1

3D Exact Analysis3D Exact Analysis ofof Functionally Graded andFunctionally Graded andLaminated Piezoelectric Laminated Piezoelectric

Plates and ShellsPlates and Shells

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

Speaker: Svetlana PlotnikovaSpeaker: Svetlana Plotnikova

Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics

3D Exact Analysis3D Exact Analysis ofof Functionally Graded andFunctionally Graded andLaminated Piezoelectric Laminated Piezoelectric

Plates and ShellsPlates and Shells

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

Speaker: Svetlana PlotnikovaSpeaker: Svetlana Plotnikova

Department of Applied Mathematics & MechanicsDepartment of Applied Mathematics & Mechanics

2

(n)i n(n)i n

n

n

3 3

33n

(1)

(2)

(3)3)(

3)(

3)(

3)()(

,)( ,, erRegeRg nnnnnn inininininin cA

Figure 1. Geometry of laminated shell

33 eer aa ,, ABase Vectors of Midsurface and SaS

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

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

)2(232

cos21

)(21

,

][3

]1[3

)(3

][3

)(3

]1[3

1)(3

n

nn

nnmn

nInnn

Im

hn

n

r(1, 2) - position vector of midsurface ; R(n)i - position vectors of SaS of the nth layer

ei - orthonormal vectors; A, k - Lamé coefficients and principal curvatures of midsurface

c = 1+k3 - components of shifter tensor at SaS

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

(n)i - transverse coordinates of SaS

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

Kinematic Description of Undeformed ShellKinematic Description of Undeformed ShellKinematic Description of Undeformed ShellKinematic Description of Undeformed Shell

3

(n)i n

(n)i n

((4)

(5)

(6)

Figure 2. Initial and current configurations of shell

Base Vectors of Deformed SaS

Position Vectors of Deformed SaS

nnn ininin )()()( uRR

)( )(3

)( nn inin uu

u (1, 2) - displacement vectors of SaS

)(,, )(33,

)()(3

)(3

)(,

)()(,

)( nnnnnnnn inininininininin uegugRg

(1, 2) - derivatives of 3D displacement vector at SaS

Kinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed ShellKinematic Description of Deformed Shell

4

Green-Lagrange Strain Tensor at SaS

Linearized Strain-Displacement Relationships

Presentation of Displacement Vectors of SaS

(7)

(8)

(9)

)(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

5

Presentation of Derivatives of Displacement Vectors of SaS

Strain Parameters

Component Form of Strains of SaS

Remark. Strains (12) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through Kulikov and Carrera (2008)

(10)

(11)

(12)

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

6

Description of Electric FieldDescription of Electric FieldDescription of Electric FieldDescription of Electric Field

Electric Field Vector at SaS

3,3,3

,)1(

1

E

kAE

– electric potential

nn

nn inin

inin

cAEE )(

,)()(

3)( 1

)(

nnn ininin EE )()(33

)(3 )(

1– electric potentials of

SaS

(n)i n

)(),( )(33,

)()(3

)( nnnn inininin

(13)

(14)

(15)

7

Displacement Distribution in Thickness Direction

Distribution of Derivatives of 3D Displacement Vector

Strain Distribution in Thickness Direction

Higher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell FormulationHigher-Order Layer-Wise Shell Formulation

(16)

(17)

(18)

(19)

][33

]1[3

)()()( , nn

i

ini

inni

n

nnuLu

nnnn

nn

ijjnin

jninL

)(3

)(3

)(33)(

nn

n

nnnn jnjn

j

jni

injnini LMuM )(

3,)()()(

3)()( ,)(

n

nn

i

nninij

innij L ][

33]1[

3)()()( ,

L (3) - Lagrange polynomials of degree In - 1 (n)i n

8

Electric Potential Distribution in Thickness Direction

Distribution of Electric Field Vector

Distribution of Derivative of Electric Potential

][33

]1[3

)()()( , nn

i

ininn

n

nnL

n

nn

i

nnini

inni ELE ][

33]1[

3)()()( ,

(20)

(21)

(22)nn

n

nnnn jnjn

j

jninjnin LMM )(3,

)()()(3

)()( ,)(

9

Variational Equation

Stress Resultants

Electric Displacement Resultants

][3

]1[3

33231)()()( )1)(1(

n

n

nn dkkLDT inni

ini

][3

]1[3

33231)()()( )1)(1(

n

n

nn dkkLH innij

inij

0

WddAAETHn i

ini

ini

inij

inij

n

nnnn

212121

W – work done by external electromechanical loads

(23)

(24)

(25)

(26)

10

Material Constants in Thickness Direction

(27)

(28)

(29)

n

nn

i

inijkl

innijkl CLC )()()(

n

nn

i

inkij

innkij eLe )()()(

n

nn

i

inik

innik L )(

Cijkl , ekij and ik – values of elastic, piezoelectric and dielectric constants on SaS of the nth layer(n)i n (n)i n (n)i n

11

Constitutive Equations

Presentations for Stress and Electric Displacement Resultants (n)

(30)

(31)

(32)

(33)

(34)

Cijk , ekij, ik – elastic, piezoelectric and dielectric constants of the nth layer (n) (n)

][33

]1[3

)()()()()( , nnnk

nkij

nk

nijk

nij EeC

][33

]1[3

)()()()( , nnnk

nik

nk

nik

ni EeD

][3

]1[3

33231)()()()( )1)(1(

n

n

nnnnnn dkkLLL knjninkjin

nnn

n

nnnn jnk

nkij

jnk

knijk

j

kjininij EeCH )()()()()()(

nnnn

n

nnnn jnk

knik

jnk

knik

j

kjinini EeT )()()()()(

12

Numerical ExamplesNumerical ExamplesNumerical ExamplesNumerical Examples1. Simply Supported Three-Layer Plate under Mechanical Loading

Analytical solution

Figure 3. PVDF [0/90/0] square plate (h = 0.01 m, p0 = 3 Pa) (r=s=1)

Table 1. Results for a piezoelectric three-ply plate with a /h = 4 under mechanical loading (Lage at al.)

sr

inrs

in

bxs

axr

uu nn

,

21)(1

)(1 sincos

sr

inrs

in

bxs

axr

uu nn

,

21)(2

)(2 cossin

sr

inrs

in

bxs

axr

uu nn

,

21)(3

)(3 sinsin

sr

inrs

in

bxs

axrnn

,

21)()( sinsin

,

,

Variable Exact In=3 In=5 In=7 In=9 In=11

u1(0, a/2, 0.005)1012, m 1.719 1.6879 1.7188 1.7188 1.7188 1.7188

u3(a/2, a/2,0.005)1011, m 1.529 1.5170 1.5285 1.5285 1.5285 1.5285

11(a/2, a/2, 0.005)101, Pa 3.371 3.3158 3.3715 3.3714 3.3714 3.3714

12(0, 0, 0.005), Pa 2.639 2.6030 2.6391 2.6391 2.6391 2.6391

13(0, a/2, 0.0023), Pa 3.081 3.1722 3.0697 3.0790 3.0789 3.0789

23(a/2,0, 0), Pa 2.614 2.2396 2.6216 2.6139 2.6140 2.6140

(a/2, a/2, 0)103, V 1.280 1.2707 1.2798 1.2798 1.2798 1.2798

D1(0, a/2, 0)1011, C/m2 2.414 2.3888 2.4139 2.4138 2.4138 2.4138

D3 (a/2, a/2, 0.005)1011, C/m2 4.970 5.4455 4.9770 4.9697 4.9696 4.9696

13

Figure 4. Distributions of transverse shear stresses, electric displacement and electric potential through the thickness of the three-ply plate subjected to mechanical loading for I1 = I2 = I3 = 7:

present analysis ( ) and Heyliger (), where z = x3/h.

14

2. Simply Supported Three-Layer Plate under Electric Loading

Analytical solution

Figure 5. PVDF [0/90/0] square plate

(h = 0.01 m, 0 = 200 V) (r=s=1)Table 2. Results for a piezoelectric three-ply plate with a /h = 4 under electric loading (Lage at al.)

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

sr

inrs

in

bxs

axrnn

,

21)()( sinsin

,

,

Variable Exact In=3 In=5 In=7 In=9 In=11

u1(0,a/2, 0.005)1010, m 3.223 3.1922 3.2226 3.2226 3.2226 3.2226

u3(a/2, a/2,0.005)109, m 3.313 3.3089 3.3131 3.3131 3.3131 3.3131

22(a/2, a/2, 0.01/6)103, Pa 2.841 2.8440 2.8408 2.8407 2.8407 2.8407

12(0, 0, 0.005) 102, Pa 5.543 5.5174 5.5427 5.5427 5.5427 5.5427

13(0, a/2, 0.003) 102, Pa 2.925 2.4660 2.9315 2.9246 2.9246 2.9246

23(a/2,0, 0.01/6) 102, Pa 2.328 1.7841

2.08342.31742.3252

2.32832.3282

2.32842.3284

2.32842.3284

33(a/2, a/2, 0)101, Pa 3.629 3.9740 3.6034 3.6292 3.6290 3.6290

D1(0, a/2, 0.005)106, C/m2 1.739 1.7393 1.7389 1.7389 1.7389 1.7389

D3 (a/2, a/2, 0.005)106, C/m2 3.100 3.0705 3.1002 3.1003 3.1003 3.1003

15

Figure 6. Distributions of transverse shear stresses, electric displacement and electric potential through the thickness of the three-ply plate subjected to electric loading for I1 = I2 = I3 = 7:

present analysis ( ) and Heyliger (), where z = x3/h.

16

3. FG Piezoelectric Square Plate under Mechanical Loading

Figure 7. PZT-4 FG square plate with grounded interfaces under

mechanical loading (r=s=1)

Analytical solution

sr

inrs

in

b

xs

a

xruu nn

,

21)(1

)(1 sincos

sr

inrs

in

bxs

axr

uu nn

,

21)(2

)(2 cossin

sr

inrs

in

bxs

axr

uu nn

,

21)(3

)(3 sinsin

sr

inrs

in

bxs

axrnn

,

21)()( sinsin

Material constants

0.5)(z0.5)(z e,e ikliklijklijkl eeCC

hxzikik /,e 30.5)(z

ik

ik

ikl

ikl

ijkl

ijkl

e

e

C

Clnlnln

17

In u1(-0.5) u3(0) (0) 11(0.5) 12(0.5) 13(0) 33(0) D1(-0.5) D3(0)

5 4.2738 -2.4856 -12.138 -7.8609 3.0219 -1.1787 -0.28738 0.91327 0.20612

7 4.2738 -2.4856 -12.138 -7.8544 3.0229 -1.1761 -0.28148 0.94503 0.21328

9 4.2738 -2.4856 -12.138 -7.8543 3.0229 -1.1762 -0.28150 0.94516 0.21323

11 4.2738 -2.4856 -12.138 -7.8543 3.0229 -1.1762 -0.28150 0.94516 0.21323

In u1(-0.5) u3(0) (0) 11(0.5) 12(0.5) 13(0) 33(0) D1(-0.5) D3(0)

5 1.1493 -0.92158 -4.4469 -15.336 6.0057 -1.1786 -0.21238 4.8146 -0.23971

7 1.1493 -0.92158 -4.4469 -15.360 6.0068 -1.1760 -0.21832 4.8517 -0.24667

9 1.1493 -0.92158 -4.4469 -15.360 6.0068 -1.1760 -0.21829 4.8519 -0.24662

11 1.1493 -0.92158 -4.4469 -15.360 6.0068 -1.1760 -0.21829 4.8519 -0.24662

Table 3. Results for FG piezoelectric plate with a/h = 10 and = 1 under mechanical loading

Table 4. Results for FG piezoelectric plate with a/h = 10 and = 1 under mechanical loading

18

Figure 8. Mechanical loading of the FG piezoelectric square plate: distributions of transverse shear stress, electric potential and electric displacement through the thickness of the plate

for I1 = 9, present analysis ( ) and Zhong and Shang () .

19

4. FG Piezoelectric Square Plate under Electric Loading

In u1(0.5) u3(0) (0) 11(0.5) 12(0.5) 13(0) 33(0) D1(0.5) D3(0)

5 208.76 -41.091 -200.62 182.37 147.65 0.18798 -0.22143 783.82 4854.2

7 208.76 -41.090 -200.68 182.80 147.66 0.14102 -0.25627 783.28 4856.0

9 208.76 -41.090 -200.68 182.80 147.66 0.14140 -0.25555 783.27 4856.0

11 208.76 -41.090 -200.68 182.80 147.66 0.14140 -0.25556 783.27 4856.0

13 208.76 -41.090 -200.68 182.80 147.66 0.14140 -0.25556 783.27 4856.0

In u1(0.5) u3(0) (0) 11(0.5) 12(0.5) 13(0) 33(0) D1(0.5) D3(0)

5 27.457 15.116 73.803 192.64 143.51 -0.18798 -0.22143 1086.0 4854.2

7 27.457 15.116 73.827 193.26 143.50 -0.14102 -0.25627 1086.7 4856.0

9 27.457 15.116 73.827 193.26 143.50 -0.14140 -0.25555 1086.7 4856.0

11 27.457 15.116 73.827 193.26 143.50 -0.14140 -0.25556 1086.7 4856.0

13 27.457 15.116 73.827 193.26 143.50 -0.14140 -0.25556 1086.7 4856.0

Table 6. Results for FG piezoelectric plate with a/h = 10 and = 1 under electric loading

Table 5. Results for FG piezoelectric plate with a/h = 10 and = 1 under electric loading

260

21033 C/m10,sinsin

q

bx

ax

qDD

20

Figure 9. Electric loading of the FG piezoelectric square plate: distributions of transverse shear stresses and electric potential through the thickness of the plate for I1 = 9,

present analysis ( ) and Zhong and Shang () .

21

In u1(-0.5) u2(-0.5) u3(-0.5) (0) 11(-0.5) 22(-0.5) 12(-0.5) 13(0) 23(0) 33(0) D3(0)

3 251.59 -539.53 740.23 2.8488 -14.325 -136.36 -82.661 56.821 -40.713 40.494 15.226

5 254.32 -543.71 742.09 2.8509 -13.457 -127.94 -83.397 57.592 -41.214 42.251 15.986

7 254.30 -543.65 742.07 2.8511 -13.433 -127.70 -83.389 57.589 -41.210 42.247 15.975

9 254.30 -543.65 742.07 2.8511 -13.431 -127.67 -83.389 57.589 -41.210 42.247 15.975

11 254.30 -543.65 742.07 2.8511 -13.430 -127.67 -83.389 57.589 -41.210 42.247 15.975

5. Piezoelectric Laminated Orthotropic Cylindrical Shell

Table 7. Results for a piezoelectric three-layer shell with S = 2 under mechanical loading

Figure 10. Three-layer [PZT4/PZT4F/PZT4] cylindrical shell under mechanical loading

(r=s=1)

1 02

1)(1

)(1 coscos

r s

inrs

in sL

ruu nn

1 02

1)(2

)(2 sinsin

r s

inrs

in sL

ruu nn

1 02

1)(3

)(3 cossin

r s

inrs

in sL

ruu nn

1 02

1)()( cossinr s

inrs

in sL

rnn

Analytical solution

hRS /

22

Figure 11. Distributions of transverse shear stresses, electric potential and electric displacement through the thickness of the three-layer shell under mechanical loading

for I1 = I2 = I3 = 7: present analysis ( ) and Heyliger ()

23

In u1(-0.5) u2(-0.5) u3(-0.5) (0) 11(-0.5) 22(-0.5) 12(-0.5) 13(0) 23(0) 33(0) D3(0)

3 -8.0471 16.158 -0.64421 2.6894 -1.5080 -12.751 2.5383 7.5733 -6.5742 0.8436 -37.197

5 -7.9780 16.144 -0.59556 2.6878 -1.6051 -13.706 2.5285 8.0537 -6.9389 1.1103 -36.807

7 -7.9778 16.143 -0.59384 2.6879 -1.6021 -13.676 2.5283 8.0504 -6.9365 1.1176 -36.808

9 -7.9778 16.143 -0.59384 2.6879 -1.6026 -13.681 2.5283 8.0503 -6.9365 1.1175 -36.808

11 -7.9778 16.143 -0.59384 2.6879 -1.6026 -13.681 2.5283 8.0503 -6.9365 1.1175 -36.808

6. Piezoelectric Laminated Orthotropic Cylindrical Shell

Table 8. Results for a piezoelectric three-layer shell with S = 2 under electric loading

1 02

1)(1

)(1 coscos

r s

inrs

in sL

ruu nn

1 02

1)(2

)(2 sinsin

r s

inrs

in sL

ruu nn

1 02

1)(3

)(3 cossin

r s

inrs

in sL

ruu nn

1 02

1)()( cossinr s

inrs

in sL

rnn

Analytical solution

hRS /Figure 12. Three-layer [PZT4/PZT4F/PZT4]

cylindrical shell under electric loading (r=s=1)

24

Figure 13. Distributions of transverse shear stresses, electric potential and electric displacement through the thickness of the three-layer shell under electric loading

for I1 = I2 = I3 = 7

25

7. FG Piezoelectric Anisotropic Cylindrical Shell

Figure 14. Four-layer FG [PZT/45/-45/PZT] cylindrical shell under mechanical loading (R/h=4) (r=1)

Analytical solution

Material constants of PZT

)(),( 10)1(

10)1( zVeezVCC ikliklijklijkl

25.05.0,)4()(),( 110)1( zzzVzVikik

)(),( 40)4(

40)4( zVeezVCC ikliklijklijkl

5.025.0,)4()(),( 440)4( zzzVzVikik

Figure 15. Through-thickness distribution of elastic constants of the top FG piezoelectric layer

1

1)(2

)(2

1

1)(1

)(1 cos,cos

r

inr

in

r

inr

in

Lr

uuL

ruu nnnn

1

1)()(

1

1)(3

)(3 sin,sin

r

inr

in

r

inr

in

Lr

Lr

uu nnnn

26

Table 9. Results for a FG piezoelectric angle-ply shell with = 1 under mechanical loading

In u1(-0.5) u2(-0.5) u3(0) (-0.5) 11(-0.5) 22(-0.5) 12(-0.5) 13(-0.125) 23(0.125) 33(0.125) D3(0.25)

5 3.3289 0.93263

7.4317 -2.6981 -3.7068 2.0244 -4.4828 62.895 -9.9884 62.693 -5.9486

7 3.3289 0.93264

7.4316 -2.6984 -3.7070 2.0244 -4.4828 62.891 -9.9855 62.693 -5.9535

9 3.3289 0.93264

7.4316 -2.6984 -3.7069 2.0242 -4.4829 62.891 -9.9855 62.693 -5.9536

11 3.3289 0.93264

7.4316 -2.6984 -3.7069 2.0242 -4.4829 62.891 -9.9855 62.693 -5.9536

In u1(-0.5) u2(-0.5) u3(0) (-0.5) 11(-0.5) 22(-0.5) 12(-0.5) 13(-0.125) 23(0.125) 33(0.125) D3(0.25)

5 1.5528 1.2698 4.1558 3.9143 -6.3030 5.2026 -2.4414 52.534 -5.3417 60.071 -6.4591

7 1.5528 1.2698 4.1558 3.9143 -6.3015 5.2042 -2.4414 52.533 -5.3404 60.070 -6.4810

9 1.5528 1.2698 4.1558 3.9143 -6.3015 5.2042 -2.4414 52.533 -5.3404 60.070 -6.4818

11 1.5528 1.2698 4.1558 3.9143 -6.3015 5.2042 -2.4414 52.533 -5.3404 60.070 -6.4819

Table 10. Results for a FG piezoelectric angle-ply shell with = 1 under mechanical loading

27

Figure 16. Distributions of stresses and electric displacement through the thickness direction of the FG piezoelectric angle-ply cylindrical shell under mechanical loading

for I1 = I2 = I3 = I4 = 9: present analysis ( ) and authors’ 3D exact solution ()

28

In u1(-0.5) u2(-0.5) u3(0) (-0.5) 11(-0.5) 22(-0.5) 12(-0.5) 13(-0.125) 23(0.125) 33(0.125) D3(0.25)

5 2.4020 -0.92166 7.4115 18.401 -2.0989 2.7989 0.44301 -2.8587 -24.129 63.358 -165.68

7 2.4019 -0.92155 7.4109 18.399 -2.0990 2.7962 0.44295 -2.8635 -24.121 63.353 -165.81

9 2.4019 -0.92155 7.4109 18.399 -2.0990 2.7962 0.44295 -2.8635 -24.121 63.353 -165.82

11 2.4019 -0.92155 7.4109 18.399 -2.0990 2.7962 0.44295 -2.8635 -24.121 63.353 -165.82

In u1(-0.5) u2(-0.5) u3(0) (-0.5) 11(-0.5) 22(-0.5) 12(-0.5) 13(-0.125) 23(0.125) 33(0.125) D3(0.25)

5 2.1244 0.58752 7.8724 26.175 -6.2103 12.716 -1.1296 -6.0595 -17.931 108.42 -363.48

7 2.1244 0.58752 7.8724 26.175 -6.2084 12.718 -1.1296 -6.0650 -17.925 108.42 -364.66

9 2.1244 0.58752 7.8724 26.175 -6.2084 12.718 -1.1296 -6.0650 -17.925 108.42 -364.70

11 2.1244 0.58752 7.8724 26.175 -6.2084 12.718 -1.1296 -6.0650 -17.925 108.42 -364.71

8. FG Piezoelectric Anisotropic Cylindrical Shell under Electric Loading

Table 11. Results for a FG piezoelectric angle-ply shell with = 1 under electric loading

Table 12. Results for a FG piezoelectric angle-ply shell with = 1 under electric loading

0V,1,sin 33301

0

ppDL

29

Figure 17. Distributions of stresses and electric displacement through the thickness direction of the FG piezoelectric angle-ply cylindrical shell under electric loading

for I1 = I2 = I3 = I4 = 9: present analysis ( ) and authors’ 3D exact solution ()

30

Thanks for your attention!Thanks for your attention!Thanks for your attention!Thanks for your attention!

ConclusionsConclusions1.1. SaS method gives the possibility to obtain exact 3D SaS method gives the possibility to obtain exact 3D

solutions of electroelasticity for thick and thin FG solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracypiezoelectric plates and shells with a prescribed accuracy.

2. New higher-order layer-wise theory of FG piezoelectric shells has been developed by using of only displacements of SaS. This is straightforward for finite element developments.

ConclusionsConclusions1.1. SaS method gives the possibility to obtain exact 3D SaS method gives the possibility to obtain exact 3D

solutions of electroelasticity for thick and thin FG solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracypiezoelectric plates and shells with a prescribed accuracy.

2. New higher-order layer-wise theory of FG piezoelectric shells has been developed by using of only displacements of SaS. This is straightforward for finite element developments.