Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
Nonlinear vibration of a nanoplate embedded on a Pasternak-type foundation using
nonlocal continuum theory
P. Soltania, V. Kamali a, O. Pashaei Narenjbon a A. Farshidianfarb
a
Department of Mechanical Engineering, Semnan branch, Islamic Azad University, Semnan, Iran
b
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
*payam.soltani@gmail.com, soltanip2@asme.org
Abstract: Nonlocal plate continuum model is utilized to investigate the nonlinear vibration behaviour of a singlelayer nanoplate. The isotropic nanoplate is assumed to be embedded on a Pasternak-type elastic foundation with the
simply supported boundary conditions. The Hamilton’s principle is applied to derive the governing equation of motion,
and the nonlinear frequency is obtained analytically using perturbation approach. The results indicate that the nonlinear
frequency is significantly dependent on the maximum amplitude. Furthermore, the nonlinear frequency increases with
an increase in the nonlocal parameter, which means that the nonlinear frequency based on the local plate theory are
underestimated. Furthermore, for arbitrary maximum amplitude, the variations of the nonlinear frequency against the
nonlocal parameter, aspect ratio, Pasternak-type foundation constants, and size effects of the nanoplate are investigated.
The present communication may be useful for designing nanomechanical devices and nano-electromechanical systems.
Keywords: Nanoplate; Nonlinear frequency; Nonlocal elasticity theory
Introduction
Since nano structures such as nanobeams, nanorods and
nanoplats demonstrate superior and exceptional
mechanical, chemical, electrical, and thermal properties,
they can be used in many novel applications in
nanomechanical devices and micro- or nanoelectromechanical systems (MEMS or NEMS) [1, 2].
Generally, three methods have been developed to
simulate the mechanical behaviour of nanostructures
namely atomistic method, hybrid atomistic-continuum
mechanics, and elastic continuum mechanics. Unlike the
atomistic method and hybrid atomistic-continuum
mechanics that are involved complex computational
processes and are still formidable and expensive,
especially for large-scale nanostructures, the continuum
mechanics approach [3] is less computationally
expensive. Moreover, the classical or local elasticity
theories are size independent. As the size of
nanostructures is very small, it is significant that smallscale effects are taken into account. Therefore, Eringen’s
nonlocal elasticity theory should be applied to represent a
more accurate continuum model in nano-scales.
Vibrating nanoplates play important roles in the thin film
elements, especially in nanosheet [4], and paddele-like
resonators[5] and understanding the dynamical
behaviours of nanoplates are the first step for designing
of these devices [6-8].
Practically, most of the structures in mechanics such as
graphen-based resonators show linear behaviour up to
displacements on the order of its thickness. However, for
the oscillations with large amplitudes, the linear treatment
may be too inaccurate and the accuracy can often be
improved sufficiently by carrying out further
approximation via geometrically nonlinear models. In the
present study, a nonlocal plate model has been developed
for an isotropic single layer graphen sheet embedded in
an elastic medium with simply supported boundary
1742
conditions. The nonlinear governing equation of motion
is derived based on classical plate theory (CLPT) and
according to the von Karman-type of kinetic nonlinearity.
The nonlinear equation is solved by the Navier-type
solution and perturbation approach. The effects of the
maximum amplitude, stiffness of the surrounding
medium, aspect ratio of the nanoplate, and the nonlocal
parameter on the dimensionless nonlinear frequency are
considered and discussed widely.
Modeling
The coordinate system used for the nanoplate in an elastic
medium is shown in Fig. 1. Origin is chosen at one corner
of the nanoplate. The x, y, z coordinates of the axes are
taken along the length a, width b and thickness h,
respectively. For the isotropic thin plane E, ν, and ρ are
the Young’s modulus, possion’s ratio, and density,
respectively. The surrounding elastic medium is
simulated as a Pasternak-type foundation with the
Winkler foundation stiffness k1 and the shearing layer
stiffness k2.
Fig.1. single layered nanoplate embedded on a Pasternak-type elastic
medium.
By using the Hamilton's principle and CLPT, and based
on the nonlocal elasticity theory, the nonlinear governing
equation of motion can be calculated as follows:
Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
Eh3 4 w( x, y, t )
2
12(1  )
2
 2 ( x, y, t )  2 w( x, y, t )  2 ( x, y, t )  2 w( x, y, t )

xy
xy
y 2
x 2
 k1w( x, y, t )  k 2 2 w( x, y, t )  h

3
W ( t )  Wmax cos(  NL .t ), NL 2     .W 2 max
8
  2 ( x, y, t )  2 w( x, y, t )
 (1  (e0 a) 2  2 ) 
x 2
y 2

h
3
12
4
(
 w( x, y, t )
2
x t
2

 2 w( x, y, t )
(1)
t 2
 w( x, y, t ) 
)  0
y 2 y 2 
In the above equation,  2 is the Laplace operator, w(x,y,t)
is the transverse displacement component in the z
direction and t is the time. In addition, e0a represents a
nonlocal parameter where e0 and a are the material
constant and an internal characteristic length, in that
order. φ(x,y,t) stands for the stress function and are
defined as:
(2)
Where Nx, Ny and Nxy are stress resultants in the indicated
directions. It should be noted that when the nonlocal
parameter e0a and the stiffness parameters of the
foundation k1 and k2 are set to be zero in Eq. 1, the local
equation of motion of a plate is easily obtained [9].
To determine the nonlinear frequency of the model, the
Navier’s approach is utilized for simply supported
boundary conditions (SSSS). The nonlinear governing
equations (Eq. 1) are satisfied by the following middle
surface displacements assumptions:

w( x, y, t ) 

nx
W (t ) sin( a
m 1 n 1
my
) sin(
)
b
Results and Discussion
In the present study, the nonlinear frequency ωNL of a
nanoplate is obtained analytically according to the
nonlocal elasticity theory through Eq. 6. The variation of
maximum amplitude Wmax can influence the nonlinear
frequency efficiently. To conduct a parametric study, the
following geometrical and mechanical properties have
been assumed: E=1.06 TPa, ρ=2250 kg/m3, h=0.34 nm,
ν=0.25, a/b=0.5 [6]. Furthermore, k1=1 Pa/m and k2=1
N/m represent the stiffness of the foundation.
Figs. 2-4 show the nonlinear frequency ratio ωNL/ ωL as a
function of the maximum amplitude Wmax, while the
impacts of a single specific parameter have been studied
in each figure. It can be seen from all these figures that
the amplitude affect the nonlinear behaviour of the model
strongly and the nonlinear frequency ratio increases with
increasing the maximum amplitude.
(3)
Eh2
2nx
 4 4
)
 m a cos(
a
32n m 2 a 2b 2 
2ny 
 n 4b 4 cos(
) W (t ) 2
b 
 ( x, y , t ) 
In which Wmax is the maximum nonlinear vibrational
amplitude. To show the net effects of the nonlinearity of
the model, the nonlinear frequency ratio can be defined as
the ratio of the nonlinear frequency ωNL to the linear
frequency ωL where L   NL Wmax0 .
4
 2
 2
 2
 Nx , 2  N y ,
  N xy
y 2
x
yx
(6)
2
(4)
(a)
m and n are the half wave numbers in the x and y
direction, respectively. Applying Eqs. 3 and 4 into Eq. 1
and multiplying both sides of the resulting equation by
“sin(nπx/a)sin(nπy/b)”, and then integrating the result
over the interval [0, a] and [0, b]; the nonlinear
differential equation of the first mode (m, n =1) can be
calculated as:
d 2W (t )
dt 2
 W (t )  W (t )3  0
(5)
Where the parameters α and β are constant values and are
not reported here for the sake of brevity. Eq. 5 is the
famous nonlinear Duffing equation and the nonlinear
natural frequency ωNL can be calculated using
perturbation method as follow [10]:
(b)
Fig. 2. The dimensionless nonlinear frequency against the maximum
amplitude for different (a) Winkler stiffness k1 (b) shearing layer
stiffness k2.
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Proceedings of the 4th International Conference on Nanostructures (ICNS4)
12-14 March, 2012, Kish Island, I.R. Iran
Fig. 3. The dimensionless nonlinear frequency against the maximum
amplitude for different nonlocal parameter e0a.
Moreover, Figs. 2 indicate that with increasing the
stiffness of the elastic foundation, the nonlinear frequency
ratio decreases and the nonlinearity of the model reduces.
The effects of the nonlocal parameter e0a on the nonlinear
frequency ratio are shown in Fig. 3. As the figure reveals,
an increase in e0a causes the model to show the nonlinear
behaviour more severely.
Fig. 4 shows that the size of the nanoplate affects the
nonlinear frequency ratio. It is apparent from the figure
that as a/b increases, the nonlinear frequency ratio
increases and the nonlinear behaviour of the model
becomes more significant in nanoplates with larger
widths.
Fig. 4. The dimensionless nonlinear frequency against the maximum
amplitude for different aspect ratio a/b.
Conclusions
On the basis of the nonlocal theory and classical plate
theory, the nonlinear free vibration of a single layered
graphene sheet with simply supported boundary
conditions is carried out. Considering the von karman
hypothesis, the nonlinear equations of motion have been
obtained using the Hamilton’s principle and solved by the
Navier type solution and perturbation method. Detailed
results demonstrated that the nonlinear frequency rises by
increasing the maximum amplitude, especially for the
large values of the nonlocal coefficient, large values of
the aspect ratio and more flexible medium foundation.
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