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A systematic approach to identify cellular auxetic materials

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Smart Materials and Structures Smart Mater. Struct. 24 (2015) 025013 (10pp)

doi:10.1088/0964-1726/24/2/025013

A systematic approach to identify cellular auxetic materials Carolin Körner and Yvonne Liebold-Ribeiro Chair of Metals Science and Technology, University of Erlangen-Nuremberg, Martensstraße 5, D-91058 Erlangen, Germany E-mail: [email protected] Received 15 July 2014, revised 2 November 2014 Accepted for publication 13 November 2014 Published 19 December 2014 Abstract

Auxetics are materials showing a negative Poisson’s ratio. This characteristic leads to unusual mechanical properties that make this an interesting class of materials. So far no systematic approach for generating auxetic cellular materials has been reported. In this contribution, we present a systematic approach to identifying auxetic cellular materials based on eigenmode analysis. The fundamental mechanism generating auxetic behavior is identified as rotation. With this knowledge, a variety of complex two-dimensional (2D) and three-dimensional (3D) auxetic structures based on simple unit cells can be identified. Keywords: auxetic material, negative Poisson’s ratio, periodic cellular structure 1. Introduction

approach is rather successful, whereas the identification of three-dimensional (3D) structures is very difficult and far from a systematic approach. Many of the 3D structures are realized by stacking two-dimensional (2D) auxetic planes or 3D reentrant unit cells, resulting in highly anisotropic or complex structures. A systematic investigation of the structures known from the literature was performed by Elipe and Lantada [22]. The realization and optimization of 3D structures especially is a challenge [24]. To what extent the available auxetic structures can be still improved is unclear, as well as the question of whether there are other auxetic structures that have not yet been discovered. In this contribution, we present a systematic approach to identify cellular auxetic materials. The starting point is the observation that the well-known 2D quadratic chiral lattice structure, which shows full auxetic behavior [24, 25], is an eigenmode of the quadratic lattice (see figure 1). Thus, the eigenmodes of basic cellular structures (triangle, square, hexagon, cube) with periodic boundary conditions are determined. Subsequently, the eigenmodes are assembled to form periodic lattices and numerically tested to determine the Poisson’s ratio. A systematic analysis of the structures showing negative Poisson’s ratios reveals the underlying mechanism leading to auxetic behavior. Nearly all auxetic structures known from the literature emerge by this simple approach. In addition, a criterion for the identification of auxetic structures based on eigenmode analysis of simple unit cells is derived.

Symmetry is a fundamental organizing principle in nature and one relevant aspect in science and technology. Hexagons, squares, and triangles, as well as other highly symmetric forms, occur in innumerable scientific problems and technological applications, including advanced structural components [1–7]. One central focus of structural engineering deals with the artificial behavior of media known as metamaterials, whose unusual properties open up a new generation of photonic, electromagnetic, and phononic applications [8, 9]. Recently, the metamaterial concept has been extended to materials showing novel mechanical behavior [10]. In particular, the so-called auxetic materials exhibit the unusual mechanical property of having a negative Poisson’s ratio, which means that the structure expands perpendicular to an applied tension instead of shrinking [11–13]. Generally, auxetic behavior is correlated with the deformation properties of reentrant or chiral structural elements. Auxetic structures have received increasing attention for applications in molecular scales of crystallizing systems or chemical reactions [14, 15] and also in larger scales for energy and sound damping structures, aerospace filler foams, and biomedical implants [16–20]. To date there is no systematic approach known from the literature to identify auxetic structures. Generally, a new structure is conceived and subsequently tested in order to demonstrate auxetic behavior [13, 21]. In two dimensions this 0964-1726/15/025013+10$33.00

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© 2015 IOP Publishing Ltd Printed in the UK

Smart Mater. Struct. 24 (2015) 025013

C Körner and Y Liebold-Ribeiro

Figure 1. (a) Quadratic lattice and chiral eigenmode, (b) Chiral quadratic lattice and its deformation behavior showing auxetic behavior.

Figure 2. Basic structures: hexagonal (a), quadratic (b), triangular (c), and cubic (d), which can be patterned in the x and y directions to form a periodic cellular lattice.

2. Methods

displacements obtained. In addition, the deformation mechanism of the respective eigenmode is analyzed.

Eigenmode analysis for four fundamental lattice types (figures 2(a)–(d)) is performed with periodic boundary conditions in order to obtain periodic lattices, using the software package Abaqus 6.13. Note that, due to the free boundary of the outer struts of unrestricted structures, simulations using this structure type lead to different results from the ones obtained with the periodic boundary conditions. The hexagonal, quadratic, and triangular structure is investigated in 2D and also quasi 3D by giving the struts a thickness in the zdirection. The geometries are defined by the distance between the nodes L and the strut thickness t (hexagon: L = 2.5 mm; square and triangle: L = 5.0 mm, t = 0.25 mm). Material parameters for titanium are used (Young’s modulus E = 110 GPa, Poisson’s ratio ν = 0.32, density ρ = 4506 kg m−3). A minimum of three quadratic triangular and hexagonal elements was chosen for the mesh of the struts in the thickness direction, using Abaqus 6.13. The dimensions of the 2D hexagonal, quadratic, and triangular lattices used for the simulated compression tests are 6 × 7 basic cell elements. The 3D cubic structure is patterned with 3 × 3 × 3 basic elements. The simulation conditions are performed in order to reproduce experiments using a uniaxial tensile machine. After eigenmode analysis, the lowest eigenmodes are assembled to form a periodic lattice whose mechanical behavior is evaluated in the x and y directions by a numerical compressing test using Abaqus 6.13. The Poisson’s ratio for the different directions is determined according to: υxy = −Δy / Δx, with ‘x’ being the direction of uniaxial compression and ‘y’ the

3. Results Figure 3 shows the first eigenmodes of the hexagonal, quadratic, and triangular 2D basic cell, assuming periodic boundary conditions. We note that the number of modes increases with the number of nodes in the basic cell element. In particular, the number of nodes for the triangular, quadratic, and hexagonal basic elements are 3, 4, and 6, respectively, which explains the dependence of the number of modes on the unit cell types observed in figure 3. For each lattice cell, these eigenmodes were assembled to form a periodic lattice structure. Numerical testing occurred by applying a negative strain in the x- or y-direction. Only lattices showing at least one negative Poisson’s value are depicted in figure 4. The absolute value of the Poisson’s ratio is a function of the strut thickness and the amplitude of the mode. These parameters are not discussed here but can be used to optimize materials properties. For the hexagonal cell, the cellular structures from the 1st three and the 10th eigenmodes show auxetic behavior (figures 4(a)–(d)). Thorough inspection of these structures shows that eigenmodes 2 and 3 are well known from the literature: the 2nd one as chiral and the 3rd one as chiral hexagonal or chiral circular cellular lattices [22]. Analyzing the quadratic unit cell results in auxetic lattices produced from eigenmodes 3–5 and 9 (figures 4(e)–(g)). 2



Figure 3. First eigenmodes of the basic 2D structures with periodic boundary conditions. (a) hexagon, (b) square, (c) triangle.

Again, these auxetic lattices are very familiar. The reentrant sinusoidal or square grid or chiral square symmetric emerges from eigenmode 3. From the 9th eigenmode we get some structure related to the lozenge grid square [22, 23, 26]. Moreover, it can be seen from figure 4(g) that a shearing deformation occurs. This effect results from the existence of a tension-shearing coupling, which is consistent with the occurrence of non-nil off-diagonal components in the compliance matrix associated with the tetrachiral structure (see reference [27]). Furthermore, as stressed by Bacigalupo and Gamboratta [28], the elastic behavior of the tetrachiral structure exhibits a mechanical performance with properties strongly dependent upon the direction and is auxetic for a narrow range of orientations. The structure from modes 4 and 5 is a mixture of the chiral square and lozenge grid square [22]. Most of the lattices produced from the eigenmodes of the triangle are not auxetic. Only the 8th eigenmode shows a negative Poisson’s ratio and is known as rotachiral [22]. The eigenmodes of the 3D cubic structure, the side views, and the Poisson’s ratios are depicted in figure 5. Some of the structures show partial auxetic behavior; i.e., the Poisson’s ratio has some negative values (figures 5(b), (f)– (h)). The eigenmodes which display full auxetic behavior with all the Poisson’s values being negative are presented in figures 5(c)–(e) and (i). These 3D structures are not known in

the literature. The compression behavior of selected partial and full auxetic structures is depicted in figure 6.

4. Discussion 4.1. Analyzing eigenmode shapes and the auxetic mechanism

It has been demonstrated that periodic lattices formed from eigenmodes of basic unit cells often show auxetic behavior. In 2D, the resulting lattices are well known in the literature, where they have been proposed. That is, a multitude of quite complex auxetic lattices follows simply by eigenmode analysis of basic unit cells. This is remarkable and opens the possibility for a systematic approach to identify auxetic lattices in 2D and also in 3D. In the following, we want to investigate why some of the eigenmodes show auxetic behavior and others do not. For this purpose, the eigenmodes of the auxetic structures from figure 4 are depicted in figure 7. From figure 7 it is apparent that the auxetic modes show a high number of centers of rotation that may be nodal points or midpoints of struts. In addition, the number of rotational centers correlates to the basis symmetry of the lattice—six for the hexagonal and four for the quadratic one. In the case of the triangular structure, six rotation centers appear in the second-order bending mode, because a first-order symmetric 3



Figure 4. Periodic lattices from the eigenmodes of figure 3 that show auxetic behavior.

structure is not possible for a triangular basic lattice [5]. It is important to note that non-auxetic modes either do not show rotational centers (e.g., Mode 7 in figure 3(b), discussed in [29]), or the number of centers does not represent the lattice symmetry. Furthermore, it is important to note that other

structures such as, for instance, the accordion cellular honeycomb structure analyzed in reference [30], display different behavior. In particular, this structure consists of a mixture of two different types of unit cells, namely, the reentrant hexagon (auxetic honeycomb) unit cell and the regular (non4



Figure 5. Eigenmodes and Poisson’s ratio of the cubic cell.

5



Figure 5.

(Continued.)

Moreover, if the rotation is restricted to one plane but has full 3D character, the eigenmode shows only partial auxetic behavior (figures 5(b) and (e)). Analyzing (b) shows that only four of the 12 struts have rotational centers at the strut midpoints. In addition, considering eigenmodes 5(f)–(h), we find that all eight nodal points are rotational centers, but the rotation is limited to one plane 5(f) and two planes 5(g) and 5 (h). The coupling between the 3D structure and the resulting negative compression is compared in table 1. The finding that rotation is a fundamental deformation mechanism observed for auxetic structures is well-known

auxetic) hexagon. The resulting deformation behavior of the structure is not auxetic, as shown by reference [30]. A similar result follows from the analysis of the 3D structures in figure 5. The structure that displays full auxetic behavior with all the Poisson’s values being negative is the one showing rotation centers at the nodes or strut midpoints in all Cartesian directions (figures 5(c)–(e) and (i)). In addition, rotation can be combined with translation, as it shows the eigenmode in figure 5(c). The rotation direction can also be along the diagonal as in the eigenmode shape of figure 5(i). 6



Figure 6. Compression behavior of selected partial and full auxetic 3D structures: 3 × 3 × 3 unit cells with free boundary conditions on the left and right side.

Figure 7. Eigenmodes leading to auxetic behavior. RL and RR denote centers of rotations where the index L and R indicates whether it is a left- or right-handed rotation with respect to the basic lattice.

from the literature [31]. Nevertheless, this finding allows us to formulate a systematic procedure to identify auxetic lattices based on simple unit cells:

(4) Assembling of the eigenmode to a periodic lattice. This method is the first systematic approach to identify auxetic structures. It is especially useful for 3D structures.

(1) Definition of a space-filling unit cell. (2) Eigenmode analysis of this unit cell with periodic boundary conditions. (3) Identification of eigenmodes with a high number of rotational centers representing the symmetry of the lattice.

4.2. Auxetic reentrant structures

Until now only rotation as an underlying deformation mechanism has been discussed. But there is another class of auxetic materials known in the literature—reentrant structures. 7



Table 1. 3D structures and their architecture, showing partial and full auxetic behavior.

Auxetic behavior in Cartesian directions

3D structure

Architecture

Figure 5(b)

2D in one plane (xy); ABAB stacking through rotation of 45° around z-axis: rotation centers only in four struts in z- direction Full 3D in three planes; equal rotation centers in all 12 struts Full 3D in three planes; equal rotation centers in all eight nodes Full 3D in three planes; different rotation directions of all eight nodes 2D quadratic chiral in one plane (yz); equal rotation of eight nodes only in yz-plane; AAA stacking in x-direction 3D in two planes (xy, yz); equal rotation of eight nodes only in xy- and yz-plane Full 3D in three planes, rotation of all eight nodes around space diagonal

figure figure figure figure

5(c) 5(d) 5(e) 5(f)

figures 5(g), (h) figure 5(i)

partial in xy-, xz- and yx-plane full full full, anisotropic partial in yz-plane partial in xy- and yz-plane full, isotropic

From simple 2D-eigenmode analysis, reentrant structures do not directly emerge. This is not astonishing, since the strut length of the reentrant structures is different for different struts. Nevertheless, reentrant cellular structures are also an output of our new approach. To demonstrate this, we consider the eigenmodes of 3D planar cellular structures. Figure 9 shows eigenmodes with fixed rotational points in the middle of the struts, where the vibration is in the 3rd direction, i.e., out of plane. Due to the out-of-plane vibration, these structures gain extension in the 3rd direction. From above, the lattices still show their basic hexagonal or quadratic structure (see figure 9, center). Only the inclined view reveals reentrant structures: the reentrant honeycomb and the double-arrow structure for the hexagonal and quadratic lattices, respectively (see figure 9, right). That is, the reentrant structures also emerge from eigenmode analysis and result from eigenmodes with fixed rotational centers. In this case, the different strut length is caused by a projection of the 3D structure onto a 2D plane.

Figure 8. Left: Inverse honeycomb; right: double arrow structure.

The most prominent one is the 2D inverse honeycomb structure, which is based on the hexagon [22]. The double arrow structure is a derivative of the quadratic structure (see figure 8).

Figure 9. 3D planar cellular structures and out-of-plane eigenmode. R denotes centers of rotations with respect to the basic lattice where the index indicated the rotation plane. (a) 3D reentrant hexagonal structure, (b) 3D reentrant quadratic structure [32]. 8



Figure 10. ABAB-stacking of the 3D reentrant (a) hexagon and (b) square.

Most of the 3D auxetic lattices known in the literature are obtained by the stacking of planes. Coupling between planes can occur in different ways. The easiest way is by straight struts. The result for ABAB stacking is depicted in figure 10 and was already described for the hexagonal one in reference [13]. Comparing the Poisson’s ratio of the different directions demonstrates an anisotropic deformation behavior. Taking the hexagon as the basis leads to partial auxetic behavior, whereas the quadratic structure is fully auxetic. Until now we have demonstrated that eigenmode analysis of basic structures and careful selection of eigenmodes leads in a very efficient way to auxetic materials. Now we want to answer why this proceeding is successful. Eigenmodes of a basic structure describe all possible deformations of the structure. That is, an arbitrary deformation can be represented as the sum of deformations, each proportional to the eigenmodes. Choosing one eigenmode as a basic structure has the consequence that this eigenmode will be a preferred deformation mode. That is, rotation will be the deformation mode if we select the eigenmode with a high number of rotational centers, representing the symmetry of the lattice.

2D reentrant structure types can be identified as a projection of a rotation. Our approach is especially useful for identifying 3D auxetic structures and should now be extended in order to further analyze Bravais lattices, thus opening up the possibility of identifying and analyzing novel materials with auxetic behavior. Moreover, additive manufacturing techniques open up the possibility to produce complex periodic auxetic structures, also in the microfield. Indeed, important challenges for the production of such structures are the reduction of the resulting strut thickness and roughness as well as the development of reliable and time-efficient manufacturing procedures, which constitute essential aspects for industrial applications.

5. Conclusion

References

Acknowledgments We would like to acknowledge funding by the German Research Council (DFG) which, within the framework of its ‘Excellence Initiative,’ supports the Cluster of Excellence ‘Engineering of Advanced Materials’ at the University of Erlangen-Nuremberg.

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