Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds (“braces”) are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is $\sim LlnL$, where $L$ is the linear size of the lattice. This transition exhibits features of both first-order and second-order transitions: The whole lattice becomes rigid at the transition, and a diverging length scale also exists. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from Maxwell's law for floppy modes, which is $\sim L$. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems, including jamming and bootstrap percolation.

Figure 1

Illustration of regular square (a), regular kagome (b), generic square (c), and generic kagome lattices (d) with nearest-neighbor (NN) bonds (black, thin) and random NNN braces (red, thick). The square lattices depicted have $L=7$ and the kagome lattices have $L=4$.

Figure 2

One example of the mapping of braced regular square lattices to bipartite graphs. In this mapping floppy modes in the “line-localized” basis, taking the form of shearing rows (columns), are mapped to black (white) nodes, and braces are mapped to edges (solid red curve) connecting the two types of nodes. The deformation of this regular square lattice illustrates how the brace locks the two floppy modes together.

Figure 3

Probability ${\mathcal{P}}_{\text{rigid}}$ for rigidity percolation in braced regular square (a) and kagome (b) lattices as a function of scaled number of braces. The curve in (a) shows the analytical asymptotic result in Eq. (3.3). The curves in (b) show the analytical result of Eq. (3.5). Error bars are smaller than the symbols.

Figure 4

Average number of floppy modes normalized by the Maxwell number $\langle F\rangle /N_M$ of braced regular square (a) and kagome (b) lattices as a function of $N_b/N_M$. The asymptotic line in (a) is from Eq. (3.9) and the theory line in (a) is from Eq. (3.10). The asymptotic line in (b) is from Eq. (3.11). Error bars are smaller than symbols.

Figure 5

Probability of rigidity percolation ${\mathcal{P}}_{\text{rigid}}$ on (a) generic square lattices and (b) generic kagome lattices as a function of $N_b/N_M$. ${\mathcal{P}}_{\text{rigid}}$ remains zero until jumping to a finite value at the Maxwell point, indicating a first-order-like transition. The solid lines show the theoretical result of Eq. (4.12). Error bars smaller than the symbols are not shown.

Figure 6

Average number of floppy modes $\langle F\rangle$ of (a) generic square and (b) generic kagome lattices as a function of $N_b/N_M$ on a semilog plot. The solid lines show the theoretical result of Eq. (4.11). There is a discontinuous change in slope at the Maxwell point $N_M$. Below $N_M,\langle F\rangle$ is proportional to $L$, but above $N_M,\langle F\rangle$ no longer scales with $L$. Error bars smaller than the symbols are not shown.

Figure 7

Snapshots of generic square [(a)–(d)] and kagome lattices [(e)–(h)] with the generic site displacements unpictured for visual clarity. Randomly placed braces are shown as red lines, rigid regions as blue areas, and stressed bonds as yellow lines. In (a) and (e), as braces are added they induce rigidity only locally. In (b) and (f) a single bond near the Maxwell point has induced rigidity in the bulk of the system, with at most $\mathcal{O}\left(L\right)$ floppy plaquettes on the edge. As additional braces are added self-stresses are generated in the bulk, as shown in (c) and (g). It is only well above the Maxwell point, as shown in (d) and (h), that the floppy modes on the edge are completely eliminated. A video of rigidity percolation in the square lattice is included as Supplemental Material [41].

Figure 8

Snapshots of a regular square lattice (a) and a regular kagome lattices (b) using the conventions of Fig. 7. In contrast to the generic lattices, the regular square lattices (a) feature multiple rigid components of intermediate size, which are separated by lines of nonrigid regions, instead of one bulk rigid region. The regular kagome lattice (b) develops a large rigid component similar to the generic lattices but rigidifies much more slowly. A video of rigidity percolation in the square lattice is included as Supplemental Material [41].

Figure 9

Regular and generic lattices differ dramatically in how individual plaquettes become rigid. As shown in (a) and (b), in either type of lattice three braced plaquettes render the fourth plaquette that shares a vertex with them rigid. Because the floppy modes of the regular lattice shear whole columns or rows, three braced plaquettes can also render a fourth distant plaquette rigid, meaning that an additional brace placed there would generate a self-stress. In (c), the plaquette with the dashed line is rigid because shearing it would require rotating the plaquettes at $(R_4,C_1)$ and $(R_1,C_3)$ to different angles, which would then shear the braced plaquette at $(R_1,C_1)$. In contrast, the generic mixing of floppy modes in (d) means that the trio of braced plaquettes do not render $(R_4,C_3)$ rigid, and so that when a brace is placed there it eliminates a floppy mode, rather than generating a self-stress as in the equivalent regular lattice.

Figure 10

Consider the possibility of Region I, measuring $l$ by $w$ plaquettes, becoming an isolated rigid region in a large generic square lattice. This would require $l+w-1$ independent braces in Region I. However, Region II would then experience $l-1$ constraints from its shared edge with Region I and so would require only $h$ additional independent braces placed in its interior to be made rigid. For large systems, this occurs with finite probability only for $h\sim \mathcal{O}\left(1\right)$. Thus, as discussed in the text, rigid regions first form with nearly $N_M=2L-3$ braces, and such regions span the entire system except possibly for a few rows and columns near the edge.

Figure 11

A floppy region on the left edge of a generic square lattice with a rigid bulk. For visual clarity, we show only a few rows and do not depict the generic displacements of vertices. There are seven random braces in seven columns, but because of their distribution, the edge is not rigid. Counting from the outer edge and treating each column as a single vertex in a graph, a brace in a column links it to the next column not already part of the rigid cluster or to the bulk, as depicted in the graph below the main diagram. Because the fourth and sixth columns would require additional braces to connect the edge to the bulk, these two columns are said to contain edge modes, indicated by green arrowheads. These two modes make the first six columns floppy, while the seventh has become part of the rigid bulk. As discussed in the text, these edge modes play an important role in the onset of rigidity.

Figure 12

The bulk rigidity probability as a function of $N_b-N_M$ on (a) generic square lattices with linear size $L=200$ and (b) generic kagome lattices with $L=100$. The solid lines are theoretical results of Eq. (4.9) in the large-system limit. The dots are simulation results. Error bars are smaller than the symbols.

Figure 13

Three line modes in a portion of the regular kagome lattice. The line modes supported on horizontal lines (red) are denoted $v_{0,i}$, those on lines with angle $2\pi /3$ (green) are denoted $v_{1,j}$ and those on lines with angle $4\pi /3$ (blue) are denoted $v_{2,k}$. On a hexagon, the indices $i,j,k$ run from 1 to $2L$.

Figure 14

The constraint from a brace (red) on the coefficients of four line modes $v_{0,i}$ (green), $v_{0,i+1}$ (orange), $v_{1,j}$ (purple), and $v_{2,k}$ (blue). Suppose that the coefficients are $c_{0,i},c_{0,i+1},c_{1,j},c_{2,k}$. Then the brace is unstretched if the $y$ component of the velocity at the upper particle is equal to the $y$ component of the velocity at the lower particle: thus $-c_{0,i}-2c_{1,j}=c_{0,i+1}+2c_{2,k}$, or, equivalently, $c_{0,i}+c_{0,i+1}+2c_{1,j}+2c_{2,k}=0$.

Figure 15

The bipartite graph percolation probability $s_{*}$ as a function of $N_b/N_M$. A giant component appears continuously at $N_b/N_M=1/2$—the singularity there is governed by the critical exponent $\beta$, which takes the mean-field percolation value 1.