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,
ð7:31Þ
J
p
3 ¼ qpμ
p
33 E 3 À qD
p
3 p ,3
ffi qp 0 μ
p
33 E 3 À qD
p
3 Δp
ð Þ ,3 ,
J
n
3 ¼ qnμ
n
33 E 3 þ qD
n
3 n ,3 ,
ffi qn 0 μ
n
33 E 3 þ qD
n
3 Δn
ð Þ ,3 ,
ð7:32Þ
S 33 ¼ u 3,3 , E 3 ¼ Àφ ,3 :
ð7:33Þ
The effective material constants in Eq. (7.31) are defined in Eq. (7.11). Successive
substitutions of Eqs. (7.31), (7.32), and (7.33) into Eqs. (7.29) and (7.30) yield four
second-order differential equations for u 3 , φ, Δp, and Δn which are functions of x 3
and time. The equations may be linear or nonlinear depending on whether the
linearized or nonlinear version of Eq. (7.32) is used.
We study the case when the rod is stress free and electrically open at its two ends
first. We use the linearized constitutive relations for currents in Eq. (7.32) to obtain
an analytical solution and then compare it with a numerical solution using the
nonlinear version of Eq. (7.32). For simplicity, we denote the axial fields and
material constants by
x ¼ x 3 , u ¼ u 3 ,
S ¼ S 33 , T ¼ T 33 ,
E ¼ E 3 , P ¼ P 3 , D ¼ D 3 ,
J
p
¼ J
p
3 , J
n
¼ J
n
3 ,
ð7:34Þ
μ
p
¼ μ
p
33 , μ
n
¼ μ
n
33 ,
D
p
¼ D
p
33 , D
n
¼ D
n
33 ,
c ¼ c 33 , e ¼ e 33 , λ ¼ λ 33 ,
ε ¼ ε 33 , p ¼ p 3 , α ¼ α 33 :
ð7:35Þ
For the linear analytical analysis, we study the special case when L ¼ 1. The
boundary and continuity conditions are
T À1
ð
Þ ¼0, D À1
ð
Þ¼ 0, J
n
À1
ð
Þ¼ 0,
ð7:36Þ
Θ 0
2a
x 3
Θ 0
Θ 0 +θ
2L
Fig. 7.4 A piezoelectric
semiconductor rod under a
local temperature change
7.2 Effects of a Local Temperature Change
183