Homogenization via unfolding in periodic elasticity with contact on closed and open cracks

We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε 1, including multiple micro-contacts between the structural components. These components can be a simply connected matrix domain with open cracks or inclusions completely surrounded by cracks, which do not touch the outer boundary. The contacts are described by the Signorini and Tresca-friction contact conditions. The Signorini condition is described mathematically by a closed convex cone, while the friction condition is a nonlinear convex functional over the interface jump of the solution on the oscillating interface. The difficulties appear when the inclusions are completely surrounded by cracks and can have rigid displacements. In this case, in order to obtain preliminary estimates for the solution in the ε-domain, the Korn inequality should be modified, first in the fixed context and then for the ε-dependent periodic case. Additionally, for all states of the contact (inclusions can freely move, or are locked/stuck to the interface with the matrix, or the frictional traction is achieved on the inclusion-matrix interface and the inclusions can slide in the tangential to the interface direction) we obtain estimates for the solution in the ε-domain, uniform with respect to ε. An asymptotic analysis (as ε → 0) for the nonlinear functionals over the growing interface is carried out, based on the application of the periodic unfolding method for sequences of jumps of the solution on the oscillating interface. This allows to obtain the homogenized limit as well as a corrector result.