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Research of Materials Science December 2013, Volume 2, Issue 4,
PP.50-57
Mechanical Properties of Graphene with
Vacancy Defects Yulin Yang
Mathematics and Physics Department, Xiamen University of
Technology, Xiamen, Fujian 361024, China
Email: [email protected]
Abstract
Defects are generally believed to degrade the mechanical
robustness and reduce the strength of graphene sheet. In this work
we
investigated the mechanical properties of monolayer graphene
sheet with randomly distributed vacancy defects. Molecular
dynamics simulations are carried out to elucidate the
atomic-level structures and tensile and shear deformations are
applied.
Ultimate strengths and fracture strains are calculated and the
effect of defect ratio is analyzed. Interestingly, super-ductility
is
observed in the high defect ratio situation. The obtained
results as demonstrated here provide new insights in understanding
the
mechanical performance of graphene based nano-materials where
defects are indispensible.
Keywords: Graphene; Mechanical Properties; Tensile Test;
Molecular Dynamics Simulation
1 INTRODUCTION
The amazing mechanical behavior and properties of graphene-based
nanomaterials has attracted significant research
interests in recent years, due to their promising prospects in
versatile branches such as micromechanics,
microelectronics, and thermal applications [1-4]
. Graphene is known to have ultra-high stiffness and strength,
yet a
wide scatter have been observed in the mechanical properties.
Pristine graphene sheet was reported to have high
Young's modulus rivaling that of graphite (~1.0 TPa), and its
superior strength (90~100 GPa for tensile load and
50~60 GPa for shear load) arises from a combination of high
stiffness and unusual flexibility and resistance to
fracture [5-7]
. However, the second law of thermodynamics dictates the
presence of a certain amount of defects and
disorders in crystalline materials [8]
. Also, the imperfections of material manufacturing process,
device or composite
production [9,10]
, chemical treatment[11]
, particle irradiation [12, 13]
and mechanical loading[14]
can all induce defects.
In most of the application situations, these unavoidable defects
can affect their material performance, especially the
mechanical properties, with the impact more or less significant
for different defect types, defect concentrations and
charilities. For single-wall carbon nanotube (SWCNT), single
vacancy defect was reported to lead to about 35%
reduction in the fracture strain [15]
. Furthermore, Sammalkorpi et al. demonstrated that the strength
reduction caused
by single vacancy depends on whether bond reconstruction occurs
prior to loading. They found that vacancies in
SWCNT can reduce the tensile strength and fracture strain by 40%
and 50%, respectively, whereas influence the
Young's modulus by only a negligible percentage [16]
. The presence of multiple defects makes nanotubes even
weaker, which reasonably explains the difference between the
comparatively low SWCNT fracture strength (13-52
GPa[17]
) observed in experiments and the high theoretical predictions
(above 100 GPa)[3,18]
. Using molecular
simulations with modified Brenner's potential, Yang et al.
predicted the normalized strength of SWCNT reduced
from 185 GPa to less than 50 GPa for defect concentration
varying from zero to 7.5% [19]
. The Stone-Wales defect
on the armchair SWCNT serves to reduce the failure stress and
strain by 20% to 40%, whereas the SW defect on the
zigzag SWCNT has negligible effect on the mechanical performance
[20]
. The role of thermodynamically
unavoidable atomistic defects in the design of carbon nanotube
based space elevator megacable was investigated and
the strength was expected to be reduced by a factor of at least
~70% [21]
. For the graphene sheet, Dettori et al. are the
first to examined the effect of point defect on the mechanical
properties of graphene and found that the defect-
induced stress field is the basin of mutual interaction between
two nearby defects. The obtained Young's modulus
and Poisson ratio showed a decreasing trend with respect to
defect density for vacancy defects [22]
. Moreover, focus
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has been placed on line defects such as grain boundaries
[23]
, which are found to reduce the strength, but the effect is
more pronounced for some boundary angles [24]
and less pronounced for others [25]
. Further continuum mechanics
theoretical improvements revealed that the detailed arrangement
of defects plays the major role in increasing or
decreasing the strength with tilt angle [26]
.
While structural defects may deteriorate the performance of
graphene-based devices, the deviation from perfection
can also be utilized and be careful engineered to achieve new
functionalities. Therefore, a good understanding of the
material performance for graphene with defects is useful for
further improvement of graphene-based nanotechnology.
In this work we focus on vacancy defects and perform molecular
dynamics (MD) simulations to investigate the
mechanical behaviour and properties of graphene sheets with
different incipient defect ratios.
2 MODELS AND METHODS
2.1 Models
The size of the monolayer graphene sheet we considered is 100
100 . The incipient vacancy defects are randomly dispersed on the
graphene basal planes according to a prescribed defect ratio ,
which is defined as the
ratio of missing atoms versus total atoms on the entire pristine
sheet. Graphene with vacancy defects at incipient
defect ratio x% will be abbreviated as VD-x%.
2.2 Methods
MD simulations are performed using the massively parallelized
modelling code LAMMPS software package[27]
, and
the atomic interactions are described by the AIREBO
potential[28]
, where the cutoff parameter of the REBO part of
the potential was modified as 2.0 to avoid nonphysical high
force [29,30]
. Periodic boundary conditions are applied in
the in-plane directions. The Velocity-Verlet integration time
step is set as 0.1 fs. Structural optimizations are
performed using the Polak-Ribire version of the conjugated
gradient algorithm [31]
. The MD simulations are
performed with a background temperature of 300 K (Nose-Hoover
thermo bath coupling [32]
). After the equilibrium
states are achieved, uniaxial tensile tests are performed under
NPT ensemble to study the mechanical properties of
the defective graphene. The engineered strain rate is 0.001
ps-1
and the strain increment is applied every 1000 time
steps. Both armchair and zigzag orientations are
investigated.
The mechanical tests are implemented to derive the stress-strain
relations and the associated parameters, namely,
maximum strength c and fracture strain F. Youngs modulus is not
analyzed because we focus on the mechanical
properties of the defective graphene sheets under heavy loads.
The macroscopic stress is obtained by averaging the
atomic virial stress over all the atoms on the sheet [33]
. Noise is reduced by averaging the results over the latter half
of
the relaxation period. The volume of graphene sheet is computed
by multiplying the in-plane area of the simulation
model with a thickness of 3.35 , which is the generally accepted
van der Waals interlayer interaction distance. Our
simulation methods are validated by calculating the maximum
strengths and fracture strains of a 100 100
pristine graphene sheet. The obtained parameters agree well with
the experimental measurements as well as other
theoretical reports as listed in Table 1.
TABLE 1 MECHANICAL PROPERTIES OF PRISTINE GRAPHENE SHEET. THE
RESULTS FROM THE PREVIOUS EXPERIMENTAL
MEASUREMENTS AND THEORETICAL REPORTS ARE ALSO LISTED FOR
COMPARISON
c (GPa) F Remarks
Arm Zig Arm Zig
Our work 91.4 107.5 0.136 0.203 MD
[5] 130 10 Nano-indenting
[34] 90 107 0.13 0.20 MD
[17] ~60 SWCNT(experiments)
3 RESULTS AND DISCUSSIONS
In this section, we will investigate the mechanical properties
of defective graphene sheets under tensile and shear
deformations. Both armchair and zigzag orientations are
discussed. Focus will be placed on the mechanical response
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under heavy loads. Furthermore, to depress the possible
fluctuation created by randomness in the arrangements of
the defects, we create 20 independent samples for each defect
ratio and perform the associated MD simulations
accordingly.
3.1 Structural deformations at the equilibrium state
0 2 4 6 8 100.00
0.01
0.02
0.03
MS
D (
An
gs
.)
Defect Ratio (%)
MSD
(a) @0K
0 2 4 6 8 10 120.3
0.6
0.9
1.2
1.5
1.8
h (
An
gs
.)
Defect Ratio (%)
(b) @300K
h
h
FIG. 1 (A) MEAN SQUARE DISPLACEMENT AS A FUNCTION OF DEFECT
RATIO FOR GRAPHENE SHEETS WITH VACANCY
DEFECTS AT ZERO TEMPERATURE AFTER EQUILIBRATION. THE INSET SHOW
THE SNAPSHOT OF GRAPHENE WITH DEFECTS,
WHERE THE DEFECTS ARE HIGHLIGHTED IN BLUE. (B) AVERAGED
OUT-OF-PLANE FLUCTUATION (RIPPLE HEIGHT h , AS
SHOWN IN THE INSET) OF DEFECTIVE GRAPHENE AT 300 K AFTER
EQUILIBRATION, AS A FUNCTION OF DEFECT RATIO.
We first analyze the structural deformations of graphene sheets
with defects. The defective graphene sheets are
created from pristine graphene membranes by removing atoms.
After structural relaxation at zero temperature, the
atoms on the sheet tend to re-arrange their local positions to
balance the spatial stress. From the mean square
displacement (MSD) of the defective graphene before and after
relaxation as presented in Fig. 1(a), one can see that
MSD increases gradually with the increasing , which can be
understood from the more disturbed atomic positions
in the larger case. However, the MSD decreases when is greater
than 8%, which indicates that in the high case
the intensive local deformation can lead to unusual atomic
interactions and therefore interesting mechanical
properties. To further analyse the intrinsic ripple structures,
the graphene sheets are equilibrated at 300 K, and the
averaged out-of-plane fluctuations h are calculated. The
amplitude of h obeys the relation h L with 0.6 0.8
for graphene. Assuming 0.6 , our estimation of the ratio of /h L
is 0.032 for pristine graphene sheet, in good
agreement with 0.035 reported in ref. [35]. Topological defects
in graphene are found to be energetically favorable to
deform out-of-plane and increase the ripple height [36]
. From Fig. 1(b) one can see that h increase dramatically
with
increasing defect ratio. Because dense vacancy defects can lead
to reduced inter-atomic confinement among adjacent
carbon atoms, thus the higher ripple amplitude in the high range
can be understood.
3.2 General Mechanical Responses
0.00 0.03 0.06 0.09 0.120
2
4
6
8
10
E
TO
T (
X1
09 J
/m3) Graphene
VD-0.05%
VD-8.5%
Tensile Strain
0.00 0.05 0.10 0.150
30
60
90
Str
es
s (
GP
a)
Tensile Strain
(a) Armchair
0.00 0.04 0.08 0.12 0.16 0.200
4
8
12
16
Graphene
VD-0.05%
VD-8.5%
E
TO
T (
X1
09 J
/m3)
Tensile Strain
0.00 0.05 0.10 0.15 0.200
30
60
90(b) Zigzag
Str
ess
(G
Pa
)
Tensile Strain
FIG.2 TOTAL ENERGY INCREMENTS PER UNIT VOLUME OF GRAPHENE SHEETS
UNDER TENSILE DEFORMATIONS ALONG THE
ARMCHAIR (A) AND ZIGZAG (B) DIRECTIONS. RESULTS OF PRISTINE
GRAPHENE, GRAPHENE SHEET WITH LOW AND HIGH
DEFECT RATIO ( =0.05% AND 8.5%) ARE PRESENTED. THE UPPER-LEFT
INSET FIGURES ARE THE CORRESPONDING STRESS-
STRAIN RELATIONS. THE MIDDLE-LEFT INSET SNAPSHOTS SHOW THE
LOADING DIRECTIONS.
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We next characterize the general mechanical response for the
defective graphene under tensile deformations. To
have a close inspect of the load-deformation rules, the total
energy method is implemented, since the increment of
the total energy should be equal to the external work. Fig.2
illustrates the total energy increment per unit volume for
pristine graphene, graphene with vacancy defects in low (single
defect) and high deformed along armchair/
zigzag directions, and the insets show the stress-strain
relations. With single defect presented, both total energy
increment rules and stress-strain relations well reproduce those
of pristine graphene but characterize much earlier
fracture points, indicating the defect-activated weakening of
the system, which will decrease the stiffness and
strength of the nanomaterial. This kind of brittle fracture has
been observed in graphene sheet with single defect [37]
or nanocrystalline grains [38]
. While similar, the responses are much complicated for high
defect ratio situations
(=8.5%). Starting with a much slower increasing rate in stress
versus strain, small reductions are occasionally
observed, leading to a serrated curve as shown in the inset
figures of Fig. 3. Failure along the weakest path is not
immediately catastrophic. The stress-strain relations exhibit
multiple stress peaks and an overall multiple fracturing
behaviour. The material becomes weaker but more ductile, with
reduced ultimate strengths and enlarged fracture
strains. The small drops in stress-strain relations are believed
to originate from geometric rearrangement on the sheet
(to dissipate the accumulated loads). This kind of fracturing
had also been observed in extended graphynes, where
secondary fracture occurs due to the mobility of the acetylene
linkages [39]
. The effect of multiple defects on strength
depends on the residual dangling bonds induced by vacancy
defects, which weakens the bond structure significantly
and enhances the mobility of carbon atoms strongly. The multiple
stress peaks pose difficulty in determining the
fracture point. However, the total energy of the system
increases with increasing strain before the final sharp drop.
Therefore, we define the fracture point as the highest energy
point. This phenomenon also indicates that for defective
graphene under large strain, although the strength remains
almost unchanged, the potential energy of the system can
still be increased.
3.3 Mechanical properties under tensile deformations
0 2 4 6 8 10 120.06
0.09
0.12
0.15
0.18
0.21
Fra
ctu
re S
train
Defect Ratio (%)
Armchair(c)
0 2 4 6 8 10 12
30
45
60
75
90
105
(b)
Maxim
um
Str
en
gth
(G
Pa)
Defect Ratio (%)
Zigzag
0 2 4 6 8 10 12
30
45
60
75
90
105
(a)
Maxim
um
Str
en
gth
(G
Pa)
Armchair
Defect Ratio (%)
0 2 4 6 8 10 120.06
0.09
0.12
0.15
0.18
0.21
Fra
ctu
re S
train
Defect Ratio (%)
Zigzag(d)
FIG.3 VARIATIONS OF THE MAXIMUM STRENGTH (A,B) AND FRACTURE
STRAIN F (C,D) WITH RESPECT TO FOR DEFECTIVE
GRAPHENE SHEETS UNDER TENSILE TESTS ALONG THE ARMCHAIR (A,C) AND
ZIGZAG (B,D) DIRECTIONS.
We now turn to analyze the mechanical properties at fracture
point for defective graphene. It should be noted that the
ultimate strength is the maximum stress in the stress-strain
curves, while the fracture strain is determined from the
spontaneous large drop of the total energy increment curves.
Fig. 3 displays the ultimate strength and fracture strain
for defective graphene with respect to defect ratio under
tensile tests, with both armchair and zigzag charilities
been considered. Without defect the ultimate tensile strength is
91.4 GPa and 107.5 GPa for armchair and zigzag
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graphene, respectively. With single vacancy defect presented,
the maximum strength is significantly degraded (arm
is 77.8 0.7GPa and zig is 86.0 1.3 GPa). The difference between
armchair and zigzag charility is narrowed from
16.1 GPa to 8.2 GPa. Similar narrowing of ultimate strength
difference between armchair and zigzag charilities has
also been observed in CNTs with single SW defect [40]
. With multiple defects appear the maximum strength
decreases gradually and saturates at higher defect ratiorange (
>7%). The strength difference between armchair and
zigzag sheet is further narrowed in high circumstances, which is
less than 2.0 GPa when is greater than 6%
(Fig.4). The strength for the armchair graphene decreases slower
than that of zigzag graphene, indicating the knock-
down effect of defects on maximum strength is more pronounced
for zigzag tests and less pronounced for armchair
tests. Similar descending-saturating trend had been observed in
exploring the failure strength of SWCNTs with
respect to defect ratio under tensile tests [41]
, wherein the normalized strength was reduced from 180 GPa to 50
GPa
with a vacancy concentration of ~ 7.6%. Also, this phenomenon
reasonably explains the difference between the
comparatively low SWCNT fracture strength (13-52 GPa [17]
) observed in experiments and the high theoretical
predictions (above 100 GPa).
Besides ultimate strength, the fracture strain is another
important parameter characterizing the mechanical properties
under heavy load. Fracture strain of defective graphene are
found to exhibit an unusual degrading-saturating-
improving trend with increasing , with the same rule holds for
both armchair and zigzag charilities (Fig.3 (c,d)).
Similar to the ultimate strength, with single VC defect presents
the fracture of the sheet is initiated much earlier. Farm
is reduced from 0.136 to 0.109 0.001, and Fzig is reduced from
0.203 to 0.124 0.003. The difference between
Farm and Fzig is significantly narrowed. The fracture strain has
been reduced by about 21%~39%, the same level as
that reported for single-wall carbon nanotube with single
vacancy defect(~35% reduction in the fracture strain[15]
).
Within all the investigated situations, F is found to decreases
gradually for low defect ratio and enlarges after further
increasing of defect ratio (Fig. 3 (c,d)). The unusual
enlargement of F in high range shows that although the
strength of the material is reduced, the ductility is greatly
improved within this range. This kind of improvement
shows a super-ductile behaviour in the defective graphene with
dense vacancy defects. Overall, for a given number
of defect ratios, the armchair configuration has less strength
and lower fracture strain, but more certainty compared
with the zigzag sheets. Similar trend has also been reported
previously for carbon nanotubes with randomly
occurring Stone-Wales defects [42]
.
0.00 2 4 6 8 10
30
45
60
75
90
105
highly defective graphene,
tiny difference
graphene with single vacancy defect,
small difference between arm
c and
zig
c
Maxim
um
Str
en
gth
(G
Pa)
Defect Ratio (%)
Armchair
Zigzag
pristine graphene,
large difference between arm
c0 and
zig
c0
FIG.4 MAXIMUM STRENGTH DIFFERENCE BETWEEN ARMCHAIR AND ZIGZAG
CHIRALITIES FOR MONOLAYER GRAPHENE SHEET
WITH VACANCY DEFECTS, AS A FUNCTION OF DEFECT RATIO.
To determine the statistical distribution of the uncertain
maximum strength and fracture strain, we fit the simulation
data by Weibull distributions. The Weibull statistics are known
to well characterize the material behaviour when
failure is governed by the weakest link (as in our situation)
and had been well demonstrated to estimate the strength
of CNTs both theoretically [42]
and experimentally [43]
. The Weibull strength distribution can be expressed as [44]
0
( ) 1 exp
m
F
(1)
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Where m is the Weibull modulus and 0 is the scaling parameter.
For sufficiently large m, the relative width of the
strength distribution decreases and 0 approximates the ensemble
average strength. From a series of pre-measured
strength i, the parameter m can be determined by maximum
likelihood method from the following equation [44]
11
1
ln 1 1ln
mNi i Ni
iimNii
m N
(2)
By iterative searching from Eq. (2) we obtained m for the
defective graphene under tensile deformations, as shown in
Fig. 5. We can see that the Weibull modulus presents a
decreasing trend with the increase of defect ratio and the
lowest m lies in the VD-8.5% defective graphene deformed along
zigzag orientation, indicating the more uncertainty
for the obtained fracture strength within this circumstance.
Generally speaking, the obtained Weibull modulus
confirms that the obtained maximum strength and fracture strain
lie within the confidence interval. Thus the
conclusions as drew above are reliable.
0 2 4 6 8 10 124
6
8
10
12
14
We
ibu
ll m
od
ulu
s
Defect Ratio (%)
Armchair
Zigzag
FIG. 5 WEIBULL MODULUS (SHAPE PARAMETER) M OF DEFECTIVE GRAPHENE
SHEETS, AS A FUNCTION OF DEFECT RATIO.
4 CONCLUSIONS
In summary, through molecular dynamics simulations we have
demonstrated the effect of multiple vacancy defects
on the mechanical properties of monolayer graphene sheets. The
maximum strength is observed to degrade with
increasing defect ratio and converges to a finite value when the
ratio is high. However, the fracture strain is observed
to decrease in the low defect ratio range and increases in the
high ratio range. For randomly and uniformly
distributed multiple defects, the fracture initiated at quite
random locations and the crack grew irregularly, super
ductility is observed in the high defect ratio range. For a
given number of defect coverage, the armchair
configuration has lower strength and smaller fracture strain,
but more certainty compared with the zigzag ones. The
difference between armchair and zigzag chiralities is
significantly narrowed with the presence of single defect and
nearly disappeared when the defect ratio is high. Our study as
demonstrated here provides valuable insights in
understanding the mechanical properties of graphene based
nanomaterials where defects are unavoidable.
ACKNOWLEDGMENT
This work was financially supported by Fujian Education Bureau
(No. GA11020).
REFERENCES
[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, et al. Electric
Field Effect in Atomically Thin Carbon Films Science, 2004,
306,
666
[2] C. D. Reddy, S. Rajendran and K. M. Liew. Equilibrium
configuration and continuum elastic properties of finite sized
graphene.
Nanotechnology, 2006, 17, 864
[3] K. Wakabayashi, M. Fujita, H. Ajiki, et al. Electronic and
magnetic properties of nanographite ribbons. Phys. Rev. B:
Condens.
Matter, 1999, 59, 8271
[4] Y. Z. He, H. Li, P. C. Si, et al. Dynamic ripples in single
layer graphene. Appl. Phys. Lett., 2011, 98, 063101
-
- 56 -
http://www.ivypub.org/rms/
[5] C. Lee, X. Wei, J. W. Kysar, et al. Measurement of the
Elastic Properties and Intrinsic Strength of Monolayer Graphene.
Science,
2008, 321, 385-8
[6] Y. Zheng, N. Wei, Z. Fan, et al. Mechanical properties of
grafold: a demonstration of strengthened graphene.
Nanotechnology,
2011, 22, 405701
[7] K. Min and N. R. Aluru. Mechanical properties of graphene
under shear deformation. Appl. Phys. Lett., 2011, 98, 013113
[8] F. Banhart, J. Kotakoski, and A. V. Krasheninnikov.
Structural defects in graphene. ACS Nano, 2011, 5, 26-41
[9] R. Andrews, D. Jacques, D. Qian, and E. C. Dickey.
Purification and structural annealing of multiwalled carbon
nanotubes at
graphitization temperatures. Carbon, 2011, 39,1681
[10] D. B. Mawhinney, V. Naumenko, A. Kuznetsova, et al. Surface
defect site density on single walled carbon nanotubes by
Titration.
Chem. Phys. Lett., 2000, 324, 213
[11] N. Pierard, A. Fonseca, Z. Konya, et al. Production of
short carbon nanotubes with open tips by ball milling. Chem. Phys.
Lett.,
2001, 335, 1
[12] G. Compagnini, F. Giannazzo, S. Sonde, et al. Ion
irradiation and defect formation in single layer graphene. Carbon,
2009, 47(14),
3201-3207
[13] B. Ni and S. B. Sinnott. Chemical functionalization of
carbon nanotubes through energetic radical collisions. Phys. Rev.
B, 2000,
61, R16343
[14] K. M. Liew, X. Q. He, and C. H. Wong. On the study of
elastic and plastic properties of multi-walled carbon nanotubes
under
axial tension using molecular dynamics simulation. Acta Mater.,
2004, 52, 2521-7
[15] Q.Wang, W. H. Duan, N. L. Richards, et al. Modeling of
fracture of carbon nanotubes with vacancy defect. Phys. Rev. B,
2007,75,201405(R)
[16] M. F. Yu, B. S. Files, S. Arepalli, et al. Tensile loading
of ropes of single wall carbon nanotubes and their mechanical
properties.
Phys. Rev. Lett., 2000, 84, 5552
[17] M. Sammalkorpi, A. Krasheninnikov, A. Kuronen, et al.
Mechanical properties of carbon nanotubes with vacancies and
related
defects. Phys Rev B, 70, 245416, 2004
[18] T.H. Liu, C. W. Pao and C. C. Chang. Effects of dislocation
densities and distributions on graphene grain boundary failure
strengths from atomistic simulations. Carbon, 2012, 50,
3465-3472
[19] M.Yang, V. Koutsos and M. Zaiser. Size effect in the
tensile fracture of single-walled carbon nanotubes with
defects.
Nanotechnology, 2007, 18, 155708
[20] K. I. Tserpes, and P. Papanikos. The effect of StoneWales
defect on the tensile behavior and fracture of single-walled
carbon
nanotubes. Composite Structures, 2007, 79(4), 581-589
[21] N. M. Pugno. The role of defects in the design of space
elevator cable: From nanotube to megatube. Acta Mater., 2007, 55,
5269-
5279
[22] R. Dettori, E. Cadelano and L. Colombo. Elastic fields and
moduli in defected graphene. J. Phys.: Condens. Matter, 2012,
24,
104020
[23] P. Y. Huang, C. S. Ruiz-Vargas, A. M. van der Zande, et al.
Grains and grain boundaries in single-layer graphene atomic
patchwork quilts. Nature, 2011, 469,389-393
[24] C. S. Ruiz-Vargas, H. L. Zhuang, P. Y. Huang, et al.
Softened Elastic Response and Unzipping in Chemical Vapor
Deposition
Graphene Membranes. Nano Lett., 2011, 11, 2259-2263
[25] R. Grantab, V. B. Shenoy, and R. S. Ruoff. Anomalous
Strength Characteristics of Tilt Grain Boundaries in Graphene.
Science,
2010, 330, 946-948
[26] Y. Wei, J. Wu, H. Yin, et al. The nature of strength
enhancement and weakening by pentagon-heptagon defects in
graphene.
Nature Mater., 2012, 11, 759-763
[27] S. Plimpton. Fast Parallel Algorithms for Short-Range
Molecular Dynamics. J. Comput. Phys., 1995, 117, 1-19
[28] W. B. Donald, O. A. Shenderova, J. A. Harrison,et al. A
second-generation reactive empirical bond order (REBO) potential
energy
expression for hydrocarbons. J Phys.: Condens. Matter, 2002, 14,
783
[29] O. A. Shenderova, D. W. Brenner, A. Omeltchenko, et al.
Atomistic modeling of the fracture of polycrystalline diamond.
Phys.
Rev. B, 2000, 61, 3877
[30] Y. P. Zheng, L. Q. Xu, Z. Y. Fan, et al. A molecular
dynamics investigation of the mechanical properties of graphene
Nanochains.
J. Mater. Chem., 2012, 22, 9798
[31] E. Polak ed. Optimization: Algorithms and Consistent
Approximations. 1997 (New York: Springer)
-
- 57 -
http://www.ivypub.org/rms/
[32] W. G.Hoover. Canonical dynamics: Equilibrium phase-space
distributions. Phys. Rev. A, 1985, 31, 1695
[33] L. Q. Xu, N. Wei, X. M. Xu et al. Defect-activated
self-assembly of multilayered graphene paper: a mechanically
robust
architecture with high strength. J. Mater. Chem. A, 2013, 1,
2002
[34] H. Zhao, K. Min and N. R. Aluru. Size and Chirality
Dependent Elastic Properties of Graphene Nanoribbons under
Uniaxial
Tension. Nano Lett., 2009, 9, 3012-5
[35] A. Fasoline, J. H. Los, M. I. Katsnelson. Intrinsic ripples
in graphene. Nat. Mater., 2007, 6 (11), 858-861
[36] O. V. Yazyev, and S. G. Louie. Topological defects in
graphene: dislocations and grain boundaries. Phys. Rev. B, 2010,
81,
195420
[37] C. Baykasoglu, and A. Mugan. Nonlinear analysis of
single-layer graphene sheets. Engineering Fracture Mechanics, 2012,
96,
241-250
[38] A. Cao, and J. Qu. Atomistic simulation study of brittle
failure in nanocrystalline graphene under uniaxial tension. Appl.
Phys.
Lett., 2013, 102, 071902
[39] S. W. Cranford, D. B. Brommer, and M. J. Buehler. Extended
graphynes: simple scaling laws for stiffness, strength and
fracture.
Nanoscale, 2012,4,7797-7809
[40] J. R. Xiao, J. Staniszewski and J. W. Gillespie Jr.
Fracture and progressive failure of defective graphene sheets and
carbon
nanotubes. Composite Structures, 2009, 88, 602-609
[41] M. Yang, V. Koutsos, and M. Zaiser. Size effect in the
tensile fracture of single-walled carbon nanotubes with
defects.
Nanotechnology, 2007, 18, 155708
[42] Q. Lu and B. Bhattacharya. Effect of randomly occurring
Stone-Wales defects on mechanical properties of carbon nanotubes
using
atomistic simulation. Nanotechnology, 2005, 16, 555-566
[43] A. Barber, I. Kaplan-ashiri, S. R. Cohen, et al. Stochastic
strength of nanotubes: an appraisal of available data. Compos.
Sci.
Technol., 2005, 65, 2380-4
[44] C. Lu, Danzer R and F. D. Fischer. Fracture statistics of
brittle materials: Weibull or normal distribution. Phys. Rev. E,
2005, 65,
067102
AUTHOR
Yulin Yang, male, was born in 1980. He obtained the Master
degree from Xiamen University in the field of theoretical
physics.
Currently his research interests including computational physics
and material science. Email: [email protected]