Evolutional contact with Coulomb friction on a periodic microstructure
We consider the elasticity problem in a heterogeneous domain with 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, but are locked to a matrix on a piece of the boundary. 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 (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005). A time discretization scheme from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005) reduces the contact problem to the frictional traction known from the previous step on each time-increment and is then solved by fixed point argument. For a fixed e, the necessary condition for the keeping the contact continuous in time (for the contraction mapping) is given in (Cocu and Rocca, Existence results for unilateral quasistaric contact problems with fricrtion and adhesion, 2000) and (Eck et al., Unilateral contact problems variational methods and existence theorems. Springer, 2005) in the form of the bound on the frictional coefficient by lower and upper bounds on the elastic tensor and norms of the direct and inverse trace operators. We further look for the spatial homogenization of the contact problems on each time-increment and introduce scaling of Sobolev-Slobodetsy norms and Bessel potentials. By shifting argument we obtain the preliminary estimates for normal tractions in a better space and proof its strong convergence. The limiting energy and the dissipation term in the stability condition obtained for the contact with Tresca's friction law in (Cioranescu et al., Asymptotic Anal. 82(3-4), 2013) are then valid also for the Coulomb one. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005), 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.