Chapter 4
Bending of Beams
Piezoelectric semiconductor beams or fibers of ZnO in bending or flexural deformation are widely used in devices [1]. Qualitatively, the bending stiffness of a beam
is smaller than its extensional stiffness. Therefore bending is suitable in
low-frequency applications. Theoretical modeling of bending is more complicated
than that of extension. This chapter begins with the establishment of a
one-dimensional model for the bending of piezoelectric semiconductor beams. For
bending, shear deformation may be important or not in different situations. We
derive beams equations for bending with shear deformation first, and then reduce
them to bending without shear deformation. For a complete treatment, while deriving the one-dimensional equations for bending, we also include extension because
bending and extension may be coupled due to anisotropy or nonlinearity. A few
static and dynamic problems are analyzed using the equations derived.
4.1 One-Dimensional Equations for Bending
Consider a piezoelectric semiconductor beam of crystal class (6mm) such as ZnO as
shown in Fig. 4.1. The lateral surface is unelectroded and is free from any electromechanical loading. The electric field in the surrounding free space is neglected as
usual. These apply to the rest of the chapter. We consider bending in the y-z plane.
The cross section and the load are symmetric about the y axis.
For linear bending with small deformation and weak fields the governing equations are Eqs. (1.1) 1 , (1.10), (1.3), (1.9) with uniform doping, and (1.5):
T ji,j ¼ ρ€ u i ,
D i,i ¼ q Δp À Δn
ð
Þ ,
ð4:1Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_4
89
Bending of Beams
Piezoelectric semiconductor beams or fibers of ZnO in bending or flexural deformation are widely used in devices [1]. Qualitatively, the bending stiffness of a beam
is smaller than its extensional stiffness. Therefore bending is suitable in
low-frequency applications. Theoretical modeling of bending is more complicated
than that of extension. This chapter begins with the establishment of a
one-dimensional model for the bending of piezoelectric semiconductor beams. For
bending, shear deformation may be important or not in different situations. We
derive beams equations for bending with shear deformation first, and then reduce
them to bending without shear deformation. For a complete treatment, while deriving the one-dimensional equations for bending, we also include extension because
bending and extension may be coupled due to anisotropy or nonlinearity. A few
static and dynamic problems are analyzed using the equations derived.
4.1 One-Dimensional Equations for Bending
Consider a piezoelectric semiconductor beam of crystal class (6mm) such as ZnO as
shown in Fig. 4.1. The lateral surface is unelectroded and is free from any electromechanical loading. The electric field in the surrounding free space is neglected as
usual. These apply to the rest of the chapter. We consider bending in the y-z plane.
The cross section and the load are symmetric about the y axis.
For linear bending with small deformation and weak fields the governing equations are Eqs. (1.1) 1 , (1.10), (1.3), (1.9) with uniform doping, and (1.5):
T ji,j ¼ ρ€ u i ,
D i,i ¼ q Δp À Δn
ð
Þ ,
ð4:1Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_4
89