T ¼ c
2
ð Þ S À λ
2
ð Þ
θ
¼ c
2
ð Þ w ,3 À x 2 v ,33
ð
ÞÀλ
2
ð Þ
θ,
D ¼ ε
2
ð Þ E ¼ Àε
2
ð Þ
φ ,3 :
ð7:112Þ
The currents exist in the semiconductor layer only and therefore the superscript 2 in
parentheses is neglected. We still have Eq. (7.85):
J
p
ffi qp 0 μ
p E À qD
p d Δp
ð Þ
dx 3
,
J
n
ffi qn 0 μ
n E þ qD
n d Δn
ð Þ
dx 3
:
ð7:113Þ
The total axial force b
T, shear force Q, bending moment M, and electric displacement
b
D are defined by the following integrations over a cross section:
b
T ¼
Z
Tdx 1 dx 2 ¼ b
Z h 1
0
Tdx 2 þ b
Z 0
Àh 2
Tdx 2
¼ c
1
ð Þ w ,3 þ e
1
ð Þ
φ ,3 À λ
1
ð Þ
θ
h 1 b
À
c
1
ð Þ v ,33
2
h
2
1 b þ c
2
ð Þ w ,3 À λ
2
ð Þ
θ
h 2 b þ
c
2
ð Þ v ,33
2
h
2
2 b,
ð7:114Þ
Q ¼
Z
T 32 dx 1 dx 2 ,
ð7:115Þ
M ¼
Z
Tx 2 dx 1 dx 2 ¼ b
Z h 1
0
Tx 2 dx 2 þ b
Z 0
Àh 2
Tx 2 dx 2
¼
c
1
ð Þ w ,3 þ e
1
ð Þ
φ ,3 À λ
1
ð Þ
θ
2
h
2
1 b
À
c
1
ð Þ v ,33
3
h
3
1 b þ
λ
2
ð Þ
θ À c
2
ð Þ w ,3
2
h
2
2 b À
c
2
ð Þ v ,33
3
h
3
2 b,
ð7:116Þ
b
D ¼
Z
Ddx 1 dx 2 ¼ b
Z h 1
0
Ddx 2 þ b
Z 0
Àh 2
Ddx 2
¼ e
1
ð Þ w ,3 À ε
1
ð Þ
φ ,3 þ p
1
ð Þ
θ
h 1 b
À
e
1
ð Þ v ,33
2
h
2
1 b þ Àε
2
ð Þ
φ ,3
h 2 b:
ð7:117Þ
The constitutive relation for the shear force Q will be provided by the moment-shear
force relationship as to be seen in the following. From the equations of motion
(Newton’s second law) of the differential element of the beam in Fig. 7.19 in the x 2
and x 3 directions as well as its moment equation, we have
7.5 Extension and Bending of Composite Beams
205