Scaling Law for Cracking in Shrinkable Granular Packings

Hydrated granular packings often crack into discrete clusters of grains when dried. Despite its ubiquity, an accurate prediction of cracking remains elusive. Here, we elucidate the previously overlooked role of individual grain shrinkage—a feature common to many materials—in determining crack patterning using both experiments and simulations. By extending classical Griffith crack theory, we obtain a scaling law that quantifies how cluster size depends on the interplay between grain shrinkage, stiffness, and size—applicable to a diverse array of shrinkable granular packings.

Figure 1

Experiments highlight the impact of bead-scale properties on cracking. (a) Schematic of a packing of shrinkable beads confined between two substrata. As the packing dries, the individual beads shrink, leading to the formation of a 2D crack network. (b) The beads are in a pendular state with capillary bridges that compress adjacent beads together (c) as observed via bright-field microscopy. Cracks form when bridges break due to bead shrinkage and friction with the substrate. (d) A packing of small, low shrinkability beads has minimal cracking. (e) A packing of larger, high shrinkability beads has more cracking. (f) A packing of even larger, high shrinkability beads with less compression has even more cracking. We use microscopy to directly measure $\mathrm{\Phi }=2.0\pm 0.1$, $8.5\pm 1.1$, and $8.5\pm 1.1$; ${\delta }_{\mathrm{wet}}=1\pm 1$, $21\pm 6$, and $15\pm 8\mu \mathrm{m}$; and $R_{\mathrm{wet}}=34\pm 9$, $59\pm 14$, and $86\pm 18\mu \mathrm{m}$ for (d)–(f), respectively. Parentheses indicate uncertainty.

Figure 2

DEM simulations isolate the influence of bead shrinkage, stiffness, and size on cracking. (a) A simulated packing of low shrinkability beads has minimal cracking. (b) A simulated packing of high shrinkability beads has a larger degree of cracking. (c) A simulated packing of high shrinkability beads with less capillary compression has an even larger degree of cracking. In (a)–(c), we set $M=613$ and hold $F_{\mathrm{fr}}/(2\pi \gamma R_{\mathrm{wet}})=0.05$ fixed to mimic the experiments. In (d)–(f), we quantify the equivalent cluster area $N$ while independently varying the (d) effective height $\mathcal{H}$, (e) shrinkability $\mathrm{\Phi }$, and (f) bead compression ${\widehat{\delta }}_{\mathrm{wet}}$. Colored lines are curves fit to the data indicating that $N$ (d) increases with height as $\propto {\mathcal{H}}^{4/3}$, (e) decreases with the shrinkability as $\propto {\mathrm{\Phi }}^{-(2m/3+2/9)}$ with $m=9/4$, and (f) increases with bead compression as $\propto {\widehat{\delta }}_{\mathrm{wet}}^{2/3}$. We do not observe a considerable difference in $N$ upon varying the packing size $M$ from 91 to 823 or using a disordered packing configuration (open symbols).

Figure 3

DEM simulation and experimental results all collapse on our scaling law [Eq. (5)]. We perform a regression on the simulation data to obtain the dashed line, with a coefficient of determination $=0.98$. For the experimental data, we fix $F_{\mathrm{fr}}=0.02\times 2\pi \gamma R_{\mathrm{wet}}$, in independent agreement with our measurement of the friction. Error bars reflect the combined uncertainties in the crack length, bead size, and packing fraction $\eta$. Different symbols show different classes of Sephadex beads with different shrinkability, compression, and bead size. We assume $m=9/4$.