WeekOneToFive

W. Warnes: Oregon State University

ME480/580: Materials Selection Lecture Notes for Week One

Winter 2009

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 Figure 1.1.

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 120,000 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:


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: COST, and ENERGY CONTENT.

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.


BUDINSKI FIGURE 18.1 NOTES:

1. Calculations in the first block! Analysis is important! 2. Analysis appears multiple times throughout design process. 3. Materials selection is in the last step. 4. Iteration?


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.


DETAILED DESIGN:

High degree of information needed about only a few materials. May require contacting specific manufacturers of materials. May require specialized testing for critical components if materials data does not already exist.

CLASS APPROACH

Two different philosophies here:

Budinski: get familiar with a set of basic materials from each category, about seventy-five in total, and these will probably handle 90% of your design needs (see Figure 18-8).

Ashby: look at all 120,000 materials initially, and narrow your list of candidate materials as the design progresses.

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 120,000 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 of selection considering shape effects. • Materials property data sources. • Specific materials classes.

SELECTION CHARTS (Ashby chapter 4)

1. 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 need to be able to compare materials based on several properties at once.


2. 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.

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.

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 information can be displayed on these charts.


EXAMPLE: the longitudinal wavespeed of sound in a material is given by the equation

V =E

!"#$

%&'

1/2

Rewrite this equation by taking the base-10 logarithm of both sides to get:

log V( ) =1

2log E( ) ! log "( )#$ %& or log E( ) = 2 log V( ) + log "( ).

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).


EXAMPLE: The selection requirement for a minimum weight design (derived next time) is to maximize the ratio of

E1/2

!= C = constant, which leads to

1

2log E( ) = log C( ) + log !( ) , or

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.



Winter 2009

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:


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.

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 f1, f2, and f3.

f1 is the place where creative design comes in.

f2 is where geometry can make a difference.

f3 is the part we're most interested in. When the factors are separable, the materials selection doesn't depend on the details of f1 or f2! This means we don't have to know that much about the design to make intelligent materials choices.


Our first step is to maximize performance by only considering f3 (selection of materials without shape effects). Later on we'll look at adding in the shape effects by maximizing f2f3.

EXAMPLE ONE: Design a light, strong tie rod.

The design requirements are:

• to be a solid cylindrical tie rod • length L • load F • safety factor Sf • 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 Sf, 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, so look up σf(steel) = 500 MPa. Now we know what the smallest area will be:


3) Now we can find the mass of the rod:

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 lower the mass by going to a lower density material. Let's use a high strength Al alloy instead of steel."

MASS (Aluminum) / MASS (Steel) = 60%. 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 (OPTIMIZATION)

Looking at this list of requirements, we start with


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-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. I prefer to always define it to be maximized, so that it maximizes performance.

3) IDENTIFY the parameters in our analytical model and MOP:

L= A= ρ= F=

4) Write an equation for the CONSTRAINED variables: (we have to safely carry the load F)


5) Rewrite the constraint equation for the free variable and substitute this into the MOP:

6) Regroup into f1, f2, and f3

To maximize p we want to choose a material that maximizes the ratio

(σ / ρ) = M = 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 • safety factor Sf • minimum mass



2) MEASURE OF PERFORMANCE (MOP), p. Minimum mass again.

3) IDENTIFY the parameters in our analytical model and MOP:

L= A= ρ= F=

4) CONSTRAINT equation: (no Euler buckling of this column)

5) Rewrite the constraint equation for the free variable and substitute this into the MOP:

6) Regroup into f1, f2, and f3

7) PERFORMANCE INDEX=


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 stiffness, etc.) 4) Identify the FREE, FIXED, PROPERTY, and CONSTRAINT parameters. 5) Develop an equation for the constraint(s). 6) Rewrite the CONSTRAINT equation for the FREE parameters in the MOP. 7) Reorganize into f1, f2, f3 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.



Winter 2009

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 stiffness, etc.) 4) Identify the FREE, FIXED, PROPERTY, and CONSTRAINT params. 5) Develop an equation for the constraint(s). 6) Rewrite the CONSTRAINT equation for the FREE parameters in the MOP. 7) Reorganize into f1, f2, f3 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:

• Circular disk shaped mirror support • Size = 2r • lightweight • deflects (δ) under own weight by less than λ.


MODEL:

MOP: minimize mass

PARAMETERS:

r = t = ρ = δ =

CONSTRAINT EQUATION: (use the helpful solutions in the appendix)

APPLY TO MOP:

CAUTION: m (mass) appears in the constraint equation. We will need to eliminate it.


Since MOP is minimum mass, p=1/m:

APPLYING PERFORMANCE INDICES TO SELECTION CHARTS

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.

End up with materials such as CFRP, and we're stuck with expensive candidate materials.


OR: we can use the performance index from above: M =E

!3"#$

%&'

1/2

=E1/2

!3/2:

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 unit in X we go up three 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.


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 useM =GPa[ ]

1/2

Mgm3

!"#

$%&

3/2.

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.


2) As we move the line, 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

These materials do not perform as well as the first set with M = 10GPa

1/2

Mgm3( )3/2

!

"

####

$

%

&&&&

.

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.


5) We stated earlier that we can use any of these ratios for M:

M =E

1/2

!3/2 or M =

E

!3 or M =

E1/3

!.

Let's check to see if that makes sense:

E1/2

!3/2

E

!3 E

1/3

!

CHART:

CHART:

CHART:

SLOPE:

SLOPE:

SLOPE:

UNITS OF M:

UNITS OF M:

UNITS OF M:


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:

For several candidate materials, we have the following property data (obtained from the Ashby chart #1):

E [GPa] ρ [Mg/m3] (ρ / E) [ s2 / m2 ] (ρ / E)1/2 [s/m] Glass 100 2.2 2.2X10-8 1.48X10-4 Composites 30 1.5 5 X10-8 2.24 X10-4 Wood Products 4 0.8 2 X10-7 4.5 X10-4 Polymer Foams 0.1 0.2 2 X10-6 1.4 X10-3


Compare these materials against a "standard" material; for instance, glass has been commonly used in this application.

Material t / t(glass), or how much thicker than glass the mirror must be. Composites Wood Products Polymer Foam

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 t = Mirror Thickness [m] Glass Composites Wood Products Polymer Foam


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.


ME480/580: Materials Selection Lecture Notes for Week Two

Winter 2009

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:


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?

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 chart 2, with log(σ) versus log(ρ), and look at a line of slope = 1.


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:

MATERIAL M[ MPa/(Mg/m3) ] CERAMICS CFRP GFRP Be alloys

Steels Ti alloys Mg alloys Al alloys Woods

Lead Cast Iron


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 =

Look at the list again:

The best performer is CERAMICS-- BUT, we need to check on the measurement of σ for ceramics used in Ashby's chart. In the description of chart 2 it says that σ 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.



Winter 2009

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=

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 =


In all of these cases, the performance index will be M1=

Look at the selection chart of modulus versus strength (Chart 4)

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?

In this case, the performance index becomes M2 =

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.


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?

The net result is that, while the selection plots are somewhat different, the materials that pass the selection are identical.

End of File.



Winter 2009

CASE STUDY: NATURAL BIOMATERIALS Reading: Ashby chapter 14.8

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, 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 14 has a set of Selection Charts that focus on natural materials.


LOW-MASS ELASTIC MATERIALS (Figure 14.11)

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 light-stiff 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:

Density [Mg/m3] Young’s Modulus [GPa] Strength [MPa] Steel 7.9 216 1900 Al alloy 2.8 80 590

Cellulose is the winner for tensile stiffness: M =E

!, beating steel by a factor of 3 to 4,

and pushing flax, hemp and cotton up pretty high. Woods, palm, and bamboo perform

very well in bending and buckling M =E1/2

!"#$

%&'

. (NOTE that the guidelines on this chart

have a typo so that the exponents are not correctly listed. What are the exponents/slopes?)


LOW-MASS HIGH-STRENGTH MATERIALS (Figure 14.12)

Again, natural materials show up nicely on this chart, with silk having the best strength-to-weight ratio. (NOTE that the guidelines on this chart also have a typo so that the exponents are not listed. What are the exponents/slopes?)


ELASTIC ENERGY STORAGE MATERIALS (Figure 14.13)

For this chart, the best materials are those with large values of σ and small values of E: in

the lower right corner. Spring materials are those with large values of M =!2

E, while

elastic hinge materials are those with large values of M =!

E.


TOUGH NATURAL MATERIALS (Figure 14.14)

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 sloping down from left to right).

The dash-dot lines show contours of constant resilience, when materials must deform elastically a large amount before failure. Skin is a particularly good material in this respect.


CASE STUDY: CERAMIC BIOMATERIALS ("Biomaterials-An Introduction", J. B. Park, R. S. Lakes, Plenum Press, 1992.)

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:


Let f = failure probability, then reliability is r = 1 - f. Total reliability is product of

individual reliabilities.

Example:

III. BIOLOGICAL APPLICATIONS


IV. BIOLOGICAL CERAMICS

Three main types, or classes:

1. INERT 2. SURFACE REACTIVE 3. RESORBABLE

IV.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.

Typical inert ceramics are:

• Al2O3 for joint prosthetics, dental applications. • LTI (low temperature isotropic) carbon for heart valves and coatings on some

prosthetics. • DLC-(Diamond-like Carbon)films because of their stability.

IV.B. 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.

IV.C. RESORBABLE BIOCERAMICS

Materials that fill space and are taken up by the body with time, presumably to be replaced with new bone growth.

Example:

V. APPLICATION EXAMPLES

Corrosion


Mixed Metals

Dental prosthetics


Mechanical Design

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.

End of File.


ME480/580: Materials Selection Lecture Notes for Week Three

Winter 2009

MULTIPLE CONSTRAINTS IN MATERIALS SELECTION: OVERCONSTRAINED DESIGN I

Reading: Ashby Chapter 9 and 10.

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:

CONSTRAINT:

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 OVER-CONSTRAINED. There are several ways we can deal with multiple constraints in the selection process. These are using DECISION MATRICES, SELECTION STAGES, COUPLING EQUATIONS, and VALUE FUNCTIONS.

I. DECISION MATRICES

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).


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. For each constraint or design goal, we develop an M value to use as one of the columns:


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).

www.cs.orst.edu/~dambrosi/edss/info.html


II. 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 again. 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. minimize the measurement uncertainty due to vibrations of the stiff structure, 2. minimize the distortions of the structure due to temperature effects, 3. keep the hardness high for good wear properties, and 4. 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


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

To minimize the thermal distortion d!

T

dx

"#$

%&'

for a given heat flow, we need to maximize

II.A.3. HIGH HARDNESS

We can treat the hardness, H, as a direct function of the yield strength:


II.A.4. COST

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:


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.


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:


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.


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:


Glasses Steel/Stone Al alloys/Composites Mg, Zn, Ni, and Ti alloys/Ceramics

Compare these in a table:

First Selection Stage Second Selection Stage Third Selection Stage Ceramics Ceramics Glasses Be Invar Steel/Stone

CFRP SiC/(pure metals) W, Si, Mo, Ag, Au, Be Al alloys/Composites

Glasses/WC/GFRP Al alloys Mg, Zn, Ni, and Ti alloys/ Ceramics

Woods/Rock, Stone, Cement/Ti, W, Mo, steel, and Al alloys

Steel


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 value functions.

End of File.


ME480/580: Materials Selection Tutorial Overview Notes on CES

Winter 2009

INTRODUCTION TO CES (CAMBRIDGE ENGINEERING SELECTOR) 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 WindowsNT account.

2) Once in WindowsNT, open and start the CES program icon.

INSIDE CES:

You will see the WELCOME screen when you startup. For now, just close this window to view the main control window. You will see that there are several toolbars along the top of the main window, and a set of "tabs" about 1/3 of the way down the window. These are how you will interact with CES.

INFORMATION:

There is a fair amount of on-line help and database information available in CES.

1) Click first on the MATERIALS 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) Now click on the RESULTS icon in the top toolbar to bring the RESULTS window to the front. You should see a list of all 377 materials records that are in the generic database.

4) You can change the database to search by choosing a different FILTER from the pull-down menu just above the set of tabs in the main window. Try several different databases. The eight databases in this program are:

1. Generic: A good starting point, with a variety of materials from all materials classes. 377 materials.

2. Megabytes on Copper: contains information on 174 different copper and copper alloys.

3. Metal: The largest database, with 1509 different metals and metal alloys. 4. Polymer: 249 materials in the polymer class. 5. Foam: a database of foam and foamable materials, about 136 of these. 6. Wood: 509 different wood and wood product materials. 7. Fibers and Particulates: 45 materials that are used as reinforcements in composite

materials. These are usually no good as stand alone materials. 8. Structural Sections: a special database for use with shape analysis.

5) Reference material is also available on-line, as well as an on-line help function (press F1 at anytime to get contextual help). Click on the BOOKS tab. The "CES InDepth" is an on-line reference book about CES and the selection process we have been using in class.

6) Click on the HELP menu item at the top of the screen and select CONTENTS. The TUTORIALS that are listed here might be a resource if you want to learn more about the capabilities of the program.

7) For the last thing to do on this part, click on the PROJECT menu and select PROJECT SETTINGS. Choose the currency you want to use for cost analysis here. This also allows you to set the units for the selection charts. (using US$ instead of UK £ would probably be a good idea).

MAKING A SELECTION CHART:

1) Click on the PROJECT tab to start. (Alternatively, you can click on the NEW PROJECT icon on the top tool bar.) Now click on the NEW GRAPHICAL STAGE icon on the second toolbar. (The toolbar buttons are, from left to right, NEW GRAPHICAL STAGE, NEW LIMIT STAGE, STAGE PROPERTIES, CURSOR, and then a set of selection tools.)

2) You should get a window with the "Graph Stage Wizard" title. Make sure that the X-axis tab is selected, and then use the ATTRIBUTES pull-down menu to choose the


material property to plot on the x-axis. Choose ELASTIC LIMIT (which is the yield strength for metals).

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.

CHANGING AND USING A SELECTION CHART:

1) Click on the STAGE PROPERTIES icon.

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 the edge of 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.

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 on the second row toolbar 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 set of materials in the box. Note that the STATUS BAR (at the bottom of the screen) gives you the X,Y location of your cursor. Note also that any material bubble that is partly inside the selection box is colored, while the others are greyed-out.

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 (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 have to 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) Click on the RESULTS icon in the top toolbar to bring the RESULTS window to the front. This now 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 button or menu item.

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. Close the RESULTS window. In the main window click on the NEW button.

2) Stage 1 will deal with M1: Click NEW GRAPHICAL STAGE, and in the X-Axis properties choose the THERMAL CONDUCTIVITY property. For the Y-Axis properties choose THERMAL EXPANSION, 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-K, and α =1 X 10-6 1/K. (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 excatly the same position that I have used (X = 10, Y = 1).

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 GPa, and ρ = 0.1 Mg/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 set 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!).

8) Stage 3 will deal with M3 and M4: Click NEW GRAPHICAL STAGE, and for the X-Axis properties we have to do something fancy. There is not a property listed for COST, but there are properties PRICE ($/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 category 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 $/volume you need for minimum cost design. We should also change the name of the axis to something sensible (like MATERIAL COST ($/m^3)) so we know what we are looking at in the selection chart.

9) In the Y-Axis properties choose HARDNESS, and click on the OK button.

10) Now click on the BOX selection tool button. Use the box to select the materials with a MATERIAL COST less than $10/m3, and a H greater than 1000 MPa.

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: Cast Al, Non-machinable Glass Ceramic, and Wrought Al Alloy.

NOTES:

You may only search one database at a time. To change databases:

1) Pull down the FILTER menu and select the database you want to search.


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 start off with the GENERIC database, and use the others as your design develops.

End of File.


ME480/580: Materials Selection Lecture Notes for Case Study

Winter 2009

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 (cost).

Use Rule of Mixtures to determine foam properties, e.g. 90% air foam:

( or, looked at another way, for hf = 2hs you can get the same deflection with 80% less mass!).

Can also laminate surface of foams with high strength layer to drive strength/weight ratio way up.


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!

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.


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 use as a primary structural material, as in this University of Michigan study in 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


FOLDED PLATE STRUCTURES WITH POLYURETHANE/PAPER BOARDS

FILAMENT WINDING ON PUR BOARD

End of File.


ME480/580: Materials Selection Lecture Notes for Week Four

Winter 2009

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 measure-ments being made. In other words, f should be larger than a critical frequency, fmin.


MODEL:

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:


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


To perform the 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 parameters and constraint parameters. First, write out the equations for the MOP using each of the constraints:

p(M1)=

p(M2)=

p(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

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 in EXCEL looks something like this, and I pulled the 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 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.


ME480/580: Materials Selection Lecture Notes for Week Four

Winter 2009

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


III.A.1. DESIGN ASSIGNMENT:

• cylindrical tie rod of length L • minimum weight • support a load F • extension less than δX • safety factor Sf

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


III.A.6. PERFORMANCE EQUATION TWO

III.A.7. DEVELOP THE COUPLING EQUATION:


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


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.


• Cantilever beam of rectangular cross section and 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: b: h: ρ: δ:

We've got two free parameters and two constraints!

CONSTRAINT ONE:


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)

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 combinations:

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.


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.


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:


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 Underconstrained; need to add a constraint. C = F Fully determined; one performance index, M. C > F 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 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:


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.


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.


ME480/580: Materials Selection Lecture Notes for Week Five

Winter 2009

CALCULATING MASSIVELY OVER-CONSTRAINED 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:


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).

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.


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 clarity in the last couple of plots.

Now, click BELOW this line so that CES selects all the materials that touch the line or are below it.

(NOTE: The position of the coupling line can be EXACTLY placed on the selection chart in CES by looking (in the "Project" Menu) for "Stage Properties". The dialog box has a tab for "Selection" that allows you to enter exact 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.


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.


ME480/580: Materials Selection Lecture Notes for Week Five

Winter 2009

DEALING WITH CONLFICTING OBJECTIVES Reading: Ashby Chapters 9 and 10.

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:


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 $/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


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!

"f2/3

= 113 * [Materials:Density] / [Materials:Elastic Limit] ^ 0.6667

Now the Y-axis will be the MASS of the beam, in kg. Similarly, we can set up the X-axis to be the COST of the beam in $. 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.


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 non-dominated. 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.


ME480/580: Materials Selection Lecture Notes for Case Study

Winter 2009

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.


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:


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


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 σ [MPa] ρ [Mg/m3] E [GPa] M1 M2

CFRP 700 1.6 30 440 0.043 GFRP 400 1.6 20 250 0.050 Steels 1800 7.8 220 230 0.122 Ti Alloys 1000 4.2 100 240 0.100 Al Alloys 430 2.6 60 165 0.140 Woods 110 0.6 1 185 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 weight-driven designs (racing yachts with CFRP).

STEEL SHIP PLATES AND FRACTURE

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.


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:


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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