Chapter 5
Extension and Bending of Plates
In addition to beams or fibers, piezoelectric semiconductor plates or thin films are
also widely used in devices in extension and bending. In this chapter, we derive
two-dimensional equations for thin plates of piezoelectric semiconductors [1, 2]. We
follow R.D. Mindlin’s approach [3–5], beginning from the three-dimensional equations with power series expansions in the plate thickness coordinate and then
truncating the expansions to obtain low-order equations for extension and/or bending. The last section is on shells.
5.1 Recapitulation of Three-Dimensional Equations
Different from rods and beams for which the piezoelectric constitutive relations in
Eq. (1.3) are convenient, for plates we use Eq. (1.2) 1,2 most of the time. For
convenience we gather the complete set of the linearized three-dimensional equation
below from Eqs. (1.1) 1 with f ¼ 0, (1.10), (1.2) 1,2 , (1.9) for uniform p 0 and n 0 , and
(1.5):
T ij,i ¼ ρ€ u j ,
D i,i ¼ q Δp À Δn
ð
Þ ,
ð5:1Þ
J
p
i,i ¼ Àq
∂
∂t
Δp
ð Þ,
J
n
i,i ¼ q
∂
∂t
Δn
ð Þ,
ð5:2Þ
T ij ¼ c ijkl S kl À e kij E k ,
D i ¼ e ikl S kl þ ε ik E k ,
ð5:3Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_5
113
Extension and Bending of Plates
In addition to beams or fibers, piezoelectric semiconductor plates or thin films are
also widely used in devices in extension and bending. In this chapter, we derive
two-dimensional equations for thin plates of piezoelectric semiconductors [1, 2]. We
follow R.D. Mindlin’s approach [3–5], beginning from the three-dimensional equations with power series expansions in the plate thickness coordinate and then
truncating the expansions to obtain low-order equations for extension and/or bending. The last section is on shells.
5.1 Recapitulation of Three-Dimensional Equations
Different from rods and beams for which the piezoelectric constitutive relations in
Eq. (1.3) are convenient, for plates we use Eq. (1.2) 1,2 most of the time. For
convenience we gather the complete set of the linearized three-dimensional equation
below from Eqs. (1.1) 1 with f ¼ 0, (1.10), (1.2) 1,2 , (1.9) for uniform p 0 and n 0 , and
(1.5):
T ij,i ¼ ρ€ u j ,
D i,i ¼ q Δp À Δn
ð
Þ ,
ð5:1Þ
J
p
i,i ¼ Àq
∂
∂t
Δp
ð Þ,
J
n
i,i ¼ q
∂
∂t
Δn
ð Þ,
ð5:2Þ
T ij ¼ c ijkl S kl À e kij E k ,
D i ¼ e ikl S kl þ ε ik E k ,
ð5:3Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_5
113