Composite Materials Research Progress

84
8.2. Non Homogeneous Laminate
Michaël Bruyneel
In this application a non homogeneous composite membrane divided in regions of constant
thickness and fibre orientations is studied. Each region is defined with an unidirectional
laminate made of a glass/epoxy material. The design over stiffness is only considered here.
The solution with respect to strength and stiffness is provided in Bruyneel (2006).
2
2
1
1
2
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1
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Figure 8.4. Initial configurations with 45 and -45 degrees plies orientations.
The quasi-unconstrained optimization problem (8.2) consists in finding the optimal
values of the plies thickness and fibers orientations in each region of the laminated composite
structure that maximize the overall stiffness (i.e. that minimize the compliance – the potential
energy of the applied loads). The vectors of the design variables are given by
θ = { θi
, i = 1,...,
n}
and t = { ti , i = 1,...,
n}
where n is the number of regions according to
Figure 8.4. The initial thicknesses are of 1 mm.
2
2
1
P
1
min Compliance
θ,t
0° ≤θ
i ≤180°
i = 1,...,
n
(8.2)
0. 01mm
≤ ti
≤ 5mm
In this problem the optimal values of the thickness is 5 mm, that is their upper bound.
Anyway this application illustrates the difficulties encountered when both kinds of design
variables appear in the design problem. The optimal values of the compliances are reported in
Figure 8.5 as a function of the number of regions. As already noticed by Foldager (1999) an
increase of the number of regions of different orientations improves the overall optimal
structural stiffness (i.e. it decreases the compliance).
The optimal fibers orientations are illustrated in Figure 8.6, for the several membrane
configurations of Figure 8.4. The iteration histories are reported in Figure 8.7. When the SAM
method is used, about 10 iterations are enough for reaching a stationary solution with respect
to a small relative variation of the objective at 2 successive iterations. The GCMMA
approximation finds this solution in a larger number of design cycles. It is observed that when
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