Interlocking hexagons model for auxetic behaviour

Abstract

A 2D ‘Rough Particle’ model consisting of interlocking hexagons is reported. Analytical expressions for the in-plane Poisson’s ratios and Young’s moduli due to particle translation along the geometrically matched male and female interlocks are derived for the model. The dependency of the mechanical properties on each of the model (geometrical and stiffness) parameters is provided, and it is shown that the assembly of interlocking hexagons deforming by particle translation along the interlocks displays auxetic (negative Poisson’s ratio) behaviour. The model predictions are compared with experimental mechanical properties for auxetic polypropylene (PP) films and fibres. The model predicts the experimental Poisson’s ratio values very well (model: ν_xy =  −1.30, ν_yx =  −0.77; experiment (PP films): ν_|| =  −1.12, $\nu_{\bot} = -0.77$). The model generally overestimates the Young’s moduli of the films, but is in reasonable agreement with the axial Young’s modulus of the fibres.

Appendix: derivation of aspect ratio of inscribed ellipse within an hexagonal cell

The inscribed ellipse touches the hexagonal cell edges at S and P (Fig. 4). Taking the origin of the x–y co-ordinate system to be,

$$ O\equiv (0,0) $$

(A1)

the co-ordinates of the hexagonal cell corners Q and R are given by,

$$ Q\equiv (l_1 +l_2 \cos \alpha ,0) $$

(A2)

$$ R\equiv (\frac{l_1}{2},l_2 \sin \alpha) $$

(A3)

The equation of the inscribed ellipse is given by,

$$ \frac{x^{2}}{(A/2)^{2}}+\frac{y^{2}}{(B/2)^{2}}=1 $$

(A4)

where, A and B are the axes of the ellipse aligned along the x and y axes, respectively.

The coordinates of P can be defined in parametric form as,

$$ P\equiv \left (\frac{A}{2}\cos T,\frac{B}{2}\sin T \right) $$

(A5)

where, T is any arbitrary angle in space.

However, it is noted from Fig. 4 that B is equal to length SS′. Thus,

$$ B=SS^{\prime}=2l_2 \sin \alpha $$

(A6)

It can be observed from Fig. 4 that the slope of the tangent to the ellipse at P is equal to the slope of RQ. Hence,

$$ -\frac{B\cot T}{A}=-\tan \alpha $$

(A7)

Since the point P lies on the line QR, the slope of line RQ will be equal to the slope of QP. Thus,

$$-\tan \alpha =\frac{B\sin T}{A\cos T-l_1 -2l_2 \cos \alpha} $$

(A8)

Therefore, from Eqs. A6–A8, the aspect ratio of an ellipse inscribed in the hexagonal cell is,

$$ \frac{A}{B}=\frac{1}{2l_2 \sin \alpha}\sqrt{l_1^2 +4l_1 l_2 \cos \alpha} $$

(A9)