66
4 Nonlinear Constitutive Relations
˘
ε p = ˘
s pq ˘
σ q + ˘
d pm ˘
E m ,
(4.45)
˘
D m = ˘
d mq ˘
σ q + ˘
mn ˘
E n .
(4.46)
For plate and shell structures, introducing the usual assumption of ˘
σ 33 = 0, the elastic
compliance constants ˘
s pq in fibrous coordinates are given as
˘
s 11 =
1
˘
Y 1
,
˘
s 12 = −
˘
ν 12
˘
Y 1
= −
˘
ν 21
˘
Y 2
, ˘
s 22 =
1
˘
Y 2
,
˘
s 44 =
1
κ ˘
G 23
, ˘
s 55 =
1
κ ˘
G 13
,
˘
s 66 =
1
˘
G 12
,
(4.47)
where ˘
Y i , ˘
ν i j and ˘
G i j are the Young’s moduli, the Poisson’s ratios and the shear
moduli, κ is the shear correction factor.
Re-arranging the constitutive equations (4.45) and (4.46) by the matrix form with
an inversed relation, one obtains
˘
σ = ˘
c˘ ε − ˘
e
T ˘
E,
(4.48)
˘
D = ˘
e ˘
ε + ˘
χ ˘
E,
(4.49)
where
˘
σ =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˘
σ 11
˘
σ 22
˘
τ 12
˘
τ 23
˘
τ 13
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, ˘
ε =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˘
ε 11
˘
ε 22
˘
γ 12
˘
γ 23
˘
γ 13
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
, ˘
D =
⎧
⎨
⎩
˘
D 1
˘
D 2
˘
D 3
⎫
⎬
⎭
, ˘
E =
⎧
⎨
⎩
˘
E 1
˘
E 2
˘
E 3
⎫
⎬
⎭
,
(4.50)
˘
c =
⎡
⎢
⎢
⎢
⎢
⎣
˘
c 11 ˘
c 12 0 0 0
˘
c 12 ˘
c 22 0 0 0
0 0 ˘
c 66 0 0
0 0 0 ˘
c 44 0
0 0 0 0 ˘
c 55
⎤
⎥
⎥
⎥
⎥
⎦
, ˘
χ =
⎡
⎣
˘
χ 11 0 0
0 ˘
χ 22 0
0 0 ˘
χ 33
⎤
⎦ ,
(4.51)
with
˘
c 11 =
˘
s 22
˘
s 11 ˘
s 22 − ˘
s 12 ˘
s 12
=
˘
Y 1
1 − ˘
ν 12 ˘
ν 21
,
˘
c 12 = −
˘
s 12
˘
s 11 ˘
s 22 − ˘
s 12 ˘
s 12
=
˘
ν 12 ˘
Y 2
1 − ˘
ν 12 ˘
ν 21
,
˘
c 22 =
˘
s 11
˘
s 11 ˘
s 22 − ˘
s 12 ˘
s 12
=
˘
Y 2
1 − ˘
ν 12 ˘
ν 21
,
˘
c 44 = κ ˘
G 23 , ˘
c 55 = κ ˘
G 13 , ˘
c 66 = ˘
G 12 .
(4.52)