Document Type : Research Paper
Authors
^{1} Department of Mathematics and Statistics, Aliah University, Kolkata700160, India
^{2} Government General Degree College, Tehatta, Nadia 741160, India
Abstract
Keywords
Main Subjects
Research Article
Tanmoy Seth ^{a } , Md Abul Kashim Molla ^{b}, Sadek Hossain Mallik ^{a * }
^{a} Department of Mathematics and Statistics, Aliah University, Kolkata700160, India
^{b} Government General Degree College, Tehatta, Nadia 741160, India
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20230929 Revised: 20240116 Accepted: 20240312 

The present article looks over thermoelastic interactions in a homogeneous, linear, transversely isotropic unbounded continuum with the aid of memorydependent derivatives in the presence of a line heat source. The exploration has been unifiedly carried out in the context of GreenLindsay and LordShulman models. A cylindrical polar coordinates system has been used to describe the problem and the eigenvalue technique has been adopted to solve the governing field equations in the transformed domain of Laplace. The solution for different thermophysical quantities is obtained in the real spacetime domain using the Stehfest method for numerical Laplace inversion. The obtained numerical data for different thermophysical quantities are plotted in graphs to investigate the impacts of the time delay parameter, and the different kernel functions, and a comparison between the considered models has been accomplished. It is worth mentioning that the results of an analogous problem for isotropic material can be easily deduced from the corresponding results of this article. The adoption of generalized thermoelastic theory with memorydependent derivative along with eigenvalue technique in analyzing the thermoelastic interactions is relatively fresh. 



Keywords: MemoryDependent derivative; Eigen value approach; VectorMatrix differential equation; Stehfest method. 


© 2024 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an open access article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
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 [19]. 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 piezophotothermoelastic materials respectively. Lotfy and Hassan [14] have investigated the propagation of thermoelastic waves within the purview of the LordShulman twotemperature generalized thermoelasticity. Yasein et al. [15] have discussed the effect of variable temperature and thermal conductivity for semiconducting elastic medium in the context of dualphaselag and LordShulman model. Mahdy et al. [16] have investigated the interaction impact between elastic waves, plasma waves, and thermal waves in hyperbolic generalized twotemperature 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 nonlocal 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 threephase lag thermoelasticity using fractional calculus. Abbas [21] has solved a thermoelastic problem for an infinite body with a spherical cavity in the context of fractionalorder 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 [2428] to name but a few.
Wang and Li [29] have launched the notion of memorydependent derivatives in place of fractional derivatives [30]. The memorydependent 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 mth order memorydependent derivative of a differentiable function f(t) relative to the delay time ω > 0 is defined as follows:

(1) 
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. mth order common derivative of
Yu et al. [31] have recently formulated a new generalized thermoelastic model using MDD [29]. Elkaramany 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 fiberreinforced cylinders using MDD by finite element method. Sarkar et al. [35] have solved a twodimensional problem in the context of memorydependent twotemperature generalized thermoelastic model. Biswas et al. [36] examined the effect of the magnetic field in an orthotropic medium with a phaselag 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 magnetothermoelastic rod. Mondal et al. [39] have studied the phase lag effect for a twotemperature generalized piezo thermoelastic problem in the context of memorydependent derivatives. For more works using memorydependent derivatives, one can go through the work of ElAttar et al. [40] Sarkar and Othman [41], Sur et al. [42], and Abouelregal et al. [43].
Memorydependent generalized thermoelasticity which provides an alternative approach to describe memorydependence 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 memorydependent 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 LordShulman [5] and GreenLindsay [1] models with the aid of memorydependent 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 spacetime domain. The obtained numerical data for the abovementioned 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.
The governing equations for generalized thermoelasticity with memorydependent effect and relaxation times for a linear, homogeneous, and anisotropic medium are [33, 47]

(2) 

(3) 

(4) 

(5) 
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:
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

(6) 


and constitutive equations can be obtained as follows
, 
(8) 


where , are the material constants, is the coupling parameter.
We introduce the following dimensionfree variables

(10) 
where
,
Using these dimension free variables we get the above equations (suppressing the asterisks for simplicity) as
, 
(11) 
, 
(12) 
where ,

(13) 

(14) 
Here we choose the kernel function as follows:

(15) 

and being constants.
To solve the problem we take the Laplace transform defined by
, Re( , 
(16) 
On equations , and to get
, 
(17) 
, 


(19) 

(20) 
where

(21) 
Using equation equation can be written as

(22) 
We now write equations and in vectormatrix differential equation as follows:

(23) 
where

(24) 
is the Bessel operator, , A=
,
and

(25) 
If and be the eigenvalues of the matrix, hen the eigenvectors and corresponding to the respective eigenvalues and are given by
, . 
(26) 
Let
then 


(27) 
where
and hence
. 
(28) 
Now equations and together yield

(29) 
which can also be written in scalar form as follows

(30) 
where we have assumed
, 
(31) 

(32) 
Let , ( = 1, 2) be two linearly independent solutions of the homogeneous equations corresponding to the equations in ( then
, , 
(33) 
and the complementary functions and of the equations in are given by
(λ_{2} r 
(34) 
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

(35) 
Particular integrals and corresponding to the equations in (30) are given by

(36) 
where
We now choose the line heat source in the form,

(37) 
which acts along ; , representing the wellknown Dirac delta function and Heaviside unit step function respectively and is a constant.
Taking Laplace transform of the equation we get

(38) 
and hence particular integrals are obtained from using and as follows

(39) 
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

(40) 
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

(41) 
Now the solution of are obtained using in the following form
. 


(42) 
From this we get
, 
(43) 
, 
(44) 
where
Integrating equation we obtain

(45) 
Using equations and in we get respectively

(46) 
and

(47) 
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.
For numerical solutions, we have considered a single crystal of zinc and the parameters are chosen as follows [21]:
N/m^{2},
N/m^{2}, , K,
W/mK, N/m^{2}K,
Kg/m^{3}.
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 abovementioned 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 .
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 abovesaid solution. The analysis of the obtained results permits us to derive the following conclusions:
The authors would like to express their sincere thanks to the referees for their helpful comments and suggestions for the improvement of this paper.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
The author declares that there is no conflict of interest regarding the publication of this manuscript.
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