We note that γ rs are Voigt’s anisotropic plate elastic constants. Equation (5.49) is the
constitutive relation suitable for extension. Its right-hand side does not have S
0
ð Þ
3j and
hence are not involved with u
1
ð Þ
a
and u
0
ð Þ
3 . When Eq. (5.49) 1 is substituted into
Eq. (5.43), mechanically it results in equations involving the extensional displacement u
0
ð Þ
a only. At the boundary of a plate edge with in-plane unit exterior normal
n and in-plane unit tangent s as shown in Fig. 5.1, the mechanical boundary
conditions for extension may be the prescription of
T
0
ð Þ
nn
or u
0
ð Þ
n , and T
0
ð Þ
ns
or u
0
ð Þ
s :
ð5:51Þ
5.5 Equations for Bending
Similar to the previous section, in this section we also mainly focus on the reduction
of the mechanical equations and will leave the electrical equations untouched. For
bending of plates with shear deformation, we want to obtain equations for the
bending displacement u
0
ð Þ
3
and fundamental thickness-shear displacements u
1
ð Þ
a .
However, in materials with strong anisotropy, u
1
ð Þ
a
may be coupled to extension.
This coupling is kept in the following, which may be eliminated through proper
stress relaxation when needed or simply neglected as an approximation. In bending,
the in-plane normal and shear stresses vary linearly along the plate thickness to
produced bending and twisting moments. Accordingly, we keep the first-order
strains as follows:
S ij ffi S
0
ð Þ
ij þ x 3 S
1
ð Þ
ij :
ð5:52Þ
From Eqs. (5.25) and (5.26), it can be seen that S
0
ð Þ
33 is involved with u
1
ð Þ
3 , and S
1
ð Þ
3j are
involved with u
2
ð Þ
j . These higher-order plate displacements will be eliminated
through two separate stress relaxation procedures in the following for the zeroand first-order constitutive relations, respectively.
For the zero-order constitutive relations, from Eq. (5.37) 1 , by setting i ¼ j ¼ 3, we
set
T
0
ð Þ
33 ¼ 2h c 33kl S
0
ð Þ
kl À e k33 E
0
ð Þ
k
¼ 0:
ð5:53Þ
This implies the following expression for S
0
ð Þ
33 :
S
0
ð Þ
33 ¼ À
1
c 3333
c 33kl S
0
ð Þ
kl À c 3333 S
0
ð Þ
33 À e k33 E
0
ð Þ
k
:
ð5:54Þ
5.5 Equations for Bending
121
constitutive relation suitable for extension. Its right-hand side does not have S
0
ð Þ
3j and
hence are not involved with u
1
ð Þ
a
and u
0
ð Þ
3 . When Eq. (5.49) 1 is substituted into
Eq. (5.43), mechanically it results in equations involving the extensional displacement u
0
ð Þ
a only. At the boundary of a plate edge with in-plane unit exterior normal
n and in-plane unit tangent s as shown in Fig. 5.1, the mechanical boundary
conditions for extension may be the prescription of
T
0
ð Þ
nn
or u
0
ð Þ
n , and T
0
ð Þ
ns
or u
0
ð Þ
s :
ð5:51Þ
5.5 Equations for Bending
Similar to the previous section, in this section we also mainly focus on the reduction
of the mechanical equations and will leave the electrical equations untouched. For
bending of plates with shear deformation, we want to obtain equations for the
bending displacement u
0
ð Þ
3
and fundamental thickness-shear displacements u
1
ð Þ
a .
However, in materials with strong anisotropy, u
1
ð Þ
a
may be coupled to extension.
This coupling is kept in the following, which may be eliminated through proper
stress relaxation when needed or simply neglected as an approximation. In bending,
the in-plane normal and shear stresses vary linearly along the plate thickness to
produced bending and twisting moments. Accordingly, we keep the first-order
strains as follows:
S ij ffi S
0
ð Þ
ij þ x 3 S
1
ð Þ
ij :
ð5:52Þ
From Eqs. (5.25) and (5.26), it can be seen that S
0
ð Þ
33 is involved with u
1
ð Þ
3 , and S
1
ð Þ
3j are
involved with u
2
ð Þ
j . These higher-order plate displacements will be eliminated
through two separate stress relaxation procedures in the following for the zeroand first-order constitutive relations, respectively.
For the zero-order constitutive relations, from Eq. (5.37) 1 , by setting i ¼ j ¼ 3, we
set
T
0
ð Þ
33 ¼ 2h c 33kl S
0
ð Þ
kl À e k33 E
0
ð Þ
k
¼ 0:
ð5:53Þ
This implies the following expression for S
0
ð Þ
33 :
S
0
ð Þ
33 ¼ À
1
c 3333
c 33kl S
0
ð Þ
kl À c 3333 S
0
ð Þ
33 À e k33 E
0
ð Þ
k
:
ð5:54Þ
5.5 Equations for Bending
121