Keywords: contact simulations, anisotropic friction law, non associated law, variational formulations, augmented Lagrangean technique, predictor-corrector scheme.

For many industrial applications, the assumption of isotropic friction is unrealistic because of directional surface machining and finishing operations. For orthotropic frictional contact, Michaowski and Mróz [3,4] pointed out the non-associated nature of the sliding rule. In the present work, their law is derived from a single non-differentiable function, called a bi-potential. On this ground, the time-integration of the contact law takes the form of a predictor-corrector scheme addressing all cases (sticking, sliding, no-contact) thanks to a projection onto the friction cone. Three-dimensional numerical applications show the strong influence of the slip rule.

The asperity model used by Michaowski and Mróz [3,4] to study anisotropic frictional contact phenomenon generates limit friction curves in the contact plane that can be slightly non-convex but very close to ellipses. Here, we consider only a convex friction cone

where is a diagonal matrix with diagonal elements and . The non associated slip rule

is defined by the convex slip potential

where is a diagonal matrix with diagonal elements and . Introducing the sliding non-associativity matrix , it is proved in [2] that the previous friction law can be put in the following compact form of a differential inclusion

where we use the dual norm and is the indicatory function of , null if and infinite otherwise. Next, we adopt an incremental approach to discretise the quasi-static evolution. Using the concept of bi-potential introduced in [1] and the implicit integration scheme, we deduce the value of the contact reaction at the end of the step from its value at the begining of the step and the relative displacement increment thanks to one very robust predictor-corrector step

where is a strictly positive number that need to be chosen in a suitable range to ensure convergence.

In this work, we propose a benchmark test to validate the algorithm for a class of non-associated anisotropic friction laws. The test of such frictional contact laws requires a 3D finite element model. The problem under consideration is a deformable elastic cylinder. The lower face is in contact with a rigid horizontal surface with anisotropic friction condition, while the upper face is subjected to a vertical rigid motion.

1

G. de Saxcé, Z.Q. Feng, "New inequality and functional for contact friction : The implicit standard material approach", Mechanics of Structures and Machines, 19,301-325, 1991. doi:10.1080/08905459108905146

2

M. Hjiaj, G. de Saxcé, Z. Mróz, "A variational-inequality based formulation of the frictional contact law with a non-associated sliding rule", European Journal of Mechanics A/Solids, 21, 49-59, 2002. doi:10.1016/S0997-7538(01)01183-4

3

R. Michalowski, Z. Mróz, "Associated and non-associated sliding rules in contact friction problems" Archives of Mechanics, 11, 259-276, 1978.

4

Z. Mróz, S. Stupkiewicz, "An anisotropic friction and wear model", International Journal of Solids and Structures, 31, 1113-1131, 1994. doi:10.1016/0020-7683(94)90167-8

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