Microporomechanics

Microporomechanics Microporomechanics Luc Dormieux Ecole Nationale des Ponts et Chaussees, ´ France Djim´ edo Kondo...

Author: Luc Dormieux | Djimedo Kondo | Franz-Josef Ulm

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Microporomechanics

Microporomechanics

Luc Dormieux Ecole Nationale des Ponts et Chaussees, ´ France

Djim´ edo Kondo Laboratoire de Mecanique ´ de Lille, France

Franz-Josef Ulm Massachusetts Institute of Technology, USA

C 2006 Copyright

John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone

(+44) 1243 779777

Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop # 02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-470-03188-9 ISBN-10: 0-470-03188-3 Typeset in 11/13pt Palatino by TechBooks, New Delhi, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents Preface

xi

Notation

xv

1 A Mathematical Framework for Upscaling Operations 1.1 Representative Elementary Volume (rev) 1.2 Averaging Operations 1.2.1 Apparent and Intrinsic Averages 1.2.2 Spatial Derivatives of an Average 1.2.3 Time Derivative of an Average 1.2.4 Spatial and Time Derivatives of e 1.3 Application to Balance Laws 1.3.1 Mass Balance 1.3.2 Momentum Balance 1.4 The Periodic Cell Assumption 1.4.1 Introduction 1.4.2 Spatial and Time Derivative of e in the Periodic Case 1.4.3 Spatial and Time Derivative of eα of in the Periodic Case 1.4.4 Application: Micro- versus Macroscopic Compatibility

1 1 3 3 6 7 7 8 8 10 14 14 16 17 18

Part I Modeling of Transport Phenomena 2 Micro(fluid)mechanics of Darcy’s Law 2.1 Darcy’s Law 2.2 Microscopic Derivation of Darcy’s Law 2.2.1 Thought Model: Viscous Flow in a Cylinder 2.2.2 Homogenization of the Stokes System 2.2.3 Lower Bound Estimate of the Permeability Tensor 2.2.4 Upper Bound Estimate of the Permeability Tensor

23 23 25 25 26 34 40

Contents

vi

2.3

2.4

2.5

2.6

2.7

Training Set: Upper and Lower Bounds of the Permeability of a 2-D Microstructure 2.3.1 Lower Bound 2.3.2 Upper Bound 2.3.3 Comparison Generalization: Periodic Homogenization Based on Double-Scale Expansion 2.4.1 Double-Scale Expansion Technique 2.4.2 Extension of Darcy’s Law to the Case of Deformable Porous Media Interaction Between Fluid and Solid Phase 2.5.1 Macroscopic Representation of the Solid–Fluid Interaction 2.5.2 Microscopic Representation of the Solid–Fluid Interaction Beyond Darcy’s (Linear) Law 2.6.1 Bingham Fluid 2.6.2 Power-Law Fluids Appendix: Convexity of π(d)

3 Micro-to-Macro Diffusive Transport of a Fluid Component 3.1 Fick’s Law 3.2 Diffusion without Advection in Steady State Conditions 3.2.1 Periodic Homogenization of Diffusive Properties 3.2.2 The Tortuosity Tensor 3.2.3 Variational Approach to Periodic Homogenization 3.2.4 The Geometrical Meaning of Tortuosity 3.3 Double-Scale Expansion Technique 3.3.1 Steady State Diffusion without Advection 3.3.2 Steady State Diffusion Coupled with Advection 3.3.3 Transient Conditions 3.4 Training Set: Multilayer Porous Medium 3.5 Concluding Remarks

42 43 43 44 45 46 48 50 50 51 52 52 54 60

63 63 64 65 66 67 71 74 75 77 82 84 86

Part II Microporoelasticity 4 Drained Microelasticity 4.1 The 1-D Thought Model: The Hollow Sphere 4.1.1 Macroscopic Bulk Modulus and Compressibility 4.1.2 Model Extension to the Cavity 4.1.3 Energy Point of View 4.1.4 Displacement Boundary Conditions 4.2 Generalization 4.2.1 Macroscopic and Microscopic Scales 4.2.2 Formulation of the Local Problem on the rev

91 91 92 94 94 95 96 97 98

Contents

vii

4.2.3 4.2.4 4.2.5 4.2.6 4.2.7

4.3

4.4 4.5

Uniform Stress Boundary Condition An Instructive Exercise: Capillary Pressure Effect Uniform Strain Boundary Condition The Hill Lemma The Homogenized Compliance Tensor and Stress Concentration 4.2.8 An Instructive Exercise: Example of an rev for an Isotropic Porous Medium. Hashin’s Composite Sphere Assemblage 4.2.9 The Homogenized Stiffness Tensor and Strain Concentration 4.2.10 Inﬂuence of the Boundary Condition. The Hill–Mandel Theorem Estimates of the Homogenized Elasticity Tensor 4.3.1 The Dilute Scheme 4.3.2 The Differential Scheme Average and Effective Strains in the Solid Phase Training Set: Molecular Diffusion in a Saturated Porous Medium 4.5.1 Deﬁnition of a Local Boundary Value Problem 4.5.2 Estimates of the Effective Diffusion Coefﬁcient

5 Linear Microporoelasticity 5.1 Loading Parameters 5.2 The 1-D Thought Model: The Saturated Hollow Sphere Model 5.2.1 Direct Solution 5.2.2 Energy Approach 5.3 Generalization 5.3.1 Deﬁnition of a Mechanical Loading on the rev 5.3.2 Homogenized State Equations 5.3.3 Symmetry of the Homogenized State Equations 5.3.4 Energy Approach 5.3.5 The Macroscopic Variable Set (E, m) 5.4 Application: Estimates of the Poroelastic Constants and Average Strain Level 5.4.1 Microscopic and Macroscopic Isotropy 5.4.2 Microscopic and Macroscopic Anisotropy 5.4.3 Average Strain Level in the Solid Phase 5.5 Levin’s Theorem in Linear Microporoelasticity 5.5.1 An Alternative Route to the Poroelastic State Equations 5.5.2 An Instructive Exercise: The Prestressed Initial State 5.6 Training Set: The Two-Scale Double-Porosity Material

100 102 103 105 106

112 116 120 122 122 125 127 130 132 133

137 137 138 139 141 143 143 145 146 148 149 151 151 153 154 156 156 160 161

Contents

viii

6 Eshelby’s Problem in Linear Diffusion and Microporoelasticity 167 6.1 Eshelby’s Problem in Linear Diffusion 167 6.1.1 Introduction 167 6.1.2 The (Diffusion) Inclusion Problem 169 6.1.3 The (Second-Order) P Tensor 171 6.1.4 An Alternative Derivation of the P Tensor (Optional) 172 6.1.5 The (Diffusion) Inhomogeneity Problem 175 6.1.6 Eshelby-Based Estimates of the Homogenized Diffusion Tensor 176 6.2 Eshelby’s Problem in Linear Microelasticity 176 6.2.1 Introduction 176 6.2.2 The (Elastic) Inclusion Problem 178 6.2.3 The Green Tensor G and the (Fourth-Order) P Tensor 180 6.2.4 G and P in the Isotropic Case 181 6.2.5 The (Elastic) Inhomogeneity Problem 183 6.2.6 An Instructive Exercise: Geometry Change of Spherical Pores in a Porous Medium Subjected to Compaction 184 6.3 Implementation of Eshelby’s Solution in Linear Microporoelasticity 185 6.3.1 Implementation of Eshelby’s Solution in the Dilute Scheme 186 6.3.2 Implementation of the Dilute Scheme with Different Pore Families 186 6.3.3 An Alternative Eshelby-Based Derivation of the Poroelastic Model 187 6.3.4 Mechanical Interaction Between Pores: The Mori–Tanaka Scheme 190 6.3.5 The Self-Consistent Approach 192 6.4 Instructive Exercise: Anisotropy of Poroelastic Properties Induced by Flat Pores 195 6.4.1 Coefﬁcients of the Eshelby Tensor 195 6.4.2 Application of the Dilute Scheme 197 6.4.3 Inﬂuence of the Mechanical Interaction 197 6.5 Training Set: New Estimates of the Homogenized Diffusion Tensor 198 6.5.1 The Mori–Tanaka Estimate of the Diffusion Coefﬁcient 199 6.5.2 The Self-Consistent Estimate of the Diffusion Coefﬁcient 200 6.6 Appendix: Cylindrical Inclusion in an Isotropic Matrix 202

Part III Microporoinelasticity 7 Strength Homogenization 7.1 The 1-D Thought Model: Strength Limits of the Saturated Hollow Sphere

207 207

Contents

7.2

7.3

7.4

7.5

ix

Macroscopic Strength of an Empty Porous Material 7.2.1 Microscopic Strength of the Solid Phase 7.2.2 Strength-Compatible Macroscopic Stress States 7.2.3 Determination of ∂G hom 7.2.4 Solid Strength Depending on the First Two Stress Invariants 7.2.5 Principle of Nonlinear Homogenization Von Mises Behavior of the Solid Phase 7.3.1 The Equivalent Viscous Behavior 7.3.2 Homogenization of the Fictitious Viscous Behavior 7.3.3 Validation The Role of Pore Pressure in the Macroscopic Strength Criterion 7.4.1 Von Mises or Tresca Solid 7.4.2 Drucker–Prager Solid Nonlinear Microporoelasticity 7.5.1 Non–Pressurized Pore Space 7.5.2 Pressurized Pore Space 7.5.3 An Alternative Approach to Strength Homogenization

210 210 212 213 214 215 216 216 218 221 222 223 224 226 226 227 229

8 Non-Saturated Microporomechanics 8.1 The Effect of Surface Tension at the Solid–Fluid Interface 8.1.1 Representation of Internal Forces at the Solid–Fluid Interface 8.1.2 Principle of Virtual Work and the Hill Lemma with Surface Tension Effects 8.1.3 State Equation with Surface Tension Effects 8.1.4 Macroscopic Strain Related to Surface Tension Effects 8.2 Microporoelasticity in Unsaturated Conditions 8.2.1 The Bishop Effective Stress in Unsaturated Porous Media 8.2.2 Surface Tension Effects in Unsaturated Porous Media 8.3 Training Set: Drying Shrinkage in a Cylindrical Pore Material System 8.3.1 The Capillary Pressure Curve 8.3.2 The State Equation 8.3.3 Strains Induced by Drying 8.4 Strength Domain of Non-Saturated Porous Media 8.4.1 Average Strain Level in a Linear Elastic Solid Phase 8.4.2 Strength in Partially Saturated Conditions

237

9 Microporoplasticity 9.1 The 1-D Thought Model: The Saturated Hollow Sphere 9.1.1 Elastic Response

267

237 238 241 243 245 246 246 248 250 250 254 257 259 260 263

267 268

Contents

x

9.2

9.3

9.4

9.1.2 Elastoplastic Response 9.1.3 The Concept of Residual Stresses 9.1.4 Energy Aspects State Equations of Microporoplasticity 9.2.1 First Approach to the Macroscopic Stress–Strain Relationship 9.2.2 Macroscopic Plastic and Elastic Strain Tensors 9.2.3 Macroscopic State Equations in Poroplasticity Macroscopic Plasticity Criterion 9.3.1 Link Between the Microscopic and the Macroscopic Plasticity Criterion 9.3.2 Arguments of the Macroscopic Yield Criterion Dissipation Analysis 9.4.1 Macroscopic Flow Rule 9.4.2 Energy Analysis 9.4.3 Effective Stress in Poroplasticity 9.4.4 On the “Effective Plastic Stress” Σ + β P1

269 271 272 274 274 275 280 281 281 282 283 283 285 287 288

10 Microporofracture and Damage Mechanics 10.1 Elements of Linear Fracture Mechanics 10.2 Dilute Estimates of Linear Poroelastic Properties of Cracked Media 10.2.1 Open Parallel Cracks 10.2.2 Randomly (Isotropic) Oriented Open Cracks 10.2.3 Anisotropic Distribution of Open Cracks 10.2.4 Effect of Total Crack Closure on the Overall Stiffness 10.3 Mori–Tanaka Estimates of Linear Poroelastic Properties of Cracked Media 10.3.1 Open Parallel Cracks 10.3.2 Closed Parallel Cracks 10.3.3 Randomly Oriented Interacting Cracks 10.3.4 Double-Porosity Model of Cracked Porous Media 10.4 Micromechanics of Damage Propagation in Saturated Media 10.4.1 LEFM–Damage Analogy 10.4.2 Extension to Multiple Cracks 10.4.3 The Role of the Homogenization Scheme in the Damage Criterion 10.5 Training Set: Damage Propagation in Undrained Conditions 10.5.1 The Case of an Incompressible Fluid 10.5.2 The Case of a Compressible Fluid 10.6 Appendix: Algebra for Transverse Isotropy and Applications

291

References

319

Index

323

291 295 296 299 300 301 303 303 304 304 305 306 306 308 310 314 315 316 317

Preface All natural composite materials (soils, rocks, woods, hard and soft tissues, etc.) and many engineered composites (concrete, bioengineered tissues, etc.) are multiphase and multiscale material systems. The multiphase composition of such materials is permanently evolving over various time and length scales, creating in the course of this process the most heterogeneous class of materials in existence, with heterogeneities that manifest themselves from the nanoscale to the macroscale. The most prominent heterogeneity of such natural composite materials is the porosity, i.e. the space left in between the different solid phases at various scales, ranging from interlayer spaces in between minerals ﬁlled by a few water molecules, to the macropore space in between microstructural units of the material in the micrometer to millimeter range. This porosity is the key to understanding and predicting macroscopic material behavior, ranging from diffusive or advective transport properties to stiffness, strength and fracture behavior. The speciﬁc nature of the mechanical behavior of multiphase porous materials was recognized early on in the groundbreaking work of M.A. Biot and K. Terzaghi, who developed the macroscopic basis of what is now known as ‘poromechanics’. Ever since, poromechanics has entered a large number of engineering applications ranging from traditional civil and environmental engineering to petroleum engineering and other geophysical applications, and more recently biomechanical engineering. In the 1970s, a breakthrough was achieved with pioneering work that relates macroscopic laws to microstructural properties. Furthermore, as new experimental techniques such as nanoindentation now provide unprecedented access to micromechanical properties and morphologies of materials, it becomes possible to trace these features from the nanoscale to the macroscale of day-to-day engineering applications, and to predict transport properties, stiffness, strength and deformation behaviors within a consistent framework of ‘micromechanics of porous media’, or in short ‘microporomechanics’. Intended as an introduction to the micromechanics of porous media, this book on ‘microporomechanics’ deals with the mechanics and physics of

xii

Preface

multiphase porous materials at the microscale. Based on the mathematical eloquence of advanced solid and ﬂuid mechanics, it develops a rigorous mathematical framework to translate microscopic properties and microbehaviors of ﬂuid and solid phases, through homogenization theories, into macroscopic constitutive relations of mass transport phenomena and poromechanical deformation of porous media, including poroelasticity, poroplasticity and porofracture and damage theory. This book is composed of three parts: following a brief introduction to the mathematical rules for upscaling operations, the ﬁrst part deals with the homogenization of transport properties of porous media within the context of asymptotic expansion techniques. This includes ﬂuid conduction (Chapter 2) which deals with the homogenization of the Newtonian and non-Newtonian ﬂuid state equations into the macroscopic Darcy’s law, and ion conduction (Chapter 3) based on the homogenization of Fick’s law including diffusion with advection, which gives rise not only to macroscopic diffusive phenomena, but also to advective and dispersive ﬂuxes. The second part deals with linear microporomechanics, and introduces linear mean-ﬁeld theories based on the concept of a representative elementary volume for the homogenization of poroelastic properties of porous materials. In particular, Chapter 4 is dedicated to a porous medium emptied of its ﬂuid phases, which corresponds to the drained situation, and for which we introduce the tools of continuum micromechanics. Chapter 5 then adds the ﬂuid pressure as a second loading parameter to the microelastic analysis. To complete this part on linear homogenization technique, Chapter 6 is devoted to Eshelby’s problem of ellipsoidal inclusions, on which many of the micromechanics techniques are based, and illustrates the application of Eshelby’s solution to linear diffusion and microporoelasticity. Finally, the third part extends the analysis to microporoinelasticity: that is, the nonlinear homogenization of a large range of frequently encountered porous material behaviors, namely strength homogenization (Chapter 7), non-saturated microporomechanics (Chapter 8), microporoplasticity (Chapter 9) and microporofracture and damage theory (Chapter 10). From the onset, our approach to writing this book on microporomechanics was nourished by the desire to deﬁne a common language for homogenization of both solid and ﬂuid behavior, which is critical when it comes to the behavior of porous media. Our approach owes much to the most gifted educators who taught us and instilled in us the beauty of solid mechanics, continuum micromechanics, periodic homogenization theory and macroscopic poromechanics: namely, Jean Salenc¸on, Andr´e Zaoui, Alain Molinari, Olivier Coussy, Jean-Louis Auriault and Patrick de Buhan. We are deeply indebted to our colleagues, former students and friends, who have been a motivation to write this book and who have put to work the microporomechanics theory in many innovative engineering applications: Christian Hellmich, Vincent Deud´e, Vincent Pens´ee, Franz Heukamp, Jean-Franc¸ois Barth´el´emy, Georgios Constantinides,

Preface

xiii

Xavier Chateau, Denis Garnier, Samir Maghous and many more. Our deepest gratitude goes to Eric Lemarchand who has accompanied us all along the way during the development and writing of this book, starting from his doctoral work at Ecole Nationale des Ponts et Chauss´ees, his postdoctoral sojourn at the Massachusetts Institute of Technology, and ﬁnally at the Laboratoire de M´ecanique de Lille. We trust that this textbook will be a source of inspiration. Luc Dormieux Paris, France Djim´edo Kondo Lille, France Franz-Josef Ulm Cambridge, Massachusetts

Notation d : characteristic size of heterogeneities ℓ : characteristic size of the rev L : characteristic size of the macroscopic structure x : position vector at the macroscopic scale z : position vector at the microscopic scale (x) : elementary volume located at macroscopic point x f : ﬂuid domain in the rev p : pore space in the rev s : solid domain in the rev ϕ : porosity (pore volume fraction), also called ‘Eulerian’ porosity ϕ0 : initial porosity (pore volume fraction) in the reference conﬁguration ϕ s = 1 − ϕ0 : solid volume fraction in the reference conﬁguration φ : Lagrangian porosity (pore volume divided by total volume of the rev in its initial conﬁguration) U : elementary cell in a periodic medium U f : ﬂuid domain in the elementary cell (U f = P f ∩ U) Us : solid domain in the elementary cell (Us = P s ∩ U) I s f : solid–ﬂuid interface within the elementary volume (resp. cell) I αβ : interface between phases α and β (α, β ∈ {s, ℓ, g}) P α (α = s, f, ℓ, g) : domain occupied by the α phase in the medium χ α (α = s, f, ℓ, g) : characteristic function of P α χ p : characteristic function of the pore space χ I : characteristic function of the domain I v(z) : microscopic ﬂuid velocity at point z vγ (z) : microscopic velocity of the γ component of the ﬂuid at point z V f (x) : macroscopic ﬂuid velocity at point x V γ (x) : macroscopic velocity of the γ component of the ﬂuid at point x 1 : second-order unit tensor 1T : second-order unit tensor of plane T I : fourth-order unit tensor

xvi

Notation

k f : ﬂuid bulk modulus k s : elastic bulk modulus of the solid phase μs : elastic shear modulus of the solid phase k hs : apparent bulk modulus of the hollow sphere k hom : homogenized bulk modulus in isotropic conditions μhom : homogenized shear modulus in isotropic conditions b : Biot coefﬁcient (isotropic case) B : tensorial Biot coefﬁcient N : Biot modulus of the solid phase (isotropic case) M : Biot modulus of the whole porous material (isotropic case) γ αβ : surface tension at I αβ ξ (z) : microscopic displacement vector at point z ε(z) : microscopic linearized strain tensor at point z d(z) : microscopic strain rate tensor at point z σ(z) : microscopic Cauchy stress tensor at point z p(z) : microscopic ﬂuid pressure P(x) : (average) ﬂuid pressure (saturated case) Pℓ : liquid pressure (unsaturated case) Pg : gas pressure (unsaturated case) Σ(x) : macroscopic Cauchy stress tensor at point x E(x) : macroscopic strain tensor at point x D(x) : macroscopic strain rate tensor at point x G(z) : Green tensor at point z A(z) : strain concentration tensor B(z) : stress concentration tensor Cs : elastic stiffness tensor of the solid Cc : ﬁctitious elastic stiffness tensor of cracks Chom : homogenized drained elastic stiffness tensor Ss = (Cs )−1 : elastic compliance tensor of the solid S : Eshelby tensor P = S : Ss a α : intrinsic average of a over the α phase a α : apparent average of a over the α phase a f : intrinsic average of a over the ﬂuid phase G s : set of microscopic stress states compatible with the solid strength f s (σ) : solid strength criterion π s (d) : support function of G s

1 A Mathematical Framework for Upscaling Operations Microporomechanics is a continuum approach that allows one to scale of physical quantities from the microscale to the macroscale. This chapter presents some of the mathematical ingredients of this micro-to-macro approach that ultimately translates into upscaling rules of physical quantities. Two approaches are presented in this regard: one based on averaging techniques on a representative elementary volume, a second one based on a periodic assumption. From an application of averaging techniques to microscopic conservation laws we derive macroscopic conservation laws, and establish the link between kinematics and internal forces employed in microporomechanics: that is, from mass balance, the link between a microscopic velocity ﬁeld and a macroscopic velocity vector; and from momentum balance, the link between microscopic stress ﬁeld and macroscopic stress tensor.

1.1 Representative Elementary Volume (rev) Continuum mechanics in general deals with the evolution of continuous material systems in three dimensions and time. Poromechanics in particular deals with the evolution of a porous continuum. One of the most critical elements of the continuum approach is the concept of an elementary volume. By deﬁnition, the elementary volume is an inﬁnitesimal part of the three-dimensional material system under consideration. More precisely, if we denote by L and ℓ the characteristic lengths of respectively the structure and the elementary volume, the condition ℓ ≪ L guarantees the relevance of the use of the tools of differential calculus offered by a continuum description. Furthermore, the elementary volume is expected to be large enough to be representative of the Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

2

A Mathematical Framework for Upscaling Operations

constitutive material,1 which explains its name as representative elementary volume (in short rev). It is intuitively understood that this property requires the characteristic size ℓ of the rev to be chosen so as to capture in a statistical sense all the information concerning the geometrical and physical properties of the physics at stake. Roughly speaking, if we denote by d the characteristic length scale of the local heterogeneities, typically the pore size in a porous medium, the condition d ≪ ℓ is expected to ensure that the elementary volume is representative. In summary, the two conditions on the size of the rev are: d≪ℓ≪L

(1.1)

Relation (1.1) is often referred to as the condition for scale separation (or scale separability condition), which is a necessary condition for the concept of rev to be valid. At the macroscopic scale, the rev is characterized by a position vector x. The characteristic order of magnitude of the variation of x is the size L of the studied material system. Furthermore, this material system is composed of different constituents, i.e. in the case of a porous continuum, one (possibly heterogeneous) solid phase and one or several ﬂuid phases. These phases represent heterogeneities. The macroscopic poromechanics theory accounts for this heterogeneous nature of the porous material system by considering each phase as a macroscopic particle. All these particles are located at the same point x. In other words, from a macroscopic point of view, the rev is regarded as the superposition of these particles in time and space. By contrast, a poromechanics approach that starts at the microscopic scale explicitly considers the heterogeneous structure of the rev. It represents the solid and the ﬂuid phases as individual separated domains in the rev. This requires a reﬁnement of the geometrical description, i.e. a change in length scale. At the microscopic scale, the position vector is now denoted by z; the characteristic order of magnitude of the variation of z is the size ℓ of the rev (Figure 1.1). Depending on z, the microscopic particle located at point z belongs to the solid phase or to a ﬂuid phase. The very existence of an rev is a key element for both micro- and macroporomechanics theories, albeit of different importance. In a pure macroscopic approach, the existence of an rev must be postulated so that the macroscopic constitutive laws derived experimentally or theoretically are representative of the response of the rev to various loadings. The principle of a micro-tomacro approach consists in replacing the real experiment that could be performed on a representative material sample by a thought experiment on the rev considered as a heterogeneous structure that is subjected to appropriate boundary conditions. The underlying idea of micromechanics is to derive the 1 The conditions for an elementary volume to be representative have been discussed by many authors. The interested reader is referred for instance to the work of Bear and Bachmat [6] or Torquato [50].

Averaging Operations

3

L

ℓ

ℓ

d

solid

fluid

fluid particle

microscopic scale

skeleton particle

macroscopic structure

macroscopic scale

Figure 1.1 Elementary volume of porous medium in a macroscopic structure. Microscopic and macroscopic scales.

macroscopic response of the sample from the microscopic one. This requires, as a ﬁrst task, clariﬁcation of the mathematical relations between the physics at each scale, the micro and the macro scales, and more particularly answers to the following two critical questions relating to the description of the kinematics and the internal forces:

r The velocity in any phase α of the porous material (e.g. α = s for the solid

phase and α = f for the ﬂuid phase) is characterized at the macroscopic scale by a vector V α (x), and by the ﬁeld v(z) at the microscopic scale. What is the link between V α (x) and v(z)? r The internal forces in the α phase are characterized at the macroscopic scale by the partial stress tensor Σα (x), and by the ﬁeld σ(z) at the microscopic scale. What is the link between Σα (x) and σ(z)?

1.2 Averaging Operations It is natural, at least for extensive physical quantities such as mass, energy, etc., to deﬁne the link between micro- and macroscales through averaging techniques deﬁned on the rev; including the links between the derivative of an average (with respect to time or spatial coordinates) and the average of the derivatives. This is the focus of this section which sets out a mathematical basis for the forthcoming application of these elements in microporomechanics. 1.2.1 Apparent and Intrinsic Averages Let 0 be a time-independent geometrical domain centered at the origin O of the coordinate system, which can be considered as an rev. For any value of x, we assume that the elementary volume (x) which is obtained from 0 by a

A Mathematical Framework for Upscaling Operations

4

translation2 of vector x is an rev for the material at the macroscopic point x, in its current conﬁguration. Furthermore, let f (z) be a C ∞ function deﬁned at the microscopic scale. It is equal to zero outside 0 and satisﬁes: (1.2) f (z) dVz = 1

The average e(x) at the macroscopic point x of a physical quantity represented in (x) by the volume density e(z) is deﬁned by: e(x, t) = e(z, t) f (z − x) dVz (1.3)

This deﬁnition depends on the choice of the function f . In practice, f can be chosen so as to tend towards the discontinuous function χ0 /|0 |, where |0 | is the volume of 0 and χ0 is the characteristic function of this domain deﬁned by χ0 (z) = 1 if z ∈ 0 and χ0 (z) = 0 if z ∈ / 0 . The deﬁnition (1.3) then asymptotically corresponds to the more familiar deﬁnition of the average: 1 e(z, t) dVz e(x, t) ≈ (1.4) |(x)| (x)

However, it will be convenient for further calculations to take advantage of the regularity of f (z) on the boundary of 0 . By way of illustration, consider the volume mass density of the porous medium ρ M(x), which at the macroscopic scale represents the total elementary mass dm contained in the rev (x) divided by its volume: ρM =

dm |(x)|

(1.5)

With the approximation f ≈ χ0 /|0 |, it is readily seen that: ρM = ρ

(1.6)

Another example is the volume fraction ϕ(x, t) of the pore space at the macroscopic point x, which can be determined from the geometry of the microstructure from: p ϕ(x, t) = χ = χ p (z, t) f (z − x) dVz (1.7)

where χ p (z, t) is the characteristic function of the pore space. ϕ(x, t) is also referred to as the porosity. From now on, P s and P f respectively denote the domains occupied by the solid and the ﬂuid phase. In addition, P p refers to the pore space. In the saturated case, we note that P f = P p . All these domains a priori depend 2 The

image of point M by this translation is point M′ such that MM′ = x.

Averaging Operations

5

on time. We also introduce the solid, ﬂuid and porous parts of the rev (x), deﬁned as α (x, t) = P α (t) ∩ (x) (α = s, f, p). For a physical quantity e(z, t) attached to the ﬂuid, and thus deﬁned on the ﬂuid domain P f only, we will encounter two different averages, namely the ‘apparent’ one, e f (x, t): f e f (x, t) = χ e(x, t) = χ f (z, t) e(z, t) f (z − x) dVz (1.8) and the ‘intrinsic’ one, e f (x, t), deﬁned by: e f (x, t) =

1 e f (x, t) ϕ

(1.9)

where χ f (z, t) is the characteristic function of the ﬂuid phase. The apparent and the intrinsic average both derive from the total amount of the physical quantity e(z, t) available in P f . However, the apparent one e f refers this total amount to the total volume of the rev, while the intrinsic average e f refers the total amount to the actual domain | f (x, t)| the ﬂuid phase occupies in the rev. Analogously, it is possible to deﬁne apparent and intrinsic averages of the solid phase. It sufﬁces indeed to replace χ f in (1.8) by the characteristic function of the solid domain P s , χ s = 1 − χ p , to obtain the apparent average e s (x, t) = χ s e(x, t), and ϕ in (1.9) by the solid volume fraction 1 − ϕ, to obtain the intrinsic average e s (x, t) = e s (x, t)/(1 − ϕ). By way of example, consider a component γ of the ﬂuid phase (for instance, a solute which together with a solvent saturates the ﬂuid domain P f ). Let ρ γ (z, t) be the microscopic mass density, and dmγ the total mass of the γ component in the rev. The macroscopic apparent and intrinsic mass densities γ γ ρa (x, t) and ρ M(x, t) are macroscopically deﬁned by: ρaγ (x, t) =

dmγ ; |(x)|

γ

ρ M(x, t) =

dmγ | f (x, t)|

(1.10) γ

Using the approximation f ≈ χ0 /|0 |, it is readily seen that ρa (x, t) and γ ρ M(x, t) are related to the ﬁeld ρ γ (z, t) by: ρaγ = ρ γ f γ

ρ M = ρaγ /ϕ = ρ γ

(1.11) f

(1.12)

Clearly enough, it is also possible to deﬁne apparent and intrinsic averages over the pore space. This amounts to replacing χ f in (1.8) by the characteristic function χ p of the pore space. The case of a saturated porous medium can be thatf the deﬁned by the condition χ p (z, t) = χ f (z, t). In this situation, we note p operators · p and · f are equivalent, as well as the operators (·) and (·) .

A Mathematical Framework for Upscaling Operations

6

1.2.2 Spatial Derivatives of an Average Let us introduce a cartesian orthonormal frame (e i ). The derivative of the apparent average e f (x) of a physical quantity e(z) deﬁned on the ﬂuid domain P f with respect to the spatial coordinate xi is directly obtained from deﬁnition (1.8): ∂f ∂ (1.13) e f (x, t) = − χ f (z, t) e(z, t) (z − x) dVz ∂ xi ∂zi where the following identity has been employed: ∂ ∂f (z − x) f (z − x) = − ∂ xi ∂zi

Integration by parts of (1.13) then yields: ∂ ∂(χ f e) e f (x, t) = (z, t) f (z − x) dVz ∂ xi ∂zi

(1.14)

(1.15)

Since χ f (z, t) is discontinuous across the solid–ﬂuid interface I s f , relation (1.15) is to be understood in the sense of the distribution theory. More precisely, let δI s f be the Dirac distribution of support I s f . It is deﬁned by: δI s f , ψ = δI s f ψ dVz = ψ dSz (1.16) Is f

where ψ is any function of D(R3 ).3 According to the deﬁnition of the derivative of a distribution, one obtains: ∂χ f ∂ψ f ∂ψ ,ψ =− χ , dVz = − =− ψni dSz (1.17) ∂zi ∂zi Is f P f ∂zi

where n = ni e i is the unit normal to I s f oriented towards the solid. Combining (1.16) and (1.17) yields: ∂χ f = −ni δI s f ∂zi The expression (1.15) can now be developed in the form: ∂e ∂ e(z, t) ni (z, t) f (z − x) dSz e f (x, t) = (z, t) − f ∂ xi ∂zi Is f

(1.18)

(1.19)

Expression (1.19) establishes the sought link between the macroscopic spatial derivative and the microscopic one in the ﬂuid domain P f . With the same reasoning applied to the apparent volume average of a physical quantity e(z) 3 D(R 3 )

is the set of C ∞ functions which are equal to zero out of a bounded domain.

Averaging Operations

deﬁned on the solid domain P s , one obtains: ∂e ∂ e s (x, t) = (z, t) + e(z, t) ni (z, t) f (z − x) dSz s ∂ xi ∂zi Is f

7

(1.20)

where the change in sign of the second term on the right-hand side (r.h.s.) stems from the opposite direction of the outward unit normal to the solid phase. 1.2.3 Time Derivative of an Average The time derivative of e f also comprises an extra term, in addition to the volume average of the time derivative ∂e/∂t f . Indeed, starting from (1.8), we obtain: ∂e ∂χ f ∂ (1.21) e f (x, t) = (z, t) + (z, t) e(z, t) f (z − x) dVz f ∂t ∂t ∂t

The second term on the r.h.s. accounts for the displacement of the boundary of the ﬂuid domain through the derivative ∂χ f /∂t. More precisely, introducing the velocity u of the interface I s f , it can be shown that: ∂e ∂ (1.22) e(z, t) (u · n)(z, t) f (z − x) dSz e f (x, t) = (z, t) + f ∂t ∂t Is f where u · n represents the normal velocity of the solid–ﬂuid interface I s f , oriented by unit normal n pointing towards the solid phase. Analogously, we obtain for the solid phase: ∂e ∂ (1.23) e(z, t) (u · n)(z, t) f (z − x) dSz e s (x, t) = (z, t) − s ∂t ∂t Is f 1.2.4 Spatial and Time Derivatives of e

We ﬁnally consider the average e in the sense of (1.3) of a physical quantity deﬁned on the whole rev; that is, on both the solid and ﬂuid phases. Using the deﬁnition (1.3), (1.14) and integrating by parts, it is readily seen that: ∂e ∂f ∂ (e) = − (z − x) e(z, t) dVz = (1.24) ∂ xi ∂zi ∂zi In turn, the determination of the time derivative of e takes advantage of the fact that the weighting function f (z) does not depend on time. It follows that: ∂ ∂e ∂e f (z − x) (z, t) dVz = (e) = (1.25) ∂t ∂t ∂t

A Mathematical Framework for Upscaling Operations

8

Expressions (1.24) and (1.25) could have been directly obtained from (1.19)– (1.20) and (1.22)–(1.23), provided that the physical quantity e(z) deﬁned on both the ﬂuid and the solid domain, P f and P s , i.e. on (x), is continuous over the solid–ﬂuid interface. In the more general case, the spatial and time derivatives of e on account of a possible discontinuity of e(z) on the solid–ﬂuid interface take the following form: ∂e ∂ e(x, t) = (1.26) + [e(z, t)] ni (z, t) f (z − x) dSz ∂ xi ∂zi Is f ∂e ∂ (1.27) (e) = − [e(z, t)] (u · n)(z, t) f (z − x) dSz ∂t ∂t Is f where [e(z, t)] = e s (z, t) − e f (z, t) denotes the jump of e over I s f oriented by the unit outward normal to the ﬂuid phase.

1.3 Application to Balance Laws Upscaling rules for several physical phenomena that are present at both the microscopic and the macroscopic scale can be derived from the balance laws. Indeed, each balance law can be formulated either at the microscopic scale or at the macroscopic one. The consistency of these two approaches of the same physical principle then delivers the upscaling rule. This technique is developed in this section.4 The starting point is the conservation laws at the microscopic scale. Taking the average of the corresponding laws in the sense of (1.3) or the apparent average in the sense of (1.8), we obtain macroscopic formulations of the balance laws from a pure upscaling reasoning. The mass balance law is considered ﬁrst; the momentum balance is addressed in Section 1.3.2. 1.3.1 Mass Balance The mathematical formulation of mass balance classically involves the concept of velocity. For a component γ of a ﬂuid phase, the velocity is represented at the microscopic scale by a ﬁeld vγ (z) deﬁned on P f . The macroscopic counterpart is a single vector V γ (x). Hence, two macroscopic mass balance equations are available: one to be derived from the average of the microscopic formulation; the second from a purely macroscopic reasoning. Since mass conservation must hold irrespective of the scale under consideration, the necessary consistency of these two approaches provides the link between the ﬁeld vγ (z) and the vector V γ (x). This is the problem we consider here. 4A

review of this technique can be found in [26] and [6].

Application to Balance Laws

9

Mass Balance of a Component of a Fluid Phase

When the ﬂuid comprises several chemical species, the macroscopic point of view describes the ﬂuid as the superposition of several macroscopic particles, each corresponding to one of these chemical species. At the microscopic scale, the approach is similar. At each point z of the ﬂuid domain P f, the ﬂuid is considered as a superposition of microscopic particles. For the γ component of the ﬂuid, the local volume unit mass and velocity are represented by the ﬁelds ρ γ (z) and vγ (z). The microscopic mass balance equation for this component thus reads: ∂ρ γ (1.28) + divz (ρ γ vγ ) = 0 ∂t where divz is the divergence operator acting on the microscopic coordinates zi . Note clearly that the mass balance equation (1.28) disregards any mass exchange between the γ component and any other components of the ﬂuid. The apparent average of (1.28) – in the sense of (1.8) – is straightforward: ∂ρ γ (1.29) + divz ρ γ vγ f = 0 ∂t f γ

We now apply (1.19) with e = ρ γ vi and (1.22) with e = ρ γ. Taking (1.9) into account, the mass balance (1.29) takes the form: ∂ f f (1.30) (ϕρ γ ) + divx (ϕρ γ vγ ) = ρ γ (u − vγ ) · n f (z − x) dSz s f ∂t I

where divx is the divergence operator acting on the macroscopic coordinates xi . The r.h.s. of (1.30) represents the mass ﬂux of the γ component across the solid–ﬂuid interface. This term is non-zero for dissolution, precipitation or adsorption phenomena that may take place at the solid–ﬂuid interface. In the absence of such mass exchange between γ and the solid, the ﬂux is zero and the following macroscopic formulation of the mass balance principle is obtained: ∂ f f (1.31) (ϕρ γ ) + divx (ϕρ γ vγ ) = 0 ∂t This macroscopic mass balance has been derived from the upscaling of its microscopic counterpart (1.28). Alternatively, a pure macroscopic approach to the mass balance principle reads: γ

∂ρa + divx ρaγ V γ = 0 ∂t

(1.32) γ

where V γ (x) is a macroscopic velocity vector of component γ , and ρa has been deﬁned in (1.11). Finally, a comparison of (1.31) and (1.32) provides the link between the microscopic and macroscopic descriptions of the kinematics.

10

A Mathematical Framework for Upscaling Operations

Indeed, the compatibility between (1.28) and (1.32) requires the macroscopic γ momentum ρa V γ to be equal to the apparent average of the microscopic mof mentum ϕρ γ vγ ; that is: Vγ =

f

ρ γ vγ ργ

(1.33)

f

where we made use of (1.11) and (1.12). Homogeneous Fluid

It is instructive to consider the case of a homogeneous ﬂuid, which by deﬁnition comprises only one single component. In this case, replacing ρ γ (z) in (1.33) by the ﬂuid mass density ρ f (z), and vγ (z) by the velocity ﬁeld v(z), the link between the velocity of the macroscopic ﬂuid particle V f (x) and the microscopic velocity ﬁeld is: f

V =

ρfv ρf

f

f

(1.34)

It is interesting to note that the macroscopic ﬂuid velocity is not a priori the volume average of the microscopic one. In fact, it is readily seen from (1.34) that this is only the case when the ﬂuid phase is incompressible; that is: f

ρ f = ρ0 ⇔ V f = v f

(1.35)

1.3.2 Momentum Balance We now turn to a second conservation law, the momentum balance, to derive upscaling rules based on the compatibility of the micro- and macroscopic expressions of the momentum balance. Average Rules for Total and Partial Stresses

We consider a porous medium saturated by a homogeneous ﬂuid. If we denote by σ(z) the symmetric microscopic stress ﬁeld,5 i.e. the stress ﬁeld deﬁned at 5 Throughout this book, all stress quantities that are introduced, whether microscopic or macroscopic, satisfy the symmetry condition:

σi j = σ ji The basis of the continuum mechanics approach employed here can be found in classical textbooks on continuum mechanics (e.g. Salenc¸on, [46]).

Application to Balance Laws

11

the microscopic scale on the rev (x), the momentum balance is: ∂σi j + ρ fi = 0 ∂z j

(1.36)

where ρ f (z) is the density of volume forces acting at the microscopic point z. For instance, for gravity forces the vector f is the acceleration due to gravity g. The mass density ρ(z) depends (through z) on the phase to which z belongs. In the particular case of incompressible solid and ﬂuid phases, ρ(z) is equal to either the intrinsic mass density ρ s of the solid or ρ f of the ﬂuid, depending on the position vector z. Using (1.24) with e = σi j , the average of (1.36) – in the sense of (1.3) – delivers a ﬁrst macroscopic expression of the momentum balance: ∂σi j ∂ + ρ fi = (σi j ) + ρ f i ∂z j ∂xj

(1.37)

In addition, a second macroscopic expression of the momentum balance is provided by the macroscopic approach: ∂ i j + ρ M Fi = 0 ∂xj

(1.38)

where Σ is a macroscopic stress tensor, while ρ M F (x) now represents the density of volume forces acting at the macroscopic scale at point x. From the very deﬁnition of f (z) and F (x), it is readily seen that the comparison of (1.37) and (1.38) provides the following remarkable results:

r The macroscopic stress tensor Σ is the average of the microscopic stress ﬁeld: Σ=σ

(1.39)

r The volume forces obey the following upscaling rule: ρF = ρ f

(1.40)

While readily veriﬁed for gravity forces, for which F = f = g, the upscaling rule (1.40) is useful in some problems related, for example, to electromagnetism, allowing for the determination of F = ρ f /ρ from its microscopic counterpart. In addition to the total stress Σ, a common stress quantity encountered in macroscopic approaches is known as partial stress Σα of the α phase (α = s or f ). In the very same way as the total stress, this partial stress tensor is a macroscopic physical quantity. Indeed, it represents the internal forces in one of the two phases. Furthermore, in such macroscopic theories, called mixture

12

A Mathematical Framework for Upscaling Operations

theories, the interaction between the solid and the ﬂuid phase is represented, at the macroscopic scale, by a volume force a (x), such that a (x) |(x)| represents the elementary force applied by the macroscopic ﬂuid particle located at point x to the macroscopic solid particle located at the same point. The principle of momentum balance applied to each phase at the macroscopic scale reads: ∂ iαj ∂xj

+ ρaα Fiα + ǫ α a i = 0

(1.41)

where ǫ s = 1 and ǫ f = −1. We want to give a microscopic derivation of both the partial stress and the interaction term in (1.41). To this end, we consider the partial average of (1.36): ∂σ ij + ρ α f iα α = 0 (1.42) ∂z j α Then, applying (1.19)–(1.20) for e = σi j yields: ∂ ∂ α σi j (z, t) n j (z, t) f (z − x) dSz σi j α = σi j (z, t) + ǫ α ∂xj ∂z j Is f

(1.43)

Finally, substituting (1.43) in (1.42) yields a second macroscopic formulation of the momentum balance of phase α, in addition to (1.41), which is obtained independently from the former by means of upscaling the microscopic formulation of the same principle: α α α (1.44) divx σ α + ρ f α − ǫ σ(z, t) · n(z, t) f (z − x) dSz = 0 Is f

Finally, from a comparison of the two macroscopic momentum balance relations – the pure macroscopic one (1.41), and the one obtained by means of upscaling (1.44) – it turns out that the partial stress Σα is the apparent average of the microscopic stress ﬁeld in the α phase: Σα = σ α

(1.45)

In turn, the volume force vector a employed in the macroscopic approach to represent the mechanical interaction between the solid and the ﬂuid phases is actually the integral of the interaction surface forces at the solid–ﬂuid interface: a =− (1.46) σ(z, t) · n(z, t) f (z − x) dSz Is f

It is interesting to note that the same physical phenomenon, e.g. the solid–ﬂuid interaction, has different representations at different scales: a surface force at the microscopic scale, and a volume force at the macroscopic scale, while the link is provided by the surface integral (1.46).

Application to Balance Laws

13

Partial Stress in an Incompressible Viscous Fluid

It is instructive to study the case of an incompressible Newtonian ﬂuid, deﬁned by a viscosity coefﬁcient μ f . For simplicity, the solid phase is assumed to be rigid and the pore volume fraction ϕ is uniform. The microscopic stress state in the ﬂuid is related to the strain rate tensor d(z), deﬁned as the symmetric part of the microscopic velocity gradient v, by the classical state equation of ﬂuid mechanics: 1 (1.47) σ = − p(z)1 + 2μ f d(z) with d = gradz v + t gradz v 2 where p stands for the (thermodynamic) ﬂuid pressure. We want to determine the macroscopic partial stress Σ f in the ﬂuid. To this end, the upscaling rule (1.45) gives: Σ f = σ f = − p f 1 + 2μ f d f Then, applying (1.19) with e = v j , we obtain: ∂v ∂ j v j f (x, t) = (z, t) − v j (z, t) ni (z, t) f (z − x) dSz f ∂ xi ∂zi Is f

(1.48)

(1.49)

This equation is simpliﬁed by the boundary condition v = 0 at the solid–ﬂuid interface I s f : ∂v ∂ j (1.50) v j f (x, t) = (z, t) f ∂ xi ∂zi

Finally, recalling (1.35), (1.50) reveals that the macroscopic strain rate D(V f ) associated with the velocity V f is the intrinsic average of the microscopic strain rate: 1 1 f D(V f ) = (1.51) gradx V f + t gradx V f = d f = d 2 ϕ

We note that the incompressibility condition tr d = 0 is upscaled in the form tr D = 0. Furthermore, substitution of (1.51) in (1.48) yields: Σ f = −ϕ p f 1 + 2ϕμ f D(V f )

(1.52)

P = pf

(1.53)

It therefore turns out that the partial stress Σ f which represents, at the macroscopic scale, the internal forces in the ﬂuid phase is identical to the stress in a homogeneous viscous ﬂuid of viscosity ϕμ f which is subjected to the pressure ϕ p f . In what follows, we will refer to p f as the macroscopic pressure denoted by P:

Finally, it is instructive to determine the solid–ﬂuid interaction volume force a for the considered case. To this end, we insert expression (1.52) for the partial

A Mathematical Framework for Upscaling Operations

14

stress Σ f into the macroscopic balance equation of the ﬂuid (1.41). As the porosity ϕ is homogeneous, we obtain: (1.54) ϕ −grad P + μ f x V f + ρ f F f − a = 0 x

where the differential operators, the Laplace operator and the gradient operator grad act on the macroscopic position vector x. We will see in Chapter 2 that grad P can be related to the macroscopic ﬂuid velocity (Darcy’s law), for which relation (1.54) will provide a means of determining a as a function of the macroscopic velocity and its derivatives.

1.4 The Periodic Cell Assumption 1.4.1 Introduction Natural porous media and particularly geomaterials are disordered materials. These materials are a priori best addressed through the rev concept introduced in Section 1.1. Nevertheless, in some situations it will prove helpful to model a microstructure in the framework of periodic media. The idea of the periodic media theory is that the fundamental information concerning the physical properties of the constituents and the morphology of the microstructure can be captured in an elementary cell. Then, a periodic model for the real material can be obtained by ﬁlling the entire space with this elementary pattern in a periodic way. The characteristic size of the elementary cell is a priori of the order of the local inhomogeneity d (see Section 1.1). For clarity and without loss of generality, the periodic framework is hereafter presented with the assumption that the cell is a cube, the edges of which are parallel to the directions (e 1 , e 2 , e 3 ) of an orthonormal frame. The edge length a is of the order of d; U denotes the cubic domain [0, a ] × [0, a ] × [0, a ]. Us and U f respectively denote the solid and ﬂuid domains in U. As in Section 1.1, we will consider a macroscopic structure of characteristic size L. The scale separation condition still reads d ≪ L, and means here that the macrostructure comprises a sufﬁcient large number of elementary cells. In the framework of the periodic assumption, the elementary cell concept is used instead of an rev. The elementary cell is not an rev, since its characteristic length is not necessarily large with respect to that of the heterogeneities.6 The periodic media theory is based on the premise that it is possible to represent spatial variations of a physical quantity as a combination of local 6 However, accurate estimates of macroscopic properties in the periodic framework require a reﬁned description of the microstructure within the elementary cell whose size may become larger.

The Periodic Cell Assumption

15

ﬂuctuations at the level of the elementary cell and a drift at the level of the macroscopic structure. This separation is represented by two dimensionless spatial variables, Z = z/a and X = z/L, which allow a representation of any physical quantity e(z) in the form e(Z, X). The mathematical way by which X is responsible for the long scale drift, and local ﬂuctuations are taken into account through the variable Z, is to introduce the condition that e(Z, X) is a periodic function of Z for any prescribed value of X: (∀ n1 , n2 , n3 ∈ N)

e(Z + n1 e 1 + n2 e 2 + n3 e 3 , X) = e(Z, X)

(1.55)

This periodicity condition actually proves that the drift δe of e on one period in the direction i is related to the derivative of e with respect to Xi : a a a ∂e (1.56) +o δe = e(z + a e i ) − e(z) = e Z, X + e i − e(Z, X) = L L ∂ Xi L

In fact, X appears as the dimensionless counterpart of the macroscopic position vector x. In particular, we note that: ∂ ∂e 1 ∂ = ⇒ δe = a L ∂ Xi ∂ xi ∂ xi

(1.57)

The dimensionless variables X and Z are now regarded as independent. Taking advantage of the periodicity of e with respect to the ‘ﬂuctuation’ variable Z, the average e of the physical quantity e is deﬁned by integration with respect to Z over any elementary cell, while X is regarded as a constant. More precisely, let UZ denote the domain obtained by applying the transformation Zi = zi /a to the elementary cell U. In other words, UZ is the cubic domain [0, 1] × [0, 1] × [0, 1]. The average e is: e(X, t) = (1.58) e(Z, X, t) dV Z UZ

By construction, we thus deﬁne e as a macroscopic quantity which depends only on the ‘drift’ variable X. In turn, the apparent average e α is deﬁned by analogy with (1.8) as χ α e: e α (X, t) = (1.59) (χ α e)(Z, X, t) dV Z UZ

Furthermore, the intrinsic average e α is deﬁned by analogy with (1.9). Hence, for the ﬂuid: 1 (χ f e)(Z, X, t) dV Z e f (X, t) = (1.60) ϕ UZ Since the reference microstructure is itself periodic, the characteristic function χ α in the reference state is periodic as well. However, in the case of a

16

A Mathematical Framework for Upscaling Operations

deformable solid phase, χ α becomes a function of time and can also depend on the drift variable X if the microscopic displacement does. The idea to deﬁne the average by integration over a period aims at capturing all the information available on the morphology of the microstructure as well as the physical properties of the constituents. The integral in (1.58), however, is performed on the period corresponding to the reference (i.e. non-deformed) state, which restricts its application to small perturbations of the geometry of the microstructure. In particular, the dependence of χ α with respect to the drift variable is disregarded. The following chain rule for differentiation is the starting point for the derivation of the periodic counterpart of (1.19), (1.22) and (1.24): 1 ∂e 1 ∂e ∂e = + ∂zi a ∂ Zi L ∂ Xi

(1.61)

Given the scale separation condition d ≪ L, and the fact that a is on the order of d, it could be appealing to consider in the total spatial derivation the ﬁrst term on the r.h.s., i.e. the ﬂuctuation term (1/a ) (∂e/∂ Zi ), as dominant over the drift term (1/L ) (∂e/∂ Xi ). This depends, however, on the function e(Z, X). To illustrate this purpose, consider in a one-dimensional setting the displacement ξ (z) = u(X) + δ n v(Z), where δ = a /L. Application of the chain rule (1.61) yields the linearized strain:

∂ξ 1 du n−1 dv ε= δ (1.62) = + ∂z L dZ dX Given a ≪ L, and provided that dv/dZ and du/d X are on the same order of magnitude, we observe indeed, for n = 0, that the ﬂuctuation term dominates the drift term in the total spatial derivative. On the other hand, for n = 1, both terms may be equally important; for higher values n ≥ 2, the drift term dominates the ﬂuctuation term. 1.4.2 Spatial and Time Derivative of e in the Periodic Case Integrating the chain rule (1.61) with respect to Z, we obtain: ∂e ∂e ∂e 1 1 dVZ = dVZ + dVZ a UZ ∂ Zi L UZ ∂ Xi UZ ∂zi

(1.63)

The ﬁrst integral on the r.h.s. is transformed into a surface integral over the cell boundary: ∂e dVZ = e ni dSZ (1.64) UZ ∂ Zi ∂UZ

The Periodic Cell Assumption

17

which is equal to zero because of the periodicity condition (1.55). On the other hand, the second integral on the r.h.s. in (1.63) can be developed in the form:

∂ ∂ ∂e (1.65) dVZ = (e) e(Z, X, t) dVZ = ∂ Xi ∂ Xi UZ UZ ∂ Xi Thus, the periodic counterpart of (1.24) is: ∂ ∂e 1 ∂ = (e) = (e) ∂zi L ∂ Xi ∂ xi

(1.66)

As in Section 1.3.2, this equation can be used to derive the link between the microscopic stress ﬁeld and the macroscopic stress tensor. Indeed, by letting e = σi j in (1.66), identical results are obtained, namely Σ = σ. Similarly, the upscaling rules (1.33), (1.34) and (1.35) concerning the ﬂuid velocities can be extended to the periodic case. Lastly, the average rule concerning the time derivative is straightforward: ∂e ∂e ∂ (e) = (Z, X, t) dV Z = (1.67) ∂t ∂t UZ ∂t 1.4.3 Spatial and Time Derivative of e α in the Periodic Case The spatial derivative e α is developed here for the ﬂuid phase. We recall that the geometry changes of the solid–ﬂuid interface are neglected. This implies that the characteristic function χ f of the ﬂuid domain is a periodic function. Hence, it depends only on Z and not on the drift variable X. Applying deﬁnition (1.59), we thus obtain: ∂e ∂ ∂ f ( e f ) = (χ e) dV Z = dV Z (1.68) χ f (Z) ∂ Xi ∂ Xi UZ ∂ Xi UZ The chain rule (1.61) integrated over the ﬂuid domain in the elementary cell UZ gives: ∂e ∂e dV Z = χ f (Z) ∂zi f ∂zi UZ ∂e ∂e 1 1 f = dV Z + dV Z (1.69) χ (Z) χ f (Z) a UZ ∂ Zi L UZ ∂ Xi In order to calculate the ﬁrst integral on the r.h.s., we observe that: χf

∂ ∂χ f ∂e = (χ f e) − e ∂ Zi ∂ Zi ∂ Zi

(1.70)

As in Section 1.2.2, the above identity must be understood in the sense of the distribution theory, since it involves the derivatives of the discontinuous function χ f . In addition, the fact that χ f e is a periodic function of the ﬂuctuation

A Mathematical Framework for Upscaling Operations

18

variable Z implies that: UZ

∂ (χ f e) dV Z = ∂ Zi

∂UZ

niZ χ f e dS Z = 0

(1.71)

sf

Let I Z be the image of the solid–ﬂuid interface I s f in U obtained by the transformation Zi = zi /a . Recalling that χ f does not depend on the drift variable X, the derivative of the discontinuous function χ f in (1.70) can be determined as in (1.18): ∂χ f = −niZ δI s f (Z) Z ∂ Zi

(1.72) sf

where niZ are the components of the unit normal to the interface I Z oriented towards the solid. Substituting (1.69), (1.70), (1.71) and (1.72) into (1.68) yields the periodic counterpart of (1.19): ∂e ∂ 1 1 ∂ e niZ dS Z ( e f ) = e f = − ∂ xi L ∂ Xi ∂zi f a I Zs f ∂e 1 = e ni dSz (1.73) − ∂zi f |U| I s f Finally, the time derivative of an apparent average is derived as in (1.21): ∂e ∂χ f ∂ e f (x, t) = e dV Z (1.74) + ∂t ∂t f UZ ∂t

The second term on the r.h.s. of (1.74) accounts for the velocity u of the solid– ﬂuid interface, and reduces to u/a in the domain of dimensionless coordinates. Hence, the periodic counterpart of (1.22) is: ∂e 1 ∂ e (u · n) dS Z e f (x, t) = + ∂t ∂t f a I Zs f ∂e 1 = (1.75) e (u · n) dSz + ∂t f |U| I s f

The previous relations, (1.73) and (1.75), are readily employed for extending the upscaling rules of partial stresses (see Section 1.3.2) to the periodic case. 1.4.4 Application: Micro- versus Macroscopic Compatibility One question which comes immediately to mind when micro-to-macro approaches are applied to deformable media is the question of geometrical compatibility on both scales. To address this question, we ﬁrst postulate7 that 7A

detailed discussion is given in Sections 4.2.3 and 4.2.5.

The Periodic Cell Assumption

19

the macroscopic linearized strain tensor E(x) is the volume average of the linearized microscopic strains ε(z): E(x) = ε(z)

(1.76)

A necessary condition for the validity of an average relation of the form (1.76) is that the geometrical compatibility of the microscopic strain ﬁeld ε(z) entails the geometrical compatibility of the macroscopic strain tensor E(x). The microscopic conditions of geometrical compatibility are: ∂2 ∂2 ∂2 ∂2 (εi j ) + (εkl ) − (εik ) − (ε jl ) = 0 ∂zk zl ∂zi z j ∂z j zl ∂zi zk

(1.77)

and their macroscopic counterparts are: ∂2 ∂2 ∂2 ∂2 (E i j ) + (E kl ) − (E ik ) − (E jl ) = 0 ∂ xk xl ∂ xi x j ∂ x j xl ∂ xi xk

(1.78)

with i,j,k,l = 1,2,3. A proof of this result can be obtained with (1.24) (resp. (1.66)). It comprises two identical steps. First, this identity is applied to e = εi j and gives: ∂εi j ∂ ∂ = (εi j ) = (E i j ) ∂zk ∂ xk ∂ xk

(1.79)

The second step consists of applying the same identity to e = ∂εi j /∂zk . We now obtain:

∂εi j ∂εi j ∂ ∂ = (1.80) ∂zl ∂zk ∂ xl ∂zk Finally, a combination of (1.79) and (1.80) yields:

∂εi j ∂ ∂2 = (E i j ) ∂zl ∂zk ∂ xk xl

(1.81)

Hence, the macroscopic compatibility (1.78) is a direct consequence of (1.81). It is instructive to note that the previous result could have been directly obtained by application of relation (1.24) with a microscopic displacement e = ξ j (z) associated with the microscopic strain ﬁeld ε(ξ ) = 12 (gradz ξ + tgradz ξ ), as follows: ε(ξ ) = ε(ξ ) = E(x)

(1.82)

where ξ (x ) is the volume average of the microscopic displacement ﬁeld. In other words, E(x) is the strain tensor associated with ξ (x ). This immediately ensures the geometrical compatibility of E(x).

Part I Modeling of Transport Phenomena

2 Micro(fluid)mechanics of Darcy’s Law This and the next chapter deal with transport phenomena at multiple scales. While this chapter is concerned with ﬂuid conduction, the next chapter will be devoted to ion conduction. The macroscopic theory of ﬂuid conduction is known as Darcy’s law. This chapter is devoted to the micromechanics derivation of Darcy’s law. The microto-macro approach is based on the periodic homogenization technique and translates the Stokes equations that describe the equilibrium and constitutive behavior of the ﬂuid at the microscale into the macroscopic Darcy’s law. In this way, the macroscopic permeability is derived as a function of the microscopic pore morphology and the ﬂuid viscosity. Upper and lower bound solution strategies based on variational principles are derived that make it possible to determine estimates of the permeability of porous media.

2.1 Darcy’s Law To motivate the forthcoming developments, consider an incompressible ﬂuid at equilibrium under gravity forces. The hydraulic head H is given by: H = P + ρ f gx2

(2.1)

where P is the (macroscopic) ﬂuid pressure deﬁned by (1.53) (Section 1.3.2), ρ f g is the ﬂuid unit weight (g is the nominal gravitational acceleration), and x2 is the vertical coordinate oriented upwards. The equilibrium of the ﬂuid implies that H is uniform throughout the ﬂuid continuum. This ﬂuid head will turn out to be an important physical quantity for the description of the ﬂow of a ﬂuid through the pore space of a porous medium. Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

Micro(fluid)mechanics of Darcy’s Law

24 Q

h1 Q

h2

Figure 2.1 Darcy’s experiment: principle of the experimental device for permeability measurement

Darcy1 was the ﬁrst to address this question. He studied the water ﬂow through a cylindrical sample of sand by controlling the head difference between the upper and lower faces of the cylinder. This head difference, H, can be achieved by means of the device depicted in Figure 2.1. If the upper and lower reservoirs R1 and R2 are subjected to the same external pressure (say, atmospheric pressure), H is equal to ρ f g(h 1 − h 2 ), where h 1 − h 2 denotes the difference between the water levels in the two reservoirs. While keeping the difference in water levels constant during the experiment, Darcy measured the ﬂow D across the section S of the sample, and obtained the remarkable result that Q = D/S is proportional to the ratio H/L of the head difference over the height L of the sample: Q=

H D =K S L

(2.2)

The coefﬁcient K which appears in (2.2) represents the permeability of the sample in the direction of the ﬂow. Darcy’s observations suggest that the ﬂuid ﬂow is proportional to the hydraulic head gradient and oriented in the opposite direction, which can be captured by the following macroscopic vector formulation of the transport

1 Henri Darcy (1803–1858), born in Dijon (France), was admitted to Ecole Polytechnique (1821) and to Ecole Nationale des Ponts et Chauss´ees (1823). In 1855 and 1856, he carried out column experiments that established the so-called Darcy’s law for ﬂow in granular media, which are described in ‘Les fontaines publiques de la ville de Dijon’. He died of pneumonia in 1858 during a trip to Paris.

Microscopic Derivation of Darcy’s Law

25

law for porous media (porosity ϕ): Q = K (−gradH) = K (−gradP + ρ f g)

(2.3)

where Q = ϕV f is referred to as the ﬁltration velocity. The validity of (2.3) is restricted to the particular case of isotropic hydraulic properties: for the same hydraulic head gradient, the ﬂow velocity is the same in all directions. The anisotropy of the geometry of the microstructure is expected to have repercussions on the macroscopic relationship between ﬂow and hydraulic head. This can be achieved, within the concept of the macroscopic Darcy ﬂow theory, by replacing the scalar permeability in (2.3) by a tensorial quantity: Q = K · (−grad P + ρ f g)

(2.4)

where K is the second-order permeability tensor. Relation (2.4) has been found to hold for a wide class of ﬂows in porous media. Finally, possible inertia effects are taken into account, in the classical manner of phenomenological approaches, through the addition of an inertia term ρ f γ f : Q = K · (−gradP + ρ f (g − γ f ))

(2.5)

2.2 Microscopic Derivation of Darcy’s Law We seek a microscopic derivation of the macroscopic Darcy’s law. This is instructive in several regards. For one thing, a microscopic derivation is able to capture the geometry of the pore space; but, more important, at the microscopic scale, the ﬂuid ﬂow is described by the Stokes equations, i.e. by another ﬂuid constitutive law than the one employed at the macroscopic scale. The challenge of the micro-to-macro approach, therefore, is to show that the Stokes equations at the microscale translate under certain conditions into Darcy’s law. 2.2.1 Thought Model: Viscous Flow in a Cylinder To motivate the forthcoming developments, we examine a 1-D thought model: a circular cylindrical pore space of radius a in a cylindrical elementary volume of cross section S (Figure 2.2). The porosity is ϕ = πa 2 /S. At the microscopic scale, r is the microscopic radial coordinate with respect to the axis of the cylindrical pore space. The ﬂuid is assumed to be an incompressible Newtonian ﬂuid of shear viscosity μ f . The velocity ﬁeld of the ﬂuid in the pore space is given by the classical Poiseuille solution: dP 1 2 2 v(r ) = − f (a − r ) ex (2.6) 4μ dx

Micro(fluid)mechanics of Darcy’s Law

26

r

S

2a

x

Figure 2.2 Viscous flow in a cylindrical pore space

where (d P/d x ) e x denotes the pressure gradient. Since the pore pressure is uniform in each section, the pressure gradient is the same at both the microscopic and the macroscopic scale. The average ﬂow through the cross section of the elementary volume, i.e. the ﬁltration velocity Q, is thus: D dP 1 π a4 Q= − (2.7) = f S μ 8S dx It is instructive to compare the result (2.7) with the phenomenological law (2.3) to ﬁnd out if both display a linear relation between the macroscopic ﬁltration velocity and the pressure gradient. Thus, from this comparison, the factor (πa 4 /8S)/μ f appears as the permeability K of this 1-D thought model, which captures the geometrical characteristics of the (no doubt too simplistic) cylindrical morphology of the pore space. The study of the 1-D thought model reveals that the permeability coefﬁcient accounts on the macroscopic scale for the geometry of the pore space, and also for the viscosity of the ﬂuid phase. In other words, the permeability coefﬁcient K is not an intrinsic property of the porous medium, but depends also on the ﬂuid. An intrinsic permeability, i.e. one that depends only on the pore geometry and not on the saturating ﬂuid phase, can be deﬁned in the following form: K = K ′ /μ f

(2.8)

For instance, for the 1-D thought model, the intrinsic permeability is K ′ = 2 (πa 4 /8S), of dimension [K ′ ] = Length . 2.2.2 Homogenization of the Stokes System We want to prove that the intrinsic permeability deﬁned by (2.8) holds irrespective of the particular geometry of the cylindrical model employed in

Microscopic Derivation of Darcy’s Law

27

Figure 2.3 Periodic cell model

Section 2.2.1, and can be generalized to the anisotropic case K = K′ /μ f . This will be achieved by employing a rigorous micromechanical interpretation of Darcy’s law within the framework of periodic homogenization.2 We consider a periodic porous medium. For simplicity, the elementary cell is assumed to be a parallelepiped denoted by U (have a quick look back to Section 1.4). It comprises all the geometrical information necessary for the determination of the microscopic velocity ﬁeld in the ﬂuid. The edges are parallel to the directions (e 1 , e 2 , e 3 ) of an orthonormal frame and their lengths are denoted by a i (i = 1, 2, 3). The boundary ∂U is included in the planes zi = 0 and zi = a i . Each cell is made up of a rigid solid in domain Us , and an incompressible viscous Newtonian ﬂuid (viscosity μ f ) in domain U f , separated by the solid–ﬂuid interface I s f . We note that S = ∂U f \ I s f is the part of the ﬂuid boundary contained in the boundary ∂U of the cell (Figure 2.3). We consider a stationary ﬂow of the ﬂuid in U f . Furthermore, the velocity is assumed to comply with the periodicity condition (1.55), which reads here: ∀ n1 , n2 , n3 ∈ N

v(z + n1 a 1 e 1 + n2 a 2 e 2 + n3 a 3 e 3 ) = v(z)

(2.9)

This periodicity condition will be removed in Section 2.4.1. Provided that (2.9) is satisﬁed, the velocity ﬁeld v(z) is entirely characterized by its value in U. The periodicity condition then reduces to: ∀z ∈ ∂U f ∩ (zi = 0)

v(z + a i e i ) = v(z)

(2.10)

Based on (1.59) and (1.60), the relevant macroscopic representations of a periodic microscopic ﬁeld b(z) correspond to the apparent or intrinsic averages of b over the ﬂuid domain in the cell: 1 1 f b(z) d Vz ; b = b(z) d Vz (2.11) b f = | U | Uf | U f | Uf 2 The

main ideas developed here are inspired by the work of Ene and Sanchez-Palencia [23].

Micro(fluid)mechanics of Darcy’s Law

28

The periodicity condition of the microscopic velocity ﬁeld v(z) implies that the average of v(z) over the ﬂuid domain U f is the same in all cells. According to (2.11), this could suggest that the periodicity condition is associated with a uniform macroscopic ﬂow. In fact, this condition must be understood at the scale of the rev, which is by deﬁnition made up of a large number of elementary cells. Its physical meaning is that the velocity ﬁeld v(z) is approximately the same in all cells contained in the considered rev, while a drift of v(z) may occur at the macroscopic scale, i.e. between two revs located at different macroscopic points x. We will see in Section 2.4.1 how to formulate these qualitative remarks in a rigorous mathematical framework by means of the technique of doublescale asymptotic expansions. Finally, in what follows we will assume that the Reynolds number is small enough so that inertia effects can be neglected in the momentum balance equation of the ﬂuid. We also disregard the gravity forces. The microscopic ﬂow is therefore subjected to the following set of Stokes equations: −grad p + μ f z vi e i = 0

(2.12)

divz v = 0

(2.13)

z

v = 0 on I s f

(2.14)

The Pore Pressure Field

From a physical point of view, the velocity v and the pressure p, which are the solutions of (2.12)–(2.14), are a priori deﬁned on U f only. However, from a mathematical point of view, it is convenient to extend these functions to the whole cell. As for the velocity, the natural extension to Us is v = 0. The continuity of the velocity ﬁeld v within U is ensured by (2.14). We look for an extension of the pressure p in the whole cell (solid and ﬂuid domains) in the form: p(z) = α · z + φ(z)

(2.15)

where α is a constant vector and φ(z) is a periodic function. A detailed proof of the existence of such an extension can be found in [23]. It is based on the fact that the periodicity of v implies the periodicity of the pressure gradient over the ﬂuid domain. The main idea of the proof is readily understood from a 1-D cell of period L. In this case, the average of the pressure derivative is: 1 L ′ p (s)ds (2.16) α= L 0

Furthermore, the periodicity of p ′ (z) implies: z+L z+L d ′ p ′ (s) ds = α L p (s) ds = 0 ⇒ dz z z

(2.17)

Microscopic Derivation of Darcy’s Law

29

We just have to consider the function φ(z) deﬁned by: z φ(z) = p(z) − α z = p ′ (s)ds − α z + p(0)

(2.18)

0

From (2.17) and (2.18), it is readily understood that the periodicity of function φ(z) is a direct consequence of (2.17). From a mathematical point of view, relation (2.16) can be easily extended to three dimensions. Indeed, differentiating (2.15) with respect to zi yields: ∂φ ∂p = αi + ∂zi ∂zi

(2.19)

Given the periodicity of φ(z), integration of (2.19) over the whole cell yields: 1 ∂p ∂p 1 dVz + dVz (2.20) φ ni dSz = αi = | U | U ∂zi | U | U ∂zi ∂U Thus, α appears as the average of the pressure gradient over the whole cell: 1 α= (2.21) grad p dVz z |U| U Two types of pressure variations are sketched in Figure 2.4 . At a length scale on the order of the cell, the pressure variation is the sum of a contribution of the linear part of the decomposition (2.15) and of a periodic ﬂuctuation. By contrast, at the macroscopic scale which is large w.r.t. the characteristic length of the cell, the variation of the pressure becomes linear as the periodic ﬂuctuation becomes negligible. The vector α, therefore, represents the macroscopic pressure gradient, grad P, which appears in (2.4). Variational Approach to the Stokes System

It is instructive to examine, for a given value of α, the existence and uniqueness of the couple (v, p) which is the solution to the problem (2.12)–(2.14) and (2.15). To this end, we derive the weak form of the Stokes system associated with a functional space H ⊂ (H1 (U f )). The classical method for the determination of the weak form consists in integrating over U f the scalar product of (2.12) and a test function u ∈ H. In the same way as the solution velocity ﬁeld v, this test function u must satisfy the periodicity condition (2.10), the incompressibility condition (2.13) and the boundary condition (2.14). The test function is a priori deﬁned on U f , but can be extended if necessary to zero over the whole cell. Thus, integration by parts yields the following weak form: ui n · grad vi dSz (2.22) ui z vi dVz = − grad ui · grad vi dVz + Uf

Uf

z

z

∂U f

z

Micro(fluid)mechanics of Darcy’s Law

30 P

a 1 f (z)

periodic cell

x, z

Figure 2.4 Sketch of the variations of the pressure field

The boundary ∂U f of the ﬂuid domain in the cell is made up of the solid–ﬂuid interface I s f and the subset S of the cell boundary. The contribution of I s f to the surface integral in (2.22) is zero, since u = 0 on I s f . The contribution of S also vanishes, since u and v are both periodic and the outward unit normals n to the cell boundary take opposite values along the planes zi = 0 and zi = a i . Expression (2.22) therefore simpliﬁes to: u · (z vi )e i dVz = − grad ui · grad vi dVz (2.23) Uf

Uf

Furthermore: u · grad p dVz = − z

Uf

z

z

p divz u dVz +

Uf

∂U f

p(u · n) dSz

(2.24)

Given the incompressibility of u, i.e. divz u = 0, the ﬁrst term on the r.h.s. is zero. Furthermore, due to the boundary condition u = 0 on the solid–ﬂuid interface, the contribution of I s f to the surface integral over ∂U f is also zero. Using the periodicity of u(z) and φ(z), (2.24) with the help of (2.15) becomes: (2.25) p(u · n) dSz = αi zi (u · n) dSz u · grad p dVz = z

Uf

S

S

or equivalently:

Uf

u · grad p dVz = z

i

αi a i

Si

u · e i dSz

(2.26)

Microscopic Derivation of Darcy’s Law

31

Si in (2.26) corresponds to the part of S that is situated in the plane zi = a i ; the integral over Si represents the outﬂux of u through the surface ∂U oriented by the unit outward normal e i . With this result, it is possible to deﬁne the ﬂuid ﬁltration velocity vector Q(u), such that the component i of the outﬂux of u per surface ∂U is: ai (2.27) Qi (u) = u · e i dSz | U | Si Since u describes a microscopic ﬂow, the ﬁltration vector Q(u) represents the apparent macroscopic ﬂow velocity. Indeed, from (2.27), it is possible to show that: 1 (2.28) z(u · n) dSz Q(u) = | U | ∂U f or equivalently: 1 Q(u) = |U|

Uf

u dVz = ϕu f

(2.29)

where the identity divz (z ⊗ u) = u for all ﬁelds u satisfying divz u = 0, as well as the divergence theorem, has been used. According to (1.35), the ﬁltration vector Q(v) associated with the microscopic solution velocity ﬁeld v is related to the macroscopic velocity vector by: Q(v) = ϕV f

(2.30)

Using (2.27), together with (2.23) and (2.26), the weak form of the problem (2.12)–(2.14) and (2.21) is: |U| ∀u ∈ H (2.31) grad ui · grad vi d Vz = − f α · Q(u) z z μ Uf Micromechanical Formulation of Darcy’s Law

Because of the ellipticity in H of the l.h.s. of (2.31), i.e. of the bilinear form A(u, v): grad ui · grad vi dVz A(u, v) = (2.32) Uf

z

z

the existence and uniqueness of the ﬁeld v, i.e. the solution of (2.31) for a given value of α, are ensured by the Lax–Milgram theorem. Furthermore, the dependence of the solution v(z) on α is linear: v(z) = −k(z) · α

(2.33)

Micro(fluid)mechanics of Darcy’s Law

32

From (2.29) it follows that: Q(v) = −K · α

with K = ϕk

f

(2.34)

Since Q(v) = ϕV f , relation (2.34) conﬁrms the general structure (2.4) of Darcy’s law. This structure has been derived here from micromechanical considerations assuming an incompressible ﬂuid ﬂow in the absence of gravity forces through a rigid solid matrix. Darcy’s law, however, remains valid for a deformable matrix (given small perturbations).3 In this case, the actual value of the permeability remains unchanged, and is determined by associating the initial (undeformed) microstructure with the rigid microstructure. In return, it is convenient then to introduce the ﬁltration vector as a function of the relative macroscopic velocity of the ﬂuid with regard to the matrix in deformation, i.e. Q(v) = ϕVr . Let vi be the solution for α = e i , so that the solution v(α) is v = αi vi . The ﬁltration velocity Q(v) that characterizes the ﬂow along α at the macroscopic scale is: Q(v) = α j Q(v j ) = −K i j α j e i

(2.35)

K i j = −Qi (v j )

(2.36)

where:

The components of the tensor K are given in (2.36) in the (e i ) base. Since functions v j (which are solutions of (2.31) for α = e j ) are inversely proportional to μ f , tensor K also obeys this inverse proportionality. This conﬁrms relation (2.8) and the general relevance of the very concept of the intrinsic permeability: K=

1 ′ K; μf

Q(v) = −

1 ′ K ·α μf

(2.37)

Moreover, considering α = e j , if we choose in problem (2.31) the test function u = vi , we obtain: A(vi , v j ) = −

|U| Q j (vi ) μf

(2.38)

Hence, the symmetry of the permeability tensor K results from the symmetry of the bilinear form A(vi , v j ) in (2.32). In a similar way, if we choose for a given vector α in (2.31) the solution v(α) as a test function, it is straightforward to show that the positivity of K results from the positivity of A(v, v): A(v, v) = −

3 This

|U| |U| α · Q(v) = α·K·α f μ μf

micromechanics proof is given in [2] (see Section 2.4.2).

(2.39)

Microscopic Derivation of Darcy’s Law

33

For forthcoming developments, it will be useful to note the following identity which comes from the incompressibility condition div v = 0 (see (2.32)): f

A(v, v) = ϕ|U|grad vi · grad vi = 4ϕ|U|J 2 (d) z

f

z

(2.40)

where d is the symmetric part of grad v and J 2 (d) its second invariant: 1 d = (grad v + t grad v); 2

1 J 2 (d) = d : d 2

(2.41)

Scale Effects on Permeability

We are interested here in the scale effects on the permeability tensor that can be found from the micromechanics theory. To this end, let ( p, v) be the solution of the problem (2.12)–(2.14) and (2.15) deﬁned in U f of the unit cell U. We now consider the domain U˜ f , which is the image of U f obtained by the homothety ˜ z) and v(˜ ˜ z) be a pressure and a H of center O and ratio λ: H(z) = λz. Let p(˜ velocity ﬁeld solutions of the Stokes equation in U˜ f . Exploring the scaling, it is readily shown that a solution ( p, ˜ v) ˜ in U˜ f can be deﬁned from the solution ( p, v) in U f by: z˜ 1 z˜ p(˜ ˜ z) = p ; v(˜ ˜ z) = v (2.42) λ λ λ In fact, it sufﬁces to develop the differential operators involved in the Stokes equations: grad p˜ = z˜

1 grad p; z λ2

z˜ v˜ =

1 z v; λ2

divz˜ v˜ = 0

(2.43)

Given the scaling property, relation (2.15) takes the following form for pressure p(˜ ˜ z): 1 1 z˜ p(˜ ˜ z) = 2 α · z˜ + φ (2.44) λ λ λ

The macroscopic pressure gradient α˜ associated with the microscopic pressure ﬁeld p˜ is therefore equal to α/λ2 . Furthermore, the ﬁltration vector associated with the microscopic velocity ﬁeld v˜ in U˜ f is identical to the one associated with v in U f ; that is: 1 1 v˜ d V˜z = v dVz = Q(v) ˜ = (2.45) Q(v) | U | Uf | U˜ | U˜ f

˜ of the porous medium built from the By deﬁnition, the permeability tensor K unit cell U˜ = H(U) relates Q(v) ˜ to α. ˜ It is therefore equal to: ˜ = λ2 K K

(2.46)

Micro(fluid)mechanics of Darcy’s Law

34

It turns out that the permeability tensor scales with the square of the scale factor λ, which actually affects the intrinsic permeability introduced in (2.8). Qualitatively, it can be concluded that the order of magnitude of the permeability is O(d 2 /μ f ), where d denotes the characteristic size of the pores and/or of the grains of the solid matrix. 2.2.3 Lower Bound Estimate of the Permeability Tensor We are interested in developing estimates of the permeability tensor. This section is dedicated to the lower bound based on the variational problem (2.31). In Section 2.2.4, an upper bound will be developed. The starting point of our investigation of lower bound estimates is the proof of a minimum principle that characterizes the solution v of the variational problem (2.31). To this end, we introduce the functional F deﬁned on H by: F(v′ ) =

1 |U| A(v′ , v′ ) + α · Q(v′ ) 2 μf

(2.47)

For a given v′ ∈ H, let δv = v′ − v. Inserting in (2.47) yields: |U| 1 grad vi · grad δvi d Vz + F(v + δv) = F(v) + A(δv, δv) + α · Q(δv) z z 2 μf Uf (2.48) Use of the variational result (2.31) with u = δv in (2.48) gives: (∀δv ∈ H) F(v + δv) = F(v) +

1 A(δv, δv) ≥ F(v) 2

(2.49)

Inequality (2.49) shows that function F (v′ ) in H realizes its minimum for the solution v′ = v of (2.31). Finally, combining this result with (2.39) shows that any v′ in H delivers a lower bound estimate of the permeability tensor K: (∀v′ ∈ H)

α·K·α ≥−

μf A(v′ , v′ ) − 2α · Q(v′ ) |U|

(2.50)

where A(v′ , v′ ) and Q(v′ ) are respectively deﬁned by (2.32) and (2.29). Ordered Relations of Permeability

It is instructive to investigate how the permeability is affected by the magnitude of the porosity. That is, two porous media that differ in porosity have different permeabilities. The question is: in which sense and order? It is intuitively expected that the permeability of a porous medium with a smaller porosity is smaller than one with a greater porosity. This result is established in the following, under some conditions.

Microscopic Derivation of Darcy’s Law

35

Consider two periodic porous media denoted by (1) and (2). The elementary cells of both media have same size (same values of the triplet (a 1 , a 2 , a 3 ); cf. Section 2.2.2). For a given value of the macroscopic pressure gradient α, let v(1) (2) and v(2) be the velocity ﬁeld solutions of (2.31) in U (1) f and U f respectively. (i) Each microstructure, deﬁned by the geometry of U f , is associated with a permeability tensor K(i) that satisﬁes (see (2.39)): α · K(i) · α = −α · Q(v(i) ) =

μ f (i) (i) (i) A (v , v ) i = 1, 2 |U|

(2.51)

We examine the particular situation in which one pore space is a subset (2) ′ (1) of the other one, say U (1) is obtained f ⊂ U f . First, an extension v of v (2) (1) ′ ′ by letting v = 0 in U f \ U f . v is clearly a test function of the Stokes problem deﬁned on U (2) f . Thus, relation (2.50) can be used in the form: α · K(2) · α ≥ −

μ f (2) ′ ′ A (v , v ) − 2α · Q(v′ ) |U|

(2.52)

(1) Using the fact that v′ = 0 in U (2) f \ U f , (2.51) is for i = 1:

A(2) (v′ , v′ ) = A(1) (v(1) , v(1) ) = −

|U| α · Q(v(1) ) μf

(2.53)

Finally, combining (2.53) and (2.51) in (2.52) yields the expected (but still remarkable) result: α · K(2) · α ≥ α · K(1) · α = −α · Q(v(1) )

(2.54)

That is, when the pore spaces are ordered in the sense of inclusions (or subsets), the permeability tensors are ordered as well, in the sense of the second-order symmetric tensors. We will summarize this result by: (2) U (1) ⇔ K(2) ≥ K(1) f ⊆ Uf

(2.55)

Instructive Exercise: Permeability of Periodic Granular Media

We now turn to the application of the variational result (2.54) to the derivation of lower bound estimates of the permeability of periodic granular media. We consider a cubic packing of spheres. Two geometrically equivalent unit cells can be chosen. One is where the unit cell is a cube with a centered spherical solid grain, the center of the solid sphere coinciding with the center of the cube and the sphere diameter 2R being equal to the edge length of the cube. The other geometrically equivalent unit cell is a cube of the same dimension comprising eight eighths of a sphere of radius R centered at each corner of the cube (Figure 2.5). The second conﬁguration is chosen in what follows.

Micro(fluid)mechanics of Darcy’s Law

36

2d

2R

Figure 2.5 Cylinder contained in the pore space of circular section

Let (e 1 , e 2 , e 3 ) be an orthonormal frame deﬁned by vectors that are parallel to the edges of the cube. A macroscopic pressure gradient α = −αe 1 is considered and we search for an estimate of K 11 . Generally speaking, the underlying idea is to consider a cylinder C that is contained in the pore space, the generatrices being parallel to e 1 . In terms of the developments of the previous section, devoted to the ordered relations, this cylinder plays the role of domain U (1) f , which is actually smaller than the ‘real’ pore space U f . The more U (1) f approximates U f , the better the permeability estimate. We search for the solution v(1) of the ﬂuid ﬂow in this cylinder of the form v1 (z2 , z3 )e 1 . The problem (2.12)–(2.14) reduces to: α in μf v1 = 0 on ∂C z v1 = −

C

(a ) (b)

(2.56)

where ∂C denotes the boundary of the cross section of C. Using (2.54), a ﬁrst lower bound estimate √ can be obtained by considering cylinder C to be circular of radius d = R( 2 − 1) and tangent to the eight spheres of the unit cell (Figure 2.6). The solution of the ﬂuid ﬂow in this cylinder is: v(1) = v1 (z2 , z3 )e 1

with v1 (z2 , z3 ) =

α 2 d − z22 − z32 f 4μ

The volume ﬂow through the cross section of the cylinder is: αd 4 π αd 4 (1) D = ≈ 0.393 v1 (r ) d S = 8 μf μf r ≤d

(2.57)

(2.58)

Microscopic Derivation of Darcy’s Law

37 z3

Z3

Z2 z2

2d

2R

Figure 2.6 Cylinder contained in the pore space of square section

Applying (2.29), this ﬂow corresponds to the following ﬁltration vector: √ √ π R2 ( 2 − 1)4 π R2 ( 2 − 1)4 (1) (2.59) e1 = − α Q(v ) = α 32μ f 32μ f Finally, from (2.54), we obtain the following lower bound estimate: √ 2 π R2 ( 2 − 1)4 −3 R ≈ 2.89 × 10 K 11 ≥ 32μ f μf

(2.60)

The found estimate can be improved by reﬁning the geometrical representation of the cylinder ﬁlling the porosity. A parallelepiped of square cross section, oriented in the Z1 direction, comes closer to√the actual pore space (Figure 2.6). The length of the square edge is 2d = 2R( 2 − 1). As before, the generatrices of C are parallel to e 1 , but the boundary of C is now made up of four plane sides, the normal vectors to the sides being ±(e 2 ± e 3 ). These sides are tangent to the eight spheres. It is readily understood that the parallelepiped with square√cross section of volume U (2) f contains the circular cylinder of di⊂ U (2) ameter 2R( 2 − 1) and volume U (1) f f . Following the ordered relations of permeability (2.55), the parallelepiped is expected to provide a better estimate of the permeability. We look for the solution v(2) in the form v1 (z2 , z3 )e 1 . It is convenient to work in the axes which√are normal to the sides √ of C. They are parallel to the vectors E 2 = (e 2 + e 3 )/ 2 and E 3 = (e 3 − e 2 )/ 2. The corresponding coordinates are denoted by Z2 and Z3 : 1 Z2 = √ (z2 + z3 ); 2

1 Z3 = √ (z3 − z2 ) 2

(2.61)

Micro(fluid)mechanics of Darcy’s Law

38

In these axes, the cross section of C in the square is deﬁned by Z2 = ±d and Z3 = ±d. To solve the problem, we note that elementary solutions of φ = 0 can be obtained in the form f (Z2 )g(Z3 ). Among these solutions, those which are even with respect to Z2 and Z3 and which are equal to zero on the planes Z2 = ±d are: π Z3 π Z2 cosh (2k + 1) (2.62) φ(Z2 , Z3 ) = A cos (2k + 1) 2d 2d In turn, the even functions which are equal to zero on the plane Z3 = ±d are: π Z2 π Z3 φ(Z2 , Z3 ) = B cos (2k + 1) cosh (2k + 1) (2.63) 2d 2d We now search for the solution v1 in the form of the following series: α v1 (Z2 , Z3 ) = − (Z2 + Z32 ) 4μ f 2 π Z3 π Z2 cosh (2k + 1) + Ak cos (2k + 1) 2d 2d k π Z3 π Z2 + Bk cos (2k + 1) cosh (2k + 1) (2.64) 2d 2d k The above expression meets condition (2.56a). Furthermore, condition (2.56b) yields: α 2 (2k + 1)π π Z3 2 cosh d + Z3 = Bk cos (2k + 1) 4μ f 2d 2 k (2.65) α 2 (2k + 1)π π Z 2 2 cosh = d + Z A cos (2k + 1) k 2 4μ f 2d 2 k

which proves that coefﬁcients Ak and Bk are equal. In fact, we have to determine the Fourier series of the functions on the l.h.s. of (2.65). If we note the following relation:

+d πx πx d if i = j cos (2i + 1) cos (2 j + 1) dx = (2.66) 0 if i = j 2d 2d −d

the coefﬁcients Ak = Bk are readily obtained in the form: Ak = Bk =

2αd 2 (−1)k (4k 2 π 2 + 4kπ 2 + π 2 − 4) μ f π 3 (2k + 1)3 cosh (2k+1)π 2

(2.67)

Finally, we are left with the determination of the ﬁltration vector associated with the ﬂuid ﬂow deﬁned by the velocity ﬁeld v1 of (2.64). The ﬂow D(2)

Microscopic Derivation of Darcy’s Law

39

through the square (Z2 = ±d, Z3 = ±d) is: +d +d v1 (Z2 , Z3 ) d Z3 d Z2 D(2) =

(2.68)

−d

−d

That is: D(2)

αd 4 = f μ

2 64(4k 2 π 2 + 4kπ 2 + π 2 − 4) (2k + 1)π − + (2.69) tanh 3 π 5 (2k + 1)5 2 k

or, numerically: D(2) ≈ 0.562

αd 4 μf

(2.70)

A comparison with (2.58) conﬁrms that the considered pore morphology ap(2) (2) > D (1) , and thus a proximation yields a greater ﬂow, i.e. U (1) f ⊂ Uf ⇔ D better estimate of the permeability. Indeed, by using Q(v(2) ) = D(2) /(4R2 ) e 1 in (2.54), an estimate of the permeability coefﬁcient which signiﬁcantly improves the estimate (2.60) is obtained: K 11 ≥ 4.14 × 10−3

R2 μf

(2.71)

As a last pore space representation, we examine a cylinder with a cross section that is made up of four circles (see Figure 2.7). This cross section is tangent to both the circle of radius d and the square of edge 2d of the two previous representations of C. The numerical solution of (2.56) is obtained by

2d

2R

Figure 2.7 Plane section of the greatest cylinder approximation contained in the porosity

Micro(fluid)mechanics of Darcy’s Law

40

a ﬁnite element calculation. The corresponding ﬂow is: D(3) ≈ 0.658

αd 4 μf

(2.72)

which in turn leads to: K 11 ≥ 4.84 × 10−3

R2 μf

(2.73)

The different cylinder representations of the pore space highlight the elements of the lower bound estimate solution strategy for permeability estimates: the key issue is the geometrical representation of the ‘real’ pore space. As the geometrical representation approaches the actual pore volume, the ordered relations of permeability (2.55) ensure that the ﬂow estimates approach the actual ﬂow through the pore space, and thus the permeability of the porous material. 2.2.4 Upper Bound Estimate of the Permeability Tensor This section develops an upper bound approach for permeability estimates based on a minimum principle. Our starting point is the Stokes problem (2.12)– (2.14). With regard to stress σ in the viscous ﬂuid, the problem is deﬁned by: divσ = 0 σ = − p1 + 2μ f d d = 12 (gradv + t gradv) divv = 0 v=0

(U f ) (U f ) (U f ) (U f ) (I s f )

(2.74)

We look for the periodic solution v of (2.74) when the pressure p(z) is speciﬁed in the form (2.15), the macroscopic pressure gradient α being the loading parameter. To this end, we introduce the stress ﬁeld σ˜ = σ + α · z1 , which in contrast to the real stress ﬁeld σ is periodic. With σ, ˜ the solution of (2.74) reduces to two periodic functions φ(z) and v(z) that represent respectively the periodic ﬂuctuation of the pressure and the velocity, satisfying: (a ) divσ˜ − α = 0 (b) σ˜ = −φ1 + 2μ f d (c) d = 12 (gradv + t gradv) (d) divv = 0 (e) v = 0

(U f ) (U f ) (U f ) (U f ) (I s f )

(2.75)

In (2.75a), the macroscopic pressure gradient α, which is the loading parameter in our problem, explicitly appears as a body force. It is interesting to note the formal analogy of (2.75) with a problem of incompressible elasticity. Indeed, it sufﬁces to replace the velocity v by a

Microscopic Derivation of Darcy’s Law

41

displacement ﬁeld ξ , and the strain rate d by the strain ε associated with ξ . Exploring this analogy, it is readily understood that an upper bound estimate of the permeability tensor can make use of the theorem of the minimum of complementary energy classically used in elasticity theory. More precisely, consider the set H∗ of admissible stress ﬁelds σ˜ ′ : H∗ = {σ˜ ′ , periodic, div σ˜ ′ − α = 0} and the functional F ∗ (σ˜ ′ ) deﬁned on H∗ such that: 1 1 ′ ∗ s′ : s′ d Vz with s′ = σ˜ ′ − tr σ˜ ′ 1 F (σ˜ ) = f 4μ U f 3 According to this deﬁnition, it is readily seen that: 1 F ∗ (σ˜ ′ ) = F ∗ (σ) ˜ : s d Vz (σ˜ ′ − σ) ˜ + F ∗ (σ˜ ′ − σ) ˜ + 2μ f U f

(2.76)

(2.77)

(2.78)

where σ˜ (resp. s) denotes the solution to (2.75) (resp. its deviatoric part). Observing from (2.75b) that s = 2μ f d, integration by parts of the last integral on the r.h.s. of (2.78) yields: 1 ′ (σ˜ − σ) ˜ : s d Vz = (σ˜ ′ − σ) ˜ : d d Vz 2μ f U f Uf ˜ d Vz = v · div(σ˜ ′ − σ) (2.79) Uf

+

v · (σ˜ ′ − σ) ˜ · ndS ∂U f

Relations (2.75a) and (2.76) indicate that div(σ˜ ′ − σ) ˜ = 0. Furthermore, the periodicity of σ, ˜ σ˜ ′ and v together with (2.75e) implies that the surface integral in (2.79) also vanishes, and relation (2.78) reduces to a minimum principle: F ∗ (σ˜ ′ ) = F ∗ (σ) ˜ + F ∗ (σ˜ ′ − σ) ˜ ≥ F ∗ (σ) ˜

(2.80)

To complete the analogy with incompressible elasticity, we are left with specifying the link between the minimum ‘complementary energy’, F ∗ (σ) ˜ realized by the solution and the permeability tensor. Recalling the fact that s = 2μ f d, it is readily seen from deﬁnition (2.77) that: ∗ f F (σ) ˜ =μ d : d d Vz (2.81) Uf

Furthermore, taking (2.75c,d) into account, we obtain: F ∗ (σ) ˜ =

μf A(v, v) 2

(2.82)

Micro(fluid)mechanics of Darcy’s Law

42

Finally, a combination of (2.39) with (2.82) yields: |U| α·K·α (2.83) 2 The minimum principle (2.80) then reveals that any element of H∗ provides an upper bound of α · K · α, i.e. formally: F ∗ (σ) ˜ =

(∀σ˜ ′ ∈ H∗ ) α · K · α ≤

2 ∗ ′ F (σ˜ ) |U|

(2.84)

Inequality (2.84) sets out a formidable optimization problem: to search for the lowest upper bound of the permeability tensor by means of admissible stress ﬁelds σ˜ ′ . 2.3 Training Set: Upper and Lower Bounds of the Permeability of a 2-D Microstructure To illustrate the use of the upper and lower bound solution strategy, we consider a 2-D microstructure in which the solid phase is a periodic set of identical parallel circular cylinders of radius R. Albeit rather academic than realistic, this simple microstructure allows the development of closed form expressions of the permeability bounds. In the orthonormal frame (e 1 , e 2 , e 3 ), the direction of the cylinders is parallel to e 3 . The ﬂuid ﬂow takes place in the plane perpendicular to the direction of the cylinders. The cross section of the elementary cell is displayed in Figure 2.8. The axes of the cylinders are located at the vertices of a square of side 2a . The scalar η = R/a is a measurement of the relative size of the cylinders in the cell. The symmetry of the geometry implies that e 1 and e 2 are eigenvectors of the permeability tensor. We are interested in developing estimates of the eigenvalues K 11 = K 22 . x2

R a x1

Figure 2.8 An academic exercise: 2-D geometry of the microstructure

Training Set: Upper and Lower Bounds of the Permeability of a 2-D Microstructure 43

2.3.1 Lower Bound The domain C ′ enclosed in between the planes x2 = ±(a − R) is a subset of the ′ pore space U f . According to (2.55), the permeability K 11 of the elementary cell ′ ′ ′ U in which the pore space is U f = C represents a lower bound of K 11 . The microscopic velocity ﬁeld v′ induced in U ′ by a macroscopic pressure gradient −αe 1 is the classical Poiseuille solution for the ﬂow between two planes: v′ =

−α 2 x2 − (a − R)2 e 1 f 2μ

(2.85)

′ : The corresponding ﬁltration velocity yields the following value of K 11 a −R 1 a2 a2 ′ Q(v′ ) = (1 − η)3 ≤ K 11 v′ d x2 = α f (1 − η)3 e 1 ⇒ K 11 = 2a R−a 3μ 3μ f (2.86)

2.3.2 Upper Bound We now employ (2.84) to derive an upper bound of K 11 . For a given macroscopic pressure gradient −αe 1 , we need to determine one admissible stress σ˜ ′ in H∗ . To this end, it is convenient to divide U f into two subdomains, |x2 | ≤ a − R and |x2 | > a − R. The components σ˜ i2 must be continuous over the interfaces x2 = ±(a − R). We now need to choose admissible stress ﬁelds that meet the conditions (2.76). The admissible stress ﬁeld we consider is given by: (a ) |x2 | ≤ a − R σ˜ ′ = −αx2 e 1 ⊗ e 2 + e 2 ⊗ e 1 −α 1 ′ x1 e 1 ⊗ e 1 + e 3 ⊗ e 3 (b) x2 > a − R σ˜ = η 2 1 − 1 (a − x2 ) e 1 ⊗ e 2 + e 2 ⊗ e 1 −α (2.87) η −α 1 x1 e 1 ⊗ e 1 + e 3 ⊗ e 3 (c) x2 < R − a σ˜ ′ = η 2 1 (a + x2 ) e 1 ⊗ e 2 + e 2 ⊗ e 1 −α 1 − η

This stress ﬁeld calls for the following comments:

′ 1. In domain |x2 | ≤ a − R, a shear stress σ˜ 12 is considered, such that divσ˜ ′ = −αe 1 . This shear stress, therefore, balances the body force αe 1 in (2.75a). Furthermore, in order to achieve the periodicity of this shear stress (same value on x2 = ±a ), the piecewise afﬁne variation depicted in Figure 2.9 is adopted. ′ 2. In domain |x2 | > a − R, the shear stress σ˜ 12 establishes the stress continu′ ity at the interface x2 = ±(a − R). In addition, the normal stress σ˜ 11 which

Micro(fluid)mechanics of Darcy’s Law

44 x2

a a –R

σ12

Figure 2.9 Shear stress variation

depends linearly on x1 , together with the shear stress σ˜ 12 , ensures that ′ ′ divσ˜ ′ = −αe 1 , thus balancing the body force αe 1 in (2.75a). Once σ˜ 11 and σ˜ 12 ′ ′ are chosen, it can be shown that the stress component σ˜ 33 = σ˜ 11 /2 minimizes ′ ′ the deviatoric stress invariant s : s in (2.77). Using σ˜ ′ , we evaluate the functional F ∗ (σ˜ ′ ). The contribution of subdomain |x2 | ≤ a − R to F ∗ (σ˜ ′ ) is: α2 ∗ F1 = x 2 d x1 d x2 (2.88) 2μ f |x2 |≤a −R 2 and the contribution of the subdomain |x2 | > a − R is: 2 x12 α2 1 ∗ F2 = +2 − 1 (a − x2 )2 d x1 d x2 2μ f x2 > a −R 2η2 η

(2.89)

Inserting (2.88) and (2.89) into (2.84) yields an upper bound of the permeability coefﬁcient; thus after integration: K 11 ≤

1 a2 16 + (64 − 12π)η − 96η2 + (64 − 15π )η3 + 24π η4 − 12π η5 f 192 ημ (2.90)

2.3.3 Comparison The upper bound (2.90) needs to be compared with the lower bound (2.86). We note a reasonable agreement of the upper and lower bound for η > 0.5 (Figure 2.10). By contrast, the two estimates signiﬁcantly diverge for small values of η. For instance, the permeability is expected to be inﬁnite for η → 0. While the upper bound estimate satisﬁes this property, the lower bound one does not. Indeed, as η ≪ 1, the upper bound (2.90) can be approximated by a 2 /(12ημ f ). By contrast, the lower bound (2.86) still predicts a ﬁnite value for η → 0. This divergence of lower and upper bound is attributed to the fact that

Generalization: Periodic Homogenization Based on Double-Scale Expansion

45

1.8 K11 1.6 a2/µf 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2

0.4

0.6

0.8

1

η

Figure 2.10 Lower (2.86) and upper (2.90) bounds of K 11

the chosen velocity v′ in (2.85) is zero at x2 = ±(a − R). Thus, as η = R/a → 0, for which the lines x2 = ±(a − R) approach the cell sides x2 = ±a , the velocity ﬁeld v′ employed in the lower bound fails to represent realistically the ﬂow through the considered (academic) 2-D microstructure. 2.4 Generalization: Periodic Homogenization Based on Double-Scale Expansion The macroscopic model of the advective ﬂux classically refers to Darcy’s law. By contrast, the periodic homogenization technique provides a rational means to derive Darcy’s law together with the macroscopic permeability tensor as a function of the geometry of the pore space. So far, our micromechanics derivation of the macroscopic permeability (see Section 2.2.2) has been based on the assumption that the microscopic ﬂow is periodic with respect to the space coordinates (2.9), and the solid matrix has been assumed to be rigid. The focus of this and the next section is to generalize the micromechanics derivation beyond these particular situations, while remaining within the framework of the periodic homogenization technique;4 that is, the porous 4 The theory presented below is inspired by the contributions of Ene and Sanchez-Palencia [23] and Auriault [2],[4].

Micro(fluid)mechanics of Darcy’s Law

46

medium itself is periodic and deﬁned by an elementary cell like the one in Figure 2.3. 2.4.1 Double-Scale Expansion Technique We employ the same notation and methodology as in Section 1.4. For simplicity, the periodic cell (see Figure 2.3) is a cube (a i = a ). Further, as already stated, since the edge length is on the order of the pore size d, we let d = a . The microscopic velocity and pressure ﬁelds are regarded as functions of both Z = z/a and X = z/L. Relation (2.13), however, now takes the form: 1 div Z v + div X v = 0 δ

with δ = a /L, and (2.12) is: 2 ∂ 2v 1 1 μ f 1 ∂ 2v ∂ 2v + grad p + grad p = 2 + Z X L δ L i δ 2 ∂ Zi2 δ ∂ Xi ∂ Zi ∂ Xi2

(2.91)

(2.92)

In order to obtain the non-dimensional form of (2.92) and (2.91), the pressure and the advective velocity have to be scaled. More precisely, considering an advective ﬂow driven by a pressure gradient which is characterized by the pressure scale P and the macroscopic length L, the advective velocity is scaled so as to balance the pressure gradient P/L. To this end, we look for the nondimensional form of (2.12), and we assume that the characteristic length of the spatial variations of the velocity ﬁeld is the pore size d = a . This reasoning yields: p = P p∗ ;

v=

Pa 2 ∗ v μf L

(2.93)

With (2.93), (2.91) and (2.92) take the form: 1 div Z v∗ + div X v∗ = 0 δ 2 ∗ ∂ 2 v∗ 1 ∂ 2 v∗ ∗ ∗ 2∂ v grad p + grad p = +δ + 2δ Z X δ ∂ Xi∂ ∂ Zi ∂ Zi2 ∂ Xi2 i

(2.94)

(2.95)

Finally, we introduce expansions of p ∗ and v∗ in powers of δ = a /L: p∗ = δ j p ∗( j) ; v∗ = (2.96) δ j v∗( j) j

j

In (2.96), all p ∗( j) (resp. v∗( j) ) are expected to be of the same magnitude. We then insert (2.96) into (2.94) and (2.95). The order of δ −1 in (2.95) yields: grad p ∗(0) = 0 Z

(2.97)

Generalization: Periodic Homogenization Based on Double-Scale Expansion

47

In other words, p ∗(0) depends only on the macroscopic variable X and does not ﬂuctuate within the cell. This result is consistent with the assumption concerning the order of magnitude of the pressure gradient. The order of δ 0 in (2.95) yields: grad p ∗(1) + grad p ∗(0) = Z v∗(0) Z

(2.98)

X

which needs to be completed by the order of δ −1 in (2.94): div Z v∗(0) = 0

(2.99)

(I s f ) v∗(0) = 0

(2.100)

and by the order of δ 0 in (2.14):

We look for a Z-periodic solution ( p ∗(1) , v∗(0) ) of (2.98)–(2.100). Let w(Z) be a Z-periodic smooth vector ﬁeld deﬁned on U f and satisfying div Z w = 0 , as well as w = 0 at I s f . The variational formulation of (2.98)–(2.100) is: ∂wi ∂vi∗(0) d VZ = (2.101) w · grad p ∗(0) dV Z (∀ w) − X ∂ Z ∂ Z j j Uf Uf The solution v∗(0) of (2.101) is unique and is a linear function of grad p ∗(0) X which can be interpreted as the macroscopic pressure gradient. Hence, if we denote by −ki∗j (Z)e i the particular solution of (2.101) for grad p ∗(0) = e j , it is X possible to obtain v∗(0) in the form: vi∗(0) = −ki∗j (Z)

∂ p ∗(0) ∂ Xj

v∗(0) = −k∗ (Z) · grad p ∗(0)

⇔

X

(2.102)

Darcy’s law is obtained by taking the average over U f of (2.102) with respect to Z, at constant X. Returning to dimensional quantities: f

ϕv(0) = −K · grad p 0

with K = ϕ

x

d2 ∗ f k μf

(2.103)

Darcy’s law needs to be completed by the macroscopic counterpart of the microscopic incompressibility condition (2.13). This condition is obtained from the order of δ 0 in (2.94) as: div Z v∗(1) + div X v∗(0) = 0

(2.104)

Then, averaging (2.104) with respect to Z over U f on account of the fact that v∗(1) is Z-periodic, and recalling (1.57), yields: f

divx v(0) = 0

(2.105)

Micro(fluid)mechanics of Darcy’s Law

48

In summary, the homogenization technique provides not only strong theoretical support for the relevance of Darcy’s law to capture microscopic ﬂow phenomena at the macroscale of porous materials, but also a rational means for the determination of the macroscopic permeability tensor K as a function of the geometry of the microstructure. This requires solving the elliptic problem (2.101). 2.4.2 Extension of Darcy’s Law to the Case of Deformable Porous Media So far, we have considered the solid matrix as rigid (see Section 2.2.2). This section is devoted to the extension of the theory to deformable porous media. For simplicity, we will assume that the solid matrix is a linear elastic material, deﬁned by the elasticity tensor Cs that relates the Cauchy stress tensor σ to the linearized strain tensor ε associated with the microscopic displacement ﬁeld ξ : ∂ξ j 1 ∂ξi s σ = C : ε; εi j = + (2.106) 2 ∂z j ∂zi There are several effects of deformation on the ﬂow through porous media. The ﬁrst possible coupling is due to the displacement of the solid–ﬂuid interface, which a priori inﬂuences the transport properties. The second one is the mechanical effect of the pore pressure on the overall deformation behavior of the porous material. The ﬁrst coupling is examined here with regard to Darcy’s law. The second coupling with the deformation behavior will be studied in Chapter 5, where we will see a consistent derivation of the macroscopic poroelasticity theory from homogenization techniques. We are already familiar with the micro-to-macro approach of Darcy’s law based on the double-scale expansion technique (see Section 2.4.1). This approach is also adopted here for the extension of the theory to deformable media, i.e. media in which the solid matrix is deformable. To start with, we normalize the elastic moduli Cisjkl , the solid displacement ξ and the pore pressure p by Es , D and P respectively: C s = Es C ∗ ;

ξ = Dξ ∗ ;

p = P p∗

The expansion of the linearized strain is: D 1 ∗(0) ∗(1) ∗(0) (Us ) ε = ε Z (ξ ) + ε Z (ξ ) + ε X (ξ ) . . . L δ

(2.107)

(2.108)

where: εβ (ξ ∗ ) denotes the symmetric part of gradβ (ξ ∗ ) (β = Z or X). Combining (2.106) and (2.108) yields the following expansion of the stress tensor in

Generalization: Periodic Homogenization Based on Double-Scale Expansion

49

the solid phase: (Us ) σ =

1 ∗(−1) σ + σ ∗(0) . . . δ

(2.109)

where = E s D/L and σ ∗(−1) = C∗ : ε Z (ξ ∗(0) );

σ ∗(0) = C∗ : ε Z (ξ ∗(1) ) + ε X (ξ ∗(0) )

(2.110)

The stress ﬁeld σ is subjected to the momentum balance equation and to the boundary condition at the solid–ﬂuid interface: (Us ) divz σ = 0;

(I s f ) σ · n = − pn + 2μ f d · n

(2.111)

where d is the strain rate of the ﬂuid phase, i.e. the symmetric part of gradz v. At the highest order in 1/δ, relation (2.111) is: (Us ) div Z σ ∗(−1) = 0;

(I s f ) σ ∗(−1) · n = 0

(2.112)

where the assumption P ≈ = Es D/L and (2.93) have been used for establishing the boundary condition in (2.112). Equations (2.110) and (2.112) indicate that ξ ∗(0) and σ ∗(−1) are Z-periodic solutions of a problem of elasticity deﬁned on the solid phase Us , in which the spatial variable is Z and the external volume and surface forces are zero. This implies that σ ∗(−1) = 0 and that ξ ∗(0) does not depend on the spatial variable Z, i.e. ξ ∗(0) = ξ ∗(0) (X). We now focus on the equations that characterize the ﬂuid ﬂow in the pore space. In contrast to (2.100), we now need to consider that the ﬂuid velocity at the solid–ﬂuid interface is equal to the solid velocity. To this end, we introduce the relative velocity v r = v − ξ˙ of the ﬂuid with respect to the solid. Since ξ ∗(0) does not depend on Z, it is readily seen that the solid deformation can be taken into account by replacing v(0) by v(0) r in (2.98), (2.99) and (2.100). Consequently, Darcy’s law in the deformable case is obtained by replacing the (absolute) ﬂuid phase velocity by the macroscopic relative velocity: f (0) (0) ˙ ϕ v (2.113) = −K · grad p (0) −ξ x

The permeability tensor K in (2.113) preserves its meaning in (2.103). It should be noted, however, that this result is based on the assumption of small displacements and inﬁnitesimal deformation of the solid matrix at the microscopic scale, which is a necessary condition for the validity of the linear elasticity approach (2.106). This is consistent with the fact that the permeability tensor deﬁned by (2.103) depends on the geometry of the microstructure through k∗ (see (2.102)). It is readily understood that the coupling between mechanical loading and permeability becomes non-negligible in the case of macroscopic ﬁnite deformation of the porous medium. However, even if the macroscopic

50

Micro(fluid)mechanics of Darcy’s Law

deformation of the porous medium is inﬁnitesimal, large local displacements of the solid may occur which change the microstructure, thus affecting the permeability. For instance, consider the case of a cracked porous medium. The application of a macroscopic stress (below the crack growth threshold) induces non-inﬁnitesimal changes of the crack geometry (width and aspect ratio), while the macroscopic strain remains in the domain of inﬁnitesimal deformation. By describing the cracks as ﬂuid inclusions surrounded by a homogeneous porous continuum, it can be shown that the permeability increase associated with crack opening is generally non-negligible,5 but does not modify the order of magnitude of the permeability.

2.5 Interaction Between Fluid and Solid Phase This section is devoted to the macroscopic and the microscopic representation of the solid–ﬂuid interaction. What we aim to derive are orders of magnitude of the macroscopic interaction force a , introduced in Section 1.3.2, and of the corresponding microscopic surface forces the ﬂuid exerts on the solid phase. 2.5.1 Macroscopic Representation of the Solid–Fluid Interaction To simplify the presentation, we assume that the solid matrix is rigid and that the porosity is homogeneous. The two relations we refer to are (1) the momentum balance equation for the ﬂuid at the macroscopic scale which we derived from upscaling (cf. (1.54)): ϕ −gradP + μ f V f + ρ f F f − a = 0 (2.114) and (2) the macroscopic Darcy’s law (2.4) on account of body forces F f : Q = ϕV f = K · −gradP + ρ f F f (2.115)

A combination of (2.115) and (2.114) provides a means of determining the body force a . We have seen in Section 1.3.2 that this body force represents at the macroscopic scale the micromechanical interaction between the solid and the ﬂuid phase (i.e. (1.46)). At the macroscopic scale, the body force a appears as a function of the macroscopic ﬂuid velocity: (2.116) a = ϕ ϕK−1 · V f + μ f V f It is interesting to compare the order of magnitude of the different terms on the r.h.s. of (2.116). The order of magnitude of K is O(d 2 /μ f ) (see Section 2.2.2), 5 See

[20]. Recent developments on the permeability of a cracked porous medium can be found in [18].

Interaction Between Fluid and Solid Phase

51

where d represents the characteristic length of the microstructure (typically the pore or grain size), which is on the same order of the characteristic length of the unit cell. If we denote by V the order of magnitude of the macroscopic velocity, the term K−1 · V f is on the order of μ f V/d 2 , while the term μ f V f is a priori on the order of μ f V/L 2 . Thus, the scale separability condition (1.1), i.e. d ≪ L, implies that the term μ f V f is generally negligible compared to ϕK−1 · V f ; that is: a ≈ ϕ 2 K−1 · V f

(2.117)

2.5.2 Microscopic Representation of the Solid–Fluid Interaction The ﬂuid ﬂow in the pore space entails a surface traction σ (z) · n (z) on the solid–ﬂuid boundary, where σ (z) is the microscopic Cauchy stress in the ﬂuid deﬁned by (1.47), and n (z) is the unit outward normal to the solid. This surface traction comprises a pressure component − p (z) n (z) and a viscous component T v = 2μ f d (z) · n (z). We want to compare the order of magnitude of these two contributions relative to the solid–ﬂuid interaction.6 For purposes of clarity, the microscopic position vector z is omitted in what follows. L denotes the characteristic length of the macroscopic material system under consideration. In Section 2.4.1, by employing the double-scale expansion technique, we showed that the microscopic and the macroscopic gradients grad p and z grad P have the same order of magnitude (see (2.97)) : x

|grad p| ≈ |grad P| = α z

x

(2.118)

The physical origin of this result can be obtained from the cylinder model developed in Section 2.2.1. In this thought model, the local and macroscopic pressure gradients are equal, since the pressure is uniform in each section. Relation (2.118) implies that the order of magnitude of the pressure is αL. This would be false if the local ﬂuctuations of the pressure were not controlled by the condition |grad p| ≈ α. Indeed, recalling the decomposition (2.15), rez lation (2.118) is equivalent to an upper bound of the gradient of the periodic pressure ﬂuctuation φ of the form |grad φ| ≤ O (α ). z In contrast to the pressure, the characteristic length of the variations of the ﬂuid velocity is the pore size d, as shown once again by the cylinder model of Section 2.2.1. The order of magnitude of the viscous stress is thus related to the order of magnitude V of the velocity v: v f V |T | = O μ (2.119) d 6 The

reasoning developed here is inspired by Auriault and Sanchez-Palencia [4].

Micro(fluid)mechanics of Darcy’s Law

52

Finally, the momentum balance equation for low Reynolds numbers relates α and V: f V (2.120) α=O μ 2 d From (2.119) and (2.120), we conclude that the ratio |T v |/ p is on the order of d/L ≪ 1, because of the scale separation condition. In other words, compared to the pressure the contribution of the viscous stress T v to the interaction surface force at the solid–ﬂuid interface is negligible. We keep this result in mind for forthcoming developments. 2.6 Beyond Darcy’s (Linear) Law Darcy’s law is a linear law between the ﬁltration velocity and the pore pressure gradient. This linear law is the macroscopic counterpart of the linearity of the Stokes equations. Deviations from Darcy’s law are therefore expected whenever the ﬂuid is not of the Newtonian type: the nonlinearity of the microscopic ﬂuid constitutive equations is expected to translate into a nonlinear relation at the macroscopic scale. By way of illustration, two cases are brieﬂy described below: a Bingham ﬂuid and a power-law ﬂuid. The focus of this section is twofold: (1) to develop the micromechanics elements of advective transport of nonlinear ﬂuids, and, by doing so, (2) to emphasize the qualitative differences that arise with respect to Darcy’s law. We remain within the framework of a purely mechanical interaction between the ﬂuid and the solid. In this case, the geometry of the solid–ﬂuid interface at the microscopic scale enters through the no-slip condition at the interface, the macroscopic advective ﬂuid transport. An enhanced theory would further account for physical–chemical interactions at the solid–ﬂuid interface, which may induce additionally nonlinear features that go beyond the scope of this section. 2.6.1 Bingham Fluid In this section, we will restrict ourselves to the idealized thought model of Section 2.2.1, i.e. the ﬂow through a cylindrical pore space. In contrast to a Newtonian ﬂuid, for which the shear stress is linearly linked to the velocity gradient, a Bingham ﬂuid can sustain a shear stress without strain rate below a certain stress threshold. Let τ = 1/2s equivalent deviatoric

: s be the ﬂuid stress, τ0 the stress threshold, and d = 1/2d : d = J 2 (d) the equivalent strain rate. For τ < τ0 , d = 0, while for τ > τ0 the Bingham ﬂuid is characterized by the following afﬁne relation: τ = τ0 + 2μ f d

if

τ > τ0

(2.121)

Beyond Darcy’s (Linear) Law

53

The tensorial form of the state equation of an incompressible Bingham ﬂuid is given as: τ0 s = d + 2μ f d if τ > τ0 d (2.122) σ = − p1 + s with d=0 if τ < τ0 Clearly, the Newtonian ﬂuid is the particular case of the Bingham model when τ0 = 0. Consider now the 1-D thought model of Section 2.2.1 (see Figure 2.2): a circular cylindrical pore space of radius a in a cylindrical elementary volume of cross section S. For an incompressible Bingham ﬂuid, we want to determine the microscopic velocity ﬁeld v that is induced in the pore space (porosity ϕ = πa 2 /S) when a uniform pressure gradient parallel to the axis of the cylinder (vector e 1 ) is applied. The pressure is expected to be an afﬁne function of z1 : p = p0 − αz1

(2.123)

The microscopic and the macroscopic pressure gradients are both equal to α = −αe 1 . In the following, we assume that α > 0. The momentum balance for (2.122) and (2.123) is: div σ = 0 ⇔ div s = −αe 1

(2.124)

and has the following solution: α s = σr z (e r ⊗ e 1 + e 1 ⊗ e r ) with σr z = − r < 0 2

(2.125)

Below a threshold gradient α < αcr = 2τ0 /a , this solution satisﬁes the Bingham threshold condition τ = −σr z < τ0 . This critical pore pressure gradient αcr = 2τ0 /a depends on both the ﬂuid through threshold τ0 and the pore geometry through the pore space radius a which represents a characteristic pore size in the 1-D thought model. In this case, according to (2.122), the pressure gradient α < αcr does not induce any ﬂuid ﬂow in the porosity. By contrast, for α > αcr , the pressure gradient induces a ﬂuid ﬂow. Indeed, considering a deviatoric stress of the form (2.125), the ﬂow velocity is expected to be parallel to the cylinder axis, i.e. v = v(r )e 1 . The strain rate therefore is: d=

1 ∂v (e ⊗ e 1 + e 1 ⊗ e r ) 2 ∂r r

(2.126)

Furthermore, substitution of (2.125) and (2.126) in (2.122) allows us to distinguish for α > 0 two radial zones with different ﬂow patterns: αcr ∂v α 1 (a ) r > a : = f τ0 − r α ∂r μ 2 (2.127) αcr ∂v : =0 (b) r < a α ∂r

Micro(fluid)mechanics of Darcy’s Law

54

The core cylindrical zone r < rcr = a αcr /α is subjected to a rigid body translation. In domain r > rcr , the velocity is obtained by integration of (2.127a) with the no-slip condition at the solid–ﬂuid interface at r = a : 1 α v(r ) = f τ0 (r − a ) − (r 2 − a 2 ) (2.128) μ 4 The uniform velocity v0 in the core zone is obtained from the continuity at r = rcr : αa 2 αcr 2 v0 = 1 − (2.129) 4μ f α It is interesting to note that v0 = 0 where the pressure gradient α > 0 approaches the threshold gradient α = αcr . Finally, with the microscopic velocity ﬁeld in hand, the ﬁltration velocity is obtained by application of (2.29) (integration over the cross section of the cylinder): a 1 2 Q= 2π v(r )r dr (2.130) v0 πrcr + S rcr Recast in the form of a nonlinear Darcy’s law, this yields after integration: Q=

k ′ (α/αcr ) α μf

(2.131)

where: πa 4 4 αcr 1 αcr 4 α k (α/αcr ) = + H −1 1− 8S αcr 3 α 3 α ′

(2.132)

with H the Heaviside function. From a comparison with (2.8), it could be appealing to refer to k ′ (α/αcr ) as an intrinsic permeability. In favor of such an interpretation is that k ′ (α/αcr ) reduces for α ≫ αcr to K ′ = (πa 4 /8 S) (Figure 2.11), which is exactly the intrinsic permeability of the linear 1-D thought model in the case of a Newtonian ﬂuid (see Section 2.2.1). In fact α ≫ αcr means that the core zone is small compared to the pore space radius, rcr ≪ a . On the other hand, if this is not the case, the main difference with the linear case is the dependence of the permeability coefﬁcient on the Heaviside term H(α/αcr − 1), which represents the fact that no ﬂuid ﬂow occurs if α < αcr , for which k ′ (α) = 0. The permeability coefﬁcient is situated in between these two limits, 0 ≤ k ′ (α/αcr ) ≤ K ′ . 2.6.2 Power-Law Fluids We consider the class of incompressible viscous materials for which the deviatoric stress derives from a potential π (d): σ = − p1 +

∂π ∂d

(2.133)

Beyond Darcy’s (Linear) Law

55

1

0.8 k′(α/αcr) K′ 0.6

0.4

0.2

0

2

4

6

8

14

10

18

α/αcr

Figure 2.11 Intrinsic permeability of the cylindrical thought model in the case of a Bingham fluid normalized by K ′ = πa 4 /(8S), as a function of the normalized pressure gradient α/αcr

The case of a Newtonian ﬂuid corresponds to π = μ f d : d = 2μ f d 2 , i.e. a quadratic form of the potential with regard to its argument. Generalizing this concept, a power-law ﬂuid is deﬁned by π(d) = βd n , for which the state equation takes the form: σ = − p1 +

nβ n−2 d d 2

(2.134)

We verify that the Newtonian ﬂuid is a power-two ﬂuid, n = 2, for which β = 2μ f . In the sequel, it is assumed that n > 1. Flow in a Cylindrical Pore

To begin with, we consider the thought model of Section 2.2.1. Proceeding as in Section 2.6.1, we look for a microscopic velocity ﬁeld v = v(r )e 1 , for which (2.126) holds. Due to the no-slip solid–ﬂuid interface condition v(r = a ) = 0, a negative value of ∂v/∂r is expected. It follows that the equivalent deviatoric strain rate is d = −dr z , and (2.134) yields: ∂v n−1 nβ nβ (2.135) σr z = − (−dr z )n−1 = − n − 2 2 ∂r

Micro(fluid)mechanics of Darcy’s Law

56

Combining (2.125) and (2.135), and taking the condition v(a ) = 0 into account, gives the solution: 1 n r n−1 1 n − 1 a n n−1 n−1 1− v(r ) = 2 (2.136) (α ) n nβ a

The ﬁltration velocity Q is obtained by integrating v = v(r )e 1 over the cross section of the cylinder: 1 2πa 3 n − 1 a α n−1 Q= (2.137) S 3n − 2 nβ

Equation (2.137) appears as an extension of (2.7)7 which is retrieved for n = 2 and β = 2μ f . It is remarkable (but not surprising) that the power-type nonlinearity of the local ﬂuid state equation translates into a macroscopic advection model between ﬁltration velocity and pressure gradient, which is of the power type as well!

An Approximate Conduction Law in the General Case

The shape of the pore space is no longer speciﬁed. The exact solution v of the nonlinear problem is the periodic velocity ﬁeld which satisﬁes: ∂π =0 −grad p + divz z ∂d d(v) = 12 gradz v + t gradz v

(U f )

(a )

(U f )

(b)

(U f )

(c)

v=0

(I s f )

(d)

divz v = 0

(2.138)

In (2.138a), the pressure ﬁeld p(z) takes the same form as in (2.15): p(z) = α · z + φ(z)

(2.139)

where φ(z) is a periodic function. In order to deal with the general case, note that the state equation (2.134) is similar to the one of an incompressible Newtonian ﬂuid: σ = − p1 + 2μ f (d)d

1 with μ f (d) = nβ d n−2 4

(2.140)

The difference lies in the fact that the viscosity μ f (d) appears as a power function of the strain rate d. Clearly, the latter varies from one point to another in the ﬂuid domain. Moreover, it depends on the macroscopic loading parameter, 7 Remember

that α = −d P/d x in the 1-D case.

Beyond Darcy’s (Linear) Law

57

namely the macroscopic pressure gradient α. In order to simplify the determination of the ﬂow, a ﬁrst-order approach consists of capturing the nonlinearity of the viscosity coefﬁcient in an average way by neglecting its heterogeneity due to that of the strain rate ﬁeld d(z). More precisely, we assume that the local viscosity coefﬁcient can be approximated by a uniform value over the ﬂuid, corresponding to the so-called reference strain rate d r : 1 (∀z ∈ U f ) μ f (d(z)) ≈ μ f (d r ) = nβ (d r )n−2 (2.141) 4 d r has to be a measure of the average amplitude of the strain rate over the ﬂuid. The most natural and convenient choice proves to be the quadratic average: f

d r = (d 2 )1/2

(2.142)

Once the heterogeneity of d(z) has been neglected, the ﬂow in the nonlinear case can be approximated by that of the Newtonian ﬂuid characterized by the viscosity coefﬁcient μ f (d r ) (velocity ﬁeld va ). In particular, recalling (2.37), the approximate ﬁltration vector Q a (v a ) takes the form: Q a (v a ) = −

1 K′ · α μ f (d r )

(2.143)

The determination of the approximate conduction law thus amounts to estimating the ‘reference’ viscosity coefﬁcient μ f (d r ). From (2.37) and (2.39), we note that: μ f (d r ) 1 A(v a , v a ) = −α · Q(v a ) = f r α · K′ · α |U| μ (d ) or, combining with (2.142) and (2.40): r f r 2 2d μ (d ) ϕ = α · K′ · α

(2.144)

(2.145)

Equations (2.140)–(2.145) provide an implicit estimate of d r and of μ f (d r ) as a function of the macroscopic pressure gradient α: 1 n2 −1 n−1 nβ 1 f r ′ μ (d ) = α·K ·α (2.146) 2n ϕ Inserting this expression of the viscosity coefﬁcient into (2.143) eventually yields: 1 2−n 2(n−1) n n−1 1 2 ′ Q a (v a ) = − α·K ·α K′ · α (2.147) nβ ϕ The above nonlinear conduction law represents a generalization of Darcy’s law to power-law ﬂuids, in the framework of the approximation (2.141). In

Micro(fluid)mechanics of Darcy’s Law

58

(2.147), the morphology of the pore space is captured by the intrinsic permeability K′ , as in the linear case. Nonetheless, the macroscopic counterpart of the nonlinearity of the ﬂuid state equation is a nonlinear relationship between macroscopic pressure gradient and ﬂux. Similar to (2.137), (2.147) predicts that: 1

(2.148)

|Qa | ∝ |α| n−1

It is interesting to compare the approximate conduction law (2.147) with the exact solution (2.137) derived in the case of cylindrical pores. Inserting α = −αe 1 into (2.147) and recalling that K ′ = πa 4 /(8S) = a 2 ϕ/8 (see Section 2.2.1) yields: 1 2πa 3 2−3n a α n−1 2(n−1) 2 (2.149) Qa = |Q a | = S nβ It appears that the ratio between Qa and the exact solution Q derived in (2.137) depends only on the power n. Of course, this ratio is equal to one for n = 2: Qa /Q =

3n − 2 2−3n 2 2(n−1) ≤ 1 n−1

(2.150)

We note that the ratio Qa /Q is greater than 0.8 for 1.4 < n < 2. The Variational Approach

We ﬁrst look for a minimum principle characterizing the exact solution v to (2.138). We refer to the same functional space H as in the linear case. Recalling (2.29) and using (2.139), we note that: (2.151) (∀u ∈ H) u · grad p d Vz = α · Q(u)|U| Uf

z

Furthermore, using the properties of H and those of v ,8 we successively obtain: ∂π (d(v)) · n d Sz = 0 u· (2.152) ∂d ∂U f and:

Uf

∂π (d(v)) : gradz u d Vz = − ∂d

Uf

u · divz

∂π d Vz ∂d

Combining (2.151) and (2.153), the weak form of (2.138a) is: ∂π (d(v)) : gradz u d Vz (∀u ∈ H) − α · Q(u)|U| = U f ∂d 8 Remember

that u and v are periodic and meet the no-slip condition on I s f .

(2.153)

(2.154)

Beyond Darcy’s (Linear) Law

59

We now introduce the functional H deﬁned on H as: π (d(u)) d Vz + α · Q(u)|U| H(u) =

(2.155)

Uf

The convexity of π(d) (for n > 1, see the Appendix, Section 2.7) implies that: ∂π (2.156) (d(v)) : gradz (u − v) ∂d Inserting this result into (2.154) yields the expected minimum principle: π(d(u)) − π(d(v)) ≥

(∀u ∈ H) H(u) ≥ H(v)

(2.157)

We now assume that n < 2. With this new assumption, we are going to establish that |α · Q a | is a lower bound of the exact value |α · Q(v)|. Recalling that π(d) = βd n (with d 2 = d : d/2), the condition n < 2 implies that:9 f n/2 f f (∀d) π (d) = βd n ≤ β d 2 (2.158) The minimum principle then reads: f n/2 + α · Q(u) (∀u ∈ H) H(v) ≤ |U| βϕ d 2 (u)

(2.159)

The idea now consists in choosing u as the solution vμ of the (linear) Stokes problem (2.12)–(2.14) associated with an arbitrary value of the viscosity coefﬁcient μ. dμ denotes the corresponding strain rate. Combining (2.39) and (2.40), we note that: f

dμ2 =

α · K′ · α 4μ2 ϕ

1 with dμ2 = dμ : dμ 2

(2.160)

and: −α · Q(vμ ) =

α · K′ · α μ

Relation (2.159) then takes the form: α · K′ · α n/2 1 − α · K′ · α (∀μ > 0) H(v) ≤ |U| ϕβ 4ϕμ2 μ

(2.161)

(2.162)

The next step consists of determining the optimal value μopt of μ, namely the one for which the r.h.s. in (2.162) realizes a minimum. Interestingly, it turns out that μopt is equal to the reference viscosity coefﬁcient μ f (d r ) introduced in (2.146): μopt = μ f (d r ) 9 Relation

¨ (2.158) is an application of the Holder inequality.

(2.163)

Micro(fluid)mechanics of Darcy’s Law

60

The corresponding upper bound on H(v) is: 1 α · K′ · α 1 H(v) ≤ · Q − 1 = −α − 1 a μ f (d r ) n n We ﬁnally focus on H(v) which is: H(v) = π (d(v)) d Vz + |U|α · Q(v)

(2.164)

(2.165)

Uf

The starting point is (2.154) applied to u = v: ∂π (d(v)) : gradz v d Vz = −α · Q(v)|U| U f ∂d Moreover, observing that ∂π/∂d = βnd n−2 d/2, we also note that: ∂π π (d(v)) d Vz (d(v)) : gradz v d Vz = n Uf U f ∂d Inserting (2.166) and (2.167) into (2.165) yields: 1 H(v) = 1 − α · Q(v) n

(2.166)

(2.167)

(2.168)

Combining (2.168) with (2.164), we eventually conclude that:10 −α · Q(v) ≥

1 μ f (d r )

α · K′ · α = −α · Q a

(2.169)

This proves that the approximate ﬁltration law (2.147) is a lower bound.

2.7 Appendix: Convexity of π(d ) We start from the identity: π(d + δd) = β

1 1 d : d + δd : δd + d : δd 2 2

p

(2.170)

where p = n/2. Introducing: J 2 (d) = d : d/2;

X=

d : δd ; J 2 (d)

Y=

J 2 (δd) J 2 (d)

(2.171)

(2.170) can be rearranged into: π(d + δd) = (1 + X + Y) p π (d) 10 Remember

that n > 1.

(2.172)

Appendix: Convexity of π(d )

61

Accordingly, the variation of π (d + δd) − π (d) is:

π(d + δd) − π(d) = ((1 + X + Y) p − 1)π (d)

Recalling that ∂π/∂d = βnd

n−2

d/2, it is readily seen that:

∂π (d) : δd = p Xπ(d) ∂d Combining (2.173) and (2.174) then yields: π (d + δd) − π (d) −

(2.173)

∂π (d) : δd = ((1 + X + Y) p − 1 − p X )π(d) ∂d

(2.174)

(2.175)

We ﬁnally observe that (d : δd)2 ≤ (d : d)(δd : δd), which implies that X2 ≤ 4Y (see (2.171)). It follows that: (1 + X + Y) p − 1 − p X ≥ (1 + X + X2 /4) p − 1 − p X = (1 + X/2)2 p − 1 − p X

(2.176)

The r.h.s. in (2.176) is a positive number provided that p > 1/2, i.e. n > 1. In other words, if n > 1, π (d) = β d n is a convex function of d.

3 Micro-to-Macro Diffusive Transport of a Fluid Component This chapter deals with diffusion in a pore space that is saturated by a ﬂuid phase composed of different chemical species. Within the framework of periodic homogenization,1 it is shown how the microscopic conduction law of a solute through a solvent, known as Fick’s law, translates into a macroscopic ion conduction model, i.e. a macroscopic Fick’s law that incorporates the available information of the geometry of the microstructure. For the case of moderate advection, the developed upscaling procedure leads to the identiﬁcation of the tortuosity tensor and the homogenized diffusion tensor. For this case, an upper and a lower bound solution strategy for the tortuosity tensor is developed. Finally, by generalizing the model within the framework of the double-scale expansion technique, we address the case of diffusion with advection, which gives rise not only to diffusive phenomena, but also to advective and dispersive ﬂuxes. 3.1 Fick’s Law We consider a porous medium in which the pore space is saturated by a liquid composed of different chemical species. For simplicity, the solid phase is assumed undeformable. Referring to the notation introduced in Chapter 1, ρ γ (z) and vγ (z) are respectively the concentration (or mass density) and the velocity ﬁelds of the chemical species γ at the microscopic scale. In turn, the velocity ﬁeld of the liquid phase v(z) is deﬁned, in a standard manner, as 1 Diffusion

in disordered porous media will be considered later on (Chapters 4 and 6).

Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

64

Micro-to-Macro Diffusive Transport of a Fluid Component

the average of the velocities of the components making up the liquid phase at point z: (3.1) f γ (z)vγ (z); f γ (z) = 1 v(z) = γ

γ

Depending on the physical meaning of f γ (volume, molar, mass fractions, etc.), different deﬁnitions are frequently encountered in the literature.2 For purposes of clarity, we restrict the presentation to the so-called dilute situation. In this case, whatever the deﬁnition of f γ in (3.1), the phase velocity can be identiﬁed as the one of the solvent, and the superscript γ , therefore, refers to the solute. Finally, for ease of presentation, we will focus on a single solute in an incompressible solvent. In the dilute situation, the phase velocity ﬁeld v(z) can be determined separately as the solution of the Stokes system (2.12). In turn, the determination of the solute velocity vγ (z) requires two further laws. One is the microscopic mass balance equation of the solute, which is of the form (1.28): ∂ρ γ + divz (ρ γ vγ ) = 0 ∂t

(3.2)

The second law relates to the physics of the problem at hand. The diffusion of a chemical species is described by Fick’s law, which relates the diffusive mass ﬂux j γ = ρ γ (vγ − v) to the gradient of the solute concentration [1]. In the dilute case, this is: (P f ) j γ = ρ γ (vγ − v) = −Dγ grad ρ γ z

(3.3)

where Dγ is the diffusion coefﬁcient which refers to the diffusion of solute γ through the solvent in an inﬁnite ﬂuid domain. From this deﬁnition, the diffusion coefﬁcient is independent of the morphology of the pore space, and only depends on the solute and the solvent. The challenge of the micro-to-macro approach is to determine the macroscopic counterpart of (3.3) in a way that incorporates the available information on the geometry on the microstructure.

3.2 Diffusion without Advection in Steady State Conditions We ﬁrst address the simplest case where the solvent velocity is negligible (no advection) in steady state conditions. The coupling of diffusion with advection in transient conditions is considered afterwards.

2 For

a review, see e.g. Bear and Bachmat [6].

Diffusion without Advection in Steady State Conditions

65

3.2.1 Periodic Homogenization of Diffusive Properties The limit of a negligible solvent velocity is v = 0. Therefore, we note that j γ = ρ γ vγ . In turn, the assumption of steady state conditions means that ∂ (.) /∂t = 0, and implies also that there is no mass exchange (precipitation or dissolution) of the chemical species γ at the solid–ﬂuid interface. In this case, the equations governing the diffusion at the microscopic scale reduce to: (a ) divz j γ = 0 (b) j γ = −Dγ grad ρ γ z (c) j γ · n = 0

(P f ) (P f ) (I s f )

(3.4)

We have already employed the periodic homogenization technique in Chapter 2: the geometry of the microstructure is periodic, and the elementary cell U contains the ﬂuid domain U f (see Figure 2.3). In turn, for this elementary cell, we look for a periodic ﬁeld of diffusive ﬂux j γ (z), such that (see (2.9)): ∀ n1 , n2 , n3 ∈ N

j γ (z + n1 a 1 e 1 + n2 a 2 e 2 + n3 a 3 e 3 ) = j γ (z)

(3.5)

According to (3.5), the diffusive ﬂux is entirely deﬁned by its value in U f , satisfying the following periodicity condition: ∀z ∈ ∂U f ∩ (zi = 0)

j γ (z + a i e i ) = j γ (z)

(3.6)

At the macroscopic scale, the diffusive ﬂux can be described by its intrinsic f average, j γ , or its apparent average j γ f (have a quick look back to Section 1.4.1, Equations (1.59) and (1.60)). In particular, we have seen in Chapter 1 that upscaling the mass balance equation (1.28) yields (1.31) which takes the following form (steady state conditions): divx ( j γ f ) = 0

(3.7)

From (3.7), it is convenient to deﬁne the macroscopic diffusive ﬂux as the apparent average J γ = j γ f . The periodicity of the microscopic ﬁeld of diffusive ﬂux together with Fick’s law (3.3) implies the periodicity of the microscopic concentration gradient grad ρ γ. Employing a similar reasoning as in Section 2.2.2 suggests z extending the concentration ﬁeld ρ γ (z) to the whole cell U as the sum of γ an afﬁne term, ρ0 + H · z , and a periodic one, r (z): γ

ρ γ (z) = ρ0 + H · z + r (z)

(3.8)

The afﬁne term in (3.8) represents the variations of the concentration at the macroscopic scale, and H the macroscopic concentration gradient. Furthermore, due to the periodicity of r (z), it is readily seen that H = gradz ρ γ . In turn, r (z) corresponds to a periodic perturbation induced by the presence of

Micro-to-Macro Diffusive Transport of a Fluid Component

66

the solid phase, which prevents the solute molecules moving in the direction of the macroscopic concentration gradient. According to (3.4)–(3.8), the diffusive ﬂux j γ appears as the sum of a constant term, −Dγ H, that would prevail in the absence of a solid phase, and a correcting periodic term that captures the effect of the solid boundary: j γ = −Dγ H + f

with

f = −Dγ grad r z

(3.9)

The challenge of the homogenization is to determine the link between the macroscopic diffusive ﬂux, J γ , and the macroscopic concentration gradient, H. 3.2.2 The Tortuosity Tensor Inserting the expression (3.8) of ρ γ into (3.4) yields the following differential problem of the ﬂuctuation r (z): (a ) zr = 0 (b) grad r · n = −H · n z

(U f ) (I s f )

(3.10)

In order to ensure the uniqueness of the solution, we further consider the solution of (3.10) for which the intrinsic average r f of r on U f is zero. Then, relations (3.10a) and (3.10b), together with the periodicity property of r (z), allow us to determine the value of r (z) as a linear function of H: r (z) = χ (z) · H

(3.11)

where the ﬁeld χ(z) deﬁned on U f depends only on the morphology of the ﬂuid domain. Once χ (z) is known, it becomes possible to relate the microscopic concentration gradient to the macroscopic one, H, by means of a linear operator, the concentration tensor A(z): (U f ) grad ρ γ = A(z) · H z

with A = 1 + t gradz χ

(3.12)

Finally, the link between the macroscopic diffusive ﬂux J γ = j γ f and the macroscopic concentration gradient H becomes: J γ = j γ f = −Dhom · H

with Dhom = ϕ Dγ T

(3.13)

Dhom is recognized as the homogenized diffusion tensor. The tensor T in (3.13) is referred to as the tortuosity tensor. It is the intrinsic average of the concentration tensor A over the ﬂuid domain: f

T = A = 1 + t gradz χ

f

(3.14)

The tortuosity tensor, through χ(z), depends only on the geometry of U f ; that is, it is intrinsic to the pore space. It captures the phenomenon that the solid phase prevents the molecules of the solute from moving parallel to the

Diffusion without Advection in Steady State Conditions

67

macroscopic concentration gradient. Its determination requires the solution of the periodic problem (3.10a)–(3.10b). It is appealing to note the formal equivalence of the diffusion tensor Dhom with the (overall) permeability tensor K. Both of them are the product of a scalar (μ f or Dγ ), which is related to the physics of the problem at hand, and of a second-order tensor (intrinsic permeability tensor K′ or ϕT) which is of purely geometrical nature. Cylindrical Porosity

By way of example, consider an elementary cell in which the pore space is a cylinder parallel to the direction of vector e 1 . The cross section is of arbitrary shape. The only possible direction for macroscopic diffusion is in the direction of the cylinder axis e 1 . We therefore consider the macroscopic concentration gradient H = he 1 . Observing that the normal to the solid–ﬂuid interface is perpendicular to e 1 , the differential problem to be solved by the ﬂuctuation term r (z) is: z r = 0 grad r · n = 0 z

(U f ) (I s f )

(3.15)

The periodicity of r (z) implies that r depends only on z2 and z3 . The solutions r (z2 , z3 ) of (3.15) are constant in U f . The condition r f = 0 then implies that r = 0. In (3.14), this result yields T = 1. The homogenized diffusion tensor is therefore Dhom = ϕ Dγ 1, and the presence of the solid phase, therefore, is entirely taken into account through the porosity. 3.2.3 Variational Approach to Periodic Homogenization Principles of Minima

We now consider the variational formulation of the elliptic differential problem. The aim is to derive an upper and a lower bound of the tortuosity tensor based on a variational approach. This variational approach consists in characterizing both the periodic ﬂuctuations f = −Dγ grad r and the solutions r (z) of z the differential problem (3.10a)–(3.10b) through principles of minima. Let F be the set of ‘admissible’ ﬂux perturbations f ′ subjected to the following conditions: (a ) f ′ piecewise C 1 (b) divz f ′ = 0 (c) f n′ = 0 (d) f ′ · n = Dγ H · n (e) ( f ′ · n) n periodic

(U f ) () (I s f ) (∂U f )

(3.16)

Micro-to-Macro Diffusive Transport of a Fluid Component

68

where denotes a possible surface of discontinuity of f ′ , and f n′ = f ′ · n is the normal component of f ′ on , which is continuous over because of the mass balance. We introduce the following functional on F: W∗ ( f ′ ) =

1 2Dγ

2

f ′ dVz

(3.17)

Uf

The solution f realizes the minimum of W∗ ( f ′ ) in F: (∀ f ′ ∈ F) W* ( f ) ≤ W* ( f ′ )

(3.18)

Indeed, let us write any ﬁeld f ′ ∈ F in the form f ′ = f + δ f . According to (3.16), we note that divz δ f = 0, δ f · n = 0 on I s f , and that (δ f · n)n is periodic. From (3.17), we have: 1 (∀ f ′ ∈ F) W∗ ( f ′ ) = W∗ ( f ) + W∗ (δ f ) + γ (3.19) f · δ f dVz D Uf Recalling that f = −Dγ grad r , and using the periodicity of r and the propz erties of δ f , it is readily seen that the last integral in (3.19) vanishes. (3.18) is a direct consequence of W∗ (δ f ) ≥ 0. Consider now the dual approach. Let R be the set of admissible concentration perturbations r ′ : R = {r ′ , C 1 , periodic}

(3.20)

and introduce the following functional deﬁned on R: Dγ W(r ) = 2 ′

Uf

′2

γ

grad r dVz + D z

Is f

r ′ H · n dSz

(3.21)

Following a similar reasoning, it can be shown that the solution r to (3.10) realizes the minimum of W(r ′ ) in R: (∀r ′ ∈ R) W(r ) ≤ W(r ′ )

(3.22)

We are left with combining the minimum principles (3.18) and (3.22). To this end, note that the minima W(r ) and W∗ ( f ) take opposite signs for the solution couple (r, f ): W(r ) + W∗ ( f ) = 0

(3.23)

Diffusion without Advection in Steady State Conditions

69

Furthermore, recalling that r and f are both periodic, integration by parts of W∗ ( f ) yields: 1 Dγ ∗ W (f) = − r H · n dSz f · grad r d Vz = − z 2 Uf 2 Is f Dγ (3.24) =− H · grad r dVz z 2 Uf Combining (3.11) and (3.24) reveals the link between W∗ ( f ) = −W(r ) and the tortuosity tensor T: 2W∗ ( f ) 2W(r ) f = − = H ·t gradz χ · H = H · (T − 1) · H Dγ |U f | Dγ |U f |

(3.25)

Finally, the two principles of minima (3.18) and (3.22), together, with (3.25), provide upper and lower bounds of the tortuosity: (∀ f ′ ∈ F)(∀r ′ ∈ R) −

2W(r ′ ) 2W∗ ( f ′ ) · T − 1 · H ≤ ≤ H ( ) Dγ |U f | Dγ |U f |

(3.26)

It is readily understood that the upper bound in (3.26) implies T ≤ 1, which is obtained for the particular choice r ′ = 0: ∀H, H · (T − 1) · H ≤ 0

(3.27)

Ordered Relations of Tortuosity

As in the study of the permeability tensor (Section 2.2.3), we compare the tortuosity tensors of two porous media which are represented by the same elementary cell except that the pore space U (1) f is smaller than the pore space (2) (1) U f . It is intuitively understood that U f ⊂ U (2) f implies that the homogenized diffusivity of the ﬁrst is smaller than that of the porous material with the greater pore space. This section provides a rigorous proof of this intuitive property that the diffusion coefﬁcient is an increasing function of the size of the pore space. The two media are subjected to the same macroscopic concentration gradi∗(1) ent H. Let j γ1 denote the diffusive ﬂux solution in U (1) f . The functional W γ γ (1) deﬁned on F realizes its minimum for f 1 = D H + j 1 , and (3.25) is: −

2W∗(1) ( f 1 ) Dγ

|U (1) f |

= H · (T(1) − 1) · H

(3.28)

Micro-to-Macro Diffusive Transport of a Fluid Component

70

Consider the extension j γ ′1 of j γ1 obtained by letting j γ ′1 (z) = 0 in domain (1) (2) (1) ′ γ U (2) f \ U f . The corresponding extension f 1 of f 1 is equal to D H in U f \ U f . (1) (2) ′ Since U f ⊂ U f , it is readily seen that f 1 is an admissible ﬂux perturbation ′ (2) of the diffusion problem in U (2) f ; that is, f 1 ∈ F . Furthermore, we obtain: W∗(2) ( f ′1 ) = W∗(1) ( f 1 ) +

1 γ 2 (1) D H (| U (2) f | − | U f |) 2

(3.29)

We now apply (3.26) to the diffusion in U (2) f : −

2W∗(2) ( f ′1 ) Dγ | U (2) f |

≤ H · (T(2) − 1) · H

(3.30)

Combining (3.28), (3.29) and (3.30) then yields:

H·

| U (1) f | |

U (2) f

|

T

(1)

− 1 · H ≤ H · (T(2) − 1) · H

(3.31)

or equivalently: | U (1) f | |

U (2) f

|

T(1) ≤ T(2) ⇔ D1hom ≤ D2hom

(3.32)

Instructive Exercise: Tortuosity and Diffusivity of a Cubic Array of Spheres

We are interested in the homogenized diffusion tensor of a cubic array of identical spheres of radius R. The elementary cell is a cube; the sides are parallel to the vectors (e 1 , e 2 , e 3 ) of an orthonormal frame; and the length of a side is 2R. The cell is subjected to a macroscopic concentration gradient H = He 1 . We want to provide a lower bound estimate based on the ordered relation (3.32). To obtain a lower bound, we seek cylindrical subdomains which ﬁt in the pore space and which are parallel to e 1 . These subdomains play the role of (2) U (1) f in the ordered relation, and the pore space plays the role of U f ≡ U f . Thus, it follows from (3.32) that the best lower bound of Dhom that can be derived with this approach is obtained for the largest cross section. The largest cross section is made up of four quarter circles of radius R, the centers of which are located on the vertices of a square of side 2R (Figure 2.7). Subject to the macroscopic gradient H = He 1 , it is readily seen that the solution of the diffusion problem in such a cylinder is: j γ1 = −Dγ H ;

f1 = 0

(3.33)

Diffusion without Advection in Steady State Conditions

71

Substituting this result in (3.28) yields T11(1) = 1; thus as a consequence of (3.32): hom (1) hom D11 = Dγ ϕ (1) ≥ D11

(3.34)

hom (1) hom the estimate obtained with is the ‘real’ diffusivity, and D11 Here, D11 (1) (1) U f ⊂ U f . Furthermore, ϕ denotes the volume fraction of the cylinder in the elementary cell, which is equal to the cross sectional ratio of the cylinder and the square of side 2R. We conclude that: π hom γ D11 ≥ D 1 − (3.35) 4

3.2.4 The Geometrical Meaning of Tortuosity It is generally admitted, in pure macroscopic theories, that the diffusion through porous media is affected by the pore volume fraction (porosity), and by the deviation of the solute molecules’ diffusion path from the ideal straight line parallel to the macroscopic concentration gradient. Both phenomena are also taken into account in (3.13), the ﬁrst through the porosity ϕ, the second through the tortuosity tensor T, here derived on the basis of a micromechanical approach. It is instructive to have a closer look at the link between the ‘classical’ macroscopic tortuosity concept and the geometry of the pore space. In order to illustrate (3.26) and to show the consistency of the micromechanical concept of tortuosity with the classical one, we consider an academic example of an elementary cell, in which the pore space of constant thickness δ is composed of a sequence of parallel horizontal and vertical planes (see Figures 3.1 and 3.2). The cell is subjected to a macroscopic concentration gradient H = He 1 . The diffusion of the solute is parallel to the plane deﬁned by e 1

δ

ey ex

di L

Figure 3.1 Approximation of the flux

Micro-to-Macro Diffusive Transport of a Fluid Component

72

δ

li

Figure 3.2 Approximation of the concentration

and e 2 . Instead of the length L along e 1 of the elementary cell, the real

length of the path of the molecules crossing the cell is on the order of D = i di . It is common practice, in macroscopic theories,3 to take this phenomenon into account through a correction of the diffusion coefﬁcient Dγ by a factor (L/D)2 . The aim of this section is to compare this macroscopic correction with estimates derived from the lower and upper bounds of the tortuosity provided by the variational approach. Lower Bound: Approximation of the Diffusive Flux

For the lower bound, it is convenient to divide the pore space into a sequence of vertical (resp. horizontal) subdomains. The intersection of two subdomains is a segment parallel to one of the bisectors of the horizontal and vertical directions (see Figure 3.1). The considered ﬂux perturbation f ′ is deﬁned as f ′ = Dγ H + j γ ′ , where the ﬂux j γ ′ is equal to Je 1 in the horizontal subdomains, and to ±Je 2 in the vertical ones. The appropriate sign (±) is prescribed by the mass balance condition (3.16c). At this stage, J is an arbitrary scalar that must be chosen so as to minimize the functional W∗ ( f ′ ). Such a ﬂux perturbation is clearly admissible in the sense of (3.16). Adding the contributions of all subdomains, we obtain: δ W∗ ( f ′ ) = di ( Dγ H + J )2 + (3.36) di Dγ 2 H 2 + J 2 2Dγ i∈I1 i∈I2 where Iα (α = 1, 2) denotes the set of the subscripts i corresponding to subdomains parallel to e α . According to (3.26), the optimal choice J opt of J minimizes

3 See,

for instance, [22].

Diffusion without Advection in Steady State Conditions

73

the expression (3.36) of W∗ ( f ′ ): dW∗ ′ L ( f ) = 0 −→ J opt = −Dγ H dJ D

(3.37)

Inserting this value into (3.36), we obtain from (3.26) a lower bound of the tortuosity:

L D

2

≤T

(3.38)

Remarkably, the macroscopic correction factor (L/D)2 turns out to be in fact a lower bound of the actual tortuosity.

Upper Bound: Approximation of the Concentration

For the upper bound, the pore space is divided into rectangular subdomains (side lengths δ and ℓi ), alternating horizontally and vertically, and in squares (side length δ) located at the corners between the vertical and horizontal rectangles (see Fig. 3.2). For a given scalar A, consider a concentration ﬁeld ρ γ ′ , which is an afﬁne function of xα in the subdomains parallel to e α (α = 1, 2); that is, ρ γ ′ = Axα + Bi in the subdomain number i, and which is uniform in the corner squares. A represents the microscopic concentration gradient. We let B1 = 0, and choose the other Bi in such a way that the continuity of ρ γ ′ is satisﬁed. Consider now the ﬁeld r ′ = ρ γ ′ − Hx. The only value of A which is compatible with the periodicity of r ′ is: A= H

L with ℓ = ℓi ℓ i

(3.39)

For this value, r ′ is an admissible concentration perturbation (r ′ ∈ R) and (3.26) holds. Next, the quadratic term of W(r ′ ) appears as the sum of the contributions of the horizontal rectangles, the vertical rectangles and the squares: Dγ 2

Uf

Dγ grad r d Vz = z 2 ′2

+

i∈I2

ℓi δ(A − H)2 +

i∈I1

H 2δ2

ℓi δ(A2 + H 2 )

i∈I2

(3.40)

74

Micro-to-Macro Diffusive Transport of a Fluid Component

and the linear term, which includes the contributions of the vertical rectangles and the squares:

Dγ I s f r ′ H · ndSz = −Dγ H 2 δ i∈I2 ℓi + Dγ − i∈I2 H 2 δ 2 − δ AH i∈I2 ℓi (3.41)

We thus obtain the following expression of W(r ′ ) (see (3.21)):

2 L 1 i∈I2 δ γ 2 ′ − 1 − 2X with X = W(r ) = D δℓH 2 ℓ ℓ

(3.42)

On observing that D = ℓ(1 + 2X), (3.42) takes the form: 2 2W(r ′ ) ( L/ℓ) − 1 − 2X = Dγ | U f | H 2 1 + 2X

(3.43)

so that (3.26) yields the following upper bound: T ≤ (1 + 2X)

L D

2

(3.44)

In summary, combining (3.44) and (3.38) yields: 1≤

T 2

( L/D)

≤ 1 + 2X

(3.45)

For the particular morphology considered here, the variational approach conﬁrms that the tortuosity is scaled by ( L/D)2 , thus conﬁrming the geometrical interpretation of the macroscopic tortuosity concept. However, it is interesting to note from (3.45) that the accuracy of the classical estimate ( L/D)2 is controlled by X which in turn depends on the thickness-to-length ratio of the rectangles.

3.3 Double-Scale Expansion Technique The extension of the theory to diffusion with advection is achieved within the framework of the double-scale expansion technique, which we have already encountered in Section 2.4.1 in the context of ﬂuid ﬂow through porous media. This and the next section employ this framework to upscale the ﬂux of chemical species in porous media. We ﬁrst revisit the steady state pure diffusive transport within the framework of the double-scale expansion technique, before extending the theory to diffusion with advection.

Double-Scale Expansion Technique

75

3.3.1 Steady State Diffusion without Advection In this section, we employ the set of assumptions of Section 3.2 related to the diffusion problem without advection (v = 0) in steady state conditions (∂ (.) /∂t = 0), for which the microscopic ﬁelds of diffusive ﬂux and concentration are governed by (3.4). Employing the same notation4 as in Sections 1.4 and 2.4.1, the concentration ρ γ is treated as a function of the ﬂuctuation variable Z = z/a and the drift variable X = z/L. Furthermore, a combination of (3.4a) and (3.4b) yields z ρ γ = 0. Applying the chain rule (1.61) of Section 1.4, z ρ γ = 0 is developed in the form: 1 1 γ γ γ grad grad div ρ + + div + Xρ γ = 0 ρ ρ X Z Z Z X δ2 δ

(3.46)

or equivalently, in terms of components:

1 ∂ 2ρ γ 2 ∂ 2ρ γ ∂ 2ρ γ + =0 + δ 2 ∂ Zi2 δ ∂ Zi ∂ Xi ∂ Xi2 i

(3.47)

where δ = a /L. Analogously, the interface condition (3.4c) takes the form: 1 γ γ (3.48) grad ρ + grad ρ · n = 0 at I s f Z X δ We seek the microscopic concentration in the form of a series expansion with respect to the powers of δ = a /L: ρ γ (Z, X) = (3.49) δ j ρ ( j) (Z, X) j

in which all ρ ( j) are Z-periodic and have the same order of magnitude.5 Inserting (3.49) into (3.47) yields: 1 ∂ 2 ρ (1) ∂ 2 ρ (0) 1 ∂ 2 ρ (0) + +2 0= 2 δ i ∂ Zi2 δ ∂ Zi ∂ Xi ∂ Zi2 i +

4 Recall 5 For

j≥2

δ j−2

∂ 2 ρ ( j) i

∂ Zi2

∂ 2 ρ ( j−1) ∂ 2 ρ ( j−2) + +2 ∂ Zi ∂ Xi ∂ Xi2

that the edge length a is equal to the pore size d. simplicity, the order j in the expansion of ρ γ is simply denoted by ρ ( j) .

(3.50)

Micro-to-Macro Diffusive Transport of a Fluid Component

76

Expression (3.50) is equivalent to a family of differential problems coupling ρ ( j) : (a ) Z ρ (0) = 0 (b) Z ρ (1) + 2 (c) Z ρ ( j) + 2

∂ 2 ρ (0) =0 ∂ Zi ∂ Xi i

(3.51)

2 ( j−1)

∂ ρ + X ρ ( j−2) = 0 ∂ Zi ∂ Xi

( j ≥ 2)

We now insert (3.49) into the interface condition (3.48). If we note that: 1 1 γ γ γ grad ρ = grad ρ + grad ρ z Z X L δ 1 1 (3.52) δ j−1 grad ρ ( j) + grad ρ ( j−1) = grad ρ (0) + Z Z X L δ j≥1 relation (3.4c) turns out to be equivalent to: (a ) grad ρ (0) · n = 0 Z

(b) (grad ρ ( j) + grad ρ ( j−1) ) · n = 0 Z

X

(3.53)

( j ≥ 1) (I s f )

Equations (3.51a) and (3.53a) combined with the Z-periodicity condition imply that ρ (0) is independent of Z; that is, ρ (0) = ρ (0) (X). Furthermore, the differential problem to be satisﬁed by the Z-periodic function ρ (1) is: Z ρ (1) = 0 (U f ) (1) (0) grad ρ · n = −grad ρ · n (I s f ) Z

(3.54)

X

The relevant spatial variable in (3.54) is Z, i.e. grad ρ (0) can be regarded as a X constant. More precisely, grad ρ (0) describes the variations of the dominating X term of the concentration gradient at the macroscopic scale. Indeed, recalling that X = z/L, it is related to the macroscopic concentration gradient H by: 1 grad ρ (0) = H X L

(3.55)

From (3.54), it is readily seen that ρ (1) linearly depends on grad ρ (0) : X

f

ρ (1) (Z, X) = ρ (1) (X) + ρ˜ (1) (Z, X) with

ρ˜ (1) (Z, X) = χ(Z) ˜ · grad ρ (0) X

(3.56) The χ˜ i (Z) component of χ˜ (Z) appears as the solution of (3.54) for grad ρ (0) = X e i . Thus, it depends on the morphology of the pore space only.

Double-Scale Expansion Technique

77

Summarizing these results, the expansion of ρ γ to ﬁrst order in δ is: f

ρ γ (Z, X) = ρ (0) (X) + δρ (1) (X) + δ ρ˜ (1) (Z, X)

(3.57)

Let us now consider the variations of ρ γ within the cell located at the macroscopic scale at X0 = z0 /L. A comparison of (3.10) and (3.54) suggests inserting the functions r (z) and χ(z) related to ρ(Z, ˜ X) and χ(Z) ˜ by: r (z) = δ ρ˜ (1) (Z, X0 );

χ (z) = a χ(Z) ˜

(3.58)

Then, in the vicinity of the macroscopic point X0 = z0 /L, the variations of ρ γ (up to ﬁrst order in δ) can be approximated by: f

ρ γ ≈ ρ (0) (X0 ) + δρ (1) (X0 ) + grad ρ (0) (X0 ) · X

z − z0 + r (z) L

(3.59)

or equivalently, using (3.55): γ

ρ γ ≈ ρ0 + H · z + r (z) with

γ

f

ρ0 = ρ (0) (X0 ) + δρ (1) (X0 ) − H · z0

(3.60)

The result obtained by means of the double-scale expansion is remarkable, as it justiﬁes (3.8) a posteriori and the results obtained in Section 3.2. In particular, the leading term of the diffusive ﬂux is: j (0) = −

Dγ (grad ρ˜ (1) + grad ρ (0) ) Z X L

On the other hand, combining (3.56) and (3.61) yields: f J γ ≈ j (0) f = −ϕ Dγ 1 + t grad Z χ˜ · H

(3.61)

(3.62)

Since (3.58) implies that grad Z χ˜ = gradz χ , the macroscopic ﬂux expression (3.62) is nothing more than the previously derived expression (3.13) with (3.14). 3.3.2 Steady State Diffusion Coupled with Advection We now turn to a more advanced application of the double-scale expansion technique to account for an advection velocity v = 0. Still, for purposes of clarity, we ﬁrst restrict ourselves to steady state conditions (∂ (.) /∂t = 0). Transient conditions will be considered in Section 3.3.3. The main difference with the pure diffusive ion transport is that (3.7) is no longer valid when ions are transported by both diffusion and advection. Returning to ﬁrst principles, it is readily seen by applying (1.31) that the macroscopic mass balance equation is now: f f divx j γ f + ϕ ρ γ v f + ρ˜ γ v˜ =0 (3.63)

78

Micro-to-Macro Diffusive Transport of a Fluid Component

where y˜ = y − y f represents the ﬂuctuation of the physical quantity y within the ﬂuid phase. The macroscopic mass ﬂux of the γ component thus appears as f the sum of three terms: the diffusive term j γ f , the advective term ϕρ γ v f , f and the so-called dispersive term ϕ ρ˜ γ v˜ . While the advective term can be macroscopically described by Darcy’s law, provided that the phase velocity v(z) satisﬁes the Stokes system (2.12)–(2.14), the diffusive and dispersive terms in (3.63) need some further attention. To this end, we return to the microscopic scale. Inserting the diffusive ﬂux j γ into (3.2), we have: divz j γ + divz ρ γ v = 0

(3.64)

Making use of Fick’s law (3.3) and of the incompressibility of the ﬂuid phase, (3.64) simpliﬁes to: −Dγ z ρ γ + v · grad ρ γ = 0 z

(3.65)

Finally, using the chain rule (1.61) in (3.65), we obtain: 1 Dγ 1 γ γ γ γ − 2 div X grad ρ + div Z grad ρ + X ρ Zρ + Z X L δ2 δ 1 1 (3.66) + v· grad ρ γ + grad ρ γ = 0 Z X L δ The ﬁrst term in (3.66) is identical to (3.46) in the pure diffusive case, while the last term is the new term related to advection. In order to discuss the effect of advection on diffusion, it is instructive to deﬁne three reference velocities v0 = Dγ /L, v1 = v0 /δ = Dγ /d and v2 = v0 /δ 2 = Dγ L/d 2 . Letting V = |v|, we will successively examine (3.66) for the cases V = O(v0 ), V = O(v1 ) and V = O(v2 ). The relative effect of advection and diffusion classically refers to the P´eclet number Pe = Vd/Dγ . The latter is obtained as the ratio of the orders of magnitude of the advection term ρ γ v and the diffusive term ρf γ (vγ − v). In the following, ρ˜ ( j) denotes the ﬂuctuation ρ ( j) − ρ ( j) . Moderate Advection: V = O(v0 = Dγ /L), Pe = O(δ)

We introduce the normalized velocity v∗ = v/v0 which, by deﬁnition, satisﬁes | v∗ | ≡ O(1). In this case, (3.66) takes the form: 1 1 − 2 Z ρ γ + (div X grad ρ γ + div Z grad ρ γ ) + X ρ γ Z X δ δ 1 (3.67) + v∗ · grad ρ γ + grad ρ γ = 0 Z X δ Then, inserting (3.49) into (3.67) yields a set of differential equations to be satisﬁed by ρ ( j) and v∗ . The boundary conditions at the solid–ﬂuid interface

Double-Scale Expansion Technique

79

expressed by (3.53) are unchanged. In particular, at the order of δ −2 and δ −1 , the following equations, to be solved by ρ (0) and ρ (1) , are derived: (a ) Z ρ (0) = 0 (b) Z ρ (1) + 2

(U f ) 2 (0)

∂ ρ + v∗ · grad ρ (0) = 0 (U f ) Z ∂ Z ∂ X i i i

(3.68)

Equation (3.68a) coincides with (3.53a). Therefore, the result discussed in Section 3.3.1, namely that the Z-periodic solution ρ (0) is independent of Z , still holds here, i.e. ρ (0) = ρ (0) (X). In turn, in this case, (3.68b) reduces to Z ρ (1) = 0, and ρ (1) is then simply the solution of the differential problem (3.54) obtained for the pure diffusive situation. In other words, for v ≤ O(v0 ), the advection has no effect on the overall description of the diffusion, which reduces to (3.13). The same holds for the dispersive term in (3.63). Indeed, the dominating term in ρ˜ γ v˜ is δ ρ˜ (1) v˜ (0) , and the dominating term in j γ is j (0) = −v0 (grad Z ρ˜ (1) + grad ρ (0) ) (see (3.61)). This implies that | ρ˜ γ v˜ | / | j γ | = O(δ) ≪ 1; that is, the X dispersive term in (3.63) is negligible with respect to the diffusive one. Advection-Dominated Transport: V = O(v1 = Dγ /d), Pe = O(1)

We consider the normalized velocity v∗ deﬁned by v = v1 v∗ . In this case, (3.65) takes the form: 1 1 − 2 Zρ γ + div X grad ρ γ + div Z grad ρ γ + X ρ γ Z X δ δ 1 1 γ γ ∗ =0 (3.69) + v · 2 grad ρ + grad ρ Z X δ δ Furthermore, the interface condition is: 1 grad ρ γ + grad ρ γ · n = 0 Z X δ

at I s f

(3.70)

We introduce the expansions for v∗ and ρ γ . Considering the order of δ −2 in (3.69) and the order of δ −1 in (3.70) yields the following differential system: − Z ρ˜ (0) + v∗(0) · grad ρ˜ (0) = 0; Z

grad ρ˜ (0) · n = 0 Z

at

Is f

(3.71)

The only solution of (3.71) is ρ˜ (0) = 0. In other words, ρ (0) depends only on the macroscopic variable X and does not ﬂuctuate within the cell. Furthermore, considering the order of δ −1 in (3.69) and the order of δ 0 in (3.70), we obtain: − Z ρ˜ (1) + v∗(0) · grad ρ˜ (1) + v∗(0) · grad ρ (0) = 0 Z

X

(3.72)

Micro-to-Macro Diffusive Transport of a Fluid Component

80

and: grad ρ˜ (1) · n = −grad ρ (0) · n at I s f Z

X

(3.73)

Taking the average of (3.72) yields: f

grad ρ (0) · v(0) = 0

(3.74)

x

f

Equation (3.74) states that the directions of the advection velocity V f = v(0) and of the macroscopic concentration gradient should be perpendicular. We note that the solution of (3.72) and (3.73) depends linearly on grad ρ (0) . ThereX fore, we look for the solution in the form ρ˜ (1) (Z , X) = χ (Z) · grad ρ (0) (X). The X differential system which characterizes the vector ﬁeld χ is:6 f

− Z χ + grad Z χ · v∗(0) + v∗(0) − v∗(0) = 0 and

grad Z χ · n + n = 0

at I s f (3.75)

It is now possible to derive the macroscopic mass ﬂuxes. If we restrict ourselves to the order of δ 0 , we obtain: jγ = −

Dγ Dγ (grad ρ˜ (1) + grad ρ (0) ) = − (1 + tgradZ χ) · grad ρ (0) Z X X L L

(3.76)

Finally, taking the average of (3.76) (with respect to Z) yields the macroscopic expression of the diffusive ﬂux: hom Ddi f = ϕ Dγ Tv f γ (0) hom γ γ with (3.77) j f = ϕ j = −Ddi f · grad ρ f x Tv = 1 + tgrad Z χ hom Both Ddi f and Tv in (3.77) display a formal similarity with the homogenized diffusivity Dhom and the tortuosity T deﬁned by (3.13) and (3.14) that were derived for the case of no advection. It could therefore be appealing to consider Tv in (3.77) as the tortuosity tensor. However, there is a major difference: in contrast to the tortuosity tensor, χ and Tv in (3.77) depend not only on the geometry of the pore space, but also on the advection velocity. In return, the tortuosity T appears as a limiting case of Tv for a zero advection velocity. Indeed, if we eliminate the advection velocity in (3.75), we obtain a periodic

6 In fact, due to (3.74), only the components of χ in the directions perpendicular to V f are physically meaningful. Accordingly, only the components of (3.75) in these directions are relevant. Nevertheless, for further use (see Section 3.3.3), it is convenient to introduce a (ﬁctitious) component of χ in the direction of Vf.

Double-Scale Expansion Technique

81

estimate of the tortuosity tensor:7 lim Tv = T0 = I + tgrad Z χ 0

f

(3.78)

v→0

where χ 0 is the scalar ﬁeld solution of: − Z χ 0 = 0 and

grad Z χ 0 · n + n = 0

at I s f

(3.79)

γ v f , we Finally, in order to derive the macroscopic dispersive mass ﬂux ρ γ = use the ﬁrst term in the expansions of ρ γ and v, i.e. ρ δ (1) + O(δ 2 ) and (0) v = v + O(δ). This leads to: ⎧ f hom ⎨ρ ργ v = −Ddisp γ v f = ϕ · grad ρ (0) x (0) γ (0) f ρ v = δv ⊗ χ · grad ρ =⇒ hom (0) X ⎩ Ddisp = −ϕd v ⊗ χ (3.80)

hom Ddisp

is a function of the As expected, the homogenized dispersive tensor (0) advection velocity. Recalling that v is on the order of Dγ/d, a comparison of (3.77) and (3.80) shows that the advective and dispersive ﬂuxes now have the same order of magnitude. This is a second major difference with the case of moderate advection for which the dispersive term is negligible compared to the diffusive one. The Non-Homogenizable Case: V = O(v2 = Dγ L/d 2 ), Pe = O(δ −1 )

The last velocity regime we consider is deﬁned by v = v2 v∗ , for which (3.66) takes the form: 1 1 div X grad ρ γ ∗ + div Z grad ρ γ ∗ + X ρ γ ∗ − 2 Zρ γ ∗ + Z X δ δ 1 1 (3.81) + v∗ · 3 grad ρ γ ∗ + 2 grad ρ γ ∗ = 0 Z X δ δ Returning to the ‘true’ spatial variable z , it is readily seen that (3.81) yields: v(0) · grad ρ γ = 0 z

(3.82)

Relation (3.82) implies that the density ρ γ is constant along the streamlines of the advective ﬂux at the microscopic scale. Hence, instead of being determined as a function of the local advective velocity and of the macroscopic gradient 7 We will revisit the case of moderate advection in the training set of Chapter 4 (Section 4.5) within the framework of homogenization of heterogeneous media based on the solution of a local boundary value problem on an rev, to obtain a micromechanical deﬁnition of the tortuosity tensor using the concept of concentration tensors.

82

Micro-to-Macro Diffusive Transport of a Fluid Component

grad ρ γ , the density ﬁeld in the elementary cell is directly determined by the x boundary conditions that prevail at the edges of the macroscopic structure; that is, there is no length scale involved other than the macroscopic one.8 As a practical consequence, an experiment performed under such conditions cannot provide intrinsic overall properties of the porous material. 3.3.3 Transient Conditions We now consider the variations in time of the concentration ρ γ . For simplicity, it is still assumed that the advection velocity ﬁeld v is time independent. We concentrate on the case of advection-dominated velocity (v1 = O(Dγ /d)). It is useful to introduce a ﬁrst characteristic time deﬁned by tv = L/v1 . It represents the order of magnitude of the time for crossing the macroscopic structure with characteristic length L. As v1 = O(Dγ /d), we note that tv = tD δ, where tD = L 2 /Dγ is the characteristic time of diffusion at the scale of the macroscopic structure. In turn, the characteristic time for describing the variations of the concentration is denoted by tc . It corresponds to the time scale of the variations of the concentration at the boundary of the macroscopic structure. Returning to (1.31), we have to insert an additional term equal to the derivative ∂ρ γ /∂t into (3.66): Dγ 1 1 1 ∂ρ γ γ γ γ γ div X grad ρ + div Z grad ρ + X ρ − 2 Zρ + Z X tc ∂τ L δ2 δ 1 1 (3.83) grad ρ γ + grad ρ γ = 0 + v· Z X L δ where τ is the non-dimensionless time t/tc . As τ is assumed to be O(1), it is important to emphasize that the evolution equations that are derived in the sequel are valid for a time interval on the order of tc . We now have to discuss the relative order of magnitude of tc /tv . To this end, we let: tc = δ α tv = δ α+1 tD

with tD =

L2 Dγ

(3.84)

Inserting (3.84) into (3.83), we obtain: 1 1 1 ∂ρ γ γ γ γ γ div X grad ρ + div Z grad ρ + X ρ − 2 Zρ + Z X δ α+1 ∂τ δ δ 1 1 γ γ ∗ =0 (3.85) + v · 2 grad ρ + grad ρ Z X δ δ 8 This

situation is called a non-homogenizable conﬁguration, and was identiﬁed by Auriault and Lewandowska [3].

Double-Scale Expansion Technique

83

which generalizes (3.69) to transient conditions. We can distinguish the following three scenarios: 1. If α ≤ −1 (tc ≫ tv ), it is readily seen that the discussion of Section 3.3.2 for the case of advection-dominated transport is still relevant. From a mathematical point of view, this comes from the fact that the time derivative in (3.85) does not arise in the ﬁrst two differential problems derived at the order of δ −2 and δ −1 respectively. In this case, (3.74) remains valid as well as the determination of the average diffusive and dispersive ﬂuxes given by (3.77) and (3.80). The physical meaning of (3.74) is that the macroscopic (i.e. average) concentration ρ (0) is uniform on a macroscopic streamline. In other words, the changes in concentration at the boundary of the macroscopic structure are instantaneously propagated within the medium. This is due to the fact that the time scale tv of advection is much smaller than the time scale of these changes. 2. If α = 0, the two characteristic times tc and tv are of the same order of magnitude. In this case, the time derivative appears in the differential problem derived at the order of δ −1 , which is now: ∂ρ (0) − Z ρ˜ (1) + v∗(0) · grad ρ˜ (1) + v∗(0) · grad ρ (0) = 0 Z X ∂τ

(3.86)

together with: grad ρ˜ (1) · n = −grad ρ (0) · n at I s f Z

X

(3.87)

Taking the average of (3.86) (with respect to Z) yields: f ∂ρ (0) + grad ρ (0) · v∗(0) = 0 X ∂τ or, in terms of dimensional variables:

(3.88)

f ∂ρ (0) + grad ρ (0) · v(0) = 0 x ∂t

(3.89)

which replaces (3.74). Equation (3.89) represents the dominating order in the mass balance equation of the solute γ . Combining (3.86) and (3.88) provides a differential problem on ρ˜ (1) in which the time does not appear explicitly: f (1) ∗(0) ∗(0) (1) ∗(0) · grad ρ (0) = 0 (3.90) − Z ρ˜ + v · grad ρ˜ + v − v Z

X

Equation (3.90) needs to be solved for the boundary condition (3.73). The difference between (3.90) and (3.72) lies in the last term, because condition (3.74) no longer holds. However, the determination of the solution to (3.90)–(3.73) follows the same reasoning as in the case α ≤ −1. We look for

Micro-to-Macro Diffusive Transport of a Fluid Component

84

ρ˜ (1) in the form ρ˜ (1) = χ · grad ρ (0) , and χ appears as the solution to (3.75). X Once again, the average diffusive and dispersive ﬂuxes are given by (3.77) and (3.80). On the other hand, in contrast to the case α ≤ −1, we recognize that the macroscopic concentration gradient grad ρ (0) may not be perX pendicular to the average velocity v(0) f . This remark is illustrated in the forthcoming training set. 3. If α = 1 (tc ≪ tv ), the time derivative appears in the differential problem derived at the order of δ −2 : ∂ρ γ (0) − Z ρ˜ (0) + v∗(0) · grad ρ˜ (0) = 0 Z ∂τ

(3.91)

together with: grad ρ˜ (0) · n = 0 Z

at I s f

(3.92)

Taking the average over the ﬂuid phase (with respect to Z) yields: ∂ (0) f (ρ ) = 0 ∂t

(3.93)

This equation states that the macroscopic concentration is not able to vary at a time scale on the order of tc . As a consequence, the changes in concentration that take place at this time scale at the boundary of a macroscopic structure do not have enough time to propagate within the structure.

3.4 Training Set: Multilayer Porous Medium To illustrate the theory presented above, consider a porous medium made up of parallel solid layers forming a periodic structure. The domain between two consecutive solid layers of thickness h is ﬁlled with ﬂuid. Let d be the period in the direction perpendicular to the layers (see Figure 3.3). The ratio ϕ = h/d is recognized as the porosity. The classical solution to the Poiseuille ﬂow between two parallel layers (differential problem (2.12)–(2.14)) takes the

z2 d

h z1

Figure 3.3 Periodic multilayer porous medium

Training Set: Multilayer Porous Medium

85 Z2

ϕ/2

Z1

1

1

Figure 3.4 Elementary cell of a multilayer porous medium

form: v(z) =

α 2μ f

h2 − z2 2 e 1 4

(3.94)

where the pressure gradient is uniform in the ﬂuid and is grad p = −αe 1 . z Introducing the non-dimensional variable Z = z/d, we consider the elementary cell of a cross section which coincides with the unit square of Figure 3.4. With the notation introduced previously, the ﬂuid velocity ﬁeld is written in the form v(Z) = v1 v∗ (Z), where: αd 2 V ϕ2 2 ∗ v (Z) = − Z2 e 1 with V = (3.95) v1 4 2μ f

We assume that V is on the order of v1 = Dγ /d (advection-dominated transf port, V/v1 = O(1)). The average ﬂuid velocity v∗ is given by: V ϕ2 1 +ϕ/2 ∗ f (3.96) v∗ = e v (Z) dZ2 = ϕ −ϕ/2 v1 6 1

We also assume that the ratio tc /tv deﬁned in (3.84) is at least on the order of O(1) (α = −1 or α = 0). We thus apply the general methodology presented in Section 3.3.3. Due to the particular morphology of the multilayer, the macroscopic concentration gradient must be parallel to e 1 . Therefore, the boundary condition (3.73) is: grad ρ˜ (1) · n = 0 at I s f Z

(3.97)

Micro-to-Macro Diffusive Transport of a Fluid Component

86

that is: ∂ ρ˜ (1) =0 ∂ Z2

ϕ Z2 = ± 2

(3.98)

Let H = ∂ ρ˜ (0) /∂ X. We look for ρ˜ (1) in the form9 ρ˜ (1) = Hχ (Z2 ). The general differential problem (3.75) reduces here to: +ϕ/2 V ϕ2 ∂χ ϕ ∂ 2χ 2 = 0 and χ dZ2 = 0 − Z2 = 0 with ± − 2+ v1 12 ∂ Z2 2 ∂ Z2 −ϕ/2 (3.99) The solution is: 7 4 V 1 2 2 4 ϕ Z2 − 2Z2 − χ= ϕ (3.100) 24 120 v1 According to (3.77), the generalized tortuosity coefﬁcient Tv is given by: 1 +ϕ/2 ∂χ Tv = 1 + d Z2 = 1 (3.101) ϕ −ϕ/2 ∂ Z2 In other words, the advection velocity does not affect the macroscopic diffusive ﬂux. Inserting the expressions (3.96) and (3.100) for v∗ and χ into (3.80) yields hom the dispersion coefﬁcient Ddisp : hom Ddisp =

d 2ϕ7 dϕ 7 V 2 = V2 7560 v1 7560Dγ

(3.102)

hom depends on the square of the advection velocity. However, We note that Ddisp it is recalled that this result holds only if V = O(v1 ).

3.5 Concluding Remarks When the P´eclet number Pe is on the order of δ, the dispersive effects are f negligible at the macroscopic scale. In other words, the term ρ˜ γ v˜ in (3.63) can be disregarded. Moreover, the macroscopic diffusive ﬂux J γ , interpreted at the microscopic scale as j γ f , can be determined from the macroscopic concentration gradient through the concept of tortuosity, as in (3.13) or (3.62). f When the P´eclet number is on the order of 1, the dispersive term ρ˜ γ v˜ is no longer negligible and can be determined from the macroscopic concentration gradient using (3.80). This equation introduces the concept of homogenized 9 Note

that the periodicity implies that χ cannot depend on Z1 .

Concluding Remarks

87

dispersion tensor which is derived from the microscopic advection velocity ﬁeld. Moreover, this advection velocity also affects the relationship between the macroscopic diffusive ﬂux and the macroscopic concentration gradient (3.77). The fact that the advection velocity is involved in the expression for Dhom dif means that the geometrical concept of tortuosity is no longer relevant. The traditional phenomenological approach introduces an effective hydrodynamic dispersion tensor Deff which is supposed to take into account both the diffusive and the dispersive ﬂux: γ v f = −Deff · grad ρ γ j γ f + ρ

(3.103)

hom Deff = Dhom dif + Ddisp

(3.104)

x

The micromechanical approach yields the following expression for Deff :

hom As stated above, both Dhom dif and Ddisp depend on the advection velocity. By contrast, the phenomenological approach consists in correcting the geometrical effects described by the tortuosity tensor by an additional term proportional to the macroscopic advection velocity. Letting V = |V f | and U = V f /V, the classical form of Deff is: 10 (3.105) Deff = ϕ Dγ T + V αT 1 + (α L − αT )U ⊗ U

where α L and αT are two constants which are supposed to capture the dispersive effects respectively in the direction of the ﬂow and in the plane perpendicular to the ﬂow. They are often referred to as longitudinal and transversal dispersivities of the porous medium. Relation (3.105) implicitly assumes that there is no coupling between geometrical effects (tortuosity) and dispersive effects. The micromechanical approach reveals that this assumption is not justiﬁed from a theoretical point of view. It is also worth observing that the dispersion coefﬁcient obtained in the training set (Section 3.4) depends on the square of the advection velocity in the domain of validity of its derivation. 10 See

Bear and Bachmat [6].

Part II Microporoelasticity

4 Drained Microelasticity The next few chapters deal with microporoelasticity; that is, the micromechanics of a porous material composed of an elastic solid and some ﬂuid phases. This chapter addresses the case of a poroelastic material ‘emptied’ of the saturating ﬂuid phases. This conﬁguration corresponds to the drained situation. For this case, the fundamental steps of the homogenization theory are developed that ultimately lead to the ‘macroscopic’ constitutive equation from the solution of a boundary value problem deﬁned on a representative elementary volume considered as a material system. Our starting point is the hollow sphere model, in which the inner sphere represents the porosity and the outer sphere the solid, and which is the simplest geometrical and mechanical model of a porous material. This 1-D thought model has all the ingredients of the homogenization theory based on the solution of a boundary value problem, which is thereafter generalized into a comprehensive microporomechanics theory. This theory is sufﬁciently general to treat a wide range of linear upscaling problems: effective elastic properties, effective diffusion coefﬁcients, and so on.

4.1 The 1-D Thought Model: The Hollow Sphere The hollow sphere model is the simplest geometrical representation of a porous material (Figure 4.1). The cavity of inner radius A represents the pore space, the outer part r ∈ [A, B] represents the solid phase, with B the outer radius. The volume fraction ϕ0 = (A/B)3 corresponds to the initial porosity. Given the spherical symmetry of this geometrical representation, it is readily understood that this thought model of the microstructure of a porous material is restricted to macroscopically isotropic porous materials. This macroscopic isotropy requires the isotropy of the solid material behavior, which will be assumed linear isotropic elastic (bulk and shear moduli k s and μs ). We hereafter consider a loading which complies with the spherical symmetry, namely Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

Drained Microelasticity

92

T = Σer

2A

2B

Figure 4.1 The hollow sphere model–stress boundary conditions

an isotropic traction. Non-isotropic loading cases will be treated in Section 4.3.1. The purpose of the homogenization approach is the determination of the macroscopic behavior from the solution of the boundary value problem deﬁned on the hollow sphere material system subjected to such an isotropic loading. 4.1.1 Macroscopic Bulk Modulus and Compressibility The isotropic load we consider is a uniform tension on the external boundary r = B. The corresponding stress tensor representing the macroscopic stress state of the hollow sphere is Σ = 1, where 1 is the second-order unit tensor (or Kronecker delta). This loading complies with the spherical symmetry, and induces only a radial deformation of the sphere, whence the 1-D thought model. If we denote by b = λB the external boundary in the deformed conﬁguration, the macroscopic deformation of the entire sphere is described by the volumetric strain tensor E = (λ − 1)1. The link between the macroscopic stress Σ = 1 and the macroscopic strain E = (λ − 1)1 is characterized by the function λ = λ(). From a purely macroscopic standpoint, this function is related to the macroscopic bulk modulus k hs of the hollow sphere: k hs =

d d = d(tr E) 3 dλ

(4.1)

By contrast, from a micromechanics standpoint, λ() is also a function of both geometrical and mechanical parameters of the hollow sphere problem, i.e. ϕ0 , k s and μs . The aim of the homogenization approach is to determine λ (, ϕ0 , k s , μs ) (and thus k hs ) from the solution of the boundary value problem

The 1-D Thought Model: The Hollow Sphere

93

of the solid phase: (a ) divσ = 0 (b) σ = Cs : ε 1 (c) ε = (grad ξ + tgrad ξ ) 2 (d) σ · e r = e r (r = B) (r = A) (e) σ · e r = 0

(4.2)

Here, σ, ε and ξ denote the stress, strain and displacement ﬁelds within the solid. We will refer to these ﬁelds as microscopic ﬁelds. Given the spherical symmetry of the problem, the displacement ﬁeld is radial. In the spherical coordinates system, the displacement ﬁeld solution to the boundary value problem (4.2) is: 1 A3 1 ξ (r ) = ξr e r with ξr (r ) = r (4.3) + 1 − ϕ0 3k s 4μs r 3

which immediately yields the sought function λ = λ (, ϕ0 , k s , μs ): ϕ0 1 ξr (B) =1+ + λ=1+ B 1 − ϕ0 3k s 4μs

(4.4)

Finally, the macroscopic bulk modulus is obtained from (4.1): k hs = k s

4μs (1 − ϕ0 ) 3ϕ0 k s + 4μs

The stress ﬁeld solution to (4.2) is given by σ = σ hs , with: A3 1 1 hs σ = 1 − 3 e r ⊗ e r − (e θ ⊗ e θ + e ϕ ⊗ e ϕ ) 1 − ϕ0 r 2

(4.5)

(4.6)

σ hs therefore corresponds to the stress solution associated with the unit macroscopic stress ( = 1). The 1-D thought model shows the principle of a homogenization approach based on the solution of a particular boundary value problem of a particular geometrical representation of the microstructure of a material. This homogenization approach allows the determination of macroscopic properties as a function of geometrical and mechanical microscopic parameters. For the particular case of a porous material represented as a hollow sphere, it is interesting to modify the expression (4.5) of k hs to highlight some particular features related to the cavity. These are the reduction of the overall stiffness due to the cavity: 3k s + 4μs hs s (4.7) k = k 1 − ϕ0 3ϕ0 k s + 4μs

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and, vice versa, the increase of the overall compressibility (bulk compliance) due to the cavity: 1 ϕ0 3 1 1 (4.8) = s + + k hs k 1 − ϕ0 k s 4μs 4.1.2 Model Extension to the Cavity In the previous developments, the boundary value problem is deﬁned on the solid phase only. For further generalization, it is convenient to regard the hollow sphere model as a heterogeneous material system composed of two elastic bodies. In addition to the solid phase, the empty spherical pore is considered also as an elastic body of stiffness C p that is much smaller than the stiffness of the solid Cs . In this case, the boundary value problem is deﬁned on the whole sphere, as follows: (a ) divσ = 0 (b) σ = C(r ) : ε; with

C(r ) =

1 (c) ε = grad ξ + tgrad ξ 2 (d) σ · e r = e r

Cs forA < r < B C p ≪ Cs for r < A

(4.9)

(r = B)

It is readily seen that the hollow sphere is a limit of the two-phase model deﬁned by (4.9). The condition σ = 0 in the cavity is obtained asymptotically through (4.9b) for C p → 0. The boundary condition (4.2e) disappears, and is implicitly expressed through the continuity of the stress vector at the surface r = A. On the other hand, while the local displacements and strains in the cavity depend on the cavity stiffness C p , the average strain in the cavity is controlled by the displacement in the solid phase at the solid–cavity interface r = A. Provided that C p ≪ Cs , this interface displacement does not depend on the stiffness C p . As a consequence, a ﬁctitious displacement ﬁeld in the cavity can be deﬁned as an extension of the solid displacement, namely the one associated with C p . Although this extension is not unique, the associated average strain is unambiguously deﬁned. We keep this in mind for the generalization of the theory. 4.1.3 Energy Point of View We have introduced the apparent bulk modulus in an ad hoc fashion through the macroscopic relation (4.1). It is useful to support this deﬁnition through an energy argument. The energy d W supplied to the hollow sphere during an incremental process is the work of the isotropic tension applied at the

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95

boundary, along the macroscopic volume change d: dW = d

(4.10)

For a linear material, the relative volume change d/ || is linearly related to the tension increase by the compressibility 1/k hs , i.e. d/ || = d/k hs , and the total energy W stored in the sphere when the tension is increased from 0 to is: || 2 W= d = || (4.11) k hs 2k hs Expression (4.11) represents the energy stored in the solid phase of the hollow sphere, but at the same time it is the energy stored in a sphere composed of a homogeneous material with bulk modulus k hs that is subjected at its boundary to a tension . Hence, instead of (4.1), we could have used (4.11) to deﬁne the macroscopic bulk modulus. Both deﬁnitions are strictly equivalent. 4.1.4 Displacement Boundary Conditions The homogenization model of the hollow sphere was obtained by solving a boundary value problem deﬁned by uniform stress boundary conditions. It is readily understood that the same result is obtained by appropriate displacement boundary conditions (Figure 4.2). Indeed, it sufﬁces to consider a uniform radial displacement ξr = E × B/3 on the surface r = B which corresponds to a macroscopic strain tensor E = E/31.1 The corresponding boundary value problem reads (a ) divσ = 0 (b) σ = C(r ) : ε;

with

C(r ) =

Cs for A < r < B C p ≪ Cs for r < A

1 (c) ε = (grad ξ + tgrad ξ ) 2 (d) ξ = 13 E × Be r (r = B)

(4.12)

The solution of (4.12) takes the form ξ = 31 EBξ hs (Figure 4.2) where ξ hs denotes the solution corresponding to a unit radial displacement e r prescribed on the boundary r = B: r ∈ [A, B] : ξ hs = ξ hs (r )e r

1

with

ξ hs (r ) =

r 3k s A3 /r 3 + 4μs B 3k s ϕ0 + 4μs

E is related to the macroscopic stretch λ according to E/3 = λ − 1.

(4.13)

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ξ = 1 EB e r 3

2A

2B

Figure 4.2 The hollow sphere model–displacement boundary conditions

Comparing (4.3) and (4.13), it appears that (4.9) and (4.12) have the same solution provided that E and are related by: = k hs tr E = k hs E

(4.14)

Finally, combining (4.11) and (4.14) yields an expression of the elastic energy W, as a function of the macroscopic strain: 1 W = k hs (tr E)2 | | 2

(4.15)

4.2 Generalization The study of the 1-D thought model of the hollow sphere has all the ingredients of a comprehensive homogenization theory based on the solution of a boundary value problem. This homogenization theory can be broken down into the following components: 1. Geometrical and mechanical modeling of the material at the microscopic scale. In the hollow sphere model, the solid is linear elastic and isotropic. The geometry captures only the porosity. A comprehensive micromechanics theory will have to reﬁne the microscopic representation of the porous material in the framework of a representative elementary volume (rev), as introduced in Section 1.1. 2. Deﬁnition of an appropriate mechanical loading on the rev in the form of a boundary value problem. In the hollow sphere model, we have already seen uniform stress boundary conditions (i.e. (4.9d)) and uniform displacement

Generalization

97

boundary conditions (i.e. (4.12d)). A more general theory, however, is required to address non-isotropic load cases. 3. Determination of the response of the rev at the microscopic scale. This is the so-called localization step. 4. Determination of the macroscopic behavior from the mechanical response of the rev subjected to this loading. 4.2.1 Macroscopic and Microscopic Scales The concept of macroscopic constitutive behavior aims at describing the geometrical transformation of an elementary volume as a consequence of forces applied to the elementary volume from the outside. The constitutive law links the macroscopic stress tensor Σ which represents these forces to the macroscopic strain tensor E which characterizes the transformation of the elementary volume at the macroscopic scale. There are two possible ways to determine the constitutive law. The purely macroscopic approach is of a phenomenological nature. It is based on experiments performed in the laboratory, in which the sample plays the role of an elementary volume. Either the stress Σ or the strain E is applied to the sample, and the other quantity is measured. An alternative approach is advanced by homogenization theory: instead of observing the mechanical response of the sample subjected to a certain evolution of Σ or E, the homogenization approach aims at deriving it from the solution of a boundary value problem deﬁned on the rev. In other words, the real experiment performed on a real sample is replaced by a theoretical experiment performed mathematically on a theoretical sample. The very use of a theoretical material sample originates from the rev concept introduced in Section 1.1. The sample is assumed to satisfy the scale separation condition (1.1). In contrast to the macroscopic approach, the homogenization theory considers the heterogeneous structure of the elementary volume, thus considering the material system at a scale that reveals its heterogeneity. We have already seen in Section 4.1 how to predict the mechanical response of a model sample (hollow sphere) with model microstructure (isotropic solid and porosity) to a model loading (uniform isotropic tension or strain), in which we made use, for purposes of clarity, of the geometrical symmetry of the hollow sphere. To remove these restrictions, we now assume that neither the geometry of the rev, nor the macroscopic loading possess such symmetries. Instead of (4.9d) and (4.12d), we look for a general formulation of a mechanical problem on the rev, which mathematically translates the fact that the rev is subjected to a macroscopic stress Σ or a macroscopic strain E. Once this boundary value problem is deﬁned, the corresponding microscopic strain and stress ﬁelds ε(z) and σ(z) are derived. Finally, the microscopic solution is

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expressed in terms of macroscopic variables. We seek an appropriate expression for the macroscopic strain E associated with the microscopic strain ﬁeld ε(z), when the rev is subjected to Σ as the loading parameter. If the loading of the rev is deﬁned by strain E, we seek an appropriate expression that links the macroscopic stress Σ to its microscopic counterpart σ(z). In this chapter, we assume that the deformation of the rev is inﬁnitesimal at both the microscopic and macroscopic scales. Moreover, the displacements in the rev are small, so that the geometry changes induced by the mechanical loading do not introduce mathematical nonlinearities. In particular, the variations of the pore volume fraction are inﬁnitesimal as well: ϕ ≈ ϕ0 . 4.2.2 Formulation of the Local Problem on the rev We developed the link between the microscopic and macroscopic internal forces in Section 1.3.2, which is brieﬂy recalled here. At the microscopic scale, the momentum balance equation is: ∂σi j (4.16) divz σ + ρ(z) f (z) = + ρ fi e i = 0 ∂z j where σ stands for the microscopic stress ﬁeld in the rev and ρ(z) f (z) is the microscopic volume density of external body forces. It is recalled that the differentiation in the divergence operator divz is performed on the microscopic coordinates zi . In turn, at the macroscopic scale, the momentum balance equation is: ∂i j (4.17) divx Σ + ρ M(x)F (x) = + ρ M Fi e i = 0 ∂xj where Σ denotes the macroscopic stress ﬁeld. It balances the macroscopic external body forces represented by the volume force density ρ M(x)F (x). The differentiation deﬁned by the divergence operator divx is performed on the macroscopic coordinates xi . By deﬁnition, the external body force which is applied on the rev located at the macroscopic point x can be expressed either as ρ M F (x)|| or ρ f (z) dVz . The vector F (x) and the vector ﬁeld f (z) are therefore related by the average rule ρ M(x)F (x) = ρ f (see (1.40)). Similarly, the stress tensor Σ(x) and the stress ﬁeld σ(z) represent, at the macroscopic and the microscopic scale, internal forces in the rev. It is on this basis that we derived in Chapter 1 the link between the microscopic stress ﬁeld in the rev and the corresponding macroscopic stress tensor through the average rule (1.39), which we recall: Σ=σ

(4.18)

Generalization

99

Asymptotic Mechanical Model of the Pores

We will adopt the same approach as in Section 4.1.2 to account for empty pores within the rev. The pore space is considered as an elastic material with vanishing stiffness values, respectively with very large compliance values. If we denote by C(z) the local elasticity tensor at the microscopic point z, and by S(z) = C(z)−1 the compliance tensor, we can write: z ∈ p z ∈ s

C(z) = C p ≪ Cs C(z) = Cs

S(z) = S p ≫ Ss S(z) = Ss = (Cs )−1

(a ) (b)

(4.19)

where for purposes of clarity the solid is assumed homogeneous. From (4.19), the real porous medium can be deﬁned as the limit case C p → 0 or S p → ∞, for which the microscopic stress σ is zero in the pore domain p . On the other hand, the local strains in p depend on the choice of C p , but not the average strain in p , which depends only on Cs and E. This is readily shown if we approximate p by a bounded domain delimited by a closed surface which constitutes the solid–pore interface. Denoting by n the outward unit normal to p , the integral of the displacement derivatives in p is: ∂ξi dVz = ξi n j dSz (4.20) p ∂z j ∂ p from which we obtain the average strain in p in the form:

1 1 p ξ ε dV = dSz ε = ⊗ n + n ⊗ ξ z | p | p 2| p | ∂ p

(4.21)

Equation (4.21) proves that the average strain in the pore domain is controlled by the value which the displacement in the solid phase takes at the solid–pore interface, which is (asymptotically) independent of C p . This generalizes the reasoning of Section 4.1.2. The Question of Volume Forces

Can one omit, or not, the body forces in the local problem formulation? That is, in the set of equations that the microscopic stress, displacement and strain ﬁelds σ, ξ and ε need to satisfy:2 divσ + ρ f = 0 σ = C(z) : ε ⇔ ε = S(z) : σ Σ=σ

(a ) (b) (c)

(4.22)

2 From now on, the divergence operator at the microscopic scale is simply denoted by div, omiting the subscript z.

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To answer this question, let us assume that the macroscopic stress ﬁeld is induced in the macroscopic structure by the body forces ρ M F . The order of magnitude of Σ, therefore, is |ρ M F | × L, where L is the characteristic length of the macroscopic structure. Let us now consider the microscopic stress ﬁeld σ. Its variations within the rev are due, on the one hand, to the heterogeneity of the rev and, on the other hand, to the existence of the body forces ρ f . More precisely, the origin of the heterogeneity is the existence of pores where the microscopic stress is σ = 0. Thus, the order of magnitude of the ﬂuctuations induced by the material heterogeneity is that of Σ. In turn, the order of magnitude of the ﬂuctuations of the stress ﬁeld σ induced at the microscopic level in the rev by the density ρ f is |ρ f | × ℓ, where ℓ is the characteristic length of the rev. Therefore, due to the scale separation rule, which implies that ǫ = ℓ/L ≪ 1, the mechanical effects of the local body forces ρ f can be neglected, at the scale of the rev, with regard to the effects of the heterogeneity, i.e. formally: ǫ = ℓ/L ≪ 1 ⇒| Σ |≈ |ρ M F | × L ≫ |ρ f | × ℓ

(4.23)

where, according to (1.40), we made use of the fact that |ρ M F | and |ρ f | have the same order of magnitude. Consequently, we can omit in the local problem formulation (4.22) the local body forces: divσ = 0 σ = C(z) : ε ⇔ ε = S(z) : σ Σ=σ

(a ) (b) (c)

(4.24)

More generally, neglecting the body forces in the local problem formulation is valid provided that | Σ | ≫ |ρ f | × ℓ. This assumption (and it is one) is not only relevant for the homogenization approach. It is frequently admitted in laboratory experiments in which the effect of gravity is omitted. 4.2.3 Uniform Stress Boundary Condition We still face serious difﬁculties if we attempt to solve (4.24). As it stands, the local problem formulation is an ill-posed problem: it has no unique solution, without additional constraints, such as periodicity conditions or other relevant boundary conditions. The underlying idea which is explored here is that the rev is subjected to the macroscopic stress state represented by the tensor Σ as a stress boundary condition. This leads to the so-called uniform stress boundary condition: (∀z ∈ ∂) σ(z) · n(z) = Σ · n(z)

(4.25)

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101

where n(z) is the outward unit normal to the boundary of the rev at point z. We have implicitly assumed such a uniform stress boundary condition in the hollow sphere thought model, for which (4.25) reduces to (4.2d), if we let Σ = 1. We ﬁrst need to check the compatibility of (4.25) with the average rule Σ = σ (see (4.18)). The starting point of this investigation is the property: ∂ z j σik = σi j ∂zk

(4.26)

which is readily derived from the momentum balance condition (4.24a). The integration of (4.26) over the whole rev yields: σi j dVz = z j σik nk dSz = z j ik nk dSz (4.27)

∂

∂

where (4.25) has been used. Integrating by parts, we obtain: ∂z j dVz = δ jk || z j nk dSz = ∂ ∂zk

(4.28)

If we note that the stress tensor Σ does not depend on the microscopic position vector z, use of (4.28) in (4.27) yields the average rule: σi j dVz = ||i j (4.29)

The intrinsic form of (4.29) is Σ = σ, which holds for any stress ﬁeld σ satisfying both the local equilibrium condition (4.24a) and the uniform stress boundary condition (4.25). It is on this basis that (4.24) can be restated as the following boundary value problem in the form of a standard problem of linear elasticity: divσ = 0 σ = C(z) : ε ⇔ ε = S(z) : σ σ(z) · n(z) = Σ · n(z) on ∂

(a ) (b) (c)

(4.30)

Solving (4.30) yields the response of the rev subjected to the stress state Σ at its boundary. This response is characterized by the microscopic stress ﬁeld σ, displacement ﬁeld ξ and strain ﬁeld ε. We are left with developing the macroscopic counterpart of this microscopic description, and seek to deﬁne the macroscopic strain tensor E which is associated with the microscopic strain ﬁeld ε(z). The link between E and ε(z) is conveniently derived from an energy approach, namely from the observa˙ supplied to the rev must coincide tion that the macroscopic rate of energy W with the microscopic energy rate. At the macroscopic scale, the rate of energy ˙ = Σ : E˙ | |, where E˙ is the macroscopic strain rate supplied to the rev is W

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associated with the considered evolution of the rev. In terms of microscopic variables, the same physical quantity is: ˙ = (4.31) W ξ˙ · σ · ndSz ∂

Furthermore, inserting the boundary condition (4.30c) in (4.31) yields: ∂ ξ˙i ˙ ξ˙i n j dSz = i j || = Σ : ε|| W = i j ˙ (4.32) ∂z j ∂ where the symmetry of Σ has been used. The compatibility condition thus reads E˙ = ε, ˙ or after integration with respect to time: (4.33)

E=ε

In summary, the stress average rule Σ = σ is required in order to ensure the compatibility between the microscopic and macroscopic momentum balance equations. It can be derived from the boundary condition (4.30c). The strain average rule E = ε, adopted as a deﬁnition, ensures the compatibility between the microscopic and macroscopic deﬁnitions of the rate of mechanical energy. 4.2.4 An Instructive Exercise: Capillary Pressure Effect The proof of the stress average rule from (4.25) developed above is based on the following assumptions:

r The momentum balance equation divσ = 0 is satisﬁed. r The stress ﬁeld is zero in p . r The stress vector is zero at the solid–pore interface ∂ p . These conditions are strictly satisﬁed in the case of a porous material with empty pores. They also hold for a porous material with pores ﬁlled with a ﬂuid at zero pressure, but only if the surface tensions at the solid–ﬂuid interface are zero as well. This is not the case when a capillary pressure develops in porous materials, with surface tensions occurring at the solid–pore interface, in which case the stress vector is not continuous along ∂ p . This question will be considered in detail in Section 8.1. Henceforth, the focus of this exercise is to derive the expression of the macroscopic stress tensor for this situation. For analytical purposes, we will consider the situation of a spherical interface of radius R, for which the stress vector discontinuity is given by Laplace’s law: [σ] · n = −σ s · n = −

2γ s f n R

(4.34)

where γ s f is the surface tension, and n is the unit normal to the interface oriented towards the pore. Relation (4.34) shows that the stress vector applied to

Generalization

103

the solid is not zero, although the ﬂuid pressure is zero. In this case, the membrane tension at the capillary interface between the solid and the ﬂuid ﬁlling the pore space is responsible for a uniform traction on the solid boundary. We assume that the pore domain p is made up of N spherical pores Sα of radii Rα . The ﬁrst part of the average rule (4.27) is still valid: σi j dVz = σi j dVz = z j σik nk dSz + z j σik nk dSz (4.35)

s

∂ p

∂

Substituting (4.25) and (4.34) in (4.35) yields: 2γ s f z j ni dSz σi j dVz = z j ik nk dSz + Rα ∂Sα ∂ α

(4.36)

Consequently, using (4.28) in (4.36) entails the following relation between the involved microscopic, macroscopic and interface stress quantities:3 σi j = i j − δi j

α

ϕα

2γ s f Rα

(4.37)

where ϕα = |Sα |/||. Finally, we deﬁne the average curvature 1/R of the interfaces as: ϕα ϕ = (4.38) R α Rα where ϕ = α ϕα is the total pore volume fraction. The macroscopic stress Σ is the sum of two terms, the volume average of the microscopic stresses σ and a macroscopic capillary tension term: ϕ Σ = σ + 2 γ sf 1 R

(4.39)

4.2.5 Uniform Strain Boundary Condition The uniform stress boundary condition (4.30c) has been introduced in order to deﬁne a well-posed boundary value problem compatible with the average rule Σ = σ. Instead of (4.30c), we explore here the idea of a regular displacement boundary condition, while keeping the stress average rule Σ = σ in order to ensure the compatibility between (4.16) and (4.17). If the microscopic strain ﬁeld ε is uniform within the rev, it is equal to the macroscopic strain E and the displacement ξ is a linear function of the microscopic coordinates, i.e. ξ (z) = E · z (+ a rigid body motion). Let us now 3 Attention should be paid to the fact that the unit normal n in (4.36) is oriented outwards with respect to the solid.

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consider a heterogeneous rev, subjected to the boundary condition: (∀z ∈ ∂) ξ (z) = E · z

(4.40)

As opposed to the homogeneous case, the material heterogeneity is responsible for microscopic ﬂuctuations of the strain ﬁeld ε(z) within the rev. In return, such ﬂuctuations are not seen in a macroscopic approach, which regards the rev from the outside. It is therefore natural to consider E in (4.40) as the macroscopic strain tensor. Furthermore, (4.40) and integration by parts yield: ∂ξi dVz = ξi n j dSz = E ik zk n j dSz (4.41) ∂z j ∂ ∂ or equivalently,4 using (4.28): ∂ξi = E i j ⇔ grad ξ = ε = E ∂z j

(4.42)

Hence, the macroscopic strain E appears as the average of the heterogeneous microscopic strain ﬁeld. This conﬁrms that the ﬂuctuations of the microscopic strain ﬁeld are not detected at the macroscopic scale. The local problem deﬁning the loading to which the rev is subjected is then: divσ = 0 σ = C(z) : ε ⇔ ε = S(z) : σ ξ (z) = E · z on ∂

(a ) (b) (c)

(4.43)

In comparison to (4.30), the macroscopic strain tensor E has replaced Σ as the loading parameter.5 As in the case of the uniform stress boundary condition, we now want to ˙ compare the microscopic and macroscopic deﬁnitions of the rate of energy W. The microscopic deﬁnition is still given by (4.31). Using (4.43c), we obtain: ˙ = W (4.44) z j E˙ i j σik nk dSz ∂

where E˙ i j does not depend on the microscopic coordinates. In this case, (4.44) together with (4.26) and (4.43a) yields: ˙ ˙ ˙ W = Ei j z j σik nk dSz = E i j σi j dVz (4.45) ∂

Finally, if we introduce the average rule Σ = σ in (4.45), we recover the macro˙ = Σ : E˙ | |. This conﬁrms that the boundary condition scopic formulation W 4 Note 5 For

that E in (4.40) is a symmetric (second-order) tensor. the hollow sphere, the boundary condition (4.43c) reduces to (4.12d) if we let E = E1/3.

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105

(4.43c) together with the stress average rule ensures the consistency between ˙ the macroscopic and microscopic deﬁnitions of the rate of energy W. In summary, the strain average rule E = ε can be derived from the uniform strain boundary condition (4.40). In this case, the stress average rule is adopted as a deﬁnition. It ensures the compatibility between the microscopic and macroscopic formulations of the momentum balance law, as well as the consistency of the microscopic and macroscopic deﬁnitions of the rate of mechanical energy. 4.2.6 The Hill Lemma Many derivations in the homogenization theory are based on a remarkable consequence of the particular structure of the uniform boundary conditions (4.30c) and (4.43c), which is generally referred to as the Hill lemma.6 The idea consists in looking for an expression of the virtual work developed by a microscopic stress ﬁeld σ ∗ in a microscopic strain ﬁeld ε′ in terms of the averages of these stress and strain ﬁelds. More precisely, we look for an appropriate set of assumptions for which the following average rule is valid: σ ∗ : ε′ = σ ∗ : ε′

(4.46)

To begin with, the stress ﬁeld σ ∗ is supposed to satisfy the momentum balance condition divσ ∗ = 0. The strain ﬁeld ε′ is geometrically compatible, which means that it is the symmetric part of the gradient of a displacement ﬁeld ξ ′ . σ ∗ and ε′ are not a priori related by any constitutive law. Two different situations are considered: 1. We assume that σ ∗ complies with a uniform stress boundary condition: (∀z ∈ ∂) σ ∗ (z) · n(z) = Σ∗ · n(z) The virtual work developed by σ ∗ in ε′ is: ∗ ′ ′ ∗ ∗ σ : ε dVz = ξi σi j n j dSz = i j

∂

Integration by parts shows that: ′ ξi n j dSz = ∂

6 This

∂

(4.47)

ξi′ n j dSz

∂ξi′ dVz ∂z j

lemma can be proved to be also valid with periodic boundary conditions [47].

(4.48)

(4.49)

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Equation (4.48) thus reads:7 ∂ξ ′ σ ∗ : ε′ dVz = i∗j i | | = Σ∗ : ε′ ∂z j

(4.50)

Finally, (4.46) is derived from (4.50) because of the stress average rule σ ∗ = Σ∗ which is a consequence of (4.47). 2. We assume that ξ ′ complies with a uniform strain boundary condition: (∀z ∈ ∂) ξ ′ (z) = E′ · z

(4.51)

No assumption is made concerning the stress vector σ ∗ · n. The virtual work developed by σ ∗ in ε′ is: ∗ ′ ′ ∗ ′ σ : ε dVz = ξi σi j n j dSz = E i j z j σik∗ nk dSz (4.52)

∂

∂

Use of (4.26) in (4.52) yields: σ ∗ : ε′ dVz = E i′ j σi∗j | | ⇒ σ ∗ : ε′ = E′ : σ ∗

(4.53)

Finally, (4.46) is derived from (4.53) because of the strain average rule ε′ = E′ which is a consequence of (4.51). In summary, given uniform boundary conditions (4.30c) or (4.43c), the volume average of the microscopic strain energy is equal to the macroscopic strain energy. 4.2.7 The Homogenized Compliance Tensor and Stress Concentration With the Hill lemma in hand, we can proceed to derive the homogenized elasticity properties of porous materials. This section deals with the case of uniform stress boundary conditions. Macroscopic State Equation

We reconsider the local problem (4.30). It is readily understood that the microscopic stress ﬁeld solution σ(z) of this elasticity problem linearly depends on the tensorial loading parameter Σ. In other words, the local stress tensor σ(z) is a linear function of the components i j . This property can be expressed through the introduction of a stress concentration tensor: a fourth-order tensor, denoted here by B(z), which relates the local stress tensor σ(z) to the 7 Note

that Σ∗ is a symmetric tensor.

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107

macroscopic one Σ: (∀z ∈ ) σi j (z) = Bi jkl (z)kl ⇔ σ(z) = B(z) : Σ

(4.54)

Given the symmetry of both microscopic and macroscopic stress tensors, σ(z) and Σ, the components of B(z) satisfy the following symmetry conditions: Bi jkl (z) = B jikl (z) = Bi jlk (z)

(4.55)

Furthermore, for given index k and l, Bi jkl e i ⊗ e j is the microscopic stress ﬁeld which takes place in the rev when subjected to the macroscopic stress state: 1 Σ = (e k ⊗ e l + e l ⊗ e k ) (4.56) 2 Hence, Bi jkl e i ⊗ e j is the solution of an elasticity problem. We have already seen that the boundary condition (4.30c) is compatible with the stress average rule. Substituting (4.54) into (4.18) yields: σ=B:Σ=B:Σ

(4.57)

which holds for any choice of the macroscopic stress tensor. It follows that: (4.58)

B=I

where I is the fourth-order (symmetric) identity tensor of components in an orthonormal frame: 1 Ii jkl = (δik δ jl + δil δ jk ) (4.59) 2 In the case of an empty porous medium (as studied throughout this chapter), the microscopic stress σ is zero in the pore domain p , as is the stress concentration tensor B (B = 0 in p ). As in Section 1.2, we denote by e α the average of the quantity e over the α phase (α = s or p). Equation (4.58) thus takes the form: s

B = (1 − ϕ0 )B = I

(4.60)

where ϕ0 denotes the initial pore volume fraction (as in Section 4.1). Combining the microscopic state equation (4.30b) with the stress concentration rule (4.54), the stress concentration tensor B(z) also provides the link between the microscopic strain state at point z and the macroscopic stress state: ε(z) = S(z) : B(z) : Σ

(4.61)

The fact that B = 0 in p , however, does not imply that ε = 0 in the pores, because the tensor of compliance is inﬁnite in p (see (4.19)). On the contrary, we intuitively expect larger strains in the pores than in the solid phase. The last step of the homogenization process is the translation of the microscopic solution into macroscopic variables. According to the strain average

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108

rule E = ε adopted as a deﬁnition within the framework of the uniform stress boundary condition, relation (4.61) yields: E = S : B : Σ = Shom : Σ

(4.62)

Shom = S : B

(4.63)

with:

This linear relationship between E and Σ represents the macroscopic state equation. Shom is referred to as the macroscopic tensor of compliance. It is readily seen that the so-called ‘direct’ average rule which holds for stress or strain8 is not valid for the tensor of compliance. Indeed, since the local tensor of compliance is inﬁnite in p , application of such a direct average rule would imply S → ∞! In other words, the deviation to the direct average rule is controlled by the phenomenon of stress concentration, i.e. by the heterogeneity of the stress ﬁeld within the rev. More precisely, substituting (4.60) into (4.63) yields: s

p

Shom = (1 − ϕ0 )Ss : B + ϕ0 S : B = Ss + ϕ0 S : B

p

(4.64) p

This relation clearly shows the additional compliance ϕ0 S : B as a consequence of the presence of an empty pore space.9 We have already encountered this additional compliance in the study of the hollow sphere model (see (4.8)). Energy Point of View

Recalling (4.32), the rate of energy supplied to the rev is: ˙ = Σ : ε˙ | | W

(4.65)

With (4.61), the above equation can also be read as: ˙ || ˙ = Σ:S:B:Σ W

(4.66)

Shom

After integration with respect to time, we obtain:

1 Σ : Shom : Σ | | (4.67) 2 which generalizes expression (4.11) of the energy stored in the hollow sphere. Expression (4.67) is nothing but the classical deﬁnition of the elastic energy in a linear elastic homogeneous material subjected to the stress state Σ, which shows the consistency of deﬁnition (4.62) of the homogenized tensor of compliance from an energy point of view. It should be emphasized, however, that W =

8 That

is, ε = E and σ = Σ. B = 0 in p , the product S : B does not vanish since S is inﬁnite in the pore space.

9 Although

Generalization

109

this result relies upon the deﬁnition of the macroscopic strain tensor E as the average ε. It is instructive to derive the homogenized elastic compliance directly from ˙ without referring to the macroscopic state equathe work rate expression W, ˙ is readily tion (4.62). From the second integral in (4.31), this work rate W understood as the work rate developed by the internal forces σ along the strain rate ε. ˙ Taking into account the microscopic state equation (4.30b), we obtain: 1 ˙ = W σ : S : σ dVz (4.68) σ : S : σ˙ dVz ⇒ W = 2 We now insert the stress concentration rule (4.54) into (4.68), and derive an expression for W as a function of the macroscopic stress state: 1 t (4.69) B : S : B dVz : Σ W = Σ: 2 where t B is the transpose of B of components t Bi jkl = Bkli j . Finally, a comparison of (4.67) and (4.69) yields: Shom = t B : S : B

(4.70)

At very ﬁrst sight, it may appear that the two approaches (volume averaging versus energy) do not deliver the same result, as (4.63) and (4.70) differ. This apparent difference, however, is readily resolved using the Hill lemma. Indeed, we apply the fourth-order tensor t B : S : B to the couple (Σ′ , Σ′′ ) of two symmetric tensors: Σ′ : t B : S : B : Σ′′ = σ ′ : S : σ ′′

(4.71)

where σ ′ = B : Σ′ (resp. σ ′′ = B : Σ′′ ) is the microscopic stress ﬁeld solution of the elasticity problem (4.30) associated with Σ′ (resp. Σ′′ ). This implies the following: 1. σ ′ complies with a uniform stress boundary condition and with the average rule σ ′ = Σ′ . 2. ε′′ = S : σ ′′ is a geometrically compatible strain ﬁeld. Its average is S : B : Σ′′ . The Hill lemma (4.46) applied to the couple (σ ′ , ε′′ ) yields: ′′ Σ′ : t B : S : B : Σ′′ = σ ′ : ε′′ = Σ′ : S

: B: Σ σ′ ε′′

(4.72)

Drained Microelasticity

110

The above identity holds for an arbitrary choice of the couple (Σ′ , Σ′′ ), and implies: t

B:S:B=S:B

(4.73)

Relation (4.73) establishes the consistency of (4.63) and (4.70). Furthermore, deﬁnition (4.70) proves the symmetry of the homogenized tensor of compliance which is not obvious from (4.63). Bounds of the Homogenized Compliance Tensor

The importance of the energy approach relates to the determination of bounds for the homogenized compliance which can be derived from the variational principles of elasticity theory. These fundamental results, namely the theorem of minimum complementary energy and the theorem of minimum potential energy, refer to the concept of statically admissible stress ﬁelds and kinematically admissible displacement ﬁelds:

r The complementary energy Ec relies on statically admissible (s.a.) stress

ﬁelds σ ∗ . These are stress ﬁelds which satisfy the momentum balance and the stress boundary conditions: divσ ∗ = 0 σ∗ · n = Σ · n

() (∂)

(4.74)

Since there is no displacement boundary condition, the complementary energy of an rev reduces to the elastic energy: 1 ∗ σ ∗ : S : σ ∗ dVz (4.75) Ec (σ ) = 2 From a variational point of view, the stress ﬁeld σ solution of the problem (4.30) realizes among all s.a. stress ﬁelds the minimum of the complementary energy: 1 1 ∗ (∀σ s.a. with Σ) σ : S : σ dVz ≤ σ ∗ : S : σ ∗ dVz (4.76) 2 2 Combining (4.67) and (4.76) yields: (∀σ ∗ s.a. with Σ) Σ : Shom : Σ ≤ σ ∗ : S : σ ∗

(4.77)

Since the local tensor of compliance S(z) is inﬁnite in p , non-trivial upper bounds of Σ : Shom : Σ are only obtained for microscopic stress ﬁelds σ ∗ that comply with the condition σ ∗ = 0 in p . The construction of such stress ﬁelds requires some additional information concerning the morphology of the rev. We will illustrate the use of (4.77) in Section 4.2.8.

Generalization

111

r The potential energy E p relies on kinematically admissible (k.a.) displace-

ment ﬁelds ξ ′ . These are displacement ﬁelds which are continuous and piecewise differentiable, and which satisfy displacement boundary conditions. The potential energy is: 1 ′ ′ ′ E p (ξ ) = (4.78) ε(ξ ) : C : ε(ξ ) dVz − ξ′ · Σ · n 2 ∂ or equivalently, after transformation of the surface integral: 1 ′ ′ ′ ′ E p (ξ ) = | | ε(ξ ) : C : ε(ξ ) − ε(ξ ) : Σ 2

(4.79)

From a variational point of view, the displacement ﬁeld ξ solution of the problem (4.30) realizes among all k.a. displacement ﬁelds the minimum of the potential energy. Using (4.62) and (4.68), we observe that: ε : C : ε = σ : S : σ = Σ : Shom : Σ;

ε = Shom : Σ

(4.80)

so that the potential energy of the solution is: 1 (4.81) E p (ξ ) = − | | Σ : Shom : Σ 2 The theorem of minimum potential energy can be recast as follows: (∀ξ ′ k.a.) Σ : Shom : Σ ≥ 2ε(ξ ′ ) : Σ − ε(ξ ′ ) : C : ε(ξ ′ )

(4.82)

Among the set of k.a. displacement ﬁelds, we explore the subset of the homogeneous transformations of the form ξ ′ (z) = E′ · z, where E′ is a symmetric tensor. This implies that the corresponding microscopic strain is uniform (ε′ = E′ ), and (4.82) takes the form: (∀E′ symm.) Σ : Shom : Σ ≥ 2Σ : E′ − E′ : C : E′

(4.83)

The optimal choice is obtained by differentiating the r.h.s. of (4.83) with −1 respect to E′ . This yields E′opt = C : Σ, with which (4.83) becomes: (∀Σ) Σ : Shom : Σ ≥ Σ : C

−1

:Σ

(4.84)

The result is summarized as follows: −1

(∀Σ) Σ : (Shom − C ) : Σ ≥ 0

(4.85)

or formally: Shom ≥ C

−1

=

1 Ss 1 − ϕ0

(4.86)

The inequality sign ‘≥’ in (4.86) is to be understood in the sense of the associated quadratic forms which appear in (4.84).

Drained Microelasticity

112

Inequality (4.85) or (4.86) is known as the Voigt bound. It is interesting to note that this bound is very general since the only required morphological parameter is the (initial) pore volume fraction ϕ0 . In return, it is expected that it can be signiﬁcantly improved if a more detailed description of the morphology is available. This will be shown in Section 4.2.8. Finally, in the particular case of isotropic macroscopic behavior, Shom can be expressed as a function of the homogenized shear and bulk moduli μhom and k hom : Shom =

1 1 J+ K hom 3k 2μhom

(4.87)

where the fourth-order tensors J and K are deﬁned as: J=

1 1 ⊗ 1; 3

K=I−J

(4.88)

J and K obey the following algebraic properties: J : J = J;

K : K = K;

K:J=J:K=0

(4.89)

If the microscopic behavior is isotropic as well, then the solid compliance is: Ss =

1 1 J+ K s 3k 2μs

(4.90)

Thus, for isotropic behavior at both the microscopic and the macroscopic scale, it is readily seen that (4.85) yields: Σd : Σd (tr Σ)2 1 1 1 1 (∀Σ) + ≥0 − − 2 μhom (1 − ϕ0 )μs 9 k hom (1 − ϕ0 )k s (4.91) where Σd = K : Σ is the deviatoric part of the macroscopic stress Σ. This leads to the following upper bounds: μhom ≤ (1 − ϕ0 )μs ;

k hom ≤ (1 − ϕ0 )k s

(4.92)

4.2.8 An Instructive Exercise: Example of an rev for an Isotropic Porous Medium. Hashin’s Composite Sphere Assemblage The hollow sphere model of Section 4.1 is more a conceptual model of porous media than a realistic geometry of an rev. In fact, the hollow sphere model does not comply with the separability-of-scale condition (1.1), d ≪ ℓ. If we let the internal radius A play the role of d, and the external radius B the role of ℓ, the condition d ≪ ℓ is only satisﬁed for small porosity. For this reason, the

Generalization

113

Si(x)

Pi

Ai z i

Bi C

Figure 4.3 Hashin’s composite sphere assemblage

apparent modulus of the hollow sphere k hs derived from the hollow sphere model cannot be considered as a homogenized bulk modulus, and one may wonder whether this estimate can be derived from homogenization theory. This is the focus of this instructive exercise. We consider an rev that is composed of spherical pores surrounded by a homogeneous isotropic linear elastic solid. Let z i (resp. Ai ) denote the center (resp. the radius) of the pore Pi . The aim of this exercise is to derive upper and lower bounds of the homogenized compressibility of this rev with the given morphology. We therefore consider as loading of the rev, in the sense of the local problem deﬁnition (4.30), the isotropic loading Σ = 1. For a given scalar x < 1, we consider the sphere Si (x) with center zi , internal radius Ai and external radius Bi = Ai /x 1/3 (see Figure 4.3). By deﬁnition, the pore Pi is identical to the cavity of the hollow sphere Si (x). Let f be the minimum value of x ensuring that, for any choice of i and j, the two spheres Si (x) and S j (x) centered at zi and z j respectively do not intersect, while being included in . So deﬁned, Si ( f ) is a hollow sphere of the type studied in Section 4.1. In particular, we note that all spheres Si ( f ) have the same apparent bulk modulus, in so far as the volume fraction of the pore in Si ( f ) is equal to f and is therefore independent of the considered pore. Its value is given by (4.7) with ϕ0 = f . Let C be the complement of the set of hollow spheres Si ( f ) in the rev : = (4.93) Si ∪ C; C ∩ Si = ∅ i

and α the volume fraction of the set of hollow spheres in : | Si ( f ) | α= i ||

(4.94)

Drained Microelasticity

114

We note that f is not equal to the porosity ϕ0 in , but related to the former by: 4π 3 | Si ( f ) | i 3 Ai ϕ0 = = f i = fα (4.95) || || To derive solutions from the complementary energy approach (4.77), we explore the following stress ﬁeld σ ∗ : z ∈ Si ( f ) : σ ∗ (z) = σ hs (z − zi , Ai , Bi ) : σ ∗ (z) = 1 z∈C

(4.96)

where σ hs (z, A, B) is the stress ﬁeld (4.6) in the hollow sphere centered at the origin, with internal and external radii A and B, subjected to the unit tension on the external boundary. The elastic energy stored in Si ( f ) when subjected to σ ∗ is immediately derived from relation (4.11), which is the solution of a hollow sphere centered at the origin, with internal and external radii A and B, and subjected to a uniform unit tension. It is readily seen that the momentum balance equation is satisﬁed in the whole rev including the continuity condition of the stress vector at the interface between Si ( f ) and C. The boundary conditions satisﬁed by σ ∗ thus ensure that it is s.a. with Σ = 1. The elastic energy stored in Si ( f ) when subjected to σ ∗ is immediately derived from (4.11): 2 1 ∗ σ : S : σ ∗ dVz = | Si ( f ) | (4.97) 2k hs Si ( f ) 2 with k hs given in (4.5). The elastic energy density in C when subjected to σ ∗ is uniform: 1 ∗ 2 σ : S : σ∗ = s 2 2k Inserting (4.97) and (4.98) into (4.77) yields: 2 1 1 |C | i | Si ( f ) | hom 2 Σ:S : Σ = hom ≤ + k k hs || ks | |

(4.98)

(4.99)

that is, the following bound: α 1−α 1 ≤ hs + k hom k ks

(4.100)

Together with this upper bound of the macroscopic compressibility, the Voigt compliance lower bound (1 − f α)/k hom ≥ 1/k s is already available from (4.86).

Generalization

115

We now look for an improved estimate in the following subset of k.a. displacement ﬁelds: z ∈ Si ( f ) z∈C

: ξ ′ (z) = E ′ zi /3 + E ′ Bi /3 ξ hs (z − zi , Ai , Bi ) : ξ ′ (z) = E ′ z/3

(4.101)

where ξ hs (z, A, B) is the displacement ﬁeld (4.13) in the hollow sphere centered at the origin, with internal and external radii A and B, and subjected to the uniform unit radial displacement e r on the external boundary r = B. It is readily seen that ξ ′ so deﬁned complies with the continuity of the displacement at the interface between C and Si ( f ), so that it is k.a. in the sense of the local problem deﬁnition (4.30). The elastic energy stored in Si ( f ) when subjected to ξ ′ is immediately derived from (4.15): 1 1 ′ 2 ε : C : ε′ dVz = k hs E ′ | Si ( f ) | (4.102) 2 2 Si ( f ) The elastic energy density in C when subjected to ξ ′ is uniform: 1 1 ′ 2 ε : C : ε′ = k s E ′ 2 2

(4.103)

Since ξ ′ is compatible with the uniform strain boundary condition, the macroscopic strain state E ′ /31 is equal to ε′ . Finally, (4.82) takes the form: Σ : Shom : Σ =

2 2 ≥ 2 E ′ − k s (1 − α) + k hs α E ′ hom k

(4.104)

which holds for any value of E ′ . The optimal choice is: E′ =

k s (1 − α) + k hs α

(4.105)

and yields the following lower bound estimate of the macroscopic compressibility: 1 k hom

≥

k s (1

1 − α) + k hs α

Putting together (4.106) and (4.100) yields:10 α 1 − α −1 ≤ k hom ≤ k s (1 − α) + k hs α + k hs ks

(4.106)

(4.107)

10 It can be easily veriﬁed that the upper bound in (4.107) is lower than the Voigt upper bound k s (1 −

f α).

Drained Microelasticity

116

Relation (4.107) shows that the combination of the two variational approaches developed above does not, in general, provide the exact value of the macroscopic bulk modulus (resp. compressibility). Nevertheless, in the limit case α → 1, we note that the apparent bulk modulus k hs is obtained as the homogenized bulk modulus. This idea is due to Hashin:11 to imagine a porous material as a ‘composite sphere assemblage’, in which the whole rev is ﬁlled with hollow spheres having the same volume fraction. This kind of fractal description of the rev of course requires the inclusion of spheres of radius decreasing towards zero. The theoretical porous material so deﬁned provides a justiﬁcation for the use of the hollow sphere bulk modulus k hs as an estimate for the macroscopic bulk modulus of an isotropic porous medium. 4.2.9 The Homogenized Stiffness Tensor and Strain Concentration Macroscopic State Equation

We now consider deﬁnition (4.43) of the local problem, and make use of the linearity of the problem. Given this linearity, it is readily understood that the microscopic strain tensor ε(z) depends linearly on the components E i j of the macroscopic strain tensor E. This link is captured by a fourth-order strain concentration tensor, denoted by A(z), which relates the local strain tensor ε(z) to the macroscopic strain: (∀z ∈ ) εi j (z) = Ai jkl (z)E kl ⇔ ε(z) = A(z) : E

(4.108)

Tensors ε(z) and E are symmetric. The components of A(z), therefore, satisfy the following symmetry relations: Ai jkl (z) = Ajikl (z) = Ai jlk (z)

(4.109)

Furthermore, making use of the fact that the boundary condition (4.43c) is compatible with the strain average rule (4.42), use of (4.108) in (4.42) yields: E=ε=A:E=A:E

(4.110)

which holds for any choice of the macroscopic strain tensor. It follows that: A=I

(4.111)

Moreover, the strain concentration tensor A(z) can be employed to link the microscopic stress state at point z to the macroscopic strain state. Indeed, combining the microscopic state equation (4.43b) with the strain concentration rule (4.108) yields: σ(z) = C(z) : A(z) : E 11 See

[29].

(4.112)

Generalization

117

Finally, we want to translate the microscopic stress ﬁeld into a macroscopic stress state. According to the stress average rule Σ = σ adopted as a deﬁnition of the macroscopic stress in the framework of the uniform strain boundary condition, (4.112) yields a linear relation between E and Σ, i.e. the macroscopic state equation: Σ = C : A : E = Chom : E

(4.113)

where Chom is recognized as the macroscopic stiffness tensor: Chom = C : A

(4.114)

It is instructive to note that (4.114) combined with (4.111) yields: s

p

Chom = (1 − ϕ0 )Cs : A = Cs : (I − ϕ0 A )

(4.115) p

which would reduce to an ‘average rule’, Chom = (1 − ϕ0 )Cs , for A = I. In this s case, note that (4.111) would imply that A = I. However, the homogenized stiffness (4.115) generally deviates from this average rule because of the hets erogeneity of the strain ﬁeld, which is responsible for the fact that A = I as p well as A = I. The stiffness reduction due to the presence of pores appears p in (4.115) through the term ϕ0 A , which can be seen as a generalization of the bulk modulus reduction in (4.7). Lastly, it is interesting to examine the expression of the energy supplied to the rev through a loading deﬁned by the uniform strain boundary condition. ˙ given by (4.45), into which We start from the expression of the rate of energy W we insert the expression (4.112) of the microscopic stress as a function E: ˙ = E˙ : C : A : E | | W

(4.116)

Integration with respect to time yields: 1 W = E : Chom : E | | 2

(4.117)

which is the generalized form of (4.15). This result deserves the same comments as (4.67). It shows that it is equivalent to deﬁning the homogenized stiffness either by (4.113) or by (4.117). Energy Point of View

As in Section 4.2.7, we could have adopted an energy approach, i.e. (4.117), as a deﬁnition for the homogenized tensor of stiffness. Starting from (4.31) combined with the microscopic state equation (4.43b), we obtain: 1 ˙ = ε : C : ε dVz (4.118) ε : C : ε˙ dVz ⇒ W = W 2

Drained Microelasticity

118

We then insert the strain concentration rule (4.108) in (4.118): 1 t W= E: A : C : A dVz : E 2

(4.119)

A comparison of (4.119) and (4.117) yields: Chom = t A : C : A

(4.120)

As in Section 4.2.7, attention must be paid to the apparent difference between the expressions (4.114) and (4.120) of the homogenized tensor of stiffness derived respectively from the deﬁnitions (4.113) and (4.117). The consistency of (4.114) and (4.120) can be established with the help of the Hill lemma. More precisely, let us apply the fourth-order tensor t A : C : A to the couple (E′ , E′′ ) of two symmetric tensors: E′ : t A : C : A : E′′ = ε′ : C : ε′′

(4.121)

where ε′ = A : E′ (resp. ε′′ = A : E′′ ) is the microscopic strain ﬁeld solution of the elasticity problem (4.43) associated with E′ (resp. E′′ ). This implies that: 1. ε′ is a geometrically compatible strain ﬁeld which complies with a uniform strain boundary condition and with the average rule ε′ = E′ . 2. σ ′′ = C : ε′′ satisﬁes the momentum balance equation div σ ′′ = 0. Its average is C : A : E′′ . Hill’s lemma (4.46) applied to the couple (σ ′′ , ε′ ) yields: ′′ E′ : C : A E′ : t A : C : A : E′′ = ε′ : σ ′′ = : E ε′ σ ′′

(4.122)

The consistency of (4.114) and (4.120) is a consequence of the fact that the above identity holds for an arbitrary choice of the couple (E′ , E′′ ): t

A:C:A=C:A

(4.123)

The energy deﬁnition (4.120) allows us to conclude on the symmetry of the homogenized stiffness tensor. Finally, recalling that Cs is a positive deﬁnite fourth-order tensor, the energy deﬁnition proves that this also holds for Chom : (∀E)

s

E : Chom : E = (1 − ϕ0 )(A : E) : Cs : (A : E) ≥ 0

(4.124)

In the case of isotropic macroscopic behavior, (4.124) yields: k hom ≥ 0;

μhom ≥ 0

(4.125)

Generalization

119

Bounds for the Homogenized Stiffness Tensor

In comparison to Section 4.2.7, the boundary condition (4.43c) has replaced (4.30c). Therefore, the deﬁnition of s.a. stress ﬁelds as well as the deﬁnition of k.a. displacement ﬁelds are modiﬁed:

r The s.a. stress ﬁelds σ ∗ must comply with the momentum balance condition: divσ ∗ = 0

()

(4.126)

r The k.a. displacement ﬁelds ξ ′ must be continuous, piecewise differentiable and comply with the uniform strain boundary conditions: (∀z ∈ ∂) ξ ′ (z) = E · z

(4.127)

In the absence of surface force boundary conditions, the potential energy reduces to the elastic energy: 1 ′ E p (ξ ) = (4.128) ε(ξ ′ ) : C : ε(ξ ′ ) dVz 2 In particular, recalling (4.120), the potential energy of the displacement solution ξ to (4.43) is: 1 1 E : t A : C : A : E dVz = | | E : Chom : E (4.129) E p (ξ ) = 2 2 The displacement ξ minimizes the potential energy among all k.a. displacement ﬁelds: (∀ξ ′ k.a. with E) E : Chom : E ≤ ε(ξ ′ ) : C : ε(ξ ′ )

(4.130)

The simplest k.a. displacement ﬁeld is the homogeneous transformation ξ ′ = E · z associated with the uniform strain ﬁeld ε(ξ ′ ) = E. Its implementation in (4.130) yields: (∀ E) E : (Chom − C) : E ≤ 0

(4.131)

which can be formally summarized by: Chom ≤ C = (1 − ϕ0 )Cs

(4.132)

This inequality12 is known as the Voigt bound. It is associated with the uniform strain boundary condition and appears as the counterpart of (4.86). The complementary energy Ec (σ ∗ ) now incorporates a term corresponding to the work of the surface forces σ ∗ · n on ∂ in the prescribed displacement 12 In

the sense of quadratic forms in the space of second-order symmetric tensors.

Drained Microelasticity

120

E · z: Ec (σ ∗ ) =

1 2

σ ∗ : S : σ ∗ dVz −

n · σ ∗ · E · z dSz

∂

= ||

1 ∗ σ : S : σ∗ − E : σ∗ 2

(4.133)

In particular, the complementary energy of the solution σ to (4.43) is (see (4.129)): 1 Ec (σ) = −E p (ξ ) = − | | E : Chom : E 2

(4.134)

The stress ﬁeld σ minimizes the complementary energy among all s.a. stress ﬁelds. Together with (4.130), this result is: (∀ξ ′ k.a. with E)(∀σ ∗ s.a.)

2 E : σ ∗ − σ ∗ : S : σ ∗ ≤ E : Chom : E ≤ ε(ξ ′ ) : C : ε(ξ ′ )

(4.135)

The example of Hashin’s composite sphere assemblage already studied in Section 4.2.8 is left as an exercise to the reader. The loading of the rev is deﬁned in the framework of (4.43) with E = E/31. Relation (4.107) can be recovered by means of (4.135) when using the following stress ﬁelds: z ∈ Si ( f ) z∈C

: σ ∗ (z) = ′ σ hs (z − zi , Ai , Bi ) : σ ∗ (z) = ′ 1

(4.136)

where ′ is an arbitrary scalar, and the displacement ﬁeld: z ∈ Si ( f ) z∈C

: ξ ′ (z) = E zi /3 + EBi /3 ξ hs (z − zi , Ai , Bi ) : ξ ′ (z) = E z/3

(4.137)

4.2.10 Influence of the Boundary Condition. The Hill–Mandel Theorem Both uniform boundary conditions (4.30c) and (4.43c) have been introduced in order to deﬁne a well-posed local problem. The question is whether they are equivalent. In contrast to the very particular situation of the hollow sphere, the displacement ﬁeld solution of the local problem deﬁned with uniform stress boundary conditions does not in general comply with the uniform strain boundary conditions. Rigorously speaking, this implies that the two approaches of the macroscopic behavior discussed in Sections 4.2.7 and 4.2.9 are a priori not consistent. In particular, this means that the compliance and stiffness tensors derived in (4.63) and (4.114) are not exactly the inverse of one another. Furthermore, in practice, neither the ‘real’ forces nor the ‘real’

Generalization

121

displacements acting on the boundary of the rev at the microscopic scale satisfy such uniform boundary conditions. These remarks actually lead to the question of whether uniform boundary conditions are relevant at all. In fact, the whole approach relies upon the condition d ≪ ℓ, which states that the rev’s characteristic length is large in comparison to that of the heterogeneities. This intuitively implies that the characteristic length w.r.t. which the solution is sensitive to the type of boundary conditions is on the order of d. More precisely, for a given value of Σ, let σ and ε denote the microscopic stress and strain ﬁeld solutions to the problem (4.30). The macroscopic strain associated with this loading is tensor E derived in (4.62). In turn, let σ E and ε E denote the solution of the problem (4.43) associated with E. The idea is that σ and ε on the one hand, and σ E and ε E on the other hand, will indeed be different, but only in a thin layer having a thickness on the order of d in the vicinity of the rev boundary. Therefore, the difference σ − σ E (resp. ε − ε E ) does not signiﬁcantly affect the average σ (resp. ε), so that σ ≈ σ E and ε ≈ ε E . This implies that the homogenized behavior can be predicted by either uniform stress or uniform strain boundary conditions. In particular, this means that the homogenized compliance derived in (4.63) and the homogenized stiffness in (4.114) are indeed the inverse of one another. Another way to address this question is to observe that the validity of the approximation σ ≈ σ E (except in the thin boundary zone) is: E = Shom : Σ B(z) : Σ ≈ C(z) : A(z) : E with (4.138)

Shom = S : B εE

σ

In other words, the stress and strain concentration tensors appear to be related by: B(z) ≈ C(z) : A(z) : Shom

(4.139)

Taking the average of B over the rev, we obtain: I = B ≈ C : A : Shom = C : A : Shom

(4.140)

C hom

More precisely, the condition for the equivalence of the uniform stress and strain boundary conditions is summarized in a classical result known as the Hill–Mandel theorem ([32],[41]), which states that: C

hom

:S

hom

3 d =I+O ℓ

(4.141)

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4.3 Estimates of the Homogenized Elasticity Tensor In this section, we will adopt the uniform strain boundary condition: the local problem is deﬁned by (4.43) with C(z) = 0 in p and C(z) = Cs in s . The starting point is the expression (4.115) of the homogenized stiffness tensor: p

Chom = Cs : (I − ϕ0 A )

(4.142)

The macroscopic elastic properties appear to be controlled by the average conp centration tensor A over the pore space. In this section, we restrict ourselves to the case of isotropy at both the microscopic and macroscopic scales. In p this case, from (4.142), A is isotropic as well and can therefore be put in the form: p

A = avJ + ad K

(4.143)

which in turn provides the corresponding estimates of the homogenized bulk and shear moduli: k hom = k s (1 − a v ϕ0 );

μhom = μs (1 − a d ϕ0 )

(4.144)

The derivation of estimates for a v and a d requires us to specify the morphology of the pore space. In the following, we assume that the morphology can be represented by a set of spherical pores embedded in the solid matrix. Furthermore, it is assumed that the connection between the spheres has no inﬂuence on the overall mechanical response. 4.3.1 The Dilute Scheme The dilute scheme assumes that the mechanical interaction between pores is negligible. When a macroscopic strain E is applied to the rev, the average strain ε p over the pore space can, therefore, be estimated as the average strain εS in a single spherical pore S embedded in an inﬁnite elastic homogeneous medium having the same stiffness tensor as the solid, i.e. Cs , and subjected to uniform strain boundary conditions on the displacement of the form ξ → E · z when z → ∞: divσ = 0 σ = Cs : ε 1 ε = (grad ξ + t grad ξ ) 2 σ · e r = 0 (r = A) ξ → E · z (z → ∞)

(a ) (b) (c) (d) (e)

(4.145)

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where A denotes the radius of the pore S. Applying (4.21) to the pore S and using (4.143), the estimate ε p ≈ εS is: 1 1 1 S ε dVz = (ξ ⊗ n + n ⊗ ξ ) dSz ≈ (tr E) a v 1 + a d Ed ε = |S| S 2|S| ∂S 3 (4.146) The coefﬁcients a v and a d can thus be determined from the values of the displacement ξ on the boundary ∂S of the sphere. We ﬁrst consider a spherical macroscopic strain E = 31 E1. The determination of the displacement is formally obtained from the solution of the hollow sphere problem in the particular case B → ∞. Letting ϕ0 → 0 in (4.13), the displacement ξ = EB/3 ξ hs is: 1 3k s A3 ξ= E r+ er (4.147) 3 4μs r 2 Using the identity:

∂S

e r ⊗ e r dSz =

4π A2 1 3

(4.148)

and substituting (4.147) into (4.146) yields: av = 1 +

3k s 4μs

(4.149)

In order to determine a d , we now examine the displacement induced by a purely deviatoric macroscopic strain: E = E(e 1 ⊗ e 1 − e 2 ⊗ e 2 )

(4.150)

In cartesian and spherical coordinates with vertical axis e 3 , the boundary condition at inﬁnity is: ⎧ ⎧ ⎨ ξr ≈ Er sin2 θ cos 2φ ⎨ ξ1 ≈ E z1 (4.151) |z| → ∞ ξ2 ≈ −E z2 ⇔ ξθ ≈ Er sin θ cos θ cos 2φ ⎩ ⎩ ξ3 ≈ 0 ξφ ≈ −Er sin θ sin 2φ

The solution is looked for in the form:13

ξr (r, θ, φ) = Dr (r ) sin2 θ cos 2φ ξθ (r, θ, φ) = Dθ (r ) sin θ cos θ cos 2φ ξφ (r, θ, φ) = Dφ (r ) sin θ sin 2φ 13 See

[39].

(4.152)

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The momentum balance equation expressed through the Navier equation reveals the general form of the functions Di (r ): 5 − 4ν s γ /r 2 1 − 2ν s Dθ (r ) = αr − 2β/r 4 + 2γ /r 2 Dφ (r ) = −Dθ (r ) Dr (r ) = αr + 3β/r 4 +

(4.153)

where ν s is the Poisson ratio of the solid phase. The scalar coefﬁcients α, β and γ are determined from the boundary condition at inﬁnity (4.151) and from the condition (4.145d), i.e. σ · e r = 0 on r = A (boundary of the spherical pore): α = E; β = −

3A5 5A3 (1 − 2ν s ) E; γ = E 14 − 10ν s 14 − 10ν s

(4.154)

The displacement ﬁeld on the sphere ∂S (r = A), which is related to a d according to (4.146), takes the form: ⎧ ⎧ ⎨ ξr = E ′ Asin2 θ cos 2φ ⎨ ξ1 = E ′ z1 r = A: (4.155) ξ = E ′ Asin θ cos θ cos 2φ ⇔ ξ2 = −E ′ z2 ⎩ θ ⎩ ′ ξ3 = 0 ξφ = −E Asin θ sin 2φ with:

E ′ = 15

3k s + 4μs 1 − νs E = 5 E 7 − 5ν s 9k s + 8μs

(4.156)

In other words, the spherical pore S is subjected to a homogeneous transformation associated with the strain tensor εS : εS = E ′ (e 1 ⊗ e 1 − e 2 ⊗ e 2 )

(4.157)

From (4.146), we conclude that: ad =

3k s + 4μs E′ =5 s E 9k + 8μs

(4.158)

We still have to insert the values of a v and a d given by (4.149) and (4.158) into (4.144): 3k s hom s k = k 1 − ϕ0 1 + 4μs (4.159) s 3k + 4μs hom s μ = μ 1 − 5ϕ0 s 9k + 8μs The derivation here is restricted to spherical pores. It will be generalized to ellipsoidal pores in Chapter 6.

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4.3.2 The Differential Scheme The purpose of the differential scheme is to overcome the limitation ϕ0 ≪ 1, which restricts the validity of the dilute estimates given by (4.159). The idea consists of starting from a homogeneous medium identical to the solid matrix (bulk and shear moduli k s and μs ), and introducing the porosity by inﬁnitesimal volume fractions in the framework of an iterative process. The ﬁrst step is actually identical to the dilute scheme. At the end of step j, the homogenized ( j) ( j) medium is characterized by bulk and shear moduli k DS and μ DS . The step j + 1 consists of removing a volume fraction dx ≪ 1 out of this medium and replacing it by the same volume of pores. After homogenization, the bulk and ( j+1) ( j+1) shear moduli of the homogenized medium thus obtained are k DS and μ DS . In this process, the incremental porosity change is dϕ0 : dϕ0 = −ϕ0 dx + dx = (1 − ϕ0 )dx

(4.160)

Since dx ≪ 1, the dilute scheme can be used in order to describe the effect of step j + 1; that is, the transition from medium j to medium j + 1. We thus ( j) ( j) apply (4.159) in which k DS , μ DS and dx now play the role of k s , μs and ϕ0 , respectively: ( j+1)

k DS

( j+1)

μ DS ( j)

( j)

( j) ( j) = k DS 1 − a v dx ( j) ( j) = μ DS 1 − a d dx

(4.161)

a v and a d are the values of a v and a d obtained by replacing k s and μs in ( j) ( j) (4.149) and (4.158) with k DS and μ DS . Equation (4.161) can be put in the form of the following differential system: dϕ0 dk DS = −a v (k DS , μ DS ) k DS 1 − ϕ0 dμ DS dϕ0 = −a d (k DS , μ DS ) μ DS 1 − ϕ0

(4.162)

where: a v (k DS , μ DS ) = 1 +

3k DS ; 4μ DS

a d (k DS , μ DS ) = 5

3k DS + 4μ DS 9k DS + 8μ DS

(4.163)

The last task is to integrate this differential system with respect to ϕ0 starting from the initial conditions k DS = k s and μ DS = μs for ϕ0 = 0. The solution denoted by k hom and μhom is obtained in the form of a relation between k hom

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and μhom together with a relation between the porosity and μhom : ⎧ ⎪ (1 + 4μs /3k s )(μhom /μs )3 ⎪ ⎪ = (1 − ϕ0 )6 ⎪ s /3k s )(μhom /μs )3/5 ⎪ 2 − (1 − 4μ ⎨ if ν s > 0.2 ⎪ ⎪ (1 − 34 μhom /k hom )5/3 μhom ⎪ ⎪ = ⎪ ⎩ μs (1 − 43 μs /k s )5/3

if ν s < 0.2

⎧ ⎪ (1 + 4μs /3k s )(μhom /μs )3 ⎪ ⎪ = (1 − ϕ0 )6 ⎪ ⎪ ⎨ 2 + (4μs /3k s − 1)(μhom /μs )3/5 ⎪ ⎪ ( 34 μhom /k hom − 1)5/3 μhom ⎪ ⎪ ⎪ ⎩ μs = ( 4 μs /k s − 1)5/3 3

(4.164)

(4.165)

As expected, it is readily shown that (4.159) can be retrieved from (4.164) if we expand k hom /k s and μhom /μs in a power series up to the ﬁrst order in ϕ0 in the vicinity of ϕ0 = 0. It can be clearly seen in Figures 4.4 and 4.5 that the validity of the dilute estimates is restricted to the domain ϕ0 ≪ 1.

1

0.8

0.6

0.4

0.2 (4.159)

(4.164) ϕ0

0 0.2

0.4

0.6

0.8

1

Figure 4.4 Estimates of k hom /k s as predicted by the dilute and differential schemes (ν s = 0.3)

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127

1

0.8

0.6

0.4

0.2 (4.159)

(4.164) ϕ0

0 0.2

0.4

0.6

0.8

1

Figure 4.5 Estimates of μhom /μs as predicted by the dilute and differential schemes (ν s = 0.3)

4.4 Average and Effective Strains in the Solid Phase For further applications, it is useful to estimate the strain level reached in the solid phase as a function of the macroscopic loading, characterized by Σ or E. The most natural way to describe this strain level is to use the intrinsic average ε s of the strain ﬁeld over the solid domain s . From the microscopic state equation in the solid, we derive the relationship between ε s and the intrinsic average σ s of the stress ﬁeld over the solid domain: εs = Cs −1 : σ s

(4.166)

Combining (4.166) with the stress average rule Σ = (1 − ϕ0 )σ s then gives the strain average as a function of the macroscopic stress: (1 − ϕ0 )ε s = Cs −1 : Σ

(4.167)

Alternatively, the macroscopic state equation (4.113) gives ε s as a function of the macroscopic strain: (1 − ϕ0 )ε s = Cs −1 : Chom : E

(4.168)

When both the microscopic and macroscopic stiffness tensors Cs and Chom are isotropic, (4.168) can be split into two equations giving the volume and

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deviatoric strain averages: (1 − ϕ0 )εv s =

k hom tr E ks

(4.169)

(1 − ϕ0 )εd s =

μhom Ed μs

(4.170)

where εd = K : ε and Ed = K : E are the deviatoric parts of the microscopic and macroscopic strains. Relations (4.169) and (4.170) are attractive because of their simplicity. However, the relevance of the deviatoric strain average is questionable. For instance, in the hollow sphere problem (Section 4.1), the macroscopic strain is non-deviatoric and (4.170) implies that εd s = 0. Still, the shear mechanism is clearly of paramount importance in the effective response of the sphere. This is shown for instance by the fact that the bulk modulus k hs in (4.7) depends on μs . In order to overcome this difﬁculty, it is convenient to refer to the concept of equivalent shear strain: 1 εd = (4.171) εd : εd 2 At the macroscopic scale, the shear strain level may be represented by an appropriate average of the equivalent shear strain which we need to relate to the macroscopic strain E. For this purpose, we will use an energy approach.14 Let us consider the strain (resp. displacement) ﬁeld solution ε (resp. ξ ) of the local problem (4.43) as a function of the coefﬁcients Cisjkl of the elastic stiffness tensor Cs of the solid. ∂C ε denotes the derivative15 of ε with respect to Cisjkl . It can be seen as the strain ﬁeld associated with the ‘displacement ﬁeld’ ∂C ξ = ∂ξ /∂Cisjkl . Note that (4.43c) implies that ∂C ξ is equal to zero on the boundary ∂. In turn, the macroscopic energy W introduced by (4.118) implicitly depends on Cisjkl through ε. The derivative16 of W with respect to Cisjkl is: ∂W 1 ∂ Cs : ε dV + = : ε dVz (4.172) ε : ∂ ε : C (z) z C ∂Cisjkl 2 s ∂Cisjkl 14 This

approach was introduced in [35]. this differentiation, note that we have to take into account all the symmetries of Cisjkl , s namely Cisjkl = C sjikl = C sjilk = Ckli j. 16 Once again, the differentiation with respect to C s i jkl in (4.172) takes into account all the symmetries of Cs . 15 In

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129

The ﬁrst integral in (4.172) takes the form ||σ : ∂C ε. Since ∂C ξ is k.a. with the macroscopic strain 0, Hill’s lemma in the form (4.51)–(4.53) shows that this term vanishes. Equation (4.117) provides an alternative way to examine the dependence of W on Cisjkl : 1 ∂ Chom ∂W = E : : E|| ∂Cisjkl 2 ∂Cisjkl

(4.173)

Combining (4.172) and (4.173) yields, for instance, for i, j, k, l = 1: E:

s ∂ Chom 2 : E = (1 − ϕ )ε 0 11 s ∂C1111

(4.174)

s = e 1 ⊗ e 1 ⊗ e 1 ⊗ e 1 has been used. Hence, if an where the identity ∂ Cs /∂C1111 s is available, estimate of the dependence of the effective stiffness Chom on C1111 s 2 the average ε11 can be determined as a function of the macroscopic strain. This is not an easy task, however, since it requires an assessment of the speciﬁc s role of C1111 on Chom , keeping all other coefﬁcients constant. By contrast, in the case of an isotropic solid phase, this technique proves to be very efﬁcient for the determination of the quadratic averages of εd and εv . More precisely, the above approach can be achieved by replacing Cisjkl with the shear modulus μs . In this case, (4.172) is: ∂W (4.175) = ε : K : ε dVz = ||(1 − ϕ0 )εd : εd s s ∂μ

and (4.174) is replaced with: s ∂ Chom 1 E: : E = 2(1 − ϕ0 )εd 2 s 2 ∂μ

(4.176)

If the macroscopic properties are isotropic as well, (4.176) simpliﬁes to: s ∂k hom ∂μhom 1 (tr E)2 + E : E = 2(1 − ϕ0 )εd 2 d d s s 2 ∂μ ∂μ

(4.177)

Similarly, replacing Cisjkl with the bulk modulus k s , we obtain: s ∂k hom ∂μhom 1 1 (tr E)2 + E : E = (1 − ϕ0 )(tr ε)2 d d s s 2 ∂k ∂k 2

(4.178)

Equations (4.177) and (4.178) suggest the introduction of the quadratic average of the shear and volume strains in the solid phase: s s s s 2 (4.179) εd = εd ; εv = (tr ε)2

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s

As opposed to εv s and εd s (see (4.169)–(4.170)), εv and εd depend on both the macroscopic shear and volume strains. In particular, for a non-deviatoric s macroscopic strain (Ed = 0), we note that εd s = 0 whereas εd = 0. These estimates of the average shear and volume strains in s will be used for the implementation of nonlinear homogenization techniques in Chapter 7. 4.5 Training Set: Molecular Diffusion in a Saturated Porous Medium From a purely mathematical point of view, the problem of the elastic response of an rev to a mechanical loading and the problem of the diffusion of a solute in a pore space are similar. The focus of this training set, therefore, is the application to the solute diffusion problem of the elements of the homogenization theory developed in this chapter for the elasticity problem. We consider a porous medium composed of a rigid solid phase and a single ﬂuid phase which ﬁlls the pore space. The volume fraction of the ﬂuid phase is denoted by ϕ. We are interested in the diffusive transport of a component of the ﬂuid phase, which will be denoted by the superscript γ . We assume steady state conditions, and advection and adsorption phenomena are neglected. The physics of diffusion at the microscopic scale has been presented in Chapter 3 (and the reader is encouraged to have a quick look back before proceeding). Chapter 3 was devoted to the periodic case. This section addresses the case of a disordered porous medium. The same notation is adopted: ρ γ , vγ and j γ = ρ γ vγ denote the density, the velocity and the diffusive ﬂux of the γ component at the microscopic scale, respectively. The diffusive ﬂux j γ and the solute concentration ρ γ are subjected to:

r The mass balance equation for the γ component: divz j γ = 0

( f )

(4.180)

r The zero mass ﬂow at the solid–ﬂuid interface17 I s f : j γ · n = 0 (I s f )

(4.181)

r Fick’s law, which involves the diffusion coefﬁcient of the γ component in an inﬁnite medium ﬁlled by the ﬂuid phase: j γ = −Dγ grad ρ γ z

(4.182)

From a physical point of view, ρ γ and j γ are deﬁned in the ﬂuid phase only. However, from a mathematical point of view, it will prove efﬁcient to extend 17 n

is the unit normal to the interface I s f .

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them as well as Fick’s law to the solid phase. We therefore introduce a diffusion coefﬁcient equal to zero in the solid phase, which implies that the diffusive ﬂux is extended to zero in the solid phase. The extended density ﬁeld as well as the normal component of the extended diffusive ﬂux are continuous over the solid–ﬂuid interface. These extended ﬁelds are subjected to the following equations: () divz j γ = 0; [ j γ ] · n = 0 at I s f

(4.183)

where [ j γ ] is the discontinuity of the diffusive ﬂux at the solid–ﬂuid interface, and: γ D in the ﬂuid γ γ () j (z) = −D(z) grad ρ with D(z) = (4.184) z 0 in the solid Relation (4.184) describes the rev as an heterogeneous medium with two different values of the diffusion coefﬁcient, depending on the location. We note that this point of view transforms condition (4.181) into a continuity condition of the diffusive ﬂux at the solid–ﬂuid interface. In turn, in the absence of advection and adsorption, the mass balance equation at the macroscopic scale is (see (1.31)): divx J γ = 0 with

J γ = jγ f = jγ

(4.185)

J γ is the macroscopic diffusive mass ﬂux of the γ component. We look for the macroscopic counterpart of the microscopic Fick’s law (4.184). To do this, we note the similarity of the mathematical equations (4.183) and (4.184), on the one hand, to (4.30a) and (4.30b), on the other hand; that is: σ

→

jγ

ξ

→

ργ

ε

→

grad ρ γ

Hooke’s law

→

Fick’s law

(4.186)

momentum balance → mass balance pores

→

solid

solid

→

pores

The homogenization of the diffusive properties can therefore be performed with the method developed in Section 4.2. First, the local boundary value problem of diffusion is deﬁned on the rev in the framework of uniform boundary conditions. This leads to the derivation of a homogenized diffusion tensor and the tortuosity tensor. Finally, estimates for these quantities are derived.

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4.5.1 Definition of a Local Boundary Value Problem According to the correspondence rule (4.186), the uniform strain boundary condition (4.43c) of the mechanical problem is replaced by a uniform density gradient boundary condition: (∀ z ∈ ∂) ρ γ (z) = H · z

(4.187)

where H represents the density gradient at the macroscopic scale. It plays the role of the macroscopic strain E in the diffusion problem. Similar to the displacement in the pores, the density of the γ component in the solid has no physical meaning. It is only required to ensure a continuous extension of the real density deﬁned in the ﬂuid into the solid phase. It is readily seen that (4.187) implies the average rule grad ρ γ = H. The boundary value problem on the rev is thus: (∀ z ∈ ∂) ρ γ (z) = H · z () divz j γ = 0 () j γ (z) = −D(z) grad ρ γ

(4.188)

z

The solution of (4.188) is a linear function of H. This property is expressed through the concept of concentration tensor, denoted here by A(z), which linearly relates the local gradient of density grad ρ γ to H: z

γ

grad ρ = A(z) · H z

(4.189)

Let H α be the average density gradient over the α constituent in (α = s, f for the solid and ﬂuid phases, respectively). We note that the consistency condition A = 1, which can be derived from the average rule grad ρ γ = H , allows us to relate H f , H s and H: s Hs = A · H (4.190) ⇒ (1 − ϕ)H s + ϕ H f = H f Hf = A · H The combination of (4.184) and (4.189) relates the macroscopic mass ﬂux J γ = j γ to the macroscopic gradient of density H: J γ = −Dhom · H

(4.191)

in which Dhom represents the homogenized macroscopic diffusion tensor:

f s (4.192) Dhom = DA = Dγ ϕA = Dγ 1 − (1 − ϕ)A Equation (4.191) is recognized as the generalized Fick’s law for the macroscopic description of diffusion in a porous medium.

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The tortuosity tensor T which is classically related to Dhom by Dhom = Dγ ϕT aims at capturing the effect of the geometry of the microstructure on the overall diffusive properties of the porous medium. Equation (4.192) yields: f

T=A =

1 s 1 − (1 − ϕ)A ϕ

(4.193)

This equation provides a micromechanical deﬁnition of the tortuosity tensor which implicitly involves the geometry of the microstructure through A(z). Note the similarity between (4.191)–(4.193), on the one hand, and (3.13)–(3.14), on the other hand. The difference lies in the fact that the averages refer in one case to the whole rev and in the other case to an elementary cell. Equations (4.192)–(4.193) show that the determination of estimates for Dhom and T res quires estimates for A (or H s ). This is the purpose of the next section. 4.5.2 Estimates of the Effective Diffusion Coefficient We explore the idea that the ﬂuid phase ﬁlling the pore space is a connected matrix in which grains (solid phase) are embedded as spherical inhomogeneities. The spatial distribution of the solid spheres is assumed to be isotropic. This s implies that A is an isotropic tensor a 1. The sphere radii are arbitrary so that any value of the porosity can be obtained. To begin with, inﬁnitesimal values of the solid volume fraction are considered. Dilute Scheme

When the volume fraction of the solid is inﬁnitesimal, it is reasonable to neglect the interaction between the solid grains. In this case, the micros scopic density and the corresponding value of A can be approximated by their values in the problem of a single spherical solid grain S embedded in an inﬁnite homogeneous ﬂuid continuum having a diffusion tensor Dγ 1, with the condition ρ γ = H · z at inﬁnity. In spherical coordinates with the origin r = 0 at the center of S, the equations of this auxiliary problem are: r r r r

> A div j γ = 0 > A j γ = −Dγ grad ρ γ =A jγ · n = 0 → ∞ ργ → H · z

(a ) (b) (c) (d)

(4.194)

where A denotes the radius of the solid sphere. The average concentration gradient within the sphere is obtained from the value of ρ γ on the boundary

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134

of the sphere: grad ρ γ

s

1 = |S|

1 grad ρ dVz = |S| γ

S

ρ γ n dSz

(4.195)

∂S

The value taken on ∂S by the solution of (4.194) thus allows us to obtain an s estimate of the tensor A = a 1: 1 s s γ grad ρ = A · H ⇔ (4.196) ρ γ n dSz = a H |S| ∂S Consider for example that H is parallel to the unit vector e 3 , i.e. H = He 3 . It can be readily seen that the solution ρ γ to (4.194) is: A3 γ (4.197) ρ = Hz3 1 + 3 2r Equation (4.196) then shows that: s

grad ρ γ =

3 3 s H ⇔ A = 1 2 2

Inserting this result into (4.192) yields an estimate of Dhom = Dhom 1: 3 hom ≈ 1 − (1 − ϕ) Dγ (1 − ϕ ≪ 1) D 2

(4.198)

(4.199)

Differential Scheme

The estimate (4.199) is not valid except for an inﬁnitesimal solid volume fraction, which of course does not correspond to real porous media. It is therefore necessary to improve the ‘dilute’ estimate and overcome this limitation. We consider hereafter the so-called differential scheme. The reasoning is similar to that presented in Section 4.3.2. Starting from ϕ = 1 (pure ﬂuid), the idea consists of introducing the solid by inﬁnitesimal increments. After each increment, the solid–ﬂuid mixture is homogenized in order to determine the estimate of Dhom corresponding to the current value of the porosity ϕ. During each step the volume fraction dx of homogenized material is removed and replaced by the same volume of solid. This process is associated with the variation dϕ = −ϕdx of the porosity. The corresponding evolution of the effective diffusion coefﬁcient can be determined by applying the dilute scheme to each inﬁnitesimal step. More j precisely, let DDS denote the homogenized diffusion coefﬁcient at the beginning of step j + 1. In turn, at the end of this step, the homogenized diffusion j+1 j j+1 coefﬁcient is DDS . DDS and DDS respectively play the role of Dγ and Dhom

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in (4.199), whereas dx plays the role of (1 − ϕ). This incremental process is therefore described by the following differential equation: dD DS =

3dϕ D DS 2ϕ

(4.200)

The solution to (4.200) with D DS = Dγ 1 for ϕ = 1 is the estimate of Dhom provided by the differential scheme, namely: Dhom = ϕ 3/2 Dγ 1

(4.201)

The corresponding tortuosity tensor is: T = ϕ 1/2 1

(4.202)

5 Linear Microporoelasticity This second chapter on microporoelasticity is devoted to the derivation – by an upscaling procedure – of the state equations of a saturated porous medium composed of a linear elastic solid phase and a pore space saturated by one single ﬂuid phase. This micromechanics theory is based on the solution of a boundary value problem on a rev which is simultaneously subjected to a uniform strain boundary condition and a uniform ﬂuid pressure at the solid–ﬂuid interface. The starting point of our investigation is the 1-D thought model of microporomechanics, the hollow sphere model, which allows a straightforward illustration of the very principles of this upscaling procedure. These principles are then translated into a rigorous upscaling theory suitable for any geometry of the pore space and for any linear elastic behavior of the solid phase. Finally, using Levin’s theorem, the state equations of linear poroelasticity are derived.

5.1 Loading Parameters The micromechanics theory we develop here combines two of the key results of Chapters 2 and 4 into a consistent upscaling theory of microporomechanics: within the limits of Darcy’s law, the mechanical action of the ﬂuid on the solid at the microscopic scale is dominated by the ﬂuid pressure, while the viscous stress is of a smaller order of magnitude (see Section 2.5.2). Furthermore, since the microscopic and macroscopic pressure gradients are on the same order of magnitude (see Section 2.4.1), the deviation of the microscopic pressure ﬁeld p(z) around its average P(x) = p f in the rev is negligible. Thus, the mechanical interaction between the ﬂuid and the solid at the microscopic level can be taken into account through a uniform pressure that is equal to the macroscopic one, i.e. P(x) = p f, and that is applied at the solid–ﬂuid interface. In this case, the macroscopic pressure P(x) and the macroscopic strain E(x) play the role of Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

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loading parameters of the rev (x), so that the microscopic stress ﬁeld σ(z) in (x) depends on P(x) and E(x). In turn, using the average rule (1.39), the macroscopic stress tensor Σ(x) can also be determined as a function of P(x) and E(x). This delivers the ﬁrst macroscopic state equation. The second one relates the pore volume change to P(x) and E(x). The focus of this chapter is the derivation of these state equations as a solution of the so-deﬁned boundary value problem. In what follows, for purposes of clarity, the position vector x at the macroscopic scale is omitted. The macroscopic stress and strain tensors and the macroscopic pressure are simply denoted by Σ, E and P, respectively.

5.2 The 1-D Thought Model: The Saturated Hollow Sphere Model To motivate the forthcoming developments, we consider the hollow sphere model displayed in Figure 5.1, which is the simplest geometrical conﬁguration of a porous material: The spherical cavity f of radius A represents the pore space, and the spherical shell surrounding the cavity (A ≤ r ≤ B) the solid phase. We consider the solid phase to be a linear elastic, homogeneous and isotropic material, characterized by the Lam´e constants μs and λs . The bulk modulus of the solid is k s = λs + 2μs /3. The volume fraction of the cavity in the undeformed reference conﬁguration represents the initial porosity ϕ0 : 3 A (5.1) ϕ0 = B We have studied the elastic response of the empty hollow sphere model in Section 4.1. The difference with Chapter 4 is that the cavity is now ﬁlled with

P

2A 2B

ξ 0 er

Figure 5.1 The saturated hollow sphere model

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139

a ﬂuid at pressure P. The stress vector acting on the solid at the solid–ﬂuid interface is: T = Pe r

at r = A

(5.2)

The pore pressure P is a controllable loading parameter in our problem. This implies that the cavity can exchange some ﬂuid with the outside, and requires the connectivity of the ﬂuid phase. In our model, this connectivity can be associated with tiny channels (dashed lines in Figure 5.1), of overall negligible mechanical effects, that connect the cavity to the outside. Furthermore, in order to represent a spherical macroscopic strain state E = E1/3, we will consider a uniform radial displacement applied on the external boundary, as already done in Section 4.1.4: ξ = ξ0 e r

at r = B

(5.3)

It is readily understood that the boundary conditions (5.2) and (5.3) induce a spatially varying strain ﬁeld in the solid, so that the local strains are heterogeneous. However, the external sphere appears to be subjected to the homothety H(z) = kz with: k =1+

ξ0 B

(5.4)

Hence, from a purely macroscopic point of view, the strain state is described by the spherical tensor E = E1/3 with: tr E = E =

|| − |0 | ξ0 =3 |0 | B

(5.5)

where |0 | (resp. ||) represents the initial (resp. current) volume of the sphere. Due to the symmetry of the problem, the loading deﬁned by (5.2) and (5.3) induces a radial distribution of stresses T = e r on the external boundary r = B. The corresponding macroscopic stress state is Σ = 1. We look for the macroscopic stress and the pore volume change induced by the loading (E, P). 5.2.1 Direct Solution The response of the saturated sphere to loading (E, P) is derived from the classical solution of the cavity expansion problem already used in Section 4.1. The solution employs a radial microscopic displacement ﬁeld ξ (z) for the solid shell: ξ (z) = ξ (r ) e r

(5.6)

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This displacement ﬁeld is associated with the microscopic strain ﬁeld ε(z): ξ (r ) (e θ ⊗ e θ + e ϕ ⊗ e ϕ ) (5.7) r Use of ε(z) in the isotropic linear elastic stress state equations, and in the momentum balance equation at the microscopic scale, yields a differential equation to be satisﬁed by ξ (r ): d 2ξ (r ) β ′ ξ (r ) + = 0 ⇒ ξ (r ) = α r + 2 (5.8) dr r r ε(z) = ξ ′ (r )e r ⊗ e r +

The microscopic radial stress σrr (r ) is: s

s

σrr (r ) = 3λ α + 2μ

β α−2 3 r

(5.9)

where the coefﬁcients α and β are determined from the boundary conditions (5.2) and (5.3): β s s σrr (A) = 3λ α + 2μ α − 2 3 = −P (5.10) A α+

E β = 3 B 3

(5.11)

That is: α=

4μs E/3 − Pϕ0 ; 3k s ϕ0 + 4μs

β P + ks E = A3 3k s ϕ0 + 4μs

(5.12)

The macroscopic stress = σrr (B) now takes the form: = k hs E − bP

(5.13)

where: k hs =

4k s μs (1 − ϕ0 ) ; 3k s ϕ0 + 4μs

b = ϕ0

3k s + 4μs 3k s ϕ0 + 4μs

(5.14)

Relation (5.13) is the ﬁrst homogenized state equation which relates the macroscopic stress to the loading parameters (E, P), which are macroscopic state variables. It appears as an extension of the macroscopic state equation of the empty porous material to the saturated pressurized porous material. In fact, k hs is the stiffness of the saturated sphere under drained conditions (P = 0), and it is readily understood that it coincides with the stiffness of the empty hollow sphere derived in Chapter 4 (see (4.5)). In turn, the coefﬁcient b takes into account the macroscopic stress induced by a pore pressure increase when the overall deformation of the porous material is blocked (E = 0). It is called the Biot coefﬁcient.

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141

State equation (5.13) is just half of the picture, as the two state variables (E, P) require two corresponding state equations. The ﬁrst is the stress state equation, the second relates to the change of the pore space. Following [12], let us introduce the so-called ‘Lagrangian’ porosity φ that is the ratio of the current volume occupied by the ﬂuid phase over the total reference volume of the sphere (rev): | f |/|0 |. φ is not equal to the so-called Eulerian porosity ϕ which represents the volume fraction of the ﬂuid in the current conﬁguration, but is related to the former by the transport formula: φ=

| f | | f | ; ϕ= =⇒ φ = (1 + trE)ϕ |0 | ||

(5.15)

Note that φ0 = ϕ0 . The advantage of the Lagrangian porosity over the Eulerian porosity stems from the fact that the variation of the Lagrangian porosity is proportional to the pore (i.e. cavity) volume change: f

| f | − |0 | φ − φ0 = |0 |

(5.16)

Hence, this porosity variation is readily derived from the displacement of the solid–ﬂuid interface: φ − φ0 =

P 4π A2 ξ (A) ⇒ φ − φ0 = bE + |0 | N

(5.17)

where: 1 3ϕ0 (1 − ϕ0 ) = s N 3k ϕ0 + 4μs

(5.18)

The coefﬁcient N is referred to as the solid Biot modulus, which quantiﬁes the pore volume change induced by a macroscopic pressure under zero macroscopic strain conditions. Equation (5.17) is the second macroscopic state equation which relates the pore volume change to the macroscopic state variables (E, P). 5.2.2 Energy Approach An alternative way of deriving the macroscopic state equation consists of ˙ denote the rate of mechanical work proevoking the energy approach. Let W ˙ P). ˙ It is the sum of the vided to the solid phase during the evolution ( E, contribution of the surface forces e r acting on the external boundary and of the contribution of pressure P acting on the solid–ﬂuid interface: ˙ ˙ + P | ˙ f | = |0 |( E˙ + P φ) ˙ W= (5.19) ξ˙ · σ · ndSz = || r =A, B

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˙ and | ˙ f | denote the rate of variation of the total volume and of the where || cavity volume, respectively. ˙ is also the work of the internal forces The increment of mechanical energy W within the solid phase along the increment of the strain ﬁeld: ˙ = W σ : ε˙ d Vz = ε : Cs : ε˙ dVz (5.20) A
A
s

where C denotes the elasticity tensor of the solid phase, which relates the microscopic strain to the stress tensor, i.e. σ = Cs : ε. We now introduce the average density of the elastic strain energy in the solid phase, which is a function of the microscopic strain ﬁeld ε . Thus, is a function of the macroscopic loading parameters E and P: 1

(E, P) = ε : Cs : ε dVz (5.21) 2|0 | A < r < B Given the non-dissipative behavior of the solid phase, it is readily understood that the mechanical energy supplied from the outside is entirely stored in the form of elastic strain energy: ˙ = |0 | ˙ W

(5.22)

Finally, let us introduce the potential energy density of the solid phase deﬁned as:

∗ = − P(φ − φ0 )

(5.23)

The use of (5.23) in (5.22) together with (5.19) delivers the macroscopic state equations from: ⎧ ∂ ∗ ⎪ ⎪ = ⎪ ⎨ ∂E

˙ ∗ = E˙ − (φ − φ0 ) P˙ ⇒ (5.24) ⎪ ⎪ ∂ ∗ ⎪ ⎩ φ − φ0 = − ∂P

According to (5.13) and (5.17), and φ − φ0 are linear functions of E and P. Relations (5.24) thus imply that ∗ is a quadratic function of these parameters: P2 1

∗ = k hs E 2 − b P E − 2 2N

(5.25)

It is instructive to note that relations (5.24) and (5.25) justify a posteriori the Maxwell symmetry observed in (5.13) and (5.17), namely the fact that the coefﬁcient of the pressure in (5.13) and that of the strain in (5.17) are both

Generalization

143

equal to the Biot coefﬁcient b: b=−

∂ ∂φ ∂ 2 ∗ = =− ∂P ∂E ∂ P∂ E

(5.26)

Finally, the expression of the elastic energy of the empty sphere already derived in (4.15) is recovered from (5.25) for P = 0, because the elastic energy and the potential energy then coincide. In this case, with the help of (5.13), we obtain for the hollow sphere model: P=0:

1 2

= ∗ = k hs E 2 = hs 2 2k

(5.27)

5.3 Generalization We now move on to the case of an arbitrary geometry of the rev (x). The generalization is based on an appropriate deﬁnition of the mechanical loading. 5.3.1 Definition of a Mechanical Loading on the rev As in Chapter 4, we assume that the macroscopic transformation of the rev is inﬁnitesimal and that the response of the rev at the microscopic scale also meets the conditions of inﬁnitesimal deformation and small displacements. We have already noted in Section 5.1 that the mechanical interaction between ﬂuid and solid is represented by a uniform pressure P in the domain f = P f ∩ which is occupied by the ﬂuid in the rev. s = P s ∩ denotes the domain occupied by the solid in the rev, so that = s ∪ f . For simplicity, we assume that the external boundary ∂ is located in the solid domain1 s . This implies that ∂s = ∂ ∪ ∂ f . The loading applied to s is deﬁned by the pressure P and the tensor E representing the strain state of the rev at the macroscopic scale, as introduced in Section 4.2.5 (see relation (4.40)). This is expressed by the following two boundary conditions: ∂ : ξ = E · z

(5.28)

∂ f : T = −P n

(5.29)

where ξ denotes the microscopic displacement ﬁeld in the solid phase and n is the unit outward normal to the solid. As in the saturated sphere model, the ﬂuid mass is able to ﬂow in or out of the rev through small channels which are assumed to have no effect on the overall mechanical behavior (see Figure 5.2). The connectedness of the pore space is taken into account through the fact that P is regarded as an independent loading parameter. 1 The

more realistic situation in which P f ∩ ∂ = ∅ has been addressed in [13].

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144

E .z Ω

Ωs Ωf P

Figure 5.2 The rev subjected to uniform strain boundary conditions with its connected pore space filled with a fluid at pressure P

The total volume change is immediately derived from (5.28): || − |0 | = ξ · ndSz = |0 | trE

(5.30)

∂0

We now generalize the deﬁnition of the Lagrangian porosity φ introduced by (5.15). As before, it is deﬁned as the ratio of the current ﬂuid volume | f | over the initial total volume of the rev |0 |, and is related to the classical Eulerian porosity ϕ by: φ=

| f | ; |0 |

φ = (1 + tr E)ϕ

(5.31)

It is important to clearly distinguish the two porosity measurements. Given the assumed inﬁnitesimal order of the macroscopic deformation, for which |tr E| ≪ 1, the approximation φ ≈ ϕ may hold. The difference between the variation of φ and that of ϕ, however, is not negligible. In fact, since φ0 = ϕ0 , relations (5.31) yield: φ − φ0 = ϕ − ϕ0 + ϕ0 tr E

(5.32)

where second-order terms are neglected. As shown by (5.31), the variation φ − φ0 of the Lagrangian porosity proves useful for the description of pore volume changes: f

| f | − |0 | φ − φ0 = |0 |

(5.33)

In the sequel, we will therefore encounter φ − φ0 rather than the Lagrangian porosity φ itself. In contrast, (5.32) shows that the variation of the Eulerian porosity does not provide a measurement of the pore volume change in so far as it is also affected by the volume change of the rev that is represented by tr E.

Generalization

145

Finally, the behavior of the solid matrix is linear elastic and homogeneous in s : σ = Cs : ε; ε = Ss : σ

(5.34)

where Ss = (Cs )−1 is the elastic compliance tensor. 5.3.2 Homogenized State Equations Under the assumptions developed in Section 5.3.1, it is readily understood that the strain and stress ﬁelds ε(z) and σ(z) deﬁned on s are linear functions of the loading parameters E and P. This property can be expressed through the introduction of the fourth-order tensor A which takes into account the local strain induced by the macroscopic strain E, and the second-order tensor A′ which takes into account the local strain induced by the pressure P: ε(z) = A(z) : E − A′ (z)P

(5.35)

A and A′ can be referred to as strain concentration tensors. The microscopic state equation (5.34) yields the expression of the microscopic stress ﬁeld σ(z):

σ(z) = Cs : A(z) : E − Cs : A′ (z)P

(5.36)

The macroscopic stress tensor Σ is now derived from the microscopic stress ﬁeld using the average rule (1.39): 1 −P1 dVz = (1 − ϕ0 )σ s − Pϕ0 1 (5.37) σ(z) dVz + Σ=σ= f |0 | s Inserting (5.36) into (5.37) yields: Σ = Chom : E − BP

(5.38)

where B and Chom are given by: s

B = ϕ0 1 + (1 − ϕ0 )Cs : A′ ; Chom = (1 − ϕ0 )Cs : A

s

(5.39)

Relation (5.38) is the ﬁrst macroscopic state equation of linear poroelasticity. Letting P = 0, it is readily seen that (5.38) reduces to state equation (4.113) derived for the empty porous media in Chapter 4. The deﬁnitions of Chom given by (5.39) and (4.115), therefore, remain valid in the saturated case, as hom well as the symmetry conditions of Chom (Cihom jkl = C kli j ) derived in Chapter 4 (see (4.123)): s

Chom = (1 − ϕ0 )Cs : A = (1 − ϕ0 )tA : Cs : A

s

(5.40)

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146

Furthermore, the variation of the pore volume f ﬁlled by the ﬂuid is related to the microscopic displacement ﬁeld ξ by: f = − ξ · n dSz (5.41) ∂ f

where n is the unit outward normal to the solid. This change in pore volume f is proportional to the change in Lagrangian porosity. Therefore, expression (5.41), through its link to ξ , implies that φ − φ0 is a linear function of the loading parameters E and P; that is: f = |0 |(φ − φ0 ) = − ξ ·n dSz + ξ ·n dSz ∂ f ∪∂

= −

s

∂

tr ε dV + |0 |tr E

(5.42)

Finally, using (5.35) in (5.42), we obtain the second macroscopic state equation of poroelasticity: φ − φ0 =

P + B′ : E N

(5.43)

where N and B′ are given by: 1 s s = (1 − ϕ0 )tr A′ ; B′ = 1 − (1 − ϕ0 )1 : A N

(5.44)

With (5.39) and (5.44) we arrive at two second-order tensor expressions B and B′ . We are left with the proof that B = B′ to complete the presentation of the macroscopic poroelastic behavior. In fact, this proof establishes the symmetry of the homogenized state equations (5.38) and (5.43). Only if this result is established will it be possible to determine a thermodynamic potential for the macroscopic behavior of a porous medium made up of an elastic solid matrix. 5.3.3 Symmetry of the Homogenized State Equations For the proof of the symmetry of the state equations (5.38) and (5.43), we apply the Maxwell–Betti theorem to the solid s subjected to the following two loadings:

r loading C1 deﬁned by E = E0 and P = 0; r loading C2 deﬁned by E = 0 and P = P0 . Let ξ (i) and σ (i) denote the displacement and stress ﬁelds deﬁned on s associated with loading Ci . σ (1) (resp. σ (2) ) can be extended into the pore space f by the condition σ (1) = 0 (resp. σ (2) = −P0 1). We also introduce the stress

Generalization

147

vector T (i) acting on ∂s . The Maxwell–Betti theorem is: (1) (2) ξ (2) · T (1) dSz ξ · T dSz =

(5.45)

∂s

∂s

Observing that ∂s = ∂ ∪ ∂ f, and taking the boundary conditions of each loading into account, we ﬁrst observe that the r.h.s. of (5.45) is equal to zero. Accordingly, this equation can be rearranged in the form: (2) (E0 · z) · T dSz = P0 (5.46) ξ (1) · n dSz ∂ f

∂

where n is the outward unit normal to the solid. Using (5.38), we obtain: σ (2) dV = −P0 B|0 | (5.47)

The integral on the l.h.s. of (5.46) thus becomes: (2) (E0 · z) · T dSz = E0 : σ (2) dV = −P0 |0 |B : E0 ∂

(5.48)

The integral on the r.h.s. of (5.46), which displays the displacement ﬂux ξ (1) · n , is related to the Lagrangian porosity change in loading C1 (see (5.41)). We therefore evaluate this integral with (5.43): P0 (5.49) ξ (1) · n dSz = −P0 |0 |B′ : E0 ∂ f

Hence, from a comparison of (5.48) and (5.49), it is readily seen that B = B′ is a consequence of the Maxwell–Betti theorem (5.46): B = B′ = 1 − (1 − ϕ0 )1 : A

s

(5.50)

The macroscopic state equations (5.38)–(5.43) are then: Σ = Chom : E − BP

(5.51)

P = N(−B : E + φ − φ0 )

(5.52)

with (from a combination of (5.39) and (5.44)): 1 = 1 : Ss : (B − ϕ0 1) N

(5.53)

For P = 0, the porous medium behavior reduces to that of a homogeneous linear elastic solid deﬁned by tensor Chom . The elasticity tensor Chom , therefore, is referred to as the tensor of elastic moduli in drained conditions, or drained elasticity tensor. Under the same conditions, the dimensionless tensor B allows us to derive the pore volume change induced under drained conditions by

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148

a given macroscopic strain. Furthermore, this second-order tensor, which is referred to as the tensor of Biot coefﬁcients, represents the macroscopic stress induced in the saturated medium by a unit pore pressure increase under zero macroscopic strain conditions. Under the same conditions, the Biot solid modulus N can be used to evaluate the pore volume change in consequence of a pressure increase. Finally, the stress quantity Σ + BP controls the value of the macroscopic strain. It is therefore referred to as macroscopic effective stress. 5.3.4 Energy Approach We established the symmetry of the state equations by evoking the Maxwell– Betti theorem. This symmetry could have been directly established by means of the energy approach developed in Section 5.2.2 for the saturated hollow ˙ provided sphere model. In the general case, the rate of mechanical work W s ˙ ˙ to the solid phase during the loading increment ( E, P) comprises two contributions:

r The work of the pore pressure acting on ∂ f = I s f . r The work of the surface forces σ · n acting on ∂ in the incremental displacement E˙ · z prescribed on this boundary.

That is: ˙ = W

∂

˙ · (σ · n) dSz + (z · E)

Is f

−Pn · ξ˙ dSz

(5.54)

where n is the unit normal vector to ∂s oriented outwards w.r.t. the solid s . The ﬂux of the velocity ξ˙ through the solid–ﬂuid interface in the second integral is equal to the pore volume rate: (5.55) −n · ξ˙ dSz = |0 |φ˙ Is f

Finally, (5.55) and (4.29)2 are inserted into (5.54) to yield the generalized form of (5.19):

˙ = |0 | i j E˙ i j + P φ˙ = |0 | Σ : E˙ + P φ˙ W (5.56)

Following the derivation of Section 5.2.2, we now deﬁne the macroscopic elastic energy density (E, P) and the potential energy density ∗ (E, P) of 2 Recall that (4.29) holds for any stress ﬁeld σ satisfying both the local equilibrium condition and the uniform stress boundary condition.

Generalization

149

the rev: 1

(E, P) = 2|0 |

s

ε : Cs : ε dVz ;

∗ = − P(φ − φ0 )

(5.57)

As in (5.21), the increment of elastic energy is related to the mechanical energy ˙ = |0 | . ˙ by W ˙ It follows that: supply W ⎧ ∂ ∗ ⎪ ⎪ Σ = ⎪ ⎨ ∂E (5.58)

˙ ∗ = Σ : E˙ − (φ − φ0 ) P˙ ⇒ ⎪ ⎪ ∂ ∗ ⎪ ⎩ φ − φ0 = − ∂P

Relations (5.58) prove that the potential energy density ∗ is the macroscopic thermodynamic potential which depends on the macroscopic strain tensor E and the pore pressure P. Given the linearity of the macroscopic state equations (5.51) and (5.52), ∗ is a quadratic function of its arguments E and P: 1 P2

∗ = E : Chom : E − −PB:E 2 2N

(5.59)

The symmetry of the macroscopic state equations (5.38) and (5.43) appears here as an immediate consequence of (5.58) and the very existence of a macroscopic thermodynamic potential, which ensures that B = B′ as a consequence of the Maxwell symmetry relations: B=−

∂ ∗ ∂Σ =− ; ∂P ∂E∂ P

B′ =

∂(φ − φ0 ) ∂ ∗ =− ∂E ∂ P∂E

(5.60)

Note that a combination of (5.57) with (5.59) also provides us with the expression for : 1 P2

= E : Chom : E + 2 2N

(5.61)

5.3.5 The Macroscopic Variable Set (E, m) The state equations (5.51) and (5.52) were derived from averaging the solid response of the porous medium only. This led to expressions of the macroscopic stress Σ and the ﬂuid pressure P as functions of the state variables E and φ − φ0 , the ﬁrst representing the macroscopic strain, the second the change in pore volume occupied by the ﬂuid phase. Some applications, however, require knowledge and control of the ﬂuid mass, rather than the ﬂuid volume. For this purpose, it is necessary to relate the ﬂuid volume to the ﬂuid mass, and develop the macroscopic counterpart of the state equations (5.51) and (5.52) as

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150

functions of the change of the ﬂuid mass currently contained in the rev. This involves inspecting the ﬂuid mass at different scales. Following the classical macroporomechanics theory,3 we introduce the ﬂuid mass content m deﬁned as the total ﬂuid mass contained in f divided by the initial volume |0 | of the rev: 1 m= ρ dVz (5.62) |0 | f Following the conclusions of Section 2.2 and the opening remarks in Section 5.1, the deviation of the microscopic pressure ﬁeld p(z) around its average P = p f in the rev is negligible. A similar argument can be employed for the deviation of the ﬂuid density ρ(z) around its intrinsic average ρ f, simply denoted by ρ f : (∀z ∈ f ) ρ(z) ≈ ρ f = ρ f

(5.63)

Using (5.63) together with (5.31), it is readily seen that the ﬂuid mass content m can be approximated by ρ f φ . The change in ﬂuid mass content per initial total volume of the rev expressed in terms of the Lagrangian porosity is then: f

m − m0 = φρ f − ϕ0 ρ0

(5.64)

In order to ensure the linearity of the macroscopic state equations with regard to the macroscopic state variables E and m − m0 , it is necessary to assume that the variations of the ﬂuid mass density around a reference value are small: f ρ f /ρ0 ≪ 1. In this case, (5.64) then yields: m − m0 f

ρ0

= φ − φ0 + ϕ0

ρ f f

ρ0

(5.65)

Furthermore, given the inﬁnitesimal order of the ﬂuid mass density variation, a linear form of the ﬂuid state equation linking the pressure and the density can be employed: P = kf

ρ f f

ρ0

(5.66)

where k f is the ﬂuid bulk modulus. Finally, using (5.66) in (5.65), and substituting the result in (5.52), yields:

m − m0 (5.67) P = M −B : E + f ρ0 3 See

[8] and [12].

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151

where M is the Biot modulus of the whole porous material – solid plus ﬂuid phase: 1 1 ϕ0 = + f M N k

(5.68)

Finally, inserting (5.67) into (5.51) yields the macroscopic stress state as a function of the state variables E and m − m0 : Σ = Cuhom : E − MB

m − m0 f

ρ0

(5.69)

where: Cuhom = Chom + MB ⊗ B

(5.70)

In the absence of ﬂuid mass exchange with the outside (m = m0 ), which is referred to as undrained conditions, the porous medium behaves like an elastic solid characterized by the elasticity tensor Cuhom . Given these particular boundary conditions, Cuhom is called the undrained elastic stiffness tensor. 5.4 Application: Estimates of the Poroelastic Constants and Average Strain Level 5.4.1 Microscopic and Macroscopic Isotropy It is an instructive exercise to investigate the structure of the state equations in the isotropic case. To this end, let us consider that the solid behavior is isotropic and deﬁned by the elasticity tensor: Cs = 3k s J + 2μs K

(5.71)

where k s and μs are the bulk modulus and the shear modulus of the solid phase, and J and K are the fourth-order tensors deﬁned by (4.88). Let us further consider an isotropic morphology of both solid and pore space. In this s case, the average strain concentration tensor A can be divided into a spherical and a deviator part:4 s

A = Asv J + Asd K

(5.72)

Use in (5.39), (5.50) and (5.53) provides the drained stiffness tensor: Chom = 3k hom J + 2μhom K; k hom = (1 − ϕ0 )k s Asv ; μhom = (1 − ϕ0 )μs Asd

(5.73)

4 Note that this property does not hold for the local strain concentration tensor A(z). This is due to the presence of the pores which locally affect the strain ﬁeld in an anisotropic way.

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152

the Biot tensor: B = b1; b = 1 − (1 − ϕ0 ) Asv = 1 −

k hom ks

(5.74)

and the Biot moduli: 1 b − ϕ0 1 b − ϕ0 ϕ0 = ; = + f s s N k M k k

(5.75)

A comparison of (5.73) to (4.92) shows that Asv and Asd capture the discrepancy between the homogenized bulk and shear moduli with respect to the Voigt bounds, which are formally identical to the so-called ‘direct average rule’. Finally, through (5.68), the undrained stiffness properties are: kuhom = k hom + Mb 2 ; μuhom = μhom

(5.76)

It is interesting (but not surprising) to note that the shear modulus μhom of the skeleton is identical in drained and undrained conditions. Relations (5.69) and (5.67) take the form: m − m0 2 hom hom tr(E)1 + 2μhom E − b M Σ = ku − μ 1 (5.77) f 3 ρ0

P = M −b tr E +

m − m0 f

ρ0

whereas (5.51) and (5.52) become: 2 hom hom Σ= k − μ tr(E)1 + 2μhom E − b P1 3 P = N(−b tr E + φ − φ0 )

(5.78)

(5.79) (5.80)

What the previous relations show is that an isotropic microscopic behavior and an isotropic geometry of the microscopic phases necessarily lead to a macroscopic isotropic poroelastic behavior. The inverse, however, is not true. For instance, a microheterogeneous porous medium composed of an anisotropic solid phase which is randomly distributed at a microscopic scale entails a macroscopic isotropic behavior. In return, in the isotropic microhomogeneous case, only one of the three poroelastic constants k hom , b and N needs to be determined from an upscaling scheme, while the others follow from (5.74) and (5.75), provided that the solid stiffness k s is known. A straightforward illustration of this remark consists of applying the Voigt bound (4.92) obtained in the isotropic case to the derivation of a bound for B = b1. Use of this result in (5.74) yields a lower bound for the Biot coefﬁcient while an upper bound is

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153

derived from (4.125): k hom ≤1 (5.81) ks The combination of (5.75) with the lower bound b ≥ ϕ0 also ensures the positivity of the solid Biot modulus N. In turn, the combination of (5.75) with the upper bound b ≤ 1 yields a lower bound for N: ϕ0 ≤ b = 1 −

N≥

ks 1 − ϕ0

(5.82)

In a similar way, any reﬁned estimate of k hom yields a reﬁned estimate of b and N. For instance, the dilute estimates of b and N are derived from (4.159): 3k s 4μs dil b dil = ϕ0 1 + ; N (5.83) = 4μs 3ϕ0 Of course, the validity of the estimates b dil and Ndil is restricted, as is that of the dilute estimate of k hom , to inﬁnitesimal values of the porosity. 5.4.2 Microscopic and Macroscopic Anisotropy We look for the link between the poromechanical coupling properties, N and B, the elastic properties of the solid phase, compliance tensor Ss or stiffness tensor Cs , and the macroscopic stiffness tensor Chom . This derivation is restricted to microhomogeneous porous media. We consider the particular loading deﬁned by a pore pressure P and the macroscopic strain E = −P Ss : 1. It is readily seen that the stress and strain solutions in s of the problem deﬁned by (5.28), (5.29) and (5.34) are uniform in s : σ = −P1; ε = −P Ss : 1; ξ (z) = −P Ss : 1 · z

(5.84)

It follows that the macroscopic stress state is equal to Σ = σ = −P1. Use of this result in (5.38) yields the following expressions for tensor B:5 B = 1 − Chom : Ss : 1 = 1 − 1 : Ss : Chom

(5.85)

in which the symmetry of the tensors Chom and Ss has been used. Note that the above relation between B, Chom and Ss can be alternatively derived from s (5.39) and (5.50). It sufﬁces to eliminate A between the expressions for Chom and B. Combining the expression (5.84) of the displacement ﬁeld with (5.41), the change in Lagrangian porosity takes the form: φ − φ0 = −Pϕ0 1 : Ss : 1 5 See

[4] for the derivation of this result in the context of periodic porous media.

(5.86)

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A comparison of (5.86) to the macroscopic state equation (5.52) allows us to recover the expression (5.53) for the Biot modulus N:

1 = (B − ϕ0 1) : Ss : 1 = (1 − ϕ0 )1 − Chom : Ss : 1 : Ss : 1 N

(5.87)

We verify that (5.85) and (5.87) reduce in the microscopic and macroscopic isotropic case to expressions (5.74) and (5.75). In summary, provided that the elastic properties of the solid are known, (5.85) and (5.87) show that the coefﬁcients N and B that characterize the poromechanical coupling can be derived from the value of the macroscopic drained stiffness tensor Chom . From a methodological as well as practical point of view, this result is very important since it implies that the homogenization of a saturated (elastic) material with a pressurized ﬂuid phase can be reduced to that of the empty porous material. Hence, the main focus of a micromechanics investigation of the poroelastic constants is Chom . 5.4.3 Average Strain Level in the Solid Phase We have studied in Section 4.4 the average strain level reached in the solid phase for an empty porous medium. It is interesting to investigate the strain level in the solid phase when the macroscopic loading includes the pressure P in the pore space. Let us ﬁrst consider the average strain ε s , which is obtained from (4.166): ε s = Ss : σ s

(5.88)

Combining (5.88) with (5.37) gives the strain average as a function of the macroscopic stress and the pore pressure: (1 − ϕ0 )ε s = Ss : (Σ + Pϕ0 1)

(5.89)

1 tr (Σ + Pϕ0 1) 3k s 1 (1 − ϕ0 )εds = Σd 2μs

(5.90)

Interestingly, we note that εs is controlled by the effective stress Σ + Pϕ0 1 . In the case of an isotropic behavior of the solid phase, (5.89) can be split into a spherical and a deviatoric part: (1 − ϕ0 )tr ε s =

where Σd = K : Σ is the deviatoric macroscopic stress. Let us then consider the equivalent shear strain εd deﬁned by (4.171). We restrict ourselves to the case of an isotropic solid phase. The quadratic average s εd2 was determined for the empty porous medium as a function of the macroscopic strain (see (4.177)). We aim at extending this result by incorporating the inﬂuence of the pore pressure. We start from the deﬁnition of the potential

Application: Estimates of the Poroelastic Constants and Average Strain Level

energy of the pressurized solid phase (see (5.57)): 1 ∗ s |0 | = ε : C : ε dVz − P tr ε dVz 2 s f

155

(5.91)

Recalling that ∂ Cs /∂μs = 2K, it is readily seen that: ∂ ∗ ∂ε ∂ε s |0 | s = : C : ε dVz + dVz (5.92) εd : εd dVz + −P1 : s ∂μ ∂μs s f s ∂μ and after rearrangement: s ∂ ∗ ∂ε = 2(1 − ϕ0 )εd2 + σ : s ∂μ ∂μs

(5.93)

As shown in Section 4.4, application of the Hill lemma makes the second term on the r.h.s. vanish. Recalling (5.59), we then obtain: s ∂ 1 P2 hom 2(1 − ϕ0 )εd2 = E : C : E − − P B : E (5.94) ∂μs 2 2N This expression can be further simpliﬁed in the case of macroscopic isotropy. Recalling (5.74) and (5.75), the derivatives of B and 1/N take the form: 1 ∂k hom 1 ∂k hom ∂ 1 ∂ = − (B) = − 1; (5.95) ∂μs k s ∂μs ∂μs N k s 2 ∂μs Substituting (5.95) into (5.94), we ﬁnally obtain: s P 2 ∂μhom 1 ∂k hom 2 2(1 − ϕ0 )εd = tr E + s + Ed : Ed 2 ∂μs k ∂μs

(5.96)

In comparison to (4.177), the pore pressure is taken into account in (5.96) by replacing the volume strain tr E in (4.177) with tr E + P/k s . Alternatively, the state equation (5.51) can be used in order to derive an s expression for εd2 as a function of the macroscopic stress and of the pore pressure. Taking the expression (5.74) for the Biot coefﬁcient b into account, (5.51) can be split into its spherical and deviatoric parts: 1 P hom (5.97) tr (Σ + P1) = k tr E + s ; Σd = 2μhom Ed 3 k Substituting (5.97) in (5.96) then yields: s 1 1 1 ∂ 1 ∂ 2 2(1 − ϕ0 )ε2d = − (tr (Σ + P1)) − Σd : Σd 18 ∂μs k hom 4 ∂μs μhom (5.98)

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Whereas the average deviatoric strain εds is independent of the pore pressure s (see (5.90)), the quadratic average εd2 appears to depend on the pore pressure through Terzaghi’s effective stress Σ + P1. The same methodology can be applied to determine the quadratic average s s εv2 =εv introduced in (4.179). In particular, it is readily found that (5.94) is now replaced by: s ∂ 1 (1 − ϕ0 )εv2 = s 2 ∂k

P2 1 hom E:C :E− −PB:E 2 2N

(5.99)

5.5 Levin’s Theorem in Linear Microporoelasticity 5.5.1 An Alternative Route to the Poroelastic State Equations Throughout this chapter, the homogenization theory focussed on the solid only: in the mechanical problem deﬁned at the microscopic scale, the material system under consideration was the solid domain s of the rev, subject to the boundary conditions (5.28) and (5.29). In particular, the strain concentration tensors A(z) and A′ (z) were deﬁned on s only. We now want to take an alternative route, which consists of deﬁning the mechanical problem on the whole rev. We have already adopted this approach in Section 4.2.2. Such an approach requires us to extend the deﬁnition of the microscopic stress, strain and displacement, σ, ε and ξ , respectively, from the solid domain s into the pore space f . Hence, in addition to the linear elastic stresses in the solid, we want to express the fact that the microscopic stress ﬁeld in the ﬂuid is uniform and equal to −P1. A convenient way of writing this in a uniﬁed way is to consider the stress tensor σ(z) everywhere in the rev in an afﬁne form: (∀z ∈ ) σ(z) = C(z) : ε(z) + σ p (z)

(5.100)

where C(z) is a heterogeneous stiffness tensor, and σ p (z) the prestress tensor: C(z) =

0 Cs

( f ) (s )

σ p (z) =

−P1 ( f ) 0 (s )

(5.101)

Strictly speaking, the displacement ξ is not deﬁned in a unique way in the pore space f if the stiffness C(z) is taken equal to zero in this domain. However, it is expected that the macroscopic response of the rev is affected by the strain ﬁeld in the pores only through its average ε p . Indeed, let us recall that the continuity of ξ across the interface between s and f implies that ε p is entirely determined by the values of ξ on this interface, and not by the

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157

value of C(z), since (see (4.21)): 1 ε = 2| f | p

∂ f ∩∂s

(ξ ⊗ n + n ⊗ ξ ) dSz

(5.102)

For a spatial distribution of C(z) and σ p (z) in the rev deﬁned by (5.101), relation (5.100) written in the solid phase reduces to the linear elastic state equation. By contrast, (5.100) written in the pore space does not represent a constitutive relation of the ﬂuid, but rather the stress state in the ﬂuid, namely a uniform pressure6 P in f . The microscopic stress, strain and displacement which characterize the response of the rev to the two loading parameters, the macroscopic strain tensor E and the microscopic prestress ﬁeld σ p , satisfy: div σ = 0 () p σ = C(z) : ε + σ (z) () ξ =E·z (∂)

(5.103)

Given the linearity of the problem at hand with respect to E and σ p , it is convenient to separate the problem in two load cases: 1. The ﬁrst load case we consider is the loading by E only, while the prestress ﬁeld is zero: div σ ′ = 0 σ ′ = C(z) : ε′ ξ′ = E · z

() () (∂)

(5.104)

Boundary problem (5.104) is nothing more than the micromechanical problem of the empty porous medium, which we addressed in Chapter 4. In particular, the microscopic strain ﬁeld is: ε′ (z) = A(z) : E

(5.105)

The corresponding macroscopic stress Σ′ is: Σ′ = σ ′ = Chom : E

(5.106)

where Chom is the homogenized elasticity tensor of the empty porous medium, which is still deﬁned by (4.115). 2. The second load case corresponds to a loading deﬁned by the prestress ﬁeld σ p only, while the macroscopic strain is zero, corresponding to a zero

6 Viscous

stresses are neglected (see Section 2.5.2).

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(microscopic) displacement on the rev boundary. In this case: div σ ′′ = 0 ′′

() ′′

p

σ = C(z) : ε + σ (z)

()

′′

ξ =0

(5.107)

(∂)

The corresponding macroscopic stress Σ′′ is the average of σ ′′ over . In order to derive Σ′′ as a function of the prestress ﬁeld, we apply Hill’s lemma (4.46) to the strain ﬁeld of ε′ (5.105) and to the stress ﬁeld σ ′′ : σ ′′ : ε′ = Σ′′ : E

(5.108)

Using (5.107), the l.h.s. of (5.108) is: σ ′′ : ε′ = ε′′ : C(z) : ε′ + σ p : ε′

(5.109)

Observing that ε′′ = 0 in this load case, a second application of Hill’s lemma to the strain ﬁeld ε′′ and to the stress ﬁeld σ ′ = C(z) : ε′ yields ε′′ : C(z) : ε′ = 0. Finally, a combination of (5.108) and (5.109) with (5.105) yields: Σ′′ = σ p : A

(5.110)

This equation is related to the classical Levin’s theorem.7 According to the deﬁnition of σ p (see (5.101)), Σ′′ is recognized as the macroscopic stress which is generated by a pressure increase under zero macroscopic strain conditions: Σ′′ = −Pϕ0 1 : A

p

(5.111) p

With the help of the consistency condition (4.111), that is ϕ0 A + s (1 − ϕ0 ) A = I, the previously derived formula (5.50) also reads: B = ϕ0 1 : A

p

(5.112)

This shows that Σ′′ is actually proportional to the tensor of Biot coefﬁcients, B: Σ′′ = −PB

(5.113)

The macroscopic stress induced by both loading parameters, E and σ p , is Σ = Σ′ + Σ′′ . Collecting the results from (5.106) and (5.110), the ﬁrst state equation of the poroelasticity theory is obtained: Σ = Chom : E + σ p : A 7 See

[38],[36].

(5.114)

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159

or, with (5.111): Σ = Chom : E − PB

(5.115)

where: B = ϕ0 1 : A

p

(5.116)

We now turn to the second state equation of poroelasticity, describing the pore volume change. The starting point here is the expression of the variation of the Lagrangian porosity as a function of the strain ﬁeld in the pore space: φ − φ0 = ϕ0 tr ε p

(5.117)

We then use the fact that the strain ﬁeld ε solution of (5.103) is the sum of ε′ (solution of (5.104)) and ε′′ (solution of (5.107)). As regards the ﬁrst solution, p tr ε′ is readily derived from a combination of (5.105) and (5.116): p

p

ϕ0 1 : ε′ = ϕ0 1 : A : E = B : E

(5.118)

In return, the solution ε′′ to the second load case satisﬁes ε′′ = 0, which implies: p

ϕ0 ε′′ = −(1 − ϕ0 )ε′′

s

(5.119)

′′ s

where ε is the mean strain ﬁeld in the solid phase. Expression (5.119) makes p it possible to determine ε′′ as a function of P. In fact, the state equation in the solid domain shows that: s

(1 − ϕ0 )ε′′ = (1 − ϕ0 )Ss : σ ′′ ′′ p

Observing that σ = −P1, we note that σ rule (5.113), i.e. Σ′′ = σ ′′ = −P B:

′′ s

s

(5.120)

can be derived from the average

s

(1 − ϕ0 )σ ′′ = P(ϕ0 1 − B)

(5.121)

′′ p

Inserting this result into (5.120)–(5.119) yields ε : p

ϕ0 1 : ε′′ = P1 : Ss : (−ϕ0 1 + B)

(5.122)

Finally, the total change in Lagrangian porosity change is obtained by a superposition of (5.118) and (5.122): φ − φ0 = B : E +

P N

(5.123)

where the expression for N already derived in (5.53) and in (5.87) is recovered: 1 = 1 : Ss : (−ϕ0 1 + B) N

(5.124)

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5.5.2 An Instructive Exercise: The Prestressed Initial State A direct application of Levin’s theorem is to study an initial prestress state of a porous medium; that is, the most common condition in many engineering applications, for instance in rock and soil mechanics, that an initial macroscopic stress Σ0 and an initial ﬂuid pressure P0 prevail in the reference conﬁguration. The porous medium, therefore, is not in a natural state, in which stress and pressure would be zero. We consider an rev subjected to a stress ﬁeld σ 0 which extends into the pore space f in the form of the pore pressure P0 . The macroscopic initial stress is Σ0 = σ 0 . The afﬁne behavior of the solid phase and the existence of a pore pressure P in the ﬂuid are summarized as follows: (∀z ∈ ) σ(z) = C(z) : ε(z) + σ p (z)

(5.125)

with: C(z) =

0 ( f ) Cs (s )

p

σ (z) =

−P1 ( f ) σ 0 (s )

(5.126)

Note that the prestress σ p can be recast in the form: σ p (z) = σ 0 (z) − (P − P0 )χ f (z)1

(5.127)

where χ f is the characteristic function of the ﬂuid phase. The boundary value problem to be satisﬁed by the microscopic stress, strain and displacement ﬁelds is identical to (5.103) and its decomposition into (5.104) and (5.107) is still valid. In order to use (5.114), we just have to evaluate the macroscopic prestress σ p : A. To this end, observe that the strain ﬁeld ε′ deﬁned in (5.105) is kinematically admissible with the macroscopic strain E and that the initial stress ﬁeld σ 0 satisﬁes the momentum balance, div σ 0 = 0. Hill’s lemma (4.46) therefore implies that: (∀E) σ 0 : ε′ = Σ0 : E ⇒ σ 0 : A = Σ0

(5.128)

Recalling (5.116) and using (5.127) in (5.114), the stress equation of state is thus: Σ = Σ0 + Chom : E − B(P − P0 )

(5.129)

The starting point for the derivation of the second state equation is (5.117): p

φ − φ0 = ϕ0 tr ε p = ϕ0 1 : ε′ + ϕ0 1 : ε′′

p

(5.130)

Training Set: The Two-Scale Double-Porosity Material

161

p

The term ϕ0 1 : ε′ is still given by (5.118). It is convenient to split (5.107) into two subproblems: div σ ′′1 = 0 σ ′′1 = C(z) ξ ′′1 = 0

:

() ε′′1

()

+ σ 0 (z)

(5.131)

(∂)

and: div σ ′′2 = 0 σ ′′2 = C(z) ξ ′′2 = 0

:

() ε′′2

f

− (P − P0 )χ (z)1 ()

(5.132)

(∂)

It is readily seen that the solution to the ﬁrst subproblem (5.131) is ξ ′′1 = 0 and σ ′′1 = σ 0 , so that: p

1 : ε′′1 = 0

(5.133)

The solution to the second subproblem (5.132) is obtained from the solution of (5.101)–(5.107), in which P is replaced by P − P0 . Equation (5.122) thus yields: p

ϕ0 1 : ε′′2 =

P − P0 N

(5.134)

Finally, combining (5.118), (5.133) and (5.134), the change in Lagrangian porosity takes the form: φ − φ0 = B : E +

P − P0 N

(5.135)

Equations (5.129) and (5.135) are the two poroelastic state equations for a prestressed initial state.

5.6 Training Set: The Two-Scale Double-Porosity Material One situation which is frequently encountered in ‘real’-life material applications is where the porosity of materials manifests itself at different scales, which are separated by at least one order of length magnitude. The focus of this training set is the derivation of the macroscopic state equations of such a two-scale double-porosity material. The rev is considered to be composed of a porous matrix M and a macropore domain P (Figure 5.3). In the sequel, the physical quantities related to M and P are denoted by the subscripts 1 and 2, respectively. The volume fraction of P is ϕ2 . It is recalled that the variations of ϕ2 w.r.t. the initial conﬁguration are negligible. The superscript 0 characterizes a quantity in the initial conﬁguration.

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P1

M

Ω Ω

P

Ω

P2

Figure 5.3 The rev of a double-porosity material

Let f 1 (z) denote the local (Lagrangian) porosity in M. Accordingly, the (normalized) pore volume φ1 located inside M is obtained by integration: φ1 =

1 ||

M

f 1 (z) dVz

(5.136)

φ1 represents the (Lagrangian) porosity associated with M, at the scale of the rev. The behavior of the porous matrix, considered to be a microhomogeneous phase, is assumed to be governed by a linear poroelastic behavior. The state equations of the matrix are then: σ = C M : ε − B M P1 P1 f 1 − f 10 = B M : ε + M N

(a )

(5.137)

(b)

(5.137)

where C M, B M, N M are the poroelastic properties of the porous matrix, while P1 is the (uniform) pressure prevailing in the microporosity of the porous matrix. By contrast, the (uniform) pressure in the macropore space P is P2 = P1 . We adopt a continuous description of the stresses in the heterogeneous rev of the form (5.100): (∀z ∈ ) σ(z) = C(z) : ε(z) + σ p (z)

(5.138)

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163

together with the distributions of C(z) and σ p (z) on account of (5.137a): ( P ) 0 ( P ) −P2 1 p (z) = σ C(z) = (5.139) M M M C ( ) −B P1 ( M) Following the approach developed in Section 5.5.1, we decompose the problem into two subproblems. The ﬁrst subproblem corresponds to overall drained conditions, for which P1 = P2 = 0. Application of a strain localization condition (5.105) yields the macroscopic stress (5.106) on account of the elasticity distribution (5.139) in the form: Σ′ = σ ′ = Chom : E C

hom

= (1 − ϕ2 ) C

M

(a ) :A

M

(5.140) (b) M

where Chom is recognized as the overall drained stiffness tensor. A is the average strain concentration tensor of the porous matrix. On the other hand, there are two contributions to the change of porosity, associated with the microporosity and the macroporosity. The microporosity change (5.137b) is deﬁned per unit of (undeformed) matrix volume M, so that the change of porosity at the macroscale in the considered subproblem is: M

′ φ1 − φ10 = (1 − ϕ2 ) ( f 1 − f 10 )′ = B1 : E (a )

B 1 = ( 1 − ϕ2 ) B M : A

M

In return, the change in macroporosity is given by (5.118):

′ φ2 − φ20 = B2 : E P M B2 = ϕ2 1 : A = 1 : I − (1 − ϕ2 )A

(5.141)

(b)

(a )

(5.142)

(b)

The second subproblem we consider is the zero-displacement boundary problem deﬁned by (5.107), for which Levin’s theorem (5.110) applies. Application of the prestress distribution (5.139) in (5.110) yields the macroscopic stress in this subproblem: ′′ = −B1 P1 − B2 P2

(5.143)

B1 and B2 obtained in the ﬁrst subproblem (5.141b) and (5.142b) are the tensors of Biot coefﬁcients associated with the pressures in the micro- and the macroporosity, respectively. Again, there are two contributions to the change of porosity to be considered: the change in microporosity:

P1 M M 0 ′′ ′′ φ1 − φ1 = (1 − ϕ2 ) B : ε + M (5.144) N

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164

and the change in macroporosity:

′′ P M φ2 − φ20 = ϕ2 1 : ε′′ = − (1 − ϕ2 ) 1 : ε′′ We need to eliminate ε′′ stress average condition:

M

(5.145)

in (5.144) and (5.145). To this end, we apply the

Σ′′ = (1 − ϕ2 ) σ ′′

M

+ ϕ2 σ ′′

= (1 − ϕ2 ) C M : ε′′

M

P

− (1 − ϕ2 ) B M P1 − ϕ2 P2 1

(5.146)

Then, we use (5.143) in (5.146) to express the average strain in the porous M matrix ε′′ as a function of P1 and P2 : (1 − ϕ2 ) ε′′

M

= SM :

(1 − ϕ2 ) B M − B1 P1 + (ϕ2 1 − B2 ) P2

(5.147)

where S M denotes the (drained) compliance tensor of the matrix. Finally, substitution in (5.144) and (5.145) yields the change of the microporosity:

′′ P2 P1 φ1 − φ10 = + (a ) N11 N12

1 − ϕ2 1 = B M : S M : (1 − ϕ2 )B M − B1 + (b) N11 NM 1 = B M : S M : (ϕ2 1 − B2 ) N12

(5.148)

(c)

and the change of the macroporosity:

′′ P1 P2 φ2 − φ20 = + N21 N22

1 = 1 : S M : B1 − (1 − ϕ2 ) B M N21 1 = 1 : S M : (B2 − ϕ2 1) N22

(a ) (b)

(5.149)

(c)

Provided that B M is an isotropic tensor (B M = b M1), the symmetry of the solid Biot moduli N12 = N21 is readily shown, by substituting (5.142b) in (5.148c) and (5.141b) in (5.149b), and by using the condition A = I.

Training Set: The Two-Scale Double-Porosity Material

165

A superposition of the two subproblems yields the macroscopic state equations of the two-scale double-porosity material: Σ = Chom : E − B1 P1 − B2 P2 (a ) P2 P1 + (b) φ1 − φ10 = B1 : E + N11 N12 P2 P1 φ2 − φ20 = B2 : E + + (c) N21 N22

(5.150)

The poroelastic constants Chom , B1 , B2 , Nij are readily determined from the poroelastic constants of the porous matrix, C M and B M (which in turn can be obtained by homogenization from the stiffness of the solid phase in the porous matrix), and an appropriate homogenization scheme.

6 Eshelby’s Problem in Linear Diffusion and Microporoelasticity This chapter presents a fundamental result of micromechanics that is at the heart of many upscaling techniques in linear diffusive transport, solid mechanics and poromechanics. It is the solution of Eshelby’s inclusion problem in an inﬁnite homogeneous domain. Eshelby’s problem is discussed ﬁrst in the context of the scalar molecular diffusion before extending the approach to linear stress–strain problems and poroelastic properties. The most important property of the inclusion problem is a constant density gradient for diffusion problems, and a constant strain (displacement gradient) for deformation problems, within an ellipsoidal inclusion that can account for a large variety of inhomogeneities found in materials in general, and in porous materials in particular: an ellipsoidal pore space within a microhomogeneous solid phase. This solution is of fundamental importance for the estimation of the concentration tensors used for micromechanical upscaling properties.

6.1 Eshelby’s Problem in Linear Diffusion 6.1.1 Introduction We have extensively studied molecular diffusion in porous media in Chapter 3 in the context of periodic media, and evoked the mathematical analogy with elasticity in the training set of Chapter 4.1 We now have a new look

1 The

reader is strongly encouraged to go back to the training set of Chapter 4 before proceeding.

Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

Eshelby’s Problem in Linear Diffusion and Microporoelasticity

168

at the molecular diffusion problem evoking a fundamental result of linear micromechanics known as Eshelby’s problem.2 We consider a porous material composed of a solid phase s and a pore space p , through which a diffusive ﬂux occurs that changes the solute concentration. Similar to the modeling approach developed in Section 4.1.2 (which we employed extensively in Section 5.5), we adopt a continuous description of the molecular diffusion throughout the porous material, solid plus pore space: the concentration ρ γ and the diffusive ﬂux j γ are extended into the solid phase, while setting the diffusion coefﬁcient to zero, Ds = 0. The physics of the molecular diffusion problem is then deﬁned by the following set of equations:3 div j γ = 0 j γ = −D(z)grad ρ γ

with D(z) =

ρ γ = H · z when z ∈ ∂

Ds = 0 Dγ

for for

(a ) z ∈ s (b) z ∈ p (c)

(6.1)

where H denotes the macroscopic concentration gradient prescribed at the boundary ∂ of the rev. The set of equations (6.1) deﬁnes a problem in a bounded domain . Instead of solving it, we may turn the problem around and deﬁne an auxiliary problem of a bounded inhomogeneity I embedded in an inﬁnite homogeneous medium ω, the former representing a solid phase inclusion, the latter the pore space. This auxiliary problem and its solution are due to Eshelby. Given the assumed inﬁnity of ω, the boundary condition (6.1c) in the original problem needs to be replaced by a condition written at inﬁnity. The set of equations that deﬁne this inhomogeneity problem is thus: div j γ = 0 j γ = −D(z)grad ρ γ

(a ) with D(z) =

ρ γ → H · z when | z |→ ∞

s

D =0 Dγ

for for

z∈I (b) z∈ω (c)

(6.2)

Furthermore, introducing δ D = Ds − Dγ = −Dγ , (6.2b) can be rearranged in the form: j γ = −Dγ grad ρ γ + j p (z) with j p (z) = −δ D χ I (z) grad ρ γ

(6.3)

where j p (z) is a ﬁctitious ﬂux that is non-zero only in the solid phase. It is recalled that χ I is the characteristic function of the domain I . 2 This 3 For

fundamental result goes back to J. Eshelby’s 1957 paper [24]. See also [25]. simplicity, we now omit the subscript z on the differential operators divz and grad . z

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169

For the purpose of analysis, let us assume that j p (z) = j I χ I (z), where j I is a constant vector. In this case, the problem deﬁned by (6.2 a), (6.2c) and (6.3) is: div j γ = 0

(a )

j γ = −Dγ grad ρ γ + j I χ I (z) γ

ρ → H · z when | z | → ∞

(6.4)

(b) (c)

This set of equations is known as Eshelby’s inclusion problem. The next three sections are devoted to solving in a step-by-step fashion Eshelby’s inclusion problem (6.4), the inhomogeneity problem (6.2), and ﬁnally the original diffusion problem (6.1). This is somewhat technical, but quite instructive. Indeed, we will see (1) that the solution of (6.4) is actually the solution of the inhomogeneity problem (6.2) for an appropriate value of j I , provided that the bounded inhomogeneity I is an ellipsoid; and (2) that the solution of this inhomogeneity problem provides estimates for the homogenized diffusion tensor Dhom that captures, at the macroscopic scale, the overall response of the microscopic physics of the molecular diffusion problem (6.1). 6.1.2 The (Diffusion) Inclusion Problem To simplify the presentation, we ﬁrst consider the case H = 0. The combination of (6.4a) and (6.4b) gives: −Dγ ρ γ + j I · grad χ I = 0

(6.5)

It is readily understood that grad χ I involves derivations of a discontinuous function, which calls for employing distribution theory. More precisely, according to the deﬁnition of the derivation of a distribution, we obtain (see (1.17)): grad χ I , ψ = −χ I , grad ψ = − grad ψ dV = − (6.6) ψn dS I

∂I

where ψ is any function of D(R3 ). In (6.6), n is the outward unit normal to I . Let us introduce the Dirac distribution δ∂ I associated with the boundary of I and deﬁned by (see (1.16)): δ∂ I , ψ = ψ dS (6.7) ∂I

From (6.6) and (6.7) it is readily seen that (see (1.18)): grad χ I = −nδ∂ I

(6.8)

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Thus, using (6.8) in (6.5) yields: −Dγ ρ γ − j I · nδ∂ I = 0

(6.9)

The solution of such an equation is conveniently achieved using the Green function concept. Denoted by G(z), the Green function is deﬁned as the elementary solution of the operator Dγ : −Dγ z G + δ0 = 0

(6.10)

where δz′ is the Dirac distribution at point z′ deﬁned by δz′ , ψ = ψ(z′ ) for any function ψ ∈ D(R3 ). Recalling that: 1 + δ0 = 0 with r =| z | (6.11) 4πr the solution of (6.10) is: G(z) = −

1 4Dγ πr

(6.12)

More generally, for any value of z′ , the solution G(z, z′ ) of: −Dγ z G + δz′ = 0

(6.13)

G(z, z′ ) = G(z − z′ )

(6.14)

is:

With the unit point solution in hand, the solution of (6.9) is obtained by superposition: γ ρ (z) = − (6.15) G(z − z′ ) j I · n dSz′ ∂I

where the integration in (6.15) is performed with respect to z′ . Using the divergence theorem also gives: I ∂ ′ ′ (6.16) ρ γ (z) = − ′ G(z − z ) j j dVz I ∂z j Observing that

∂ (G(z ∂z′j

− z′ )) = − ∂z∂ j (G(z − z′ )), we obtain:

∂ ρ (z) = ∂z j γ

′

G(z − z ) dVz′ I

j jI

(6.17)

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171

In order to obtain the concentration gradient, an additional derivation is performed: ∂2 ∂ρ γ ′ ′ (6.18) (z) = G(z − z ) dVz j jI ∂zi ∂zi ∂z j I This result can be recast in the form: grad ρ γ = P · j I

(6.19)

where: ∂2 Pi j (z) = ∂zi ∂z j

′

G(z − z ) dVz′ I

(6.20)

The solution (6.19) holds for H = 0. However, since the problem (6.4) is linear with respect to j I and H , the concentration gradient in the general case (H = 0) takes the form: grad ρ γ (z) = P(z) · j I + H

(6.21)

In summary, provided that j I is constant, the solution of Eshelby’s inclusion problem (6.4) reduces to the determination of the expression of the P tensor deﬁned by (6.20), which is shown next. 6.1.3 The (Second-Order) P Tensor The form (6.12) of the Green function (or its more general expression (6.14)) motivates the introduction of a potential (z) of the domain I , which depends only on the geometry of I : 1 (6.22) dVz′ (z) = ′ I | z−z | Using the potential (z), relations (6.12) and (6.20) immediately lead to: Pi j (z) = −

∂ 2 1 (z) 4π Dγ ∂zi ∂z j

(6.23)

It thus sufﬁces to specify the geometry of the inclusion I in order to obtain the P tensor. It is instructive to investigate speciﬁc geometries of I . The easiest geometry is the case where I is a sphere of radius a , centered at the origin. In this case, the restriction of (z) to I is: r2 2πa 2 with r =| z | (z) = 3− 2 (6.24) 3 a

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The potential (z) in the spherical case turns out to be a quadratic function of the coordinate r = |z|. According to (6.23), Pi j in I does not depend on z: (∀z ∈ I ) P =

1 1 3Dγ

(6.25)

This property of the spherical inclusion, namely that P is uniform over it, holds for any inclusion I of ellipsoidal geometry. This is one of the fundamental results of Eshelby’s inclusion problem. Indeed, for z ∈ I , the potential (z) of any ellipsoid is recognized to be a quadratic polynomial function of the coordinates zi . From (6.23), the P tensor, therefore, is uniform throughout the inclusion z ∈ I . In return, for z ∈ / I , the P tensor depends on the location, i.e. P = P (z). 6.1.4 An Alternative Derivation of the P Tensor (Optional) The presented derivation of the P tensor is based on the isotropy of Fick’s law at the microscale. Although Fick’s law is isotropic, it will turn out to be useful for some applications (for instance, multiscale homogenization, differential scheme applications, etc.) to have an expression for the P tensor for the case where the diffusivity tensor at the microscopic scale is anisotropic. While no doubt mathematically more demanding, this generalization will also turn out to be useful for the transposition of the diffusion problem to microelasticity.4 In the anisotropic case, the microscopic diffusive ﬂux j γ is related to the microscopic concentration gradient by: j γ = −D · grad ρ γ

with

γ

D = Di j e i ⊗ e j

(6.26)

A generalization of (6.10) leads to the following deﬁnition of the Green function G(z): γ

−Dkl G ,kl + δ0 = 0

(6.27)

where subscript , i refers to the derivation with respect to zi . As in (6.14), the solution of the more general equation: γ

−Dkl G,kl + δz′ = 0 is G(z, z′ ) = G(z − z′ ). For a given value of z with r =| z |, we have: 2π = δ0 (ξ · z) dSξ r |ξ |=1 4 The

following presentation is inspired by [52].

(6.28)

(6.29)

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173

In order to prove (6.29), it is convenient to let z be parallel to the θ = 0 axis. The variable transformation ζ = ξ · z = r cos θ on the r.h.s. of (6.29) is: 2π +r dζ 2π δ0 (ξ · z) dSξ = dϕ δ0 (ζ ) = (6.30) r r 0 −r |ξ |=1 Observing that: ∂ δ0 (ξ · z) = ξi δ0′ (ξ · z) ∂zi

(6.31)

and recalling that | ξ | = 1, we take the Laplacian of both sides in (6.30): −1 δ ′′ (ξ · z) dSξ (6.32) δ0 (z) = 8π 2 |ξ |=1 0 The Green function G(z) therefore satisﬁes (see (6.27)): 1 γ δ ′′ (ξ · z) dSξ = 0 Dkl G ,kl + 8π 2 |ξ |=1 0

(6.33)

For a given value of ξ on the unit sphere, relation (6.33) motivates a search for the solution G ξ of: γ

ξ

Dkl G ,kl + δ0′′ (ξ · z) = 0

(6.34)

−1 γ G ξ (z) = −δ0 (ξ · z) Dkl ξk ξl

(6.35)

An immediate solution is:

By superposition, one obtains the Green function in the form: −1 γ −1 δ (ξ G(z) = dSξ · z) D ξ ξ 0 k l kl 8π 2 |ξ |=1

(6.36)

We are now ready to derive Pi j . Substituting: −1 ′′ γ ∂2G 1 δ0 (ξ · z) dSξ ξi ξ j Dkl ξk ξl (z) = − 2 ∂zi ∂z j 8π |ξ |=1

(6.37)

in (6.20) yields:

1 Pi j (z) = − 2 8π

ξi ξ j |ξ |=1

−1 γ Dkl ξk ξl

I

δ0′′ (ξ

′

· (z − z )) dVz′

dSξ

(6.38)

We still have to determine the value of the integral over I in (6.38). To this end, we introduce function I(ζ ) and its derivative I ′′ (ζ ), which depends on

Eshelby’s Problem in Linear Diffusion and Microporoelasticity

174

the geometry of the inclusion I : ′′ ′ ′′ δ0 (ζ − ξ · z ) dVz′ = I (ζ ) with I(ζ ) = δ0 (ζ − ξ · z′ ) dVz′

(6.39)

I

I

Equation (6.38) can be recast in the form: −1 ′′ γ 1 I (ξ · z) dSξ ξi ξ j Dkl ξk ξl Pi j (z) = − 2 8π |ξ |=1

(6.40)

Expression (6.40) provides an alternative means of determining the P tensor for the anisotropic diffusion case. It is particularly efﬁcient for numerical implementation. By way of application, we reconsider the case of a spherical inclusion S(O, a ) of radius a , centered at the origin O. (r ′ , θ ′ , ϕ ′ ) denote the spherical coordinates of z′ in S(O, a ). It is readily understood that I(ζ ) does not depend on the orientation of the unit vector ξ . It is therefore possible to assume that ξ is parallel to the θ ′ = 0 axis, so that ξ · z′ = r ′ cos θ ′ . The variable transformation (r ′ , θ ′ , ϕ ′ ) → (r ′ , ζ ′ , ϕ ′ ) with ζ ′ = ζ − r ′ cos θ ′ yields: r ′ +ζ a 2π ′ ′ ′ δ0 (ζ ′ )dζ ′ (6.41) r dr dϕ I(ζ ) = When | ζ |< a , this yields: 2π ′ I(ζ ) = dϕ 0

−r ′ +ζ

0

0

a

r ′ dr ′ = π(a 2 − ζ 2 ) ⇒ I ′′ (ζ ) = −2π

(6.42)

|ζ |

which gives:5 (∀z ∈ S(O, a )) I ′′ (ξ · z) = −2π

(6.43)

Finally, returning to (6.40): (∀z ∈ I = S(O, a ))

1 Pi j (z) = 4π

|ξ |=1

−1 γ dSξ ξi ξ j Dkl ξk ξl

(6.44)

To complete the application, it is instructive to check that (6.44) reduces to (6.25) in the case of an isotropic diffusion case, for which D = Dγ 1. In this case: 1 1 (∀z ∈ I ) P(z) = 1 (6.45) ξ ⊗ ξ dSξ = γ 4π D |ξ |=1 3Dγ

5 Remember

that |ξ | = 1.

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175

where we made use of the following relation: 1 4π

ξ ⊗ ξ dS = |ξ |=1

1 1 3

(6.46)

6.1.5 The (Diffusion) Inhomogeneity Problem We now return to the inhomogeneity problem (6.2), which with (6.3) is restated in the form: div j γ = 0 γ

(a ) γ

γ

j = −D grad ρ − δ Dχ I (z) grad ρ ρ γ → H · z when | z | → ∞

γ

(b)

(6.47)

(c)

In order for the inhomogeneity problem (6.47) to coincide with the inclusion problem (6.4), it is a necessary and sufﬁcient condition that the concentration gradient is uniform in the inclusion domain I . In this case, the solution of the inhomogeneity problem is given by solution (6.21) of the inclusion problem, γ and it appears that vector j I in (6.4b) is related to grad ρ I by: γ

grad ρ I = P(z) · j I + H I

j =

γ −δ D grad ρ I

(6.48)

γ

Eliminating grad ρ I between the two equations in (6.48) yields the following expression for j I : (∀z ∈ I )

−1 j I = −δ D 1 + δ DP(z) ·H

(6.49)

The identity between (6.4) and (6.47) is achieved only if j I is a constant. This proves to be true when the domain I is an ellipsoid since P is constant in I for this particular geometry. The corresponding value of the concentration gradient in I is then: γ

grad ρ I = (1 + δ D P I )−1 · H

(6.50)

By way of application, consider a spherical inhomogeneity I with DI = 0 and δ D = −Dγ (i.e. solid inclusion). Using expression (6.25) for P I in (6.50) immediately yields the concentration gradient in the spherical inhomogeneity: γ

grad ρ I =

3 H 2

(6.51)

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

6.1.6 Eshelby-Based Estimates of the Homogenized Diffusion Tensor γ

Not surprisingly, given the constant value of grad ρ I in I , we retrieve in (6.51) the result (4.198) obtained from the direct solution (4.197) of a single spherical solid grain embedded in an inﬁnite homogeneous ﬂuid continuum. This comparison also hints at the use of Eshelby’s solution for estimating the effective diffusive properties of porous media in the case of non-spherical grains. Indeed, (4.192) suggests that estimates of the homogenized diffusion tensor Dhom s can be obtained from estimates of A which represent the average concentration tensor over the solid phase of the real rev. The dilute scheme already presented in Section 4.5.2 is based on the assumption that the interactions between the solid grains in the rev are negligible. In this case, the solution of Eshelby’s inhomogeneity problem turns out to be of great interest. Indeed, following our developments in Section 4.5.2, the average concentration gradient in the solid phase of the rev can be approximated by the uniform concentration gradient within an ellipsoidal solid grain embedded in an inﬁnite pore space, with the condition ρ γ = H · z at inﬁnity. Combining (4.196) with (6.50) yields the following dilute estimate of the concentration tensor in such an ellipsoidal inclusion phase: s

Adil = A I = (1 − Dγ P I )−1

(6.52)

where P I is given by (6.23) or (6.40). Use of this result in (4.192) provides a dilute estimate of the homogenized diffusion tensor: hom Ddil = Dγ (1 − (1 − ϕ) (1 − Dγ P I )−1 )

(6.53)

It is readily shown that use of (6.25) in (6.52) and (6.53) yields expressions (4.198) and (4.199) derived for the single spherical solid grain within the assumption of an inﬁnitesimal solid volume fraction. We will see later on (Section 6.5) how Eshelby’s fundamental solution also applies to non-dilute solutions.

6.2 Eshelby’s Problem in Linear Microelasticity 6.2.1 Introduction The original problem treated by Eshelby in his 1957 paper was a mechanical one, not a diffusion one: an elastic inclusion in an inﬁnite elastic homogeneous medium. This original application of Eshelby’s inhomogeneity– inclusion problem to linear microelasticity is the focus of this section. The steps we developed in Section 6.1 for the scalar diffusive transport problem, however, are readily adapted to the higher order microelasticity problem, by reversing the role played by the pore phase p and the solid phase s , and

Eshelby’s Problem in Linear Microelasticity

177

by evoking the mathematical similarity (4.186) between the two problems: jγ → σ γ ρ → ξ γ grad ρ → ε ′ Fick s law → Hooke′ s law mass balance → momentum balance solid s → pores p pores p → solid s

(6.54)

By analogy with (6.1), we adopt a continuous description of the deformation throughout the porous material, solid plus pore space: the displacement ξ and the stress tensor σ are extended into the pore space p , while prescribing a zero elastic stiffness to p ; C p = 0. In the case of the empty (or drained) porous medium (have a quick look back to Section 4.2.5), the mechanical problem deﬁned in a bounded domain, the rev , subjected to a macroscopic strain tensor E is: div σ = 0 σ = C(z) : ε with C(z) = ξ = E · z when z ∈ ∂

(a )

Cp = 0 Cs

for for

z ∈ p z ∈ s

(b)

(6.55)

(c)

Adapting the deﬁnition introduced in Section 6.1.1, Eshelby’s elastic inhomogeneity problem consists of turning the problem (6.55) around, by considering a closed pore I embedded in an inﬁnite solid domain ω, which is subjected at inﬁnity to uniform strain boundary conditions. The governing equations of this ‘ersatz’ problem are: div σ = 0 σ = C(z) : ε with C(z) = ξ = E · z when z → ∞

(a )

I

C =0 Cs

for for

z∈I z∈ω

(b)

(6.56)

(c)

where E denotes the uniform strain tensor at inﬁnity. Furthermore, introducing δ C = C I − Cs = −Cs , (6.56b) is rewritten in the form: σ = Cs : ε + σ p (z); with σ p (z) = δ C : εχ I (z)

(6.57)

where σ p (z) is a ﬁctitious stress that is non-zero only in the inclusion I . It is readily seen that (6.55), (6.56) and (6.57) are the mechanical analogs, in the sense of the mathematical similarity (6.54), of the molecular diffusion equations (6.1), (6.2) and (6.3). The solution procedure we therefore adapt follows faithfully the one developed in Section 6.1. First we will derive the solution

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

of (6.56a), (6.56c) and (6.57) by assuming that σ p (z) = σ I χ I (z) where σ I is constant within I . This assumption yields Eshelby’s inclusion problem in the form: div σ = 0 σ = Cs : ε + σ I χ I (z) ξ = E · z when z → ∞

(a ) (b) (c)

(6.58)

Then, we will see that the solution of the inhomogeneity problem (6.56) can be derived from that of the inclusion problem (6.58) provided that the pore inclusion I is an ellipsoid. Finally, the solution of the inhomogeneity problem will allow us to derive estimates for the homogenized elasticity tensor Chom which represents the overall response to the initial problem (6.55). 6.2.2 The (Elastic) Inclusion Problem We begin with the case E = 0. Substituting (6.58b) into (6.58a) yields the momentum balance equation in the form: div(Cs : ε) + σ I · grad χ I = 0

(6.59)

where grad χ I involves the derivation of a discontinuous function, which calls for the application of (6.8). Equation (6.59) thus becomes: div(Cs : ε) − σ I · nδ∂ I = 0

(6.60)

where δ∂ I stands for the Dirac distribution associated with the boundary of I . The solution of (6.60) is derived using the Green function concept. For the mechanics problem at hand, this means the elementary displacement ﬁeld solution ξ ( p) induced in an inﬁnite homogeneous elastic continuum (stiffness tensor Cs ) by a unit point force f located at the origin and parallel to e p , i.e. f = δ0 (z)e p (Kelvin problem): div Cs : grad ξ ( p) + δ0 (z)e p = 0

(6.61)

Cisjkl G kp, jl + δi p δ0 (z) = 0

(6.62)

The second-order Green tensor G is deﬁned by G · e p = ξ ( p) . Use in (6.61) entails the following conditions to be satisﬁed by the components of G:

where subscript , jl denotes the differential operation ∂ 2 /∂z j ∂zl . A generalization of (6.61) and (6.62) consists of considering the displacement induced at point z by a unit point force δz′ e p located at point z′ , in the form

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179

G(z, z′ ) = G(z − z′ ) · e p , which satisﬁes: div(Cs : grad G ) + δz′ e p = 0 Cisjkl G kp, jl + δi p δz′ = 0

(6.63)

Using the unit point–load solution, the solution of (6.60) is obtained by superposition: ξ (z) = − (6.64) G(z − z′ ) · σ I · n(z′ ) dSz′ ∂I

That is: ξi (z) = −

∂I

G il (z − z′ )nk (z′ ) dSz′ σlkI

(6.65)

where the integration in (6.64) and (6.65) is performed with respect to z′ . Using the divergence theorem also gives: ∂ ∂ I ′ ′ ′ ξi (z) = − G il (z − z ) dVz′ σlkI (6.66) ′ G il (z − z ) dVz σlk = ∂z ∂z k I I k Finally, the strain tensor is obtained by an additional derivation:6 ∂2 ′ εi j (z) = σlkI G il (z − z ) dVz′ ∂z j ∂zk I (i j)

(6.67)

or equivalently, making use of the symmetry of σ I : ε(z) = −P(z) : σ I

(6.68)

where: ∂2 Pi jkl (z) = − ∂z j ∂zk

′

G il (z − z ) dVz′ I

(6.69)

(i j),(kl)

Finally, by superposition, the complete solution of (6.58) as a function of σ I and E is: ε(z) = −P(z) : σ I + E

(6.70)

With (6.70), which is the mechanics analog of the diffusion solution (6.21), and which is based on the assumption σ I = const, we arrive at reducing the solution of (6.58) to the determination of the fourth-order P tensor, and of the second-order Green function G(z − z′ ). 6 The

subscript (i j), (kl) stands for a double symmetrization, Pi jkl = P jikl = Pi jlk = Pkli j .

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6.2.3 The Green Tensor G and the (Fourth-Order) P Tensor The determination of the Green tensor in microelasticity is less straightforward than in the diffusion case, even in the isotropic case (see Section 6.1.3). On the other hand, we have already encountered, in Section 6.1.4, an alternative means of determining the Green function for anisotropic diffusion problems. The focus of this section, therefore, is a transposition of this method developed for diffusion problems to microelasticity, based on the similarity (6.54) of both problems. The starting point is (6.33). Written for the elasticity problem, relation (6.62) becomes: Cisjkl G kp, jl −

1 8π 2

|ξ |=1

δ0′′ (ξ · z) dSξ δi p = 0

(6.71)

By analogy with (6.34), for a given value of ξ on the unit sphere, we search for the solution Gξ of the following differential equation: Cisjkl G kp, jl + δ0′′ (ξ · z) δi p = 0

(6.72)

We look for Gξ (z) in the form g(ξ · z). We then obtain: ξ

′′ G kp, jl = ξ j ξl gkp (ξ · z)

(6.73)

We now introduce the second-order tensor K(ξ ) = ξ · Cs · ξ of components: K ik (ξ ) = Cisjkl ξ j ξl

(6.74)

With this notation, the intrinsic form of (6.72) is: K(ξ ) · g′′ (ζ ) + δ0′′ (ζ )1 = 0 with

ζ =ξ ·z

(6.75)

An immediate solution of (6.75) is g(ζ ) = −(K(ξ ))−1 δ0 (ζ ). We now return to (6.71). By superposition, we obtain: 1 G(z) = (K(ξ ))−1 δ0 (ξ · z) dSξ (6.76) 2 8π |ξ |=1 Expression (6.76) is the elasticity counterpart of the Green function expression (6.36). We are now ready to derive Pi jkl . We obtain: 1 ∂ 2 G il (z) = ∂z j ∂zk 8π 2

|ξ |=1

ξ j ξk (K(ξ ))il−1 δ0′′ (ξ · z) dSξ

(6.77)

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181

Thus, by application of (6.69): 1 −1 ′′ ′ δ0 (ξ · (z − z )) dVz′ dSξ Pi jkl (z) = − 2 ξ j ξk (K(ξ ))il (i j),(kl) 8π |ξ |=1 I (6.78) ′′ Similarly to (6.39), we introduce function I(ζ ) and its derivative I (ζ ): I(ζ ) = δ0 (ζ − ξ · z′ ) dVz′ and I ′′ (ζ ) = δ0′′ (ζ − ξ · z′ ) dVz′ (6.79) I

I

Substituting (6.79) into (6.78) yields the following general expression for the P tensor (the counterpart of (6.40)): 1 ξ j ξk (K(ξ ))il−1 (6.80) I ′′ (ξ · z) dSξ Pi jkl (z) = − 2 (i j),(kl) 8π |ξ |=1 This expression of the P tensor has an advantage over expression (6.69) in that it does not require knowledge of the Green tensor, but only of the derivative of function I(ζ ) which depends on the geometry of the pore inclusion I . Due to the symmetry of K(ξ ), it also proves the symmetry of P (Pi jkl = Pkli j ). By way of illustration, consider the inclusion to be a sphere of radius a , centered at the origin. For z ∈ I , we obtain I ′′ (ξ · z) = −2π (see (6.43)), and the P tensor is: −1 1 I ξ j ξk (K(ξ ) )(i j),(kl) d Sξ (6.81) Pi jkl = il 4π |ξ |=1 6.2.4 G and P in the Isotropic Case It is instructive to explore the theory developed above for the idealized case of a spherical inclusion I embedded in an isotropic solid. Using the deﬁnition of the fourth-order tensors J and K deﬁned by (4.88), the isotropic stiffness tensor Cs is: Cs = 3k s J + 2μs K

(6.82)

The determination of both G and P ﬁrst requires the calculation of K = ξ · Cs · ξ . For this calculation, it is useful to note that: 1 ξ ·J·ξ = ξ ⊗ξ 3

(6.83)

1 1 ξ ·K·ξ = ξ ⊗ξ + 1 6 2

(6.84)

and:

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182

where we took into account that | ξ | = 1. Thus, by application of (6.74), we obtain: s s s 4μ + 3k ξ ⊗ξ +1−ξ ⊗ξ (6.85) K=μ 3μs The determination of P requires the inverse of K. Recalling that | ξ | = 1, we note that: (1 − ξ ⊗ ξ ) · (1 − ξ ⊗ ξ ) = 1 − ξ ⊗ ξ (1 − ξ ⊗ ξ ) · ξ ⊗ ξ = 0

(6.86)

ξ ⊗ξ ·ξ ⊗ξ =ξ ⊗ξ from which we obtain the following identity: (∀ m, n = 0) (m(1 − ξ ⊗ ξ ) + nξ ⊗ ξ )−1 =

1 1 (1 − ξ ⊗ ξ ) + ξ ⊗ ξ m n

(6.87)

The inverse K−1 is thus: −1

K

1 = s μ

μs + 3k s 1− ξ ⊗ξ 4μs + 3k s

(6.88)

For a spherical inclusion, the P tensor is given by (6.81). Using (6.88), the integrand of (6.81) can be developed in the form: −1 1 ξ j ξk K(ξ ) (6.89) = s ξ j (1 + a ξ ⊗ ξ )il ξk (i j)(kl) il μ (i j),(kl) with: a =−

μs + 3k s 4μs + 3k s

(6.90)

The two terms on the r.h.s. of (6.89) can be developed as follows: 1 ξ j (1)il ξk (i j)(kl) = ξi δ jk ξl + ξ j δik ξl + ξi δ jl ξk + ξ j δil ξk 4 ξ j (ξ ⊗ ξ )il ξk = ξi ξ j ξk ξl (i j)(kl)

(6.91) (6.92)

In order to perform the integration of the above expressions over the unit sphere, (6.46) and the following identity are useful: 2 1 1 (6.93) ξ ⊗ ξ ⊗ ξ ⊗ ξ dSξ = J + K 4π |ξ |=1 3 15

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183

Finally, using (6.89) in (6.81) yields after integration: 2 1 1 μs + 3k s 1 (J + K) − J+ K P= s μ 3 4μs + 3k s 3 15

(6.94)

or equivalently: P=

β α J+ K s 3k 2μs

(6.95)

with: α=

3k s ; 3k s + 4μs

β=

6(k s + 2μs ) 5(3k s + 4μs )

(6.96)

Note again that the expression of the Green tensor is not required for the determination of P. Nevertheless, for the purpose of completeness, we derive the expression of the Green tensor based on (6.76). Considering (6.88), this requires calculation of the following two integrals: J1 = δ0 (ξ · z) dSξ ; J2 = ξ ⊗ ξ δ0 (ξ · z) dSξ (6.97) |ξ |=1

|ξ |=1

Using (6.30) yields J 1 = 2π/r . Then, using the same variable transformation as in (6.30), it is readily seen that: π (6.98) J2 = (1 − Z ⊗ Z) with Z = z/ | z | r Inserting these results together with (6.88) into (6.76) ﬁnally yields the Green tensor for the isotropic case: s 1 μs + 3k s 7μ + 3k s G(z) = (6.99) ⊗ Z 1 + Z 8π μs r 4μs + 3k s 4μs + 3k s 6.2.5 The (Elastic) Inhomogeneity Problem We now return to the inhomogeneity problem (6.56), which with the help of (6.57) is written in the form: div σ = 0 (a ) s σ = C : ε + δ C : εχ I (z) (b) ξ = E · z when z → ∞ (c)

(6.100)

Following the line of argument developed in Section 6.1.5 for the diffusion problem, the solution of the elastic inhomogeneity problem (6.100) coincides with the solution of the elastic inclusion problem (6.58), if and only if the strain tensor is uniform throughout the pore inclusion I . In this case, using (6.57), (6.58b) and (6.70) yields the following two relations between prestress

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σ I and strain ε I : σ I = δC : εI

ε I = −P : σ I + E

(6.101) (6.102)

Eliminating ε I between (6.101) and (6.102) allows us to link σ I to the macroscopic strain E: σ I = (I + δ C : P)−1 : δ C : E

(6.103)

In order for (6.58) and (6.100) to be identical, σ I must be a constant. This is satisﬁed only by relation (6.103) if P = const. Akin to the diffusion problem, this property is satisﬁed when the inhomogeneity I is an ellipsoid. The corresponding value of the strain tensor in I is: ε I = (I + P : δ C)−1 : E

(6.104)

It is worth noting that the above result holds for any constant value of δ C. In the particular case of the empty porous medium considered here, we let δ C = −Cs . It is then convenient to introduce the so-called Eshelby tensor: S = P : Cs

(6.105)

The micro–macro link (6.104) between the (pore) inclusion strain ε I and the macroscopic strain E takes the form: ε I = (I − S)−1 : E

(6.106)

By way of application, we reconsider the case of a spherical inhomogeneity embedded in an isotropic medium. Using expression (6.95) for P and (6.82) for Cs , the Eshelby tensor S is readily obtained from (6.105) (with the help of (4.89)): S = αJ + β K

(6.107)

where α and β are given by (6.96). It is straightforward to verify that the use of (6.107) in (6.106) yields the same result as (4.146) where a v and a d are given by (4.149) and (4.158). In other words, relation (6.106) is an extension of the spherical pore strain solution (4.146) to the case of an ellipsoidal pore. 6.2.6 An Instructive Exercise: Geometry Change of Spherical Pores in a Porous Medium Subjected to Compaction How does the shape of spherical pores embedded in an isotropic solid phase change when subjected to vertical compaction? The intuitive answer is that they become ellipsoidal. With the results of Section 6.2.5, however, we now have the tools in hand to quantify this geometry change. To this end, we

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185

consider an rev of a porous medium subjected to vertical compaction, deﬁned by the macroscopic strain tensor: E = Ee 3 ⊗ e 3

with

E< 0

(6.108)

where e 3 denotes the unit vector of the vertical direction. The result of Section 6.2.5 is that the deformation of each individual pore is homogeneous and characterized by the strain tensor ε I given in (6.106): ε I = (I − S)−1 : E

(6.109)

In terms of the terminology introduced in Section 4.2.9, the tensor (I − S)−1 can be seen as the strain concentration tensor linking the macroscopic strain to the microscopic strain in the pore. For spherical pores, using (6.107), this strain concentration relation reads: 1 1 εI = J+ K :E (6.110) 1−α 1−β Using the deﬁnition (4.88) of J and K with (6.108) yields: J:E=

E 1; 3

K:E=

Thus: E ε = 3 I

1 2 + 1−α 1−β

E 2e 3 ⊗ e 3 − e 1 ⊗ e 1 − e 2 ⊗ e 2 3

e3 ⊗ e3 +

1 1 − 1−α 1−β

(6.111)

(e 1 ⊗ e 1 + e 2 ⊗ e 2 )

(6.112) As expected, the spherical pore of initial radius a which is subjected to the vertical compaction (6.108) deforms into the ellipsoid: z12 + z22 z32 + = a2 I 2 I 2 (1 + ε11 (1 + ε33 ) )

(6.113)

While it is readily understood that the length of the vertical axis of the pore I I I I < 0, and that ε33 < ε11 = ε22 , the sign of the horizontal decreases, since ε33 I strain ε11 and the associated horizontal length change are maybe less intuitive, as they relate to Poisson’s effects. In fact, it is observed that the horizontal length decreases as well if β < α. From (6.96), this is the case if ν s > 1/5.

6.3 Implementation of Eshelby’s Solution in Linear Microporoelasticity We have already encountered the concept of estimates of poroelastic constants in Chapter 4. Consistent with (4.142), the estimation of the poroelastic conp stants relies on estimating the average of the strain concentration tensor A in

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

the pore space. This requires an appropriate representation of the pore space. We brieﬂy revisit the concept of estimates in the context of the dilute scheme, for which the initial porosity ϕ0 ≪ 1. In Sections 6.3.4 and 6.3.5, we will see how to deal with a non-inﬁnitesimal porosity. 6.3.1 Implementation of Eshelby’s Solution in the Dilute Scheme In the dilute scheme developed in Section 4.3.1, the pore space is represented as a set of separated spherical pores which do not interact with one another. More generally, akin to Section 6.1.6, the pores can be approximated by ellipsoids of appropriate shape and orientation embedded in an inﬁnite elastic medium which has the same stiffness as the solid phase and which is subjected to uniform strain boundary conditions at inﬁnity. For the empty porous medium, the estimate of the average strain in the pore space is then provided by the solution (6.104)–(6.106) of Eshelby’s inhomogeneity problem; that is, p the average strain concentration tensor A is estimated by: p

A ≈ (I + P : δ C)−1 = (I − S)−1

(6.114)

Using (6.114) directly in (4.142) and (5.116) yields the dilute estimates of the poroelastic constants: Chom = Cs : I − ϕ0 (I − S)−1 (6.115) B = ϕ0 1 : (I − S)−1

(6.116)

This highlights how the appropriate estimates of the average strain concentrap tion tensor A are readily implemented to obtain estimates of the poroelastic constants. 6.3.2 Implementation of the Dilute Scheme with Different Pore Families In some material applications, the total pore space can be modeled by a set of different pore families that differ in pore shape and porosity. In order to estimate the poroelastic constants from (4.142) and (5.116), it is necessary to

divide p the pore space p into different domains i (volume fraction ϕ0i , ϕ0 = i ϕ0i ). The average strain concentration tensor in the total pore domain is then estii mated from the weighted sum of the averages A of the strain concentration p tensor over i : p

A =

ϕ0i i A ϕ0

(6.117)

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187

p

Each pore family i constitutes a set of pores of the same shape and orientation, represented by an ellipsoid Ii . Within the context of a dilute scheme which neglects the mechanical interaction between pores of each pore family p and in between pore families, the average strain over i is readily estimated from the uniform strain ε Ii = (I − Si )−1 : E in Ii embedded in an inﬁnite elastic medium of stiffness Cs and subjected to the uniform strain boundary condition ξ = E·z at inﬁnity. In other words, the average strain concentration tensor i of each pore family A may be approximated by (I − Si ) −1 , where Si denotes the Eshelby tensor relative to the ellipsoid Ii . Within the limits of the dilute approximation (ϕ0 ≪ 1), the solution of this problem, therefore, appears as a straightforward extension of the single pore family model (same shape and orientation). The average strain concentration tensor in the pore domain is thus: p

A =

ϕ0i −1 (I − Si ) ϕ0

(6.118)

In turn, the estimate (4.142) of the homogenized stiffness Chom takes the form:

−1 hom s i C =C : I− (6.119) ϕ0 (I − Si ) i

and the Biot tensor B follows from (5.85). 6.3.3 An Alternative Eshelby-Based Derivation of the Poroelastic Model The classical Eshelby inhomogeneity problem, as presented in Section 6.2.5, takes into account the heterogeneity of the elastic properties but disregards the pore pressure in the pores. It is natural to look for an extension of Eshelby’s problem to poroelasticity taking this pore pressure directly into account. An Extended Eshelby Problem: Coupled Prestress and Inhomogeneity

The idea consists of considering a uniform prestress π I in the pore inclusion I . As before, the pore inclusion is embedded in an inﬁnite linear elastic solid medium. This is readily achieved by adding π I χ I (z) to stress σ p (z) in (6.57). The problem to be solved is then: div σ = 0 (a ) s I σ = C : ε + δ C : ε + π χ I (z) (b) ξ = E · z when z → ∞ (c)

(6.120)

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

where δ C = C I − Cs = −Cs . The solution of (6.120) is readily obtained from that of the inclusion problem if we let the quantity δ C : ε + π I in (6.120b) play the role of a constant prestress tensor σ I in (6.58b). This condition, σ I = const, requires both ε I and π I to be uniform throughout I . In this case, a straightforward extension of (6.101) and (6.102) yields the following relations between E, π I and ε I : σ I = δC : εI + π I ε I = −P : σ I + E

(6.121)

Eliminating σ I in (6.121) yields the expression for ε I : ε I = (I + P : δ C)−1 : (E − P : π I )

(6.122)

Given that E and π I are constant, the inclusion strain ε I is constant as well provided that I is an ellipsoid for which P = const. Application to the Derivation of the Poroelastic Model

The extended Eshelby problem so deﬁned paves the way for a new approach to the poroelasticity state equations. Indeed, it sufﬁces to consider the pores of the real rev as pressurized inclusions of identical shape and of zero stiffness. The extended Eshelby inhomogeneity problem (6.120) can be implemented within the framework of a dilute scheme: a single ellipsoidal inclusion I embedded in an inﬁnite solid matrix subjected to the macroscopic strain tensor E at inﬁnity, and a constant prestress π I in the inclusion that is equal to the pressure tensor −P1. The key idea of this derivation is to estimate the average strain ε p in the pores of the real rev by the dilute approximation ε I of Eshelby’s inhomogeneity problem. In fact, a straightforward application of (6.122), using δ C = −Cs and the Eshelby tensor (6.105), i.e. S = P : Cs and P = S : (Cs )−1 = S : Ss , yields the strain induced in the pore by the loading E and −P1: ε I = (I − S)−1 : (E + P S : Ss : 1) or equivalently, using the identity (I − S)−1 : S = (I − S)−1 − I ε I = (I − S)−1 : E + P (I − S)−1 : Ss : 1 − Ss : 1

(6.123)

(6.124)

Furthermore, using the dilute estimate ε I for ε p in (5.117), the change in Lagrangian porosity φ − φ0 is: φ − φ0 = ϕ0 1 : ε I

(6.125)

That is, using (6.124) in (6.125): φ − φ0 = Bdil : E +

P Ndil

(6.126)

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189

with: Bdil = ϕ0 1 : (I − S)−1 ;

1 = (Bdil − ϕ0 1) : Ss : 1 Ndil

(6.127)

It is readily seen that (6.127) is consistent with (6.116). Equation (6.126) with (6.127) is the second macroscopic state equation of poroelasticity within the limits of the dilute approximation (no interaction between pore inclusions). The ﬁrst state equation of linear poroelasticity is obtained by applying the average rule (5.37) together with a linear elastic constitutive behavior for the solid material: Σ = (1 − ϕ0 )Cs : εs − Pϕ0 1

(6.128)

If we express the average solid strain εs in (6.128) using the micro–macro strain compatibility condition E = (1 − ϕ0 )εs + ϕ0 ε p

(6.129)

and estimate the average pore strain ε p by the dilute estimate ε I given by (6.124), we obtain: Σ = Cs : I − ϕ0 (I − S)−1 : E − Pϕ0 Cs : (I − S)−1 : Ss : 1 (6.130)

To derive a more compact expression of this state equation, we note the identity: X:Y:1=1:Y:X

(6.131)

which holds for any fourth-order tensors X (resp. Y) that meet the symmetry condition Xi jkl = Xkli j (resp. Yi jkl = Ykli j ). This result is then applied with X = Cs and Y = (I − S)−1 : Ss . Indeed, we observe that (I − S)−1 : Ss can be replaced by (Cs : (I − S))−1 , and that Cs : (I − S) is equal to Cs − Cs : P : Cs . The symmetry of Cs and P thus guarantees that of (I − S)−1 : Ss . Applying (6.131), we then obtain: ϕ0 Cs : (I − S)−1 : Ss : 1 = ϕ0 1 : (I − S)−1 : Ss : Cs = ϕ0 1 : (I − S)−1 = Bdil (6.132) Finally, (6.130) takes the more familiar form of the ﬁrst state equation of Biot’s poroelasticity theory (6.133): hom Σ = Cdil : E − PBdil

(6.133)

hom Cdil = Cs : (I − ϕ0 (I − S)−1 )

(6.134)

with:

which is identical to (6.115).

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

6.3.4 Mechanical Interaction Between Pores: The Mori–Tanaka Scheme The dilute scheme is restricted to an inﬁnitesimal porosity, in which case the mechanical interaction between pores can be neglected. The so-called differential scheme presented in Section 4.3.2 is one way of overcoming this restriction and accounting for the mechanical interaction between pores. An alternative scheme that deals with mechanically interacting pores using Eshelby’s result is the focus of this section. The case of a single pore family (same shape and orientation) is considered ﬁrst, before extending the approach to more general situations. By its very deﬁnition, the single ellipsoidal pore I in Eshelby’s inhomogeneity problem of an inclusion in an inﬁnite solid domain does not ‘sense’ the other pores of the pore space. In the framework of Eshelby’s inhomogeneity problem, one way to capture the interaction between pores consists of changing the uniform strain boundary condition at inﬁnity. This is the idea: replace the condition E · z by E0 · z, and deﬁne E0 appropriately to meet the micro–macro strain compatibility condition, ε = E. As in the dilute scheme, the uniform strain in the ellipsoid I is adopted as an estimate for the average strain ε p in the pore space of the real rev. In addition, the average strain εs in the solid phase of the rev is set equal to the average strain in the homogeneous continuum surrounding I in the inhomogeneity problem, which is equal to E0 .7 These conditions are summarized as follows: ε p = ε I = (I + P : δ C)−1 : E0 εs = E0

(6.135)

where δ C = −Cs for the (empty) pore. We still have to relate E and E0 , which is readily achieved using the strain average condition ε = E: ϕ0 ε p + (1 − ϕ0 )εs = E Inserting (6.135) into (6.136) yields: −1 E0 = (1 − ϕ0 )I + ϕ0 (I + P : δ C)−1 :E

(6.136)

(6.137)

Relations (6.135) and (6.137) entail the following estimate of the average strain concentration tensor in the pore space: −1 p A = (I + P : δ C)−1 : (1 − ϕ0 )I + ϕ0 (I + P : δ C)−1 (6.138)

7 The dilute scheme does not meet this condition, since the average strain in the homogeneous continuum surrounding I is equal to E, and not E0 .

Implementation of Eshelby’s Solution in Linear Microporoelasticity

191

Finally, inserting (6.138) into (4.142) yields the following expression for the so-called Mori–Tanaka estimate8 of the drained stiffness tensor: −1 (6.139) Cmt = Cs : I − ϕ0 (I − S)−1 : (1 − ϕ0 )I + ϕ0 (I − S)−1

or:

−1 Cmt = (1 − ϕ0 )Cs : (1 − ϕ0 )I + ϕ0 (I − S)−1

(6.140)

It is readily seen that the dilute scheme estimate (6.115) corresponds to an expansion to the ﬁrst order of (6.140) with respect to the pore volume fraction. The expression for the Biot tensor B that accounts for pore interactions is obtained from a combination of (6.138) (with P : δ C = −S) and (5.116): −1 −1 −1 mt B = ϕ0 1 : (I − S) : (1 − ϕ0 )I + ϕ0 (I − S) (6.141) −1 = ϕ0 1 : I − (1 − ϕ0 )S

In the same way as in Section 6.3.2, it is interesting to extend the Mori–Tanaka estimate to the case where several families of pores have to be considered. Relations (6.137) and (6.138) are now respectively replaced by:

−1 −1 E0 = (1 − ϕ0 )I + ϕ0i (I + Pi : δ Ci ) :E (6.142) i

and: j

A = I + P j : δC j

−1

: (1 − ϕ0 )I +

i

ϕ0i (I + Pi : δ Ci )−1

−1

(6.143)

Inserting (6.143) and (6.117) into (4.142) and recalling that δ Ci = −Cs yields the sought Mori–Tanaka estimate which generalizes (6.140):9

−1 −1 Cmt = (1 − ϕ0 )Cs : (1 − ϕ0 )I + ϕ0i (I − Si ) (6.144) i

By way of application of (6.140), we consider the case of a single family of spherical pores embedded in an isotropic solid phase. Use of (6.107) in (6.140) 8 Denoted

by the subscript mt. can be veriﬁed that the estimate derived in (6.144) is symmetric. This is strongly related to the fact that the heterogeneities considered here are pores; that is, inclusions with zero stiffness. 9 It

Eshelby’s Problem in Linear Diffusion and Microporoelasticity

192

yields: k mt =

(1 − ϕ0 )k s ϕ0 ; (1 − ϕ0 ) + 1−α

μmt =

(1 − ϕ0 )μs ϕ0 (1 − ϕ0 ) + 1−β

(6.145)

or equivalently, using expressions (6.96) for α and β: k mt = k s

4(1 − ϕ0 )μs ; 3ϕ0 k s + 4μs

μmt = μs

(1 − ϕ0 )(9k s + 8μs )

9k s (1 + 23 ϕ0 ) + 8μs (1 + 23 ϕ0 )

(6.146)

Interestingly, the estimate of the homogenized bulk modulus in (6.146) turns out to coincide with the bulk modulus of the hollow sphere derived in Section 4.1, also retrieved in section 4.2.8 for Hashin’s composite sphere assemblage (see (4.107) for α → 1). It proves to be equal to the upper Hashin– Shtrikman bound.10 The same upper bound interpretation holds true for the estimate of the homogenized shear modulus. 6.3.5 The Self-Consistent Approach The Mori–Tanaka estimate is based on a generalized Eshelby problem in which the pore is surrounded by the solid phase. Accordingly, this scheme is expected to be relevant for a morphology of the microstructure where the pores can be regarded as inclusions embedded in a solid matrix. In contrast, we hereafter examine the disordered morphology where neither the solid phase nor the pore space can be regarded as the matrix. The idea of the so-called self-consistent scheme consists of assuming that each particle of a given phase (pore or solid) reacts as if it were embedded in the equivalent homogeneous medium which is looked for. Let Csc denote the stiffness tensor of this equivalent homogeneous medium (the superscript ‘sc’ stands for self-consistent). In the case of an isotropic morphology, the average strain in the pore space (resp. in the solid) is estimated by the uniform strain in a spherical pore (resp. spherical solid particle) surrounded by an inﬁnite medium with stiffness Csc , subjected to the uniform strain boundary condition ξ → E0 · z at inﬁnity: −1 ε p = I + Psc : (C p − Csc ) : E0 = (I − Psc : Csc )−1 : E0 (6.147) −1 εs = I + Psc : (Cs − Csc ) : E0

When the self-consistent estimate of the stiffness tensor is isotropic, Psc is the P tensor of a spherical inclusion in an isotropic medium: Psc = 10 See

[30],[31].

α sc β sc J + K 3k sc 2μsc

(6.148)

Implementation of Eshelby’s Solution in Linear Microporoelasticity

193

Psc depends on the unknown self-consistent estimates k sc and μsc of the homogenized bulk and shear moduli. Relations (6.147) are now inserted into (6.136) and (6.137) is then replaced by: −1 −1 E0 = ϕ0 (I + Psc : (C p − Csc ))−1 + (1 − ϕ0 ) I + Psc : (Cs − Csc ) :E

(6.149)

that is: E0 = (I + Psc : (C − Csc ))−1

−1

:E

(6.150) α

Returning to (6.147), the average strain concentration tensors A (α = s, p) take the form: −1 −1 α (6.151) A = (I + Psc : (Cα − Csc ))−1 : I + Psc : (C − Csc ) The self-consistent estimate Csc is eventually derived from (4.114): Csc = C : (I + Psc : (C − Csc ))−1 : (I + Psc : (C − Csc ))−1

−1

(6.152)

Taking advantage of the fact that Psc is identical for both phases (it is given by (6.148)), it is useful to note that (6.152) can also be expressed as: (C − Csc ) : (I + Psc : (C − Csc ))−1 = 0

(6.153)

From (6.153), it is readily seen that: (I + Psc : (C − Csc ))−1 = I

(6.154)

This further implies that E0 = E (see (6.150)), and yields the simpliﬁed form of (6.152): Csc = C : (I + Psc : (C − Csc ))−1

(6.155)

In the case of an isotropic solid, the equivalent homogenized medium is isotropic as well.11 Equation (6.155) thus provides the following two scalar equations: ks 1 + α sc (k s − k sc )/k sc

(6.156)

μs 1 + β sc (μs − μsc )/μsc

(6.157)

k sc = (1 − ϕ0 ) and: μsc = (1 − ϕ0 )

In the general case, these equations are coupled because α sc and β sc both depend on k sc and μsc according to (6.96). Let us illustrate the self-consistent 11 Remember

that the morphology was assumed to be isotropic as well.

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Eshelby’s Problem in Linear Diffusion and Microporoelasticity

approach in the particular case of an incompressible solid phase (k s → ∞), for which (6.156) is: k sc = (1 − ϕ0 )

k sc 1 − ϕ0 sc = (3k + 4μsc ) sc α 3

(6.158)

that is: k sc 4(1 − ϕ0 ) = sc μ 3ϕ0

(6.159)

Combining (6.159) with (6.157) eventually yields: k sc = 4μs

(1 − 2ϕ0 )(1 − ϕ0 ) ; ϕ0 (3 − ϕ0 )

μsc = 3μs

(1 − 2ϕ0 ) (3 − ϕ0 )

(6.160)

Let us compare the estimates provided by the Mori–Tanaka scheme (see (6.146)) and the self-consistent one. Except in the case of dilute pore volume fraction (ϕ0 ≪ 1), strong differences are observed (see Figure 6.1). The Mori–Tanaka scheme predicts strictly positive effective bulk and shear moduli for any value of the porosity ϕ0 < 1, even in the vicinity of ϕ0 = 1. This is due to the fact that the Mori–Tanaka scheme corresponds to a matrixinclusion morphology, in which the pores are embedded in the solid phase. 1

µhom 0.8 µs

0.6

0.4

(MT) 0.2 (SC) 0 0.2

0.4

0.6 ϕ0

0.8

1

Figure 6.1 Mori–Tanaka and self-consistent estimates of the homogenized shear modulus for a porous medium with incompressible solid phase

Instructive Exercise: Anisotropy of Poroelastic Properties Induced by Flat Pores

195

As in the Mori–Tanaka scheme, the self-consistent scheme (6.160) predicts that the effective stiffness is a decreasing function of ϕ0 . However, the selfconsistent estimate vanishes for ϕ0 = 1/2. This level of porosity is classically interpreted as a percolation threshold of the pores. Another point of view might consist in increasing the solid volume fraction: the self-consistent scheme would then predict that an effective stiffness appears beyond a solid volume fraction equal to 1/2.

6.4 Instructive Exercise: Anisotropy of Poroelastic Properties Induced by Flat Pores In the case of an isotropic matrix, the anisotropy of the homogenized stiffness and the Biot tensor proves to be directly related to the anisotropy of the pore geometry. Such an anisotropy of the pore shape is found for instance in sedimentary rocks as a consequence of the compaction process. To illustrate this effect, we consider the pore space as a set of oblate ellipsoidal pores with the same orientation and same aspect ratio, which are embedded in an isotropic solid matrix (bulk and shear moduli k s and μs ). The pores are symmetric with respect to the vertical axis which corresponds to the small axis of the ellipsoid (unit vector e 3 ). At the macroscopic scale, the homogenized poroelastic medium is transversely isotropic. The Biot tensor B is therefore characterized by two scalar coefﬁcients b v and b h , respectively, for the vertical and horizontal directions: B = b h (e 1 ⊗ e 1 + e 2 ⊗ e 2 ) + b v e 3 ⊗ e 3

(6.161)

The anisotropy of this tensor is described by the ratio a = b h /b v . The purpose of this exercise is to relate the anisotropy coefﬁcient a to the aspect ratio of the pores, for a given value of the porosity. To this end, we will make use of the estimates of Chom and B derived in the dilute scheme and in the case of mechanical interaction as modeled by the Mori–Tanaka scheme. We therefore need the Eshelby tensor of the oblate ellipsoid which captures the inﬂuence of the pore geometry on the overall anisotropy. 6.4.1 Coefficients of the Eshelby Tensor We list below the components of the Eshelby tensor for an oblate ellipsoidal pore in an isotropic matrix. The equation of the oblate ellipsoid is: z12 + z22 z32 + =1 (6.162) a2 c2 The aspect ratio is X = c/a < 1. For the sake of clarity, we introduce: 2 1 − X2 Si jkl (6.163) Si′ jkl = (ν s − 1) 1 − X2

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196

With this notation, the coefﬁcients of the Eshelby tensor in the frame (e 1 , e 2 , e 3 ) are: 1 ′ S1111 X (19 − 8ν s ) X 1 − X2 + (8ν s − 13) arccos ( X) = 16 + (4 − 8 ν s ) arccos ( X) X2 (6.164) + (8 ν s − 10) X3 1 − X2 1 (2 − 2ν s ) 1 − X2 + (2ν s − 1)X arccos ( X) 2 + (2 ν s − 5) X2 1 − X2 + (4 − 2 ν s ) X3 arccos ( X)

′ S3333 = −

1 X −(1 + 8ν s ) X 1 − X2 16 + (8 ν s − 1) arccos ( X) + (4 − 8 ν s ) arccos ( X) X2 + (8 ν s − 2) X3 1 − X2

(6.165)

′ = − S1122

1 ′ = − X − 2(1 + ν s ) arccos ( X) X2 S1133 4 + (2 ν s − 1) arccos ( X) + (3 − 2 ν s ) X 1 − X2 + 2 ν s X3 1 − X2

1 2(1 − 2ν s ) arccos(X) X3 4 + (1 + 4ν s ) arccos X − (3 − 4ν s )X2 1 − X2 − 4ν s 1 − X2

(6.166)

(6.167)

′ ′ = S3322 = S3311

′ = S1313

′ = S1212

1 (2 − ν s ) X arccos ( X) + (2ν s − 2) 1 − X2 4 + (1 + ν s ) X3 arccos ( X) − 3ν s X2 1 − X2 + (ν s − 1) X4 1 − X2

1 X (9 − 8ν s ) X 1 − X2 + (8ν s − 6) X3 1 − X2 16 + (4 − 8 ν s ) arccos ( X) X2 + (8 ν s − 7) arccos ( X)

Note that Si jkl = S jikl = Si jlk .

(6.168)

(6.169)

(6.170)

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197

6.4.2 Application of the Dilute Scheme Using a dilute scheme, for which B is given by (6.116), both coefﬁcients b h and b v are proportional to the porosity, so that a depends only on the aspect ratio X. The ‘exact’12 analytical expression of the anisotropy ratio a could be derived from (6.116) using the Eshelby tensor S for an oblate pore geometry (see Section 6.4.1). In return, a ﬁrst approximation consists of using a Taylor series expansion of a around a given aspect ratio. In particular, a Taylor series development to second order around X = 1 (spherical pore) gives: a =1−

12μs 18(35k s + 8μs ) (1 − X) − (1 − X)2 + O(1 − X)2 9k s + 8μs 7(9k s + 8μs )2

(6.171)

or alternatively, in terms of Poisson’s ratio: a =1−

3(1 − 2ν s )(47 + 11ν s ) 6(1 − 2ν s ) (1 − X) − (1 − X)2 + O(1 − X)2 7 − 5ν s 7(7 − 5ν s )2 (6.172)

For ν s = 0.3, Figure 6.2 compares the approximation (6.171) to the exact value of a . It is remarkable that the accuracy of (6.171) goes beyond the domain X ≈ 1, and can be used almost over the entire interval ]0, 1]. In fact, the relative error between (6.171) and the exact value is less than 6% for X ∈ [0.2, 1]. In the domain X ∈ ]0, 0.2] of ﬂat ellipsoids, the accuracy of (6.171)–(6.172) is less satisfactory. A better approximation can be obtained with a Taylor series expansion of a around X = 0: a=

νs π(1 − 2ν s )2 + X 1 − νs 8(1 − ν s )2 +X2

(1 − 2ν s )((8π 2 − 64)ν s 2 − 14π 2 ν s + 5π 2 + 64) 64(1 − ν s )3

(6.173)

Unlike (6.171), the validity of (6.173) is restricted to a smaller interval (see Figure 6.2). Nonetheless, for ν s = 0.3, the relative error between (6.173) and the exact value of a is smaller than 2% for X ∈ ]0, 0.2]. 6.4.3 Influence of the Mechanical Interaction It is interesting to investigate the inﬂuence of the mechanical interaction on the anisotropy of tensor B induced by ﬂat pores. The anisotropy ratio a = b h /b v is derived by application of (6.141). The Taylor series expansion of a to second

12 Within

the framework of the dilute scheme.

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198

1.8

(1) : Taylor expansion (6.171)

1.6

(2) : Taylor expansion (6.172)

a 1.4

1.2

1 (2) 0.8

0.6

0.4

(1) 0

0.2

0.4

0.6

0.8

X

Figure 6.2 Comparison between the dilute estimate of a and its Taylor series (ν s = 0.3)

order around X = 1 is: a =1−

12(1 − ϕ0 )μs (1 − X) 6ϕ0 (k s + 2μs ) + 9k s + 8μs

(6.174) 18(1 − ϕ0 )(ϕ0 (52μs + 10k s ) + 35k s + 8μs ) 2 2 − (1 − X) + O(1 − X) 7(6ϕ0 (k s + 2μs ) + 9k s + 8μs )2

The accuracy of this estimate is remarkably good (see Figure 6.3). Over the whole interval X ∈ ]0, 1], the relative error between (6.174) and the exact value for ϕ0 = 0.4 and ν s = 0.3 is less than 5%.

6.5 Training Set: New Estimates of the Homogenized Diffusion Tensor This training set complements the one described in Chapter 4 (Section 4.5). The aim is to investigate the effect of interaction between solid inclusions on the homogenized diffusion tensor. To this end, we will adapt the Mori–Tanaka scheme (Section 6.3.4) and the self-consistent scheme (Section 6.3.5) to the molecular diffusion problem.

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199

0.95 a 0.9

0.85

0.8

(1) (2) : estimate of a from (6.141) (1) : Taylor expansion (6.173)

0.75

0.7

(2)

0

0.2

0.4

0.6

0.8

X

Figure 6.3 Estimate of a taking account of pore interaction and its Taylor series (ν s = 0.3, ϕ0 = 0.4)

6.5.1 The Mori–Tanaka Estimate of the Diffusion Coefficient As in Section 6.3.4, the ﬁrst task is to generalize the inhomogeneity problem (6.47) in order to capture the interaction between solid inclusions. In analogy to the ﬁctitious macroscopic strain tensor E0 , the boundary condition at inﬁnity involves a ﬁctitious macroscopic concentration gradient H 0 , instead of the real one H: div j γ = 0 γ

γ

(a ) γ

j = −D grad ρ − δ Dχ I (z) grad ρ

ρ γ → H 0 · z when | z | → ∞

γ

(b)

(6.175)

(c)

In the context of molecular diffusion, (6.135) is replaced by: s

γ

grad ρ γ = grad ρ I = (1 + δ DP I )−1 · H 0 p grad ρ γ = H 0

(6.176)

with δ D = Ds − Dγ = −Dγ . In this equation, P I is the P tensor for an inclusion I embedded in an inﬁnite medium with diffusion coefﬁcient Dγ .

200

Eshelby’s Problem in Linear Diffusion and Microporoelasticity

Due to the condition grad ρ γ = H , (6.137) is replaced by: −1 ·H H 0 = ϕ1 + (1 − ϕ)(1 + δ DP I )−1

(6.177)

A combination of (6.176) and (6.177) yields an estimate of the average concens tration tensor A : −1 s A = (1 + δ DP I )−1 · ϕ1 + (1 − ϕ)(1 + δ DP I )−1 (6.178) s

Recalling that Ds = 0 and that δ D = −Dγ , the above expression of A can also be expressed as: s

A = (1 − ϕ Dγ P I )−1

(6.179)

The last step consists of inserting (6.179) into (4.192): Dhom = Dγ 1 − (1 − ϕ)(1 − ϕ Dγ P I )−1

(6.180)

Let us illustrate this result in the case of spherical solid particles for which P I is given by (6.25). Equation (6.180) then takes the form: Dhom =

2ϕ Dγ 1 3−ϕ

(6.181)

6.5.2 The Self-Consistent Estimate of the Diffusion Coefficient As in Section 6.3.5, the idea of the self-consistent scheme consists of assuming that each particle of a given phase (pore or solid) reacts as if it were embedded in the equivalent homogeneous medium which is looked for. Let us examine the isotropic case where the homogenized medium is characterized by the homogenized diffusion coefﬁcient Dsc . For an isotropic morphology, we consider a spherical inclusion I of the α phase (α = s, p), embedded in an inﬁnite medium with diffusion coefﬁcient Dsc . The corresponding inhomogeneity problem is: div j γ = 0 j γ = −Dsc grad ρ γ − (Dα − Dsc )χ I (z) grad ρ γ ρ γ → H 0 · z when | z |→ ∞

(a ) (b) (c)

(6.182)

Relations (6.176) and (6.177) are respectively replaced by: −1 α γ grad ρ γ = grad ρ I = 1 + (Dα − Dsc )Psc · H0

(6.183)

and:

−1 H 0 = ϕ(1 + (Dγ − Dsc )Psc )−1 + (1 − ϕ)(1 + (Ds − Dsc )Psc )−1 · H (6.184)

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201 s

Note that Psc = 1/3Dsc 1 (see (6.25)). The average concentration tensor A is then: α

A = (1 + (Dα − Dsc )Psc )−1 · (1 + (D − Dsc )Psc )−1

−1

(6.185)

The last step consists of inserting (6.185) into Dhom = DA (see (4.192)): Dhom = D(1 + (D − Dsc )Psc )−1 · (1 + (D − Dsc )Psc )−1

−1

(6.186)

A similar reasoning to the one employed in Section 6.3.5 shows that (6.186) reduces to: Dhom = D(1 + (D − Dsc )Psc )−1

(6.187)

That is, for the isotropic case: Dhom =

3ϕ − 1 γ D 1 2

(6.188)

As for the elastic properties (see Figure 6.1), strong differences are observed between (6.181) and (6.188) (Figure 6.4). The Mori–Tanaka scheme predicts a strictly positive value of Dhom even in the vicinity of ϕ = 0. This is due to the fact that this scheme corresponds to a Dhom Dγ

1

0.8

0.6

0.4 (MT) 0.2 (SC) 0 0.2

0.4

0.6 ϕ

0.8

1

Figure 6.4 Mori–Tanaka and self-consistent estimates of the homogenized diffusion coefficient

Eshelby’s Problem in Linear Diffusion and Microporoelasticity

202

morphology in which the ﬂuid surrounds the solid particles. In other words, the Mori–Tanaka scheme provides a way to account for the connectedness of the ﬂuid phase, when this property is achieved. In contrast, the self-consistent estimate of the homogenized diffusion coefﬁcient vanishes for porosity smaller than 1/3. Therefore, this value of the porosity can be interpreted as the percolation threshold for molecular diffusion in the ﬂuid phase.

6.6 Appendix: Cylindrical Inclusion in an Isotropic Matrix Cylindrical inclusions in an isotropic matrix are to be understood as inclusions of prolate spheroid type with one axis of the ellipsoid (direction z3 ) being very much longer than the other two axes. The equation of the cross section is: z22 z12 + =1 a 2 c2

(6.189)

In the case of a circular section (a = c), the non-zero components of the Eshelby tensor, Si jkl = S jikl = Si jlk , are given by: S1111 = S2222 =

9 k s + μs 5 − 4ν s = 8(1 − ν s ) 4 3k s + 4μs

S1122 = S2211 =

4ν s − 1 1 3k s − 5μs = 8(1 − ν s ) 4 3k s + 4μs

S1133 = S2233 =

1 3k s − 2μs νs = 2(1 − ν s ) 2 3k s + 4μs S1313 = S2323 =

S1212 = S2121 =

(6.190)

1 4

1 3k s + 7μs 3 − 4ν s = 8(1 − ν s ) 4 3k s + 4μs

In the case of an elliptic section (X = c/a ): S1111 =

X (3 + 2 X − 2 ν s (1 + X))

S1122 =

2 (1 + X)2 (1 − ν s )

X (−1 + 2 ν s (1 + X)) 2 (1 + X)2 (1 − ν s )

; S2222 = ; S2211 =

2 + 3 X − 2 ν s (1 + X)

(6.191)

−X + 2 ν s (1 + X)

(6.192)

2 (1 + X)2 (1 − ν s )

2 (1 + X)2 (1 − ν s )

Appendix: Cylindrical Inclusion in an Isotropic Matrix

S1133 =

203

νs X νs ; S = 2233 (1 − ν s ) (1 + X) (1 − ν s ) (1 + X)

(6.193)

1 + X2 + X − ν s (1 + X)2

(6.194)

1 X ; S2323 = 2(1 + X) 2 (1 + X)

(6.195)

S1212 = S1313 =

2 (1 + X)2 (1 − ν s )

Part III Microporoinelasticity

7 Strength Homogenization The next few chapters deal with the microporomechanics of the nonlinear elastic and inelastic behavior of porous materials. This chapter addresses the question of how to deﬁne a macroscopic strength criterion of a porous material system based on the strength behavior of the solid phase. We start with the 1-D hollow sphere model in order to identify the essential ingredients necessary for strength homogenization. We then study the case of an empty porous material, for which we implement a nonlinear homogenization technique. In a second step, we introduce a ﬂuid pressure loading at the solid–ﬂuid interface, and investigate how this pressure affects the formulation of macroscopic poromechanics strength criteria.

7.1 The 1-D Thought Model: Strength Limits of the Saturated Hollow Sphere To motivate the forthcoming developments, we investigate the strength domain of a saturated porous medium in the context of the 1-D thought model of porous materials, the hollow sphere model (see Chapters 4 and 5). The cavity is ﬁlled with a ﬂuid at pressure P which acts on the solid–ﬂuid interface (r = A). The external boundary is subjected to a radial distribution of surface forces T = n, where 1 represents the (isotropic) macroscopic stress state. For a given value of pressure P, we look for the set G hs (P) of values of which are compatible with the strength of the solid phase. Let us ﬁrst assume that the solid is a von Mises material, deﬁned by the strength criterion: 1 s s 2 ij ij s

f s (σ) = J 2 − k 2 ≤ 0

(7.1)

where J 2 = is the stress deviator invariant, and k is the cohesion. The convexity of f (σ) implies that G hs (P) is an interval characterized by the Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

Strength Homogenization

208

minimum and maximum admissible values of , denoted by − and + . Given the spherical symmetry, we look for a statically admissible stress ﬁeld σ deﬁned on the solid domain A < r < B, of the form: σ = σrr (r )e r ⊗ e r + σθ θ (r )(e θ ⊗ e θ + e ϕ ⊗ e ϕ )

(7.2)

which meets the yield condition f s (σ) = 0: √ |σrr − σθθ | = k 3

(7.3)

Furthermore, the stress ﬁeld satisﬁes the boundary conditions on the surfaces r = A and r = B: σrr (A) = −P;

σrr (B) =

(7.4)

According to the sign of σrr − σθ θ in (7.3), the momentum balance equation div σ = 0 takes the form: √ ∂σrr 2k 3 ± =0 (7.5) ∂r r A combination of (7.4) and (7.5) yields the domain G hs (P) = [ − , + ]: 2k + = −P − √ ln ϕ 3 2k − = −P + √ ln ϕ 3

(7.6)

where ϕ = (A/B)3 denotes the cavity volume fraction. An interesting observation is that the strength of the hollow sphere as deﬁned by (7.6) is controlled by Terzaghi’s effective stress + P: ∈ G hs (P)

⇔

2k 2k √ ln ϕ ≤ + P ≤ − √ ln ϕ 3 3

(7.7)

Let us then consider a Drucker–Prager solid, deﬁned by the strength criterion: 1 s tr σ − h + J 2 f (σ) = α (7.8) 3 in which h represents the isotropic tensile strength (or cohesion pressure). In what follows we will assume that P > −h. Considering again a stress ﬁeld of the form (7.2), the stress σ meets the yield condition f s (σ) = 0 if σrr and σθ θ are related by: 1 |σrr − σθ θ | (σrr + 2σθ θ ) − h = 0 (7.9) +α √ 3 3

The 1-D Thought Model: Strength Limits of the Saturated Hollow Sphere

209

Let us introduce two auxiliary parameters β and γ : β=

2α/3

√ ; 2α/3 − 1/ 3

γ =

2α/3

√ 2α/3 + 1/ 3

(7.10)

√ We start by considering the case α < 3/2 for which β < 0. If σrr > σθ θ , the momentum balance equation div σ = 0 is: 3β ∂σrr + (σrr − h) = 0 ∂r r In turn, if σrr < σθ θ , (7.11) is replaced by:

(7.11)

∂σrr 3γ + (σrr − h) = 0 (7.12) ∂r r Combining (7.4) and (7.11) (resp. (7.4) and (7.12)), the upper and lower limits of G hs (P) for the Drucker–Prager porous material are: + = h − (P + h)ϕ γ − = h − (P + h)ϕ β

(7.13)

The stress is compatible with the strength domain of the solid provided that: +P ≤ h(1 − ϕ γ ) (7.14) ∈ G hs (P) ⇔ h(1 − ϕ β ) ≤ 1 + P/ h Relation (7.14) reveals that the strength of the saturated hollow sphere composed of a Drucker–Prager solid is controlled by the following effective stress expression: e f f =

+P 1 + P/ h

(7.15)

√ Let us now have a closer look at the case α > 3/2. We consider the stress ﬁeld deﬁned by (7.2) and (7.4) together with the condition: σrr − σθ θ ′ 1 (σrr + 2σθ θ ) − h = 0 (7.16) +α √ 3 3 √ where α ′ can take any value on the interval [0, 3/2[. From the momentum balance condition div σ = 0, it is readily found that: −3β ′ r σrr = h − (P + h) (7.17) A where: β′ =

2α ′ /3

√ 2α ′ /3 − 1/ 3

(7.18)

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210

Combining (7.16) and √ (7.17), it is readily found that σrr > σθ θ and tr σ/3 < h. Recalling that α ′ < 3/2 < α, this implies: |σrr − σθ θ | 1 s f (σ) = tr σ − h < 0 (7.19) +α √ 3 3 ′

Accordingly, the stress = σrr (B) = h − (P + h)ϕ β is admissible. Observing that limα′ →√3/2 = −∞, we conclude that the strength of the hollow sphere in compression is inﬁnite, while the maximum stress in tension, + , is still given by (7.13). In other words, the triaxial strength is√open on the negative hydrostatic axis, and closed on the tension side: α > 3/2, G hs (P) = ]−∞, + ]. 7.2 Macroscopic Strength of an Empty Porous Material For the upscaling strength behavior, a good starting point is the empty or nonpressurized pore space. In this case, the determination of the overall strength of a porous material requires a description of the strength of the solid phase only, together with morphological information concerning the geometry of the microstructure. To this end, we recall some classical results of convex analysis that will turn out to be useful in the forthcoming developments. 7.2.1 Microscopic Strength of the Solid Phase There are two equivalent ways to deﬁne the strength of the solid phase, which can be referred to as the direct deﬁnition and the dual one. The direct approach consists of deﬁning the convex set G s of strengthcompatible (microscopic) stress states. This is the approach we took in the study of the hydrostatic strength domain of the hollow sphere in Section 7.1. From a mathematical point of view, this is achieved by means of a (convex) strength criterion f s (σ): G s = {σ, f s (σ) ≤ 0}

(7.20)

The boundary ∂G s is characterized by the condition f s (σ) = 0 and the zerostress state σ = 0 is assumed to be strength compatible, i.e. f s (0) ≤ 0. In contrast to the direct approach, a dual deﬁnition of the strength criterion consists of introducing the support function π s (d) of G s , which is deﬁned on the set of symmetric second-order tensors d and which is convex w.r.t. d: π s (d) = sup(σ : d, σ ∈ G s )

(7.21)

π s (d) represents the maximum ‘plastic’ dissipation capacity that the material can afford. The fact that the zero stress is strength compatible, i.e. 0 ∈ G s ,

Macroscopic Strength of an Empty Porous Material

211 σ : d > π s(d)

d σ : d = π s(d)

σk l

σ : d < π s(d) H (d )

σij G

s

Figure 7.1 Geometrical interpretation of the support function

implies the non-negativity of π s (d) ≥ 0. Furthermore, it is readily seen that: (∀ t ∈ R+ ) π s (td) = t π s (d)

(7.22)

The dual deﬁnition of the solid strength thus takes the form: σ ∈ Gs

⇔

(∀ d) σ : d ≤ π s (d)

(7.23)

For a given value of d, we recognize that the condition σ : d = π s (d) deﬁnes a hyperplane H(d) in the stress space. This hyperplane is tangent to the boundary ∂G s at the point σ at which the normal to ∂G s is parallel to d (see Figure 7.1). Moreover, differentiating (7.22) with respect to t > 0 yields: ∂π s (d) : d = π s (d) ∂d

(7.24)

It follows that the stress state σ ≡ ∂π s /∂d(d) is located on H(d). Furthermore, the convexity of the support function reads: π s (d′ ) − π s (d) ≥ Combining (7.24) and (7.25) yields:

∂π s (d) : d′ − d ∂d

(∀ d′ ) π s (d′ ) ≥

∂π s (d) : d′ ∂d

(7.25)

(7.26)

According to the dual deﬁnition (7.23) of G s , relation (7.26) ensures that σ ≡ ∂π s /∂d(d) is located at the intersection of H(d) with G s .

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212

7.2.2 Strength-Compatible Macroscopic Stress States A microscopic stress ﬁeld σ(z) deﬁned on the rev is statically compatible with a given macroscopic stress state Σ provided it satisﬁes:

r the momentum balance condition div σ = 0; r the average rule Σ = σ; and r a zero stress in the pore space, (∀z ∈ p ) σ = 0. In turn, a macroscopic stress state Σ is compatible with the material strength if a microscopic stress ﬁeld σ(z) exists that is statically admissible with Σ and compatible with the strength of the solid. Let G hom denote the set of such strength-compatible macroscopic stress states: G hom = {Σ, ∃ σ s. a. with Σ, (∀z ∈ s ) σ(z) ∈ G s }

(7.27)

For a given macroscopic strain rate tensor D, let us deﬁne the set V(D) of kinematically admissible microscopic velocity ﬁelds v(z): V(D) = {v, (∀z ∈ ∂) v(z) = D · z}

(7.28)

For Σ ∈ G hom , let σ comply with the conditions of (7.27). Furthermore, let us consider an arbitrary element v ∈ V(D). The Hill lemma presented in Section 4.2.6 states that:1 Σ : D = σ : d = (1 − ϕ)σ : d

s

(7.29)

where d denotes the microscopic strain rate associated with the velocity ﬁeld v. Recalling that σ is compatible with the strength of the solid, it follows from (7.23) and (7.29) that: Σ : D ≤ hom (D) with

hom (D) = (1 − ϕ) inf π s (d) v∈V(D)

s

(7.30)

Relation (7.30) shows that G hom is located in a half-space bounded by the hyperplane Σ : D = hom (D). In particular, if Σ belongs to both this hyperplane and G hom , it is located on the boundary ∂G hom at a point at which the normal to ∂G hom is parallel to D: Σ : D = hom (D) ⇒ Σ ∈ ∂G hom (7.31) Σ ∈ G hom 1 In the sections devoted to the failure criterion, the pore volume fraction is denoted by ϕ instead of ϕ0 . This notation points to the fact that the conﬁguration of the microstructure which is relevant for the analysis of failure is not the initial one but the current conﬁguration.

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213

7.2.3 Determination of ∂Ghom We present here a strategy for the determination of ∂G hom . This strategy is based on a systematic method2 for deriving microscopic stress ﬁelds σ associated in the sense of (7.27) with the macroscopic stresses located on ∂G hom . More precisely, in the dual deﬁnition of the solid strength (7.23), we have seen that the microscopic stress ﬁeld σ = ∂π s /∂d(d) is located on the boundary ∂G s . It is intriguing to explore this property in the sense of a nonlinear viscous behavior of the solid phase in the rev, by deﬁning this property σ = ∂π s /∂d(d) as a viscous state equation, which is non-zero only in the solid phase and σ = 0 in the pore space. For a given macroscopic strain rate D, consider the microscopic stress and velocity ﬁelds σ(z) and v(z) which are solutions of the mechanical problem deﬁned on the rev by the Hashin boundary conditions v(z) = D · z on ∂: div σ = 0 ∂π s (d) σ= ∂d σ=0

()

(a )

(s )

(b)

( p )

(c)

(7.32)

d = 12 (grad v + tgrad v) () (d) (∂) (e) v(z) = D · z

According to the conclusion of Section 7.2.1, the stress ﬁeld solution to (7.32) is compatible with the strength of the solid phase. Equation (7.27) implies that Σ = σ ∈ G hom . In particular, (7.30) holds. Let us now combine (7.24) and (7.32b) to obtain: π s (d) =

∂π s (d) : d = σ : d ∂d

(7.33)

Taking the average of (7.33) over the solid phase yields: s

s

(1 − ϕ)π s (d) = (1 − ϕ)σ : d = Σ : D

⇒ hom (D) ≤ Σ : D

(7.34)

The combination of (7.30) and (7.34) proves that hom (D) = Σ : D, which means (see (7.31)) that Σ is located on the boundary ∂G hom . The determination of ∂G hom therefore reduces to ﬁnding the effective behavior of a porous medium made up of a nonlinear viscous solid phase (see (7.32b,c)).

2 This

presentation is inspired by [37].

Strength Homogenization

214

7.2.4 Solid Strength Depending on the First Two Stress Invariants From now on we assume that the strength of the solid phase is controlled by the mean stress and the equivalent deviatoric stress: I1 = tr σ s f (σ) = F(I1 , J 2 ) with (7.35) J 2 = 12 s : s where s = σ − I1 1/3 is the deviatoric stress tensor. Let us the volume introduce ′ ′ strain rate I1 and the equivalent deviatoric strain rate J 2 associated with the strain rate tensor d: ′ I1 = tr d 1 ′ d = I1 1 + δ with (7.36) J 2′ = 21 δ : δ 3 According to deﬁnition (7.21), the support function is now: 1 ′ s π (d) = sup I1 I + s : δ, F(I1 , J 2 ) ≤ 0 3 1

(7.37)

For a given value of J 2 , the choice of s which maximizes s : δ is parallel to δ, namely s = δ J 2 /J 2′ . Relation (7.37) thus takes the form: 1 ′ s ′ (7.38) π (d) = sup I1 I + 2 J 2 J 2 , F(I1 , J 2 ) ≤ 0 3 1 It then turns out that the support function depends only on the invariants I1′ and J 2′ of d: π s (d) = π s I1′ , J 2′ (7.39)

The state equation (7.32b) is therefore: σ=

∂π s ′ ′ ∂π s ′ ′ 1 + I I , J δ = Cs (d) : d , J ∂ I1′ 1 2 ∂ J 2′ 1 2

(7.40)

The ﬁctitious viscous behavior of the solid phase is found to be deﬁned by an isotropic secant ‘stiffness’ tensor Cs (d); that is, by secant bulk and shear moduli k s (I1′ , J 2′ ) and μs (I1′ , J 2′ ): ⎧ ′ ′ 1 ∂π s ′ ′ ⎪ s ⎪ I , J = k I ,J ⎪ 1 2 ⎨ I1′ ∂ I1′ 1 2 ′ ′ ′ ′ s s s C (d) = 3k I1 , J 2 J + 2μ I1 , J 2 K with ⎪ ∂π s ′ ′ ⎪ ⎪ I ,J ⎩ 2μs I1′ , J 2′ = ∂ J 2′ 1 2 (7.41)

Macroscopic Strength of an Empty Porous Material

215

7.2.5 Principle of Nonlinear Homogenization Taking (7.40) into account, we note that (7.32b,c) can be summarized as follows: z ∈ s : C(z) = Cs (d(z)) σ(z) = C(z) : d(z) with (7.42) z ∈ p : C(z) = 0 Accordingly, the boundary value problem (7.32) is now: div σ = 0 σ(z) = C(z) : d(z)

() (a ) () (b)

d = 21 (grad v + tgrad v) () (c) v(z) = D · z (∂) (d)

(7.43)

In this form, relations (7.43) are formally identical to the problem (4.43) introduced in Chapter 4, provided that the strain ε (resp. the displacement ξ ) in (4.43) is replaced by the strain rate d (resp. the velocity v). However, two essential differences exist between (4.43) and (7.43). In (4.43), the elastic stiffness is homogeneous in the solid phase and is independent of the loading. By contrast, like the strain rate d(z), the tensor Cs (d(z)) which appears in (7.42)–(7.43b) is heterogeneous and depends on the load level. The so-called secant methods in nonlinear homogenization aim at capturing the dependence of C(z) = Cs (d(z)) on the loading level in an average way. The idea consists of introducing a reference strain rate ﬁeld3 dr in s and to approximate the ‘real’ heterogeneous stiffness by a uniform value in the whole solid phase: (∀z ∈ s )

C(z) = Cs (d(z)) ≈ Cs (dr )

(7.44)

Accordingly, dr is looked for in the form of an average of the strain rate ﬁeld d(z) over s , which of course should depend on the load level. Indeed, there are various ways to implement (7.44) that differ in the choice of the reference strain rate.4 The simplest choice consists of deﬁning dr as the intrinsic average of the strain rate over the solid phase: s

dr = d(z)

(7.45)

Second-order moments introduced in Section 4.4 constitute an interesting alternative to (7.45) for deﬁning the reference strain. Transposing (4.179), we introduce a reference deviatoric strain rate, deﬁned as: 1 s ′r J2 = δ:δ (7.46) 2 3 This strategy has already been encountered in Section 2.6.2 while looking for an extension of Darcy’s law for power-law ﬂuids. 4 For a comprehensive presentation of nonlinear homogenization, see [48].

Strength Homogenization

216

With these elements in hand, let us summarize the successive steps of the homogenization procedure: 1. With the approximation (7.44), (7.43) reduces to a standard problem of heterogeneous linear elasticity. It is therefore possible to determine the macroscopic stress Σ = σ as in (4.113)–(4.115): Σ = Chom : D

with

p

Chom = Cs (dr ) : (I − ϕ A (dr ))

(7.47)

which represents the ﬁrst step of the nonlinear homogenization problem. 2. The next step consists of determining the reference strain as a function of the loading level, according to the adopted deﬁnition. This step can be performed using the results of Section 4.4 concerning the ﬁrst- and secondorder moments of the strain ﬁeld, applied here to the strain rate ﬁeld. Formally, they yield dr as a function of D: dr = dr (D)

(7.48)

It is worth emphasizing that these developments have been obtained in a linear framework. 3. The last step consists of solving the nonlinearity of the problem (7.47)–(7.48) which comes from the dependence of Chom on dr , with dr being a function of D. Combining these equations, the macroscopic state equation takes the form: Σ = Chom (D) : D

(7.49)

It is important to note that the result of this nonlinear homogenization technique depends on the linear homogenization scheme which is chosen for relating Chom to Cs (dr ) (ﬁrst step). In particular, this choice incorporates morphological assumptions concerning the geometry of the microstructure (matrix-inclusion concept or polycrystal-like microstructure). We have already observed that these assumptions yield very different estimates of the effective stiffness (see for instance Figure 6.1). Similar results are therefore expected as regards the effective strength.

7.3 Von Mises Behavior of the Solid Phase 7.3.1 The Equivalent Viscous Behavior We want to apply the method of Sections 7.2.3 and 7.2.5 to the case of a solid √ of the von Mises type, deﬁned by the strength criterion (7.1). In the (I1 , J 2 ) space, the set G s of strength-compatible stress states associated with (7.1) is a circular cylinder. Its axis is the I1 axis and its radius is equal to k.

Von Mises Behavior of the Solid Phase

217

The deﬁnition (7.1) suggests application of the results of Section 7.2.4. It can be easily veriﬁed that the support function π s (d) associated with (7.1) is: √ k 2d : d if tr d = 0 π (d) = ∞ if tr d = 0 s

(7.50)

The inﬁnite value of π s (d) in the case tr d = 0 is due to the fact that G s is not a closed domain. Indeed, stress states of the form λ1 with arbitrary high values of |λ| clearly meet the condition f s (σ) ≤ 0 since f s (σ) = −k 2 . Hence, if tr d = 0, λ1 : d = λ tr d can take arbitrary large values as λ tends towards inﬁnity. π s (d) therefore is a highly singular function. This unfortunately prevents a straightforward application of (7.40)–(7.41). In order to avoid this mathematical difﬁculty, we introduce a sequence of closed domains G sj that tends towards the cylinder G s deﬁned in (7.1) when j → ∞. For instance, let us deﬁne G sj as the ellipse centered at the origin of √ the (I1 , J 2 ) space (Figure 7.2): f js (σ)

=

I1 Lj

2

2 J 2 − k2 ≤ 0 +

(7.51)

We verify that G sj tends towards G s as the half-length L j k of the horizontal axis of the ellipse tends towards inﬁnity. Recast in this form, the support function π sj (d) is differentiable and as a consequence the corresponding secant stiffness Csj (d) can be derived easily by means of (7.41). Let G hom denote the set j of macroscopic stress states compatible with the strength of the ﬁctitious solid deﬁned by (7.51). A stress state Σ located on ∂G hom is the average σ of a stress j

√

Gs

J2

k

Gsj

I1

Figure 7.2 Elliptic approximation of the von Mises criterion

Strength Homogenization

218

ﬁeld σ solution to the following problem: div σ = 0 σ(z) = C j (z) : d(z)

() (a ) () (b)

d = 21 (grad v + tgrad v) () (c) v(z) = D · z (∂) (d)

(7.52)

The methodology of Section 7.2.5 then yields an estimate for the domain G hom j . The real domain G hom is obtained asymptotically from lim L j →∞ G hom . j 7.3.2 Homogenization of the Fictitious Viscous Behavior We ﬁrst have to derive the support function π sj (d) of G sj . For a given value of the strain rate d, we recall (see Section 7.2.1 and Figure 7.1) that the maximum value of σ : d is reached at the point σ ∗ where d is normal to ∂G sj . d is therefore parallel to ∂ f js /∂σ(σ ∗ ): ∂ f js

d = λ˙

∂σ

(σ ∗ )

(7.53)

where λ˙ is a positive scalar. Using (7.51), we successively obtain:

(7.54)

˙ 2 (σ ∗ ) = 2λk

(7.55)

I∗ d = λ˙ 2 12 1 + s∗ Lj and: ∂ f js

˙ ∗: π sj (d) = λσ

∂σ

With the same notation as in (7.36), we now observe from (7.54) that: I1′ = λ˙

6I1∗ ; L 2j

J 2′ = λ˙ 2 J 2∗

(7.56)

Finally, a combination of the previous equations with (7.51) allows us to elim˙ and the support function becomes: inate λ,

π sj (d) = 2k

L 2j 36

I1′ 2 + J 2′

(7.57)

Von Mises Behavior of the Solid Phase

219

The secant stiffness Csj (d) is then given by (7.41), with the following secant bulk and shear moduli: k sj (I1′ , J 2′ ) = k

2μsj (I1′ ,

J 2′ )

= k

L 2j /18 J 2′ + L 2j I1′ 2 /36 1 J 2′ + L 2j I1′ 2 /36

(a ) (7.58) (b)

Let us now apply the method proposed in Section 7.2.5 to the porous material composed of the ﬁctitious solid with stiffness Csj (d) (problem (7.52)). We start by writing the macroscopic behavior in the form (7.47): 1 tr Σ = k hom tr D; j 3

Σd = 2μhom j ∆

(7.59)

where Σd (resp. ∆) is the macroscopic deviatoric stress (resp. strain rate): 1 Σ = (tr Σ) 1 + Σd ; 3

1 D = (tr D) 1 + ∆ 3

(7.60)

We then note from (7.58) that k sj /μsj = O(L 2j ). As L j tends towards inﬁnity, the ﬁctitious solid can be regarded as asymptotically incompressible. We now need to relate k hom and μhom to μsj . This requires us to select a linear j j homogenization scheme. As already stated, the Mori–Tanaka estimate of the effective behavior is implicitly associated with a matrix-inclusion morphology, in which the pores play the role of the inclusion phase. In contrast, the perfectly disordered microstucture can be addressed within the framework of a self-consistent approach. In both cases, recalling that we deal with an incompressible solid, the macroscopic bulk and shear moduli can be expressed in the form: k hom = K μsj ; j

μhom = Mμsj j

(7.61)

Recalling (6.146), the Mori–Tanaka estimates of M and K are: K mt (ϕ) =

4(1 − ϕ) ; 3ϕ

Mmt (ϕ) =

1−ϕ 1 + 2ϕ/3

(7.62)

In turn, the self-consistent estimates of M and K are derived from (6.160): K sc (ϕ) = 4

(1 − 2ϕ)(1 − ϕ) ; ϕ(3 − ϕ)

Msc (ϕ) = 3

(1 − 2ϕ) (3 − ϕ)

(7.63)

The second step of the nonlinear homogenization procedure deals with the determination of the reference strain as a function of the macroscopic loading. It is recalled that this step is performed within the framework of

Strength Homogenization

220

linear elasticity. Let us focus on the reference deviatoric strain rate J 2′ r that is given by (4.177) as a function of the macroscopic strain rate5 D: ∂μhom ∂k hom 1 1 j j ′r 2 +∆:∆ (7.64) (tr D) J2 = 2(1 − ϕ) 2 ∂μsj ∂μsj Using (7.61), this equation can also be expressed as: 1 1 2 ′r (tr D) K + ∆ : ∆ M J2 = 2(1 − ϕ) 2

(7.65)

Note that (7.65) is the particular form of (7.48) in the present problem. The asymptotic incompressibility is I1′ r ≈ 0. More precisely, let us deﬁne I1′ r as s tr d . Equation (4.169)6 readily yields: s μj k hom j ′r ′r (1 − ϕ)I1 = s tr D ⇒ I1 = O s tr D = O(L −2 (7.66) j )tr D kj kj 2

It then follows from (7.64) that J 2′ r /I1′ r = O(L 4j ). The expression (7.58b) for μsj thus reduces to: k r (7.67) 2μsj (J 2′ ) ≈ J 2′ r

The third and last step consists of dealing with the nonlinearity of the state equation (7.59). This nonlinearity comes from the fact that μsj depends on J 2′ r which itself is a function of D. Combining (7.59), (7.61) and (7.67) yields: 2 m ′ r J2 tr D = K k (7.68) 1 1 d ′ r J2 ∆:∆= 2 M k with: 1 1 Σd : Σd (7.69) m = tr Σ; d = 3 2 Finally, we insert (7.68) into (7.65): 1 m 2 1 d 2 + (7.70) (1 − ϕ) = K k M k

This equation represents the asymptotic locations in the stress space of the macroscopic stress state solutions of (7.52), for arbitrary values of D. In other 5 The

macroscopic strain E in (4.177) is formally replaced by D. is, εv and E in (4.169) are replaced by tr d and D, respectively.

6 That

Von Mises Behavior of the Solid Phase

221

words, it deﬁnes the boundary of G hom which is found to be a closed elliptic domain centered at the origin of the (m , d ) plane. Let us now discuss (7.70) w.r.t. the morphology of the rev and to the corresponding homogenization scheme. In particular, for the matrix-inclusion morphology, use of the Mori–Tanaka scheme yields7 (see (7.62)): 2ϕ 3ϕ 2 d2 = k 2 (1 − ϕ)2 (7.71) m + 1 + 4 3 First, we note that (7.1) is retrieved for ϕ = 0. The other limit case corresponds to ϕ → 1 for which we observe that the effective strength vanishes. Conversely, some strength is available even for high values of the porosity, provided that ϕ < 1. This should be attributed to the matrix-inclusion morphology which has been considered here through the use of the Mori–Tanaka estimate. Consider next the self-consistent (or polycrystal) scheme which captures the morphology of a perfectly disordered solid phase intermixed with porosity. Inserting (7.63) into (7.70) yields the following self-consistent estimate of the homogenized strength criterion: ϕ(3 − ϕ) 2 ϕ 2 d = k 2 (1 − ϕ)(1 − 2ϕ) (7.72) m + 1 − 4(1 − ϕ) 3

As in the previous case, (7.1) is retrieved for ϕ = 0. However, the homogenized strength now vanishes for ϕ ≥ 1/2. As for the stiffness (see Section 6.3.5), the macroscopic strength exhibits a percolation threshold of the pore space at ϕ = 1/2. The domains of admissible macroscopic stress states corresponding to (7.71) and (7.72) are shown in Figure 7.3. 7.3.3 Validation It is instructive to compare the results obtained with the Mori–Tanaka scheme in Section 7.3.2 to the one of the 1-D hollow sphere model of Section 7.1. In fact, the geometry of the hollow sphere in which the cavity is surrounded by the solid is a particular form of the matrix-inclusion morphology captured by the Mori–Tanaka scheme. Furthermore, despite its limitation, the hollow sphere model provides a reasonable estimate of the strength under hydrostatic compression or traction of both a microscopic and macroscopic isotropic material. Letting P = 0 in (7.6), the hydrostatic strength domain of the hollow sphere is: 2k ± hs = ± √ ln ϕ 3

7 For

(7.73)

a discussion on this type of criterion as compared to the one derived by Gurson [28], see [27].

Strength Homogenization

222 1 Σ d /k

ϕ = 0.3

(SC)

(MT)

−2

−1

0

Σ m /k

1

2

−1

Figure 7.3 Mori–Tanaka and self-consistent estimates of the domain of admissible macroscopic stress states (von Mises solid)

In turn, the Mori–Tanaka estimate (7.71) yields the following hydrostatic strength limits of the empty porous material: 2k 1 − ϕ ± mt = ±√ √ ϕ 3

(7.74)

Figure 7.4 displays an excellent agreement between the estimates (7.73) and (7.74), except for inﬁnitesimal values of the porosity. For such small values, high strain rates are expected to concentrate around the pores, which cannot be captured by the reference strain rate concept (7.44). In fact, an average value over the whole solid phase fails to provide an accurate estimate of the local strain rate level. This is why we observe a divergence of the estimate (7.74) from the more accurate estimate (7.73).

7.4 The Role of Pore Pressure in the Macroscopic Strength Criterion We now investigate the role of a ﬂuid pressure P in the macroscopic strength criterion. The presence of such a ﬂuid pressure does not affect the strengthcompatible stress state deﬁnition (7.27). On the other hand, the conditions for a microscopic stress ﬁeld σ to be statically admissible with the macroscopic stress Σ are now: div σ = 0 Σ=σ (∀z ∈ p ) σ = −P1

(7.75)

The Role of Pore Pressure in the Macroscopic Strength Criterion

223

3 (mt)

2

Σ± k

(hs) 1

0

ϕ

1

Figure 7.4 Von Mises solid: hydrostatic strength predicted by the hollow sphere model (hs) and the Mori–Tanaka scheme (mt)

Let G hom (P) denote the set of strength-compatible macroscopic stress states for the value P of the pore pressure. In particular, G hom (0) is the domain obtained in the non-pressurized case, which has been studied in Section 7.2. Let us consider a given macroscopic stress state Σ ∈ G hom (P) and a microscopic stress ﬁeld σ complying with (7.75) and with the strength criterion of the solid. We then introduce σ ˜ = σ + P1. It follows that: ⎧ div σ ˜ =0 ⎪ ⎪ ⎨ Σ + P1 = σ ˜ Σ ∈ G hom (P) ⇔ ∃σ (7.76) ˜ p (∀z ∈ ) σ ˜ =0 ⎪ ⎪ ⎩ ˜ ∈ G s + P1 (∀z ∈ s ) σ

where G s + P1 is the set obtained from G s by application of the translation a → a + P1. 7.4.1 Von Mises or Tresca Solid

In the case of a von Mises or Tresca solid, the strength is not inﬂuenced by the hydrostatic stress, i.e. G s + P1 = G s . Accordingly, if Σ ∈ G hom (P), the properties of σ ˜ ensure that Σ + P1 ∈ G hom (0). This reasoning can be summarized

Strength Homogenization

224

by: G hom (0) = G hom (P) + P1

(7.77)

Σ ∈ G hom (P) ⇔ Σ + P1 ∈ G hom (0)

(7.78)

or, alternatively:

Expression (7.78) can be seen as a generalization of (7.7) obtained for the hollow sphere model. In fact, it shows that the macroscopic strength criterion of the pressurized porous material can be formulated as a function of Terzaghi’s effective stress Σ + P1. For a von Mises solid, the strength domain is obtained from (7.70) by replacing the mean stress of the empty porous material, m , with the mean effective stress of the pressurized medium, m + P: 1 1 2 (m + P)2 + = k 2 (1 − ϕ) (7.79) K M d where K and M are still deﬁned by (7.62) for a Mori–Tanaka morphology and by (7.63) for a (self-consistent) polycrystal morphology. Expression (7.77) allows for the following straightforward geometrical interpretation: G hom (P) is obtained from G hom (0) by a translation parallel to the m axis in the (m , d ) plane (Figure 7.5). 7.4.2 Drucker–Prager Solid It is interesting to check whether the Terzaghi effective stress concept still holds in the case of a solid strength criterion that is sensitive to the mean stress. To this end, let us assume that the set of admissible stress states for the solid is a cone. Its apex lies on the line σ1 = σ2 = σ3 in the space of principal stresses and represents an isotropic tensile stress state h1. This set is denoted by G sh . G hom(0)

Σd

Σm

P<0

P>0

G

hom

(P )

Figure 7.5 Ghom(0) and Ghom(P ) in the case of a von Mises solid

The Role of Pore Pressure in the Macroscopic Strength Criterion

225

The scalar h > 0 can be referred to as the tensile strength. The corresponding set of macroscopic stress states in drained condition (P = 0) is denoted by G hom h (0). This representation is a characteristic example of a Drucker–Prager material (see (7.8)). Interestingly, we observe that the sets G sh and G sh ′ associated with two different values h > 0 and h ′ > 0 of the tensile strength can be deduced from one another by either homothety or translation of G sh :8 (a )

G sh ′ =

h′ s G ; h h

(b)

G sh ′ = G sh + (h ′ − h)1

(7.80)

According to the deﬁnition given in (7.27), (7.80a) implies that the set of admissible macroscopic stress states in drained condition linearly depends on the microscopic tensile strength h: G hom h ′ (0) =

h ′ hom G (0) h h

(7.81)

It is readily seen from (7.80b) that G sh + P1 = G sh+P

(7.82)

Introducing this result into (7.76) shows that hom Σ ∈ G hom h (P) ⇔ Σ + P1 ∈ G h+P (0)

(7.83)

Assuming that P > −h, a combination of (7.81) and (7.83) then yields: P hom G hom (7.84) G h (P) + P1 = 1 + h (0) h That is: Σ ∈ G hom h (P) ⇔

Σ + P1 ∈ G hom h (0) 1 + P/ h

(7.85)

The previous relations show that the macroscopic strength is controlled by the following effective stress: Σe f f =

Σ + P1 1 + P/ h

(7.86)

Clearly, the deﬁnition of the effective stress depends on the solid behavior.9 In contrast to Terzaghi’s effective stress relevant for a von Mises solid, the effective stress Σe f f deﬁned by (7.86) for a Drucker–Prager solid does not linearly depend on the pore pressure P. This result generalizes the 1-D result 8 λG s is the image of G s by the homothety of which the center is located at the origin, with a ratio equal to λ: z → λz. 9 For further examples, see [14] and [15].

Strength Homogenization

226

(7.14) obtained for the hollow sphere model. In other words, it sufﬁces to estimate the strength domain G hom (0) for the empty porous material, and determine G hom (P) from a straightforward application of (7.84). Estimates for G hom (0) in the case of a Drucker–Prager solid will be developed in Section 7.5.3 where the derivation of G hom (P) from G hom (0) will be considered further (See Figure 7.7).

7.5 Nonlinear Microporoelasticity 7.5.1 Non–Pressurized Pore Space Section 7.2 was devoted to the failure of a porous material induced by that of the solid phase. We now address the nonlinear behavior of the solid in reversible conditions. In this case, the elastic energy stored in the solid is characterized by a potential ψ s (ε) which deﬁnes the stress equation of state of the solid phase in the classical form: σ=

∂ψ s ∂ε

(7.87)

It is convenient to split the strain tensor into a spherical part and a deviatoric part εd : ′ 1 ′ I1 = tr ε = εv (7.88) ε = I1 1 + εd with J 2′ = εd2 = 21 εd : εd 3 We further assume that the behavior is isotropic, and that ψ s depends only on the two invariants I1′ and J 2′ . The state equation (7.87) of the solid thus takes the form σ = Cs (ε) : ε where the secant stiffness tensor Cs (ε) is deﬁned by: ⎧ 1 ∂ψ s ⎪ s ⎪ = k ⎪ ⎨ I1′ ∂ I1′ (7.89) Cs (ε) = 3k s (I1′ , J 2′ )J + 2μs (I1′ , J 2′ )K with ⎪ ∂ψ s ⎪ s ⎪ ⎩ 2μ = ∂ J 2′

The introduction of a potential to derive the state equation of the nonlinear elastic behavior of the solid has some important implications. In fact, if the nonlinear behavior of the solid phase is given in the form σ = Cs (ε) : ε, the secant stiffness tensor must meet some conditions in order to ensure the existence of a potential ψ s (ε) in the sense of (7.87). In the isotropic case, i.e. Cs = Cs (I1′ , J 2′ ), a necessary and sufﬁcient condition for the existence of ψ s is readily obtained from (7.89): 2

s ∂μs ′ ∂k = I 1 ∂ I1′ ∂ J 2′

(7.90)

Nonlinear Microporoelasticity

227

We are left with specifying the link between the microscopic strain invariants and the macroscopic loading of the rev deﬁned by the macroscopic strain tensor E. The microscopic stress, strain and displacement ﬁelds σ, ε and ξ , respectively, in the rev are solutions of the following boundary value problem: div σ = 0 () (a ) () (b) σ(z) = C(z) : ε(z) ε = 12 (grad ξ + tgrad ξ ) () (c) (∂) (d) ξ (z) = E · z

(7.91)

where:

z ∈ s : z ∈ p :

C(z) = Cs (ε(z)) C(z) = 0

(7.92)

In (7.91), ξ and ε play the role of v and d in (7.42). This formal identity of (7.42)– (7.43) and (7.91)–(7.92) allows us to employ the tools developed in Section 7.2.5 for the homogenization of the nonlinear elastic behavior. In particular, the strain level in the solid phase is represented by a reference strain εr that depends on the macroscopic loading. In analogy to (7.44), a linear homogeneous approximation of the solid state equation is adopted: (∀z ∈ s )

σ(z) = Cs (εr ) : ε(z)

(7.93)

Following (7.47), this technique yields the macroscopic state equation in the form: p Σ = Chom : E with Chom = Cs (εr ) : I − ϕ0 A (εr ) (7.94)

where Chom is the homogenized elastic stiffness corresponding to the linear elastic solid deﬁned by (7.93) with stiffness Cs (εr ). 7.5.2 Pressurized Pore Space We want to assess the inﬂuence of the pore pressure P on the macroscopic nonlinear poroelastic state equations. The behavior of the solid phase is still deﬁned by (7.87) and (7.89). The nonlinear boundary value problem of the rev is deﬁned by: div σ = 0 () σ = C(z) : ε + σ p (z) () ξ =E·z (∂)

(7.95)

where: C(z) =

0 ( p ) s C (ε(z)) (s )

p

σ (z) =

−P1 ( p ) 0 (s )

(7.96)

Strength Homogenization

228

It is interesting to note the formal identity of (7.95)–(7.96) and (5.101)–(5.103) which have been introduced in the study of the linear poroelastic behavior. The only difference is that Cs is now a function of the local strain. According to the concept of reference strain, the approximation (7.93) Cs = Cs (εr ) of the solid behavior is adopted. For purposes of clarity, we restrict ourselves to an isotropic behavior for which the reference strain εr is characterized by two invariants: I1′ r = tr εs ;

J 2′ r = εd2

s

(7.97)

In this isotropic case, the approximate solid stiffness reads: Cs (εr ) = 3k s I1′ r , J 2′ r J + 2μs I1′ r , J 2′ r K

(7.98)

In analogy to the linear theory (5.115) and (5.123), the nonlinear state equations are then: Σ = Chom I1′ r , J 2′ r : E − PB I1′ r , J 2′ r (7.99) P φ − φ0 = B I1′ r , J 2′ r : E + ′ r ′ r N I1 , J 2

Furthermore, I1′ r and J 2′ r can be derived as functions of the macroscopic stress Σ and the pore pressure P from (5.90) and (5.98), respectively. Relations (7.99) then take the form: Σ = Chom (Σ, P) : E − PB(Σ, P) φ − φ0 = B(Σ, P) : E +

P N(Σ, P)

(7.100)

where the expressions of B(Σ, P) as a function of Chom (Σ, P) and of N(Σ, P) as a function of B(Σ, P) are the same as in the linear case: B(Σ, P) = 1 : (I − Cs (Σ, P)

−1

: Chom (Σ, P))

1 −1 = 1 : Cs (Σ, P) : (−ϕ0 1 + B(Σ, P)) N(Σ, P)

(7.101)

It is important to recall from Section 5.4.3 that the reference strains I1′ r and J 2′ r introduced in (7.97) depend on the macroscopic stress and the pore pressure, respectively, through Σ + Pϕ0 1 and Σ + P1. Through k s (I1′ r , J 2′ r ) and μs (I1′ r , J 2′ r ), the same dependence exists for the macroscopic stiffness Chom , as well as for B and N. Hence, in the general case, there is no effective stress controlling the macroscopic non linearity. Two particular cases deserve attention:

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r k s and μs do not depend on J ′ . In this case, (7.90) shows that μs is necessarily 2

a constant. According to (5.90), I1′ r is determined from the solution of: (1 − ϕ0 )I1′ r =

3k s

1 ′ tr (Σ + Pϕ0 1) I1 r

(7.102)

It turns out that I1′ r is a function of tr (Σ + Pϕ0 1), which in turn controls the macroscopic non linearity. A linear scheme such as (6.146) or (6.160) then provides k hom and μhom as functions of I1′ r . r k s and μs do not depend on I ′ . In this case, (7.90) shows that k s is necessarily 1 a constant. According to (5.98), J 2′ r is determined from the solution of: 1 ∂ 1 ∂ 1 1 2 ′r 2(1 − ϕ0 )J 2 = − (tr (Σ + P1)) − Σd : Σd (a ) 18 ∂μs k hom 4 ∂μs μhom k hom = K(k s , μs ); μhom = M(k s , μs ) s

μ =μ

s

(J 2′ r )

(b) (c) (7.103)

It turns out that J 2′ r is a function of tr (Σ + P1) and Σd . The effective stress Σ + P1 thus controls the macroscopic non linearity. Note that the determination of J 2′ r requires the linear scheme (7.103b). 7.5.3 An Alternative Approach to Strength Homogenization In order to illustrate the far reaching possibilities of the nonlinear elastic model (7.87), the focus of this section is the development of an alternative method for the derivation of the macroscopic strength criterion for a von Mises solid and a Drucker–Prager solid. For both materials, G hom (P) can be derived from G hom (0) through application of (7.78) and (7.84), respectively (see Section 7.4). We therefore restrict ourselves to the determination of G hom (0). Von Mises Solid

In the framework of an isotropic potential ψ s (I1′ , J 2′ ), the idea consists of choosing the secant shear modulus μs in such a way that the condition (7.1), i.e. f s (σ) = 0, is asymptotically satisﬁed at the microscopic level, for large values of the second invariant J 2′ : f s lim σ =0 (7.104) ′ J 2 →∞

Consider a radial macroscopic strain path E(λ) parallel to the macroscopic strain state E0 : λ → E = λE0 . It is expected that large local deviatoric strains (J 2′ ≫ 1) are induced by large macroscopic strains (λ ≫ 1). Accordingly, the

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230

corresponding macroscopic stress state Σ = σ should lie on the boundary of G hom : lim Σ ∈ ∂G hom

λ→∞

(7.105)

In other words, the boundary ∂G hom is made up of the extremities of the macroscopic stress paths. In contrast, for small values of λ, the macroscopic stress path is expected to lie inside G hom . Let us look for ψ s (I1′ , J 2′ ) in the form: 1 2 ψ s I1′ , J 2′ = k s I1′ + F J 2′ (7.106) 2 where k s is a constant. With (7.89), the microscopic state equation is: (7.107) σ = k s I1′ 1 + 2μs J 2′ εd with 2μs (J 2′ ) = F ′ J 2′ ′ ′ ′ J2 The condition (7.104) with f s (σ) given by (7.1) is satisﬁed if F (J ) ≈ k/ 2 when J 2′ ≫ 1. This leads us to choose F(J 2′ ) ≈ 2k J 2′ . In other words, the secant shear modulus μs (J 2′ ) is a decreasing function of the deviatoric strain that is equivalent to k/(2 J 2′ ) for large values of the deviatoric strain. In particular, for large values of the deviatoric strain, we note that μs /k s ≪ 1. This means that the solid behaves as an incompressible material. Let us now consider the response of a rev made up of such a solid. Given the apparent incompressibility, the homogenized stiffness predicted by the Mori–Tanaka and the self-consistent schemes is: k hom = K (ϕ)μs J 2′ r hom hom hom C = 3k J + 2μ K with (7.108) μhom = M(ϕ)μs J 2′ r where J 2′ r is the reference deviatoric invariant introduced in (7.97). According to the chosen scheme, K (ϕ) and M(ϕ) are given by (7.62) or (7.63). J 2′ r is derived from (4.177) as a function of the macroscopic strain E: 1 K 2 ′r (tr E) + MEd : Ed (7.109) J2 = 2 2(1 − ϕ)

The macroscopic state equation (7.94) then yields: 2 m ′ r J2 tr E = K k (7.110) 1 d ′ r 1 E : Ed = J2 2 d M k where Ed is the deviatoric part of E. Finally, inserting (7.110) into (7.109), we obtain: 1 m 2 1 d 2 (1 − ϕ) = + (7.111) K k M k

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which is identical to the macroscopic strength criterion (7.70). Equations (7.71) and (7.72) are then retrieved respectively with the Mori–Tanaka and the selfconsistent estimates of K (ϕ) and M(ϕ). Although the calculations performed in this section and in Section 7.3 are very similar, the approach is quite different. The solution Σ of (7.32) depends on D only through its orientation. Indeed, the same solution Σ is obtained for the macroscopic strain rate tD for any value of t. This is due to the fact that: (∀t > 0)

∂π s ∂π s (td) = (d) ∂d ∂d

(7.112)

which is readily derived from (7.22). Hence, in contrast to the nonlinear elastic approach, the concept of stress path is not relevant when the equivalent viscous behavior deﬁned by (7.32b) is used (Section 7.3). As a consequence, there is no need to consider large values of the strain rate D in (7.32) for the determination of ∂G hom . Drucker–Prager Solid

It is interesting to observe that the existence of a potential of the form (7.106) is actually unimportant in the strength homogenization approach based on nonlinear elasticity. The crucial point is that the nonlinear behavior σ = Cs (ε) : ε of the solid phase deﬁned by (7.107) meets the condition (7.104). This remark allows us to extend this technique to a wide class of strength criteria. This is illustrated here for the Drucker–Prager solid for which the strength criterion is given by (7.8). We look for a ﬁctitious nonlinear solid characterized by a secant stiffness tensor, Cs (ε), which should asymptotically satisfy f s (σ) = 0 for large deviatoric strains (see (7.104)). Once again, we assume that Cs (ε) is an isotropic function of I1′ and J 2′ : (7.113) Cs (ε) = 3k s I1′ , J 2′ J + 2μs I1′ , J 2′ K This yields:

= k s I1′ , J 2′ I1′ ′ √ J 2 = 2μs I1′ , J 2′ J2

1 tr σ 3

(7.114)

The asymptotic condition f s (σ) = 0 at large deviatoric strain is: ′ ′ h − k s I1′ , J 2′ I1′ s (7.115) μ I1 , J 2 ≈ α 2 J 2′ In contrast to the von Mises solid for which μs (J 2′ ) ≈ k/(2 J 2′ ), the shear modulus μs in (7.115) now depends on both I1′ and J 2′ , and the shear strength (represented by 2μs J 2′ ) increases with the conﬁning pressure (represented by −k s I1′ ). As regards the bulk modulus, the simplest choice is a constant value

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232

for k s . It is worth observing that the corresponding model does not meet the condition (7.90) which ensures the existence of a potential, but which is not necessary for implementing the technique. The nonlinear problem to be solved for the effective strain now comprises the two unknowns I1′ r and J 2′ r . Using (5.90) and (5.98) for P = 0, we obtain: m ks ∂ 1 1 ∂ 2 ′r d2 4(1 − ϕ)J 2 = − s hom m − ∂μ k ∂μs μhom k hom = K (ϕ)μs ; μhom = M(ϕ)μs h − ks I ′r μs I1′ r , J 2′ r = α ′ 1 2 J 2r (1 − ϕ)I1′ r =

(a ) (b)

(7.116)

(c) (d)

As for the von Mises case, we have used in (7.116c) the fact that μs /k s tends towards zero for large deviatoric strains. The combination of (7.116b) and (7.116c) ﬁrst yields: ′ r ′r 2 1 1 ′r s (1 − ϕ) 2 J 2 μ I1 , J 2 = m2 + d 2 (7.117) K M We then insert (7.116a) and (7.116d) into the l.h.s. of (7.117): 1−ϕ 1−ϕ 2 2 2 2 2 α h (1 − ϕ) = m − α + 2α 2 h(1 − ϕ)m + d2 K M

(7.118)

which also takes the form: α 2 h 2 (1 − ϕ)2 = (m − mc )2 1 − ϕ − α2 K

1 α2 − K 1−ϕ

+ d2

1 M

(7.119)

with: mc = −

α2h 1/K − α 2 /(1 − ϕ)

(7.120)

The domain G hom h (0) of admissible macroscopic stress states is an ellipse centered at the point (mc , 0) in the (m , d ) plane. In contrast to the von Mises solid, G hom h (0) is not symmetric w.r.t. the origin. This is due to the fact that the Drucker–Prager solid phase is sensitive to the conﬁning pressure. Inserting (7.62) into (7.119) yields the Mori–Tanaka estimate of G hom (P). It predicts that the radii of the ellipse are proportional to 1 − ϕ. The macroscopic strength therefore only vanishes for ϕ → 1. In turn, use of (7.63) in (7.119) yields the self-consistent estimate of G hom (P). The radii of the ellipse are now √ proportional to 1 − 2ϕ. The percolation threshold ϕ = 1/2 is thus retrieved. We note that these conclusions are identical to those obtained for the von Mises solid (see (7.71) and (7.72)).

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σd

α

Σd

1

(DP)

(VM)

h

2k

σm

Σm

Figure 7.6 Drucker–Prager (DP) and von Mises (VM) solids with αh = k: the macroscopic shear strength for m = 0 is identical

It is interesting to note that (7.70) is retrieved as the limit of (7.119) for h → ∞ and α → 0 with αh = k. Indeed, in this case, the Drucker–Prager criterion (7.8) tends towards the von Mises expression (7.1). Note also that the shear strength do predicted for m = 0 by (7.70) is identical to that predicted by (7.119) provided that αh = k (see Figure 7.6):10 (7.121) do = k M(1 − ϕ)

Let us ﬁnally examine the strength of the porous medium when the pore space is saturated by a ﬂuid at pressure P. According to (7.84), G hom h (P) can be obtained from G hom (0) by a combination of two transformations. First, the h homothety centered at the origin with ratio 1 + P/ h > 0 is applied to G hom h (0). The domain thus obtained is then translated along the m axis (Figure 7.7). The analytical expression of the strength criterion, i.e. the equation of ∂G hom (P) in the (m , d ) plane, is readily obtained by replacing m and d in (7.119) with ef f ef f m and d deﬁned according to (7.86): me f f =

m + P ; 1 + P/ h

ef f

d

=

d 1 + P/ h

(7.122)

As in Section 7.3.3, let us now compare the estimate (7.119) with the hollow sphere result of a Drucker–Prager solid. Since the strength is controlled by an effective stress, we can restrict the analysis to the case P = 0. For the domain deﬁned by (7.62)–(7.119) to be bounded (ﬁnite strength in compression), it is √ necessary that α < 3ϕ/2. The following Mori–Tanaka-based estimates limit the hydrostatic strength under compression or traction: + mt = αh

1−ϕ ; α + 3ϕ/4

− mt = −αh

1−ϕ −α + 3ϕ/4

(7.123)

10 The strength criterion (7.119) is identical to the result obtained by [5] by means of the nonlinear homogenization technique of Section 7.2.3, in which the solid phase is represented as a ﬁctitious nonlinear viscous constitutive behavior.

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234

Σd

P

Σm

G hom (P )

G hom (0)

Figure 7.7 Ghom(0) and Ghom(P ) in the case of a Drucker–Prager solid

In turn, recalling (7.10)–(7.13), the hollow sphere estimates are: + hs = h(1 − ϕ γ );

− hs = h(1 − ϕ β )

(7.124)

+ Figure 7.8 shows an excellent agreement in the tensile strength between mt + and hs for any value of porosity ϕ and friction coefﬁcient α. The agreement − − is also very good for the compressive strength mt and hs for values of ϕ 2 and α far from the critical value ϕcr = 4α /3, which marks the transition from

1

Σ+ h

0 ϕ

1

Figure 7.8 Drucker–Prager material: hydrostatic strength in traction predicted by the hollow sphere model and the Mori–Tanaka scheme (α = 0.3)

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10

--

|Σ |

(mt)

h

(hs) 0 ϕ cr

ϕ

1

Figure 7.9 Drucker–Prager material: hydrostatic strength in compression predicted by the hollow sphere model and the Mori–Tanaka scheme (α = 0.3)

ﬁnite to inﬁnite strength in compression according to the Mori–Tanaka scheme (Figure 7.9). However, the proposed nonlinear homogenization scheme fails to correctly predict the strength in compression for ϕ near to or less than ϕcr . This shortcoming of the method is due to the fact that a single reference strain for the whole solid phase is not accurate enough for low porosities. The technique could be improved by dividing the solid into several domains, with a reference strain for each of them [7].

8 Non-Saturated Microporomechanics This chapter examines how the microporomechanics theory can accommodate surface tension effects which become important in unsaturated conditions. We start with the representation of internal forces at the solid–ﬂuid interface as membrane stresses, which makes it possible to extend the tools of continuum micromechanics, namely the principle of virtual work and the Hill lemma, to account for surface tension. Based on this extended theory, we derive the macroscopic state equations of non-saturated poroelasticity. By way of illustration, we discuss the effective stress concept in nonsaturated poroelasticity, with and without surface tension. Finally, we show how non-saturated conditions affect the strength criteria of porous media.

8.1 The Effect of Surface Tension at the Solid–Fluid Interface We have already seen in Section 4.2.4 that the presence of surface tension at the solid–ﬂuid interface induces a stress discontinuity. Even in saturated conditions, surface tensions exist at the solid–ﬂuid interface, although they are (and can be) neglected in many practical applications. In return, surface tensions dominate the response of non-saturated porous materials. By non-saturated we mean that the pore domain is not ﬁlled by one single ﬂuid phase, but by several. The surface tensions that are generated along ﬂuid–ﬂuid and solid– ﬂuid interfaces induce deformations which affect the macroscopic behavior. It is readily understood (see e.g. Section 4.2.4) that such surface tensions will affect the upscaling rules developed in Chapter 5 and Chapter 7 for the saturated case. This section presents the necessary steps required to incorporate the surface tensions into the microporoelasticity theory.

Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

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8.1.1 Representation of Internal Forces at the Solid–Fluid Interface The mechanical modeling of surface tension usually consists of introducing a membrane stress tensor of the form γ s f 1T at the interface I s f between the solid phase and the ﬂuid phase. Since it is deﬁned on a surface, the membrane stress concept is two-dimensional by nature. γ s f denotes the surface tension of the surface, while 1T is the identity tensor of the plane T tangent to the interface at the considered (microscopic) point: 1T = 1 − n ⊗ n

(8.1)

where n denotes the unit normal to the interface. We need to be aware of the physical meaning of such a membrane stress. Let I1 and I2 be two subdomains of I s f separated by the curve L. t denotes the unit tangent vector to L and τ = t × n is the normal to L located on the plane T. Assuming that τ is oriented towards I2 , the elementary force applied by I2 on I1 on the elementary segment tds is γ s f τ ds (Figure 8.1). The concept of interface is a two-dimensional idealization of the transition zone between two phases which can be thought as a thin three-dimensional layer. Accordingly, the elementary segment tds corresponds to a ﬂat rectangle with inﬁnitesimal thickness ε. The elementary force γ s f τ ds appears as the integral of the surface forces acting on this rectangle: +ε/2 sf γ τ = lim σ m · τ dz (8.2) ε→0

−ε/2

z n τ

ε

n

t

τ

ds I2 I1 L I sf

Figure 8.1 The membrane stress concept

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239

L (∏)

t L

ez

n

Figure 8.2 Cylindrical solid–fluid interface

where the superscript m stands for ‘membrane’. As opposed to the 2-D concept of membrane stress, σ m is a 3-D stress ﬁeld. The above equation holds for any orientation of τ in the tangent plane T and yields the following tensorial identity: +ε/2 sf γ 1T = lim σ m dz (8.3) ε→0

−ε/2

Note that there is a mathematical singularity of the membrane stress, which stems from the fact that the integral over a layer of vanishing thickness yields a non-zero limit. The mathematical concept which takes this kind of singularity into account is the Dirac distribution δI s f (see (1.16)), which makes it possible to represent the internal forces located at the interface in a 3-D way:1 σ m = γ s f 1T δI s f

(8.5)

Instead of dealing with two different mathematical representations of internal forces (3-D in the solid and the ﬂuid, and 2-D in the membrane), we are now able to develop a uniﬁed 3-D stress model: σ(z) = σ s (z)χ s (z) − P1χ f (z) + γ s f 1T δI s f

(8.6)

where σ s (z) is the classical Cauchy stress ﬁeld deﬁned on the solid phase, and functions χ α (z) denote the characteristic functions of the α domain (α = s or f ). The presence of surface tension at the solid–ﬂuid interface is responsible for a discontinuity the stress vector across I s f . This can be understood easily in the case of a cylindrical interface for which the curvature is equal to zero in the cylinder axis, e z (see Figure 8.2). Let ρ > 0 denote the radius of curvature of the curve L obtained in the cross section of the interface with a plane 1 Indeed,

following (8.3), it is readily seen that: σ m ψ(z) d V = γ s f 1T ψ(z) dS = γ s f 1T (z)δI s f (z)ψ(z) d V Is f

(8.4)

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240

−n − γ sf t

ds

dℓ γ sf (t + dt) +n

ρ

Figure 8.3 Discontinuity of the stress vector at the solid–fluid interface

perpendicular to e z . The unit normal and tangent vectors n (oriented towards the center of curvature) and t to L are related by: 1 dt n= ρ ds

(8.7)

Let us now consider an elementary ﬂat rectangular body (dℓ ≪ ds). The upper and lower faces of this body lie respectively above and below L. Denoting by σ ± the stress ﬁelds on both sides of the interface and neglecting the vanishing contribution of the stress vector acting on the right and left side of the rectangle (Figure 8.3), the equilibrium of the body is: (σ + · n + σ − · (−n))ds + γ s f dt = 0

(8.8)

that is: [σ] · n +

γ sf n=0 ρ

(8.9)

where [σ] = σ + − σ − (according to the orientation of n). If the center of curvature of the solid–ﬂuid interface is located in the ﬂuid, then σ + = −P1 (see Figure 8.4). In this case, the stress vector σ − · n that acts on the solid above the interface is: γ sf γ sf + − σ ·n= σ ·n+ (8.10) n n=− P− ρ ρ The solid is now not subjected to the ﬂuid pressure P, but to the corrected pressure P − γ s f /ρ which is smaller than P. Conversely, if the center of curvature is located in the solid, then σ − = −P1 (Figure 8.4). In this case, the stress vector σ + · (−n) that acts on the solid is: γ sf γ sf + − σ · (−n) = σ · (−n) + n (8.11) n= P+ ρ ρ

Instead of the ﬂuid pressure P, it is now the corrected pressure P + γ s f /ρ to which the solid is subjected, and which is greater than P. The 3-D

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241

σ – = –P 1 S F

F S

n

n

σ + = –P 1

Figure 8.4 Influence of the location of the center of curvature

generalization of (8.9) requires the introduction of the tensor of curvature of the interface, deﬁned by2 b = −grad n: [σ] · n + γ s f (1T : b)n = 0

(8.12)

We verify that (8.12) is consistent with (8.9), since (1T : b)n reduces to n/ρ for a cylindrical interface. 8.1.2 Principle of Virtual Work and the Hill Lemma with Surface Tension Effects Following the representation of internal forces at the solid–ﬂuid interface, we need to revisit the key tools of micromechanics, namely the principle of virtual work and the Hill lemma. To this end, let us consider a virtual displacement ﬁeld ξ ′ (resp. strain ﬁeld ε′ ) deﬁned on the solid and extended into the pore space. Using (8.6), the work of the internal forces along the strain ﬁeld ε′ comprises not only the contribution of the solid and the ﬂuid phases, but also the contribution of the solid–ﬂuid interface: ′ s ′ ′ σ : ε dV = σ : ε dV + −P1 : ε dV + γ s f 1T : ε′ dS (8.13)

s

p

Is f

where (8.5) and the deﬁnition (1.16) of the Dirac distribution have been used. The strain work in the solid can be rearranged as usual:3 ′ s s ′ (8.14) ξ ′ · σ s · n dS σ : ε dV = ξ · σ · n dS +

s

∂

Is f

where the r.h.s. represents the work of the external forces applied to the solid on the external boundary ∂ and on the solid–ﬂuid interface. Equation (8.14) is the standard formulation of the principle of virtual work for the solid phase. In turn, the virtual work of the ﬂuid pressure can be expressed in the form of 2 This deﬁnition is based on the orientation of the surface, which is given by the orientation of the unit normal n. 3 Recall that the boundary of the rev is assumed to be located in the solid domain. This implies that ∂ s = ∂ ∪ I s f .

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242

a surface integral:

′

p

−P1 : ε dV =

Is f

Pn · ξ ′ dS

(8.15)

In (8.14) as well as in (8.15), n denotes the unit normal to the solid–ﬂuid interface oriented towards the ﬂuid. Combining these equations yields: ′ s s ′ ′ ξ ′ · [σ] · n dS σ : ε dV + −P1 : ε dV = ξ · σ · n dS −

s

p

Is f

∂

(8.16)

where [σ] = −P1 − σ s . Let us now examine the work of the membrane stress. It will prove useful to split the displacement into a component ξ ′t in the tangent plane T and a normal component ξn′ : γ s f 1T : grad (ξn′ n) dS γ s f 1T : ε′ dS = γ s f 1T : grad ξ ′t dS + Is f

Is f

Is f

(8.17)

Recalling (8.12), the contribution of the normal displacement also takes the form: sf ′ ′ sf γ 1T : grad (ξn n) dS = − ξn γ 1T : b dS = ξ ′ · [σ] · n dS Is f

Is f

Is f

(8.18)

where we make use of the fact that the discontinuity [σ] · n of the stress vector has no component in the tangent plane T. Finally, if we consider the solid– ﬂuid interface as a closed surface (remember that ∂ f = I s f ), it can be shown that the contribution of the tangent displacement in (8.17) disappears4 . The strain work of the membrane stress then reduces to: (8.19) γ s f 1T : ε′ dS = ξ ′ · [σ] · n dS Is f

Is f

Equation (8.19) is recognized as the principle of virtual work for the solid– ﬂuid interface. Indeed, on the r.h.s., [σ] · n represents the surface density of external forces applied to the membrane. We now combine (8.16) and (8.19) into (8.13): s σ (z)χ s (z) − P1χ f (z) + γ s f 1T δI s f : ε′ dV = (8.20) ξ ′ · σ s · n dS

∂

This equation states that the total strain work in the rev is equal to the work of the surface forces σ s · n applied to its boundary. To derive this result, note 4 To get an idea of the proof, just replace the closed surface by a closed line, with ξ ′ tangent to the latter. t In this case, the result becomes obvious.

The Effect of Surface Tension at the Solid–Fluid Interface

243

that it was essential to incorporate the contribution of the membrane stresses to the work of internal forces. We further assume that the displacement ﬁeld ξ ′ is kinematically admissible (k.a.) with the macroscopic strain E (i.e. meets the boundary condition (4.40)). Equation (8.20) shows that the l.h.s. of this equation does not depend on the particular choice of ξ ′ , since it is controlled by the values of ξ ′ on the boundary ∂ . Thus, any choice is possible, and we choose the uniform strain boundary condition ξ ′ = E · z. This yields: ′

(∀ξ k.a. E)

s σ (z)χ s (z) − P1χ f (z) + γ s f 1T δI s f : ε′ dV = Σ : E (8.21)

where the quantity: Σ = (1 − ϕ0

s )σ s

1 − Pϕ0 1 + | |

Is f

γ s f 1T dS

(8.22)

appears as the macroscopic stress in the rev . Relation (8.21) is the extension of the Hill lemma (see Section 4.2.6) that takes surface tension effect into account, whereas (8.22) is the corresponding generalization of (1.39). As a particular virtual displacement ﬁeld, the displacement ﬁeld solution to the loading deﬁned by E with P = γ s f = 0 can be considered. The corresponding strain ﬁeld is ε′ = A : E (see (4.108)), so that (8.21) becomes: s σ s (z)χ s (z) − P1χ f (z) + γ s f 1T δI s f : A = (1 − φ0)σ s − Pϕ0 1 1 γ s f 1T dS + | | I s f

(8.23)

This result will prove to be useful in the next section.

8.1.3 State Equation with Surface Tension Effects We look for the macroscopic strain E induced by the macroscopic stress Σ and the pore pressure P in the presence of surface tension effects. For this purpose, it is necessary to deﬁne the initial state of the rev, which we will characterize by the macroscopic conditions Σ = P = 0. The microscopic stress ﬁeld in the solid which exists in this initial state, denoted by σ s0 , is not deﬁned in a unique way. Nevertheless, it cannot be the natural state (σ s0 = 0), since it has to balance the surface forces induced by the surface tension in the interface I s f (see (8.12)): (I s f ) :

σ s0 · n = γ s f (1T : b)n

(8.24)

244

Non-Saturated Microporomechanics

where n is the unit normal oriented towards the ﬂuid. In addition, it is subjected to the average condition (see (8.22) with P = 0): 1 ss (1 − ϕ0 )σ 0 + γ s f 1T dS = 0 (8.25) | | I s f For these reasons, the microscopic stress σ s0 χ s + γ s f 1T δI s f can be regarded as a self-balanced stress ﬁeld. Note also that (8.23) can be applied to the initial state (σ s = σ s0 , P = 0) and is: (σ s0 χ s + γ s f 1T δI s f ) : A = 0

(8.26)

This initial state is taken as the reference conﬁguration for the deﬁnition of microscopic as well as macroscopic strains. The microscopic state equation in the solid phase is therefore: σ s = σ s0 + Cs : ε

(8.27)

The methodology of Section 5.5.1 can be easily adapted in order to take into account the initial stress in the solid, as well as the membrane stress at the interface. In particular, (5.101) is now replaced by: ⎧ ⎧ f 0 ( ( f ) ) ⎪ ⎪ ⎨ ⎨ −P1 (8.28) C(z) = 0 (I s f ) σ p (z) = γ s f 1T δI s f (I s f ) ⎪ ⎪ ⎩ Cs ( s ) ⎩ σs s ( ) 0 or:

C(z) = Cs χ s (z) σ p (z) = −P1χ f (z) + σ s0 (z)χ s (z) + γ s f 1T δI s f

(8.29)

In order to derive the macroscopic state equation from (5.114), we need to introduce the above expression for σ p into the macroscopic prestress σ p : A, according to Levin’s theorem. Note that the strain concentration tensor A to be used in this matter is strictly identical to that of Section 5.5.1 and is still deﬁned by (5.105). Recalling (5.112) as well as (8.26) yields: p

σ p : A = (σ s0 χ s + γ s f 1T δI s f ) : A − Pϕ0 1 : A = −PB

(8.30)

The macroscopic state equation is found to be formally identical to (5.115): Σ = Chom : E − PB

(8.31)

However, it must be kept in mind that the reference conﬁguration used in (8.31) is not identical to the reference conﬁguration employed in the absence of surface tension.

The Effect of Surface Tension at the Solid–Fluid Interface

245

8.1.4 Macroscopic Strain Related to Surface Tension Effects In line with the developed arguments, it is interesting to evaluate the macroscopic strain Eγ associated with the surface tension. More precisely, Eγ can be deﬁned as the macroscopic strain induced by the loading Σ = P = 0 and γ s f = 0. The reference conﬁguration of the rev w.r.t. which Eγ is deﬁned is the natural state (no initial microscopic stress in the solid). According to (5.114), Eγ is: Eγ = −Shom : σ p : A p

(8.32)

sf

Inserting σ = γ 1T δI s f into (8.32) yields: 1 sf γ hom E = −S γ 1T : A dS :

Is f

(8.33)

As expected from the linear behavior of the solid, Eγ is proportional to the surface tension. A similar reasoning can be used for evaluating the macroscopic strain induced in a porous medium when the ﬂuid that initially ﬁlls the pore space (subscript 1) is replaced by another one (subscript 2). For instance, this is the case after complete5 drying. The surface tension at the interface between ﬂuid i and the solid is desf noted by γi . Starting from the prestressed reference conﬁguration of Section sf 8.1.3 associated with γ s f = γ1 , the macroscopic strain induced by the ﬂuid exchange is: 1 sf sf 1T : A dS (8.34) Eγ = −(γ2 − γ1 ) Shom :

Is f

A more accurate derivation of Eγ would need to specify the morphology of the solid–ﬂuid interface. For purposes of illustration, we assume that the pore space can be modeled as a set of spherical pores. In the framework of the dilute scheme (see Sections 4.3.1 and 6.3), the local value of A in (8.34) can be p approximated by an estimate of A : 1 p γ s f hom E = −δγ S : (8.35) 1T dS : A

Is f sf

sf

where δγ s f = γ2 − γ1 . Furthermore, integration of the surface tension over a sphere clearly yields an isotropic tensor. Indeed, denoting by S(r ) a sphere of radius r , and by V(r ) = 4πr 3 /3 the sphere volume, we observe that: 1T dS = 2 dS = 8πr 2 (8.36) tr S(r )

S(r )

5 When the liquid phase is not completely replaced by gas, the strain induced by drying comprises an additional term. We will examine this situation in Section 8.2.2.

Non-Saturated Microporomechanics

246

It is then readily seen that: 1 V(r )

S(r )

1T dS =

2 1 r

(8.37)

Let us now characterize the statistical distribution of pore radii by the function α(r ) such that | |α(r ) dr is the volume of pores with radius between r and r + dr . According to this deﬁnition, we note that: ∞ α(r ) dr = ϕ0 (8.38) 0

Using (8.37), the integral of the surface tension over the whole solid–ﬂuid interface takes the form: ∞ 2α(r ) 1 dr 1 (8.39) 1T dS =

Is f r 0 We then insert (8.39) into (8.35): ∞ δγ s f 2α(r ) p γ E =− dr Shom : (ϕ0 1 : A ) ϕ0 r 0 Recalling (5.85) and (5.112), Eγ reduces to: ∞ δγ s f 2α(r ) γ E =− dr Shom − Ss : 1 ϕ0 r 0

(8.40)

(8.41)

For example, if the distribution of pore radii is uniform (r = r0 ), the statistical function α(r ) is ϕ0 δr0 . Assuming that the microscopic and macroscopic elasticity tensors are isotropic, Eγ takes the form: 2δγ s f b 1 1 2δγ s f γ 1 = − − 1 (8.42) E =− 3r0 k hom k s 3r0 k hom It is interesting to remark that the sign of the strain induced by the ﬂuid exchange is positive if δγ s f < 0 and negative if δγ s f > 0: the ﬁrst case corresponds to swelling, the latter to shrinkage. Furthermore, for a given value of the porosity, the strain magnitude is proportional to the inverse of the pore radius since b as well as k hom does not depend on the pore radius. As expected, the swelling or shrinkage phenomenon associated with ﬂuid exchange is enhanced for small pores.

8.2 Microporoelasticity in Unsaturated Conditions 8.2.1 The Bishop Effective Stress in Unsaturated Porous Media The poroelastic response of an unsaturated porous medium can be ﬁrst addressed as an application of Levin’s theorem. We consider a simpliﬁed

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247

mechanical model in which the pore space p is ﬁlled by two immiscible ﬂuids, a liquid and a gas. The pore space is divided into two parts: the dop p main ℓ ﬁlled by liquid and the domain g ﬁlled by gas. The liquid (resp. p gas) pressure is denoted by pℓ (resp. pg ) and is assumed to be uniform in ℓ p p p (resp. g ). The volume fractions of ℓ and g in the rev are denoted by ϕℓ and ϕg . The saturation ratio Sr is deﬁned as the volume fraction of the liquid in the pore space: p

| | ϕℓ = ℓ ; | |

p

| g | ϕg = ; | |

Sr =

ϕℓ ; ϕ

ϕ = ϕℓ + ϕ g

(8.43)

As usual, the assumption of small displacements implies that | | ≈ | 0 | and that ϕ ≈ ϕ0 . By contrast, it is interesting to note that large variations of the gas and liquid volume fractions may occur when the pressure difference pc = pg − pℓ , called macroscopic capillary pressure, is modiﬁed. This is the case during drying or wetting experiments. ϕℓ and ϕg therefore refer to the current conﬁguration of the ﬂuid phases. For purposes of clarity, we will ﬁrst set the surface tension phenomenon and the corresponding membrane stresses derived in Section 4.2.4 aside in this model. The general case will be addressed in Sections 8.2.2 and 8.3. This simpliﬁed model thus only takes into account the pressure difference in the two ﬂuid phases, i.e. the macroscopic capillary pressure pc . The loading parameters of the micromechanical problem are E, pℓ and pg , which deﬁne the boundary value problem: div σ = 0 ( ) p σ = C(z) : ε + σ (z) ( ) ξ =E·z (∂ )

(8.44)

together with the appropriate deﬁnition of the spatial distribution of the heterogeneous elasticity tensor C(z) and of the prestress σ p (z) in the rev: ⎧ ⎧ p p 0 ( ) ⎪ ⎪ ℓ ⎨ ⎨ − pℓ 1 ( ℓ ) p p C(z) = 0 ( g ) σ p (z) = − pg 1 ( g ) (8.45) ⎪ ⎪ ⎩ Cs ( s ) ⎩0 s ( )

While Σ′ is still deﬁned by (5.106), the calculation of Σ′′ based on (5.110) now takes into account the values of σ p (z) in the two subdomains: ℓ

Σ′′ = σ p : A = −ϕℓ pℓ 1 : A − ϕg pg 1 : A

g

(8.46)

The total macroscopic stress generated by the loading parameters E, pℓ and pg is the sum of Σ′ and Σ′′ deﬁned by (8.46), which yields: Σ = Chom : E − pℓ Bℓ − pg Bg

(8.47)

Chom is the homogenized elasticity tensor of the empty porous medium. The Biot coefﬁcients Bℓ and Bg respectively associated with the liquid and gas

Non-Saturated Microporomechanics

248

pressures are: ℓ

B ℓ = ϕℓ 1 : A ;

Bg = ϕg 1 : A

g

(8.48)

The unsaturated case a priori requires two (tensorial) Biot coefﬁcients, for the liquid and for the gas, respectively. However, recalling (5.116), it is readily seen that these coefﬁcients are not independent in so far as their sum is equal to the Biot coefﬁcient in the fully saturated case: ℓ g p Bℓ + B g = 1 : ϕ ℓ A + ϕ g A = ϕ0 1 : A = B (8.49)

Using (8.49), it is interesting to note that (8.47) can be rewritten so as to introduce the capillary pressure: Σ + pg B − pc Bℓ = Chom : E

(8.50)

In some cases, it is reasonable to assume that there is no morphological difp p ference between ℓ and g , i.e. between the liquid-ﬁlled and the gas-ﬁlled ℓ g domains. In this case, the averages A and A are expected to be equal, and p therefore equal to A . Using (8.48) and (8.49), this implies that: Bℓ = Sr B;

Bg = (1 − Sr )B

(8.51)

In this case, (8.47) becomes: Σ + B( pg − Sr pc ) = Σ + B p = Chom : E

(8.52)

in which p = (1 − Sr ) pg + Sr pℓ . The l.h.s. of (8.52) is known as the Bishop effective stress. Equation (8.52) reduces the non-saturated case to an equivalent saturated one, in which the pore pressure P is equal to the average p. The derivation provides a micromechanical basis for the interpretation of this concept. However, rather than justifying the concept, the micromechanical model actually helps us to identify its domain of validity. In particular, the Bishop effective stress can become a crude approximation when surface tension effects are not negligible. We will address this point quantitatively in Section 8.3. 8.2.2 Surface Tension Effects in Unsaturated Porous Media The mechanical equilibrium between a gas and a liquid phase at different pressures is not possible without the development of surface tensions at the liquid–gas interface (unless the interface is a plane, in which case the gas and the liquid pressure must be equal). These surface tensions are disregarded in the Bishop model. Moreover, surface tensions also exist at the gas–solid and liquid–solid interfaces and are responsible for a surface loading at the boundary of the solid. For these reasons, a more comprehensive mechanical model of unsaturated porous media is due. Such a model needs to take into account membrane stresses γ αβ at the interfaces I αβ (α, β ∈ {s, ℓ, g}), in addition to p p pressures pℓ and pg in ℓ and g .

Microporoelasticity in Unsaturated Conditions

249

We assume that the initial state is fully liquid-saturated and deﬁned by the conditions Σ = pℓ = 0.6 Following the developments of Section 8.1, the initial microscopic stress ﬁeld σ s0 must comply with the loading induced by the surface tension γ sℓ acting on the whole boundary of the liquid. The corresponding conﬁguration of the rev is chosen as the reference conﬁguration. It is worth recalling that pℓ and pg are related by the condition of mechanical equilibrium of the liquid–gas interface as given by (8.12). With σ α = − pα 1 (α = ℓ, g), we retrieve Laplace’s law (with n oriented towards the gas): pc = pg − pℓ = γ ℓg tr b

(8.53)

In order to adapt the Levin-based approach of Section 8.2.1, we introduce a prestress σ p of the form: σ p = − pℓ 1χ ℓ (z) − pg 1χ g (z) + σ s0 χ s (z) + γ αβ 1T δI αβ

(8.54)

which generalizes (8.6). Relation (8.54) can also be written in the form: σ p = − pℓ 1χ ℓ (z) − pg 1χ g (z) + σ s0 χ s (z) + γ sℓ 1T δI s f + γ˜ αβ 1T δI αβ

(8.55)

where we let I s f = I s f ∪ I sg , and:

γ˜ sg = γ sg − γ sℓ γ˜ sℓ = 0 γ˜ ℓg = γ ℓg

(8.56)

The macroscopic prestress Σ′′ of (8.46) is now replaced by: ℓ

Σ′′ = σ p : A = −ϕℓ pℓ 1 : A − ϕg pg 1 : A

g

+ (σ s0 χ s (z) + γ sℓ 1T δI s f ) : A + γ˜ αβ 1T δI αβ : A

(8.57)

From the properties of σ s0 (see (8.26)) and with the same notation as in Section 8.2.1, Σ′′ reduces to: σ p : A = − pg B + pc Bℓ + γ˜ αβ 1T δI αβ : A

(8.58)

Inserting (8.58) into (5.114) yields the macroscopic state equation: Σ + pg B − pc Bℓ − γ˜ αβ 1T δI αβ : A = Chom : E

(8.59)

The comparison with (8.50) reveals that the surface tension effects can be captured through the last term on the l.h.s. of (8.59). It is interesting to observe that the presence of γ˜ αβ instead of γ αβ amounts to replacing γ sg by γ sg − γ sℓ on I sg and γ sℓ by 0 on I sℓ . 6 Clearly,

the initial state could also be fully gas-saturated.

Non-Saturated Microporomechanics

250

It is worth noting that the surface tensions γ αβ are not independent. They are related by the Young equation:7 γ sg = γ ℓg cos θ + γ sℓ

(8.60)

Σ + pg B − pc Bℓ − γ ℓg (1T δI sg + 1T δI ℓg ) : A = Chom : E

(8.61)

where θ is the contact angle between the solid–liquid interface and the liquid– gas interface. Note that (8.60) can be seen as a mechanical equilibrium of the triple junction line subjected to the three different membrane stresses. In particular, when the liquid perfectly wets the solid (θ = 0), (8.60) shows that γ˜ ℓg = γ ℓg = γ˜ sg . In this case: The liquid–gas surface tension γ ℓg also involved in (8.53) thus becomes the relevant mechanical parameter for dealing with the membrane stresses at the interfaces. 8.3 Training Set: Drying Shrinkage in a Cylindrical Pore Material System The Bishop effective stress model does not account for surface tension effects, as it accounts for the presence of two immiscible ﬂuids in the pore space only through the difference in pressure in the gas phase and in the liquid phase. This translates at the macroscopic scale into the pressure term −B pc Sr in (8.52). It is useful to investigate the domain of application of this model w.r.t. to the comprehensive model (8.59), by comparing the order of magnitude of this pressure term pc Sr to the order of magnitude of the surface tension term. This requires investigating the effect of surface tensions for a speciﬁc pore morphology. For purposes of illustration, we consider that the porous network is composed of long parallel cylindrical pores within a solid matrix. For reasons to be discussed, it is convenient to obtain such a cylinder pore morphology from prolate ellipsoids with aspect ratio α = a /c ≪ 1 (see Figure 8.5). Furthermore, we assume that the hydraulic connection and the thermodynamic equilibrium between pores is achieved. This ensures that the same gas and capillary pressure prevails in all pores. Finally, the liquid is assumed to perfectly wet the solid (contact angle θ = 0), so that the macroscopic state equation reduces to (8.61). 8.3.1 The Capillary Pressure Curve It is common practice in geomechanics to quantify capillary effects through the so-called capillary pressure curve pc = pc (Sr ), which is thought to be a 7 See

[22].

Training Set: Drying Shrinkage in a Cylindrical Pore Material System z

M

′

ω+ ℓg

m

r∗

M

η

ρ

C

251

z∗

M

C

Z O

r

Figure 8.5 Prolate pore in unsaturated conditions

macroscopic material property. It is the aim of this section to derive this curve for the particular geometry of a cylindrical pore deﬁned in a cylindrial coordinate system by the pore symmetry axis Oz (see Figure 8.5). In the unsaturated regime, it is assumed that the liquid is trapped in two distinct regions which are symmetric w.r.t. the plane z = 0. These regions are separated from the gas domain by two spherical menisci. The cylindrical coordinates r∗ = mM and + z∗ = Om deﬁne the location of the meniscus ωℓg in the domain z > 0 and are related by: z∗2 r∗2 + =1 a 2 c2

(8.62)

where a and c are ellipse half-lengths, from which the cylinder derives for α = a /c ≪ 1. The normal n to the ellipsoid is therefore: r∗ /a 2 e + z∗ /c 2 e z n= r r∗2 /a 4 + z∗2 /c 4

(8.63)

+ Let C(O, Z) denote the center of the spherical meniscus ωℓg . Since C M is parallel to n, the center coordinate of the meniscus is readily obtained:

CM × n = 0 ⇒ Z = (1 − α 2 )z∗

(8.64)

Non-Saturated Microporomechanics

252

as well as the meniscus radius ρ: ρ = |C M| = a 1 + (α 2 − 1)u2∗

with u∗ =

z∗ c

(8.65)

and the angle η between C M and the symmetry axis: cos η =

αu∗ Cm = ρ 1 + (α 2 − 1)u2∗

(8.66)

Recalling that the meniscus is spherical, the capillary pressure pc is immediately derived from ρ according to Laplace’s law 2γ ℓg /ρ: pc =

2γ ℓg /a

(8.67)

1 + (α 2 − 1)u2∗

We note that pc increases as the meniscus moves away from the midplane z = 0. The critical value of the capillary pressure at which the unsaturated regime is initiated is obtained for u∗ = 0 in (8.67): pccr = 2γ ℓg /a . This critical value corresponds to the formation of a gas bubble of radius a centered at the origin O. In order to determine the whole capillary pressure curve pc = pc (Sr ), we need to express the saturation degree Sr as a function of u∗ . For this purpose, + the volume Vℓ+ of the liquid delimited by the meniscus ωℓg can be determined by geometrical construction, as displayed in Figure 8.6. V1 denotes the volume of the pore in the domain z > z∗ : c a √1−z2 /c 2 r dr = πa 2 c(2/3 − u* + 1/3u3∗ ) (8.68) dz V1 = 2π z∗

0

V2 denotes the part of the sphere centered at point C, with radius ρ, deﬁned by the condition δ < η: 2π 3 V2 = r 2 dr sin δ dδdϕ = ρ (1 − cos η) (8.69) 0< r < δ 3 0< δ< η

z∗

δ η

Vℓ

=

V1

--

V2

+

Figure 8.6 Decomposition of the liquid volume

V3

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253

Finally, V3 is the cone with apex C and height Cm: π 1 V3 = πr∗2 Cm = α 2 a 2 c u∗ 1 − u2∗ 3 3

(8.70)

This decomposition yields: Sr =

2Vℓ+ 3(V1 − V2 + V3 ) = 2 4πa c/3 2πa 2 c

(8.71)

From the fact that the aspect ratio of cylindrical pores α ≪ 1, Sr can be approximated by an expansion to ﬁrst order in α or even by its limit for α → 0 (see Figure 8.7): 1 Sr ≈ (1 − u∗ )2 (2 + u∗ ) − α 2

3 2 1 − u∗

(8.72)

The same remark holds true for pc as well: pc ≈

pccr

(8.73)

1 − u2∗

1 Sr

(1) Zero-order expansion

0.8

(2) First-order expansion and exact solution

0.6 (1)

0.4 (2)

0.2

0

0.2

0.4

0.6

0.8

1

u∗

Figure 8.7 Saturation ratio as a function of u∗ (α = 0.1)

Non-Saturated Microporomechanics

254

6 pc pcr c 5

4

3

2

0

0.02

0.04

0.06

Sr

Figure 8.8 Capillary pressure curve for prolate pores (α = 0.05)

However, the correct asymptotic value of pc when Sr → 0 cannot be derived from (8.73), but from (8.67): lim pc =

Sr →0

pccr α

(8.74)

The capillary pressure curve for a cylindrical pore morphology is obtained from the combination of (8.72) and (8.67), and is displayed in Figure 8.8. 8.3.2 The State Equation We want to evaluate orders of magnitudes of the different terms in the state equation (8.59) that characterizes the unsaturated regime. The ﬁrst term we consider is the capillary pressure term pc Bℓ . Recalling (8.48), we note that: ℓ (8.75) pc Bℓ = pc Sr ϕ0 1 : A

Using (8.67) and (8.72), the term pc Sr for α → 0 is readily found to tend to zero as Sr → 0: pc Sr ≈

γ ℓg (1 − u∗ )3/2 (2 + u∗ ) ; √ a 1 + u∗

lim pc Sr = 0

Sr →0

(8.76)

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255

This limit indicates that the capillary pressure increase is not sufﬁcient to balance the decrease of the saturation ratio. Furthermore, to evaluate the capillary ℓ pressure term (8.75), we need an estimate for the concentration tensor A . In a ﬁrst approach, we will adopt a dilute approximation of the strain concentration tensor: an ellipsoid embedded in a matrix subjected at inﬁnity to a uniform strain boundary condition. Following the developments presented in Section 6.3 and similarly to (4.146), it is found that: ℓ

A≈A ≈A

p

(8.77)

Recalling (5.116), (8.77) implies that Bℓ = Sr B. Inserting this conclusion into (8.61) yields the following formulation of the macroscopic state equation: Σ + B( pg − pc Sr ) − γ ℓg (1T δI sg + 1T δI ℓg ) : A = Chom : E

(8.78)

We thus recover the Bishop effective stress (8.52) corrected by an additional term which takes surface tension effects into account. To evaluate the contribution of this term, we consider the symmetry conditions of the problem, which + allow us to restrict ourselves to the half-system z > 0. We denote by ωsg the + + + . solid–gas interface in the domain 0 < z < z∗ and we introduce ω = ωsg ∪ ωℓg First, from (8.77), the membrane stress term takes the form: ℓg

γ (1T δI sg

2 γ ℓg N + 1T δI ℓg ) : A = | |

ω+

1T dS : A = ϕ0 Γ : A

(8.79)

where N denotes the number of pores in the rev , so that the total porosity is ϕ0 = Vp N /| |, with Vp = 4πa 2 c/3 the individual pore volume, while Γ appears as the macroscopic membrane stress: 2γ ℓg Γ= Vp

ω+

1T dS

The macroscopic state equation (8.78) thus becomes: Σ + ϕ0 ( pg − pc Sr )1 − Γ : A = Chom : E

(8.80)

(8.81)

From (8.81), it is readily understood that the quantitative assessment of surface tension effects w.r.t. the capillary pressure terms amounts to determining the macroscopic membrane stress Γ as a function of Sr or u∗ . We adopt cylindrical coordinates, for which e θ = e z × e r . The symmetry of the problem shows that Γ is a transversely isotropic tensor: Γ = G e z ⊗ e z + λ(e r ⊗ e r + e θ ⊗ e θ ) (8.82)

Non-Saturated Microporomechanics

256

+ + Let |ω+ | denote the total area of the interfaces ωsg and ωℓg . Observing that ℓg + tr Γ = 4γ |ω |/Vp , we note that G and λ are related by:

4γ ℓg |ω+ | = G(1 + 2λ) Vp

(8.83)

We ﬁrst focus on the determination of G. Let t denote the unit vector tangent to ω+ in the plane normal to e θ : r/a 2 e − z/c 2 e r + : t= z ωsg ; r 2 /a 4 + z2 /c 4

+ ωℓg : t = − cos δe r + sin δe z

(8.84)

The identity tensor 1T of the plane tangent to the interface is: 1T = t ⊗ t + e θ ⊗ e θ

(8.85)

Observing that G = e z · Γ · e z , a combination of (8.80) and (8.85) yields: 2γ ℓg G= (8.86) (t · e z )2 dS Vp ω+ + On the ellipsoidal interface ωsg , it may be seen

that the elementary surface is dS = δ(u)dθ dz with u = z/c and δ(u) = a 1 + (α 1 − 1)u2 . In turn, on the + spherical interface ωℓg , we have dS = ρ 2 sin δ dδ dθ , where δ is the angle deﬁned in Figure 8.6. From (8.84), we then obtain:

2

+ ωsg

(t · e z ) dS = 2π ac

+ ωℓg

(t · e z )2 dS = 2πρ 2

u∗ 0

η

1 − u2

1 + (α 2 − 1)u2

du (8.87)

sin3 δ dδ

0

Finally, an expansion of G to ﬁrst order in α yields: 3γ 2γ log u∗ 1 − u2∗ + arcsin u∗ + log 1 − u2∗ α G= 2a a

(8.88)

The ﬁrst and the second terms on the r.h.s. correspond to the contributions of + + ωsg and ωℓg , respectively. Hence, it turns out that surface tension effects are not only associated with the liquid–gas interface (as often believed), but also due to the solid–gas interface. More precisely, expression (8.88) for G shows that ± the contribution of the menisci ωℓg (second term on r.h.s.) is negligible with respect to the contribution of the solid–gas interface, except in the vicinity of u∗ = 0, i.e. near the saturated state.

Training Set: Drying Shrinkage in a Cylindrical Pore Material System

257

+ For similar reasons, the contribution of the meniscus ωℓg to the area |ω+ | is negligible: + |ω+ | ≈ ωsg = ρ δ(u) dz dθ dz dθ ≈ πa c u∗ 1 − u2∗ + arcsin u∗ 0< z< z∗

(8.89)

Comparing (8.88) and (8.89), and recalling (8.83), while keeping in mind that α ≪ 1, it is found that: 4γ ℓg |ω+ | 1 =2 ⇒ λ= Vp G 2

(8.90)

Eventually, the following expression for Γ can be used for prolate ellipsoids meeting the condition α ≪ 1: G(u∗ ) 3γ 2 u∗ 1 − u∗ + arcsin u∗ (8.91) 1 + ez ⊗ ez with G(u∗ ) ≈ Γ ≈ 2 2a

From (8.72), G can alternatively be regarded as a function G(Sr ) of the saturation ratio. It is also useful to note from (8.91) that: 3π cr p (8.92) lim G = Sr →0 8 c In analogy to the Biot tensor B = ϕ0 1 : A, it is convenient to introduce: ϕ0 (8.93) B′ = 1 + ez ⊗ ez : A 2 Then, recalling (8.81)–(8.80), the macroscopic state equation (8.81) can be rewritten in the compact form: Σ + B( pg − pc Sr ) − G(Sr )B′ = Chom : E

(8.94)

It is now possible to compare the contributions of the capillary pressure term and that of the surface tension to the effective stress (l.h.s. in (8.94)). To this end, the variations of pc Sr and G as functions of u∗ are displayed in Figure 8.9. The capillary pressure term dominates for high values of Sr , whereas the surface tension becomes the leading phenomenon for low saturation ratios. For intermediate values of Sr , both terms have the same order of magnitude. In other words, the Bishop effective stress which corresponds to the ﬁrst two terms on the l.h.s. of (8.94) is relevant only in the vicinity of the saturated state, but it is expected to fail to describe linear elastic deformation of a nonsaturated porous medium over the entire saturation range. 8.3.3 Strains Induced by Drying We want to give an estimate of the strains induced by drying in a stress-free macroscopic setting. We start from the initial state described in Section 8.1.3:

Non-Saturated Microporomechanics

258 1.2 (1) pcSr/pccr 1

(2) G/p cr given by (8.91) c

0.8 (2) 0.6

0.4

(1)

0.2

0

0.2

0.4 u∗

0.6

0.8

1

Figure 8.9 Variations of the capillary pressure and the surface tension terms with the location of the liquid–gas meniscus (α ≪ 1)

the pore space is liquid-saturated, and both the macroscopic stress Σ and the liquid pressure pℓ are zero. The liquid pressure is now decreased from 0 to − pccr . Just before gas entry, the macroscopic strain Ecr induced by this loading is obtained from (5.115): Ecr = − pccr Shom : B

(8.95)

where Chom = Cs : (I − ϕ0 A) and B = ϕ0 1 : A. In the framework of the dilute approximation (ﬁrst-order expansion with respect to the volume fraction ϕ0 ), we note that the difference between Chom and Cs is O(ϕ0 ). Since B is O(ϕ0 ) as well, it is possible to replace Shom by Ss in (8.95). Furthermore, if we assume an isotropic solid behavior, we obtain: pccr trB (8.96) 3k s Since tr Ecr < 0, the decrease in liquid pressure induces shrinkage. We now consider the unsaturated regime, during which the gas pressure is kept constant at pg = 0. From (8.94) we obtain:8 (8.97) E = −Shom : pc Sr B + GB′ ≈ −Ss : pc Sr B + GB′ Ecr ≈ − pccr Ss : B;

tr Ecr = −

8 Note that the continuity between (8.95) and (8.97) is ensured by the fact that the transition between saturated and unsaturated regimes is characterized by Sr ≈ 1 and G ≈ 0.

Strength Domain of Non-Saturated Porous Media

259

where B′ is deﬁned by (8.93). For Sr → 0, the contribution of the capillary pressure vanishes, and G is given by (8.92), so that for an isotropic solid: E0 = −

3π cr s p S : B′ ; 8 c

tr E0 = −

3π pccr trB′ 8 3k s

(8.98)

At this stage a comparison of the results (8.96) and (8.98) is due. This amounts to comparing tr B and tr B’. Based on a dilute extimate of A, it appears that tr Ecr < tr E0 for any (positive) value of the Poisson’s ratio.9 The decrease of the saturation ratio induces some macroscopic swelling with respect to the conﬁguration reached at the time of gas entry. However, with respect to the initial conﬁguration, the asymptotic volume strain is strictly negative. In other words, drying induces a shrinkage phenomenon. However, the maximum of this shrinkage is not reached at the end of drying but at the beginning of the unsaturated phase. More precisely, (8.97) yields: pc Sr pccr G ′ tr E = −E with E = (8.99) tr B + tr B pccr pccr 3k s The variations of tr E as a function of Sr are displayed in Figure 8.10. For ϕ0 on the order of 0.1, it is found that the volume strain is on the order of E.

8.4 Strength Domain of Non-Saturated Porous Media The aim of this section is to investigate the effect of non-saturated conditions on the strength domain of porous media. Based on the developments in Chapter 7, we have two methods at our disposal to study such effects, one based on the representation of the solid phase as a ﬁctitious viscous material (Section 7.2.5), the other based on a nonlinear elastic secant representation of the shear strength (Section 7.5.3). Here we will employ the secant method for our investigation. This requires ﬁrst of all the speciﬁcation of the average strain level in microhomogeneous phases. Once this is speciﬁed, it becomes possible to study the strength in partially saturated conditions. 9A

more quantitative analysis requires us to estimate the strain localization tensor for prolate ellipsoids in the limit case of α ≪ 1. Such estimates were developed in Chapter 6. A dilute estimate of A for prolate ellipsoids in an isotropic matrix, and with α ≪ 1, yields the following expressions for tensors B and B′ : 2(1 − ν s ) Bdil = ϕ0 1 − ez ⊗ ez s 1 − 2ν B′dil = ϕ0 where ν s is the Poisson ratio of the solid.

1 − νs 1 1 − 2ν s

Non-Saturated Microporomechanics

260

− tr E 9 ϕ0 E 8.5

8

7.5

7

6.5

0

0.2

0.4

0.6

0.8

Sr

Figure 8.10 Volume strain induced by drying (ν s = 0.3)

8.4.1 Average Strain Level in a Linear Elastic Solid Phase We have seen that the speciﬁc mechanical effects to be considered in unsatup p rated conditions are two different pressures pℓ and pg prevailing in ℓ and g respectively, as well as membrane tension effects at the interfaces (see (8.59)). For the sake of simplicity, we will focus on the effects of the liquid and gas pressures on the macroscopic strength, and will disregard the effects of the membrane tensions. We look for an estimate of the deviatoric effective strain as deﬁned in (7.97). As in Section 5.4, the starting point is the potential energy ∗ of the solid phase. The potential energy is the difference between the elastic energy of the solid and the work of the ﬂuid pressures. Generalizing (5.91), we obtain: 1 s ∗ | 0 | = tr ε dVz − pg ε : C : ε dVz − pℓ tr ε dVz (8.100) p p 2 s

g

ℓ Equation (5.92) is now replaced by: ∂ ∗ ∂ε : Cs : ε dVz | 0 | s = εd : εd dVz + s ∂μ

s

s ∂μ ∂ε ∂ε dVz + dVz + − pg 1 : − pℓ 1 : s p p ∂μ ∂μs

ℓ

g

(8.101)

Strength Domain of Non-Saturated Porous Media

261

Observing that the microscopic stresses in the different domains are Cs : ε in p p

s , − pg 1 in g and − pℓ 1 in ℓ , we observe that (8.101) can be rewritten in the form: ∂ ∗ ∂ε | 0 | s = : σ dVz (8.102) εd : εd dVz + s ∂μ

s

∂μ Similar to (5.93), the Hill lemma then yields: s ∂ ∗ = 2(1 − ϕ0 )εd2 s ∂μ

On this basis we rewrite (8.100) in the form: 1 | 0 | ∗ = σ : ε dVz − pℓ tr ε dVz − pg tr ε dVz p p 2

ℓ

g

(8.103)

(8.104)

and, applying the Hill lemma once more, we obtain: ∗ =

1 Σ : E − pℓ ϕ ℓ 1 : εℓ − pg ϕ g 1 : εg 2

(8.105)

In order to take advantage of (8.103), it appears that we need estimates for the average volume strains in both the liquid and gas domains. These average strains can be derived from the solution of Eshelby’s problem for prestressed inclusions (see Section 6.3.3) within the framework of an appropriate linear homogenization scheme. We assume in the sequel that the morphology of the microstructure is compatible with the use of the Mori–Tanaka scheme, the solid phase playing the role of the matrix and the pores that of the inclusion phase. Accordingly, the average strain in α (α = ℓ, g) is assumed to be equal to the uniform strain that is induced in an ellipsoidal pore saturated by a ﬂuid at pressure pα , embedded in an inﬁnite solid matrix subjected to uniform strain boundary conditions at inﬁnity: ξ → E0 · z when z → ∞ (see (6.122) and Figure 8.11).E0 represents the average strain εs in the solid matrix (Section 6.3.4): εℓ = (I − S)−1 : (E0 + pℓ P : 1) εg = (I − S)−1 : (E0 + pg P : 1) εs = E0

(8.106)

where S is the Eshelby tensor of an ellipsoidal inclusion in the solid matrix (see (6.107)) and P = S : (Cs )−1 . We now take advantage of the average rule E = ε that provides E0 as a function of E, pℓ and pg : −1 : E − ϕ0 p(I − S)−1 : P : 1 E0 = (1 − ϕ0 )I + ϕ0 (I − S)−1

(8.107)

Non-Saturated Microporomechanics

262

E 0 . z at ∞

pα

Figure 8.11 Eshelby’s problem with pressurized cavity

Inserting (8.107) into (8.106), the identity E = ε then yields: εα = pα (I − S)−1 : P : 1 + (I − (1 − ϕ0 )S)−1 : E − ϕ0 p(I − S)−1 : P : 1 (8.108)

with α = ℓ or g. Returning to (8.105), it appears that we have to estimate the total work of the ﬂuid pressures. Using (8.108) and (6.144), we obtain: pα ϕ α 1 : εα = pB : E − ϕ0 p(I − S)−1 : P : 1 + ϕ0 p 2 1 : (I − S)−1 : P : 1

(8.109)

where repeated sub- and superscripts indicate summation, and where p 2 = (1 − Sr ) pg2 + Sr pℓ2 . After some algebra, the above quantity can also be rearranged as follows: p2 + ϕ0 p 2 − p 2 1 : (I − S)−1 : P : 1 pα ϕ 1 : ε = pB : E + N α

α

(8.110)

where the expression (5.87) for N has been used. The last step consists of substituting the state equation (8.47) and (8.110) into (8.105). This yields the following estimate of the potential energy in non-saturated conditions: ϕ0 2 p2 1 p − p 2 1 : (I − S)−1 : P : 1 (8.111) − ∗ = E : Chom : E − pB : E − 2 2N 2

Interestingly, it is readily seen that p 2 − p 2 = Sr (1 − Sr ) pc2 . Considering spherical pores and recalling (6.107), the above expression therefore reduces to: 3ϕ0 p2 1 − Sr (1 − Sr ) pc2 ∗ = E : Chom : E − pB : E − 2 2N 8μs

(8.112)

This expression appears as the extension of (5.59) to non-saturated conditions, the homogeneous pore pressure P being replaced by the average pressure p. On the other hand, from an energy point of view, the last term, in (8.112) shows that unsaturated conditions are not equivalent to saturated conditions

Strength Domain of Non-Saturated Porous Media

263

with P = p. The correcting term, which depends on the capillary pressure, disappears in saturated conditions (Sr = 1 or Sr = 0). It takes into account the mechanical interaction between liquid-ﬁlled and gas-ﬁlled pores. We now return to (8.103) and assume macroscopic isotropy. Using (5.95) and (8.112), we obtain: s 1 ∂k hom p 2 ∂μhom 3ϕ0 2 2(1 − ϕ0 )εd = tr E + s + Ed : Ed + Sr (1 − Sr ) pc2 2 s s s 2 ∂μ k ∂μ 8μ (8.113) which extends (5.96) to unsaturated conditions. Alternatively, the state equas tion (8.71) can be used to derive an expression for εd2 as a function of Σ, p and pc . If membrane tension effects are disregarded, this amounts to replacing P in (5.98) by p and to adding a correcting term controlled by Sr (1 − Sr ) pc2 : s ∂ ∂ 3ϕ0 1 1 2 2 4(1 − ϕ0 )εd = − s hom (m + p) − d2 + Sr (1 − Sr ) pc2 s hom ∂μ k ∂μ μ 4μs 2 (8.114) 8.4.2 Strength in Partially Saturated Conditions We now investigate the inﬂuence of partially saturated conditions on the strength of a porous medium using the secant method of Section 7.5.3. As in Section 8.4.1, we disregard membrane tension effects, and assume that the morphology of the microstructure considered is a matrix-inclusion one, which justiﬁes the use of the Mori–Tanaka scheme. From now on, the porosity is simply denoted by ϕ, as in Chapter 7, since we deal with the strength in the current conﬁguration of the rev. Von Mises Solid

We ﬁrst consider the case of a solid matrix of the von Mises type (see (7.1)). Following the approach developed in Section 7.5.3, we look for the limit states of an rev whose matrix material behavior is deﬁned by the potential (7.106) with F(J 2′ ) ≈ 2k J 2′ . In the framework of the reference strain concept, the macroscopic stress then is: r

r

Σ + pB(J 2′ ) = Chom (J 2′ ) : E (8.115)

′ r The reference strain εd = J 2 r deﬁned in (7.97) is obtained from (8.114) and the solution of the following nonlinear problem: ∂ 3ϕ 1 1 ∂ 2 ′r Sr (1 − Sr ) pc2 d2 + 4(1 − ϕ)J 2 = − s hom (m + p) − s hom ∂μ k ∂μ μ 4μs 2 (8.116)

Non-Saturated Microporomechanics

264

k hom = K (k s , μs , M); μhom = M(k s , μs , M) r k μs = μs J 2′ ≈ 2 J 2′ r

(8.117) (8.118)

We look for the stress states reached asymptotically on radial strain paths λ → λE˜ (see (7.105)). For large values of λ, the reference deviatoric strain is expected to be large as well, and (8.117) can be replaced by the simpliﬁed expressions (7.62)–(7.108) which are valid if μs /k s ≪ 1: k hom =

4(1 − ϕ) s ′ r μ J2 ; 3ϕ

μhom =

r 1−ϕ μs J 2′ 1 + 2ϕ/3

(8.119)

We then combine (8.116) and (8.119): ′r 2 1 + 32 ϕ 2 3ϕ Sr (1 − Sr ) 2 3ϕ 2 ′r s ( + + pc p) + = 2 J2 μ J2 m 4(1 − ϕ)2 (1 − ϕ)2 d 4(1 − ϕ) (8.120) Recalling that μs (J 2′ r ) ≈ k/(2 J 2′ r ) in the domain of large strains (see (8.118)), the macroscopic stress state asymptotically lies on an elliptic curve in the (m , d ) plane: 1 + 32 ϕ 2 3ϕ 3ϕ Sr (1 − Sr ) 2 2 p) + ( + d = k 2 − pc m 2 2 4(1 − ϕ) (1 − ϕ) 4(1 − ϕ)

(8.121)

Strictly speaking, the strength predicted by the above criterion is not controlled by the effective stress Σ + p1 since the capillary pressure pc also appears on the r.h.s. of (8.121). On the other hand, compared to the empty situation (7.71) and the saturated situation (7.79), it is observed that unsaturated conditions can be taken into account by replacing the pressure P in Σ + P1 by the average pressure p, provided that the shear strength k of the solid is itself replaced by the corrected strength k ′ given by: 3ϕ Sr (1 − Sr ) pc2 ′ k =k 1− (8.122) 4(1 − ϕ) k 2 In other words, the strength in non-saturated conditions involves the Bishop effective stress Σ + p1, but it is not controlled by it, as the strength k ′ depends on saturation and capillary pressure. From a geometrical point of view, the strength criterion in the empty case (i.e. (7.71)) is an ellipse centered at the origin as in Figure 8.12. The strength criterion corresponding to the saturated state with ﬂuid pressure p < 0 (suction) is obtained by a translation

Strength Domain of Non-Saturated Porous Media Σd

Ghom (0)

265

Ghom (p, Sr = 1)

–p

Σm

Ghom (p, Sr ) Figure 8.12 Macroscopic strength criterion in non-saturated conditions (von Mises solid matrix)

along the positive hydrostatic axis. The strength criterion for the unsaturated state is then obtained from the previous ellipse by the homothety with ratio k ′ /k < 1. Drucker–Prager Solid

In the case of a Drucker–Prager solid matrix (7.8), the same method can be used except for the fact that the shear modulus is now given by (7.115) in the range of large deviatoric strains. We then need an estimate for both I1′ r and J 2′ r , still deﬁned as in (7.97). Disregarding membrane tension effects, the average stress rule is Σ = (1 − ϕ)σ s − ϕ p1. As regards I1′ r , we just have to replace P by p in (5.90). Hence, instead of the set of equations (7.116) obtained in the saturated case, the nonlinear problem now becomes: m + ϕ p ks ∂ 1 1 ∂ 2 d2 = − s hom (m + p) − ∂μ k ∂μs μhom

(1 − ϕ)I1′ r = 4(1 − ϕ)J 2′ r

3ϕ Sr (1 − Sr ) pc2 2 s 4μ 1−ϕ 4(1 − ϕ) s μ ; μhom = μs k hom = 3ϕ 1 + 2ϕ/3 α r μs = (h − k s I1′ ) ′r 2 J2 +

(a )

(b) (c) (d) (8.123)

As in the von Mises case, (8.120) is retrieved from the combination of (8.123b) and (8.123c). From (8.123a) and (8.123d), the following yield envelope is

Non-Saturated Microporomechanics

266

obtained: 1 + 23 ϕ 2 3ϕ Sr (1 − Sr ) 2 3ϕ m + ϕ p 2 2 2 (m + p) + + pc = α h − 4(1 − ϕ)2 (1 − ϕ)2 d 4(1 − ϕ) 1−ϕ

(8.124)

Generalizing (7.122), we now introduce: Σ′′ =

Σ + p1 ; 1 + p/ h

m′′ =

m + p ; 1 + p/ h

d′′ =

d 1 + p/ h

(8.125)

With these notations, (8.124) becomes: 2 3ϕ 2 2 2 2 ′′ 2 2 − α m + 1 + ϕ d′′ + 2α 2 h(1 − ϕ)m′′ α h (1 − ϕ) − Q( pc ) = 4 3 (8.126) where: 3ϕ(1 − ϕ)Sr (1 − Sr ) 2 Q( pc ) = pc (8.127) 4(1 + p/ h)2 with P being replaced by p, the above criterion is identical to (7.118), except for the correcting term Q( pc ).

9 Microporoplasticity When the stress in the solid phase of an elastoplastic porous material reaches a strength threshold, plastic deformation takes place in addition to the elastic deformation which is partially recovered upon unloading. While the classical plasticity theory captures the behavior of the solid, the macroscopic behavior of the material exhibits a pronounced poroplastic behavior due to the expansion of the solid phase into the pore space, or due to the pressure at the solid–ﬂuid interface generating deformation of the solid. In particular, a part of the energy which is provided to the material system by a macroscopic stress and a ﬂuid pressure is dissipated in the form of heat, and another part is frozen into the microstructure and is neither recovered upon unloading nor dissipated. The latter phenomenon is known as hardening. The focus of this chapter is the development of the micro–macro relations for such a poroplastic hardening material in saturated conditions, including the poroplastic state equations, the yield criterion, and the ﬂow and hardening rule.

9.1 The 1-D Thought Model: The Saturated Hollow Sphere To motivate the forthcoming developments, consider the hollow sphere model, which we recall is the simplest geometrical conﬁguration of a porous material. As throughout this book, the solid phase is subjected to a uniform normal stress applied on the external boundary (r = B), and to the cavity pressure P applied on the cavity wall (r = A). The corresponding boundary conditions are the same as applied in Section 7.1, from which we recall: σrr (A) = −P;

σrr (B) =

(9.1)

In what follows, only positive values of P and are considered. In addition, it is assumed that + P > 0. Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

Microporoplasticity

268

The solid phase situated in r ∈ [A, B] is considered an ideal elastoplastic material (no hardening). The admissible stress states are described by a yield criterion of the von Mises type: 1 f s (σ) = s : s − k 2 2

(9.2)

where k is a constant. The ﬂow rule is assumed to be associated, so that the plastic strain rate ε˙ pl is proportional to the deviatoric stress s: ˙ ε˙ pl = λs

(λ˙ ≥ 0)

(9.3)

where λ˙ is the plastic multiplier. The elastic properties of the solid are assumed to be homogeneous. For simplicity, we assume the solid phase to be incompressible, both elastically (through a Poisson’s ratio ν s = 1/2) and plastically through the chosen ﬂow rule (9.3). As a consequence, the radial displacement ξr (r ) during elastic and plastic evolutions is the solution of the following differential equation: div ξ =

ξr ∂ξr β + 2 = 0 ⇒ ξr = 2 ∂r r r

(9.4)

where β is a constant. β does not vary spatially within the solid, and depends a priori on the mechanical properties of the solid and on the loading deﬁned by (9.1). We investigate the response of the material system for both the elastic and the elastoplastic regimes, and look for micro–macro relations that allow us to capture the overall poro–elastoplastic behavior. 9.1.1 Elastic Response The initial state of the material is assumed to be stress free. Hence, small values of and P entail an elastic response of the hollow sphere, which we derived in Section 5.2. For ν s = 1/2, the stress ﬁeld, strain ﬁeld and displacement ﬁeld solutions, σ EL , εEL and ξ EL , are: 3 A 1 + ϕ0 P − ( + P) σrrEL = 1 − ϕ0 r 3 1 1 A EL σθELθ = σφφ = (9.5) + ϕ0 P + ( + P) 1 − ϕ0 2 r ξrEL =

+ P A3 1 4(1 − ϕ0 ) μs r 2

EL where ϕ0 = ( A/B )3 . Recalling that + P > 0, we note that σrrEL < σθθ . As long s EL as f (σ ) ≤ 0 is satisﬁed throughout the solid domain, the response of the

The 1-D Thought Model: The Saturated Hollow Sphere

269

hollow sphere remains elastic and is characterized by (9.5). That is: 2 1 1 EL EL s : s − k 2 = σrrEL − σθELθ − k 2 ≤ 0 2 3

In other words, the yield condition f s (σ EL ) = 0 takes the form: √ σrrEL = σθELθ − k 3

(9.6)

(9.7)

This yield condition is reached ﬁrst at the cavity wall r = A, which deﬁnes the onset of plasticity that occurs for a critical value ′e of + P: √ 2k 3 ′e = (1 − ϕ0 ) (9.8) 3 As long as this threshold is not reached, the macroscopic strain E = 13 E E L 1 is derived from (9.5): E EL = 3

ξrEL (B) +P 3ϕ0 = B 4(1 − ϕ0 ) μs

(9.9)

9.1.2 Elastoplastic Response We now assume that + P increases monotonically beyond the threshold ′e , thus entailing plastic deformation. The solution is based on the assumption that the yield condition f s (σ) = 0 is reached in the spherical domain A < r < c. The sphere r = c is the boundary between the elastic part of the solid domain (ε pl = 0) and the elastoplastic one (ε pl = 0). Since we expect that σrr < σθθ , the stress components in the plastic zone satisfy: √ σrr = σθ θ − k 3 (9.10) It follows that the momentum balance equation in the region A < r < c takes the form: √ k 3 ∂σrr −2 =0 (9.11) ∂r r Due to the boundary condition σrr = −P at the cavity wall r = A, the stress ﬁeld in the elastoplastic zone is deﬁned by: √ r for r ∈ [A, c] : σrr = −P + 2k 3 ln (9.12) A In turn, in the domain c < r < B which is free of plastic deformation, the stress and displacement are readily obtained from the elastic solution associated with the boundary conditions: √ c σrr (c) = −P + 2k 3 ln ; σrr (B) = (9.13) A

Microporoplasticity

270

The position of the boundary r = c between the plastic and elastic zones relates to the loading through the fact that the yield condition f s (σ) = 0 in the domain r ≥ c is only reached at r = c: c 3 √ 1 c + P = 2k 3 + ln 1 − ϕ0 (9.14) 3 A A √ We readily verify that the limit load ′+ = −2k/ 3 ln ϕ0 derived in (7.6) is retrieved when the spherical interface r = c reaches the external boundary r = B. Using (9.14), the displacement solution of the elastic problem deﬁned on the interval r ∈ [c, B] by the boundary conditions (9.13) takes the form (9.4) valid throughout the solid domain: √ k 3 3 β ξr = 2 ; β = c (9.15) r 6μs The macroscopic strain E = 13 E1 induced by the loading is related to ξr by: √ ξr (B) ϕ 0 k 3 c 3 E =3 (9.16) = B 2μs A

Relations (9.14) and (9.16) characterize the response of the hollow sphere in the elastoplastic regime. The = (E) curve is plotted in Figure 9.1 for two ϕ0 = 0.3 0.8 Σ +P √ k 3 0.6

0.4

ϕ0 = 0.5 0.2

0

0.05

0.1

0.15

µs E √ 0.2 3k 3

Figure 9.1 Elastoplastic response of the hollow sphere

The 1-D Thought Model: The Saturated Hollow Sphere

271

porosity values, ϕ0 = 0.3 and ϕ0 = 0.5. An interesting observation is that the macroscopic stress and the pore pressure P affect the macroscopic strain E through Terzaghi’s effective stress + P. This will turn out to be a general result for porous media made up of a von Mises solid phase. 9.1.3 The Concept of Residual Stresses We now consider an unloading of the effective stress, ′ = + P, from some value lying between ′e and ′+ . We seek the condition for the unloading phase to be elastic until complete removal of the macroscopic stress on the boundary r = B and of the pore pressure P on the cavity wall r = A. If the unloading process is indeed elastic, the residual stress ﬁeld σ R is the sum of the stress ﬁeld σ reached prior to unloading and of the elastic solution obtained by replacing and P with − and −P in (9.5). The latter is clearly equal to −σ EL . In other words: σ = σ R + σ EL

(9.17)

For (9.17) to be valid, the residual stress state σ R must meet the yield condition f s (σ R ) ≤ 0. It can be readily seen that this requirement is fulﬁlled regardless of the value of ′ in the interval [ ′e , ′+ ], provided that the porosity ϕ0 satisﬁes the condition: 2(1 − ϕ0 ) + ln ϕ0 ≥ 0

(9.18)

This approximately corresponds to ϕ0 ≥ 0.2, which is assumed in the sequel. Accordingly, the displacement ﬁeld ξ R in the residual state is the sum of the displacement ﬁeld ξ prior to unloading and of the elastic component −ξ EL : ξ R = ξ − ξ EL

(9.19)

The macroscopic strain E R in the residual state represents the response of the sphere after removal of the loading. It is therefore referred to as macroscopic plastic strain and denoted by E pl . It is related to the corresponding displacement ﬁeld as in (9.9) and (9.16): E pl = E R =

ξrR (B) B

(9.20)

The macroscopic counterpart to (9.19) thus becomes (see Figure 9.2): E pl = E − E EL

(9.21)

Recalling (9.14), we obtain: E

pl

√ kϕ0 3 c 3 c = − 1 − 3 ln 6μs (1 − ϕ0 ) A A

(9.22)

Microporoplasticity

272 Σ+ P

Σ

′+

′e

Σ

S1

S2

E el

E pl

E

Figure 9.2 Macroscopic elastic and plastic strains

which reveals that the macroscopic plastic strain is related to the location of the boundary between the plastic and elastic zones in the solid domain. We note that the counterpart to (9.19) in terms of strains is: ε = ε R + εEL

(9.23)

This is a reminder of another decomposition of the microscopic strain, namely: ε = ε pl + εel

(9.24)

However, it is important to observe that ε pl = ε R . Indeed, it is recalled that ε pl = 0 for r > c. In contrast, we have: εrrR = −2E R

B3 r3

(9.25)

The fact that ε pl = ε R immediately implies that εel = εEL . 9.1.4 Energy Aspects We consider a monotonic increase of the effective stress ′ = + P and we start from the stress-free initial state ( = P = 0). The elastic energy stored in

The 1-D Thought Model: The Saturated Hollow Sphere

273

the solid domain is: 1

= 2||

s

σ : Ss : σ dVz

(9.26)

Recalling the two different expressions for the stress ﬁeld in the elastic and plastic zones yields: c 3 c 6 ϕ0 k 2

= 2 − ϕ0 − 1 (9.27) 2μs A A

Alternatively, it is interesting to insert the decomposition (9.17) of the stress ﬁeld into (9.26):

1 EL

= +R+ σ EL : Ss : σ R dVz (9.28) || s where:

EL

1 = 2||

σ s

EL

s

EL

: S : σ dVz ;

1 R= 2||

s

σ R : Ss : σ R dVz

(9.29)

EL represents the energy stored in the solid domain when the latter behaves elastically. It has already been determined in Section 5.2.2:

EL =

3ϕ0 ′2 8(1 − ϕ0 ) μs

(9.30)

R represents the elastic energy which remains stored in the solid when the macroscopic loading has been removed ( = P = 0). Interestingly, it is found that:

= EL + R Comparing (9.31) to (9.28) implies:

σ EL : Ss : σ R dVz = 0

(9.31)

(9.32)

s

This will be shown later to be a general property. Equation (9.31) in fact reveals that EL represents the component of the elastic energy that is recovered on unloading. Graphically it corresponds to the triangular area S2 in Figure 9.2: S2 = EL

(9.33)

The total mechanical energy W supplied to the solid is the area S = S1 + S2 located under the = (E) curve1 in Figure 9.2. Furthermore, W appears 1 This

result uses the incompressibility of the solid phase.

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as the sum of the elastic energy stored in the solid and of the dissipated energy D: W = + D = S1 + S2

(9.34)

Combining (9.31), (9.33) and (9.34), we conclude that S1 = D + R. In other words, the energy S1 that is not recovered during unloading is not entirely dissipated. A part of it remains stored in the solid after complete removal of the loading. The latter, i.e. the elastic energy in the residual state, is therefore often referred to as ‘frozen’ energy.

9.2 State Equations of Microporoplasticity This section is devoted to a generalization of the hollow sphere model of saturated poroplasticity. 9.2.1 First Approach to the Macroscopic Stress–Strain Relationship When the constitutive behavior is elastoplastic, the local state equation in the solid becomes afﬁne: σ(z) = Cs : (ε(z) − ε pl (z))

(9.35)

where ε pl (z) is the microscopic plastic strain. The component of the strain tensor ε that is linearly related to the stress is called the elastic strain, and is denoted hereafter by εel : εel = ε − ε pl = Ss : σ

(9.36)

where Ss is the elastic compliance tensor. Extending the technique employed in Section 5.5 to the poroelastic case, the stress ﬁeld in the rev is conveniently rewritten in the form: σ(z) = C(z) : ε(z) + σ p (z)

(9.37)

with: C(z) =

Cs in s 0 in p

p

σ (z) =

−Cs : ε pl (z) in s −P1 in p

(9.38)

Indeed, according to (5.106) and (5.110), the macroscopic state equation relating the macroscopic stress Σ to the macroscopic strain E takes the form: Σ = Chom : E + σ p : A

(9.39)

In the poroplastic case, the macroscopic ‘prestress’ σ p : A is not only due to the contributions of the pore pressure P as in the poroelastic case, but also

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275

due to the microscopic ‘plastic prestress’ in the solid: σ p : A = −PB − (1 − ϕ0 )ε pl : Cs : A

s

(9.40)

s where the symmetry of the homogenized elasticity tensor Cs (Cisjkl = Ckli j ) has been used. The second-order tensor B in (9.40) is the same as in the poroelastic state equation (5.115) and is given by (5.116). Recalling (4.139) relating the strain and stress concentration tensors A and B, (9.40) also becomes: s

σ p : A = −PB − (1 − ϕ0 )Chom : ε pl : B

(9.41)

hom where the symmetry of the elasticity tensor Chom (Cihom jkl = C kli j ) has been used. Finally, inserting (9.41) into (9.39) yields the macroscopic stress–strain state equation of poroplasticity:

Σ = Chom : (E − E pl ) − PB

(9.42)

with: s

E pl = (1 − ϕ0 )ε pl : B

(9.43)

Recalling that B is equal to zero in the pore space, it is interesting to note that E pl can be recast in the form ε pl : B for any extension of the microscopic plastic strain ﬁeld ε pl into the pore space. Relation (9.42) suggests that E pl represents the macroscopic strain that is locked into the microstructure when the loading (Σ, P) is removed, if such an unloading is reversible. This permanent strain tensor is therefore referred to as the macroscopic plastic strain tensor. 9.2.2 Macroscopic Plastic and Elastic Strain Tensors Interestingly, Section 9.2.1 introduces the concept of plastic strain at the macroscopic scale, without any reference to an elastic component of the macroscopic strain. We now want to give an in-depth analysis of the macroscopic plastic and elastic strain tensors and to clarify their micromechanical origin.2 This will allow us not only to retrieve the state equation (9.42) and the relation (9.43) between E pl and ε pl , but also to derive the second state equation involving the pore volume change, as well as to provide an energy approach to the elastoplastic evolution at both the microscopic and macroscopic scales. Let us consider that the microscopic plastic strain ﬁeld is given. It is convenient for our purposes to deﬁne uniform stress boundary conditions on ∂ (see Section 4.2.10 for the equivalence between stress and strain boundary conditions). The response of the rev is characterized by a stress ﬁeld σ in 2 This

section extends the pioneering work of H.D. Bui [10] to saturated porous media.

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and a displacement ﬁeld ξ in s . These ﬁelds are the solution of: div σ = 0 () σ = −P1 ( p ) σ = Cs : (ε − ε pl ) (s ) (9.44) σ·n=Σ·n (∂) 1 ε = grad ξ + tgrad ξ (s ) 2 From a mathematical point of view, we can consider the macroscopic stress tensor Σ and the pore pressure P on the one hand, and the microscopic plastic strain ﬁeld ε pl on the other hand, as two different loading parameters in the mechanical problem deﬁned by (9.44). The idea then is to split (9.44) into two auxiliary problems each of which is associated with one of these two components of the mechanical loading.3 First Auxiliary Problem: (Σ, P) (No Plastic Strain)

In the ﬁrst auxiliary problem, the loading is deﬁned by the couple (Σ, P) only (no plastic strain). It therefore corresponds to the poroelastic response of the rev and its solution (σ EL , ξ EL ) complies with the following conditions: div σ EL = 0 () EL σ = −P1 ( p ) σ EL = Cs : εEL (s ) (9.45) σ EL · n = Σ · n (∂) 1 εEL = grad ξ EL + tgrad ξ EL (s ) 2 As usual, the displacement can be extended into the pore space. The solution to (9.45) has been studied in Chapter 5 within the equivalent framework of uniform strain boundary conditions. Using (5.115), we can immediately relate the macroscopic strain EEL = εEL to the loading parameters Σ and P: EEL = Shom : (Σ + PB) with EL

Shom = Chom

−1

(9.46)

The evolution δφ of the Lagrangian porosity that takes place during the poroelastic response of the rev has been derived in (5.123): δφ EL =

P + B : EEL N

(9.47)

3 However, from a physical point of view, it should be emphasized that (Σ, P) are indeed ‘real’ loading parameters in the sense that they can be controlled during an experiment, whereas ε pl is rather the consequence of the loading than a loading itself: ε pl cannot be prescribed!

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If the unloading process which consists of removing the macroscopic stress Σ and the pore pressure P is reversible, i.e. does not induce additional plasticity, −σ EL and −εEL are recognized as the microscopic stress and strain changes that are induced in the rev by the unloading. It follows that the corresponding macroscopic strain EEL is identical to the classical concept of (macroscopic) elastic strain. It will therefore be denoted as Eel : Eel = EEL = εEL

(9.48)

For the same reason, δφ EL introduced in (9.47) represents the reversible or elastic part of the Lagrangian porosity change induced by the loading (Σ, P). It will therefore be denoted as δφ el : δφ el = δφ EL = ϕ0 tr εEL

p

(9.49)

It will prove useful to relate the poroelastic response σ EL to the macroscopic loading (Σ, P). To this end, for a second time we split the problem (9.45) into two subproblems. In both of them, the solid constitutive behavior is purely elastic. In the ﬁrst subproblem, the pore pressure is equal to P and the macroscopic stress is −P1: div σ ′ = 0 () ′ σ = −P1 ( p ) σ ′ = Cs : ε′ (s ) (9.50) (∂) σ ′ · n = −Pn 1 ε′ = grad ξ ′ + tgrad ξ ′ (s ) 2 It is readily seen that the stress solution of (9.50) is uniform and is σ ′ = −P1. In the second subproblem of (9.45), the macroscopic stress Σ + P1 is applied to the rev under drained conditions: div σ ′′ = 0 () ′′ σ =0 ( p ) σ ′′ = Cs : ε′′ (s ) (9.51) (∂) σ ′′ · n = (Σ + P1) · n 1 ε′′ = grad ξ ′′ + tgrad ξ ′′ (s ) 2 This problem has been studied in Section 4.2.3. In particular, the local stress σ ′′ is given by (4.54): σ ′′ = B(z) : (Σ + P1)

(9.52)

Finally, the microscopic poroelastic stress takes the form: σ EL (z) = B(z) : (Σ + P1) − P1

(9.53)

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278

Second Auxiliary Problem: Plastic Loading εpl

In the second auxiliary problem of (9.44), the loading parameter is the microscopic plastic strain ﬁeld ε pl . The response of the rev to this loading is deﬁned by a microscopic stress ﬁeld σ R and a microscopic displacement ﬁeld ξ R that are solutions of: div σ R = 0 () R σ =0 ( p ) σ R = Cs : (ε R − ε pl ) (s ) (9.54) σR · n = 0 (∂) 1 ε R = grad ξ R + tgrad ξ R (s ) 2 According to (9.54), the macroscopic stress tensor balanced by σ R is Σ R = σ R = 0. In turn, the macroscopic strain induced by this loading is E R = ε R . σ R and ε R are recognized as the microscopic stress and strain states that remain in the rev after removing the macroscopic stress Σ and the pore pressure P, provided that the unloading process is reversible. They are therefore called ‘residual’ stress and strain states, which is indicated by the superscript R. The elastic energy R stored in the unloaded rev is: s 1 1 R = (1 − ϕ0 )σ R : Ss : σ R = σ R : Ss : σ R 2 2

(9.55)

where we made use of σ R = 0 in p (pores). E R appears as the residual macroscopic strain state of the rev after unloading, which is classically referred to as the macroscopic plastic strain. E R therefore coincides with the tensor E pl introduced in (9.42). It is then appealing to look for the link between E R and the microscopic plastic strain ε pl ; that is, to retrieve (9.43) directly. Let us introduce the poroelastic solution associated with P = 0 EL (drained condition), deﬁned at the microscopic scale by σ EL 0 = σ (Σ, 0) and EL EL ε0 = ε (Σ, 0). We ﬁrst apply the Hill lemma (see Section 4.2.6) to the couple EL R (σ EL 0 , ε ). Recalling that σ 0 is statically admissible with the macroscopic stress Σ, we obtain: R EL R R σ EL 0 : ε = σ0 : ε = Σ : E

(9.56)

We now make use of the fact that ε R = Ss : σ R + ε pl in s , and that σ EL 0 = 0 in p: s

EL R s R + (1 − ϕ )σ EL : ε pl σ EL 0 0 : ε = (1 − ϕ0 )σ 0 : S : σ 0

s

(9.57)

Recalling that σ R = 0 in p and that σ R = 0, we apply the Hill lemma, a R second time but now to the couple (εEL 0 , σ ): s

s R = εEL : σ R = εEL : σ R = 0 (1 − ϕ0 )σ EL 0 :S :σ 0 0

(9.58)

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279

Applying (4.54) to σ EL 0 in the second term of the r.h.s. of (9.57) then shows that: pl R σ EL 0 :ε =ε :B:Σ

(9.59)

E R = ε pl : B

(9.60)

where we have used B = 0 in p . Combining (9.56) and (9.59) eventually yields the link between the macroscopic residual (or plastic) strain tensor E R and the microscopic plastic strain ﬁeld ε pl :

Finally, comparing (9.43) and (9.60) conﬁrms that E pl and E R are indeed identical, which justiﬁes the superscript pl: s

E pl = E R = ε R = ε pl : B = (1 − ϕ0 )ε pl : B

(9.61)

An important result of these developments is that the macroscopic plastic strain is not the average of the microscopic plastic strain ﬁeld ε pl . Furthermore, the residual or plastic Lagrangian porosity change, denoted by δφ R or by δφ pl , that remains after unloading is related to the average of the volume strain tr ε R over the pore space: δφ R = δφ pl = ϕ0 tr ε R

p

(9.62)

Relations (9.48)–(9.49) on the one hand, and (9.61)–(9.62) on the other hand, show that EEL (resp. δφ EL ) and E R (resp. δφ R ) are identical to the concepts of elastic and plastic macroscopic strains (resp. porosity changes) that are classically encountered in macroscopic poroplasticity theories. In what follows, we will use the more familiar notation E pl and Eel (resp. δφ pl and δφ el ) rather than E R and EEL (resp. δφ R and δφ EL ). In contrast to the macroscopic strains, it is emphasized that the microscopic strains εEL and ε R are not equal to the microscopic elastic and plastic components εel and ε pl of the microscopic strain ε. This is due to the fact that ε R is the sum of the plastic component ε pl and an elastic component Ss : σ R . In particular, as opposed to εEL and ε R , the strain ﬁelds εel and ε pl are not geometrically compatible: εel = εEL ;

ε pl = ε R

(9.63)

pl

It will be useful to relate δφ directly to the microscopic plastic strain ε pl . To this end, we ﬁrst note that a combination of (9.61) and (9.62) yields: δφ pl = tr E pl − (1 − ϕ0 )tr ε R

s

(9.64)

We then use the fact that σ R = 0 together with the condition σ R = 0 in p . s s s This implies that σ R = 0 and ε R = ε pl (see (9.54)). Equation (9.64) eventually becomes: s s s δφ pl = (1 − ϕ0 ) tr ε pl : B − tr ε pl = tr E pl − (1 − ϕ0 )tr ε pl (9.65)

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Relations (9.61) and (9.65) allow us to determine the macroscopic counterpart of the plastic microscopic strain. As expected, we note that: pl E˙ = 0 pl (9.66) ε˙ = 0 ⇒ δ φ˙ pl = 0 This intuitive result expresses that the macroscopic behavior has no plastic component if the microscopic evolution is elastic. This result will prove useful for the derivation of the macroscopic yield criterion in Section 9.3. 9.2.3 Macroscopic State Equations in Poroplasticity By construction, the solution (σ, ε) of (9.44) is the sum of the solutions of (9.45) and(9.54): σ = σ EL + σ R ε = εEL + ε R

(a ) (b)

(9.67)

Alternatively to (9.67b), we may also split the microscopic strain into its elastic and plastic components εel and ε pl . Nevertheless, decomposition (9.67b) will prove more adaptable for the forthcoming developments: el el ε = εEL ε + ε pl (9.68) ε= ε pl = ε R εEL + ε R Taking the average of (9.67b) over the whole rev yields (see (9.48) and (9.61)): ε = εEL + ε R ⇔ E = Eel + E pl

(9.69)

In turn, the average of (9.67b) over the pore space p leads to a decomposition of the total Lagrangian porosity change φ − φ0 into the elastic and plastic contributions introduced by (9.49) and (9.62): φ − φ0 = ϕ0 tr ε p = δφ el + δφ pl

(9.70)

The two state equations of poroplasticity4 are obtained from the combination of (9.46) with (9.69) and of (9.47) with (9.70): Σ + BP = Chom : (E − E pl ) P + B : (E − E pl ) δφ − δφ pl = N

(a ) (b)

(9.71)

To complete the model, we are left with: (1) investigating the mathematical formulation of the macroscopic yield criterion, which speciﬁes when plastic deformation occurs in the micro solid phase; and (2) specifying the evolution 4 The state equations (9.71) are identical to those derived in [12] within a macroscopic thermodynamic framework.

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281

of E pl and δφ pl , i.e. the macroscopic ﬂow rule. This will be the purpose of Sections 9.3 and 9.4.1, respectively. 9.3 Macroscopic Plasticity Criterion 9.3.1 Link Between the Microscopic and the Macroscopic Plasticity Criterion We assume that the solid phase is an elastic perfectly plastic material. The microscopic elasticity domain G s is deﬁned by the yield criterion f s (σ) < 0. Both function f s (σ) and the domain G s are assumed to be convex. We deﬁne an elastic state of the rev by the condition that the response to any inﬁnitesimal loading (dΣ, dP) is purely elastic. Such a deﬁnition implies that the microscopic stress state lies strictly inside G s everywhere in s : (∀z ∈ s )

f s (σ(z)) < 0

(9.72)

This suggests a possible formulation of the macroscopic yield criterion F as follows:5 F = sup f s (σ(z))

(9.73)

z∈s

˙ P), ˙ the condition F < 0 ensures that the rate Indeed, for any loading rate (Σ, of the microscopic plastic strain is ε˙pl = 0. Relation (9.66) then shows that the rates of the macroscopic plastic strain and of the plastic Lagrangian porosity change are zero as well: pl E˙ = 0 (9.74) F < 0 ⇒ δ φ˙ pl = 0 In turn, if F = 0 and F˙ < 0, it is readily seen from (9.73) that: ⎧ s s ⎪ ⎨ f (σ(z)) = 0 and f˙ < 0 (∀z ∈ s ) or ⎪ ⎩ s f (σ(z)) < 0

This further implies that ε˙pl = 0. It follows from (9.66) that: E˙ pl = 0 F = 0 and F˙ < 0 ⇒ δ φ˙ pl = 0

(9.75)

(9.76)

By contrast, if F = 0 and F˙ = 0, a plastic evolution of the rev is possible (ε˙pl = 0) with its macroscopic counterpart ( E˙ pl = 0, δ φ˙ pl = 0). Together with (9.74) 5 Note

that the previous deﬁnition does not deﬁne the macroscopic criterion in a unique way.

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282

and (9.76), this shows that the state function F deﬁned in (9.73) meets the conditions that classically deﬁne a yield criterion. We still have to identify the arguments of F, however. 9.3.2 Arguments of the Macroscopic Yield Criterion Let us observe that the plastic state of the rev at time t is fully characterized by the microscopic plastic strain ﬁeld ε pl (t) in the solid phase, or alternatively, by the solution σ R (t) to (9.54). According to (9.53) and (9.67), the microscopic stress state is: σ = σ R (t) + B : (Σ + P1) − P1

(9.77)

It should be emphasized that the microscopic stress ﬁeld σ that prevails in when subjected to the loading deﬁned by Σ and P is not a one-to-one function of Σ and P. We refer here to the stress ﬁeld that is compatible in the sense of (9.44) with the microscopic plastic strain ﬁeld ε pl (t) together with the macroscopic loading (Σ, P). Recall that ε pl (t) as well as σ R (t) depends not only on (Σ, P), but on the whole loading history until the current time. Inserting (9.77) into (9.73) reveals that F can be regarded as a function of the loading parameters (Σ, P) on the one hand, and of the microscopic stress ﬁeld σ R (t) on the other hand. We recall that this microscopic stress ﬁeld represents the current plastic state of the rev: F (Σ, P, σ R (t)) = sup f s −P1 + B(z) : (Σ + P1) + σ R (z, t) (9.78) z∈s

As expected, the macroscopic yield criterion depends through (Σ, P) on the current mechanical load applied to the rev. In turn, the stress ﬁeld σ R (t) in (9.78) is recognized as a macroscopic hardening parameter in the sense that it controls the inﬂuence of the history of the rev on the current value of the macroscopic criterion. Since this hardening parameter is a ﬁeld, it should be observed that we come up with an inﬁnite number of scalar hardening parameters. However, in a highly simpliﬁed way, the evolution of the elastic energy R stored in the residual state (see (9.55)) can be used to describe the macroscopic hardening. According to (9.54), σ R (t) appears as the stress counterpart of the microscopic strain ﬁeld ε pl . Interestingly, if ε pl were geometrically compatible, then σ R (t) = 0, and no macroscopic hardening would take place. Actually, the fact that ε pl is not geometrically compatible is deeply related to the heterogeneity of the rev; that is, the presence of a pore space. Indeed, the presence of the pore space is recognized to be responsible for the hardening behavior of porous materials. In the (Σ, P) space, the macroscopic ‘elastic’ domain G hom (σ R (t)) corresponding to the hardening parameter σ R (t) is characterized by the condition

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283

F (Σ, P, σ R (t)) ≤ 0. According to (9.78), it is made up of the macroscopic loadings (Σ, P) for which the microscopic stress ﬁeld σ R (t) + σ EL (Σ, P) is compatible with the plasticity criterion of the solid everywhere in s : s s R (Σ, P) ∈ G hom (σ R (t)) ⇔ (∀z ∈ ) f −P1 + B(z) : (Σ + P1) + σ (z, t) ≤ 0 (9.79) We now want to prove the convexity of the macroscopic elastic domain G hom (σ R (t)) as a consequence of the convexity of G s (resp. of the yield criterion f s (σ)). We therefore consider two macroscopic loadings Li = (Σi , Pi ) that both belong to G hom (σ R (t)), satisfying the condition F (Σi , Pi , σ R (t)) ≤ 0. In the hardening state deﬁned by σ R (t), the microscopic stress ﬁeld σ i induced by Li is given by (9.77): σ i = σ R (t) + B : (Σi + Pi 1) − Pi 1

The condition F (Σi , Pi , σ R (t)) ≤ 0 is equivalent to (see (9.79)): (∀z ∈ s ) f s σ i (z) ≤ 0 (i = 1, 2)

(9.80)

(9.81)

We now consider a point Lλ on the segment L1 L2 of the (Σ, P) space: Lλ = (Σλ , Pλ ) = λ(Σ1 , P1 ) + (1 − λ)(Σ2 , P2 )

(λ ∈ [0, 1])

(9.82)

Since the expression for σ in (9.77) depends linearly on Σ and P, the microscopic stress ﬁeld σ λ induced by Lλ is: σ λ = λσ 1 + (1 − λ)σ 2

(9.83)

In turn, the convexity of the microscopic yield criterion f s (σ) together with (9.81) and (9.83) yields: (∀z ∈ s ) f s σ λ (z) ≤ 0 ⇔ F Σλ , Pλ , σ R (t) ≤ 0 (9.84)

This means that the segment L1 L2 is a subset of the macroscopic elastic domain G hom (σ R (t)), which proves the convexity of this domain. In other words, when the rev in the hardening state deﬁned by σ R (t) is subjected to the macroscopic stress path from L1 to L2 on the segment L1 L2 , its response is purely elastic. 9.4 Dissipation Analysis 9.4.1 Macroscopic Flow Rule Let σ be the current microscopic stress state in some location z of s , while ε˙ pl represents the rate of plastic strain induced by some stress rate σ. ˙ We now assume that the ﬂow rule of the solid in s satisﬁes the normality condition: ε˙ pl = χ˙

∂f s (σ) ∂σ

(χ˙ ≥ 0)

(9.85)

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284

This property can be expressed alternatively by the so-called ‘principle of maximum plastic work’: (∀σ ∗ ∈ G s )

(σ − σ ∗ ) : ε˙ pl ≥ 0

(9.86)

We are going to establish an extension of the principle of maximum plastic work at the macroscopic scale. To begin with, let us introduce the strain rate ε˙ R associated with the plastic strain rate ε˙ pl considered as given. According to (9.54), it is the solution of: (a ) div σ˙ R = 0

()

R

( p )

(b) σ ˙ =0

(c) σ˙ R = Cs : (ε˙ R − ε˙ pl )

(s )

(9.87)

R

(∂) (d) σ˙ · n = 0 1 R R (e) ε˙ R = grad ξ˙ + tgrad ξ˙ (s ) 2

p

Recalling (9.61) and (9.62), we have ε˙ R = E˙ pl and δ φ˙ pl = ϕ0 tr ε˙ R . We then apply the Hill lemma (see Section 4.2.6) to the stress ﬁeld σ (Σ = σ) and the strain rate ε˙ R : s

Σ : E˙ pl = σ : ε˙ R = (1 − ϕ0 )σ : ε˙ R − Pδ φ˙ pl

(9.88)

In turn, application of the Hill lemma to σ R and ε˙ R gives: s

σ R : ε˙ R = (1 − ϕ0 )σ R : ε˙ R = 0

(9.89)

Recalling that σ = σ R (t) + σ EL (see (9.67a)), the combination of (9.88) and (9.89) yields: Σ : E˙ pl + Pδ φ˙ pl = (1 − ϕ0 )σ EL : ε˙ R

s

(9.90)

Let us now introduce any other loading (Σ∗ , P∗ ). Let σ EL ∗ denote the corresponding ‘elastic’ stress ﬁeld: σ EL ∗ = B : (Σ∗ + P∗ 1) − P∗ 1

(9.91)

Following the same reasoning as above, it is readily seen that: Σ∗ : E˙ pl + P∗ δ φ˙ pl = (1 − ϕ0 )σ EL ˙R ∗ :ε

s

(9.92)

Combining (9.90) and (9.92), we obtain: (Σ − Σ∗ ) : E˙ pl + (P − P∗ )δ φ˙ pl = (1 − ϕ0 )(σ EL − σ EL ˙R ∗ ):ε

s

(9.93)

Recalling that ε˙ R = ε˙ pl + Ss : σ˙ R (see (9.87c)), it is interesting to split the r.h.s.

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285

according to: s

(1 − ϕ0 )(σ EL − σ EL ˙ R = (1 − ϕ0 )(σ EL − σ EL ˙ pl ∗ ):ε ∗ ):ε

s

˙R + (εEL − εEL ∗ ):σ

(9.94)

The second term on the r.h.s. of (9.94) vanishes by application of the Hill lemma: s

˙ R = (1 − ϕ0 )(σ EL − σ EL ˙ pl (1 − ϕ0 )(σ EL − σ EL ∗ ):ε ∗ ):ε

s

(9.95)

We now assume that (Σ∗ , P∗ ) ∈ G hom (σ R (t)). Accordingly, the microscopic R stress ﬁeld σ ∗ = σ EL ∗ + σ (t) is compatible with the yield criterion of the solid, so that (9.86) holds: pl σ(z) − σ ∗ (z) : ε˙ pl = σ EL (z) − σ EL ˙ ≥0 (9.96) (∀z ∈ s ) ∗ (z) : ε

Inserting (9.96) into (9.93) and using (9.95) eventually proves that: ∀(Σ∗ , P∗ ) ∈ G hom σ R (t) (Σ − Σ∗ ) : E˙ pl + (P − P∗ )δ φ˙ pl ≥ 0

(9.97)

This result constitutes the ‘principle of maximum plastic work’ at the macroscopic scale; that is, for the porous material. It is convenient to visualize this result geometrically in the (Σ, P) space. In this space, G hom (σ R (t)) is a convex domain which – by deﬁnition – contains the current macroscopic stress– pressure state (Σ, P). If (Σ, P) lies strictly inside G hom (σ R (t)), the ‘vector’ (Σ − Σ∗ , P − P∗ ) can take any orientation in R7 . This implies that E˙ pl = 0 and δ φ˙ pl = 0. By contrast, if (Σ, P) ∈ ∂G hom (σ R (t)), (9.97) prescribes that (E˙ pl , δ φ˙ pl ) is oriented like the outward normal to G hom (σ R (t)) at the point (Σ, P): ∂F Σ, P, σ R (t) E˙ pl = λ˙ ∂Σ ˙ pl

δφ

∂F Σ, P, σ R (t) = λ˙ ∂P

(λ˙ ≥ 0)

(9.98)

In other words, the macroscopic plastic ﬂow rule involves both the macroscopic plastic strain rate and the plastic component of the rate of Lagrangian porosity. The normality rule in the (Σ, P) space holds and appears as a consequence of the normality rule at the microscopic scale. 9.4.2 Energy Analysis During plastic evolutions (ε˙ pl = 0), a part of the energy supplied to the rev is ˙ dissipated in the form of heat. Denoted by D||, the dissipation is obtained

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by integration over the solid phase:6 D˙ = (1 − ϕ0 )σ : ε˙ pl

s

(9.99)

Recalling that ε˙ pl = ε˙ R − Ss : σ˙ R (see (9.87)) and using (9.88), the above equation can also be expressed as: D˙ = Σ : E˙ pl + Pδ φ˙ pl − (1 − ϕ0 )σ : Ss : σ˙ R

s

(9.100)

We now recall that σ R = 0 in p . Using (9.67a), the third term on the r.h.s. becomes: s

(1 − ϕ0 )σ : Ss : σ˙ R = εEL : σ˙ R + σ R : Ss : σ˙ R

(9.101)

D˙ = Σ : E˙ pl + Pδ φ˙ pl − R˙

(9.102)

An application of the Hill lemma to the couple (σ˙ R , εEL ) shows that the ﬁrst term on the r.h.s. of (9.101) vanishes. The second term is the time derivative of the residual elastic energy R (see (9.55)). The (macroscopic) dissipation is therefore: and is subjected to the condition D˙ ≥ 0. The part of the energy supplied to the solid phase s of the rev that is not dissipated is stored in the form of elastic strains. In the isothermal evolutions considered here, this non-dissipated energy is identical to the free energy of the solid. This free energy density7 is denoted by :

s 1 1

= (9.103) εel : Cs : εel dVz = (1 − ϕ0 )εel : Cs : εel 2|0 | s 2

Recalling that εel = ε − ε pl and ε = εEL + ε R (see (9.68)), also becomes:

= 21 (1 − ϕ0 )εEL : Cs : εEL

s

+ 21 (1 − ϕ0 )(ε R − ε pl ) : Cs : (ε R − ε pl )

+(1 − ϕ0 )εEL : Cs : (ε R − ε pl )

s

(9.104)

s

The ﬁrst term in (9.104) is the free energy of the rev in the poroelastic case. It has been studied in Chapter 5 and is given by (5.61) (in which we replace E by Eel = εEL ). For the second and third terms in (9.104), we introduce σ R = Cs : (ε R − ε pl ) (see (9.54)). Recalling (9.55), we recognize in the second term the elastic energy R in the residual state. The third term can be rewritten in the form σ R : εEL 6 Recall that the solid is a perfectly plastic material. This justiﬁes the expression of the dissipation given by (9.99). 7 We have already encountered this free energy density in (5.57) in the purely elastic case.

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287

and is therefore equal to zero (the Hill lemma). eventually takes the form: P2 1 + R(σ R )

= Eel : Chom : Eel + 2 2N

(9.105)

This result justiﬁes the structure of the macroscopic free energy often postulated in macroscopic theories of poroplasticity [12]. The ﬁrst two (quadratic) terms in (9.105) represent the recoverable elastic energy: that is, the component of the free energy that is recovered when the loading (Σ, P) is removed, if the unloading process is elastic. By contrast, R is the component of the energy supplied to the rev that is neither dissipated nor recovered during an elastic unloading. Interestingly, R provides a way to measure the hardening parameter σ R . It is left as an exercise to verify that the total work rate of external forces can be decomposed into the (rate of) elastic energy and dissipation: σ : ε˙ = ˙ + D˙

(9.106)

9.4.3 Effective Stress in Poroplasticity First, let us consider the particular case of solids for which the (microscopic) yield criterion is not affected by the conﬁning pressure: (∀ σ) (∀ α ∈ R)

f s (σ) = f s (σ + α1)

(9.107)

This property holds, for instance, for Tresca and von Mises materials. For such materials, we note from (9.53) that: (9.108) f s σ R + σ EL = f s (σ R + B : (Σ + P1))

Using the deﬁnition (9.78) of the macroscopic criterion, it is readily seen that: (9.109) F Σ, P, σ R = sup f s σ R (z) + B(z) : (Σ + P1) z∈s

In other words, the macroscopic yield criterion depends on Σ and P through Terzaghi’s ‘effective’ stress Σ + P1: (9.110) F Σ, P, σ R = F˜ Σ + P1, σ R

Note that this reasoning does not evoke any assumption concerning the microscopic ﬂow rule. More generally, for solids that do not meet condition (9.107), the macroscopic effective stress should be looked for as a function of Σ′′ depending on both Σ and P. In other words, we assume that the macroscopic criterion depends on Σ and P through the ‘effective’ stress Σ′′ : F (Σ, P, σ R ) = F˜ (Σ′′ , σ R )

with

Σ′′ = Σ′′ (Σ, P)

(9.111)

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288

˙ P). ˙ From a mathematical point of Let us consider a macroscopic loading (Σ, ˙ ′′ = 0 while view, (9.111) means that the value of the criterion is not affected if Σ keeping σ R constant. That is, if: ′′ −1 ∂Σ′′ ˙ ∂Σ′′ ∂Σ′′ ∂Σ ′′ ˙ ˙ ˙ ˙ :Σ+ P = 0 ⇔ Σ = −P (9.112) : Σ = ∂Σ ∂P ∂Σ ∂P ˙ P) ˙ meeting condition (9.112), we have: Hence, for any loading (Σ, ˙ ˙ σ R ) = F (Σ, P, σ R ) F (Σ + Σdt, P + Pdt,

R EL EL s sup f σ (z) + σ (z) + σ˙ dt = sup f s σ R (z) + σ EL (z) z∈s

(9.113)

z∈s

˙ + P1) ˙ − P1. ˙ Assuming that F = 0, we denote by spl the where σ˙ EL = B : (Σ subdomain of s where the microscopic yield condition f s (σ) = 0 is achieved. A necessary condition for (9.113) to be valid is: ∂f s ∂f s ˙ + P1) ˙ ≤0 ˙ − P1 (9.114) (σ(z)) : σ˙ EL (z) = : B : (Σ ∂σ ∂σ From a mechanical point of view, (9.114) means that the incremental micro˙ P) ˙ is purely elastic: σ˙ = σ˙ EL . Recalling scopic response σ˙ to the loading (Σ, (9.112), and observing that the sign of P˙ can be either positive or negative, (9.114) yields: ′′ −1 ∂f s ∂Σ ∂f s ∂f s ∂Σ′′ s (∀ z ∈ pl ) :B: = :B:1− :1 (9.115) : ∂σ ∂Σ ∂P ∂σ ∂σ (∀ z ∈ spl )

9.4.4 On the ‘Effective Plastic Stress’ Σ + β P1 We now discuss the validity of an effective stress formulation of the form Σ′′ = Σ + β P1, for which (9.115) reduces to: ∂f s ∂f s :B:1= :1 (9.116) (∀ z ∈ spl ) (1 − β) ∂σ ∂σ First, if (9.107) is satisﬁed, the r.h.s. of (9.116) is zero and this leads to β = 1. We thus retrieve the result derived at the beginning of Section 9.4.3. By contrast, if β = 1, (9.116) implies that (9.107) holds. In other words, the effective stress Σ + P1 is relevant if and only if the solid yield criterion does not depend on the conﬁning pressure. The question is whether it was possible that β = 1. In this case, the above conclusion shows that the microscopic yield criterion is necessarily sensitive to the conﬁning pressure. We investigate the class of criteria of the form: f s (σ) = J 2 − g(σm ) ≤ 0 (9.117)

Dissipation Analysis

where

√

289

J 2 and σm denote the microscopic deviatoric and mean stress:

1 1 (9.118) J 2 = σ d : σ d ; σm = tr σ 2 3 The von Mises criterion corresponds to g(σm ) equal to a constant. We hereafter assume that g(σm ) is a decreasing concave function (g ′ (σm ) < 0, g ′′ (σm ) ≤ 0). It is also assumed that g(0) > 0, which physically means that the solid phase has a cohesion, i.e. it can sustain a deviatoric stress without being conﬁned. For instance, in the Drucker–Prager criterion, g(σm ) is an afﬁne function: g(σm ) = α(h − σm )

(h, α > 0)

(9.119)

where h is the strength of the solid subjected to an isotropic traction. Let us start from the natural state (σ = 0 in s ). We analyze the response of the rev to a stress path of the form Σ = (t)1 and P = 0 where (t) is a monotonic function, either decreasing (compression) or increasing (traction) in time, starting from (0) = 0 (spherical macroscopic loading in dry conditions). At the onset of plasticity, the microscopic stress ﬁeld is still given by the elastic stress concentration tensor, σ = ± B : 1, where + > 0 (resp. − < 0) denotes the macroscopic elastic threshold in tension (resp. compression). Multiplying (9.116) by ± yields: ∂f s ∂f s ∂f s : B : 1 = (1 − β) : σ = ± :1 ∂σ ∂σ ∂σ It is readily seen from (9.117) that: ∂f s : σ = J 2 − σm g ′ (σm ) = g(σm ) − σm g ′ (σm ) (∀ z ∈ spl ) ∂σ so that (9.120) is: (∀ z ∈ spl ) (1 − β) g(σm ) − σm g ′ (σm ) = − ± g ′ (σm ) (∀ z ∈ spl )

(1 − β) ±

(9.120)

(9.121)

(9.122)

This result is now implemented for a Drucker–Prager solid. With the notations h and α introduced in (9.119), (9.122) becomes: β =1−

± h

(9.123)

Relation (9.123) yields different values for β depending on the loading path (traction vs. compression), which contradicts the assumption that the macroscopic yield criterion is a function of an effective stress of the form Σ + β P1 , except for β = 1. More generally, we note that:8 (∀ σm )

g(σm ) − σm g ′ (σm ) ≥ g(0) > 0

(9.124)

8 Just consider the function γ (σ ) = g(σ ) − g(0) − σ g ′ (σ ). Clearly, it is found that γ ′ (σ ) = −σ g ′′ (σ ) m m m m m m m and γ (0) = 0. According to the concavity of g(σm ) (g ′′ (σm ) ≤ 0), it follows that γ (σm ) ≥ γ (0) = 0.

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where g(0) > 0 can be associated with the cohesion of the solid. Since + and − have opposite signs, and recalling that g ′ (σm ) is negative, we observe from (9.122) that two different values for β are found for an isotropic compression path (β > 1) and for an isotropic tension path (β < 1). This cannot be so, if the macroscopic yield criterion is a unique function of an effective plastic stress of the form Σ + β P1. Hence, it clearly appears that this stress concept is not relevant to the class of materials described by (9.117). It has sometimes been proposed9 that the plastic porosity change and the macroscopic plastic strain could be related by a phenomenological rule of the form: δ φ˙ pl = βtr E˙ pl

(9.125)

If the normality rule is valid at the microscopic scale, we know from Section 9.4.1 that the macroscopic ﬂow rule is associated in the sense of (9.98). Combining (9.98) with (9.125) yields: ∂F ∂F (Σ, P, σ R ) = (Σ, P, σ R ) (9.126) ∂Σ ∂P We recognize that (9.126) is actually equivalent to stating that the macroscopic yield criterion depends on the effective stress Σ + β P1. Indeed, it is sufﬁcient to verify that (9.126) implies the following result: βtr

∂F ˙ ∂F ˙ + β P1 ˙ =0 ⇒ P˙ = 0 (9.127) Σ :Σ+ ∂Σ ∂P Thus, by way of conclusion, a phenomenological rule of the form (9.125) with β = 1 cannot be applied to materials described by (9.117). In contrast, β = 1 in (9.125) implies that the macroscopic rule depends on the Terzaghi effective stress Σ + P1 which in turn implies that tr ∂ f /∂σ = 0 and tr ε pl = 0. In other words, β = 1 is related to the plastic incompressibility of the solid phase. This is consistent with the link previously derived between δφ pl , tr ε pl and E pl (see (9.65)). 9 See

for instance[12].

10 Microporofracture and Damage Mechanics This chapter investigates the linear and nonlinear poroelastic behavior of cracked porous media. We start with some elements of linear elastic fracture mechanics (LEFM) applied to an rev whose solid is subjected to a uniform strain boundary condition and a ﬂuid pressure. The analysis shows that the key to the description of crack propagation is an appropriate representation of the overall drained elasticity of cracked porous media, which allows us to determine the energy release rate and to derive fracture propagation criteria. This is achieved by means of an Eshelby-type continuum micromechanics representation of cracks in saturated porous media. It is on this basis that we identify the crack density parameter as the governing state variable for the description of damage propagation. Several geometrical conﬁgurations of cracks in porous media are considered (parallel cracks, randomly oriented cracks and anisotropic crack orientation). In analogy to LEFM, strain-based and stress-based damage propagation criteria are derived for different homogenization schemes. By way of application, the criteria are tested for the undrained situation.

10.1 Elements of Linear Fracture Mechanics Consider a three-dimensional material system made up of a linear elastic material and a single plane crack. C ± denote the upper and lower lips of the crack C. N is the unit normal to C ± oriented from C − toward C + (Figure 10.1). The crack of initial (resp. current) volume V0 (resp. V) is saturated by a ﬂuid at pressure P. The displacement is prescribed on the external boundary ∂. In order to scale the magnitude of this prescribed displacement, a scalar loading

Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

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292

∂Ω N C+ C–

P

Figure 10.1 Elastic structure with single pressurized crack

parameter E(t) is introduced: (∀z ∈ ∂) ξ (z, t) = E(t)ξ o (z)

(10.1)

where ξ o (z) is a time-independent function deﬁned on ∂. ˙ Elasticity is based on the assumption that the rate of mechanical work W provided to is entirely stored in the form of elastic energy density |0 | ˙ in the solid, where |0 | denotes the volume of the considered structure. By contrast, when the work cannot be stored entirely in the solid system, irreversible deformation takes place, dissipating a part of the supplied work into the form of heat. This is expressed by the Clausius–Duhem inequality, which for isothermal evolutions reads: ˙ − |0 | ˙ ≥ 0 D˙ = W

(10.2)

˙ is the sum of the contributions of (1) the surface forces T = σ · n The work W acting on ∂ in the velocity ξ˙ , and (2) the ﬂuid pressure P acting on the crack lips C ± : ˙ ˙ W= (10.3) T · ξ d Sz + P[ξ˙ ] · N d Sz C

∂

+ − where [ξ˙ ] = ξ˙ − ξ˙ is the velocity jump across C: that is, the crack opening vector. Substitution of (10.1) in (10.3) yields: ˙ ˙ (10.4) T · ξ o d Sz + P V˙ W=E ∂

where V˙ is the rate of the crack volume change. Equivalently, introducing the potential energy density ∗ = − P(V − V0 )/|0 |, the dissipation rate

Elements of Linear Fracture Mechanics

becomes: D˙ = E˙

∂

293

T · ξ o d Sz − P˙ (V − V0 ) − ˙ ∗ |0 | ≥ 0

(10.5)

Among all kinematically admissible (k.a.) displacement ﬁelds ξ ′ that are compatible with the boundary condition (10.1), the displacement ﬁeld ξ on which characterizes the response of the cracked structure for the loading parameters E and P and for the geometry deﬁned by the crack surface ℓ minimizes the potential energy: 1 1 ′ s ′ ′ ∗ inf ε : C : ε d Vz − P(V − V0 ) (10.6) = |0 | ξ ′ k.a.E 2 (ℓ) in which ε′ is the strain ﬁeld associated with ξ ′ , and V ′ is the corresponding crack volume change. From (10.6), it is apparent that ∗ depends on E and P, as well as on ℓ; that is, formally: ∗ = ∗ (E, P, ℓ)

(10.7)

In the case of reversible evolutions, D˙ = 0, ℓ is a constant which intervenes only as a ﬁxed geometry parameter in the evaluation of ∗ , but which does not affect the change of the potential energy ˙ ∗ . By contrast, when the crack propagates, creating in the course of this process additional crack the surfaces, ˙ ˙ potential energy changes not only with the incremental loading E, P , but also due to the increase of fracture surface dℓ. Hence, applying (10.7) to (10.5): ∂ ∗ ∂ ∗ ˙ V − V0 ∂ ∗ 1 D˙ o ˙ ˙ −P − ℓ≥0 =E + T · ξ d Sz − |0 | |0 | ∂ ∂E |0 | ∂P ∂ℓ (10.8) ˙ P˙ , in which the crack does not propagate In an incremental loading E, (ℓ˙ = 0), the response of the system is elastic, D˙ = 0, and (10.8) yields the global response of the structure: 1 ∂ ∗ T · ξ o d Sz = |0 | ∂ ∂E (10.9) V − V0 ∂ ∗ =− |0 | ∂P

It is then readily recognized from (10.8) that fracture propagation, ℓ˙ > 0, dissipates energy through the creation of additional crack surfaces, and that the driving force of this process is the energy release rate G: ˙ D˙ = G ℓ;

G = −|0 |

∂ ∗ ≥0 ∂ℓ

(10.10)

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A fracture propagates when the energy release rate G reaches a threshold, the fracture energy G f , which is a material property characterizing the solid energy per unit surface that is irreversibly dissipated upon fracture propagation: G (ℓ) − G f ≤ 0; ℓ˙ ≥ 0; G − G f ℓ˙ = 0 (10.11)

and:

ℓ˙ > 0

⇒

D˙ = G f ℓ˙

(10.12)

If – due to incremental fracture propagation ℓ → ℓ + dℓ – the energy release decreases from G(ℓ) = G f to G(ℓ + dℓ) < G(ℓ), the crack propagation will stop (see (10.11)). This corresponds to stable fracture propagation: ∂ 2 ∗ ∂G = −|0 | < 0 ∂ℓ ∂ℓ2

(10.13)

In the case G(ℓ + dℓ) > G(ℓ), the fracture process is said to be unstable. Making use of (10.6), it is readily seen that: ∂ (V − V0 ) 1 ∂ ∗ ∂T o G = −|0 | P |d P=0 = −E · ξ d Sz (10.14) ∂ℓ dE=0 2 ∂ℓ ∂ ∂ℓ The previous relations are the classical equations of linear elastic fracture mechanics (LEFM), here extended to a pressurized crack. It becomes apparent that the key to LEFM analysis is to evaluate the energy release rate, respectively the potential energy of , in the presence of a propagating crack. In fact, once the expression for ∗ (E, P, ℓ) is established, a straightforward application of (10.10) allows the derivation of the energy release rate. This reduces the analysis in essence to the study of the elastic response of the cracked structure. Although the analysis established here is for a structure with a single crack C, it is interesting to make a formal analogy to homogenization theory. In particular, the uniform strain boundary conditions (4.40) can be seen as a particular case of the boundary condition (10.1). In a given cartesian orthonormal frame (e 1 , e 2 , e 3 ), it corresponds to E → E i j and ξ o → z j e i . Then, taking advantage of the identity: z j e i · σ · n d Sz = |0 |σi j (10.15) ∂

and introducing the normalized crack volume φ = V/|0 |, Equations (10.9), which characterize the global response of the structure, become: ∂ ∗ ∂E ∂ ∗ φ − φ0 = − ∂P σ =

(10.16)

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where σ is the average of the stress ﬁeld over the structure. While the structure with a single crack C is not the representative elementary volume of a porous material, the formal identity of (10.17) with the state equations (5.58) of poroelasticity should be noted. Under such controlled deformation and ﬂuid pressure boundary conditions the energy release can be directly derived from (10.14): ∂ ∗ ∂ (φ − φ0 ) ∂ 1 G = −|0 | (10.17) |d P=0 = |0 | P −E: ∂ℓ dE=0 2 ∂ℓ ∂ℓ Similarly, due to the linearity of the solid behavior, ∗ (E, P, ℓ) is expected to be a quadratic function of E and P, and the potential energy density can be put in a form similar to (5.59): P2 1 − P Bstr (ℓ) : E ∗ = E : Cstr (ℓ) : E − 2 2Nstr (ℓ)

(10.18)

where Cstr (ℓ), Bstr (ℓ), Nstr (ℓ) are the ‘poroelastic’ constants characterizing the cracked structure (the superscript str stands for ‘structure’). The purpose of this chapter is to investigate the macroscopic response of a representative elementary volume of a saturated cracked material, which comprises not only a single crack but also a disordered distribution of cracks. The next two sections deal with the linear poroelasticity of such cracked media within the context of continuum micromechanics, and seek expressions for ∗ (E, P, ℓ) generalizing the form (10.18). The sought expressions will form the basis for later developments of crack propagation criteria in saturated porous media.

10.2 Dilute Estimates of Linear Poroelastic Properties of Cracked Media A convenient way to represent cracks is in the form of oblate spheroids. For a crack Ci , we introduce an orthonormal frame (Ci , t i1 , t i2 , ni ), in which ni denotes the unit normal to the crack plane and Ci the center of the crack. This oblate spheroid occupies a domain deﬁned by: z12 + z22 z32 + ≤1 a2 c2

(10.19)

with z1 = z · t i1 , z2 = z · t i2 and z3 = z · ni ; a is the crack radius and c the halfopening. The aspect ratio X = c/a of such a penny-shaped microcrack is subject to the condition X ≪ 1. We employ a continuum micromechanics approach in which a crack represents an inhomogeneity with an elastic stiffness tensor Cc . Cc = 0 corresponds to an open crack, which ensures that the stress

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vector on the crack surface is zero. In turn, Cc = 0 can be used to represent a closed crack. In this section, the pore space is assumed to be made up of cracks alone. In a ﬁrst approach, we will employ a dilute scheme to derive estimates of the homogenized poroelastic properties of cracked media. This amounts to assuming that there is no interaction between cracks. 10.2.1 Open Parallel Cracks The simplest case of a cracked medium corresponds to the one with parallel cracks deﬁned by the same radius and crack aspect ratio. The determination of Chom (and subsequently of the Biot tensor B) requires an estimate of the c average strain concentration tensor for cracks, denoted by A : c

Chom = Cs : (I − ϕ c A )

(10.20)

where ϕ c represents the volume fraction of cracks present in the medium, which for oblate spheroids of radius a and aspect ratio X is: 4 4 ϕ c = πN a 3 X = π ǫ X (10.21) 3 3 where N denotes the crack density (number of cracks per unit volume) of the considered set of parallel cracks, and ǫ = N a 3 is the crack density parameter.1 In line with the Eshelby-based estimates (see Chapter 6, Section 6.3), c application of (6.114) yields the dilute estimate of A : c

A = (I − S(X, n))−1

(10.22)

where S(X, n) is the Eshelby tensor associated with the considered crack family, which depends on the aspect ratio X of the oblate spheroid and its orientation n. As we have seen in Section 6.2.5, the Eshelby tensor S(X, n) depends also on the elasticity of the uncracked solid phase: S(X, n) = P(X, n) : Cs

(10.23)

where P(X, n) is deﬁned by (6.69) and (6.80). Finally, a combination of (10.20), (10.21) and (10.22) yields the following dilute estimate of the overall stiffness tensor: 4 Chom = Cs : I − π ǫ X(I − S(X, n))−1 (10.24) 3 The dependence of the homogenized stiffness tensor (10.24) on the current value of the aspect ratio X may seem puzzling at ﬁrst sight. Indeed, since the crack aspect ratio X is expected to vary signiﬁcantly as the crack propagates, and hence as a function of the applied loading, this dependency could appear 1 This

parameter was introduced in [9].

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297

to be in contradiction with the assumption of linearity which is at the very heart of the use of the strain concentration tensor in the homogenization procedure. However, a closer look at the mathematical properties of the quantities involved in expression (10.24) for Chom partly resolves this contradiction. In fact, for X ≪ 1, which corresponds to the geometrical model of penny-shaped microcracks, (I − S(X, n))−1 is singular, of the order of 1/ X. In turn, this implies that X(I − S(X, n))−1 has a ﬁnite limit T when X → 0. A constant value of Chom is therefore obtained, as long as the crack remains open (X > 0): 4 hom s (10.25) C = C : I − π ǫ T(n) 3 where: T(n) = lim X(I − S(X, n))−1 = lim X(I − P(X, n) : Cs )−1 X→0

X→0

(10.26)

On the other hand, one may object to this reasoning that the strain concenc tration tensor A given by (10.22) itself depends on the aspect ratio X, and hence via X on the applied loading. In other words, the problem appears to be geometrically nonlinear. To resolve the problem related to geometry changes, we consider a loading increment deﬁned by E˙ on the current conﬁguration. Due to the linearity of the relation between the microscopic strain rate ε˙ and ˙ we introduce the strain rate concentration tensor: E, ˙ = A(z) : E˙ (∀ z ∈ ) ε(z)

(10.27)

On this basis, it is then readily seen that the macroscopic stress rate and strain rate tensors are linearly related: ˙ = Cthom : E˙ Σ

c

with Cthom = C : A = Cs : (I − ϕ c A )

(10.28)

The resulting expression of the homogenized stiffness tensor strictly has the same form as (10.20), but Cthom is now the tangent macroscopic stiffness tensor, which is indicated by subscript t, and which depends on the current conﬁguration: ϕ c is the current volume fraction of the cracks. The average strain rate c concentration tensor At is estimated by: c

At = (I − S(X, n))−1

(10.29)

which is formally identical to (10.22), but which now depends on the current crack aspect ratio X and orientation n. Finally, using (10.21) and taking advantage of (10.26), it is readily seen that Cthom is constant, independent of X, and equal to the effective stiffness tensor Chom given by (10.25): 4 hom s Ct = C : I − π ǫ T(n) (10.30) 3

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Equation (10.30) proves that the macroscopic state equation is indeed linear, despite variations of the crack aspect ratio and of the non linearity of the strain concentration rule. The only condition for the linearity of the macroscopic behavior is that the cracks remain open.2 In what follows, the homogenized properties of cracked media are related to the tangent state equations. For simplicity, the subscript t will be omitted. In the case of an isotropic solid, (10.30) deﬁnes the elasticity of a transversely isotropic macroscopic behavior. The components of the Eshelby tensor of an oblate spheroid can be found in handbooks.3 For X ≪ 1 (penny-shaped cracks), it is sufﬁcient to expand the components Si jkl to ﬁrst order in X in the formula given in Section 6.4.1. In the local orthonormal frame (e 1 , e 2 , e 3 ) with the unit normal of the crack plane n = e 3 , we obtain the following non zero components of Si jkl : S1111 = S2222 =

13 − 8ν s π X; 32(1 − ν s )

S3333 = 1 −

1 − 2ν s π X 1 − νs 4

8ν s − 1 2ν s − 1 π X; S = S = πX 1133 2233 32(1 − ν s ) 8(1 − ν s ) 4ν s + 1 7 − 8ν s νs 1 − π X ; S = πX S3311 = S3322 = 1212 1 − νs 8ν s 32(1 − ν s ) νs − 2 π 1 1+ X S1313 = S2323 = 2 1 − νs 4 S1122 = S2211 =

(10.31)

(with Si jkl = S jikl = Si jlk ). ν s is the Poisson’s ratio of the elastic solid matrix. Note that the Eshelby tensor is non-symmetric. The non-zero components of tensor T are: T3311 = T3322 =

4ν s (1 − ν s ) ; (1 − 2ν s )π

T1313 = T2323 =

T3333 =

2(1 − ν s ) (2 − ν s )π

4(1 − ν s )2 ; (1 − 2ν s )π (10.32)

with Ti jkl = Tjikl = Ti jlk . This yields the following intrinsic expression for T(n) (see Section 10.6): 4(1 − ν s ) νs T(n) = n⊗n⊗1 π 1 − 2ν s νs 1 + (1⊗n ⊗ n + n ⊗ n⊗1) − n ⊗ n ⊗ n ⊗ n (10.33) 2 − νs 2 − νs 2 Crack closure and a quantitative description of the evolution of the crack aspect ratio can be found in [16] and [19], and will be treated in Section 10.2.4. 3 See [33],[44],[42].

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where the tensor operation a⊗b between two second-order tensors, a and b, stands for: 1 (10.34) a ik b jl + a il b jk a⊗b i jkl = 2 Next, the combination of (10.25) and (10.33) yields the following expression for the homogenized stiffness tensor Chom : Chom = Cs − a 1 ǫ 2a 2 1 ⊗ 1 + a 3 1 ⊗ (n ⊗ n) + (n ⊗ n) ⊗ 1 (10.35) +1⊗(n ⊗ n) + (n ⊗ n)⊗1 − ν s (n ⊗ n) ⊗ (n ⊗ n)

Coefﬁcients a i depend only on the Poisson’s ratio of the solid matrix, except a 1 which depends also on the Young’s modulus E s : a1 =

16E s (1 − ν s ) (2 − ν s )ν s 2 (2 − ν s )ν s ; a = ; a = 2 3 3(2 − ν s )(1 + ν s ) 2(1 − 2ν s )2 (1 − 2ν s )

(10.36)

c

According to deﬁnitions (10.22) and (10.26) of A and T, we note that the c property Ti jkl = 0 implies that Ai jkl = O(1/ X). Returning to (10.27), this means that the shear strain rates ε˙ 13 and ε˙ 23 as well as the normal strain rate ε˙ 33 are ˙ X ≫ |E|. ˙ In other words, a crack is responsible for shear and of order |E|/ normal strain ampliﬁcation. Such ampliﬁcation effects may be induced by macroscopic stretch rates E˙ 11 , E˙ 22 and E˙ 33 , or by macroscopic shear strain rates E˙ 13 and E˙ 23 . Finally, the dilute estimates of the Biot tensor and Biot modulus are readily derived from (6.127): 1 4π ǫ 16(1 − ν s 2 ) 16(1 − ν s ) s s ⊗ n + ν 1 ; B= )n ǫ 1:T= ǫ (1 − 2ν = 3 3(1 − 2ν s ) N 3E s (10.37) where E s is the Young’s modulus of the solid. We note that B is anisotropic and that this anisotropy induces an anisotropic contribution of the pore ﬂuid pressure to the macroscopic stress state.4 Note that the ratio b 33 /b 11 is equal to ν s /(1 − ν s ). This result is consistent with (6.172) in the domain X ≪ 1. 10.2.2 Randomly (Isotropic) Oriented Open Cracks We consider a random distribution of the crack orientations within the solid phase. To simplify the presentation, we assume that all cracks have the same radius. The crack network is divided into crack families corresponding to 4 A micromechanical analysis of anisotropic poroelastic properties can also be found in [49]. See also [11] in the context of rock mechanics.

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speciﬁc orientations. N again denotes the crack density. Starting from (6.119), and transforming the sum into an integral over the spherical angular coordinates θ ∈ [0, π ] and ψ ∈ [0, 2π] deﬁning the orientation of the unit normal n, yields: π 2π −1 sin θdψ 4πa 3 Chom = Cs : I − N dθ (10.38) X I − S(X, θ, ψ) 3 4π 0 0

Considering the limit5 of (10.38) for X → 0, it is found that the determination of Chom reduces to evaluating the average T of T(θ, ψ) over crack orientations: 4 hom s (10.39) C = C : I − π ǫT 3

where: T =

π

dθ 0

2π

T(θ, ψ) 0

sin θdψ 4π

(10.40)

For deriving T from (10.33), relations (6.46) and (6.93) are useful. It is convenient to introduce the (isotropic) fourth-order tensor Q deﬁned by: Q=

4π T = Q1 J + Q2 K 3

(10.41)

with: Q1 =

16 1 − ν s 2 ; 9 1 − 2ν s

Q2 =

32 (1 − ν s )(5 − ν s ) 45 2 − νs

(10.42)

Equation (10.39) together with (10.41) and (10.42) yields: Chom = Cs : (I − ǫ Q);

B = ǫ1 : Q = b1 with

b = ǫ Q1 =

16 1 − ν s 2 ǫ 9 1 − 2ν s

(10.43)

10.2.3 Anisotropic Distribution of Open Cracks A generalization of the case of randomly oriented cracks is obtained by considering the density function f (θ, ψ) of the distribution of crack orientation: f (θ, ψ) sin θ dθ dψ/4π represents the number of cracks per unit volume for which the angular coordinates belong to [θ, θ + dθ ] × [ψ, ψ + dψ]. The density function f (θ, ψ) meets the condition: dS N = (10.44) f (θ, ψ) 4π |n|=1 5 In

fact, it is not necessary for the aspect ratios X to be identical for all cracks, provided that X ≪ 1.

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In this case, relation (10.38) is: −1 d S 4πa 3 f (θ, ψ) Chom = Cs : I − X I − S(X, θ, ψ) 3 4π |n|=1

(10.45)

(10.46)

Its limit X → 0 becomes: C

hom

s

=C : I−

|n|=1

4π 3 dS a f (θ, ψ)T(θ, ψ) 3 4π

One interesting observation is that the micromechanical parameter ǫ = f (θ, ψ)a 3 controls the inﬂuence of the defects on the overall elasticity. More precisely, (10.46) implicitly assumes that the crack radius is either a constant or a function of θ and ψ. This suggests replacing (10.39) by: 4 dS 4π s hom s = C : I − π ǫ T (10.47) ǫ(θ, ψ)T(θ, ψ) C =C : I− 3 |n|=1 4π 3 where ǫ(θ, ψ) = ǫ(n) represents the crack density parameter in the direction of n.6 10.2.4 Effect of Total Crack Closure on the Overall Stiffness We now consider the case of a solid matrix containing a set of closed cracks with unit normal n. It is assumed that the contact between the two crack lips is frictionless. In contrast to open cracks, we must now take into account the fact that a compressive normal stress can be transmitted through the contact of the crack lips, while the shear stress in the crack plane remains zero. Adopting a continuum micromechanics approach, we represent the closed crack by a ﬂat ellipsoidal inhomogeneity ﬁlled with an isotropic ﬁctitious material having a zero shear modulus, μc = 0, and a non-zero bulk modulus, k c = 0. Hence: Cc = 3k c J

(10.48)

It appears natural to choose for k c a stiffness value related to the solid phase stiffness, for instance k c = k s in the case of an isotropic solid. However, it can be shown that the value of k c does not affect the homogenized properties of the composite made of the solid and the ﬁctitious inhomogeneities, provided that k c = 0. This is due to the fact that the volume fraction ϕ cf and the corresponding aspect ratio X f of the ﬁctitious inhomogeneities tend asymptotically towards zero, and therefore do not appear in the expression for the homogenized properties.7 6 A statistical description of the radius distribution can be found in [19]. Representation of the directional distribution of cracks by means of a second-order tensor can be found in [40] and [34]. This provides a link with the description of damage by macroscopic tensorial variables [43],[21]. 7 A validation of the concept of ﬁctitious material by means of a fracture mechanics-based approach can be found in [17].

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To illustrate the effect of crack closure, we consider an elastic solid matrix and parallel closed cracks. The homogenized stiffness is: s

c

Chom = C : A = (1 − ϕ cf )Cs : A + ϕ cf Cc : A = Cs + ϕ cf (Cc − Cs ) : A

c

(10.49)

where Cc is given by (10.48). Since the value k c = 0 does not affect the homogenization result, we let Cc = 3k s J, in which case we obtain: c

Chom = Cs : (I − ϕ cf K : A ) with K = I − J

(10.50)

Finally, employing a dilute scheme, an estimate of the strain concentration c tensor A is derived from (6.104): −1 c A = I + P : (C c − C s ) = (I − S(X f , n) : K)−1 (10.51)

where S(X f , n) is the Eshelby tensor for the considered set of inhomogeneities (see (10.31)). The dilute estimate of Chom obtained from (10.50) and (10.51) then becomes: 4 (10.52) Chom = Cs : I − π ǫ X f K : (I − S(X f , n) : K)−1 3 For ﬂat ellipsoids (X f ≪ 1), K : (I − S(X f , n) : K)−1 is found to be of order O(1/ X f ). Accordingly, the quantity X f K : (I − S(X f , n) : K)−1 can be replaced by its limit T′ when X f → 0. The expression for Chom for a set of parallel pennyshaped closed cracks is then: 4 ′ hom s with T′ = lim X f K : (I − S(X f , n) : K)−1 (10.53) C = C : I − π ǫT X f →0 3 A comparison of (10.53) and (10.25) shows that a total crack closure is captured by replacing T in (10.25) with T′ deﬁned in (10.53). For parallel cracks normal to n = e 3 , we obtain the following non-zero components of tensor T′ : ′ ′ T1313 = T2323 =

2(1 − ν s ) (2 − ν s )π

(10.54)

′ = Ti′jlk ). For parallel closed cracks with normal n, T′ is: (with Ti′jkl = Tjikl 4(1 − ν s ) T′ (n) = ⊗ n) + (n ⊗ n)⊗1 − 2(n ⊗ n) ⊗ (n ⊗ n) (10.55) 1⊗(n π(2 − ν s )

The corresponding expression for Chom derived from (10.53) is: Chom = Cs − a 1 ǫ 1⊗(n ⊗ n) + (n ⊗ n)⊗1 − 2(n ⊗ n) ⊗ (n ⊗ n)

(10.56)

where the coefﬁcient a 1 is given in (10.36). In contrast to open cracks, closed hom hom and C2323 : cracks normal to n = e 3 only affect the shear moduli C1313 16(1 − ν s ) hom hom C1313 = C2323 = μs 1 − ǫ (10.57) 3(2 − ν s )

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′ ′ However, since T1313 = T1313 and T2323 = T2323 , these shear moduli are identical to those obtained for open cracks. For materials weakened by parallel closed cracks, B is obtained by replacing T in (10.37) by expression (10.55) for tensor T′ . This leads to B = 4π3 ǫ 1 : T′ = 0. Consequently, it appears that the ﬂuid pressure has no effect on the poroelastic behavior when cracks are closed. Wondering about the role of the ﬂuid in a closed crack may seem surprising. In fact, at a microscopic scale, the crack may be closed in a mechanical sense, while some ﬂuid remains trapped between the two perfectly smooth crack lips. Therefore, a residual pore space may exist even if the lips are in contact. The approach also holds for more complicated situations, such as randomly oriented closed cracks requiring an averaging process on crack orientations. In this case, it sufﬁces to replace T in (10.39) by T′ . Starting from (10.55) and using (6.46) and (6.93), we obtain: 32 (1 − ν s ) hom s s ǫ K (10.58) C = 3k J + 2μ 1 − 15 (2 − ν s )

As expected, only the homogenized shear modulus is affected by crack closure. Compared to (10.43), this effect is different from the one produced by randomly oriented open cracks. Finally, without difﬁculty we verify that a combination of (10.58) and (5.85) yields B = 0. 10.3 Mori–Tanaka Estimates of Linear Poroelastic Properties of Cracked Media The dilute scheme neglects any interaction between cracks present in the rev. In a reﬁned approach, such interactions can be captured by the Mori–Tanaka scheme (see Section 6.3.4), and the estimate (6.140) which we recall: −1 (10.59) Chom = (1 − ϕ)Cs : (1 − ϕ)I + ϕ (I − S)−1 where ϕ denotes the current porosity.

10.3.1 Open Parallel Cracks When the aspect ratio X of the cracks tends to zero, the crack porosity ϕ = 4π ǫ X/3 tends to zero as well, whereas ϕ(I − S)−1 tends to 4π T/3 (see (10.26)). Accordingly, the Mori–Tanaka estimate of the effective stiffness takes the form: −1 4 hom s (10.60) C = C : I + π ǫT 3

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The corresponding homogenized compliance tensor is: 4 hom S = I + π ǫ T : Ss 3

(10.61)

that is: Shom = Ss +

16(1 − ν s 2 ) s ⊗ n) + (n ⊗ n)⊗1 − ν n ⊗ n ⊗ n ⊗ n ǫ 1⊗(n (10.62) 3(2 − ν s )E s

The compliance turns out to be a linear function of the crack density parameter ǫ. Interestingly, despite the fact that ϕ → 0 as X → 0, the Mori–Tanaka estimate is different from the dilute estimate (10.30). This emphasizes that the appropriate measurement of damage induced by cracks is the crack density parameter ǫ and not the crack porosity. Indeed, the Mori–Tanaka and the dilute estimates are only equivalent when ǫ ≪ 1. In other words, the fact that the crack porosity is inﬁnitesimal is not sufﬁcient for the dilute scheme to be valid. 10.3.2 Closed Parallel Cracks The homogenized stiffness of an elastic medium containing closed cracks is given by (10.50). Using a similar reasoning as in Section 6.3.4, the Mori–Tanaka estimate of Ac becomes: −1 Ac = (I − S : K)−1 : (1 − ϕ cf )I + ϕ cf (I − S : K)−1 (10.63)

Recalling that ϕ cf = 4π ǫ X f /3 and considering the limit when X f goes to zero, we obtain: −1 4 4 ′ ′ c c ϕ f A = π ǫT : I + π ǫT (10.64) 3 3

where T′ is the tensor introduced in (10.53). Using (10.50), the effective stiffness tensor and compliance tensor take the form: −1 4 4 hom s hom ′ ′ C = C : I + π ǫT (10.65) ; S = I + π ǫ T : Ss 3 3 Hence, in order to derive the stiffness properties for closed cracks from the solution of open cracks, it sufﬁces to replace T by T′ in (10.60) and (10.61). 10.3.3 Randomly Oriented Interacting Cracks In the case of randomly oriented open cracks, the homogenized stiffness tensor Chom is obtained from (6.144). When the aspect ratio of each crack tends to zero, ϕ tends to zero as well, and the limit of ϕ j (I − S j )−1 is equal to ǫ Q, where Q

Mori–Tanaka Estimates of Linear Poroelastic Properties of Cracked Media

305

is the tensor deﬁned by (10.41). This yields the following expression for Chom and B: Chom = Cs : (I + ǫ Q)−1 ; B = 1 : I − (I + ǫ Q)−1 (10.66) or equivalently:

Chom =

2μs 3k s J+ K; 1 + ǫ Q1 1 + ǫ Q2

B=ǫ

Q1 1 1 + ǫ Q1

(10.67)

A similar reasoning can be implemented for closed cracks as well and yields: Chom = 3k s J +

2μs 1+

32 (1−ν s ) ǫ 15 (2−ν s )

K;

B=0

(10.68)

10.3.4 Double-Porosity Model of Cracked Porous Media To this point, we have considered that the cracks form the entire porosity. We now examine the situation where the pore space is composed of connected cracks and pores of similar size (no scale separation between pores and cracks)8 and investigate their inﬂuence on the effective behavior using the Mori–Tanaka scheme. For analytical purposes, we assume that the cracks are open. The total porosity in the current conﬁguration is divided into the pore volume fraction ϕ p and the crack volume fraction ϕ c . We note that ϕ c ≪ 1 so that ϕ p ≈ ϕ. Furthermore, in contrast to changes in ϕ c , the change in pore volume fraction ϕ p is negligible. For simplicity, we assume that the pore shape is spherical and we recall the corresponding Eshelby tensor Ssph (see (6.95)): Ssph = α s J + β s K

with α s =

Analogous to (6.144), we have: C

hom

s

3k s 6 k s + 2μs s ; β = 3k s + 4μs 5 3k s + 4μs

−1

= (1 − ϕ)C : (1 − ϕ)I + ϕ(I − Ssph )

+

i∈C

ϕic (I

−1

− Si )

−1

(10.69)

(10.70)

where C formally denotes the set of cracks. We then consider the limit of (10.70) for X → 0:

−1 4π ǫ i Ti (10.71) Chom = (1 − ϕ)Cs : (1 − ϕ)I + ϕ(I − Ssph )−1 + 3 i∈C 8 Otherwise, the effects of the cracks and those of the pores can be addressed in distinct homogenization steps. See Section 5.6.

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from which the homogenized compliance tensor is obtained:

4π ǫ i 1 hom −1 i S = (1 − ϕ)I + ϕ(I − Ssph ) + T : Ss 1−ϕ 3 i∈C

(10.72)

As in Sections 10.2, 10.3.1, 10.3.2 and 10.3.3, the next step requires us to specify the distribution of the crack orientation. In the case of parallel cracks, we have: −1 4π ǫ −1 hom s T (10.73) C = (1 − ϕ)C : (1 − ϕ)I + ϕ(I − Ssph ) + 3

Similarly, application of the Mori–Tanaka estimate for the case of spherical pores and randomly oriented cracks yields: −1 Chom = (1 − ϕ)Cs : (1 − ϕ)I + ϕ(I − Ssph )−1 + ǫ Q (10.74) 10.4 Micromechanics of Damage Propagation in Saturated Media Among the two dimensionless parameters that are candidates for characterizing the crack network, namely the crack volume fraction ϕ c = 43 π ǫ X and the crack density parameter ǫ = N a 3 , it is the crack density parameter that is the most relevant one as regards the elasticity and hence the potential energy of cracked porous media (10.18). It is therefore natural to consider ǫ – instead of the crack surface ℓ – as the state variable governing the effect of fracture propagation, and derive constitutive laws for damage propagation in porous media in analogy to LEFM as seen in section 10.1. This is the focus of this section. 10.4.1 LEFM–Damage Analogy To make the analogy to LEFM, consider one single family of identical parallel cracks9 (surface ℓ = πa 2 , density N ) in an rev. In this case, ǫ = N (ℓ/π )3/2 ; hence, assuming the crack density N is constant: ǫ 2/3 2π −1/3 ; dℓ = ǫ dǫ (10.75) ℓ=π N 3N 2/3 which shows that ℓ and ǫ are actually equivalent state variables. Hence, as an alternative to (10.7) and (10.18) we can let: 1 P2 ∗ (E, P, ǫ ) = E : Chom (ǫ ) : E − − PB (ǫ ) : E 2 2N (ǫ )

(10.76)

9 The reasoning is also relevant for a family of identical cracks with orientations that are isotropically distributed. However, in this case, an isotropic loading must be considered.

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Inserting the above equation into (10.9) together with (10.17) allows us to identify the state equations of the cracked poroelastic medium: Σ=

∂ ∗ = Chom (ǫ ) : E − B (ǫ ) P ∂E

(10.77)

∂ ∗ P + B (ǫ ) : E = ∂P N (ǫ )

(10.78)

φ − φ0 = −

Note that φ in (10.78) represents the ratio V/|0 |, in which V is the sum of the crack volumes. Relation (10.10) now takes the form: ∂ ∗ ∂ ∗ D˙ =− ℓ˙ = − ǫ˙ ≥ 0 |0 | ∂ℓ ∂ǫ

(10.79)

The driving force of crack propagation (resp. damage propagation) is therefore: ∂ ∗ ∂B 1 1 ∂ Chom P2 ∂ +P =− E: :E+ :E (10.80) Gǫ = − ∂ǫ 2 ∂ǫ 2 ∂ǫ N ∂ǫ Indeed, in analogy to the energy release rate G in (10.10) (of dimension [G] = M/T 2 ) deﬁned by (10.10), which is the driving force of fracture propagation ℓ˙ ≥ 0, we can consider Gǫ (of dimension [Gǫ ] = ML−1 T2 ) to be the driving force of damage propagation associated with an increase of the crack density parameter (ǫ˙ ≥ 0). Furthermore, given this analogy, it is appealing to adopt a criterion of the form (10.11) to describe damage propagation: Gǫ − G c ≤ 0;

ǫ˙ ≥ 0;

(Gǫ − G c ) ǫ˙ = 0

(10.81)

where G c is analogous to the fracture energy G f . On the other hand, in contrast to Section 10.1, we now deal with a distribution of cracks, and not with a single one. The total dissipation is therefore the sum of the contribution of each crack in the rev. With the latter being estimated by (10.12), it is found that: D˙ = N G f ℓ˙ = G c ǫ˙ |0 |

(10.82)

Recalling (10.75) then yields: Gc =

2π Gf 3

N ǫ

1/3

=

2π G f 3 a

(10.83)

Hence, in contrast to the fracture energy, G c is not a scale-independent material property, but depends on the scale of observation and the crack size, and

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hence on the loading history.10 Nonetheless, it is emphasized that (10.82) and (10.83) are based on the assumption that the dissipation occurring during the propagation of one of the cracks can be estimated as if it were alone. In other words, the effect of the interactions between cracks on dissipation is disregarded. Finally, analogous to (10.13), we seek a criterion for stable damage propagation. If the dependence of G c on ǫ is disregarded, the same reasoning as in Section 10.1 shows that stability is ensured if: ∂ 2 ∗ ∂Gǫ (E, P, ǫ) = − (E, P, ǫ) < 0 (10.84) ∂ǫ ∂ǫ 2 This criterion is no longer valid if G c depends on ǫ, as suggested by (10.83), but can be readily generalized with a similar reasoning. More precisely, let us assume that the condition (10.81) for crack propagation is reached for the loading level (E, P) and the crack density parameter ǫ: Gǫ (E, P, ǫ) = G c (ǫ)

(10.85)

The crack propagation is stable if Gǫ (E, P, ǫ + dǫ) < G c (ǫ + dǫ). That is, if: ∂Gǫ (E, P, ǫ) < G ′c (ǫ) ∂ǫ which can also be expressed as:

(10.86)

∂ 2 ∗ (E, P, ǫ) + G ′c (ǫ) > 0 (10.87) ∂ǫ 2 When G c (ǫ) is a decreasing function of ǫ, the above condition turns out to be more restrictive than (10.84). 10.4.2 Extension to Multiple Cracks The previous LEFM–damage analogy holds as well for parallel or randomly oriented open cracks of same crack radius a . A straightforward extension to 10 The onset of damage propagation in a perfectly elastic brittle material can be determined by noting that damage starts to propagate when a macroscopically applied stress state induces a critical stress intensity √ equal to the fracture toughness K I c . In uniaxial tension, K I = K I c = f t πa cr , where f t is the macroscopic tensile strength and a cr is the critical half-length of the crack. Use of a cr in (10.83) yields: a |r e f |ǫcr 1/3 cr G c = G c0 = G c0 a |0 |ǫ

where |r e f | is the reference rev (specimen size) on which the tensile test is performed, and G c0 the damage threshold: G c0 = 2π 2

G f f t2 K I2c

= 2π 2

where we make use of the plane stress LEFM relation K I c =

f t2 E

EG f .

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multiple cracks of different radii and density requires a statistical analysis (see Section 10.2.3). We now consider the situation where the crack network is represented by several crack families, each one being associated with a crack density parameter ǫ i . The radius is assumed to be uniform in each family, but can be different from one family to the next. An immediate extension of the methodology presented in Section 10.4.1 consists of introducing a damage criterion (10.81) for each crack family (note that G ic is a priori different for each crack family because of its dependence on the crack size (see (10.83)): (∀ i) Gǫi (E, P, ǫ 1 , . . . , ǫ n ) − G ic ≤ 0

where Gǫi is the driving force of the damage growth dǫ i ≥ 0: ∂B 1 ∂ Chom P2 ∂ 1 ∂ ∗ i +P i :E :E+ Gǫ = − i = − E : i i ∂ǫ 2 ∂ǫ 2 ∂ǫ N ∂ǫ

(10.88)

(10.89)

We now take advantage of (5.85): ∂ ∂ (B) = − i (Chom ) : Ss : 1 i ∂ǫ ∂ǫ

(10.90)

and of (5.87): ∂ ∂ǫ i

1 N

= −1 : Ss :

∂ (Chom ) : Ss : 1 ∂ǫ i

(10.91)

Incorporating (10.90) and (10.91) into (10.89), the damage criterion associated with crack family i is: 1 ∂ Gǫi = − (E + P Ss : 1) : i (Chom ) : (E + P Ss : 1) ≤ G ic 2 ∂ǫ

(10.92)

Expression (10.92) turns out to be a strain formulation of the damage criterion, in which E + P Ss : 1 appears as an ‘effective strain’ controlling the damage propagation dǫ i ≥ 0. Alternatively, a stress formulation is obtained from a combination of (10.92) with the state equations. Indeed, inserting (5.85) into (5.38) reveals that: Σ + P1 = Chom : (E + P Ss : 1)

(10.93)

Eliminating E + P Ss : 1 between (10.92) and (10.93) yields a stress formulation for the damage criterion: ∂ 1 Gǫi = − (Σ + P1) : Shom : i (Chom ) : Shom : (Σ + P1) ≤ G ic 2 ∂ǫ

(10.94)

It is interesting to note that the damage criterion (10.94) is controlled by Terzaghi’s effective stress Σ + P1. By way of illustration, let us evaluate Gǫi using the Mori–Tanaka estimate (10.71) of the double-porosity model of cracked porous media, which

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combines the effects of spherical pores (volume fraction ϕ p ≈ ϕ) and various crack families. From (10.72), we ﬁrst note that: ∂ hom 4π (S ) = Ti : Ss i ∂ǫ 3(1 − ϕ)

(10.95)

∂ (Chom ) = −Chom : Hi : Chom ∂ǫ i

(10.96)

Using the fact that Shom : Chom = I, it is readily seen that:

with Hi = 4π/(3(1 − ϕ))Ti : Ss . Inserting the above result into (10.94) yields:

1 (∀ i) Gǫi = (Σ + P1) : Hi : (Σ + P1) ≤ G ic (10.97) 2 We observe that the propagation of the ith crack family is described by an elliptical stress criterion. The orientation of the crack family is taken into account through the anisotropic tensor Hi . 10.4.3 The Role of the Homogenization Scheme in the Damage Criterion As we have seen in Sections 10.2 and 10.3, the homogenized poroelastic properties of a cracked medium depend on the choice of the homogenization scheme, i.e. on the estimate of the strain (rate) concentration tensor. This choice should therefore also impact the expression of the damage criterion. For purposes of clarity, we restrict this discussion to the isotropic case: the distribution of cracks is isotropic and the macroscopic strain is of the form E = tr E 1/3. The loading deﬁned by tr E and P is expected to induce an isotropic stress state Σ = tr Σ 1/3. The damage growth is therefore controlled by a single crack density parameter ǫ. We also assume that the pore space is made up of cracks only. Both stability criteria (10.84) (G ′c (ǫ) = 0) and (10.87) (G c (ǫ) being given by (10.83)). will be considered. Dilute Scheme

The dilute estimates of the poroelastic coefﬁcients are given by (10.43) with 1/N = b/k s . Accordingly, the onset of damage is characterized by the following criterion: P 2 k s Q1 tr E + s ≤ Gc (10.98) Gǫ (E, P) = 2 k where Q1 is given by (10.42). Interestingly, from the second poroelastic state equation (10.78), it follows that the quantity ǫ Q1 (tr E + P/k s ) is equal to the

Micromechanics of Damage Propagation in Saturated Media

311

normalized pore volume change φ − φ0 , and (10.98) then becomes: ks Gǫ (φ, ǫ) = 2Q1

φ − φ0 ǫ

2

3 Es = 32(1 − ν s 2 )

φ − φ0 ǫ

2

≤ Gc

(10.99)

Evoking the criterion of stable damage propagation (10.84) which disregards the dependence of G c on ǫ, it appears from (10.99) that ∂Gǫ /∂ǫ (φ, ǫ) < 0. Hence, a loading in which the evolution φ(t) of the pore volume is prescribed appears to induce a stable damage process. It is readily seen that the same result is obtained with the stability criterion (10.87), with G c (ǫ) given by (10.83). These conclusions are no longer valid in the case of a loading deﬁned by arbitrary combinations of E(t) and P(t). In fact, the poroelastic constants (10.43) derived from the dilute scheme depend linearly on ǫ, as does the dilute estimate of the potential energy ∗ (E, P, ǫ ). As a consequence, Gǫ as deﬁned by (10.98) does not depend on ǫ. It follows that ∂Gǫ /∂ǫ E(t), P(t) = 0: the stability of the damage process cannot be ensured. Furthermore, (10.98) states that loading parameters E(t) and P(t) are no longer independent during the damage process, Gǫ = G c . These two observations reveal some serious shortcomings in the use of the dilute scheme for damage prediction. The Mori–Tanaka Scheme

The Mori–Tanaka estimate (10.67) leads to: k hom (ǫ) =

ks ; 1 + ǫ Q1

B(ǫ) = b1 =

ǫ Q1 1; 1 + ǫ Q1

1 b = s N k

(10.100)

The damage criterion (10.92) is then: k s Q1 P 2 Gǫ (E, P, ǫ) = tr E + s ≤ Gc 2(1 + Q1 ǫ)2 k

(10.101)

Let us ﬁrst assume that G c is a constant. From (10.101) and (10.84), it is readily found that the damage process is stable, as ∂Gǫ /∂ǫ < 0 in any loading E(t) and P(t). Furthermore, at the onset of damage propagation, Gǫ = G c , (10.101) allows us to derive the value of the crack density parameter ǫ as a function of the macroscopic strain E and the pore pressure P. This suggests that the Mori– Tanaka scheme removes the shortcomings of the dilute scheme. Finally, the damage criterion (10.101) can also be formulated as a function of the macroscopic stress tensor Σ = tr Σ 1/3: Q1 Gǫ = s 2k

1 tr Σ + P 3

2

≤ Gc

(10.102)

Microporofracture and Damage Mechanics

312 Σ ′m

Dilute Mori-Tanaka PCW

E ′ = tr E +P/k s

Figure 10.2 Effective stress response of a cracked material when Gc is regarded as a constant

From a purely phenomenological point of view, this stress-based formulation of the damage criterion has some similarities to a perfect elastoplastic behavior as long as no unloading is considered (see Figure 10.2). Let us now adopt the stability criterion (10.87) with (10.83). It can be shown that the conclusion of the stability analysis depends on the value of the initial crack density parameter ǫ0 which should be compared to the critical value mt mt ǫcr = 1/(5Q1 ): the damage propagation is only stable if ǫ0 > ǫcr . Furthermore, inserting (10.83) into (10.101) provides the link between ǫ and the ‘effective’ loading parameter E ′ = tr E + P/k s : 1 + ǫ Q1 ǫ0 1/6 (10.103) E ′ = E 0′ 1 + ǫ0 Q1 ǫ

where ǫ0 denotes the crack density parameter at the onset of propagation and E 0′ the corresponding value of the effective loading parameter. mt When the stability condition ǫ > ǫcr is satisﬁed, it is readily seen from (10.103) that ǫ is an increasing function of E ′ . In turn, (10.102) reveals that the mean effective stress m′ = 31 tr Σ + P decreases as the propagation goes on. That is, as ǫ increases: ǫ 1/6 0 m′ = m′0 (10.104) ǫ

where m′0 denotes the value of m′ at the onset of propagation. The typical shape of the stress–strain curve is displayed in Figure 10.4 below. The Ponte-Castaneda and Willis Estimate

An original way of incorporating the spatial distribution of cracks in the micromechanical analysis was proposed by Ponte-Castaneda and Willis (PCW) [45], which consists of separating the effects of inclusion shape and spatial

Micromechanics of Damage Propagation in Saturated Media

313

distribution. In the case of an isotropic distribution of crack orientation and of an isotropic spatial distribution of cracks, this line of reasoning leads to the following expression for the effective bulk modulus: k pcw Q1 ǫ =1− s k 1 + Q′1 ǫ

(10.105)

with: Q′1 = Q1

1 + νs 3(1 − ν s )

(10.106)

From (10.105), estimates of the Biot coefﬁcient b and modulus N are readily derived using (5.85) and (5.87). Along the same line of argument employed before, we determine the driving force of the damage process for this scheme using a strain-based formulation: P 2 (1 − 2ν s )(1 − ν s 2 ) Gǫ = 648k s tr E + ≤ Gc (10.107) (27(1 − 2ν s ) + 16(1 + ν s )2 ǫ)2 ks and a stress-based formulation: (1 − ν s 2 ) Gǫ = 648k (1 − 2ν s )(27 − 32(1 + ν s )ǫ)2 s

1 tr Σ + P 3

2

≤ Gc

(10.108)

If G c is regarded as a constant, it is readily seen from (10.107) that the stability condition (10.84) is satisﬁed, as for the Mori–Tanaka scheme. However, in contrast to the Mori–Tanaka scheme, the effective stress m′ = 13 tr Σ + P predicted by the PCW scheme decreases as cracks propagate. This highlights the sensitivity of the damage criterion w.r.t. the chosen homogenization scheme. Figure 10.2 displays the predictions of the dilute, MT and PCW schemes. With the stability criterion (10.87) and (10.83), the conclusion of the stability analysis is found to depend on the value of the initial crack density parampcw eter ǫ0 which should now be compared to the critical value ǫcr = 1/(5Q′1 ). pcw The damage propagation is only stable if ǫ0 > ǫcr . Figure 10.3 presents the pcw mt variations of ǫcr and ǫcr as functions of the Poisson coefﬁcient ν s . As in pcw the Mori–Tanaka case, when the stability condition ǫ0 > ǫcr is satisﬁed, the crack density parameter proves to be an increasing function of the loading parameter E ′ = tr E + P/k s . We just have to replace Q1 by Q′1 in (10.103): 1 + ǫ Q′1 ǫ0 1/6 (10.109) E ′ = E 0′ 1 + ǫ0 Q′1 ǫ As expected, the effective mean stress m′ is a decreasing function of ǫ as well as of E ′ : m′ ∝

27 − 32(1 + ν s )ǫ ǫ 1/6

(10.110)

Microporofracture and Damage Mechanics

314

0.3

0.25 pcw

ǫcr 0.2

0.15

0.1 mt

ǫ cr

0.05

νs 0

0.1

0.2

0.3

0.4

0.5

Figure 10.3 Effect of the Poisson ratio ν s on ǫcrmt and ǫcrpcw

In contrast to (10.104), (10.110) predicts that the macroscopic effective stress m′ vanishes when the damage parameter ǫ reaches the value 27/(32(1 + ν s )) (see Figure 10.4). However, the latter lies outside the domain of validity of the PCW estimate, which is bounded by ǫ = 3/(4π ). 10.5 Training Set: Damage Propagation in Undrained Conditions It is an instructive exercise to investigate the response of the cracked material under undrained conditions. As discussed in Section 5.3.5, undrained conditions correspond to the absence of ﬂuid mass exchange with the outside. In this case, the ﬂuid pressure P is related to the macroscopic strain E by means of (5.67) and (5.68). Consequently, the macroscopic response is described by (5.69) for which the undrained elastic stiffness tensor is deﬁned by (5.70): Cuhom = Chom + MB ⊗ B

(10.111)

where M is the overall Biot modulus (5.68): 1 ϕ0 1 = + f M N k with k f the ﬂuid bulk modulus.

(10.112)

Training Set: Damage Propagation in Undrained Conditions

315

1

′

Σm ′0

Σm 0.8

MT 0.6

DIL

0.4

P CW 0.2

′

0

1

0.5

1.5

2

2.5

3

′

E /E 0

Figure 10.4 Effective stress response of a cracked material when Gc is given by (10.83) (ǫ = ǫcr , ν s = 1/3)

10.5.1 The Case of an Incompressible Fluid We consider an isotropic solid matrix weakened by a set of open parallel cracks normal to n = e 3 , which are ﬁlled with an incompressible ﬂuid (k f → ∞ and M = N). Irrespective of the homogenization scheme employed, the macroscopic compliance tensor of the cracked material takes the following form in Voigt notation: ⎛ 1 ⎞ νs νs ⎜ Es ⎜ ⎜ νs ⎜− ⎜ Es ⎜ ⎜ s hom ν Sijkl = ⎜ ⎜− E s ⎜ ⎜ ⎜ 0 ⎜ ⎜ 0 ⎝ 0

−

Es

1 Es

−

−

Es

0

0

0

s

ν Es

0

0

0

hom S3333 (ǫ)

0

0

0

0

0

hom (ǫ) 2S3131

0

0

0

0

hom 2S3131 (ǫ)

0

0

0

0

νs Es

−

0 0 1+ν Es

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ s

(10.113)

where ǫ is the crack density parameter. It is readily found that parallel cracks normal to e 3 only affect the Young’s modulus in directions normal to the cracks and the shear moduli in the crack plane. It follows that the Biot tensor (5.85)

Microporofracture and Damage Mechanics

316

is here: (Bi j ) =

b 11 0 0

0 b 11 0

0 0 b 33

with: b 11 = −

hom s E − 1) ν s (S3333 ; H

b 33 =

hom s E − 1) (1 − ν s )(S3333 H

(10.114)

hom s E (1 − ν s ). Similarly, the corresponding Biot modulus with H = (ν s )2 − S3333 N is obtained from (5.87):

N = −E s

(1 +

ν s )(1

H hom s − 2ν s )(S3333 E − 1)

(10.115)

The macroscopic undrained stiffness of the cracked medium Cuhom is derived by substituting (10.113), (10.114) and (10.115) in (10.111). It is then possible to investigate damage propagation under undrained conditions and derive the corresponding damage development. To do so, we employ (10.94) for which we evaluate the effective stress Σ + P1. By way of illustration, consider a uniaxial loading in the direction e 3 , Σ = e 3 ⊗ e 3 , being a positive scalar (in order to maintain cracks open). Recalling that E = (Cuhom )−1 : Σ in undrained evolutions, the resulting ﬂuid pressure is P = −NB : E = − . The effective stress is therefore: Σ + P1 = e 3 ⊗ e 3 + P1 = − (1 − e 3 ⊗ e 3 )

(10.116)

Moreover, the damage criterion (10.94) requires us to perform the derivation of the macroscopic compliance Shom (in the form (10.113)) w.r.t. the crack density parameter ǫ. It follows from this calculation that: 1 ∂ hom Gǫ = ( e 3 ⊗ e 3 + P1) : (S ) : ( e 3 ⊗ e 3 + P1) = 0 2 ∂ǫ

(10.117)

Hence, for an incompressible ﬂuid, the driving force of damage propagation is always zero, and the damage criterion (10.94) cannot be reached for any value of stress . This conclusion holds for any isotropic loading state on a cracked porous medium weakened by an isotropic distribution of cracks. Indeed, since Chom and B are isotropic, it is readily found that Σ + P1 = 0, so that Gǫ = 0. 10.5.2 The Case of a Compressible Fluid We now examine the more general situation of a compressible ﬂuid. The developments of Section 10.5.1 remain valid except for the fact that N is now replaced by the Biot modulus M deﬁned by (10.112). For convenience, let us

Appendix: Algebra for Transverse Isotropy and Applications

317

introduce the scalar α such that α/E s = ϕ0 /k f . We again consider a set of open parallel cracks normal to n = e 3 . Following the same line of argument that led to (10.115), the Biot modulus M is: H hom M = −E s (10.118) (1 + ν s )(1 − 2ν s ) S3333 E s − 1 + α H Substituting (10.113), (10.114) and (10.118) in (10.111) yields the undrained macroscopic stiffness tensor of the cracked medium Cuhom . For the uniaxial loading Σ = e 3 ⊗ e 3 , the ﬂuid pressure P corresponding to undrained conditions is: P=−

hom s S3333 E −1

hom s α + S3333 E −1

(10.119)

Note that P = − is naturally retrieved for an incompressible ﬂuid by letting α = 0 in (10.119). In return, from a derivation of the macroscopic compliance Shom (in the form (10.113)) w.r.t. ǫ, the driving force of the damage propagation is not zero as in the incompressible ﬂuid case, but reads: 1 ∂ hom Gǫ = ( e 3 ⊗ e 3 + P1) : (S ) : ( e 3 ⊗ e 3 + P1) 2 ∂ǫ hom ∂ S3333 α 2 ∂ǫ 2 = 2 hom s 2 α + S3333 E − 1

(10.120)

hom Since S3333 is a increasing function of ǫ, Gǫ ≥ 0: the added compliance induced by the compressibility of the ﬂuid phase may entail damage propagation as the stress increases. This conclusion holds for cracked porous media with an isotropic distribution of cracks, which is subjected to an isotropic macroscopic loading.

10.6 Appendix: Algebra for Transverse Isotropy and Applications The homogenized properties of the medium made up of an isotropic solid matrix with parallel cracks exhibit transversely isotropic symmetry. For this reason, it is interesting to introduce standard notation and the corresponding simpliﬁed algebra [51] for fourth-order transverse isotropic tensors. This algebra is particularly useful for easily obtaining the inner product of transversely isotropic tensors or tensor inversion. Given a unit vector n, let us introduce i N = n ⊗ n and i T = 1 − n ⊗ n, as well as: 1 E1 = i T ⊗ i T ; E2 = i N ⊗ i N ; E3 = i T ⊗i T − E1 ; E4 = i N ⊗i T + i T ⊗i N 2 (10.121) where the notation (10.34) is used.

Microporofracture and Damage Mechanics

318

It is readily seen that: (10.122)

E1 + E2 + E3 + E4 = I

and: if p = q ;

E p : Eq = E p

E p : Eq = 0

if p = q

(10.123)

Two other elementary tensors are also deﬁned: E5 = i N ⊗ i T ;

E6 = i T ⊗ i N

(10.124)

From these identities, it is shown that any transversely isotropic (not necessarily symmetric) fourth-order tensor can be decomposed as: L = c E1 + d E2 + e E3 + f E4 + g E5 + h E6

(10.125)

which may be expressed in the symbolic form: L = [c, d, e, f, g, h]

(10.126)

If L is symmetric (L = t L), as for the stiffness or compliance tensor, then g = h. With this notation, considering L′ = [c ′ , d ′ , e ′ , f ′ , g ′ , h ′ ], the tensor product L : L′ is: L : L′ = [cc ′ + 2hg ′ , dd ′ + 2gh ′ , ee ′ , f f ′ , gc ′ + dg ′ , hd ′ + ch ′ ]

(10.127)

and the inverse of L is: −1

L

d c 1 1 g h = , , , ,− ,− l l e f l l

(10.128)

with l = cd − 2gh. We note that the non-symmetric tensor (I − S(X, n)) can be recast in the form [1 − c, 1 − d, 1 − e, 1 − f, −g, −h] with, for n = e 3 : c = S1111 + S1122 ; f = 2S3131 ;

d = S3333 ;

g = S3311 ;

e = S1111 − S1122 ;

(10.129)

h = S1133

These components of Si jkl are deﬁned in (10.31). Then, taking the inverse of (I − S(X, n)) and performing the limit calculation (X → 0) in (10.26), we obtain the following expression for tensor T(n) associated with the set of open parallel cracks normal to n: 1 νs 4(1 − ν s ) 1 − ν s (10.130) E2 + E4 + E5 T(n) = π 1 − 2ν s 2 − νs 1 − 2ν s which is equivalent to the explicit form given in (10.33).

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Index 1-D Thought Model (see Hollow Sphere Model) Advection, 56, 63, 64, 77 Moderate, 78 Dominated Transport, 79, 82, 83 Anisotropy Poroelastic properties, 195, 299 Average, 3 Apparent, 5, 15, 65 Capillary pressure effect on Stress, 102–103 Flow, 26 Intrinsic, 5, 15, 65, 66, 127, 215 Pore Strain, 99, 157 Pressure ﬁeld, 137, 150 Quadratic, 57, 129, 154–156, 215 Second-order, (see Quadratic) Spatial Derivative, 6, 16, 17 Strain level, 127–130, 154–156, 260–263 Time Derivative, 7, 17, 18 Averaging Operations, (see Average) Balance Laws, 8 Mass, 8, 64, 65, 77, 83 Momentum, 10, 98, 178, 208, 209, 269 Bishop effective stress (see Effective Stress) Biot coefﬁcient, 140, 143, 152 Cracked porous media, 299, 300, 303, 316 Dilute Estimate, 186, 188, 299 Nonlinear, 228 Non-saturated case, 247–248, 257 Mori-Tanaka Estimate, 191, 305 Poroplasticity, in, 275

Tensor of, 148, 158, 296, 316 Transversely isotropic, 195 Biot modulus, Cracked porous media, 299, 316–317 Dilute Estimate, 189, 299 Nonlinear, 228 Overall, 151, 314 Solid, 141, 152, 159, 164, 316–317 Body forces, (see Volume Forces) Bounds of, Biot coefﬁcient, 152–153 Solid Biot modulus, 153 Bulk modulus of Hashin’s sphere assemblage, 112–116 Compliance Tensor, 110–112 Stiffness Tensor, 119–120 Permeability, 34, 43, 40, 43–44 Tortuosity, 67–69, 72, 67–69, 73 Capillary Pressure Curve, 250–254 Drying Shrinkage, 257–259 Effect on Stress Average, 102–103 Strength Homogenization, 264 Macroscopic, 247–248 State equation, 254–257 Clausius-Duhem Inequality, 292 Cohesion, 207, 268 Pressure, 208 Compaction, 184–185 Compliance, (see Stiffness) Complimentary Energy, 41, 110, 114, 119

Microporomechanics L. Dormieux, D. Kondo and F.-J. Ulm C 2006 John Wiley & Sons, Ltd

324 Concentration Tensor Eshelby estimate, 186 Isotropy, 151 Molecular Diffusion, in, 66, 132, 200 Mori-Tanaka Estimate, 190, 199–200 Pore Families, 186–187 Self-Consistent estimate, 193, 200–202 Strain, 116, 145, 185, 259, 275 Open Parallel Cracks, 296 Strain Rate, 297 Stress, 106, 275, 289 Conduction Fluid, 23 Approximate law for power-laws ﬂuids, 56–58 Contact angle, 250 Crack, 291 Anisotropic Distribution of, 300–301 Aspect Ratio, 295, 296, 298, 303 Closure, 301–303, 304, 305 Density parameter, 296, 300, 301, 304, 306 Open Parallel, 296–299, 303–304, 306 Penny-Shaped, 295 Porosity (or volume fraction), 296, 301, 305, 306 Randomly Oriented Open, 299, 304–305, 306 Damage Criterion, 310–314 Mechanics, 291 Propagation, 306–308, 314–317 Darcy’s Law, 23, 78 Deformable porous media, 48–50 Generalization to power-law ﬂuids, 57 Microscopic derivation of, 25, 27, 31–33 Phenomenological, 23, 26 Non-linear, 52–60 Differential Scheme, Diffusion, 134–135, 172 Elasticity, 125–127, 190 Diffusion, 63, 167 Analogy with elasticity, 131, 177 Characteristic Time of, 82 Coefﬁcient, 64, 168 Disordered porous medium, in, 130–135 Estimates of Diffusion Coefﬁcient, 198–202 Inhomogeneity Problem (see Eshelby’s Problem) Microscopic equation of, 65

Index Percolation Threshold, 202 Tensor, 66, 80, 132, 169, 176 With Advection, 77–84 Without Advection, 64–74, 75–77 Diffusivity (see Diffusion Tensor) Dilute Situation, 64 Dilute Scheme, Biot coefﬁcient, 153 Biot modulus, 189, 299 Biot tensor, 186, 189, 197, 299 Damage Criterion, 310–311 Diffusion, 133–134, 176 Elasticity, 122–124, 186–187, 189 Microporofracture mechanics, in, 295–303 Non-saturated poroelasticity, in, 245, 255, 258, 259 Poroelastic model, 187–189, 190 Transverse Isotropy, 197 Dirac distribution (see Distribution Theory) Dispersion, 63, 78 Coefﬁcient, 86, 87 Hydrodynamic dispersion tensor, 87 Multilayer porous medium, in, 84–86 Tensor, 81 Dissipation, Damage, 307 Fracture, 292–294, 307 Plastic, 210, 274, 285–286 Distribution Theory, 6, 169, 178, 239, 241 Double-Scale Expansion, 45–48, 74–84 Double Porosity Model Cracked Porous media, 305–306, 309–310 Two-Scale, 161–165 Drained Conditions, 91, 163, 177, 277–278 Drucker-Prager solid, 208–210, 289 Pore pressure effect, 224–226 Strength homogenization, 231–235, 265–266 Drying, 245, 247 Drying Shrinkage, (see Shrinkage) Effective Stress, 148, 152, 272, 316 Bishop, 246–248, 250, 255, 264 Poroplasticity, in, 287–290 Terzaghi, 156, 208, 224, 225, 229, 287, 290, 309, 312 Elasticity Analogy with diffusion, 177 Domain, 281, 282–283 Incompressible (Analogy with), 40

Index

325

Effect on ﬂow through porous media, 48 Stiffness, (see Stiffness) Elastoplasticity (see Microporoplasticity) Energy Approach Hollow Sphere, 94–96, 141–143, 272–274 Drained Poroelasticity, in, 101, 104 Microporoplasticity, in, 285–287 Strain Concentration, in, 117 Stress Concentration, in, 108 Energy Release Rate, 293–295 Eshelby’s Problem, 167, 261 Implementation in Linear Microporoelasticity, 185 Linear Diffusion, in, 167–176 Linear Microelasticity, in, 176–184 Eshelby Tensor, 184, 261, 302 Cylindrical inclusion in isotropic matrix, 202–203 Isotropy (spherical), 184, 305 Oblate ellipsoidal pore, 195–196 Oblate spheroid (cracks), 296, 298

Hill Lemma, The, 105–106, 109, 118, 155, 158, 160, 212, 237, 261, 278, 284–287 With surface tension effects, 241–243 Hill-Mandel Theorem, The, 120–121 Hollow Sphere Model Drained Poroelasticity, in, 91–96 Hardening Poroplasticity, 267–274 Hashin’s Sphere Assemblage, in, 112–114 Linear Microporoelasticity, in, 138–143 Strength Homogenization, in, 207–210, 221–222, 233–235 Homogenization Scheme Dilute, (see Dilute Scheme) Differential Scheme, (see Differential Scheme) Effect on Damage Criterion, 310–314 Mori-Tanaka, (see Mori-Tanaka) Polycrystal (see Self-Consistent) Ponte-Castaneda and Willis, 312–314 Self-Consistent (see Self-Consistent) Hydraulic Head, 23

Fick’s Law, 64, 65, 172 Fictitious (or equivalent) viscous behavior, 213–214, 216, 218–219, 259 Flow Rule, 268, 283–285, 290 Fluid Bingham, 52 Immiscible, 247, 250 Linear state equation, 150 Mass content, 150 Newtonian, 13, 25, 54, 55, 56 Power-Law, 54 Fracture Energy, 294, 307–308 Mechanics, (see LEFM) Propagation, 293–294 Free Energy, 286, 292 Frozen (hardening) Energy, 274, 287

Inclusion Problem (see Eshelby’s Problem) Inelastic behavior, 207 Inhomogeneity Problem (see Eshelby’s Problem) Interaction (mechanical) between pores (see Mori-Tanaka scheme) Isotropy, Elasticity tensor, 151 Morphology, 151

Green Function, 170, 172–173, 178, 180 Green Tensor, 178, General deﬁnition, 180–181 Isotropic case, in, 181–183 Hardening, (see Microporoplasticity) Hashin’s composite sphere model, 112–116, 120 Helmholtz Energy (see Free Energy)

Kinematics, 9 Laplace’s Law, 102, 249, 252 LEFM, 291–295, 306 Damage Analogy, 306–308 Levin’s Theorem, 156–159, 160, 163, 244, 246–247, 249, 274 Loading parameters, Drained Microelasticity, in, 97, 104, 106, 113, Drying induced, 258 Fracture Mechanics, in, 291–292 Linear Microporoelasticity, in, 137, 139, 140, 143, 146 Microporoplasticity, in, 276, 282, 288 Localization Tensor (see Concentration Tensor)

326 Maxwell-Betti Theorem, 146–148 Maxwell Symmetry, 142 Membrane stress (stresses), 237, 248–250, 260 Macroscopic, 255 Representation 238–241 Work of, 242–243 Meniscus, i, 251–252, 256–257 Microporoelasticity, 89 Cracked Porous Media, of, 295–306 Drained, 91 Nonlinear, 226–229 Non-saturated, 237, 246–250 Microporofracture Mechanics, 291 Energy Release Rate in, 294–295, 307–314 LEFM-Damage Analogy, 306–308 Propagation, 294, 307 Stability, 294, 308 State equations of, 307 Undrained Conditions, 314–317 Microporoinelasticity, 205 Microporoplasticity, 267 Dissipation Analysis, 285–287 Effective Stress in, 287–290 Flow Rule, 268, 283–285 Hardening, 282, 287 State equations, in, 274–281 Yield (or Plasticity) Criterion, 268, 281–283 Mixture Theories, 11 Mori-Tanaka Scheme, 190–192, 261–262 Cracked Porous Media, 303–306 Damage Criterion, 311–312 Diffusion Tensor, 198–200 Strength upscaling, in, 219, 220, 224, 232, 233, 263 Transversely Isotropic Elasticity, 197–198 Morphology, Matrix-Inclusion (see Mori-Tanaka) Perfectly disordered material (see Self-Consistent) Polycrystal (see Self-Consistent) Solid-ﬂuid interface, of, 245 Multiscale homogenization, 165, 172 Natural State, 160, 243, 289 Non-Homogenizable Case, 81 Nonlinear Homogenization, (see Secant Methods) Non-saturated conditions, 237 Péclet Number, 78, 86 Percolation Threshold, 195, 202, 221, 232

Index Periodic Media Theory, 14, 45, 167 Cell Assumption in, 14, 27, 65 Darcy’s Law, 27 Diffusive Properties, 65 Periodicity (see Periodic Media Theory) Permeability Analogy with incompressible elasticity, 40–42 Coefﬁcient, 24 Intrinsic, 26–27, 54–55, 58 Lower Bound, of, 34, 43 Ordered relations, 34 Periodic Granular Material, 35–40 Scale effects on, 33–34 Second-order tensor, 25 Tensor symmetry, 32 Upper Bound, of, 40, 43–44 Plasticity (see Microporoplasticity) Plastic Work, principle of maximum, 284–285 Poiseuille ﬂow Cylinder, 25 Parallel layers, 84 Pore Families, 186–187, 191 Porosity, Double, 161–165 Eulerian, 141, 144 Initial, 138 Lagrangian, 141, 144, 146, 150, 159, 161, 162, 188, 247 Elastic, 276–277, 280 Plastic, 279, 280, 281, 285 Porous Matrix, 162 Potential Convexity of, 60–61 Nonlinear Elasticity, 226 Power-Law Fluids, 54–55 Thermodynamic, 146 Potential Energy, 111, 119–120, 155 Density, 142, 148–149 Microfracture Mechanics, in, 292–293, 295, 306 Non-saturated conditions, in, 260–262 Prestress, (see Levin’s Theorem) Virtual work principle, 237, 241–243 P-Tensor, Anisotropic Diffusion, for 174, 199 Fourth-order, 179, 181, 296 Isotropy, 181–183, 192 Second-order, 171–175 Spherical inclusion, 172, 174, 175, 181–183, 200

Index Reference strain, 227, 235, 263 Rate, 57, 215, 220 Representative Elementary Volume (rev), 1 Deﬁnition, 1–3 Drained Poroelasticity, in, 97 Linear Microporoelasticity, in, 143–144 Two-Scale Double Porosity Material, 162 Reynolds number, 28, 52 Saturation, 247, 252, 255, 257 Scale effects on Damage Threshold, 307 Permeability, 33–34 Secant Method, 215, 229–235, 259–266 Secant Stiffness, 214, 226 Self-Consistent Scheme, Microporoelasticity, in, 192–195 Diffusion problem, in, 198–201 Strength Upscaling, in, 219, 221, 224, 232 Shrinkage, 246, 250–259 Solid-ﬂuid interaction, 11–13 Macroscopic representation of, 50–51 Microscopic representation of, 51–52 Solvent, 64 Solute, 64, 66 Steady State Conditions, 64, 65, 75–82 Stiffness, 48 Bounds, 110–112, 114–116, 119 Cracked Media, of, 295–306 Estimates, 122 Differential Scheme, 125–127 Dilute Scheme, 122–124, 186–187, 189, 295–302 Mori-Tanaka Scheme, 191–192, 303–306, 311 Ponte-Castaneda and Willis, 313 Self-Consistent Scheme, 193–194 Heterogeneous, 100–101, 104, 156–157, 160, 162, 214–215, 227, 244, 247, 274 Homogenization, Boundary Condition Effects (see Hill-Mandel Theorem) Double-Porosity Material, 163, 305–306 Strain Concentration, 116–118, 145, 157, 244 Stress Concentration, 108–110 Isotropic, 92–94, 112, 118, 124, 126, 151–152, 192, 193–194, 299–300

327 Secant, (see Secant Stiffness) Tangent, 297 Transverse Isotropic, 315 Undrained, 151, 152, 314 Strain Boundary Condition, 103–105, 106, 116, 177 Compatibility, 18, 190 Concentration, 116–118 Drying (see Shrinkage) Effective, 127 Elastic, 272 Macroscopic, 276–277, 280 Microscopic, 274, 280 Equivalent Shear, 128, 154 Inclusion, 184 Linearized, 48 Plastic (or permanent), 271 Macroscopic, 275, 278–279, 281, 285 Microscopic, 274, 278–279 Rate tensor, 33, 53, 215 Surface Tension, 245–246 Stokes Equations, 28, 78 Variational approach to, 29–31 Strength Domain, 207 Drucker-Prager criterion, 208–210 Dual deﬁnition, 210, 213 Empty porous material, of, 210–216 Homogenization, 207, 231 Drucker-Prager solid, 231–235 Von-Mises solid, 216–222, 229–231 Isotropic tensile, 208 Non-saturated porous media, of, 259–266 Drucker-Prager solid, 265–266 Von-Mises solid, 263–265 Pore Pressure effect, 222–226 Von Mises criterion, 207–208 Stress, stresses, 10 Bingham ﬂuid, 53 Boundary Conditions, 92, 97, 100–102, 105, 113 Concentration, 106–110 Discontinuity, 102–103, 240, 242 Newtonian ﬂuid, 40 Partial, 10, 11, 13 Power-Law Fluids, 54–55 Residual, 271–272, 278 Total, 11 Support function, 210, 214 Von Mises solid, 217

Index

328 Surface Tension, 102–103, 237, 238–240, 255, 257 Liquid-Gas interface, at, 248–250 State equation with, 243–244 Strain induced by, 245–246, 250 Swelling, 246, 259 Terzaghi, (see Effective Stress) Tortuosity Cubic Array of spheres, of, 70–71 Cylindrical porosity, of, 69 Estimates, 133–135 Geometrical meaning of, 71 Limit of concept of, 87 Lower bound of, 67–69, 72 Multilayer Porous Medium, of, 86 Ordered relations, 69–70 Tensor, 66, 80–81, 133 Upper Bound of, 67–69, 73 Transient Conditions, 82–84 Transport Phenomena, 21 Pure diffusive, 74 Diffusive and Advective, 77 Advection-Dominated, 79 Transverse Isotropy, 195, 255–256, 317–318 Tresca solid, 223, 287 Undrained conditions, 151 Damage Propagation in, 314–317

Uniform Boundary Condition, Strain, 103, 120–121, 122, 139, 177, 243, 255, 261, 276, 291 Strain Rate, 297 Stress, 100, 120–121, 275 Variational Approach to Compliance Tensor, 110–112 Periodic Homogenization, 67–69 Power-Law Fluid Conduction, 58–60 Stiffness Tensor, 119–120 Stokes system, 29–33 Viscosity, 13, 25, 57, 59 Viscous Flow In a cylinder, 25 In a cylindrical pore, 26, 55, 58 Viscous Stress, 52, 137, 157 Strength homogenization, in, 213 Voigt bound, 112, 114–115, 119, 152 Volume Forces, 11–12, 40, 99–100 Von Mises solid, 207, 287 Flow Rule, 268 Pore pressure effect, 223–224 Strength homogenization, 216–222, 229–231, 263–265 Yield Criterion, 268, 289 Wetting, 247, 250 Yield (see Strength) Young equation, 250

Microporomechanics

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