Low-energy non-linear excitations in sphere packings

Abstract

We study theoretically and numerically how hard frictionless particles in random packings can rearrange. We demonstrate the existence of two distinct unstable non-linear modes of rearrangement, both associated with the opening and the closing of contacts. Mode one, whose density is characterized by some exponent θ , corresponds to motions of particles extending throughout the entire system. Mode two, whose density is characterized by an exponent θ = θ , corresponds to the local buckling of a few particles. Mode one is shown to yield at a much higher rate than mode two when a stress is applied. We show that the distribution of contact forces follows P (f ) ∼ f min(θ ,θ) , and that imposing that the packing cannot be densified further leads to the bounds γ ≥ 1 2+θ, where γ characterizes the singularity of the pair distribution function g(r) at contact. These results extend the theoretical analysis of [1] where the existence of mode two was not considered. We perform numerics that support that these bounds are saturated with γ ≈ 0.38, θ ≈ 0.17 and θ ≈ 0.44. We measure systematically the stability of all such modes in packings, and confirm their marginal stability. The principle of marginal stability thus allows to make clearcut predictions on the ensemble of configurations visited in these out-of-equilibrium systems, and on the contact forces and pair distribution functions. It also reveals the excitations that need to be included in a description of plasticity or flow near jamming, and suggests a new path to study two-level systems and soft spots in simple amorphous solids of repulsive particles.

Pushing on a Nonlinear Material

“…Its scaling with the distance to jamming does not match any scaling reported before for length scales of linear origin, such as l à or l c [4,11]. This suggests that r c could encompass crucial information about the density of the low energy nonlinear excitations reported recently for sphere packings [25]. Further insights in this matter could come from simulations of pointlike response of the kind reported in [7] albeit in the nonlinear regime.…”

“…At finite shear strain amplitude γ, nonlinear effects become dominant [9,25,26] and the mechanical response of the system is no longer relevantly described exclusively by Δz. Finally, while dilatancy effects-namely, the increase of volume or pressure under shear-are important in sheared granular experiments [27][28][29], they are systematically missed in numerical and theoretical studies of soft spheres near jamming.…”

Unifying Suspension and Granular flows near Jamming

“…Numerically it is found that θ e ≈ 0.44 both in two and three dimensions [32,33], suggesting that this quantity may be independent of dimension. Moreover, its value does not depend on the preparation protocol of the isostatic state: up to error bars, equal values are found from compression of hard spheres [32], shear-jammed hard disks [35], and decompression of soft spheres [33,36] 1 .…”

“…When an additional stress ratio is imposed to a marginal solid, naively one expects the contacts carrying [32,33]. In a packing only those contacts lead to plasticity when stress is increased, or when a shock (say a collision) occurs in the bulk of the material [32,34].…”

“…In a packing only those contacts lead to plasticity when stress is increased, or when a shock (say a collision) occurs in the bulk of the material [32,34]. The density of extended contacts as a function of the force f in the contact follows…”