Homogenization Methods for Multiscale Mechanics

by Mei, Chiang C.; Vernescu, Bogdan

• ISBN13: 9789814282444
• ISBN10: 9814282448
• Format: Hardcover
• Copyright: 2010-05-30
• Publisher: Textstream

Summary

In many physical problems several scales present either in space or in time, caused by either 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 novel 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.

Table of Contents

Dedication	p. v
Acknowledgments	p. vii
Preface	p. ix
Introductory Examples of Homogenization Method	p. 1
Long Waves in a Layered Elastic Medium	p. 1
Short Waves in a Weakly Stratified Elastic Medium	p. 6
Dispersion of Passive Solute in Pipe Flow	p. 10
Scale Estimates	p. 11
Multiple-Scale Analysis	p. 12
Dispersion Coefficient for Steady Flow	p. 17
Dispersion Coefficient for Oscillatory Flow	p. 18
Typical Procedure of Homogenization Analysis	p. 19
References	p. 20
Diffusion in a Composite	p. 23
Basic Equations for Two Components in Perfect Contact	p. 23
Effective Equation on the Macroscale	p. 24
Effective Boundary Condition	p. 29
Symmetry and Positiveness of Effective Conductivity	p. 33
Laminated Composites	p. 35
Bounds for Effective Conductivity	p. 38
First Variational Principle and the Upper Bound	p. 38
Dual Variational Principle and the Lower Bound	p. 41
Hashin-Shtrikman Bounds	p. 44
Results and Implications	p. 44
Derivation of Hashin-Shtrikman Bounds	p. 46
Other Approximate Results for Dilute Inclusions	p. 50
Thermal Resistance at the Interface	p. 52
Laminated Composites with Thermal Resistance	p. 58
Effective Coefficients	p. 58
Application to Thermal Barrier Coatings	p. 61
Bounds for the Effective Conductivity	p. 63
Variational Principles and Bounds	p. 63
Application to a Particulate Composite	p. 66
Chemical Transport in Aggregated Soil	p. 70
Heat Transfer in a Two-Slab System	p. 79
References	p. 82
Seepage in Rigid Porous Media	p. 85
Equations for Seepage Flow and Darcy's Law	p. 85
Uniqueness of the Cell Boundary-Value Problem	p. 90
Symmetry and Positiveness of Hydraulic Conductivity	p. 91
Numerical Computation of the Permeability Tensor	p. 92
Seepage of a Compressible Fluid	p. 96
Two-Dimensional Flow Through a Three-Dimensional Matrix	p. 99
Governing Equations	p. 100
Homogenization	p. 103
Numerical Results	p. 107
Porous Media with Three Scales	p. 109
Effective Equations	p. 110
Properties of Hydraulic Conductivity	p. 113
Macropermeability of a Laminated Medium	p. 114
Brinkman's Modification of Darcy's Law	p. 118
Effects of Weak Fluid Inertia	p. 123
Spatial Averaging Theorem	p. 130
References	p. 132
Dispersion in Periodic Media or Flows	p. 135
Passive Solute in a Two-Scale Seepage Flow	p. 135
The Solute Transport Equation and Scale Estimates	p. 136
Macroscale Transport Equation	p. 138
Numerical Computation of Dispersivity	p. 145
Macrodispersion in a Three-Scale Porous Medium	p. 149
From Micro- to Mesoscale	p. 151
Mass Transport Equation on the Macroscale	p. 152
Second-Order Seepage Velocity	p. 156
Dispersion and Transport in a Wave Boundary Layer Above the Seabed	p. 158
Depth-Integrated Transport Equation in the Boundary Layer	p. 159
Effective Convection Velocity	p. 164
Correlation Coefficients $$$ and Dispersivity Tensor	p. 166
Dispersion Under a Standing Wave in a Lake	p. 169
Derivation of Convection-Dispersion Equation	p. 173
An Alternate Form of Macrodispersion Tensor	p. 175
References	p. 176
Heterogeneous Elastic Materials	p. 179
Effective Equations on the Macroscale	p. 180
The Effective Elastic Coefficients	p. 184
Application to Fiber-Reinforced Composite	p. 185
Elastic Panels with Periodic Microstructure	p. 186
Order Estimates	p. 189
Two-Scale Analysis and Effective Equations	p. 190
Homogeneous Plate - A Limiting Case	p. 196
Variational Principles and Bounds for the Elastic Moduli	p. 199
First Variational Principle and the Upper Bound	p. 199
Second Variational Principle and the Lower Bound	p. 201
Hashin-Shtrikman Bounds	p. 203
Partially Cohesive Composites	p. 208
Effective Equations on the Macroscale	p. 210
Variational Principles	p. 212
Bounds for Particulate Composites	p. 219
Size Effects for Particulate Composites	p. 226
Critical Radii for Particulate Composites	p. 227
Properties of a Tensor of Fourth Rank	p. 234
References	p. 235
Deformable Porous Media	p. 239
Basic Equations for Fluid and Solid Phases	p. 240
Scale Estimates	p. 242
Quasi-Static Poroelasticity	p. 242
Dynamic Poroelasticity	p. 244
Multiple-Scale Expansions	p. 246
Averaged Total Momentum of the Composite	p. 248
Averaged Mass Conservation of Fluid Phase	p. 251
Averaged Fluid Momentum	p. 252
Quasi-Static Case	p. 252
Dynamic Case	p. 253
Time-Harmonic Motion	p. 254
Properties of the Effective Coefficients	p. 257
Three Identities for General Media	p. 257
Homogeneous and Isotropic Grains	p. 259
Computed Elastic Coefficients	p. 262
Boundary-Layer Approximation for Macroscale Problems	p. 263
The Outer Approximation	p. 264
Boundary-Layer Correction	p. 267
Plane Rayleigh Wave in a Poroelastic Half Space	p. 272
Properties of the Compliance Tensor	p. 274
Variational Principle for the Elastostatic Problem in a Cell	p. 275
References	p. 276
Wave Propagation in Inhomogeneous Media	p. 279
Long Wave Through a Compact Cylinder Array	p. 279
Bragg Scattering of Short Waves by a Cylinder Array	p. 286
Envelope Equations	p. 287
Dispersion Relation for a Detuned Wave Train	p. 291
Scattering by a Finite Strip of Periodic Cylinders	p. 292
Sound Propagation in a Bubbly Liquid	p. 294
Scale and Order Estimates	p. 295
Near Field of a Spherical Bubble	p. 296
The Intermediate Field	p. 298
The Macroscale Equation	p. 300
One-Dimensional Sound Through a Weakly Random Medium	p. 302
Weakly Nonlinear Dispersive Waves in a Random Medium	p. 306
Envelope Equation	p. 306
Modulational Instability	p. 311
Harmonic Generation in Random Media	p. 312
Long Waves in Shallow Water	p. 313
Harmonic Amplitudes	p. 315
Gaussian Disorder	p. 319
References	p. 321
Additional References on Homogenization Theory	p. 325
Subject Index	p. 327
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