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Homogenization of a linear elastic body with rigid inclusions and a Robin type boundary conditions

Lazarus Signing

Abstract

This paper is devoted to study of the limiting behaviour of an elastic material with periodically distributed rigid inclusions of size ε, as the small parameter ε goes to zero. We address here the case with inclusions of the same size as the period of the structure. The body in consideration here is suppose to be clamped on one part of its exterior boundary and submitted to given tractions on the other. By means of the well known two-scale convergence techniques, one convergence result is proved.

Homogenization of a linear elastic body with rigid inclusions and a Robin type boundary conditions

Abstract

This paper is devoted to study of the limiting behaviour of an elastic material with periodically distributed rigid inclusions of size ε, as the small parameter ε goes to zero. We address here the case with inclusions of the same size as the period of the structure. The body in consideration here is suppose to be clamped on one part of its exterior boundary and submitted to given tractions on the other. By means of the well known two-scale convergence techniques, one convergence result is proved.

Paper Structure

This paper contains 3 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. A convergence result for the homogenization process

Key Result

Proposition 1

For each real $\varepsilon >0$, there exists an operator $\mathcal{P}_{\varepsilon }$ of $H^{1}\left( \Omega ^{\varepsilon };\mathbb{R}\right) ^{N}$ into $H^{1}\left( \Omega ;\mathbb{R}\right) ^{N}$ with the following properties: and for all $\mathbf{v}\in H^{1}\left( \Omega ^{\varepsilon };\mathbb{R}\right) ^{N}$, where the constant $c>0$ depends solely on $Y$ and $T$.

Theorems & Definitions (22)

  • Proposition 1
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Theorem 3
  • Remark 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • ...and 12 more