Constitutive Equations, Reduced Constitutive Equations, and Internal Constraints

Abstract

The general principles of mechanics apply to all bodies and motions, and the diversity of materials in nature is represented in the theory by constitutive equations. A constitutive equation is a relation between forces and motions. In popular terms, forces applied to a body “cause” it to undergo a motion, and the motion “caused” differs according to the nature of the body. In continuum mechanics the forces of interest are contact forces, which are specified by the stress tensor T. Just as different figures are defined in geometry as idealizations of certain important natural objects, in continuum mechanics ideal materials are defined by particular relations between the stress tensor and the motion of the body. Some materials are important in themselves, but most of them are of more interest as members of a class than in detail. Thus a general theory of constitutive equations is needed. The material presented here draws heavily from the work of Noll.