# A simple weak form for contact problems with coulomb friction

## This paper proposes a weak form for the contact problem with Coulomb friction, written as extension of the standard virtual work principle and involving both the displacements and the multipliers defined on the reference contact surface. The mixed relationship is shown to be equivalent to the strong form of the initial/boundary value contact problem, and it can be discretized by means of the finite element method in a simple way. | Vietnam Journal of Mechanics, VAST, Vol. 33, No. 4 (2011), pp. 259 – 282 A SIMPLE WEAK FORM FOR CONTACT PROBLEMS WITH COULOMB FRICTION 1 Nguyen Huynh Tan Tai1 and Le Van Anh2 Faculty of Civil Engineering, Thu Dau Mot University, Vietnam 2 Faculty of Science, University of Nantes, France Abstract. This paper proposes a weak form for the contact problem with Coulomb friction, written as extension of the standard virtual work principle and involving both the displacements and the multipliers defined on the reference contact surface. The mixed relationship is shown to be equivalent to the strong form of the initial/boundary value contact problem, and it can be discretized by means of the finite element method in a simple way. Typical numerical examples are given to assess the efficiency of the formulation in statics and dynamics. Key words: Contact, friction, finite element method. 1. INTRODUCTION Contact problems are of practical significance in mechanics and their solution is complex because of the nonlinear and non-smooth laws governing the interface response between the contacting bodies. When large deformations and large slips are involved, the problem becomes highly nonlinear and robust formulations must be designed with a view to efficiently solve it. Over more than three decades, a great number of formulations and solution methods have been proposed in the literature in order to deal with different kinds of contact problems encountered, from frictionless contact in linear elastostatics up to frictional contact in large deformation with complex material constitutive and interfacial contact laws, either in statics or dynamics. A complete review can be found in the books of Wriggers [1] and Laursen [2], where one can find a great deal of references related to the existing methods for the treatment of the contact constraints, in particular the classical Lagrange multiplier, the penalty and the augmented Lagrangian methods. The augmented Lagrangian methods are .

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