Transforming Mesoscale Granular Plasticity Through Particle Shape

When an amorphous material is strained beyond the point of yielding, it enters a state of continual reconfiguration via dissipative, avalanchelike slip events that relieve built-up local stress. However, how the statistics of such events depend on local interactions among the constituent units remains debated. To address this we perform experiments on granular material in which we use particle shape to vary the interactions systematically. Granular material, confined under constant pressure boundary conditions, is uniaxially compressed while stress is measured and internal rearrangements are imaged with x rays. We introduce volatility, a quantity from economic theory, as a powerful new tool to quantify the magnitude of stress fluctuations, finding systematic, shape-dependent trends. In particular, packings of flatter, more oblate shapes exhibit more catastrophic plastic deformation events and thus higher volatility, while rounder and also prolate shapes produce lower volatility. For all 22 investigated shapes the magnitude $s$ of relaxation events is well fit by a truncated power-law distribution $P(s)\sim s^{-\tau }\exp (-s/s^{*})$, as has been proposed within the context of plasticity models. The power-law exponent $\tau$ for all shapes tested clusters around $\tau =1.5$, within experimental uncertainty covering the range 1.3–1.7. The shape independence of $\tau$ and its compatibility with mean-field models indicate that the granularity of the system, but not particle shape, modifies the stress redistribution after a slip event away from that of continuum elasticity. Meanwhile, the characteristic maximum event size $s^{*}$ changes by 2 orders of magnitude and tracks the shape dependence of volatility. Particle shape in granular materials is therefore a powerful new factor influencing the distance at which an amorphous system operates from scale-free criticality. These experimental results are not captured by current models and suggest a need to reexamine the mechanisms driving mesoscale plastic deformation in amorphous systems.

Popular Summary

When stressed, materials typically fail either by developing cracks that lead to clean breaks or by reconfiguring the internal arrangement of constituent subunits (plastic deformation). An astounding variety of disordered, amorphous materials exhibit statistically similar plastic deformation behavior, with constituent subunits ranging from the scale of atoms in metallic glasses, to millimeters in foams, to truly macroscopic dimensions in earthquakes. In all of these cases, the internal restructuring gives rise to a sequence of sudden movements when subunits squeeze past one another. The statistical distribution of such plastic deformation events, most of them small and becoming increasingly rare at large sizes, is thought to be generic. Whether this distribution is, in fact, universal and thus independent of the makeup of the material is debated, largely because systematic tests are difficult. Our paper addresses this issue with the first detailed investigation of how the shape of individual subunits affects the statistics of plastic deformation events.

Our experiments on amorphous granular systems composed of 3D-printed particles demonstrate that the likelihood of observing events of a particular size is indeed remarkably robust and compatible with universal behavior, even for strikingly different particle shapes. However, there is a characteristic magnitude that limits the largest event cascades. Here, our experiments show a strong and systematic shape dependence, which we relate to the way particles can move with respect to their neighbors.

These results provide new insight into how general amorphous systems deform plastically and how to minimize catastrophic failure events.

Figure 1

Triaxial compression into the plastic regime. (a) A 10-cm-tall, 5-cm-diameter packing of icosahedra in a rubber membrane and evacuated to ${\sigma }_{\mathrm{conf}}=20\text{kPa}$ before uniaxial stress $q$ is applied. (b) Raw stress-strain data for a single run each for the five particle shapes shown on the right (see Table 1 for shape definitions). All data in this paper are from the range $\epsilon =[0.1,0.2]$. Insets: Magnified portions of the compression data for (left to right) the 3:1 disk, sphere, and $m=4$ supercube particles show the same sudden stress drops, though at different stress scales.

Figure 2

Localized slips revealed by x-ray videography. Stress-strain data (left) for representative particle shapes, with numbered vertical bars that indicate the strains at which the x-ray data shown on the right were taken. Subtracting successive x-ray frames reveals the particle rearrangements that cause sudden stress drops in $q(\epsilon )$. We show this explicitly for the first labeled stress drop event, where 1a and 1b are the raw x-ray frames subtracted to obtain difference image 1. The dark horizontal band at the top of each x-ray difference image reflects the top plate’s motion during compression. For a sense of scale, the dark blue regions in the difference images for events 5 and 10 each correspond to a single particle shifting position by roughly a particle diameter. For clarity, the shaded bars in the stress-strain curves are plotted 5 times wider than the strain window captured by a single x-ray frame. The x-ray images for events 2–10 are all shown at the same scale, enabling direct comparison.

Figure 3

Plastic deformation phase space. The jerkiness of the plastic regime and the shear strength are mapped out by plotting the annualized plastic volatility $V$ versus the angle of internal friction $\psi$. Cross bars for each particle shape are the standard deviation of $V$ and $\psi$ across runs.

Figure 4

Shear strength versus deviations from a sphere. The angle of internal friction data from Fig. 3 are plotted against the inverse of sphericity. As a measure of the compactness, sphericity is calculated as the ratio of a particle’s surface area to that of a sphere with the same volume.

Figure 5

Stress drop magnitude distributions. The drop distributions were fit by maximum likelihood estimation to a truncated power law of the form $D(s)\sim s^{-\tau }\exp (-s/s^{*})$. The values cited for the uncertainty on $\tau$ are $2\sigma$ for the marginal distribution, and only the best-fit value of $s^{*}$ is listed. All plots are over the same range. Data points are binned logarithmically (blue) after fitting (black). The distributions are organized by (from left to right) the first column containing spheres, simple deviations from a sphere, and tetrahedra, the second column cubes and simple deviations from a cube, the third column the lens family, and the fourth column the disk family.

Figure 6

Parameter space for the truncated power-law fits to $D(s)$. The dotted (solid) ellipses around each best fit $(\tau ,s^{*})$ are the $1\sigma$ ($2\sigma$) confidence regions as determined by the nonparametric bootstrap method.

Figure 7

Strain rate dependence of $\tau$. Uncertainties are calculated as before with the bootstrap method. The gray region contains the points which were taken at the same strain rate as the rest of the paper, $\dot{\epsilon }=8\times {10}^{-4}$. These points for both particle shapes and the $\dot{\epsilon }=3\times {10}^{-4}$ data for the lenses are offset slightly to display multiple sets of runs taken at the same strain rate.

Figure 8

Detectable event rate versus slip event magnitude upper cutoff. Uncertainty for the event rate is calculated as the standard error of the mean count per run.