Composite Materials Research Progress

⎧σ
x ⎫ ⎡ mE
⎪ ⎪ ⎢
x
⎨σ
y ⎬ = ⎢mν
xy E y
⎪ ⎪ ⎢
⎩
σ xy ⎭ ⎣
0
Optimization of Laminated Composite Structures… 57
mν
yxE
x
mEy
0
0 ⎤⎧
ε x ⎫ ⎡Qxx
⎥⎪
⎪
0 =
⎢
⎥⎨
ε y ⎬ ⎢
Qyx
G ⎥⎪
⎪
xy ⎢
⎦⎩
γ xy ⎭ ⎣ 0
Qxy
Qyy
0
0 ⎤⎧
ε x ⎫
⎪ ⎪
0
⎥
⎥⎨
ε ⎬ ,
1
y m =
Q ⎥⎪
⎪ 1 −ν
xyν
ss ⎦⎩
γ xy ⎭
yx
(3.1)
The stresses and strains can be written in the structural coordinates (1,2,3) as in (3.2) and
(3.3) where θ is the angle between the local and structural axes, defined in Figure 3.1.
⎡ 2
⎧ε1
⎫ cos θ
⎪ ⎪ ⎢ 2
⎨ε
2 ⎬ = ⎢ sin θ
⎪ ⎪ ⎢
⎩ε
6 ⎭ ⎢
2 cosθ
sinθ
⎣
⎡ 2
⎧σ1
⎫ cos θ
⎪ ⎪ ⎢ 2
⎨σ
2 ⎬ = ⎢ sin θ
⎪ ⎪ ⎢
⎩σ
6 ⎭ ⎢
cosθ
sinθ
⎣
sin
cos
2
2
θ
θ
− 2 cosθ
sinθ
sin
cos
2
2
θ
θ
− cosθ
sinθ
− cosθ
sinθ
⎤ ⎧ ε x ⎫
⎥ ⎪ ⎪
cosθ
sinθ
⎥ ⎨ ε y ⎬
2 2
cos θ − sin θ ⎥ ⎪ ⎪
⎥⎦
⎩
γ xy ⎭
− 2cosθ
sinθ
⎤ ⎧σ
x ⎫
⎥ ⎪ ⎪
2 cosθ
sinθ
⎥ ⎨σ
y ⎬
2 2
cos θ − sin θ ⎥ ⎪ ⎪
⎥⎦
⎩
σ xy ⎭
(3.2)
(3.3)
For a ply with an orientation θ with respect to the structural axes, the constitutive
relations write:
⎧σ1
⎫ ⎡Q
⎪ ⎪
=
⎢
⎨σ
2 ⎬ ⎢
Q
⎪ ⎪
⎩σ
6 ⎭ ⎢⎣
Q
11
12
16
Q
Q
Q
12
22
26
Q
Q
Q
16
26
66
⎤⎧ε1
⎫
⎥⎪
⎪
⎥⎨ε
2 ⎬
⎥⎪
⎪
⎦⎩ε
6 ⎭
where the matrix of the stiffness coefficients in the structural axes takes the form:
⎡
(3.4)
⎧Q
4 4 2 2
2 2
11 ⎫ c s 2c
s 4c
s
⎪
Q
⎪ ⎢ 4 4 2 2
2 2 ⎥
⎪ 22 ⎪ ⎢ s c 2c
s 4c
s ⎥ ⎧Qxx
⎫
⎪Q
2 2 2 2 4 4
2 2
12 ⎪ ⎢
⎪
4 Q
⎪
c s c s c + s − c s ⎥ ⎪ yy ⎪
Q ( 1,
2,
3)
= ⎨ ⎬ = ⎢
⎥
2 2 2 2 2 2 2 2 2 ⎨ ⎬ (3.5)
⎪Q66
⎪ ⎢c
s c s − 2c
s ( c − s ) ⎥ ⎪Qxy
⎪
⎪Q
⎪ ⎢ 3 3 3 3 3 3 ⎥
16 c s cs cs c s 2(
cs c s)
⎪
⎩Q
⎪
⎢
− −
−
⎥ ss ⎭
⎪ ⎪
( x,
y,
z)
⎪Q
3 3 3 3 3 3
⎩ 26 ⎪⎭
⎢
( 1,
2,
3)
⎣ cs − c s ( c s − cs ) 2(
c s − cs ) ⎥⎦
with
c = cos θ s = sinθ
The variation of the Q’s with respect to the angle θ is plotted in Figure 3.2. It is observed
that the stiffness coefficients are highly non linear in terms of the fibers orientation.
⎤