Emergent Strain Stiffening in Interlocked Granular Chains

Granular chain packings exhibit a striking emergent strain-stiffening behavior despite the individual looseness of the constitutive chains. Using indentation experiments on such assemblies, we measure an exponential increase in the collective resistance force $F$ with the indentation depth $z$ and with the square root of the number $\mathcal{N}$ of beads per chain. These two observations are, respectively, reminiscent of the self-amplification of friction in a capstan or in interleaved books, as well as the physics of polymers. The experimental data are well captured by a novel model based on these two ingredients. Specifically, the resistance force is found to vary according to the universal relation $\log F\sim \mu \sqrt{\mathcal{N}}{\mathrm{\Phi }}^{11/8}z/b$, where $\mu$ is the friction coefficient between two elementary beads, $b$ is their size, and $\mathrm{\Phi }$ is the volume fraction of chain beads when semidiluted in a surrounding medium of unconnected beads. Our study suggests that theories normally confined to the realm of polymer physics at a molecular level can be used to explain phenomena at a macroscopic level. This class of systems enables the study of friction in complex assemblies, with practical implications for the design of new materials, the textile industry, and biology.

Figure 1

(a) Description of the experimental setup (see the main text for details). (b) Snapshots showing the final states of dense ($\mathrm{\Phi }=1$) granular chain assemblies after the removal of their cylindrical container, for various numbers $\mathcal{N}$ of beads per chain as indicated in the top left corner of each snapshot.

Figure 2

(a) Indentation force $F$ as a function of indentation depth $z$, for granular chain assemblies with various numbers $\mathcal{N}$ of beads per chain as indicated. (b) Slope of the log-linear representation of the $\mathcal{N}\ge 2$ data (see the current inset) as a function of $\mathcal{N}$. The solid line indicates a $\sim \sqrt{N}$ behavior, while the dashed one corresponds to the best power-law fit: $0.03{\mathcal{N}}^{0.46\pm 0.06}$. (Inset) Log-linear representation of the $\mathcal{N}\ge 2$ data, with $F_0=1$ N an arbitrary reference force. The solid lines indicate linear trends in this representation.

Figure 3

(a) Log-linear representation of the indentation force $F$ as a function of the indentation depth $z$, for granular chain assemblies with $\mathcal{N}=30$ beads per chain, semidiluted in a medium of unconnected beads, with selected (for clarity) volume fractions $\mathrm{\Phi }$ of chain beads as indicated. (b) Slope calculated in the log-linear representation (a) as a function of the volume fraction $\mathrm{\Phi }$ of chain beads. The solid line indicates a $\sim {\mathrm{\Phi }}^{11/8}$ behavior, while the dashed one corresponds to the best power-law fit: $\sim {\mathrm{\Phi }}^{1.1\pm 0.2}$.

Figure 4

Universal log-linear representation of all the dense and semidilute granular chain indentation experimental data of Figs. 2(inset) and 3, corresponding to $\mathcal{N}\ge 2$ and $\mathrm{\Phi }>{\mathrm{\Phi }}^{*}\sim 0.06$, as predicted by Eq. (5). The ensemble corresponds to various numbers $\mathcal{N}$ of beads per chain and volume fractions $\mathrm{\Phi }$ of chain beads as indicated. Note that the dimensionless friction coefficient $\mu$ is not included in the $x$ axis, as it is neither varied nor measured, and since there is anyway a missing prefactor in Eq. (5).