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in the aluminum plate. This eddy current experiences Lorentz forces under the
interaction with the dynamic magnetic field
B
d
from the coil itself and also with
F
d
resulted from the
dynamic magnetic field always acts like a repulsive force between the aluminum
plate and the coil, whereas the Lorentz force
B
s
.TheLorentzforce
the external static magnetic field
F
s
due to the static magnetic field
is equal to the cross product of the eddy current vector and the external static
magnetic field. These two forces work together to generate vibrations under the
coil. The Cartesian coordinate system is shown in Fig. 1, and the origin locates on
the surface of the aluminum plate. Finite element method (FEM) is an effective
way to calculate Lorentz forces generated by EMATs [6-9,11-13].
In isotropic homogeneous elastic material, the propagation of elastic wave is
)+
ρ∂
2
/∂t
2
=
μ
×
(
×
u
)
−
(
λ
+2
μ
)
(
u
u
F
.
(1)
u
is the displacement vector of elastic wave;
λ
and
μ
are Lam
e
constants;
where,
F
is the body forces.
Driven by the Lorentz forces, Rayleigh waves will be generated in the alu-
minum plate. Rayleigh waves' displacements consist of two components: the in-
plane displacement and the out-of-plane displacement. Rayleigh waves generated
by surface point forces have been documented in early researches [10, 11-16].
Lorentz forces mainly exist under the meander-line coil within the skin depth.
Assuming Lorentz forces have been decomposed into
N
point forces and the
Rayleigh wave displacement at (
X
,
Y
,
Z
) generated by the
i
th
point Lorentz
force component
F
Li
is
u
i
, the total displacement
u
for the whole force compo-
nents of a meander-line coil EMAT is given by
N
i
=1
u
i
(
t
;
X,Y,Z
;
u
(
t
;
X,Y,Z
)=
F
Li
)
.
(2)
3 Modeling Process of Rayleigh Wave EMATs
Rayleigh wave displacements generated by Lorentz force due to dynamic mag-
netic field
f
s
should be calculated
separately [13, 17]. In this paper, two EMAT models are built: 1) a whole model
which consists of a coil, a magnet and an aluminum plate; and 2) a dynamic
magnetic field model (DMF model) which consists of a coil and an aluminum
plate. The whole model describes a complete working process of an EMAT which
contains the influence of both
f
d
and that due to static magnetic field
f
d
and
f
s
, and the Lorentz force
f
L
in the whole
model is the composite force of
f
s
. The DMF model describes a working
process of an EMAT without the static magnetic field, and the Lorentz force is
completely generated by the dynamic magnetic field. As a result, the Lorentz
force in DMF model is
f
d
and
f
s
can be acquired by (7). The 3-D modeling
process of Rayleigh wave EMATs is illustrated in Fig. 2.
This modeling method is valid and effective to calculate Rayleigh waves gen-
erated by EMATs working on Lorentz force mechanism [13].
f
d
.Thus,
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