## Dynamic Problem in Thermoelastic Solid Using Dual-Phase-Lag Model with Internal Heat Source

**Praveen Ailawalia**^{1,}, **Shilpy Budhiraja**^{2}

^{1}Department of Applied Sciences, Baddi University of Emerging Sciences and Technology, Makhnumajra, Baddi, Solan, H.P(INDIA)

^{2}Research Scholar Punjab Technical University, Jalandhar, Punjab(INDIA)

### Abstract

The dual-phase lag heat transfer model is employed to study the problem of isotropic generalized thermoelastic medium with internal heat source. The force is acting along the interface of isotropic generalized thermoelastic medium and the elastic layer of depth h. The normal mode analysis is used to obtain the exact expressions for displacement components, force stress and temperature distribution. The variations of the considered variables through the horizontal distance are illustrated graphically. The results are discussed and depicted graphically.

### At a glance: Figures

**Keywords:** Dual-phase-lag model, thermoelasticity, temperature distribution, normal-mode

*Journal of Mathematical Sciences and Applications*, 2014 2 (1),
pp 10-16.

DOI: 10.12691/jmsa-2-1-3

Received September 18, 2013; Revised February 28, 2014; Accepted March 04, 2014

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Ailawalia, Praveen, and Shilpy Budhiraja. "Dynamic Problem in Thermoelastic Solid Using Dual-Phase-Lag Model with Internal Heat Source."
*Journal of Mathematical Sciences and Applications*2.1 (2014): 10-16.

- Ailawalia, P. , & Budhiraja, S. (2014). Dynamic Problem in Thermoelastic Solid Using Dual-Phase-Lag Model with Internal Heat Source.
*Journal of Mathematical Sciences and Applications*,*2*(1), 10-16.

- Ailawalia, Praveen, and Shilpy Budhiraja. "Dynamic Problem in Thermoelastic Solid Using Dual-Phase-Lag Model with Internal Heat Source."
*Journal of Mathematical Sciences and Applications*2, no. 1 (2014): 10-16.

Import into BibTeX | Import into EndNote | Import into RefMan | Import into RefWorks |

### 1. Introduction

Thermoelasticity theories which involve finite speed of thermal signals (second sound) have created much interset during the last three decades. The conventional coupled dynamic thermoelasticity theory (CTE) based on the mixed parabolichyperbolic governing equations of (Biot ^{[1]}; Chadwick ^{[2]}) predicts an infinite speed of propagation of thermoelastic disturbance. To remove the paradox of infinite speed for propagation of thermoelastic disturbance, several generalized thermoelasticity theories have been developed, which involve hyperbolic governing equations. Among these generalized theories, the extended thermoelasticity theory (ETE) proposed by Lord and Shulman^{[3]} involving one relaxation time (called single-phase-lag-model) and the temperature-rate-dependent theory of thermoelasticity (TRDTE) proposed by Green and Lindsay ^{[4]} involving two relaxation times are two important models of generalized theory of thermoelasticity. Experimental studies (Kaminski ^{[5]}, Mitra et al. ^{[6]}, Tzou ^{[7, 8]}) indicate that the relaxation times can be of relevance in the cases involving a rapidly propagating crack tip, a localized moving heat source with high intensity, shock wave propagation, laser technique etc. Because of the experimental evidence in support of finiteness of heat propagation speed, the generalized thermoelasticity theories are considered to be more realistic than the conventional theory in dealing with practical problems involving very large heat fluxes at short intervals like those occurring in laser units and energy channels. For a review of the relevant literature, see (Chandrasekharaiah ^{[9]}, Ignaczak ^{[10]}).

Green and Naghdi ^{[11, 12, 13]} formulated three different models of thermoelasticity among which, in one of these models, there is no dissipation of thermoelastic energy. This model is referred to as the G-N model of thermoelasticity without energy dissipation (TEWOED). Problems concerning generalized thermoelasticity theories and G-N theory have been studied by many authors (RoyChoudhuri and Debnath ^{[14]}, RoyChoudhuri ^{[15, 16, 17]}, Dhaliwal and Rokne ^{[18, 19]}, RoyChoudhuri ^{[20]}, Chandrasekharaiah and Murthy ^{[21]}, Chandrasekharaiah and Srinath ^{[22]}, RoyChoudhuri and Banerjee ^{[23]}, RoyChoudhuri and Bandyopadhyay ^{[24]}, Roy- Choudhuri and Dutta ^{[25]}). Tzou ^{[7, 8]} and Ozisik and Tzou ^{[26]} have developed a new model called dual-phase-lag model for heat transport mechanism in which Fourier’s law is replaced by an approximation to a modification of Fourier’s law with two different time translations for the heat flux and the temperature gradient. According to this model, classical Fourier’s law has been generalized as , where the temperature gradient at a point P of the material at time corresponds to the heat flux vector at the same point at the time . Here K^{∗} is the thermal conductivity of the material. The delay time is interpreted as that caused by the microstructural interactions (small-scale heat transport mechanisms occurring in microscale) and is called the phase-lag-of the temperature gradient. The other delay time is interpreted as the relaxation time due to the fast transient effects of thermal inertia (small scale effects of heat transport in time) and is called the phase-lag of the heat flux. If and , Tzou ^{[7, 8]} refers to the model as the single phase-lag model. The case corresponds to the dual phase-lag model of the constitutive equation connecting the heat flux vector and the temperature gradient. The case becomes identical with the classical Fourier’s law. Further for materials with , the heat flux vector is the result of a temperature gradient and for materials with , the temperature gradient is the result of a heat flux vector. For a review of the relevant literature, see (Chandrasekharaiah ^{[27]}). A hyperbolic thermoelastic model was developed in this same reference, taking into account the phase-lag of both temperature gradient and heat flux vector and also the second order term in in Taylor’s expansion of the heat flux vector and the first order term in in Taylor’s expansion of the temperature gradient in the generalization of classical Fourier’s law. It may be pointed out that ETE was formulated by taking into account the thermal relaxation time, which is in fact the phase-lag of the heat flux vector (single-phase-lag model).

Abouelregal ^{[28]} studied Rayleigh waves in a thermoelastic solid half-space using dual-phase-lag model. Kumar and Chawla ^{[29]} investigated plane wave propagation in anisotropic three-phase-lag and two-phase-lag model. Mukhopadhyay et al. ^{[30]} discussed the theory of two-temperature thermoelasticity with two-phase lags. Chakravorty and Chakravorty ^{[31]} discussed the transient disturbances in a relaxing thermoelastic half space due to moving stable internal heat source. Kumar and Devi ^{[32]} studied thermomechanical interactions in porous generalized thermoelastic material permeated with heat source. Lotfy ^{[33]} studied the transient disturbance in a half-space under generalized magneto-thermoelasticity with a stable internal heat source. Lotfy ^{[34]} discussed the transient thermo-elastic disturbances in a visco-elastic semi-space due to moving internal heat source. Othman ^{[35]} studied the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources.

The present paper is concerned with the investigation related to the effect of dual phase-lag subjected to a normal force at the interface of elastic layer and isotropic generalized thermoelastic medium with internal heat source. The normal mode method is used to obtain the exact expressions for the considered variables. The distributions of the considered variables are represented graphically.

### 2 Formulation of the Problem and Fundamental Equations

We consider a elastic layer (depth h) lying over the surface of a linear homogeneous isotropic, thermally conducting elastic half space. All quantities considered are functions of the time variable t and of the coordinates x and y. A rectangular coordinate system (x, y, z) with y-axis pointing vertically downward is considered. The region y > 0 is accupied by isotropic generalized thermoelastic elastic medium(medium I) and the region −h < y < 0 represents the elastic layer(medium II). The plane y = 0 represents the interface of medium I and medium II.

The field equations and constitutive relations for a homogeneous, generalized thermoelastic solid in the absence of incremental body forces and heat sources

(1) |

(2) |

(3) |

Then the heat conduction equation in the context of dual phase lag thermoelasticity proposed by Tzou in this case takes the form

(4) |

For medium I (isotropic generalized thermoelastic medium) replace by I and for medium II (elastic layer of depth h) replace by II.

### 3. Solution of the Problem

If we restrict our analysis parallel to xy plane and , the displacement components have the following form

(5) |

From Equations (2) and (5), we obtain the strain components

(6) |

To facilitate the solution, following dimensionless quantities are introduced:

(7) |

where

Equation (3), with the help of equations (1) and (5)- (7) may be recast into the dimensionless form after suppressing the primes as:

(8) |

(9) |

The heat conduction equation is given by

(10) |

Using the expression relating displacement components , , to the scalar potential functions and

(11) |

in equations (8) - (10), we obtain

(12) |

(13) |

(14) |

where

### 4. Normal Mode Analysis

The solution of the considered physical variable can be decomposed in terms of normal modes as the following form

(15) |

(16) |

where are the magnitude of the functions, ω is the complex time constant and a is the wave number in x-direction and Q_{0} is the magnitude of stable internal heat source.

Using (15) - (16), in equations (12) - (14) we obtain,

(17) |

(18) |

(19) |

where

Eliminating T from equations (17) - (18), we obtain,

(20) |

where,

The solution of equation(20) is given by:

(21) |

In a similar way, we get

(22) |

The solution of equation (19) is given by:

(23) |

where

(24) |

where , are some parameters depending on a and ω. are the roots of the characteristic equation (20).

Neglecting thermal effect in equation (1), we obtain elastic layer of depth h. Adopting the same approach, the displacement components and stresses , in the elastic layer (i.e for −h ≤ y ≤ 0) is given by

(25) |

(26) |

(27) |

(28) |

where

### 5. Applications

The boundary conditions at the interface y = 0 subjected to an arbitrary normal force P1 are

(29) |

where P_{1} is the magnitude of mechanical force. Using equations(1) and (7) on the non-dimensional boundary conditions and then using (21) - (23), we get the expressions of displacement, force stress and temperature distributions for isotropic generalized thermoelastic medium as,

(30) |

(31) |

(32) |

(33) |

(34) |

where

Invoking the boundary conditions (29) at the surface y = 0, we obtain a system of seven equations, and applying the inverse of matrix method, we have the value of seven constants S_{j}, j = 1, 2, 3, ..... 7.

where , , i = 1, 2, 3, .., 7. are defined in appendix A.

### 6. Numerical Results

For computational work, to illustrate the analytical procedure presented earlier, we consider now a numerical example. The results depict the variations of displacements, force stress and temperature distribution. For this purpose, sand stone is considered as the thermoelastic material body for which we have the physical constants as follows

= 2.30*gm*/(*cm*)3, = 0.4 × 10^{−}^{5}/(*degC*),

= 0.8 × 1011 *dyne*/*cm*^{2},

K^{*}= 0.6 × 10^{−}^{2}*cal*/*cmsecdegC*, C_{E} = 0.23*cal*/*gmdegC*, T_{0} = 296*k*.

The physical constants for elastic layer are given by Bullen ^{[36]},

= 0.884×1011*dyne*/*cm*^{2},

= 1.2667×1011*dyne*/*cm*^{2}, = 2.6*gm*/*cm*^{3}.

The computations are carried out in the range 0 ≤ x ≤ 10 and on the surface y = 1.0. The numerical values for normal displacement u_{2} , normal force stress t22 and temperature distribution T are shown in figure 1- figure 6 for mechanical force with , *a* = 2.1 for *a*

a. Isotropic generalized thermoelastic medium with internal heat source ( = 2 and =2) by solid line.

b. Isotropic generalized thermoelastic medium with internal heat source ( = 4 and =2) by dashed line.

c. Isotropic generalized thermoelastic medium with internal heat source ( = 6 and = 2) by solid line with centered symbol (*).

d. Isotropic generalized thermoelastic medium with internal heat source ( = 8 and = 2) by dashed line with centered symbol (*).

e. Isotropic generalized thermoelastic medium with internal heat source ( = 1 and = 4) by solid line.

f. Isotropic generalized thermoelastic medium with internal heat source ( = 2 and = 4) by dashed line.

g. Isotropic generalized thermoelastic medium with internal heat source ( = 3 and = 4) by solid line with centered symbol (*).

h. Isotropic generalized thermoelastic medium with internal heat source ( = 4 and = 4) by dashed line with centered symbol (*).

**Figure 1.**Variations of Normal displacement u

_{2}with dlstance x for phase-lag of heat flux

**Figure**

**2**

**.**Variations of Normal force t

_{22}with dlstance x for phase-lag of heat flux

**Figure**

**3**

**.**Variations of Temperture distribution T with dlstance x for phase-lag of heat flux

**Figure**

**4**

**.**Variations of Normal displacement u

_{8}with dlstance x for phase-lag of temperature gradient

**Figure**

**5**

**.**Variations of Normal force stress t

_{22}with dlstance x for phase-lag of temperature gradient

**Figure**

**6**

**.**Variations of Temperture distribution T with dlstance x for phase-lag of temperature gradient

### 7. Discussions

**7.1. Effect of Phase Lag of the Heat Flux (τ**

_{q})Figure 1 to Figure 3 gives the effect of the phase-lag of the heat flux (with fixed value of = 2).

Figure 1 depicts the variations of normal displacement u_{2} with distance x. The behaviour of normal displacement u_{2} with reference to x is same i.e. oscillatory for = 2, = 4 and = 8 with difference in their magnitude, whereas for = 4 and = 6 show opposite oscillatory pattern in the entire range which show the impact of phase-lag of the heat flux.

The variations of normal force stress t_{22} with distance x is depicted in figure 2.

The values of normal force stress t_{22} for = 2 and = 4 show similar patterns with different degree of sharpness. i.e. the values for = 2 and = 4 increases and decreases alternately with distance x. It is also noticed that the variations of normal force stress t_{22} for = 6 show oscillatory pattern. Figure 3 shows the variations of temperature distribution T with distance x.

The pattern observed for = 2 and = 4 are opposite in nature with fluctuating values which clearly reveals the effect of phase lag of the heat flux. The trend of variations for = 6 and = 8 are similar in nature in the entire interval, i.e. the values for = 6 and = 8 increases and decreases alternately with distance x.

**7.2. Effect of Phase-lag of Temperature Gradient (τ**

_{θ})Figure 4 to Figure 6 gives the effect of the phase-lag of temperature gradient (with fixed value of =4).

Figure 4 depicts the variations of normal displacement u_{2} with distance x. The variations of normal displacement u_{2} are comparable amongst themselves for = 1 and = 3. These variations are oscillatory in nature and opposite to the variations of normal displacement u_{2} for = 2 and = 4.

Figure 5 depicts the variations of normal force stress t22 with distance x. The pattern observed for = 1 and = 4 are opposite in nature near the point of application of the source but with increase in x, both assume similar pattern with fluctuating values. It is noticed that the variations of normal force stress t_{22} for 10 = 2 show oscillatory pattern about the origin. Also the pattern observed for = 3 and = 4 are opposite in nature with fluctuating values.

The variations of temperature distribution T with distance x is depicted in figure 6. The pattern observed for = 3 and = 4 are opposite in nature with fluctuating values. which clearly reveals the effect of phase-lag of temperature gradient, whereas for = 1 and = 3 shows similar oscillatory pattern with different degree of sharpness in magnitude. Further temperature distribution T shows small variations close to zero value in the whole range for = 2.

### 8. Conclusion

Appreciable effect of dual-phase-lag (DPL) model i.e. effect of phase-lag of heat flux () and effect of phase-lag of temperature gradient () is observed on the components of displacement, force stress and temperature distribution. The variations of Normal displacement u_{2} are uniform in nature in comparison to Normal force stress t_{22} and Temperature distribution T under the effect of . The normal mode analysis used in this article is applicable to wide range of problems in different branches. This method gives exact solutions without any assumed restrictions on either the temperature or stress distributions.

### Appendix A

### References

[1] | M.N. Biot, (1956). Thermoelasticity and irreversible thermodynamics. J. Appl. Phys.,Vol. 27 (3), pp. 240-253. | ||

In article | CrossRef | ||

[2] | P. Chadwick, (1960). In progress in solid mechanics, vol. I, edited by R. Hill and I.N.Sneddon, North Holland, Amsterdam. | ||

In article | |||

[3] | H.W. Lord and Y. Shulman, (1967). A generalized dynamical theory of thermoelasticity,J. Mech. Phys. Solids, Vol. 15 (5), pp. 299-309. | ||

In article | CrossRef | ||

[4] | A.E. Green and K.A. Lindsay, (1972). Thermoelasticity. J. Elasticity, Vol. 2 (1), pp. 1-7. | ||

In article | CrossRef | ||

[5] | W. Kaminski, (1990). Hyperbolic heat conduction equation for materials with a nonhomogenous inner structure. J. Heat Transf., Vol. 112, pp. 555-560. | ||

In article | CrossRef | ||

[6] | K. Mitra, S. Kumar and A. Vedaverz, (1995). Experimental evidence of hyperbolic heat conduction in processed meat. J. Heat Transf., Vol. 117, pp. 568-573. 12 | ||

In article | |||

[7] | D.Y. Tzou, (1995). Experimental support for the lagging behaviour in heat propagation. J. Thermophys. Heat Transf., Vol. 9(4), pp. 686-693. | ||

In article | |||

[8] | D.Y. Tzou, (1995). A unified approach for heat conduction from macro to microscale. J. Heat Transf., Vol. 117, pp. 8-16. | ||

In article | CrossRef | ||

[9] | D.S. Chandrasekharaiah, (1986). Thermo-elasticity with second sound. Appl. Mech. Rev., Vol. 39 (3), pp. 355-375. | ||

In article | CrossRef | ||

[10] | J. Ignaczak, (1989). In thermal stresses, vol. III, chap. 4, edited by R.B. Hetnarski, Elsevier, Oxford. | ||

In article | |||

[11] | A.E. Green and P.M. Naghdi, (1977). On thermodynamics and the nature of the second law. Proc. R. Soc. Lond. A, Vol. 357, pp. 253-270. | ||

In article | CrossRef | ||

[12] | A.E. Green and P.M. Naghdi, (1992). On undamped heat waves in an elastic solid. J. Therm. Stresses, Vol. 15, pp. 253-264. | ||

In article | CrossRef | ||

[13] | A.E. Green and P.M. Naghdi, (1993). Thermoelasticity without energy dissipation. J. Elasticity, Vol. 31 (3), pp. 189-208. | ||

In article | CrossRef | ||

[14] | S.K. Roy Chaudhuri and L. Debnath, (1983). Magneto- thermo-elastic plane waves in rotating media. Int. J. Eng. Sci., Vol. 21 (2), pp. 155-163. | ||

In article | CrossRef | ||

[15] | S.K. Roy Chaudhuri, (1984). Electro-megneto-thermo-elastic plane waves in rotating media with thermal relaxation. Int. J. Eng. Sci. Vol. 22 (5), pp. 519-530. | ||

In article | CrossRef | ||

[16] | S.K. Roy Chaudhuri, (1985). Effect of rotation and relaxation times on plane waves in generalized thermoelasticity. J. Elasticity, Vol. 15 (1), pp. 59-68. | ||

In article | CrossRef | ||

[17] | S.K. Roy Chaudhuri, (1987). On magneto thermo-elastic plane waves in infinite rotating media with thermal relaxation, pp. 361-366 in proceedings of the IUTAM Symposium on the Electromagnetomechanical Interactions in Deformable Solids and Structures (Tokyo, 1986), edited by Y. Yamamoto and K. Miya, North-Holland, Amsterdam. | ||

In article | |||

[18] | R.S. Dhaliwal and J.G. Rokne, (1988). One-dimensional generalized thermo-elastic problem for a half-space. J. Therm. Stresses, Vol 11, pp. 257-271. | ||

In article | CrossRef | ||

[19] | R.S. Dhaliwal and J. G. Rokne, (1989). One-dimensional thermal shock problem with two relaxation times. J. Therm. Stresses, Vol 12, pp. 259-279. | ||

In article | CrossRef | ||

[20] | S.K. Roy Chaudhuri, (1990). Magneto-thermo-micro-elastic plane waves in finitely conducting solids with thermal relaxation, pp. 461-468 in Proceedings of the IUTAM Symposium on Mechanical Modeling of New Electromagnetic Materials (Stockholm), edited by R.K.T. Hsieh, Elsevier, Amsterdam. | ||

In article | |||

[21] | D.S. Chandrasekharaiah and H.N. Murthy, (1993). Thermoelastic interactions in an unbounded body with a spherical cavity. J. Therm. Stresses, Vol. 16, pp. 55-70. | ||

In article | CrossRef | ||

[22] | D.S. Chandrasekharaiah and K.S.Srinath, (1996). One-dimensional waves in a thermoelastic half-space without energy dissipation. Int. J. Eng. Sci., Vol. 34 (13), pp. 1447-1455. 13. | ||

In article | |||

[23] | S.K. Roy Chaudhuri and M. Banerjee, (2004). Magnetoelastic plane waves in rotating media in thermoelasticity of Type II (G-N model). Int. J. Math. Math. Sci., Vol. 71, pp. 3917-3929. | ||

In article | CrossRef | ||

[24] | S.K. Roy Chaudhuri and N. Bandyopadhyay, (2005). Thermoelastic wave propagation in a rotating elastic medium without energy dissipation. Int. J. Math. Math. Sci., Vol. 1, pp. 99-107. | ||

In article | CrossRef | ||

[25] | S.K. Roy Chaudhuri and P.S. Dutta, (2005). Thermo-elastic interaction without energy dissipation in an infinite solid with distributed periodically varing heat sources. Int. J. Solids Struct., Vol. 42 (14), pp. 4192-4203. | ||

In article | CrossRef | ||

[26] | M.N. Ozisik and D.Y. Tzou, (1994). On the wave theory of heat conduction. J. Heat Transf.(ASME), Vol. 116, pp. 526-535. | ||

In article | CrossRef | ||

[27] | D.S. Chandrasekharaiah, (1998). Hyperbolic thermo-elasticity: a review of recent literature. Appl. Mech. Rev., Vol. 51 (12), pp. 705-729. | ||

In article | CrossRef | ||

[28] | A. E Abouelregal, (2011). Rayleigh waves in a thermoelastic solid half space using dual-phase-lag model. Int. J. Engg. Sci., Vol. 49, Issue 8, 781-791. | ||

In article | CrossRef | ||

[29] | Rajneesh Kumar and Vijay Chawla, (2011). A study of wave propagation in anisotropic three-phase-lag and two-phase-lag model. Int. Comm. Heat and Mass Transfer, Vol 38 (9), 1262-1268. | ||

In article | CrossRef | ||

[30] | SantwanaMukhopadhyay, Rajesh Prasad and Roushan Kumar, (2011). On the theory of Two-Temperature Thermoelasticity with Two phase-Lags. J. Thermal Stresses, Vol. 34 (4), 352-365. | ||

In article | CrossRef | ||

[31] | S. Chakravorty and A. Chakravorty, (1998). Transient disturbances in a relaxing thermoelastic half space due to moving stable internal heat source. Int. J. Math. And Math. Sci., Vol. 21, pp. 595-602. | ||

In article | CrossRef | ||

[32] | R. Kumar and S. Devi, (2008). Thermomechanical interactions in porous generalized thermoelastic material permeated with heat source. Multidiscipline Modeling in Mat. and Str., Vol. 4, pp. 237-254. | ||

In article | |||

[33] | Kh. Lotfy, (2010). Transient disturbance in a half-space under generalized magnetothermoelasticity with a stable internal heat source under three theories. Multidiscipline Modeling in Mat. and | ||

In article | |||

[34] | Kh. Lotfy, (2011). Transient thermo-elastic disturbances in a visco-elastic semi-space due to moving internal heat source. International Journal of Struc. Int., Vol. 2, pp. 264 - 280. | ||

In article | CrossRef | ||

[35] | M.I.A. Othman, (2011). State space approach to the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources. Journal of Appl. Mech. and Tech. Phys., Vol. 52, pp. 644-656. | ||

In article | CrossRef | ||

[36] | K.E. Bullen, (1985). An introduction to the theory of seismology. Combridge University Press. | ||

In article | |||