Mechanics of elastic networks

TL;DR

This paper develops a theoretical framework to determine the effective elastic properties of periodic lattice structures in 2D and 3D, including specific formulas for various lattice configurations and applications to pentamode materials.

Contribution

It introduces a general algebraic formula for the effective elasticity of elastic networks, including explicit expressions for multiple lattice types and applications to pentamode materials.

Findings

Derived algebraic formulas for effective elastic constants of various 3D lattices.

Provided explicit formulas for isotropic and anisotropic pentamode materials.

Identified the 'stiffest' lattice among proposed configurations.

Abstract

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 $Z=d+1$.