CTf =-
CI
where C is an integer, I is the second moment of area, and Ym is the distance from the centerline of the beam to the surface of the beam. This is, of course, equal to the radius for a circular beam. You may also wish to show that the same Material Index, i.e. Equation 18, is obtained regardless of whether the beam cross section is assumed to be circular, square, rectangular, etc. By taking the logarithm of both sides of Equation 18, you can show that the appropriate graph to use is a plot of log strength (y-axis) versus log
P
(x-axis), and the guideline will have a slope of 1.5.CASE STUDY: ENERGY EFFICIENT FURNACE WALLS
An important materials selection problem for ceramic engineers is choosing the best material for use in constructing the walls of h a c e s , i.e. kilns.
Assume that the goal is to minimize the energy, i.e. cost, of heating the finace from the ambient temperature, Ti, to the operating temperature, Tf, where it is held for a time, t. There are two sources of energy loss per unit area: 1) loss by steady state conduction, Q1, through the walls of the furnace, and 2) the energy required to raise the temperature of the walls, 4 2 . These losses are given by the following two equations:
(20)
where: k = thermal conductivity
c,
= specific heatp
= density w = wall thicknessIf we were only trying to minimize the energy loss via conduction, we would pick a material that had the smallest value for the thermal conductivity and we would make the walls very thick. If we only considered the energy lost in raising the temperature of the walls, we would select a material having a low value for the volumetric specific heat, C,p, plus we would make the walls very thin. These are conflicting objectives; how are we to pick the best material and wall thickness?
To minimize the total energy loss, we must minimize the sum of Q1 and 4 2 .
To determine the optimum thickness, we differentiate Equation 22 with respect to the wall hckness, w, and equate the result to zero, obtaining:
w=(e)
112Substituting Equation 23 into Equation 22 in order to eliminate the wall thickness, w, gives:
Q =(Tf - T0)(2t)”2 (kC,p)1’2
(24)Since for a given process, Tf, To, and t are fixed, Q can be minimized by minimizing (kCpp)1’2, i.e. by maximizing the Material Index, M,
M
=(kC,p)-1'2
By using the relation between thermal conductivity and thermal diffusivity,
k
where
a
is the thermal diffusivity, it is possible to rewrite Equation 25 asTaking the logarithm of both sides of this equation yields
This shows that Figure 8 can be used to select the best material(s) for the walls of the furnace. The design guideline has a slope of !4. The Material Index,
M y
ismaximized by moving the design guideline toward the lower right corner of Figure 8. We look toward the lower right corner this time because we desire to pick materials having low conductivities. Actually, our goal is to maximize the Material Index, given by Equation 27. In other words, the best materials are located below the design guideline shown in Figure 8. This shows the best materials to be polymer foams, cork, plaster, wood, and polymers. These materials work only as long as low temperatures (45OOC) are involved.
For higher temperatures, porous ceramics are, of course, the appropriate choice.
Either Equation 25 or 27 can be used to calculate values of the Material Index for specific porous ceramics. The material with the largest value is best. Equation 23 can be used to calculate the optimum wall thickness. It varies from one material to another.
For some applications, there may not be sufficient space for very thick walls. By using Equation 26, it is possible to rewrite Equation 23 as
.=[E)
1'2 =(2at)'I2
Figure 8. Thermal conductivity versus thermal diffusivity.
This shows that materials having large thermal diffusivities require thick walls for a fixed value oft. If limited space is available and we must also fix the wall thickness, then Equation 29 shows that only materials with thermal diffusivities less than w2/2t are permissible. We still use Equations 25 or 27 to select the most energy efficient material fiom this restricted set of materials. Clearly, mechanical and chemical properties as well as cwt should be considered before making a final selection.
Ashby tabulates Material Indices for a large number of other design problems.
*
These include deflection, strength, vibration, and fiacture of ties, shafts, beams, columns, plates, flywheels, and internally pressurized tubes and spheres. Several thermal, thenno-mechanical, and electro-mechanical problems are also included.INFLUENCE OF COMPONENT SHAPE ON MATERIAL SELECTION So far, we have compared materials of the same cross sectional shape.
The cross section could have any shape, but it had to be the same for the different materials being considered. This is not “fair” since some materials are better suited than others for being shaped into efficient cross sections. For example, a steel beam is most efficient when shaped like an I, i.e. an I-beam. On the other hand, a beam made of rubber (an unlikely choice) would likely buckle if it had the same cross section.
We treat the fact that different materials have different practical limits for the extent to which they can be shaped by introducing a quantity called the shape factor,
$,
into the Material Index. Without going through the derivation,’ the reader is asked to accept that for the case of the light, stiff beam considered previously, the Material Index including shape is:(E$; y2
M = P
where
4;
is the shape factor for the elastic deflection of a beam. Compare this equation to Equation 16 where shape was not considered.To get a “feel” for the shape factor, consider two beams both having rectangular corss sections. Suppose one beam has a cross section of 2x4 while the other has a cross section of 1x8. They both have the same cross sectional area, but the 1x8 has a larger value for
$; ,
and, not coincidentally, is also more susceptible to buckling. The general fomula for the shape factor4:
is:where I is the second moment of area as before and A is the cross sectional area.
Using this equation and Equation 32, which gives I for a rectangular cross section of height, h, and width, b, we can calculate
4;
for the two beams.I=- bh3
12
These calculations show that the 2x4 has a shape factor of 2.1, while the shape factor is 8.4 for the 1x8.
The shape factor for the 1x8 is 4 times larger than that for the 2x4.
This means that the 1x8 is 4 times stiffer than the 2x4, where stifess is defined as force divided by deflection. That is, the load on the 1x8 could be 4 times that on the 2x4 for the same amount of deflection.
Alternatively, for the same load, the 1x8 would deflect a fourth as much as the 2x4.
Let us think about how the shape factor can be incorporated into the selection of materials using a light, stiff beam as an example.
First, using Figure 1 as we did before when we were not considering shape, a few promising
Log E
Figure 9. The materials A and €3 being considered for a light, stiff beam.
materials are selected. For example, assume we have narrowed our selection to two materials, A and B, shown in Figure 9. Material B appears superior to material A since it is closer to the guideline shown in the figure. But we have not yet considered shape.
If we take the logarithm of both sides of Equation 30 we obtain
logE$i =210gp+ 2 10gM
(33)This equation suggests that we can use a log E versus log
p
plot, with guidelines having a slope of 2, to select materials if we multiply the shape factor,$;,
times the Young’s modulus, E. To show the result graphically, we simply move the points up vertically on the graph. Suppose material A is capable of a shape factor of 20 while materialB
is capable of only a shape factor of 2. We multiply the value of E for material A by 20 obtaining 5x20 = 100. We show the new position of material A after accounting for its shape factor as pointA*
in Figure 10. Similarly, we multiply the modulus of B by 2, obtaining 1x2 = 2. The new position, B*, is shown in Figure 10. Now, after considering shape, material A isLog E
100 50
I 0
5
I
b
I 0 0 0 I
I 5 I 0
Log P
Figure 10. Positions of materials A and B after considering shape.
found to be a better material for the beam. Point A* is on a new guideline that is located nearer the upper left corner of the graph. Point
B*
is located below this new line.An alternative non-graphical procedure is to first pick several promising materials and then calculate the Material Index. For the light, stiff beam example we have just considered, the procedure would be to plug E,
$:,
and f3 values into Equation 30. The best material would be the one with the largest calculated value of M. For the materials A and B considered above, the calculated value of M are (5~20)”~/10 = 1 and (1~2)”~/5 =0.28, respectively. Once again, material A is shown to be the best material when shape is considered.
The reader is warned that the graphical approach requires the use of the Cambridge Engineering Selector software’ (CES) for some design problems, while the approach of calculating M by plugging in values will always work.
The reader should also be aware that there are four types of shape factors.
We have considered only
4;
which is appropriate for the elastic deflection of beams. For design of a light, strong beam, the appropriate shape factor is @B.For torsional loading, other shape factors are required as described by Ashby.’
f
MULTIPLE OBJECTIVES AND CONSTRAINTS
The design problems we have considered so far have been oversimplified.
Real design problems often involve more than one objective and constraint. For example, we may wish to select a material for an aerospace application, where we have the goals of minimizing both mass and cost and we have the constraints that the component must not deflect more than some fixed amount and also must not fail when subjected to a given load. If it is a part for an engine, we also are constrained by the material’s melting point, creep and oxidation rates, etc. Such
problems typically do not have a unique solution. Nevertheless, methods have been developed for attacking such problems, two of which are explained
Successive Treatment
For illustrative purposes, consider a fairly simple design problem where we wish a light, stiff, strong beam. We could first select several potential materials for a light, stiff beam using the Material Index
E”2 / p
as before. We could then select a set of materials for a light, strong beam using the Material Index o ~ ’ ~/ p
. Materials that are common to both sets would likely be good choices. This general procedure could even be applied if we were trying to solve a design problem that had more than two constraints. We would just identify multiple sets of materials and again look for a material that appeared in each set.We may have to enlarge our initial sets if a common material does not occur in each set. This method is far from perfect, and two people may not select the same material.
Weight-Factors
Another method, which also involves some subjectivity, utilizes weight- factors. The key properties or Material Indices are first identified and then assigned weighting factors. The most important properties are assigned higher weight-factors based on the judgement of the designer or design team. Clearly, this step is subjective. The weight-factors are selected so that the sum of all the weight-factors is 1. A list of materials of interest is generated. The next step is to normalize each property or Material Index. For example, suppose fiacture toughness is a property of interest. A normalized value for fracture toughness is obtained for each material of interest by dividing each fiacture toughness value by the value for the material having the largest fracture toughness. The normalized values for each property are then multiplied by the appropriate weight-factor and the sum calculated for each material. The material having the largest value for the sum is selected as the best material. Clearly, the “best” material depends on the values selected for the weight-factors. For example, one team may believe that strength is more important than density and would therefore assign a higher value to strength, while another team assigns a higher value to density. The two teams likely will not select the same material. Skill, i.e. experience, in assigning weight- factors is very important. Ashby’ and Bourel16 provide good descriptions of several other useful methods for the selection of materials when multiple goals and constraints are present.
CAMBRIDGE ENGINEERING SELECTOR SOFTWARE
Computer software called the Cambridge Engineering Selector’ has been developed specifically for use in graphical selection of materials. It readily permits construction of numerous graphs such as those shown previously where one property is plotted against another. Any property that appears in the database can be plotted against any other property. Thus, the software can be used to plot unusual but usefbl graphs such as that shown in Figure 11 , where the price of various materials is plotted versus their hardness. The software also provides tabulations of physical, mechanical, thennal, and electrical properties for important ceramics, metals, polymers, and composites. Some unusual materials are included such as ice, leather, bambo, etc. The software also provides addresses of vendors for specific materials and shapes. In addition, the s o h a r e describes many material fabrication processes.
10000 lOd000 1&07 1 e k 1 0
Hardness (Pa)
Figure 1 1. Price versus hardness graph created using the Cambridge Engineering Selector software.
One of the disadvantages of the CES software is that the database does not contain as many specific grades of materials as desired. This has been recognized by the vendor, Granta Limited, and many more materials have been added over
the past couple of years. Recently, ASM International has agreed to provide fbnding and make their material database available for inclusion in the CES sohare. This wilt significantly advance this user-friendly s o h a r e and will likely result in it being widely used.
PROCESS SELECTION
Ashby also describes a graphical approach for process selection. Figure 12 shows one of several graphs that are used. In this example, it is assumed that the tolerance and surface finish of the part to be fabricated are known. Tolerance and surface finish are termed process attributes. Their values for a specific component define a point or small region on the graph. Processes capable of meeeting these requirements contain, i.e. enclose, this region. For example, Figure 12 shows that investment casting, as well as many other processes, have the potential for making a part having a surface finish of 1 pm and a tolerance of 0.1 mm. The designer then sequentially utilizes several other graphs to narrow the selection. A second graph that might be used is shown in Figure 13. Here the process attributes are hardness and melting point. Note that most of the casting processes are not feasible for materials such as ceramics that have very high melting points. For these materials, possible processes include powder methods, chemical vapor deposition, etc. Paper copies of many of the graphs required for process selection are available,* but the multiple steps required to select the preferred process are simplified by use of the CES software.'
Others have described alternative methods for process selection with emphasis on the interaction between manufacturing and desigd'
'
It is important to realize that the h c t i o n of a component, its shape, materials selection, and process selection are certainly interdependent and should be performed concurrently in order to avoid costly iterations. It is not productive to select a material and, some time later, realize that an economical process for fabricating that material into the desired shape is not available. That is, function, shape, material, process, and cost must be considered together throughout the design process.
SOURCES OF PROPERTY DATA
A very large number of databases for material properties and specifications perfonnance are available. Westbrook7 provides an extensive compilation subdivided by material types. More than 30 sources are given for ceramics, glasses, and cements. Many more sources are provided for metals, polymers, and composites. Sources of materials purchasing information are also provided.
1 o - ~ 1 o-2 1 0-' 1 10
RMS Surface Roughness, R (lm)
Figure 12. Example of a process selection graph.'
Guidance in evaluation, interpretation, and estimation data is available from a number of sources.195 Clearly, since many properties of ceramics are very sensitive to microstructure, great care must be exercised in choosing property values. For conceptual design, handbook and database values may be acceptable.
For detailed design, data from vendors or in-house test data are preferable.
1
200 500 too0 2000 3 o 0 O 4 o 0 o M ) o
Melting Temperature (K)
Figure 13. Hardness versus melting point graph used for process selection.'
1 ii
Hardness vs..
104
10s
g
5 b
loo
p
Y I
10
A c ~ o w ~ e d ~ ~ ~ n t :
Permission of Michael F. Ashby to use graphs from his text is appreciated.
REFERJZNCES 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Michael F. Ashby, Materials Selection in Mechaxal Design, Seconc Edition, Butterworth Heinemann, Boston, 1999.
J. A, Charles and F. A. A. Crane, Selection and Use of E n ~ n e e ~ n g Materials, Second Edition, Butterworth Heinemann, Boston, 1989.
Materials Selection and Design, ASM Handbook, Volume 20, ASM
~ntemational, Materials Park, Ohio, 1997.
G. T. Murray, Introduction to Engineering Materials: Behavior, Properties, and Selection, Marcel Dekker, Inc., New York, 1993.
Cambridge Engineering Selector, S o h a r e available fi-om: Granta Design Limited, Rustat House, 62 Clifton Road, Cambridge CB1 7EG, United Kingdom. See www.grantadesign.com. Also available from ASM International: See www.asminternational.org.
David L. Bourell, Decision Matrices in Materials Selection, pp. 291-296 in Reference 3.
Jack H. Westbrook, Sources of Materials Property Data and Info~ation, pp.
49 1-506 in Reference 3.
Henry W. Stoll, Introduction to Manufacturing and Design, pp. 669-675 in Reference 3.
Geofiey Boothroyd, Design for Manufacture and Assembly, pp. 676-686 in Reference 3.
John A. Schey, M a n u f a c ~ ~ g Processes and Their Selection, pp. 687-704 in Reference 3.
David P. Hoult and Lawrence Meador, Manufacturing Cost Estimating, pp.
7 16-722 in Reference 3.
PROBLEMS 1.
2.
3.
4.
5.
Describe, using your own words, the general procedure for deriving a Material Index.
Describe how graphs, such as Figure 1, are used to select materials.
Use Figure 5 to select three promising materials for use in fabricating a low
cost, strong tie.
Using Figure 5, calculate the cost of teflon (PTFE) in dollars per kilogram.
You can obtain a value for the density of teflon fiom Figure 2.
Derive the Material Index for a light, stiff tie, i.e. a tie that has minimum mass and elongates no more than some given, fixed value. Assume that the length of the tie and the applied force are fixed and known.
STATISTICAL DESIGN
Hassan El-Shall, and Kimberly G. Christmas Department of Materials Science and Engineering
~ ~ v e r s i ~ of Florida Gainesville, F1.32611 Af3STMCT
Industrial products are the results of several studies including: bench scale investigations in academic and industrial research laboratories, in pilot plant studies, and quality control to check design specifications through out production.
All of these activities require experimen~tion and collection of data. Statistics should play a role in every facet of data collection and analysis, h m initial problem formulation to the drawing of final conclusions.
Statistical experimental design is an effective tool that ensures precise
~ f o ~ a ~ o n about the responses of interest, and
ant^
economic use of resources. In addition, statistical data analysis techniques aids in clearly and concisely summarizing salient features of experimental data.In this paper, various statistical designs including screening, full factorial and optimization designs are explained. Analysis of variance “ANOVA” is introduced and utilized together with practical examples to illustrate the e~ectiveness of these methodologies in better designs of ceramics and other materials.
~TRODUCTION
Engineers and scientists are usually well trained in the mechanics of experimentation. In most of the cases, however, the strategy typically used is the one-variable-at-a-time strategy. This is not only ineficient strategy; it suffm fiom several disadvantages as discussed later.
The best experimental designs result from the combined knowledge of science, engineering, technology, and statistics in the area of interest. Statistics help the experimenter to perceive and understand the real world with far greater precision than would otherwise be possible.
The f o l l o ~ n g is a c o m p ~ s o n between the statistical design methods and one- variable-at-a time strategy of e x ~ e n t a t i o n .