130
6 Nonlinear Analysis of Piezoceramic Laminated Structures
Θ
3
Θ
2
Θ
1
35 × 7 × 1 m m 3
Fig. 6.33 Cantilevered bimorph beam
Table 6.7 Material properties
Properties
3203HD a,b
G-1195 b,c
T300/976
Aluminum
Y 1 (GPa)
60.24
63
150
70
Y 2 (GPa)
60.24
63
9
70
ν 12
0.253
0.28
0.3
0.3
ν 23
0.494
0.28
0.3
0.3
G 12 (GPa)
20.04
24.61
7.1
26.92
G 23 (GPa)
19.084
24.61
2.5
26.92
G 13 (GPa)
19.084
24.61
7.1
26.92
d 31
(×10 −12 m/V)
−320
−254
d 32
(×10 −12 m/V)
−320
−254
33 // 0
d
3800
1694.9
β 331
(×10 −18 m 2 V −2 )
−520
−165
β 332
(×10 −18 m 2 V −2 )
−520
−165
a CTS Corporation: http://www.ctscorp.com/
b Kapuria and Yasin [24]
c Yao et al. [25]
d Electrical permittivity of air, 0 = 8.85 × 10 −12 F/m
node doubly curved thick shell elements) and under equivalent moments distributed
at the corresponding nodes. The equivalent moments due to driving electric field
imposed on piezoelectric patches are approximated by integral of the product of
longitudinal stress and moment arm over the cross-section, where the electroelastic
materially nonlinear constitutive relations given in (4.73) are considered at different
levels of electric fields. The present results of LIN5WE and LIN5SE models agrees
excellently with those in the literature and obtained by Abaqus.
The next example for validation study is a simply supported smart plate bonded
with two piezoelectric substrate layers, as shown in Fig. 6.35. The host plate is
made of aluminum and piezoelectric material is chosen as G-1195. The dimensions
6 Nonlinear Analysis of Piezoceramic Laminated Structures
Θ
3
Θ
2
Θ
1
35 × 7 × 1 m m 3
Fig. 6.33 Cantilevered bimorph beam
Table 6.7 Material properties
Properties
3203HD a,b
G-1195 b,c
T300/976
Aluminum
Y 1 (GPa)
60.24
63
150
70
Y 2 (GPa)
60.24
63
9
70
ν 12
0.253
0.28
0.3
0.3
ν 23
0.494
0.28
0.3
0.3
G 12 (GPa)
20.04
24.61
7.1
26.92
G 23 (GPa)
19.084
24.61
2.5
26.92
G 13 (GPa)
19.084
24.61
7.1
26.92
d 31
(×10 −12 m/V)
−320
−254
d 32
(×10 −12 m/V)
−320
−254
33 // 0
d
3800
1694.9
β 331
(×10 −18 m 2 V −2 )
−520
−165
β 332
(×10 −18 m 2 V −2 )
−520
−165
a CTS Corporation: http://www.ctscorp.com/
b Kapuria and Yasin [24]
c Yao et al. [25]
d Electrical permittivity of air, 0 = 8.85 × 10 −12 F/m
node doubly curved thick shell elements) and under equivalent moments distributed
at the corresponding nodes. The equivalent moments due to driving electric field
imposed on piezoelectric patches are approximated by integral of the product of
longitudinal stress and moment arm over the cross-section, where the electroelastic
materially nonlinear constitutive relations given in (4.73) are considered at different
levels of electric fields. The present results of LIN5WE and LIN5SE models agrees
excellently with those in the literature and obtained by Abaqus.
The next example for validation study is a simply supported smart plate bonded
with two piezoelectric substrate layers, as shown in Fig. 6.35. The host plate is
made of aluminum and piezoelectric material is chosen as G-1195. The dimensions