Characterization of force chains in granular material

It has been observed that the majority of particles in a granular material carries less than the average load and that the number of particles carrying larger than the average load decreases exponentially with increasing contact force. The particles carrying above average load appear to form a strong network of forces while the majority of particles belong to a weak network. The strong network of forces appear to have a spatial characteristic whereby the stronger forces are carried though chainlike particle groups referred to as force chains. There is a strong case for a connection between force chains of the discrete medium and the trajectory of the most compressive principal stress in its continuous idealization. While such properties seem obvious from descriptive analysis of physical and numerical experiments in granular media, progress in quantification of the force chain statistics requires an objective description of what constitutes a force chain. A procedure to quantify the occurrence of force chains is built on a proposed definition having two parts: first, the chain is a quasilinear arrangement of three or more particles, and second, along the chain, stress concentration within each grain is characterized by the vector delineating the most compressive principal stress. The procedure is incorporated into an algorithm that can be applied to large particle assemblies to compile force chain statistics. The procedure is demonstrated on a discrete element simulation of a rigid punch into a half space. It was found that only approximately half of the particles within the group of so-called strong network particles are part of force chains. Throughout deformation, the average length of force chains varied slightly but the number of force chains decreased as the punch advanced. The force chain lengths follow an exponential distribution. The procedure provides a tool for objective analysis of force chains, although future work is required to incorporate branching of force chains into the analysis.

Figure 1

The initial setup and dimensions of the flat punch system.

Figure 2

(Color online) Contact forces between particles for a section of the particle assembly. The color of the force represents the magnitude, with light (red online) corresponding to large forces down to dark (blue online) representing small forces.

Figure 3

Particles in an idealized portion of a force chain. The pathway of force transmission (force chain) is indicated by a gray line. The double-sided arrow through the particle center represents the direction of most compressive principal stress.

Figure 4

(Color online) Particles colored by $\mid {\sigma }_3\mid$.

Figure 5

(Color online) Particles represented by arrows, where the direction of the arrow is the direction of ${\sigma }_3$, and the color of the arrow is the magnitude of ${\sigma }_3$. Note that only those particles with $\mid {\sigma }_3\mid$ greater than the average are shown.

Figure 6

Two assemblies of highly stressed particles that (a) do not and (b) do form a force chain.

Figure 7

A simplified flowchart of the algorithm (see Sec. 5 for more details).

Figure 8

(Color online) The algorithm used to find force chains. Double-sided arrows through particle centers indicate the direction of the minor principal stress. Single-sided arrows joining particle centers indicate branch vectors. (a) Start by choosing a particle to start at, particle $A$. (b) Look in the forward direction of particle $A$’s minor principal stress, ${\sigma }_3^A$. Here two possibilities for the next particle in the chain are found ($B$ and $C$). The angle $\alpha$ defines how far either side of ${\sigma }_3^A$ to look. (c) Next, perform a back check: ensure that particle $A$ lies within $\alpha$ deg of particle $B$’s minor principal stress. In this case, it does not, so $B$ is rejected as the next particle. (d) Performing a back checkfrom particle $C$ is successful, so particle $C$ is the next particle in the chain.

Figure 9

A case where there exists two possibilities for force chains found by the algorithm.

Figure 10

(Color online) Force chains found by the algorithm—particles shown are part of a force chain.

Figure 11

(Color online) Force chains found underneath the flat punch—particles colored light (red online) are part of a force chain, while dark (blue online) particles are not.

Figure 12

(Color online) Evolution of the percentage of particles with $\mid {\sigma }_3\mid$ greater than the average (“high stress”), as well as the percentage of particles actually part of force chains.

Figure 13

(Color online) Evolution of the number of force chains found.

Figure 14

(Color online) Evolution of the average length of force chains.

Figure 15

(Color online) The distribution of lengths of force chains, for five different stages of indentation (%). Inset: Logarithmic plot demonstrating an exponential distribution.