Local and global avalanches in a two-dimensional sheared granular medium

We present the experimental and numerical studies of a two-dimensional sheared amorphous material composed of bidisperse photoelastic disks. We analyze the statistics of avalanches during shear including the local and global fluctuations in energy and changes in particle positions and orientations. We find scale-free distributions for these global and local avalanches denoted by power laws whose cutoffs vary with interparticle friction and packing fraction. Different exponents are found for these power laws depending on the quantity from which variations are extracted. An asymmetry in time of the avalanche shapes is evidenced along with the fact that avalanches are mainly triggered by the shear bands. A simple relation independent of the intensity is found between the number of local avalanches and the global avalanches they form. We also compare these experimental and numerical results for both local and global fluctuations to predictions from mean-field and depinning theories.

Figure 1

(a) Schematic view of the evolution of the shear stress $\tau$ in a sheared granular system at constant volume, for slowly loaded stiff grains (the $x$ axis increases to the left). After a transient regime where the stress grows elastically, the shear stress fluctuates about the yield stress ${\tau }_c$, as observed in several experiments and simulations [2, 5, 6, 22, 23] . (b) Shear jamming phase diagram (adapted from [21]) including jammed (J) [24], unjammed (uJ), fragile (F) [25], and shear-jammed (SJ) regimes. A frictional granular system slowly sheared at constant packing fraction $\varphi$ follows the vertical line with arrows until reaching the yield stress ${\tau }_c$, where it will fluctuate between jammed and unjammed states below and above ${\tau }_c$. The fluctuations above ${\tau }_c\left(\varphi \right)$ give rise to intermittent dynamics [see (a)].

Figure 2

(a) From top to bottom, evolution of the dimension of the pure shear cell during one cycle. An initial $44\times 40{\mathrm{cm}}^2$ rectangle is progressively shrunk in one direction and expanded in the other one to form a $55\times 32{\mathrm{cm}}^2$ rectangle [green (dark) rectangles] and the motion is then reversed [red (clear) rectangles] to achieve a 20% shear amplitude. (b) Three-dimensional schematic of the biaxial experimental cell. Moving walls shear, step by step, a set of bidisperse photoelastic particles with a UV-ink bar on them. At each step the system is imaged with white light, between crossed polarizers and with UV light.

Figure 3

Top view of the granular system (a) in transmitted white light, (b) between crossed polarizers, and (c) in UV light. (d) From left to right, small and large $\mathrm{Teflon}$® wrapped particles and small and large bare particles.

Figure 4

(a) The plain blue (dark) line is the global energy $\mathcal{E}$ stored in the granular media measured experimentally as the sum of $G^4$ over the particles. The red (clear) dashed line is the power $\mathcal{P}$ dissipated in the system computed as minus the negative part of the derivative of the energy with respect to strain. Both quantities are plotted as a function of the shear step. Each gray shaded box marks shearing in one direction (shearing is reversed periodically). (b) Cartoon of the threshold power (energy released per strain unit) $\mathcal{P}$ signal to extract global avalanches. For each $i$ avalanche we measure the beginning strain $S_{b_i}$, ending strain $S_{e_i}$, duration $D_i$, and accumulated dissipated energy $E_i$.

Figure 5

(a) Absolute value of the particle rotation $\mathrm{\Delta }\theta$ measured from step $j_0$ to $j_0+1$ in experiment I. The clusters of large rotation corresponds to local grain rearrangements. (b) Nonaffine displacement of grains between step $j_0$ and $j_0+1$ (motion due to the boundary is removed). Particle positions after removing the affine displacements, interpolated from the boundary motion. The direction of nonaffine motion is indicted by arrows and the magnitude of nonaffine motion is indicated by the color scale.

Figure 6

(a) Energy $G^4$ (arbitrary units) stored in each particle before (step $j_0)$ and after (step $j_0+1)$ an avalanche for experiment I. Step $j_0$ corresponds to Fig. 5. We notice that some force chains break. (b) Variation of the energy from step $j_0+1$ to $j_0$. Blue (dark) particles are unloaded (lose energy), whereas red (clear) are loaded (gain energy). (c) Schematic of the evolution of a particle undergoing a strong energy drop after a shear step. Colors and signs stand for the same particles from step to step. (d) Grains involved in an avalanche at shear strain step $j_0$. The same colors stand for the same avalanches. We see that local avalanches detected with energy follow the force chain structure (see the text for details).

Figure 7

(a) Probability density function of the energy of global avalanches $P\left(E\right)$ for different threshold values ${\mathcal{P}}_{th}$ (color from blue to brown) for experiment I and with a low threshold value $\left(\overline{{\mathcal{P}}_{th}}=10\right)$ for simulation 2 (black). For experimental data, the statistics follow a power law over 3 decades with exponent $\beta =-1.24\pm 0.11$. For simulations, a power law is also observed over 2.5 decades with $\overline{\beta }=-1.43\pm 0.14$. (b) Probability density function of the energy stored in the system $\mathcal{E}$ when an avalanche is triggered (depinning energy ${\mathcal{E}}_{\mathrm{dep}})$ for different threshold values for experiment I.

Figure 8

(a) Average shape of the global avalanches measured for different avalanche duration. Although avalanches are symmetric for short duration, they become progressively clearly asymmetric: a strong acceleration at the beginning and then a slow deceleration. (b) Average coordination number $Z$ when an avalanche begins as a function of the energy $E$ of the avalanche. Avalanches happen equally in the shear and fragile regimes, at both low and high coordination numbers. Both figures are plotted with data from experiment $I$.

Figure 9

Probability density function of the energy of global avalanches $P\left(E\right)$ measured in (a) experiments and (b) numerical simulations. The particle packing fraction $\varphi$ and the static friction coefficient between particles $\nu$ are varied over the different experiments and simulations. The energy $P\left(E\right)$ follows a power law with a constant exponent $\beta =-1.24\pm 0.11$ for experimental data and $\overline{\beta }=-1.43\pm 0.14$ for the numerical simulations. Here $\varphi$ has no effect on the statistics, while the upper cutoffs of the power laws decrease with the friction coefficient $\nu$. In order to characterize the effect of the friction, $P\left(\overline{E}\right)$ is rescaled using the method described in [58] (c) to collapse curves for different friction and upper cutoff. The inset shows the scaling of the upper cutoff plotted as a function of $\nu$.

Figure 10

(a) Probability density functions $P\left(N\right)$ of the avalanche sizes measured at the local scale in terms of the number of particles involved in each avalanche for experiment I. Avalanches are detected using a threshold on rotation, displacement, and pressure or energy drop or increase. Power laws are found with exponents varying from $-2$ to $-3$ (see the text for details). (b) Probability density functions $P\left(\overline{N}\right)$ of the avalanche sizes measured in terms of the number of particles involved in each avalanche for simulation 2. Avalanches are detected using a threshold on rotation, displacement, and energy drop. Power laws are found with exponents varying from $-1.8$ to $-2.5$. In all cases the probability density function of the avalanche energies $P\left(E\right)$ detected from the energy drop is also plotted for comparison. A power law spanning almost three decades is observed with exponent ${\beta }_l=-2.05\pm 0.09$ for experiment and ${\overline{\beta }}_l=-2.08\pm 0.1$ for numerical simulation.

Figure 11

For experiment I, we plot the ratio of the number of avalanches based on the rotation to the number of avalanches based on the displacement (or energy drop) provided the avalanches have an overlap in space and time (see the text for more details).

Figure 12

Probability density function of the avalanche energy $P\left(E\right)$ detected from the energy drop for different packing fractions $\varphi$ and particle friction coefficients $\nu$ in (a) experiments and (b) simulations. Within the error bars, power laws with similar exponents are observed. Also shown is the variation of the power-law upper cutoff ${\overline{E}}_0$ as a function of (c) the interparticle friction $\nu$ at $\varphi =0.788$ and (d) the packing fraction $\varphi$ at $\nu =0.7$.

Figure 13

(a) For experiment I, probability density function of the avalanche energy $P\left(E\right)$ detected from the energy drop for different coordination numbers $Z$. The inset shows the evolution of the energy upper cutoff $E_0$ as a function of the coordination number $Z$. A sharp increase in $E_0$ is observed near $Z=3$, which is the minimal number of contacts for large-scale mechanical stability. (b) For experiment I, probability density function of the number of local avalanches detected during a global avalanche for different ranges of global avalanche energy $E$.

Figure 14

Number of local avalanches detected with particle displacement per unit area per cycle for (a) experiment I and (b) simulation 2. Also shown is the average local avalanche shape for avalanches detected with the energy drop with size $N$ between eight and ten particles for (c) experiment I and (d) simulation 2. See the text for more details.