On homogenization of a variational inequality for an elastic body with periodically distributed fissures

We study the problem of small deformations of an elastic body with periodically distributed fissures, where one-sided constraints are imposed on the sides of the fissures; this problem is equivalent to a variational inequality. We prove that if the linear size of the period of the distribution of the fissures tends to zero, then the solutions of this problem converge in the $L^2$-norm to the solution of the homogenized problem, which is a non-linear boundary-value problem of elasticity theory for a domain without fissures.