From Granular Matter to Generalized Continuum

Abstract

Summary. Following a cursory review and synthesis of multipolar continua, the rudiments of graph theory, and granular mechanics, a graph-theoretic, energy-based homogenization is proposed for the systematic derivation of multipolar stress and kinematics in granular media. This provides a weakly non-local hierarchy of multipolar field equations for quasi-static mechanics based on polynomial representations of the kinematics of the type employed in past works. As an improvement on those works, a method is proposed for avoiding "overfitting" of fluctuations based on the so-called "Generalized Additive Method" of statistics. Among other results, it is shown that the standard formula for Cauchy stress in granular media may break down owing to multipolar effects, and that granular rotations in the typical granular medium should not lead to Cosserat effects, as the lowest-order departure from the simple-continuum model.

Granular hypoplasticity with Cosserat effects

“…The resulting block-diagonal form for the matrix in (31) leads to a similar form for the inverse, and the first relation in (31) reduces to the uncoupled form…”

“…It should be noted that (19) [43]. 7 More general continua [31] are obtained by assigning additional kinematic variables to material points, the simplest example being Eringen's microstretch continuum [30] involving dilatation as well as rotation. Although sometimes proposed as a model of granular dilatancy [32,33], the physical significance is not clear.…”

“…Quotation marks are employed here to emphasize that the terminology should, strictly speaking, be reserved for matrix representations like that in (31), which are necessary because of the well-known limitation on complex-variable representations 11 . With X = D = Λ Λ Λ + ıK, (31) defines a pseudo-linear kinematic form, with associated (pseudo-)quadratic form:…”

“…The requirement that M be self-adjoint implies symmetry of the matrix in (31), with the consequence that…”

A general rotation averaging method for granular materials

“…The overall mechanical behaviors of granular materials can be reasonably well described by continuum approaches [3,9,11,22,23,29]. Since granular materials are discrete in nature, some attempts have been made to obtain the continuum properties by averaging methods [2,5,6,16,25,39], which are also referred as homogenization methods [7,19] or coarse grain methods [30,38].…”