The problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimensional elastic body

We consider the equilibrium problem for a two-dimensional elastic body containing two contacting thin inclusions of a rectilinear shape. The inclusions are elastic and are modeled within the framework of the theory of Timoshenko beams. The inclusions intersect at a right angle, and one of the inclusions delaminates from the elastic matrix, forming a crack. The problem is posed as a variational one and a complete differential formulation is obtained in the form of a boundary value problem, including junction conditions at a common point of inclusions. On the edges of the cut, boundary conditions of the form of inequalities are specified. The equivalence of the variational and differential formulations of the problem is proved under the condition of sufficient smoothness of the solutions. The passage to the limit with respect to the stiffness parameter of one of the inclusions is substantiated.