where
ρb ω
2
1 ¼ ρω
2
1 þ ρ
3κ
2 c 44
ρh
2
e
2
15
ε 11 c 44
¼ ρω
2
1 1 þ k
2
15
À
Á
ð5:106Þ
is the piezoelectrically stiffened infinite plate thickness-shear frequency when there
exists a coupling to the electric filed in the x 1 direction. When the semiconduction is
small, Eq. (5.105) can be solved by an iteration or perturbation procedure. As the
lowest or zero-order of approximation, we neglect the small semiconduction. Equation (5.105) reduces to the dispersion relation of long thickness-shear waves in a
piezoelectric dielectric plate:
ρω
2
0
ð Þ ¼ ρb ω
2
1 þ b c 11 ξ
2
1 ,
ð5:107Þ
which is dispersive but not dissipative. For the next or first-order of approximation,
we substitute Eq. (5.107) into the right-hand side of Eq. (5.105) and obtain
ρω
2
1
ð Þ ¼ ρb ω
2
1 þ b c 11 ξ
2
1 À
p 0 qμ
p
11
ε 11 d
p
11 ξ
2
1 À iω 0
ð Þ
À
Á
 1 þ
1
3
h
2 k
2
15 ξ
2
1
ρ ω
2
0
ð Þ À ω
2
1
À ξ
2
1 b c 11
h
i
þ κ
2 k
2
15 c 44 ξ
2
1
n
o
,
ð5:108Þ
which is complex and describes waves that are both dispersive and dissipative.
5.11 Equations for Shells
Shell structures are widely used in devices. The recent development of flexible
electronics results in more devices with shell structures. In this section, we establish
two-dimensional equations for piezoelectric semiconductor shells [8]. Consider a
differential shell element (see Fig. 5.3). (α 1 ,α 2 ,α 3 ) is an orthogonal curvilinear
coordinate system. α 1 and α 2 are the middle surface principal coordinates, and α 3
is the thickness coordinate. The thickness of the shell, 2h, is much smaller than the
principal radii of curvatures R 1 and R 2 of the middle surface. Let A 1 and A 2 be the
Lamè coefficients corresponding to α 1 and α 2 at the middle surface. The metric in
(α 1 ,α 2 ,α 3 ) is given by A 1 (1 + α 3 /R 1 ), A 2 (1 + α 3 /R 2 ), and 1. Equations (5.1), (5.2),
(5.3), (5.4), (5.5), (5.6), (5.7), and (5.8) are still valid when they are considered as
tensor equations and generalized into curvilinear coordinate systems. They are used
to derive two-dimensional shell equations in this section.
The strain-displacement relations take the following form [9]:
132
5 Extension and Bending of Plates
ρb ω
2
1 ¼ ρω
2
1 þ ρ
3κ
2 c 44
ρh
2
e
2
15
ε 11 c 44
¼ ρω
2
1 1 þ k
2
15
À
Á
ð5:106Þ
is the piezoelectrically stiffened infinite plate thickness-shear frequency when there
exists a coupling to the electric filed in the x 1 direction. When the semiconduction is
small, Eq. (5.105) can be solved by an iteration or perturbation procedure. As the
lowest or zero-order of approximation, we neglect the small semiconduction. Equation (5.105) reduces to the dispersion relation of long thickness-shear waves in a
piezoelectric dielectric plate:
ρω
2
0
ð Þ ¼ ρb ω
2
1 þ b c 11 ξ
2
1 ,
ð5:107Þ
which is dispersive but not dissipative. For the next or first-order of approximation,
we substitute Eq. (5.107) into the right-hand side of Eq. (5.105) and obtain
ρω
2
1
ð Þ ¼ ρb ω
2
1 þ b c 11 ξ
2
1 À
p 0 qμ
p
11
ε 11 d
p
11 ξ
2
1 À iω 0
ð Þ
À
Á
 1 þ
1
3
h
2 k
2
15 ξ
2
1
ρ ω
2
0
ð Þ À ω
2
1
À ξ
2
1 b c 11
h
i
þ κ
2 k
2
15 c 44 ξ
2
1
n
o
,
ð5:108Þ
which is complex and describes waves that are both dispersive and dissipative.
5.11 Equations for Shells
Shell structures are widely used in devices. The recent development of flexible
electronics results in more devices with shell structures. In this section, we establish
two-dimensional equations for piezoelectric semiconductor shells [8]. Consider a
differential shell element (see Fig. 5.3). (α 1 ,α 2 ,α 3 ) is an orthogonal curvilinear
coordinate system. α 1 and α 2 are the middle surface principal coordinates, and α 3
is the thickness coordinate. The thickness of the shell, 2h, is much smaller than the
principal radii of curvatures R 1 and R 2 of the middle surface. Let A 1 and A 2 be the
Lamè coefficients corresponding to α 1 and α 2 at the middle surface. The metric in
(α 1 ,α 2 ,α 3 ) is given by A 1 (1 + α 3 /R 1 ), A 2 (1 + α 3 /R 2 ), and 1. Equations (5.1), (5.2),
(5.3), (5.4), (5.5), (5.6), (5.7), and (5.8) are still valid when they are considered as
tensor equations and generalized into curvilinear coordinate systems. They are used
to derive two-dimensional shell equations in this section.
The strain-displacement relations take the following form [9]:
132
5 Extension and Bending of Plates