Differential equation of hysteresis: Application to partial martensitic transformation in shape-memory alloys

Abstract

Many unusual thermomechanical properties of shape-memory alloys are directly connected with martensitic type phase transitions in these systems. Because the martensitic transformations, as a rule, are the first order transitions, a special attention should be given to a hysteretic behavior of shape-memory alloys. The most important characteristics of the temperature- or stress-induced martensitic transformation, have been previously studied in detail. It has been shown that such macroscopic state variables as inelastic strain or volume fraction of the martensite are always complex multi-valued functions of the temperature and external stress. Therefore, the shape-memory alloys should be considered as systems having an infinite number of state equations, representing inelastic strain and volume fraction of martensite as functions of the external stress and temperature, correspondingly. Some of the phenomenological approaches for the thermomechanical state equations for shape memory alloys were recently published. In particular, a special type of differential equation describing evolution of the inelastic macroscopic strain and volume fraction of martensite as a function of the temperature has been proposed. Its application to partial temperature cycling processes in shape-memory alloys and some other problems associated with the irreversible processes caused by hysteresis are discussed in the present paper.