A formulism of two-phase equilibrium and phase diagram for elastic-plastic deformed system under non-hydrostatic stress conditions: Formulations and verification

Abstract

Phase equilibrium, phase diagram and phase transformation of a solid that undergoes elastic-plastic deformation under non-hydrostatic stress conditions have technical and scientific significance, and numerous experimental and theoretical studies have been performed on this issue; however, a comprehensive theoretical description remains an unresolved problem. The crucial obstacle lies in seeking the phase equilibrium conditions, especially the formula of the general chemical potential. Duvall and Graham (1977) specified that it is impossible to construct a general Gibbs function, i.e. a general chemical potential, to define the equilibrium conditions for phase transformation when shear stress is present, except for some particular cases. This work focuses on the phase equilibrium problem of a solid-solid two-phase system, which permits elastic-plastic deformation under non-hydrostatic stress loading. We found the expression of the general chemical potential, which was often used in previous is not rigorous in logic.On the basis of a widely used approximation that the constitutive relationships between deviatoric stress and deviatoric elastic strain are linear, and the elastic coefficients depend only on pressure and temperature, a formula of the general chemical potential was derived in this paper. The result was found to be similar to that in hydrostatic thermodynamics as long as we substitute temperature T with an effective quantity Teff, which has the unit of temperature, and pressure P with another effective quantity Peff, which has the unit of pressure. The expressions of Teff and Peff are correlated with deviatoric elastic strain, elastic coefficients and the pressure and temperature partial derivatives of elastic coefficients. It is through the correlation between Teff and Peff and deviatoric quantities that the influence of deviatoric stress on phase equilibrium and phase transformation is imposed. In classical solid mechanics, the mechanical properties of a solid body are usually divided into two classes: volumetric properties and deviatoric properties. In addition, the deviatoric properties are thought to be affected by volumetric properties, but the volumetric properties are not influenced by deviatoric properties. Thus, the deviatoric and volumetric properties are standing unequal in classical solid mechanics. In contrast, the irrational inequality between volumetric and deviatoric properties has been successfully removed in the formulism proposed in this paper. For an isotropic solid system, the general Clausius-Clapyron relation was further deduced, which can be used to quantitatively calculate the shift in transition pressure caused by deviatoric stress. As an example of application, as well as verification, the formulism proposed in this paper was employed to quantitatively interpret the scatter in transition pressure of the α-ω transformation of titanium observed in experiments, which was commonly thought to have a relationship with non-hydrostatic stresses, but comprehensive quantitative theoretical interpretations are still needed.