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Scale invariance and universality of force networks in static granular matter


Force networks form the skeleton of static granular matter1,2. They are the key factor that determines mechanical properties such as stability3, elasticity4,5 and sound transmission6,7, which are important for civil engineering and industrial processing. Previous studies have focused on investigations of the global structure of external forces8,9,10,11 (the boundary condition) and on the probability distribution of individual contact forces4,12. So far, however, precise knowledge of the disordered spatial structure of the force network has remained elusive. Here we report that molecular dynamics simulations of realistic granular packings reveal scale invariance of clusters of particles interacting by means of relatively strong forces. Despite visual variation, force networks for various values of the confining pressure and other parameters have identical scaling exponents and scaling function, thereby determining a universality class. Unexpectedly, the flat ensemble of force configurations13,14,15 (a simple generalization of equilibrium statistical mechanics) belongs to this universality class, whereas some widely studied simplified models16,17,18 do not. This implies that the elasticity of the grains and their geometrical disorder do not affect the universal mechanical properties.

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Figure 1: Dependence of the force network on pressure and polydispersity.
Figure 2: Force chains at different scales.
Figure 3: The universal scaling function.


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We thank B. Behringer, W. Ellenbroek, C. Goldenberg, M. van Hecke, W. van Saarloos and K. Shundyak for discussions. S.O. is financially supported by the Dutch research organization FOM (Fundamenteel Onderzoek der Materie). E.S. is supported by the PHYNECS training network of the European Commission.

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Correspondence to Bernard Nienhuis.

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Supplementary Figure 1

This file contains a Supplementary Figure displaying probability distributions of force magnitudes in all the systems studied. (PDF 69 kb)

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Ostojic, S., Somfai, E. & Nienhuis, B. Scale invariance and universality of force networks in static granular matter. Nature 439, 828–830 (2006).

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