Keywords: contact, friction, Coulomb's law, bipotential, constitutive relation error estimator, finite elements.

Unilateral contact problems are found in many fields of structural calculation. For these nonlinear simulations, it is essential to evaluate the errors made due to the numerical approximations and iterative algorithms involved in nonlinear calculations. For this kind of problem, few research works have proposed a posteriori error estimators to evaluate space discretization errors. An error estimator based on stress recovery was proposed in [7]. However, its computation involves a penalty parameter inside the estimator, which is a major disadvantage. More recently, an extension of the error in constitutive relation to frictionless contact problems was proposed in [1].

The aim of this paper is to extend this study to the case of contact involving Coulomb friction. This extension is based on a formulation of the constitutive relation for unilateral contact with friction using a bipotential [2] and on a specific modeling of the contact zone as a whole mechanical entity, with its own variables designated by , its kinematic constraints, its equilibrium equations and its constitutive relations [4,5].

First, we consider the static problem of two elastic bodies in unilateral contact with friction along their common boundary . Our study assumes small transformations. Contact is represented by means of Lagrange multipliers for two different discretizations of the non-interpenetration condition. The friction problem is solved by using the iterative scheme described in [6], in which a contact problem with Tresca friction is solved at each iteration. However, the techniques which are described below can be applied with other methods.

The concept of error in constitutive relation is based on the partitioning of the equations of the reference problem into two sets. The first set contains the admissibility conditions, i.e. the kinematic constraints and the equilibrium equations. The second set contains the constitutive relations:

where "" is the bipotential defined by:

and where and are the indicator functions of the convex cones:

Then, the error in constitutive relation is defined by:

In order to compute the error in constitutive relation, we need to derive a set of admissible fields from the finite element solution. The techniques proposed in [1] were extended to the case of contact involving friction. In particular, and are constructed so that the bipotential remains finite.

In this paper, the estimator is computed on several examples.

Finally, we show how the concept of error in constitutive relation enables us to build an error indicator to evaluate the quality of the numerical solution to the problem and, thus, to separate the global error into an error due to this iterative scheme and an error due to the spatial discretization. Such a separation is necessary to develop methods of adaptive calculation which are robust.

1

P. Coorevits, P. Hild, J.P. Pelle, "Contrôle et adaptation des calculs éléments finis pour les problèmes de contact unilatéral", Revue européenne des éléments finis, 8, 1, 7-29, 1999.

2

De Saxce, "Une généralisation de l'inégalité de Fenchel et ses applications aux lois constitutives", Comptes Rendus Académie des Sciences, Paris.

3

P. Ladevèze, "Comparaison de modèles continus", Thèse d'État, Université P. et M. Curie, Paris.

4

P. Ladevèze, "Mécanique non linéaire des structures" Hermes, Paris.

5

P. Ladevèze, J.P. Pelle "La maîtrise du calcul en mécanique linéaire et non linéaire", Hermes (à paraître).

6

M. Raous, L. Chabrand, J. Lebon "Numérical methods for frictionnal contact problems and applications" Journal de Mécanique Théorique et Appliquée, numéro spécial, Vol 1 à 7, 111-128, 1988.

7

Wriggers "An adaptative finite element algorithm for contact problems in plasticity" Computational Mechanics, 17, 88-97. doi:10.1007/BF00356481

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