## Abstract

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.

## Authors

Mircea **Sofonea**

Laboratoire de Mathématiques et Physique, Université de Perpignan

Flavius **Patrulescu**

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

## Cite this paper as:

M. Sofonea, F. Pătrulescu, *Analysis of a history-dependent frictionless contact problem*, Math. Mech. Solids, 18 (2013) no.4, pp. 409-430.

## About this paper

##### Journal

Mathematics and Mechanics of Solids

##### Publisher Name

SAGE Publications, Thousand Oaks, CA

##### Print ISSN

1081-2865

##### Online ISSN

1741-3028

## MR

3179460

## ZBL

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soon