FatigueBy 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 materialCrack 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 cycleFracture�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 processStage I: Initiation/nucleationStage II:Stable growthStage 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 stressesd) Optimizing geometry - avoid internal corners, notches etc.

RangeFactorIntensityStressKKK

CyclePerRateGrowthCrackNa

MinMax ___

____

=−=Δ

=δδ

Stage II Power law regime (Paris law); influence of microstructure, R, environment, not as strong as for Stage I.

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• “Hard”surfaces• Residual compressive stresses (a compressive stress helps to keep a crack closed)

• Rough surfaces (deep scratches, dents…)• Stress concentrations• Corrosive environments

Stage III

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)

Similar to failure under static mode (cleavage, microvoid coalescence, etc). Microstructure, R, important; environment not so important

σmax=maximum stress in the cycleσmin=minimum stress in the cycleσmean=mean stressσa=alternating stress amplitudeΔσ=range of stress R=stress ratio

Max

Min

MinMax

MinMaxa

MinMaxMean

Min

Max

Rσσ

σσσ

σσσ

σσσ

σσ

=

−=Δ

−=

+=

2

2

The fatigue strength (Sf) initially starts at a value of Sut at N=0 and declines logarithmically with increasing cyclesIn some materials at 106–107cycles, the S-N diagram plateaus and the fatigue strength remains constantthis 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.

S-N Diagram

Fatigue data is highly variable and must be described in an statistical manner. Fatigue failure is an statistical event.

104 105 106 107

N

S

The S-N Curves are really showing 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)

• Low-cycle fatigue (LCF) less than 1000 cycles• High-cycle fatigue (HCF) more than 1000 cycles

Fatigue Regimes

High Cycle Fatigue Failure of a transmission shaft

Crack origin

(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

Stress-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

Strain-Life Approach

Service Equipment, e.g., automobiles

When the cyclic load level varies during the fatigue process, a cumulative damage model is often hypothesized. To illustrate, take the lifetime to be N1 cycles at a stress level 1 and N2 at 2. If damage is assumed to accumulate at a constant rate during fatigue and a number of cycles n1 is applied at stress 1, where n1 < N1 , then the fraction of lifetime consumed will be

Miner's law for cumulative damage

1

1

Nn

12

2

1

1 =+Nn

NnTo determine how many additional cycles the specimen

will survive at stress 2, an additional fraction of life will be available such that the sum of the two fractions equals one:

Note that absolute cycles and not log cycles are used here. Solving for the remaining cycles permissible at 2: ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

1

122 1

NnNn

The generalization of this approach is called Miner's Law, and can be written :where nj is the number of cycles applied at a load corresponding to a lifetime of Nj .

1=ΣJ

j

Nn

Example 1Consider 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 n1 = 105 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.

SolutionFrom the S-N relationship, we know the lifetime at S = 0.6σf = constantwould be N1 = 398107 and the lifetime at S = 0.7σf = constant would be N2 = 15849.

118683981071000001158491

1

122 =⎟

⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

NnNn

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. Nf=107.

• *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 polymersNo endurance limit– Aluminum– Magnesium– Copper– Nickel– Some stainless steels– Some High strength carbon and alloy steels

For SteelsFor steels with an ultimate strength greater than 200 kpsi, endurance does not increase so we just set a limit at 50% of 200kpsi, i. e., Se’= 100 kpsi.Other factors

Crack GrowthFatigue cracks nucleate and grow when stresses vary. The stress intensity factor under static stress is given by:

For a stress range, the stress intensity range per cycle is: aYKI πσ=

( ) aYaYK MinMaxI πσπσσ Δ=−=Δ

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 PropagationThree stages of crack growth, I, II and III.Stage I: Crack Initiation: transition to a finite

crack growth rate from no propagation below a threshold value of ∆K.

Stage II: Crack Propagation, “power law”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 da/dN

Log ∆K∆Kth

∆Kc

III

III

For Stage II:( )mIKC

Na

Δ=δδ

Paris Equation: Where C and m are empirical constants

Combined Mean and Alternating Stresses

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 σmby the ultimate tensile strength Sut of the material.

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 presence of a mean-stress component has a significant effect on failure.

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 CriteriaSimilar 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 oryielding.

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´andmean σm,o´ stressapply 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 MethodTo 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.

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:

Fatigue Strength and Endurance Limit

�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 specimenand the actual part being analyzed:

• 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• Basquin eq: σa = σf’(Nf)b

• Endurance limit: 107 cycles (no failures

Stress-Life Method

Application of Correction Factors1. 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�for rotating cylindrical parts:

�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:

097.0

097.0

189.1.........:2508_

869.0.........:.103.0_

1.....:)8_(3.0_

−

−

=≤≤

=≤≤

=≤

dkmmdmmfor

dkindinfor

kmmindfor

size

size

size

�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 Kloadfor 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

Celsius

Fahrenheit

( )

( )8400032.011020840

0.1840

4500058.01550450

0.1450

−−=⇒<<

=⇒≤

−−=⇒<<

=⇒≤

TkT

kT

TkT

kT

Temp

Temp

Temp

Temp

kreliabilitycollected 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: