where Eq. (4.7) has been used. Then the axial force N, the bending moment M, the
transverse shear force Q, the zero- and first-order moments of the relevant electric
displacement and current components can be expressed as
N ¼
Z
A
T 3 dA ¼ c 33 Aw ,3 þ e 33 Aϕ
0
ð Þ
,3 ,
M ¼
Z
A
x 2 T 3 dA ¼ c 33 Iψ ,3 þ e 33 Iϕ
1
ð Þ
,3 ,
Q ¼
Z
A
T 4 dA ¼ c 44 A v ,3 þ ψ
ð
Þþe 15 Aϕ
1
ð Þ ,
ð4:13Þ
D
0
ð Þ
2 ¼
Z
A
D 2 dA ¼ e 15 A v ,3 þ ψ
ð
ÞÀε 11 Aϕ
1
ð Þ ,
D
0
ð Þ
3 ¼
Z
A
D 3 dA ¼ e 33 Aw ,3 À ε 33 Aϕ
0
ð Þ
,3 ,
D
1
ð Þ
3 ¼
Z
A
x 2 D 3 dA ¼ e 33 Iψ ,3 À ε 33 Iϕ
1
ð Þ
,3 ,
ð4:14Þ
J
p 0
ð Þ
2
¼
Z
A
J
p
2 dA ¼ Àqp 0 μ
p
11 Aϕ
1
ð Þ
À qD
p
11 Ap
1
ð Þ ,
J
p 0
ð Þ
3
¼
Z
A
J
p
3 dA ¼ Àqp 0 μ
p
33 Aϕ
0
ð Þ
,3 À qD
p
33 Ap
0
ð Þ
,3 ,
J
p 1
ð Þ
3
¼
Z
A
x 2 J
p
3 dA ¼ Àqp 0 μ
p
33 Iϕ
1
ð Þ
,3 À qD
p
33 Ip
1
ð Þ
,3 ,
ð4:15Þ
J
n 0
ð Þ
2
¼
Z
A
J
n
2 dA ¼ Àqn 0 μ
n
11 Aϕ
1
ð Þ
þ qD
n
11 An
1
ð Þ ,
J
n 0
ð Þ
3
¼
Z
A
J
n
3 dA ¼ Àqn 0 μ
n
33 Aϕ
0
ð Þ
,3 þ qD
n
33 An
0
ð Þ
,3 ,
J
n 1
ð Þ
3
¼
Z
A
x 2 J
n
3 dA ¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ,
ð4:16Þ
where I and A are the moment of inertia and the area of the beam cross section. For a
circular cross section, they are
I ¼
Z
A
x
2
2 dA ¼
πa
4
4
, A ¼ πa
2
:
ð4:17Þ
The one-dimensional equations of motion, the charge equation of electrostatics, and
the conservation of charge for holes and electrons are obtained by integrating
Eqs. (4.1) and (4.2) as well as their products with x 2 over the fiber cross section
and using integration by parts or the two-dimensional divergence theorem over the
cross section. The results are
92
4 Bending of Beams