we obtain three equations for the bending displacement u
0
ð Þ
3 and shear displacements
u
1
ð Þ
a (a ¼ 1, 2). In addition to the couplings to electric fields, Eq. (5.63) 1 may be
coupled to the extensional displacements u
0
ð Þ
a as explained earlier. At the boundary
of a plate with in-plane unit exterior normal n and in-plane unit tangent s as shown in
Fig. 5.1, mechanically, we may prescribe the following boundary conditions:
T
0
ð Þ
nn
or u
0
ð Þ
n , T
0
ð Þ
ns
or u
0
ð Þ
s , T
0
ð Þ
n3
or u
0
ð Þ
3 ,
T
1
ð Þ
nn
or u
1
ð Þ
n , and T
1
ð Þ
ns
or u
1
ð Þ
s :
ð5:64Þ
5.6 Reduction to Bending Without Shear Deformation
For the bending of very thin plates in which the flexural motion is dominant and the
shear deformation is small, the Kirchhoff classical or elementary theory of bending
without shear deformation is sufficient. To reduce the theory for bending with shear
deformation in the previous section to the theory of bending without shear deformation, two approximations are needed. First we take the plate thickness-shear
strains S
0
ð Þ
a3 to vanish, which implies, from Eq. (5.25),
u
1
ð Þ
a ¼ Àu
0
ð Þ
3,a ) S
1
ð Þ
ab ¼ Àu
0
ð Þ
3,ab :
ð5:65Þ
Equation (5.65) enables us to express u
1
ð Þ
a in terms of u
0
ð Þ
3 in Eqs. (5.26) and (5.62).
Another approximation needed is to neglect the rotatory inertia ρ2h
3 /3 in Eq. (5.63) 2
to obtain
T
1
ð Þ
ab,a À T
0
ð Þ
3b þ t
1
ð Þ
b ¼ 0,
ð5:66Þ
from which we can solve for T
0
ð Þ
3b and substitute the result into Eq. (5.63) 1 to obtain
the equation for elementary flexure:
T
1
ð Þ
ab,ab þ t
1
ð Þ
b,b þ t
0
ð Þ
3 ¼ 2hρ € u
0
ð Þ
3 :
ð5:67Þ
With substitutions from Eqs. (5.62) 1 and (5.65), we can write Eq. (5.67) as an
equation for the bending displacement u
0
ð Þ
3 . At the boundary of a plate with an
in-plane unit exterior normal n and an in-plane unit tangent s (see Fig. 5.1), the
boundary conditions for elementary bending without shear deformation may be
T
0
ð Þ
n3 þ T
1
ð Þ
ns,s
or u
0
ð Þ
3 , and T
1
ð Þ
nn
or u
0
ð Þ
3,n :
ð5:68Þ
124
5 Extension and Bending of Plates
0
ð Þ
3 and shear displacements
u
1
ð Þ
a (a ¼ 1, 2). In addition to the couplings to electric fields, Eq. (5.63) 1 may be
coupled to the extensional displacements u
0
ð Þ
a as explained earlier. At the boundary
of a plate with in-plane unit exterior normal n and in-plane unit tangent s as shown in
Fig. 5.1, mechanically, we may prescribe the following boundary conditions:
T
0
ð Þ
nn
or u
0
ð Þ
n , T
0
ð Þ
ns
or u
0
ð Þ
s , T
0
ð Þ
n3
or u
0
ð Þ
3 ,
T
1
ð Þ
nn
or u
1
ð Þ
n , and T
1
ð Þ
ns
or u
1
ð Þ
s :
ð5:64Þ
5.6 Reduction to Bending Without Shear Deformation
For the bending of very thin plates in which the flexural motion is dominant and the
shear deformation is small, the Kirchhoff classical or elementary theory of bending
without shear deformation is sufficient. To reduce the theory for bending with shear
deformation in the previous section to the theory of bending without shear deformation, two approximations are needed. First we take the plate thickness-shear
strains S
0
ð Þ
a3 to vanish, which implies, from Eq. (5.25),
u
1
ð Þ
a ¼ Àu
0
ð Þ
3,a ) S
1
ð Þ
ab ¼ Àu
0
ð Þ
3,ab :
ð5:65Þ
Equation (5.65) enables us to express u
1
ð Þ
a in terms of u
0
ð Þ
3 in Eqs. (5.26) and (5.62).
Another approximation needed is to neglect the rotatory inertia ρ2h
3 /3 in Eq. (5.63) 2
to obtain
T
1
ð Þ
ab,a À T
0
ð Þ
3b þ t
1
ð Þ
b ¼ 0,
ð5:66Þ
from which we can solve for T
0
ð Þ
3b and substitute the result into Eq. (5.63) 1 to obtain
the equation for elementary flexure:
T
1
ð Þ
ab,ab þ t
1
ð Þ
b,b þ t
0
ð Þ
3 ¼ 2hρ € u
0
ð Þ
3 :
ð5:67Þ
With substitutions from Eqs. (5.62) 1 and (5.65), we can write Eq. (5.67) as an
equation for the bending displacement u
0
ð Þ
3 . At the boundary of a plate with an
in-plane unit exterior normal n and an in-plane unit tangent s (see Fig. 5.1), the
boundary conditions for elementary bending without shear deformation may be
T
0
ð Þ
n3 þ T
1
ð Þ
ns,s
or u
0
ð Þ
3 , and T
1
ð Þ
nn
or u
0
ð Þ
3,n :
ð5:68Þ
124
5 Extension and Bending of Plates