The total axial force and electric displacement in the composite rod are defined by
integrations of T and D over the entire cross section and are found to be
b
T ¼ b cS À b eE À b λθ,
b
D ¼ b eS þ b εE þ b pθ,
ð7:86Þ
where
b c ¼ c
1
ð Þ A
1
ð Þ
þ c
2
ð Þ A
2
ð Þ , e ¼ e
1
ð Þ A
1
ð Þ ,
b λ ¼ λ
1
ð Þ A
1
ð Þ
þ λ
2
ð Þ A
2
ð Þ ,
b ε ¼ ε
1
ð Þ A
1
ð Þ
þ ε
2
ð Þ A
2
ð Þ , b p ¼ p
1
ð Þ A
1
ð Þ
:
ð7:87Þ
A
(1) and A
(2) are the cross-sectional areas of the piezoelectric and semiconductor
layers, respectively. The one-dimensional equation of motion is obtained by applying Newton’s second law to a differential element of the rod with length dz as shown
in Fig. 7.14:
∂ b
T
∂z
þ f z, t
ð Þ ¼ 2b ρ
1
ð Þ h þ ρ
2
ð Þ c
€ u,
ð7:88Þ
where f(z,t) is the axial load per unit length of the rod. Similarly, the one-dimensional
charge equation can be obtained by considering the differential element in Fig. 7.14
under electrical loads:
d b
D
dz
¼ A
2
ð Þ q Δp À Δn
ð
Þ :
ð7:89Þ
The one-dimensional continuity equations are simply
z
dz
ˆ
T
ˆ
ˆ
T dT

fdz
Fig. 7.14 A differential
element of the rod
7.4 Extension of Composite Rods
199