Force networks and the dynamic approach to jamming in sheared granular media

Diverging correlation lengths on either side of the jamming transition are used to formulate a rheological model of granular shear flow, based on the propagation of stress through force chain networks. The model predicts three distinct flow regimes, characterized by the shear rate dependence of the stress tensor, that have been observed in both simulations and experiments. The boundaries separating the flow regimes are quantitatively determined and testable. In the limit of jammed granular solids, the model predicts the observed anomalous scaling of the shear modulus and a new relation for the shear strain at yield. Copyright c © EPLA, 2007 Jamming occurs when an amorphous collection of particles spontaneously develops rigidity and supports weight like a solid instead of flowing like a liquid [1]. The transition takes place without static spatial ordering, but is accompanied by long-range dynamical correlations arising from the collective motion of groups of particles [2–4]. In the case of sheared granular materials, the transition from jammed to flowing phases occurs as the applied stress Σ is increased or the packing fraction φ is decreased, and there is a packing fraction φc at which the system jams in the limit of zero stress [5–7]. Diverging correlation lengths are observed on each side of the transition, and are related to the average size of force chain networks for φ< φc [2,8] and the average size of isostatic clusters for φ> φc [3,9]. In the jammed state many macroscopic observables exhibit power law scalings in (φ−φc) [5–7] and the flowing state rheology changes dramatically as the packing fraction is increased above φc. Although the jamming transition controls both dynamic and static properties near φc, theories tend to focus mainly on the latter [9,10]. Here we theoretically explore the jamming transition by first considering flows with non-zero shear rate γ̇ and then taking the limit of γ̇→ 0 to access the static case. This procedure leads to quantitative predictions for the flowing rheology that match observations in ref. [11]. It also predicts the anomalous static scaling of the shear modulus measured in refs. [5,6] and a new scaling relation for the static yield strain, γ∗ ∝ (φ−φc)1/2, which provides a testable prediction of the theory. Granular materials behave as peculiar liquids. This can be clearly demonstrated by measuring the shear rate dependence of the stress tensor Σ(γ̇). Depending on various parameters, the system exhibits either Bagnold scaling Σ∝ γ̇, elastic-inertial scaling Σ∝ γ̇, or quasistatic scaling Σ∝ γ̇ (i.e. constant) [11]. The rheology of the flow is dependent on φ, with Bagnold scaling for φ< φc, quasi-static scaling for φ> φc, and elasticinertial scaling in between. This is in marked contrast to Newtonian fluids where Σ∝ γ̇ and only the proportionality constant depends on φ. Although the phase diagram of granular shear flow has been extensively studied in simulations [11–13] and experiments [14], the origins of the rheological crossovers remain to be explained. Arriving at a solution to this problem requires a well-developed theory for the stress tensor in dense granular flows. The stress tensor ultimately depends on the amorphous microscopic arrangement of grains. Forces are transmitted via contacts between grains and a perturbation on one grain can have long-range effects. Indeed, both experiments and simulations indicate that contact forces are correlated [2,15] and tend to form quasi–one-dimensional filaments, or force chains, that permeate the material [16]. Since contact between grains is the only form of force transfer, Σ is fully determined by properties of the networks [17,18]. Here we investigate the role of force chain