Anisotropic particles strengthen granular pillars under compression

We probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar, acting as a proxy for a disordered solid, under uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particle trajectories. In general, drawing connections between local structure and local dynamics can be challenging in amorphous materials due to lower precision of atomic positions, so this study aims to elucidate such connections. We vary local interactions by using three different particle shapes: discrete circular grains (monomers), pairs of grains bonded together (dimers), and groups of three bonded in a triangle (trimers). We find that dimers substantially strengthen the pillar and the degree of this effect is determined by orientational order in the initial condition. In addition, while the three particle shapes form void regions at distinct rates, we find that anisotropies in the local amorphous structure remain robust through the definition of a metric that quantifies packing anisotropy. Finally, we highlight connections between local deformation rates and local structure.

Figure 1

Photographs of the particle shapes used in this study: (a) monomers, (b) dimers, and (c) trimers. In (a), large and small rods are labeled with their diameters and the height of each rod is 19 mm.

Figure 2

Schematic of the procedure for constructing dimers. (a) A small drop of adhesive is placed near the top of a rod along its side. A second rod is carefully moved toward the first, both standing on a horizontal table. To ensure both rods are standing upright, they are guided using a vernier caliper set to the rod diameter. (b) The rods are brought nearly into contact, adhesive now adhering to both surfaces. The callout shows a top view. (c) The adhesive spreads between the rods due to capillarity, while simultaneously curing and forming a strong bond. The final separation is exaggerated for clarity. In the top view, the rods are in contact, also shown in Fig. 1. For trimers, a similar procedure is repeated, adding a third rod to form a triangular composite.

Figure 3

(a) A photograph of the apparatus from the top-down, slightly off-line from the actual camera used to image the system. The acquisition system and the stepper motor that drives the compression bar are not pictured. (b) A raw unprocessed image of the granular pillar comprised of dimers. The compression and static bars are visible in the top and bottom of the image, respectively. The direction of gravity is also labeled. (c) A close-up image of a neighborhood of dimers within the interior of the pillar.

Figure 4

Main: Stress-strain curve for a single trial of compression of a pillar with dimers. The stress is the force applied by the moving bar divided by the current width of the pillar, $\sigma =F/W$. The strain is the change of the pillar height divided by the initial pillar height, $\gamma =\mathrm{\Delta }H/H_0=(H_0-H)/H_0$. Raw images of the pillar are interspersed along the curve, corresponding to points when the pillar is in its initial condition ($\gamma =0$), at yield ($\gamma \sim 0.01$), and undergoing long-term deformation and failure ($\gamma >0.01$). Inset: A zoomed-in area of the failure portion of the stress-strain curve (outlined with a black box in the main plot) exhibiting several avalanches. The largest avalanche in this window is labeled $S$.

Figure 5

Stress-strain curves for (a) pillars comprised of randomly packed shapes: monomers (closed circles), dimers (open circles), and trimers (triangles), and (b) different preparation protocols of dimers: random (open circles), horizontal ($+$'s), and vertical (diamonds). The results shown in these plots are derived from five trials averaged together. Strain is shown on a logarithmic scale, in order to highlight both the modulus at strains below yield and the long-term material strengths for strains well beyond yield. Before averaging, the point of zero strain $\gamma =0$ is adjusted to minimize contributions from individual particle motions at the top of the pillar.

Figure 6

Evolution of a two-dimensional order parameter, $\langle 2{cos}^2\theta -1\rangle$, for every dimer packing. $\theta$ is the angle between the orientation of each dimer and the horizontal compressing bar and $\langle ·\rangle$ is the ensemble average. The order parameter value at $\gamma =0$ indicates the amount of order present in the initial state, either along or perpendicular to the compression bar.

Figure 7

Snapshots of pillars comprised of dimers undergoing deformation at the same strain ($\gamma \sim 0.1$). Each picture corresponds to a different orientational packing protocol: (a) random, (b), horizontal, and (c) vertical. Note the buckling behavior in (b), as well as the smooth pillar boundaries in (c). Full movies including overlays with $D_{\text{min}}^2$ can be found in the Supplemental Material [51].

Figure 8

Distributions of avalanche sizes $S=\mathrm{\Delta }\sigma /(mg\mu /D)$ across all trials for monomers, dimers, and trimers. An increasing portion of the distributions, for values below the noise level of stress, is not shown. It is difficult to assign fits given the lack of range, although it seems all curves would fit a power law with exponent $-3/2$ reasonably well. We also see monomers have a smaller maximum $S$ compared to dimers and trimers.

Figure 9

Illustration of two approaches for probing the local structure of the pillar, using data from a packing of dimers. Voronoi tesselation and Delaunay triangulation are drawn with respect to either (left) individual rod positions (“Atoms”) or (right) composite particle centroids (“Molecules”). The “molecular” approach takes the same Voronoi tessellation produced using the “atomistic” approach and cuts out edges drawn across dimer and trimer bonds. In both, the arrows $\mathbf{C}$ point from the center of the rod or particle to the centroid of its Voronoi cell (magnified $10\times$).

Figure 10

Linear plot of the probability density functions of $Q_k$, defined as the weighted divergence of the particle center-to-cell centroid vector field in Eq. (1), derived from the “molecular” approach, for deformed random packings of the different particle shapes. A triangle over which $Q_k$ is calculated is considered deformed if at least one of its constituent particles has moved at least one large monomer radius $R$. A dashed line is drawn at $Q_k=0$, separating underpacked ($Q_k<0$) and overpacked ($Q_k>0$). The solid line is an ideal Gaussian curve for monomers, given the mean and standard deviation of $Q_k$. The semilogarithmic version of this plot is shown in Fig. 11.

Figure 11

Semilogarithmic plots of the probability density functions of $Q_k$, defined as the weighted divergence of the particle center-to-cell centroid vector field in Eq. (1), for (a),(b) deformed random packings of the different particle shapes and (c),(d) deformed dimer packings with different initial ordering protocols. The “atomistic” approach is used in (a) and (c), while the “molecular” approach is used in (b) and (d). A triangle over which $Q_k$ is calculated is considered deformed if at least one of its constituent particles has moved at least one large monomer radius $R$. In every plot, a dashed line is drawn at $Q_k=0$, separating underpacked ($Q_k<0$) and overpacked ($Q_k>0$). Solid lines are ideal Gaussian curves for monomers in (a) and (b) and random dimers in (c) and (d), given the mean and standard deviation of $Q_k$.

Figure 12

Cumulative distribution functions of the ratio of tangential rotational velocity to translational velocity for dimers and trimers. For each dimer, $r$ is the diameter of one of its constitutive rods; for each trimer, $r$ is $(1+2/\sqrt{3})$ times the radius of one its constitutive rods. $\omega$ is the rotational velocity of a particle at a particular time, while $v$ is its translational velocity at the same time. Only particles that have moved at least one monomer radius $R$ from its initial position are considered. For a large majority of dimers and trimers, motion is primarily associated with translations rather than rotations. Rotational motion is not measured for monomers.

Figure 13

(a) For the same trial of dimers as Fig. 4, a plot of both stress $\sigma$ and the average deviatoric strain rate $\langle J_2\rangle$ throughout the pillar. Note that many of the peaks and troughs of $\langle J_2\rangle$ correspond to those of $\sigma$. (b)–(d) $\sigma$ plotted versus $\langle J_2\rangle$ using data from all recorded trials, demonstrating varying levels of correlation between the global material response and local deformation profile for the different shapes. In (b) and (c), we see a general positive correlation for monomers and dimers, suggesting that local deformations are a primary mechanism for stress relaxation. In (d), we see this trend is mostly absent for trimers, suggesting that the particles are now so large that stick-slip motion due to body friction between particles and the substrate is contributing more to fluctuations in stress.

Figure 14

Bivariate histograms of $J_{2,\text{rel}}$, defined in Eq. (5), versus $Q_k$ for (a) monomers, (b) dimers, and (c) trimers. The plots are colored by the logarithm (base 10) of counts within each bin.

Figure 15

Binned averages of $J_{2,\text{rel}}$ versus $Q_k$ for all particle shapes as measured in (a) “atoms” and (b) “molecules.” The error bars are standard errors of the mean of $J_{2,\text{rel}}$ values within each $Q_k$ bin. In general, there is a negative correlation between the two, aside from dilatancy effects at maximally packed regions (max $Q_k>0$) for monomers and trimers. However, dimers do not exhibit this dilatancy effect, while regions of dimers that are highly underpacked $[Q_k\lesssim -1.0$ in (a), $Q_k\lesssim -0.3$ in (b)] show a lower likelihood to deform compared to monomers and trimers.