Auxetic behavior on demand: a three steps recipe for new designs
Daniel Acuna, Francisco Gutiérrez, Rodrigo Silva, Humberto Palza, Alvaro S. Nunez, Gustavo Düring
AAuxetic behavior on demand: a three steps recipe for new designs
Daniel Acuna , , ∗ Francisco Guti´errez , , Alvaro S. Nunez , , , † and Gustavo D¨uring , ‡ Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago, Chile ANID - Millenium Nucleus of Soft Smart Mechanical Metamaterials, Santiago, Chile and CEDENNA, Avda. Ecuador 3493, Santiago, Chile (Dated: February 1, 2021)Auxetic behavior is a fascinating mechanical property which probably represents the paradigm ofmechanical metamaterials. Despite that nowadays it is a widely known phenomenon, a fundamentalmicro-mechanical understanding which allows the proper control and design of new auxetic materialremains mystified. In this letter we show an important step forward setting a unified framework fora large class of auxetic materials composed of 2D rotating rigid units. In addition, a simple pathwayfor the design of new auxetic material is established based on three simple rules. In particular, weconstruct for the first time exotic crystals, quasi-crystal and isotropic materials with negative Poissonratio, which in an ideal design reach the perfect auxetic limit. At the core of this behavior is a lowenergy mode reminiscent of a non trivial floppy mode that exists in the idealized scenario. A naturalconnection between 2D rotating rigid auxetic with an antiferromagnetic spin is established whichallows to identify general properties of the material. Finally, we show that the auxetic response isrobust under small perturbation in the design.
I. INTRODUCTION
The design of new materials with unusual mechani-cal properties and advanced functionalities has become avery active field of research in soft matter physics. Theseso-called mechanical metamaterials acquire their myste-rious behavior from the particular inner architecture andnot from the constituent materials properties. In re-cent years, metamaterials have been engineered to dis-play topological protection[1–6], programmable shapes[7–11], nonlinear response [2, 10–14] and negative elasticconstants [12, 13, 15–17] among others. Auxetic materi-als are probably the epitome of mechanical metamateri-als, which were for the first time intentionally designedby Lakes in 1987 [15]. An auxetic material, unlike com-mon elastic materials, when compressed (expanded) ina given direction, its response is to compress (expand)in the perpendicular direction. This unusual property ischaracterized by a negative Poisson’s ratio ν , the ratiobetween the strain in one direction and the strain in itsperpendicular direction.A negative Poisson’s ratio has been found in natu-ral bioauxetics [18, 19] and molecular auxetics [20, 21].Nowadays, with the onset of 3D printing, a wide range ofauxetic materials are being developed [11, 13, 16, 22–24],with interest in their enhanced mechanical properties,like increased energy absorption[25], enhanced indenta-tion resistance[26], high fracture toughness[27], synclasticcurvature in bending[28], and variable permeability[29],with applications in bio-medicine[30] and textiles[31] assome examples.A variety of shapes and geometries have been identi- ∗ [email protected] † alnunez@dfi.uchile.cl ‡ [email protected] fied as prototypical auxetics. The list ranges from re-entrant structures[34] to rotating units[35, 36]. Chiralstructures[37] and others [38] complete the list. Despitethe extensive literature and enormous progress describ-ing different types of auxetic materials no fundamentalmicroscopic principles for a unified description exist. Thedistinction between types of auxetics relies mainly on em-pirical observation rather than in fundamental principles,and no general prescription exists to build them. In thisletter, we present a unified framework for the descriptionsof bi-dimensional auxetic materials with rotating units.These structures are generically made out of polygonsconnected through their vertices (see Fig. 1 for exam-ples). Under external loads the stresses focalize on thevertices leaving almost undeformed the bulk of the poly-gons [22]. The auxetic behavior arise because neighborpolygons tend to rotate in opposite directions along aparticular low energy mode. This mode is reminiscent ofa non-trivial floppy mode, or mechanism, that exists inthe ideal case with zero bending energy (i.e. the polygonsare connected through ideal hinges). The emergence ofthis “auxetic” floppy mode is due to the network’s par-ticular topology and exists even for overconstrained net-works. In this limit the bulk modulus vanishes while theshear modulus remains finite implying a perfect auxeticbehavior, i.e. with a Poisson ratio ν = − a r X i v : . [ c ond - m a t . s o f t ] J a n - - + + a b c d FIG. 1:
Auxetic behavior on demand . Our proposed algorithm generated three instances of auxetic materials.The three systems correspond to the top row a) Random Lattice (isotropic lattice), b) Penrose’s quasicrystal, and c)Exotic Crystal. The structures were simulated using the commercial software Ansys Mechanical[32] with a finiteelement method. Their uniaxial compression results are depicted in the bottom row where the auxetic behavior isapparent, the color shows the intensity of the stress in the material. This unusual property’s basic mechanism is thecoordination and synchronization of the buckling instability at each of the weak links that provide the structure itsstability. The effective collective pattern that emerges is analog to an antiferromagnetic arrangement. Each of thetwo interconnected lattices that fit the bipartite system rotates in opposite senses, as illustrated in the inset of figurec). Out of this analogy, we infer that several properties of the anisotropic XY antiferromagnet[33] are inherited intothe context of auxetic systems. Finally, in d), we display the Poisson’s ratios calculated from the simulations.cal antiferromagnetic model. Out of this mapping it ispossible to create a large variety of auxetic structures,including the somewhat elusive isotropic auxetic mate-rial [39, 44]. In Fig. 1 one can see three different exam-ples; a new type of auxetic crystals, a quasicrystal andan isotropic (disordered) structure. Considering a build-ing elastic material with a Poisson ratio ν = 0 . G = 0 . M P a , the mechanical responseof the designed materials were obtained using the soft-ware Ansys (for more details see Appendix D). After afinite initial load, they display a clear auxetic behaviorwith a Poisson’s ratio reaching values between − .
65 and − . N polygons and N h hinges betweenpolygons, the system is jammed when the number of de-grees of freedom 3 N is less than the number of constraints2 N h . Expressing N h as a function of the coordination(the average number of hinges per polygon) N h = zN ,one gets a critical coordination z c = 3.Above critical coordination polygon networks typicallyguaranteed mechanical stability, due to the absence oftrivial floppy modes. Therefore, the existence of an “aux-etic” floppy mode must be related to a very precise geo-metrical construction which also implies the appearanceof a non trivial self stress state mode following the ranktheorem [46]. The starting point is the observation thatin an ideal polygon network the free hinges connectingdifferent rigid units only allow the rotation of the poly-gons. Then any floppy mode requires that all the neigh-bors of each polygon have the same rotation rate (as afunction of strain). If the neighbors of a given polygonare also neighbors between them the system will thenjam. This observation sets the key ingredient for rotatingunit auxetic theory, which requires the system to be bi-partite. Although most bipartite polygon networks showsome level of auxetic response under various conditions,as we will discuss later, additional conditions are neces-sary to obtain a perfect auxetic behavior. II. A SIMPLE MODEL FOR PERFECTAUXETICS
A minimal bi-dimensional model for rotating unitsauxetics, consists of a series of polygons connected bysprings of zero natural length [41]. The springs actas ideal hinges as long as they are not compressed orstretched. Replacing ideal hinges with springs not onlysimplifies construction in terms of energy, but also in-troduces elasticity into our materials to study non idealscenarios. Each rigid unit has three degrees of freedom,two translational (cid:126)x i = ( x i , y i ) and one rotational θ i , notnecessarily measured from the centroid of each polygon.We are interested in the behavior of perfect auxetics,to build them we establish 3 requirements.1. The network must be bipartite.
This allows the units to counter rotate respect toeach other, like cogs in a machine. The units ar-ranged in a bipartite network can be separated intwo sets A and B , i.e. each connected to the otherbut not to itself, see Fig. 2.2. Initially and at rest, every pair of neighbor-ing polygons position’s ( (cid:126)x i , (cid:126)x j ) and the ver-tex between them have to be collinear. This initial setting, matches a maximum extensionconfiguration. Furthermore, it establishes a rela-tionship between the internal angles of every pairof neighboring polygons | α ij + β ji | = π , see Fig. 2.3. The ratio between the distance of a polygonto one of its vertex and the distance of hisneighbor to the same vertex must be a con-stant in the network.
Each vertex of a rigid unit is characterized bya vector (cid:126)a ij = a ij (cos( θ i + α ij ) , sin( θ i + α ij )) or (cid:126)b ji = b ji (cos( − θ j − β ji ) , sin( − θ j − β ji )), corre-sponding to sets A or B respectively. The index i will be used for polygons in the set A and theindex j for polygons in the set B . Vectors (cid:126)a ij ( (cid:126)b ji )point from the position (cid:126)x of the polygon i ( j ) intothe vertex connecting with polygon j ( i ), as seen inFig. 2. Therefore, the ratio C = b ji /a ij must be aconstant through the network.Creating a polygon network that fulfills these rules isquite simple. Starting from a planar bipartite graph onecan always build a perfect auxetic. To understand theorigin of this behavior we turn to the energy of the poly-gon network V = k (cid:88)
1. The second one is found when all thepolygons of each set rotate at the same rate θ i = θ A and θ j = θ B , wherecos( θ A ) = 1 + C + λ (1 − C )2 λ , (3)and cos( θ B ) = 1 + C − λ (1 − C )2 λC . (4)These are a minimum in the range 1 < λ < (cid:12)(cid:12) C − C (cid:12)(cid:12) only if both θ A and θ B have the same sign, i.e. poly-gons counter rotate respect to each other. Evaluat-ing the potential energy in this minimum we find that V ( θ i = θ A , θ j = θ B ) = 0 (for a detailed derivation seeAppendix E), thus this solution describes a zero energymode of the system. This floppy mode corresponds to asystem with zero bulk modulus, meaning that the mate-rial expands and contracts equally in all directions, fora direct calculation of the bulk modulus see AppendixF. As the Poisson’s ratio is defined as ν = − d(cid:15) x d(cid:15) y , with (cid:15) being the strain in each direction, if the system expands a cb FIG. 2:
The 3 steps recipe . a) First step, we start with a bipartite network, the blue and red colors stand for theA and B sets respectively. We showcase a random bipartite network to demonstrate the versatility of this recipe.The network zoom in represents the second step. Every node of the graph becomes the position of each polygon andwe place each polygon’s vertex on top of each corresponding segment of the network, where neighbor’s polygonsshare a vertex position. At rest the polygon’s position is collinear with its neighbor’s and the vertex between them. b) All the polygons are then connected by zero natural length springs at the common vertices. The zoom in displaysthe distance a ij from the node to each vertex in the A set polygons and b ji which is the analogue but for the B set.Then the third step consists of moving the vertices of each polygon such that the ratio C = b ji /a ij remains constantalong the network. In addition the angles α ij and β ji are displayed at the undeformed initial stage. Note that | α ij + β ji | = π . c) A compression shows that the polygon network is a perfect auxetic. In the zoom in we see howeach polygon counter rotates with respect to its neighbors. θ i and θ j show the rotation of the A and B setrespectively.equally in both directions then (cid:15) x = (cid:15) y and ν = − III. RANDOM PERFECT AUXETICS
Recently, several isotropic auxetics materials with aPoisson’s ratio close to − C : 1 ratio, see Fig. 2. IV. BEYOND PERFECT AUXETICS
The bipartite condition seems to be fundamental tohave an auxetic material composed of rotating units. In-troducing defects on the network will frustrate the ro-tation of the units affecting the auxetic behavior, thiseffect will be addressed elsewhere. Here we will considerthe effect of breaking the other conditions, by modify-ing the angle that connects two polygons, thus making | α ij + β ji | = π + δ ij . For the sake of simplicity we used aperiodic polygon network and a fixed δ that will changesign from one link to the next, see Fig. 3. By perturb-ing the polygon network the floppy mode is destroyedand the polygon network jams. Under compression thesystem now increases its energy by keeping initially aPoisson’s ratio close to zero (see Fig 3). However at afinite strain an elastic instability occurs and the systemjumps to a rotated configuration recovering its auxeticbehavior as seen in Fig. 3.Although the latter behavior seem to be generic un-der small perturbations of the perfect auxetic networkit is not always the case. It is known that rectangularnetworks [23], which are not perfect auxetics, preserve afloppy mode and the Poisson’s ratio moves continuouslyfrom positive to negative values. To understand the con-dition for the existence of this floppy mode we need amore general description on polygon bipartite networks. a b FIG. 3: a) In blue an unperturbed perfect auxeticlattice, in red the same lattice but perturbed by anangle δ . The perturbation is applied such that each pairof polygons is no longer collinear between them andtheir vertex. The angle δ measures how much the vertexhas been displaced. Notice that this turns thepreviously parallelogram shaped holes into trapezoidalholes. b) The Poisson’s ratio ν of the perturbednetwork as a function of a vertical compression,measured by the Cauchy strain (cid:15) y . Each curve has adifferent perturbation angle δ . Each perturbed systemstarts with a ν = 0 until a point where it suddenlyrecovers its auxetic behavior. For more informationabout the measurements see Appendix C. V. FLOPPY MODES ON BIPARTITEPOLYGON NETWORKS
Bipartite graphs have only even sided cycles, these areclosed paths that start and end at the same node. Thesimplest of cycles are the faces of the graph, which arethe regions bounded by edges. When transforming a bi-partite graph into a polygon network, even cycles arereflected at the geometry of empty spaces between thepolygons which are also enclosed by the same numberof sides. We will refer to this empty spaces as holes,and to the vertex between polygons as hinges. Noticethat no odd sided holes can exist in bipartite systems.In the case of our perfect auxetic materials, one can use the Varignon’s quadrilateral theorem[48] and Thales the-orem to show that the constant ratio C = b ji /a ij directlyimplies that all 4 sided holes will be parallelograms.For a system to be deformed while at zero energy, itneeds the opposite angles at each hinge to have an op-posite deformation rate. This condition is extremely dif-ficult to fulfill if the inner angles inside a hole have anon-linear behavior as a function of the deformation, asis the case with trapezoidal 4 sided holes. Now parallel-ograms have the special property that when deformed,all of their inner angles share the same deformation rate,except for the sign. All 4 sided holes in perfect auxeticsare parallelograms, this allows every inner angle in eachhole to be linearly related to the deformation, fulfillingthe restriction at each hinge. This mechanism suggeststhat any bipartite polygon network with only parallelo-gram holes will have a non-trivial floppy mode, however,it will not necessarily behave like a perfect auxetic.In particular, when geometrically perturbing the poly-gons of a perfect auxetic with only 4 sided holes, whilepreserving their parallelogram shape, the floppy modepersists. The simplest example is the rectangular net-work studied in [23] where the Poisson’s ratio was ob-served to change its sign depending on the strain. Sim-ilar results are observed in a more general case where 4sided holes remain parallelograms after a perturbation ofan isotropic perfect auxetic, showing a continuous changefrom positive to negative Poisson’s ratios under strain asseen in Fig. 4. VI. CONCLUSION
We have presented a simple model within this workthat builds the necessary framework to create, design,and characterize rotating unit auxetics. Such frameworkis built upon a simple analogy between the rotating unitauxetics and an anisotropic XY antiferromagnetic sys-tem. As shown in Fig. 1, we have applied those ideasto generate novel auxetic structures in the form of acrystal, a quasicrystal, and a random lattice. Each de-sign can be represented, within our theory, by a minimalmodel, based upon polygons and springs that capturesits essential collective response to external loads. Thesemodels can be simulated straightforwardly to test mate-rials properties while ignoring bending forces. However,if needed, bending can easily be added to the model. Aswe have seen, this model correctly describes the behaviorof all rotating-unit systems and could be used to pre-dict new behaviors. In particular, we have generalizedthe behavior of auxetic domain walls, which are naturaltextures that these systems have because of the analogywith magnetic systems. More phenomena related to thisanalogy remain to be seen and encourage further inves-tigation. As a major tangible result, our work leads usto establish the ground rules to create never seen be-fore isotropic perfect auxetics. With current 3D printingtechnology, it should be relatively simple to realize anyFIG. 4: The Poisson’s ratio ν of a geometricallyperturbed random auxetic with parallelogram 4 sidedholes (orange circles) and the original unperturbedperfect auxetic (green triangles), both as a function ofthe Cauchy strain (cid:15) y . Both systems display anon-trivial floppy mode. The perfect auxetic behaves asexpected with a ν = −
1. The perturbed auxetic startswith a positive ν which changes sign as the deformationincreases. The blue curve shows the theoreticalPoisson’s ratio of a rectangular auxetic, the rectanglesin it have a 1 . VII. ACKNOWLEDGMENTS
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FIG. 5: An isotropic perfect auxetic with periodicboundary conditions and a domain wall. The system iscompressed with δλ = 0 .
2. The color of each polygonrepresents their normalized angle of rotation. Thesystem has two domains one in red and the other inblue, both separated by a yellow domain wall.
Appendix A: Domain Walls
We can see that Eq. 2 is analogous to that of an an-tiferromagnetic spin system. In such case, θ representsthe spin direction, J ij is a symmetric coupling constantand the sum over neighbours H i = (cid:80) n H in is a magneticfield as a function of space.Therefore, we can find magnetic related phenomena,like domain walls, in polygon networks. These appearbetween two stable solutions of the system which dif-fer in the turning direction of the polygons, as seen inFig. 5. These domain walls have been previously studiedin auxetics[23, 41].The polygon angle defines the state of the polygonnetwork, when 0 < λ ≤ < λ < (cid:12)(cid:12) C − C (cid:12)(cid:12) the angle is given by Eq. 3 and Eq. 4.Then θ can be used as the order parameter of this systemwhich is controlled by the external parameter δλ = λ − H = (cid:90) dx (cid:18) t θ ( x ) + uθ ( x ) + K ∇ θ ( x )) (cid:19) (A1)Here t , u and K are analytical functions of δλ . To keepthe system stable K and u are positive constants close tothe critical point, and t = − t δλ + O ( δλ ) .The stable solution for an homogeneous system is givenby: FIG. 6: The normalized rotational angle of thepolygons as a function of distance, revealing theexistence of a domain wall in the middle. Each curverepresents a polygon network with different levels ofcompression, measured by δλ . In the inset the x-axishas been rescaled by δλ − / , all the curves collapse intoa single one showing that it’s the correct scaling. Theaverage distance from a polygon’s position to itsvertices is a = 1 /
2. The normalization coefficient θ corresponds to the maximum rotation angle of thesystem. We used a periodic polygon network as the onein Fig. 3 with a vertical domain wall.¯ θ = (cid:114) t u δλ / . (A2)To determine the length scale of a domain wall wesearch for the minimum energy in a system with θ ( ∞ ) =¯ θ , θ ( −∞ ) = − ¯ θ . The differential equation for such sys-tem is: K d θdx = tθ + 4 uθ , (A3)whose well known solution is: θ ( x ) = ¯ θ tanh (cid:16) x ∆ (cid:17) . (A4)Where the domain wall width is defined as:∆ = (cid:114) Kt δλ − / . (A5)Thus we see that the domain wall width scales like∆ ∼ δλ − / . To check this, we performed numericalsimulations where we minimized the energy of a polygonnetwork with periodic boundary conditions until it hada couple of stable domain walls, the results can be seenin Fig. 6.In Fig. 6 if we consider that the polygon angle as afunction of position is θ ( x ) = θ f ( x ) with | f ( x ) | ≤
1, theexpansion of this function around the origin is f ( x ) = mx + O ( x ), as f is an even function. Now throughsimple trigonometry we can relate the slope at the origin m to the domain wall length, approximately ∆ = 2 /m ,and from the rescaled inset in Fig. 6 we can determinethat m ∼ δλ / , therefore we obtained the same result aspredicted where ∆ ∼ δλ − / . Appendix B: Building Random Bipartite PlanarGraphs
One of the main problems in creating a random perfectauxetic material is the construction of a random bipartiteplanar graph, from which we can construct the polygonnetwork. The graph must be bipartite so that the poly-gons can counter rotate with respect to each other, and itmust be planar so that we can build the polygon networkwithout overlapping them.We propose two methods to create these graphs. Thefirst is a heuristic pruning algorithm, which takes advan-tage of the property that a graph with only even cycleswill be bipartite. The second is a general transformationthat can quickly create bipartite graphs by combining agraph with its dual graph.
Pruning Algorithm
A bipartite graph has only even cycles, where a cycleis the shortest path between a node and itself. We call acycle even or odd depending on the number of bonds inits path. Here we prune a graph in such a way that allthe cycles of the resulting graph are even, transformingthe graph into a bipartite graph.If we have two neighboring cycles that share a bond,and we prune this bond, we will end up with a singlecycle. We can think of this operation as an addition ofcycles. Where if the starting cycles are both even or odd,the resulting cycle will be even. And if one cycle is evenand the other is odd, the resulting cycle is odd. We canextrapolate this property to a pair of separate odd cycleswith only even cycles between them. If we remove a lineof bonds between the odd cycles, we will end up with asingle even cycle. Then if we prune the bonds betweenall pairs of odd cycles, we will end up with a bipartitegraph with only even cycles.The pruning algorithm we implemented follows somesimple steps, we start with a random planar graph withan even number of odd cycles, ideally with a high coor-dination number, e.g. z = 6. Next, we place a marker atan odd cycle and use a breadth-first search algorithm onthe dual graph, to find the path to its closest odd cycle.We remove the bonds in this path, transforming bothodd cycles into a single even cycle. Finally we move the marker to another odd cycle and repeat the procedureuntil all cycles are even. An example of the initial andfinal graphs is in Fig. 7. While removing bonds we preferpaths that leave each node with at least three bonds, togive the system more stability and to avoid generatingbig holes. If a node is left with less than two links, weeliminate the node.This algorithm works well on small graphs, but on big-ger ones it may eliminate a huge amount of bonds leavingbig holes. To avoid this problem a more sophisticatedpath optimization algorithm is needed, where the pathsbetween all pairs of odd cycles are calculated beforehand,minimizing the distance of each path and making surethey avoid each other. Once all of the paths are com-puted, the bond elimination process can be performed,minimizing the size of the holes. Furthermore, this algo-rithm doesn’t necessarily work for graphs with periodicboundary conditions. As having only even sided cyclesguarantees bipartivity if the graph has free boundary con-ditions, but it doesn’t if the graph has periodic boundaryconditions. Bipartite Transformation
A bipartite graph is made out of two independent sets,each one connected to the other but note to itself. Herewe connect two independent sets, a graph and its dualgraph, transforming both into a single planar bipartitegraph.Given a graph, we first determine its dual graph. Thenwe connect each node of the graph with each neighboringnode in the dual graph, by neighbor node we mean thenode in the dual graph that represents a face in contactwith the node in the original graph. At last, we eliminateall the starting bonds of the graph and its dual graph,leaving only the new bonds connecting both graphs. Theresulting graph will be bipartite, and if the original graphwas planar, the resulting graph will be planar too. Thefurther understand this procedure, see Fig. 8.This transformation can be performed on any kind ofgraph and several times in a row creating a graph withmore nodes each time. We can reverse the transforma-tion, though we may not know if we obtained the graphor its dual graph when performing the inverse transfor-mation, unless we keep track of at least one node fromthe starting graph.
Appendix C: Numerical Methods
To obtain the data shown in Fig. 3 and Fig. 4, we per-formed numerical simulations of polygon networks. Tomodel these networks we used the potential energy inEq. 1. Depending on the problem and the boundaryconditions we used different methods to compress andtest the polygon networks. Regardless of the bound-ary conditions the Poisson’s ratio was calculated using ν = − d(cid:15) x d(cid:15) y = − dL x dL y L y L x , where L x and L y are the approxi-mate dimensions of the system in each axis. Periodic Boundary Conditions a b FIG. 7:
Pruning Algorithm. a)
Isotropic contact amorphous network [14, 47] with a high coordination z = 6. b) Pruned network, the bonds between pairs of odd cycles have been removed, adding them together into even cycles.The result is a bipartite planar graph.The polygon networks in Fig. 3 were set in a periodicboundary box. To compress the material uniaxially weshrank the box in the vertical direction, and we let thesystem relax in the horizontal direction by minimizing itsenergy. At each step we measured the dimensions of thebox, L x and L y . Free Boundary Conditions
The system in Fig. 4 has free boundary conditions andan internal floppy mode. To efficiently deform the ma-terial, we applied a deformation in the direction of thefloppy mode and minimized its energy afterwards. Toobtain the floppy mode, we fixed 3 degrees of freedom inthe system and found a non-trivial solution for M ˙ (cid:126)q = 0,where M is the Hessian and ˙ (cid:126)q is the floppy mode. Ateach step we approximated the system by a rectangle ofdimension L x and L y . Appendix D: Finite Element Simulations
For the simulations in Fig. 1, we used three bipar-tite networks which were created using the methods de-scribed in Appendix B, for the exotic crystal we modifieda tetrakis tiling by skewing it and applying the bipartitetransformation; for the quasi-crystal we used a Penrosetiling which was cut in a suitable square shape; lastly therandom network was created using the pruning method.All the materials were built with the same ratio betweenaverage polygon size and bond thickness, such that theyexhibit a similar behavior.For our static finite elements simulations, we used thecommercial software ANSYS [32] and a Neo-Hookean en-ergy density as a material model, with an initial shear modulus, G = 0 . M P a , and Poisson’s ratio ν = 0 . elements approximately. To compress thematerial uniaxially, we applied a vertical displacement ofthe top row of polygons, and fixed the position of thebottom row of polygons. We imposed a free boundarycondition in the horizontal direction and we imposed ano-penetration condition, therefore the material can con-tact itself.To measure the Poisson’s ratio, we approximated thewhole system as a rectangle with dimensions L x and L y . Then the strains are ∆ (cid:15) x = L x − L (0) x L (0) x and ∆ (cid:15) y = L y − L (0) y L (0) y [16], where L x and L y are the dimensions of thematerial at rest. Finally we used the engineering strainPoisson’s ratio ν = − ∆ (cid:15) x ∆ (cid:15) y . (D1) Appendix E: Perfect Auxetic Detailed Derivation
In this section we will find the zero energy auxeticmode of a perfect auxetic polygon network, this is a poly-gon network that follows 3 requirements.1.- The underlying graph of the connected polygonsmust be bipartite, this means that we can split the graphinto two sets, where each set connects to the other but1 a bc d
FIG. 8:
Planar Bipartite Transformation. a)
Start with a random planar contact network, in this case thecoordination is z = 5. b) Compute the dual graph of the network (blue nodes and dashed lines), place each node ofthe dual graph inside its corresponding cycle. c) Disconnect each node of the dual graph (blue to blue nodes), andreconnect them to the nodes corresponding to the vertices of its cycle in the original graph (red and blue nodesconnected by dashed lines). d) Remove the connections of the original graph (red to red nodes) and keep theconnections between the dual graph and the original graph. As both initial graphs are independent sets that don’tconnect to themselves, the final result is a random planar bipartite graph.not to itself, we will call each set A and B .2.- At rest and without prestress the system must haveat least one configuration where every pair of neighboringpolygons positions ( (cid:126)x i , (cid:126)x j ) are collinear with the vertexbetween them.3.- If we set a point in space representing each polygonposition (cid:126)x i , not necessarily the centroid of each polygon, the distance from this position to each vertex will be a ij in polygons of set A , and b ji in polygons of set B , the firstindex indicates the origin polygon and the second indexrepresents the neighboring polygon, finally the ratio ofthis distances between neighboring polygons must remainconstant, so C = b ji a ij is constant.The following demonstration applies to undercon-2trained and overconstrained systems, though the laterare of higher interest as this zero mode is non-trivial inthem.To show the existence of this zero mode, we will ex-pand its potential energy and find its minimum under theassumption that the distance between polygons remainsconstant and that all polygons in each set rotate equally.Finally we will show that the minimum of energy is zerofor any value of the expansion coefficient λ .Using restriction 1, as the system is bipartite, we canwrite its potential energy as an interaction of each set A and B . V = k (cid:88)
1. Leaving Eq. E9 like C cos( θ B ) + cos( θ A ) = (1 + C ) λ . (E11)We simplify this equation using sin( θ A ) = C sin( θ B ).cos( θ A ) = 1 + C + λ (1 − C )2 λ (E12)3cos( θ B ) = 1 + C + λ ( C − λC (E13)This solution is a minimum of energy for 1 < λ < (cid:12)(cid:12) C − C (cid:12)(cid:12) . We will now replace it into the potential energyin Eq. E8, selecting the solutions where both θ A and θ B have the same sign and using that sin( θ A ) = C sin( θ B ),we can write Eq. E8 as, V = (( C + 1) + λ (1 − C + 2 C cos( θ B )(cos( θ A )+ C cos( θ B ))) − λ ( C + 1)(cos( θ A ) + C cos( θ B ))) k (cid:88)