1. Reference problem#
The analysis consists in determining the average damage suffered by a part subjected to a random loading.
Random loading is entirely characterized by the values of the spectral moments of order 0, 2 and 4: \({\lambda }_{0}\), \({\lambda }_{2}\) and \({\lambda }_{4}\) which are introduced under the keywords MOMENT_SPEC_0, MOMENT_SPEC_2 and MOMENT_SPEC_4.
To calculate the damage, you must choose one of the two counting methods available in Code_Aster:
method for counting stress peaks,
method for counting when a given level has been exceeded.
It is also necessary to introduce the Wöhler curve of the material, which can be defined in three distinct mathematical forms:
point by point function, which gives the value of the number of cycles at break, as a function of the alternating stress \({S}_{\mathrm{alt}}\),
Basquin’s analytic form: \(D=A{S}_{\mathrm{alt}}^{\beta }\)
analytical form « current area »
\({S}_{\mathrm{alt}}\) = alternating stress = \(1/2({E}_{C}/E)\Delta \sigma\)
\(X={\mathrm{log}}_{10}({S}_{\mathrm{alt}})\)
\(N={10}^{\mathrm{a0}+\mathrm{a1}X+\mathrm{a2}{X}^{2}+\mathrm{a3}{X}^{3}}\)
\(D=\) |
|
if \({S}_{\mathrm{alt}}\ge {S}_{l}\) |
\(0.\) |
otherwise |
where \({E}_{C}\) = Young’s modulus associated with the fatigue curve of the material,
\(E\) = Young’s modulus used to determine the constraints,
the material constants \(\mathrm{a0}\), \(\mathrm{a1}\), \(\mathrm{a2}\), and \(\mathrm{a3}\),
and \({S}_{l}\) the endurance limit of the material.
In addition, it is possible to take into account an elastoplastic concentration coefficient \(\mathrm{Ke}\), defined by:
\(\{\begin{array}{ccc}{K}_{e}=1& \mathrm{si}& \Delta \sigma <3{S}_{m}\\ {K}_{e}=1+(1-n)/(\Delta \sigma /3{S}_{m}-1)/(n(m-1))& \mathrm{si}& 3{S}_{m}<\Delta \sigma <3m{S}_{m}\\ {K}_{e}=1/n& \mathrm{si}& 3m{S}_{m}<\Delta \sigma \end{array}\)
where \({S}_{m}\) is the maximum allowable stress,
and \(n\) and \(m\) two constants that depend on the material.
In this test, for a single given random loading, the average damage is determined in ten distinct configurations, according to the shape of the Wöhler curve and the cycle counting method.
1.1. Material properties for fatigue studies#
The properties of the material relate to the data of a Wöhler curve making it possible to determine the number of cycles at break for a given loading level.
1.1.1. Wöhler curve in Basquin analytical form#
Setup 1 |
A |
\(\beta\) |
1.0017309939 E—14 |
4.065 |
Setup 2 |
A |
\(\beta\) |
|
1.1.2. Wöhler curve in « current zone » form#
Configuration definition parameters 3:
\(\mathrm{a0}\) |
|
|
|
|
|
|
|
11.495 |
—5. |
0.25 |
—0.07 |
Configuration definition parameters 4:
\(\mathrm{a0}\) |
|
|
|
|
|
|
|
11.495 |
—5. |
0.25 |
—0.07 |
In addition, an elasto-plastic concentration coefficient \(\mathrm{Ke}\) defined by the parameters for this configuration is taken into account.
\(\mathrm{Sm}\) |
|
|
0.6 |
1.4 |
1.1.3. Wöhler curve in point by point function form (configuration 5)#
\({S}_{\mathrm{alt}}\) |
|||||||
\(N\) |
3.125E+11 |
976562.5E+4 |
1.E+8 |
12860.09 |
5949.899 |
3051.76 |
\({S}_{\mathrm{alt}}\) |
|||||||||
\(N\) |
1693.51 |
1000.0 |
1000.0 |
620.921 |
620.921 |
401.8779 |
269.329 |
185.934 |
131.6869 |
\({S}_{\mathrm{alt}}\) |
|||||||||
\(N\) |
95.3674 |
70.4296 |
70.4296 |
52.9221 |
40.3861 |
31.25 |
24.4852 |
24.4852 |
19.40379 |
\({S}_{\mathrm{alt}}\) |
|||||||||
\(N\) |
15.5368 |
12.55869 |
12.55869 |
10.23999 |
8.41653 |
6.96917 |
5.81045 |
5.81045 |
4.8754 |
\({S}_{\mathrm{alt}}\) |
165 |
||||||||
\(N\) |
4.11523 |
3.49294 |
3.49294 |
2.98023 |
2.55523 |
2.20093 |
1.90397 |
1.90397 |
1.65382 |
\({S}_{\mathrm{alt}}\) |
|||||
\(N\) |
1.44209 |
1.26207 |
1.26207 |
1.10835 |
0.976562 |
1.2. Loading history#
Random loading is entirely characterized by the values of the spectral moments:
\({\lambda }_{0}\) |
|
|
182.5984664 |
96098024.76 |
6.346193569E+13 |