Material covariant constitutive laws for continua with internal structure

In this work, we study the transformation properties of the local form of the material (referential) balance of energy equation under the superposition of arbitrary material diffeomorphisms. For this purpose, the tensor analysis on manifolds is utilized. We show that the material balance of energy equation, in general, cannot be invariant; in fact an extra term appears in the transformed balance of energy equation, which is directly related to the work performed by the configurational stresses. By making the fundamental assumption that the body and the ambient space manifolds are always related in the course of deformation and by utilizing the metric concept, we determine this extra term. Building on this, we derive several constitutive equations for the material stress tensor. The compatibility of these constitutive equations with the second law of thermodynamics is evaluated. Finally, we postulate that the material balance of energy equation is covariant, and we study this case in detail, as well.