materials

Article Tuning the Performance of Metallic Auxetic Metamaterials by Using Buckling and Plasticity

Arash Ghaedizadeh 1, Jianhu Shen 1, Xin Ren 1,2 and Yi Min Xie 1,*

Received: 28 November 2015; Accepted: 8 January 2016; Published: 18 January 2016 Academic Editor: Geminiano Mancusi

1 Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia; [email protected] (A.G.); [email protected] (J.S.); [email protected] (X.R.) 2 Key Laboratory of Traffic Safety on Track, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China * Correspondence: [email protected]; Tel.: +61-3-9925-3655

Abstract: Metallic auxetic metamaterials are of great potential to be used in many applications because of their superior mechanical performance to elastomer-based auxetic materials. Due to the limited knowledge on this new type of materials under large plastic deformation, the implementation of such materials in practical applications remains elusive. In contrast to the elastomer-based metamaterials, metallic ones possess new features as a result of the nonlinear deformation of their metallic microstructures under large deformation. The loss of auxetic behavior in metallic metamaterials led us to carry out a numerical and experimental study to investigate the mechanism of the observed phenomenon. A general approach was proposed to tune the performance of auxetic metallic metamaterials undergoing large plastic deformation using buckling behavior and the plasticity of base material. Both experiments and finite element simulations were used to verify the effectiveness of the developed approach. By employing this approach, a 2D auxetic metamaterial was derived from a regular square lattice. Then, by altering the initial geometry of microstructure with the desired buckling pattern, the metallic metamaterials exhibit auxetic behavior with tuneable mechanical properties. A systematic parametric study using the validated finite element models was conducted to reveal the novel features of metallic auxetic metamaterials undergoing large plastic deformation. The results of this study provide a useful guideline for the design of 2D metallic auxetic metamaterials for various applications.

Keywords: mechanical metamaterial; auxetic; buckling; large deformation; plasticity

1. Introduction Four fundamental mechanical properties of materials in isotropic elasticity are Poisson’s ratio (ν), Young’s modulus (E), shear modulus (G), and bulk modulus (K). Note that these four parameters are interrelated, and the applicability of the theory of elasticity is limited to stress-strain conditions wherein the stress is below the yield point. Young’s modulus is the measure of stiffness in the linear elastic range. Poisson’s ratio is the least studied among these elastic constants. However, it governs the deformation feature of materials under various loading conditions [1,2]. This property is represented by the negative of the ratio between transverse and longitudinal strains [3]. The majorities of materials have a positive Poisson’s ratio that is about 0.5 for rubber and 0.3 for glass and steel [3]. The thermodynamic requirement in the theory of elasticity for a conservative system demonstrates that for homogeneous isotropic materials, the theoretical bound of Poisson’s ratio is from ´1 to 0.5. Therefore, the existence of materials with negative Poisson’ ratio (NPR) has long been accepted and they are known as “auxetic” materials [3,4]. The deformation feature of auxetic materials can be

Materials 2016, 9, 54; doi:10.3390/ma9010054 www.mdpi.com/journal/materials Materials 2016, 9, 54 2 of 17 described as an anomalous behavior that indicates that materials expand (contract) in a transverse direction when uniaxially stretched (compressed) [3–7]. Based on their origin, auxetic materials can be classified as naturally occurring or artificial. The exotic mechanical properties of the natural auxetic materials inspired plenty of researchers to search for the underlying mechanisms that cause auxetic behavior. Those findings were applied to the design of artificial materials [1,8]. Many cellular structures were designed to have exceptional properties. When these properties are superior to these found in nature, the artificial cellular material is termed as a metamaterial. Metamaterials gain their uncommon and unique properties from geometrical configurations rather than chemical composition [9,10]. Furthermore, the behavior of metamaterials was normally explained in terms of the intricate interplay between their microstructures and their deformation mechanisms [11,12]. Over the past decades, continuous efforts were put in towards the development of auxetic metamaterials because of their unconventional behavior under uniaxial loading. Their unique properties opened a window towards a wide range of applications such as the design of novel fasteners [13], biomedical applications [14], energy-absorbing devices [15], acoustic dampers [16], membrane filters with variable permeability [17], and the design of composites [18]. Macroscopic auxetic cellular structures with 2D re-entrant honeycombs cells were firstly presented by Gibson and Ashby [19]. Then an artificially designed auxetic foam with a re-entrant cell was reported in the seminal research of Lakes in 1987 [3]. Following that, the interest of researchers was concentrated on periodic 2D auxetic materials through modifying the geometry of microstructures. These consist of composites with star-shaped inclusions [20], a structure formed from lozenge grids [21], a formed structure from square grids [22], square lattice of circular holes [11], and many other 2D topological patterns. Due to the notable challenge in the fabrication of 3D auxetic materials that consist of a microstructure with complicated geometries, only a small number of synthetic 3D auxetic metamaterials have been fabricated [10]. Recently an auxetic cube using “buckliball” as the building cell by Babaee et al.[23] and auxetic cube with simple spherical by Shen et al.[10] were invented, manufactured, and tested. It is worth noting that, most of these 2D and 3D auxetic metamaterials were constructed with soft elastic base materials. They deformed elastically under uniaxial loading and the deformation was reversible and repeatable. However, the weak stiffness and strength and the low softening temperature of these elastomeric metamaterials prevented them from many applications requiring high stress and high temperature. In contrast to elastomers, metals have higher stiffness, strength, density, and melting point. Copper, gold, platinum, silver, and brass are especially ductile and might be suitable for fabricating novel auxetic metamaterials. There is a great potential for metallic auxetic metamaterials to be used in engineering products. Only a few researchers studied the auxetic performance of metallic auxetic metamaterials undergoing large plastic deformation. Therefore, knowledge of the deformation features and auxetic performance of metallic metamaterials is very limited. The first work in this area was done by Friis et al.[5], who proposed a polymeric and metallic foam with auxetic behavior. Recently, a 2D metallic auxetic periodic structure with low porosity was reported by Taylor et al.[24]. Another study in this area was carried out by Dirrenberger et al.[25] on the auxetic behavior of the metamaterials undergoing plastic deformation by using an anisotropic compressible plasticity framework as a macroscopic model. Moreover, the behavior of elastoplastic auxetic microstructural arrays was assessed employing a continuum-based micromechanical model by Gilat et al.[26]. Inspired by previous works on buckling-induced auxetic elastic and elastoplastic metamaterials [10,11,23,27–30], we expanded the similar design approach to metallic structures. However, the buckling-induced geometrical design lost its auxetic behavior under large plastic deformation when the base material was changed to metals. Therefore, the specific aim of this research is to develop a general approach to designing and tuning the metallic metamaterials with auxetic performance undergoing large plastic deformation. The auxetic performance of metamaterials on any length of scale is dependent on the geometrical features and deformation mechanisms of their Materials 2016, 9, 54 3 of 17 microstructures. Hence, the general approach of the metamaterial starts at the microstructure level by employing an existing deformation mechanism that leads to auxetic behavior [10,11]. In this approach, the geometry of microstructure of a regular structure is modified then it is altered by the desired buckling modes. A new category of 2D metallic auxetic metamaterials over a large strain range can be obtained by using different scale factors. The new metallic metamaterial exhibits auxetic behavior with almost constant negative values of Poisson’s ratio over a large strain range. This feature is very useful for practical applications in macroscopic metamaterials as well as in nanoscale structures [31,32]. A set of parametric studies using validated finite element (FE) simulation have been done to show the possibility of providing an effective control method to tune the mechanical properties of the 2D metallic auxetic metamaterial during the designing process.

2. A General Approach to Tuning the Performance of 2D Metallic Auxetic Metamaterial In order to obtain a novel class of 2D metallic auxetic metamaterials with the potential to retain tuneable auxetic behavior under a wide range of applied strains, a general systematic approach was developed based on our previous work [10]. This general approach leads to the possibility of tuning the auxetic behavior and mechanical properties for the targeted applications. The key idea of this methodology is the exploitation of a pattern from the linear buckling analysis to create microstructures of metamaterials with an auxetic response. This methodology can be divided into four steps as described in the following sections. Moreover, the magnitude of alteration of microstructure and plastic properties of the base material were studied as the primary key factors in tuning the performance of 2D metallic auxetic metamaterial. These results are presented in the parametric study section.

2.1. Initial Geometric Design of Microstructures Modified from a Regular Pattern The auxetic performance of a metamaterial was determined by the geometrical features and deformation mechanisms of its microstructure. According to the geometry of microstructure of metamaterials that exhibited auxetic behavior induced by elastic instability, the ribs of their building cell were slightly thicker in the proximity of their connecting points rather than the middle parts of the ribs [10,11,23,27]. Consequently, rotation occurred at the middle parts of these ribs in clockwise and counter-clockwise directions, to form a deformation mechanism after the buckling of the ribs [33,34]. These observations and mechanisms led us to think that buckling-induced auxetic cellular metamaterials could be produced from a conventional regular cellular lattice through geometrical modification of their microstructures. In this research the microstructure of a new buckling-induced 2D auxetic metamaterial was produced from an existing conventional regular lattice through the proposed modification. This method consists of moving a small portion of the middle part of their ribs to the proximity of the connecting joints. This geometrical modification on the microstructure was based on the hypothesis that, under deformation, joints behaved as “rigid joints” and rotated relatively to each other after the ribs buckled. The geometric modification of the unit cell of the regular lattice through the proposed method is illustrated in Figure1b,c. The new buckling-induced 2D auxetic metamaterial was similar to the buckling-induced auxetic metamaterials investigated by Bertoldi et al.[11]. According to the buckling pattern, the representative volume element (RVE) contained four unit cells as shown in Figure1d. Similar to the designing approach of elastic metamaterials [ 10], the bulk 2D metamaterial was constructed by replicating RVEs along two planar directions, as shown in Figure1e. Materials 2016, 9, 54 4 of 17 Materials 2016, 9, 54 4 of 17

(a) (b) (c) (d) (e)

Figure 1. (a) Conventional regular lattice; (b) building unit cell of initial regular lattice; (c) modified Figure 1. (a) Conventional regular lattice; (b) building unit cell of initial regular lattice; (c) modified unit building cell of microstructure; (d) representative volume element; (e) designed bulk unit building cell of microstructure; (d) representative volume element; (e) designed bulk metamaterial metamaterial with cross‐shaped microstructures for experiments. with cross-shaped microstructures for experiments. 2.2. Investigation of the Instability of Initial 2D Auxetic Metamaterial Using a Linear Perturbation Procedure 2.2. Investigation of the Instability of Initial 2D Auxetic Metamaterial Using a Linear Perturbation Procedure (Linear(Linear Buckling Buckling Analysis) Analysis) The elastic instability in elastomeric metamaterials triggered a pattern switching phenomenon The elastic instability in elastomeric metamaterials triggered a pattern switching phenomenon above a critical load, which led to auxetic behavior under compression [28]. In order to identify the above a critical load, which led to auxetic behavior under compression [28]. In order to identify the desired buckling mode, finite element analysis (FEA) was performed on the existing cross lattice and desired buckling mode, finite element analysis (FEA) was performed on the existing cross lattice and the modified periodic cellular metamaterial. The stability of these microstructures was evaluated the modified periodic cellular metamaterial. The stability of these microstructures was evaluated using using commercial finite element software ABAQUS with standard solver (Simulia, Providence, RI, commercialUSA) and finite a linear element perturbation software procedure ABAQUS was with conducted standard to solver find the (Simulia, critical Providence, loads (eigenvalues) RI, USA) for and a lineardifferent perturbation buckling modes procedure (Eigen was modes). conducted The numerical to find the models critical were loads constructed (eigenvalues) with linear for different solid bucklingelements modes of the (Eigen secondary modes). accuracy The numerical (element C3D8 models with were a mesh constructed sweeping with seed linear size solidof 0.4 elements mm). ofIt the should secondary be noted accuracy that the (element applied C3D8boundary with conditions a mesh sweeping influence seed the buckling size of 0.4 modes mm). of It the should finite be notedsized that numerical the applied model. boundary In this conditionsstudy, all degrees influence of the freedom buckling on modesthe top ofand the bottom finite sized nodes numerical of the model.numerical In this model study, were all degrees constrained of freedom except for on the nodal top and movement bottom nodes of the oftop the surface numerical in the model direction were constrainedof the applied except load. for The the analysis nodal movement was carried of out the to top simulate surface the in thestandard direction uniaxial of the compression. applied load. It is The analysisworth noting was carried that for out purely to simulate elastic models the standard the buckling uniaxial modes compression. were independent It is worth of the noting modulus that of for purelyelasticity; elastic thus models any value the buckling of modulus modes could were be independent used to conduct of the the modulus buckling of analysis. elasticity; Therefore, thus any the value ofproposed modulus coulddesign be approach used to conduct was independent the buckling of analysis.the influences Therefore, of the the elastic proposed base designmaterial. approach The wasYoung’s independent modulus of theof elasticity influences and of Poisson’s the elastic ratio base of material. brass were The used Young’s as the modulus properties of elasticityof the base and Poisson’smaterial ratio in numerical of brass weresimulations. used as the properties of the base material in numerical simulations.

2.3.2.3. Identification Identification of of the the Desired Desired Buckling Buckling Pattern Pattern fromfrom an Elastic Instability Instability Analysis Analysis AsAs mentioned mentioned before, before, the the linear linear buckling buckling analysis yields yields a a number number of of eigenvalues eigenvalues and and buckling buckling modemode eigenvectors eigenvectors corresponding corresponding toto thesethese eigenvalues.eigenvalues. Normally Normally the the lowest lowest eigenvalue eigenvalue with with the the correspondingcorresponding buckling buckling mode mode is of of interest interest because because the the buckling buckling modes modes with with higher higher eigenvalues eigenvalues are difficult to trigger under quasi‐static loading. The desired buckling modes, termed alternating are difficult to trigger under quasi-static loading. The desired buckling modes, termed alternating periodic buckling mode, were identified through the deformation features of buckling‐induced periodic buckling mode, were identified through the deformation features of buckling-induced auxetic auxetic metamaterials previously investigated [10,11,23,35]. In most cases, the auxetic behavior of metamaterials previously investigated [10,11,23,35]. In most cases, the auxetic behavior of this type of this type of metamaterials was featured by such a mode with alternating mutually orthogonal metamaterials was featured by such a mode with alternating mutually orthogonal ellipses. According ellipses. According to this identified principle, the seventh mode shape of initial metamaterial to this(regular identified conventional principle, lattice) the seventh is similar mode to the shape observed of initial buckling metamaterial pattern (regular that leads conventional to the auxetic lattice) is similarbehavior. to In the contrast, observed the buckling first buckling pattern mode that shape leads of to the the newly auxetic designed behavior. metamaterial In contrast, (with the the first bucklinglowest modeeigenvalue) shape can of the be adopted newly designed as the desired metamaterial buckling (with mode the shape. lowest eigenvalue) can be adopted as the desired buckling mode shape. 2.4. Design New Metamaterials through Superposition of Desired Buckling Mode 2.4. Design New Metamaterials through Superposition of Desired Buckling Mode Previous research on designing elastic buckling‐induced auxetic metamaterials revealed that alteringPrevious the initial research geometry on designing of the microstructure elastic buckling-induced was a way to trigger auxetic the metamaterials deformation to revealed a specific that alteringpattern the that initial led geometryto auxetic ofbehavior the microstructure [10,34]. One wasset of a the way output to trigger of the the buckling deformation analysis to a was specific a patternnumber that of lednon‐ todimensional auxetic behavior buckling [10 modes.,34]. OneThus, set in order of the to output define ofthe the magnitude buckling of analysis alteration was of a numbermicrostructure, of non-dimensional a scale factor buckling is required. modes. In this Thus, study, in the order pattern to define scale thefactor magnitude (PSF) was of proposed alteration to of microstructure,add the desired a scalebuckling factor mode is required. shape to Inthe this initial study, geometry the pattern of the scale metamaterial. factor (PSF) To quantify was proposed the

Materials 2016, 9, 54 5 of 17 to add the desired buckling mode shape to the initial geometry of the metamaterial. To quantify the magnitudeMaterials 2016 of, 9 the, 54 alternation, the PSF was determined using the special geometric feature of an5 RVEof 17 of the microstructure initially designed. The innermost RVE of the bulk metamaterial was selected to magnitude of the alternation, the PSF was determined using the special geometric feature of an RVE quantify the PSF. The final microstructures for metallic auxetic metamaterials were obtained by adding of the microstructure initially designed. The innermost RVE of the bulk metamaterial was selected to thequantify desired the mode PSF. shape The multiplied final microstructures by the specified for metallic design scaleauxetic factor metamaterials (DSF) to the were original obtained coordinates. by DSFadding is an independentthe desired mode factor shape that consistsmultiplied of scalingby the specified the maximum design displacement scale factor (DSF) amplitude to the to original visualize a deformedcoordinates. shape. DSF Whenis an independent the walls at thefactor center that of consists the RVE of touchedscaling the each maximum other, the displacement corresponding DSFamplitude (in this caseto visualize 0.0092) a was deformed defined shape. as PSF When = 100%, the walls as shown at the in center Figure of2 c.the Other RVE touched magnitudes each of PSFother, were the defined corresponding accordingly DSF as(in 0%this and case 20%, 0.0092) corresponding was defined as to PSF DSFs = 100%, of 0 and as shown 0.00184, in respectively.Figure 2c. TheOther normalized magnitudes desired of PSF buckling were defined mode accordingly and its innermost as 0% and RVE 20%, with corresponding different PSFs to DSFs are of presented 0 and in0.00184, Figure2 .respectively. Through changing The normalized the PSF, desired a new buckling class of auxetic mode and metamaterials its innermost with RVE different with different auxetic performancePSFs are presented was produced in Figure [10 2.]. Through The tuneable changing auxetic the performance PSF, a new class was of a auxetic special metamaterials feature of this with class of auxeticdifferent metamaterials auxetic performance and potentially was produced could be [10]. implemented The tuneable in particular auxetic performance applications was with a special a specific negativefeature valueof this of class Poisson’s of auxetic ratio. metamaterials It should be notedand potentially that, without could this be alternation,implemented the in deformationparticular ofapplications this type of with auxetic a specific metamaterial negative was value uniform of Poisson’s at the beginningratio. It should stage be under noted compression, that, without whichthis exhibitedalternation, a positive the deformation Poisson’s ratioof this until type buckling of auxetic occurred metamaterial [10]. In was our uniform previous at workthe beginning [10], the stage desired bucklingunder compression, mode of the bulkwhich metamaterial exhibited a positive was superposed Poisson’s toratio the until original buckling designed occurred microstructures [10]. In our of metamaterial.previous work It resulted [10], the in desired a non-periodic buckling microstructure. mode of the bulk The metamaterial periodicity was was achieved superposed by patterning to the original designed microstructures of metamaterial. It resulted in a non‐periodic microstructure. The the altered central RVE of the bulk metamaterial in this study. The altered bulk metamaterial was periodicity was achieved by patterning the altered central RVE of the bulk metamaterial in this formed by repeating this altered RVE as the unit cell. study. The altered bulk metamaterial was formed by repeating this altered RVE as the unit cell.

(a) (b) (c)

Figure 2. Definition of PSF to alter the initial microstructure of a buckling‐induced metamaterial. The Figure 2. Definition of PSF to alter the initial microstructure of a buckling-induced metamaterial. normalized desired buckling mode shapes scaled by different corresponding DSFs and their most The normalized desired buckling mode shapes scaled by different corresponding DSFs and their most central RVEs. (a) PSF = 0, DSF = 0; (b) PSF = 20%, DSF = 0.00184; (c) PSF = 100%, DSF = 0.0092. central RVEs. (a) PSF = 0, DSF = 0; (b) PSF = 20%, DSF = 0.00184; (c) PSF = 100%, DSF = 0.0092. 3. Experiments 3. Experiments 3.1. Fabrication of 2D Metamaterials for Experiments 3.1. Fabrication of 2D Metamaterials for Experiments The metallic specimens were fabricated using a high‐resolution 3D printing technique. In the The metallic specimens were fabricated using a high-resolution 3D printing technique. In the first first step, the metamaterial was printed in wax then the wax model was put inside liquid plaster. step, the metamaterial was printed in wax then the wax model was put inside liquid plaster. After After setting the plaster, it was placed in an industrial oven to melt out the wax. The wax was used setting the plaster, it was placed in an industrial oven to melt out the wax. The wax was used as a as a supporting material to make a plaster mold. Eventually, the molten metal was poured into the supportingplaster mold material and was to make placed a plaster at the mold.fixed position Eventually, to become the molten solid. metal Raw was brass poured was selected into the plasteras a moldrepresentative and was placed base material at the fixed of printed position specimens to become because solid. Rawof its brass excellent was ductility selected among as a representative available basematerials material for of 3D printed printing. specimens The mechanical because properties of its excellent of brass ductility were obtained among directly available from materials a standard for 3D printing.tensile test The on mechanical six dog‐bone properties specimens of brassusing werean MTS obtained (material directly test system) from a machine standard (MTS tensile Company, test on six dog-boneEden Prairie, specimens MN, USA), using anwhich MTS is (material shown in test Figure system) 3. The machine average (MTS mechanical Company, properties Eden Prairie, of brass MN, USA),tests which are summarized is shown in in Figure Table3 .1. The The average influence mechanical of the potential properties anisotropy of brass of tests base are material summarized is not in Tablestudied1. The here. influence This potential of the potential may be anisotropycaused by ofthe base inevitable material inclusion is not studied of air bubbles here. This into potential brass mayspecimens be caused during by the the inevitable manufacturing inclusion process. of air bubbles The penetration into brass specimensof tiny bubbles during causes the manufacturing very small process.holes inside The penetration the specimens of tiny and bubbles changes causes the mechanical very small properties holes inside and the density specimens of different and changes printed the mechanicalspecimens. properties However, and the density excellent of differentagreements printed between specimens. the numerical However, and the experimental excellent agreements results betweenrevealed the that numerical the influence and experimentalof potential anisotropy results revealed is negligible that thein this influence study. of potential anisotropy is negligible in this study.

Materials 2016, 9, 54 6 of 17 Materials 2016, 9, 54 6 of 17

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(a) (b)

Figure 3. Tensile test of 3D printed brass dog‐bone using MTS machine. (a) Front view of brass Figure 3. Tensile(a) test of 3D printed brass dog-bone using MTS(b) machine. (a) Front view of brass dog‐bone. (b) Schematic curve of stress‐strain for brass. dog-bone. (b) Schematic curve of stress-strain for brass. Figure 3. Tensile test of 3D printed brass dog‐bone using MTS machine. (a) Front view of brass dog‐bone. (b) Schematic curve ofTable stress 1.‐strain Material for brass.properties of brass. Table 1. Material properties of brass. Modulus of Elasticity (GPa) YieldTable Stress 1. Material (MPa) propertiesStrain Hardening of brass. Modulus (GPa) Density (GPa) Modulus of86.76 Elasticity (GPa) Yield Stress140.34 (MPa) Strain Hardening1.65 Modulus (GPa) Density8720 (GPa) Modulus of 86.76Elasticity (GPa) Yield Stress 140.34 (MPa) Strain Hardening 1.65 Modulus (GPa) Density 8720 (GPa) The 2D86.76 periodic bulk metamaterials140.34 for original buckling‐induced1.65 design (PSF of 0%)8720 and for an alteredThe one 2D with periodic unit cells bulk with metamaterials a PSF of 20% for are original shown in buckling-induced Figure 4.The first design3D printed (PSF brass of 0%) sample and foris a The 2D periodic bulk metamaterials for original buckling‐induced design (PSF of 0%) and for an anbuckling altered‐induced one with metamaterial unit cells withwith PSF a PSF = 0, of as 20% shown are in shown Figure in 4a, Figure and the4.The other first is the 3D altered printed one brass that altered one with unit cells with a PSF of 20% are shown in Figure 4.The first 3D printed brass sample is a samplewas superposed is a buckling-induced by the desired buckling metamaterial mode withwith a PSF PSF = of 0, 20%, as shown is shown in in Figure Figure4a, 4b. and The the dimensions other is buckling‐induced metamaterial with PSF = 0, as shown in Figure 4a, and the other is the altered one that theof the altered first and one second that was test superposed specimens bywere the height desired × width buckling × depth mode = 92.8 with mm a PSF × 88 of mm 20%, × 10 is shownmm. Two in was superposed by the desired buckling mode with a PSF of 20%, is shown in Figure 4b. The dimensions Figureplates 4wereb. The added dimensions to the bulk of the materials first and to secondconstrain test the specimens degrees of were freedom height of ˆthewidth top andˆ depth bottom = of the first and second test specimens were height × width × depth = 92.8 mm × 88 mm × 10 mm. Two 92.8nodes’ mm surfacesˆ 88 mm exceptˆ 10 for mm a. degree Two plates of freedom were added in the to direction the bulk of materials load. To to check constrain the accuracy the degrees of the of plates were added to the bulk materials to constrain the degrees of freedom of the top and bottom freedommanufactured of the topsamples, and bottom the weights nodes’ surfacesof printed except models for a were degree measured of freedom and in thecompared direction with of load. the nodes’ surfaces except for a degree of freedom in the direction of load. To check the accuracy of the Toweight check of the our accuracy initial design of the obtained manufactured from the samples, numerical the mode. weights The of weights printed of models printed were samples measured were manufactured samples, the weights of printed models were measured and compared with the andslightly compared heavier with than the the weight designed of our initialones. designA numerical obtained comparison from the numerical of models mode. with The the weights different of weightweight ofobtained our initial by designchanging obtained the void from of theRVE numerical confirmed mode. that Thethese weights small ofweight printed differences samples werewere printedslightly samplesheavier werethan slightlythe designed heavier ones. than A the numerical designed comparison ones. A numerical of models comparison with the of different models withnegligible. the different In order weight to compare obtained the bydeformation changing thepatterns void ofof RVErubber confirmed and metallic that metamaterials these small weight with weightinitial obtainedgeometric by design changing (PSF the =void 0%), of aRVE 2D confirmed elastomeric that specimen these small was weight manufactured differences usingwere differences were negligible. In order to compare the deformation patterns of rubber and metallic negligible.silicon‐based In orderrubber to (Tangocompare plus, the In’Tech,deformation Ramsey, patterns MN, of USA), rubber as and shown metallic in the metamaterials top‐most row with of metamaterials with initial geometric design (PSF = 0%), a 2D elastomeric specimen was manufactured initialFigure 5.geometric The material design properties (PSF of= silicon0%), ‐abased 2D rubberelastomeric (Tango specimen plus) were was measured manufactured in reference using [10]. siliconusing silicon-based‐based rubber rubber (Tango (Tango plus, plus, In’Tech, In’Tech, Ramsey, Ramsey, MN, MN, USA), USA), as asshown shown in in the the top top-most‐most row row of Figure5 5.. TheThe materialmaterial propertiesproperties ofof silicon-basedsilicon‐based rubber (Tango plus)plus) werewere measuredmeasured inin referencereference [[10].10].

(a) (b)

(a) Figure 4. Cont. (b)

Figure 4. Cont.

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(c) (d)

Figure 4. The final design of metamaterial employed for FEA and experimental investigation. Figure 4. The final design( ofc) metamaterial employed for FEA and experimental(d) investigation. (a) Front (a) Front view and perspective view of buckling‐induced metamaterial (PSF = 0%; density: 2798.05 kg m−3, view and perspective view of buckling-induced metamaterial (PSF = 0%; density: 2798.05 kg m´3, overallFigure mass: 4. 228.5The final g, mass design error: of 1.08%,metamaterial height employed × width × fordepth: FEA 92.8 and mm experimental × 88 mm ×investigation. 10 mm, Scale bar: overall mass: 228.5 g, mass error: 1.08%, height ˆ width ˆ depth: 92.8 mm ˆ 88 mm ˆ 10 mm, Scale 20 mm);(a) Front (b) viewfront and view perspective and perspective view of buckling view ‐ofinduced the metamaterial metamaterial (PSFwith = altered0%; density: geometry 2798.05 kg(PSF m− 3=, 20%, bar: 20 mm); (b) front view and perspective view of the metamaterial with altered geometry (PSF = 20%, overalloverall mass: mass: 222.57 228.5 g,g, mass density: error: 1.08%,2725.43 height kg × mwidth−3, mass × depth: error: 92.8 mm2.06%, × 88 mmheight × 10 mm,× width Scale bar:× depth: ´3 overall92.8 20mm mm); mass: × 88 (b mm) 222.57front × 10view g,mm, and density: Scale perspective bar: 2725.43 20 viewmm); kg of ( cthe m) schematic metamaterial, mass diagram error: with 2.06%,altered of central geometryheight regionˆ (PSFwidth with = 20%, ˆ16depth nodes : overall mass: 222.57 g, density: 2725.43 kg m−3, mass error: 2.06%, height × width × depth: 92.8(PSF mm = 0%);ˆ 88 (d mm) schematicˆ 10 mm diagram, Scale bar:of central 20 mm); region (c) schematic with 16 nodes diagram (PSF of = central 20%). The region image with processing 16 nodes 92.8 mm × 88 mm × 10 mm, Scale bar: 20 mm); (c) schematic diagram of central region with 16 nodes (wasPSF used = 0% );to ( dmeasure) schematic the diagramhorizontal of and central vertical region center with‐to 16‐center nodes distances (PSF = 20% of). nodes. The image processing (PSF = 0%); (d) schematic diagram of central region with 16 nodes (PSF = 20%). The image processing was used to measure the horizontal and vertical center-to-center distances of nodes. was used to measure the horizontal and vertical center‐to‐center distances of nodes.

((aa) )

(b) (b)

(c)

Figure 5. Comparison of deformation patterns of three new designed metamaterials between Figure 5. Comparison of deformation patterns of three new designed metamaterials between numerical numerical and experimental results. (a) Experimental(c) and numerical results of initial design of and experimental results. (a) Experimental and numerical results of initial design of auxetic elastic Figureauxetic 5. Comparisonelastic metamaterial of deformation (rubber) with patterns PSF = 0%; of (b ) threeexperimental new designed and numerical metamaterials results of initial between metamaterialdesign of (rubber)metallic metamaterial with PSF = 0%; with (b )PSF experimental = 0%; (c) experimental and numerical and results numerical of initial results design of the of new metallic numerical and experimental results. (a) Experimental and numerical results of initial design of metamaterialdesign of withauxetic PSF metallic = 0%; metamaterial (c) experimental with andPSF numerical= 20% (Scale results bar: 20 of themm, new load design direction of auxeticY, auxetic elastic metamaterial (rubber) with PSF = 0%; (b) experimental and numerical results of initial strain rate: 5 × 10−3 S−1). ˆ ´3 ´1 metallicdesign of metamaterial metallic metamaterial with PSF = with 20% (ScalePSF = bar:0%; 20(c) mm, experimental load direction and Y,numerical strain rate: results 5 10of theS new). design of auxetic metallic metamaterial with PSF = 20% (Scale bar: 20 mm, load direction Y, strain rate: 5 × 10−3 S−1).

Materials 2016, 9, 54 8 of 17

3.2. Experimental Setup for Uniaxial Compression Testing of 2D Metallic Metamaterials Quasi-static uniaxial compression tests were carried out to investigate the performance of 2D metallic auxetic metamaterials under compression. The tensile tests were conducted by a Shimadzu machine (Shimadzu Company, Kyoto, Japan) at a fixed strain rate of 5 ˆ 10´3 s´1. The deformation process was captured by two cameras. The first camera was used to capture photos in the lateral direction of our 2D test specimens every 30 s. The second camera was employed to record a performance video of the test process. The nonlinear response of metamaterials was considered to be in the range of effective auxetic strain. In other words, the focus of this research was on the behavior of metamaterials, whereas the value of nominal applied strain was below the densification strain. Densification strain was defined as the maximum value of applied strain that satisfies the condition of maximum energy efficiency, as used by Shen et al.[36]. The experimental value of Poisson’s ratio was calculated through image processing from the nodes in the central parts of ligaments, as shown in Figure4c,d. Indeed, at a specific value of nominal strain, the xi,j, yi,j coordinates of the centroid point of each unit cell were determined. The row and column indices are represented by 1 ! i !8 and 1 ! j !8, respectively. ` ˘ The central area under consideration consists of 16 nodes; these are designated with a red dashed line indicating 3 ! i ! 6 and 3 ! j ! 6, as shown in Figure4c,d. The horizontal and vertical centroid-to-centroid distances were calculated by using coordinates xi,j, yi,j , ∆xi,j “ xi`1,j ´ xi´1,j and ∆y “ y ´ y . Also, the center to center distances between the points of undeformed unit i,j i,j`1 i,j´ 1 ` ˘ cells before compression were defined by ∆x p0q “ ∆y p0q “ 11 mm. Equation (1) was employed to calculate the local values of the engineering strain Poisson’s ratio.

∆Xi,j ∆x p0q υi,j “ ´ (1) ∆Yi,j ∆y p0q Eventually, four values of Poisson’s ratio were calculated from the central nodes under consideration, and the average of them was computed at each specific value of nominal strain. The deformation pattern of elastic buckling-induced auxetic metamaterial with a rubber base material is shown in Figure5a, which is similar to the pattern observed by Bertoldi et al. [11].

3.3. Experimental Results As expected, the deformation pattern of a brass metamaterial undergoing plastic flow was different from that of a rubber sample with only elastic deformation. When the base material was changed to metal, the auxetic behavior disappeared, and a new non-auxetic pattern similar to the global buckling pattern for elastic bulk materials [10] was observed, as shown in Figure5b. This pattern usually yielded a zero or positive Poisson’s ratio. In contrast, the test specimen with altered microstructures (PSF = 20%) exhibited obvious auxetic behavior, as shown in Figure5c. It developed from a localized buckling pattern and was similar to mutual ellipses with long orthogonal axes. The results for all samples are presented in Figure6 as a function of nominal strain. These experimental results revealed the significant influence of base materials with nonlinear properties or metal plasticity on the auxetic behavior of the buckling-induced auxetic materials. They also proved the effectiveness of designing the metallic auxetic metamaterials using a buckling mode to alter the initial geometry of buckling-induced metamaterials. By applying different values of PSF, a family of metallic metamaterials were generated, which exhibited obvious auxetic behavior nearly in the full range of its deformation process, as illustrated in Figure6. A detailed analysis on the deformation process of the metallic specimen was conducted to reveal the reason behind the loss of auxetic behavior in Figure5b. At the beginning of the compression process ( ε < 0.06), the initial localization of plastic deformation occurred within the bottom layer of cells, and they were crushed firstly. As compression progressed, more localization took place within the central layers until all layers of the specimen were crushed and a non-auxetic deformation pattern was formed, as shown in Figure5b. The observed Materials 2016, 9, 54 9 of 17 crushing process also showed that the localized plastic deformation did not propagate layer by layer. This deformationMaterials 2016, 9, 54 localization triggered a different pattern without auxetic behavior. Thus,9 of the 17 key feature of the observed pattern in the brass specimen with PSF = 0% was the localization of plastic Thus, the key feature of the observed pattern in the brass specimen with PSF = 0% was the deformation. Inherently, deformation localization in one layer was caused by the sudden decrease of localization of plastic deformation. Inherently, deformation localization in one layer was caused by loadingthe modulus sudden decrease of the metallic of loading base modulus material. of Thethe metallic starting base layer material. for this The localization starting layer was determinedfor this by thelocalization manufacture was imperfectiondetermined by in the the manufacture brass specimen. imperfection in the brass specimen.

Figure 6. Comparison of auxetic and non‐auxetic behavior of new design of auxetic metallic Figure 6. Comparison of auxetic and non-auxetic behavior of new design of auxetic metallic metamaterial with PSF = 20%, initial design of metallic metamaterial with PSF = 0 and initial design metamaterial with PSF = 20%, initial design of metallic metamaterial with PSF = 0 and initial design of of auxetic elastic metamaterial (rubber) with PSF = 0% (Scale bar: 20 mm, load direction Y, strain rate: auxetic5 × elastic 10−3 S−1). metamaterial (rubber) with PSF = 0% (Scale bar: 20 mm, load direction Y, strain rate: 5 ˆ 10´3 S´1). 4. FE Simulations 4. FE Simulations 4.1. Post‐Buckling Analysis and Validation of Numerical Models 4.1. Post-Buckling Analysis and Validation of Numerical Models FE simulations were conducted to further clarify the loss of auxetic behavior of metallic FEbuckling simulations‐induced weremetamaterials conducted and toto furtherreveal the clarify features the observed loss of in auxetic experiments behavior on the of new metallic buckling-inducedmetallic auxetic metamaterials metamaterials. andA nonlinear to reveal post the‐buckling features analysis observed on all in of experiments those metamaterials on the new metallicunder auxetic uniaxial metamaterials. compression was A carried nonlinear out using post-buckling ABAQUS/Explicit analysis (Simulia, on all Providence, of those metamaterials RI, USA). underIt uniaxial should be compression noted that the was results carried of post out‐buckling using ABAQUS/Explicit analyses were affected (Simulia, by the Providence,effects of complex RI, USA). self‐contact and large deformations. Remarkably, the complex self‐contact was an unavoidable It should be noted that the results of post-buckling analyses were affected by the effects of complex factor for large deformation analysis [37]. To overcome this obstacle, the explicit simulation was self-contact and large deformations. Remarkably, the complex self-contact was an unavoidable factor carried out with a prescribed velocity profile. The mechanical properties of brass were set as a for largebilinear deformation material model analysis in FEA. [37]. In To numerical overcome models, this obstacle, the classical the metal explicit plasticity simulation model was used carried to out with adefine prescribed the yield, velocity hardening profile. rule, The mechanicaland inelastic properties flow of the of brassmetamaterials were set asat arelatively bilinear materiallow modeltemperatures. in FEA. In numerical The plasticity models, behavior the of classical this model metal was plasticity simplified model as a bilinear was used isotropic to define hardening the yield, hardeningbehavior. rule, The and von inelastic Mises yield flow surface of the metamaterialsis used to measure at relatively isotropic lowyielding temperatures. by defining Thethe exact plasticity behaviorvalue of of this yield model stress was as simplified a function as of a uniaxial bilinear isotropicequivalent hardening plastic strain. behavior. The inertia The von effect Mises was yield surfaceminimized is used to by measure defining isotropic amplitude yielding to apply by the defining velocity the on exacttop of value the FE of model yield stressas it was as a used function in of uniaxialreference equivalent [37]. The plastic velocity strain. was The applied inertia gradually effect was and minimizedthe acceleration by defining at the beginning amplitude and to ending apply the of compression process was zero [10]. velocity on top of the FE model as it was used in reference [37]. The velocity was applied gradually At the first stage of post‐buckling analysis, the FE simulations were carried out utilizing 3D and the acceleration at the beginning and ending of compression process was zero [10]. solid elements, the same as the previous buckling analysis. The deformation process of the newly Atdesigned the first metallic stage ofauxetic post-buckling metamaterial analysis, with PSF the = FE 20% simulations is shown in were Figure carried 5c. A comparison out utilizing of 3D the solid elements,deformation the same process as the between previous the buckling experiments analysis. and The the deformationnumerical model process revealed of the an newly excellent designed metallicagreement auxetic between metamaterial them, as with shown PSF in = Figure 20% is 5. shown in Figure5c. A comparison of the deformation process betweenThree‐dimensional the experiments shell andelements the numerical were used model to reduce revealed the an computational excellent agreement costs. Four between them,nodes—standard, as shown in Figure linear5. square element with second order accuracy (S4R)—were used with at least Three-dimensionalfive integration points shell at the elementsminimum links were in usedthe FE tomodel. reduce The FE the simulation computational results with costs. shell Four nodes—standard,elements were linear validated square by elementcomparing with with second experimental order accuracy results and (S4R)—were also with used the result with atof least previous post‐buckling analyses with 3D solid elements, as shown in Figure 7a. They agreed with five integration points at the minimum links in the FE model. The FE simulation results with shell each other in general trend and features. However, the level of stress corresponding to FE results elements were validated by comparing with experimental results and also with the result of previous was higher than the stress level from the experimental results. These differences were attributed to post-buckling analyses with 3D solid elements, as shown in Figure7a. They agreed with each other in

Materials 2016, 9, 54 10 of 17 general trend and features. However, the level of stress corresponding to FE results was higher than the stressMaterials level 2016 from, 9, 54 the experimental results. These differences were attributed to the imperfection10 of 17 of the printed specimen in our experiments, which was supported by the large variation of properties, the imperfection of the printed specimen in our experiments, which was supported by the large listed in our previous work [29]. variation of properties, listed in our previous work [29].

Nominal strain Nominal strain 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 20 0.200 0.0 0.200 Exp_PSF= 20% Energy efficiency 18 -0.1 FE _3D shell elements-PSF=20% FE _3D shell elements-PSF= 20% 0.175 0.175 FE _3D solid elements-PSF= 20% -0.2 16 Energy efficiency 0.150 Starting point of 0.150

14 y -0.3 Effective auxetic strain range=5.7876E-10 0.125 0.125 12 -0.4

10 0.100 -0.5 0.100 efficienc

8 gy -0.6

Stress (MPa) 0.075 0.075

Poisson's ratio Effective auxetic strain range Ener 6 -0.7 efficiency Energy 0.050 Densification Strain 0.050 Densification Strain 4 -0.8 0.2040 0.2040 0.025 0.025 2 -0.9

0 0.000 -1.0 0.000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Nominal strain Nominal strain (a) (b)

Figure 7. (a) Comparison of nominal stress‐strain curves of the auxetic metamaterial between experiment Figure 7. (a) Comparison of nominal stress-strain curves of the auxetic metamaterial between and FE results (FE models with 3D shell elements and 3D solid elements) and using energy efficiency experimentmethod and to FEfind results densification (FE models strain; ( withb) defining 3D shell the lower elements bound and and 3D upper solid bound elements) of effective and using energy efficiencyauxetic strain method range. to find densification strain; (b) defining the lower bound and upper bound of effective auxetic strain range. 4.2. Defining a Strain Range for Auxetic Metamaterials Using the Energy Efficiency Method 4.2. DefiningIn a this Strain study, Range the foreffective Auxetic auxetic Metamaterials strain range Using was defined the Energy as the Efficiency strain between Method the starting point for negative value of Poisson’s ratio shown in Figure 7b and the densification point from In this study, the effective auxetic strain range was defined as the strain between the starting point stress‐strain shown in Figure 7a,b. The starting point of effective auxetic strain range was defined as for negativethe first value point of that Poisson’s Poisson’s ratio ratio shownbegan to in decline Figure as7 shownb and in the Figure densification 7b. point from stress-strain shown in FigureThe auxetic7a,b. The behavior starting of the point metallic of effectivemetamaterial auxetic started strain after rangeapplication was of defined a very small as the strain first point that Poisson’sin the compression ratio began test, to and decline this value as shown was close in Figure to zero.7 Theb. endpoint of the effective auxetic range Theis auxeticdensification behavior strain; of this the physically metallic corresponds metamaterial to the started start point, after applicationfrom which a of sharply a very rise small of strain in the compressionstress is observed test, in and the this stress value‐strain was relationship close to zero.as shown The in endpoint Figure 7a. of It the is shown effective that auxeticfrom the range is densification strain onwards, the stress level in stress‐strain curve rises sharply. The term “effective” densification strain; this physically corresponds to the start point, from which a sharply rise of stress is is used to indicate two important features of the metallic auxetic metamaterials within the described observedstrain in the range stress-strain for their potential relationship applications. as shown One is in that Figure the 7nominala. It is shownPoisson’s that ratio from is negative the densification and strain onwards,similar, and the the stress other is level that inthe stress-strain stress level over curve the entire rises deformation sharply. The range term is similar “effective” and below is used to indicatethe two densification important strain. features A relative of the constant metallic plateau auxetic stress metamaterials and similar Poisson’s within the ratio described over the entire strain range for theirdeformation potential applications. range from the One FE results is that was the nominalcalculated Poisson’s using an energy ratio is efficiency negative method, and similar, as was and the other isused that in the references stress level [36,38]. over In thethis entiremethod, deformation firstly the absorbed range isenergy similar is defined and below as the the total densification area under the stress‐strain curve. Then, the energy efficiency parameter is found by dividing the strain. Aabsorbed relative energy constant by the plateau stress itself, stress as and shown similar below: Poisson’s ratio over the entire deformation range from the FE results was calculated using an energy efficiency method, as was used in references [36,38]. In this method, firstly the absorbed energy is defined as the total area under the stress-strain(2) curve. Then, the energy efficiency parameter is found by dividing the absorbed energy by the stress itself, as When the E(ε) reaches the maximum point, the corresponding strain is the densification strain. shown below: The densification strain is marked by a dashed line in Figure 7a,b. As a special feature, the fact that ε the densification strains from the experimental dataσ pareεq d similarε to those from the FE results also E pεq “ 0 (2) proved the accuracy of the FE models. ş σ When the E(ε) reaches the maximum point, the corresponding strain is the densification strain. 4.3. Parametric Studies The densification strain is marked by a dashed line in Figure7a,b. As a special feature, the fact that the After the numerical models were validated, parametric studies were carried out to confirm the densification strains from the experimental data are similar to those from the FE results also proved cause of loss of auxetic behavior for buckling‐induced metallic specimens identified in the the accuracyexperiments. of the As FE for models. the pattern‐altered specimen, the dominant factors to control the performance of metallic auxetic metamaterials were identified and qualified using the validated FE models. Based 4.3. Parametric Studies After the numerical models were validated, parametric studies were carried out to confirm the cause of loss of auxetic behavior for buckling-induced metallic specimens identified in the experiments. As for the pattern-altered specimen, the dominant factors to control the performance of metallic auxetic Materials 2016, 9, 54 11 of 17 metamaterialsMaterials 2016, were 9, 54 identified and qualified using the validated FE models. Based on experimental11 of 17 observations, the reason for the loss of auxetic behavior for metallic specimens was related to the plastic on experimental observations, the reason for the loss of auxetic behavior for metallic specimens was deformation of the buckling-induced auxetic metamaterials. The localization of the plastic deformation related to the plastic deformation of the buckling‐induced auxetic metamaterials. The localization of caused by buckling at an RVE length scale resulted in a global deformation mode rather than the the plastic deformation caused by buckling at an RVE length scale resulted in a global deformation desiredmode buckling rather than pattern the desired in all RVEs. buckling It was pattern difficult in all toRVEs. verify It was this difficult assumption to verify directly. this assumption However, two methodsdirectly. to preventHowever, the two localization methods to of prevent plastic the strain localization could be of appliedplastic strain to buckling-induced could be applied design.to The firstbuckling method‐induced is eliminating design. The buckling first bymethod introducing is eliminating a large imperfection;buckling by theintroducing second methoda large is the increaseimperfection; of strain hardeningthe second method modulus is the of itsincrease base materialof strain hardening close to the modulus elastic of modulus. its base material If those close methods shouldto bethe effective,elastic modulus. the assumption If those methods would beshould proved be indirectly.effective, the As assumption mentioned would before, be the proved buckling modeindirectly. alternation As ofmentioned the microstructure before, the buckling could be mode used alternation as an innovative of the microstructure approach to design could newbe used metallic auxeticas an metamaterials. innovative approach To enhance to design and facilitatenew metallic its potential auxetic metamaterials. applications, anTo individualenhance and tuning facilitate method its potential applications, an individual tuning method should be provided to assist with the design should be provided to assist with the design of a metallic metamaterial with a prescribed performance. of a metallic metamaterial with a prescribed performance. Based on the feature of metamaterials, the Based on the feature of metamaterials, the change of Poisson’s ratio can be achieved by adjusting the change of Poisson’s ratio can be achieved by adjusting the geometrical configuration of geometricalmicrostructures, configuration while other of microstructures, mechanical properties while othercan be mechanical achieved by properties changing the can elastoplastic be achieved by changingproperties the elastoplasticof its base material. properties It is ofdifficult its base to individually material. It change is difficult those to features individually by experimental change those featuresmethods, by experimental so the validated methods, FE model so was the used validated to do the FE parametric model was study. used to do the parametric study. In thisIn study,this study, the magnitudethe magnitude of alterationof alteration was was represented represented by by PSF. PSF. As As mentioned mentioned before, the the PSF was aPSF key was parameter a key parameter to alter to the alter initial the initial topology topology of RVEof RVE with with desired desired buckling buckling mode. mode. The The range range of variationof variation of PSF of was PSF between was between 0% and 0% 100%. and 100%. A series A series of systematic of systematic simulations simulations were were conducted conducted to to study the influencestudy the of influence PSF on auxetic of PSF on performance auxetic performance of the new of metallic the new auxetic metallic metamaterials. auxetic metamaterials. The elastoplastic The elastoplastic properties of brass were defined identically to the material properties in the experiment. properties of brass were defined identically to the material properties in the experiment. The results The results are presented in Figure 8, in which the effective auxetic strain range is marked with are presented in Figure8, in which the effective auxetic strain range is marked with round dots. round dots.

0.1 FE result(PSF=0%) 0.0 FE result(PSF=10%) -0.1 20 FE result(PSF=0%) FE result(PSF=20%) -0.45 18 FE result(PSF=10%) FE result(PSF=30%) -0.50 FE result(PSF=40%) FE result(PSF=20%) 16 -0.55 FE result(PSF=50%) FE result(PSF=30%) 14 FE result(PSF=60%) -0.60 FE result PSF=40% ( ) 12 FE result(PSF=80%) -0.65 FE result(PSF=50%) 10 -0.70 FE result(PSF=60%) -0.75 FE result(PSF=80%) 8 Stress (MPa) Poisson's ratio Poisson's -0.80 6 -0.85 4 -0.90 2 -0.95 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Nominal strain Nominal strain (a) (b) PSF 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 Poisson's ratio 0.225 -0.1 Effective auxetic strain range 0.200 -0.2

-0.3 0.175

-0.4 0.150

-0.5 0.125 -0.6 0.100 Poisson's ratio -0.7 Ideal region of effecitve 0.075 -0.8 auxetic strain design Effective strain range 0.050 -0.9

-1.0 0.025 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PSF (c)

Figure 8. Parametric study on influence of PSF on auxetic behavior. (a) Evaluations of Poisson’s ratio Figure 8. Parametric study on influence of PSF on auxetic behavior. (a) Evaluations of Poisson’s ratio as function of applied strain corresponding to different PSF; (b) stress‐strain curves corresponding to as function of applied strain corresponding to different PSF; (b) stress-strain curves corresponding to different PSF; (c) average values of Poisson’s ratio and effective auxetic strain ranges corresponding c differentto different PSF; ( PSF.) average values of Poisson’s ratio and effective auxetic strain ranges corresponding to different PSF.

Materials 2016, 9, 54 12 of 17

According to these results, it was determined that the increase of PSF resulted in the decrease of the densificationMaterials 2016 strain, 9, 54 and thus the effective auxetic strain range for auxetic behavior. This phenomenon12 of 17 was obvious,According as shown to these in Figure results,8 c.it was By increasing determined the that PSF, the increase the mutual of PSF ellipses resulted in RVEin the became decrease smaller, of thusthe the densification internal walls strain of the and elliptical thus the parts effective touched auxetic each strain other range at a for smaller auxetic strain. behavior. Therefore, This the densificationphenomenon strains was decreased obvious, as accordingly. shown in Figure 8c. By increasing the PSF, the mutual ellipses in RVE becameThe nonlinear smaller, response thus the internal of Poisson’s walls ratioof the with elliptical respect parts to touched compressive each other strain at is a shownsmaller in strain. Figure 8a withTherefore, different PSFthe densification values. The strains negative decreased values accordingly. of Poisson’s ratio did not change much with respect to PSF, especiallyThe nonlinear when response the PSF of are Poisson’s in the range ratio of with 10% respect to 60%. to Thiscompressive finding revealedstrain is shown that the in value of Poisson’sFigure 8a ratio with isdifferent dominated PSF values. by the The deformation negative values patterns of Poisson’s initiated ratio by did the not buckling change much mode with applied. respect to PSF, especially when the PSF are in the range of 10% to 60%. This finding revealed that the In this PSF range, the effective auxetic strain range can be individually designed with a prescribed value of Poisson’s ratio is dominated by the deformation patterns initiated by the buckling mode negative value of Poisson’s ratio required by the aimed application, as illustrated in Figure8c. An ideal applied. In this PSF range, the effective auxetic strain range can be individually designed with a regionprescribed for individually negative value altering of Poisson’s the effective ratio auxetic required strain by the is definedaimed application, and illustrated as illustrated with a dashedin rectangleFigure in 8c. Figure An ideal8c. Withinregion for this individually region, Poisson’s altering the ratio effective maintained auxetic a strain constant is defined negative and illustrated value and the effectivewith auxetica dashed strain rectangle can in be Figure individually 8c. Within changed this region, by Poisson’s the value ratio of PSF.maintained a constant negative valueAs mentioned and the effective previously, auxetic thestrain strain can be hardening individually of changed the base by materialthe value of could PSF. be used to prevent the localizationAs mentioned of the previously, plastic deformation the strain hardening caused by of bucklingthe base material at RVEs. could Plastic be used strain to prevent hardening the is a commonlocalization phenomenon of the plastic for metallic deformation materials, caused which by buckling can be at simplified RVEs. Plastic as a strain linear hardening relationship is a with respectcommon to plastic phenomenon strain. It isfor represented metallic materials, by the slopewhich of can this be line,simplified termed as the a linear strain relationship hardening with modulus respect to plastic strain. It is represented by the slope of this line, termed the strain hardening (Ep), in the stress-strain curves shown in Figure9c. In most cases, the stress-strain curve of a metallic modulus (Ep), in the stress‐strain curves shown in Figure 9c. In most cases, the stress‐strain curve of materiala metallic is determined material is by determined the elastic by region the elastic (Es) as region well as(Es) the as plastic well as region the plastic (Ep). region In this (Ep). study, In this a simple ratiostudy, was defined a simple as ratio Ep/Es was to characterizedefined as Ep/Es the effectto characterize of plastic the hardening effect of on plastic the auxetic hardening performance on the of our designedauxetic performance metamaterials. of our This designed ratio wasmetamaterials. termed the This “plastic ratio was hardening termed ratio.”the “plastic The variationhardening range of thisratio.” ratio The is between variation 0range and 1.of this ratio is between 0 and 1.

(PSF=0%, Elastic perfectly plastic) 0.10 (PSF=0%, Brass, Ep/Es=0.02 ) 0.05 150 (PSF=0%, Ep/Es=0.1) 0.00 (PSF=0%, Ep/Es=0.2) -0.05 -0.10 (PSF=0%, Ep/Es=0.4) 125 -0.15 (PSF=0%, Ep/Es=0.6) -0.20 (PSF=0%, Elastic perfectly plastic) (PSF=0%, Ep/Es=0.8) -0.25 (PSF=0%, Brass, Ep/Es=0.02) 100 (PSF=0%, Ep/Es=1) -0.30 (PSF=0%, Ep/Es=0.1) -0.35 -0.40 (PSF=0%, Ep/Es=0.2) 75 -0.45 (PSF=0%, Ep/Es=0.4) -0.50 (PSF=0%, Ep/Es=0.6) -0.55 Poisson's ratio Poisson's Stress (MPa) 50 -0.60 (PSF=0%, Ep/Es=0.8) -0.65 (PSF=0%, Ep/Es=1) -0.70 25 -0.75 -0.80 -0.85 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Nominal strain Nominal strain (a) (b)

(c)

Figure 9. The results of a parametric study on the influence of plastic hardening on the recovery of Figure 9. The results of a parametric study on the influence of plastic hardening on the recovery of auxetic behavior. (a) Evaluations of Poisson’s ratio as a function of applied strain corresponding to auxetic behavior. (a) Evaluations of Poisson’s ratio as a function of applied strain corresponding to different hardening ratios for metamaterials with PSF = 0%.; (b) stress‐strain curves corresponding to b differentdifferent hardening hardening ratios ratios for for metamaterials metamaterials withwith PSF = = 0%; 0%.; (c) ( schematic) stress-strain curve curvesof stress corresponding‐strain for to differentmetallic hardening materials (strain ratios hardening for metamaterials ratio). with PSF = 0%; (c) schematic curve of stress-strain for metallic materials (strain hardening ratio).

Materials 2016, 9, 54 13 of 17

MaterialsA ratio 2016 of, 9, 0 54 represents perfectly plastic material and a ratio of 1 represents purely13 of 17 elastic material. The result for a plastic hardening ratio of 1 with buckling-induced metamaterials confirmed A ratio of 0 represents perfectly plastic material and a ratio of 1 represents purely elastic that increasing the plastic hardening ratio will restore the auxetic behavior of buckling-induced material. The result for a plastic hardening ratio of 1 with buckling‐induced metamaterials metamaterials. As shown in Figure5a, the experimental rubber specimen with PSF = 0% exhibited confirmed that increasing the plastic hardening ratio will restore the auxetic behavior of auxeticbuckling behavior‐induced under metamaterials. compression As asshown a purely in Figure elastic 5a, material the experimental or when rubber Ep/Es specimen = 1. This with finding wasPSF further = 0% validated exhibited byauxetic a numerical behavior investigationunder compression on the as initiala purely geometry elastic material of buckling-induced or when metamaterialEp/Es = 1. withThis finding PSF = 0%.was Thefurther FE validated results for by differenta numerical values investigation of Ep/Es on at the PSF initial = 0 geometry are presented of in Figurebuckling9a,b. It‐induced is shown metamaterial that the response with PSF of = 0%. a metamaterial The FE results changed for different from values non-auxetic of Ep/Es at to PSF auxetic = 0 by increasingare presented the plastic in Figure hardening 9a,b. ratio.It is shown Based onthat this the evidence, response theof a reason metamaterial behind changed the loss offrom auxetic behaviornon‐auxetic was attributed to auxetic toby theincreasing localization the plastic of the hardening plastic ratio. deformation Based on caused this evidence, by buckling the reason at RVEs. To evaluatebehind the the loss effect of auxetic of plasticity behavior on was the auxeticattributed behavior to the localization of our newly of the designed plastic deformation metallic auxetic caused by buckling at RVEs. To evaluate the effect of plasticity on the auxetic behavior of our newly metamaterials, the hardening ratio was varied in the range of 0 to 1. The results of this numerical designed metallic auxetic metamaterials, the hardening ratio was varied in the range of 0 to 1. The parametric study for the newly designed buckling-induced metamaterial with PSF = 20% are presented results of this numerical parametric study for the newly designed buckling‐induced metamaterial in Figurewith PSF10 and = 20% for are other presented values in of Figure PSF are 10 shownand for inother Figure values 11 .of Also, PSF are it should shown bein Figure mentioned 11. Also, that it when Ep/Esshould tends be to mentioned zero, the that response when isEp/Es closer tends to theto zero, rotation the response of rigid is squares closer to and the so rotation the Poisson’s of rigid ratio tendssquares to be ´ and1 [39 so]. the According Poisson’s to ratio the tends numerical to be − results1 [39]. shownAccording in Figuresto the numerical 10 and 11 results it can shown be concluded in that theFigures difference 10 and of 11, the it calculatedcan be concluded negative that value the ofdifference Poisson’s of ratiothe calculated for models negative with different value of strain hardeningPoisson’s ratios ratio were for models negligible. with different Also, the strain effective hardening auxetic ratios strain were ranges negligible. for modelsAlso, the with effective different hardeningauxetic ratios strain wereranges similar for models and the with difference different amonghardening these ratios ranges were can similar be neglected. and the difference However, the stressamong level these increased ranges considerably can be neglected. when However, the hardening the stress ratios level increased increased fromconsiderably 0 to 1. when Based the on this hardening ratios increased from 0 to 1. Based on this observation, metallic auxetic metamaterials observation, metallic auxetic metamaterials undergoing large deformation were dominated by the undergoing large deformation were dominated by the topology of the microstructures; this was topology of the microstructures; this was independent of the properties of the base material. independent of the properties of the base material.

(PSF=20%,Ep/Eps=0.1) 200 (PSF=20%,Ep/Es=0.1) (PSF=20%,Ep/Eps=0.2) 0.0 (PSF=20%,Ep/Es=0.2) (PSF=20%,Ep/Eps=0.4) 175 (PSF=20%,Ep/Es=0.4) -0.1 (PSF=20%,Ep/Eps=0.6) (PSF=20%,Ep/Es=0.6) (PSF=20%,Ep/Eps=0.8) -0.2 150 (PSF=20%,Ep/Es=0.8) (PSF=20%,Ep/Eps=1) (PSF=20%,Ep/Es=1) -0.3 125 -0.4 100 -0.5 75 -0.6 Stress (MPa)

Poisson's ratio -0.7 50

-0.8 25 -0.9 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Nominal strain Nominal strain (a) (b) Ep/Es 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.300 0.275 -0.1 0.250 -0.2 0.225 -0.3 0.200 -0.4 0.175

-0.5 0.150 0.125 -0.6 Poisson's ratio 0.100

Poisson's ratio -0.7 Effective auxetic strain range 0.075 Effective strain range -0.8 0.050 -0.9 0.025

-1.0 0.000 0.00.10.20.30.40.50.60.70.80.91.01.1 Ep/Es (c)

Figure 10. The results of a parametric study on the influence of plastic hardening on the auxetic Figure 10. The results of a parametric study on the influence of plastic hardening on the auxetic behavior of a metamaterial with PSF = 20%. (a) Evaluations of Poisson’s ratio as a function of applied behavior of a metamaterial with PSF = 20%. (a) Evaluations of Poisson’s ratio as a function of applied strain corresponding to different hardening ratios; (b) stress‐strain curves corresponding to different strain corresponding to different hardening ratios; (b) stress-strain curves corresponding to different hardening ratios; (c) average values of Poisson‘s ratio and effective auxetic strain ranges hardeningcorresponding ratios; ( cto) averagedifferent valuesstrain hardening of Poisson‘s ratios ratio for andmetamaterials effective auxetic with PSF strain = 20%. ranges corresponding to different strain hardening ratios for metamaterials with PSF = 20%.

Materials 2016, 9, 54 14 of 17

Materials 2016, 9, 54 14 of 17

-0.700 0.275

-0.725 0.250

-0.750 0.225 -0.775 0.200 -0.800 0.175 -0.825 0.150 -0.850 0.125 -0.875 Poisson's ratio-PSF=10% Effective strain range PSF=10% Poisson's ratio-PSF=20% 0.100 -0.900 Effective strain range PSF=20% Poisson's ratio Poisson's Poisson's ratio-PSF=30% 0.075 -0.925 Effective strain range PSF=30% Poisson's ratio-PSF=40% Effective strain range -0.950 0.050 Effective strain range PSF=40% Poisson's ratio-PSF=50% Effective strain range PSF=50% -0.975 Poisson's ratio-PSF=60% 0.025 Effective strain range PSF=60% -1.000 0.000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Ep/Es Ep/Es (a) (b)

Figure 11. The results of a parametric study on the influence of pattern scale factors on auxetic Figure 11. behaviorThe results and ofeffective a parametric auxetic strain study range. on the (a) influenceEvaluations of of pattern Poisson’s scale ratio factors as the onfunction auxetic of behavior and effectivehardening auxetic ratio strain corresponding range. (a )to Evaluations pattern scale offactors; Poisson’s (b) evaluations ratio as of the effective function auxetic of hardening strain ratio correspondingrange toas the pattern function scale of hardening factors; ratio (b) evaluationscorresponding of to different effective pattern auxetic scale strain factors. range as the function of hardening ratio corresponding to different pattern scale factors. 5. Discussion and Conclusions 5. DiscussionThe and findings Conclusions of this study demonstrated a fundamental way to design a new class of 2D metallic auxetic metamaterials. The new design approach consisted of two important improvements The findingscompared of with this the study previous demonstrated approach for elastomer a fundamental [10,11]. The way first toimprovement design a newwas the class inclusion of 2D metallic auxetic metamaterials.of a geometric Themodification new design of the approach conventional consisted structure of to two generate important a new improvementsbuckling‐induced compared metamaterial. The second improvement was altering the initial geometry of a non‐metallic auxetic with the previous approach for elastomer [10,11]. The first improvement was the inclusion of a metamaterial at the RVE level. The FE models were validated by experiments. A systematic geometric modificationparametric study of was the carried conventional out using validated structure FE to models generate on the a effects new buckling-inducedof the plastic properties metamaterial. of The secondthe improvement base material and was the alteringpattern scale the factor. initial geometry of a non-metallic auxetic metamaterial at the RVE level.It The is interesting FE models to note were that validateda similar approach by experiments. to that of the PSF A systematic has been also parametric used by other study was researchers to direct the deformation path of soft metamaterials [40]. The effect of boundary carried outconditions using validated on the auxetic FE modelsperformance on theof our effects 2D auxetic of the metamaterial plastic properties was also investigated. of the base From material and the patternthe scale results factor. of those investigations, the following conclusions can be drawn: It is interesting(1) The deformation to note process that a similarof the metallic approach auxetic to thatmetamaterial of the PSFis influenced has been by also boundary used by other researchers toconditions; direct the however, deformation the auxetic path ofbehavior soft metamaterials and its performance [40]. Thecorresponding effect of boundary to different conditions on the auxeticdeformation performance processes of our are 2D insensitive auxetic to metamaterial boundary conditions. was also investigated. From the results of those investigations,(2) The auxetic the behavior following of our conclusions designed metallic can metamaterial be drawn: remains within the effective auxetic strain range. (1) The deformation(3) The effective processauxetic strain of can the be metallic controlled auxetic individually metamaterial by PSF while maintaining is influenced a similarly by boundary negative value of Poisson’s ratio, especially when the PSF is in the range of 10% to 60%. conditions;(4) The stiffness however, and strength the auxetic of the behaviormetallic auxetic and metamaterials its performance can be individually corresponding controlled to different deformationthrough processes adjustment are of insensitivethe properties to of the boundary base materials, conditions. while the negative value of Poisson’s (2) The auxeticratio behavior remains relatively of our constant. designed metallic metamaterial remains within the effective auxetic

strain(5) range.The loss of auxetic behavior in the metallic buckling‐induced metamaterial is attributed to the localization of plastic collapse of RVEs. (3) The effective(6) The results auxetic from strain FE simulation can be confirmed controlled that individually the buckling‐induced by PSF metamaterial while maintaining would have a similarly negativeauxetic value behavior of Poisson’s after the ratio, plastic especially hardening modulus when the is large PSF than is in a thecertain range value of determined 10% to 60%. by (4) The stiffnessthe microstructures and strength of ofmetamaterials. the metallic These auxetic results metamaterials confirm that the effectiveness can be individually of increasing controlled plastic hardening ratio will restore the auxetic behavior of buckling‐induced metamaterials. through adjustment of the properties of the base materials, while the negative value of Poisson’s ratio remainsExcept relatively for those solid constant. conclusions based on our numerical and experimental results, there are some concerns relating to the 2D microstructures of our metallic auxetic metamaterials. The mechanism (5) The lossof the of resultant auxetic metallic behavior auxetic in metamaterials the metallic is buckling-induced the rotating of joints/squares. metamaterial The analytical is attributed model to the localizationof rotating of interconnected plastic collapse squares of RVEs.was proposed independently and simultaneously by Ishibashi & (6) The resultsIwata [41] from and FE Grima simulation & Evans confirmed[39]. Thus, most that of the the buckling-induced auxetic behavior of the metamaterial metallic auxetic would have auxetic behavior after the plastic hardening modulus is large than a certain value determined by the microstructures of metamaterials. These results confirm that the effectiveness of increasing plastic hardening ratio will restore the auxetic behavior of buckling-induced metamaterials.

Except for those solid conclusions based on our numerical and experimental results, there are some concerns relating to the 2D microstructures of our metallic auxetic metamaterials. The mechanism of the resultant metallic auxetic metamaterials is the rotating of joints/squares. The analytical model of rotating interconnected squares was proposed independently and simultaneously Materials 2016, 9, 54 15 of 17 by Ishibashi & Iwata [41] and Grima & Evans [39]. Thus, most of the auxetic behavior of the metallic auxetic metamaterial can be predicted by their model. To some extent, the PSF used to define the topology of the RVE can be equivalent to the rotation angle of the square units in the analytical model. The insensitive negative value of Poisson’s ratio of ´0.9 with respect to PSF can be approximately explained by the negative value of Poisson’s ratio of ´1 in a model of the rotating of joints/squares. However, thereMaterials is 2016 a, noticeable 9, 54 advantage to using PSF to design the microstructures of metallic15 of 17 auxetic metamaterials,metamaterial besides can tuneability. be predicted by The their volume model. To fraction some extent, does the not PSF vary used significantly to define the topology through of changing the PSF, asthe shown RVE can in Figure be equivalent 12a. Thus, to the itrotation can be angle revealed of the that square a set units of in 2D the auxetic analytical metallic model. metamaterials The that are obtainedinsensitive with negative different value PSFsof Poisson’s have similar ratio of volume−0.9 with fractions. respect to PSF can be approximately − It shouldexplained be noted by the thatnegative the value Poisson’s of Poisson’s ratio ratio was of measured 1 in a model in of the the centerrotating region of joints/squares. of the specimens However, there is a noticeable advantage to using PSF to design the microstructures of metallic both in experimentsauxetic metamaterials, and FE simulations. besides tuneability. The nominalThe volume strain fraction of thedoes overall not vary specimen significantly was through using to plot all strain-relatedchanging curves. the PSF, Thisas shown method in Figure was 12a. used Thus, in it all can recent be revealed experimental-related that a set of 2D auxetic research metallic on auxetic structuresmetamaterials and materials that [are10 ,obtained11,23,29 with]. This different measurement PSFs have similar resulted volume in fractions. the insensitivity of the measured Poisson’s ratioIt with should respect be noted to that strain the Poisson’s localization. ratio was It willmeasured enlarge in the the center negative region of strain the specimens range for auxetic both in experiments and FE simulations. The nominal strain of the overall specimen was using to behavior, evidencedplot all strain by‐related the constantcurves. This negative method was Poisson’s used in all ratio recent of experimental´0.9 after its‐related effective research strain on of 0.2 in Figures6 andauxetic7. Based structures on the and observed materials [10,11,23,29]. deformation This of measurement a metallic auxetic resulted metamaterialin the insensitivity that of isthe presented in Figure5cmeasured ( ε = 0.2040), Poisson’s the ratio walls with of respect central to rowsstrain localization. of unit building It will enlarge cells are the touching. negative strain However, range the unit − building cellsfor auxetic of the behavior, top and evidenced bottom by rows the constant are not negative compacted Poisson’s completely. ratio of 0.9 As after mentioned its effective before, the strain of 0.2 in Figures 6 and 7. Based on the observed deformation of a metallic auxetic metamaterial base martialthat of is thepresented specimen in Figure was 5c made(ε = 0.2040), of ductile the walls metal. of central Beyond rows of the unit auxetic building straincells are range, touching. the metallic metamaterialHowever, can stillthe unit be compressedbuilding cells of further the top byand plasticbottom rows deformation are not compacted of those completely. not fully As compacted layers. Whilementioned the local before, strains the base in themartial central of the region specimen remain was made nearly of ductile constant metal. or Beyond are slightly the auxetic compressed, the Poisson’sstrain ratio range, will the remain metallic negative.metamaterial Thus, can still the be measurement compressed further of the by plastic Poisson’s deformation ratio in of the center those not fully compacted layers. While the local strains in the central region remain nearly constant region mayor not are slightly be representative compressed, the of thePoisson’s whole ratio specimen will remain when negative. the strainThus, the localization measurement occurs, of the as shown in Figure5Poisson’sb. We present ratio in the those center data region just may for not comparison be representative purposes; of the whole the accuracyspecimen when of this the strain measurement should belocalization checked further.occurs, as Forshown a similarin Figure reason,5b. We present we used those thedata densification just for comparison strain purposes; to define the the end point for anaccuracy effective of this strain measurement range. should be checked further. For a similar reason, we used the densification strain to define the end point for an effective strain range.

0.45

0.40

0.35

0.30

0.25

0.20

0.15 Volume Fraction Volume Fraction Volume 0.10

0.05

0.00 0 1020304050607080 PSF (a) (b)

Figure 12. (a) Relation between PSF and corresponding volume fraction; (b) comparison of auxetic Figure 12.behaviors(a) Relation between between buckling PSF‐induced and design corresponding with PSF = 0%, volume buckling fraction; pattern‐altered (b) comparisondesign without of auxetic behaviorsplates between (Scale buckling-induced bar: 20 mm, load direction design Y, strain with rate: PSF 5 =× 10 0%,−3 S− buckling1). pattern-altered design without plates (Scale bar: 20 mm, load direction Y, strain rate: 5 ˆ 10´3 S´1). In order to study the influence of the boundary on the auxetic performance of a designed In orderspecimen, to study the thetop and influence bottom ofplates the were boundary removed on so the that auxetic the top and performance bottom surfaces of a of designed a new 2D specimen, designed metamaterial with PSF = 20% were not constrained. An experiment was conducted to the top andreveal bottom its effect plates on the were deformation removed process. so that At thethe beginning top and bottomof the test, surfaces a localized of deformation a new 2D designed metamaterialoccurred with near PSF the = 20%loading were end. not With constrained. further compression, An experiment a localized waslateral conducted inwards deformation to reveal its effect on the deformationdeveloped from process. the top At moving the beginning surface toward of the the test, bottom a localized fixed surface. deformation Interestingly, occurred this near the loading end.deformation With further process compression, led to obvious a auxetic localized behavior. lateral The inwards auxetic versus deformation strain curves developed of two from the specimens were put together and the results were presented in Figure 12b. Interestingly, the overall top movingtrend surface and the toward average thevalues bottom of this fixedset of Poisson’s surface. ratio Interestingly, curves were similar. this deformation These observations process led to obvious auxeticillustrated behavior. that although The auxetic the deformationversus strain process curves of our of twometallic specimens auxetic metamaterial were put together under and the results were presented in Figure 12b. Interestingly, the overall trend and the average values of this set of Poisson’s ratio curves were similar. These observations illustrated that although the deformation Materials 2016, 9, 54 16 of 17 process of our metallic auxetic metamaterial under compression was significantly influenced by boundary conditions, the auxetic performance was insensitive to the boundary conditions. It was revealed that the enhancement of the plastic-hardening ratio leads to restoring the auxetic behavior of buckling-induced auxetic metamaterials. According to Figure9b, this effect is obvious when the hardening modulus is in the range of Ep/Es = 0 and Ep/Es = 0.3. Out of this range, the auxetic performance of models with different hardening ratios was similar and their differences could be neglected. This result is in accordance with the finding that the auxetic effect persists and becomes even stronger when the hardening modulus is in the range of h = 100 MPa and h = 1000 MPa by Dirrenberger et al.[25]. However, regarding the metallic auxetic metamaterials with altered microstructures, the influence of an enhancement of the plastic-hardening ratio on auxetic performance is negligible.

Acknowledgments: This work was supported by the Australian Research Council (DP140100213). Author Contributions: Yi Min Xie and Jianhu Shen initiated and supervised this study. Arash Ghaedizadeh and Jianhu Shen carried out the analysis and experiments. Xin Ren provided technical assistance for FE Simulations. Arash Ghaedizadeh wrote the draft manuscript. Yi Min Xie, Jianhu Shen, and Xin Ren edited and revised the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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