We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.

Sufficient conditions ensuring that the solution set of some operator equations involving a duality mapping is non-empty, compact and convex are given.

We are interested in the dynamic evolution of a thermoviscoelastic body which is on frictional contact with a rigid foundation. The contact is modeled by a general normal damped response condition with friction law and heat exchange. We establish the existence and uniqueness of the weak solution, under the condition that the viscosity is sufficiently strong. Finally the numerical analysis of a fully discrete scheme is presented.

This book includes a self-contained theory of inequality problems and their applications to unilateral mechanics. Fundamental theoretical results and related methods of analysis are discussed on vario

The aim of this Chapter is to discuss an approach based on the use of topological tools (Leray-Schauder degree and continuation results) in a way that is suitable in the setting of unilateral analysis. Particular attention is paid to some nice and fundamental theorems. The material developed in this Chapter will be used later in various directions. In particular, the topological methods constitute powerful devices for the study of unilateral eigenvalue problems (see Chapter 10).

The aim of this paper is to discuss the mathematical strategies permitting the treatment of second order unilateral systems involving singular mass, damping, and stiffness matrices. Reduction methods are used here to transform second order differential inclusions in first order and classical results on differential inclusions are considered in order to obtain solution. Frictions and impact problems arising in Unilateral Mechanics are studied so as to illustrate the theoretical approach.

We describe and analyze a frictional problem for a system with a compressed spring which behaves as if it has a spring constant that is negative over a part of its extension range. As a result, the problem has three critical points. The friction is modeled by the Coulomb law. We show that there are three separate stick regions for some values of the parameters, centered on the critical points. We model three other versions of the process. Then we describe a numerical scheme for the models and present a number of computer simulations.

The aim of this paper is to discuss a mathematical solution procedure to solve a Ramsay-type growth model that explains the fundamentals of consumption and capital accumula-tion in a dynamic equilibrium setting. The problem is formulated as a system of recursive equations and studied through some numerical experiments for the time path of the different variables of the model under some alternative assumption for the steady-state equilibrium of the labour market conditioning the possible singularity of the model.

We analyze a nonlinear abstract evolution problem describing a class of frictional contact processes between a viscoelastic body and a foundation. The problem is set as a time-dependent differential inclusion. The existence of a unique solution is established using the theory of elliptic variational inequalities and Banach's fixed point theorem. A dual formulation of the problem is also introduced and an equivalence result between the two problems is proved. Finally, the abstract results obtained are used to solve some frictional contact problems for viscoelastic materials.

Dynamic frictional contact with adhesion of a viscoelastic body and a foundation is formulated as a hemivariational inequality. This may model the dynamics of rock layers. The normal stress–displacement relation on the contact boundary is nonmonotone and nonconvex because of the adhesion process. A sequence of regularized problems is considered, the necessary a priori estimates are obtained, and the existence of a weak solution for the hemivariational inequality is established by passing to the limit as the regularization parameter vanishes.

We prove the existence and uniqueness of the weak solution for a quasistatic thermoviscoelastic problem which describes bilateral frictional contact between a deformable body and a moving rigid foundation. The model consists of the heat equation for the temperature, the elliptic viscoelasticity system for the displacements, the SJK-Coulomb law of friction and frictional heat generation condition. The proof is accomplished in two steps. First, the existence of solutions for a regularized problem is established and a priori estimates obtained. Then the limit function, which is the weak solution of the original problem, is shown to be the unique fixed point of the solution operator when the friction coefficient is small.

We prove the existence of a unique weak solution to the quasistatic problem of frictional contact between a deformable body and a rigid foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We establish the continuous dependence of the solution on the normal compliance function. Moreover, we prove the existence of a unique solution to the problem of sliding contact with wear.

The quasistatic contact of a viscoelastic body with a rigid foundation is studied. The material behavior is modeled by a general nonlinear viscoelastic constitutive law. The contact is with directional friction and the foundation's resistance is proportional to the normal velocity. The existence of a unique weak solution to the problem is proved. The sliding frictional contact problem with wear is introduced, too, and the existence of its unique weak solution established. The proofs are based on fixed point theorems and elliptic variational inequalities, and the results hold when the friction and damping coefficients are small. Keywords: Quasistatic contactdirectional frictiondamped normal responseviscoelastic constitutive lawexistence and uniquenessweak solutionselliptic variational inequalitiesfixed pointssliding contactwear Additional informationNotes on contributors M. Rochdi† M. Shillor† M. Sofonea†

In this paper we introduce the abstract concept of "linear boundary conditions" in the study of deformable bodies. We establish two existence and uniqueness results concerning respectively quasistatic and dynamic problems involving such type of boundary conditions. We also apply these existence results in the study of viscoplastic problems involving classical boundary conditions.