Laboratory and Classroom Study of Low Cycle Fatigue and Linear Elastic
Fracture Mechanics
Abstract
Low cycle fatigue theory and linear elastic fracture mechanics are important topics for
mechanical engineering students to learn and understand. Essential in broadening the scope and
depth of students’ knowledge of mechanics, these topics create a better versed engineer with
experience in topics necessary in many industries. Current required coursework in Mechanical
Engineering at *** is insufficient in that it only covers high cycle fatigue theory and bypasses
fracture mechanics altogether. Fatigue and fracture specimens and tests were designed to create
demonstrations that could be easily incorporated into an existing course to show the contrast with
the fatigue life prediction methods currently being taught, as well as introduce students to
fracture mechanics. The results of the fatigue tests were favorable, with good correlation
between theoretical and experimental results while the fracture tests also proved successful in
that results were consistent and repeatable. These successful results were instrumental in
creating a lesson plan to be presented to students in the form of hands-on experience in
conjunction with classroom instruction in the theory. This paper presents an outline of the
current topics taught, the design, implementation, and results of the new instructional laboratory
tests, and the future plans for implementation in a classroom setting.
Background
This paper covers the creation of laboratory exercises in conjunction with classroom instruction
in order to better expose engineering students to topics that are currently not covered in the
curriculum. Specifically, the topics of low cycle fatigue and linear elastic fracture mechanics are
not well represented within the coursework for mechanical engineering students. The current
course that covers fatigue, Design of Machine Elements, focuses only on high cycle theories –
specifically stress based fatigue theories – with no mention of fracture mechanics whatsoever.
While high cycle fatigue theory is generally acceptable in many applications for materials like
steel under low stress conditions, the theory is insufficient in describing some of the more
complex phenomena witnessed under more intense loading in different materials, or crackgrowth-based life prediction methods that require the use of fracture mechanics.
High cycle fatigue theory focuses mainly on specimens that are subjected to relatively low stress
situations where the cyclic deformation is entirely elastic. A power relationship is usually
derived between the stress a specimen is subjected to and the amount of cycles required to fail
the part. The stress-life plot (S-N plot) is perhaps the most recognizable portion of this theory.
With the x and y axes logarithmically scaled in order to linearize the power relationship between
stress and cycles to failure, a clear relationship becomes apparent. Equations 1 through 3 outline
this relationship, where S1000 is the stress that would yield 1000 cycles to failure, Se is the
endurance limit – the stress for “infinite” life, S is the applied stress, and N is the number of
cycles to failure1. This theory, however, is rather basic and does not cover material that may be
subjected to loading that results in plastic deformation.
𝐶 1
𝑁 = 10−𝑏 𝑆 𝑏
(1)
1
𝑆1000
𝑏 = − log10
3
𝑆𝑒
(2)
(𝑆1000 )2
𝑆𝑒
(3)
𝐶 = log10
Strain Amplitude (in/in)
Cyclic loading in the plastic region results in low cycle count to failure. Low cycle fatigue is a
strain based theory that better represents the behavior of materials subjected to cyclic loading
resulting in plastic deformation. Low cycle fatigue theory takes into account both the elastic and
plastic strain, while high cycle fatigue theory ignores plasticity. Similar to the S-N plot of stresslife theory, strain life theory has a strain-life plot (ϵ-N plot) that has logarithmic axes to linearize
the power relationships given to elastic and plastic strain. For low life, plastic strain is dominant
over elastic strain. For high life - where stress based theories apply - the opposite is true and
elastic strain dominates plastic strain. The point at which these two curves cross - and the theory
changes - is known as the transition life. For an amount of cycles greater than the transition life,
where elastic strain dominates, stress life theory may be used. For a number of cycles less than
the transition life, plastic strain is dominant and a low cycle, strain-life approach must be used
for accurate life prediction. A sample strain-life plot is seen in Figure 1, with the elastic strain,
plastic strain, and total strain curves shown, along with a line marking the transition life – which
is detailed in Equation 41.
Δϵe/2
Δϵp/2
Δϵ/2
Transition Life
Reversals to Failure, 2Nf (reversals
Figure 1. Generic strain-life curves.
1
1 𝜖𝑓 ′𝐸 (𝑏−𝑐)
𝑁𝑡 = (
)
2 𝜎𝑓 ′
(4)
Linear elastic fracture mechanics (LEFM) is an important topic for many of the same reasons
low cycle fatigue is. Stress-based failure models often use the applied load and cross-sectional
area only in determining when a specimen will fail. This is insufficient, however, when the
geometry may be variable throughout a specimen. Though the cross-sectional area may stay
constant, intensity of loading may change due to the shape. This referenced geometrical change
is often due to notches or cracks in a specimen. Mathematical models exist for different
geometrical conditions based off of empirical data, which allows for more accurate fracture
predictions than traditional stress-based methods. Additionally, whereas yield strength and
ultimate strength are commonly used in said methods, LEFM uses a material’s fracture
toughness for more accurate failure predictions. A material’s fracture toughness is a property
which quantifies the ability of a material with a crack to resist fracture.
The need to characterize a material’s basic properties is also an important lesson. Published
values are generally for very specific variations of materials. Depending on manufacturing
practices and tolerances, properties such as the elastic modulus, ultimate strength, and yield
strength may be very different from the published values.
The exclusion of these theories from the current curriculum creates a gap in a mechanical
engineer’s knowledge of failure theory. With a switch from a quarter based academic schedule
to a semester based schedule slated for the Fall of 2013, the extra weeks per term create the
necessary time to teach more than just the high cycle fatigue that is covered now. With a large
portion of mechanical engineers at *** concentrating their coursework in specialized areas such
as aerospace and biomedical engineering, the need to describe the behavior of complex alloys
that may be subjected to higher stresses becomes even greater.
Experiment
The experiment was created to meet a set of design constraints that will allow easy incorporation
of this activity into an existing course. To clearly illustrate high cycle fatigue and low cycle
fatigue, the design constraints included:




Load capacity of the test system (±22,000 lbf)
Grip size (0.39 – 0.63 in diameter)
Time to run test (Approximately 2 hours)
Sample Length (Approximately 2 – 6 in)
Focusing first on the geometrical and material design considerations of the fatigue specimens, it
was first necessary that the specimen be the correct dimensions to fit within the grips of the
Instron 8801 servo-hydraulic fatigue test system. Maximum force, and thus maximum stress,
would determine whether a specimen could be loaded to fail in the low cycle region. Material
characteristics, such as the elastic modulus, were also important for this reason. Time proved to
be only a small consideration, as the Instron’s 100Hz capability would allow for even high cycle
tests to be completed within a reasonable amount of time. After reviewing a variety of
alternatives, a 0.25 in diameter hourglass 1018 Steel fatigue specimen (with 0.5 in grip diameter)
was chosen for this series of experiments. Tensile tests were performed in order to verify the
elastic modulus, E, the tensile strength, SU, and the 0.2% offset yield strength, SY. Figure 2
below shows the results from this test. Shown on this plot is the actual stress-strain plot for the
tensile test, as well as the 0.2% offset line in order to characterize the yield strength. Table 1
summarizes the important values found from this test.
100000
90000
80000
Stress (psi)
70000
60000
50000
40000
30000
20000
10000
0
0
2
4
6
8
Strain (%)
10
12
Figure 2. 1018 steel tensile test results
Table 1. Results from tensile test of fatigue specimen.
Published2
Experimental Difference (%)
E (ksi)
29,700
28,670
3.6%
Su (psi)
63,800
92,000
30.7%
Sy(psi)
53,700
77,000
30.3%
This showcases a prime example of the necessity to test a material instead of simply trusting
published values. Depending on manufacturing processes (such as cold drawing to diameters
smaller than tested values) properties may change dramatically from what may be found in
published sources. The experimental values were used for the modeling of the both stress-life
fatigue curve and strain-life fatigue curves. Expected fatigue behavior of this material using the
above experimental values is shown in Figure 3 and 4.
Alternating Stress, S (psi)
1E+05
Theoretical
Transition Life
1E+04
1E+02
1E+03
1E+04
1E+05
Life to Failure, N (Cycles)
1E+06
Figure 3. Expected stress-life fatigue behavior.
Strain Amplitude (in/in)
1E-01
1E-02
Δϵe/2
Δϵp/2
Δϵ/2
1E-03
Transition Life
1E-04
1E+03
1E+04
1E+05
Reversals to Failure, 2Nf (Reversals)
1E+06
Figure 4. Expected strain-life fatigue behavior
Fatigue tests were conducted in load control under R = -1 loading (i.e., mean stress = 0). Load
amplitude would be calculated depending on whether a test were to be run for high cycle fatigue
or low cycle fatigue, and frequency of cycling was decided upon at 10Hz, which would provide
relatively short tests for low cycle fatigue, and manageable tests for high cycle fatigue. 9
samples were available, and 3 were selected to be run to high cycles, with the other 6 to be
cycled to fail in the low cycle region. A summary of the planned tests is shown in Table 2,
showing expected life to failure at each applied load.
Table 2. Expected life based on applied load.
Load Amplitude (lbf)
3000 (2 tests)
Stress Amplitude (psi)
61,115
N (Cycles)
11,980
57,041
55,000
52,967
50,930
48,892
44,818
40,744
2800
2700
2600
2500
2400
2200
2000
21,000
28,245
38,373
52,795
73,589
149,355
324,299
A fracture specimen was designed in order to illustrate the difference between failure by net
section yield and that by fracture. Three different crack types were placed onto a sample, each
with the same total area. Computation of theoretical Mode I stress intensity factors for a
centered crack, single edged crack, and double edged crack could be done in order to predict
where the specimen would fail. This demonstrates the contrast with stress-based failure analysis
– where each crack geometry would be equal as load and cross-sectional area are constant
throughout the specimen. Constraints included the thickness of the plate (so as to be
accommodated by the test system), as well as other geometrical constraints related to the
mathematical theory pertaining to each notch type. A material with published values for fracture
toughness was imperative. 6061 Aluminum was chosen due to its availability and extensive
published material properties. Figure 5 shows a basic schematic of the fracture specimen used,
Equation 5 outlines the stress intensity factor calculation, and Equations 6 through 8 detail the
specific calculations for the centered notch, single edge notch, and double edge notch
respectively1.
𝐾𝐼 = 𝑓(𝑔)𝜎√𝜋𝑎
𝑓(𝑔) = √sec (
𝜋𝑎
)
2𝑛
(5)
(6)
𝑎
𝑎 2
𝑎 3
𝑎 4
𝑓(𝑔) = 1.12 − 0.231 ( ) + 10.55 ( ) − 21.72 ( ) + 30.39 ( )
𝑏
𝑏
𝑏
𝑏
(7)
𝑎
𝑎 2
𝑎 3
𝑓(𝑔) = 1.12 + 0.203 ( ) − 1.197 ( ) + 1.930 ( )
𝑏
𝑏
𝑏
(8)
Figure 5. Schematic of fatigue specimen.
Prototype Results
The fatigue specimens were cycled using the different peak loads described above. These results
were compared to the theoretical stress-life curve, as seen in Figure 6. Immediately evident is
that the specimens cycled above the transition life correlate well with the high cycle fatigue
curve, whereas those that failed at lives less than the transition life deviate from the curve
dramatically. This underscores the fact that high cycle fatigue theory is insufficient in describing
the behavior of material under similar loading conditions.
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
Figure 6. Fatigue Results plotted on stress-life curve.
When these data points are plotted against the theoretical strain-life plot, however, they all
correlate quite well (Figure 7). Both the high cycle and low cycle specimens match up well with
the total strain curve. Of note is the fact that the 3 highest life data points are those that are
dominated by elastic strain, and thus match with both the stress-life and strain-life plots.
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
Figure 7. Fatigue results plotted on strain-life curves.
Comparing Figures 6 and 7, it’s clear to see the very reason why low cycle fatigue theory exists.
Overall, better correlation is achieved using strain-life theory rather than stress-life theory. Most
of the data points correlate with the total strain curve in Figure 7, whereas the data points deviate
from the stress-life curve in Figure 6 below the transition life. The specimens in this study with
the lowest life deviate even from strain-life theory. During testing, buckling of the material was
observed, due to a probable misalignment of the test system. This misalignment caused bending
in the high load specimens, which hastened their failure and led to a discrepancy with the
theoretical calculations.
Magnified inspection of the fracture surfaces of these fatigue specimens reveal that they failed as
expected. Beach marks are present on the surfaces, which are indicative of a crack propagating
through the specimen causing failure. These lines, running parallel to the direction of crack
growth, can be seen in Figure 8. Towards the right of the photo, the specimen’s edge and
fracture surface can be seen, with Beach marks leading up to that point.
Figure 8. Magnified fracture surface of fatigue specimen.
The fracture tests proved to be equally successful. Two tests were performed on similar
specimens at different extension rates – 0.4 in/min and 0.05 in/min. Both specimens failed at the
same point in the specimen (at the point of the single edge notch) and at roughly the same force
(approximately 6600 lbf). Figures 9 and 10 below show the fractured specimens while Figure 11
is a plot of the loading results from the two specimens. The specimen extended at the slower rate
exhibited added bending. As the crack propagated across the single edge notch and the load
concentrated even more to one side, the double edge notch had time to widen and begin to crack.
While this started on the specimen extended at the faster rate, it is not nearly as evident.
Figure 9. Failed specimen, extension rate 0.4 in/min
Figure 10. Failed specimen, extension rate 0.05 in/min
7000
6000
Load (lbf)
5000
4000
0.4 in/min
3000
0.05 in/min
2000
1000
0
0
200
400
600
Time (s)
800
1000
Figure 11. Load results of fracture specimens.
The fracture specimen loaded at 0.4 in/min also exhibits a typical fracture surface expected from
a specimen in the plane stress region. Figure 12 attempts to capture the angled fracture across
the width of the material, expected from this specimen’s geometry and this material’s properties.
While the other specimen exhibits a similar surface, it’s difficult to capture due to the irregular
path the crack followed due to the extra bending it had experienced.
Figure 12. Fracture surface of specimen, extension rate 0.4 in/min.
While the model predicted that the specimen would fail at 2600 lbf, it used the plane strain value
for the fracture toughness. Since fracture toughness is dependent on the thickness of the
specimen, and empirical data is necessary in order to characterize this relationship, this value
cannot be directly computed. It is, however, to be expected that the specimen would fail above
the modeled value since the fracture toughness should increase as the thickness decreases into
the plane stress region.
Classroom Implementation
I can discuss a bit here, but I think this one is up to Dr. Boedo to hammer out exactly how things
end up being presented in DME.
Discussion and Future Plans
Again, I think this is going to be dependent on how well it’s received by students in class.
Summary
Outline a quarter’s worth of work, yada yada yada, leave open ended for how it goes in the fall.
Acknowledgements
Bibliography
1 – BANNANTINE, S-N equation (bottom of page 1)
2 – 1018 material properties (page 4)
Keep for reference
1.
ASME, "Vision 2030―Creating the Future of Mechanical Engineering Education,"
American Society of Mechanical Engineers, New York 2010.
2.
Felder, R., Brent, R. [2004], “The intellectual development of science and engineering
students part 1. Models and challenges”, J. Eng. Educ., Vol. 93, No. 4, pp. 269-277.
3.
Felder, R., Brent, R. [2004], “The intellectual development of science and engineering
students part 2. Teaching to promote growth”, J. Eng. Educ., Vol. 93, No. 4, pp. 279-291.
4.
DeBartolo, E., Zaczek, M., Hoffman, C. [2006], “Student-faculty partnerships”, Proc.
Amer. Soc. for Eng. Educ. Conf. and Expo., Chicago, IL.
5.
Bailey, M. [2007] “Enhancing life-long learning and communication abilities through a
unique series of projects in thermodynamics”, Proc. Amer. Soc. for Eng. Educ. Conf. and Expo.,
Honolulu, HI.
6.
Bailey, M. [2011] “Studying the impact on mechanical engineering students who
participate in distinctive projects in thermodynamics”, Proc. Amer. Soc. for Eng. Educ. Conf.
and Expo., Vancouver, BC.
7.
T.A. Angelo and K. P. Cross, 1993. Classroom Assessment Techniques, 2nd ed. San
Francisco: Jossey-Bass.
8.
Christopher H. Conley, Stephen J. Ressler, Thomas A. Lenox, and Jerry W. Samples,
“Teaching Teachers to Teach Engineering – T4E”, Journal of Engineering Education, January
2000, pp 31-38.