J
p
i,i ¼ Àq
∂Δp
∂t
,
J
n
i,i ¼ q
∂Δn
∂t
,
ð4:2Þ
S ij ¼ s
E
ijkl T kl þ d kij E k ,
D i ¼ d ikl T kl þ ε
T
ik E k ,
ð4:3Þ
J
p
i ¼ qp 0 μ
p
ij E j À qD
p
ij Δp
ð Þ ,j ,
J
n
i ¼ qn 0 μ
n
ij E j þ qD
n
ij Δn
ð Þ ,j ,
ð4:4Þ
S ij ¼ u i,j þ u j,i
À
Á
=2,
E i ¼ Àφ ,i :
ð4:5Þ
To develop a one-dimensional theory for the extension and bending in the y-z plane
with shear deformation, we make the following approximations of the relevant
mechanical displacements, electric potential, and carrier concentration perturbations
[2]:
u 2 x, t
ð Þ ffi v x 3 , t
ð
Þ,
u 3 x, t
ð Þ ffi w x 3 , t
ð
Þþx 2 ψ x 3 , t
ð
Þ,
φ x, t
ð Þ ffi ϕ
0
ð Þ x 3 , t
ð
Þþx 2 ϕ
1
ð Þ x 3 , t
ð
Þ,
Δp x, t
ð Þ ffi p
0
ð Þ x 3 , t
ð
Þþx 2 p
1
ð Þ x 3 , t
ð
Þ,
Δn x, t
ð Þ ffi n
0
ð Þ x 3 , t
ð
Þþx 2 n
1
ð Þ x 3 , t
ð
Þ,
ð4:6Þ
where w(x 3 , t) is the extensional displacement, v(x 3 , t) the bending displacement, and
ψ(x 3 , t) the shear deformation accompanying bending. Then, from Eq. (4.5), we
calculate the relevant strains, electric fields, and carrier concentration perturbation
gradients as
S 3 ¼ S 33 ¼ u 3,3 ¼ w ,3 þ x 2 ψ ,3 ,
S 4 ¼ 2S 23 ¼ u 2,3 þ u 3,2 ¼ v ,3 þ ψ,
E 2 ¼ Àφ ,2 ¼ Àϕ
1
ð Þ , E 3 ¼ Àφ ,3 ¼ Àϕ
0
ð Þ
,3 À x 2 ϕ
1
ð Þ
,3 ,
Δp
ð Þ ,2 ¼ p
1
ð Þ ,
Δp
ð Þ ,3 ¼ p
0
ð Þ
,3 þ x 2 p
1
ð Þ
,3 ,
Δn
ð Þ ,2 ¼ n
1
ð Þ ,
Δn
ð Þ ,3 ¼ n
0
ð Þ
,3 þ x 2 n
1
ð Þ
,3 :
ð4:7Þ
x1, x
x2, y
L
c
x3, z
Fig. 4.1 A ZnO beam with
its c-axis along x 3
90
4 Bending of Beams