first- and second-order solutions shown in Fig. 3.10 become significantly different.
The first-order solution represented by the solid line is antisymmetric about the
middle of the rod. However, the second-order solution loses the antisymmetry. This
is a fundamental and qualitative difference. In fact, the electromechanical fields
described by the second-order solution lose the symmetry or antisymmetry of the
first-order solution in general [5]. A finite element numerical solution using the
commercial software COMSOL is also provided in the figure which is very close to
the second-order perturbation solution.
From Eq. (3.21), the polarization and the effective polarization charge density can
be calculated from
P 3 ¼ D 3 À ε 0 E 3 ,
ρ
P
¼ ÀP k,k ¼ ÀP 3,3 :
ð3:106Þ
They are found to be
P
1
ð Þ
¼D
1
ð Þ
À ε 0 E
1
ð Þ ,
ρ
P 1
ð Þ
¼ À ∇ Á P
1
ð Þ
¼
kef
c cosh kL
sinh kx À ε 0
kef
ε T
33 c cosh kL
sinh kx,
ð3:107Þ
ρ
P 2
ð Þ
¼ À ∇ Á P
2
ð Þ
¼ À
qm cosh kL
6 sinh kL
sinh k x þ L
ð
ÞÀ sinh k x À L
ð
Þ
½
Š
þ
ε 0 qm cosh kL
6ε T
33 sinh kL
sinh k x þ L
ð
ÞÀ sinh k x À L
ð
Þ
½
Š
þ
qm
3
cosh 2kx À
ε 0 qm
3ε T
33
cosh 2kx:
ð3:108Þ
Fig. 3.10 Comparison of
first-order (linear), secondorder (nonlinear), and
COMSOL (nonlinear)
solutions of electron
concentration distribution
3.7 Electrically Nonlinear Extension by End Forces
53