Generalities on constitutive models

The balance equations of a body form an under-determined differential system, insufficient to yield specific results unless further relations are supplied. The balance equations for a continuum (free from internal structures) are the continuity equation, the equation of motion, and the balance of energy, namely five equations, for the fourteen unknowns (mass density, velocity, symmetric stress, energy density, heat flux) in the pertinent space-time domain. The insufficiency of the balance equations to solve a dynamic problem is conceptual and is consistent with the fact that different material properties are expected to provide different responses. Mathematically the material properties of the body are expressed by constitutive equations, or constitutive assumptions, which provide a model of the material behaviour. Constitutive equations are not a mere mathematical model. They have to be physically admissible, and this is ascertained through the compatibility with the objectivity principle and the second law of thermodynamics.