w 0
ð Þ ¼ 0, ψ 0
ð Þ ¼ 0, b
D 0
ð Þ ¼ 0, J
p
1 0
ð Þ ¼ 0, J
n
1 0
ð Þ ¼ 0,
M L
ð Þ ¼ 0, Q L
ð Þ ¼ F, b
D L
ð Þ ¼ 0, J
p
1 L
ð Þ ¼ 0, J
n
1 L
ð Þ ¼ 0:
ð6:121Þ
Δp and Δn must satisfy the following global charge neutrality conditions:
Z L
0
Δndx 1 ¼ 0,
Z L
0
Δpdx 1 ¼ 0:
ð6:122Þ
Only one of Eq. (6.122) is independent. To determine the electric potential uniquely,
we set
φ 0
ð Þ ¼ 0:
ð6:123Þ
Mathematically, we have a system of linear ordinary differential equations with
constant coefficients. The solution can be obtained in a systematic manner. From
Eq. (6.119), with the use of Eq. (6.115), we can write the charge equation as
e 15 A
1
ð Þ w ,11 þ ψ ,1
À
Á À b ε b
Aφ ,11 ¼ q Δp À Δn
ð
Þ A
2
ð Þ
:
ð6:124Þ
From Eqs. (6.117) 1 and (6.113), the bending equation for u 3 becomes
b c b
A w ,11 þ ψ ,1
À
Á þ e 15 A
1
ð Þ
φ ,11 ¼ 0:
ð6:125Þ
Equations (6.112), (6.120) and the current boundary conditions in Eq. (6.121) lead to
Àqp 0 μ
p
11 φ ,1 À qD
p
11 Δp
ð Þ ,1 ¼ 0,
Àqn 0 μ
p
11 φ ,1 þ qD
p
11 Δn
ð Þ ,1 ¼ 0:
ð6:126Þ
Integrating Eqs. (6.126), we obtain
qΔp ¼ À
qp 0 μ
p
11 φ
D
p
11
þ C 1 ,
ÀqΔn ¼ À
qn 0 μ
n
11 φ
D
n
11
þ C 2 ,
ð6:127Þ
where C 1 and C 2 are undetermined constant. With the use of Eqs. (6.125) and
(6.127), we can write the charge equation in Eq. (6.124) as
φ ,11 À k
2
φ ¼ À
C 1 þ C 2
ð
Þ A
2
ð Þ
e ε
,
ð6:128Þ
where
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6 Composite Structures