Embed Size (px)
Citation preview
ORIGINAL ARTICLE
Vibration of gold nanobeam with variable thermal conductivity:state-space approach
H. M. Youssef
Received: 25 April 2012 / Accepted: 17 September 2012 / Published online: 7 October 2012
� The Author(s) 2012. This article is published with open access at Springerlink.com
Abstract The non-Fourier effect in heat conduction and the
coupling effect between temperature and strain rate are the
two significant effects in the nanoscale beam. In the present
work, the solution of vibration of gold nanobeam resonator
induced by thermal shock is developed in the context of
generalized thermoelasticity with variable thermal conduc-
tivity. State-space and Laplace transform methods are used
to determine the lateral vibration, the temperature, the dis-
placement, the strain, the stress, and the strain–stress energy.
The numerical results have been studied and represented
graphically with some comparisons to stand on the effects of
the variability of thermal conductivity.
Keywords Thermoelasticity � Euler–Bernoulli equation �Gold nanobeam � State-space approach
Introduction
Many attempts have been made to investigate the elastic
properties of nanostructured materials using atomistic
simulations. Diao et al. (2004) studied the effect of free
surfaces on the structure and elastic properties of gold
nanowires by atomistic simulations. Although the atomistic
simulation is a good way to calculate the elastic constants
of nanostructured materials, it is only applicable to
homogeneous nanostructured materials (e.g., nanoplates,
nanobeams, nanowires, etc.) with limited number of atoms.
Moreover, it is difficult to obtain the elastic properties of
the heterogeneous nanostructured materials using atomistic
simulations. For these and other reasons, it is prudent to
seek a more practical approach. One such approach would
be to extend the classical theory of elasticity down to the
nanoscale by including in it the hitherto neglected surface/
interface effect. For this it is necessary first to cast the latter
within the framework of continuum elasticity.
Recently and due to their many important technological
applications, nanomechanical resonators have attracted
considerable attention. Accurate analysis of various effects
on the characteristics of resonators, such as resonant fre-
quencies and quality factors, is crucial for designing high-
performance components. Many authors have studied the
vibration and heat transfer process of beams. Kidawa
(2003) has studied the problem of transverse vibrations of a
beam induced by a mobile heat source. The analytical
solution to the problem was obtained using the Green’s
functions method. However, Kidawa did not consider the
thermoelastic coupling effect. Boley (1972) analyzed the
vibrations of a simply supported rectangular beam sub-
jected to a suddenly applied heat input distributed along its
span. Manolis and Beskos (1980) examined the thermally
induced vibration of structures consisting of beams,
exposed to rapid surface heating. They have also studied
the effects of damping and axial loads on the structural
response. Al-Huniti et al. (2001) investigated the thermally
induced displacements and stresses of a rod using the
Laplace transformation technique. Ai Kah Soh et al. (2008)
studied the vibration of micro/nanoscale beam resonators
induced by ultra-short-pulsed laser by considering the
thermoelastic coupling term in Sun and Fang (2008). The
propagation characteristics of the longitudinal wave in
H. M. Youssef (&)
Mechcanical Department, Faculty of Engineering,
Umm Al-Qura University, P.O. Box 5555, Makkah,
Saudi Arabia
e-mail: [email protected]
H. M. Youssef
Mathematical Department, Faculty of Education,
Alexandria University, Alexandria, Egypt
123
Appl Nanosci (2013) 3:397–407
DOI 10.1007/s13204-012-0158-9
nanoplates with small-scale effects are studied by Wang
et al. (2010).
The uncoupled theory of thermoelasticity (the classical
thermoelasticity) predicts two phenomena not compatible
with physical observations. First, the equation of heat con-
duction of this theory does not contain any elastic terms.
Second, the heat equation is of a parabolic type, predicting
infinite speeds of propagation for heat waves. Biot (1956)
introduced the theory of coupled thermoelasticity to over-
come the first shortcoming. The governing equations for this
theory are coupled, eliminating the first paradox of the
classical theory. However, both theories share the second
shortcoming since the governing equations for the coupled
are of the mixed: parabolic-hyperbolic. When very fast
phenomena and small structure dimensions are involved, the
classical law of Fourier becomes inaccurate. Modern tech-
nology has enabled the fabrication of materials and devices
with characteristic dimensions of a few nanometers. Exam-
ples are super lattices, nanowires, and quantum dots. At these
length scales, the familiar continuum Fourier law for heat
conduction is expected to fail due to both classical and
quantum size effects (Rao 1992). The generalizations to the
coupled theory were introduced by Lord and Shulman
(1967), who obtained a wave-type heat equation by postu-
lating a new law of heat conduction to replace the classical
Fourier’s law. Since the heat equation of this theory is of the
wave-type, it automatically ensures finite speeds of propa-
gation for heat and elastic waves. The remaining governing
equations for this theory, namely, the equations of motion
and constitutive relations, remain the same as those for the
coupled and the uncoupled theories. Among many applica-
tions, the studying of the thermoelastic damping in MEMS/
NEMS has been improved in Sun et al. (2006), Fang et al.
(2006), and Duwel et al. 2003). A more sophisticated model
is then needed to describe the thermal conduction mecha-
nisms in a physically acceptable way.
The physical property of a solid body related to applica-
tion of heat energy is defined as a thermal property. Thermal
properties explain the response of a material to the applica-
tion of heat, and thermal conductivity is one of the most
important thermal properties. Thermal conductivity K is
ability of a material to transport heat energy through it from
high-temperature region to low-temperature region and it is a
microstructure sensitive property whose values range from
20 to 400 for metals, from 2 to 50 for ceramics, and for
polymers an order of 0.3. Heat is transported in two ways:
electronic contribution and vibrational (phonon) contribu-
tion. In metals, electronic contribution is very high. Thus
metals have higher thermal conductivities. It is same as
electrical conduction. Both conductivities are related
through Wiedemann–Franz law: K ¼ t LT where L is Lor-
entz constant and t is the electrical conductivity. As different
contributions to conduction vary with temperature, the above
relation is valid to a limited extension for many metals. With
increase in temperature, both number of carrier electrons and
contribution of lattice vibrations increase. Thus thermal
conductivity of a metal is expected to increase. However,
because of greater lattice vibrations, electron mobility
decreases. The combined effect of these factors leads to very
different behavior for different metals. For example, thermal
conductivity of iron initially decreases and then increases
slightly; thermal conductivity decreases with increase in
temperature for aluminum, whereas it increases for platinum
and gold (Zhang 2007). The question as to the effects of these
variations on the lateral vibration, the temperature, the dis-
placement, the strain, the stress, and the strain–stress energy
distributions in nanobeam resonator arises.
Youssef used the state-space approach to solve a prob-
lem of generalized thermoelasticity for an infinite material
with a spherical cavity and variable thermal conductivity
subjected to ramp-type heating (Zhang 2007). The effect of
variability of the thermal conductivity on all the studied
fields is discussed in Youssef and El-Bary (2006a, b, c) and
Ezzat and Youssef (2009). Youssef and Elsibai discussed
the vibration of gold nanobeam induced by different types
of thermal loading (Youssef and Elsibai 2010).
In particular, the state-space approach is useful because
(1) linear systems with time-varying parameters can be
analyzed in essentially the same manner as time-invariant
linear systems, (2) problems formulated by state-space
methods can easily be programmed on a computer, (3) high-
order linear systems can be analyzed, (4) multiple input—
multiple output systems can be treated almost as easily as
single input–single output linear systems, and (5) state-space
theory is the foundation for further studies in such areas as
nonlinear systems, stochastic systems, and optimal control
(Ezzat 2008). For solving coupled thermoelastic problems
using the state-space approach in which the problem is
rewritten in terms of state-space variables, namely, the
temperature, the displacement and their gradients, has been
developed by Bahar and Hetnarski (1977a, b, 1978).
In the present work, the solution of vibration of gold
nanobeam resonator induced by thermal shock is devel-
oped in the context of generalized thermoelasticity with
variable thermal conductivity. State-space and Laplace
transform methods are used to determine the lateral
vibration, the temperature, the displacement, the strain, the
stress, and the strain–stress energy. The effect of the var-
iability of thermal conductivity has been studied and rep-
resented graphically with some comparisons.
Formulation of the problem
Although challenging, creating a beam with a rectangular
cross section is the easiest when compared with other cross
398 Appl Nanosci (2013) 3:397–407
123
sections. Consider small flexural deflections of a thin
elastic beam of length ‘ð0� x� ‘Þ , width b � b2� y� b
2
� �
and thickness h � h2� z� h
2
� �as in Fig. 1 for which the x,
y, and z axes are defined along the longitudinal, width, and
thickness directions of the beam, respectively. In equilib-
rium, the beam is unstrained, unstressed, no damping
mechanism present, and at temperature T0 everywhere
(Youssef and Elsibai 2010).
In the present study, the usual Euler–Bernoulli assump-
tion (Youssef and Elsibai 2010) is adopted, i.e., any plane
cross-section, initially perpendicular to the axis of the beam
remains plane and perpendicular to the neutral surface dur-
ing bending. Thus, the displacement components u; v;wð Þare given by
u x; y; z; tð Þ ¼ �zow x; tð Þ
o x; v x; y; z; tð Þ ¼ 0
and w x; y; z; tð Þ ¼ w x; tð Þ: ð1Þ
Hence, the differential equation of thermally induced
lateral vibration of the beam may be expressed in the
following form (Youssef and Elsibai 2010):
o4 w
o x4þ qA
EI
o2 w
o t2þ aT
o2 MT
o x2¼ 0; ð2Þ
where E is Young’s modulus, I [= bh3/12] the inertial
moment about x axis, q the density of the beam, aT the
coefficient of linear thermal expansion, w x; tð Þ the lateral
deflection, x the distance along the length of the beam,
A ¼ hb is the cross section area and t the time, and MT is
the thermal moment, which is defined as
MT x; z; tð Þ ¼ 12
h3
Zh=2
�h=2
h x; z; tð Þ z dz; ð3Þ
where h x; z; tð Þ ¼ T � T0ð Þ is the dynamical temperature
increment of the resonator, in which T x; z; tð Þ is the tem-
perature distribution and T0 the environmental temperature.
According to Lord–Shulman model (L–S), the non-
Fourier heat conduction equation has the following form
(Youssef and Elsibai 2010):
o
o xK hð Þ o h
o x
� �þ o
o zK hð Þ o h
o z
� �
¼ 1 þ so
o
o t
� �qCE hð Þ _h þ bT0 _e
� �; ð4Þ
where e ¼ ouoxþ ov
oyþ ow
ozis the volumetric strain, CE hð Þ is
the specific heat with variable temperature at constant
volume, so the thermal relaxation time, K hð Þ the thermal
conductivity with variable temperature, q the density, b ¼EaT
1�2t in which t is Poisson’s ratio, and the dot above it
means partial derivative with respect to time.
Since we have the relation (Zhang 2007)
qCE hð Þ ¼ K hð Þj
; ð5Þ
where j is the thermal diffusivity,
Eq. (4) will take the following form:
o
o xK hð Þ o h
o x
� �þ o
o zK hð Þ o h
o z
� �
¼ 1 þ so
o
o t
� �K hð Þ
j_h þ bT0 _e
� �: ð6Þ
Considering the following mapping which converts the
nonlinear terms of the temperature to be linear (Zhang
2007; Youssef and El-Bary 2006a, b, c; Ezzat and Youssef
2009)
# ¼ 1
Ko
Zh
0
K nð Þ d n; ð7Þ
where Ko is the thermal conductivity at the normal case.
Differentiating Eq. (7) with respect to the coordinates x
and z, respectively, we get
K hð Þ o ho x
¼ Ko
o#
o xand K hð Þ o h
o z¼ Ko
o#
o z: ð8Þ
Differentiating the above equations with respect to the
coordinates x and z, respectively, we obtain
Ko
o2 #
o x2¼ o
o xK hð Þ o h
o x
� and Ko
o2 #
o z2¼ o
o zK hð Þ o h
o z
� :
ð9Þ
Differentiating Eq. (7) with respect to time, we get
Ko_# ¼ K hð Þ _h; ð10Þ
Appling the Eq. (8)–(10) in Eq. (6), we get
o2#
o x2þ o2#
o z2¼ o
o tþ so
o2
o t2
� �1
j#þ bT0
Ko
e
� �; ð11Þ
where there is no heat flow across the upper and lower
surfaces of the beam, so that ohoz¼ o#
oz¼ 0 at z ¼ �h=2 : For
a very thin beam and assuming the temperature varies in
x
y
z
Fig. 1 The gold nanobeam resonator
Appl Nanosci (2013) 3:397–407 399
123
terms of a sin pzð Þ function along the thickness direction,
where p ¼ p=h , gives (Youssef and Elsibai 2010):
# x; z; tð Þ ¼ #1 x; tð Þ sin pzð Þ: ð12Þ
Hence, Eq. (2) gives
o4 w
o x4þ qA
EI
o2 w
o t2þ 12aT
h3
o2#1
o x2
Zh=2
�h=2
z sin pzð Þdz ¼ 0: ð13Þ
To get Eq. (14), we used the linearity condition of
thermoelasticity such thathj jT0� 1, which gives
o2h1
o x2� o2#1
o x2: ð14Þ
Equation (11) takes the following form:
o2#1
o x2sin pzð Þ � p2#1 sin pzð Þ
¼ o
o tþ so
o2
o t2
� �1
j#1 sin pzð Þ � bT0
Ko
zo2w
o x2
� �: ð15Þ
After doing the integrations, Eq. (13) gives
o4 w
o x4þ qA
EI
o2 w
o t2þ 24aT
hp2
o2#1
o x2¼ 0: ð16Þ
In Eq (15), we multiply the both sides by z and integrate
with respect to z from � h2
to h2; then we obtain
o2#1
o x2� p2#1
� �¼ o
o tþ so
o2
o t2
� �e#1 �
bT0p2h
24Ko
o2w
o x2
� �;
ð17Þ
where e ¼ 1j :
Now, for simplicity we will use the following non-
dimensional variables (Youssef and Elsibai 2010):
x0;w0; h0ð Þ ¼ e co x;w; hð Þ ; t0; s0o� �
¼ e c2o t; soð Þ;
r0 ¼ rE; h01 ¼ h1
To
; c2o ¼ E
q: ð18Þ
Hence, we have
o4 w
o x4þ A1
o2 w
o t2þ A2
o2#1
o x2¼ 0; ð19Þ
and
o2#1
o x2� A3#1 ¼ o
o tþ so
o2
o t2
� �#1 � A4
o2w
o x2
� �; ð20Þ
where
A1 ¼ 12
h2; A2 ¼ 24atTo
p2h; A3 ¼ p2; A4 ¼ p2bh
24Koe:
We have dropped the prime for convenience.
Formulation the problem in the Laplace transform
domain
Application of the Laplace transform to Eq. (19) and (20)
results in the following formula:
�f sð Þ ¼ L f ðtÞ½ � ¼Z1
0
f tð Þe�stdt:
Hence, for the zero initial conditions of all the states
functions, we obtain the following system:
d4�w
dx4þ A1s2�w þ A2
d2 �#1
dx2¼ 0; ð21Þ
and
d2#1
dx2� A3
�#1 ¼ s þ sos2� �
�#1 � A4
d2�w
dx2
� �: ð22Þ
We will consider the function �g as follows:
d2�w
dx2¼ �g; ð23Þ
then, we have
d2 �#1
dx2¼ a1
�#1 � a2�g; ð24Þ
and
d2�gdx2
¼ �a3�w � a4�#1 þ a5�g; ð25Þ
where
a1 ¼ A3 þ s þ sos2� �
; a2 ¼ A4 s þ sos2� �
; a3 ¼ A1s2;
a4 ¼ A2 A3 þ s þ sos2� �
; a5 ¼ A2A4 s þ sos2� �
:
State-space formulation
Choosing as a state variable, the functions �w, �#1, �g, d�wdx
¼�w0; d �#1
dx¼ �#0
1 and d�gdx
¼ �g0 in the x direction, Eq. (23)–(25)
can be written in matrix form using the Bahar–Hetnarski
method (Youssef and Elsibai 2010; Ezzat 2008; Bahar and
Hetnarski 1977a, b, 1978):
d�Vðx; sÞdx
¼ AðsÞ�Vðx; sÞ; ð26Þ
where
�Vðx; sÞ ¼
�wðx; sÞ�#1ðx; sÞ�gðx; sÞ�w0ðx; sÞ�#0
1ðx; sÞ�g0ðx; sÞ
2
6666664
3
7777775
; ð27Þ
400 Appl Nanosci (2013) 3:397–407
123
and
AðsÞ ¼
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 1 0 0 0
0 a1 �a2 0 0 0
�a3 �a4 a5 0 0 0
2
6666664
3
7777775
: ð28Þ
The formal solution of Eq. (26) is given by
Vðx; sÞ ¼ exp½AðsÞ:x�Vð0; sÞ; ð29Þ
where
�Vð0; sÞ ¼
�wð0; sÞ�#1ð0; sÞ�gð0; sÞ�w0ð0; sÞ�#0
1ð0; sÞ�g0ð0; sÞ
2
6666664
3
7777775
:
The characteristic equation of the matrix A(s) has the
form
k6 � lk4 þ mk2 � n ¼ 0; ð30Þ
where
l ¼ a1 þ a5;m ¼ a1a5 � a2a4 þ a3; and n ¼ a1a3:
k21; k2
2 and k23 are the roots of the characteristic Eq. (30) and
satisfy the following relations:
k21 þ k
22 þ k
23 ¼ l; ð31aÞ
k21 k
22 þ k
22 k
23 þ k
21 k
23 ¼ m; ð31bÞ
k21 k
22 k
23 ¼ n: ð31cÞ
The Taylor series expansion for the matrix exponential
is given by
exp½AðsÞx� ¼X1
i¼0
½AðsÞx �i
i!: ð32Þ
Using the Cayley–Hamilton theorem (Youssef and
Elsibai 2010; Ezzat 2008; Bahar and Hetnarski 1977a, b,
1978), this infinite series can be truncated to
exp½AðsÞ:x� ¼ Lðx; sÞ ¼ a0I6 þ a1A þ a2 A2 þa3 A
3
þ a4 A4 þa5 A
5; ð33Þ
where I6 is the unit matrix of order 6 and a0 - a5 are some
parameters depending on s and x to be determined.
Using the Cayley–Hamilton theorem again, we obtain
expð�kixÞ ¼X5
j¼0
aj �kið Þ j; i ¼ 1; 2; 3: ð34Þ
The solution of this system gives aj; j ¼ 0; 1; 2; 3; 4; 5:
See the ‘‘Appendix’’.
Substituting from Eq. (32) into Eq. (33), we obtain the
matrix exponential in the form
exp½AðsÞ:x� ¼ Lðx; sÞ ¼ ½Lijðx; sÞ�; i; j ¼ 1; 2; 3; 4; 5; 6;
ð35Þ
where Lijðx; sÞ; i; j ¼ 1; 2; 3; 4; 5; 6 are defined in the
‘‘Appendix’’.
Now, we will consider the first end of the nanobeams
x = 0 is clamped and loaded thermally, which gives
(Youssef and Elsibai 2010):
w 0; tð Þ ¼ g 0; tð Þ ¼ 0; ð36Þ
and
h 0; tð Þ ¼ h0f tð Þ; ð37Þ
where h0is constant.
Substituting from Eq. (37) into the mapping in (7), we
obtain
#1 0; tð Þ ¼ 1
Ko
Zh0f tð Þ
0
K nð Þdn: ð38Þ
After using Laplace transform, the above conditions
take the forms
�w 0; sð Þ ¼ �g 0; sð Þ ¼ 0; ð39Þ
and
�#1 0; sð Þ ¼ GðsÞ: ð40Þ
where
GðsÞ ¼ 1
Ko
Z1
0
Zh0f tð Þ
0
K nð Þdn
2
64
3
75e�stdt: ð41Þ
Applying the conditions (39) and (40) into Eq. (29), we
obtain
Vð0; sÞ ¼
0
GðsÞ0
�w0ð0; sÞ�#0
1ð0; sÞ�g0ð0; sÞ
2
6666664
3
7777775
: ð42Þ
To get �w0ð0; sÞ; �#01ð0; sÞ and �g0ð0; sÞ, we will consider the
other end of the beam x ¼ ‘ is clamped and remains at zero
increment of temperature as follows:
wð‘; tÞ ¼ #1ð‘; tÞ ¼ gð‘; tÞ ¼ 0: ð43Þ
After using Laplace transform, we have
�wð‘; sÞ ¼ �#1ð‘; sÞ ¼ �gð‘; sÞ ¼ 0: ð44Þ
Hence, we obtain
Appl Nanosci (2013) 3:397–407 401
123
�w0ð0; sÞ�h01ð0; sÞ�g0ð0; sÞ
2
64
3
75
¼ �GðsÞL14 ‘; sð Þ L15 ‘; sð Þ L16 ‘; sð ÞL24 ‘; sð Þ L25 ‘; sð Þ L26 ‘; sð ÞL34 ‘; sð Þ L35 ‘; sð Þ L36 ‘; sð Þ
2
64
3
75
�1L12 ‘; sð ÞL22 ‘; sð ÞL32 ‘; sð Þ
2
64
3
75
0
B@
1
CA:
ð45Þ
After some simplifications by using MAPLE software,
we get the final solutions in the Laplace transform domain
as follows:
�w x; sð Þ ¼ D sinh k1 ‘� xð Þð Þk2
1 � k22
� �k2
1 � k23
� �sinh k1‘ð Þ
þ D sinh k2 ‘� xð Þð Þk2
2 � k21
� �k2
2 � k23
� �sinh k2‘ð Þ
þ D sinh k3 ‘� xð Þð Þk2
3 � k21
� �k2
3 � k22
� �sinh k3‘ð Þ
; ð46Þ
�# z; x; sð Þ ¼ � a2k21D sin pzð Þ sinh k1 ‘� xð Þð Þ
k21 � a1
� �k2
1 � k22
� �k2
1 � k23
� �sinh k1‘ð Þ
� a2k22D sin pzð Þ sinh k2 ‘� xð Þð Þ
k22 � a1
� �k2
2 � k21
� �k2
2 � k21
� �sinh k2‘ð Þ
� a3k22D sin pzð Þ sinh k3 ‘� xð Þð Þ
k23 � a1
� �k2
3 � k21
� �k2
3 � k22
� �sinh k3‘ð Þ
; ð47Þ
�u z; x; sð Þ ¼ � zDk1 cosh k1 ‘� xð Þð Þk2
1 � k22
� �k2
1 � k23
� �sinh k1‘ð Þ
� z D k2 cosh k2 ‘� xð Þð Þk2
2 � k21
� �k2
2 � k21
� �sinh k2‘ð Þ
� zDk3 cosh k3 ‘� xð Þð Þk2
3 � k21
� �k2
3 � k22
� �sinh k3‘ð Þ
; ð48Þ
�e z; x; sð Þ ¼ zDk21 sinh k1 ‘� xð Þð Þ
k21 � k2
2
� �k2
1 � k23
� �sinh k1‘ð Þ
þ zDk22 sinh k2 ‘� xð Þð Þ
k22 � k2
1
� �k2
2 � k21
� �sinh k2‘ð Þ
þ zDk23 sinh k3 ‘� xð Þð Þ
k23 � k2
1
� �k2
3 � k22
� �sinh k3‘ð Þ
; ð49Þ
where D ¼ G sð Þa1a2
a1 � k21
� �a1 � k2
2
� �a1 � k2
3
� �:
To determine the function K hð Þ, we will consider the
thermal conductivity depends on the temperature with
linear function in the form
K hð Þ ¼ Ko 1 þ K1hð Þ; ð50Þ
where for the gold, we have K1 0 (Zhang 2007).
By using the mapping in (7), we get
#1 ¼ h þ K1
2h2; ð51Þ
and
#1 0; tð Þ ¼ h0f tð Þ þ K1
2h0f tð Þð Þ2: ð52Þ
where h0 is constant.
For thermal shock problem, we will consider
f tð Þ ¼ H tð Þ ¼ 1 t [ 0
0 t\0
�; ð53Þ
where H(t) is called the Heaviside unite step function; then
we get
#1 0; tð Þ ¼ h0H tð Þ þ K1
2h0H tð Þð Þ2; ð54Þ
Hence, we obtain
G sð Þ ¼ h0 þK1
2h2
0
� �1
s; ð55Þ
After obtaining 0, the temperature increment h can be
obtained by solving Eq. (51) to get
h ¼ �1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 2K1#
p
K1
where 1 þ 2K1#ð Þ[ 0: ð56Þ
The Strain–stress energy
The stress on the x axis, according to Hooke’s law is
rxx x; z; tð Þ ¼ E e � aThð Þ: ð57Þ
By using the non-dimensional variables in (18), we
obtain the stress in the form
rxx x; z; tð Þ ¼ e � aT T0h: ð58Þ
After using Laplace transform, the above equation takes
the following form:
�rxx x; z; sð Þ ¼ �e � aT T0�h: ð59Þ
The strain energy which is generated on the beam is
given by
W x; z; tð Þ ¼X3
i;j¼1
1
2rijeij ¼
1
2rxxexx ¼ � 1
2zrxxg; ð60Þ
or, we can write it in the form
W x; z; tð Þ ¼ � 1
2z L�1 �rxxð Þ �
L�1 �gð Þ �
; ð61Þ
where L�1 �½ � is the inversion of Laplace transform.
Numerical inversion of the Laplace transform
In order to determine the solutions in the time domain, the
Riemann-sum approximation method is used to obtain the
402 Appl Nanosci (2013) 3:397–407
123
numerical results. In this method, any function in Laplace
domain can be inverted to the time domain as
f ðtÞ ¼ egt
t
1
2�f gð Þ þ Re
XN
n¼1
�1ð Þn�f g þ inpt
� �" #
: ð62Þ
where Re is the real part and i is imaginary number unit.
For faster convergence, numerous numerical experiments
have shown that the value of f satisfies the relation g t �4:7 Tzou (1996).
Numerical results and discussion
Now, we will consider a numerical example for which
computational results are given. For this purpose, gold (Au)
is taken as the thermoelastic material for which we take
the following values of the different physical constants
(Youssef and Elsibai 2010):
Ko ¼ 318 W= mKð Þ; aT ¼ 14:2 10ð Þ�6K�1;
q ¼ 1;930 kg=m3; T0 ¼ 293 K;
Ct ¼ 130 J= kg Kð Þ; E ¼ 180 GPa; t ¼ 0:44:
The aspect ratios of the beam are fixed as ‘=h ¼ 10 and
b=h ¼ 1=2, when h is varied, and ‘ and b change
accordingly with h. For the nanoscale beam, we will take
the range of the beam length ‘ 1 � 100ð Þ 10�9 m, the
original time t will be considered in the picoseconds
1 � 100ð Þ 10�12 sec and the relaxation time so in the
range 1 � 100ð Þ 10�14 sec:
The figures were prepared using the non-dimensional
variables which are defined in (18) for beam length ‘ ¼1:0; h0 ¼ 1:0; z ¼ h=6; t ¼ 0:15; and K1 ¼ 0:0; 0:2 and 0:5:
The Figs. 2, 3, 4, 5, 6, 7 show that the variability of the
thermal conductivity plays a vital role on the speed of the
wave propagation of all the studied fields, and increasing of
Fig. 2 The temperature distribution
Fig. 3 The lateral vibration distribution
Fig. 4 The stress distribution
Fig. 5 The displacement distribution
Appl Nanosci (2013) 3:397–407 403
123
the parameter K1 causes increasing on all the state func-
tions distributions. The damping of the strain–stress energy
decreases when the parameter K1 increases.
Also, when K1 = 0.0 all the results coincide with the
results on (Youssef and Elsibai 2010).
Figures 8, 9, 10, 11, 12 and 13 represent the temperature,
the lateral vibration, the stress, the displacement, the volu-
metric strain, and the stain-stress energy distribution when
Fig. 6 The strain distribution
Fig. 7 The strain–stress energy distribution
Fig. 8 The temperature distribution at K1 = 0.2
Fig. 9 The lateral vibration distribution at K1 = 0.2
Fig. 10 The stress distribution at K1 = 0.2
404 Appl Nanosci (2013) 3:397–407
123
K1 = 0.2 in three-dimension figures with wide range of the
coordinate x 0:0� x� ‘ð Þ and the time t 0:0\t� 0:5ð Þ to
stand on the behavior of the wave propagation of all the
studied fields with respect to the coordinate x and the time t
together. The peak points of the temperature, the lateral
vibration, and the displacement increase when the time
increases, whereas the strain, the stress, and the strain–stress
energy decrease when the time increases.
Conclusion
This paper is dealing with the effect of the thermal con-
ductivity on vibration of the deflection, the temperature, the
displacement, the strain, the displacement, the stress, and
the strain–stress energy. Euler–Bernoulli gold nanobeam
with variable thermal conductivity induced by thermal
shock has been considered. State-space approach and
numerical technique, based on Laplace transformation,
have been used to calculate the numerical results of all the
studied fields. We get the following results:
1. Thermal conductivity has significant effects on the
speed of the wave propagation of all the studied fields.
2. Thermal conductivity depends on the temperature with
linear function in the form K hð Þ ¼ Ko 1 þ K1hð Þ where
for gold K1 0
3. The effect of the variability of the thermal conductivity
decreases when the length of the beam increases.
4. The maximum value of the strain–stress energy
increases when the changing in the thermal conduc-
tivity with respect to the temperature increases.
5. The changing in the thermal conductivity with respect
to the temperature effects on the damping of the
strain–stress energy of the beam.
Open Access This article is distributed under the terms of the
Creative Commons Attribution License which permits any use, dis-
tribution, and reproduction in any medium, provided the original
author(s) and the source are credited.
Fig. 11 The displacement distribution at K1 = 0.2
Fig. 12 The strain distribution at K1 = 0.2
Fig. 13 The strain–stress energy distribution at K1 = 0.2
Appl Nanosci (2013) 3:397–407 405
123
Appendix
F ¼ 1
ðk21 � k2
2Þðk22 � k2
3Þðk23 � k2
1Þ;
c1 x; sð Þ ¼ ðk22 � k
23Þ coshðk1xÞ; c2 x; sð Þ
¼ ðk23 � k
21Þ coshðk2xÞ;
c3 x; sð Þ ¼ ðk21 � k
22Þ coshðk3xÞ; s1 x; sð Þ
¼ ðk22 � k2
3Þk1
sinhðk1xÞ;
s2 x; sð Þ ¼ ðk23 � k2
1Þk2
sinhðk2xÞ; s3 x; sð Þ
¼ ðk21 � k2
2Þk3
sinhðk3xÞ:
a0 x; sð Þ ¼ �Fðk22 k
23 c1 þ k
21 k
23 c2 þ k
21 k
22 c3Þ;
a1 x; sð Þ ¼ �Fðk22 k
23 s1 þ k
21 k
23 s2 þ k
21 k
22 s3Þ;
a2 x; sð Þ ¼ F½ðk22 þ k
23Þc1 þ ðk2
3 þ k21Þc2 þ ðk2
1 þ k22Þc3�;
a3 x; sð Þ ¼ F½ðk22 þ k
23Þs1 þ ðk2
3 þ k21Þs2 þ ðk2
1 þ k22Þs3�;
a4 x; sð Þ ¼ �Fðc1 þ c2 þ c3Þ; a5 x; sð Þ ¼ �Fðs1 þ s2 þ s3Þ:
L11 x; sð Þ ¼ a0 � a4a3; L12 x; sð Þ ¼ � a4a4; L13 x; sð Þ¼ a2 þ a4a5;
L14 x; sð Þ ¼ a1 � a5a3; L15 x; sð Þ ¼ � a5a4; L16 x; sð Þ¼ a3 þ a5a5;
L21 x; sð Þ ¼ a4a2a3; L22 x; sð Þ¼ a0 þ a2a1 þ a4 a2
1 þ a2a4
� �;
L23 x; sð Þ ¼ � a2a2 � a4a2 a1 þ a5ð Þ; L24 x; sð Þ ¼ a5a2a3;
L25 x; sð Þ ¼ a1 þ a3a1 þ a5ða21 þ a2a4Þ; L26 x; sð Þ
¼ �a3a2 � a5a2ða1 þ a5Þ;
L31 x; sð Þ ¼ �a2a3 � a4a3a5; L32 x; sð Þ¼ �a2a4 � a4a4ðal þ a5Þ;
L33 x; sð Þ ¼ a0 þ a2a5 þ a4ða2a4 � a3 þ a25Þ; L34 x; sð Þ
¼ �a3a3 � a5a3a5;
L35 x; sð Þ ¼ �a3a4 � a5a4ða1 þ a5Þ; L36 x; sð Þ¼ a1 þ a3a5 þ a5ða2a4 � a3 þ a2
5Þ;
L41 x; sð Þ ¼ �a3a3 � a5a3a5; L42 x; sð Þ¼ �a3a4 � a5a4ða1 þ a5Þ;
L43 x; sð Þ ¼ a1 þ a3a5 þ a5ða2a4 � a3 þ a25Þ; L44 x; sð Þ
¼ a0 � a4a3; L45 x; sð Þ ¼ �a4a4;
L46 x; sð Þ ¼ a2 þ a4a5; L51 x; sð Þ¼ a3a2a3 þ a5a2a3ða1 þ a5Þ;
L52 x; sð Þ ¼ a1a1 þ a3ða21 þ a2a4Þ þ a5ða3
1 þ 2a1a2a4
þ a2a4a5Þ;
L53 x; sð Þ ¼ �a1a2 � a3a2ða1 þ a5Þ � a5a2ða21 þ a1a5
þ a2a4 � a3 þ a25Þ;
L54 x; sð Þ ¼ a4a2a3; L55 x; sð Þ ¼ a0 þ a2a1 þ a4ða21 þ a2a4Þ;
L56 x; sð Þ ¼ �a2a2 � a4a2ða1 þ a5Þ; L61 x; sð Þ¼ �a1a3 � a3a3a5 � a5a3ða2a4 � a3 þ a2
5Þ;
L62ðx; sÞ ¼ �a1a4 � a3a4ða1 þ a5Þ � a5a4ða21 þ a1a5
þ a2a4 � a3 þ a25Þ;
L63 x; sð Þ ¼ a1a5 þ a3ða2a4 � a3 þ a25Þ þ a5ða1a2a4
þ a5ð2a2a4 � 2a3 þ a25ÞÞ;
L64 x; sð Þ ¼ � a2a3 � a4a3a5;
L65 x; sð Þ ¼ �a2a4 � a4a4ða1 þ a5Þ;
L66 x; sð Þ ¼ a0 þ a2a5 þ a4ða2a4 � a3 þ a25Þ;
References
Al-Huniti NS, Al-Nimr MA, Naij M (2001) Dynamic response of a
rod due to a moving heat source under the hyperbolic heat
conduction model. J Sound Vib 242:629–640
Bahar LY, Hetnarski RB (1977) State space approach to thermoelas-
ticity. In: Proceedings of 6th Canadian Congress of Applied
Mechanics, University of British Columbia, Vancouver, British
Columbia, Canada, pp 17–18
Bahar LY, Hetnarski RB (1977) Transfer matrix approach to
thermoelasticity. In: Proceedings of 15th Midwest mechanical
conference, University of Illinois at Chicago Circle, pp 161–163
Bahar LY, Hetnarski RB (1978) State space approach to thermoelas-
ticity. J Therm Stresses 1:135–145
Biot M (1956) Thermoelasticity and irreversible thermo-dynamics.
J Appl Phys 27:240–253
Boley BA (1972) Approximate analyses of thermally induced
vibrations of beams and plates. J Appl Mech 39:212–216
Diao JK, Gall K, Dunn ML (2004) Atomistic simulation of the
structure and elastic properties of gold nanowires. J Mech Phys
Solids 52:1935–1962
Duwel A, Gorman J, Weinstein M, Borenstein J, Ward P (2003)
Experimental study of thermoelastic damping in MEMS gyros.
Sens Actuators, A 103:70–75
Ezzat MA (2008) State space approach to solids and fluids (review).
Can J Phys 86:1241–1250
Ezzat MA, Youssef HM (2009) Stat-space approach of conducting
magneto-thermoelastic medium with variable thermal conduc-
tivity subjected to ramp-type heating. J Therm Stresses 32:414–
427
Fang DN, Sun YX, Soh AK (2006) Analysis of frequency spectrum of
laser-induced vibration of microbeam resonators. Chin Phys Lett
23(6):1554–1557
Kidawa-Kukla J (2003) Application of the Green functions to the
problem of the thermally induced vibration of a beam. J Sound
Vib 262:865–876
Lord H, Shulman Y (1967) A generalized dynamical theory of
thermoelasticity. Mech Phys Solid 15:299–309
Manolis GD, Beskos DE (1980) Thermally induced vibrations of
beam structures. Comput Methods Appl Mech Eng 21:337–355
Rao JS (1992) Advanced theory of vibration (Nonlinear Vibration and
One Dimensional Structures). Wiley, New York
406 Appl Nanosci (2013) 3:397–407
123
Soh AK, Sun Y, Fang D (2008) Vibration of microscale beam induced
by laser pulse. J Sound Vib 311:243–253
Sun YX, Fang DN, Soh AK (2006) Thermoelastic damping in micro-
beam resonators. Int J Solids Struct 43:3213–3229
Sun Y, Fang D, Saka M, Soh AK (2008) Laser-induced vibrations of
micro-beams under different boundary conditions. Int J Solids
Struct 45:1993–2013
Tzou D (1996) Macro-to –micro heat transfer. Taylor & Francis,
Washington DC
Wang YZ, Li FM, Kishimoto K (2010) Scale effects on the
longitudinal wave propagation in nanoplates. Physica E 42:
1356–1360
Youssef HM, El-Bary AA (2006a) Thermal shock problem of a
generalized thermoelastic layered composite material with
variable thermal conductivity. J Math Prob Eng 87940:1–14
Youssef HM, El-Bary AA (2006b) Mathematical model for thermal
shock problem of a generalized thermoelastic layered composite
material with variable thermal conductivity. J Comput Methods
Sci Technol 12(2):165–171
Youssef HM, El-Bary AA (2006c) Generalized thermoelasticity for
an infinite material with variable thermal conductivity. J Adv
Math Sci Appl 16(1):339–353
Youssef HM, Elsibai KA (2010) Vibration of gold nano-beam
induced by different types of thermal loading- a state-space
approach. J Nanoscale Microscale Thermophys Eng 15(1):48–69
Zhang ZM (2007) Nano/Microscale heat transfer, McGraw-Hill,
USA. doi:10.1036/007143674X
Appl Nanosci (2013) 3:397–407 407
123
