As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R² is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

We propose a model for a 2D elastic body with a thin elastic inclusion in which delamination of the inclusion may take place, thus forming a crack. Non‐linear boundary conditions at the crack faces are imposed to prevent mutual penetration. We prove existence and uniqueness of the equilibrium configuration, considering both the variational and the differential formulations. Moreover, we study the dependence of solutions on the rigidity of the beam and we prove that in the limit corresponding to infinite and zero rigidity, we recover the case of a semi‐rigid inclusion and the case of a crack with non‐penetration conditions, respectively. The convergence of solutions is proved both using variational inequalities and Γ‐convergence.