u i x 1 , x 2 , x 3 , t
ð
Þ¼
X 1
m¼0
x
m
3 u
m
ð Þ
i
x 1 , x 2 , t
ð
Þ,
φ x 1 , x 2 , x 3 , t
ð
Þ¼
X 1
m¼0
x
m
3 φ
m
ð Þ x 1 , x 2 , t
ð
Þ,
ð5:9Þ
Δp x 1 , x 2 , x 3 , t
ð
Þ¼
X 1
m¼0
x
m
3 p
m
ð Þ x 1 , x 2 , t
ð
Þ,
Δn x 1 , x 2 , x 3 , t
ð
Þ¼
X 1
m¼0
x
m
3 n
m
ð Þ x 1 , x 2 , t
ð
Þ:
ð5:10Þ
Our goal is to obtain two-dimensional equations for u
m
ð Þ
i
, φ
(m)
, p
(m) and n
(m) .
Corresponding to Eqs. (5.9) and (5.10), from Eqs. (5.5) and (5.6), we obtain
S ij ¼
X 1
m¼0
x
m
3 S
m
ð Þ
ij , E i ¼
X 1
m¼0
x
m
3 E
m
ð Þ
i ,
ð5:11Þ
P i ¼
X 1
m¼0
x
m
3 P
m
ð Þ
i , N i ¼
X 1
m¼0
x
m
3 N
m
ð Þ
i ,
ð5:12Þ
where
S
m
ð Þ
ij ¼
1
2
u
m
ð Þ
j,i þ u
m
ð Þ
i,j þ m þ 1
ð
Þ δ i3 u
mþ1
ð
Þ
j
þ δ 3j u
mþ1
ð
Þ
i
h
i
,
E
m
ð Þ
i
¼ Àφ
m
ð Þ
,i À m þ 1
ð
Þδ 3i φ
mþ1
ð
Þ ,
ð5:13Þ
P
m
ð Þ
j ¼ p
m
ð Þ
,j þ m þ 1
ð
Þδ 3j p
mþ1
ð
Þ ,
N
m
ð Þ
j ¼ n
m
ð Þ
,j þ m þ 1
ð
Þδ 3j n
mþ1
ð
Þ
:
ð5:14Þ
Then we multiply Eqs. (5.1) and (5.2) by x
m
3 and integrate the equations through the
plate thickness. This gives the following plate equations of motion, electrostatics,
and continuity:
T
m
ð Þ
aj,a À mT
mÀ1
ð
Þ
3j
þ t
m
ð Þ
j ¼
X 1
n¼0
ρB
m,n
ð Þ
€ u
n
ð Þ
j , a ¼ 1, 2,
D
m
ð Þ
a,a À mD
mÀ1
ð
Þ
3
þ d
m
ð Þ
¼
X 1
n¼0
qB
m,n
ð Þ p
n
ð Þ
À n
n
ð Þ
,
ð5:15Þ
J
p m
ð Þ
a,a À mJ
p mÀ1
ð
Þ
3
þ j
p m
ð Þ
¼ À
X 1
n¼0
qB
m,n
ð Þ
_
p
n
ð Þ ,
J
n m
ð Þ
a,a À mJ
n mÀ1
ð
Þ
3
þ j
n m
ð Þ
¼
X 1
n¼0
qB
m,n
ð Þ
_
n
n
ð Þ ,
ð5:16Þ
5.2 Hierarchy of Two-Dimensional Equations
115