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Numerical Investigation of the Fracture Mechanism of Defective Graphene Sheets Na Fan 1 , Zhenzhou Ren 1 , Guangyin Jing 2 , Jian Guo 1,3 , Bei Peng 1,4, * and Hai Jiang 1, * 1 2 3 4

*

School of Mechatronics Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China; [email protected] (N.F.); [email protected] (Z.R.); [email protected] (J.G.) National Key Laboratory and Incubation Base of Photoelectric Technology and Functional Materials, School of Physics, Northwest University, Xi’an 710069, China; [email protected] School of Materials Science and Engineering, Xiangtan University, Xiangtan 411105, China Center for Robotics, University of Electronic Science and Technology of China, Chengdu 611731, China Correspondence: [email protected] (B.P.); [email protected] (H.J.); Tel.: +86-28-61831723 (H.J.)

Academic Editor: Martin O. Steinhauser Received: 23 December 2016; Accepted: 8 February 2017; Published: 11 February 2017

Abstract: Despite the unique occurrences of structural defects in graphene synthesis, the fracture mechanism of a defective graphene sheet has not been fully understood due to the complexities of the defects. In this study, the fracture mechanism of the monolayer graphene with four common types of defects (single vacancy defect, divacancy defect, Stone–Wales defect and line vacancy defect) were investigated systematically for mechanical loading along armchair and zigzag directions, by using the finite element method. The results demonstrated that all four types of defects could cause significant fracture strength loss in graphene sheet compared with the pristine one. In addition, the results revealed that the stress concentration occurred at the carbon–carbon bonds along the same direction as the displacement loading due to the deficiency or twist of carbon–carbon bonds, resulting in the breaking of the initial crack point in the graphene sheet. The fracture of the graphene sheet was developed following the direction of the breaking of carbon–carbon bonds, which was opposite to that of the displacement loading. Keywords: graphene; defect; dynamic fracture; finite element method; stress concentration

1. Introduction Graphene has attracted extensive interests in recent years since its experimental discovery in 2004 [1]. Its two-dimensional (2D) hexagonal monolayer network of carbon atoms exhibits extraordinary mechanical, electrical and thermal properties [2], allowing broad applications in a variety of areas such as electrodes, chemical nanosensors, nanocomposites and nanooscillators [3]. To date, a number of methods have been developed for the synthesis of graphene, including chemical vapor deposition (CVD) [4], mechanical exfoliation [5] and ion implantation [6]. However, it has been proven that the structural defects are unavoidable during the growth process regardless of the synthesis method [7,8]. Vacancy defects are defined as the absence of carbon atoms on the graphene sheet, including single and multiple vacancies. Stone–Wales (SW) defect is the nonhexagonal rings formed by reconstruction of graphenic lattice. These structural defects are inevitably introduced during the synthesis or functionalization process of graphene, and may affect the mechanical properties of graphene dramatically [9] by lowering the fracture toughness of the graphene sheets [10]. Therefore, the fracture mechanical behaviors of graphene sheets, in particular the dynamic fracture, are extremely important for the development of graphene-based toughening nano-composites. Computational simulation is a powerful tool and has been widely used in the study of the fracture behaviors of graphene sheets due to the difficulties of precise defect control during the experimental Materials 2017, 10, 164; doi:10.3390/ma10020164

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measurements. For example, Zhang et al. [11] revealed the nanofracture in graphene sheet under complex mechanical stresses using molecular dynamics (MD) simulation. Wang et al. [12] studied the mechanical properties of the graphene with a family of 5-8-5 defects by using MD simulation. Xu et al. [13] proposed a coupled quantum/continuum mechanics approach to study the crack propagation from the edge of the graphene sheet. Georgantzinos et al. [14] reported a nonlinear structural mechanics approach examined the impact of pinhole defects on the fracture of graphene. He et al. [15] investigated the effects of the orientation and tilting angles of the Stone-Thrower-Wales (STW, 5-7-7-5) defects based on MD simulation. They found that the breaking strength of graphene decreased with the increasing tilting angle. However, the MD simulations are limited in both length scales and the time consumption. Alternatively, finite element (FE) models based on continuum theory have been adopted to overcome the weakness in the MD simulations. For instance, Xiao et al. [16] studied the fracture and progressive failure of a defective (SW defect) graphene sheet by FE simulation. Baykasoglu et al. [17] developed an atomistic based finite element model for predicting the fracture in both SW and single vacancy defective graphene sheets. Their results showed that the graphene sheets exhibited an orthotropic fracture behavior. Canadija et al. [18] studied the effect of vacancy location and the density of vacancies on the bending behavior of graphene sheets by using a FE-based structural mechanics approach. Recently, Zhang et al. [19] used FE method to predict the instability of dynamic fracture in a line defective graphene sheet. As mentioned above, many efforts have been made to investigate the fracture of the defective graphene sheets, however, a comprehensive and comparative study of mechanical properties contributed from all the typical defects have not been systematically studied, especially the process of the dynamic fracture of graphene sheet. In this study, the FE model based on molecular structural mechanics was established with the geometric nonlinear effects taken into account, to study the mechanical properties and dynamic fracture of graphene sheet with four different types of defect (single vacancy (SV) defect, divacancy (DV) defect, SW defect and line vacancy (LV) defect). This numerical study revealed the fracture mechanism of these four types of defective graphene sheets, which could serve as a benchmark to search for an effective way for inhibiting the fracture of the graphene sheets. 2. Model and Methodology 2.1. Equivalent Nonlinear Timoshenko Beams for Covalent Carbon–Carbon (C–C) Bonds In order to simulate the fracture of graphene, an equivalent model of graphene must be established. The large deformation and nonlinear geometric effects shall not be ignored in the fracture analysis process. Considering the realistic distributions of thickness and equilibrium length of the C–C bonds, the covalent bond could be equated to nonlinear Timoshenko beams in this study, as presented by the linear planar beam elements (Beam 21) in Abaqus 6.10. The constants of the beam elements were computed and listed in Table 1 [19,20]. Belytschko et al. [21] noted that once the length of a C–C bond exceeded the cut-off distance rc , the interatomic force would drop rapidly to zero, and the bond was regarded as broken. The interaction range of carbon atoms was confined within a distance rc = 0.17 nm to avoid unphysical spurious bond forces [21]. The dynamic fracture process would be shown by the successional breakages of C–C bonds. Besides, the accurate nonlinear constitutive relation of the equivalent beam for a C–C bond could be revised as follow [19]:   σ = 20.852 −e−4.658ε + e−3.817ε

(1)

where σ is the stress of the C–C bond, and ε is the strain of the C–C bond. According to the stress–strain curve of the equivalent beam, the critical strain εc and critical stress σc were 0.197 and 1501 Gpa, respectively. The C–C bond broke when it reached the critical stress σc . The material constitutive relation of beam (Equation (1)) was coded as VUMAT subroutine in ABAQUS/Explicit.

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Table 1. Constants of the beam elements [19,20]. Materials 2017, 10, 164

Bond cross-sectional diameter, d

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0.089 nm

1. Constants of the beam elements0.142 [19,20]. BondTable length, r0 nm –3 nm2 Cross-sectional area, A 6.22 × 10 Bond cross-sectional diameter, d 0.089 nm Moment of inertia, Ib 3.08 × 10 –3 nm4 Bond length, r0 0.142 nm Young’s modulus, Eb 19.5 TPa Cross-sectional Aµ 6.22 × 10 –3 nm2 Poisson’sarea, ratio, 0.23 b Moment of inertia, I b 3.08 × 10 –3 nm4 Shear modulus, Gb 7.93 TPa Young’s modulus, Eb 19.5 TPa Poisson’s ratio, μb 0.23 2.2. FE Model of the Pristine GrapheneGand Verifications Shear modulus, b 7.93 TPa

2D FE models were constructed by Beam21 (shear) and Mass elements in ABAQUS. In the model 2.2. FE Model of the Pristine Graphene and Verifications of graphene, a carbon atom was represented by a node at either end of the beam element, while the 2D FE the models werebetween constructed by Beam21 (shear) and Mass process elementswould in ABAQUS. In the beam reflected reaction nodes. The dynamic fracture be shown by the modelbreaking of graphene, a carbon following of C–C bonds.atom was represented by a node at either end of the beam element, while the reflected reaction between nodes. Thepristine dynamicgraphene fracture process be shown of Figure 1abeam illustrates thethe equivalent FE model of the sheet.would The dimensions by the following breaking of C–C bonds. the defect-free model of graphene were 12.5520 nm (lx ) × 12.6470 nm (ly ). It contained 6180 carbon Figure 1a illustrates the equivalent FE model of the pristine graphene sheet. The dimensions of atoms (nodes) and 9159 bonds (elements). To obtain the mechanical properties and dynamic fracture the defect-free model of graphene were 12.5520 nm (lx) × 12.6470 nm (ly). It contained 6180 carbon of theatoms graphene model, uniform normal loadings of displacement were applied to nodes on one edge, (nodes) and 9159 bonds (elements). To obtain the mechanical properties and dynamic fracture as shown in Figure In order to facilitate subsequent analysis, were we defined configurations of the graphene1.model, uniform normalthe loadings of displacement appliedthe to nodes on one of the C–C bonds as shown in Figure 1c. Bond A denoted that the C–C bond paralleled to the edge, as shown in Figure 1. In order to facilitate the subsequent analysis, we definedarmchair the direction (AC direction). Bondbonds B denoted thatinthe C–C 1c. bond intersected thethat zigzag direction configurations of the C–C as shown Figure Bond A denoted the C–C bond (ZZ paralleled the◦ armchair direction (ACstructural direction).configuration Bond B denoted that the C–C bond intersected direction) with to a 30 angle. The armchair consisting of one Bond A and two the zigzag direction (ZZ direction) with a 30° angle. The armchair structural configuration consisting Bond B was denoted as Type A, while the zigzag structural configuration consisting of two Bond B of one Bond A and was denoted as Type Z. two Bond B was denoted as Type A, while the zigzag structural configuration consisting of two Bond B was denoted as Type Z.

Figure 1. (a)1. The equivalent FEFEmodel graphenesheet. sheet.(b)(b) Two directions of the Figure (a) The equivalent modelofofthe the pristine pristine graphene Two directions of the displacement loading, AC direction and and ZZ (c) Schematics of theof configurations of the C–C displacement loading, AC direction ZZdirection. direction. (c) Schematics the configurations of the bonds. C–C bonds.

To validate the FE model, the mechanical properties of a pristine graphene was firstly analyzed

To FE model, the mechanical properties a pristine graphene was firstly analyzed by validate using thethe loading parameters. All of the models wereofapplied a uniaxial displacement loading by using the loading parameters. All of the models were applied a uniaxial displacement loading along AC and ZZ direction, respectively (Figure 1). The loading time was longer than the minimum alongnatural AC and ZZ direction, respectively (Figure 1). The loading time was longer than the minimum period (the natural period could be obtained through the simulation) due to the quasi-static natural period (the natural period could be through the simulation) due toloading the quasi-static fracture process of graphene. Therefore, theobtained loading time was at least 6.75 ps while the time must also beof sufficient to ensure the occurrence of fracture. In the the strain rate the ε was set at ε time fracture process graphene. Therefore, the loading time was at study, least 6.75 ps while loading –1 with a loading time of 18 ps. Along AC direction (l = ly = 12.6470 nm), the Δl/(l·t) 9.84 × 10–4tofsensure must =also be =sufficient the occurrence of fracture. In the study, the strain rate ε was set at displacement loading Δl on the ends oftime the model wasAlong 1.1200 AC nm. direction At the ZZ (ldirection (l = lx = nm), –4 –1 ε = ∆l/(l·t) = 9.84 × 10 fs with both a loading of 18 ps. = ly = 12.6470

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the displacement loading ∆l on the both ends of the model was 1.1200 nm. At the ZZ direction (l = lx = 12.5520 nm), the displacement loading ∆l was 1.1116 nm. Figure 2 showed the calculated stress–strain curves of the pristine graphene under the loading along AC and ZZ directions, respectively. The fracture stress (σf ) and fracture strain (εf ) along the AC and ZZ directions were 104.1 GPa, 122.8 GPa and 0.149, 0.164, respectively. Both of the curves fell within the previous simulation study [17,22–24], Materials 2017, 10, 164 4 of 12 i.e., σf : 90–125 GPa (AC direction), 110–125 GPa (ZZ direction) and εf : 0.14–0.21 GPa (AC direction), 0.14–0.24 GPa (ZZthe direction). The loading Young’sΔlmodulus E and ratios µthe of the pristine graphene 12.5520 nm), displacement was 1.1116 nm.Poisson’s Figure 2 showed calculated stress– of the pristine graphene under the loading along AC and ZZ directions, respectively. sheetstrain couldcurves be calculated as follow: The fracture stress (σf) and fracture strain (εf) along the AC and ZZ directions were 104.1 GPa, 122.8 mFpfell ly within the ε GPa and 0.149, 0.164, respectively. Both of theE curves AC direction = , µ = − lx previous simulation study tl ∆l εly and εf: 0.14–0.21 GPa (AC x y [17,22–24], i.e., σf: 90–125 GPa (AC direction), 110–125 GPa (ZZ direction) direction), 0.14–0.24 GPa (ZZ direction). The Young’s modulus E and Poisson’s ratios μ of the εly pristine graphene sheet could be calculated as follow:mFp l x ZZ direction E= , µ=− εlx (2) AC direction = tly ∆l x, μ = −

(2)

(3)

where m is number of loading points (120 for AC direction and the force of ,μ = −103 for ZZ direction), Fp is (3) ZZ direction = the single loading point and t is the thickness of the model (0.334 nm). The obtained Young’s modulus where m is number of loading points (120 for AC direction and 103 for ZZ direction), Fp is the force of and Poisson’s the pristine sheet alsomodel corresponded closely to the experimental the single ratios loadingofpoint and t is graphene the thickness of the (0.334 nm). The obtained Young’s values (Young’s modulus of E = 1.0 ± 0.1 TPa [25]) as shown in Table 2. modulus and Poisson’s ratios of the pristine graphene sheet also corresponded closely to the experimental values (Young’s modulus of E = 1.0 ± 0.1 TPa [25]) as shown in Table 2. Table 2. Young’s modulus and Poisson’s ratios of pristine graphene sheet Table 2. Young’s modulus and Poisson’s ratios of pristine graphene sheet

Direction

Direction

AC direction AC direction ZZ direction ZZ direction

Young’s modulus (TPa) Young’s modulus (TPa) 1.075 1.075 1.096 1.096

Poisson’s Ratio Poisson’s Ratio 0.172 0.172 0.162 0.162

Figure 2. 2.Stress–strain ofdefect-free defect-free graphene. Figure Stress–strain curves curves of graphene.

2.3.Model FE Model of the Defective Graphene 2.3. FE of the Defective Graphene structural defectswere were generated generated inevitably inevitably during or chemical The The structural defects duringthetheproduction production or chemical functionalization process of graphene. In this study, we investigated four typical types of defects (SV functionalization process of graphene. In this study, we investigated four typical types of defects defect, DV defect, SW defect and LV defect) in the graphene sheet, as shown in Figure 3. In all (SV defect, DV defect, SW defect and LV defect) in the graphene sheet, as shown in Figure 3. In all models, the defect was located at the center of the graphene sheet. The SV defect was formed by models, the one defect wasatom located at the center theBond graphene sheet. The formed missing carbon and three C–C bondsof(one A and two Bond B),SV anddefect the DVwas defect was by missing one carbon atom and three C–C bonds (one Bond A and two Bond B), and the DV defect formed by missing two carbon atoms and five C–C bonds (one Bond A and four Bond B). The SW was formed by was missing twobycarbon atoms androtation five C–C bonds (one Bond A and Bond B).was The SW defect formed the result of 90° of the central C–C bond, and four the LV defect formed by a line of DV defect (12 Bond A and 13 Bond B) along the ZZ direction.

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defect was formed by the result of 90◦ rotation of the central C–C bond, and the LV defect was formed by a line of DV defect (12 Bond A and 13 Bond B) along the ZZ direction. Materials 2017, 10, 164 5 of 12

Figure 3. Equivalent FE model of the defective graphene (the dash line indicates the missing or

Figure 3. Equivalent FE model of the defective graphene (the dash line indicates the missing or rotating rotating C–C bonds). C–C bonds).

3. Results and discussion

3. Results and Discussion 3.1. Dynamic Fracture of Graphene with Different Defects

3.1. Dynamic Fracture of Graphene with Different Defects

The dynamic fracture and propagation in graphene by FE analysis with the four different types of defects under loading stress along inAC and ZZ shown in different Figures 4–7 The dynamic fracture and propagation graphene by directions FE analysisare with the four types of (Supplementary Videos andAC Figures 8–11directions (Supplementary Videos 5–8), respectively. As the defects under loading stress1–4) along and ZZ are shown in Figures 4–7 (Supplementary displacements of both ends of the grapheneVideos increased the C–C bond broke when of it both Videos 1–4) and Figures 8–11 (Supplementary 5–8),gradually, respectively. As the displacements reached the critical stress σc = 1501 GPa, resulting in the crack propagation instantly. As could be ends of the graphene increased gradually, the C–C bond broke when it reached the critical stress seen in these figures, the propagation of the crack presented was radically symmetrical due to the σc = 1501 GPa, resulting in the crack propagation instantly. As could be seen in these figures, the geometric symmetry of the model. The occurrence of the branches was observed during the propagation of the crack presented was radically symmetrical due to the geometric symmetry of the propagation of crack, which agreed with the previous studies by Omeltchenko [26] and Zhang [27]. model. Thealloccurrence the branches wasthat observed during the propagation crack, agreed From the fractureofanalysis, we found the cracks were always initiated of from the which defective with area, the previous studies by Omeltchenko [26] and Zhang [27]. From all the fracture analysis, therefore all these four typical types of defects were considered had various influence on the we found that the cracksofwere always initiated from the defective area, therefore all these four typical dynamic fracture graphene. It was obvious that the deficiency of the C–C bondson occurred in the SVfracture defect, DV defect and LV types of defects were considered had various influence the dynamic of graphene. defect. When applying stressofalong the AC direction as shown the stress It was obvious that theloading deficiency the C–C bonds occurred in theinSVFigures defect,4,5,7, DV defect and LV concentration generated around the defective area but only on the Bond A of the Type A (Figure 1c). defect. When applying loading stress along the AC direction as shown in Figures 4, 5 and 7, the stress For example, in Figure 4, there were two symmetrical Type A, the stress concentration occurred on concentration generated around the defective area but only on the Bond A of the Type A (Figure 1c). the two Bond A of the Type A. The breaking firstly occurred on the two Bond A due to the stress For example, in Figure 4, there were two symmetrical Type A, the stress concentration occurred on concentration, which we called it “initial crack point”. With the time, the crack was propagating the two Bond A of the Type A. The breaking firstly occurred on the two Bond A due to the stress until the fracture of the whole graphene sheet, resulting in the crack propagating to the ZZ direction concentration, which we calledalso it “initial crack point”.5With crack wasdirection propagating in a line. This phenomenon was found in Figures and 7.the Thetime, crackthe propagation of SW until the fracture the whole graphene resulting in the crack propagating the ZZ direction defectiveofgraphene (Figure 6) wassheet, different from those with the vacancy defectstosince there was no in a line. missing This phenomenon alsoshown was found in Figures and bearing 7. The crack propagation direction of SW C–C bond. It was in Figure 6 that the5stress area was at the edge of the model rathergraphene than the defective Hence, the initial crack point of the defect was not atsince the defect defective (Figure site. 6) was different from those with theSW vacancy defects theresite was no but C–C at thebond. cornersItof theshown graphene This wasthe because C–C bond was hard to break missing was in sheet. Figure 6 that stressthe bearing area was at the edgeunder of thethe model loading force along the AC direction due to the complete C–C bonds. The stress concentration rather than the defective site. Hence, the initial crack point of the SW defect was not at the defect site appeared at the corners.

but at the corners of the graphene sheet. This was because the C–C bond was hard to break under the loading force along the AC direction due to the complete C–C bonds. The stress concentration appeared at the corners.

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Figure The Figure4. Thecrack crackpropagation propagationof ofthe theSV SVdefect defectunder underloading loadingalong alongAC ACdirection. direction. Figure 4.4.The crack propagation of the SV defect under loading along AC direction.

Figure 5. The crack propagation of the DV defect under loading along AC direction. Figure Figure5.5.The Thecrack crackpropagation propagationof ofthe theDV DVdefect defectunder underloading loadingalong alongAC ACdirection. direction.

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Figure 6.6.The crack propagation of defect under loading along AC direction. Figure ofthe theSW SWdefect defectunder under loading along direction. Figure 6.The Thecrack crackpropagation propagation of the SW loading along ACAC direction.

Figure7.7.The Thecrack crackpropagation propagation of along ACAC direction. Figure of the theLV LVdefect defectunder underloading loading along direction. Figure 7. The crack propagation of the LV defect under loading along AC direction. Under the loading stress along the ZZ direction (as shown in Figures 8–11), due to the missing Under the loading stress along the ZZ direction (aslocal shown inconcentration Figures 8–11), due generate to the missing Bond B inthe theloading SV defect, DValong defect,the and LV defect, the stress could at Under stress ZZ direction (as shown in Figures 8–11), due to the missing Bond B inB the SV defect, DV defect, defect, the local stressstress concentration could generate Bond Bond around the defective area.and As LV shown in Figure 8, local concentration occurred at at the Bond B in thedefective SV defect, DVAs defect, and LV defect,local the local stress concentration could generateBat B around shown Figure at the Bond Bthe in the Type Z.area. Since there wasin only one 8, Type Z stress on theconcentration top but two atoccurred the bottom, the Bond stress in Bond B around the defective area. As shown inthe Figure 8, local stress concentration occurred at the theconcentration Type Z. Since there was only one Z on top the but two at the bottom, concentration initially generated onType the top and then bottom, followed bythe thestress occurrence of a Bond B in the Type Z. Since there was only one Type Z on the top but two at the bottom, the stress initially generated onthe theDV topdefective and thengraphene the bottom, followed theconcentration occurrence ofwould a Y-shaped crack. Y-shaped crack. In (Figure 9), the by stress generate at In concentration initially generated on and then thecause bottom, by the occurrence of of a B at the both of the top and bottom Type Z and two followed initial crack points. crack theBond DV defective graphene (Figure 9),the thetop stress concentration would generate at Bond B The at the both Y-shaped crack. In the DV defective graphene (Figure 9), the stress concentration would generate at was along armchair andcrack formed an I-shaped crack. As shown inwas Figure 11, the thepropagation top and bottom Typethe Z and causedirection two initial points. The crack propagation along Bond B at the both of the top and bottom Type Z and cause two initial crack points. The crack in the LV defective graphene sheet, the stress concentration was found occurred at the four Bond B armchair direction and formed an I-shaped crack. As shown in Figure 11, in the LV defective graphene

propagation was along the armchair direction and formed an I-shaped crack. As shown in Figure 11, in the LV defective graphene sheet, the stress concentration was found occurred at the four Bond B

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sheet, the stress concentration was found occurred at the four Bond B in Type Z thus four initial cracks Type Z initial cracks and crack formed. In SW andin H-shaped crack were formed. In H-shaped the SW defective graphene, shown in Figuregraphene, 10, although inan Type Z thus thus four four initial cracks and an an H-shaped crack were were formed. as In the the SW defective defective graphene, as shown in Figure 10, although there was no lack of the C–C bond, there was one C–C bond as was shown Figure 10,C–C although lack of bond the C–C there wasdirection one C–C as bond that there noin lack of the bond,there therewas wasno one C–C thatbond, had the same the that loading had the same direction as the loading stress due to the rotation of the C–C bonds. This C–C bond haddue the to same stress This due to thebond rotation of the C–Cbreak bonds. This C–C bond stress thedirection rotation as of the theloading C–C bonds. C–C could easily caused by the stress could break caused by stress concentration under the stress along the could easily easilyunder break the caused by the the stress concentration under Afterwards, the loading loading four stressinitial alongcrack the ZZ ZZ concentration loading stress along the ZZ direction. points direction. Afterwards, four initial crack points generated at Bond B both the top direction. four points generated at the the four four B at at in both top and and generated at Afterwards, the four Bond B atinitial both crack the top and bottom defective area,Bond resulting an the X-shaped crack. bottom bottom defective defective area, area, resulting resulting in in an an X-shaped X-shaped crack. crack.

Figure The crack propagation along ZZ direction. Figure 8.8. of SV SVdefect defectunder underloading loading along direction. Figure 8.The Thecrack crackpropagation propagation of of SV defect under loading along ZZZZ direction.

Figure 9. The crack propagation of DV defect under loading along ZZ direction.

Figure Thecrack crackpropagation propagation of along ZZZZ direction. Figure 9.9.The of DV DVdefect defectunder underloading loading along direction.

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Figure 10. The crack propagation of SW defect under loading along ZZ direction. Figure10. 10.The Thecrack crackpropagation propagationof ofSW SWdefect defectunder underloading loadingalong alongZZ ZZdirection. direction. Figure

Figure 11. The crack propagation of LV defect under loading along ZZ direction. Figure Figure11. 11.The Thecrack crackpropagation propagationofofLV LVdefect defectunder underloading loadingalong alongZZ ZZdirection. direction.

3.2. Fracture Strength of Graphene with Different Defects 3.2. 3.2.Fracture FractureStrength StrengthofofGraphene Graphenewith withDifferent Different Defects Defects In order to give a comprehensive and comparison studies of mechanical properties of all the In give aa comprehensive and comparison studies of properties of the In order order to tothe givefracture comprehensive comparison of mechanical mechanical of all allThe the typical defects, strength ofand graphene with studies those four defects hasproperties been studied. typical defects, the fracture strength of graphene with those four defects has been studied. The typical defects, the of fracture strength of graphene withloading those four has along been studied. The stress–strain curves these defective graphene under weredefects calculated both the AC stress–strain curves of these thesedefective defectivegraphene grapheneunder underloading loading were calculated along both the AC stress–strain curves of were calculated along both the AC and and ZZ directions, as shown in Figure 12. As a comparison, the stress–strain curve of the pristine and ZZ directions, as shown in Figure 12. As a comparison, the stress–strain curve of thegraphene pristine ZZ directions, as shown in Figure 12. As a comparison, the stress–strain curve of the pristine graphene was also included in the plots. graphene was also in included in the plots. was Figure also included the plots. 12a demonstrated the effect of these defects on the fracture strength of graphene along Figure 12a demonstrated the effect effectof ofthese these defectsononthe the fracture strength graphene along Figure 12a demonstrated the strength of of graphene along the the AC loading direction by comparing that ofdefects the pristine fracture graphene. The fracture stress of the the AC loading direction by comparing that of the pristine graphene. The fracture stress of the AC loading direction by comparing that of the pristine graphene. The fracture stress of the pristine pristine graphene was 104.1 GPa under the loading along AC direction, as shown in Figure 12a, pristine graphene was 104.1 GPa under the loading along AC direction, as shown in Figure 12a, while it was only 43.2 GPa for the graphene with LV defect, which was decreased by 60.4%. In this while it was only 43.2 GPa for the graphene with LV defect, which was decreased by 60.4%. In this case, the stress concentration occurred on the two Bond A of the Type A located at the edge of the case, the stress concentration occurred on the two Bond A of the Type A located at the edge of the

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graphene was 104.1 GPa under the loading along AC direction, as shown in Figure 12a, while it was only 43.2 GPa for the graphene with LV defect, which was decreased by 60.4%. In this case, the stress Materials 2017, 10,occurred 164 10 of as 12 concentration on the two Bond A of the Type A located at the edge of the defective area shown in Figure 7, the cracks of these two rows propagated coinstantaneous in opposite directions. defective as shown in line Figure 7, the of theseeffect two rows coinstantaneous in This resultarea showed that the defect hadcracks a substantial on thepropagated fracture strength of graphene opposite directions. This result showed that the line defect had a substantial effect on the fracture along the AC direction. strength of graphene the AChad direction. It was found thatalong SW defect very little impact on the fracture strength of graphene under It was found that SW defect had very little impact on the fracture strength of graphene the loading along AC direction. The fracture stress of graphene sheet with SW defect was 103.7under GPa, the loading along AC direction. The fracture stress of graphene sheet with SW defect was 103.7 which was only 0.3% less than that of the defect-free graphene sheet. Because the Bond A did not GPa, miss which was only 0.3%concentration less than that of the defect-free graphene sheet. the Because the Bond A did The not in this case, no stress occurred on the defective area along AC loading direction. miss in this case, no stress concentration occurred on the defective area along the AC loading initial crack point of the SW defect was not at the defect site but the corners of the graphene sheet. direction. The initial crack point that of the SWthe defect wasstress not at butby theAC corners ofwas the This also could be used to explain why fracture in the SW defect defect site caused loading graphene sheet. also could be used to explain that why the fracture stress in SW defect caused similar to that in This the pristine graphene. by AC loading was similar to that pristinewith graphene. In addition, the fracture stressinofthe graphene SV defect and DV defect were almost identical, In addition, the fracture stress of graphene with defect and DV defect were almost 80.0 GPa and 79.6 GPa, which were decreased by 23.2%SV and 23.5%, respectively. As shown in identical, Figures 4 80.0 GPa and 79.6 GPa, which were decreased by 23.2% and 23.5%, respectively. As shown in and 5, both of them missed the structure, one Bond A, the stress concentration occurred at the same Figures 4 and 5, both of them missed the structure, one Bond A, the stress concentration occurred at sites, thus they had the closed locations of the initial crack points and propagation as well as the the samestress. sites, thus they had the closed locations of the initial crack points and propagation as well fracture as theHowever, fracture stress. under the loading along ZZ direction, as shown in Figure 12b, the LV defect made the However, the loading along by ZZ10.5% direction, as shown in Figureto12b, LVofdefect made fracture stress ofunder graphene only decrease to 105.4 GPa compared 122.8the GPa the pristine the fracturewhich stresswas of graphene only decrease by 10.5% 105.4ofGPa to 122.8 GPa of the graphene, quite different to the decrease of to 60.4% the compared fracture stress in AC loading pristine graphene, which was quite different to the decrease of 60.4% of the fracture stress in AC direction. The LV defect missed 12 Bond A and 13 Bond B, however there were seven Type Z (14 Bond loading direction. The LV defect missed 12 Bond A and 13 Bond B, however there were seven Type Z B) located at the top and bottom of the defective area. This might reduce the stress concentration at (14 Bond B) located at theB would top and bottom of the stress defective area. mightmore reduce the stress Type Z because these Bond share the loading evenly andThis it needed loading stress concentration at Type Z because these Bond B would share the loading stress evenly and it needed to generate the stress concentration at the left and right of the edge of the defective area. However, more AC loading stressonly to generate stress concentration at top the and left bottom and right of defective the edge area, of the along direction, one Type the A individually exists at the of the it defective area. However, along AC direction, only one Type A individually exists at the top and is more likely to have local stress concentration thus resulting in the fracture. The DV defect made the bottom of the of defective area, it isdecrease more likely to have localGPa stress resulting in the fracture stress graphene sheet by 29.8% to 86.2 andconcentration the SV defectthus made it decrease by fracture. The DV defect made the fracture stress of graphene sheet decrease by 29.8% to 86.2 GPa and 14.2% to 109.9 GPa, respectively. This was because the SV defect had one more Type Z to share the the SV defect it decrease by DV 14.2% to 109.9 GPa, respectively. This was because the SV defect loading stress made compared with the defect, which meant that the SV defect needed more loading had one more Type Z to share the loading stress compared with the DV defect, which meant that the stress to generate the stress concentration. The SW defect made the fracture stress decrease by 21.2% SV96.8 defect needed loading stress to generate stress effects concentration. Thedirection SW defect made the to GPa. Thesemore are radically different from the the decrease on the AC (decrease of fracture stress decrease by 21.2% to 96.8 GPa. These are radically different from the decrease effects 0.3%). This was because once applied the loading along the ZZ direction, the stress concentration was on the ACon direction of 0.3%). was because generated one C–C(decrease bond (Figure 10) atThis the defective area.once applied the loading along the ZZ direction, the stress concentration was generated on one C–C bond (Figure 10) at the defective area.

Figure 12. 12. Stress–strain curves of the pristine pristine and and defective defective graphene graphene under under loading loading along: along: (a) AC ZZ direction. direction. direction; and (b) ZZ

4. Conclusions In this study, the molecular structural mechanics based finite element models of monolayer graphene with SV defect, DV defect, SW defect and LV defect were developed to study the fracture strength and dynamic fracture, thus revealing the mechanism of fracture failure of the defective graphene. The numerical simulation results demonstrated that the missing and rotation of the C–C

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4. Conclusions In this study, the molecular structural mechanics based finite element models of monolayer graphene with SV defect, DV defect, SW defect and LV defect were developed to study the fracture strength and dynamic fracture, thus revealing the mechanism of fracture failure of the defective graphene. The numerical simulation results demonstrated that the missing and rotation of the C–C bonds may lead to a stress concentration at the defective area along the same direction as the displacement loading, resulting in the breaking of the covalent C–C bonds, and generating the initial crack points. The crack propagated along the crack of the C–C bonds and caused the fracture of the graphene sheet. The breaking of the C–C bonds by the stress concentration significantly reduced the fracture strength of the graphene. The LV defect had the largest effect on the fracture strength of graphene under loading along the AC direction which the DV defect presented the largest effect along the ZZ direction. Overall, the fracture of defective graphene would be attributed to the stress concentration caused by the breaking of C–C bonds and the initial crack point at the defective site along the displacement loading direction. An effective solution to control the fracture is to decrease the stress concentration in the defective area, for example, layer stack to stagger the defective area. Zhi-Min Liao et al. [28] found that the fracture force distribution of the stacked graphene was very different from that of monolayer graphene. The membrane of stacked graphene became less sensitive to the defects during nanoindentation. Stacked graphene may reduce the stress concentration compared with the monolayer graphene. All the results in the present study provide researchers more valuable information, thus helping to prevent unexpected catastrophic failure in various graphene applications. Supplementary Materials: The following are available online at www.mdpi.com/1996-1944/10/2/164/s1, Videos 1–8. Acknowledgments: The authors would like to acknowledge the partial supports provided by the National Natural Science Foundation of China (No. 51575090), the Fundamental Research Funds for the Central Universities (No. ZYGX2014Z004, No. ZYGX2016KYQD118 and No. ZYGX2015J084), the China Postdoctoral Science Foundation Grant (No. 2016M590873) and the National Youth Top-Notch Talent Support Program. The authors would like to acknowledge Bin Zhang, Haifeng Xiao and Gang Yang for giving the help of establishment of the FE model. Author Contributions: N.F. and Z.R. performed all the calculations and simulations; N.F., Z.R. and H.J. performed the data analysis; and all authors discussed the results and participated in writing and revising of the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8.

Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666–669. [CrossRef] [PubMed] Chen, H.; Müller, M.B.; Gilmore, K.J.; Wallace, G.G.; Li, D. Mechanically strong, electrically conductive, and biocompatible graphene paper. Adv. Mater. 2008, 20, 3557–3561. [CrossRef] Geim, A.K. Graphene: Status and Prospects. Science 2009, 324, 1530–1534. [CrossRef] [PubMed] Somani, P.R.; Somani, S.P.; Umeno, M. Planer nano-graphenes from camphor by CVD. Chem. Phys. Lett. 2006, 430, 56–59. [CrossRef] Green, A.A.; Hersam, M.C. Solution phase production of graphene with controlled thickness via density differentiation. Nano Lett. 2009, 9, 4031–4036. [CrossRef] Garaj, S.; Hubbard, W.; Golovchenko, J.A. Graphene synthesis by ion implantation. Appl. Phys. Lett. 2010, 97, 183103. [CrossRef] [PubMed] Wang, C.; Han, Q.; Xin, D. Fracture analysis of single-layer graphene sheets with edge crack under tension. Mol. Simul. 2015, 41, 325–332. [CrossRef] Daniels, C.; Horning, A.; Phillips, A.; Massote, D.V.; Liang, L.; Bullard, Z.; Sumpter, B.G.; Meunier, V. Elastic, plastic, and fracture mechanisms in graphene materials. J. Phys. Condens. Matter 2015, 27, 373002. [CrossRef] [PubMed]

Materials 2017, 10, 164

9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

12 of 12

Rajasekaran, G.; Narayanan, P.; Parashar, A. Effect of Point and Line Defects on Mechanical and Thermal Properties of Graphene: A Review. Crit. Rev. Solid State Mater. Sci. 2015, 41, 47–71. [CrossRef] Daly, M.; Reeve, M.; Singh, C.V. Effects of topological point reconstructions on the fracture strength and deformation mechanisms of graphene. Comput. Mater. Sci. 2015, 172–180. [CrossRef] Zhang, B.; Mei, L.; Xiao, H. Nanofracture in graphene under complex mechanical stresses. Appl. Phys. Lett. 2012, 101, 123915. [CrossRef] Wang, S.; Yang, B.; Yuan, J.; Si, Y.; Chen, H. Large-Scale Molecular Simulations on the Mechanical Response and Failure Behavior of a defective Graphene: Cases of 5-8-5 Defects. Sci. Rep. 2015, 5, 14957. [CrossRef] [PubMed] Xu, M.; Tabarraei, A.; Paci, J.T.; Oswald, J.; Belytschko, T. A coupled quantum/continuum mechanics study of graphene fracture. Int. J. Fract. 2012, 173, 163–173. [CrossRef] Georgantzinos, S.K.; Katsareas, D.E.; Anifantis, N.K. Limit load analysis of graphene with pinhole defects: A nonlinear structural mechanics approach. Int. J. Mech. Sci. 2012, 55, 85–94. [CrossRef] He, L.; Guo, S.; Lei, J.; Sha, Z.; Liu, Z. The effect of Stone-Thrower-Wales defects on mechanical properties of graphene sheets—A molecular dynamics study. Carbon 2014, 75, 124–132. [CrossRef] Xiao, J.R.; Staniszewski, J.; Gillespie, J.W. Fracture and progressive failure of defective graphene sheets and carbon nanotubes. Compos. Struct. 2009, 88, 602–609. [CrossRef] Baykasoglu, C.; Mugan, A. Nonlinear fracture analysis of single-layer graphene sheets. Eng. Fract. Mech. 2012, 96, 241–250. [CrossRef] Canadija, M.; Brˇci´c, M.; Brni´c, J. Bending behaviour of single-layered graphene nanosheets with vacancy defects. Eng. Rev. 2013, 33, 9–14. Zhang, B.; Xiao, H.; Yang, G.; Liu, X. Finite element modelling of the instability in rapid fracture of graphene. Eng. Fract. Mech. 2015, 141, 111–119. [CrossRef] Scarpa, F.; Adhikari, S.; Srikantha, P.A. Effective elastic mechanical properties of single layer graphene sheets. Nanotechnology 2009, 20, 2643–2646. [CrossRef] [PubMed] Belytschko, T.; Xiao, S.P.; Schatz, G.C.; Ruoff, R.S. Atomistic simulations of nanotube fracture. Phys. Rev. B Condens. Matter 2002, 65, 121. [CrossRef] Wang, M.C.; Yan, C.; Ma, L.; Hu, N.; Chen, M.W. Effect of defects on fracture strength of graphene sheets. Comput. Mater. Sci. 2012, 54, 236–239. [CrossRef] Liu, F.; Ming, P.; Li, J. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys. Rev. B 2007, 76, 471–478. Ansari, R.; Ajori, S.; Motevalli, B. Mechanical properties of defective single-layered graphene sheets via molecular dynamics simulation. Superlattices Microstruct. 2012, 51, 274–289. [CrossRef] Lee, C.; Wei, X.; Kysar, J.W.; Hone, J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 2008, 321, 385–388. [CrossRef] [PubMed] Omeltchenko, A.; Yu, J. Crack front propagation and fracture in a graphite sheet: A molecular-dynamics study on parallel computers. Phys. Rev. Lett. 1997, 78, 2148–2151. [CrossRef] Zhang, B.; Yang, G.; Xu, H. Instability of supersonic crack in graphene. Phys. Rev. B Condens. Matter 2014, 434, 145–148. [CrossRef] Lin, Q.Y.; Zeng, Y.H.; Liu, D.; Jing, G.Y.; Liao, Z.M.; Yu, D. Step-by-Step Fracture of Two-Layer Stacked Graphene Membranes. ACS Nano 2014, 8, 10246–10251. [CrossRef] © 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).