64
4 Nonlinear Constitutive Relations
where
c = T
T
˘
cT , e = Q
T
˘
eT , = Q
T
˘
Q .
(4.40)
The transformation matrix Q is an identity matrix if the electrical coordinate axes
are parallel to the curvilinear coordinate lines, and T is given by
T =
⎡
⎢
⎢
⎢
⎢
⎣
t
2
11
t
2
21
t 11 t 21
0 0
t
2
12
t
2
22
t 12 t 22
0 0
2t 11 t 12 2t 21 t 22 t 11 t 22 + t 12 t 21 0 0
0
0
0
t 22 t 12
0
0
0
t 21 t 11
⎤
⎥
⎥
⎥
⎥
⎦
,
(4.41)
with
t 11 = g
1
· ˘
i 1 =
g 11 cos θ ,
t 12 = g
1
· ˘
i 2 = −
g 11 sin θ ,
t 21 = g
2
· ˘
i 1 =
g 22 sin θ ,
t 22 = g
2
· ˘
i 2 =
g 22 cos θ .
(4.42)
4.4 Constitutive Relations for Macro-fiber Composites
4.4.1 Configurations of Macro-fiber Composites
Macro-fiber composites mainly consist of piezoceramic fibers, epoxy matrix and
electrodes. They have two different modes of structures, namely d31 or d33 modes.
The first mode of MFC material is abbreviated as MFC-d31. The piezoelectric fiber is
oriented in the in-plane direction, and the polarization is pointing along the thickness
direction. Thus in the first mode the d 31 effect is dominating the actuation forces.
The second type of MFC material is denoted as MFC-d33. MFC-d33 is arranged in
a specific manner such that the polarization of the piezoelectric material is along the
piezo-fiber direction. Therefore, MFC-d33 mainly uses the d 33 effect for generation
of actuation forces. Because the coefficient d 33 is usually much larger (about 2 times
larger) than d 31 . MFC-d33 patches have larger actuation forces than MFC-d31 ones.
Additionally, actuation voltages for MFC-d31 patches can be applied in the range
from only −60 to 360 V (with the electrode distance of 0.18 mm), while those
for MFC-d33 patches can vary between −500 and 1500 V (with center-to-center
interdigitated electrode spacing of 0.5 mm) [10]. The schematic of these two kinds
of MFCs are shown in Fig. 4.4a and b, respectively.
The special arrangement of MFC material increases the structural flexibility. The
interdigitated electrodes reduce the impact on structural performance due to damage
or fracture of the piezoceramics or electrode. Any damage on piezoceramics or
electrode will not influence significantly on the overall actuation effect.
In order to present clearly the modeling procedure, three coordinate systems are
defined, as can be seen in Fig. 4.4, namely the curvilinear coordinate system repre-
4 Nonlinear Constitutive Relations
where
c = T
T
˘
cT , e = Q
T
˘
eT , = Q
T
˘
Q .
(4.40)
The transformation matrix Q is an identity matrix if the electrical coordinate axes
are parallel to the curvilinear coordinate lines, and T is given by
T =
⎡
⎢
⎢
⎢
⎢
⎣
t
2
11
t
2
21
t 11 t 21
0 0
t
2
12
t
2
22
t 12 t 22
0 0
2t 11 t 12 2t 21 t 22 t 11 t 22 + t 12 t 21 0 0
0
0
0
t 22 t 12
0
0
0
t 21 t 11
⎤
⎥
⎥
⎥
⎥
⎦
,
(4.41)
with
t 11 = g
1
· ˘
i 1 =
g 11 cos θ ,
t 12 = g
1
· ˘
i 2 = −
g 11 sin θ ,
t 21 = g
2
· ˘
i 1 =
g 22 sin θ ,
t 22 = g
2
· ˘
i 2 =
g 22 cos θ .
(4.42)
4.4 Constitutive Relations for Macro-fiber Composites
4.4.1 Configurations of Macro-fiber Composites
Macro-fiber composites mainly consist of piezoceramic fibers, epoxy matrix and
electrodes. They have two different modes of structures, namely d31 or d33 modes.
The first mode of MFC material is abbreviated as MFC-d31. The piezoelectric fiber is
oriented in the in-plane direction, and the polarization is pointing along the thickness
direction. Thus in the first mode the d 31 effect is dominating the actuation forces.
The second type of MFC material is denoted as MFC-d33. MFC-d33 is arranged in
a specific manner such that the polarization of the piezoelectric material is along the
piezo-fiber direction. Therefore, MFC-d33 mainly uses the d 33 effect for generation
of actuation forces. Because the coefficient d 33 is usually much larger (about 2 times
larger) than d 31 . MFC-d33 patches have larger actuation forces than MFC-d31 ones.
Additionally, actuation voltages for MFC-d31 patches can be applied in the range
from only −60 to 360 V (with the electrode distance of 0.18 mm), while those
for MFC-d33 patches can vary between −500 and 1500 V (with center-to-center
interdigitated electrode spacing of 0.5 mm) [10]. The schematic of these two kinds
of MFCs are shown in Fig. 4.4a and b, respectively.
The special arrangement of MFC material increases the structural flexibility. The
interdigitated electrodes reduce the impact on structural performance due to damage
or fracture of the piezoceramics or electrode. Any damage on piezoceramics or
electrode will not influence significantly on the overall actuation effect.
In order to present clearly the modeling procedure, three coordinate systems are
defined, as can be seen in Fig. 4.4, namely the curvilinear coordinate system repre-