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Asystematicapproachtoidentifycellularauxeticmaterials

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A systematic approach to identify cellular auxetic materials

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2015 Smart Mater. Struct. 24 025013

(http://iopscience.iop.org/0964-1726/24/2/025013)

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A systematic approach to identify cellularauxetic materials

Carolin Körner and Yvonne Liebold-Ribeiro

Chair of Metals Science and Technology, University of Erlangen-Nuremberg, Martensstraße 5, D-91058Erlangen, Germany

E-mail: [email protected]

Received 15 July 2014, revised 2 November 2014Accepted for publication 13 November 2014Published 19 December 2014

AbstractAuxetics are materials showing a negative Poisson’s ratio. This characteristic leads to unusualmechanical properties that make this an interesting class of materials. So far no systematicapproach for generating auxetic cellular materials has been reported. In this contribution, wepresent a systematic approach to identifying auxetic cellular materials based on eigenmodeanalysis. The fundamental mechanism generating auxetic behavior is identified as rotation. Withthis knowledge, a variety of complex two-dimensional (2D) and three-dimensional (3D) auxeticstructures based on simple unit cells can be identified.

Keywords: auxetic material, negative Poisson’s ratio, periodic cellular structure

1. Introduction

Symmetry is a fundamental organizing principle in nature andone relevant aspect in science and technology. Hexagons,squares, and triangles, as well as other highly symmetricforms, occur in innumerable scientific problems and techno-logical applications, including advanced structural compo-nents [1–7]. One central focus of structural engineering dealswith the artificial behavior of media known as metamaterials,whose unusual properties open up a new generation of pho-tonic, electromagnetic, and phononic applications [8, 9].

Recently, the metamaterial concept has been extended tomaterials showing novel mechanical behavior [10]. In parti-cular, the so-called auxetic materials exhibit the unusualmechanical property of having a negative Poisson’s ratio,which means that the structure expands perpendicular to anapplied tension instead of shrinking [11–13]. Generally,auxetic behavior is correlated with the deformation propertiesof reentrant or chiral structural elements. Auxetic structureshave received increasing attention for applications in mole-cular scales of crystallizing systems or chemical reactions[14, 15] and also in larger scales for energy and sounddamping structures, aerospace filler foams, and biomedicalimplants [16–20].

To date there is no systematic approach known from theliterature to identify auxetic structures. Generally, a newstructure is conceived and subsequently tested in order todemonstrate auxetic behavior [13, 21]. In two dimensions this

approach is rather successful, whereas the identification ofthree-dimensional (3D) structures is very difficult and farfrom a systematic approach. Many of the 3D structures arerealized by stacking two-dimensional (2D) auxetic planes or3D reentrant unit cells, resulting in highly anisotropic orcomplex structures. A systematic investigation of the struc-tures known from the literature was performed by Elipe andLantada [22]. The realization and optimization of 3D struc-tures especially is a challenge [24]. To what extent theavailable auxetic structures can be still improved is unclear, aswell as the question of whether there are other auxeticstructures that have not yet been discovered.

In this contribution, we present a systematic approach toidentify cellular auxetic materials. The starting point is theobservation that the well-known 2D quadratic chiral latticestructure, which shows full auxetic behavior [24, 25], is aneigenmode of the quadratic lattice (see figure 1). Thus, theeigenmodes of basic cellular structures (triangle, square,hexagon, cube) with periodic boundary conditions are deter-mined. Subsequently, the eigenmodes are assembled to formperiodic lattices and numerically tested to determine thePoisson’s ratio. A systematic analysis of the structuresshowing negative Poisson’s ratios reveals the underlyingmechanism leading to auxetic behavior. Nearly all auxeticstructures known from the literature emerge by this simpleapproach. In addition, a criterion for the identification ofauxetic structures based on eigenmode analysis of simple unitcells is derived.

Smart Materials and Structures

Smart Mater. Struct. 24 (2015) 025013 (10pp) doi:10.1088/0964-1726/24/2/025013

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protected]://dx.doi.org/10.1088/0964-1726/24/2/025013http://crossmark.crossref.org/dialog/?doi=10.1088/0964-1726/24/2/025013&domain=pdf&date_stamp=2014-12-19http://crossmark.crossref.org/dialog/?doi=10.1088/0964-1726/24/2/025013&domain=pdf&date_stamp=2014-12-19https://www.researchgate.net/publication/281549254_Influence_of_negative_Poisson's_ratio_on_stent_applications?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/262305658_Design_of_materials_with_prescribed_nonlinear_properties?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/262305658_Design_of_materials_with_prescribed_nonlinear_properties?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/261031897_Complex_Ordered_Patterns_in_Mechanical_Instability_Induced_Geometrically_Frustrated_Triangular_Cellular_Structures?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/258060700_Surfactant_free_most_probable_TiO2_nanostructures_via_hydrothermal_and_its_dye_sensitized_solar_cell_properties?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/245373727_Structural_and_Drug_Diffusion_Models_of_Conventional_and_Auxetic_Drug-Eluting_Stents?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/235029695_Acoustic_Behaviour_of_Negative_Poisson's_Ratio_Materials?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/232722486_Vortex_dynamics_in_triangular-shaped_confining_potentials?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/231007652_Effects_of_inclusion_shapes_on_the_band_gaps_in_two-dimensional_piezoelectric_phononic_crystals?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/229919561_Auxetic_cellular_structures_through_selective_electron-beam_melting?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/229919561_Auxetic_cellular_structures_through_selective_electron-beam_melting?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/228594417_Selective_Laser_Melting_of_Honeycombs_with_Negative_Poisson's_Ratio?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/223052587_Heat_Transfer_Efficiency_of_Metal_Honeycombs?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/51622034_An_Auxetic_structure_configured_as_oesophageal_stent_with_potential_to_be_used_for_palliative_treatment_of_oesophageal_cancer_Development_and_in_vitro_mechanical_analysis?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/50908536_Honeycomb-Structured_Silicon_Remarkable_Morphological_Changes_Induced_by_Electrochemical_DeLithiation?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/50248152_Performance_Evaluation_of_Auxetic_Molecular_Sieves_with_Re-Entrant_Structures?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/27254680_Literature_Review_Materials_with_Negative_Poisson's_Ratios_and_Potential_Applications_to_Aerospace_and_Defence?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/profile/Carolin_Koerner?el=1_x_100&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/profile/Yvonne_Liebold-Ribeiro?el=1_x_100&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==

2. Methods

Eigenmode analysis for four fundamental lattice types(figures 2(a)–(d)) is performed with periodic boundary con-ditions in order to obtain periodic lattices, using the softwarepackage Abaqus 6.13. Note that, due to the free boundary ofthe outer struts of unrestricted structures, simulations usingthis structure type lead to different results from the onesobtained with the periodic boundary conditions. The hex-agonal, quadratic, and triangular structure is investigated in2D and also quasi 3D by giving the struts a thickness in the z-direction. The geometries are defined by the distance betweenthe nodes L and the strut thickness t (hexagon: L= 2.5 mm;square and triangle: L = 5.0 mm, t= 0.25 mm). Material para-meters for titanium are used (Young’s modulus E= 110 GPa,Poisson’s ratio ν= 0.32, density ρ= 4506 kg m−3). A mini-mum of three quadratic triangular and hexagonal elementswas chosen for the mesh of the struts in the thickness direc-tion, using Abaqus 6.13. The dimensions of the 2D hex-agonal, quadratic, and triangular lattices used for thesimulated compression tests are 6 × 7 basic cell elements. The3D cubic structure is patterned with 3 × 3 × 3 basic elements.The simulation conditions are performed in order to repro-duce experiments using a uniaxial tensile machine.

After eigenmode analysis, the lowest eigenmodes areassembled to form a periodic lattice whose mechanical behavioris evaluated in the x and y directions by a numerical compres-sing test using Abaqus 6.13. The Poisson’s ratio for the dif-ferent directions is determined according to: υxy=−Δy /Δx, with‘x’ being the direction of uniaxial compression and ‘y’ the

displacements obtained. In addition, the deformation mechan-ism of the respective eigenmode is analyzed.

3. Results

Figure 3 shows the first eigenmodes of the hexagonal,quadratic, and triangular 2D basic cell, assuming periodicboundary conditions. We note that the number of modesincreases with the number of nodes in the basic cell element.In particular, the number of nodes for the triangular, quad-ratic, and hexagonal basic elements are 3, 4, and 6, respec-tively, which explains the dependence of the number ofmodes on the unit cell types observed in figure 3.

For each lattice cell, these eigenmodes were assembled toform a periodic lattice structure. Numerical testing occurredby applying a negative strain in the x- or y-direction. Onlylattices showing at least one negative Poisson’s value aredepicted in figure 4. The absolute value of the Poisson’s ratiois a function of the strut thickness and the amplitude of themode. These parameters are not discussed here but can beused to optimize materials properties.

For the hexagonal cell, the cellular structures from the 1stthree and the 10th eigenmodes show auxetic behavior(figures 4(a)–(d)). Thorough inspection of these structuresshows that eigenmodes 2 and 3 are well known from theliterature: the 2nd one as chiral and the 3rd one as chiralhexagonal or chiral circular cellular lattices [22].

Analyzing the quadratic unit cell results in auxetic lat-tices produced from eigenmodes 3–5 and 9 (figures 4(e)–(g)).

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 aperiodic cellular lattice.

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Smart Mater. Struct. 24 (2015) 025013 C Körner and Y Liebold-Ribeiro

https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==

Again, these auxetic lattices are very familiar. The reentrantsinusoidal or square grid or chiral square symmetric emergesfrom eigenmode 3. From the 9th eigenmode we get somestructure related to the lozenge grid square [22, 23, 26].Moreover, it can be seen from figure 4(g) that a shearingdeformation occurs. This effect results from the existence of atension-shearing coupling, which is consistent with theoccurrence of non-nil off-diagonal components in the com-pliance matrix associated with the tetrachiral structure (seereference [27]). Furthermore, as stressed by Bacigalupo andGamboratta [28], the elastic behavior of the tetrachiralstructure exhibits a mechanical performance with propertiesstrongly dependent upon the direction and is auxetic for anarrow range of orientations.

The structure from modes 4 and 5 is a mixture of thechiral square and lozenge grid square [22].

Most of the lattices produced from the eigenmodes of thetriangle are not auxetic. Only the 8th eigenmode shows anegative Poisson’s ratio and is known as rotachiral [22].

The eigenmodes of the 3D cubic structure, the sideviews, and the Poisson’s ratios are depicted in figure 5. Someof the structures show partial auxetic behavior; i.e., thePoisson’s ratio has some negative values (figures 5(b), (f)–(h)). The eigenmodes which display full auxetic behaviorwith all the Poisson’s values being negative are presented infigures 5(c)–(e) and (i). These 3D structures are not known in

the literature. The compression behavior of selected partialand 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 fromeigenmodes of basic unit cells often show auxetic behavior. In2D, the resulting lattices are well known in the literature,where they have been proposed. That is, a multitude of quitecomplex auxetic lattices follows simply by eigenmode ana-lysis of basic unit cells. This is remarkable and opens thepossibility for a systematic approach to identify auxetic lat-tices in 2D and also in 3D. In the following, we want toinvestigate why some of the eigenmodes show auxeticbehavior and others do not. For this purpose, the eigenmodesof the auxetic structures from figure 4 are depicted in figure 7.

From figure 7 it is apparent that the auxetic modes showa high number of centers of rotation that may be nodal pointsor midpoints of struts. In addition, the number of rotationalcenters correlates to the basis symmetry of the lattice—six forthe hexagonal and four for the quadratic one. In the case ofthe triangular structure, six rotation centers appear in thesecond-order bending mode, because a first-order symmetric

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

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https://www.researchgate.net/publication/261673038_Homogenization_of_periodic_hexa-_and_tetrachiral_cellular_solids?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/257376345_Equivalent_mechanical_properties_of_auxetic_lattices_from_discrete_homogenization?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==

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

structures such as, for instance, the accordion cellular hon-eycomb structure analyzed in reference [30], display differentbehavior. In particular, this structure consists of a mixture oftwo different types of unit cells, namely, the reentrant hexa-gon (auxetic honeycomb) unit cell and the regular (non-

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

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https://www.researchgate.net/publication/261031897_Complex_Ordered_Patterns_in_Mechanical_Instability_Induced_Geometrically_Frustrated_Triangular_Cellular_Structures?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==https://www.researchgate.net/publication/249358146_Zero_Poisson's_Ratio_Cellular_Honeycombs_for_Flex_Skins_Undergoing_One-Dimensional_Morphing?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==

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

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auxetic) hexagon. The resulting deformation behavior of thestructure is not auxetic, as shown by reference [30].

A similar result follows from the analysis of the 3Dstructures in figure 5. The structure that displays full auxeticbehavior with all the Poisson’s values being negative is theone showing rotation centers at the nodes or strut midpointsin all Cartesian directions (figures 5(c)–(e) and (i)). Inaddition, rotation can be combined with translation, as itshows the eigenmode in figure 5(c). The rotation directioncan also be along the diagonal as in the eigenmode shape offigure 5(i).

Moreover, if the rotation is restricted to one plane but hasfull 3D character, the eigenmode shows only partial auxeticbehavior (figures 5(b) and (e)). Analyzing (b) shows that onlyfour of the 12 struts have rotational centers at the strut mid-points. In addition, considering eigenmodes 5(f)–(h), we findthat all eight nodal points are rotational centers, but therotation is limited to one plane 5(f) and two planes 5(g) and 5(h). The coupling between the 3D structure and the resultingnegative compression is compared in table 1.

The finding that rotation is a fundamental deformationmechanism observed for auxetic structures is well-known

Figure 5. (Continued.)

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

(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 thelattice.

(4) Assembling of the eigenmode to a periodic lattice.

This method is the first systematic approach to identifyauxetic structures. It is especially useful for 3D structures.

4.2. Auxetic reentrant structures

Until now only rotation as an underlying deformationmechanism has been discussed. But there is another class ofauxetic materials known in the literature—reentrant structures.

Figure 6. Compression behavior of selected partial and full auxetic 3D structures: 3 × 3 × 3 unit cells with free boundary conditions on the leftand 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 aleft- or right-handed rotation with respect to the basic lattice.

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The most prominent one is the 2D inverse honeycombstructure, which is based on the hexagon [22]. The doublearrow structure is a derivative of the quadratic structure (seefigure 8).

From simple 2D-eigenmode analysis, reentrant structures donot directly emerge. This is not astonishing, since the strut lengthof the reentrant structures is different for different struts. Never-theless, reentrant cellular structures are also an output of our newapproach. To demonstrate this, we consider the eigenmodes of3D planar cellular structures. Figure 9 shows eigenmodes withfixed rotational points in the middle of the struts, where thevibration is in the 3rd direction, i.e., out of plane.

Due to the out-of-plane vibration, these structures gainextension in the 3rd direction. From above, the lattices stillshow their basic hexagonal or quadratic structure (see figure 9,center). Only the inclined view reveals reentrant structures: thereentrant honeycomb and the double-arrow structure for thehexagonal and quadratic lattices, respectively (see figure 9,right). That is, the reentrant structures also emerge fromeigenmode analysis and result from eigenmodes with fixedrotational centers. In this case, the different strut length iscaused by a projection of the 3D structure onto a 2D plane.

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

3D structure ArchitectureAuxetic behavior in Cartesiandirections

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

partial in xy-, xz- and yx-plane

figure 5(c) Full 3D in three planes; equal rotation centers in all 12 struts fullfigure 5(d) Full 3D in three planes; equal rotation centers in all eight nodes fullfigure 5(e) Full 3D in three planes; different rotation directions of all eight nodes full, anisotropicfigure 5(f) 2D quadratic chiral in one plane (yz); equal rotation of eight nodes only in yz-plane;

AAA stacking in x-directionpartial in yz-plane

figures 5(g), (h) 3D in two planes (xy, yz); equal rotation of eight nodes only in xy- and yz-plane partial in xy- and yz-planefigure 5(i) Full 3D in three planes, rotation of all eight nodes around space diagonal full, isotropic

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

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

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https://www.researchgate.net/publication/241780228_Comparative_study_of_auxetic_geometries_by_means_of_computer-aided_design_and_engineering?el=1_x_8&enrichId=rgreq-e423983ba0f4e55f0082f171c2dd1ff5-XXX&enrichSource=Y292ZXJQYWdlOzI2OTc3NjM2ODtBUzoyOTQ2MTU3NjU3OTg5MThAMTQ0NzI1MzI4MDY4Mw==

Most of the 3D auxetic lattices known in the literature areobtained by the stacking of planes. Coupling between planescan occur in different ways. The easiest way is by straightstruts. The result for ABAB stacking is depicted in figure 10and was already described for the hexagonal one in reference[13]. Comparing the Poisson’s ratio of the different directionsdemonstrates an anisotropic deformation behavior. Taking thehexagon as the basis leads to partial auxetic behavior,whereas the quadratic structure is fully auxetic.

Until now we have demonstrated that eigenmode analysisof basic structures and careful selection of eigenmodes leadsin a very efficient way to auxetic materials. Now we want toanswer why this proceeding is successful. Eigenmodes of abasic structure describe all possible deformations of thestructure. That is, an arbitrary deformation can be representedas the sum of deformations, each proportional to the eigen-modes. Choosing one eigenmode as a basic structure has theconsequence that this eigenmode will be a preferred defor-mation mode. That is, rotation will be the deformation modeif we select the eigenmode with a high number of rotationalcenters, representing the symmetry of the lattice.

5. Conclusion

In this paper we demonstrate how eigenmode analysis of simpleunit cells—hexagon, square, and triangle—can be used toidentify a variety of 2D and 3D auxetic cellular materials. Themain mechanism for auxetic behavior is identified as a collectiverotation of either nodal points or midpoints of struts. For rea-lizing full auxetic behavior, these rotational points must repre-sent the symmetry of the lattice. Even the folding mechanism of

2D reentrant structure types can be identified as a projection of arotation. Our approach is especially useful for identifying 3Dauxetic structures and should now be extended in order to furtheranalyze Bravais lattices, thus opening up the possibility ofidentifying and analyzing novel materials with auxetic behavior.Moreover, additive manufacturing techniques open up the pos-sibility to produce complex periodic auxetic structures, also inthe microfield. Indeed, important challenges for the productionof such structures are the reduction of the resulting strut thick-ness and roughness as well as the development of reliable andtime-efficient manufacturing procedures, which constituteessential aspects for industrial applications.

Acknowledgments

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

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

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1. Introduction2. Methods3. Results4. Discussion4.1. Analyzing eigenmode shapes and the auxetic mechanism4.2. Auxetic reentrant structures

5. ConclusionAcknowledgmentsReferences

a systematic approach to identify cellular auxetic materials€¦ · present a systematic approach...

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