Extended granular micromechanics approach: a micromorphic theory of degree n

For many problems in science and engineering, it is necessary to describe the collective behavior of a very large number of grains. Complexity inherent in granular materials, whether due the variability of grain interactions or grain-scale morphological factors, requires modeling approaches that are both representative and tractable. In these cases, continuum modeling remains the most feasible approach; however, for such models to be representative, they must properly account for the granular nature of the material. The granular micromechanics approach has been shown to offer a way forward for linking the grain-scale behavior to the collective behavior of millions and billions of grains while keeping within the continuum framework. In this paper, an extended granular micromechanics approach is developed that leads to a micromorphic theory of degree n. This extended form aims at capturing the detailed grain-scale kinematics in disordered (mechanically or morphologically) granular media. To this end, additional continuum kinematic measures are introduced and related to the grain-pair relative motions. The need for enriched descriptions is justified through experimental measurements as well as results from simulations using discrete models. Stresses conjugate to the kinematic measures are then defined and related, through equivalence of deformation energy density, to forces conjugate to the measures of grain-pair relative motions. The kinetic energy density description for a continuum material point is also correspondingly enriched, and a variational approach is used to derive the governing equations of motion. By specifying a particular choice for degree n, abridged models of degrees 2 and 1 are derived, which are shown to further simplify to micro-polar or Cosserat-type and second-gradient models of granular materials.