Ensemble theory for slightly deformable granular matter

Abstract

Given a granular system of slightly deformable particles, it is possible to obtain different static and jammed packings subjected to the same macroscopic constraints. These microstates can be compared in a mathematical space defined by the components of the force-moment tensor (i.e. the product of the equivalent stress by the volume of the Voronoi cell). In order to explain the statistical distributions observed there, an athermal ensemble theory can be used. This work proposes a formalism (based on developments of the original theory of Edwards and collaborators) that considers both the internal and the external constraints of the problem. The former give the density of states of the points of this space, and the latter give their statistical weight. The internal constraints are those caused by the intrinsic features of the system (e.g. size distribution, friction, cohesion). They, together with the force-balance condition, determine which the possible local states of equilibrium of a particle are. Under the principle of equal a priori probabilities, and when no other constraints are imposed, it can be assumed that particles are equally likely to be found in any one of these local states of equilibrium. Then a flat sampling over all these local states turns into a non-uniform distribution in the force-moment space that can be represented with density of states functions. Although these functions can be measured, some of their features are explored in this paper. The external constraints are those macroscopic quantities that define the ensemble and are fixed by the protocol. The force-moment, the volume, the elastic potential energy and the stress are some examples of quantities that can be expressed as functions of the force-moment. The associated ensembles are included in the formalism presented here.

Boussinesq-like problems in discrete media

“…Although it was originally developed for thermal systems, it can be applied to granular media. Several approaches [9,8,7,14,13,25] have been proposed since first Edward's model in 1989. That based on the extensive stress [14,13,25] has been followed in this research.…”

“…Several approaches [9,8,7,14,13,25] have been proposed since first Edward's model in 1989. That based on the extensive stress [14,13,25] has been followed in this research.…”

Protocol-independent granular temperature supported by numerical simulations

“…al. [48] pointed out that the size of the time step in DEM simulations may influence the width and the shape (but not the mean) of the determined probability distributions, even if the time step appears sufficiently small using common criteria. In order to make sure that this effect does not influence our results, we repeated some of the simulations with a time step of ∆t = 10 −5 s. We found that the mean volume fraction, the volume fraction variance and also the shape of the volume fraction distribution did not change on reduction of the time step.…”