Unjamming due to local perturbations in granular packings with and without gravity

We investigate the unjamming response of disordered packings of frictional hard disks with the help of computer simulations. First, we generate jammed configurations of the disks and then force them to move again by local perturbations. We study the spatial distribution of the stress and displacement response and find long range effects of the perturbation in both cases. We record the penetration depth of the displacements and the critical force that is needed to make the system yield. These quantities are tested in two types of systems: in ideal homogeneous packings in zero gravity and in packings settled under gravity. The penetration depth and the critical force are sensitive to the interparticle friction coefficient. Qualitatively, the same nonmonotonic friction dependence is found both with and without gravity, however the location of the extrema are at different friction values. We discuss the role of the connectivity of the contact network and of the pressure gradient in the unjamming response.

Figure 1

Schematic picture of the two types of granular packings confined by an external pressure bath (a) and by gravity (b). The dashed lines mark periodic boundaries. Some typical perturbations are illustrated with gray particles and arrows showing the direction of perturbation.

Figure 2

(a),(b) Displacement response fields in the laboratory frame for two different locations of the perturbation. The resulting vector field can be relatively localized (a) or more widespread (b) depending on the perturbed contact. (c) Displacement response field in the contact frame averaged over several thousand perturbations. The system contained 3000 disks with friction coefficient 0.5. The unit of the length, in which $x$ and $y$ are measured, is set to the maximum grain radius. For clarity, the magnitude of the displacements has been increased by a factor of ${10}^{11}$ in all figures.

Figure 3

The magnitude of the average displacements $d$ in terms of the distance from the perturbed contact $r$. Here $\xi$ stands for the gap generated at the perturbed contact. Different slopes correspond to different friction coefficients $\mu$. For each value of friction four systems of different sizes are investigated. The total number of particles is between 500 and 8000. The error bars remain below $5\%$ of the values $d\left(r\right)∕\xi$ for the whole range of $r$.

Figure 4

Dependence of the penetration exponent $\alpha$ (a) and the mean critical force $⟨F_{\mathrm{crit}}⟩$ (b) on the system size $N$ for three different friction coefficients $\mu$.

Figure 5

Average critical force $⟨F_{\mathrm{crit}}⟩$ (open circles) scaled by the average normal contact force $⟨F_{\mathrm{cont}}⟩$ and the average penetration depth $⟨\delta ⟩$ (full circles) as functions of the friction coefficient $\mu$. The vertical dashed line emphasizes that the two extrema are located at the same $\mu$.

Figure 6

Each data point represents one contact in the frame of the critical force $F_{\mathrm{crit}}$ and the normal component of the original contact force $F_{\mathrm{cont}}$. The four figures correspond to four packings constructed with different frictions. $⟨F_{\mathrm{cont}}⟩$ is the average normal contact force in each packing. The dashed lines correspond to $F_{\mathrm{crit}}=F_{\mathrm{cont}}$.

Figure 7

Probability distribution of the critical forces $F_{\mathrm{crit}}$ (a) and the critical forces normalized with respect to their mean in each packing $F_{\mathrm{crit}}∕⟨F_{\mathrm{crit}}⟩$ (b) for different friction coefficients. The inset displays a semilogarithmic plot of probability distribution of the normalized critical forces when averaged over all friction coefficients. The dashed line is an exponential fit of the tail of the curve.

Figure 8

The correlation between the critical force $F_{\mathrm{crit}}$ (full circles) or the normal component of the original contact force $F_{\mathrm{cont}}$ (open circles) and the penetration depth $\delta$ in terms of the friction coefficient $\mu$.

Figure 9

Comparison of the results of the fixed pressure (solid circles) and the fixed volume (open circles) perturbation methods. (a) The penetration depth $\delta$ and (b) the critical force $F_{\mathrm{crit}}$ for several contacts subjected to the perturbation. (c) The relative change in the size of the system $\lambda$ (solid circles) and the pressure $P$ (open circles) obtained during the perturbation of each contact with the fixed pressure and fixed volume methods, respectively. Here, the friction coefficient is 0.5.

Figure 10

The average magnitude of the change in the normal component of contact forces due to the perturbation $⟨\mid \mathrm{\Delta }F\mid ⟩$ scaled by $F_0$ in terms of the distance from the perturbed contact $r$. Different curves correspond to different friction coefficients $\mu$.

Figure 11

(a) The profile of the pressure change $\mathrm{\Delta }P$. Shading is logarithmic in the amplitude of the pressure change, with black indicating a pressure increase, and white, a pressure decrease. Dots show where the sign of $\mathrm{\Delta }P$ is changed. (b) $\mathrm{\Delta }P$ along the $x$ and $y$ axes. (c) The maximum angle of friction mobilization ${\alpha }_{\max }$. The $x$ axis indicates the direction of separation at the perturbed contact. The friction coefficient is 0.5.

Figure 12

The average penetration depth $⟨\delta ⟩$ (solid circles), and the average critical force $⟨F_{\mathrm{crit}}⟩$ (open circles) with respect to the average effective weight $⟨F_y⟩$ as functions of the friction coefficient $\mu$, for the perturbation of topmost (a) and lowermost (b) particles. The vertical dashed lines are reminiscent of the position of the extrema in the homogeneous case.

Figure 13

Influence of the friction on the average coordination number $z$ for homogeneously confined packings (open circles) and packings settled under gravity (full circles). The middle curve (stars) corresponds to the gravity case, when the contribution of the rattler particles, that transmit no load except their own weight, is subtracted.