Shakedown theorems in Contact Mechanics

Abstract

In this Note, the phenomenon of cumulative slip of solids under cyclic loads is considered. The topic concerns with the relative displacements of two solids maintained in contact by friction. This problem of the daily-life mechanics can be compared to the shakedown of elastic plastic solids under cyclic loads. A transcription of the plastic shakedown theorems is given here for the problem of frictional contact. The statement of Melan theorem is first given for the cumulative-slip problem under some restrictive assumptions. It suggests again the introduction of a safety coefficient with respect to slips. The safety coefficient can be computed from two static and kinematic approaches in min–max duality and leads again to Koiter theorem. As in plasticity, these results are available only for associated laws and do not hold for Coulomb law of friction, except in some very particular situations. These results can be understood mathematically as a particular case of shakedown theorems in plasticity, when the plastic strain is localized on a surface. For Coulomb friction, numerical simulations by direct step-by-step calculations show that the asymptotic behaviour of the response with or without slip-shakedown could be obtained very quickly after some cycles. To cite this article: N. Antoni, Q.-S. Nguyen, C. R. Mecanique 336 (2008).