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On the frictionless unilateral contact of two viscoelastic bodies. (English) Zbl 1121.74422

Summary: We consider a mathematical model which describes the quasistatic contact between two deformable bodies. The bodies are assumed to have a viscoelastic behavior that we model with Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the classical Signorini condition with zero-gap function. We derive a variational formulation of the problem and prove the existence of a unique weak solution to the model by using arguments of evolution equations with maximal monotone operators. We also prove that the solution converges to the solution of the corresponding elastic problem, as the viscosity tensors converge to zero. We then consider a fully discrete approximation of the problem, based on the augmented Lagrangian approach, and present numerical results of two-dimensional test problems.

MSC:

74M15 Contact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
74D10 Nonlinear constitutive equations for materials with memory
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
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