ME480/580: Materials Selection
Lecture Notes for Week One
Winter 2011
MATERIALS SELECTION IN THE DESIGN
PROCESS
Reading: Ashby Chapters 1, 2, and 3.
Reference: Kenneth G. Budinski, Engineering Materials: Properties and Selection
Fifth edition, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996.
HISTORICAL CONTEXT
Ashby does a nice job of setting the historical context of the development of materials
over the years with the cover illustration for Chapter One.
Note: Change from NATURAL materials (on left) toward MANUFACTURED materials
(on right) toward ENGINEERED materials (near future). We have an increasingly large
number of materials to deal with, on the order of 160,000 at present!
Other books for general reading on the history and development of materials science and
engineered materials are:
J. E. Gordon, The New Science of Strong Materials, or Why You Don't Fall Through the
Floor, Princeton University Press, Princeton, NJ.
M. F. Ashby and D. R. H Jones, Engineering Materials Parts 1, 2, and 3, Pergamon
Press, Oxford, UK.
F. A. A. Crane and J. A. Charles, Selection and Use of Engineering Materials,
Butterworths, London, UK.
P. Ball, Made to Measure: New Materials for the 21st Century, Princeton University
Press, 1997.
MATERIALS PROPERTIES
Before we can discuss the appropriate selection of materials in design, we have to have a
foundation of what we mean by "materials properties". Both Budinski and Ashby provide
lists of these in the texts. For example:
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This is a pretty complete list. Budinski discusses these in his chapter 2, and Ashby has his
own definitions and discussion in chapter 3 in which he breaks the materials down into
six categories: METALS, POLYMERS, CERAMICS, ELASTOMERS (which Budinski
groups with plastics), GLASSES (which Budinski groups with ceramics), and HYBRIDS
(or composite materials). In all, about 120,000 different materials with property values
ranging over 5 orders of magnitude!
The importance of these chapters is that unless you have a clear idea of how a property
value is measured (see Homework One), you cannot properly use the property for
calculations in mechanical design. To these properties, we will add two other important
materials properties: PRICE, and EMBODIED ENERGY (and other environmental
materials properties.)
MATERIALS IN THE DESIGN PROCESS
Different authors have different ideas about how the design process should work.
Budinski's design strategy is found in figure 18-1.
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BUDINSKI FIGURE 18.1 NOTES:
1. Calculations in the first block! Analysis is important! (NOTE: ALGEBRA is a
critical skill for success in this class, as is UNIT ANALYSIS. You’ve been
warned!)
2. Analysis appears multiple times throughout design process.
3. Materials selection is in the last step.
4. Iteration?
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In my opinion, Ashby uses a better design strategy, especially in terms of materials
selection. He breaks the design flow path into three stages, called CONCEPTUAL
DESIGN, EMBODIMENT DESIGN, and DETAIL DESIGN (see figure 2.1).
CONCEPTUAL DESIGN:
All options are kept open.
Consideration of alternate working principles.
Assess the functional structure of your design.
EMBODIMENT DESIGN:
Use the functional structure to ANALYSE the operation.
Sizing of components.
Materials down-selection.
Determination of operational conditions.
DETAIL DESIGN:
Specifications written for components.
Detailed analysis of critical components.
Production route and cost analysis performed.
How does materials selection enter into Ashby's process? (Figure 2.5)
Materials selection enters at EVERY STAGE, but with differing levels of
CONSTRAINT and DETAILED INFORMATION.
CONCEPTUAL DESIGN:
Apply PRIMARY CONSTRAINTS (eg. working temperature, environment, etc.).
(Budinski figure 18-2 has a good list of primary constraints to consider.)
100% of materials in, 10-20% candidates come out.
EMBODIMENT DESIGN:
Develop and apply optimization constraints.
Need more detailed calculations and
Need more detailed materials information.
10-20% of materials in, 5 candidate materials out.
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DETAILED DESIGN:
High degree of information needed about only a few materials.
May require contacting specific suppliers of materials.
May require specialized testing for critical components if materials data does not
already exist.
CLASS APPROACH
Two different philosophies have been presented here:
Budinski: get familiar with a set of basic materials from each category, about seventyfive in total, and these will probably handle 90% of your design needs (see Figure 18-8).
Ashby: look at all 160,000 materials initially, and narrow your list of candidate materials
as the design progresses using some technique to narrow your choices.
Materials selection has to include not only properties, but also SHAPES (what standard
shapes are available, what shapes are possible), and PROCESSING (what fabrication
route can or should be used to produce the part or raw material, eg. casting, injection
molding, extrusion, machining, etc.). It can also include ENVIRONMENTAL IMPACT.
The point is that the choice of materials interacts with everything in the engineering
design and product manufacturing process (see Ashby Figure 2.6).
In the remainder of this course we will develop a systematic approach to dealing with all
these interactions and with looking at the possibilities of all 160,000 of these materials
based on the use of MATERIALS SELECTION CHARTS as developed by Ashby.
Flow of the course:
•
•
•
•
•
Optimization of selection without considering shape effects.
Optimization under multiple constraints.
Optimization of selection considering shape effects.
Considerations of environmental impact.
Optimization of material process selection.
SELECTION CHARTS (Ashby chapter 4)
1. Materials don't exhibit single-valued properties, but show a range of properties, even
within a single production run (see Ashby, Figure 4.1 for example.)
EXAMPLES: The elastic modulus of copper varies over a few percent
depending on the purity, texture, grain size, etc.
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The mechanical strength of alumina (Al2O3) varies by more than a factor
of 100 depending on its porosity, grain size, etc.
Metal alloys show large changes in their mechanical and electrical
properties depending on the heat treatments and mechanical working they
have experienced.
NOTE: Because the properties of materials may vary over large ranges, it
will be critical to be able to interpret property data using SEMI-LOG and
LOG-LOG plots. If you aren’t comfortable with logarithmic math and
making and reading log axes on plots REVIEW IT!
2. Performance is seldom limited by only ONE property.
EXAMPLE: in lightweight design, it is not just strength that is important,
but both strength and density. We will need to be able to compare
materials based on several properties at once.
Because of these facts, we can produce charts such as this selection chart from Ashby:
There is a tremendous amount of information and power in these charts. First of all, they
provide the materials property data as "balloons" in an easy to compare form. Secondly,
other physical information can be displayed on these charts.
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EXAMPLE: the longitudinal wavespeed of sound in a material is given by the equation
" E%
V =$ '
# !&
log (V ) =
1/2
Rewrite this equation by taking the base-10 logarithm of both sides to get:
1
# log ( E ) ! log ( " ) %& or log ( E ) = 2 log (V ) + log ( " ) .
2$
This is an equation of the form Y = A + BX, where:
Y = log(E),
A = constant = 2log(V) = y-axis intercept at X = 0,
B = slope = 1, and
X = log(ρ).
This appears as a line of slope = 1 on a plot of log(E) versus log(ρ). Such a line connects
materials that have the same speed of sound (constant V).
NOTE: X = 0 means what for the value of density?
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EXAMPLE: The selection requirement for a particular minimum weight design (derived
next time) is to maximize the ratio of
E 1/2
1
= C = constant, which leads to log ( E ) = log ( C ) + log ( ! ) , or
!
2
log ( E ) = 2 log ( C ) + 2 log ( ! ) ,
Y
=
A
+ BX.
This is a straight line of slope = 2 on a plot of log(E) versus log(ρ).
Such a line connects materials that will perform the same in a minimum weight design,
that is, all the materials on this line have the same value of the constant, C.
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week One
Winter 2011
PERFORMANCE INDICES
Reading: Ashby Chapters 4 and 5.
Materials Selection begins in conceptual design by using PRIMARY CONSTRAINTS non-negotiable constraints on the material imposed by the design or environment.
Examples might include "must be thermally insulating", or "must not corrode in
seawater".
These take the form of "PROPERTY > PROPERTYcritical", and appear as horizontal or
vertical lines on the selection charts.
NOTE: Don't go overboard on primary constraints. They are the easiest to apply and
require the least thought and analysis, but they can often be engineered around, for
example, by active cooling of a hot part, or adding corrosion resistant coatings.
After initial narrowing, you should develop PERFORMANCE INDICES.
DEFINITIONS:
PERFORMANCE:
OBJECTIVE FUNCTION:
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CONSTRAINT:
PERFORMANCE INDEX:
In the following, we will assume that performance (P) is determined by three factors:
FUNCTIONAL REQUIREMENTS (carry a load, store energy, etc.)
GEOMETRICAL REQUIREMENTS (space available, shape, size)
MATERIALS PROPERTIES
What we want to do is OPTIMIZE our choice of materials to maximize the performance
of the design subject to the constraints imposed on it, so that we will try to
make P ! Pmax .
We will further assume that these three factors are SEPARABLE, so that the performance
equation can be written as:
If this is true, then maximizing performance will be accomplished by independently
maximizing the three functions fn1, fn2, and fn3.
fn1 is the place where creative design comes in.
fn2 is where geometry can make a difference.
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fn3 is the part we're most interested in. When the factors are separable, the materials
selection doesn't depend on the details of fn1 or fn2! This means we don't have to know
that much about the design to make intelligent materials choices.
Our first step in this class will be to maximize performance by only considering fn3
(selection of materials without shape effects). Later on we'll look at adding in the effect
of shape on performance by maximizing the product of fn2 x fn3.
EXAMPLE ONE: Design a light, strong tie rod.
The design requirements are:
•
•
•
•
to be a solid cylindrical tie rod
length L
load F, which may include a safety factor
minimum mass
Let's start by doing this the "old" way:
OLD WAY: PART ONE
1) CALCULATIONAL MODEL to use in the analysis
(pretty simple for this example).
2) We know an equation for the failure strength of a tie rod:
We know F, and we can always look up σf, so we can find the right cross sectional area,
A. In the past, this part has always been made in our company from STEEL, which we
know if a good high strength material, so we can look up in a database somewhere the
σf(steel). Now we know what the smallest area will be:
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3) And now we can find the mass of the rod, the measure of performance:
From our analysis, we can see that by choosing a higher strength steel, we can use a
smaller A and thereby reduce our mass. Our recommendation: use a high strength steel.
OLD WAY: PART TWO
A new engineer comes along and she says "Wait...the design constraint says minimum
mass, and your analysis shows that we can ALSO lower the mass by going to a lower
density material. Let's use a high strength Al alloy instead of steel."
From the materials properties I found that the MASS (Aluminum) / MASS (Steel) = 60%
which means we can get an increase in performance of 1.67 times. Our recommendation:
use a high strength aluminum.
What's wrong with these two approaches? Nothing really. They both rely on established
tradition in the company, and the use of "comfortable" materials. They both also
ASSUME a material essentially at the outset.
ASHBY APPROACH (LINEAR OPTIMIZATION THEORY)
Looking at this list of requirements, we start with
1) CALCULATIONAL MODEL to use in the analysis
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2) Determine the MEASURE OF PERFORMANCE (MOP), P.
In this case, we have been told that the goal is to get a part that has a minimum mass.
NOTE: P is defined so that the larger it is the better our performance; we want to
maximize P. This is our OBJECTIVE FUNCTION. NOTE also that Ashby defines P to
be either minimized or maximized, just so long as you keep track of which one it is. We
could also write the MOP as
3) IDENTIFY the parameters in our analytical model and MOP:
L=
A=
F=
ρ=
4) Write an equation for the CONSTRAINED variables: (we have to safely carry the load
F)
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5) Rewrite the constraint equation for the free variable and substitute this into the MOP:
6) Regroup into the three functional groups fn1, fn2, and fn3
To maximize P we want to choose a material that maximizes the ratio
#!f &
%$ " (' = M = MATERIALS PERFORMANCE INDEX.
NOTE: We don't need to know anything about F, or A, to choose the best material for the
job!
EXAMPLE TWO: Design a light, stiff column.
The design requirements are:
•
•
•
•
slender cylindrical column
length L fixed
compressive load F
minimum mass
1) CALCULATIONAL MODEL to use in the analysis
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2) MEASURE OF PERFORMANCE (MOP), P.
Minimum mass again.
3) IDENTIFY the parameters in our analytical model and MOP:
L=
A=
F=
ρ=
4) CONSTRAINT equation: (no Euler buckling of this column)
where n is a constant that depends on the end conditions, and E is the Young’s Modulus.
(NOTE: There are a number of convenient mechanics equations in the appendix in the
back of Ashby, appendix B, which I will use almost exclusively. You may use any
analytical equations you like as long as you understand them!)
5) Rewrite the constraint equation for the free variable and substitute this into the MOP:
6) Regroup into fn1, fn2, and fn3
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7) PERFORMANCE INDEX = MMAX =
RECIPE FOR OPTIMIZATION
1) Clearly write down the design assignment/goal.
2) Identify a model to use for calculations.
3) Determine the measure(s) of performance with an equation (weight, cost,
energy content, stiffness, etc.)
4) Identify the FREE, FIXED, PROPERTY, and CONSTRAINT parameters.
5) Develop an equation for the constraint(s).
6) Solve the CONSTRAINT equation for the FREE parameters and substitute into
the MOP.
7) Reorganize into the fn1, fn2, fn3 functions to find M.
NOTES:
i) M is always defined to be maximized in order to maximize performance.
ii) A full design solution is not needed to find M! You can do a lot of materials
optimization BEFORE your design has settled into specifics.
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week One
Winter 2011
MATERIALS OPTIMIZATION WITHOUT SHAPE
Reading: Ashby Chapter 5, 6.
RECIPE FOR OPTIMIZATION
1) Clearly write down the design assignment/goal.
2) Identify a model to use for calculations.
3) Determine the measure(s) of performance with an equation (weight,
cost, energy content, stiffness, etc.)
4) Identify the FREE, FIXED, PROPERTY, and CONSTRAINT params.
5) Develop an equation for the constraint(s).
6) Solve the CONSTRAINT equation for the FREE parameters in the
MOP.
7) Reorganize into fn1, fn2, fn3 functions to find M.
NOTES:
i) M is always defined to be maximized in order to maximize
performance.
ii) A full design solution is not needed to find M! You can do a lot of
materials optimization BEFORE your design has settled into specifics.
EXAMPLE THREE: Mirror support for a ground based telescope. Typically these
have been made from glass with a reflective coating--the glass is used only as a stiff
support for the thin layer of silver on the top surface. Most recent telescopes have
diameters in the 8-10 m range, and are typically limited by the mirror being out of
position by more than one wavelength of the light it is reflecting (λ). The design
requirements are that the mirror be large, and that it not sag under it's own weight by
more than 1-λ when simply supported. Since the mirror will need to be moved around to
point it in the right direction, it needs to be very light weight.
DESIGN ASSIGNMENT:
•
•
•
•
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Circular disk shaped mirror support
Size = 2r
lightweight
deflects (δ) under own weight by less than λ.
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MODEL:
MOP: minimize mass
PARAMETERS:
r=
t=
ρ=
δ=
CONSTRAINT EQUATION: (use the helpful solutions in appendix B of Ashby or a
mechanics textbook)
For a simply supported disk under its own weigh, the center deflection is:
(NOTE: The “less-than-or-equal” sign is a good way to identify a constraint parameter.)
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APPLY TO MOP: Solve for the free parameter, t.
CAUTION: m (mass) appears in the constraint equation. We will need to eliminate it as
we plug into the objective function, P.
Since the MOP is minimum mass, Pmax =
1
:
m
E
E 1/ 3
,
or
M
=
; maximizing any of these will
!3
!
maximize our performance. For reasons that will become clear later, it is always best to
use the performance index that comes directly from the optimization analysis, in this
NOTE: This is equivalent to an M =
" E%
case, M = $ 3 '
#! &
1/2
APPLYING PERFORMANCE INDICES TO SELECTION CHARTS
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Use the telescope mirror support as an example. We use Ashby's CHART 1 (E versus ρ).
We could apply PRIMARY CONSTRAINTS and say that, in order for the design to
work, the modulus must be E > 20 [GPa], and the density, ρ < 2 [Mg/m3].
Our selection region will be in the upper left, and we end up with expensive candidate
materials such as CFRP.
" E%
OR: we can use the performance index from above: M = $ 3 '
#! &
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1/2
=
E 1/2
:
! 3/2
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Which gives us a line of slope = 3 on a log(E) versus log(ρ) plot (Chart 1).
How do we plot this on the chart? Start with an X-Y point, say
X = log(density) = log(ρ) = log(0.1 [Mg/m3]) = -1
Y = log(modulus) = log(E) = log(0.1 [GPa]) = -1
Now, for every decade unit in X we go up three decade units in Y (slope = 3).
one unit in X gives X = log(ρ) = 0, which gives ρ = 1.0 [Mg/m3], and
three units in Y gives Y = log(E) = 2, which gives E = 100 [GPa].
This is a line of slope = 3. Ashby helps us out with some guide lines for common design
criteria.
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NOTES:
1) This line connects materials with the SAME PERFORMANCE INDEX for this design
(same value of M).
What are the units of the performance index? It will be different for every design
situation, but for the telescope example:
Let's just use M =
[GPa ]1/2
. It will always be easiest to leave the units of the
3/2
! Mg $
m 3 %&
"#
performance index in the scale units of the plot of materials properties.
Look at our line-- it passes through the point E = 0.1 [GPa], ρ = 0.1 [Mg/m3].
It also passes through the point
E = 100 [GPa,] ρ = 1 [Mg/m3].
This means that all materials on this line will perform the same, and should be considered
as equal candidates for the job.
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2) As we move the line (keeping the slope the same), we change the value of the
performance index (M), and thus the PERFORMANCE of the material in this design. For
example, if we move the line to the lower right to the point
E = 1000 [GPa], ρ = 5 [Mg/m3], then
These materials do not perform as well as the first set of materials with
!
$
# GPa1/2 &
M = 10 #
3/2 & .
# Mg 3
&
#"
&%
m
(
)
3) As we move to larger E and smaller ρ, does M increase or decrease?
Remember, we have to keep the line slope equal to three, or we won't have an equi-index
line.
We want high performance, so we keep shifting the line to the upper left until we only
have a small set of materials above the line -- THESE are the CANDIDATE materials for
this design.
We find a lot of materials that perform AS WELL AS OR BETTER THAN the
composites!
4) As M changes, what does that mean?
so a material with an M = 4 weighs HALF that of a material with an M = 2, but TWICE
an M = 8. By maximizing M, we minimize the mass... just what our design calls for.
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5) We stated earlier that we can use any of these ratios for M:
M=
E 1/2
E
E 1/ 3
or
M
=
or
M
=
.
! 3/2
!3
!
Let's check to see if that makes sense:
E
!3
E 1/2
! 3/2
E 1/ 3
!
CHART:
CHART:
CHART:
SLOPE:
SLOPE:
SLOPE:
UNITS OF M:
UNITS OF M:
UNITS OF M:
The VALUE of the M will be different in each case, the UNITS of M will be different,
but because the SLOPE and the SELECTION CHARTS are the same, the MATERIALS
SELECTED WILL BE THE SAME!
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6) Reality Check Number One:
This optimization procedure has given us the BEST PERFORMING MATERIALS given
our stated objective (measure of performance) and constraint. But are the answers
sensible? How do we know?
Look back at the derivation. All but one of the parameters are known (FIXED) or are
determined by the optimization process (MATERIAL PROPERTIES). To check the
design, it is important to use the materials that have been suggested to determine the
value of the free parameter, t in this case, to see if it is indeed sensible.
For a first check, let's compare the relative thicknesses needed for the different materials
to function in the design:
From the derivation, we know:
Solve for t to get the free parameter as a function of the other parameters in the design:
The relative thickness of two competing materials is given by:
NOTE: The advantage of comparing the relative thickness is that a lot of the design
parameters cancel out, so that we don’t need to know a lot about the details of the design
to look at the relative values of the free parameter.
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For several candidate materials, we have the following property data (obtained from the
Ashby chart #1):
Glass
Composites
Wood Products
Polymer Foams
E [GPa]
100
30
4
0.1
ρ [Mg/m3]
2.2
1.5
0.8
0.2
(ρ / E) [Mg/m3GPa]
0.022
0.05
0.2
2
(ρ / E)1/2 [Mg/m3GPa]1/2
0.148
0.224
0.447
1.4
Compare these materials against a "standard" material; for instance, glass has been
commonly used in this application.
Material
Composites
Wood Products
Polymer Foam
t / t(glass), or how much thicker than glass the mirror must be.
NOTE that all four of these materials PERFORM the same—they have the same value of M and the same
performance (MASS). But, because they have different properties, they have different values of the free
parameter needed to make them work.
7) Reality Check Number Two:
So, we know the relative thicknesses, but what about the actual thicknesses? To find
these, we need to have values for all of the FIXED and CONSTRAINT parameters—we
need to know more about the design. Let's pick some reasonable values:
r=
g = 9.8 [m/s2]
λ=
From the analysis, we know t =
Material
Glass
Composites
Wood Products
Polymer Foam
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t = Mirror Thickness [m]
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WOW! These are HUGE!!! What went wrong?
Two important points here. FIRST, the optimization process tells you the best materials
for the job. It doesn't guarantee that your design will work. It is quite possible that the
design cannot be built to work using existing materials. If this is the case, what are your
options?
GIVE UP, or REDESIGN.
SECOND, the design requirements, calculational model, or constraint equations may be
wrong or too simple to accurately describe the design. Your options are:
GIVE UP, CHECK YOUR ASSUMPTIONS, or REDESIGN.
In this case we know an 8 [m] mirror has been constructed from glass that is only about 1
[m] thick and that it works. How could we redesign to reduce the thickness needed for the
mirror?
One option:
Now the model must change, perhaps to a simple beam like this, or something more
complex.
(NOTE - for the simple beam model shown above, the performance index turns out to be
the same, which yields the same materials for the selection process. Changing to a more
realistic model or design changes the constants in the equations, but not the best choice of
materials. Woooo...cool!)
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Two
Winter 2011
MATERIALS SELECTION OPTIMIZATION
WITHOUT SHAPE EFFECTS -- II
Reading: Ashby Chapters 5 and 6.
EXAMPLE: Materials for Flywheels
DESIGN ASSIGNMENT: Design a flywheel to store as much energy per unit weight as
possible and not fail under centripetal loading.
MODEL: Solid disk of diameter 2R and thickness t rotating with angular velocity ω.
MOP: Maximize energy per unit mass
Kinetic energy of spinning disk:
Mass of flywheel:
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PARAMETERS:
R=
ω=
t=
To complete our list of constraint and materials property parameters, we'll need to look at
the "no failure" constraint. Basically, we'll keep increasing the rotational velocity until
the flywheel comes apart. What is the maximum stress in the flywheel?
Our constraint is that the maximum stress must be less than the yield strength, so
CONSTRAINT EQUATION: rewrite in terms of the free parameters as
APPLY TO MOP:
APPLY TO SELECTION CHART: Given our performance index, we probably want to
use a selection chart like log(σ) versus log(ρ), and look at a line of slope = 1.
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MATERIALS SELECTION: We want to consider materials above and to the left of our
line, as these have larger values of σ / ρ. What materials do we get?
First, let's examine the units of M, and then make a table from data in the selection chart:
MATERIAL
M[ MPa/(Mg/m3) ]
CERAMICS
CFRP
GFRP
Be alloys
Steels
Ti alloys
Mg alloys
Al alloys
Woods
Lead
Cast Iron
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WOW! Why did we not select lead and cast iron in favor of low density materials?
Our design requirement was maximum U/m, which led us to increase ω up to the failure
constraint--a strength limited design.
For lead and cast iron flywheels, the design statement is different. If we just want to
maximize U, then the
MOP =
(The message is: the design statement is critical to getting the right answer!)
Look at the list again:
The best performer is CERAMICS-- BUT, we need to check on the measurement of σf
for ceramics used in Ashby's chart. In the description of the selection chart it says that σf
means:
•
•
•
0.2% offset tensile yield strength for METALS
non-linear stress point for POLYMERS
compressive crushing strength for CERAMICS
The flywheel is in tensile loading, so ceramics are not such good performers. The best
performers are:
CFRP
GFRP
Be
Other alloy systems
To down-select, we need another constraint criterion (COST?). This brings up an
important issue about MULTIPLE CONSTRAINTS, which we'll postpone until a future
time.
End of File.
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W. Warnes: Oregon State University
ME480/580: Materials Selection
Lecture Notes for Week Two
Winter 2011
MATERIALS SELECTION OPTIMIZATION
WITHOUT SHAPE EFFECTS -- III
Reading: Ashby Chapter 5 and 6.
A DIFFERENT EXAMPLE: Spring design
We use springs for storing elastic energy. We usually want to maximize the
energy/volume, or the energy/mass. Stored elastic energy is found from the stress-strain
curve for the material as the work done by the applied stress:
ENERGY/VOL=
Because we are in the elastic region.
This is the area under the stress-strain curve
up to the yield stress, and gives us
[ENERGY/VOL]axial =
Leaf springs and torsion bars are less efficient in storing energy than axially loaded
springs because not all the material is loaded to the yield point, so
[ENERGY/VOL]torsion =
[ENERGY/VOL]leaf =
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In all of these cases, the performance index will be M1-MAX=
Look at the selection chart of modulus versus strength:
BUT!! This time better materials appear to the LOWER RIGHT (increasing σ and
decreasing E).
We find lots of conventional materials for springs (elastomers, steels) but also many
others:
•
•
•
Ceramics- good in compression
Glass- often used in high precision instrumentation
Composites-look interesting
WHAT ABOUT ENERGY/MASS SPRINGS?
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In this case, the performance index becomes M2-MAX =
What do we use for a selection chart? Since the mass is a key consideration in these
spring designs, we want to have ρ represented in both the axes:
Now we can use selection chart 5:
The selection leads us to elastomers, ceramics and polymers, but the metals lose out
because of their high density.
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This raises the question of "how do I know which selection chart to use?"
Two ideas to keep in mind at this point:
1. As already stated, the mass is important, so keeping ρ in the axes is a good idea.
2. We don't have selection charts for the other ones.
But what if we did?
We can construct the other selection plots using the CES software, and the net result is
that, while the selection plots are somewhat different, the materials that pass the selection
are identical.
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Two
Winter 2011
CASE STUDY: NATURAL BIOMATERIALS
Reading: Ashby chapter 12.9
BIOMIMETICS
Natural materials have continually been used by mankind in the development of new
engineering applications. Many natural materials continue into the present day as useful
materials, including wood, bamboo, and natural fibers, such as cotton, hemp, and silk.
Especially recently, as materials engineers have become increasingly facile at building
new materials from the ground up (composites, multilayers and heterostructures,
functionally gradient materials, quantum well structures, etc.), natural materials have
become a focus for developing materials for engineering applications. This field of
research has become known as “biomimetics”—using natural materials as models for
new engineered materials.
Recognizing why biomimetics is such an exciting research area starts by looking at the
materials that nature has developed for its use. In almost all cases, natural materials are
composite, or “hybrid”, materials, often displaying structural features over large range of
dimensional scales. Ashby has collected the physical properties of many natural materials
in his book and in the CES software (which we’ll look at next week). Chapter 12 has a set
of Selection Charts hidden away that focus on natural materials.
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LOW-MASS ELASTIC MATERIALS (Figure 12.13)
This selection chart is used for selecting materials for applications involving stiffness per
unit mass. We’ve already analyzed a couple of different applications which require lightstiff materials under different loading conditions, leading to the three guidelines shown
on the plot.
Let’s put STEEL and Al on the figure, just for reference:
Steel
Al alloy
Density [kg/m3]
7900
2800
Young’s Modulus [GPa]
216
80
Strength [MPa]
1000
500
E
, beating steel by a factor of 3 to 4,
!
and pushing flax, hemp and cotton up pretty high. Woods, palm, and bamboo perform
"
E 1/2 %
very well in bending and buckling $ M =
.
! '&
#
Cellulose is the winner for tensile stiffness: M =
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LOW-MASS HIGH-STRENGTH MATERIALS (Figure 12.14)
Again, natural materials show up nicely on this chart, with silk having the best strengthto-weight ratio.
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ELASTIC ENERGY STORAGE MATERIALS (Figure 12.15)
For this chart, the best materials are those with large values of σ and small values of E: in
!2
the upper left corner. Spring materials are those with large values of M =
, while
E
!
elastic hinge materials are those with large values of M = .
E
What’s the best natural material for springs?
What’s best for elastic hinge applications?
Interesting!
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TOUGH NATURAL MATERIALS (Figure 12.16)
Materials with large values of toughness are at the top of the chart (antler, bamboo, and
bone), good for impact loading.
The criterion for carrying a load safely when a crack is present is shown by the lines of
constant fracture toughness (the dashed lines at 45 degrees.)
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CASE STUDY: IMPLANTABLE BIOMATERIALS
("Biomaterials-An Introduction", J. B. Park, R. S. Lakes, Plenum Press, 1992, and “Biomedical
Engineering and Design Handbook”, Volumes 1&2, Myer Kutz, editor, McGraw-Hill Co. Inc., 2009 )
I. HISTORY
•
•
•
•
•
•
•
•
1860's: Aseptic surgical techniques developed by Lister.
1890's: Bone repair using plaster of Paris.
early 1900's: Metallic plates used for bone fixation during skeletal repair.
Problems with corrosion and failure.
1930's: Development of stainless and Co-Cr alloys. First successful joint
replacements.
1940's: WWII pilots-PMMA shows low bioreactivity, and leads to development
of PMMA as an adhesive and skull bone replacement.
1950's: Blood vessel replacements.
1960's: Cerosium- an epoxy filled porous ceramic used as a direct bone
replacement.
1970's: Bioglasses.
II. DESIGN CONCERNS
1. Material properties (strength, fatigue, toughness, corrosion).
2. Design (load distribution, stress concentrators).
3. Biocompatibility (immune system, toxicity, inflammation, cancer).
Other effects on success rate include surgical technique, patient health, and patient
activity.
Relative importance of these issues changes with time:
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III. A PARTIAL LIST OF BIOLOGICAL APPLICATIONS (from Park)
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IV. BIOPOLYMERS
Five types and three reactivities:
1. WATER SOLUBLE (used in solution for lubrication, improving hydrophyllic
interactions at surfaces, reduce thrombogenesis)
2. HYDROGELS (poly hydroxyethylmethacrylate [PHEMA] used in soft contact
lenses, others used for drug delivery systems)
3. GELS (react in-situ to form soft structures; natural fibrin cross-links to form clots)
4. ELASTOMERS (principally silicones and polyurethanes, PUR)
a. Silicones more bio-inert than PUR and is oxygen permeable
b. Silicones are thermosets, while PUR are thermoplastics
c. PUR can be processed to larger range of properties
d. Silicones: artificial finger joints, blood vessels, heart valves, catheters,
implants (breast, nose, chin, ear)
e. PUR: pacemaker leads, angioplasty balloons, heart membranes
5. RIGID (main ones are Nylons [significant water absorbance issues], PET, PEEK,
PMMA, PVC [external uses], PP. PE [especially UHMWPE in hip and knee
prosthetics as a low friction and wear surface, tricky in a metal-PE wear couple
though][LDPE can’t be autoclaved since Tm is too low so only outside body uses],
and PTFE [bioinert])
The reactivities are described as BIOINERT, BIOERODABLE, and
BIODEGRADABLE.
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V. BIOLOGICAL CERAMICS
Four main types, or classes, determined by reaction rate in body:
1.
2.
3.
4.
INERT
POROUS INGROWTH
BIOACTIVE
RESORBABLE
Initial research on ceramic biomaterials was fueled by an interest in the chemical
inertness of them as a class, but over the past 25-30 years there has been a definite shift
toward the bioactive ceramics.
V.A. INERT BIOCERAMICS
Oxides (chemically stable), Carbon. In general these are characterized as: no change is
found in tissue, or the degradation product is easily handled by the body's natural
regulation process. In inert ceramics, the body tissue forms micron sized fibrous
membranes around the insert, and it is locked into place by mechanical interlocking of
the rough surfaces.
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Typical inert ceramics are:
•
•
•
•
Al2O3 for joint prosthetics, dental applications. Alumina has a very low abrasion
rate, about 10X less than a PE/metal wear surface in joints replacements.
ZrO2 is an alternative to alumina, with a higher fracture toughness and even better
wear resistance.
LTI (low temperature isotropic) carbon for heart valves and coatings on some
prosthetics.
DLC-(Diamond-like Carbon) films because of their stability.
V.B. POROUS INGROWTH BIOCERAMICS
Surface preparation of the ceramics is a critical part of the functionality of the implant.
Made with a porous or roughened surface can allow essentially inert bioceramics to
establish a strong mechanical “bond” with natural tissue by allowing tissue growth into
pores and rough surfaces.
V.C. SURFACE REACTIVE BIOCERAMICS
Small amount of selective chemical reactivity with tissue leads to a CHEMICAL BOND
between the tissue and the implant. Implant is protected from further degradation due to
the reacted "passivation" layer.
•
•
BIOGLASS: Na2O-CaO-CaF2 – P2O5 – SiO2
APATITE: Ca10 (PO4) - 6OH2
Used for small bone replacements (low stress) and as coatings on other inserts to enhance
bonding.
Surface coatings often experience failure due to fatigue of the substrate, and the coatings
are not so good in tension.
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V.D. RESORBABLE BIOCERAMICS
Materials that fill space and are taken up by the body with time, presumably to be
replaced with new bone growth. The goal is to provide a “scaffold” on which new healthy
tissue can grow and eventually replace the implanted ceramic.
Example:
VI. APPLICATION EXAMPLES
Corrosion Issues
Mixed Metals (Galvanic Corrosion)
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Dental prosthetics: the most successful area of application of composite materials in the
bio-applications is in the dental prosthetic area, particularly involving ceramic
composites. (Dr. Kruzic: Research on mechanical failure in dental composites.)
Mechanical Design
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Hip/Knee Prosthetics
Ti shaft in bone fixed by glue (PMMA) or cement. High density Al2O3 ball and socket
joint. Better than Ti on HDPE because no release of metallic and polymeric wear
particles (toxicity).
State of the art: replace the Ti with C fiber reinforced graphite, or with thermoplastic
matrix/carbon fiber composite and protective coating.
VII. FDA REGULATIONS
Biomaterials is one of the engineering areas most involved in government regulation. The
definitions are specific but not always obvious. For instance, an example given in the
Kutz book (vol. 2, p. 22, emphasis added by me):
“How does this affect your morning toothbrushing? When you brush your teeth you are
using a MEDICAL DEVICE—the brush. The brush works in a mechanical manner on
your teeth to remove unwanted material. The toothpaste you use could be a COSMETIC
in that it is applied to the teeth to cleanse. However, if you choose a fluoride toothpaste
you are using a DRUG, since the fluoride is metabolized in the body in order to prevent
tooth decay. If you choose to use an oral rinse to reduce adhesion of plaque to your teeth
before you brush, you are using a MEDICAL DEVICE. The oral rinse loosens plaque
that is then removed by your brushing.”
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Three
Winter 2011
MULTIPLE CONSTRAINTS IN MATERIALS
SELECTION: OVERCONSTRAINED DESIGN I
Reading: Ashby Chapter 7 and 8.
Most design problems are more complex than those examples we've discussed so far.
Let's look at a more complex design:
EXAMPLE
DESIGN ASSIGNMENT:
•
•
•
•
Cantilever beam of square cross section and fixed length L.
Support an end load, F, without failing.
End deflection must be less than δ.
Minimum mass.
MODEL:
MOP: minimum mass:
PARAMETERS:
L:
F:
t:
ρ:
δ:
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Okay…let’s tackle them as if they were each a separate constraint, using the optimization
recipe.
FIRST CONSTRAINT: No failure under the end load, F.
SUBSTITUTE INTO THE MOP:
WHAT ABOUT THE OTHER CONSTRAINT (ON DEFLECTION)?
SUBSTITUTE INTO MOP:
Uh-oh... we've got two constraints, and now we have two materials performance indices
(M) and they're DIFFERENT! What do we do?
This type of design is called an OVERCONSTRAINED design-- That is, we have more
constraints than free parameters. Most materials selection problems are OVERCONSTRAINED. There are several ways we can deal with multiple constraints in the
selection process, by using DECISION MATRICES, MULTIPLE SELECTION
STAGES, COUPLING EQUATIONS, and PENALTY FUNCTIONS.
I. DECISION MATRICES
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Commonly used and presented in other design classes. One version comes from Crane
and Charles (see syllabus for reference).
In simplest form, a matrix is developed with the DESIGN REQUIREMENTS along the
columns and the CANDIDATE MATERIALS along the rows:
I.A.
Materials are rated in a GO-NO GO fashion as either acceptable (a), under-value (U),
overvalue (O), or excessive (E).
PROBLEMS:
1.
2.
3.
Next best (but still not very good) approach is to inject some quantitative measure by
replacing U, a, O, E with numbers 1-5 (increasing is better).
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I.B.
This provides a quantifiable selection criterion, but
PROBLEMS:
1.
2.
To eliminate concern #2, we could add WEIGHTING FACTORS,
I.C.
but this just adds another level of subjectivity. How can you back-up the assertion that the
rigidity is 2.5 times more important than cracking resistance?
One significant improvement we can add here is to use PERFORMANCE INDICES
rather than materials properties. This essentially takes us beyond primary constraints into
the realm of optimization. For each constraint or design goal, we develop an M value to
use as one of the columns:
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I.D.
Crane and Charles convert these to dimensionless numbers (relative values) by dividing
by the largest property value, and then sum these to determine the overall rating of the
material.
This is better (we're selecting based on performance indices) but now we're back to
treating all of these with the same importance. The last act of Crane and Charles is to
apply weighting factors to the performance indices:
I.E.
This is pretty good except that it is STILL SUBJECTIVE because there is no justification
for the weighting factors that are used.
The difficulty with most of the decision matrix approaches is simply this subjectivity.
There are some schemes for improving that, and Dr. Ullman's group at OSU has been
studying the design methodology and has developed an approach that has resulted in a
computer program called the Engineering Decision Support System (EDSS).
http://www.cs.orst.edu/~dambrosi/edss/info.html
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II. MULTIPLE SELECTION STAGES
A second approach to the multiple constraints problem is the use of selection stages. Each
of the constraints is used to develop a performance index as we did in the earlier
example:
M1, M2, M3, ..., Mn.
These are rank ordered in order of importance (uh-oh...) from most important to least. We
use the first performance index on the appropriate selection chart, and select a large
enough (uh-oh...) group of materials to leave something for the other constraints to work
with. Repeat with the other performance indices.
(Why the "uh-oh"s? How do we decide on the rank ordering? Subjective decision. How
do we decide on the number of materials to leave in the pool at each stage? Subjective
decision again.)
II.A. EXAMPLE: Multiple Stage Selection for a Precision Measurement System
(micrometer).
There are several design goals we want to meet with this design:
1.
2.
3.
4.
minimize the measurement uncertainty due to vibrations of the stiff structure,
minimize the distortions of the structure due to temperature effects,
keep the hardness high for good wear properties, and
keep the cost low.
Let's tackle these one at a time-II.A.1. VIBRATIONS
We want to drive the natural frequency of the main structure as high as possible. The
useful approximations give us the natural frequency as
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II.A.2. THERMAL DISTORTION
The strain due to a change in the temperature of the structure is determined by
If we want to know how the thermal strain changes along the length of our structure due
to a temperature gradient, we take the derivative to find
We also know (for a 1-D heat flow approximation) that the heat flux is given by
" d! %
To minimize the thermal distortion $ T ' for a given heat flow, we need to maximize
# dx &
II.A.3. HIGH HARDNESS
We can treat the hardness, H, as a direct function of the yield strength:
II.A.4. COST
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Finally to keep the cost low, we want to maximize
So, to summarize, we have FOUR design goals, each of which gives us a different
performance index:
Minimize vibrations:
Minimize thermal distortion:
Maximize hardness:
Minimize cost:
With the multiple stage selection approach we will take each of these individually and
make a series of selection charts.
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II.B. First Selection Stage: We will use Ashby's chart 1, with a slope of 1, and the
selection area above and left of the line:
We don't want to eliminate too many materials, otherwise, there'd be nothing left for the
other criteria to do.
Rank ordered list of materials that "passed" this selection stage, from highest performers
to lowest:
Ceramics
Be
CFRP
Glasses/WC/GFRP
Woods/Rock, Stone, Cement/Ti, W, Mo, steel, and Al alloys.
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II.C. Second Selection Stage: We will use Ashby's chart 10, with a slope of 1, and the
selection area below and right of the line:
Rank ordered list of materials that "passed" this selection stage, from highest performers
to lowest:
Ceramics
Invar
SiC/W, Si, Mo, Ag, Au, Be (pure metals)
Al alloys
Steel
Notice that there is some overlap between materials that passed the first stage and those
that passed the second. That’s good.
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II.D. Third Selection Stage: We will use Ashby's chart 15, and apply the last two
constraints as primary constraints. We want to search in a selection area in the upper left
of the chart:
Rank ordered list of materials that "passed" this selection stage, from highest performers
to lowest:
Glasses
Steel/Stone
Al alloys/Composites
Mg, Zn, Ni, and Ti alloys/Ceramics
Compare these in a table:
First Selection Stage
Ceramics
Be
CFRP
Second Selection Stage
Ceramics
Invar
SiC/(pure metals)
W, Si, Mo, Ag, Au, Be
Glasses/WC/GFRP
Al alloys
Woods/Rock, Stone,
Cement/Ti, W, Mo, steel,
and Al alloys
Steel
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Third Selection Stage
Glasses
Steel/Stone
Al alloys/Composites
Mg, Zn, Ni, and Ti alloys/
Ceramics
W. Warnes: Oregon State University
The candidate materials that make it through all three stages are STEELS and Al
ALLOYS.
We might want to relax the selection criteria a bit to take another look at ceramic
materials, which appear in two of the lists.
The main advantage of this multiple stage selection process is that the assumptions are
simple and clearly stated regarding the rank ordering of the performance indices. The
disadvantage is that it is still subjective in determining the rank ordering and the position
of the selection lines on each of the charts.
The quantitative approach to multiple constraints combines the decision matrices and
selection stages with coupling equations and/or penalty functions. These are topics we’ll
look at next.
End of File.
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ME480/580: Materials Selection
Tutorial Overview Notes on CES-Edupack
Winter 2011
INTRODUCTION TO CES (CAMBRIDGE
ENGINEERING SELECTOR)-EDUPACK
SOFTWARE FOR WINDOWS
Nomenclature:
Throughout these notes references to buttons or icons that should be clicked will be given
in BOLD and pull-down menu items will be given in ITALICS .
1) Log onto your Engineering account.
2) Once in Windows, open the MIME Apps and start the CES-EduPack 2010 program.
(We still have the older version, CES Selector 3.1, on-line. Don’t use it by mistake!)
INSIDE CES:
You will see the WELCOME screen when you startup, and a “Choose Configuration”
window.
There are three “levels” of material and process database information in this version of
the software:
Level 1: about 70 materials and 70 processes, with a limited set of property data;
Level 2: about 100 materials and 110 processes, with an extended set of property
data;
Level 3: about 3000 materials with a comprehensive data set for each.
1) For now, choose “English -- Level 1” until you are used to the program. Later on we’ll
switch to Level 3 to use all the information for doing problems and the design project.
2) You should now be in the main program control window. At the top on the left, you
should see the database you are using, along with a pull down menu for the TABLE
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(MaterialUniverse), and SUBSET (Edu Level 1). You will also see a toolbar with several
icons along the top of the main window. These are how you will interact with CES.
CES INFORMATION:
There is a large amount of on-line help and database information available in CES.
1) You should automatically be in the browsing tool, but if not click on the BROWSE
tab to see the information in the materials database.
2) Double click on a folder to open it. Eventually you'll work your way through the
hierarchy to an individual material record. Take a look at the materials record. This is the
database information that has been developed for each material in the database.
3) You can change the database to browse by choosing a different TABLE or SUBSET
from the pull-down menus. Try several different tables to see what they offer.
4) You may also change databases by clicking on the CHANGE button. If you change to
Level 3, you will find SEVEN different TABLES, and a larger number of SUBSETS.
Within the Level 3 MaterialsUniverse, for example, there are a number of SUBSETS,
including All Bulk Materials, Ceramics, Foams, Magnetic Materials, Metals, Polymers,
and Woods. You might use these to narrow down a selection process to a smaller class of
materials.
5) You can also SEARCH the database using the SEARCH button. ‘Nuf said.
6) Reference material is also available on-line, as well as an on-line help function. Click
on the HELP button or the menu item. The "CES InDepth" is an on-line reference book
about CES and the selection process we have been using in class. In fact, all of the
appendices form the textbook can be found in here (if you know where to look!)
7) There are also video tutorials and getting started guides that you can access if you want
to learn more about the capabilities of the program.
8) For the last thing to do on this part, click on the TOOLS button and select OPTIONS.
Click on the UNITS tab to set the preferred units of the data. Choose the currency you
want to use for cost analysis here. This also allows you to set the units for the selection
charts. Choose “SI (consistent)” for the unit system. (HINT: using USD [$] instead of
Myanmar Kyat would probably be a good idea).
MAKING A SELECTION CHART (the cool stuff):
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1) Click on the SELECT tab to start. (Alternatively, you can choose the NEW PROJECT
menu item in the File Menu.) You will need to choose a database and subset to use in the
selection project.
2) Now click on the NEW GRAPH STAGE icon on the toolbar or from the SELECT
menu. (The toolbar buttons are, from left to right, NEW GRAPH STAGE, NEW LIMIT
STAGE, NEW TREE STAGE.)
3) You should get a window with the "Graph Stage Wizard" title. Make sure that the XAXIS tab is selected, and then use the ATTRIBUTES pull-down menu to choose the
material property to plot on the x-axis. Choose YIELD STRENGTH (ELASTIC LIMIT)
from the pull down list.
3) Click on the Y-AXIS tab to set the material property for the Y-axis.
4) Select YOUNG'S MODULUS from the ATTRIBUTES menu.
5) Click OK.
6) You should now have a new window labeled "Stage: 1" with a graph of your selection
chart, showing on the right side of the screen, along with a new tool bar row with about
16 icons on it.
CHANGING AND USING A SELECTION CHART:
1) Click on the STAGE PROPERTIES icon (the first icon on the left of the new tool
bar.)
2) You can now change the axes of the active stage. Change the SCALES to be LINEAR
in both X and Y. Click OK. Now you know why the data is usually plotted on a log-log
plot.
3) Click on any bubble on the chart to find out what the material is. Drag the pop-up label
around, and it should leave a connecting line behind pointing to the bubble. Doubleclicking on a bubble brings up the materials data sheet for that material.
4) Delete the label by selecting it with the mouse and pushing the DELETE key.
5) Change the axes back to log-log.
6) There are three types of selection tools you can use: point-line, gradient-line, and box.
These are the icons that follow the CURSOR icon.
7) For simple or primary constraints, you should use the BOX selection tool. Click on the
BOX button. Then click on a point in the selection chart and drag the mouse to enclose a
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set of materials in the box. Note that the STATUS BAR (at the bottom left of the screen)
gives you the X,Y location of your cursor in the plot units. Note also that any material
bubble that is partly inside the selection box is colored, while the others are greyed-out.
The colored bubbles have been selected by the selection process, and now show up as a
list in the RESULTS section on the left side.
8) Now click on the GRADIENT-LINE selection tool button. Type in the slope of the
selection criterion (line slope) you want (use 1 for now), and click OK.
9) Click on some X-Y position to position the line, and the line will be drawn for you at
that location. Notice that the STATUS BAR shows you the value of the selection
criterion for the line position you have chosen, along with the value of M for that line (be
wary of the units, though!). The final step is to click either ABOVE or BELOW the line
to tell the program which region is the selection region. Again, colored materials have
passed this selection, and greyed materials have failed.
10) Moving the cursor onto the selection line allows you to reposition the selection line
for higher or lower M values. If you want to change the slope, you can start over by
clicking the GRADIENT-LINE selection tool button again.
11) Note that you may have only ONE selection criterion operating at a time on a single
selection chart. If you want more than one criterion for a particular set of x-y axes, you
need to make-up additional STAGES with the same axes and apply the other selection
criteria to those.
12) The RESULTS section in the left of the window shows you the materials passing
your selection criterion. You can modify the results section by using the pull-down menu
to choose what results to view. This is especially helpful when using multiple stages.
13) Finally, you can save this set of selection criterion to disk and recall it later using the
SAVE PROJECT menu item. In the FILE menu
A MULTIPLE STAGE EXAMPLE:
We want to do a materials selection for a high quality precision measuring system,
essentially a top line micrometer (we did this one in class as our example of a multiple
stage selection process). After extensive analysis, we have found that we need a material
that will produce a LOW THERMAL DISTORTION (M1 = λ / α), LOW VIBRATION
(M2 = (E / ρ)1/2), maximize the HARDNESS (M3 = H), and minimize the cost
(M4 = 1/C ρ).
1) First you will need to start with a clean project. In the FILE menu click on the NEW
PROJECT item. We will use the “EduLevel 1: Materials” database for this example.
Make sure this is set up in the selection data section.
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2) Stage 1 will deal with M1: Click NEW GRAPHICAL STAGE selection, and in the
X-Axis properties choose the THERMAL CONDUCTIVITY property. For the Y-Axis
properties choose THERMAL EXPANSION COEFFICIENT, and click on the OK
button.
3) Now click on the GRADIENT-LINE selection tool button. Type in the slope of the
selection criterion you want (use 1), and click OK. Locate the point for λ = 10 [W/m-°C],
and α =1 X 10-6 [1/°C]. (Remember that you can use the Status Bar at the bottom of the
window to tell you the X-Y position of the cursor.) Click BELOW the line (since we
want large λ and small α).
4) Now that you have a selection criterion on the graph, click on the STAGE
PROPERTIES icon. A new tab is available that lets you change the details of your
selection- slope, side of the line, and exact location! Use this to place your selection line
in exactly the same position that I have used (X = 10, Y = 1). If you’ve done everything
the same as I have, you should see SEVEN candidate materials in results list.
5) Stage 2 will deal with M2: Click NEW GRAPHICAL STAGE, and in the X-Axis
properties choose the DENSITY. In the Y-Axis properties choose YOUNG'S
MODULUS, and click on the OK button.
6) Now click on the GRADIENT-LINE selection tool button. Type in the slope of the
selection criterion you want (use 1), and click OK. Locate the point for E = 2 x 109 [Pa],
and ρ = 100 [kg/m3]. Click ABOVE the line (since we want large E and small ρ).
7) If you click on the STAGE PROPERTIES button while in Stage 2 you can choose to
turn off or on the display to show the RESULT INTERSECTION, those materials that
have passed all the stages so far. If you only want to see the materials that pass, choose to
HIDE FAILED RECORDS. (I don't recommend this at the beginning!). Your results list
should show SIX materials now that pass both selection criterion.
8) Stage 3 will deal with M3 and M4: Click NEW GRAPHICAL STAGE, and for the XAxis properties we have to do something fancy. There is not a property listed for COST,
but there are properties PRICE [USD/kg] and ρ [kg/m3]. First, for the x-axis, click on the
ADVANCED button. You should see a hierarchical list of all the materials properties
available. Click on the GENERAL PROPERTIES in the pull-down menu and you will
see a list of the general properties. By choosing properties from the list and using the
math function buttons, you can set up quite complicated materials selection axes. Wow!
Isn't this cool? Select PRICE and multiply it by DENSITY to get the X-axis to be the
[USD/volume] you need for minimum cost design. Click OK. We should also change the
name of the axis to at least include the UNITS!!!! (something like MATERIAL COST
[$/m^3]) so we know what we are looking at in the selection chart.
9) In the Y-Axis properties choose HARDNESS-VICKERS, and click on the OK button.
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10) Now click on the BOX selection tool button. Use the box to select the materials with
a MATERIAL COST less than 10,000 [USD/m3], and a HARDNESS greater than 1 x 109
[Pa].
11) Go to the RESULTS window and check your results. You can view the selection
criteria you have used here, as well as the materials that have passed each stage. If you
have done this problem the same way I have, you will end up with three materials
passing: Al Alloys, Silicon, and Silicon Carbide.
NOTES:
You may only search one database at a time. To change databases:
1) Click the CHANGE button in the “Selection Data” section and select the database you
want to search.
2) Then choose the subset of materials you want to “Select From…” You can fiddle with
this, for instance, by choosing to look only at ceramics or metals.
Once you have developed a selection stage, changing databases does not change your
selection stage(s) or selection criteria. CES will automatically run through the selection
process using the new database whenever you change databases. It's easy to search the
other databases this way. My advice is to always start off with the ALL BULK
MATERIALS subset in the LEVEL 3 database, and use the others as your design
develops.
(If you do this now, you should have 14 candidate materials from the three stage
selection, using the Level 3 database.)
End of File.
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ME480/580: Materials Selection
Lecture Notes for Case Study
Winter 2011
CASE STUDIES IN MATERIALS SELECTION:
POLYMER FOAMS
("Polymeric Foams", Klempner and Frisch, 1991; "Plastic Foams" Frisch and Saunders, 1973)
Why look at foams? EXAMPLE: Simply supported beam in bending- minimum mass (or
cost). Assume b, L, are fixed, h is free, and the center deflection under load, F, is limited.
Use Rule of Mixtures to determine foam properties, e.g. 90% air foam:
SO…this means the foam material, which is 90% nothing with no properties, has a
performance nearly FIVE TIMES the solid polymer beam ( or, looked at another way, for
hf = 2hs you can get the same deflection with 80% less mass!).
One can also laminate the surface of foams with a high strength layer to drive
strength/weight ratio even farther up.
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Foams also have energy absorption properties due to the compressibility of the gas in the
cells.
New materials class—Foamed metals (Al and steel) behave exactly the same way! Foams
are nifty!
TYPES OF POLYMER FOAMS:
•
•
•
•
Gas-dispersed foams, using "blowing agents"
Syntactic foams, using hollow spheres of glass or plastic.
Open-cell vs. closed-cell.
POLYURETHANE FOAMS:
Most widely used. Depending on chemistry can vary their properties from flexible
cushions to rigid foams for structural applications, with density ranging from 0.0096-0.96
Mg/m3.
Can be made in a continuous process as a "bun" 2-8 feet wide X 1-5 feet thick X 10-60
feet long.
Can be processed as "integral skin" foams.
POLYSTYRENE FOAMS: Also very widely used in the form of extruded blocks. Formed
by:
1. Force volatile liquid (neopentane) into crystalline spheres of PS (ρ ~ 0.96 Mg/m3)
2. Pre-expansion done with steam, spheres expand to 0.016-0.16 Mg/m3.
3. Final-expansion in a mold with steam heat, spheres fuse together.
ABS FOAMS: Used in pallets, and as structural material in furniture.
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SYNTACTIC FOAMS: Use hollow microspheres (30 micron diameter) of glass, ceramic,
or plastic for difficult to foam materials, such as epoxies.
ARCHITECTURAL USES OF FOAMS:
Besides insulating properties (PUR foams among the lowest thermal conduction
materials), can also be used as a primary structural material, as in this University of
Michigan study from the late 60's.
Major controlling factor: keeping within small elastic and creep deformation limits.
Looked at double-curved shells. Several different approaches:
POLYSTYRENE SPIRAL GENERATION
POLYURETHANE SPRAY APPLICATION
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FOLDED PLATE STRUCTURES WITH POLYURETHANE/PAPER BOARDS
FILAMENT WINDING ON PUR BOARD
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Four
Winter 2011
OVERCONSTRAINED DESIGN PART II:
ACTIVE CONSTRAINT METHOD
Let's look at an example with a single design objective (MOP), but several constraints, an
OVERCONSTRAINED problem. One way to approach it is to use a multiple selection
stage process, as we did for the precision micrometer example in the last lecture. A
difficulty with the approach is the ordering of the constraints and selection stages, and the
subjective placement of the selection line in the multiple stages. A more systematic
approach uses the "active constraint" approach. An example:
EXAMPLE: The support rod for an infrared-electronics cooling cryogenic fluid container
in a spacecraft is to be designed. The most important characteristic of this tie rod is that it
should carry a minimum amount of conductive heat into the cryogenic container. The
conductive heat flow equation tells us that the conductive heat flow along this support rod
is:
where C is a constant (the temperature gradient), λ is the thermal conductivity of the rod,
and A is the cross sectional area of the rod.
There are three constraints on the rod:
First, that the loading due to the mass of the cryogenic fluid and container should not
exceed the failure strength of the tie rod (ignore the mass of the rod).
Second, the deflection, δ, should be less than a critical value, δmax.
Third, the vertical frequency of vibration must be high enough to not affect the measurements being made. In other words, f should be larger than a critical frequency, fmin.
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MODEL:
Assume a SOLID CYLIDER for the rod, A = ! r 2 .
MOP: minimum heat flow into the cryogen:
PARAMETERS
L=
λ=
A (or r) =
δmax =
fmin =
F=
q=
PERFORMANCE EQUATION ONE : Start with the load constraint:
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PERFORMANCE EQUATION TWO : Now use the deflection constraint:
PERFORMANCE EQUATION THREE : And finally the vibration constraint: For a
vibrating rod with a mass at the end, the fundamental (lowest) frequency is
where K is the elastic stiffness, given by
and m is the mass of the cryogenic container, mc. Then
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To perform the multiple selection stage process, we would set up two stages, one for M1
and one for M2, M3.
For the active constraint approach, we have to know more about the design, especially the
details of the values of the fixed and constraint parameters. First, write out the equations
for the MOP using each of the constraints:
PMAX(M1)=
PMAX (M2)=
PMAX (M3)=
If we know, or can estimate, the values of the fixed and constraint parameters, we can
calculate numerical values of the measures of performance for each material. Let's put
some numbers down for this design:
F (= mc*a) = 196 [N]
mc = 20 [kg]
δmax = 0.01 [m]
fmin = 100 [Hz] = 100 [1/s]
L = 0.1 [m]
C = temperature gradient = (300 [K])/(0.1 [m]) = 3000 [K/m]
Now we can set up a spreadsheet table of values for the material properties of the
materials we're interested in and calculate the measures of performance. My spreadsheet
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in EXCEL looks something like this, and I pulled the rough values from the Ashby
selection charts:
For each individual material, we look at the SMALLEST value of P. WHY?
In order to satisfy all the constraints, we must satisfy the one that most limits our
performance. If we can satisfy that one (by choosing a particular value of r, the free
parameter), we will satisfy all of them,
In this example, the minimum performance for all of the materials is the vibration
constraint-- it is the ACTIVE CONSTRAINT for all of the materials we have examined.
If we don't satisfy it, the design will fail.
Now, we can pick the material with the LARGEST value of the active constraint
performance (P3 in this example) to be the optimal performer for the design, in this case
CFRP.
What have we learned by going through this active constraint analysis that we didn't
know before?
1) To become more objective and quantitative in the selection process for
OVERCONSTRAINED designs, we need to know more detailed information about the
design.
2) It's a lot more work and time to do all of the quantitative calculations, but...
3) We now know what the limiting constraints on the materials are. With the spreadsheet,
we can play some "what if" games with the fixed parameters-- how do the P's change if
you decrease the cutoff frequency, or increase the mass, or allow less deflection? These
trade-offs can be used to tune up the design and go back to your boss/client with a
quantitative reason to consider changing one of the fixed parameters. As the values of
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these parameters change, at some point one of the other constraints will become the
active constraint for a given material.
The Last Step: REALITY CHECK: Let's plug back into the constraint equations to find
the value of the cylinder radius in each case:
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Four
Winter 2011
OVERCONSTRAINED DESIGN PART III: COUPLING
EQUATIONS
Consider a design situation in which we have one design goal (MOP), two constraints,
and one free parameter. We are over-constrained in this situation.
By calculating a performance index analysis using the first constraint, we end up with:
With the second constraint, we have:
Now, the MOP is the same, so we can equate these two (we only have one design, which
will perform at a given level, P):
The relative weighting of the two performance indices is DETERMINED BY THE
DESIGN and not by our subjective judgments!
III.A. EXAMPLE: A Light Tie Rod
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III.A.1. DESIGN ASSIGNMENT:
•
•
•
•
cylindrical tie rod of length L
minimum weight
support a load F
extension less than δX
III.A.2. MODEL: Same as before
III.A.3. MOP: minimum mass:
III.A.4. PARAMETERS
L=
F=
δX =
A=
ρ=
III.A.5. PERFORMANCE EQUATION ONE (derived previously in the example from
Week One)
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III.A.6. PERFORMANCE EQUATION TWO
III.A.7. DEVELOP THE COUPLING EQUATION:
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Result
The best performing material will be one in which E/ρ is maximized, σ/ρ is maximized,
and their ratio is held at L/δX.
How do we apply this to a selection chart? We want to use a chart for this example like
chart 5.
We want both of our performance indices to be maximized, so we'll be looking at
materials in the upper right hand corner of the chart. For our particular design, we'll have
a given value of L/δX that is determined by the constraints of the design. Lets say it is
100.
We will look at a straight line of slope 1 on the plot, and we want the line for which the
ratio of the performance indices is 100
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By moving along this line of constant L/δX, we improve our performance by increasing
the values of the performance index, and we simultaneously maintain the weighting
factor determined by the design.
Our best choice of material is...
We probably want to open up the search region a bit, to allow some materials other than
diamond in the mix, so for this case we can use a rectangular search region centered on
the line of L/δX = 100.
By moving away from the line, we shift toward STIFFNESS DOMINATED designs (to
the upper left) or toward STRENGTH DOMINATED design (to the lower right).
NOTE: If you use coupling equations, you don't need to use multiple stage selection
processes, but you may have to generate your own Ashby Selection Charts!
MULTIPLE CONSTRAINT DESIGN: FULLY
DETERMINED DESIGNS
Look back at previous lectures -- we had an example of OVERCONSTRAINED design
with the cantilever beam. We had two constraints (no failure under an end load, F, and
deflection less than δ), and only one free parameter (square cross section of t X t). We
ended up with two materials performance indices, M1 and M2, which we could use in a
two-stage selection process.
Alternatively, we could use a coupling equation to couple the two M values together and
do a one stage process, as just described (check out HW3...).
A last possibility is to revise the design statement to increase the number of free
parameters. This will give us two free parameters and two constraints--fitting the
definition of a FULLY DETERMINED design.
EXAMPLE: Light cantilever beam
DESIGN ASSIGNMENT: Let's change it slightly, from a square beam of t X t to a
rectangular beam of b X h.
•
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Cantilever beam of rectangular cross section and length L.
W. Warnes: Oregon State University
•
•
•
Support an end load, F, without failing.
End deflection must be less than δ.
Minimum mass.
MODEL:
MOP: minimum mass:
PARAMETERS:
L:
F:
b:
h:
ρ:
δ:
We've got two free parameters and two constraints!
CONSTRAINT ONE:
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CONSTRAINT TWO:
SOLVE FOR b AND h:
SUBSTITUTE INTO MOP:
This type of design is called FULLY DETERMINED design. We can get a complete
solution (with one M value), because we have the same number of free parameters as
constraints.
MULTIPLE CONSTRAINT OPTIMIZATION
(A GENERAL DESCRIPTION)
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The first part of the optimization process is writing out the following:
1. Measure(s) of Performance: quantitative functions to maximize the relative
success of different designs. (P)
2. Constraining Equation(s): functions that set acceptable limits on the behavior of
the design in use. (C)
3. Design-fixed Parameters: parameters that appear in the P and/or C equations that
are not changeable under the conditions of the design. (D)
4. Free Parameters: the additional parameters from P and/or C that are not fixed.
(F)
There are several possible scenarios:
I.A. SINGLE MOP DESIGNS
For designs with a single measure of performance, we can imagine several possibilities:
I.A.1. Zero Free Parameters
This is a pretty unusual situation, but it is conceivable when an existing
design is to be used and only requires a change in material. Not a lot of
opportunity here for optimization.
I.A.2. One Free Parameter
I.A.2.a. One Constraint Equation (1C1F)
With one C and one F we are FULLY DETERMINED, and the
constraint, C, is applied to the measure of performance, P, through
the free parameter, F, to develop a single performance index, M.
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I.A.2.b. Two Constraints (2C1F)
Now the design is OVERCONSTRAINED. We treat each
constraint separately as in the 1C1F case. In so doing, we end up
with two performance indices:
Since we still have only one P, these two functions can be equated
to find a coupling equation (or a relative weighting factor) of
M1 / M2:
I.A.2.c. Three (and more) Constraints
The design is definitely overconstrained. We start the same way
we did for the 2C1F design:
Using the pairs of performance index functions, we can determine
the relative weightings of these M's.
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I.A.3. Two Free Parameters
I.A.3.a. One Constraint (1C2F)
In this case the design in UNDERCONSTRAINED. We need to
find a way of either fixing one of the parameters, or come up with
another constraint. In some cases we may be able to change
variables to reduce to 1F.
EXAMPLE: A minimum mass connecting rod of rectangular cross
section with heat flow larger than some value qo.
Convert from 2F to 1F using A = bh, since both constraint and
MOP depend only on the area A.
I.A.3.b. 2C2F
Fully determined design. Solve the two constraining equations for
the two unknowns (F1, F2) and plug into the P.
We end up with ONE performance index, M.
I.A.3.c. 3C2F
Overconstrained, and can be treated as three independent 2C2F
problems:
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Since we are still talking about 1P designs, we can generate
coupling equations here as well:
I.A.4. General Results for 1P Designs
C<F
C=F
C>F
Underconstrained; need to add a constraint.
Fully determined; one performance index, M.
Overconstrained; multiple M's, coupling equation(s).
I.B. MULTIPLE MOP DESIGNS
The first step will be to rank order the P's. Remember how to determine
whether you are dealing with a P or a C:
•
•
If the feature is to be MINIMIZED or MAXIMIZED, then it is a P.
If the feature must be GREATER THAN or LESS THAN a
reference value, then it is a C.
With a rank ordered list of the P's, we can treat each one separately as a
single MOP problem:
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NOTES:
1. The DESIGN will generally have a single set of constraints that
can be applied to all of the P's, but the free parameters may be
different in different P equations. In this example, P1 depends only
on F1 and F2, while P2 depends on F1 and F3.
2. If a constraint equation doesn't involve any of the F's in a particular
P, then the constraint can't be used to optimize this measure of
performance. In this example, C3 does not apply to P1.
3. You can't get coupling equations between M's determined from
different P's. In this example:
We can form coupling equations by coupling the three P2 equations, but
we can't find a coupling equation relating M(2)12 and M(1)12 because P1 is
not equal to P2.
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WHAT NEXT?
What do you do with all of these performance indices and coupling
equations? Two choices:
Rank order the performance indices by order of importance and perform a
multiple stage selection process, or;
Get more information about the design and determine the active constraint
for each material in a tabular matrix, or:
Set up a decision matrix based on the performance index values for each
material. the decision matrix can be rank ordered, or can be set up with
weighting factors as determined from the coupling equations.
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Five
Winter 2011
CALCULATING MASSIVELY OVERCONSTRAINED DESIGNS
It is often a good idea at the beginning of a design project to figure out how intense the
analysis is going to be by calculating the total number of performance indices and
coupling equations you are likely to end up with. Just a quick word about how to do this
as a combinatorial problem.
EXAMPLE: You have a design with
•
•
•
ONE measure of performance,
THREE free parameters, and
EIGHT constraints.
To find a materials performance index, we need to have a FULLY DETERMINED
design, so we'll want to take three of the eight constraints at a time to solve for the three
free parameters. How many combinations of the eight constraints do we have in sets of
three?
Now, the problem with this counting is that it counts the combination of constraint 1+2+3
as different from the combination of 1+3+2 and 3+2+1. We need to divide the total by the
number of combinations of three constraints (in any order) that we can have. This
overcounting factor is found by:
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So, the total number of UNIQUE combinations of the eight constraints in sets of three is
56! WOW...that's 56 M-values in this problem!
BUT WAIT...THAT'S NOT ALL! How many combinations of the M-values can we have
in groups of two to create unique coupling equations do we have? Using the same
process, we get:
That's a LOT of coupling plots to make up...even with CES.
Here is a plot showing the rapid increase in the number of M-values and coupling
equations with the number of constraints for a single MOP, four free parameter design.
Ouch!
What can we do to make this better? One choice is to do what I told you not to
do...change one of the constraints into a measure of performance (for example, instead of
having a maximum allowable cost, set up a minimum cost measure of performance).
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This changes the problem. We now have a design with
•
•
•
TWO measures of performance,
THREE free parameters, and
SEVEN constraints.
Using the combinatorial calculations above, we would have (for each of two MOPs)
This is still pretty ugly, but is a lot more tractable than the original problem. Of course,
we've replaced the original problem with needing to (subjectively) determine which MOP
is the most important.
The main message is that, when you are massively OVER-CONSTRAINED it is best to
try to reduce the number of constraints you have to being only one or two larger than the
number of free parameters you have, and the way to do this is to turn some of the
constraints into MOPs.
REMEMBER: If you have the same number of free parameters as constraints, you will
have only ONE M-value, no matter how many free parameters you have!
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CES AND COUPLING CHARTS
You may have wondered by now how to do the coupling charts in CES and have CES tell
you the best materials in the RESULTS section. The answer is that you have to trick it,
and here's how.
FIRST: Make up a coupling chart the
way that you usually do, with M1 on the
Y-axis and M2 on the X-axis. This will
be STAGE ONE of a multiple stage
selection process. Find the correct
location of the coupling equation line
(slope of one and at the correct location
for the design parameters) and set the
line on the chart at the correct location.
This is shown schematically in the
figure to the right with the correct value
of the coupling constant shown as the
dotted line. The solid line is slightly
offset above the correct position for (NOTE: The position of the coupling line can
clarity in the last couple of plots.
be EXACTLY placed on the selection chart in
CES by looking (in the "Project" Menu) for
Now, click BELOW this line so that "Stage Properties". The dialog box has a tab
CES selects all the materials that touch for "Selection" that allows you to enter exact
the line or are below it.
values for where you want the line placed.)
SECOND: Make a COPY of the first
selection stage by clicking on it in the
PROJECT window, choosing COPY,
and PASTE. This will make an identical
version of your first stage as STAGE
TWO. In stage two, choose the
selection region to be the area ABOVE
the line.
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By setting the conditions of Stage Two
to show only the subset of materials that
have passed BOTH stage one and two,
you will see only those materials that
touch the selection line (shown
schematically to the right). The
RESULTS window will also list only
these materials.
To select the BEST MATERIALS,
make one more copy of the original
stage, setting it as STAGE THREE. For
this stage, use a selection line that is
approximately at right angles to the
coupling equation line and select the
region ABOVE the line to be the active
region. By moving this selection line up
and down you can pick off the materials
that give you the MAXIMUM values of
M1 and M2, AND are on the coupling
line. The RESULTS window now lists
the materials passing all three stages.
You can move the third stage selection
back and forth to determine the rank
order of the materials as well.
If you have a design that has more than one coupling equation, you will have to make a
number of selection stage sets, three stages for each coupling equation. If you use the
copy and paste functions, though, this is not too tough to do, and you only have to fiddle
with the selection line in the third stage of each coupling equation set.
End of File.
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ME480/580: Materials Selection
Lecture Notes for Week Five
Winter 2011
DEALING WITH CONLFICTING OBJECTIVES
Reading: Ashby Chapters 7 and 8.
So far we have talked about the myriad techniques for dealing with overconstrained
designs, using the active constraint method, or the coupling equation approach. In all of
these designs, we have been very careful to define only one measure of performance. But,
there are some design situations in which you find yourself with two or more design
objectives; multiple measures of performance. In many cases, these multiple objectives
are conflicting; you can’t satisfy them both with the same material. This requires a
different approach taken from optimization theory called TRADE-OFF PLOTS, and
PENALTY FUNCTIONS.
NOTE: The biggest difference in our process and thinking from what we have done so far
is that the objective function must be defined such that we want to MINIMIZE it in order
to get the best performance. I have been careful to require everything to be defined in
terms of MAXIMIZING PERFORMANCE, but for this type of optimization analysis, we
need to define a minimizing function that will maximize performance. Ashby calls these
objective equations P, as before, so we just need to be careful that we know whether the P
requires maximizing or minimizing to bring success.
A SIMPLE EXAMPLE: We’ll go back to a previous problem–the simple cantilever. The
design statement has been:
•
•
•
•
MINIMUM MASS (our measure of performance),
Fixed length, L,
Square cross section, b X b,
Not fail plastically under end load F.
This is a FULLY DETERMINED design, and the only change we need to make from
previous analysis is that the measure of performance, Pmin, will be a minimizing function:
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Going through the usual routine with the constraint gives us the following expression for
the measure of performance:
In the normal analysis, we’d want to pull out the materials selection index, in this case,
and then use it to make a selection plot. (NOTE: Do we want to find materials that have
large or small values of M?)
For our design, we are also told that we must MINIMIZE THE COST. This is clearly a
second design objective, and the chance that the material that minimizes mass will also
minimize cost is pretty slight. Here we have a case of multiple and conflicting objectives.
We will go ahead and analyze the design using the second constraint, which we write as
where C is the material property of “Price”, having units of [USD/kg]. Pushing through
with the analysis (using the load constraint) gives us the following result for P:
Two objectives, two M-values, can we couple them? NO! The P’s are different, so we
can’t set them equal to find a coupling equation.
To proceed, we need to know more about the design. As in other complex designs, we
need to know the values of the fixed parameters to carry on. Let’s assume
F=
L=
Then the constant factor in the first measure of performance is
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Plugging this into the measure of performance to be minimized, we get:
In this case, the constant factor in the second measure of performance is the same, so our
second measure of performance to be minimized is:
We can use CES to make a plot for us of the two MOP’s, mass and cost. Using the
ADVANCED axis option, we can write out the equation
m = 113
!
= 113 * [Density] / [Yield Strength (Elastic Limit)] ^ 0.6667
" 2/3
f
Now the Y-axis will be the MASS of the beam, in [kg]. Similarly, we can set up the Xaxis to be the COST of the beam in [USD]. Schematically, the plot looks like this:
Okay…time for some optimization theory terminology. Each of the bubbles on this plot
is called a SOLUTION, because it represents, for a particular material, the cost and mass
of a beam that will satisfy the constraint.
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Look at the bubble labeled “A”—we can see that there are many other solutions that have
either a smaller value of mass or a smaller cost, represented by the vertical and horizontal
lines on the plot. The materials between the lines have BOTH a smaller mass and lower
cost. A is said to be a DOMINATED solution, because there is at least one other solution
that outperforms it on BOTH performance metrics.
The B bubble, on the other hand, is a NON-DOMINATED solution, because there are no
other solutions that have both a smaller mass and a lower cost. But is B the OPTIMUM
solution?
Looking at the plot shows that there are, in fact a whole variety of solutions that are nondominated. We can draw a line through them all and we arrive at a boundary, which is
called the TRADE-OFF SURFACE, along which all the non-dominated solutions lie.
We can, at this point, use our expertise or intuition to choose the best materials from all
of the candidate, non-dominated, solutions, but there must be some quantitative way of
dealing with this. The answer is to develop PENALTY FUNCTIONS.
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ME480/580: Materials Selection
Lecture Notes for Case Study
Winter 2011
CASE STUDIES IN MATERIALS SELECTION:
SHIPBUILDING
(REFERENCE: "Brittle Behavior of Engineering Structures", E. R. Parker, John Wiley
and Sons, NY, 1957.)
There are two basic parts to a
ship- the hollow HULL, and
the SUPERSTRUCTURE.
The hull is subjected to two
forces:
1) gravity due to the
mass of the ship and
the
cargo,
and
2) buoyancy of the
hull.
While these forces balance,
they are not always uniformly
distributed, and can be
strongly affected by cargo
loading.
For shorter cargo ships,
"HOGGING" is common, as
the buoyancy in the center is
larger per unit length than it
is at the ends.
Longer ships tend to "SAG", even in still water, but the worst case comes from riding the
waves.
The hull is subjected to a large bending moment, and so tends to fail in panel buckling.
The superstructure is used as a panel stiffener to prevent hull buckling.
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We can analyze this using simple beam bending:
The first performance measure will be to minimize the mass of the ship:
subject to the constraint of no failure:
The second performance measure is to minimize the deflection subject to the failure
constraint:
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Since these are two DIFFERENT MOP's, we can't generate a coupling equation. Look at
potential materials using a multi-stage selection.
SELECTION STAGE 1) σ versus ρ = CHART 2, slope = 1, upper left
CANDIDATE MATERIALS:
•
•
•
•
•
•
CFRP
GFRP
Steels
Ti alloys
Al alloys
Wood
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SELECTION STAGE 2) E versus σ = CHART 4, slope = 1, upper left
CANDIDATE MATERIALS:
•
•
•
•
Al alloys
Steels
Ti alloys
CFRP, GFRP, Wood
PERFORMANCE:
CANDIDATE MATL
CFRP
GFRP
Steels
Ti Alloys
Al Alloys
Woods
σ
[MPa]
700
400
1800
1000
430
110
ρ [Mg/m3]
E [GPa]
M1
M2
1.6
1.6
7.8
4.2
2.6
0.6
30
20
220
100
60
1
440
250
230
240
165
185
0.043
0.050
0.122
0.100
0.140
0.009
For M1 the best performers are polymer composites, but they lose out to steel in M2 for
which they show deflections three times larger than the steels. Ti and Al look pretty
good, but they lose out when we throw cost into the equation. HIGH TENSILE
STRENGTH STEEL is the commonly used material, except in high performance weightdriven designs (racing yachts with CFRP).
STEEL SHIP PLATES AND FRACTURE
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In the early 1900's, ship plates were completely riveted together. At the end of WWI a
push for faster construction times drove shipbuilders toward using substantially welded
ship plates, but as the war stopped, the money for development dried up. In 1921 a small
merchant ship (the FULLAGAR, 150 ft. long) was the first fully welded ship to hit the
water, and worked in England for many years.
At the start of WWII, the push came on to rapidly produce ships for the merchant marine
fleet to supply the war effort, and welding technology was again pushed. The approach
was a "cookie cutter" one, with a small number of ship plans, and many shipyards
producing the same design. The construction was begun in 1941, and in total,
2500 Liberty Ships
500 T-2 tankers
400 Victory ships
were constructed. Shortly after these ships entered service, they began breaking apart,
sometimes spectacularly! The rapid and massive scale-up required by the war meant that
unskilled laborers and inadequate welding practice were used, and blamed for what
happened.
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Two major causes of the failures were found:
1) STRESS RAISERS: access holes through the decking plates and structural plates were
cut for ladderways and cargo loading. These were initially cut as rectangular holes. Many
cracks initiated at the corners of these holes. By changing the design to rounded holes,
many fewer failures were reported.
2) UNKNOWN EFFECTS: (at the time)
No correlation was found between failure and the tensile strength of the steel samples
taken from various parts of the failed ship plates. Loading at failure was typically around
700 [MPa], well within the design load.
Extensive study of the brittle fracture energy (toughness) using the Charpy impact test
found the following:
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Ductile to Brittle Transition Temperature (DBTT) is arbitrarily set as about 15 [ft-lbs] of
fracture energy. ANSWER: the DBTT was too high (the steel was brittle at the
temperatures of the North Sea).
End of File.
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