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A frictionless contact problem for viscoelastic materials. (English) Zbl 1035.74040
Summary: We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a viscoelastic behavior that we model with Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the Signorini condition with zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.

MSC:
74M15 Contact in solid mechanics
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
65N15 Error bounds for boundary value problems involving PDEs
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