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This paper is devoted to numerical analysis of a new class of elliptic variational–hemivariational inequalities in the study of a family of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modeled by a nonmonotone multivalued subdifferential relation allowing slip dependence. The problem involves a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set for the locking constraints and a nonconvex locally Lipschitz friction potential. Solution existence and uniqueness result on the inequality can be found in Migórski and Ogorzaly (2017). In this paper, we introduce and analyze a finite element method to solve the variational–hemivariational inequality. We derive a Céa type inequality that serves as a starting point of error estimation. Numerical results are reported, showing the performance of the numerical method. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions. [ABSTRACT FROM AUTHOR]

We consider a class of elliptic variational-hemivariational inequalities in an abstract Banach space for which we introduce the concept of wellposedness in the sense of Tykhonov. We characterize the well-posedness in terms of metric properties of a family of associated sets. Our results, which provide necessary and sufficient conditions for the well-posedness of inequalities under consideration, are valid under mild assumptions on the data. Their proofs are based on arguments of monotonicity, lower semicontinuity and properties of the Clarke directional derivative. For well-posed inequalities we also prove a continuous dependence result of the solution with respect to the data. We illustrate our abstract results in the study of one-dimensional examples, then we focus on some relevant particular cases, including variationalhemivariational inequalities with strongly monotone operators. Finally, we consider a model variational-hemivariational inequality which arises in Contact Mechanics for which we discuss its well-posedness and provide the corresponding mechanical interpretations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a dynamic frictional contact problem for viscoelastic materials with long memory is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality for the velocity. A result on the unique weak solvability to this inequality is proved through a recent contribution on evolutionary subdifferential inclusions and a fixed point approach. [ABSTRACT FROM AUTHOR]

Abstract The goal of the paper is to study the behavior of solutions to the history-dependent variational–hemivariational inequalities under perturbation of the data. First, we give an existence and uniqueness result which has been obtained by using previous works. Second, we deliver a continuous dependence result when all the data are subjected to perturbations. Finally, a quasistatic viscoelastic frictional contact problem is presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this article, we study the variational-hemivariational inequalities with Neumann boundary condition. Using a nonsmooth critical point theorem, we prove the existence of infinitely many solutions for boundary-value problems. Our technical approach is based on variational methods.

Abstract A static contact problem for inhomogeneous elastic materials is studied with a non-polynomial growth of the elasticity under the Coulomb’s law of dry friction and the normal compliance condition. We demonstrate the results on existence and uniqueness of a solution to an abstract subdifferential inclusion and a variational–hemivariational inequality in the reflexive Orlicz–Sobolev space which are applied to the static elastic frictional problem. [ABSTRACT FROM AUTHOR]

We consider a class of history-dependent variational-hemivariational inequalities with constraints. Besides the unique solvability of the inequalities, we study the behavior of the solution with respect to the set of constraints and prove a continuous dependence result. The proof is based on various estimates, monotonicity arguments and the properties of the Clarke subdifferential. Then, we consider a mathematical model which describes the equilibrium of a locking material with memory, in contact with an obstacle. We comment the model and state its weak formulation, which is in a form of a history-dependent variational-hemivariational inequality for the displacement field. We prove the unique weak solvability of the model, then we use our abstract result to prove the continuous dependence of the solution with respect to the set of constraints. We apply this convergence result in the study of an optimization problem associated to the contact model. [ABSTRACT FROM AUTHOR]

The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity. [ABSTRACT FROM AUTHOR]

We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interface law which involves unilateral constraints and subdifferential conditions. We describe the model and derive its weak formulation, which is in the form of an elliptic variational–hemivariational inequality for the displacement field. Then, we establish the existence of a unique weak solution to the problem. Next, we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a particular version of the model for which we prove the continuous dependence of the solution on the bounds which govern the locking and the normal displacement constraints, respectively. We apply this convergence result in the study of an optimization problem associated to the contact model. [ABSTRACT FROM AUTHOR]

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]

Abstract: In this paper, we provide results on existence, uniqueness and convergence for a class of variational‐hemivariational inequalities of elliptic type involving a constraint set and a nondifferentiable potential. We introduce a penalized and regularized problem without constraints and with G a ̂teaux differentiable potential. We prove that the solution to the original problem can be approached, as a parameter converges, by the solution of the approximated problem. An application to frictional contact problem with the Signorini contact condition and a static version of the Coulomb friction law illustrates the results. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of three solutions to the p(x)-Kirchhoff type equations in RN: By means of nonsmooth three critical points theorem and the theory of the variable exponent Sobolev spaces, we establish the existence of three critical points for the problem. Moreover, we study the existence of three radially symmetric solutions for a class of quasilinear elliptic inclusion problem with discontinuous nonlinearities in RN: Our approach is based on critical point theory for locally Lipschitz functionals due to Iannizzotto. [ABSTRACT FROM AUTHOR]

We study a new class of elliptic variational-hemivariational inequalities in reflexive Banach spaces. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. We deliver a result on existence and uniqueness of a solution to the inequality. Next, we show the continuous dependence of the solution on the data of the problem and we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a mathematical model which describes the equilibrium of an elastic body in unilateral contact with a foundation. The model leads to a variational-hemivariational inequality for the displacement field, that we analyse by using our abstract results. [ABSTRACT FROM AUTHOR]

This paper mainly analyzes the general elliptic variational–hemivariational inequalities with or without constraints by using the virtual element method. The approximations can be internal or external and a C e ́ a's type inequality is derived for a priori error estimates. Then, we apply the results to a variational–hemivariational inequality arising in frictional contact problems, and the optimal order error estimate is obtained for the linear virtual element solution under appropriate solution regularity assumptions. Finally, numerical simulation results are reported to show the performance of the proposed method, in particular, numerical convergence orders are in good agreement with the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper we study a class of quasi-variational–hemivariational inequalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a solution-dependent set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan–Glicksberg fixed point theorem. Two examples of the interior and boundary semipermeability models illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a new class of first order evolutionary variational-hemivariational inequalities for which we prove an existence and uniqueness result. The proof is based on a timediscretization method, also known as the Rothe method. It consists of considering a discrete version of each inequality in the class, proving its unique solvability, and recovering the solution of the continuous problem as the time step converges to zero. Then we introduce a quasi-static frictionless problem for Kelvin-Voigt viscoelastic materials in which the contact is modeled with a nonmonotone normal compliance condition and a unilateral constraint. We prove the variational formulation of the problem cast in the abstract setting of variational-hemivariational inequalities, with a convenient choice of spaces and operators. Further, we apply our abstract result in order to prove the unique weak solvability of the problem. [ABSTRACT FROM AUTHOR]

In this article, we study the variational-hemivariational inequalities with Neumann boundary condition. Using a nonsmooth critical point theorem, we prove the existence of infinitely many solutions for boundaryvalue problems. Our technical approach is based on variational methods. [ABSTRACT FROM AUTHOR]

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum ofL-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions. [ABSTRACT FROM AUTHOR]

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law. [ABSTRACT FROM PUBLISHER]

The goal of this paper is to study a mathematical model which describes the adhesive contact between a deformable body and a foundation. The body consists of a viscoelastic material with long memory and the process is assumed to be quasistatic. The adhesive contact condition on the normal plane is modeled by a version of normal compliance condition with unilateral constraint in which adhesion is taken into account, the adhesive contact condition on the tangential plane is described by an adhesive Clarke subdifferential condition, and the evolution of the bonding field is described by an ordinary differential equation. We derive the variational formulation of this problem which is a system of a variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. Then, we obtain existence and uniqueness results on abstract inclusions and abstract variational–hemivariational inequalities. Finally, we apply the abstract results to prove the existence of a unique weak solution to the contact problem. [ABSTRACT FROM AUTHOR]

A system of a first order history-dependent evolutionary variational– hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality. [ABSTRACT FROM AUTHOR]

The paper is devoted to the Galerkin method and Finite Element Method for a stationary variational–hemivariational inequality modelling unilateral adhesive and frictionless contact of an elastic body with a foundation. Adhesion is modelled by a simplified Winkler-type law. An abstract theorem on the convergence of the Galerkin method for a class of nonlinear and pseudomonotone elasticity operators is proved. The theorem generalizes the result of Haslinger et al. (1999, Finite Element Method for Hemivariational Inequalities, Theory, Methods and Applications. Boston: Kluwer Academic Publishers). The problem is solved numerically on a mesh of linear triangles, by minimization of an associated energy functional for the linear, coercive and symmetric case. For nonsmooth optimization the Proximal Bundle Method (PBM) is used. For the benchmark problem of 2D linear elasticity, the numerical assessment of the method convergence rate is done. Moreover, tests are performed to establish the speed of PBM convergence. [ABSTRACT FROM AUTHOR]

In this paper we present global a priori bounds for a class of variational inequalities involving general elliptic operators of second-order and terms of generalized directional derivatives. Based on Moser's and De Giorgi's iteration technique we prove the boundedness of solutions of such inequalities under certain criteria on the set of constraints. In our proofs we also use the localization method with a certain partition of unity and a version of a multiplicative inequality estimating the boundary integrals. Some sets of constraints satisfying the required conditions are stated as well. [ABSTRACT FROM AUTHOR]

A class of variational-hemivariational inequalities is studied in this paper. An inequality in the class involves two nonlinear operators and two nondifferentiable functionals, of which at least one is convex. An existence and uniqueness result is proved for a solution of the inequality. Continuous dependence of the solution on the data is shown. Convergence is established rigorously for finite element solutions of the inequality. An error estimate is derived which is of optimal order for the linear finite element method under appropriate solution regularity assumptions. Finally, the results are applied to a variational-hemivariational inequality arising in the study of some frictional contact problems. [ABSTRACT FROM AUTHOR]

• A new unilateral constraint frictional contact model for a version of normal velocity is studied. • A coupled system that consists of a hemivariational inequality and a variational fihemivariational inequality is derived. • The unique weak solvability of the contact problem is obtained. A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations. [ABSTRACT FROM AUTHOR]

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive support. We model the material's behavior with an electro-elastic constitutive law; the frictional contact is described with a boundary condition involving Clarke's generalized gradient and the electrical condition on the contact surface is modelled using the subdifferential of a proper, convex and lower semicontinuous function. We derive a variational formulation of the model and then, using a fixed point theorem for set valued mappings, we prove the existence of at least one weak solution. Finally, the uniqueness of the solution is discussed; the investigation is based on arguments in the theory of variational-hemivariational inequalities. [ABSTRACT FROM AUTHOR]

Abstract: The goal of the present paper is twofold. First, we study the -dependence of the solution set of the variational–hemivariational inequality depending on a parameter of the form where is the -Laplacian, and is a nonempty closed, convex set of the Sobolev space . Assuming an ordered pair of appropriately defined sub–supersolutions, we are going to show by variational methods that is compact-valued and possesses extremal single-valued selections, which depend monotonously on provided that satisfies a certain monotonicity assumption. Second, the results of the first part along with regularity results for the -Laplacian allow us to characterize the solution behavior of an unstable degenerate elliptic free boundary problem (for and ) of the form: [Copyright &y& Elsevier]

We consider a mathematical model which describes the adhesive contact between a linearly elastic body and an obstacle. The process is static and frictionless. The normal contact is governed by two laws. The first one is a Signorini law, representing the fact that there is no penetration between a body and an obstacle. The second one is a Winkler type law signifying that if there is no contact, the bonding force is proportional to the displacement below a given bonding threshold and equal to zero above the bonding threshold. The model leads to a variational-hemivariational inequality. We present the numerical results for solving a simple two-dimensional model problem with the Proximal Bundle Method (PBM). We analyze the method sensitivity and convergence speed with respect to its parameters. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational-hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational-hemivariational inequality and give some conditions under which the variational-hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational-hemivariational inequality and the well-posedness of corresponding inclusion problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a class of obstacle problems for variational–hemivariational inequalities, by using nonsmooth version of three points critical theory in [S.A. Marano, D. Motreanu, On a three critical points theorem for non-differentiable functions and application to nonlinear boundary value problems, Nonlinear Anal. 48 (2002) 37–52], the existence of three solutions for the problem is obtained. [Copyright &y& Elsevier]

Abstract: The well-known method of sub- and supersolutions is a powerful tool for proving existence and comparison results for initial and/or boundary value problems of nonlinear ordinary differential equations as well as for nonlinear partial differential equations of elliptic and parabolic type. The main goal of this paper is to extend the idea of the sub- and supersolution method in a natural and systematic way to quasilinear elliptic variational–hemivariational inequalities. Owing to the intrinsic asymmetry of the latter (where the problems are stated as inequalities rather than as equalities) an appropriate generalization of the notion of sub- and supersolutions to variational–hemivariational inequalities is by no means straightforward. The obtained results of this paper complement the development of the sub- and supersolution method for nonsmooth variational problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to treat general nonlinear eigenvalue problems for variational–hemivariational inequalities which depend on a parameter. A result ensuring the existence and location information for the solutions is given by using a new minimax method. [Copyright &y& Elsevier]

In this paper, we consider a model which describes the quasistatic contact between a viscoplastic body and a foundation. The material's behavior is modeled with a rate-type viscoplastic constitutive law with an internal state variable. The contact is modeled with normal compliance, unilateral constraint, memory term, and friction which is under a total slip-dependent version of Coulomb's law. For the weak formulation of the problem, which is in the form of a system coupling two nonlinear integral equations with a history-dependent variational–hemivariational inequality, we introduce a fully discrete scheme and derive an error estimate. Under appropriate regularity assumptions, we obtain an optimal-order error estimate in finite element spaces. Finally, numerical results are reported to show the performance of the numerical method. [ABSTRACT FROM AUTHOR]