MD-6A

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Fatigue

By far the majority of engineering design projects involve machineparts 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 differentmechanisms):

Low-Cycle fatigue: Domain associated with high loads and short

service life. Significant plastic strain occurs during each cycle. Lownumber of cycles to produce failure. 1

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Fatigue is a progressive failure

phenomena associated with the

initiation and propagation ofcracks to an unstable size.

When the crack reaches a critical

dimension, one additional cyclecauses 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 thatare not only lower than the UTS

but they can be lower than the

Yield Strength.

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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 canoccur at a stress concentration even though the nominal stresses are below y 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 y 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 formcracks 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).

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Crack Propagation

a large stress concentration is developed around thecrack 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 y of the materialcrack growth is due to TENSILE stresses and grows

along planes normal to the maximum tensile stress

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cycle stresses that are always compressive will

not elicit crack propagation

the rate of crack growth is very small (10 -9to

10-5

mm/cycle) but after numerous cycles thecrack 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.

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Typical fatigue

fracture surface

Each clamshell marking might

represent hundreds or

thousands of cycles.

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Stages I, II, and III of fatigue fracture process

Stage I:Initiation/nucleation

Stage II:

Stable growthStage III:

Final Fracture

Stage I

Cracks can initiate internally or externally (most often); surfacetreatment 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

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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).

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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.

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RangeFactorIntensityStressKKK

CyclePerRateGrowthCrackN

a

MinMax ___

____

==

=

Stage II Power law regime (Paris law);influence of microstructure,R,

environment, not as strong as forStage I.

A and m are parameters that depend on the material

environment, frequency, temperature, stress ratio.

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Factors in Fatigue Life

Fatigue failure is controlled by how difficult it is to start andpropagate a crack (Stage I and II).

Anything that makes this process easier will reduce a components

fatigue life.

Good Things Bad Things

Smooth surfaces Hardsurfaces

Residual compressive stresses (a

compressive stress helps to keep acrack closed)

Rough surfaces (deep scratches,dents)

Stress concentrations

Corrosive environments

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Stage III

As the crack grows, and if theplastic zone size becomes

comparable to the specimen

thickness (provided fracturedoesnt take place earlier), the

crack can begin to reorient

itself 45 to the tensile stress

axis (plane stress conditions)

Similar to failure understatic mode (cleavage,

microvoid coalescence, etc).

Microstructure,R,important; environment not

so important

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max=maximum stress in the cycle

min=minimum stress in the cycle

mean=mean stress

a=alternating stress amplitude

=range of stress

R=stress ratio

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Max

Min

MinMax

MinMaxa

MinMaxMean

Min

Max

R

=

=

=

+=

2

2

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The fatigue strength (Sf) initially

starts at a value ofSutatN=0 anddeclines logarithmically with

increasing cycles

In some materials at 106107cycles,theS-Ndiagramplateaus and the

fatigue strength remains constant

this plateau is called the endurancelimit (Se) and is very important since

stresses below this limit can be

cycled indefinitely without causing a

fatigue failure.

S-N Diagram

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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 showingthe probability of failure.

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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

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(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 causelocal yielding

(e) Most complicated to use

Strain-Life Approach

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Service Equipment, e.g., automobiles

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When the cyclic load level varies during the fatigue process, a

cumulative damage model is often hypothesized. To illustrate,

take the lifetime to beN1 cycles at a stress level 1 andN2 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, wheren1 < N1 , then the fraction of lifetime consumed will be

Miner's law for cumulative damage

1

1

N

n

12

2

1

1=+

Nn

NnTo determine how many additional cycles the specimenwill 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

N

nNn

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The generalization of this approach is calledMiner's Law,

and can be written :

where njis the number of cycles applied at a loadcorresponding to a lifetime ofNj .

1=J

j

N

n

Example 1

Consider a hypothetical material in which the S-N curve is linear from a

value equal to the fracture stress f at one cycle (log N = 0), falling to a

value of f/2 at log N = 7as shown. This behavior can be described by

the equation

The material has been subjected to

n1 = 105 load cycles at a levelS =

0.6f, and we wish to estimate how

many cycles n2 the material can

now withstand if we raise the loadtoS = 0.7f.

Solution

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Solution

From the S-N relationship, we know the lifetime atS = 0.6f= constant

would beN1 = 398107and the lifetime atS = 0.7f= constantwould beN2 = 15849.

11868398107

1000001158491

1

122 =

=

= N

nNn

Design Philosophy: Damage Tolerant Design

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Design Philosophy: Damage Tolerant Design

S-N (stress-cycles) curves = basic characterization.

Old Design Philosophy =Infinite Life design: accept empiricalinformation 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.

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Endurance Limit

Low strength carbon and alloy steel

Some stainless steels, irons, Titanium alloys

Some polymers

No endurance limit

Aluminum Magnesium

Copper

Nickel

Some stainless steels Some High strength carbon and alloy steels

For Steels

For steels with an ultimate strength greater

than 200 kpsi, endurance does not increase sowe just set a limit at 50% of 200kpsi, i. e., Se

= 100 kpsi.

Other factors

Crack Growth

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Crack Growth

Fatigue 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 =

( ) aYaYK MinMaxI ==Cracks grow as a function of the number of stress cycles (N), stress

range ( I) and stress intensity factor range (KI). For a KI belowsome threshold value (KI)threshold a crack will not grow.

Fatigue Crack Propagation Log da/dN

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Fatigue Crack Propagation

Three stages of crack growth, I, II and III.

Stage I: Crack Initiation: transition to a finitecrack growth rate from no propagationbelow a threshold value of K.

Stage II: Crack Propagation, power lawdependence of crack growth rate on K.This is linear in log-log coordinates.

Stage III: Crack Unstable, acceleration of

growth rate with K, approachingcatastrophic fracture.

Log da/dN

Log Kth

KcI

II

III

For Stage II:

( )mI

KCN

a=

Paris Equation: Where C and m are

empirical constants

Combined Mean and

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Alternating Stresses

The plots are normalized by dividing the

alternating stress a by the fatigue

strengthSfof the material under fullyreversed stress (at the same number of

cycles) and dividing the mean stress mby the ultimate tensile strengthS

ut

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

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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. a only) the part should fail at

Se (orSf) whereas, under pure static stress (i.e. m only) the part

should fail atSut.Thus, the failure envelope is constructed on a a-m plot by

connectingSe (orSf) on the a-axis withSuton the m-axis:

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The two most common failure criteria.

Both of these are used in conjunction withthe Langer first-cycle yield criterion:

If l th t th S d S ith th t d

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If we replace the strengthsSa andSm with the stresses na and nm(where n is the factor of safety), the factor of safety can be solved

for:

G l S l i P d

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General Solution Procedure:

determine the fully corrected endurance (or fatigue) limit Se

(orSf) determine nominal stresses a,o and m,o at the site of interest

apply stress concentrations KfandKfm to determine a and

m 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:

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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,axialdo 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 a,oand

mean m,ostressapply the fatigue stress concentration factorsKfandKfmuse the product of the stress concentration factors if more than one are present

at the site of interest

calculate the factor of safety ( nfor ny) as before

St Lif M th d

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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.

Fatigue Strength and Endurance Limit

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The fatigue strength (Sf) and the endurance limit (Se) for some

materials can be found (refer to text appendices) or can be estimatedfrom the following relations:

g g

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

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, y pp p y

physical and environmental differences between the test specimen

and the actual part being analyzed:

Stress-Life Method

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In fatigue testing, the applied stress, a, is typically described bythe stress amplitude of the loading cycle and is defined as:

a = (max - min )/2 = /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:a = f(Nf)

b

Endurance limit: 107 cycles (no failures

Stress Life Method

Application of Correction Factors

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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 thestress.

2. Components that are loaded axially will have all their fibers see

this maximum stress, therefore, we should adjust the fatiguestrength to reflect this condition.

Surface Factor (ksurface)

Rotating beam specimens are polished to avoidadditional stress concentrations and thus rougher

surfaces need to be accounted for:

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Size Factor (ksize)

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rotating beam specimens are small and larger diameter beam tend to

fail at lower stresses due to the increased probability of the materialcontaining microscopic flaws

for rotating cylindrical parts:

for non-rotating parts, an equivalent diameter

obtained by equating the volume of material stressedabove 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

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for axial loading, there is no size effect

Load Factor (kload)

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( load) 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 stressesKloadfor torsion is 1.

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kreliabilityll t d d t l h i bilit i t d ith it d

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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: