We solve Eq. (3.1) for T 3 and D 3 and obtain the following one-dimensional
piezoelectric constitutive relations for thin rods:
T 3 ¼ c 33 S 3 À e 33 E 3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ,
ð3:2Þ
where c 33 , e 33 , and ε 33 are the effective one-dimensional elastic, piezoelectric, and
dielectric constants. They are related to the usual three-dimensional material constants s
E
pq , ε
T
ij , and d ip through
c 33 ¼ 1=s
E
33 ,
e 33 ¼ d 33 =s
E
33 ,
ε 33 ¼ ε
T
33 À d
2
33 =s
E
33 :
ð3:3Þ
It can be verified that ε 33 þ e
2
33 =c 33 ¼ ε
T
33 : We will write ε
T
33 as ε 33 in the rest of this
chapter whenever there is no ambiguity. The nonlinear one-dimensional constitutive
relations for currents are taken from Eq. (1.2) 3,4 :
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
:
ð3:4Þ
The one-dimensional field equations are from Eq. (1.1):
∂T 3
∂x 3
þ f 3 ¼ ρ€ u 3 ,
∂D 3
∂x 3
¼ q p À n þ N
þ
D À N
À
A
À
Á
,
q _
p ¼ À
∂J
p
3
∂x 3
,
q _
n ¼
∂J
n
3
∂x 3
:
ð3:5Þ
The relevant strain-displacement relation and the electric field-potential relation are
from Eq. (1.5):
x3
x2
x1
c
Fig. 3.1 A piezoelectric
semiconductor rod of
crystals of class (6mm)
32
3 Extension of Rods