describe shear deformation in bending but may be present in the extensional
constitutive relations in Eq. (5.44) because of anisotropy. Therefore Eq. (5.44) is
not ready to be used for extension yet. We take Poisson’s effect into consideration by
the usual plate thickness stress relaxation condition T
0
ð Þ
33 ¼ 0. We also relaxed the
plate shear stresses associated with bending by setting T
0
ð Þ
31 ¼ 0 and T
0
ð Þ
32 ¼ 0. In
summary, we take the following to be approximately true:
T
0
ð Þ
3j ¼ 0, j ¼ 1, 2, 3:
ð5:45Þ
Equation (5.45) is the usual stress relaxation for extension. According to the compact
matrix notation, with the range of p and q as 1, 2, . . . and 6, Eq. (5.45) can be written
as
T
0
ð Þ
q ¼ 0, q ¼ 3, 4, 5:
ð5:46Þ
For convenience ,we introduce another index convention in which subscripts u, v,
and w take the values 3, 4, and 5 while subscripts r and s take the remaining values
1, 2, and 6. Then Eq. (5.44) can be written as
T
0
ð Þ
r ¼ 2h c rs S
0
ð Þ
s þ c ru S
0
ð Þ
u À e kr E
0
ð Þ
k
,
T
0
ð Þ
v ¼ 2h c vs S
0
ð Þ
s þ c vw S
0
ð Þ
w À e kv E
0
ð Þ
k
¼ 0,
D
0
ð Þ
i ¼ 2h e is S
0
ð Þ
s þ e iu S
0
ð Þ
u À ε ij E
0
ð Þ
j
,
ð5:47Þ
where Eq. (5.46) has been used. From Eq. (5.47) 2 we solve for
S
0
ð Þ
u ¼ Àc
À1
uv c vs S
0
ð Þ
s þ c
À1
uv e kv E
0
ð Þ
k :
ð5:48Þ
The substitution of Eq. (5.48) into Eq. (5.47) 1,3 gives
T
0
ð Þ
r ¼ 2h γ rs S
0
ð Þ
s À ψ kr E
0
ð Þ
k
,
D
0
ð Þ
i ¼ 2h ψ is S
0
ð Þ
s þ ζ ij E
0
ð Þ
j
,
ð5:49Þ
where the effective plate material constants relaxed for the extension of thin plates
are
γ rs ¼ c rs À c rv c
À1
vw c ws , r, s ¼ 1, 2, 6,
ψ ks ¼ e ks À e kw c
À1
wv c vs , v, w ¼ 3, 4, 5,
ζ kj ¼ ε kj þ e kv c
À1
vw e jw , j, k ¼ 1, 2, 3:
ð5:50Þ
120
5 Extension and Bending of Plates