Thermodynamically consistent approach for one-dimensional phenomenological modeling of shape memory alloys

The present work investigates the thermodynamic inconsistency in the definition of constant and nonconstant material functions for the one-dimensional shape-memory alloy constitutive models, with respect to the first principles. Thermodynamic consistency for the one-dimensional shape memory alloy differential equation is also investigated within the framework of one-dimensional elasticity at different length scales of stress and martensite fraction. It is shown that the previously proposed improvements in constitutive models using compatible nonconstant material functions cannot be derived from the first principles, yielding inconsistencies in the definition of the differential form of the constitutive relations. Additionally, the compatibility conditions on stress due to the previously defined compatible material functions in terms of constant and nonconstant material functions are also discussed. Derivations are provided to highlight the inconsistencies in the definition of differential form of constitutive relation due to previously proposed expressions for material functions. Finally, in this work new expressions for the differential equation with constant material function and corresponding transformation tensor are derived from the first principles. Subsequently, a consistent form of a differential constitutive model for shape-memory alloys is proposed. The discussions highlight that there is further requirement to propose compatible forms of nonconstant material functions through consistent definition of differential form of constitutive relation, which may help to further rebuild the 2D and 3D SMA models based on multiscale modeling.