On the collapse of locally isostatic networks

Abstract

We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

Self-assembly of three-dimensional open structures using patchy colloidal particles

“…They possess striking properties such as negative Poisson's ratio [1][2][3][4], negative thermal expansion [5,6], and beyond [7,8], leading to many interesting applications in engineering.…”

Topological boundary modes in isostatic lattices

“…There are many periodic lattices, including the square and kagome lattices in d = 2 dimensions and the cubic and pyrochlore lattices in d = 3, that are locally isostatic with coordination number z = 2d for every site under periodic boundary conditions. These lattices, which are the subject of this paper, have a surprisingly rich range of elastic responses and phonon structures [15][16][17][18][19] that exhibit different behaviors as bending forces or additional bonds are added.The analysis of such systems dates to an 1864 paper by James Clerk Maxwell 20 that argued that a lattice with N s mass points and N b bonds has N 0 = dN s − N b zero modes. Maxwell's count is incomplete, though, because N 0 can exceed dN s − N b if there are N ss states of self-stress, where springs can be under tension or compression with no net forces on the masses.…”

“…There are many periodic lattices, including the square and kagome lattices in d = 2 dimensions and the cubic and pyrochlore lattices in d = 3, that are locally isostatic with coordination number z = 2d for every site under periodic boundary conditions. These lattices, which are the subject of this paper, have a surprisingly rich range of elastic responses and phonon structures [15][16][17][18][19] that exhibit different behaviors as bending forces or additional bonds are added.…”

Finite auxetic deformations of plane tessellations

“…without constraint symmetry), namely the (3.6.3.6) called the trihexagonal tessellation or kagome structure, already discussed in earlier studies [42][43][44][45][46]. Kapko et al [43] have noted that the number of collapse mechanisms grows with the size of the unit cell and have also considered crystallographic symmetry constraints for the deformation of this tessellation.…”