A thermomechanical framework for constitutive models for rate-independent dissipative materials

A formulation of elastic–plastic theory for rate independent materials is described, based on the use of thermodynamic potentials. The four energy functions commonly used in thermodynamics (internal energy, Helmholtz free energy, enthalpy and Gibbs free energy) are used to provide descriptions depending on which combinations of the stress, strain, temperature and entropy are taken as the independent variables. Much use is made of Legendre transformations to establish the links between the different energy functions. Dissipative behaviour is introduced through the use of kinematic internal parameters, and their conjugate variables, which are termed generalised stresses. A dissipation function or a yield function is used to describe the irreversible behaviour, and these are related by a degenerate case of the Legendre transformation. A central theme is that the constitutive behaviour is entirely determined by the knowledge of two scalar potentials. A systematic presentation is made of 16 possible ways of formulating constitutive behaviour within this framework. From four of these forms it is possible to establish the incremental response entirely by differentiation of the two potentials and by standard matrix manipulation. Examples are provided of the forms of the potentials for certain simple cases. The paper builds on previous work by Ziegler and other authors, and extends and generalises work by Collins and Houlsby to include thermal effects.