On the connection between two equilibrium problems for cracked bodies in the cases of thin and volume rigid inclusions

Abstract

Abstract We consider equilibrium problems for an inhomogeneous two-dimensional body with a crack and a rigid inclusion. The matrix of the body is assumed to be elastic. The boundary condition on the crack curve is an inequality describing mutual nonpenetration of the crack faces. We study two different equilibrium models. For the first model, we assume that a volume rigid inclusion is described by a domain. The second one describes a body containing a set of connected thin rigid inclusions, each corresponding to a curve. The crack is given by the same curve in both models. We prove that the solutions of equilibrium problems corresponding to the second model strongly converge to the problem solution for the first model as the number of inclusions tends to infinity.

Keywords

• Variational inequality
• Nonpenetration
• Non-linear boundary conditions
• Crack
• Rigid inclusion