Stress-Life Fatigue Modeling:
All equations are referenced on the Excel spreadsheet and can be found in Fundamentals of
Metal Fatigue Analysis by Bannantine, Comer, and Handrock.
Material:
1018 Steel, 0.5 in diameter
Experimental values:
SU = 92,147 psi
SY = 77,000 psi
E = 28,670 ksi
Published data (matweb):
RA = 40%
Calculated values:
𝑆1000 ≈ 0.9 ∗ 𝑆𝑈 = 46,073 psi
𝑆𝑒′ ≈ 0.5 ∗ 𝑆𝑈 = 82,932 psi
𝐶𝑆𝑖𝑧𝑒 = 1
𝐶𝐹𝑖𝑛𝑖𝑠ℎ ≈ 0.77
𝑆𝑒 = 𝐶𝐹𝑖𝑛𝑖𝑠ℎ ∗ 𝐶𝑆𝑖𝑧𝑒 ∗ 𝑆𝑒 ′= 35,476 psi
1
𝑏 = − 3 log10
𝐶 = log10
𝑆1000
𝑆𝑒
(𝑆1000 )2
𝑆𝑒
= -0.1229
= 5.28751
𝐶 1
𝑁 = 10−𝑏 𝑆 𝑏
On this spreadsheet, stress is the independent variable, and cycles to failure the dependent
variable.
Focusing on the S-N plot, there are two basic sets of data – experimental and theoretical – with a
vertical line for the transition life (to be explained later, when discussing strain-life). Plotted on
the normal log-log scale, it’s easy to see the theoretical relationship between stress and life, as
well as the experimental results.
Alternating Stress, S (psi)
1E+05
Experimental
Theoretical
Transition Life
1E+04
1E+02
1E+03
1E+04
1E+05
Life to Failure, N (Cycles)
1E+06
Strain-Life Fatigue Modeling:
Again, all necessary equations are referenced and can be found in the same source.
Material properties are the same as found on the previous tab.
1
𝜖𝑓 ′ ≈ ln 1−𝑅𝐴 = 0.51083
𝜎𝑓 ′ = Coefficient of S-N plot power relationship (trendline) = 193,868
𝑏 = Exponent of S-N plot power relationship (trendline) = -0.12293
𝑐 = -0.5 (Approximate value)
1
𝑁𝑡 =
1 𝜖𝑓 ′𝐸 (𝑏−𝑐)
(
)
2 𝜎𝑓 ′
= 47,862 Cycles
∆𝜖 𝜎𝑓 ′
𝑏
𝑐
=
(2𝑁𝑓 ) + 𝜖𝑓 ′(2𝑁𝑓 )
2
𝐸
The transition life is the life at which the plastic strain is equal to the elastic strain. For lives
greater than the transition life, elastic strain dominates plastic strain. In this region, stress-based
fatigue models are sufficient, as the plastic strain becomes increasingly less influential as the
cycles to failure increase. For lives less than the transition life, plastic strain dominates elastic
strain. In this region, strain-based fatigue models are necessary, as stress-based theories fail to
account for plastic deformation. Transition life may be represented as both reversals or cycles,
and is shown as both on this spreadsheet. The vertical line seen on the plot represents the
transition life for this material and specimen geometry.
The final plot shows all three theoretical curves – elastic strain, plastic strain, and total strain –
along with the experimental data and a demarcation for the transition life.
Strain Amplitude (in/in)
1E-01
1E-02
Experimental
Δϵe/2
Δϵp/2
1E-03
Δϵ/2
Transition Life
1E-04
1E+02
1E+03
1E+04
1E+05
Reversals to Failure, 2Nf (Reversals)
1E+06
Comparison:
Looking first at the stress-life plot, it becomes immediately clear that specimens cycled to fail
higher than the transition life match quite well with the S-N curve, while those below the
transition life deviate. This is expected, as the S-N curve does not take into account the plastic
strain experienced by the specimens and thus cannot accurately predict life in this region.
The strain-life plot, however, shows better overall correlation. Experimental data points are
expected to match with the total strain curve. The specimens that were cycled to a high amount
of cycles match well with the total strain curve. Those below the transition life match well, too,
save for a few outliers.
These outliers that do not match well with the strain-life plot exhibited buckling – well below the
theoretical value of force needed for buckling. A probably misalignment of the test system is to
blame for this. Since the specimens buckled, they were subjected to bending instead of just axial
loading. All theoretical calculations were done assuming axial loading, and thus these buckled
specimens cannot be expected to correlate with theoretical values.
Specimens:
Photograph of failed fatigue specimens.
Photograph of failed fatigue specimens, with buckling.
Fracture Modeling:
All necessary equations are from Mechanical Behavior of Materials, 3rd Edition by Dowling,
pages 327, 351, and 364.
Three notch types were examined in this study, a centered notch, a double edged notch, and a
single edged notch. In stress based fracture models, all three notches (assuming the notch areas
were the same) would act the same. That is, given a specified load, all three would have the
same cross sectional area. With these three notch geometries included on the same specimen, the
intensity factors could be calculated and it could be predicted at which point the specimen would
fail.
Schematic of fracture specimen.
Material Properties
𝐾𝐼𝐶 = 26,400 ksi*sqrt(in)
𝑆𝑌 = 40,000 psi
Centered notch (CEN):
𝑲 = 𝑺𝒈 √𝝅𝒂 for (a/b ≤ 0.4)
𝐾 = 𝐹𝑆𝑔 √𝜋𝑎 for (a/b ≥ 0.4)
𝐹=
1−0.5𝛼+0.326𝛼2
√1−𝛼
for (h/b ≥ 1.5) and where (α = a/b)
Double edged notch (DEN):
𝑲 = 𝟏. 𝟏𝟐𝑺𝒈 √𝝅𝒂 for (a/b ≤ 0.6)
𝐾 = 𝐹𝑆𝑔 √𝜋𝑎 for (a/b ≥ 0.6)
𝐹 = (1 + 0.122 cos 4
𝜋𝛼
2
) √𝜋𝛼 tan
2
𝜋𝛼
2
for (h/b ≥ 2) and where (α = a/b)
Single edge notch (SEN):
𝐾 = 1.12𝑆𝑔 √𝜋𝑎 for (a/b ≤ 0.13)
𝑲 = 𝑭𝑺𝒈 √𝝅𝒂 for (a/b ≥ 0.13)
𝑭 = 𝟎. 𝟐𝟔𝟓(𝟏 − 𝜶)𝟒 +
𝟎.𝟖𝟓𝟕+𝟎.𝟐𝟔𝟓𝜶
(𝟏−𝜶)𝟑/𝟐
for (h/b ≥ 1) and where (α = a/b) = 1.77553
Therefore, the specimen would be expected to fail at SEN, since it’s modifying factor, F, is the
greatest among the three geometries.
Plane Stress and Plane Strain must be taken into consideration as well. If the thickness of the
specimen obeys the following equation, it can be said to be in the plane strain region. Otherwise,
it is somewhere in the plane stress region.
2
𝐾
𝑡 ≥ 2.5 ( 𝑆𝐼𝐶 ) = 1.089
𝑌
Finally, plasticity limitations were taken into account. The zone in which plastic deformation
occurs around a crack or notch time can be estimated. For LEFM theory to be valid, a region
around the notch tip where the intensity factor is dominant is necessary. For this purpose, the Kfield – which is dependent on the plastic zone – needs to be calculated. For a dominant K-field,
the following must hold true:
𝑎, (𝑏 − 𝑎), ℎ ≥
4 𝐾 2
( )
𝜋 𝑆𝑌
That is, the distance from the notch tip to all edges or other notches must be greater than four
times the plastic zone – a general factor of safety to assure a region of K-dominance.
As expected, the specimens failed at the single edged notch. Both specimens failed at a force of
approximately 6600 lbf – greater than the value of 2500 lbf calculated using the critical fracture
strength. It makes sense that this force would be greater, as the fracture strength should increase
as the thickness decreases (and thus movement into the plane stress region increases).
Specimens:
Failed specimen, extension rate 0.4 in/min
Failed specimen, extension rate 0.05 in/min
Note that the specimen run with a slower extension rate had time to bend – leading to a less
regular failure region, as well as more defined failure at DEN.