differential element of the rod near its ends, per unit θ/Θ 0 , we have Δn(2cbdz)
ffiγ(2cbdz)n 0 whose absolute value is plotted in Fig. 7.17 versus h/c when h + c is
fixed. The curves all have a maximum when h and c are not very different.
7.5 Extension and Bending of Composite Beams
In this section we study the behavior of the composite beam in Fig. 7.18 under a
uniform temperature change [5]. It consists of a piezoelectric dielectric layer (layer
(1)) such as ceramics poled along the axial direction and a nonpiezoelectric semiconductor layer (layer (2)) such as silicon. The cross-sectional areas of the piezoelectric and semiconductor layers are denoted by A
(1) and A
(2) . The beam is
unelectroded. The x 3 axis is at the interface which is in both extension and bending
under a uniform temperature change because of the lack of structural symmetry
about the interface.
Equations (7.1), (7.2), (7.3), (7.4), and (7.5) are still valid, with piezoelectric
dielectrics and nonpiezoelectric semiconductors as special cases. We derive a set of
one-dimensional equations first. For extension and bending without shear deformation in the x 3 -x 2 plane, the bending and axial displacements are approximated by
u 2 ffi v x 3 , t
ð
Þ, u 3 ffi w x 3 , t
ð
ÞÀx 2 v ,3 :
ð7:107Þ
The axial strain S 3 is expressed in terms of w and v through
Si and LiNbO 3
Si and BaTiO 3
Si and PZT-5A
0
1
2
3
4
5
0.0
0.5
1.0
1.5
h/c
| |(2cbdz)n
g
0
×10
4
Fig. 7.17 jγ j (2cbdz)n 0 at the end of the rod versus h/c for different combinations of materials.
dz ¼ 0.1μm. Θ 0 ¼298 K. p 0 ffi 0. n 0 ¼ 10
23
/m
3
7.5 Extension and Bending of Composite Beams
203
ffiγ(2cbdz)n 0 whose absolute value is plotted in Fig. 7.17 versus h/c when h + c is
fixed. The curves all have a maximum when h and c are not very different.
7.5 Extension and Bending of Composite Beams
In this section we study the behavior of the composite beam in Fig. 7.18 under a
uniform temperature change [5]. It consists of a piezoelectric dielectric layer (layer
(1)) such as ceramics poled along the axial direction and a nonpiezoelectric semiconductor layer (layer (2)) such as silicon. The cross-sectional areas of the piezoelectric and semiconductor layers are denoted by A
(1) and A
(2) . The beam is
unelectroded. The x 3 axis is at the interface which is in both extension and bending
under a uniform temperature change because of the lack of structural symmetry
about the interface.
Equations (7.1), (7.2), (7.3), (7.4), and (7.5) are still valid, with piezoelectric
dielectrics and nonpiezoelectric semiconductors as special cases. We derive a set of
one-dimensional equations first. For extension and bending without shear deformation in the x 3 -x 2 plane, the bending and axial displacements are approximated by
u 2 ffi v x 3 , t
ð
Þ, u 3 ffi w x 3 , t
ð
ÞÀx 2 v ,3 :
ð7:107Þ
The axial strain S 3 is expressed in terms of w and v through
Si and LiNbO 3
Si and BaTiO 3
Si and PZT-5A
0
1
2
3
4
5
0.0
0.5
1.0
1.5
h/c
| |(2cbdz)n
g
0
×10
4
Fig. 7.17 jγ j (2cbdz)n 0 at the end of the rod versus h/c for different combinations of materials.
dz ¼ 0.1μm. Θ 0 ¼298 K. p 0 ffi 0. n 0 ¼ 10
23
/m
3
7.5 Extension and Bending of Composite Beams
203