φ ffi φ x 3 , t
ð
Þ, E 3 ¼ Àφ ,3 :
ð6:74Þ
Under the usual stress approximation for a beam in bending, we assume
T 1 ffi 0, T 2 ffi 0:
ð6:75Þ
Then the relevant constitutive relations for the upper ceramic layer can be written as
S 3 ¼ s
1
ð Þ
33 T
1
ð Þ
3 þ d
1
ð Þ
33 E 3 ,
D
1
ð Þ
3 ¼ d
1
ð Þ
33 T
1
ð Þ
3 þ ε
1
ð Þ
33 E 3:
ð6:76Þ
From Eq. (6.76) we solve for the stress and electric displacement to obtain
T
1
ð Þ
3 ¼ c
1
ð Þ
33 S 3 À e
1
ð Þ
33 E 3 ,
D
1
ð Þ
3 ¼ e
1
ð Þ
33 S 3 þ ε
1
ð Þ
33 E 3 ,
ð6:77Þ
where
c
1
ð Þ
33 ¼ 1=s
1
ð Þ
33 , e
1
ð Þ
33 ¼ d
1
ð Þ
33 =s
1
ð Þ
33 ,
ε
1
ð Þ
33 ¼ ε
1
ð Þ
33 1 À k
2
33
À
Á
, k
2
33 ¼ d
1
ð Þ
33
2 = ε
1
ð Þ
33 s
1
ð Þ
33
:
ð6:78Þ
Similarly, for the lower ceramic layer with opposite poling, formally Eqs. (6.76),
(6.77), and (6.78) are still valid but d
1
ð Þ
33 changes its sign. For the semiconductor layer
in the middle which is nonpiezoelectric, the constitutive relations are
T
2
ð Þ
3 ¼ c
2
ð Þ
33 S 3
D
2
ð Þ
3 ¼ ε
2
ð Þ
33 E 3 ,
ð6:79Þ
where c
2
ð Þ
33 ¼ 1=s
2
ð Þ
33 . The bending moment M is defined by the following integral
over a cross section of the beam:
M ¼
Z
x 2 T 3 dx 1 dx 2 ¼ ÀDu 2,33 þ 2Ge
1
ð Þ
33 φ ,3 ,
ð6:80Þ
where Eqs. (6.77) and (6.79) have been used, and
D ¼
2
3
c
2
ð Þ
33 c
3
þ
2
3
c
1
ð Þ
33
c þ h
ð
Þ
3 À c
3
h
i
n
o
b,
G ¼ c þ
h
2
hb:
ð6:81Þ
156
6 Composite Structures