Homogenization

The following sections are included:

• PREFACE

• SERGUEI KOZLOV ‒ A BRIEF BIOGRAPHICAL SKETCH

• SOME RECOLLECTIONS

• SERGUEI KOZLOV ‒ A REVIEW OF SCIENTIFIC CONTRIBUTIONS

• LIST OF PUBLICATIONS OF S. KOZLOV

• CONTENTS

In problems of conduction and fluid flow through complex random media, systems with a broad distribution in the local properties are often encountered. Here we introduce a continuum percolation model for such media, which is exactly solvable for the effective transport properties in the high disorder limit. The model represents such systems as fluid flowing through consolidated granular media and fractured rocks, as well as electrical conduction in some matrix-particle composites near critical volume fractions. Moreover, the results provide a rigorous basis for the widely used Ambegaokar, Halperin, and Langer critical path analysis [1]. This method rests on the proposition that transport in media with a broad range of local conductances g is dominated by a critical conductance g_c, which is the smallest conductance such that the set {g | g > g_c} percolates, and in our model this proposition is rigorously established.

This paper develops the ideas which were first appeared in S. Kozlov's paper [5].

Herein we discuss the effective thermoconductivity and effective shear modulus of the rectangular (quadratic) lattice structure with the width of the bond equal to a small parameter. We give the asymptotic expansion of the effective thermoconductivity and effective shear modulus which are determined in terms of homogenization theory. First order approximation for the effective thermoconductivity is well known [1,2]. It follows from the principle of splitting of the homogenized operator. The main goal of the first part of this work is to find the correctors to this approximation. We give an explicit form of the second order approximation and of the complete asymptotic expansion. It corresponds to the influence of the orthogonal strips on the effective coefficient. In the second section we calculate the effective shear modulus of the same structure. Even the first order approximation was unknown. We obtain here this approximation as well as the complete asymptotic expansion.

Mathematical homogenization theory is used to calculate the effective moduli of composites and mixtures if the ratio ε of the length scale of inhomogeneity to the macroscopic length scale for the problem is small. In many problems there are additional small parameters γi, besides ε. In this paper structures and materials are considered under the assumptions that certain parameters γi formed by the ratios of different phases concentrations and moduli, or the ratios of coefficients connected with different properties of a phase are small. The approximate explicit formulae for the effective moduli are proposed and the errors of these formulae are estimated.

This paper deals with the homogenization of a sub-differential equation in randomly perforated domain with Dirichlet boundary condition on its numerous "holes" A physical interpretation of this problem corresponds to flows of viscous-plastic liquids through a porous medium. In this asymptotic problem the leading term of the solution satisfies the Darcy law that links the mean value of velocity and the pressure drop. The mean velocity depends on the pressure drop in a very special way due to plasticity of the liquid. Usually there is a vicinity of zero point on the axis of pressure drop where the velocity equals identically zero. It takes place for any periodically perforated domain at least. Here the random checkerboard is considered as an example of a porous structure. In this case the vicinity degenerates, and the mean velocity does not vanish for any positive pressure drop. The explicit asymptotic formula for the Darcy law in the neighborhood of zero point is the main result of this paper.

Many problems in mechanical engineering and physics are reduced to studying the following question. There is a variational problem to minimize a sum of quadratic and linear functionals. The linear functional is stochastic, therefore the minimum value is random. How to find the probability density function of the minimum value? Such type of problem appears in statistical mechanics of fluid motion, Kosterlitz-Throuless phase transition, brittle-to-ductile transition in solids, reaction of cracked bodies to random excitation, lubrication theory, etc. It is shown in this paper that the probability density function of minimum values is determined by some variational problem. The features of this variational problem are discussed.

An exactly solvable model of elastic crystal with random point defects is suggested. It is shown that if the defects are distributed according to Poisson's law then the strain-stress distribution function is described by a shifted Cauchy distribution (an asymmetric Lorentz curve). The asymmetry is calculated analytically in terms of the Frenel integral. A simple explanation of the an illusory contradiction between the asymmetry and the distribution's zero mean is presented.

Optimal design problems are usually formulated as problems of minimization of the energy, stored in the design under a prescribed loading. Solutions of these problems are unstable to perturbations of the loading. We suggest a new min-max formulation of the optimal design problem which has a stable solution. The loading is not prescribed, but only a set of admissible loadings is given. The stable optimal design problem is formulated as minimisation of the stored energy of the project under the most unfavorable loading. This most dangerous loading is one that maximizes the stored energy over the class of admissible functions. The problem is reduced to minimization of Steklov eigenvalues. Several stable solutions of various optimal design problems are demonstrated; among them are the optimal structure of a material stable to variations in uniaxial loading, the optimal specific stiffness of an uncertainly loaded beam, and the stable design of an optimal wheel.

The paper is concerned with the bounds on the effective properties of planar isotropic elastic composites. Specifically, coupled bulk-shear moduli bounds are presented that improve Hashin-Shtrikman-Walpole uncoupled bounds. They are given by the combination of the Hashin-Shtrikman-Walpole bulk modulus bounds, and the two new curves (hyperbolas) in the bulk-shear moduli plane. The bounds are identical to those obtained earlier by Cherkaev and Gibiansky[1]. However, the derivation is new and the form of the bounds is simpler. Derivation involves an analysis of quasiaffine quadratic forms of generalized fourth-order strain and stress tensors associated with eighth-order rotationally invariant tensors. Open questions, attainability of the bounds, and generalization to the three-dimensional problem are discussed.

We consider the Laplace equation in a bounded domain consisting of a porous medium, a nonperforated domain and an interface between them. The homogeneous Dirichlet condition is prescribed on the boundaries of the perforations. We suppose the porous medium having a periodic structure with a period ε and consider the limit ε → 0. The effective behaviors inside the porous medium and the nonperforated domain are obtained and in addition the boundary conditions to be imposed at the interface between two media are determined. Using the boundary layers describing interaction between two very different media we estimate the L²-norm of the difference between the solution u^ε of the ε-problem and the effective behavior terms and show that it is bounded from above by Cε^1/2. Finally, we calculate the weak L²-limit of the renormalized function u^ε/ε² and weak M(Ω)-limit of the renormalized ∇u^ε/ε².

In this paper, we give illustrations of some specific features of control over the solutions of linear hyperbolic equations produced by the variation of their senior coefficients.

The paper contains the spectral theory of one-dimensional SchrÖdinger operator with deterministic sparse potential. It generalizes and extends the classical results by D. Pearson and the recent results by B. Simon and his group (Caltech). The technical tools are: phase formalism, martingale methods, multiscaling averaging.

The following sections are included:

• Introduction

• Kozlov's estimate

• The Strange exponent

• The change of metic

• The physics of V

• The invariance of d(x,y)

• The homegeneous dimension

• Scaled Poincaré inequalities

• References

We prove the central limit theorem for one nonlinear stochastic partial differential equation with weak nonlinearity and study the spectral asymptotics of the infinitesimal generator of the corresponding Markov process.

Series on Advances in Mathematics for Applied Sciences