| Under suitable programs of mechanical or thermal loading, many solid materials are capable of undergoing phase transformations from one crystal structure to another. The austenite-martensite transformation that occurs in a variety of metallic alloys, including the so-called shape-memory materials, provides an example. The present paper represents an effort to model coupled thermo-mechanical effects in the macroscopic response of solids that arise from the occurrence of phase transformations. A Helmholtz free energy potential is constructed to describe the thermo-mechanical response of the hypothetical material to be considered here. As a function of strain, the potential is non-convex in a certain range of temperature this feature is essential for the modeling of phase transformations. Apart from some general preliminary considerations pertaining to finite thermoelasticity, the analysis is carried out in the context of a simple problem, idealized from an experiment, in which an annular cylinder is deformed to a state of radially symmetric, finite anti-plane shear in the presence of differing inner and outer surface temperatures. After constructing all radially symmetric web solutions involving at most a single surface of discontinuity of strain or temperature gradient, we determine the implications for quasi-static motions of the second law of thermodynamics. The thermo-mechanical phase transformation-induced hysteresis, residual deformation and stress relaxation effects exhibited by this model are discussed and the results concerning creep rate as predicted by the present model are in qualitative agreement with the laboratory observation. Finally, the shape-memory effect as predicted by the present thermo-mechanical model in the setting of finite anti-plane shear is illustrated. |
