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Xiaoyang Zheng, Xiaofeng Guo, [Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675)

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[A mathematically defined 3D auxetic metamaterial with tunable mechanical and conduction properties](https://mdr.nims.go.jp/datasets/74d7c161-b404-4688-ab61-5697031a67df)

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A mathematically defined 3D auxetic metamaterial with tunable mechanical and conduction propertiesMaterials and Design 198 (2021) 109313Contents lists available at ScienceDirectMaterials and Designj ourna l homepage: www.e lsev ie r .com/ locate /matdesA mathematically defined 3D auxetic metamaterial with tunablemechanical and conduction propertiesXiaoyang Zheng a,b, Xiaofeng Guo c, Ikumu Watanabe a,b,⁎a Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japanb Research Center for Structural Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Japanc School of Materials Science and Engineering, Southwest University of Science and Technology, Mianyang 621010, ChinaH I G H L I G H T S G R A P H I C A L A B S T R A C T• We propose a novel 3D auxeticmetamaterial derived from a mathe-matically defined triply periodic mini-mal surface.• Nickel plating enhanced the stiffness,strength, and conductivity of themetamaterial without the auxeticityand resilience.• The relationships between geometricparameters and material properties ofthe metamaterial were discussed.• These relationships provide insight intotuning its performance over a broadrange.⁎ Corresponding author at: Graduate School of Pure andTsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan.E-mail address: WATANABE.Ikumu@nims.go.jp (I. Wahttps://doi.org/10.1016/j.matdes.2020.1093130264-1275/© 2020 The Authors. Published by Elsevier Ltda b s t r a c ta r t i c l e i n f oArticle history:Received 20 September 2020Received in revised form 4 November 2020Accepted 8 November 2020Available online 17 November 2020Keywords:Negative Poisson’s ratioMetamaterialMinimal surface3D printingElectroless platingAn auxetic metamaterial is a type of mechanical metamaterial that has a negative Poisson's ratio. Most auxeticmetamaterials are truss-based or originate from Boolean operations of simple geometries. Herein, we introducea new 3D auxetic metamaterial that is mathematically generated from an implicit expression. Further, thismetamaterial is fabricated by 3Dprinting using a flexiblematerial, which allows it to recover from large deforma-tions. The buckling-induced auxetic behavior of the metamaterial was first evaluated via compression tests andfinite element analyses. A nickel layerwas then plated onto the surface to enhance its stiffness, strength, and con-ductivity without loss of auxeticity and resilience. The integration of 3D printing and electroless plating enabledaccurate control over themechanical and conduction properties of the auxeticmetamaterial; these properties arepresented as contour maps for guidance in functional applications.© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).1. IntroductionAuxetic metamaterials constitute a class of mechanical metamateri-als that have topology-controlled properties. They have attracted wide-spread attention because of their inherently unique mechanicalApplied Sciences, University oftanabe).. This is an open access article underperformance in terms of their negative Poisson's ratio [1–4]. Whenauxeticmetamaterials are uniaxially compressed, they contract in all di-rections, whereas common structural materials, such as polymers andpolycrystalline metals, whose Poisson's ratios are generally in therange of 0.25–0.35, expand in a direction perpendicular to the appliedload. Further, when bent, a plate made of an auxetic metamaterial de-forms into a convex shape, whereas a plate made of a material with apositive Poisson's ratio deforms into a saddle shape. Because of specificcharacteristics such as synclastic behavior, shear resistance, fracturethe CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).http://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2020.109313&domain=pdfhttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/https://doi.org/10.1016/j.matdes.2020.109313mailto:WATANABE.Ikumu@nims.go.jphttps://doi.org/10.1016/j.matdes.2020.109313http://creativecommons.org/licenses/by-nc-nd/4.0/http://www.sciencedirect.com/science/journal/www.elsevier.com/locate/matdesX. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313resistance, indentation resistance, and variable permeability, they areconsidered as promising candidates in strain sensing [5,6], actuation[7,8], biomedicine [9,10], aerospace applications [11,12], and electro-chemical energy storage and conversion [13–15]. Consequently, consid-erable efforts have been devoted to the development and fabrication ofnew auxetic metamaterials.Auxetic metamaterials are typically classified into re-entrant[16–23], chiral [24–28], rotating [29–34], and hierarchical laminatestructures [35–37] according to their deformation mechanisms. In gen-eral, the re-entrant, chiral, and rotating structures are porous and com-posed of a single component, whereas the hierarchical laminatestructures are solid and consist of two or more components with differ-ent Poisson's ratios. The empty spaces in the interior structures of thesematerials are necessary for rotation, bending, and torsion of the liga-ments and nodes in the former three analogies, resulting in low stiff-nesses and strengths compared with their matrix materials. Mostauxetic metamaterials have periodic array structures according totheir design principles in terms of the components, porosities, and peri-odicities. These basic arrays generally comprise beams, trusses, or shells,or are generated from Boolean operations of simple geometries, such ascylinders, cones, spheres, and boxes (Fig. 1). The modeling methodgreatly limits its applications, especially where optimal performancesare needed.However, triply periodicminimal surfaces (TPMSs), which are a typeof metamaterial, have recently gained much attention owing to theirFig. 1. Three common methods used to generate auxetic metamaterials: Boolean operationsgenerate serveral complicated structures based on implicit expressions.2mathematically controlled and fascinating topologies [38,39], such astheir bioinspired morphology, smooth surfaces without edges and cor-ners [40], and higher stiffness and strength than their disordered coun-terparts [41]. Extensive efforts have been invested to expand the designspaces of TPMS-basedmaterials, such as skeletal, sheet, graded, and hy-brid lattice structures [42–46]; however, thus far, only a few studieshave focused on achieving negative Poisson's ratios [47,48]. Herein,we extend the application of TPMSs to auxetic metamaterials byexploiting their particular characteristics.3D printing or additive manufacturing is a standard method usedto fabricate delicately designed auxetic metamaterials. Rubber-likematerials, such as flexible resins and thermoplastic polyurethane,are the typical raw materials used for 3D-printed auxetic metamate-rials, which render them resilient but result in inadequate stiffnessand strength [49–51]. Metal-based 3D printing techniques allowthe fabrication of auxetic metamaterials made of metals, such asstructural steel and titanium alloy [52,53]; however, the plasticityhighly restricts their application where large deformations are re-quired. Although some fabrication methods have been developed toensure both resilience and stiffness, the feasible structures are highlylimited. For instance, wire-woven metals manufactured by forminghelical wires can achieve high energy absorption and fracture resis-tance capabilities. However, their basic geometries are mainly 3Dtruss-like cells, which do not allow fabricating complex internalshapes [34,54]., lofting, and mathematical control. Mathematical control is used in this work and couldX. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313In the present work, we combine 3D printing with electroless platingto derive a new type of auxeticmetamaterial fromTPMSs. First, the geom-etries of the TPMS-based auxeticmetamaterial aremathematically gener-ated with an arbitrary relative density. We then investigate the auxeticbehavior of the metamaterial under uniaxial compression using a 3D-printed specimen, followed by verification via finite element method(FEM) simulations. Next, we metalize the 3D-printed models by platinga layer of nickel on their surfaces and evaluate their performance undermultiple compression tests. Finally, we quantify the effective mechanicaland conduction properties andpresent them in the formofmaps depend-ing on the relative densities and nickel layer thicknesses.2. Methods2.1. Generation of microstructuresA minimal surface is a smooth surface that has zero mean curvatureand locally minimizes the surface area for a given boundary [55]. Whena minimal surface is translationally symmetric in three dimensions, it iscalled a TPMS. TPMSs can be modeled using different level-set approxi-mation equations. In thiswork, the geometry of the auxeticmetamaterialis derived fromSchwarz Primitiveminimal surface [56], and amore com-plicated implicit equation is used to determine the isosurface of the de-signed structure [46]:F x, y, zð Þ ¼ cos xð Þ þ cos yð Þ þ cos zð Þð Þð−0:4 cos xð Þ cos yð Þ þ cos yð Þ cos zð Þ þ cos zð Þ cos xð Þð Þ þ c ð1ÞHere, the level-set constant c regulates the volume fractions of the twophases separated by the isosurface. A skeletal lattice is built after cappingthe open borders of the isosurface. The proposed auxetic metamaterialpossesses a cubic symmetry with a single unit cell within the domain x,y, z ∈ [−π,π]. Its relative density, denoted by ρ (i.e., the volume fractionof the solid phase), has an almost linear relationship with c, as shown inFig. 2a. In this work, each isosurfacewas generated on the basis of the im-plicit expression using MATLAB (MathWorks, USA). After capping theisosurface borders, the geometry of the TPMS-based auxeticmetamaterialwas exported as standard triangle language (.STL) files for 3D printingand FEM simulations. Fig. 1 compares three different methods used toFig. 2. TPMS-based auxetic metamaterial. (a) Different relative densities determined by the levcells. (c,d) 3D-printed specimen before and after electroless nickel plating. (e,f) Optical microcoated specimen after multicle compression cycling. (h,i) Scanning electron micrographs show3generate auxetic metamaterials: Boolean operations of simple geome-tries, lofting operations that create 3D objects using 3D lattices as frame-work with arbitrary cross-sectional shapes such as cycle and square, andmathematical control. It demonstrates the flexibility of mathematicalcontrol to build complicated structures based on TPMS-based unit cells.2.2. Specimen preparationTo reduce the effects of the dimensions on the experiments, eachmodel used in the experiments consisted of 6 × 6 × 6 unit cells(Fig. 2b). The specimens of the proposed auxetic metamaterial were fab-ricated using an elastic photopolymer resin (Elastic 50A resin, Formlabs,USA) and a 3D printer (Form 3, Formlabs, USA). To achieve a subtle sur-face finish, no additional support structures were used, and the layerthickness and length of each specimen were set to 0.05 mm and24 mm, respectively. After washing with isopropanol for 10 min, these3D-printed specimens were fully cured at 60 °C for 20 min.We used an electroless nickel plating method to metalize the 3D-printed specimens. First, each sample was neutrally degreased in acleaning solution containing 50 g/L Na2CO3, 35 g/L Na2SiO3, and 3 g/LC12H25NaO4S for 5 min. After rinsing thoroughly with distilled waterfor 2 min, the sample was etched in a 3 M NaOH solution for 30 min,followed by rinsing again with distilled water for 2 min. In the nextstage, the sample was sensitized in an aqueous solution containing20 g/L SnCl2 and 20 mL/L 37% HCl for 5 min, and thereafter activatedin an aqueous solution containing 0.1 g/L PdCl2 and 20 mL/L 37% HClfor 5 min. After rinsing with distilled water, nickel was deposited in anelectroless manner using a nickel plating bath containing 32 g/LNiSO4⋅6(H2O), 20 g/L Na3C6H5O7, 25 g/L NH4Cl, and 28 g/L NaPO2H2.This step was performed at 90 °C, and the plating time was variedfrom 5 to 30 min to achieve different nickel layer thicknesses. Finally,the nickel-coated sample was rinsed with distilled water and driedunder a nitrogen stream. All chemicals used in the electroless nickelplating process were purchased from Sigma-Aldrich.2.3. CharacterizationOptical microscope images were obtained using a 3D digital micro-scope (DSX1000, OLYMPUS, Japan). Scanning electron micrographsel-set constant c. (b) A rendering of the auxetic metamaterial consisting of 6 × 6 × 6 unitscope images of the nickel-coated specimen. (g) Optical microscope images of the nickel-ing the cross-sectional and surface morphologies of the nickel-coated specimen.X. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313were acquired using a scanning electron microscope (Gemini500, Zeiss,UK) with an 8 mmworking distance and 5 kV accelerating voltage. Thesurface of nickel layer was captured under Inlens detection signal,whereas the cross-section the nickel layer and the interface of the nickellayer and resin were captured under SE2 detection signal. The nickelthickness was determined by measuring the width of the nickel layersin the SEM images.2.4. Uniaxial compression tests and current-voltage characteristicsThe mechanical properties of the prepared specimens were evalu-ated by static compression tests using a motorized test stand(ESM303, Mark-10, USA). The specimens were uniaxially compressedbetween two plates at a constant displacement rate of 1 mm/min forboth loading and unloading. These results were independent of the dis-placement rate, which is an excellent approximation to the stationaryconditions in FEM simulations. The load and displacement datawere re-corded to produce stress-strain curves, and the deformation patterns ofthe specimens were captured using two cameras. With regard to thedata analyses, the Poisson's ratios of each specimen were evaluated byextracting the displacements of the nodes of the deformed geometriesfrom the recorded videos via postprocessing in MATLAB. The Young'smoduli were measured by linearly fitting the initial linear portions ofthe stress-strain curves during loading. The critical buckling strainsand stresses were defined by the 0.2% offset strain based on the shapeof the stress-strain curves during loading.The conduction properties of the specimens were also evaluatedduring the compression tests, as illustrated schematically in Fig. 3. Thetwo compression plates were coveredwith insulating tape on the insideand copper foil tape on the outside. The copper foil tapes were con-nected to a digital sourcemeter (Series 2400 SourceMeter SMU,Keithley, USA) to determine the current-voltage characteristics. A con-stant voltage (U) of 0.2 V was applied to obtain the electric currents(I) under different strains. Thus, the electrical conductivity (κ) is deter-mined by the equation κ ¼ IL=UL02, where L is the height of the 3D-printed models during the compression, L0 = 24 mm that is the lengthof the 3D-printed models, and S ¼ L02 that is the sectional area of the3D-printed models.2.5. Finite element method simulationsThe deformation behavior of the new auxetic metamaterial werefurther quantitatively investigated using an FEM simulation platform(COMSOL Multiphysics Ver. 5.4, COMSOL, Sweden). As the 3D-printedresin can be considered as a hyperelastic material, we defined the ma-trix material in the simulations using the incompressible neoHookeanmodel for simplicity. The material model was a single parametermodel with a Young's modulus of 0.6615 MPa that was fitted from thecompression tests. As the deformation behavior under compressionFig. 3. Schematic illustrating the stressstrain and conductivitystrain curves by uniaxialcompression tests.4was subject to buckling instability, a linearized buckling analysis wasfirst performed to compute the first-order buckling mode shape,followed by a post-buckling analysis using the computed mode shape.For the linearized buckling analysis, a fixed external load was appliedalong the z-axis to compute the shape of the buckling mode. For thepost-buckling analysis, a parametric sweep of the displacement alongthe z-axis was used with a stop condition when adjacent boundarieswere in contact. The post-buckling analysis helped to obtain thestress-strain curves in the three-dimensional Euclidean space. Specifi-cally, models were built using approximately 2.5 × 105 second-ordertetrahedral solid elements. All simulations were performed under peri-odic boundary conditions. The method applies these periodic boundaryconditions to the three pairs of faces of the unit cell, which is based on arepresentative volume element (RVE) technique [57,58].3. Results and discussion3.1. Auxetic behaviorThe auxetic behavior and deformation mechanism of the newly fab-ricated metamaterial were investigated under uniaxial compression. Ascan be seen from the 3D visualization of the metamaterial (Fig. 2a), asingle unit cell consists of three ligaments, which are orthogonal tothe center node in 3D Euclidean space. This forms a single continuouslattice structure with smooth surfaces. We first conducted a uniaxialcompression test on the 3D-printed specimen with ρ = 0.16 (Fig. 2c).The high resolution of the 3D printing technique ensured a good surfacefinish for the 3D-printed samples.Then, we performed a systematical analysis of the auxetic behaviorby considering the deformed patterns, strain-stress curves, andPoisson's ratios, as shown in Figs. 4 and 5. Fig. 4a shows a sequence ofprogressively deformed shapes of the specimen under seven differentlevels of longitudinal engineering strain (i.e., change in the height ofthe sample divided by its original height, εzz = (L − L0)/L0). Here, thespecimen witnesses a dramatic contraction in all three directions,along with changes to its interior structure (the holes are changedfrom circles to ellipses). The lateral shrinkage proves that it is a 3Dma-terial with negative Poisson's ratio. The structural transformation fromstraight to bent ligaments demonstrates that the deformation behavioris subject to buckling instability. It is noted that the periodicity of theauxetic metamaterial can be considered as a new RVE comprising2 × 2 × 2 unit cells upon buckling.To further probe the buckling-induced auxetic behavior, we ana-lyzed the stresses and Poisson's ratios under different compressionstrains. Fig. 5a shows the stress-strain curve of the 3D-printed specimenas a function of the longitudinal strain εzz. It is observed that it is a typicalelastic buckling curve consisting of a linear elastic regime and a stressplateau. The transition occurs at εzz ≈ − 0.03 when the ligamentsbegin to bend, which further demonstrates the buckling-inducedauxetic behavior. Because the structure is difficult to shrink whenthere are no empty interior spaces, we set a stop conditionwhen the lig-aments around the holes come intomutual contact. As a result, the den-sification region of stress-strain curves is missed, and the overalldeformation strain is approximately εzz ≈ − 0.3.A more quantitative evaluation of the negative Poisson's ratios isshown in Fig. 5b, where these ratios are calculated from the engineeringstrain as νij= − εjj/εii, for i, j= x, y, z. The transverse strains εxx and εyyare obtained from the average transverse strains of the four nodes of theinner-most RVE to reduce the influence of boundary conditions, includ-ing that of the friction on the up and down surfaces and the freedom ofthe exterior traction-free surfaces. This is attributable to the approxima-tion that the innermost RVE can be assumed as an infinitely periodicstructure [31]. The calculated Poisson's ratios (νzx and νzy) first mono-tonically decrease and then reach a plateau after εzz ≈ − 0.1. Each pla-teau lasts over a wide range from εzz ≈ − 0.1 to −0.3 for νzx ≈ − 0.5and νzy ≈ − 0.25.Fig. 4.Buckling-induced auxetic behavior of the proposed auxeticmetamaterial and a sequence of progressively deformed configurations of the (a) 3D-printed specimen, (b) nickel-coatedspecimen, and (c) FEMmodel. The deformed degree is defined as the displacement norm of each meshed element divided by the original height of the model.X. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313In the next stage, we conducted FEM simulations using an RVEconsisting of 2 × 2 × 2 unit cells to investigate the auxetic behaviormore qualitatively and quantitatively. The stress-strain curve of theFEM model shows excellent agreement with the result from the 3D-printed specimen (Fig. 5a). Moreover, the progressively deformed con-figurations from the FEM simulation are substantially alike compared tothe experimental results (Fig. 4c). The geometry shrinks in all three di-rections accompanying the bending of ligaments and the rotation ofFig. 5. (a) Stress-strain curves fromuniaxial compression tests and FEMsimulations. (b) Evolutiostrain. The FEM results are observed to be in good agreement with the experimental data.5nodes, demonstrating that the proposed auxetic metamaterial belongsto rotation structures. Fig. 5b presents the transverse strains εxx andεyy and the Poisson's ratios plotted as a function of the longitudinalstrain εzz. It is clear that the transverse strains decrease with εzz undercompression, exhibiting lateral constriction behavior. The Poisson's ra-tios calculated from the transverse strains also fit the experimentaldata well. The Poisson's ratios are positive during the initial compres-sion and become negative after the structure undergoes buckling;n of the transverse engineering strains and Poisson's ratios as a function of the longitudinalX. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313then, this trend gradually decreases and reaches a plateau with furtherloading. To clarifywhether the periodic boundary conditionwill removeall possible size effects, we performed FEM simulations on a 6 × 6 × 6model using periodic boundary condition (Fig. 6). It shows the same re-sult in terms of stressstrain curve, Young's modulus, and critical buck-ling stress and strain compared with the 2 × 2 × 2 model. Further,Fig. 6 also reveals the deformation mechanism that leads to the auxeticeffect; the compression leads to the bending of ligaments, the rotationof nodes, and the transformation of holes from circles to ellipses. Over-all, the excellent agreement of the deformed patterns, stress-straincurves, and Poisson's ratioswith the experimental observations demon-strate the accuracy and efficacy of our FEM model.3.2. Influence of relative densityGiven the outstanding qualitative and quantitative agreement be-tween the experiment and the FEM simulation, we further explore theinfluence of the relative density of the new auxetic metamaterial onthe mechanical properties based on FEM models. Five different relativedensities ranging from 0.12 to 0.28 are considered. The stress-straincurves from the FEM simulation results are shown in Fig. 7a, wherethe stresses are normalized using the Young's modulus of the matrixmaterial, E0 = 0.6615 MPa. This shows that both the Young's modulus(E) and critical buckling stress (σc) increase with the relative density.The critical buckling strain (εc) also rises from about 0.009 to 0.101, pro-viding evidence that a higher relative density suppresses the bucklingbehavior.Fig. 7b presents the transverse strains εxx and εyy versus the longitu-dinal strain εzz from the FEM simulation results. The evolution of thetransverse strains displays a similar tendency: a short plateau followedby a continuous decline. It is noteworthy that the short plateaus are ex-tended, and the ultimate transverse strains are lessened as the relativedensity increases.We also calculated the Poisson's ratios from the trans-verse strains, as shown in Fig. 7c. Similarly, the Poisson's ratios at thesame εzz increase with relative density, for example, the Poisson's ratiosνzx increases from−0.58 to−0.29 at εzz= − 0.3when the relative den-sity increases from 0.12 to 0.28. These results demonstrate that the pro-posed auxeticmetamaterial is harder to shrink if it is denser. In addition,this auxetic metamaterial displays a transversely asymmetric behavioras νzx ≠ νzy.Fig. 6. Auxetic behavior of a 6 × 6 × 6 model showing agreement with the 2 × 2 × 2 mod6The performance of the proposed auxetic metamaterial, includingYoung's modulus, critical buckling strain and stress, and Poisson'sratio, is compared with several typical auxetic metamaterials, asshown in Fig. 8. The Young's moduli and critical stresses are normalizedrespectively by the Young's modulus of their constitute materials. Itshows that the proposed auxetic metamaterial has higher normalizedYoung's modulus and critical stress than others owing to its higher rel-ative density. Further, the structure possesses a longer buckling strainup to −0.1 at ρ = 0.28 and a larger negative Poisson's ratio up to−0.63 at ρ = 0.12. Overall, varying relative density enables to tunethe mechanical properties over a broad range.3.3. Influence of nickel layerTo further functionalize the auxetic metamaterial, we coated a con-formal nickel layer onto the surfaces of the 3D-printed specimen withρ = 0.16 by electroless plating. After plating a nickel layer of 0.57 μmthickness, the overall structure remains unchanged without apparentvolume shrinkage or expansion (Fig. 2d). Moreover, the dense nickellayer, having a grain size of hundreds of nanometers, adheres to theresin surface well, thereby ensuring good structural stability afterlarge deformation (Figs. 2e–i). Note that we assumed the relative den-sity keeps unchanged after plating as the nickel layer thickness isquite small compared to the unit cell size of 3D-printed models.Fig. 4b displays the continuous deformed patterns of the nickel-coated specimen under uniaxial compression, which exhibits a similarresult as that of the 3D-printed specimen and FEM model. In addition,compared with the 3D-printed specimen, the nickel-coated specimenhas nearly twice the values of the Young's modulus and critical bucklingstress, but its critical buckling strain remains almost unchanged(εc ≈ − 0.03), as shown in Fig. 5a. Note that there are some decreasesin the stresses after buckling owing to local plastic deformation andcracks in the nickel layers (Fig. 2g). Interestingly, the 3D-printed andnickel-coated specimens possess almost identical Poisson's ratiosunder the same transverse strains (Fig. 5b). The results prove that thenickel layer has a negligible impact on the critical buckling strain andPoisson's ratio, but significantly enhances its Young's modulus and crit-ical buckling stress.To investigate the resilience and stability of the specimens, we per-formed multicycle compression tests. Figs. 9a and b compare the 3Dprinted specimen with the nickel-coated specimen with respect toel and revealing the deformation mechanism of the proposed auxetic metamaterial.Fig. 7. Influence of relative density onmechanical properties. Evolutions of the (a) normalized engineering stresses, (b) transverse engineering strains, and (c) Poisson's ratios as functionof the longitudinal strain. The methods of calculating Young's moduli E, critical buckling strains εc, and critical buckling stresses σc are also illustrated in the first figure.X. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313their stress-strain curves during loading and unloading over multiplecompression cycles. The small decreases in stresses for the nickel-coated specimen during consecutive compression cycles are a result ofthe local yield and crack of nickel layers. On the other hand, the Young'smoduli decrease only marginally because the buckling behavior is a re-sult of local deformation. When the load exceeds the critical bucklingstress, the ligaments begin to bend, resulting in the plastic deformationand cracking of the nickel layers at the center of the ligaments wherethey bend. This part of the nickel layers contributes mainly to enhancethe critical buckling stress. However, the Young's modulus is deter-mined by entire nickel layer; the deformation and cracks of the nickellayers in these local regions reduce the Young's modulus only slightly.Nevertheless, both kinds of specimens reach peak stress whencompressed to εzz = − 0.3, and recover their original heights afterload removal. The stress-strain curves are also nearly identical evenafter 20 compression cycles, demonstrating excellent resilience and sta-bility for both classes of specimens.To explore how the mechanical and conduction properties are en-hanced by increasing the thickness of the nickel layers (t), we depositednickel layers having different thicknesses on the 3D-printed specimenswith ρ=0.16 by varying the electroless plating time. The thicknesses ofFig. 8. Comparison of the proposed auxetic metamaterial with several typical auxetic met(c) Normalized critical strain, and (d) Poisson's ratio.7the nickel layers are nearly linearly dependent on the plating time, and anickel layer of about 1.71 μm thickness was formed by plating 30 min(Fig. 9c). The stress-strain curves from the first cycle of compressiontests are shown in Fig. 9d, revealing a mechanical enhancement thatthe stresses increase with the thickness at the same εzz. Comparedwith the non-plated specimen, the one with the 1.71 μm nickel layerhas an approximately ten-fold increase in terms of its stiffness andstrength, increasing from 35 kPa to 375 kPa in terms of Yongs modulusand from0.9 kPa to 9 kPa in terms of critical buckling stress, respectively(Figs. 9e and f). Surprisingly, the critical buckling strains are almost in-dependent of the thickness of the nickel layer, and have a constantvalue of εzz ≈ − 0.03.After plating the nickel layer, these specimens become conductive.Fig. 10a displays the conductivity-strain curves of a sample with t =0.57 μm and ρ = 0.16 during multiple compression tests. Likewise,the conductivity-strain curves also have excellent repeatability after20 cycles of compression, without apparent losses in their conductivi-ties. Under compression, the electrical conductivity increases linearlyfrom 128 S/m initially to 230 S/m until the specimen suffers buckling;thereafter, it decreases immediately and then increases during furthercompression. After unloading, the electrical conductivity is almostamaterials. (a) Normalized Young's modulus, (b) Normalized critical buckling stress,Fig. 9. Influence of nickel layer thickness onmechanical and conduction properties. Stress-strain curves of (a) a 3D-printed specimenwith ρ=0.16 and (b) a nickel-coated specimenwithρ=0.16 and t=0.57 μm frommultiple compression tests. (c) Evaluation of the nickel layer thickness that is linearly dependent on plating time. (d) Stress-strain curves of nickel-coatedspecimens with different thicknesses of nickel layers from the first cycle of compression tests. Dependence of (e) the Young's modulus and (f) critical buckling stress on the nickel layerthickness. In the stress-strain curves, solid lines represent loading and dot lines represent unloading.X. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313restored to its original point, in pace with the recovery of the specimen.The results further prove the remarkable resilience and stability of thesespecimens under multiple large compressions.We also explored the influence of the nickel layers on the electricalconductivities by coating different thicknesses of the nickel layers onthe 3D-printed specimens with ρ = 0.16. Fig. 10b presents theconductivity-strain curves of these specimens from the first cycle ofcompression tests. Analogous to the stress-strain curves in Fig. 9d,each of these conductivity-strain curves consist of three parts duringloading: a steep incline followed by a drop and a gradual rise. As ex-pected, the conduction is enhanced remarkably by plating thicker nickellayers; for example, when the thickness increases from 0.28 μm to1.71 μm, the conductivities increase from approximately 58 S/m to371 S/m at the initial shape (Fig. 10c).3.4. Data maps of mechanical and conduction propertiesWe further conducted amore comprehensive study by investigatingthemechanical and conduction properties of thematerial dependent onFig. 10. Influence of nickel layer thickness on conduction properties. (a) Conductivity-stracompression tests. (b) Conductivity-strain curves of nickel-coated specimens with differentconductivity on the nickel layer thickness at the initial shapes. In the conductivity-strain curve8the relative density and nickel layer thickness. We fabricated a range ofspecimens by electroless plating of nickel layers with different thick-nesses on 3D printed models with ρ= 0.12− 0.28. Their effective me-chanical and conduction properties, including Young'smoduli, Poisson'sratios, critical buckling strains, critical buckling stresses, and electricalconductivities, were calculated from the first cycle of uniaxial compres-sion tests. These results are presented as contour maps in Fig. 11, thusproviding a method to tune the mechanical and conduction properties.Figs. 11a and b respectively present the Young's moduli and criticalstresses from the experimental results, which are normalized by theYoung's modulus of the 3D-printed resin, E0 = 0.6615 MPa. Increasingeither the relative density or the nickel layer thickness can tremen-dously enhance the Young's moduli and critical stresses. After platinga 1.71 μm thick nickel layer on the 3D-printed specimen with ρ =0.28, the normalized Young's modulus increases from 0.02 to 1.02 andthe critical stress increases from 1.79 × 10−4 to 8.60 × 10−2 comparedwith the green 3D-printed specimen with ρ = 0.12.Regarding the tuning of the Poisson's ratios for νxy (data were takenfrom the points at εzz = − 0.3) and critical buckling strains, changingin curves of the nickel-coated specimen with ρ = 0.16 and t = 0.57 μm from multiplethicknesses of nickel layers from the first cycle of compression tests. (c) Dependence ofs, solid lines represent loading and dot lines represent unloading.Fig. 11. 3D Contourmaps of mechanical and conduction properties, including (a) normalized Young's modulus E/E0, (b) normalized critical buckling stressesσc/E0, (c)minimumPoisson'sratiosνzx at εzz= − 0.3, (d) critical buckling strains εc, (e) electrical conductivities κ at the initial shapes, and (f) electrical conductivities κ at critical buckling shapes. The results are built byintegration of the results obtained by plating t=0.28,0.57,0.85,1.14,1.42, and 1.71 μm nickel layers onto the 3D printed specimenswith ρ=0.12,0.16,0.20,0.24, and 0.28, as indicated bythe gray dots in the first map.X. Zheng, X. Guo and I. Watanabe Materials and Design 198 (2021) 109313only the relative densities appears to be a suitable solution (Figs. 11cand d). It is worth noting that the Poisson's ratios and critical bucklingstrains are nearly independent of the nickel layer thicknesses, but aredominated by the relative densities. The Poisson's ratios increase fromapproximately−0.78 to−0.29 and the critical buckling strains increasefrom 0.01 to 0.1 when the relative densities increase from 0.12 to 0.28.However, the electrical conductivities are hugely determined by thenickel layer thicknesses. Figs. 11e and f show the electrical conductivi-ties at the initial and critical buckling shapes, respectively. This showsthat the 3D-printed specimens become conductive after plating withnickel layers of hundreds of nanometers thickness on the surfaces, anda 1.71 μm nickel layer achieves a conductivity of 700 S/m. Moreover,these results also indicate that the conductivities are difficult to manip-ulate by varying the relative density; although, for the same nickel layerthickness, the 3D-printed specimen with a higher relative density has alarger conductivity than the one with a lower relative density at criticalbuckling shapes.4. ConclusionsWe presented a novel auxetic metamaterial originating from a typeof TPMS structure. The mathematically defined modeling method en-ables the designing of more complicated systems using themetamaterial as basic unit cells. The relative density of themetamaterialis determined by varying the level-set parameter of implicit expression.The compression tests, along with the FEM simulations, demonstratethat the auxetic behavior of the new metamaterial is dominated bybuckling instability and is retained over a broad range of longitudinalstrain values up to 0.3. The metamaterial was 3D printed using arubber-like material, followed by plating with nickel nanolayers; thisproduces a fully reversible and resilient substrate, whose mechanicaland conduction properties are highly tunable by varying the relativedensity and nickel layer thickness. More importantly, data maps of theYoung's moduli, critical buckling stresses and strains, Poisson's ratios,and conductivities provide guidance for functional applications of thematerial. Finally, although this work focused only on a single TPMS9structure and a single plated metal, it is highly recommended that astructural gallery combining other TPMSs and electroless plating tech-niques (e.g., Cu, Ag, and Au) be developed in future work, thereby pav-ing a way towards functional applications, such as strain sensors,actuators, and electrochemical energy storage and conversion.Declaration of Competing InterestThe authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.References[1] G.N. Greaves, A. Greer, R.S. Lakes, T. Rouxel, Poisson’s ratio and modern materials,Nat. Mater. 10 (11) (2011) 823–837.[2] R.S. Lakes, Negative-poisson’s-ratio materials: Auxetic solids, Annu. Rev. Mater. Res.47 (2017) 63–81.[3] Z. Wang, C. Luan, G. Liaom, J. Liu, X. Yao, J. 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A mathematically defined 3D auxetic metamaterial with tunable mechanical and conduction properties 1. Introduction 2. Methods 2.1. Generation of microstructures 2.2. Specimen preparation 2.3. Characterization 2.4. Uniaxial compression tests and current-voltage characteristics 2.5. Finite element method simulations 3. Results and discussion 3.1. Auxetic behavior 3.2. Influence of relative density 3.3. Influence of nickel layer 3.4. Data maps of mechanical and conduction properties 4. Conclusions Declaration of Competing Interest References