Quantifying Interparticle Forces and Heterogeneity in 3D Granular Materials

Interparticle forces in granular materials are intimately linked to mechanical properties and are known to self-organize into heterogeneous structures, or force chains, under external load. Despite progress in understanding the statistics and spatial distribution of interparticle forces in recent decades, a systematic method for measuring forces in opaque, three-dimensional (3D), frictional, stiff granular media has yet to emerge. In this Letter, we present results from an experiment that combines 3D x-ray diffraction, x-ray tomography, and a numerical force inference technique to quantify interparticle forces and their heterogeneity in an assembly of quartz grains undergoing a one-dimensional compression cycle. Forces exhibit an exponential decay above the mean and partition into strong and weak networks. We find a surprising inverse relationship between macroscopic load and the heterogeneity of interparticle forces, despite the clear emergence of two force chains that span the system.

Figure 1

Experiment setup. (a) Configuration of the specimen and equipment. (b) 3D image of segmented sample reconstructed from x-ray data. (c) Example diffraction pattern from a single acquisition. (d) Macroscopic load measured by the load cell and volume-averaged grain stresses, described in the text.

Figure 2

Evolution of system responses with load step. (a) Macroscopic force from the load cell. (b) Number of total contacts in the system. (c) Stiffness from load cell data. Values at steps 11, 16, and 21 were not computed due to device issues.

Figure 3

(a) Force inference results at six load steps on a translucent rendering of grains colored by strain ellipsoids. (b) Force chains found by plotting only forces whose normal magnitude exceeds twice the mean. Corresponding grains are also plotted with 30% opacity. Floating grains occur where force chains diverge. (c) Comparison of vertical forces from various sources.

Figure 4

(a) Average spatial autocorrelation coefficient $⟨R(r){⟩}_r$ for forces aligned vertically as illustrated in the inset. (b) Slope of fit to exponential decay of normal force magnitudes. (c) Gini coefficient computed using all normal force magnitudes in the system. (d) Engaged friction coefficient for grain-grain contacts and grain-boundary contacts. Dashed lines correspond to maximum and minimum loads.