get_002.pdf

Journal of Mechanical Science and Technology 28 (9) (2014) 3645~3652

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-014-0826-7

Mode I fracture toughness analysis of a single-layer grapheme sheet

Minh-Nguyen Ky and Young-Jin Yum*

School of Mechanical Engineering, University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan, 680-749, Korea

(Manuscript Received November 5, 2013; Revised May 21, 2014; Accepted May 23, 2014)

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Abstract To predict the fracture toughness of a single-layer graphene sheet (SLGS), analytical formulations were devised for the hexagonal

honeycomb lattice using a linkage equivalent discrete frame structure. Broken bonds were identified by a sharp increase in the position of the atoms. As crack propagation progressed, the crack tip position and crack path were updated from broken bonds in the molecular dy-namics (MD) model. At each step in the simulation, the atomic model was centered on the crack tip to adaptively follow its path. A new formula was derived analytically from the deformation and bending mechanism of solid-state carbon-carbon bonds so as to describe the mode I fracture of SLGS. The fracture toughness of single-layer graphene is governed by a competition between bond breaking and bond rotation at a crack tip. K-field based displacements were applied on the boundary of the micromechanical model, and FEM results were obtained and compared with theoretical findings. The critical stress intensity factor for a graphene sheet was found to be

2.63 ~ 3.2 mICK MPa= for the case of a zigzag crack.

Keywords: Critical stress intensity factor; Fracture of graphene; Fracture toughness; Stress intensity factor ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

A single-layer graphene sheet (SLGS) is a one-atom-thick, two-dimensional layer of sp2-bonded carbon that is densely packed so as to form a honeycomb crystal lattice [1]. Youngs modulus and the thermal conductivity of graphene rival those measured for graphite (1.06 TPa and 3000 Wm-1K-1, respec-tively) [2, 3]. Graphene is the strongest known material in the world, and while its density is only 2.2 g/cm3, it has been ob-served that graphene is harder than diamond and about 100 times stronger than the best steels in the world.

Fracture toughness, a critically important property for de-sign applications, describes the ability of a material to resist fracture when a crack is present. A coupled quan-tum/continuum mechanics study of graphene fracture was previously conducted to examine fracture in nanomaterials by coupling quantum mechanics (QM) and continuum mechanics (CM) [4]. In the method, the continuum domain encompasses the entire system, while quantum calculations are restricted to a subdomain in the vicinity of the crack tip. Molecular dynam-ics simulations have been performed to investigate the me-chanical properties of hydrogen functionalized graphene for H-coverage spanning the entire range from graphene (H-0%) to graphane (H-100%). It was found that the Youngs

modulus, tensile strength, and fracture strain of functionalized graphene deteriorate drastically with increasing H-coverage up to about 30% [5]. Large-scale molecular dynamics calcula-tions based on a reactive bond-order potential reveal that, for a certain graphite sheet orientation where some of the C-C bonds are parallel to the applied strain, the system undergoes cleavage fracture [6]. The modeling and simulations outlined in previous work [7] highlight the atomistic mechanisms for the nonlinear mechanical behavior of graphenenanoribbons with edge effects, which is potentially important for develop-ing integrated graphene-based devices. Approaches to model-ing fracture at the atomic scale range from the conceptually simple but computationally challenging fully atomistic ap-proach, to a variety of mixed atomistic/continuum models that include atomistic effects through either an approximate scheme or by embedding an atomistic crack-tip region in a bulk continuum [8]. The atomistic structures as well as the continuum model of graphene sheet with central crack were established [9] and the corresponding near tip stress field and strain energy release rate were evaluated from the finite ele-ment analysis. Earlier findings [10] show that the strain energy release rates obtained from the continuum model are in good agreement with those derived from a discrete atomistic model. Therefore, the strain energy release rate is an appropriate pa-rameter that can be employed in both the atomistic and con-tinuum models to describe the fracture of a covalently bonded graphene sheet. Methods of computing the J-integral in an

*Corresponding author. Tel.: +82 522592132, Fax.: +82 522591680 E-mail address: [email protected]

Recommended by Editor Maenghyo Cho KSME & Springer 2014

3646 M.-N. Ky and Y.-J. Yum / Journal of Mechanical Science and Technology 28 (9) (2014) 3645~3652

atomic system have been successfully developed [11]. Simula-tions reveal that, under small-strain deformation, the values of the J-integral agree well with the energy release rates. An equivalent continuum sheet model for the mechanical behav-ior of SLGS under external loading has also been proposed [12]. For this purpose, the Young's modulus, shear modulus, and Poisson's ratio of the continuum model are calculated through an atomistic analysis of SLGS behavior. Previous attempts by Gibson et al. to model the fracture toughness of a brittle honeycomb structure have yielded well-known results [13, 14]. Lim et al. [15] investigated the fracture of brittle cellular materials, in which the maximum stress on the strut surface is evaluated, and they found that it is necessary to consider not only the bending moment but also axial and shear forces. The atomistic model consists of equivalent structural beams, where the beam properties are expressed in terms of the covalent bond stiffnesses so as to simulate the interatomic forces of the SLGS carbon atoms under imposed loadings. The mechanisms of deformation and fracture of graphene sheets under uniaxial tension were studied on the nanoscale, while strain characteristics were determined using microme-chanical relations [16]. Research on the nonlinear mechanical properties of graphenenanoribbons [17] showed that the nominal strain to fracture is considerably lower for armchair graphenenanoribbons than for zigzag ribbons. Macroscopic fracture parameters were investigated for 2D graphene sys-tems containing atomic-scale cracks. The atomic stress distri-butions matched quite well with those obtained from linear elastic solutions. Jack et al. [18] carried out systematic studies to ascertain the influence of vacancy patterns in graphene on the dynamics of crack propagation. Recently, new results and models on the fracture toughness of an open-cell were pre-sented [19, 20], and the findings were used to calculate the fracture toughness of foams. Terdalkar et al. [21] proposed that the fracture of a monolayer of graphene is governed by the competition between bond breaking and bond rotation at a crack tip. While the researchers have repeated most of the simulations using the Tersoff-Brenner potential, K-field analy-sis has not yet been used to examine the behavior of cracks in graphene.

Our main objective was to investigate the fracture tough-ness of SLGS. Quasi-MD simulations were first used in a systematic analysis to examine the stress intensity factors con-trolling the elongation process of SLGS. We then developed a theoretical framework to model the fracture toughness ( ICK ) of a 2D graphene sheet for both zigzag and armchair cases using the same approach outlined by Gibson and Ashby [14]. From simple mechanics, the maximum stress was calculated in each bond near the crack tip in SLGS and thus, the entire network could be determined. This in turn allowed for the formulation of expressions pertaining to bond breaking and bond rotation at a crack tip. The validity of the devised theo-retical model was confirmed by studying several open-cells with short cracks.

2. Atomistic simulation

MD simulations were performed to investigate the me-chanical properties of hydrogen functionalized SLGS. Using the large-scale atomic molecular massively parallel simulator (LAMMPS) [22], a classical molecular dynamics code, atom-istic simulations were conducted to investigate the propaga-tion crack growth rate and the evolution of the associated atomic positions near the crack tip during the development of crack growth in SLGS. Interatomic bonds of atoms were de-scribed using the adaptive intermolecular reactive bond order (AIREBO) potential [23], the parameters of which were set as suggested to terminate the unphysical high bond force arising from artificial switching functions [6]. This potential allows for covalent bond breaking and creation with associated changes in the hybridization of atomic orbitals within a classi-cal potential, thereby enabling simulations of bond formation, bond breaking, and failure of functionalized graphene.

The graphene structure consists of carbon layers where the carbon atoms are arranged in a hexagonal pattern. The geome-try of SLGS was proposed in AMBER [12] with a bond angle of 120o between carbon atoms. A reasonable domain size was chosen so that its outer boundary falls in the K-dominant zone, which can make all-atom simulations computationally effi-cient. Our simulation method was validated by calculations involving 4032 carbon atoms with dimensions of around 100x100, as shown in Fig. 1.

To simulate the crack growth process as accurately as pos-sible, the displacement associated with crack-opening (mode I) is applied to the boundary of a pristine specimen. Specifi-cally, we used a displacement field [24] given by:

(1)

where m is the elastic shear modulus of the material, E (= 1.05

Fig. 1. Geometry of a graphene sheet with a center crack. The dimen-sion of the graphene sheet with center crack where W (width) and H(height); (O) represents carbon atoms.

M.-N. Ky and Y.-J. Yum / Journal of Mechanical Science and Technology 28 (9) (2014) 3645~3652 3647

TPa) is the elastic modulus, 31

nkn

-=+

for plane stress,n (=

0.186) is Poissons ratio, and KI is the mode I stress intensity factor.

In the simulations, the displacement is applied to every atom by scaling the atomic positions via the application of boundary conditions along the appropriate coordinates. Asymmetric cleavage in Fig. 1 for a zigzag crack under pure tension demonstrates that brittle fracture via bond breaking prevails at room temperature. To obtain the fracture behavior, the tensile loading in terms of the incremental strain rate is applied along one of the coordinate directions. Loading in the y-direction means that zigzag graphene is undergoing tensile deformation.

Cracks are labeled according to the surfaces adjacent to their tips. For example, a zigzag crack has zigzag surfaces near its tip. As a bond breaks, the unstable geometry is associ-ated with a sudden decrease in area and an abrupt increase in the length of the bond at the crack tip. Fig. 2 shows the atomic positions in a system with a zigzag crack at several different load steps. With an available small enough increment of load-ing *IK = 0.005 within 60 steps and performed by step by step, until fracture occurs, the first bond is broken abruptly between two atoms at crack tip that are most stretched, subse-quently accompanied with disruption of adjacent bonds exhib-iting a dynamic fracture process.

To visualize the evolution of the atomic structure, visual molecular dynamics (VMD) [25, 26] open source software is employed. One of the drawbacks of MD is the a priori esti-

mate of atomic interactions, the interactive forces between various atoms. This interaction is generally represented in the form of an interatomic potential energy model, meaning it is imperative that the interatomic potential is accurately quanti-fied.

The stress intensity factor ( )IK plays a key role in the field of fracture mechanics, especially in the area of linear elastic fracture mechanics. The underlying assumption of lin-ear elastic fracture mechanics is that the growth of a crack is controlled by the stress field at the crack tip. Thus, it follows that crack growth is characterized by IK , the stress intensity factor in opening mode.

It was found that a load step number of 52 corresponds to an abrupt change in the positions of atoms.

The value of the normalized stress intensity factor is

(2)

Consequently, the critical stress intensity factor for gra-

phene was found to be 3.2 MPa m .ICK =

3. Equivalent honeycomb analysis for the graphene

sheet model

3.1 Fracture toughness behavior of a graphene sheet

If a graphene sheet is loaded to a point near its fracture stress, one cell wall will fail. This in turn causes the stress on neighboring walls to increase until they also fail. The failed cluster is like a crack, where the stress concentration at the periphery causes further walls to fail as the crack propagates across the section. The problem is best approached using the methods of fracture mechanics. Consider a brittle graphene sheet containing a crack cluster of broken cells. When the sheet is loaded in tension, the cell walls will at first bend elas-tically. The load is transmitted through the graphene sheets as a set of discrete forces and moments acting on each of the cell walls. This condition defines the fracture toughness of the grapheme sheets ( )ICK , which we will calculate. The method requires a number of assumptions. First, if the graphene sheet is to be treated as a continuum, the crack length must be large relative to the cell size. Second, the contribution of axial forces in the cell walls to the internal stress ahead of the crack tip is not neglected. Third, the cell wall material has a constant modulus of rupture, .fsd Crack deflection is the process by which an initial crack tilts and twists when it encounters a rigid inclusion. This in turn causes an increase in the total fracture surface area.

In the present section, we explore the tensile fracture re-sponse of a graphene sheet (Fig. 3). The fracture behavior of a brittle graphene sheet was examined through the use of linear elastic fracture mechanics (LEFM) concepts to estimate the fracture toughness of the hexagonal lattice. The stress field of an equivalent linear elastic continuum was employed to calcu-late the stresses on the cell walls of the lattice directly ahead of

Fig. 2. Atomic position at different load steps; the red line indicates a crack.


the crack tip. A crack with a length of 2c in an elastic solid lying normal

to a remote tensile stress s1 creates a singular local stress field, sl [13]:

(3)

where r is the distance ahead of the crack tip (half of the width of the cell) and s1 is the remote stress perpendicular to the plane of the crack.

3.2 The geometry and deflections for loading in the arm-

chair direction (X1)

In tension, the failure process is different. Upon the failure of a cell wall in tension, the load is transferred to adjacent walls. Failure of adjacent walls then ensues, leading to the formation a macroscopic crack, an example of which is shown Fig. 4.

A bending moment exists in the vertical beam ahead of the crack due to the gradient in stress ahead of the crack tip. This in turn results in a larger force pulling on the first column

above the end of the crack than that acting on the second and third columns. The moments in the two angled beams do not cancel out, and a dual moment rotates the vertical beam ahead of the crack tip. This moment creates a tensile stress on the crack side of the vertical cell wall.

The bending moment on a cell wall just ahead of the crack tip may be expressed as follows:

Where 1M and 2M are the bending moments exerted by the force on the wall.

If we assume that 1 2 ,M M? then,

The resulting force on the cell wall is:

(4)

where d is the thickness (diameter) of the bonded graphene sheet.

The above approach can be used for crack propagation when a stress is applied in the armchair (X1) direction. The stress on the cell wall is:

where M is the moment about the neutral axis, y is the perpen-dicular distance from the neutral axis, and I is the second mo-ment of inertia area about the neutral axis.

We subsequently obtain:

(5)

Fig. 3. Graphene sheet with a crack.

Fig. 4. The local stress field supplemented by the actual tensile stress in the cell wall.

Fig. 5. Forces in the armchair cell wall to the internal stress ahead of the crack tip.


Using cosr l q= and Eq. (3), we have:

Substitution into Eq. (4) yields:

From Eq. (5), we have:

With graphene sheets of 30q =

If fracture occurs when the stress exceeds the fracture

strength of the cell wall material, the following expression is obtained:

It is helpful to interpret this result using the terminology of

fracture mechanics. Tensile fracture will occur when the frac-ture toughness is reached:

If we re-write the above equation in terms of the fracture

toughness in the armchair direction (X1), then we have:

(6)

3.3 The geometry and deflections for loading in the zigzag direction (X2)

A bending moment in the inclined beam exists ahead of the crack.

The bending moment of the beam may be expressed as:

If it is assumed that, on average, the crack occupies half the

width of the unit cell, then:

(7)

The stress in the zigzag (X2) direction may be given as:

(8)


(a)

(b)

Fig. 6. Crack propagation leading to brittle tensile failure in a graphene sheet: (a) the geometry and deflections for loading in the zigzag direc-tion (X2); (b) the local stress field supplemented by the actual tensile stress in the cell wall.

Fig. 7. Forces and moments in the zigzag cell walls.


where sin .2

l lr q+=

From Eq. (7):


With a graphene sheet of 30q =

If fracture occurs when the stress exceeds the fracture

strength of the cell wall material,

If we rewrite the above equation in terms of the fracture

toughness in the zigzag direction(X2):

(9)

4. Finite element analysis for graphene sheets

For the finite element analysis, a grid of 100x60 cells was employed (Fig. 8). Here, ANSYS 12.1 was used to model the struts with the zigzag model. The stress distribution along the outermost layer of cells is uniform and the stress field far away from the crack tip is undisturbed. The analysis gives the forces and moments at both ends of each strut from which the critical skin stress of the first unbroken cell edge ahead of the crack tip can be calculated. Failure occurs when the critical skin stress reaches the modulus of rupture of the cell wall, which is assumed to be constant at this point in the Ref. [14]. The thickness ( d ) and length ( l ) of a graphene sheet vary depending on the study, and different values have been re-ported by Duplock [2], Zhou [27], Tu and Ou-Yang [28], Pan-tano [29], Kudin [30], Goupalov [31], and Reddy [32]. We used AMBER force constants [12], where the thickness of the C-C bond is 0.84 and Poissons ratio is 0.186 for an equilib-rium length l of 1.38 .

Twenty-five different crack lengths (c = 1,2,3,25 cells) are used in the present work. However, an experimental value for the constant modulus of rupture (sfs) of SLGS was not available, so we calculated sfs from Eq. (9) and obtained a value of 29.11MPa.

Fracture occurs when the internal stress in the cell wall ahead of the crack tip reaches sfs of the cell wall, i.e., at an applied stress of .fss s= The fracture toughness is then cal-culated as

To verify the accuracy of the modeling procedure and for-

mulation developed in this study, we constructed a finite ele-ment model of a graphene sheet in ANSYS [33]. Since the fundamental aspect of the developed formulation relies on the axial and bending stresses of a carbon-carbon bond, a built-in two-dimensional beam element is employed for the finite ele-ment analysis on the basis of the same assumptions used in the analytical solution. The geometrical properties of the beam elements are derived using AMBER and subsequently fed into the model as input data.

Gibson and Ashby [13] provided an analytical formula for the fracture toughness of open-cell foam. For a typical honey-comb with a central crack, the expression for the mode I frac-ture toughness is found to be [34]:

where C1 is a microstructure coefficient that was numerically found to be 0.18 by Huang and Chiang, fss is the modulus of rupture of the solid cell walls, and *r and sr are the densities of the honeycomband solid material from which they are made, respectively.

Ultimately, the fracture toughness obtained from MD simu-lations for the zigzag direction case was 17% higher than the FEM result and 21% smaller than the Ashby and Gibson

Fig. 8. Loading geometry for the finite element analysis of SLGS hav-ing a central crack with a length of 2c.


value. The good agreement observed between the MD simula-tion and numerical results indicates that micromechanics can be a powerful tool in predicting the fracture behavior of SLGS and other cellular solids.

From the MD simulation, the stress intensity factor IK that will cause failure of the crack-tip elements was deter-mined; this value was taken as the fracture toughness of the SLGS shown in Table 1.

5. Conclusions

The results of a theoretical study on the prediction of the SLGS fracture toughness under uniaxial tension were pre-sented. The fracture toughness of a graphene sheet with a cen-ter crack was characterized using atomistic simulations and an analytical model. A new MD method was also developed to investigate the fracture of graphene sheets. The fracture toughness values predicted by our devised models are in rea-sonable agreement with those from molecular dynamics simu-lations based on the AIREBO potential. The analytical models

proposed in this study allow for a determination of the equiva-lent homogenized mechanical properties of a single graphene sheet with different deformation and bending mechanisms of the C-C bonds. We showed that the fracture of a single gra-phene sheet involves both bond rotation and bond breaking. The fracture equation can be accurately evaluated from the local stress field in the immediate vicinity of the crack tip. Moreover, the local stress distributions around the crack tip were studied and calculated by applying remote K-field de-formations. A comparison of atomistic simulation and finite element results revealed that the fracture toughness obtained from the continuum model differs by17% with that attained from the discrete model associated with the same boundary condition. As a result, the concept of the strain energy release rate is regarded as a physical quantity that can establish con-nections between atomistic simulations and analytical models for modeling the fracture of a covalently bonded graphene sheet. Our results suggest that the fracture toughness of SLGS is independent of the crack length, and the predicted toughness of grapheme is 3.2 MPa mICK = .

Acknowledgment

This work was supported by the 2013 Research Fund of University of Ulsan.

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Table 1. Comparison of mode I fracture toughness values.

No. Authors Fracture toughness of SLGS (MPa m )

1 FEM (present work) 2.63 2 MD Theory (present work) 3.2

3 Mei Xu [4] 3.71

4 Ashby and Gibson [34] 4.04 5 Nicola M. Pugnoy [35] 3.21

6 Shi Weichen [36] 2.6454 (AC)

7 Bin Zhang [37] 3.38

Fig. 9. Comparison of the mode I SLGS fracture toughness values obtained in this work with those reported in previous research.


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Minh-Nguyen Ky received his B.Sc. degree from NhaTrang University and M.Sc. degree from University of Tech-nology in Ho Chi Minh, Vietnam, both degrees in manufacturing engineering. Dr. Kys Ph.D. is from University of Ulsan and his research interests include manufacturing and fracture mechanics.

Young-Jin Yum received M.S. and Ph.D. degrees in Aeronautical Engineer-ing from Korea Advanced Institute of Science and Technology, Korea in 1981 and 1989. He is a professor of the School of Mechanical Engineering at the University of Ulsan, Korea.

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