Elasticity and Response in Nearly Isostatic Periodic Lattices

The square and kagome lattices with nearest-neighbor springs of spring constant k are isostatic with a number of zero-frequency modes that scale with their perimeter. We analytically study the approach to this isostatic limit as the spring constant k^' for next-nearest-neighbor bonds vanishes. We identify a characteristic frequency ω*~√k^' and length l*~√k/k^' for both lattices. The shear modulus C₄₄=k^' of the square lattice vanishes with k^', but that for the kagome lattice does not.