Composite Materials Research Progress

and
γ
2
γ
0
Optimization of Laminated Composite Structures… 59
⎡ U1
=
⎢
⎢
⎢⎣
sym
⎡ U3
=
⎢
⎢
⎢⎣
sym
U
U
−U
U
3
3
4
1
0 ⎤
0
⎥
⎥
U5⎥⎦
0 ⎤
0
⎥
⎥
−U
3⎥⎦
1
U1
= ( 3Qxx
+ 3Q
yy + 2Qxy
+ 4Qss
)
8
1
U2
= ( Qxx
− Qyy
)
2
1
U3
= ( Qxx
+ Qyy
− 2Qxy
− 4Qss
)
8
⎡
⎢
0
⎢
γ 3 = ⎢
⎢
⎢sym
⎢⎣
3.1.2. Constitutive Relations for a Laminate
⎡ U2
0 0⎤
γ
⎢
⎥
1 = −U
0
⎢ 2
(3.8)
⎥
⎢⎣
sym 0⎥⎦
0
0
U2
⎤
2 ⎥
U ⎥
2
⎥
2 ⎥
0 ⎥
⎥⎦
⎡ 0
γ
⎢
4 =
⎢
⎢⎣
sym
0
0
U3
⎤
−U
⎥
3⎥
0 ⎥⎦
1
U4
= ( Qxx
+ Qyy
+ 6Qxy
− 4Qss
)
8
1
U5
= ( Qxx
+ Qyy
− 2Qxy
+ 4Qss
)
8
Composite structures are thin membranes, plates or shells made of n unidirectional
orthotropic plies stacked on the top of each other. Such structures can support in and out-of
plane loadings. In the following the constitutive relations for a laminate made of several
individual plies are derived. The notations are defined in Figure 3.3. In the case of plane
stress, i.e. the effects of transverse shear is neglected, in-plane normal and shear loads N, as
well as the flexural and torsional moments M are applied to the laminate. Those loadings are
computed by considering the stress state in each ply with the relations (3.9):
⎧N1
⎫ ⎧σ
⎫
h / 2 1
⎧M1
⎫ ⎧σ
⎫
h / 2 1
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
N = ⎨N
2 ⎬ = ∫ ⎨σ
2 ⎬dz
M = ⎨M
2 ⎬ = ∫ ⎨σ
2 ⎬zdz
(3.9)
⎪N
⎪ −h / 2 ⎪ ⎪
⎩ 6 ⎭ ⎩σ
⎪
6 ⎭
M ⎪ −h / 2 ⎪ ⎪
⎩ 6 ⎭ ⎩σ
6 ⎭
For a first order cinematic theory, where the displacement through the laminate’s
thickness is linear in the z coordinate measured with respect to the mid-plane of the plate/shell
(Figure 3.3), the vector of laminate’s strains εl is linked to the in-plane strains and the
curvatures via the relation εl = ε + zκ
0
. With this definition it turns that the constitutive
relations for a laminate are given by (3.10) where A, B and D are the in-plane, coupling and
bending stiffness matrices of the laminate.