Evolutional contact with Coulomb's friction on a periodic microstructure
We consider the elasticity problem in a heterogeneous domain with an e-periodic micro-structure, e ≪ 1, including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile-like material. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in . A time discretization scheme from  reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time-increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in  for closed cracks and can be extended to textile-like materials. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from , for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro-level lead to a kind of an evolutional plastic behavior of the homogenized medium.