Dionysian Hard Sphere Packings are Mechanically Stable at Vanishingly Low Densities

TL;DR

This paper introduces Dionysian hard sphere packings that are mechanically stable at extremely low densities, enabling the creation of lightweight materials with finite strength and no lower density limit.

Contribution

The authors demonstrate the existence of stable, low-density sphere packings with finite strength, challenging the notion that stability requires higher density.

Findings

Packings have asymptotically zero density but finite bulk and shear moduli.

Structures are linearly resistant to all applied deformations.

No lower bound exists on the density for stable structures.

Abstract

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.