Study on Generalized Thermoelastic Problem of Semi‐infinite Plate Heated Locally by the Pulse Laser

International Journal of Engineering Practical Research (IJEPR) Volume 3 Issue 4, November 2014 www.seipub.org/ijepr doi: 10.14355/ijepr.2014.0304.06

Study on Generalized Thermoelastic Problem of Semi‐infinite Plate Heated Locally by the Pulse Laser Ronghou Xia*1, Yanfeng Guo2, Weiqin Li3 Xi’an University of Technology, Xi’an 710048, People’s Republic of China *1

xiaronghou@xaut.edu.cn; 2guoyf@xaut.edu.cn; 3wqlee@126.com

Abstract A generalized thermoelastic coupled problem for the semi‐ infinite plane induced by pulsed laser heating locally is studied by adopting L‐S generalized thermoelasticity. In order to avoid the loss of the precision in general integral transformation method, the finite element control equations are solved directly in the time domain. Temperatures and displacements distributions are obtained and represented graphically under pulsed laser heating in the semi‐infinite plane. The results show that the maximum temperature on the structure always locates near the thermal wave front and reduces gradually alone with time evolution.

Keywords L‐S Generalized Thermoelastic Theory; Pulsed Laser Heating; Thermoelasticity Coupled; Finite Element Method; Time Domain

Introduction The Fourier heat conduction law based on the classic thermalelastic theory is sufficiently accurate under usual conditions. However, this theory considers that the heat balance is set up instantaneously, and it implies that the thermal signal propagates with the infinite speed. Under some extreme conditions, such as the case of a very short time, we find that the results are inconsistent with the experimental observations, which means that the Fourierʹs law is not applicable to describe the relationship between the heat flux and the temperature gradient under a very short time. In order to correct above deviations, Lord and Shulman, Green and Lindsay established the generalized thermoelastic theory which introduced one (L‐S theory) and two (G‐ L theory) thermal relaxation time into the classic thermalelastic theory. Above two theories are able to characterize thermal disturbance transfers with a limited speed in the cases of the macro space and a very short time, and show the second sound effect of the solid. In the early 1990, Green and Naghdi proposed another new generalized thermoelastic theory (G‐N theory) on the basis of energy balance and

entropy balance. This theory considers that the energy does not dissipate, so it is an ideal and no energy dissipation thermoelastic theory. Sherief studied G‐L generalized thermoelastic problem using the state space method. Chen and Weng studied the transient generalized thermoelastic plane problem and dynamic response of elastic porous materials. Sherief and Megahed studied two‐dimensional thermoelastic problem of the semi‐infinite plane suffered the heat source. Sherief and Helmy studied magneto‐thermo‐ elastic coupled problems of two dimensional semi‐ infinite planes. Tian and Shen solved the generalized magneto‐thermo‐elastic problem first time using the finite element method by direct integration in the time domain.

Chandrasekharaish described and summarized aforementioned three kinds of generalized theory respectively.

In recent years, the laser pulse technology has been widely used in the material processing and non‐ destructive testing. Thus, the thermoelastic waves induced by the laser pulse heating are concerned by majority scholars. Qiu and Tien studied the heat conduction mechanism of metal under the conditions of the ultrafast laser heating. Tang et al combined Green function method and the integral transform method, and analyzed the thermal transfer behavior of finite stiffness thick plates under ultrashort laser pulses heating. Wang and Xu analyzed the thermoelastic wave problems of semi‐infinite rod induced by nanosecond, picosecond and femtosecond laser pulse heating, respectively. The paper proves that thermal transfer with a limited speed in the medium, but in solving problem, the numerical inverse transformation method are adopted and introduced discrete error and truncation error, making the temperature steps effect are not fully demonstrated on heat wave front, in addition, the research work is only limited to one‐dimensional case.

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In this paper, we analysis boundary value problem of a homogeneous, isotropic two‐dimensional semi‐ infinite plate suffered laser pulse heated locally using L‐S generalized thermoelastic theory.To solve this problem, we adapt finite element method by direct integration in time domain and obtain temperature distributions as well as displacement of semi‐infinite plate when laser pulse is heated locally.

Basic Equations

    T  e   e  (4)  T,ii =c1  T   T 0 kk kk Where c1  (  CE 

 2 T0 ) .   2

In the above equations, a comma followed by a suffix denotes material derivative, and a superposed dot denotes the derivative with respect to time. In corresponding equations i, j , k  1, 2 . T is the absolute temperature of elastic medium and T0 is the reference

We consider a homogeneous, isotropic semi‐infinite plate ( x  0,    y   ). At time t = 0, the local region of boundary (edge DE) suffers a sudden laser pulse heating, as shown in Fig. 1. Because the direction of the laser pulse heating is parallel to the oxy plane, it can be regard as a plane stress problem and all the variables are independent of z, the component of the displacement is only along the x and y directions.

temperature.  ij denotes the components of the stress tensor, and eij is the components of the strain tensor. ui is the components of the displacement vector. 

and  are Lame’s constants.  and  are the coefficients of linear thermal expansion and thermal conductivity, respectively. cE is the specific heat at constant strain, and  is the mass density. The thermal relaxation time  ensures that the heat conduction equation can predict finite velocity of heat propagation.

From equation (3), equation (2) and equation (1), we can get the governing equation of motion expressed by displacements and temperatures

 ui , jj        u j ,ij   T,i   ui (5) For numerical convenience, the following non‐ dimensional quantities are introduced x   x FIG. 1 SEMI‐INFINITE PLANE INDUCED BY LOCALLY PULSED LASER HEATING

The equation of motion without considering external force is

 ij , j   ui (1) Constitutive equation is

 ij = ekk  ij +2 eij -   T-T0   ij (2) 2 2 Where    ,     and  =  3  2     2   2 for plane stress problem.

Geometric equation is eij =





1 ui , j  u j ,i (3) 2

Energy equation expressed with the displacements and the temperatures

96

y   y u    u v   v

t    2 t     2

Where  

 ij =

 ij 

=

T-T0  (6) = T0 T0

c    2 and   1 .  k

In order to facilitate the writing, the asterisk of the quantities will be eliminated. The governing equations of motion and energy equation can be written as follows

2

2  2u  2 v  2u  2  u (7) d b      xy y 2 x x 2 t 2

2

2 2v  2u  2 v  2  v d b (8)      xy x 2 y y 2 t 2

 2  2   2  2   2 2 t x y t   2u  2 v    3u  3v  g      g   2 yt 2   xt yt   xt

(9)

International Journal of Engineering Practical Research (IJEPR) Volume 3 Issue 4, November 2014 www.seipub.org/ijepr

 T0  +2     , b= ,d  , g= .    c1

Where  0 is a given dimensionless temperature, H (t ) and H ( y ) are Heaviside unit step function.

For x  , y  

Solutions of the Problems Governing equations (7‐9) are coupled with each other.

In order to overcome the shortcomings of the conventional analytical methods which adapts Laplace transformation and Fourier transform method and avoid the errors caused by numerical inverse transformation, we apply finite element analysis software package Flex PDE issued by the United States PDE Solutions Inc and solve the problem using finite element method by direct integration in time domain.

By adjusting computation error limit ERRLIM in the mesh refinement module of FlexPDE software package, the finite element mesh can be re‐divided and improved with the passage of time, so it can improve the accuracy in solving the problem and reflect the fluctuations effect of thermal accurately.

Numerical Examples and Discussions Using above finite element methods solved directly in time domain, we analyze above two‐dimensional semi‐infinite plate suffered by the laser pulse heating on the AB edge locally (‐0.3 ≤ DE ≤ 0.3). According to the symmetry of geometries and load conditions, we only analyze the semi‐infinite plane where x‐axis is the symmetric edge. Analysis model OCFB is shown in Fig. 1, OC = 1, OB = 1.2. Dimensionless width of the laser pulse is taken as 0.02, the dimensionless heating region OD = 0.3. In this work, we adopt nickel materials in calculation. Material constants are given as follows.

Density:   8900 kg/m3 . Heat transfer coefficient:

k  90.7 w/mK. Specific heat under constant strain: cE  444 J/kgK. Linear thermal expansion coefficient: 10   1.26  105 1/K. Shear modulus: G  7.58  10 kg/s2m.

Bulk modulus: B  1.72  1011 kg/s2m. Dimensionless thermal relaxation time:   0.05 .

u ( x, y, t )  v( x, y, t )   ( x, y, t )  0 (18) 1.5

t=0.05 t=0.10 t=0.15

1.0

x

Where  2 =

0.5

0.0

-0.5 0.0

0.2

0.4

0.6

0.8

1.0

X

FIG. 2 TEMPERATURE DISTRIBUTIONS ON EDGE OF OC AT DIFFERENT TIME

Fig. 2 presents temperature distributions on the edge of OC at the different time. It can be seen that there is a distinct temperature step on thermal wave front in temperature distribution on OC. After the laser pulse removeing, heat affected region continues to expand, the highest temperature locates on the wave front.

According to the nature of the wave equation, the velocity of dimensionless heat wave is 1  1  4.472 in L‐S the generalized 1 0.05 thermoelastic theory. When time t=0.05,0.10,0.15 , the position of heat wave front on the edge OC are lx  0.2236, 0.4472, 0.6708 , respectively, which are consistent with the theoretical values, which means that the propagation of thermal with finite speed.

After the laser pulse heating (t>0.02), the maximum temperature no longer lies in the original location (edge OD) and moves along with the passage of time, and simultaneously the highest temperature decreases.

1.5

Initial conditions

t=0.05 t=0.10 t=0.15

1.0

 ( x, y, t )  ( x, y, t )  u ( x, y, t )  u ( x, y, t )  v( x, y, t )  v( x, y, t )  0

y

At t  0 0.5

(16) 0.0

Boundary conditions For x  0

-0.5 0.0

Q ( x, y , t )   ( x, y , t )   0  H (t )  H  t  0.02   H  0.3  y 

0.2

0.4

0.6

0.8

1.0

1.2

Y

(17)

FIG. 3 TEMPERATURE DISTRIBUTIONS ON EDGE OF OB AT DIFFERENT TIME

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Fig. 3 indicates temperature distributions on the edge of OB at different time.it can be seen from the figure that the wave front step gradually deceases along with the passage of time.

0.00015

t=0.05 t=0.10 t=0.15

0.00010 0.00005 0.00000

Ux

-0.00005 -0.00010 -0.00015 -0.00020

distributions are obtained and represented graphically.

The results show that: (1) the temperatures are restricted in a limited region, and are not changed beyond this region, which means propagation of heat at finite speed. (2) Under the solution of finite element method directly in time domain, the obtained location of the heat wave front agrees with the theoretical value. (3) For the laser pulse heating, the temperature gradually decreases after the laser pulse heating, highest temperature always locates on the heat wave front.

-0.00025

ACKNOWLEDGMENT

-0.00030 -0.00035 -0.00040 0.0

0.2

0.4

0.6

0.8

1.0

X

FIG. 4 HORIZONTAL DISPLACEMENT DISTRIBUTIONS ON EDGE OF OC AT THE DIFFERENT TIME

Fig. 4 presents displacement distributions in x direction on the edge OC at the different time. Due to restrictions on the right boundary, when the OD is heated, firstly, displacement arises negative on OC, secondly shifts from the negative into the positive gradually, and lastly becomes zero at the heat wave front.

0.00010

This work was supported by the Funds of Xi’an University of Technology (Grant 104‐211002 and 104‐ 211104) and of Shaanxi Province Natural Science Foundation (Grant 2014JM1024).

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Conclusions In this paper, we study semi‐infinite plate suffered laser pulse heating locally using L‐S generalized thermoelastic theory. In order to avoid the loss of precision in general integral transformation method, the finite element control equations are solved directly in the time domain. Temperatures and displacements

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International Journal of Engineering Practical Research (IJEPR) Volume 3 Issue 4, November 2014 www.seipub.org/ijepr

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