T 1 ¼ c
2
ð Þ
11 S 1 ¼ c
2
ð Þ
11 x 3 ψ ,1 ,
T 5 ¼ c
2
ð Þ
55 S 5 ¼ c
2
ð Þ
55 w ,1 þ ψ
ð
Þ ,
D 1 ¼ ε
2
ð Þ
11 E 1 ¼ Àε
2
ð Þ
11 φ ,1 ,
ð6:109Þ
where
c
2
ð Þ
11 ¼ 1=s
2
ð Þ
11 , c
2
ð Þ
55 ¼ 1=s
2
ð Þ
55 :
ð6:110Þ
The carrier concentration perturbations in the semiconductor layers are approximated by
Δp x, t
ð Þ ffi Δp x 1 , t
ð
Þ,
Δn x, t
ð Þ ffi Δn x 1 , t
ð
Þ:
ð6:111Þ
The relevant one-dimensional linear constitutive relations for the current densities
are
J
p
1 ¼ qp 0 μ
p
11 E 1 À qD
p
11 Δp
ð Þ ,1
¼ Àqp 0 μ
p
11 φ ,1 À qD
p
11 Δp
ð Þ ,1 ,
J
n
1 ¼ qn 0 μ
n
11 E 1 þ qD
n
11 Δn
ð Þ ,1
¼ Àqn 0 μ
n
11 φ ,1 þ qD
n
11 Δn
ð Þ ,1 :
ð6:112Þ
The shear force Q, bending moment M, and total axial electric displacement b
D are
given by the following integrals over a typical cross section of the beam:
Q ¼
Z
T 13 dx 2 dx 3 ¼ b c b
A w ,1 þ ψ
ð
Þþe 15 A
1
ð Þ
φ ,1 ,
ð6:113Þ
M ¼
Z
x 3 T 1 dx 1 dx 3 ¼ Dψ ,1 ,
ð6:114Þ
b
D ¼
Z
D 1 dx 2 dx 3 ¼ e 15 A
1
ð Þ w ,1 þ ψ
ð
ÞÀb ε b
Aφ ,1 ,
ð6:115Þ
where
A
1
ð Þ
¼ 2bc, A
2
ð Þ
¼ 2bh, b
A ¼ A
1
ð Þ
þ A
2
ð Þ ,
b c b
A ¼ c
1
ð Þ
55 A
1
ð Þ
þ c
2
ð Þ
55 A
2
ð Þ , b ε b
A ¼ ε
1
ð Þ
11 A
1
ð Þ
þ ε
2
ð Þ
11 A
2
ð Þ ,
D ¼
2
3
bc
3 c
1
ð Þ
11 þ
2
3
b c þ h
ð
Þ
3 À c
3
h
i
c
2
ð Þ
11 :
ð6:116Þ
164
6 Composite Structures