Energetic formulation for rate-independent processes: remarks on discontinuous evolutions with a simple example

This work aims to discuss, in the simplest way, some open issues concerning the energetic formulation for the modeling of rate-independent processes by means of a simple one-dimensional example. Early and unphysical jumps are avoided taking advantage of a time reparametrization and enforcing an energy imbalance during time-discontinuous transitions. A new energetic selective criterion is then proposed, able to predict the final state after a jump and to represent intrinsically the energy contribution of rate-dependent phenomena as inertia and viscosity that may arise during fast discontinuous transitions. An improved version of the energetic formulation is finally proposed, preserving completely the rate-independent setting. All concepts and results are accompanied by explicative pictures which, hopefully, will help the non-specialist reader, to whom this paper is addressed, in the understanding of the discussion.