motion, the charge equation of electrostatics, the heat equation, and the conservation
of charge for holes and electrons:
T ji,j ¼ ρ€ u i ,
D i,i ¼ q p À n þ N
þ
D À N
À
A
À
Á
,
Θ 0 _
η ¼ Àh i,i þ γ,
q _
p ¼ ÀJ
p
i,i ,
q _
n ¼ J
n
i,i ,
ð1:43Þ
where h is the heat flux vector, γ the body heat source, Θ 0 a reference temperature,
and η the entropy density. γ includes body heat from various origins and may have a
nonlinear expression such as the Joule or Ohmic heating due to conduction. Constitutive relations accompanying Eq. (1.43) can be written in the following form:
T ij ¼ c ijkl S kl À e kij E k À λ ij θ,
D i ¼ e ikl S kl þ ε ik E k þ p i θ,
h i ¼ Àκ ij θ ,j þ κ
E
ij E j ,
η ¼ λ kl S kl þ p k E k þ αθ,
J
p
i ¼ qpμ
p
ij E j À qD
p
ij p ,j À qpD
Tp
ij θ ,j
¼ qpμ
p
ij E j À qD
p
ij p ,j À σ
Tp
ij θ ,j ,
J
n
i ¼ qnμ
n
ij E j þ qD
n
ij n ,j þ qnD
Tn
ij θ ,j
¼ qnμ
n
ij E j þ qD
n
ij n ,j þ σ
Tn
ij θ ,j ,
ð1:44Þ
where |θ| < < Θ 0 is a small temperature change from Θ 0 . λ ij are the thermoelastic
constants, p j the pyroelectric constants, κ ij the heat conduction coefficients, D
Tp
ij and
D
Tn
ij the thermal diffusion constants [9], κ
E
ij , σ
Tp
ij , and σ
Tn
ij the thermoelectric constants
[12]. D
Tp
ij and D
Tn
ij may be related to D
p
ij and D
n
ij through [9]:
D
Tp
ffi
D
p
2T
, D
Tn
ffi
D
n
2T
:
ð1:45Þ
The heat flux and the currents in Eq. (1.44) are further restricted by the ClausiusDuhem inequality [12]. Similar to Eq. (1.3), the constitutive relations for
thermopiezoelectric materials can also be written in other forms. In addition, there
are other nonlinear forms of thermoelectric constitutive relations which can be found
in [13].
10
1 Macroscopic Theory
of charge for holes and electrons:
T ji,j ¼ ρ€ u i ,
D i,i ¼ q p À n þ N
þ
D À N
À
A
À
Á
,
Θ 0 _
η ¼ Àh i,i þ γ,
q _
p ¼ ÀJ
p
i,i ,
q _
n ¼ J
n
i,i ,
ð1:43Þ
where h is the heat flux vector, γ the body heat source, Θ 0 a reference temperature,
and η the entropy density. γ includes body heat from various origins and may have a
nonlinear expression such as the Joule or Ohmic heating due to conduction. Constitutive relations accompanying Eq. (1.43) can be written in the following form:
T ij ¼ c ijkl S kl À e kij E k À λ ij θ,
D i ¼ e ikl S kl þ ε ik E k þ p i θ,
h i ¼ Àκ ij θ ,j þ κ
E
ij E j ,
η ¼ λ kl S kl þ p k E k þ αθ,
J
p
i ¼ qpμ
p
ij E j À qD
p
ij p ,j À qpD
Tp
ij θ ,j
¼ qpμ
p
ij E j À qD
p
ij p ,j À σ
Tp
ij θ ,j ,
J
n
i ¼ qnμ
n
ij E j þ qD
n
ij n ,j þ qnD
Tn
ij θ ,j
¼ qnμ
n
ij E j þ qD
n
ij n ,j þ σ
Tn
ij θ ,j ,
ð1:44Þ
where |θ| < < Θ 0 is a small temperature change from Θ 0 . λ ij are the thermoelastic
constants, p j the pyroelectric constants, κ ij the heat conduction coefficients, D
Tp
ij and
D
Tn
ij the thermal diffusion constants [9], κ
E
ij , σ
Tp
ij , and σ
Tn
ij the thermoelectric constants
[12]. D
Tp
ij and D
Tn
ij may be related to D
p
ij and D
n
ij through [9]:
D
Tp
ffi
D
p
2T
, D
Tn
ffi
D
n
2T
:
ð1:45Þ
The heat flux and the currents in Eq. (1.44) are further restricted by the ClausiusDuhem inequality [12]. Similar to Eq. (1.3), the constitutive relations for
thermopiezoelectric materials can also be written in other forms. In addition, there
are other nonlinear forms of thermoelectric constitutive relations which can be found
in [13].
10
1 Macroscopic Theory