Keywords: unilateral contact, friction, dual formulation, equilibrium finite elements, block-relaxation.

The aim of this lecture is to present a formulation and associated numerical methods to solve unilateral contact problems with dry friction. Unilateral contact and friction are modeled by Signorini and Coulomb laws. Signorini's law of unilateral contact and Coulomb's law of friction constitute a simple and useful framework for the analysis of unilateral frictional contact problems of a linearly elastic body with a rigid support [5]. We have developed techniques based on primal or dual formulations. It seems useful, from a mechanical point of view, to develop techniques based on dual formulations, because one computes directly the stresses which are quantities of interest. The continuous and discrete dual formulations of this problem lead to a quasi-variational inequalities, whose unknowns, after condensation, are the normal and tangential contact forces at points/nodes of the initial contact area [1,3,4,6]. New numerical solution methods, based on iterative relaxation and block-relaxation techniques, are proposed [4,7]. The relaxation procedure is a succession of local minimizations in given convexes. The definition of the convex of constraints, which is a cylinder, varies for tangential or normal components. This algorithm seems very robust. At the typical step of the block-relaxation iteration, two sub-problems are solved one after the other: the former is a problem of friction with given normal forces, and the latter is a problem of unilateral contact with prescribed tangential forces. Both of them are standard problems of quadratic programming. This method is the dual version of the famous PANA algorithm [2]. We have proved that for sufficiently small friction coefficients, the typical step of the iteration is shown to be a contraction. The contraction principle is used to establish the well-posedness of the discrete dual condensed formulation, to prove the convergence of the proposed algorithm, and to obtain an estimate of the convergence rate.

1

G. Duvaut, J.L. Lions "Inequalities in mechanics and physics", Springer Verlag, Berlin, 1976.

2

P.D. Panagiotopoulos "A nonlinear programming approach to the unilateral and friction boundary value problem in the theory of elasticity", Ingenier Archiv, 44, 421-432, 1975. doi:10.1007/BF00534623

3

P. Bisegna, F. Lebon, F. Maceri "The unilateral frictional contact of a piezoelectric body with a rigid support", in J.M.C. Martins, M.M. Marques Eds., Contact Mechanics International Symposium, Proceedings of the 3rd International Conference, Peniche, Portugal, Kluwer, 2002.

4

P. Bisegna, F. Lebon, F. Maceri "D-PANA : a convergent block-relaxation solution method for the discretized dual formulation of the Signorini-Coulomb contact problem", Comptes Rendus Académie des Sciences Paris, Série I, 333, 1053-1058, 2001. doi:10.1016/S0764-4442(01)02153-X

5

F. Lebon, M. Raous "Friction modelling of a bolted junction under internal pressure loading", Computers and Structures, 43, 925-933, 1992. doi:10.1016/0045-7949(92)90306-K

6

J.J. Telega "Topics on unilateral contact problems of elasticity and inelasticity", In: Nonsmooth mechanics and applications, Eds. J.J. Moreau, P.D. Panagiotopoulos, Springer Verlag, Wien-New York, 341-462, 1988.

7

J. Haslinger, Z. Dostal, R. Kucera "On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction", preprint, 2001.

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