The Global Behaviour Of Elasto Plastic Materials With Hysteresis Type State Equations

Mathematics Research Report MRR95-081

The Global Behaviour Of Elasto Plastic Materials With Hysteresis Type State Equations

Robert S. Anderssen, Ivan G. Gotz and Karl-Heinz Hoffmann

Abstract: A one-dimensional model is derived in order to study how the elasticity of a material, such as a biopolymer, changes due to the action of external forces. It takes the form of an initial-boundary value problem for a viscous elasto-plastic material which is coupled to an auxiliary (stress-strain) state equation which characterizes the interaction between the material and the external forces. In the oscillatory loading of muscles and the mixing of grain flour, the state equation must model a situation where the stress depends on the earlier history of the strain and exhibits, at least approximately, a rate-independent character. If it is assumed that the interaction is rate-independent, one is led to model the auxiliary stress-strain relationship as 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 the internal energy of the material increases during the interaction, and, hence, the associated Duhem-Madelung hysteron has negative spin.

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