## Abstract

We report here structures, constructed with regular polygonal prisms, that exhibit negative Poisson’s ratios. In particular, we show how we can construct such a structure with regular *n*-gonal prism-shaped unit cells that are again built with regular *n*-gonal component prisms. First, we show that the only three possible values for *n* are 3, 4 and 6 and then discuss how we construct the unit cell again with regular *n*-gonal component prisms. Then, we derive Poisson’s ratio formula for each of the three structures and show, by analysis and numerical verification, that the structures possess negative Poisson’s ratio under certain geometric conditions.

## 1. Introduction

When we stretch an object in one direction, we usually expect it to shrink in its perpendicular direction. Similarly, we expect an object to expand in the perpendicular direction when we compress it in a direction. Poisson’s ratio, which is defined as the negative ratio of axial strain in the direction of deformation to the normal strain, is usually used to quantify a material deformation behaviour. In general, most materials exhibit positive Poisson’s ratios. However, materials with negative Poisson’s ratios, so-called auxetic materials, have also been reported in cubic metals [1,2], silicates [3] and zeolites [4].

Owing to potentially beneficial properties such as energy absorption [5], fracture toughness [6] and shear resistance [7,8], considerable research effort has been made in recent years. As a result, various auxetic structures have been proposed such as honeycomb structures [9–12], rotating unit networks [13–18] and chiral structures [19–21]. Auxetic materials have various practical applications in sound and shock absorption [22], biomedicine [23], sports [24] and smart transformation optics [25–29].

Attard and Grima described a three-dimensional structure constructed with cuboid-shaped unit cells at the end of Ref. [30], predicting the auxeticity of the structure. The cuboid-shaped unit in this structure is again constructed with eight component cuboids. In fact, as Attard and Grima predicted, the structure exhibits auxetic behaviour. In this paper, we explore the possibility of constructing three-dimensional auxetic structures not only with cuboids but also with other regular polygonal prisms. To be more specific, we attempt to construct a structure with regular *n*-gonal polygonal prism-shaped unit cells, each of which is again constructed with *n*-gonal polygonal prisms as its components. In particular, we show that the only feasible values for *n* is 3, 4 and 6 and investigate analytically the auxeticity of the three structures constructed with regular triangular prisms, regular square prisms and regular hexagonal prisms one by one. We also provide numerical evaluations to illustrate the tendency of auxeticity of the structures upon the variation of geometrical shapes of the structures.

## 2. Three-dimensional auxetic structure with regular prisms

In this paper, we propose three-dimensional auxetic structures constructed with regular prisms. Let us recall that the structure proposed in Ref. [30] is constructed by connecting cuboids, as illustrated in figure 1*a*. To be more specific, the structure in figure 1*a* is constructed by repeatedly stacking up unit cells depicted in figure 1*b*. The cuboid, illustrated in 1*b*, indicates the virtual territory of a unit cell constructed with eight component square prisms. Since each of these eight square prisms do not alone form a functional block, we shall regard them, in the remainder of this paper, as components of the unit cell depicted in figure 1*b*. We note that the cross-sectional area shall look like the one in figure 1*c*, if we view the unit cell from the above.

In this paper, we shall regard the cuboid depicted in figure 2*b* as a regular square prism and attempt to consider the unit cell geometry of other regular polygonal prisms in addition to the square prism. We note that only specific regular *n*-gonal prisms can fill 360^{°} compactly. For example, we note that it is impossible to fill 360^{°} compactly with regular pentagonal prisms since the internal angle of a regular pentagonal prism which is 108^{°} does not divide 360^{°}. In fact, the only regular polygonal prisms that can compactly fill 360^{°} are regular triangular prism, regular square prism and regular hexagonal prism as illustrated in figure 2. In this paper, we shall only discuss three-dimensional structures constructed solely with triangular-, square- or hexagonal-prism unit cells.

We recall that the square prism unit cell in figure 1*b* is constructed again with eight component square prisms. In a similar manner, we can construct a triangular prism unit cell with component triangular prisms and a hexagonal prism unit cell with component hexagonal prisms as illustrated in figure 3*a*,*b*. In figure 3*a*, the bigger triangle indicates the triangular cross section of the triangular prism unit cell and the three smaller triangles indicate the cross sections of the upper three component triangular prisms. Similarly, in figure 3*b*, the bigger hexagon indicates the hexagonal cross section of the hexagonal prism unit cell and the six smaller hexagons represent the cross sections of the upper six component hexagonal prisms. In the following three sections, we study analytically the auxetic behaviours of the three-dimensional structures constructed with regular triangular prisms, regular square prisms and regular hexagonal prisms one by one with the help of Solidworks 2016.

## 3. Regular square prism configuration

In this section, we first consider the three-dimensional structure, depicted in figure 4*a*, constructed in regular square-prism configuration. As described in §2, the structure is constructed by stacking up regular square prism-shaped unit cells shown in figure 4*b*. As depicted in figure 4*c*, each unit cell is constructed by suitably connecting eight regular square prisms. The upper four component square prisms are connected at the four vertices depicted as small circles in figure 4*c* and that the lower four components prisms are similarly connected. The upper four and the lower four prisms are then connected at another four vertices, depicted as small squares in figure 4*c*, forming a unit cell. Therefore, constructed unit cells are then connected at still another four vertices depicted as small squares in figure 4*c*, which finally results in the desired structure in figure 4*a*.

We note that the geometrical shape of the unit cell is determined by two variables *a* and *b* as illustrated in figure 4*d*. We denote by *A*,*B*,*C*,*D*,*E*,*F*,*G*,*H* the vertices of the unit cell in its fully compressed position as illustrated in figure 4*b*. Similarly, we denote by *A*′,*B*′,*C*′,*D*′,*E*′,*F*′,*G*′,*H*′ the vertices of the unit cell in stretched position, as illustrated in figure 4*c*. We note that the deformation is determined by the characteristic angle *θ*, which is defined as the angle between a cross section of cuboids and the *x*′-axis as shown in figure 5*b*. The eight component prisms in each unit cell are assumed to deform symmetrically. We observe that the base lengths *L*_{x} and *L*_{z} in figure 5 are determined by the angle *θ*. In particular, we note that

When the angle gets bigger from *θ* to *θ*+Δ*θ*, the strains *ϵ*_{x} and *ϵ*_{z} are given by
*ν*_{zx} is given by

We next discuss the behaviour of Poisson’s ratio as a function of geometrical parameters *a* , *b* and angle *θ*. We observe in equation (3.5) that Poisson’s ratio of the structure is scale independent as in the previous section. Consequently, we plot the Poisson’s ratio with respect to *θ* for various values of *b*/*a*. Figure 6*a* illustrates the Poisson’s ratio as a function of *θ* for five different values of *b*/*a*. We note that the right-hand side of equation (3.5) is a product of two fractions. Here, the first fraction is positive for any *a*,*b* and *θ* values. Therefore, the auxeticity of the structure is determined by the second fraction. By investigating the value of the second fraction, we can observe that the Poisson’s ratio falls below zero and hence exhibits auxetic behaviour for *θ* values between 0 and *θ*_{U} gets smaller when *b*/*a* becomes larger. We also note that the absolute value of the Poisson’s ratio *ν*_{zx} for the same *θ* gets bigger for larger value of *b*/*a* in the auxetic region. The volume fraction, namely, the material proportion in a square-prism unit cell is illustrated in figure 6*b*.

Next, to quantify the stiffness of the structure, we derive the on-axis Young’s moduli. We assume that the prisms are rigid and hence the deformation of the structure is induced solely by rotation of the prisms. Then, the stiffness of the structure depends on the stiffness of the springs with which prisms are put together. We model the connecting points as torsional springs. Each of the torsional springs satisfies the angular form of Hooke’s law
*τ* is the torque applied to the prism, Δ*θ* is the angular displacement from its initial equilibrium position and *κ* is the torsion coefficient of the spring. We assume that torsional springs are located at the edges, respectively, small squares in figure 4*c*. The vertices represented as small circles are assumed only to support, without torsional springs, the connections between prisms. Therefore, per each unit cell, the total number of vertices working as torsional springs is 8. We now note that the energy Δ*W* required to change the angle from *θ* to *θ*+Δ*θ* is given by
*κ*_{p} indicates the torsion coefficient of the springs located at the vertices represented as square in figure 4*c*. The amount of energy stored per unit volume due to an infinitesimally small strain *ϵ*_{i} for loading in the *i*-direction (*i*=*x*,*z*) is given by

By the conservation of energy, the following equation must hold:
*V* is the volume of the unit cell. We note that the volume *V* is given by
*E*_{i}(*i*=*x*,*z*) are given by

Consequently, the Young’s moduli in the *x*- and *z*-axes are given by
*c*,*d*, respectively, with *a*=1, *κ*_{p}=1.

In the regular square prism structure, the Poisson’s ratios have transitions from negative to positive or vice versa. From the definition of Poisson’s ratio, *ν*_{zx}=−*ϵ*_{x}/*ϵ*_{z}=−(Δ*L*_{x}/*L*_{x})/(Δ*L*_{z}/*L*_{z})=−(*L*_{z}/*L*_{x})(*dL*_{x}/*dL*_{z})=−(*L*_{z}/*L*_{x})((*dL*_{x}/*dθ*)/(*dL*_{z}/*dθ*)), the Poisson’s ratio *ν*_{zx} changes the sign when Δ*L*_{i}, i.e. one of the two derivatives, *dL*_{i}/*dθ* are equal to zero because the length *L*_{x} and *L*_{z} are always positive for angle *θ* [14]. In this structure, only the case of *dL*_{z}/*dθ*=0 is possible, not *dL*_{x}/*dθ*=0. If we consider the case of *dL*_{z}/*dθ*=0, which is in the denominator of *ν*_{zx}, the value of Poisson’s ratio approaches to *a*). For the Young’s moduli (*E*_{x}, *E*_{z}), (*dL*_{z}/*dθ*=0)^{2} is in the denominator of *E*_{z} (see equation (3.12)), thus *d*), where the more stretching in the *z*-direction cannot make any further change in angle *θ*.

To illustrate the feasibility of the method of construction described above, we built a prototype and compressed it in one direction to see the result, which are briefly summarized in figure 7. The prototype was manufactured by three-dimensional printer with thermoplastic polyurethane (TPU). Connecting points of unit prisms are fabricated as a small (less than 1 mm) cylinder.

We compressed our three-dimensional printed demonstrator and measured the auxetic behaviour. The video for auxetic compression is provided as electronic supplementary material. By measuring the *L*_{x}, *L*_{z} from our measurement, we can find out the experimentally observed Poisson’s ratio. We plotted *L*_{x} versus Poisson’s ratio for theoretical calculation and experimental measurement and inserted as figure 7*e*. The differences between theoretical and experimental results are from the following reasons. Our demonstrated sample is made of TPU, which is flexible. If the prototype is not flexible, i.e. rigid body, it is not compressible but broken while we compress it. Connecting points of unit cubes are printed as small cylinder, not exactly zero but less than 1 mm, differing from our theoretical model. While we compress the sample vertically, the frictional force between the compressing plate and sample cubes prevents the sample auxetically compress in the lateral direction.

## 4. Regular triangular prism configuration

Next, we consider the three-dimensional structure, constructed in regular triangular-prism configuration as depicted in figure 8*a*. The structure is constructed by stacking up regular triangular prism-shaped unit cells shown in figure 8*b*. Each unit cell is constructed by connecting eight regular triangular prisms as depicted in figure 8*c*. We note that the upper three component triangular prisms are connected at three vertices, depicted as small circles in figure 8*c*, and that the lower three components prisms are similarly connected. The upper three and the lower three prisms are then connected at another three vertices, depicted as small triangles in figure 8*c*, forming a unit cell. So constructed unit cells are then connected on the three lines depicted as small rectangles in figure 8*c*, in the process of stacking up to form the desired structure in figure 8*a*.

We note that the geometrical shape of the unit cell is described with two lengths *a* and *b* as illustrated in figure 8*d*. We denote by *A*,*B*,*C*,*D*,*E*,*F* the vertices of the unit cell in its fully compressed position as illustrated in figure 8*b*. In the same manner, we denote by *A*′,*B*′,*C*′,*D*′,*E*′,*F*′ the vertices of the unit cell in stretched position as illustrated in figure 8*c*. We note that the deformation is determined by the characteristic angle *θ*, which is defined as the angle between a cross-section of triangular prism and the *x*-axis as shown in figure 9*b*. The six component prisms in each unit cell are assumed to deform symmetrically as shown in figures 8*c* and 9*a*. We observe, in figure 9, that the base lengths *L*_{x} and *L*_{z} are determined by the angle *θ*. In particular, we note that

When the angle gets bigger from *θ* to *θ*+Δ*θ*, the strains *ϵ*_{x} and *ϵ*_{z} are given by
*ν*_{zx} is given by

We next discuss the behaviour of Poisson’s ratio as a function of geometrical parameters *a* , *b* and angle *θ*. We observe, in equation (4.5), that Poisson’s ratio of the structure is scale independent. Consequently, we plot the Poisson’s ratio with respect to *θ* for various values of *b*/*a*. Figure 10*a* illustrates the Poisson’s ratio as a function of *θ* for five different values of *b*/*a*. We note that the right-hand side of (4.5)is a product of two fractions. Since the first fraction is positive for any *a*,*b* and *θ* values, the auxeticity of the structure is determined by the second fraction. We note that the structure exhibits auxetic behaviour for *θ* values between 0 and *θ*_{U} gets smaller when *b*/*a* becomes larger and that the absolute value of the Poisson’s ratio *ν*_{zx} for the same *θ* gets bigger for larger value of *b*/*a* in the auxetic region. For additional information about the result of the deformation, the volume fraction, which is defined as the proportion of the material in a triangular-prism unit, is illustrated in figure 10.

Next, to quantify the stiffness of the structure, we derive the on-axis Young’s moduli. For simplicity, we assume that torsional springs are located at the vertices and edges, respectively, depicted as small triangles and small rectangles (figure 8*c*). The vertices represented as small circles are assumed only to support, without torsional springs, the connections between prisms. Therefore, per each unit cell, the number of vertices and the number of edges working as torsional springs are both three. The amount of energy Δ*W* required to change the angle from *θ* to *θ*+Δ*θ* is given by
*κ*_{l} and *κ*_{p} indicate the torsion coefficient of the springs located, respectively, at the edges depicted as small rectangles and at the vertices depicted as small triangles in figure 8*c*. For regular triangular prism, the volume of the unit cell depicted in figure 9 is given by *E*_{i}(*i*=*x*,*z*), which are given by

Consequently, the Young’s moduli in the *x*- and *z*-axes are given by
*c*,*d*, respectively, with *a*=1, *κ*_{l}=*κ*_{p}=1.

In the regular triangular prism structure, the Poisson’s ratios have transitions from negative to positive or vice versa. In this structure, only the case of *dL*_{z}/*dθ*=0 is possible, not *dL*_{x}/*dθ*=0. If we consider the case of *dL*_{z}/*dθ*=0, which is in the denominator of *ν*_{zx}, the value of Poisson’s ratio approaches to *a*). For the Young’s moduli (*E*_{x}, *E*_{z}), (*dL*_{z}/*dθ*=0)^{2} is in the denominator of *E*_{z} (see equation (4.7)), thus *d*), where the more stretching in the *z*-direction cannot make any further change in angle *θ*.

## 5. Regular hexagonal prism configuration

Finally, we consider the three-dimensional structure, constructed in regular hexagonal-prism configuration as depicted in figure 11*a*. The structure is constructed by stacking up regular hexagonal prism-shaped unit cells shown in figure 11*b*. Each unit cell is constructed by connecting 12 regular hexagonal prisms as depicted in figure 11*c*. The upper six component hexagonal prisms are connected at six vertices red in figure 11*c* and the lower six components prisms are similarly connected. The upper six and the lower six prisms are then connected at another six vertices depicted as small hexagons in figure 11*c*, forming a unit cell. So constructed unit cells are then connected at still another six vertices depicted as small hexagons in figure 11*c*, which finally results in the desired structure in figure 11*a*.

The geometrical shape of the unit cell is again described by two parameters *a* and *b* illustrated in figure 11*d*. We denote by *A*,*B*,*C*,*D*,*E*,*F*,*G*,*H*,*I*,*J*,*K*,*L* the vertices of the unit cell in its fully compressed position as illustrated in figure 11*b*. Similarly, we denote by *A*′,*B*′,*C*′,*D*′,*E*′,*F*′,*G*′,*H*′,*J*′,*K*′,*L*′ the vertices of the unit cell in the stretched position as illustrated in figure 11*c*. We note that the deformation is determined by the characteristic angle *θ*, which is defined as the angle between a cross-section of the hexagonal prism and the *x*-axis as shown in figure 12*b*. The 12 component prisms in each unit cell are assumed to deform symmetrically. We observe that the base lengths *L*_{x} and *L*_{z} in figure 12 are determined by the angle *θ*. In particular, we note that

When the angle gets bigger from *θ* to *θ*+Δ*θ*, the strains *ϵ*_{x} and *ϵ*_{z} are given by
*ν*_{zx} is given by

We next discuss the behaviour of Poisson’s ratio as a function of geometrical parameters *a* , *b* and angle *θ*. We observe in equation (5.5) that Poisson’s ratio of the structure is again scale independent. Hence, we plot the Poisson’s ratio with respect to *θ* for various values of *b*/*a*. Figure 13*a* illustrates the Poisson’s ratio as a function of *θ* for six different values of *b*/*a*. We note that the right-hand side of equation (5.5) is a product of two fractions. Again, the first fraction is positive for any *a*,*b* and *θ* values and hence the auxeticity of the structure is determined by the second fraction. By comparing the second fraction of the right-hand side of equation (5.5) with the corresponding fractions in equations (4.5) and (3.5), we can easily predict that the structure considered in this section has slightly different auxetic behaviour compared with the structures considered in the previous two sections. For example, if *θ*. The volume fraction, which is defined as the proportion of the material in a hexagonal-prism unit, is illustrated in figure 13*b*.

Next, to quantify the stiffness of the structure, we derive the on-axis Young’s moduli. For simplicity, we assume that torsional springs are located at the vertices depicted represented as a small hexagon in figure 11*c*. The vertices represented as small circles are assumed only to support, without torsional springs, the connections between prisms. Therefore, per each unit cell, the total number of vertices working as torsional springs is 12. The amount of energy Δ*W* required to change the angle from *θ* to *θ*+Δ*θ* is given by
*E*_{i}(*i*=*x*,*z*) as

Consequently, the Young’s moduli in the *x*- and *z*-axes are given by
*c*,*d* respectively, with *a*=1, *κ*_{p}=1.

In the regular hexagonal prism structure, the Poisson’s ratios have transitions when Δ*L*_{i}, i.e. one of the two derivatives, *dL*_{i}/*dθ* are equal to zero. In this structure, both cases of *dL*_{z}/*dθ*=0 and *dL*_{x}/*dθ*=0 are possible. If we consider the case of *dL*_{z}/*dθ*=0, which is in the denominator of *ν*_{zx}, the value of Poisson’s ratio approaches to *a*). For the Young’s moduli (*E*_{x}, *E*_{z}), (*dL*_{z}/*dθ*=0)^{2} is in the denominator of *E*_{z} (see equation (5.7)), thus *d*), where the more stretching in the *z*-direction cannot make any further change in angle *θ*. For *E*_{x}, (*dL*_{x}/*dθ*=0)^{2} is in the denominator of *E*_{x} (see equation (5.7)), thus *c*). In particular, when *dL*_{x}/*dθ*=0, *dL*_{z}/*dθ*=0.

## 6. Conclusion

In this paper, we proposed three-dimensional structures constructed with regular triangular prisms and regular hexagonal prims in addition to regular square prisms. We derived Poisson’s ratio formula for each of the three structures and showed, by analysis and numerical verification, that the structures exhibit auxetic behaviours under certain geometrical conditions. We suggested various shapes for auxetic structures, such as regular triangular, square and hexagonal prism configurations. For the high-impact resistance, for example, higher volume fraction after compression may be more useful. If we fully compress our auxetic structures (*θ*=0), the final volume fractions are 0.75, 1 and 0.67 for regular triangular, square and hexagonal prism configurations, respectively, independently of base length *a* or height *b*. In the view of the benefits associated with the negative Poisson’s ratio behaviour, we hope that this research will stimulate future work and be applied to new exciting technological applications.

## Data accessibility

All data are completely provided in the paper.

## Authors' contributions

K.K. conceived the idea of this study. J.K. designed and performed numerical evaluation and experiments. D.-S.Y. assisted in designing the structure. D.-S.Y. and K.K. helped in the analysis of the structures. J.K., D.-S.Y. and K.K. edited the manuscript.

## Competing interests

We declare we have no competing interests.

## Funding

This research was supported by Low Observable Technology Research Center program of the Defense Acquisition Program Administration and Agency for Defense Development.

## Acknowledgements

The authors would like to express their gratitude to referees for providing helpful comments.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3790000.

- Received December 20, 2016.
- Accepted May 18, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.