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
E = −gradφ = −∇φ
(6. 4)
The formulations in Eq.(6. 3) are also clearly rewritten following matrix forms
as:
137
σ xx c11 c12 0 ε xx 0 0 e31 0
12 c22 0 ε yy − 0 0 e31 0 Ez =
σb =
c
Cb ε b − Cbc Eb
σ yy =
σ 0
0 c66 ε xy 0 0 0 Ez
xy
τ xz c55 0 γ xz e15
τs =
=
0 c γ − 0
τ
yz
44 yz
0 Ex
Cs γ − Ccs E s
=
e14 E y
Dx e15 0 γ xz k11
Dp =
=
0 e γ +
D
y
14 yz
0
0 Ex
Ccs γ + Ck E s
=
k22 E y
(6. 5)
D z = e31ε x + e32ε y + k33 Ez
where cij , eij and kij define the reduced constants of FGPMP plates and they are
expressed by:
c11 =
c11 −
c132
c2
, c12 =
c12 − 13 , c66 =
c66
c33
c33
(6. 6)
2
33
c e
e
e31 =
e31 + 13 33 , k11 =
k11 , k33 =
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 ) e31ε x + e32ε y + k33 Ez dVˆ
(
Vˆ
)
(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
Ω
Ω
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
=
∫
∫
Ω
Ω
c
∫
c
Ω
∫
Ω
(6. 11)
2
3
ε x0e131δφ z dΩ + ε 1x e31
δφ z dΩ + ∫ ε x2e31
δφ z dΩ + ∫ ε y0e132δφ z dΩ + ∫ ε 1y e322 δφ z dΩ +
∫
∫
Ω
Ω
Ω
Ω
Ω
ε 2e 3 δφ z dΩ + φ z kˆ δφ z dΩ − eiωt h /2 g ′ ( z ) dz 2V0 k δφ z dΩ
33
33
∫
∫
∫
∫ y 32
− h /2
h
Ω
Ω
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
Ω
δΠ 2
=
b T
Ω
=
δΠ 3
s T
Ω
Ω
∫ (φ )
s T
Ω
=
δΠ 6
s T
s
ˆ sδ=
C
c ε dΩ; δΠ 5
∫ (ε )
s T
ˆ sδφ s dΩ;
C
c
(6. 13)
Ω
Ω
Ω
δΠ 7
=
∫ε
Ω
∫ε
Ω
2
3
2
e δφ z dΩ + ∫ ε 1x e31
δφ z dΩ + ∫ ε x2e31
δφ z dΩ + ∫ ε y0e132δφ z dΩ + ∫ ε 1y e32
δφ z dΩ +
0 1
x 31
Ω
Ω
ˆ
e δφ z dΩ + ∫ φ z k33δφ z dΩ − eiωt ∫
2 3
y 32
h /2
− h /2
Ω
Ω
Ω
2V0
k33δφ z dΩ
h
Ω
g ′ ( z ) dz ∫
For details, we need to rewrite the above terms as follows:
δΠ1
=
∫ ( εˆ )
b T
ˆ bδεˆ b dΩ
C
Ω
where
εˆ b = {ε 0
ε1
Ab
T
ˆ b = Bb
ε2 ; C
Eb
}
Bb
Db
Fb
Eb
Fb
H b
c11
h /2
( Ab , Bb , Db , Eb , Fb , Hb ) = ∫− h/2 (1, z, z 2 , f ( z ), z f ( z ), f 2 ( z ) ) c12
0
=
δΠ 2
∫ (φ )
b T
Ω
T
ˆ b1δε 0dΩ + ( φb )T C
ˆ b 2δε1dΩ + ( φb=
ˆ b 3δε 2dΩ
C
C
)
c
c
c
∫
∫
Ω
Ω
where
140
c12
c22
0
0
0 dz
c66
∫ (φ )
b T
Ω
(6. 14)
ˆ bδ εˆ b dΩ
C
c
0 0 e31
ˆ
ˆ
ˆ
ˆ ;C
ˆ
=
=
C
C
C
∫− h/2 g ' ( z ) 0 0 e31 dz;
C
0 0 0
0 0 e31
0 0 e31
h /2
h /2
2
3
b
b
ˆ
ˆ
C
zg ' ( z ) 0 0 e31 dz; C
f ( z ) g ' ( z ) 0 0 e31 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
Ω
where
=
εˆ s
ˆ
ε
ε } ;C
{=
s0
s1 T
s
As
B
s
Bs
;
Ds
(6. 16)
0
dz
c44
c55
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
Ω
Ω
where
ˆs
C
c
=
δΠ 6
A cs B cs
; A s ; B cs ; Dcs )
=
s
s ( c
B c Dc
∫ (φ )
s T
e15
2
′
(
)[1,
'(
),
(
)]
g
z
f
z
f
z
0
∫− h/2
h /2
0
dz
e14
(6. 17)
ˆ k δφ s dΩ
C
Ω
where
ˆ k = h /2 g ( z ) g ( z ) k11
C
∫− h/2
0
=
δΠ 7
∫ε
Ω
(6. 18)
dz
22
0
k
2
3
e δφ z dΩ + ∫ ε 1x e31
δφ z dΩ + ∫ ε x2e31
δφ z dΩ + ∫ ε y0e132δφ z dΩ +
0 1
x 31
Ω
Ω
ˆ
Ω
1 2
z
2 3
z
z
z
iωt
∫ ε y e32δφ dΩ + ∫ ε y e32δφ dΩ + ∫ φ k33δφ dΩ − e ∫
Ω
Ω
h /2
− h /2
Ω
in which
141
2V0
k33δφ z dΩ
h
Ω
g ′ ( z ) dz ∫
e131 = ∫
h /2
e132 = ∫
h /2
h /2
h /2
3
2
= ∫ f ( z ) g ′ ( z ) e31dz
= ∫ zg ′ ( z ) e31dz e31
g ′ ( z ) e31dz e31
−
h
/2
− h /2
− h /2
;
;
h /2
h /2
2
3
= ∫ zg ′ ( z ) e32dz e32
= ∫ f ( z ) g ′ ( z ) e32dz
g ′ ( z ) e32dz e32
−
− h /2
− h /2
h
/2
;
;
(6. 19)
h /2
ˆ
k33 = ∫ g ′ ( z ) g ′ ( z ) k33dz
− 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
∫
Ω
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
Ω
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
e31
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
=
∫
Ω
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
Ω
(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
Ω
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
;
Ω
=
−
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
Ω
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
Ω
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 ) k33B z dΩ;
Ω
Ω
T
2
e32
3
e32
;
T
B 2y
B3y
2
3
1
ˆ
ˆ
E
=
e131 e31
e31
x
=
;E y e32
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 )
Ω
I
T
0
−2e31V0
B dΩ ; B I
0=
−2e31V0 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
uφ
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 [165].
(**): Material properties are given in [166].
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. [82] using a refined four-variable plate theory. The relative
error percentages compared with the analytical solution [82] are also given in the
parentheses. Table 6. 2 reveals that the obtained results correlate well with the
147
