Time-dependent variational inequalities for viscoelastic contact problems

AbstractWe consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using standard arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximations of the problem. We use the finite element method to discretize the spatial domain and we introduce spatially semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution, and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximations of a viscoelastic contact problem with normal compliance and friction

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

In this paper an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem we provide the optimal error estimate

Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics

In the paper we deliver a new existence and uniqueness result for a class of abstract nonlinear variational-hemivariational inequalities which are governed by two operators depending on the history of the solution, and include two nondifferentiable functionals, a convex and a nonconvex one. Then, we consider an initial boundary value problem which describes a model of evolution of a viscoelastic body in contact with a foundation. The contact process is assumed to be dynamic, and the friction is described by subdifferential boundary conditions. Both the constitutive law and the contact condition involve memory operators. As an application of the abstract theory, we provide a result on the unique weak solvability of the contact problem

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

In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate

Optimal control of history-dependent evolution inclusions with applications to frictional contact

In this paper, we study a class of subdifferential evolution inclusions involving history-dependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example of an optimal control of a dynamic frictional contact problem in mechanics, whose weak formulation is the evolution variational inequality

A new class of history-dependent quasi variational-hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational-hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational-hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.Comment: 15

Well-posedness of history-dependent evolution inclusions with applications

In this paper, we study a class of evolution subdifferential inclusions involving history-dependent operators. We improve our previous theorems on existence and uniqueness and produce a continuous dependence result with respect to weak topologies under a weaker smallness condition. Two applications are provided to a frictional viscoelastic contact problem with long memory, and to a nonsmooth semipermeability problem