ρ
P is plotted in Fig. 3.11. It is induced by the extensional force through piezoelectric
coupling and is screened by the mobile charges in a semiconductor. There also exist
concentrated effective polarization charges at the two ends of the rod.
3.8 Local Extension
Consider the piezoelectric semiconductor rod of crystals of class (6mm) shown in
Fig. 3.12. Its c-axis is along x ¼ x 3 . The rod is unbounded along x and is under a local
pair of concentrated forces [6]. We consider linear extension.
We follow the notation of Eq. (3.80). p 0 and n 0 are defined by Eq. (1.8). The
governing equations are from Eqs. (1.1) 1 with i ¼ 3, (1.10), (3.2), (1.9) with i ¼ 3,
and (3.6):
∂T
∂x
þ f ¼ ρ€ u,
∂D
∂x
¼ q Δp À Δn
ð
Þ ,
ð3:109Þ
q
∂ Δp
ð Þ
∂t
¼ À
∂J
p
∂x
,
q
∂ Δn
ð Þ
∂t
¼
∂J
n
∂x
,
ð3:110Þ
T ¼ cS À eE,
D ¼ eS þ εE,
ð3:111Þ
Fig. 3.11 Comparison of
first-order (linear) and
second-order (nonlinear)
solutions of effective
polarization charge density
54
3 Extension of Rods