Quantitative Understanding of the Onset of Dense Granular Flows

The question of when and how dense granular materials start to flow under stress, despite many industrial and geophysical applications, remains largely unresolved. We develop and test a simple equation for the onset of quasistatic flows of granular materials which is based on the frictional aging of the granular packing. The result is a nonmonotonic stress-strain relation which—akin to classical friction—is independent of the shear rate. This relation suffices to understand the quasistatic deformations of aging granular media, and its solid-to-liquid transition. Our results also elucidate the (flow) history dependence of the mechanical properties, and the sensitivity to the initial preparation of granular media.

Figure 1

Schematic of the experimental apparatus. A rheometer (Anton Paar MCR 302) is used to rotate a small cut of an aluminum cylindrical tube (outer diameter 29.7 mm, wall thickness of 2.3 mm) around its symmetry axis. The tube is laying on a plane covered by a $\sim 1\mathrm{mm}$ layer of PMMA spherical particles of $\sim 40\mathrm{\mu }\mathrm{m}$ diameter. Both the tube cross section and the bottom plate surfaces are sandblasted to have a surface roughness of $2–3\mathrm{\mu }\mathrm{m}$ (measured with a Keyence optical profilometer). The contact between the rheometer tool and the tube is through a frictionless bearing so that the tube is free to move in normal direction. In this way, the rheometer doesn’t apply any normal load to the granular media.

Figure 2

Stress overshoot of the onset of granular flow. (a) Friction as a function of time for three typical static to steady flow transitions of the same granular media and system with nominally identical experimental conditions, but prepared at different times. The constant imposed velocity is $0.86\mathrm{\mu }\mathrm{m}/\mathrm{s}$. The red dotted lines are the best fits of Eq. (3) with $a$, $\alpha v$, and $t_0$, respectively, as 3.31, 0.036, 0.23 for blue; 2.99, 0.051, 0.39 for gray; and 2.67, 0.058, 0.55 for black curve, with the units of mN, ${\mathrm{s}}^{-1}$, s. (b) Friction transients as a function of deformation at different sliding velocities for a granular sample that is not changed anymore after the preparation, just rests for 1 min between the tests. The red dotted line is the best fit of Eq. (3) with $a=3.21\mathrm{mN}$, $\alpha =0.055\mathrm{\mu }{\mathrm{m}}^{-1}$, and $x_0=v{t}_0=0.28\mathrm{\mu }\mathrm{m}$. The insets show the details of the behaviors at short times. (c) The parameters $a$ (left axis) and $\alpha$ (right axis) against the total normal load $F_N$ (the weight of the tube) on the granular system. The error bars indicate the standard error of at least eight independent measurements. The inset shows the steady-state friction as a function of the normal load. The standard errors in this case are smaller than the marker size.

Figure 3

Solid-to-liquid failure or unjamming transition of granular matter. (a) A stress ramp (black, right axis) is imposed on a granular sample; its transient stress overshoot under a constant shear rate ($0.86\mathrm{\mu }\mathrm{m}/\mathrm{s}$) has been determined before (grey, right axis). The granular material flows suddenly at a point slightly above the peak frictional strength (blue, left axis). The inset magnifies the microscale deformations before the solid-to-liquid failure. (b) Creep deformation of the granular material subjected to constant shear stresses below the threshold friction. The characteristic stress overshoot of the granular sample under a constant shearing ($0.86\mathrm{\mu }\mathrm{m}/\mathrm{s}$) and the applied shear stresses are shown in the inset. When the applied stress is above the steady-state friction (blue dashed line), the creep deformation is larger. However, the granular structure remains stable and resists the flow. (c) After the peak of the friction overshoot, at time $t=80\mathrm{s}$, the imposed constant shear rate is instantly switched to a constant shear stress by the rheometer (black, right axis). The constant stress in the second stage is slightly higher than the frictional strength at $t=80\mathrm{s}$. This immediately leads to the failure and flow of the granular material (blue, left axis). Again, the complete stress overshoot of the sample under a constant shear has been determined before (grey, right axis).