Systematic validation of experimental data usable for verifying the multiaxial fatigue prediction methods

The paper discusses some of the issues, the researchers interested in verifying various multiaxial fatigue limit estimation solutions are facing to. Even recently, newly proposed criteria have been or are tested on dozens of experimental inputs. Papuga in [1] pointed out, that applicability of the most often used test batch is limited and only half of these data items is worth using for such purposes. This paper extends that analysis by describing the weak points of various data sets used in this domain for validating new proposals on multiaxial fatigue limit estimates. The conclusion from the extensive analysis is that the researchers should adopt other test sets only if they very well know their background.


INTRODUCTION
hen Papuga wrote the paper [1] on various multiaxial fatigue limit estimation methods and their validation, he referred to a set of 407 experimental items on various materials and obtained from different papers.At that moment, he cared little about the quality of the data obtained in that way.The reason for including a data item W into the validation set was a simple stating of the experimenter or of any other researcher, that the data items are usable for such a validation.Already during preparation of that data set, some bad practice or misleading data interpretation was uncovered and discussed in [1].Anyhow, only a proper analysis [4] of the most often cited and reused data set defined by Papadopoulos et al. [2] and of other often used data items from the paper by Nishihara and Kawamoto [3] made a start point for an extensive research on evaluating the data quality.

METHOD OF EVALUATION
his paper further extends the analyses provided in [4] by updating them and enlarging the scope of the data check.Due to the multitude of papers available for validation, it can be in no way complete, and some other such papers can be found.On the other hand, it is covering now all data sources found, which were used by the cited authors in their effort to validate new or older multiaxial fatigue limit estimation methods, and which were used for such a purpose by at least one researcher not related to the team of experimenters.Some newer sets from fresh experiment campaigns were reported meantime, but the scope of a conference paper cannot give a thorough overview.Data presented here are quickly touched and only their weak points are commented.The experiments are very valuable, their realization took a lot of time, a lot of work and a lot of funds.Anyhow, only short evaluation can get here.Thus the sets are criticized when necessary, so that the other researchers were aware of potential problems if accepting them, while the detailed description or appraisal cannot be provided here.The measure of acceptability of experimental results for validation purposes has been already discussed in [4].ASTM [5] e.g.recommends between 6 and 12 specimens to be used for an S-N curve applicable for an exploratory research.Papuga in [4] proposed a system of weights, where 8 and more finished experiments per the S-N curve result in a full weight (1.0) of the item, while the weight decreases linearly to 0.2 for the case of 4 specimens per the S-N curve, and experiments with fewer test points are simply discarded.In this paper, we do not evaluate the appropriateness to the validation in another way than by the check if the S-N curve is described with minimum 4 specimens.The question of the data dispersion, applicability of experiments tested on one load level only, of the importance of the basic material curves in fully reversed push-pull and fully reversed torsion, the comparison between the statistical representativeness of the fatigue limit derived from an S-N curve and from the staircase analyses are only a few of the questions that should be further raised, but cannot be discussed in such a short paper.The basic reference for the next evaluation is Tab. 1, where the individual papers describing validation of some multiaxial fatigue limit estimation method(s) are referred to, and the data sets used for this purpose are marked by "x" symbol.The papers often just reuse data from a second-hand source, so the fact that some paper uses Nishihara's and Kawamoto's data [3] need not mean that there are not error in interpreting it or in the transcription.The table also does not describe, if the authors referring to a particular data set used in a complete form or if they selected only its parts.The individual test sets described in Tab. 1 are evaluated in the next section including also the test set marks extensively used in FatLim and Finliv databases [6][7].Due to the limited space available for Tab. 1, these marks could not be preserved there as well, but they can be deduced from the context.

Nishihara and Kawamoto (NKc, NKd, NKh, NKm -[3])
he fact that the data items of this data set on fatigue limits obtained for hard steel, mild steel, cast iron and duralumin are the 1st, 5th, 8th and 9th as regards the frequency of their use shows its importance in validating of the multiaxial fatigue limit estimation methods.The careful analysis of the set in [4] anyhow uncovers that only two S-N curves obtained for the hard steel material can be seen as applicable, if accepting the low coefficient of determination R 2 < 0.75 obtained during the regression analysis of data by the Basquin formula.For the other materials, either the material curves (mild steel, cast iron) or the multiaxial experiments (duralumin) dispose of too few specimens, in some cases limited even to only one finished and one unfinished experiment.Though the set is so broadly used for validation practice, the analysis of the original paper shows that applicability of so poorly documented S-N curves is inadequate.[14] x x XC48 [39] x x x 25CrMo4 [18] x x x cast iron [36] x En24T [37] x GG30 [34] x

Neugebauer (NbA, NbB -[8])
The research report of Neugebauer on GGG-40 and GTS-45 was extracted by McDiarmid [9] and later by Stefanov [10] to support their findings.The data are reported in graphs of the S-N curves only (mostly two load levels up to 6-13 data points in total) and have to be derived by an image analysis.The weak point of the test set is hidden in the text [8], where Neugebauer notes that all pure torsion experiments were run with deformation control.The set thus mixes outputs of these deformation-controlled experiments with the common load-controlled experiments for all other load variants.
Froustey and Lasserre (FLA, FLB - [11], [12]) The test sets have been already discussed in [4], and only the major conclusions are provided here.Froustey and Lasserre ran the tests on three different lots of 30NCD16, while only one of these lots (used for FLB test set here) is complete as regards the uniaxial experiments, while the FLA test set lacks such material data.This is probably the reason, why Dubar [13] later attempted to prepare a unified test set on 30NCD16 by weighting the results of individual batches in an undefined way, while thus confusing other researchers by seeming wholly new set.

Heidenreich, Zenner et al. (HeG, Hei, HRZ, HZ -[14], [15])
The authors repeatedly published results of their ongoing experimental campaign mainly on three different lots of 34Cr4 steel.They also reported tests on GGG-60 cast iron (HeG), but these are not anyhow present in Tab. 1.The test set has been already commented in [4] as being acceptable for validation purposes, and only some individual curves should be sorted out (the authors used lower numbers of specimens for the staircase method).The only unacceptable exception is the HZ test set, where no reversed torsion experiments were run, and the test set is not thus applicable to testing for most of the fatigue limit estimation criteria.

Lempp (Lem -[16])
Lempp's tests on 42CrMo4V steel are commonly reused for validation purposes thanks to Papadopoulos' test set [2], and thus they have been already evaluated in [4].The originally reported fatigue limit estimates are mostly based on extrapolations to cycle counts far exceeding the last finished experiment.Even if the number of cycles is decreased to define another section point for fatigue strength determination, other issues become obvious: • Lempp describes pronounced anisotropy of the material (83% of fatigue strength of specimens oriented in the transverse direction compared to longitudinal ones).• The alternate torsion fatigue curve is defined by 7 data points, but the scatter of data is so high, that it leads to the coefficient of determination R 2 = 0.624.• The test case of repeated axial loading is not covered at all.The prediction methods including also the fatigue limit in repeated axial loading cannot be validated by this test set.

Mielke (Mie -[17]), Troost, Akin and Klubberg (TAK -[18]), Kaniut (Kan -[19])
Mielke's set of fatigue limits on hardened 25CrMo4 steel tested by the staircase method is also often reproduced in various validation campaigns.Its weak point can be found in the intrinsic unequal size effect involved in various load combinations.The push-pull tests were realized on tiny bars of 1.9 mm diameter, while the torsion test sets on hollow specimens with outer diameter 20 mm and 2 mm wall thickness and the multiaxial tests on hollow specimens with 34 mm outer diameter and 1 mm wall thickness.
From further analysis of reports, it can be found that Troost, Akin and Klubberg [18] continued on tests on the same batch of material, thus forming the TAK set.They tested both specimen size configurations and nicely confirmed the size effect.Thanks to that, they results can be combined with Mielke's multiaxial data, so that Mielke's set could be used for validation purposes.Kaniut studied at the same university, and he reported experiments realized on the same material with identical material parameters.It can be deduced so that he also worked on the same lot of 25CrMo4.His main focus was nevertheless set to determining the fatigue limit values for two-channel loading with unequal frequencies applied in each load channel.
Because he used the staircase method for obtaining the fatigue limits, there are no material curves available for fatigue damage accumulation, while the unequal frequencies on individual load channels are likely to form more than one load cycle per a complete period of the total load cycle.Papuga highlighted this point in [1], and decided not to include similar load conditions into the validation test set, but other researchers [20][21] do not find such reasoning important and work with this test set as well.
Bennebach (Ben - [22]), Palin-Luc (PaL - [23]) As usually in the case of tests realized at ENSAM in Bordeaux, staircase philosophy was adopted for fatigue limit determination for these both cases.Because the first international publication of the test set was found in the paper by Morel and Palin-Luc [24], test sets were originally marked MPC and MPB in [1].
Bennebach's PhD thesis [22] allows the reader to define just two fatigue limits.Such a low number differs from the MPC set, where four items are available.One more fatigue limit can be traced back to Palin-Luc [23], where unfortunately no information is present, whether the same lot of material was used.The origin of the last item in MPC test set has not been found yet.
Findley (Fin - [25]) Papuga reported in [1] this test set three times, because he defined three different S-N curve section cuts at 10 6 , 10 7 and 10 8 cycles in order to increase the number of experiments realized on aluminum alloys.Though the same strategy was accepted e.g. in [26], this simple multiplication of the same data set is not consequent, because it was not adopted while processing the other data sets.
Gough and Pollard (GPx - [27][28]), Gough (Ggh - [29]) Gough and Pollard published in the 30 th of the 20 th century several papers (see e.g.[27][28]) describing results of their multiaxial tests on bar specimens manufactured from various steels and cast irons.The combination of plane bending and torsion without any phase shift and without any mean stress can be found in all data sets.Apparently, the focus of their effort was mostly devoted to covering of various materials extensively, than to determining the fatigue limits precisely.Therefore, many of these experiments do not conform to the lowest standards defined here before, and big scatter of data in some curves can be found in many cases.As a result of such practice, one half of the defined multiaxial fatigue limit items could be included into the validation data set at maximum.After the WWII, Gough published further papers (e.g.[29]), mostly devoted to more complex testing on S65A steel.Nevertheless, no information on the way the reported fatigue limits were set can be found in any of his papers from that period, and only his personal classification describing the reliability of the outputs is available.

Rotvel (Rot -[30])
Rotvel's data were used for a validation study even recently [31], though Papuga wrote in [1], that the experiments miss the case of pure torsion loading.This value has to be estimated, if the test set should be admitted for validation purposes, and this is apparently the method of determination used in [31].
Bhongbhibhat (BhO, BhA - [32]) These tests realized on specimens on large hollow specimens manufactured from 42CrMo4 (BhO) and St35 (BhA) are affected by substantially different cross-sections of specimens used for uniaxial and multiaxial fatigue tests.Additionally, the lifetime regions of individual curves in these data groups differ so substantially, that only small intersections can be found.

Issler (Iss -[33])
Issler's data set suffer from the doubtful way the fatigue limits were determined.No clue is provided by the author, but based on an analysis of his results, it seems that he either used the section cut at 2 000 000 cycles or the stress level of the last finished experiment with the longest lifetime.Section cuts at lower number of cycles are recommended to avoid potential extrapolations.The S-N curves for multiaxial load combinations are described by graphs with data points only.Unfortunately, the uniaxial fatigue curves are not present at all, and the fatigue limits derived by the author are provided solely.Because of the way the fatigue limits were determined in other cases, it cannot be anticipated how precise these values are.
Baier (BaB, Bai, BaG - [34]) The data on both steels (BaB -Ck35, Bai -34CrMo4) are very interesting for validation purposes.The tests were realized as S-N curve tests on sufficient numbers of hollow specimens in most cases.Anyhow, the tests on GG30 cast iron (BaG test set), for which any fully reversed torsion tests are missing, should not be used.

Table 1 :
List of papers referring to individual test sets.