Nonlocal Elasticity near Jamming in Frictionless Soft Spheres

We use simulations of frictionless soft sphere packings to identify novel constitutive relations for linear elasticity near the jamming transition. By forcing packings at varying wavelengths, we directly access their transverse and longitudinal compliances. These are found to be wavelength dependent, in violation of conventional (local) linear elasticity. Crossovers in the compliances select characteristic length scales, which signify the appearance of nonlocal effects. Two of these length scales diverge as the pressure vanishes, indicating that critical effects near jamming control the breakdown of local elasticity. We expect these nonlocal constitutive relations to be applicable to a wide range of weakly jammed solids, including emulsions, foams, and granulates.

Figure 1

(a) Sinusoidal forcing applied to a soft sphere packing, and (b) the resulting displacements. (c) A parametric plot of the constitutive relation is linear with a slope that increases with wave number $q$, violating classical elasticity (dashed line).

Figure 2

(a) Longitudinal compliance ${\widehat{S}}_{\parallel }(q)$ vs inverse wavelength $\lambda$ for pressures ${10}^{-5.5}\le p\le {10}^{-2}$ in half decade steps. (inset) Nonlocal length scale determined from ${\widehat{S}}_{\parallel }(2\pi /{\ell }_{\parallel })/{\widehat{S}}_{\parallel }(0)=2$. (b) Data collapse of the excess compliance $\mathrm{\Delta }{\widehat{s}}_{\parallel }(q)$ for mixed (main panel) and pure power law rescaling (inset).

Figure 3

(a) The transverse compliance shows no pressure-dependent crossover for ${10}^{-5.5}\le p\le {10}^{-2}$. (b) The excess compliance at low $p$ is nonmonotonic, even at large system size $N$. Here: $p={10}^{-4}$ and $2^{10}\le N\le 2^{16}$. (inset) Nonlocal length scale determined from ${\widehat{S}}_{\perp }(2\pi /{\ell }_{\perp })/{\widehat{S}}_{\perp }(0)=2$.

Figure 4

(a) Data collapse of the transverse Fourier spectrum before (inset) and after (main panel) rescaling $q^{\prime }$ with $p^{-0.25}$. (b) The longitudinal Fourier spectrum shows similar collapse before (inset) and after (main panel) rescaling $q^{\prime }$ with $p^{-0.5}$.