Stick-slip instabilities in sheared granular flow: The role of friction and acoustic vibrations

We propose a theory of shear flow in dense granular materials. A key ingredient of the theory is an effective temperature that determines how the material responds to external driving forces such as shear stresses and vibrations. We show that, within our model, friction between grains produces stick-slip behavior at intermediate shear rates, even if the material is rate strengthening at larger rates. In addition, externally generated acoustic vibrations alter the stick-slip amplitude, or suppress stick-slip altogether, depending on the pressure and shear rate. We construct a phase diagram that indicates the parameter regimes for which stick-slip occurs in the presence and absence of acoustic vibrations of a fixed amplitude and frequency. These results connect the microscopic physics to macroscopic dynamics and thus produce useful information about a variety of granular phenomena, including rupture and slip along earthquake faults, the remote triggering of instabilities, and the control of friction in material processing.

Figure 1

A key ingredient for the emergence of flow instabilities is the nonmonotonic variation of the steady-state compactivity ${\chi }^{\text{ss}}$, as a function of the imposed shear rate $\dot{\gamma }$. (The quantity ${\chi }^{\text{ss}}$ increases monotonically with the layer thickness or volume, so the layer thickness follows the same qualitative behavior.) For clarity, this is shown here for a pressure $p=10$ MPa, which lies below the critical pressure necessary for stick-slip to occur (see Fig. 8 below). It is implicitly assumed that physical insight gained from laboratory experiments at lower pressures can be extrapolated to higher pressures in the MPa range.

Figure 2

The parameter ${\mu }_0$, which provides a measure of the apparent yield stress, as a function of the compactivity $\chi$. This quantity must be a rapidly decreasing function of $\chi$ whenever $\chi <{\widehat{\chi }}_0$, and cease to be so above ${\widehat{\chi }}_0$, as indicated by the vertical dashed line. (${\widehat{\chi }}_0$ is the steady-state compactivity in the slow shear limit.) This condition conforms with the intuition that the stress needed to unjam a granular medium increases with the packing fraction. This, combined with the rapid increase of the steady-state compactivity ${\chi }^{\text{ss}}$ below ${\widehat{\chi }}_0$ for frictional grains shown in Fig. 1, provides the apparent weakening that permits stick-slip instabilities at high-enough pressure. Here we have used the interpolation Eq. (4.13) with ${\mu }_1=0.3$ and ${\mu }_2=3.7\times {10}^{-3}$.

Figure 3

Variation of (a) shear stress $s$ and (b) compactivity $\chi$ with shear strain in the presence (solid curves) and absence (dashed curves) of interparticle friction and in the absence of external vibration. In each case the curve for $p=60$ MPa (red) lies above that for $p=40$ MPa (blue). Observe that stick-slip occurs only in the presence of interparticle friction, which provides a means for the compactivity to fall below ${\widehat{\chi }}_0=0.3$, shown with the dotted line in (b), in the ideal hard-sphere model, and facilitates the emergence of instabilities. The slip phase of stick-slip coincides with a sharp increase in the compactivity or volume dilatation.

Figure 4

Variation of (a) shear stress $s$ and (b) compactivity $\chi$ with the plastic strain rate ${\dot{\gamma }}^{\text{pl}}$ in the presence (solid curves) and absence (dashed curves) of interparticle friction and in the absence of external vibration. The pressure values are $p=40$ MPa (blue) and 60 MPa (red), and the imposed strain rate is $\dot{\gamma }={10}^2{\mathrm{s}}^{-1}$, indicated by the vertical dotted line. Stick-slip occurs for frictional grains, and each of the corresponding phase curves converges to a limit cycle. The limit cycle for $p=60$ MPa encloses a bigger area than for $p=40$ MPa. For frictionless grains, each of the phase curves converges to a fixed point, indicated by each of the filled circles, on the phase diagram. (The fixed points of $\chi$ versus ${\dot{\gamma }}^{\text{pl}}$ for $p=40$ and 60 MPa overlap with one another.) In the case of frictional grains, the applied load “sticks” left of the vertical dotted line, and creeps at a slower but nonzero rate than the imposed strain rate, while the compactivity decreases. Once the shear stress exceeds that needed to overcome frictional resistance and initiate slip, both the compactivity and the plastic strain rate increase dramatically, and the stress drops. Then the applied load sticks, and the shear stress builds up again during the new stick phase. Thus the stick-slip cycle repeats itself. This representation of stick-slip dynamics can be directly compared to that in Fig. 3.

Figure 5

Variation of (a) shear stress $s$, (b) plastic strain rate ${\dot{\gamma }}^{\text{pl}}$, and (c) accumulated plastic strain with total shear strain, in sheared, unvibrated angular sand, a granular material composed of frictional grains. As before, the pressure values are $p=40$ MPa (blue, with earlier onset of slip) and 60 MPa (red, with later onset of slip). Observe from (b) that ${\dot{\gamma }}^{\text{pl}}$ is slowly increasing during the stick phase of each stick-slip cycle; this accounts for the the apparent softening seen towards the end of each stick-slip cycle in (a). Panel (c) shows that while much of the plastic strain is accumulated during the slip phase, there is plastic strain during the stick phase as well.

Figure 6

Vibrations suppress or amplify stick-slip in different parameter regimes. (a) Shear stress $s$ a function of shear displacement $\gamma$ for sheared angular sand, for different magnitudes of the normal stress $p$, at applied shear rate $\dot{\gamma }=2.4\times {10}^2{\mathrm{s}}^{-1}$. External acoustic vibration is turned on (with intensity $\rho =5\times {10}^{-4}$) for $6<\gamma <8$, and off otherwise. At the lower pressure of $p=45$ MPa (green curve), vibrations amplify stick-slip amplitude before atttenuating it. At the higher pressure of $p=50$ MPa (red curve), vibrations amplify the stick-slip amplitude and period. The red curve for $p=50$ MPa has been offset by 10 MPa for clarity. (b) Variation of shear stress $s$ with plastic shear rate ${\dot{\gamma }}^{\text{pl}}$ under the conditions in (a) above, for shear strains $4<\gamma <8$. The dashed lines denote the behavior under vibrations of fixed intensity, while the lighter shades indicate transients before approaching a limit cycle or fixed point. At the lower pressure of $p=45$ MPa, except for a short transient the phase curve traces a limit cycle when the sand is not vibrated. The phase curve spirals towards a single fixed point, denoted by the filled green circle, when the material is vibrated. At the higher pressure of $p=50$ MPa, except for short transients immediately following the switching on or off of fixed-intensity external vibrations, the phase curve traces a smaller limit cycle in the clockwise direction when the material is not subject to vibrations and a bigger limit cycle when vibrations are switched on.

Figure 7

Shear stress $s$ as a function of accumulated shear strain $\gamma$, at applied shear rate $\dot{\gamma }=2.4\times {10}^2{\mathrm{s}}^{-1}$, with and without external vibrations at fixed intensity. The onset of slip occurs earlier in the presence of vibrations, as shown by the location of the first stress drop on the dashed curve for vibrated sand. This observation is reminiscent of the triggering of an earthquake by seismic waves arriving from elsewhere.

Figure 8

Phase diagram showing the parameter space where stick-slip occurs, in the absence $(\rho =0)$, and at two different, fixed vibration intensities $(\rho =5\times {10}^{-4}), 2\times {10}^{-3}$. Stick-slip operates only at imposed shear rates above $\dot{\gamma }\sim 80{\mathrm{s}}^{-1}$ and pressures above $p\sim 30$ MPa. Vibrations suppress stick-slip for a small range of pressure at all shear rates for which stick-slip is possible (pink) but amplify the stick-slip amplitude for a wide range of shear rates and pressure (yellow). There is a narrow range of shear rates and pressure (magenta and green) for which stick-slip is possible in the absence of vibrations and for only one, but not both, of the two vibration intensities $\rho$. The blue circles and red squares denote the shear rate and pressure values corresponding to the stress-strain curves in Figs. 3 and 6.