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Thermoelastic response of a one-dimensional semi-infinite

rod heated by a moving laser pulse

Journal: Canadian Journal of Physics

Manuscript ID cjp-2016-0057.R1

Manuscript Type: Article

Date Submitted by the Author: 13-Apr-2016

Complete List of Authors: Sun, Yuxin; Beihang University Press Ma, Jingxuan; Beihang University Press Wang, Xin; Monash University - Malaysia Campus Soh, Ai Kai; Monash University - Malaysia Campus Yang, Jialing; Beihang University, The Solid Mechanics Research Center

Keyword: movable heat source, laser pulse, thermoelastic response, semi-infinite

rod, non-Fourier effect

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

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Thermoelastic Response of a One-dimensional Semi-infinite

Rod Heated By a Moving Laser Pulse

Yuxin Sun1∗, Jingxuan Ma1, Xin Wang2, Ai Kah Soh2, Jialing Yang1

1 School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, P. R. China

2 School of Engineering, Monash University Malaysia, Bandar Sunway, 46150, Malaysia

Abstract

In the present study, thermoelastic behavior of a semi-infinite rod which is subjected to

a time exponentially decaying laser pulse is formulated. The rod is free at the left end and

the laser pulse moves along the axial direction from the left end. The non-Fourier effect of

heat conduction equation is considered and the Laplace transformation method is employed

in solving the governing equations. The temperature, displacement, strain and stress in the

rod are derived and the distributions of the parameters at different positions are analyzed.

Also the influence of the laser speed is investigated.

Key words: movable heat source; laser pulse; thermoelastic response; semi-infinite rod;

non-Fourier effect.

PACS Nos.: 44.05.+e, 44.10.+i, 46.25.Hf, 46.70.Hg.

* Corresponding author. E-mail address: [email protected].

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1．．．．Introduction

Boeing Company has developed a kind of high-energy laser weapon, which have

successfully destroyed more than 150 target drones in three series of tests. This technology

can be used to intercept missiles, cannonballs, rocket and satellites which can’t be attacked

by normal weapons. So it’s significant to find the way to defend against laser weapon.

To study the thermoelastic behavior of a structure induced by a moving heat source, it

is important to obtain the temperature distribution firstly [1-4]. For example, Kim [5]

evaluated the temperature distribution around a rectangular shape source moving at a

constant speed along the axis of a bar. Elsen et al. [6] described the analytical and numerical

solution of the heat conduction equation for a localized moving heat source in a 3D

semi-infinite medium.

In laser pulse heating, the high-intensity energy flux and short duration will introduce a

very large thermal gradient in the structure. In such cases, as pointed out by many researchers,

the non-Fourier effect of heat conduction must be considered, because the classical Fourier

model, which leads to an infinite propagation speed of the thermal energy, is no longer valid

[7, 8]. With this motivation, Lord and Shulman [9], and Green and Lindsay [10], established

the L-S and G-L generalized thermoelasticity theory, by introducing one or two relaxation

time parameters into the classical thermoelasticity theory, respectively. He et al. [11] solved a

boundary value problem of one-dimensional semi-infinite piezoelectric rod with the left

boundary subjected to a sudden heat flux using the theory of generalized thermoelasticity

with one relaxation time. Xia et al. [12] investigated dynamic response of an isotropic

semi-infinite plate subjected to a moving heat source based on the L-S generalized

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thermoelasticity theory.

Up to date, the thermoelastic behavior induced by a moving heat source has attracted

many attentions [13-15]. Huniti et al. [16] investigated the dynamic thermal and elastic

behavior of a rod due to a moving heat source. Yevtushenko and Ukhanska [17] derived the

solution of a quasi-steady thermoelastic problem for the elastic convective half-space when

a heat source is moving on its surface. Shuja and Yilbas [18] investigated thermal stress

analysis of laser multi-beam heating of a moving steel sheet. It is found that presence of

multi-spots at the surface modifies temperature and stress fields in the heated region, which

is more pronounced with increasing intensity at the irradiated spots. However, the heat

source is assumed to have constant power in most works.

Laplace transform method is efficient for solving partial differential equations [19].

However, the expressions of the solutions in the transformation domain are usually

complicated and cannot be inverted to the physical domain analytically. As an alternative, the

numerical inversion is applied. Durbin [20] presented a numerical inversion method which is

the most acceptable, but the inversion may become highly oscillatory or get away from the

right solution with the increase of time in some cases. Sun et al. [21] developed a general

algorithm of the inverse Laplace transformation, which overcomes the drawback of the

Durbin’s method and can achieve more reliable results for long time inversion.

In the present study, the mechanical response of a rocket in flight irradiated by a time

decaying laser pulse is researched. Since the time duration of laser pulse is very short, the

rocket is regarded as a one-dimensional semi-infinite rod.

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2. Problem formulations

Consider a semi-infinite rod subjected to time exponentially decaying laser pulse. The

original point of x -axis is put on the left end, and the laser pulse moves towards right from

the left end with constant speed v . The environmental temperature is 0T . The heat transfer

equation considering the non-Fourier effect can be written as [22]:

2 2

0 0

2 2

1t tT q q T T

x k k t t tα α∂ ∂ ∂ ∂

+ + = +∂ ∂ ∂ ∂

(1)

where, (0 )x x≤ <∞ is the axial coordinate, t is time, T is the temperature, k is the

thermal conductivity, 0t is the relaxation time, α is the thermal diffusivity.

The laser pulse is decaying with time, so the heat source q is expressed as

( )0 expp

tq q x vt

tδ

= − −

(2)

where, 0q is the power intensity of the laser pulse,

pt is the time duration of the laser

pulse, and ( )δ

is the Dirac delta function.

The vibration equation of a rod is:

2 2

2 2

u u T

x E t x

ρβ

∂ ∂ ∂− =

∂ ∂ ∂ (3)

where u is the displacement, ρ is the density, E is the Young’s modulus, β is the

coefficient of thermal expansion.

The stress of the rod is:

( )0x xE E T Tσ ε β= − − (4)

Where /x u xε = ∂ ∂ is the strain.

By introducing the following dimensionless variables,

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0 00

0

, , , ,p p p

T T tt x

T t t tθ τ τ ξ

α−

= = = = p

vV

tα= ，

0

,u E

WT

ρβα

=0

xxS

E T

σβ

=

From Eqs. (1)- (4), it can be obtained

( ) ( )2 2

1 02 2exp ,A

θ θ θτ ψ ξ τ τ

ξ τ τ∂ ∂ ∂

+ − = +∂ ∂ ∂

(5)

2 2

2 32 2

W WA A

θξ τ ξ∂ ∂ ∂

− =∂ ∂ ∂

(6)

4x

WS A θ

ξ∂

= −∂ (7)

where ( ) ( ) ( )0, V Vψ ξ τ δ ξ τ τ δ ξ ττ∂

= − + −∂

,

0 01

0

1p

p

q tA t

kT tα

= −

, 2

p

AEt

ρα= ,

3

2

1A

A= ,

4 2A A= .

The rod has the same temperature with the evironment initially, and the left end is

adiabatic, so the dimensionless initial and boundary conditions related to the heat

conduction equation are

0

0

0, 0τ

τ

θθ

τ==

∂= =

∂ (8)

0

0ξ

θξ =

∂=

∂ (9)

On the other hand, the left end of the rod is stress free, so the dimensionless initial and

boundary conditions related to the vibration equation are

0

0

0, 0W

Wτ

ττ==

∂= =

∂ (10)

0| 0xS ξ= = (11)

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3. Analytical Solutions

Taking the Laplace transformation of Eqs. (5) and (8) with respect to the τ variable,

the following equations are obtained:

( )

2

1 2 32expM M M

θξ θ

ξ∂

− + =∂

%%

(12)

where ( )1 0

1

1 1A sM

V

τ+ + = , 2

1 sM

V

+= , 2

3 0M s sτ= + .

It is obtained from Eq. (12) that

( ) ( ) ( )11 3 2 3 22

2 3

exp exp expM

C M C M MM M

θ ξ ξ ξ= − + − −−

% (13)

where 1C and 2C are constants to be determined from the boundary conditions Eq. (9).

Since the temperature is finite in the rod, 2 0C = . Thus, the temperature in the Laplace

transformation domain can be obtained from Eqs. (12) and (9) as

( ) ( )1 3 2 2exp expB M B Mθ ξ ξ= − − −% (14)

where ( )

1 2 11 2 22

2 32 3 3

,M M M

B BM MM M M

= =−−

.

The Laplace transformation of Eqs. (6) and (10) with respect toτ variable gives

22

2 32

WA s W A

θξ ξ∂ ∂

− =∂ ∂

%%% (15)

Substitution of Eq. (14) into Eq. (15) results in

( ) ( )2

2

2 3 1 3 3 2 2 22exp exp

WA s W A B M M B M Mξ ξ

ξ∂ − = − − + −

∂

%%

(16)

Similarly, the displacement is finite in the rod, so the displacement in the Laplace

transformation domain can be obtained from Eqs. (16) and (11) as

( ) ( ) ( )1 2 1 2 2 3exp exp expW D A s h M h Mξ ξ ξ= − + − + −% (17)

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where1 3 1 2 1 2 2 3

2

1[ ( ) ]D A B B hM h M

A s= − − + + , 2 3 2

1 2 2

2 2

B A Mh

A s M

−=

−,

1 3 3

2 2

2 3

B A Mh

A s M=

−.

Substitution of Eqs. (14) and (17) into Eq. (7), the stress in the Laplace transformation

domain can be obtained as

( ) ( ) ( )2 1 2 3 2 4 3exp exp expxS A D s A s h M h Mξ ξ ξ= − − + − − −% (18)

where 2 2

2 2 1 23 42 2 2

2 2 2 3

,B A s B A s

h hA s M A s M

= =− −

.

The strain can be obtained as

( ) ( ) ( )

0 4

0 4 1 2 2 1 2 2 2 3 3exp exp exp

x

WT A

T A D A s A s hM M h M M

ε βξ

β ξ ξ ξ

∂=

∂

= − − − − − −

%%

(19)

4. Numerical inversion Laplace transform

In order to get the time history of the temperature, displacement and stress of the rod,

the inversion of Laplace transform of Eqs. (14), (17)-(19) are taken with respect to the

variable τ . However, the expressions are complicated and it’s difficult to get the inverse

explicitly. Usually, a numerical calculation method is applied to take the inverse Laplace

transformation [21]. The algorithm is described concisely in the following.

Suppose that ( )f t is a real function of t , with ( ) 0f t = for 0t < . The Laplace

transform of the function and its inversion are defined as follows:

( ) ( ) ( )0

stf s L f t e f t dt∞ −= = ∫% (20)

( ) ( ) ( )1exp

2

i

if t f s st ds

i

η

ηπ

+ ∞

− ∞= ∫ % (21)

where s is the complex transform parameter and η is a real number greater than the real

parts of all singularities of ( )f s% .

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When ( )f s% is known, the inversion may be obtained with the theory of complex

analysis. However, the expression is usually complicated and the integral in Eq. (21) cannot

be analytically evaluated. As an alternative, a numerical inversion is applied.

Let s iη ϖ= + , then we have

( ) ( )0

10 Re df f iη ϖ ϖ

π

∞ = + ∫ % , for 0t = (22)

( ) ( ) ( ) ( ) ( ){ }0

Re cos Im sin dt

ef t f i t f i t

η

η ϖ ϖ η ϖ ϖ ϖπ

∞ = + − + ∫ % % , for 0t > . (23)

Now the integration variable is changed into ϖ . According to the discretization

approach, the integration interval is divided into small sub-spaces, and the nodes are

denoted as kϖ ( 1,2,...,k = ∞ ). Applying the linear interpolating functions to ( )f iα ϖ+%

in the interval1[ , ]k kϖ ϖ + Eqs. (22) and (23) can be changed into

( ) ( )11

02

k k k

k

F Ff

ϖπ

∞+

=

+ ∆=∑ , for 0t = (24)

( ) ( )( ) ( )( )

( ) ( )( )

11

21 1

1

cos cosexp

sin sin

k kk k

k

k k kk k

k

F Ft t

tf t

G Gtt t

ϖ ϖϖη

πϖ ϖ

ϖ

++∞

= ++

− − ∆ = − − − ∆

∑ for 0t > (25)

where,

1k k kϖ ϖ ϖ+∆ = −, (26)

( )Rek kF f iη ϖ = + % , ( )1 1Rek kF f iη ϖ+ +

= + % , (27)

( )Imk kG f iη ϖ= + , ( )1 1Imk kG f iη ϖ+ + = + % . (28)

5. Results and discussions

In the present work, the thermoelastic problem is analyzed by considering a rod made

of copper. The material parameters [16] are: 38960kg / mρ = ,4 21.1283 10 m / sα −= × ,

13

0 4.348 10 st −= × , 111.19 10 PaE = × , 5 11.76 10 Kβ − −= × , ( )385J/ kg Kk = ⋅ . The parameters

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of the laser pulse are: 6

0 1 10 Jq = × ,31 10 spt−= × .

Fig. 1 shows the dimensionless temperature at different position when the laser pulse

moves at the dimensionless speed 1V = . The influence of the motion of the laser pulse can

be seen clearly. At the position 0ξ = , the temperature increases to the peak value in a short

time and then drops quickly. However, at the position 10ξ = , the temperature is 0 at the

beginning and increases slowly from the time about 4τ = . This is induced by the motion of

the laser pulse and the relaxation of heat. In addition, it takes longer time to reach the peak

value of temperature as the distance from the free point increases.

Fig. 2 shows the dimensionless temperature at the position 5ξ = with different

moving speed of the laser pulse. The influence of the moving speed can be seen clearly. The

motion of heat also can be observed clearly. It is shown that the temperature decreases as

the speed of laser pulse increases. When the laser pulse moves at low speed, more energy is

absorbed at the specified position. So the temperature increases to a higher value. However,

the heat energy is distributed in a larger region at higher speed and the heat source intensity

is lower. As a result, the temperature is lower.

Fig. 3 shows the dimensionless displacement at different position when the laser pulse

moves at the speed 1V = . Since the rod is free, it expands after irradiated by the laser pulse,

and the left end extends towards left, which leads to negative displacement. The magnitude

of displacement decreases along the axial direction.

Fig. 4 shows the dimensionless displacement at the position 5ξ = with different

moving speed of the laser pulse. It is shown that the displacement decreases as the speed of

laser pulse increases.

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Since strain is a nondimensional parameter, it is obtained directly. Fig. 5 shows the

strain at different position when the laser pulse moves at the speed 1V = . Fig. 6 shows the

strain at the position 5ξ = with different moving speed of the laser pulse. It can be found

that the tendency of strain is the same with that of temperature.

Fig. 7 shows the dimensionless stress at different position when the laser pulse moves

at the speed 1V = . The transpose of stress wave along the axial direction is shown clearly.

Fig. 8 shows the dimensionless stress at the position 5ξ = with different moving speed of

the laser pulse. Because the rod is free, the stress induced by the constraint on the boundary

and deformation is smaller than that induced by temperature variation.

6. Conclusions

A rocket in flight was modeled as a one-dimensional semi-infinite free rod. The

temperature, displacement, strain and stress in the rod induced by a moving time decaying

laser pulse were derived and the distributions of the parameters at different positions were

analyzed. The transpose of heat and stress along the axial direction can be observed clearly

from the results. Also the influence of the laser speed was investigated. As the laser speed

increases, the temperature and displacement decreases, however, the stress increases.

Acknowledgement

The work described in this paper is financially supported by the National Natural

Science Foundation of China under grant number 11002017. The authors would like to

gratefully acknowledge the support.

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References

[1] N.T. Nguyen, A. Otha, K. Matsuoka, N. Suzuki, Y. Maeda. Weld. Res. Suppl. 8, 265-s

(1999).

[2] B.S. Yilbas, M. Sami, H.I. AbuAlHamayel. Appl. Surf. Sci. 134, 159 (1998).

[3] Y.M. Ali, L.C. Zhang. Int. J. Heat Mass Trans. 48, 2741 (2005).

[4] J. Kidawa-Kukla. Int. J. Heat Mass Trans. 51, 865 (2008).

[5] C.K. Kim. J. Mech. Sci. Tech. 25, 895 (2011).

[6] M.V. Elsen, M. Baelmans, P. Mercelis, J.P. Kruth. Int. J. Heat Mass Trans. 50, 482

(2007).

[7] D.W. Tang, N. Araki N. J. Phys. D: Appl. Phys. 29, 2527 (1996).

[8] M.N. Ozisik, D.Y. Tzou. J. Heat Trans. ASME 116, 526 (1994).

[9] H.W. Lord, Y. Shulman. J. Mech. Phys. Solids. 15, 299 (1967).

[10] A.E. Green, K.A. Lindsay. J. Elasticity 2, 1 (1972).

[11] T.H. He, X.G. Tian, Y.P. Shen. Int. J. Engng. Sci. 40, 1081 (2002).

[12] R.H. Xia, X.G. Tian, Y.P. Shen. Acta Mechanica Solida Sinica. 27, 300 (2014).

[13] G. Lykotrafitis, H.G. Georgiadis. Int. J. Solids. Struc. 40, 899 (2003).

[14] N. Sumi, R.B. Hetnarski. Nec. Eng. Des. 117, 159 (1989).

[15] T.H. He, L. Cao. Math. Comp. Model. 49, 1710 (2009).

[16] N.S. Al-Huniti, M.A. Al-Nimr, M. Naji. J. Sound Vib. 242, 629 (2001).

[17] A.A. Yevtushenko, O.M. Ukhanska. Int. J. Heat Mass Trans. 37, 2737 (1994).

[18] S.Z. Shuja, B.S. Yilbas. Optics Lasers Eng. 51, 446 (2013).

[19] A. Magdy, S. Ahmed, A. Angail. Appl. Math. Comput. 147, 169 (2004).

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[20] F. Durbin. Comput. J. 17, 371 (1974).

[21] Y.X. Sun, D.N. Fang, M. Saka, A.K. Soh. Int. J. Solids Struct. 45, 1993 (2008).

[22] J. Xu, X.W. Wang. Physica B. 351, 213 (2004).

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Captions of Figures

Fig. 1 Dimensionless temperature at different positions ( 1V = ).

Fig. 2 Dimensionless temperature at 5ξ = with different laser speed.

Fig. 3 Dimensionless displacement at different positions ( 1V = ).

Fig. 4 Dimensionless displacement at 0ξ = with different laser speed.

Fig. 5 Strain at different positions ( 1V = ).

Fig. 6 Strain at 5ξ = with different laser speed.

Fig. 7 Dimensionless stress at different positions ( 1V = ).

Fig. 8 Dimensionless stress at 5ξ = with different laser speed.

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Fig. 1

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Fig. 2

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Fig. 3

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Fig. 4

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Fig.5

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Fig. 6

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Fig. 7

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Fig. 8

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