1

3D Exact Thermoelectroelastic Analysis of Piezoelectric

Laminated Plates

G.M. Kulikov and S.V. PlotnikovaG.M. Kulikov and S.V. Plotnikova

Speaker: Gennady KulikovSpeaker: Gennady Kulikov

Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics

3D Exact Thermoelectroelastic Analysis of Piezoelectric

Laminated Plates

G.M. Kulikov and S.V. PlotnikovaG.M. Kulikov and S.V. Plotnikova

Speaker: Gennady KulikovSpeaker: Gennady Kulikov

Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics

2

Description of Temperature Field

Description of Temperature Field

(1)

(2)

Temperature Gradient in Orthonormal Basis

N - number of layers; In - number of SaS of the nth layer

n = 1, 2, …, N; in = 1, 2, …, In; mn = 2, 3, …, In - 1

)2(232

cos21

)(21

,

][3

]1[3

)(3

][3

)(3

]1[3

1)(3

n

nn

nnmn

nInnn

Im

hn

n

(n)1, (n)2, …, (n)I - sampling surfaces (SaS)

(n)i - transverse coordinates of SaS

[n-1], [n] - transverse coordinates of interfaces

n

n

3 3

33

3,3, ,1

TTcA

T(1, 2, 3) – temperature; c= 1+k3 – components of shifter tensor at SaS

A(1, 2), k(1, 2) – Lamé coefficients and principal curvatures of midsurface

3

Temperature Distribution in Thickness Direction

Temperature Gradient Distribution in Thickness Direction

][33

]1[33

)()()( ,, nninin

i

ininn nn

n

nn TTTLT

n

nnnn

i

nnini

ini

ini

inni L ][

33]1[

3)(

3)()()()( ),(,

nnnn

nn

ijjnin

jninL

)(3

)(3

)(33)(

T(n)i (1, 2) – temperatures of SaS; L(n)i (3) – Lagrange polynomials of degree In - 1n n

(3)

(4)

(5)

(n)i (1, 2) – temperature gradient at SaSn

i

4

Relations between Temperature Gradient and Temperature

nn

n inin

in TcA

)(,)(

)( 1

n

nnnn

j

jninjnin TM )()(3

)()(3 )(

nnjik

knjn

knin

injninjn ijM

nnnnn

nn

nn

nn

for

1)(

,)(

3)(

3

)(3

)(3

)(3

)(3

)(3

)(

nn

nn

nnnn jnjn

ij

injninin LMMM )(3,

)()(3

)()(3

)( ,)()(

nnn ininin kcc 33 1

c(n)i (1, 2) – components of shifter tensor at SaSn

(6)

(7)

(8)

Derivatives of Lagrange Polynomials at SaS

5

Description of Electric FieldDescription of Electric FieldElectric Field E in Orthonormal Basis

Electric Potential Distribution in Thickness Direction

Electric Field Distribution in Thickness Direction

n

nnnn

i

nnini

ini

ini

inni EEELE ][

33]1[

33)()()( ,,

n

nnnnnn

n

j

jninjnininin

in MEcA

E )()(3

)()(3

)(,)(

)( )(,1

3,3, ,1

EcA

E

][33

]1[33

)()()( ,, nninin

i

ininn nn

n

nnL

Relations between Electric Field and Electric Potential

(9)

(10)

(11)

(12)

(n)i (1, 2) – electric potentials of SaSn

E(n)i (1, 2) – electric field at SaSn

i

6

n

3 3

33n

(13)

(14)

(15)3)(

3)()(

,)( , egeRg

nnnn inininin cA

3)(

3)( erR nn inin

Position Vectors and Base Vectors of SaS

(n)1, (n)2, …, (n)I - sampling surfaces (SaS)

(n)i - transverse coordinates of SaS

[n-1], [n] - transverse coordinates of interfaces

)2(232

cos21

)(21

,

][3

]1[3

)(3

][3

)(3

]1[3

1)(3

n

nn

nnmn

nInnn

Im

hn

n

Kinematic Description of Undeformed ShellKinematic Description of Undeformed Shell

N - number of layers; In - number of SaS of the nth layer

n = 1, 2, …, N; in = 1, 2, …, In; mn = 2, 3, …, In - 1

r(1, 2) – position vector of midsurface ; ei(1, 2) – orthonormal base vectors of midsurface

7

u (1, 2) – displacement vectors of SaS(n)i n

((16)

(17)

(18)

Base Vectors of Deformed SaS

Position Vectors of Deformed SaS

)(,, )(33,

)()(3

)(3

)(,

)()(,

)( nnnnnnnn inininininininin uegugRg

(n)i n (1, 2) – values of derivative of displacement vector at SaS

Kinematic Description of Deformed ShellKinematic Description of Deformed Shell

nnn ininin )()()( uRR

)( )(3

)( nn inin uu

u (1, 2, 3) – displacement vector

8

Green-Lagrange Strain Tensor at SaS

Linearized Strain Tensor at SaS

Displacement Vectors of SaS in Orthonormal Basis

(19)

(20)

(21)

)(1

2 )()()()()()(

)( nnnn

nn

n inj

ini

inj

iniin

jin

iji

inij

ccAAgggg

3)()(

333)(

,)()()(

3

)(,)(

)(,)(

)(

,1

2

112

eeue

eueu

nnnn

nn

n

nn

n

n

inininin

inin

inin

inin

in

cA

cAcA

iin

iin

iin

iin nnnn u ee )()()()( , u

9

Derivatives of Displacement Vectors in Orthonormal Basis

Strain Parameters of SaS

Linearized Strains of SaS

Remark. Strains (24) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through results of Kulikov and Carrera (2008)

(22)

(23)

(24)

nnn

nnn

n

n

n

n

n

inininin

inin

inin

inin

in

c

cc

)(3

)(33

)(3)(

)()(3

)()(

)()(

)(

,1

2

112

,)()(

,3)(

3

)()(,

)()(3

)()(,

)(

1,

1

1,

1

AAA

BukuA

uBuA

ukuBuA

nnn

nnnnnnn

ininin

ininininininin

iin

iin nn

Aeu )()(

,1

10

Displacement Distribution in Thickness Direction

Strain Distribution in Thickness Direction

Presentation for Derivative of Displacement Vector

(25)

(26)

(27)

][33

]1[3

)()()( , nn

i

ini

inni

n

nnuLu

nn

n

nnnn jnjn

j

jni

injnini LMuM )(

3,)()()(

3)()( ,)(

n

nn

i

nninij

innij L ][

33]1[

3)()()( ,

11

Variational Formulation of Heat Conduction Problem

0J

dTQdddccAAqJ

n

ni

ni

n

nn3212121

)()(

][3

]1[3

21

][3

]1[3

321)()()(

n

n

nn dccLqQ inni

ini

Variational Equation

q(n) – heat flux of the nth layer; Qn – specified heat fluxi

Heat Flux Resultants

(28)

(29)

(30)

12

(31)

(32)

(33)

dTQddAAQJn i

ini

ini

n

nnn2121

)()(

21

][33

]1[3

)()()( ,Γ nnnj

nij

ni kq

][3

]1[3

321)()()()()()()( ,

n

n

nnnnn

n

nnn dccLLkQ jninjinjnj

nij

j

jinini

Fourier Heat Conduction Equations

Basic Functional of Heat Conduction Theory

k(n) – thermal conductivity tensor of the nth layerij

Heat Flux Resultants

13

(n) = T (n) - T0 – temperature rise; W – work done by external loads

(34)

(35)

(36)

(37)

(38)

Stress Resultants

Electric Displacement Resultants

Entropy Resultants

Formulation of Thermopiezoelectric Problem

][3

]1[3

321)()()(

n

n

nn dccLH innij

inij

2121Ω

)()()()()()( Θ21

Π ddAASERHn i

ininini

ini

inij

inij

n

nnnnnn

][3

]1[3

321)()()(

n

n

nn dccLDR inni

ini

][3

]1[3

321)()()(

n

n

nn dccLS innin

0 W

14

Constitutive Equations

(39)

(40)

(41)

(42)

(43)

(44)

Stress, Electric Displacement and Entropy Resultants

][33

]1[3

)()()()()()()( ,Θ nnnnij

nk

nkij

nk

nijk

nij EeC

][33

]1[3

)()()()()()( ,Θ nnnni

nk

nik

nk

nik

ni rEeD

][33

]1[3

)()()()()()( ,Θ nnnnnk

nk

nkl

nkl

n Er

(n)Cijk , ekij , ij , ik , ri , (n) – physical properties of the nth layer (n) (n)

(n) (n)

nnn

n

nnn jnnij

jnk

nkij

jnk

nijk

j

jininij EeCH )()()()()()()()( ΘΛ

nnn

n

nnn jnni

jnk

nik

jnk

nik

j

jinini rEeR )()()()()()()( ΘΛ

nnn

n

nnn jnnjnk

nk

jnkl

nkl

j

jinin ErS )()()()()()()( ΘΛ

15

Numerical ExamplesNumerical Examples1. Elastic Rectangular Plate under Mechanical Loading

Analytical solution

Table 1. Results for a single-layer square plate with E = 107 Pa, = 0.3, a = b = 1 m and a/h = 3

sr

inrs

in

b

xs

a

xruu nn

,

21)(1

)(1 sincos

sr

inrs

in

b

xs

a

xruu nn

,

21)(2

)(2 cossin,

sr

inrs

in

bxs

axr

uu nn

,

21)(3

)(3 sinsin

Dimensionless variables in crucial points

))1(2/(,/),2/,2/(,/),2/,0( 033011 EGhpzaaGuuhpzaGuu

hxzpzaapzapzaa /,/),2/,2/(,/),2/,0(,/),2/,2/( 3033330131301111

In -u1(0.5) u3(0) 11(0.5) 13(0) 33(0)

3 0.3986372494504680 1.278878243389591 1.999302146854837 0.4977436151168003 0.4752837277628003

70.4358933937131948 1.342554953466513 2.124032428288413 0.7022762666060094

0.4943950643281928

110.4358933942603120 1.342554689543095 2.124018410314048 0.7023022083223538

0.4944039935419638

150.4358933942603121 1.342554689542491 2.124018410193782 0.7023022084767580

0.4944039936052152

190.4358933942603120 1.342554689542491 2.124018410193780 0.7023022084767578

0.4944039936052150

Exact 0.4358933942603120 1.342554689542491 2.124018410193781 0.7023022084767578 0.4944039936052149

Exact results have been obtained by authors using Vlasov's closed-form solution (1957)

16

In -u1(0.5) u3(0) 11(0.5) 13(0) 33(0)

3 16.69442044348862 112.4420178073405 19.91690124596434 1.706202131658078 0.4978329738513218

7 16.81014395019634 113.1720941024937 20.04394635519068 2.383373139214406 0.4999497585886086

11 16.81014395019644 113.1720941004914 20.04394633189751 2.383373791176881 0.4999497657893389

15 16.81014395019644 113.1720941004913 20.04394633189750 2.383373791176909 0.4999497657893389

19 16.81014395019645 113.1720941004914 20.04394633189751 2.383373791176910 0.4999497657893400

Exact 16.81014395019644 113.1720941004914 20.04394633189751 2.383373791176911 0.4999497657893391

Table 2. Results for a single-layer square plate with E = 107 Pa, = 0.3, a = b = 1 m and a/h = 10

Exact results have been obtained by authors using Vlasov's closed-form solution (1957)

Figure 1. Through-thickness distributions of transverse stresses of the plate with a/h = 1: present

analysis ( ─ ) for I1 = 11 and Vlasov's closed-form solution ()

17

2. Piezoelectric Plate in Cylindrical Bending for Heat Flux Boundary Conditions

Analytical solution

Dimensionless variables in crucial points

C/N109238.3W/mK,9K,/110396.4K,293,m1 1200

600

dkTa

002

003

02

0 /),2/(10,/),2/(10 qazadkqazahk

hxzqazaukuqazuhku /,/),2/(100,/),0(100 3002

303003

101

r

inr

in

r

inr

in

axr

axr nnnn 1)()(1)()( sin,sinΘΘ

0,cos )(2

1)(1

)(1

nnn in

r

inr

in ua

xruu

r

inr

in

a

xru nn 1)(

3)(

3 sinu

18

Table 4. Results for a single-layer Cadmium-Selenide plate in case of heat flux boundary conditions with a/h = 10

In (0.5) (0.5) u1(0.5) u3(0)

3 1.035337086088057 -1.329936076735578 -4.750069515595288 4.830450207594048

7 1.035337187913652 -1.329961621213969 -4.749949149330288 4.829776282323792

11 1.035337187913652 -1.329961621213969 -4.749949149330288 4.829776282323684

15 1.035337187913653 -1.329961621213969 -4.749949149330288 4.829776282323684

19 1.035337187913652 -1.329961621213969 -4.749949149330288 4.829776282323683

Table 3. Results for a single-layer Cadmium-Selenide plate in case of heat flux boundary conditions with a/h = 2

In (0.5) (0.5) u1(0.5) u3(0)

3 1.514830974804894 -1.276585132031578 -6.992090213777479 4.039437427219185

7 1.516041850892555 -1.287176996356866 -6.936228360092725 4.045924923554766

11 1.516041850893569 -1.287176996676169 -6.936228359388562 4.045924921944983

15 1.516041850893570 -1.287176996676170 -6.936228359388562 4.045924921942870

19 1.516041850893570 -1.287176996676169 -6.936228359388563 4.045924921942869

23 1.516041850893570 -1.287176996676170 -6.936228359388563 4.045924921942869

19

3. Piezoelectric Plate in Cylindrical Bending for Convective Boundary Conditions

Dimensional variables in crucial points

00330 /),2/(10,/),2/( Skzaaqqza

000032

30003 /),2/(,/),2/(10 EdzaDSDhzad

002

330011 /),2/(100,/),0(10 hSzauuShzuu

0001333

130001123

11 /),0(10,/),2/(10 EzSEzaS

hxzhaSEzaEzaS /,/,/),2/(10,/),2/(10 30200

300033

4333

W/mK9K,/110396.4,2,2.0K,293,m1 06

00 khhhhTa

GPa785.42C/N,109238.3 012

0 Ed

Analytical solution

r

inr

in

r

inr

in

axr

axr nnnn 1)()(1)()( sin,sinΘΘ

0,cos )(2

1)(1

)(1

nnn in

r

inr

in ua

xruu

r

inr

in

a

xru nn 1)(

3)(

3 sinu

20

Table 5. Results for a single-layer Cadmium-Selenide plate in case of convective boundary conditions with a/h = 2

In (-0.5) (0.5) (0.5) u1(0.5) u3(0.5) 11(0.5) 13(-0.25) 33(0) D3(0) q3(0.5) (0.5)

5 0.28881 0.63570 -5.2859 -2.9088 9.9580 -15.682 3.1848 -2.8412 -1.3398 -10.920 3.9685

7 0.28881 0.63570 -5.2859 -2.9087 9.9580 -13.098 3.6968 3.9052 -1.3255 -10.929 3.9685

9 0.28881 0.63570 -5.2859 -2.9087 9.9580 -13.089 3.6938 3.8899 -1.3256 -10.929 3.9685

11 0.28881 0.63570 -5.2859 -2.9087 9.9580 -13.089 3.6938 3.8898 -1.3256 -10.929 3.9685

13 0.28881 0.63570 -5.2859 -2.9087 9.9580 -13.089 3.6938 3.8898 -1.3256 -10.929 3.9685

Exact 0.2888 0.6357 -5.286 -2.909 9.958 -13.09

Table 6. Results for a single-layer Cadmium-Selenide plate in case of convective boundary conditions with a/h = 10

In (-0.5) (0.5) (0.5) u1(0.5) u3(0.5) 11(0.5) 13(-0.25) 33(0) D3(0) q3(0.5) (0.5)

5 0.72894 0.90045 -10.953 -4.1311 2.8608 -6.9137 1.7532 -9.9104 -3.3167 -2.9865 5.6212

7 0.72894 0.90045 -10.953 -4.1311 2.8608 -6.7182 1.9635 2.0573 -3.3154 -2.9865 5.6212

9 0.72894 0.90045 -10.953 -4.1311 2.8608 -6.7182 1.9634 2.0579 -3.3154 -2.9865 5.6212

11 0.72894 0.90045 -10.953 -4.1311 2.8608 -6.7182 1.9634 2.0579 -3.3154 -2.9865 5.6212

Exact 0.7289 0.9004 -10.95 -4.131 2.861 -6.718

Exact results have been obtained by Dube, Kapuria, Dumir (1996)

Exact results have been obtained by Dube, Kapuria, Dumir (1996)

21

Figure 2. Through-thickness distributions of temperature, heat flux, electric displacement, transverse stresses and entropy of the single-layer Cadmium-Selenide

plate in case of convective boundary conditions for In = 11

22

4. Laminated Piezoelectric Rectangular Plate under Temperature Loading

sr

inrs

in

b

xs

a

xrnn

,

21)()( ,sinsinΘΘ

sr

inrs

in

bxs

axrnn

,

21)()( sinsin

,sincos,

21)(1

)(1

sr

inrs

in

b

xs

a

xruu nn

sr

inrs

in

bxs

axr

uu nn

,

21)(2

)(2 cossin

sr

inrs

in

bxs

axr

uu nn

,

21)(3

)(3 sinsin

Analytical solution

Dimensionless variables in crucial points

00330 Θ/),2/,2/(10,/Θ),2/,2/(ΘΘ Skzaaaqqzaa

0000330003 Θ/),2/,2/(,Θ/),2/,2/(10 EdzaaDDazaad

00330011 Θ/),2/,2/(100,Θ/),2/,0(10 Sazaauuazauu

00012120001111 Θ/),0,0(10,Θ/),2/,2/(10 EzEzaa

000332

330001313 Θ/),2/,2/(10,Θ/),2/,0(10 EzaaSEzaS

hxzhaSEzaa /,/,Θ/),2/,2/(10 30200

3

GPa785.42C/N,109238.3 012

0 Ed

W/mK9K,/110396.4K,293,m1 06

00 kTba

23

Table 7. Results for a two-layer PZT5A/Cadmium-Selenide plate with a/h = 2

In (0) (0) u1(0.5) u3(0.5) 11(0.5) 12(0.5) 13(0) 33(0) D3(0) q3(0) (0.5)

3 0.60597 3.3660 -2.3924 10.317 -4.8826 -5.0768 1.64751.4076

-2.3110-3.8790

10.220-0.38923

-3.0893-2.6675

6.2427

5 0.60583 3.3689 -2.3801 10.215 -4.6437 -5.0506 1.34211.3515

-1.7144-1.6464

-0.17554-0.31134

-3.3443-3.3422

6.2428

7 0.60583 3.3689 -2.3801 10.215 -4.6431 -5.0506 1.35491.3500

-1.6483-1.6475

-0.30981-0.31035

-3.3465-3.3465

6.2428

9 0.60583 3.3689 -2.3801 10.215 -4.6432 -5.0506 1.35011.3501

-1.6477-1.6477

-0.31034-0.31035

-3.3465-3.3465

6.2428

11 0.60583 3.3689 -2.3801 10.215 -4.6432 -5.0506 1.35011.3501

-1.6477-1.6477

-0.31035-0.31035

-3.3465-3.3465

6.2428

Table 8. Results for a two-layer PZT5A/Cadmium-Selenide plate with a/h = 10

In (0) (0) u1(0.5) u3(0.5) 11(0.5) 12(0.5) 13(0) 33(0) D3(0) q3(0) (0.5)

3 0.86748 6.6728 -2.8460 6.2762 -1.7075 -6.0394 1.67353.3784

-6.5647-8.2434

1.8313-1.8259

-3.5125-3.5162

6.2434

5 0.86748 6.6726 -2.8455 6.2730 -1.6988 -6.0384 2.71172.7139

-6.3918-6.3860

-1.8001-1.8020

-3.5268-3.5268

6.2434

7 0.86748 6.6726 -2.8455 6.2730 -1.6989 -6.0384 2.71322.7132

-6.3876-6.3876

-1.8020-1.8020

-3.5268-3.5268

6.2434

9 0.86748 6.6726 -2.8455 6.2730 -1.6989 -6.0384 2.71322.7132

-6.3876-6.3876

-1.8020-1.8020

-3.5268-3.5268

6.2434

24

Figure 3. Through-thickness distributions of temperature, heat flux, electric potential, transverse stresses and entropy of the two-layer

PZT5A/Cadmium-Selenide plate for I1 = I2 = 9

25

In u1(-0.5) u3(-0.5) 11(-0.5) 12(-0.5) 13(0.125) 23(0.125) (-0.5) (-0.5) D3(0.25)

3 1.1318 6.4450 -4.8241 1.9546 7.3267 8.5498 2.5402 -1.5100 -4.1738

5 1.1328 6.4493 -4.8174 1.9567 7.4820 8.8193 2.5433 -1.5156 -1.8940

7 1.1328 6.4493 -4.8174 1.9567 7.4805 8.8168 2.5433 -1.5156 -1.8911

9 1.1328 6.4493 -4.8174 1.9567 7.4805 8.8168 2.5433 -1.5156 -1.8911

5. Antisymmetric Angle-Ply Plate Covered with PZT Layers

Figure 4. [PZT/60/60/PZT] square plate with grounded interfaces under mechanical loading for r = s = 1

Table 9. Results for the unsymmetric angle-ply plate with a/h = 4 under mechanical loading

�

bxs

axr

ubxs

axr

uu nnn ininin 21)(10

21)(10

)(1 cossin~sincos

bxs

axr

ubxs

axr

uu nnn ininin 21)(20

21)(20

)(2 sincos~cossin

bxs

axr

ubxs

axr

uu nnn ininin 21)(30

21)(30

)(3 coscos~sinsin

bxs

axr

bxs

axr nnn ininin 21)(

021)(

0)( coscos~sinsin

Analytical solution

26

Figure 5. Mechanical loading of the unsymmetric angle-ply plate: distributions of transverse shear stresses and electric displacement through the thickness of the plate for I1 = I2 = I3 = I4 =7

27

In u1(-0.5) u3(-0.5) 11(-0.5) 12(-0.5) 13(0.125) 13(0.125) 23(0.125) 23(0.125) D3(0.25)

5 35.268 90.078 25.657 52.285 1.8363 -2.6948 18.563 -7.2148 -17212

7 35.268 90.078 25.657 52.285 1.7915 -2.6507 18.487 -7.1405 -17213

9 35.268 90.078 25.657 52.285 1.7917 -2.6508 18.488 -7.1407 -17213

6. Antisymmetric Angle-Ply Plate Covered with PZT Layers

Table 10. Results for the unsymmetric angle-ply plate with a/h = 4 under electric loading

�

�

bxs

axr

ubxs

axr

uu nnn ininin 21)(10

21)(10

)(1 cossin~sincos

bxs

axr

ubxs

axr

uu nnn ininin 21)(20

21)(20

)(2 sincos~cossin

bxs

axr

ubxs

axr

uu nnn ininin 21)(30

21)(30

)(3 coscos~sinsin

bxs

axr

bxs

axr nnn ininin 21)(

021)(

0)( coscos~sinsin

Analytical solution

Figure 6. [PZT/60/60/PZT] square plate with grounded interfaces under electric

loading for r = s = 1

28

Figure 7. Electric loading of the unsymmetric angle-ply plate: distributions of transverse stresses through the thickness of the plate for I1 = I2 = I3 = I4 =7

29

Thanks for your attention!Thanks for your attention!