M. Rochdi

Quasistatic Viscoelastic Contact with Normal Compliance and Friction

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 Coulombs 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.

A quasistatic contact problem with directional friction and damped response

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 foundations 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.

Quasi-static thermoviscoelastic contact problem with slip dependent friction coefficient

We prove existence and uniqueness of the weak solution for a quasi-static thermoviscoelastic problem which describes bilateral frictional contact between a deformable body and a moving rigid foundation. Friction is modeled with slip rate dependent friction coefficient, and it may depend either on the current slip rate or on the accumulated slip over the contact history. The frictional heat generated in the process is taken into account. The proof is based on the existence of solutions for a regularized problem, a priori estimates and a fixed-point argument, which provides the solution when the friction coefficient is sufficiently small.

Frictional contact of a nonlinear spring

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.

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.

Minimax Methods for Inequality Problems

We know from Chapter 2 that, if we intend to consider concrete problems in unilateral Mechanics involving both monotone and nonmonotone unilateral boundary (or interior) conditions, then we have in general to deal with a nonsmooth and nonconvex energy functional — expressed as the sum of a locally Lipschitz function \(\Phi :X \to \mathbb{R}\) and a proper, convex and lower semi-continuous function \(\psi :X \to \mathbb{R} \cup \left\{ { + \infty } \right\}\) — whose critical points are defined as the solutions of the variational-hemivariational inequality

Fundamental Existence Theory of Inequality Problems

The first purpose of this Chapter is to list and prove the fundamental existence theorems applicable to the study of inequality problems. Variational and hemivariational inequalities are studied for several important classes of operators among which monotone and hemicontinuous operators, semicoercive operators, nonlinear perturbations of semicoercive operators, maximal monotone operators and pseudomonotone perturbations of maximal monotone operators. The second purpose of this Chapter is to draw from the aforementioned abstract theorems the basic methods which can be used to study inequality problems. For instance, the monotonicity method (Sections 3.3, 3.11), the projection method (Section 3.2), the Fichera’s approach (Section 3.4), the recession approach (Section 3.5), the method of lower and upper solutions (Section 3.6), the method of maximal monotone operators and semigroup of contractions (Sections 3.7, 3.9), the Brezis approach (Section 3.8) are here discussed. The results of this chapter will be used later in Chapters 5, 6 and 7 so as to study various classes of elliptic, parabolic and hyperbolic unilateral problems.

Topological Methods for Inequality Problems

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).

Rock’s Interface Problem Including Adhesion

A rock’s dynamic contact model taking into account friction and adhesion phenomena is discussed. It consists of a hemivariational inequality because of the adhesion process. A weak solution is obtained as a limit of a sequence of solutions to some regularized problems after establishing the necessary estimates.