# American Institute of Mathematical Sciences

April  2016, 9(2): 409-425. doi: 10.3934/dcdss.2016004

## Modelling contact with isotropic and anisotropic friction by the bipotential approach

 1 Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, bâtiment Boussinesq, Cité Scientifique, 59655 Villeneuve d'Ascq cedex

Received  May 2015 Revised  October 2015 Published  March 2016

Based on an extension of Fenchel's inequality, the bipotential approach is a non smooth mechanics tool used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Generally, such constitutive laws are given by a graph $M$. We propose a simple necessary and sufficient condition for the existence of a bipotential $b$ for which $M$ is the set of couples $(x,y)$ of dual variables such that $b(x,y) = \langle x,y \rangle$, and a method to construct such a bipotential by covering $M$ with cyclically monotone graphs which are not necessarily maximal (bipotential convex cover). As application, we show how to obtain the bipotential of the law of unilateral contact with Coulomb's friction by a bipotential convex cover. Introduced to extend the classical calculus of variation, the bipotential concept is also useful to construct numerical schemes for friction contact laws. In recents works, we extended the bipotential approach to a certain class of orthotropic frictional contact with a non-associated sliding rule proposed by Michałowski and Mróz. The bipotential suggests a predictor-corrector numerical scheme.
Citation: Géry de Saxcé. Modelling contact with isotropic and anisotropic friction by the bipotential approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 409-425. doi: 10.3934/dcdss.2016004
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