Solution Procedures

4.4 Coupled Thermopiezoelectricity in One-Dimensional

4.4.3 Solution Procedures

The following non-dimensional parameters are introduced to streamline the solution procedure:

g¼q0C K0

;c ¼ ffiffiffiffiffic0

q0

r

;ðx;uÞ ¼cgðx;uÞ;ðt;s;vÞ ¼c²gðt;s;vÞ

h¼ h h0

;/¼ 20

e₀L/;R¼ R

K₀h0c²g²;rxx¼rxx

c₀ ;D_x¼Dx

e₀;l¼Lcg

ð4:53Þ

in which, K0,c0,q0,e0 and 20 are the thermal conductivity, elastic coefficients, density, piezoelectric and dielectric coefficients atx¼0, respectively. The material properties of the rod vary exponentially along the x-axis, following the relation shown below:

v¼v0e^kx ð4:54Þ

where v is an arbitrary material property, andk is an arbitrary non-homogeneity index. Finally, the moving thermal disturbance is defined in the following form:

R¼R₀dðxttÞ ð4:55Þ whereR0andtare the intensity and velocity of the heat source whiledis the Dirac delta function.

a₉h;xxþa₁a₉h;xa₁₁ha₆a₁₀u_;xþa₇a₁₀/_;x¼ a5a10

s e^stx ð4:56dÞ where

a₁¼ k

cg;a₂¼ e²₀L 20c0

;a₃¼b0h⁰ c0

;a₄¼p0h⁰ e0

;a₅¼q0R0

t ;a₆¼ b0

qC a7¼p0ce0L

20K0 ;a8¼ð1þsvÞ;a9 ¼1;a10¼sð1þn0ssÞ a11 ¼s 1þssð1þn0Þ þs²s²n0

ð4:57Þ

For convenience, overbar and tilde signs have been omitted in the above equations.

4.4.3.2 Coupled Thermopiezoelectricity Analysis

In this section, the coupled thermopiezoelectrical response of the FGPM rod is analyzed based on the classical and generalized L-S theories. As such, in Eq. (4.56), only the energy equation based on the L-S theory is considered. The solution of the linear ordinary differential equations of Eq. (4.56) contains two components; the particular solution and the general solutions. The former can be written in the following format:

u_p;/p;hp

¼P_u;P_/;P_h

e^stx ð4:58Þ Here, subscript p is used to denote the particular solution. By substituting Eq. (4.58) into Eq. (4.56), we can solve the algebraic equation and obtain:

s t

2a1s

ts² I ^s_t ²a1s t

a3a₈^s_tþa₁

s t

2a1s

t J ^s_t ²a1s t

a₄ a8s tþa₁a₈

a6a10s

t a7a10s

t a9 s

t 2a1s

t

a10

2 66 64

3 77 75

Pu

P_/ P_h 8<

: 9=

;¼ 0 0 ^a5_s^a10 8<

:

9=

;

ð4:59Þ where,I¼a2cgand J¼Lcg. To obtain the general solution, we successively eliminate/andu in the governing equations, which results in an ordinary differ- ential equation containing h. Then we find hg;x from the third equation of the homogenous form of Eq. (4.56):

/g;x¼ a9

a7a10hg;xx a1a9

a7a10hg;xþ 1

a7hgþa6

a7

ug;x ð4:60Þ

4.4 Coupled Thermopiezoelectricity in One-Dimensional… 151

where subscript g is used to indicate the general solution. Using Eq. (4.60), the governing equations are reduced to:

A₁u_g;xxþa₁A₁u_g;xþA₂hg;xxxþ2a₁A₂hg;xxþA₃hg;xþa₁A₄hg¼0 ð4:61aÞ B1ug;xxþa1B1ug;xs²ugB2hg;xxx2a1B2hg;xxþB3hg;xþa1B4hg¼0 ð4:61bÞ where

A₁¼1Ja₆ a7

;A₂¼ Ja₉ a7a10

;A₃¼ J a7

a²₁a₉ a10

1

þa₄a₈;A₄¼ J a7

þa₄a₈

B1¼1þIa6

a₇ ;B2¼ Ia9

a₇a₁₀;B3¼ I

a₇ 1 a²₁a9

a₇a₁₀

a3a8;B4¼ I a₇a3a8

ð4:62Þ To obtain the following equation forug, we can multiply Eq. (4.61b) by^A_B¹

1and add this result with (4.61a):

u_g¼D₁hg;xxxþ2a₁D₁hg;xþD₂hg;xþa₁D₃hg ð4:63Þ in which,

D1¼ 1

s² B2þB1A2

A1

;D2¼ 1

s² B3B1A3

A1

;D3¼ 1

s² B4B1A4

A1

ð4:64Þ

Substituting Eq. (4.63) and its derivatives into Eq. (4.61a) leads to the ordinary differential equation with constant coefficients below:

E1hg;xxxxxþ3a1E1hg;xxxxþE2hg;xxxþa1E3hg;xxþE4hg;xþa1A4hg¼0 ð4:65Þ where

E1¼A1D1;E2¼A1 D2þ2a²₁D1

þA2;E3¼A1ðD2þD3Þ þ2A2;E4

¼a²₁A1D3þA3 ð4:66Þ Solving Eq. (4.65) allows us to obtain the following characteristic equation:

E1f⁵þ3a1E1f⁴þE2f³þa1E3f²þE4fþa1A4¼0 ð4:67Þ Analytical methods have been proposed for solving quintic equations, such as using the Hermit-Kronecker method and the Mellin method [36,37]. However, it can be noted that one of the characteristic roots of Eq. (4.67) isf1¼ a1, and as such we can factor this term from Eq. (4.67). Therefore, it is possible to analytically

solve the resulting fourth order algebraic equation [38] and the solution for the temperature can be written as follows:

hðx;sÞ ¼hgþhp¼C_hie^fixþP_he^stx ð4:68Þ Substituting Eq. (4.68) into Eqs. (4.63) and (4.60) leads to the following equations for displacement and electric potential:

u xð ;sÞ ¼Cu_iC_hie^fixþPue^stx ð4:69aÞ /ðx;sÞ ¼C_/iC_hie^fixþC⁰þP_/e^stx ð4:69bÞ whereC_hi andC⁰are integration constants.C_uiandC_/iare defined in the following forms:

Cu_i ¼D1f³i þ2a1D1f²iþD2fiþa1D3 ð4:70aÞ

C_/i¼ 1 a7

a₉

a10fia₁a₉ a10

þ 1

fiþa6Cui

ð4:70bÞ

Applying the boundary conditionu;/;^@h_@xx¼0

;l¼ð0;0;0Þon the system results in the following system of equations which can be solved for the integration constants:

Cu1 Cu2 Cu3 Cu4 Cu5 Cu6

Cu1e^f1l Cu2e^f2l Cu3e^f3l Cu4e^f4l Cu5e^f5l Cu6e^f6l C_/1 C_/2 C_/3 C_/4 C_/5 C_/6 C_/1e^f1l C_/2e^f2l C_/3e^f3l C_/4e^f4l C_/5e^f5l C_/6e^f6l

f1 f2 f3 f4 f5 f6

f1e^f1l f2e^f2l f3e^f3l f4e^f4l f5e^f5l f6e^f6l 0

BB BB BB

@

1 CC CC CC A

C_h1 C_h2 C_h3 C_h4 C_h5 C⁰ 8>

>>

>>

><

>>

>>

>>

: 9>

>>

>>

>=

>>

>>

>>

;

¼

Pu

Pue^svl P_/ P_/e^svl

P_h^s_v P_h^s_ve^svl 8>

>>

>>

><

>>

>>

>>

:

9>

>>

>>

>=

>>

>>

>>

;

ð4:71Þ

Moreover, the normalized stress and electric displacement can be found in the Laplace domain as shown below:

rx¼ C_hi fiCuiþ^a_L^2JC_/i

a3ð1þvsÞ

e^{fix s}_tP_uþ^a_L^2JP_/

þa3ð1þvsÞP_h

e^a1x ð4:72aÞ

D1¼ C_hi fiC_uiJC_/i

þa4ð1þvsÞ

e^fix _t^sP_uJP_/

a4ð1þvsÞP_h

e^a1x ð4:72bÞ

4.4 Coupled Thermopiezoelectricity in One-Dimensional… 153

In the case where the strain rate or the time rate of change of thermal sources is relatively low, the displacement effects are ignored in the energy equation. Then we can solve the governing equations with a similar approach by using the following values:

a6 ¼0;a8¼1;a9¼1;a10 ¼s;a11 ¼s ð4:73Þ

4.4.3.3 Uncoupled Thermopiezoelectricity Analysis

In this section, the FGPM rod is studied on the basis of classical uncoupled ther- mopiezoelectricity. The classical uncoupled theory does not take into account the coupling effect of strain and electric potential on temperature. Despite thisflaw, it remains accurate enough to successfully model many engineering applications, especially if the rate of strain and electricfield and relatively small [39]. When this occurs, the governing equations from Eq. (4.56) are simplified as follows:

u_;xxþa₁u_;xs²uþa₂cg/_;xxþa₁a₂cg/_;xa₃h;xa₁a₃h¼0 ð4:74aÞ u_;xxþa1u_;xLcg/_;xxa1Lcg/_;xþa4h;xþa1a4h¼0 ð4:74bÞ h;xxþa1h;xsh¼ a5e^stx ð4:74cÞ Multiplying the second equation of Eq. (4.74) by^a_L²and summing it to thefirst equation:

F1u_;xxþa1F1u_;xs²uþF2h;xþa1F2h¼0 ð4:75aÞ h;xxþa₁h;xsh¼ a5e^stx ð4:75bÞ where

F₁¼1þa2

L F₂¼a2a4

L a₃ ð4:76Þ

It can be noted that the second equation of Eq. (4.75) is an ordinary differential equation. As such, the solution can be obtained as:

h¼C₁e^r1xþC₂e^r2xþhp ð4:77aÞ hp¼ a₅

sv 2a1 s

v se^svx ð4:77bÞ

r1;2¼a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²₁þ4s p

2 ð4:77cÞ

Substituting the temperature relation from Eq. (4.77) into Eq. (4.75a) leads to:

F1u_;xxþa1F1u_;xs²u¼ F2C1ðr1þa1Þe^r1xF2C2ðr2þa1Þe^r2x F2hp s

vþa1

ð4:78Þ

The solution of Eq. (4.78) contains two parts; the general solution and the particular solution:

u¼C3e^n1xþC4e^n2xþup1þup2þup3 ð4:79aÞ

n1;2¼a1F1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²₁F₁²þ4F1s² p

2F1

ð4:79bÞ

u_p1¼ F2C1ðr1þa1Þ

F1r²₁þa1F1r1s²e^r1x ð4:79cÞ u_p2¼ F2C1ðr2þa1Þ

F1r²₂þa1F1r2s²e^r2x ð4:79dÞ

u_p3¼ F₂C_h1^s_vþa₁

F₁ ^s_v ²a1F₁ ^s_v s²e^svx ð4:79eÞ Inserting the displacement and the temperature equations into Eq. (4.74b) results in the ordinary differential equation for electric potential written below:

Lcg/_;xxþa1Lcg/_;x¼C1a4ðr1þa1Þe^r1xþC2a4ðr2þa1Þe^r2x þhpa4 s

v

þa1

þC3 n²1þa1n1

e^n1xþC4 n²2þa1n2

e^n2x þup1 r²₁þa1r1

þup2 r₂²þa1r2

þup3

s v

2a1

s v

ð4:80Þ By solving Eq. (4.80), we complete the solution procedure for the uncoupled thermoelasticity analysis:

/¼C5e^a1xþC6þ/p₁þ/p₂þ/p₃þ/p₄þ/p₅ ð4:81aÞ

/p₁¼ C3 n²1þa1n1

Lcgn²1þa₁Lcgn1

e^n1x ð4:81bÞ

4.4 Coupled Thermopiezoelectricity in One-Dimensional… 155

/p2¼ C4 n²2þa1n2

Lcgn²2þa1Lcgn2

e^n2x ð4:81cÞ

/p₃¼up₁ r²₁þa1r1

þC1a4ðr1þa1Þe^r1x Lcgr²₁þa1Lcgr1

ð4:81dÞ

/p4¼up₂ r²₂þa1r2

þC2a4ðr2þa1Þe^r2x

Lcgr²₂þa₁Lcgr₂ ð4:81eÞ

/p5 ¼up3

sv 2a1 s

v

þhpa4 _v^sþa1

Lcg ^s_v ²a1Lcg ^s_v ð4:81fÞ

4.4.3.4 Numerical Inversion of the Laplace Transform

Having obtained the coupled and uncoupled solutions in the Laplace domain, we now employ the so-called fast Laplace inverse transform [14]. Performing this numerical inversion will transform the results into the time domain for later anal- ysis. The form of this inversion process have been previously stated in Sect.4.2.3 and is omitted here. The only change concerns the values of the constraint parameters, which are stated below:

aT¼100

15 ;Ln¼5;N¼900