The control variational method for beams in contact with deformable obstacles

Abstract

We consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. We state the variational formulation of the problem and prove an existence and uniqueness result for the weak solution. Then, we provide an alternative approach to the model, based on the control variational method. Necessary and sufficient optimality conditions are derived, together with an approximation property. Finally, we extend our results to some versions of the model which describe the contact with a single obstacle, including a timedependent case.Mathematics Subject Classification: 74M15, 49J15, 49S05, 49J40, 74K10.

History-Dependent Problems with Applications to Contact Models for Elastic Beams

“…Its aim is to complete [13] with a new existence and uniqueness results in the study of a class of subdifferential inclusions and hemivariational inequalities, and to apply these results in the analysis of a quasistatic contact model for elastic beams, which extends the contact model considered in [2]. A brief comparison between the results obtained in this current paper and those in [2,13] is the following.…”

“…This paper is a continuation of [2,13]. Its aim is to complete [13] with a new existence and uniqueness results in the study of a class of subdifferential inclusions and hemivariational inequalities, and to apply these results in the analysis of a quasistatic contact model for elastic beams, which extends the contact model considered in [2].…”

“…Moreover, one may use such models as tests and benchmarks for computer schemes meant for simulation of complicated multidimensional contact problems. Models, analysis and simulations of contact problems for beams can be found in [2,6,10,11,18] and the references therein. In [2], a mathematical model which describes the unilateral contact of a beam between two deformable obstacles was considered.…”

“…Models, analysis and simulations of contact problems for beams can be found in [2,6,10,11,18] and the references therein. In [2], a mathematical model which describes the unilateral contact of a beam between two deformable obstacles was considered. The unique weak solvability of the model was obtained by using the control variational method and numerical simulation related to this method were presented, as well.…”

“…With respect to [2], the main novelty of the model studied in this paper lies in the contact condition we use. As a consequence, the problem we study here is time-dependent and, therefore, neither the arguments on stationary variational inequalities nor the arguments on the control variational method used in [2] work in this case. For this reason we use the arguments on history-dependent hemivariational inequalities we develop previously in that paper.…”