Numerical solution of the equilibrium problem for a two-dimensional elastic body with a thin semirigid inclusion

Abstract

The equilibrium problem for a two-dimensional elastic body containing a thin semirigid inclusion is considered. The inclusion delaminates from the elastic matrix, forming a crack; therefore, the problem is posed in a nonsmooth domain with a cut. The mathematical model of the delaminated thin semirigid inclusion was developed on the assumption that the rigidity of the material differs in different directions. The problem statement is presented both in the form of a variational inequality and in the form of a boundary value problem. The boundary condition on the crack faces has a form of inequality and, as a result, the problem is non-linear. Consequently, the construction of an algorithm for the numerical solution of the problem requires the use of additional analytical methods. The methods of domain decomposition, the method of Lagrange multipliers, and the finite element method are used. An algorithm for the numerical solution of the problem is constructed and a computational example is provided.