Constitutive models for granular materials including quasi-static frictional behaviour: Toward a thermodynamic theory of plasticity

This work deals with the thermodynamic formulation of constitutive models for materials whose quasi-static behaviour is governed by internal friction, e.g., dry granular materials. The process of internal friction is represented here phenomenologically with the help of a second-order, symmetric-tensor-valued internal variable. A general class of models for the evolution of this variable is considered, including as special cases a hypoelastic-like form for this relation as well as the hypoplastic form of Kolymbas (1991). The thermodynamic formulation is carried out in the context of the Müller-Liu entropy principle. Among other things, it is shown that for the hypoelastic-type models, a true equilibrium inelastic Cauchy stress exists. On the other hand, such a stress does not exist for the hypoplastic model due to its rate-independence and incremental non-linearity. With the help of a slight generalization of the notion of thermodynamic equilibrium, i.e., to thermodynamic “quasi-equilibrium,” however, such a Cauchy stress can be formulated for the hypoplastic model. As it turns out, this quasi-equilibrium for the Cauchy stress represents a thermodynamic generalization of the so-called quasi-static stress postulated for example by Goddard (1986) in the context of his viscoplastic model for a frictional-dissipative, and in particular for granular, materials.