From solid to granular gases: the steady state for granular materials

Abstract

SUMMARYThis paper aims at extending the well‐known critical state concept, associated with quasi‐static conditions, by accounting for the role played by the strain rate when focusing on the steady, simple shear flow of a dry assembly of identical, inelastic, soft spheres. An additional state variable for the system, the granular temperature, is accounted for. The granular temperature is related to the particle velocity fluctuations and measures the agitation of the system. This state variable, as is in the context of kinetic theories of granular gases, is assumed to govern the response of the material at large strain rates and low concentrations. The stresses of the system are associated with enduring, frictional contacts among particles involved in force chains and nearly instantaneous collisions. When the first mechanism prevails, the material behaves like a solid, and constitutive models of soil mechanics hold, whereas when inelastic collisions dominate, the material flows like a granular gas, and kinetic theories apply. Considering a pressure‐imposed flow, at large values of the normal stress and small values of the shear rate, the theory predicts a nonmonotonic shear rate dependence of the stress ratio at the steady state, which is likely to govern the evolution of landslides. Copyright © 2013 John Wiley & Sons, Ltd.

A visco‐elasto‐plastic model for granular materials under simple shear conditions

“…As in Vescovi et al [40], in this paper, an ideal assembly of identical spherical particles of diameter d and density ρ p is taken into consideration. Simple shear conditions are imposed ( Figure 1) and thus the components of the stress vector σ governing the response of the material are the shear stress τ and the normal stress σ, whereas the components of the strain vector are the shear strain γ and the normal strain v. The state variables taken into account are the void ratio e (or, alternatively, the material concentration ν = 1/(1 + e)), and the granular temperature T , defined as…”

“…In contrast, the theoretical constitutive model hereafter presented, as in Johnson and Jackson [34,35], Savage [36], Louge [37], Lee and Huang [38], Berzi et al [39] and Vescovi et al [40], assumes a parallel scheme according to which the stress tensor is evaluated as the sum of two contributions: one "ratedependent" and another "rate-independent". The model hereafter illustrated can be interpreted as the extension to unsteady conditions of the model discussed in Berzi et al [39] and Vescovi et al [40].…”

“…In contrast, the theoretical constitutive model hereafter presented, as in Johnson and Jackson [34,35], Savage [36], Louge [37], Lee and Huang [38], Berzi et al [39] and Vescovi et al [40], assumes a parallel scheme according to which the stress tensor is evaluated as the sum of two contributions: one "ratedependent" and another "rate-independent". The model hereafter illustrated can be interpreted as the extension to unsteady conditions of the model discussed in Berzi et al [39] and Vescovi et al [40]. The parallel scheme was already proposed by some authors [34,35,36,37,38], but the novelty of the approach introduced by Berzi et al [39] and Vescovi et al [40] concerns the role of critical state [41,42,43,44,28]: this is here interpreted as the limit for granular stationary flows, under simple shear conditions, when the granular temperature nullifies.…”

“…The model hereafter illustrated can be interpreted as the extension to unsteady conditions of the model discussed in Berzi et al [39] and Vescovi et al [40]. The parallel scheme was already proposed by some authors [34,35,36,37,38], but the novelty of the approach introduced by Berzi et al [39] and Vescovi et al [40] concerns the role of critical state [41,42,43,44,28]: this is here interpreted as the limit for granular stationary flows, under simple shear conditions, when the granular temperature nullifies.…”

“…In Berzi et al [39] and Vescovi et al [40], as it is here in the following, a revised kinetic theory is adopted: this allows to account, in the collisional contribution, for both the clusters formation occuring at high concentrations [21,22] and the contact duration during a collision. In this paper the quasi-static contribution is computed by assuming a perfect elasto-plastic constitutive relationship including the critical state concept.…”