Continuum approach to wide shear zones in quasi-static granular matter

TL;DR

This paper investigates the continuum modeling of wide shear zones in slow, dense granular flows, proposing constitutive relations and highlighting the importance of variable effective friction for different geometries.

Contribution

It introduces a set of testable constitutive assumptions for wide shear zones and demonstrates the necessity of a variable effective friction coefficient in complex geometries.

Findings

Wide shear zones in symmetric flows align with constant effective friction.

Split-bottom flows require a variable effective friction coefficient.

Flow geometry influences the constitutive relations in granular flows.

Abstract

Slow and dense granular flows often exhibit narrow shear bands, making them ill-suited for a continuum description. However, smooth granular flows have been shown to occur in specific geometries such as linear shear in the absence of gravity, slow inclined plane flows and, recently, flows in split-bottom Couette geometries. The wide shear regions in these systems should be amenable to a continuum description, and the theoretical challenge lies in finding constitutive relations between the internal stresses and the flow field. We propose a set of testable constitutive assumptions, including rate-independence, and investigate the additional restrictions on the constitutive relations imposed by the flow geometries. The wide shear layers in the highly symmetric linear shear and inclined plane flows are consistent with the simple constitutive assumption that, in analogy with solid friction, the effective-friction coefficient (ratio between shear and normal stresses) is a constant. However, this standard picture of granular flows is shown to be inconsistent with flows in the less symmetric split-bottom geometry - here the effective friction coefficient must vary throughout the shear zone, or else the shear zone localizes. We suggest that a subtle dependence of the effective-friction coefficient on the orientation of the sliding layers with respect to the bulk force is crucial for the understanding of slow granular flows.