where we have introduced an index convention that subscripts a, b, c, and d assume
1 and 2 only but not 3. In Eqs. (5.15) and (5.16), the plate electromechanical
resultants are defined by the following integration along the plate thickness:
T
m
ð Þ
ij , D
m
ð Þ
i , J
p m
ð Þ
i
, J
n m
ð Þ
i
n
o
¼
Z h
Àh
x
m
3 T ij , D i , J
p
i , J
n
i
È
É
dx 3 :
ð5:17Þ
The electromechanical loads at the plate top and bottom surfaces are represented by
t
m
ð Þ
j ¼ x
m
3 T 3j
Â
à h
Àh
, d
m
ð Þ
¼ x
m
3 D 3
Â
à h
Àh
,
j
p m
ð Þ
¼ x
m
3 J
p
3
Â
à h
Àh
, j
n m
ð Þ
¼ x
m
3 J
n
3
Â
à h
Àh
:
ð5:18Þ
In addition,
B
m,n
ð Þ
¼
Z h
Àh
x
m
3 x
n
3 dx 3 ¼
2h
mþnþ1
= m þ n þ 1
ð
Þ , m þ n even,
0, m þ n odd:
(
ð5:19Þ
Next we derive plate constitutive equations. Substituting Eqs. (5.3) and (5.4) into
Eq. (5.17), using Eqs. (5.11), (5.12), and (5.19), we obtain the plate constitutive
equations as follows:
T
m
ð Þ
ij ¼
X 1
n¼0
B
m,n
ð Þ c ijkl S
n
ð Þ
kl À e kij E
n
ð Þ
k
,
D
m
ð Þ
i
¼
X 1
n¼0
B
m,n
ð Þ e ijk S
n
ð Þ
jk þ ε ij E
n
ð Þ
j
,
ð5:20Þ
J
p m
ð Þ
i
¼
X 1
n¼0
B
m,n
ð Þ
μ
p
ij E
n
ð Þ
j À D
p
ij P
n
ð Þ
j
,
J
n m
ð Þ
i
¼
X 1
n¼0
B
m,n
ð Þ
μ
n
ij E
n
ð Þ
j þ D
n
ij N
n
ð Þ
j
:
ð5:21Þ
With successive substitutions from Eqs. (5.20), (5.21), (5.13), and (5.14), we can
write Eqs. (5.15) and (5.16) as equations for u
n
ð Þ
i , φ
(n) , p
(n) , and n
(n) .
116
5 Extension and Bending of Plates