c
0
ð Þ
pq
h i
¼
c 11
c 12 0 κ 2 c 14
κ 1 c 15
c 16
c 21
c 22 0 κ 2 c 24
κ 1 c 25
c 26
0
0
0
0
0
0
κ 2 c 41 κ 2 c 42 0 κ 2 κ 2 c 44 κ 1 κ 2 c 45 κ 2 c 46
κ 1 c 51 κ 1 c 52 0 κ 1 κ 2 c 54 κ 1 κ 1 c 55 κ 1 c 56
c 61
c 62 0 κ 2 c 64
κ 1 c 65
c 66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ð5:59Þ
e
0
ð Þ
iq
h i
¼
e 11 e 12 0 κ 2 e 14 κ 1 e 15 e 16
e 21 e 22 0 κ 2 e 24 κ 1 e 25 e 26
e 31 e 32 0 κ 2 e 34 κ 1 e 35 e 36
2
6
4
3
7
5:
ð5:60Þ
The two correction factors appear as products or squares at a few places in Eq. (5.59)
because they are introduced in the energy density function (the electric enthalpy) for
the zero-order constitutive relations, which is a quadratic form [5, 6]. In the case of
piezoelectric dielectric plates, the correction factors are determined by requiring the
two fundamental thickness-shear resonance frequencies obtained from the
two-dimensional plate equations to be equal to the corresponding exact frequencies
predicted by the three-dimensional equations. With shear correction factors thus
determined, the two-dimensional plate equations and the exact three-dimensional
equations yield the same frequencies for a particular motion, i.e., the thickness-shear
vibration of a plate in the two fundamental thickness-shear modes. This is useful in
the analysis of plate piezoelectric devices operating with thickness-shear modes.
For the first-order constitutive relations, in Eq. (5.39) 1 , we set
T
1
ð Þ
q ¼ 0, q ¼ 3, 4, 5:
ð5:61Þ
Then, similar to the derivation of Eqs. (5.46) through (5.50), we obtain the relaxed
first-order constitutive relations:
T
1
ð Þ
r ¼
2h
3
3
γ rs S
1
ð Þ
s À ψ kr E
1
ð Þ
k
,
D
1
ð Þ
i ¼
2h
3
3
ψ is S
1
ð Þ
s þ ζ ij E
1
ð Þ
j
,
ð5:62Þ
where the effective plate material constants γ rs , ψ ks , and ζ kj are given by Eq. (5.50).
The right-hand side of Eq. (5.62) does not depend on S
1
ð Þ
3j or u
2
ð Þ
j . When the relevant
ones of Eqs. (5.58) 1 and (5.62) 1 are substituted into
T
0
ð Þ
a3,a þ t
0
ð Þ
3 ¼ 2hρ€ u
0
ð Þ
3 ,
T
1
ð Þ
ab,a À T
0
ð Þ
3b þ t
1
ð Þ
b ¼
2h
3
ρ
3
€ u
1
ð Þ
b ,
ð5:63Þ
5.5 Equations for Bending
123
0
ð Þ
pq
h i
¼
c 11
c 12 0 κ 2 c 14
κ 1 c 15
c 16
c 21
c 22 0 κ 2 c 24
κ 1 c 25
c 26
0
0
0
0
0
0
κ 2 c 41 κ 2 c 42 0 κ 2 κ 2 c 44 κ 1 κ 2 c 45 κ 2 c 46
κ 1 c 51 κ 1 c 52 0 κ 1 κ 2 c 54 κ 1 κ 1 c 55 κ 1 c 56
c 61
c 62 0 κ 2 c 64
κ 1 c 65
c 66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ð5:59Þ
e
0
ð Þ
iq
h i
¼
e 11 e 12 0 κ 2 e 14 κ 1 e 15 e 16
e 21 e 22 0 κ 2 e 24 κ 1 e 25 e 26
e 31 e 32 0 κ 2 e 34 κ 1 e 35 e 36
2
6
4
3
7
5:
ð5:60Þ
The two correction factors appear as products or squares at a few places in Eq. (5.59)
because they are introduced in the energy density function (the electric enthalpy) for
the zero-order constitutive relations, which is a quadratic form [5, 6]. In the case of
piezoelectric dielectric plates, the correction factors are determined by requiring the
two fundamental thickness-shear resonance frequencies obtained from the
two-dimensional plate equations to be equal to the corresponding exact frequencies
predicted by the three-dimensional equations. With shear correction factors thus
determined, the two-dimensional plate equations and the exact three-dimensional
equations yield the same frequencies for a particular motion, i.e., the thickness-shear
vibration of a plate in the two fundamental thickness-shear modes. This is useful in
the analysis of plate piezoelectric devices operating with thickness-shear modes.
For the first-order constitutive relations, in Eq. (5.39) 1 , we set
T
1
ð Þ
q ¼ 0, q ¼ 3, 4, 5:
ð5:61Þ
Then, similar to the derivation of Eqs. (5.46) through (5.50), we obtain the relaxed
first-order constitutive relations:
T
1
ð Þ
r ¼
2h
3
3
γ rs S
1
ð Þ
s À ψ kr E
1
ð Þ
k
,
D
1
ð Þ
i ¼
2h
3
3
ψ is S
1
ð Þ
s þ ζ ij E
1
ð Þ
j
,
ð5:62Þ
where the effective plate material constants γ rs , ψ ks , and ζ kj are given by Eq. (5.50).
The right-hand side of Eq. (5.62) does not depend on S
1
ð Þ
3j or u
2
ð Þ
j . When the relevant
ones of Eqs. (5.58) 1 and (5.62) 1 are substituted into
T
0
ð Þ
a3,a þ t
0
ð Þ
3 ¼ 2hρ€ u
0
ð Þ
3 ,
T
1
ð Þ
ab,a À T
0
ð Þ
3b þ t
1
ð Þ
b ¼
2h
3
ρ
3
€ u
1
ð Þ
b ,
ð5:63Þ
5.5 Equations for Bending
123