Shakedown theorems and asymptotic behaviour of solids in non-smooth mechanics

Non-smooth mechanics is concerned with systems for which constraints are imposed on the physical quantities or their time derivatives. This article addresses the asymptotic behaviour (i.e. as time tends towards infinity) of such systems when they are submitted to a given loading history. A special emphasis is laid on shape-memory alloys structures, which are a typical example of systems for which an analysis in non-smooth mechanics is required. Extending the approach introduced by Koiter in plasticity, we state sufficient conditions for the energy dissipation to remain bounded in time, independently on the initial state. Concerning the asymptotic behaviour in the particular case of cyclic loadings, we also point out the fundamental differences that exist between the framework of plasticity and that of non-smooth mechanics.