S ij ¼ s
E
ijkl T kl þ d kij E k þ α ij θ,
D i ¼ d ikl T kl þ ε
T
ik E k þ p i θ,
ð7:4Þ
J
p
i ffi qp 0 μ
p
ij E j À qD
p
ij Δp
ð Þ ,j ,
J
n
i ffi qn 0 μ
n
ij E j þ qD
n
ij Δn
ð Þ ,j :
ð7:5Þ
In Eq. (7.5), thermoelectric couplings disappear because of the uniform temperature.
For a thin rod, we have, approximately
u 3 ffi u 3 x 3 , t
ð
Þ, φ ffi φ x 3 , t
ð
Þ,
ð7:6Þ
which are uniform over a cross section. We also assume
Δp ffi Δp x 3 , t
ð
Þ, Δn ffi Δn x 3 , t
ð
Þ:
ð7:7Þ
From Eq. (7.6), the axial strain and electric field are
S 3 ¼ S 33 ¼ u 3,3 , E 3 ¼ Àφ ,3 :
ð7:8Þ
For thin rods the stress is approximately uniaxial, i.e.,
T 1 ¼ T 2 ffi 0:
ð7:9Þ
Substituting Eq. (7.9) into the relevant ones of Eq. (7.4) and solving for T 33 and D 3 ,
we obtain
T 33 ¼ c 33 S 33 À e 33 E 3 À λ 33 θ,
D 3 ¼ e 33 S 33 þ ε 33 E 3 þ p 3 θ,
ð7:10Þ
where the effective one-dimensional material constants are defined by
x 3
x 2
x 1
c
2L
Fig. 7.1 A free
piezoelectric semiconductor
rod
178
7 Thermal Effects