Biomedical Engineering Reference
In-Depth Information
Because charge is provided by the integration of the electrical displacement over the
surface of a body, for a piezoelectric film (including PVDF), the only practical way in
which charge,
Q
, can be collected is on its surface (thickness direction). Therefore, for
PVDF films in sensing applications (given a negligible electrical field), we are able to
expand Equation 3.15 and combine it with Equation 3.17 to arrive at:
Q
3
=
d
31
σ
1
+
d
32
σ
2
+
d
33
σ
3
(3.19)
3.2 IEEE Notation
Another popular alternative form of Equations 3.9 and 3.10 (when the change in temper-
ature
≈
0) has been suggested by IEEE and adopted by ANSYS:
=
c
ijkl
S
kl
−
e
kij
E
k
T
ij
(3.20)
D
i
=
e
ikl
S
kl
+
ε
ij
E
k
(3.21)
where:
T
ij
= stress components, (
σ
ij
is adopted in this document)
c
ijkl
= strain components
e
kij
= piezoelectric coefficients
E
k
= electric led components
D
i
= electric displacement components
ε
ij
= permittivity components.
Implementing a compressed matrix notation, the above equation can be expressed in a
more familiar form by replacing
ij
or
kl
by
p
or
q
, where
i, j, k
,and
l
take the values 1,
2, 3 and
p, q
take the values 1, 2,
...
, 6 according to Table 3.1. Then
c
ijkl
,
e
ikl
,and
T
ij
can be replaced by
c
pq
,
e
ip
,and
T
p
, respectively.
The constitutive Equations 3.20 and 3.21 can be written as:
T
p
=
c
pq
S
q
−
e
kp
E
k
(3.22)
D
i
=
e
iq
S
q
+
ε
ik
E
k
(3.23)
where
S
ij
=
S
p
when
i
=
j
,
p
=1,2,3, and2
S
ij
=
S
p
when
i
=
j
,
p
=4,5,6.
Table 3.1
Conversion table for replacing tensor
indices with matrix indices
ij
or
kl
p
or
q
11
1
22
2
33
3
23 or 32
4
31 or 13
5
12 or 21
6
