Jammed frictionless disks: Connecting local and global response

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 80, Issue 6, 2009, Pages

  Ellenbroek, Wouter G ^a,b   Van Hecke, Martin ^c   Van Saarloos, Wim ^a  

^a UNIVERSITEIT LEIDEN   (Netherlands)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

^c LEIDEN UNIVERSITY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

COARSE GRAINING; CRITICAL SCALING; DISPLACEMENT ANGLE; DIVERGING LENGTH SCALE; ELASTIC PROPERTIES; EXTERNAL PERTURBATIONS; FRICTIONLESS DISKS; GLOBAL RESPONSE; LENGTH SCALE; LINEAR RESPONSE; NON-AFFINE DEFORMATION; PACKING DISORDER; QUASI-STATIC; RELATIVE DISPLACEMENT; SMALL SCALE; STRESS FIELD; VIBRATIONAL MODES;

DISKS (MACHINE COMPONENTS); DISKS (STRUCTURAL COMPONENTS); ELASTIC MODULI; ELASTICITY; JAMMING;

BOUNDARY LAYER FLOW;

EID: 77149149423     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.80.061307     Document Type: Article

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54. The point which deviates from the scaling of G is obtained for the lowest pressure from the point response. We believe this large deviation to be due to the fitting procedure: we fit the Poisson ratio ν from the stress field Δ σαβ (r). As p→0, we observe ν→1, its maximum value in two dimensions. In this limit, the fitting of the stress tensor becomes less accurate for numerical reasons, which is made even worse by the fact that the data become more noisy at low p. In the bulk response we see a similar effect: the fluctuations around the average values are much larger at lower pressure.
55. See page 21 in Ref.
56. For a general potential Vij ∼ δ ij α the factor 2/3 in Eq. becomes 1/ (α-1).
57. This is easily seen in the case of a simple shear in the x direction: all ui are in the x direction, hence so are the uij, while the rij are isotropic.
58. Because of the π -periodic nature of the angle α, we add copies of the peak at -π/2 and at 3π/2. This improves the fit for larger pressures.
59. In addition, it is natural to eliminate systematic dependence of u∥ and u∥ on the particle radii, and we have divided each u∥,ij and u∥,ij by the actual length rij of the contact-this would also allow us to compare the results to the case of affine displacements, which have local u∥,ij and u∥,ij which are proportional to rij, but we have already seen that the displacements are strongly nonaffine, so we do not show the corresponding plot here.
60. Note that because the distribution of u∥ under compression is not centered around zero, we are collapsing the average of the distributions, not the width.
61. This holds in general for floppy modes in systems with radial interactions between the particles.