Phase transitions in materials with thermal memory and a Fourier term

A model for thermally induced phase transitions in rigid materials with thermal memory was recently proposed, both for the case where the phases have the same conductivity properties and where they are different. In this work, the model is generalized to the case of heatflow relations which include instantaneous contributions of the Fourier type as well as memory terms. The temperature gradient is decomposed into two parts, each zero on one phase and equal to the temperature gradient on the other. However, they vary smoothly over the transition zone. Asymptotic analysis is carried out which shows that, to leading order, temperature is continuous across the transition zone and the normal derivatives of the temperature on each phase boundary obey a condition of the classical form with no explicit dependence on the memory terms. This latter result emerges out of first order terms in the asymptotic analysis. Effects explicitly related to thermal memory only begin to play a role in the analysis of second order terms. These results contrast sharply with those for materials without the instantaneous terms.