Mechanics of elastic networks

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 $Z=d+1$.