(0) ) = ρ∗b Cb∗

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = ∗(2)

λ∗b (grady∗ Tb

∗(1)

+ gradx∗ Tb

)·n

over Γ∗

(4.100)

Integrating the first two equations over their volume of definition, using the divergence theorem, equation (4.100) and the periodicity properties lead to the following macroscopic description: divx∗ (λIIeff∗ gradx∗ T ∗(0) ) = (ρ∗ C ∗ )eff

∂T ∗(0) ∂t∗

(4.101)

where: λIIeff∗ = ij

1 |Ω∗ |

Ω∗ a

λ∗a (Iij +

(ρC)eff∗ = ρ∗ C ∗ =

1 |Ω∗ |

∂χII∗ 1 aj ) dΩ∗ + ∗ ∗ ∂yi |Ω |

Ω∗

Ω∗ b

λ∗b (Iij +

∂χII∗ bj ) dΩ∗ ∂yi∗

ρ∗ C ∗ dΩ∗

The structure of model II is identical to that of model I. It is however important to note that λIIeff∗ depends on h∗ and: λIeff∗ = λIIeff∗ (λ∗a , λ∗b , h∗ → ∞) In dimensional variables, the model becomes: divX (λIIeff gradX T ) = (ρC)eff

∂T ¯ + O(ε) ∂t

4.3.3. Model III – high contact resistance We increase the contact resistance (1/h) still further, by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(1). The first order in temperature,

136


Ω∗ -periodic in y∗ , now satisfies: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)


∗(0)

·n=0

(4.102)

in Ω∗b

) = 0,

λ∗a grady∗ Ta∗(0) · n = 0 λ∗b grady∗ Tb

in Ω∗a

(4.103)

over Γ∗

(4.104)

over Γ∗

(4.105) ∗(0)

The boundary value problems for Ta before, it is easy to show that: Ta∗(0) (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , t∗ ),

∗(0)

and Tb

∗(0)

and

Tb

are decoupled. Proceeding as ∗(0)

(x∗ , y∗ , t∗ ) = Tb

(x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisfies: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) = 0 ∗(1)


∗(0)

+ gradx∗ Tb

)) = 0

in Ω∗a

(4.106)

in Ω∗b

(4.107) ∗(0)

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = −h∗ (Ta∗(0) − Tb

) over Γ∗ (4.108)

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n ∗(1)

= λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)·n ∗(1)

over Γ∗

(4.109)

∗(1)

Thus we have two decoupled linear problems for Ta and Tb . System (4.1064.109) imposes a continuity condition obtained by integrating (4.106) over Ω∗a . Using the divergence theorem and equation (4.108), we find: ∗(0) ∗(0) ∗ ∗(0) ∗ ∗(0) h (Ta − Tb )dS = (Ta − Tb ) h∗ dS ∗ = 0 Γ∗

Γ∗

The condition h∗ > 0 then implies: ∗(0)

Ta∗(0) = Tb

= T ∗(0) (x∗ , t∗ ) ∗(1)

(4.110) ∗(1)

∗(1)

The system for Tb gives the same condition. The unknowns Ta and Tb are both linear functions of gradx∗ T ∗(0) , up to an arbitrary function of x∗ and t∗ . ∗ ∗(0) + T¯a∗(1) (x∗ , t∗ ) Ta∗(1) (x∗ , y∗ , t∗ ) = χIII∗ a (y ) · gradx∗ T


Tb

∗(1)

∗ ∗(0) (x∗ , y∗ , t∗ ) = χIII∗ + T¯b b (y ) · gradx∗ T

137

(x∗ , t∗ )

and χIII∗ are independent of h∗ . They are solutions We note that the vectors χIII∗ a b ∗ ∗ to the canonical problems P(λa , λb = 0) and P(λ∗a = 0, λ∗b ) respectively (see equations 4.26-4.30), so that: divy∗ (λ∗a (grady∗ χIII∗ + I) = 0 in Ω∗a a divy∗ (λ∗b (grady∗ χIII∗ + I) = 0 b

in Ω∗b

λ∗a (grady∗ χIII∗ + I) · n = 0 a

over Γ∗

λ∗b (grady∗ χIII∗ + I) · n = 0 b

over Γ∗

III∗ χIII∗ a = χb = 0

Macroscopic behavior is obtained from the continuity equation which forms part of the boundary value problem (4.98-4.99) for T ∗(2) . It is clear that, just as was the case for model II, this model also has the same structure as model I: divx∗ (λIIIeff∗ gradx∗ T ∗(0) ) = (ρC)eff∗

∂T ∗(0) ∂t∗

(4.111)

where: λIIIeff∗ = λIIIeff∗ + λIIIeff∗ a b λIIIeff∗ aij

1 = ∗ |Ω |

λIIIeff∗ = bij

1 |Ω∗ |

Ω∗ a

Ω∗ b

∂χIII∗ aj ) dΩ∗ ∂yi∗

λ∗a (Iij +

∂χIII∗ bj

λ∗b (Iij +

(ρC)eff∗ = ρ∗ C ∗ =

1 |Ω∗ |

∂yi∗

Ω∗

) dΩ∗

ρ∗ C ∗ dΩ∗

The effective conductivities λIIIeff∗ and λIIIeff∗ of constituents a and b can be a b obtained by considering each constituent as perfectly insulated from its neighbor. Thus we have: λIIIeff∗ = λIIeff∗ (λ∗a , λ∗b , h = 0)

138


In the specific case where phase b is dispersed (non-connected) within matrix a, the condition of zero flux at the interface Γ∗ requires that χIII∗ = −y∗ . As a result, b phase b does not participate in the conductive heat transfer: λIIIeff∗ =0 b In dimensional variables, the model can be written: divX (λIIIeff gradX T ) = (ρC)eff

∂T ¯ + O(ε) ∂t

4.3.4. Model IV – model with two coupled temperature fields We increase the contact resistance (1/h) still further by a factor of ε−1 . We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε). The first order in temperature, Ω∗ -periodic in y∗ , satisfies, as in case II: divy∗ (λ∗a grady∗ Ta∗(0) ) = 0 ∗(0)


in Ω∗a

) = 0 in Ω∗b

λ∗a grady∗ Ta∗ · n = λ∗b grady∗ Tb∗ · n = 0

over Γ∗ ∗(0)

Once again, the boundary value problems for Ta this we deduce that:

∗(0)

Ta∗(0) (x∗ , y∗ , t∗ ) = Ta∗(0) (x∗ , t∗ ) and Tb

∗(0)

and Tb

are decoupled. From ∗(0)

(x∗ , y∗ , t∗ ) = Tb

(x∗ , t∗ )

The second order in temperature, Ω∗ -periodic in y∗ , satisfies: divy∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = 0 in Ω∗a ∗(1)


∗(0)

+ gradx∗ Tb

)) = 0

λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) ) · n = 0 ∗(1)

λ∗b (grady∗ Tb

∗(0)

+ gradx∗ Tb

)·n=0

in Ω∗b

over Γ∗ over Γ∗ ∗(1)

∗(1)

We obtain two decoupled linear problems for Ta and Tb . These two problems do not have any continuity requirements, as can be seen by integrating the partial ∗(1) differential equations over their respective domains of definition. The unknowns Ta


∗(0)

and Tb are linear functions of gradx∗ Ta include an arbitrary function of x∗ and t∗ .

∗(0)

and gradx∗ Tb

139

respectively, and

∗ ∗(0) Ta∗(1) (x∗ , y∗ , t∗ ) = χIV∗ + T¯a∗(1) (x∗ , t∗ ) a (y ) · gradx∗ Ta ∗(1)

Tb

∗(0)

∗ (x∗ , y∗ , t∗ ) = χIV∗ b (y ) · gradx∗ Tb

∗(1) + T¯b (x∗ , t∗ )

= χIII∗ and χIV∗ = χIII∗ are independent of h∗ and are solutions Vectors χIV∗ a a b b to the canonical problems P(λ∗a , λ∗b = 0) and P(λ∗a = 0, λ∗b ) respectively. Finally, the temperature T ∗(2) , Ω∗ -periodic in y∗ , satisfies: divy∗ (λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ))+ divx∗ (λ∗a (grady∗ Ta∗(1) + gradx∗ Ta∗(0) )) = ρ∗a Ca∗ ∗(2)


∗(1)


∗(1)

+ gradx∗ Tb

∗(0)

+ gradx∗ Tb

∗(0)

∂Ta ∂t∗

)) = ρ∗b Cb∗

∗(0)

∂Tb ∂t∗

∗(0)

∗(2)

∗(1)

+ gradx∗ Tb

(4.112)

in Ω∗b

(4.113)

))+

λ∗a (grady∗ Ta∗(2) + gradx∗ Ta∗(1) ) · n = −h∗ (Ta∗(0) − Tb λ∗b (grady∗ Tb

in Ω∗a

∗(0)

) · n = h∗ (Ta∗(0) − Tb

)

) over Γ∗ (4.114) over Γ∗

(4.115)

Integrating (4.112) over Ω∗a , making use of (4.114), gives our first macroscopic equation (4.116). We obtain our second macroscopic equation (4.117) by treating (4.113) and (4.115) in the same way: ∗(0)

(ρC)eff∗ a

∂Ta ∂t∗

(ρC)eff∗ b

∂Tb ∂t∗

∗(0)

= divx∗ (λIVeff∗ gradx∗ Ta∗(0) ) − H ∗ (Ta∗(0) − Tb a

∗(0)

∗(0)

= divx∗ (λIVeff∗ gradx∗ Tb b

where: λIVeff∗ = λIIIeff∗ , a a (ρC)eff∗ a

1 = ∗ |Ω |

λIVeff∗ = λIIIeff∗ b b

Ω∗ a

ρ∗a Ca∗ dΩ∗

∗(0)

) + H ∗ (Ta∗(0) − Tb

)

(4.116)

)

(4.117)

140


(ρC)eff∗ = b 1 H = ∗ |Ω | ∗

1 |Ω∗ | Γ∗

Ω∗ b

ρ∗b Cb∗ dΩ∗

h∗ dΓ∗

In the same way as for model III, in the specific case where phase b is dispersed (non-connected) within matrix a, we have: = λIIIeff∗ =0 λIVeff∗ b b The macroscopic behavior now involves two temperature fields. In dimensional form, the model becomes: (ρC)eff a

∂Ta ¯ = divX (λIVeff gradX Ta ) − H(Ta − Tb ) + O(ε) a ∂t

(ρC)eff b

∂Tb ¯ = divX (λIVeff gradX Tb ) + H(Ta − Tb ) + O(ε) b ∂t

4.3.5. Model V – model with two decoupled temperature fields The contact resistance (1/h) is very large. We have: PL = O(1), L = O(1), C = O(1) and BL = O(ε2 ). The analysis used for model IV can be applied step by step as before, except for the right-hand side of equations (4.114) and (4.115) which are now zero. This leads to decoupled macroscopic equations: ∗(0)

(ρC)eff∗ a

∂Ta ∂t∗

(ρC)eff∗ b

∂Tb ∂t∗

= divx∗ (λVeff∗ gradx∗ Ta∗(0) ) a

∗(0)

∗(0)

= divx∗ (λVeff∗ gradx∗ Tb b

)

(4.118)

(4.119)

with: λVeff∗ = λIVeff∗ = λIIIeff∗ , a a a

λVeff∗ = λIVeff∗ = λIIIeff∗ b b b

It can immediately be seen that if we assume BL = O(εp ), p > 2 then we end up with model V . In physical quantities the model can be written in the form: (ρC)eff a

∂Ta ¯ = divX (λVeff a gradX Ta ) + O(ε) ∂t

(ρC)eff b

∂Tb ¯ = divX (λVeff gradX Tb ) + O(ε) b ∂t


141

4.3.6. Example: bilaminate composite Consider again the composite depicted in Figure 4.4. As noted above, the symmetries of the problem require that for any model: eff λeff 22 = λ33 ,

eff eff λeff 12 = λ13 = λ23 = 0

For model I, we have the classical results: λIeff 22 =

λb λa cb λa + (1 − cb )λb

Ieff λIeff 11 = λ33 = (1 − cb )λa + cb λb

Now consider model II. Conductivity in the plane of layers remains the same as that of model I: IIeff λIIeff 11 = λ33 = (1 − cb )λa + cb λb

In the direction perpendicular to the layers contact resistance takes effect, and as a result we find: −1 2 λb λa 1 IIeff λ22 = + , with λIeff 22 = Ieff lh c λ + (1 − cb )λb λ22 b a which approaches the conductivity λeff 22 as h rises. Model III introduces a discontinuity in the medium in the direction perpendicular to the layers: the macroscopic equivalent material does not conduct in this direction, at least to the first order of approximation that we are considering here. (The conductivity is hl/2, a quantity which is negligible to first order.) Conductivity in the plane of the layers is that of model I, with the contact resistance playing no role in this direction: λIIIeff = 0, 22

λIIIeff = λIIIeff = (1 − cb )λa + cb λb 11 33

Finally, models IV and V introduce the following effective parameters: IVeff Veff Veff λIVeff a11 = λb11 = λa11 = λb11 = 0 Veff λIVeff a22 = λa33 = (1 − cb )λa Veff λIVeff b22 = λb33 = cb λb

H=

2h l

Model IV approaches model V as h approaches zero. In contrast, there is no direct transition from model IV to model III.

142


4.3.7. Choice of model Here we look for a dimensionless number which can characterize the different models, independent of details of perturbation they are subjected to. It is enough to consider Bl in this case: Bl =

h c lc = εBL λc Composite B

O(3 )

O(2 )

Two temperature fields

Composite A

O()

O(1)

O(1 )

Bl =

hc lc c

One single temperature field

Figure 4.6. Composite materials with contact resistance: different models as a function of the number Bl

The different models are characterized by: – model I: Bl O(ε−1 ) ; – model II: Bl = O(1) ; – model III: Bl = O(ε) ; – model IV: Bl = O(ε2 ) ; – model V: Bl O(ε3 ). Thus we can split the composite materials into two categories: – Composites A such that Bl 1. The only possible estimates of Bl are Bl O(ε−1 ) and O(1). These materials can only be described by models I or II. – Composites B such that Bl 1. The only possible estimates of Bl are Bl = O(1), O(ε), O(ε2 ) and O(ε3 ). These materials can only be described by models II, III, IV and V. The different models and their domains of validity are summarized in Figure 4.6. The scale separation parameter ε depends on the applied disturbance and on the point under consideration in the material. A given material can thus be described by different models depending on the nature of the disturbance and the spatial position.

Chapter 5

Diffusion/Advection in Porous Media

5.1. Introduction The transport of a solute by diffusion-convection in a porous medium is involved in a number of fields such as environmental engineering, process engineering, and even metallurgy. The works of Taylor [TAY 54] and Aris [ARI 56] were undoubtedly the first to demonstrate coupling between the flow and solute transport equation in elementary geometries. These works were extended to porous media by Brenner [BRE 80] using the method of moments and with the help of periodic homogenization [MAU 91; MEI 92; AUR 93b; AUR 95; AUR 96]. This chapter concerns the study of diffusive and advective transport of a lowconcentration solute in a saturated porous medium. The macroscopic transport equations are derived from the Fick and Navier-Stokes equations at the microscopic scale. 5.2. Diffusion-convection on the pore scale and estimates Consider a rigid porous Galilean medium with connected pores. We assume it to be periodic with period Ω (Figure 5.1). The pores Ωf are saturated with a viscous, incompressible Newtonian fluid containing a low concentration of solute c. The fluid is in slow steady-state isothermal flow, so that the solute is transported by diffusion and convection. We ignore adsorption of the solute onto the surface Γ of the pores, and absorption into the solid matrix Ωs . We denote as lc and Lc the characteristic lengths of the pores and macroscopic length. The solute transport is described by conservation of mass: ∂c + divX (−Dmol gradX c + vc) = 0 ∂t

in Ωf

(5.1)

144


and the condition on the surface of the pores that: Dmol gradX c · n = 0

over Γ

(5.2)

where c is the solute concentration (mass of solute per unit volume of fluid), Dmol is the (positive) molecular diffusion tensor which is in general isotropic, t is the time, n the unit vector giving the normal to Γ exterior to Ωf , and v is the fluid velocity. This is analyzed in section 7.2 in Chapter 7 and here we will make use of the results obtained in that section. Introducing into (5.1-5.2): c = cc c∗ ,

v = vc v ∗ ,

t = t c t∗ ,

Dmol = Dc Dmol∗

where the quantities with subscript c and exponent ∗ are the characteristic and dimensionless quantities respectively, and then adopting the microscopic viewpoint X = x∗ lc , we can write the microscopic description in dimensionless form as: Pl

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + Pel v∗ c∗ ) = 0 ∂t∗

Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

over Γ∗

(5.3) (5.4)

Lc

lc

n

s

f (a)

(b)

Figure 5.1. Porous medium: (a) Macroscopic structure, (b) Representative Elementary Volume (REV) with period Ω

This system introduces two dimensionless numbers: ∂c | lc2 ∂t = Pl = |divX (−Dmol gradX c)| Dc tc |

(5.5)


145

and the Péclet number defined by: Pel =

|divX (vc)| v c lc = mol |divX (−D gradX c)| Dc

(5.6)

where Dc and vc are the characteristic values of molecular diffusion and the velocity of the fluid. The characteristic time tc is the time over which we intend to describe the solute transport: tc is the characteristic time of the observation. On the macroscopic scale these numbers become: PL =

L2c , Dc tc

PeL =

vc Lc Dc

It is useful to introduce the characteristic times associated with advection and diffusion [AUR 95], which on the scale of the pores are: = tadv l

lc , vc

tdif = l

lc2 Dc

and on the macroscopic scale are: = tadv L

Lc , vc

tdif L =

L2c Dc

The ratio of characteristic times for diffusion and advection is the Péclet number: Pel =

tdif l , tadv l

PeL =

tdif L tadv L

These considerations lead us to consider the following situations: adv – Diffusion dominates at the macroscopic scale: tdif L = εtL . To study this case dif we use the macroscopic diffusion time tc = tL as our characteristic time. This means that Pl = O(ε2 ) and Pel = O(ε2 ). – Diffusion and advection are of the same order of magnitude on the macroscopic adv dif adv scale: tdif L = tL . In this case we use tc = tL = tL as our characteristic time. This 2 means that Pl = O(ε ) and Pel = O(ε). = εtdif – Advection dominates on the macroscopic scale: tadv L L . In this case we adv use tc = tL as our characteristic time. This means that Pl = O(ε) and Pel = O(1). – We will also consider the case where advection is overwhelmingly dominant on adv = ε2 tdif the macroscopic scale: tadv L L . In this case the characteristic time is tc = tL −1 and Pl = O(1) or Pel = O(ε ).

146


5.3. Diffusion dominates at the macroscopic scale The characteristic time is the time take for diffusion at the macroscopic scale: adv tc = tdif L = εtL

then Pel = O(ε2 ) and Pl = O(ε2 )

Equations (5.3) and (5.4) take the following dimensionless form: ε2

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + ε2 v∗ c∗ ) = 0 ∂t∗


within Ω∗f

(5.7)

over Γ∗

(5.8)

c∗ = c∗(0) (x∗ , y∗ , t∗ ) + εc∗(1) (x∗ , y∗ , t∗ ) + ε2 c∗(2) (x∗ , y∗ , t∗ ) + · · ·

(5.9)

We look for c∗ and v∗ in the form:

v∗ = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · ·

(5.10)

where c∗(i) (x∗ , y∗ , t∗ ) and v∗(i) (x∗ , y∗ ) are y∗ -periodic and where x∗ = εy∗ . The method consists of substituting these expansions into the dimensionless system (5.75.8) and identifying the powers of ε. We note that due to the two spatial variables and choice of microscopic viewpoint, the spatial derivative takes the following form: grady∗ −→ ε gradx∗ + grady∗ 5.3.1. Homogenization 5.3.1.1. Boundary value problem for c∗(0) After substituting in the expansions (5.9) and (5.10), we obtain at the lowest order in ε: divy∗ (Dmol∗ grady∗ c∗(0) ) = 0

in Ω∗f

(5.11)

Dmol∗ grady∗ c∗(0) · n = 0

over Γ∗

(5.12)

where c∗(0) is periodic in y∗ . Evidently the solution is not unique: c∗(0) is only determined up to an additive function of x∗ . We introduce the space C of regular functions α defined over Ω∗f , with a mean of zero over Ω∗f : 1 α = ∗ |Ω |

1 α dΩ = φ ∗ ∗ |Ω Ωf f| ∗

Ω∗ f

α dΩ∗ = 0


147

This condition, which is not required by the stated problem, is introduced to ensure the Hilbert character of C and to avoid the indeterminate nature of c∗(0) . The bilinear, symmetric, positive definite form: ∂α ∂β Dij ∗ ∗ dΩ∗ = Dmol∗ grady∗ α · grady∗ β dΩ∗ (α, β)C = ∗ ∂y ∂y Ω∗ Ω i j f f defines a scalar product over C. We multiply the two members of (5.11) by α ∈ C, and integrate over Ω∗f . Integrating by parts, and using boundary conditions over Γ∗ and periodicity, we obtain the following equivalent weak formulation: Dmol∗ grady∗ α · grady∗ c∗(0) dΩ∗ = 0 (5.13) ∀α ∈ C, (α, c∗(0) )C = Ω∗ f

We can use the Lax-Milgram theorem to show that this formulation ensures the existence and uniqueness of c∗(0) ∈ C, in other words satisfying the condition of zero mean. As a result, the solution of (5.11)-(5.12) must be: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ )

(5.14)

The concentration c∗(0) is constant over the period. 5.3.1.2. Boundary value problem for c∗(1) The second order c∗(1) satisfies: divy∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) = 0

within Ω∗f

(5.15)

Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ) · n = 0

over Γ∗

(5.16)

where c∗(1) is periodic in y∗ and gradx∗ c∗(0) appears as a source term. After multiplication of the two members of (5.15) by α ∈ C, and integration over Ω∗f , we obtain the weak formulation: ∀α ∈ C, (α, c∗(1) )C = Dmol∗ grady∗ α · grady∗ c∗(1) dΩ∗ Ω∗ f

=

Ω∗ f

Dmol∗ grady∗ α · gradx∗ c∗(0) dΩ∗

(5.17)

As was the case for c∗(0) , this formulation shows the existence and uniqueness of c∗(1) ∈ C, which means we can write the solution to linear problem (5.15)-(5.16) in the form: c∗(1) (x∗ , y∗ , t∗ ) = χdif∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ )

(5.18)

148


where χdif∗ is a periodic vector with a mean of zero over Ω∗f . The function c¯∗(1) is an arbitrary function of x∗ and t∗ introduced by the external condition χdif∗ = 0. The vector χdif∗ satisfies: divy∗ (Dmol∗ (grady∗ χdif∗ + I)) = 0 within Ω∗f Dmol∗ (grady∗ χdif∗ + I) · n = 0

over Γ∗

χdif∗ = 0 5.3.1.3. Boundary value problem for c∗(2) Finally, bearing in mind the results above, the boundary value problem for c∗(2) can be reduced to: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

within Ω∗f (5.19)

over Γ∗

(5.20)

where c∗(2) is y∗ -periodic. In contrast to the boundary condition for c∗(0) and c∗(1) , the boundary value problem for c∗(2) introduces a compatibility condition obtained by integration of (5.19) over Ω∗f . Bearing in mind the boundary conditions over Γ∗ , the periodicity, and after division by |Ω∗ |, we find: φ

∂c∗(0) − divx∗ (Ddif∗ gradx∗ c∗(0) ) = 0 ∂t∗

(5.21)

where φ is the porosity and the effective diffusion tensor Ddif∗ is defined as: dif∗ = Dij

1 |Ω∗ |

Ω∗ f

mol∗ Dik (Ijk +

∂χdif∗ j ) dΩ∗ ∂yk∗

(5.22)

5.3.2. Macroscopic diffusion model Returning to physical quantities, we find: φ

∂c ¯ − divX (Ddif gradX c) = O(ε), ∂t

Ddif = Dc Ddif∗

(5.23)

whose relative precision is O(ε). The tensor Ddif is purely diffusive. It is possible to show, starting from equation (5.17), that the tensor Ddif is positive and symmetric.


149

In macroscopic model (5.23), c is the mass of solute per unit volume of fluid. Using the mass of solute per unit of volume of the porous medium c = φ c, the model becomes: ∂c ¯ − divX (φ−1 Ddif gradX c) = O(ε) ∂t 5.4. Comparable diffusion and advection on the macroscopic scale The characteristic time is now: adv tc = tdif L = tL

which implies

Pel = O(ε)

and Pl = O(ε2 )

In dimensionless form, equations (5.3) and (5.4) can be written: ε2

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + εv∗ c∗ ) = 0 ∂t∗


within Ω∗f

over Γ∗

(5.24) (5.25)

5.4.1. Homogenization 5.4.1.1. Boundary value problems for c∗(0) and c∗(1) We introduce expansions (5.9) and (5.10) into the system of equations (5.34-5.35). It is easy to see that the boundary value problems for c∗(0) and c∗(1) are unchanged. Solutions (5.14) and (5.18) obtained in the preceding section remain valid: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ ) c∗(1) (x∗ , y∗ , t∗ ) = χdif∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ ) 5.4.1.2. Boundary value problem for c∗(2) In contrast, the boundary value problem for c∗(2) becomes: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ))+ divy∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) + divx∗ (c∗(0) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

over Γ∗

within Ω∗f

(5.26) (5.27)

150


where c(2) is y∗ -periodic, and v∗(0) and v∗(1) are the fluid velocities for the first two orders of approximation. These velocities are given by independent consideration of slow, steady-state flow of an incompressible Newtonian fluid in a porous medium (see Chapter 7). This leads to the following results. Conservation of mass of the fluid to the first two orders can be written: divy v∗(0) = 0,

divy v∗(1) + divx v∗(0) = 0

(5.28)

The no-slip boundary condition at Γ∗ gives: v∗(0) = v∗(1) = 0

(5.29)

In addition, to first order, the velocity of the fluid on the microscopic scale is proportional to the pressure gradient (see Chapter 7), v∗(0) (x∗ , y∗ ) = −

k∗ (y∗ ) gradx∗ p∗(0) , η∗

with

p∗(0) = p∗(0) (x∗ )

(5.30)

At the macroscopic scale, these results lead to Darcy’s law: divx v∗(0) = 0,

v∗(0) = −

K∗ gradx∗ p∗(0) η∗

(5.31)

The compatibility condition is obtained by integration of (5.26) over Ω∗f . Using the divergence theorem, the boundary conditions over Γ∗ (5.27), the periodicity and finally the no-slip condition at the boundary Γ∗ (5.29), we find: φ

∂c∗(0) − divx∗ (Ddif∗ gradx∗ c∗(0) − v∗(0) c∗(0) ) = 0 ∂t∗

(5.32)

where Ddif∗ is the effective diffusion tensor introduced in the previous case and v∗(0) is the Darcy velocity given by equation (5.31). 5.4.2. Macroscopic diffusion-advection model In dimensional variables, the model (5.32) becomes: φ

∂c ¯ − divX (Ddif gradX c − vc) = O(ε), ∂t

Ddif = Dc Ddif∗

(5.33)

with a relative precision of O(ε). The tensor Ddif , purely diffusive, is the same as that defined for the case where diffusion dominates: it is not modified by advection. In terms of the concentration per unit volume of the porous material, this model can be written: ∂c ¯ − divX (φ−1 Ddif gradX c − φ−1 vc) = O(ε) ∂t


151

5.5. Advection dominant at the macroscopic scale The characteristic time is now: dif tc = tadv L = εtL

then Pel = O(1)

and Pl = O(ε)

Equations (5.3) and (5.4) take the dimensionless form: ε

∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + v∗ c∗ ) = 0 ∂t∗


within Ω∗f

(5.34)

over Γ∗

(5.35)

5.5.1. Homogenization 5.5.1.1. Boundary value problem for c∗(0) The boundary value problem for c∗(0) is now: divy∗ (Dmol∗ .grady∗ c∗(0) − v∗(0) c∗(0) ) = 0 Dmol∗ grady∗ c∗(0) · n = 0

within Ω∗f

(5.36)

over Γ∗

(5.37)

where c∗(0) is a periodic function of y∗ . With the help of equations (5.28) and (5.29) to first order, the associated weak formulation is similar to (5.13): ∗(0) ∀α ∈ C, (α, c )C = Dmol∗ grady∗ α · grady∗ c∗(0) dΩ∗ = 0 (5.38) Ω∗ f

As a consequence, equation (5.14) remains applicable: c∗(0) (x∗ , y∗ , t∗ ) = c∗(0) (x∗ , t∗ )

(5.39)

The concentration c∗(0) is constant across the period. 5.5.1.2. Boundary value problem for c∗(1) The boundary value problem for c∗(1) is strongly modified: ∂c∗(0) − divy∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) )) ∂t∗ +divy∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) + divx∗ (c∗(0) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ) · n = 0

in Ω∗f

over Γ∗

(5.40) (5.41)

152


where c∗(1) is y∗ -periodic and gradx∗ c∗(0) appears as a source term. This system introduces a compatibility condition for the existence of c∗(1) , which can be obtained by taking the volume mean of (5.40) over Ω∗ . Using the divergence theorem, the boundary condition over Γ∗ (5.41), periodicity and finally no-slip condition at the wall Γ∗ (5.29), we find: φ

∂c∗(0) + divx∗ (v∗(0) c∗(0) ) = 0 ∂t∗

(5.42)

where the velocity v∗(0) is the Darcy velocity, the mean of v∗(0) over Ω∗ . As might be expected, advection alone is present at the first order of approximation. We need to find the first correction to this macroscopic model which brings diffusion into play. For that, substituting (5.42) into equation (5.40), the latter becomes: divy∗ (Dmol∗ (gradx∗ c∗(0) + gradx∗ c∗(1) )) − v∗(0) · grady∗ c∗(1) = (v∗(0) − φ−1 v∗(0) ) · gradx∗ c∗(0)

in Ω∗f

(5.43)

The solution to the boundary value problem (5.43)–(5.41) can be described using the space C. Linearity allows us to write: c∗(1) (x∗ , y∗ , t∗ ) = χdis∗ (y∗ ) · gradx∗ c∗(0) + c¯∗(1) (x∗ , t∗ )

(5.44)

where χdis∗ is a periodic vector with a mean of zero over Ω∗f . Its components χdis∗ k are the solutions to three independent problems (k = 1, 2, 3): dis∗ ∂χdis∗ ∂ ∗(0) ∂χk mol∗ k (Ijk + = ∗ (Dij ∗ )) − vi ∂yi ∂yj ∂yi∗ ∗(0)

vk mol∗ Dij (Ijk +

∂χdis∗ k )ni = 0 ∂yj∗

− φ−1 v ∗(0) k

within Ω∗f

over Γ∗

(5.46)

which can be written in the form: divy∗ (Dmol∗ (grady∗ χdis∗ + I)) − v∗(0) · grady∗ χdis∗ = v∗(0) − φ−1 v∗(0) Dmol∗ (grady∗ χdis∗ + I) · n = 0 χdis∗ = 0

over Γ∗

(5.45)

within Ω∗f


153

We note that χdis∗ = χdif∗ , since χdis∗ depends on the velocity field to first order, and hence on the pressure gradient gradx∗ p(0) . It is interesting to note that the above system is the same as that obtained by Brenner [BRE 80] using the method of moments. 5.5.1.3. Boundary value problem for c∗(2) The macroscopic relationship satisfied by the corrector c∗(1) is obtained through the compatibility condition associated with the boundary value problem for c∗(2) : ∂c∗(1) − divy∗ (Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ))− ∂t∗ divx∗ (Dmol∗ (gradx∗ c∗(0) + grady∗ c∗(1) ))+ divy∗ (c∗(0) v∗(2) + c∗(1) v∗(1) + c∗(2) v∗(0) )+ divx∗ (c∗(0) v∗(1) + c∗(1) v∗(0) ) = 0 Dmol∗ (gradx∗ c∗(1) + grady∗ c∗(2) ) · n = 0

in Ω∗f

(5.47) over Γ∗

(5.48)

where c∗(2) is y∗ -periodic. If we take the volume mean of (5.47), we find: ∂c(1) − divx∗ (Dmol∗ (I + grady∗ χdis∗ ).gradx∗ c∗(0) − ∂t∗ v∗(1) c∗(0) − c∗(1) v∗(0) ) = 0

(5.49)

which represents the condition for the existence of c∗(2) . This equation can be slightly modified by introducing equations (5.44) for c∗(1) and (5.30) for v∗(0) : ∂c(1) − divx∗ (Ddis∗ gradx∗ c∗(0) − v∗(1) c∗(0) − c¯∗(1) v∗(0) ) = 0 ∂t∗

(5.50)

where Ddis∗ is the effective dispersion tensor: dis∗ = Dij

1 |Ω∗ |

Ω∗ f

mol∗ Dik (Ijk +

∂χdis∗ k ∗ dis∗ ∂p∗(0) j ) + ik χ dΩ∗ ∗ ∂yk η∗ j ∂x∗k

(5.51)

Equation (5.50) represents the macroscopic behavior of the first-order correction.

154


5.5.2. Dispersion model To second order, the concentration and velocity can be written: c∗ ≈ c∗(0) + εc∗(1) = φ c∗(0) + εφ c¯∗(1)

(5.52)

v∗ ≈ v∗(0) + εv∗(1)

(5.53)

Adding equations (5.42) and (5.50) multiplied by ε, term by term, we find: ∂c∗ − divx∗ (εφ−1 Ddis∗ .gradx∗ c∗ − φ−1 v∗ c∗ ) = O(ε2 ) ∂t∗

(5.54)

where we note that the precision of this model is O(ε2 ). The diffusion term is small compared to the advection term, PeL = ε−1 . Returning now to physical quantities, we find: ∂c tc Lc −1 dis φ−1 ¯ 2) − divX Lc (ε φ D gradX c − vc) = O(ε ∂t cc Dc cc vc cc

(5.55)

We recall that tc = tadv = Lc /vc and vc Lc /Dc = PeL = ε−1 . We obtain: L ∂c ¯ 2) − divX (φ−1 Ddis gradX c − φ−1 vc) = O(ε ∂t

(5.56)

As shown by Auriault and Adler [AUR 95], the dispersion tensor Ddis is positive, but is not generally symmetrical. Using the concentration per unit volume of fluid cΩf = c(0) + ε c¯(1) = c/φ, the model becomes: φ

∂cΩf ¯ 2) − divX (Ddis gradX cΩf − vcΩf ) = O(ε ∂t

One important question concerns the physical meaning of the volume average v, which contains the first correction v(1) . It turns out that v(1) is not irrotational – see (7.11b). As a result v(1) is not generally a flux (a similar reasoning can be found for v(0) in section 7.2). As a consequence, v itself is not generally a flux. Another consequence of the presence of v(1) is that v does not generally satisfy Darcy’s law. (For more details, see [AUR 05b].) 5.6. Very strong advection The characteristic macroscopic advection time tadv L becomes very large compared to the macroscopic diffusion time tdif L . The characteristic time of the observer is now: 2 dif tc = tadv L = ε tL

which implies

Pel = O(ε−1 ) and Pl = O(1)


155

Equations (5.3) and (5.4) can be written in dimensionless form as: ∂c∗ + divy∗ (−Dmol∗ grady∗ c∗ + ε−1 v∗ c∗ ) = 0 ∂t∗ Dmol∗ grady∗ c∗ · n = 0

within Ω∗f

over Γ∗

(5.57) (5.58)

We need to find a suitable macroscopic model. We substitute the asymptotic expansions of c∗ and v∗ into the above system. At the lowest order in c∗ we obtain: divy∗ (c∗(0) v∗(0) ) = 0

within Ω∗f

(5.59)

The incompressibility of the fluid, divy∗ v∗(0) = 0, means we can write the preceding relationship in the form: v∗(0) · grady∗ c∗(0) = 0 Dmol∗ grady∗ c∗(0) · n = 0

within Ω∗f

(5.60)

over Γ∗

(5.61)

To make this clearer, consider a constant concentration c0 applied from the outset at the entrance to the macroscopic sample, and maintained constant from then onwards. Equation (5.60) shows that c∗(0) remains constant along a streamline across a given period: c∗(0) = c0 if the fluid particle which was at the entrance to the sample at time t∗ = 0 has reached this period, or c∗(0) = 0 if not. This situation applies throughout the macroscopic sample, since over the time tc diffusion has only developed over a length O(lc ). At the next order we have: ∂c∗(0) + divy∗ (−Dmol∗ grady∗ c∗(0) + c∗(0) v∗(1) + c∗(1) v∗(0) ) = 0 ∂t∗ Which, through integration over Ω∗f gives the following compatibility condition: ∂c∗(0) =0 ∂t∗ This result is impossible because this dimensionless term is O(1) by construction. The situation Pl = O(1) is therefore not homogenizable: a macroscopically equivalent continuum does not exist. This last case illustrates the consistency of the method mentioned in section 2.4.1 of Chapter 2. 5.7. Example: parallel slits

porous medium consisting of a periodic lattice of narrow

In order to illustrate the above results, we will consider a porous medium consisting of a periodic lattice of narrow parallel slits, whose characteristic dimensions are given

156


e2 L2

e1

(a)

L1 e2

l2

f

h

e1

(b)

Figure 5.2. Porous medium consisting of a periodic lattice of narrow slits:(a) macroscopic structure, (b) dimensionless period Ω∗

in Figure 5.2. The molecular diffusion tensor is isotropic, Dmol∗ = D mol∗ I, and flow is driven by a macroscopic pressure gradient in the direction e1 : gradx∗ p∗(0) =

dp∗(0) e1 dx∗1

5.7.1. Analysis of the flow As we will see in Chapter 7, the velocity v∗(0) and the pressure p∗(1) , both y -periodic, are obtained by solving over Ω∗ the boundary value problem which takes the following dimensionless form: ∗

η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ (v∗(0) ) = 0

within

v∗(0) = 0

Γ∗

over

within

Ω∗f

Ω∗f

The specific geometry of the porous medium necessarily implies that: ∗(0)

v∗(0) = v1

(y2∗ ) e1

and p∗(1) = p∗(1) (y2∗ )


157

Thus the preceding problem reduces to: η∗

∗(0)

dp∗(0) d2 v 1 − =0 dy2∗2 dx∗1

∗(0)

v1

within

Ω∗f

= 0 over Γ∗

Solving this system of equations leads us to: ∗(0) v1

1 dp∗(0) = ∗ 2η dx∗1

y2∗2

h∗2 − 4

whose mean can be written: ∗(0)

v1

=−

φh∗2 dp∗(0) 12η∗ dx∗1

5.7.2. Determination of the dispersion coefficient The dispersion coefficient is obtained by solving over Ω∗ the boundary value problem (5.45-5.46). In this particular case, the components χdis∗ only depend on k y2∗ , and this boundary value problem becomes: d dy2∗

dis∗ ∗(0) ∗(0) mol∗ dχ1 D = v1 − φ−1 v1 dy2∗

d dy2∗

dχdis∗ 2 Dmol∗ 1 + =0 dy2∗

d dy2∗

dis∗ mol∗ dχ3 D = 0 within Ω∗f dy2∗

Dmol∗

D

Dmol∗

within Ω∗f

dχdis∗ 1 .n2 = 0 over Γ∗ dy2∗

mol∗

dχdis∗ 1 1+ dy2∗

.n2 = 0

dχdis∗ 3 .n2 = 0 over Γ∗ dy2∗

dis∗ dis∗ χdis∗ 1 = χ2 = χ3 = 0

within Ω∗f

over Γ∗

158


Solving this system, we find: 1 = 24η ∗ Dmol∗

χdis∗ 1

y2∗2

h∗2 1− 2

7h∗4 + 240

dp∗(0) dx∗1

χdis∗ = −y2∗ 2 χdis∗ =0 3 Using equation (5.51), the two non-zero components of the dispersion tensor are: dis∗ D11

= φD

φh∗6 + 30240η ∗2 Dmol∗

mol∗

dp∗(0) dx∗1

2

dis∗ D33 = φDmol∗ dis∗ dis∗ We note that D33 is purely diffusive and that D11 is the sum of a diffusive part and a dispersive part arising from the coupling between the concentration at the first order c∗(1) and the advection (v∗(0) ). In addition, it can be shown that:

Pe∗2 dis∗ D11 = φDmol∗ 1 + 210 where Pe∗ is the Péclet number defined by: Pe∗ =

∗(0)

v1 h∗ φDmol∗

Finally, in dimensional form, the dispersion tensor for the porous medium consisting of a periodic lattice of narrow parallel slits can be written: ⎛ dis ⎞ D11 0 0 0 0 ⎠ Ddis = ⎝ 0 dis 0 0 D33 with: dis D11

= φD

mol

Pe2 1+ , 210

dis D33 = φDmol

and

Pe =

v1 h φDmol

Here we recover the results established by Wooding [WOO 60]. When the geometries are more complex, the determination of the dispersion tensor requires the numerical solution of the boundary value problem (5.45-5.46).


159

For example, numerical estimates can be found in [EDW 91; SAL 93; COE 97; SOU 97] of the dispersion tensor for porous two-dimensional and three-dimensional media (both periodic and random).

5.8. Conclusion The different models obtained, as well as their domains of validity, are summarized in Table 5.1. There is a continuum, as the Péclet number decreases, from the dispersion model to the diffusion-advection model and then to the diffusion model. Pel Pl

Time tc of observation

Diffusion/dispersion tensor

Model

ε2 ε2

2 tdif L = Lc /Dc

Diffusion: Ddif

Diffusion

Diffusion: Ddif

Diffusion-advection

Dispersion: Ddis

Advection Diffusion: Dispersion

ε

2 adv ε2 tdif L = Lc /Dc = tL

ε

tadv L = Lc /vc

ε−1 1

tadv L = Lc /vc

1

Not homogenizable

Table 5.1. Macroscopic models of solute transport in a porous medium

The experimental measure of the dispersion coefficient raises various difficulties due to the size L of the macroscopic lattice being used [AUR 97]. The macroscopic models obtained above are useful in regions of the macroscopic domain where there is a separation of length scales. This introduces two types of limitation on the range of validity of the models. First this condition must be fulfilled both by the geometry of the porous medium: εg =

lc 1 Lc

(5.62)

where Lc is the characteristic size of the sample, and also by the physical process under consideration. In the case of solute transport in a test column, we can write that

160


condition as: εph =

lc 1, Lph

Lph ≈ |

∂ c −1 ( )| ∂X c0

(5.63)

where c0 is the concentration (assumed constant) at the entrance to the column. These limitations are analyzed in [AUR 97] for column experiments. In addition, the condition of separation of length scales is not met close to macroscopic boundaries: in general a boundary layer must be introduced to connect the solution obtained by homogenization and the conditions at the boundaries of the lattice [BEN 78]. Determination of the effective coefficients from the experimental parameters will be valid if the thickness of this boundary layer is small compared to the size of the lattice and the measurement is performed outside this boundary layer [LEW 98]. It has been observed that for elliptical or parabolic problems such as problems of elasticity or pure diffusion, the thickness of the boundary layer is approximately lc [SAN 87], with lc ≈ 5 to 10 grain sizes [BEA 72; ANG 94]. Thus in these cases the limitation on the domain of validity of the macroscopic models is given by the conditions εg 1 and εph 1. When advection is present at the macroscopic scale, the thickness of the boundary layer changes from lc to δc > lc . Now we will evaluate δc . The characteristic time tdif l is the time taken by the solute to diffuse over a distance lc : tdif l =

lc2 Dc

When advection is present, the solute particles are also convected at velocity vc . Over the time tdif l , the solute particles are transported over a distance: vc tdif l =

lc2 vc Dc

The thickness of the boundary layer in the presence of advection is therefore: δ c = lc +

lc2 vc Dc

Compare δc and lc : vc lc δc =1+ = 1 + Pel lc Dc The thickness of the boundary layer increases with the Péclet number. The length Lc of the macroscopic lattice must be increased in consequence of this, particularly in the presence of dispersion. The numerical investigations of Salles et al. [SAL 93] confirm the above results.

Chapter 6

Numerical and Analytical Estimates for the Effective Diffusion Coefficient

6.1. Introduction The two preceding chapters have highlighted different models of macroscopic transfer depending on the properties of the constituents and their interfaces on the microscopic scale. Each of these models – valid for various periodic geometries obeying certain connectivity conditions – has been elaborated in an analytical manner for bilaminate composites. This chapter will extend and complete these results, shifting the focus onto fibrous, cellular or granular materials: – the numerical procedure of homogenization is illustrated through determination of the macroscopic transfer coefficients for composite media with spherical inclusions (connected or disconnected) and fibrous media; – for comparison, common self-consistent analytical estimates are recalled, with this approach also being extended to three-constituent materials and those with contact resistance; – finally, comparison with experimental results allows us to judge the suitability of these models for describing the properties of real materials. The aim is mostly illustrative. The main purpose is to present some basic results, numerical or analytical, for periodic or random microstructures. The calculation of effective properties has been the subject of a rich literature which explores complex random three-dimensional geometries, or geometries corresponding to real materials. On this subject, outside the scope of the present work, we refer the reader for example to: [PER 79; SAN 82a; THO 90; ADL 92a; b; COE 97; and SAH 03].

162


6.2. Effective thermal conductivity for some periodic media In this first part, we determine effective thermal conductivities for periodic composite media with spherical inclusions, either connected or disconnected, as well as for fibrous media, by solving the canonical boundary value problem resulting from the process of homogenization as presented in Chapter 4 (section 4.2.2) where the contact between the constituents is assumed to be perfect. In addition, on the microscopic scale, the thermal conductivities of each of the constituents are assumed to be constant, isotropic, and of the same order of magnitude as each other: λa = λa I,

λb = λb I,

λa = O(λb )

and

6.2.1. Media with spherical inclusions, connected or non-connected 6.2.1.1. Microstructures Here we consider composite media consisting of a periodic lattice of spherical inclusions (phase b), connected or non-connected, in a matrix (phase a). The volume fractions of the matrix and the inclusions are ca and cb = 1 − ca respectively. The radius of the inclusions is denoted R. The periodic lattices we will consider are simple cubic (SC) and body centered cubic (BCC) lattices. The characteristic dimensions of the period corresponding to each of these microstructures under study are shown in Figure 6.1. Phase b for the SC lattice and the BCC lattice is connected when the

l

l e2

e3

(a)

b R

e2

a

e1

e3

b R

a e1

(b)

Figure 6.1. Composite media consisting of a periodic lattice of inclusions, connected or non-connected. Characteristic dimensions of the period. (a) Simple cubic lattice, (b) body centered cubic lattice

volume fraction of the inclusion cb is greater than π/6 ≈ 0.523 and √ π 3/8 ≈ 0.68 respectively. Above this volume fraction, or percolation threshold, region Ωb consists of connected, truncated spheres. For the simple cubic lattice, the contact number between the spheres is 6, whereas for the centered cubic lattice the contact number is 8, and then 12 when cb > 0.93.


163

Given the symmetries of each of these microstructures, the effective thermal conductivity tensor is isotropic: λeff = λeff I

(6.1)

6.2.1.2. Solution to the boundary value problem over the period In dimensionless form, the effective thermal conductivity for the materials in question is defined in equation (4.82): 1 ∂t∗ai ∂t∗bi eff∗ ∗ ∗ ∗ ∗ λa (δij + ) dΩ + λb (δij + ∗ ) dΩ λij = ∗ (6.2) |Ω | ∂yj∗ ∂yj Ω∗ Ω∗ a b where the y∗ -periodic vector t∗ (y∗ ) = (t∗1 , t∗2 , t∗3 ) is the solution to the canonical problem P(λ∗a , λ∗b ) (4.26-4.30) over the period Ω∗ . This boundary value problem can be written in the following compact form: divy∗ (λ∗ (grady∗ t∗ + I)) = 0 in Ω∗ [t∗ ]Γ∗ = 0 over Γ∗ [λ∗ (grady∗ t∗ + I)]Γ∗ · n = 0

over Γ∗

t∗ = 0 The three elementary solutions t∗i (y∗ ) are associated with unit macroscopic gradients in the three directions gradx∗ T ∗(0) = ei (i = 1, 2, 3). Given the symmetries of the microstructures, all we need to determine is t∗1 (y∗ ). This boundary value problem has been solved by finite element analysis [COM 08]. Figures 6.2 (a) and (b) present respectively, for each of these microstructures under study, the magnitude of vector t∗ = t∗3 e3 solving the canonical problem P(λ∗a , λ∗b ) for λa /λb = 10 and ca = 0.6. The symmetries of the microstructures imply that t∗3 is symmetric with respect to the axis e3 and non-symmetric with respect to axes e1 and e2 . These comments are valid whatever the volume fraction ca and the ratio of the conductivities λa /λb . 6.2.1.3. Effective thermal conductivity Figure 6.3 presents, for two conductivity ratios λa /λb (=10 or 100), the evolution of effective thermal conductivity λeff /λa obtained from numerical simulations as a function of the volume fraction of the matrix ca . These figures show that (i) for the two microstructures under consideration, and whatever the conductivity ratio λa /λb (with λa > λb ), λeff is a monotonically increasing function of ca , and (ii) whatever the conductivity ratio λa /λb , λeff becomes sensitive to the arrangement of the inclusions when the volume fraction cb of the inclusions is greater than 0.5.

164


e2

e2

e3

e1

(a)

e3

e1

(b)

Figure 6.2. Elementary t∗3 solution to the canonical problem P(λ∗a , λ∗b ), (ca = 0, 6, λa /λb = 10). (a) Simple cubic lattice (SC) (black = -0.09, white = 0.09), (b) body centered cubic lattice (black = -0.073, white = 0.073)

6.2.1.3.1. Comparison with the Voigt bounds (V), Reuss (R) and Hashin and Shtrikman (HS+ and HS-) The upper Voigt bounds [VOI 87] (V) and lower Reuss bounds [REU 29] (R) are, for λeff : λR =

λa λb λeff ca λa + (1 − ca )λb = λV (1 − ca )λa + ca λb

(6.3)

If we make the additional assumption that the material is isotropic on the macroscopic scale, improved bounds have been established by Hashin and Shtrikman [HAS 63] (Chapter 1, equation 6.4). These bounds, denoted HS+ and HS-, are for λa > λb : λHS− = λb +

ca +

1 λa −λb

cb 3λb

λeff λa +

cb +

1 λb −λa

ca 3λa

= λHS+

(6.4)

Figure 6.3 shows that numerical estimates of the effective conductivity of the two microstructures under study lie between bounds whatever the conductivity ratio and the volume fraction of the matrix ca . We can also note that the numerical results are very close to the upper Hashin and Shtrickman bound (6.4) over a large range of volume fraction of the matrix ca . The differences are considerably more pronounced when both phases are connected. 6.2.1.3.2. Comparison with the self-consistent estimate with simple inclusions The original self-consistent scheme involves considering simple homogenous spherical inclusions (Chapter 1, section 1.4). In this particular case, the effective conductivity is given implicitly by equation (1.30). As Figure 6.3 shows, this selfconsistent estimate (SCE) underestimates the conductivity of the two periodic media


165

1

a / b = 10

e& / a

0.8

V

0.6

HS+ 0.4

HS-

SCE

R 0.2

0 0

0.2

0.4

ca

0.6

0.8

1

1

a / b = 100

e& / a

0.8

V

0.6

HS+ 0.4

0.2

HS-

SCE

R 0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.3. Evolution of the effective thermal conductivity λeff /λa of simple cubic () and centered cubic () lattices with spherical inclusions, as a function of the volume fraction of the matrix, ca . Comparison, for two conductivity ratios λa /λb (=10 or 100), of the numerical results with the Voigt (V) and Reuss (R) bounds (equation (6.3)), of Hashin and Shtrickman (HS+ and HS-) (equation (6.4)) and the self-consistent estimate (SCE) with simple inclusions (equation (1.30))

in question, whatever the volume fraction of the inclusions. The differences are more marked when the volume fraction ca of the matrix is low, i.e. when the inclusions are connected. As has already been emphasized, the traditional self-consistent scheme

166


with simple spherical inclusions does not allow us to capture the connected or dispersed nature of the phases independently of their concentration. We will see later (section 6.3.1) that this can be achieved by applying the self-consistent scheme with bi-composite substructures, for example. 6.2.1.3.3. Effects of the thermal conductivity contrast λa /λb Evolution of the effective thermal conductivity λeff /λa , as derived from numerical simulations as a function of the thermal conductivity contrast λa /λb , for a fixed volume fraction of the matrix ca = 0.5, is shown in Figure 6.4. 6

HS+ V

ca = 0.5

5

e& / a

4

HS-

3

2

R

1

HS0 10-4

HS+

V

R 10-2

100

102

104

a / b Figure 6.4. Evolution of the effective thermal conductivity λeff /λa of simple cubic lattices () et and centered cubic lattices () with spherical inclusions as a function of the conductivity ratio λa /λb (ca = 0.5).Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.4))

This figure emphasizes the high sensitivity of conductivity λeff to the conductivity contrast λa /λb , when 10−3 < λa /λb < 103 . For weaker or stronger conductivity ratios, effective conductivity becomes nearly constant. In addition, we find a good agreement between the numerical results for the centered cubic lattice and the upper (if λa /λb > 1) and lower (if λa /λb < 1) Hashin and Shtrickman bounds (1.24). 6.2.1.3.4. High conductivity contrast When the conductivity ratio λa /λb = ∞ (or λb = 0), the effective thermal conductivity reduces to that of matrix a on its own: λeff a (Chapter 4, section 4.2.3.2).


167

It is obtained by solving the canonical problem P(λ∗a , 0). The evolution of λeff a /λa as a function of the volume fraction ca is shown in Figure 6.5. It can again be seen that the upper Hashin and Shtrikman bound [HAS 63] is still a good approximation to λeff a across a wide range of volume fractions of the matrix ca . These results clearly apply to all diffusive phenomena, such as solute diffusion, of weak concentration in a fluid saturating the porous skeleton consisting of inert dif mol spherical inclusions. By analogy, λeff , where Ddif is a (ca )/λa becomes D (φ)/D mol is the molecular diffusion coefficient and φ is the the effective diffusion tensor, D porosity (Chapter 5, section 5.3). 1

a / b =

e&/ a

0.8

0.6

V HS+

0.4

0.2

HS-

R

0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.5. Evolution of the effective thermal conductivity λeff /λa of simple cubic () and centered cubic () lattices with spherical inclusions when λb = 0 (ca = 0.5). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.4))

When the conductivity ratio λa /λb tends to zero, and the spherical inclusions are non-connected, this situation corresponds to that of highly conductive inclusions embedded in a matrix (Chapter 4, section 4.2.4). The effective conductivity λ+eff is obtained by solving the specific boundary value problem defined by equations (4.63– 4.64). When the inclusions are connected, the effective thermal conductivity tends towards that of the network of inclusions (phase b) λeff b , given by solving the canonical problem P(0, λ∗b ). Figure 6.6 shows the evolution of λeff /λa when λa /λb = 10−3 . This figure underlines the importance of the connectivity of the phases on the effective

168


1000

a / b = 0 . 0 01 HS+

e& / a

100

10

HS-

1 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.6. Evolution of the effective thermal conductivity λeff /λa of simple cubic () and body centered cubic () lattices of spherical inclusions when λa /λb = 0, 001. Comparison of numerical results with the Hashin and Shtrikman (HS+ and HS-) bounds (equation 6.4)

conductivity. When the volume fraction of the inclusions is below the connectivity threshold (cb = 0.48 for the simple cubic lattice and cb = 0.32 for the body centered cubic lattice), the transfer and effective coefficient (λ+eff ) are controlled by the weakly-conducting matrix. However, the influence of highly conducting inclusions on the effective conductivity is not negligible. For a volume fraction of the matrix ca = 0.6, the effective conductivity is typically three times higher than that of the matrix. Finally, we observe that in this case the effective conductivities of the two microstructures are very close to the lower Hashin and Shtrikman bound. On the other hand, above the connectivity threshold the transfers are dictated by the connected network of highly conducting inclusions (≈ λeff b ). In this case, the effective conductivities of the two materials are very close to the upper bound of Hashin and Shtrikman. The differences, very obviously close to the connectivity threshold, reflect the influence of the geometry of the contacts between spherical particles on the transfer processes. 6.2.2. Fibrous media consisting of parallel fibers 6.2.2.1. Microstructures We will consider fibrous media consisting of a periodic square or triangular lattice of parallel fibers (phase b) buried in a matrix (phase a). The fibers have a circular cross-section, and their radius is denoted R (Figure 6.7).


l

l 2R b

169

l

a

b 2R

$ l 3

a

e2 (a)

e1

(b)

Figure 6.7. Fibrous media consisting of (a) square and (b) triangular arrangements of parallel fibers with a circular cross section, embedded in a matrix. Characteristic dimensions of the period

The fibrous networks we will consider are not connected. For the square network, the volume fraction of the fibers cb = πR2 /l2 , varies between 0 and cbmax = π/4 ≈ 0.785. For of the fibers is given by cb = √ the triangular lattice, the volume fraction √ 2πR2 /( 3l2 ) and varies between 0 and cbmax = π/(2 3) ≈ 0.907. Given the symmetries of each of these microstructures, the effective thermal conductivity tensor is transversely isotropic and can be written: eff λeff = λeff T (e1 ⊗ e1 + e2 ⊗ e2 ) + λL e3 ⊗ e3

(6.5)

eff eff eff eff where λeff T (= λ11 = λ22 ) and λL (= λ33 ) are the transverse and longitudinal effective thermal conductivities respectively.

6.2.2.2. Solution to the boundary value problem over the period As for composites with spherical inclusions, the effective thermal conductivity of the materials under consideration, defined by equation (6.2), is obtained by determining the y∗ -periodic vector t∗ (y∗ ) = (t∗1 , t∗2 , t∗3 ) which solves the canonical problem P(λ∗a , λ∗b ) (4.26–4.30) over the period. For the microstructures we are considering, all components t∗i are independent of y3∗ and it must be the case that t∗3 = 0. As a result, determination of the vector t∗ reduces to solving the canonical problem P(λ∗a , λ∗b ) in the plane (e1 , e2 ). This problem has been solved by finite element analysis [COM 08]. Figures 6.8 and 6.9 show respectively, for each of the microstructures studied, the mesh used for the numerical simulations as well as the solutions t∗1 and t∗2 . The symmetries of the microstructure imply that t∗1 (t2 ) is symmetric and symmetric with respect to the axes e1 and e2 (e2 and e1 ).

170


e2

e1

(a)

(b)

(c)

Figure 6.8. Fibrous media consisting of a square lattice of parallel fibers (ca = 0.5, λa /λb = 100): (a) example of the mesh used for finite element numerical simulation, (b) t∗1 and (c) t∗2 (black = -0.125, white = 0.125), solutions to the canonical problem P(λ∗a , λ∗b )

e2

(a)

e1

(b)

(c)

Figure 6.9. Fibrous media consisting of a triangular lattice of parallel fibers (ca = 0.5, λa /λb = 10) (a) example of the mesh used for finite element numerical simulation, (b) t∗1 and (c) t∗2 (black = -0.277, white = 0.277), solutions to the canonical problem P(λ∗a , λ∗b )

6.2.2.3. Effective thermal conductivity Since t∗3 = 0, equation (6.2) implies that the longitudinal conductivity λeff L (= ) of the two microstructures under consideration is the arithmetic mean of the λeff 33 conductivities of constituents a and b: λeff L = ca λa + (1 − ca )λb

(6.6)


171

This result can easily be seen from the fact that the transfer parallel to the axis of the fibers occurs at the same constant temperature gradient. This does not apply for transfer perpendicular to the axis of the fibers. Figure 6.10 shows, for two conductivity ratios λa /λb (=10 or 100), evolution of the effective transverse thermal conductivity λeff T /λa determined from numerical simulations as a function of the volume fraction of the matrix ca . As for composites with spherical inclusions, these figures show that (i) for the two microstructures under consideration, and whatever the conductivity ratio λa /λb (with λa > λb ), λeff T is a monotonically increasing function of ca , and (ii) whenever the conductivity ratio λa /λb , λeff T becomes sensitive to the arrangement of the fibers when the volume fraction of the matrix ca falls below 0.4, i.e. when the volume fraction of the fibers cb > 0.6. 6.2.2.3.1. Comparison with the Voigt (V), Reuss (R) and Hashin and Shtrikman (HS+ and HS-) bounds eff The upper Voigt (V) bounds and lower Reuss (R) bounds for λeff T and λL are given by equation (6.3). These bounds do not include the macroscopic anisotropy induced by the fiber arrangement. As we have already emphasized, when the transfer occurs parallel to the axis of the fibers, each phase is subject to the same constant temperature V gradient, and for this reason the Voigt bound is reached: λeff L =λ .

Despite the anisotropy caused by the arrangement of the fibers, it is interesting to compare the numerical values of λeff T with the bounds established by Hashin and Shtrikman [HAS 63] for a two-dimensional isotropic material. For λa > λb these bounds become: λHS− 2D = λb +

ca +

1 λa −λb

cb 2λb

λeff T λa +

cb +

1 λb −λa

ca 2λa

= λHS+ 2D

(6.7)

As shown in Figure 6.10, the effective conductivities of the two microstructures lie between these bounds whatever the conductivity ratio and the volume fraction of the matrix ca . As for composites with spherical inclusions, it can be seen that the numerical results are very close to the upper bound of Hashin and Shtrikman (6.7) across a wide range of volume fractions. The differences become more pronounced close to the percolation threshold of the fibers. 6.2.2.3.2. Influence of the thermal conductivity contrast λa /λb In the same way as for composites with spherical inclusions, Figure 6.11 emphasizes the high sensitivity of conductivity λeff T to the thermal conductivity contrast λa /λb , when 10−2 < λa /λb < 102 . A good agreement can also be seen between the numerical results and the upper bounds (if λa /λb > 1) and lower bounds (if λa /λb < 1) of Hashin and Shtrikman.

172


a / b = 10 0.8

e& / T a

0.6

V HS+

0.4

HS-

R 0.2

0 0

0.2

0.4

ca

0.6

0.8

1

1

a / b = 100

e& / T a

0.8

0.6

V HS+

0.4

0.2

HS-

R 0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.10. Evolution of the effective transverse thermal conductivity λeff T /λa of fibrous media consisting of a square () or triangular () arrangement of parallel fibers as a function of the volume fraction of the matrix ca . Comparison, for two conductivity ratios λa /λb (=10 or 100), of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

6.2.2.3.3. Exchange of properties between the phases Keller [KEL 64] showed that the effective transverse conductivities of any parallel fiber network which has two orthogonal axes of symmetry e1 and e2 , have the following property: eff λeff 11 (λa , λb ) λ22 (λb , λa ) = λa λb

(6.8)


3

HS+ V

HS-

2

173

ca = 0.5

e& / T a

R

1

V R 0 10-4

10-2

100

HS-

102

HS+

104

a / b Figure 6.11. Evolution of the effective transverse thermal conductivity λeff T /λa of fibrous media consisting of a square () or triangular () lattice of parallel fibers as a function of the conductivity ratio λa /λb (ca = 0.5). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

eff where λeff ii (., .) = λii (λmatrix , λfiber ) (without summation). This property was then generalized by Mendelson [MEN 75] to any microstructure whose axes e1 and e2 are principal axes of the effective conductivity tensor. For the square and triangular lattices we are considering here, axes e1 and e2 are axes of symmetry and, conveniently, are also principal axes of λeff . Because of this, property (6.8) becomes:

λeff T (λa , λb ) = λa

λeff T (λb , λa ) λb

−1 (6.9)

This property is satisfied by the numerical results presented in Figure 6.11. If we know the transverse conductivity λeff T (λa , λb ) of a given microstructure, this equation means we can easily determine the effective transverse conductivity of this same microstructure if the properties of the phases are interchanged – in other words λeff T (λb , λa ). 6.2.2.3.4. High conductivity contrast When the conductivity ratio λa /λb = ∞ (or λb = 0), the effective thermal conductivity tensor reduces to that of the matrix a alone: λeff = λeff a (Chapter 4,

174


a / b =

e& / Ta a

0.8

0.6

V HS+ 0.4

0.2

HS-

R

0 0

0.2

0.4

ca

0.6

0.8

1

Figure 6.12. Evolution of the effective transverse thermal conductivity λeff Ta /λa of fibrous media consisting of a square () or triangular () lattice of parallel non-conducting fibers as a function of the volume fraction of the matrix (λa /λb = ∞). Comparison of numerical results with the bounds of Voigt (V), Reuss (R) (equation (6.3)) and Hashin and Shtrikman (HS+ and HS-) (equation (6.7))

section 4.2.3.2). The effective longitudinal conductivity for the two microstructures considered reduces to: eff λeff L = λLa = ca λa eff The transverse conductivity λeff T = λTa can be obtained as before by numerical solution of the canonical boundary value problem P(λ∗a , 0) (4.26–4.30) over the period. The evolution of λeff Ta /λa as a function of the volume fraction ca is shown in Figure 6.12.

The trends observed above for a finite conductivity ratio λa /λb are still valid. The upper bound of Hashin and Shtrikman [HAS 63] remains a good approximation to λeff Ta across a wide range of concentration ca . Finally we note again that these results can be applied to all types of diffusion phenomena, in particular to solute diffusion, diluted in a fluid saturating the fibrous medium. By analogy, λeff a (ca )/λa becomes Ddif (φ)/Dmol (Chapter 5, section 5.3). Finally, if the conductivity ratio λa /λb tends to zero, the effective longitudinal conductivity (6.6) reduces to: eff λeff L = λLb = (1 − ca )λb


175

In the transverse direction, this situation corresponds to highly conductive fibers (inclusions) embedded in a matrix. In this particular case, the effective thermal conductivity tensor λ+eff is obtained by solving the boundary value problem across the period (4.63–4.64) (Chapter 4, section 4.2.4). For the microstructures considered, the property (6.8) implies that: λ+eff T = λa

λeff Ta λa

−1

The ratio λ+eff T /λa is clearly very close to the lower bound of Hashin and Shtrikman [HAS 63] across a wide range of the volume fraction ca (in contrast with the ratio λeff Ta /λa , Figure 6.11). 6.3. Study of various self-consistent schemes The principle of self-consistent schemes was introduced in Chapter 1, taking as an example of application the traditional scheme where each constituent is treated independently of the others using a spherical substructure. We emphasized the morphological restrictions associated with this scheme (which leads to the fact that any phase with a concentration below one-third is dispersed). Here we will consider both procedures which can be applied to generic composite substructures, and also combinations of self-consistent schemes. These approaches, which do not require the periodicity of the medium, and lead to simple analytical formulations, can be justified in the context of random media and are often used for the representation of materials. 6.3.1. Self-consistent scheme for bi-composite inclusions 6.3.1.1. Granular or cellular media Consider a bi-composite medium where the a constituent, with concentration ca , is connected and the b constituent, with concentration cb = 1 − ca , is dispersed. Following the concept developed by [BRU 35; KER 56b; LAN 52; HAS 63], a single substructure Ω – consisting of a sphere Ωb of radius Rb of the b constituent surrounded by a concentric sphere Ωa of external radius Ra of medium a, following the same volume proportions as the real material (Figure 6.13) – is sufficient to describe the material. We recall the reasoning developed in Chapter 1 (section 1.4) when embedding the composite sphere in a medium of conductivity λeff . Subjecting this system to a uniform temperature gradient G = |G|ej = Gej at infinity, the macroscopic (isotropic) conductivity satisfies general relation (1.29): λeff =

N α=1

cα βα

176


G = G ez

ez

er

M

r Ra ( b

ey

O Rb a

ex

Homogeneous medium

Figure 6.13. Generic spherical composite substructure Ω of a bi-composite granular medium where the a constituent is connected

which for a single substructure reduces to: λeff = β(λeff , λa , λb )

(6.10)

where we recall that: β(λeff , λa , λb )G =

1 ej · Ω

λb gradTb dΩ

λa gradTa dΩ + Ωa

Ωb

is the projection in the direction of G of the mean flux crossing the substructure. We note that since there is only one substructure, the imposed gradient G is also the mean gradient over the substructure (for proof see section 1.4) so that: 1 G= Ω

λb gradTb dΩ = gradT

λa gradTa dΩ + Ωa

Ωb

We can easily express the problem of the bi-composite spherical substructure in equation form if we use spherical coordinates (r, θ, ϕ), with the origin at the center of the substructure Ω defined by r < Ra , and the θ = 0 axis given by G. First of all, to encapsulate the isotropy of the space, the temperatures Tα in each medium (α = a, b), and the temperature T in the homogenous medium, must necessarily take the form: Tα = fα (r · r, r · G, G · G),

T = f (r · r, r · G, G · G)

Taking account of the linear dependence on G, Tα becomes: Tα = r · GFα (r) = Gcos(θ)fα (r)


177

whence: gradTα = G(fα cos(θ)er −

fα sin(θ)eθ ) r

Additionally, the Fourier equation can be written: λα (Tα ) = 0,

so that

r−2 ((r2 fα ) − 2fα ) = 0

and, integrating: fα (r) = Aα

Ra2 r + Bα 2 r Ra

Since the temperature takes a finite value at the center of the substructure, Ab = 0. This means that the flux is uniform in the internal sphere. Following the same methodology, we have in the homogenous medium: f (r) = A

Ra2 r +B 2 r Ra

Also, since G is the uniform gradient applied at infinity, we require that B = G. From continuity of temperature: gradTa dΩ + gradTb dΩ = Ta ndS = T ndS = Ω(A + B)ej Ωa

Ωb

∂Ω

∂Ω

and since G = gradT it follows that A = 0. Hence the flux is also uniform in the medium outside the substructure, so that at its boundary the substructure is subject to homogenous conditions of temperature gradient and flux. Because of this, we know that (for proof see Chapter 1, Reuss bound): 1 λa gradTa dΩ + λb gradTb dΩ = λeff gradT = λeff G Ω Ωa Ωb an equality which is consistent with (6.10). We still need to express the conditions of continuity of normal flux and of temperature at the two interfaces (r = Rb , r = Ra ). Thus we obtain the following system of four equations in four unknowns: Bb

=

λb Bb

=

Aa + Ba R3 λa −2Aa a3 + Ba Rb

(6.11) (6.12)

178


Aa + Ba

=

G

(6.13)

λa (−2Aa + Ba ) = λeff G

(6.14)

which only has non-trivial solutions for: λeff = λa (1 +

1 − ca ) 1 ca − λb 3 1− λa

(6.15)

6.3.1.2. Fibrous media For fibrous media we use a generic substructure consisting of a bi-composite cylinder (Figure 6.14), to which we apply the same calculation, bearing in mind that this geometry requires us to treat the transfers both along the axis and perpendicular to the axis of the fibers. The solution for transfer parallel to the axis of the fibers is obvious since they are subjected to a constant gradient: the macroscopic longitudinal conductivity is thus the arithmetic mean of the conductivities of the constituents. λeff L = ca λa + (1 − ca ).λb The treatment of transfer perpendicular to the axis of the cylindrical substructure closely follows that for the spherical substructure. In the plane, r and θ designate the polar coordinates, Tα is the temperature in media α = a, b and T is the temperature in the homogenous medium. These take the form: Tα = Gcos(θ)gα (r),

T = Gcos(θ)g(r),

gα (r) = Aα

with

with

g(r) = A

Ra r + Bα r Ra

r Ra +B r Ra

The system to be solved is thus: Bb

=

Aa + Ba

λ b Bb

=

R2 λa (−Aa a2 Rb

Aa + Ba

=

G

(6.18)

λeff T G

(6.19)

λa (−Aa + Ba ) =

(6.16) + Ba )

(6.17)


G = G ey

ey

179

er M r

Ra ( b

O

ex

Rb a Homogeneous medium

Figure 6.14. Generic composite cylindrical substructure Ω for a bi-composite fibrous medium where the a constituent is connected

which gives: λeff T = λa (1 +

1 − ca ) 1 ca − λb 2 1− λa

(6.20)

6.3.1.3. General remarks on bi-composite models We have seen that the bi-composite spherical (or cylindrical) substructure is neutral with respect to the equivalent medium, in the sense that replacing sphere (or a cylinder) in the equivalent medium with a substructure in no way disturbs the exterior field, which remains uniform. This operation can therefore be repeated until the point where the whole space is occupied by homothetic substructures. As a result, the value λeff is an exact value for this particular class of materials, whether the organization of the substructures is periodic, random, or anything else, and is independent of their size distribution (see Figure 6.15). As we have already mentioned several times, Hashin and Shtrikman [HAS 63] showed that for any bi-composite (random or otherwise) which has macroscopic isotropy, the value (6.15) determined using the more conductive connected constituent (λa > λb ) defines an upper bound (HS+), and conversely, (6.15) with the less conductive part (λa < λb ) defining a lower bound (HS-). It can easily be verified that, in these models, the external phase of the substructure is necessarily connected: whether the conductivity of the internal phase is zero or infinite, the effective conductivity remains finite. Thus, in contrast to traditional schemes, we can specify the connected or disperse nature of the phases, independent

180


Figure 6.15. Class of media consisting of homothetic composite spherical substructures for which the self-consistent result (6.15) is exact

of their concentration. It can also be observed that in the bi-composite inclusions the interaction between phases is explicit, whereas in the traditional scheme it is only expressed by way of the effective medium. These results apply to all phenomena governed by a diffusion process. For example, for diffusion of a solute with a diffusion coefficient Dmol , diluted in a fluid saturating a porous, granular, isotropic and inert medium, of porosity φ, the effective diffusion Ddif would be: 1 − φ 2φ (6.21) Ddif = Dmol 1 + φ = Dmol 3 −φ 3 −1 As for pressure diffusion, the permeability of mixtures consisting of clay with permeability ka and concentration ca , associated either with impermeable sand grains or with highly permeable pockets of water, would be respectively: 1 − ca eff = ka 1 + ca kas , 3 −1

1 − ca eff = ka 1 + 3 kaw ca

(6.22)

For macroscopically isotropic media, with the help of (6.15) we obtain estimates of the effective conductivities using models developed in Chapter 4: – bi-composites with conductivities of the same order of magnitude: 1 − ca ) λeff = λa (1 + 1 ca − λb 3 1− λa

(6.23)


181

– highly conducting inclusions in a matrix: if we take the limit λb → ∞ we obtain: 1 − ca ) (6.24) λ+eff = λa (1 + 3 ca – weakly conducting inclusions: in the limit λb → 0 we obtain: 1 − ca ) (6.25) λeff a = λa (1 + ca 3 −1 We note that the term expressing the local non-equilibrium situation is calculated over the internal sphere only, and in the harmonic regime gives [AUR 83]: λb 1 3ωt iω iω (1 − τ (ω)dΩ = cb 1 + coth ) with ωt = 2 Ω Ωb iω ωt ωt Rb ρb Cb which concludes the formulation of the macroscopic parameters for this model. In Figures 6.3 and 6.10, the analytical values (6.15) and (6.20) (denoted HS+) are compared to those obtained numerically for periodic assemblies of spheres or cylinders, as a function of their concentration ca . We note the very good agreement of the two self-consistent and numerical results for the bi-composite substructure (HS+) as long as the inclusions are dispersed in the matrix. When this is not the case, the values diverge. In fibrous media, contact between fibers leads to an inversion of connectivity of the phases, which explains the observed differences. In a granular medium, contact between the spheres implies connectivity of both phases, and thus only a small difference. Approximations for other substructures close to their connectivity thresholds have been established by Andrianov et al. [AND 99]. Since in addition we know that the self-consistent values are exact regardless of the assembly (random or otherwise) of homothetic substructures, it is reasonable to assume that, other than for very specific morphologies, the self-consistent values (for a bi-composite spherical or cylindrical substructure) offer an acceptable estimate for bi-composites (cellular or fibrous) where the matrix is connected and the inclusions dispersed. 6.3.2. Self-consistent scheme with multi-composite substructures 6.3.2.1. n-composite substructure The same principles can be used to treat n-composite substructures formed of concentric embedded spheres (or cylinders). Such a structure represents a medium formed of composite inclusions dispersed in a connected matrix, represented by the external surface of the substructure (Figure 6.16). For a medium with n constituents, numbered starting with constituent 1 at the center to constituent n on the outside of the substructure, the effective conductivity

182

Homogenization of Coupled Phenomena G = G ez

ez

er

( Rn

M

r ey

O 1

2 n1 n Homogeneous medium

ex

Figure 6.16. n substructure-composite

can be found by solving a 2nx2n system of equations. However, since boundary conditions at the interfaces are all of the same nature we can proceed by recurrence [BOU 96b]. Thus for an n-composite medium, defining: γj = 1 − j

cj

j = 1, n

k=1 ck

the expression for the effective conductivity takes for form of a finite continued fraction: λeff = λn (1+

γn 1 − γn − 3

1−

γn−1

λn−1 (1 + λn 1 − γn−1 − 3

where the term associated with layer i is: )

1 1−

..

. λi

)

1 .. 1−

γi λi (1 + .. 1 − γi . − 3

)(6.26)

1

.

..

.

γ2 λ2 (1 + λ3 1 − γ2 − 3

)

1 1−

λ1 λ2


183

For example, for a tri-composite medium, the effective conductivity would be given by: λeff = λ3 (1 +

γ3 1 − γ3 − 3

)

1 1−

γ2 λ2 (1 + λ3 1 − γ2 − 3

)

1 1−

(6.27)

λ1 λ2

6.3.2.2. Treatment of a contact resistance In a multi-composite substructure, suppose one of the layers i has infinitesimal thickness d = Ri+1 − Ri−1 and surface area Γ, so that the concentration ci of this constituent is linked to its specific surface area σ = Γ/Ω by ci = σd. How can we express the conditions between constituents i − 1 and i + 1? In layer i the normal components of the temperature gradient and flux are effectively constant around the values: 1 (Ti−1 (Ri−1 ) − Ti+1 (Ri+1 )) , d

and

Q=

λi (Ti−1 (Ri−1 ) − Ti+1 (Ri+1 )) d

Thus, to a first approximation, we can deduce from the two continuity equations for the flux on the two surfaces of layer i: λi−1 grad(Ti−1 ) · er = Q

and

Q = λi+1 grad(Ti+1 ) · er

firstly the flux continuity between constituents i − 1 and i + 1: λi−1 grad(Ti−1 ) · er = λi+1 grad(Ti+1 ) · er

(6.28)

and secondly, setting h = λi /d, the relationship between the flux and the temperature differential between these two constituents: λi−1 grad(Ti−1 ) · er = h(Ti−1 (Ri−1 ) − Ti+1 (Ri+1 ))

(6.29)

Equations (6.28) and (6.29) define the boundary conditions encapsulating the presence of a contact resistance h−1 : when the resistance h−1 → ∞ the flux is zero, and the surface introduces a perfect insulation; when h−1 → 0, the temperatures are continuous and the contact is perfect. Thus we can make direct use of the results for the multi-composite medium to estimate the effect of a contact resistance between the two constituents i − 1 and i + 1,

184


i by replacing λi with hd, and γi − 1 with σd/ 1 ck . Since d ≈ 0 and γi ≈ 1 the term associated with the contact resistance takes the form: λi h γi 1 1 1 ). )= (1+ (d+ ( )≈ .. .. .. 1 1 1 1 − γi σ σ . . . − + − i i .. .. 3 .. 3 1 ck 3h 1 ck . . . 1− d− λi h Thus for bi-composite media that are macroscopically isotropic, consisting of inclusions (medium b) dispersed in a matrix (medium a), with a specific surface area of contact σ and a resistance h−1 , the values of effective conductivity for the models developed in Chapter 4 (section 4.3) can be estimated as follows: – media with very low contact resistance (h−1 → 0) (model I), we recover the value obtained with perfect contact: 1 − ca ) (6.30) λIeff = λeff = λa (1 + 1 ca − λb 3 1− λa – media with moderate contact resistance (model II), in other words: h = O(λb /(3(1 − ca ))) σ we obtain an effective conductivity which depends on the contact resistance: 1 − ca λIIeff = λa (1 + ) (6.31) 1 ca − λb 1 3 1− ( ) λb σ λa 1 + 3h(1−c a) λeff a

– media with high contact resistance (h−1 → ∞) (model III), we recover the value where only the matrix contributes to the conduction: 1 − ca ) ; λIIIeff = λeff =0 (6.32) λIIIeff = λIIIeff a a = λa (1 + ca b − 1 3

As established through periodic homogenization, the dispersed inclusions, insulated by the contact resistance, do not participate in the transfer; – models with two temperature fields (coupled and decoupled): the same values (6.32) apply and H = hσ. 6.3.3. Combined self-consistent schemes Starting from the two traditional self-consistent schemes for composite substructures, it is possible to construct other schemes which are adapted to the morphology of the material.


185

6.3.3.1. Mixed self-consistent schemes A first possibility involves applying a traditional scheme with composite inclusions. For example, for a medium with three constituents a, b and c, media b and c both being dispersed in matrix a with concentration ca < 1/3, it is clear that neither the traditional scheme nor the tri-composite sphere scheme are acceptable: in the traditional scheme phase a – with a concentration less than 1/3 – will be dispersed, and in the tri-composite scheme one of the constituents b or c will not be in contact with matrix a. If on the other hand we consider two bi-composite schemes {a, b} and {a, c}, and if we apply the traditional scheme to these, we are then able to respect the morphological charateristics of connectivity. We note however that the model is not unique since the respective concentration of the two substructures is an adjustable parameter. 6.3.3.2. Multiple self-consistent schemes Another possibility is to apply a self-consistent scheme several times. Consider for example a medium with three constituents, consisting of inclusions c in a matrix α, which itself consists of inclusions b in a matrix a. If a significant difference in size exists between constituents a, b and inclusions c – the condition of separation of length scales – the heterogenous medium a, b behaves like a homogenous medium on the scale of the inclusions c. We can then proceed with a double homogenization, determining first of all the effective conductivity λα of the matrix α with the help of a bi-composite inclusion {a, b}, and then the effective properties of the whole system using a substructure {α, c}. In this way the model we have constructed is unique, and its morphological structure is explicit (Figure 6.17), with phase a being connected, and phases b and c dispersed, but with very different sizes.

Figure 6.17. Morphology associated with a double homogenization procedure with a bi-composite inclusion

186


Applying result (6.15), and observing that in substructure {a, b} the concentrations of constituents a, b are respectively ca /(1 − cc ) and cb /(1 − cc ), and that in the substructure {α, c} the concentrations are cα = 1−cc and cc , we obtain in succession: λα = λa (1 +

cb ca − 3

1−

and λeff = λα (1 +

),

1 λb λa

cc 1 − cc − 3

)

1 λc λα

1−

With the same concentrations ca , cb , cc we could also propose another model where the phase a remains connected, swapping the role of phases b and c, which then gives: λeff = λβ (1 +

cb 1 − cb − 3

),

1 1−

with

λb λβ

λβ = λa (1 +

cc ca − 3

)

1 1−

λc λa

By way of example, we will consider the case of the permeability of a soil consisting of saturated clay (ca , ka ), grains of sand (cs , ks = 0) and pockets of water (cw , kw = ∞) [IBR 02]. The model where the sand/clay mixture surrounds the water pockets gives: λeff = λa (1 + 3 ca 3

cs −1

)(1 + 3

cw ) 1 − cw

whereas the model where the clay and the water pockets surround the impermeable grains gives: λeff = λa (1 + 3

cw cs )(1 + ) 1 − cs ca −1 3

Figure 6.18 illustrates the significant difference between the two models and demonstrates the impact of the choice of morphology. Still on the subject of multiple combined schemes, we mention the recursive (or differential) model which proceeds by successive homogenizations with the traditional scheme [HAS 88; BER 02]. Its morphological interpretation is difficult because of the fact that, at each stage, interactions between substructures occur through the intermediary of an effective medium, which assumes a very wide range of scales for the heterogenities. These examples should serve to demonstrate that the self-consistent approach with a composite substructure makes it fairly easy to describe media with inclusions dispersed in a matrix. On the other hand, media where several constituents are connected are clearly less easy to model.


187

cw

cs cw ez

cs

Figure 6.18. Impact of the morphology on the effective permeability of clay-sand-water mixtures. Isovalues normalized by the permeability of clay as a function of the concentration of sand cs and free water cw . (a) pockets of water dispersed in a sand-clay mixture (b) inclusions of sand dispersed in a macroporous saturated clay

Furthermore, we see that by altering the combination of self-consistent schemes used, many different models are possible. However, the difficulty here is not in establishing the value of the effective coefficient, but rather in establishing what type of material it is reasonable to apply the model to. This is one of the important limitations of the self-consistent approach as compared to periodic homogenization.

188


6.4. Comparison with experimental results for the thermal conductivity of cellular concrete In this section we will compare self-consistent estimates with experimental results in order to judge the appropriateness of these models for describing the properties of real materials. We will concentrate here on the thermal properties of dry and damp cellular concrete, as studied in Boutin [BOU 96a]. Autoclaved cellular concrete is a construction material known for its insulating properties. Its weak thermal conductivity – lying between 0.08 and 0.3 W/mK – is of course higher than that of air (λair = 0.026 W/mK), but considerably lower than that of the minerals which it is made from (λs = 0.894 W/mK). This is a result of its high porosity – 0.65 < φ < 0.85 – which also leads to a low density of between 350 kg/m3 and 800 kg/m3 , clearly lower than that of the minerals it is made from ρs = 2650 kg/m3 . Its insulating properties depend strongly on its density, and we also know that for a given density the thermal conductivity increases significantly with the presence of water in the material (λw = 0.602 W/mK). Based on experimental studies, empirical relationships have been established to link the conductivity to these two parameters, density and water level. We will compare these trends with selfconsistent estimates.

Figure 6.19. Macro-porosity of aerated concrete


189

The self-consistent approach with a composite substructure seems well suited to cellular concrete. This consists of a combination of porous structures on a series of very different length scales. We distinguish: – quasi-spherical macropores (with a diameter of the order of a millimeter) surrounded by a solid membrane (Figure 6.19); – a mesoporosity of the solid membrane formed of pores (with a size of around 10 μm) separating the crystalline aggregates; – finally, the microporosity of the crystalline aggregates (≈ 0.1μm). This scheme is not however entirely adequate, because although the unconnected spherical pores give a good description of the macroporosity, these only give a partial approximation to the porosity of the crystalline aggregates. This material, whose constituents are in perfect contact, is described by the canonical model (see Chapter 4, section 4.2.2). Although the air is a poor conductor compared to the solid, the memory effect behavior is never achieved, because the heat capacity of the air (ρCair = 10−3 J/Km3 ) is clearly smaller than that of the solid (ρCs = 1.6 J/Km3 ). To convince ourselves of this, we calculate in the harmonic regime the frequency ωnel at which the dynamic regime (local non-equilibrium) applies in the pores. It is enough to express that the thermal boundary layer thickness is of the same order as the size of the macropores lpore = 1 mm so that: λair = lpores , ρCair ωnel

numerically

ωnel ≈ 2.5 × 107 Hz

The macroscopic conduction model is only correct if the separation of length scales is respected. Let ωse be the frequency fulfilling this condition. Given the properties of the material, this condition leads us to: λbeton lpores , (1 − φ)ρCs ωse

numerically

ωse 1.2 × 104 Hz ωnel

As a consequence, when the problem is homogenizable, the air in the pores is still in the quasi-static regime, in other words in local equilibrium.

6.4.1. Dry cellular concrete To treat dry cellular concrete, we make direct use of result (6.15), considering that the material consists of bubbles of air surrounded by a solid membrane. Given the weak conductivity of the air with respect to that of the solid particles (λair λs ) and

190


expressing total porosity φ as a function of densities of the solid constituent ρs and the material ρ: ρ = ρs (1 − φ) we obtain, starting from equation (6.15): λeff (ρ) = λs (1 +

φ 1−φ 3

−1

) = λs

2 2 = λs 3ρs −1 ρ −1

3 1−φ

(6.33)

Given the values of ρs and λs , in the normal range of densities the curvature of the function λeff (ρ) is very slight, which means we can replace it by its gradient at the point ρ = 500 kg/m3 : λeff (ρ) ≈ (−8.05 + 0.261ρ)10−3

(6.34)

Figure 6.20 shows the very good agreement between this theoretical expression, experimental results of Frey [FRE 92], and the empirical formula proposed by Millard [MIL 92]: λeff (ρ) ≈ (−10.87 + 0.266ρ)10−3

Figure 6.20. Model-experiment comparison for the conductivity of dry cellular concrete as a function of its density. The continuous lines, which are almost coincidental, represent the empirical correlation of Millard [MIL 92] and the bi-composite inclusion model (equation 6.34). The squares () are the experimental values of Frey [FRE 92]

6.4.2. Damp cellular concrete Cellular concrete can take up water by condensation of water vapor in the smallest pores. Following the conventions established by experimental results, conductivity in


191

the damp state λeff (ρ, u) follows from conductivity in the dry state λeff (ρ), and from the content of water u by mass, with the empirical relationship: λeff (ρ, u) ≈ λeff (ρ)(1 + 4, 00u) For this damp material, we will examine two models. The first model uses the self-consistent scheme with tri-composite inclusions with a layer of water, whose thickness depends on u, lying between the air and the solid membrane. In this description, the concentration of the water is the same whatever the size of the pores. The concentration of water in the pores cw can be expressed as a function of the content by mass of water u: u = cw φ

ρw ρ

where:

cw =

u ρw

1 1 − ρ ρs

−1

Due to the weak conductivity of the air compared to those of the two other constituents, the conductivity of cellular concrete with dry density ρ and a water content u can be written with the help of (6.27 ) as: λeff (ρ, u) = λs (1 +

φ 1−φ − 3

1− 1−

cw 3

)

cw λw (1 + 2 ) 3 λs

and, in linearized form close to u = 0: λw ρs 1 eff eff λ (ρ, u) ≈ λ (ρ) 1 + u 1 − 3ρρs λs ρw a form which allows us to compare these results to those in the literature. For example, for ρ = 500 kg/m3 , we find: λeff (ρ, u) ≈ λeff (ρ) (1 + 1.91u) Compared to the empirical relation, it is clear that the tri-composite sphere approach agrees qualitatively with the experiment, but underestimates the effect of the water content. This can be attributed to a poor description of the distribution of water in the pores. In the second approach, we will proceed by “double homogenization”. In order to better describe the distribution of water, which mostly occupies the small pores, we will consider that the air bubbles are surrounded by a porous medium saturated with water. We will then proceed by first determining the effective conductivity of the

192


saturated medium, and then the conductivity of the ensemble. This approach assumes implicitly that the saturated pores are much smaller than the dry pores, so that on the length scale of the latter the saturated membrane can be considered as a homogenous medium. Following this description, the density of damp cellular concrete is that of the saturated medium. The volume density of water cw in this medium is thus directly linked to the water content by mass by the relationship: cw = u

ρs ρw

With a bi-composite structure where water is trapped in the solid sphere – which assures the connectivity of the solid phase – conductivity of the saturated membranes can be written: λsw = λs (1 +

cw

1 − cw 1 − 3 1 − λλws

)

and, with the presence of saturated pores of air with concentration cair , we obtain (neglecting λa ): λeff (ρ, u) = λsw (1 +

cair 1−cair − 3

1

)

In order that this description with double homogenization should also reflect the properties of the dry material (u = 0), we must have: (1 +

cair cw φ ) = (1 + )(1 + ) 1 − cair 1−φ 1 − cw −1 −1 −1 3 3 3

from which it follows that: λeff (ρ, u) = λeff (ρ)(1 +

cw

1 − cw 1 − 3 1 − λλws

) (1 +

cw )−1 1 − cw −1 3

For water contents u < 0.2, approximating the curves by their tangent at the origin, we obtain the following approximation: λeff (ρ, u) ≈ λeff (ρ)(1 + u

9 2(2 +

λw λs )

λw ρ s ) λs ρw


193

which gives numerically: λeff (ρ, u) ≈ λeff (ρ)(1 + 3.45u) This equation is in good agreement with experimental results, particularly since the coefficient obtained from the slope at the origin underestimates the dependence on u. Thus, for normal water contents, the approach by “double homogenization” appears to better describe the microstructure and gives a satisfactory description of the thermal properties of damp cellular concrete. In conclusion, despite simplifying the assumptions, we have obtained acceptable estimates for the conductivity of dry or damp cellular concrete in the form of simple analytical expressions which clearly show the essential parameters. This result is not specific to cellular concrete; other examples can be found which have a good agreement between theory and experiment, particularly for materials based on particles of vegetable matter [ARN 00; ARN 04], and for the permeability of heterogenous soils [IBR 02; KAC 04].


PART THREE

Newtonian Fluid Flow Through Rigid Porous Media


Chapter 7

Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium

7.1. Introduction The slow flow of a Newtonian fluid in an isotropic porous medium is traditionally described using Darcy’s law [DAR 56], Darcy having carried out the first experimental study of flow through porous media: v = −

K gradX p η

(7.1)

where v is the fluid flux across a surface of the porous medium, K the permeability, η the dynamic viscosity and p the pressure. This is an example of the laws which had been discovered some time previously to model the effect of a viscous fluid on an obstacle. Indeed, the physics described by (7.1) is that of the interaction of a Newtonian fluid with an obstacle: a porous medium is an obstacle just like any other! Newton [NEW 87; NEW 26] analyzed the resistance R of a fluid on a moving body and modeled it with: R = Av + Cv 2

(7.2)

where A and C are constants. The v 2 term is recognized as having an inertial origin, and Newton indicated that it is only an approximation – something with which all his successors would concur (Newton also proposed R = Av + Bv 3/2 + Cv 2 ). The laws of (7.1) and (7.2) describe a one-dimensional flow. Clearly Darcy’s law (7.1) is equivalent to that of Newton (7.2) when the velocity is small. de Coulomb

198


gradP/hvi

[deC 01] also obtained equation (7.2) based on meticulous experiments. Equation (7.2) would then be rediscovered by many scientists [deP 04; WEI 45; DAR57 (DarcyWeisbach law); FOR 01 (Forchheimer law)]. Reynolds [REY 83] was the first to have indicated the ranges of validity of the v and v 2 behavior. Chézy’s law [CHÉ 75; MOU 21], which can be written as R = Cv 2 , also appears as a special case of (7.2). It is hard to believe that none of these authors were aware of Newton’s work. Subsequently, many generalizations of Darcy’s law have been applied to more complex

hvi Figure 7.1. Evolution of the resistance to flow as a function of velocity. The points correspond to the experimental results of Rasoloarijoana and Auriault [RAS 94]. Darcy’s law, a constant resistance obtained for very low velocities, can be corrected by weak inertial effects up to a velocity of 0.06 m/s (parabolic variation in resistance), beyond which inertial effects dominate (linear variation of resistance)

situations. In theoretical terms, we refer the reader to Marˇusić-Paloka and Mikelić [MAR 00] for the treatment of non-linearities which are neglected in the laminar regime. Equation (7.2) is only an approximation. We will not concern ourselves with investigation of this here. In this chapter, we intend to define the laws describing the flow of an incompressible viscous Newtonian fluid, using the method of multiple scale expansions, in a rigid porous medium in various different situations: – slow steady-state flow of an incompressible Newtonian fluid in a rigid matrix: Darcy’s law; – linear dynamics of an incompressible Newtonian fluid in a matrix; – steady-state flow of an incompressible Newtonian fluid in a rigid matrix: the appearance of inertial non-linearities.

Incompressible Newtonian Fluid Flow

199

The work of Matheron [MAT 67] offers an interesting and different approach to that presented here, one which relies on probabilities. 7.2. Steady-state flow of an incompressible Newtonian fluid in a porous medium: Darcy’s law Here we will determine the macroscopic description of an isothermal flow of a Newtonian fluid in a rigid porous matrix. This topic has been the subject of much research in the context of the homogenization of periodic structures, initially by Ene and Sanchez-Palencia [ENE 75]. Other macroscopization techniques can also be applied [BEA 72; WHI 86]. A good review of early works can be found in Scheidegger [SCH 74]. We will consider a rigid porous matrix which is periodic with period Ω. The fluid occupies the pores Ωf , and Γ represents the surface of the solid (Figure 7.2). To

Lc

lc

f

s n

(a)

(b)

Figure 7.2. (a) Porous medium, (b) period

simplify things, we will assume that the viscosity of the fluid is constant. A variable viscosity, i.e. one which depends on the fast variable y∗ and which is periodic, would not introduce any further complications. The temperature is constant. The equations governing velocity v, pressure p and density ρ of an incompressible viscous Newtonian fluid of viscosity η are in general terms the following, for a flow in a matrix: – dynamics equation (Navier-Stokes): ∂v + (v grad ηΔX v − gradX p = ρ X )v ∂t – conservation of mass: divX (v) = 0

(7.3)

(7.4)

200


– no-slip condition: v|Γ = 0

(7.5)

Gravity is included here in the pressure term. Equations (7.3) and (7.4) introduce three dimensionless numbers: – the Reynolds number Re: |ρ(v gradX )v| Re = |ηΔX v| – the transient Reynolds number Rt : ∂v |ρ | ∂t Rt = |ηΔX v| – the number Q defined by: |gradX p| Q= |ηΔX v| Since the flow here is steady-state, Rt is zero. We also assume the movements to be slow: the local Reynolds number is small so that non-linearities do not appear on the macroscopic scale, at least to the first order of approximation. Because of this, we can consider that: Rel = O

ρc vc lc ηc

= O(ε),

or

ReL = O

ρc vc Lc ηc

= O(1)

Thus the dominant terms in the Navier-Stokes equation are the viscous term and the pressure term. We will now evaluate ratio QL between these two quantities. In a filtration experiment, the flow is driven by a macroscopic pressure gradient: gradX p = O

pc Lc

At the same time, v varies inside the pores with a characteristic size lc : ηΔX v = O

ηc vc lc2

It follows that: pc =O Lc

ηc vc lc2


201

The dimensionless number QL is therefore: QL = O

pc L2c Lc ηc vc

= O(ε−2 )

The dimensionless equations describing the local situation for an incompressible fluid can be written from the macroscopic viewpoint as: – dynamics equation: ε2 η ∗ Δx∗ v∗ − gradx∗ p∗ = ε2 ρ∗ (v∗ gradx∗ )v∗

(7.6)

– the conservation of mass reads: divx∗ v∗ = 0

(7.7)

– the no-slip condition: v∗ |Γ∗ = 0

(7.8)

7.2.1. Darcy’s law Having adopted the macroscopic viewpoint, we look for the unknowns v∗ and p∗ in the form: v∗ (x∗ , y∗ ) = v∗(0) (x∗ , y∗ ) + εv∗(1) (x∗ , y∗ ) + ε2 v∗(2) (x∗ , y∗ ) + · · · p∗ (x∗ , y∗ ) = p∗(0) (x∗ , y∗ ) + εp∗(1) (x∗ , y∗ ) + ε2 p∗(2) (x∗ , y∗ ) + · · ·

(7.9)

with y∗ = ε−1 x∗ , v∗(i) and p∗(i) being Ω∗ -periodic in y∗ . We will introduce these expansions into equations (7.6, 7.7, 7.8). Observing that the derivatives appearing in this system must be considered with respect to x∗ (macroscopic viewpoint), and that due to the double scale, the derivative operator becomes: ∂ ∂ + ε−1 ∗ ∂x∗ ∂y we obtain, identifying the powers of ε: grady∗ p∗(0) = 0

(7.10a)

η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 ———————-

(7.10b)

202


divy∗ v∗(0) = 0

(7.11a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.11b) ———————-

v∗(0) = 0 v∗(1) = 0

(7.12a) over Γ∗

(7.12b) ———————-

Equation (7.10a) gives p∗(0) = p∗(0) (x∗ ). Equations (7.10b), (7.11a) and (7.12a) then represent the fundamental problem which must be solved over the period, where the unknowns v∗(0) and p∗(1) are Ω∗ -periodic in y∗ . We can see that v∗(0) and p∗(1) are linear functions of gradx∗ p∗(0) : v∗(0) = −

k∗ gradx∗ p∗(0) η∗

p∗(1) = a∗ · gradx∗ p∗(0) + p∗(1) (x∗ ) where the tensor k∗ is a function of the variable y∗ . Since the pressure p∗(1) is only defined up to a constant value, we take the volume mean of the vector a∗ to be zero. Now consider (7.11b). This involves the local volume conservation of v∗(1) where divx∗ v∗(0) appears as a source term. Also, v∗(1) is Ω∗ -periodic and (7.12b) shows that it is zero over Γ∗ . The source term divx∗ v∗(0) satisfies a compatibility condition: its volume mean must be zero. This can be shown by integrating (7.11b) over Ω∗f . Using: 1 . dΩ∗ . = ∗ |Ω | Ω∗f we obtain: divx∗ v∗(0) = divx∗ v∗(0) = −

1 |Ω∗ |

=−

1 |Ω∗ |

Ω∗ f

divy∗ v∗(1) dΩ∗

∂Ω∗ f

v∗(1) · n dS ∗ = 0

(7.13)

where n is the unit normal to Γ∗ exterior to Ω∗f . Finally we have: divx∗ v∗(0) = 0, v∗(0) = −

K∗ gradx∗ p∗(0) , K∗ = k∗ η∗

(7.14)


203

which represents a macroscopic volume conservation, but also a conservation of momentum. Tensor K∗ is the permeability tensor. Equation (7.14b) is Darcy’s law (as long as v∗(0) is a flux, i.e. a surface mean – see later). For a medium with a random structure, locally stationary, Ω∗ is the representative elementary volume, and the line of reasoning which leads to (7.14) is still valid. If we then include the uniqueness of the problem (7.10b, 7.11a, 7.12a) in the steady state case, all representations lead to K∗ and (7.14) follows. When viscosity varies across the period, the flow law (7.14b) becomes v∗(0) = −Λ∗ gradx∗ p∗(0) , where Λ∗ is a hydraulic conductivity. Returning to dimensional variables with: x∗ =

X , Lc

v∗(0) =

η∗ =

η ηc

v(0) v ¯ = + O(ε), vc vc

p∗(0) =

p(0) p ¯ = + O(ε) pc pc

it follows that: ¯ divX v = O(ε),

v = −

ηc Lc vc 1 ∗ ¯ K gradX p + O(ε) pc η

But: μc Lc vc ∗ 2 ∗ 2 ∗ K = Q−1 L Lc K = lc K pc Thus we finally obtain: ¯ divX v = O(ε),

v = −

K ¯ gradX p + O(ε), η

K = lc2 K∗

¯ where O(ε) is a small term, of order ε relative to the other terms in the equality. 7.2.2. Comments on macroscopic behavior 7.2.2.1. Physical meaning of the macroscopic quantities The pressure p∗(0) does not raise any issues because, since it is independent of the local variable, its macroscopic definition is the same as that introduced at the microscopic scale. This is not a priori the case for v∗(0) , defined as a volume mean, whereas a Darcy velocity is a flux, i.e. a surface mean. In fact the two means are indeed equal here. This follows from the solenoidal character of v∗(0) . We start from the identity: ∗(0)

∂v ∂ ∗(0) ∗(0) (vk yi∗ ) = k ∗ yi∗ + vk Iik ∗ ∂yk ∂yk

204


Integrating over Ω∗f , with the divergence theorem and the no-slip condition (7.12a), we find: 1 ∗(0) ∗(0) ∗ vi = ∗ v yi nk dS ∗ |Ω | ∂Ω∗f ∩∂Ω∗ k where ∂Ω∗f and ∂Ω∗ are the boundaries of Ω∗f and Ω∗ , and n is the unit normal vector exterior to Ω∗f . Let li∗ be the dimensionless length of the period along the yi∗ axis and Σ∗i the cross-section of the period at yi∗ = li∗ (see Figure 7.3).

l2 s1

f1

n

f s

y l1 Figure 7.3. Period Ω∗ of a porous medium (two-dimensional case) ∗(0)

Σ∗fi is the fluid part of Σ∗i . Because vk yi∗ is Ω∗ -periodic in the yj∗ , j = i direction, and is zero for yi∗ = 0, we are left with: 1 ∗(0) ∗(0) ∗(0) vi = ∗ vi li∗ dS ∗ = |Σ∗i |−1 vi dS ∗ |Ω | Σ∗f Σ∗ f i

i

(without summation over i). As a result, v∗(0) is indeed a flux. 7.2.2.2. Structure of the macroscopic law The mathematical structure of Darcy’s law is different to that of the Navier-Stokes equations from the outset. It still nevertheless involves the basic principles of mechanics, which can more easily be seen when Darcy’s law is written in the form: gradx∗ p∗(0) = −H∗ v∗(0) ,

H∗ = η ∗ K∗−1

which is permissible given the invertibility of K∗ (see section 7.2.2.4). The value of dimensionless number Q defined above, or equation (7.6), gives an estimate of v(0) and thus of k and thence K: |K| = O(|k|) = O(lc2 )


205

This estimate is very approximate, since the geometry of the pores can alter the value of K to a very significant extent, as we will see in Chapters 9 and 10. The equation above is often used to define lc starting from the permeability. Finally, in order to explicitly include gravity, all we need to do is to replace gradx∗ p∗(0) with gradx∗ p∗(0) + ρ∗ g∗ , where g∗ is the gravitational acceleration. 7.2.2.3. Study of the underlying problem In order to study the boundary-value problem (7.10b, 7.11a, 7.12a) where the unknowns v∗(0) and p∗(1) are Ω∗ -periodic in y∗ , we introduce the Hilbert space V of Ω∗ -periodic vectors with zero divergence, defined over Ω∗f , with a value of zero on Γ∗ and possessing the scalar product: (α, v)V =

Ω∗ f

η∗

∂αi ∂vi ∗ dΩ ∂yj∗ ∂yj∗

Multiplying the two members of (7.11b) by α ∈ V and integrating them, we find: Ω∗ f

∗

η αΔy∗ v

∗(0)

∗

dΩ =

∗(0)

Ω∗ f

αgradx∗ p

∗

dΩ +

Ω∗ f

αgrady∗ p∗(1) dΩ∗

Taking account of the equations: αΔv = div(α grad v) − grad α grad v α grad p = div(α p) − p divα of (7.11a), of the divergence theorem, and of (7.12a), it follows that: ∀α ∈ V,

∂αi η∗ ∗ ∗ ∂y Ωf j

∗(0)

∂vi dΩ∗ = − ∂yj∗

Ω∗ f

αi

∂p∗(0) ∗ dΩ ∂x∗i

(7.15)

Similarly, it is possible to show [SAN 80] that this formulation is equivalent to (7.10b), (7.11a) and (7.12a). The existence and uniqueness of the solution then follows from the Lax-Milgram theorem [NEC 67]. 7.2.2.4. Properties of K∗ Permeability tensor K∗ is positive and symmetric when the pores are connected in all three spatial directions. Let k∗j /η ∗ be the specific solution to (7.15) for velocity v∗(0) when: 1 if i = j, gradx∗i p∗(0) = 0 otherwise.

206


Consider the form (7.15) with on one hand v∗(0) = k∗q /η ∗ and α = k∗p /η ∗ , and on the other hand v∗(0) = k∗p /η∗ and α = k∗q /η. It follows, given the symmetry of the scalar product, that: ∗ ∗ ∂kpi ∂kqi ∗ ∗ ∗ ∗ dΩ = − kqp dΩ = − kpq dΩ∗ ∂yj∗ ∂yj∗ Ω∗ Ω∗ Ω∗ f f f The tensor K∗ is thus symmetric. We note that this symmetry follows from the conservation equations on the scale of the pores. Now consider (7.15) with α = v∗(0) : 1 |Ω∗ |

Ω∗ f

η∗

∗(0)

∗(0)

∂vi ∂vi dΩ∗ = ∂yj∗ ∂yj∗

−

1 |Ω∗ |

Ω∗ f

∗(0) ∗(0) ∂p dΩ∗ ∂x∗i

vi

=

∗ ∂p∗(0) ∂p∗(0) Kij ∂x∗i η ∗ ∂x∗j

(7.16)

The first expression is positive. The same applies to the final expression, and so K∗ is positive. 7.2.2.5. Energetic consistency The first part of (7.16) represents the mean of the local dissipation density. We have: η ∗ Δy∗ v∗(0) = 2η ∗ divy∗ Dy∗ (v∗(0) ) where D is the strain rate tensor. The symmetry of D means we can write the weak formulation (7.15) in the form: ∀α ∈ V, 2η ∗ Dy∗ (α) : Dy∗ (v∗(0) )dΩ∗ = − α · gradx∗ p∗(0) dΩ∗ Ω∗ f

Ω∗ f

and (7.16) becomes: K 1 2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) )dΩ∗ = gradx∗ p∗(0) · ∗ gradx∗ p∗(0) ∗ |Ω | Ω∗f η Thus the mean of the local dissipation density equals the macroscopic dissipation density. 7.2.3. Non-homogenizable situations Now consider situations where QL = O(ε−2 ), with all other things remaining unchanged. It is sufficient to study QL = O(ε−1 ) and QL = O(ε−3 ), with the other cases following immediately.


207

7.2.3.1. Case where QL = O(ε−1 ). The normalized Navier-Stokes equation can now be written: εη ∗ Δx∗ v∗ − gradx∗ p∗ = ερ∗ (v∗ gradx∗ )v∗ and we obtain for the different orders: η ∗ Δy∗ v∗(0) − grady∗ p∗(0) = 0

(7.17a)

η ∗ Δy∗ v∗(1) − gradx∗ p∗(0) − grady∗ p∗(1) = ρ∗ (v∗(0) grady∗ )v∗(0) ———————-

(7.17b)

divy∗ v∗(0) = 0

(7.18a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.18b)

divy∗ v∗(2) + divx∗ v∗(1) = 0

(7.18c) ———————-

v∗(0) = 0

(7.19a)

v∗(1) = 0

(7.19b)

v(∗2) = 0

over Γ∗

(7.19c) ———————-

Equations (7.17a, 7.18a, 7.19a), with v∗(0) and p∗(0) Ω∗ -periodic, represents a homogenous boundary-value problem. It is easy to show the existence and uniqueness of the solution. This follows directly from the discussion in 7.2.2.3. We obtain: v∗(0) = 0,

p∗(0) = p∗(0) (x∗ )

which is an acceptable because v∗(0) = O(1). The following problem involves Ω∗ -periodic v∗(1) and p∗(1) . It can now be written: ηΔy∗ v∗(1) − grady∗ p∗(1) − gradx∗ p∗(0) = 0 divy∗ v∗(1) = 0 v∗(1) = 0

over Γ∗

208


This is the boundary-value problem (the underlying problem over the unit cell) which we studied in 7.2.1., where v∗(0) has been replaced with v∗(1) . Thus: v∗(1) =

k∗ gradx∗ p∗(0) η∗

Finally (7.18c), along with (7.19c) and Ω∗ -periodicity, gives the compatibility condition: divx∗ v∗(1) = 0,

v∗(1) = −

K∗ gradx∗ p∗(0) η∗

As we will see, this is again an equivalent macroscopic description, but the first nonzero term of v∗ is now εv∗(1) , so that the value of QL is ipso facto reduced to O(ε−2 ). Thus we have established by contradiction that a situation with QL = O(ε−1 ) does not exist in practice. 7.2.3.2. Case where QL = O(ε−3 ) The normalized Navier-Stokes equation is now written as: ε3 η ∗ Δx∗ v∗ − gradx∗ p = ε3 ρ∗ (v∗ gradx∗ )v∗ and we obtain for the different orders: grady∗ p∗(0) = 0

(7.20a)

gradx∗ p∗(0) + grady∗ p∗(1) = 0

(7.20b)

η∗ Δy∗ v∗(0) − grady∗ p∗(2) − gradx∗ p∗(1) = 0

(7.20c)

Equation (7.20a) again gives p∗(0) = p∗(0) (x∗ ). But (7.20b), with Ω∗ -periodic p∗(1) , introduces the compatibility condition gradx∗ p∗(0) = 0, which is not acceptable because gradx∗ p∗(0) = O(1). Thus p∗(0) is independent of x∗ and (7.20b) leads to p∗(1) = p∗(1) (x∗ ). Finally, (7.20c) gives v∗(0) in the form of a linear vectorial function of gradx∗ p∗(1) . Combining p∗(0) and p∗(1) into p∗(0) + εp∗(1) with: gradx∗ (p∗(0) + εp∗(1) ) = ε gradx∗ p∗(1) we again obtain a Darcy’s law with a smaller pressure gradient. But the value of QL is therefore reduced to the value O(ε−2 ) used in 7.2.1. A situation where QL = O(ε−3 ) is not homogenizable: an intrinsic macroscopic description does not exist. Here we have again arrived at a situation of non-homogenizability which is similar to the one we examined for the dynamics of elastic composites in section 3.3.2.3 of Chapter 3.


209

Thus the method of multiple scale expansions can be described as consistent with itself in that it only gives a result when homogenization is possible. To be rigorous, it is clear that a number QL = O(ε−3 ) indicates a strong pressure gradient, which implies a local non-linear description, with a different evaluation of the Reynolds number to that adopted here.

7.3. Dynamics of an incompressible fluid in a rigid porous medium The force exerted on a rigid body with velocity v in slow transient motion in a Newtonian liquid is classically written as [NAV 22; LAN 71]: R=

t −∞

M (t − τ ) v(τ ) dτ

(7.21)

where M is a memory function which depends on the density and viscosity of the fluid. (7.21) is a dynamic equation with memory, and the conservation of body momentum leads to an increased mass: its apparent mass is greater than its actual mass. The transient flows in porous media are of the same nature [BIO 56b]. With respect to change of length scale studies, the results in this section can be found in Levy [LEV 79] and Auriault [AUR 80]. The study of dynamic behavior of a fluid in a rigid porous matrix can at first glance seem purely academic, since few applications can be envisaged where the fluid can vibrate but the skeleton remains immobile, unless the fluid density is much lower than that of the solid. Nevertheless it will become clear later (Chapters 8, 12 and 14) that this study is the cornerstone of treatment of the acoustics of deformable saturated porous media. We will therefore consider a rigid porous matrix. The notations are the same as those used previously, and we will adopt the microscopic viewpoint. The threedimensional period (the cell) is referred to as Ω∗ , with local variable y∗ . The period consists of a rigid solid part Ω∗s and of the pores Ω∗f . The boundary common to Ω∗s and Ω∗f will still be denoted Γ∗ . The pores are saturated by an incompressible fluid of viscosity η. Such a period is again illustrated in Figure 7.2 for a two-dimensional problem.

7.3.1. Local description and estimates The local description is given by the Navier-Stokes equation, the incompressibility condition, and no-slip condition over Γ: ηΔX v − gradX p = ρ

∂v + (v gradX )v ∂t

(7.22)

210


divX v = 0

(7.23)

v|Γ = 0

(7.24)

The conservation of momentum (7.22) introduces three dimensionless numbers, and the macroscopic description clearly depends on the values of these numbers. As we have an acoustic vibration in mind, the perturbations are small and the same applies to the local Reynolds number, which we will take to be O(ε): Rel =

|ρv gradX )v| = O(ε) |ηΔX v|

The dynamics bring into play the transient inertial term on the local scale. The local transient Reynolds number is therefore O(1): ∂v | ∂t = O(1) Rtl = |ηΔX v| |ρ

This number can be expressed in different ways: ρc lc2 ωc Rtl = = ηc

lc δb

2 = (ωc τd )

2

where: δb =

ηc ωc ρc

gives the thickness of the boundary layer which appears along the fluid-solid interface Γ, and: τd = lc

ρc ωc ηc

is the time taken by a diffusive shear wave to cross the period. A local transient Reynolds number O(1) corresponds to a boundary layer thickness of the same order of magnitude as the size of the pores, and to a vibration whose period is of the same order of magnitude as the time taken by the diffusive shear wave to cross the period. Finally, the estimate made in section 7.2 for the number Q still applies: Ql =

pc lc |gradX p| = = O(ε−1 ) |ηΔX v| ηc vc


211

This value of Ql is, as in the other situations considered up to now, the only one which leads to a macroscopic description. Thus, adopting the microscopic viewpoint, (7.22) can be written: ∗ ∂v ∗ ∗ η ∗ Δy∗ v∗ − ε−1 grady∗ p∗ = ρ∗ (7.25) + ε(v grad )v ∗ y ∂t∗ 7.3.2. Macroscopic behavior: generalized Darcy’s law We want to find v∗ and p∗ in the form: v∗ (x∗ , y∗ , t∗ ) = v∗(0) (x∗ , y∗ , t∗ ) + εv∗(1) (x∗ , y∗ , t∗ ) + · · · p∗ (x∗ , y∗ , t∗ ) = p∗(0) (x∗ , y∗ , t∗ ) + εp∗(1) (x∗ , y∗ , t∗ ) + · · · x∗ = εy∗ , v∗(i) and p∗(i) are Ω∗ -periodic. Substituting these expansions into (7.25), (7.23) and (7.24), we obtain for the successive orders: grady∗ p∗(0) = 0 ρ∗

(7.26a)

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗ ———————-

(7.26b)

divy∗ v∗(0) = 0

(7.27a)

divy∗ v∗(1) + divx∗ v∗(0) = 0

(7.27b) ———————-

v∗(0) = 0 v∗(1) = 0

(7.28a) over Γ∗

(7.28b) ———————-

Thus a series of boundary-value problems are defined, which make it possible to obtain the successive terms v∗(i) and p∗(i) . The first problem is defined by (7.26a) and gives p∗(0) in the form: p∗(0) = p∗(0) (x∗ , t∗ ) The pressure is, to first order, uniform in the pore. The second and fundamental problem of interested involves unknowns v∗(0) and p∗(1) . The pressure p∗(0) is at this

212


stage considered to be an arbitrary function of x∗ . With (7.26b), (7.26b) and (7.27a), it follows that: ρ∗

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗

(7.29)

divy∗ v∗(0) = 0

(7.30)

v∗(0) = 0

(7.31)

where v∗(0) and p∗(1) are Ω∗ -periodic. It is convenient to study this problem in Fourier space. If ω ∗ is the frequency, (7.29) can be written in the form: ρ∗ iω ∗ v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0)

(7.32)

Problem (7.30), (7.31) and (7.32) is linear. The existence and uniqueness of v∗(0) and p∗(1) are investigated later in section 7.3.3. Unknowns v∗(0) and p∗(1) are linear functions of gradx∗ p∗(0) . In what follows, we will only require v∗(0) : ∗(0)

vi

= −λ∗ij (x∗ , ω∗ )

∂p∗(0) ∂x∗j

(7.33)

The λ∗ij components here are functions of ω ∗ , and are complex because of the inertial term contained in (7.32). The viscosity η ∗ and density ρ∗ are taken into consideration in λ∗ , but the separation of viscous and inertial effects in the macroscopic description turns out to be very difficult (see section 7.3.3). Now we will look for the macroscopic description. This is obtained from (7.27b) and (7.28b): divy∗ v∗(1) + divx∗ v∗(0) = 0 v∗(1) = 0

over Γ∗

where v∗(1) is Ω∗ -periodic. As we have already seen several times, this volume conservation implies a compatibility equation: divx∗ v∗(0) = 0 From this it follows that: divx∗ v∗(0) = 0

(7.34)


213

with: v∗(0) = −Λ∗ (ω ∗ )gradx∗ p∗(0) ,

Λ∗ = λ∗

(7.35)

where Λ∗ is the dynamic hydraulic conductivity. System (7.33-7.34) represents the macroscopic description we are looking for. Relation (7.35) is a generalized Darcy’s law. The dynamic permeability K∗ can be introduced by: Λ∗ =

K∗ η∗

where K∗ depends on η ∗ ω ∗ /ρ∗ . In dimensional variables, the macroscopic model becomes: ¯ divX v = O(ε)

¯ v = −Λ(ω)gradX p + O(ε)

or alternatively: ¯ divX v = O(ε)

v = −

K(ω) ¯ gradX p + O(ε) η

7.3.3. Discussion of the macroscopic description Here we will restrict ourselves to the main points. (The reader is referred to [AUR 80; AVA 81; BOR 83; AUR 85b] for more details.) 7.3.3.1. Physical meaning of macroscopic quantities The physical meaning of macroscopic quantities v∗(0) and p∗(0) is the same as discussed in 7.2.2.1. We will not return to that here. 7.3.3.2. Energetic consistency We are now in a position to tackle energetic consistency. This involves showing that the macroscopic densities of viscous dissipation and kinetic energy are equal to the volume means of the microscopic densities of these same respective quantities. First of all, with φ representing porosity, and H∗ the inverse of Λ∗ (which exists, see below): H∗ = HR∗ + iHI∗ = Λ∗−1 we rewrite the generalized Darcy’s law in the form: gradx∗ φp∗(0) = −φHR∗ v∗(0) − φHI∗ ω ∗−1 v˙ ∗(0)

(7.36)

214


where v˙ ∗(0) is the mean acceleration and represents the time derivative of the velocity at constant frequency. Equation (7.36) is a generalized Darcy’s law when the frequency is constant. It is written in the form of a conservation of momentum equation. It introduces a dissipative term φHR∗ v∗(0) and an inertial term φHI∗ ω ∗−1 v˙ ∗(0) . We will now look for an variational form equivalent to the system (7.29a,b-7.32). Let W be the space of vectors with zero divergence, which are Ω∗ -periodic and take complex values, defined over Ω∗f , zero over Γ∗ , and satisfying the scalar product: (α, β)W =

Ω∗ f

(

∂αi ∂ β˜i + αi β˜i ) dΩ∗ ∂yj∗ ∂yj∗

˜ is the complex conjugate of β. Thus the equivalent variational formulation where β can be written, proceeding as in section 7.2.2.3, as: ∀α ∈ W,

Ω∗ f

η∗(

∗(0)

∂vi ∂ α ˜i ∗(0) +iω ∗ ρ∗ vi α ˜ i ) dΩ∗ = − ∂yj∗ ∂yj∗

Ω∗ f

∂p∗(0) α ˜ i dΩ∗ (7.37) ∂x∗i

This formulation ensures the existence and uniqueness of v∗(0) [LEV 79]. We will take α = v∗(0) as real in (7.37). It follows for the right-hand side that:

∂p∗(0) ∗(0) vi dΩ∗ = |Ω∗ |gradx∗ p∗(0) · v∗(0) = −|Ω∗ |H∗ v∗(0) · v∗(0) ∂x∗i

Ω∗ f

Taking the real and imaginary parts of (7.37) after transformation in this way, we obtain: HR∗ v∗(0) · v∗(0) =

1 |Ω∗ |

HI∗ ω ∗−1 v∗(0) · v∗(0) =

Ω∗ f

1 |Ω∗ |

η∗

∗(0)

∗(0)

∂vi ∂vi dΩ∗ ∂yj∗ ∂yj∗

Ω∗ f

ρ∗ v∗(0) · v∗(0) dΩ∗

(7.38)

(7.39)

Introducing strain rate tensor D and using (7.29a), the right-hand side of (7.38) can be easily transformed into: 1 |Ω∗ |

∗(0)

∗(0)

∂vi ∂vi 1 ∗ η ∗ ∗ dΩ = |Ω∗ | ∗ ∂y ∂y Ωf j j ∗

Ω∗ f

2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) ) dΩ∗

so that (7.38) becomes: H

R∗

v

∗(0)

· v

∗(0)

1 = ∗ |Ω |

Ω∗ f

2η ∗ Dy∗ (v∗(0) ) : Dy∗ (v∗(0) ) dΩ∗

(7.40)


215

Equations (7.39) and (7.40) show the energetic consistency for kinetic and dissipated energy respectively. 7.3.3.3. The tensors H∗ and Λ∗ are symmetric Consider two real velocity fields v∗p and v∗q , corresponding respectively to = δip and vi∗q = δiq , given p and q, p = q. We will write the expression (7.37) for v∗(0) = v∗p , α = v∗q and then subsequently for v∗(0) = v∗q , α = v∗p . We then obtain: ∂v ∗q ∂vi∗p ∂p∗(0) ∗ η ∗ ( i∗ + iω ∗ ρ∗ vi∗q vi∗p ) dΩ∗ = − |Ω |Iip = |Ω∗ |H∗pq ∗ ∂yj ∂yj ∂x∗i Ω∗ f vi∗p

and: Ω∗ f

η∗ (

∂vi∗p ∂vi∗q ∂p∗(0) ∗ + iω ∗ ρ∗ vi∗p vi∗q ) dΩ∗ = − |Ω |Iiq = |Ω∗ |H∗qp ∗ ∗ ∂yj ∂yj ∂x∗i

The symmetry of the scalar product in the left-hand sides of these two above equations implies the equality of the right-hand sides: H∗pq = H∗qp 7.3.3.4. Low-frequency behavior The inertial term in (7.32) tends to zero as the frequency tends to zero. The generalized Darcy’s law then approaches the classical Darcy’s law: K∗ (0) K∗ = η∗ η∗

Λ∗ (ω ∗ ) → Λ∗ (0) =

when

ω∗ → 0

where K∗ is the steady state or intrinsic permeability tensor. 7.3.3.5. Additional mass effect Inertial density φHI∗ ω ∗−1 introduced in the generalized Darcy’s law takes the form of a tensor. It is easy to demonstrate an additional mass effect, familiar in fluid mechanics, particularly in the isotropic case. We will start from (7.39) with HI∗ ij = I∗ ∗q H Iij and we will consider the velocity field v introduced above: I∗

H ω

∗−1

vi∗q

1 =ρ |Ω∗ |

1 = ∗ |Ω |

∗

Ω∗ f

Ω∗ f

ρ∗ vi∗q vi∗q dΩ∗

vi∗q dΩ∗ = δiq

216


The Schwarz inequality applied to the integral of velocity v∗q leads to: 1 1 |Ω∗ | = | v∗q dΩ∗ | |Ω∗ | 2 | vi∗q vi∗q dΩ∗ | 2 Ω∗ f

Ω∗ f

Thus: φHI∗ ω ∗−1 ρ∗ We refer to Chapter 10 section 10.3.1.2 for a proof of the following properties [BOU 08]: dHR∗ 0 dω ∗

;

d(HI∗ /ω ∗ ) 0 dω ∗

7.3.3.6. Transient excitation: Dynamics with memory effects When the porous medium is subject to a transient excitation, the generalized Darcy’s law (for example (7.36)) must be replaced with: gradx∗ φp∗(0) = −F ∗−1 (φH∗ ) ∗ v∗(0)

(7.41)

where F ∗−1 (φH∗ ) is the inverse Fourier transform of φH∗ , and ∗ indicates a convolution product. Thus the generalized Darcy’s law for transient motion describes dynamics with memory effects, as in (7.21). 7.3.3.7. Quasi-periodicity When periodicity varies slowly, in other words in the case of quasi-periodicity, where the geometry depends on macroscopic variable x∗ , all the results obtained above remain valid. During the process of homogenization, x∗ then plays the role of an independent parameter. Thus macroscopic coefficients Λ∗ and H∗ become dependent on x∗ . 7.3.4. Circular cylindrical pores The mean behavior of a viscous incompressible Newtonian fluid through a porous matrix whose pores are circular cylinders has been known for a long time (see for example [BIO 56a; 56b]), in any case well before the formulation of the method of asymptotic expansions. This is because in this particular case, the periodicity is arbitrary along the pore axis. Thus the fundamental cellular boundary-value problem to be solved when homogenization is used is identical to the problem posed directly on the macroscopic scale: due to the arbitrary periodicity, the different terms in the asymptotic expansions are independent of the fast variable along the pore axis. This property of course applies regardless of the cross-section of the cylindrical pores.


217

This explains why this particular type of pore can be taken as a reference, and the results obtained can be generalized to other porous media [BIO 56a; b; AVA 81]. Significant differences exist in the size of H(ω) however, notably for the behavior of the inertial coefficient at high frequencies [BOR 83; AUR 85b]. Here we will present

e2

e2

a

e1

e3

a

Figure 7.4. Porous medium consisting of circular cylindrical pores: macroscopic sample and representative elementary volume

the derivation of the generalized Darcy’s law for a porous medium with parallel circular cylindrical pores, in the context of the method of asymptotic expansions. The medium is shown in Figure 7.4. The pores have radius a∗ in terms of the slow variable. The dimensionless problem to be solved is given by (7.29), (7.30) and (7.31): ρ∗

∂v∗(0) = η ∗ Δy∗ v∗(0) − grady∗ p∗(1) − gradx∗ p∗(0) ∂t∗

divy∗ v∗(0) = 0 v∗(0) = 0

over

within

within

Ω∗f

Ω∗f

Γ∗

The calculation is carried out in dimensional cylindrical variables (X, r, θ), with X pointing along the pore axis (e3 ). Bearing in mind pore geometry and incompressibility, v(0) points along the pore axis and is independent of X and θ, and p(1) is independent of θ. Also, the arbitrary periodicity in X direction implies that p(1) does not depend on X. Since the problem is macroscopically one-dimensional, the macroscopic pressure gradient is taken along the axis of the pore. The system to be solved can thus be simplified significantly. The equations recalled above are written in dimensionless form. We will now return to the physical quantities (remember that the same notations were used in both cases in order to simplify matters). By projection onto the axes, we obtain, for constant pulsation: iω 1 dv (0) 1 dp(0) d2 v (0) − v (0) = + 2 dr r dr νf η dX

218


1 dp(1) =0 η dr where νf is the kinematic viscosity, νf = η/ρ. Over the pore boundary, the velocity satisfies: v (0) (a) = 0 It follows from this that p(1) = p(1) (X), and also that the solution v (0) can be written: 1

1

v (0) = AJ0 (i(iωνf−1 ) 2 r) + BN0 (i(iωνf−1 ) 2 r) −

1 dp(0) iωρ dX

where A and B are two constants of integration which must be determined, and J0 and N0 are Bessel functions. Since v (0) is finite at the origin, B = 0, and A can be obtained using the no-slip condition at r = a. It follows that: A=

1 1 iωρJ0 (i(iωνf−1 ) 2 a)

dp(0) dX

and the velocity distribution in the pore is given by: v (0) =

1

−1 +

J0 (i(iωνf−1 ) 2 r) 1 2

J0 (i(iωνf−1 ) a)

1 dp(0) iωρ dX

Finally, taking the mean over the cross-section S = πa2 of the pore, we obtain the generalized Darcy’s law for constant pulsation: v

(0)

φ = S

a

2πrv (0) dr 0

To first order we have: v = −Λ

dp , dX 1

φ J2 (i(iωνf−1 ) 2 a) Λ=− iωρ J0 (i(iωνf−1 ) 12 a) where J2 is the Bessel function. The values of Λ and H = Λ−1 are shown in Figures 7.5 and 7.6 as a function of the dimensionless pulsation.


2/*a2 1

2

Log(a2 /2") Figure 7.5. Variation of Λ1 and Λ2 as a function of frequency

*HI /(')

0.1*HR /2 Log(a2 /2") Figure 7.6. Variation of the real (HR ) and imaginary (HI ) parts of H as a function of frequency

In the case of transient motion, the dynamic Darcy’s law becomes: φ v = − ρ

t −∞

G(t − τ )

dp (τ ) dτ dX

where memory function G is obtained via the inverse Fourier transform of Λρ/φ: G=4

−2 νf t λ−2 exp −λ n n a2 n=1 ∞

for

t>0

219

220


G=0

t0

t

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