Abstract
We consider a quasistatic problem which models the contact between a deformable body and an obstacle called foundation. The material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. In Contact Mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions. The mathematical analysis of contact models leads to the study of variational and hemivariational inequalities. For this reason a large number of contact problems lead to inequalities which involve history dependent operators, called history dependent inequalities. Such inequalities could be variational or hemivariational and variational hemivariational. In this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses, we stablish an existence and uniqueness result. The proof of the result is based on arguments of variational inequalities monotone operators and Banach fixed point theorem.
Highlights
In Contact Mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions
Contact mechanics still remain a rich domain of research, and the literature devoted to various aspects of the subject is growing
Further extensions to non convex contact conditions with non-monotone and possible multi-valued constitutive laws led to the active domain of non-smooth mechanic within the framework of the so-called hemivariational inequalities, for a mathematical as well as mechanical treatment we refer to [10]
Summary
Contact mechanics still remain a rich domain of research, and the literature devoted to various aspects of the subject is growing. A class of variational inequalities with history dependent operators was considered in [15], where abstract existence, uniqueness and regularity results were proved. We introduce a new model of frictional contact for viscoelastic materials and to illustrate the use of history dependent variational hemivariational inequality in its variational analysis. It is in a form of a historydependent variational-hemivariational inequality in which the unknown is the displacement field. in Section 4 we state our main existence and uniqueness result, Theorem (4.2) the proof of the theorem is obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for history dependent operators
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