n ffi n
0
ð Þ
þ n
1
ð Þ
þ n
2
ð Þ ,
φ ffi φ
1
ð Þ
þ φ
2
ð Þ ,
u ffi u
1
ð Þ
þ u
2
ð Þ
:
ð3:93Þ
Substituting Eq. (3.93) into the above equations, we obtain a series of perturbation
problems. The reference state with n
0
ð Þ
¼ N
þ
D is the solution to the zero-order
problem in which all other fields vanish. The first-order problem is
n
1
ð Þ
¼
ε
T
33
q
d
2
φ
1
ð Þ
dx
2
,
d
2 u
1
ð Þ
dx
2
¼ À
e
c
d
2
φ
1
ð Þ
dx
2
,
Àn
0
ð Þ
μ
n dφ
1
ð Þ
dx
þ D
n dn
1
ð Þ
dx
¼ 0,
ð3:94Þ
and
T
1
ð Þ
ÆL
ð Þ ¼ f , D
1
ð Þ
ÆL
ð Þ ¼ 0,
Z L
ÀL
n
1
ð Þ dx ¼ 0,
u
1
ð Þ 0
ð Þ ¼ 0, φ
1
ð Þ 0
ð Þ ¼ 0:
ð3:95Þ
The solution to the first-order problem was obtained in the previous section. The
electric potential, electron concentration, and mechanical displacement fields are
φ
1
ð Þ
¼
ef
ε T
33 ck cosh kL
sinh kx,
ð3:96Þ
n
1
ð Þ
¼ N
þ
D
μ
n
D
n
ef
ε T
33 ck cosh kL
sinh kx,
ð3:97Þ
u
1
ð Þ
¼ À
e
2 f
ε T
33 c 2 k cosh kL
sinh kx þ
f
c
x,
ð3:98Þ
where
k
2
¼
q
ε T
33
μ
n
D
n N
þ
D :
ð3:99Þ
Clearly, the first-order perturbation solution represents a linear solution proportional
to f. The second-order problem is given by
3.7 Electrically Nonlinear Extension by End Forces
51