Code_Aster ®
Version
7.4
Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
A. Mr. DONORE, F. MEISSONNIER Key
:
R7.04.01-C Page
: 1/38
Organization (S): EDF-R & D/AMA
Handbook of Référence
R7.04 booklet: Evaluation of the damage
R7.04.01 document
Estimate of the fatigue life to large
a number of cycles and in fatigue oligocyclic
Summary:
The majority of the industrial structures are subjected to variable efforts in the time which, repeated one
great number of times can lead to their rupture by fatigue. One presents in this note the principal ones
functionalities of commands POST_FATIGUE [U4.83.01] and/or CALC_FATIGUE [U4.83.02] and/or CALC_ELEM
[U4.81.01] which makes it possible to estimate the limit of endurance and the office plurality of damage of a part.
The various methods available are:
·
linear office plurality: methods based on uniaxial tests (methods of Wöhler, Manson-Coffin
and Taheri).
These methods have as a common point to determine a value of damage starting from the evolution with
run from the characterizing time of a scalar component, for the calculation of the damage, the amplitude of
constraints or of structural deformations.
With this intention, it is necessary to extract by a method of counting of cycles, the elementary cycles of
loading undergone by the structure, to determine the elementary damage associated with each cycle and
to determine the total damage by a linear rule of office plurality;
·
nonlinear office plurality: method of Lemaître and method of Lemaître-Sermage
These methods make it possible to calculate the damage D at every moment T, starting from the data of
tensor of the constraints (T) and the cumulated plastic deformation p (T);
·
limit of endurance: criteria of Crossland and Dang Van Papadopoulos
These criteria apply to uniaxial or multiaxial loadings in periodic constraints. They
provide a value of criterion indicating if there is fatigue or not. Definite equivalent constraints
for these criteria can also be used for to calculate the office plurality of damage.
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
A. Mr. DONORE, F. MEISSONNIER Key
:
R7.04.01-C Page
: 2/38
Count
matters
1 Introduction ............................................................................................................................................ 3
2 Methods of Wöhler, Manson-Coffin and Taheri ..................................................................................... 5
2.1 Extraction of the peaks ........................................................................................................................... 5
2.2 Methods of counting of cycles ................................................................................................... 6
2.2.1 Method RAINFLOW .............................................................................................................. 6
2.2.2 Method RCC_M .................................................................................................................. 11
2.2.3 Method “naturalness” ................................................................................................................. 13
2.3 Calculation of the damage: method of Wöhler .................................................................................... 14
2.3.1 Diagram of endurance ...................................................................................................... 15
2.3.2 Influence geometrical parameters on the endurance .................................................... 17
2.3.2.1 Coefficient of stress concentration ............................................................. 17
2.3.2.2 Elastoplastic coefficient of concentration .......................................................... 18
2.3.3 Influence average constraint ..................................................................................... 18
2.4 Calculation of the damage: method of Manson-Coffin ........................................................................ 19
2.5 Calculation of the damage: method of Taheri ..................................................................................... 21
2.5.1 Method Taheri-Manson ...................................................................................................... 21
2.5.2 Taheri-Mixed method ........................................................................................................... 22
2.6 Calculation of the total damage ............................................................................................................... 23
2.7 Conclusion ..................................................................................................................................... 23
3 Calculation of the damage of Lemaître generalized ....................................................................................... 24
3.1 The law of Lemaître generalized ...................................................................................................... 24
4 Criteria of Crossland and Dang Van Papadopoulos ............................................................................. 26
4.1 Criterion of Crossland ...................................................................................................................... 26
4.2 Criterion of Dang Van Papadopoulos .............................................................................................. 27
4.3 Calculation of a value of damage .................................................................................................. 29
5 Conclusion ........................................................................................................................................... 30
6 Bibliography ........................................................................................................................................ 31
Appendix 1 ................................................................................................................................................. 32
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
A. Mr. DONORE, F. MEISSONNIER Key
:
R7.04.01-C Page
: 3/38
1 Introduction
The industrial experiment shows that the ruptures of structure or machine components in
normal operation are generally due to fatigue. Its masked progressive character
conduit very often with a brutal rupture.
One understands by fatigue the consecutive modification of the properties of materials to the application of
cycles of efforts, cycles whose repetition can lead to the rupture of the parts made up with these
materials [bib1].
Various methods are available for the evaluation of the damage. The second part of it
document is devoted to the presentation of oldest which is methods based on
uniaxial tests: method of Wöhler, method of Manson-Coffin and more recently methods
proposed by S. Taheri (EDF-R & D/AMA).
These methods have as a common point to determine a value of damage starting from the evolution with
run from the characterizing time of a scalar component, for the calculation of the damage, the state of
constraints or of structural deformations.
The evaluation of the damage is based on the use of curves of fatigue of the material (Wöhler or
Manson-Coffin), associating a variation of constraint of amplitude given to a number of cycles
acceptable.
To use these curves starting from a real uniaxial loading, it is necessary to treat the history of
constraints or of the deformations by identifying elementary cycles (cf [§2.2]).
The difficulty in defining a cycle for a complex signal explains the profusion of the methods of
counting appeared in the literature [bib2].
Two methods among most usually used were introduced into Code_Aster:
·
counting of extended in cascade or method RAINFLOW,
·
regulate RCC_M.
One adds to it a third method which we will call method of “natural” counting and which
respect the command of application of the cycles of loading.
For each elementary cycle, one evaluates an elementary damage using methods founded on
curves of Wöhler, Manson-Coffin or both simultaneously.
For the method of Wöhler (cf [§2.3]) the user can correct the constraint to be integrated in the curve
of Wöhler by:
·
a concentration factor of constraints KT, to take account of the geometry of
part,
·
an elastoplastic coefficient of concentration Ke,
·
a correction of Goodman or Gerber in the diagram of Haigh to take account of
average constraint of the cycle.
In addition, one proposes to define the curve of Wöhler in three different forms, a form
discretized point by point and two analytical forms.
The method of Manson-Coffin (cf [§2.4]) applies to loadings in deformations. The curve
of Manson-Coffin is defined in a single form, forms discretized point by point.
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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Author (S):
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:
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The methods of Taheri (cf [§2.5]) also apply to loadings in deformations and
require the data of the curve of Manson-Coffin and possibly of the curve of Wöhler. Their
characteristic is to hold account about application of the elementary cycles of loading with
structure, contrary to the two other methods.
Note:
Three methods of extraction of the elementary cycles are available: method of Rainflow,
regulate RCC_M and “natural” counting.
The first two methods do not hold account about application of the cycles what
is of no importance for the calculation of the damage by the methods of Wöhler or Manson-Coffin.
For the calculation of the damage by the methods of Taheri, it is necessary to use the method of extraction of
cycles by “natural” counting [§2.2.3] which respects the command of application of the cycles.
For the whole of these methods calculation of the total damage undergone by the structure is determined by
a method of office plurality, the rule of Miner.
The third part of this document presents the methods of Lemaître and Lemaître-Sermage which are
“analytical” methods allowing to calculate the damage D (in each moment T) from
data of the tensor of constraints (T) and of the cumulated plastic deformation p (T). These two
methods apply to loadings in unspecified constraints (uniaxial or multiaxial).
A linear rule of office plurality can be used to determine the total damage undergone by the structure.
Lastly, the criteria of Crossland and Dang Van Papadopoulos are presented in fourth and last
part of this document. They apply to unspecified loadings (uniaxial or multiaxial) in
constraints and periodicals. They provide a value of criterion indicating if there is fatigue or not.
From the value of the criterion, one can specify a scalar component characterizing the state of
structure for calculation of the damage and to determine a value of damage by using the curve of
Wöhler of material.
Handbook of Référence
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Code_Aster ®
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
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:
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2
Methods of Wöhler, Manson-Coffin and Taheri
2.1
Extraction of the peaks
The user provides to Code_Aster a function which defines the history (scalar) loading in one
not given. For that, it has key word HISTOIRE.
On this history of the loading, which can be complex, a first operation of extraction of the peaks
is realized. This operation consists in reducing the history of loading to the only fundamental peaks.
Note:
In fatigue, one names loading in a point given the value of the response of the structure in
this point.
In the use of the curves of Wöhler, it is about constraint in this point.
In the use of the curves of Manson-Coffin, it is about deformation in this point.
The history of loading is thus the evolution in the course of the time of a constraint, or one
deformation.
If the function remains increasing or decreasing on more than two consecutive points, they are removed
intermediate points to keep only the two extreme points.
One also removes history of the loading the points for which variation of the value
constraint or deformation is lower than a certain level chosen by the user. That
amounts applying a filter to the history of the loading. The value of the level of the filter is introduced by
the user under key word DELTA_OSCI.
For illustration let us consider the following history of loading:
N°
not
1 2 3
4 5 6
7 8
9
10
11
12
13
14
Moment 0.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. 11. 12.
13.
Loading 4. 7. 2. 10. 9.6
9.8
5.
9.
3.
4.
2. 2.4 2.2
12.
N°
not
15 16 17 18 19 20
21 22
23 24 25 26 27 28 29
Moment 14.
15.
16.
17.
18.
19. 20.
21.
22.
23.
24. 25. 26.
27. 28.
Loading 5. 11. 1. 4. 3. 10.
6.
8.
12.
4.
8. 1. 9.
4. 6.
The extraction of the peaks of this history of loading, with a value of delta of 0.9 conduit with
to destroy all the oscillations of amplitude lower than 0.9. What leads to the history of loading
following:
N°
not
1 2 3 4 7 8 9
10
11 14 15 16 17
Moment 0.
1.
2.
3.
6.
7. 8.
9.
10. 13. 14. 15. 16.
Loading 4. 7. 2. 10. 5. 9. 3.
4.
2. 12. 5. 11. 1.
N°
not
18 19 20 21 23 24 25 26 27 28 29
Moment 17.
18.
19.
20.
22. 23. 24.
25.
26. 27. 28.
Loading 4. 3. 10. 6.
12. 4. 8.
1.
9. 4. 6.
Note:
Let us note CH the value of the loading; CH can be a constraint or a deformation.
Handbook of Référence
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Titrate:
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Date:
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:
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History of loading was removed:
·
item 5 bus CH =
CH
)
5
(
- CH (4) <0.9,
·
item 6 bus CH =
CH (6)
- CH (4) <0.9,
·
item 12 bus CH =
CH 12
(
)
- CH)
11
(
<0.9,
·
the point 13 bus CH =
CH
)
13
(
- CH)
11
(
0
< .9.
In the same way one removes the point 22 bus the history of loading is increasing between the items 21, 22 and
23 and thus one keep only the extreme points.
2.2
Methods of counting of cycles
During their life, the industrial structures are generally subjected to loadings
complexes whose levels of stresses are variable.
The methods of counting of cycles make it possible to extract from the history of loading, of the cycles
elementary according to various criteria.
Code_Aster proposes three distinct methods including two nonstatistical methods among
methods most usually used.
2.2.1 Method
RAINFLOW
Method of counting of extended in cascade more often called method of RAINFLOW,
defines cycles which physically correspond to loops of hysteresis in the plan
stress-strains. In the literature, one counts several alternatives of this method.
The algorithm implemented in Code_Aster is essentially that proposed by
recommendation AFNOR A 03-406 of November 1993 [bib3] (with characteristics which is
specified during the presentation of the detail of the algorithm) and breaks up into three stages:
·
A first stage which consists in rearranging the history of the loading (T) or (T) of such
left that the loading begin with the maximum value, in absolute value, of the loading.
Note:
In recommendation AFNOR A 03-406, it is not made state of a rearrangement of
history of loading. This rearrangement is however carried out in the software
POSTDAM [bib2] and included in Code_Aster.
·
The second stage consists in extracting the elementary cycles from the history of loading
thus rearranged.
The method consists in being based on four successive points of the history of loading
(CH (I), i=, 1Nbpoint).
One notes:
X = CH (I +)
1
- CH (I) and Y = CH (I +)
2
- CH (I +)
1
and Z = CH (i+
)
3
- CH (i+)
2.
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Titrate:
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As long as Y is strictly higher than X
with
or Z, one traverses the history of the loading in
moving of a point towards the line (what amounts incrementing the value of I).
As soon as Y is lower or equal than X and inferior or equal to Z, it is considered that one has
met an elementary cycle which is defined by the two points (I +)
1 and (I + 2).
The amplitude of the cycle is given by CH = CH (I +
)
1
- CH (I + 2).
When the cycle is extracted one removes the two points of the history of loading and one
the algorithm continues.
·
The third stage consists in treating the residue, i.e. the remaining history of loading
after the stage of extraction of the cycles.
With this intention, one adds the same residue with his continuation possibly realizing some
precautions on the level of connection following the values of the extrema considered thus that
value of the first and the last slope of the residue.
The last point of the residue the first point of the cycle succeeds. So points considered
can not seem extrema more. If that occurs, it is appropriate
to eliminate. Eight different cases are encountered. To treat them explicitly, let us call
1
R and R2 the first two points of the residue and RN and R
1
-
N its the last two points.
Note:
Recommendation AFNOR A 03-406 fact also state of a possible preprocessing of
signal, which would consist of a filtering of the signal (suppression of the parasites) and of one
quantification of the history of loading.
The filtering of the signal is possible, at the request of the user (see [§2.1]. Extraction of
peaks).
The quantification of the signal can be useful for the speed of the analysis of the results of
analysis of fatigue. Practically, the quantification of the signal consists in cutting out
maximum extent of the signal in classes of intervals of constant width called not,
and to bring back to a value representative of a given class (its average value in
General) all values located in this class. This possibility of preprocessing
signal as for it, is not available in Code_Aster.
Handbook of Référence
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Titrate:
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Date:
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:
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Encountered case
Connection
1) (R
).(
)
).(
)
N
Rn1 R2
R1 > 0 and (RN
Rn1 R1
RN < 0
R
RN
R
N
R
2
2
Rn1
R
R
n1
1
R1
R
R
n1
1
R
R
n1
1
R
R
N
R
2
RN
2
has) Raccordement without problem: transition (RN, R1)
Encountered case
Connection
2) (R
).(
)
).(
) > 0
N
Rn1 R2
R1 > 0 and (RN
Rn1 R1
RN · 0
R
R
2
2
RN
R1
Rn1
Rn1
Rn1
Rn1
R1
RN
R
R
2
2
b) Raccordement transition (Rn1, R2), one eliminates R1 and RN
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Date:
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Encountered case
Connection
3) (R
).(
)
).(
)
N
Rn1 R2
R1 < 0 and (RN
Rn1 R1
RN < 0
R
R
N
N
R1
R
R
n1
n1
R2
R2
R
R
2
2
Rn1
Rn1
R1
RN
RN
c) Raccordement transition (RN, R2), one eliminates R1
Encountered case
Connection
4) (R
).(
)
).(
)
N
Rn1 R2
R1 < 0 and (RN
Rn1 R1
RN · 0
> 0
R
R
1
1
RN
R
R
2
2
R
Rn1
n1
Rn1
Rn1
R
R2
2
RN
R
R
1
1
D) Raccordement transition (Rn1, R1), one eliminates RN
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
A. Mr. DONORE, F. MEISSONNIER Key
:
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In order to illustrate the method and to clarify the points which would remain obscure, one considers the history of
loading following (which for the example is considered of type forced):
N° not
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Moment 0.
1.
2.
3.
4.
5.
6. 7.
8.
9.
10.
11. 12. 13. 14.
Loading 0. 40. 10. 60. 20. 50.
30. 80.
70.
30.
50. 20. 30. 25. 0.
(T)
8
60.
4
6
2
10
7
14
12
5
1
15
T
3
13
11
9
The method of RAINFLOW thus leads, on this example, (see [§Annexe1], for detail of the stages of
the algorithm) with the determination of 7 elementary cycles defined by the maximum value and the value
minimal of the loading, for each cycle.
Cycle 1:
VALMAX = 20.
VALMIN = 30.
Cycle 2:
VALMAX = 25.
VALMIN = 0.
Cycle 3:
VALMAX = 30.
VALMIN = 50.
Cycle 4:
VALMAX = 40.
VALMIN = 10.
Cycle 5:
VALMAX = 50.
VALMIN = 30.
Cycle 6:
VALMAX = 60.
VALMIN = 20.
Cycle 7:
VALMAX = 80.
VALMIN = 70.
Note:
·
The calculation of the damage not holding account about appearance of the elementary cycles
of loading, it is without consequence to rearrange the history of the loading.
·
For the methods of Taheri, the command of application of the elementary cycles of loading is
taken into account, also very vigilant with the use of such a method of counting is necessary it to be
cycles. It is advised, for the calculation of the damage by the methods of Taheri, to use
method of “natural” counting known as [§2.2.3].
Handbook of Référence
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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Author (S):
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2.2.2 Method
RCC_M
This method consists in forming the elementary cycles of stress while starting with those which
the greatest variations cause.
Thus for a history of loading comprising NR points, one determines NR/2 cycles elementary
if NR is even and NR/2+1 if NR is odd.
The algorithm breaks up into two stages. The first stage consists in ordering the history of
loading of smallest with the greatest value of the constraint, or the deformation.
The second stage consists, as for it, to form the elementary cycles with the greatest variation
value of the constraint, or deformation.
On the history of loading CH (T) rearranged, the elementary cycles are defined by:
VALMAX = CH
for I =,
1 NR/2
NR + -
1 I
VALMIN =
I
CH
If NR is odd one determines a definite additional cycle by:
VALMAX = CH
if CH
> CH
NR/2+1
NR/2+1
m
VALMIN
= CH
+
NR/2+
2 *
1
m
CH
and
VALMAX = CH
if not
NR/2+1
VALMIN
= CH
+
NR/2+
2 *
1
m
CH
NR
1
where
m
CH = constraint average or average deformation of the loading =
I
CH.
NR 1
To illustrate method RCC_M let us consider the same example as that used for the method
RAINFLOW (of which the loading was considered of type forced).
N° not
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Moment 0.
1.
2.
3.
4.
5.
6. 7.
8. 9.
10.
11. 12. 13. 14.
Loading 0. 40. 10. 60. 20. 50.
30. 80.
70. 30.
50. 20. 30. 25. 0.
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(T)
8
4
6
2
10
7
14
12
5
E 7
cycl
E 6
E 5
E 4
1
cycl
cycl
cycl
E 3
T
15
E 2
3
cycl
E 1
cycl
cycl
13
11
9
The first stage which consists in ordering the history of the loading, of smallest with largest
value of the loading, conduit to following storage:
N° not
9
11
13
3
1 15
5 12
14
7
10
2
6
4
8
Loading 70. 50. 30. 10. 0. 0.
20. 20. 25. 30.
30. 40. 50. 60. 80.
The history of loading being composed of 15 points, method RCC_M determines 8 cycles
elementary:
Cycle 1:
VALMAX = 80.
and
VALMIN = 70.
Cycle 2:
VALMAX = 60.
and
VALMIN = 50.
Cycle 3:
VALMAX = 50.
and
VALMIN = 30.
Cycle 4:
VALMAX = 40.
and
VALMIN = 10.
Cycle 5:
VALMAX = 30.
and
VALMIN = 0.
Cycle 6:
VALMAX = 30.
and
VALMIN = 0.
Cycle 7:
VALMAX = 25.
and
VALMIN = 20.
NR
1
Cycle 8:
VALMAX = 20.
and
VALMIN = 6.
because
=
m
= 6.
I
NR 1
Note:
This method of counting of cycles does not hold absolutely account about appearance of
cycles, and systematically orders the elementary cycles by decreasing amplitude. This
method must be used with vigilance for the calculation of the damage by the methods of Taheri
whose characteristic is to hold account about application of the cycles of loading. For
calculation of the damage by the methods of Taheri, it is strongly advised to use the method of
“natural” counting of cycles known as [§2.2.3].
Handbook of Référence
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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2.2.3 Method
“natural”
This method consists in generating the cycles in the order of their appearance in the history of
loading.
Thus for a history of loading of NR +1 points, one determines NR/2 cycles elementary if NR
elementary par and NR/2+1 cycles if NR odd.
The method consists in being based on three successive points of the history of loading.
One notes X = CH (I +)
1 - CH (I) and Y = CH (I + 2) - CH (I +)
1.
If X Y one considers that one met an elementary cycle which is defined by the two items (I)
and (I +)
1.
The amplitude of the cycle is given by CH
= CH (I +)
1 - CH (I).
If X < Y one considers that one met an elementary cycle which is defined by the two points
(I +) 1 and (I + 2).
The amplitude of the cycle is given by CH = CH (I + 2) - CH (I +)
1.
When the cycle is extracted one removes the two items (I) and (I +)
1 of the history of loading and one
the algorithm continues.
If the number of points (NR +)
1 of the history of loading is odd, the algorithm described
previously allows to discuss all the items.
If the number of points (NR +)
1 of the history of loading is even, it remains to discuss the two items
remainders.
It is considered that these two points form a cycle defines by the two points NR and (NR +)
1.
The amplitude of the cycle is given by CH
= CH (NR +)
1 - CH (NR).
To illustrate this method let us consider the same example as that used for the methods
RAINFLOW and RCC_M.
N° not
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Moment 0.
1.
2.
3.
4.
5.
6. 7.
8. 9.
10.
11. 12. 13. 14.
Loading 0. 40. 10. 60. 20. 50.
30. 80.
70. 30.
50. 20. 30. 25. 0.
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(T)
8
4
6
2
10
7
14
12
5
1
T
15
3
13
11
9
The history of loading being composed of 15 points, the method “naturalness” determines 7 cycles
elementary:
Cycle 1:
VALMAX = 40.
and
VALMIN = 10.
Cycle 2:
VALMAX = 60.
and
VALMIN = 10.
Cycle 3:
VALMAX = 50.
and
VALMIN = 20.
Cycle 4:
VALMAX = 80.
and
VALMIN = 70.
Cycle 5:
VALMAX = 30.
and
VALMIN = 70.
Cycle 6:
VALMAX = 30.
and
VALMIN = 50.
Cycle 7:
VALMAX = 25.
and
VALMIN = 30.
Note:
This method is that which it is strongly recommended to use in the case of the calculation of
damage by the methods of Taheri.
2.3
Calculation of the damage: method of Wöhler
The number of cycles to the rupture is determined by interpolation of the curve of Wöhler of material
for a level of alternate constraint given (to each elementary cycle a level corresponds
of amplitude of constraint
= max -
min and an alternate constraint S
= 1/2
alt
).
The damage of an elementary cycle is equal contrary to the number of cycles to the rupture D = 1/NR.
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In the case of a uniaxial homogeneous test with an alternate constraint pure (or symmetrical), it
a many cycles to the rupture are given starting from a diagram of endurance, still called
curve of Wöhler or curve S-N.
In the case of geometrical defects or of elementary cycles of nonnull average constraint, of
corrections of the curve of Wöhler are necessary before the determination of the number of cycles to
rupture and thus of the elementary damage.
2.3.1 Diagram
of endurance
The diagram of endurance, also called curve of Wöhler or curve S-N (curve forced
a number of cycles to the rupture) is obtained in experiments by subjecting test-tubes to
periodic cycles of efforts (generally sinusoidal) of normal amplitude and frequencies
constants, and by noting the number of cycles NR to the end of which the rupture occurs.
The curve of Wöhler is thus defined for a given material and is presented in the form:
Salt
zone 1
zone 2
zone 3
105
106
107
ln NR
NR: Numbers of cycle
with the rupture
1
where Salt = the alternate constraint of the cycle = max -
min
2
One distinguishes three zones on this curve:
·
a zone of oligocyclic fatigue, under strong constraint, where the rupture occurs after one very
small number of alternations,
·
a zone of fatigue or limited endurance, where the rupture is reached after a number of
cycles which grows when the constraint decrease,
·
a zone of unlimited endurance or zone of security, under low constraint, for which
rupture does not occur before a number given of cycles superior to the lifespan
under consideration for the part.
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There are many expressions of the diagram of endurance:
·
Oldest is that of Wöhler:
ln (NR) = has - B Salt éq
2.3.1-1
where NR is the number of cycles to the rupture,
Salt the alternate constraint applied,
has and B two characteristics of material.
This analytical expression does not return account well, of a horizontal branch or
asymptotic of complete curve S-N, but it often gives a representation very
good of average part of the curve.
·
Since 1910, Basquin proposes the formula:
ln (NR) = has - B ln (Salt) éq
2.3.1-2
to take account of the curvature of the curve of Wöhler which connects the branch
downward with the horizontal branch.
D = damage of an elementary cycle = 1/NR =
S
With
where
With e-a
=
and B
alt
=
·
Another analytical shape of the curve of Wöhler is proposed in POSTDAM to hold
count curve out of the singular zone:
S
=
alt
1/2 (E/E
C
)
éq
2.3.1-3
where
E =
of
Modulate
material,
tire
of
curve
with
associated
Young
C
E =
of
Modulate
for
used
Young
to determine
constraint
S.
X =LOG10 (Salt)
NR = a0+ 1
X has +
2
a2X +
3
10
a3X
1/NR
if S
S
where S
of
limit
is
material
endurance
D =
alt
L
L
.
0
if not
Note:
If one takes a2 = a3 = 0 and E/E =1
C
one finds the formula of Basquin.
The user can introduce the curve of Wöhler into operator DEFI_MATERIAU [U4.43.01] under
three distinct forms:
·
a point by point discretized form (key word WOHLER under the key word factor FATIGUE
in DEFI_MATERIAU).
The curve of Wöhler is in this case a function which gives the number of cycles to the rupture
NR according to the alternate constraint Salt and for which the user chooses the mode
of interpolation:
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- “LOG” ----> interpolation logarithmic curve on the number of cycles to the rupture and on
alternate constraint (formula of Basquin per pieces),
- “LIN” ----> linear interpolation on the number of cycles to the rupture and on the constraint
alternated (this interpolation is disadvised because the curve of Wöhler is not absolutely
not linear in this reference mark).
- “LIN”, “LOG” interpolation in logarithmic curve on the number of cycles to the rupture and in
linear on the alternating load, which leads to the expression given by Wöhler.
The user must also choose the type of prolongation of the function on the right and on the left (if it is
necessary to interpolate the function in an unauthorized point by the definition of the function there is stop of
program by fatal error).
·
an analytical form of Basquin (key words A_BASQUIN and BETA_BASQUIN under the word
key factor FATIGUE in DEFI_MATERIAU)
D = ASalt They are constant A and used in this formula which are with
to introduce by the user (in accordance with code POSTDAM).
·
an analytical form except singular zone
S
=
alternated
constraint
=1/
alt
(
2nd/
C E)
X = LOG10 (Salt)
NR =
a0+ 1
X has +
2
a2X +
3
10
a3X
1/NR
if S
S
where S
of
limit
is
material
endurance
D =
alt
L
L
0.
if not
The user must introduce:
EC. = Module of Young associated with the curve with fatigue with the material (key word E_REFE under the word
key factor FATIGUE in DEFI_MATERIAU)
E =
Young modulus used to determine the constraints (key word E under the key word factor
ELAS in DEFI_MATERIAU),
constants of the material a0, a1, a2 and a3 (key words A0, A1, A2 and A3 under the key word factor
FATIGUE in DEFI_MATERIAU)
and SSL limit of endurance of the material (key word SSL under the key word factor FATIGUE in
DEFI_MATERIAU).
Note:
This expression of the damage is available in the same form in the software
POSTDAM.
2.3.2 Influence geometrical parameters on the endurance
2.3.2.1 Coefficient of stress concentration
According to the geometry of the part, it can be necessary to balance the value of the constraint
applied by the coefficient of stress concentration KT .KT is a coefficient function of
geometry of the part, the geometry of the defect and the type of loading.
This coefficient is given by the user under key word KT of the key word factor COEF_MULT.
It is used to apply to the history of the loading, a homothety of report/ratio KT, which returns to
to multiply all the values of the history of loading by coefficient KT.
(The calculation of the damage will be made on a history of loading (T) =KT × (T)).
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2.3.2.2 Elastoplastic coefficient of concentration
It can also be necessary to balance the value of the pressure applied by the coefficient of
elastoplastic concentration Ke.
The elastoplastic coefficient of concentration Ke (aimed to the B3234.3 articles and B3234.5 of the RCC_M
[bib4]) is defined as being the relationship between the amplitude of real deformation and the amplitude of
fictitious deformation determined by the elastic analysis.
An acceptable value of the Ke coefficient can be determined by [bib4]:
K =1
if
<
S
3
E
m
K =1+
E
(1 - N) (/S
3
-
m
) 1/(N (m -) 1 if S
3
< <
m
m
3 Sm
K =1/N
if
m
3 S
<
E
m
where Sm is the acceptable maximum constraint,
and N and m two constants depending on material.
The elastoplastic factor Ke is a report/ratio of homothety of the loading. This factor dependant on
the amplitude of the loading. It is applied, cycle by cycle to the values of the maximum constraint and
minimal of each cycle.
Data S, N
m
m
and
are introduced under key words SM_KE_RCCM, N_KE_RCCM and
M_KE_RCCM under the key word factor FATIGUE in DEFI_MATERIAU.
The user asks for the taking into account of the elastoplastic concentration factor while indicating
CORR_KE: “RCCM” in POST_FATIGUE [U4.83.01].
2.3.3 Influence average constraint
If the part is not subjected to pure or symmetrical alternate constraints, i.e. if
average constraint of the cycle is not null, resistance to the dynamic stresses of material
(its limit of endurance) decreases.
One thus balances the curve of Wöhler to calculate the number of effective cycles to the rupture with the assistance
various diagrams.
The diagram of Haigh makes it possible to determine the evolution of the limit of endurance according to
average constraint m and of the alternate constraint Salt.
Salt
parabola of Gerber
straight line
Diagram of
of
HAIGH
Goodman
Known
m
Starting from a cycle (Salt
, m) identified in the signal one calculates the value of the alternate constraint
corrected '
Salt.
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'
If
Goodman
of
straight line
use
one
S = Salt
alt
1 - m
Known
'
S =
Salt
If
To stack
of
parabola
use
one
alt
2
1
m
Known
It is noticed that this last does not differentiate the average constraint in traction and compression.
where Su is the limit with the rupture of material.
The influence of the average constraint is taken into account only on request of the user (key word
CORR_HAIG).
Note:
If the curve of Wöhler is defined by the analytical form except singular zone [éq 2.3.1-3], of
extended from variation of constraints being in lower part of the limit of endurance can
to find higher than this one. To avoid that, one corrects the limit of endurance SSL while taking
a limit of corrected endurance [bib5]:
'
S = SSL
for
Goodman
of
straight line
L
1 - m
Known
'
S =
SSL
L
for
To stack
of
parabola
2
1
m
Known
2.4
Calculation of the damage: method of Manson-Coffin
The applicability of the method of Manson-Coffin [bib1] is oligocyclic plastic fatigue,
who as his name indicates it shows two fundamental characteristics:
·
it is plastic, i.e. a significant plastic deformation occurs with each
cycle,
·
it is oligocyclic, i.e. the materials have an endurance finished with this type of
stress.
To describe the behavior of materials in fatigue oligocyclic plastic, one uses tests with
alternate imposed deformation.
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In the case, of a uniaxial homogeneous test with an alternated deformation, the number of cycles with
rupture is given starting from a diagram of resistance, which connects the variation of deformation to
a number of cycles involving the rupture.
In the diagram of resistance, one separates the deflections total, elastic and plastic. These
diagrams are still known under the name of Whetstone sheath-Manson which proposed them in 1950.
Variation of deformation
10
(%)
T
p
E
0,1
ln (a number of cycles with rupture NR)
2
3
4
5
6
10
10
10
10
10
10
p
Relations
E ln
- (NR) and
ln
- (NR) are lines. The relation
T ln
- (NR) presents, as
2
2
2
with it, a curvature towards the positive deformations.
It was shown that a relation power connected the plastic deformation (
p) and deformation
rubber band (
E) with the number of cycles to the rupture, which leads to the following relations:
- has
p=AN
B
-
e=BN
- has
B
-
t=AN +BN
where has and B are two characteristics of material (in general A is close to 0,5 and B close to
0,12); With and B, two constants of material.
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The user can introduce the curve of Manson-Coffin in a single mathematical form: form
discretized point by point. It is a function which gives the number of cycles to the rupture NR in
function of the amplitude of deformation T.
2
As for the curve of Wöhler, the user can choose the mode of interpolation on the number of
cycles with the rupture and on the amplitude of deformation.
The type of prolongation of the function on the right and on the left is also with the choice of the user.
The damage of an elementary cycle is equal contrary to the number of cycles to the rupture D 1
=/NR.
2.5
Calculation of the damage: method of Taheri
The methods of calculation of the damage proposed by Taheri [bib12] are two: one them
will name respectively Taheri-Manson and Taheri-mixed. These methods apply to
loadings characterized by a scalar component of deformation type.
These methods have as a characteristic to hold account about application of the elementary cycles
of loading to the structure. For this reason, it is advisable to be vigilant for the choice of the method of
counting of the cycles. It is strongly advised to use the method of counting known as method
“natural” [§2.2.3].
2.5.1 Method
Taheri-Manson
Are N cycles elementary of half-amplitude
1,
N
L
.
2
2
The value of the elementary damage of the first cycle is determined by interpolation on the curve of
Manson-Coffin of material.
The calculation of the elementary damage of the following cycles is carried out by the algorithm:
+
·
I 1
if
I
2
2
the value of the elementary damage of the cycle (I)
1
+ is determined by interpolation on
curve of Manson-Coffin of material.
+
·
I 1
if
I
<
2
2
one determines:
i+1
i+1
J
=F
, Max
NAPPE
2
2
j<i 2
then
*
i+1
i+1
= FONC
F
.
2
2
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FNAPPE is the cyclic curve of cyclic work hardening with préécrouissage of material.
FONC
F
is the cyclic curve of work hardening of material.
*
I 1
+
The value of the damage of the cycle (I)
1
+ is determined by interpolation of
on the curve of
2
Manson-Coffin of material.
Note:
If all the cycles applied are arranged by ascending value of the amplitude of deformation,
this method is identical to the method of Manson-Coffin.
2.5.2 Method
Taheri-Mixte
Are N cycles elementary, of half-amplitude
1,
N
L
.
2
2
The value of the elementary damage of the first cycle is determined by interpolation on the curve of
Manson-Coffin of material.
The calculation of the elementary damage of the following cycles is carried out by the algorithm:
+
·
I 1
if
I
2
2
the value of the elementary damage of the cycle (I)
1
+ is determined by interpolation on
curve of Manson-Coffin of material.
+
·
I 1
if
I
<
2
2
one determines:
i+1
i+1
J
=F
, Max
NAPPE
2
2
j<
I 2
where FNAPPE is the cyclic curve of cyclic work hardening with préécrouissage of material.
I 1
+
The value of the damage of the cycle (I)
1
+ is obtained by interpolation of
on the curve
2
of Wöhler of material.
Note:
If all the cycles applied to the structure are arranged by ascending value of the amplitude of
deformation, this method is identical to the method of Manson-Coffin.
The damage of an elementary cycle is equal contrary to the number of cycles to the rupture D 1
=/NR.
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2.6
Calculation of the total damage
The simplest approach and most known to determine the total damage of a part subjected to
I
N cycles of alternate constraint Salt or alternate deformation Ealt is the linear rule of
damage proposed by Miner:
I
N
Di=
Ni
Under operation, the structures are subjected to various loadings of different amplitudes.
tire undergone is due to the accumulation of the elementary damage and the total damage is calculated with
assistance of the rule of office plurality of Miner [bib6]:
N
D
= I
total
NR
I
I
In the case of Wöhler and Manson-Coffin, this law supposes that the damage increases linearly
with the number of imposed cycles and which it is independent of the level of loading and the command
of application of the levels of loading (whereas in experiments, it is shown that the command
of application of the loading is an important factor for the lifespan of material).
The calculation of the total damage is required by the user with key word CUMUL.
The methods suggested by Taheri hold account about application of the loading, in
calculation of the elementary damage associated each cycle.
2.7 Conclusion
For the methods based on uniaxial tests, the calculation of the total damage undergone by a part
subjected to a history of loading breaks up into several stages:
·
extraction of the peaks of the history of loading, to lead to a simpler history,
·
extraction of the elementary cycles of the history of loading by a method of counting
cycles,
·
calculation of the elementary damage associated each elementary cycle resulting from the real history
loading,
- possibly (and for the method of Wöhler), correction of the loading by one
coefficient of stress concentration KT,
- possibly (and for the method of Wöhler), correction of the loading by one
elastoplastic coefficient of concentration Ke,
-
possibly (and for the method of Wöhler), correction of Haigh to take account of
the nonnull value of the average constraint,
·
calculation of the total damage, by a linear rule of office plurality.
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3
Calculation of the damage of generalized Lemaître
This law of damage relates to the study of the starting of a macroscopic fissure, using one
post-processor of mechanics of damage based on a unified formulation of the laws
of evolution of the damage. This one uses, on the one hand, of the laws of evolution of the damage
specific to the various mechanisms considered, and, in addition, a more general model based on
a micromechanical analysis of the phenomenon of starting.
This law offers a single formalism which supposes that the various damage mechanisms
all are controlled by the plastic deformations, elastic deformation energy and one
process of instability.
3.1
The law of Lemaître generalized
The law of Lemaître generalized consists of an enrichment of the method of calculation of damage of
Lemaitre [bib7] by the introduction of a law in power (model of Lemaître-Sermage). It is written
[bib14]:
S
Y
D & =
p & Si p>
p
S
D
éq 3.1-1
D =
0
if not
with:
2
2
eq
2
Y =
R and R =
(
1+) + (
3 1-2) H.
2nd (1-D) 2
3
eq
Y is the rate of refund of density of elastic deformation energy.
H
R is related to triaxiality,
the rate of triaxiality.
eq
3D D
eq = ij ij is the equivalent constraint of von Mises.
2
D
1
ij
= ij - kk Sij is the diverter of the constraint.
3
p is the threshold of damage, S and S of the characteristics material.
D
p (T) cumulated plastic deformation.
This law thus makes it possible to calculate the damage D (T) starting from the data of the tensor of the constraints
(T) and of the cumulated plastic deformation p (T).
The integration of the equations [éq 3.1-1] led to:
1
D (
2s 1 2s
1
S
S
2s 1
ti+1)
+
+
=
-
1 (-
1 D (Ti)
-
(+
C) + (-
C)
+
(p (ti+1) - p (Ti))
if p > Pd
D (
2
ti+1)
=
0
if not
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with:
+
1
2
3
2
C =
+ T + T
+
-
+
T + T
3rd (
I
eq I
I
H I
T
+
I 1
+) S (Ti 1
+) (1
(1) (1) 2nd (Ti 1+) S (Ti 1+) (1 2 (1) (1) éq 3.1-2
-
1
2
3
2
C =
+ T
T +
- T
T
3rd (Ti) S (Ti) (1 (I) eq (I)
2nd (Ti) S (Ti) (1 2 (I) H (I).
It is supposed that D (T) 0
=.
O
Note:
·
It is considered that the characteristics material E (Young modulus), (coefficient of
fish) and S (parameter material) depend on the temperature T.
·
The value of the Young modulus and the value of the Poisson's ratio are defined in
DEFI_MATERIAU [U4.43.01] under the key word factor ELAS_FO.
·
The values of S, Pd and of S are defined in DEFI_MATERIAU under the key word
factor DOMMA_LEMAITRE and operands S, ESPS_SEUIL and EXP_S. Les parameters
S and Pd can depend on temperature TEMP.
·
The law of Lemaître is obtained by assigning the value S = 1
Knowing the value of the damage D (Ti), i= N
,
0, one can calculate a value of total damage:
N
D= D (Ti).
I 1
=
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Version
7.4
Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
A. Mr. DONORE, F. MEISSONNIER Key
:
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4
Criteria of Crossland and Dang Van Papadopoulos
The criteria [bib9] and [bib13] allow for metal structures subjected to constraints
forced following a great number of cycles to distinguish the loadings damaging from
others.
One can classify the criteria in two categories according to nature of their approach:
·
macroscopic approach: criterion of Crossland,
·
microscopic approach: criterion of Dang Van Papadopoulos.
The criteria of Crossland and Dang Van Papadopoulos apply to uniaxial loadings or
multiaxial periodicals.
The goal of these criteria is not to determine a value of damage, but a value of criterion
Rcrit such as:
R
0 step of damage
crit
R
> 0 damage
(fatigue).
possible
crit
4.1
Criterion of Crossland
The criterion of Crossland is empirical and is written only starting from variables
macroscopic.
In fact, starting from trial runs, one could note that the amplitude of cission as well as the pressure
hydrostatic played a fundamental part in the mechanisms of fatigue of the structures.
This is why, Crossland postulated the criterion:
R
= + aP
- B
crit
has
max
where
= 1
has
Max Max (Dt
amplitude of cission
1) -
(Dt0) =
2 0t T
T T
0
0 1
with
D
diverter of the tensor of the constraints.
1
P
= max
max
trace = maximum hydrostatic pressure.
0 tT 3
d0
d0
has =
0 -
and B =
0
3
3
with:
=
0 limit of endurance in alternated pure shearing,
D =
0
limit of endurance in alternate pure traction and compression.
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Code_Aster ®
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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:
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4.2
Criterion of Dang Van Papadopoulos
It appeared that the crack initiation of fatigue is a microscopic phenomenon occurring with
a scale about the grain. This is why, of the criteria of fatigue, starting from variables
microscopic local were postulated.
The implemented criterion [bib8], [bib9] and [bib10] in Code_Aster is the criterion of Dang Van
Papadopoulos, which is written in the form:
Rcrit = K * + has Pmax - B
where:
R
D
D
K * =
if
R = Max ((T) - C)
*: ((T) - C)
*
2
0tT
K * = R
1
if
R = Max J (T) = Max
D
T - C
D
T - C
2
(()
) *:(()
) *
0tT
0tT
2
with:
·
R, the radius of the smallest sphere circumscribed with the way of loading in the space of
diverters of the constraints;
·
J (T), the second invariant of the diverters of the constraints;
2
·
C * = Min max (D
T () - C): (D
T () - C), the center of the hypersphère.
C
T
Note:
It is the definition of R which uses J (T) which is programmed.
2
1
P
max
= maximum hydrostatic pressure =
Max
trace
0 T T3
D
D
0
0
has = -
and B =
0
3 3
0
with:
0 = limit of endurance in alternated pure shearing,
d0 = limit of endurance in alternate pure traction and compression.
The basic idea of Papadopoulos is to write that the grain obeys a criterion of plasticity of the type
von Mises instead of the criterion of plasticity of the Tresca type used by Dang Van.
Papadopoulos conducted a campaign of comparisons between the results provided by its criterion and
experimental results, which shows that the predictions of the criterion of Papadopoulos are
excellent for the loadings closely connected; they are a little less precise for the ways not
closely connected.
In its thesis [bib10] Papadopoulos shows that the criterion of Crossland and the criterion of Dang Van
Papadopoulos give the same results for radial loadings.
The algorithm employed for the calculation of the radius of the smallest sphere circumscribed with the way of
loading in the space of the diverters of constraints, is that proposed in [bib11]. It is about one
recurring algorithm which rests on the second invariant of the diverters of the constraints.
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
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:
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Let us note If the value of the diverter of the constraints at the moment Ti, One centers it hypersphère with
the iteration N, RN the radius of the hypersphère to iteration N and X the “isotropic parameter of work hardening” of
the algorithm.
·
Phase of initialization of the algorithm:
1 NR
1
O
=
S
NR
I
I 1
=
R
= 0.
1
·
Iteration of stage N at the stage
1
+
N:
one supposes One and RN known. One calculates then:
D =
Si-O
+1
N
P = D - RN
- If
P 0
>
RN 1 = R +x.
+
N
P
O -
N If
O
N 1
+ = If 1
+ + RN 1
+
O -
N If 1
+
- If 0
<
P
N
R =
+1
N
R
N
O =
+1
N
O
The algorithm ends when all the points If are in the hypersphère of center One and of
RN.
Ti
P
if
If
O
O
N
N + 1
Ti +1
RN
S i+1
- of VI
If +1
- of vi+1
RN +1
0
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Code_Aster ®
Version
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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Author (S):
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:
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4.3
Calculation of a value of damage
These two criteria applicable to multiaxial periodic loadings make it possible to say if there is
damage or not:
R
0 step of damage
crit
R
> 0 damage
.
(fatigue)
possible
crit
These criteria do not provide a value of damage. It can however be interesting to calculate
a value of damage by using the curves of Wöhler of material. With this intention, it is necessary to define
an equivalent constraint *, value to be interpolated on the curve of Wöhler.
The curves of Wöhler can be built starting from shear tests in which case the limit
of endurance is 0, but are more generally built starting from tests of traction and compression
for which the limit of endurance is d0 (d0
< 0).
So that there is coherence between the criterion and the curve of Wöhler it is necessary that:
*
no damage
0
for
Wöhler
of
curve
one
cisailleme
in
defined
NT,
* >
too bad
0
* D
no damage
0
for
Wöhler
of
curve
one
traction
in
defined
- compression.
* > D
too bad
0
It thus seems possible to us to take:
* = R
+
for
Wöhler
of
curve
one
cisailleme
in
rare),
enough
is
who
(it
NT
crit
0
* = (R
crit + 0) (d0/0) for
Wöhler
of
curve
one
traction
in
- compression.
In a general way, the user can take * = (Rcrit + 0) corr where corr is a coefficient of
correction introduces by the user.
By defect, this coefficient corr is taken equal to (d0/0) (case of the curve of Wöhler introduced in
traction and compression).
Note:
In the literature, one does not find presentation of a step of use of a criterion
to calculate a value of damage. It is known however that certain industrialists use one
such step, but without knowing the adopted form of it.
The step implemented in Code_Aster is proposed by department AMA.
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
14/04/05
Author (S):
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:
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5 Conclusion
In this note the various methods of calculation from the damage available are exposed is in
operator POST_FATIGUE either in operator CALC_FATIGUE, or in the two commands
at the same time.
One can classify these methods in two great classes:
·
estimate of the damage to great numbers of cycles,
·
estimate of the damage in fatigue oligocyclic plastic.
In the first class of problems, one finds the method of Wöhler, based on tests
uniaxial, and which applies to loadings in constraint. One also finds in this class,
the criterion of Crossland, which is an empirical criterion being based on macroscopic sizes and
the criterion of Dang Van Papadopoulos which is based on microscopic phenomena.
The two criteria are addressed to loadings in constraints which can be uniaxial or
multiaxial but periodic.
In the second class of problems, one finds the method of Manson-Coffin and the methods of
Taheri, which applies to loading in deformations.
The whole of the methods based on uniaxial tests (method of Wöhler, method of
Manson-Coffin and methods of Taheri) are available in two operators POST_FATIGUE and
CALC_FATIGUE.
The criteria, as for them, are only available in POST_FATIGUE.
Handbook of Référence
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Code_Aster ®
Version
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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6 Bibliography
[1]
C. BATHIAS, J.P. MUZZLE: The fatigue of materials and the structures. Collection University
from Compiegne - PUM Presses of Université of Montreal MALOINE S.A. Editeur Paris.
[2]
I. BAKER: Algorithm of fatigue: comparison of various methods of counting of
cycles of constraints - Note HP/169/88/44.
[3]
Tire under stresses of variable amplitude: Rainflow method of counting of the cycles.
AFNOR A normalizes November 03-406, 1993.
[4]
RCC_M. Edition January 1983.
[5]
E. VATIN: Schedule of conditions of version 2 of software POSTDAM - Note HP/14/94/017/A.
[6]
F. WAECKEL: Estimate of fatigue to great numbers of cycles - Note HP/62/94/128/A.
[7]
J. LEMAITRE: Unified formulation of the laws of evolution of damage. CR Academy of
Sciences, Paris, T.305, series II, 1987.
[8]
P. BALLARD, DANG VAN KY, H. MAITOURNAM: Calculation of the metal parts to fatigue.
Support of course Collège Polytechnique (5, February 6, and 7 1996).
[9]
E. LORENTZ: Implementation of the criteria of fatigue. Creation of a post-processor for
Systus (GDF).
[10]
V. PAPADOPOULOS: Polycyclic fatigue of metals. A new approach. Thesis of
Ioannis V. PAPADOPOULOS 1987.
[11]
K. DANG VAN, B. GRIVEAU, O. HOUSEHOLD: There is new Multiaxial Fatigue Limit Criterior theory
and applications. Mechanical Engineering Publications, London 1989.
[12]
S. TAHERI: With low cycle ramming cumulation rule for not proportional loading tires. Note
HI-74/94/082/0.
[13]
S. TAHERI: Bibliography on multiaxial fatigue with great number of cycles, HI-
74/94/086/0.
[14]
PH. SERMAGE: Tire thermal multiaxial at variable temperature, thesis of doctorate
ENS-Cachan, Dec. 1998.
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Code_Aster ®
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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Author (S):
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:
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Appendix 1
The following history of loading is considered (which for the example is considered of type forced):
N° not
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Moment 0.
1.
2.
3.
4.
5.
6. 7.
8.
9.
10.
11. 12. 13. 14.
Loading 0. 40. 10. 60. 20. 50.
30. 80.
70.
30.
50. 20. 30. 25. 0.
(T)
8
60.
4
6
2
10
7
14
12
5
1
15
T
3
13
11
9
The stage of rearrangement of the history of loading leads to the following loading:
N°
not
8 9
10 11
12 13
14 15 2 3 4 5 6 7
Loading 80. 70. 30. 50. 20. 30.
25. 0. 40.
10.
60. 20. 50. 30.
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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(T)
8
4
6
2
10
14
7
12
5
1
15
T
3
13
11
9
The second stage consists in extracting the elementary cycles. The first extracted cycle is the cycle defined by
items 12 and 13 since 12
(
)
-
)
13
(
is lower than 14
(
)
-
)
13
(
and 12
(
)
-
)
13
(
is lower than
12
(
)
-
)
11
(
.
Cycle 1: VALMAX = 20. and VALMIN = 30.
The cycle having been extracted one removes these two points of the history of the loading, and one starts again on
remaining history.
(T)
8
4
6
2
10
14
7
5
1
=
15
T
3
11
9
The following cycle extract is the cycle defined by items 14 and 15.
Cycle 2: VALMAX = 25. and VALMIN = 0.
The remaining history, after suppression of these two points is:
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Titrate:
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Date:
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(T)
8
4
6
2
10
7
5
T
3
11
9
One extracts then the cycle defined by items 10 and 11.
Cycle 3: VALMAX = 30. and VALMIN = 50.
One sets out again on the following history of loading:
(T)
8
4
6
2
7
5
T
3
9
The following cycle extract is defined by items 2 and 3.
Cycle 4: VALMAX = 40. and VALMIN = 10.
The remaining history of loading is (it is the residue of the history of the loading):
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(T)
8
4
6
7
5
T
9
One cannot extract any more from cycles, because all the history of the loading was traversed.
One thus passes at the third stage, which consists in treating the residue:
(T)
8
8
4
4
6
6
7
7
5
5
T
9
9
One adds the same residue with his continuation, and one starts again the second stage on this loading.
The following cycle extract is defined by items 6 and 7.
Cycle 5: VALMAX = 50. and VALMIN = 30.
The remaining history of loading is:
Handbook of Référence
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Titrate:
Estimate of the fatigue life to great number of cycles
Date:
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(T)
8
8
4
4
6
7
5
5
T
9
9
The following cycle extract is defined by items 4 and 5.
Cycle 6: VALMAX = 60. and VALMIN = 20.
The remaining history of loading is:
(T)
8
8
4
6
7
5
T
9
9
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Date:
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The last extracted cycle is a cycle defined by items 8 and 9.
Cycle 7: VALMAX = 80. and VALMIN = 70.
(T)
8
4
6
7
5
T
9
It is noticed well that when one applies counting RAINFLOW to the unit made up of the two residues, one
obtains in end counting again the initial residue.
Handbook of Référence
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