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

Authors: Mahmoud Ragab; S. M. Abo-Dahab; Ahmed E. Abouelregal; A. A. Kilany
Journal: Complexity
 (2021)

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## Abstract

Piezoelectric ceramics are used more in the active control of noise and vibration with sensors and actuators. As a result, a one-dimensional homogeneous thermoelastic fixed piezoelectric rod subjected to a moving heat flow has been considered. The heating rod is affected by three fields, namely, thermal, mechanical, and electric potential. In order to design a reliable smart structure, this study is necessary and the thermopiezoelectric behavior of piezoelectric ceramics must be understood clearly. The Laplace transform method is applied to acquire the distributions of stress, deformation, and temperature fields. The effect of heat source velocity and phase lag parameters on the considered physical fields is displayed graphically, and the results are compared with other models of thermoelasticity. The present findings could find possible applications for designing receiving portions of transformers of Rosen type for voltage magnification.


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## Body

## 1. 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. The 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. The 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 [7–12].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.To illustrate the thermopiezoelectric interaction, various works have been carried out [13–25]. Abo-Dahab et al. [26] discussed the electromagnetothermoelastic medium with diffusion and voids in the context of Lord-Shulman or dual-phase-lag models under influence of rotation and gravity.A reverse problem of including interfaces in a piezoelectric structure was proposed by Nanthakumar et al. [27] in order to overcome this algorithm. The material interfaces are implied by the level sets defined by regularization with total variation penalty words. The 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. The 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). The 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). The governing flexoelectricity equations are modeled on a NURBS-based IGA formula that uses their higher-order continuity and thus avoids a complex combined formulation. The 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. [33–37] 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 dual-phase-lag models.The 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. This 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. This 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. Therefore, 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. The problem is expressed in the context of Tzou [5] model. By using the Laplace transform and the Laplace numerical inversion, we solved the problem. The influence of speed of the moving heat source and the phase lags on displacement, stress, and temperature is explained.


## 2. Basic Equations

The governing equations of thermopiezoelectric materials without body forces and free charge are as follows [16, 21]:Strain-displacement relations:(1)εij=12ui,j+uj,i,i,j=1,2,3.Stress-strain-temperature and electric field relations:(2)σij=cijklεkl−γijT−T0−ejikDk,(3)Ei=eijkεkj+ϵikDk+piT−T0,i,j,k=1,2,3.The motion equation:(4)σji,j=ρu..i,i,j=1,2,3.The heat equation proposed by Tzou [6]:(5)1+τθ∂∂tKijTji=δ+τq∂∂tρCE∂T∂t+T0γij∂ui,j∂t−T0pi∂ϕi∂t−Q.Electric and Gauss equations:(6)Di,i=0,Ei=−ϕ˙i,whereσij represents the stress tensor components, εij represents the strain tensor components, Ei represents the electric field vector components, Kij represents the thermal conductivity tensor, cijkl represents the elastic constants, γij represents the thermal moduli, ϕ represents the function of electric potential, ejik represents the piezoelectric moduli, T represents the temperature increment, T0 represents the initial temperature, Di represents the electric displacement components, ui represents the displacement vector components, ρ represents the mass density, and ɛik represents the electric permittivity. Also, CE refers to the specific heat, Q refers to the applied heat source, pi refers to the pyroelectric constants, τq refers to the phase lag of heat flux, and τθ refers to the dual-phase-lag parameter. Equations (1)–(5) are the field equations of the generalized linear thermoelastic-piezoelectric applicable for the following:(i)
Dynamical coupled thermoelastic-piezoelectric (CD theory), whenτθ=0=τq(ii)
Generalized piezothermoelasticity (LS), whenτθ=0,τq=τ0>0, where τ0 is the relaxation time(iii)
Generalized linear piezothermoelasticity without energy dissipation (GN model), whenτq=1 and τθ=δ=0(iv)
Generalized piezothermoelasticity with two different phase lags (DPL model), whenτq≥τθ>0


## 3. Problem Formulation

We consider the direction of polarization of a piezoelectric rod finite lengthL 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. The x-axis coincides with the axial of the rod. The temperature T and the displacement component u, for a one-dimensional problem, can be written as(7)T=Tx,t,ux=ux,t.The strain componente is given by(8)e=exx=∂u∂x.We consider that there are no free charges within the piezoelectric rod. Then, Gauss’s law gives(9)divD=0,which can be rewritten as(10)∂D∂x=0.This leads toD=Dt. For simplification, we will consider D as constant along the piezoelectric rod. Equations (2), (3), and (5) may be reduced to(11)σxx=σ=λ+2μ∂u∂x−γT−T0−e1D,(12)λ+2μ∂2u∂x2−γ∂T∂x=ρ∂2u∂t2,(13)K1+τθ∂∂t∂2T∂x2=1+τq∂∂tρCE∂T∂t+T0γ∂u∂x∂t−Q,where γ=3λ+2μαt,αt is the thermal expansion coefficient, K is the thermal conductivity, and e1 is the piezoelectric constant. To obtain the dimensionless form of the basic equations, we introduce the following variables:(14)x∗,u∗=c0ηx,u,t∗=c02ηt,τq∗,τθ∗=c02ητq,τθ,T∗=T−T0T0,σ∗=σλ+2μ,D∗=e1Dλ+2μ,η=ρCEKQ∗=γQKc02η2λ+2μ,c02=λ+2μρ.Introducing equation (14) into (11)–(13), we get (dropping the asterisks)(15)σ=∂u∂x−bT−D,∂2u∂x2−b∂T∂x=∂2u∂t2,1+τθ∂∂t∂2T∂x2=1+τq∂∂t∂T∂t+g∂u∂x∂t−Q,where(16)b=γT0λ+2μ,g=γρCE.The nondimensional form of the moving heat sourceQ can be taken as(17)Q=Q0δx−vt,where δ is the delta function, Q0 is the heat source strength, and v is the velocity of Q along the positive direction of the x-axis.


## 4. Initial and Boundary Conditions

We consider the initial conditions to be(18)ux,t|t=0=∂ux,t∂t|t=0=0,0≤x≤l,Tx,t|t=0=∂Tx,t∂t|t=0=0,t=0,0≤x≤l.As the rod is fixed and thermally insulated, the boundary conditions are as follows:(19)u0,t=ul,t=0,(20)∂T0,t∂x=∂Tl,t∂x=0.


## 5. Laplace Transform

The Laplace transform of any functionft is given by(21)£ft=f¯s=∫0∞e−stftdt,Res>0.After using the Laplace transform, we obtain(22)σ¯=dudx−bT¯−D¯s,(23)d2dx2−s2u¯=bdT¯dx,(24)d2dx2−qsT¯=qsgdudx−αqexp−svx,where(25)α=Q0v,q=1+τqs1+τθs.The boundary conditions (19) and (20) are also transformed to(26)u¯0,s=u¯l,s=0,(27)dT¯0,sdx=dT¯l,sdx=0.


## 6. Solution in the Transformed Field

EliminatingT¯ between (23) and (24), we have(28)d4dx4−s2+sq1+gbd2dx2+qs3u¯bαsqvexp−svx.The characteristic equation of (28) takes the form(29)m4−s2+sq1+gbm2+qs3=0.The rootsm12 and m22 of equation (29) satisfy the relations(30)m12+m22=s2+sq1+gb,m12m22=qs3,where m1 and m2 are given by(31)m1,22=12s2+sq1+gb±s2+sq1+gb2−4qs3.Solving equation (28), we get(32)u¯=A1e−m1x+A2em1x+A3e−m2x+A4em2x+A5exp−svx,where Ai,i=1,2,3,4 all are parameters and(33)A5=bαsqv3s4−s2+sq1+gbs2+v4qs3.Likewise, the functionT¯ satisfies the following equation:(34)d4dx4−s2+sq1+gbd2dx2+qs3T¯=αs2q1−v2v2exp−svx.The solution toT¯ is given from (34) as(35)T¯=B1e−m1x+B2em1x+1B3e−m2x+1B4em2x+B5exp−svx,where Bi,i=1,2,3,4 are some parameters that can be determined from the boundary conditions.Substitutingu¯ from equation (32) and T¯ from equation (35) into (23), we can find the following relationship:(36)Bi=−1imi2−s2mibAi,i=1,2,3,4,B5=−s1−v2bvA5.In order to determine the parametersAi,i=1,2,3,4, we need to apply boundary conditions (26) and (27).(37)A1+A2+A3+A4=−A5,(38)A1e−m1l+A2em1l+A3e−m2l+A4em2l=−A5exp−svl,(39)−m1B1+m1B2−m2B3+m2B4=svB5,(40)−m1B1e−m1l+m1B2em1l−m2B3e−m2l+m2B4em2l=svB5exp−svl.Solving equations (40)–(43), Aii=1,2,3,4 can be expressed as(41)A1=m22−s/v2expm1l−exp−s/vxl2m12−m22sinhmilA5,A2=−m22−s/v2exp−m1l−exp−s/vxl2m12−m22sinhm1lA5,A3=−m12−s/v2expm2l−exp−s/vxl2m12−m22sinhm2lA5,A4=m12−s/v2exp−m2l−exp−s/vxl2m12−m22sinhm2lA5.Substitutingu¯ and T¯ into equation (22), we obtain(42)σ¯=−s2m1A1e−m1x+s2m1A2em1x+−s2m2A3e−m2x+s2m2A4em2x−svA5exp−svx−D¯s.


## 7. 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(43)gx,t=ectT1g¯x,c2+Re∑k=1NexpikπtT1g¯x,c+ikπT1,2t≥T1≥0,where N is a sufficiently large integer which was chosen such that(44)ectReexpikπtT1g¯c+ikπT1≤ε1,where ε1 is a small positive and the parameter c is positive-free as stated by Honig [31].


## 8. Numerical Results and Discussion

To illustrate the above results, the material constants of the piezoelectric rod were taken as follows given in Table1 [8].Table 1
Physical constants of piezoelectric material.
ParameterValueK386NK−1s−1μ3.86×1010Nm−2ρ8954kgm−3αt1.78×10−5K−1T0293Kε0.0168l5η0.003887Q010D10−6α0.0104λ7.76×1010Nm−2The numerical calculation and the discussion of the problem are carried out in two different groups for different positions ofx and time t. The 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. The second case shows the study of variations of different fields in models of thermoelasticity and the numerical results were also compared in different models. The last case is the study of the dependence of different physical distributions on the time when the parameter v remains constant. The figures display that there are two separate phases in the history of electrical potential: before and after heat source leaves the rod. The 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.Figure 1
Schematic diagram for the piezoelectric finite rod.Figures2–4 display the variants of temperature, displacement, and pressure with various values of the parameter vv=1,2,3 in which we detect the following.Figure 2
The variation of temperature distributionT with moving heat source velocity v.Figure 3
The variation of displacement distributionu with moving heat source velocity v.Figure 4
The variation of stress distributionσ with moving heat source velocity v.Figure2 displays that when the parameter v increases, there is an increase in the profile of temperature.The distribution of temperatureT 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 Figure2 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. This means that the thermal wave speed is greater than the simplified solution in the current solution. More 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 Figure3, we can observe that the distribution of displacement u decreases with increasing the parameter v. This results in a reduction in the density of thermal energy along the length of the rod.It can also be found from Figure3 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.The displacement variationu increases monotonically and it reaches a peak value at another wave front.From Figure4, 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.The distributions of physical fields are graphically depicted in Figures5–7 for different theories of thermal elasticity. This group illustrates the influence of two-phase-lag parameters on the numerical results, and we observed a slight difference in the value of the distributions in the three models.Figure 5
Temperature distribution for different theory of thermoelasticity.Figure 6
Displacement distribution for different theory of thermoelasticity.Figure 7
Stress distribution for different theory of thermoelasticity.Figures8–10 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:Figure 8
Temperature distribution for different values of time.Figure 9
Displacement distribution for different values of time.Figure 10
Thermal stress distribution for different values of time.The temperature profile increases with increasing value of time during certain periods.The maximum values of the distributions of displacement and stress increase as time increases.Finally, through Figures2–10, 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. This 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 Figure3, we can observe that the distribution of displacement u decreases with increasing the parameter v. This results in a reduction in the density of thermal energy along the length of the rod.It can also be found from Figure3 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.The displacement variationu increases monotonically and it reaches a peak value at another wave front.From Figure4, 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.The temperature field stabilizes as the heat source moves away from the rod, contrary to the acceleration, electrical potential, and stress fields. There can, however, be some variations before the temperature achieves a constant value, depending on the thermal rest time.The distributions of physical fields are graphically depicted in Figures5–7 for different theories of thermal elasticity. This group illustrates the influence of two-phase-lag parameters on the numerical results, and we observed a slight difference in the value of the distributions in the three models. The propagation of thermal waves is very sensitive to the speed of heat sources. Higher speed allows the thermal disturbance to spread more rapidly. The 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. The 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.Figures8–10 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:The temperature profile increases with increasing value of time during certain periods.The maximum values of the distributions of displacement and stress increase as time increases.Finally, through Figures2–10, 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. This 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.


## 9. Conclusion

The response of a thermoelastic piezoelectric rod exposed to a heat source is studied using the model of thermoelasticity with phase lags. The 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. The 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.The phenomenon of limited propagation speeds is manifested in all these forms. The heat wave interface is moving forward at a limited speed over time. This indicates that the mechanism of heat wave propagation in the case of generalized theories of thermoelasticity differs from that in the classical theory.The values of temperature, exclusion, and nondimensional stress increase with the increase of the moving heat source. The value ofv has a significant impact on all areas of distribution in the case of the three theories.



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