An equilibrium problem for a 2D elastic body with thin inclusions and defects is analyzed. The presence of defects means that the problem is formulated in a non-smooth domain. The defects are characterized by a positive damage parameter. Nonlinear boundary conditions at the defect faces are imposed to prevent a mutual penetration between the faces. An existence of solutions is proved, and different formulations of the problem are proposed. We study an asymptotics of solutions with respect to the damage parameter and analyze the limit models. Moreover, we study the dependence of the solution on the rigidity parameter of the inclusions. In particular, passages to infinity and to zero of the rigidity parameter are investigated.