 Open Access
 Total Downloads : 24
 Authors : Arvind Kumar
 Paper ID : IJERTCONV4IS03039
 Volume & Issue : RACEE – 2015 (Volume 4 – Issue 03)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Laser Pulse in Micropolar Thermoelastic Medium with Three Phase Lag Model
Arvind Kumar Department of Mathematics Punjab Technical University,
Kapurthala (India)
Abstract The present investigation deals with the deformation in micropolar generalized thermoelastic medium with three phase lag subjected to thermomechanical loading due to thermal laser pulse. Normal mode analysis technique is used to solve the problem. Concentrated normal force and thermal source are taken to illustrate the utility of approach. The closed form expressions of normal stress, tangential stress, couplestress, and temperature distribution are obtained.. Some particular cases of interest are deduced from the present investigation.
Keywords Micropolar Thermoelastic, Three phase lag, Pulse Laser, concentrated normal force and concentrated thermal source.

INTRODUCTION
Modern engineering structures are often made up of materials possessing internal structure. Polycrystalline materials, materials with fibrous or coarse grain structure come in this category. Classical theory of elasticity is inadequate to represent the behavior of such materials. The analysis of such materials requires incorporating the theory of oriented media. The linear theory of micropolar elasticity was developed by Eringen [1]. A micropolar continuum is a collection of interconnected particles in the form of small rigid bodies undergoing both translational and rotational motions. Typical examples of such materials are granular media and multimolecular bodies whose microstructures act as an evident part in their macroscopic responses. Rigid chopped fibers, elastic solids with rigid granular inclusions and other industrial materials such as liquid crystals are examples of such materials.
The generalized theory of thermoelasticity is one of the modified versions of classical uncoupled and coupled theory of thermoelasticity and has been developed in order to remove the paradox of physical impossible phenomena of infinite velocity of thermal signals in the classical coupled thermoelasticity. Hetnarski and Ignaczak [4] examined five generalizations of the coupled theory of thermoelasticity. The first generalization is due to Lord and Shulman [2] who formulated the generalized thermoelasticity theory involving one thermal relaxation time. Green and Lindsay [3] developed a temperature ratedependent thermoelasticity that includes two thermal relaxation times. One can refer to Hetnarski and Ignaczak [4] for a review and presentation of generalized theories of thermoelasticity.
The third generalization of the coupled theory of thermoelasticity is developed by Hetnarski and Ignaczak and is known as low temperature thermoelasticity. The fourth generalization to the coupled theory of thermoelasticity
introduced by Green and Nagdhi [5] and this theory is concerned with the thermoelasticity theory without energy dissipation. The fifth generalization to the coupled theory of thermoelasticity is developed by Tzou [6] and Chanderashekhariah [7] and is referred to dual phaselag thermoelasticity. He introduced two phase lags to both the heat flux vector and the temperature gradient and considered constitutive equations to describe the lagging behavior in the heat conduction in solids. Roychoudhuri [8] has recently introduced the threephaselag heat conduction equation in which the Fourier law of heat conduction is replaced by an approximation to a modification of the Fourier law with the introduction of three different phaselags for the heat flux vector, the temperature gradient and the thermal displacement gradient. The stability of the threephaselag heat conduction equation is discussed by Quintanilla and Racke [9]. Quintanilla has studied the spatial behavior of solutions of the threephaselag heat conduction equation.
Laser technology has a vital application in nondestructive materials testing and evaluation. When a solid is heated with a laser pulse, it absorbs some energy which results in an increase in localized temperature. This cause thermal expansion and generation of the ultrasonic waves in the material. The irradiation of the surface of a solid by pulsed laser light generates wave motion in the solid material. There are generally two mechanisms for such wave generation, depending on the energy density deposited by the laser pulse. At high energy density, a thin surface layer of the solid material melts, followed by an ablation process whereby particles fly off the surface, thus giving rise to forces that generates ultrasonic waves. At low energy density, the surface material does not melt, but it expands at a high rate and wave and wave motion is generated due to thermoelastic processes.
Very rapid thermal processes (e.g., the thermal shock due to exposure to an ultrashort laser pulse) are interesting from the stand point of thermoelasticity, since they require a coupled analysis of the temperature and deformation fields. A thermal shock induces very rapid movement in the structural elements, giving the rise to very significant inertial forces, and thereby, an increase in vibration. Rapidly oscillating contraction and expansion generates temperature changes in materials susceptible to diffusion of heat by conduction [10]. This mechanism has attracted considerable attention due to the extensive use of pulsed laser technologies in material processing and nondestructive testing and characterization [11, 12]. The socalled ultra short lasers are those with pulse durations ranging from nanoseconds to femto seconds. In the
case of ultra short pulsed laser heating, the high intensity energy flux and ultra short duration lead to a very large thermal gradients or ultrahigh heating may exist at the boundaries. In such cases, as pointed out by many investigators, the classical Fourier model, which leads to an infinite propagation speed of the thermal energy, is no longer valid [13]. Researchers have proposed several models to describe the mechanism of heat conduction during short pulse laser heating, such as the parabolic onestep model [14], the hyperbolic onestep model [15], and the parabolic twostep and hyperbolic twostep models [16, 17]. It has been found that usually the microscopic twostep models, i.e., parabolic and hyperbolic twostep models, are useful for thin films. Simulation on laser ultrasound wave form in non metallic materials was discussed by Wang et al [18].
Scruby et al. [19] considered the point source model to study the ultrasonic generation by lasers. He studied the heated surface by laser pulse irradiation in the thermoelastic system as a surface center of expansion (SCOE). He also discussed the applications of laser technology in flaw detection and acoustic microscopy. Rose [20] later presented
computed numerically. The resulting expressions are then applied to the problem of a micropolar thermoelastic three phase lag medium whose boundary is subjected to two types of loads i.e. mechanical load and thermal load. The resulting quantities are shown graphically to show the effect of laser irradiation

BASIC EQUATION
Following Roychoudhuri [8], the basic equations in a homogeneous, isotropic micropolar generalized thermoelastic medium with three phase lag model in the absence of body forces and body couples are given by:
(,1 + Âµ)(. u) + (Âµ + K)2u + K Ã— </ /31T = pu ,
(1.1)
(y2 2K)</ + (a + /3)(. </) + K Ã— u = pj</ , (1.2)
K 1 + a ) + K1 a 1 + t a ) 2T = 1 + q a +
a more exact mathematical basis. Point source model explain 2 2 at a at at
main features of lasergenerated ultrasoud waves but this rq a )(pc T + yT e pQ ), (1.3)
model fails to explain precursor in epicenter waves. Later
2 at2 E
O kk
introducing the thermal diffusion McDonald [21] and Spicer
[22] proposed a new model known as lasergeneratedtij
= (,1Ocp + ,1u
r,r
)oij
+ Âµ(u
i,j
+ uj,i
) + K(uj,i
ultrasound model. This model reported excellent agreement Eijkcpk) /31 1 + 1 a ) oijT , (1.4)
between theory and experiment for metal materials. But due at
to the optical penetration effect, this model cannot be applied to the study of lasergenerated ultrasound in nonmetallic material directly. The optical absorption occurs at the surface layer in metallic materials, and the heat penetration is resulted due to heat diffusion. In nonmetallic materials, the laser beam can penetrate the specimen to some finite depth and induced a buried bulk thermal source, so the features of the lasergenerated ultrasound will be significantly different from that in metallic materials.
Dubois [23] experimentally demonstrated that penetration depth play a very important role in the laserultrasound generation process. Ezzat et al. [24] discussed the thermo elastic behavior in metal films by fractional ultrafast laser.
mij = acpr,roij + /3cpi,j + ycpj,i + bOEmjicp,m ,
(1.5)
The plate surface is illuminated by laser pulse given by the heat input
Q = !Of(t)g(x1)(x3) (1.6)
Where, !O is the energy absorbed. The temporal profile f(t)
is represented as,
t – t )
t
f(t) = 2 e to (1.7)
o
Here tO is the pulse rise time. The pulse is also assumed to
have a Gaussian spatial profile in x
AlHuniti and AlNimr [25] investigated the thermoelastic
behavior of a composite slab under a rapid dualphase lag heating. The comparison of onedimensional and two
2
g(x) = 1 2rrr
1
X
2
– l
e r2 (1.8)
Q =
dimensional axisymmetric approaches to the thermomechanical response caused by ultrashort laser heating was studied by Chen et al. [26]. Kim et al. [27] studied thermoelastic stresses in a bonded layer due to pulsed laser radiation. Thermoelastic material response due to laser pulse heating in context of four theorems of thermoelasticity was discussed by Youssef and AlBary [28]. Theoretical study of the effect of enamel parameters on laser induced surface
where r is the beam radius, and as a function of the depth x3
the heat deposition due to the laser pulse is assumed to decay exponentially within the solid,
3
(x ) = ye (1.9)
X
Equation (1.7a) with the aid of (1.7b, 1.7c and 1.7d) takes the form
acoustic waves in human incisor was studied by Yuan et al [29]. A two dimensional generalized thermoelastic diffusion problem for a thick plate under the effect of laser pulse
o 2rrr2t2
– t )
te to
2
e
– l
r2 e
– , (1.10)
o
thermal heating was studied by Elhagary [30].
In this research, taking into account the radiation of ultra short laser, we have established a model for micropolar thermoelastic medium with three phase lag model. The stress components and temperature distribution have been
Here ,1, , a, /3, y, K, are material constants, p is mass density, u = (u1, u2, u3) is the displacement vector and
</ = (cp1, cp2, cp3) is the microrotation vector, T is temperature and TO is the reference temperature of the body chosen, tij are components of stress, mij are components of
couple stress, eij are components of strain, ekk is the
a1 ae + a22u3 + a3 acp2 /31 a = pu3
dilatation, oij is Kroneker delta function, O, 1 are the diffusion relaxation times and , are thermal relaxation
a
a l
(2.8)
a
times with
O 1
0. Here 1
a8 aul au ) + a62cp2 2a3cp2 = cp2
O 1 O = = O = 1 = y1 = 0
a a l
for Coupled Thermoelastic theory (CT) model. 1 = 1 = 0, E = 1, y1 = O For LordShulman (LS) model and E = 0,
(2.9)
K 1 + a ) + K1w a 1 + t a ) 2T = 1 + q a +
y1 = O where O > 0 for GreenLindsay (GL) model.
at a at at
r 2
q a2 )(a T + a e a Q ) (2.10)
In the above equations symbol (,) followed by a
2 at2 13
14 kk 15
suffix denotes differentiation with respect to spatial coordinates and a superposed dot ( ) denotes the derivative
Using the potential functions cp and 1/ as:
with respect to time respectively.

FORMULATION OF THE PROBLEM
u = acp + al/,
1
a l a
, u = acp al/, , (2.11)
3
a a l
We consider a micropolar generalized thermoelastic solid
The equations (2.7)(2.10) reduce to:
V2cp cp T = 0, (2.12)
with rectangular Cartesian coordinate system OX1X2X3
K 1 +
a ) + K w a 1 +
a ) 2 a
1 +
having origin on x3axis with x3axis pointing vertically
at

a
t at 13
downward the medium. A normal force/thermal source is
a + rq2 a2 ) T a
a2 1 +
a + rq2 a2 ) 2cp =
assumed to acting on the origin of the rectangular Cartesian coordinate system.
q at

at2
14 at2
q at
X2
2 at2
a15 a
+ q
a2 +
rq2 a
) te
– t )
to
– l
e
r2 e
– , (2.13)
If we restrict our problem for plane strain parallel to x1x3 plane with
u = (u1, 0, u3), </ = (0, cp2, 0), (2.1)
at at2 2 at
a221/ 1/ + a3cp2 = 0, (2.14) a62cp2 2a3cp2 a821/ = cp2, (2.15) Where,
a = J+Âµ , a
= K+Âµ , a
= K , a
= , a =
l
Then the field equations in micropolar generalized
1 pc2 2
l
l
l
K
pc2 3
pjW2 6
jpc2 1
pc2
thermoelastic solid in the absence of body forces and body
, a13 = pc2cE , a14 = /31TOy , a15 = l
couples the equations of motion can be written as:
pc2 1
{3lW
(,1 + Âµ) ae + (Âµ + K)2u
K acp2 /3
a = pu


SOLUTION OF THE PROBLEM
a l
1
(2.2)
a
1 a l 1
The solution of the considered physical variables can be decomposed in terms of the normal modes as in the following
(,1 + Âµ) ae + (Âµ + K)2u3 + K acp2 /31 a = pu3
form:
a
(2.3)
a l
a
K aul au ) + y2cp
2Kcp
= jpcp
{cp, 1/, T, cp2, c}(x1, x3, t) = {cp , 1/ , T , cp 2 , c}(x3)ei(k lWt)
a
a l
2 2 2
(2.4)
(3.1)
K 1 + a ) + K1 a 1 + t a ) 2T = 1 + q a +
Here w is the angular velocity and k is wave number. After
at a at at
rq2 a2 )(pc T + yT e pQ ) (2.5)
some simplifications the general solution of the above system
2 at2 E
O kk
satisfying the radiation conditions that (cp, T, cp2, 1/) 0 as
For further consideration it is convenient to introduce in equations (1.1)(1.5) the dimensionless quantities defined as:
(x , x ) = W (x , x ), (u , u ) = W (u , u ), (t, , V) =
x3 are given as following:
1 2 3 1
cp = B eml + B em2 + B em + L e (3.9)
pc
1 3 cl 1 3 1 3 cl 1 3 O
T = d B eml + d B em2 + d B em + L e
w(t, O, V), T =
{3l
, =
rij
2
, cp = l cp2 ,
1 1 2 2
(3.10)
3 3 2
l
pc2
l
l
W
ij
{32
{3l o
2 {3l o
pCc2
m =
mij , Q = l Q , w = l , y1 = (3,1 + 2Âµ +
1/ = B em4 + B em5 (3.11)
ij
K)a
pc
, c2 = (J+2Âµ+K)
pc2
K 4 5
(2.6)
t 1 p
cp2 = e4B4em4 + e5B5e
m5 (3.12)
Usig (2.6), the equations (2.2)(2.5) reduce to:
a1 ae + a22u1 a3 acp2 /31 a = pu1
a l
a
(2.7)
a l

BOUNDARY CONDITIONS
We consider concentrated normal force and concentrated thermal source at the boundary surface x3 = 0, mathematically, these can be written as:
t33 = F1, t31 = 0 , m32 = 0, T = F2 ,
(4.1)
where F1 is the magnitude of the applied force and F2 is the constant temperature applied on the boundary.
Case 1: for the normal force: F2 = 0
Case 2: for the thermal source: F1 = 0
Substituting the values of cp, cp, T, 1/, cp2 from the equations (3.9)(3.12) in the boundary condition (4.1) and using (1.4)(1.5), (2.1), (2.11), (3.1) and solving the resulting equations, we obtain:

M.N. Ozisik, D.Y. Tzou, On the wave theory in heatconduction, ASME J. Heat Transfer, 116, 526535, 1994.

W.S. Kim, L.G. Hector, M.N. Ozisik, Hyperbolic heat conduction due to axisymmetric continuous or pulsed surface heat sources, J Appl. Phys., 68, 547885, 1990.

T.Q. Qiu, C.L. Tien, Heat transfer mechanisms during shortpulse laser heating of metals. ASME J Heat Transfer, 115, 83541, 1993.

D.Y. Tzou, Macro to micro scale heat transfer: The lagging behavior, Bristol, Taylor & Francis, 1997.

J. Wang, Z. Shen, B. Xu, X. Ni, J. Guan, J. Lu , Simulation on thermoelastic stress field and laser ultrasound wave form in non metallic materials by using FEM, Applied Physics A, 84, 301307, 2006.

C. B. Scruby, L. E. Drain, Laser Ultrasonics Techniques and Applications, Adam Hilger, Bristol, UK, 1990.

L.R.F. Rose, Pointsource representation for lasergenerated ultrasound, Journal of the Acoustical Society of America, 75, 3, 723, 1984.

F. A. McDonald, On the Precursor in LaserGenerated Ultrasound Waveforms in Metals, Applied Physics Letters, 56, 3, 230232, 1990.

J. B. Spicer, A. D. W. Mckie, and J. W. Wagner, Quantitative Theory for Laser Ultrasonic Waves in a Thin Plate, Appl. Phys. Lett., 57, 18821884, 1990.

M. Dubois, F. Enguehard, L. Bertrand, M. Choquet, J.P. Monchalin, Appl. Phys. Lett., 64, 554, 1994.
i=1
t33 = 4 G1ie
mi + M1e
– (4.3)

M.A. Ezzat, A. Karamany and M.A. Fayik, Fractional ultrafast laser
induced thermoelastic behavior in metal films, Journal of Thermal Stresses, 35, 637651, 2012.
31 i=1 2i 2
t = 4 G emi + M e (4.4)
4
32 i=1 3i 3
m = 4 G emi + M e (4.5) T =

N.S. AlHuniti , M.A. AlNimr , Thermoelastic behavior of a composite slab under a rapid dualphaselag heating, Journal of
Thermal Stresses, 27, 607623, 2004.
4
i=1
G4i
emi + M e , (4.6)


SPECIAL CASES

J.K. Chen, J.E. Beraun, C.L. Tham, Comparison of onedimensional and twodimensional axisymmetric approaches to the thermomechanical response caused by ultrashort laser heating, Journal of Optics, 4, 650661, 2002.

W.S. Kim, L.G. Hector, R.B. Hetnarski, Thermoelastic stresses in a bonded layer due to repetitively pulsed laser heating, Acta Mechanica,

Micropolar Thermoelastic Solid

In absence of three phase lag effect in Equations (4.3) – (4.7), we obtain the corresponding expressions of stresses, displacements and temperature for micropolar generalized thermoelastic half space.


REFERENCES

A.C. Eringen, Linear theory of micropolar elasticity, Journal of Mathematics and Mechanics, 15, 909923, 1966.

H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solid, 15, 299306, 1967.

A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elasticity, 2, 1 5, 1972.

Hetnarski, R.B. and Ignaczak J., Solutionlike waves in a low temperature nonlinear thermoelastic solid, Int. J. Eng. Sci. 34(1996), 17671787.

Green, A.E. and Nagdhi, P.M., A reexamination of the basic postulates of thermo mechanics, Proceedings of the Royal Society of London, 432(1991), 171194.

Tzou, D.Y., A unified field approach for heat conduction from macro to micro Scales, ASME J. Heat Transf. 117(1995), 816.

Chandersekharaiah, D.S., Hyperbolic Thermoelasticity: A review of recent literature, Appl. Mech. Rev. 51(1998), 705729.

Roychoudhuri, S.K., On thermoelastic three phase lag model, J. Therm. Stress. 30(2007), 231238.

Quintanilla, R. and Racke, R., A note on stability in threephaselag heat conduction, Int. J.

D. Trajkovski, R. Cukic, A coupled problem of thermoelastic vibrations of a circular plate with exact boundary conditions, Mech. Res Commun., 26, 21724, 1999.

X. Wang, X. Xu, Thermoelastic wave induced by pulsed laser heating. Appl. Phys. A, 73, 10714, 2001.

X. Wang, X. Xu, Thermoelastic wave in metal induced by ultrafast laser pulses. J. Thermal Stresses, 25, 45773, 2002.

D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys., 61, 4173, 1989.
125, 107128, 1997.

H.M. Youssef, A.A. ElBary, Thermoelastic material response due to laser pulse heating in context of four theorems of thermoelasticity, Journal of thermal stresses, 37, 137989, 2014.

L. Yuan, K. Sun, Z. Shen, X. Ni, J. Lu, Theoretical study of the effect of enamel parameters on laser induced surface acoustic waves in human incisor, Int. J. Thermophys., July 2014.

M.A. Elhagary, A twodimensional generalized thermoelastic diffusion problem for a thick plate subjected to thermal loading due to laser pulse, Journal of thermal stresses, 37, 14161432, 2014.

A.C. Eringen, Microcontinuum field theories I: Foundations and Solids, SpringerVerleg, New York 1999.

A.C. Eringen, Plane waves in nonlocal micropolar elasticity, Int. J. Eng. Sci., 22, 11131121, 1984.

R. Kumar , L. Rani, Elastodynamic response of mechanical and thermal source in generalized thermoelastic half space with voids, Mechanics and Mechanical Engineering, 9, no. 2, 2945, 2005.