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A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions
Accepted Manuscript A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions Lubin Song, Zhangxin Guo, Gin Boay Chai, Zhihua Wang, Yongcun Li, Yunbo Luan PII:
S0008-6223(18)30792-9
DOI:
10.1016/j.carbon.2018.08.058
Reference:
CARBON 13413
To appear in:
Carbon
Received Date: 24 August 2018 Accepted Date: 27 August 2018
Please cite this article as: L. Song, Z. Guo, G.B. Chai, Z. Wang, Y. Li, Y. Luan, A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions, Carbon (2018), doi: 10.1016/j.carbon.2018.08.058. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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n=24(4%) n=12(2%)
n=30(5%)
n=6(1%)
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n=36(6%)
x L’y L’x y Pillared graphene sheet with different volume fraction
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z
L’z
ACCEPTED MANUSCRIPT A finite element method to investigate the elastic properties of
pillared graphene sheet under different conditions Lubin Songa,c,d, Zhangxin Guoa,b,c,*, Chai Gin Boayb, Zhihua Wanga,d, Yongcun Lia,d, Yunbo Luana,d a
College of Mechanics, Taiyuan University of Technology, Taiyuan 030024, China School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore c State Key Laboratory for strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China d Shanxi Key Laboratory of Material Strength and Structure Impact, Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
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b
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ABSTRACT
In this paper, an investigation was carried out to understand the mechanical elastic properties of a newly developed 3D nanocarbon structure material known as PGS (pillared graphene sheet). The effect of various parameters such as the pillared
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distance, the chirality, the volume fraction and the inclination angle of carbon nanotube on the elastic moduli was studied using the finite element method. The
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commercially available finite element software ABAQUS was utilized for the modelling and analysis of this new structure. Several different models were developed
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to study the effect of the various parameters mentioned earlier. Some interesting conclusions were deduced from the finite element investigation. It was found that the pillared distance, chirality and volume fraction affected the modulus significantly. The change in volume fraction impacted a bigger influence on the Young’s modulus Ez and shear modulus Gxy. In some instances, Ez was increased by approximately 40 1 *Corresponding authors: E-mail address: [email protected] (Zhangxin Guo), Tel. number: 13994231172
ACCEPTED MANUSCRIPT times. In addition, the Young’s modulus Ez was also found to be affected by the change in the inclination angle and was shown to increase with increasing angle. 1. Introduction
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A new-type of material based on nanocarbon structure enormously aroused investigator’s interest since the material fullerene with 0-dimensional structure was discovered in 1985 by Kroto et al [1]. Nanocarbon material is an extended family of
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fullerene which contains multi-type allotropic carbon structure such as fullerene (0D),
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carbon nanotube (1D), graphene (2D) and nanodiamond (3D) et [1,2]. These newly discovered materials have been shown to play a vital role in the fields of physical, chemical, mechanical, optoelectronic and many others due to their many outstanding properties. For example, carbon nanotubes (CNTs) have excellent superconducting
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properties and high stiffness in the axial direction. High stiffness, low density, excellent electronic and thermal properties have been discovered in graphene sheet (GS) in planar directions over the past decades [3-6]. However, the excellent
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performance in nanocarbon material is limited by their nanostructure. CNTs have low
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conduction in uniaxial direction and weak shear properties. And graphene has weak out-of-plane mechanical properties [7,8]. For that reason, hybrid materials which combine the advantages of two or more
nanocarbon structure, have been synthesized and investigated by various field researchers. Lee et al [9] synthesized the nano peapod structure for studying the bandgap properties. Okada et al [10] and Shotono et al [11] also discussed the nano 2
ACCEPTED MANUSCRIPT peapod structure on the aspects of energetics and electron-phonon coupling strength [10,11]. Feng et al [12] studied a new kind of carbon foam on the welding of single-walled carbon nanotube and this hybrid material included carbon nanobuds
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[13], sp2 and sp3 -hybridized nanocarbon [14] and pillared graphene [15]. Pillared graphene is now gaining popularity and is being studied by more and more scholars owing to its new 3D structure as shown in Figure 1. A review of the
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published literature showed that the comparison of monolayer and multilayered graphene sheets was investigated quite intensively in recent years, as did with
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research on the single-walled and multi-walled CNTs. Their conclusion seemed to suggest that the monolayered graphene and single-walled CNT gave favourable mechanical properties. To exploit these favourable mechanical properties, this new 3D
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structure PGS (pillared graphene sheet) is usually combined with single-walled CNTs
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and monolayer graphene sheets.
Figure 1. The elementary unit of pillared graphene sheet
This new hybrid structure material PGS owns multifunctional properties such as 3
ACCEPTED MANUSCRIPT thermal and mechanical properties [16,17] relative to its constituents – CNT and graphene. As this PGS research is still at its infant stage and with a promising future, researchers are rushing in to fabricate, characterize and simulate the PGS. The effect of inter-tube bridging and bridging type on the mechanical properties of pillared
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graphene were investigated by Deniz Bilgili et al [18]. Sangwook Sihn [19] effectively predicted the 3D elastic moduli and Poisson ratio of pillared graphene
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nanostructures. Navid Sakhavand [18,20] studied the thermal transport and the effect of the junction configuration on the deformation and mechanical properties of the
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PGS nanostructure.
Many simulation efforts have been spent on predicting the mechanical and thermal properties, and most of the analysis used the molecular dynamics (MD),
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diverse function theory (DFT), and tight-binding molecular dynamics (TBMD) [20-23]. Although these theories can effectively simulate the movement of the molecular system but it is usually limited by the system size and is confined to study
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relatively short-lived phenomenon at the timescale [19]. In other words, these theories
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cannot completely simulate and solve the static problem of models with larger dimensions. For static mechanical problem with a large dimension system, it is necessary to use other methods to solve. Therefore many researchers utilized the shell model based on the classical continuum mechanics and this to a large extent helps in solving the tricky dimension problem [24]. But the accuracy of the shell model is a problem due to the loss of the interatomic interaction among C atom and the effect of
4
ACCEPTED MANUSCRIPT the C-C bond. Li and Chou [25,26] presented a molecular structure mechanics (MSM) approach at the atomistic scale which is not perplexed in time scale. MSM is a semi-atomistic approach which is based on the structural mechanism and molecular mechanism. The fundamental notion of MSM is that nanostructure like CNT or
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graphene is treated as a frame-like structure where the C-C bond is equivalently replaced with a beam member and the node is treated as the C atom. From then on,
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many published works using the MSM have successfully predicted the elastic modulus, bulking, deformation and stress concentration of CNT and GS [26-29].
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An effective finite element approach for modeling the structure of PGS using the MSM method is proposed and presented in this paper. Though some research findings on PGS have been reported, most of them focused on the thermal and
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electric properties. Sangwook Sihn et al [19] investigated the effect of the PL (pillared length) and PD (pillared distance) on the elastic modulus of PGS. Rouzbeh Shahsavari et al [20] studied the stiffness, stress-strain relations, bond strains and
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inelastic deformation mechanism for PGS. But they only investigated a single and
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ideal model. In this paper, elastic properties of the PGS under different conditions are investigated using a commercial finite element software ABAQUS. The contributing effects of the chirality, the pillared distance, volume fraction, and the inclination angle of carbon nanotubes are evaluated in details.
2. Numerical simulation 2.1 Finite Element Model 5
ACCEPTED MANUSCRIPT Matsumoto et al [30] reported work on a vertically aligned CNT-graphene structure (VACNT-graphene). Later, both of Dimitrakakis et al [15] and Duangkamon Baowan et al [31] discussed the design and configurations of the PGS junction. Their main idea was to create a hole in graphene sheet with a hole size that of the diameter
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of the CNT radial. Subsequently, the CNT were vertically conjunct near the hole by combining the corresponding atoms. An effective simulation using this method were
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developed by Wang et al [31,32].
In this contribution, one of our studies focuses on the combination of different
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chiral CNT and graphene. Monolayer graphene is a layer of carbon atoms arranged in a hexagonal lattice and single wall carbon nanotubes are constituted by a monolayer graphene sheet wrapped into a cylinder. Graphene and carbon nanotubes were divided
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into zigzag and armchair types according to their chirality. Figure 2 shows the specific classification of CNT and graphene and the formation process of CNT. Therefore, two main holes in tow chiral graphene were created for combining two chiral CNT which
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can be seen in Figure A1 (Appendix A). Through that, four different models of PGS
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were created and they were named as PGS-ZA, PGS-AA, PGS-AZ, PGS-ZZ as shown in Table 1. Each model is divided into five categories according to the pillared distance (2.84nm, 2.5nm, 2nm, 1.5nm, 1.25nm). The details of the mechanical properties caused by chirality can be found in several publications [32-34]. In addition, Figure A1 exhibits the actual junctions in PGS. In our simulations, (6,6) CNT and (10,0) CNT were used for ensuring the similar diameter in two 6
ACCEPTED MANUSCRIPT different sizes of graphene. It is worth noting that the junctions between the CNT and graphene should satisfy the Euler’s polygon theorem which states: F+V-E=2(1-G), where F, V and E represents the respective number of faces, vertices and edges for the given polyhedron and G is an Euler characteristic. For PGS junctions, G=2. Hence, a
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total bond spare of 12 was shared by CNT and GS. In this contribution, the configuration in the different four model types contained six hexagons which have a
(Graphene) Zigzag sheet T
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surplus of +1 and this satisfy the criterion [20].
Armchair sheet
Armchair Sheet
Zigzag Sheet (b)
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(a)
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(CNT)Zigzag（ （n,0） ）
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Figure 2. Specific classification of CNT(a) and graphene and the formation process of CNT(b)
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ACCEPTED MANUSCRIPT Table 1.
Four different FE model of the PGS
FEM model style
Graphene
+
SWCNT
=
3D carbon nanostructure
PSG-ZA
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PSG-AA
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PSG-ZZ
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PSG-AZ
In fact, the growth of carbon nanotubes is not always vertical [37]. So, different angles of inclination (bending growth of carbon nanotubes) applied on the PGS-ZA
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model were made to study the angle effect on the elastic properties of PGS. The
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middle CNT was set to spin at a certain angle along the x axis. The inclined angle is measured from the Z-axis with reference to the middle CNT. To reduce the number of variables, the spacing between two layers of graphene was maintained constant. And to ensure that the beam elements at the junction points were as long as possible, the lattice number of carbon nanotubes near the hole was increased appropriately as shown in Figure A3. 8
ACCEPTED MANUSCRIPT 2.2 Computational Method A finite element approach based on the mechanical molecular structure is presented in this paper. As can be seen in Figure A4 (Appendix A), the chemical bond
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was simulated with an elastic beam element. The C atom was replaced with an equivalent node. After which the PGS was established as a frame-like structure. These different models were then implemented in ABAQUS. The ABAQUS beam element
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has six degrees of freedom (U1, U2, U3, U12, U13, U23) corresponding to three nodal forces and three nodal moment components. The details of the modelling and From these results, the stiffness
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computational steps are described in Appendix A.
matrix were extracted to compute the elastic moduli. To verify the validity of the calculation, the elastic modulus of the CNT, graphene and PGS were respectively
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simulated and compared with published results as shown in Tables 2 and 3.
Table 2. Comparison of Young’s modulus for Graphene and SWCNT
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Investigators
Young’s modulus(TPa)
Graphene 1.042
Mahmood M. Shokrieh et al[35].
1.04
Kudin KN et al[36].
1.029
Present work (armchair)
0.989
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A. Sakhaee-Pour et al[32].
SWCNT
Ehsan Mohammadpour[27].
0.881
Mahmood M. Shokrieh[35].
1.033-1.042
Present work
0.956 9
ACCEPTED MANUSCRIPT Table 3. The effective modulus contrast with other research(PSG-ZA) PL(nm)
PD
Ex(Gpa)
Sihn et al[19].
1.21
2.36
155.02
Sinh et al.
3.25
2.36
2.45
Present work Present work
Shahsavari
et
Ey
Ez
Gxy
Gxz
Gyz
155.45
11.48
41.72
15.59
15.33
57.81
57.82
26.97
14.56
3.84
3.78
2.66
77.0
72.25
10.17
29.88
3.08
2.32
1.21
2.36
136.89
141.11
8.02
38.90
14.01
12.26
1.474
2.0
98.99
101.05
10.1
42.21
4.80
3.91
3. Results and discussion 3.1 The effect of chirality
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al[20].
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Investigators
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In this paper, four sets of models were created for investigating the chiral effect on the elastic properties. The Young’s moduli and shear moduli were computed by
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simulating two different areas – effective area and equivalent area. The effective area is the overall sectional area of the PGS and the equivalent area is the net atomic area
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which discarded the empty part of the carbon PGS structure. The equivalent area is usually for the comparison of the elastic properties of the subgrade components of CNT and graphene. For predicting the elastic properties on the periodic geometry, the periodic boundary constraints were applied. Six different displacement and loads were imposed respectively on corresponding boundary nodes. Then, a linear elastic expression of stress-strain was adopted in the computation of the Young’s modulus 10
ACCEPTED MANUSCRIPT and shear modulus. For each of graphene and CNT, chirality has an important effect on their elastic properties, so it is necessary to investigate if chirality has the same influence on the
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new 3D nanocarbon structure PGS. Figures 3 to 5 show the change in the effective and equivalent Young’s modulus Ex, Ey, Ez and Shear modulus Gxy, Gxz, Gyz with pillared distance. As can be seen in Figures 3 and 4, the Young’s moduli Ex and Ey of
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all the four types of combination showed a downward trend within the range of the increase in PD. One possible reason of this decrease is that the defect in graphene is
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getting closer to the boundary with the increase in the distance. It can also be seen that there is a decreasing trend in the equivalent modulus but this trend is higher than that of the effective modulus. This suggested that the equivalent Young’s modulus Ex and
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Ey were strongly affected by the PD and were closer to the CNT and graphene. The results obtained from the four models showed two main significant deductions. The first one is that the in-plane modulus Ex of PGS-AA is larger than the
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other three types while the in-plane moduli Ey of PGS-ZA is the largest. From the
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chirality of graphene alone, the result conforms to the effect of chirality on graphene sheet [38]. And the second one is that PGS with an armchair CNT has a bigger in-plane modulus than the PGS with a zigzag CNT. This means that in the in-plane direction, the deformation of CNT with a zigzag sheet is smaller due to having more beam elements in the perpendicular to the load direction. And Figure 5 shows that the Young’s modulus Ez is more consistent. When the PD is kept constant at 1.25nm, the 11
ACCEPTED MANUSCRIPT Young’s modulus Ex of PGS-AA is ~4.9% higher than the lowest modulus Ex of PGS-ZZ and the Young’s modulus Ey and Ez are ~6.1% and ~19.4% higher. In addition, noting that the Young’s moduli Ez of PGS-AA is reduced in total by ~84% as the PD changed from 1.25nm to 2.84nm, which is more significant than the ~12.2%
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reduction of Ex and ~13.6% reduction of Ey. This means that the PD or chirality has a greater impact on Ez. On the whole it can be observed that though the maximum
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equivalent Young’s moduli Ez of PGS-ZA (~200GPa) is lower than the maximum equivalent Young’s moduli Ex (~720GPa) and Ey (~735GPa), but it showed a
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99 98 97 96 95 94 1.2
1.4
1.6
1.8
2.0
2.2 2.4
2.6 2.8
3.0
equivalnet Young's modulus E x (Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
100
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effective Young's modulus Ex(Gpa)
significant improvement in the out-of-plane relative to graphene sheet.
720 700 680 660 640
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
PD(nm)
PD(nm)
(b)
(a)
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PGS-ZZ PGS-AZ PGS-ZA PGS-AA
740
Figure 3. Effective (a) and equivalent (b) Young’s modulus Ex of four types of PGSs against
effective Young's modulus Ey(Gpa)
96.0
equivalent Young's modulus Ey(Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
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PD
95.5 95.0 94.5 94.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
740 720 700 680 660 640 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
PD(nm)
(a)
(b)
Figure 4. Effective (a) and equivalent (b) Young’s modulus Ey of four types of PGSs against PD
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PGS-ZZ PGS-AZ PGS-ZA PGS-AA
18 16 14 12 10 8 6 4 2 0 1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
200 150 100 50 0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
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1.2
equivalent Y oung's m odulus E Z (G pa)
effective Young's modulus Ez(Gpa)
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PD(nm)
PD(nm)
(b)
(a)
Figure 5. Effective (a) and equivalent (b) Young’s modulus Ez of four types of PGSs against
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PD
Figures 6 to 8 show the variation of the effective and equivalent shear moduli (Gxy,
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Gxz, Gyz) with increasing PD. The shear modulus Gxy shows an upward trend while shear moduli Gxz and Gyz show a downward trend with increasing PD. Interestingly, for the same pillared distance, each of the three shear moduli presented different
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performance in terms of chirality. The shear moduli Gxy and Gyz of PGS-AA is little higher than that of the other three PGS model with PGS-ZA yielding the highest shear moduli Gxz. As can be seen in Figures 6 and 8, the shear modulus Gxy of PGS-ZA is
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higher than that of PGS-AZ, but in the case of Gyz, this is lower. Comparing the four
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PGS models with the same pillared distance of 1.25nm, it can be seen that the maximum shear modulus Gxy (equivalent modulus) of PGS-AA is ~13.33% higher than the minimum Gxy of PGS-ZZ and Gxz, Gyz are ~34.2% and ~34.6% respectively between the maximum and minimum. This suggested the shear modulus of Gxz and Gyz are easily influenced by the chirality. Looking at the general trend of the shear moduli of PGS-AA with increasing PD, 13
ACCEPTED MANUSCRIPT the shear modulus Gxy has a growth of ~56.7% while the Gxz and Gyz has a respective reduction of ~78.8% and~79.3%. Similarly for the other three models, the shear moduli Gxy, Gxz and Gyz of PGS-AZ has a respective change rate of 55.2%, 77.8% and 79.4% while the respective change rate of the shear moduli of PGS-ZA are 57.5%,
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78.8% and 81.2%. Finally, PGS-ZZ has ~42.3%, ~79.3% and ~76.5 % change rate to Gxy, Gxz and Gyz respectively. That means that all the shear moduli are significantly
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affected by the pillared distance. Comparing the four different models together, the rate of change in the maximum shear modulus Gxy is ~14.4% more than the least rate
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of change in PGS-ZZ. And for shear moduli Gxz and Gyz, there is no significant difference. It can be deduced that the chirality does have a certain effect on the shear modulus Gxy of PGS models. In summary, both the pillared distance and chirality
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
52 50 48 46 44 42 40 1.4
1.6
1.8
2.0
2.2
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1.2
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effective shear modulus GXY(Gpa)
54
2.4
2.6
2.8
3.0
PD(nm)
equivalent shear modulus GXY(Gpa)
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affected the elastic properties of the new 3D structure significantly.
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
240 220 200 180 160 140 120 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
(a) (b) Figure 6. effective (a) and equivalent (b) shear modulus Gxy of four types of PGSs against PD
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4.0 3.5 3.0 2.5 2.0 1.5 1.0 1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
35 30 25 20 15 10 5 1.2
3.0
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
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1.2
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
40
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
equivalent shear modulus Gxz(Gpa)
effective shear modulus Gxz(Gpa)
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PD(nm)
PD(nm)
(a)
(b)
Figure 7. Effective (a) and equivalent (b) shear modulus Gxz of four types of PGSs against
2.0 1.5 1.0 0.5 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
PD(nm)
(a)
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2.5
30 25
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3.0
equivalent shear modulus Gyz(Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
20 15 10 5
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
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effective shear Young's modulus Gyz (Gpa)
PD
(b)
Figure 8. Effective (a) and equivalent (b) shear modulus Gyz of four types of PGSs against
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PD
3.2 The effect of volume fraction
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The volume fraction of CNT is another parameter that affects the elastic properties
of PGS. A little larger zigzag graphene (10nm*10nm) model and armchair CNT (2.214nm) model were set up for this investigation. One of the four PGS models presented earlier, PGS-ZA was chosen. By keeping the same size of graphene and changing the number of CNT, five different volume fractions were analyzed i.e – 1%，
15
ACCEPTED MANUSCRIPT 2%，4%，5% and 6%. It can be seen in Figure 9(b) that when the volume fraction varies from 1% to 5%, the equivalent Young’s moduli of Ex and Ey show declining trend while the modulus Ez clearly show increasing trend. The Young’s modulus Ez (8Gpa) is far smaller than Ex (589GPa) and Ey (600GPa) when the volume fraction is
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1%. But when the volume fraction is increased to 6%, Ez is 350GPa and Ex, Ey are 511GPa and 520GPa respectively. This shows that the change in volume fraction has a
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significant influence on the Young’s modulus Ez which can be increased by approximately 40 times.
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In Figure 10(b), it can be seen that all the equivalent three shear moduli of Gxy, Gxz and Gyz increased gradually with increasing volume fraction. And the rate of change in Gxz and Gyz seem to be higher. When the volume fraction is 6% (n=36), Gxy
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improved by ~ 13.4%, while Gxz has risen 3.5-fold and Gyz has risen 10-fold. This shows that the volume fraction has minor effect on the Gxy, but has a greater impact on the shear moduli relative to the axial z-direction. Thus, by controlling the number
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of CNT rationally with the same volume, PGS with different moduli can be tailored
Ex Ey
60
equaivelent Young's modulus (Gpa)
effective Young's modulus(Gpa)
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according to the needs.
Ez
50 40 30 20 10
0 0.01
0.02
0.03
0.04
0.05
0.06
Vf
Ex Ey
600
Ez
500 400 300 200 100 0 0.01
0.02
0.03
0.04
0.05
0.06
Vf
(b)
(a) 16
Figure 9. Effective (a) and equivalent (b) Young’s modulus of four types of PGSs against volume fraction
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Gxz 5
Gxy
4 3 2 1 0 0.01
0.02
0.03
0.04
0.05
120
Gyz Gxz
100
Gxy
80 60 40 20 0 0.01
0.06
0.02
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effective shear modulus(Gpa)
equaivelent shear modulus(Gpa)
Gyz
6
0.03
0.04
0.05
0.06
Vf
Vf
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(a) (b) Figure 10. Effective (a) and equivalent (b) shear modulus of four types of PGSs against volume fraction
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3.3 Angle of Inclination
This section discussed the effect of the angle of inclination on the moduli of PGS. By appropriately increasing the length of the carbon nanotubes and creating a similar
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hole in the graphene, a new PGS model was constructed for investigating the elastic properties affected by the change in the angle of inclination. Table 4 provides a
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summarized tabulation of the obtained data results.
Table 4. Comparison of equivalent Young’s modulus of PGS with different angle of
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inclination
Equivalent modulus (Gpa)
Ex
0o
650.21
Ez
Gxy
Gxz
Gyz
666.54
26.6
122.7
8.01
7.23
641.30
655.6
27.2
120
7.98
7.1
12o
638.45
646.2
28.6
121.3
7.96
7.01
24o
635.40
640.2
30.3
122.1
7.95
6.94
6o
Ey
17
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The equivalent Young’s modulus Ex and Ey remains relatively constant with the increase in the angle of carbon nanotubes but Ez increases. The main reason for this
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could be that the increase in the inclined angle helps to reduce the distance between the two adjacent holes that grow on the graphene. As for the shear moduli in general, there is no obvious change in moduli by changing the angle of the inclination of the
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carbon nanotubes. Bear in mind that the current results are based on a simple condition, there are many other conditions to analyze such as the axis of rotation, the
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location of rotation, the number of the rotation and the order of arrangement etc.
4.Conclusion
In this paper, a finite element approach based on the molecular structure
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mechanical is proposed for investigating the relevant elastic properties of the PGS under different conditions. In total, three main parameters of the PGS were studied in
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our paper – different chirality with various PD, volume fraction and inclination angle of the carbon nanotube.
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It was found that both chirality and PD have a great influence on the elastic
moduli of PGS. Chirality has a greater impact on moduli of Ez (19.4%), Gxz (34.2%)and Gyz (34.6%). PGS-AA have a higher Ex while PGS-ZA have a higher Ez. This indicates that different chirality has a different effect on the different directional moduli. Moreover, the Young’s modulus of Ez (200GPa) is lower than the Young’s moduli of Ex (720GPa) and Ey (735GPa), but PGS yielded a significantly improved 18
ACCEPTED MANUSCRIPT out-of-plane moduli relative to graphene sheet and in-plane relative to CNT. By changing the number of the CNT, five different volume fractions were studied. It can be concluded that the larger the volume fraction, the higher the moduli of Ez,
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Gxz and Gyz. Young’s moduli of Ex and Ey reduced respectively by ~13.24% and ~13.33% when the volume fraction changed from 1% to 6% while the shear modulus Gxy improves by ~13.4%.
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Lastly by rotating CNT along the X axis, five different PGS-AZ models were
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created. The effects of the angle of inclination on the elastic moduli were not obvious but in general, the modulus of Ez is affected by the change in the angle of inclination and improves with the increasing angle.
The results presented in this paper show the enormous potential of the newly 3D
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carbon structure PGS as an enhancer of the resin. Critical issues related to different angles of inclination of CNTs in PGS and their effects on the mechanical properties of
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PGS-reinforced polymer-matrix composites are currently under investigation. Their
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results will be published in due course.
Acknowledgments
This work was supported by the National Nature Science Foundation of China[grant numbers
11602160,11572214,11390362,11402160,21501129];the
Scientific
and
Technological Innovation Programs of Higher Education Institutions in Shanxi[grant 19
ACCEPTED MANUSCRIPT number 2017117]; and the opening foundation for state key laboratory for strength and vibration of mechanical structures[grant number SV2017-KF-01]; and the
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“1331project” Key Innovation Teams of Shanxi Province.
References
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ACCEPTED MANUSCRIPT Elastic Constants of Graphene Sheet. Structure 2018; 13:131-138. [7] Min-Feng Yu, Oleg Lourie, Mark J. Dyer, Katerina Moloni, Thomas F. Kelly and Rodney S. Ruoff. Strength and breaking mechanism of multiwalled carbon
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Energetics and electronic structures of encapsulated C_(60) in a carbon nanotube. Phys. Rev. Lett 2001; 86(17):3835-3838.
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ACCEPTED MANUSCRIPT AS et al. A novel hybrid carbon material. Nature Nanotechnology 2007; 2(3):156-161. [14] Yangming Lin, Zhenbao Feng, Linhui Yu, Qinging Gu, Shuchang Wu, Dang Sheng Su. Insights into the surface chemistry and electronic properties of sp2 and sp3
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[18] Deniz Bilgili, Levent Kirkayak, Mesut Kirca. The effects of intertube bridging through graphene nanoribbons on the mechanical properties of pillared graphene. Composites Part B: Engineering 2017; 120:1-9. [19] Sihn, Sangwook, Varshney. Vikas, Roy. Ajit K, Farmer. Barry L. Prediction of 3D elastic moduli and Poisson’s ratios of pillared graphene nanostructures. Carbon 2012; 50(2):603-611. 22
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[24] Shokrieh M.M, Zeinedini A. Effect of CNTs debonding on mode I fracture
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International
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40(10):2487-2499.
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ACCEPTED MANUSCRIPT Physics A: Materials Science and Processing 2012; 106(3): 581-588. [28] S.K. Jalali, M.J. Beigrezaee, N.M. Pugno. Atomistic evaluation of the stress concentration factor of graphene sheets having circular holes. Physica E:
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Low-dimensional Systems and Nanostructures 2017; 93:318-323. [29] S.F. Asbaghian Namin, R. Pilafkan. Vibration analysis of defective graphene sheets using nonlocal elasticity theory. Physica E: Low-dimensional Systems and
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Nanostructures 2017; 93:257-264.
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[30] Takanori Matsumoto, Susumu Saito. Geometric and electronic structure of new carbon network materials: nanotube array on graphite sheet. Journal of the Physical Society of Japan 2002; 71(1):2765-2770.
[31] Baowan D, Cox B.J., Hill J.M. Joining a carbon nanotube and a graphene sheet.
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Nanoscience and Nanotechnology, 2008. ICONN 2008. [32] Sakhaee-Pour A. Elastic properties of single-layered graphene sheet. Solid State
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Communications 2009; 149(1-2):91-95.
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[33] YongChao Wang, YinBo Zhu, FengChao Wang, XiaoYi Liu, HengAn Wu. Super-elasticity and deformation mechanism of three-dimensional pillared graphene network structure. Carbon 2017; 118:588-596. [34] Navid Sakhavand, Rouzbeh Shahsavari. Synergistic Behavior of Tubes, Junctions, and Sheets Imparts Mechano Mutable Functionality in 3D Porous Boron Nitride Nanostructures. The Journal of Physical Chemistry C 2014; 118(39):22730-22738. 24
ACCEPTED MANUSCRIPT [35] Mahmood M. Shokrieh, Roham Rafiee. Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Materials and Design 2010; 31(2): 790-795.
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[36] Konstantin N. Kudin, Gustavo E. Scuseria, Boris I. Yakobson. C2F, BN, and C nanoshell elasticity from ab initio computations. Physical Review. Series B 2001; 64(23):235406.
Electrocatalyst:
Cobalt-Embedded
Nitrogen-Doped
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Structured
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[37] Wei C, Wang H ,Eid K, Kim J, Kim JH, Alothman ZA et al.Three-Dimensionally Carbon
Nanotubes/Nitrogen-Doped Reduced Graphene Oxide Hybrid for Efficient Oxygen Reduction. Chemistry 2017; 23(3):637-643.
[38] H Zhao 1, K Min, N R Aluru. Size and Chirality Dependent Elastic Properties of
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Graphene Nanoribbons under Uniaxial Tension. Nano Letters 2009; 9(8):3012-3015.
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ACCEPTED MANUSCRIPT APPENDIX A SUPPLEMENTARY INFORMATION The PGS model to study the effects of the volume fraction (different number of carbon nanotubes growing in the same size graphene) on the elastic properties is
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modelled with a larger size graphene sheet of length (L′x), width (L′y), height (L′z). Different number of CNT (armchair sheet) will be associated with graphene (Zigzag sheet). The total volume is: VT=L′x⋅ L′y⋅ L′z and the volume of CNT is: Vc=nπD2/4⋅Ls.
heptagon
hexagon
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Armchair hole
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schematic diagrams are shown in Figures A1 to A4.
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Thus, expression V= Vc/ VT can be taken as the volume fraction. The various
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zigzag hole
Figure A1. Schematic of different hole types in graphene and junctions in PGS.
26
ACCEPTED MANUSCRIPT
n=24 n=12
n=6
L’z L’y
z
L’x
n=36
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x
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n=30
y
z
z y
x
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y
θ
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θ
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Figure A2. The schematic diagram of different volume fraction.
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Figure A3. The schematic diagram of the inclination angle of carbon nanotube
Atom C
C-C Bond
Equivalent node Equivalent beam element
Figure A4. The equivalent FE Model of single lattice 27
ACCEPTED MANUSCRIPT As presented by Li and Chou[A1]，nanocarbon structure can be treated as a large molecules including ordered atoms which are controlled by electron-nucleus and nucleus-nucleus interactions. The force field can be expressed by steric potential energy. Because C-C bond is a covalent bond, so the main contribution to the steric
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energy are bond stretching, bond angle bending, dihedral angle torsion, out-of- plane torsion [A2]. In the context of the structure mechanic, c-c bond can be regarded as a
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beam element and the potential energies between C atoms can be represented as below:
=
(( − (* )%=
&
&
%
%
(- − -* )%=%
&
=% =
+
∅
(∆-)%
(∆∅)%
& 0 =%
where
&
(∆()%
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is for a bond stretch interaction,
(A.1)
is bond angle in bending,
is the
sum of a dihedral angle in in-plane torsion and out-of-plane torsion in the molecular ,
,
are the force field constants for the beam axial stretching, flexural
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and
bending and torsional twisting in structure mechanics, respectively. For simulation in
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ABAQUS, some of the required beam element parameters such as Young’s modulus(E), shear modulus (G), length (Lb), cross section diameter D (assume that C-C bond has a circular beam section) need to be known, which can be obtained by: = d=4
,
=
E=
=
,
G=
A=
!
28
(A.2) !"
I= #
!"
J=
%$ ACCEPTED MANUSCRIPT
In this paper, all finite element models presented have a periodic geometry with a periodic boundary so that consistent results can be obtained regardless of the model
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size. For predicting the mechanical properties using ABAQUS, one would apply various loading and constraint conditions to boundaries of the finite element model and obtain the corresponding stresses and strains. As such the new 3D structure has an
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effective area and equivalent area. The actual FE model and the periodic boundary applied in ABAQUS are shown in Figures A5 and A6.
Equivalent area(A)
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Effective area(A)
D 8
6
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PD
x
7
PL(Ls) z
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y
t
x
(a)
7
(b)
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Figure A5. Concrete FE model of PGS (a)Top view and (b) side view 15,2=13
12,2 =13
(x=24 2 )
y
29
x (a)
(b)
ACCEPTED MANUSCRIPT Figure A6. Boundaries for simulating Young’s modulus (a) and shear modulus (b)
Six constraint conditions were established on the periodic boundary for simulating three Young’s moduli and three shear moduli. Displacement and load were applied on
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the boundaries and the strain were then simulated, and then after the total resultant forces on all nodes were summed up for computing the stress. Each mode of the constrain-strain loading gave a column component in 6 by 6 stiffness matrix. After six
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constrain conditions, a complete 6 by 6 matrix were obtained and the corresponding
9: =?:= @: @: =>:= 9:
9: =A9& ;: =A@&
( i, j =1, … ,6) 9%
@%
9$
@$
B%$
E%$
B&$
E&$
B&% CD&
E&% CD&
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Young’s and shear moduli were then computed. The following equations (A.3) to (A.7)
(A.3)
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clearly shows the calculation process.
where 9: , ;: , <:= , >:= are stress component, strain components, stiffness matrix and
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flexibility matrix respectively. Noting that CNT and GS are intrinsic anisotropy, and
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with numerical precision, the stiffness matrix is symmetric and off-diagonal terms [A3]. The main assumption here is that the PGS is an orthotropic material for predicting its elastic properties. As a result, the ?:= and >:= can be expressed as ?&& ?&% I ?%% H C := = H >KL H H G
D& S := = C:=
?&$ 0 ?%$ 0 ?$$ 0 ?
>&& >&% I >%% H = H >KL H H G
0 0 0 0 ?MM
>&$ 0 >%$ 0 >$$ 0 >
0 0 P O 0 O 0 O 0 O ?## N
0 0 0 0 >MM
0 0 P O 0 O 0 O 0 O >## N
30
(A.4)
ACCEPTED MANUSCRIPT below:
The corresponding stress was then calculated by
B&$ =
W^ | ^Z^[\] %
WX | _Z_[\] $
9$ =
W_ | _Z_[\]
B&% =
and
&
= `⋅ a
%
= 2 `⋅d
%
= b⋅ a
= 2 b⋅d
$
= b⋅ `
%
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&
$
WX | ^Z^[\]
$
(A.5)
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W^ | _Z_[\] U TB%$ = $ S
9% =
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V 9 = WX | XZX[\] T & &
= 2⋅πed
(A.6)
After obtaining the 9: and ∈: , the >:= was calculated by inverting ?:= . Finally,
U S
& &%
1 , >&& 1 = , >##
=
%
1 , >%% 1 = >MM
=
&$
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V
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Young’s moduli and shear moduli were obtained through:
,
$
=
%$
1 >$$
1 = >##
(A.7)
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The material constants of beam element and the model size used in the finite element models are shown in Table A1 below.
31
ACCEPTED MANUSCRIPT Table A1. The material constants of beam element and the model size Material
i gh( ) jk
jk ) hmn
g o ( i
d(nm)
L(nm)
0.34
0.146
0.142
Lx
Ly
Lz
L′′x
L′′y
L′′z
D
4.544
4.674
5.0
9.514
9.838
9.4
0.8
constants
Model
g l ( i
t(nm)
6.52s Dt
8.75s D&*
jk ) hmn
2.78s D&*
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REFERENCES
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size(nm)
carbon
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[A1] Li Chunyu, Chou Tsu-Wei. A structural mechanics approach for the analysis of International
nanotubes.
40(10):2487-2499.
Journal
of
Solids
and Structures 2003;
[A2] Ansari R, Rouhi S. Atomistic finite element model for axial buckling of
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single-walled carbon nanotubes. Physica. Section E 2010; 43(1): 58-69. [A3] Sihn, Sangwook, Varshney. Vikas, Roy. Ajit K, Farmer. Barry L. Prediction of
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3D elastic moduli and Poisson’s ratios of pillared graphene nanostructures. Carbon
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2012; 50(2):603-611.
32
S0008-6223(18)30792-9
DOI:
10.1016/j.carbon.2018.08.058
Reference:
CARBON 13413
To appear in:
Carbon
Received Date: 24 August 2018 Accepted Date: 27 August 2018
Please cite this article as: L. Song, Z. Guo, G.B. Chai, Z. Wang, Y. Li, Y. Luan, A finite element method to investigate the elastic properties of pillared graphene sheet under different conditions, Carbon (2018), doi: 10.1016/j.carbon.2018.08.058. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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n=24(4%) n=12(2%)
n=30(5%)
n=6(1%)
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n=36(6%)
x L’y L’x y Pillared graphene sheet with different volume fraction
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z
L’z
ACCEPTED MANUSCRIPT A finite element method to investigate the elastic properties of
pillared graphene sheet under different conditions Lubin Songa,c,d, Zhangxin Guoa,b,c,*, Chai Gin Boayb, Zhihua Wanga,d, Yongcun Lia,d, Yunbo Luana,d a
College of Mechanics, Taiyuan University of Technology, Taiyuan 030024, China School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore c State Key Laboratory for strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China d Shanxi Key Laboratory of Material Strength and Structure Impact, Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
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b
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ABSTRACT
In this paper, an investigation was carried out to understand the mechanical elastic properties of a newly developed 3D nanocarbon structure material known as PGS (pillared graphene sheet). The effect of various parameters such as the pillared
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distance, the chirality, the volume fraction and the inclination angle of carbon nanotube on the elastic moduli was studied using the finite element method. The
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commercially available finite element software ABAQUS was utilized for the modelling and analysis of this new structure. Several different models were developed
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to study the effect of the various parameters mentioned earlier. Some interesting conclusions were deduced from the finite element investigation. It was found that the pillared distance, chirality and volume fraction affected the modulus significantly. The change in volume fraction impacted a bigger influence on the Young’s modulus Ez and shear modulus Gxy. In some instances, Ez was increased by approximately 40 1 *Corresponding authors: E-mail address: [email protected] (Zhangxin Guo), Tel. number: 13994231172
ACCEPTED MANUSCRIPT times. In addition, the Young’s modulus Ez was also found to be affected by the change in the inclination angle and was shown to increase with increasing angle. 1. Introduction
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A new-type of material based on nanocarbon structure enormously aroused investigator’s interest since the material fullerene with 0-dimensional structure was discovered in 1985 by Kroto et al [1]. Nanocarbon material is an extended family of
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fullerene which contains multi-type allotropic carbon structure such as fullerene (0D),
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carbon nanotube (1D), graphene (2D) and nanodiamond (3D) et [1,2]. These newly discovered materials have been shown to play a vital role in the fields of physical, chemical, mechanical, optoelectronic and many others due to their many outstanding properties. For example, carbon nanotubes (CNTs) have excellent superconducting
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properties and high stiffness in the axial direction. High stiffness, low density, excellent electronic and thermal properties have been discovered in graphene sheet (GS) in planar directions over the past decades [3-6]. However, the excellent
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performance in nanocarbon material is limited by their nanostructure. CNTs have low
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conduction in uniaxial direction and weak shear properties. And graphene has weak out-of-plane mechanical properties [7,8]. For that reason, hybrid materials which combine the advantages of two or more
nanocarbon structure, have been synthesized and investigated by various field researchers. Lee et al [9] synthesized the nano peapod structure for studying the bandgap properties. Okada et al [10] and Shotono et al [11] also discussed the nano 2
ACCEPTED MANUSCRIPT peapod structure on the aspects of energetics and electron-phonon coupling strength [10,11]. Feng et al [12] studied a new kind of carbon foam on the welding of single-walled carbon nanotube and this hybrid material included carbon nanobuds
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[13], sp2 and sp3 -hybridized nanocarbon [14] and pillared graphene [15]. Pillared graphene is now gaining popularity and is being studied by more and more scholars owing to its new 3D structure as shown in Figure 1. A review of the
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published literature showed that the comparison of monolayer and multilayered graphene sheets was investigated quite intensively in recent years, as did with
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research on the single-walled and multi-walled CNTs. Their conclusion seemed to suggest that the monolayered graphene and single-walled CNT gave favourable mechanical properties. To exploit these favourable mechanical properties, this new 3D
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structure PGS (pillared graphene sheet) is usually combined with single-walled CNTs
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and monolayer graphene sheets.
Figure 1. The elementary unit of pillared graphene sheet
This new hybrid structure material PGS owns multifunctional properties such as 3
ACCEPTED MANUSCRIPT thermal and mechanical properties [16,17] relative to its constituents – CNT and graphene. As this PGS research is still at its infant stage and with a promising future, researchers are rushing in to fabricate, characterize and simulate the PGS. The effect of inter-tube bridging and bridging type on the mechanical properties of pillared
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graphene were investigated by Deniz Bilgili et al [18]. Sangwook Sihn [19] effectively predicted the 3D elastic moduli and Poisson ratio of pillared graphene
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nanostructures. Navid Sakhavand [18,20] studied the thermal transport and the effect of the junction configuration on the deformation and mechanical properties of the
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PGS nanostructure.
Many simulation efforts have been spent on predicting the mechanical and thermal properties, and most of the analysis used the molecular dynamics (MD),
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diverse function theory (DFT), and tight-binding molecular dynamics (TBMD) [20-23]. Although these theories can effectively simulate the movement of the molecular system but it is usually limited by the system size and is confined to study
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relatively short-lived phenomenon at the timescale [19]. In other words, these theories
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cannot completely simulate and solve the static problem of models with larger dimensions. For static mechanical problem with a large dimension system, it is necessary to use other methods to solve. Therefore many researchers utilized the shell model based on the classical continuum mechanics and this to a large extent helps in solving the tricky dimension problem [24]. But the accuracy of the shell model is a problem due to the loss of the interatomic interaction among C atom and the effect of
4
ACCEPTED MANUSCRIPT the C-C bond. Li and Chou [25,26] presented a molecular structure mechanics (MSM) approach at the atomistic scale which is not perplexed in time scale. MSM is a semi-atomistic approach which is based on the structural mechanism and molecular mechanism. The fundamental notion of MSM is that nanostructure like CNT or
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graphene is treated as a frame-like structure where the C-C bond is equivalently replaced with a beam member and the node is treated as the C atom. From then on,
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many published works using the MSM have successfully predicted the elastic modulus, bulking, deformation and stress concentration of CNT and GS [26-29].
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An effective finite element approach for modeling the structure of PGS using the MSM method is proposed and presented in this paper. Though some research findings on PGS have been reported, most of them focused on the thermal and
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electric properties. Sangwook Sihn et al [19] investigated the effect of the PL (pillared length) and PD (pillared distance) on the elastic modulus of PGS. Rouzbeh Shahsavari et al [20] studied the stiffness, stress-strain relations, bond strains and
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inelastic deformation mechanism for PGS. But they only investigated a single and
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ideal model. In this paper, elastic properties of the PGS under different conditions are investigated using a commercial finite element software ABAQUS. The contributing effects of the chirality, the pillared distance, volume fraction, and the inclination angle of carbon nanotubes are evaluated in details.
2. Numerical simulation 2.1 Finite Element Model 5
ACCEPTED MANUSCRIPT Matsumoto et al [30] reported work on a vertically aligned CNT-graphene structure (VACNT-graphene). Later, both of Dimitrakakis et al [15] and Duangkamon Baowan et al [31] discussed the design and configurations of the PGS junction. Their main idea was to create a hole in graphene sheet with a hole size that of the diameter
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of the CNT radial. Subsequently, the CNT were vertically conjunct near the hole by combining the corresponding atoms. An effective simulation using this method were
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developed by Wang et al [31,32].
In this contribution, one of our studies focuses on the combination of different
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chiral CNT and graphene. Monolayer graphene is a layer of carbon atoms arranged in a hexagonal lattice and single wall carbon nanotubes are constituted by a monolayer graphene sheet wrapped into a cylinder. Graphene and carbon nanotubes were divided
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into zigzag and armchair types according to their chirality. Figure 2 shows the specific classification of CNT and graphene and the formation process of CNT. Therefore, two main holes in tow chiral graphene were created for combining two chiral CNT which
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can be seen in Figure A1 (Appendix A). Through that, four different models of PGS
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were created and they were named as PGS-ZA, PGS-AA, PGS-AZ, PGS-ZZ as shown in Table 1. Each model is divided into five categories according to the pillared distance (2.84nm, 2.5nm, 2nm, 1.5nm, 1.25nm). The details of the mechanical properties caused by chirality can be found in several publications [32-34]. In addition, Figure A1 exhibits the actual junctions in PGS. In our simulations, (6,6) CNT and (10,0) CNT were used for ensuring the similar diameter in two 6
ACCEPTED MANUSCRIPT different sizes of graphene. It is worth noting that the junctions between the CNT and graphene should satisfy the Euler’s polygon theorem which states: F+V-E=2(1-G), where F, V and E represents the respective number of faces, vertices and edges for the given polyhedron and G is an Euler characteristic. For PGS junctions, G=2. Hence, a
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total bond spare of 12 was shared by CNT and GS. In this contribution, the configuration in the different four model types contained six hexagons which have a
(Graphene) Zigzag sheet T
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surplus of +1 and this satisfy the criterion [20].
Armchair sheet
Armchair Sheet
Zigzag Sheet (b)
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(a)
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(CNT)Zigzag（ （n,0） ）
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Figure 2. Specific classification of CNT(a) and graphene and the formation process of CNT(b)
7
ACCEPTED MANUSCRIPT Table 1.
Four different FE model of the PGS
FEM model style
Graphene
+
SWCNT
=
3D carbon nanostructure
PSG-ZA
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PSG-AA
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PSG-ZZ
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PSG-AZ
In fact, the growth of carbon nanotubes is not always vertical [37]. So, different angles of inclination (bending growth of carbon nanotubes) applied on the PGS-ZA
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model were made to study the angle effect on the elastic properties of PGS. The
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middle CNT was set to spin at a certain angle along the x axis. The inclined angle is measured from the Z-axis with reference to the middle CNT. To reduce the number of variables, the spacing between two layers of graphene was maintained constant. And to ensure that the beam elements at the junction points were as long as possible, the lattice number of carbon nanotubes near the hole was increased appropriately as shown in Figure A3. 8
ACCEPTED MANUSCRIPT 2.2 Computational Method A finite element approach based on the mechanical molecular structure is presented in this paper. As can be seen in Figure A4 (Appendix A), the chemical bond
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was simulated with an elastic beam element. The C atom was replaced with an equivalent node. After which the PGS was established as a frame-like structure. These different models were then implemented in ABAQUS. The ABAQUS beam element
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has six degrees of freedom (U1, U2, U3, U12, U13, U23) corresponding to three nodal forces and three nodal moment components. The details of the modelling and From these results, the stiffness
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computational steps are described in Appendix A.
matrix were extracted to compute the elastic moduli. To verify the validity of the calculation, the elastic modulus of the CNT, graphene and PGS were respectively
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simulated and compared with published results as shown in Tables 2 and 3.
Table 2. Comparison of Young’s modulus for Graphene and SWCNT
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Investigators
Young’s modulus(TPa)
Graphene 1.042
Mahmood M. Shokrieh et al[35].
1.04
Kudin KN et al[36].
1.029
Present work (armchair)
0.989
AC C
A. Sakhaee-Pour et al[32].
SWCNT
Ehsan Mohammadpour[27].
0.881
Mahmood M. Shokrieh[35].
1.033-1.042
Present work
0.956 9
ACCEPTED MANUSCRIPT Table 3. The effective modulus contrast with other research(PSG-ZA) PL(nm)
PD
Ex(Gpa)
Sihn et al[19].
1.21
2.36
155.02
Sinh et al.
3.25
2.36
2.45
Present work Present work
Shahsavari
et
Ey
Ez
Gxy
Gxz
Gyz
155.45
11.48
41.72
15.59
15.33
57.81
57.82
26.97
14.56
3.84
3.78
2.66
77.0
72.25
10.17
29.88
3.08
2.32
1.21
2.36
136.89
141.11
8.02
38.90
14.01
12.26
1.474
2.0
98.99
101.05
10.1
42.21
4.80
3.91
3. Results and discussion 3.1 The effect of chirality
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al[20].
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Investigators
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In this paper, four sets of models were created for investigating the chiral effect on the elastic properties. The Young’s moduli and shear moduli were computed by
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simulating two different areas – effective area and equivalent area. The effective area is the overall sectional area of the PGS and the equivalent area is the net atomic area
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which discarded the empty part of the carbon PGS structure. The equivalent area is usually for the comparison of the elastic properties of the subgrade components of CNT and graphene. For predicting the elastic properties on the periodic geometry, the periodic boundary constraints were applied. Six different displacement and loads were imposed respectively on corresponding boundary nodes. Then, a linear elastic expression of stress-strain was adopted in the computation of the Young’s modulus 10
ACCEPTED MANUSCRIPT and shear modulus. For each of graphene and CNT, chirality has an important effect on their elastic properties, so it is necessary to investigate if chirality has the same influence on the
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new 3D nanocarbon structure PGS. Figures 3 to 5 show the change in the effective and equivalent Young’s modulus Ex, Ey, Ez and Shear modulus Gxy, Gxz, Gyz with pillared distance. As can be seen in Figures 3 and 4, the Young’s moduli Ex and Ey of
SC
all the four types of combination showed a downward trend within the range of the increase in PD. One possible reason of this decrease is that the defect in graphene is
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getting closer to the boundary with the increase in the distance. It can also be seen that there is a decreasing trend in the equivalent modulus but this trend is higher than that of the effective modulus. This suggested that the equivalent Young’s modulus Ex and
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Ey were strongly affected by the PD and were closer to the CNT and graphene. The results obtained from the four models showed two main significant deductions. The first one is that the in-plane modulus Ex of PGS-AA is larger than the
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other three types while the in-plane moduli Ey of PGS-ZA is the largest. From the
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chirality of graphene alone, the result conforms to the effect of chirality on graphene sheet [38]. And the second one is that PGS with an armchair CNT has a bigger in-plane modulus than the PGS with a zigzag CNT. This means that in the in-plane direction, the deformation of CNT with a zigzag sheet is smaller due to having more beam elements in the perpendicular to the load direction. And Figure 5 shows that the Young’s modulus Ez is more consistent. When the PD is kept constant at 1.25nm, the 11
ACCEPTED MANUSCRIPT Young’s modulus Ex of PGS-AA is ~4.9% higher than the lowest modulus Ex of PGS-ZZ and the Young’s modulus Ey and Ez are ~6.1% and ~19.4% higher. In addition, noting that the Young’s moduli Ez of PGS-AA is reduced in total by ~84% as the PD changed from 1.25nm to 2.84nm, which is more significant than the ~12.2%
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reduction of Ex and ~13.6% reduction of Ey. This means that the PD or chirality has a greater impact on Ez. On the whole it can be observed that though the maximum
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equivalent Young’s moduli Ez of PGS-ZA (~200GPa) is lower than the maximum equivalent Young’s moduli Ex (~720GPa) and Ey (~735GPa), but it showed a
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99 98 97 96 95 94 1.2
1.4
1.6
1.8
2.0
2.2 2.4
2.6 2.8
3.0
equivalnet Young's modulus E x (Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
100
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effective Young's modulus Ex(Gpa)
significant improvement in the out-of-plane relative to graphene sheet.
720 700 680 660 640
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
PD(nm)
PD(nm)
(b)
(a)
EP
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
740
Figure 3. Effective (a) and equivalent (b) Young’s modulus Ex of four types of PGSs against
effective Young's modulus Ey(Gpa)
96.0
equivalent Young's modulus Ey(Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
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PD
95.5 95.0 94.5 94.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
740 720 700 680 660 640 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
PD(nm)
(a)
(b)
Figure 4. Effective (a) and equivalent (b) Young’s modulus Ey of four types of PGSs against PD
12
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
18 16 14 12 10 8 6 4 2 0 1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
200 150 100 50 0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
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1.2
equivalent Y oung's m odulus E Z (G pa)
effective Young's modulus Ez(Gpa)
ACCEPTED MANUSCRIPT
PD(nm)
PD(nm)
(b)
(a)
Figure 5. Effective (a) and equivalent (b) Young’s modulus Ez of four types of PGSs against
SC
PD
Figures 6 to 8 show the variation of the effective and equivalent shear moduli (Gxy,
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Gxz, Gyz) with increasing PD. The shear modulus Gxy shows an upward trend while shear moduli Gxz and Gyz show a downward trend with increasing PD. Interestingly, for the same pillared distance, each of the three shear moduli presented different
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performance in terms of chirality. The shear moduli Gxy and Gyz of PGS-AA is little higher than that of the other three PGS model with PGS-ZA yielding the highest shear moduli Gxz. As can be seen in Figures 6 and 8, the shear modulus Gxy of PGS-ZA is
EP
higher than that of PGS-AZ, but in the case of Gyz, this is lower. Comparing the four
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PGS models with the same pillared distance of 1.25nm, it can be seen that the maximum shear modulus Gxy (equivalent modulus) of PGS-AA is ~13.33% higher than the minimum Gxy of PGS-ZZ and Gxz, Gyz are ~34.2% and ~34.6% respectively between the maximum and minimum. This suggested the shear modulus of Gxz and Gyz are easily influenced by the chirality. Looking at the general trend of the shear moduli of PGS-AA with increasing PD, 13
ACCEPTED MANUSCRIPT the shear modulus Gxy has a growth of ~56.7% while the Gxz and Gyz has a respective reduction of ~78.8% and~79.3%. Similarly for the other three models, the shear moduli Gxy, Gxz and Gyz of PGS-AZ has a respective change rate of 55.2%, 77.8% and 79.4% while the respective change rate of the shear moduli of PGS-ZA are 57.5%,
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78.8% and 81.2%. Finally, PGS-ZZ has ~42.3%, ~79.3% and ~76.5 % change rate to Gxy, Gxz and Gyz respectively. That means that all the shear moduli are significantly
SC
affected by the pillared distance. Comparing the four different models together, the rate of change in the maximum shear modulus Gxy is ~14.4% more than the least rate
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of change in PGS-ZZ. And for shear moduli Gxz and Gyz, there is no significant difference. It can be deduced that the chirality does have a certain effect on the shear modulus Gxy of PGS models. In summary, both the pillared distance and chirality
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
52 50 48 46 44 42 40 1.4
1.6
1.8
2.0
2.2
AC C
1.2
EP
effective shear modulus GXY(Gpa)
54
2.4
2.6
2.8
3.0
PD(nm)
equivalent shear modulus GXY(Gpa)
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affected the elastic properties of the new 3D structure significantly.
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
240 220 200 180 160 140 120 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
(a) (b) Figure 6. effective (a) and equivalent (b) shear modulus Gxy of four types of PGSs against PD
14
4.0 3.5 3.0 2.5 2.0 1.5 1.0 1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
35 30 25 20 15 10 5 1.2
3.0
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
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1.2
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
40
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
equivalent shear modulus Gxz(Gpa)
effective shear modulus Gxz(Gpa)
ACCEPTED MANUSCRIPT
PD(nm)
PD(nm)
(a)
(b)
Figure 7. Effective (a) and equivalent (b) shear modulus Gxz of four types of PGSs against
2.0 1.5 1.0 0.5 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
PD(nm)
(a)
SC
2.5
30 25
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3.0
equivalent shear modulus Gyz(Gpa)
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
3.0
PGS-ZZ PGS-AZ PGS-ZA PGS-AA
20 15 10 5
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PD(nm)
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effective shear Young's modulus Gyz (Gpa)
PD
(b)
Figure 8. Effective (a) and equivalent (b) shear modulus Gyz of four types of PGSs against
EP
PD
3.2 The effect of volume fraction
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The volume fraction of CNT is another parameter that affects the elastic properties
of PGS. A little larger zigzag graphene (10nm*10nm) model and armchair CNT (2.214nm) model were set up for this investigation. One of the four PGS models presented earlier, PGS-ZA was chosen. By keeping the same size of graphene and changing the number of CNT, five different volume fractions were analyzed i.e – 1%，
15
ACCEPTED MANUSCRIPT 2%，4%，5% and 6%. It can be seen in Figure 9(b) that when the volume fraction varies from 1% to 5%, the equivalent Young’s moduli of Ex and Ey show declining trend while the modulus Ez clearly show increasing trend. The Young’s modulus Ez (8Gpa) is far smaller than Ex (589GPa) and Ey (600GPa) when the volume fraction is
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1%. But when the volume fraction is increased to 6%, Ez is 350GPa and Ex, Ey are 511GPa and 520GPa respectively. This shows that the change in volume fraction has a
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significant influence on the Young’s modulus Ez which can be increased by approximately 40 times.
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In Figure 10(b), it can be seen that all the equivalent three shear moduli of Gxy, Gxz and Gyz increased gradually with increasing volume fraction. And the rate of change in Gxz and Gyz seem to be higher. When the volume fraction is 6% (n=36), Gxy
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improved by ~ 13.4%, while Gxz has risen 3.5-fold and Gyz has risen 10-fold. This shows that the volume fraction has minor effect on the Gxy, but has a greater impact on the shear moduli relative to the axial z-direction. Thus, by controlling the number
EP
of CNT rationally with the same volume, PGS with different moduli can be tailored
Ex Ey
60
equaivelent Young's modulus (Gpa)
effective Young's modulus(Gpa)
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according to the needs.
Ez
50 40 30 20 10
0 0.01
0.02
0.03
0.04
0.05
0.06
Vf
Ex Ey
600
Ez
500 400 300 200 100 0 0.01
0.02
0.03
0.04
0.05
0.06
Vf
(b)
(a) 16
Figure 9. Effective (a) and equivalent (b) Young’s modulus of four types of PGSs against volume fraction
ACCEPTED MANUSCRIPT
Gxz 5
Gxy
4 3 2 1 0 0.01
0.02
0.03
0.04
0.05
120
Gyz Gxz
100
Gxy
80 60 40 20 0 0.01
0.06
0.02
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effective shear modulus(Gpa)
equaivelent shear modulus(Gpa)
Gyz
6
0.03
0.04
0.05
0.06
Vf
Vf
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(a) (b) Figure 10. Effective (a) and equivalent (b) shear modulus of four types of PGSs against volume fraction
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3.3 Angle of Inclination
This section discussed the effect of the angle of inclination on the moduli of PGS. By appropriately increasing the length of the carbon nanotubes and creating a similar
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hole in the graphene, a new PGS model was constructed for investigating the elastic properties affected by the change in the angle of inclination. Table 4 provides a
EP
summarized tabulation of the obtained data results.
Table 4. Comparison of equivalent Young’s modulus of PGS with different angle of
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inclination
Equivalent modulus (Gpa)
Ex
0o
650.21
Ez
Gxy
Gxz
Gyz
666.54
26.6
122.7
8.01
7.23
641.30
655.6
27.2
120
7.98
7.1
12o
638.45
646.2
28.6
121.3
7.96
7.01
24o
635.40
640.2
30.3
122.1
7.95
6.94
6o
Ey
17
ACCEPTED MANUSCRIPT
The equivalent Young’s modulus Ex and Ey remains relatively constant with the increase in the angle of carbon nanotubes but Ez increases. The main reason for this
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could be that the increase in the inclined angle helps to reduce the distance between the two adjacent holes that grow on the graphene. As for the shear moduli in general, there is no obvious change in moduli by changing the angle of the inclination of the
SC
carbon nanotubes. Bear in mind that the current results are based on a simple condition, there are many other conditions to analyze such as the axis of rotation, the
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location of rotation, the number of the rotation and the order of arrangement etc.
4.Conclusion
In this paper, a finite element approach based on the molecular structure
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mechanical is proposed for investigating the relevant elastic properties of the PGS under different conditions. In total, three main parameters of the PGS were studied in
EP
our paper – different chirality with various PD, volume fraction and inclination angle of the carbon nanotube.
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It was found that both chirality and PD have a great influence on the elastic
moduli of PGS. Chirality has a greater impact on moduli of Ez (19.4%), Gxz (34.2%)and Gyz (34.6%). PGS-AA have a higher Ex while PGS-ZA have a higher Ez. This indicates that different chirality has a different effect on the different directional moduli. Moreover, the Young’s modulus of Ez (200GPa) is lower than the Young’s moduli of Ex (720GPa) and Ey (735GPa), but PGS yielded a significantly improved 18
ACCEPTED MANUSCRIPT out-of-plane moduli relative to graphene sheet and in-plane relative to CNT. By changing the number of the CNT, five different volume fractions were studied. It can be concluded that the larger the volume fraction, the higher the moduli of Ez,
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Gxz and Gyz. Young’s moduli of Ex and Ey reduced respectively by ~13.24% and ~13.33% when the volume fraction changed from 1% to 6% while the shear modulus Gxy improves by ~13.4%.
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Lastly by rotating CNT along the X axis, five different PGS-AZ models were
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created. The effects of the angle of inclination on the elastic moduli were not obvious but in general, the modulus of Ez is affected by the change in the angle of inclination and improves with the increasing angle.
The results presented in this paper show the enormous potential of the newly 3D
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carbon structure PGS as an enhancer of the resin. Critical issues related to different angles of inclination of CNTs in PGS and their effects on the mechanical properties of
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PGS-reinforced polymer-matrix composites are currently under investigation. Their
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results will be published in due course.
Acknowledgments
This work was supported by the National Nature Science Foundation of China[grant numbers
11602160,11572214,11390362,11402160,21501129];the
Scientific
and
Technological Innovation Programs of Higher Education Institutions in Shanxi[grant 19
ACCEPTED MANUSCRIPT number 2017117]; and the opening foundation for state key laboratory for strength and vibration of mechanical structures[grant number SV2017-KF-01]; and the
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“1331project” Key Innovation Teams of Shanxi Province.
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[38] H Zhao 1, K Min, N R Aluru. Size and Chirality Dependent Elastic Properties of
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Graphene Nanoribbons under Uniaxial Tension. Nano Letters 2009; 9(8):3012-3015.
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ACCEPTED MANUSCRIPT APPENDIX A SUPPLEMENTARY INFORMATION The PGS model to study the effects of the volume fraction (different number of carbon nanotubes growing in the same size graphene) on the elastic properties is
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modelled with a larger size graphene sheet of length (L′x), width (L′y), height (L′z). Different number of CNT (armchair sheet) will be associated with graphene (Zigzag sheet). The total volume is: VT=L′x⋅ L′y⋅ L′z and the volume of CNT is: Vc=nπD2/4⋅Ls.
heptagon
hexagon
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Armchair hole
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schematic diagrams are shown in Figures A1 to A4.
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Thus, expression V= Vc/ VT can be taken as the volume fraction. The various
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zigzag hole
Figure A1. Schematic of different hole types in graphene and junctions in PGS.
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ACCEPTED MANUSCRIPT
n=24 n=12
n=6
L’z L’y
z
L’x
n=36
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x
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n=30
y
z
z y
x
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y
θ
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θ
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Figure A2. The schematic diagram of different volume fraction.
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Figure A3. The schematic diagram of the inclination angle of carbon nanotube
Atom C
C-C Bond
Equivalent node Equivalent beam element
Figure A4. The equivalent FE Model of single lattice 27
ACCEPTED MANUSCRIPT As presented by Li and Chou[A1]，nanocarbon structure can be treated as a large molecules including ordered atoms which are controlled by electron-nucleus and nucleus-nucleus interactions. The force field can be expressed by steric potential energy. Because C-C bond is a covalent bond, so the main contribution to the steric
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energy are bond stretching, bond angle bending, dihedral angle torsion, out-of- plane torsion [A2]. In the context of the structure mechanic, c-c bond can be regarded as a
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beam element and the potential energies between C atoms can be represented as below:
=
(( − (* )%=
&
&
%
%
(- − -* )%=%
&
=% =
+
∅
(∆-)%
(∆∅)%
& 0 =%
where
&
(∆()%
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is for a bond stretch interaction,
(A.1)
is bond angle in bending,
is the
sum of a dihedral angle in in-plane torsion and out-of-plane torsion in the molecular ,
,
are the force field constants for the beam axial stretching, flexural
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and
bending and torsional twisting in structure mechanics, respectively. For simulation in
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ABAQUS, some of the required beam element parameters such as Young’s modulus(E), shear modulus (G), length (Lb), cross section diameter D (assume that C-C bond has a circular beam section) need to be known, which can be obtained by: = d=4
,
=
E=
=
,
G=
A=
!
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(A.2) !"
I= #
!"
J=
%$ ACCEPTED MANUSCRIPT
In this paper, all finite element models presented have a periodic geometry with a periodic boundary so that consistent results can be obtained regardless of the model
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size. For predicting the mechanical properties using ABAQUS, one would apply various loading and constraint conditions to boundaries of the finite element model and obtain the corresponding stresses and strains. As such the new 3D structure has an
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effective area and equivalent area. The actual FE model and the periodic boundary applied in ABAQUS are shown in Figures A5 and A6.
Equivalent area(A)
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Effective area(A)
D 8
6
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PD
x
7
PL(Ls) z
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y
t
x
(a)
7
(b)
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Figure A5. Concrete FE model of PGS (a)Top view and (b) side view 15,2=13
12,2 =13
(x=24 2 )
y
29
x (a)
(b)
ACCEPTED MANUSCRIPT Figure A6. Boundaries for simulating Young’s modulus (a) and shear modulus (b)
Six constraint conditions were established on the periodic boundary for simulating three Young’s moduli and three shear moduli. Displacement and load were applied on
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the boundaries and the strain were then simulated, and then after the total resultant forces on all nodes were summed up for computing the stress. Each mode of the constrain-strain loading gave a column component in 6 by 6 stiffness matrix. After six
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constrain conditions, a complete 6 by 6 matrix were obtained and the corresponding
9: =?:= @: @: =>:= 9:
9: =A9& ;: =A@&
( i, j =1, … ,6) 9%
@%
9$
@$
B%$
E%$
B&$
E&$
B&% CD&
E&% CD&
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Young’s and shear moduli were then computed. The following equations (A.3) to (A.7)
(A.3)
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clearly shows the calculation process.
where 9: , ;: , <:= , >:= are stress component, strain components, stiffness matrix and
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flexibility matrix respectively. Noting that CNT and GS are intrinsic anisotropy, and
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with numerical precision, the stiffness matrix is symmetric and off-diagonal terms [A3]. The main assumption here is that the PGS is an orthotropic material for predicting its elastic properties. As a result, the ?:= and >:= can be expressed as ?&& ?&% I ?%% H C := = H >KL H H G
D& S := = C:=
?&$ 0 ?%$ 0 ?$$ 0 ?
>&& >&% I >%% H = H >KL H H G
0 0 0 0 ?MM
>&$ 0 >%$ 0 >$$ 0 >
0 0 P O 0 O 0 O 0 O ?## N
0 0 0 0 >MM
0 0 P O 0 O 0 O 0 O >## N
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(A.4)
ACCEPTED MANUSCRIPT below:
The corresponding stress was then calculated by
B&$ =
W^ | ^Z^[\] %
WX | _Z_[\] $
9$ =
W_ | _Z_[\]
B&% =
and
&
= `⋅ a
%
= 2 `⋅d
%
= b⋅ a
= 2 b⋅d
$
= b⋅ `
%
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&
$
WX | ^Z^[\]
$
(A.5)
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W^ | _Z_[\] U TB%$ = $ S
9% =
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V 9 = WX | XZX[\] T & &
= 2⋅πed
(A.6)
After obtaining the 9: and ∈: , the >:= was calculated by inverting ?:= . Finally,
U S
& &%
1 , >&& 1 = , >##
=
%
1 , >%% 1 = >MM
=
&$
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V
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Young’s moduli and shear moduli were obtained through:
,
$
=
%$
1 >$$
1 = >##
(A.7)
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The material constants of beam element and the model size used in the finite element models are shown in Table A1 below.
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ACCEPTED MANUSCRIPT Table A1. The material constants of beam element and the model size Material
i gh( ) jk
jk ) hmn
g o ( i
d(nm)
L(nm)
0.34
0.146
0.142
Lx
Ly
Lz
L′′x
L′′y
L′′z
D
4.544
4.674
5.0
9.514
9.838
9.4
0.8
constants
Model
g l ( i
t(nm)
6.52s Dt
8.75s D&*
jk ) hmn
2.78s D&*
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REFERENCES
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size(nm)
carbon
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[A1] Li Chunyu, Chou Tsu-Wei. A structural mechanics approach for the analysis of International
nanotubes.
40(10):2487-2499.
Journal
of
Solids
and Structures 2003;
[A2] Ansari R, Rouhi S. Atomistic finite element model for axial buckling of
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single-walled carbon nanotubes. Physica. Section E 2010; 43(1): 58-69. [A3] Sihn, Sangwook, Varshney. Vikas, Roy. Ajit K, Farmer. Barry L. Prediction of
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3D elastic moduli and Poisson’s ratios of pillared graphene nanostructures. Carbon
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2012; 50(2):603-611.
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