Equilibrium problem for an inhomogeneous two-dimensional elastic body with two interacting thin rigid inclusions

Abstract

A new nonlinear mathematical model is proposed that describes an equilibrium of a two-dimensional elastic body with two interacting thin rigid inclusions. Thin rigid inclusions are defined by straight line segments. In the initial state of the body, both inclusions are in contact at one breaking point. Both inclusions can move, subject to the conditions in the form of systems of inequalities describing a possible contact of the inclusions. These three systems of inequalities for infinitesimal rigid displacements correspond to three possible cases of mutual configurations of inclusions in equilibrium state. Inclusions exfoliate near the breaking point from the elastic matrix, in other words, there are cracks of a given length on both sides of thin inclusions. Nonpenetration conditions of the Signorini type are imposed on the curves defining the cracks. On a part of the outer boundary, clamping conditions are prescribed. The problem is formulated as a minimization of an energy functional over a non-convex set of possible displacements defined in an appropriate Sobolev space. The existence of a solution to the problem is proved. Optimality conditions of the solution have been obtained under the condition of sufficient smoothness of the solution.