5.7 Equations for Crystals of Class (6mm)
Plates of crystals of class (6mm) such as ZnO are widely used in devices. Under the
compact matrix notation, with the c-axis along the plate thickness x 3 , the material
matrices of these crystals are
c pq
 à ¼
c 11 c 12 c 13 0
0
0
c 21 c 11 c 13 0
0
0
c 31 c 31 c 33 0
0
0
0
0
0 c 44 0
0
0
0
0
0 c 44 0
0
0
0
0
0 c 66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ð5:69Þ
e ip
 à ¼
0
0
0
0 e 15 0
0
0
0 e 15 0 0
e 31 e 31 e 33 0
0 0
0
B
@
1
C
A, ε ij
 à ¼
ε 11 0
0
0 ε 11 0
0
0 ε 33
0
B
@
1
C
A, ð5:70Þ
where c 66 ¼ (c 11 –c 12 )/2. μ ij and d ij have the same structure as ε ij . For these materials
there is no coupling between extension and shear. After the stress relaxation in
Eq. (5.53), only some of the elements of Eqs. (5.69) and (5.70) are modified.
Eq. (5.56) takes the following specific form:
c pq
 à ¼
c 11 c 12 0 0
0
0
c 21 c 11 0 0
0
0
0
0 0 0
0
0
0
0 0 c 44 0
0
0
0 0 0 c 44 0
0
0 0 0
0 c 66
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ð5:71Þ
e iq
 à ¼
0
0 0 0 e 15 0
0
0 0 e 15 0 0
e 31 e 31 0 0
0 0
2
6
4
3
7
5, ε kl
½ Š ¼
ε 11 0
0
0 ε 11 0
0
0 ε 33
2
6
4
3
7
5,
ð5:72Þ
where
c 11 ¼ c 11 À c 13
ð Þ
2 =c 33 ,
c 12 ¼ c 12 À c 13
ð Þ
2 =c 33 ,
e 31 ¼ e 31 À e 33 c 13 =c 33 ,
ε 33 ¼ ε 33 þ e
2
33 =c 33 :
ð5:73Þ
5.7 Equations for Crystals of Class (6mm)
125