J
p
¼ qp 0 μ
p E À qD
p ∂ p 0 þ Δp
ð
Þ
∂x
,
J
n
¼ qn 0 μ
n E þ qD
n ∂ n 0 þ Δn
ð
Þ
∂x
,
ð3:112Þ
S ¼
∂u
∂x
, E ¼ À
∂φ
∂x
,
ð3:113Þ
where f ¼ f 3 . The definitions of Δp and Δn are given in Eq. (1.7). We consider
uniform p 0 and n 0 . With successive substitutions, Eqs. (3.109) and (3.110) can be
written as
c
∂
2 u
∂x 2 þ e
∂
2 φ
∂x 2 þ f ¼ ρ€ u,
e
∂
2 u
∂x 2 À ε
∂
2 φ
∂x 2 ¼ q Δp À Δn
ð
Þ ,
ð3:114Þ
∂
∂t
Δp
ð Þ ¼ p 0 μ
p ∂
2 φ
∂x 2 þ D
p ∂
2 Δp
ð Þ
∂x 2 ,
∂
∂t
Δn
ð Þ ¼ Àn 0 μ
n ∂
2 φ
∂x 2 þ D
n ∂
2 Δn
ð Þ
∂x 2 :
ð3:115Þ
For static problems, Eqs. (3.114) and (3.115) reduce to
c
d
2 u
dx
2
þ e
d
2
φ
dx
2
þ f ¼ 0,
e
d
2 u
dx
2
À ε
d
2
φ
dx
2
¼ q Δp À Δn
ð
Þ ,
ð3:116Þ
0 ¼ p 0 μ
p d
2
φ
dx
2
þ D
p d
2
Δp
ð Þ
dx
2
,
0 ¼ Àn 0 μ
n d
2
φ
dx
2
þ D
n d
2
Δn
ð Þ
dx
2
:
ð3:117Þ
Equations (3.116) and (3.117) can be partially decoupled into
F
F
2a
x
Fig. 3.12 A piezoelectric semiconductor rod under a local pair of concentrated forces
3.8 Local Extension
55