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Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

In-plane compressive behavior of graphene-coated aluminum nano-honeycombs

Yu Zhoua, Wu-Gui Jianga,⁎, Xi-Qiao Fengb, Duo-Sheng Lic, Qing-Hua Qind, Xiao-Bo Liua

a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 330063, ChinabDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, Chinac School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang 330063, Chinad Research School of Engineering, The Australian National University, Acton, ACT 2601, Australia

A R T I C L E I N F O

Keywords:Graphene-coatingNano-honeycombIn-plane compressive behaviorMolecular dynamicsTheoretical method

A B S T R A C T

The in-plane compressive behaviors of aluminum (Al) nano-honeycombs (ANHCs) and graphene-coated ANHCs(GANHCs) are investigated by using the molecular dynamics (MD) and theoretical methods. The dependences oftheir mechanical properties on the crystalline direction and the relative mass density are examined. Our MDsimulations show that the tensile strength, tensile energy absorption, in-plane elastic modulus and compressivestrength of ANHCs can be significantly reinforced by graphene-coating, whereas the compressive energy ab-sorption capacity of GANHCs until densification is lower than that of ANHCs. A modified Gibson-Ashby model isdeveloped to estimate the in-plane Young’s modulus and compressive strength of GANHCs, which shows a goodagreement with the MD results.

1. Introduction

As a type of novel lightweight materials, nano-honeycomb materialshave attracted much attention in different fields such as chemistry [1],physics [2] and engineering [3] due to the combination of structuraloptimization of honeycomb materials and the excellent properties ofnanomaterials [4,5]. Nano-honeycomb materials can be used for na-nowire fabrication and as structural supports for devices [6,7]. Masudaand Fukuda [8], and Chu et al. [9] have successfully produced highlyordered alumina nanofilms with cylindrical pores using the templatingmethod.

The discovery of graphene [10] has stimulated enormous studies oncarbon (C) nanomechanics because of their extraordinary strength,stiffness and low-density properties. Graphene holds promise for im-portant applications as, for example, reinforcements in polymers [11],ceramics [12] and metal [13–15]. Wang et al. [13] fabricated alu-minum composites reinforced with graphene nanosheets through afeasible methodology based on flake powder metallurgy, and found thatthe tensile strength of 249MPa was achieved in the Al composites re-inforced with only 0.3 wt% graphene nanosheets, which is 62% en-hancement over the unreinforced Al matrix. By using mechanical mil-ling and hot rolling, Shin et al. [14] found that with an addition of only0.7 vol% few-layer graphene (FLG), the composite exhibits a tensilestrength of ∼440MPa, about twice higher than that of monolithic Al.

Al always severs in a complex environment. Coating is a popularway to protect Al products. Compared with alumina coatings, graphenecoatings possess various unique properties that are more suitable forlightest and thinnest protective barriers of metal aviation componentsagainst stress corrosion as well as overheating and lightning stroke, dueto their excellent electrical conductivity, heat resisting property, che-mical inertness and transparency [16–19]. Chen et al. [16] and Prasaiet al. [17] found that graphene coating can prevent the formation ofany oxide on the protected metal surfaces (Cu, Ni, Cu/Gr, and Cu/Ni).This protection method offers significant advantages and can be used onmany metals that catalyze graphene growth. Topsakal et al. [18] stu-died Al (1 1 1) surface with graphene coating by first-principles calcu-lations and found that continuous coating of pristine graphene on re-active surfaces can provide an excellent protection from oxidation ofreactive surfaces at nanoscale, and the weak protection caused by thedefect of graphene can be solved by increasing the number of coatinglayers. Sharma et al. [19] studied graphene-Cu composites by using theMD simulation method and found that graphene can improve bothYoung’s modulus and thermal conductivity of the nanocompositescompared with pure Cu. Chemical vapor deposition (CVD) can producehigh-quality graphene films on a metal (e.g., Cu, Ni, Pt or alloy) surfaceat high temperatures, and the films are then transferred to other sub-strates [20]. It should be noticed that the chemical reaction between Aland graphene leads to the formation of Al carbide (Al4C3) at a higher

https://doi.org/10.1016/j.commatsci.2018.10.011Received 6 July 2018; Received in revised form 18 September 2018; Accepted 9 October 2018

⁎ Corresponding author.E-mail addresses: jiangwugui@nchu.edu.cn (W.-G. Jiang), qinghua.qin@anu.edu.au (Q.-H. Qin).

Computational Materials Science 156 (2019) 396–403

0927-0256/ © 2018 Elsevier B.V. All rights reserved.

T

temperature, which weakens the mechanical properties of graphene-Alcomposites [21]. However, the transferring method is difficult to fab-ricate for graphene-coated Al nanohoneycombs (ANHCs). Fortunately,the development of low-temperature graphene growth [22] has pro-vided a possibility for graphene coating on inner walls of ANHCs.

Molecular dynamics (MD) simulations have been widely used topredict the mechanical properties of nanomaterials [23,24]. For tradi-tional porous materials, classical expressions have been given by Gibsonand Ashby [7] to predict the Young’s modulus and yield stress in termsof the relative density of the material, and the scaling constants can befitted by finite element (FE) analysis [25]. It is important to notice thatthese constants are not suitable well in naonoporous materials due tothe large surface-to-volume ratio [26,27]. Yuan and Wu [28] found thatthe Gibson-Ashby’s scaling law is still applicable in nanoscale, but thescaling constants should be determined from experimental or simula-tion results.

It is of great interest to explore the reinforcing effect of graphene insuch nanostructured materials as nanohoneycombs. In this work, the in-plane compressive behaviors of GANHCs including single-layer gra-phene (SG) coated ANHCs (SGANHCs) and double-layer graphene (DG)coated ANHCs (DGANHCs) as well as pure ANHCs with different re-lative densities are firstly investigated via MD simulations. We alsopropose a modified Gibson-Ashby model to predict the in-plane Young’smodulus and compressive strength of GANHCs.

2. Methodology

When coated on the inner wall of cylindrical pores, a sheet of gra-phene can be curved into a one-dimensional structure, that is, a carbonnanotube (CNT). Hence, the graphene coating is substituted by a CNT inour model. Although the mechanical properties of CNT/Al compositeswere investigated by Silvestre et al. [29] and Choi et al. [30] via MDsimulations, both studies focused on the prediction of the out-of-planemechanical behavior. A representative volume element (RVE) (seeFig. 1) of ANHCs, SGANHCs and DGANHCs is chosen respectively forMD simulations to investigate the uniaxial compressive behavior of theRVE in the in-plane direction. Fig. 1 shows the perspective and topviews of the RVEs for polycrystal ANHCs, SGANHCs and DGANHCs. Thex and y axes are considered as the in-plane directions, and the z axis isthe out-of-plane direction, as shown in Fig. 1. The sizes of all RVEs are12 nm×12 nm×12 nm (Fig. 1a), with a through cylindrical pore inthe center (as shown in Fig. 1b), coated with a single-walled carbonnanotube (SWCNT) (as shown in Fig. 1c and d) or a double-walledcarbon nanotube (DWCNT) (as shown in Fig. 1e and f). The chirality ofthe CNTs in our simulations is armchair. The inner-outer combinationof DWCNTs is accord with (n, n)@(n+ 5, n+5), which is the moststable structure among armchair CNTs [31]. The length of CNTs alongthe z axis is 12 nm. The compression direction is along the y axis whichcorresponded to three crystal orientations (monocrystal [1 0 0], [1 1 1]directions and polycrystalline). For the [1 0 0] compression direction,the x, y and z axes correspond to [0 1 0], [1 0 0], [0 0 1] directions,respectively. For [1 1 1] direction, the x, y, z axes correspond to [1 1̄ 0],[1 1 1] and [1 1 2̄] directions respectively. As shown in Fig. 1, 12 grainsare randomly distributed inside the bulk in the polycrystalline Almodel. To investigate the influence of relative mass density, 7 classes ofANHCs with a center cylindrical pore of different radii are built. In eachcompressive direction, there are 7 groups, which have identical innerradius and coating type, as listed in Table 1. The relative density ρr ofthe RVEs is calculated by ρr =MR/MB, where MB is the relative atomicmass of the bulk and MR is the relative atomic mass of the RVE. Forsimplicity, the mass of the graphene coating is not taken account intowhen ρr is calculated in GANHCs. All models are constructed by usingAtomsk [32]. Atomic images are visualized via Ovito [33].

MD simulations are performed by means of the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [34]. The Al-Al in-teraction is commonly described by an embedded atom method (EAM)

potential [35], and the binding interaction of C-C is described by thewidely used adaptive intermolecular reactive empirical bond order(AIREBO) potential [36]. The Lennard-Jones (LJ) 12-6 potential isadopted to simulate interatomic forces between C and Al atoms andlong rang van der Waals (vdW) interaction between C atoms. The LJinteraction is written as = −U r ε σ r σ r( ) 4 [( / ) ( / ) ]ij ij ijLJ

12 6 , ≤r r( ),ij c where rijis the distance between those atoms not forming bonds, ε is the welldepth and σ is the size parameter, and rc is the cutoff distance. The LJparameters εAl-C and σAl-C are determined by Lorentz-Berthelot (LB)mixing rule [37]. The value of εAl-C is estimated from geometric average(εAl-C= ε εAl C ) between εAl and ε ,C and the value of σAl-C is estimatedfrom the arithmetic average ((σAl + σC)/2) between σAl and σC. Fornon-bonded C atoms Kutana and Giapis [38] recommended the LJparameters εC =0.00296 ev and σC= 3.407 Å, and Munilla et al. [39]suggested the LJ parameters εAl= 0.4157 ev and σAl= 2.62 Å. Hence,according to the LB rule, the LJ parameters for the C-Al interface are εAl-C= 0.035078 ev and σAl-C= 3.0135 Å. The equilibrium distance be-tween the Al substrate and graphene in our tests is about 2.6 Å, whichconforms to the cohesive law proposed by Jiang et al. [40] that theequilibrium distance is 0.858σAl-C, i.e. 2.59 Å. So the distance betweenthe graphene coating and the inner wall of the Al matrix is taken as2.59 Å (Fig. 1d). The LJ cutoff radius is chosen as 2.5σ in both C-Alinteraction and long range C-C interaction, beyond which vdW inter-action is very weak and can be neglected.

The periodic boundary conditions are applied in all three directions.Prior to loading, all models are relaxed using NPT ensemble (the totalnumber of particles, the system’s volume, and the system’s pressure areconstant) for 100 ps to minimize the total free energy. Then a NVTensemble (the total number of particles, the system’s volume, and theabsolute temperature are constant) is adopted to simulate the uniaxialcompressive behavior by applying a constant strain rate of 0.01 ps−1.Young’s moduli E are calculated by the slope of the stress-strain curves(E= σ/ε) at a strain of 2%. All simulations are carried out at thetemperature of 300 K. The time step in the MD simulations is taken as1 fs.

3. Results and discussion

3.1. Stress-strain curves

Fig. 2 shows the tensile and compressive stress-strain curves of the1st group of samples (as listed in Table 1) for polycrystalline ANHCs,SGANHCs and DGANHCs, respectively. It can be seen from Fig. 2a thatthe in-plane tensile Young’s moduli, tensile strengths and tensile energyabsorption of both SGANHCs and DGANHCs are much larger than thoseof the pure ANHCs without graphene coating. The in-plane compressivestress-strain curves of all samples including ANHCs, SGANHCs andDGANHCs in our simulations follow a similar trend with traditionalfoam and porous materials [7].

Fig. 3 gives the atomic configurations at different stages. Taking the1st group of polycrystalline DGANHCs as the example, the compressivedeformation can be divided into four stages (as shown in Fig. 2b, blueline): (I) initial linear elastic stage, (II) instability of pore, (III) collapsepropagation, and (IV) densification stage. At the beginning of com-pression, RVEs enter into the linear elastic stage. As compared to thepure ANHCs, their Young’s moduli are increased by graphene coatingsand the Young’s moduli of the DGANHCs are higher than those of theSGANHCs. Fig. 3(a–c) shows that, prior to reaching the peak stress, afew regions of the RVEs transform into plastic deformation from elasticdeformation (Stage I). It also can be seen from Fig. 2b that the com-pressive strength is enhanced by the SG coating and the reinforced ef-fect of DG coating is better than that of the SG coating. The in-planecompressive strengths of ANHCs, SGANHCs and DGANHCs are0.70 GPa, 1.69 GPa and 2.10 GPa, respectively. The compressivestrength of the DGANHCs is higher than that of the SGANHCs due to theincrease in the cross-sectional area of the coating. Fig. 2b also shows

Y. Zhou et al. Computational Materials Science 156 (2019) 396–403

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that, compared with the critical buckling compressive strain (corre-sponding to the peak stress) of the SGANHCs, the reduced criticalbuckling compressive strain of the DGANHCs is attributed to the in-coherent interface between the inner and outer graphenes which allowsfor free interlayer slips. This result is in contrast to the intuitive ideathat the critical buckling strain could increase significantly as thecoating thickness increases, but consistent with the statement that thecritical axial buckling strain of a double-walled nanotube embedded inan elastic matrix is lower than that of an embedded single-walled na-notube under otherwise identical conditions reported by Ru [41]. After

reaching the peak stress (i.e. the compressive strength), the pores be-come unstable and begin to collapse (Stage II). For the pure ANHC, thepore collapses gradually (as shown in Fig. 3d) and the in-plane com-pressive stress enters the plateau stage without obvious softening stage.But for the GANHCs, the graphene coatings begin to curl up and attractAl atoms to move with them under the compressive loading (as shownin Fig. 3e and f), which facilitates the collapse of the pore and results insignificant softening (Stage II). Then the stress enters the plateau stage(Stage III) up to the densification stage (Stage IV). In stage IV, thestrain-hardening effect becomes noticeable in the stress-strain curves,

(e)

(c)

(f)

(d)

(a) (b)

x

12nm

12nm 12nm

z

x

y y

z

2.59ÅCNT(68,68)

CNT(63,63)@(68,68)

Fig. 1. (a, c and e) Perspective views and (b, d and f) In-plane (top) views (x-y plane) of RVEs for ANHCs, SGANHCs and DGANHCs.

Table 1Parameters of RVEs in the study.

Group Inner radii of RVEs (Å) Chirality of SWCNT Chirality of DWCNT ρrAl of ANHC ρr

c of SWCNT ρrc of DWCNT

1 49.45 (68,68) (63,63)@(68,68) 47% 13% 26%2 46.00 (63,63) (58,58)@(63,63) 54% 14% 27%3 42.56 (58,58) (53,53)@(58,58) 61% 15% 29%4 39.12 (53,53) (48,48)@(53,53) 67% 17% 32%5 35.67 (48,48) (43,43)@(48,48) 72% 18% 35%6 32.22 (43,43) (38,38)@(43,43) 77% 20% 38%7 28.78 (38,38) (33,33)@(38,38) 82% 22% 42%

Y. Zhou et al. Computational Materials Science 156 (2019) 396–403

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as shown in Fig. 2b. The densification strain εd is determined roughly bythe point of the tangent intersection of the stress-strain curve in thedensification stage with strain axis [42]. Fig. 2b shows that the densi-fication strain εd of the DGANHCs is lower than that of the SGANHCs,and εd of the SGANHCs is lower than that of the ANHCs, which ismainly caused by the difference in the non-bonded Al-Al, Al-C, and C-Cequilibrium distances. The non-bonded equilibrium distance is about0.858σ, where σ is the size parameter used in LJ 12-6 potential. Asdescribed in Section 2, the size parameters are σC= 3.407 Å, σAl-C= 3.0135 Å, and σAl= 2.62 Å. The distance between atoms graduallydecreases during densification and meanwhile the repulsion betweenatoms increases dramatically. That is, first the DGANHCs, then theSGANHCs, and finally the ANHCs enter the densification stage. The in-plane compressive energy absorption W is calculated by ∫=W σdεε

0d

and can be acquired from our MD results. Obviously, the in-planecompressive energy absorptions of the GANHCs are lower than those ofthe ANHCs. Yi et al. [42] reported a novel graphene-based carbonhoneycomb (GCH) with giant energy absorption capacity, and theyfound that the GCH has a special three-dimensional structure which isdifferent from graphene coating in the present study. The atoms alongthe junction line in the GCHs are relaxed into lower energy config-uration with alternating distances, leading to a transition of the sp2 C-Cbonds along the joints into the sp3 C-C bonds, which provides a highplateau stress in a longer strain range and then results in the giantenergy absorption capacities of GCHs.

Fig. 3(g–i) shows the atomic configurations of GANHCs, SGANHCsand DGANHCs, respectively, when they are fully densified. It is seen

that for the ANHCs, the void can be filled by Al atoms, but for GANHCs,the gap is difficult to be filled by carbon atoms because the graphenecoatings are still perfect until densification due to the strong σ bondsbetween carbon atoms.

3.2. In-plane Young’s modulus

The effective Young’s modulus and yield stress of metallic foamsmaterials have been systematically analyzed by Gibson and Ashby [7].They derived the scaling laws correlating Young’s modulus with therelative density of the porous materials:

= +∗E E C ρ C ρ[ ]s 1 r2

1'

r (1)

where E* and Es are the Young’s moduli of the porous and bulk mate-rials respectively; ρr is the relative density which is defined as the massdensity ∗ρ of the foam divided by the mass density ρ of the solid ma-terial. FE analysis suggested that =C C1 1

' ∼0.32 [25]. Through MD si-mulations on indentation process of Cu monocrystals with the (0 0 1)crystallographic oriented plane, Ojos et al. [26] introduced the indenterdiameter into the classical Gibson-Ashby expression to properly takeinto account the influence of the pore size on the Young’s modulus withthe relative density of the nanoporous metal. Considering the indenterdiameter is inconvenient to be used in compression tests, here theclassical Gibson-Ashby expression, i.e. Eq. (1), is still applied to fit ourMD results of ANHCs.

To predict the in-plane Young’s modulus E* of GANHCs, based onthe classical mechanics theory for composite materials [7,43], E* can beexpressed as,

=⎡

⎣⎢⎢

− − +−

− − −

⎤

⎦⎥⎥

∗E E ρρ

E E ρ(1 1 )

1

1 (1 / ) 1s r

Al rAl

s fT

rAl

(2)

where ρrAl is the relative density of ANHCs, which can be seen in

Table 1, and E Tf is the transverse Young’s modulus of the reinforcement

phase. The values of Es of bulk monocrystal Al along [1 0 0], [1 1 1]directions and polycrystalline Al are 105 GPa, 115 GPa and 95 GPa re-spectively which have been evaluated through uniaxial compressiontests via NVT ensemble MD simulations. For the graphene coatedANHC, we assume that Ef

T in Eq. (2) equals to the radial Young’smodulus Er of the CNTs. Unlike the axial Young’s modulus Ez of theCNTs, the radial Young’s modulus Er is very difficult to be determined.Reported values of the radial Young’s modulus Er vary by up to 3 ordersof magnitude [44]. Yang and Li [45] measured the effective radialmodulus Er of straight SWCNTs by using well-calibrated tapping modeand contact mode atomic force microscopies. It is found that the mea-sured Er decreases from 57 GPa to 9 GPa as the diameter of the SWCNTsincreases from 0.92 to 1.91 nm. Zheng et al. [46] investigated experi-mentally the transverse deformability of DWCNTs using an ultrathinnanomembrane covering scheme, and they reported that Er of DWCNTsdecreases from 40 GPa to 6 GPa as the nanotube outer diameter in-creases from about 2 nm to 3.6 nm. Using the molecular structuralmechanics (MSM) method, Li and Chou [47] found that the radialYoung’s modulus of SWCNTs decreased from 1.3 TPa to 0.3 TPa as thediameter of SWCNTs increased from 0.5 nm to 2.5 nm. Chen et al. [48]presented a modified MSM model for determining the mechanicalproperties in the radial direction of SWCNTs. They found that the radialelastic modulus of the armchair SWCNTs with a radius of 0.48–2.38 nmis about 30–0.3 GPa. Wang et al. [49] used MD simulations to examinethe relation between the radial compressive Young’s modulus Er ofCNTs and their MD results showed that the armchair SWCNTs with aradius of 1–3.5 nm is about 0.65–0.03 GPa and DWCNTs with a radiusof 1.2–3.7 nm is about 2.8–0.24 GPa.

Here we assume that the CNTs is a nanoporous material of graphenematrix and void, as shown in Fig. 4a and b for SWCNTs and DWCNTsrespectively. Through the Eq. (1) we can calculate the Young’s modulus

0.00 0.05 0.10 0.15 0.200

1

2

3

4

Polycrystal: group 1

In-p

lane

tens

ile st

ress

(GPa

)

Strain

ANHC SGANHC DGANHC

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0

2

4

6

8

10

In-p

lane

com

pres

sive

stre

ss(G

Pa)

Strain

ANHC SGANHC DGANHC

Polycrystal: group 1

(I)(II) (III)

(IV)

d

(b) Fig. 2. In-plane (a) tensile and (b) compressive stress-strain curves of group 1for polycrystalline ANHCs, SGANHCs and DGANHCs. (I) elastic stage, (II) col-lapse of pore, (III) collapse propagation, and (IV) densification stage.

Y. Zhou et al. Computational Materials Science 156 (2019) 396–403

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EfT of CNTs and then substitute it into Eq. (2) to get the theoretical

solutions of the in-plane Young’s modulus of GANHCs. The relativedensity ρr

cof graphenes can be calculated by (2rt− t2)/r2, where r is theinner radius of RVE, and t is the thickness of CNTs, whose values ofSWCNTs and DWCNTs are assumed to be 3.4 Å and 6.8 Å respectively,as listed in Table 1. The Egraphene is chosen as 669 GPa in our theoreticalmodels, which was figured out by Reddy et al. [50] using MD simula-tions with Brenner’s potential.

The in-plane Young’s moduli E* of the ANHCs along the [1 0 0],[1 1 1] directions, and polycrystals with different relative densities as

well as their corresponding GANHCs (i.e. groups 1–7 in Table 1) ob-tained from the MD simulations are listed in Fig. 5. The results esti-mated from the modified Gibson-Ashby approach are also shown inFig. 5. Compared to the MD results, the classical Gibson-Ashby ex-pression can give an accurate prediction of in-plane Young’s moduli E*of the ANHCs at a low relative density, but it becomes less accurate for ahigher relative density.

As for ANHCs and GANHCs, the in-plane E* linearly increases withthe increase in the relative mass density. The reinforcing effect ofgraphene coating on Young’s moduli can be measured by (EGANHC-

Fig. 3. Atomic snapshots at a specified macro loading strain, (a–c) critical buckling compressive strain, (d–f) stage of pore collapse and (g–i) densification for group 1of polycrystalline ANHC, SGANHC and DGANHC, respectively. Identification of atomic configurations is based on common neighbor analysis (CNA).

Fig. 4. Theoretical diagrammatic sketches of (a) SGANHCs and (b) DGANHCs.

Y. Zhou et al. Computational Materials Science 156 (2019) 396–403

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EANHC)/EANHC, where EGANHC and EANHC are the in-plane Young’smoduli of GANHCs and ANHCs in case of the same group, respectively.As compared to the ANHCs with the same relative density, the Young’smoduli E* can be increased by the SG coating ranging from 52% to177% and the DG coating ranging from 65% to 276% in [1 0 0] direc-tion; by the SG coating ranging from 49% to 130% and the DG coatingrange from 64% to 197% in [1 1 1] direction; by the SG coating rangingfrom 72% to 163% and the DG coating ranging from 88% to 253% inpolycrystals.

3.3. Compressive strength

The plateau stress for a porous metallic material is reached when itscells begin to collapse plastically. This plateau stress is often defined asthe compressive strength of the porous material. For porous materials,the compressive strength can be estimated by the following Gibson-Ashby expression [7]:

= +∗σ σ C ρ C ρ[ ]s 2 r3/2

2'

r (3)

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.8520

40

60

80

100

120ANHC

SGANHC DGANHC Equation 1 for ANHC Equation 2 for SGANHC Equation 2 for DGANHC

E* (G

Pa)

Relative density

[100] direction

(a)

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.8520

40

60

80

100

120

E* (GPa

)

Relative density

[111] direction

(b)

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.8520

40

60

80

100

120Polycrystalline

E* (GPa

)

Relative density(c)

Fig. 5. Young’s moduli E* of ANHCs and GANHCs in (a) [1 0 0] direction, (b)[1 1 1] direction, and (c) polycrystalline Al.

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.851

2

3

4

5

6

7

8[100] direction

MD result of ANHC MD result of SGANHC MD result of DGANHC Equation 3 of ANHC Equation 4 of SGANHC Equation 4 of DGANHC

(GPa

)

Relative density

C2=1.429 C2'=-0.651C3=0.350 C4=0.742

(a)

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.851

2

3

4

5

6

7

8 [111] direction

(GPa

)

Relative density

C2=1.606 C2'=-0.753C3=0.409 C4=0.651

(b)

1

2

3

4

5

6Polycrystalline

(GPa

)

Relative density

C2=1.584 C2'=-0.902C3=0.317 C4=0.696

(c)

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

Fig. 6. Compressive strength σ* of ANHCs and GANHCs in (a) [1 0 0] direction,(b) [1 1 1] direction, and (c) polycrystalline Al.

Y. Zhou et al. Computational Materials Science 156 (2019) 396–403

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where σ* and σs are the yield strengths of the porous and the solid cellwall materials, respectively, = =C C0.33, and 0.44,2 2

' which weredetermined by FE analysis [25]. Through MD simulations of singlecrystalline nanoporous cooper under adiabatic uniaxial compression in[0 0 1] direction, Yuan and Wu [27] found that the Gibson-Ashby’sscaling law is still applicable in nanoscale, but the constants C1, C1

', C2

and C2' in Eqs. (1) and (3) should be determined from experimental or

simulation results. In this study, we will determine these four constantsby fitting the MD results of ANHCs, which are listed in Fig. 6.

To get the in-plane compressive strength σ* of GANHC analytically,since the radial Young’s moduli of CNTs can be analytically obtainedfrom Eq. (1), here we assume the σ* of GANHCs as,

⎜ ⎟= ⎡

⎣⎢ + + ⎛

⎝⎞⎠

⎤

⎦⎥

∗σ σ C ρ C ρ CEEs

s

C

2 r3/2

2'

r 3fT 4

(4)

where σs of bulk Al in [1 0 0], [1 1 1] directions and polycrystalline bulkAl are 8.1 GPa, 7.4 GPa and 7.1 GPa respectively, which have beenfirstly determined by using unaxial loading in an NVT ensemble. TheYoung’s moduli Es of bulk Al has been described in Section 3.2. Ef

T isthe radial Young’s modulus of SG coating or DG coating which can bepredicted by Eq. (1); C2, C2

' are acquired by fitting ANHC data, and C3,C4 are obtained by fitting SGANHC data. When Eq. (4) is used to predictthe σ* of DGANHCs, none of any parameters are needed to fit.

For nanomaterials, surface effect may induce a size dependency inthe size-independent classical elasticity theory [27,51,52], and thissurface effect depends upon the crystallographic orientation [53].Therefore, different parameters should be fitted for different crystal-lographic orientation. The compressive in-plane strengths σ* in dif-ferent crystal orientations of ANHCs and GANHCs are shown inFig. 6a–d, respectively. From Fig. 6 we can see that, the compressivestrength σ* of GANHC can be well predicted by Eq. (4) effectively.

The reinforced effect of graphene coatings on compressive strengthcan be evaluated by (σGANHC− σANHC)/σANHC, where σGANHC and σANHCare the compressive strength of GANHCs and ANHCs, respectively. Incomparison to the ANHCs with the same relative density, the com-pressive strength σ* can be increased by the SG coating ranging from38% to 110% and the DG coating ranging from 73% to 119% in [1 0 0]direction; by the SG coating ranging from 35% to 105% and the DGcoating range from 83% to 146% in [1 1 1] direction; by the SG coatingranging from 48% to 128% and the DG coating ranging from 78% to183% in polycrystals.

4. Conclusions

In summary, we have performed MD simulations and theoreticalanalysis to study the in-plane compressive behavior of aluminum nano-honeycombs (ANHCs) and graphene coated ANHCs (GANHCs). Thecompressive energy absorption capacity of GANHCs until densificationis lower than that of pure ANHCs, which is mainly caused by the greaternon-bonded Al-C, and C-C equilibrium distances due to the graphenecoatings compared to the Al-Al equilibrium distance. It is found that thegraphene coatings can significantly reinforce the tensile strength, thetensile energy absorption, the in-plane elastic modulus and compressivestrength of aluminum nano-honeycombs.

In addition, we proposed a modified Gibson-Ashby model to de-termine the in-plane elastic modulus and compressive strength ofGANHCs. For ANHCs, we found that at a low relative density, theclassical Gibson-Ashby scaling laws is still applicable to predict the in-plane Young’s moduli, but as the relative density increases, theGibson–Ashby scaling laws underestimates the values due to thestronger surface effect of the pore. Our study of the in-plane compres-sive strength of ANHCs shows that the constants in the classical Gibson-Ashby expression should be re-fitted from MD data and they are de-pendent on the loading crystal orientation. The compressive strength ofGANHCs can be well predicted by the proposed expression.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Grant Nos. 11772145 and 11372126). D.S. Li isgrateful for the support of the National Natural Science Foundation ofChina (Grant No. 51562027) and X.B. Liu is grateful for the support ofthe Science and Technology Project of Jiangxi Province EducationDepartment (Grant No. KJLD12073).

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