Stiffest Elastic Networks

TL;DR

This paper identifies and characterizes a class of isotropic elastic networks that are optimally stiff, providing explicit bounds and criteria, with examples in 2D and 3D, advancing understanding of network rigidity.

Contribution

It introduces the concept of the stiffest elastic networks, deriving explicit bounds and criteria, and providing examples in multiple dimensions.

Findings

Stiffest networks have explicitly given elastic moduli.

They serve as upper bounds surpassing traditional bounds.

Displacement fields are affine at microscopic scales.

Abstract

The rigidity of a network of elastic beams crucially depends on the specific details of its structure. We show both numerically and theoretically that there is a class of isotropic networks which are stiffer than any other isotropic network with same density. The elastic moduli of these \textit{stiffest elastic networks} are explicitly given. They constitute upper-bounds which compete or improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks, and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions.