Strain stiffening due to stretching of entangled particles in random packings of granular materials

Stress-strain relations for random packings of entangling chains under triaxial compression can exhibit strain stiffening and sustain stresses several orders-of-magnitude beyond typical granular materials. X-ray tomography reveals the transition to this strong strain stiffening occurs when chains are long enough to entangle an average of about one chain each, which results in system-filling clusters of entangled chains, similar to the Erdös-Rényi model for randomly connected graphs. The number of entanglements is nearly proportional to the area surrounded by entangling particles with an excluded volume effect, thus the existence of system-filling clusters of entanglements can be predicted assuming random particle positions and orientations with an excluded volume effect if the particle shapes in the packing are known. A tendency was found for chain links to stretch when the packing was strained. This suggests that the strength of these packings comes from stretching of the links of chains, but only when the system-filling network of entanglements provides constraints that prevents failure by shear banding, so that particles must be deformed to move further under strain. The slope of the stress-strain relation of a packing can be calculated from a mean-field model consisting of the product of the effective extensional modulus of the chain, packing fraction, probability of stretched links, and the ratio of strain of stretched links to packing strain. In this model, the increasing slope of the stress-strain curve is mainly due to the fraction of stretched links increasing with strain, and assuming the fraction of stretched links is proportional to strain results in a quadratic prediction for the stress-strain curve. The stress-strain model requires as input measurements of the ratio between local particle deformation and global average strain, and the probability of stretching for nonrigid particles, resulting in a quadratic curvature that agrees with experiments within the run-to-run variation (30%). This model for the stress-strain relation is shown to be generalizable to different shapes of entangling particles by applying it to staples, where the packing strength comes from the bending of staples instead of stretching links. The permanent plastic deformation of staples allows measuring statistical quantities from inspection of a poured-out sample after a triaxial compression, without the need for in situ imaging. Both the probability of staples bending and the average bend angle of the arms were found to increase with strain, and these inputs into the model result in a quadratic curvature of the stress-strain that agrees with experiments within the model uncertainties (37%).