Asme short course eastern nc section fatigue and fracture mechanics

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008ASME Short CourseEastern NC SectionFatigue and Fracture MechanicsJ W Eischen, PhD, PEDept of Mechanical and Aerospace EngineeringNC State UniversitySaturday April 21, 2007NC State Campus

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008I. Introduction and Overview1. Historical perspective2. Economic consequences of fracture and fatigue3. Case histories4. Concept of the stress intensity factor (SIF), relationship tostress analysisII. Stress Intensity Factors1. Practical applications2. Stress Analysis of Cracks Handbook3. Numerical calculation of SIF’s using FEA and BoundaryIntegral Method4. Concept of influence functions5. Modes of fractureIII. Fracture Toughness Testing and Fractography1. Plastic zone effects2. Plane strain fracture toughness test, J integral test3. FractographyIV. Fatigue1. Classic fatigue crack initiation2. Fatigue crack growth rate models3. Effect of overloads, retardation models4. Cycle countingV. Introduction to Elastic Plastic Fracture Mechanics1. Introduction to plasticity theory2. Practical applications- use of the Ductile Fracture Handbook

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008I. Fracture Mechanics- Introduction and OverviewWhy Study Fracture Mechanics?• The presence of cracks or crack-like defects inengineering structures cannot be precluded• Current energy and material conservationconsiderations require that structures be designed withless safety margin, and hence greater tolerance todefects• Requirement to determine residual strength and life ofcracked structure1.) Historical Perspective• 15th century- DaVinci conducted experiments on ironwire, longer wires failed at lower loads than shorterwires, any ideas why?• 1638- Galileo investigated the strength of a column intension- referred to this as “resistance to fracture”• 1843- Rankine first discussed fatigue in relation torailroad axles• 1922- A. A. Griffith was the first to make quantitativeconnection between strength and crack size• 1950’s- G. Irwin set the stage to make fracturemechanics an engineering discipline, focused mainlyon linear elastic fracture mechanics (LEFM)• 1968- J. Rice introduced the J integral-beginning ofelastic plastic fracture mechanics

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008• Currently- very active research area, particularlyfracture of composites and ceramics, dynamic fracture,molecular and atomistic simulations of the fractureprocess, fracture of: thin films, interfaces, andmicroelectronic devices2.) Economic Consequences of Fatigue and Fracture1982 NBS studyCosts:-$120billion annually in US alone, direct andimputed costsPossible savings:-$35billion annually if all currently knownfracture control technology were applied, many designsbased on stress analysis and not failure analysis.3.) Case Histories- some famous failures involving fracture• 1944 Cleveland LNG storage tank totally destroyed 79houses, 2 factories, 217 autos, heavily damaged 35houses and 13 factories, $6-7million (1944), due towelding defect and fatigue crack growth caused byvibration of passing trains.• World War II Liberty ships (EC-2)- hulls literallybroke in half, 145/2500 totally destroyed, 700/2500damaged, cracking due to poor welding and brittlesteel in cold weather• 1952 DeHavilland Comet commercial airplane- firstjet powered pressurized passenger plane, 3 aircraftsuffered in-flight breakup due to fatigue cracks nearstress-risers at window openings

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008• 1979 Chicago, AA DC-10 crash, 271 fatalities, enginepylon fracture, suspect maintenance procedures• 1985 Japan, JAL B-747 crash, aft pressure bulkheadfatigue failure, 500 plus fatalities• 1988 Hawaii, Aloha Airlines B-737, in-flight partialfuselage breakup, fatigue cracking along a lap joint.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Comet

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Liberty Ship

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Aloha Airlines

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20084.) Concept of Stress Intensity Factor (SIF) - Relationshipto Basic Stress AnalysisConsider the following defect free cantilever beam. Inorder to design the beam according to traditional methods,we insist that the maximum stress be less than theallowable stress. Let us assume that the allowable stress isthe yield strength σ ys .Design philosophy:σσσmaxallowmax bending 2allow< σ= σ6PL=bh= σ (no safety factor)ys2bh⇒ P < σ6Lys(allowable load)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Next consider a cracked cantilever beam.The best estimate of the maximum stress is nowFor a sharp crackσ⇒maxσ =Kt6PLmax 2bhstress concentration factorK →∞and thent→∞P →0 (zero allowable load, no load carrying capability)The traditional design approach concludes that the structurewill fail for any value of the load P since the stressconcentration factor is infinite. We know from experiencethat the structure will not necessarily fail if the load is lowenough and/or the crack size is not too large. Fracturemechanics can resolve this dilemma through use of aparameter called the stress intensity factor (SIF), K I . Forthis problem the SIF is

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20086PLKI= 1.12 π a = 1.12σ 2maxπabhDesign philosophy:KI

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The following numerical example deals with the so-called“center-cracked plate”.The material is 4340 steel with a yield strength σ ys =240ksiand a fracture toughness K Ic =50ksi-in 1/2 . The plate width is2W=10in, the thickness is 1in, and the crack length is2a=0.5in. The applied stress is σ 0 =60ksi. The SIF for thisloading/geometrical configuration isKI= σ0 πa= 60 π(0.25)= 53ksi in⇒ K > K (unstable fracture predicted)IIcA common additional check is a calculation of the “netsection”stress

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008σσnetnetFAnet 0 end= = = =< σnetysσ AAnet(60)(10×1)(10 − 0.5)(1)63.2ksiThus, we say that the failure mode will be brittle fracture,and not yield overload.References:[1.] Broek, D., The Practical Use of Fracture Mechanics,Kluwer, 1989.[2.] Barsom, R., and Rolfe, S., Fracture and Fatigue Controlin Structures, 3 rd Ed., ASTM, 1999.[3.] Kanninen, M., and Popelar, C., Advanced FractureMechanics, Oxford University Press, 1985.[4.] Sanford, R., Principles of Fracture Mechanics, PrenticeHall, 2003.[5.] Tada, H., Paris, P, and Irwin, G, The Stress Analysis ofCracks Handbook, ASME Press, 3 rd Ed.[6.] Zahoor, A., Ductile Fracture Handbook, Vols. 1-3,Electric Power Research Institute, 1989.[7.] ASM Handbook Volume 12- Fractography, 1989.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008II. Stress Intensity FactorsIt is instructive to examine the concentrated stresses at theedge of an elliptical hole in a thin plate, when the plate issubjected to tensile stressesThe maximum stress at the long end of the elliptical hole isσa= σ (1+2 ) bmax 0K t= SCFNote that when a=b (circular hole), the stress concentrationfactor is 3, i.e. a well-known result.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The radius of curvature at the end of the major axis isρ =2ba⇒ K = 1+2Thus, as ρ →0, K t →∞ and σ max →∞. This marks thetransition from a smooth defect to a sharp crack.tThe formal definition of the stress intensity factor is:aρK ≡ lim σ ( r,0) 2πrI r→0yy• The SIF depends on the intensity of the stress near thecrack tip. It will depend only on the geometry andloading (material also if a composite)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20081.) Stress Analysis of Cracks Handbook• Center-cracked plate• Double-edge crack plate• Single edge crack plate• Edge-cracked beam• Edge-cracked simply supported beam• Cracks emanating from a thru-holes in a plate• Crack emanating from an elliptical notch• Buried penny-shaped crack• Edge-cracked solid shaft• Semi-elliptical surface crack• Circumferentially cracked thin-walled pressure vessel• Longitudinally cracked thin-walled pressure vessel

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J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20082.) Practical applicationsDesign of a high strength steel pressure vessel-Data: p 0 =5000psi, D=2R=30in, t>0.5in, γ steel =489lb/ft 3σ ys K Ic $/lbA 260ksi 80ksi-in 1/2 $1.40B 220 110 1.40C 180 140 1.00D 180 220 1.20E 140 260 0.50F 110 170 0.15Problem: Design for satisfactory performance (SF=2),minimum cost and weight are also important

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Stress based design:σσmaxmaxminallowhoopσysσallow=2pR σ0 ys=t 2t= σ= σ =pDpR0t5000(30) 150,0000⇒min= = =σys σys σysσ allow t min Cost-$/ft Weight-lb/ftA 130ksi 0.58in 255$/ft 182lb/ftB 110 0.68 298 213C 90 0.83 258 258D 90 0.83 310 258E 70 1.07 165 330F 55 1.36 62 416

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Fracture mechanics based design:The leak-before-break (LBB) failure mode assumes a thruwallcrack with length equal to twice the wall thickness.This crack is designed to leak before causing a failure byfractureKIcKI=2pR0πttminminK=20⇒ tmin= =⎜ K ⎟IcIc2 2⎛ pD π ⎞ ⎛265,868⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ KIc⎠

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Κ Ιc /2 t min Cost-$/ft Weight-lb/ftA 40ksi-in 1/2 11.04in 3126 $/ft 2233 lb/ftB 55 5.84 2107 1505C 70 3.61 1016 1016D 110 1.46 533 444E 130 1.05 162 324F 85 2.45 108 720

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Hot Isostatic Press failure-HIP Specs:• Pressure= 15,000psi• Temperature=2400F• Up to 24hr process times• Q&T steel, σ uts =185ksi, σ ys =165ksi• OD=97in, ID=76in, L=14ft (huge!)• Failed at cycle 1898, 507 cycles after change incooling jacket

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Failure Analysis• Failed pieces reassembled- crack map points tomultiple origin sites, presence of corrosion pits onvessel OD• Final fracture origin sites indicate semi-circularsurface flaws in the r-θ plane on vessel OD neardiameter transition• Possible temper embrittlement of steel during heattreatment at manufacture• Problems with water treatment chemistry

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Stress Analysis:• Force on lid:F = pπr = 68,000,000lbs• Axial stress (due to pressure only):2i2σ = prizz23,700 psi2 2r − r=• Axial stress (thermal, 300F ΔT):oiσzzEαΔT=± =+ 42,000 psi (OD)2(1 −υ)• Total stress: σ zz =65,700psi• Stress concentration at diameter taper neglected sofar, FEA only way to calculate• Concentrated total stress with FEA σ zz =118,000psi• Various heat flux models yield a range 90,600< σ zz

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Fracture Mechanics Analysis:• Flaw model:rIDOD2aaθ• SIFKI= 1.12σzzπ aQ• Critical crack size measurements 0.857

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20083.) Numerical Calculation of SIF’s Using Finite ElementAnalysisThe distribution of stress near a sharp crack tip under ModeI (symmetric) loading conditions is:σσσxxyyxyKIθ ⎡ θ 3θ⎤= cos 1 sin sin2πr 2 ⎢−⎣ 2 2 ⎥⎦KIθ ⎡ θ 3θ⎤= cos 1 sin sin2πr 2 ⎢+⎣ 2 2 ⎥⎦KIθ θ 3θ= cos sin cos2πr 2 2 2The definition of the stress intensity factor actually followsfrom these expressionsK ≡ lim σ ( r,0) 2πrI r→0yy

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Thus if we know the distribution of stress ahead of cracktip in a real structure we can obtain an estimate of the SIFby graphing σ yy(,0) r 2 π r vs. r .The SIF is just the intercept of this plot. This is a quick, butnot very accurate way to compute the SIF. Another way todo this with FEA data is to use the crack openingdisplacements behind the crack tip, which are for Mode IwhereuyKIr θ ⎡κ + 1 2 θ ⎤= sin cosG 2π2 ⎢ −2 2⎥⎣⎦κ = 3 −4υ for plane strain (thick plates)3-ν= for plane stress (thin plates)1+ ν

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008An alternative definition of the SIF is thenIn this case we would graph2G2πKI ≡ limr→0uy( r, π ) κ + 1 r2G2πuy(,0) rκ + 1 rvs. r . This willtypically yield better results for so-called “displacementbased finite element methods.Yet another more efficient and accurate way to use the FEAmethod to calculate SIF’s is to work with the strain energyof the structure, which is defined as:1U = σεij ijdV2∫VolIt turns out the rate of change of the total strain energy Uwith respect to crack length is related to the stress intensityfactorG21 dU KI= J = = (for plane stress)t da E2KI= (for plane strain)2E /(1 −ν)The quantities G and J (which happen to be equivalent) arethe strain energy release rate and J integral (more later inEPFM), respectively. In practice, this requires the solutionof two FEA problems, with slightly differing crack lengths,then

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20081dU 1 U ( a + Δa) −U ( a)=t da t ΔaandKI=EG

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The following example, a finite width center-cracked panel,has been solved using the FEA method.The reference SIF for this problem is based on Isida (1971),International Journal of Fracture, Vol. 7, No.3 , pp. 301-316. This solution includes a finite height, as well as afinite width correction.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008FEA results:KI=0.306

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Stress and Displacement Extrapolation:Stress Extrapolation0.40.35y = 0.0604x + 0.2978K I0.30.250.20 0.2 0.4 0.6 0.8rDisplacement Extrapolation0.40.35y = -0.0959x + 0.3498K I0.30.250.20 0.2 0.4 0.6 0.8 1r

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Strain Energy Method:a U(a) 2ΔU/tΔa * K I2 0.162019 - -2.01 0.162483 0.0928 0.3192.1 0.166906 0.0977 0.328*- factor of 2 due to symmetry, account for material belowthe modeled area for the right crack tip

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20084.) Concept of Influence FunctionsThe following configuration doesn’t have direct practicalimport, but can be used to develop SIF’s for more complexsituations.KIAP a + ξ, KP a −ξ=IB=πaa− ξ πaa+ξThese are referred to as influence functions for the stressintensity factors. If the loads are centered, thenKIA=Pπ a

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Now consider the following problem, a “pressurized crack”KIAa1 a+ ξ σ0dξa+ξ= ∫σ yy( ξ)dξ=aaa πa−ξ∫a πa−ξ−−IFaCarrying out the integration givesKIA= σ0πaWe will now learn why this result is not a surprise. For ageneral non-uniform loading on a center-cracked panel, asshown in the figure below, it turns out the stress intensityfactor can be determined from the related probleminvolving pressure loading (non-uniform in this case)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008That is, for any general far field loading,KIA=a∫−aσyy( xdx ) a+xπ a a−x• This integral applies only for the center-cracked panelgeometry• Every other geometry and loading configurationrequires a different influence functionThis method of calculating SIF’s is often referred to as theinfluence function (IF) method. It depends on having a SIFsolution for point loads on the crack surfaces for a

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008particular structural configuration. Another example is theedge-crack geometryThe IF function for this case is2Pc aKI= F( , )π a a bcc3.52(1 − ) 4.35 −5.28F = a − a +3/2 1/2(1 −a/ b) (1 −a/ b)⎧ c 3/2⎫⎪1.30 − 0.30( )ac⎪⎧ c a⎫⎨ + 0.83 −1.76 ⎬⎨1 −(1 − ) ⎬2⎪ 1 − ( a/ b)a⎪⎩a b⎭⎩⎭

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Then for uniform loading on the edges of a finite widthedge cracked plate,aσ0c aKI= ∫ F dc=σ0πaa b0 π aa( , ) 1.12 if is very smallbGeneral purpose fracture mechanics software is often basedon the IF method in order to treat general loadings onstructures. All that is required is the stress distribution onthe un-cracked structure.Exponent Inc. NASCRAC software:http://www.exponent.com/about/nascrac.htmlwww.exponent.com/about/products/geolib.htmStructural Integrity Associates Inc. pc-CRACK software:http://www.structint.com/products/pc-crack/pccrack.html

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20085.) Modes of FractureWe have discussed what has been referred to as Mode Ifracture, where the loading is symmetrical about the crackplane and tends to only open the crack surfaces. It is alsopossible to have loading that leads to the other “modes”shown below. In most practical applications, the crack willtend to develop and orient itself so there is only Mode Ideformation; hence what we have learned is appropriate formost situationsMode I Mode II Mode IIIOpening Sliding Tearing

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008III. Fracture Toughness Testing and FractographyThe fracture toughness, K Ic , of a given material must bedetermined experimentally, i.e. it is a material property.The fracture toughness test is designed so that there isminimal yielding of the material near the crack tip duringthe test, ensuring Linear Elastic conditions. Therefore,before delving into the details of the test, we mustunderstand plasticity effects.1.) Plastic Zone EffectsThe so-called Irwin estimate of the plastic zone size, r y , issimply obtained by setting the normal stress on the crackplane equal to the yield strength.KI2πry= σys1 ⎛ K ⎞I⇒ ry= 2π⎜σ ⎟⎝ ys ⎠2(based on elastic stresses- ad hoc)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008A more elaborate analysis, including the effect of plasticityresults in an estimate of the plastic zone size twice thisvalueIrwin proposed the concept of an effective crack length toaccount for plastic effects, i.e.a = a+reffy

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008KI⎛1 ⎛ K ⎞I= σ0π⎜a+ ⎜ 2π⎜σ⎟⎝ ys⎝⎠2⎞⎟⎟⎠solve for K IKI=σ0πa1 ⎛ σ ⎞01− 2 ⎜σ ⎟⎝ ys ⎠2The denominator is referred to as the plastic zonecorrection factor.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20082.) Plane Strain Fracture Toughness Test, J Ic Integral TestThe American Society for Testing and Materials has putforward standards for measuring K Ic (and J Ic ). Someremarks on the E399-06 (following page)• Possible test specimens are compact tension, 3pt bend,and arc• Specimen dimensions are given in the standard (toensure minimal r y )• Test specimen is loaded to failure while monitoringforce and crack opening displacement• Size requirements are often restrictive, for A533Bpressure vessel steel the required thickness is 30in ormore!• Invalid tests with insufficient thickness may containuseful information

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20083.) ASM Fractography Tutorial

T • U • T • O • R • I • A • LFractography of Metals and Plastics(continued)by Ronald J. Parrington, P.E.Fractography of Metals and PlasticsFractography is critical to failure analysis of metals and plastics. Fractography of plastics is a relativelynew field withmany similarities tometals. Using case histories,various aspects of failure analysisand fractography of metals andplastics are compared andcontrasted.Failure modes common toboth metals and plastics includeductile overload, brittle fracture,impact, and fatigue. Analogiescan also be drawn betweenstress-corrosion cracking (SCC)of metals and stress cracking ofpolymers. Other metal/plasticfailure analogies include corrosion/chemicalaging, dealloying/scission, residual stress/frozen-instress, and welds/knit lines. Stress raisers, microstructure, material defects, and thermomechanical historyplay important roles in both types of materials. The key fractographic features for metals and plasticsare described in this paper.Historical PerspectivePlastics have been in existence forapproximately 130 years. John Hyattpatented nitrocellulose, the firstcommercial plastic, in 1869. However,full-scale development and use ofplastics is only approximately 50 yearsold. In contrast, metals have been inuse for hundreds of years.The application of engineering materialsis unavoidably accompanied bythe occurrence of failures, many ofwhich have been catastrophic. Theconsequences of material failures, includingdeaths, financial losses, andlegal ramifications, have encouragedthe development of effective failureanalysis methods. Although the costof failure analysis may exceed thevalue of the part, the cost of servicefailures usually far exceeds the costof failure analysis. Many of the techniquesused over the years for the evaluationof metals have been successfullyapplied to plastics, with onlyminor modifications.Fractography is arguably the mostvaluable tool available to the failureanalyst. Fractography, a term coinedin 1944 to describe the science of examiningfracture surfaces, has actuallybeen used for centuries as part of thefield of metallurgy. Even before that,however, Stone Age man possessed aworking knowledge of fracture. Archeologicalfindings of lithic implements,weapons, and tools shaped fromstone by controlled fracture indicatethat prehistoric man knew how to selectrocks with favorable fracture behavior,use thermal spalling to detach bedrockfrom the working core, and shape stoneby pressure flaking.Fractography, as we know it today,developed in the 16th century as aquality-control practice employed forferrous and nonferrous metalworking.De La Pirotechnia, published by VannoccioBiringuccio in 1540, [1] is oneof the first documents to detail fractographictechniques.66 of 10916 Volume 2(5) October 2002Practical Failure Analysis

Invention of the optical microscopein 1600 provided a significant newtool for fractography, yet it was notused extensively by metallurgists untilthe eighteenth century. In 1722, R.A.de Réaumur [2] published a book withengravings that depicted macroscopicand microscopic fracture surfaces ofiron and steel. Interestingly, the categoriesof macroscopic features developedby de Réaumur have remainedessentially unchanged through thecenturies.Partly due to the development ofmetallographic techniques for examiningcross sections of metals, interestin microfractography waned duringthe nineteenth century. Metalworkerscontinued to use fractographic techniquesfor quality-assurance purposes,but, for the most part, researchers andpublications ignored fractography.Several technological developmentsin the twentieth century revitalizedinterest in fractography. Carl A.Zapffe [3] developed and extensivelyused fractographic techniques tostudy the hydrogen embrittlement ofsteels. His work led to the discoveryof techniques for photographingfracture surfaces at high magnifications.The first fractographs werepublished by Zapffe in 1943.An even more revolutionary developmentwas the invention of thescanning electron microscope (SEM).The first SEM appeared in 1943.Unlike the transmission electron microscope,which was developed a fewyears earlier, it could be used for fracturesurface examination. An SEMwith a guaranteed resolution of approximately500 Å became commerciallyavailable in 1965. Comparedwith the optical microscope, the SEMexpands resolution by more than oneorder of magnitude and increases thedepth of focus by more than twoorders of magnitude. The tools formodern fractography were essentiallyin place before plastics achieved widespreaduse.Failure Analysis OverviewThe general procedure for conductinga sound failure analysis is similarfor metallic and nonmetallic materials.The steps include: (1) informationgathering; (2) preliminary, visualexamination; (3) nondestructive testing;(4) characterization of materialproperties through mechanical, chemical,and thermal testing; (5) selection,preservation, and cleaning of fracturesurfaces; (6) macroscopic examinationof fracture surfaces, secondary cracking,and surface condition; (7) microscopicexamination; (8) selection,preparation, and examination of crosssections; (9) identification of failuremechanisms; (10) stress/fracturemechanics analysis; (11) testing tosimulate failure; and (12) data review,formulation of conclusions, and reporting.Although the basic steps of failureanalysis are nearly identical, somedifferences exist between metals andplastics. Nondestructive testing ofmetals includes magnetic-particle,eddy-current, and radiographic inspectionmethods that are not generallyapplicable to plastics, forobvious reasons. However,ultrasonic and acoustic emissiontechniques find applicationsfor both materials.Similarly, different chemicaltest methods are necessary.Typical test methods formetals are optical emissionspectrometry, inductivelycoupled plasma, and combustion.Fourier transform infraredspectroscopy is usedextensively to identify plasticsFig. 1by molecular bonding, and thermaltesting, differential scanning calorimetry,and thermogravimetric analysisare also very important for polymercharacterization. Energy-dispersive x-ray spectroscopy, used in conjunctionwith the SEM, is a very practical toolfor elemental chemical analysis ofboth metals and plastics. Also noteworthyis that different chemical solutionsare required for metals and plasticsto clean and/or protect fracturesurfaces and to etch cross sections toreveal microstructure.Causes of FailureOf course, the primary objective ofa materials failure analysis is to determinethe root cause of failure.Whether dealing with metallic ornonmetallic materials, normally, theroot cause can be assigned to one offour categories: design, manufacturing,service, or material. Often, severaladverse conditions contribute to thepart failure. Many of the potentialroot causes of failure are common tometallic and nonmetallic materials.Improper materials selection, overlyhigh stresses, and stress concentrationsare examples of design-relatedproblems that can lead to prematurefailure. Materials selection must takeinto account environmental sensitivi-Fracture of a glass-filled polyamide threaded partdue to stress concentration at the thread root67 of 109Practical Failure Analysis Volume 2(5) October 200217

Fractography of Metals and Plastics (continued)ties as well as requisite mechanicalproperties and welding/joining characteristics.Stress raisers are frequentlya preferred site for fracture origin,Fig. 2Fig. 3Fig. 4Cross section showing fracture along the knit line ofa perfluoralkoxyethylene-lined impellerCross section of a polyacetal hinge that fractured(arrow) through an area of porosityMicrobiologically induced corrosion of a 304 SSTvessel weld, characterized by pitting and selectiveleaching (arrow)particularly in fatigue. Stress raisersinclude thread roots (Fig.1), sharpradii of curvature, through holes, andsurface discontinuities (e.g., gatemarks in molded plastic parts).Similarly, many manufacturingand material problemsfound in metals also areobserved or have a corollaryin plastics. Weldments are atrouble-prone area for metals,as are weld lines or knit linesin molded plastics (Fig. 2).High residual stresses canresult from metalforming,heat treatment, welding, andmachining. Similarly, highfrozen-in stresses in injection-moldedplastic partsoften contribute to failure.Porosity and voids are commonto metal castings andplastic molded parts (Fig. 3).Pores and voids serve asstress raisers and reduceload-carrying capability.Other manufacturing- andmaterial-related problemsthat may lead to failureinclude adverse thermomechanicalhistory, poor microstructure,material defects,and contamination.Fig. 5Hollowing out of a polyacetal hinge due to acidcatalyzedhydrolysisEnvironmental degradation is oneof the most important service-relatedcauses of failure for metals and plastics.Other degradation processes includeexcessive wear, impact, overloading,and electrical discharge.Failure MechanismsAnother key objective of failure analysisis to identify the failure mechanism(s).Once again, some failure modesare identical for metals and plastics.These modes include ductile overload,brittle fracture, impact, fatigue, wear,and erosion.Analogies also can be drawn betweenmetals and plastics with regardto environmental degradation.Whereas metals corrode by an electrochemicalprocess, plastics arevulnerable to chemical changes fromaging or weathering. Stress-corrosioncracking (SCC), a specific form ofmetallic corrosion, is similar in manyways to stress cracking of plastics.Both result in brittle fracture due tothe combined effects of tensile stressand a material-specific aggressive environment.Similarly, dealloying orselective leaching in metals (Fig. 4),the preferential removal of oneelement from an alloy by corrosion,is somewhat similar to scission ofpolymers (Fig. 5), aform of aging thatcan cause chemicalchanges by selectivelycutting molecularbonds.Analogies can alsobe drawn betweenmetals and anothertype of polymer:rubber. Internal hydrogenin steels canprecipitate and causehydrogen damage,which is frequently68 of 10918 Volume 2(5) October 2002Practical Failure Analysis

Fig. 6Fig. 7Fig. 8Hydrogen damage of induction-hardened steelpiston rod displaying fisheyesExplosive decompression fracture of rubber O-ring,characterized by fisheye-like patternsBeach and radial marks visible on torsional fatiguefracture of a 6 in. diameter 4340 shaftcharacterized by localized brittle areasof high reflectivity, known as flakesor fisheyes, on otherwise ductile fracturesurfaces (Fig. 6). Similarly, explosivedecompression in rubber O-ringsproduces fisheye-like ovular patternson the fracture surfaces (Fig.7). Explosive decompression isthe formation of small rupturesor embolisms when an elastomericseal, saturated with highpressuregas, experiences anabrupt pressure reduction.This failure mechanism isanalogous to the “bends” thatafflict divers who surface tooquickly.FractographyWhen material failure involvesactual breakage, fractographycan be employed toidentify the fracture origin,direction of crack propagation,failure mechanism,material defects, environmentalinteraction, and the natureof stresses. Some of the macroscopicand microscopicfeatures employed by thefailure analyst to evaluatefracture surfaces of metalsand plastics are describedsubsequently. Note, however,that many of the fractographicfeatures described forplastics are not observable forFig. 9Brittle fracture of an epoxy layer displaying amirror zone, rib marks, and hacklesreinforced plastics and plasticscontaining high filler content.Macroscopically VisibleFractographic FeaturesOn a macroscopic scale, all fractures(metals and plastics) fall into one oftwo categories: ductile and brittle.Ductile fractures are characterized bymaterial tearing and exhibit grossplastic deformation. Brittle fracturesdisplay little or no macroscopicallyvisible plastic deformation andrequire less energy to form. Ductilefractures occur as the result of appliedstresses exceeding the material yieldor flow stress. Brittle fractures mayoccur at stress levels below the materialyield stress. In practice, ductile fracturesoccur due to overloading orunderdesigning and are rarely thesubject of a failure analysis. However,the unexpected brittle failure of normallyductile materials is frequentlythe subject of a failure analysis.Many macroscopically visible fractographicfeatures serve to identifythe fracture origin(s) and direction ofcrack propagation. Fractographic featurescommon to metals and plasticsare radial marks and chevron patterns.Radial marks (Fig. 8) are lines on a fracturesurface that radiate outward fromthe origin and are formed by the intersectionof brittlefractures propagatingat different levels.Chevron or herringbonepatterns areactually radial marksresembling nestedletter V’s and pointingtoward the origin.Fatigue failures inmetals display beachmarks and ratchetmarks that serve to(continued on page 44)69 of 109Practical Failure Analysis Volume 2(5) October 200219

Fractography of Metals and Plasticsidentify the origin and the failuremode. Beach marks (Fig. 8) are macroscopicallyvisible semielliptical linesrunning perpendicular to the overalldirection of fatigue crack propagationand marking successive positions ofthe advancing crack front. Ratchetmarks are macroscopically visible linesrunning parallel to the overalldirection of crack propagation andformed by the intersection of fatiguecracks propagating from multipleorigins.Brittle fractures in plastics also exhibitcharacteristic features, several ofwhich are macroscopically visible (Fig.9). These features may include a mirrorzone at the origin, a mist region,and rib marks. The mirror zone is aflat, featureless regionsurrounding theorigin and associatedwith the slow crackgrowthphase offracture. The mistregion is locatedimmediately adjacentto the mirror zoneand displays a mistyappearance. This areais a transition zonefrom slow to fastcrack growth. Ribmarks are semiellipticallines resemblingbeachmarks in metallicfatigue fractures.MicroscopicallyVisibleFractographicFeaturesOn a microscopicscale, ductile fracturein metals (Fig. 10)displays a dimpledsurface appearancecreated by microvoidFig. 10Fig. 11(continued) from page 19)coalescence. Ductile fracture in plastics(Fig. 11) is characterized by materialstretching related to the fibrillarnature of the polymer response tostress. Although a part may fail ina brittle manner, ductile fracturemorphology is frequently observedaway from the origin. For example,the final fast fracture by ductileoverload produces the shear lip inmany metal failures, even when thecrack originated and was propagatedby SCC, fatigue, or hydrogen embrittlementprocesses. The extent ofthis overload region is an indicationof the stress level. Generally, the largerthe overload region, the higher thestress level on the failed component.Brittle fracture of metallic materialsDimpled appearance typical of ductile fracture ofmetallic materialsFracture of a polyethylene tensile-test specimenexhibiting material stretchingFig. 12 Brittle fracture of an FC-0205 powder metalcontrol rod displaying cleavage facetsFig. 13may result from numerous failuremechanisms, but there are only a fewbasic microfractographic features thatclearly indicate the failure mechanism.These features are cleavage facets(Fig. 12), intergranular facets (Fig.13), and striations (Fig. 14). Cleavagefacets form in body-centered cubic(bcc) and hexagonal close-packedmetals when the crack path follows awell-defined transgranular crystallographicplane (e.g., the {100} planesin bcc metals). Cleavage is characteristicof transgranular brittle fracture.Intergranular fracture, recognizableby its “rock candy” appearance, occurswhen the crack path follows grainboundaries. Intergranular fracture istypical of many forms of SCC, hy-Intergranular fracture of an embrittled cast steelpneumatic wrench70 of 10944 Volume 2(5) October 2002Practical Failure Analysis

Fig. 14Fig. 15Fig. 16Fatigue striations visible on type 302 stainless steelspring fractureFatigue striations emanating from fracture origin ofpolycarbonate latch handleSEM micrograph of fatigue striations shown inFig.15drogen embrittlement, and temperembrittledsteel. Fatigue failures ofmany metals exhibit striations at highmagnifications. (Normally, magnificationsof 500 to 2,500× are required.)Striations are semiellipticallines on a fatigue fracture surface thatemanate outward from the origin andmark the crack-front positionwith each successive stresscycle. The spacing of fatiguestriations is usually veryuniform and can be used tocalculate the crack growthrate, if the cyclic stressfrequency is known. Striationsare discriminated fromstriation-like artifacts on thefracture surface in that truefatigue striations never crossor intersect one another.Plastics do not displaycleavage and intergranularfracture. However, similar tometals, striations are foundon fatigue fracture surfaces(Fig. 15, 16). Striations inplastics typically are observableat much lower magnifications(50 to 200×). However,local softening andmelting due to hystereticheating can obliterate fatiguestriations in less rigid plastics.In addition to mirrorzones, mist regions, and ribmarks, which are normallyvisible without the aid of amicroscope, brittle fracture ofplastics may display hackles,Wallner lines, and conicmarks. Hackles (Fig. 9) aredivergent lines radiatingoutward from the fractureorigin. They are analogous toriver patterns observed on thecleavage facets of transgranularbrittle fractures ofmetals. Wallner lines arefaint, striation-like markingsformed by the interaction ofstress waves reflected fromphysical boundaries with the advancingcrack front. Conic marks areparabolic-shaped lines pointing backtoward the origin. Hackles andWallner lines may or may not be visiblewithout the aid of a microscope.Closing RemarksFractographic techniques, developedand applied to metal failures forcenturies, have been readily adaptedto the fracture analysis of plasticssince their emergence as a key engineeringmaterial over the last 50 years.However, more work remains to bedone to advance fractography ofplastics. One notable area for researchis fracture analysis of composites,reinforced plastics, and plastics containinghigh filler content. Fracturesof these materials too often aredismissed as inherently lackingmeaningful fractographic features.Finally, there is a definite need for anauthoritative publication on fracturein plastics.AcknowledgmentsThe author gratefully acknowledgesthe contributions of Dave Christieand Steve Ruoff of IMR Test Labs.References1. V. Biringuccio: De La Pirotechnia, Venice,1540; see translation by C.S. Smith andM.T. Gnudi: The ‘Pirotechnia’ of VannoccioBiringuccio, American Institute of Mining,Metallurgical and Petroleum Engineers,New York, 1942.2. R.A. de Réaumur: L’Art de Convertir le FerForgé en Acier, et L’Art d’Adocir le Fer Fondu,(in French), Michel Brunet, Paris, 1722;see translation by A.G. Sisco: Réaumur’sMemoirs on Steel and Iron, University ofChicago Press, 1956.3. C.A. Zapffe and G.A. Moore: Trans.AIME, 1943, 154, pp. 335-59.Selected References• W. Brostow and R.D. Corneliussen: Failureof Plastics, Hanser Publishers, Munich, 1986.71 of 109Practical Failure Analysis Volume 2(5) October 200245

Fractography of Metals and Plastics (continued)• T.J. Davies and I. Brough: “General Practicein Failure Analysis,” Metals Handbook (9 thed.), vol. 11, Failure Analysis and Prevention,American Society for Metals, Metals Park,OH, 1986.• M. Ezrin: Plastics Failure Guide: Cause andPrevention, Hanser Publishers, New York,1996.• F.R. Larson and F.L Carr: “How FailuresOccur…Topography of Fracture Surfaces,”Source Book in Failure Analysis, AmericanSociety for Metals, Metals Park, OH, 1974.• Metals Handbook (8 th ed.), vol. 9, Fractographyand Atlas of Fractographs, AmericanSociety for Metals, Metals Park, OH, 1974.• U. Portugall and K. Steinlein: Prac. Metallography,1999, 36(8), pp. 446-62.Credit InfoRonald J. Parrington, IMR Test Labs, 131Woodsedge Drive, Lansing, NY 14882. Contacte-mail: ron@imrtest.com.Failure Analysis Resources from ASM InternationalASM Handbooks in print and CD-ROMVolume 19: Fatigue and Fracture1996 • 1,057 pages• ISBN: 0-87170-385-8 • ASM PublicationThis volume contains the essential data necessary for you to makeinformed decisions on alloy design and material selection to controlfatigue and fracture. This Handbook is especially useful in evaluatingtest data and helping you understand the key variables that affectresults.Sections include: Fatigue Mechanisms, Crack Growth, Testing,Engineering Aspects of Fatigue Life, Fracture Mechanics ofEngineering Materials, Fatigue and Fracture Control, Castings,Weldments, Wrought Steels, Aluminum Alloys, Titanium Alloys andSuperalloys, Other Structural Alloys, Solders, Advanced Materials.Appendices contain coverage of fatigue strength parameters and stressintensityfactors.Order #06197G-RPADPFAJPrice: $198.00 ASM Member $159.00Volume 11: Failure Analysis and Prevention1986 • 843 pages • ISBN: 0-87170-017-4 • ASM PublicationProvides coverage of every aspect of this critical field–from areview of the engineering aspects of failure analysis to the major typesof failures to in-depth analyses of failures of specific components andassemblies. Also included are failure analysis of composites, ceramics,polymers, and integrated circuits. Detailed case histories guide youthrough the investigative procedures used and the corrective measuresimplemented.Sections include: Engineering Aspects of Failure and Failure Analysis,Failure Mechanisms and Related Environmental Factors, Products ofPrincipal Metalworking Processes, Manufactured Components andAssemblies, Engineered and Electronic Materials.Order #06366G-RPADPFAJPrice: $198.00 ASM Member $159.00Volume 12: Fractography1987 • 517 pages • ISBN: 0-87170-018-2 • ASM PublicationProvides engineers with enhanced capability for recognizing andinterpreting the various features of a fracture, enabling you to performimproved failure analyses and to better determine the relationship ofthe fracture mode to the microstructure. The Atlas of Fractographscontains more than 1,300 fractographs invaluable for your understandingof the causes and mechanisms of the fracture of engineering materials.Sections include: History of Fractography, Modes of Fracture, Preparationand Preservation of Fracture Specimens, Photography ofFractured Parts and Fracture Surfaces, Visual Examination and LightMicroscopy, Scanning Electron Microscopy, Transmission ElectronMicroscopy, Quantitative Fractography, Fractal Analysis of FractureSurfaces.Order #06365G-RPADPFAJPrice: $198.00 ASM Member $159.0072 of 10946 Volume 2(5) October 2002Practical Failure Analysis

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008IV. Fatigue1.) Fatigue crack initiation• Total fatigue life is composed of two parts- initiationand propagation- N = Ni + Np• We are focusing on propagation- how far and at whatrate an existing crack goes• The number of cycles to failure is Nf= Ni+ Np• Initiation life is typical based on the stress range (orrange of strain when the stress level is near or aboveyield level)Δ σ = σ −σmaxmaxmin1σmean = ( σmax + σmin)2σminR =σ

!!!J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008• Material, surface prep, environment, residual stress,temperature, stress concentration all effect theseresults• Flat part of the curve is usually at 10 6 -10 8 cycles• Endurance limitσe≡ maximum stress level at R =−1 below whichcrack initiation is absent• For steels, the ratio of the endurance limit to theultimate strength is about ½.• The stress that is used in the S-N curve must be theconcentrated stressσ max= K σ tSCF−notK Inomnominal stress

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20082.) Fatigue crack growth rate modelsWe know that for a general cracked structure with anapplied load PK I= f ( a,geometry)PCrack Propagation Mechanisms1.) Brittle fracture K I> K Ic⇒ unstable crack growth2.) Slow stable crack growth due to time varying loadswhen K < KIIc

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008As P varies in a cyclic manner the crack length mayincrease in a stable manner. If we know P max , P min , we alsocan calculate K max , K min , ΔK I , K mean . For many materials thecrack growth is observed to depend on ΔK I and K mean . Thatis, for each loading cycleΔ a = f ( ΔK I, Kmean)The most common way to express the crack growth “rate”isdadN=f ( ΔKI, Kmean)determined experimentally

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Typical resultsThe straight line portion of the data is usually fit to theParis Lawda= C( ΔK ) nIdNRemarks:• Does not account for threshold, mean stress, oracceleration effects, which are important• For A36 structural steelda10 33.6 10 ( KI) in / cycledN= × − Δ ksiin

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Example- edge cracked stripProblem: Given an initial flaw size, a 0 , predict the numberof cycles to grow the crack to a length a fΔKda= C(1.12Δσπa)dNThis equation can then be integrated∫I= 1.12Δσπada N= C(1.12σ )n / 2a∫ Δ0a f n n / 2πa0The result is1−n/ 2 1−n/ 2af− a0N =(1 − n / 2) C(1.12Δσ)Data: for (PMMA)nnπdNn / 2C=2.4x10 -24 , n=6.64, a 0 =0.010in, Δσ=4000psi,K Ic =1000psi-in 1/2

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008What is the critical crack size? (this will be a f )KacIc= 1.12σmaxπa1 ⎛ KIc⎞= 0.016inπ⎜ =1.12σ⎟⎝ ⎠2Then to grow the crack from 0.01in to 0.016 in is N=66cycles. The following table shows the effect of changingthe magnitude of the alternating stress.Δσ (psi) N (cycles)6000 4.55000 15.14000 663000 4482000 66111000 660,000500 65,000,000Note the very strong dependence on the alternating stresslevel. This is typical in fatigue. It means accurate stresslevels are needed as input for predictions of lifeMean Stress EffectsThe Paris law does not include the effect of mean stress, forexample the following two stress spectrums would result inthe same crack growth rate,

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008It is reasonable to expect more rapid crack growth for thecase when R=0.67.Walker’s lawnda C( ΔKI)= for -1 ≤ R < 1mdN (1 − R)• accelerates crack growth for high R ratiosForman’s law( ΔK)nda CI= for -1 ≤ R < 1dN (1 −R)K −Δ KIc• accelerates crack growth for high R ratios and asK I →K Ic• threshold effects are not included• mean stress effects are usually importantI

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Variable Amplitude Loading:So far we have addressed constant amplitude loading. Morerealistic stress spectrums would appear as followsThe methodology for predicting crack growth due to aspectrum like this would be to break it down into a series ofconstant amplitude “transients”

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Boeing Wing Skin Fatigue Example:• We have strain-gage data (converted to stress) for alocation near a rivet hole on the underside of the wing• We will assume the initial flaw size is 1/8 in.• We will estimate the number of flights to failure( K ) 2.2−6da 3× 10 ΔI=dN (1 −R)K −ΔKKIc= 70ksi inIcIin/cycleWe can use case 19.1 from the SACH for the SIF.⎛ a ⎞KI= σnomπaF0 ⎜ ⎟=1.45σnomπa⎝R+a⎠Δ K = 1.45ΔσπaInomIn order to proceed, make the strong assumption that da/dNin the first flight remains constant in all future flights(discuss implications of this)Δ K = 1.45 Δ σ π(0.125) = 0.91ΔσInomnomAccumulate da/dN for each transient during a typical flightΔ a =−63 10 0.91( σ ) 2.2× Δnom(1 −R)(70) −0.91ΔσnomN

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Transient σ max σ min Δσ nom R N ΔK I ΔaT-off/land 46 0 46 0 1 42 3.94x10 -4Climb 15 o 46 20 26 0.44 2 24 4.06Climb/cruise 40 24 16 0.6 3 15 2.43Cruise 36 24 12 0.67 17 11 7.99Descent 42 18 24 0.43 3 22 4.41Transition 44 18 26 0.41 1 24 1.79Approach 44 28 16 0.64 2 15 2.04Total: 0.0027inThis is the increase in crack length in the first flight. Tocalculate the total number of flights to failure, we need thecritical crack size21 ⎛ K Ic⎞ 1cr= ⎜ ⎟ 2π σmax F0a⎝where σmax⎠= 46ksiIt is necessary to iterate to find the critical crack sizebecause F 0 depends on a.a cr F 00.35 (guess) 1.20.5 1.10.61 1.080.63 close enough

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The number of flights to failure is thenN f0.63in − 0.125in= = 187flights0.0027in/flight• This is not conservative because it does not take intoaccount an increasing K I as a increases.• Updating the SIF after each flight only gives N f =60flights• Updating the SIF after each transient only gives N f =59flights

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008HIP Failure Fatigue Example:• Why did the vessel fail in 1898 cycles or 507 cyclesafter wet-wall cooling was installed?• Fatigue crack propagation governed by da/dN=C(ΔK I ) n• Integrate backwards from a cr , predict a i consistent witha given number of cycles• Results:• Air environment: a i =0.446-0.776in for 1898cycles• Hot water corrosive environment: a i =0.042-0.081in for 507 cycles (consistent with observedpit sizes)

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20083.) Effect of overloads, retardation modelsProblematic load sequence:• Tension overloads (shown) tend to produce retardationof crack growth or crack arrest. Tensile overloadsleave compressive residual stress ahead of the cracktip• Compressive overloads have the opposite effect, i.e.acceleration of crack growthThere are several models for the retardation effect; one ofthe popular ones is Wheeler’s (Trans. ASME J. BasicEngr., Vol. 94, 1972) model.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008wherea0dyaidy=crack length when overload occurs0= plastic zone induced by overload (Irwin calc)= instantaneous crack length after overload (varies)i= plastic zone induced cyclic stresses following overload (Irwin calc)δ = distance from instantaneous crack tip to edge of overload plastic zoneThe crack growth rate law while the crack tip is imbeddedin the overload plastic zone is⎛ da ⎞ ⎛dy i⎞ ⎛ da ⎞⎜ ⎟ = ⎜ ⎟ ⎜ ⎟⎝dN⎠retarded⎝ δ ⎠ ⎝dN⎠mFor high strength steels, m=1.3 and for titanium alloysm=3.4.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Another model is Elber’s (see ASTM STP 486, 1971),which employs the concept of an opening stress, σ op , astress level that must be overcome before additional crackgrowth occurs. After an overload, the opening stress isgreater than 0. ThenΔ σ = σmax−σminΔ σeff = σmax − σopempiricalThus, effectively, this acts to reduce ΔK I

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20084.) Cycle countingThere are several algorithms for converting a variableamplitude load spectrum to a series of constant amplitudetransients. A popular one is called the rainflow method.Assuming we have a series of max-min (peak-valley) pairsin sequence the rules are:1.) Start water drop from each peak and valley insuccession from left to right2.) Flow from a peak will stop if it sees a higher peak thanthe one from it started, similar for deeper valleys3.) Flow will stop to avoid collision with an earlier drop4.) Match up min-max and max-min pairs, these are thecycles

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Min-Max Pairs σ max σ min NAH/HA 30 -36 1CB/BC 18 -18 1EF/FE 3 -12 1GD/DG 12 -30 1

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008From Fuchs and Stephens, Metal Fatigue in Engineering, Wiley, 1980.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008V. Introduction to Elastic Plastic Fracture Mechanics1.) Introduction to Plasticity TheoryLinear Elastic Fracture Mechanics (LEFM) assumes alinear relationship between stress and strainCase i.) linear elastic material responseThe are alternative models of nonlinear response

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Case ii.) nonlinear elastic responseCase iii.) Elastic-plastic responsewhereε = total strainEε = elastic strainPε = plastic strain

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Elastic Plastic Fracture Mechanics (EPFM), for the mostpart, is based on Case ii.), assuming the structure is in aloading mode only, no unloading. A Ramberg-Osgoodstress-strain curve is often employed.• α, n, σ ys , and ε ys are constants used to fitexperimental stress-strain dataNow we need to have a stress-strain model that accountsfor multiaxial stresses in the vicinity of the crack tip (e.g.σ xx , σ yy , σ xy , etc. Then,

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008ε = ε + εijEijPijelasticplasticn−11+ν ν 3 ⎛ σ ⎞ sij= σij − δijσkk+ αεysE E 2 ⎜σ ⎟ ⎝ ys ⎠ σyswheres= σ −δσij ij ij kkσ =3ssij2ij/ 3 (deviatoric stress)(effective stress)The so-called HRR stress distribution at a crack tipassuming nonlinear stress-strain response is⎛n 1J ⎞ +σ ij= σ ij ( θ)⎜αεysσysIr⎟⎝n ⎠1

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Compare with LEFM,σ KI EJ( ) ij= fijfij( )2 rθ =π2πrθEPFM,nn1= 1: σij∼ (same as LEFM)r>> 1: σ ∼ constant (no singularity at all)ijRemark:• No K I in EPFM, the J integral assumes this role.

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 20082.) Practical Applications- use of the Ductile FractureHandbookThis handbook (three 2in thick 3-ring binders) provides atabulation of EPFM solutions for a wide range of crackconfigurations and loadings for commonly encounteredgeometry• Compact tension specimen• Center cracked panel• Single edge notched specimen (tension andbending)• Double edge notched specimen in tension• Internally pressurized cylinder with an axial crack• Cylinder under tensile load with an internalcircumferential crackFor these cases several quantities of interest are available:limit load P 0 , J integral, crack mouth openingdisplacement, load line displacement, crack tip openingdisplacement

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008The failure criterion in EPFM is a two-step one:Onset of crack extensionJ > J = JContinued unstable crack growthIcRdJdadJ>daR

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Example: circumferentially cracked cylinderData: Geometry- R i =90in, R o =99in., b=9in.,2L=180in.(length), a 0 =2.25in.Material- 304SS, E=30x103ksi, ν=0.3, σ ys =30ksi,α=1.69, n=5.42, ε ys =0.001

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J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Elastic J e :Jewhere2( 1 ν )2I∞ aeRiKI= σ πaeF( , ) F tabulatedb RP = π R −Rye2 2(o i)a = a+φrr1 n −1⎛K ⎞= ⎜ ⎟βπ nI+ 1⎜σ⎟ ⎝ ys ⎠1φ =2⎛ P ⎞1+ ⎜ ⎟⎝Po⎠2P = σ πR −R3wherey2 2o ys o cR = R + ac= −iKEσ∞2oElastic J p :a ⎛J c h⎝Pp= ασysεys 1 ⎜ ⎟b Po⎞⎠n+1

J W Eischen, Fatigue and Fracture Mechanics Short Course Notes, © February 2008Total J:eJ = J + JpRemarks:• For σ ∞ =40ksi, crack size would increase to 2.4in andarrest• For σ ∞ =44ksi, crack would be unstable