A variational problem for an elastic body with periodically located cracks

We consider a nonlinear problem of equilibrium of an elastic body with periodically located cracks. On the edges of these cracks, non-penetration conditions are given. The nonlinear problem is formulated in the form of variational inequality. The period of distribution of the cracks, as well as their sizes, depends on a small parameter. The behavior of the solution to the problem with periodically located cracks is determined by the first two terms $\mathbf{u}^0(x)$ and $\mathbf{u}^1(x, y)$ of the asymptotic expansion. In this paper, we study the solution of the variational inequality on a periodicity cell (a local problem). For the first corrector $\mathbf{u}^1(x, y)$, we construct a penalty equation and a linear iterative equation in integral form. We prove that the sequence of solutions of the problem with penalty converges to the solution of the problem on the cell when the small regularization parameter tends to zero. We show that the approximate solution of the iteration equation converges strongly to the solution of the penalty equation.