ME Web Module

Online Support ModuleMECHANICAL ENGINEERING

1

CONTENTS

Learning Objectives 2M.1 Introduction 2

Materials Selection for a Torsionally Stressed Cylindrical Shaft (Case Study) 2

M.2 Strength ConsiderationsTorsionallyStressed Shaft 3

M.3 Other Property Considerations and the FinalDecision 8

Fracture 9

M.4 Principles of Fracture Mechanics 9M.5 Flaw Detection Using Nondestructive

Testing Techniques 21M.6 Fracture Toughness Testing 25M.7 Impact Fracture Testing 29

Fatigue 33

M.8 Cyclic Stresses 33M.9 The S-N Curve 35M.10 Crack Initiation and Propagation 37M.11 Crack Propagation Rate 41M.12 Factors That Affect Fatigue Life 45M.13 Environmental Effects 47

Automobile Valve Spring (Case Study) 48

M.14 Mechanics of Spring Deformation 48M.15 Valve Spring Design and Material

Requirements 50M.16 One Commonly Employed Steel Alloy 52

Investigation of Engineering Failures 54

M.17 Reasons for Failure 55M.18 Root Causes 56

The Failure Analysis 57

M.19 What Exactly Is the Failure Problem? 57M.20 What Is the Root Cause of the Failure

Problem? 57M.21 What Are Possible Solutions? 70M.22 Which of These Is the Best

Solution? 71M.23 Effective Evaluation of Corrective

Actions 71M.24 The Final Report 71

Failure of an Automobile Rear Axle (Case Study) 72

M.25 Introduction 72M.26 Testing Procedure and Results 73M.27 Discussion 79

Summary 80Equation Summary 84Important Terms and Concepts 85References 86Questions and Problems 86Design Problems 90Problem Duplication Guide 95Glossary 96Answers to Selected Problems 97Index for Support Module 98

Due to constraints on book length, several topics especially suited to the discipline ofmechanical engineering were either not discussed in sufficient detail or omitted from theprint textbook.Therefore, it was decided to provide this online web module supplement,which includes the following:

A materials selection case studyMaterials Selection for a Torsionally StressedCylindrical Shaft.

Alternative (and more detailed) versions of Sections 9.5 and 9.8 Principles ofFracture Mechanics and Fracture Toughness Testing.

An alternative (and more detailed) treatment of the topic of fatigueSections 9.9through 9.14.

A case study on constraints and materials used for an automobile valve spring.

A submodule Investigation of Engineering Failures that outlines a protocol thatmay be used to analyze the failure of engineered components.

Another case study that details an investigation that was conducted to determinethe cause of failure of an automobile rear axle.

Learning ObjectivesAfter studying this web module, you should be able to do the following:

M.1 INTRODUCTION

1. Describe the manner in which materials selection charts are employed in the materialsselection process.

2. Explain why the strengths of brittle materialsare much lower than predicted by theoreticalcalculations.

3. Define fracture toughness in terms of (a) abrief statement and (b) an equation; define allparameters in this equation.

4. Make distinctions between stress intensity factor, fracture toughness, and plane strainfracture toughness.

5. In a qualitative manner, describe how a condi-tional value of plane-strain fracture toughnessis determined using ASTM Standard E 399-09.

6. Name and briefly describe the two techniquesthat are used to measure impact energy (ornotch toughness) of a material.

7. Define fatigue and specify the conditions under which it occurs.

8. From a fatigue plot for some material, determine (a) the fatigue lifetime (at a speci-fied stress level) and (b) the fatigue strength(at a specified number of cycles).

9. Cite five measures that may be taken to improve the fatigue resistance of a metal.

10. Briefly describe the steps that are used to ascertain whether a particular metal alloy issuitable for use in an automobile valve spring.

11. List and briefly explain the three root causes offailure.

12. List the four questions that a typical failureinvestigation seeks to answer.

13. Make a list of procedures/analyses that were usedto determine the cause of failure described in theFailure of an Automobile Rear Axle case study.

Materials Selection for a Torsionally StressedCylindrical Shaft (Case Study)

We begin this web module for mechanical engineers by presenting a case study on ma-terials selection. This process of materials selection involves, for some specified applica-tion, choosing a material having a desirable or optimum property or combination of

2

M.2 Strength ConsiderationsTorsionally Stressed Shaft 3

M.2 STRENGTH CONSIDERATIONSTORSIONALLY STRESSED SHAFTFor this portion of the design problem, we will establish a criterion for selection oflight and strong materials for this shaft. We will assume that the twisting moment andlength of the shaft are specified, whereas the radius (or cross-sectional area) may bevaried. We develop an expression for the mass of material required in terms of twist-ing moment, shaft length, and density and strength of the material. Using this expres-sion, it will be possible to evaluate the performancethat is, maximize the strengthof this torsionally stressed shaft with respect to mass and, in addition, relative tomaterial cost.

Consider the cylindrical shaft of length L and radius r, as shown in Figure M.1. Theapplication of twisting moment (or torque), Mt, produces an angle of twist . Shear stress at radius r is defined by the equation

(M.1)

Here, J is the polar moment of inertia, which for a solid cylinder is

(M.2)

Thus,

(M.3)

A safe design calls for the shaft to be able to sustain some twisting moment without frac-ture. In order to establish a materials selection criterion for a light and strong material,we replace the shear stress in Equation M.3 with the shear strength of the material fdivided by a factor of safety N, as1

(M.4)tf

N

2Mtpr 3

t 2Mtpr 3

J pr 4

2

t Mtr

J

properties. Selection of the proper material can reduce costs and improve performance.Elements of this materials selection process involve deciding on the constraints of theproblem and, from these, establishing criteria that can be used in materials selection tomaximize performance.

The component or structural element we have chosen to discuss is one that has rele-vance to a mechanical engineer: a solid cylindrical shaft that is subjected to a torsionalstress. Strength of the shaft will be considered in detail, and criteria will be developed formaximizing strength with respect to both minimum material mass and minimum cost.Other parameters and properties that may be important in this selection process are alsodiscussed briefly.

1The factor of safety concept as well as guidelines for selecting values are discussed in Section 7.20.

Figure M.1 A solid cylindrical shaft thatexperiences an angle of twist in response tothe application of a twisting moment Mt.

fr

L

Mt

4 Online Support Module: Mechanical Engineering

It is now necessary to take into consideration material mass. The mass m of anygiven quantity of material is just the product of its density () and volume. Since thevolume of a cylinder is r 2L, then

(M.5)

or, the radius of the shaft in terms of its mass is

(M.6)

Substituting this r expression into Equation M.4 leads to

(M.7)

Solving this expression for the mass m yields

(M.8)

The parameters on the right-hand side of this equation are grouped into three sets of paren-theses. Those contained within the first set (i.e., N and Mt) relate to the safe functioning ofthe shaft. Within the second parentheses is L, a geometric parameter. Finally, the materialproperties of density and strength are contained within the last set.

The upshot of Equation M.8 is that the best materials to be used for a light shaft thatcan safely sustain a specified twisting moment are those having low ratios. In termsof material suitability, it is sometimes preferable to work with what is termed a performanceindex, P, which is just the reciprocal of this ratio; that is

(M.9)

In this context we want to use a material having a large performance index.At this point it becomes necessary to examine the performance indices of a variety of

potential materials. This procedure is expedited by the use of materials selection charts.2

These are plots of the values of one material property versus those of another property.Both axes are scaled logarithmically and usually span about five orders of magnitude, soas to include the properties of virtually all materials. For example, for our problem, thechart of interest is logarithm of strength versus logarithm of density, which is shown inFigure M.2.3 It may be noted on this plot that materials of a particular type (e.g., woods,and engineering polymers) cluster together and are enclosed within an envelope delin-eated with a bold line. Subclasses within these clusters are enclosed using finer lines.

P t23fr

rt23f

m 12NMt 223 1p13L 2 a

r

t23fb

2Mt BpL3r3

m3

tf

N

2Mt

p aBmpLr

b3

r BmpLr

m pr2Lr

2A comprehensive collection of these charts may be found in M. F. Ashby, Materials Selection in Mechanical Design,4th edition, Butterworth-Heinemann, Woburn, UK, 2011.3Strength for metals and polymers is taken as yield strength; for ceramics and glasses, compressive strength; forelastomers, tear strength; and for composites, tensile failure strength.

For a cylindricalshaft of length L andradius r that isstressed in torsion,expression for massin terms of densityand shear strength ofthe shaft material

Strengthperformance indexexpression for atorsionally stressedcylindrical shaft

performance index

materials selectionchart

Now, taking the logarithm of both sides of Equation M.9 and rearranging yields

(M.10)

This expression tells us that a plot of log tf versus log r will yield a family of straight andparallel lines all having a slope of each line in the family corresponds to a different per-formance index, P.These lines are termed design guidelines, and four have been included inFigure M.2 for P values of 3, 10, 30, and 100 All materials that lie on one ofthese lines will perform equally well in terms of strength-per-mass basis; materials whose

(MPa)23m3/Mg.

32;

log tf 32

log r 32

log P


0.1 0.3 1 3 10 30

10,000

1000

100

10

1

0.1

MgAlloys

Ash

SiGlasses

Density (Mg /m3)

Engineeringceramics

Engineeringcomposites

Engineeringalloys

Porousceramics

Engineeringpolymers

Woods

Elastomers

Polymerfoams

Str

engt

h (M

Pa)

P = 100

P = 30

P = 10

P = 3

Diamond

Cermets

Sialons

B MgO

Al2O3 ZrO2

Si3N4SiC

Ge

NylonsPMMA

PS

PPMELPVC

EpoxiesPolyesters

HDPE

PU

PTFE

SiliconeLDPE

SoftButyl

WoodProducts

Ash

Balsa

Balsa

Oak

Oak

Pine

Pine

Fir

Fir

Parallelto Grain

Perpendicularto Grain

Cork

CementConcrete

Engineeringalloys

KFRPCFRPBe

GFRPLaminates

KFRP

Pottery TiAlloys

Steels

W Alloys

Mo Alloys

Ni Alloys

Cu Alloys

CastIrons

ZnAlloys

Stone,Rock

LeadAlloys

CFRPGFRP

UNIPLY

Al Alloys

Figure M.2 Strength versus density materials selection chart. Design guidelines for performance indices of 3,10, 30, and 100 (MPa)23m3/Mg have been constructed, all having a slope of (Adapted from M. F. Ashby, Materials Selection in Mechanical Design. Copyright 1992. Reprinted by permission ofButterworth-Heinemann Ltd.)

32.

positions lie above a particular line will have higher performance indices, whereas thoselying below will exhibit poorer performances. For example, a material on the P 30 linewill yield the same strength with one-third the mass as another material that lies alongthe P 10 line.

The selection process now involves choosing one of these lines, a selection linethat includes some subset of these materials; for the sake of argument let us pick P 10 m3/Mg, which is represented in Figure M.3. Materials lying along this line orabove it are in the search region of the diagram and are possible candidates for this1MPa 2 23


Engineeringalloys

KFRPCFRPBe

GFRPLaminates

KFRP

Pottery

MgAlloys

TiAlloys

Steels

W Alloys

Mo Alloys

Ni Alloys

Cu Alloys

CastIrons

ZnAlloys

Stone,Rock

Al Alloys

LeadAlloys

CFRPGFRP

UNIPLY

10,000

1000

100

10

1

Glasses

Engineeringcomposites

Engineeringalloys

Porousceramics

Engineeringpolymers

Woods

Elastomers

Polymerfoams

Str

engt

h (M

Pa)

P = 10

Diamond

Cermets

Sialons

B MgO

Al2O3 ZrO2

Si3N4SiC

Si Ge

NylonsPMMA

PS

PPMELPVC

EpoxiesPolyesters

HDPE

PU

PTFE

Silicone

CementConcrete

LDPE

SoftButyl

WoodProducts

Ash

AshBalsa

Balsa

Oak

Oak

Pine

Pine

Fir

Fir

Parallelto Grain

Perpendicularto Grain

Cork

300 MPa

(MPa)2/3 m3/Mg

0.1 0.3 1 3 10 300.1

Density (Mg /m3)

Engineeringceramics

Figure M.3 Strength versus density materials selection chart. Materials within the shaded region are acceptablecandidates for a solid cylindrical shaft that has a mass-strength performance index in excess of 10 and a strength of at least 300 MPa (43,500 psi).(Adapted from M. F. Ashby, Materials Selection in Mechanical Design. Copyright 1992. Reprinted by permission ofButterworth-Heinemann Ltd.)

(MPa)23m3/Mg


rotating shaft. These include wood products, some plastics, a number of engineeringalloys, the engineering composites, and glasses and engineering ceramics. On the basis offracture toughness considerations, the engineering ceramics and glasses are ruled out aspossibilities.

Let us now impose a further constraint on the problemnamely that the strength ofthe shaft must equal or exceed 300 MPa (43,500 psi).This may be represented on the ma-terials selection chart by a horizontal line constructed at 300 MPa, see Figure M.3. Nowthe search region is further restricted to the area above both of these lines.Thus, all woodproducts, all engineering polymers, other engineering alloys (viz., Mg and some Al alloys),and some engineering composites are eliminated as candidates; steels, titanium alloys,high-strength aluminum alloys, and the engineering composites remain as possibilities.

At this point we are in a position to evaluate and compare the strength performancebehavior of specific materials. Table M.1 presents the density, strength, and strength per-formance index for three engineering alloys and two engineering composites, whichwere deemed acceptable candidates from the analysis using the materials selectionchart. In this table, strength was taken as 0.6 times the tensile yield strength (for thealloys) and 0.6 times the tensile strength (for the composites); these approximationswere necessary because we are concerned with strength in torsion, and torsionalstrengths are not readily available. Furthermore, for the two engineering composites, itis assumed that the continuous and aligned glass and carbon fibers are wound in a helicalfashion (Figure 15.15) and at a 45 angle referenced to the shaft axis. The five materialsin Table M.1 are ranked according to strength performance index, from highest to lowest:carbon fiber-reinforced and glass fiber-reinforced composites, followed by aluminum,titanium, and 4340 steel alloys.

Materials cost is another important consideration in the selection process. In real-life engineering situations, economics of the application often is the overriding issue andnormally will dictate the material of choice. One way to determine materials cost is bytaking the product of the price (on a per-unit mass basis) and the required mass ofmaterial.

Cost considerations for these five remaining candidate materialssteel, aluminum,and titanium alloys, and two engineering compositesare presented in Table M.2. Inthe first column is tabulated The next column lists the approximate relativecost, denoted as this parameter is simply the per-unit mass cost of material divided bythe per-unit mass cost for low-carbon steel, one of the common engineering materials.The underlying rationale for using is that although the price of a specific material willvary over time, the price ratio between that material and another will, most likely,change more slowly.

c

c;r/t23f .

f f23/ P

Material (Mg/m3) (MPa) [(MPa)2/3m3/Mg]

Carbon fiber-reinforced composite 1.5 1140 72.8(0.65 fiber fraction)a

Glass fiber-reinforced composite 2.0 1060 52.0(0.65 fiber fraction)a

Aluminum alloy (2024-T6) 2.8 300 16.0

Titanium alloy (Ti-6Al-4V) 4.4 525 14.8

4340 Steel (oil-quenched 7.8 780 10.9and tempered)

a The fibers in these composites are continuous, aligned, and wound in a helical fashion ata 45 angle relative to the shaft axis.

Table M.1Density (), Strength(f), and StrengthPerformance Index (P)for Five EngineeringMaterials

Finally, the right-hand column of Table M.2 shows the product of and . Thisproduct provides a comparison of these materials on the basis of the cost of materialsfor a cylindrical shaft that would not fracture in response to the twisting moment Mt. Weuse this product inasmuch as is proportional to the mass of material required(Equation M.8) and is the relative cost on a per-unit mass basis. Now the most econom-ical is the 4340 steel, followed by the glass fiber-reinforced composite, the carbon fiber-reinforced composite, 2024-T6 aluminum, and the titanium alloy. Thus, when the issue ofeconomics is considered, there is a significant alteration within the ranking scheme. Forexample, inasmuch as the carbon fiber-reinforced composite is relatively expensive, it issignificantly less desirable; in other words, the higher cost of this material may not out-weigh the enhanced strength it provides.

crt23f

rt23fc


Table M.2 Tabulation of the f23 Ratio, Relative Cost (c), and Product of f23 and c for Five EngineeringMaterialsa

f23 c c(f23 )Material 102 [Mg(MPa)23m3] ($$) 102($$)[Mg(MPa)23m3]

4340 Steel (oil-quenched 9.2 3.0 27and tempered)

Glass fiber-reinforced composite 1.9 28.3 54(0.65 fiber fraction)b

Carbon fiber-reinforced composite 1.4 43.1 60(0.65 fiber fraction)b

Aluminum alloy (2024-T6) 6.2 12.4 77

Titanium alloy (Ti-6Al-4V) 6.8 94.2 641aThe relative cost is the ratio of the price per unit mass of the material and a low-carbon steel.bThe fibers in these composites are continuous, aligned, and wound in a helical fashion at a angle relative to theshaft axis.

45

M.3 OTHER PROPERTY CONSIDERATIONS AND THE FINAL DECISIONTo this point in our materials selection process we have considered only the strength ofmaterials. Other properties relative to the performance of the cylindrical shaft may beimportantfor example, stiffness, and, if the shaft rotates, fatigue behavior (Sections M.9through M.13). Furthermore, fabrication costs should also be considered; in our analysisthey have been neglected.

Relative to stiffness, a stiffness-to-mass performance analysis similar to the one just dis-cussed could be conducted. For this case, the stiffness performance index is

(M.11)

where G is the shear modulus. The appropriate materials selection chart (log G versuslog r) would be used in the preliminary selection process. Subsequently, performanceindex and per-unit-mass cost data would be collected on specific candidate materials;from these analyses the materials would be ranked on the basis of stiffness perform-ance and cost.

In deciding on the best material, it may be worthwhile to make a table employing theresults of the various criteria that were used.The tabulation would include, for all candidatematerials, performance index, cost, and so forth for each criterion, as well as comments rel-ative to any other important considerations. This table puts in perspective the importantissues and facilitates the final decision process.

Ps 1Gr

Ps

M.4 Principles of Fracture Mechanics 9

FractureM.4 PRINCIPLES OF FRACTURE MECHANICS4

Brittle fracture of normally ductile materials, such as that shown in the chapter-openingFigure b (of the oil barge) for Chapter 9, has demonstrated the need for a better under-standing of the mechanisms of fracture. Extensive research endeavors over the last cen-tury have led to the evolution of the field of fracture mechanics. This subject allowsquantification of the relationships between material properties, stress level, the presenceof crack-producing flaws, and crack propagation mechanisms. Design engineers are nowbetter equipped to anticipate, and thus prevent, structural failures. The present discus-sion centers on some of the fundamental principles of the mechanics of fracture.

Stress ConcentrationThe fracture strength of a solid material is a function of the cohesive forces that existbetween atoms. On this basis, the theoretical cohesive strength of a brittle elastic solidhas been estimated to be approximately E/10, where E is the modulus of elasticity. Theexperimental fracture strengths of most engineering materials normally lie between 10and 1000 times below this theoretical value. In the 1920s, A. A. Griffith proposed thatthis discrepancy between theoretical cohesive strength and observed fracture strengthcould be explained by the presence of microscopic flaws or cracks that always exist un-der normal conditions at the surface and within the interior of a body of material.Theseflaws are a detriment to the fracture strength because an applied stress may be ampli-fied or concentrated at the tip, the magnitude of this amplification depending on crackorientation and geometry. This phenomenon is demonstrated in Figure M.4, a stressprofile across a cross section containing an internal crack. As indicated by this profile,

fracture mechanics

Figure M.4 (a) The geometry of surface and internal cracks. (b) Schematic stress profile along the line X-X in (a),demonstrating stress amplification at crack tip positions.

rt

s0

s0

X'X

x'x2a

a

Position along XX'

s0

sm

Str

ess

x'x

4This section is an expanded and more detailed version of Section 9.5.

(a) (b)

the magnitude of this localized stress decreases with distance away from the crack tip.At positions far removed, the stress is equal to the nominal stress 0, or the applied loaddivided by the specimen cross-sectional area (perpendicular to this load). Because oftheir ability to amplify an applied stress in their locale, these flaws are sometimes calledstress raisers.

If it is assumed that a crack has an elliptical shape (or is circular) and is oriented per-pendicular to the applied stress, the maximum stress at the crack tip, , is equal to

(M.12a)

where 0 is the magnitude of the nominal applied tensile stress, t is the radius of curvatureof the crack tip (Figure M.4a), and a represents the length of a surface crack, or half of thelength of an internal crack. For a relatively long microcrack that has a small tip radius ofcurvature, the factor may be very large (certainly much greater than unity); underthese circumstances Equation M.12a takes the form

(M.12b)

Furthermore, m will be many times the value of 0.Sometimes the ratio m0 is denoted the stress concentration factor Kt:

(M.13)

which is simply a measure of the degree to which an external stress is amplified at thetip of a crack.

Note that stress amplification is not restricted to these microscopic defects; it mayoccur at macroscopic internal discontinuities (e.g., voids or inclusions), sharp corners,scratches, and notches. Figure M.5 shows theoretical stress concentration factor curves forseveral simple and common macroscopic discontinuities.

The effect of a stress raiser is more significant in brittle than in ductile materials. Fora ductile metal, plastic deformation ensues when the maximum stress exceeds the yieldstrength. This leads to a more uniform distribution of stress in the vicinity of the stressraiser and to the development of a maximum stress concentration factor less than thetheoretical value. Such yielding and stress redistribution do not occur to any apprecia-ble extent around flaws and discontinuities in brittle materials; therefore, essentially thetheoretical stress concentration will result.

Griffith then went on to propose that all brittle materials contain a population ofsmall cracks and flaws, which have a variety of sizes, geometries, and orientations.Fracture will result when, upon application of a tensile stress, the theoretical cohesivestrength of the material is exceeded at the tip of one of these flaws. This leads to the for-mation of a crack that then rapidly propagates. If no flaws were present, the fracturestrength would be equal to the cohesive strength of the material. Very small and virtu-ally defect-free metallic and ceramic whiskers have been grown with fracture strengthsthat approach their theoretical values.

Griffith Theory of Brittle FractureDuring the propagation of a crack, there is a release of what is termed the elastic strainenergy, some of the energy that is stored in the material as it is elastically deformed.

Kt sm

s0 2 a

artb

1 2

sm 2s0 aartb

1 2

1art 2 1 2

sm s0 c1 2 aartb

1 2

d

sm


stress raiser

For tensile loading,computation ofmaximum stress at a crack tip


Furthermore, during the crack extension process, new free surfaces are created at thefaces of a crack, which give rise to an increase in surface energy of the system. Griffithdeveloped a criterion for crack propagation of an elliptical crack (Figure M.4a) by perform-ing an energy balance using these two energies. He demonstrated that the critical stress screquired for crack propagation in a brittle material is described by the expression

(M.14)sc a 2Egspab 1 2

Str

ess

conc

entr

atio

nfa

ctor

Kt

Str

ess

conc

entr

atio

n fa

ctor

Kt

Str

ess

conc

entr

atio

n fa

ctor

Kt

w

dw

3.4

3.0

2.6

2.2

1.8

3.8

3.4

3.0

2.6

2.2

1.8

1.4

1.0

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.00 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8 1.0

d

2r

b

b

hw

w

h

r+

rh

rh

br

12=

br 1=

wh

2.00=

wh

1.25=

wh

1.10=

br 4=

Figure M.5Theoretical stressconcentration factorcurves for three simplegeometrical shapes.(From G. H. Neugebauer,Prod. Eng. (NY), Vol. 14,pp. 8287, 1943.)

Critical stress forcrack propagation ina brittle material

(a)

(b)

(c)

where

a 5 one-half the length of an internal crack

Worth noting is that this expression does not involve the crack tip radius rt, as does thestress concentration equations (Equations M.12a and M.12b); however, it is assumed thatthe radius is sufficiently sharp (on the order of the interatomic spacing) so as to raise thelocal stress at the tip above the cohesive strength of the material.

The previous development applies only to completely brittle materials, for whichthere is no plastic deformation. Most metals and many polymers do experience someplastic deformation during fracture; thus, crack extension involves more than producingjust an increase in the surface energy. This complication may be accommodated by re-placing gs in Equation M.14 by gs gp, where gp represents a plastic deformation en-ergy associated with crack extension. Thus,

(M.15a)

For highly ductile materials, it may be the case that gp >> gs such that

(M.15b)

In the 1950s, G. R. Irwin chose to incorporate both gs and gp into a single term, Gc, as

(M.16)

Gc is known as the critical strain energy release rate. Incorporation of Equation M.16into Equation M.15a after some rearrangement leads to another expression for theGriffith cracking criterion as

(M.17)

Thus, crack extension occurs when exceeds the value of Gc for the particularmaterial under consideration.

ps2aE

Gc ps 2a

E

Gc 21gs gp 2

sc a2Egp

pab

1 2

sc c2E1gs gp 2

pad

12

gs specific surface energy

E modulus of elasticity


EXAMPLE PROBLEM M.1

Maximum Flaw Length Computation

A relatively large plate of a glass is subjected to a tensile stress of 40 MPa. If the specific sur-face energy and modulus of elasticity for this glass are 0.3 J/m2 and 69 GPa, respectively, deter-mine the maximum length of a surface flaw that is possible without fracture.

Solution

To solve this problem it is necessary to employ Equation M.14. Rearranging this expressionsuch that a is the dependent variable, and realizing that and

, leads to

m 8.2 106 m 0.0082 mm 8.2

12 2 169 109 N/m2 2 10.3 N/m 2

p 140 106 N/m2 2 2

a 2Egsps2

E 69 GPags 0.3 J/m

2,s 40 MPa,


Stress Analysis of CracksAs we continue to explore the development of fracture mechanics, it is worthwhile toexamine the stress distributions in the vicinity of the tip of an advancing crack.There arethree fundamental ways, or modes, by which a load can operate on a crack, and each willaffect a different crack surface displacement; these are illustrated in Figure M.6. Mode Iis an opening (or tensile) mode, whereas modes II and III are sliding and tearing modes,respectively. Mode I is encountered most frequently, and only it will be treated in the en-suing discussion on fracture mechanics.

For this mode I configuration, the stresses acting on an element of material areshown in Figure M.7. Using elastic theory principles and the notation indicated, tensile(sx and sy)

5 and shear (txy) stresses are functions of both radial distance r and the angle uas follows:6

(M.18a)

(M.18b)

(M.18c)

If the plate is thin relative to the dimensions of the crack, then sz 0, or a condition ofplane stress is said to exist. At the other extreme (a relatively thick plate), sz n(sx sy), and the state is referred to as plane strain (since 0); n in this expression isPoissons ratio.

In Equations M.18, the parameter K is termed the stress intensity factor; its use pro-vides for a convenient specification of the stress distribution around a flaw. It should benoted that this stress intensity factor and the stress concentration factor Kt in EquationM.13, although similar, are not equivalent.

Pz

txy K

22prfxy 1u 2

sy K

22prfy 1u 2

sx K

22prfx1u 2

5This y denotes a tensile stress parallel to the y-direction and should not be confused with the yield strength(Section 7.6), which uses the same symbol.6The f() functions are as follows:

fxy1u 2 sin u

2 cos

u

2 cos

3u2

fy1u 2 cos u

2 a1 sin

u

2 sin

3u2b

fx1u 2 cos u

2 a1 sin

u

2 sin

3u2b

Figure M.6 The threemodes of crack surfacedisplacement. (a) Mode I,opening or tensile mode;(b) mode II, sliding mode; and(c) mode III, tearing mode.

(a) (b) (c)

stress intensity factor

plane strain

The stress intensity factor is related to the applied stress and the crack length by thefollowing equation:

(M.19)

Here Y is a dimensionless parameter or function that depends on both the crack andspecimen sizes and geometries, as well as on the manner of load application. More willbe said about Y in the discussion that follows. Moreover, it should be noted that K hasthe unusual units of [alternatively ]).

Fracture ToughnessIn the previous discussion, a criterion was developed for the crack propagation in a brittlematerial containing a flaw; fracture occurs when the applied stress level exceeds some criticalvalue sc (Equation M.14). Similarly, since the stresses in the vicinity of a crack tip can bedefined in terms of the stress intensity factor, a critical value of K exists that may be used tospecify the conditions for brittle fracture; this critical value is termed the fracture toughnessKc, and, from Equation M.19, is defined by

(M.20)

Here,sc again is the critical stress for crack propagation, and we now represent Y as a func-tion of both crack length (a) and component width (W)i.e., as Y(aW).

Relative to this Y(aW) function, as the aW ratio approaches zero (i.e., for very wideplanes and short cracks), Y(a/W) approaches a value of unity. For example, for a plate ofinfinite width having a through-thickness crack, Figure M.8a, Y(a/W) 1.0; whereas for aplate of semi-infinite width containing an edge crack of length a (Figure M.8b), Y(aW)

1.1. Mathematical expressions for Y(aW) (often relatively complex) in terms of aWare required for components of finite dimensions. For example, for a center-cracked(through-thickness) plate of width W (Figure M.9),

(M.21)Y1aW 2 aWpa

tan paWb

12

Kc Y1aW 2 sc1pa

ksi1in.MPa1m 1psi1in.

K Ys 1pa


Fracture toughnessdependence oncritical stress forcrack propagationand crack length

fracture toughness

Figure M.7 The stresses actingin front of a crack that is loadedin a tensile mode I configuration.

y

r

x

z

q

s y

s x

s z

txy

Stress intensityfactordependenceon applied stress andcrack length


Here the pa/W argument for the tangent is expressed in radians, not degrees. It is oftenthe case for some specific component-crack configuration that Y(a/W) is plotted versusa/W (or some variation of a/W). Several of these plots are shown in Figures M.10a, b, andc; included in the figures are equations that are used to determine Kcs.

By definition, fracture toughness is a property that is the measure of a materialsresistance to brittle fracture when a crack is present. Its units are the same as for thestress intensity factor (i.e., or

For relatively thin specimens, the value of Kc depends on and decreases with increas-ing specimen thickness B, as indicated in Figure M.11. Eventually, Kc becomesindependent of B, at which time the condition of plane strain is said to exist.7 The constantKc value for thicker specimens is known as the plane strain fracture toughness KIc, whichis also defined by8

(M.22)

It is the fracture toughness normally cited since its value is always less than Kc.The I sub-script for KIc denotes that this critical value of K is for mode I crack displacement, asillustrated in Figure M.6a. Brittle materials, for which appreciable plastic deformation isnot possible in front of an advancing crack, have low KIc values and are vulnerable tocatastrophic failure. On the other hand, KIc values are relatively large for ductile mate-rials. Fracture mechanics is especially useful in predicting catastrophic failure in materi-als having intermediate ductilities. Plane strain fracture toughness values for a number

KIc Ys1pa

psi1in. 2 .MPa1m

Figure M.8 Schematic representations of (a) an interiorcrack in a plate of infinite width, and (b) an edge crack in aplate of semi-infinite width.

2a a

Figure M.9 Schematicrepresentation of a flatplate of finite widthhaving a through-thickness center crack.

W2

W

2a

B

s

s

(a) (b)

Minimum specimenthickness for acondition of planestrain

plane strain fracturetoughness

Plane strain fracturetoughness for mode I crack surfacedisplacement

7Experimentally, it has been verified that for plane strain conditions

(M.23)

where sy is the 0.002 strain offset yield strength of the material.8In the ensuing discussion we will use Y to designate Y(a/W) in order to simplify the formof the equations.

B 2.5 aKIcsyb

2


W

F

Y

F

a

B

F

B

a

W

S

1.00

2.00

3.00

4.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6

a/W

Kc YFWB a

Y

1.10

1.12

1.14

1.16

1.18

1.20

1.22

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2a/W

Kc YFWB a

Y

0.9

1.1

1.3

1.5

1.7

1.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6

a/W

Kc

S/W 8

S/W 4

3FSY4W2B

a

W

F

F

a a

B

Figure M.10 Y calibration curves for three simple crack-plate geometries.(Copyright ASTM. Reprinted with permission.)

(a)

(b)

(c)


of different materials are presented in Table M.3; a more extensive list of KIc values isgiven in Table B.5 of Appendix B.

The stress intensity factor K in Equations M.18 and the plane strain fracture toughnessKIc are related to one another in the same sense as are stress and yield strength.A materialmay be subjected to many values of stress; however, there is a specific stress level at whichthe material plastically deformsthat is, the yield strength. Likewise, a variety of Ks arepossible, whereas KIc is unique for a particular material, and indicates the conditions of flawsize and stress necessary for brittle fracture.

Several different testing techniques are used to measure KIc; one of these is describedlater in Section M.6.Virtually any specimen size and shape consistent with mode I crack dis-placement may be utilized, and accurate values will be realized, provided that the Y scaleparameter in Equation M.22 has been properly determined.

Figure M.11 Schematic representation showingthe effect of plate thickness on fracture toughness.

Frac

ture

tou

ghne

ss K

c

Plane stressbehavior

Thickness B

Plane strainbehavior

KIc

Yield Strength KIc

Material MPa ksi MPa ksi .

Metals

Aluminum alloya (7075-T651) 495 72 24 22

Aluminum alloya (2024-T3) 345 50 44 40

Titanium alloya (Ti-6Al-4V) 910 132 55 50

Alloy steela 1640 238 50.0 45.8(4340 tempered @ 260C)

Alloy steela 1420 206 87.4 80.0(4340 tempered @ 425C)

Ceramics

Concrete 0.21.4 0.181.27

Soda-lime glass 0.70.8 0.640.73

Aluminum oxide 2.75.0 2.54.6

Polymers

Polystyrene (PS) 25.069.0 3.6310.0 0.71.1 0.641.0

Poly(methyl methacrylate) 53.873.1 7.810.6 0.71.6 0.641.5(PMMA)

Polycarbonate (PC) 62.1 9.0 2.2 2.0

1in1m

Table M.3Room-TemperatureYield Strength andPlane Strain Fracture Toughness Data forSelected EngineeringMaterials

aSource: Reprintedwith permission,Advanced Materialsand Processes, ASMInternational, 1990.

The plane strain fracture toughness KIc is a fundamental material property thatdepends on many factors, the most influential of which are temperature, strain rate, andmicrostructure. The magnitude of KIc decreases with increasing strain rate and decreasingtemperature. Furthermore, an enhancement in yield strength wrought by solid solutionor dispersion additions or by strain hardening generally produces a correspondingdecrease in KIc. Furthermore, KIc normally increases with reduction in grain size as com-position and other microstructural variables are maintained constant.Yield strengths areincluded for some of the materials listed in Table M.3.

Design Using Fracture MechanicsAccording to Equations M.20 and M.22, three variables must be considered relative tothe possibility for fracture of some structural componentviz. the fracture toughness(Kc) or plane strain fracture toughness (KIc), the imposed stress (s), and the flaw size(a), assuming, of course, that Y has been determined. When designing a component, it isfirst important to decide which of these variables are constrained by the application andwhich are subject to design control. For example, material selection (and hence Kc orKIc) is often dictated by factors such as density (for lightweight applications) or the cor-rosion characteristics of the environment. Alternatively, the allowable flaw size is eithermeasured or specified by the limitations of available flaw detection techniques. It isimportant to realize, however, that once any combination of two of the preceding param-eters is prescribed, the third becomes fixed (Equations M.20 and M.22). For example,assume that KIc and the magnitude of a are specified by application constraints; therefore,the design (or critical) stress sc is given by

(M.24)

On the other hand, if stress level and plane strain fracture toughness are fixed by thedesign situation, then the maximum allowable flaw size ac is given by

(M.25)ac 1p

aKIcsYb

2

sc KIc

Y1pa


Computation ofdesign stress

Computation ofmaximum allowableflaw length

EXAMPLE PROBLEM M.2

Determination of the Possibility of Critical Flaw Detection

A structural component in the form of a very wide plate, as shown in Figure M.8a, is to be fab-ricated from a 4340 steel. Two sheets of this alloy, each having a different heat treatment andthus different mechanical properties, are available. One, denoted material A, has a yieldstrength of 860 MPa (125,000 psi) and a plane strain fracture toughness of 98.9 MPa

. For the other, material Z, y and KIc values are 1515 MPa (220,000 psi) andrespectively.

(a) For each alloy, determine whether or not plane strain conditions prevail if the plate is 10mm (0.39 in.) thick.

(b) It is not possible to detect flaw sizes less than 3 mm, which is the resolution limit of the flawdetection apparatus. If the plate thickness is sufficient such that the KIc value may be used,determine whether or not a critical flaw is subject to detection.Assume that the design stresslevel is one half of the yield strength; also, for this configuration, the value of Y is 1.0.

60.4 MPa1m 155 ksi1in. 2 ,190 ksi1in. 2

1m


Solution

(a) Plane strain is established by Equation M.23. For material A,

0.033 m 33 mm (1.30 in.)

Thus, plane strain conditions do not hold for material A because this value of B is greaterthan 10 mm, the actual plate thickness; the situation is one of plane stress and must betreated as such.

And for material Z,

which is less than the actual plate thickness, and therefore the situation is one of plane strain.

(b) We need only to determine the critical flaw size for material Z because the situation formaterial A is not plane strain, and KIc may not be used. Employing Equation M.25 andtaking s to be sy/2,

0.002 m 2.0 mm (0.079 in.)

Therefore, the critical flaw size for material Z is not subject to detection since it is lessthan 3 mm.

ac 1p

a 60.4 MPa1m11 2 11515/2 2 MPa b2

B 2.5 a 60.4 MPa1m1515 MPa

b 2 0.004 m 4.0 mm 10.16 in. 2

B 2.5 aKIcsyb 2 2.5 a 98.9 MPa1m

860 MPab 2

DESIGN EXAMPLE M.1

Material Specification for a Pressurized Spherical Tank

Consider a thin-walled spherical tank of radius r and thickness t (Figure M.12) that may be usedas a pressure vessel.

(a) One design of such a tank calls for yielding of the wall material prior to failure as a result ofthe formation of a crack of critical size and its subsequent rapid propagation.Thus, plastic distor-tion of the wall may be observedand the pressure within the tankreleased before the occurrence ofcatastrophic failure. Consequently,materials having large criticalcrack lengths are desired. On thebasis of this criterion, rank themetal alloys listed in Table B.5,Appendix B, as to critical cracksize, from longest to shortest.

(b) An alternative design that isalso often utilized with pressurevessels is termed leak-before-break. On the basis of principlesof fracture mechanics, allowanceis made for the growth of a crackthrough the thickness of the

t

p

pp

p

p

p

pp

r

2a

s

s

Figure M.12 Schematic diagram showing the cross sectionof a spherical tank that is subjected to an internal pressure p,and that has a radial crack of length 2a in its wall.


vessel wall prior to the occurrence of rapid crack propagation (Figure M.12).Thus, the crack willcompletely penetrate the wall without catastrophic failure, allowing for its detection by the leak-ing of pressurized fluid. With this criterion the critical crack length ac (i.e., one-half the totalinternal crack length) is taken to be equal to the pressure vessel thickness t. Allowance for ac t instead of ac t/2 ensures that fluid leakage will occur prior to the buildup of dangerouslyhigh pressures. Using this criterion, rank the metal alloys in Table B.5, Appendix B as to themaximum allowable pressure.

For this spherical pressure vessel, the circumferential wall stress s is a function of the pressurep in the vessel and the radius r and wall thickness t according to

(M.26)

For both parts (a) and (b), assume a condition of plane strain.

Solution

(a) For the first design criterion, it is desired that the circumferential wall stress be less thanthe yield strength of the material. Substitution of sy for s in Equation M.22, and incorporationof a factor of safety N leads to

(M.27)

where ac is the critical crack length. Solving for ac yields the following expression:

(M.28)

Therefore, the critical crack length is proportional to the square of the KIc-sy ratio, whichis the basis for the ranking of the metal alloys in Table B.5. The ranking is provided in TableM.4, where it may be seen that the medium carbon (1040) steel with the largest ratio has thelongest critical crack length and, therefore, is the most desirable material on the basis of thiscriterion.

(b) As stated previously, the leak-before-break criterion is just met when one-half the internalcrack length is equal to the thickness of the pressure vessel (i.e., when a t). Substitution ofa t into Equation M.22 gives

ac N 2

Y 2p a

KIcsyb

2

KIc Y asy

Nb 1pac

s pr

2t

Material(mm)

Medium carbon (1040) steel 43.1

AZ31B magnesium 19.6

2024 aluminum (T3) 16.3

Ti-5Al-2.5Sn titanium 6.6

4140 steel (tempered @ ) 5.3


Ti-6Al-4V titanium 3.7

17-7PH stainless steel 3.4

7075 aluminum (T651) 2.4


4340 steel (tempered @ ) 0.93260C

370C

425C

482C

aKIcSyb

2Table M.4Ranking of Several MetalAlloys Relative to CriticalCrack Length (YieldingCriterion) for a Thin-WalledSpherical Pressure Vessel

M.5 Flaw Detection Using Nondestructive Testing Techniques 21

(M.29)

And, from Equation M.26,

(M.30)

The stress is replaced by the yield strength because the tank should be designed to contain thepressure without yielding; furthermore, substitution of Equation M.30 into Equation M.29, af-ter some rearrangement, yields the following expression:

(M.31)

Hence, for some given spherical vessel of radius r, the maximum allowable pressure con-sistent with this leak-before-break criterion is proportional to The same severalmaterials are ranked according to this ratio in Table M.5; as may be noted, the mediumcarbon steel will contain the greatest pressures.

Of the eleven metal alloys listed in Table B.5, the medium carbon steel ranks first ac-cording to both yielding and leak-before-break criteria. For these reasons, many pressurevessels are constructed of medium carbon steels when temperature extremes and corro-sion need not be considered.

K 2Icsy.

p 2

Y 2pr a

K 2Icsyb

t pr

2s

KIc Ys1pt

Material (MPa-m)

Medium carbon (1040) steel 11.2


Ti-5Al-2.5Sn titanium 5.8

2024 aluminum (T3) 5.6


17-7PH stainless steel 4.4

AZ31B magnesium 3.9

Ti-6Al-4V titanium 3.3

4140 steel (tempered @ C) 2.4


7075 aluminum (T651) 1.2

260C

370

425C

482C

KIc2

Sy

Table M.5Ranking of Several MetalAlloys Relative to MaximumAllowable Pressure (Leak-Before-Break Criterion) for aThin-Walled Spherical PressureVessel

M.5 FLAW DETECTION USING NONDESTRUCTIVE TESTING TECHNIQUESA number of nondestructive testing (NDT) techniques have been developed that per-mit detection and measurement of both internal and surface flaws.9 Such techniques areused to examine structural components currently in service for defects and flaws thatcould lead to premature failure; in addition, NDTs are used as a means of quality con-

9Sometimes the terms nondestructive evaluation (NDE) and nondestructive inspection (NDI) are also used for thesetechniques.

nondestructivetesting

trol for manufacturing processes.As the name implies, these techniques must not destroyor damage the material/structure being examined nor impair its future serviceability.Some testing techniques are capable of detecting only surface defects, some only subsur-face (interior) defects, while other tests detect defects at both surface and subsurfacesites. Furthermore, in some instances location of testing is important. Some testing meth-ods must be conducted in a laboratory setting; others may be adapted for use in the field.Several commonly employed NDT techniques are visual inspection, optical microscopy,scanning electron microscopy, dye (or liquid) penetrant, magnetic particle, radiographic,ultrasonic, and acoustic emission. A listing of these techniques and their characteristicsare presented in Table M.6.

The following discussions on some of these techniques are very brief and con-densed. More detailed treatments are provided in the end-of-module references.

Visual InspectionVisual inspection is probably the most common detection technique and the easiest toconduct; of course, only cracks and defects found on surfaces may be observed visually.Only relatively large cracks/defects are subject to detection with the unaided eye or amagnifying glass. For inspection of inaccessible/remote regions, mirrors, fiberscopes,and borescopes may be used. Fiberscopes and borescopes are optical devices composedof an eyepiece on the inspection end and a lens on the observation end, which arelinked by either a rigid or flexible tube (normally of an optical fiber bundle and a pro-tective outer sheath), which acts as the optical relay system, and in some cases is usedto illuminate the remote object. An imaging device (e.g., video camera) may also beincorporated.

Portable video inspection camera systems are used to inspect the interiors of largestructures (e.g., containers, railroad tank cars, sewer lines) that are inaccessible and/orhazardous. A video camera (with a zoom lens) is mounted on a pole, cable, or trolley fordeployment into the structure that is to be inspected.

Visual inspection of some confined (and normally horizontal) and long structures(such as pipelines, air ducts, and reactors) is possible using robotic crawlers. Thesedevices typically include a sensor (or video camera) mounted on a mobile support car-riage that is capable of traveling through the system to be inspected. The crawler mustbe steerable, capable of both forward and reverse motions, as well as acceleration,deceleration, and stopping; in addition, it must have the ability to negotiate bends inpiping and to pass through different diameters of pipes. An illumination system is alsoprovided, and the camera must deliver full-directional viewing as well as have a remoteadjustable focus.


Defect SizeTechnique Defect Location Sensitivity (mm) Testing Location

Visual inspection Surface 0.1 Laboratory/in-field

Optical microscopy Surface 0.10.5 Laboratory

Scanning electron Surface 0.001 Laboratorymicroscopy (SEM)

Dye penetrant Surface 0.0350.25 Laboratory/in-field

Magnetic particle Surface 0.5 Laboratory/in-field

Radiography Subsurface 2% of Laboratory/in-field(x-ray/gamma ray) specimen thickness

Ultrasonic Subsurface 0.50 Laboratory/in-field

Acoustic emission Surface/ 0.1 Laboratory/in-fieldsubsurface

Table M.6A List of SeveralCommonNondestructive TestingTechniques

Source: Adapted fromASM Handbook,Vol. 19, Fatigue andFracture. Reprintedwith permission ofASM International. Allrights reserved. www.asminternational.org.

M.5 Flaw Detection Using Nondestructive Testing Techniques 23

Optical and Scanning Electron Microscopy InspectionsFor detection of small surface cracks (less than about 0.5 mm in size), employment ofoptical and/or scanning electron microscopic techniques is necessary. Normally inspec-tions of these types are conducted in a laboratory setting (as opposed to in the field).Brief discussions of these two techniques are presented in Section 5.12.

Dye (Liquid) Penetrant InspectionThis common and low-cost technique is used to detect surface cracks in nonporous ma-terials. In essence, a liquid is used to enhance the visual contrast between a defect andthe bulk solid material. This liquid must have high surface wetting characteristics (i.e., alow surface tension) and is applied (by spraying, brushing, or dipping) onto the surfaceof the part to be inspected. After adequate time has been allowed for this liquid to pen-etrate (by capillary action) into any surface-breaking defects that are present, excess liq-uid is removed, and a powder developer is applied that draws the penetrant out of anydefects and to the surface, making possible their observation. Visual inspection is per-formed using a white light. The penetrant may also be loaded with a fluorescent dye toincrease detection sensitivity; under these circumstances an ultraviolet (or black) light(in a darkened environment) is used to reveal the defects.

Figure M.13 shows a surface crack that was exposed using this technique.

Magnetic Particle InspectionA variation of the dye penetrant technique is used to detect both surface and near-surfacedefects in ferrous alloys that may be magnetized. For such a material that has been magnet-ized, in the vicinity of a surface flaw (or discontinuity) there is a distortion of the magnetic flux(or field) such that the flux leaks from (or passes out of) the solid.This inspection techniqueutilizes fine iron particles that are suspended in a suitable liquid carrier (e.g., kerosene); theseparticles are often coated with a fluorescent dye. The article to be inspected is first magnet-ized, and when such a suspension is applied (sprayed or painted) onto its surface, iron parti-cles become attracted to regions where any leakage fields are presenti.e., they cluster in thevicinity of any surface defects. Visual detection of these clusters (and defects) is possibleunder proper lighting conditions (e.g., a fluorescent light and a dark ambient).A surface crackthat was made visible using magnetic particles is presented in Figure M.14.

After observation, demagnetization of the inspected object is possible.

Figure M.13 Photograph showing a surface crack inan automobile engine connecting rod that was madevisible using a dye penetrant.(Photograph courtesy of Center for NDE, Iowa State University.)

Figure M.14 Photograph showing a surface crack ina crane hook that was revealed using the magneticparticle inspection technique.(Photograph courtesy of Center for NDE, Iowa State University.)

The final three nondestructive testing techniques (radiographic, ultrasonic [pulseecho], and acoustic emission) are utilized to detect subsurface (interior) defects. The firsttwo techniques (radiographic and ultrasonic testing) employ some type of signal or energysource (e.g., ultrasonic waves, x-rays) to probe the object to be examined. An interactionbetween the incoming signal and a defect or crack causes some type of signal disruption,and a response in the form of an image or signature that may be sensed and recorded (suchas a photograph or a blip on a screen).

Radiographic TestingThe radiographic testing technique utilizes x-rays or gamma radiation (from a radioac-tive isotope such as iridium-192 or cobalt-60) as a signal source.This radiation is directedupon, penetrates, and passes through the object to be inspected. An image is generatedon the remote side by radiation that is transmitted through the object. Photographic film(that is sensitive to the type of radiation) is most frequently employed as the detectiondevice to record this image (Figure M.15); fluoroscopic screens and digitized systemswith video monitors may also be used. An image results from variations in the transmit-ted radiation intensity over the cross-section of the object. Defects and cracks will ap-pear as part of this image inasmuch as transmission intensity will be different throughregions containing defects than those regions that are defect-free.This technique is com-monly used to assess the integrity of welds.

It should be noted that health hazards are associated with radiographic testing; ex-posure to both x-rays and gamma rays must be avoided since both are forms of ionizingradiation.

Ultrasonic (Pulse-Echo) InspectionFor this inspection technique, the input energy (or signal) is in the form of ultrasonicwavesi.e., sound waves having high frequencies, normally in the range of 0.1 to 50 MHz.These waves are emitted from a transducer as intermittent pulses, which are introducedinto and propagate through the object that is being inspected. Normally both transducerand test object are immersed in a coupling medium (a liquid such as water or oil) so asto promote the transfer of ultrasonic waves. These waves experience reflection (orecho) whenever they encounter some type of interface or discontinuity, such as theback face of the test object or the surface of an interior defect. A reflected wave isreceived by the same transducer, which then converts the wave signal into an electricalsignal. Test results are displayed on a screen as reflected signal strength amplitude versustravel time (i.e., the time between when the pulsed signal was sent and when thereflected signal was received). A high intensity peak of signal strength represents the


Figure M.15 Photograph generated using x-radiationthat shows an internal defect (dark wedge-shaped region)in an automobile motor mount casting.(Photograph courtesy of Center for NDE, Iowa State University.)

M.6 Fracture Toughness Testing 25

time at which a reflected wave was received. Signal travel time may be converted intotravel distance inasmuch as the velocity of the ultrasonic waves is known. Measurementof travel time is important in order to distinguish between back-surface and defectechos, and, in addition, to determine the location (depth) of a defect.A complete inspec-tion involves passing the transducer probe over the entire region of the test object.

Aerospace, aviation, and automotive industries utilize this NDT technique extensively.

Acoustic EmissionAcoustic emission testing also utilizes ultrasonic waves to detect the presence of cracksusually in the frequency range between 30 kHz and 1MHz. However, unlike conventionalultrasonic testing, this technique monitors sound (acoustic) waves that are emitted duringthe failure process (i.e., as a crack forms and then propagates) and while a structure is inservicethat is, no ultrasonic signal is artificially generated and then collected.Associatedwith the formation and extension of cracks is the release of elastic strain energy, in theform of sound waves, that propagate throughout the material and ultimately to its surfacewhere they may be recorded using some type of sensor (i.e., a transducer). This signal isconverted into an electrical signal and then displayed on a screen for analysis.

One advantage of acoustic emission testing (over other NDT techniques) is that itmonitors failure processes that are dynamic (i.e., crack formation and growth). As such,instantaneous information relative to the status of and risk of failure is provided. Thistechnique is frequently used on aircraft. For example, a group of transducers mountedin a highly stressed area can detect the presence of a crack the instant that it forms, and,in addition, very precisely determine its origin by measurement of the time it takes forthe signal to reach different transducers.

M.6 FRACTURE TOUGHNESS TESTINGAs noted earlier, fracture toughness is defined as a materials resistance to crack propaga-tion and ultimately to brittle fracture. In Section M.4 we used the symbol KIc to represent thefracture toughness for the condition of plane strain (i.e., when specimen thickness is greaterthan crack length) and also when stress application is such as to promote mode I crack sur-face displacement (Figure M.6). Inasmuch as KIc is such an important material property withregard to fracture prevention, it seems reasonable to explore the manner in which it is meas-ured.A variety of standardized tests have been devised. In the United States, these test meth-ods are developed by ASTM; for the international marketplace, standards are established bythe International Organization for Standardization (ISO). Most of these techniques are de-signed for measuring fracture toughness values for metals and their alloys; in addition, somehave also been developed for ceramics, polymers, and composite materials.

In essence, a typical fracture toughness test is conducted on a standard specimen thatcontains a preexisting crack. A testing apparatus loads the specimen at a prescribed rate,and continuously records load magnitude and crack displacement. Resulting data are ana-lyzed and fracture toughness parameters are determined. These parameters are then sub-jected to qualification procedures in order to ensure they meet established criteria beforethe fracture toughness values are deemed acceptable. We have chosen to describe one ofthe earliest and least complicated fracture toughness test standards that was developed:ASTM Standard E 399-09,Standard Test Method for Linear-Elastic Plane-Strain FractureToughness KIc of Metallic Materials.

First of all, as the title of this standard suggests, the test is used to measure KIc whenthe crack-tip region is exposed to a condition of plane-strain upon load application. In ad-dition, the material being tested should exhibit linear-elastic behaviorthat is, a plot ofload versus crack displacement is linear, and virtually all deformation to the point of frac-ture is elastic (i.e., the material has limited ductility). Furthermore, it may be recalled(Section M.4), that the elastic stress field near a crack tip can be described in terms of the

stress intensity factor K; as will be seen we use this parameter in the development of themethodology for this testing technique.

Two specimen geometries permitted by Standard E 399single-edge notched bendand compact tensionare represented in Figures M.16a and M.16b, respectively. Asnoted in these illustrations, a three-point loading scheme is used for the bend specimen,whereas the compact specimen is loaded in tension. Specimen size is not specified by thisstandard, which must be selected. Test validity is dependent on specimen size, which isnot subject to evaluation until after the conclusion of the test; therefore, unless an ade-quate size is chosen, the test will need to be repeated. A notch is machined in each spec-imen, after which a very sharp precrack of length a is introduced at the notch root usingcyclic fatigue-loading.As noted in Figures M.16a and M.16b, initial crack length includesboth notch depth as well as precrack length. Details relating to specimen size selection,geometrical tolerances, notch configuration, and precracking procedures are containedin the ASTM standard.

During testing, load is applied at a specified rate and measured using a load cell, whichis one component of the testing apparatus. Furthermore, a clip gage, mounted on the testspecimen across the open end of the notch (Figure M.17), monitors crack displacement.Results are plotted as load (P) versus displacement (v).The test is continued until fracture,after which the initial crack length (a) (Figures M.16a and M.16b) is physically measuredon the broken specimen halves. From these data a conditional load PQ is measured, fromwhich a conditional KIc may be determined (and labeled KQ); this KQ is then evaluated asto its validity as we explain below.

Three different types of load versus displacement curves have been observed, whichare presented in Figure M.18 (and labeled I, II, and III). The procedure for determina-tion of this conditional PQ value is described as follows: For each curve type, a tangent isconstructed at the initial linear portion of the curve (OA) and its slope is determined. Astraight-line segment having a slope 5% less than this initial tangent is then constructedfrom the origin; the intersection of this secant (OP5) with the load-displacement curve isindicated by the point labeled P5 for each of the curves shown in this plot. If, on a load-displacement curve, every force point that precedes P5 is less than P5 (as is the case foronly curve I in Figure M.18), then PQ P5. On the other hand, when there is a sharpdrop in load just past the termination of the linear-elastic region, such that maximum


S

B

P

W

Support pin

aPrecrack

Loading pin

B

W

Precrack

P

P

a

Figure M.16 Configuration of (a) single-edge notched bend and (b) compact-tension specimens used forfracture toughness tests (ASTM Standard E 399-09).(From V. J. Colangelo and F. A. Heiser, Analysis of Metallurgical Failures, 2nd edition. Copyright 1987 by John Wiley & Sons,New York. Reprinted with permission of John Wiley & Sons, Inc.)

(a) (b)

M.6 Fracture Toughness Testing 27

load on the resultant cusp precedes and is greater than P5 (Figure M.18, curves II andIII), then this maximum load is taken as PQ.

At this time it becomes necessary to impose the first validity criterionto determinewhether the specimen is too ductile to be tested using this technique. This criterion isexpressed quantitatively by the expression

(M.32)

where Pmax is the maximum force that the test specimen is able to sustain (Figure M.18).If this criterion is not satisfied, then another fracture toughness testing technique mustbe employed.10

However, if the criterion specified by Equation M.32 is realized, the next step is todetermine a value for the conditional KQ. For the single-edge notched bend specimen,the following equation is employed:

(M.33)

In this expression (and from Figure M.16a)PQ the conditional load value, determined as described aboveS distance between support pointsB specimen thicknessW specimen width (or depth)a precrack length

KQ PQ S

BW32 f a a

Wb

PmaxPQ

1.10

W

Test fixture

Testspecimen

Displacementgage

P

O O ODisplacement,

Load

, P

Type I

A Pmax

P5 PQ Pmax PQP5 P5

PmaxA A

Type II Type III

95% Secant

PQ

Figure M.17 Schematic diagram showing adisplacement gage that has been installed on asingle-edge notched bend specimen inpreparation for a fracture toughness test.(Adapted with permission from Figure A2.1 in ASTME 399-09 Standard Test Method for Linear-ElasticPlane-Strain Fracture Toughness KIc of MetallicMaterials. Copyright ASTM International, 100 BarrHarbor Drive, West Conshohocken, PA 19428. A copyof this standard may be obtained from ASTMInternational, www.astm.org.)

Figure M.18 Three principal types of load versusdisplacement curves that may be generated during a fracturetoughness test. (ASTM Standard E 399-09).(Adapted with permission from Figure 7 in ASTM E 399-09 StandardTest Method for Linear-Elastic Plane-Strain Fracture Toughness KIcof Metallic Materials. Copyright ASTM International, 100 BarrHarbor Drive, West Conshohocken, PA 19428. A copy of thisstandard may be obtained from ASTM International, www.astm.org.)

10For example, ASTM Standard E 1820.

KIc validitycriterionmaximumdegree of ductility

In Equation M.33, is a calibration function that depends on the a/W ratio as

(M.34)

Similarly, the following equation is used to compute KQ for the compact-tensionspecimen configuration:

(M.35)

in which B and W are the specimen thickness and width (depth), respectively, and a isthe precrack length (Figure M.16b). In this case

(M.36)

Again, KQ is conditional, and before it can be accepted as a valid KIc value, verifica-tion of a condition of plane-strain must be established. Such is possible when the follow-ing criterion is satisfied (for both specimen geometries)11:

(M.37)

Here a and B are, respectively, crack length and specimen thickness, and sy is the 0.2%offset yield strength (measured in tension).

By way of summary:

When the criteria specified by Equations M.32 and M.37 are met, then KQ is avalid value for KIc, and may be reported as such.

If the condition of Equation M.32 is not satisfied, then another testing techniquemust be employed.

And, finally, when the criterion of Equation M.32 is met, while at the same timeEquation M.37 is not realized, then the test must be repeated using a thickerspecimen. A new specimen thickness may be estimated by incorporating themeasured value of KQ into Equation M.37.

a and B 2.5 aKQsyb

2

a2 aWb c0.866 4.64 a

aWb 13.32 a

aWb

2

14.72 aaWb

3

5.6 aaWb

4

d

a1 aWb

32

f aaWb

KQ PQ

B2W f a

aWb

3BaW

1.99 aaWb a1

aWb c2.15 3.93 a

aWb 2.7 a

aWb

2

d

2 c1 2 aaWb d c1 a

aWb d

32

f aaWb

f aaWb


KIc validitycriterionminimumcrack length andminimum specimenthickness

11Note the similarity between Equations M.23 and M.37, the former of which was cited earlier as a minimumthickness for the condition of plane strain.

M.7 Impact Fracture Testing 29

M.7 IMPACT FRACTURE TESTING12

12 This section is virtually identical to Section 9.8.13ASTM Standard E 23, Standard Test Methods for Notched Bar Impact Testing of Metallic Materials.

Prior to the advent of fracture mechanics as a scientific discipline, impact testing techniqueswere established to ascertain the fracture characteristics of materials at high loading rates.It was realized that the results of laboratory tensile tests (at low loading rates) could not beextrapolated to predict fracture behavior. For example, under some circumstances normallyductile metals fracture abruptly and with very little plastic deformation under high loadingrates. Impact test conditions were chosen to represent those most severe relative to the po-tential for fracturenamely, (1) deformation at a relatively low temperature, (2) a highstrain rate (i.e., rate of deformation), and (3) a triaxial stress state (which may be introducedby the presence of a notch).

Impact Testing TechniquesTwo standardized tests,13 the Charpy and the Izod, are used to measure the impactenergy (sometimes also termed notch toughness).The Charpy V-notch (CVN) techniqueis most commonly used in the United States. For both the Charpy and the Izod, the spec-imen is in the shape of a bar of square cross section, into which a V-notch is machined(Figure M.19a). The apparatus for making V-notch impact tests is illustrated schemati-cally in Figure M.19b. The load is applied as an impact blow from a weighted pendulumhammer that is released from a cocked position at a fixed height h. The specimen ispositioned at the base as shown. Upon release, a knife edge mounted on the pendulumstrikes and fractures the specimen at the notch, which acts as a point of stress concentra-tion for this high-velocity impact blow. The pendulum continues its swing, rising to amaximum height which is lower than h. The energy absorption, computed from thedifference between h and is a measure of the impact energy. The primary differencebetween the Charpy and Izod techniques lies in the manner of specimen support, asillustrated in Figure M.19b.These are termed impact tests because of the manner of loadapplication. Variables including specimen size and shape as well as notch configurationand depth influence the test results.

Both plane strain fracture toughness and these impact tests determine the fractureproperties of materials. The former are quantitative in nature, in that a specific propertyof the material is determined (i.e., KIc).The results of the impact tests, on the other hand,are more qualitative and are of little use for design purposes. Impact energies are ofinterest mainly in a relative sense and for making comparisonsabsolute values are oflittle significance.Attempts have been made to correlate plane strain fracture toughnessesand CVN energies, with only limited success. Plane strain fracture toughness tests arenot as simple to perform as impact tests; furthermore, equipment and specimens aremore expensive.

Ductile-to-Brittle TransitionOne of the primary functions of the Charpy and the Izod tests is to determine whethera material experiences a ductile-to-brittle transition with decreasing temperature and, ifso, the range of temperatures over which it occurs.As may be noted in the chapter-openingphotograph (of the oil barge) for Chapter 9, widely used steels can exhibit this ductile-to-brittle transition with disastrous consequences. The ductile-to-brittle transition isrelated to the temperature dependence of the measured impact energy absorption. Thistransition is represented for a steel by curve A in Figure M.20. At higher temperaturesthe CVN energy is relatively large, corresponding to a ductile mode of fracture. As the

h,h,

Charpy test, Izod test

impact energy

ductile-to-brittle transition

temperature is lowered, the impact energy drops suddenly over a relatively narrow tem-perature range, below which the energy has a constant but small value; that is, the modeof fracture is brittle.

Alternatively, appearance of the failure surface is indicative of the nature of frac-ture and may be used in transition temperature determinations. For ductile fracture, thissurface appears fibrous or dull (or of shear character) as in the steel specimen of Fig-ure M.21 which was tested at 79C. Conversely, totally brittle surfaces have a granular


10 mm(0.39 in.)

Izod

ScaleCharpy

Starting positionPointer

End of swing

Specimen

Anvil

8 mm(0.32 in.)

10 mm(0.39 in.)

Notch

h'

Hammer

h

Figure M.19 (a)Specimen used for

Charpy and Izodimpact tests. (b) A

schematic drawing ofan impact testing

apparatus. Thehammer is releasedfrom fixed height h

and strikes thespecimen; the energyexpended in fracture

is reflected in thedifference between hand the swing height

Specimenplacements for both

Charpy and Izodtests are also shown.

[Figure (b) adaptedfrom H. W. Hayden,W. G. Moffatt, and J.

Wulff, The Structure andProperties of Materials,

Vol. III, MechanicalBehavior, p. 13.

Copyright 1965 byJohn Wiley & Sons,

New York. Reprintedby permission of John

Wiley & Sons, Inc.]

h.

(a)

(b)

M.7 Impact Fracture Testing 31

(shiny) texture (or cleavage character) (the 59C specimen in Figure M.21). Over theductile-to-brittle transition, features of both types will exist (in Figure M.21, displayedby specimens tested at 12C, 4C, 16C, and 24C). Frequently, the percent shear frac-ture is plotted as a function of temperaturecurve B in Figure M.20.

For many alloys there is a range of temperatures over which the ductile-to-brittle transition occurs (Figure M.20); this presents some difficulty in specifying a singleductile-to-brittle transition temperature. No explicit criterion has been established, and sothis temperature is often defined as the temperature at which the CVN energy assumessome value (e.g., 20 J or 15 ft-lbf), or corresponding to some given fracture appearance(e.g., 50% fibrous fracture). Matters are further complicated by the fact that a differenttransition temperature may be realized for each of these criteria. Perhaps the most conser-vative transition temperature is that at which the fracture surface becomes 100% fibrous;on this basis, the transition temperature is approximately 110C (230F) for the steel alloythat is shown in Figure M.20.

Structures constructed from alloys that exhibit this ductile-to-brittle behaviorshould be used only at temperatures above the transition temperature to avoid brittleand catastrophic failure. Classic examples of this type of failure occurred with disastrous

Figure M.20 Temperaturedependence of the Charpy V-notchimpact energy (curve A) and percentshear fracture (curve B) for an A283steel.(Reprinted from Welding Journal. Used bypermission of the American WeldingSociety.)

Impa

ct e

nerg

y (J

)

100

80

60

40

20

0

100

80

60

40

20

0

40 0 40 80 120 160 200 240 280

40 20 0 4020 80 12060 100 140

Temperature (F)

Temperature (C)

She

ar f

ract

ure

(%)

Impactenergy

Shearfracture

A

B

Figure M.21 Photograph offracture surfaces of A36 steel CharpyV-notch specimens tested at indicatedtemperatures (in C).(From R. W. Hertzberg, Deformation andFracture Mechanics of EngineeringMaterials, 3rd edition, Fig. 9.6, p. 329.Copyright 1989 by John Wiley & Sons,Inc., New York. Reprinted by permissionof John Wiley & Sons, Inc.)

59 12 4 16 24 79

consequences during World War II when a number of welded transport ships away fromcombat suddenly split in half.The vessels were constructed of a steel alloy that possessedadequate ductility according to room-temperature tensile tests. The brittle fracturesoccurred at relatively low ambient temperatures, at about 4C (40F), in the vicinity ofthe transition temperature of the alloy. Each fracture crack originated at some point ofstress concentration, probably a sharp corner or fabrication defect, and then propagatedaround the entire girth of the ship.

In addition to the ductile-to-brittle transition represented in Figure M.20, two othergeneral types of impact energy-versus-temperature behavior have been observed; theseare represented schematically by the upper and lower curves of Figure M.22. Here itmay be noted that low-strength FCC metals (some aluminum and copper alloys) andmost HCP metals do not experience a ductile-to-brittle transition (corresponding to theupper curve of Figure M.22) and retain high impact energies (i.e., remain ductile) withdecreasing temperature. For high-strength materials (e.g., high-strength steels and tita-nium alloys), the impact energy is also relatively insensitive to temperature (the lowercurve of Figure M.22); however, these materials are also very brittle, as reflected by theirlow impact energy values. The characteristic ductile-to-brittle transition is representedby the middle curve of Figure M.22. As noted, this behavior is typically found in low-strength steels that have the BCC crystal structure.

For these low-strength steels, the transition temperature is sensitive to both alloycomposition and microstructure. For example, decreasing the average grain size resultsin a lowering of the transition temperature. Hence, refining the grain size both stren-gthens (Section 8.9) and toughens steels. In contrast, increasing the carbon content,although it increases the strength of steels, also raises their CVN transition, as indicatedin Figure M.23.

Izod or Charpy tests are also conducted to assess the impact strength of polymericmaterials. As with metals, polymers may exhibit ductile or brittle fracture under impactloading conditions, depending on the temperature, specimen size, strain rate, and modeof loading, as discussed in the preceding section. Both semicrystalline and amorphouspolymers are brittle at low temperatures and both have relatively low impact strengths.However, they experience a ductile-to-brittle transition over a relatively narrow temper-ature range, similar to that shown for a steel in Figure M.20. Of course, impact strengthundergoes a gradual decrease at still higher temperatures as the polymer begins tosoften. Typically, the two impact characteristics most sought after are a high impactstrength at the ambient temperature and a ductile-to-brittle transition temperature thatlies below room temperature.

Most ceramics also experience a ductile-to-brittle transition, which occurs only atelevated temperatures, ordinarily in excess of 1000C (1850F).


Impa

ct e

nerg

y

Low-strength (FCC and HCP) metals

Low-strength steels (BCC)

High-strength materials

Temperature

Figure M.22 Schematic curvesfor the three general types ofimpact energy-versus-temperaturebehavior.

M.8 Cyclic Stresses 33

Fatigue

Temperature (C)

Temperature (F)Im

pact

ene

rgy

(J)

Impa

ct e

nerg

y (f

t-lb

f)

200 100

200 0 200 400

0 100 200

100

0

40

80

120

160

0.01 0.11

0.22

0.310.43

0.53

0.63

0.67

200

240

0

200

300

Figure M.23 Influence of carboncontent on the Charpy V-notch energy-versus-temperature behavior for steel.(Reprinted with permission from ASMInternational, Metals Park, OH 44073-9989,USA; J. A. Reinbolt and W. J. Harris, Jr.,Effect of Alloying Elements on NotchToughness of Pearlitic Steels, Transactions ofASM, Vol. 43, 1951.)

Fatigue is a form of failure that occurs in structures subjected to dynamic and fluctuat-ing stresses (e.g., bridges, aircraft, and machine components). Under these circumstancesit is possible for failure to occur at a stress level considerably lower than the tensile oryield strength for a static load.The term fatigue is used because this type of failure normallyoccurs after a lengthy period of repeated stress or strain cycling. Fatigue is importantinasmuch as it is the single largest cause of failure in metals, estimated to be involved inapproximately 90% of all metallic failures; polymers and ceramics (except for glasses)are also susceptible to this type of failure. Furthermore, fatigue failure is catastrophicand insidious, occurring very suddenly and without warning.

Fatigue failure is brittlelike in nature even in normally ductile metals, in that thereis very little, if any, gross plastic deformation associated with failure. The process occursby the initiation and propagation of cracks, and typically the fracture surface is perpen-dicular to the direction of an applied tensile stress.

M.8 CYCLIC STRESSES14

The applied stress may be axial (tension-compression), flexural (bending), or torsional(twisting) in nature. In general, three different fluctuating stress-time modes are possi-ble. One is represented schematically by a regular and sinusoidal time dependence inFigure M.24a, where the amplitude is symmetrical about a mean zero stress level, forexample, alternating from a maximum tensile stress (smax) to a minimum compressivestress (smin) of equal magnitude; this is referred to as a reversed stress cycle. Anothertype, termed a repeated stress cycle, is illustrated in Figure M.24b; the maxima and min-ima are asymmetrical relative to the zero stress level. Finally, the stress level may varyrandomly in amplitude and frequency, as exemplified in Figure M.24c.

14 This section is virtually identical to Section 9.9.

fatigue

Also indicated in Figure M.24b are several parameters used to characterize the fluc-tuating stress cycle. The stress amplitude alternates about a mean stress sm, defined asthe average of the maximum and minimum stresses in the cycle, or

(M.38)

The range of stress sr is the difference between smax and smin, namely,

(M.39)

Stress amplitude sa is one-half of this range of stress, or

(M.40)sa sr

2smax smin

2

sr smax smin

sm smax smin

2


0

min

max

Time

+

Str

ess

Tens

ion

Com

pres

sion

0

min

max

Time

+

Str

ess T

ensi

onC

ompr

essi

onm

a

r

Time

+

Str

ess T

ensi

onC

ompr

essi

on

Figure M.24 Variation of stress with time thataccounts for fatigue failures. (a) Reversed stresscycle, in which the stress alternates from amaximum tensile stress (+) to a maximumcompressive stress of equal magnitude.(b) Repeated stress cycle, in which maximum andminimum stresses are asymmetrical relative tothe zero stress level; mean stress rangeof stress and stress amplitude are indicated.(c) Random stress cycle.

sasr,sm,

()

Mean stress for cyclicloadingdependenceon maximum andminimum stress levels

Computation ofrange of stress forcyclic loading

Computation ofstress amplitude forcyclic loading

(a)

(b)

(c)

M.9 The S-N Curve 35

Finally, the stress ratio R is the ratio of minimum and maximum stress amplitudes:

(M.41)

By convention, tensile stresses are positive and compressive stresses are negative.For example, for the reversed stress cycle, the value of R is 1.

R smin

smax

Computation ofstress ratio

Bearing housing Bearing housing

Load

Specimen

Load

Flexible coupling

High-speedmotor

Counter

+

Figure M.25 Schematic diagram of a fatigue-testing apparatus for making rotating-bending tests.(From KEYSER, MATERIALS SCIENCE IN ENGINEERING, 4th, 1986. Printed and Electronically reproduced bypermission of Pearson Education, Inc., Upper Saddle River, New Jersey.)

M.9 THE S-N CURVE15

As with other mechanical characteristics, the fatigue properties of materials can bedetermined from laboratory simulation tests.16 A test apparatus should be designed toduplicate as nearly as possible the service stress conditions (stress level, time frequency,stress pattern, etc.).A schematic diagram of a rotating-bending test apparatus commonlyused for fatigue testing is shown in Figure M.25; the compressi

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