J
p
3 ffi qp 0 μ
p
33 E 3 À qD
p
33 Δp
ð Þ ,3 ,
J
n
3 ffi qn 0 μ
n
33 E 3 þ qD
n
33 Δn
ð Þ ,3 :
ð6:88Þ
The one-dimensional conservation of holes and electrons are
q
∂
∂t
Δp
ð Þ ¼ ÀJ
p
3,3 ,
q
∂
∂t
Δn
ð Þ ¼ J
n
3,3 :
ð6:89Þ
We need to solve the boundary-value problem for the static bending of a
cantilever beam under a constant end force F as shown in Fig. 6.8. The boundary
conditions are
u 2 0
ð Þ ¼ 0, u 2,3 0
ð Þ ¼ 0, b
D 3 0
ð Þ ¼ 0, J
p
3 0
ð Þ ¼ 0, J
n
3 0
ð Þ ¼ 0,
M L
ð Þ ¼ 0, Q L
ð Þ ¼ F, b
D 3 L
ð Þ ¼ 0, J
p
3 L
ð Þ ¼ 0, J
n
3 L
ð Þ ¼ 0:
ð6:90Þ
From Eqs. (6.83) 1 , (6.84), and the mechanical boundary condition on the shear force
at x 3 ¼ L in Eq. (6.90), we have
ÀDu 2,333 þ 2Ge
1
ð Þ
33 φ ,33 ¼ F:
ð6:91Þ
From Eq. (6.91) we solve for
u 2,333 ¼
2Ge
1
ð Þ
33
D
φ ,33 À
F
D
:
ð6:92Þ
From Eq. (6.87), with the use of Eq. (6.85), we obtain
x 3
dx 3
M
M+dM
Q
Q+dQ
f 2 dx 3
Fig. 6.9 A differential
element of the beam under
shear forces, bending
moments, and transvers load
158
6 Composite Structures