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Oct 1, 2018 - However, in the open literature, only a negative Poisson's ratio ... tuned simultaneously through adjusting the chord height to span ratio .... in the y direction and Es = 2500 MPa is the Young's modulus of the solid of which the CCF are ...... Shan, Y.; Nian, G.; Xu, Q. Strength analysis of syntactic foams using a ...
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Mechanical Metamaterials Foams with Tunable Negative Poisson’s Ratio for Enhanced Energy Absorption and Damage Resistance Shaohua Cui 1 , Baoming Gong 1 , Qian Ding 2 , Yongtao Sun 2,3,4,5, *, Fuguang Ren 2 , Xiuguo Liu 1, *, Qun Yan 6 , Hai Yang 6 , Xin Wang 6 and Bowen Song 2 1

2

3 4 5 6

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Department of Materials Science and Engineering and Tianjin Key Laboratory of Advanced Joining Technology, Tianjin University, Road Weijin 92, Tianjin 300072, China; [email protected] (S.C.); [email protected] (B.G.) Department of Mechanics and Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Yaguan Road 135, Tianjin 300350, China; [email protected] (Q.D.); [email protected] (F.R.); [email protected] (B.S.) State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, China Key Laboratory of Aeroacoustics and Dynamics, Aircraft Strength Research Institute, Xi’an 710065, China; [email protected] (Q.Y.); [email protected] (H.Y.); [email protected] (X.W.) Correspondence: [email protected] (Y.S.); [email protected] (X.L.); Tel.: +86-151-0226-2170 (Y.S.)

Received: 2 August 2018; Accepted: 28 September 2018; Published: 1 October 2018

 

Abstract: Systematic and deep understanding of mechanical properties of the negative Poisson’s ratio convex-concave foams plays a very important role for their practical engineering applications. However, in the open literature, only a negative Poisson’s ratio effect of the metamaterials convex-concave foams is simply mentioned. In this paper, through the experimental and finite element methods, effects of geometrical morphology on elastic moduli, energy absorption, and damage properties of the convex-concave foams are systematically studied. Results show that negative Poisson’s ratio, energy absorption, and damage properties of the convex-concave foams could be tuned simultaneously through adjusting the chord height to span ratio of the sine-shaped cell edges. By the rational design of the negative Poisson’s ratio, when compared to the conventional open-cell foams of equal mass, convex-concave foams could have the combined advantages of relative high stiffness and strength, enhanced energy absorption and damage resistance. The research of this paper provides theoretical foundations for optimization design of the mechanical properties of the convex-concave foams and thus could facilitate their practical applications in the engineering fields. Keywords: convex-concave foams; tunable negative Poisson’s ratio; enhanced energy absorption and damage resistance

1. Introduction Metamaterials are rationally designed artificial materials whose effective properties arise not from the bulk behavior of the materials that compose them, but from their deliberate structuring [1–4]. Unusual mechanical properties of the metamaterials, such as unique negative Poisson’s ratio (NPR), superlight weight, high stiffness, strong strength, high specific energy absorption, excellent fracture toughness, and vibration reduction characteristics, play a vital role for their multifunctional Materials 2018, 11, 1869; doi:10.3390/ma11101869

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applications in the fields of mechanical and aerospace engineering [5,6]. NPR foams, which get fatter when stretched and thinner when compressed [7], are a typical kind of mechanical metamaterials. When compared with the conventional foams with a positive Poisson’s ratio (PPR), the NPR foams could have the advantages, such as the ability to undergo synclastic curvature, enhanced toughness, increased indentation resistance, improved shear stiffness, sound dampening properties, self-adaptive vibrational damping, and shock absorption, etc. [8–12]. To explore mechanical properties of the NPR foams and thus promote their applications in the fields of mechanical and aerospace engineering, a variety of NPR foams have been proposed by researchers in the past several decades. Prof. Rod Lakes is the scientist who first proposed the idea of man-made NPR foams, i.e., the reentrant foams [13], which showed superior indentation resistance [8] and fracture toughness [14] to the conventional foams and were very suitable to be made as the polymer seat cushions [15,16]. Evans et al. [17] proposed the reentrant three-dimensional elongated dodecahedron NPR foams. The group of Prof. Scarpa systematically studied mechanical properties of the NPR polyurethane foams, which showed better dynamic crushing, damping and acoustic behaviors [18], superior tensile fatigue performance [19,20], higher dynamic stiffness, and enhanced viscous dissipations [21] than the conventional foams of comparable density. In recent years, many new kinds of NPR foams of various cell geometries have been proposed. Based on the very simple initial geometric shapes, i.e., spherical voids in a cubic matrix, Shen et al. [22] proposed a series of cubic foams with NPR over a large strain range. Through three-dimensional (3D) printing and investment casting technology, Xue et al. [23] fabricated the Al-based auxetic re-entrant foams with sound surface and interior qualities and investigated their compressive properties. Adopting the interlocking assembly concept, Wang et al. [24] introduced the interlocking assembled 3D re-entrant NPR foams. Ai and Gao [25] designed four types of 3D bi-material lattice foams with both NPR and non-positive coefficient of thermal expansion. In recent years, the groups of Prof. Gambarotta and Prof. Bacigalupo systematically studied the overall mechanical properties (e.g., equivalent elastic properties and wave propagation characteristics etc.) of several kinds of metamaterials, such as hexachiral [26,27], tetrachiral [26,28–30] and anti-tetrachiral [31–33] materials, periodic beam lattice materials [34–36], rigid periodic blocky materials [37], and so on. In this paper, mechanical properties of the orthogonal isotropic NPR convex-concave foams (CCF) [38] are investigated through the experimental and finite element methods. The CCF are constructed by replacing cell edges of the conventional open cell foams (COF), whose straight cell edges of square cross sections are arranged in planes at forty-five degrees relative to each other [38], with the sine-shaped cell edges [39–44] of equal mass but different cross sections. In the open literature, only NPR effect of the CCF is simply mentioned [38]. In practice, optimization design of the mechanical properties of the CCF plays a vital role for their engineering applications. To facilitate their practical applications in the engineering fields, mechanical properties of the CCF need to be deeply and systematically understood. Therefore, in this paper, based on the polymer materials VeroWhite Plus, mainly choosing one COF and three kinds of CCF of equal mass as examples, the effects of geometrical morphology on mechanical properties of the CCF, including elastic moduli, energy absorption, and damage characteristics, are systematically investigated combining the experimental and finite element methods. 2. Geometrical Models of the CCF Cubic unit cell models of the COF and the CCF are depicted in Figure 1a,b, respectively. t0 and l0 are the edge length and edge thickness of the COF, and the CCF have sine-shaped cell edges with chord span l0 , chord height h, and side thickness t. Geometrical parameters play a key role for mechanical properties of the foams [45]. To show the effects of geometrical topology on mechanical behaviors, including elastic moduli, energy absorption, and fracture characteristic, of the CCF, here one COF and three kinds of CCF of equal mass are taken as the examples for illustration. As shown in Figure 2a, the four kinds of samples are named COF0, CCF1, CCF2, and CCF3, respectively. They are constructed

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3 of 18 of (Figure 2b) of their cubic unit cell (Figure 1a,b). The materials the samples are VeroWhite Plus, whose density and Poisson’s ratio are 1.18 kg/m3 and 0.33. by × 4 × 4 arrays of their cubic unit cell (Figure 1a,b). The materials ofThe the samples are4constructed by 4(Figure × 4 × 42b) arrays (Figure 2b) of their cubic unit cell (Figure 1a,b). materialsare of 3 and 0.33. 3 VeroWhite Plus, whose density and Poisson’s ratio are 1.18 kg/m the samples are VeroWhite Plus, whose density and Poisson’s ratio are 1.18 kg/m and 0.33.

(a)

(b)

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(b)

(c) Figure 1. (a) Cubic unit cell model of the conventional open-cell foams with straight cell edges, of (c) square cross sections, arranged in planes at forty-five degrees relative to each other, in which the edge 1. is the Figure 1. (a)l Cubic unit cellthickness model of is the open-cell foams straight cell edges, ratio of length and the edge ;conventional (b) Cubic unit cell model ofwith the negative Poisson’s t0 conventional 0 at at forty-five degrees relative to each other, in which the edge square cross cross sections, sections,arranged arrangedininplanes planes forty-five degrees relative to each other, in which the convex-concave foams, which are constructed by replacing cell edges of the conventional open cell lengthlength is l0 and thickness is t is ; (b) Cubic unit cell edge is l0 the andedge the edge thickness t0 ; (b) Cubic unit cellmodel modelofofthe thenegative negative Poisson’s Poisson’s ratio foams of (a) with sine-shaped cell 0edges of equal mass but different square cross sections; (c) convex-concave foams, which are constructed by replacing cell edges of the conventional open cell conventional Schematic diagram of the sine-shaped cell edges of the convex-concave foams (CCF) with chord span foams ofof(a) with sine-shaped cell cell edges of equal mass but different square cross sections; (c) Schematic foams (a)height with l0 , chord h,sine-shaped and thickness t. edges of equal mass but different square cross sections; (c) diagram ofdiagram the sine-shaped cell edges cell of the convex-concave foams (CCF) with chord span l0 , chord Schematic of the sine-shaped edges of the convex-concave foams (CCF) with chord span height h, and thickness l0 , chord height h, and t. thickness t.

(a)

(b)

Figure Front views COF sample and corresponding three kinds CCF samples Figure 2. 2(a)(a) Front views thethe COF sample and thethe corresponding three kinds of of CCF samples of of (a)ofof (b) equal mass: (a1) COF0 (t0 (t =01=mm, l0 = mm); (a2) CCF1 (h (h = 1= mm); (a3) CCF2 (h (h = 2= mm); (a4) CCF3 1 mm, l0 10 = 10 mm); (a2) CCF1 1 mm); (a3) CCF2 2 mm); (a4) CCF3 equal mass: (a1) COF0 Figure (a)(b) Front views of the COF sample and the corresponding three of CCF samples of (h = 3= mm); Schematic diagram ofof the samples constructed 44 × 4××4 4arrays arrays the cubic (h 32mm); (b) Schematic diagram theCCF CCF samples constructed by by × 4kinds ofof the cubic unit 0 = 1 mm, l0 = 10 mm); (a2) CCF1 (h = 1 mm); (a3) CCF2 (h = 2 mm); (a4) CCF3 equal mass: (a1) COF0 (t unit cell shown Figure cell shown in in Figure 1b.1b. (h = 3 mm); (b) Schematic diagram of the CCF samples constructed by 4 × 4 × 4 arrays of the cubic unit cell shown in Figure 1b.

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Here, the edge length and edge thickness of the COF0 are t0 = 1 mm, l0 = 10 mm. The chord height h of CCF1, CCF2, and CCF3 are 1 mm, 2 mm, and 3 mm, respectively (Figure 2a). Apparently, when h = 0, the CCF becomes the COF. The curve of the sine-shaped cell edges (Figure 1c) of the CCF samples can be mathematically described as y = h sin(π/lo x ) (x ∈ [0, l0 ]). To avoid the intersection of the sine-shaped cell edges, here h/l0 < 0.5 is assumed. Define the curve length of the sine-shaped cell edges as s, then s is expressed as: s=

Z l0 q 0

1 + (y0 )2 dx =

Z l0 q 0

1 + h2 π 2 /l02 cos2 (π/l0 x )dx

(1)

According to the equal-mass principle, we have st = t0 l0 , which gives the side thickness t of the sine-shaped cell edges of the CCF samples t = t 0 l0 /

Z l0 q 0

1 + h2 π 2 /l02 cos2 (π/l0 x )dx

(2)

Substituting, t0 = 1 mm, l0 = 10 mm, h = 1,2,3 mm into Equation (2) gives a side thickness t of CCF1, CCF2, and CCF3. They are 0.9975, 0.99, and 0.9779, respectively. At the same time, apparently, the chord height to span ratio h/l0 of the COF0, CCF1, CCF2, and CCF3 are 0, 0.1, 0.2, and 0.3, respectively. 3. Effect of Geometrical Morphology on Elastic Moduli of the NPR CCF In this part, effects of geometrical topology on elastic moduli, including Poisson’s ratio and relative Young’s modulus, of the COF0 (h/l0 = 0), CCF1 (h/l0 = 0.1), CCF2 (h/l0 = 0.2), and CCF3 (h/l0 = 0.3) are investigated. Poisson’s ratio νxy of the four samples are calculated numerically by exerting periodic boundary conditions [46] on their own cubic unit cell in the orthogonal x, y and z directions. Due to the geometrical symmetry of the COF and CCF, it is not difficult to imagine that νxy = νyx = νxz = νzx = νyz = νzy . From Figure 3a, it is easy to see that Poisson’s ratio νxy of COF0, CCF1, CCF2, and CCF3 are 0.06, −0.13, −0.31, and −0.35, respectively. Apparently, the COF, whose cell edges are straight, have PPR, but the CCF, whose cell edges are sine-shaped curved, have NPR and with the increase of the chord height to span ratio h/l0 of the CCF, the effect of NPR accelerates. In other words, with the cell edges changing from straight shape (COF) to curved sinoidal shape (CCF), Poisson’s ratios transform from positive to negative and in some extent, the more curved the cell edges of the CCF, the more obvious the effect of NPR. This indicates that the NPR of the CCF can be tuned through changing the degree of curvature, i.e., the chord height to span ratio h/l0 , of the CCF. As shown in Figure 3b, relative Young’s modulus Ey /Es of the four samples are investigated through the numerical and experimental methods. Here, Ey is the Young’s modulus of the CCF in the y direction and Es = 2500 MPa is the Young’s modulus of the solid of which the CCF are made of. Due to the geometrical symmetry of the COF and CCF, it is also not difficult to imagine that Ex = Ey = Ez . Like the Poisson’s ratio νxy , Ey of the four samples by numerical simulations are also calculated through exerting periodic boundary conditions on their own cubic unit cell in the orthogonal x, y, and z directions [46]. On the other hand, Ey by the experiments are obtained from the linear elastic stage of quasi-static uniaxial compressive tests, the details of which will be discussed in Section 4. From Figure 3b, we can see that, as a whole, the relative Young’s modulus Ey /Es , as calculated by numerical simulations, agree well with that of the experiments. The relative Young’s modulus Ey /Es decreases with the increase of the chord height to span ratio h/l0 of the cell edges. In other words, stiffness of the studied four kinds of samples decreases with the increase of the chord height to span ratio.

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Figure 3. (a) Poisson’s ratio ν xy ( ν xy =ν yx =ν xz =ν zx =ν yz =ν zy ) and (b) relative Young’s modulus Ey Es ( Ex =Ey =Ez ) of the COF0, CCF1, CCF2 and CCF3, in which Ey is the Young’s modulus of the CCF in the y direction and Es is the Young’s modulus of the solid of which the CCF are made of.

(a) (b) Except the examples of COF0, CCF1, CCF2, and CCF3, to further investigate the effect of =ν ν=xzν = =ννzxyz ==ν zyνyz Ey Es Figure3.3.(a) (a)Poisson’s Poisson’sratio ratioνxyν xy ( ν xy==ν ) andνzy (b) relative Young’s modulus Figure (νxy νyx ) and (b) relative Young’s modulus yx = xz geometrical morphology on elastic moduli of zxthe CCF, = the following parameters are taken as E( yE/E (E Eythe = ECOF0, COF0, CCF1, CCF2 andinCCF3, modulus of the x = z ) of the y is the Young’s =sEfor =E ) of CCF1, CCF2 and CCF3, whichinEwhich is the EYoung’s modulus of the CCF in examples x y zmore illustration: h l = 0, 0.1, 0.2, 0.3 , t l y = 0.02 : 0.02 : 0.1 . The relative Young’s 0 0 CCF in the y direction and Es is the Young’s modulus of the solid of which the CCF are made of.

the y direction whichin theFigure CCF are4a,b, maderespectively. of. Ey Es and l0 solid andEPoisson’s ratiomodulus are of shown From modulus ν xy vs oft the s is the Young’s Except the examples of COF0, CCF1, CCF2, and CCF3, to further investigate the effect of Figure 4a, it is easy to see that for the fixed h l0 , the relative Young’s modulus Ey Es increases with Exceptmorphology the examples of COF0, CCF1, CCF2, CCF3, to parameters further investigate effect of geometrical on elastic moduli of the CCF,and the following are taken the as examples E E t l0 . For t l the increase ofmorphology fixed , as a whole, the relative Young’s modulus decreases geometrical on elastic moduli of the CCF, the following parameters are taken as for more illustration: h/lthe = 0, 0.1, 0.2, 0.3, t/l = 0.02 : 0.02 : 0.1. The relative Young’s modulus 0 y s 0 0 ht/ll00 =to 0,the 0.1, 0.2,in0.3 = 0.02 :4b), 0.02for : 0.1h. lThe E Poisson’s ratio νxy vs. are shown Figure 4a,b, respectively. From 4a, it is , t l0(Figure Young’s examples for more y /Ethe s and l0 . With = 0 relative respect Poisson’s ratio , Figure with increase of hillustration: ν xy is positive 0 easy to see that for the fixed h/l , the relative Young’s modulus E /E increases with the increase of s 0 ratio ν Ey increases Es and Poisson’s areh shown 4a,b, respectively. From modulus l0 = 0.1yin , νFigure and it slightly with the increasexy ofvst lt0 .l0For xy is negative and the effect of t/l0 . For the fixed t/l0 , as a whole, the relative Young’s modulus Ey /Es decreases with the increase Ey and Es and h l0t , lthe relative modulus increases with Figure 4a, it is easy to see with that for fixed of l0 Young’s = 0.2 = . For , ν0, negative keeps almost NPR obviously thethe increase xy νis 0 of h/ldecreases 4b), hfor h/l it slightly xy is positive 0 . With respect to the Poisson’s ratio (Figure 0 increases with t/lfrom =0.06. 0.1,Then, νxy relative is the negative and the effect ofEdecreases NPR l.0For Es decreases t lt0increase l0 0.02 increases effect of NPR slightly when athe constant when increase of the theoffixed , h/l as ato the Young’s modulus decreases 0 . tFor 0 whole, y obviously with from the increase of t/l h/l =, 0.2, is negative keeps almost a constant when 0 . For twith l0 increases h lto = 00.3 0.1. For andand almost the ν xy νisxynegative h l0 .toWith h l0 a= constant 0 , ν xy iswith respect ratio (Figure 4b), forkeeps positive the increase of 0.06 0 the Poisson’s t/l0 increases from 0.02 to 0.06. Then, the effect of NPR slightly decreases when t/l0 increases from t lh/l l0 , in and t l0 the halconstant = 0.1 increase ofFor . For thewith fixedthet increase general, effect of NPR of CCF increases . keeps For , νwith andto it 0.1. slightly increases ofalmost isthe negative and thewith effectthe of xy the 0 0 0.06 increase of t/l 0 = 0.3, νxy is negative 0 . For the l0 obviously . increase t l0 increases = 0.2the . For h l0with , νincrease NPR t/l decreases with of theNPR increase ofCCF fixed , inhgeneral, the effect of the of h/l0and . keeps almost 0of xy is negative

a constant when t l0 increases from 0.02 to 0.06. Then, the effect of NPR slightly decreases when

t l0 increases from 0.06 to 0.1. For h l0 = 0.3 , ν xy is negative and almost keeps a constant with the increase of t l0 . For the fixed t l0 , in general, the effect of NPR of the CCF increases with the increase of h l0 .

(a)

(b)

Figure 4.4. (a) relative Young’s modulus Ey /EEs yand ratio νxyratio vs. νt/l = l0 0 for Es (b) (a)The The relative Young’s modulus andPoisson’s (b) Poisson’s vsfor th/l xy 0 0, 0.1, 0.2, 0.3. h l0 = 0, 0.1, 0.2, 0.3 .

4. Effect of Geometrical Morphology on Energy Absorption Properties the NPR CCF

a) (b) To show the effect of (geometrical morphology on energy absorptions of the CCF, uniaxial quasi-static compression tests (MTS E46 (MTS Systems Corporation, Eden Prairie, l0 forwith Figure 4. (a) The relative Young’s modulus Ey Es and (b) Poisson’s ratio ν xy MN, vs tUSA) a loading rate of 1 mm/min) are performed on the four kinds of 3D-printed samples COF0, CCF1, h l0 = 0, 0.1, 0.2, 0.3 . CCF2, and CCF3. For each kind of sample, the experiments are repeated three times. Specific energy

4. Effect of Geometrical Morphology on Energy Absorption Properties the NPR CCF

To show the effect of geometrical morphology on energy absorptions of the CCF, uniaxial quasistatic compression tests (MTS E46 (MTS Systems Corporation, Eden Prairie, MN, USA) with a loading rate of 1 mm/min) are performed on the four kinds of 3D-printed samples COF0, CCF1, CCF2, and Materials 2018, 11, 1869 6 of 18 CCF3. For each kind of sample, the experiments are repeated three times. Specific energy absorption (SEA) [47–51] is used to evaluate the energy absorption capacity of the samples, which is expressed as absorption (SEA) [47–51] is used to evaluate the energy absorption capacity of the samples, which is expressed as RL x EA 0LFF((xx)d )dx EA (3) SEA SEA = = == 0 (3) m m m m where EA EA is is the the energy energy absorption, absorption, L L is is the the effective effective total total crushing crushing length length (that (that is is to to say, say, the the crushing crushing where length of of the the regime regime of of densification, densification, in in which which the the crushing crushing force force rises rises steeply, steeply, is is not not included) included) [45], [45], length and F is the crushing force and m is the mass of the structure. and F is the crushing force and m is the mass of the structure. Typical axial axial load-longitudinal load-longitudinaldeformation deformationcurves curvesofofthe thefour four samples given in Figure 5, Typical samples areare given in Figure 5, in in which curves NPR CCF1,CCF2, CCF2,and andCCF3 CCF3are areapproximately approximatelytruncated truncatedat at the the beginning beginning which thethe curves forfor thethe NPR CCF1, of the final stage of densification where the crushing force begins to rise steeply [45]. It is obvious that of the final stage of densification where the crushing force begins to rise steeply [45]. It is obvious underunder uniaxial quasi-static compression the PPR COF0 is brittle crushing, without the phenomenon of that uniaxial quasi-static compression the PPR COF0 is brittle crushing, without the collapse plateau. However, for CCF1, CCF2, and CCF3, the long collapse plateau exists in the loading phenomenon of collapse plateau. However, for CCF1, CCF2, and CCF3, the long collapse plateau process, meansprocess, that the which CCF could have much higher energy absorption capacities than the COF. exists in which the loading means that the CCF could have much higher energy absorption At the same time, a whole, crushing forces larger those means capacities than theasCOF. At the same time, asof a CCF1 whole,are crushing forcesofofCCF2 CCF1and areCCF3. larger Itthose of that energy absorption capacity of the CCF1, whose NPR effect is weaker than CCF2 and CCF3, is CCF2 and CCF3. It means that energy absorption capacity of the CCF1, whose NPR effect is weaker larger than those of CCF2 and CCF3. than CCF2 and CCF3, is larger than those of CCF2 and CCF3.

Figure 5. Typical Typical crushing crushing force-deformation curves of the four sample COF0, CCF1, CCF2, and CCF3 under uniaxial quasi-static compression in the y direction.

peak crushing crushing forces forces and and SEA SEA of of the the four four samples samplesare areshown shownin inFigure Figure6a,b, 6a,b,respectively. respectively. Initial peak From Figure 6a, it is easy to see that the initial peak crushing forces, i.e., the strengths, of the four samples decrease with the increase of the chord height to span ratio. Figure 6b shows that the SEA of larger than that of the PPRPPR COF0, and and SEASEA of CCF1, whose NPR NPR CCF1, CCF1, CCF2, CCF2,and andCCF3 CCF3are aremuch much larger than that of the COF0, of CCF1, whose effecteffect is weaker than CCF2 and CCF3, is larger than those the of CCF2 CCF3. From the above NPR is weaker than CCF2 and CCF3, is larger than of those the and CCF2 and CCF3. From the mentioned analysis, it can come the conclusion that energy absorption capacity of the CCF can be above mentioned analysis, it canto come to the conclusion that energy absorption capacity of the CCF tuned enhanced through through the rational of design the chord height to span ration, i.e., the geometrical can beand tuned and enhanced thedesign rational of the chord height to span ration, i.e., the morphology morphology or NPR, of the slightly introducing the introducing effect of NPR, theeffect energy geometrical orCCF. NPR,Inofother the words, CCF. Inbyother words, by slightly the of absorption abilities of the CCF could be greatly enhanced with only slightly reducing the stiffness and strength of the CCF.

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NPR, absorption abilities of the CCF could be greatly enhanced with only slightly NPR,the theenergy energy Materials 2018, 11, 1869 absorption abilities of the CCF could be greatly enhanced with only slightly 7 of 18 reducing reducingthe thestiffness stiffnessand andstrength strengthofofthe theCCF. CCF.

(a) (a)

(b) (b)

Figure 6.6. (a)(a) Initial peak crushing forces and (b) energy (SEA) ofofthe Figure6. Initial peak crushing forces and (b)specific specificabsorption energyabsorption absorption (SEA) thethreethreeFigure (a) Initial peak crushing forces and (b) specific energy (SEA) of the three-dimensional dimensional (3D)-printed four samples COF0, CCF1, CCF2, and CCF3. dimensionalfour (3D)-printed four samples COF0,and CCF1, CCF2, and CCF3. (3D)-printed samples COF0, CCF1, CCF2, CCF3.

Effect ofofGeometrical Geometrical 5.5.Effect Effectof GeometricalMorphology Morphologyon onCollapse CollapseModes Modesofofthe theNPR NPRCCF CCFunder underQuasi-Static Quasi-Static Uniaxial Compressions Compressions Uniaxial Uniaxial Compressions In this this part,effect effect of geometrical morphology collapse on collapse modes of the CCF NPR CCF quasiunder In In thispart, part, effectofofgeometrical geometricalmorphology morphologyon on collapsemodes modesofofthe theNPR NPR CCFunder under quasiquasi-static uniaxial compressions are investigated. static staticuniaxial uniaxialcompressions compressionsare areinvestigated. investigated. Under quasi-static uniaxial compressions, collapse mode of the PPRPPR COF0 is abrupt layer by layer Under quasi-static uniaxial Under quasi-static uniaxialcompressions, compressions,collapse collapsemode modeofofthe the PPRCOF0 COF0isisabrupt abruptlayer layerby by brittlebrittle fracture in a very short time. time. It is very difficult to taketophotos of the of collapse process using layer fracture in a very short It is very difficult take photos the collapse process layer brittle fracture in a very short time. It is very difficult to take photos of the collapse process the common camera. So, here, collapse modes of the PPR COF0 are not discussed. Typical collapse using usingthe thecommon commoncamera. camera.So, So,here, here,collapse collapsemodes modesofofthe thePPR PPRCOF0 COF0are arenot notdiscussed. discussed.Typical Typical modes ofmodes the NPR CCF1, CCF2 and CCF3 underCCF3 uniaxial quasi-static compressive tests (Figure 5) are collapse collapse modesofofthe theNPR NPRCCF1, CCF1,CCF2 CCF2and and CCF3under underuniaxial uniaxialquasi-static quasi-staticcompressive compressivetests tests shown in Figures 7a, 8a and 9a,7a–9a, respectively. Numerical simulations of collapse process untiluntil 25% (Figure (Figure5)5)are areshown shownininFigures Figures 7a–9a,respectively. respectively.Numerical Numericalsimulations simulationsofofcollapse collapseprocess process until strainstrain of the NPR NPR CCF1, CCF2,CCF2, and CCF3CCF3 under quasi-static compressions in the yindirection are shown 25% 25% strainofofthe the NPRCCF1, CCF1, CCF2,and and CCF3under underquasi-static quasi-staticcompressions compressions inthe they ydirection directionare are in Figures 7b, 8b and 9b, respectively. Details of the numerical simulations are are as follows: two two rigid shown shownininFigures Figures7b–9b, 7b–9b,respectively. respectively.Details Detailsofofthe thenumerical numericalsimulations simulations areasasfollows: follows: tworigid rigid plates with with size 40 mm × 40 mm ××2 2mm, mm, elasticmodulus modulus 210,000MPa, MPa, andPoisson’s Poisson’s ratio 0.33, 0.33, have plates plates withsize size4040mm mm× ×4040mm mm × 2 mm,elastic elastic modulus210,000 210,000 MPa,and and Poisson’sratio ratio 0.33,have have been put on the top and bottom of the COF or CCF along the y direction. Elastic modulus and Poisson’s been beenput puton onthe thetop topand andbottom bottomofofthe theCOF COFororCCF CCFalong alongthe they ydirection. direction.Elastic Elasticmodulus modulusand and ratio of the materials, materials, of which the COF and CCF are made of, are 2500 of, MPa and 0.33, respectively. Poisson’s Poisson’sratio ratioofofthe the materials,ofofwhich whichthe theCOF COFand andCCF CCFare aremade made of,are are2500 2500MPa MPaand and0.33, 0.33, The rigid plates are connected with the COF or CCF usingor the tie constrains. The bottom rigid plate is respectively. respectively.The Therigid rigidplates platesare areconnected connectedwith withthe theCOF COF orCCF CCFusing usingthe thetietieconstrains. constrains.The Thebottom bottom fixed.plate An uniaxial displacement of 10 mm is exerted on theistop rigid platethe in the yrigid direction. The COF rigid rigid plateisisfixed. fixed.An Anuniaxial uniaxialdisplacement displacementofof1010mm mm isexerted exertedon on thetop top rigidplate plateininthe they y and CCF are constituted by 64 cubic unit cells (Figure 1a,b). B31 beam element is used for the COF direction. direction.The TheCOF COFand andCCF CCFare areconstituted constitutedby by6464cubic cubicunit unitcells cells(Figure (Figure1a,b). 1a,b).B31 B31beam beamelement elementisis and CCF. The beam element length is element 0.1 mm and theisnumber of thethe beam elements isbeam 76,800. For the used for the COF and CCF. The beam length 0.1 mm and number of the elements used for the COF and CCF. The beam element length is 0.1 mm and the number of the beam elements top and bottom rigidand plates, the rigid C3D8R element is utilized. The length of the element of is 0.5 mm and is is76,800. 76,800.For Forthe thetop top andbottom bottom rigidplates, plates,the theC3D8R C3D8Relement elementisisutilized. utilized.The Thelength length ofthe theelement element the corresponding elements number for each rigid plate is 400. In addition, the default self-contact isis0.5 0.5mm mmand andthe thecorresponding correspondingelements elementsnumber numberfor foreach eachrigid rigidplate plateisis400. 400.InInaddition, addition,the thedefault default boundary conditions given by Abaqus have been used to dealbeen with used the contact interactions between self-contact boundary conditions given by Abaqus have to deal with the self-contact boundary conditions given by Abaqus have been used to deal with thecontact contact the cell edges. interactions interactionsbetween betweenthe thecell celledges. edges.

(a) (a) Figure 7. Cont.

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(b) (b) Figure Frontview viewofofcollapse collapsemodes modes of of the the NPR CCF1 compressions in in Front Figure 7. 7. CCF1under underquasi-static quasi-staticuniaxial uniaxial compressions y direction:(a)(a)experiment; experiment;(b) (b)numerical numerical simulation simulation (until (until25% 25%strain). strain). thethe y direction: (until 25% strain).

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Figure Frontview viewofofcollapse collapsemodes modes of of the the NPR NPR CCF2 compressions in in Figure 8. 8. Front CCF2under underquasi-static quasi-staticuniaxial uniaxial compressions the y direction: (a) experiment; (b) numerical simulation (until 25% strain). Figure 8. Front (a) view of collapse(b) modes of the simulation NPR CCF2(until under25% quasi-static the y direction: experiment; numerical strain). uniaxial compressions in the y direction: (a) experiment; (b) numerical simulation (until 25% strain).

(a)

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(b) Figure Frontview viewofofcollapse collapsemodes modes of of the the NPR CCF3 compressions in in Figure 9. 9. Front CCF3under underquasi-static quasi-staticuniaxial uniaxial compressions (b) y direction:(a)(a)experiment; experiment;(b) (b)numerical numerical simulation simulation (until thethe y direction: (until25% 25%strain). strain).

Figure 9. Front view of collapse modes of the NPR CCF3 under quasi-static uniaxial compressions in the y direction: (a) experiment; (b) numerical simulation (until 25% strain).

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As a whole, experimental results and numerical simulations of collapse modes of the NPR CCF1 (Figure 7) agree very well with each other, as well as the NPR CCF2 (Figure 8). Experimental results (Figure 9a) and numerical simulations (Figure 9b) of the collapse modes of the NPR CCF3 are different. Due to the fabrication errors produced in the 3D-printed process, in the experiment (Figure 9a), the initial collapse starts from the bottom end of the CCF3. In the numerical simulation (Figure 9b), the initial collapse starts from the middle layer of the CCF3. But, their final collapse modes (Figure 9a,b) are similar. From Figures 7–9, it is obvious that under uniaxial compression in the y direction, the CCF1, CCF2, and CCF3 all shrink in the transverse x direction, indicating their NPR characteristics. However, their collapse modes are different. For the CCF1 (Figure 7) whose NPR effect is relatively weak, in the uniaxial quasi-static compressive loading process, stresses are mainly concentrated on the convex and concave parts of the vertical curved cell edges (Figure 7a,b), so it collapses layer by layer from the peak points of the convex and concave vertical cell edges. For the CCF2 (Figure 8), whose cell edges are more curved than CCF1 (i.e., the NPR effect is more obvious than CCF1), under uniaxial quasi-static compressions, the adjacent curved cell edges will contact with each other. It renders a collapse mode of ‘X’ shape for CCF2, as that shown in Figure 8a,b. For the CCF3 (Figure 9), whose cell edges are much more curved (h/l0 = 0.3), under compressions the adjacent curved cell edges will be more easier to contact with each other and the horizontal slipping shears occur in the middle layers. Finally, it exhibits the oblique quadrilateral collapse mode for CCF3 (Figure 9a,b). 6. Effect of Geometrical Morphology on Damage Properties of the NPR CCF In this part, effect of geometrical morphology on damage properties of the NPR CCF is investigated through the plastic-damage constitute model that was proposed by Senaz [52], which is initially developed for concrete. 6.1. Plastic-Damage Constitute Model for the 3D-Printed VeroWhite Plus Materials First, plastic-damage constitute model for the 3D-printed VeroWhite Plus materials is introduced. It is well known that for the heterogeneous materials there is an important relation between the damage field and deformations of the microstructures [53–56]. Increase of the damage means the deterioration of the structural integrity of the intrinsic microstructures. Due to the heterogeneous characteristics of the 3D-printed VeroWhite Plus materials, damage distributions of the NPR CCF could be studied through the plastic-damage constitute model [53,57–60], thus predicting and explaining macro deformations and fracture behaviors of the NPR CCF. To establish the plastic-damage constitute model for the 3D-printed VeroWhite Plus materials, tensile and compressive tests are performed on the 3D-printed VeroWhite Plus specimens (Figure 10a). Tensile tests (Figure 10a,b) are performed on two kinds of dog-bone shaped specimens, one kind is the transversely printed tensile specimen (TPT) and the other kind is the longitudinally printed tensile specimens (LPT). Tensile tests, according to the ASTM D638 standard [61], of each kind of specimens are repeated two times on MTS E46 with a loading rate of 2 mm/min. The compressive test is performed on the cubic specimen with size 1 mm × 1 mm × 1 mm (Figure 10a). The corresponding stress-strain curves of the tensile and compressive tests are shown in Figure 10c.

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Figure10. 10.(a) (a)3D-printed 3D-printedVeroWhite VeroWhitespecimens specimensfor fortensile tensileand andcompressive compressivetests: tests:transversely transverselyprinted printed Figure tensile specimen means transversely printedprinted tensile dog-bone specimens, tensile specimen(TPT) (TPT) means transversely tensile shaped dog-bone shapedlongitudinally specimens, printed tensileprinted specimens (LPT) means longitudinally printed tensileprinted dog-bone shaped specimens and longitudinally tensile specimens (LPT) means longitudinally tensile dog-bone shaped the cubic specimen is for compressive test; (b) Tensile tests of the VeroWhite TPT and LPT specimens; specimens and the cubic specimen is for compressive test; (b) Tensile tests of the VeroWhite TPT and (c) Stress-strain curves of the 3D-printed Plus VeroWhite TPT, LPT, and cubic specimens; LPT specimens; (c) Stress-strain curves of VeroWhite the 3D-printed Pluscompressive TPT, LPT, and compressive (d) Ratios of elongation andofreduction of and areareduction of the TPTofand LPT. cubic specimens; (d) Ratios elongation area of the TPT and LPT.

Compressive stress-strain 10c) of the 3D-printed VeroWhite Plus materials is similar Compressive stress-straincurve curve(Figure (Figure 10c) of the 3D-printed VeroWhite Plus materials is to that of the pure concrete, so the plastic-damage model, proposed by Senaz [52], for pure concrete similar to that of the pure concrete, so the plastic-damage model, proposed by Senaz [52], for pure could becould used be for used the compressive damage analysis the 3D-printed VeroWhite Plus materials. concrete for the compressive damage of analysis of the 3D-printed VeroWhite Plus The plastic-damage model could alsocould be used forbetensile damage analysis ofanalysis the 3D-printed VeroWhite materials. The plastic-damage model also used for tensile damage of the 3D-printed Plus materials. reasons arereasons that theare ratios elongation and reduction area of the 3D-printed VeroWhite Plus The materials. The thatofthe ratios of elongation andof reduction of area of the VeroWhite Plus TPT and LPT are very small (Figure 10d), and, at the same time, tensile stress-strain 3D-printed VeroWhite Plus TPT and LPT are very small (Figure 10d), and, at the same time, tensile curves of thecurves TPT and LTPTPT are similar toare thatsimilar of the to compressive which also show obvious linear stress-strain of the and LTP that of thetest, compressive test, which also show elastic stage and plastic damage stage. From the above analysis, the plastic-damage model that was obvious linear elastic stage and plastic damage stage. From the above analysis, the plastic-damage proposed Senaz [52], could be used both compressive tensile damage analysis of the model thatby was proposed by Senaz [52],for could bethe used for both theand compressive and tensile damage 3D-printed VeroWhite Plus materials. analysis of the 3D-printed VeroWhite Plus materials. Inthe thelinear linearelastic elasticstage, stage,the therelation relation between between the the stress stress and and the the strain strain is is expressed expressed as: as: In

= EE0ε0 ε σσ = In the plastic damage stage it is given by

(4)(4)

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E0ε

σ=

E ε ε ε (αgiven 2)( ) + (1 − 2α )( )2 + α ( )3 + 0 −by In the plastic damage stage1it+ is E ε ε ε s

0

0

(5)

0

E0 ε elastic modulus, ε 0 is the strain corresponding in which σ is the stress, εσis= the strain, EE0 is the initial (5) ε 1 + (α + E0s − 2)( ε 0 ) + (1 − 2α)( εε0 )2 + α( εε0 )3 to the peak stress σ 0 , Es is the secant modulus, and α is the characteristic coefficient which has the

form in which σ is the stress, ε is the strain, E0 is the initial elastic modulus, ε 0 is the strain corresponding to the peak stress σ0 , Es is the secant modulus, and α is the characteristic coefficient which has the form E σ0 （ -1） EE0 ( σσ0 u− 1) ε u E σ −− ε u (6) α α== 0εε u (6) ε 0ε 2 2 0 ((εu0u − 1 ) − 1)

ε0

Here, σu is the ultimate stress and ε u is the strain corresponding to σu [52]. Here, σ u is the ultimate stress and ε u is the strain corresponding to σ u [52]. The total damage D, summation of the compressive and tensile damages, is expressed as: The total damage D, summation of the compressive and tensile damages, is expressed as: r σ (7) D = 1− σ E0 ε D =1− (7) E0 ε Substituting Equations (5) and (6) into Equation (7) gives the formula of damage evolution for the Substituting Equations (5) and (6) into Equation (7) gives the formula of damage evolution for 3D-printed VeroWhite Plus materials the 3D-printed VeroWhite Plus materials   0 0 Elastic stage Elastic stage  1 1 (8) D = 1 − r Plasticdamage damage stage Plastic stage D = 1－ (8)  E0  ε )+(1−2α )( ε )+ α ( ε ) E 1 +( α + − 2 )( ε ε ε  0 α ( ε 0) S 1+(α + 0 －E2)( ) +ε 0(1－2α )( ε)+  ε0 ε0 ε0 ES  In Section damage properties of theof3D-printed VeroWhite Plus NPRPlus CCFNPR will be numerically In Section6.2, 6.2, damage properties the 3D-printed VeroWhite CCF will be investigated through Abaqus. The corresponding tensile and compressive plastic-damage parameters numerically investigated through Abaqus. The corresponding tensile and compressive plasticfor the numerical are shown in Figure These parameters are damage parameterssimulations for the numerical simulations are 11a,b, shownrespectively. in Figure 11a,b, respectively. These calculated by curves, of the 3D-printed VeroWhite Plus TPT, LPT Plus and parameters arecombining calculatedthe by stress-strain combining the stress-strain curves, of the 3D-printed VeroWhite compressive cubic specimens (Figure 10c), and Equation (7). To verify these plastic-damage parameters, TPT, LPT and compressive cubic specimens (Figure 10c), and Equation (7). To verify these plasticthe stress-strain curvesthe of the tensile dog-bone and compressive specimen damage parameters, stress-strain curves shaped of the specimen tensile dog-bone shaped cubic specimen and are numerically simulated. The related numerical results and the experimental results are shown in compressive cubic specimen are numerically simulated. The related numerical results and the Figure 11c. Obviously, they agree very well with each other, indicating that the damage properties of experimental results are shown in Figure 11c. Obviously, they agree very well with each other, the NPR CCF be analyzed using the andCCF compressive parameters shown indicating thatcould the damage properties oftensile the NPR could beplastic-damage analyzed using the tensile and in Figure 11a,b. compressive plastic-damage parameters shown in Figure 11a,b.

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Figure 11. (a)(a) Tensile and (b) compressive plastic-damage parameters of the3D-printed 3D-printed VeroWhite Figure Tensile and(b) (b)compressive compressiveplastic-damage plastic-damage parameters VeroWhite Figure 11.11. (a) Tensile and parametersofofthe the 3D-printed VeroWhite Plus materials; (c) Comparison ofofstress-strain curves of the the tensiledog-bone dog-bone shapedspecimen specimen and Plus materials; Comparisonof stress-strain curves curves of of and Plus materials; (c)(c) Comparison stress-strain the tensile tensile dog-boneshaped shaped specimen and compressive cubic specimen: experiments and numerical simulations. compressive cubic specimen:experiments experimentsand and numerical numerical simulations. compressive cubic specimen: simulations.

Damage 6.2.6.2. Properties Damage Propertiesofofthe theNPR NPRCCF CCF Utilizing the periodic boundary conditions andsimply simplychoosing choosing222× ×2 22arrays arrays the cubic Utilizing the periodic boundaryconditions conditionsand simply choosing ××222×× ofof the cubic unit Utilizing the periodic boundary arrays unit cellcell (Figure 1b) asas the COF0,CCF1, CCF1,CCF2, CCF2,and andCCF3 CCF3 under uniaxial (Figure 1b) themodel, model,damage damageproperties properties of the COF0, under uniaxial compressive loadings inthe direction are numerically numerically the compressive loadings yy direction are studied through theplastic-damage plastic-damage compressive loadings in in the ythe direction are numerically studiedstudied throughthrough the plastic-damage constitute constitute that isgiven given Figure11a,b. 11a,b. constitute that ininFigure model thatmodel ismodel given in is Figure 11a,b. Fixing thebottom bottomend, end,aacompressive compressive displacement displacement of is is Fixing the of 0.75 0.75 mm mm(uniaxial (uniaxialstrain strainofof3.75%) 3.75%) exerted on the top of the COF0 in the y direction. The corresponding compressive and tensile damage exerted on the top of the COF0 in the y direction. The exerted The corresponding corresponding compressive and tensile damage distributions (front view) of the COF0 are shown in Figure 12a,b, respectively. From Figure 12a, it is distributions (front view) of the COF0 are shown in Figure 12a,b, respectively. From Figure 12a, it is (front view) of the COF0 are shown in Figure 12a,b, respectively. From Figure 12a, easy to see that severe buckling deformations occur at the intersections between the vertical cell edges easy to see buckling deformations occuroccur at theat intersections between the vertical cell edges it is easy tothat see severe that severe buckling deformations the intersections between the vertical cell and the horizontalcell cell edges.From Fromthe theview view of of A-A A-A crosscuttings, cross sections ofofthe vertical cellcell and the horizontal crosscross sections the edges and the horizontaledges. cell edges. From the view of crosscuttings, A-A crosscuttings, sections of vertical the vertical edges show mainly compressive damage and almost no tensile damage. From the view of B-B edges show mainly compressive damage and almost nonotensile cell edges show mainly compressive damage and almost tensiledamage. damage.From Fromthe the view view of B-B crosscuttings, tensile damage occurs near the intersections of cross sections of the horizontal cell crosscuttings, tensile tensiledamage damageoccurs occurs near intersections of cross sections the horizontal cell crosscuttings, near thethe intersections of cross sections of theofhorizontal cell edges. edges. edges.

Figure 12. Damage distributions of the COF0 under the compressive displacement of 0.75 mm (uniaxial Figure 12. Damage distributions of the COF0 under the compressive displacement of 0.75 mm strain of 3.75%) in the y direction: (a) compressive damages; (b) tensile damages. (uniaxial strain of 3.75%) in the y direction: (a) compressive damages; (b) tensile damages. Figure 12. Damage distributions of the COF0 under the compressive displacement of 0.75 mm (uniaxial strain of 3.75%) in the y direction: (a) compressive damages; (b) tensile damages.

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Likewise, displacements of of 11 mm mm (uniaxial (uniaxial strain strain of of 5%) 5%) is is exerted on top Likewise, aa compressive compressive displacements exerted on top of of the the CCF1 in the y direction. As shown in Figure 13a, compressive damages are concentrated on the CCF1 in the y direction. As shown in Figure 13a, compressive damages are concentrated on the inner inner concave of vertical the vertical curved edges. shown Figure13b, 13b,tensile tensile damages damages are concave parts parts of the curved cellcell edges. As As shown ininFigure are concentrated on the outer convex parts of the vertical curved cell edges. Almost no damage occurs concentrated on the outer convex parts of the vertical curved cell edges. Almost no damage occurs for The maximum maximum damages damages occur occur on on edges edges of of cross cross sections sections of of peak peak for the the horizontal horizontal curved curved cell cell edges. edges. The points points of of the the inner inner concave concave (view (view of of AA AA crosscutting crosscutting in in Figure Figure 13a) 13a) and and outer outer convex convex (view (view of of BB BB crosscutting in Figure 13b) parts, which will firstly lead to the initial fracture of the structures. crosscutting in Figure 13b) parts, which will firstly lead to the initial fracture of the structures.

Figure 13. Damage distributions of the CCF1 under the compressive displacement of 1 mm (uniaxial strain of 5%) in the y direction: (a) compressive damages; (b) tensile tensile damages. damages.

Compared (uniaxial strain ofof 7.5%) is Compared with withthe theCCF1, CCF1,aalarger largercompressive compressivedisplacement displacementofof1.5 1.5mm mm (uniaxial strain 7.5%) exerted onon CCF2 in the y direction. However, the the compressive (Figure 14a)14a) and and the tensile (Figure 14b) is exerted CCF2 in the y direction. However, compressive (Figure the tensile (Figure damages of the CCF2 are obviously reduced. From the views of AA (Figure 14a) and BB (Figure 14b) 14b) damages of the CCF2 are obviously reduced. From the views of AA (Figure 14a) and BB (Figure crosscuttings, it is also that both area peak the compressive and tensile 14b) crosscuttings, it isapparent also apparent thatthe both theand areathe and thevalue peak of value of the compressive and damages are decreased. TheseThese indicate thatthat the the introducing of of NPR could tensile damages are decreased. indicate introducing NPR couldenhance enhancethe the damage damage resistance resistance capacity capacity of of the the CCF. CCF. When compared with the CCF2, a much larger compressive displacement of 1.65 mm (uniaxial strain of 8.25%) is exerted on CCF3 in the y direction. Apparently, compressive (Figure 15a) and tensile (Figure 15b) damages of CCF3 are much more reduced, indicating that damage resistance capacity of the CCF3 is much more enhanced. From the view of AA crosscutting, the maximum compressive damage still occurs on edges of cross sections of peak points of the inner concave. But, the maximum tensile damage deviates a distance from the edges and is not on edges of cross sections of peak points of the outer convex anymore.

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Figure 14. Damage Damage distributions distributions of the CCF2 CCF2 under under the the compressive compressive displacement of 1.5 mm (uniaxial Materials 2018, x FOR strain of11, 7.5%) in PEER the yy REVIEW direction: (a) compressive damages; (b) tensile damages.

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When compared with the CCF2, a much larger compressive displacement of 1.65 mm (uniaxial strain of 8.25%) is exerted on CCF3 in the y direction. Apparently, compressive (Figure 15a) and tensile (Figure 15b) damages of CCF3 are much more reduced, indicating that damage resistance capacity of the CCF3 is much more enhanced. From the view of AA crosscutting, the maximum compressive damage still occurs on edges of cross sections of peak points of the inner concave. But, the maximum tensile damage deviates a distance from the edges and is not on edges of cross sections of peak points of the outer convex anymore.

15.Damage Damagedistributions distributions of the CCF3 under the compressive displacement of 1.65 mm Figure 15. of the CCF3 under the compressive displacement of 1.65 mm (uniaxial strain of 8.25%) in 8.25%) the y direction: (a) compressive damages; damages; (b) tensile(b) damages. (uniaxial strain of in the y direction: (a) compressive tensile damages.

To further explain the effect of geometrical morphology on damage properties of CCF, the maximum damages of the cross sections of the vertically placed cell edges under different compressive displacements have been calculated. The corresponding maximum compressive and tensile damages versus the compressive displacements are given in Figure 16a,b, respectively. It is obvious that the increase of the effect of NPR could postpone the occurrence of structural damage of the CCF and it decrease the damage degree.

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(uniaxial strain of 8.25%) in the y direction: (a) compressive damages; (b) tensile damages.

To further further explain explainthe theeffect effectof of geometrical morphology damage properties of CCF, To geometrical morphology on on damage properties of CCF, the the maximum damages of the cross sections of the vertically placed cell edges under different maximum damages of the cross sections of the vertically placed cell edges under different compressive displacements displacements have have been been calculated. calculated. The The corresponding corresponding maximum maximum compressive compressive and and compressive tensile damages versus the compressive displacements are given in Figure 16a,b, respectively. is tensile damages versus the compressive displacements are given in Figure 16a,b, respectively. ItIt is obvious that the increase of the effect of NPR could postpone the occurrence of structural damage of obvious that the increase of the effect of NPR could postpone the occurrence of structural damage of the CCF CCF and and itit decrease decrease the the damage damage degree. degree. the

(a)

(b)

Figure Figure 16. 16. The Themaximum maximumdamages damagesof ofthe the cross cross sections sections of of the the vertically vertically placed placed cell cell edges edges under under different different compressive compressive displacements: displacements: (a) (a) compressive compressive damages; damages; (b) (b) tensile tensile damages.

7. Conclusions Conclusions 7. The orthogonal orthogonal isotropic isotropic NPR NPR CCF CCF are are aa typical typical kind kind of of mechanical mechanical metamaterials, metamaterials, which which have have The great potential potential applications applications in in the the fields fields of of mechanical mechanical and and aerospace aerospace engineering. engineering. Systematic Systematic and and great deep understanding of their mechanical properties plays a vital role for their practical engineering deep understanding of their mechanical properties plays a vital role for their practical engineering applications. In this paper, based on the polymer materials VeroWhite Plus and mainly choosing one COF and three kinds of CCF of equal mass as examples, the effects of the geometrical morphology on elastic moduli, energy absorption, and damage characteristics of the CCF, are systematically investigated through the experimental and finite element methods. Results show that NPR, energy absorption, and damage properties of the CCF could be tuned simultaneously through adjusting the chord height to span ratio of the sine-shaped cell edges. By rational design of the NPR, as compared to the COF of equal mass, the CCF could have combined advantages of relative high stiffness and strength, enhanced energy absorption, and damage resistance. The study of this paper provides the theoretical foundations for optimization design of mechanical properties of the NPR CCF and thus could promote their practical applications in the engineering fields. However, one thing noteworthy is that this paper has only focused on mechanical properties of the polymer (VeroWhite Plus) CCF, due to their easy fabrication through the 3D-printed method. In fact, the stiffness and strength of the VeroWhite Plus materials are very low. For further enlarging, their practical engineering applications, the method to make much stiffer and higher strength metal CCF should be explored. Author Contributions: Conceptualization, S.C., Y.S. and B.G.; Methodology, S.C. and B.G.; Software, S.C.; Validation, Q.D. and X.L.; Formal Analysis, S.C., Y.S., F.R. and B.S.; Investigation, S.C. and B.G.; Resources, Q.Y., H.Y. and X.W.; Writing-Review & Editing, S.C. and Y.S.; Funding Acquisition, Y.S. Funding: This research was funded by National Natural Science Foundation of China No. 11502162, National Natural Science Foundation of Tianjin No. 15JCQNJC05100, State Key Laboratory for Strength and Vibration of Mechanical Structures (Xi’an Jiaotong University) No. SV2015-KF-02, State Key Laboratory of Mechanical System and Vibration (Shanghai Jiaotong University) No. MSV201611 and State Key Laboratory of Structural Analysis for Industrial Equipment (Dalian University of Technology) No. GZ1408.

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Acknowledgments: Jianqiao Sun of UC Merced and Tianjin University is appreciated for the insightful and helpful discussions. Conflicts of Interest: There is no conflicts of interests.

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