J
p
3 ¼ qpμ
p
33 E 3 À qD
p
33
∂p
∂x 3
,
J
n
3 ¼ qnμ
n
33 E 3 þ qD
n
33
∂n
∂x 3
,
ð3:217Þ
S 3 ¼
∂u 3
∂x 3
, E 3 ¼ À
∂φ
∂x 3
:
ð3:218Þ
Let
u 3 x 3 , t
ð
Þ ¼ u x 3
ð Þ þ e u x 3 , t
ð
Þ, φ x 3 , t
ð
Þ ¼ φ x 3
ð Þ þ e
φ x 3 , t
ð
Þ,
S 3 x 3 , t
ð
Þ ¼ S x 3
ð Þ þ e S x 3 , t
ð
Þ, E 3 x 3 , t
ð
Þ ¼ E x 3
ð Þ þ e
E x 3 , t
ð
Þ,
T 3 x 3 , t
ð
Þ ¼ T x 3
ð Þ þ e
T x 3 , t
ð
Þ, D 3 x 3 , t
ð
Þ ¼ D x 3
ð Þ þ e
D x 3 , t
ð
Þ,
ð3:219Þ
p x 3 , t
ð
Þ ¼ p x 3
ð Þ þ e p x 3 , t
ð
Þ, n x 3 , t
ð
Þ ¼ n x 3
ð Þ þ e n x 3 , t
ð
Þ,
J
p
3 x 3 , t
ð
Þ ¼ J
p x 3
ð Þ þ e J
p x 3 , t
ð
Þ, J
n
3 x 3 , t
ð
Þ ¼ J
n x 3
ð Þ þ e J
n x 3 , t
ð
Þ,
ð3:220Þ
where we have used an over bar to describe the finite static initial fields and a tilde for
the small dynamic incremental fields. For one-dimensional problems of the extension of a rod, to simplify the notation, we denote
x 3 ¼ x,
c 33 ¼ c, e 33 ¼ e, ε 33 ¼ ε,
μ
p
33 ¼ μ
p , D
p
33 ¼ D
p ,
μ
n
33 ¼ μ
n , D
n
33 ¼ D
n
:
ð3:221Þ
Substituting Eqs. (3.219), (3.220), and (3.221) into Eqs. (3.214), (3.215), (3.216),
(3.217), and (3.218), collecting zero- and first-order terms of the small incremental
fields, we obtain two sets of equations. One is for the static initial fields:
∂T
∂x
¼ 0,
∂D
∂x
¼ q p À n þ N
þ
D À N
À
A
À
Á
,
ð3:222Þ
x3
x2
x1
c
Fig. 3.33 A piezoelectric
semiconductor rod of
crystals of class (6mm)
86
3 Extension of Rods