Shakedown of discrete systems involving plasticity and friction

In associated plasticity, systems subjected to cyclic loading are sometimes predicted to shake down, meaning that, after some dissipative cycles, the response goes back to a purely elastic state, where no plastic flow occurs. Frictional systems show a similar behaviour, in the sense that frictional slips due to the external loads may cease after some cycles. It has been proved that, for complete contacts with elastic behaviour and Coulomb friction, Melan's theorem gives a sufficient condition for the system to shake down, if and only if there is no normal-tangential coupling at the interfaces. In this paper, the case of discrete systems combining elastic-plastic behaviour and Coulomb friction is considered. In particular, it is proved that Melan's theorem still holds for contact-wise uncoupled systems, i.e., the existence of a residual state, comprised of frictional slips and plastic strains, is a sufficient condition for the system to shake down.