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In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational-hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem. [ABSTRACT FROM AUTHOR]

We consider a dynamic process of frictional contact between a non-clamped viscoelastic body and a foundation. We assume that the normal contact response depends on the depth of penetration of the foundation by the considered body, and the dependence between these two quantities is governed by normal compliance conditions. On the other hand, the friction force is assumed to be a nonmonotone function of the slip rate where the friction threshold also depends on the depth of the penetration. Our aim in this paper is twofold. The first one is to prove the existence and the uniqueness of a weak solution for the contact problem under consideration. The second one is to provide the numerical analysis of the process involving its semi-discrete and fully discrete approximation as well as estimation of the error for both numerical schemes and the validation of such a result. [ABSTRACT FROM AUTHOR]

We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper the sensitivity of optimal solutions to control problems described by second order evolution subdifferential inclusions under perturbations of state relations and of cost functionals is investigated. First we establish a new existence result for a class of such inclusions. Then, based on the theory of sequential $$\Gamma $$ -convergence we recall the abstract scheme concerning convergence of minimal values and minimizers. The abstract scheme works provided we can establish two properties: the Kuratowski convergence of solution sets for the state relations and some complementary $$\Gamma $$ -convergence of the cost functionals. Then these two properties are implemented in the considered case. [ABSTRACT FROM AUTHOR]

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the mathematical modelling of viscoelastic frictional contact processes in mechanics which involve adhesion. The model consists of a coupled system of the hemivariational inequality of hyperbolic type for the displacement and the ordinary differential equation for the bonding field. The frictional forces are derived from nonconvex superpotentials through the generalized Clarke subdifferential. The properties of the body are described by a modified Kelvin–Voigt constitutive law. The existence of weak solution to the problem is proved by embedding it into a class of second order evolution inclusions and by applying a surjectivity result for multivalued pseudomonotone operators. We also establish a result on the regularity of weak solution to the model. Finally, examples of subdifferential boundary conditions which include the functions of d.c. type are provided. [Copyright &y& Elsevier]