Elastic contact problem with Coulomb friction and normal compliance in Orlicz spaces

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.     

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].     

Analysis of a history-dependent frictional contact problem

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation. The material’s behaviour is modelled with a constitutive law with long memory. The contact is frictional and is modelled with normal compliance and memory term, associated to the Coulomb’s law of dry friction. We present the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. Then we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities. We also study the dependence of the weak solution with respect to the data and prove a convergence result.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

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 frictionless contact problem for elastic-viscoplastic materials with internal state variables

This paper deals with an initial and boundary value problem describing the quasistatic evolution of a rate-type viscoplastic material with internal state variables which is in frictionless contact. Two variational formulations of the problem are proposed, and existence and uniqueness results established. The equivalence of the variational formulation is studied and a strong convergence result involving penalized problems is proved.     

Dynamic contact problem for viscoelastic piezoelectric materials with normal damped response and friction

In this paper, we deal with a class of inequality problems for dynamic frictional contact between a piezoelectric body and a foundation. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential. 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. The existence of a weak solution to the model is proved by embedding the problem into a class of second-order evolution inclusions and by applying a surjectivity result for multivalued operators.     

Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces

We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces. To do this, we prove the weak–weak type modular inequality of the Hardy–Littlewood maximal operator with respect to the Young function. Orlicz–Morrey spaces contain $L^p$ spaces ($1\le p\le \infty$), Orlicz spaces, and generalized Morrey spaces as special cases. Hence, we get necessary and sufficient conditions on these function spaces as corollaries.     

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

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.     

An electro-viscoelastic frictional contact problem with damage

We consider a quasistatic frictional contact problem between a piezoelectric body and a foundation. The contact is modeled with normal compliance and friction is modeled with a general version of Coulomb's law of dry friction; the process is quasistatic and the material's behavior is described by an electro-viscoelastic constitutive law with damage. We derive a variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, and the damage field. Then we provide the existence of a unique weak solution to the model. The proof is based on arguments of evolutionary variational inequalities and fixed point.     

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.     

Nonhomogeneous eigenvalue problem with indefinite weight

This paper, following the theory of partial differential equations on the Orlicz–Sobolev spaces, is mainly concerned with the nonhomogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. The results show that the spectrum of such problems contains a continuous family of eigenvalues.     

A quasistatic viscoplastic contact problem with normal compliance and friction

We consider a quasistatic contact problem between a viscoplasticbody and an obstacle, the so-called foundation. The contactis modelled with normal compliance and the associated versionof Coulomb's law of dry friction. We derive a variational formulationof the problem and, under a smallness assumption on the normalcompliance functions, we establish the existence of a weak solutionto the model. The proof is carried out in several steps. Itis based on a time-discretization method, arguments of monotonicityand compactness, Banach fixed point theorem and Schauder fixedpoint theorem.     

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.     

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.     

A viscoelastic frictionless contact problem with adhesion

We consider a model for the quasistatic, bilateral, adhesive and frictionless contact between a viscoelastic body and a rigid foundation. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and an integro-differential equation for the bonding field. The existence of a unique weak solution for the problem is established by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space. We also consider the problem describing the bilateral contact between two viscoelastic bodies, and establish similar results.     

Strong convergence of a hybrid steepest descent method for the split common fixed point problem

We study the convergence properties of an iterative method for a variational inequality defined on a solution set of the split common fixed point problem. The method involves Landweber-type operators related to the problem as well as their extrapolations in an almost cyclic way. The evaluation of these extrapolations does not require prior knowledge of the matrix norm. We prove the strong convergence under the assumption that the operators employed in the method are approximately shrinking.     

Smooth dependence on parameters of solutions and contact regions for an obstacle problem

We consider a class of variational inequalities which includes a model of a beam compressed by a force λ and unilaterally supported by a connected obstacle at the height h. A smooth dependence on parameters λ and h of solutions and intervals of their contact with the obstacle is proved in a neighbourhood of a given solution. The basic idea is to show local equivalence of the inequality with a suitable equation and to apply the implicit function theorem to this equation.     

Regularization for a class of quasi-variational-hemivariational inequalities

The aim of the present paper is to study the solvability and regularization for a class of multivalued quasi-variational–hemivariational inequalities in reflexive Banach spaces. By applying the Kluge fixed point theorem and the Minty technique, we prove the solvability of the considered multivalued quasi-variational–hemivariational inequality, based on which some convergence results are obtained by introducing its regularization problem with the help of regularization operator. The applicability of the obtained abstract results is established by a mathematical model of a frictional contact problem with a class of elastic material, where the existence and stability results for the weak solution of contact problem are studied.     

A coupling of FEM-BEM for a kind of Signorini contact problem

In this paper, we consider a kind of coupled nonlinear problem with Signorini contact conditions. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive an asymptotic error estimate of the approximation to the coupled FEM-BEM variational inequality. Then we design an iterative method for solving the coupled system, in which only three standard subproblems without involving any boundary integral equation are solved. It will be shown that the convergence speed of this iteration method is independent of the mesh size.