Unsteady flow, clusters, and bands in a model shear-thickening fluid

We analyze the flow curves of a two-dimensional assembly of granular particles which are interacting via frictional contact forces. For packing fractions slightly below jamming, the fluid undergoes a large scale instability, implying a range of stress and strain rates where no stationary flow can exist. Whereas small systems were shown previously to exhibit hysteretic jumps between the low and high stress branches, large systems exhibit continuous shear thickening arising from averaging unsteady, spatially heterogeneous flows. The observed large scale patterns as well as their dynamics are found to depend on strain rate: At the lower end of the unstable region, force chains merge to form giant bands that span the system in the compressional direction and propagate in the dilational direction. At the upper end, we observe large scale clusters which extend along the dilational direction and propagate along the compressional direction. Both patterns, bands and clusters, come in with infinite correlation length similar to the sudden onset of system-spanning plugs in impact experiments.

Figure 1

Flow curves for different system sizes $N$ and volume fractions $\varphi$: $\varphi =0.78,N=3600$ (black), $\varphi =0.7975,N=8000$ (blue), and $N=32000$ (green). Inset: different starting configurations ($\varphi =0.801,N=80000$); the yield-stress branch is metastable.

Figure 2

Left: stress $\sigma$ as a function of time for $N=8000$, $\varphi =0.7975$, and $\dot{\gamma }=1.143\times {10}^{-4}$; right: corresponding distribution of stress $P\left(\sigma \right)$. The average over this distribution, restricted to the regions around the two peaks, gives the two data points in the blue curve of Fig. 1 that carry an error bar. The error is obtained by varying the stress value at which the distribution is split.

Figure 3

Snapshots of local shear stress for (a) $\dot{\gamma }=3.297\times {10}^{-5}$, (b) $\dot{\gamma }=1.143\times {10}^{-4}$, (c) $\dot{\gamma }={10}^{-3}$, and (d) $\dot{\gamma }=0.05$. Particle $i$ is colored in black (gray) whenever its local shear stress ${\sigma }_i$ is above (below) average. (e) Snapshot of local shear stress for a high aspect ratio sample: $L_x=10,L_y=260$ ($\dot{\gamma }=1.732\times {10}^{-4},\varphi =0.79875$).

Figure 4

Correlation function vs distance $r$ taken along (a) the compressive and (b) dilational directions. Data are averaged over the corresponding perpendicular direction. $N=80000$, $\varphi =0.79750$.

Figure 5

Correlation length $\xi$ vs strain rate $\dot{\gamma }$. Length scale extracted from the decay of the stress correlation function $C_{\sigma }\left(r\right)$ with $r$ taken along (a) the compressive and (b) the tensile directions.

Figure 6

Profiles of local shear stress $\sigma$ and number of contacts per particle $z$, taken along the (tensile) diagonal direction $r$. Values are averaged over the compressive direction. $N=80000$, $\varphi =0.79875$.

Figure 7

Average contact lifetime $\langle \tau \rangle$ vs strain rate $\dot{\gamma }$. Highlighted are the strain rate ${\dot{\gamma }}_l$ at the onset and at the upper end ${\dot{\gamma }}_u$ of shear thickening. $N=80000$, $\varphi =0.79750$.

Figure 8

Global pressure $p$ (black) and global stress $\sigma$ (red) as a function of strain $\gamma$. $N=80000$, $\varphi =0.79875$.

Figure 9

(a) Time evolution of stress bands at the lower end of ST ($\dot{\gamma }=1.1\times {10}^{-4}$). The $x$ axis represents the spatial coordinate along the dilational direction, i.e., along the propagation direction of the band; values are averaged over perpendicular stripes. The $y$ axis is time. The color code is shear stress. Four system-spanning bands move along the dilational direction but in opposite directions, collide, and move on. (b) The same event, now with particle accelerations. (c) Stress at the upper end of ST ($\dot{\gamma }=1.0\times {10}^{-3}$). The $x$ axis now represents the spatial coordinate along the compressional direction; values are averaged over perpendicular stripes.