We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The contact is frictionless and is modeled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational formulation of the problem which is the form of a history-dependent variational inequality for the displacement field.
Then, using a recent result obtained by Sofonea and Matei, we prove the unique weak solvability of the problem. Next, we study the continuous dependence of the weak solution with respect the data and prove a first convergence result. Finally, we prove that the weak solution of the problem represents the limit of the weak solution of a contact problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.
Laboratoire de Mathématiques et Physique, Université de Perpignan
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
M. Sofonea, F. Pătrulescu, Analysis of a history-dependent frictionless contact problem, Math. Mech. Solids, 18 (2013) no.4, pp. 409-430.
Mathematics and Mechanics of Solids
SAGE Publications, Thousand Oaks, CA