A Thermoelastic Piezoelectric Fixed Rod Exposed to an Axial Moving Heat Source via a Dual-Phase-Lag Model

Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt Department of Computer Science, Faculty of Computers and Information, Luxor University, Luxor, Egypt Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt


Introduction
Normally, in recent decades, more consideration has been given to the thermoelasticity field due to its applied aspects in various fields, especially geology, geophysics, structures, biology, physics, acoustics, plasma, and so forth. e generalized thermoelasticity theory is one of the improved forms of the classical thermoelasticity and has been modified to remove the defects of the classical coupled thermoelasticity. e modified thermoelasticity models were introduced by Lord and Shulman [1], Green and Lindsay [2], and Green and Naghdi [3] as well as by Tzou [4], Tzou [5], and Tzou [6]. On the other hand, many works, in literature, have applied generalized thermoelasticity theories to investigate the thermoelastic response of solid materials [7][8][9][10][11][12].
Piezoelectric materials are an important kind of smart material, as they are able to cause strain or generate electrical energy. Piezoelectric materials are essential materials that have the ability to convert electrical energy into mechanical energy to stimulate strain and act as an engine when these materials are powered by an electric field. Mostly, piezoelectric devices operate in a thermoelastic coupling medium. Often, piezoelectric devices and piezoelectric materials operate in a medium with pyroelectric and thermoelastic coupling.
A reverse problem of including interfaces in a piezoelectric structure was proposed by Nanthakumar et al. [27] in order to overcome this algorithm. e material interfaces are implied by the level sets defined by regularization with total variation penalty words. e reverse issue is iteratively solved, and, for the study of each iteration, the extended finite element approach is used. For three-dimensional structures, the wording is presented, and inclusions of different materials are recognized by multiple level sets.
In order to mathematically model various phenomena in science and engineering, partial differential equations (PDEs) are important. To solve these problems is a key step in achieving accurate understanding of the conduct of natural and engineered structures. In general, analytical methods are typically not sufficient to solve PDEs representing real systems to an appropriate degree. One must use methods of choice. e most popular alternative is likely the finite element approach for engineering problems (FEM). But strong alternatives are also available, such as mesh-free methods and the Isogeometric Analysis (IGA). e basic idea is to approximate the PDE solution with specially constructed functions with desired properties. Samaniego et al. [28] have explored an alternative for approximation Deep Neural Networks (DNNs). In that work, the emphasis is on mechanical problems, and the PDE energy format is analyzed. For a computer learning to solve a mechanical problem, the energy of a mechanical system seems to be a normal loss function.
Hamdia et al. [29] have used sensitivity analysis to define crucial parameters affecting flexoelectric materialsʼ energy conversion factor (ECF). e governing flexoelectricity equations are modeled on a NURBS-based IGA formula that uses their higher-order continuity and thus avoids a complex combined formulation. e input parameters examined include model and material characteristics, and sampling has taken place in the likelihood region using the Latin hypercube sampling (LHS) method. Khalil et al. [30] investigated electromagnetic field and initial stress on a photothermal semiconducting voids medium under thermoelasticity theories. Abo-Dahab et al. [33][34][35][36][37] studied the effect of relaxation times in different mediums with several variables. Alotaibi et al. [38] used fractional calculus of thermoelastic P-Waves reflection under influence of gravity and electromagnetic fields. Abd-Alla et al. [39] illustrated the effect of several fields on a generalized thermoelastic medium with voids in the context of Lord-Shulman or dualphase-lag models.
e Laplace transform method is simple to describe. Given an initial value problem (IVP), apply the Laplace transform operator to both sides of the differential equation.
is will transform the differential equation into an algebraic equation whose unknown, F (s), is the Laplace transform of the desired solution. Once you solve this algebraic equation for F (s), take the inverse Laplace transform of both sides; the result is the solution to the original IVP.
In addition, the method used in this research is conceptually simple and easy to program. is method is effective in meeting the requirements of automatic digital computation in an efficient manner for two reasons. First, it is fast (economical) on digital computers now available in contrast to other more complicated methods of numerical quadrature. Second, the method is conceptually simple and requires a minimum of programming effort because the resultant inverse function is given as a Fourier cosine series with the coefficients being appropriate values of the forward transform. erefore, one avoids the use of involved algorithms which are encountered in methods using orthogonal functions.
In the current investigation, the response of a thermoelastic piezoelectric fixed rod is investigated. Also, the rod exposed to a heat source moves over its axis. e problem is expressed in the context of Tzou [5] model. By using the Laplace transform and the Laplace numerical inversion, we solved the problem. e influence of speed of the moving heat source and the phase lags on displacement, stress, and temperature is explained.

Basic Equations
e governing equations of thermopiezoelectric materials without body forces and free charge are as follows [16,21]: Strain-displacement relations: Stress-strain-temperature and electric field relations: e motion equation: e heat equation proposed by Tzou [6]: Electric and Gauss equations: where σ ij represents the stress tensor components, ε ij represents the strain tensor components, E i represents the electric field vector components, K ij represents the thermal conductivity tensor, c ijkl represents the elastic constants, c ij represents the thermal moduli, ϕ represents the function of electric potential, e jik represents the piezoelectric moduli, T represents the temperature increment, T 0 represents the initial temperature, D i represents the electric displacement components, u i represents the displacement vector components, ρ represents the mass density, and ε ik represents the electric permittivity. Also, C E refers to the specific heat, Q refers to the applied heat source, p i refers to the pyroelectric constants, τ q refers to the phase lag of heat flux, and τ θ refers to the dual-phaselag parameter. Equations (1)-(5) are the field equations of the generalized linear thermoelastic-piezoelectric applicable for the following: where τ 0 is the relaxation time (iii) Generalized linear piezothermoelasticity without energy dissipation (GN model), when τ q � 1 and τ θ � δ � 0 (iv) Generalized piezothermoelasticity with two different phase lags (DPL model), when τ q ≥ τ θ > 0

Problem Formulation
We consider the direction of polarization of a piezoelectric rod finite length L that is parallel to the axial direction of the rod. Also, the two ends of rod are assumed to be fixed and thermally insulated. Initially, the rod is not deformed and at rest. e x-axis coincides with the axial of the rod. e temperature T and the displacement component u, for a one-dimensional problem, can be written as e strain component e is given by We consider that there are no free charges within the piezoelectric rod. en, Gauss's law gives div(D) � 0, which can be rewritten as is leads to D � D(t). For simplification, we will consider D as constant along the piezoelectric rod. Equations (2), (3), and (5) may be reduced to where c � (3λ + 2μ)α t , α t is the thermal expansion coefficient, K is the thermal conductivity, and e 1 is the piezoelectric constant. To obtain the dimensionless form of the basic equations, we introduce the following variables: , Introducing equation (14) into (11)-(13), we get (dropping the asterisks) where e nondimensional form of the moving heat source Q can be taken as where δ is the delta function, Q 0 is the heat source strength, and v is the velocity of Q along the positive direction of the x-axis.

Initial and Boundary Conditions
We consider the initial conditions to be As the rod is fixed and thermally insulated, the boundary conditions are as follows:

Laplace Transform
e Laplace transform of any function f(t) is given by After using the Laplace transform, we obtain where e boundary conditions (19) and (20) are also transformed to

Solution in the Transformed Field
Eliminating T between (23) and (24), we have e characteristic equation of (28) takes the form e roots m 2 1 and m 2 2 of equation (29) satisfy the relations where m 1 and m 2 are given by Solving equation (28), we get where A i , (i � 1, 2, 3, 4) all are parameters and Likewise, the function T satisfies the following equation: e solution to T is given from (34) as where B i , (i � 1, 2, 3, 4) are some parameters that can be determined from the boundary conditions. Substituting u from equation (32) and T from equation (35) into (23), we can find the following relationship:

Laplace Transform Inversion
We will apply a numerical technique to calculate the inversion of Laplace transform to get the physical fields of the problem in the actual field. To reverse the Laplace transform of various fields, we espouse a method based on the expansion of the Fourier series [31]. In this technique, any field g(x, s) in the inverse field of the Laplace transform is reversed in the physical domain using the relation where N is a sufficiently large integer which was chosen such that where ε 1 is a small positive and the parameter c is positivefree as stated by Honig [31].

Numerical Results and Discussion
To illustrate the above results, the material constants of the piezoelectric rod were taken as follows given in Table 1 [8]. e numerical calculation and the discussion of the problem are carried out in two different groups for different positions of x and time t. e first group is studying how the variations of displacement, temperature, and stress differ with various velocities of the heat source v when the other parameters are constants. e second case shows the study of variations of different fields in models of thermoelasticity and the numerical results were also compared in different models.
e last case is the study of the dependence of different physical distributions on the time when the parameter v remains constant. e figures display that there are two separate phases in the history of electrical potential: before and after heat source leaves the rod. e effect of a heat source prevails in the first stage, while the relation between electrical power and displacement plays an important role in the second stage and contributes to the vibration of the electrical potential as shown in Figure 1. Figures 2-4 display the variants of temperature, displacement, and pressure with various values of the parameter v (v � 1, 2, 3) in which we detect the following. Figure 2 displays that when the parameter v increases, there is an increase in the profile of temperature. e distribution of temperature T begins at its greatest value at x � 0 and gradually decreases until the zero wave reaches the direction of propagation of the heat wave.
As Figure 2 shows, although still the wavefront of the simplified solution travels to the right end of the pole, the wave front is on its way back to the left end of the rod. is means that the thermal wave speed is greater than the simplified solution in the current solution. More Complexity importantly, while in [32] the authors believed thermal wave fronts were present, they did not actually exist for a longer rod with L � 10, as shown in [32].
From Figure 3, we can observe that the distribution of displacement u decreases with increasing the parameter v.
is results in a reduction in the density of thermal energy along the length of the rod.
It can also be found from Figure 3 that the displacement at x � 0, 5 is kept at a zero value, which corresponds to the boundary conditions of the problem where the piezoelectric rod is fixed. e displacement variation u increases monotonically and it reaches a peak value at another wave front.
From Figure 4, it is clear to us that the absolute stress decreases with increasing speed due to the decrease in the energy of the applied heat source along the axial of the rod. e distributions of physical fields are graphically depicted in Figures 5-7 for different theories of thermal elasticity. is group illustrates the influence of two-phaselag parameters on the numerical results, and we observed a slight difference in the value of the distributions in the three models. Figures 8-10 are presented to illustrate the distributions of the physical fields at various values of time t to show the influence of time in each field. We note the following: e temperature profile increases with increasing value of time during certain periods.
e maximum values of the distributions of displacement and stress increase as time increases.
Finally, through Figures 2-10, it becomes clear that once the immediate time is given, the nonzero values of temperature, displacement, and thermal stress are only in a limited region and, beyond this area, gradually fade away.
is explains why the heat dissipation speed is limited in the piezoelectric rod, as this is completely different as compared to the classical thermal models, which predict unlimited velocity. Due to the limited heat diffusion, the heat-disturbed region is limited when the appropriate time is given, resulting in the pressure and displacement caused by the heat also in a specific region.
From Figure 3, we can observe that the distribution of displacement u decreases with increasing the parameter v.
is results in a reduction in the density of thermal energy along the length of the rod.
It can also be found from Figure 3 that the displacement at x � 0, 5 is kept at a zero value which corresponds to the boundary conditions of the problem where the piezoelectric rod is fixed. e displacement variation u increases monotonically and it reaches a peak value at another wave front.
From Figure 4, it is clear to us that the absolute stress decreases with increasing speed due to the decrease in the energy of the applied heat source along the axial of the rod. e temperature field stabilizes as the heat source moves away from the rod, contrary to the acceleration, electrical potential, and stress fields.
ere can, however, be some variations before the temperature achieves a constant value, depending on the thermal rest time.
e distributions of physical fields are graphically depicted in Figures 5-7 for different theories of thermal elasticity. is group illustrates the influence of two-phaselag parameters on the numerical results, and we observed a slight difference in the value of the distributions in the three models. e propagation of thermal waves is very sensitive to the speed of heat sources. Higher speed allows the thermal disturbance to spread more rapidly. e magnitudes of the nondimensional variables increase as the speed of the moving heat resource increases. As the largest values of curves indicate obviously, the effect of the thermal source transmission speed on all considered quantities is quite high.   8 Complexity e influence of the heat source is an inherent consideration not to be overlooked in the evaluation of heat stress and fields at the heat source. Figures 8-10 are presented to illustrate the distributions of the physical fields at various values of time t to show the influence of time in each field. We note the following: e temperature profile increases with increasing value of time during certain periods.
e maximum values of the distributions of displacement and stress increase as time increases.
Finally, through Figures 2-10, it becomes clear that once the immediate time is given, the nonzero values of temperature, displacement, and thermal stress are only in a limited region and, beyond this area, gradually fade away.
is explains why the heat dissipation speed is limited in the piezoelectric rod, as this is completely different as compared to the classical thermal models, which predict unlimited velocity. Due to the limited heat diffusion, the heat-disturbed region is limited when the appropriate time is given, resulting in the pressure and displacement caused by the heat also in a specific region.

Conclusion
e response of a thermoelastic piezoelectric rod exposed to a heat source is studied using the model of thermoelasticity with phase lags. e coupled governing equations of piezoelectricity and thermoelasticity are expressed. Based on the Laplace transform and an approximation inversion technique, the system of equations is solved. e results of the physical variables are displayed graphically and discussed.
From the graphs and theoretical discussion, we can conclude the following: With a comparison between the different models of thermoelasticity, that is, coupled theory (CD), Lord and Shulman (LS), and Tzou (DPL) models, there are differences in the results between different models. e phenomenon of limited propagation speeds is manifested in all these forms. e heat wave interface is moving forward at a limited speed over time. is indicates that the mechanism of heat wave propagation in the case of generalized theories of thermoelasticity differs from that in the classical theory. e values of temperature, exclusion, and nondimensional stress increase with the increase of the moving heat source. e value of v has a significant impact on all areas of distribution in the case of the three theories.
Data Availability e data are supported from the authors' published papers, and all the results are new.

Conflicts of Interest
e authors declare that they have no conflicts of interest.