Homogenization of contact problem with Coulomb's friction on periodic cracks

We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb-friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix-point argument in the Sobolev spaces with a little higher regularity, H1+alpha. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed-point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth-order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro-polar elasticity.