A variational inequality and applications to quasistatic problems with Coulomb friction

International audienceThe aim of this paper is to study an evolution variational inequality that generalizes some contact problems with Coulomb friction in small deformation elasticity. Using an incremental procedure, appropriate estimates and convergence properties of the discrete solutions, the existence of a continuous solution is proved. This abstract result is applied to quasistatic contact problems with a local Coulomb friction law for nonlinear Hencky and also for linearly elastic materials

A class of implicit evolution inequalities and applications to dynamic contact problems

International audienceThis paper deals with the analysis of a class of implicit variational inequalities which generalizes some dynamic contact problems coupling adhesion and friction between two viscoelastic bodies of Kelvin-Voigt type. Existence and uniqueness results are proved for a general system of evolution inequalities that constitutes a unified approach to study some complex dynamic surface interactions, including rebonding, debonding and friction conditions. The proofs are based on incremental formulations, several estimates, compactness arguments and a fixed point technique. These results are applied to dynamic frictional contact conditions with reversible adhesion and the coefficient of friction depending on the slip velocity

Internal and subspace correction approximations of implicit variational inequalities

International audienceThe aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established

Approximation results and subspace correction algorithms for implicit variational inequalities

International audienceThis paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established

Sur la modélisation mathématique de conditions unilatérales en mécanique du contact

Dans ce travail on considère quelques problèmes quasi-statiques de contact avec frottement local en élasticité linéarisée, concernant le contact unilatéral et une condition de contact modifiée. On décrit le cadre fonctionnel et des formulations variationnelles, associées aux problèmes continus et incrémentaux, qui permettent d'obtenir des modèles adaptés aux différentes applications. On présente également quelques propriétés remarquables des solutions variationnelles qui vérifient la condition de contact modifiée. Ces résultats peuvent être étendus à des cas plus généraux comme, par exemple, le contact entre deux corps (visco)élastiques ou le couplage adhésion-frottement

A dynamic unilateral contact problem for a cracked body

CDROMIn this work we investigate a class of dynamic contact problems for cracked viscoelastic and elastic bodies, when Signorini’sconditions between the two faces of the crack are considered. Firstly, using a penalty method we study a variational formulationof a unilateral contact problem with nonlocal friction for a cracked viscoelastic body. Several estimates on the penalized solutionsare obtained which enable us to analyze the time and spatial discretizations of the problem. Then we consider the correspondingelastic problem, for which a fictitious domain formulation is proposed with Lagrange multipliers representing the normal jump of thedisplacements. Numerical examples, based on the fictitious domain method for solving the diffraction of elastic waves by cracks, arepresented

A dynamic viscoelastic problem with friction and rate-depending contact interactions

International audienceThe aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems

Variational formulation of a dynamic surface interaction problem in viscoelasticity

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A class of dynamic contact problems with Coulomb friction in viscoelasticity

International audienceThe aim of this work is to study a class of nonsmooth dynamic contact problem which model several surface interactions, including relaxed unilateral contact conditions, adhesion and Coulomb friction laws, between two viscoelastic bodies of Kelvin-Voigt type. An abstract formulation which generalizes these problems is considered and the existence of a solution is proved by using Ky Fan's fixed point theorem, suitable approximation properties, several estimates and compactness arguments

A variational analysis of some dynamic surface interaction problems in viscoelasticity

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