History-dependent contact models for viscoplastic materials

Mikael Barboteu (1) , Flavius Pătrulescu (2) , Ahmad Ramadan (1) , Mircea Sofonea (1)

1 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations
2 Tiberiu Popoviciu Institute of Numerical Analysis

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.

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

Domaines: Mathématiques [math]

Mikael Barboteu, Flavius Pătrulescu, Ahmad Ramadan, Mircea Sofonea. History-dependent contact models for viscoplastic materials. IMA Journal of Applied Mathematics, 2014, 79 (6), pp.1180-1200. ⟨10.1093/imamat/hxt024⟩. ⟨hal-01142140⟩