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
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 Growth ____ CyclePerRate δN
Max KKK Min =−=Δ Stress Intensity Factor ___ Range
A and m are parameters that depend on the material 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) =mean stress =maximum stress in the cycle =minimum stress in the cycle =range of stress =alternating stress amplitude max min mean a
σ σ σ σ Δσ R=stress ratio σ Max
σ Min +σσ σ = Max Min Mean 2 −σσ σ = Max Min a 2
Max −=Δ σσσMin 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: Nn 22 ⎜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 ⎟ Nn 22 ⎜1−= ⎟ ⎜115849 −= ⎟ =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: I = πσaYK For a stress range, the stress intensity range per cycle is:
I YK ()Max −=Δ Min Δ= πσπσσaYa Cracks grow as a function of the number of stress cycles (N), stress range (ΔσI ) and stress intensity factor range (ΔKI ). For a ΔKI below some threshold value (ΔKI)threshold a crack will not grow. Fatigue Crack Propagation Log da/dN Three stages of crack growth, I, II and III. Stage I: Crack Initiation: transition to a finite I crack growth rate from no propagation below a threshold value of ∆K. II ∆Kc Stage II: Crack Propagation, “power law” III dependence of crack growth rate on ∆K. This is linear in log-log coordinates. Stage III: Crack Unstable, acceleration of growth rate with ∆K, approaching catastrophic fracture. Log ∆K ∆Kth
For Stage II: δa m Paris Equation: Where C and m are ()Δ= KC I empirical constants δN Combined Mean and Alternating Stresses The presence of a mean-stress component has a significant effect on failure. When a mean component of stress is added to the alternating component, (b) and (c) the material fails at lower alternating stresses than it does under fully reverse loading. The plots are normalized by dividing the alternating stress σa by the fatigue strength Sf of the material under fully reversed stress (at the same number of cycles) and dividing the mean stress σm by the ultimate tensile strength Sut of the material. A parabola that intercepts 1 on each axis is called the Geber Line. A straight line connecting 1 on each axis is called the Goodman line The Goodman line is often used as a design criterion, since it is more conservative than the Geber line.
Fatigue Failure Criteria Similar to the static failure analysis, a failure envelope is constructed using the mean and amplitude stress components.
Under pure alternating stress (i.e. σa only) the part should fail at Se (or Sf) whereas, under pure static stress (i.e. σm only) the part should fail at Sut. Thus, the failure envelope is constructed on a σa-σm plot by connecting Se (or Sf) on the σa-axis with Sut on the σm-axis: The two most common failure criteria.
Both of these are used in conjunction with the Langer first-cycle yield criterion: If we replace the strengths Sa and Sm with the stresses nσa and nσm (where n is the factor of safety), the factor of safety can be solved for: General Solution Procedure:
determine the fully corrected endurance (or fatigue) limit Se (or Sf) determine nominal stresses σa,o and σm,o at the site of interest apply stress concentrations Kf and Kfm to determine σa and σm calculate the factor of safety against fatigue (nf) calculate the factor of safety against first-cycle yield (ny) determine whether the part is at risk for failure by fatigue or yielding. Combination of Loading Modes: Assuming that all of the loading modes are in-phase with one another:
use the fully corrected endurance (or fatigue) limit for bending
multiple any alternating axial loads by the factor 1/kload,axial •do not have to adjust torsion loads since this is taken care of when determining the von Mises effective stress determine the principal stresses at the site of interest
determine the nominal von Mises alternating stress σa,o´and mean σm,o´ stress apply the fatigue stress concentration factors Kf and Kfm •use the product of the stress concentration factors if more than one are present at the site of interest
calculate the factor of safety (nf or ny) as before Stress-Life Method To determine the strength of materials under the action of fatigue loads, specimens are subjected to repeated or varying forces of specified magnitudes while the cycles or stress reversals are counted to destruction. S-N Diagram The ordinate of the S-N diagram is called the fatigue strength. Fatigue Strength and Endurance Limit The fatigue strength (Sf´) and the endurance limit (Se´) for some materials can be found (refer to text appendices) or can be estimated from the following relations:
the fatigue strength or endurance limit are typically determined from the standard material tests (e.g. rotating beam test) however, they must be appropriately modified to account for the physical and environmental differences between the test specimen and the actual part being analyzed: Stress-Life Method
• In fatigue testing, the applied stress, σa, is typically described by the stress amplitude of the loading cycle and is defined as:
• σa = (σmax - σmin )/2 = Δσ/2 • The stress amplitude is generally plotted against the number of cycles to failure on a linear-log scale. S-N plots • Tests performed on unnotched specimens • Constant amplitude
• Cycles to failure (Nf) monitored for each stress amplitude level (S) • Plotted linear-log b • Basquin eq: σa = σf’(Nf) • Endurance limit: 107 cycles (no failures Application of Correction Factors 1. Loading Effects: The tests are conducted on a specimen that is in pure bending. Only the outer fibers see the full magnitude of the stress. 2. Components that are loaded axially will have all their fibers see this maximum stress, therefore, we should adjust the fatigue strength to reflect this condition.
Surface Factor (ksurface) Rotating beam specimens are polished to avoid additional stress concentrations and thus rougher surfaces need to be accounted for:
Size Factor (ksize) rotating beam specimens are small and larger diameter beam tend to fail at lower stresses due to the increased probability of the material containing microscopic flaws ≤ kmmindfor size =1.....:)8_(3.0_ for rotating cylindrical parts: − 097.0 ≤≤ ...... :.103.0_ kindinfor size = 869.0 d − 097.0 8_ ≤≤ ...... :250 kmmdmmfor size = 189.1 d
for non-rotating parts, an equivalent diameter obtained by equating the volume of material stressed above 95% of the maximum stress to the same volume in a rotating beam specimen: and then the previous set of equations can be used to calculate ksize
for axial loading, there is no size effect Load Factor (kload) fatigue tests are carried using rotating bending tests and thus a strength reduction factor is required for other modes of cyclic loading: NOTE: If one uses von Mises effective stresses, thus adjusting for shear vs. normal stresses Kload for torsion is 1. ktemperature standardized tests are conducted at room temperature and higher temperatures tend to cause a reduction in Sy making crack propagation easier
two types of problems arise when temperature is a consideration: i) if Sf´or Se´ is known (i.e. from tables), use: ii) if Sf´ or Se´ were estimated (from previous relations), temperature correct the tensile strength of material (using table 7-6) before estimating Sf´or Se´and then use:
Or T ≤ 450 ⇒ k = 0.1 Celsius Temp
450 550 kT Temp −=⇒<< 0058.01 ()T − 450
Fahrenheit 840 kT Temp =⇒≤ 0.1
840 T 1020 kTemp −=⇒<< 0032.01 ()T −840 kreliability collected data always has some variability associated with it and depending on how reliable one wishes that the samples met (or exceeded) the assumed strength, we use the following correction factor: