Microsoft Word - numero_37_art_20 M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 146 Focussed on Multiaxial Fatigue and Fracture Multiaxial fatigue criterion based on parameters from torsion and axial S-N curve M. Margetin Slovak university of technology, Faculty of mechanical engineering, Slovakia matus.margetin@stuba.sk R. Ďurka VAKUUMTECHNIK s.r.o., Slovakia durka@vakuumtechnik.sk V. Chmelko Slovak university of technology, Faculty of mechanical engineering, Slovakia vladimir.chmelko@stuba.sk ABSTRACT. Multiaxial high cycle fatigue is a topic that concerns nearly all industrial domains. In recent years, a great deal of recommendations how to address problems with multiaxial fatigue life time estimation have been made and a huge progress in the field has been achieved. Until now, however, no universal criterion for multiaxial fatigue has been proposed. Addressing this situation, this paper offers a design of a new multiaxial criterion for high cycle fatigue. This criterion is based on critical plane search. Damage parameter consists of a combination of normal and shear stresses on a critical plane (which is a plane with maximal shear stress amplitude). Material parameters used in proposed criterion are obtained from torsion and axial S-N curves. Proposed criterion correctly calculates life time for boundary loading condition (pure torsion and pure axial loading). Application of proposed model is demonstrated on biaxial loading and the results are verified with testing program using specimens made from S355 steel. Fatigue material parameters for proposed criterion and multiple sets of data for different combination of axial and torsional loading have been obtained during the experiment. KEYWORDS. Fatigue; Multiaxial; S355, Criterion, Material properties. INTRODUCTION atigue life time prediction plays an important part in mechanical equipment design, regarding operating safety as well as equipment's reliability and economical design. The ongoing increase of machines' operating parameters and the pursuit of both effective material use and operating reliability make the analysis of fatigue process significant in the area of constructions' mechanical endurance calculation. The critical point of a construction that determines the life time of the whole equipment is often localized on a component that is exposed to a complex loading of external forces. Whether it's high-pressure piping systems [1] or F M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 147 mobile working machines [2], components of both are subjected to combined loading dependent on external conditions. Such a loading causes a stresses in the critical point, and this stress state is nearly always multiaxual. In the process of life time estimation in multiaxial stress state, it is not sufficient to transform this state into uniaxial stress state according to static strength hypotheses, as they especially don't consider the cyclical properties of materials and the different effects of normal and shear stresses on the fatigue life time. Therefore, it's necessary to use a mathematical model that is both able to reduce the multiaxial stress state to uniaxial stress state and that respects the mentioned problems at the same time. The methodology of transformation into uniaxial stress state then needs to be able to include also the change in the direction of damage, hence to respect the directional characteristic of the fatigue process. Nearly a century passed since first attempts to tackle the problem of multiaxial fatigue have been made, and as for the situation today, there are plenty of criteria that consider component's multiaxial stress state. According to the methodology of assessment of loading process in the critical point, these criteria can be divided into stress-based criteria [3,4,5,6], strain-based criteria [7,8,9] and criteria based on fracture mechanics [10,11,12]. This text presents a new stress-based criterion that transforms the multiaxial stress state of a cyclic loading into an equivalent uniaxial stress. This criterion is based on the critical plane approach. After presentation in the text, the criterion is subsequently verified using proportional tension/compression and torsion loading in an experiment. MULTIAXIAL FATIGUE CRITERION oday's most used stress-based criteria that transform multiaxial stress state into equivalent stress amplitude in critical plane are in the form the following linear or non-linear combination:     c ea fbσ dτ f N (1) Findley [3], McDiarmid [4] and Matake [5] have derived criteria for the calculation of the equivalent amplitude of shear stress as a linear combination of amplitude of shear stress and normal stress in the critical plane in the following form    eq fτ σ kτ f N (2) On the other hand, Carpintieri with Spagnoli [6] and Papuga with Ruzicka [13] have derived criteria for the calculation of the equivalent amplitude of normal stress in the critical plane as a non-linear combination in the following form    eq 1 2 fσ k σ k τ f N (3) The difference between the respective criteria is in the definition of the critical plane and in the form of material parameters k that consider the effect of normal and shear stresses. The resultant amplitude of the equivalent stress is then compared with the adequate fatigue life time curve in order to determine the finite fatigue life time or with fatigue limit in order to determine the infinite fatigue life time. The results achieved by presented hypotheses more or less correlate with the experimental results, however, there are some commonly known and well documented problems: Parameters weighting the effect of normal and shear stress in hypotheses are independent on the loading level (i.e. number of cycles to failure), which is not true universally [14,15]. Material parameters are based on conventional values (yield stress, fatigue limit) that are strongly dependent on the methodology of determination and sometimes their existence itself is questionable (fatigue limit being the example - there's no agreement on whether there is an actual amplitude of stress that isn't damaging). Neither one of the criteria provides correct results for both boundary loading conditions (pure torsion and pure tension/compression loading). T M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 148 Based on the previous analysis of the problem, authors have decided to present their own criterion for transformation of multiaxial stress state into equivalent uniaxial stress state. Presented criterion results from the following theoretical premises: The criterion is based on the critical plane approach and it assumes that the critical plane is the plane with the maximal shear stress amplitude. The premise that the plane with the maximal shear stress amplitude plays a key role in the process of the crack initiation is well documented in work [16]. The criterion reckons with non-linear combination of shear stress amplitude and normal stress amplitude in the critical plane. 2 2 eq , ,τ kσ τa cr a cr  (4) The criterion doesn't use conventional values of material parameters. Parameters that represent cyclical characteristics of the tested material are in the form of Baskin equation parameters for pure axial loading Eq. 5 and pure torsion loading Eq. 6.   b' σa f fσ σ 2N (5)   b' τa f fτ τ 2N (6) The criterion is derived so that it provides correct results for both boundary loading conditions (pure torsion and pure tension/compression loading represented by Eqs. 5,6). The parameter weighting normal stress is then in Eq. 7 and the resulting form of the criterion is shown in Eq. 8.   max 2 b bτ σ2, b f , , f f 2τ 2σ k σ σ 1                    (7)    ma 2 b bτ σ2, b b2 2 ,x max f τ a ff, , f f -1 2τ 2σ τ σ τ τ 2N σ σ r                               (8) EXPERIMENTAL ASSESSMENT o verify the function of the proposed criterion, an experiment was conducted in which experimental specimens were tested at different levels of proportional multiaxial loading. Experiment was carried out in the Strength and Elasticity Laboratory of the Faculty of Mechanical Engineering STU, using two experimental stands: Inova EDYZ testing system (tension/compression test) and MTS Bionix 370.02 Axial/Torsion testing system (torsion test and tension/torsion test). Two sets of experimental specimens were manufactured from steel S355J2+C (chemical composition is in Tab. 1 and specimens' geometry in Fig. 1.). σy02 [MPa] σu [MPa] A5 [%] 655 680 11.2 C [%] P [%] S [%] Mn [%] Si [%] Cu [%] Al [%] Mo [%] Ni [%] Cr [%] 0.16 0.014 0.025 1.31 0.18 0.12 0.018 0.01 0.06 0.07 Table 1: Mechanical and chemical properties of S355J2+C steel T M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 149 Figure 1: Geometry of the experimental specimens. In the first part of the experiment, Baskin equation parameters for pure axial loading and pure torsion loading were acquired. Experimental specimens were loaded in the force control mode. Failure condition of the experimental specimen was defined by the moment of the so-called “technical initiation of fatigue crack” (0,5–1 mm). The number of cycles prior to the initiation of the fatigue crack was determined on the basis of a continuous measurement of the deformation response to the loading regime of the test specimen σa (or τa) = const.. Completion of the test was defined either by the increase of the deformation (or by the angle of the distortion) by 1% in reference to the mean value or by the achievement of the life time of 2.106 cycles. The values of material parameters for the regression line and for the upper and lower prediction intervals of reliability for pure axial loading and pure torsion loading are shown in Tab. 2. P=50% P=2.5% P=97.5% τf [MPa] bτ [-] τf [MPa] bτ [-] τf [MPa] bτ [-] R 550 -0.0736 565 -0.0739 536 -0.0732 -0.9876 σf [MPa] bσ [-] σf [MPa] bσ [-] σf [MPa] bσ [-] R 636 -0.0531 655 -0.0531 619 -0.0531 -0.9779 Table 2: Fatigue properties. To verify the validity of the proposed criterion, the experimental program was carried under multiaxial stress state. The experimental specimen was subjected to a proportional combination of axial and torsion loading using controlled loading force and torque. The test was completed by achieving the same conditions as in the uniaxial loading (see above). CONCLUSIONS esults of the experiment are tabularly summarized in Tab. 4. Life time estimated by the help of the presented hypothesis Eq. 4 is Nf_com and the actual measured life time is Nf_exp. For the purposes of comparison, Tab. 4 includes also estimated life times with the help of the well known hypotheses based on stresses in critical plane R M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 150 presented by Findley (Nf_find) [3] and McDiarmid (Nf_mcd) [4] Eqs 9 and 10. For the calculation of the fatigue life time, material parameters for regression line of fatigue curves were used (Tab. 2). Material parameters used in Eqs 9 and 10 are shown in Tabs. 1, 2 and 3.       b* τeq a fin n f fmaxτ τ k σ τ 2N (9)           b'mcd τ eq a n,max f f u k τ τ σ τ 2N 2σ (10) kfin [-] τf* [MPa] kmcd [MPa] 0.131 555 244 Table 3: Material parameters for Eqs 9 and 10 n. τa [MPa] σa [MPa] Nf_exp Nf_com Nf_find Nf_mcd 1 159 204 650800 643902 485504 291483 2 180 204 172700 202990 154283 94670 3 167 204 597300 415031 313706 189936 4 209 163 105300 96899 65261 42766 5 183 163 536120 449372 283251 181345 6 196 163 198810 205010 133788 86699 7 209 122 193800 165247 113010 77586 8 201 122 301140 267920 178905 122110 9 193 122 375630 441567 287341 194911 10 185 122 996040 740543 468583 315769 11 209 82 231490 226301 178946 129725 12 201 82 572500 375732 291264 210195 13 193 82 952600 635892 482358 346420 14 185 82 2000000* 1098588 813741 581393 Table 4: Experimental data. Looking at the table, it's evident that estimations using Findley's or McDiarmid's criteria are too conservative for the specimen material and the loading levels we used. At the same time, comparing these two criteria, Findley's criterion has smaller deviation from the experimentally acquired data. Figure 2 shows comparison of calculated and experimentally acquired life times for the proposed hypothesis. For each specimen, Nf_cal are listed in the chart shown in a probabilistic form for the regression line and the lower and the upper prediction reliability intervals of the material parameters (Tab. 2). Chart shows that the proposed hypothesis correlates well with the experimentally acquired values of the fatigue life time. Majority of the experimentally acquired life times are placed within the reliability interval of the estimated life times. Experimentally acquired life times of specimens number 3,10,12, 13 and 14 were outside of the reliability interval. For these cases, the criterion provided conservative results. The specimen number 14 was put aside after going through 2 .106 loading cycles. Based on the experimental verification of the proposed hypothesis (Figure 2) and on its comparison with the well known hypotheses (Tab. 4), following can be stated:  The hypothesis correlates well with the experimentally acquired data - the majority of measured life times are placed within the prediction interval of the calculated life times.  In case the hypothesis doesn't provide correct results (experimentally acquired life time is outside of the prediction interval of the calculated life time), the results are on the conservative side of the calculation. M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 151  In comparison with Findley's and McDiarmid's hypotheses, the life times calculated for the particular experimental program are closer to the measured values. At the same time, both hypotheses provide significantly more conservative results then proposed criterion. Figure 2: Comparison of experimentally acquired and calculated life times. REFERENCES [1] Garan, M., Sulko, M., Analysis of the Service Straining on the Beam-Axle of Lorry, Int J A Sci Tech, 3(7) (2013) 44- 47. [2] Garan, M., Monitoring of fatigue damague by sensing the deformation state, PhD. Thesis, Slovak University of Technology, Slovakia, (2010). [3] Findley, W.N., Fatigue of metals under combinations of stresses, Trans ASME, 79 (1957) 1337-1338. [4] McDiarmid, D.L., A general criterion for high cycle multiaxial fatigue failure, Fatigue Fract Eng M, 14(4) (1991) 429- 453. DOI: 10.1111/j.1460-2695.1991.tb00673.x [5] Dang Van K., Sur la résistance a la fatigue des métaux, PhD. Thesis, Sci Techniq l´Armement, France, (1973). [6] Carpinteri A, Spagnoli A., Multiaxial high-cycle fatigue criterion for hard metals, Int J Fatigue, 23(2) (2001) 135-45. DOI: 10.1016/S0142-1123(00)00075-X [7] Fatemi, A., Socie, D.F., A critical plane approach to multiaxial fatigue damage including out-of-phase loading, Fatigue Fract Eng M, 11(3) (1988) 149-166. DOI: 10.1111/j.1460-2695.1988.tb01169.x [8] Brown, M.W., Miller, K.J., A Theory for Fatigue Failure under Multiaxial Stress-Strain Conditions, P I Mech Eng, 187 (1973) 745-756. [9] Smith, R.N., Watson, P., Topper, T.H., A stress-strain parameter for the fatigue of metals, J Mater, 5 (1970) 767-778. [10] Tanaka, K., Fatigue crack propagation from a crack inclined to the cyclic tensile axis, Eng Fract Mech, 6(3) (1974) 493-507. DOI: 10.1016/0013-7944(74)90007-1 [11] Sih, G.C., Barthelemy, B.M., Mixed mode fatigue crack growth predictions, Eng Fract Mech, 13(3) (1980) 439-451. DOI: 10.1016/0013-7944(80)90076-4 [12] Hoshide, T., Socie, D.F., Crack nucleation and growth modeling in biaxial fatigue, Eng Fract Mech, 29(3) (1988) 287- 299. DOI: 10.1016/0013-7944(88)90018-5 [13] Papuga, J., Ruzicka, M., Two new multiaxial criteria for high cycle fatigue computation, Int J Fatigue, 30(1) (2008) 58- 66. DOI: 10.1016/j.ijfatigue.2007.02.015 7 11 9 1 2 3 4 5 6 8 10 12 13 14 2,E+04 2,E+05 2,E+06 2,E+04 2,E+05 2,E+06 N f_ ca l Nf_exp M. Margetin et alii, Frattura ed Integrità Strutturale, 37 (2016) 146-152; DOI: 10.3221/IGF-ESIS.37.20 152 [14] Karolczuk, A., Kluger, L., Lagoda, T., A correction in the algorithm of fatigue life calculation based on the critical plane approach, Int J Fatigue, 83(2) (2016) 174-183. DOI: 10.1016/j.ijfatigue.2015.10.011 [15] Durka, R., Contribution to fatigue life time evaluation of structures under multiaxial loading, PhD. Thesis, Slovak University of Technology, Slovakia, (2012). [16] Polak, J., Man, J., Vystavel, T., Petranec, M., The shape of extrusions and intrusions and initiation of stage I fatigue cracks, Mater Sci Eng A, 517 (2009) 204-211. DOI: 10.1016/j.msea.2009.03.070 NOMENCLATURE σ, τ - normal and shear stress respectively σa, τa, σa,eq - normal, shear and equivalent stress respectively σf’, τf’ - normal and shear fatigue strength coeficient respectively bσ, bτ - normal and shear fatigue strength exponent σn - normal stress in computed plane Nf - cycles to failure σy02 - yield strength σu - ultimate strength ki - material parameters used to weight normal and shear stress respectively R - correlation coefficient P - probability of occurrence << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice