A mixture of elastic materials with different constituent temperatures and internal constraints

A novel treatment of the 2nd Law of Thermodynamics and the development of general thermomechanical constraints are introduced for a mixture of two elastic materials in which the constituents may have different temperatures. First, a homothermal quasi-static process at a common mixture temperature is introduced. Part I of the 2nd Law of Thermodynamics is invoked to assert that the Clausius integrals are path-independent, which leads to a prescription, or an identification, of the partial entropy functions. Then, two assumptions are introduced that establish the values of the partial entropy functions for general processes, including those for which the constituent temperatures are not equal. Constitutive restrictions are derived for path-independent processes from the mixture energy equation, and further constitutive restrictions are derived for general processes upon invoking the Clausius–Duhem inequality as a statement of Part II of the 2nd Law of Thermodynamics. The complete set of constitutive restrictions are then shown to equal those derived by other authors, a result which supports the adopted assumptions concerning the partial entropy functions for general processes. Then, an internal constraint involving the deformation gradient tensors and the constituent temperatures is represented by a constraint manifold, and an internally constrained mixture of elastic materials is associated with each unique equivalence class of unconstrained mixtures. The examples of a mixture constrained to have a common temperature and a mixture constrained by temperature-dependent intrinsic compressibility are discussed.