Àc 33 Iv ,3333 þ e 33 Iϕ
1
ð Þ
,333 À Pv ,33 ¼ 0,
ð4:86Þ
Àe 33 Iv ,333 À ε 33 Iϕ
1
ð Þ
,33 þ ε 11 Aϕ
1
ð Þ
¼ qI Àn
1
ð Þ
,
ð4:87Þ
Àn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ D
n
33 In
1
ð Þ
,33 þ n 0 μ
n
11 Aϕ
1
ð Þ
À D
n
11 An
1
ð Þ
¼ 0:
ð4:88Þ
We consider a simply-supported beam with zero deflection and zero bending
moment at both ends, which is the simplest case for buckling analysis. The boundary
conditions are that at x 3 ¼ 0 and L:
v ¼ 0,
M ¼ Àc 33 Iv ,33 þ e 33 Iϕ
1
ð Þ
,3 ¼ 0,
D
1
ð Þ
3 ¼ Àe 33 Iv ,33 À ε 33 Iϕ
1
ð Þ
,3 ¼ 0,
J
n 1
ð Þ
3
¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ¼ 0:
ð4:89Þ
We need to solve Eqs. (4.86), (4.87), and (4.88) under Eq. (4.89). Let
v ¼ A sin
mπ
L
x 3 , ϕ
1
ð Þ
¼ B cos
mπ
L
x 3 ,
n
1
ð Þ
¼ C cos
mπ
L
x 3 , m ¼ 1, 2, 3Á Á Á,
ð4:90Þ
where A, B, and C are undetermined constants. Equation (4.90) satisfies Eq. (4.89).
Substituting Eq. (4.90) into Eqs. (4.86), (4.87), and (4.88), we obtain three linear
homogeneous equations for A, B, and C. For nontrivial solutions, the determinant of
the coefficient matrix of the linear equations has to vanish, which leads to an
equation for P that determines the critical buckling load denoted by P cr . The
corresponding nontrivial solutions of A, B, and C determine the buckling modes
for the deflection, electric potential, and carrier concentration perturbation.
The buckling load P cr is found to be
P cr
P
e
cr
¼ 1 þ
k
2
33
1 þ
ε 11 A
ε 33 I
L
mπ
À Á 2 þ
qn 0 μ n
33
ε 33 D
n
33
L
mπ
À Á 2 ,
ð4:91Þ
where we have denoted
P
e
cr ¼ c 33 I
mπ
L
2
, k
2
33 ¼
e
2
33
ε 33 c 33
:
ð4:92Þ
P
e
cr is the buckling load of the corresponding elastic dielectric beam when electromechanical couplings are neglected. k
2
33 is the axial electromechanical coupling
factor. Its role in Eq. (4.91) represents the piezoelectric stiffening effect which tends
4.6 Buckling
109
1
ð Þ
,333 À Pv ,33 ¼ 0,
ð4:86Þ
Àe 33 Iv ,333 À ε 33 Iϕ
1
ð Þ
,33 þ ε 11 Aϕ
1
ð Þ
¼ qI Àn
1
ð Þ
,
ð4:87Þ
Àn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ D
n
33 In
1
ð Þ
,33 þ n 0 μ
n
11 Aϕ
1
ð Þ
À D
n
11 An
1
ð Þ
¼ 0:
ð4:88Þ
We consider a simply-supported beam with zero deflection and zero bending
moment at both ends, which is the simplest case for buckling analysis. The boundary
conditions are that at x 3 ¼ 0 and L:
v ¼ 0,
M ¼ Àc 33 Iv ,33 þ e 33 Iϕ
1
ð Þ
,3 ¼ 0,
D
1
ð Þ
3 ¼ Àe 33 Iv ,33 À ε 33 Iϕ
1
ð Þ
,3 ¼ 0,
J
n 1
ð Þ
3
¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ¼ 0:
ð4:89Þ
We need to solve Eqs. (4.86), (4.87), and (4.88) under Eq. (4.89). Let
v ¼ A sin
mπ
L
x 3 , ϕ
1
ð Þ
¼ B cos
mπ
L
x 3 ,
n
1
ð Þ
¼ C cos
mπ
L
x 3 , m ¼ 1, 2, 3Á Á Á,
ð4:90Þ
where A, B, and C are undetermined constants. Equation (4.90) satisfies Eq. (4.89).
Substituting Eq. (4.90) into Eqs. (4.86), (4.87), and (4.88), we obtain three linear
homogeneous equations for A, B, and C. For nontrivial solutions, the determinant of
the coefficient matrix of the linear equations has to vanish, which leads to an
equation for P that determines the critical buckling load denoted by P cr . The
corresponding nontrivial solutions of A, B, and C determine the buckling modes
for the deflection, electric potential, and carrier concentration perturbation.
The buckling load P cr is found to be
P cr
P
e
cr
¼ 1 þ
k
2
33
1 þ
ε 11 A
ε 33 I
L
mπ
À Á 2 þ
qn 0 μ n
33
ε 33 D
n
33
L
mπ
À Á 2 ,
ð4:91Þ
where we have denoted
P
e
cr ¼ c 33 I
mπ
L
2
, k
2
33 ¼
e
2
33
ε 33 c 33
:
ð4:92Þ
P
e
cr is the buckling load of the corresponding elastic dielectric beam when electromechanical couplings are neglected. k
2
33 is the axial electromechanical coupling
factor. Its role in Eq. (4.91) represents the piezoelectric stiffening effect which tends
4.6 Buckling
109