Micromechanically based constitutive relations for two-dimensional flow of granular materials are presented. First, overall stresses are related to the interparticle forces and microstructural parameters. Then, the overall velocity gradient is related to measures of relative sliding and rotation of granules. The notion of the
class
of granules with continuously evolving distribution of contact normals, is introduced. Simple local constitutive relations are considered for the rate of change of the contact forces, the evolution of the contact normals, the mechanism of local failure, and the density of contacts in a particular class. This leads to macroscopic rate constitutive equations through a Taylor averaging method. Due to nonlinearity of the rate constitutive equations, the response is computed by an incremental procedure. As an illustration, the overall response of a two-dimensional assembly of discs subjected to an overall shearing deformation is determined. In addition, explicit results are presented for the evolution of fabric, contact forces, and the history of active and inactive classes of contacts. The stress-strain relations and the evolution of fabric and contact forces are in qualitative agreement with the observed behaviour of granular materials. In light of these results, the mechanisms of failure and inelastic deformation of dense as well as loose granular materials are discussed.
A constitutive model for granular materials with evolving contact structure and contact forces—part II: constitutive equations