Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets
A molecular structural mechanics approach is used to study the effect of double-atom vacancy defects (DAVD) on the elastic properties of zigzag and armchair single-layered graphene sheets (SLGS). To this end, the space frame structure is also adopted to model the interatomic forces of the C–C bonds....
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Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. 2020-12-19T15:38:30Z 2020-12-19T15:38:30Z 2017 Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets / Z.Q. Wang, Z.W. Yu, X.Y. Sun, H. Li, Y.J. Wang // Проблемы прочности. — 2017. — № 6. — С. 157-164. — Бібліогр.: 15 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173748 539.4 A molecular structural mechanics approach is used to study the effect of double-atom vacancy defects (DAVD) on the elastic properties of zigzag and armchair single-layered graphene sheets (SLGS). To this end, the space frame structure is also adopted to model the interatomic forces of the C–C bonds. The numerical simulation results obtained via the finite element method strongly suggest that double-atom vacancy defects reduce the elastic module of SLGS, which effect weakens with graphene size. Finally, the elastic modulus and Poisson’s ratio of SLGS are found to decrease with the number of DAVDS. В рамках подхода молекулярной структурной механики исследуется влияние двойных атомарных вакансионных дефектов кристаллической решетки на упругие свойства однослойных листов графена с зигзагообразной и плетеной структурой. Для моделирования межатомных сил связей типа углерод–углерод используется пространственная структурная сетка. Результаты численного моделирования, полученные методом конечных элементов, подтверждают, что наличие указанных вакансионных дефектов снижает модуль упругости графена, что приводит к уменьшению его несущей способности. Установлено, что увеличение количества вакансионных дефектов обусловливает снижение модуля упругости и коэффициента Пуассона однослойных листов графена. У рамках підходу молекулярної структурної механіки досліджується вплив подвійних атомарних вакансійних дефектів кристалічної гратки на пружні властивості одношарових листів графена із зигзагоподібною та плетеною структурою. Для моделювання міжатомних сил зв’язків типу вуглець–вуглець використовується просторова структурна сітка. Результати чисельного моделювання, отримані методом скінченних елементів, підтверджують, що наявність указаних вакансійних дефектів знижу є модуль пружності графена, що призводить до зменшення його несівної здатності. Установлено, що збільшення кількості вакансійних дефектів призводить до зниження модуля пружності та коефіцієнта Пуассона одношарових листів графена. This work was supported by the National Natural Science foundation of China (Nos. 11602066, 11472086, and 11532013) and the China Postdoctoral Science Foundation on the 56th bath of surface founds the project (2014M561327) and the National Science Foundation of Heilongjiang Province of China (QC2015058 and 42400621-1-15047), the Foundation Research Funds for the Central Universities. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets Влияние двойных атомарных вакансионных дефектов кристаллической решетки на упругие свойства однослойных листов графена Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets |
| spellingShingle |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. Научно-технический раздел |
| title_short |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets |
| title_full |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets |
| title_fullStr |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets |
| title_full_unstemmed |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets |
| title_sort |
effect of double-atom vacancy defects on the elastic properties of single-layered graphene sheets |
| author |
Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. |
| author_facet |
Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. |
| topic |
Научно-технический раздел |
| topic_facet |
Научно-технический раздел |
| publishDate |
2017 |
| language |
English |
| container_title |
Проблемы прочности |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| format |
Article |
| title_alt |
Влияние двойных атомарных вакансионных дефектов кристаллической решетки на упругие свойства однослойных листов графена |
| description |
A molecular structural mechanics approach is used to study the effect of double-atom vacancy defects (DAVD) on the elastic properties of zigzag and armchair single-layered graphene sheets (SLGS). To this end, the space frame structure is also adopted to model the interatomic forces of the C–C bonds. The numerical simulation results obtained via the finite element method strongly suggest that double-atom vacancy defects reduce the elastic module of SLGS, which effect weakens with graphene size. Finally, the elastic modulus and Poisson’s ratio of SLGS are found to decrease with the number of DAVDS.
В рамках подхода молекулярной структурной механики исследуется влияние двойных атомарных вакансионных дефектов кристаллической решетки на упругие свойства однослойных листов графена с зигзагообразной и плетеной структурой. Для моделирования межатомных сил связей типа углерод–углерод используется пространственная структурная сетка. Результаты численного моделирования, полученные методом конечных элементов, подтверждают, что наличие указанных вакансионных дефектов снижает модуль упругости графена, что приводит к уменьшению его несущей способности. Установлено, что увеличение количества вакансионных дефектов обусловливает снижение модуля упругости и коэффициента Пуассона однослойных листов графена.
У рамках підходу молекулярної структурної механіки досліджується вплив подвійних атомарних вакансійних дефектів кристалічної гратки на пружні властивості одношарових листів графена із зигзагоподібною та плетеною структурою. Для моделювання міжатомних сил зв’язків типу вуглець–вуглець використовується просторова структурна сітка. Результати чисельного моделювання, отримані методом скінченних елементів, підтверджують, що наявність указаних вакансійних дефектів знижу є модуль пружності графена, що призводить до зменшення його несівної здатності. Установлено, що збільшення кількості вакансійних дефектів призводить до зниження модуля пружності та коефіцієнта Пуассона одношарових листів графена.
|
| issn |
0556-171X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/173748 |
| citation_txt |
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-Layered Graphene Sheets / Z.Q. Wang, Z.W. Yu, X.Y. Sun, H. Li, Y.J. Wang // Проблемы прочности. — 2017. — № 6. — С. 157-164. — Бібліогр.: 15 назв. — англ. |
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2025-11-27T01:45:00Z |
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UDC 539.4
Effect of Double-Atom Vacancy Defects on the Elastic Properties of Single-
Layered Graphene Sheets
Z. Q. Wang, Z. W. Yu, X. Y. Sun,
1
H. Li, and Y. J. Wang
2
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, China
1 sunxiaoyu520634@163.com
2 906871278@qq.com
ÓÄÊ 539.4
Âëèÿíèå äâîéíûõ àòîìàðíûõ âàêàíñèîííûõ äåôåêòîâ êðèñòàëëè÷åñêîé
ðåøåòêè íà óïðóãèå ñâîéñòâà îäíîñëîéíûõ ëèñòîâ ãðàôåíà
Ç. Ê. Âàíã, Ç. Â. Þ, Ê. ß. Ñóí, Õ. Ëè, È. Æ. Âàíã
Êîëëåäæ àýðîêîñìè÷åñêîãî è ãðàæäàíñêîãî ñòðîèòåëüñòâà, Õàðáèíñêèé èíæåíåðíûé óíèâåð-
ñèòåò, Õàðáèí, Êèòàé
 ðàìêàõ ïîäõîäà ìîëåêóëÿðíîé ñòðóêòóðíîé ìåõàíèêè èññëåäóåòñÿ âëèÿíèå äâîéíûõ àòîìàð-
íûõ âàêàíñèîííûõ äåôåêòîâ êðèñòàëëè÷åñêîé ðåøåòêè íà óïðóãèå ñâîéñòâà îäíîñëîéíûõ
ëèñòîâ ãðàôåíà ñ çèãçàãîîáðàçíîé è ïëåòåíîé ñòðóêòóðîé. Äëÿ ìîäåëèðîâàíèÿ ìåæàòîìíûõ
ñèë ñâÿçåé òèïà óãëåðîä–óãëåðîä èñïîëüçóåòñÿ ïðîñòðàíñòâåííàÿ ñòðóêòóðíàÿ ñåòêà. Ðåçóëü-
òàòû ÷èñëåííîãî ìîäåëèðîâàíèÿ, ïîëó÷åííûå ìåòîäîì êîíå÷íûõ ýëåìåíòîâ, ïîäòâåðæäàþò,
÷òî íàëè÷èå óêàçàííûõ âàêàíñèîííûõ äåôåêòîâ ñíèæàåò ìîäóëü óïðóãîñòè ãðàôåíà, ÷òî
ïðèâîäèò ê óìåíüøåíèþ åãî íåñóùåé ñïîñîáíîñòè. Óñòàíîâëåíî, ÷òî óâåëè÷åíèå êîëè÷åñòâà
âàêàíñèîííûõ äåôåêòîâ îáóñëîâëèâàåò ñíèæåíèå ìîäóëÿ óïðóãîñòè è êîýôôèöèåíòà Ïóàñ-
ñîíà îäíîñëîéíûõ ëèñòîâ ãðàôåíà.
Êëþ÷åâûå ñëîâà: ãðàôåí, äâîéíûå àòîìàðíûå âàêàíñèîííûå äåôåêòû, óïðóãèå ñâîé-
ñòâà, õèðàëüíîñòü, ìîëåêóëÿðíàÿ ñòðóêòóðíàÿ ìåõàíèêà.
Introduction. Graphene is a new type of low-dimensional carbon material, which
emerged after the discovery of fullerenes and carbon nanotubes. Due to its single atomic
layer thickness, graphene is considered to be a two-dimensional material, which consists of
carbon atoms in a honeycomb lattice structure. Since its news-breaking appearance in 2004
[1], graphene has received significant attention due to its outstanding physical properties
such as high thermal conductivity [2], stiffness, and strength [3]. However, the issue of
mechanical properties of single-layered graphene sheets (SLGS) being impaired by atom
vacancy defects has received scarce attention yet. However, due to limitations in the
graphene production, imperfections, such as adatoms, Stone–Wales defects, line defects,
and vacancies of atoms cannot be avoided in graphene nanostructures [4]. Of these, the
vacancy defects are the most typical for graphene and strongly affect the mechanical
properties of SLGS. Therefore, it is of high theoretical significance and practical importance
to study the influence of double-atom vacancy defects (DAVD) on the mechanical
properties of SLGS.
Numerous researchers have investigated the elastic properties of graphene using
various theoretical methods. Thus, Pei et al. applied the molecular dynamics approach to
estimate the elastic modulus values for different chirality graphenes, including the so-called
© Z. Q. WANG, Z. W. YU, X. Y. SUN, H. LI, Y. J. WANG, 2017
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 157
armchair and zigzag graphenes, for which the respective values of 0.89 and 0.83 TPa were
reported in [5]. The application of another molecular dynamics method (namely, the
empirical Tersoff potential) by Bu et al. [6] yielded the graphene elastic modulus of up to
1.24 TPa. Within framework of the ab initio density functional theory (via DFT method)
Liu et al. [7] estimated the elastic modulus and Poisson’s ratio of graphene as 1.05 TPa and
0.86, respectively. Also, Reddy adopted a continuum mechanics approach to determine the
elastic modulus of non-equilibrium and equilibrium graphene, obtaining values of 1.0 and
0.7 TPa, respectively [8]. Li applied the structural mechanics approach to monolayer
graphene, with the simulation of graphene sheets by space frame structures, and predicted
that SLGS elastic modulus reaches 1.02 TPa [9]. However, there are scarce research reports
on the effect of DAVDs on the elastic properties of SLGS.
A molecular structural mechanics approach was adopted in this paper to study the
effect of DAVDs on the elastic properties of the zigzag and armchair SLGS with account of
their sizes and the number of DAVDs.
1. Theoretical Details. In terms of the molecular mechanics, graphene as a large
molecule contains many regularly arranged carbon atoms with each atom in a force field [9]
that is generated by interactions between electron-nucleus and nucleus-nucleus and controls
the atom movements. The steric potential energy corresponds to the force field, which
controls only the relative location of the nuclei. In Fig. 1, different kinds of potential energy
terms of carbon–carbon (C–C) bonds are illustrated.
For the system, the total potential energy functions are given by
U U U U U Utotal r vdw� � � � �� � � � �� � � , (1)
where Ur , U� , U� , U� , and Uvdw are energies, related to the bond stretching, angle
variation, inversion, angle variation, and van der Waals nonbonding interaction, respectively.
The last term is negligible, in comparison with the other four terms that make the dominant
contributions to the total steric energy. Furthermore, as compared to the other terms, the
torsion and inversion energies are the most critical for the graphene subjected to in-plane
tension loading. Consequently, the sum of angle variation and bond stretching terms,
represented by harmonic functions under the assumption of small linearly elastic
deformations, can be used to calculate the total energy of monolayer graphene with a fairly
high accuracy [10–12]. The bond stretching and angle variation harmonic functions are
derived via the following equations:
Fig. 1. Potential energy of C–C bonds.
158 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
U k r r k rr r r� � �
1
2
1
2
0
2 2( ) ( ) , (2)
U k k� � �� � �� � �
1
2
1
2
0
2 2( ) ( ) . (3)
Here k� and kr represent the force constant for the angle variation and bond stretching,
respectively, � and �0 are the bond angles after and before deformation, respectively,
while r and r0 are the interatomic distances after and before the deformation, respectively.
Within framework of the structural mechanics, the beam potential energy can be
expressed via twisting and stretching terms, which correspond to bending and tensile axial
load schemes depicted in Fig. 2. Under the axial load N , a uniform beam’s strain energy
can be derived as
U
N
EA
dl
N L
EA
EA
L
LA
L
� � �
1
2
1
2
1
2
2
0
2
2( ) , (4)
where U A is tensile load energy, E is the elastic modulus of the uniform beam, A is the
beam cross-sectional area, L is the elongation, and L is the length of beam. In the beam.
In case of application of a pure bending moment M (Fig. 2), the beam bending potential
energy, U M , is derived as follows:
U
M
EI
dl
EI
L
EI
L
M
L
� � �
1
2
2 1
2
2
2
0
2 2� �( ) , (5)
where � is the rotation angle of the beam edges and I is the moment of inertia for this
rotational motion.
That is to say, the equivalence of the relevant terms of molecular and structural
mechanics systems is assumed due to the independence of their potential energy terms. This
implies the equivalence of the molecular system and the structural beam, which is used to
simulate the interatomic forces acting on the covalently bonded carbon atoms. The beam
stiffness values correspond to the constants of covalent force field terms, which are derived
as follows:
EA
L
kr� ,
EI
L
k� � , (6)
Fig. 2. Stretching and bending of a uniform beam.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 159
Effect of Double-Atom Vacancy Defects on the Elastic Properties ...
where kr and k� are the bending and stretching/tensile constants of the equivalent beams,
respectively. These can be assessed by the stiffness matrix method. Alternatively, the
structural beam simulation by atomistic finite elements was proposed [13].
It is noteworthy that specifying the diameter and elastic modulus of the beam allows
one to compare the atomistic limited approach results with those of the stiffness matrix
method. By substituting parameters A
d
�
� 2
4
and I
d
�
� 4
64
into Eq. (6), the beam
characteristics can be derived via the force field constants as
d
k
kr
� �
4 , E
k L
k
r�
2
4� �
, (7)
where E is the elastic modulus, d is the cross-sectional diameter, and L is the length of
beam, which is assumed to be equal to the carbon covalent bond interatomic distance.
Application of the AMBER force field of molecular dynamics simulation [12] yields
k� �
�878 10 10. N
nm/rad 2 and kr �
�652 10 7. N/nm. The distance between two
contiguous carbon atoms is estimated as 0.1412 nm [14]. Solving Eq. (7) yields
E �
�5488 10 6. N/nm2 and d � 0.146618 nm. Additionally, according to [15], Poisson’s
ratio � � 0.3 is assumed.
To simulate the monolayer graphene nanofilms, the space frame structures, consisting
of beam elements, are used in the theoretical analysis, which predicts that the mechanical
characteristics of SLGS are size-dependent and exhibit a chirality. The latter refers to a
property of asymmetry, due to which chiral objects are distinguishable from their mirror
images. Figure 3 shows the chiral SLGS studied in this work, namely, zigzag and armchair
sheets of different side width and length values, which are designated as b and a,
respectively.
To examine the elastic properties of the SLGS under study, a space frame structure
model is used. The normal stress � acting at the atomistic model’s free edge is used for the
exploration of the SLGS elastic properties under tension. As a result, the continuum sheet’s
normal stress is calculated as
� �
nf
bt
, (8)
a b
Fig. 3. Zigzag (a) and armchair (b) graphene with DAVDs.
160 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
where n is the number of nodes exposed to the external forces, f is the magnitude of the
tensile forces, t and b are thickness and width of the SLGS, respectively. According to
strain and tensile stress, the elastic modulus and Poisson’s ratio are calculated as
E
nf
bt
a
a
a
� �
�
�
, �
�
�
� �b
a
b
b
a
a
, (9)
where �a and �b are tensile strains of SLGS, equal to the elongation ratios, while a
and b are increments of the initial SLGS side length a and width b, respectively.
2. Results and Discussion. According to the theoretical analysis presented in the
above section, the space frame structures consisting of beam elements are used to simulate
the monolayer graphene nanofilms. To consider the influence of DAVDs on the elastic
properties of SLGS with various sizes, the original model with seven sets of dimensions
( )a b� was constructed: (a) 0.985�1.137 nm, (b) 1.969�1.990 nm, (c) 2.953� 2.842 nm,
(d) 3.938�3.695 nm, (e) 4.430�4.547 nm, (f) 5.415�5.400 nm, and (g) 6.399�6.252 nm.
The evolution of the elastic modulus and Poisson’s ratios of zigzag and armchair SLGS
with the increase in the number of DAVDs was also investigated in this study. It is
considered that DAVDs located at the central line can eliminate the boundary effect of
SLGS. The elastic moduli and Poisson’s ratio of the zigzag and armchair SLGS were
calculated for the above graphene sizes. These results are depicted in Figs. 4 and 5,
respectively.
2.1. The Influence of DAVDs on the Elastic Properties of SLGS. According to Figs. 4
and 5, the elastic moduli and Poisson’s ratios of SLGS with DAVDs are lower than those of
perfect SLGS, but with an increase in the SLGS size, this deterioration effect on the elastic
properties is weakened, which finding proves that the elastic properties of SLGS are
size-dependent. For the same size of graphene, the elastic modulus of zigzag SLGS is lower
than that of armchair SLGS, which implies that the elastic properties of SLGS are also
dependent on the chirality pattern.
For the dimensions of zigzag graphene with DAVDs, which correspond to the above
cases (a)–(g) for, the respective reductions of Poisson’s ratio and the elastic modulus
corresponded to: (a) 9.96 and 8.38%, (b) 2.70 and 5.23%, (c) 1.02 and 2.42%, (d) 0.51 and
1.41%, (e) 0.50 and 0.96%, (f) 0.25 and 0.64%, and (g) 0.13 and 0.32%, respectively. For
a b
Fig. 4. The elastic modulus vs. size of zigzag (a) and armchair (b) SLGS.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6 161
Effect of Double-Atom Vacancy Defects on the Elastic Properties ...
the armchair graphene, the respective reduction values were as follows: (a) 32.84 and
9.07%, (b) 11.34 and 2.80%, (c) 5.61 and 1.43%, (d) 3.22 and 0.72%, (e) 2.31 and 0.74%,
(f) 1.71 and 0.49%, and (g) 1.05 and 0.24%. The above patterns depicted in Figs. 4 and 5
confirm that the effect DAVDs on the elastic properties of SLGS of two types of chirality
weakens with their size. Moreover, the effect of DAVDs on the elastic properties of
armchair SLGS is stronger than that on the zigzag one.
2.2. Number of DAVDs vs. Elastic Properties. The evolution of the elastic modulus
and Poisson’s ratios of zigzag and armchair SLGS with the increase in the number of
DAVDs was also investigated in this study. For this purpose, the model of type (a b� ) with
the size of 6.399�6.252 nm was constructed. DAVDs located at the central line are
assumed to eliminate the boundary effect of SLGS. The elastic properties of zigzag and
armchair SLGS with the number of DAVDs varying from zero to seven are depicted in
Fig. 6, which demonstrates that the elastic modulus and Poisson’s ratio of the armchair
SLGS exceed those of the respective zigzag SLGS. In addition, an increase in the number
of DAVDs leads to a continuous deterioration of the elastic properties of both types of
SLGS. The deterioration of elastic properties of zigzag and armchair SLGS due to the
availability of numerous DADS can be reduced to the elastic modulus decrease by 0.5 and
0.1% per defect, and that of Poisson’s ratio by about 0.5 and 0.25% per defect, respectively.
a b
Fig. 5. Poisson’s ratio vs. size of zigzag (a) and armchair (b) SLGS.
a b
Fig. 6. Evolution of the elastic modulus (a) and Poisson’s ratio (b) of SLGS for the number of
DAVDs varying from zero to seven.
162 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
Conclusions. In this work, the molecular structural mechanics method was adopted to
simulate the elastic properties of SLGS with DAVDs. The sizes of graphene sheets and the
numbers of DAVDs were taken into account in the evaluation of the elastic moduli and
Poisson’s ratios of SLGS with two kinds of chiralities: zigzag and armchair ones. The
results obtained show that the above parameters of the SLGS with DAVDs are lower than
those of perfect SLGS. However, with increasing graphene size, the influence of DAVDs
on the elastic properties of SLGS becomes weaker. DAVDs with the increasing graphene
nanofilm size exert a stronger effect on the elastic properties of armchair SLGS, as
compared to zigzag ones. For the constant graphene nanofilms’ sizes, the increased amount
of DAVDs leads to a continuous decline in the elastic moduli and Poisson’s ratios of zigzag
and armchair chirality SLGS.
Acknowledgments. This work was supported by the National Natural Science
foundation of China (Nos. 11602066, 11472086, and 11532013) and the China Postdoctoral
Science Foundation on the 56th bath of surface founds the project (2014M561327) and the
National Science Foundation of Heilongjiang Province of China (QC2015058 and
42400621-1-15047), the Foundation Research Funds for the Central Universities.
Ð å ç þ ì å
Ó ðàìêàõ ï³äõîäó ìîëåêóëÿðíî¿ ñòðóêòóðíî¿ ìåõàí³êè äîñë³äæóºòüñÿ âïëèâ ïîäâ³é-
íèõ àòîìàðíèõ âàêàíñ³éíèõ äåôåêò³â êðèñòàë³÷íî¿ ãðàòêè íà ïðóæí³ âëàñòèâîñò³
îäíîøàðîâèõ ëèñò³â ãðàôåíà ³ç çèãçàãîïîä³áíîþ òà ïëåòåíîþ ñòðóêòóðîþ. Äëÿ ìîäå-
ëþâàííÿ ì³æàòîìíèõ ñèë çâ’ÿçê³â òèïó âóãëåöü–âóãëåöü âèêîðèñòîâóºòüñÿ ïðîñòî-
ðîâà ñòðóêòóðíà ñ³òêà. Ðåçóëüòàòè ÷èñåëüíîãî ìîäåëþâàííÿ, îòðèìàí³ ìåòîäîì ñê³í-
÷åííèõ åëåìåíò³â, ï³äòâåðäæóþòü, ùî íàÿâí³ñòü óêàçàíèõ âàêàíñ³éíèõ äåôåêò³â çíè-
æóº ìîäóëü ïðóæíîñò³ ãðàôåíà, ùî ïðèçâîäèòü äî çìåíøåííÿ éîãî íåñ³âíî¿ çäàò-
íîñò³. Óñòàíîâëåíî, ùî çá³ëüøåííÿ ê³ëüêîñò³ âàêàíñ³éíèõ äåôåêò³â ïðèçâîäèòü äî
çíèæåííÿ ìîäóëÿ ïðóæíîñò³ òà êîåô³ö³ºíòà Ïóàññîíà îäíîøàðîâèõ ëèñò³â ãðàôåíà.
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Received 14. 08. 2017
164 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 6
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
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/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/MultimediaHandling /UseObjectSettings
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]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|