On the Auxetic Properties of `Rotating Rectangles' with Different Connectivity

Auxetic materials and structures exhibit the unusual property of becoming wider when stretched and thinner when compressed, i.e., they have negative Poisson’s ratios. In recent years, this unusual behaviour has been predicted or experimentally measured in a number of naturally-occurring and man-made materials ranging from foams where the auxetic effect arises from the particular microstructure of the foams to silicates and zeolites where the auxetic behaviour occurs at themolecular level. In these auxetic systems, the negative Poisson’s ratios can be explained in terms of models based on the geometry of the system (i.e., the geometry of the material’s internal structure) and the way this geometry changes as a result of applied loads (deformationmechanism). In recent years various two and three dimensional theoretical models and structures which can lead to negative Poisson’s ratio have been proposed including, two and threedimensional ‘re-entrant’ systems, models based on rigid ‘free’ molecules, chiral structures and systems made from ‘rotating rigid units’ such as squares, triangles, rectangles or tetrahdera. In all of these systems, the Poisson’s ratio does not depend on scale although it can depend on the relative dimensions of certain features in the geometry. In particular we have recently shown that whilst a two-dimensional system constructed from perfectly rigid squares connected together through simple hinges at the vertices of the squares will always maintain its aspect ratio when stretched or compressed see Fig. 1(a), i.e., it will exhibit constant Poisson’s ratios equal to 1 irrespective of the size of the square or direction of loading, the equivalent structure built from hinged rigid rectangles as illustrated in Fig. 1(b) will exhibit in-plane Poisson’s ratios which depend on the shape of the rectangles (the ratio of the lengths of the two sides) and the relative orientation of the rectangles (i.e., the angles that two adjacent rectangles make with respect to each other). This means that for such a system, the Poisson’s ratios will be strain dependent and dependent on the direction of loading. This note is aimed at highlighting the fact that there exist two types of ‘rotating rectangles’ structures, and that two systems based on the same ‘building block’ (rigid rectangle) and same deformation mechanism (rectangle rotation), but different connectivity, exhibit very different mechanical properties. More specifically, for rectangles of the same size (a b), tessellating corner-sharing rectangular networks in which each corner is shared between two rectangles can only be formed from two connectivity schemes, which we shall refer to as Type I and Type II. The Type I network refers to the system where four rectangles are connected in such a way that the empty spaces between the rectangles form rhombi of size (a a) and (b b) as illustrated in Fig. 1(b). The Type II network refers to the system with a connectivity where the empty spaces between the rectangles form parallelograms of size (a b) as illustrated in Fig. 1(c). If the four rectangles are connected in any other way (for example, with the empty spaces between the rectangles forming a ‘kite’ of side lengths ‘a, a, b, b’) the resulting unit cannot form a tessellating structure. The Type I ‘rotating rectangles’ structure has been extensively studied and it has been shown that this structure exhibits properties which are dependent on the shape and size of the rectangles and are strain dependent and anisotropic. In particular it has been shown that such Type I ‘rotating rectangles’ structures are capable of exhibiting both positive and negative Poisson’s ratio where, for example, the on-axis Poisson’s ratios are dependent on the ratio of the lengths (a=b) and on the angle between the rectangles since: v21 1⁄4 ðv12Þ 1 1⁄4 a sin 2 ð Þ b cos 2 ð Þ a2 cos2 2 ð Þ b2 sin 2 ð Þ Here we study, for the first time, the behaviour of the Type II ‘rotating rectangles’ which as we will show exhibits very different properties. As illustrated in Fig. 1(c), a rectangular unit cell with cell sides are parallel to the Ox1 and Ox2 axis may be used to describe the Type II network. This unit cell contains two (a b) rectangles with projections in the Oxi directions given by: