T
1
ð Þ
ij ¼
2h
3
3
c ijkl S
1
ð Þ
kl À e kij E
1
ð Þ
k
,
D
1
ð Þ
i ¼
2h
3
3
e ijk S
1
ð Þ
jk þ ε ij E
1
ð Þ
j
,
ð5:39Þ
J
p 1
ð Þ
i
¼
2h
3
3
μ
p
ij E
1
ð Þ
j À D
p
ij P
1
ð Þ
j
,
J
n 1
ð Þ
i
¼
2h
3
3
μ
n
ij E
1
ð Þ
j þ D
n
ij N
1
ð Þ
j
,
ð5:40Þ
where T
0
ð Þ
ab represent the plate in-plane extensional and shear forces, T
0
ð Þ
b3 the
transverse shear forces in bending, and T
1
ð Þ
ab the bending and twisting moments.
The zero- and first-order surface loads are
t
0
ð Þ
j ¼ T 3j h
ð Þ À T 3j Àh
ð Þ, d
0
ð Þ
¼ D 3 h
ð Þ À D 3 Àh
ð Þ,
j
p 0
ð Þ
¼ J
p
3 h
ð Þ À J
p
3 Àh
ð Þ, j
n 0
ð Þ
¼ J
n
3 h
ð Þ À J
n
3 Àh
ð Þ,
ð5:41Þ
t
1
ð Þ
j ¼ h T 3j h
ð Þ þ T 3j Àh
ð Þ
Â
Ã
, d
1
ð Þ
¼ h D 3 h
ð Þ þ D 3 Àh
ð Þ
½
,
j
p 1
ð Þ
¼ h J
p
3 h
ð Þ þ J
p
3 Àh
ð Þ
Â
Ã
, j
n 1
ð Þ
¼ h J
n
3 h
ð Þ þ J
n
3 Àh
ð Þ
Â
à :
ð5:42Þ
5.4 Equations for Extension
In this section, we reduce the zero-order equations in the previous section to
equations for extension. This is mainly a treatment of the mechanical fields and
equations. The electrical fields and equations are not reduced. Both the zero- and
first-order electrical fields are kept so that they are available in different situations for
electroded or unelectroded plates. For the extensional motion of a thin plate, the
relevant in-plane extensional displacements are u
0
ð Þ
b where b ¼ 1 and 2. The in-plane
extensional and shear stresses as well as strains are approximately uniform along the
plate thickness. The zero-order equations of motion and constitutive relations are
sufficient:
T
0
ð Þ
ab,a þ t
0
ð Þ
b ¼ 2hρ€ u
0
ð Þ
b , a, b ¼ 1, 2,
ð5:43Þ
T
0
ð Þ
ij ¼ 2h c ijkl S
0
ð Þ
kl À e kij E
0
ð Þ
k
,
D
0
ð Þ
i ¼ 2h e ikl S
0
ð Þ
kl þ ε ij E
0
ð Þ
j
:
ð5:44Þ
From Eq. (5.25), it can be seen that S
0
ð Þ
33 is involved with u
1
ð Þ
3 , and it describes the
Poisson’s effect in extension. S
0
ð Þ
31 and S
0
ð Þ
32 are involved with u
1
ð Þ
a and u
0
ð Þ
3 . They
5.4 Equations for Extension
119
1
ð Þ
ij ¼
2h
3
3
c ijkl S
1
ð Þ
kl À e kij E
1
ð Þ
k
,
D
1
ð Þ
i ¼
2h
3
3
e ijk S
1
ð Þ
jk þ ε ij E
1
ð Þ
j
,
ð5:39Þ
J
p 1
ð Þ
i
¼
2h
3
3
μ
p
ij E
1
ð Þ
j À D
p
ij P
1
ð Þ
j
,
J
n 1
ð Þ
i
¼
2h
3
3
μ
n
ij E
1
ð Þ
j þ D
n
ij N
1
ð Þ
j
,
ð5:40Þ
where T
0
ð Þ
ab represent the plate in-plane extensional and shear forces, T
0
ð Þ
b3 the
transverse shear forces in bending, and T
1
ð Þ
ab the bending and twisting moments.
The zero- and first-order surface loads are
t
0
ð Þ
j ¼ T 3j h
ð Þ À T 3j Àh
ð Þ, d
0
ð Þ
¼ D 3 h
ð Þ À D 3 Àh
ð Þ,
j
p 0
ð Þ
¼ J
p
3 h
ð Þ À J
p
3 Àh
ð Þ, j
n 0
ð Þ
¼ J
n
3 h
ð Þ À J
n
3 Àh
ð Þ,
ð5:41Þ
t
1
ð Þ
j ¼ h T 3j h
ð Þ þ T 3j Àh
ð Þ
Â
Ã
, d
1
ð Þ
¼ h D 3 h
ð Þ þ D 3 Àh
ð Þ
½
,
j
p 1
ð Þ
¼ h J
p
3 h
ð Þ þ J
p
3 Àh
ð Þ
Â
Ã
, j
n 1
ð Þ
¼ h J
n
3 h
ð Þ þ J
n
3 Àh
ð Þ
Â
à :
ð5:42Þ
5.4 Equations for Extension
In this section, we reduce the zero-order equations in the previous section to
equations for extension. This is mainly a treatment of the mechanical fields and
equations. The electrical fields and equations are not reduced. Both the zero- and
first-order electrical fields are kept so that they are available in different situations for
electroded or unelectroded plates. For the extensional motion of a thin plate, the
relevant in-plane extensional displacements are u
0
ð Þ
b where b ¼ 1 and 2. The in-plane
extensional and shear stresses as well as strains are approximately uniform along the
plate thickness. The zero-order equations of motion and constitutive relations are
sufficient:
T
0
ð Þ
ab,a þ t
0
ð Þ
b ¼ 2hρ€ u
0
ð Þ
b , a, b ¼ 1, 2,
ð5:43Þ
T
0
ð Þ
ij ¼ 2h c ijkl S
0
ð Þ
kl À e kij E
0
ð Þ
k
,
D
0
ð Þ
i ¼ 2h e ikl S
0
ð Þ
kl þ ε ij E
0
ð Þ
j
:
ð5:44Þ
From Eq. (5.25), it can be seen that S
0
ð Þ
33 is involved with u
1
ð Þ
3 , and it describes the
Poisson’s effect in extension. S
0
ð Þ
31 and S
0
ð Þ
32 are involved with u
1
ð Þ
a and u
0
ð Þ
3 . They
5.4 Equations for Extension
119