An isostatic lattice is one at the threshold of mechanical stability.
The square and kagome lattices (see Figure 1a-b) in two dimensions are
examples of isostatic lattices. A 2D kagome lattice of N sites has of
order N1/2 zero-energy bulk modes under periodic boundary conditions.
Theoretical study shows that when neighboring triangles are counter
rotated through an arbitrary angle α shown in Figure 1c, the bulk
modulus vanishes, making the Poisson's ratio equal to -1, and all of the
bulk zero modes of the α =0 lattice disappear. The study of rigidity
and its restoration in these lattices as a function of bending forces or
next-nearest-neighbor springs will improve our understanding of jamming
of hard spheres systems and of networks of semi-flexible polymers, and
offer new paradigms for the microscopic mechanics of disordered media.
Taking the general concept of network instability, we expect to develop a
new opto-mechano based materials platform to dynamically tune
the photonic and phononic properties.
Fig. 1. (a) Square and (b) kagome lattices with NN springs of spring
constant k and NNN springs of spring constant k'. White circles in (a)
and white triangles in (b) show a zero-energy distortion. (c) A
sequence of twisted kagome lattices. Pulling the purple lattice along
the horizontal axis cause it to expand in both the horizontal and
vertical directions to the pink and grey lattices, the defining
characteristic of auxetic materials. Twisted kagome lattices with
triangles at angle a to horizontal.