On general principles of the theory of constitutive relations in classical continuum mechanics

The traditional principles of the theory of constitutive relations in classical continuum mechanics are discussed. In light of the approaches by A. A. Ilyushin and by W. Noll, the equivalence and the completeness of their general reduced forms of constitutive relations for simple classical media are noted. First, it is mentioned that a possible presence of internal kinematic constraints is rarely taken into account and needs special modifications in formulating the principles and relations. Secondly, a systematic study of internal body forces in constitutive relations has not been done before. Here a unified approach to the theory of constitutive relations is proposed to describe properties of deformation resistance of a body including both internal kinematic constraints and internal body forces. The general reduced forms of the system of constitutive relations are derived in terms of different definitions of a dynamical process in a body. The case of a simple body is considered in detail in view of the theory of objective tensors, their diagrams and frame-independent relations between objective tensors. The completeness of Ilyushin’s and Noll’s types of relations for the most general constitutive formulations is noted and confirmed by examples.