Transcript of Combining materials for composite-material cars Ford initiated research at a time when they took a...
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- Combining materials for composite-material cars Ford initiated
research at a time when they took a look at making cars from
composite materials. Graphite-epoxy is too expensive, glass- epoxy
is not stiff enough. Grosset, L., Venkataraman, S., and Haftka,
R.T., Genetic optimization of two-material composite laminates,
Proceedings, 16th ASC Technical Meeting, Blacksburg, VA, September
2001
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- Multi-material laminate Materials: one material = 1 ply (
matrix or fiber materials) E.g.: glass-epoxy, graphite-epoxy,
Kevlar-epoxy Use two materials in order to combine high efficiency
(stiffness) and low cost Graphite-epoxy: very stiff but expensive;
glass-epoxy: less stiff, less expensive Objective: use
graphite-epoxy only where most efficient, use glass-epoxy for the
remaining plies
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- Multi-criterion optimization Two competing objective functions:
WEIGHT and COST Design variables: number of plies ply orientations
ply materials No single design minimizes weight and cost
simultaneously: A design is Pareto-optimal (non-dominated) if there
is no design for which both Weight and Cost are lower Goal:
construct the trade-off curve between weight and cost (set of
Pareto-optimal designs, also called Pareto front)
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- Material properties Graphite-epoxy Longitudinal modulus, E 1 :
20.01 10 6 psi Transverse modulus, E 2 : 1.30 10 6 psi Shear
modulus, G 12 : 1.03 10 6 psi Poisson s ratio, 12 : 0.3 Ply
thickness, t: 0.005 in Density, : 5.8 10 -2 lb/in 3 Ultimate shear
strain, ult : 1.5 10 -2 Cost index: $8/lb Glass-epoxy Longitudinal
modulus, E 1 : 6.30 10 6 psi Transverse modulus, E 2 : 1.29 10 6
psi Shear modulus, G 12 : 6.60 10 5 psi Poisson s ratio, 12 : 0.27
Ply thickness, t: 0.005 in Density, : 7.2 10 -2 lb/in 3 Ultimate
shear strain, ult : 2.5 10 -2 Cost index: $1/lb
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- Optimization problem Minimize weight and cost of a 30x36 plate
By changing ply orientations and material m i subject to a
frequency constraint:
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- Constructing the Pareto trade- off curve Simple method:
weighting method. A composite function is constructed by combining
the 2 objectives: W: weight C: cost : weighting parameter (0 1) A
succession of optimizations with varying from 0 to 1 is solved. The
set of optimum designs builds up the Pareto trade-off curve This is
not a fool-proof approach (Why?), and it is time consuming. Genetic
algorithms provide a shortcut.
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- Multi-material Genetic Algorithm Two variables for each ply:
Fiber orientation Material Each laminate is represented by 2
strings: Orientation string Material string Example: [45/0/30/0/90]
is represented by: Orientation:45-0-30-0-90 Material: 2-2-1-2-1 GA
maximizes fitness: Fitness = -F 1: graphite- epoxy 2:
glass-epoxy
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- Simple vibrating plate problem Minimize the weight (W) and cost
(C) of a 36x30 rectangular laminated plate 19 possible ply angles
from 0 to 90 in 5- degree step Constraints: Balanced laminate (for
each + ply, there must be a - ply in the laminate) first natural
frequency > 25 Hz Frequency calculated using Classical
Lamination Theory
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- How constraints are enforced Balance constraint hard coded in
the strings: stacks of are used Example: (45-0-30-25-90) represents
[45/0/30/25/90] s Frequency constraint is incorporated into the
objective function by a penalty, which is proportional to the
constraint violation
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- Genetic operators Roulette wheel selection based on rank
Two-point crossover Mutation and permutation Ply deletion and
addition Operators apply to each chromosome individually. Best
individual passed to the next generation (elitist)
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- Pareto Trade-off curve ($) (lb) A (16.3,16.3) B (6.9,55.1) C
point C 64% lighter than A; 17% more expensive 53% cheaper than B;
25% heavier
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- Optimum laminates Cost minimization: [50 10 /0] s, cost =
16.33, weight = 16.33 Weight minimization: [50 5 /0] s, cost =
55.12, weight = 6.89 Intermediate design: [50 2 /50 5 ] s, cost =
27.82, weight = 10.28 Glass-epoxy in the core layers to increase
the thickness Graphite-epoxy as outer plies for a high frequency
Midplane Intermediate optimum laminates: sandwich-type
laminates
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- Problems two-material laminate Check the reasonableness of the
weight ratio of the two extreme laminates (16.3/6.9) with simple
calculation. Illustrate the difference between crossover of two
laminates when (a) the materials and angles are segregated into two
chromosomes and (b) there is one string with the two numbers for
each ply being together. Why does the figure of the Pareto front
not show values of alpha between 0 and 0.7?