Civil Engineering Reference
In-Depth Information
the PVDF was bonded onto it using double-sided tape. This method allows for a durable
and controlled bonding. This bonding layer adds mass and stiffness to the PVDF film,
and thus was accounted for in the finite element model as an isotropic thin elastic layer.
The properties of this layer (Young's modulus, Poisson ratio, structural loss factor) were
identified using a parametric study comparing numerical results and experimental mea-
surements of the radiated pressure from a smart foam placed in a small rectangular cavity
(interior dimensions: height 20 mm; width 64 mm; depth 78 mm). These parameters have
been found to have a negligible influence on the passive absorption of the smart foam.
The modelling of the PVDF film is plate-like. The associated weak formulation, in
discretized form, is given by (Leroy 2008):
T
[
H
m
]
{
T
[
H
f
]
{
work of the in-plane elastic forces
+{
{
δε
m
}
ε
m
}
}
work of the bending elastic forces
δχ
}
χ
d
S
pi
T
[
H
c
]
pi
ω
2
T
[
ρI
]
+{
}
work of the shearing elastic forces
δγ
}
{
γ
d
S
−
{
δu
}
{
u
}
d
S
(13.76)
work of the inertial forces
T
T
−
{
δε
m
}
{
e
c
}
E
z
+
δE
z
{
e
c
}
{
ε
m
}+
δE
z
ε
33
E
z
]
h
d
S
pi
[
work of the piezoelectric and dielectric forces
T
−
{
σ
}
n
d
S
work of the elastic external forces
pi
{
δu
}
+
δD
z
d
S
=
0
∀
(δu,δ)
work of the electrical external forces
where
pi
denotes the surface of the piezoelectric domain and
the surface over which
the electric charge is applied. The first line of this equation depicts the elastic, plate-like,
behaviour of the film.
ε
m
is the in-plane strain field of in-plane type,
χ
is the curvature
vector and
γ
is the shear strain vector. In order to use this plate element in 3D problems,
a drilling degree of freedom of rotation is added, resulting in six degrees of freedom for
the elastic variables. Matrices [
H
m
]
,
[
H
f
]and[
H
c
] represent the in-plane stiffness matrix,
the bending stiffness matrix and the shearing stiffness matrix, respectively.
h,ρ
pi
,E
and
ν
represent the thickness, density, Young's modulus and Poisson ratio of the PVDF film,
respectively. The third line of Equation (13.76) describes the piezoelectric behaviour of
the film. It assumes that the electric field is applied across the film's thickness,
E
z
/h
where
is the electrical potential. This potential is the selected electrical variable in this
2D formulation. The dielectric permittivity matrix [
ε
d
] reduces to the component of the
permittivity along the z axis
ε
33
. The matrix of the piezoelectric coupling coefficients [
e
]
becomes a column vector
=
$
'
e
31
e
32
0
{
e
c
}=
(13.77)
%
(
Finally, the last line of Equation (13.76) describes the external loads. Again, because
of 2D modeling, the electrical displacement vector reduces to a scalar, noted
D
z
.Itis
