World Journal of Nano Science and Enginee ring, 2011, 1, 37-44
doi:10.4236/wjnse.2011.12006 Published Online June 2011 (http://www.SciRP.o rg/journal/wjnse)
Vibration of Nano Beam Induced by Ramp Type Heating
Hamdy M. Youssef1, Khaled A. Elsibai2
1Faculty of Engineering, Umm Al-Qura University, Makkah, Saudi Arabia
2Mathematic s De partment, Faculty Science, Umm Al-Qura University, Makkah, Saudi Arabia
Received March 23, 2011; revised April 18, 2011; accepted May 8, 2011
Abstract
The non-Fourier effect in heat conduction and the coupling effect between temperature and strain rate, be-
came the most significant effects in the nano -scale beam. In the present study, a generalized solution for the
generalized thermoelastic vibration of a bounded nano-beam resonator induced by ramp type of heating is
developed and the solutions take into account the above two effects. The Laplace transforms and direct me-
thod are used to determine the lateral vibration, the temperature, the displacement, the stress and the energy
of the beam. The effects of the relaxation time and the ramping time parameters have been studied with some
comparisons.
Keywords: Thermoelasticity, Nano-Beam, Ramp-Type Heating, Non-Fourier Heat Conduction
1. Introduction
The generalized thermoelasticity theories have been de-
veloped with the aim of removing the paradox of infinite
speed of heat propagation inherent in the classical dy-
namical coupled thermoelasticity theory (Biot-CTE) [1],
Lord and Shulman (L-S) [2] obtained a wave-type heat
equation by postulating a new law of heat conduction to
replace the classical Fourier’s law. Since the heat equa-
tion of this theory is of the wave-type, it automatically
ensures finite speeds of propagation 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
theory.
Many attempts have been made r ecently to investigate
the elastic properties of nanostructured materials by ato-
mistic simulations. Diao et al. [3] studied the effect of
free surfaces on the structure and elastic properties of
gold nanowires by atomistic simulations. Although the
atomistic simu la tion is a good wa y to c a lc ulate the elastic
constants of nanostructured materials, it is only applica-
ble to homogeneous nanostructured materials (e.g., na-
noplates, nanobeams, nanowires, etc.) with limited num-
ber of atoms. More over, it is difficult to ob tain the elastic
properties of the heterogeneous nanostructured materials
using atomistic simulations. For these and other reasons,
it is prudent to seek a more practical approa ch. One suc h
approach would be to extend the classical theory of elas-
ticity down to the nanosca le by including in it the hither-
to neglected surface/interface effect. For this it is neces-
sary first to cast the latter within the framework of con-
tinuum elasticity.
Nano-mechanical resonators have attracted considera-
ble attention recently due to their many important tech-
nological applications. Accurate analysis of various ef-
fects on the characteristics of resonators, such as reso-
nant frequencies and quality factors, is crucial for de-
signing high-performance components. Many authors
have studied the vibration and heat transfer process of
beams. Kidawa [4] has studied the problem of transverse
vibrations of a beam induced by a mobile heat source.
The analytical solution to the problem was obtained us-
ing the Green ’s functi ons met hod. H owever, Kidawa di d
not consider the thermoelastic coupling effect. Boley [5]
analyzed the vibrations of a simply supported rectangular
beam subjected to a suddenly applied heat input distri-
buted along its span. Manolis and Beskos [6] 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. [7] investigated the
thermally induced displacements and stresses of a rod
using the Laplace transformation technique. Ai Kah Soh
et al. studied the vibration of micro/nanoscale beam re-
sonators induced by ultra-short-pulsed laser by consider-
ing the thermoel asti c coupling term i n [ 8,9].
When very fast phenomena and small structure di-

H. M. YOUSSEF ET AL.
38
mensions are involved, the classical law of Fourier be-
comes inaccurate. A more sophisticated model is then
needed to describe the thermal conduction mechanisms
in a physically acceptable way. Modern technology has
enabled the fabrication of materials and devices with
characteristic dimensions of a few nanometers. Examples
are super-lattices, na nowires, and quantum do ts. At these
length scales, the familiar continuum Fo urie r law for heat
conduction is expected to fail due to both classical and
quantum size effects [10]. Among many applications, the
stud yin g of t he t he rmoelastic damping in MEMS /NEM S
has been improved in [11,13].
It is worthwhile to mention here that in most of the
earlier studies, mechanical or thermal loading on the
bounding surface is considered to be in the form of a
shock. However, the sudden jump of the load is merely
an idealized situation because it is impossible to realize a
pulse described mathematically by a step function; even
very rapid rise-time (of the order of 10–9 s) may be slo w
in terms of the contin uum. This is partic ularly true in the
case of second sound effects when the thermal relaxation
times for typical metals are less than 10–9 s [14]. It is thus
felt that a finite time of rise of external load (mechanical
or thermal) applied on the surface should be considered
while studying a practical problem of this nature. Most
ultrafast heat sources (such as certain lasers) involve the
emission of a pulse (for example) that heats a material
over a finite time due to the finite rise time of the pulse.
Considering the aspect of rise of time, Misra et al. [15]
and Youssef with many authors investigated many ap-
plications in which the ramp-type heating is used [16-
22].
In this paper, the non-Fouri er effe c t in hea t co nd uc ti o n,
and the coupling effect between temperature and strain
rate in nanoscale beam will be studied. In the present
work, a generalized solution for the generalized ther-
moelastic vibration of a nano beam resonator i nduced by
ramp type of heating will be developed. The solution
takes into account the above two effects. The Laplace
transformation method will be used to determine the lat-
eral vibration, the temperature, the displacement, the
stress and the energy of the beam. The effects of the re-
laxation time and the ramping time parameters will be
studied and represented graphically.
2. Problem Formulation
Since beams with rectangular cross-sections are easy to
fabricate, such cross-sections are commonly adopted in
the design of NEMS resonators. Consider small flexural
deflections o f a th in elastic beam of length
( )
2LLxL−≤≤
, width
22
bb
by

−≤≤


and thickness
22
hh
hz

−≤≤


, for which the x, y and z axes are de-
fined along the longitudinal, width and thickness direc-
tions of the beam, respectively. In equilibrium, the beam
is unstrained, unstressed, and at temperature T0 every-
where [8].
In the present study, the usual Euler–Bernoulli as-
sumption [8,9] is adopted, i.e., any plane cross-section,
initially perpendicular to the axis of the beam, remains
plane and perpendicular to the neutral surface during
bending. Thus, the di s plac ements ar e given by
( )()( )
,,0,,, ,,
w xt
uzvwxyztwxt
x
∂
=−= =
∂
(1)
Hence, the differential equation of thermally induced
lateral vibration of the beam may be expressed in the
form:
2 42
42 2
0
T
T
M
w Aw
EI
xtx
ρα
∂
∂∂
++ =
∂∂∂
, (2)
where E is Young’s modulus, I [= bh3 /12] is the inertial
moment about x-axis,
ρ
is the density of the beam,
T
α
is the coefficient of linear thermal expansion,
( )
,w xt
the lateral deflection, x is the distance along the
lengt h o f t he b eam,
A hb=
is the cross section area, t is
the time, and
T
M
is the thermal moment, which is de-
fined as
/2
3/2
12 d
h
Th
M zz
h
θ
−
=
∫
, (3)
where 0
TT
θ = − is the dynamical temperature incre-
ment of the resonator, in which T(x, z, t) is the tempera-
ture distribution and T0 is the environ mental temperature.
The non-Fourier heat conduction equation has the fol-
lowing form [16-18]:
22 20
22 2
o
CT
e
t kk xz t
υ
ρβ
θθ
τθ

∂∂∂ ∂

+=+ +


∂ ∂∂ ∂

 , (4)
where
uvw
e
xyz
∂∂∂
=++
∂∂∂
is the volumetric strain, where
C
υ
is the specific heat at constant volume,
o
τ
is the
thermal relaxat ion ti me, k is t he t he r mal c onduc ti vi t y a nd
12
T
E α
βν =−
in which
ν
is Po i s son’s ratio .
Where there is no heat flow across the upper a nd lower
surfaces of the beam, so that
0
z
θ ∂=
∂
at
2zh= ±
, for
a very thin beam and assuming that the temperature va-
ries in terms of a
( )
sin pz
function along the thickness
direction, where
πph=
, gives:
()()( )
1
, ,,sinxz tx tpz θθ =
.

H. M. YOUSSEF ET AL.
39
Hence, Equation ( 2) gives
( )
2/2
42 1
42 32
/2
12 sind 0
h
T
h
w Awzpz z
EI xt hx
αθ
ρ
−
∂
∂∂
++ =
∂∂ ∂
∫
(5)
and Equation (4) gives
( )( )
( )
22
11
2
22
0
1
22
sin sin
sin
o
pz ppz
x
CT
w
pz z
tk k tx
υ
θθ
ρβ
τθ
∂−
∂
 
∂∂ ∂
=+−
 
∂∂∂
 
(6)
After doing the integrations, Equation (5) takes the
form
2 42 1
42 22
24 0
T
w Aw
EI
xth x
αθ
ρ
π
∂
∂∂
++ =
∂∂ ∂
. (7)
In Equation (6), we multiply the both sides by z and
integrating with respect to z from 2h− to 2h, then
we obtain
2 2
22
20 1
11
2 22
π
24
o
Thw
p
tk
x tx
β θθτ ηθ

 
∂∂∂ ∂
−=+ −



∂
∂ ∂∂
 

, (8)
where
C
k
υ
ρ
η =
.
Now, for simplicity we will use the following non-
dimensional variables:
()()
()( )
2
2
1
1
, ,,,,,,,
,,
oo oo
o
o
xw hcxwhtct
E
c
ET
ητη τ
θ
σ
σθ ρ
′ ′′′′ = =
′′
= ==
(9)
The n, we have
2 42 1
12
42 2
0
ww
AA
xt x
θ ∂
∂∂
++ =
∂∂∂
, (10)
and
2
22
1311 4
2 22
ow
AA
t
x tx
θθ τθ
 
∂∂∂ ∂
−=+ −
 
∂ ∂ ∂∂
 , (11)
where
2
2
12 34
22
24
12 ,,,
24
to
Th
AAApAk
hh
απβ
η
π
=== =
,
and we have canceled the prime for convenient.
3. Formulations the Problem in the Laplace
Transform Domain
Applying the Laplace transform for Equations (10) and
(11) defined by the formula
()( )( )
0
ed
st
fs Lftftt
∞−
= =


∫
.
Hence, we obtain the following system of differential
equations
2 421
12
42
d
d0
dd
wAs wA
xx
θ
++ =
, (12)
and
( )
22
2
1311 4
22
dd
dd
o
w
A ssA
xx
θθ τθ

−=+−


. (13)
We have considered all the initial states of variables
were zero, i.e.
( )()( )()
1
1
d,0d ,0
,0,0 0
dd
x wx
x wx
xx
θ
θ == ==
(14)
We can re-write the above system of equations in the
form
2 421
12
42
d d0
dd
Asw A
xx
θ

++=


, (15)
and
22
11 2
22
dd
dd
w
xx
αθ α

−=−

 , (16)
where
( )( )
22
132 4
,
oo
AssssA ατα τ = ++=+
By el i mi nat in g
w
from the above system of equation,
we get
( )
64 222
12 2111
0DADAs DAsw αα α

−++ −=

, (17)
where
1
θ
satisfies the same equation, i.e.
( )
64222
12 21111
0DADAs DAs ααα θ

−++ −=

. (18)
The characteristic equation can be presented as
642
0mn λλ λ − +−=
(19)
The roots of this equation, namely, 1 λ ±, 2 λ ± and
3
λ ±, satisfy the following rela tions
222
12 3122
A λλλαα ++=+=
,
22 22 222
1223 131
As m λλ λλ λλ
++==
,
222 2
12 311
As n λλλα ++= =
.
We can consider the solution take the form
( )
6
1exp
ii
i
wC x λ
=
=
∑
, (20)
and
( )
6
11exp
ii
i
Ex θλ
=
=∑, (21)

H. M. YOUSSEF ET AL.
40
where i C and i E (i = 1, 2,

, 6) are some parame-
ters d epend, only, on s.
Using Equation (16) and (21), we get
( )
2
2
21
ii
i
i
C
E
αλ
λα
= −−
,
this gi ves
( )
( )
2
6
122
11
exp
iii
ii
Cx
λ
θα λ
λα
=
= −−
∑
. (22)
Now, to get the values of the constants i C and i E
we will consider the two ends of the micro-beams are
clamped, then the boundary conditions are [8,9]:
( )( )
2
2
d,
,0
d
w Lt
w Ltx
±
±= =
, (23)
and loaded thermally by ramp-type hea ti ng, which give
( )
10 0
0
0
0 for0
,for 0
1 for
t
t
Ltt t
t
tt
θθ

≤


± =<<


≥


, (24)
where
0
t
is non-negative constant and is called ramp-
type parameter and 0
θ is co ns tant [18] .
After using Laplace transform, the above conditions
take the forms
( )
( )
2
2
d,
,0
d
w Ls
w Lsx
±
±= =
, (25)
and
( )( )
0
0
12
0
1
ts
e
s Fs
ts
θ
θ
−

−
= =


. (26)
Applying the conditions (25) and (26) into Equations
(20) and (22), we obtain
( )
3
1
cosh 0
ii
i
CL λ
=
=
∑
, (27)
( )
32
1
cosh 0
ii i
i
CL λλ
=
=
∑
, (28)
( )
( )
( )
2
3
2
12
1
cosh
iii
ii
Fs
CL
λλα
λα
=
= −
−
∑
. (29)
By solving the above system of linear algebraically
equations, we get
The lateral deflection
( )( )()
( )( )
( )()
( )( )
( )()
( )( )
33
112 2
2222 22222222
1213 12321323
cosh cosh cosh coshcoshcosh
,
xL
xL xL
wxs G
λλ
λλ λλ
λλλλ λλλλλλλλ


= ++

−− −−−−

(30)
The tempe rature
()( )
( )()
()( )( )
( )()
()( )( )
( )()
()( )( )
22
111 222
22 222222222
111 21 3211 232
2
33 3
2 2222
31132 3
cosh coshcoshcosh
, ,sin
cosh cosh
xL xL
x z sGpz
xL
λλλ λλλ
θα λαλλλλλαλλλλ
λλ λ
λαλλλλ


=−+
−−− −−−



+
−−−

, (31)
The displacement
( )( )()
( )( )
( )()
( )( )
( )()
( )( )
33 3 1 11222
222222222222
1213 12321323
sinh cosh sinh coshsinhcosh
,,
xL
xL xL
ux z szG
λλλ λ λλλλλ
λλλλ λλλλλλλλ


=− ++

−− −−−−

, (32)
where
( )
( )( )( )
222
1112 13
12
Fs
G
αλαλαλ
αα
−−−
=
.
4. The Stress and the Strain Energy
The stres s on the x-axis, according to Hooke’s law is
( )
,,
xx T
u
xztE x
σ αθ
∂

= −

∂

, (33)
By using the non-dimensional variables in (9), we ob-
tain the stress in the form
( )
0
,,
xx T
u
xzt T
x
σ αθ
∂
= −
∂. (34)
After using Laplace transform, the above equation
gives

H. M. YOUSSEF ET AL.
41
( )
0
,,
xx T
u
xzs T
x
σ αθ
∂
= −
∂
. (35) By using Equations (31) and (32) with equation (35),
we get
( )( )()
( )( )
( )()
( )( )
( )()
( )( )
( )( )()
()( )( )
( )()
( )
22
11 1222
2222 2222
12131232
2
33 3
2222
1323
22
111 222
02 222222
111213211
cosh coshcoshcosh
,,
cosh cosh
cosh coshcoshcosh
sin
xx
T
xL xL
x z szG
xL
xL xL
T Gpz
λλ λλλλ
σλλλλ λλλλ
λλ λ
λλλλ
λλλ λλλ
αα λαλλλλ λαλ


=−+
−−−−



+
−−

++
−−−−
( )( )
( )()
()( )( )
2222
232
2
33 3
2 2222
31132 3
cosh cosh
xL
λλλ
λλ λ
λαλλλλ



−−



+
−−−

(36)
The energy which is generated on the beam is given
by
2
3
2
,1
11 1
22 2
ij ijxx xxxx
ij
w
We ez
x
σσσ
=
∂
=== −∂
∑
, (37)
or, we can write as follows:
( )
2
11
2
1
2
xx
w
WzLL x
σ
−−


∂

= −


∂



, (38)
where
( )( )
1
L fsft
−

=

is the Laplace inverse trans-
form.
5. Numerical Inversion of the Laplace
Transform
In order to determine the solutions in the time domain,
the Riemann-sum approximation method is used to ob-
tain the numerical results. In t his me t ho d , a n y function in
Laplace domain can be inverted to the time domain as
( )( )()
1
1Re 1
2
tNn
n
e in
ft ff
tt
κ
π
κκ
=


=+− +
 


∑
(39)
where Re is the real part and i is imagi nary numb er unit.
For faster convergence, numerous numerical experiments
have shown that the value of
κ
satisfies the relation
4.7t κ ≈
[23].
6. Numerical Results and Discussion
Now, we will consider a numerical example for which
computatio nal results are given. For thi s purpose, Silico n
is taken as the thermoelastic material for which we take
the following values of the different physical constants
[24]:
( )
156KW mK=
,
( )
61
2.59 10
T
k α
−−
=
,
3
2330kgm ρ =
,
0
293Tk=
,
( )
713CJ kgK
υ
=
169E GPa=
,
0.22 υ =
.
The aspect ratios of the beam are fixed as
10Lh=
and
12
bh=, when h is varied, L and b change accor-
dingl y wit h h.
For the nanoscale beam, we will take the range of the
beam length
( )
12
1 10010Lm
−
−×
. The original time t
and the ramping time parameter
0
t
will be considered
in the picoseconds
( )
12
1 10010sec
−
−×
and the relaxa-
tion time 0 τ
in the r ange
( )
14
1 10010sec
−
−×
.
The figures were prepared by using the non-di men-
sional variables which are defined in (9) for a wide range
of beam length 2L when
1.0L=
, 01.0
θ = 6zh=
and
0.15t=
.
In Figures 1-5, we represented the lateral vibration,
the temperature, the displacement, the stress and the
energy of the beam at different values of the relaxation
time when 00.0 τ = (Biot model) and 00.02 τ = (L-S
model) and we have found that, the relaxation time has
significant ef fects on all the studied fields. In the conte xt
of L-S model, the relaxation time gives values of the
lateral vibration, the temperature, the displacement, the
stress and the ener gy les s than their va lue s in the c onte xt
of Biot model which is very obvious in the peek points.
We can say that, in the context of the generalized
thermoelasticity the speed of the wave propagation of all
the studied fields ar e finite and the damping of the strain
ener gy increasing.
In Figures 6-10, we represented the lateral vibration,
the temperature, the displacement, the stress and the
energy of the beam at different values of the ramping
time parameter when
() ()
0
0.10 0.15tt= <=
,
0
0.15tt= =
and
() ()
0
0.20 0.15tt=>=
in the c ontext

H. M. YOUSSEF ET AL.
42
Figure 1. The lateral deflection w for L-S and Biot theories.
Figure 2. The temper at ure for L-S and Biot theories.
Figure 3. The displacement for L-S and Biot theories.
Figure 4. The stress for L-S and Biot theories.
Figure 5. The energy for L-S and Biot theories.
Figure 6. The lateral deflection w at different time of
ramping par ameter.

H. M. YOUSSEF ET AL.
43
Figure 7. The temperature at different time of ramping
parameter.
Figur e 8. The displacement at different time of ramping
parameter.
Figure 9. The stress at different time of ramping parameter.
Figure 10. The energy at different time of ramping para-
meter.
of L -S model.
We have found that, the ramping time parameter has
significant effects on all the studied fields. The increas-
ing in the value of the ramping time parameter causes
decreasing in the values of the lateral vibration, the tem-
perature, the displacement, the stress and the energy
which are very obvious in the peek points of the curves.
Also , the da mping o f the st rain ene rgy is increases when
the ramping time parameter increases.
7. Conclusions
This paper has investigated the vibration characteristics
of the deflection, the temperature, the displacement, the
stress and the strain energy of an Euler–Bernoulli beam
induced by a ramp type heating. An analytical direct
method and numerical technique based on the Laplace
transformation has been used to calculate the vibration of
the deflection, the temperature, the displacement, the
stress and the strain energy. The effects of the relaxation
time and the ramping time parameter on all the studied
fields have been shown and represented graphically. The
non-Fourier law of heat conduction gives a finite speed
of wave propagation and increases the damping of the
strain ene rgy.
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