Size of heterogeneous deformations in sheared granular flows

Sheared granular materials spontaneously develop heterogeneous and transient kinematic fields which resemble turbulence. We focus here on characterizing the length scale of these heterogeneities. We simulate plane-shear flows and measure the local grain velocity, velocity fluctuation, shear rate, vorticity, and dilation rate. The results show that these quantities are widely distributed, even more so at low inertial numbers. We relate this effect to spatial correlations that develop at low inertial numbers, where the flow becomes partitioned into “cold” patches with lower than average strain rate and “hot” zones where the strain rate is concentrated. We finally extract a characteristic length scale for the size of these heterogeneities, corresponding to the size of a representative elementary volume. This size is found to vary from a few grain diameters at high inertial number to a few dozen grain diameters at low inertial number, following a characteristic power law. These insights into. representative elementary volume (REV) size may be of importance to further develop continuum models for granular flows, capturing local and nonlocal effects.

Figure 1

Velocity fluctuations in plane-shear flows. (a) Schematic of the plane-shear flow configuration, featuring a snapshot of the grain packing under applied pressure $P$ and shear velocity $u$. The system dimension is $100d\times 80d$. (b),(c) Snapshots of grain velocity fluctuations for inertial numbers $I=0.01$ and $I=0.5$, respectively. The arrows represent the velocity fluctuations $\delta v$ in both the $x$ and $y$ directions for each grain, with the arrow base located at the center of the grain.

Figure 2

Velocity and relative velocity distributions in plane-shear flows. (a),(b) Distributions of grain velocity $\left(v_y^i\right)$ and grain relative velocity $\left(v_y^{ij}\right)$ in the $y$ direction for two different inertial numbers $I=0.01$ and $I=1$, respectively. Black lines are normal distribution fits; red line is an exponential distribution fit. Means and standard deviation of the distributions are given in the legends. (c),(d) Distributions of $v_y^i$ and $v_y^{ij}$ for a range of inertial numbers. (a)–(d) Velocities are in units of $d/t_i$. (e) Standard deviation of velocity fluctuations $\sigma \left(\delta v_y^i\right)$ (blue) and $\sigma \left(\delta v_y^{ij}\right)$ (green) for different inertial numbers $I$, and power law in Eq. (3) (black line). (f) Kurtosis of $v_y^i$ (blue) and $v_y^{ij}$ (green) for different inertial numbers.

Figure 3

Velocity gradient and decomposition into deformation rates. Top: Illustration of how the velocity gradient tensor $F$ is calculated for a single grain (green) including close neighbors (blue) using Eq. (4); the calculation involves relative position $\left(r^{ij}\right)$ and relative velocity $\left(v_y^{ij}\right)$. The cutoff distance used to define neighboring grains in this study is $|r^{ij}|\le 2d$, which is highlighted as the dashed circle. This cutoff distance is chosen to evaluate the velocity gradient on the smallest possible zone while ensuring that enough neighbors are present to determine $F$. Bottom: Decomposition of the velocity gradient tensor into volumetric strain rate, shear strain rate, and vorticity, corresponding to the trace, symmetric, and antisymmetric parts of $F$.

Figure 4

Distribution of local deformation rates. (a) Shear strain rate $\left({\dot{\epsilon }}_s\right)$, (b) volumetric strain rate $\left({\dot{\epsilon }}_v\right)$, and (c) vorticity $\left(w\right)$ for different inertial number $I$. (a)–(c) Means and standard deviations of the distributions are indicated in the legends. (d) Means $\langle {\dot{\epsilon }}_s\rangle$ (green line), $\langle {\dot{\epsilon }}_v\rangle$ (red line), and $\langle w\rangle$ (blue line) of these distributions normalized by the macroscopic shear rate $\dot{\gamma }$, including a black line at 0.5 for visual reference.

Figure 5

Fluctuations of deformation rates, represented by the standard deviation of the local shear strain rate $\sigma \left({\dot{\epsilon }}_s\right)$, volumetric strain rate $\sigma \left({\dot{\epsilon }}_v\right)$, and vorticity $\sigma \left(w\right)$ in unit $t_i^{-1}$ at different inertial number. The black line is the power law in Eq. (6) with $\beta =0.23,m=0.70$.

Figure 6

Spatial distribution of kinematic fields. (a)–(i) Normalized shear strain rate, volumetric strain rate, and vorticity fields at three different inertial numbers $I=0.001, I=0.01$, and $I=0.1$, respectively. (j)–(l) Spatial distribution of extreme values in each kinematic field at the same inertial numbers.

Figure 7

REV size. (a),(b) Illustrations of the window size ($\mathrm{W}=2d$ and $40d$) used to measure standard deviations $\sigma$ of kinematic quantities, and the mean of these standard deviations $\langle \sigma \rangle$ including all windows and all times. (c)–(e) Evolution of the mean standard deviation as a function of the window size $W$, normalized by its maximum value defined as ${\sigma }_{max}=\langle \sigma \rangle (W=40d)$. Dashed lines on (c) show the method used to extract a minimum REV size $W_0$ for different inertial numbers. (d) Minimum REV size $W_0$ obtained from shear strain rate, volumetric strain rate, and vorticity velocity fluctuation in the $\mathit{y}$ direction, as a function of the inertial number. The green and black lines correspond to the power law in Eq. (7).

Figure 8

Effect of criterion on REV. REV size of shear strain rate for two criteria of $95\%$ and $90\%$. Lines are best fit using Eq. (7).

Figure 9

Spatial correlation of $v_y$. (a) The dependence of the $v_y$ spatial correlation to the controlling parameter $\mathrm{\Delta }\left(d\right)$, where $d$ is the average diameter of the grains. (b) The comparison between the extracted minimum REV size $W_0/d$ and correlation length ($l$) obtained from $v_y$ for different inertial numbers.