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.