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A13 –Materials Selection in Design 1

Materials Selection in Design

References1. Ashby, Michael F., Materials Selection in Design, Butterworth-

Heinemann, 2nd Edition, 1999.2. Cambridge Engineering Selector v3.1, Granta Design Limited,

Cambridge, UK, 2000.

Introduction

How does an engineer choose, from a vast menu, the material best suited to his design purpose? Is it based on experience? Is there a systematic procedure that can be formulated to make a rational decision? There is no definitive answer to these questions, however the procedure can be somewhat approached in a systematic manner. Ashby's book and the Cambridge Engineering Selector (CES) software shed some light on the materials selection decision making process.


From Ashby: "Materials selection inherently must be based on at least 5 inter-related criteria:

Function of structural component Materials available and their properties Shape and size of structural component Process used to manufacture structural component Cost and Availability (both of material and process)

Function typically dictates the choice of both material and shape. Process is influenced by the material selected. Process also interacts with shape -- the process determines the shape, the size, the precision and, of course, the cost. The interactions are two-way: specification of shape restricts the choice of material and process; but equally the specification of process limits the materials you can use and the shapes they can take. The more sophisticated the design, the tighter the specifications and the greater the interaction. The interaction between function, material,


shape and process lies at the heart of the materials selection process."

Engineering Materials and their Properties

One can generally divide materials into classes, for example:


When one or more these materials types are combined we obtain a composite material. In what follows, we will not consider the metallurgy and chemistry of materials; rather we will focus on the various properties of the material types that are of importance to engineers.

Metals have relatively high moduli of elasticity and high strength. Strength is usually accomplished by alloying and by mechanical and heat treatment, but they remain ductile, allowing them to be formed by deformation processes. Typically strength is measured by the stress at yielding. Tensile and compressive strength is typically quite close. Ductility of metals may be as low as 2% (high strength steel) but may be quite high. Metals are subject to fatigue and typically are the least resistant to corrosion. Some hard metals may be difficult to machine.

Ceramics and glasses also have high moduli, but, unlike metals, they are brittle. Their strength in tension means the brittle fracture


strength; in compression it is the brittle crushing strength, which about 15 times higher then the tensile strength. Ceramics have no ductility and therefore have a low tolerance to stress concentrations. Ductile metals tolerate stress concentration by deforming inelastically (so that load is redistributed); but ceramics are unable to do this. Brittle materials tend to have a high scatter in strength properties. Ceramics are stiff, hard, retain their strength at high temperatures, are abrasion resistant, and are corrosion resistant.

Polymers and elastomers are completely different. They have moduli that are low, roughly 50 times less than those of metals do, but they can often be nearly as strong as metals. Consequently, elastic deformations can be very large. They can creep, even at room temperature, and their properties tend to very greatly with temperature. Polymers are corrosion resistant. They are easy to shape through moulding processes.


Composites can be developed which combine the attractive properties of the other classes of materials while avoiding some of their drawbacks. They tend to be light, stiff and strong, and can be tough. Most readily available composites have a polymer matrix (usually epoxy or polyester) reinforced by fibers of glass, carbon or Kevlar. They typically cannot be used above 250C because the polymer matrix softens. Metal matrix composites can be utilized at much higher temperatures. Composite components are expensive and they are relatively difficult to form and join. Thus, while having attractive properties, the designer will use them only when the added performance justifies the added cost.

Some important definitions for material properties

Elastic modulus (units: psi, MPa) - the slope of the linear-elastic part of the stress-strain curve. Young's modulus, E, describes tension or compression. The shear modulus, G, describes shear loading.


Poisson's ratio, , is dimensionless and is the negative ratio of the lateral strain to the axial strain in axial loading.

Accurate moduli are often measured dynamically by exciting the natural vibrations of a beam or wire, or by measuring the speed of sound waves in the material.

Strength, (units: psi, MPa) For metals, the strength is identified by the 0.2% offset yield

strength, . For polymers, the strength is identified as the stress at

which the stress-strain curve becomes significantly nonlinear; typically a strain of 1%.

Strength for ceramics and glasses depends strongly on the mode of loading - in tension strength means the fracture strength

while in compression it means the crushing strength which is 10 to 15 times larger.


For metals, yield under multiaxial loads are related to that in simple tension by a yield function; for example the vonMises yield function:

Ultimate Tensile Strength, (units: psi, MPa)The nominal stress at which a round bar of the material, loaded in tension, separates. For brittle materials (ceramics, glasses and brittle polymers) it is the same as the failure strength in tension. For metals, ductile polymers and most composites, it larger than the strength by a factor of 1.1 to 3 because of the work hardening, or, for composites because of load transfer to the reinforcing fibers.

A12 - Design for Column and Plate Buckling 9


Resilience, R (units )The maximum energy stored elastically without any damage to the material, and which is released again on unloading, i.e., the area under the elastic portion of the stress-strain curve.


Hardness, H (units: psi, MPa)A measure of its strength. It is measured by pressing a pointed diamond or hardened steel ball into the surface of the material; defined as the indenter force divided by the projected area of the indent.

Toughness, (units: ) and fracture toughness, (units: , )

A measure of the resistance of the material to the propagation of a crack. The fracture toughness is measured by loading a sample in tension that contains a deliberately introduced crack of length 2c (which is perpendicular to load), and the determining the tensile stress at which the crack propagates. Fracture toughness is

defined by , and the toughness is , where

Y is a geometric factor, near 1, which depends on the sample geometry.


Loss coefficient, (dimensionless)A measure of the degree to which a material dissipates energy in cyclic loading. Essentially, the ratio of energy dissipated to the elastic energy (for a given stress that the material is loaded to). Related to the damping capacity of a material (how much damping a material has). If the loss coefficient is zero, there is no damping. Depends on the frequency of the loading.


Materials Selection Charts

Mechanical, thermal and other properties for materials may be displayed in a variety of ways. What is needed is a way to compare materials in a useful way for properties that are important for the design problem under consideration.

For example, If we want a structure to stiff but light, then we want to choose a

material that has a high stiffness (E) to density () ratio. If we want a structure to be strong but light, then we want to

choose a material that has a high strength ( ) to density () ratio.

If we want a structure that is tough (resistant to crack formation or propagation) and light, then we want to choose a material that has a high fracture toughness ( ) to density () ratio.

Here are some charts from Ashby and the Cambridge Engineering Selector v3.1 software (CES).


Materials Selection - the basics

Lets take a look at the basics of material selection. First we need to define the concept of material indices. The design of any structural element is specified by three things: the functional requirement (F), the geometry (G) and the properties of the material of which it is made (M). The performance (P) is described functionally by an equation of the form:

The quantity describes some aspect of the performance of the component: its mass, or volume, or cost, or life, etc. Optimum design is the selection of the material and geometry that maximize or minimize according to its desirability or otherwise.


In many cases, the three groups of parameters are separable so that can be written as:

When the groups are separable, the optimum choice of material becomes independent of the other details of the design, i.e., it is the same for all geometries, G, and for all the values of the functional requirement, F. The optimum subset of materials can now be identified without solving the complete design problem, i.e., without considering or even know all the details of and . The function is called the material efficient coefficient, or material index. Lets take a look at an example to see how this works.


Example 1: Material index for a light, strong axial bar (rod)We want to design a bar of length to carry a tensile force without failure; and to be of minimum mass. Thus, maximizing performance means minimizing the mass while still carrying the load F safely. Function, objective and constraints may be listed as:

Function: Axial rodObjective: Minimize the massConstraints: (a) Length ( ) specified

(b) Support tensile load without material failure

We need an equation describing the quantity to be maximized or minimized. This is the mass of the rod. This equation, called the objective function, is given by:


where is the cross-sectional area of the rod and is the density of the material out of which it is made. The length and force are specified and are therefore fixed; the cross-sectional area A is free to choose.

We could obviously reduce the mass by reducing the cross-sectional area , but there is a constraint; the area must be sufficient to carry the load and not fail, i.e.,

where is the failure strength. Eliminating from the last two equations gives:


Notice that the first term contains the specified load while the second term contains the specified length . The last term contains the material properties. Hence, the lightest bar which will carry safely is that made of the material with smallest value of

. [Note: we should be including the safety factor here so that becomes . However, if the same safety factor is used for each material in a problem, its value does not enter into the material selection.]

It might be easier, or more natural, to ask what must be maximized in order to maximize performance. We therefore invert the material properties in and define the material index as


The lightest bar that will safely carry the load without failing is that with the largest value of the material index . This index is sometimes called the specific strength.

How do we determine the candidate materials with the best

(largest) ratio? We use the chart in Fig. 4.4 from Ashby,

or generate the chart using the CES software.

Note: The material index for stiff, light bar is similarly obtained as the largest value for the following material index .

We now use the chart in Fig. 4.3 from Ashby (or CES).


Example 2: Material index for a light, stiff simply supported beam

Consider a simply supported beam of length , square cross-section (bxb), and subjected to a transverse force at mid-span. We want to design a beam which must meet a constraint on its stiffness , i.e., it must not deflect more than under the load .

Function: BeamObjective: Minimize the massConstraints: (a) Length ( ) specified

(b) Support bending load without deflecting too

much

What does the term stiffness mean? Recall that for a cantilever

beam with a load at its end, the deflection is given by


which can be written as . The term is called

the stiffness and is similar to a stiffness coefficient in a finite

element analysis. Hence, for the cantilevered beam: .

For the problem at hand (simply supported beam with point load at the center), beam theory gives the maximum deflection (at the center of the beam) as:

where ="stiffness" of the simply supported beam (for a

point load at the center). The constraint equation than requires that


The moment of inertia is given by:

Note that the length ( ) is specified and the stiffness is specified by equation . The area is free to be determined.

The mass of the beam (objective function) is given by:

The mass can be reduced by reducing the area, but only so far that the stiffness constraint [equation ] is still met.

Substituting from equation into gives


Substituting from equation into equation gives:

Note that we have separated the design problem into the three parameters: function (F), geometry (G) and material (M). The best materials for a light, stiff beam are those which maximize the material index M:

It will turn out that the above result is valid for beams with any support condition and with any type of bending load location or

distribution. We now use the guideline in the chart in Fig.

4.3 from Ashby (or CES) to determine the best candidate materials.


Example 3: Material index for a light, strong simply supported beam

In stiffness-limited applications, it is elastic deflection that is the active constraint, i.e., deflection limits performance. In strength-limited applications, deflection is acceptable provided the component does not fail, i.e., strength is the active constraint.

Consider the selection of a simply supported beam (square cross-section) for a strength-limited application. The dimensions are as in the previous case. The design requirements are summarized by:

Function: BeamObjective: Minimize the massConstraints: (a) Length ( ) specified

(b) Support bending load without failing byyield or fracture


The objective function is still on mass [equation ] but the constraint is now that of strength, i.e., the beam must the support the load without failing. The bending moment is a maximum at the center and is equal to . The stress at the top surface ( ) is given by . Hence F is given by . The failure load occurs when

, or

where is a constant depending upon support conditions and load application/distribution, and is the distance between the neutral (centroidal) axis of the beam and its outer most fiber. Note that for the simply supported beam with point load at the center,

(as derived above) and (half the height). Using equation and equation to eliminate from the objective function


in equation gives the mass of the beam that will support the load :

Note that is typically for ductile metals. The mass is now minimized by selecting materials that maximize the material index M:

As stated before, the design requirement is characterized by: function, an objective and constraints. What is the difference between constraints and an objective? A constraint is a feature of the design that must be met at a specified level (for example,


deflection or stiffness). An objective is a feature for which a maximum or minimum is sought (mass in the last few cases).

The objective function is sometimes not easy to choose because there may be many options. For example, the objective function might be cost, it might be corrosion resistance, it might be elastic energy storage (for a spring), it might be thermal efficiency for an insulation system, and the list goes on.

We note from these three cases, that the satisfaction of the objective function requires choosing materials where is a ratio is maximized. Plotting these stiffness to mass (weight) or strength to mass ratios for broad classes of materials allows one to very quickly see which materials are "better." Note also that, like the last example, it is often not a simple ratio like but something more complex like , or for the deflection-limited case .


Example 4: Material index for a cheap, stiff column

We desire the cheapest cylindrical column of length and diameter 2r that will safely support a compressive load without buckling. The requirements are:Function: ColumnObjective: Minimize the costConstraints: (a) Length ( ) specified

(b) Support compressive force without buckling

The objective function is the cost C defined by

where is the cost/kg of the material of the cylindrical column (we consider the cost of the processed material only and ignore fabrication and other costs). A long slender column will buckle


elastically if exceeds the Euler critical load . The solution is safe if

where n is a constant that depends on the end conditions (n=1 for pinned-pinned condition, n=4 for clamped-clamped condition). For the cylindrical cross-section , the moment of inertia is

where A is the cross-sectional area. Note that the load and the length are specified; the free variable is the cross-sectional area

. Substituting into gives . Substituting A

from into this last result gives


As before, we obtain the functional, geometry and material parameters. The cost of the column is minimized by choosing materials with largest value of the material index given by:


Table 6.6 Procedure for deriving material indices (from Ashby)

Step Action

1 Define the design requirements:(a) Function: what does the component do?(b) Objective: what is to be maximized or minimized?(c) Constraints: essential requirements which must be met:

stiffness, strength, corrosion resistance, forming characteristics

2 Develop an equation for the objective in terms of the functional requirements, the geometry and the material properties (the objective function).

3 Identify the free (unspecified) variables.


4 Develop equations for the constraints (no yield; no fracture; no buckling, etc.).

5 Substitute for the free variables from the constraint equations into the objective function.

6 Group the variables into three groups: functional requirements (F), geometry (G), and material properties (M); thus

Performance characteristic

or Performance characteristic

7 Read off the material index, expressed as a quantity M, which optimizes the performance characteristic.


Table 5.7 Examples of material indices (from Ashby)

Function, Objective and Constraint Index

Bar, minimum weight, stiffness prescribed

Beam, minimum weight, stiffness prescribed

Beam, minimum weight, strength prescribed

Beam, minimum cost, stiffness prescribed

Beam, minimum cost, strength prescribed


Column, minimum cost, buckling load prescribed

Spring, minimum weight for given energy storage

Thermal insulation, minimum cost, heat flux prescribed

Electromagnet, maximum field, temperature rise prescribed

( = density; E = Young's modulus; = elastic limit; = cost/kg; = thermal conductivity; electrical conductivity; = specific heat)


In the material selection charts shown previously (from CES), note that the various material properties are plotted on log-log scales. The reason for this is as follows. For a condition like

where C is a constant; we can take the log of each side to obtain

or

When plotted as vs. (or E vs. on log-log scale), this equation represents a family of straight parallel lines with a slope of 2 and an intercept on the -axis of ; and each line corresponds to a value of the constant C. These lines are referred to as selection guide lines in CES. Any material falling on a given straight line will have equal values of , i.e., be of equal


"goodness" in satisfying a material index for stiffness to weight ratio. Selecting a higher curve (greater C) in the family of curves is equivalent to selecting a family of materials with higher stiffness to weight ratio.

There are many go-no go limits that may limit the values of specific properties. For example, in Example 4 (column buckling), if the diameter is constrained to , this will require a material with a modulus greater than [found by inverting equation ]

Property limits will plot as horizontal or vertical lines on material selection charts. The restriction on r* leads to a lower bound for E then given by the equation above. It might also be a design requirement that the column diameter lie within certain limits (for example, the column diameter must satisfy ). In this case, we would have both upper and lower limits on the diameter and thus upper and lower limits on the modulus E.

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