New Inequality and Functional for Contact with Friction: The Implicit Standard Material Approach∗

Abstract

This paper is devoted to the analysis of the two-or three-dimensional elastic contact problem with Coulomb friction. quasi-static equilibrium, and small displacements. The classical approach is based on two minimum principles, or variational inequalities: the first for unilateral contact and the second for friction. In practical applications, this leads to an algorithm of alternately solving the two problems until convergence is achieved. A coupled approach using one principle or one inequality only is presented. This new approach, based on a model of material called implicit standard, allows for extension of the notion of a normality law to dissipative be

Bipotentials for Non-monotone Multivalued Operators: Fundamental Results and Applications

“…The graph of the law of unilateral contact with Coulomb's dry friction is the graph of the following bipotential, previously given in [33]:…”

“…A possible way to study non-associated constitutive laws by using convex analysis, proposed first in [33], consists in constructing a "bipotential" function b of two variables, which physically represents the dissipation. See definition 3.1 and the section 3 for the introduction into the subject of bipotentials.…”

A comparative study between two smoothing strategies for the simulation of contact with large sliding

“…Usuellement, trois méthodes sont utilisées dans le cadre de la méthode éléments finis pour traîter des problèmes de contact: (1) la méthode de pénalité [3,15], (2) la méthode des multiplicateurs de Lagrange [5,14] et, (3) la méthode du Lagrangien augmenté [34,2,30]. La simplicité d'implémentation de la méthode de pénalité est malheureusement contre-balancée par le fait qu'elle permet, par construction, des pénétrations entre les structures.…”

Bi-First: A simple and efficient algorithm to identify dissipated energy in impact problems

“…In order to avoid nondifferentiable potentials that occur in nonlinear mechanics, such as in contact problems, it is convenient to use the Augmented Lagrangian Method [9][10][11][12][13]. For the contact bi-potential b c , given by (17), provided thatu n ≥ 0 and r ∈ K μ , we have:…”

“…Since friction problems are not minimization problems, the formulation needs to be extended. Alart and Curnier [9], Simo and Laursen [10] and De Saxcé and Feng [11] have obtained some extensions in mutually independent works. The first two works consist of applying Newton's method to the saddle-point equations of the augmented Lagrangian.…”