Rheology of granular materials with size distributions across dense-flow regimes
Graphical abstract
Introduction
Dense flows of granular materials occur in numerous natural and industrial processes, and a variety of distinct rheological behaviors are observed. For non-cohesive, monodisperse particles, three dense-flow regimes have been identified – namely, the quasistatic, inertial, and intermediate regimes [1], [2], [3], [4] – each of which manifests different scalings of the mean stresses with shear rate and volume fraction. Numerous constitutive stress models have been constructed with these scalings in mind [2], [5], [6], [3], [7], [8], [9], [10]. However, real particles can be cohesive and often have particle size distributions (PSD). Constitutive models have been constructed to take account of cohesion [11], [12], where a new rate-independent regime below critical (jamming) volume fraction is identified. To take account of polydispersity, new kinetic theory has been developed for granular flow (e.g. [13], [14], [15], [16], [17]). However, it is limited to dilute systems and does not account for friction. Although a few studies have appeared in the literature on dense-flow rheology with polydispersity, they are limited to binary mixtures [18], [19], [20], [21], or mixtures with four particle sizes [6]. It is not known at this time whether a single rheological model that can describe dense flows of granular mixtures of any size distribution with fair accuracy. The present study demonstrates that it is indeed possible.
In this paper, we investigate the rheology of granular materials with various size distributions through discrete element method (DEM) simulations of simple shear flows of frictional particles. Most of the simulations are performed using Hertzian contact model. To test the robustness, a few simulations with Hookean contact model have also been performed. We perform simulations for four types of PSDs (binary, linear, Gaussian, and lognormal), and demonstrate that the different regimes observed for monodisperse particles persist even when size distribution is included. The jamming volume fraction that separates the quasistatic and inertial regimes is found to increase when particles manifest size distribution. We further demonstrate that pressure can be scaled by powers of the distance to the jamming volume fraction, in the same manner as was done for the case of uniformly sized particles [2], provided that we recognize that the jamming volume fraction and the effective particle size change with PSD. We then show that these jamming volume fractions used for collapse of pressure can be expressed as a function of polydispersity and skewness of the PSD. Finally, for shear stress ratio (ratio between shear stress and pressure), we demonstrate that the inertial number model is still valid for particles with size distributions, provided that the yield stress ratio is adjusted to account for PSD. Thus, we are able to extend the stress model proposed by Chialvo et al. [2] for dense flows of frictional particles with same size to account for frictional particles with any size distribution.
Section snippets
Simulation methods
Following the study on jamming behaviors in a static system [22], four types of distributions P(r) of radii r are studied: binary, linear, Gaussian, and lognormal. P(r) is a number-based continuous distribution, and is normalized to unity, .. The distributions are illustrated in Fig. 1. We also include a monodisperse case as the base case.
For monodisperse case, all particles have the same scaled radius of 0.5. The binary distribution consists of particles with two distinct radii, r1 and r
Flow regimes
In most of the simulations reported here, particle friction coefficient μ = 0.5 is used, representing a typical value for granular materials. As discussed later, the conclusions drawn in the paper are applicable for other μ values as well, only requiring slight modifications to certain model parameters as detailed previously for monodisperse particles [2], [11]. For each size distribution, simulations are performed for various shear rates and volume fractions.
Fig. 2(a) plots the scaled pressure p/
Pressure
It has been shown in experimental [7] and computational [2], [11], [5], [6], [29], [3] studies of monodisperse particles that pressure vs. shear rate data at several different volume fractions will collapse onto two curves (one above ϕc and one below) upon scaling the pressure and shear rate by powers of | ϕ − ϕc |, the distance to jamming. As seen in Fig. 4(a), pressure vs. shear rate data for monodisperse particles from Fig. 2(a) collapse onto two curves with p * = (p/k)/| ϕ − ϕc |ϵ and
Shear stress ratio
As both shear stress τ and pressure p vary over several orders of magnitude, shear stress ratio η ≡ τ/p, which varies over a smaller range, is studied. Successful rheological models for dense granular flows have been developed that employ a dimensionless parameter called the inertial number [8], [9], [10]. This inertial number, , serves a basis to collapse stresses over a range of volume fractions and shear rates for granular flows of monodisperse, hard particles. Specifically, several
Summary
We have investigated shear flows of dense, frictional granular materials of various size distributions via DEM simulations. The inertial, quasistatic, and intermediate regimes observed for monodisperse particles persist for polydisperse particles. However, the critical volume fraction that separates the quasistatic and inertial regimes is increased when particles have size distributions. This critical volume fraction is found to be the same for both Hookean and Hertzian contact models, and can
Acknowledgment
We thank Professor Jin Sun (University of Edinburgh) for his help with simulations.
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