On rotating rigid parallelograms and their potential for exhibiting auxetic behaviour

Abstract

Auxetic systems have the anomalous property of becoming wider when uniaxially stretched, i.e. exhibit a negative Poisson's ratio. One of the mechanisms which can give rise to this property is based on rotating rigid units, in particular 2D rigid polygons which are connected together at their corners through hinges and rotate relative to each other when uniaxially stretched. This work extends earlier preliminary work on connected rigid parallelograms and presents expressions for the mechanical properties for all the types of planar systems that can be constructed from rigid parallelograms of equal size connected at their vertices through flexible hinges. In particular, we derive and discuss the mechanical properties for the Type Iα, Iβ and IIβ rotating parallelograms which were not previously analysed and compare them with the properties of the Type IIα systems. We show that despite being rather similar to each other, the different types of ‘rotating parallelograms’ have very different mechanical properties and different abilities to exhibit auxetic behaviour.