A general convergence result for a functional related to the theory of homogenization
SIAM Journal on Mathematical Analysis
An extension theorem from connected sets, and homogenization in general periodic domains
Nonlinear Analysis: Theory, Methods & Applications
Homogenization and two-scale convergence
SIAM Journal on Mathematical Analysis
Homogenization and porous media
Homogenizing the acoustic properties of the seabed: part I
Nonlinear Analysis: Theory, Methods & Applications - Lakshmikantham's Legacy: A tribute on his 75th birthday
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We undertake a rigorous derivation of the diphasic Biot's law, describing small deformations of a seabed of the characteristic size L"0/@e^2 and containing a pore structure of the characteristic size @e. The solid part of the seabed (the matrix) is elastic and the pores contain a viscous fluid. The fluid is supposed incompressible or slightly compressible. In this case, the contrast of property is of order @e^2, i.e., the normal stress of the elastic matrix is of the same order as the fluid pressure. We suppose a periodic matrix and obtain the a priori estimates. Then we let the characteristic size of the inhomogeneities tend to zero and pass to the limit in the sense of the two-scale convergence. The obtained effective equations represent a two-scale system for three velocities and two pressures. We prove uniqueness for the homogenized two-scale system. Then we introduce several auxiliary problems and obtain a problem without the fast scale. This new system is diphasic and corresponds to the diphasic effective behavior already observed in papers by Biot. In the effective equations, it is possible to distinguish the velocities of the fluid and the solid part, respectively. The effective stress tensor contains an instantaneous elasticity tensor and there are double porosity terms. We give a detailed study of the effective equations and compare them with the original Biot's poroelasticity equations.