Biomedical Engineering Reference
In-Depth Information
In the matrix format they can be written [3]:
{
T
}
6
×
1
=
[
c
]
6
×
6
{
S
}
6
×
1
−
[
e
]
6
×
3
{
E
}
3
×
1
(3.24)
[
e
]
6
×
6
{
S
}
6
×
1
+
{
D
}
6
×
1
=
[
ε
]
6
×
3
{
E
}
3
×
1
(3.25)
Normally the units are: [
e
]:pCm
−
2
;[
c
]:Nm
−
2
and [
d
]:pCN
−
11
A typical set of data for the piezoelectric analysis comprises [
e
], [
c
], and [
ε
s
]ortheset
of [
d
], [
s
], and [
ε
T
], where [
ε
s
] is the dielectric permittivity matrix at constant strain while
[
ε
T
] is the dielectric permittivity matrix at constant stress. Between these, the following
relationships can be found:
ε
S
=
ε
T
−
[
e
]
T
[
d
]
(3.26)
The orthotropic dielectric matrix [
ε
] uses electrical permittivity and the following values
for the dielectric permittivity matrix at constant stress are used in this topic [4, 5]:
⎡
⎣
⎤
⎦
7.35 0 0
0 . 70
0
ε
T
=
0
8.05
The following values are used for the piezoelectric strain matrix [4, 5]:
⎡
⎣
⎤
⎦
pC N
−
1
00 0 0
d
15
0
00 0
d
24
00
20 2
[
d
]
=
−
18 0
0 0
and for the stiffness matrix:
⎡
⎣
⎤
⎦
4.7 2.92 2.14 0 0 0
2.43 4.83 1.99 0 0 0
2.20 2.38 4.60 0 0 0
0 0 0 0.106 0 0
0000 .40
0
[
d
]
=
Gpa
0
0
0
0
2.66
In presenting research results for two-dimensional analyses throughout this topic,
(unless otherwise stated) data was provided by Goodfellow, the manufacturer of the
uniaxial and biaxial PVDF film.
3.3 Fundamentals of PVDF
The piezoelectric polymer, PVDF film, exhibits an extremely large piezoelectric
and pyroelectric response, which makes it ideally suitable for the design of highly
sensitive sensors used in both the robotics and endoscopic environments [6-10]. The
piezoelectric applications of the PVDF film vary from robotic (e.g., matrix sensors,
1
Recall from section 3.1 that
D
=
d
σ
and
ε
=
d
T
E
,where
d
is the piezoelectric coefficient,
σ
is the stress,
D
the electric displacement and
E
electric field. On the other hand the stress strain is related by:
σ
c
ε
or
alternatively
ε
=
s
σ
where
c
and
s
are the elastic stiffness and compliance matrices, respectively. The matrix
d
is called
piezoelectric strain matrix
as it initially related mechanical strain to the electric field. In addition, from
ε
=
d
T
E
we can write
c
ε
=
cd
T
E
and then using the Hooks relation:
σ
=
eE
. Therefore, the piezoelectric matrix
can also be defined in [
e
] from (
piezoelectric stress matrix
, since it relates the electrical field with stress) where
the relationship between them can be defined as: [
e
][
c
][
d
].
=
