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

Abstract

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 ∼L ln L, 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 ∼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.

Fiber networks below the isostatic point: Fracture without stress concentration

“…The crossover is controlled by the central-force isostatic point (CFIP), at which the degrees of freedom and central-force constraints balance and the system is at the verge of mechanical instability. The CFIP occurs when z = 2d where z is the average coordination number at the crosslinks and d is the spatial dimension [23][24][25][26][27][28][29][30][31][32]. In a network where at most two fibers meet at a crosslink, z < 2d (dangling ends are removed because they don't contribute to elasticity) and the linear elastic moduli depend on the bending stiffness.…”

Rigidity percolation and geometric information in floppy origami

“…Additionally, as in purely 2D systems, floppy modes in origami are more likely to involve boundary quads (22,23); indeed, corner quads are an extreme example as they can bend without involving any quads in the bulk. Therefore, rigidifying the boundaries first might be the most efficient way to rigidify the whole system.…”

Topological Boundary Floppy Modes in Quasicrystals

“…As we show below, all parallelogram tilings have a mean coordination hzi ¼ 2d and are thus Maxwell networks (note that we limit ourselves to tilings where all edges are "complete," i.e., nodes of a parallelogram merge with nodes of the neighboring parallelogram when they are tiled together, instead of sitting in the middle of the edge of other parallelograms). Here we extend the notion of Maxwell lattices to "Maxwell networks" to include aperiodic networks with hzi ¼ 2d [42][43][44][45][46][47][48][49]. Jammed packings of frictionless spheres [50] and Mikado fiber networks [51,52] are both examples of Maxwell networks.…”