Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example

In the context of a special problem, this paper investigates the possibility of modeling dissipative mechanical response in solids on the basis of the equilibrium theory of finite elasticity for materials that may lose ellipticity at large strains. Quasi-static motions for such materials are in general dissipative if the associated equilibrium fields involve discontinuous displacement gradients. For the problem treated, consideration of such deformations is shown to lead naturally to an internal variable formalism similar to those used to describe macroscopic plastic behavior arising from microstructural effects. For quasi-static motions which are maximally dissipative in a specified sense, this formalism leads to a mechanical response which resembles that associated with the pseudo-elastic effect in shape-memory alloys