Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

History-dependent variational–hemivariational inequalities in contact mechanics

We consider an abstract class of variational–hemivariational inequalities which arise in the study of a large number of mathematical models of contact. The novelty consists in the structure of the inequalities which involve two history-dependent operators and two nondifferentiable functionals, a convex and a nonconvex one. For these inequalities we provide an existence and uniqueness result of the solution. The proof is based on arguments of surjectivity for pseudomonotone operators and fixed point. Then, we consider a viscoelastic problem in which the contact is frictionless and is modeled with a new boundary condition which describes both the instantaneous and the memory effects of the foundation. We prove that this problem leads to a history-dependent variational–hemivariational inequality in which the unknown is the displacement field. We apply our abstract result in order to prove the unique weak solvability of this viscoelastic contact problem.     

A class of history-dependent systems of evolution inclusions with applications

This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, we prove that the system has a unique solution. We conclude with two examples of an evolutionary differential variational–hemivariational inequality and of a dynamic frictional contact problem in mechanics, which illustrate the abstract results.     

A quasistatic viscoplastic contact problem with normal compliance,unilateral constraint,memory term and friction

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint and memory term. We present the classical formulation of the problem together with the list of assumptions on the data. Then we derive the variational–hemivariational formulation of the model and we prove its unique weak solvability. The proof is based on a recent abstract result of a class of history-dependent variational–hemivariational inequalities.     

A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the Kelvin–Voigt viscoelastic law, include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using a surjectivity result for operators of pseudomonotone type. The uniqueness is obtained for a large class of operators of subdifferential type satisfying a relaxed monotonicity condition.     

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.     

Adhesive contact problem for viscoplastic materials

We examine a mathematical model that describes a quasistatic adhesive contact between a viscoplastic body and deformable foundation. The material’s behaviour is described by the rate-type constitutive law which involves functions with a non-polynomial growth. The contact is modelled by the normal compliance condition with limited penetration and adhesion, a subdifferential friction condition also depending on adhesion, and the evolution of bonding field is governed by an ordinary differential equation. We present the variational formulation of this problem which is a system of an almost history-dependent variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. The results on existence and uniqueness of solution to an abstract almost history-dependent inclusion and variational–hemivariational inequality in the reflexive Orlicz–Sobolev space are proved and applied to the adhesive contact problem.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Analysis of a general dynamic history-dependent variational–hemivariational inequality

This paper is devoted to the study of a general dynamic variational–hemivariational inequality with history-dependent operators. These operators appear in a convex potential and in a locally Lipschitz superpotential. The existence and uniqueness of a solution to the inequality problem is explored through a result on a class of nonlinear evolutionary abstract inclusions involving a nonmonotone multivalued term described by the Clarke generalized gradient. The result presented in this paper is new and general. It can be applied to study various dynamic contact problems. As an illustrative example, we apply the theory on a dynamic frictional viscoelastic contact problem in which the contact is modeled by a nonmonotone Clarke subdifferential boundary condition and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the total slip.     

Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction

We examine a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the slip displacement dependent friction. First, we derive a weak formulation of the model in the form of a hemivariational inequality. Then we embed the hemivariational inequality into a class of second-order evolution inclusions for which we provide a result on the existence of a solution. We conclude with examples of the subdifferential boundary conditions for contact with normal compliance and the slip dependent friction.     

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.     

A class of history-dependent variational-hemivariational inequalities

We consider a new class of variational-hemivariational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities, since each inequality of the class involve unilateral constraints, a history-dependent operator and two nondifferentiable functionals, of which at least one is convex. We prove an existence and uniqueness result of the solution. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. Then, we prove a convergence result which shows the continuous dependence of the solution with respect to the data. Finally, we present a quasistatic frictionless problem for viscoelastic materials in which the contact is modeled with normal compliance and finite penetration and the elasticity operator is associated to a history-dependent Von Mises convex. We prove that the variational formulation of the problem cast in the abstract setting of history-dependent quasivariational inequalities, with a convenient choice of spaces and operators. Then we apply our general results in order to prove the unique weak solvability of the contact problem and its continuous dependence on the data.     

Vanishing viscosity for hemivariational inequalities modeling dynamic problems in elasticity

In this paper we consider a mathematical model describing a dynamic linear elastic contact problem with nonmonotone skin effects. The subdifferential multivalued and multidimensional reaction–displacement law is employed. We treat an evolution hemivariational inequality of hyperbolic type which is a weak formulation of this mechanical problem. We establish a result on the existence of solutions to the Cauchy problem for the hemivariational inequality. This result is a consequence of an existence theorem for second order evolution inclusion. For the latter, using the parabolic regularization method, we obtain the solution as a limit when the viscosity term tends to zero.     

A Class of Subdifferential Inclusions for Elastic Unilateral Contact Problems

We consider a class of stationary subdifferential inclusions in a reflexive Banach space. We reformulate the problem in terms of a variational inequality with multivalued term and prove an existence result using the Kakutani-Fan-Glicksberg fixed point theorem. This approach allows to consider, in a natural way, a dual variational formulation of the problem. Next, we study the link between the primal and dual formulations and provide an equivalence result. Then, we consider a new mathematical model which describes the contact of an elastic body with a foundation. We apply the abstract formalism to derive the primal and the dual variational formulations of the problem, in terms of displacement and stress, respectively. Finally, we present existence and equivalence results in the study of this contact model.     

Variational analysis of fully coupled electro‐elastic frictional contact problems

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a foundation. The material behavior is described with a nonlinear electro‐elastic constitutive law. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive and both the frictional contact and the conductivity on the contact surface are described with subdifferential boundary conditions which involve a fully coupling between the mechanical and electrical variables. We derive a variational formulation of the problem which is in the form of a system coupling two hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on recent results for inclusions of subdifferential type in Sobolev spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response

In this paper we deal with a viscoelastic unilateral contact problem with normal damped response. The process is assumed to be dynamic and frictionless. Normal damping function is modeled by the Clarke subdifferential of a nonconvex and nonsmooth function. First, the variational formulation of this problem is provided in the form of a nonlinear first order variational–hemivariational inequality for the velocity field. Then, based on the surjectivity results for pseudomonotone and maximal monotone operators, we obtain the unique solvability for a new class of abstract evolutionary variational-hemivariational inequalities. Finally, we apply our abstract results to prove the existence of a unique weak solution to the corresponding contact problem.     

A convergence result for history-dependent quasivariational inequalities

In this paper, we consider a general class of history-dependent quasivariational inequalities with constraints. Our aim is to study the behavior of the solution with respect to the set of constraints and, in this matter, we prove a continuous dependence result. The proof is based on various estimates and monotonicity arguments. Then, we consider two mathematical models which describe the equilibrium of a viscoplastic and viscoelastic body, respectively, in contact with a deformable foundation. The variational formulation of each model is in a form of a history-dependent quasivariational inequality for the displacement field, governed by a set of constraints. We prove the unique weak solvability of each model, then we use our abstract result to prove the continuous dependence of the solution with respect to the set of constraints.     

Homogenization of boundary hemivariational inequalities in linear elasticity

In this paper we consider a mathematical model describing static elastic contact problems with the Hooke constitutive law and subdifferential boundary conditions. We treat boundary hemivariational inequalities which are weak formulations of contact problems. We establish existence and uniqueness of solutions to hemivariational inequalities. Using the notion of H-convergence of elasticity tensors we investigate the limit behavior of the sequence of solutions to hemivariational inequalities.     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.