Auxetic deformations and elliptic curves

TL;DR

This paper introduces a new, simplified algorithm for detecting auxetic behavior in certain 3D periodic frameworks, linking the problem to properties of elliptic curves and classical invariants, with applications in materials science.

Contribution

It presents a novel, efficient method for identifying auxetic deformations in 3D frameworks using elliptic curve invariants, simplifying previous approaches.

Findings

Algorithm applicable to most zeolite structures

Auxetic behavior characterized by elliptic curve properties

Fast recognition method using Aronhold invariants

Abstract

The problem of detecting auxetic behavior, originating in materials science and mathematical crystallography, refers to the property of a flexible periodic bar-and-joint framework to widen, rather than shrink, when stretched in some direction. The only known algorithmic solution for detecting infinitesimal auxeticity is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of 3D periodic bar-and-joint frameworks with 3 degrees-of-freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve.