N ,3 ¼ ρA€ w,
D
0
ð Þ
3,3 ¼ qA p
0
ð Þ
À n
0
ð Þ
,
J
p 0
ð Þ
3,3 ¼ ÀqA _
p
0
ð Þ ,
J
n 0
ð Þ
3,3 ¼ qA _
n
0
ð Þ ,
ð4:18Þ
Q ,3 ¼ ρA€ v,
M ,3 À Q ¼ ρI €
ψ,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI p
1
ð Þ
À n
1
ð Þ
,
J
p 1
ð Þ
3,3 À J
p 0
ð Þ
2
¼ ÀqI _
p
1
ð Þ ,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ qI _
n
1
ð Þ
:
ð4:19Þ
The substitution of Eqs. (4.13), (4.14), (4.15), and (4.16) into Eqs. (4.18) and (4.19)
yields two sets of second-order linear ordinary differential equations. One is for
extension involving w, ϕ
(0) , p
(0) , and n
(0) which is the same as the equations for
extension used in multiple places in the previous chapter. The other is for bending
with shear deformation involving v, ψ, ϕ
(1) , p
(1) , and n
(1) which is gathered below:
c 44 A v ,33 þ ψ ,3
À
Á þ e 15 Aϕ
1
ð Þ
,3 ¼ ρA€ v,
c 33 Iψ ,33 þ e 33 Iϕ
1
ð Þ
,33 À c 44 A v ,3 þ ψ
ð
ÞÀe 15 Aϕ
1
ð Þ
¼ ρI €
ψ,
e 33 Iψ ,33 À ε 33 Iϕ
1
ð Þ
,33 À e 15 A v ,3 þ ψ
ð
Þþε 11 Aϕ
1
ð Þ
¼ qI p
1
ð Þ
À n
1
ð Þ
,
Àqp 0 μ
p
33 Iϕ
1
ð Þ
,33 À qD
p
33 Ip
1
ð Þ
,33 þ qp 0 μ
p
11 Aϕ
1
ð Þ
þ qD
p
11 Ap
1
ð Þ
¼ ÀqI _
p
1
ð Þ ,
Àqn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ qD
n
33 In
1
ð Þ
,33 þ qn 0 μ
n
11 Aϕ
1
ð Þ
À qD
n
11 An
1
ð Þ
¼ qI _
n
1
ð Þ
:
ð4:20Þ
Shear deformation in a beam needs to be considered when the beam is not very long
and thin, or when the wavelength is not much larger than the characteristic dimension of the cross section of the beam in a dynamic problem. The above equations are
valid when the effect of shear deformation cannot be neglected in bending. Its
accuracy can be improved with the use of a shear correction factor which is
necessary when shear deformation is strong. Shear correction factors will be
discussed in Chap. 5 on plates in a systematic manner.
4.2 Reduction to Bending Without Shear Deformation
For the beam in Fig. 4.1, we separate and collect the equations for bending with shear
deformation from the previous section below, without the equations for extension.
The mechanical displacements, electric potential, and carrier concentration perturbations are approximated by
4.2 Reduction to Bending Without Shear Deformation
93
D
0
ð Þ
3,3 ¼ qA p
0
ð Þ
À n
0
ð Þ
,
J
p 0
ð Þ
3,3 ¼ ÀqA _
p
0
ð Þ ,
J
n 0
ð Þ
3,3 ¼ qA _
n
0
ð Þ ,
ð4:18Þ
Q ,3 ¼ ρA€ v,
M ,3 À Q ¼ ρI €
ψ,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI p
1
ð Þ
À n
1
ð Þ
,
J
p 1
ð Þ
3,3 À J
p 0
ð Þ
2
¼ ÀqI _
p
1
ð Þ ,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ qI _
n
1
ð Þ
:
ð4:19Þ
The substitution of Eqs. (4.13), (4.14), (4.15), and (4.16) into Eqs. (4.18) and (4.19)
yields two sets of second-order linear ordinary differential equations. One is for
extension involving w, ϕ
(0) , p
(0) , and n
(0) which is the same as the equations for
extension used in multiple places in the previous chapter. The other is for bending
with shear deformation involving v, ψ, ϕ
(1) , p
(1) , and n
(1) which is gathered below:
c 44 A v ,33 þ ψ ,3
À
Á þ e 15 Aϕ
1
ð Þ
,3 ¼ ρA€ v,
c 33 Iψ ,33 þ e 33 Iϕ
1
ð Þ
,33 À c 44 A v ,3 þ ψ
ð
ÞÀe 15 Aϕ
1
ð Þ
¼ ρI €
ψ,
e 33 Iψ ,33 À ε 33 Iϕ
1
ð Þ
,33 À e 15 A v ,3 þ ψ
ð
Þþε 11 Aϕ
1
ð Þ
¼ qI p
1
ð Þ
À n
1
ð Þ
,
Àqp 0 μ
p
33 Iϕ
1
ð Þ
,33 À qD
p
33 Ip
1
ð Þ
,33 þ qp 0 μ
p
11 Aϕ
1
ð Þ
þ qD
p
11 Ap
1
ð Þ
¼ ÀqI _
p
1
ð Þ ,
Àqn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ qD
n
33 In
1
ð Þ
,33 þ qn 0 μ
n
11 Aϕ
1
ð Þ
À qD
n
11 An
1
ð Þ
¼ qI _
n
1
ð Þ
:
ð4:20Þ
Shear deformation in a beam needs to be considered when the beam is not very long
and thin, or when the wavelength is not much larger than the characteristic dimension of the cross section of the beam in a dynamic problem. The above equations are
valid when the effect of shear deformation cannot be neglected in bending. Its
accuracy can be improved with the use of a shear correction factor which is
necessary when shear deformation is strong. Shear correction factors will be
discussed in Chap. 5 on plates in a systematic manner.
4.2 Reduction to Bending Without Shear Deformation
For the beam in Fig. 4.1, we separate and collect the equations for bending with shear
deformation from the previous section below, without the equations for extension.
The mechanical displacements, electric potential, and carrier concentration perturbations are approximated by
4.2 Reduction to Bending Without Shear Deformation
93