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

International audienceThe work presented in this communication is at the crossroad between direct methods and non-smooth mechanics. One objective is to derive shakedown theorems in situations where constraints are prescribed on the internal variables. A motivation is the study of shape memory alloys (SMAs) : in those materials, inelastic deformation occurs as the result of a solid/solid phase transformation between different crystallographic structures. Much effort has been devoted to developing constitutive laws for describing the behaviour of SMAs. The phase transformation is typically tracked by an internal variable which - depending on the complexity of the material model - may be scalar or vectorial. A fundamental observation is that, in most of SMA models, that internal variable must comply with some a priori inequalities, resulting from the mass conservation in the phase transformation process. As a consequence, the internal variable is constrained to take values in a set T that is not a vectorial space. The presence of such constraints constitutes a crucial difference with standard plasticity models, and calls for special attention when the structural evolution problem is considered. Non-smooth mechanics offer a sound mathematical framework for handling constraints on state variables. This communication is devoted to studying the asymptotic behaviour (i.e. as time tends to infinity) of solids in the framework of non-smooth mechanics