maximum near h ¼ c, which is reasonable. When h is very small, the piezoelectric
layers are very thin and cannot produce a lot of polarization charges to drive the
mobile charges. When h is very large, the semiconductor layer is very thin, and there
are not many mobile charges.
6.3 Bending of Beams with e 33
In this section we consider bending of the composite beam in Fig. 6.8 [4]. The beam
consists of a nonpiezoelectric semiconductor layer such as Si and a pair of piezoelectric dielectric layers such as ceramics poled along the axial direction. The two
ceramic layers are identical except that their poling directions are opposite to each
other. The left end of the beam is fixed. The right end is under a transverse shear
force F. The piezoelectric layers are unelectroded on their lateral surfaces. The
electric field in the surrounding free space is neglected as usual as an approximation.
Consider bending without shear deformation in the x 3 -x 2 plane. The flexural and
axial displacements are approximately described by
u 2 ffi u 2 x 3 , t
ð
Þ, u 3 ffi Àx 2 u 2,3 :
ð6:72Þ
Thus the axial strain S 3 can be expressed in terms of u 2 as
S 3 ¼ Àx 2 u 2,33 :
ð6:73Þ
The electric potential and field are approximated by
x3
(2) Si
(1) PZT
L
P
P
F
(1) PZT
x2
x1
x2
b
2c
h
h
Fig. 6.8 Side view and
cross section of a composite
beam of piezoelectric
dielectrics and
nonpiezoelectric
semiconductors
6.3 Bending of Beams with e 33
155
layers are very thin and cannot produce a lot of polarization charges to drive the
mobile charges. When h is very large, the semiconductor layer is very thin, and there
are not many mobile charges.
6.3 Bending of Beams with e 33
In this section we consider bending of the composite beam in Fig. 6.8 [4]. The beam
consists of a nonpiezoelectric semiconductor layer such as Si and a pair of piezoelectric dielectric layers such as ceramics poled along the axial direction. The two
ceramic layers are identical except that their poling directions are opposite to each
other. The left end of the beam is fixed. The right end is under a transverse shear
force F. The piezoelectric layers are unelectroded on their lateral surfaces. The
electric field in the surrounding free space is neglected as usual as an approximation.
Consider bending without shear deformation in the x 3 -x 2 plane. The flexural and
axial displacements are approximately described by
u 2 ffi u 2 x 3 , t
ð
Þ, u 3 ffi Àx 2 u 2,3 :
ð6:72Þ
Thus the axial strain S 3 can be expressed in terms of u 2 as
S 3 ¼ Àx 2 u 2,33 :
ð6:73Þ
The electric potential and field are approximated by
x3
(2) Si
(1) PZT
L
P
P
F
(1) PZT
x2
x1
x2
b
2c
h
h
Fig. 6.8 Side view and
cross section of a composite
beam of piezoelectric
dielectrics and
nonpiezoelectric
semiconductors
6.3 Bending of Beams with e 33
155