Derivation of compatibility conditions and nonconstant material function for one-dimensional constitutive relations of shape memory alloys

The present work investigates the thermodynamic inconsistencies in the definition of the compatibility conditions on stress for constant and nonconstant material functions in one-dimensional modeling of shape memory alloys based on the first principles. In this work, simplifications are provided validating inconsistencies in the earlier proposed non-constant material functions used to satisfy compatibility conditions. It is presented that the inconsistencies originate due to an incorrect definition of the compatibility conditions on stress. In the first step, it is shown that, due to inconsistent definitions of the compatibility conditions, the material functions cannot be derived from the first principles. Consequently, it is presented that the material functions result in an incorrect form of the differential constitutive equation. Furthermore, it is also analyzed that these incorrect definitions on the compatibility conditions result in an inconsistent form of nonconstant material functions as well as the differential equation, which are proposed in earlier models. As a result, in the present work the consistent definition of the compatibility conditions for one-dimensional shape memory alloy models is derived. Next, the new and correct definition for the compatibility conditions is proposed, which is used to derive a new and consistent form of nonconstant material function. Finally, a consistent form of non-constant material function and differential equation are derived from first principles, which satisfy the new definition of compatibility conditions on stress.