Chapter 3
Extension of Rods
Piezoelectric semiconductor rods or fibers of ZnO in extensional deformation are
widely used in devices [1]. This chapter begins with the establishment of a
one-dimensional model for the extension of thin rods, followed by a series of static
and dynamic problems.
3.1 One-Dimensional Equations
Consider the piezoelectric semiconductor rod shown in Fig. 3.1. The shape of the
cross section A may be arbitrary. The rod is assumed to be long and thin, i.e., its
length is much larger than the characteristic dimension of the cross section. It is
made from a piezoelectric semiconductor crystal of class (6mm) such as ZnO. The
c-axis of the crystal is along the axis of the rod. The lateral surface of the rod is
traction-free and is unelectroded. The electric field in the surrounding free space is
neglected. These apply to the rest of the chapter.
We are interested in the extensional deformation of the rod which can be
described approximately by a set of one-dimensional equations for the axial displacement u 3 and the axial stress T 3 ¼ T 33 . An important approximation for
extension of thin rods is the relaxation of the lateral stress components, i.e.,
T 1 ¼ T 2 ffi 0. In this case, from Eq. (1.3),
S 3 ¼ s
E
33 T 3 þ d 33 E 3 ,
D 3 ¼ d 33 T 3 þ ε
T
33 E 3 :
ð3:1Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_3
31
Extension of Rods
Piezoelectric semiconductor rods or fibers of ZnO in extensional deformation are
widely used in devices [1]. This chapter begins with the establishment of a
one-dimensional model for the extension of thin rods, followed by a series of static
and dynamic problems.
3.1 One-Dimensional Equations
Consider the piezoelectric semiconductor rod shown in Fig. 3.1. The shape of the
cross section A may be arbitrary. The rod is assumed to be long and thin, i.e., its
length is much larger than the characteristic dimension of the cross section. It is
made from a piezoelectric semiconductor crystal of class (6mm) such as ZnO. The
c-axis of the crystal is along the axis of the rod. The lateral surface of the rod is
traction-free and is unelectroded. The electric field in the surrounding free space is
neglected. These apply to the rest of the chapter.
We are interested in the extensional deformation of the rod which can be
described approximately by a set of one-dimensional equations for the axial displacement u 3 and the axial stress T 3 ¼ T 33 . An important approximation for
extension of thin rods is the relaxation of the lateral stress components, i.e.,
T 1 ¼ T 2 ffi 0. In this case, from Eq. (1.3),
S 3 ¼ s
E
33 T 3 þ d 33 E 3 ,
D 3 ¼ d 33 T 3 þ ε
T
33 E 3 :
ð3:1Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_3
31