Nearly Isostatic Periodic Lattices

In 1864, James Clerk Maxwell showed that a system of N point particles in d-dimensions is mechanically stable only if the number, z, of two-point contacts between particles exceeds z_c = 2d. Systems with z=z_c are isostatic. Recent work confirms that randomly packed spheres are isostatic at the point J where the volume fraction \phi reaches the critical value \phi_c necessary to support shear and that the mechanics of this isostatic state determine behavior at volume fractions above \phi_c. Infinite square and kagome lattices with nearest neighbor springs are isostatic, but their finite counterparts have N^{1/2} ``floppy" modes of zero frequency. This talk will discuss the mechanical properties and phonon spectrum of nearly isostatic versions of these lattices in which next-nearest-neighbor springs with a variable spring constant are added either homogeneously or randomly. In particular, it will show that these lattices exhibit characteristic lengths that diverge as 1/(z-z_c) and frequencies that vanish as (z-z_c) in agreement with general arguments by the Chicago group. The shear elastic modulus depends on the geometry of the isostatic network and is not universal. Response near z=z_c in the random case is highly nonaffine. This talk will also discuss an isostatic variant of the kagome lattice that has a vanishing bulk modulus and a negative Poisson ratio and whose floppy isostatic modes are not found in the phonon spectrum with periodic boundary conditions. Finally, if time permits, the application of some of these ideas to networks of semi-flexible polymers will be discussed.