## Abstract

We consider two mathematical models which describe the frictionless process of contact between a rate-type viscoplastic body and a foundation. The contact is modelled with normal compliance and memory term such that penetration is not restricted in the first problem, but is restricted with unilateral constraint in the second one.

For each problem, we derive a variational formulation in terms of displacements, which is in a form of a history-dependent variational equation and a history-dependent variational inequality. Then we prove the unique weak solvability of each model. Next, we prove the convergence of the weak solution of the first problem and the weak solution of the second problem, as the stiffness coefficient of the foundation converges to infinity.

Finally, we provide numerical simulations which illustrate this convergence result.

## Authors

Mikael **Barboteu**

(Laboratoire de Mathématiques et Physique, Université de Perpignan)

Flavius **Patrulescu**

(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Ahmad **Ramadan**

(Laboratoire de Mathématiques et Physique, Université de Perpignan)

Mircea **Sofonea**

(Laboratoire de Mathématiques et Physique, Université de Perpignan)

## Keywords

viscoplastic material; frictionless contact; normal compliance; unilateral constraint; memory term; history-dependent variational inequality; weak solution; numerical simulations

## Cite this paper as:

M. Barboteu, F. Pătrulescu, A. Ramadan, M. Sofonea, *History-dependent contact models for viscoplastic materials*, IMA J. Appl. Math., 79 (2014) no. 6, pp. 1180-1200.

## About this paper

##### Publisher Name

Oxford University Press, Oxford

##### Print ISSN

0272-4960

##### Online ISSN

1464-3634

## MR

3286321

## ZBL

1307.74055

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