Fatigue Analysis - Mechanical Engineering

Strain-Life Approach

For Extreme Environments

Fatigue Analysis

Strain-Life ApproachPresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Spring 2021

Outline

• Monotonic Tension

• Strain-Controlled Fatigue

• Cyclic Stress-Strain Behavior

• Strain-Life Approach

• Life Estimation and Transition Life

• Mean Stress

• Advantages and Limitations of Approaches

• Summary

• Questions

Monotonic Tension

Monotonic Tension

• Monotonic tension stress-strain properties are usually reported in handbooks and are used in many specifications.

• Monotonic behavior is obtained from a tension test where a specimen with circular or rectangular cross section within the uniform gage length is subjected to a monotonically increasing tensile force until it fractures.

• They are easy tests to perform and provide information that has become conventionally accepted.

• However, their relation to fatigue behavior may be remote.

Monotonic Tension

• ASTM E8/E8M - Standard Test Methods for Tension Testing of Metallic Materials• Displacement Controlled.

• A Fixed Displacement Rate is applied.

• Used to obtain material characteristics and strengths

• Tensile stress σ slowly increasing P

• Load and deflection are recorded

Monotonic Tension

• Monotonic uniaxial stress-strain behavior can be based on "engineering" stress-strain or "true" stress-strain relationships.

• The difference is in using original versus instantaneous gage section dimensions.

• Engineering Stress and Strain

• True Stress and Strain

0

P

A = 0

0

l l

l

−=

P

A0

l0

l

A

t

i

P

A =

00

lnl

tl

dl l

l l

= =

l0

Monotonic Tension

• For small strains, <2%, • Engineering stress, σ ≈ True stress, σt

• Engineering strain, ε, ≈ True strain, εt.

• No distinction between "engineering" and "true" components is needed for these small strains.

• For larger strains, the differences become appreciable.

Monotonic Tension

Tensile properties of some engineering alloys are also given in Table A.1 and A.2.

Note: Stress-strain behavior of a material can be sensitive to the strain rate, particularly at elevated temperatures

Monotonic Tension

• The subscript f represents fracture. The true fracture strength, σf, can be calculated from Pf/Af but is usually corrected for necking. The Bridgman correction factor is used to compensate for this triaxial state of stress and applies to cylindrical specimens [3]. The corrected true fracture strength, σf is then given by

• R is the radius of curvature of the neck and Dmin is the diameter of the cross section in the thinnest part of the neck.

Monotonic Tension

• Inelastic/plastic strain results in permanent deformation which is not recovered upon unloading.

• The unloading curve is elastic and parallel to the initial elastic loading line.

• Total strain, ε, is

Straine p = +

Monotonic Tension

• For many metals, a plot of true stress versus true plastic strain in log-log coordinates results in a linear curve.

• An example of such a plot is shown for AISI 11V41 steel.

To avoid necking influence, only data between the yield strength and ultimate strength portions of the stress-strain curve are used to generate this plot.

This curve is represented by the power function:

( )n

pK =

Monotonic Tension

• This type of true stress-true strain relationship is often referred to as the Ramberg-Osgood relationship.

• K is the strength coefficient (stress intercept at εp= 1)

• n is the strain hardening exponent (slope of the line).

• The total true strain is given by:

( )n

pK =

1/n

e pE K

= + = +

• Value of n gives a measure of the material’s work hardening behavior.

• K and n for some engineering alloys are also given in Table A.2.

Table A.1

Monotonic Tension

• Key monotonic properties• Modulus of Elasticity, E• Ultimate Tensile Strength, SUT

• Yield Strength, Sy

• corrected true fracture strength, σf

• True fracture strain, εf

• Strength coefficient, K• Strain hardening exponent, n

• We will find fatigue version of some of these monotonic properties, specifically, S’y, σ’f, ε’f,K’, n’, later.

Strain-Controlled Fatigue


• In strain-controlled fatigue testing, a constant strain amplitude, εa is applied to the material or component until it has reached failure.

• Attributes:

• Strain amplitude is plotted against cycles to failure (e.g. ε-N diagram)

• Accounts for actual stress-strain response of the material

• Can be extrapolated to multiaxial loading and notched geometries

• Can be combined with creep relations for high temperature analysis


• Types of Strain

• Mechanical Strain History• Most widely used

• Thermal Strain History• For non-isothermal fatigue testing

• Plastic Strain History• For isothermal testing

• For test programs involving multiple temperature levels


• An important aspect of the fatigue process is plastic deformation.

• Fatigue cracks usually nucleate from plastic straining in localized regions.

• Therefore, cyclic strain-controlled tests can better characterize fatigue behavior of a material than cyclic stress-controlled tests, particularly in the low cycle fatigue region and/or in notched members.

• Strain-controlled fatigue testing has become very common, even though the testing equipment and control are more complicated than the traditional load or stress-controlled testing.

Strain-Controlled Test Methods

• Strain-controlled testing is usually conducted on a servo-controlled closed-loop testing machine.

• A uniform gage section smooth specimen is subjected to axial straining.

• An extensometer is attached to the uniform gage length to control and measure strain over the gage section.

• A standard strain-controlled test consists of constant amplitude completely reversed straining at a constant or nearly constant strain rate.

• The most common strain-time control signals used are triangular (sawtooth) and sinusoidal waveforms.

• Stress response generally changes with continued cycling and is measured with a load cell. Stress and plastic strain variations are usually recorded periodically throughout the test and cycling is continued until fatigue failure occurs.

• ASTM Standard E606: Strain-Controlled Fatigue Testing.

Strain-Controlled Test Methods

• An important consideration in axial fatigue testing is uniformity of stress and strains in the specimen gage section.

• A major source of non-uniformity of gage section stress and strains is a bending moment resulting from specimen misalignment that can significantly shorten the fatigue life. Specimen misalignment can result from:• eccentricity and/or tilt in the load-train components (including load cell, grips,

and load actuator),

• improper specimen gripping,

• lateral movement of the load-train components during the test due to their inadequate stiffness.

Often fatigue tests are operated in pulsating tension (R>0) to mitigate misalignment. (Typically R=0.1)

Strain History Nomenclature• Def’s:

– Strain range

– Strain amplitude

–Mean strain

– Strain ratio

– Amplitude ratio

max min

2a

−=

max min

2m

+=

max min = −

min

max

R

=

m

aA

=

Period

Frequency

1

f =

1f

=

m

min

max

a

a

Stra

in

Cyclic Stress-Strain Curve

Cyclic σ-ε Curve

• The stress-strain behavior obtained from a monotonic test can be quite different from that obtained under cyclic loading.

• This was first observed by Bauschinger (Effect). His experiments indicated the yield strength in tension or compression was reduced after applying a load of the opposite sign that caused inelastic deformation.

• Thus, one single reversal of inelastic strain can change the stress-strain behavior of metals.

0

1

2

3

Cyclic σ-ε Curve

Fatigue failure almost always begins at a local discontinuity

When stress at discontinuity exceeds elastic limit, plastic strain occurs

The figure shows true stress-true strain hysteresis loops of the first five stress reversals

The area within a hysteresis loop is energy dissipated during a cycle (usually in the form of heating). This energy represents the plastic work from the cycle.

Appreciable progressive change in stress-strain behavior during inelastic cycling.

(a) Fully annealed,

showing cyclic

hardening

(c) Cold-worked, showing

cyclic softening

(b) Partially annealed, showing small cyclic hardening and softening

Cyclic Softening

Cyclic σ-ε Curve

• In essence, under strain-control,• Cyclic Hardening –more stress is required each cycle to produce the same

amount of strain

• Cyclic Softening –less stress is needed to produce the same amount of strain

Stabilized Stage III (Unstable)Cyclic Softening

Notice these two are only slightly different

Cyclic σ-ε Curve

• Strain amplitude is held constant throughout the test

• Peak and valley stresses are plotted for each cycle

• Both crack initiation life and fatigue life can be measured

• Strain is later plotted versus fatigue life

Cyclic σ-ε Curve

• After a certain number of cycles, the hysteresis loop of a material can stabilize (i.e. not change significant from cycle to cycle) Typically the hysteresis loop at Nf/2

• This means any further loops are almost identical up to stage III unstable crack propagation

• Total strain range can be decomposed into plastic and elastic components

e p pE

= + = +

Cyclic σ-ε Curve

• Note: Even though true stress and strains are used, no distinction is usually made between the true and the engineering values, because:• the differences between true and engineering values during the tension and

compression parts of the cycle are opposite to each other, and therefore, cancel out,

• strain levels in cyclic loading applications are often small (typically less than 2%), compared to strain levels in monotonic loading.

Cyclic Stress-Strain CurveStabilized Hysteresis Loops

Stabilized Hysteresis Loops

• A family of stabilized hysteresis loops at different strain amplitudes is used to obtain the cyclic stress-strain curve of a material.

• The tips from the family of multiple loops can be connected to form the cyclic stress-strain curve.

• This curve does not contain the monotonic upper and lower yield points.


• Three methods commonly used to obtain the cyclic stress-strain curve are: • the companion test method

• the incremental step test method

• the multiple step test method

• Even though some differences exist between the results from the three methods, they are small in most cases.


• Companion Test Method – a series of companion samples are tested at various strain levels until the hysteresis loops stabilize. Half-life or near half-life hysteresis loops from each specimen and strain amplitude are used to obtain the cyclic stress-strain curve.

• Incremental Step Test Method – a single specimen is subjected to a series of blocks of gradually increasing and decreasing strain amplitudes. After the material has stabilized (usually after several strain blocks), the hysteresis loops from half a stable block ar then used to obtain the cyclic stress-strain curve.

( ) ( )1 2

...

( )f t =

Incremental Step Test Method


• Multiple Step Test Method - It is similar to the incremental step test method, except rather than incrementally increasing and decreasing strain in each block the strain amplitude is kept constant.

• Once cyclic stability is reached at the constant strain amplitude, the stable hysteresis loop is recorded, and strain amplitude is increased to a higher level.

• This process is repeated until enough stable hysteresis loops are recorded to construct the cyclic stress-stain curve.

Strain History of Multiple Step Test Method

Tension, Compression, and Cyclic Stress-Strain Response

➢ Cyclic Softening

➢ Cyclic Hardening

➢ Cyclically stable

➢ Mixed Hardening

cyclic stress-strain curve of some engineering alloys


• Soft materials will harden

• Material will cyclically harden if

• Dislocation density is initially low and increases rapidly

• Hard materials will soften

• Material will cyclically soften if

• Dislocations rearrange and offer less resistance to deformation

1.4u

y

S

1.2u

y

S

Stabilized Hysteresis Loops: Temperature

Stabilized Hysteresis Loops: Rate Effects


• Similar to monotonic tension, a plot of true stress amplitude, σa, versus true plastic strain amplitude, Δεp/2, in log-log coordinates for is represented by the power function:

• The cyclic stress-strain curve is represented by a Ramberg-Osgoodrelationship as follows:

2

n

p

a K

=

K’ - Cyclic strength coefficientn’ - Cyclic strain hardening exponent. Usually between 0.10 and 0.25.

11

2 2 2 2 2

nnpe a a

aE K E K

= = + = + = +


• Massing’s hypothesis states that the stabilized hysteresis loop branch may be obtained by doubling the basic material stress-strain curve.

• The equation for the stabilized hysteresis loop is written as follows:

1

22

n

E K

= +

1

22

n

E K

= +

1

22

n

E K

= +

1

22

n

e pE K

= + = +


• The cyclic yield strength, Sy′, is defined at 0.2% strain offset which corresponds to a plastic strain of 0.002 on the cyclic stress-strain curve. It can be estimated by substituting Δεp/2 = 0.002 into

• Values of K’ and n’ for selected engineering alloys are given in Table A.2.

2

n

p

a K

=

( )' 0.002n

yS K

=

Cyclic Stress-Strain CurveExample

Example

• Given the results of a companion test series for an engineering alloy find

(a) the true stress amplitude versus true plastic strain amplitude plot

(b) the Ramberg-Osgood relationship to obtain the cyclic stress-strain curve

(c) What type of hardening behavior is observed?

Log-Log true stress amplitude versus true plastic strain amplitude plot

Linear-Linear true stress amplitude versus true plastic strain amplitude plot

Mixed Behavior Observed



• The strain-based approach to fatigue problems:• is widely used at present.• Strain can be directly measured.• Application of this approach is common in notched member fatigue.

• Strain-life design method is based on relating the fatigue life of notched parts to the life of small unnotched specimens cycled to the same strains as the material at the notch root.• Since fatigue damage is assessed directly in terms of local strain, this approach is also

called local strain approach.• Expected fatigue life can be determined knowing the strain-time history at the notch

root and smooth strain-life fatigue properties of the material.• The remaining fatigue crack growth life of a component can be analyzed using

fracture mechanics concepts (Ch 6).



• Steady-state hysteresis loops can be reduced to elastic and plastic strain ranges or amplitudes.

• Cycles to failure can involve from about 10 to 107cycles and frequencies can range from about 0.1 to 10 Hz.

• Beyond 106 cycles, load or stress-controlled tests at higher frequencies can often be performed because of the small plastic strains and the greater time to failure.

• The strain-life curves are often called low cycle fatigue data because much of the data are for less than 105 cycles.


• Failure criteria include • the life to a small detectable crack,

• life to a certain percentage decrease in load amplitude (50% drop level is recommended by the ASTM standard E606),

• life to a certain decrease in the ratio of unloading to loading moduli, or

• life to fracture.

• Strain-life fatigue curves plotted on log-log scales are shown schematically in Fig. 5.11.

Material Constants

• The total strain amplitude can be resolved into elastic and plastic strain components from the steady-state hysteresis loops.

• Both the elastic and plastic curves can be approximated as straight lines.

• At large strains or short lives, the plastic strain component is predominant, and at small strains or longer lives the elastic strain component

Modeling the Strain-Life (ε-N) Curve

• Elastic - Basquin (1910)

• Plastic - Coffin-Manson (1954,53)

• The intercepts of the two straight lines at 2Nf= 1 are σf’/E for the elastic component and εf’ for the plastic component.

( )22

bfel

fNE

=

( )22

cpl

f fN

=

Relation of Fatigue Life to Strain

• Applying the equations, the total-strain amplitude is

• Known as Manson-Coffin relationship between fatigue life and total strain

• Some values of coefficients and exponents given in Tables

• Equation has limited use for design since values for total strain at discontinuities are not readily available

( ) ( )2 22 2 2

b cpl fel

a f f fN NE

= = + = +

Modeling the Strain-Life (ε-N) Curve

• Nomenclature


• Basquin’s Law [1910]

• Fatigue strength coefficient, 'F is approximately equal to the true fracture strength at fracture, F (in MPa or ksi)

• Fatigue strength exponent, b is the slope of the elastic-strain line, and is the power to which the life 2N must be raised to be proportional to the true-stress amplitude. Ranges between 0.05 to -0.12


• Coffin-Manson Law [1954, 1953]

• Fatigue ductility coefficient, 'Fis approximately equal to te true fracture ductility, F

• Fatigue ductility exponent, c is the slope of plastic-strain line, and is the power to which the life 2N must be raised to be proportional to the true plastic-strain amplitude. Ranges from -0.5 to -0.7


Typical Complete Strain-life Curve With Data Points For Annealed 4340 Steel

Fatigue life can be plotted as cycles or reversals

Data plotted on semi-log or log-log

Data includes a bend.

Elastic Line

Plastic Line

Definition of the Strain-Life Curve

• Def: Transition Life, Nt – point at which elastic and plastic contributions to the life are equal.

• For lives <2Nt, the deformation is mainly plastic,

• For lives >2Nt, the deformation is mainly elastic.

( )1

2

b c

f

t

f

EN

−

=

Strain-Life Curves: Different Materials• Many materials have similar life at

a total strain amplitude of about 0.01.

• At larger strains, increased life is dependent more on ductility, while at smaller strains longer life is obtained from higher strength materials.

• The optimum overall strain-life behavior is for tough metals, which are materials with good combinations of strength and ductility.

Monotonic Tension Test

• Strain Energy• Resilience is the elastic energy

absorbed by the specimen and is equal to the area under the elastic portion of the stress-strain curve.

• Tensile toughness is the total energy density or energy per unit volume absorbed during deformation (up to fracture) and is equal to the total area under the engineering stress-strain curve

( )2

2

ys

RUE

=

U d =

Strain-Life Curves: Thermomechanical Processing

Medium Carbon Steel in quenched and normalized condition

Strain-Life Curves: Ni Alloys

Strain-Life Curves: Al Alloys


• Strain-life fatigue data for selected engineering alloys are included in Table A.2.

• These properties are obtained from small, polished, unnotched axial fatigue specimens under constant amplitude fully reversed cycles of strain.

• Material properties in Table A.2 also omit influences of surface finish, size, stress concentration, temperature, and corrosion.

Table of Strain-Life Properties

Table of Strain-Life Properties

Shigley’s Mechanical Design

Strain-Life ApproachExample

Solving Problems

Plots

Plots

• stabilized hysteresis loop

1

22

n

E K

= +

Equations

Properties

• The fatigue strength coefficient, σ’f and fatigue strength exponent, b, are the intercept and slope of the linear least squares fit to stress amplitude, Δσ/2, versus reversals to failure, 2Nf, using a log-log scale

• Similarly, the fatigue ductility coefficient, ε’f, and fatigue ductility exponent, c, are the intercept and slope of the linear least squares fit to plastic strain amplitude,Δεp/2, versus reversals to failure, 2Nf, using a log-log scale.

Strain-Life Properties

• Plastic strain amplitudes can either be measured directly from half of the width of stable hysteresis loops at σ = 0 or calculated from

• A difference usually exists between the measured and calculated values, which results from the difference between monotonic and cyclic moduli of elasticity and rounding of the hysteresis loops near the strain axis.


Fatigue Strength Coefficient, σf′

σf′ ≈ σf

where σf is the true fracture strength measured from a monotonic tensile tests

Note: For steels with BHN<500: σf =Su+50ksi

Fatigue Strength Exponent, b

-0.13 < b < -0.05

Fatigue Ductility Coefficient, εf′

εf′ ≈ εf

Note:

Fatigue Ductility Exponent, c

-1.0 < c < -0.5

1ln

1f

RA

=

−


2 1

510 0.25

P Pb

−=

−

4 3

410 10

P Pc

−=

−

Cyclic hardening exponent, n’

Cyclic hardening coefficient, K’

Strain-Life to Cyclic Properties

• The cyclic strength coefficient, K’, and cyclic strain hardening exponent, n’, are obtained from fitting stable stress amplitude versus plastic strain amplitude data.

• Rough estimates of K’ and n’ can also be calculated from the low cycle fatigue properties by using:

Fatigue strength exponent, b

Fatigue ductility exponent, c

bn

c =

( )f

n

f

K

=

( )1 5

nb

n

−=

+

( )

1

1 5c

n

−=

+

Monotonic Approximation

Textbook

• When the subject material is known, use Table A2 to obtain monotonic, cyclic, and strain-life properties.

On Failure,

• Note that life to failure may be defined in several ways. These Include:

• Separation of specimen

• Development of a given crack length (often 1mm)

• Loss of specified load carrying capability (often 10 or 50% load drop)

• Specimen separation in the most common failure criteria for uniaxial loading. However, in many cases, there is not a large difference between these criteria.

Be Careful…

• Cycles versus Reversals. The strain-life approach measures life in terms of reversals (2N), whereas the stress-life method uses cycles (N). A reversal is one-half of a full cycle.

• Amplitude versus range. The strain-life approach uses both strain range, ∆ε and amplitude, εa which differ by a factor of 2.

• Cyclic σ-ε curve versus hysteresis curve. Massing’s hypothesis states that the hysteresis curve can be modeled as twice the cyclic stress strain curve.

• In essence, Find Nf !!! Not 2Nf !!!

2a =

Mean Stress

Mean Stress

Increase life

Decrease life

Mean Stress: SAE 1045 Hardened Steel

Mean Stress: Relaxation

• Mean stress tends toward zero for large plastic strain amplitudes

• This is true for cases having either compressive or tensile mean stress

• Note this is not cyclic softening.

• Mean stress relaxation can occur in materials that are cyclically stable.

Morrow’s Mean Stress Correction

• Morrow’s Mean Stress Correction

• The predictions made with Morrow’s mean stress correction model are consistent with the observations that mean stress effects are significant at low values of plastic strain, where the elastic strain dominates.

• This relation incorrectly predicts that the ratio of elastic to plastic strain range is dependent on mean stress.

• Predictions for compressive mean stress cases can be inaccurate.

( ) ( )2 22

b cf m

a f f fN NE

− = = +

Mean Stress Correction

( )1

2

b c

f

t

f m

EN

−

= −

Manson and Halford Model

• Manson and Halford Model [1981]

• here both the elastic and plastic terms are affected by the mean stress is given by

• This equation tends to predict too much mean stress effect at short lives or where plastic strains dominate. At high plastic strains, mean stress relaxation occurs.

Manson and Halford Model

In Manson-Halford, Both the elastic and plastic portion of the Strain-life Curve are modified by the mean stress.

Smith, Watson, and Topper Model

• Smith, Watson, and Topper Model [1970]

This equation assumes that for different combinations of strain amplitude, εa, and mean stress, σm, the product σmax εa

remains constant for a given life.

If σmax is zero, this Eq. predicts infinite life, which implies that tension must be present for fatigue fractures to occur.

The SWT Eq. has been shown to correlate mean stress data better for a wide range of materials and is regarded to be more promising for general use.

Smith, Watson, and Topper Model

Fatigue Design Considerations

• In many design situations transient material behavior can be critical and must be considered during material selection

• Ductility useful for LCF conditions (Ex: annealed 4340 steel)

• High strength for HCF conditions (Ex: maragning steel)

• If mechanical loading is very different from benchmark conditions, then conduct additional LCF experiments

Comparison of Stress-Life and Strain-Life

Advantages of Stress-Life

• Empirical parameters for a lot of materials has been determined• Marin factors• Fatigue Strengths

• Easy to use for design applications

• Back of the envelope calculations allow good accuracy

• Works well for long life components

Disadvantages for Stress-Life

• Cannot be used for LCF conditions

• If loads are fluctuating in a pseudo-random way, stress-life can yield non-conservative results

• Mostly empirical

Advantages of Strain-Life

• Takes into account actual stress-strain response of material during cycling.

• More conservative of the two approaches

• Widely used in industry

• Accurately predicts crack initiation stage

• Closer connection to physical nature of material

• Distinguishes between initiation and propagation

Disadvantages for Strain-Life

• Analysis depends on testing, strain data must be available

• Analysis is more complicated

• Less data is available

Stages of Component Fatigue

• Force-controlled and strain-controlled fatigue are relevant to crack initiation and early propagation

• LEFM Methods are appropriate otherwise

Summary

➢Basic material mechanical properties such as strength and ductility can be obtained from simple monotonic tensile tests. However, The stress-strain behavior obtained from simple monotonic tensile tests can be quite different from that obtained under cyclic loading.

➢Cyclic loading can cause hardening and/or softening of the material. Using a monotonic stress-strain curve of a cyclic softening material in a cyclic loading application can significantly underestimate the extent of plastic deformation present.

➢Changes in cyclic deformation behavior are more pronounced at the beginning of cyclic loading, as the material usually gradually stabilizes with continued cycling.

Summary

➢stable cyclic stress-strain curve can be represented by the following equation for many metals:

➢where K’ and n’ are material cyclic deformation properties.

➢Fatigue cracks usually nucleate from plastic straining in localized regions. Strain-based approach to fatigue problems is widely used at present.

11

2 2 2 2 2

nnpe a a

aE K E K

= = + = + = +

Summary

➢The strain-life equation is expressed as:

➢The Strain-life approach is a comprehensive approach which can be applied for the treatment of both low cycle and high cycle fatigue.

➢In the low cycle region plastic strain is dominant, whereas in the high cycle region elastic strain is dominant.

( ) ( )2 22 2 2

b cpl fel

a f f fN NE

= = + = +

Summary

➢At large strains better fatigue resistance depends more on ductility, while at small strains it depends more on strength.

➢Strain-controlled cycling with a mean strain results in a mean stress that usually relaxes at large strain amplitudes due to the presence of plastic deformation.

➢A non-relaxing mean stress can significantly affect the fatigue life with tensile mean stress having detrimental effect and compressive mean stress having beneficial effect.

➢Other synergistic effects of loading, environment, and component or material processing can also influence the strain-life behavior.

Homework 5

• Problem 5.3 from the text.






References

• Stephens, R.I., Fatemi, A., Stephens, R.R., Fuchs, H.O., 2000, Metal Fatigue in Engineering 2nd Edition, Wiley.

• Suresh, S., 1998, Fatigue of Materials 2nd Edition, Cambridge University Press.

• Bannantine, J., 1989, Fundamentals of Metal Fatigue Analysis, Pearson.

• Schijve, J., 2008, Fatigue of Structure and Materials 2nd Edition, Springer.

• Shigley’s Mechanical Design

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

[email protected]

me.utep.edu/cmstewart/

mailto:[email protected]

http://me.utep.edu/cmstewart/

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Fatigue Analysis - Mechanical Engineering