A
(1) and A
(2) are the cross sectional areas of the piezoelectric and semiconductor
layers, respectively. Similarly, the total electric displacement over the cross section
of the composite rod is
b
D ¼ D
1
ð Þ A
1
ð Þ
þ D
2
ð Þ A
2
ð Þ
¼ e
1
ð Þ S þ ε
1
ð Þ E
A
1
ð Þ
þ ε
2
ð Þ E
A
2
ð Þ
¼ e
1
ð Þ A
1
ð Þ S þ ε
1
ð Þ A
1
ð Þ
þ ε
2
ð Þ A
2
ð Þ
E
¼ b eS þ b εE,
ð6:50Þ
where
b ε ¼ ε
1
ð Þ A
1
ð Þ
þ ε
2
ð Þ A
2
ð Þ
:
ð6:51Þ
For extension, the equation of motion in the axial direction can be obtained by
considering a differential element of the rod with length dz as shown in Fig. 6.3:
∂ b
T
∂z
þ f z, t
ð Þ ¼ 2b ρ
1
ð Þ h þ ρ
2
ð Þ c
€ u,
ð6:52Þ
where f(z,t) is the axial load per unit length of the rod. Similarly, the charge equation
of electrostatics and the conservation of holes and electrons of the composite rod are
∂ b
D
∂z
¼ q Δp À Δn
ð
Þ A
2
ð Þ ,
ð6:53Þ
q
∂
∂t
Δp
ð Þ ¼ À
∂J
p
∂z
,
q
∂
∂t
Δn
ð Þ ¼
∂J
n
∂z
:
ð6:54Þ
We consider static extension of an electrically isolated rod. The boundary conditions are
z
dz
ˆ
T
ˆ
ˆ
T dT
+
fdz
Fig. 6.3 A differential
element of the
composite rod
150
6 Composite Structures