Force cycles and force chains

We examine the coevolution of $N$ cycles and force chains as part of a broader study which is designed to quantitatively characterize the role of the laterally supporting contact network to the evolution of force chains. Here, we elucidate the rheological function of these coexisting structures, especially in the lead up to failure. In analogy to force chains, we introduce the concept of force cycles: $N$ cycles whose contacts each bear above average force. We examine their evolution around force chains in a discrete element simulation of a dense granular material under quasistatic biaxial loading. Three-force cycles are shown to be stabilizing structures that inhibit relative particle rotations and provide strong lateral support to force chains. These exhibit distinct behavior from other cycles. Their population decreases rapidly during the initial stages of the strain-hardening regime—a trend that is suddenly interrupted and reversed upon commencement of force chain buckling prior to peak shear stress. Results suggest that the three-force cycles are called upon for reinforcements to ward off failure via shear banding. Ultimately though, the resistance to buckling proves futile; buckling wins under the combined effects of dilatation and increasing compressive load. The sudden increase in three-force cycles may thus be viewed as an indicator of imminent failure via shear bands.

Figure 1

(Color online) (a) Contact forces on a force chain in black arrows and (b) corresponding particle load vectors whose direction and magnitude are indicated by the alignment and thickness of the (black) solid lines. (c) The force chain buckles during $\left[{\epsilon }^A,{\epsilon }^B\right]=\left[0.0388,0.0392\right]$, for a buckling threshold of ${\theta }^{\ast }=1°$; the angle ${\theta }^B$ is shown for segment 1-2-3. The change in the orientation of the (blue) dashed line from center to edge of particles in (b) and (c) indicates rotations.

Figure 2

(Color online) Evolution of the population of minimal cycles $C_N$ with respect to axial strain. Lines with dots in order of upper to lower represent cycles $C_3$ through $C_{13}$. Inset shows corresponding frequency distribution. Peak shear stress for this sample occurs at axial strain $\left|{\epsilon }_{yy}\right|=0.034$ [6]. Initial stage is indicated by thick black solid line, final stage is indicated by (red) dashed line, and intermediate stages go from dark to light (green) as strain increases.

Figure 3

(Color online) Transition of low-order cycle populations to higher-order cycles. A large rise in the number of 3 cycles (dashed red line) and 4 cycles (solid black line) becoming higher-order cycles due to dilatation correlates with drops in shear stress. Conversely, the number of 4 cycles collapsing to 3 cycles (solid gray line) is much less.

Figure 4

(Color online) An example cycle evolution in a particle cluster taken from the sample. (a) 3 and 4 cycles in a particle cluster. These open up and combine to form (b) 6 cycles in the lower left portion of the particle cluster and (c) 5 cycles in the upper right portion of the cluster, ultimately leading to (d) single 9 cycles.

Figure 5

(Color online) (a) The unbuckled force chain (red solid particles) shown in Fig. 1a and its confining neighbors (blue outlined particles). (b) Distribution of cycles around this force chain; rattlers which bear only one contact, particles delineated in dashed outline in the figure, are excluded in a minimal cycle basis. (c) Buckled force chain configuration, as in Fig. 1c, with confining neighbors and cycle membership.

Figure 6

(Color online) (a),(b) Strain evolution of the average ratio of the number of $C_N$ to the total number of cycles around force chains. (c) shows strain evolution of the average concentration of 3 cycles in a force chain together with shear stress.

Figure 7

(Color online) (a) Left: ratio of 3 cycles to all cycles during force chain lifetime classified by nonbuckling (NBFC) and buckling (BFC). Right: concentration of 3 cycles in a force chain (NBFC and BFC). (b) Box plot of average ratio of number of 3 cycles per force chain particle over force chain lifetime during shear band development. Red lines inside the boxes show the median of the data, dots are the mean, and blue line is a straight line fit to the mean. Red crosses represent outliers [41].

Figure 8

(Color online) Frequency distribution of force cycles $C_N^f$ for various strain states through to critical state. Inset shows corresponding distribution of force chains (schemes are as inset of Fig. 2).

Figure 9

(Color online) (a) Strain evolution of the population of three-force cycles together with the shear stress. (b) Strain evolution of the population of buckling force chains at three different threshold buckling angles together with the shear stress. Vertical green dashed line indicates the onset of buckling force chains detected away from the boundaries.

Figure 10

(Color online) Strain evolution of the population of three-force cycles together with the shear stress for two systems identical in every respect to that considered in Fig. 9 except for the following aspects: (a) the rolling resistance is higher, i.e., rolling friction ${\mu }^r=0.2$ [22]; (b) the loading condition is strain-controlled biaxial compression under constant volume [21]. Vertical green dashed line indicates onset of buckling force chains detected away from the boundaries.

Figure 11

(Color online) Spatial distribution of three-force cycles (gray) together with force chains that have buckled (red) during the development of shear band.

Figure 12

(Color online) (a) Histograms of relative particle rotations at contacts in three-force cycles (black bar) and all other contacts (white bar) within and during the development of the shear band. Rotations are measured in radians. Inset: the empirical cumulative distribution functions. (b) Equivalent to (a) but plotted on a log-log scale. Contacts within three-force cycles clearly exhibit lower relative particle rotations (frustrated) than other contacts. This is confirmed by a Kolmogorov-Smirnov test whose null hypothesis of samples that are drawn from the same distribution can be rejected at the 5% level of significance.

Figure 13

(Color online) (a) Three-force cycles provide a dual resistance to force chain buckling: by impeding rotation which tend to dominate during buckling and by providing strong lateral support with above global average forces propping up the force chain. (b) Prevalent supporting three-force-cycle configurations of force chains. Red particles (light) form a force chain; blue particles (dark) may or may not form another force chain.