The resultants of the electric displacement and surface charge are, for m ¼ 1 and 2,
D
m
ð Þ
k
¼
Z h
Àh
D k α
m
3 dα 3 , D
m
ð Þ
¼ α
m
3 D 3 A 1 A 2
Â
à h
Àh
:
ð5:140Þ
The current resultants and surface currents are
J
p m
ð Þ
k
¼
Z h
Àh
J
p
k α
m
3 dα 3 , J
p m
ð Þ
¼ α
m
3 J
p
3 A 1 A 2
Â
à h
Àh
,
J
n m
ð Þ
k
¼
Z h
Àh
J
n
k α
m
3 dα 3 , J
n m
ð Þ
¼ α
m
3 J
n
3 A 1 A 2
Â
à h
Àh
:
ð5:141Þ
For shell constitutive relations, we substitute the three-dimensional constitutive
relations in Eqs. (5.3) and (5.4) into Eqs. (5.139), (5.140), and (5.141), and perform
the following stress relaxations:
T
0
ð Þ
33 ¼
Z h
Àh
T 33 dα 3 ¼ 0,
T
1
ð Þ
3j ¼
Z h
Àh
T 33 α 3 dα 3 ¼ 0, j ¼ 1, 2, 3,
ð5:142Þ
so that S
0
ð Þ
33 and S
1
ð Þ
3j , and hence u
1
ð Þ
3 and u
2
ð Þ
j are eliminated in the same procedure as
that in Sect. 5.5. Shear correction factors can be introduced in the manner of
Eq. (5.57). Then the following two-dimensional constitutive relations are obtained
(see Eqs. (5.58), (5.62), (5.38), and (5.40)):
T
0
ð Þ
ij ¼ 2h c
0
ð Þ
ijkl S
0
ð Þ
kl À e
0
ð Þ
kij E
0
ð Þ
k
,
D
0
ð Þ
k ¼ 2h e
0
ð Þ
kij S
0
ð Þ
ij þ ε kl E
0
ð Þ
l
,
ð5:143Þ
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
, r, s ¼ 1, 2, 6,
ð5:144Þ
J
p 0
ð Þ
i
¼ 2h μ
p
ij E
0
ð Þ
j À D
p
ij P
0
ð Þ
j
,
J
n 0
ð Þ
i
¼ 2h μ
n
ij E
0
ð Þ
j þ D
n
ij N
0
ð Þ
j
,
ð5:145Þ
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
:
ð5:146Þ
138
5 Extension and Bending of Plates