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.
Highlights
Mathematical modeling of the processes associated with deformations of solids with inhomogeneities in the form of inclusions or cracks is an actively studied research topic
For inhomogeneous bodies with a crack along the boundary of a rigid or elastic inclusion, problems are further complicated by relations describing mechanical interaction of the inclusion and the supporting matrix
We investigate the connection between two different twodimensional models describing the equilibrium of an elastic body with a rigid inclusion
Summary
Mathematical modeling of the processes associated with deformations of solids with inhomogeneities in the form of inclusions or cracks is an actively studied research topic. For inhomogeneous bodies with a crack along the boundary of a rigid or elastic inclusion, problems are further complicated by relations describing mechanical interaction of the inclusion and the supporting matrix. We follow an approach that uses inequality-type boundary conditions on the crack faces [14,15,16,17,18,19,20,21,22].
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