2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Since the loading history is very simple, the reference results can be obtained manually by applying the algorithms presented in the reference document [R7.04.01].

2.2. Benchmark results#

2.2.1. Modeling A#

\(t\)	\(D(t)\) ** (Too bad)
43.11
	0.000848907
	0.014474925
	0.178374238
	0.524693005
	0.602827469
	0.73829052
	0.792149807
	0.967604351

2.2.2. B modeling#

The reference results for test case number 2 are obtained using a spreadsheet in which the Lemaître-Sermage damage expression was implemented according to a numerical integration diagram identical to that used in routine POST_FATIGUE of*Code_Aster*.

First, we check that the uncertainty in the results obtained for the value \(s=1.0\) via the spreadsheet is acceptable:

Damage (Excel calculation)	Damage (reference solution)	Difference ( \(\text{\%}\) )
0.0000000000	0.0000000000	0.00000%
0.0008489062	0.0008489070	-0.00010%
0.0144749268	0.0144749250	0.00001%
0.1783742841	0.1783742380	0.00003%
0.5246932887	0.5246930050	0.00005%
0.6028278917	0.6028274690	0.00007%
0.7382915411	0.7382905200	0.00014%
0.7921514337	0.7921498070	0.00021%
0.9676720845	0.9676043510	0.00700%

In a second step, a reference solution is generated for a value of

= 1.003:

\(t\)	\(D(t)\) (Damage — Excel solution)
43.11	0.0
	0.004742198
	0.083020455
	1.809947268
	2.003578566
	0.083020455
	0.178700399
	0.199207053
	0.220252827

2.3. Uncertainty about the solution#

Analytical 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.