D
1
ð Þ
a ¼ À
2
3
h
3
ζ 11 φ
1
ð Þ
,a ,
ð5:79Þ
and semiconduction:
J
p 0
ð Þ
a
¼ 2hq Àp 0 μ
p
11 φ
0
ð Þ
,a À d
p
11 p
0
ð Þ
,a
,
J
p 0
ð Þ
3
¼ 2hq Àp 0 μ
p
33 φ
1
ð Þ
À d
p
33 p
1
ð Þ
,
ð5:80Þ
J
p 1
ð Þ
a
¼
2
3
h
3 q Àp 0 μ
p
11 φ
1
ð Þ
,a À d
p
11 p
1
ð Þ
,a
,
ð5:81Þ
J
n 0
ð Þ
a
¼ 2hq Àn 0 μ
n
11 φ
0
ð Þ
,a þ d
n
11 n
0
ð Þ
,a
,
J
n 0
ð Þ
3
¼ 2hq Àn 0 μ
n
33 φ
1
ð Þ
þ d
n
33 n
1
ð Þ
,
ð5:82Þ
J
n 1
ð Þ
a
¼
2
3
h
3 q Àn 0 μ
n
11 φ
1
ð Þ
,a þ d
n
11 n
1
ð Þ
,a
,
ð5:83Þ
where φ
(2) , p
(2) , and n
(2) have been neglected.
The substitution of the above constitutive relations into Eqs. (5.33) 1 , (5.63) 2 ,
(5.34), (5.35), and (5.36) yields the following equations for extension described by
u
0
ð Þ
a :
c 11 u
0
ð Þ
1,11 þ c 66 u
0
ð Þ
1,22 þ c 12 þ c 66
ð
Þ u
0
ð Þ
2,21 þ e 31 φ
1
ð Þ
,1 þ
1
2h
t
0
ð Þ
1 ¼ ρ€ u
0
ð Þ
1 ,
c 66 u
0
ð Þ
2,11 þ c 11 u
0
ð Þ
2,22 þ c 12 þ c 66
ð
Þ u
0
ð Þ
1,12 þ e 15 φ
1
ð Þ
,2 þ
1
2h
t
0
ð Þ
2 ¼ ρ€ u
0
ð Þ
2 ,
ð5:84Þ
flexure described by u
0
ð Þ
3 :
κ
2 c 44 u
0
ð Þ
3,aa þ u
1
ð Þ
a,a
þ κe 15 φ
0
ð Þ
,aa þ
1
2h
t
0
ð Þ
3 ¼ ρ€ u
0
ð Þ
3 ,
ð5:85Þ
thickness-shear described by u
1
ð Þ
a :
c 11 u
1
ð Þ
1,11 þ c 66 u
1
ð Þ
1,22 þ c 12 þ c 66
ð
Þ u
1
ð Þ
2,21
À3h
À2
κ
2 c 44 u
1
ð Þ
1 þ u
0
ð Þ
3,1
þ κe 15 φ
0
ð Þ
,1
h
i
þ
3
2h
3
t
1
ð Þ
1 ¼ ρ€ u
1
ð Þ
1 ,
c 66 u
1
ð Þ
2,11 þ c 11 u
1
ð Þ
2,22 þ c 12 þ c 66
ð
Þ u
1
ð Þ
1,12
À3h
À2
κ
2 c 44 u
1
ð Þ
2 þ u
0
ð Þ
3,2
þ κe 15 φ
0
ð Þ
,2
h
i
þ
3
2h
3
t
1
ð Þ
2 ¼ ρ€ u
1
ð Þ
2 ,
ð5:86Þ
electrostatics described by φ
(0) and φ
(1) :
5.7 Equations for Crystals of Class (6mm)
127