Equilibrium problem for a Timoshenko plate with a geometrically nonlinear condition of nonpenetration for a vertical crack

Under consideration are the variational problems concerning the equilibrium of plates containing a crack. Two new mathematical models are proposed in which the nonpenetration conditions define the corresponding nonconvex sets of admissible functions. The first model describes the equilibrium of a Timoshenko plate with a crack, and the second corresponds to a composite plate containing a crack along a Kirchhoff—Love elastic inclusion. The proposed approach is substantiated by an explicit example. We prove the existence of solutions for the corresponding variational problems and show that the equilibrium equations are satisfied for each of the problems.