3 Objective • The objective of this lecture is to explain the phenomenon of fatigue and also to show howObjective resistance to fatigue failure depends onCrackInitiation microstructure.S-N • For 27-302, Fall 2002: this slide set containscurvesCyclic more material than can be covered in the timestress-strn available. Slides that contain material overCrackPropagate and above that expected for this course areMicrostr. marked “*”.effectsDesign
4 References • Mechanical Behavior of Materials (2000), T. H. Courtney, McGraw-Hill, Boston.Objective • Phase transformations in metals and alloys, D.A.Crack Porter, & K.E. Easterling, Chapman & Hall.Initiation • Materials Principles & Practice, ButterworthS-Ncurves Heinemann, Edited by C. Newey & G. Weaver.Cyclic • Mechanical Metallurgy, McGrawHill, G.E. Dieter, 3rdstress-strn Ed.CrackPropagate • Light Alloys (1996), I.J. Polmear, Wiley, 3rd Ed.Microstr. • Hull, D. and D. J. Bacon (1984). Introduction toeffects Dislocations. Oxford, UK, Pergamon.Design
5 σa := Alternating stress σm := Mean stress Notation R := Stress ratio ε := strain Nf := number of cycles to failure A := Amplitude ratioObjective ∆ εpl := Plastic strain amplitudeCrack ∆ εel := Elastic strain amplitudeInitiation K’ := Proportionality constant, cyclic stress-strainS-N n’ := Exponent in cyclic stress-straincurves c := Exponent in Coffin-Manson Eq.;Cyclic also, crack lengthstress-strn E := Young’s modulusCrack b := exponent in Basquin Eq.Propagate m := exponent in Paris LawMicrostr. K := Stress intensityeffects ∆ K := Stress intensity amplitudeDesign a := crack length
6 Fatigue • Fatigue is the name given to failure in response to alternating loads (as opposed to monotonicObjective straining).Crack • Instead of measuring the resistance to fatigueInitiation failure through an upper limit to strain (as inS-Ncurves ductility), the typical measure of fatigue resistanceCyclic is expressed in terms of numbers of cycles tostress-strn failure. For a given number of cycles (required inCrack an application), sometimes the stress (that can bePropagate safely endured by the material) is specified.Microstr.effectsDesign
7 Fatigue: general characteristics • Primary design criterion in rotating parts. • Fatigue as a name for the phenomenon based on theObjective notion of a material becoming “tired”, i.e. failing atCrack less than its nominal strength.Initiation • Cyclical strain (stress) leads to fatigue failure.S-Ncurves • Occurs in metals and polymers but rarely inCyclic ceramics.stress-strn • Also an issue for “static” parts, e.g. bridges.CrackPropagate • Cyclic loading stress limit<static stress capability.Microstr.effectsDesign
8 Fatigue: general characteristics • Most applications of structural materials involve cyclic loading; any net tensile stress leads to fatigue.Objective • Fatigue failure surfaces have three characteristicCrack features: [see next slide, also Courtney figs. 12.1, 12.2]Initiation – A (near-)surface defect as the origin of the crackS-N – Striations corresponding to slow, intermittent crack growthcurves – Dull, fibrous brittle fracture surface (rapid growth).Cyclicstress-strn • Life of structural components generally limited byCrack cyclic loading, not static strength.Propagate • Most environmental factors shorten life.Microstr.effectsDesign
9 S-N Curves • S-N [stress-number of cycles to failure] curve defines locus of cycles-to-failure for given cyclic stress.Objective • Rotating-beam fatigue test is standard; alsoCrack alternating tension-compression.Initiation • Plot stress versus the [Hertzberg]S-Ncurves log(number of cyclesCyclic to failure), log(Nf).stress-strn [see next slide,Crack also Courtney figs. 12.8, 12.9]Propagate • For frequencies < 200Hz,Microstr. metals are insensitive toeffects frequency; fatigue life inDesign polymers is frequency dependent.
10 Fatigue testing, S-N curve σmean 3 > σmean 2 > σmean 1 σa The greater the number of cycles in the loading history,Objective σmean 1 the smaller the stress thatCrackInitiation σmean 2 the material can withstandS-N σmean 3 without failure.curves log NfCyclicstress-strn Note the presence of aCrackPropagate fatigue limit in manyMicrostr. steels and its absenceeffects in aluminum alloys.Design [Dieter]
11 Endurance Limits • Some materials exhibit endurance limits, i.e. a stress below which the life is infinite: [fig. 12.8]Objective – Steels typically show an endurance limit, = 40% ofCrack yield; this is typically associated with the presenceInitiation of a solute (carbon, nitrogen) that pinesS-N dislocations and prevents dislocation motion atcurves small displacements or strains (which is apparentCyclic in an upper yield point).stress-strn – Aluminum alloys do not show endurance limits;CrackPropagate this is related to the absence of dislocation-pinningMicrostr. solutes.effects • At large Nf, the lifetime is dominated by nucleation.Design – Therefore strengthening the surface (shot peening) is beneficial to delay crack nucleation and extend life.
14 Fatigue Crack Propagation • Crack Nucleation → stress intensification at crack tip.Objective • Stress intensity → crack propagation (growth);CrackInitiation - stage I growth on shear planes (45° ),S-N strong influence of microstructure [Courtney: fig.12.3a]curves - stage II growth normal to tensile load (90° )Cyclic weak influence of microstructure [Courtney: fig.12.3b].stress-strn • Crack propagation → catastrophic, or ductile failureCrackPropagate at crack length dependent on boundary conditions,Microstr. fracture toughness.effectsDesign
15 Fatigue Crack Nucleation • Flaws, cracks, voids can all act as crack nucleation sites, especially at the surface.Objective • Therefore, smooth surfaces increase the time toCrack nucleation; notches, stress risers decrease fatigueInitiation life.S-Ncurves • Dislocation activity (slip) can also nucleate fatigueCyclic cracks.stress-strnCrackPropagateMicrostr.effectsDesign
16 Dislocation Slip Crack Nucleation • Dislocation slip -> tendency to localize slip in bands. [see slide 10, also Courtney fig. 12.3]Objective • Persistent Slip Bands (PSB’s) characteristic ofCrack cyclic strains.Initiation • Slip Bands -> extrusion at free surface. [see next slideS-N for fig. from Murakami et al.]curves • Extrusions -> intrusions and crack nucleation.Cyclicstress-strnCrackPropagateMicrostr.effectsDesign
17 Slip steps and theObjective stress-strainCrackInitiation loopS-NcurvesCyclicstress-strnCrackPropagateMicrostr.effectsDesign
18 Design Philosophy: Damage Tolerant Design • S-N (stress-cycles) curves = basic characterization. • Old Design Philosophy = Infinite Life design: acceptObjective empirical information about fatigue life (S-N curves);Crack apply a (large!) safety factor; retire components orInitiation assemblies at the pre-set life limit, e.g. Nf=107.S-Ncurves • *Crack Growth Rate characterization ->Cyclic • *Modern Design Philosophy (Air Force, not Navystress-strn carriers!) = Damage Tolerant design: acceptCrackPropagate presence of cracks in components. Determine lifeMicrostr. based on prediction of crack growth rate.effectsDesign
19 Definitions: Stress Ratios • Alternating StressObjective • Mean stress ≡ σm = (σmax +σmin)/2.CrackInitiation • Pure sine wave ≡ Mean stress=0.S-N • Stress ratio ≡ R = σmax/σmin.curvesCyclic • For σm = 0, R=-1stress-strnCrack • Amplitude ratio ≡ A = (1-R)/(1+R).PropagateMicrostr. • Statistical approach shows significanteffects distribution in Nf for given stress.Design • See Courtney fig. 12.6; also following slide.
21 Mean Stress • Alternating stress ≡ σa = (σmax-σmin)/2. • Raising the mean stress (σm) decreases Nf. [see slide 19,Objective also Courtney fig. 12.9]Crack • Various relations between R = 0 limit and the ultimateInitiation (or yield) stress are known as Soderberg (linear toS-N yield stress), Goodman (linear to ultimate) andcurves Gerber (parabolic to ultimate). [Courtney, fig. 12.10, problemCyclic 12.3]stress-strnCrackPropagate endurance limit at zero mean stressMicrostr. σaeffectsDesign tensile strength σmean
22 Cyclic strain vs. cyclic stress • Cyclic strain control complements cyclic stress characterization: applicable to thermalObjective fatigue, or fixed displacement conditions.CrackInitiation • Cyclic stress-strain testing defined by aS-N controlled strain range, ∆ εpl. [see next slide,curves Courtney, figs. 12.24,12.25]Cyclicstress-strn • Soft, annealed metals tend to harden;CrackPropagate strengthened metals tend to soften.Microstr. • Thus, many materials tend towards a fixedeffects cycle, i.e. constant stress, strain amplitudes.Design
23 Cyclic stress-strain curveObjectiveCrack [Courtney]InitiationS-NcurvesCyclicstress-strnCrackPropagateMicrostr.effects • Large number of cycles typically needed to reachDesign asymptotic hysteresis loop (~100). • Softening or hardening possible. [fig. 12.26]
24 Cyclic stress-strain • Wavy-slip materials [Courtney] generally reach asymptote in cyclic stress-Objective strain: planar slipCrack materials (e.g. brass)Initiation exhibit history dependence.S-Ncurves • Cyclic stress-strain curve defined by the extrema,Cyclic i.e. the “tips” of thestress-strnCrack hysteresis loops. [Courtney fig. 12.27]Propagate • Cyclic stress-strain curvesMicrostr. tend to lie below those foreffects monotonic tensile tests.Design • Polymers tend to soften in cyclic straining.
25 Cyclic Strain Control • Strain is a more logical independent variable for characterization of fatigue. [fig. 12.11]ObjectiveCrack • Define an elastic strain range as ∆ εel = ∆σ/E.Initiation • Define a plastic strain range, ∆ εpl.S-Ncurves • Typically observe a change in slope betweenCyclic the elastic and plastic regimes. [fig. 12.12]stress-strnCrack • Low cycle fatigue (small Nf) dominated byPropagateMicrostr. plastic strain: high cycle fatigue (large Nf)effects dominated by elastic strain.Design
26 Strain control of fatigueObjectiveCrackInitiationS-N [Courtney]curvesCyclicstress-strnCrackPropagateMicrostr.effectsDesign
27 Cyclic Strain control: low cycle • Constitutive relation for cyclic stress-strain:ObjectiveCrack • n’ ≈ 0.1-0.2Initiation • Fatigue life: Coffin Manson relation:S-NcurvesCyclic • εf ~ true fracture strain; close to tensilestress-strnCrackPropagateMicrostr. ductilityeffects • c ≈ -0.5 to -0.7Design • c = -1/(1+5n’ ); large n’ → longer life.
28 Cyclic Strain control: high cycle • For elastic-dominated strains at high cycles, adaptObjectiveCrack Basquin’s equation:Initiation • Intercept on strain axis of extrapolatedS-Ncurves elastic line = σf/E.Cyclic • High cycle = elastic strain control:stress-strnCrack slope (in elastic regime) = b = -n’ /Propagate (1+5n’ ) [Courtney, fig. 12.13] • The high cycle fatigue strength, σf,Microstr.effectsDesign scales with the yield stress ⇒ high strength good in high-cycle
30 Total strain (plastic+elastic) life • Low cycle = plastic control: slope = c • Add the elastic and plastic strains.ObjectiveCrackInitiationS-Ncurves • Cross-over between elastic and plastic control isCyclic typically at Nf = 103 cycles.stress-strn • Ductility useful for low-cycle; strength for high cycleCrackPropagate • Examples of Maraging steel for high cycleMicrostr. endurance, annealed 4340 for low cycle fatigueeffects strength.Design
31 Fatigue Crack Propagation • Crack Length := a. Number of cycles := N Crack Growth Rate := da/dNObjective Amplitude of Stress Intensity := ∆ K = ∆ σ√ c.Crack • Define three stages of crack growth, I, II and III,Initiation in a plot of da/dN versus ∆ K.S-N • Stage II crack growth: application of linear elastic fracturecurves mechanics.Cyclic • Can consider the crack growth rate to be related to the appliedstress-strn stress intensity.Crack • Crack growth rate somewhat insensitive to R (if R<0) in Stage IIPropagate [fig. 12.16, 12.18b]Microstr. • Environmental effects can be dramatic, e.g. H in Fe, ineffects increasing crack growth rates.Design
32 Fatigue Crack Propagation • Three stages of crack da/dN growth, I, II and III. • Stage I: transition to aObjective finite crack growth rateCrack from no propagation I ∆KcInitiation below a threshold valueS-N of ∆K. IIcurves • Stage II: “power law” dependence of crack IIICyclic growth rate on ∆K.stress-strn • Stage III: accelerationCrack of growth rate with ∆K,Propagate approachingMicrostr. catastrophic fracture.effectsDesign ∆Kth ∆K
33 *Paris Law • Paris Law:ObjectiveCrack • m ~ 3 (steel); m ~ 4 (aluminum).Initiation • Crack nucleation ignored!S-Ncurves • Threshold ~ Stage ICyclicstress-strn • The threshold represents an enduranceCrackPropagate limit.Microstr. • For ceramics, threshold is close to KIC.effectsDesign • Crack growth rate increases with R (for R>0). [fig. 12.18a]
34 *Striations- mechanism • Striations occur by development of slip bands in each cycle, followed by tip blunting,Objective followed by closure.CrackInitiation • Can integrate the growth rate to obtain cyclesS-N as related to cyclic stress-strain behavior. [Eqs.curves 12.6-12.8]Cyclicstress-strnCrackPropagateMicrostr.effectsDesign
35 *Striations, contd. • Provided that m>2 and α is constant, can integrate.ObjectiveCrackInitiationS-Ncurves • If the initial crack length is much less than the finalCyclic length, c0<cf, then approximate thus:stress-strnCrackPropagateMicrostr.effectsDesign • Can use this to predict fatigue life based on known crack
36 *Damage Tolerant Design • Calculate expected growth rates from dc/dN data.ObjectiveCrack • Perform NDE on all flight-critical components.Initiation • If crack is found, calculate the expected life ofS-Ncurves the component.Cyclic • Replace, rebuild if too close to life limit.stress-strnCrack • Endurance limits.PropagateMicrostr.effectsDesign
37 Geometrical effects • Notches decrease fatigue life through stress concentration.Objective • Increasing specimen size lowers fatigue life.Crack • Surface roughness lowers life, again through stressInitiation concentration.S-N • Moderate compressive stress at the surfacecurves increases life (shot peening); it is harder to nucleate aCyclic crack when the local stress state opposes crackstress-strn opening.CrackPropagate • Corrosive environment lowers life; corrosion eitherMicrostr. increases the rate at which material is removed fromeffects the crack tip and/or it produces material on the crackDesign surfaces that forces the crack open (e.g. oxidation). • Failure mechanisms
38 Microstructure-Fatigue Relationships • What are the important issues in microstructure- fatigue relationships?Objective • Answer: three major factors.Crack 1: geometry of the specimen (previous slide); anything on theInitiation surface that is a site of stress concentration will promoteS-N crack formation (shorten the time required for nucleation ofcurves cracks).Cyclic 2: defects in the material; anything inside the material that canstress-strn reduce the stress and/or strain required to nucleate a crackCrack (shorten the time required for nucleation of cracks).Propagate 3: dislocation slip characteristics; if dislocation glide is confinedMicrostr. to particular slip planes (called planar slip) then dislocationseffects can pile up at any grain boundary or phase boundary. TheDesign head of the pile-up is a stress concentration which can initiate a crack.
39 Microstructure affects Crack Nucleation • The main effect of da/dN microstructure (defects, surface treatment, etc.)Objective is almost all in the lowCrack stress intensity regime, I i.e. Stage I. Defects,InitiationS-N for example, make it II ∆Kc easier to nucleate acurves crack, which translates into a lower threshold IIICyclicstress-strn for crack propagationCrack (∆ Kth).Propagate • Microstructure alsoMicrostr. affects fractureeffects toughness and therefore Stage III.Design ∆Kth ∆K
40 Defects in Materials • Descriptions of defects in materials at the sophomore level focuses, appropriately on intrinsic defects (vacancies, dislocations). For the materials engineer, however, defectsObjective include extrinsic defects such as voids, inclusions, grainCrack boundary films, and other types of undesirable second phases.Initiation • Voids are introduced either by gas evolution in solidification or by incomplete sintering in powder consolidation.S-Ncurves • Inclusions are second phases entrained in a material during solidification. In metals, inclusions are generally oxides from theCyclic surface of the metal melt, or a slag.stress-strn • Grain boundary films are common in ceramics as glassyCrack films from impurities.Propagate • In aluminum alloys, there is a hierachy of names for secondMicrostr. phase particles; inclusions are unwanted oxides (e.g. Al2O3);effects dispersoids are intermetallic particles that, once precipitated,Design are thermodynamically stable (e.g. AlFeSi compounds); precipitates are intermetallic particles that can be dissolved or precipiated depending on temperature (e.g. AlCu compounds).
41 Metallurgical Control: fine particles • Tendency to localization of flow is deleterious to the initiation of fatigue cracks, e.g. Al-7050 with non-Objective shearable vs. shearable precipitates (Stage I in a da/Crack dN plot). Also Al-Cu-Mg with shearable precipitatesInitiation but non-shearable dispersoids, vs. only shearableS-N ppts.curvesCyclicstress-strnCrackPropagateMicrostr.effects graph courtesy of J.Design Staley, Alcoa
42 Coarse particle effect on fatigue • Inclusions nucleate cracks → cleanliness (w.r.t. coarse particles) improves fatigue life, e.g. 7475Objective improved by lower Fe+Si compared to 7075:Crack 0.12Fe in 7475, compared to 0.5Fe in 7075;Initiation 0.1Si in 7475, compared to 0.4Si in 7075.S-NcurvesCyclicstress-strnCrackPropagateMicrostr.effects graph courtesy of J.Design Staley, Alcoa
43 Alloy steel heat treatment • Increasing hardness tends to raise the endurance limit for high cycle fatigue. This is largely a functionObjective of the resistance to fatigue crack formation (Stage I in a plot of da/dN).CrackInitiationS-Ncurves Mobile solutes that pinCyclic dislocations → fatiguestress-strn limit, e.g. carbon in steelCrackPropagateMicrostr.effectsDesign [Dieter]
44 Casting porosity affects fatigue Gravity cast versusObjective squeeze cast [Polmear]Crack versusInitiation wroughtS-N Al-7010curvesCyclicstress-strnCrackPropagate • Casting tends to result in porosity. Pores are effective sites forMicrostr. nucleation of fatigue cracks. Castings thus tend to have lower fatigueeffects resistance (as measured by S-N curves) than wrought materials.Design • Casting technologies, such as squeeze casting, that reduce porosity tend to eliminate this difference.
45 Titanium alloys [Polmear]ObjectiveCrackInitiationS-NcurvesCyclicstress-strnCrack • For many Ti alloys, the proportion of hcp (alpha) and bcc (beta) phasesPropagate depends strongly on the heat treatment. Cooling from the two-phase region results in a two-phase structure, as Polmear’s example, 6.7a. Rapid coolingMicrostr. from above the transus in the single phase (beta) region results in a two-effects phase microstructure with Widmanstä tten laths of (martensitic) alpha in a beta matrix, 6.7b.Design • The fatigue properties of the two-phase structure are significantly better than the Widmanstä tten structure (more resistance to fatigue crack formation). • The alloy in this example is IM834, Ti-5.5Al-4Sn-4Zr-0.3Mo-1Nb-0.35Si-0.6C.
46 *Design Considerations • If crack growth rates are normalized by the elastic modulus, then material dependence is mostlyObjective removed! [Courtney fig. 12.20]Crack • Can distinguish between intrinsic fatigue [use Eq.Initiation 12.4 for combined elastic, plastic strain range] forS-Ncurves small crack sizes and extrinsic fatigue [use Eq. 12.6 for crack growth rate controlled] at longer crackCyclicstress-strn lengths. [fig. 12.21….]Crack • Inspection of design charts, fig. 12.22, shows thatPropagate ceramics sensitive to crack propagation (highMicrostr.effects endurance limit in relation to fatigue threshold).Design
47 *Design Considerations: 2 • Metals show a higher fatigue threshold in relation to their endurance limit. PMMA andObjective Mg are at the lower end of the toughnessCrackInitiation range in their class. [Courtney fig. 12.22]S-N • Also interesting to compare fracturecurves toughness with fatigue threshold. [Courtney fig.Cyclic 12.23]stress-strnCrack • Note that ceramics are almost on ratio=1 line,PropagateMicrostr. whereas metals tend to lie well below, i.e.effects fatigue is more significant criterion.Design
50 *Variable Stress/Strain Histories • When the stress/strain history is stochastically varying, a rule for combiningObjective portions of fatigue life is needed.CrackInitiation • Palmgren-Miner Rule is useful: ni is theS-N number of cycles at each stress level, and NficurvesCyclic is the failure point for that stress.stress-strn [Ex. Problem 12.2]CrackPropagateMicrostr.effectsDesign * Courtney’s Eq. 12.9 is confusing; he has Nf in the numerator also
51 *Fatigue in Polymers • Many differences from metals • Cyclic stress-strain behavior often exhibitsObjectiveCrack softening; also affected by visco-elasticInitiation effects; crazing in the tensile portionS-N produces asymmetries, figs. 12.34, 12.25.curvesCyclic • S-N curves exhibit three regions, with steeplystress-strn decreasing region II, fig. 12.31.CrackPropagate • Nearness to Tg results in strong temperatureMicrostr.effects sensitivity, fig. 12.42Design
52 Fatigue: summary • Critical to practical use of structural materials. • Fatigue affects most structural components,ObjectiveCrack even apparently statically loaded ones.Initiation • Well characterized empirically.S-Ncurves • Connection between dislocation behavior andCyclic fatigue life offers exciting researchstress-strnCrack opportunities, i.e. physically based modelsPropagate are lacking!Microstr.effectsDesign