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A class of integro-differential variational inequalities with applications to viscoelastic contact. (English) Zbl 1080.47052
Summary: We consider a class of abstract evolutionary variational inequalities arising in the study of frictional contact problems for linear viscoelastic materials with long-term memory. First, we prove an abstract existence and uniqueness result, by using arguments of evolutionary variational inequalities and Banach’s fixed point theorem. Next, we study the dependence of the solution on the memory term and derive a convergence result. Then, we consider a contact problem to which the abstract results apply. The problem models a quasistatic process, the contact is bilateral and the friction is modeled with Tresca’s law. We prove the existence of a unique weak solution to the model and we provide the mechanical interpretation of the corresponding convergence result. Finally, we extend these results to the study of a number of quasistatic frictional problems for linear viscoelastic materials with long-term memory.

MSC:
47N60Applications of operator theory in biology and other sciences
47J20Inequalities involving nonlinear operators
47J35Nonlinear evolution equations
74D99Materials of strain-rate type and history type, other materials with memory
74M10Friction (solid mechanics)
74M15Contact (solid mechanics)