According to the compact matrix notation, with the range of p, q as 1, 2, . . . and
6, Eq. (6.19) can be written as
T q ¼ 0, q ¼ 2, 4, 6:
ð6:20Þ
For convenience we introduce an index convention that subscripts u, v, and w take
the values 2, 4, and 6 while subscripts r, s, and t take the remaining values 1, 3, and
5. Then, the constitutive relations for a nonpiezoelectric semiconductor plate can be
written as
T r ¼ c
0
rs S s þ c
0
ru S u ,
T v ¼ c
0
vs S s þ c
0
vw S w ¼ 0,
D i ¼ ε
0
ij E j ,
ð6:21Þ
where we have used a prime to indicate the elastic and dielectric constants of the
plate. From Eq. (6.21) 2 , we obtain
S u ¼ Àc
0 À1
uv c
0
vs S s :
ð6:22Þ
The substitution of Eq. (6.22) into Eq. (6.21) 1 gives the constitutive relations needed:
T r ¼ c
0
rs S s ,
D i ¼ ε
0
ij E j ,
ð6:23Þ
where we have kept Eq. (6.21) 3 for convenience. In Eq. (6.23) 1 , the thin-plate
effective elastic constants are
c
0
rs ¼ c
0
rs À c
0
rv c
0 À1
vw c
0
ws :
ð6:24Þ
With subscripts a, b, c, and d assuming 1 and 3 but not 2, Eq. (6.23) can be written as
T ab ¼ c
0
abcd S cd ,
D i ¼ ε
0
ij E j :
ð6:25Þ
Integrating Eq. (1.1) 1 for i ¼ 1, 3 when f ¼ 0 with respect to x 2 through the plate
thickness, and also integrating Eq. (1.10) 1,2 along the plate thickness, we obtain the
following equations for the plate:
144
6 Composite Structures