The problem on T-shape junction of thin inclusions

T. S. Popova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. 1578-1593

Résumé

In this paper, we study the problem of equilibrium of a two-dimensional elastic body containing two contacting thin inclusions. One of the inclusions is elastic and is modeled within the Timoshenko beam theory. The other inclusion is rigid and is characterized by a given structure of displacement functions. The inclusions intersect, forming a T-shaped system in an elastic matrix. Two cases of junction are considered: in the absence of a connection between the inclusions and for the case of perfect adhesion between them. It is assumed that the elastic inclusion delaminates from the elastic matrix forming a crack. Due to the presence of a crack, the elastic body occupies a domain with a cut, while on the cut edges, as on a part of the boundary, boundary conditions of the form of inequalities are set. The problem is posed as a variational one, and a complete differential formulation in the form of a boundary value problem is also obtained, including the junction conditions at a common point of inclusions. The equivalence of the variational and differential formulations of the problem is proved under the condition of sufficient smoothness of the solutions.

Détail

Keywords: variational inequality, Timoshenko inclusion, thin elastic inclusion, thin rigid inclusion, crack, non-penetration conditions, nonlinear boundary conditions, junction problem.

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