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Evolutional contact with Coulomb friction on a periodic microstructure. (English) Zbl 1456.74138
Constanda, Christian (ed.) et al., Integral methods in science and engineering. Theoretical and computational advances. Papers based on the presentations at the international conference, IMSE, Karlsruhe, Germany, July 21–25, 2014. Cham: Birkhäuser/Springer. 455-470 (2015).
Summary: We consider the elasticity problem in a heterogeneous domain with $$\varepsilon$$-periodic micro-structure, $$\varepsilon\ll 1$$, 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 [A. Mielke and R. Rossi, Math. Models Methods Appl. Sci. 17, No. 1, 81–123 (2007; Zbl 1121.34052)]. A time discretization scheme from [Mielke and Rossi, loc. cit.] 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 $$\varepsilon$$, the necessary condition for the keeping the contact continuous in time (for the contraction mapping) is given in [M. Cocu and R. Rocca, M2AN, Math. Model. Numer. Anal. 34, No. 5, 981–1001 (2000; Zbl 0984.74054); C. Eck et al., Unilateral contact problems. Variational methods and existence theorems. Boca Raton, FL: Chapman & Hall/CRC (2005; Zbl 1079.74003)] 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 [D. Cioranescu et al., Asymptotic Anal. 82, No. 3–4, 201–232 (2013; Zbl 1273.74413)] 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, loc. cit.], 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.
For the entire collection see [Zbl 1330.65004].

##### MSC:
 74M25 Micromechanics of solids 74M15 Contact in solid mechanics 74M10 Friction in solid mechanics
##### Keywords:
periodic microstructure; micro-contact; Coulomb friction
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