6.4 Bending of Beams with e 15
The bending of a composite beam in the previous section was based on e 33 and the
normal stress over a cross section. In this section, we analyze the bending of a
composite beam based on e 15 and the shear stress over a cross section [5]. Consider
the composite beam in Fig. 6.12. It consists of a piezoelectric dielectric middle layer
labeled by “(1)” such as ceramics poled along the x 3 direction, and two identical
nonpiezoelectric semiconductor layers labeled by “(2)” such as silicon at the top and
bottom. Different from Fig. 6.8, the piezoelectric layer in Fig. 6.12 is in the middle
because in bending the shear stress over a cross section is large in the middle. The
left end of the beam is fixed. The right end is under a static transverse shear force F.
Fig. 6.10 Distributions of
(a) hole concentration
perturbation Δp, and (b)
electron concentration
perturbation Δn for different
values of the shear force F in
a composite beam of
PZT-5A and silicon. x ¼ x 3
6.4 Bending of Beams with e 15
161
The bending of a composite beam in the previous section was based on e 33 and the
normal stress over a cross section. In this section, we analyze the bending of a
composite beam based on e 15 and the shear stress over a cross section [5]. Consider
the composite beam in Fig. 6.12. It consists of a piezoelectric dielectric middle layer
labeled by “(1)” such as ceramics poled along the x 3 direction, and two identical
nonpiezoelectric semiconductor layers labeled by “(2)” such as silicon at the top and
bottom. Different from Fig. 6.8, the piezoelectric layer in Fig. 6.12 is in the middle
because in bending the shear stress over a cross section is large in the middle. The
left end of the beam is fixed. The right end is under a static transverse shear force F.
Fig. 6.10 Distributions of
(a) hole concentration
perturbation Δp, and (b)
electron concentration
perturbation Δn for different
values of the shear force F in
a composite beam of
PZT-5A and silicon. x ¼ x 3
6.4 Bending of Beams with e 15
161