In Eq. (5.54), S
0
ð Þ
33 has been eliminated on the right-hand side because when i ¼ j ¼ 3,
the two terms containing S
0
ð Þ
33 cancel with each other. From Eq. (5.54), the thickness
expansion or contraction accompanying the in-plane extension of the plate due to
Poisson’s effect can be found if interested. Substituting Eq. (5.54) back into
Eq. (5.37), we obtain
T
0
ð Þ
ij ¼ 2h c ijkl S
0
ð Þ
kl À e kij E
0
ð Þ
k
,
D
0
ð Þ
i ¼ 2h e ikl S
0
ð Þ
kl þ ε ij E
0
ð Þ
j
,
ð5:55Þ
where the relaxed material constants are defined by
c ijkl ¼ c ijkl À c ij33 c 33kl =c 3333 ,
e kij ¼ e kij À e k33 c 33ij =c 3333 ,
ε ij ¼ ε ij þ e i33 e j33 =c 3333 :
ð5:56Þ
We note that the right-hand side of Eq. (5.55) does not depend on S
0
ð Þ
33 and T
0
ð Þ
33 ¼ 0
is automatically satisfied by Eq. (5.55). However, for the transverse shear forces in
bending, the right-hand side of T
0
ð Þ
3b from Eq. (5.55) may depend on the extensional
strains S
0
ð Þ
ab and hence the extensional displacements u
0
ð Þ
a
in materials with high
anisotropy.
Since we are using a linear function x 3 u
1
ð Þ
a as an approximation for the fundamental thickness-shear deformation which in dynamic problems has a sinusoidal
variation along the plate thickness, we introduce two shear correction factors κ 1 and
κ 2 to improve the accuracy of the plate equations by modifying the following zeroorder plate shear strains:
S
0
ð Þ
31 ! κ 1 S
0
ð Þ
31 , S
0
ð Þ
32 ! κ 2 S
0
ð Þ
32 :
ð5:57Þ
This modifies the c ijkl and e ijk in Eq. (5.55) into c
0
ð Þ
ijkl and e
0
ð Þ
ijk . Then Eq. (5.55) takes
the following form in the compact matrix notation:
T
0
ð Þ
p ¼ 2h c
0
ð Þ
pq S
0
ð Þ
q À e
0
ð Þ
kp E
0
ð Þ
k
,
D
0
ð Þ
i ¼ 2h e
0
ð Þ
iq S
0
ð Þ
q þ ε ij E
0
ð Þ
j
:
ð5:58Þ
The structures of the c
0
ð Þ
pq and e
0
ð Þ
iq matrices are [6]
122
5 Extension and Bending of Plates