Ferroelectrics – Physical Effect
282
4. ME effect at bending modes of EMR
A key drawback for ME effect at longitudinal modes is that the frequencies are quite high,
on the order of hundreds of kHz, for nominal sample dimensions. The eddy current losses
for the magnetostrictive phase can be quite high at such frequencies, in particular for
transition metals and alloys and earth rare alloys such as Terfenol-D, resulting in an
inefficient magnetoelectric energy conversion. In order to reduce the operating frequency,
one must therefore increase the laminate size that is inconvenient for any applications. An
alternative for getting a strong ME coupling is the resonance enhancement at bending
modes of the composite. The frequency of applied ac field is expected to be much lower
compared to longitudinal acoustic modes. Recent investigations have showed a giant ME
effect at bending modes in several layered structures (Xing et al., 2006; Zhai et al., 2008;
Chashin et al., 2008). In this section, we focus our attention on theoretical modeling of ME
effects at bending modes. (Petrov et al., 2009)
An in-plane bias field is assumed to be applied to magnetostrictive component to avoid the
demagnetizing field. The thickness of the plate is assumed to be small compared to
remaining dimensions. Moreover, the plate width is assumed small compared to its length.
In that case, we can consider only one component of strain and stress tensors in the EMR
region. The equation of bending motion of bilayer has the form:
2
22
2
0
b
w
w
D
; (11)
where
2
2
is biharmonic operator, w is the deflection (displacement in z-direction), t and ρ
are thickness and average density of sample, b=
p
t+
m
t, ρ=(
p
ρ
p
t +
m
ρ
m
t)/b,
p
ρ,
m
ρ,
and
p
t,
m
t, are
densities and thicknesses of piezoelectric and piezomagnetic, correspondingly, and D is
cylindrical stiffness.
The boundary conditions for x=0 and x=L have to be used for finding the solution of above
equation. Here L is length of bilayer. As an example, we consider the plate with free ends.
At free end, the turning moment M
1
and transverse force V
1
equal zero: M
1
=0 and V
1
=0 at
x=0 and x=L, where
111
A
zT dz
,
1
1
V
x
, and A is the cross-sectional area of the
sample normal to the
x-axis. We are interested in the dynamic ME effect; for an ac magnetic
field H applied to a biased sample, one measures the average induced electric field and
calculates the ME voltage coefficient. Using the open circuit condition, the ME voltage
coefficient can be found as
0
0
3
3
31
1
133
p
z
p
zt
E
p
Edz
E
H
tH
; (12)
where
E
3
and H
1
are the average electric field induced across the sample and applied
magnetic field. The energy losses are taken into account by substituting
for complex
frequency
+i
with
/
=10
-3
.
As an example, we apply Eq. 12 to the bilayer of permendur and PZT. Fig. 2 shows the
frequency dependence of ME voltage coefficient at bending mode for free-standing bilayer