Variational and reciprocal principles on the temperature-rate dependent two-temperature thermoelasticity theory

The motive of the present work is to establish some theorems under a recent theory of thermoelasticity proposed by Shivay and Mukhopadhyay (2019). The governing equations for the temperature-rate dependent two-temperature (TRDTT) thermoelasticity theory were stated by Youssef (2006). Shivay and Mukhopadhyay (2019) employed generalized thermodynamics laws and presented a modified TRDTT thermoelasticity theory. The generalized two-temperature relation obtained in view of this modified TRDTT thermoelasticity theory for an anisotropic material reduces to the two-temperature relation provided by Youssef in a specific situation. They further discussed the uniqueness of solution under this theory. In the present work, we consider the basic equations of this new model for the homogeneous and anisotropic material. Later, we present an alternative formulation of a mixed boundary-initial value problem incorporating initial data into the field equations. The variational principle of the convolution type by using this formulation is derived. Lastly, we derive a reciprocal principle for this theory.