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.     

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.     

Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

Numerical Algorithms - In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution...     

Bifunction hemivariational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the bifunction hemivariational inequality. This new class includes several classes of variational inequalities as special cases. A number of iterative methods for solving bifunction hemivariational inequalities are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.     

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.     

Elliptic variational hemivariational inequalities

This paper is devoted to the existence of solutions for elliptic variational hemivariational inequalities. The operators involved are taken to be multivalued and noncoercive. Using the notion of the generalized gradient of Clarke and recession method, some existence results of solutions have been proved.     

Semicoercive variational hemivariational inequalities

The aim of this paper is the mathematical study of a general class of semicoercive variational hemivariational inequalities introduced by P.D. Panagiotopoulos in order to formulate problems of mechanics involving nonconvex and nonsmooth energy function. Our approach is based on the asymptotic behavior of the functions which are involved in the variational problems.     

Generalized quasi-variational-like hemivariational inequalities

In this paper, we introduce and study a new class of generalized quasi-variational-like hemivariational inequalities with multi-valued η$\eta$-pseudomonotone operators in Banach spaces. Some new existence theorems of solutions for this class of generalized quasi-variational-like hemivariational inequalities are proved. The results presented in this paper generalize and extend some known results.     

Evolution hemivariational inequalities with hysteresis

In this paper we study evolution hemivariational inequalities containing a hysteresis operator. For such problems we establish an existence result by reducing the order of the equation and then by the use of the time-discretization procedure.     

Analysis and control of stationary inclusions in contact mechanics

We start with a mathematical model which describes the frictionless contact of an elastic body with an obstacle and prove that it leads to a stationary inclusion for the strain field. Then, inspired by this contact model, we consider a general stationary inclusion in a real Hilbert space, governed by three parameters. We prove the unique solvability of the inclusion as well as the continuous dependence of its solution with respect to the parameters. We use these results in the study of an associated optimal control problem for which we prove existence and convergence results. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact and provide the corresponding mechanical interpretations.     

On stably quasimonotone hemivariational inequalities

We establish some existence results for variational-hemivariational inequalities of the Hartman-Stampacchia type involving stably quasimonotone set-valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive a sufficient condition for the existence and boundedness of solutions.     

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ₁ of (−Δ,H ₀ ¹ (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.     

On constrained hemivariational inequalities and their approximation

We prove the existence of a solution of a constrained hemivariational inequality, and, develop a fully discrete approximation of it. The relation between the constrained hemivariational inequality and a problem of finding substationary points of the corresponding potential function is also studied.     

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.     

Strong convergence results for hemivariational inequalities

The purpose of this paper is to study a regularization method of solutions of ill-posed problems involving hemivariational inequalities in Banach spaces. Under the assumption that the hemivariational inequality is solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method. Our results generalize and extend the previously known theorems.