Homogenization of Elasticity Problems with Boundary Conditions of Signorini type

G. A. Iosif'yan

Matematičeskie zametki, Tome 75 (2004) no. 6, pp. 818-833

Résumé

In a perforated domain $\Omega^\varepsilon =\Omega\cap\varepsilon \omega$ formed of a fixed domain $\Omega$ and an $\varepsilon$-compression of a 1-periodic domain $omega$, we consider problems of elasticity for variational inequalities with boundary conditions of Signorini type on a part of the surface $S^\varepsilon _0$ of perforation. We study the asymptotic behavior of solutions as $\varepsilon\to0$ depending on the structure of the set $S^\varepsilon _0$. In the general case, the limit (homogenized) problem has the two distinguishing properties: (i) the limit set of admissible displacements is determined by nonlinear restrictions almost everywhere in the domain $\Omega$, i.e., in the limit, the Signorini conditions on the surface $S^\varepsilon _0$ can turn into conditions posed at interior points of $\Omega$ (ii) the limit problem is stated for an homogenized Lagrangian which need not coincide with the quadratic form usually determining the homogenized elasticity tensor. Theorems concerning the homogenization of such problems were obtained by the two-scale convergence method. We describe how the limit set of admissible displacements and the homogenized Lagrangian depend on the geometry of the set $S^\varepsilon _0$ on which the Signorini conditions are posed.