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The closed form solutions for Cattaneo and stress equations due to step input pulse heating
Physica B 405 (2010) 3869–3874
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Physica B journal homepage: www.elsevier.com/locate/physb
The closed form solutions for Cattaneo and stress equations due to step input pulse heating H. Al-Qahtani, B.S. Yilbas n King Fahd University of Petroleum & Minerals, Box 1913, Dhahran 31261, Saudi Arabia
a r t i c l e in fo
abstract
Article history: Received 24 February 2010 Received in revised form 15 May 2010 Accepted 8 June 2010
The analytical solution for the temperature and stress fields in a solid subjected to a step input pulse heating is presented. The closed form solutions of Cattaneo and thermal stress equations are obtained using the Laplace transformation technique. The predictions obtained from the closed form solutions are presented in nondimensional form. It is found that temperature rise is rapid in the early heating period and as the time progresses, temperature rise becomes gradual. The stress generated in the heating cycle is tensile while it becomes compressive in the cooling cycle. & 2010 Elsevier B.V. All rights reserved.
Keywords: Cattaneo Pulse Heating Thermal stress
1. Introduction In short pulse heating of solid surfaces, heat wave is generated and the energy transport to the solid bulk takes place through heat wave at finite speed. In this case, heat is conducted in the solid due to the near neighborhood excitation via changing of momentum and energy on a microscopic scale in a wave form. The average communication time between the neighborhoods is associated with the relaxation time. The thermal communication takes place in a dissipative nature resulting in the thermal resistance in the solid medium. If the heating duration is longer than the relaxation time, the speed of the heat wave propagation approaches to infinity and the heat wave equation reduces to the classical Fourier heat equation. However, the Fourier heating model fails to predict the temperature propagation speed and suffers anomalies when the heating duration is comparable or less than the relaxation time. Therefore, Cattaneo’s heating model becomes appropriate to account for the temperature propagation speed while eliminating the anomalies arise from the Fourier heating model. Cattaneo’s heating model is governed by the hyperbolic temperature equations. Moreover, the heating of submicron sized solid devices during the short heating period results in non-equilibrium heating situation in the solid; in which case, the governing equation takes the hyperbolic form. The heating situation can also be modeled using the Cattaneo’s equation. Although the numerical solution is possible for such heating situation, the analytical solution of Cattaneo’s heating equation is
n
Corresponding author. Tel.: +966 38604481; fax: + 966 38602949. E-mail address: [email protected] (B.S. Yilbas).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.06.019
fruitful to generalize the temperature field in terms of the heat source parameters and the material properties. The application of the Laplace transformation method provides the exact solution to the Cattaneo’s heating equation. In addition, temperature field developed in the solid causes the thermal expansion of the substrate, which, in turn, generates the thermal stress field within the heated region. Consequently, investigation into the analytical solution of the hyperbolic heat and the thermal stress equations becomes necessary. Considerable research studies were carried out to examine the hyperbolic heating situation of solids by the externally applied sources. The hyperbolic heat conduction due to the mode locked laser pulse train was investigated by Hector et al. [1]. They presented the semi-analytical solution to the heating problem and indicated that the parabolic Fourier heating model significantly underestimated the surface temperature, since the rate of energy propagation was assumed to be infinite. Ready [2] introduced the closed form solution for temperature rise due to pulse heating of solid surfaces. He used the Fourier heating model to analyze temperature distribution in the solid. Blackwell [3] developed an analytical solution for temperature rise inside the solid heated by a pulsed laser. He adopted a Fourier heating model when formulating temperature distribution during the heating process. The wave diffusion and parallel non-equilibrium heat conduction were investigated by Honner and Kunes [4]. They introduced a dimensionless criterion for microscale heat transfer due to the equilibrium and the non-equilibrium diffusion conduction problems. Malinowski [5] examined the heat conduction and the generation in solids after considering finite velocity of the heat propagation and the inertia of the internal heat source. He showed that unlike the classical hyperbolic model, the relaxation
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solutions generally did not tend to approach the corresponding parabolic solution. A material-invariant version of the Maxwell– Cattaneo law was proposed by Christov [6]. He showed that the new formulation allowed for the elimination of the heat flux yielding a single equation for the temperature field. The convection-diffusion equation based on Cattaneo’s law was examined by Gomez et al. [7]. They introduced a finite element solution of the problem and the performance of the algorithm developed was verified by solving some 2-D test cases. The thermal wave oscillations and thermal relaxation time determination in a hyperbolic heat transport model were examined by OrdonezMirech and Alvarado-Gil [8]. They presented the simple analytical expressions for the values of the maxima and minima of the oscillations as well as the frequencies, at which they occurred. The new finite integral transformation method to the wave model of conduction was introduced by Duhamel [9]. He tested the Cattaneo Vornetto model through the comparison of the results with the transient molecular dynamics solutions. He indicated that when the used parameters of the continuous model were near their equilibrium values, the agreement both results remained weakly qualitative. Yilbas [10], and Yilbas and Kalyon [11] introduced the closed form solutions for temperature distribution inside the solid substrate subjected to a laser heating. The studies were limited to the Fourier heat diffusion model and may not be applicable for the short heating durations. Moreover, Yilbas and Pakdemir [12] presented a semi-analytical solution for the hyperbolic heat conduction problem. They introduced a perturbation method to obtain the analytical solutions. However, the approach introduced provides only the approximate solutions for temperature distribution. Considerable research studies were carried out to examine thermal stress development in solids subjected to a diffusive heating. Wang et al. [13] studied thermal stress development in solids due to the laser heating process. They used the effects of phase transition and formulate the thermal stress equations during the heating cycle. Yuan et al. [14] investigated the thermal stress development during the laser heating of two sheet metals. They validated the predictions with the experimental data. Some perspectives of macroscale and microscale thermal transport and thermo-mechanical interactions were presented by Tamma and Zhou [15]. They explained the significance of constitutive models for both macroscale and microscale heat conduction in conjunction with generalizations drawn concerning the physical relevance and the role of relaxation and retardation times emanating from the Jefferys type heat flux constitutive model, with consequences to the Cattaneo heat flux model and subsequently to the Fourier heat flux model. Thermoelastic wave formation due to parabolic and hyperbolic heat conduction in a helix was studied by Ostoja-Starzewski [16]. He showed that the thermal diffusivity and the relaxation time of the Maxwell–Cattaneo model had minor effects, whereas the thermoelastic coupling constant was dominant for the harmonic motions. Yilbas et al. [17] investigated thermal stress development during the laser heating pulse. They presented a closed form solution for the stress distribution in the laser irradiated region. Kalyon and Yilbas [18] studied thermal stress development in a semi-infinite solid subjected to a laser step input pulse. They formulated temperature and stress fields after considering the Fourier heating model and the equilibrium conditions. Yilbas and Al-Ageeli [19] formulated the thermal stress development in solids during the laser heating pulses. The analytical solutions developed were limited with the Fourier diffusion analysis and extension of the work including Cattaneo diffusion becomes necessary. In the present study, the analytical solution for temperature and thermal stress distributions in a solid subjected to a pulse heating is presented. The closed form solutions for the Cattaneo
diffusion and corresponding thermal stress equations are obtained by using the Laplace transformation method.
2. Mathematical analysis The assumption of equilibrium heating between quantized electronic excitations and quanta of lattice vibrations, the phenomenological model was proposed by Fourier [20]. The model proposed is parabolic one-step heat conduction with the notion of infinite speed of propagation of thermal disturbances. However, to account for a finite temperature propagation speed and to avoid the anomalies associated with the Fourier heat conduction model, the Cattaneo model was developed on the notion of relaxing the heat flux [21], which is expressed as
E
@q ¼ qK rT @t
ð1Þ
where q¼ krT, e is the relaxation time, K the conductivity tensor, q the heat flux vector, and T the temperature. The Cattaneo equation yields to the hyperbolic heat conduction equation for the temperature pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field, which is propagative with a speed c ¼ ðk=rCp eÞ, where r is the density and Cp the specific heat capacity. ! @2 T @T @2 T ð2Þ k 2 ¼ rCp þ E 2 @t @x @t
2.1. Nondimensionalization The preceding heat equation can be put in a nondimensional form using the following nondimensional parameters: t¼
t , 2E
x x ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 4Ek=rCp
Tk T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 4Ek=rCp
ð3Þ
where the P is the pulse intensity. The resulting nondimensional equation is thus @2 T @T @2 T þ 2 ¼2 2 @t @x @t
ð4Þ
with nondimensional initial and boundary conditions T ¼0 q ¼ f ðtÞ
at at
t¼0
ð5Þ
x¼0
ð6Þ
The closed form solution of Eq. (4) can be possible by adapting the Laplace transformation method. Taking the Laplace transformation of the governing Eq. (4), initial and boundary conditions (5),(6) results in @2 T^ ^ T^ ¼ 0 s2 T2s @x2
ð7Þ
@T^ sq^ ¼ FðsÞ at x ¼ 0 @x The last equation can be rewritten as
q^ ¼
@T 1 ¼ FðsÞðs þ 1Þ @x s
at
x ¼ 0,
ð8Þ
ð9Þ
2.1.1. Pulse shape The step input pulse intensity is considered, which can be written as f ðtÞ ¼ P½uðttÞuðtÞ Since the model is linear, one can use the superposition principle to compute the temperature distribution due to the two terms of the pulse function separately. It is convenient to get the
H. Al-Qahtani, B.S. Yilbas / Physica B 405 (2010) 3869–3874
solution due to the second term of the f(t) first and then use the mathematical formula L1 ðets FðsÞÞ ¼ uðttÞf ðttÞ
ð10Þ
to get the solution due to the first term.
2.1.2. Temperature distribution The solution of the governing equation in the sdomain can be represented as pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 Tðx,sÞ ¼ C1 ex s þ 2s þ C2 ex s þ 2s ð11Þ For the solution to be bounded C2 must vanish and therefore pffiffiffiffiffiffiffiffiffiffi 2 Tðx,sÞ ¼ C1 ex s þ 2s ð12Þ By applying the transformed boundary condition, the constant C1 can be determined and the solution in the transformed domain can be finally written as pffiffiffiffiffiffiffiffiffiffi 2 1þ 2s ex s þ 2s p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tðx,sÞ ¼ ð13Þ sðs þ 2Þ The first term of the solution can be inversely transformed as ! hpffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffi 1 1 x sðs þ 2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e t 2 x2 u½tx ð14Þ ¼ et I0 L sðs þ2 where I0 is the modified Bessel function of the first kind and u is the unit step function. The second term can be inversed using the rule Z t 1 FðsÞ ¼ L1 f ðlÞdl ð15Þ s 0 Therefore, the final solution in the time domain can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Tðx,tÞ ¼ u½tx et I0 ½ t 2 x2 þ 2 el I0 ½ l x2 dl ð16Þ
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and c1 and c2 are the nondimensional wave speed and the nondimensional thermal modulus, respectively. pffiffiffiffiffiffiffiffiffiffiffi sx sx c2 M e sðs þ 2Þx ð21Þ s^ ¼ B1 ec1 þ B2 ec1 þ 1 22 sðc1 ðs þ 2ÞsÞ For boundness of the solution B1 ¼0, and the condition of free stress (i.e. s ¼0) on the surface is imposed as this stage to calculate the other coefficient B2 as B2 ¼
c12 M2 2 sðc1 sþ 2c12 sÞ
ð22Þ
s^ ðx,sÞ ¼ s^ h ðx,sÞ þ s^ p ðx,sÞ
ð23Þ
where csx
c2 c ðsþ 2Þe
1 2 s^ h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
s
c2 c ðs þ2Þe
1
2 s^ p ¼ p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
þ2sðc12 s þ 2c12 sÞ
s
pffiffiffiffiffiffiffiffiffiffiffi
sðs þ 2Þx
þ 2sðc12 ðs þ 2ÞsÞ
ð24Þ
^ h can be written as ^ h . The stress component s 2.1.3.1. Inversion of s a multiplication of two subcomponents as ^ ð2Þ s^ h ðx,sÞ ¼ s^ ð1Þ h ðx,sÞsh ðx,sÞ
ð25Þ
where
s^ ð1Þ h ¼
c12 c2 ðs þ2Þ csx e 1 ðc12 s þ2c12 sÞ
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s^ ð2Þ h ¼ 2
s þ2s
ð26Þ
The inverse Laplace transform of the two subcomponents is carried out as the following: 1 x 1 ^ ð1Þ 2 sð1Þ ðx,tÞ ¼ L ð s Þ ¼ c c d t 2 1 h h c1 c12 1 ! 2c1 ðxc1 tÞ 2 x c2 1 2 e 1 U t ð27Þ c1 ðc1 1Þ2
0 t ^ ð2Þ sð2Þ ðx,tÞ ¼ L1 ðs h Þ ¼ e J0 ½it h
2.1.3. Thermal stresses To solve for the stress distribution inside the substrate material, equation governing the momentum in a one-dimensional solid for a linear elastic case can be considered, i.e. @2 s 1 @2 s @2 T 2 2 ¼ c2 2 2 @x @t c1 @t
ð17Þ
The preceding equation is rewritten using the following nondimensional quantities along with the previously described ones (Eq. (3)) rffiffiffiffiffiffiffiffiffiffiffi 2c2 Pk ErCp s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 ¼ c1 s¼ c2 ¼ ð18Þ 2 E k Cp Er2 Ek=Cp r
0
^ p . Similarly, the second component of the 2.1.3.2. Inversion of s ^ p can be written can be written as a multiplication of stresses s two subcomponents ^ ð2Þ s^ p ðx,sÞ ¼ s^ ð1Þ p ðx,sÞsp ðx,sÞ
Therefore
where
@2 s 1 @2 s @2 T 2 2 ¼ c2 2 2 @t @x c1 @t
s^ ð1Þ p ¼
ð19Þ
By applying the Laplace transformation, the nondimensional stress field equation in the sdomain is given by
s^ 00 ðxÞ
pffiffiffiffiffiffiffiffiffiffiffi ^ ðxÞ s2 s ¼ M2 e sðs þ 2Þ x c12
where c2 sðs þ 2Þ M2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 þ 2s
ð20Þ
ð28Þ
The further details of the Laplace inversion are given in the Appendix A. ^ h ðx,tÞ by Having obtained sð1Þ ðx,tÞ and sð1Þ ðx,tÞ one can get s h h employing the convolution theorem Z t ^ ð2Þ sh ðx,tÞ ¼ s^ ð1Þ ð29Þ h ðtÞsh ðttÞdt
c12 c2 ðs þ 2Þ c12 ðs þ2Þs
ð30Þ pffiffiffiffiffiffiffiffiffiffiffi e sðs þ 2Þx
s^ ð2Þ p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ð31Þ
s þ2s
The inverse Laplace transform is carried out for both subcomponents 0 1 2c2 t
ð1Þ p ðx,tÞ ¼
s
1
L
1 c2 1
B d½t 2e 1 C ð1Þ C ð ^ p Þ ¼ c12 c2 B @c2 1 ðc2 1Þ2 A 1 1
s
pffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ^ ð2Þ t 2 2 sð2Þ p ðx,tÞ ¼ L ðsp Þ ¼ e I0 ð t x ÞU½tx
ð32Þ
ð33Þ
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ð1Þ ^ ð1Þ Having obtained sð1Þ p ðx,tÞ and sp ðx,tÞ one can get s p ðx,tÞ by employing the convolution theorem Z t ^ ð2Þ sp ðx,tÞ ¼ s^ ð1Þ ð34Þ p ðtÞsp ðttÞdt 0
The further details of the Laplace inversion are given in the Appendix A. All components of the stresses have been transformed to (x,t) domain, and we are ready to get the stresses as
sðx,tÞ ¼ sh ðx,tÞ þ sp ðx,tÞ
ð35Þ
3. Results and discussion The closed form solution for temperature and stress distribution in solid subjected to a step impulse heating from the external source. The analytical solutions of Cattaneo and the thermal stress equations are obtained using the Laplace transformation method with the appropriate boundary conditions. Fig. (1) shows nondimensional step input power intensity distribution, while Fig. 2 shows temporal variation of nondimensional temperature at different locations inside the solid substrate. Temperature increases rapidly in the early heating period (0rtr50). The rapid rise of temperature is attributed to the internal energy gain from the heat source, since the heat conduction from the surface vicinity to the solid bulk is small, due to low temperature gradient, in the early heating period [10]. As the heating progresses, temperature rise becomes gradual due to the enhancement of the diffusional energy transfer from the surface region to the solid bulk with progressing time. The rapid rise of temperature in the early heating period is more pronounced in the surface vicinity (xrx2) due to the high rate of internal energy gain from the heat source. However, @T/@t variation becomes almost constant at some depths below the surface (x4x3) during the heating period (tr400). In this case, the term @2T/@t2 approaches to zero and the Cattaneo equation reduces to the classical diffusion equation (Fourier equation). Consequently, the energy transfer is governed by the temperature gradient due to diffusion at some depth below the surface where @T/@t becomes almost constant. Moreover, once the cooling cycle initiates (t4400, when heating pulse ends) temperature reduces rapidly. The rate of temperature decay in the early cooling cycle (400rtr430) is high and as the cooling period progresses, temperature decay becomes gradual. The rapid decay of temperature produces large values of @T/@t which, in turn,
Fig. 1. Step pulse.
Fig. 2. Temperature distribution vs. time at different depths. x1 ¼ 100; x2 ¼200; x3 ¼ 300; x4 ¼ 400; and x5 ¼500.
Fig. 3. Temperature distribution vs. depth at different times. t1 ¼70; t2 ¼ 124; t3 ¼ 179; and t4 ¼233.
enhances the value of the term @2T/@t2. In this case, Cattaneo equation governs the energy transfer in the surface vicinity due to presence of large @2T/@t2. Moreover, the gradual decay of temperature at some depths below the surface (xZx3) results in low values of @2T/@t2, in which case, Fourier heat diffusion dominates the energy transfer in this region in the cooling period. Fig. 3 shows dimensionless temperature variation inside the substrate material for different durations. Temperature decays sharply in the surface region and it becomes gradual as the distance increases from the surface (x Z10). This is more pronounced in the early heating period (t ¼t1). This situation can also be seen from Fig. 4, in which the temperature gradient is shown for different times as similar to Fig. 3. The sharp change in the temperature gradient indicates that the internal energy gain from the heat source is high in the surface region. It should be noted that increasing temperature gradient results in high heat flux from the surface region to the solid bulk. Consequently, the internal energy gain increases in the surface region while the diffusional energy loss from the surface vicinity to the solid bulk is small for the heating period t ¼t1. However, as the distance
H. Al-Qahtani, B.S. Yilbas / Physica B 405 (2010) 3869–3874
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Fig. 6. Nondimensional stresses vs. space at different times. t1 ¼70; t2 ¼124; t3 ¼179; and t4 ¼233. Fig. 4. Temperature gradient vs. depth at different times. t1 ¼70; t2 ¼124; t3 ¼179; and t4 ¼233.
Fig. 5. Nondimensional stresses vs. time at different locations. x1 ¼ 100; x2 ¼ 200; x3 ¼300; and x4 ¼ 400.
increases from the surface towards the solid bulk, temperature gradient reduces gradually. In addition, as the time increases t Zt3, temperature gradient reduces gradually irrespective of the locations below the surface. This indicates that temperature decay inside the substrate material is almost at a steady rate resulting in steadily decreasing gradient. Moreover, as the time progresses, heat diffusion from the surface region enhances the temperature gradient towards the solid bulk, i.e. temperature gradient is in the order of 0.2 at x¼20 for t Zt3 while it is in the order of 0.02 at x ¼20 for t ¼t1. This is attributed to the internal energy gain of the substrate material at the depth x¼20 due to the diffusional heating during the time t ¼t3. Fig. 5 shows temporal variation of dimensionless stress distribution at different locations below the surface. Thermal stress developed appears to be in wave form such that it decays first sharply and later the decay becomes gradual in the amplitude. The stress wave propagates into the substrate material with a constant speed and with almost constant peak amplitude, provided that the amplitude decays incrementally due to the damping effect of the substrate material. During the heating cycle, the wave generated is tensile with positive amplitude. However, in the cooling cycle, it becomes compressive with negative amplitude. The generation of the
tensile wave in the heating cycle is attributed to the thermal expansion of the substrate material in the surface region. This generates a positive thermal displacement at the surface. Consequently, the stress wave generated in the surface vicinity becomes tensile. Once the thermal stress wave is generated, it propagates into the substrate material with the speed of sound, c, [19]. Consequently, at depth below the surface and at the time it reaches any location, the wave appears as repeating at this particular location. In the case of the cooling cycle (t Z400), the sudden cooling of the surface results in contraction in the surface region and generates a compression wave propagating into the substrate material. Since the tensile wave is generated in an earlier stage than the compressive wave, both waves do not meet at any location in the substrate material. Consequently, canceling of the amplitudes of the tensile and the compression waves in the substrate is less likely to occur. The compression wave behaves similar to the tensile wave; in which case, the wave propagates at constant speed and the wave amplitude decays slowly as it propagates into the substrate material due to the damping effect of the substrate material. Fig. 6 shows dimensionless thermal stresses developed inside the substrate material for different times in the heating period. The thermal stress developed during the heating cycle appears as the stress wave propagating at a constant wave speed. The wave front is sharp and a tail is formed at the back of the wave. The same behavior is observed in the previous study [18]. The tail of the wave is due to the temperature gradient developed inside the substrate material. In this case, the thermal strain developed inside the substrate material during the heating cycle is responsible for the formation of the wave tail, which decays gradually towards the surface. As the wave further propagates into the substrate material the tail of the wave extends into the substrate material. Moreover, the amplitude of the tail of the wave reduces in the substrate material due to the change in the temperature gradient along the depth below the surface.
4. Conclusion Heating of a semi-infinite solid with a step input pulse is considered and the analytical solutions for temperature and thermal stress fields are obtained using the Laplace transformation method. The closed form solutions for Cattaneo equation and the thermal stress equation with the appropriate boundary
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conditions are presented. It is found that temperature increases rapidly in the early heating period and as the heating progresses, the rate of increase in temperature becomes gradual. The high rate of increase in temperature is attributed to the internal energy gain of the substrate material from the heat source and small amount of energy transfer by diffusion from the surface vicinity to the solid bulk in the early heating period. In this case, @T/@t, and @2T/@t2 attain high values and varies sharply with progressing time. Consequently, the heat diffusion is not governed by the classical Fourier law in the early heating period. This situation is also observed onset of the initiation of the cooling period for which temperature decays rapidly with progressing time. However, as the heating progresses, the rate of temperature rise becomes gradual and the term @2T/@t2 becomes small in Cattaneo equation. The heat diffusion is governed mainly by the Fourier law. This is also true during the long cooling periods. Thermal stress developed in the surface vicinity is in the form of a stress wave which propagates into the substrate material with a constant speed. Moreover, the stress wave is tensile in the heating cycle due to the thermal expansion of the surface while it is compressive in the cooling cycle because of the thermal contraction of the surface region during the cooling period. The thermal wave generated has a tail with decaying amplitude and it extends over the heated region.
Acknowledgment The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for this work.
Appendix A ^h Inversion of s !
c12 c2 ðs þ 2Þ csx e 1 c12 ðs þ 2Þs ! 2c12 c2 c12 c2 csx csx 1 1 ¼ L1 e e ðc12 1Þðc12 s þ2c12 sÞ c2 1 ! 1 ! 2c12 c2 c12 c2 csx csx 1 1 e 1 L e 1 ¼L ðc12 1Þðc12 s þ2c12 sÞ c12 1 ! ! 2c1 ðxc1 tÞ c 2 c2 2c12 c2 x x c2 1 1 ¼ 21 d t e U t þ c1 c1 c1 1 ðc12 1Þ2 ! 2c1 ðxc1 tÞ 1 x 2 x c2 1 1 d t e U t 2 ¼ c12 c2 2 c1 c1 c1 1 ðc1 1Þ2 ! 1 1 ^ ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ et J0 ½it sð2Þ ðx,tÞ ¼ L1 ðs h Þ¼L h sðs þ 2Þ 1 ^ ð1Þ sð1Þ ðx,tÞ ¼ L1 ðs h Þ¼L h
^p Inversion of s 1 ^ ð1Þ sð1Þ ðsp Þ ¼ L1 p ðx,tÞ ¼ L
! c12 c2 ðs þ 2Þ c12 ðs þ2Þs !
c12 c2 2c12 c2 c12 1 ðc12 1Þðc12 s þ 2c12 sÞ ! ! c2 c2 2c12 c2 ¼ L1 21 L1 c1 1 ðc12 1Þðc12 s þ 2c12 sÞ 1 ! 0 2c2 t 1 c12 c2 2c12 c2 c2 1 @ A 1 d½t e ¼ 2 c1 1 ðc12 1Þ2 pffiffiffiffiffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffiffiffiffiffiffiffii sðs þ 2Þx 1 ^ ð2Þ 1 e ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ et U ½txI0 t 2 x2 sp ðx,tÞ ¼ L ðsp Þ ¼ L sðs þ2Þ ¼ L1
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
L.G. Hector Jr., W.S. Kim, M.N. Ozisik, Int. J. Eng. Sci. 30 (1992) 1731. J.F. Ready, J. Appl. Phys. 36 (1963) 462. F.J. Blackwell, ASME J. Heat Transfer 112 (1990) 567. M. Honner, J. Kunes, ASME J. Heat Transfer 121 (1999) 702. L. Malinowski, J. Phys. D: Appl. Phys. 26 (1993) 1176. C.I. Christov, Mech. Res. Commun. 36 (2009) 481. H. Gomez, I. Colominas, F. Navarrina, M. Casteleiro, Comput. Methods Appl. Mech. Eng. 196 (2007) 1757. J. Ordonez-Miranda, J.J. Alvarado-Gil, Int. J. Therm. Sci. 48 (2009) 2053. P. Duhamel, Int. J. Heat Mass Transfer 47 (2004) 573. B.S. Yilbas, Int. J. Mech. Sci. 39 (1997) 671. B.S. Yilbas, M. Kalyon, J. Phys. D: Appl. Phys. 34 (2001) 222. B.S. Yilbas, M. Pakdemir, Phys. B: Condens. Matter 382 (2006) 213. H.G. Wang, Y.H. Guan, T.L. Chen, J.T. Zhang, J. Mater. Process. Technol. 63 (1997) 550. G.F. Yuan, C.S. Ge, X.Y. Zeng, Acta Metall. Sin. 17 (2004) 43. K.K. Tamma, X. Zhou, J. Therm. Stresses 21 (1998) 405. M. Ostoja-Starzewski, J. Therm. Stresses 26 (2003) 1205. B.S. Yilbas, M. Sami, S.Z. Shuja, S, Opt. Lasers Eng. 30 (1998) 25. M. Kalyon M., B.S. Yilbas, Proc. Inst. Mech. Eng.: Part C J. Mech. Eng. Sci. 215 (2001) 1429. B.S. Yilbas, N. Al-Ageeli, Int. J. Mech. Sci. 46 (2004) 57. J.B.J. Fourier, Theorie Analytique De La Chaleur. Paris, 1822. C. Cattaneo, C. R. 247 (1958) 431.
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Physica B journal homepage: www.elsevier.com/locate/physb
The closed form solutions for Cattaneo and stress equations due to step input pulse heating H. Al-Qahtani, B.S. Yilbas n King Fahd University of Petroleum & Minerals, Box 1913, Dhahran 31261, Saudi Arabia
a r t i c l e in fo
abstract
Article history: Received 24 February 2010 Received in revised form 15 May 2010 Accepted 8 June 2010
The analytical solution for the temperature and stress fields in a solid subjected to a step input pulse heating is presented. The closed form solutions of Cattaneo and thermal stress equations are obtained using the Laplace transformation technique. The predictions obtained from the closed form solutions are presented in nondimensional form. It is found that temperature rise is rapid in the early heating period and as the time progresses, temperature rise becomes gradual. The stress generated in the heating cycle is tensile while it becomes compressive in the cooling cycle. & 2010 Elsevier B.V. All rights reserved.
Keywords: Cattaneo Pulse Heating Thermal stress
1. Introduction In short pulse heating of solid surfaces, heat wave is generated and the energy transport to the solid bulk takes place through heat wave at finite speed. In this case, heat is conducted in the solid due to the near neighborhood excitation via changing of momentum and energy on a microscopic scale in a wave form. The average communication time between the neighborhoods is associated with the relaxation time. The thermal communication takes place in a dissipative nature resulting in the thermal resistance in the solid medium. If the heating duration is longer than the relaxation time, the speed of the heat wave propagation approaches to infinity and the heat wave equation reduces to the classical Fourier heat equation. However, the Fourier heating model fails to predict the temperature propagation speed and suffers anomalies when the heating duration is comparable or less than the relaxation time. Therefore, Cattaneo’s heating model becomes appropriate to account for the temperature propagation speed while eliminating the anomalies arise from the Fourier heating model. Cattaneo’s heating model is governed by the hyperbolic temperature equations. Moreover, the heating of submicron sized solid devices during the short heating period results in non-equilibrium heating situation in the solid; in which case, the governing equation takes the hyperbolic form. The heating situation can also be modeled using the Cattaneo’s equation. Although the numerical solution is possible for such heating situation, the analytical solution of Cattaneo’s heating equation is
n
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0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.06.019
fruitful to generalize the temperature field in terms of the heat source parameters and the material properties. The application of the Laplace transformation method provides the exact solution to the Cattaneo’s heating equation. In addition, temperature field developed in the solid causes the thermal expansion of the substrate, which, in turn, generates the thermal stress field within the heated region. Consequently, investigation into the analytical solution of the hyperbolic heat and the thermal stress equations becomes necessary. Considerable research studies were carried out to examine the hyperbolic heating situation of solids by the externally applied sources. The hyperbolic heat conduction due to the mode locked laser pulse train was investigated by Hector et al. [1]. They presented the semi-analytical solution to the heating problem and indicated that the parabolic Fourier heating model significantly underestimated the surface temperature, since the rate of energy propagation was assumed to be infinite. Ready [2] introduced the closed form solution for temperature rise due to pulse heating of solid surfaces. He used the Fourier heating model to analyze temperature distribution in the solid. Blackwell [3] developed an analytical solution for temperature rise inside the solid heated by a pulsed laser. He adopted a Fourier heating model when formulating temperature distribution during the heating process. The wave diffusion and parallel non-equilibrium heat conduction were investigated by Honner and Kunes [4]. They introduced a dimensionless criterion for microscale heat transfer due to the equilibrium and the non-equilibrium diffusion conduction problems. Malinowski [5] examined the heat conduction and the generation in solids after considering finite velocity of the heat propagation and the inertia of the internal heat source. He showed that unlike the classical hyperbolic model, the relaxation
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solutions generally did not tend to approach the corresponding parabolic solution. A material-invariant version of the Maxwell– Cattaneo law was proposed by Christov [6]. He showed that the new formulation allowed for the elimination of the heat flux yielding a single equation for the temperature field. The convection-diffusion equation based on Cattaneo’s law was examined by Gomez et al. [7]. They introduced a finite element solution of the problem and the performance of the algorithm developed was verified by solving some 2-D test cases. The thermal wave oscillations and thermal relaxation time determination in a hyperbolic heat transport model were examined by OrdonezMirech and Alvarado-Gil [8]. They presented the simple analytical expressions for the values of the maxima and minima of the oscillations as well as the frequencies, at which they occurred. The new finite integral transformation method to the wave model of conduction was introduced by Duhamel [9]. He tested the Cattaneo Vornetto model through the comparison of the results with the transient molecular dynamics solutions. He indicated that when the used parameters of the continuous model were near their equilibrium values, the agreement both results remained weakly qualitative. Yilbas [10], and Yilbas and Kalyon [11] introduced the closed form solutions for temperature distribution inside the solid substrate subjected to a laser heating. The studies were limited to the Fourier heat diffusion model and may not be applicable for the short heating durations. Moreover, Yilbas and Pakdemir [12] presented a semi-analytical solution for the hyperbolic heat conduction problem. They introduced a perturbation method to obtain the analytical solutions. However, the approach introduced provides only the approximate solutions for temperature distribution. Considerable research studies were carried out to examine thermal stress development in solids subjected to a diffusive heating. Wang et al. [13] studied thermal stress development in solids due to the laser heating process. They used the effects of phase transition and formulate the thermal stress equations during the heating cycle. Yuan et al. [14] investigated the thermal stress development during the laser heating of two sheet metals. They validated the predictions with the experimental data. Some perspectives of macroscale and microscale thermal transport and thermo-mechanical interactions were presented by Tamma and Zhou [15]. They explained the significance of constitutive models for both macroscale and microscale heat conduction in conjunction with generalizations drawn concerning the physical relevance and the role of relaxation and retardation times emanating from the Jefferys type heat flux constitutive model, with consequences to the Cattaneo heat flux model and subsequently to the Fourier heat flux model. Thermoelastic wave formation due to parabolic and hyperbolic heat conduction in a helix was studied by Ostoja-Starzewski [16]. He showed that the thermal diffusivity and the relaxation time of the Maxwell–Cattaneo model had minor effects, whereas the thermoelastic coupling constant was dominant for the harmonic motions. Yilbas et al. [17] investigated thermal stress development during the laser heating pulse. They presented a closed form solution for the stress distribution in the laser irradiated region. Kalyon and Yilbas [18] studied thermal stress development in a semi-infinite solid subjected to a laser step input pulse. They formulated temperature and stress fields after considering the Fourier heating model and the equilibrium conditions. Yilbas and Al-Ageeli [19] formulated the thermal stress development in solids during the laser heating pulses. The analytical solutions developed were limited with the Fourier diffusion analysis and extension of the work including Cattaneo diffusion becomes necessary. In the present study, the analytical solution for temperature and thermal stress distributions in a solid subjected to a pulse heating is presented. The closed form solutions for the Cattaneo
diffusion and corresponding thermal stress equations are obtained by using the Laplace transformation method.
2. Mathematical analysis The assumption of equilibrium heating between quantized electronic excitations and quanta of lattice vibrations, the phenomenological model was proposed by Fourier [20]. The model proposed is parabolic one-step heat conduction with the notion of infinite speed of propagation of thermal disturbances. However, to account for a finite temperature propagation speed and to avoid the anomalies associated with the Fourier heat conduction model, the Cattaneo model was developed on the notion of relaxing the heat flux [21], which is expressed as
E
@q ¼ qK rT @t
ð1Þ
where q¼ krT, e is the relaxation time, K the conductivity tensor, q the heat flux vector, and T the temperature. The Cattaneo equation yields to the hyperbolic heat conduction equation for the temperature pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field, which is propagative with a speed c ¼ ðk=rCp eÞ, where r is the density and Cp the specific heat capacity. ! @2 T @T @2 T ð2Þ k 2 ¼ rCp þ E 2 @t @x @t
2.1. Nondimensionalization The preceding heat equation can be put in a nondimensional form using the following nondimensional parameters: t¼
t , 2E
x x ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 4Ek=rCp
Tk T ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 4Ek=rCp
ð3Þ
where the P is the pulse intensity. The resulting nondimensional equation is thus @2 T @T @2 T þ 2 ¼2 2 @t @x @t
ð4Þ
with nondimensional initial and boundary conditions T ¼0 q ¼ f ðtÞ
at at
t¼0
ð5Þ
x¼0
ð6Þ
The closed form solution of Eq. (4) can be possible by adapting the Laplace transformation method. Taking the Laplace transformation of the governing Eq. (4), initial and boundary conditions (5),(6) results in @2 T^ ^ T^ ¼ 0 s2 T2s @x2
ð7Þ
@T^ sq^ ¼ FðsÞ at x ¼ 0 @x The last equation can be rewritten as
q^ ¼
@T 1 ¼ FðsÞðs þ 1Þ @x s
at
x ¼ 0,
ð8Þ
ð9Þ
2.1.1. Pulse shape The step input pulse intensity is considered, which can be written as f ðtÞ ¼ P½uðttÞuðtÞ Since the model is linear, one can use the superposition principle to compute the temperature distribution due to the two terms of the pulse function separately. It is convenient to get the
H. Al-Qahtani, B.S. Yilbas / Physica B 405 (2010) 3869–3874
solution due to the second term of the f(t) first and then use the mathematical formula L1 ðets FðsÞÞ ¼ uðttÞf ðttÞ
ð10Þ
to get the solution due to the first term.
2.1.2. Temperature distribution The solution of the governing equation in the sdomain can be represented as pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 Tðx,sÞ ¼ C1 ex s þ 2s þ C2 ex s þ 2s ð11Þ For the solution to be bounded C2 must vanish and therefore pffiffiffiffiffiffiffiffiffiffi 2 Tðx,sÞ ¼ C1 ex s þ 2s ð12Þ By applying the transformed boundary condition, the constant C1 can be determined and the solution in the transformed domain can be finally written as pffiffiffiffiffiffiffiffiffiffi 2 1þ 2s ex s þ 2s p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tðx,sÞ ¼ ð13Þ sðs þ 2Þ The first term of the solution can be inversely transformed as ! hpffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffi 1 1 x sðs þ 2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi e t 2 x2 u½tx ð14Þ ¼ et I0 L sðs þ2 where I0 is the modified Bessel function of the first kind and u is the unit step function. The second term can be inversed using the rule Z t 1 FðsÞ ¼ L1 f ðlÞdl ð15Þ s 0 Therefore, the final solution in the time domain can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Tðx,tÞ ¼ u½tx et I0 ½ t 2 x2 þ 2 el I0 ½ l x2 dl ð16Þ
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and c1 and c2 are the nondimensional wave speed and the nondimensional thermal modulus, respectively. pffiffiffiffiffiffiffiffiffiffiffi sx sx c2 M e sðs þ 2Þx ð21Þ s^ ¼ B1 ec1 þ B2 ec1 þ 1 22 sðc1 ðs þ 2ÞsÞ For boundness of the solution B1 ¼0, and the condition of free stress (i.e. s ¼0) on the surface is imposed as this stage to calculate the other coefficient B2 as B2 ¼
c12 M2 2 sðc1 sþ 2c12 sÞ
ð22Þ
s^ ðx,sÞ ¼ s^ h ðx,sÞ þ s^ p ðx,sÞ
ð23Þ
where csx
c2 c ðsþ 2Þe
1 2 s^ h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
s
c2 c ðs þ2Þe
1
2 s^ p ¼ p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
þ2sðc12 s þ 2c12 sÞ
s
pffiffiffiffiffiffiffiffiffiffiffi
sðs þ 2Þx
þ 2sðc12 ðs þ 2ÞsÞ
ð24Þ
^ h can be written as ^ h . The stress component s 2.1.3.1. Inversion of s a multiplication of two subcomponents as ^ ð2Þ s^ h ðx,sÞ ¼ s^ ð1Þ h ðx,sÞsh ðx,sÞ
ð25Þ
where
s^ ð1Þ h ¼
c12 c2 ðs þ2Þ csx e 1 ðc12 s þ2c12 sÞ
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s^ ð2Þ h ¼ 2
s þ2s
ð26Þ
The inverse Laplace transform of the two subcomponents is carried out as the following: 1 x 1 ^ ð1Þ 2 sð1Þ ðx,tÞ ¼ L ð s Þ ¼ c c d t 2 1 h h c1 c12 1 ! 2c1 ðxc1 tÞ 2 x c2 1 2 e 1 U t ð27Þ c1 ðc1 1Þ2
0 t ^ ð2Þ sð2Þ ðx,tÞ ¼ L1 ðs h Þ ¼ e J0 ½it h
2.1.3. Thermal stresses To solve for the stress distribution inside the substrate material, equation governing the momentum in a one-dimensional solid for a linear elastic case can be considered, i.e. @2 s 1 @2 s @2 T 2 2 ¼ c2 2 2 @x @t c1 @t
ð17Þ
The preceding equation is rewritten using the following nondimensional quantities along with the previously described ones (Eq. (3)) rffiffiffiffiffiffiffiffiffiffiffi 2c2 Pk ErCp s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c1 ¼ c1 s¼ c2 ¼ ð18Þ 2 E k Cp Er2 Ek=Cp r
0
^ p . Similarly, the second component of the 2.1.3.2. Inversion of s ^ p can be written can be written as a multiplication of stresses s two subcomponents ^ ð2Þ s^ p ðx,sÞ ¼ s^ ð1Þ p ðx,sÞsp ðx,sÞ
Therefore
where
@2 s 1 @2 s @2 T 2 2 ¼ c2 2 2 @t @x c1 @t
s^ ð1Þ p ¼
ð19Þ
By applying the Laplace transformation, the nondimensional stress field equation in the sdomain is given by
s^ 00 ðxÞ
pffiffiffiffiffiffiffiffiffiffiffi ^ ðxÞ s2 s ¼ M2 e sðs þ 2Þ x c12
where c2 sðs þ 2Þ M2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 þ 2s
ð20Þ
ð28Þ
The further details of the Laplace inversion are given in the Appendix A. ^ h ðx,tÞ by Having obtained sð1Þ ðx,tÞ and sð1Þ ðx,tÞ one can get s h h employing the convolution theorem Z t ^ ð2Þ sh ðx,tÞ ¼ s^ ð1Þ ð29Þ h ðtÞsh ðttÞdt
c12 c2 ðs þ 2Þ c12 ðs þ2Þs
ð30Þ pffiffiffiffiffiffiffiffiffiffiffi e sðs þ 2Þx
s^ ð2Þ p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ð31Þ
s þ2s
The inverse Laplace transform is carried out for both subcomponents 0 1 2c2 t
ð1Þ p ðx,tÞ ¼
s
1
L
1 c2 1
B d½t 2e 1 C ð1Þ C ð ^ p Þ ¼ c12 c2 B @c2 1 ðc2 1Þ2 A 1 1
s
pffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ^ ð2Þ t 2 2 sð2Þ p ðx,tÞ ¼ L ðsp Þ ¼ e I0 ð t x ÞU½tx
ð32Þ
ð33Þ
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ð1Þ ^ ð1Þ Having obtained sð1Þ p ðx,tÞ and sp ðx,tÞ one can get s p ðx,tÞ by employing the convolution theorem Z t ^ ð2Þ sp ðx,tÞ ¼ s^ ð1Þ ð34Þ p ðtÞsp ðttÞdt 0
The further details of the Laplace inversion are given in the Appendix A. All components of the stresses have been transformed to (x,t) domain, and we are ready to get the stresses as
sðx,tÞ ¼ sh ðx,tÞ þ sp ðx,tÞ
ð35Þ
3. Results and discussion The closed form solution for temperature and stress distribution in solid subjected to a step impulse heating from the external source. The analytical solutions of Cattaneo and the thermal stress equations are obtained using the Laplace transformation method with the appropriate boundary conditions. Fig. (1) shows nondimensional step input power intensity distribution, while Fig. 2 shows temporal variation of nondimensional temperature at different locations inside the solid substrate. Temperature increases rapidly in the early heating period (0rtr50). The rapid rise of temperature is attributed to the internal energy gain from the heat source, since the heat conduction from the surface vicinity to the solid bulk is small, due to low temperature gradient, in the early heating period [10]. As the heating progresses, temperature rise becomes gradual due to the enhancement of the diffusional energy transfer from the surface region to the solid bulk with progressing time. The rapid rise of temperature in the early heating period is more pronounced in the surface vicinity (xrx2) due to the high rate of internal energy gain from the heat source. However, @T/@t variation becomes almost constant at some depths below the surface (x4x3) during the heating period (tr400). In this case, the term @2T/@t2 approaches to zero and the Cattaneo equation reduces to the classical diffusion equation (Fourier equation). Consequently, the energy transfer is governed by the temperature gradient due to diffusion at some depth below the surface where @T/@t becomes almost constant. Moreover, once the cooling cycle initiates (t4400, when heating pulse ends) temperature reduces rapidly. The rate of temperature decay in the early cooling cycle (400rtr430) is high and as the cooling period progresses, temperature decay becomes gradual. The rapid decay of temperature produces large values of @T/@t which, in turn,
Fig. 1. Step pulse.
Fig. 2. Temperature distribution vs. time at different depths. x1 ¼ 100; x2 ¼200; x3 ¼ 300; x4 ¼ 400; and x5 ¼500.
Fig. 3. Temperature distribution vs. depth at different times. t1 ¼70; t2 ¼ 124; t3 ¼ 179; and t4 ¼233.
enhances the value of the term @2T/@t2. In this case, Cattaneo equation governs the energy transfer in the surface vicinity due to presence of large @2T/@t2. Moreover, the gradual decay of temperature at some depths below the surface (xZx3) results in low values of @2T/@t2, in which case, Fourier heat diffusion dominates the energy transfer in this region in the cooling period. Fig. 3 shows dimensionless temperature variation inside the substrate material for different durations. Temperature decays sharply in the surface region and it becomes gradual as the distance increases from the surface (x Z10). This is more pronounced in the early heating period (t ¼t1). This situation can also be seen from Fig. 4, in which the temperature gradient is shown for different times as similar to Fig. 3. The sharp change in the temperature gradient indicates that the internal energy gain from the heat source is high in the surface region. It should be noted that increasing temperature gradient results in high heat flux from the surface region to the solid bulk. Consequently, the internal energy gain increases in the surface region while the diffusional energy loss from the surface vicinity to the solid bulk is small for the heating period t ¼t1. However, as the distance
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Fig. 6. Nondimensional stresses vs. space at different times. t1 ¼70; t2 ¼124; t3 ¼179; and t4 ¼233. Fig. 4. Temperature gradient vs. depth at different times. t1 ¼70; t2 ¼124; t3 ¼179; and t4 ¼233.
Fig. 5. Nondimensional stresses vs. time at different locations. x1 ¼ 100; x2 ¼ 200; x3 ¼300; and x4 ¼ 400.
increases from the surface towards the solid bulk, temperature gradient reduces gradually. In addition, as the time increases t Zt3, temperature gradient reduces gradually irrespective of the locations below the surface. This indicates that temperature decay inside the substrate material is almost at a steady rate resulting in steadily decreasing gradient. Moreover, as the time progresses, heat diffusion from the surface region enhances the temperature gradient towards the solid bulk, i.e. temperature gradient is in the order of 0.2 at x¼20 for t Zt3 while it is in the order of 0.02 at x ¼20 for t ¼t1. This is attributed to the internal energy gain of the substrate material at the depth x¼20 due to the diffusional heating during the time t ¼t3. Fig. 5 shows temporal variation of dimensionless stress distribution at different locations below the surface. Thermal stress developed appears to be in wave form such that it decays first sharply and later the decay becomes gradual in the amplitude. The stress wave propagates into the substrate material with a constant speed and with almost constant peak amplitude, provided that the amplitude decays incrementally due to the damping effect of the substrate material. During the heating cycle, the wave generated is tensile with positive amplitude. However, in the cooling cycle, it becomes compressive with negative amplitude. The generation of the
tensile wave in the heating cycle is attributed to the thermal expansion of the substrate material in the surface region. This generates a positive thermal displacement at the surface. Consequently, the stress wave generated in the surface vicinity becomes tensile. Once the thermal stress wave is generated, it propagates into the substrate material with the speed of sound, c, [19]. Consequently, at depth below the surface and at the time it reaches any location, the wave appears as repeating at this particular location. In the case of the cooling cycle (t Z400), the sudden cooling of the surface results in contraction in the surface region and generates a compression wave propagating into the substrate material. Since the tensile wave is generated in an earlier stage than the compressive wave, both waves do not meet at any location in the substrate material. Consequently, canceling of the amplitudes of the tensile and the compression waves in the substrate is less likely to occur. The compression wave behaves similar to the tensile wave; in which case, the wave propagates at constant speed and the wave amplitude decays slowly as it propagates into the substrate material due to the damping effect of the substrate material. Fig. 6 shows dimensionless thermal stresses developed inside the substrate material for different times in the heating period. The thermal stress developed during the heating cycle appears as the stress wave propagating at a constant wave speed. The wave front is sharp and a tail is formed at the back of the wave. The same behavior is observed in the previous study [18]. The tail of the wave is due to the temperature gradient developed inside the substrate material. In this case, the thermal strain developed inside the substrate material during the heating cycle is responsible for the formation of the wave tail, which decays gradually towards the surface. As the wave further propagates into the substrate material the tail of the wave extends into the substrate material. Moreover, the amplitude of the tail of the wave reduces in the substrate material due to the change in the temperature gradient along the depth below the surface.
4. Conclusion Heating of a semi-infinite solid with a step input pulse is considered and the analytical solutions for temperature and thermal stress fields are obtained using the Laplace transformation method. The closed form solutions for Cattaneo equation and the thermal stress equation with the appropriate boundary
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conditions are presented. It is found that temperature increases rapidly in the early heating period and as the heating progresses, the rate of increase in temperature becomes gradual. The high rate of increase in temperature is attributed to the internal energy gain of the substrate material from the heat source and small amount of energy transfer by diffusion from the surface vicinity to the solid bulk in the early heating period. In this case, @T/@t, and @2T/@t2 attain high values and varies sharply with progressing time. Consequently, the heat diffusion is not governed by the classical Fourier law in the early heating period. This situation is also observed onset of the initiation of the cooling period for which temperature decays rapidly with progressing time. However, as the heating progresses, the rate of temperature rise becomes gradual and the term @2T/@t2 becomes small in Cattaneo equation. The heat diffusion is governed mainly by the Fourier law. This is also true during the long cooling periods. Thermal stress developed in the surface vicinity is in the form of a stress wave which propagates into the substrate material with a constant speed. Moreover, the stress wave is tensile in the heating cycle due to the thermal expansion of the surface while it is compressive in the cooling cycle because of the thermal contraction of the surface region during the cooling period. The thermal wave generated has a tail with decaying amplitude and it extends over the heated region.
Acknowledgment The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for this work.
Appendix A ^h Inversion of s !
c12 c2 ðs þ 2Þ csx e 1 c12 ðs þ 2Þs ! 2c12 c2 c12 c2 csx csx 1 1 ¼ L1 e e ðc12 1Þðc12 s þ2c12 sÞ c2 1 ! 1 ! 2c12 c2 c12 c2 csx csx 1 1 e 1 L e 1 ¼L ðc12 1Þðc12 s þ2c12 sÞ c12 1 ! ! 2c1 ðxc1 tÞ c 2 c2 2c12 c2 x x c2 1 1 ¼ 21 d t e U t þ c1 c1 c1 1 ðc12 1Þ2 ! 2c1 ðxc1 tÞ 1 x 2 x c2 1 1 d t e U t 2 ¼ c12 c2 2 c1 c1 c1 1 ðc1 1Þ2 ! 1 1 ^ ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ et J0 ½it sð2Þ ðx,tÞ ¼ L1 ðs h Þ¼L h sðs þ 2Þ 1 ^ ð1Þ sð1Þ ðx,tÞ ¼ L1 ðs h Þ¼L h
^p Inversion of s 1 ^ ð1Þ sð1Þ ðsp Þ ¼ L1 p ðx,tÞ ¼ L
! c12 c2 ðs þ 2Þ c12 ðs þ2Þs !
c12 c2 2c12 c2 c12 1 ðc12 1Þðc12 s þ 2c12 sÞ ! ! c2 c2 2c12 c2 ¼ L1 21 L1 c1 1 ðc12 1Þðc12 s þ 2c12 sÞ 1 ! 0 2c2 t 1 c12 c2 2c12 c2 c2 1 @ A 1 d½t e ¼ 2 c1 1 ðc12 1Þ2 pffiffiffiffiffiffiffiffiffiffiffi ! hpffiffiffiffiffiffiffiffiffiffiffiffiffii sðs þ 2Þx 1 ^ ð2Þ 1 e ð2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ et U ½txI0 t 2 x2 sp ðx,tÞ ¼ L ðsp Þ ¼ L sðs þ2Þ ¼ L1
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
L.G. Hector Jr., W.S. Kim, M.N. Ozisik, Int. J. Eng. Sci. 30 (1992) 1731. J.F. Ready, J. Appl. Phys. 36 (1963) 462. F.J. Blackwell, ASME J. Heat Transfer 112 (1990) 567. M. Honner, J. Kunes, ASME J. Heat Transfer 121 (1999) 702. L. Malinowski, J. Phys. D: Appl. Phys. 26 (1993) 1176. C.I. Christov, Mech. Res. Commun. 36 (2009) 481. H. Gomez, I. Colominas, F. Navarrina, M. Casteleiro, Comput. Methods Appl. Mech. Eng. 196 (2007) 1757. J. Ordonez-Miranda, J.J. Alvarado-Gil, Int. J. Therm. Sci. 48 (2009) 2053. P. Duhamel, Int. J. Heat Mass Transfer 47 (2004) 573. B.S. Yilbas, Int. J. Mech. Sci. 39 (1997) 671. B.S. Yilbas, M. Kalyon, J. Phys. D: Appl. Phys. 34 (2001) 222. B.S. Yilbas, M. Pakdemir, Phys. B: Condens. Matter 382 (2006) 213. H.G. Wang, Y.H. Guan, T.L. Chen, J.T. Zhang, J. Mater. Process. Technol. 63 (1997) 550. G.F. Yuan, C.S. Ge, X.Y. Zeng, Acta Metall. Sin. 17 (2004) 43. K.K. Tamma, X. Zhou, J. Therm. Stresses 21 (1998) 405. M. Ostoja-Starzewski, J. Therm. Stresses 26 (2003) 1205. B.S. Yilbas, M. Sami, S.Z. Shuja, S, Opt. Lasers Eng. 30 (1998) 25. M. Kalyon M., B.S. Yilbas, Proc. Inst. Mech. Eng.: Part C J. Mech. Eng. Sci. 215 (2001) 1429. B.S. Yilbas, N. Al-Ageeli, Int. J. Mech. Sci. 46 (2004) 57. J.B.J. Fourier, Theorie Analytique De La Chaleur. Paris, 1822. C. Cattaneo, C. R. 247 (1958) 431.