Elasticity of frictionless particles near jamming

Abstract

We study the linear elastic response of harmonic disk packings near jamming via three types of probes: (i) point forcing, (ii) constrained homogeneous deformation of subregions of large systems, and (iii) unconstrained deformation of the full system subject to periodic boundary conditions. For the point forcing, our results indicate that the transverse component of the response is governed by a lengthscale ξT, which scales with the confining pressure, p, as ξT∼p-0.25, while the longitudinal component is governed by ξL, which scales as ξL∼p-0.4. The former scaling is precisely the transverse lengthscale, which has been invoked to explain the structure of normal modes near the density of states anomaly in sphere packings, while the latter is much closer to the rigidity length, l*∼p-0.5, which has been invoked to describe the jamming scenario. For the case of constrained homogeneous deformation, we find that μ(R), the value of the shear modulus measured in boxes of size R, gives a value much higher than the continuum result for small boxes and recedes to its continuum limit only for boxes bigger than a characteristic length, which scales like p-0.5, precisely the same way as l*. Finally, for the case of unconstrained homogeneous deformation, we find displacement fields with power spectra, which are consistent with independent, uncorrelated Eshelby transformations. The transverse sector is amazingly invariant with respect to p and very similar to what is seen in Lennard-Jones glasses. The longitudinal piece, however, is sensitive to p. It develops a plateau at long wavelength, the start of which occurs at a length that grows in the p→0 limit. Strikingly, the same behavior is observed both for applied shear and dilation.

Nonlocal Elasticity near Jamming in Frictionless Soft Spheres

“…A similar p-independent crossover was noted in Ref. [6] without making the connection to nonlocality. The transverse compliance is nonmonotonic, with an initial dip that appears to survive in the infinite system size limit [ Fig.…”

“…In the transverse case the fall off comes later and more gradually. We note that prior work has related l à [4], l c [5], or both [6] to (deviations from) classical elastic Green's functions [4][5][6].…”

“…Soft sphere packings, a standard model for emulsions, foams, and granular materials, provide an important example of this effect. Response functions in simulated packings depart significantly from elasticity [4][5][6] when the packings are close to the (un) jamming transition at zero confining pressure p [7,8].…”

“…Details remain controversial, but there is consensus that the breakdown of classical elasticity near jamming is governed by l à [4], l c [5], or both [6].…”

“…[6,45] and especially [31,32], which explicitly make the connection to nonlocality. Packings are subjected to longitudinal and transverse force densities…”