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2 Functionally graded piezoelectric material plate formulation based on Bézier extraction for NURBS

# 2 Functionally graded piezoelectric material plate formulation based on Bézier extraction for NURBS

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exhibited in section 3.2.5 is used in order to compute the free vibration frequencies

of the plate.

The function of the electrical potential is chosen so that the distribution of

electric and magnetic potentials through the plate thickness is fulfilled Maxwell’s

equation in the quasi-static approximation by [82-83]:

=

Φ ( x, y , z , t ) g ( z ) φ ( x, y , t ) +

2z

V0 eiω t

h

(6. 1)

where V0 is the applied electric voltage, g ( z ) is an arbitrary distributed function of zcoordinate, φ ( x, y, t ) expresses the function of the electrical potential in reference

plane and ω is the eigen value. In this paper, g ( z ) is given as g ( z ) = − cos(π z ) .

h

According to Eq.(6. 1), the electric fields ( Ex , E y and Ez ) become:

−Φ, y =

− g ( z ) φ, y ;

Ex =

−Φ, x =

− g ( z ) φ, x ; E y =

−Φ, z =

− g ′ ( z )φ −

Ez =

2V0 iω t

e

h

(6. 2)

For a piezo-electrically actuated FG piezoelectric porous plate, the constitutive

relations are described by:

=

σ ij Cijkl ε kl − ekij Ek

(6. 3)

=

Di eikl ε kl + kik Ek

where σ ij , ε kl , Di and Ek are stress, strain, electric displacement and electric field

components, respectively; Cijkl , eijk and kik define elastic, piezoelectric and dielectric

constants, respectively.

The electric field vector E can be expressed as

(6. 4)

The formulations in Eq.(6. 3) are also clearly rewritten following matrix forms

as:

137

σ xx   c11 c12 0  ε xx  0 0 e31   0 

  

 

 

12 c22 0  ε yy  − 0 0 e31   0  Ez =

σb =

c

Cb ε b − Cbc Eb

σ yy  =

σ   0

0 c66  ε xy  0 0 0   Ez 

 xy  

τ xz  c55 0  γ xz  e15

τs =

 =

 0 c  γ  −  0

τ

yz

44   yz 

  

0   Ex 

Cs γ − Ccs E s

 =

e14   E y 

 Dx  e15 0  γ xz   k11

Dp =

 =

 0 e  γ  + 

D

y

14   yz 

  

 0

0   Ex 

Ccs γ + Ck E s

  =

k22   E y 

(6. 5)

D z = e31ε x + e32ε y + k33 Ez

where cij , eij and kij define the reduced constants of FGPMP plates and they are

expressed by:

c11 =

c11 −

c132

c2

, c12 =

c12 − 13 , c66 =

c66

c33

c33

(6. 6)

2

33

c e

e

e31 =

e31 + 13 33 , k11 =

k11 , k33 =

k33 +

c33

c33

Now, Hamilton’s principle is used to obtain the governing equations of free

vibration for FGPMP plates:

t

∫ (δΠ

0

S

− δΠ K + δΠ I )dt =0

(6. 7)

where Π S , Π K and Π I are strain energy, kinetic energy and potential energy from

initial stress which is generated from applying electric voltage, respectively.

The strain energy δΠ S is defined as

 σ xxδε xx + σ yyδε yy + τ xyδγ xy + τ xzδγ xz + τ yzδγ yz − ... 

dVˆ

Dxδ Ex − Dyδ E y − Dzδ Ez

Vˆ 

δΠ S =∫ 

(6. 8)

Substituting Eq. (6. 5) into Eq. (6. 8), the discrete Galerkin weak form can be

rewritten as

138

 (δε b )T Cb ε b − (δε b )T Cb Eb + δγ T Cs γ − δγ T Cs E s − 

c

c

dVˆ −

δΠ S ∫ 

T

T



s

s

s

k s

Vˆ  ( δ E ) C γ − ( δ E ) C E

c

T

∫ (δ Ez ) e31ε x + e32ε y + k33 Ez dVˆ

(

)

(6. 9)

in which

ε x =ε x0 + zε 1x + f ( z )ε x2 ; ε y =ε y0 + zε 1y + f ( z )ε y2

(6. 10)

Eq. (6. 9) can be split into two independent integrals following to middle

surface and z-axis direction as:

ˆ bδεˆ b dΩ +  ( φb )T C

ˆ b1ε 0dΩ + ( φb )T C

ˆ b 2ε1dΩ + ( φb )T C

ˆ b 3ε 2dΩ  +

C

c

c

c

Ω

T

T

T

T

( εˆ s ) Cˆ sδε sdΩ + ( φs ) Cˆ sδε sdΩ + ( ε s ) Cˆ sδφsdΩ + ( φs ) Cˆ kδφsdΩ +

∫ ( εˆ )

b T

δΠ S

=

c

c

(6. 11)

2

3

 ε x0e131δφ z dΩ + ε 1x e31

δφ z dΩ + ∫ ε x2e31

δφ z dΩ + ∫ ε y0e132δφ z dΩ + ∫ ε 1y e322 δφ z dΩ + 

Ω

 ε 2e 3 δφ z dΩ + φ z kˆ δφ z dΩ − eiωt h /2 g ′ ( z ) dz 2V0 k δφ z dΩ

33

33

 ∫ y 32

− h /2

h

Ω

The left side of Eq. (6. 11) can be rewritten under compact forms as:

δ Π S = δΠ1 + δΠ 2 + δΠ 3 + δΠ 4 + δΠ 5 + δΠ 6 + δΠ 7

where

139

(6. 12)

δΠ1

=

∫ ( εˆ )

ˆ bδεˆ b dΩ;

C

∫ (φ )

ˆ b1ε 0dΩ + ( φb )T C

ˆ b 2ε1dΩ + ( φb )T C

ˆ b 3ε 2dΩ;

C

c

c

c

∫ ( εˆ )

ˆ sδ=

C

ε s dΩ; δΠ 4

∫ (φ )

ˆ k δφ s dΩ;

C

b T

δΠ 2

=

b T

=

δΠ 3

s T

∫ (φ )

s T

=

δΠ 6

s T

s

ˆ sδ=

C

c ε dΩ; δΠ 5

∫ (ε )

s T

ˆ sδφ s dΩ;

C

c

(6. 13)

δΠ 7

=

∫ε

∫ε

2

3

2

e δφ z dΩ + ∫ ε 1x e31

δφ z dΩ + ∫ ε x2e31

δφ z dΩ + ∫ ε y0e132δφ z dΩ + ∫ ε 1y e32

δφ z dΩ +

0 1

x 31

ˆ

e δφ z dΩ + ∫ φ z k33δφ z dΩ − eiωt ∫

2 3

y 32

h /2

− h /2

2V0 

k33δφ z dΩ

h

g ′ ( z ) dz ∫

For details, we need to rewrite the above terms as follows:

δΠ1

=

∫ ( εˆ )

b T

ˆ bδεˆ b dΩ

C

where

εˆ b = {ε 0

ε1

 Ab

T

ˆ b =  Bb

ε2 ; C

 Eb

}

Bb

Db

Fb

Eb 

Fb 

H b 

 c11

h /2

( Ab , Bb , Db , Eb , Fb , Hb ) = ∫− h/2 (1, z, z 2 , f ( z ), z f ( z ), f 2 ( z ) ) c12

 0

=

δΠ 2

∫ (φ )

b T

T

ˆ b1δε 0dΩ + ( φb )T C

ˆ b 2δε1dΩ + ( φb=

ˆ b 3δε 2dΩ

C

C

)

c

c

c

where

140

c12

c22

0

0

0  dz

c66 

∫ (φ )

b T

(6. 14)

ˆ bδ εˆ b dΩ

C

c

0 0 e31 

ˆ

ˆ

ˆ

ˆ  ;C

ˆ

=

=

C

C

C

∫− h/2 g ' ( z ) 0 0 e31  dz;

C

0 0 0 

0 0 e31 

0 0 e31 

h /2

h /2

2

3

b

b

ˆ

ˆ

C

zg ' ( z ) 0 0 e31  dz; C

f ( z ) g ' ( z ) 0 0 e31  dz

c

c

∫=

− h /2

− h /2

0 0 0 

0 0 0 

b1

c

b

c

=

δΠ 3

b2

c

∫ ( εˆ )

s T

b3

c

h /2

b1

c

(6. 15)

ˆ sδε s dΩ

C

where

=

εˆ s

ˆ

ε

ε } ;C

{=

s0

s1 T

s

As

B

 s

Bs 

;

Ds 

(6. 16)

0

dz

c44 

c55

0

h /2

( A s ; B s ; Ds ) = ∫− h/2 (1, f '( z ), f ′2 ( z )) 

=

δΠ 4

∫ (φ )

s T

∫ (ε )

s T

ˆ sδε s dΩ=

C

and δΠ 5

c

ˆ sδφ s dΩ

C

c

where

ˆs

C

c

=

δΠ 6

 A cs B cs 

; A s ; B cs ; Dcs )

=

s

s ( c

 B c Dc 

∫ (φ )

s T

e15

2

(

)[1,

'(

),

(

)]

g

z

f

z

f

z

0

∫− h/2

h /2

0

dz

e14 

(6. 17)

ˆ k δφ s dΩ

C

where



ˆ k = h /2 g ( z ) g ( z )  k11

C

∫− h/2

 0

=

δΠ 7

∫ε

(6. 18)

 dz

22 

0

k

2

3

e δφ z dΩ + ∫ ε 1x e31

δφ z dΩ + ∫ ε x2e31

δφ z dΩ + ∫ ε y0e132δφ z dΩ +

0 1

x 31

ˆ

1 2

z

2 3

z

z

z

iωt

∫ ε y e32δφ dΩ + ∫ ε y e32δφ dΩ + ∫ φ k33δφ dΩ − e ∫

h /2

− h /2

in which

141

2V0 

k33δφ z dΩ

h

g ′ ( z ) dz ∫

e131 = ∫

h /2

e132 = ∫

h /2

h /2

h /2

3

2

= ∫ f ( z ) g ′ ( z ) e31dz

= ∫ zg ′ ( z ) e31dz e31

g ′ ( z ) e31dz e31

h

/2

− h /2

− h /2

;

;

h /2

h /2

2

3

= ∫ zg ′ ( z ) e32dz e32

= ∫ f ( z ) g ′ ( z ) e32dz

g ′ ( z ) e32dz e32

− h /2

− h /2

h

/2

;

;

(6. 19)

h /2

ˆ

k33 = ∫ g ′ ( z ) g ′ ( z ) k33dz

− h /2

Note that when g(z) =0 at z = ± h / 2 and g(z) is an even function,

h /2

− h /2

g ′ ( z ) dz = 0 . Hence, δΠ 7 is not affected by the applied electric voltage V0 .

The variation of the kinetic energy of the mass system can be written as

=

δΠ K

ˆ ˆ dΩ

δ uˆ T mu

(6. 20)

where

 u1 

 I1

 2 ˆ 

uˆ = u  ; m = I 2

u 3 

I 4

 

I2

I3

I5

I4 

I 5 

I 6 

(6. 21)

in which the mass inertia terms I i (i =1:6) are calculated as Eq.(6. 22):

1 0 0 

( I1 , I 2 , I3 , I 4 , I 5 , I 6 ) = ∫ ρ ( z ) (1, z, z , f ( z ), zf ( z ), f ( z ) ) 0 1 0 dz

− h /2

0 0 1 

h /2

2

2

(6. 22)

The potential energy obtained from external applied electric voltage can be

written by

T

0

 w   N

=

δΠ I ∫ δ  0, x   0x

 w0, y   N xy

N xy0   w0, x 

∂w ∂δ w 

I  ∂w ∂δ w

=

+

 dΩ N ∫ 

 dΩ

0 

N y   w0, y 

x

x

y

y

Ω

(6. 23)

where

N xy0 = 0, N x0 = N y0 = N I = −eiωt ∫

h /2

− h /2

e31

2V0

dz

h

(6. 24)

6.2.2 Approximated formulation

By using the Bézier extraction of NURBS, the displacement field u of the plate

is approximated as follows

142

m×n

u (ξ ,η ) = ∑ RAe (ξ ,η )d A

(6. 25)

A

where n×m is the number of basis functions, RAe (ξ ,η ) is a NURBS basis function

which is written in the compact form of the linear combination of Bézier extraction

operator

d A = {u0 A

and

v0 A

Bernstein

polynomials

and

β xA β yA θ xA θ yA } is the vector of nodal degrees of

T

w0 A

freedom associated with control point A.

The electric field E in Eq. (6. 4) can be rewritten as

E = −∇Nφ φA = −Bφ φA

(6. 26)

in which φA is electric potential related to control point A and Nφ is the shape

functions for the electric potential.

Substituting Eq.(6. 25) to Eq.(3. 11), the first component of Eq. (6. 12), δΠ1

is approximated based on Bézier extraction of NURBS as

(6. 27)

δΠ1 =δ dT K1d

where

K1

=

ˆ b Bˆ dΩ=

Bˆ Tu C

; Bˆ u

u

[B1

(6. 28)

B 2 B3 ]

T

and

 RA, x

 0 0 0 RA, x

0 0 0 0 0 0

mxn

B1 = ∑  0

R A , y 0 0 0 0 0 ; B 2 = − ∑  0 0 0 0

A 1=

A 1

 RA, y RA, x 0 0 0 0 0 

 0 0 0 RA, y

mxn

 0 0 0 0 0 RA, x

mxn

B3 = ∑ 0 0 0 0 0 0

A=1

 0 0 0 0 0 RA, y

0 

RA, y 

RA, x 

0

RA, y

RA, x

0 0

0 0 ;

0 0 

(6. 29)

The second component of Eq. (6. 12), δΠ 2 is obtained by substituting Eqs.(6.

25) and (6. 26) to Eqs. (3. 11) and (6. 13).

δΠ 2 =δφT K 2d

where

143

(6. 30)

K2 =

∫ ( Bφ )

b T

(6. 31)

0

b

ˆ bB

ˆ

C

−∑  0 

c u dΩ ; Bφ =

A=1

 RA 

mxn

The third component of Eq. (6. 12), δΠ 3 , is given as

(6. 32)

δΠ 3 =δ dT K 3d

where

K3

=

∫ (B )

s T

s

ˆ s B=

C

dΩ; B s B1s

B 2s 

(6. 33)

T

and

B1s

0 0 0  2 mxn 0 0 0 0 0 RA

 0 0 RA, x − RA

=

0 0 R

; B s ∑ 

R

0

0

0

0 0 0 0 0

A 1=

A

,

y

A

A 1 0

mxn

0

RA 

(6. 34)

The fourth term of Eq. (6. 12), δΠ 4 , becomes

δΠ 4 =δφT K 4d

(6. 35)

where

mxn  R

A, x 

s T ˆs s

s

d

;

=

K5 =

B

C

B

B

(

)

R 

φ

∫Ω φ c

A=1  A, y 

(6. 36)

The next term of Eq. (6. 12), δΠ 5 is similar to δΠ 4 , in which K 5 is the

transpose matrix of K 5

=

K5

∫ (B )

s T

ˆ s B s dΩ;

C

c φ

(6. 37)

The one more term of Eq. (6. 12), δΠ 6 has form

δΠ 6 =δφT K 6φ

=

K6

∫ (B )

s T

φ

ˆ k B s dΩ;

C

φ

(6. 38)

(6. 39)

The final term of Eq. (6. 12), δΠ 7 is given as

=

δΠ 7 δ dT K 71φ + δ dT K 72φ + δφT K 73φ

where

144

(6. 40)

(6. 41)

K 7 =K 7 + K 72 + K 73 =

∫Ω ( B x )

T

T ˆ

ˆ B dΩ + ( B )T E

E

x z

∫ y ˆ y B z dΩ + ∫ ( B z ) k33B z dΩ;

T

2

e32

3

e32

 ;

T

B 2y

B3y 

2

3

1

ˆ

ˆ

E

=

e131 e31

e31

x

=

 ;E y e32

B x =

B1x B 2x B3x  ; B y B1y

=

T

T

and

B1x =

B3x

mxn

∑  RA, x

mxn

0 0 0 0 0 0 ; B 2x = −∑ 0 0 0 RA, x

0 0 0 ;

A 1=

A 1

mxn

1

=

0 0 0 0 0 RA, x 0 ; B y

mxn

∑ 0

A 1=

A 1

mxn

RA, y

(6. 42)

0 0 0 0 0 ;

mxn

−∑ 0 0 0 0 RA, x 0 0 ; B3y =

B 2y =

∑ 0 0 0 0 0 0 RA, y ;

A 1=

A 1

mxn

B z = − ∑ RA

A=1

The potential energy in Eq. (6. 23) is also approximated as

(6. 43)

δΠ I =δ dT K I d

where

KI

∫ (B )

I

T

0 

 −2e31V0

B dΩ ; B I

 0=

−2e31V0  I

 0 0 RA, x

 0 R

A=1  0

A, y

mxn

0 0 0 0

0 0 0 0 

(6. 44)

The elementary governing equation of motion can be generally derived by

substituting Eqs.(6. 9),(6. 20),(6. 23) into Eq.(6. 7) as:

   K uu

 M 0  d

 0 0    + K

  φ   φu

K uφ  d  0 

=

K φφ   φ  0 

(6. 45)

where

K uu =

K1 + K 3 - K I

K φu =

− (K 2 + K 4 )

; K uφ =

− ( K 5 + K 71 + K 72 ) ;

;

K φφ =

− ( K 6 + K 73 )

145

(6. 46)

Substituting the second line of Eq. (6. 45) into the first line, the shortened

form is obtained as

(

)

 + K − K K −1K d =

Md

0

uu

φu

φφ

(6. 47)

The governing equation of free vibration problem can be formulated by the

following form

0

(K − ω M )d =

2

(6. 48)

where the global stiffness matrix K is given as

=

K K uu − K uφ K −φφ1K φu

(6. 49)

and the global mass matrix M is described like as Eq.(5. 13).

6.3

Numerical examples and discussions

Since the square and circular plates are important structural parts in

engineering structures, four examples for square and circular porous functionally

graded piezoelectric plates with various geometric from simple to complex, different

boundary conditions and two types of porous distribution using isogeometric finite

elements based on Bézier extraction are considered. In addition, the cubic function

f ( z )= z −

4z3

[112,158] is employed in the following examples, but the use of other

3h 2

forms of f(z) in the present formulation is straightforward.

Two types of Dirichlet boundary conditions are used as:

 Simply supported:

 Rectangular plate

u=

w=

θ=

β=

0 at =

y 0, b and v=

θ=

β=

w=

0 at =

x 0, a

0

0

x

x

0

0

y

y

 Circular plate

u=

v=

w=

0 at boundaries

0

0

0

 Fully clamped:

u=

v=

w=

β=

β=

θ=

θ=

0 at boundaries .

0

0

0

x

y

x

y

146

Table 6. 1. Material properties [165-166].

Properties

PZT-4(*)

PZT-5A(*)

PZT-4(**)

PZT-5H(**)

c11=c22 (GPa)

138.499

99.201

139

126

c12

77.371

54.016

77.8

79.1

c13

73.643

50.778

74

83.9

c33

114.745

86.856

115

117

c55

25.6

21.1

25.6

23

30.6

22.6

30.6

23.5

e31 (Cm )

-5.2

-7.209

-5.2

-6.5

e33

15.08

15.118

15.1

23.3

c66

-2

12.72

e15

2

-2

-1

12.322

k11 (C m N )

1.306 x10

-9

k33

1.115 x10-9

12.7

-9

17

1.53 x10

-9

6.46 x10

15.05 x10-9

1.5 x10-9

5.62 x10-9

13.02 x10-9

(*): Material properties are given in .

(**): Material properties are given in .

6.3.1

Square plates

6.3.1.1 The square FGPMP plate

A square FGPM plate (a/b =1) subjected to various electric voltages and

boundary conditions is condsidered. The length to thickness ratio is given as a/h =

100. The accuracy and reliability of the present method are verified by analytical

solutions which are given in the literature. Material properties are given in Table 6.

1.

The

ω = ωb 2 / h

non-dimensional

frequency

parameter

ω

is

defined

as

( ρ / c11 ) PZT −4 . First, the convergence and accuracy of solutions using

quadratic (p = 2) Bézier elements at mesh levels of 7x7, 11x11, 15x15 and 17x17

elements are investigated as depicted in Table 6. 2 for a perfect FGPM plate with

simply supported boundary conditions. Figure 6. 1 illustrates Bézier control mesh of

a square FGPM plate using 7x7, 11x11, 15x15 and 17x17 quadratic Bézier elements.

The first non-dimensional frequency is compared with the analytical solution

reported by Barati et al.  using a refined four-variable plate theory. The relative

error percentages compared with the analytical solution  are also given in the

parentheses. Table 6. 2 reveals that the obtained results correlate well with the

147 ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

2 Functionally graded piezoelectric material plate formulation based on Bézier extraction for NURBS

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