Fatigue

Fatigue / Introduction
Theory of Elasticity CE-527 (Fatigue) submitted to Dr. Nildem TAYŞİ By Hussein Ghanim HASAN (PhD. Student)
Fatigue was discovered when several
investigators noted that bridge and railroad
components were cracking when subjected
to repeated loading.
Fatigue is caused by repeated application of
stress to the metal. It is the failure of a
material by fracture when subjected to a
cyclic stress .
Fatigue occurring suddenly and without
warning. There is very little gross plastic
deformation associated with fatigue failure
and so is brittle like.
railroad
bridge
aircraft

Fatigue / Introduction
Fatigue is an important form of behaviour in all materials
including metals, plastics, rubber and concrete when rotating
and subjected to alternating stresses.
There are many other situations where fatigue failure will be
very dangerous.
Examples: door springs, aircraft wings are subjected to
repeated loads, oil and gas pipes are often subjected to static
loads will cause fatigue.

Fatigue / DEFINITIONS
• A fatigue cycle is defined as the time between two successive max (or min) in
the stress. The number of fatigue cycles per second is defined as the cyclic
frequency.
𝑆 𝑚: mean stress
𝑆 𝑎 : Amplitude stress
R : ratio stress

Type of Fatigue loading
I. Completely reversed cycle of
stress:
Explains the type of fatigue
loading where a member is
subjected to opposite loads
alternately with a mean stress
(𝑆 𝑚) is zero.
For example bending of steel
wire continuously in other
direction leads to alternate
tensile and compressive stresses
on its surface layers and failure
fatigue.
(a) Completely reversed stress cycle

ii. Repeated stress cycles:
where a member is subjected to
only tension but to various
degrees.
For example: A spring subjected to
repeated tension as in a toy would
lead to fatigue failure.
(b) repeated stress cycle

iii. Irregular or random stress
cycle:
This type of fatigue loading
where a member could be
subjected to irregular loads just as
in the case of aircraft wings
subjected to wind loads.
(c) random stress cycle
i.e. if the load changes from one magnitude to another the load is said to be
fluctuating load. (the direction does not necessarily change)

Under the strain-controlled fatigue test, a metal specimen may
exhibit the cyclic stress-strain response as following:
a.) Cyclically hardening
If the stress required to enforce the strain increases on subsequence
reversals, the material undergoes cyclic hardening as shown in Fig. In
this case, the yield and ultimate strength of the material are increased.
The example of the metal that exhibits this response is
the annealed pure metal.
b.) Cyclically softening
If the stress required to enforce the strain decreases on subsequence
reversals, the material undergoes cyclic softening as shown in Fig. In
this case, the yield and ultimate strength of the material are decreased.
The example of the metal that exhibits this response is
the cold worked pure metal.
Exhibit the cyclic stress-strain

c.) Cyclically stable
Through the cyclic hardening and softening, some intermediate
strength levels are attained which represents a cyclically stable
condition. The stable condition is usually reached in about 20-40% of
total fatigue life.
d.) Mixed behavior
A material exhibits cyclic softening at the early stage of fatigue life and
then cyclic hardening at the later stage of fatigue life.
Exhibit the cyclic stress-strain

Cyclic Stress-Strain Behavior
Consider a stress-strain response curve
from a fatigue test of a specimen as
shown, When the strain is increased from
0 to Ԑ max , the stress is also increased
from 0 to σmax.
Then, when we unload the specimen
from the strain Ԑmax to Ԑmin the stress-
strain curve follows the unloaded line,
and the stress is decreased from σ max to
σ min. Finally, if we reload the specimen
from Ԑ min to Ԑ max , the stress is
increased from σ min to σ max by
following the reload curve.

Cyclic Stress-Strain Behavior
It can be seen that there is a loop occurred
due to the unloading and reloading the
specimen. This loop is called hysteresis
loop.
It represents measurement of plastic
deformation work done on the material.
The area within the loop is the energy per
unit volume dissipates (reduced) during a
cycle.
During loading, the area under the stress-
strain curve is the strain energy per unit
volume absorbed by the material.

Distinguish between the stress at a point and the nominal
stress

Fatigue failure
The crack is developed from a small flaw until it reaches the
critical size.
It is usually developed at high stress gradient area. The stress
gradient becomes larger and larger, and the crack progresses
more rapidly.

Fatigue Crack Propagation
Normally, Crack propagation can be caused by cyclic
loading. Typical constant amplitude crack propagation as
shown in Fig

The crack propagation rate
𝑑𝑎
𝑑𝑁
versus stress
intensity range ∆𝑘
Region i : Crack growth rate is slow and its
behavior is associated with fatigue crack growth
edge value Δk below which the crack growth is
negligible.
Region II: The relationship between log
𝑑𝑎
𝑑𝑁
versus
logΔk is linear and steeper than the curve in Region I.
This is due to unstable crack growth of the test
specimen.

Region III: The crack growth rate is very high and
little fatigue life is involved.
The relationship in this line is
The value of m is important since it indicates the degree of
sensitivity of the growth rate of the stress.

Fatigue - Failures
 m
KC
dN
da

if we integrate between the initial size of a crack (𝑎 𝑜) and
the crack size required for fracture to occur (𝑎 𝑓), we find
that the number of cycles to failure is given by
From the steady state crack growth relationship of
Y = dimensionless geometric constant that varies depending on the
geometry of the crack, but usually taken as 1.

Example I
to

Example I

Example II

“S-N” curve is mean the
amplitude stress versus cycles to
failure, which when plotted using
the stress amplitude on the
vertical axis and the number of
cycle to failure on the horizontal
axis.
S - N Diagram and Stress Life Relation
S-N curve is A very useful way to visual the failure for a specific
material with the S-N curve.

S - N Diagram and Stress Life Relation
Se = endurance limit of the specimen (infinite life > 106
)
For (ferrous) materials exhibiting a knee in the S-N curve at 106
cycles.
For most nonferrous metal such as aluminum, there have not endurance
limit and the S - N curve has a continuous slope. An endurance limit for
these materials is taken on the stress value corresponding to 5×108
cycles.
Sf = fatigue strength of the specimen (infinite life > 5x 108
)

Fatigue Strength Diagram
Mean Stress Correction for Endurance Limit

Modified Goodman Diagram
Mean stress
Alternating
stress
m
a
Sut
Goodman line
Sy
Yield line
Sy
Se
Safe zone
C

Factors Affecting the Fatigue Crack Growth
Stress ratio effect
The increasing of R ratio will lead to increase a crack growth
rate.
Frequency effect
At normal environmental condition, frequency has little effect
on fatigue life for metallic structure. However, the growth rate
will be significantly affected if under an adverse environment.
Temperature effect
Fatigue life will be reduced if the temperature is increased.
Surface Roughness:
Smoothly polished specimens have higher fatigue strength.

Stress-based approach is based on the nominal stresses in
the region of the component being analyzed. The nominal
stress that can be resisted under cyclic loading is
determined by considering mean stresses and by making
adjustments for the effects of stress risers such as holes and
fillet.
Strain-based approach involves more detailed analysis of
the localized yielding that may occur at stress risers during
cyclic loading.
Fracture mechanic approach is used to treat growing crack
due to cyclic loading by using the method of fracture
mechanics.
The major approaches to analyzing and designing against
fatigue failure

Example III
Where the magnitude of the stress varies from σ min =-0.6 σ max to max σ . The
shaft is made of stress-relieved, the ultimate stress of 830 MPa , yielding
stress of 660 MPa , the endurance limit of 410 MPa. Determine the magnitude
of max σ based on a factor of safety of 1.80 against the failure at 𝑁𝑓 =
107
cycles.
Since the steel is a ductile material, we will use the Gerber relation to determine the magnitude of σ max.

Endurance Limit Modifying Factor
However, in most structural applications, the endurance limit of a
structural member is obtained from modifying the data from the
tests. There are many factors that affect the endurance limit. The
followings are significant factors:

Example III

Video about a Fatigue test
Theory of Elasticity
CE-527

References
Advanced mechanics of materials, Dr. Sittichai Seangatith. May
(2001)
Experimental Techniques In Materials And Mechanics, C.
Suryanarayana.(2011).
Modern Metal Fatigue Analysis, Draper John (2008)..
Fatigue Damage, Lalanne, C. (2009).

Fatigue / Discussion
Thank You for
your attention

Fatigue

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