Homogenization of Coupled Phenomena in Heterogenous Media

Both naturally-occurring and man-made materials are often heterogeneous materials formed of various constituents with different properties and behaviours. Studies are usually carried out on volumes of materials that contain a large number of heterogeneities. Describing these media by using appropriate mathematical models to describe each constituent turns out to be an intractable problem. Instead they are generally investigated by using an equivalent macroscopic description - relative to the microscopic heterogeneity scale - which describes the overall behaviour of the media.

Fundamental questions then arise: Is such an equivalent macroscopic description possible? What is the domain of validity of this macroscopic description? The homogenization technique provides complete and rigorous answers to these questions.

This book aims to summarize the homogenization technique and its contribution to engineering sciences. Researchers, graduate students and engineers will find here a unified and concise presentation.

The book is divided into four parts whose main topics are

• Introduction to the homogenization technique for periodic or random media, with emphasis on the physics involved in the mathematical process and the applications to real materials.
• Heat and mass transfers in porous media
• Newtonian fluid flow in rigid porous media under different regimes
• Quasi-statics and dynamics of saturated deformable porous media

Each part is illustrated by numerical or analytical applications as well as comparison with the self-consistent approach.

Main notations 17

Introduction 21

PART ONE. UPSCALING METHODS 27

Chapter 1. An Introduction to Upscaling Methods 29

Chapter 2. Heterogenous Medium: Is an Equivalent Macroscopic Description Possible? 55

Chapter 3. Homogenization by Multiple Scale Asymptotic Expansions 75

PART TWO. HEAT AND MASS TRANSFER 107

Chapter 4. Heat Transfer in Composite Materials 109

Chapter 5. Diffusion/Advection in Porous Media 143

Chapter 6. Numerical and Analytical Estimates for the Effective Diffusion Coefficient 161

PART THREE. NEWTONIAN FLUID FLOW THROUGH RIGID POROUS MEDIA 195

Chapter 7. Incompressible Newtonian Fluid Flow Through a Rigid Porous Medium 197

Chapter 8. Compressible Newtonian Fluid Flow Though a Rigid Porous Medium 229

Chapter 9. Numerical Estimation of the Permeability of Some Periodic Porous Media 257

Chapter 10. Self-consistent Estimates and Bounds for Permeability 275

PART FOUR. SATURATED DEFORMABLE POROUS MEDIA 337

Chapter 11. Quasi-statics of Saturated Deformable Porous Media 339

Chapter 12. Dynamics of Saturated Deformable Porous Media 367

Chapter 13. Estimates and Bounds for Effective Poroelastic Coefficients 389

Chapter 14. Wave Propagation in Isotropic Saturated Poroelastic Media 407

Bibliography . 453

Index 473

Jean-Louis Auriault received a civil engineer degree from Ecole Nationale des Ponts et Chaussées, Paris. He served as a Professor of continuum mechanics at University Joseph Fourier, Grenoble.

Claude Boutin is civil engineer. He received Habilitation at University Joseph Fourier, Grenoble. He serves as a Professor at école Nationale des Travaux publics de l'Etat, Lyon.

Christian Geindreau, after ENS Cachan, received a Ph.D in mechanics at the University Joseph Fourier. He serves as a Professor in mechanics at the University Joseph Fourier, Grenoble.