Collisional Stress in Granular Flows: Bagnold Revisited

Abstract

A formulation by Bagnold for the constitutive relations in dense, rapidly sheared granular flows is reexamined in light of more recent developments based on the kinetic theory of dense gases. Bagnold's formulation expressed stress as the product of three averaged quantities: the number density of solids per unit area; frequency of intergranular collisions per particle; and momentum transfer per collision. These three terms are derived from the kinetic theory formulation in order to make clear the relationship between the two approaches. The resulting terms are more accurately modeled than those from Bagnold's original work. In particular, the tensorial nature of the stresses is preserved, whereas the previous formulation yielded a traction force. A model of the particle kinematics is incorporated into the kinetic formulation, which is simpler than those previously used and proves to be, in this case, a reasonable approximation. The stress tensor is computed for the simple shear flow of smooth, identical, inelastic disks. A reasonable agreement is found between these results and the results of numerical simulations by other researchers.