Coupled thermopiezoelectric behaviour of a one ... - SAGE Journals

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Coupled thermopiezoelectric behaviour of a onedimensional functionally graded piezoelectric medium based on C–T theory A H Akbarzadeh1, M H Babaei2, and Z T Chen1,* 1 Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, Canada 2 Center for Smart Materials & Structures, Royal Military College of Canada, Kingston, Canada The manuscript was received on 27 September 2010 and was accepted after revision for publication on 23 March 2011. DOI: 10.1177/0954406211406954

Abstract: In this article, the transient thermopiezoelectric behaviour of a one-dimensional (1D) functionally graded piezoelectric medium subjected to a moving heat source is investigated. The formulation is given based on the Chandrasekhariah and Tzou (C–T) generalized thermoelasticity theory to consider the details of energy transport in the material in comparison with the Lord– Shulman (L–S) generalized theory. All material properties are taken to vary exponentially along the length of the medium except for phase lags, the relaxation time, and the specific heat, which are taken to be constant. The governing partial differential equations are given in the three coupled fields of displacement, temperature, and electric potential based on the C–T theory. Using Laplace transform to eliminate the time dependency of the problem, an analytical method is presented to obtain the coupled fields in the Laplace domain. The solutions are then derived in time domain by employing the fast Laplace inversion technique. Numerical results are shown to display the effects of discontinuities on the temperature and stress distribution, non-homogeneity index and the phase-lag constants of heat flux and temperature gradient on the wave propagation of temperature and stress fields based on the dual-phase-lag model of the C–T. Furthermore, the results are compared between the C–T and L–S thermoelasticity theories. Finally, the results are validated with those reported in the literature. Keywords: functionally graded piezoelectric materials, generalized thermoelasticity, dualphase-lag model, moving heat source, discontinuity, Laplace inversion method

1

INTRODUCTION

Piezoelectric materials are being used in a wide variety of smart structures because of their excellent coupled electro-mechanical properties. They have extensive applications in micro-electromechanical systems, acoustic and pressure sensing, precision position control, sensors for monitoring, ultrasonic *Corresponding author: Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB E3B5A3, Canada. email: [email protected]

transducers, piezoelectric composite structures, loudspeakers and microphones, medical industry, and aerospace explorations [1, 2]. An overview of applications of piezoelectric materials for intelligent and aerospace structures has been reported by Crawley [3]. Piezoelectric laminates suffer from abrupt changes of material properties in different layers, which cause high-stress concentration, creep fatigue proneness, and failure at the layer interfaces. Therefore, a new type of piezoelectric materials called functionally graded piezoelectric materials (FGPMs) was developed in which material properties change Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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A H Akbarzadeh, M H Babaei, and Z T Chen

continuously in some directions [4]. A review of principal developments and diverse areas relevant to functionally graded materials has been reported in references [5, 6]. Furthermore, some available processes for fabricating of functionally graded materials have been reported such as: powder metallurgy, plasma spraying, centrifugal casting, and physical vapour decomposition [7, 8]. In some applications, the system undergoes low- or high-temperature gradients, where the pyroelectric effect should be considered; for instance, moving and stationary heat sources are regularly used in many manufacturing processes or occur at contact surfaces. Hence, understanding the coupled thermopiezoelectricity responses of an FGPM plays a significant role in design of a smart structure. The classical coupled thermoelasticity was introduced by Biot [9] to consider the effects of elastic terms in the heat equation. Since the heat equation in this theory is parabolic type, it predicts an unrealistic infinite speed for heat propagation. Therefore, some generalized theories were presented to consider the second sound effect of the heat wave, a phenomena that the thermal disturbances are found to propagate as progressive waves with little dispersion [10]. L–S [11] employed a new law of heat conduction with a relaxation time, namely, the time that the temperature field needs to adjust itself to thermal disturbances. Green and Lindsay (G–L) [12] introduced another version of generalized thermoelasticity using two relaxation times for constitutive relations of stress tensor and entropy. Recently, a generalized theory has been proposed by Chandrasekharaiah and Tzou (C–T) [13, 14] to modify classical thermoelasticity by considering the dual-phase-lag (DPL) of heat flux and temperature gradient. This theory establishes the idea that the temperature gradient may precede the heat flux or vice versa. Due to the close agreements of this theory with experiments for a wide range of length and time scales which includes microscale and macroscale ranges, interest in this theory has grown [15, 16]. Chrzeszczyk [17] proved the local existence, uniqueness, and continuous dependency for the solution of the initial-value problem for the generalized thermoelastic materials. In the analysis, he wrote the field equations as a quasi-linear hyperbolic system. Tzou [18] studied the significance of relaxation time thermodynamically and mechanically. He verified that the relaxation time comes from the rate equation in the second law of non-equilibrium, irreversible thermodynamics. Furthermore, it can be interpreted as the phase lag between the heat flux vector and the temperature gradient. Antaki [15] obtained a solution for a semi-infinite slab using the DPL model. The slab Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

was subjected by a constant surface temperature and constant surface heat flux. The results showed the difference between the transient temperature responses based on the classical and DPL models. Thermally induced displacement of a rod due to a moving heat source was investigated by Al-Huniti et al. [19]. The thermal analysis was based on the hyperbolic heat conduction model. They presented the effects of the moving heat speed and the convection heat transfer coefficient on the results. Generalized thermoelasticity of a one-dimensional (1D) semi-infinite piezoelectric rod subjected to a sudden heat source was investigated by He et al. [20] based on the L–S theory. Employing the Laplace transform and the state space approach, they solved the governing partial differential equations to find two discontinuity points in both stress and temperature solutions. The 3D fundamental solution for a generalized thermoelastic infinite medium subjected to a continuous heat source was given by Aouadi [21]. Using Laplace transform and Helmholtz theorem, he analysed the exact expressions of discontinuities in the field functions. In addition, Aouadi [22] presented the coupled 2D thermopiezoelctric analysis for a thick infinite plate using the hybrid Laplace transform-finite element method. He considered generalized thermoelasticity as well as classical coupled thermoelasticity and found the wave-type heat propagation in the piezoelectric plate. He et al. [23] studied the dynamic thermoelastic response of a piezoelectric rod subjected to a moving heat source based on the L–S generalized theory. Choudhuri [24] introduced a three-phase-lag model of coupled thermoelasticity to extend the C–T generalized theory. In his study, the Fourier’s law was replaced by a modified formulation which includes three different phase lags in heat flux, temperature gradient, and the thermal displacement gradient. The three-phase-lag heat conduction leads to hyperbolic partial differential equations with a fourth-order derivative with respect to time. This model contains all the previous theories at the same time. Following his work, Quintanilla [25] studied the spatial behaviour of temperature for the three-phase-lag heat conduction. He obtained the spatial evolution of solutions of an initial-value problem on the lateral surface of a cylinder with zero boundary conditions. The generalized thermopiezoelectric response of a 1D FGP medium based on the L–S theory was investigated in reference [26]. However, in some cases such as pulsed laser heating and ultra-fast heating source, there exists a delay in the response of the medium with respect to the heat source and the delay of heat flux may differ from that of temperature gradient. Therefore, thermoelastic analysis based on

One-dimensional functionally graded piezoelectric medium

the DPL model is still beneficial to understand the dynamic thermoelastic behaviour of piezoelectric materials. This article presents the behaviour of a 1D FGP medium based on C–T generalized theory. The medium is subjected to a moving heat source. Governing equations for the FGP medium excited by a moving heat source is written based on the most general form of the C–T theory which will reduce to the L–S theory in a special case. Three coupled fields, namely displacement, temperature, and electric potential are solved in the Laplace domain using the successive decoupling method. Then, the solutions are inverted to time domain employing the fast Laplace inverse transform. Numerical results are calculated to display the effect of the non-homogeneity indices and phase-lag constants on the thermopiezoelectric response. Finally, the results are verified with those reported in the literature. 2

BASIC EQUATIONS

Consider a 1D thermopiezoelectric medium of length L, aligned along the x-axis as shown in Fig. 1. The origin of the coordinate is put at the left end. Both ends are fixed, thermally insulated, and electrically grounded. The medium is subjected to a moving heat source that locates at the left end at t ¼ 0 and moving towards the right end at a constant speed of . The constitutive equations in generalized linear thermoelasticity for a piezoelectric medium are given as follows [10, 27, 28] ij ¼ cijkl "kl ij eijk Ek Di ¼ eijk "jk þ pi þ 2ij Ej 2 @ @ 2 @ 1 þ þ t2 2 qi ¼ Kij 1 þ t1 ,j @t @t @t C S ¼ ij "ij þ þ pi Ei 0

ð1Þ

In the above equations, ij ,"kl ,Ek ,Di ,qi ,S and are, respectively, stress tensor, strain tensor, electric field vector, electric displacement vector, heat flux vector, entropy per unit mass, and temperature change, ¼ T T0 , where T is the absolute temperature and T0 is the ambient temperature; cijkl , ij , eijk , pi , 2ij , Kij , , C , and are elastic coefficients, thermal moduli, piezoelectric coefficient, pyroelectric and dielectric

Fig. 1

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coefficients, heat conduction coefficients, density, specific heat and relaxation time, respectively; t1 and t2 are defined according to two approximations of the modification of Fourier law based on C–T theory as follows [10, 29] ðaÞt1 ¼ i0, ¼ q i0, t22 ¼ 0, q i i0 1 ðbÞt1 ¼ i0, ¼ q i0, t22 ¼ q2 2

ð2Þ

in which, q is a phase lag of the heat flux and a phase lag of the temperature gradient. This formulation allows us to analyse the generalized thermoelasticity based on the L–S theory as well. Governing equations for the thermopiezoelectricity problem in the absence of body force and volume charges are ij,j ¼ u€ i Di,i ¼ 0 _ 0 R þ qi,i ¼ 0 S

ð3Þ

where, a superposed dot represents differentiation with respect to time, while a comma denotes partial differentiation with respect to the space variable. The quasi-stationary electric field equation and the linear strain–displacement relations are given as follows Ei ¼ ,i 1 "ij ¼ ui,j þ uj,i 2

ð4Þ

where is the electric potential. In the current 1D problem, all aforementioned field variables are functions of only x and t. Therefore, the constitutive and governing equations (1) to (3) are simplified in the following form xx ¼ cu,x þ e,x Dx ¼ eu,x þ p 2 ,x @ @2 @ 1 þ þ t22 2 qx ¼ K 1 þ t1 ,x @t @t @t S ¼ u,x þ

C p,x 0

xx,x ¼ u€ Dx,x ¼ 0 _ 0 RÞ þ qx,x ¼ 0 ðS

ð5Þ

The 1D thermopiezoelectric medium subjected to a moving heat source

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in which the summation convention is not applied; and the subscript of material properties have been omitted for convenience; u is the displacement component in the x direction. The medium is nonhomogenous and all material properties are assumed to be functions of x, except specific heat, and the phase lags of heat flux and temperature gradient that are taken to be constant. Using equation (5), the governing differential equations are derived as c,x u,x þ cu,xx u€ ,x ,x þ e,x ,x þ e,xx ¼ 0 e,x u,x þ eu,xx þ p,x þ p,x ",x ,x ",xx ¼ 0

u,xx þ a1 u,x s 2 u þ a2 c ,xx þ a1 a2 c ,x

K,x ,x þ K ,xx þ K,x t1 _,x þ Kt1 _,xx € C ð_ þ € þ t 2 Þ € þ ðR þ R_ þ t 2 RÞ

u,xx þ a1 u,x Lc ,xx a1 Lc ,x þ a4 a9 ,x þ a1 a4 a8 ¼ 0

2

2

0 ðu_ ,x þ u€ ,x þ t22 u€ ,x Þ þ 0 pð_,x þ €,x þ t 2 €,x Þ ¼ 0

ð6Þ

2

To simplify the analysis, we introduce the following non-dimensional parameters rffiffiffiffiffi c0 0 C uÞ ¼ c ðx, uÞ, c ¼ , ðx, , ¼ K0 0 20 2 t1 , t2 Þ ¼ c ðt , , t1 , t2 Þ, ¼ , ¼ ðt , , 0 e0 L R xx Dx xx ¼ R ¼ , Dx ¼ , l ¼ Lc 2 2 , c0 e0 K0 0 c ð7Þ where, K0 , c0 , 0 , e0 , and 20 are the thermal conductivity, elastic coefficients, density, piezoelectric and dielectric coefficients at x ¼ 0, respectively. The material properties vary exponentially along the x-axis according to the following formulation

¼ 0 e x

ð8Þ

in which, is the generalized material property except relaxation time, phase lags, and specific heat; is an arbitrary non-homogeneity index. Furthermore, the moving heat source is specified in the following form R ¼ R0 ðx t Þ

ð9Þ

where R0 and are the intensity and velocity of the heat source while, the Dirac delta function. 3

obtained in the Laplace domain, a numerical Laplace inversion is applied to obtain the final solution in time domain. The initial conditions for the displacement, electric potential, and temperature change are assumed to be zero. In the following equations, overbar and tilda signs have been removed for convenience. The coupled equations in equation (6) are written in the non-dimensional form in the Laplace domain as follows

SOLUTION PROCEDURES

The time dependency of the problem is treated using the Laplace transform, defined as follows f~ ðsÞ ¼

Z1

e st f ðt Þdt

ð10Þ

0

where, f~ is the Laplace transform of function f ðt Þ and s the Laplace transform variable. Once the solution is Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

a3 a8 ,x a1 a3 a8 ¼ 0 ð11Þ

a9 ,xx þ a1 a9 ,x a10 a6 a10 u,x þ a7 a10 ,x a5 a10 s x e ¼ s where e02 L 0 0 , a ¼ , a3 ¼ , 2 c 20 c0 c0 p0 0 0 R0 0 a4 ¼ , a6 ¼ , , a5 ¼ e0 0 C p0 c e0 L a7 ¼ 20 K 0 a8 ¼ 1, a9 ¼ ð1 þ st1 Þ, a10 ¼ sð1 þ s þ t22 s 2 Þ a1 ¼

ð12Þ

The solution of the linear ordinary differential equation system of (11) has got two parts, particular solution and a general solution. The particular solution can be written in the following form s

ðup , p , p Þ ¼ ðPu , P , P Þe x

ð13Þ

where subscript ‘p’ indicates the particular solution. The unknown functions Pu , P , and P can be found by substituting equation (13) into equation (11) and solving the algebraic equation as follows 2 3 a3 a8 s þ a1 ðs Þ2 a1 s s 2 I ðs Þ2 a1 s 6 s 2 7 6 ð Þ a1 s J ðs Þ2 a1 s a4 a9 s þ a1 a8 7 5 4 a6 a10 s a7 a10 s a9 ðs Þ2 a1 s a10 9 8 9 8 > = > = < Pu > < 0 > 0 P ¼ ð14Þ > ; > ; : > : a5 a10 > s P where, I ¼ a2 c and J ¼ Lc . The general solution of the governing equations is obtained by eliminating and u successively in the corresponding homogenous governing equations. First, we find g ,x from the third equation of the homogenous form of equation (11) g ,x ¼

a9 a1 a9 1 a6 g ,xx g ,x þ g þ ug ,x a7 a7 a10 a7 a10 a7

ð15Þ


where subscript ‘g’ indicates the general solution. Using equation (15), the governing equations of (11) leads to

B1 ug ,xx þ a1 B1 ug ,x s 2 ug B2 g ,xxx

analytically the fourth degree algebraic equation [32]. The solution for the temperature is eventually obtained as s

ðx, sÞ ¼ g þ p ¼ Ci e i x þ P e x

A1 ug ,xx þ a1 A1 ug ,x þ A2 g ,xxx þ 2a1 A2 g ,xx þ A3 g ,x þ a1 A4 g

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ð16Þ

2a1 B2 g ,xx þ B3 g ,x þ a1 B4 g

Substituting equation (23) into equations (18) and (15), we can find the solutions for the displacement and the electric potential as follows s

In which,

uðx, sÞ ¼ Cui Ci e i x þ Pu e x

a6 Ja9 J a12 a9 A1 ¼ 1 J , A2 ¼ , A3 ¼ 1 a7 a10 a7 a7 a10 J þ a4 a8 , A4 ¼ þ a4 a8 a7 Ia6 Ia9 I a12 a9 B1 ¼ 1 þ , B2 ¼ , B3 ¼ 1 a7 a7 a7 a10 a7 a10 I a3 a8 , B4 ¼ a3 a8 ð17Þ a7

ðx, sÞ ¼ Ci Ci e i x þ C 0 þ P e x

We can find ug via multiplying the second equation of (16) by BA11 and summing with the first equation of (16) as follows u,g ¼ D1 g ,xxx þ 2a1 D1 g ,xx þ D2 g ,x þ a1 D3 g

ð18Þ

where 1 B 1 A2 1 B 1 A3 B þ ¼ B , D , 2 2 3 s2 s2 A1 A1 1 B 1 A4 D3 ¼ 2 B4 ð19Þ s A1 D1 ¼

Substituting equation (18) into the first equation of (16) gives the following ordinary differential equation with constant coefficients E1 g ,xxxxx þ 3a1 E1 g ,xxxx þ E2 g ,xxx þ a1 E3 g ,xx þ E4 g ,x þ a1 A4 g ¼ 0

ð23Þ

ð20Þ

ð24Þ

s

where Ci and C 0 are integration constants and Cui and Ci are obtained in the following form Cui ¼ D1 i3 þ 2a1 D1 i2 þ D2 i þ a1 D3 1 a9 a1 a9 1 Ci ¼

i þ þ a6 Cui a7

i a10 a10

ð25Þ

Then, we apply the boundary condition @ u, , @x ¼ ð0, 0, 0Þ on the system that leads to x¼0,l the following system of equations for the unknown integration constants 3 2 Cu1 Cu2 Cu3 Cu4 Cu5 0 6 Cu e 1 l Cu e 2 l Cu e 3 l Cu e 4 l Cu e 5 l 0 7 7 6 1 2 3 4 5 7 6 6 C1 C2 C3 C4 C5 17 7 6 6 C e 1 l C e 2 l C e 3 l C e 4 l C e 5 l 1 7 2 3 4 5 7 6 1 7 6 4 1

2

3

4

5 05

1 e 1 l

2 e 2 l

3 e 3 l

4 e 4 l 9 8 9 8 Pu > C1 > > > > > > > > > > > > s l > > > > > P e > > > > C u > > > 2 > > > > > > > > = < C = < P > 3 ¼ s l P e > > > > > > > > > > > C4 > > > > > s > > > > > > > > P C > > > 5 > > > > > > : 0; > s : s l C P e ;

5 e 5 l

0

ð26Þ

E1 5 þ 3a1 E1 4 þ E2 3 þ a1 E3 2 þ E4 þ a1 A4 ¼ 0 ð22Þ

Furthermore, the normalized stress and electric displacement can be found in Laplace domain as a2 J x ¼ Ci i Cui þ Ci a3 ð1 þ vs Þ e i x L

s a2 J Pu þ P þ a3 ð1 þ vs ÞP e a1 x L D1 ¼ Ci i Cui JCi þ a4 ð1 þ vs Þ e i x hs io Pu JP a4 ð1 þ vs ÞP e a1 x ð27Þ

There are some analytical approaches for solving a quintic equation, a polynomial of fifth degree, using the Hermit–Kronecker method and the Mellin method [30, 31]. However, since one of the characteristic roots of equation (22) is 1 ¼ a1 , we can factor this term from equation (22) and solve

Till now, we have obtained the solutions in the Laplace domain. Then, we transform them to the time domain using a numerical Laplace inversion method so called fast Laplace inverse transform [33]. According to this method, the numerical Laplace inversion of function f~ ðx, sÞ at time tk ,

In which E1 ¼ A1 D1 , E2 ¼ A1 ðD2 þ 2a12 D1 Þ þ A2 , E3 ¼ A1 ðD2 þ D3 Þ þ 2A2 , E4 ¼ a12 A1 D3 þ A3

ð21Þ

The characteristic equation of equation (20) is given as follows


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f ðx, tk Þ, has the following form aþi1 Z 1 f ðx, tk Þ ¼ e stk f~ ðx, sÞds 2i ai1 ( " N 1 o X 1 n~ GðkÞ Re f ðx, aÞ þ Re ðAðx, nÞ: 2 n¼0 )#

þ iBðx, nÞÞW nk

ð28Þ

k ¼ 0, 1, . . . , N 1

pffiffiffiffiffiffiffi 1, and Ln X 2 Aðx, nÞ ¼ Re f~ ðx, a þ iðn þ mN Þ , T m¼0 Ln X 2 ~ Bðx, nÞ ¼ Im f ðx, a þ iðn þ mN Þ T m¼0

where i ¼

GðkÞ ¼

ð29Þ

2 akt T 2 e , t ¼ , W ¼ e i N T N

In which, T is the total time interval over which the numerical Laplace inversion is performed, a an arbitrary real number larger than any of the real parts of the singularities of f~ ðx, sÞ, t the time increment, Ln and N the two arbitrary parameters which affect the accuracy of the solution. Furthermore, Korrectur method [34] has been used o reduce the discretization and truncation errors. 4

RESULTS

Assume the left end of the FGPM rod is cadmium selenide with the material properties listed in Table 1 [35]. The medium is initially at ambient temperature 0 ¼ 293K when it is subjected to the moving heat source with constant velocity and the intensity of R0 ¼ 10 0 . The non-dimensional phase lags of heat flux and temperature gradient are

Fig. 2

taken to be 0.05 and 0.04, respectively. The constant velocity of the moving heat source is 0.5, and it is assumed for all of the following results. In addition, the analysis based on C–T theory is performed for both t2 ¼ 0 and t2 ¼ pqffiffi2 for comparison and the results are given in non-dimensional form as defined in equation (7). Since the solution procedure was performed based on the general approximation of C–T theory, we can investigate the solution of the FGPM medium based on the L–S theory by setting the value of t1 and t2 equal to zero. The results of this article are exactly reduced to the analytical solution for transient thermopiezoelectric response of a 1D medium based on the L–S theory reported in reference [26] that verifies our solution procedure. Figures 2 through 5 compare the response of a homogenous 1D medium based on two approximations of C–T theory and the L–S theory for displacement, temperature change, stress, and electric potential subjected to a moving heat source. The results are shown at non-dimensional time t ¼ 0:1333 at which time the non-dimensional location of the heat source is x ¼ t ¼ 0:0667. It should be mentioned that the heat source reaches the end of the medium at non-dimensional time texit ¼ 2. For L–S theory the maximum of temperature occurs at this location, but the C–T theories with those values selected for the phase lags does not show this

Table 1

Material properties of the left side of the 1D medium

c0 ¼ 74:1 109 N=m2 0 ¼ 0:621 106 N=m2 K e0 ¼ 0:347C=m2 20 ¼ 90:3 1012 C=Nm2 p0 ¼ 2:94 106 C=km2 CE ¼ 420J=kgK 0 ¼ 7600kg=m3 K0 ¼ 12:9W=mK

Comparison of the displacement distribution based on C–T and L–S theories



Fig. 3

Comparison of the temperature distribution based on C–T and L–S theories

Fig. 4

Fig. 5

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Comparison of the stress distribution based on C–T and L–S theories

Comparison of the electric potential distribution based on C–T and L–S theories

phenomena. Nonetheless, when the phase lag of temperature gradient decreases, the maximum temperature happens at the location of heat source, as will be shown later in Fig. 6.

The above results were based on the non-Fourier heat conduction in generalized thermoelasticity. Figures 3 and 4 show that the wave front in stress and temperature distribution based on the C–T Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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theory for t2 6¼ 0 is ahead of the wave front based on the L–S theory. However, when t2 ¼ 0 in the C–T theory, the wave front is not observed in stress and temperature distribution. Figures 6 and 7 investigate the effect of the phase lag of ¼ t1 on the temperature and the stress distribution for the C–T theory with t2 6¼ 0. The medium is assumed to be homogenous and the results are depicted at non-dimensional time t ¼ 0:1333. When the phase lag of increases, the finite speed of wave

propagation for the stress and the temperature increases and the wave fronts move ahead further. Furthermore, decreasing the value of , weakens the wave front, and eventually at t1 ¼ 0:01, the waterfronts disappear. The effect of on the stress and the temperature distribution for the C–T theory with t2 ¼ 0 is depicted in Figs 8 and 9. No wavefront is observed in these figures for the temperature and stress. According to Figs 6 through 9, we realize that a decrease in t1 as the

Fig. 6

Effect of the phase lag of temperature gradient on the temperature distribution for C–T theory with t2 6¼ 0

Fig. 7

Effect of the phase lag of temperature gradient on the stress distribution for C–T theory with t2 6¼ 0



Fig. 8

Effect of the phase lag of temperature gradient on the temperature distribution for C–T theory with t2 ¼ 0

Fig. 9

Effect of the phase lag of temperature gradient on the stress distribution for C–T theory with t2 ¼ 0

Fig. 10

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Effect of non-homogeneity index on the distribution of elastic constant Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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Fig. 11

Effect of non-homogeneity index on the displacement history for C–T theory

Fig. 12

Effect of non-homogeneity index on the temperature history for C–T theory

phase lag of temperature gradient increases the absolute values of extremums for temperature and stress distribution. This conclusion is valid for the two different approximation of the C–T theory for t2 6¼ 0 and t2 ¼ 0. The material properties of the thermopiezoelectric medium changes exponentially with . For instance, changes of the elastic coefficient through the length of the medium are depicted in Fig. 10. Figures 11 through 14 show the non-homogeneity effect of on the displacement, temperature, stress, and electric Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

potential history according to the C–T theory (t2 6¼ 0). The results are depicted for a non-dimensional location x ¼ 0:5. It is seen that an increase in rises the absolute mean value of fluctuation for the displacement and the stress. Furthermore, the amplitude of oscillation for the electric potential increases when increases. From Fig. 12, we can deduce that the temperature at each location increases until the time that the moving heat source leaves the rod at texit ¼ 2; then the temperature reaches to its constant value after some small oscillation. However, after the exit time


Fig. 13

Fig. 14

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Effect of non-homogeneity index on the electric displacement history for C–T theory

Effect of non-homogeneity index on the electric potential history for C–T theory

of the heat source, the displacement, stress, and electric potential fluctuate with constant amplitude. Although only the results of the C–T theory for t2 6¼ 0 are given here for brevity, similar results can be found for the C–T theory with t2 ¼ 0, and the L–S theory. It should be asserted that the results for L–S theory for time history response of a 1D homogenous medium is exactly same as the analytical results in reference [26].

Figure 15 shows the effect of non-homogeneity index on the electric displacement history based on the C–T theory (t2 6¼ 0). It is shown that the absolute value of electric displacement rises when the value of increases. Furthermore, the electric displacement smoothly increases until the moving heat source exits and then remains in a constant value. Similar results can be observed for the C–T theory with t2 ¼ 0 and the L–S theory using our solution procedure. Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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Fig. 15

Effect of non-homogeneity index on the electric displacement history for C–T theory

Fig. 16

Effect of non-homogeneity index on the displacement distribution

Figures 16 through 19 depict the effect of nonhomogeneity index on the distribution of displacement, temperature, electric potential, and stress based on the C–T theory with ¼ t1 ¼ 0:04 and t2 6¼ 0 at non-dimensional time t ¼ 0:1333. Increasing decreases the absolute value of displacement, temperature, and electric potential. Although increasing reduces the height of wavefront, the location of wavefront does not change for different values of . In addition, when increases, the absolute value of stress before its maximum decreases, Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

while increases after the maximum. Similar results can be observed for C–T theory with t2 ¼ 0 and the L–S theory as well. 5 CONCLUSIONS Thermopiezoelectric response of a 1D FGP medium is considered. The medium is subjected to a moving heat source; the analysis is performed based on the generalized thermoelasticity C–T theory. The theory shows a good agreement with experiments in


Fig. 17

Fig. 18

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Effect of non-homogeneity index on the temperature distribution

Effect of non-homogeneity index on the electric potential distribution

microscale and macroscale ranges. Specifically in high-rate heating applications, the non-equilibrium thermodyanamic transition and the microscopic effects in the energy exchange are significant issues which the C–T theory addresses. The formulation includes the generalized thermoelasticity based on L–S theory as well. The material properties of the FGP medium are assumed to vary continuously through the length of the medium following an exponential function. Solution of the problem is found

analytically in Laplace domain, and then numerically is inverted to time domain by fast Laplace inverse transform. The results are exactly reduced to those reported for the coupled thermoelasticity analysis based on L–S theory when t1 ¼ t2 ¼ 0. According to the type of loading, the geometry of a medium, and the thermal properties of materials, the heat flux vector can precede of temperature gradient or vice versa. Using the C–T theory with the two phase lags of and q supports us to consider the Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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Fig. 19

Effect of non-homogeneity index on the stress distribution

aforementioned characteristic. Our investigation verifies that the results based on the C–T theory with t2 ¼ 0 reduces to those results due to classical coupled thermoelasticity, when q ¼ . In C–T theory, q and can be interpreted as two relaxation time, while in L–S theory, we have got only one relaxation time. However, the C–T theory with t2 ¼ 0 cannot observe the wave front in the stress and the temperature distribution. Furthermore, the location of wave front in the temperature and the stress distribution according to C–T theory with t2 6¼ 0 is ahead of the location based on L–S theory. In addition, increasing the phase lag of rises the finite speed of wave propagation in the stress and the temperature, therefore it leads to wave front moves ahead further. It should be mentioned that the maximum temperature at each time occurs at the location of heat source for L–S theory but this phenomena only can be detected for C–T theory when the value of decreases. Carrying out the analysis provides us to study the effects of non-homogeneity index on the response of an FGP medium for C–T and L–S theories as well. The results show that an increase of decreases the absolute value of displacement, temperature, and electric potential at each time based C–T and L–S theories. Using the DPL theories provide a general macroscopic description of the multiphysical behaviour of functionally graded materials that can consider the microscopic effects and include the other generalized thermoelasticity theories. ß Authors 2011 Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

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APPENDIX Notation cijkl cijkl Di eijk Ek Kij pi qi R0 S t t1 and t2 T T0 u ij "kl 2ij ij q

elastic coefficients specific heat electric displacement piezoelectric coefficient electric field heat conduction coefficient pyroelectric coefficient heat flux vector intensity of the heat flux entropy per unit mass time parameters of C–T theory absolute temperature ambient temperature displacement component in the x direction thermal moduli strain tensor dielectric coefficient non-homogeneity index electric potential density stress tensor relaxation time phase lag of heat flux phase lag of temperature gradient temperature change generalized material property generalized material property
