n
2
ð Þ
¼
ε
T
33
q
d
2
φ
2
ð Þ
dx
2
,
d
2 u
2
ð Þ
dx
2
¼ À
e
c
d
2
φ
2
ð Þ
dx
2
,
À n
0
ð Þ
μ
n dφ
2
ð Þ
dx
þ D
n dn
2
ð Þ
dx
¼ n
1
ð Þ
μ
n dφ
1
ð Þ
dx
,
ð3:100Þ
and
T
2
ð Þ
ÆL
ð Þ ¼ 0, D
2
ð Þ
ÆL
ð Þ ¼ 0,
Z L
ÀL
n
2
ð Þ dx ¼ 0,
u
2
ð Þ 0
ð Þ ¼ 0, φ
2
ð Þ 0
ð Þ ¼ 0:
ð3:101Þ
Through some algebra the second-order solution is found to be
φ
2
ð Þ
¼ À
qm cosh kL
6k
2
ε T
33 sinh kL
sinh k x þ L
ð
ÞÀ sinh k x À L
ð
Þ
½
Š
þ
qm
12k
2
ε T
33
cosh 2kx À
q
k
2
ε T
33
m
12
À
m cosh kL
3
!
,
ð3:102Þ
n
2
ð Þ
¼ À
m cosh kL
6 sinh kL
sinh k x þ L
ð
ÞÀ sinh k x À L
ð
Þ
½
Š
þ
m
3
cosh 2kx,
ð3:103Þ
u
2
ð Þ
¼
e
c
qm cosh kL
6k
2
ε T
33 sinh kL
sinh k x þ L
ð
ÞÀ sinh k x À L
ð
Þ
½
Š
À
e
c
qm
12k
2
ε T
33
cosh 2kx þ
e
c
q
k
2
ε T
33
m
12
À
m cosh kL
3
!
,
ð3:104Þ
where
m ¼ N
þ
D
μ
n
D
n
2
ef
ε T
33 ck cosh kL
2
:
ð3:105Þ
It can be seen that the second-order solution is quadratic in f through m.
As a numerical example, consider a ZnO rod with 2L ¼ 1.2 μm and n 0 ¼
N
þ
D ¼ 10
21 m
À3 . Figure 3.10 shows the comparison of the first-order linear solution
described by n ffi n
(0) + n
(1) with the second-order nonlinear solution when the
applied axial stress f ¼ 2 Â 10
6 N/m
2 . When f is small, the first-order solution is
expected to be accurate. Our numerical tests indeed show that when f < 0.6 Â 10
6
N/m
2 , the first- and second-order solutions become indistinguishable when plotted in
the scale of Fig. 3.10. When the axial stress is increased to f ¼ 2 Â 10
6 N/m
2 , the
52
3 Extension of Rods