P 3 ¼ D 3 À ε 0 E 3 ,
D 3 ¼ b
D 3 = A
1
ð Þ
þ A
2
ð Þ
,
ρ
P
¼ ÀP 3,3 :
ð6:139Þ
We are interest in the formation of the PN junction through diffusion and the
associated electromechanical fields. It is sufficient to consider the static bending of
a free beam whose bending moment M and the shear force Q are identically zero
along the beam. In this case, Eqs. (6.92), (6.93), (6.95), and (6.96) take the following
form:
u 2,33 ¼
2Ge
1
ð Þ
33
D
φ ,3 ,
ð6:140Þ
À2Ge
1
ð Þ
33 u 2,333 À b εφ ,33 ¼ q Δp À Δn
ð
Þ A
2
ð Þ ,
ð6:141Þ
qΔp ¼ À
qp 0 μ
p
33
D
p
33
φ þ C 1 ,
ÀqΔn ¼ À
qn 0 μ
n
33
D
n
33
φ þ C 2 ,
ð6:142Þ
φ ,33 À k
2
φ ¼ À
1
e ε
C 1 þ C 2
ð
Þ A
2
ð Þ ,
ð6:143Þ
where
k
2
¼
1
e ε
qp 0 μ
p
33
D
p
33
þ
qn 0 μ
n
33
D
n
33
A
2
ð Þ ,
e ε ¼ b ε þ
2Ge
1
ð Þ
33
2
D
,
ð6:144Þ
and C 1 and C 2 are undetermined constants. We need to treat the two halves of the
beam in Fig. 6.15a separately. We use a prime for the material parameters and
undetermined constants of the left half and a double prime for those of the right half.
For x 3 < 0, we write
φ ¼ k
0 C
0
3 exp k
0 x 3
ð
ÞÀk
0 C
0
4 exp Àk
0 x 3
ð
Þþ
1
k
0
ð Þ
2
e ε
0
C
0
1 þ C
0
2
À
Á
A
2
ð Þ ,
ð6:145Þ
u 2 ¼
2Ge
1
ð Þ
33
D
C
0
3 exp k
0 x 3
ð
ÞþC
0
4 exp Àk
0 x 3
ð
Þ
Â
à þ C
0
5 x 3 þ C
0
6 ,
ð6:146Þ
170
6 Composite Structures