Fatigue By far the majority of engineering design projects involve machine parts subjected to fluctuating or cyclic loads. Such loading induces fluctuating or cyclic stresses that often results in failure by fatigue. There are two domains of cyclic stresses (two different mechanisms): Low-Cycle fatigue: Domain associated with high loads and short service life. Significant plastic strain occurs during each cycle. Low number of cycles to produce failure. 1<N<103 High-cycle Fatigue: Domain associated with low loads and long service life. Strains are mostly confined to the elastics range. High number of cycles to produce fatigue failure. N>103 Fatigue is a progressive failure phenomena associated with the initiation and propagation of cracks to an unstable size. When the crack reaches a critical dimension, one additional cycle causes sudden failure. From a designer point of view, fatigue can be a particularly dangerous form of failure because: it occurs over time and it occurs at stresses levels that are not only lower than the UTS but they can be lower than the Yield Strength. There are three stages of fatigue failure: •Crack initiation •Crack propagation and •Fracture due to unstable crack growth. Crack Initiation (Ductile Materials) under cyclic loading, that contains a tensile component, localized yielding can occur at a stress concentration even though the nominal stresses are below σy this distorts the material and creates slip (or shear) bands (localized regions of intense deformation due to shearing) as the stress cycles, additional slip bands are created and coalesce into microcracks this mechanism dominates as long as σy is exceeded somewhere in the material Crack Initiation (Brittle Materials) materials that are less ductile, do not have the same ability to yield and thus form cracks more easily (i.e. notch-sensitive) most brittle materials completely skip this stage and proceed directly to crack propagation at sites of pre-existing flaws (e.g. voids, inclusions). Crack Propagation a large stress concentration is developed around the crack tip and each time the stress becomes tensile the crack grows a small amount when the stress becomes compressive, zero or to a lower tensile state, the growth of the crack stops (momentarily) this process will continue as long as the stresses at the crack tip cycle below and above the σy of the material crack growth is due to TENSILE stresses and grows along planes normal to the maximum tensile stress cycle stresses that are always compressive will not elicit crack propagation the rate of crack growth is very small (10-9to 10-5mm/cycle) but after numerous cycles the crack can become quite large If the fracture surface is viewed at high magnification, striations can be observed due to each stress cycle Fracture cracks will continue to grow if tensile stresses are high enough and at some point, the crack becomes so large that sudden failure occurs patterns can be seen on the fracture surface which indicate that failure was due to fatigue. Typical fatigue fracture surface Each clamshell marking might represent hundreds or thousands of cycles. Stages I, II, and III of fatigue fracture process Stage I: Initiation/nucleation Stage II: Stable growth Stage III: Final Fracture Stage I •Cracks can initiate internally or externally (most often); surface treatment important, especially for high cycle fatigue. •Average crack growth can be less than lattice spacing. •microstructure, R, environment have big effects. •Plastic zone smaller than grain size Persistent slip bands (Suresh, Ch 4) Factors that affect fatigue life Magnitude of stress (mean, amplitude...) Quality of the surface (scratches, sharp transitions and edges). Solutions: a) Polishing (removes machining flaws etc.) b) Introducing compressive stresses (compensate for applied tensile stresses) into thin surface layer by “Shot Peening”- firing small shot into surface to be treated. Ion implantation, laser peening. c) Case Hardening - create C- or N- rich outer layer in steels by atomic diffusion from the surface. Makes harder outer layer and also introduces compressive stresses d) Optimizing geometry - avoid internal corners, notches etc. Stage II Power law regime (Paris law); influence of microstructure, R, environment, not as strong as for Stage I. δa = Crack____GrowthRate Per Cycle δN ΔKKK =Max −Min Stress =___Intensity Factor Range A and m ers that depend on the material are paramet environment, frequency, temperature, stress ratio. Factors in Fatigue Life • Fatigue failure is controlled by “how difficult it is to start and propagate a crack” (Stage I and II). • Anything that makes this process easier will reduce a components fatigue life. Good Things Bad Things • Smooth surfaces • Rough surfaces (deep scratches, • “Hard”surfaces dents…) • Residual compressive stresses (a • Stress concentrations compressive stress helps to keep a • Corrosive environments crack closed) Stage III Similar to failure under static mode (cleavage, microvoid coalescence, etc). Microstructure, R, important; environment not so important As the crack grows, and if the plastic zone size becomes comparable to the specimen thickness (provided fracture doesn’t take place earlier), the crack can begin to reorient itself 45° to the tensile stress axis (plane stress conditions) σmax=maximum stress in the cycle σmin=minimum stress in the cycle σmean=mean stress σa=alternating stress amplitude Δσ=range of stress R=stress ratio σ Max σ Min σ+ σ σ = Max Min Mean 2 σ− σ σ = Max Min a 2 Δσ = σMax −Min σ σ R = Min σ Max S-N Diagram The fatigue strength (Sf) initially starts at a value of Sut at N=0 and declines logarithmically with increasing cycles In some materials at 106–107cycles, the S-N diagram plateaus and the fatigue strength remains constant this plateau is called the endurance limit (Se) and is very important since stresses below this limit can be cycled indefinitely without causing a fatigue failure. Fatigue data is highly variable and must be described in an statistical manner. Fatigue failure is an statistical event. S The S-N Curves are really showing 104 105 106 107 N the probability of failure. Fatigue Failure Mode or Fatigue-Life Methods •Stress-Life (S-N) •Strain-Life (e-N) •Linear Elastic Fracture Mechanics Approach (LEFM) Fatigue Regimes • Low-cycle fatigue (LCF) less than 1000 cycles • High-cycle fatigue (HCF) more than 1000 cycles Crack origin High Cycle Fatigue Failure of a transmission shaft Stress-Life Approach (a) Load amplitudes are predictable and consistent over the life of the part (b) Stress-based model - determine the fatigue strength and/or endurance limit (c) Keep the cyclic stress below the limit Strain-Life Approach (a) Gives a reasonably accurate picture of the crack-initiation stage (b) Accounts for cumulative damage due to variations in the cyclic load (c) Combinations of fatigue loading and high temperature are better handled by this method (d) LCF, finite-life problems where stresses are high enough to cause local yielding (e) Most complicated to use Service Equipment, e.g., automobiles Miner's law for cumulative damage When the cyclic load level varies during the fatigue process, a cumulative damage model is often hypothesized. To illustrate, n take the lifetime to be N1 cycles at a stress level 1 and N2 at 2. 1 If damage is assumed to accumulate at a constant rate during N 1 fatigue and a number of cycles n1 is applied at stress 1, where n1 < N1 , then the fraction of lifetime consumed will be To determine how many additional cycles the specimen n1 n2 will survive at stress 2, an additional fraction of life will + =1 be available such that the sum of the two fractions equals N1 N2 one: Note that absolute cycles and not log cycles are used ⎛ n ⎞ ⎜ 1 ⎟ here. Solving for the remaining cycles permissible at 2:n2= N 2⎜1 − ⎟ ⎝ N1 ⎠ The generalization of this approach is called Miner's Law, n j and can be written : Σ =1 N where nj is the number of cycles applied at a load J corresponding to a lifetime of Nj . Example 1 Consider a hypothetical material in which the S-N curve is linear from a value equal to the fracture stress σf at one cycle (log N = 0), falling to a value of σf /2 at log N = 7 as shown. This behavior can be described by the equation The material has been subjected to 5 n1 = 10 load cycles at a level S = 0.6σf, and we wish to estimate how many cycles n2 the material can now withstand if we raise the load to S = 0.7σf. Solution From the S-N relationship, we know the lifetime at S = 0.6σf = constant would be N1 = 398107 and the lifetime at S = 0.7σf = constant would be N2 = 15849. ⎛ n ⎞ ⎛ 100000 ⎞ ⎜ 1 ⎟ n2= N 2⎜1 − ⎟=15849⎜ − 1 ⎟ =11868 ⎝ N1 ⎠ ⎝ 398107 ⎠ Design Philosophy: Damage Tolerant Design • S-N (stress-cycles) curves = basic characterization. • Old Design Philosophy = Infinite Life design: accept empirical information about fatigue life (S-N curves); apply a (large!) safety factor; retire components or assemblies at the pre-set life limit, e.g. 7 Nf=10 . • *Crack Growth Rate characterization -> • *Modern Design Philosophy (Air Force, not Navy carriers!) = Damage Tolerant design: accept presence of cracks in components. Determine life based on prediction of crack growth rate. Endurance Limit – Low strength carbon and alloy steel – Some stainless steels, irons, – Titanium alloys – Some polymers For Steels No endurance limit For steels with an ultimate strength greater –Aluminum than 200 kpsi, endurance does not increase so –Magnesium we just set a limit at 50% of 200kpsi, i. e., Se’ – Copper = 100 kpsi. –Nickel Other factors – Some stainless steels – Some High strength carbon and alloy steels Crack Growth Fatigue cracks nucleate and grow when stresses vary. The stress intensity factor under static stress is given by: KI = σ Y π a For a stress range, the stress intensity range per cycle is: ΔKYσI =()Max σ −Min a π= Y Δ σ a π Cracks grow as a function of the number of stress cycles (N), stress range (ΔσI ) and stress intensity factor range (ΔKI ).