Analysis of a Contact Problem With Normal Compliance, Finite Penetration and Nonmonotone Slip Dependent Friction

Abstract

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example.

Modeling and simulations for quasistatic frictional contact of a linear 2D bar

“…Then, we present the approximation of the problem by using a uniform discretization of the time interval and the finite element method in the plane. We use arguments similar to those used in Barboteu et al (2012aBarboteu et al ( , 2014Barboteu et al ( , 2016 and, for this reason, we skip many of the details.…”

Numerical solution of a contact problem with unilateral constraint and history-dependent penetration

“…In this section, we discuss the numerical solution of Problem P. To this end, we use arguments similar to those used in [15,24,25], based on an adapted combination of the penalty method and the augmented Lagrangian method for the numerical treatment of the specific contact condition (2.5). The starting point of our method is an alternative variational formulation of Problem P by considering a Lagrange multiplier associated with the normal contact stress.…”

“…For the numerical examples in the next section, we consider a "node-to-rigid" contact element, which is composed of one node of 3 and one Lagrange multiplier node. Details on this construction can be found in [15,24,25,28]. Then, the numerical approximation of Problem P V leads at each time step n to the solution of a system of nonlinear equations of the form:…”

A Dynamic Contact Problem with History-Dependent Operators

“…Another novel feature of this paper is the analysis of the dynamics. In contrast to other contributions in the field, cf., e.g., [1,12,17,27] and [31], we treat a dynamic contact problem for which the mathematical techniques are less developed than for quasistatic evolutionary models. We underline that there are no results on existence, uniqueness and convergence of solutions to the dynamic hemivariational inequality in Problem 17, which models the contact problem under consideration.…”