Microsoft Word - numero_35_art_27 C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 232 Focussed on Crack Paths Fretting fatigue crack propagation rate under variable loading conditions C. Gandiolle, S. Fouvry LTDS, Ecole Centrale de Lyon, 36 avenue Guy de Collonges, 69134 Ecully Cedex, France camille.gandiolle@ec-lyon.fr, siegfried.fouvry@ec-lyon.fr ABSTRACT. Fretting fatigue experiments aim to represent industrial problems and most of them endure variable loading. Being able to assess lifetime of assemblies, especially for low propagation rate conditions, is essential as experimental validation is often too expensive. Both experimental and numerical approaches are proposed to follow the crack propagation rate of steel on steel cylinder/plane fretting fatigue contact submitted to variable loading conditions. An original experimental monitoring has been implemented on the fretting-fatigue test device to observe crack propagation using a potential drop technique. A calibration curve relating crack length and electrical potential was established for the studied contact. It allows direct knowledge of the crack length and crack propagation rate. It was applied to mixed load test showing crack arrest for the last loading condition. To explain this behavior, a 2-dimensional FE modeling was implemented to simulate the complexes multi-axial contact stressing. The crack propagation rate was formalized using an effective stress intensity factor amplitude ΔKeff coupled with Paris law of the material. The crack arrest condition for a given loading was related to ΔKeff along the expected crack path crossing the material crack arrest threshold ΔK0. The failure was related to ΔKeff reaching the critical stress intensity factor KIC. A good correlation with experiments was observed allowing to predict the crack arrest condition although the model tends to overestimate the final crack length extension. KEYWORDS. Fretting fatigue; FEM; Cracking; Variable loading. INTRODUCTION retting is defined as a small oscillatory movement between two bodies in contact which induce relative displacement between the two surfaces. Combined with cyclic bulk fatigue loading, the so called fretting-fatigue loading can induce catastrophic damages such as wear or cracking, which critically reduce the endurance of assemblies. In addition, fretting fatigue experiments aim to represent industrial problems and most of them endure variable loadings. Being able to assess lifetime of assemblies, especially for variable loading applied for very high number of cycles, is essential as experimental validation is often too expensive. This study concentrates on cracking damage and is restricted to the partial slip domain. Crack nucleation risk is usually investigated by applying multi-axial fatigue criterion. Predictions were improved by considering the severe stress gradients imposed by the contact loading, using non-local process volume stress averaging strategy [1] or equivalent critical distance [2]. F C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 233 Once a crack is nucleated, depending on the loading amplitude, it will propagate or reach a crack arrest condition. The prediction of crack arrest was addressed by Araujo et al. [3] applying a short crack regime strategy. Crack propagation and crack propagation rate are usually addressed using Paris law. Numerous models exist to predict the lifetime of assemblies. Some consider that total lifetime can be approximated by the crack nucleation life and that crack propagation is negligible [4]. Others consider only the crack propagation phase [5]. Some authors also estimate the total lifetime as the sum of the crack nucleation life and the crack propagation life [6]. In the frame of this research work we investigated how to predict the crack propagation rate and the crack arrest condition of fretting fatigue test subjected to variable loading conditions. EXPERIMENTS Materials he studied material is a 32C1 steel (E=200GPa, ν=0.3). It shows low yield stress and is thus described by an elastic-plastic law. Because this study was conducted for industrial purposes, the industrial monotonic material law with isotropic hardening was used to describe the hardening of the material. Fig. 1 plots the monotonic hardening of the studied steel which was obtained from a simple tensile test and normalized by the yield strength σy,flat. The material was tested under various fretting fatigue conditions, using a cylinder/plane contact configuration, with a cylinder of R=4.6 mm radius applied with a normal force P on the flat material. The cylinder was a FM35 steel (E=200GPa, v=0.3), but with higher yield stress (σy_cylinder >> σy,flat), to investigate cracking on the plane specimen only. A similar monotonic elastic-plastic law with isotropic hardening was considered to describe its behavior (Fig. 1). Figure 1: Monotonic elastic-plastic material laws of the flat and cylinder components (R=4.6mm). Conventional 4 points bending tests were used to identify the crack propagation law of the study material. It follows Paris law:  mΔKC dN db . (1) With b, the crack length and N the number of cycles. The parameter C and m were really close to the British Standard (BS) parameters disclosed in Tab. 1. The crack arrest condition was obtained for ΔKth_10-7=ΔK0=5.7 MPa.m1/2, and KIC=212 MPa.m1/2. As several high stress ratios were applied, an effective stress intensity factor range ΔKeff was preferred to describe the overall crack propagation behavior. ΔKeff was established considering a simplification of Elber approximation proposed by V. Gros [7]: T C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 234 For RK>0; ΔKeff = K1max-4 with RK = KImin/KImax (2) For -110-9cycles-1: crack propagation. T C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 236 Experimental identification of fretting fatigue crack propagation rate The potential drop technique was applied on the following solicitation: P, σF,moy/σy,flat=0.78, RF=0.85, Q*/P=0.30. Fig. 4 plots the obtained potential and the crack propagation calculated from the calibration curve. The crack nucleates around NCN=105 cycles, and propagates slowly until b=300µm. then the propagation rate increases until failure at NT=1.73x106 cycles for a final crack length of bT=2.8 mm. The propagation life is easily deduced: NP=NT-NCN=1.63x106 cycles. The crack initiation time is less than 6% of the total lifetime of the contact. The crack initiation life will thus be neglected in the lifetime prediction. (a) (b) Figure 4: (a) Potential as a function of the number of cycles, (b) crack length as a function of the number of cycles calculated with the calibration curve. (R=4.6mm, P, σF,moy/σy,flat=0.78, RF=0.85, Q*/P=0.30). PREDICTION OF CRACK PROPAGATION RATE Finite Element analysis inite element (FE) analysis was carried out using Abaqus 6.10 software. A 2D plain strain model of the fretting fatigue test was generated (Fig. 5a). The dimensions and boundary conditions matched the parameters of the physical experiment. The model was meshed with CPE3-type linear triangular elements, except in the contact zone where CPE4R-type linear quadrilateral elements were used; this zone was also meshed more densely than the other regions (5µm squares). F C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 237 W=8mm σ Q P x z 33 11 y 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 25 50 75 100 C um ul at ed p la st ic s tr ai n fretting fatigue cycles Elastic shakedown Asymptotic evolution (a) (b) Figure 5: (a) Abaqus model of the test. (b) Cumulative plastic strain evolution simulated at the contact border hot spot for a crack arrest condition using the monotonic elastic-plastic law for the study material (P, σF,moy/σy,flat=0.78, RF=0.85, Q*/P=0.30). Surface-to-surface discretization with small sliding was adopted for contact accommodation. The Lagrange multiplier was selected as the contact algorithm. The friction coefficient of the contact µ was determined experimentally using the variable displacement technique described by Voisin et al. [14], µ=1.0. The cylinder and fatigue plane sample behaviors were described by the monotonic plastic laws introduced in Fig. 1. The normal force with which the cylinder was applied to the plane was high enough to generate plasticity. The added fatigue loading contributed to extend the plastic state. For each simulation, the most highly strained integration point was monitored and its cumulative plastic strain evolution was plotted as a function of the fretting fatigue loading cycles (Fig. 5b). The level of activated plasticity decreased after each cycle, due partly to material hardening but mostly to plastic accommodation of the contact geometry. So the cumulative plastic strain increased until reaching an asymptotic evolution, i.e. a stable state corresponding to elastic shakedown. Numerical analysis showed that elastic shakedown was achieved after around 80 loading cycles for fretting fatigue. Fatigue analysis was therefore performed on the stable elastic shakedown state. Crack propagation rate identification A decoupled approach was used to predict the crack propagation. First the contact stress state was obtained by finite elements modeling (FEM), then the normal stressing along the expected crack path are extracted at the contact border for the maximum and minimum loading conditions as schematized in Fig. 6. x P Q σ11 h t Figure 6. Stress extraction along crack path for a fretting fatigue case. Then the mode I stress intensity factor (SIF) was calculated using Bueckner weight function approach [13]: C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 238  h xI dtttMK 0 ).().( 2   (7) with                2 21 2/1 ..1.)( h t m h t mttM (8) and 62 rCrBAm iiii  (9) with A1=Aref, B1=27.9558Aref, C1=14.2870Aref, A2=0.4070Aref, B2=5.3504Aref and C2=113.9489Aref. The contribution of mode II was neglected [1]. Finally the effective stress intensity range is obtained following the Eqs. 2 to 4. Crack nucleation life was neglected following the observation of the previous paragraph. The total lifetime was thus equivalent to the propagation life. The loading cycles related to the propagation stage were computed using ΔKeff integrated from b=0 up to failure:      Tb b mPT KC db NN 0 (10) Failure was related to KImax=KIC with KIC=212MPa. Alternatively, if ΔKeff(b) crosses the crack arrest condition ΔK0 then the crack stops propagating and crack arrest is reached. Fig. 7 compares the crack length obtained from the potential drop technique with the crack length determined from the predictive method. A rather good correlation is observed. In addition the model tends to overestimate the crack extension rate which indirectly provides a conservative and safe estimation of the fretting fatigue cracking risk. Figure 7: Comparison of experimental and theoretical crack propagation rates. (R=4.6mm, P, σF,moy/σy,flat=0.78, RF=0.85, Q*/P=0.30). Application of prediction method to a mixed load test Three blocks of different fretting fatigue loading were applied successively on the fretting fatigue contact (Tab. 2 and Fig. 8). Individually, block 1 led to failure (Fig. 7), block 2 led to crack arrest and block 3 generated no detectable crack. Fig. 9a plots potential evolution of the mixed load test and Fig. 9b plots the crack propagation calculated with the calibration curve. It shows that a crack is generated at the first block and propagates throughout the first and second bloc. The crack stops propagating when the loading changes for the third block. C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 239 Bloc σF,moy/σy-flat σF,alt/σy,flat Q*/P N (cycles) 1 0.78 0.85 0.3 70000 2 0.78 1 0.3 670000 3 0.78 0.85 0.15 1000000 Table 2: Fretting fatigue loadings and durations of the mixed load test blocks Figure 8: Sketch of loading blocks. (a) (b) Figure 9: (a) Potential evolution during mixed load test, (b) Crack propagation rate established with the calibration curve. In order to explain this behavior, the mixed load test was simulated. Blocks were applied one after the other on the FE model as in the experimental test. Loading history, that is the residual stress from each loading block, is thus taken into account by the next block. For each loading block, ΔKeff is plotted as a function of depth in Fig. 10. Knowing crack lengths obtained at the end of each block from Fig. 9b, it is possible to shift from one curve to the other. At the end of the second bloc, crack length was equal to b2=170µm. and at this depth, ΔKeff (block 3) passes below the crack arrest threshold condition ΔK0=5.7MPa.m1/2. Combining these crack propagation paths evolutions, the final crack arrest condition achieved when these three loading sequences were imposed can be understood. C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 240 Figure 10 : Evolutions of ΔKeff of each loading blocks as a function of depth. The predictive method was applied for the studied mixed load test condition. Eq. 10 is incremented step by step and depending on N, the relevant ΔKeff is considered. Fig. 11 plots the predicted crack propagation extension compared to the experimental crack propagation extension. Prediction is conservative as the predicted crack length is longer than the experimental crack length. However the predictive method recognizes the crack nucleation on the first block and the crack arrest at the third block. Hence even if the model tends to overestimate the final crack extension, it well predicts the fretting fatigue crack arrest condition. Figure 11: Comparison of theoretical crack propagation rate calculated with the predictive method and experimental crack propagation rate. Loading: R=4.6mm, P, Block 1 : σF,moy/σy,flat=0.78, Rσ=0.85, Q*/P=0.3, N=70000; block 2 : σF,moy/σy,flat=0.78, Rσ=1, Q*/P=0.3, N=670000; block 3 : σF,moy/σy,flat=0.78, Rσ=0.85, Q*/P=0.15, N=1000000. CONCLUSION he crack propagation rate of fretting fatigue loading was investigated both experimentally and numerically. The potential drop technique method was implemented on the fatigue fretting test device and a calibration curve was established. With this curve, the crack length was known throughout a fretting fatigue test. A decoupled approach was applied to estimate the stress intensity factor evolution with the fretting fatigue crack extension thus to predict the crack propagation risk. The contact stress state was obtained by finite elements modeling, then the mode I stress intensity factor was calculated using a weight function approach and finally the Paris law was applied on the deduced effective stress intensity range along the crack. It allowed for good estimation of the crack propagation rate. T C. Gandiolle et alii, Frattura ed Integrità Strutturale, 35 (2016) 232-241; DOI: 10.3221/IGF-ESIS.35.27 241 Finally these strategies were applied to a mixed load test, more representative of an industrial loading case. The predictive method was adjusted to consider the successive ΔKeff of each loading, depending on the number of cycles. This simple method allowed really good prediction of crack nucleation and crack arrest condition. The predicted crack extension was slightly too conservative; however it is consistent with the security coefficient needed in industry. Better predictions may be achieved using a more representative cyclic plastic law and more elaborate description of the ΔKeff parameter. REFERENCES [1] Fouvry, S., Kapsa, P., Vincent, L., A multiaxial fatigue analysis of fretting contact taking into account the size effect, ASTM STP., 1367 (2000) 167-182. [2] Araújo, J., Nowell, D., The effect of rapidly varying contact stress fields on fretting fatigue, Int. J. Fatigue, 24 (2002) 763-775. [3] Araujo, J.A., Nowell, D., Analysis of pad size effects in fretting fatigue using short crack arrest methodologies, Int. J. Fatigue, 21 (1999) 947-956. [4] Ruiz, C., Boddington, P.H.B., Chen, K.C., An Investigation of Fatigue and Fretting in a Dovetail Joint, Exp. Mech., 24 (1984) 208-217. [5] Giannakopoulos, A.E., Lindley, T.C., Suresh, S., Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue, Acta Mater., 46 (1998) 2955-2968. [6] Navarro, C., Munoz, S., Dominguez, J., On the use of multiaxial fatigue criteria for fretting fatigue life assessment, Int. J. Fatigue, 30 (2008) 32-44. [7] Gros, V., Etude de l’amorçage et de la propagation des fissures de fatigue dans les essieux-axes ferroviaires, Ecole centrale Paris, 1996. [8] Barnett, W., Troiono, A., Crack Propagation in Hydrogen Induced Brittle Fracture of Steel, J Met., 9 (1952) 94. [9] Kondo, Y., Sakae, C., Kubota, M., Yanagihara, K., Non-propagating crack behaviour at giga-cycle fretting fatigue limit, Fatigue Fract. Eng. Mater. Struct., 28 (2005) 501-506. [10] Meriaux, J., Fouvry, S., Kubiak, K.J., Deyber, S., Characterization of crack nucleation in TA6V under fretting-fatigue loading using the potential drop technique, Int. J. Fatigue, 32 (2010) 1658-1668. [11] Proudhon, H., Fouvry, S., Yantio, G.R., Determination and prediction of the fretting crack initiation : introduction of the (P, Q, N) representation and definition of a variable process volume, Int. J. Fatigue, 28 (2006) 707-713. [12] Gandiolle, C., Fouvry, S., Experimental Analysis and Modeling of the Crack Arrest Condition Under Severe Plastic Fretting Fatigue Conditions, Procedia Eng., 66 (2013) 783-792. [13] Bueckner, H.F., Weight functions and fundamental fields for the penny shaped and the half plane crack in three spaces, Int. J. Solids Struct., 23 (1987) 57-93. [14] Voisin, J.M., Vannes, A.B., Vincent, L., Daviot, J., Giraud, B., Analysis of a tube-grid oscillatory contact: methodology selection of superficial treatments, Wear, 181-183 (1995) 826-832. << /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. 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