We consider an equilibrium problem for a 2D elastic body with two thin closely located elastic inclusions. Inclusions are in a contact with each other which means a presence of a crack between them. Nonlinear boundary conditions of inequality type are imposed at the crack faces providing a mutual non-penetration. Moreover, the inclusions cross the external boundary of the elastic body. The unique solvability of the problem is proved. Passages to limits are investigated as rigidity parameters of the inclusions tend to infinity, and limit models are analyzed. Inverse problems for finding the rigidity parameter and Lamé parameters of the elastic body are investigated with a boundary measurement of the tip point displacement of the inclusion.