Laser Pulse in Micropolar Thermoelastic Medium with Three Phase Lag Model

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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, couple-stress, 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.


    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 multi-molecular 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 rate-dependent 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 phase-lag 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 three-phase-lag 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 phase-lags for the heat flux vector, the temperature gradient and the thermal displacement gradient. The stability of the three-phase-lag heat conduction equation is discussed by Quintanilla and Racke [9]. Quintanilla has studied the spatial behavior of solutions of the three-phase-lag 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 ultra-short 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 non-destructive testing and characterization [11, 12]. The so-called 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 ultra-high 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 one-step model [14], the hyperbolic one-step model [15], and the parabolic two-step and hyperbolic two-step models [16, 17]. It has been found that usually the microscopic two-step models, i.e., parabolic and hyperbolic two-step 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


    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 + µ)(. u) + (µ + K)2u + K × </ /31T = pu ,


    (y2 2K)</ + (a + /3)(. </) + K × 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 laser-generated 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 laser-generated


    = (,1Ocp + ,1u



    + µ(u


    + 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 laser-generated ultrasound in non-metallic material directly. The optical absorption occurs at the surface layer in metallic materials, and the heat penetration is resulted due to heat diffusion. In non-metallic materials, the laser beam can penetrate the specimen to some finite depth and induced a buried bulk- thermal source, so the features of the laser-generated ultrasound will be significantly different from that in metallic materials.

    Dubois [23] experimentally demonstrated that penetration depth play a very important role in the laser-ultrasound 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 ,


    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 )


    f(t) = 2 e to (1.7)


    Here tO is the pulse rise time. The pulse is also assumed to

    have a Gaussian spatial profile in x

    Al-Huniti and Al-Nimr [25] investigated the thermoelastic

    behavior of a composite slab under a rapid dual-phase lag heating. The comparison of one-dimensional and two-


    g(x) = 1 2rrr




    – 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 Al-Bary [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,


    (x ) = ye- (1.9)


    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



    – l

    r2 e

    – , (1.10)


    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 l



    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 Lord-Shulman (LS) model and E = 0,


    K 1 + a ) + K1w a 1 + t a ) 2T = 1 + q a +

    y1 = O where O > 0 for Green-Lindsay (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.


    u = acp + al/,


    a l a

    , u = acp al/, , (2.11)


    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 x3-axis with x3-axis pointing vertically


    1. 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 co-ordinate system.

      q at

    2. at2

    14 at2

    q at


    2 at2

    a15 a

    + q

    a2 +

    rq2 a

    ) te

    – t )


    – l


    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+µ , a

    = K+µ , a

    = K , a

    = , a =


    Then the field equations in micropolar generalized

    1 pc2 2





    pc2 3

    pjW2 6

    jpc2 1


    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


    (,1 + µ) ae + (µ + K)2u

    K acp2 /3

    a = pu


    a l




    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 + µ) ae + (µ + K)2u3 + K acp2 /31 a = pu3




    a l


    K aul au ) + y2cp


    = jpcp

    {cp, 1/, T, cp2, c}(x1, x3, t) = {cp , 1/ , T , cp 2 , c}(x3)ei(k l-Wt)


    a l

    2 2 2



    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, cp-2, 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 e-ml + B e-m2 + B e-m + L e- (3.9)


    1 3 cl 1 3 1 3 cl 1 3 O

    T = d B e-ml + d B e-m2 + d B e-m + L e-

    w(t, O, V), T =


    , =



    , cp = l cp2 ,

    1 1 2 2


    3 3 2








    {3l o

    2 {3l o


    m =

    mij , Q = l Q , w = l , y1 = (3,1 + 2µ +

    1/ = B e-m4 + B e-m5 (3.11)




    , c2 = (J+2µ+K)


    K 4 5


    t 1 p

    cp-2 = e4B4e-m4 + 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 l


    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 ,


    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/, cp-2 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:

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      t-33 = 4 G1ie

      -mi + M1e

      – (4.3)

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      31 i=1 2i 2

      t- = 4 G e-mi + M e- (4.4)


      32 i=1 3i 3

      m- = 4 G e-mi + M e- (4.5) T =

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      e-mi + M e- , (4.6)


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      1. 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.


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