Homogenization methods for multiscale mechanics

In many physical problems several scales are present in space or time, caused by inhomogeneity of the medium or complexity of the mechanical process. A fundamental approach is to first construct micro-scale models, and then deduce the macro-scale laws and the constitutive relations by properly averaging over the micro-scale. The perturbation method of multiple scales can be used to derive averaged equations for a much larger scale from considerations of the small scales. In the mechanics of multiscale media, the analytical scheme of upscaling is known as the Theory of Homogenization The authors share the view that the general methods of homogenization should be more widely understood and practiced by applied scientists and engineers. Hence this book is aimed at providing a less abstract treatment of the theory of homogenization for treating inhomogeneous media, and at illustrating its broad range of applications. Each chapter deals with a different class of physical problems. To tackle a new problem, the approach of first discussing the physically relevant scales, then identifying the small parameters and their roles in the normalized governing equations is adopted. The details of asymptotic analysis are only explained afterwards.

Dedication	p. v
Acknowledgments	p. vii
Preface	p. ix
1 Introductory Examples of Homogenization Method	p. 1
1.1 Long Waves in a Layered Elastic Medium	p. 1
1.2 Short Waves in a Weakly Stratified Elastic Medium	p. 6
1.3 Dispersion of Passive Solute in Pipe Flow	p. 10
1.3.1 Scale Estimates	p. 11
1.3.2 Multiple-Scale Analysis	p. 12
1.3.3 Dispersion Coefficient for Steady Flow	p. 17
1.3.4 Dispersion Coefficient for Oscillatory Flow	p. 18
1.4 Typical Procedure of Homogenization Analysis	p. 19
References	p. 20
2 Diffusion in a Composite	p. 23
2.1 Basic Equations for Two Components in Perfect Contact	p. 23
2.2 Effective Equation on the Macroscale	p. 24
2.3 Effective Boundary Condition	p. 29
2.4 Symmetry and Positiveness of Effective Conductivity	p. 33
2.5 Laminated Composites	p. 35
2.6 Bounds for Effective Conductivity	p. 38
2.6.1 First Variational Principle and the Upper Bound	p. 38
2.6.2 Dual Variational Principle and the Lower Bound	p. 41
2.7 Hashin-Shtrikman Bounds	p. 44
2.7.1 Results and Implications	p. 44
2.7.2 Derivation of Hashin-Shtrikman Bounds	p. 46
2.8 Other Approximate Results for Dilute Inclusions	p. 50
2.9 Thermal Resistance at the Interface	p. 52
2.10 Laminated Composites with Thermal Resistance	p. 58
2.10.1 Effective Coefficients	p. 58
2.10.2 Application to Thermal Barrier Coatings	p. 61
2.11 Bounds for the Effective Conductivity	p. 63
2.11.1 Variational Principles and Bounds	p. 63
2.11.2 Application to a Particulate Composite	p. 66
2.12 Chemical Transport in Aggregated Soil	p. 70
Appendix 2A Heat Transfer in a Two-Slab System	p. 79
References	p. 82
3 Seepage in Rigid Porous Media	p. 85
3.1 Equations for Seepage Flow and Darcy's Law	p. 85
3.2 Uniqueness of the Cell Boundary-Value Problem	p. 90
3.3 Symmetry and Positiveness of Hydraulic Conductivity	p. 91
3.4 Numerical Computation of the Permeability Tensor	p. 92
3.5 Seepage of a Compressible Fluid	p. 96
3.6 Two-Dimensional Flow Through a Three-Dimensional Matrix	p. 99
3.6.1 Governing Equations	p. 100
3.6.2 Homogenization	p. 103
3.6.3 Numerical Results	p. 107
3.7 Porous Media with Three Scales	p. 109
3.7.1 Effective Equations	p. 110
3.7.2 Properties of Hydraulic Conductivity	p. 113
3.7.3 Macropermeability of a Laminated Medium	p. 114
3.8 Brinkman's Modification of Darcy's Law	p. 118
3.9 Effects of Weak Fluid Inertia	p. 123
Appendix 3A Spatial Averaging Theorem	p. 130
References	p. 132
4 Dispersion in Periodic Media or Flows	p. 135
4.1 Passive Solute in a Two-Scale Seepage Flow	p. 135
4.1.1 The Solute Transport Equation and Scale Estimates	p. 136
4.1.2 Macroscale Transport Equation	p. 138
4.1.3 Numerical Computation of Dispersivity	p. 145
4.2 Macrodispersion in a Three-Scale Porous Medium	p. 149
4.2.1 From Micro- to Mesoscale	p. 151
4.2.2 Mass Transport Equation on the Macroscale	p. 152
4.2.3 Second-Order Seepage Velocity	p. 156
4.3 Dispersion and Transport in a Wave Boundary Layer Above the Seabed	p. 158
4.3.1 Depth-Integrated Transport Equation in the Boundary Layer	p. 159
4.3.2 Effective Convection Velocity	p. 164
4.3.3 Correlation Coefficients $$$ and Dispersivity Tensor	p. 166
4.3.4 Dispersion Under a Standing Wave in a Lake	p. 169
Appendix 4A Derivation of Convection-Dispersion Equation	p. 173
Appendix 4B An Alternate Form of Macrodispersion Tensor	p. 175
References	p. 176
5 Heterogeneous Elastic Materials	p. 179
5.1 Effective Equations on the Macroscale	p. 180
5.2 The Effective Elastic Coefficients	p. 184
5.3 Application to Fiber-Reinforced Composite	p. 185
5.4 Elastic Panels with Periodic Microstructure	p. 186
5.4.1 Order Estimates	p. 189
5.4.2 Two-Scale Analysis and Effective Equations	p. 190
5.4.3 Homogeneous Plate - A Limiting Case	p. 196
5.5 Variational Principles and Bounds for the Elastic Moduli	p. 199
5.5.1 First Variational Principle and the Upper Bound	p. 199
5.5.2 Second Variational Principle and the Lower Bound	p. 201
5.6 Hashin-Shtrikman Bounds	p. 203
5.7 Partially Cohesive Composites	p. 208
5.7.1 Effective Equations on the Macroscale	p. 210
5.7.2 Variational Principles	p. 212
5.7.3 Bounds for Particulate Composites	p. 219
5.7.4 Size Effects for Particulate Composites	p. 226
5.7.5 Critical Radii for Particulate Composites	p. 227
Appendix 5A Properties of a Tensor of Fourth Rank	p. 234
References	p. 235
6 Deformable Porous Media	p. 239
6.1 Basic Equations for Fluid and Solid Phases	p. 240
6.2 Scale Estimates	p. 242
6.2.1 Quasi-Static Poroelasticity	p. 242
6.2.2 Dynamic Poroelasticity	p. 244
6.3 Multiple-Scale Expansions	p. 246
6.4 Averaged Total Momentum of the Composite	p. 248
6.5 Averaged Mass Conservation of Fluid Phase	p. 251
6.6 Averaged Fluid Momentum	p. 252
6.6.1 Quasi-Static Case	p. 252
6.6.2 Dynamic Case	p. 253
6.7 Time-Harmonic Motion	p. 254
6.8 Properties of the Effective Coefficients	p. 257
6.8.1 Three Identities for General Media	p. 257
6.8.2 Homogeneous and Isotropic Grains	p. 259
6.9 Computed Elastic Coefficients	p. 262
6.10 Boundary-Layer Approximation for Macroscale Problems	p. 263
6.10.1 The Outer Approximation	p. 264
6.10.2 Boundary-Layer Correction	p. 267
6.10.3 Plane Rayleigh Wave in a Poroelastic Half Space	p. 272
Appendix 6A Properties of the Compliance Tensor	p. 274
Appendix 6B Variational Principle for the Elastostatic Problem in a Cell	p. 275
References	p. 276
7 Wave Propagation in Inhomogeneous Media	p. 279
7.1 Long Wave Through a Compact Cylinder Array	p. 279
7.2 Bragg Scattering of Short Waves by a Cylinder Array	p. 286
7.2.1 Envelope Equations	p. 287
7.2.2 Dispersion Relation for a Detuned Wave Train	p. 291
7.2.3 Scattering by a Finite Strip of Periodic Cylinders	p. 292
7.3 Sound Propagation in a Bubbly Liquid	p. 294
7.3.1 Scale and Order Estimates	p. 295
7.3.2 Near Field of a Spherical Bubble	p. 296
7.3.3 The Intermediate Field	p. 298
7.3.4 The Macroscale Equation	p. 300
7.4 One-Dimensional Sound Through a Weakly Random Medium	p. 302
7.5 Weakly Nonlinear Dispersive Waves in a Random Medium	p. 306
7.5.1 Envelope Equation	p. 306
7.5.2 Modulational Instability	p. 311
7.6 Harmonic Generation in Random Media	p. 312
7.6.1 Long Waves in Shallow Water	p. 313
7.6.2 Harmonic Amplitudes	p. 315
7.6.3 Gaussian Disorder	p. 319
References	p. 321
Additional References on Homogenization Theory	p. 325
Subject Index	p. 327

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