Developments in Ceramic Materials Research

220
ε
S
33
R. Ramesh, H. Kara, Ron Stevens and C. R. Bowen
⎡
p 2
c S
2v
e
⎤
31
= v ⎢ε
33 +
v
c
p E E ⎥ +
⎢⎣
v ( c11
+ c12
) + v ( c11
+ c12
) ⎥⎦
The piezoelectric coefficients are given by,
e
d
d
33
31
33
= v
= v
c
C
⎡
⎢e
⎢⎣
33
⎡
⎢d
⎢⎣
d
=
C
v s
33
11
31
v
E ( c − c )
( ) ( )⎥ ⎥ p
2v
e
⎤
31 13 12
−
c
p E E
v c11
+ c12
+ v c11
+ c12
⎦
+ d
33
v
C
s11
P E
v s33
+
v
C
P
E ( s ) ⎤
12 − s13
P E ⎥
11 + v s33
⎥⎦ The hydrostatic coefficients are given by,
d h
g
h
= d
d
=
T
ε
33
h
33
+ 2d
31
s
The thickness-mode electromechanical coupling coefficient is given by,
k
2
t
=
ε
2
e33
S D
33c
33
The electrical impedance of the composite plate vibrating in thickness-mode is given by,
1 ⎡ 2 tan
Z =
⎢
1−
k t
jωC
0 ⎣ αt
( αt)
⎤
⎥⎦
where, t is half thickness of the composite plate
α = ω
ρ
D
c 33
ω = 2πf
(13)
and C 0 is the clamped capacitance as given by,
p
ε
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)