Force distributions in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces. For periodic boundary conditions corresponding to isotropic compressive stress, we find that the probability distribution for single-contact forces decays faster than exponentially. This superexponential decay persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges in the force distribution, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.

Figure 1

An $n\times n$ triangular lattice (with $n=6$). Under periodic boundary conditions light edges on the left are identified with dangling edges on the right and light edges on top are identified with dangling edges on the bottom. The macroscopic stress state is fixed by imposing compressive forces $F_1$, $F_2$, and $F_3$ along each lattice direction.

Figure 2

A slab of material and set of edges used to prove that each layer of edges in $L_k$ must have a sum of forces equal to $F_k$.

Figure 3

The wheel move associated with the node marked by a disk. Forces are shifted by an amount $\Delta f$, chosen to be small enough that no edge becomes negative. Force on each edge marked with a thick solid segment is increased and on each edge marked with a thick dashed segment is decreased, or vice versa.

Figure 4

A two-dimensional convex space, outlined in gray, with two different sets of basis vectors. The basis in (a) is such that it is possible to move off the lower-lefthand corner along either basis direction, while in (b) this is not the case.

Figure 5

$3\times 3$ triangular lattice under isotropic stress. The curve gives the force distribution, points represent numerical simulations of the same quantity. $P\left(f\right)$ vanishes at $F$, the force imposed along each lattice direction. Here $F=3$, fixing $⟨f⟩=1$.

Figure 6

A typical $15\times 15$ lattice. The force on an edge is redundantly mapped to color and width (stronger forces are darker and thicker).

Figure 7

Force distributions for $n\times n$ lattices under isotropic stress for $n=5$, 10, 15, and 20. For all cases the decay is faster than exponential. The $n=15$ and $n=20$ cases are nearly identical, indicating convergence to a universal curve for large systems. The straight line is drawn as a guide to the eye. For each curve, $⟨f⟩=1$.

Figure 8

Above, correlations are calculated in the longitudinal and transverse directions. Below, correlation function $g\left(r\right)=⟨f_i{f}_{i+r}⟩-⟨f_i^2⟩$ on a $20\times 20$ lattice. The correlation is computed for edges that have the same orientation. The longitudinal correlation refers to displacements by $r$ lattice constants along the direction of the edge. The transverse correlation refers to displacements perpendicular to the edge. The plots above are averaged over multiple reference edges. The transverse correlations are in fact negative, but we take their absolute value to facilitate plotting on a log scale.

Figure 9

A $15\times 15$ lattice at rigidity percolation. Deleted edges are covered with disks. The force on an edge is redundantly mapped to color and width (stronger forces are darker and thicker).

Figure 10

The ratio of average number of degrees of freedom in threshold lattices to the number in the undiluted case $⟨\delta ⟩=(⟨N_m⟩-1)∕(n^2-1)$ vs the ratio of deleted edges in the threshold lattice to the total number of edges $\varphi =n_d∕3n^2$. For each plot a number of lattices at the rigidity-percolation threshold were generated and the degrees of freedom and deleted edges counted. For large enough lattices, the system has a large number of degrees of freedom available as soon as rigidity percolation is reached. Note that the observed interval of $\varphi$ values shifts as the lattice size is increased.

Figure 11

$P\left(f\right)$ for $20\times 20$ triangular lattices diluted to the rigidity-percolation threshold. The plus signs represent an average over $2.5\times {10}^6$ moves on a single randomly diluted lattice. The open circles represent an average over 75 lattices, each run for $5\times {10}^4$ moves. The straight line is a guide to the eye.

Figure 12

$\alpha \to \infty$ limit of the $n\times n$ triangular lattice. Dashed edges represent attachments under periodic boundary conditions. The macroscopic stress state is fixed by imposing compressive force $\Delta$ along the strong lattice direction, the only one carrying force. Force is transmitted along $n$ chains, each composed of $n$ edges, in any way such that a sum over chains yields $\Delta$ and all forces remain compressive.

Figure 13

$3\times 3$ triangular lattice under anisotropic stress. The force distribution $P_s\left(f\right)$ for forces in the strong direction is plotted for increasing anisotropy. Each distribution is scaled to $⟨f_s⟩=1$. The limiting form is a piecewise linear function of $f$. $P_s$ for finite anisotropy has a linear region in the middle; this region grows in width and approaches the limiting slope as anisotropy is increased.

Figure 14

A $15\times 15$ anisotropic lattice with $\alpha =5$. The force on an edge is redundantly mapped to color and width (stronger forces are darker and thicker). The stronger stress is imposed in the horizontal direction.

Figure 15

$P\left(f\right)$ for strong, weak, and all lattice directions on a $15\times 15$ lattice with $\alpha =3$. The strong direction(s) is entirely responsible for the tail in $P\left(f\right)$. Top: Two weak $F_k$’s and one strong. Bottom: One weak $F_k$ and two strong. On both plots, $⟨f⟩=1$.

Figure 16

$P\left(f\right)$ on a $15\times 15$ lattice for increasing anisotropy. Portions of the tail for stronger anisotropies are difficult to distinguish from exponential decay, and the limiting behavior is a true exponential (gray line). The dashed line indicates the interval in $f$ over which the $\alpha =8$ curve is approximately exponential. $⟨f_s⟩=1$ for every curve.

Figure 17

Three of the nine basis elements employed to evaluate Eqn. (1) for the $3\times 3$ isotropic lattice. Solid bars indicate a positive contribution to the force on an edge; dashed bars indicate a negative contribution. Double bars indicate a contribution with twice the weight. The honeycomb in (a) is one of three. The elements in (b) and (c) are each repeated for the other two lattice directions.

Figure 18

Useful references for the proof in Appendix .