2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is generated from option POST_FATIGUE. The methodology adopted consists in defining a history of stress loading and recovering the evolution of the associated cumulative plastic deformation from a \(\mathrm{3D}\) tensile test in thermo-viscoplasticity.
The loading history \(\sigma (t)\) and \(p(t)\) is then used in a POST_FATIGUE calculation with the material parameters presented in paragraph [§1.1] to define a reference solution.
2.2. Benchmark results#
The Lemaître damage reference result is obtained for a tensile test with imposed deformation and at a constant temperature. The stress state and the cumulative plastic deformation resulting from this test are as follows:
Time \([s]\) |
Constraint \(\mathrm{Sxx}(t)[\mathrm{Pa}]\) |
Cumulative Plastic Deformation \(P(t)\) |
50 |
7.15030E+06 |
0.000000E+00 |
100 |
1.43006E+07 |
0.000000E+00 |
150 |
2.14509E+07 |
0.000000E+00 |
200 |
2.86012E+07 |
0.000000E+00 |
250 |
3.57515E+07 |
0.000000E+00 |
300 |
4.29018E+07 |
0.000000E+00 |
350 |
5.00521E+07 |
0.000000E+00 |
400 |
5.72024E+07 |
0.000000E+00 |
450 |
6.43527E+07 |
0.000000E+00 |
500 |
7.15030E+07 |
0.000000E+00 |
550 |
7.86533E+07 |
0.000000E+00 |
600 |
8.58036E+07 |
0.000000E+00 |
650 |
9.29539E+07 |
0.000000E+00 |
700 |
1.00091E+08 |
9.547120E-08 |
750 |
1.06433E+08 |
5.747160E-06 |
800 |
1.10614E+08 |
2.650910E-05 |
850 |
1.12888E+08 |
6.060610E-05 |
900 |
1.14130E+08 |
1.019250E-04 |
950 |
1.14913E+08 |
1.464460E-04 |
1000 |
1.15508E+08 |
1.922890E-04 |
This loading history is then used with the operator POST_FATIGUE option LEMAIT_S to estimate the damage over time with the material properties defined in paragraph [§1.1]. The temperature is assumed to be constant and equal to \(20°C\). Depending on the value of the parameter \(s\) used, the following damages are found:
Damage (reference) |
||
Time \([s]\) |
Case \(s=0.8\) |
Case \(s=1.003\) |
50 |
0.00000E+00 |
0.00000E+00 |
100 |
0.00000E+00 |
0.00000E+00 |
150 |
0.00000E+00 |
0.00000E+00 |
200 |
0.00000E+00 |
0.00000E+00 |
250 |
0.00000E+00 |
0.00000E+00 |
300 |
0.00000E+00 |
0.00000E+00 |
350 |
0.00000E+00 |
0.00000E+00 |
400 |
0.00000E+00 |
0.00000E+00 |
450 |
0.00000E+00 |
0.00000E+00 |
500 |
0.00000E+00 |
0.00000E+00 |
550 |
0.00000E+00 |
0.00000E+00 |
600 |
0.00000E+00 |
0.00000E+00 |
650 |
0.00000E+00 |
0.00000E+00 |
700 |
0.00000E+00 |
0.00000E+00 |
750 |
5.43732E-03 |
3.19264E-02 |
800 |
2.75450E-02 |
1.90334E-01 |
850 |
6.75939E-02 |
1.00000E+00 |
900 |
1.21543E-01 |
1.00000E+00 |
950 |
1.87318E-01 |
1.00000E+00 |
1000 |
2.66202E-01 |
1.00000E+00 |
2.3. Uncertainty about the solution#
Digitally generated solution.
2.4. Bibliographical references#
A.M. DONORE: Estimation of lifespan in fatigue with large numbers of cycles and in oligocyclic fatigue. Note [R7.04.01] Index B.