MECH 5390 – Fatigue Analysis for Extreme Environments

Dr. Calvin M. Stewart

Department of Mechanical Engineering

The University of Texas at El Paso

The Stress-Life (S-N) Approach

• Fatigue Loading• Fatigue Test Machines• Test Specimen and Standards• The Stress-Life Approach• Endurance Limit• Effect of Mean Stress• Putting it all together

• Advantages and Limitations of S-N Approach• Design Considerations• Relevant Testing Standards• Questions

Fatigue Loading

Fatigue Loading

• Some load histories may be simple and repetitive, while in other cases they may be completely random.

• The randomness may contain substantial portions of more deterministic loading.

• For example, the ground-air-ground cycle of an aircraft has substantial similarity from flight to flight.

Fatigue Loading

Offshore Structures

Aircraft

Fatigue Loading

Ground-Air-Ground Cycle of Air Craft

Fatigue Loading

Typical Load Histories From Actual Ground Vehicle Components

Fatigue Loading

A Typical Load History of Short-Span Bridge

Fatigue Loading

• These load histories are typical of those found in real-life engineering situations.

• Fatigue from variable amplitude loading involving histories such as these is discussed in Chapter 9. Constant amplitude loading is introduced in this chapter.

• Constant amplitude loading is used:• To obtain material fatigue behavior/properties for use in fatigue design,• Some real-life load histories can occasionally be modeled as essentially

constant amplitude.

Sinusoidal Fluctuating Stress

Pulsating Tension

Completely Reversed (S-N Curve)

Fig. 6–23

Fluctuating stress with high frequency ripples

Nonsinusoidalfluctuating stress

Nonsinusoidalfluctuating stress

LaboratoryReal World

Fatigue Loading• Def’s:

– Stress range

– Stress amplitude

–Mean stress

– Stress ratio, R

– Amplitude ratio, A

max min

2a

max min

2m

max min

min

max

R

m

aA

Period

Frequency

1

f

1f

Fatigue Loading

• Fully Reversed: R = -1; A = ∞

• Zero to Tension: R = 0; A = 1

• Zero to Compression: R = ∞; A = -1

• R w.r.t A

• A w.r.t. R

1

1

AR

A

1

1

RA

R

Fatigue Loading

• R= -1 and R= 0 are two common reference test conditions used for obtaining fatigue properties.

• R= -1 is called the fully reversed condition since Smin= -Smax

• R = 0, where Smin = 0, is called pulsating tension.

• One cycle is the smallest segment of the stress versus time history which is repeated periodically.

• Under variable amplitude loading, the definition of one cycle is not clear and hence reversals of stress are often considered.

• In constant amplitude loading, one cycle equals two reversals.

Fatigue Loading

• Tensile and/or compressive mean loads and fully reversed loads are prevalent in all fields of engineering.

• Examples of different mean loadings.

• The transmission history indicates significant tensile mean stress.

• The suspension history shows significant compressive mean stress loading.

• The bracket history is dominated by essentially fully reversed, R= -1, loading.

Fatigue Loading

• A thin or thick-walled pressure vessel subjected to cyclic internal pressure represents a component subjected to mean tensile stresses.

• Helical compression springs are actually under torsion, but the applied cyclic forces involve compressive mean forces.

• A cantilever beam deflected at the free end and then released to vibrate represents a damped vibration with essentially zero mean stress.

Fatigue Test Machines

Fatigue Test Machines

• Rotating Cantilever Bending Fatigue Test Machine

Constant load amplitude Non-uniform bending moment along the specimen length

RR Moore Tester Close up

Fatigue Test Machines

• Rotating Bending Fatigue Test Machine

Constant load amplitude Uniform bending moment along the specimen length

Fatigue Test Machines

• Constant Deflection Amplitude Cantilever Bending

• Load amplitude changes with specimen cyclic hardening or softening and decreases as cracks in the specimen nucleate and grow.

• The eccentric crank test machines do have an advantage over the rotating bending test machines in that the mean deflection, and hence the initial mean stress, can be varied.

Fatigue Test Machines

• Axially Loaded Fatigue Test Machine

• Capable of applying both mean and alternating axial loads in tension and/or compression

Fatigue Test Machines

• A Test Setup For Combined In-phase Torsion and Bending With or Without Mean Stress

• Uniform torque and a non-uniform bending moment along the specimen length

Fatigue Test Machines

• A Modern Servo-hydraulic Test System

• Principle of operation includes: • generating an input signal of load, strain, or displacement

using a function generator,

• applying this input through a hydraulic actuator, measure the specimen response via a load cell, clip gage, or an LVDT,

• compare this output with the input. The difference drives the system.

• Control and test data outputs are usually through a PC and software.

• Test frequency can range from mHz to kHz.

Fatigue Test Machines

• These test systems can perform: • constant or variable amplitude load, strain, deformation, or stress intensity

factor controlled tests on small specimens or

• can be utilized with hydraulic jacks for components, subassemblies, or whole structures.

• Two or more control systems are used for multiaxial testing.

Fatigue Test Machines

• Biaxial – Tension Torsion Fatigue Rig

Fatigue Test Machines

• Pressurized Specimen

Fatigue Test Machines

• Biaxial Cruciform Testing

Fatigue Test Machines

Electromagnetic

(Linear Motor)Electro-Mechanical

(Screw)Servo-Hydraulic

Moderate Frequency

High Force

High Power

Dynamic

Low Frequency

Moderate Force

Moderate Power

Quasi-Static

High Frequency

Low Force

Low Power

Dynamic

Servo-Electric

Low ↔ Moderate Frequency

Low ↔ High Force

Low ↔ High Power

Static ↔ Dynamic

Fatigue Test Machines: Sensors

Linear Variable Differential Transformer

(LVDT)

Potential Drop Method

Clip gauge / ExtensometerLoad Cell

Digital Image Correlation

Strain Gauge

Fatigue Test Specimens

Fatigue Test Specimens

• These specimens are usually used for axial or bending tests.

• These specimens usually have finely polished surfaces to minimize surface roughness effects.

• No distinction between crack nucleation and growth is normally made with these specimens.

• Careful alignment is needed for axial loaded specimens to minimize bending.

Fatigue Test Specimens

• Stress concentration influence can be studied with most of these specimens by machining in notches, holes, or grooves.

Bridgman Notched Specimens and specific parameters

Fatigue Test Specimens

• A thin-walled tube specimen designed for torsion and combined axial/torsion with the possibility of adding internal and/or external pressure.

• This multiaxial loading can be performed in-phase or out-of-phase.

• The thin-walled tube allows for essentially uniform normal and shear stresses in the cross-sectional area.

Hollow specimen for Gigacyclefatigue tests

Fatigue Test Specimens

• Fatigue specimens come in many form factors.

• Sheet/plate specimens

Photograph of EBM Ti-6Al-4V flat dog bone specimen, with EBM growth and fatigue loading

directions indicated

Fatigue Test Specimens

• Specimens shown have been used for obtaining fatigue crack growth data.

• In all cases a thin slit, notch, or groove with a very small root radius is machined into the specimen.

• Fatigue crack growth testing is covered in Section 6.4.

Fatigue Test Specimens

Cruciform Specimen

Standards

• Standard• A set of specifications for parts, materials, or processes• Intended to achieve uniformity, efficiency, and a specified quality• Limits the multitude of variations

• Code• A set of specifications for the analysis, design, manufacture, and construction

of something• To achieve a specified degree of safety, efficiency, and performance or quality• Does not imply absolute safety

• Various organizations establish and publish standards and codes for common and/or critical industries

Standards

• ASTM International –American Society of Testing and Materials

• ISO - International Organization for Standardization

• BS – British Standards

• EN – European Standards

• ASME – American Society of Mechanical Engineers

• SAE – Society of Automotive Engineers

ASTM E399 -12E3

Organization Standard No. Publication Year Edition

Standards: ASTM

• E647 Measurement of Fatigue Crack Growth Rates.

• E739 Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life ( -N) Fatigue Data.

• E1012 Verification of Specimen Alignment Under Tensile Loading

• E1049 Cycle Counting in Fatigue Analysis.

• E1823 Standard Terminology Relating to Fatigue and Fracture Testing.

Standards: ISO

• ISO/DIS 12106 Metallic Materials-Fatigue Testing-Axial Strain-Controlled Method.

• ISO/DIS 12107 Metallic Materials-Fatigue Testing-Statistical Planning and Analysis of Data.

• ISO/DIS 12108 Metallic Materials-Fatigue Testing-Fatigue Crack Growth Method.

Stress-Life Approach

The Stress-Life (S-N) Approach

• The Stress-Life, S-N, method was the first approach used in an attempt to understand and quantify metal fatigue.

• It was the standard fatigue design method for almost 100 years.

• The S-N approach is still widely used in design applications where the applied stress is primarily within the elastic range of the material and the resultant lives (cycles to failure) are long, such as power transmission shaft.

• The Stress-Life method does not work well in low-cycle fatigue, where the applied strain have a significant plastic component.

• In low-cycle fatigue the Strain-Life method is more appropriate.

Stress-Life Approach: Typical Curve

Stress-Life Approach

• Typical schematic S-N curve obtained under axial load or stress control test conditions with smooth specimens.

• Constant amplitude S-N curves of this type are plotted on semi-log or log-log coordinates.

• S-N curves obtained under torsion or bending load-control test conditions often do not have data at the shorter fatigue lives (say 10^3 or 10^4 cycles and less) due to significant plastic deformation.

• Torsion and bending stress equations τ=T r / J and σ= M y /I can only be used for nominal elastic behavior.

Stress-Life Approach: Variability

• Typical variability with less variability at shorter lives and greater variability at longer lives.

• Variability in life for a given stress level can range from less than a factor of two to more than an order of magnitude.

• Variability and statistical aspects of fatigue data are discussed in Ch. 13.

Stress-Life Approach: Endurance Limit

• Fig. (a) shows a continuous sloping curve, while Fig. (b) shows a discontinuity or “knee” in the S-N curve which indicates an endurance limit.

• This endurance limit has been found in only a few materials (i.e. low and medium strength steels) between 10^6 and 10^7 cycles in non-corrosive conditions.

• Most materials do not contain the endurance limit even under controlled environments.

• Under corrosive environments all S-N data have a continuous sloping curve.

• When sufficient data are available, S-N curves are usually drawn through median points and thus represent 50 percent expected failures.

S-N Diagram for Steel in Laboratory Air

010 , utS

Stress-Life Approach: Definitions

• Definitions

• Cycles to Failure, Nf – the number of cycles a material (or a component) can resistance fatigue failure connoted by fatigue life

• Fatigue Strength, Sf – the value of fully reversed alternating stress, σa that corresponds to a particular number of Nf

• Endurance/Fatigue Limit, Se – a stress level below which a material can be said to have infinite life

• Low Cycle Fatigue (LCF) - when Nf is in the range of 100 to 103.

• High Cycle Fatigue (HCF) - when the number of Cycles to failure, Nf , is in the range of 103 to 106.

• Infinite Life - when Nf is equal to or greater than 106.

Stress-Life Approach: Endurance Limit

• Fatigue typically consists of crack nucleation, growth, and final fracture, as it was emphasized in Chapter 3.

• The Stress-Life Approach cannot distinguish the individual stages.

• It only gives information concerning final fracture.

Stress-Life Approach: Basquin’s Equation

• Basquin in 1910 suggest a log-log straight line S-N relationship

B

ATri-Slope Model

Basquin Model

B

f fS A N

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes

Basquin Equation

uA S

1 6 log e uB S S

How to Find A & B Constants

Stress-Life Approach

• The S-N diagram is applicable for completely reversed stresses

• Other fluctuating stresses exist!

• Sinusoidal loading patterns are common, but not necessary

Stress-Life Approach

• Assumptions:– High cycle fatigue regime

– Strain range is dominated by elastic rather than plastic strain

– Peak and valley stresses are constant

– Only one component of loading (e.g., axial, torsional, bending) acting at a time

– Stress amplitude is plotted against cycles to failure (e.g. S-N diagram)

Endurance Limit, Se

The Endurance Limit, Se

• The Endurance limit, Se is due to interstitial elements, such as carbon or nitrogen in iron which pin dislocations.

• This prevents the slip mechanism that leads to the formation of microcracks.

• Care must be taken when using the endurance limit since it can disappear due to

1. Periodic overloads (which unpin dislocations)

2. Corrosive environments (due to fatigue corrosion interaction)

3. High temperature (which mobilize dislocations)

Typical S-N Curve for Steel

S-N Diagram for Nonferrous Metals

• Nonferrous metals often do not have an endurance limit.• Fatigue strength Sf is reported at a specific number of cycles• Plot shows typical S-N diagram for aluminums

Endurance Limit for Nonferrous Alloys

S-N Curves for Polymers

Relationship of Se and SuSteel Wrought & Cast Iron

Aluminum Alloys Copper Alloys

Estimates for Steel

• Steels data clustered near the fatigue ratio Se/Su≈0.5 for the low- and medium-strength steels. The data however, actually falls between 0.35 and 0.6 for Su <1400MPa. For Su > 1400 MPa, Se does not increase significantly.

• For unnotched, highly polished, small bending specimen fatigue limits for steels are.

• The UTS can be approximated using Brinell Hardness

0.5 for 1400 MPa 200ksi

700 MPa for 1400 MPa 200ksi

e u u

e u

S S S

S S

3.45 for MPa units

0.5 for ksi units

u

u

S HB

S HB

Typical Endurance Limits

Unmodified endurance limit, S’e

• Test specimens are subjected to repeated stress while counting cycles to failure

• Most common test machine is R. R. Moore high-speed rotating-beam machine

• Subjects specimen to pure bending with no transverse shear

• As specimen rotates, stress fluctuates between equal magnitudes of tension and compression, known as completely reversed stress cycling

• Specimen is carefully machined and polished

Fig. 6–9

Unmodified endurance limit, S’e

The Endurance Limit, Se

• Representative monotonic tensile properties and bending fatigue limits of selected engineering alloys obtained under the above conditions are given in Table A.1.

• The fatigue limits given in Table A.1 must be substantially reduced in most cases before they can be used in design situations.

• For example, 10 to 25 percent reductions in these values for just size effect alone is not unreasonable for bend specimens greater than 10 mm (0.4 in.) in diameter.

The Endurance Limit, Se

• For a given material the fatigue limit has an enormous range depending on:

• surface finish, • size, • type of loading, • Cyclic rate (frequency),• temperature,• corrosive, and other aggressive environments, • mean stresses,• residual stresses, and • stress concentrations.

Endurance Limit Modifiers

• Correction factors are used to account for various effects on the endurance limit

• The unmodified endurance limit, S’e, is based on the rotating bending tests

• Marin factors are unitless, positive, and have a maximum value of unity.

Surface Condition Factor, CSurf

• Unmodified endurance limit determined from highly polished samples

• Data is typically quite scattered

• Small variation of surface leads to large variation of life

Surface Condition Factor, CSurf

Surface Condition Factor, CSurf

Example 6–3

Size Factor, CSize

Size Factor, CSize• Note: Because most of fatigue life is spent in making a very small crack a

little bigger, larger bodies have shorter fatigue lives because of the larger spatial extent of the high stresses in their notch-root stress fields.

• For bending and torsion loads in round specimen, the trend of the size factor data is given by

• The fatigue limit for axial load can range between 0.75 to 0.9 of the bending fatigue limits for small specimens.

Rotating Round

Example 6–4

Type of Load

• Accounts for changes in endurance limit for different types of fatigue loading.

• Only to be used for single load types. Use Combination Loading method when more than one load type is present.

• Bending is a more favorable loading condition than axial loading because (surface initiating) fatigue cracks propagate into lower stress environments.

Temperature Factor, CTemp

• At high temperatures, endurance limit declines compared to RT

• Equations based on steel (not for Al, Mg, and Cu alloys)

Environment, CEnv

• Note: Chemical attack can create pits at which fatigue cracks start.

• Corrosion can greatly reduce the portion of fatigue life devoted to fatigue crack initiation and growth and thus greatly reduce the fatigue strength at long lives.

Reliability Factor, CR

• S-N data has a statistical nature

• Most endurance data reported as mean values

• Reliability = 1 - Probability of Failure

• Model based on data Haguenand Wirching (1975)

Reliability Factor CR

• Simply obtain CR for desired reliability from the table below

Table 6–5

Miscellaneous Factor, CMisc

• Quantity designated for various factors that are not easy to account for:• Residual stresses• Directional characteristics from cold working• Case hardening• Corrosion• Surface conditioning, e.g. electrolytic plating and metal spraying• Cyclic Frequency• Frettage Corrosion• Radiation effects

• In these situations, failure is likely to be more accelerated and the endurance limit lower than expected from separate estimates

• The value of CMisc should be lower near stress concentrations

Mean Stress

Effect of Mean Stress

min

max

R

max min

2m

Fully Reversed: R = -1; A = ∞Zero to Tension: R = 0; A = 1Min equal to Max: R = 1; A = 0

Zero Mean Stress

Increasing Mean Stress

The mean stress, Sm, can have substantial influence on fatigue behavior.

In general, tensile mean stresses are detrimental and compressive mean stresses are beneficial.

Effect of Mean Stress

• At intermediate or high stress levels under load control test conditions, substantial cyclic creep (also referred to as cyclic ratcheting) which increases the mean strain, can occur in the presence of mean stresses.

• This cyclic creep adds to the detrimental effects of tensile mean stress on fatigue life and results in additional undesirable excess deformation.

Effect of Mean Stress

• There exist many methods to plotting the relationship between alternating stress, Sa and mean stress, Sm

• S-N curve generated frome fully reversed, (Sm= 0, R = -1), fatigue limit of smooth specimens

• Similar behavior exists for other alloys.

• The general trend indicates that tensile mean stresses are detrimental.

• Much of the data fall between the straight and curved lines.

Effect of Mean Stress

Tensile Mean Stress Influence on Long-life (5x10^7Cycles) Fatigue Strength For Aluminum Alloys

Add Tensile Mean Stress

Endurance Strength Decreases

Fatigue Failure Criteria

a m

a

m

Fatigue Failure Criteria

• Gerber passes through the data

• ASME-elliptic passes through data and incorporates rough yielding check

• Modified Goodman is linear, so simple to use for design. It is more conservative than Gerber.

• Soderberg provides a very conservative single check of both fatigue and yielding. Rarely Used.

• Langer line represents standard yield check. It is equivalent to comparing maximum stress to yield strength.

Fatigue Failure Criteria

INFINITELIFE

FINITELIFE

a m

a

m

Fatigue Failure Criteria

Fatigue Failure Criteria

• To determine factor of safety against fatigue failure, replace stress amplitude and mean stress with nfσa and nfσm, respectively; solve for nf

• Modified Goodman

• Gerber

• ASME-Elliptic and Langer

How to Use Fatigue Criteria

• Find Cycles to Failure given Mean Stress

• If we have fatigue with mean stress, Sm we need to use a fatigue criteria todetermine the equivalent fully-reversed alternating stress, SNf condition.The SNf is found by replacing Se with SNf.

• Determine the onset of Yielding

• Check for yielding by replace Su with the monotonic yield strength Sy. And Se with the cyclic yield strength SNf.

• No yielding if <1

1

1

a m

e ut

a m

Nf ut

S S

S S

Example: Modified Goodman

1

1

a m

e ut

a m

y y

S S

S S

Putting it all together

Using the Book

• In our Book, Basquin’s equation using σf' is based on reversals, 2Nf, rather than cycles, Nf.

• Sa or SNf= σf' (2Nf)b

• Values of the σf', and b, defined as the fatigue strength coefficient and the fatigue strength exponent respectively, are given in Table A.2 for a few selected engineering alloys.

• When doing your HW you will need to keep this in mind!

Using the Book

• The slope, B, depends upon many factors and for unnotched parts could vary from about –0.05 to –0.2.

• For example, the varying slope can be indicated by using surface effects with Sf taken from Fig. 4.15.

• Surface effects are dominant at long fatigue lives and less significant at short lives with convergence of the S-N curves at Suat Nf= 1.

• This gives different values of slope B for each surface condition.

• The convergence at Nf= 1 is reasonable because surface finish does not have an appreciable affect on monotonic properties for most smooth metal specimens.

Using the Book

• Oftentimes the slope, B, for smooth unnotched specimens is about -0.1.

• This suggests that for unnotched specimens the fatigue life is approximately inversely proportional to the 10thpower of alternating stress.

• Thus, a 10% increase or decrease in alternating stress will cause about a factor of three decrease or increase, respectively, in fatigue life.

• For notched parts the slope of the S-N curve on logarithmic scales is steeper yielding more extreme changes.

• Thus, even small changes in applied alternating stress can have a significant effect on fatigue life.

How to Plot S-N Curve without Data

• Sometimes you need to approximate the S-N Curve when only the Su and/or Seare known!

• Basquin Approach• Simple

• More Conservative

• Tri-Slope Approach• Slightly more complex

• More Accurate

B

ATri-Slope Model

Basquin Model

Basquin Approach

• Basquin uses the Ultimate Tensile Strength as a reference point.

B

A

Basquin Model

B

f fS A N

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes

Basquin Equation

uA S

1 6 log e uB S S

How to Find A & B Constants

Tri-Slope Approach (for HCF)

• Tri-Slope uses the S1000 point corresponding to 1000 cycles as a reference point

B

ATri-Slope Model

B

f fS A N

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes

Basquin Equation

2

1000

e

SA

S 10001 3 log eB S S

How to Find A & B Constants

3

100010 , S

Tri-Slope Approach (for S1000)

Generally only modification factors for S1000 are load type and reliability

3

100010 , S

1000 load US C S

Cload

S1000

Finite Life• We can make finite life predictions by replacing the endurance limit,

Se with a finite life such as S10,000.

10,000S

DOS AND DON’TS IN DESIGN

• Test systems are available to perform fatigue and durability tests for almost every conceivable situation from a small highly polished laboratory specimen to that of a large scale complex structure.

• The endurance limit under constant amplitude loading conditions occurs for a few metals (notably low and medium strength steels), but under in-service variable amplitude loading with corrosive, temperature, or other environmental conditions, the fatigue limit is rare.

• Do consider the Marin Factors !!!

• Do note, the Marin factors do not account for interaction of affects

DOS AND DON’TS IN DESIGN

• Yielding, finite or long life, and mean stress effects can be approximated using the five fatigue failure criterion.

• Don't neglect the advantages of compressive mean or compressive residual stresses in improving fatigue life and the detrimental effect of tensile mean or tensile residual stresses in decreasing fatigue life.

• Do attempt to use actual fatigue data in design, but if this is not possible or reasonable, approximate estimates of median fatigue behavior can be made.

DOS AND DON’TS IN DESIGN

• Consider a critical location in component

• Consider the mechanical (and thermal) loads experienced by that component

• Use Solid Mechanics analysis or FEA to convert external loads to internal stresses

• Determine the loading rate and the cyclic stress history

Ex: Hip prosthesis

(generic Ti alloy)

S-N Examples

Assignments

• Problem 4.1 from the text.

• Problem 4.5 from the text.

• Problem 4.6 from the text.

• Problem 4.11 from the text

Contact Information

Calvin M. StewartAssociate ProfessorDepartment of Mechanical EngineeringThe University of Texas at El Paso500 W. University Blvd.Suite A126El Paso, Texas 79968-0717

Email: cmstewart@utep.eduURL: http://me.utep.edu/cmstewart/Phone: 915-747-6179Fax: 915-747-5019

References

• Shigley’s Mechanical Design.

Fatigue Design Considerations

• Consider actual stresses, including stress concentrations, rather than to nominal average stresses.

• Use gradual changes in section and symmetry of design to reduce secondary flexure

• Provide suitable protection against corrosion

• Avoid metallic plating with widely different properties than underlying material

• Consider pre-stressing when feasible, to include shot peening and cold working

• Consider temperature effects