Research Article

Generalized Thermoelastic Interactions Using an Eigenvalue Technique in a Transversely Isotropic Unbounded Medium with Memory Having a Line Heat Source

Tanmoy Seth ^a , Md Abul Kashim Molla ^b, Sadek Hossain Mallik ^{a *}

^a Department of Mathematics and Statistics, Aliah University, Kolkata-700160, India

^b Government General Degree College, Tehatta, Nadia- 741160, India

1.     Introduction

The heat conduction equation of the classical theory of thermoelasticity does not accommodate any elastic term and it is parabolic in nature. It denies the practical observation of heat generation due to elastic changes and it also recommends boundless speed of the thermal wave propagation. The generalized theory of thermoelasticity prevails over these major imperfections of the classical theory of thermoelasticity. Towards the formulation of generalized theory of thermoelasticity, many pioneers have their valuable contributions of which we mention here some of them [1-9]. Ezzat [10] has solved a thermoelastic problem with two relaxation times in cylindrical regions. Youssef [11] has studied a thermoelastic problem in an infinite medium with a cylindrical cavity containing a moving heat source. Lotfy et al. [12, 13] have solved generalized thermoelastic problems for functionally graded and piezo-photo-thermoelastic materials respectively. Lotfy and Hassan [14] have investigated the propagation of thermoelastic waves within the purview of the Lord-Shulman two-temperature generalized thermoelasticity. Yasein et al. [15] have discussed the effect of variable temperature and thermal conductivity for semiconducting elastic medium in the context of dual-phase-lag and Lord-Shulman model. Mahdy et al. [16] have investigated the interaction impact between elastic waves, plasma waves, and thermal waves in hyperbolic generalized two-temperature theory. Abouelregal et al. [17–19] have solved various generalized thermoelastic problems considering Moore–Gibson–Thompson generalized thermoelastic theory. The generalized theory of thermoelasticity is widely implemented to tackle various problems with high heat flux for short time intervals which generally appears in nuclear reactors, energy channels, LASER units, etc.

Fractional calculus is an important tool for solving different physical problems in the field of generalized thermoelasticity because of the non-local property of the fractional operator. The fractional operator carries the previous state together with the current state of a dynamical system. Ezzat et al. [20] have introduced a unified mathematical model of heat conduction with three-phase lag thermoelasticity using fractional calculus. Abbas [21] has solved a thermoelastic problem for an infinite body with a spherical cavity in the context of fractional-order thermoelasticity. For an unbounded generalized thermoelastic media with a spherical cavity, Molla et al. [22] have studied the effects of fractional parameters on stress distributions. Lotfy et al. [23] have studied the effects of variable thermal conductivity in a magnetophotothermal elastic medium in the context of fractional calculus. Sufficient works in the field of generalized thermoelasticity have been performed by several researchers using fractional calculus, of which we mention [24-28] to name but a few.

Wang and Li [29] have launched the notion of memory-dependent derivatives in place of fractional derivatives [30]. The memory-dependent derivative (MDD) is defined in an integral form of a common derivative with a kernel function on a slipping interval. The delayed time intervals and the kernel function for MDD can be chosen freely whereas they are fixed in fractional derivatives. The order of the derivative in fractional calculus is necessarily a fraction but in MDD the order of the derivative is simply an integer which leads to easier numerical calculations. Taking all such points into consideration it is easy to guess that this kind of definition is better than a fractional one for reflecting the memory effect and this definition leads to recognizing a more intuitionistic physical explanation. According to them m-th order memory-dependent derivative of a differentiable function f(t) relative to the delay time ω > 0 is defined as follows:

where  is the kernel function which can be chosen freely and the kernel  must be a differentiable function with respect to its arguments. When =1 and then  i.e. m-th order common derivative of

Yu et al. [31] have recently formulated a new generalized thermoelastic model using MDD [29]. El-karamany and Ezzat [32] have derived the variational principle, reciprocity theorem, and uniqueness of solutions in a thermodiffusive medium in the context of MDD. Ezzat et al. [33] have established a new generalized thermoelasticity model based on MDD. Sur et al. [34] have solved a physical problem in fiber-reinforced cylinders using MDD by finite element method. Sarkar et al. [35] have solved a two-dimensional problem in the context of memory-dependent two-temperature generalized thermoelastic model. Biswas et al. [36] examined the effect of the magnetic field in an orthotropic medium with a phase-lag model based on MDD. Abouelregal and Dargail [37] have introduced a new mathematical model for functionally graded thermoelastic nanobeams (FGNB) with MDD due to a periodic heat flux. Mondal and Sur [38] have studied the memory effects in a functionally graded magneto-thermoelastic rod. Mondal et al. [39] have studied the phase lag effect for a two-temperature generalized piezo thermoelastic problem in the context of memory-dependent derivatives. For more works using memory-dependent derivatives, one can go through the work of El-Attar et al. [40] Sarkar and Othman [41], Sur et al. [42], and Abouelregal et al. [43].

Memory-dependent generalized thermoelasticity which provides an alternative approach to describe memory-dependence that has been commonly depicted by fractional generalized thermoelasticity is comparatively new. To the best of our knowledge, only a few works have been reported in the literature to date that adopt generalized thermoelasticity theory with memory-dependent derivative and eigenvalue approach in solving the generalized thermoelastic problems.

The intent of this article is to look over thermoelastic interactions in a homogeneous, linear, transversely isotropic unbounded continuum in the presence of a line heat source. The exploration has been unified and carried out in the context of Lord-Shulman [5] and Green-Lindsay [1] models with the aid of memory-dependent derivatives. A cylindrical polar coordinates system has been used to trace the problem and the eigenvalue technique [44] is employed to solve the governing field equations in the transformed domain of Laplace. Using the Stehfest method [45] for numerical Laplace inversion the solution for stress, displacement, and temperature is brought to the real space-time domain. The obtained numerical data for the above-mentioned thermophysical quantities are plotted in graphs to look over the impacts of the time delay parameter and the different kernel functions and a comparison between the considered models has been accomplished. Results of analogous problems with integer order thermoelasticity theory realizable from [46] can also be achievable as a special case of our results. Also, results of analogous problems for isotropic material can be easily deduced from the corresponding results of this article.

2.     Basic Governing Equations

The governing equations for generalized thermoelasticity with memory-dependent effect and relaxation times for a linear, homogeneous, and anisotropic medium are [33, 47]

• strain-displacement relations:

• the constitutive equations:

• stress-equations of motion:

• the heat conduction equation:

where  respectively stand for the body force per unit mass, the mass density of the medium, the mechanical displacement, the strain tensor, the stress tensor, elasticity tensor and  is the thermal elastic coupling tensor;  is the specific heat at constant strain, Q is the heat source per unit mass, , ,  are thermal relaxation parameters,  is the thermal conductivity tensor, θ is the change in temperature above the reference temperature ,  denotes first order memory dependent derivative, subscript comma represents material derivative with respect to space variable and the superscript dot represents derivative with respect to time.

Special Cases:

1. If we consider = =  = 0 then the governing equations  -  reduce to the governing equations of classical coupled thermoelasticity (CCTE) [48].
2. When , then the governing equations  -  reduce to the governing equations of Lord- Shulman model which is also known as extended thermoelasticity (ETE) [5]. does 0, 0,  = 0 then the governing equations  -  reduce to the governing equations of Green-Lindsay model which is also known as temperature rate dependent thermoelasticity (TRDTE) [1].

3.     Formulation of the Problem

We consider an unbounded, linear thermoelastic continuum that is homogeneous and transversely isotropic in nature and is influenced under the action of a line heat source acting along the -axis of the cylindrical polar coordinates system  used for the description of the problem. We assume that the problem is axisymmetric and hence the displacement, and temperature are functions of  and  only. Therefore the displacement vector possesses only radial component  =  and the stress tensor possesses only radial component  and transverse component .

As a consequence, the equations of motion, and heat conduction equation take the following forms respectively

and constitutive equations can be obtained as follows

where ,  are the material constants,  is the coupling parameter.

We introduce the following dimension-free variables

where

,

Using these dimension free variables we get the above equations (suppressing the asterisks for simplicity) as

where   ,

Here we choose the kernel function  as follows:

 and  being constants.

4.     Solution of the Problem

To solve the problem we take the Laplace transform defined by

On equations ,    and  to get

where

Using equation  equation  can be written as

We now write equations  and  in vector-matrix differential equation as follows:

where

is the Bessel operator, , A=
,

and

If  and  be the eigenvalues of the matrix, hen the eigenvectors  and  corresponding to the respective eigenvalues and are given by

Let

where

and hence

Now equations  and  together yield

which can also be written in scalar form as follows

where we have assumed

Let ,  ( = 1, 2) be two linearly independent solutions of the homogeneous equations corresponding to the equations in (  then

and the complementary functions  and  of the equations in  are given by

where   = (  = 1, 2) and ,  respectively denote the modified Bessel functions of first and second kind of order one.

In absence of body force (i.e when ), the equation  gives

,

which implies

Particular integrals  and  corresponding to the equations in (30) are given by

where

We now choose the line heat source in the form,

which acts along ; ,  representing the well-known Dirac delta function and Heaviside unit step function respectively and  is a constant.

Taking Laplace transform of the equation  we get

and hence particular integrals are obtained from  using  and  as follows

where we have disregarded the second term in  for the sake of bounded solution. Again, since the heat source is acting along the  axis, taking  the particular integrals of equations in  are obtained from  as follows

Furthermore, since the medium is infinte and  become unbounded as  and as  respectively, we set  in equation (34). Thus the solutions for the equations in (30) are given by

Now the solution of  are obtained using  in the following form

From this we get

where

Integrating equation  we obtain

Using equations  and  in    we get respectively

and

All results of [46] corresponding to the fractional parameter  = 1 can be derived from our results by taking  → 0 and .

Again the results for isotropic material can be derived from our results by simply taking  and  in our calculations, where ,  are Lame’s constant,  is the coefficient of linear thermal expansion for isotropic material.

5.     Numerical Results and Discussion

For numerical solutions, we have considered a single crystal of zinc and the parameters are chosen as follows [21]:

N/m²,  

N/m², , K,

W/mK, N/m²K,

Kg/m³.

Using MATLAB software numerical solutions for displacement, temperature and radial stress are obtained in the time domain by the Stehfest method [45]. The numerically obtained solutions are plotted in graphs to study the influence of the delay time parameter and the kernel functions on the above-mentioned thermophysical quantities.

Fig. 1. Variation of displacement  with  for different theories at  = 0.5,  = 0.1,  = 1,  = 1.

Figure 1 exhibits the variation of displacement u along r for different theories in the range 0 <  < 0.8 for  = 0.5,  = 0.1,  = 1,  = 1 and it shows that the magnitude of the displacement component  corresponding to TRDTE model is larger compare to ETE model which has again greater magnitude than CCTE model. In all the curves magnitude of  gradually increases in 0 <  < 0.1 (approx) and then decreases to vanish.

Fig. 2. Variation of displacement  with  for different theories at  = 0.5,  = 0.3,  = 0,  = 0.

Figure 2 indicates the variation of displacement u with the time t for different theories when  = 0.5,  = 0,  = 0,  = 0.3 in the range 0 <  < 1. We see that the initial value of u for each of the three models is the same. With the increase of time t the values of u gradually increase to achieve larger values in the TRDTE model compared to the ETE model which again has larger values compared to the CCTE model and finally becomes stable in all cases.

Fig. 3. Variation of temperature  with  for different theories at  = 0.5,  = 0.1,  = 1,  = 1.

Figure 3 shows the variation of temperature  along r for different theories in the range 0 <  < 0.8 when  = 0.5,  = 1,  = 1,  = 0.1. From the graph, it is clear that the magnitude of  corresponding to the ETE model is the greatest and that corresponding to the TRDTE model is the least.

Fig. 4. Variation of temperature  with  for different theories at  = 0.5,  = 0.3,  = 0,  = 0.

Figure 4 represents the variation of temperature  with the time t for different theories when  = 0.5, = 0.3,  = 0,  = 0 in the range 0 <  < 1. From the figure, we see that the magnitude of the corresponding to the ETE model is the greatest and that corresponding to the TRDTE model is the least.

Fig. 5. Variation of radial stress  with  for different theories at  = 0.5,  = 0.1,  = 1,  = 1.

Figure 5 indicates the variation of radial stress  with  for different theories at  = 0.5,  = 0.1,  = 1,  = 1 in the range 0 <  < 0.8. We observe that the magnitude of radial stress  for each model is decreasing in nature and eventually vanishes. The maximum of the radial stress  for each model is attained near the center, which is the largest for the TRDTE model and the smallest for the CCTE model.

Fig. 6. Variation of radial stress  with  for different theories at  = 0.5,  = 0.3,  = 0,  = 0.

Figure 6 exhibits the variation of radial stress   with  for different theories when  = 0.5,
 = 0.3,  = 0,  = 0 in the interval 0 <  < 1. We see that the magnitude of  corresponding to the CCTE model is the least and that corresponding to the TRDTE model is the greatest. Graphs corresponding to different thermoelastic models are distinguishable in the interval 0.1 ≤  ≤ 0.8 (approx) and after which they become stable.

Fig. 7. Variation of displacement  with  for different values of ω in TRDTE model at  = 0.1,  = 0,  = 0.

Figure 7 represents the variation of displacement  along  for different values of  in the TRDTE model at  = 0.1,  = 0,  = 0 in the range 0 <  < 0.8. For any choice of  distribution of  have increasing nature near the origin and after that they decay to vanish.

Fig. 8. Variation of displacement  with for different kernel functions in TRDTE model at  = 0.5, = 0.1.

Figure 8 demonstrates that the variation of displacement u with r for different kernel functions in the TRDTE model at  = 0.5,  = 0.1 in 0 <  < 0.8. For every choice of the kernel function, the distribution of  has a uniform increasing nature in 0 <  ≤ 0.1 (approx) and has a uniform decreasing nature in 0.1 ≤  ≤ 0.66 (approx). It is observed that the curve corresponding to the nonlinear kernel (  = 1,
 = 1) has greater magnitude compared to all linear kernels. For graphs corresponding to linear kernels, we see that magnitudes are greater for smaller values of the parameter .

Fig. 9. Variation of temperature  with  for different delay time parameter  in TRDTE model at  = 0.1,  = 0,  = 0.

Figure 9 indicates the variation of temperature  along  for different delay time parameter  in the TRDTE model when  = 0.1,
 = 0,  = 0 for the interval 0 <  < 0.8. We see that the value of  for each chosen delay time parameter is the same near origin and with larger value of  the curve travels the larger distance vanishes.

Fig. 10. Variation of temperature  with  for different kernel functions in TRDTE model at  = 0.5,  = 0.3.

Figure 10 exhibits the variation of temperature  along  for different kernel functions in the TRDTE model at  = 0.5,  = 0.3 for the interval 0 <  < 0.8. It is clear that for every chosen kernel function, the temperature  attains the greatest value near the origin and uniformly decreases nature. For the nonlinear kernel
(  = 1,  = 1)  travels the smaller distance to vanish. Among the linear kernels the magnitudes are greater for greater value of the parameter  and travels the larger distance to vanish for larger value of the parameter .

Fig. 11. Variation of the radial stress  with  for different delay time parameter  in TRDTE model at  = 0.1,  = 0,  = 0.

Figure 11 shows that the variation of the radial stress  versus radial distance r for different delay time parameter  in TRDTE model at  = 0.1,  = 0,  = 0 in the range
0 <  < 0.8. We see that the radial stress  is compressive in nature for all considered . Each values of  curve has oscillatory behavior near the origin and then uniformly decrease to vanish. With larger values of  the length of the intervals in which oscillatory behavior appears diminishes.

Fig. 12. Variation of the radial stress  with  for different kernel functions in TRDTE model at  = 0.5,  = 0.1.

Figure 12 indicates the variation of the radial stress  versus radial displacement  for different kernel functions in TRDTE model at
 = 0.5,  = 0.1 in the range 0 <  < 0.8. From the graph we see that for any choice of the kernel function magnitude of the radial stress is decreasing in nature and eventually vanishes. Its magnitude has larger value for non linear kernel (  = 1,  = 1). For linear kernels magnitudes are greater for smaller values of the parameter .

6.     Conclusions

The problem of investigating the displacement  temperature  and the radial stress  in linear, homogeneous and transversely isotropic infinitely extended thermoelastic continuum under the influence of line heat source has been studied for different models in the context of memory dependent derivative with aid of eigenvalue approach. Cylindrical polar coordinates system  has been employed for the description of the problem and solution is obtained in transformed domain of Laplace. Stehfest's [45] method is used for numerical Laplace inversion of the above-said solution. The analysis of the obtained results permits us to derive the following conclusions:

1. Significant effects of different considered models, delay time parameters, and different choices of kernel functions are observed in the distribution of displacement component temperature  and the radial stress component
2. In the region of consideration of the magnitude of displacement component  corresponding to the TRDTE model is greatest and that corresponding to the CCTE model is the smallest. A similar observation appears in the considered interval of time.
3. The magnitude of temperature corresponding to the ETE model is the largest and that corresponding to the TRDTE model is the smallest and this observation is true in both the region of consideration as well as in the interval of consideration.
4. Radial stress component is compressive in nature. The magnitude of corresponding to the TRDTE model is the largest and that corresponding to the CCTE model is the smallest. This observation is true in the region of consideration as well as in the interval of consideration.
5. Displacement component and the radial stress component  travels longest distance to vanish for the nonlinear kernel  but this phenomenon happens for temperature distribution  for the case of linear kernel function when  = 0,
6. Displacement component and the radial stress component  travels longest distance to vanish for the smallest value of ω, but for temperature distribution  the curve corresponding to largest value of  vanishes at longest distance.
7. As the paper deals with thermoelastic interactions in the context of memory-dependent derivative, it describes the behavior of an elastic material more realistically than the theory of thermoelasticity with fractional as well as integer one.
8. All results of [46] corresponding to the fractional parameter = 1 can be derived from our results by taking  → 0 and .
9. The results for isotropic material can be derived from our results by simply taking
   and  in our calculations, where λ, μ are Lame’s constant,  is the coefficient of linear thermal expansion for isotropic material.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their helpful comments and suggestions for the improvement of this paper.

Funding Statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this manuscript.

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