On the relationship between alternative variational formulations of a frictional contact model

Abstract

We consider a frictional contact model, mathematically described by means of a nonlinear boundary value problem in terms of PDE. We draw the attention to three possible variational formulations of it. One of the variational formulations is a variational inequality of the second kind and the other two are mixed variational formulations with Lagrange multipliers in dual spaces. As main novelty, we establish the relationship between these three variational formulations. We also pay attention to the recovery of the formulation in terms of PDE starting from the mixed variational formulations.Key words: boundary value problem, frictional contact, variational inequality of the second kind, mixed variational formulation, dual Lagrange multipliers, weak solutions.

Weak Solutions via Two-Field Lagrange Multipliers for Boundary Value Problems in Mathematical Physics

“…In this particular case, according to [12], the unique solution of Problem 4 coincides with the first component of the unique solution of Problem 5 as well as with the first component of each solution of Problem 3.…”

Three Weak Formulations for an Obstacle Model and Their Relationship

“…As it is known, the variational inequality of the first kind can be seen as a variational inequality of the second kind governed by the indicator function of the set K = {r ∈ R | r ≤ 0}. However, even that the primal variational formulations of both models rely on the theory of the variational inequalities of the second kind, the study in the present paper is not a particularization of the results obtained in [20]. The obstacle problem is a physical model completely different from the bilateral frictional contact model, the obstacle model being a unilateral frictionless contact problem.…”

“…The present paper can be seen as a continuation of [20] by considering a different class of contact models; unlike [20], where a bilateral frictional contact model was considered, herein a particular unilateral frictionless contact model is under consideration. In [20], the first variational formulation was a variational inequality of the second kind involving a convex functional governed by a positive friction bound. In the present paper, the first formulation is a variational inequality of the first kind.…”

Exponential stabilization of a dynamic frictional contact problem for viscoelastic materials with normal damped response and infinite memory

“…One can also find some results concerning optimal control for antiplane contact problems in [28,15,30]. More results on a dynamic contact problem can be found in [8,10,12,24,25,26,42,44,37,39], etc.…”

“…Multiplying the first equation in (25) by u(t), integrating by parts over Ω by recalling Green's formula (4) and boundary conditions and using assumption (18), it is inferred that…”