The rod is in extension along z. We denote the relevant axial fields by
u 3 ¼ u, S 33 ¼ S, T 33 ¼ T,
E 3 ¼ E, D 3 ¼ D,
J
p
3 ¼ J
p , J
n
3 ¼ J
n
:
ð6:38Þ
For extension of thin rods we have, approximately,
u ffi u z, t
ð Þ, φ ffi φ z, t
ð Þ:
ð6:39Þ
The relevant strain-displacement relation and electric field-potential relation are
S ¼
∂u
∂z
, E ¼ À
∂φ
∂z
:
ð6:40Þ
For polarized ceramics, the relevant constitutive relations are
S ¼ s
E
33 T þ d 33 E,
D ¼ d 33 T þ ε
T
33 E,
ð6:41Þ
where T 1 ¼ T 2 ¼ 0 has been assumed (stress relaxation for thin rods). Equation (6.41)
can be rewritten as
T
1
ð Þ
¼ c
1
ð Þ S À e
1
ð Þ E,
D
1
ð Þ
¼ e
1
ð Þ S þ ε
1
ð Þ E,
ð6:42Þ
where c
(1) , e
(1) , and ε
(1) are the effective one-dimensional elastic, piezoelectric, and
dielectric constants. They are related to the usual three-dimensional material constants through
(2) Si
(1) PZT
z
2L
P
P
F
(1) PZT
F
x
y
b
2c
h
h
Fig. 6.2 Side view and
cross section of a composite
rod of piezoelectric
dielectrics and
nonpiezoelectric
semiconductors
148
6 Composite Structures