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 elasticit y (internal elastic energy) of viscoelastic and elastoplastic material s, such as biopolymers (muscles and grain our dough) or metals, change s 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 betwe en 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 m etals, the state equation must model how the stress depends on the ear lier 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 there by led to model the auxiliary stress-strain relationship as a constitu tive relationship involving a Duhem-Madelung hysteresis operator. As w ell as discussing the formulation of such models along with the proper ties of Duhem-Madelung hysteresis operators, this paper examines the e xistence and uniqueness for the solutions of such coupled systems. In addition, some global estimates are derived for these solutions, and t heir asymptotic behavior, as the time increases, is studied under the assumption that a part of the internal (elastic) energy dissipates dur ing the interaction and, hence, the associated Duhem-Madelung hysteron has negative spin.