Symmetry detection of auxetic behaviour in 2D frameworks

TL;DR

This paper develops a symmetry-based method to identify equiauxetic auxetic behaviour in 2D frameworks, characterized by Poisson's ratio -1, using group theory and Maxwell's approach.

Contribution

It introduces a symmetry-extended Maxwell framework to detect equiauxetic mechanisms in 2D periodic structures based on their symmetry groups.

Findings

Frameworks with certain rotational symmetries can exhibit equiauxetic behaviour.

Mechanisms preserving full rotational symmetry are identified as equiauxetic.

For n=6, mechanisms halving rotational symmetry are also equiauxetic.

Abstract

A symmetry-extended Maxwell treatment of the net mobility of periodic bar-and-joint frameworks is used to derive a sufficient condition for auxetic behaviour of a 2D material. The type of auxetic behaviour that can be detected by symmetry has Poisson's ratio -1, with equal expansion/contraction in all directions, and is here termed equiauxetic. A framework may have a symmetry-detectable equiauxetic mechanism if it belongs to a plane group that includes rotational axes of order n = 6, 4, or 3. If the reducible representation for the net mobility contains mechanisms that preserve full rotational symmetry (A modes), these are equiauxetic. In addition, for n = 6, mechanisms that halve rotational symmetry (B modes) are also equiauxetic.