E 1 ¼ À
1
A 1 1 þ
α 3
R 1
∂φ
∂α 1
, E 2 ¼ À
1
A 2 1 þ
α 3
R 2
∂φ
∂α 2
, E 3 ¼ À
∂φ
∂α 3
:
ð5:113Þ
The divergence of a vector field, e.g., the electric displacement vector, in the shell
coordinates is
∇ Á D ¼
1
A 1 1 þ
α 3
R 1
A 2 1 þ
α 3
R 2
∂
∂α 1
A 2 1 þ
α 3
R 2
D 1
!
&
þ
∂
∂α 2
A 1 1 þ
α 3
R 1
D 2
!
þ
∂
∂α 3
A 1 1 þ
α 3
R 1
A 2 1 þ
α 3
R 2
D 3
! '
:
ð5:114Þ
For thin shells, if the dependence of A 1 (1 + α 3 /R 1 ) and A 2 (1 + α 3 /R 2 ) on α 3 is
neglected after their derivatives in the above formulas with respect to α 3 have been
carried out, Eqs. (5.109) through (5.114) reduce to
S 11 ¼
1
A 1
∂u 1
∂α 1
þ
u 2
A 2
∂A 1
∂α 2
þ
A 1 u 3
R 1
!
,
S 22 ¼
1
A 2
∂u 2
∂α 2
þ
u 1
A 1
∂A 2
∂α 1
þ
A 2 u 3
R 2
!
, S 33 ¼
∂u 3
∂α 3
,
ð5:115Þ
2S 23 ¼
∂u 2
∂α 3
À
u 2
R 2
þ
1
A 2
∂u 3
∂α 2
,
2S 31 ¼
∂u 1
∂α 3
À
u 1
R 1
þ
1
A 1
∂u 3
∂α 1
,
2S 12 ¼
A 1
A 2
∂
∂α 2
u 1
A 1
!
þ
A 2
A 1
∂
∂α 1
u 2
A 2
!
,
ð5:116Þ
E 1 ¼ À
1
A 1
∂φ
∂α 1
, E 2 ¼ À
1
A 2
∂φ
∂α 2
, E 3 ¼ À
∂φ
∂α 3
,
ð5:117Þ
∇ Á D ¼
1
A 1 A 2
∂
∂α 1
A 2 D 1
½
Šþ
∂
∂α 2
A 1 D 2
½
Šþ
∂
∂α 3
A 1 A 2 D 3
½
ŠþA 1 A 2
1
R 1
þ
1
R 2
D 3
&
'
:
ð5:118Þ
The power series expansion method for deriving plate equations can also be
applied to shells. For a first-order shear deformation theory, we use the expansions in
Eqs. (5.22) and (5.23) in which where u
1
ð Þ
3 , u
2
ð Þ
j , φ
(2) , p
(2) , and n
(2) will be eliminated
through stress relaxation or simply neglected later. Under Eqs. (5.22) and (5.23), the
strains, electric fields, and the gradients of carrier concentration perturbations for
thin shells can be written as
134
5 Extension and Bending of Plates