Then Eqs. (4.23) 1 and (4.24) reduce to
M ¼ Àc 33 Iv ,33 þ e 33 Iϕ
1
ð Þ
,3 ,
ð4:28Þ
D
0
ð Þ
2 ¼ Àε 11 Aϕ
1
ð Þ ,
D
1
ð Þ
3 ¼ Àe 33 Iv ,33 À ε 33 Iϕ
1
ð Þ
,3 :
ð4:29Þ
In bending without shear deformation, the moment of inertia I on the right-hand side
of Eq. (4.22) 2 can be neglected. Then Eq. (4.22) 2 implies that
Q ¼ M ,3 ¼ Àc 33 Iv ,333 þ e 33 Iϕ
1
ð Þ
,33 ,
ð4:30Þ
where Eq. (4.28) has been used. Effectively, Eq. (4.30) serves as the constitutive
relation for Q, and Eq. (4.23) 2 is abandoned. When Eq. (4.30) is substituted into
Eq. (4.22) 1 , we obtain the equation for bending without shear deformation as
Àc 33 Iv ,3333 þ e 33 Iϕ
1
ð Þ
,333 ¼ ρA€ v:
ð4:31Þ
Equation (4.31) is accompanied by Eq. (4.22) 3–5 to produce four equations needed
for v, ϕ
(1) , p
(1) , and n
(1) .
4.3 Static Bending of a Cantilever
Consider the ZnO beam shown in Fig. 4.2. The left end is fixed. The right end is
under the action of a transverse shear force f y [2].
We consider bending with shear deformation governed by Eqs. (4.22)–(4.26) and
(4.20). We limit ourselves to the case of an n-type semiconductor so that p
(1)
ffi 0.
The four remaining fields are v, ψ, ϕ
(1) , and n
(1) . For static bending, the relevant
equations reduce to
x1, x
x2, y
L
a
c
x3, z
fy
Fixed
end
Shear force
Fig. 4.2 A ZnO beam
under a static shear force f y
4.3 Static Bending of a Cantilever
95