The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function

This chapter examines the derivation of constitutive relations from the free energy and the dissipation function. Continuum mechanics allows one to establish constitutive relations, deducing them from a single pair of scalar functions characterizing the material. The simplest materials dealt with in continuum mechanics are elastic. More general processes and those taking place in more general materials are irreversible and require more constitutive relations, connecting the dissipative forces with the velocities. The orthogonality condition and the equivalent extremum principles have been established for velocities in the form of vectors or symmetric tensors. It is found that if the deformation of an elastic body is neither isothermal nor adiabatic, the strain tensor has to be supplemented by the additional independent state variable. The connection between stress and elastic strain is given by the generalized Hooke's law and connects the stress with the plastic strain and its time rate. It is found that orthogonality in velocity space, which is essentially responsible for the results, does not necessarily imply orthogonality in force space.