Surface Effect on the Nanowire Forest Indentation
The surface effect on the mechanical behavior of a nanowire forest indentation is theoretically studied. The use of a large-radius spherical indenter revealed a complex deformation pattern of the nanowire forest. The nanowire forest compression, buckling and postbuckling were analyzed. The effect of...
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| Cite this: | Surface Effect on the Nanowire Forest Indentation / F. Yang // Проблемы прочности. — 2018. — № 1. — С. 130-137. — Бібліогр.: 17 назв. — англ. |
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| citation_txt | Surface Effect on the Nanowire Forest Indentation / F. Yang // Проблемы прочности. — 2018. — № 1. — С. 130-137. — Бібліогр.: 17 назв. — англ. |
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| description | The surface effect on the mechanical behavior of a nanowire forest indentation is theoretically studied. The use of a large-radius spherical indenter revealed a complex deformation pattern of the nanowire forest. The nanowire forest compression, buckling and postbuckling were analyzed. The effect of surface and packing density on the nanowire forest behavior with the indentation depth is discussed. The results show that the surface effect is of importance for in-depth hardness. The analysis is instrumental in measuring the mechanical properties of the nanowire forest and in designing nanowire-forest devices for various applications
Изучено теоретически влияние поверхности на механическое поведение нанопроволочного леса при индентировании. Выявлен сложный характер деформирования нанопроволочного леса при использовании сферического индентора большого радиуса, выполнен анализ характеристик сжатия, потери устойчивости и состояния после потери устойчивости. Обсуждено влияние поверхности и плотности упаковки на поведение нанопроволочного леса в зависимости от глубины индентирования. Показано, что поверхность оказывает влияние на изменение твердости по глубине. Полученные результаты найдут применение при измерении механических свойств и разработке устройств из нанопроволочного леса, используемых в различных областях.
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UDC 539.4
Surface Effect on the Nanowire Forest Indentation
F. Yang
a,b
a Department of Mechanics, Xi’an University of Science and Technology, Xi’an, China
b Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, China
yangfan0832@163.com
The surface effect on the mechanical behavior of a nanowire forest indentation is theoretically
studied. The use of a large-radius spherical indenter revealed a complex deformation pattern of the
nanowire forest. The nanowire forest compression, buckling and postbuckling were analyzed. The
effect of surface and packing density on the nanowire forest behavior with the indentation depth is
discussed. The results show that the surface effect is of importance for in-depth hardness. The
analysis is instrumental in measuring the mechanical properties of the nanowire forest and in
designing nanowire-forest devices for various applications.
Keywords: surface effects, nanowire forests, indentation.
Introduction. Vertically aligned nanowires and nanotubes (nanoforests) have many
potential applications, such as field emission, super-hydrophobic surfaces, optoelectronics,
solar cells and energy absorption materials etc [1, 2]. On vertically aligned carbon nanotube
forest, a facile yet efficient route has been successfully developed to fabricate well
dispersed Ni nanoparticles [3]. In addition to their electronic, optical and wetting properties,
the mechanical properties of these nanoforests are important to their functional performance.
For nanoforests, it is of more interest to estimate the overall statistical properties of
nanowires and nanotubes rather than those of individual wires. Using flat punch nano-
indentation, Maschmann et al. [4] examined the axial compressive mechanical behavior of
carbon nanotube arrays. Xiao et al. [5] used electro-discharge machining with different
electrode sizes to study the machining mechanism of a carbon nanotube forest. Using a
sharp atomic force microscopy (AFM) tip to conduct nanoindentation into a nanoforest, Qi
et al. [6] determined the collective behavior of nanotubes, in which each nanotube was
modeled as a cantilever subjected to bending. When a comparatively larger-radius indenter
is employed, individual nanotubes are directly compressed and then buckle upon reaching a
critical load. Wang et al. [7] investigated the force-depth behavior of indentation into a
nanoforest by both the finite element method and an analytical model.
It is known that the mechanical behaviors of nanowires such as their elastic modulus
and yield stress etc. display strongly size dependence on its cross-sectional size. Surface
effects play an important role to contribute noticeably to the mechanical behavior of
nanowires, owing to the remarkably large surface area to volume ratio. For examples, Chen
et al. [8] proposed a core-shell model to explain the size-dependent elastic modulus of ZnO
nanowires. Miller and Shenoy [9] adopted the surface elasticity theory to analyze the
stretching and bending of nanobeams, which shows a good agreement with direct atomic
simulations. Wang and Feng [10, 11] studied both the impact of residual surface stress and
of surface elasticity on the buckling and vibration of the nanobeams. He and Lilley [12]
addressed the static bending of nanowires and proved the size dependence of the elastic
modulus with this model. Qiu et al. [13] have demonstrated the rational design and
fabrication of meso-/macroporous ZnCo2O4/MnO2 hierarchical core/shell nanocone forests
using a facile hydrothermal approach. Zheng et al. [14] found the effect of surface elasticity
on the elastic moduli can be well characterized by the Cauchy–Born surface model.
© F. YANG, 2018
130 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1
It is noteworthy that the surface effect on the overall behavior of nanowire forest
indentation has not been accounted for. Meanwhile, the post-buckling of nanowire forests
has not been considered during the indentation process in many indentation studies,
despite the fact that it often occurs and has great research significance in the indentation
test.
Hence, the objective of the current work is to study the effects of surface elasticity and
residual surface stress on the mechanics of indentation into nanowire forests. During the
whole indentation process, the complicated deformation behavior of nanowire forests with
the large spherical indenter in this analysis includes compression, buckling and post-
buckling.
The paper is organized as follows. In Section 2, we address surface effects on the
deformation of a single nanowire under compression. Based on the force-displacement
relationship of a nanowire that we obtain, the indentation of nanoforest is then analyzed in
Section 3.
1. Deformation of a Nanowire under Compression. During the indentation of
nanowire forests, each nanowire under the indenter experiences linearly elastic compression,
buckling, and post-buckling. As a preparation, we first analyze the influence of surface
effects on the deformation of a single nanowire. Surface effects on the mechanical response
of nano-structured elements can be considered for using surface stresses � s . In accordance
with the assumption of Cammarate [15], the relation between surface stresses � s and strain
tensor � in a one-dimensional linear model is given by
� � �s sE� �0 , (1)
where �0 is the residual surface stress, while E s is the surface elastic modulus, which can
be obtained through atomistic simulations or experiments [9].
When surface effects are considered, the effective flexural rigidity ( )*EJ of a
nanowire is given by [10, 11]
( )
( ),
*EI
Ebt E bt E t
ED
s s
�
� �
�
1
12
1
2
1
6
64
3 2 3
4
rectangular
� �
8
3E Ds ( ),circular
�
�
(2)
and the effective stiffness ( )*EA is [9]
( )
( ) ( ),
(
*EA
Ebt E b t
ED E D
s
s�
� �
�
2
4
2
rectangular
circula
�
� r ),
�
�
(3)
where b is the width and t is the height for a rectangular cross section while D is the
diameter of a circular cross section, and E is the bulk elastic modulus of the nanowire.
Based on the Laplace–Young equation, the influence of residual surface stress is
described by a distributed normal pressure qn depending on the current surface curvature
�:
q Hn �
* ,� (4)
where H* expresses a constant determined by the residual surface stress and the
cross-sectional shape
Surface Effect on the Nanowire Forest Indentation
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1 131
H
b
D
* ( ),
( ).
�
�
�
2
2
0
0
�
�
rectangular
circular
(5)
In the process of compression, when the axial load is less than the critical load, the
nanowire experiences a pure elastic compression, while the compression force is related to
the end displacement h by
P EA h l� ( ) ,*
(6)
where l is the length of nanowire.
When the axial load P reaches the critical value Pcr , the nanowire may buckle. With
the account of the surface effect, the critical load for a fixed-hinged beam is given by
P
EI
l
Hcr � �
2 2
2
� ( )
.
*
*
(7)
For a fixed-free nanowire, the critical load is derived implicitly as in [16],
H P
P H
EI
lcr
cr*
*
*
cos
( )
.�
�
�
�
�
�
�
�
� (8)
Meanwhile, for the conventional Euler beam the critical axial load is
P
EI
l
cr
0
2
0
2
�
�� ( )
, (9)
with �� 2 for a fixed-hinged beam, �� 1/4 for a cantilever beam, and
( )
( ),
( ).
EI
Ebt
ED
0
3
4
1
12
64
�
�
�
rectangular
circular
� (10)
When the compression load is beyond the critical value Pcr , postbuckling takes place.
It is necessary to employ the finite deformation to determine the force displacement
relationship. For convenience, the arc length s and the slope angle � are denoted in the
deformation of an nanowire. The displacements along the axial and transverse directions of
the nanowire are used by u x( ) and w x( ). Denoting ds dx as �, the geometric equations
can be expressed as
du
dx
�
� cos ,� 1
dw
dx
�� sin ,�
d
dx
d
ds
� �
� � . (11)
The curvature � of the deformed central axis and the strain of central axis �0 are
given by
�0 1�
� , �
�
�
d
ds
. (12)
Consequently, the moment M and the axial resultant force N of the same cross
section are obtained as
F. Yang
132 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1
M EI� ( ) ,* � N EA� ( ) ,* �0 (13)
where ( )*EA and ( )*EI are derived via Eqs. (2) and (3), respectively.
For a differential element of length ds between two cross sections normal to the
deflected axis of the beam, the equilibrium conditions yield:
dH
ds
qn� sin ,�
dV
ds
qn� cos ,�
dM
ds
H V� �sin cos ,� � (14)
where H is the component of the resultant force along the x-axis and V is the components
of the resultant force along y-axis. Using H and V, the axial resultant force N can be
gives
N V H�
sin cos .� � (15)
Considering the differential equations (11) and (14) with specified boundary
conditions, shooting method is employed to solve the two-point boundary value problem
numerically.
To illustrate surface effects on the compression of nanowires, we consider a silver
nanowire with E� 76 GPa, �0 � 0.89 J/m2, and E s � 1.22 N/m [17]. For nanowires with
circular cross section of radius D and l� 200 nm, Fig. 1 displays the force–displacement
relationship for a fixed-hinged and a fixed-free nanowires, respectively. According to the
conventional elastic analysis, the normalized load P Pcr
0 expresses independent of the
absolute size of beam D. However, when surface effects are involved, the normalized axial
load P Pcr
0 depends evidently on the diameter D, especially when D reduces to dozens of
nanometers.
It is seen that surface effects enhance the critical load for a fixed-hinged nanowire,
while the axial force will decrease after buckling. Conversely, surface effects decrease the
critical load for a cantilever, while it needs to increase the axial force for further
compression. It is noticed that for h l�0.3, the force displacement relationship can be
described by a uniform expression for both cases as
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1 133
Surface Effect on the Nanowire Forest Indentation
a b
Fig. 1. The applied load vs the displacement of a nanowire for the fixed-hinged (a) and cantilever (b)
cases.
P P
h h h h
h hcr
cr cr
cr cr
0
1
�
�
�
�
�
�
�
� � � �
, ,
[ ( )], ,
(16)
where �� P Pcr cr
0 , �� h l, �cr crP EA� ( ) ,* h lcr cr� � .
Thus the influence of surface effects on the force displacement relationship is
completely reflected by two factors � and �. Figure 1 plots the dependence of � and �
on the diameter D of nanowires.
2. Analytical Indentation into Nanowire Forests Model. Now, we consider the
indentation on nanowire forests with a spherical indenter as illustrated in Fig. 2. Assume
each nanowire has the same length l, and the radius R of the indenter is much larger
than l. For a simplification, we neglect the intermolecular interaction between nanowires,
and we assume that the nanowire contacts the indenter only at its top end after post-
buckling, which can be ensured for h l�0.3.
The friction between the indenter and the nanowires affects the critical buckling load
of an individual tube. We consider two extreme cases. When the indenter/nanowire
interface is frictionless, the nanowires are modeled by fixed-free beams. When the
indenter/nanowire interface is completely rough, and no slip is permitted, the nanowires are
simulated by clamped-pinned beams.
When the indent depth h hcr� , nanowires under the indenter experience a pure
compression:
P Pi
i
n
�
�
� ,
1
(17)
where n is the number of nanowires encountering the indenter and Pi is the indentation
force for the ith individual nanowire and its magnitude is determined by its indentation
depth hi in terms of Eq. (6).
Assuming a uniform distribution of nanowires with m nanowires per unit area, then
the total indentation force is obtained by
P P
h h
l
m R h dh m l P
h
h
cr
h i
i i cr cr
cr
�
�
�
�
�
�
�
�
�
�
���
0
22 � � �( )
�
�
��
�
�
�
�
�
�
2
3
R
l
h
l
. (18)
134 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1
F. Yang
a b
Fig. 2. Schematic of indentation on nanowire forests.
When the indent depth h hcr� , parts of nanowires around the indenter tip will take
place buckling as indicated by the penetration area A1 (Fig. 3), while parts of nanowires
around the contact fringe are still under simple compression. The boundary between them
is determined by h hi cr� . Thus, the resultant force P is transferred to nanowires under
postbuckling in area A1 and nanowires under elastic compression in area A2.
Then,
F P P p pi i
AA
� � � ���1 2
21
, (19)
P p
h h
l
m R h dhcr
ih h
i i
cr
1 0
1 2� �
�
�
�
�
�
�
� � �
( )
( ) , (20)
P p
h h
l
m R h dhcr
i
h h
h
i i
cr
2 2�
�
�
�
�
�
�
�
( )
( ) .� (21)
Then, we have
P
P m l
R
l
h
l
h
l
R
l
h
l
cr
cr cr cr
�
� � � �
2
2
3
2�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
R
l
h
l
�
�
�
� �
�
�
�
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
!
"
1
2
3
2
� � � � � �
R
l
h
l
h
l
h
l
cr cr cr #. (22)
For the case of R l�� , Eq. (22) simplifies as Eq. (19).
When h hcr� ,
P
P m Rl
h
lcr cr� �
�
�
�
�
�
�
�
1
2
. (23)
When h hcr� ,
P
P m Rl
h
l
h
lcr
cr cr�
� � �� �
�
�
�
�
�
� �
�
�
�
�
�
�2 . (24)
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1 135
Surface Effect on the Nanowire Forest Indentation
Fig. 3. Schematic of penetration area on nanowire forests.
Results of indentation into forests of coiled carbon nanotubes show a force–depth
relationship of F h
2, which accounts for the individual elastic contribution of the carbon
nanotube and the contact geometry.
3. Results and Discussion. As an example, consider a spherical indenter with radius
R l� 50 and each individual nanowire in nanowire forests having the same length,
diameter, and material properties as the nanowire in Fig. 2. With these parameters and
Eq. (16), a density m�7.77 for pinned-fixed nanowire forests and m�25 for cantilever
nanowire forests can avoid adhesion instability of two neighboring nanowires.
Figures 4 and 5 compare the load–depth behavior while considering compression,
buckling, and post-buckling, between the solution with surface effect and the classical
solution in which surface effect is neglected. For pinned-fixed nanowire forests in Fig. 4,
compared to the classical solution, the dimensionless applied load F Pcr
0 accounted the
surface effects are much significantly larger and trend to show a stiffer behavior when areal
density m� 7, while the cantilever nanowire forests show the opposite trends in Fig. 5. The
results show that the surface effects play an important role in depth-dependent hardness.
Evidently, for the different areal densities of m� 5 and 7, the dimensionless applied load
F Pcr
0 becomes larger as the increase of m for both pinned-fixed nanowire forests and
cantilever nanowire forests.
Conclusions. In this paper, the surface effects of surface elasticity and residual
surface stress on mechanics of indentation into well-ordered forests of nanowires by a
relatively large spherical indenter (R l�� ) is analyzed and discussed. During the whole
indentation process, the complicated deformation behavior of nanowire forest with a
large-radius spherical indenter includes compression, buckling, and postbuckling. The
surface effect contribute noticeably to the mechanical behavior of nanowire forests due to a
remarkably large surface area/volume ratio. The surface and packing density effect on the
in-depth behavior of indentation of nanowire forest is discussed. When the surface effect is
allowed for, pinned-fixed nanowire forests tend to be stiffer than classical beam forests,
while the cantilever case tends towards a softer behavior. The study is instrumental in
measuring the mechanical properties of the nanowire forest and in designing
nanowire-forest devices for various applications.
Acknowledgments. Project (11502197) supported by the National Natural Science
Foundation of China; Project (2016JQ1032) supported by the Science and Technology
Department of Shaanxi Province; Project (16JK1504) supported by the Education
Department of Shaanxi Province; Post-doctoral program in Shaanxi Province.
136 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1
F. Yang
Fig. 4 Fig. 5
Fig. 4. The load–depth behavior of pinned-fixed nanowire forests with different area densities.
Fig. 5. The load–depth behavior of cantilever nanowire forests with different area densities.
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Received 15. 09. 2017
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 1 137
Surface Effect on the Nanowire Forest Indentation
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| id | nasplib_isofts_kiev_ua-123456789-173821 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T13:37:06Z |
| publishDate | 2018 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Yang, F. 2020-12-21T18:43:22Z 2020-12-21T18:43:22Z 2018 Surface Effect on the Nanowire Forest Indentation / F. Yang // Проблемы прочности. — 2018. — № 1. — С. 130-137. — Бібліогр.: 17 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173821 539.4 The surface effect on the mechanical behavior of a nanowire forest indentation is theoretically studied. The use of a large-radius spherical indenter revealed a complex deformation pattern of the nanowire forest. The nanowire forest compression, buckling and postbuckling were analyzed. The effect of surface and packing density on the nanowire forest behavior with the indentation depth is discussed. The results show that the surface effect is of importance for in-depth hardness. The analysis is instrumental in measuring the mechanical properties of the nanowire forest and in designing nanowire-forest devices for various applications Изучено теоретически влияние поверхности на механическое поведение нанопроволочного леса при индентировании. Выявлен сложный характер деформирования нанопроволочного леса при использовании сферического индентора большого радиуса, выполнен анализ характеристик сжатия, потери устойчивости и состояния после потери устойчивости. Обсуждено влияние поверхности и плотности упаковки на поведение нанопроволочного леса в зависимости от глубины индентирования. Показано, что поверхность оказывает влияние на изменение твердости по глубине. Полученные результаты найдут применение при измерении механических свойств и разработке устройств из нанопроволочного леса, используемых в различных областях. Project (11502197) supported by the National Natural Science Foundation of China; Project (2016JQ1032) supported by the Science and Technology Department of Shaanxi Province; Project (16JK1504) supported by the Education Department of Shaanxi Province; Post-doctoral program in Shaanxi Province. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Surface Effect on the Nanowire Forest Indentation Влияние поверхности на индентирование нанопроволочного леса Article published earlier |
| spellingShingle | Surface Effect on the Nanowire Forest Indentation Yang, F. Научно-технический раздел |
| title | Surface Effect on the Nanowire Forest Indentation |
| title_alt | Влияние поверхности на индентирование нанопроволочного леса |
| title_full | Surface Effect on the Nanowire Forest Indentation |
| title_fullStr | Surface Effect on the Nanowire Forest Indentation |
| title_full_unstemmed | Surface Effect on the Nanowire Forest Indentation |
| title_short | Surface Effect on the Nanowire Forest Indentation |
| title_sort | surface effect on the nanowire forest indentation |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173821 |
| work_keys_str_mv | AT yangf surfaceeffectonthenanowireforestindentation AT yangf vliâniepoverhnostinaindentirovanienanoprovoločnogolesa |