# Quenched Stresses And Linear Elastic Response Of Random Packings Of Frictionless Particles Near Jamming

thesis

posted on 01.02.2014, 00:00 by Kamran KarimiWe study stress correlations and elastic response in large-scale computer
simulations of particle packings near jamming. We show that there are characteristic lengths in both the stresses and elastic response that diverge in
similar ways as the confining pressure approaches zero from above. For the
case of the stress field, we show that the power spectrum of the hydrostatic
pressure and shear stress agrees with a field-theoretic framework proposed
by Henkes and Chakraborty [15] at short to intermediate wavelengths (where
the power is flat in Fourier space), but contains significant excess power at
wavelengths larger than about 50 to 100 particle diameters, with the specific
crossover point going to larger wavelength at decreasing pressure, consistent
with a divergence at p=0.These stress correlations were missed in previous
studies by other groups due to limited system size. For the case of the elastic
response, we probe the system in three ways: i) point forcing, ii) constrained
homogeneous deformation where the system is driven with no-slip boundary
conditions, and iii) free periodic homogeneous deformation. For the point
force, we see distinct characteristic lengths for longitudinal and transverse
modes each of which diverges in a different way with decreasing pressure with
ET⇠p

^{-1/4}and EL⇠p^{-0.4}respectively. For the constrained homogeneous deformation we see a scaling of the local shear modulus with the size of the probing region consistent with E⇠p^{-1/2}similar to the EL⇠p^{-0.4}observed in the longitudinal component of the point response and in perfect agreement with the rigidity length discussed in recently proposed scenarios for jamming. Finally, we show that the transverse and longitudinal contributions to the strain field in response to unconstrained deformation (either volumetric or shear) have markedly different behavior. The transverse contribution is surprisingly invariant with respect to p with localized shear transformations dominating the response down to surprisingly small pressures. The longitudinal contribution develops a feature at small wavelength that intensifies with decreasing p but does not show any appreciable change in length. We interpret this pressure-invariant length as the characteristic shear zone size.