Fragility and hysteretic creep in frictional granular jamming

The granular jamming transition is experimentally investigated in a two-dimensional system of frictional, bidispersed disks subject to quasistatic, uniaxial compression without vibrational disturbances (zero granular temperature). Three primary results are presented in this experimental study. First, using disks with different static friction coefficients ($\mu$), we experimentally verify numerical results that predict jamming onset at progressively lower packing fractions with increasing friction. Second, we show that the first compression cycle measurably differs from subsequent cycles. The first cycle is fragile—a metastable configuration with simultaneous jammed and unjammed clusters—over a small packing fraction interval (${\varphi }_1<\varphi <{\varphi }_2$) and exhibits simultaneous exponential rise in pressure and exponential decrease in disk displacements over the same packing fraction interval. This fragile behavior is explained through a percolation mechanism of stressed contacts where cluster growth exhibits spatial correlation with disk displacements and contributes to recent results emphasizing fragility in frictional jamming. Control experiments show that the fragile state results from the experimental incompatibility between the requirements for zero friction and zero granular temperature. Measurements with several disk materials of varying elastic moduli $E$ and friction coefficients $\mu$ show that friction directly controls the start of the fragile state but indirectly controls the exponential pressure rise. Finally, under repetitive loading (compression) and unloading (decompression), we find the system exhibits pressure hysteresis, and the critical packing fraction ${\varphi }_c$ increases slowly with repetition number. This friction-induced hysteretic creep is interpreted as the granular pack's evolution from a metastable to an eventual structurally stable configuration. It is shown to depend on the quasistatic step size $\Delta \varphi$, which provides the only perturbative mechanism in the experimental protocol, and the friction coefficient $\mu$, which acts to stabilize the pack.

Figure 1

(a) Experimental schematic: The system consists of 950 large and 950 small disks (ratio of radii 1:1.5). Two movable boundaries at opposite ends control the system's packing fraction $\varphi$ and are used to provide uniaxial, quasistatic compression. Force sensors labeled A through F in the schematic measure the boundary pressure. The image is contrast enhanced data for $\varphi =0.8113$. (b) Side view schematic of force sensor placement within the compression boundary. When the micrometer (left) pushes on the boundary plate to its right, the force sensor registers the cumulative force of the boundary plate and granular pack. (c) Schematic of the static friction coefficient measurement apparatus. Four disks of diameter $d_L=0.9525$ cm are placed in contact with each other as shown in the schematic. The three outer disks are held fixed and have no translational or rotational degrees of freedom. The middle disk can be rotated by means of an external lever. A normal force $F_N=M_{N}g$ is applied by a suspended weight of mass $M_N$ on the top disk. This force is transferred to the disk sandwiched in the middle, which in turn transfers it to the two bottom disks at an angle of ${30}^{\circ }$. A tangential force $F_T=M_{T}g$ applied on the middle disk allows it to slip at a critical force, which allows determination of the static friction coefficient $\mu$ for the disks.

Figure 2

$P$ vs $\varphi$ (quasistatic step size $\Delta \varphi =1\times {10}^{-4}$). The packing fraction $\varphi$ at which system pressure starts increasing monotonically falls with increasing static friction coefficient $\mu$. The change in pressure slopes arises from different elastic moduli of the materials used.

Figure 3

Deviation of fractional ${\varphi }_c$ with respect to ${\varphi }_{\mathrm{RCP}}$ vs $\mu$ for materials described in Fig. 2. The error bars are estimates as opposed to statistical averages over many runs.

Figure 4

(a) $P$ vs $\varphi$ (linear scale) for the first (black circles) and second (red squares) compression cycles. The pressure scaling is gradual for the first cycle as compared to the more abrupt transition during the second cycle where ${\varphi }_c$ is indicated. The inset plots the pressure from all four boundaries (same vertical and horizontal scale). No anisotropy is observed in the pressure signal. (b) $P$ vs $\varphi$ for the first jamming cycle (log-linear scale) shows the existence of an intermediate regime where pressure scales exponentially. The solid line is an exponential fit to the data $P\propto e^{\varphi /{\chi }_P}$ with ${\chi }_P=0.00195$, and the dashed line is a linear fit. The values of ${\varphi }_1$ and ${\varphi }_2$ are indicated in the plot.

Figure 5

(a) Fraction of disks moving $N_m/N_T$ vs $\varphi$ showing a constant region for $\varphi <{\varphi }_1$ and exponential decay for the fragile regime (${\varphi }_1<\varphi <{\varphi }_2$) with ${\chi }_N\approx 0.001$. (b) Normalized displacement variance $\sigma /\tilde{d}$ relative to a state near ${\varphi }_2$ showing a linear decrease for $\varphi <{\varphi }_1$ and an exponential decrease with ${\chi }_{\sigma }=0.0007$ for the fragile regime (${\varphi }_1<\varphi <{\varphi }_2$). (c) Image of the superposition of the difference in the stress network (black lines) and the magnitude of disk displacements between $\varphi =0.8095$ and $\varphi =0.8105$. Arrows indicate ${\varphi }_1$ and ${\varphi }_2$.

Figure 6

Bar graph shows the number of displaced disk centers $N_d$ normalized by total number of disks $N_c$ counted within an annulus of width $\Delta r$ at a radial distance $r$ from the center of a newly formed stress chain (see schematic in the inset) vs the dimensionless radial distance $(r+\Delta r)/\tilde{d}$ from the center of a newly formed stress chain [$\tilde{d}=(d_L+d_S)/2$ is the average diameter of the large and small disk]. The solid circles (red) present the cumulative fraction $\langle N_d/N_c\rangle$ within the radius $r+\Delta r$. The vertical dotted line is a marker indicating the radial distance beyond which an abrupt fall in $\langle N_d/N_c\rangle$ is observed.

Figure 7

Log-linear plot of the mean ($•$, black solid circles in lower curve) and variance ($\circ$, red circles in upper curve) of stress chain domain area normalized by total area, $a_m$, $a_v$ (left axis) or by the mean area formed by connections between different combinations of three disks in a close packed triangular array, ${\tilde{a}}_m$, ${\tilde{a}}_v$ (right axis), respectively. Insets show stress networks and corresponding domains for $\varphi =0.8104$ (left) and $\varphi =0.8148$ (right). The dashed (lower curve) and dotted (upper curve) lines are fits to a combined linear dependence with a decaying exponential (see main text) for $a_m$ and $a_v$, respectively. The grayscale region highlights the transition from exponential to linear behavior under compression. The solid red and black lines are markers included to indicate the end of transition from exponential to linear behavior near ${\varphi }_2=0.8124$.

Figure 8

$P$ vs $\varphi$ for first compression cycle where the system is gently tapped (albeit with no systematic control) after each quasistatic step.

Figure 9

Pressure $P$ from the compression sensor for the first compression cycle versus $\varphi$ for PE disks ($\mu =0.19$) jams at higher packing fractions with increasing step magnitude of quasistatic compression $\Delta \varphi =1\times {10}^{-4}$, $3.5\times {10}^{-4}$, and $7\times {10}^{-4}$.

Figure 10

$P$ vs $\varphi$ (equivalent to stress-strain measurement) for PE disks at $\Delta \varphi =1\times {10}^{-4}$ exhibits hysteresis.

Figure 11

Top: Difference between ${\varphi }_c^n$ and its initial onset value ${\varphi }_c^0$ as a function of the number of cycles of loading and unloading $n$ for different materials. Bottom: Same quantities for the PE material and for different step sizes $\Delta \varphi$. Materials and step sizes are labeled in the plot legends as are the coefficients in the solid-line exponential fits.

Figure 12

$P$ vs $\varphi$ in log-linear scale for the first compression cycle. All plots have been horizontally shifted for coincidence of ${\varphi }_1$. The exponential pressure scaling ($P\propto e^{\varphi /{\chi }_P}$) for PE and PEG exhibits same slope (${\chi }_P=0.00195$) as demonstrated by the exponential fit (solid black line), implying variation in friction coefficient has no effect on ${\chi }_P$. On the other hand, PEA disks with same friction coefficient as PE, but lower modulus exhibit a shallower exponential scaling (with ${\chi }_P=0.0063$, long dashed blue line) over approximately same range of pressure as PE and PEG.

Figure 13

$P$ vs $\varphi$ for the second compression cycle in linear scale shows the predicted power law scaling for granular jamming transition for all four boundaries. The pressure rise is abrupt and is in contrast with the gradual (exponential) rise observed for the first compression cycle. The solid line is the best power-law fit to the experimental data with a power law exponent of 1.15. The long-dashed line representing the best linear fit to the data demonstrates that the experimental data does not follow linear scaling.