Cauchy’s stress theorem and its influence on the first elasticity tensor

The well-known Clausius–Planck inequality for solids is derived from the second law of thermodynamics and provides the important relation between stress and strain tensors according to the Coleman–Noll procedure. For the detailed derivation of this inequality, mass continuity equation, the equation of motion and the first law of thermodynamics in local form are required. The transfer of integral to local formulation can be calculated using Cauchy’s stress theorem and the divergence theorem. In the literature, however, there are two definitions for both, the Cauchy’s stress theorem and the divergence theorem for second-order tensors. The aim of this article is to show the resulting differences in the first principles for polar solids up to the first elasticity tensor for hyperelastic materials.