The Global Behavior of Elastoplastic and Viscoelastic Materials with Hysteresis-Type State Equations

A one-dimensional model is derived in order to study how the elasticity (internal elastic energy) of visco-elastic and elasto-plastic materials, such as biopolymers (muscles and grain our dough) or metals, changes due to the action of external forces. For such materials, the model takes the form of an initial-boundary value problem, corresponding to Newton's second law, which is coupled to an auxiliary (stress-strain) state equation which characterizes the nature of the interaction between the material and the external forces. In the oscillatory loading of muscles and the mixing of grain our as well as of the fatiguing of metals, the state equation must model how the stress depends on the earlier history of the strain as well as describe how the material gains or loses elastic energy due to the action of the loading. One is thereby led to model the auxiliary stress-strain relationship as a constitutive relationship involving a Duhem-Madelung hysteresis operator. As well as discussing the formulation of such models along with the properties of Duhem-Madelung hysteresis operators, the paper examines the existence and uniqueness for the solutions of such coupled systems. In addition, some global estimates are derived for these solutions, and their asymptotic behaviour, as the time increases, is studied under the assumption that a part of the internal (elastic) energy dissipates during the interaction, and, hence, the associated Duhem-Madelung hysteron has negative spin. 2