Generalized Lagging Response of Thermoelastic Beams

Research ArticleGeneralized Lagging Response of Thermoelastic Beams

Ibrahim H. El-Sirafy,1 Mohamed A. Abdou,2 and Emad Awad2

1 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt2 Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

Correspondence should be addressed to Emad Awad; [email protected]

Received 24 October 2013; Accepted 23 March 2014; Published 30 April 2014

Academic Editor: Oleg V. Gendelman

Copyright 2014 Ibrahim H. El-Sirafy et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The vibrations of Euler-Bernoulli metal beam are accommodated in the present study by taking into account the possibility ofactivating the microstructural effects, captured by the temperature gradient phase lag, in the fast transient process captured by theheat flux phase lag. The thermal moment is approximated as the difference between the top and the bottom surface temperatures(Massalas and Kalpakidis 1983).Three generalizations of the Biot model of thermomechanics are considered: Lord-Shulman, dual-phase-lag (Tzou 1997), andmodified dual-phase-lag (Awad 2012). It is found that when the response time is shortened, the materialdimensions are small, or when the method of heating is changed, the dual-phase-lag model records a significant decrease in thelattice variables. The spurious serrations of the classical thermoelastic wave are smoothed in the dual-phase-lag wave. The dual-phase-lag thermoelasticity is the closest macroscopic approach to the ultrafast model (Chen et al. 2002).

1. Introduction

Theheat transport inmetals, on themacroscopic level, can bemodeled mathematically via the classical Fourier law for heatconduction:

( , ) = ( , ) , ()

where is the heat flux vector, is the position vector, is the time, is the thermal conductivity, is the absolutetemperature, and denotes the spatial gradient operator.

When the problem of heat transfer at relatively smallvalues of time is considered, it is found that the classicalFourier law () results in a nonzero temperature values forlarge values of distance, which contradicts the experimentalresults. For solving this paradox in the Fourierian model,Morse and Feshbach [1], Cattaneo [2], and Vernotte [3,4] hypothesized the existence of thermal relaxation duringthe temperature propagation and expressed their hypothesisthrough the non-Fourierian law:

( , ) + 0

( , )

= ( , ) , ()

where 0is a characteristic constant expressing the thermal

relaxation time in the energy carriers collision and /

is the differentiation with respect to the time variable .The energy equation, yielded from (), is of the hyperbolictype that predicts a wave nature for temperature propa-gation with finite speed; see Joseph and Preziosi [5] andTzou [6].

Based on the non-Fourier law (), Lord and Shulman[7] established a linear theory of isotropic thermoelasticmaterials, Lord-Shulman model, to replace the classical onedue to Biot [8]. Sherief and his colleagues [911] have takeninto account the Lord-Shulman model in details, and moreinformative reviews can be found in Chandrasekharaiah[12], Hetnarski and Ignaczak [13], and Ignaczak and Ostoja-Starzewski [14].

On the microscopic level, because of the rapid develop-ment and growing interests of the short-pulse laser (laserwith pulse duration ranging from 109 to 1015 seconds),great efforts were theoretically and experimentally directed toevaluate the energy exchange between electrons and phononsduring laser heating process and introduce a governinglaw which measures the electron temperature and latticetemperature; see [1519]. When the response time is shortand the material thickness is small so that it becomescomparable with the atomic distances in the scale sense, thenon-Fourier law () fails to predict the correct experimental

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 780679, 13 pageshttp://dx.doi.org/10.1155/2014/780679

2 Mathematical Problems in Engineering

values; see Qiu and Tien [20, 21]. As a result, a novel law ofheat transfer was proposed to replace the Cattaneo-Vernotteequation () and the conventional energy equation / =V(/); see Anisimov et al. [16]:

()

= () (

) +

()

= ( ) ,

()

where and

are the heat capacities of electron and

lattice, respectively, and

are the electron and lattice

absolute temperatures, respectively, is the laser source term,and denotes the electron-phonon coupling factor. Thesystem () was phenomenological not based on a rigorousthermodynamical basis. Qiu and Tien [21] introduced athermodynamic derivation for the phenomenological system() and its hyperbolic version.

Till 1995, all the works mentioned above were confinedto the knowledge of specialists of Statistical Mechanicsand Quantum Mechanics. Tzou [23, 24] introduced thelagging behavior idea in order to incorporate themicroscopicinteractions into the macroscopic formulae and, hence,to involve as many researchers as possible in the rapidgrowth of microscale heat transfer achieved in this time.The generalized lagging behavior (generalized dual-phase-lag, or DPL) is briefly proposing a non-Fourier law ofheat conduction which replaces () and () and takes theform

( , + ) = ( , +

) , ()

where and

are, respectively, the heat flux and the

temperature gradient phase lags. When the Taylor expansionis used with neglecting the terms (3

) and (2

), () reads

(1 +

+

2

2

2

2) ( , ) = (1 +

) ( , ) ,

()

where combining () with the energy equation / =V(/) yields the same equation governing the latticetemperature and resulted from the hyperbolic version of ();refer also to Tzou and Xu [25].

Awad [26] has proposed a refined aspect to () to be onthe form

(1 +

+

2

2

2) ( , ) = (1 +

) ( , ) ,

()

where 2= 0.5

2

. In ()(),

captures the microstructural

effects (/), and 2

(or 2) captures the ballistic behavior

of electron transport [25].Tzou [27] and Chandrasekharaiah [12], based on (), have

derived the linear theory of dual-phase-lag thermoelasticity,DPLT, that has been also reviewed in the survey due toHetnarski and Ignaczak [13]. In [26], another version of

DPLT has been proposed using the law (); we will referto this version as MDPLT. The non-Fourier law () hasbeen extended to thermoelectric materials by Abdou et al.[28].

The coupled equations of thermoelastic beams, includingthe effects of shear deformation and rotatory inertia, werederived by Jones [29]. Jones [29] referred to the paperof Ignaczak and Nowacki [30] that presented a similaranalysis; however, he did not supply any appraisal to thework because the reference was not available. Massalas andKalpakidis [31] introduced an analytical solution to a simplysupported Euler-Bernoulli beam without rotatory inertia.They expressed the thermal moment as a linear combinationof the upper and bottom surface temperatures of the beam.Sankar and Tzeng [32] have analytically solved the thermoe-lastic equilibrium equation of a functionally graded Euler-Bernoulli beam, by generalizing Sankar [33] work. They sug-gested that the temperature changes exponentially throughthe beam thickness. Babaei et al. [34] have numerically solvedthe coupled thermoelastic equations of a functionally gradedEuler-Bernoulli beam by employing the Galerkin finite ele-ment method [35]. They made use of a combination of theideas proposed by [31, 32] to express the temperature. Sun etal. [22, 36] have studied the laser-induced vibrations of a gen-eralized thermoelastic Euler-Bernoulli beam. They adoptedthe Lord-Shulman model in their analysis and followedthe Massalas-Kalpakidis assumption [31] with sinusoidalrepresentation of the temperature through the thickness ofthe beam obeying zero-temperature gradient on its upperand bottom surfaces. Khisaeva and Ostoja-Starzewski [37]have studied the thermoelastic damping of a generalizedEuler-Bernoulli microscale beam under the Lord-Shulmanmodel. Youssef and Elsibai [3841] have explored the vibra-tion of Euler-Bernoulli gold nanobeam due to ramp typeheating on one end of the beam under various theories ofthermoelasticity. They adopted the same assumption of [22].We refer to some analytical methods that have been intro-duced by el-Sirafy and Abdou [42] and Abdou and Aseeri[43].

It appears that when the ultrashort pulsed laser andthe micro/nanostructures are collected in one subject, it isquite natural to give a great deal to the microstructuraleffects during the fast-transient process. Motivated by theabove works, we will reconsider the micro/nanostructuralEuler-Bernoulli beams responses in this study by taking intoaccount the DPLT/MDPLT models. The problem will bemathematically modeled in the next section. Two differentmethods of heating will be considered on the upper surfaceof the beam: ultrashort pulsed laser heating and constant heatflux. The solutions will be derived in the third section byusing Laplace transform and the direct approach techniques.A numerical technique will be adopted in the fourth sectionto recover the solutions in the physical domain, and thena microscale gold beam is chosen to study the essentialdifferences between themathematicalmodels and the heatingmethods. The lattice variables are plotted against the spatialand temporal variables to show the way in which these vari-ables decay and the domain in which the heat influences; see[4447].

Mathematical Problems in Engineering 3

Thermal source on the upper surface z = h/2z

y

0

h

b

x

L

Figure 1: Problem description.

2. Problem Modeling

Consider an isotropic homogeneous rectangular thermoelas-tic beam of length (0 ), width (/2 /2), andthickness (/2 /2), where , , and are the usualCartesian axes lying, respectively, along the length,width, andthickness of the beam so that -axis coincides with the beamaxis and and axes coincide with the end ( = 0) withorigin located at the axis of the beam; see Figure 1.

According to the Euler-Bernoulli assumption, see Jones[29] and Hetnarski and Eslami [48], we assume that thedisplacement components are given by

(, , , ) =

V (, , , ) = 0

(, , , ) = (, ) ,

(1)

where (, ) is the lateral deflection of the beam. From nowon, all the variables are assumed to be independent of .

The (axial) stress along the beam axis, , is given by themodified Hookes law:

= (

1 2]) , (2)

where is the Young modulus, is the coefficient of linear

thermal expansion, ] is the Poisson ratio, = / is

the dilatation along the beam axis, and (, , ) = 0

denotes the deviation of absolute temperature (, , ) fromthe ambient temperature

0.

The bending moment resultant of the beam, , can beevaluated via the following relation:

=

/2

/2

d. (3)

Making use of Euler-Bernoulli assumption (1), andHookes

law (2) in (3), we arrive at

= (

2

2+ ) (4a)

= (

2

2+

) , (4b)

where = /(12]), is the secondmoment of the cross-

section area of the beam andis the thermal moment, and

andare given by

=

/2

/2

2d = 3

12

, =

(1 2])

/2

/2

d.

(4c)

The equation governing transverse deflections of thebeam, neglecting the rotatory inertia effects, is [29]

2

2+

2

2= 0,

(5)

where denotes the beam density and = is the cross-sectional area of the beam.

In combining (4b) and (5), we have

4

4+

2

2+

2

2= 0.

(6)

In addition to (6), the energy equation including thehyperbolic theories, considered in [26], is given as

(1 +

)

2 = [1 +

+ (1

2

+ 2

2

2

)

2

2]

(V

+ 0

)

(1 + 12

+ (2+ 3)

) ,

(7)

where 2 = (2/2) + (2/2) is the two-dimensionalLaplace operator,

= 1+ 2, =12, 1=

(1/)((1/) + (1/

))

1, 2= , is the relaxation time

at the Fermi level, V is the specific heat at constant volume,= (, ) denotes the initial heating source, to be defined

later, and 1, 2, and

3are controlling numbers which take

their values from the set {0, 1}.Equation (7), in essence, is considered a unified energy

equation in the linear theory of thermoelasticity with finitewave speed. Although it involves various thermoelastic theo-ries, we will focus on the following cases.

(i) When 1= 2= 0,

3= 1,

= 0, (7) reduces

the energy equation of the Lord-Shulman ther-moelasticity [7]. This model has been employed tomicroscale beams by Sun et al. [22, 36].

(ii) When 1= 3= 0,

2= 1, (7) reduces the energy

equation of the hyperbolic dual-phase-lag thermoe-lasticity, DPLT, [12, 27].

(iii) When 2= 3= 0,

1= 1, (7) reduces the energy

equation of the modified hyperbolic dual-phase-lagthermoelasticity, MDPLT, [26].

In view of the Euler-Bernoulli assumption (1), (7) couldtake the following form:

1

2 =

2(

V

0

3

2

) , (8a)


where

= (, ) = 3(, ) ,

1= 1 +

,

2= 1 +

+ 1

2

2

2+ 2

2

2

2

2,

3=

1

(1 + 12

+ (2+ 3)

) .

(8b)

Following Massalas and Kalpakidis [31], we multiply (8a)by d and integrate over the interval (/2, /2); this leadsto

1{(

2

2

1

2) (, ) +

2 (1 2])

[

(,

2

, ) +

(,

2

, )]}

= 2(

V

0

2

(1 2])2

3

2

)

(, ) () ,

(9a)

whereis mathematically approximated as the difference

between the temperatures at the upper and bottom surfacesof the beam, or, alternatively, the temperature is assumed tovary linearly through the thickness of the beam, thus, we have

(,

2

, ) (,

2

, ) =

(1 2])

2(, ) (9b)

and () is given by

() =

(1 2])

/2

/2

(, ) d. (9c)

(Assumption (9b) was successfully used by Massalas andKalpakidis [31] for beams with infinitesimal thickness.)

The problem, so far, is not emphasized in the small-scalelength. Introducing of the following dimensionless quantitiesseems to be crucial for the paper aim:

=

, =

, =

V

,=

2,

=

, V =

.

(10)

Accordingly, (6) and (9a), (9b), and (9c) read, after droppingthe dashes for convenience,

4

4+ 2

2

2+

12

2

1

2

2= 0

1{(

2

2

1

2

1

) (, )

+1[

(,

2

, ) +

(,

2

, )]}

= 2(2

3

3

2

) (, )

4

() ,

(11)

where

1=

, 2=

, 1=

6

2

12 (1 2])

,

2=

VV

, 3=

0

2

V

2(1 2])2

,

1= 1 +

1

, 2= 1 +

2

+ (1

2

3+ 2

2

2

2

)

2

2,

1=

V

, 2=

V

, 3=

V

(12)

and () is the dimensionless form of

(). Thus, (11) are

the governing equations of a vibrating Euler-Bernoulli beamwith length and aspect ratios

1and 2, under two transverse

thermal causes: (i) temperature gradient on the upper andbottom surfaces and (ii) thermal source applied in a normaldirection to the beam axis. Solving (11) leads to finding thebeam lateral deflection , the thermal moment

, and the

axial stress . The thermal momentcan be considered as

having the same behavior of temperature.

3. Solution in the Laplace Domain

The direct approach was first applied to coupled thermoe-lastic problems by Bahar and Hetnarski [49]. It has recentlyshowed successful results; see Ezzat and Awad [50, 51]. TheLaplace transform is defined by

{ ()} =

0

() d = () . (13)

By assuming zero initial conditions for , , and

their rates {/, /,

2/

2,

2/

2,

3/

2,

4/

2

2}, and in view of (11) and (13), we obtain

(

4+

12

2

2

1

) (, ) = 2

2(, ) (14)

(

2 1)(, ) =

2

2 (, )

() , (15)


where

=

, 1 () =

1

2

1

+

22

3

2 () =

32

1

,

1= 1 +

1,

2= 1 +

2 + (

1

2

3+ 2

2

2

2

)

2,

() = 1[

(,

2

, ) +

(,

2

, )] +

4

()

1

(16)

and () denotes the Laplace transform of the heating source

().The differential equation of the lateral deflection and

the thermal momentare

{

6

4+

2 } [

] =[

[

0

12

2

2

1

]

]

, (17a)

where

() = 1+ 22, () =

12

2

2

1

, () = 1.

(17b)

The differential equation governing the lateral deflection can take the form

(

2

2

1) (

2

2

2) (

2

2

3) = 0, (18)

where 1, 2, and

3are the characteristic roots of the

equation:

6

4+

2 = 0 (19a)

and satisfy the well-known relations:

2

1+

2

2+

2

3=

2

1

2

2+

2

2

2

3+

2

1

2

3= .

2

1

2

2

2

3=

(19b)

See (A.1) in the Appendix.Then the lateral deflection is given by

(, ) =

3

=1

(

+

) , (20)

where the coefficients and

, = 1, 2, 3, to be determined

later, are depending on the Laplace variable .The thermal moment is given by

(, ) =

3

=1

(

+

) +

1

, (21)

where and

, = 1, 2, 3, are depending on the Laplace

variable . If a relation is established between { ,} and

{, }, it will be needed six boundary conditions in order to

determine the unknowns {, }. Proceed in this direction

and substitute (20) and (21) into (15), it yields

(

) =

2

2

1

2

(

) , = 1, 2, 3, (22)

which helps us to rewrite (21) to be

(, ) =

3

=1

2

2

1

2

(

+

) +

1

. (23)

By holding similar arguments, the axial stresses can bewritten as

avg (, ) = 1

3

=1

2

(

22

12 (

2

1)

1) (

+

)

12

121

,

(24a)

where

avg (, ) = (,

2

, ) (,

2

, ) (24b)

is used and we recalled (4a), the approximation of thermalmoment (9b), (10) and (13), and (20) and (23).

In order to determine the unknown coefficients and

, = 1, 2, 3, we need to specify the thermal influence().

In the next section, we will discuss two different applicationsof thermal loads over the upper surface of the beam.

4. Applications

In this section, we consider the vibrations of a goldmicroscalebeam, of length = 0.1 m and aspect ratios

1= 0.1,

2=

0.05, induced by two methods of heating:

(i) laser source with non-Gaussian temporal profile setalong the upper surface = /2,

(ii) constant heat flux normal to the upper surface =/2.

First, consider the following profile:

() =

0

exp(

) , (25)

where is the laser-pulse duration and

0is the laser-pulse

intensity, namely, the total energy carried by the laser beamper unit cross-section and unit time. The profile (25) hassuccessfully been studied in the recent literatures [22, 26, 27,36, 52], in the case of microscale beams with single thermalrelaxation and microfilm with dual-phase-lag.

The laser source , according to case (i) of the present

section, can be therefore defined as

(, ) =

1

() exp( (/2)

) , (26)


0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0 0.2

0

0.000004

0.0000035

0.000003

0.0000025

0.000002

0.0000015

0.000001

0.0000005

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4 0.6 0.8 1 1.2

DPLTMDPLT

Lord-Shulman

0.00005

0.0001

0.00015

x

xw

w

Figure 2: Laser-induced lateral deflection at dimensionless time = 0.1 under the three models.

where and have their usual meaning, the surface reflec-tivity and the penetration depth of the beam material.

Furthermore, the beam is considered to be simply sup-ported with ends ( = 0, = ) kept at the ambient tem-perature. Thereby, the boundary conditions are determinedthrough

(0, ) = 0

2 (0, )

2= 0

(0, ) = 0

(27a)

(, ) = 0

2 (, )

2= 0

(, ) = 0.

(27b)

Moreover, the temperature gradient on the upper and bottomsurfaces are assumed null (no heat losses); namely,

(,

2

, ) =

(,

2

, ) = 0. (27c)

Next, we can evaluate the thermal influence given by(16), and the result is

() =

6 (1 ) 05[1 + (

14+ 22+ 32) ]

2

12(1 2]) (1 +

5)

21

,

(28a)

where

= [(1

2

) + (1 +

2

)

/] ,

0=

0

,

4=

V2

, 5=

V

.

(28b)

The nondimensional forms of the boundary conditions(27a), (27b), and (27c) in the Laplace domain read

(0, ) = 0

d2 (0, )d2

= 0 (0, ) = 0

(29a)

(1, ) = 0

d2 (1, )d2

= 0 (1, ) = 0.

(29b)

By the aid of (29a) and (29b) and the thermal influence(28a), and (28b), the solutions (20) and (23) lead to

3

=1

(+ ) = 0 (30a)


0

0.0005

0.001

0.0015

0.002

0 20 40 60 80 100 120 140

0

0.000006

0.000004

0.000002

00.2 0.4 0.6 0.8 1 1.2

DPLTMDPLT

Lord-Shulman

0.0005

0.001

0.0015

0.000002

0.000004

0.000006

0.000008

0.002

t

t

w

w

Figure 3: Laser-induced midpoint lateral deflection under the three models.

3

=1

2

(+ ) = 0 (30b)

3

=1

2

2

1

(+ ) =

12

(30c)

3

=1

(

+

) = 0 (30d)

3

=1

2

(

+

) = 0 (30e)

3

=1

2

2

1

(

+

) =

12

. (30f)

Then, by solving (30a), (30b), (30c), (30d), (30e), and(30f) one can obtain the required coefficients; see (A.3) in theAppendix.

Second, as stated in assumption (ii), we replace the lasersource term with a constant heat flux {

0} normal to the

upper surface of the beam, with keeping the bottom surfaceat zero temperature gradient.Theboundary conditions on theupper and bottom surface are then given through theDPL lawof heat conduction as

0= (1 +

)

(, /2, )

,

(, /2, )

= 0.

(31)

Applying (10) and (13), (31) yields

dd(,

2

, ) =

0

1

,

dd(,

2

, ) = 0. (32)

The other boundary conditions, (29a) and (29b), hold true inthe second application. In view of (16) and (32), the thermalinfluence of the second problem is given by

() =

10

1

(33)

and (30a), (30b), (30c), (30d), (30e), and (30f) hold theirforms with replacing from (28a) and (28b) to (33).

This completes the solution of the mathematical modelin the Laplace transformed domain. In order to obtain thesolution of the present application in the physical domain, wefirstly apply the well-known formula:

() =

1

2

+

()

d, (34)

to (20), (23) and (24a), and (24b) and use the results of (28a)and (28b), (30a), (30b), (30c), (30d), (30e), and (30f) and (33).Secondly, we adopt a numerical inversion method based on


0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

0.1

0

0.05

0.15

0.2

1.2

2

4

x

MT

MT 0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

DPLTMDPLT

Lord-Shulman

Figure 4: Laser-induced thermal moment at dimensionless time = 0.1 under the three models.

the Fourier series expansion, by which the integral (34) canbe approximated as a series

() =

1

[

[

1

2

Re ()

+

=0

Re(( +

1

)) cos(

1

)

=0

Im(( +

1

)) sin(

1

)]

]

=1

21 (2

1+ )

(35)

for 0 21. The above series (35) is called the

Durbin formula and the last term in which is called thediscretization error. Honig and Hirdes [53] developed amethod for accelerating the convergence of the Fourier seriesand a procedure that computes approximately the best choiceof the free parameters. Further, they established a FORTRANsubroutine that helps in implementing series (35). For furthernumerical methods, see pp. 317322 in [27].

5. Numerical Results and Discussion

The numerical values of the material parameters of gold (Au)are given as follows, see [23, 24, 26, 54, 55]:

= 79 10

9 Pa = 1930Kg m3 ] = 0.44

= 1.4 10

5 K1 = 315W m1 K1

V = 2.492 106 J m3 K1

= 89.286 ps

= 0.7838 ps

2

= 0.04 ps

= 0.1725 ps = 0.93

= 0.02 ps

0= 3.14 10

15 = 0.1 h

0= 300K

0= 2.25 10

11W m2.

(36)

Figure 2 shows the laser-induced lateral deflections atan early instant of time = 0.1. A similar graph (for theLord-Shulman model) has been obtained by Sun et al. [22]at = 0.4, 2 with the symmetry shape about the axis =/2. The DPLT and MDPLT models record values lowerthan that recorded in the Lord-Shulman model. In the insetplot of Figure 2, the curves inside the dotted rectangle areillustrated describing both the DPLT and MDPLT models ofthermomechanics. It is notable that the same result shown


0

5

10

15

20

25

0.01 0.51 1.01 1.51 2.01 2.51

0.15

0.05

0

0.1

0.2

0.25

21.951.91.851.81.751.71.651.61.5 1.55

5

10

t

t

MT

MT

0.05

0.1

0.15

0.2

0.25

x

(a)

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25

1.5

0.5

0

1

2

2.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.05

0.1

t

t

MT

MT

0.5

1

1.5

2

2.5

DPLTMDPLT

(b)

Figure 5: (a) Laser-inducedmidpoint dimensionless thermalmoment under the Lord-Shulmanmodel; see Figure 4 in [22]. (b) Laser-inducedmidpoint dimensionless thermal moment under the dual-phase-lag models.


0

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 1 1.20.01

0.02

0.03

av

g

t = 0.1

t = 0.5

t = 1

x

(a)

0

0.00005

0.0001

0.00015

0 0.2 0.4 0.6 0.8 1 1.2

0.00005

0.0001

av

g

x

DPLTMDPLT

(b)

Figure 6: (a) Laser-induced axial stress average at different values of dimensionless time according to the Lord-Shulman model. (b) Laser-induced axial stress at = 0.1, according to the dual-phase-lag models.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 50 100 150 200 250

w

DPLTMDPLT

Lord-Shulman

t

Figure 7: Heat flux-induced midpoint dimensionless lateral deflec-tion under the three models.

in the thermal shock problem, see [26], has been obtainedhere. Namely, the MDPLT model has a weak effect on thedisplacement of particles compared with the DPLT model,especially at the small values of time.

Figure 3 focuses on the temporal history of the midpointvibrations ( = /2). Obviously, the Lord-Shulman modelpredicts thermomechanical waves propagating with velocityfaster than the DPLT andMDPLTmodels, having amplitudeshigher than those recorded by theDPLT andMDPLTmodels.The midpoint returns to the quiescent mode in the Lord-Shulman model before the other models. The inset plotof Figure 3 clarifies the differences between the DPLT andMDPLT lateral deflections. As stated previously [26], themain difference between the DPL andMDPLmodels appearsin the early response times, and the difference betweenDPLT and MDPLT lateral deflections is apparent in thedimensionless time interval [0, 1]. TheMDPLTmodel lowersthe deflection values and increases smoothly within [0, 0.38].This trend is reversed in the subsequent interval [0.38, 0.76].DPLT thermomechanical waves propagate with speed faster

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40 45

MT

DPLTMDPLT

Lord-Shulman

t

Figure 8: Heat flux-induced midpoint dimensionless thermalmoment under the three models.

than MDPLT waves if relatively small values of time areconsidered.

Figures 4 and 5 describe the thermal moment variationswith beam length and time. The smooth curves of the DPLTand MDPLT models, which are incomparable with the Lord-Shulman curve, are clear in Figure 4. The DPLT and MDPLTcurves in the rectangular frame are illustrated in the insetplot of Figure 4. The difference between DPLT and MDPLTcurves drawn in inset-plot 4 is temporarily at = 0.1.Inset-plot 5b shows the

vibrations of the midpoint. At

= 0.1, the two inset-plots 4 and 5b coincide; however, ina subsequent instant = 0.15, one finds the two curves toreverse their positions near the midpoint, and at = 0.2 thereis no significant difference between the DPLT and MDPLTthermal moments. Furthermore, in the dimensionless timeinterval [0, 0.2], the thermal moment

of DPLT model

has frequency values higher than that in the MDPLT model.Figure 5(a), actually, supports the current study. It is notdissimilar from Figure 4, which is drawn by Sun et al. [22]for a Silicon beam, except for Figure 5(a), which begins from


00.000010.000020.000030.000040.000050.000060.00007

0 0.2 0.4 0.6 0.8 1 1.20.00001

0.00002

L = 1m

L = 10m

w

x

DPLTMDPLT

Lord-Shulman

(a)

00.20.40.60.8

11.21.41.61.8

0 0.2 0.4 0.6 0.8 1 1.2

MT

L = 1mL = 10m

x

DPLTMDPLT

Lord-Shulman

(b)

Figure 9: (a) Laser-induced lateral deflection, for a beam with length = 1 and 10m, at dimensionless time = 0.1 under the three models.(b) Laser-induced thermal moment, for a beam with length = 1 and 10m, at dimensionless time = 0.1 under the three models.

= 0.01, and the decaying rate in the gold beam is faster. Itwas difficult to draw

at earlier times than 0.01. However, it

is predicted to begin from null obeying the initial condition.In comparison between Figures 5(a) and 5(b), the DLPT andMDPLT thermalmoments have peaksmuch lower than peaksof Lord-Shulman curve.Themidpoint

-variation of Lord-

Shulman model has many fluctuations and short time zoneof effect, namely, the midpoint thermal moment variation ofDPLT and MDPLT models remains for a long time interval.The different velocities of the wave propagation of the threemodels may be a reason beyond the distinct fluctuations inFigure 5(a) and inset-plot of 5b. The thermal wave velocitiesof the Lord-Shulman and DPLT and MDPLT models, LS,, and MD, respectively, are given by

LS = V

=

2V

2

MD = V

1

2

V =

V.

(37)

For gold (Au), LS = 1.268 104m/s,

= 1.914 10

5m/s,MD = 6.149 10

5m/s; namely, LS < < MD.The thermal stress variations are plotted in Figures 6(a)

and 6(b) along the beam. In accordance with the previousresults, the stress curve of the Lord-Shulmanmodel recordedpeaks higher than that recorded by the DPLT and MDPLTmodels. In Figure 6(a), the stress curve along the beam underthe Lord-Shulman model is plotted at three different instants = 0.1, 0.5, and 1. The low values of stress curve of theMDPLT model are clear in Figure 6(b).

In Figures 7 and 8, the temporal history of the midpointdeflection and thermal moment are represented, respectively,for the heat flux problem. The midpoint lateral deflectionof the heat flux application vibrates about the dimensionlessvalue = 0.057, while it vibrates about the value = 0 inthe laser application. In the heat flux problem, the DPLT andMDPLT models reach steady state before the Lord-Shulman

model in the heat flux application unlike what is obtainedin the laser problem. The history of the midpoint thermalmoment variation is represented in Figure 8 for the threemodels.The sudden increase in the Lord-Shulman curve andthe gradual increase in the DPLT and MDPLT curves arecharacteristic features of the constant heat flux application.

Figures 9(a) and 9(b) show the effect of the beam lengthon the lattice variables, deflection, and thermal moment, forthe laser application. But the result can be generalized to thesecond application. It can be readily seen from Figures 9(a)and 9(b) that (i) the longer the beam is the lower the lateraldeflection and thermal moment are. (ii) The longer the beamis the less difference between models is.

6. Conclusion

The macroscopic vibrations of a gold Euler-Bernoulli ther-moelastic rectangular microscale beam have been studiedduring two different ways of heating, the laser-pulse heating,and the constant heat flux on the upper surface. The directapproach method has been employed to obtain the exactsolutions in the Laplace transformed domain. A numericaltechnique has been adopted to recover the solutions in thephysical domain. The paper focuses on the distinguishablefeatures of the Lord-Shulman model of thermomechanics,based on the Cattaneo-Vernotte law, and the Tzou-model ofthermomechanics (DPLT and MDPLT), based on the Qiu-Tien hyperbolic two-step model.

The dual-phase-lag models (DPLT and MDPLT), basi-cally, depend on a certain mechanism of heat transport. Itsubdivides the heat transport to three processes [20, 21]:(i) the deposition of radiation energy on electron, (ii) thetransport of energy by electron subsystem, and (iii) theheating of material lattice via electron-phonon interactions.The single relaxation time model (Lord-Shulman), on theother hand, neglects the electron-phonon interactions duringthe fast transient processes. This may be the major reasonbeyond the behavior of DPLT andMDPLT.The results of this


paper can be supported by the evident differences betweenthe electron and lattice temperatures during ultrafast laserheating; see Figures 1 and 9 in Chen et al. [54] and Figure2 in Tzou and Pfautsch [55]. There is a material rangeof picoseconds needed to reach the thermal equilibriumbetween phonons and electrons (thermalization time), whilstboth the electrons and the lattice are heated simultaneouslyin the Lord-Shulman thermomechanics. Upon reaching thethermal equilibrium between electrons and the lattice, theDPLT andMDPLT become immaterial. Furthermore, as seenfrom Figures 4, 5(a), and 5(b), the meaningless V-shape hasdisappeared from the DPLT models, which indicates theimproved numerical results of the DPLT models.

One can conclude that whenever the response time isrelatively short and the material dimensions are relativelysmall, the dual-phase-lag thermoelastic models could notbe disregarded and could be considered as the most closemacroscopic approach to study the general behavior (but notaccurate because of the constant thermophysical properties)rather than encountering the strong nonlinearity of theultrafast model [54].

Appendix

Upon solving (19b), the characteristic roots of the problemare given by

1=

1

3

{ + 21cos(

cos1 (2)

3

)}

1/2

,

2=

1

3

{ 21sin( sin1 (

2)

3

)}

1/2

3=

1

3

{ 21sin(

sin1 (2)

3

)}

1/2

,

(A.1)

where

1=

2 3,

2=

2

3 9 + 27

2

3

1

. (A.2)

The problem coefficients, and

, = 1, 2, 3, are

determined through

1=

(

2

1 1) (

2

2 1) (

2

3 1)

2

12(

2

1

2

2) (

2

1

2

3) (

1+ 1)

,

2=

(

2

3

2

1) (

1+ 1)

(

2

2

2

3) (

2+ 1)

1

3=

(

2

1

2

2) (

1+ 1)

(

2

2

2

3) (

3+ 1)

1,

1=

11,

2=

22,

3=

33.

(A.3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Alexandria University financial support for this work isacknowledged. The authors are indebted to the referees fortheir helpful comments.

References

[1] P. M. Morse and H. Feshbach, Methods of Theoretical Physics,McGraw-Hill, New York, NY, USA, 1st edition, 1953.

[2] C. Cattaneo, A form of heat conduction equation whicheliminates the Paradox of instantaneous propagation, ComptesRendus, vol. 247, pp. 431433, 1958.

[3] P. Vernotte, Les paradoxes de la theorie continue de lequationde la chaleur, Comptes Rendus, vol. 246, pp. 31543155, 1958.

[4] P. Vernotte, Some possible complications in the phenomena ofthermal conduction, Comptes Rendus, vol. 252, pp. 21902191,1961.

[5] D. D. Joseph and L. Preziosi, Heat waves, Reviews of ModernPhysics, vol. 61, no. 1, pp. 4173, 1989.

[6] D. Y. Tzou, Thermal shock phenomena under high-rateresponse in solids, in Annual Review of Heat Transfer, C.-L.Tien, Ed., pp. 111185,Hemisphere Publishing,Washington,DC,USA, 1992.

[7] H.W. Lord and Y. Shulman, A generalized dynamical theory ofthermoelasticity, Journal of the Mechanics and Physics of Solids,vol. 15, no. 5, pp. 299309, 1967.

[8] M. A. Biot, Thermoelasticity and irreversible thermodynam-ics, Journal of Applied Physics, vol. 27, no. 3, pp. 240253, 1956.

[9] R. S. Dhaliwal and H. H. Sherief, Generalized thermoelasticityfor anisotropic media, Quarterly of Applied Mathematics, vol.38, no. 1, pp. 18, 1980.

[10] H. H. Sherief, Fundamental solution of the generalized ther-moelastic problem for short times, Journal of Thermal Stresses,vol. 9, no. 2, pp. 151164, 1986.

[11] M. N. Anwar and H. H. Sherief, State space approach togeneralized thermoelasticity, Journal of Thermal Stresses, vol.11, no. 4, pp. 353365, 1988.

[12] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: areview of recent literature, Applied Mechanics Reviews, vol. 51,no. 12, pp. 705729, 1998.

[13] R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity,Journal of Thermal Stresses, vol. 22, no. 4, pp. 451476, 1999.

[14] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity withFinite Wave Speeds, OxfordMathematical Monographs, OxfordUniversity Press, Oxford, UK, 2010.

[15] M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov, Relaxationbetween electrons and crystalline lattices, Soviet Physics JETP,vol. 4, pp. 173178, 1957.

[16] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, Electronemission from metal surfaces exposed to ultra-short laserpulses, Soviet Physics JETP, vol. 39, pp. 375377, 1974.

[17] S. D. Brorson, J. G. Fujimoto, and E. P. Ippen, Femtosecondelectronic heat-transport dynamics in thin gold films, PhysicalReview Letters, vol. 59, no. 17, pp. 19621965, 1987.

[18] S. D. Brorson, A. Kazeroonian, J. S. Moodera et al., Femtosec-ond room-temperature measurement of the electron-phononcoupling constant inmetallic superconductors,Physical ReviewLetters, vol. 64, no. 18, pp. 21722175, 1990.


[19] H. E. Elsayed-Ali, T. Juhasz, G. O. Smith, andW. E. Bron, Fem-tosecond thermoreflectivity and thermotransmissivity of poly-crystalline and single-crystalline gold films, Physical Review B,vol. 43, no. 5, pp. 44884491, 1991.

[20] T. Q. Qiu and C. L. Tien, Short-pulse laser heating on metals,International Journal of Heat and Mass Transfer, vol. 35, no. 3,pp. 719726, 1992.

[21] T. Q. Qiu and C. L. Tien, Heat transfer mechanisms duringshort-pulse laser heating of metals, Journal of Heat Transfer,vol. 115, no. 4, pp. 835841, 1993.

[22] Y. Sun, D. Fang, M. Saka, and A. K. Soh, Laser-induced vibra-tions of micro-beams under different boundary conditions,International Journal of Solids and Structures, vol. 45, no. 7-8,pp. 19932013, 2008.

[23] D. Y. Tzou, The generalized lagging response in small-scaleand high-rate heating, International Journal of Heat and MassTransfer, vol. 38, no. 17, pp. 32313240, 1995.

[24] D. Y. Tzou, Unified field approach for heat conduction frommacro- to micro-scales, Journal of Heat Transfer, vol. 117, no. 1,pp. 816, 1995.

[25] D. Y. Tzou and J. Xu, Nonequilibrium transport: the laggingbehavior, in Advances in Transport Phenomena, L. Q. Wang,Ed., pp. 93170, Springer, Berlin, Germany, 2011.

[26] E. Awad, On the generalized thermal lagging behavior: refinedaspects, Journal of Thermal Stresses, vol. 35, no. 4, pp. 293325,2012.

[27] D. Y. Tzou, Macro- to Microscale Heat Transfer: The LaggingBehavior, Taylor and Francis, New York, NY, USA, 1997.

[28] M. A. Abdou, I. H. El-Sirafy, and E. Awad, Thermoelectriceffects ofmetals within small-scale regimes: theoretical aspects,Chinese Journal of Physics, vol. 51, no. 5, pp. 10511066, 2013.

[29] J. P. Jones, Thermoelastic vibrations of a beam, Journal of theAcoustical Society of America, vol. 39, no. 3, pp. 542548, 1966.

[30] J. Ignaczak and W. Nowacki, Transversal vibration of a plate,produced by heating, ArchiwumMechaniki Stosowanej, vol. 13,pp. 651667, 1961.

[31] C. V. Massalas and V. K. Kalpakidis, Coupled thermoelasticvibrations of a simply supported beam, Journal of Sound andVibration, vol. 88, no. 3, pp. 425429, 1983.

[32] B. V. Sankar and J. T. Tzeng, Thermal stresses in functionallygraded beams,AIAA Journal, vol. 40, no. 6, pp. 12281232, 2002.

[33] B. V. Sankar, An elasticity solution for functionally gradedbeams, Composites Science and Technology, vol. 61, no. 5, pp.689696, 2001.

[34] M. H. Babaei, M. Abbasi, and M. R. Eslami, Coupled ther-moelasticity of functionally graded beams, Journal of ThermalStresses, vol. 31, no. 8, pp. 680697, 2008.

[35] M. R. Eslami and H. Vahedi, Coupled thermoelasticity beamproblems, AIAA journal, vol. 27, no. 5, pp. 662665, 1989.

[36] Y. Sun, D. Fang, and A. K. Soh, Thermoelastic damping inmicro-beam resonators, International Journal of Solids andStructures, vol. 43, no. 10, pp. 32133229, 2006.

[37] Z. F. Khisaeva andM.Ostoja-Starzewski, Thermoelastic damp-ing in nanomechanical resonators with finite wave speeds,Journal of Thermal Stresses, vol. 29, no. 3, pp. 201216, 2006.

[38] H. M. Youssef and K. A. Elsibai, Vibration of nano beaminduced by Ramp type heating,World Journal of Nano Scienceand Engineering, vol. 1, pp. 3744, 2011.

[39] H. M. Youssef and K. A. Elsibai, Vibration of gold nanobeaminduced by different types of thermal loadinga state-space

approach, Nanoscale and Microscale Thermophysical Engineer-ing, vol. 15, no. 1, pp. 4869, 2011.

[40] K. A. Elsibai and H. M. Youssef, State-space approach tovibration of gold nano-beam induced by ramp type heatingwithout energy dissipation in femtoseconds scale, Journal ofThermal Stresses, vol. 34, no. 3, pp. 244263, 2011.

[41] H. M. Youssef, Vibration of gold nanobeam with variablethermal conductivity: state-space approach, Journal of AppliedNanoscience, 2013.

[42] I. H. el-Sirafy andM. A. Abdou, First and second fundamentalproblems of infinite plate with a curvilinear hole, Journal ofMathematical and Physical Sciences, vol. 18, no. 2, pp. 205214,1984.

[43] M. A. Abdou and S. A. Aseeri, Closed forms of goursatfunctions in presence of heat for curvilinear holes, Journal ofThermal Stresses, vol. 32, no. 11, pp. 11261148, 2009.

[44] S. Chirita and R. Quintanilla, Spatial decay estimates of Saint-Venant type in generalized thermoelasticity, International Jour-nal of Engineering Science, vol. 34, no. 3, pp. 299311, 1996.

[45] E. S. Awad, A note on the spatial decay estimates innon-classical linear thermoelastic semi-cylindrical boundeddomains, Journal ofThermal Stresses, vol. 34, no. 2, pp. 147160,2011.

[46] B. Carbonaro and R. Russo, Energy inequalities and thedomain of influence theorem in classical elastodynamics,Journal of Elasticity, vol. 14, no. 2, pp. 163174, 1984.

[47] E. Awad, Domain of influence theorem and uniqueness resultsin magneto-thermoelasticity, International Journal of AppliedElectromagnetics and Mechanics, vol. 37, no. 1, pp. 18, 2011.

[48] R. B. Hetnarski and M. R. Eslami, Thermal StressesAdvancedTheory and Applications, vol. 32, Springer, New York, NY, USA,2009.

[49] L. Y. Bahar and R. B. Hetnarski, Direct approach to thermoe-lasticity, Journal of Thermal Stresses, vol. 2, no. 1, pp. 135147,1979.

[50] M. A. Ezzat and E. S. Awad, Micropolar generalized magneto-thermoelasticity with modified Ohms and Fouriers laws,Journal of Mathematical Analysis and Applications, vol. 353, no.1, pp. 99113, 2009.

[51] M. A. Ezzat and E. S. Awad, Constitutive relations, uniquenessof solution, and thermal shock application in the linear theoryof micropolar generalized thermoelasticity involving two tem-peratures, Journal of Thermal Stresses, vol. 33, no. 3, pp. 226250, 2010.

[52] D. Y. Tzou, Experimental support for the lagging behavior inheat propagation, Journal of Thermophysics and Heat Transfer,vol. 9, no. 4, pp. 686693, 1995.

[53] G. Honig and U. Hirdes, Amethod for the numerical inversionof Laplace transforms, Journal of Computational and AppliedMathematics, vol. 10, no. 1, pp. 113132, 1984.

[54] J. K. Chen, J. E. Beraun, and D. Y. Tzou, Thermomechanicalresponse of metals heated by ultrashort-pulsed lasers, Journalof Thermal Stresses, vol. 25, no. 6, pp. 539558, 2002.

[55] D. Y. Tzou and E. J. Pfautsch, Ultrafast heating and thermome-chanical coupling induced by femstosecond lasers, Journal ofEngineering Mathematics, vol. 61, no. 2-4, pp. 231247, 2008.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014


Stochastic AnalysisInternational Journal of

Documents

Generalized Lagging Response of Thermoelastic Beams