Continuum Modeling of Secondary Rheology in Dense Granular Materials

Abstract

Recently, a new nonlocal granular rheology was successfully used to predict steady granular flows, including grain-size-dependent shear features, in a wide variety of flow configurations, including all variations of the split-bottom cell. A related problem in granular flow is that of mechanicallyinduced creep, in which shear deformation in one region of a granular medium fluidizes its entirety, including regions far from the sheared zone, effectively erasing the yield condition everywhere. This enables creep deformation when a force is applied in the nominally quiescent region through an intruder such as a cylindrical or spherical probe. We show that the nonlocal fluidity model is capable of capturing this phenomenology. Specifically, we explore creep of a circular intruder in a two-dimensional annular Couette cell and show that the model captures all salient features observed in experiments, including both the rate-independent nature of creep for sufficiently slow driving rates and the faster-than-linear increase in the creep speed with the force applied to the intruder.Cooperativity is a hallmark of quasi-static, dense granular deformation. At a microscopic level, deformation in granular materials takes place through the rearrangement of clusters of grains. These rearrangement events lead to long-range fluctuations, or agitations, that influence the behavior of nearby clusters, leading to macroscopic manifestations of cooperativity, which are quite varied. Most familiarly, the length-scales associated with velocity fields in dense granular flows depend crucially upon the grain size [1], with grain-size dependent shear band widths being observed in many geometries [2][3][4][5][6][7][8]. Other manifestations of cooperativity include the dependence of volumetric outflow rate on grain size in drainage flows [9,10] and the so-called H stop -effect, in which thin granular layers require greater tilt to flow down an inclined surface [11,12]. A more recently-observed example of cooperativity in dense granular flows is that of mechanicallyinduced creep [13][14][15], understood as follows. When an intruder, such as a sphere, rod, or vane, is placed in a dense granular material, one must apply a force to the intruder that exceeds a critical value in order to move it through the granular media. However, shear deformation far from the intruder enables it to move, or creep, through the granular media for any non-zero value of the applied force -even when this force is less than the critical value. That is to say, flow anywhere in a granular media erases the yield condition everywhere! These cooperative phenomena provide stringent tests for a continuum model of granular flow. Local approaches to continuum modeling of granular materials, such as granular rheology [16][17][18] or soil mechanics [19,20], which relate the stress at a point to the local strain, strain-rate, or locally evolved state variables, are not equipped to address size-dependent manifestations of cooperativity. Recently, we proposed a nonlocal rheology for ...

Local rheological measurements in the granular flow around an intruder

“…This spatial dependence demonstrates that local rheology may be not sufficient to describe strong non-parallel flows such as the present flow around a cylinder. While such spatial variations of µ have been recently predicted by non-local modeling in intruder geometry [10], the finite level of dilatancy D of stationary flows has not been reported yet theoretically or numerically. The present finding may thus put further constraints on the formulation of continuum models by opening up novel challenge in modeling and understanding non-parallel compressible granular flows using compressible non-local rheology.…”

“…The present finding may thus put further constraints on the formulation of continuum models by opening up novel challenge in modeling and understanding non-parallel compressible granular flows using compressible non-local rheology. Experimentally, checking for non-local rheology is challenging as the computation of Laplacian terms -at the core of these non-local models [8,10] -is hampered by finite resolution and large fluctuations.…”

A finite element implementation of the nonlocal granular rheology

“…It bears noting that we have observed no evidence of numerical instabilities or spurious oscillation patterns in our calculations. Since the model has shown promise in predicting granular flows [14,16], we expect that the ability to robustly solve these equations will be of use to researchers working in the field, and hence, we have made all Abaqus UEL subroutines 27 and input files available as supporting information. One possible limitation of the approach is that it does not smoothly transition to the local case.…”

“…However, the mathematical system of equations is non-standard, and solving these equations in arbitrary flow configurations presents a new challenge. The purpose of this paper is to report on our finite-element-based approach to solving boundary-value problems with the nonlocal granular rheology, which we have utilized in our past work [14,16], and show that our approach is robust. Several specific questions that arise in formulating a finite-element implementation are…”

Mechanical fluctuations suppress the threshold of soft-glassy solids: The secular drift scenario

“…Note that in spite of resemblances, this phenomenon is a priori different from another fluidization process occurring when a granular packing is placed in contact with a fluidized shear band [38,39]. In the last case, theoretical analysis and numerical simulations show that the induced creeping process comes from a non-local stress relaxation, from the flowing part to the material bulk [40][41][42][43]. The generality of the scenario, mixing two generic features of glassy system make it suitable to be tested experimentally on many other practical situation like colloidal glasses, pastes, clays or even glass-former molecular systems, which actually may turn out to be of practical importance to assess the stability and reliability of structures strained externally in their environment over very long time scales.…”