We conducted molecular dynamics simulations to investigate the atomistic edge crack vacancy interactions in graphene. We demonstrate that the crack tip stress field of an existing crack in graphene can be effectively tailored (reduced by over 50% or increased by over 70%) by the strategic placement of atomic vacancies of varied shapes, locations, and orientations near its tip. The crack vacancy interactions result in a remarkable improvement (over 65%) in the fracture strength of graphene. Moreover, at reduced stiffness of graphene, due to a distribution of atomic vacancies, a drastic difference (~60%) was observed between the fracture strengths of two principal crack configurations (i.e. armchair and zigzag). Our numerical simulations provide a remarkable insight into the applicability of the well-established continuum models of crack microdefect interactions for the corresponding atomic scale problems. Furthermore, we demonstrate that the presence of atomic vacancies in close proximity to the crack tip leads to a multiple stage crack growth and, more interestingly, the propagating cracks can be completely healed even under a significantly high applied tensile stress level (~5 GPa). Our numerical experiments offer a substantial contribution to the existing literature on the fracture behavior of two dimensional nanomaterials.

1. Tailoring Fracture Strength of Graphene
M. A. N. Dewapriya and S. A. Meguid
Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial
Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada
Abstract
We conducted molecular dynamics simulations to investigate the atomistic edge crack-vacancy
interactions in graphene. We demonstrate that the crack-tip stress field of an existing crack in
graphene can be effectively tailored (reduced by over 50% or increased by over 70%) by the
strategic placement of atomic vacancies of varied shapes, locations, and orientations near its tip.
The crack-vacancy interactions result in a remarkable improvement (over 65%) in the fracture
strength of graphene. Moreover, at reduced stiffness of graphene, due to a distribution of atomic
vacancies, a drastic difference (~60%) was observed between the fracture strengths of two
principal crack configurations (i.e. armchair and zigzag). Our numerical simulations provide a
remarkable insight into the applicability of the well-established continuum models of
crack-microdefect interactions for the corresponding atomic scale problems. Furthermore, we
demonstrate that the presence of atomic vacancies in close proximity to the crack-tip leads to a
multiple-stage crack growth and, more interestingly, the propagating cracks can be completely
healed even under a significantly high applied tensile stress level (~5 GPa). Our numerical
experiments offer a substantial contribution to the existing literature on the fracture behavior of
two-dimensional nanomaterials.
Keywords: Crack-tip stress field, edge crack, fracture of graphene, crack healing, crack-vacancy
interaction, tailoring of fracture properties

2. 2
1. Introduction
The extraordinary stiffness, fracture strength, and flexibility of graphene [1], a two-dimensional
(2D) hexagonal lattice of sp2
bonded carbon, demonstrate great potentials in a rich variety of
applications such as flexible electronics [2–4], nanoelectromechanical systems [5,6], and
nanocomposites [7–9]. Even though an atomically perfect graphene sample has an extraordinary
fracture strength of 130 GPa [1], pristine graphene samples of large area are yet to be developed
for the proposed applications. Even with a pristine graphene sample, atomic flaws such as
vacancies and adatoms are difficult to avoid in real engineering fabrication of graphene-based
nanodevices [5,10–15]. A recent experiment revealed that the fracture toughness of graphene is
~4 MPa√m [16], which is significantly lower than conventional engineering materials [17]. This
relatively small fracture toughness poses serious limitations in the use of graphene for numerous
structural applications [18]. Therefore, it is important to study the effect of the presence of and
interactions between atomistic cracks and defects upon the fracture behavior of graphene. In
addition, investigating potential toughening mechanisms for graphene is highly desirable in its
commercialization for real life engineering applications.
In continuum fracture mechanics, it has been well established that the interaction between
crack and a micro defect in close proximity plays an important role in the overall failure
mechanism of quasi-brittle materials [19–25]. It has also been demonstrated that the crack-tip
stress field in a linear elastic continuum can be controlled by strategically placing a microdefect
near the crack-tip [26,27]. Recent nanoindentation tests of graphene containing vacancies have
revealed that the catastrophic failure of graphene can be transformed into a local failure by
controlling its defect concentration [28]. This observation suggests that existing cracks in
graphene can be arrested by introducing a controlled distribution of topological defects, which
has been realized for some other nanomaterials [29,30].
Most of the recent efforts have been focused on enhancing the fracture strength of
graphene by introducing topological defects such as pentagon-heptagons [31–35] and grain
boundaries [36] without paying attention to the complex stress state surrounding crack-defect
interactions, which could unveil rich atomistic mechanisms of 2D materials. In fact, many
existing molecular dynamics (MD) studies on fracture properties of graphene have focused on
the central crack problem partly due to the convenience of using periodic boundary conditions in

3. 3
MD. In addition, advanced continuum-based design tools for characterizing crack-defect
interaction [19–27] have not been tested at the atomic scale, which is critically important
considering the limited applicability of continuum concepts at the nanoscale [37–43]. If
applicable, the continuum tools such as design envelopes to ascertain crack-tip stress shielding
and amplification zones due to the crack-defect interactions [26,27] can be very useful for
nanoscale design of 2D materials [44].
In this work, we focus our attention on exploring the influence of interactions between
existing edge nanocracks in graphene and arbitrarily located atomic vacancies at the crack-tip.
Our atomistic simulations demonstrate that vacancies of varied shapes, locations, and
orientations can be effectively used to shield the existing cracks from high stress levels leading
to crack arrest. Moreover, our numerical uniaxial tensile tests reveal that the crack-vacancy
interactions in graphene lead to a multiple-stage crack growth and crack healing under
remarkably high applied tensile stress levels (~5 GPa), providing a new route for strain
engineering of graphene-based devices.
2. Molecular dynamics simulations
We conducted numerical uniaxial tensile tests of graphene samples using LAMMPS MD
simulator [45] with AIREBO potential [46], where the cutoff distance was modified to be 2 Å
[32,34,47,48]. At the beginning of all MD simulations, energy of the graphene samples was
minimized using the conjugate gradient algorithm. Then, before applying strain, the samples
were allowed to relax over 25 ps and the time step was selected to be 0.5 fs. Canonical (NVT)
ensemble was used for the simulations, where temperature was maintained at 300 K using
Nośe-Hoover thermostat. An initial random displacement perturbation (maximum of 0.01 Å)
along x, y, and z directions was imposed on carbon atoms to facilitate reaching the equilibrium
configuration. In fact, inducing out-of-plane displacement perturbations is essential in these
simulations because the Nośe-Hoover thermostat induces artificial thermal expansion in the
absence of out-of-plane deformation [49]. After the graphene samples reached equilibrium, strain
was applied to the samples along the y-direction (εyy) at a rate of 0.001 ps-1
. The planar
dimensions of the simulated samples were selected to be 60 nm × 60 nm, and the length of edge
crack was 10 nm. The selected simulation domain ensures that its boundaries do not fall within
the K-dominant region which is at and near the crack-tip region, where the stress field decays

4. 4
quite rapidly. Virial stress [50] was used for fracture characterization. Visual Molecular
Dynamics package [51] was used to visualize deformations and fracture of graphene.
Figure 1 shows a typical MD simulation sample of graphene containing an atomic vacancy
interacting with an edge crack. The origins of the two rectangular coordinate systems xy and x0y0
are taken at the tip of the edge crack and at the center of the vacancy, respectively. The
orientation angle of the vacancy is ϕ, and its major and minor diameters are 2c and 2b,
respectively. The distance between the tip of the crack and the center of the vacancy is taken to
be r. The inclination angle between the x-axis and the line joining the tip of the crack and the
center of the vacancy is θ. The value of 2c is selected to be 3.6 nm (see figure 1).
Figure 1 A typical MD simulation sample of graphene containing an edge crack and an
interacting atomic vacancy.
3. Results and discussion
3.1 Crack tip stress distribution
The stress distribution of individual carbon atoms at the crack-tip can be very informative in
characterizing the fracture behavior of graphene. In order to obtain the time averaged stress of
atoms at the incipient crack propagation, we conducted two sequential MD simulations as

5. 5
outlined in one of our earlier publications [52]. Figures 2a and 2b show the stress distributions at
the armchair and zigzag crack-tips, respectively. These stress distributions resemble the ones
predicted by the continuum linear elastic fracture mechanics (LEFM) [40]. The peak stress at the
tip of the armchair crack at the incipient crack propagation is 17% higher than the corresponding
stress of the zigzag crack. This is due to the high far field (or applied) stress σ0 level of the
armchair crack configuration and, more importantly, the different bond arrangements at the
crack-tips. In contrast to the isolated bond perpendicular to the crack at the zigzag crack-tip (see
inset of figure 2a), the two inclined bonds at the armchair crack-tip (inset of figure 2b)
accommodate part of the applied tensile strain by adjusting the bond angles, which allows the
atom at the crack-tip to carry a higher strain prior crack growth leading to a higher atomic stress.
According to LEFM, the critical stress intensity factor (SIF) of a single edge-cracked sample
under mode-I loading (where the crack surfaces are displaced directly apart) KIC can be defined
as follows [17]:
aK πσfIC 12.1= (1)
where a is the initial crack length, and σf is the fracture stress, i.e. the critical far field stress at
the incipient crack propagation. The computed KIC for armchair and zigzag cracks are 4.04 and
3.97 MPa√m, respectively, which are in excellent agreement with the experimentally measured
value 4 MPa√m [16].
Earlier, Gong and Meguid studied the interaction between a semi-infinite crack and an
elliptical vacancy located near its tip (see figure 1) under mode I loading [26]. In the absence of
the vacancy, the singular stress field near the crack-tip can be described by using the
corresponding SIF aK πσ0I 12.1= . However, the presence of the elliptical vacancy in close
proximity to the crack tip influences the crack-tip stress field and leads to a modified SIF; we
will call v)-(c
IK . When a collinear elliptical vacancy is located ahead of the crack, i.e. θ = 0 and
ϕ = 0, the solution for the normalized SIF under mode I loading I
v)-(c
I KK can be explicitly
expressed up to the order (c/r)4
as follows [26]:

6. 6
( ) ( ) ...49124623
128
1
1
4
1
1
4
432
22
2
2
I
v)-(c
I
+⎟
⎠
⎞
⎜
⎝
⎛
−++⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
+=
r
c
r
c
K
K
ββββ (2)
where β is b/c. For the case of a circular vacancy (i.e. β = 1), equation (2) reduces to
...
4
1
2
1
1
42
I
v)-(c
I
+⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+=
r
c
r
c
K
K
(3)
Considering the leading order solution up to order (c/r)2
, a general solution for any combination
of r, θ, and ϕ can be given as,
( ) ( )C
r
c
G
r
c
K
K 2
2
2
2
I
v)-(c
I
1
2
1
8
1 ββ +⎟
⎠
⎞
⎜
⎝
⎛
+−⎟
⎠
⎞
⎜
⎝
⎛
+= (4)
where C and G are explicitly expressed as follows:
(5a)
(5b)
In order to characterize the crack-vacancy interaction, we use normalized crack-tip stress
c
tip
v)-(c
tip σσ , where v)-(c
tipσ is the crack-tip stress along the y-direction in the presence of an
interacting vacancy and c
tipσ is the crack-tip stress in the absence of any interacting vacancy. The
normalized crack-tip stress c
tip
v)-(c
tip σσ was computed at an applied tensile strain εyy level of 1% for
various arrangements of the interacting vacancies. The values of c
tipσ for the armchair and zigzag
cracks are 63.9 and 56 GPa, respectively. Figures 2c-2g demonstrate that the presence of
vacancies greatly influences the stress field of zigzag crack compared to that of the armchair
crack, as a result of their underlying crystal structures at the crack-tips. It can be seen in Figures
2c and 2f that the collinear (i.e. θ = ϕ = 0) nanocracks and circular vacancies result in an increase
in the crack-tip stress field (known as stress amplification effect, i.e. c
tip
v)-(c
tip σσ > 1). The oriented
vacancies (see figures 2d and 2g) with the orientation angle θ > 60°, result in a decrease in the
crack-tip stress field (known as stress shielding effect, i.e. c
tip
v)-(c
tip σσ < 1). The stress shielding
C = cos
3θ
2
⎛
⎝
⎜
⎞
⎠
⎟cos
θ
2
⎛
⎝
⎜
⎞
⎠
⎟
G = 2cos 2φ + θ( ) + 4cos 2φ − θ( ) + 8cos 2φ − 2θ( ) − 6cos 2φ − 3θ( )
− 8cos 2φ − 4θ( ) − 3cos 3θ( ) + 3cos θ( )

7. 7
and stress amplification effects are due to the fact that the presence of vacancies leads to a
change in the density of the stress trajectories at the crack-tip. Increased and decreased densities
of the stress trajectories correspond to the amplification and shielding effects, respectively. The
maximum shielding effect that can be achieved by means of the studied arrangements of
vacancies is 55%, which is in the case of the zigzag crack interacting with a circular vacancy
located at r = 2.8 nm and θ = 118°. Figure 2e shows that the amplification effect due to a
nanocrack can be successfully tailored (~20%) by regulating the obliquity of the nanocrack.
More remarkably, Figures 2c-2g show that the continuum-based analytical solutions given in
equation (3) and (4) are able to accurately capture the trends of the crack-tip stress fields
obtained from the atomistic simulations. Here, we compare I
v)-(c
I KK given by the analytical
solutions with c
tip
v)-(c
tip σσ obtained from the atomistic simulations. The normalized crack-tip
stress c
tip
v)-(c
tip σσ is a comparable quantity to the corresponding normalized stress intensity factors
[17]. It should be noted that the higher order terms of equation (3) and (4) have been neglected,
which limits their accuracy. As demonstrated in the Supporting Information (figure S1), the
accuracy improves when (c/r) becomes smaller. However, the continuum expressions are unable
to predict the influence of the underlying crystal structures (i.e. armchair and zigzag) on the
crack-tip stress field, which sets a limit on developing a unified continuum fracture mechanics
framework at the atomic scale. However, due to the remarkable accuracy and the extremely high
computational efficiency, the analytical solutions can be employed to create detailed design
envelopes to ascertain the crack-tip shielding and amplification zones as a result of the presence
of atomic vacancies ahead of the crack-tip in graphene (see figure S2 in the Supporting
Information).

8. 8
Figure 2 The effect of atomic vacancies on the crack-tip stress field. Figures (a) and (b) show
averaged stress σyy distributions at the tips of armchair and zigzag cracks at the incipient crack
propagation, respectively. Figures (c-g) show variation of the normalized crack-tip stress
c
tip
v)-(c
tip σσ with r, θ, and ϕ. Figures (c), (d), and (e) are for the collinear (θ = ϕ = 0), oriented (r is
fixed and ϕ = 0), and oblique (r and θ are fixed) nanocracks, respectively. Figures (f) and (g) are
for the collinear and oriented circular vacancies, respectively. Insets in Figures (c-g) show the
stress σyy distribution at an armchair crack-tip due to an applied tensile strain εyy of 1% in the
presence of an atomic vacancy at the specified location.

9. 9
3.2. Crack arrest
We further characterized the atomistic crack-vacancy interaction by computing the critical SIF
for the crack-vacancy arrangements considered in figure 2. We defined the normalized critical
SIF IC
v)-(c
C KK , where v)-(c
CK is the critical mode I SIF of a sample containing an edge crack
interacting with a vacancy, which is defined as
aK πσ v)-(c
f
v)-(c
C 12.1= (6)
where v)-(c
fσ is the fracture stress of the sample. It should be noted that equation (6) is analogous
to equation (1), and v)-(c
fσ obtained from the MD simulations takes into account the influence of
atomic vacancies.
In spite of the fact that vacancies have a greater influence on the crack-tip stress field of the
zigzag crack than the armchair crack (see Figures 2c-2g), Figures 3a-3e show that the vacancies
have a comparatively higher influence on v)-(c
CK of armchair crack compared to the zigzag crack.
This observation is due to the fact that the vacancies are mostly located along the propagation
path of the armchair crack, which results in a crack-vacancy coalescence leading to possible
crack arrest. Conversely, the vacancies are not aligned along the propagation path of the zigzag
crack except for the collinear vacancies. Figures 3d and 3e show that the presence of circular
vacancies at both the armchair and zigzag cracks leading to a remarkable crack arrest, i.e.
IC
v)-(c
C KK > 1, for almost all of the studied vacancy arrangements. Specifically, our results reveal
that the presence of strategically located holes in close proximity to the crack tip can enhance the
fracture strength of armchair and zigzag cracks by 56% and 67%, respectively. Moreover, a
two-stage crack propagation is observed in both the zigzag and armchair cracks in the presence
of atomic vacancies as shown in Figures 3f and 3g, respectively.

10. 10
Figure 3 Fracture characterization of the crack-vacancy systems considered. Figures (a-e) depict
the variation of the normalized critical SIF with r, θ, and ϕ. Figures (a), (b), and (c) are for the
collinear, oriented, and oblique nanocracks, respectively. Figures (f) and (g) are for the collinear
and oriented circular vacancies, respectively. Insets in Figures (a-e) depict the relative position of
the corresponding atomic vacancy with respect to the crack-tip. Figures (f) and (g) show the
stress σyy distribution and the crack propagation of graphene containing armchair and zigzag
cracks, respectively.

11. 11
3.3. Crack healing
Self-healing of cracks and other defects in graphene [53–55] provide a potential route for
developing higher strength and greater resistance to fracture. Our numerical experiments reveal
that the partially propagated cracks (see Figures 3f and 3g) can heal completely though it is
under applied tensile stresses that can be as high as 5 GPa. Figure 4a shows the variation of the
spatially averaged stress of the process zone (an area of 2 nm × 2 nm ahead of the crack-tip
containing 170 atoms) during the loading and unloading of a graphene sample containing an
armchair crack and an interacting circular vacancy. The loading and unloading were carried out a
strain rate of 0.001 ps-1
. The sudden decrease in the stiffness at the end of region I is due to a
partial crack propagation, and it continues to propagate in the region II (see Figures 4c and 4d).
The unloading was initiated at time T = 33 ps. The stiffness of the sample in region III is reduced
by 23% compared to the stiffness in region I due to the partial crack propagation. Crack healing
initiates at the beginning of region IV, where the number of broken bonds decreases from six to
five (see Figures 4e and 4f), which slightly increases the effective stiffness of the process zone. It
should be noted that the initial crack healing occurs under σ0 of 10.5 GPa, and the partially
propagated crack has completely healed when σ0 is reduced to 5 GPa during the subsequent
unloading (see figure 4g). As shown in figure S3 of Supporting Information, crack healing was
observed even under continuous unloading.

12. 12
Figure 4 Healing of a partially propagating armchair crack in the presence of an interacting
circular vacancy located at r = 2.9 nm and θ = 88°. (a) Variation of the spatially averaged stress
σyy of the process zone during the loading and unloading of graphene. The unloading was paused
at time T = 54 ps and then the sample was equilibrated for 15 ps (region IV). The far field stress
σ0 during the equilibration was 10.5 GPa. Subsequent unloading was carried out for 6 ps
(region V) and the sample was then equilibrated for another 15 ps (region VI). Figures (b-d)
depict snapshots of the crack-tip during the loading stage. The arrows demonstrate the broken
bonds at the crack surface due to the partial crack propagation. Figures (e-g) depict the stress σyy
distributions at the corresponding points during the unloading stage.

13. 13
3.4. The effects of an extended distribution of vacancies
Finally, we evaluated the influence of an array of vacancies on the crack-tip stress field and the
failure mechanism of graphene. We considered four distributions of atomic vacancies arranged
in a 2D array of 10 nm × 10 nm with various spacing among the individual vacancies along x
and y directions (dx, dy). Figures 5a and 5b show two vacancy distributions corresponding to
armchair crack-tip. In order to obtain the stiffness of the area in the vicinity of the crack-tip
containing multiple atomic vacancies (i.e. the damaged zone), we conducted numerical uniaxial
tensile test of graphene samples with dimensions of 10 nm × 10 nm containing the corresponding
periodic distributions of vacancies (see figure S4 in Supporting Information for details). Table 1
gives the stiffness of graphene containing different distributions of vacancies representing the
damaged zone. Figure 5c shows linear variations of the normalized crack-tip stress c
tip
v)-(c
tip σσ
with the stiffness of the damaged zone as predicted by the continuum damage mechanics models
[23–25]. However, the normalized critical SIF IC
v)-(c
C KK in figure 5d shows completely different
relationships with the stiffness due to the unique atomistic crack propagation pattern involving
crack deflection, crack branching, and crack bridging as shown in figure 5e. More interestingly, a
dramatic difference (~60%) exists between the critical SIFs of armchair and zigzag cracks at
lower stiffness values (~0.7 TPa) and, in fact, the stiffness-SIF relationships of the armchair and
zigzag cracks are completely different. Figure 5e demonstrates that the catastrophic failure of
graphene [16,40,56] can be transformed into a number of sequential local failures as a result of
varied atomistic vacancy distributions, which agrees with a recent experiment [28]. This result
further demonstrates the rich potential of employing the combination of atomic vacancies and the
underlying crystal structure of graphene in controlling its fracture strength and crack propagation
pattern.

14. 14
Figure 5 The effect of an array of single vacancies at the crack-tip on the fracture characteristics
of graphene. Figures (a) and (b) show stress σyy distribution at an armchair crack-tip due to an
applied tensile strain εyy of 1%. Figures (c) and (d) show the variations of the normalized crack-
tip stress and the critical SIF with the stiffness of the process zone, respectively. Figure (e)
depicts the propagation of an armchair crack in the presence of an array of single vacancies.

15. 15
Table 1 Stiffness of graphene samples containing various distributions of single atomic
vacancies.
armchair zigzag
dx
(nm)
dy
(nm)
S
(TPa)
dx
(nm)
dy
(nm)
S
(TPa)
0.8 0.7 0.69 0.7 0.8 0.71
1.3 1.0 0.76 1.0 0.8 0.78
1.3 1.3 0.78 1.2 1.3 0.85
1.7 1.5 0.82 1.5 1.7 0.90
4. Conclusions
In summary, our atomistic simulations, complemented by a continuum-based analytical model,
shed light on previously overlooked aspect concerning the tailoring of the fracture properties of
graphene. Our results demonstrate that crack-vacancy interaction can be effectively used to
transform the crack-tip stress field, fracture strength, crack propagation pattern, and even crack
healing of graphene; thus, providing another dimension in the design space of graphene. In this
regards, the atomistic crack-vacancy interactions supersede the other recently studied methods of
improving crack arrest in graphene. For example, the atomic vacancies can lead to a reduction in
the crack-tip stress field by as much as 55% and ultimately increase the fracture strength by 67%.
In addition, our numerical experiments provide a remarkable insight into the conditions leading
to crack arrest and the potential toughening mechanisms of graphene due to the crack-vacancy
interactions. In addition, our numerical experiments elucidate the pronounced effects of the
underlying crystal structure on the atomistic fracture characteristics of 2D materials.
Supporting Information
Information on the accuracy of the analytical model, design envelops for the crack-vacancy
interactions, crack healing under continuous unloading, and stress-strain relations of the
damaged zones are provided.

16. 16
Acknowledgements
The authors wish to thank NSERC and the Discovery Accelerator Supplement for their kind
support of this research. Computing resources were provided by WestGrid and Compute/Calcul
Canada.
References
[1] Lee C, Wei X, Kysar J W and Hone J 2008 Measurement of the Elastic Properties and Intrinsic Strength of Monolayer
Graphene Science 321 385–8
[2] Eda G, Fanchini G and Chhowalla M 2008 Large-area ultrathin films of reduced graphene oxide as a transparent and
flexible electronic material Nat. Nanotechnol. 3 270–4
[3] Kim K S, Zhao Y, Jang H, Lee S Y, Kim J M, Kim K S, Ahn J-H, Kim P, Choi J-Y and Hong B H 2009 Large-scale
pattern growth of graphene films for stretchable transparent electrodes. Nature 457 706–710
[4] Jo G, Choe M, Lee S, Park W, Kahng Y H and Lee T 2012 The application of graphene as electrodes in electrical and
optical devices Nanotechnology 23 112001
[5] Eichler A, Moser J, Chaste J, Zdrojek M, Wilson-Rae I and Bachtold A 2011 Nonlinear damping in mechanical resonators
made from carbon nanotubes and graphene. Nat Nanotechnol 6 339–342
[6] Singh V, Sengupta S, Solanki H S, Dhall R, Allain A, Dhara S, Prita Pant and Deshmukh M M 2010 Probing thermal
expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators
Nanotechnology 21 165204
[7] Sainsbury T, Gnaniah S, Spencer S J, Mignuzzi S, Belsey N A, Paton K R and Satti A 2017 Extreme mechanical
reinforcement in graphene oxide based thin-film nanocomposites via covalently tailored nanofiller matrix compatibilization
Carbon 114 367–76
[8] Ramanathan T, Abdala A, Stankovich S, Dikin D, Herrera-Alonso M, Piner R, Adamson D, Schniepp H, Chen X, Ruoff R
and others 2008 Functionalized graphene sheets for polymer nanocomposites Nat. Nanotechnol. 3 327–331
[9] Kim Y, Lee J, Yeom M S, Shin J W, Kim H, Cui Y, Kysar J W, Hone J, Jung Y, Jeon S and Han S M 2013 Strengthening
effect of single-atomic-layer graphene in metal–graphene nanolayered composites Nat. Commun. 4
[10] Banhart F, Kotakoski J and Krasheninnikov A V 2011 Structural Defects in Graphene ACS Nano 5 26–41
[11] Chen C, Rosenblatt S, Bolotin K I, Kalb W, Kim P, Kymissis I, Stormer H L, Heinz T F and Hone J 2009 Performance of
monolayer graphene nanomechanical resonators with electrical readout Nat. Nanotechnol. 4 861–7
[12] Cançado L G, Jorio A, Ferreira E H M, Stavale F, Achete C A, Capaz R B, Moutinho M V O, Lombardo A, Kulmala T S
and Ferrari A C 2011 Quantifying Defects in Graphene via Raman Spectroscopy at Different Excitation Energies Nano
Lett. 11 3190–6
[13] Eckmann A, Felten A, Mishchenko A, Britnell L, Krupke R, Novoselov K S and Casiraghi C 2012 Probing the Nature of
Defects in Graphene by Raman Spectroscopy Nano Lett. 12 3925–30
[14] Vlassiouk I, Smirnov S, Ivanov I, Fulvio P F, Dai S, Meyer H, Chi M, Hensley D, Datskos P and Lavrik N V 2011
Electrical and thermal conductivity of low temperature CVD graphene: the effect of disorder Nanotechnology 22 275716
[15] Hwang J-Y, Kuo C-C, Chen L-C and Chen K-H 2010 Correlating defect density with carrier mobility in large-scaled
graphene films: Raman spectral signatures for the estimation of defect density Nanotechnology 21 465705
[16] Zhang P, Ma L, Fan F, Zeng Z, Peng C, Loya P E, Liu Z, Gong Y, Zhang J, Zhang X and others 2014 Fracture toughness
of graphene Nat. Commun. 5
[17] Meguid S A 1989 Engineering fracture mechanics (London ; New York : New York: Elsevier Applied Science ; Sole
distributor in the USA and Canada, Elsevier Science Pub)
[18] Zhang T, Li X and Gao H 2015 Fracture of graphene: a review Int. J. Fract. 1–31
[19] Thompson J Y, Anusavice K J, Balasubramaniam B and Mecholsky J J 1995 Effect of Micmcracking on the Fracture
Toughness and Fracture Surface Fractal Dimension of Lithia-Based Glass-Ceramics J. Am. Ceram. Soc. 78 3045–9
[20] Rose L R F 1986 Effective Fracture Toughness of Microcracked Materials J. Am. Ceram. Soc. 69 212–4

17. 17
[21] Wu C C, Freiman S W, Rice R W and Mecholsky J J 1978 Microstructural aspects of crack propagation in ceramics J.
Mater. Sci. 13 2659–70
[22] Rühle M, Evans A G, McMeeking R M, Charalambides P G and Hutchinson J W 1987 Microcrack toughening in
alumina/zirconia Acta Metall. 35 2701–10
[23] Evans A G and Fu Y 1985 Some effects of microcracks on the mechanical properties of brittle solids—II. Microcrack
toughening Acta Metall. 33 1525–31
[24] Ortiz M 1987 A Continuum Theory of Crack Shielding in Ceramics J. Appl. Mech. 54 54
[25] Hutchinson J W 1987 Crack tip shielding by micro-cracking in brittle solids Acta Metall. 35 1605–19
[26] Gong S X and Meguid S A 1992 Microdefect interacting with a main crack: A general treatment Int. J. Mech. Sci. 34 933–
45
[27] Meguid S A, Gong S X and Gaultier P E 1991 Main crack-microcrack interaction under mode I, II and III loadings:
Shielding and amplification Int. J. Mech. Sci. 33 351–9
[28] López-Polín G, Gómez-Herrero J and Gómez-Navarro C 2015 Confining Crack Propagation in Defective Graphene Nano
Lett. 15 2050–4
[29] Lu K, Lu L and Suresh S 2009 Strengthening Materials by Engineering Coherent Internal Boundaries at the Nanoscale
Science 324 349–52
[30] Wang S, Qin Z, Jung G S, Martin-Martinez F J, Zhang K, Buehler M J and Warner J H 2016 Atomically Sharp Crack Tips
in Monolayer MoS 2 and Their Enhanced Toughness by Vacancy Defects ACS Nano 10 9831–9
[31] Zhang T, Li X and Gao H 2014 Defects controlled wrinkling and topological design in graphene J. Mech. Phys. Solids 67
2–13
[32] Zhang T, Li X and Gao H 2014 Designing graphene structures with controlled distributions of topological defects: A case
study of toughness enhancement in graphene ruga Extreme Mech. Lett. 1 3–8
[33] Meng F, Chen C and Song J 2015 Dislocation Shielding of a Nanocrack in Graphene: Atomistic Simulations and
Continuum Modeling J. Phys. Chem. Lett. 6 4038–42
[34] Meng F, Chen C and Song J 2017 Lattice trapping and crack decohesion in graphene Carbon 116 33–9
[35] Rajasekaran G and Parashar A 2017 Enhancement of fracture toughness of graphene via crack bridging with stone-
thrower-wales defects Diam. Relat. Mater. 74 90–9
[36] Jung G, Qin Z and Buehler M J 2015 Molecular mechanics of polycrystalline graphene with enhanced fracture toughness
Extreme Mech. Lett. 2 52–9
[37] Luan B and Robbins M O 2005 The breakdown of continuum models for mechanical contacts Nature 435 929–32
[38] Tapasztó L, Dumitrică T, Kim S J, Nemes-Incze P, Hwang C and Biró L P 2012 Breakdown of continuum mechanics for
nanometre-wavelength rippling of graphene Nat. Phys. 8 739–42
[39] Zhang D-B, Akatyeva E and Dumitrică T 2011 Bending Ultrathin Graphene at the Margins of Continuum Mechanics
Phys. Rev. Lett. 106
[40] Yin H, Qi H J, Fan F, Zhu T, Wang B and Wei Y 2015 Griffith Criterion for Brittle Fracture in Graphene Nano Lett. 15
1918–24
[41] Dewapriya M A N and Rajapakse R K N D 2016 Development of a homogenous nonlinear spring model characterizing
the interfacial adhesion properties of graphene with surface defects Compos. Part B Eng. 98 339–49
[42] Dewapriya M A N, Rajapakse R K N D and Nigam N 2015 Influence of hydrogen functionalization on the fracture
strength of graphene and the interfacial properties of graphene-polymer nanocomposite Carbon 1 6991–7000
[43] Dewapriya M A N and Rajapakse R K N D 2014 Molecular Dynamics Simulations and Continuum Modeling of
Temperature and Strain Rate Dependent Fracture Strength of Graphene With Vacancy Defects J. Appl. Mech. 81 081010
[44] Akinwande D, Brennan C J, Bunch J S, Egberts P, Felts J R, Gao H, Huang R, Kim J-S, Li T, Li Y, Liechti K M, Lu N,
Park H S, Reed E J, Wang P, Yakobson B I, Zhang T, Zhang Y-W, Zhou Y and Zhu Y 2017 A review on mechanics and
mechanical properties of 2D materials—Graphene and beyond Extreme Mech. Lett. 13 42–77
[45] Plimpton S 1995 Fast parallel algorithms for short-range molecular dynamics J Comput Phys 117 1–19
[46] Stuart S J, Tutein A B and Harrison J A 2000 A reactive potential for hydrocarbons with intermolecular interactions J.
Chem. Phys. 112 6472
[47] Zhang T, Li X, Kadkhodaei S and Gao H 2012 Flaw Insensitive Fracture in Nanocrystalline Graphene Nano Lett. 12
4605–10
[48] Dilrukshi K G S, Dewapriya M A N and Puswewala U G A 2015 Size dependency and potential field influence on
deriving mechanical properties of carbon nanotubes using molecular dynamics Theor. Appl. Mech. Lett. 5 167–72

18. 18
[49] Dewapriya M A N, Phani A S and Rajapakse R K N D 2013 Influence of temperature and free edges on the mechanical
properties of graphene Model. Simul. Mater. Sci. Eng. 21 065017
[50] Tsai D H 1979 The virial theorem and stress calculation in molecular dynamics J. Chem. Phys. 70 1375–82
[51] Humphrey W, Dalke A and Schulten K 1996 VMD – Visual Molecular Dynamics J. Mol. Graph. 14 33–8
[52] Dewapriya M A N and Meguid S A 2017 Atomistic modeling of out-of-plane deformation of a propagating Griffith crack
in graphene Acta Mech.
[53] Yoon T, Kim J-H, Choi J H, Jung D Y, Park I-J, Choi S-Y, Cho N S, Lee J-I, Kwon Y-D, Cho S and Kim T-S 2016
Healing Graphene Defects Using Selective Electrochemical Deposition: Toward Flexible and Stretchable Devices ACS
Nano 10 1539–45
[54] Hong J, Lee J-B, Lee S, Seo J, Lee H, Park J Y, Ahn J-H, Il Seo T, Lee T and Lee H-B-R 2016 A facile method for the
selective decoration of graphene defects based on a galvanic displacement reaction NPG Asia Mater. 8 e262
[55] Aramfard M and Deng C 2016 Interaction of shear-coupled grain boundary motion with crack: Crack healing, grain
boundary decohesion, and sub-grain formation J. Appl. Phys. 119 085308
[56] Dewapriya M A N, Rajapakse R K N D and Phani A S 2014 Atomistic and continuum modelling of temperature-
dependent fracture of graphene Int. J. Fract. 187 199–212

19. 19
Supporting Information for
Tailoring Fracture Strength of Graphene
M. A. N. Dewapriya and S. A. Meguid
Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial
Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada
1. The effects of higher order terms on the analytical solution given by eq. 4
We computed the percentage error of the normalized stress intensity factor I
v)-(c
I KK (see
equation 4) when it is compared with the normalized crack-tip stress (c)
tip
v)-(c
tip σσ obtained from
MD simulations as follows:
100errorPercentage
)c(
tip
)vc(
tip
I
)vc(
I
)c(
tip
)vc(
tip
×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= −
−−
σ
σ
σ
σ
K
K
(s1)
As shown in figure S1a, the agreement between the analytical solution and the MD simulations is
excellent for smaller holes (i.e. for smaller c/r) demonstrating the influence of the higher order
terms on the accuracy of the analytical solution.
In order to improve the accuracy of the existing leading order solution in equation 4, a
correction factor (κ) can be introduced into the solution using a few representative values of
(c)
tip
v)-(c
tip σσ obtained from the MD simulations. As evidenced by figure S1a, the correction factor
depends on the orientation θ and the diameter of the hole d. For example, when d = 1.4 nm, the
correction factor κ can be computed using the four MD simulation results given in Figure S1a as
follows:

20. 20
05.101.122 2
I
)vc(
I
)c(
tip
)vc(
tip
++−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= −
−
αα
σ
σ
κ
K
K
(s2) where α = θ×10-3
, and the corrected normalized stress intensity factor can be expressed as
( )I
v)-(c
IcorrectedI
v)-(c
I KKKK κ= .
Figure S1 The effects of the hole diameter d on the accuracy of the leading order solutions given
in eq. 4. (a) Variation of the percentage error for three hole diameters, where the inset depicts the
relative position of the hole with respect to the crack-tip and r is kept fixed at ~3.2 nm. Figures
(b-d) show the stress σyy distribution at the crack-tip due to an applied tensile strain εyy of 1% for
the holes located at θ = 31°.

21. 21
2. Design envelops developed using the analytical solution given by equation 4
Figure S2 Design envelops depicting the influence of the atomic vacancies on the crack-tip
stress field of an armchair crack: for the (a) oriented holes and (b) oriented and oblique
nanocracks.

22. 22
3. Crack healing under continuous unloading
Figure S3a shows the variation of the averaged stress of the process zone (2 nm × 2 nm
containing 170 atoms) during the loading and the continuous unloading of a graphene sample
with an armchair edge crack and an interacting circular vacancy. Partial crack propagation occurs
at the end of the region I, and it continues to propagate as shown in Figures S3c and 3d. The
unloading was started at T = 33 ps. During the unloading, crack healing first occurs at T = 54 ps
and it continues through the region IV, however there is no significant increase in the effective
stiffness of the process zone due to the crack healing. Half of the initially propagated crack has
been healed when the applied stress is completely removed at T = 66 ps, and the crack is entirely
healed during the subsequent relaxation of the graphene sample as shown in figure S3g.

23. 23
Figure S3 Healing of the initially propagated armchair crack in the presence of a circular
vacancy of diameter 3.6 nm located at r = 2.9 nm and θ = 88°. (a) Variation of the spatially
averaged stress σyy of the process zone during the loading and unloading of the graphene sample.
Figures (b-g) show snapshots of the crack-tip at various growth stages. The arrows in Figures
(c-e) indicate the broken bonds at the crack surface due to the partial crack propagation. Figures
(f) and (g) show considerable crack healing with time.
4. Stress-strain relations of graphene samples containing an edge crack and a distribution
of single vacancies
Figure S4 Stress distribution of graphene samples. Figures (a) and (b) show the stress-strain
relations for three graphene samples with various vacancy configurations as demonstrated in
Figures (c-e). In addition, Figures (c-e) show stress σyy distribution of the graphene samples due

24. 24
to an applied tensile strain εyy of 1%. Size of the sample-1 is 10 nm × 10 nm and it is
60 nm × 60 nm for the sample-2 and -3.