Éléments sur le comportement des matériaux

1. CONTENTS
INTRODUCTION
1
STRESS – LIFE (S-N)
2
DESIGN S-N CURVES WITH ULTIMATE
TENSILE STRENGH
4
NOTCH EFFECT
5
MEAN STRESS EFFECT
6
COMBINED PROPORTIONAL LOADS
7
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STRESS-LIFE AND FATIGUE LIMIT
TESTING
3

2. 1 INTRODUCTION
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What is Fatigue ?
Fatigue is a localized damage process of a component produced by cyclic
loading. It is the result of the cumulative process consisting of crack initiation,
propagation, and final fracture of a component.
What is Stress ?
Stress is a physical quantity that expresses the internal forces that
neighboring particles of a continuous material exert on each other or Stress is
defined as the force per unit area of a material.
What is the relation between them ?
To treat this topic, we are going to show how fatigue analysis and design can be
based on stress, by focusing on how to establish fatigue data and determine
fatigue life based on a stress-based damage.
We have to note that cyclic stresses are the governing parameter for fatigue
failure.

3. 2 STRESS-LIFE (S-N)
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The cyclic loading applied to the specimen is defined by either a constant stress range (𝑆𝑟)
or a constant stress amplitude (𝑆𝑎).
The stress range is the algebraic difference between the maximum stress and the minimum
stress and the stress amplitude is equal to one-half of the stress range.
𝑆𝑟 = 𝑆𝑚𝑎𝑥 − 𝑆𝑚𝑖𝑛 and 𝑆𝑎= 𝑆𝑟 2
Also, S-N fatigue testing is conducted using a loading that alternate about a zero mean
stress. We call it fully reversed loading.
The mean stress is given by :
𝑆𝑚 =
𝑆𝑚𝑎𝑥 + 𝑆𝑚𝑖𝑛
2
The fatigue life of the specimen at a combination of stress amplitude and mean stress
levels is defined by the number of cycles to failure 𝑁𝑓 .
Here, the fatigue life (cycles or reversals) refers to the life required to nucleate and grow
a small crack to visible crack length.
NOTE : each cycle is equal to two reversals

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The typical relation between stress and fatigue life is
𝑆𝑎 = 𝑆′
𝑓(2𝑁𝑓)𝑏
Where b is the fatigue strength and 𝑆′
𝑓 the fatigue strength coefficient.
Plot of this relation is called S-N curves.
S–N curve can be generated for standard smooth material specimens, for individual
manufactured structural components, for subassemblies, or for complete structures.
S–N curve provides the baseline fatigue data on a given geometry, loading condition, and
material processing for use in subsequent fatigue life and strength analyses. This baseline
data can be adjusted to account for realistic component conditions such as notches,
size, surface finish, surface treatments, temperature, and various types of loading.
How to plot S-N curves for a given specimen ?

5. 3 STRESS-LIFE AND FATIGUE LIMIT TESTING
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S-N fatigue tests tends to give how much time a specimen can resist up to its complete
failure under a cyclic stress.
Tests consist of a specified number of specimen that would be subjected individually to a
specified stress level and counting how much cycle is necessary to get the failure of the
specimen.
For the same stress level, two specimens can break up at different number of cycle. Also,
observation show that we can have two specimens under different stress level and have the
same fatigue life. So the statistical nature of the fatigue must be considered.
Many statistical methods are used to conducted testing and each methods is particularly
different of the other.
S–N test methods presented by the Japan Society of Mechanical Engineers (1981),
Nakazawa and Kodama (1987), ASTM (1998), Shen (1994), Wirshing (1983), and
Kececioglu (2003) are widely used by researchers for S–N testing and fatigue life
predictions.

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3 STRESS-LIFE AND FATIGUE LIMIT TESTING
It is necessary to understood requirements, limitations, and approaches to the
construction of an S–N curve.
requirements, limitations, and approaches
We are going to focuses on the median S–N test method with a small sample size
presented by the Japan Society of Mechanical Engineer.
Requirements
This method requires 14 specimens. Eight specimens are used to determine the finite
fatigue life region, and six specimens are used to find the fatigue limit.
Approaches
The curve for the finite life region is determined by testing two samples at each of four
different levels of stress amplitude and the fatigue limit is tested by the staircase method
with six specimens. The recommended test sequence is shown in Figure 4.3, in which the
number next to the data point represents the order in which the specimen is to be
tested.
The finite life region data is assumed linear in the log–log coordinates, and the data are
analyzed by the least-squares method.

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3 STRESS-LIFE AND FATIGUE LIMIT TESTING
b. Design s–n curves in the finite life region
As the S-N curves have been obtained, in practice, we are considering its value is
associated with an error. So, we have to define a reliability and confidence on this value.
a. Analysis of fatigue data in the finite
life region
Analysis is based on the least-squares
method for generating a line of best fit
from the data.
.
The regression line is
𝒀 = 𝑨 + 𝑩𝑿
Where
𝑩 = 𝒊=𝟏
𝒏𝒔
(𝑋𝑖 − 𝑿)(𝑌𝑖 − 𝒀)
𝒊=𝟏
𝒏
(𝑋𝑖 − 𝑿) ²
𝑨 = 𝒀 − 𝑩𝑿

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3 STRESS-LIFE AND FATIGUE LIMIT TESTING
reliability is possibility of survival of the specimen tested and confidence traduce how
much we can believe in the reliability value .
This permit to have an interval or bounds around the median S-N curves plotted earlier.
The equation of this bounds or the design S-N curves are equation is
𝒀𝑳 𝑿𝒊 = 𝒀 𝑿𝒊 − 𝑲 ∗ 𝒔
where 𝑠 is the sample standard error of 𝑌 on 𝑋𝑖, 𝐾 is a multiplier equal to 2 or 3.
Limitations
As K doesn’t depend of the sample size( K is constant), this method fail to account for the
statistical distribution of fatigue life.
To overcome this, another methodology is used : the one-side lower-bound tolerance
limit method (Lieberman, 1958) , double-sided confidence intervals approach (ASTM, 1998)
and the approximate Owen one-side tolerance limit (Shen et al., 1996).

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3 STRESS-LIFE AND FATIGUE LIMIT TESTING
For double-sided confidence intervals approach (ASTM, 1998) and the approximate Owen
one-side tolerance limit (Shen et al., 1996), the value of K depends on the sample size.
c. Fatigue strength testing
The objective here is to estimate a statistical distribution of the fatigue strength at a specific
high-cycle fatigue life. The staircase method is widely used for this test.
Description
A mean fatigue limit has to be first estimated, and a fatigue life test is then conducted at a
stress level a little higher than the estimated mean. If the specimen fails prior to the life of
interest, the next specimen has to be tested at a lower stress level. If the specimen does not
fail within this life of interest, a new test has to be conducted at a higher stress level.
Therefore, each test is dependent on the previous test results, and the test continues with
a stress level increased or decreased.
To determine statistical parameters of the test results two typical data reduction are used
: techniques, the Dixon-Mood (1948) and the Zhang-Kececioglu (1998) methods.

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4 DESIGN S-N CURVES WITH ULTIMATE TENSILE STRENGH
Methodology
1- estimate the ultimate tensile strength
For low and medium strength of steels with low Brinell hardness (<500 BHN), the ultimate
tensile strength of a material can be linearly approximated as :
2- estimate the fatigue strength at 1000 cycles
For low and medium strength of steels with low Brinell hardness (<500 BHN), the ultimate
tensile strength of a material can be linearly approximated as :
where 𝑆1000 is the fatigue strength at 1000 cycles based on the standard test specimens
and 𝐶𝑅 is the modifying factor at a specified reliability level.

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3- estimate the fatigue limit
The fatigue limit 𝑆𝑒 can be estimated by modifying the bending fatigue limit 𝑆𝑏𝑒 with the
four factors for the type of loading (𝐶𝐿), surface finish (𝐶𝑆), size (𝐶𝐷), and reliability level
(𝐶𝑅):
4 DESIGN S-N CURVES WITH ULTIMATE TENSILE STRENGH
Note that these modifying factors are empirically based and usually range from 0.0 to 1.0
fatigue strength at a specific fatigue life can be determined by the following equation:

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4 DESIGN S-N CURVES WITH ULTIMATE TENSILE STRENGH

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5 NOTCH EFFECT
Notch have effect on the fatigue Failure of a specimen.
It is dependent of a factor called fatigue strengh reduction factor and defined by
For engineering applications, the fatigue strength reduction factor can be empirically related
to the elastic stress concentration factor by a notch sensitivity factor q
Value of q is given by

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5 NOTCH EFFECT
We can use relative stress gradient instead of using the Notch radius to calculate the stress
gradient effects on the fatigue strength reduction
Notch effect at intermediate and short lives
where 𝐾′𝑓 is the fatigue strength reduction factor at 1000 cycles.
Estimate of fatigue life for a notched component
Where 103 ≤ 𝑁 ≤ 106.
And using
RSG is dependent of the radius and the type of loadings.

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6 MEAN STRESS EFFECT
Mean stresses have a significant effect on fatigue behavior of components.
Normal mean stresses are responsible for the opening and closing state of microcracks :
tensile normal mean stresses are detrimental and compressive normal mean stresses are
beneficial in terms of fatigue strength.
The shear mean stress does not influence the opening and closing state of microcracks,
and, not surprisingly, has little effect on crack propagation.
Gerber’s mean stress correction
There are other mean stress
correction equation.

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7 COMBINED PROPORTIONAL LOADS
Previously, specimen were subjected only to a uni-axial loading,
Here, multi-axial loadings are used to conduct fatigue test of the specimen.
As for each loading we have a stress , the effective stress can be found by using the von
Mises criterion with the stresses calculated from the loads and the corresponding fatigue
strength reduction factors. For brittle materials, the maximum principal stress theory is
recommended
Approach are fully described in the textbooks Fatigue Testing and Analysis. Theory and
Practice (2004, Butterworth-Heinemann) by Yung-Li Lee, Jwo Pan, Richard Hathaway,
Mark Barkey.

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End