68
4 Nonlinear Constitutive Relations
4.4.4 Piezo Constants for MFC-d33 Type
Compared to MFC-d31 material, even though MFC-d33 material has similar arrangement, but it has different polarization direction in piezoelectric fiber. The polarization
direction of MFC-d33 is pointing along the piezoelectric fiber reinforcement. Thus
the driving electric field must align in the same or opposite direction of polarization.
The piezoelectric constant matrix for MFC-d33 will be organized as
˘
e =
⎡
⎣
˘
e 11 ˘
e 12 0 0 0
0 0 ˘
e 26 0 0
0 0 0 0 ˘
e 35
⎤
⎦ .
(4.59)
In MFC-d33 material, both the polarization and electric field directions align
with the piezo fiber orientation. This will lead to MFC-d33 mainly using d33 effect.
Because only one pair of electrodes existing in MFC-d33 patches, the electric field
can be applied only in the polarization direction. Therefore, in a similar way, the
constitutive equation for the direct effect reduces to
˘
D 1 =
˘
e 11 ˘
e 12 0 0 0
˘
ε + ˘
χ 11 ˘
E 1 ,
(4.60)
with
˘
e 11 =
˘
d 11 ˘
s 22 − ˘
d 12 ˘
s 12
˘
s 11 ˘
s 22 − ˘
s 12 ˘
s 12
= ˘
d 11 ˘
c 11 + ˘
d 12 ˘
c 12 ,
(4.61)
˘
e 12 =
˘
d 11 ˘
s 12 − ˘
d 12 ˘
s 11
˘
s 12 ˘
s 12 − ˘
s 11 ˘
s 22
= ˘
d 11 ˘
c 12 + ˘
d 12 ˘
c 22 ,
(4.62)
˘
χ 11 = ˘
11 − ˘
d 11 ˘
e 11 − ˘
d 12 ˘
e 12 ,
(4.63)
The mode of interdigitated electrodes in MFC-d33 patches are very different
with that in MFC-d31. In this arrangement, the electric field is very complex and
distributed non-uniformly, in which a certain volume of piezoelectric fiber is inactive.
The real electric field distribution along the piezoelectric fiber was deeply investigated
by Bowen et al. [16]. For simplicity, the paper follows the work of Williams [17] that
the electric field is assumed to be uniform and constant between two electrodes and
distributed perfectly through the material, which yields
˘
E 1 = −
˘
Φ 1
h E
.
(4.64)
Here h E denotes the distance between two electrodes, which is not equal to the
thickness of the MFC-d33 layer, as can be seen in Fig. 4.4b, and ˘
Φ 1 is the electric
voltage applied along the ˘
Θ
1 -axis.
4 Nonlinear Constitutive Relations
4.4.4 Piezo Constants for MFC-d33 Type
Compared to MFC-d31 material, even though MFC-d33 material has similar arrangement, but it has different polarization direction in piezoelectric fiber. The polarization
direction of MFC-d33 is pointing along the piezoelectric fiber reinforcement. Thus
the driving electric field must align in the same or opposite direction of polarization.
The piezoelectric constant matrix for MFC-d33 will be organized as
˘
e =
⎡
⎣
˘
e 11 ˘
e 12 0 0 0
0 0 ˘
e 26 0 0
0 0 0 0 ˘
e 35
⎤
⎦ .
(4.59)
In MFC-d33 material, both the polarization and electric field directions align
with the piezo fiber orientation. This will lead to MFC-d33 mainly using d33 effect.
Because only one pair of electrodes existing in MFC-d33 patches, the electric field
can be applied only in the polarization direction. Therefore, in a similar way, the
constitutive equation for the direct effect reduces to
˘
D 1 =
˘
e 11 ˘
e 12 0 0 0
˘
ε + ˘
χ 11 ˘
E 1 ,
(4.60)
with
˘
e 11 =
˘
d 11 ˘
s 22 − ˘
d 12 ˘
s 12
˘
s 11 ˘
s 22 − ˘
s 12 ˘
s 12
= ˘
d 11 ˘
c 11 + ˘
d 12 ˘
c 12 ,
(4.61)
˘
e 12 =
˘
d 11 ˘
s 12 − ˘
d 12 ˘
s 11
˘
s 12 ˘
s 12 − ˘
s 11 ˘
s 22
= ˘
d 11 ˘
c 12 + ˘
d 12 ˘
c 22 ,
(4.62)
˘
χ 11 = ˘
11 − ˘
d 11 ˘
e 11 − ˘
d 12 ˘
e 12 ,
(4.63)
The mode of interdigitated electrodes in MFC-d33 patches are very different
with that in MFC-d31. In this arrangement, the electric field is very complex and
distributed non-uniformly, in which a certain volume of piezoelectric fiber is inactive.
The real electric field distribution along the piezoelectric fiber was deeply investigated
by Bowen et al. [16]. For simplicity, the paper follows the work of Williams [17] that
the electric field is assumed to be uniform and constant between two electrodes and
distributed perfectly through the material, which yields
˘
E 1 = −
˘
Φ 1
h E
.
(4.64)
Here h E denotes the distance between two electrodes, which is not equal to the
thickness of the MFC-d33 layer, as can be seen in Fig. 4.4b, and ˘
Φ 1 is the electric
voltage applied along the ˘
Θ
1 -axis.