05224G_Chapter14

CHAPTER 14

Fatigue

FATIGUE FAILURES OCCUR due to theapplication of fluctuating stresses that are muchlower than the stress required to cause failureduring a single application of stress. It has beenestimated that fatigue contributes to approxi-mately 90% of all mechanical service failures.Fatigue is a problem that can affect any part orcomponent that moves. Automobiles on roads,aircraft wings and fuselages, ships at sea, nu-clear reactors, jet engines, and land-based tur-bines are all subject to fatigue failures. Fatiguewas initially recognized as a problem in the early1800s when investigators in Europe observedthat bridge and railroad components werecracking when subjected to repeated loading. Asthe century progressed and the use of metalsexpanded with the increasing use of machines,more andmore failures of components subjectedto repeated loads were recorded. Today, struc-tural fatigue has assumed an even greaterimportance as a result of the ever-increasing useof high-strength materials and the desire forhigher performance from these materials.

14.1 Stress Cycles

There are three basic factors necessary tocause fatigue: (1) a maximum tensile stress ofsufficiently high value, (2) a large enough var-iation or fluctuation in the applied stress, and (3)a sufficiently large number of cycles of theapplied stress. There are many types of fluctu-ating stresses. Several of the more commontypes encountered are shown in Fig. 14.1. Afully reversed stress cycle, shown in the topgraph of Fig. 14.1, where the maximum andminimum stresses are equal, is commonly usedin testing. This is the type of stress produced, forexample, by the R.R. Moore rotating-beamfatigue machine shown in Fig. 14.2, which issimilar to what a shaft may encounter duringservice. Since this was the original type ofmachine used to generate fatigue data, quite a bit

of the data in the literature is for fully reversedbending with no mean stress applied on top of it.Another common stress cycle is the repeatedstress cycle, in which there is a mean stress (sm)applied on top of the maximum and minimum

Fig. 14.1 Typical loading cycles

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stresses. Themiddle graph in Fig. 14.1 shows thecondition where both stresses (cyclic and ap-plied) are tensile (greater than zero), but it is alsopossible to test with both stresses in compres-sion. In addition, the maximum and minimumstresses in the cycle do not necessarily have to beequal in value. The last type of loading cycle isthe random or irregular stress cycle, in which thepart is subjected to random loads during service,as shown in the bottom graph of Fig. 14.1.Although a majority of the fatigue data in theliterature is for fully reversed bending, there arealso axial test machines (Fig. 14.3) that arecapable of tension and compression loading inboth the high- and low-cycle fatigue ranges.These modern test frames are closed-loopservohydraulically controlled and can be pro-grammed with almost any desired fatigue spec-trum.

A fluctuating stress is made up of two com-ponents: a mean or steady stress, sm, and analternating or variable stress, sa. The stressrange, sr, is the difference between the max-imum and minimum stress in a cycle:

sr=smax7smin (Eq 14.1)

The alternating stress is one-half the stressrange:

sa=sr2=

smax7smin

2(Eq 14.2)

The mean stress is the algebraic average of themaximum and minimum stress in the cycle:

sm=smax+smin

2(Eq 14.3)

Two ratios frequently used in presentingfatigue data are:

Stress ratio R=smin

smax(Eq 14.4)

Amplitude ratio A=sasm

=17R

1+R(Eq 14.5)

14.2 High-Cycle Fatigue

High-cycle fatigue involves a large number ofcycles (N4105 cycles) and an elasticallyapplied stress. High-cycle fatigue tests areusually carried out for 107 cycles and sometimes5 · 108 cycles for nonferrous metals. Althoughthe applied stress is low enough to be elastic,plastic deformation can take place at the cracktip. High-cycle fatigue data are usually pre-sented as a plot of stress, S, versus the number ofcycles to failure, N. A log scale is used for thenumber of cycles. The value of stress, S, can bethe maximum stress, smax, the minimum stress,smin, or the stress amplitude, sa. The S-N rela-tionship is usually determined for a specifiedvalue of the mean stress, sm, or one of the tworatios, R or A.

The fatigue life is the number of cycles tofailure at a specified stress level, while the fati-gue strength (also referred to as the endurancelimit) is the stress below which failure does notoccur. As the applied stress level is decreased,the number of cycles to failure increases.Normally, the fatigue strength increases as thestatic tensile strength increases. For example,

Fig. 14.2 Schematic of R.R. Moore reversed-bending fatigue machine. Source: Ref 1

244 / Elements of Metallurgy and Engineering Alloys



high-strength steels heat treated to over1400 MPa (200 ksi) yield strengths have muchhigher fatigue strengths than aluminum alloyswith 480 MPa (70 ksi) yield strengths. A com-parison of the S-N curves for steel and aluminumis shown in Fig. 14.4. Note that steel not only hasa higher fatigue strength than aluminum, but italso has an endurance limit. Below a certainstress level, the steel alloy will never fail dueto cyclic loading alone. On the other hand,aluminum does not have a true endurance limit.It will always fail if tested to a sufficient numberof cycles. Therefore, the fatigue strength ofaluminum is usually reported as the stress level itcan survive at a large total number of cycles,usually 5 · 108 cycles. It should be noted thatthere is a considerable amount of scatter infatigue test results. It is therefore important totest a sufficient number of specimens to obtainstatistically meaningful results.

For a large number of steels, there is a directcorrelation between tensile strength and fatiguestrength; higher-tensile-strength steels havehigher endurance limits. The endurance limit isnormally in the range of 0.35 to 0.60 of thetensile strength. This relationship holds upto a hardness of approximately 40 HRC(~1200 MPa, or 180 ksi tensile strength), and

then the scatter becomes too great to be reliable(Fig. 14.5). This does not necessarily mean it iswise to use as high a strength steel as possibleto maximize fatigue life because, as the tensilestrength increases, the fracture toughness de-creases and the environmental sensitivityincreases. The endurance limit of high-strengthsteels is extremely sensitive to surfacecondition, residual-stress state, and the presenceof inclusions that act as stress concentrations.

Fatigue cracking can occur quite early in theservice life of the component by the formation ofa small crack, generally at some point on theexternal surface. The crack then propagatesslowly through the material in a directionroughly perpendicular to the main tensile axis(Fig. 14.6). Ultimately, the cross-sectional areaof the member is reduced to the point that it canno longer carry the load, and the member fails intension. The fracture surface of a fatigued high-strength part is shown in Fig. 14.7. The portionof the fracture surface due to fatigue crackgrowth and the portion finally cracked due tooverload are clearly evident.

As previously mentioned, much of the fatiguedata in the literature has been determined forcompletely reversed bendingwithsm= 0.How-ever, the effects of mean stress are important,

Alignment fixture

Load cell

Top grip

Heat shield

Extensometer

Bottom grip

Heating coilAssembly

Specimen

Fig. 14.3 Modern fatigue test frame

Chapter 14: Fatigue / 245



and an increase in mean stress will always causea reduction in the fatigue life (Fig. 14.8). Anumber of mathematical models have beendeveloped that allow the effects of mean stresson stress amplitude to be predicted from fullyreversed-bending data. Goodman developed alinear model, while Gerber used a parabolicmodel (Fig. 14.9). Test data for ductile metals

usually fall closer to the Gerber parabolic curve;however, because of the scatter in fatigue dataand the fact that notched data fall closer to theGoodman line, the more conservative Goodmanrelationship is often used in practice. If thecomponent design is based on yield rather thanultimate strength, as most are, then the evenmore conservative Soderberg relationship canbe used. Mathematically, the three relationshipscan be expressed by:

sa=se 17smsu

! "x# $(Eq 14.6)

where x= 1 for the Goodman line, x= 2 for theGerber curve, su= sy for the Soderberg curve,and se is the fatigue limit for completelyreversed bending.

14.3 Low-Cycle Fatigue

During cyclic loading within the elasticregime, stress and strain are directly relatedthrough the elastic modulus. However, for cyclic

80

1045 Steel EnduranceLimit

No EnduranceLimit

2024-T6 Al

60

40

20

0

500

400

300

200

100

0103 104 105 106 107 108 109 1010

Number of Cycles

Stre

ss A

mpl

itude

(ksi

)

Stre

ss A

mpl

itude

(MPa

)

Fig. 14.4 Comparison of steel and aluminum fatigue beha-vior. Source: Ref 2

140

130

400

500

600

700

800

900

120

All test results fromone laboratory

All specimens 0 to 2micro-in. finish

110

100

Fatig

ue li

mit,

(ksi

)

Fatig

ue li

mit,

(MPa

)

90

80

70

6020 25 30 35 40 45

Rockwell C hardness

Quenched and tempered steels10542340403240424053

40634068413041404340

51405150516086409262

50 55 60 65 70

Fig. 14.5 Endurance limit versus hardness for steels. Source: Ref 3




loading that produces plastic strains, the re-sponses are more complex and form a hysteresisloop (Fig. 14.10). From point O up to point A,the component is in tension. On unloading,the strain response of the specimen followsthe curve from A to D. At D, the componentis under no stress. As the component is subjectedto compressive stress, the strain responsefollows the curve from D to B. Releasing thecompressive stress from B and reapplyingtensile stress, the component stress-strain con-dition returns to point A along the curve de-fined byB, C, and A. PointsA and B represent thecyclic stress and strain limits. The total strain,

De, consists of both elastic and plastic compo-nents:

De=Dee+Dep (Eq 14.7)

where Dee is the elastic strain and Dee=Ds/E,and Dep is the plastic strain and Dep is the widthof the loop at its center, that is, the distance CDin Fig. 14.10. The area of the hysteresis loop isequal to the work done or the energy loss percycle.

In cyclic strain-controlled fatigue, the strainamplitude is held constant during cycling.Since plastic deformation is not completely

Nucleation Crack Growth Final Failure

Fig. 14.6 Typical propagation of a fatigue crack. Source: Ref 4

Fig. 14.7 Fatigue crack growth in a high-strength steel part




reversible, the stress-strain response duringcycling can change, largely depending on theinitial condition of the metal. The metalcan either undergo cyclic strain hardening,cyclic strain softening, or remain stable. Cyclicstrain hardening and softening are illustratedschematically in Fig. 14.11. In both cases, thehysteresis loops change with successivecycles. Cyclic hardening leads to increasingpeak strain with increasing cycles, while cyclicsoftening results in decreasing strain levelswith increasing cycles. In general, strong metalstend to cyclically soften, and low-strengthmetals tend to cyclically harden. However,the hysteresis loop tends to stabilize after afew hundred cycles, when the materialattains a stable condition for the imposed strainlevel.

Stre

ss A

mpl

itude

(σa)

Incr

easi

ng M

ean

Stre

ss

σm = 240 MPa (35 ksi)

σm = 0 MPa (0 ksi)

σm = 480 (70 ksi)

Cycles to Failure (Nf)

Fig. 14.8 Effect of mean stress on fatigue life

σ

σ

σ σ

σσσσ

σ

(σm)

Fig. 14.9 Comparison of Goodman, Gerber, and Soderberg models for relating mean stress to stress amplitude




Typically, metals harden if su=sy ! 1:4 andsoften if su=sy " 1:2.

The reason for hardening or softening isrelated to the dislocation microstructure of themetal. When the metal is highly work hardenedand the dislocation density is high, cyclic strainallows the rearrangement of dislocations intomore stable networks, thereby reducing thestress at which plastic deformation occurs.Conversely, when the initial dislocation densityis low, the cyclic strain increases the dislocationdensity, increasing the amount of elastic strainand stress on the material.

Low-cycle fatigue test data are often pre-sented as a plot of the plastic strain range, Dep,versus cycles to failure, N. When plotted on log-log coordinates, a straight line is obtained that isdescribed by the Coffin-Manson relation:

Dep2

=e0f(2N)c (Eq 14.8)

where Dep=2 is the plastic strain amplitude, ande0f is the fatigue ductility coefficient defined bythe strain intercept at 2N= 1. For many metals,

e0f is approximately equal to the true fracturestrain, ef. 2N is the number of strain reversals tofailure, where one cycle is two reversals, and c isthe fatigue ductility exponent, which usuallyvaries between #0.5 and #0.7. A smaller valueof c results in longer fatigue lives.

The Basquin equation, which describes thehigh-cycle, low-strain regimewhere the nominalstrains are elastic, is:

sa=Dee2

E=s0f(2N)b (Eq 14.9)

where sa is the alternating stress amplitude,Dee=2 is the elastic strain amplitude, E is themodulus of elasticity, and s0f is the fatiguestrength coefficient, defined as the stress inter-cept at 2N= 1. s0f is approximately equal to thetrue fracture stress, sf . 2N is the number of loadreversals to failure, and b is the fatigue strengthexponent, which varies for most metals between#0.05 and #0.12. A smaller b results in a longerfatigue life.

Combining Basquin’s equation and theCoffin-Manson equation gives an equation

∆σ/2E

∆ε

∆εp ∆εe = ∆σ/E

∆σ

OStrain

Stress A

B

C D

∆σ/2E

σa = ∆σ/2

Fig. 14.10 Stress-strain hysteresis loop for cyclic loading




that can be used to estimate the entire range offatigue lives:

De

2=

Dee2

+Dep2

(Eq 14.10)

De

2=

s0fE(2N)b+e0f(2N)

c (Eq 14.11)

The fatigue strain-life curve tends toward theplastic curve at large strain amplitudes and

Fig. 14.11 Cyclic hardening and cyclic softening




toward the elastic curve at small strain am-plitudes (Fig. 14.12). For high-cycle strainconditions, ductile metals have the longest lives,while at low-cycle strain conditions, strongmetals have the longest lives.

14.4 Cumulative Damage

Fatigue tests are often conducted under sim-ple conditions, such as constant amplitude andconstant frequency. However, in real structures,the loading conditions are rarely simple. Manystructures are subjected to a range of fluctuatingloads, mean stress levels, and variable fre-quencies. Thus, it is important to be able topredict the life of a component subjected tovariable amplitude loading using data generatedin constant amplitude laboratory tests. Cumula-tive damage theories consider the fatigue pro-cess to be one of damage accumulation until thelife of the component is exhausted. Consider thefatigue life diagram in Fig. 14.13. At a constantstress, s1, the life is 200 cycles, while at s2, thelife is 400 cycles. According to cumulativedamage theory, fatigue life is gradually exhau-sted. That is, at points A and C, 100% of the lifeis available, while at points B and D, the re-spective lives are completely exhausted. If fati-gue damage accumulates in a linear manner,each cycle contributes the same amount ofdamage at a given stress level. For example, ats1, on cycling the metal from A to E, one-fourthof the fatigue life is consumed (50 of 200cycles). If the stress level is reduced to s2, thepercentage of life exhausted at s2 is equivalentto the percentage of life exhausted at s1. That is,if one-fourth of the fatigue life is consumed ats1, the component still will have lost one-fourth

of its life even if the stress is reduced to s2.For example, at point E, 50 cycles of 200 areconsumed. If, when the component reaches thenumber of cycles defined by point E, the stress isreduced to s2, the component will be in thecondition defined by point F, with 100 of its 400cycles consumed. The same kind of change canbe described for a low-to-high stress traverse.

Cumulative damage during fatigue is oftendetermined using the Palmgren-Miner rule,which assumes that the total life of a part can beestimated by adding up the percentage of lifeconsumed by each stress level:

n1N1

+n2N2

+ $ $ $ $ $ $ $+ nkNk

=1 orXj=k

j=1

njNj

=1

(Eq 14.12)

where n1, n2 $ $ $ $ nk represent the number ofcycles at a specific stress level, and N1,N2 $ $ $ $ $ Nk represent the fatigue life incycles at the same stress level.

This rule should be used with caution. It is alinear relationship that does not consider theeffects of understressing or overstressing.Understressing, in which relatively low stresslevels are initially applied, can result in longerfatigue lives at a higher stress level, due tolocalized strain hardening. On the other hand,overstressing, in which high stresses are initiallyapplied, can result in shorter fatigue lives at alow stress level, due to damage induced by thehigher stress levels.

Fig. 14.12 Fatigue life in terms of total strainFig. 14.13 Cumulative damage during high-to-low loading

sequence. Source: Ref 5




14.5 Fatigue Crack Nucleationand Growth

Fatigue cracks almost always initiate at freesurfaces, usually external surfaces but alsointernal surfaces if the metal contains defectssuch as voids and cracked second-phase parti-cles. Common external surface defects includegeometric notches and surface roughness.

Fatigue crack nucleation and growth occursin the following stages.

Stage I. Crack initiation usually starts ata notch or other surface discontinuity. Even inthe absence of a surface defect, crack initiationwill eventually occur due to the formation ofpersistent slip bands (PSBs), so called becausetraces of the bands persist even when the surfacedamage is polished away. Slip bands are a resultof the systematic buildup of fine slip movementson the order of only 1 nm. However, the plasticstrain within the PSB can be as much as 100times greater than that in the surroundingmaterial. The back-and-forth movement of theslip bands leads to the formation of intrusionsand extrusions at the surface, eventually leadingto the formation of a crack (Fig. 14.14). Theinitial crack propagates parallel to the slip bands.The crack propagation rate during stage I is verylow, on the order of 1 nm per cycle, and producesa practically featureless fracture surface. Thecrack initially follows the slip bands atapproximately 45% to the principal stress direc-tion. When the crack length becomes sufficientfor the stress field at the tip to become dominant,the overall crack plane changes and becomesperpendicular to the principal stress, and thecrack enters stage II.

Stage II crack growth occurs when the stage Icrack changes direction and propagates in a

direction normal to the applied stress. Crackgrowth proceeds by a continual process of cracksharpening followed by blunting, as shown inthe Fig. 14.15 illustration. Crack propagationduring crack growth often produces a pattern offatigue striations (Fig. 14.16), with each striationrepresenting one cycle of fatigue. Althoughstriations are indicative of fatigue, fatigue fail-ures can occur without the formation of stria-tions. Striations are microstructural details thatare best examined with a scanning electronmicroscope and are not visible to the naked eye.Frequently, visible examination of a fatiguedsurface will reveal a series of concentric mark-ings on the surface, referred to as beach marks(Fig. 14.6). These are present as a result of stresschanges during fatigue, for example, the startingand stopping of a rotating shaft. Each of thebeach marks can contain thousands or even tensof thousands of fatigue cycles.

Stage III. Ultimate failure occurs when thefatigue crack becomes long enough that theremaining cross section can no longer supportthe applied load.

14.6 Fatigue Crack Propagation

Linear elastic fracture mechanics assumesthat all structures contain flaws. Cracks growfrom an initial size, ao, to a critical size, ac, cor-responding to failure as a function of the numberof load cycles (Fig. 14.17). The crack growthrate, da/dn, can be determined from the slope ofthe curve. Initially, the crack growth rate is slowbut increases with increasing crack length. Ofcourse, the crack growth rate is also higher forhigher applied stresses. If one can characterizethe crack growth, it is then possible to estimate

Fig. 14.14 Development of extrusions and intrusions during fatigue. Source: Ref 4




the service life or inspection intervals requiredunder specific loading conditions and serviceenvironment. In the fracture mechanics ap-proach to fatigue crack growth, the crack growthrate, or the amount of crack extension per load-ing cycle, is correlated with the stress-intensityparameter,K. This approach makes it possible toestimate the useful safe lifetime and to establish

inspection intervals. An idealized da/dn versusDK curve is shown in Fig. 14.18. In region I,DKth is the fatigue crack growth threshold,which is at the lower end of theDK range, wherecrack growth rates approach zero. In region II,the crack growth rate is stable and essentiallylinear and can be modeled by power law equa-tions, such as the Paris equation:

da=dn=C(DK)m (Eq 14.13)

where: a is the flaw or crack size in inches; n isthe number of cycles; C and m are constantparameters and are related to material variables,environment, temperature, and fatigue stressconditions; and DK=DKmax#DKmin is thestress-intensity parameter range. The constantsC and m are material parameters that must bedetermined experimentally. Typically, m is inthe range of 2 to 4 for metals and 4 to 100 forceramics and polymers.

During stage II growth in the linear crackgrowth region, the Paris law can be used todetermine the number of cycles to failure. DKcan be expressed in terms of Ds:

DK=DKmax7DKmin=Ysmax

·ffiffiffiffiffiffipa

p7Ysmin

ffiffiffiffiffiffipa

p=YDs


p(Eq 14.14)

1

2

3

4

5

6

Striation

Fig. 14.15 Fatigue crack propagation. Source: Ref 6

2 µm

Striation

Fig. 14.16 SEM image showing fatigue striations




where Y depends on the specific specimen geo-metry. Thus, the Paris law becomes:

da

dn=C(YDs


p)m (Eq 14.15)

One of the goals of fatigue analysis is to beable to predict the fatigue life of structures. Thefatigue life, n, can be solved for by rearrangingEq 14.13:

dn=da

C(DK)m(Eq 14.16)

which may then be integrated to give:

nf=ðnf

0

dn=ðac

ao

da

C(DK)m(Eq 14.17)

Substitution of the expression forDK (Eq 14.14)gives:

nf=ðac

ao

da

C(YDsffiffiffiffiffiffipa

p)m=

1

Cp m=2(Ds)m

ðac

ao

da

Yma m=2

(Eq 14.18)

It is assumed that Ds (or smax#smin) isconstant and that Y depends on the crack lengthand therefore cannot be removed from withinthe integral. For cases where Y depends on cracklength, these integrations will generally beperformed using numerical techniques. Duringstage II crack growth, some metals are sensitiveto the load ratio, R, as shown in Fig. 14.19 for

7075-T651 aluminum plate. Since the Parisequation does not account for the load ratio,there are other expressions that do account forthe sensitivity of crack growth rate to the loadratio. One of the most utilized is the one devel-oped by Foreman and his associates:

da

dn=

C(DK)m

(17R)Kc7DK(Eq 14.19)

In these expressions, it is necessary to deter-mine the initial crack length, ao, and the finalor critical crack length, ac. Crack lengths canbe detected using a number of nondestructivetesting techniques. If no cracks are detected, itmust be assumed that a crack exists below theresolution of the detection system being used. Asubcritical crackwill eventually grow to a lengthat which the metal immediately fails, that is:

Kmax?Kc (Eq 14.20)

or

Ysmaxffiffiffiffiffiffiffiffipaf

p! Kc (Eq 14.21)

Solving for ac gives:

ac=K2c

pY2s2max

(Eq 14.22)

ac

ao

ao

nf

ac

Cra

ck L

engt

h a

Fatigue Life

dadn

Inspection Retirement

Number of Cycles

Fig. 14.17 Crack length as a function of cycles. Source:Ref 5 Fig. 14.18 Crack propagation curve for fatigue loading




Note that even if a component contains adetectable crack, it can remain in service,provided that it is periodically inspected. Thisphilosophy forms the basis for what is known asthe damage-tolerance design approach.

Finally, in region III, the crack growth rateaccelerates, since the fracture toughness of thematerial is approached, and there is a local ten-sile overload failure.

14.7 Crack Closure

During fatigue cycling, the crack growth ratecan slow due to crack closure. For example, if thestructure is subjected to a series of high-tensileoverloads that are followed by loads at lowertensile stress levels, the crack may temporarilystop propagating, leading to a decrease in thecrack growth rate, da/dn, and consequently have

a longer fatigue life than would normally beexpected. Crack closure mechanisms effectivelyreduce the DK during fatigue cycling, thusretarding the crack growth. Since DK is equal toKmax#Kmin and the actual DK is reduced be-cause crack closure is hindered,DK is effectivelysmaller, and crack growth rate is then reduced.

Four mechanisms that can cause crack closureare shown in Fig. 14.20. Plasticity-inducedclosure results from compressive residual stres-ses developing in the plastic wake. Roughness-induced closure is due to crack meandering, inwhich the crack deviates from mode I dis-placements and is subject to mode II shear dis-placements. These displacements cause amismatch between the upper and lower crackfaces, which results in contact between the crackfaces during cycling. Oxide debris-inducedclosure results from corrosion products becom-ing wedged in the crack interface. Finally, fluid

1 10 (12) 100 (120)

R = 0.1

R = +0.5

R = –0.5

R = σmin/σmax

10–2

10–3

10–4

10–5

10–6

3×10–2

3×10–3

3×10–4

3×10–5

3×10–6

da/d

N (in

./cyc

le)

da/d

N (c

m/c

ycle

)

Center cracked panels, L-T Direction7075-T651, 0.6 cm (0.25 in.) Thickness

∆K, ksi·√in. (MPa √m)

Fig. 14.19 Stress ratio effects on an aluminum alloy




pressure can act as a wedge, preventing totalcrack closure.

14.8 Geometrical Stress Concentrations

In most structures, fatigue cracking usuallyinitiates at a stress concentration. The stressconcentration may by inherent in the design,such as a fillet, hole, thread, or other geometricalfeature, or the stress concentration can resultfrom a manufacturing process, such as a roughsurface finish or residual tensile stresses intro-duced by heat treatment.

The effect of geometrical stress concentra-tions on fatigue is often studied by testing not-ched specimens. When a notch is present in aspecimen under uniaxial loading, three effectsare present: (1) there is an increase or con-centration of stress at the root of the notch, (2)there is a stress gradient from the notch towardthe center of the specimen, and (3) a triaxial stateof stress exists. The dramatic reduction in

fatigue life of normalized 4340 steel sheet con-taining different types of stress concentrations isshown in Fig. 14.21.

The effect of notches on fatigue strength isdetermined by comparing the S-N curves ofnotched and unnotched specimens. The data forthe notched specimens are usually plotted interms of nominal stress based on the net crosssection of the specimen. The effect of the notchin decreasing the fatigue strength is reported asthe fatigue strength reduction factor, or thefatigue notch factor, Kf:

Kf=Fatigue limit unnotched

Fatigue limit notched(Eq 14.23)

For metals that do not have a definite fatiguelimit, the fatigue notch factor is based on thefatigue strength at a specified number of cycles.Values of the fatigue notch factor vary with theseverity of the notch, the type of notch, thematerial, the type of loading, and the appliedstress level.

Plasticity-Induced Closure Roughness-Induced Closure

Mode II Displacement

Oxide-Induced Closure Fluid-Induced Closure

Corrosion Product

Fluid

Fig. 14.20 Fatigue crack closure mechanisms in metals. Source: Ref 7




Notched fatigue data are also reported using anotch sensitivity factor, q:

q=Kf71

Kt71(Eq 14.24)

This relationship compares the theoreticalstress-concentration factor, Kt, to the fatiguenotch factor, Kf. In this relationship, a materialthat experiences no reduction in fatigue due to anotch will have a notch sensitivity factor ofq= 0, while one that experiences a reduction infatigue up to the full theoretical value will have anotch sensitivity factor of q= 1. The value of qis dependent on the material and the radiusof the notch root, as illustrated by the plotsshown in Fig. 14.22 for steels. The notch sensi-tivity factor, q, significantly decreases withsmaller notch radii. Although this seems coun-terintuitive, it occurs because Kf increases moreslowly than Kt with decreasing notch radius.For notches with large radii, Kf is almost equalto Kt. For materials with small notches, Kf isless than Kt. In addition, lower-strength metalsare less affected than high-strength metals bygeometric discontinuities that reduce the fatigueresistance, because high-strength metals have alimited capacity for deformation and crack tipblunting.

14.9 Manufacturing StressConcentrations

Because almost all fatigue failures start on asurface, the surface finish and residual-stressstate near the surface can have a profound effect

on the fatigue strength. Since the early daysof fatigue investigations, it was recognizedthat the fatigue life of a component is verydependent on the surface finish produced bymachining or grinding operations. As shownin Fig. 14.23, highly polished steel specimensperform much better in fatigue than even care-fully machined surfaces. Further degradationin fatigue strength occurs for hot rolled andforged surfaces. Finally, when corrosion isintroduced, the fatigue life can be seriouslydegraded.

Parts that are formed at room temperature willcontain residual stresses. For example, the sur-face of a part that was formed in tension willcontain residual compressive stresses, and asurface that was in compression during formingwill contain residual tensile stresses. Since fati-gue always occurs under tensile loading, thesurface with the residual tensile stresses will bethe most prone to fatigue cracking. Similar toforming, some quenching operations during heattreatment can result in a tensile residual-stresspattern on the surface that will adversely affectfatigue strength.

The fatigue data shown in Table 14.1 for 4340steel forgings tested in fully reversed bendingillustrate two important points. First, vacuummelting, which removes inclusions, improvesboth the longitudinal and transverse endurancelimits. Second, since forging tends to elongateinclusions in the longitudinal or working direc-tion, the fatigue endurance limit is higher in thelongitudinal direction. The lower transverseendurance limit in steels containing inclusions isattributed to stress concentrations at the inclu-sions, which can be quite high when an elon-gated stringer is oriented transverse to the tensilestress. For the case of the vacuum-melted steel,the transverse limit is still somewhat lower than

Unnotched

Hole, Kt = 2Fillet, Kt = 4Edge Notch, Kt = 4

104 105 106 107 108 1090

20

40

60

80

100

200

300

400

500

Cycles to Failure

Stre

ss A

mpl

itude

(ksi)

Stre

ss A

mpl

itude

(MPa

)

4340 Steel Normalized

Fig. 14.21 Effect of geometrical stress concentrations onfatigue life. Source: Ref 8

Fig. 14.22 Notch sensitivity versus notch radius for steels.Source: Ref 9




for the longitudinal direction, indicating theextreme sensitivity of transverse fatigue prop-erties to microstructure.

Due to their extreme strength and hardness,high-strength steels are often finished by grind-ing to provide precise dimensions, remove anynicks or scratches, and provide smooth surfaces.However, great care must be taken whengrinding these ultrahigh-strength steels. Impro-per or abusive grinding can result in grindingburns, in which the surface is heated above theaustenitizing temperature and the austeniteformed converts to untempered martensite oncooling (Fig. 14.24). This untempered marten-sitic surface layer is brittle and susceptible toforming a network of fine cracks that can reducethe fatigue strength by as much as 30%. Even ifthe grinding temperature is not high enough toproduce austenite, it may cause the originaltempered martensite microstructure on the sur-face to become overtempered, thereby reducingthe strength and hardness.

14.10 Environmental Effects

Corrosion Fatigue. The simultaneous actionof fatigue and corrosive attack is known ascorrosion fatigue. In the absence of fatigueloading, corrosive attack can often cause pittingof a metal surface. These pits can then act asstress concentrations that will initiate fatiguecracking. When corrosion and fatigue occursimultaneously, the chemical attack greatlyaccelerates fatigue crack growth. Materials suchas steels, which show a definite endurance limitwhen tested in air, do not exhibit a definiteendurance limit when tested in a corrosiveenvironment (Fig. 14.25). Frequently, fatiguewill corrupt the protective films that wouldnormally slow the corrosive attack, allowingcontinued and accelerated attack. Since corro-sion is time dependent, slow cycling rates willresult in greater life reductions than acceleratedfatigue cycling. Another possible effect of acorrosive environment on fatigue life is the

0.1

100 140 180 220

500Ultimate Tensile Stress (MPa)

1000 1500

260060

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ultimate Tensile Strength (ksi)

Surfa

ce R

eten

tion

Fact

or

Mirror Polished

Fine Ground orCommercially Polished

Machined

Hot Rolled

As-Forged

Corroded inTap Water

Corroded in Salt Water

Fig. 14.23 Effect of surface finish on fatigue of steels. Source: Ref 10




absence of the stable stage II crack growththat occurs according to the Paris law. Areduction in fatigue life of 50% or more is notunusual.

There are a number of methods used to com-bat corrosion fatigue. First, the choice of amaterial for an aggressive environment shouldprobably be based on its corrosion resistancerather than its fatigue properties. For example,stainless steel, bronze, or beryllium-copperwould probably yield better service than heattreated steels. Corrosion-resistant coatings thatisolate the surface from the environment aresuccessful as long as the coating does not ruptureduring cycling. Zinc and cadmium coatings onsteel and aluminum alclad coatings on alumi-num generally produce better fatigue lives inaggressive environments, even though thecoatings may reduce fatigue lives somewhatwhen tests are conducted in air. The introductionof surface compressive residual-stress patternsby shot peening or nitriding can be very useful.

Low-Temperature Fatigue. Fatigue lifenormally increases with decreasing tempera-tures. Although steels become more notchsensitive at low temperatures, there is noexperimental evidence indicating that any sud-den change in fatigue properties occurs at tem-peratures below the ductile-to-brittle transitiontemperature.

High-Temperature Fatigue. In general, thefatigue strength of metals decreases with in-creasing temperatures. An exception is mildsteel, which exhibits a maximum in fatiguestrength between 205 and 300 %C (400 and575 %F) due to strain aging. As the temperatureexceeds approximately half the melting point,Tm, creep can become the dominant cause offailure. Ferrous alloys, which usually exhibitan endurance limit at room temperature, willno longer have an endurance limit when thetemperature exceeds approximately 425 %C(800 %F). In general, the higher the creepstrength of an alloy, the higher will be its high-temperature fatigue strength. However, there are

conflicting requirements, with fine grain sizesbeing beneficial to fatigue properties, whilecoarse grain sizes are preferred in certain creepregimes. In addition, the beneficial effectsof fatigue improvement processes such as shotpeening, which are effective at room tempera-ture, may be annealed out at elevated tempera-tures.

Table 14.1 Effect of inclusions in 4340 forgedsteel

PropertyElectric furnace

meltedVacuummelted

Longitudinal fatigue limit MPa (ksi) 800 (116) 960 (139)Transverse fatigue limit MPa (ksi) 545 (79) 830 (120)Ratio transverse/longitudinal 0.68 0.86Hardness (HRC) 27 29

Source: Ref 11

Fig. 14.24 Effects of grinding burns (untempered marten-site) on high-strength steel. Source: Ref 12




Thermal Fatigue. Thermal stresses resultwhen dimensional changes in response to tem-perature changes are restricted by some type ofconstraint imposed on the component. For thecase of a simple bar constrained rigidly at bothends, the thermal stress, s, developed by atemperature change, DT, is:

s=aEDT (Eq 14.25)

where a is the linear coefficient of thermal ex-pansion, and E is the modulus of elasticity.

If failure occurs during one cycle of thermalstress because of a rapid temperature change, thephenomenon is known as thermal shock, whilefailures occurring after multiple thermal stresscycles caused by a cyclic temperature change arereferred to as thermal fatigue. An example ofthermal fatigue occurs in ceramic thermal bar-rier coatings that are applied to nickel-basesuperalloys used in jet engines. The large dif-ferences in the coefficients of thermal expansionof the ceramic coating and the underlying nickelblade result in cracking and spalling of thecoating. Another example of thermal fatigueoccurs in electronics. Heat generated duringoperation of electronic devices can cause fatigueof the soldered joints.

The tendency for thermal fatigue has beenrelated to the parameter:

sfk=Ea (Eq 14.26)

where sf is the mean fatigue strength, and k is thethermal conductivity. A high value of thisparameter indicates good resistance to thermalfatigue.

14.11 Fatigue Life Improvement

One of the most effective methods ofimproving fatigue life is to induce residualcompressive stresses on the surface of the part.This is often accomplished by shot peening or bysurface rolling with contoured rollers. Shotpeening involves propelling fine steel or castiron shot into the surface at high velocities. Theseverity of the stress produced by shot peening ismonitored by measuring the residual deforma-tion of shot-peened specimens called Almenstrips. In addition to inducing compressive stresson the surface, the surface layers are alsostrengthened by the cold working that occursduring shot peening. Important variables in shotpeening include the hardness of the shot, thesize, the shape, and the velocity. Shot peeningmust be carefully controlled so as not to intro-duce surface damage to the part. Shot peening

Fig. 14.25 Effect of corrosion on fatigue performance

Fatigue Technology, Inc.

Zone of compressive residual stress extends one radius fromedge of hole

Surface surrounding hole isyielded beyond elastic limit

Tension zone beyondcompression zone

Fig. 14.26 Residual-stress state around cold-worked hole




imparts greater fatigue life during high-cyclefatigue than it does for higher-stress low-cyclefatigue. In low-cycle fatigue, large stresses inthe plastic range cause “fading” of the residual-stress pattern.

Steel parts subjected to wear conditions areoften carburized or nitrided to harden the surfacefor greater wear resistance. Since both of theseprocesses also introduce residual compressivestresses on the surface, there is an improvementin fatigue life. The greatest improvement occurswhere a high stress gradient occurs, as in bend-ing or torsion, with less improvement in axialloading. Nitriding is extremely effective inimproving the fatigue strength of notched spec-imens. Other surface-hardening methods, suchas flame or induction hardening, produce similareffects. Case hardening is more effective forcomponents that have a stress gradient, as inbending, or where there is a notch. For axiallyloaded components, fatigue cracks can formunder the case in the softer, weaker core at theboundary between the surface and bulk.

Many large structures are assembled withmechanical fasteners. In the aerospace industry,fatigue life improvement of aluminum structuresis often accomplished with cold working offastener holes and/or installing interference fitfasteners. Since fatigue cracks often initiate at

fastener holes in metallic structures, methodssuch as cold working fastener holes andinterference fit fasteners have been developedto improve fatigue life. Both cold working andinterference fit fasteners set up residual com-pressive stress fields in the metal immediatelyadjacent to the hole (Fig. 14.26). The appliedtension stress during fatigue loading must thenovercome the residual compressive stress fieldbefore the hole becomes loaded in tension. Thefatigue improvement due to cold working in2024-T851 aluminum is shown in Fig. 14.27.

Cold working of holes is usually conductedusing either the split-sleeve or split-mandrelmethod. Both methods involve pulling a man-drel through the hole that expands the hole dia-meter, creating plastic deformation of materialaround the hole and a resulting residual com-pressive stress field. The residual-stress field,depending on the material and the amount ofexpansion, will extend approximately one radiusfrom the edge of the hole. In the split-sleeveprocess (Fig. 14.28), a stainless steel split sleeveis placed over a tapered mandrel and insertedinto the hole. The hole is cold worked when thelargest part of themandrel is drawn back throughthe sleeve. After cold working, the sleeve isremoved and discarded. In the split-mandrelprocess, a collapsible mandrel is placed in the

20

25

30

35

40

45

150

200

250

300Metal WithNo Hole

Cold WorkedOpen Hole

Tight Fit Fastener inCold Worked Hole

ReamedOpen Hole

Tight Fit Fastener inReamed Hole

104 105 106 107

Cycles to Failure

Max

imum

Net

Stre

ss (k

si)

Max

imum

Net

Stre

ss (M

Pa)

2024-T851 Al

Fig. 14.27 Fatigue life improvement with cold working. Source: Ref 13




hole, and as the mandrel is withdrawn, itexpands to cold work the hole.

Interference fit fasteners can also be used inmetallic structures to improve fatigue life.Whenan interference fit fastener is installed in metal, italso plastically deforms a small zone around thehole, setting up a compressive stress field, whichagain is beneficial when fatigue loading is pri-marily tensile. The amount of interference canvary, depending on structural requirements, butit is usually in the range of 0.07 to 0.10 mm(0.003 to 0.004 in.). In some highly loaded holes,both cold working and interference fit fastenersare used. While cold working and the use ofinterference fit fasteners are proven methods ofimproving fatigue resistance, both increaseassembly costs and should only be specifiedwhen they are really needed.

Castings can be hot isostatic pressed (HIP) tohelp reduce internal porosity. The HIP of alu-minum castings is usually conducted using

argon pressure at 105 MPa (15 ksi) and tem-peratures in the range of 480 to 525 %C (900 to980 %F). The use of HIP usually results inimproved mechanical properties, especiallyfatigue strength, but, of course, it adds to the costand cycle time. The improvement in fatigue lifefor the aluminum casting alloy A201.0-T7 as aresult of HIP is shown in Fig. 14.29.

14.12 Fatigue Design Methodologies

Since the 1800s, a number of design philo-sophies or methodologies have evolved to dealwith design against fatigue failures.

Infinite-Life Design. This is the oldest of thedesign philosophies and is based on maintainingthe stresses at some fraction below the fatiguestrength of the metal. The initial material isassumed to be free of flaws. This methodology isbased on the classic S-N curve. It is most

Fig. 14.28 Split-sleeve cold working process. Reprinted with permission from SAE Paper # 982145 # 1998 SAE International.Source: Ref 14




applicable when the component is only stressedin the elastic range and displays a distinctendurance limit, such as for steels. Although ithas largely been superseded by later methodol-ogies, it is simple to use and can be usefulwhere periodic inspection is either impracticalor not warranted. However, since it is a con-servative methodology, it usually results in aheavier design than some of the later meth-odologies.

Safe-life design is based on the assumptionthat the part is initially flaw-free and has a finelife in which to develop a critical crack size. Likethe infinite-life methodology, safe-life assumesan initial flaw-free component. This methodol-ogy was developed to account for parts that weresubjected to higher loads that produced plasticstrains. Under these conditions, the descriptionof local events in terms of strain made moresense and resulted in the development ofassessment techniques that used strain as adetermining quantity, that is, log strain (e) ver-sus log number of cycles (N). The failure cri-terion is usually the detection of a small crack orsome equivalent measure related to a substantialchange in load-deflection response, althoughfailure may also be defined as fracture.

Fail-Safe Design. The fail-safe design phi-losophy assumes that fatigue cracks willbe detected and repaired before they lead tofailure. This methodology was developed inthe aircraft industry where large safety marginswere weight-prohibitive. Fail-safe designs

incorporate multiple load paths and crackstoppers in the structure. In other words, if aprimary load path fails, the load will be pickedup by an alternate load path to prevent the struc-ture from failing. A major part of this metho-dology is rigid certification criteria along withthe capability to detect and inspect cracks.

Damage-tolerant design is the latest designmethodology and is an extension of fail-safedesign. The refinement of the damage-tolerantmethodology is the assumption that structuresdo contain cracks and that fracture mechanicsapproaches can be used to determine crackgrowth rates. If the crack growth rate can be cal-culated, it is then possible to inspect the structureat various intervals and either repair or retire itprior to the crack reaching critical dimensions.

Although much progress has been made in thedesign against fatigue failures, they still occurwith disturbing frequency. For any design, it isimperative that part or component testing beconducted prior to placing the part in service. Inaddition, it is important that the test conditionsare representative of the actual environment thatthe component will experience in service.

REFERENCES

1. Fatigue Failures, Failure Analysis andPrevention, Vol 11, ASM Handbook, ASMInternational, 2002

Fig. 14.29 Effect of hot isostatic pressing (HIP) on fatigue life of A201.0-T7 aluminum casting. Source: Ref 15




2. S. Kalpakjian, Manufacturing Engineeringand Technology, 3rd ed. Addison-WesleyPublishing Co., 1995

3. Properties and Selection of Metals, Vol 1,Metals Handbook, 8th ed., AmericanSociety for Metals, 1961

4. R.A. Higgins, Engineering Metallurgy—Applied Physical Metallurgy, 6th ed.,Arnold, 1993

5. M.A.Meyers and K.K. Chawla,MechanicalMetallurgy—Principles and Applications,Prentice-Hall Inc., 1984

6. C.M. Laird, STP 415, American Society forTesting and Materials, 1966, p 130

7. T.L. Anderson, Fracture Mechanics: Fun-damentals and Applications, 3rd ed., Taylor& Francis, 2005

8. J. Collins and S. Daniewicz, FailureConsiderations, Mechanical EngineersHandbook, John Wiley & Sons, Inc.,1998

9. M.R. Mitchell, Fundamentals of ModernFatigue Analysis for Design, Fatigue andFracture, Vol 19, ASM Handbook, ASMInternational, 1996

10. R.C. Juvinall, Stress, Strain, and Strength,McGraw-Hill Book Co., 1967

11. G.E. Dieter, Mechanical Metallurgy,3rd ed., McGraw-Hill Book Co., 1986,p. 375–431

12. G.E. Toten, et al., Factors Relating to HeatTreating Operations, Failure Analysis andPrevention, Vol 11, ASM Handbook, ASMInternational, 2002

13. F.C. Campbell, Manufacturing Technologyfor Aerospace Structural Materials, Else-vier Scientific, 2006

14. A. Leon, “Developments in AdvancedColdworking,” SAE 982145, Society ofAutomotive Engineers, 1998

15. J.R. Davis, ASM Specialty Handbook:Aluminum and Aluminum Alloys, ASMInternational, 1993

SELECTED REFERENCES

& M.F. Ashby and D.R.H. Jones, EngineeringMaterials 1 — An Introduction to TheirProperties and Applications, 2nd ed., But-terworth Heinemann, 1996

& T.H. Courtney, Mechanical Behavior ofMaterials, 2nd ed., McGraw-Hill Book Co.,2000, p 566–629

& H.W. Hayden, W.G. Moffatt, and J. Wulf,The Structure and Properties of Materials,Vol III, John Wiley, 1965

& M.P. Kaplan and T.A. Wolff, Fatigue-LifeAssessment, Failure Analysis and Preven-tion, Vol 11, ASM Handbook, ASM Inter-national, 2002

& E. Krempl, Design for Fatigue Resistance,Materials Selection and Design, Vol 20,ASM Handbook, ASM International, 1997

& R.A. Lund and S. Sheybany, Fatigue Frac-ture Appearances, Failure Analysis andPrevention, Vol 11, ASM Handbook, ASMInternational, 2002




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