# Geometrical Elaboration of Auxetic Structures

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## Abstract

This paper analyzes the basic geometric and kinematic characteristics of auxetic structures. The geometric principles are then transferred to a family of new possible forms. We investigate and elaborate auxetic behavior in a purely geometric way which is based on the kinematic movement of different frameworks. We then demonstrate its usefulness by analyzing the involved geometry with computer software but without computer simulations or numerical approximations. Instead, using cut flat material and, depending on the cuts and the material used, we enable the kinematical movement of the structures. We also analyze monostable auxetic structures whose movements can be described as geometrically precise, as well as bistable ones. Based on geometric considerations, we combine rigid materials and composites and select appropriate joint connections to allow the application of this system in an architectural scale in, for example, facades, screens or shading systems.

## Keywords

Auxetics Kinematics Fabrication## Introduction

When elastic material is stretched, it becomes longer in the direction of stretching and is typically thinner in the orthogonal direction. The behavior of material under such a deformation is described by one of the fundamental mechanical terms—Poisson’s ratio (Evans et al. 2000). Poisson’s ratio (µ) defines how a material expands (or contracts) transversely when being compressed longitudinally. Most natural materials have a positive µ indicative of our intuitive understanding that material becomes thinner when it is stretched. Auxetic materials however, can exhibit a negative Poisson’s ratio. Such materials undergo lateral expansion when stretched longitudinally and become thinner when compressed (Ellul et al. 2009).

The behavior of auxetic materials is of great interest for material sciences and most auxetic research is done on microscale and mesoscale. The application of such materials includes medical implants of expandable stents (Gatt et al. 2015; Najabat et al. 2014; Tan et al. 2011) through to applications in sports fabrics (Sanami et al. 2014) to programmable materials that can form different shapes autonomously by folding (Hawkes et al. 2010). Such material concepts are experimentally realized by perforating various cut motifs into a sheet of material creating a network of solids and voids. Thin parts of material that connect solids act as hinges and allow auxetic behavior of the whole system. Furthermore, we can distinguish between monostable and bistable auxetic mechanical materials. The main mechanical properties of a monostable material is that such auxetics cannot maintain the transformed shape upon load removal whereas bistable auxetic materials have two stable positions—exhibiting a switchable expandability (Rafsanjani and Pasisi 2016).

The existing research in the material sciences is strongly based on experiments, computer simulations and numerical approximations of auxetic capabilities. Only a small number of research studies give a precise geometric description of auxetic structures (Borcea and Streinu 2015; Elipe and Lantada 2012) or describe possible applications for design purposes (Konakovic et al. 2016; Mesa 2016). For example, the research of Borcea and Streinu (2015) presents purely geometric notions of auxetic one-parameter deformations of periodic 2D and 3D frameworks. But from a design point of view, this gives no framework for further development or a hint of how to find new auxetic designs with one-parameter deformations. Elipe and Lantada (2012) records the results of a comparative study of planar auxetic geometries by means of computer-aided design and engineering. They develop a library of existing different auxetic structures and simulate the behavior of different auxetic geometries and elaborate properties of the auxetic structures.

Our work will go further because we will analyze the geometry of auxetic structures in order to define classes of possible auxetic patterns. Based on the seventeen wall paper groups—especially on Platonic and Archimedean patterns and the associated Euclidian transformations—we will analyze periodical patterns and develop additional a-periodical patterns in order to construct a network with a continuous change of the auxetic volume. Our parametric approach will enable generation of such patterns performed in an appropriate CAD system. With a series of parametric design studies and physical prototypes we will demonstrate that our approach encompasses a rich class of periodical and a-periodical patterns that, apart from potential use in material science, can be applied as kinematic systems in engineering and design for performative architecture.

## Auxetics

_{1}and F

_{2}are applied in two opposite (longitudinal) directions on the material, it is compressed in the transverse direction. The new volume V

_{1}usually becomes smaller or at most the same as V which means V

_{1}≤ V. The ratio between the negative transverse strain divided by the longitudinal one is called Poisson’s ratio. Since typical materials contract in transverse direction, when stretched in longitudinal one, Poisson’s ratio is positive and approximately between 0 and 0.5 (Lakes 1987).

## Monostable and Bistable Auxetics

From our geometric point of view it is important to distinguish between two auxetic forms, since we want to describe the motion of the structure in a precise kinematic way (Bottema and Roth 1979).

### Monostable Auxetics

One of the drawbacks of the system in Fig. 5 is that the hinges collide in the closed position. Therefore, a practical implementation is restricted to open positions of the systems or it must be realized by means of special hinges (see later section on hinges, and Fig. 14) or a design shown in Fig. 7.

In order to enable the system to close, we have to ensure that the short edge of the parallelogram is half of the length of the longer one. For practical purposes, we have to cut gaps into the pattern parts in order to create an empty place for the hinges (Fig. 7). Depending on the material used and the pattern design, this scheme has to be adapted accordingly.

### Bistable Auxetics

The term bistable auxetics is used to describe systems of linked elements which are not moveable as long as rigid parts are used. In order to make the system moveable, elastic or bendable materials are used. The system always has a start position where all the parts are in one plane. If forces are applied the elements are bent and move out of the plane until they reach a stable end position, again in the plane. Due to the two stable positions, the name bistable is used.

## Joints

## From Kinematic Structure to Parametric Design

In the first step, for the design of a variety of auxetic structures we choose a “simple” kinematic system which is moveable with at least one degree of freedom. Simple means that the system usually consists of one or two types of congruent polygons, not necessarily regular. The polygons are connected by congruent parallel (diamond-shaped) four-bar linkages. Figures 2 and 5 show quadrangles and diamond-shaped four-bar linkages. Figure 6 shows rectangles and parallel four-bar linkages. The patterns in Fig. 8 include triangles, quadrangles, hexagons and octagons.

In a second step, the polygon-outlines are replaced by curves; in most cases these are NURBS curves (Farin 2001). This can be done arbitrarily in a certain range, whereby it must be ensured that the individual elements fit together and the mechanism can be opened and closed. This can be done by means of the theory of the wallpaper groups. In order to vary the curved outlines, the control points of the NURBS curves should be defined parametrically. This leads to examples like those in Figs. 3 and 9. So the underlying kinematic system of such an auxetic structure always stays the same, only the outline shape of the individual forms change.

As long as the underlying kinematic system is moveable and planar, we get moveable and planar auxetic forms. Conversely, the geometric pattern in the Figs. 10, 11, 12 and 13, is not moveable in a kinematic sense, since kinematic always deals with rigid bodies. The way out for this situation is use flexible materials and to bend the individual parts.

## Design Examples

In order to make the system moveable we did not cut the whole border of each element but left a small amount of material in place of the connections. Since these connections are very small they can act like revolute joins (Fig. 16, right side).

## Conclusion

Auxetic materials and auxetic behavior is very effective not just for practical use but also for design purposes in architecture, such as façade elements, screens or shading systems. Since our approach to the matter is a geometric one we have explained the structure of such materials in a precise way, without computer simulation or approximation. This works most of all for monostable auxetics, whose behavior can be descripted in a precisely geometric way. As a base we used Platonic and Archimedean patterns as well as the wallpaper groups. The examples shown in this paper, which can be described precisely in a geometric way, can be scaled to any size. This is the great advantage compared to auxetic structures used in material sciences, which use elastic material properties which they explore in a more experimental way. For application purposes in a bigger scale, the joint system has to be considered and solved, since adjacent parts often interfere. Finally, there is still an open question as to which of the wallpaper groups can be used to construct an efficient auxetic structure, either monostable or bistable. Also three-dimensional auxetic structures, which are not discussed in this paper, could be a valuable subject for further research. There already exist examples of such structures and a deeper analysis would be worthwhile. Future work will be to investigate applications for architectural design, especially in shading systems and façade elements.

## Notes

### Acknowledgements

Open access funding provided by Graz University of Technology.

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