Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

David Cebon and Mike Ashby

University of Cambridge, Cambridge, United Kingdom 

Materials Selection for Aerospace Systems

NASA/TM—2012-217411

January 2012
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Steven M. Arnold

Glenn Research Center, Cleveland, Ohio

David Cebon and Mike Ashby

University of Cambridge, Cambridge, United Kingdom 

Materials Selection for Aerospace Systems

NASA/TM—2012-217411

January 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217411 iii

Contents

Summary........................................................................................................................................................................1

1.0 Introduction ..........................................................................................................................................................1

2.0 A Systematic Approach to Materials Selection ....................................................................................................4

2.1 Methodology................................................................................................................................................4

2.1.1 Step 1: Translation ..........................................................................................................................4

2.1.2 Step 2: Screening/Rejection ............................................................................................................7

2.1.3 Step 3: Ranking ...............................................................................................................................7

2.1.4 Step 4: Research..............................................................................................................................7

2.1.5 Step 5: Specific Cultural Constraints ..............................................................................................8

2.2 Material Performance Indices......................................................................................................................8

2.2.1 Case Study 1a: Blades...................................................................................................................10

2.2.2 Case Study 2a: Pressure Vessels...................................................................................................14

2.3 Material Property Charts............................................................................................................................16

2.4 Material Choices for Case Studies.............................................................................................................18

2.4.1 Case Study 1b: Blades...................................................................................................................18

2.4.2 Case Study 2b: Pressure Vessels...................................................................................................22

3.0 Advanced Selection ............................................................................................................................................24

3.1 Materials for High Temperature Applications...........................................................................................24

3.1.1 Maximum and Minimum Service Temperature ............................................................................28

3.2 Design for Creep........................................................................................................................................30

3.2.1 Constitutive Equation for Creep Deflection ..................................................................................31

3.2.2 Constitutive Equation for Creep Fracture......................................................................................32

3.2.3 Constitutive Equation for Relaxation ............................................................................................32

3.2.4 Material Selection in the Creep Regime........................................................................................33

3.2.5 The Selection Procedure................................................................................................................37

3.2.6 Case Study.....................................................................................................................................37

3.3 Engineered Materials.................................................................................................................................40

3.3.1 Holes in Material-Property Space .................................................................................................41

3.3.2 Synthesizing the Properties of Cellular Structures: Foams and Lattices.......................................42

3.3.3 Mechanical Properties...................................................................................................................44

3.3.4 Thermal Properties........................................................................................................................47

3.3.5 Electrical Properties......................................................................................................................47

3.3.6 Example: Aluminum Metal Matrix Composite (MMC) Foams and Lattices................................48

3.3.7 The CES Hybrid Synthesizer ........................................................................................................49

4.0 Data Issues..........................................................................................................................................................49

5.0 Conclusions ........................................................................................................................................................51

Appendix A.—Table of Material Indices....................................................................................................................53

A.1 Creep Fracture ...........................................................................................................................................54

A.2 Creep Deflection........................................................................................................................................55

A.3 Creep Relaxation .......................................................................................................................................56

Appendix B.—Design Strength and Indices For Creep Deflection .............................................................................57

B.1 Simple Tension or Compression: Minimum Weight Design.....................................................................57

B.2 Bending of Beams, Minimum Weight Design...........................................................................................58

B.2.1 Torsion of Shafts, Minimum Weight Design ................................................................................60

References ...................................................................................................................................................................63

NASA/TM—2012-217411 1

Materials Selection for Aerospace Systems

Steven M. Arnold

National Aeronautics and Space Administration

Glenn Research Center 

Cleveland, Ohio 44135

David Cebon and Mike Ashby

University of Cambridge

Cambridge CB2 1PZ United Kingdom

Summary

A systematic design-oriented, five-step approach to material selection is described: 1) establishing

design requirements, 2) material screening, 3) ranking, 4) researching specific candidates and 5) applying 

specific cultural constraints to the selection process. At the core of this approach is the definition 

performance indices (i.e., particular combinations of material properties that embody the performance of a 

given component) in conjunction with material property charts. These material selection charts, which 

plot one property against another, are introduced and shown to provide a powerful graphical environment 

wherein one can apply and analyze quantitative selection criteria, such as those captured in performance 

indices, and make trade-offs between conflicting objectives. Finding a material with a high value of these 

indices maximizes the performance of the component. Two specific examples pertaining to aerospace 

(engine blades and pressure vessels) are examined, both at room temperature and elevated temperature 

(where time-dependent effects are important) to demonstrate the methodology. The discussion then turns

to engineered/hybrid materials and how these can be effectively tailored to fill in holes in the material 

property space, so as to enable innovation and increases in performance as compared to monolithic 

materials. Finally, a brief discussion is presented on managing the data needed for materials selection, 

including collection, analysis, deployment, and maintenance issues.

1.0 Introduction

The importance of weight reduction in aerospace systems has been a major factor from the very 

beginning. Consequently, the use of new materials (e.g., dual alloy turbine disks and composite materials 

(ceramic, metallic and polymer based)) and new types of structural concepts, particularly the thin-walled 

type (sized primarily based on buckling) have dominated. Clearly, from a designer’s perspective the 

primary function of a structure is to transmit forces through space with the minimum possible weight and 

cost to the customer. Typically the job of a designer is to balance a variety of functional requirements 

(i.e., types of loading conditions (tension, compression, bending, vibration, cyclic, etc.) with constraints 

(manufacturability, geometric limits, environmental aspects, maintainability, to name a few) so as to 

arrive at the “optimum” choice of structural concept and material selection for a given weight and/or cost

(Figure 1).

This task can often be a daunting one for both the inexperienced as well as experience practitioner 

due to the wide range of choices available. Further, depending upon the engineer's background, for 

example a materials versus mechanics oriented professional, their answer to the question of designing for 

suitable deflection, for example, maybe quite different. The materials oriented person would typically 

think they need a material with high stiffness (i.e., Young’s modulus (E)), whereas the structures oriented 

person would naturally think of high rigidity (e.g., EI, the product of Young’s modulus and the moment 

of inertia (I) given a beam in bending); the first being material property oriented and the other 

performance oriented. This diversity in perspective is precisely why a robust and systematic methodology 

for connecting material and application is required as will be presented and illustrated in this chapter.
NASA/TM—2012-217411 2

Figure 1.—Schematic indicating the intersection of material, 

manufacturability and structural concept which determines 

the engineering utility (sweet spot) of a given “system”.

Figure 2.—Life cycle cost (Larson and Pranke, 1999).

Further, design is not a linear process as it involves multiple iterations, since the consequences of choices 

(both structural and material) made at the preliminary concept or embodiment stages may not become apparent 

until much later downstream when the details are examined more fully. Consequently, iteration (looping back 

to explore alternatives) is an essential part of the design process as is material selection which enters each stage 

of the design process. The importance of incorporating as much physics and understanding of the problem as 

early in the design process is dramatically illustrated in Figure 2, wherein it has been proposed that over 

50 percent of the life cycle cost of a given system is locked in by the end of the preliminary design phase, with 

85 percent by the end of the embodiment phase, and almost 100 percent by the time one goes to production, 

yet only 10 percent of the life cycle cost has been expended. This figure strongly suggests, once again, the need 

for a robust and systematic methodology for connecting material selection and application requirements as 

early in the process as possible to reduce the overall life cycle cost.

Clearly, material information is central to all aspects of engineering products. However, the nature of 

the data needed in the early stages differs greatly in its level of precision and breadth from that needed 

later on. At the concept-stage, the designer requires approximate property-values, but for the widest 

possible range of materials. All options are open: a polymer may be the best choice for one concept, a 

metal for another, even though the function is the same. The problem, at this stage, is not precision and 
NASA/TM—2012-217411 3

detail; it is breadth and speed of access: how can the vast range of data be presented to give the designer 

the greatest freedom in considering alternatives?

At the embodiment stage the landscape has narrowed. Here we need data for a subset of materials, but 

at a higher level of precision and detail. These are found in the more specialized handbooks and software 

that deal with a single class or sub-class of materials—metals, or just aluminum alloys, for instance. The 

risk now is that of losing sight of the bigger spread of materials to which we must return if the details do 

not work out; it is easy to get trapped into a single line of thinking (or set of “connections”) when other 

combinations of connections offer a better solution to the design problem.

The final stage of detailed design requires a still higher level of precision and detail, but for only one 

or a very few materials. Such information is best found in the data-sheets issued by the material producers 

themselves, and in detailed databases for restricted material classes. A given material (polyethylene, for 

instance) has a range of properties that derive from differences manufacturing by different producers 

make it. At the detailed design stage, a supplier must be identified, and the properties of his product must 

be used in the design calculations, given the slight properly differences observed in a material for 

different material producers. And sometimes even this is not good enough. If the component is a critical 

one (meaning that its failure could, in some sense or another, be disastrous) then it may be prudent to 

conduct in-house tests to measure the critical properties, using a sample of the material that will be used 

to make the product itself.

At the core of the material selection process is the inherent interaction between function, material, 

shape and process (see Figure 3). Function dictates the choice of both material and shape; and yet the 

selection of a material and process cannot be separated from the choice of shape.

1 Since in order to 

manufacture a shape, the material is subjected to processes2 and these processes are influenced by the 

material’s formability, machinability, weldability, heat-treatability, and so on. Also, process obviously 

interacts with shape—the process determines the shape, the size, the precision and, of course, the cost. 

And, of course, the processing route can dramatically change the properties of the material. These 

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 greater the interactions and consequently, the tighter the specifications.

Figure 3.—At the core of material selection in mechanical design is 

the interaction between function, material, shape and process.

 1

The word “shape” is used here to include the external, macro-shape, and— when necessary—the internal, or micro-shape, as in a 

honeycomb or cellular structure. 

2

Here processes include: primary forming processes (like casting and forging), material removal processes (machining, drilling), 

finishing processes (such as polishing) and joining processes (e.g., welding). 
NASA/TM—2012-217411 4

2.0 A Systematic Approach to Materials Selection

To ‘select’ means ‘to choose’. But from what? Behind the concept of selection lies that of the 

permissible set of entities from which the choice is to be made. Within an engineering enterprise, the 

permissible set of materials will probably be the ‘preferred’ list, specified by the “Materials Authority”.

However, for truly ‘optimal’ materials selection, the permissible set of entities must include all materials: 

all metals, all polymers, all ceramics and glasses, all composites. We refer to this as the ‘Universe’ of 

materials. If some of these materials are left out, the selection is no longer optimal over all materials but is 

only optimal over some subset of them. If, for example, the choice is limited to metals from the start, then 

the selection can only be optimal over the single family of materials, that of metals.

There is a second implication to the concept of selection. It is that all members of the permissible set 

must be regarded as candidates—they are, after all, viable choices—until, by a series of selection stages, 

they are shown to be otherwise. From this arise several key requirements of a data structure for the 

Selection database. The selection table must be comprehensive (includes all members of the permissible 

set). It should be structured—that is it must contain attributes that are universal (apply to all members of 

the permissible set of entities) and the attributes should be discriminating (have recognizably different 

values for different members of the permissible set). Similar considerations apply to any selection 

exercise (Cebon and Ashby 2000).

In the universe of materials, many attributes are universal and discriminating: density, bulk modulus and 

thermal conductivity are examples. Universal attributes can be used for screening and ranking, the initial 

stage of any selection exercise. But if the values of one or more screening attributes are grossly inaccurate or 

missing, that material will be eliminated by default. It is important, therefore, that the database be complete

having no holes or gaps in the records which would make a material fail a selection by default due to 

absence of data and be of high quality, meaning that the data in it can be trusted. This creates the need for 

data checking and estimation, which are tackled by methods described in Ashby (1998).

There are four main steps, which we here call translation, screening/rejection, ranking, and supporting 

information (research and organizational cultural constraints) see Figure 4. The steps can be likened to those 

in selecting a candidate for a job. The job is first analyzed (i.e., essential skills and required experience of 

the candidate identified; “translation”) and advertised. Some of these are simple go/no go criteria like the 

requirement that the applicant “must have a valid driving license”, or “a degree in materials science”, 

eliminating anyone who does not (“screening”). Others imply a criterion of excellence, such as “typing 

speed and accuracy are priorities”, or “preference will be given to candidates with a substantial publication 

list”, implying that applicants will be ranked by these criteria (“ranking”). Finally references and interviews 

are sought for the top ranked candidates, building a file of supporting information (research and local 

cultural constraints)—an opportunity to probe deeply into character and potential.

2.1 Methodology

2.1.1 Step 1: Translation

How are the design requirements for a component (defining what it must do) translated into a 

prescription for a material? Any engineering component has one or more functions: to support a load, to 

contain a pressure, to transmit heat, and so forth. This must be achieved subject to constraints: that certain 

dimensions are fixed; that the component must carry the design loads or pressures without failure; that it 

insulates or conducts; that it can function in a certain range of temperature and in a given environment; 

and many more. In designing the component, the designer has an objective: to make it as cheap as

possible, perhaps, or as light, or as safe, or perhaps some combination of these. Certain parameters can be 

adjusted to optimize the objective giving the designer the freedom to vary dimensions that have not been
NASA/TM—2012-217411 5

Figure 4.—Schematic illustrating a material selection strategy with the five 

specific steps being: Design requirements, material screening/rejection, 

ranking, research and organizational cultural constraints are shown.

constrained by design requirements and, most importantly, free to choose the material for the component. 

We refer to these as free variables. Function and constraints, objective and free variables (see Table 1) 

define the boundary conditions for selecting a material and—in the case of load-bearing components—a 

shape for its cross section. The first step in relating design requirements to material properties is a 

clear statement of function, constraints, objective, and free variables.

TABLE 1.—FUNCTION, CONSTRAINTS, OBJECTIVES AND FREE VARIABLES

Function What does component do?

Constraints* What non-negotiable conditions must be met?

What negotiable but desirable conditions?

Objective What is to be maximized or minimized?

Free variables What parameters of the problem is the designer free to change?

*It is sometimes useful to distinguish between “hard” and “soft” constraints. Stiffness and strength might be 

absolute requirements (hard constraints); cost might be negotiable (a soft constraint).

2.1.1.1 Function

Structural elements are components that perform a physical function (i.e., they carry loads, transmit 

heat, store energy, and so on) in short, they satisfy functional requirements. The functional requirements 

are specified by the design; for example a tie rod must carry a specified tensile load; a spring must 

provide a given restoring force or store a given energy; a heat exchanger must transmit heat at a given 

heat flux. Furthermore the loading on a given component/structure can generally be decomposed into 

some combination of axial tension or compression, bending and torsion; wherein almost always, one 

mode dominates. So common is this that the functional name given to a component often describes the 

way it is loaded: ties carry tensile loads; beams carry bending moments; shafts carry torques; and columns 

carry compressive axial loads. The words “tie”, “beam”, “shaft”, and “column” each imply a function; 

such that many simple engineering functions can be described by single words or short phrases, saving 

the need to explain the function in detail. Table 2 itemizes some aerospace components and their 

associated simple idealization that can be extremely useful for material selection purposes.
NASA/TM—2012-217411 6

TABLE 2.—AEROSPACE APPLICATIONS/IDEALIZATIONS

Application Idealization Loading Constraints

Truss framework Tie Tension Strength

Fuselages Cylinder

Compression

Tension

Bending

Buckling

Strength

Deflection

Shafts Thin walled tube/Beam

Torque

Rotation

Bending

Bending

Shear

Vibration

Combustion Chambers Thin walled Cylinder,

Internal pressure

Pressure

Noise

Thermal

Oxidation

Blades 

(Fan, Compressor, 

Turbine)

Tie

Beam

Bending

Tension

Fatigue

Thermal

Creep

Deflection, Strength, Vibration, 

Erosion, Temperature

Disks 

(Compressor, Turbine) Disk Rotation, Thermomechanical Fatigue

Strength, Ductility, Toughness, 

Oxidation

Space Truss 

Framework Tie rod Axial Loading Strength, Buckling

Nozzles Thin walled Cylinder

Pressure

Noise

Thermal

Strength, Vibration

Wing Plate

Bending

Tension

Twist

Vibration

Fatigue

Buckling,

Vibration, Strength

Deflection

Cryogenic Tanks Spherical Shell Internal Pressure

Thermal

Strength, low temperature

Yield before break

Leak before break

2.1.1.2 Constraints

Constraints are design requirements that must be satisfied: for example, the minimum working 

temperature of a material must be ≤ –20 °C; the strength must be ≥ 400 MPa; the fracture toughness must 

be ≥ 15 MPa√m, and so on. Constraints screen out unsuitable choices or enable rejection of specific 

materials for cause.

2.1.1.3 Objectives

Objectives are design criteria that must be maximized or minimized to optimize the performance of a 

component. Their function is to rank the materials and facilitate selection of the best candidates. For 

materials selection, the objectives can be used to generate ‘material performance indices,’ which are 

combinations of material properties that characterize performance in a given application (see Sec. 2.2.2).

Typical examples are the specific stiffness of a material E/ρ, and the specific strength σf /ρ (E is the 

Young’s modulus, σf is the failure strength and ρ is the density). These particular indices can be used to 

select the optimum material for a light, stiff tie rod, or a light, strong tie rod, respectively. Many material 

performance indices have been derived and tabulated for standard design cases in mechanical, structural, 

thermo-mechanical, and electro-mechanical engineering (Ashby 2005).

2.1.1.4 Free Variables

Free variables are any parameters that are left open to the designer’s free choice. Examples are: crosssectional area, length, thickness, material type, color, etc.
NASA/TM—2012-217411 7

2.1.2 Step 2: Screening/Rejection

Unbiased selection requires that all permissible materials are considered to be candidates until shown 

to be otherwise. Rejection eliminates candidates that cannot do the job at all because one or more of their 

attributes lies outside the limits set by the constraints. As examples, the requirement that “the component 

must function in boiling water,” or that “the component must be transparent” imposes obvious limits on 

the attributes of maximum service temperature and optical transparency that successful candidates must 

meet. We refer to these as attribute limits. One should not be too hasty in applying attribute limits 

however, as it may be possible to engineer around them; for example, a component that gets too hot 

maybe cooled (or the geometry modified to enable use of the same material at lower stress, EI versus E), 

or one that corrodes may be coated with a protective film. An aerospace specific case in point is the desire 

by many an engineer to impose a minimum limit on fracture toughness and strain to failure (e.g., KIC >

15 MPa/m2 and ef >2 percent) in order to ensure adequate tolerance to stress concentrations. This is most 

likely to be a rash step this early in the design process, as it would eliminate all polymers, ceramics and 

many composite materials from consideration. Since a more innovative designer may be able to put these 

materials to good use. Consequently, at this stage in the process the engineer needs to try to keep as many 

options open as possible.

2.1.3 Step 3: Ranking

Attribute limits do not, however, help with ordering the candidates that remain after the screening 

constraints have been applied. To do this we need optimization criteria. They are found in the material 

indices, a subset of the performance indices3 described in Section 2.2.2, which measure how well a 

candidate can do the job. Performance is sometimes limited by a single property, sometimes by a 

combination of them. For example, the best materials for buoyancy are those with the lowest density, ρ; 

those best for thermal insulation the ones with the smallest values of the thermal conductivity, λ. Here 

maximizing or minimizing a single property maximizes performance. But as we shall see—the best 

materials for a light stiff tie-rod are those with the greatest value of the specific stiffness, E/ρ, where E is 

Young’s modulus. The best materials for a spring are those with the greatest value of σ2

f / E where σf is 

the failure stress. The property or property-group that maximizes performance for a given design is called 

its material index. There are many such indices, each associated with maximizing some aspect of 

performance. These indices provide significant insight into the types of optimization strategies needed as 

well as selection guidelines for the given problem at hand.

To summarize: material rejection (or screening) isolates candidates that are capable of doing the job; 

ranking identifies those among them that can do the job best.

2.1.4 Step 4: Research

The outcome of Steps 1 through 3 produces a ranked short-list of candidates that meet the constraints 

and that maximizes or minimizes the criterion of excellence; whichever is required (this is why in 

Figure 4 we have used a funnel.

You could just choose the top-ranked candidate, but what bad secrets might it hide? What are its 

strengths and weaknesses? Does it have a good reputation? What, in a word, is its pedigree? To proceed 

further we seek a detailed profile of each candidate: it’s supporting information (see Step 4 in Figure 4). 

Supporting information differs greatly from the structured property data used for screening. Typically, it 

is descriptive, graphical or pictorial: case studies or experience of previous uses of the material, details of 

its corrosion behavior in particular environments, information of availability and pricing, experience of its 

environmental impact. Such information is found in corporate documentation, handbooks, suppliers’ data 

sheets, CD-based data sources and the world-wide web. Supporting information helps narrow the shortlist to a few final choices, allowing a definitive match to be made between design requirements and 

 3

They provide criteria of excellence that allow ranking of materials by their ability to perform well in the given application.
NASA/TM—2012-217411 8

material attributes. Why are all these steps necessary? Without screening and ranking (material rejection), 

the candidate-pool can be enormous and the volume of supporting information overwhelming. Dipping 

into it, hoping to stumble on a good material, gets you nowhere. But once a small number of potential 

candidates has been identified by the screening-ranking steps, detailed supporting information can be 

sought for these few candidates, and the task becomes tractable.

2.1.5 Step 5: Specific Cultural Constraints

The final choice between competing candidates will, often, depend on local conditions: on in-house 

expertise or equipment, on the availability of local suppliers, and so forth. A systematic procedure cannot 

help here—the decision must instead be based on local/institutional knowledge. This does not mean that 

the result of the systematic procedure (describe above) is irrelevant. It is always important to know which 

material is best, even if, for local reasons (e.g., specific processing method for a possible material is to 

expensive), you decide not to use it. 

2.2 Material Performance Indices

The screening and ranking process can be made quantitative by linking the technical (e.g., functional, 

geometric, material) and economic requirements of the design to the attribute profiles stored in a given 

database. The foundations of a robust and systematic ranking procedure are particular combinations of 

material properties that embody the performance of the component. These property combinations are 

called performance indices, as put forth by Ashby (2005). The performance, P, of a structural element is 

determined by three, typically independent, aspects: the functional requirements, F, the geometry, G, and 

the properties of the material, M, of which it is made. The performance P of the element can often be 

described by an equation of the form

P = f (F, G, M) (1)

where P, the performance metric, describes some aspect of the performance of the component; (for 

example its mass, volume, cost, or life) and “f ( )” means “a function of”. Optimum design is the selection 

of the material and geometry that maximize or minimize P, according to its functional requirements.

When this group of parameters can be assumed to be separable (that is, M, F and G are independent 

of each other), then the performance index is merely a product of three functions, f1, f2, and f3;

P = f1 (F) ⋅ f2 (G) ⋅ f3 (M) (2)

where the product f1(F) ⋅ f2(G) has been defined as the structural efficiency coefficient (and incorporates 

both functional requirements and geometry), and f3(M) is defined as the material efficiency coefficient. 

With this significant simplification, the overall performance index can be maximized or minimized by 

selecting a material to minimize f3(M), independent of the details of the design. This enables the optimal 

subset of materials to be identified without solving the complete design problem. While this is clearly a 

simplification of the full coupled design problem, it can provide a great deal of insight in the preliminary 

design stage of a project. This powerful and general method is simple, provided one is able to clearly 

identify, from the outset, the objective(s) (what you are trying to maximize or minimize), the constraints, 

and which parameters are specified and which are free. 

When any component function is combined with specific constraints and objectives, a specific 

material index (Mp = f3(M)) is produced as illustrated in Figure 5. Therefore, it is possible to develop 

numerous material and structural indices that enable one to efficiently and accurately perform material 

selection for a variety of design functions (e.g., stiffness, strength, and vibration-limited designs, as well 

as damage-tolerant, electro-mechanical, thermal, and thermal-mechanical designs) on a number of 

fundamental structural components described in Table 2 (i.e., tie, beam, shaft, column, panel, 

cylinder/shell with internal pressure and rotating disks), see Appendix A and Ashby (2005) for additional 
NASA/TM—2012-217411 9

details. The power of this approach is clearly illustrated by examining the differences in the resulting 

material index for a tie, beam and plate (see Figure 6) with a specified stiffness; that is 1 2 1 3 ; ; E E E

ρ ρ ρ , 

respectively. Clearly, the only difference in the index is the exponent on Young’s modulus where for a tie 

its one, a beam its 1⁄2 and a plate is 1/3. Consequently, the best materials for a light, stiff tie rod, beam or 

plate are those with the smallest values of 1 2 1 3 ; ; E E E

ρ ρ ρ , respectively. The significance of this difference in 

exponents on the choice is best seen by creating a plot of Young’s modulus versus density (i.e., a material 

property chart) as will be described in Section 2.2.3.

Figure 5.—The specification of function, objective, and constraint leads to a material index (Ashby 

2005).

Figure 6.—(a) A tie with cross-section area A, loaded in tension. Its stiffness is S = F/δ where F is 

the load and δ is the extension. (b) A panel loaded in bending. Its stiffness is S = F/δ, where F is 

the total load and δ is the bending deflection. (c) A beam of square section, loaded in bending. 

Its stiffness is S = F/δ, where F is the load and δ is the bending deflection.
NASA/TM—2012-217411 10

Figure 7.—Defines the influence of 

the end constraint the elastic 

flexural stiffness for beams and 

plates, see Table 3.

TABLE 3.—DESIGN REQUIREMENTS LIGHT STIFF: TIE, BEAM, PLATE.

Function Tie Beam

(A = b · b)

Plate 

(A = h · W)

Constraint

Functional

Tie must support axial 

tensile load F without 

deflecting too much; 

(S = F/δ ≥ A·E/Lo)

Beam must support a bending load 

F without deflecting too much, 

meaning that the bending stiffness 

S is specified (S ≥ C1·E·I/L3)

Panel must support a bending load F

without deflecting too much, i.e., S is 

specified (S ≥ C1·E·I/L3

)

Geometric Length Lo is specified Length L is specified Length L and width W are specified

Objective Minimize mass (m = A·L·ρ) Minimize mass (m = A·L·ρ) Minimize mass (m = W·h·L·ρ)

Free Variables Cross section area, A

Choice of material

Cross section area, A

Choice of material

Panel thickness, h

Choice of material

Performance 

equation m ≥ (S) (L2

) ( ) E

ρ m ≥ ( )

1 2 12

C1L

S

C L

S

1

12 (L3

) ( 1 2 ) E

ρ m ≥ ( )

1 3 12

C1L

S

C W

S

1

12 (WL2

) ( 1 3 ) E

ρ

Before continuing we want to introduced two types of applications that will be utilized throughout 

this chapter to demonstrate the various concepts discussed herein; these are those of blades be they in the 

fan (low temperature), compressor (mid temperature) or turbine (high temperature) sections of a jet 

engine (see Figure 8) and that of an internal pressure vessel (e.g., cryogenic tank, fuel lines or 

combustion chambers within a jet engine, see Figure 10). Here we will establish the indices that will be 

useful for material selection.

2.2.1 Case Study 1a: Blades

An important aeronautic application that one can utilize to illustrate the material selection process is 

that of fan blades, compressor blades or turbine blades in a gas turbine engine, as shown in Figure 8. The 

main functional distinction in the blades used in the different location is size, shape and temperature 

capacity. Figure 8(c) illustrates how a typical blade can be idealized as a cantilever beam subjected to a 

combination of axial tension (due to centrifugal loads), bending (due to pressure variations across the 

blade) and vibration loadings (due to the motion of the rotor and rapidly changing air flow field in the 

engine). 
NASA/TM—2012-217411 11

Figure 8.—Schematics of turbine jet engines (a) with and (b) without high by 

pass ratios, along with a typical blade response (c) due to engine 

operations..

The length L and force F are specified and are therefore fixed; the cross-section A, is free. We can 

reduce the mass by reducing the cross-section, but there is a constraint: the section-area A must be 

sufficient to carry the tensile load F, given a failure strength σf; thus requiring that

f

F A

σ

≥ (3)

Consequently, combining this constraint with the objective function we arrive at: 

( )( ) 















σ

ρ ≥

f

m F L (4)

Note the form of this result. The first bracket contains the specified load F. The second bracket contains 

the specified geometry (the length L of the tie/beam) which together makeup the structural index 

previously discussed. The last bracket contains a specific relationship of material properties which were 

previously defined as the material index. The lightest beam that will carry F safely then is that one made 

of the material with the smallest value of ρ/σf. We could define this as the material index of the problem, 

seeking a minimum, but it is more usual, when dealing with specific properties, to express them in a form 

for which a maximum is sought. We therefore invert the material properties in Eq. (4) and define the 

material index M1, as

















ρ

σ

≥ f M1 (5)
NASA/TM—2012-217411 12

Therefore, the lightest beam or tie-rod that will safely carry the load F without failing is that with the 

largest value of this index, the “specific strength”. A similar calculation for a light stiff tie (one for which 

the stiffness S rather than the strength σf is specified) leads to the previous index, given in Table 3.

2.2.1.1 Performance Index for Bending at Room Temperature

Consider the case of a beam of section-area, A, length, L, subjected to bending, it must support a 

specified load F without failing and be as light as possible, see Figure 6. From strength of materials 

theory, the stress due to bending is

Z

M

I

My σmax = = (6)

where I is the moment of inertia, y the distance from the neutral axis to outer edge of the beam, and Z is 

the section modulus. Failure occurs than if the load exceeds the moment, i.e.,

M = Zσmax = Zσ f (7)

where again σf is the stress at which failure of the material will occur. To enable the section shape to 

change for the same area of material, the section modulus Z is replaced by φBZo where Zo is the section 

modulus for a square beam, i.e., Zo = A3/2/6, and φB is the shape efficiency factor. The maximum shape 

factor can be considered to be a material property and used profitably in the selection of materials. Note, 

solid equiaxed sections (circles, squares, hexagons, etc.) all have efficiency factors close to 1; whereas 

efficient shapes like thin-walled tubes or I-sections can have shape factors of 50 or more. The physical 

limit to φB is usually set by local buckling of the component. 

As before in the case of the tie rod, the mass can be decreased by reducing the cross-section, but there is 

a constraint: the section-area A must be sufficient to carry the bending stress, given a failure strength σf; thus 

requiring that

2 3

o

6 















/ σ

≥

B f

M A (8)

Finally, combining this constraint with the objective of minimum weight, m = ρLA, one obtains the 

following expression:

( )

2 3

3 2 2 3

o

6 















/ σ

ρ ≥

B f

m M L (9)

Consequently, the best material and shape combination is that with the greatest value of the material 

index

( )

ρ

/ σ

≥

2 3

2

oB f M (10)

2.2.1.2 Performance Index for Vibration at Room Temperature

There are two ways to mitigate vibration of a mechanical system: by ensuring that the input does not 

excite a natural frequency, or, if this is not possible, by ensuring that the system is sufficiently well damped to 

avoid excessive motion at resonance. Here we consider the first of these. In many cases, the need is to avoid 
NASA/TM—2012-217411 13

exciting the lowest natural frequency of the system, and this can be achieved by ensuring that it occurs at high 

frequency, relative to the input.

Any undamped system vibrating at one of its natural frequencies can be reduced to the simple 

problem of a mass m attached to a spring of stiffness k—the restoring force per unit displacement. The 

natural frequency of such a system is

m

k f

π = 2

1 (11)

Different geometries require appropriate estimates of the effective stiffness, k, and mass, m—often these 

can be estimated with sufficient accuracy by approximate modeling. Depending on the geometry of the 

beam and the end constraints, the higher natural frequencies of rods and beams can be simple multiples of 

the lowest. Figure 9 shows the lowest natural frequencies of the flexural modes of uniform beams or 

plates of length L with various end constraints. The spring stiffness k is equivalent to the stiffness S in 

Table 3 for beam/plate bending, so the natural frequencies can be written:

4

2

2 m L

C EI f

π o

= (12)

where C2 (given in Figure 9) depends on the end constraints and mo is the mass of the beam per unit 

length. The mass per unit length is just the area times the density, Aρ, so the natural frequency becomes

ρ

/

π = E B

L

C A f o

2 12 4

2 (13)

where I = φBIo has been utilized to enable the most efficient shape to be considered. Thus, frequencies 

scale as M3 = Eo/ B ρ when all dimensions are specified or when the stiffness is specified

ρ

/ 



 





π = E B

L

C S f o

2 12

1 2

5

2 (14)

the frequencies scale with ρ

Eo/ B .

Figure 9.—The natural vibration 

modes of beams clamped in 

different ways.
NASA/TM—2012-217411 14

2.2.2 Case Study 2a: Pressure Vessels

Multiple applications/structures can be classified as cylindrical or spherical in nature, particularly in 

the case of space structures. Often these structures are internally pressurized as well, for example the feed 

lines on rocket engines, composite overwrap pressure vessels, cryogenic fuel tanks, and 

expendable/reusable launch vehicles, as shown in Figure 10. Consequently, in this case study,

performance indices pertinent to cylindrical/ spherical pressure vessels will be identified.

In the case of cryogenic storage tanks, multiple design configurations can be, and have been, 

envisioned from a single tank with insulation to hybrid tanks with either insulating materials or pure 

vacuum in between walls or various combinations thereof. The overall objective of the designs is to have

a safe, lightweight, thermally-efficient cryogenic storage system. Some important tank system parameters 

relative to flight durations are presented in Table 4. The materials, tank structural configurations, and 

insulation system options are numerous and interdependent.

Figure 10.—Various space structures/components/vessels; a) Space Shuttle Main 

Engine, SSME, b) schematic of Ares V heavy lift system, c) composite over wrap 

pressure vessels, d) double walled cryogenic storage tank. .

TABLE 4.—IMPORTANT TANK SYSTEM PARAMETERS

RELATIVE TO FLIGHT DURATION

[Listed in order of importance.]

Short flight 

duration

Mass density

Strength and toughness

Coefficient of thermal expansion

Stiffness

Thermal diffusivity

Thermal conductivity

Long flight

duration

Mass density

Thermal conductivity

Strength and toughness

Coefficient of thermal expansion

Stiffness

Thermal diffusivity
NASA/TM—2012-217411 15

The key material indices applicable for both thermal and mechanical issues of interest in the case of 

cryogenic tanks are shown in Table 5. Clearly it is desirable to use materials that possess high strength, 

high fracture toughness, and high stiffness, as well as low density and low permeation to liquid and 

gaseous hydrogen; however, no single material provides all these attributes simultaneously. 

Consequently, material performance indices associated with these properties, such as those given in 

Table 5, must be used to identify the best material candidates for hybrid tank wall construction. Among 

these parameters, strength and density tend to dominate the design criteria.

TABLE 5.—PERFORMANCE INDICES FOR THERMAL AND

MECHANICAL COMPONENTS OF CRYOGENIC STORAGE TANK

Function and constraints Performance index, 

maximize

Thermal 

Minimum heat flux at steady state, fixed thickness 1/k

Minimum temperature rise in specified time, fixed thickness 1/a

Maximum energy stored for given temperature rise and time k/α1/2

Minimum thermal distortion k/α

Mechanical

Strength-limiting design with minimum mass σf/ρ

Damage-tolerant design with minimum mass KIc/ρ

Deformation-limiting design with minimum mass E/ρ

k = thermal conductivity 

a = thermal diffusivity (k/ρCp)

ρ = mass density 

Cp = specific heat 

α = coefficient of thermal expansion 

σf = strength 

KIc = mode I fracture toughness 

E = Young’s modulus 

Time, t = w2/2a with w = wall thickness. 

In the case of thin-walled spherical pressure vessels of radius R, the stress in the wall is given by 

t

PR

2

σ = (15)

And the mass of the thin-walled spherical tank is given by

m = πR tρ2 4 (16)

From fracture mechanics we know that a crack will propagate by fast fracture when the stress intensity 

factor, which is a combination of the applied far field stress, σ, and crack length, 2c, reaches the fracture 

toughness of the material, KIc. This condition can be expressed by the following relation:

KIc = Yσ πc (17)

where Y is a parameter that takes on various values depending upon the changes in crack geometries and 

loading conditions (see Broek (1984) and Anderson (1995)). 

Consequently the internal pressure at which the working stress of the tank is below the critical stress 

that would cause a crack to propagate can be obtained by rearranging Eqs. (15) and (17) and substituting 

the results:

















π

≤

Y c

K

R

t P 2 Ic (18)
NASA/TM—2012-217411 16

Clearly, the largest pressure (for a given R, t and c) is obtained for the material with the largest fracture 

toughness since the pressure in Eq. (18) is proportional to KIc. Such a design, however, is not fail-safe, so 

additional conditions have been traditionally introduced to provide a greater level of safety. These are 

known as “yield-before-break” and “leak-before-break”.

Yield-before-break requires that the stress to cause fracture be greater than that to yield the material, 

thus providing a state of detectable deformation prior to fracture. This condition is expressed as:

2

I

2 max

1

















π σ

≤

y

K c

Y

C (19)

where the stress in Eq. (17) has been replaced by the yield stress, σy, of the material. Consequently, the 

tolerable crack size, and thus integrity of the pressure vessel, is maximized by maximizing the following 

material index:

y

M K c

σ = I

4 (20)

Leak-before-break, typically utilized for large pressure vessels, requires that a stable crack be just 

large enough to penetrate both the inner and outer surfaces of the vessel (i.e. without fast facture) so that 

the leak caused by the presence of the crack can be detected. This condition is obtained by requiring that 

the unstable crack size exceeds the thickness (i.e., 2c > t) and that the wall thickness, t, contain the 

pressure, P, without yielding. This condition leads to the following:

















π σ

≤ y

K c

RY

P

2

I

2

4

(21)

Consequently, the pressure is carried most safely when the material index

y

K c M

σ =

2

I

5 (22)

is the largest possible.

Both of these material performance indices, M4 and M5, can be maximized by choosing materials with 

low yield stresses, however, that could be problematic since the thickness and therefore the mass of the 

tank is inversely related to the yield stress, see Eqs. (15) and (16). Therefore for aerospace applications, 

which are typically weight critical, this demands that σy/ρ be as large as possible since

σ ρ

π ≥

y

R P

m

3 2

(23)

These indices will be used latter to down select materials for use with lightweight pressure vessels.

2.3 Material Property Charts 

Material property charts enable the condensing of a large body of information into a compact but 

accessible form. They reveal correlations between material properties that aid in checking and estimating data; 

and in conjunction with performance indices, constitute the backbone for tackling real design problems. They 

provide a convenient way of mapping out the areas of 'property space' occupied by each material class, and the 

sub-areas occupied by individual materials and provide a mechanism for surveying design-limiting properties 

(be it a single value or a combination of properties). 
NASA/TM—2012-217411 17

Utilizing the performance indices discussed in the previous section, enables the engineer to choose the 

chart axes so that function specific information can be displayed. For example, a chart of Young’s modulus 

(E) verses density (ρ), see Figure 11, not only guides the best choice of material for the stiffness limited 

design of a tie rod, beam or a plate, (see Appendix A) but also shows the longitudinal wave velocity (E/ρ)

1/2

and design for vibration. Similarly a plot of fracture toughness KIC against modulus E shows the critical 

strain energy release rate (toughness) GIC; a diagram of thermal conductivity (κ) verses diffusivity (a) also 

gives the volume specific heat ρCv; tensile strength (σf) verses Young’s modulus (E) shows the elastic 

energy-storing capacity, σf

2

/E, to name a few. 

The most striking feature of the charts, as pointed out by Ashby (2005) is the way in which members of 

a material class cluster together, despite the wide range in various properties. For example, the modulus and 

density of metals (denoted by dark gray contour labeled Metals) in Figure 11, occupy a field that is distinct 

from that of polymers (denoted as lighter gray), or that of ceramics (lightest gray ellipse), or that of 

composites (hashed ellipse). The same is true of strength, toughness, thermal conductivity and the rest. Even 

though the fields sometimes overlap, they always have a characteristic place within the whole picture, due to 

the physical make up of the various materials. All charts have one thing in common, that some areas within 

the given material space are not populated. Some of these areas are inaccessible for fundamental reasons 

that relate to fundamental physics: the size and packing of atoms and the nature of the forces that bind their 

atoms together. But other areas are empty even though, in principle, they are accessible and if accessed 

would enable novel design possibilities. It is precisely this clustering, in combination with the material (or 

more generally the performance) indices that enable the rejection and ranking of various materials with 

respect to structural function.

Material property charts have numerous applications. One is the checking and validation of data, 

another concerns the development of, and identification of uses for, new materials which can fill in empty 

regions, see Section 2.3.2. But most important of all, the charts form the basis for the previously presented 

procedure for materials selection. 

On a log-log plot, a contour with a constant value of a material index plots as a straight line. For 

example, the performance index for a light-stiff plate is given by (Appendix A):

= ρ 1 3 M6 E

Taking logs of both sides:

( ) = log( )− log(ρ) 3

1 log M6 E

or

( ) ( ) ( ) 6 log E = 3log ρ + 3log M

Consequently each line in the family of straight parallel lines of slope 3, on a plot of log(E) verses 

log(ρ), corresponds to a fixed value of M6. All materials that lie on a line of constant E1/3/ρ perform 

equally well as a light, stiff panel, while those above the line perform better (are lighter for a specified 

stiffness) than those below the line.

Examining Figure 11 it can be seen that Mg-alloys, Al-alloys, Ti-alloys, steels and W-alloys all fall near a 

line with the same value of E/ρ (the index for displacement limited tie rod subjected to tensile loading, see the 

heavy solid line). This raises the question: Why do we use aluminum in the manufacture of airplanes and not 

steel? This is immediately obvious when one takes into account the appropriate material index for plates/shells 

subjected to bending, i.e., E1/3/ρ, see Appendix A. This is plotted as a dashed line of slope 3, passing through 

the aluminum bubble on Figure 11. It can be seen that aluminum has a much higher value of E1/3/ρ than steel 

and therefore has superior performance for a light, stiff plate. Additional attribute limits (e.g., horizontal or 

vertical lines) can also be added to narrow the associated search window (e.g., E > 10 GPa or ρ > 2 Mg/m2

), 

thereby producing a short list of candidate materials. Material property charts will be utilized in the next 

section for two case studies to illustrate the utility of these charts.
NASA/TM—2012-217411 18

Figure 11.—Modulus versus density property chart.

2.4 Material Choices for Case Studies

2.4.1 Case Study 1b: Blades

Returning to our first case study related to jet engine blades (be they fan, compressor or turbine) one 

can quickly identify the various materials that will be best in the case of resisting bending loads. As 

indicated in Eq. (10) the appropriate material index is 

( )2 3

2

o

ρ

/ σ

≥ B f M (24)

for temperatures wherein time-dependent behavior can be ignored. This index can also be utilized to indicate 

which material would be useful for each region of the engine if plotted against maximum service temperature, 

see Figure 12. For example, the fan section of jet engine experiences temperatures up to 290 °C, with the 

compressor section experiencing temperatures up to 700 °C, and the turbine experiencing 1600°C or less. 

Since the actual material index, M2, is plotted as the ordinate in Figure 12, a horizontal line becomes our 

selection guideline. Therefore, considering the bending strength alone, it becomes immediately obvious, which 

materials could be the best choice for low mass blades. For fan blades, Al, Ti, PMCs; for compressor blades, 

Ti-6-4, Ti-6-2-4-6, B/Al, beryllium; and for turbine blades Inconel 718, Waspaloy, and SiC/SiC. Note that 

beryllium would be eliminated for toxicity reasons rather mechanical performance. (This elimination would 

occur in the research stage of the selection process in Figure 4). These are but a few of our choices (depending 

upon the required value of the material index) that have been substantiated over the years through direct

experience. 

It is also clear from Figure 12 that if the choice is limited, apriori, to metals, then as temperature 

increases (as the application region moves toward the rear of engine) the material performance index must 

be lowered. This indicates that the mass of higher temperature blades must be greater than lower 

temperature blades, for a similar duty. However, if the design space is opened-up to include non-metals 

(particularly ceramics) the index can be maintained (in fact, doubled from 10 to 20) even at elevated 

temperatures. It is evident then why significant investment is being placed upon the development of 

composite alternatives to enable significantly lighter weight components for elevated temperatures. 
NASA/TM—2012-217411 19

Figure 12.—Flexural strength versus maximum service temperature.

Bending stress is but one type of loading required to be sustained by a given blade design (albeit fan, 

compressor or turbine), the others being axial tension (due to centrifugal loads) and vibrational loadings 

(due to rotational motion and fluctuating gas flows through the engine). As shown previously, to 

minimize mass (or weight) of a given blade requires that a material with a high specific strength (σf/ρ) in 

tension be selected and a high value of E1/2/ρ to maximize its fundamental vibrational frequency. Note 

any weight savings in blades has a multiplying effect on other components such as significant reduction 

in compressor and turbine disks as well as shaft loads.

To assist in making the trade between several constraints the software package 'CES Selector' from 

Granta Design can be utilized (Granta, 2011). This software enables engineers to depict graphically the 

various material indices and overlay them to arrive at a subset of candidate materials that can be 

investigated further. For a discussion on how to deal with multiple constraints and objectives the reader is 

referred to Chapter 9 of Ashby (2005).

Figure 13 shows the subset of materials after specifying values for the three material indexes given 

tensile loading, bending, and vibration. The values assumed were: ( ) ( ) 60, 10, 2 3 o

1 = ≥ 2 = ≥

ρ

/ σ

ρ

σ f B f M M

and M3 = Eo/ B ρ ≥10,000 . Note that o/ B = 1 in both M2 and M3. Figure 13(a) shows the subset of 

materials when each index is treated independently, while Figure 13(b) shows the corresponding results 

when interaction between indexes are accounted for, that is if a given material is eliminated due to failure 

to meet anyone criterion then it is removed in all three property chart spaces whether or not it fails in that

particular space or not. Comparing Figure 13(a) to (b) it is immediately obvious that by considering all 

loading criteria at once (i.e., Figure 13(b)) leads to a significantly smaller subset of workable material 

systems for the various temperature ranges or blade types. This assumes the value of the material index 

(requirement) would be the same in all three temperature regimes. It is interesting to note that Ti-6-4, Ti-6-2-4-

6, Inconel 718, and Waspoloy are equivalent with respect to specific stiffness but titaniums are significantly 

better in both bending and specific strength thus substantiating their use in fan and compressor blade 

applications. A key factor yet to be considered is inelastic effects that come into play at elevated temperatures; 

this will be briefly considered in Section 2.3 Advanced Selection.
NASA/TM—2012-217411 20

Figure 13.—Illustration of the interaction between three material indices for a blade subjected to 

tension, bending and vibration. (a) Independent; and (b) Interacting constraints.
NASA/TM—2012-217411 21

Figure 13.—Concluded.
NASA/TM—2012-217411 22

2.4.2 Case Study 2b: Pressure Vessels 

The storage of liquid hydrogen in a lightweight tank provides significant challenges. The low density of 

the hydrogen fuel results in the need for a larger volume storage vessel relative to other fuels. Mechanical tank 

loads are derived from 1) the difference in pressure within the tank and the ambient conditions, 2) fuel weight, 

3) vehicle acceleration loads, 4) fuel slosh due to aircraft maneuvers, and 5) the weight of the tank system and 

its supports. Fuel slosh is bound to be encountered as the aircraft maneuvers or as it encounters air turbulence 

during the flight. Furthermore, the internal tank pressure has to be maintained at a constant absolute pressure in 

order to maintain the hydrogen in a liquid state. The tank weight increases with increasing operating pressure. 

Typically, gaseous hydrogen is used as the pressurant for liquid hydrogen. Two important criteria for tank wall 

design include material selection and wall architecture, which are not mutually exclusive. Many options are 

available and their advantages and disadvantages are discussed in Mital et al. (2006) where guidelines are 

provided for choosing an optimum system. Here the discussion will be limited to selection of the tank wall 

material, based upon mechanical loading alone to illustrate the selection process. 

Clearly it is desirable to use materials that possess high strength, high fracture toughness, and high stiffness, 

as well as low density and low permeation to liquid and gaseous hydrogen; however, no single material provides 

all these attributes simultaneously. Consequently, once again material performance indices, associated with the 

functional requirements, such as those given in Table 5, must be used to identify the best material candidates for 

tank wall construction. Among the material parameters, strength and density tend to dominate the design criteria 

as the mass of the tank is inversely related to specific strength (see Eq. (23)). Figure 14 shows strength versus 

mass density for various engineering materials. In this case, materials in the upper left corner are preferable. 

Composite materials exhibit high specific strength relative to metals and are well suited to aerospace 

applications: continuous fiber reinforced polymer (CFRP) composites provide the highest strength yet lightest 

choice. However, the use of continuous fiber reinforced composite materials most likely will involve higher 

initial manufacturing costs and their permeability to hydrogen is a potential complication. As per Figure 14, the 

materials that have the sufficient strength and acceptably low densities are PMCs, CMCs and metallic materials.

Ceramic materials offer high specific strength, but due to their low fracture toughness (see Figure 15), are not 

viable for a tank wall material. A potential lower cost alternative to CFRP may be discontinuous reinforced 

metallic composites (DRX), specifically discontinuous reinforced aluminum (DRA) as described by Miracle 

(2001). DRAs are essentially isotropic and can be manufactured using less expensive techniques such as 

casting. The DRX materials have the added benefit of extremely low (if not negligible) hydrogen gas 

permeability issues typically associated with PMC systems. However, during a detailed study, Arnold et al. 

(2007) showed that minimum gage thickness requirements kept DRA materials from outperforming PMC 

materials for this application.

Two key material performance indices that are generally applicable to the design of high pressure vessels 

but may also be applicable to low pressure cryogenic storage tanks where described previously. These indices 

are 'yield-before break', M4 = KIc/σf, and the 'leak-before-break', M5 = KIc

2

/σf and are illustrated in Figure 15. 

Utilizing the first index ensures that the stress required to propagate a critical flaw is greater than that to yield 

the material and the second criterion, ensures that the maximum pressure carried will result in a stable crack 

that will just enable a leak to occur prior to catastrophic failure. Figure 16 once again clearly illustrates the 

difference between treating material indices as independent quantities (a) versus considering the interaction of 

the material indices during the selection process (b); the latter resulting in a reduction in potential material 

candidates. In Figure 16 only a subset (aerospace materials) of the 'Material Universe' database available 

within CES Selector is examined for illustrative purposes. Here we see that if the assumed values of the 

material indices are 1 = ( )≥ 20

ρ

σ f M and 10 2 I

5 = ≥ σ y

M K c only a finite set of viable materials; for example Ti, 

PMCs remain. 
NASA/TM—2012-217411 23

Figure 14.—Strength versus mass density for various engineering materials as per Ashby (2005).

Figure 15.—Fracture toughness versus elastic limit.
NASA/TM—2012-217411 24

Figure 16.—Illustrates the impact of allowing interaction of material indices for a pressurized vessels. 

(a) Independent; and (b) Interacting constraints. 

3.0 Advanced Selection

3.1 Materials for High Temperature Applications

The behavior of real materials to thermal and mechanical stimuli can vary greatly depending upon the 

magnitude and multiaxiality of loading and the magnitude of the homologous temperature (TH = T/Tm); 

that is the ratio of applied temperature (measured on an absolute scale, e.g., Kelvin K) to that of the 

melting point of a given material. For example at room temperature material response is typically timeindependent and either reversible (i.e., linear elastic) or irreversible (inelastic), depending upon whether 

or not the yield stress of the material has been exceeded. Alternatively, when TH > 0.25 time-dependent 

behavior, both reversible and irreversible, can be commonly observed as illustrated schematically in 

Figure 17, wherein i) strain rate sensitive , ii) creep, iii) relaxation, iv) thermal recovery, v) dynamic 

recovery and vi) creep/plasticity interaction response behavior are all shown. Other complex time and 

path dependent behavior such as cyclic ratcheting, creep/fatigue interaction, thermal mechanical fatigue 

are also often times observed depending upon the magnitude and type of loading (e.g., thermal or 

mechanical) being applied, see Lemaitre and Chaboche (1990). Note in the case of Ni based superalloys 

this ratio is significantly higher (TH ≥ 0.6 to 0.9), see Chester et al. (1987))
NASA/TM—2012-217411 25

A prerequisite for meaningful assessment of component durability and life, and consequently design of 

structural components, is the ability to accurately predict the stress and strains occurring within a loaded 

structure, composed of a given material. As constitutive material models (be they simple or complex) 

provide the required mathematical link between stress and strain, this necessitates the development and 

characterization of an appropriate constitutive model for any material before that material can be certified 

for use by a designer. Thus constitutive models with varying levels of idealizations have been proposed and 

utilized, each with its own shortcomings/limitations.

Figure 17.—Schematics showing representative hereditary material behavior at 

elevated temperature.
NASA/TM—2012-217411 26

The most well-known and widely used constitutive relation/model is the generalized Hooke’s law

σij = Cijkl ekl (25)

which describes multiaxial time-independent, linear (proportional) reversible material behavior. Up to 

now we have only utilized uniaxial Hooke’s law: σ = E e. Extension into the irreversible regime is 

accomplished by assuming an additive decomposition of the total strain tensor, εij , 

th

ij I εij = eij + εij + ε (26)

into three components, that is a reversible mechanical, eij, (i.e., elastic/viscoelastic); an irreversible, I εij , 

(i.e., inelastic or viscoplastic); and a reversible thermal strain, th εij component.

Equation (26) can also be rearranged...

th

ij I eij = εij − εij − ε (27)

Substituting expression (27) into Eq. (25) gives a stress strain relation that incorporates irreversible 

strains as well as reversible ones, that is:

( ) th

ij I σij = Cijkl εij − εij − ε (28)

Numerous models describing the evolution of the inelastic strain have been proposed in the literature. 

Table 6 contains a few representative examples with a brief description of their required material 

parameters.

The well known Norton-Bailey creep model (Skrzypek and Hetnarski, 2000) and its multiaxial 

generalization proposed by Odqvist (1936) followed by a coupled fully-associative unified 

viscoelastoplastic model has been formulated (Arnold and Saleeb (1994) and Saleeb et al. (2001)) with 

sufficient generality to permit systematic introduction of multiple mechanisms for both viscoelastic (timedependent reversible) and viscoplastic (time-dependent irreversible) response components. This general, 

multi-mechanism, hereditary deformation model has been shown to accurately represent a wide spectrum 

of material response under different loading conditions for the case of titanium alloys (Saleeb and Arnold 

(2001) and Arnold et al. (2001 and 2004). Examples include 1) rate-dependent (effective) material tangent 

stiffness during initial loading or any subsequent reversed loading, 2) pure transient response (e.g., in 

creep or relaxation) within the reversibility region, 3) an elastic behavior upon stress reversal, irrespective 

of the load level, as well as, 4) other response features common to ‘unified viscoplastic’ formulations 

(e.g., rate-sensitivity, creep-plasticity interaction, thermal recovery, etc.). The following generalized 

anisotropic material behavior constitutive model results; see Table 6 for definition of the associated flow 

and evolutionary laws: 

E q I σ = (ε − ε ) +  , (29)

All of this is said to emphasize that although for mathematical convenience materials are often 

classified into specific idealized groups (e.g., linear elastic, plastic, viscoelastic, and viscoplastic), nothing 

can compel real materials, however, to behave according to these idealized models. And typically, when 

dealing with actual complex material behavior as represented in Figure 17, it is not sufficient to describe 

this behavior with simple single parameter relationships or point wise values (e.g., E, ν, σy, H, etc.) which 

have worked so nicely for material selection at low homologous temperatures (e.g., typically at room 

temperature) where such complex path dependent behavior is suppressed.
NASA/TM—2012-217411 27

TABLE 6.—EXAMPLES OF VARIOUS CONSTITUTIVE MODELS AND THEIR REQUIRED MATERIAL PARAMETERS

Model Idealized 

material 

behavior

Mathematical representation Required isotropic 

material parameters

RambergOsgood

(1943)

Elastic 

Plastic

n

y

I H 















σ

σ ε =

σy, H, n

NortonBailey

[Skrzypek 

and Hetnarski 

(2000)]

Uniaxial 

Creep

ε= σ I n A

A, n

Odqvist

(1936)

Multiaxial 

Creep

'

2

( 1)/2

'

2

where 1 / 2( 1) and 1 / 2(3 )

1 / 2 with 1 / 3

I m

ij ij

n

ij ij ij ij kk ij

BJ S

mn B A

J SS S

+

ε =

=− =

= =σ − σ δ

 A, n

GVIPS

Arnold et al. 

(1994,2004); 

Saleeb et al. 

(2001); 

Saleeb and 

Arnold 

(2004)

Viscoelastoplastic ( ) (a) a( ) I () () a a 1 (a) qM M q ij ijkl kl kl ijkl klrs rs −  = ε −ε + η  

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

if F 0, ;; 1

0 otherwise,

b n ij I b b ij b

f F F f F gG H G  γ  ≥ β

ε =  = = −  μ 



[ ]

( ) ( ) ( )

( )

( ) ( )

( ) ( )

if 0

1

b

b

m b (b) (b) b (b) (b) I ij ijkl kl kl kl kl kl

b b

R G

Q

H G

β

 

  α = ε − π π − σ −α ≥

−  

 

Where

1 1

,

M Ν

(a) (b)

ij ij ij ij

a b

q q

= =

= α= α ∑ ∑

( ) ( ) ( ) , b

ijkl ij

b γij = ijkl σkl −αkl πkl =  α

( ) ( ) 1

2

1

2 σ −α σ −α − κ F = ij ij ijkl ij ij ,

( ) ( ) ( )

2

( )

( )

2

1 b

ijkl ij

b

ij

b

G b α α κ = 

And all three forth-order tensors, Eijkl, ijkl and (a) ηijkl , are taken to be 

coaxial, that is, Eijkl = EsNijkl , a ijkl

a Mijkl M( )N ( ) = ,and ( ) (a)

a ijkl

a ηijkl = ρ M

where

( )( ) ( )

( )











 δ δ + δ δ

+ ν δ δ ++ ν − ν

ν Nijkl = ij kl ik jl il jk 1

1

1 1 2

Finally, 

( ) { }

( ) ( ) () () ( ) ( ) 2 () ( ) ( )

1

2 11

b b b b b

ijkl b m ij kl b b b

QH G Z

G G

β   β = −   − α α κ − −β      

Note Zm is the “generalized” inverse of ijkl ; see Saleeb and Wilt

(1993); for further elaboration on this.

2+ 2M reversible 

constants:

Es, M(a), ν, ρ(a)

and

3+5N irreversible 

constants

κ, n, μ

H(b), R(b), m(b), κ(b),

β(b)

where N defines the 

number of 

viscoplastic 

mechanisms and M

the number of 

viscoelastic

So what should we do when the application operates at elevated temperatures—throw up our hands 

and just select materials based on linear elastic behavior? Not necessarily as there are simplified (albeit 

less accurate) approaches for high-temperature performance that allow some inelastic (e.g., creep) 

behavior to be accounted for and therefore influence the material selection process. Four such approaches 

are: 1) maximum service temperature (as in Sec. 2.2.4), 2) allowable stress, 3) design for creep, and 4) 

deformation mechanism maps. In this section, a brief description of the maximum service temperature 

and allowable stress approaches will be discussed. This will be followed by a more detailed description of 
NASA/TM—2012-217411 28

how to account for creep behavior in the material selection process. It is strongly suggested, however, that 

a rigorous nonlinear analysis, with a proper constitutive model, e.g., the unified viscoplastic GVIPS 

model (Arnold et al., 1994, 2004; Saleeb et al., 2001, 2004), be conducted for the top material candidates 

to verify the validity/suitability of the chosen material for a given design application. 

3.1.1 Maximum and Minimum Service Temperature

The simplest measures of tolerance to temperature are the maximum and minimum service 

temperatures, Tmax and Tmin. The former tells us the highest temperature at which the material can 

reasonably be used without oxidation, chemical change or excessive deflection or ‘creep’ becoming a 

problem (the continuous use temperature, or CUT, is a similar measure). The latter is the temperature 

below which the material becomes brittle or otherwise unsafe to use. These are empirical, with no 

universally accepted definitions; therefore caution should be used to fully understand the underlying 

assumptions and implications. For example, the minimum service temperature for carbon steels is the 

ductile-to-brittle-transition temperature—a temperature below which the fracture toughness falls steeply. 

For elastomers it is about 0.8 Tg, where Tg is the glass temperature. Below Tg elastomers cease to be 

rubbery and become hard and brittle. Examining Figure 18 wherein strength versus maximum 

temperature is plotted, it is apparent that polymers and low melting metals like the alloys of zinc, 

magnesium and aluminum offer useful strength at room temperature but by 300 °C they cease to be useful 

(as inelastic/softening behavior is exhibited)—indeed, few polymers have useful strength above 135 °C. 

Titanium alloys and low-alloy steels can have useful strength up to 600 °C; above this temperature highalloy stainless steels and more complex superalloys based on nickel, iron and cobalt are needed. The 

highest temperatures require refractory metals like tungsten or technical ceramics such as silicon carbide 

(SiC) or alumina (Al2O3). 

In Figure 12, the flexure strength versus maximum service temperature was utilized to identify 

potential candidate materials for three types of gas turbine engine blades: fan blades T < 300 °C; 

compressor blades, T < 700 °C; and turbine blades that operate above 700 °C. In case study 1b, Ti-6-4 

was identified as a potential fan or compressor blade material whose maximum service temperature was 

given a range of approximately 350 to 420 °C, see Figure 13. However, in a recent study (see Arnold et 

al. 2009) the complete time-dependent and rate-dependent temperature regimes for this material were 

experimentally mapped out. Results demonstrated that this maximum service temperature is too 

aggressive and should really be limited to 300 °C, since only below this temperature can it be assumed 

that no rate-dependence and only a minor amount of time-dependence are present, see Figure 19. In 

Figure 19 the deformation response of Ti-6-4 over a wide range of temperatures is documented. Here, the 

variation of the moduli and experimentally determined threshold stress, κ, (that stress which truly 

delineates between the reversible and irreversible strain regimes) are plotted as a function of temperature

(see Arnold et al. 2001). The modulus ES represents the “infinitely slow” modulus, i.e., the elastic 

modulus of the material if it was loaded at an infinitely slow strain rate, whereas the modulus ED

represents the “dynamic modulus”, which is the modulus of the material if it is loaded “infinitely fast” 

(e.g., very high, ≥ 1×10–3

) rate (i.e., all time dependence is locked into this modulus). As can be seen in 

Figure 19, at elevated temperatures there is a significant difference between the two modulus values, 

indicating that even in the so-called “elastic” range, there is significant time-dependence (reflected as rate 

dependent apparent moduli). Below a temperature of about 300 °C, the two moduli are approximately 

equal; indicating the response of the Ti-6-4 material is rate-independent but not necessarily timeindependent. Above 300 °C, the response is rate-dependent and time-dependent even below the threshold 

stress. To appreciate the practical significance of this fact, the operating temperatures typically 

encountered in aircraft engines are also noted below the horizontal axis in Figure 19, as well as the 

occasional, higher-temperature regime encountered during over-temp maneuvers. Clearly, even when one 

is within the typical engine design range and expecting the material response to be reversible, or “elastic”, 

the material behavior (at least in the case of Ti-6-4) would in fact be rate-dependent and would generate 

an additional strain of σ∗/Es over time, where σ∗ is the current applied stress that is less than or equal to
NASA/TM—2012-217411 29

the threshold stress κ denoted by the square symbol line. Consequently, at temperatures greater than 300

°C and stresses below κ, if classical elasticity methods were used in the design, then this rate 

dependence, i.e., viscoelastic response, would not be captured. Furthermore, at stresses above κ, the 

material response is viscoplastic due to the rate-and-time-dependence, and similarly using the classical 

methods of plasticity in analysis and design, would also not be accurate.

Figure 19 further indicates that the value of the threshold stress, κ , decreases significantly as a 

function of temperature. More importantly, Figure 19 demonstrates that at elevated temperatures the 

threshold stress (κ) is significantly below the traditional “proportional limit” or apparent “yield stress” of 

the material (obtained at a total strain rate of 0.001/sec) indicated by the dashed line (and x symbols). 

However, if one obtains the proportional limit from tensile test conducted at 1.0×10–6

/sec strain rate the 

values would correspond very closely to those of the threshold stress. This comparison then leads to the 

conclusion that irreversible material behavior can take place at stress levels well below those considered 

using traditional methods for the analysis of metals. If classical design methods were used, the material 

response below the “proportional limit” would be assumed to be fully reversible, and only the response 

after the proportional limit would be assumed to be irreversible. However, as is shown in the figure, that 

assumption could lead to significant inaccuracies in the predictions of the material response.

All of this is said to reinforce the fact that although simplifications can be made to aid the material 

selection/rejection process it remains extremely important, when dealing with material behavior at 

elevated temperatures, that significant care be taken to fully understand the various candidate materials 

that one will be suggesting at the end of any selection process.

Figure 18.—A chart showing the strength (elastic limit) versus maximum service temperature of the 

material.
NASA/TM—2012-217411 30

Figure 19.—Variation of modulus and threshold stress, κ, as a function of temperature 

for Ti-6-4 material.

3.2 Design for Creep

How are materials chosen to avoid failure by creep at high temperatures (Figure 20)? Traditionally

material selection has been based largely upon experience. Polymers can be used at room temperature, 

but—with only a few exceptions—not above 100 °C. The most creep resistant of aluminum alloys are 

good to about 200 °C; titanium alloys to 600 °C; stainless steels to 850 °C and so on. But optimal 

selection requires much more than this. The choice depends not only on material properties, but on the 

mode of loading (tension, bending, torsion, internal pressure), on the failure criterion (excessive 

deflection, fracture, relaxation of stress, buckling, etc.) and on the optimization objective (minimizing 

weight, or cost, or maximizing life). A designer, not himself a specialist on creep, has no easy way to 

identify the subset of materials best suited to his needs, or to predict the ways in which a change in the 

design might influence the choice.

In design against creep, one seeks the material and the shape which will carry the design loads 

without failure, for the design life at the design temperature. The meaning of “failure” depends on the 

application. We distinguish four types of failure, illustrated in Figure 20. 

a. Displacement-limited applications, in which precise dimensions or small clearances must be 

maintained (as in the disks and blades of turbines), when design is based on creep rates ε or 

displacement rates δ

 .

b. Rupture-limited applications, in which dimensional tolerance is relatively unimportant, but 

fracture must be avoided (as in pressure-piping), when design is based on time-to-failure tf.

c. Stress-relaxation-limited applications in which an initial tension relaxes with time (as in the pretensioning of cables or bolts) when design is based on a characteristic relaxation time tr.

d. Buckling-limited applications, in which slender columns or panels carry compressive loads (as in 

the upper wing skin of an aircraft, or an externally-pressurized tube), when design is based on 

critical time-to-instability tb.
NASA/TM—2012-217411 31

Figure 20.—Creep is important in four classes of design: (a) 

displacement-limited; (b) failure limited, (c) relaxation-limited, 

(d) buckling limited.

Figure 21.—A typical creep curve showing the tensile strain 

ε plotted against time t. It shows primary, secondary and 

tertiary stages, and the quantities εss, εf, tf.

To tackle any of these situations constitutive equation, as previously discussed, are required which relate 

the strain-rate ε or time-to-failure tffor a material to the stress σ and temperature T to which it is exposed.

3.2.1 Constitutive Equation for Creep Deflection

When a material is loaded at constant stress and homologous temperature TH, above approximately 

0.25, it typically will experience creep, as schematically depicted in Figure 21. Creep response typically 

involves a primary extension stage, followed by a steady-state creep stage, and ends with an accelerating 

tertiary stage. The important parameters are marked: the steady state creep rate εss  , the time to fracture, tf, 

and the creep ductility, εf.

The characteristics of the curve and the way it changes with temperature and stress are described 

mathematically by a constitutive equation (Finnie and Heller, 1959; Hult, 1966; Penny and Marriott, 
NASA/TM—2012-217411 32

1971; Gittus, 1975; Frost and Ashby, 1982; Evans and Wilshire, 1985). Many constitutive equations for 

creep-rate have been suggested, some purely empirical, some science-based; most are a mix of the two. 

Those most widely used in engineering design when deflection is important, relate the steady-state strainrate εss  to the tensile stress σ and the temperature T thus: 

( ) 



 





ε = σ − RT

Q Af ss exp , (30)

where A is a kinetic constant, Q an activation energy, R the gas constant and f (σ) means “ a function of 

stress σ”. The function f (σ) can be approximated, over restricted ranges of stress, by the Norton-Bailey 

power law given in Table 6 and slightly rewritten here:

n

o

o

n

o

ss RT

Q A 















σ

σ − = ε 















σ

σ

ε = .

 exp  , (31)

where the constant A, the activation energy Q, the exponent n and the characteristic strength constant σo

are material properties. Considerable experience has accumulated in the use of Norton’s Law, which has 

the appeal that it allows analytical solutions to a wide range of engineering problems (see, for instance, 

Finnie and Heller, 1959; Hult, 1966; Penny and Marriott, 1971). For this reason we shall use it here even 

though, from a scientific point of view, it lacks a completely respectable pedigree. Nor is it capable of 

accounting for creep/plasticity effects that more sophisticated unified viscoplastic models were developed 

to handle.

3.2.2 Constitutive Equation for Creep Fracture

When fracture rather than deflection is design-limiting, creep is characterized instead by the time to 

fracture, tf. It, too, can be described by a constitutive equation with features like those of Eq. (31). Here, 

again, a power law gives an adequate description over a restricted range of σ and T:

q

f

f

f

q

f

f t

RT

Q

t B

− −

















σ

σ =































σ

σ = exp 0 , (32)

With its own values of kinetic constants B, activation energy Qf, exponent q and characteristic strength σf.

3.2.3 Constitutive Equation for Relaxation

Relaxation requires a constitutive equation which combines creep (inelastic) and elastic response when 

total strain rate is held constant at zero. Consequently, from Eq. (29), and assuming tension it takes the form:

I σ = −Eεss   , (33)

where E is Young’s modulus, εss  is given by Eq. (31) and σ is the rate of change of stress with time. It is 

important to realize however that although the form of the inelastic strain is the same as Eq. (33) the 

material constants take on completely different values since the relaxation spectrum is typically 

significantly shorter than that of steady state creep. For the bending of a beam (as in Figure 22) the 

equation becomes instead

c S

F = δ

 

, (34)

where F is the rate of change of force F, S is the bending stiffness and δc

 is the creep deflection rate of 

the beam. Similar expressions describe torsion and compression.
NASA/TM—2012-217411 33

Figure 22.—A cantilever beam loaded with an end load F. 

(a) At room temperature a plastic hinge forms where the 

bending moment is highest. (b) At high temperature, creep 

plasticity is distributed. (c) Creep fracture starts where the 

local tensile stress is highest.

3.2.4 Material Selection in the Creep Regime

The objective here is to derive indices which parallel those used for room temperature design (Ashby, 

1992), but now tailored for design when creep takes place. The immediate difficulty is that the 'strength' is 

now no longer a fixed material property, but depends on temperature and on the strain-rate. The treatment 

below is kept as brief as possible. Fuller details can be found elsewhere (e.g., Abel and Ashby, 1994).

3.2.4.1 Deflection-Limited Design

Consider first the simple case of a tensile member—a tie rod—of minimum weight, designed to carry 

a load F for a life time, t, without deflecting more than δ at a temperature T. If the tie has length L, the 

steady strain-rate must not exceed

Lt

δ

ε = , (35)

Inserting this into the constitutive Eq. (32) for tensile creep and inverting gives

n

A Lt

F

1/

0

0 















ε

δ

σ = = σ  , (36)

The objective is to minimize the mass of the tie. Solving for A and substituting this into the equation 

m = ALρ for the mass gives

D

L F

m

σ

ρ = ,

with
NASA/TM—2012-217411 34

n

D Lt

1/

0

0 















ε

δ

σ = σ  , (37)

Thus the lightest tie which meets the constraints of F, T, t and σ is that made of the material with the 

largest value of

















ρ

σ = D M6 , (38)

This has the same form as the previous elastic material index for a tie rod

ρ

σ = y M1 , (39)

with σy replaced by σD, defined above. Consequently it contains both temperature and time.

The analysis of beams, shafts, pressure vessels (and the like) is a little more complex, but follows the 

same pattern. Consider, as an illustration, the cantilever beam of Figure 22(b) carrying a load F, but now 

at a temperature such that it creeps. The objective, as before, is to make the beam as light as possible; the 

constraints (again as before) are that its length L and the proportions, of its cross-section are fixed, and 

that it must support the load F for a time t at temperature T without deflecting more than δ. The design 

specification constrains the deflection rate, ε : it must not exceed δ/t. The deflection rate ε for a 

cantilever beam with end load F, creeping according to the constitutive Eq. (32) is 

n

n

n

bh

n

FL n L

n 























 



 +

σ = ε+

ε = 2 +1

0

2

0

1

2

4 2 1

( 2)

2   , (40)

See Appendix B for complete derivation. Eliminating b and h between the equations m = ALρ and (40), 

using b = αh and A = αh2 gives

3 1

2

3 1

1

3 1

3 3

4

. 2

+ 2 1 +

+

+

+

















σ





 



 + = ρ α n

n

D

n

n

n

n

FL

n

n

m L ,

(41)

with σD, which we call the design strength, given by

n

D L

n

1

0

0 2

2

















ε

+ δ

σ = σ 



, (42)

Equation (41) looks messy, but it is not as bad as it seems. For example the fully-plastic limit can be

found by setting n = ∞, thereby simplifying Eq. (41) to

2/3

1/3 4













σ = ρ α

D

FL

m L , (43)
NASA/TM—2012-217411 35

which is identical with the mass of the plastic beam { }2/3 1/3 4

y

FL m L σ = ρ α if σD is replaced by σy. The 

design strength in creep, then, plays the same role as the yield strength in room-temperature plasticity. In 

fact, Eq. (43) is a good approximation to the more complex result of Eq. (41) over the entire range of 

values of the exponent n (3 < n < 20) normally encountered in metals and alloys. Inspection shows that 

the mass is minimized by maximizing the index M2 given earlier ρ

σ = 2 3

2 y M with σy replaced by σD

(remember that σD contains temperature and deflection-rate).

A parallel calculation for a panel (flat plate in bending) gives an equation ρ

σ = 2 / 3

7

y M again with σy

replaced by σD, appropriately defined. Similar expressions are derived for torsion and for internal 

pressure (Appendix B). Selection with the objective of minimizing cost rather than weight lead to 

identical results with ρ replaced by Cmρ, where Cm is the material cost per kg; and the objective of 

minimizing energy content is achieved by replacing this by qmρ, where qm is the energy content per kg.

3.2.4.2 Fracture-Limited Design

Consider next an application in which fracture, not deflection-rate, is design-limiting (Figure 20(b)). 

For a bending beam the largest stresses appear—and creep-fracture starts—in the outer fibers at the place 

where the bending moment M is greatest (Figure 22(c)). The time t to the onset of failure of the 

cantilever, using the constitutive relation of Eq. (32), is

q

fo

fo bh

FL

q

q t t 















σ

+ = 2

4 . 22 1

, (44)

Writing b = αh, solving for the area A = bh and substituting in m = ALρ gives

2 3

1 3 4

2

2 1

















σ 













 + = ρ α

D

FL

q

q m L , (45)

with the design strength

q fo

D fo t

t

1

















σ = σ ,

(46)

The parallel with deflection-limited design is obvious; and once again the result reduces to that for full 

plasticity in the limit q = ∞. The mass is minimized, as before, by maximizing the index of equation 

ρ

σ = 2 3

7

y M with σy replaced by this new, fracture-related σD which depends on design life, t, and on 

temperature, T. Analogous calculations for ties and panels give equations 

ρ

σ = y M1 and 

ρ

σ = 1 3

3

y M —only 

the definition of σD is different. Similar expressions describe torsion and internal pressure (Appendix 

B.1), and are modified for cost or energy content by replacing ρ by Cmρ or qmρ as before.

3.2.4.3 Relaxation-Limited Design

A tensile cable, or a bolt, that is pretensioned to provide a bearing or clamping force F at an elevated 

temperature, relaxes with time by creep. The calculation is a standard one; elastic strain σ/E is replaced 

over time by creep. The total change in strain in the cable or bolt is zero, since its ends are fixed. The 

governing equation for the stress in the component was given earlier as Eq. (33). Integrating this with the 

boundary condition σ = σi at t = 0, gives
NASA/TM—2012-217411 36

( )

0

0

1 1

1

σ

ε = − 















σ

σ  − 

 





σ

σ − − E t

n

n

i

o

n

o 

, (47)

where σi is the stress to which the cable or bolt was originally tightened, and σ is the stress to which it has 

relaxed in time t. In this case a constraint is specified by defining a characteristic relaxation time, tr, as the 

time required for the stress to relax to a specified fraction of its initial value. Inverting Eq. (47) gives:

( ) 1

0

1

1

1 −

−

− ε σ

σ

































σ

σ = − n

n

o

n

i

r n E

t  , (48)

Clearly the quantity in the square bracket can be thought of as merely a scaling factor, consequently 

the result is independent of the choice of σ/σi. For illustrative purposes the ratio σ/σi is taken to be 0.5 

and for n > 3 the term in square brackets is then close to unity; thus for simplicity it shall be neglected. 

The design specifies the minimum bearing or clamping load F. Writing σ = F/A and substituting for A in 

equation m = ALρ, gives the mass of the cable or bolt which will safely provide a clamping load greater 

than F for a life tr;

D

L F

m

σ

ρ = , (49)

with

( )

1

1

1 0

−

















− ε

σ

σ = σ n

r

o D o n E t ,

(50)

The mass is minimized by selecting the material with the greatest value of σD/ρ, that is, the material index 

is once more that of equation ρ

σ = y M1 replaced by this new σD.

Springs, too, relax their tension with time. Most are loaded in bending, when the constitutive behavior 

is that of Eq. (34). Taking a beam of length L as an example, we write, for the stiffness S:

3

2

L

C EI S = , (51)

where I is the second moment of its area and C2 is a constant. Integrating with the boundary conditions F

= Fi at t = 0 gives a result with the form of Eq. (47). Proceeding as before, we find for the minimum 

weight design of a leaf spring (or any spring loaded in bending), which must not relax its restoring force 

in time tr at temperature T, the index M7 is 

ρ

σ = 2 3

7

y M , with σD (when n > 3) given by

1

1

.

−





















ε

σ

σ ≈ σ

n

o r

o D o

E  t ,

(52)

Abel and Ashby (1994) list results for other modes of loading. Similar calculations for ties and panels 

give equations ρ

σ = y M1 and 

ρ

σ = 1 3

8

y M again—only the definition of σD is different. The earlier 

adaptations to cost or energy apply here as well.
NASA/TM—2012-217411 37

3.2.5 The Selection Procedure

Expressions for the indices M and the associated design strengths σD are summarized in Appendix 

A.1 to A.3. The close parallel between these results and those for room temperature plasticity (equations

ρ

σ = y M1 , ρ

σ = 2 3

7

y M and 

ρ

σ = 1 3

8

y M ) suggests a selection procedure. The design temperature T and 

acceptable deflection-rate δ

 or life, t, or relaxation time tr are identified. Using this information, values 

for the appropriate σD are calculated from a database of creep properties for materials (it is necessary to 

capture the value of σD at the value σy to allow for the change of deformation mechanism to yielding at 

low temperatures). These are used to construct a chart of log (σD) against log (ρ). This is the creepequivalent of Figure 14, but is specific to the particular temperature, deflection-rate or life required by the 

design since these appear in the definition of σD. The indices M1, M7 and M8 can be plotted onto it, 

allowing optimum selection for each application.

This is best understood through examples. Those of the next section are deliberately simplified to 

avoid unnecessary digression. The method remains the same when the complexity is restored.

3.2.6 Case Study

The example below illustrates the selection of materials for structures loaded at elevated 

temperatures, and which are limited by deflection. Components limited by fracture or by stress relaxation

can be handled in a similar manner. Remember many other considerations enter the selection of materials 

for high-temperature use as well; for example resistance to oxidation, to thermal shock, and so on. Here 

we consider the selection for the initial short-list of candidates, of which these additional considerations 

would then be investigated.

3.2.6.1 Case Study 1c: Fan and Turbine Blades for Gas Turbines

A rotating blade of an aircraft turbine is self-loaded (Figure 20(a) and Figure 23); the centrifugal 

force caused by its own mass is much larger than that exerted by the gases which propel it. Adiabatic 

compression of the intake air can heat the compressor-fan blade to 400 °C or more. The dominant mode 

of steady loading, therefore, is tensile, and proportional (for fixed blade proportions) to the density of the 

blade material. It could, then, be anticipated that the appropriate index is that for tensile loading, M1 of 

equation 

ρ

σ = y M1 , with the design strength for tensile loading, σD, replacing σy, see Eq. (38). 

More detailed analyses add complexity, but confirm this result (Able and Ashby, 1994). The turbine 

blade is loaded in the same way, but is hotter: designers would like to go to 1000 °C. The task is to select 

materials to maximize the safe angular velocity of the compressor or turbine blade, designed to operate 

for a life t of 1000 hr without extending by more than δ, which is required to be 1.0 percent of its length, 

for each of these temperatures, and at the same time to minimize the weight.

The profile and section are determined by the blade design—neither is free. The mass is minimized 

by minimizing 

ρ = 1 M9 , (53)

Figure 24 through Figure 26 show σD creep strength, calculated for 400, 1000, and 1500 °C, 

respectively, with a value of L

δ

 and T corresponding to the design specification, plotted against density, 

ρ. Selection lines plotting the appropriate indices are shown and the corresponding potential material 

selections are listed in Table 7 through Table 9. The sweet spot (desired location) for material selection 

being the upper left hand corner of the figures.
NASA/TM—2012-217411 38

Figure 23.—A schematic of a turbine blade

Figure 24.—A selection chart for deflection limited design at 400 °C for a life 

of 1000 hr.
NASA/TM—2012-217411 39

Figure 25.—A selection chart for deflection limited design at 1000 °C for a life 

of 1000 hr

Figure 26.—A selection chart for deflection limited design at 1500 °C for a life 

of 1000 hr.

TABLE 7.—CANDIDATE MATERIALS FOR DEFLECTION 

LIMITED BLADE DESIGN AT 400 °C FOR A LIFE OF 1000 hr

Material Comment

Ti-matrix composites (Ti/Si/Cf) The ultimate in performance, but very expensive

Ti alloys (e.g., Ti 685, 6242) High creep strength and low density makes these the best choice

Ni-based superalloys (e.g., IN738) Excellent creep strength, but both indices M1 and M4 inferior to Ti alloys

Iron based superalloys, stainless steels Less good than nickel-based alloys, but cheaper

Ceramics and CMC’s Excellent values of M1 and M4 but brittleness is a problem
NASA/TM—2012-217411 40

TABLE 8.—CANDIDATE MATERIALS FOR DEFLECTION 

LIMITED BLADE DESIGN AT 1000 °C FOR A LIFE OF 1000 hr

Material Comment

Ni-based superalloys Among metallic alloys, these have the highest values of M1 but heavy

Refractory metals W-Re alloys offer high M1 but very heavy

Oxide ceramics and silicon-based ceramics, CMS’s Large weight saving possible if design can accommodate brittleness

TABLE 9.—CANDIDATE MATERIALS FOR DEFLECTION 

LIMITED BLADE DESIGN AT 1500 °C FOR A LIFE OF 1000 hr

Material Comment

Oxide and silicon-based ceramics (SiC, Al203) Offer high values of M1 and M4 but design must accommodate brittleness

Refractory metals Tungsten – rhenium creep resistant, but heavy

Ceramic composites (SiC-SiC; carbon-carbon) Potential for gains in performance, but brittleness and chemical stability 

require attention

3.3 Engineered Materials

Engineered or 'hybrid' materials are combinations of two or more materials, or of materials and space, 

assembled in such a way as to have attributes not offered by any one material alone (Figure 27).

Particulate and fiber reinforced composites are examples of one type of hybrid, but there are many others: 

sandwich structures, lattice structures, segmented structures and more. Hybrids and composites differ 

from bulk materials in providing an opportunity to tailor the properties of the ‘material’ to the 

application—in effect designing the material, while designing the component. This requires the ability to 

synthesize properties of candidate hybrid materials during the design process. This in turn requires the

engineer to choose the components of the hybrid, their configuration and their relative fraction. The new 

variables expand the design space, allowing the creation of new “materials” with specific property 

profiles. Consequently, one can view these new “materials” as mini-structures since internal to the

Figure 27.—Hybrid materials combine the properties of two (or 

more) monolithic materials, or of one material and space. 

They include fibrous and particulate composites, foams and 

lattices, sandwiches and almost all natural materials.
NASA/TM—2012-217411 41

material the individual constituents will redistribute the local stress and strain fields to carry the globally

applied fields most effectively.

But how is one to compare a hybrid—a sandwich structure for example—with monolithic materials 

such as polycarbonate or titanium? To do this we must think of the sandwich not only as a hybrid with 

faces of one material bonded to a core of another, but as a “material” in its own right, with its own set of 

effective properties; it is these that allow the comparison. 

The approach adopted here is one of breadth rather than precision. The aim is to assemble methods to 

allow the properties of alternative hybrids to be scanned and compared with those of monolithic materials, 

seeking those that best meet a given set of design requirements. Once materials and configuration have 

been chosen, standard methods—optimization routines, finite-element analyses—can be used to refine 

them. But what the standard methods are not good at is the quick scan of alternative combinations. That is 

where the approximate methods discussed here pay off.

The broad classes of hybrid can be considered in this way:

• Cellular structures are combinations of material and space giving precise control of density, 

stiffness, strength and thermal conductivity.

• Sandwich structures have outer faces of one material supported by a core of another, usually a 

low density material—a configuration that can offer a flexural stiffness per unit weight that is 

greater than that offered by either component alone. 

• Composites combine two solid components, one, the reinforcement, contained in the other, the 

matrix. 

• Coated materials enable the surface properties of the bulk material to be enhanced by addition of 

a thin surface layer.

Continuum and micro-mechanical models can be used to estimate the equivalent properties of each 

configuration. These can then be plotted on material selection charts, which become comparison-tools for 

exploring unique combinations of configuration and material. Some examples follow later.

3.3.1 Holes in Material-Property Space

As previously described, material properties can be “mapped” as material property charts of which 

Figure 28 is an example (for an up-to-date survey see Ashby et al. 2010). All the charts have one thing in 

common: parts of them are populated with materials but other parts are not. Some parts of the holes are 

inaccessible for fundamental reasons that relate to the size of atoms and the nature of the forces that bind 

them together. But others are empty even though, in principle, they could be filled.

3.3.1.1 Criteria of Excellence

Is anything to be gained by developing materials (or material-combinations) that lie in these holes?

To answer this a criteria of excellence is needed to assess the merit of any given hybrid. These are 

provided by the material indices, described above. If a possible hybrid has a value of any one of these that 

exceed those of existing materials, it achieves our goal.

The axes of Figure 28 are Young’s modulus, E, and density ρ. The property-combinations E/ρ, E1/2/ρ

and E1/3/ρ are measures of the excellence or material indices for selecting materials for light, stiff 

structures. A grid of lines of one index—E/ρ—is plotted on the figure. The arrow lies normal to the index 

lines. If the filled areas can be expanded in the direction of the arrow (i.e., to greater values of E/ρ) the 

materials so created will enable lighter, stiffer structures to be made. The arrow thus defines a vector for 

material development.

One approach to filling holes—the long-established one—is that of developing new metal alloys, new 

polymer chemistries and new compositions of glass and ceramic so as to create monolithic materials that 

expand the populated areas of the property charts. But developing new materials can be expensive and
NASA/TM—2012-217411 42

Figure 28.—A material property chart showing modulus-density space, with contours of 

specific modulus, E/ρ. Part of the space is occupied by materials, part it empty (the 

“holes”). Material development that extended the occupied territory in the direction of 

the arrow (the “vector for material development”) allows components with greater 

stiffness-to-weight than is at present possible.

uncertain, and the gains tend to be incremental rather than step-like. An alternative is to combine two or 

more existing materials so as to allow a superposition of their properties—in short, to create hybrids. The 

great success of carbon and glass-fiber reinforced composites at one extreme, and of foamed materials at 

another (hybrids of material and space) in filling previously empty areas of the material property charts is 

encouragement to explore ways in which such hybrids can be designed. 

When is a hybrid a “material”? There is a certain duality about the way in which hybrids are thought 

about and discussed. Some, like filled polymers, composites or wood are treated as materials in their own 

right, each characterized by its own set of macroscopic material properties. Others—like galvanized steel—

are seen as one material (steel) to which a coating of a second (zinc) has been applied, even though it could 

be regarded as a new material with the strength of steel but the surface properties of zinc. Sandwich panels 

illustrate the duality, sometimes viewed as two sheets of face-material separated by a core material, and 

sometimes—to allow comparison with monolithic materials—as a “material” with their own density, axial 

and flexural stiffness and strength, thermal conductivity, expansion coefficient, etc. To call any one of these 

a “material” and characterize it as such is a useful shorthand, allowing designers to use existing methods 

when designing with them. But if we are to design the hybrid itself, we must deconstruct it, and think of it as 

a combination of materials (or of material and space) in a chosen configuration.

3.3.2 Synthesizing the Properties of Cellular Structures: Foams and Lattices

This section provides an example of the synthesizing properties of foams and cellular structures. Some 

simple mathematical models for synthesizing homogenized properties of these materials are discussed first.

This is followed by a numerical example. The principal sources of the following models are: Gibson and 

Ashby (1997) and Ashby et al. (2000). For a similar discussion of other classes of hybrids: composites and 

sandwich panels see Ashby et al. (2010).
NASA/TM—2012-217411 43

Figure 29.—A cell in a low density foam. When the foam is loaded, the 

cell edges bend, giving a low-modulus structure.

Figure 30.—A micro-truss structure and its unit cell. The 

cell edges stretch when the structure is loaded, 

making it stiffer and stronger than a foam of the same 

density.

Cellular structures—foams and lattices—are hybrids of a solid and a gas. The properties of the gas 

might at first sight seem irrelevant, but this is not so. The thermal conductivity of low-density foams of the 

sort used for insulation is determined by the conductivity of the gas contained in its pores; and the dielectric 

properties, and even the compressibility, can depend on the gas properties. There are two distinct species of 

cellular solid. The first, typified by foams, are bending-dominated structures; the second, typified by 

triangulated lattice structures, are stretch dominated—a distinction explained more fully below. 

Foams are cellular solids made by expanding polymers, metals, ceramics or glasses with a foaming 

agent—a generic term for one of many ways of introducing gas, much as yeast does in bread-making. 

Figure 29 shows an idealized cell of a low-density foam. It consists of solid cell walls or edges 

surrounding a void space containing a gas or fluid. Foams have the characteristic that, when loaded, the 

cell walls bend. Lattice-structures (Figure 30) are configured to suppress bending, so the cell edges have 

to stretch instead. 

3.3.2.1 Density

Cellular solids are characterized by their relative density, which for the structure shown here (with 

t << L) is 

2

1

~





 



 = ρ

ρ

L

t C s

(54)

where ρ

~ is the density of the foam, ρs is the density of the solid of which it is made, L is the cell size, t is 

the thickness of the cell edges and C1 is a constant, approximately equal to 1.
NASA/TM—2012-217411 44

3.3.3 Mechanical Properties 

In Figure 31 the compressive stress-strain curve of a cellular solid is depicted. The material is linear 

elastic, with modulus E

~ up to its elastic limit, at which point the cell edges yield, buckle or fracture. The 

foam continues to collapse at a nearly constant stress (the “plateau stress”, σpl ~ ) until opposite sides of the 

cells impinge (the “densification strain” εd

~ ), when the stress rises rapidly. The mechanical properties are 

calculated in the ways developed below.

Elastic Moduli of Bending-Dominated Foams: A remote compressive stress σ exerts a force F ∝ σL2

on the cell edges, causing them to bend and leading to a bending deflection δ, as shown in Figure 29. For 

the open-celled structure shown in the figure, the bending deflection scales as

E I

F L

s

3

δ∝ (55)

where Es is the modulus of the solid of which the foam is made and 12

4t I = is the second moment of area 

of the cell edge of square cross section, t

2

. The compressive strain suffered by the cell as a whole is then ε

= 2δ/L. Assembling these results gives the modulus = σ/ ε ~

E of the foam as

s

s

E C E

2

2

~ ~ 















ρ

ρ = (bending-dominated behavior) (56)

Since E = Es

~ when ρ = ρs ~ , we expect the constant of proportionality C2 to be close to unity—a 

speculation confirmed both by experiment. Numerical simulation gives C1 = 0.7, the value used in the 

examples below. The quadratic dependence means that a small decrease in relative density causes a large 

drop in modulus. When the cells are equiaxed in shape, the foam properties are isotropic with shear 

modulus, bulk modulus and Poisson’s ratio via

3

~ ~ ~ 1

8

~ 3 G = E K = E ν = (57)

Elastic Moduli of Stretch-Dominated Lattices: The structure shown in Figure 30, is fully triangulated.

This means that the cell edges must stretch when the structure is loaded elastically. On average one third 

of its edges carry tension when the structure is loaded in simple tension, regardless of the loading 

direction. Thus

s

s

E C E















ρ

ρ =

~ ~ 3 (stretch-dominated behavior) (58)

with C3 = 1/3. The modulus is linear, not quadratic, in density giving a structure that is stiffer than a foam

of the same density. The structure of Figure 32 is almost isotropic, so we again approximate the shear 

modulus, bulk modulus and Poisson’s ratio by Eq. (57).

3.3.3.1 Yield Strength, Flexural Strength and Compressive Strength

Strength of bending-dominated foams. When the structure of Figure 29 is loaded beyond the elastic 

limit, its cell walls may yield, buckle elastically or fracture as shown in Figure 32. Consider yielding first 

(Figure 32(a)). Cell edges yield when the force exerted on them exceeds their fully plastic moment.
NASA/TM—2012-217411 45

4

3 , t

M y s

f

σ = (59)

where σy,s is the yield strength of the solid of which the foam is made. This moment is related to the 

remote stress by M ∝ FL ∝ σL3

. Assembling these results gives the compressive failure strength when 

yield dominates, σc ~

y s

s

c C ,

3/ 2

4

~ ~ σ 















ρ

ρ σ = (yield of foams) (60)

where the constant of proportionality, C4 ≈ 0.3, has been established both by experiment and by numerical 

computation.

Elastomeric foams collapse not by yielding but by elastic bucking; brittle foams by cell-wall fracture 

(Figure 32(b) and (c)). As with plastic collapse, simple scaling laws describe this behavior well. Collapse 

by buckling occurs when the stress exceeds

s

s

c E

~ 2

0.05 ~ 















ρ

ρ σ ≈ (buckling of foams) (61)

We identify the compressive strength σc ~ with the lesser of Eqs. (60) or (61). We further set the yield 

strength σy ~ the flexural strength flex ~

σ equal to σc ~ .

Figure 31.—The modulus is determined by cell edge 

bending or stretching. The plateau stress is 

determined by buckling, plastic bending or 

fracturing of the cell walls.
NASA/TM—2012-217411 46

Figure 32.—Collapse of foams. (a) When a foam made of a plastic 

materials loaded beyond its elastic limit, the cell edges bend 

plastically. (b) An elastomeric foam, by contrast, collapses by the 

elastic buckling of its cell edges. (c) A brittle foam collapses by the 

successive fracturing of cell edges.

Strength of stretch-dominated lattices: Collapse occurs when the cell edges yield, giving the collapse 

stress 

y s

s

c ,

~

3

~ 1

σ















ρ

ρ σ ≈ (yield of lattices) (62)

This is an upper bound since it assumes that the struts yield in tension or compression when the structure 

is loaded. If the struts are slender, they may buckle before they yield. They do so at the stress

s

s

c E

~ 2

0.2 ~ 















ρ

ρ σ ≈ (63)

We identify the compressive strength σc ~ with the lesser of Eqs. (62) and (63).We further identify the 

yield strength σy ~ , the flexural strength flex ~

σ with σc ~ .

3.3.3.2 Fracture Toughness

Fracture toughness of bending-dominated foams: Foams that contain long crack-like flaws compared 

to the cell size  fail by fast fracture, meaning that the crack propagates unstably, if the stress intensity 

factor exceeds the critical value KIc

~ , which is the lesser of 
NASA/TM—2012-217411 47

c s c s

s

c K K

a

K I , I ,

1/ 2 3/ 2

I and

~

0.5 ~ 















ρ

ρ 



 



 =  (64)

Here a is the intrinsic flaw size of the cell edge material, KIc,s is its fracture toughness and σts,s is its tensile 

strength; wherein typically  / a ≈ 10 . 

This behavior is not confined to open-cell foams. Most closed-cell foams also follow these scaling 

laws. At first sight, this is an unexpected result because the cell faces must carry membrane stresses when 

the foam is loaded, and these should lead to a linear dependence of both stiffness and strength on relative 

density. The explanation lies in the fact that the cell faces are very thin; they buckle or rupture at stresses 

so low that their contribution to stiffness and strength is small, leaving the cell edges to carry most of the 

load.

Fracture toughness of stretch-dominated lattices: Lattices that contain long crack-like flaws 

compared to the cell size  fail by fast fracture if the stress intensity factor exceeds the critical value KIc

~ , 

which is the lesser of 

c s c s

s

c K K

a

K I , I ,

1/ 2

I and

~

0.5 ~ 















ρ

ρ 



 



 =  (65)

 / a is the ratio of the cell size of the foam to the flaw size in the material—again typically having a 

value of approximately 10.

3.3.4 Thermal Properties

Specific heat and thermal expansion: The specific heat Cp

~ of foams and lattices (units: J/kg.K) and 

the expansion coefficient α

~ (units K–1

) are the same as those of the solid of which they are made.

Thermal conductivity: The cells in most cellular structures are sufficiently small that convection of 

the gas within them is completely suppressed. The thermal conductivity of the foam is then the sum of 

that conducted thought the cell walls and that through the still air (or other gas) they contain. To an 

adequate approximation

g

s

s

s s

λ































ρ

ρ λ + − 































ρ

ρ +















ρ

ρ λ =

~

1

~

2

~

3

~ 1

3/ 2

(66)

where λs is the conductivity of the solid and λg that of the gas (for dry air it is 0.025 W/m.K). The term 

associated with the gas is important: blowing agents for foams intended for thermal insulation are chosen to 

have a low value of λg.

3.3.5 Electrical Properties

Resistivity: The electrical resistivity, ρelec

~ of foam is given by

elec s

s s

elec , ~ 3/ 2

2

~

~ 3

ρ

































ρ

ρ +















ρ

ρ

ρ =

(67)

Dielectric properties: Insulating foams are attractive for their low dielectric constant, εr

~ , falling 

towards 1 (the value for air or vacuum) as the relative density decreases:
NASA/TM—2012-217411 48

( ) 















ρ

ρ ε = + ε − s

r r s

~

1 1 ~ , (68)

where εr,s is the dielectric constant of the solid of which the foam is made. The dielectric loss tangent is 

independent of foam density and just equal to that of the solid of which the foam is made.

3.3.6 Example: Aluminum Metal Matrix Composite (MMC) Foams and Lattices

The starting point is aluminum 20 percent SiC(p),

4 a mix that is the basis of one of the Cymat range 

of metal foams. The properties of two hybrids are compared a “Foam” and an “Octet lattice”, that 

generates two sets of new records. They appear on the Modulus-Density chart of Figure 33 as lines of 

yellow and green ellipses, respectively. The starting material, Al 20 percent SiC(p), is identified at the 

upper right. The modeled foams, plotted yellow, are labeled in red with their relative densities in brackets.

Measured data for real aluminum-SiC(p) foams, also with relative densities in brackets, are labeled in 

black, allowing a comparison. Note that the lattices, plotted in green, lie at higher values of modulus for 

the same density. At a relative density of 3.5 percent (0.035) the lattice is predicted to be ten times stiffer 

than a foam of that density. 

In Figure 34 many other engineering materials have been added. The lattice structures out-perform all of 

them, expanding the populated area of the chart. So an AlSiC(p) lattice provides one possible way to find a 

material with superior modulus/density performance than conventional materials. Further examples can be 

found in Ashby et al. (2010). 

Figure 33.—The input window of the synthesizer, set for Cellular structures – Octet lattice. The chart shows some of 

the output. The starting material, Al 20 percent SiC(p) appears at the upper right. The line of yellow ellipses with red 

labels show the modulus and density of foams made from Al-SiC(p). They should be compared with the measured 

values for real aluminum SiC(p) foams, shown in red with black labels. The relative densities are listed in brackets.

The modulus and density of lattices made of the same material are shown in green for comparison. Chart 

generated by the CES Hybrid Synthesizer, (Ashby et al., 2010).

 4 (p) means “particulate”.
NASA/TM—2012-217411 49

Figure 34.—The moduli of Al-SiC lattices compared to those of the other materials of engineering. The lattices lie 

outside the currently populated areas of the chart. Numbers in brackets are relative densities. Chart generated by 

the CES Hybrid Synthesizer, (Ashby et al., 2010).

3.3.7 The CES Hybrid Synthesizer

The approach described has been implemented in the CES Selector (2011), as a scoping tool that 

allows fast exploration of hybrid structures (see Ashby et al., 2010). It presents approximate analyses of 

the properties that might be achieved by forming a single material into a cellular structure or by 

combining two materials to form a sandwich or a composite. The purpose of the tool is to encourage 

innovation by allowing estimated property-profiles for virtual novel hybrids to be explored and compared 

with the property-profiles of established engineering materials.

If a suitable innovative material can be found in this way, the next steps are more detailed analyses, 

e.g., using finite element methods, followed by manufacturing, testing and qualifying the new material. 

4.0 Data Issues

To accomplish the material selection processes described above, the quantity and quality of the 

material information is central to all aspects of the process and most importantly to the final engineering 

products. The needs for materials information change throughout the lifecycle of an engineering 

component see Table 10. Likewise, the informatics infrastructure required to handle the potentially 

massive amounts of material data should not be overlooked nor underestimated.

Early in the design process, the ‘conceptual design’ phase, information is needed for all possible 

materials, but at a relatively low level of precision, since the objective at this stage is to select a small 

subset of materials that can optimally perform the function of the component with acceptable cost and 

environmental impact. ‘Typical’ values of the engineering properties and environmental impact data are 

best suited to this task. Narrowing this subset to one or two preferred candidates requires encyclopedic 

data sources that can inform the designer of a wide range of (often unquantifiable) characteristics of the 

material. This ‘supporting information’ includes details of corrosion, heat treatment, environmental 

impact and joining characteristics, as well as previous in-house experience in using the material. As the 
NASA/TM—2012-217411 50

design of the component is finalized, detailed stress analyses and lifing calculations are needed, which in 

turn require very precise ‘allowable’ design values of the properties of a single material.

TABLE 10.—MATERIAL INFORMATION NEEDS THROUGH A PRODUCT LIFE CYCLE

Product life phase Materials information Engineering operation

Design Selection data (typical props)

Supporting information

Environmental impact information

Design ‘allowables’

Specific constitutive and failure 

models for detailed design

Materials selection

Environmental analysis

Life cycle assessment

Design calculations

Stress analysis

Lifing, simulation

Manufacture Processing data Process modeling,

Manufacturing simulation

In-service Durability information Maintenance

Predict remaining life

Investigate failures

End of life Environmental impact information

Hazardous materials regulations

Recycling information

End of life policy implementation

Next the component must be manufactured, which has its own data requirements for information 

about the processing characteristics of the raw material—the viscosity when molten, the strength at high 

temperature, the hardness, toughness, etc. Then when the component is in service, there is a need for 

material data for maintenance purposes (e.g., for manufacturing spare parts), for predicting the remaininglife and for investigating service failures when they arise. Finally, at the end of life, the component may 

be recycled or disposed-of in some other way, therefore requiring knowledge of the environmental 

characteristics of the material—for example information about toxicity or degradation in landfill and 

associated legislative requirements—can be essential. 

All this is to make the point that having a multifaceted, “complete” material data source in a well 

organized, easily retrievable, information management system is a critically important and often 

unappreciated aspect of the entire engineering process. The value of a given database depends on its 

precision and its completeness, or in other words on its quality. Consequently, numerous procedures of 

checking and validating data have been devised, see for example Ashby (1998), Cebon and Ashby (2000) 

and Cebon and Ashby (2006) for discussion of checking methods, including property range and 

dimensionless correlations concepts. 

The material data lifecycle is described in Figure 35 wherein data is captured and consolidated from 

external sources, legacy databases as well as internal (possibly proprietary) testing programs. Next data is 

analyzed and integrated to create/discover useful information and then deployed (disseminated) to the 

people who need and use it. With the continual maintenance of the whole system (the data and 

information generated as well as the relationships, or links, between them) being the last yet essential 

stage of the data lifecycle. To support the various required activities throughout this data lifecycle 

requires the integration (preferably seamless) of a variety of software tools. These range from data input, 

reduction/analysis, visualization, reporting tools; material parameter estimation tools; product life 

management tools (PLM); to structural analysis codes that utilize a central database. These tools should 

enable material and structural engineers throughout the enterprise to input, manage and utilize 

information in as an efficient, reliable and user-friendly way as possible. An information management 

system that was conceived of by the Material Data Management Consortium [www.mdmc.net] and built 
NASA/TM—2012-217411 51

Figure 35.—Four aspects of 

material data lifecycle.

by Granta Design provides just such an environment and information management toolset. The toolset is 

composed of two primary components: 

1. GRANTA MI: is a comprehensive materials information system accessible from the web. Its unique 

database is specifically designed to store materials properties, response curves and other data relating 

to materials. It comprises a series of powerful software tools which help you to control, analyze, and 

apply your data.

2. CES Selector: is a PC software application that offers advanced graphical analysis of materials data, 

plus specialist eco design and modeling tools, in support of materials selection and substitution 

decisions.

5.0 Conclusions

Trade-offs are endemic in both everyday life and materials selection. Typically the job of a designer 

is to balance a variety of functional requirements (i.e., types of loading conditions (tension, compression, 

bending, vibration, cyclic, etc.) with constraints (manufacturability, geometric limits, environmental 

aspects, maintainability, to name a few) so as to arrive at the 'optimum' choice of structural concept and 

material selection for a given weight and/or cost. Herein a systematic, design-oriented, five step approach

to material selection has been described: 1) establishing design requirements, 2) material screening, 

3) ranking, 4) researching specific candidates and 5) applying specific cultural constraints to the selection 

process. At the core of this approach was the introduction of performance indices, through mathematical 

analysis of idealizations representing various engineering problems. Finding a material with a high value 

of these indexes maximizes the performance of the component. Material selection charts, which plot one 

property against another, were also introduced and shown to provide a powerful graphical environment 

wherein one could apply and analyze quantitative selection criteria, such as those captured in performance 

indices, and make trade-offs between conflicting objectives. Two specific examples pertaining to 

aerospace (engine blades and pressure vessels) where examined both at room temperature and elevated 

temperature (wherein time dependent effects were accounted for) to demonstrate the methodology. A

discussion with respect to engineered/hybrid materials and how these can be effectively tailored to fill in 

holes in the material property space so as to enable innovation and increases in performance over 

monolithic materials. Finally, a brief discussion relative to the data collection, analysis, deployment, and 

maintenance issues was addressed.

NASA/TM—2012-217411 53

Appendix A.—Table of Material Indices

Function and constraints Stiffness-limited 

design

Strength-limited

design

Vibration-limited

design

Tie (tensile strut)

Stiffness, length specified; section area free E/ρ ρ

σf E/ρ

Shaft (loaded in torsion)

Stiffness, length, shape specified, section area free G1/2/ρ ρ

σ2 3

f

Stiffness, length, outer radius specified; wall thickness free G/ρ ρ

σf

Stiffness, length, wall-thickness specified; outer radius free G1/3/ρ ρ

σ1 2

f

Beam (loaded in bending)

Stiffness, length, shape specified; section area free E1/2/ρ ρ

σ2 3

f E1/2/ρ

Stiffness, length, height specified; width free

All dimensions specified

E/ρ

----- ρ

σf

-----

-----

E/ρ

Stiffness, length, width specified; height free E1/3/ρ ρ

σ1 2

f

Column (compression strut, failure by elastic buckling)

Buckling load, length, shape specified; section area free E1/2/ρ ρ

σf

Panel (flat plate, loaded in bending)

Stiffness, length, width specified, thickness free

All dimensions specified

E1/3/ρ

----- ρ

σ1 2

f

-----

E1/3/ρ

E/ρ

Plate (flat plate, compressed in-plane, buckling failure)

Collapse load, length and width specified, thickness free E1/3/ρ ρ

σ1 2

f

Cylinder with internal pressure

Elastic distortion, pressure and radius specified; wall 

thickness free E/ρ ρ

σf

Spherical shell with internal pressure

Elastic distortion, pressure and radius specified; wall 

thickness free E/(1 – v)ρ ρ

σf
NASA/TM—2012-217411 54

A.1 Creep Fracture

Creep fracture. Time to fracture (“life”) = tf; n >> 1

Mode of loading Geometry Design strength Index

Tension; centrifugal loading

Any b and h q

f

f

D f t

t

1

0

0 













 σ = σ ρ

σ = D M

Bending b = αh

b fixed

h free

h fixed

b free 

q

f

f

D f t

t

1

0

0 













 σ = σ

ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

Torsion Solid 

R free

t = αR

t fixed

R free

R fixed

t free

q

f

f

D f t

t

1

0

0 













 σ = σ

ρ

σ =

2 3

D M

ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

Pressure tubes and vessels

R fixed

t free

q

f

f

D f t

t

1

0

0 













 σ = σ ρ

σ = D M

To minimize cost, replace ρ by Cmρ in the expressions for M; to minimize energy content, replace ρ by qmρ.
NASA/TM—2012-217411 55

A.2 Creep Deflection

Creep deflection. Time to deflect through δ (“life”) = t; δ ∝ δ /t  n >> 1

Mode of loading Geometry Design strength Index

Tension; centrifugal loading

Any b and h n

o

D o L

1

















ε

δ σ = σ 



ρ

σ = D M

Bending b = αh

b fixed

h free

h fixed

b free

n

o

D o L

1

















ε

δ σ = σ 



ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

Torsion Solid

R free

t = αR

t fixed

R free

R fixed

t free

n

o

D o

1

3 















ε

Θ

σ = σ 



n

o

D o

L

R

1

3 















ε

Θ σ = σ 



ρ

σ =

2 3

D M

ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

Pressure tubes and vessels

R fixed

t free

n

o

D o R

R

1

















ε σ = σ 



ρ

σ = D M

To minimize cost, replace ρ by Cmρ in the expressions for M; to minimize energy content, replace ρ by qmρ.
NASA/TM—2012-217411 56

A.3 Creep Relaxation

Creep relaxation. Time to relax to Fi/2 or Ti/2 (“life”) = tr; n >> 1

Mode of loading Geometry Design strength Index

Tension

Any b and h 1

1

−

















ε

σ σ = σ n

o r

o

D o E t ρ

σ = D M

Bending b = αh

b fixed

h free

h fixed

b free

1

1

−

















ε

σ σ = σ n

o r

o

D o E t

ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

Torsion Solid

R free

t = αR

t fixed

R free

R fixed

t free

1

1

2 3

−

















πε

σ

σ = σ n

o r

o D o  Gt

ρ

σ =

2 3

D M

ρ

σ =

2 3

D M

ρ

σ =

1 2

D M

ρ

σ = D M

To minimize cost, replace ρ by Cmρ in the expressions for M; to minimize energy content, replace ρ by qmρ.
NASA/TM—2012-217411 57

Appendix B.—Design Strength and Indices For Creep Deflection 

Here the design strength σD is defined and the performance indices M are derived for the creep 

fracture problem defined in Figure 36. First, the objective function is established; as it describes the 

quantity (like mass or cost) to be maximized or minimized. Then the constraints are formulated—the 

acceptable deflection rate or life. Next, the constraints are substituted into the objective function, 

eliminating free variables, and the design strength, σD, is identified and lastly, the performance index M,

is extracted.

B.1 Simple Tension or Compression: Minimum Weight Design

A tie of length L and section A carries a tensile load F (Figure 36(a)). The objective function is

m = ALρ (B1)

The extension rate of δ

 of the tie is

n n

A

F L L L 















σ = ε 















σ

σ δ = ε = ε

0

0

0

0     (B2)

The free variable is the section area A. The mass is reduced by reducing A, but A is constrained by the 

need to carry F without deflecting faster than δ

 . For a prescribed maximum deflection rate, δ the section 

must equal or exceed

0

1

0

σ





 





δ

ε = L F A n



 (B3)

This is the constraint. Substituting it into the objective function (B69) to eliminate A gives

L F L n m AL

1

0

0





 





δ

ε

σ

ρ = ρ = 

 (B4)

Comparing this with the fully plastic solution

y

L F

m

σ

ρ = (B5)

we can extract the design strength, σD:

n

D L

1

0

0 















ε

δ

σ = σ 

 (B6)

which plays the role, at high temperature, of the yield strength at low. Thus the lightest materials which 

will support a load F at temperature T without deflecting faster than δ

 are those with the greatest value of 

ρ

σ = D M (B7)

with σD given by Eq. (B74). To minimize cost instead of mass, the density ρ in this equation is replaced 

by Cmρ, where Cm is the cost per kg of the material.
NASA/TM—2012-217411 58

Figure 36.—Modes of loading: (a) tension, (b) bending, (c) a cantilever and (d) torsion.

B.2 Bending of Beams, Minimum Weight Design

Consider a beam loaded in bending, as in Figure 36(b). The objective function is given, as before, by 

Eq. (B69). To develop the constraint we must analyze the creep deflection rate, δ

 , of the beam. Three 

standard assumptions are made: the bending displacements are small; the neutral axis lies in the plane of 

symmetry of the beam section; and plane sections remain plane. The bending moment M carried by the 

beam is related to the stress within it, σ(y) by:

( ) ∫ = σ

2

0

2

h

M b y y dy (B8)

The strain rate ε(y) at any point in the section is related to the change of curvature, K by

(y) y K ε = (B9)

Substituting this into Norton’s Law (Eq. (B74)) gives the stress distribution

( ) ( ) n n

y yK

y

1

0

0

1

0

0 















ε = σ 















ε

ε

σ = σ 





 (B10)

Inserting this into Eq. (B76), integrating and inverting gives the key equation for any creeping-beam 

problem:
NASA/TM—2012-217411 59

2 1

0

0

2

2

2 1 +





 





















σ





 



 + = ε

n n

b h

M

n

n K   (B11)

We shall apply it to the cantilever of Figure 36(c). The curvature rate, K(x)  at the point x (Figure 36(c)) 

contributes

d( ) Xd xK(x)dx   =  δ = θ (B12)

to the end deflection rate. The moment at x is

M = Fx (B13)

Combining Eqs. (B79) and (B81), and integrating along the beam from x = 0 to x = L gives

n

n

n

bh

n

FL n L

n 























 



 +

σ

ε

+

δ = 2 +1

0

2

0

1

2

4 2 1

2

2   (B14)

If the section is rectangular, with b = αh its area A is 

2 A = bh = αh (B15)

Solving Eq. (B82) for A gives an explicit equation for the constraint on A; and inserting this into Eq. 

(B69) for the mass m gives

3 1

2

3 1

1

3 1

3 3 4

2

2 1 + +

+

+

+

















σ





 



 + = ρ = ρ α n

n

D

n

n

n

n FL

n

n

m AL L (B16)

As explained in the main text, the design strength for bending is defined as

n

D L

n

1

0

0 2

2

















ε

δ 



 



 +

σ = σ 

 (B17)

because in the limit n = ∞, Eq. (B84) reduces to the result for perfect plasticity with σD = σy; that is, σD

once again plays the role, for high temperatures, of the yield strength σy in room temperature design. 

Results for other configurations are listed in Section A.1. 

The quantity 3

1

3 1

1

3 1

3 3

1, , 2

2 1 ≈ ≈ α ≈ α+ +

+

+

+

n

n

n

n

L L

n

n and 1.3 2 1

 ≈ 

 



 + n

n

n for all n between 3 and 10, 

allowing the good approximation:

2 3

1 3 4







 







σ = ρα

D

FL

m L (B18)

with
NASA/TM—2012-217411 60

n n

D L tL

1

0

0

1

0

0 















ε

δ = σ 















ε

δ

σ = σ  

 (B19)

This is the same as the tensile result of the last section although δ

 now refers to a lateral displacement, 

not an extension. Thus the lightest materials which will support a given bending moment at temperature T

without a bending deflection rate greater than δ

 are those with the greatest value of

ρ

σ =

2 3

D M (B20)

When b is fixed and only h is free the definition of σD is unchanged, but the performance index becomes:

ρ

σ =

1 2

M D (B21)

and when h is fixed and only b varies it becomes instead:

ρ

σ = M D (B22)

To minimize cost instead of mass, ρ in the last three equations is replaced by Cmρ where Cm is the 

cost per kg of the material.

B.2.1 Torsion of Shafts, Minimum Weight Design

Torsion appears simpler, but there is a wrinkle. We seek the highest shaft which will carry a torque T

without rotating faster than θ

 . The objective function for a light shaft is given, as before, by Eq. (B69). A 

rotation rate θ

 of a thin-walled tube of length L, radius R and wall thickness t (Figure 36(d)) gives a 

strain rate in the wall of

L

Rθ

γ =   (B23)

This generates a shear stress in the wall which is found by rewriting Norton’s Law in terms of shear stress 

τ and shear strain-rate γ (using τ = σ 3 and γ = 3ε ). Combining this with (B91) gives

n

L

R

1

0

0

3 3 3 















ε

σ θ = σ

τ = 

 (B24)

The torque Tq is given by 2πR2t τ

n

q L

R t R T

1

0

0

2

3 3

2

















ε

π σ θ = 

 (B25)

It is tempting to identify the design strength with the quantity
NASA/TM—2012-217411 61

n

D

L

R

1

0

0 3 















ε

θ

σ = σ 

 (B26)

but this is valid only when R is held fixed and t is allowed to vary in optimizing the mass or the cost. 

When R is a free variable it cannot be included in σD since—in the optimization to minimize mass—R

must be eliminated. When R is fixed, we can write (using the definition above of σD):

3

2 0 π 2 σ = R t Tq (B27)

The minimum section which will carry a torque Tq without rotating faster than θ

 is

D

q

R

T

A Rt

σ = π = 3

2 (B28)

This gives the constraint on A. Substituting it into Eq. (B69) gives

D

q

R

T

m AL

σ

ρ = ρ = 3 (B29)

The mass is minimized by maximizing

ρ

σ = D M (B30)

When, instead, t is fixed and R can be varied, we must redefine the design strength as

n

D

1

0

0 3 















ε

θ

σ = σ  (B31)

Replacing this in Eq. (B93), solving for A and using this to calculate the mass M, as before, gives:

2 1

2 1

2 2

2 1

1 3

2 +

+

+

+

+

















σ = π ρ n

n

D

q n

n

n

n tT

m t L (B32)

which, when n > 3, reduces to

1 2

3

2 















σ = π ρ

D

q tT

m L (B33)

The mass is minimized by choosing materials with high values of 

ρ

σ =

1 2

M D (B34)
NASA/TM—2012-217411 62

Repeating the calculation for a solid torsion bar gives

















ε

θ 



 





+

π σ =

0

0

3

3 3 1 3

2





L

R

n

R n T (B35)

The definition of design strength remains the same as Eq. (B99). Inserting this in the last equation and 

solving for the section A = πR2 gives, via Eq. (B79)

3 1

2

3 1

3 3

3 2

3 3 +

+

+

















πσ = πρ n

n

D

q n

n T

m C L (B36)

with

3 1

2

3 3

3 1 +





 



 + = n

n

n

n C (B37)

For all n > 3, C3 ≈ 1 and the exponents simplify to give

















σ

ρ = 















πσ = πρ 2 3

2 3

2 3

3 2.8

2

3 3

D

q

D

q T L

T

m L (B38)

The lightest shaft is that with the greatest value of

ρ

σ =

2 3

M D (B39)

A similar calculation for a tube in which the thickness scales as the radius, such that t = αR gives the 

same result, with the same definition for σD.
NASA/TM—2012-217411 63

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Materials Selection for Aerospace Systems 

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14. ABSTRACT

A systematic design-oriented, five-step approach to material selection is described: 1) establishing design requirements, 2) material 

screening, 3) ranking, 4) researching specific candidates and 5) applying specific cultural constraints to the selection process. At the core of 

this approach is the definition performance indices (i.e., particular combinations of material properties that embody the performance of a 

given component) in conjunction with material property charts. These material selection charts, which plot one property against another, are 

introduced and shown to provide a powerful graphical environment wherein one can apply and analyze quantitative selection criteria, such 

as those captured in performance indices, and make trade-offs between conflicting objectives. Finding a material with a high value of these 

indices maximizes the performance of the component. Two specific examples pertaining to aerospace (engine blades and pressure vessels) 

are examined, both at room temperature and elevated temperature (where time-dependent effects are important) to demonstrate the

methodology. The discussion then turns to engineered/hybrid materials and how these can be effectively tailored to fill in holes in the 

material property space, so as to enable innovation and increases in performance as compared to monolithic materials. Finally, a brief 

discussion is presented on managing the data needed for materials selection, including collection, analysis, deployment, and maintenance 

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15. SUBJECT TERMS

Material performance indices; Material properties; Materials for high temperature applications; Corrosion and oxidation; Material data 

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