The stress relaxation in Eqs. (5.46) or (5.61) yields the following results:
γ rs
½ Š ¼
c 11 c 12 0
c 12 c 11 0
0
0 c 66
2
6
6
4
3
7
7
5 , ψ ks
½ Š ¼
0
0 0
0
0 0
e 31 e 31 0
2
6
6
4
3
7
7
5 ,
ζ kj
 à ¼
ζ 11 0
0
0 ζ 11 0
0
0 ε 33
2
6
6
4
3
7
7
5 , r, s ¼ 1, 2, 6, k, j ¼ 1, 2, 3,
ð5:74Þ
where
ζ 11 ¼ ε 11 þ e
2
15 =c 44 :
ð5:75Þ
For plates made from crystals of class (6mm), the two thickness-shear correction
factors are equal, i.e., κ 1 ¼ κ 2 ¼ κ. The value of κ will be determined in the next
section.
The constitutive relations in Eqs. (5.58), (5.62), (5.38), and (5.40) take the
following form for extension:
T
0
ð Þ
11 ¼ 2h c 11 u
0
ð Þ
1,1 þ c 12 u
0
ð Þ
2,2 þ e 31 φ
1
ð Þ
,
T
0
ð Þ
22 ¼ 2h c 12 u
0
ð Þ
1,1 þ c 11 u
0
ð Þ
2,2 þ e 31 φ
1
ð Þ
,
T
0
ð Þ
12 ¼ 2hc 66 u
0
ð Þ
1,2 þ u
0
ð Þ
2,1
,
ð5:76Þ
bending:
T
0
ð Þ
3a ¼ 2h κ
2 c 44 u
0
ð Þ
3,a þ u
1
ð Þ
a
þ κe 15 φ
0
ð Þ
,a
h
i
,
T
1
ð Þ
11 ¼
2
3
h
3 c 11 u
1
ð Þ
1,1 þ c 12 u
1
ð Þ
2,2
,
T
1
ð Þ
22 ¼
2
3
h
3 c 12 u
1
ð Þ
1,1 þ c 11 u
1
ð Þ
2,2
,
T
1
ð Þ
12 ¼
2
3
h
3 c 66 u
1
ð Þ
1,2 þ u
1
ð Þ
2,1
,
ð5:77Þ
electrostatics:
D
0
ð Þ
a ¼ 2h κe 15 u
0
ð Þ
3,a þ u
1
ð Þ
a
À ε 11 φ
0
ð Þ
,a
h
i
,
D
0
ð Þ
3 ¼ 2h e 31 u
0
ð Þ
a,a À ε 33 φ
1
ð Þ
,
ð5:78Þ
126
5 Extension and Bending of Plates