Constitutive Structures: Basic Aspects

In this chapter, we explain the necessity of introducing what we commonly call constitutive structures and indicate in an isothermal setting a mechanical dissipation inequality as a source of a priori constitutive restrictions. We focus attention essentially on the elastic behavior, and when there are large strains, we show the physical incompatibility between objectivity of the elastic energy and the convexity of the energy itself with respect to the deformation gradient. Then we discuss the elastic behavior in the small-strain regime. We present the notions of material isomorphism and material symmetries, indicating how they allow us to distinguish between simple solids and fluids. Among material symmetries, we discuss isotropy at length. We then include some digressions on viscous materials, showing how the introduction of the incompressibility internal constraint in this case leads to the Navier–Stokes equations.