This work aims to formulate the temperature-rate dependent two-temperature (TRDTT) theory of thermoelasticity. The two-temperature thermoelasticity theory and the temperature-rate dependent thermoelasticity theory are two well-established thermoelasticity theories, which are developed from the generalized thermodynamic principles independently. Although the constitutive equations for TRDTT theory have been introduced, the formulation for the theory from the thermodynamical principles is not yet derived. Therefore, this work is an attempt to establish the theory from the generalized laws of thermodynamics and derive all the governing equations and constitutive relations for the theory. We derive a new and more general two-temperature relation that involves the temperature-rate terms of conductive and thermodynamic temperatures. We observe that this relation is different from the two-temperature relation reported in the literature. Further, we prove the uniqueness of solution for a general mixed initial boundary value problem in the context of linear modified TRDTT thermoelasticity theory for anisotropic medium. To investigate the effect of the present modified TRDTT theory, we solve a one-dimensional half-space problem and highlight the significance of the present theory.

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