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INCORPORATED EFFECTS OF SURFACE STRESS
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AND NONLOCAL ELASTICITY ON BENDING
3
ANALYSIS OF NANOPLATES EMBEDDED IN AN
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ELASTIC MEDIUM
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Running head: Incorporated effects of surface stress and nonlocal elasticity on
6
bending analysis of nanoplates
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Monchai Panyatong*, Boonme Chinnaboon, and Somchai
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Chucheepsakul
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Department of Civil Engineering, Faculty of Engineering,
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King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand
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Tel. 0-2470-9146; Fax. 0-2427-9063; E-mail: art_gear33@hotmail.com
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*Corresponding author
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Abstract
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This paper studies the bending behavior of nanoplates, incorporating the effects
17
of surface stress and nonlocal elasticity. Thin and moderately thick nanoplates
18
embedded in an elastic medium are analyzed. The complete governing
19
equations, including both the effects of surface stress and nonlocal elasticity, are
20
derived, while the solution for bending is solved by Navier’s approach. The
1
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analytical solution in this study could serve as a benchmark in the evaluation of
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future research. Based on the results of this study, the opposite influence of
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surface stress and nonlocal elasticity on the bending of nanoplates is observed.
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Various parametric studies are investigated to elucidate the combined effects of
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surface stress and nonlocal elasticity.
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Keywords: Nanoplates, Surface Stress, Nonlocal Elasticity, Bending, Analytical
Solution
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Introduction
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Due to the dominant mechanical, thermal, chemical and electronic performances of
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nanostructures, nanoplates have widely been used as an important part of nanosensors
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and nanoactuators in the development of nanoelectromechanical systems (NEMS) in
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recent decades (Craighead, 2000). A thorough understanding of the bending behavior
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of nanoplates is important in the design of NEMS materials. The structure at
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nanoscale shows a significant size-dependent behavior. It is a well-known fact that
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the classical continuum theory does not take into account the so-called size effects,
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which are manifested in the response of nanostructures. The high surface-to-bulk
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ratio of nanostructures leads to the exhibition of different behaviors compared with
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the conventional structures in macroscopic elements because of the considerable
41
influence of surface stress.
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The effect of surface stress on the behavior of nanoplates has been investigated by
2
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many researchers. Lim and He (2004) developed a continuum model to analyze the
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bending, buckling and vibration behaviors of thin elastic films with nanoscale
45
thickness. Lu et al. (2006) proposed a continuum model including surface effects for
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plate-like thin film structures, which could be used for size-dependent static and
47
dynamic analysis. Assadi and Farshi (2011) studied the stability and self-instability of
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circular nanoplates, including surface effects. Ansari and Sahmani (2011)
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investigated the free vibration characteristics of nanoplates, including surface stress
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effects, based on the continuum model. Assadi (2013) examined the effect of surface
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properties on forced vibration of rectangular nanoplates. Wang and Wang (2013)
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evaluated the influence of surface energy on the post-buckling behavior of nanoplates
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by applying Galerkin’s method to solve the problems. Shaat et al. (2013) studied the
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bending behavior of ultra-thin functionally graded plates, focusing on the influence of
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surface energy.
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Considering small-scale effects, nonlocal elastic theory has presented reliable and
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accurate results in revealing the mechanical behaviors of nanostructures. For
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examples, Pradhan and Murmu (2009) studied the buckling of single-layer graphene
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sheets subjected to biaxial compression load, and solved the problems using the
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differential quadrature method. Murmu and Pradhan (2009) investigated the influence
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of small-scale effect on free in-plane vibration by employing nonlocal continuum
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model. Aksencer and Aydogdu (2011) analyzed the buckling and vibration of
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nanoplates using Navier- and Levy-type solutions. Malekzadeh et al. (2011)
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determined the thermal buckling behavior of orthotropic arbitrary straight-sided
3
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quadrilateral nanoplates in an elastic medium by employing the differential
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quadrature method. Farajpour et al. (2011) investigated the buckling behavior of
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variable thickness nanoplates under biaxial compression, and solved the problems by
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Galerkin’s method. Satish et al. (2012) analyzed the thermal vibration of orthotropic
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nanoplates by using the two variable refined plate theory and nonlocal continuum
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mechanics. Wang and Li (2012) studied the bending behavior of a nanoplate
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embedded in an elastic matrix. Pouresmaeeli et al. (2013) examined the vibration of
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viscoelastic orthotropic nanoplates embedded in a viscoelastic medium. Zenkour and
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Sobhy (2013) investigated thermal buckling of nanoplates resting Winkler-Pasternak
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elastic medium, based on the sinusoidal shear deformation plate theory. Chakraverty
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and Behera (2014) studied the free vibration of rectangular nanoplates, and solved the
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problems by Rayleigh–Ritz method. Moreover, some researchers considered a
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combination of both surface effects and nonlocal elasticity. For examples, Wang and
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Wang (2011) studied the vibration of rectangular nanoplates, which combined the
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effects of surface energy and nonlocal elasticity. Narendar and Gopalakrishnan
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(2012) investigated the wave propagation characteristics of nanoplates, using
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nonlocal plate theory together with surface effects. Juntarasaid et al. (2012) analyzed
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the bending and buckling load of nanowires, including surface stress and the nonlocal
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elasticity effect with various boundary conditions. Farajpour et al. (2013) investigated
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the influence of temperature change, surface parameters and nonlocal effects on
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buckling of single-layer graphene sheets, using differential quadrature method. Asemi
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and Farajpour (2014) studied thermo-mechanical vibration of circular graphene
4
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sheets, both surface and nonlocal effects are taken into account, and solved
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formulation by using Galerkin's method.
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To the best of authors’ knowledge, based on a review of past literature, no studies
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have been performed on the bending behavior of nanoplates embedded in an elastic
91
medium and incorporating both the surface effect and nonlocal elasticity. Therefore,
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the main purpose of this study is to investigate the bending behavior of nanoplates,
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combining the two above-mentioned small-scale effects, based on classical plate
94
theory and Mindlin plate theory. The elastic medium is characterized as a two-
95
parameter elastic foundation. Thus, the complete governing equations for bending
96
analysis of nanoplates are achieved. The present work also studies the influence of
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the surface effect and nonlocal parameter on the displacement ratio when the sizes of
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the nanoplates are varied.
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Formulation of the problem
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The formulation presented in this study is concerned with the application of nonlocal
102
constitutive relations and surface stress at the upper and lower surfaces of the
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nanoplates. Thus, it is limited to the two dimensional cases and the plate theories are
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employed to derived governing equations for bending problem with the displacement
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at any material point depend only the displacement middle plane of nanoplates.
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Nonlocal elasticity theory
5
108
In nonlocal elasticity theory, it is assumed that the stress at a point depends not only
109
on the strain at that point but also on the strains at all other points in the body.
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According to Eringen (2002), the nonlocal constitutive relations of a Hookean body
111
can be expressed as
112
1   
nl
xx

E
 xx   yy  ,
1  2
(1a)
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1   
nl
yy

E
 yy   xx  ,
1 2
(1b)
114
1   
nl
xy
 2G xy ,
(1c)
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1   
nl
xz
 2G xz ,
(1d)
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1   
nl
yz
 2G yz ,
(1e)
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where E , G and  are the elastic modulus, the shear modulus and Poisson’s ratio,
118
respectively. The scale factor
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internal characteristic lengths (such as lattice spacing, granular distance, distance
120
between C-C bonds) and e0 is a material constant which determined to calibrate the
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nonlocal model with experimental results or the results of molecular dynamics (MD)
122
simulations.
2
2
2
2
2
   e0li  is nonlocal parameter, where li is an
2
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Surface stress
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For nanostructures, the surface-to-bulk ratio is significant. Therefore, the surface
126
effect cannot be ignored. Recently, the Gurtin–Murdoch theory of an elastic solid
6
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surface (Gurtin and Murdoch, 1975, 1978) has been widely employed for the
128
investigation of various mechanical responses of the nanoplates. Based on this theory,
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the original surface stress tensor is expressed by the following equation:
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 s   0 I   0   0   trε sur  I  2  u0   0  ε sur   0  u,
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where  0 is the residual surface tension, 0 and u0 are the surface Lame constants,
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I is
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  u is the surface gradient of the displacement field describing its deformation. Eq.
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(2) refers to the constitutive a tensor tangent to the surface and a component normal
135
to this surface. In previous works (Sharma and Ganti, 2002; Sharma et al., 2003;
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Duan et al., 2005a, 2005b), the original surface stress tensor has been simplified by
137
ignored the last term in Eq. (2). However, if the residual surface tension is important,
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the last term in Eq. (2) should be included (Mogilevskaya et al., 2008). Especially,
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for the bending problem, the residual surface tension is significant which contributes
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the bending stiffness of structures (Miller and Shenoy, 2000). Therefore, in the
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present work, the original surface stress equation is developed without any
142
simplifications. Let the upper and lower surfaces of the nanoplates be denoted by
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S and S , respectively. Using Eq. (2), the surface stresses at the upper and lower
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surfaces of the nanoplates can be expressed as follows:
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 xxs    0   2u0  0 

(2)
the unit tangent tensor, trε sur is the trace of the surface strain tensor ε sur , and

u
v
  0   0  ,
x
y
(3a)
7

u
  0   0  ,
y
x
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 yys    0   2u0  0 
147
 xys   u0 
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 yxs   u0 
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 xzs    0
w
,
x
(3e)
150
 yzs    0
w
.
y
(3f)
(3b)
 u 

 y x


,
  0
x

(3c)
 u 

 y x

u
,
  0
y

(3d)
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Classical plate theory
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The displacement field of the classical or Kirchhoff plate theory can be written as
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u ( x, y , z )  u ( x, y )  z
w
,
x
(4a)
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v ( x, y , z )  v ( x, y )  z
w
,
y
(4b)
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w( x, y, z )  w( x, y ) ,
157
where u ( x, y ), v( x, y ) and w( x, y ) are the displacement components of the material
158
point at the meddle plane of the plate. The strain-displacement relations can be
159
expressed as
160
u
2w
 xx   z 2 ,
x
x
(4c)
(5a)
8
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v
2w
 yy   z 2 ,
y
y
(5b)
162
1  u v 
2w
,
 xy      z
2  y x 
xy
(5c)
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 xz  0 ,  yz  0 ,  zz  0 .
164
The bending moment resultants incorporating the effects of surface stress and
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nonlocal elasticity can be written as
166
h
 xx   xxs    xxs      xxnl zdz ,
2  h /2
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 yy  
168
h
 xy   xys    xys      xynl zdz .
2  h /2
169
Considering Eqs. (6a)–(6c), the first and second parts of the right-hand side result
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from the surface stress effect and the nonlocal elasticity, respectively. Let the upper
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and lower surfaces of the nanoplates have the same properties. Then using Eqs. (1a)–
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(1c), (3a)–(3c), (4a)–(4c), (5a)–(5c) and (6a)–(6c) we obtain the bending moment
173
resultants, as follows:
174
1    


 2w
h 2
2w
2w 
2w 
1   2   2u0  0  2   0   0  2   D  2   2  ,

2
x
y 
y 

 x
(7a)
175
1  2   yy 
 2w
h 2
2 w
2 w 
2w 
2 
1



2
u







D







  0 0 y 2 0 0 x2   y 2 x2  ,
2




(7b)
(5d)–(5f)
h /2
s
yy
(6a)
h /2
h
      yynl zdz ,
2  h /2
s
yy
(6b)
h /2
2
xx
(6c)
9
176
h 2
2w 
2w
2 
1     xy  2 1     2u0  0  xy   D 1    xy ,


177
where D  Eh3 /12(1   2 ) is the flexural rigidity of the nanoplates. To derive the
178
equilibrium equations of classical plate theory, it is necessary to find the shear forces
179
( S x , S y ) acting on the cross sections; both lower and upper surfaces of the nanoplates
180
must be considered, as follows (Assadi, 2013):
181
Sx 
182
Sy 
183
From the sum of the resultant forces in the transverse direction, which contain the
184
external loading q ( x, y ) and reaction force of a two-parameter elastic foundation (an
185
elastic medium), and then applying Eqs. (7a)–(7c) and (8a)–(8b), the following
186
governing equation can be obtained:
187


 h2
h2
 2u0  0  6 w   D   2u0  0   2 0   Gb   4 w
2
2


188
where k w and Gb are the Winkler foundation stiffness and the shear layer stiffness of
189
the foundation, respectively.
2
M xx M xy
M xx M xy
w
,

  xzs    xzs  

 2 0
x
y
x
y
x
M yy
y

M xy
x
  yzs    yzs  
M yy
y

M xy
x
w
.
y
  2 0  Gb   kw   2 w  kw w  1   2  q,
190
191
 2 0
Mindlin plate theory
10
(7c)
(8a)
(8b)
(9)
192
The Mindlin plate theory is known as the first-order shear deformation theory, and is
193
based on the displacement field that can be expressed as
194
u( x, y, z)  u( x, y)  z x ( x, y) ,
(10a)
195
v( x, y, z )  v( x, y )  z y ( x, y ) ,
(10b)
196
w( x, y, z )  w( x, y ) ,
(10c)
197
where u ( x, y ), v( x, y ) and w( x, y ) are the displacement components of the material
198
point at the middle plane, and x and  y are the rotations about the y and x axes,
199
respectively. Furthermore, the strain can be expressed as
200
 xx 
 x
u
,
z
x
x
(11a)
201
 yy 
 y
v
,
z
y
y
(11b)
202
 
1  u v  z  
 xy       x  y  ,
2  y x  2  y
x 
203
 xz  
 x  ,
2  x

204
 yz    y  .
2  y

205
The bending moment and shear force resultants, including surface stress and nonlocal
206
effects, can be written by following equations:
207
 xx  
1  w

1  w

s
xx
(11c)
(11d)
(11e)
h /2
h
      xxnl zdz ,
2  h /2
s
xx
(12a)
11
h /2
208
h
 yy   yys    yys      yynl zdz ,
2  h /2
209
h
 xy   xys    xys      xynl zdz ,
2  h /2
210
 yx  
211
S x   xzs    xzs    k 2
(12b)
h /2
s
yx
(12c)
h /2
h
      yxnl zdz ,
2  h /2
s
yx
h /2

(12d)
 xznl dz ,
(12e)
 yznl dz ,
(12f)
 h /2
212
S y   yzs    yzs    k 2
h /2

 h /2
213
where k 2 is the shear correction factor. Consequently, by using Eqs. (1a)–(1e), (3a)–
214
(3f), (10a)–(10c), (11a)–(11e) and (12a)–(12f), assuming that the top and bottom
215
surfaces have the same material properties, the bending moment and shear force
216
resultants can be simplified to the following relations:
217
1    

 y 
 y

  x
 x
h2
1  2   2u0  0 
  0   0 


 D
2
x
y 
y

 x

 ,(13a)

218
1  2   yy 
 y
 y

 
 
h2
1   2   2u0  0 
  0   0  x   D  x  

2
y
x 
x

 y

 , (13b)

1    
   x  y
h2

1   2  u0 


2
x
  y

2
xx
2
219
xy

 y 



  0
x 


  x  y 
1
 D 1    


2
x 
 y
12
,
(13c)
1    
2
yx
220

   x  y 
 x 
h2

1   2  u0 



  0
2
x 
y 

  y

  x  y 
1
 D 1    


2
x 
 y
,
(13d)
221
1    S
x
 2 0 1   2 
w
 w

 k 2Gh 
 x  ,
x
 x

(13e)
222
1    S
y
 2 0 1   2 
 w

w
 k 2Gh 
 y  .
y
 y

(13f)
223
The equilibrium equations of the Mindlin plate resting on two-parameter elastic
224
foundation are given as
225
S x S y

 q  kw w  Gb2 w ,
x y
(14a)
226
 xx  yx

 Sx  0 ,
x
y
(14b)
227
2
2
 yy
y

 xy
x
 Sy  0 .
(14c)
228
Finally, by substituting Eqs. (13a)–(13f) into Eqs.(14a)–(14c), the governing
229
equations of the nanoplates, including the surface and nonlocal effects based on
230
Mindlin plate theory, can be expressed as
231
  x  y

k 2Gh 

  2 w   2 0 1   2   2 w
y
 x

 1   2  q  1   2  kw w  Gb 2 w 
13
,
(15a)
232
233

 2 y
 2 x
 2 
h2
2 
1     2u0  0  2   u0  0   0 
  u0   0  2 x  

2
x
xy
y 

  2 x 1
 2 y 
 2 x 1
 w

2 w
D  2  1    2  1   
 k 2Gh 
  x   0,
  2 0 1   
2
y
2
xy 
x
 x

 x

 2 y
 2 y 
 2 x
h2
2 
1     2u0  0  y 2   u0  0   0  xy   u0   0  x 2  
2


  2 y 1
 2 y 1
 w

 2 x 
2 w
D  2  1    2  1   
 k 2Gh 
  y   0.
  2 0 1   
2
x
2
xy 
y
 y

 y
(15b)
(15c)
234
Note that if the surface effect is neglected (  0 , u0 and 0 are all set to zero), Eqs.
235
(15a)–(15c) agree with the work of Wang and Li (2012).
236
237
Solution of the problem
238
In this section, the governing differential equations for bending behavior of the
239
nanoplates, including surface stress and nonlocal effects, have been solved by
240
Navier’s approach for simply supported boundary conditions. The simply supported
241
boundary conditions for a rectangular plate are
242
for the classical plate:
b) w  0 and M yy  0 at y  0 and b ,
243
244
245
a) w  0 and M xx  0 at x  0 and a ,
for the Mindlin plate:
c) w  0, M xx  0 and  y  0 at x  0 and a ,
d) w  0, M yy  0 and  x  0 at y  0 and b ,
246
where a and b are the length and width of the nanoplates, respectively. The
247
displacement solutions can be expressed as
14
248
for the classical plate:
249
C
wC   wmn
sin  m x sin  n y ,


(16)
m 1 n 1
250
for the Mindlin plate:
251
M
wM   wmn
sin  m x sin  n y ,


(17a)
m 1 n 1

252

 x   X mn cos  m x sin  n y ,
(17b)
m 1 n 1

253

 y   Ymn sin  m x cos  n y ,
(17c)
m 1 n 1
254
where  m  m / a and n  n / b , m and n denote the half-wave numbers in the x
255
and y directions, respectively. From Eqs. (16) and (17a)–(17c), the simply supported
256
boundary conditions for the nanoplates will be satisfied automatically. Furthermore,
257
the external load can be also expressed by the Fourier series as
258
q   qmn sin  m x sin  n y .


(18)
m 1 n 1
259
260
The displacement coefficient of the classical plate
261
Substituting Eqs. (16) and (18) into Eq. (9), the displacement coefficient of the
262
classical plate is represented by
263
w 
C
mn


qmn 1    m2   n2 
C1   
2
m

2 3
n
 C2   
2
m

2 2
n
 C3   
2
m
15
2
n
k
,
w
(19)
264
where C1   h 2 (2u0  0 ) / 2 , C2  D  h 2 (2u0  0 ) / 2  2  0  Gb
265
and C3  2 0  Gb   kw .
266
From Eq. (19), by setting 0 , u0 , 0 and  to zero, the coefficient of the traditional
267
solution is obtained, which does not include the effects of surface stress and nonlocal
268
elasticity:
269
CO
wmn

270
271
D   
2
m

2 2
n
qmn
 Gb   
2
m
2
n
k
.
(20)
w
C
The displacement ratio ( Rmn
) in the classical plate theory is defined as
C
wmn
.
CO
wmn
C
Rmn

(21)
272
273
The displacement coefficient of the Mindlin plate
274
Substituting Eqs. (17a)–(17c) and (18) into Eqs. (15a)–(15c) generates a linear
275
system
276
 a11
a
 21
 a31
277
where aij , i, j  1, 2,3 , as given in the Appendix. Then, by solving the above equation,
278
the displacement coefficient of the Mindlin plate can be written as
279
M
wmn

a12
a22
a32
M
 qmn 
a13   wmn

  

a23   X mn    0 
a33   Ymn   0 
(22)
qmn  a22 a33  a23a32 
a11  a22 a33  a23a32   a12  a23 a31  a21a33   a13  a21a32  a22 a31  .
16
(23)
280
The displacement coefficient without the effects of surface stress and nonlocal
281
elasticity can be calculated by setting 0 , u0 , 0 and  in the expressions of aij to zero.
282
By excluding the effect of size-dependent behavior in Eq. (23), the traditional
283
M
MO
coefficient wmn
is obtained. Furthermore, the displacement ratio ( Rmn
) in Mindlin
284
plate theory is also defined as
285
M
Rmn

M
wmn
MO .
wmn
(24)
286
287
Numerical example and discussion
288
For the present study, an aluminum nanoplates is embedded in an elastic medium
289
with the following material properties (Assadi, 2013);
290
 0  0.910 N / m , 0  5.26 N / m , u0  2.26 N / m . It is assumed that h  2 nm , the
291
shear correction factor is k 2   2 / 12 , and the elastic medium properties (Wang and
292
Li, 2012) are the Winkler foundation modulus, kw  1.13 1018 Pa / m , and the stiffness
293
of the shearing layer, Gb  2 N / m . Nanoplates with the effects of either nonlocal
294
elasticity or surface stress are considered separately. Meanwhile, nanoplates with a
295
combination of these effects are also examined.
296
C
M
The relationships between the displacement ratio ( Rmn
or Rmn
) and the non-
297
dimensional length ( a / h ) of nanoplates with different half-wave numbers are shown
298
in Fig. 1(a)–(h) and Fig. 2(a)–(h) for classical and Mindlin plate theory, respectively.
17
E  68.5 GPa,   0.35 ,
299
It is apparent that all graphs show a similar trend. For low values of the non-
300
dimensional length ( a / h ), there is a large difference in the displacement ratio
301
between each small-scale effect; whereas the ratio gradually converges to 1, namely
302
to the solution of classical plate theory, with increasing non-dimensional length
303
( a / h ). Based on these results, both surface stresses and nonlocal elasticity have a
304
greater influence on nanoplates with smaller sizes. Furthermore, if only the effect of
305
surface stress is considered, the displacement ratio will always be less than 1. This
306
means that the surface stresses improve the bending stiffness of nanoplates. In
307
contrast, the displacement ratio will always be higher than 1 when only the effect of
308
nonlocal elasticity is taken into account. It is evident that the influence of the
309
nonlocal parameter is predominant when high values of the half-wave numbers
310
( m, n ) are examined. From the aforementioned results, it is interesting to highlight
311
the incorporate effects of surface stress and nonlocal elasticity from the proposed
312
formulations. Due to the opposite results of each individual effect, the displacement
313
ratio tapers off when the combination of both effects is considered.
314
To evaluate the influence of shear deformation on nanoplates, Fig. 2(a)–(h), plotted
315
according to Mindlin plate theory, can be compared with Fig. 1(a)–(h), based on
316
M
classical plate theory. It is obvious that the displacement ratio Rmn
seems to have a
317
C
value less than the displacement ratio Rmn
when the same parameters, the half-wave
318
numbers ( m, n ), the non-dimensional length ( a / h ) and the parameters of small scale
319
are concerned. This implies that the small scale of nanoplates has less effect on
18
320
Mindlin plates than on classical plates.
321
To demonstrate the effect of residual surface tension, Fig.3 and Fig. 4 are established
322
to compare the results between including and excluding the residual surface effects
323
for both classical and Mindlin plates, respectively. It is evidently observed that
324
including the residual surface has significantly affected in increasing the
325
C
M
displacement ratios Rmn
and Rmn
. The effect on the bending of nanoplates has been
326
also reported by Miller and Shenoy (2000).
327
Finally, the relationships between the displacement ratio and the elastic foundation
328
parameters are presented in Figs. 5 and 6 and Figs. 7 and 8 for classical and Mindlin
329
plate theory, respectively. A decreasing trend can be clearly observed in the
330
C
M
displacement ratios Rmn
and Rmn
as the values of Winkler foundation stiffness ( k w )
331
and shear layer stiffness ( Gb ) increase.
332
333
Conclusions
334
In this study, the governing equations for the bending analysis of nanoplate models,
335
including both surface stress and nonlocal elasticity effects, are derived. Thin and
336
moderately thick nanoplates and the two-parameter elastic foundation are also
337
considered in the formulations. The analytical solution from the present study could
338
be applied as a benchmark for comparison with other numerical solutions in future
339
work.
340
The results of this investigation can be summarized as follows:
19
341

When the size of a structure is in the order of nanometers, the surface stress
342
and nonlocal elasticity have great significance on bending behavior, which
343
cannot be neglected.
344

If the surface stress effect is examined in particular, the displacement ratio
345
will always be less than 1. This means that the surface stresses improve the
346
bending stiffness of nanoplates. In contrast, the displacement ratio will always
347
be higher than 1 when only the effect of nonlocal elasticity is taken into
348
account.
349

Because opposite results of each individual effect are observed, the
350
displacement ratio will have a value between those obtained from each effect
351
when the combined surface stress and nonlocal elasticity are considered.
352

classical plates.
353
354

357
The effect of residual surface tension leads to improvement the bending
stiffness of nanoplates.
355
356
The small scale of nanoplates has less effect on Mindlin plates than on

The influence of surface stress and nonlocal elasticity decreases when
nanoplates rest on an elastic foundation.
358
359
Acknowledgement
360
This work was supported by the Higher Education Research Promotion and National
20
361
Research University Project of Thailand, Office of the Higher Education
362
Commission.
363
364
Appendix
365
Coefficients a11 through a33 :
 k Gh  2 

2
0
2
m
  n2   2 0   m2   n2 
2
 Gb  m2   n2   kw ,
(A1)
366
a11
367
a12 
k 2Gh m
,
1    m2   n2 
(A2)
368
k 2Gh n
a13 
,
1    m2   n2 
(A3)
369
a21  2 0  m    m3   m  n2   k 2Gh m ,
1     
2
m

a22 
370
2
n



(A4)



h2
h2
2
4
2 2
2
u










 0 0 m  m m n 
 u0   0   n2     n4   m2  n2 
2
2
 1    2

 D
 n   m2   k 2Gh,
 2



h2
D
 u0  0   0   m n   m3  n   m  n3   1    m  n ,
2
2
(A5)
371
a23 
372
a31  2 0  n    m2  n   n3   k 2Gh n ,
(A7)
373
a32  a23 ,
(A8)


21
(A6)
a33 
374




h2
h2
2
4
2 2
2
u










 0 0 n  n m n 
 u0   0   m2    m4   m2  n2 
2
2
 1    2

 D
 m   n2   k 2Gh.
 2

(A9)
375
376
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25
1.4
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
1.3
1.2
a / b  1, m  1, n  1
1.1
1.0
0.9
0.8
a)
0.7
5
6
7
8
9
454
10 11 12 13 14 15 16 17 18 19 20
a/h
2.4
C
The displacement ratio, Rmn
2.2
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
2.0
1.8
a / b  1, m  2, n  1
1.6
1.4
1.2
1.0
0.8
0.6
455
b)
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
26
3.5
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
3.0
2.5
a / b  1, m  2, n  2
2.0
1.5
1.0
0.5
c)
5
6
7
8
9
456
10 11 12 13 14 15 16 17 18 19 20
a/h
4.5
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
4.0
3.5
3.0
a / b  1, m  3, n  1
2.5
2.0
1.5
1.0
0.5
457
d)
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
27
2.4
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
2.2
2.0
1.8
a / b  2, m  1, n  1
1.6
1.4
1.2
1.0
0.8
0.6
e)
5
6
7
8
9
458
10 11 12 13 14 15 16 17 18 19 20
a/h
3.5
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
3.0
2.5
a / b  2, m  2, n  1
2.0
1.5
1.0
0.5
459
f)
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
28
8.0
C
The displacement ratio, Rmn
7.0
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
6.0
5.0
a / b  2, m  2, n  2
4.0
3.0
2.0
1.0
0.0
g)
5
6
7
8
9
460
10 11 12 13 14 15 16 17 18 19 20
a/h
6.0
Surface effect only
Surface and nonlocal effects,   2 nm 22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm 2
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
5.0
4.0
a / b  2, m  3, n  1
3.0
2.0
1.0
h)
0.0
461
462
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
Figure 1(a)-(h). Variation of the displacement ratio as a non-dimensional length of
29
463
nanoplates for the classical plate which different length-to-width ratio and half-wave
464
numbers
465
1.4
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
1.3
1.2
a / b  1, m  1, n  1
1.1
1.0
0.9
a)
0.8
466
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
30
2.2
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
2.0
1.8
a / b  1, m  2, n  1
1.6
1.4
1.2
1.0
0.8
b)
5
6
7
8
9
M
The displacement ratio, Rmn
467
468
3.4
3.2
3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
10 11 12 13 14 15 16 17 18 19 20
a/h
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
a / b  1, m  2, n  2
c)
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
31
3.0
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
2.8
2.6
2.4
2.2
a / b  1, m  3, n  1
2.0
1.8
1.6
1.4
1.2
d)
1.0
0.8
5
6
7
8
9
469
10 11 12 13 14 15 16 17 18 19 20
a/h
2.2
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
2.0
1.8
a / b  2, m  1, n  1
1.6
1.4
1.2
e)
1.0
0.8
470
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
32
3.0
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
2.8
2.6
2.4
2.2
a / b  2, m  2, n  1
2.0
1.8
1.6
1.4
1.2
f)
1.0
0.8
5
6
7
8
9
M
The displacement ratio, Rmn
471
472
5.8
5.4
5.0
4.6
4.2
3.8
3.4
3.0
2.6
2.2
1.8
1.4
1.0
0.6
10 11 12 13 14 15 16 17 18 19 20
a/h
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
a / b  2, m  2, n  2
g)
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
33
4.2
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
3.8
3.4
3.0
a / b  2, m  3, n  1
2.6
2.2
1.8
1.4
h)
1.0
0.6
473
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
474
Figure 2(a)-(h).Variation of the displacement ratio as a non-dimensional length of
475
nanoplates for the Mindlin plate which different length-to-width ratio and half-wave
476
numbers.
34
1.05
m  1, n  1
C
The displacement ratio, Rmn
1.00
0.95
0.90
0.85
m  2, n  1
0.80
0.75
0.70
477
Including residual surface effect
Without residual surface effect
a / b  1,   0
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
478
Figure 3. The effect of residual surface tension on the displacement ratio for the
479
classical plate which different half-wave numbers.
35
1.05
m  1, n  1
The displacement ratio, RmnM
1.00
0.95
0.90
m  2, n  1
0.85
0.80
480
Including residual surface effect
Without residual surface effect
a / b  1,   0
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
a/h
481
Figure 4. The effect of residual surface tension on the displacement ratio for the
482
Mindlin plate which different half-wave numbers.
36
2.0
Surface effect only
Surface and nonlocal effects,   2 nm
Surface and nonlocal effects,   4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
1.8
1.6
2
2
2
2
1.4
1.2
1.0
0.8
a  b  10 nm, h  2 nm,
0.6
0.4
m  n  1, Gb  0 N / m
0.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
k w  1.13  1018 Pa / m
483
484
Figure 5. Variation of the displacement ratio as Winkler foundation stiffness for the
485
classical plate.
37
2.0
Surface effect only
Surface and nonlocal effects,   2 nm
Surface and nonlocal effects,   4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm
C
The displacement ratio, Rmn
1.8
2
2
2
1.6
2
1.4
1.2
1.0
0.8
a  b  10 nm, h  2 nm,
0.6
0.4
m  n  1, k w  0 Pa / m
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Gb  2.0 N / m
486
487
Figure 6. Variation of the displacement ratio as shear layer stiffness for the classical
488
plate.
38
1.8
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
1.6
1.4
1.2
1.0
0.8
a  b  10 nm, h  2 nm,
m  n  1, Gb  0 N / m
0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
k w  1.13  10 Pa / m
18
489
490
Figure 7. Variation of the displacement ratio as Winkler foundation stiffness for the
491
Mindlin plate.
39
1.8
Surface effect only
Surface and nonlocal effects,   2 nm22
Surface and nonlocal effects,  2 4 nm
Nonlocal effect only,   2 nm
Nonlocal effect only,   4 nm2
M
The displacement ratio, Rmn
1.6
1.4
1.2
1.0
0.8
a  b  10 nm, h  2 nm,
m  n  1, k w  0 Pa / m
0.6
492
0.0
0.5
1.0
1.5
2.0
Gb  2.0 N / m
2.5
3.0
3.5
4.0
493
Figure 8. Variation of the displacement ratio as shear layer stiffness for the Mindlin
494
plate.
40