Asymptotic analysis of junction problem for Euler – Bernoulli and Timoshenko inclusions in elastic body

We consider the equilibrium problem for a 2D elastic body with two thin elastic inclusions with a junction at a point. It is assumed that a crack exists between the inclusions and the body. Inequality-type boundary conditions are imposed at the crack faces to prevent mutual penetration. The problem depends on rigidity parameter of one of the inclusions: we are talking about family of problems. A weak convergence of solutions of the family of problems in suitable functional spaces is proved. By this convergence, we pass to the limit in the problems and establish the form of limit problem. Strong convergence of solutions of family of problems is also established. On its basis, the existence of a solution of the optimal control problem is proved. The optimal control problem is formulated in accordance with the Griffiths failure criterion, the control parameter is the rigidity parameter of the inclusion.