Thin inclusion at the junction of two elastic bodies: non-coercive case.

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Junction problem for thin elastic and volume rigid inclusions in elastic body.

The article concerns a junction problem for two-dimensional elastic body with a thin elastic inclusion and a volume rigid inclusion. It is assumed that the inclusions have a common point. A delamination of the thin inclusion from the surrounding elastic body is assumed thus forming an interfacial crack. Constraint-type boundary conditions are imposed at the crack faces to prevent interpenetration between the faces. Moreover, a connection between the crack faces is characterized by a positive damage parameter. Limit transitions are justified as the damage parameter tends to infinity and to zero. In addition to this, a transition to limit is analysed as a rigidity parameter of the thin inclusion tends to infinity. Limit models are investigated. In particular, junction conditions at the common point are found for all cases considered. This article is part of the theme issue 'Non-smooth variational problems and applications'.

On the topological derivative due to kink of a crack with non-penetration. Anti-plane model.

A topological derivative is defined, which is caused by kinking of a crack, thus, representing the topological change. Using variational methods, the anti-plane model of a solid subject to a non-penetration condition imposed at the kinked crack is considered. The objective function of the potential energy is expanded with respect to the diminishing branch of the incipient crack. The respective sensitivity analysis is provided by a Saint-Venant principle and a local decomposition of the solution of the variational problem in the Fourier series.