Issue 50

J. Papuga et alii, Frattura ed Integrità Strutturale, 50 (2019) 163-183; DOI: 10.3221/IGF-ESIS.50.15 164 stress amplitudes are proportional. Non-proportional loading can be characterized above all by the fact that the stress (strain) tensor components at any moment are not simple multiples of the tensor at another moment. This condition causes the principal stress directions to rotate during time. For in-phase loading, there are various equivalent values that can be used. However, the question arises: What is the appropriate equivalent stress for non-proportional loading? Before dealing with this question, it is necessary to specify the research domain focused on in this paper. The area under investigation is the high-cycle fatigue region, where the effect of plasticity is almost negligible in the macroscopic scale. In order to avoid the potential effect of non-homogeneous stress distribution over the critical cross-section, and to skip the potential need to involve its effect in the analysis, only experiments related to unnotched specimens will be studied here. The target of the paper is not just to show the phase shift effect within various experimental data sets, but above all to investigate how well various multiaxial fatigue strength criteria respond to the phase shift effect, and whether they follow the observable and sufficiently proven experimental trends. One subgroup of non-proportional load cases is the class of out-of-phase loadings. Here, a phase shift can be found between individual load channels that have the same period (and usually correspond to harmonic loading). Many experiments on this topic have been reported in the high-cycle fatigue domain, and they will be discussed later on. One of the most explicit summaries of the findings of previous researchers, in stress-based and also strain-based fatigue life prediction was provided by Sonsino [3], and [4]. His conclusions are that the switch from in-phase loading to out-of-phase loading while keeping the same stress amplitudes on individual load channels causes:  An increase in the fatigue life in the case of load-controlled loading  A decrease in the fatigue life in the case of deformation-controlled fatigue tests of unnotched specimens, or in the case of load-controlled loading of notched specimens. This behavior of notched components is attributed to the deformation-controlled loading of the volume around the notch, where the yield stress can be exceeded. Sonsino states that the life reduction is caused by the induced multiaxial hardening. He admits that there are materials that show a decreased fatigue life in this situation, though no multiaxial hardening can be observed, and that some other mechanism may cause this behavior. To prove this, Sonsino [4] refers to multiple papers and research works. If only the load-controlled fatigue experiments referred to by him in [4] are summarized, it can be found that:  The data related to the unpublished report by Löwisch and Bomas [5] seem to provide clear proof of the stated behavior.  Experiments performed by Hug [6] on three different materials, two of which are cited here, also conform to the stated behavior under load control. The behavior is assessed in the region between 1000 and 10000 cycles, in the domain classically referred to as the low-cycle fatigue region. Even the lowest stress levels applied within those tests exceed the yield stress of the virgin material. It is therefore very questionable, whether the reported output confirms Sonsino’s conclusions on load-controlled experiments, or whether it undermines his statements concerning the deformation-controlled tests.  When reporting on Simbürger’s experiments in [7], Sonsino refers to the experiments performed in bending and torsion loading of notched specimens, where the observed decrease in the fatigue life confirms his conclusion concerning the behavior of notched components. However, Sonsino did not draw attention to the experiments on hollow specimens loaded by push-pull and torsion, which are described in the same PhD thesis [7]. These experimental results are reprinted here in Fig. 3, and they show behavior that contradicts Sonsino’s summary. Other comparative analyses of various computational criteria in the HCF domain can be found, see e.g. [1], [7]-[12]. The difference in the response to in-phase loading and to out-of-phase loading is not usually treated separately from other load cases, and in many cases there is only an analysis of the overall prediction errors of all experiments together. The only exception that the authors are aware of is provided by the papers by Papuga [1] and Papuga and Halama [13], where groups of tests conforming to in-phase (IP) and out-of-phase (OP) loading without other partial effects are consistently separated and analyzed. In order to be able to characterize the typical response to the loading case that is being evaluated, the benchmark test set should contain enough data items reflecting the tested property, so that the typical scatter of fatigue results will not affect the overall output too much. The quality of the benchmark data set of experiments, i.e. a sufficiently proven fatigue characteristic (the whole S-N curve, the fatigue limit, etc.), is a basic requirement for any such analysis. The widely-used benchmark test sets for analyzing the quality of new fatigue strength estimation criteria have some deficiencies. The most widely cited test set, investigated by Nishihara and Kawamoto [14], provides a suitable example. Papuga [15] gives reasons why this test set should be disregarded altogether. The main reason is a general lack of experimental data that enable the fatigue strength (the fatigue limit) to be set with acceptable accuracy – the maximum number of data points used in [14] to define the fatigue limit is 5, and this number of data points is reached just once in the whole testing campaign on three