finding auxetic frameworks in periodic tessellations
TRANSCRIPT
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Holger Mitschke , Jan Schwerdtfeger , Fabian Schury , Michael Stingl , Carolin Körner , Robert F. Singer , Vanessa Robins , Klaus Mecke , and Gerd E. Schröder-Turk *
Finding Auxetic Frameworks in Periodic Tessellations
EWS
It appears that most models for micro-structured materials with auxetic defor-mations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent approach of fi nding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms, which have Pois-son’s ratio v ss = −1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices. One of these, baptized Triangle-Square Wheels , has been produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective Poisson’s ratio v LE ≈ −0.75, also in agreement with fi nite element modeling. The similarity between the Poisson’s ratios v SS of the skeletal structure and v LE of the linear-elastic cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.
Auxetic structures, those with a negative Poisson’s ratio, were once incorrectly considered a theoretical possibility without relevant experimental realization. [ 1 ] To date, auxetic structures
© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheiAdv. Mater. 2011, 23, 2669–2674
DOI: 10.1002/adma.201100268
H. Mitschke , Prof. K. Mecke , Dr. G. E. Schröder-Turk Institut für Theoretische PhysikFriedrich-Alexander Universität Erlangen-NürnbergStaudtstr. 7, 91058 Erlangen, Germany E-mail: [email protected] F. Schury , Prof. M. Stingl Department MathematikLehrstuhl Angewandte Mathematik IIFriedrich-Alexander Universität Erlangen-NürnbergMartenstr. 3, 91058 Erlangen, Germany Dr. J. Schwerdtfeger Friedrich-Alexander Universität Erlangen-NürnbergInstitute of Advanced Materials and Processes (ZMP)Dr.-Mack-Str. 81, 90762 Fürth, Germany Dr. C. Körner , Prof. R. F. Singer Friedrich-Alexander Universität Erlangen-NürnbergInstitute of Materials Science and Technology (WTM)Martensstr. 5, 91058 Erlangen, Germany Dr. V. Robins Applied Maths, Research School of PhysicsThe Australian National UniversityCanberra, 0200 ACT, Australia
have been found in polymeric and metal foams, [ 2 , 3 ] human cancellous bone, [ 4 ] liquid crystals, [ 5 ] carbon “buckypaper” nanotube sheets, [ 6 ] polypropylene fi lms, [ 7 ] polymer gels, [ 8 ] semi-crystalline elastomeric poly-propylene fi lms, [ 9 ] crystalline networks, [ 10 ] coulombic crystals in crystallized ion plasmas, [ 11 ] tetrahedral framework sili-cates, [ 12 ] micro-porous polymers, [ 13 , 14 ] α -cristobalites, [ 15 ] cubic metals, [ 16 , 17 ] cow skin, [ 18 ] self-avoiding membranes, [ 19 , 20 ] and in periodic planar structures realized by soft-lithography. [ 21 ] A very early report of negative Poisson’s ratio in iron pyrite is given in Voigt. [ 22 ] Related phenomena are the negative normal stress in bio-polymer networks, [ 23 ] and the dilatancy of granular media. [ 24 ] Models leading to auxetic defor-mations have also been suggested for RNA gels, [ 25 ] polymer composites with hexag-onal symmetry, [ 26 ] three-dimensional net-work or fi ber models, [ 6 , 27 , 28 ] isotropic strut frameworks, [ 29 ] string networks under ten-sion [ 30 ] and many-body systems even with
isotropic pair-wise potentials. [ 31 ] Possible technological appli-cations include enhanced shock-absorption, [ 32 ] self-cleaning fi lters, [ 33 ] tunable photonic crystal devices [ 11 ] and, if the mag-nitude of Poisson’s ratio exceeds one, strain amplifi cation. [ 34 ] Complex physical behavior beyond the mechanical properties results, e.g., phonon-dispersion [ 35 ] and wave propagation or attenuation. [ 36 , 37 ]
Auxetic materials share the common property that below a certain length scale the material is inhomogeneous. In fact, Blumenfeld’s statement “Auxeticity is the result of internal structural degrees of freedom that get in the way of affi ne defor-mations” [ 38 ] illustrates the origins of auxetic behavior. The inho-mogeneities responsible for auxetic behavior may be classifi ed loosely into two types: First, the structure may be a composite material consisting of two phases, one of which is often void and hence not load-bearing and the other load-bearing phase has an inhomogeneous micro-structure. This type is realized e.g. in skeletal structure models and solid cellular structures. The prototypical auxetic deformation in the inverted honey-comb is of this type, see Figure 1 . Importantly, because of the inhomogeneous nature of the material the effective Poisson’s ratio ν can attain smaller values ν < − 1 than the theoretical limit for isotropic homogeneous structures. [ 39 ] Second, auxetic behavior may result in materials where molecular interactions result in higher symmetry, e.g., cubic or hexagonal [ 15 – 17 ] or even full rotational isotropy [ 31 ] of the atomic confi gurations
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Figure 1 . The prototypical planar auxetic structure is the inverted honey-comb with re-entrant elements, depicted here (a) as a skeletal structure of stiff struts loosely jointed at the common vertices, and (b) realized by lithography techniques. When stretched in horizontal direction, an aux-etic structure also expands in vertical direction. The lithography image is reproduced from Figure 3 of Xu et al. [ 21 ]
and, importantly, where this symmetry remains in place even when the material is strained. In these cases, auxeticity results as a consequence of the fact that hexagonal or cubic symmetry imposes that all perpendicular strains are equal, therefore leading to a value of ν = − 1 for the Poisson’s ratio. While the symmetry constraints that lead to the auxetic behavior in these latter cases are relevant to our discussion of periodic tessella-tions below, the focus of this article are structures of the fi rst type consisting in a micro-structured composite material.
In addition to mechanisms based on polygons with re-entrant angles as their essential structural motif, other mechanisms based on rotating elements, [ 40– 42 ] interlocking elements of fi xed shape [ 43 ] or stretching of squares [ 44 ] have been proposed.
1. Skeletal Structure Models of Periodic Tessellations
A convenient geometric model for studying auxetic defor-mations are so-called skeletal structures [ 45 ] of perfect rigid rods that pivot freely at mutual joints of two or more rods, cf. Figure 1 . The deformation behavior of skeletal structures has been studied widely, [ 46 – 48 ] in particular also in the context of determi-nacy and rigidity. [ 49 – 52 ] The majority of skeletal structure models discussed as auxetic models in the literature are periodic and often also symmetric; that is, they are tessellations of the plane by polygons with straight edges.
Technically, a skeletal structure S is a graph with coordinates assigned to the nodes. A graph , denoted ( K , E ), consists of a fi nite set K of nodes and a set E of edges that is a subset of all pairs of nodes from K . Every node i corresponds to a joint, and has coordinates P i = { x i , y i }. Every edge e = { i , j }, where i and j are the indices to points in K , corresponds to a rigid bar. The length of bar { i , j } is denoted l { ij } , see also Figure 1 . The rigidity of the bars defi nes distance equations
∣∣Pi − P j
∣∣ − l{i j } = 0 ∀ {i, j } ∈ E
(1)
that must be fulfi lled. Solutions for the vertices coordinates p of the multivariate
system of quadratic equations (1) defi ne permissible confi gurations that are compatible with the stiffness of the struts. A deformation is a one-dimensional hyper-path through the space of permissible confi gurations. In this nomenclature, an applied strain along one of the periodic lattice translation vectors, say a , corresponds to cal-culating the permissible confi gurations if the lattice translation
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length a = | a | is changed by ε . Assuming that the unstrained initial tessellation is a permissible confi guration, the following scenarios for the structure upon an applied small strain are possible: (a) there is no solution for small ε ; the skeletal structure is rigid. This is e.g. the case for a triangular lattice. (b) There are multiple solutions to Equations (1); several deformation pathways exist. Poisson’s ratio is not well-defi ned. (c) A unique solution p ( ε ) exists; the structure can only be deformed in a single way.
Numerical solutions to the Equations (1) for a given skeletal structure and given applied strain ε are easily obtained by basic Newton-Raphson methods, possibly combined with singular value decomposition to deal with degeneracy.
Poisson’s ratio, that characterizes the contraction or exten-sion of a material in the vertical direction to an applied uni-axial strain, is a uniquely defi ned property of the skeletal structure only if the deformation behavior is unique. If the deformation is ambiguous, Poisson’s ratio can still be defi ned for a given deformation pathway p ( ε ) out of the continuum of solutions, e.g. p ( ε ) may be determined by minimization of an energy functional, e.g. harmonic angular springs. Poisson’s ratio is defi ned as the negative of the ratio of the vertical strain to the applied strain. If the portion of material considered is a rectangle of size l 0 × h 0 the Cauchy strain δ = ( l − l 0 )/ l 0 and X = ( h − h 0 )/ h 0 are commonly used, with Poisson’s ratio defi ned as ν = − X / δ . For periodic skeletal structures with translational unit cells that are not rectangular, the generalized defi nition given, e.g., by Mitschke et al. [ 53 , 54 ] is required.
The concepts described here have been applied to a system-atic identifi cation of new auxetic mechanisms; [ 53 , 54 ] the basic idea is to systematically analyze the deformation behavior of large classes of tessellations (which are a widely studied topic in mathematics with large numbers of tessellations enumer-ated in structural archives [ 55 ] ) with the aim of identifying those with a negative Poisson’s ratio and a unique deformation mode. This idea has to date been pursuit for two classes of planar tes-sellations, namely the 35 one- and two-uniform tessellations of the plane by regular polygons or star polygons where all cor-ners are vertices, as listed in Grünbaum and Shephard. [ 55 ] The result of this study is that the tessellation Triangle-Square-Wheels (3 6 ;3 2 .4.3.4) [ 55 ] has a unique deformation mode; Figure 2 shows this tessellation as a skeletal structure (top right) and in an alternative form with the rigid hexagonal units fi lled (top left). A further new auxetic tessellation, Hexagonal-Wheels (3 6 ;3 2 .6 2 ), with equivalent properties has been found, [ 53 , 54 ] but has not yet been realized experimentally and is not discussed here. The Poisson’s ratio is ν SS = − 1 and is constant for all strains ε that yield permissible confi gurations, due to the fact that the bar equations constrain the system to have hexagonal symmetry. Another structure with unique auxetic deformation behavior found among the above 35 tessellations is the trihexagonal tes-sellation or kagome structure (3.6.3.6) composed of rotating trian-gles that has already been discussed by Grima et al. [ 56 ]
2. Realization as Linear-Elastic Cellular Solids by Selective Electron Beam Melting
For engineering applications, the ultimate goal is the iden-tifi cation of spatial structures that are auxetic when realized
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Figure 2 . The auxetic Triangle-Square-Wheels , [ 53 , 54 ] modeled by a skeletal structure (top) and manufactured as a linear-elastic cellular structure by selective electron beam melting (middle). The difference between the left and right images are the solid hexagons. The strain-strain curves showing applied strain ε l versus resulting transversal strain ε t are determined by tensile experiments on the SEBM models (data points). The linear fi ts to these data clearly demonstrate the negative Poisson’s ratio of approxi-mately − 0.75 ± 0.1.
as frameworks of struts of a given width composed of homo-geneous linear-elastic material without explicit hinge mecha-nisms; this structure is henceforth called a linear-elastic cellular solid. The use of skeletal structures, with infi nitely-thin solid incompressible struts joint loosely at common vertices, is a means to this end that reduces the task to a simpler, more geo-metric problem that allows the systematic search. To validate this approach, we now show that the linear-elastic cellular solid corresponding to one of the identifi ed auxetic skeletal struc-tures, is indeed also auxetic.
Samples of the linear-elastic cellular solid representing the Triangle-Square-Wheels tessellation were produced using an Arcam A2 selective electron beam melting system (SEBM), see Figure 2 . In SEBM objects are built up from metal powder layer
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by layer. The geometric information for the parts is given in the form of CAD-fi les from which layer maps detailing the loca-tion of solid material are extracted. In the actual process thin layers of metal powder ( ∼ 0.1 mm) are deposited on a steel start plate and an electron beam is used to locally melt the material. A new powder layer is deposited and the process repeated until the part is completed. This approach gives almost complete freedom for the design of parts and also allows building com-plex cellular structure such as the ones discussed in the present work in a highly controlled manner. For details see e.g. Heinl et al. [ 57 ] and Schwerdtfeger et al. [ 58 ]
Poisson’s ratio was determined by mechanical tensile testing. The bone-shaped samples shown in Figure 2 , built from Ti-6Al-4 V, have approximate dimensions of 2 mm thickness, 20 mm width and 52 mm gauge length. Measurements were performed on an Instron tensile testing machine. The strain normal to the force direction was measured by an extensiometer mounted on the sample using sharp blades. The cross head position was used to determine the strain in force direction. For the low forces necessary to deform the relatively thin samples (approx. 1 kN) it can be assumed to be a good approximation of the actual elongation of the sample.
The measured Poisson’s ratio is approximately − 0.75. This is a clear indication that even when realized as a linear-elastic cel-lular solid, the Triangle-Square-Wheels tessellation is auxetic.
3. Finite Element Methods for Linear Elastic Properties
Linear elastic properties of the triangle-square wheel cellular structure were also determined by a standard fi nite element method (FEM) in combination with the so-called solid isotropic material with penalization method (SIMP). [ 59 ] In the core of these methods we assign every fi nite element a pseudo density value of 1 if a strut of the structure intersects the fi nite element or a small positive value (e.g. 10 − 6 ) otherwise. This allows for the representation of many different structures using a discreti-zation of the design domain by rectangular fi nite elements.
The full (macroscopic, or effective) linear elastic stiffness tensor is computed approximately by means of the homogeni-zation method; analytically, the homogenized elastic tensor is the result of an asymptotic limiting process, see Bendsøe and Sigmund, [ 60 ] Allaire [ 61 ] and references therein. Numerically, its properties are derived directly from the solution of three par-tial differential equations on a rectangular base cell with peri-odic boundary conditions, in which the microscopic reaction of the linear-elastic cellular structure w.r.t. given macroscopic test strains is computed.
The solutions of these partial differential equations are approximated by associated fi nite element problems in which the test strains show up as right hand sides. Thus, the test strains correspond, in a certain sense, to the forces applied in real world measurements. The solutions of the fi nite element problems are the periodic responses to these forces on the base cell.
Finally, the entries of the homogenized stiffness tensor C Hi j
are computed from these solutions by evaluation of a strain type energy formula. Poisson’s ratios ν FEM and Young’s moduli
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Figure 3 . Effect of symmetry constraints on the deformation behavior of periodic skeletal structures: The skeletal structure (shown in black thick lines and dots) is the Great Rhombitrihexagonal Tiling (4.6.12) . (a,b) It is rigid in its most symmetric embedding ( p 6 mm ) and in one of its sub-groups ( p 3 m 1). (c,d) Constraining symmetry to p 6 and p 31 m yields two distinct deformations, each unique and with ν ( δ ) = − 1. (e & f) Some sub-groups, including cm , p 2 and p 1, yield ambiguous deformations; in (e,f) those minimizing the harmonic functional
∑["i j k − "0
i j k ]2 of the edge angles "i j k at vertices are shown. In (c-f) the left images represent δ = 0 and the right ones δ = − 0.2. Green lines and symbols represent symmetry elements as in Hahn et al. [ 63 ]
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δ = 0 δ =(c) p6 – unique
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δ = 0 δ = − 0.2(f) p1 – ambiguous
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0.2
0.2−
E FEM are a simple function of the entries of C Hi j and can be
computed in a straightforward way. [ 62 ] Applying this procedure to the two structures presented
in Figure 2 with microscopic elastic moduli E = 114 GPa and ν = 0.369 corresponding to Ti-6Al-4V, this FEM analysis yields almost isotropic material behavior. The Poisson’s ratio is found to be ν FEM = –0.72635 for the structure with hollow hexagonal wheels, and ν FEM = –0.73026 for the structure with solid hexa-gons, both in approximate agreement with the measurements on the SEBM models.
4. Skeletal Structures with Symmetry-Constraint Deformations
The analysis of Poisson’s ratio ν SS of a skeletal structure is only well-defi ned if the deformation mode is unambiguous. Unfor-tunately, the deformation of most tessellations is not unique. On the other hand, when realized as a linear-elastic cellular solid these tessellations have unambiguous deformations and well-defi ned Poisson’s ratio. It is hence an interesting exercise to investigate the imposition of additional constraints on the deformation of skeletal structures that make the deformations unique. We here review the possibility to enforce unique defor-mation behavior of skeletal structures by constraining sym-metries during the deformation.
As a striking example, the inverted honeycomb in Figure 1 is a structure with an ambiguous deformation mode. If the hori-zontal strain is applied by pulling horizontally on a single one of the left-most and a single one of the right-most vertical edges, the deformation is ambiguous and may yield deformed structures without the horizontal mirror symmetry. The com-monly depicted deformation [ 13 , 19 , 21 , 32–34 ] is obtained if one enforces that the structure retains the horizontal mirror planes or, equivalently, if one prohibits shear modes of the structure.
The symmetry of a skeletal structure is encoded by its sym-metry group G that contains all symmetry operations that map the structure onto itself. If the symmetry group G is chosen to be p 1 (in the common notation of Hahn et al. [ 63 ] ) the skeletal structure is only constraint to maintain the periodicity of the tessellation. On the other hand, G can be chosen as the maxi-mally symmetric group G max that includes all symmetry opera-tions of the initial tessellation. In between these extremes, G can be chosen as one of the subgroups of G max , that is, during the deformation some but not all symmetry relations remain in place. Often highly symmetric skeletal structures are rigid when constraining most or all of the symmetries, have one or a number of subgroups where the deformation is unique, and have ambiguous deformation modes if too many of the sym-metry constraints are relaxed, see Figure 3 .
Symmetry can impose immediate constraints on Poisson’s ratio. In particular, for hexagonal and square symmetry groups (where the lattice vectors are of equal length and at a fi xed angle to each other) Poisson’s ratios is by defi nition ν SS = –1. Thus if the deformation is constraint to retain square or hexagonal symmetry, the skeletal structure is always auxetic.
Symmetry constraints are here introduced to enlarge the set of tessellations with a well-defi ned Poisson’s ratio, motivated by the (as yet largely untested) assumption that the deformation
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of a realization as a linear-elastic solid may bear some resem-blance to the symmetry-constraint deformation of the skeletal structure. However, symmetry constraints can also model the effect of specifi c interactions or constraints of a physical system. For example, they may result if the struts represent molecular bonds of polyphenylacetylene networks. [ 56 ] A similar constraint resulting in auxetic behavior was also reported in elastomeric polypropylene where a fi xed angle of approximately 80 0 between connected crystalline lamellae arises from the epitaxial growth mechanism of crystalline lamellae. [ 9 ]
5. Conclusions
The subject of this article is the systematic identifi cation of periodic auxetic materials. The approach is based on two ingre-dients: the fi rst is to use repositories of symmetric planar tessel-lations as sets of candidate geometries, and identify those with auxetic deformations among them. The second is the study of the simple geometric problem of skeletal structures of stiff rods rather than the problem of a linear-elastic cellular solids. This search, as yet only performed on a small set of planar tessella-tions, has produced novel auxetic skeletal structures.
For the Triangle-Square-Wheels tessellation which is one of the newly found skeletal structures fi nite element calculations and tensile experiments with samples produced by rapid pro-totyping have shown that this structure is also auxetic when realized as a linear-elastic cellular solid.
It is not a priori clear that the Poisson’s ratio ν SS of a skeletal structure relates to the Poisson’s ratio ν LE of the linear-elastic material. In fact, the sets of equations underlying these two dif-ferent problems have distinct differences in their mathematical structure. [ 38 ] In particular, the interpretation of Poisson’s ratio as the ratio of elastic moduli in linear-elastic theory does not apply to the skeletal structures.
For the identifi cation of auxetic linear-elastic cellular solids, the presented approach hinges on the assumption or belief that geometry is an important, if not even the dominant, determi-nant of deformation modes. This assumption is, for the single Triangle-Square-Wheels tessellation, validated by the results pre-sented here. More scrutiny and further detailed studies, e.g. by FEM, are however necessary to provide wider justifi cation for this assumption and to demonstrate its limits.
From an engineering point of view, it is attractive to fur-ther improve the mechanical properties of the linear-elastic cellular solids derived from the Hexagonal-Wheels and the Triangle-Square-Wheels tessellations, that is, minimize the Pois-son’s ratio. Modern structural optimization techniques provide the methodology for it; e.g. in shape optimization the topology of the structure is maintained while the outer shape is altered until an (at least local) minimum of the Poisson’s ratio is reached. [ 60 , 64 ]
The arguably most important aspect of this work is the possible generalization to three-dimensional structures. In 3D only few auxetic mechanisms have been described, [ 2 , 12 , 17 , 58 , 65–68 ] most of which have planar equivalents. Inherently three-dimen-sional auxetic mechanisms are rare, if not unknown. Given the enormous number of known three-dimensional networks (e.g. approx. 14500 listed in the EPINET network repository, [ 69 ] 1700 in the RCSR, [ 70 ] ) known and conjectured zeolite structures
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(e.g. Treacy et al., [ 71 ] Baerlocher et al. [ 72 ] ), crystals (e.g. Grazulis et al., [ 73 ] Belsky at al. [ 74 ] ) and spatial tessellations (e.g. O’Keeffe et al. [ 70 ] ), it appears likely that there are some truly three-dimensional auxetic mechanisms among these that have not yet been identifi ed as such. Once implemented, the method-ology described here and in ref. [ 53 , 54 ] will be suffi ciently fast to allow for systematic searches within this plethora of candidate structures. It is likely to yield new insight into three-dimensional deformation mechanisms. Combined with modern methods of topological optimization [ 75 , 76 ] and rapid-prototyping methods it represents a new approach to the design of auxetic structures.
Acknowledgements The authors gratefully acknowledge fi nancial support by the Deutsche Forschungsgemeinschaft (DFG) through grant ME 1363/9 and support of the Cluster of Excellence “Engineering of Advanced Materials” at the University of Erlangen-Nuremberg, which is funded by the DFG within the framework of its “Excellence Initiative”. We also acknowledge support by the German Academic Exchange Service and the Group of Eight (Australia) for travel to Australia.
Received: January 21, 2011 Published online: April 21, 2011
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