Some Unilateral Boundary-Value Problems for Elastic Bodies with Rugged Boundaries

Abstract

The system of linear elasticity is considered in a domain whose boundary depends on a small parameter ε > 0 and has a part with a rugged structure. The rugged part of the boundary may bend sharply and embrace cavities or channels, and as ε → 0, it approaches a limit surface on the boundary of the limit domain. On the rugged part of the boundary, conditions of two types are considered: (I) contact with rigid obstacles (conditions of Signorini type); (II) reaction forces involving the parameter ε and nonlinearly depending on displacements. We investigate the asymptotic behavior of weak solutions to such boundary-value problems as ε → 0 and construct the limit problem, according to the geometric structure of the rugged part of the boundary and the external surface forces and their dependence on the parameter ε. In general, the limit problem has the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone characterizes the boundary conditions of the limit problem and is described in terms of the functions involved in the nonlinear boundary conditions on the rugged boundary. As shown by examples, in the limit, the type of boundary condition may change. To justify these asymptotic results, we give a detailed exposition of some facts about extensions, Korn's inequalities, traces, and nonlinear boundary conditions in partially perforated domains with Lipschitz continuous boundaries. Bibliography: 16 titles.