  We consider a periodic lattice structure in $d=2$ or $3$ dimensions with unit
cell comprising $Z$ thin elastic members emanating from a similarly situated
central node. A general theoretical approach provides an algebraic formula for
the effective elasticity of such frameworks. The method yields the effective
cubic elastic constants for 3D space-filling lattices with $Z=4$, $6$, $8$,
$12$ and $14$, the latter being the "stiffest" lattice proposed by Gurtner and
Durand (2014). The analytical expressions provide explicit formulas for the
effective properties of pentamode materials, both isotropic and anisotropic,
obtained from the general formulation in the stretch dominated limit for
$Z=d+1$.
