JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 731-749, Warsaw 2006 APPLICATION OF EXPERIMENTAL RESULTS TO NUMERICAL MODELS OF FATIGUE CRACKS PROPAGATING IN THE ROLLING CONTACT ZONE Paweł Pyrzanowski Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology e-mail: pyrzan@meil.pw.edu.pl The paper presents numerical models of the ”squat” type crack serving as an example of the RCF (Rolling Contact Fatigue) crack that appears in railheads. In developing these models, one took into account the re- sults of experimental investigations. Several types of experiments were done: in situ measurements of the crack growing rate and the real sha- pe of the contact zone between the worn rail and wheel, measurements of shapes of the real crack in macro- and micro-scales and crack thick- ness distribution, interactions between crack faces. The results obtained by means of the developed numerical models facilitate understanding of fracture processes and contribute to more precise predictions of the lifetime. Key words: experimental mechanics, fracture mechanics, rolling contact fatigue, numerical calculations 1. Introduction Thepaper is devoted to the problemof fatigue crack propagation in railheads. This type of cracks is often called the Rail Contact Fatigue (RCF) one. The present paper deals with cracks called ”squat” or ”black spots” that were ob- served in rails in Poland and in other countries all over the world at the end of the twentieth century. Cracks of this type appeared in tracks in which the rails were made of steel of improved strength properties (resistant to abrasive wear) and had been used about ten years earlier (Kondo et al., 1996; Sawley andReiff, 2000). The cracksmay cause catastrophic accidents of trains, some- times very dangerous as they can be found in high-speed lines. Figure 1 shows 732 P. Pyrzanowski a top view of a typical large-size singular ”squat” type crack, initiated on the rolling strip. The length of this crack on a raceway is about 70mm. Fig. 1. Top view of a typical large size singular ”squat” type crack These cracks can serve as a good example of a problem to be solved in the field of contact mechanics. The main task is to explain the fatigue crack initiation and propagation under conditions of rolling contact. Inmanyworks, propagation of such cracks was described using both two-dimensional models (Bogdański, 2002;DubourgandVillechaise, 1992;Olzak andStupnicki, 1999b) and three-dimensional ones (Bogdański et al., 1998; Bogdański and Brown, 2003; Neves et al., 1997). However, some problems still need attention and detailed examination, particularly in view of the fact that the results of recent experimental investigations have not been taken into account in thesemodels. 2. Parameters used in numerical models Each numericalmodel should include someparameters the values ofwhich can be assumed, calculated analytically or determined experimentally. Some of the parameters used in the analysis of crack propagation in the contact zone are briefly described below: • Material properties. Some material constants of the rail steel, like Young’s modulus or Poisson’s ratio, can be accepted in the sameway as for other grades of steel, another parameters must be measured. First of all, these are parameters of stress-strain curves used in elasto-plastic calculations and fracturemechanics parameters like the rate of the crack growth. • Distribution of the wheel-rail contact stresses. For a new (not exploited yet) rail and wheel, which have nominal shapes of rolling surfaces, geo- Application of experimental results to numerical models... 733 metry of these surfaces is known, and distribution of contact stresses can be calculated form Hertz’s equations. For worn rails and wheels, however, the contact problem must be solved numerically or measured experimentally. • Crack shape. The shape of crack faces both in micro- and macro-scales and the crack thickness distributionmust be determined experimentally. • Crack faces interaction. The interactions between crack faces comprise friction, micro-slip of the crack faces as well as stress concentration ne- ar the crack face asperities. They must be measured and they can be represented by a friction coefficient in numerical models. • Liquid penetration into the crack. The presence of a liquid in the crack brings about significantly different results in comparison with those ob- tained without the liquid. The presence of the liquid should be verified experimentally. 3. Experimental determination of parameters Since some of the parameters necessary for numerical models cannot be assu- med in onother way, they should be determined experimentally. 3.1. Material properties In railsmade of a steel of improved strength properties used in Polish rail- roads900A (St90AP) steel hasbeenused.The steel hasa chemical constitution given in Table 1 (Żak et al., 2003). Table 1.Chemical constitution of 900A steel Element C Mn Si P S Al Mass fraction [%] 0.7-0.76 1.0-1.25 0.2-0.4 < 0.03 < 0.025 < 0.004 Table 2 contains material properties that weremeasured by the Author in accordance with the PN-EN Standard for specimens cut off from a railhead. The results obtained by the Author are similar to those obtained by other researchers, see e.g. Bochenek (1999). 734 P. Pyrzanowski Table 2.Material properties of 900A steel Parameter Symbol Unit Value Uncertainty Young’s modulus E MPa 192000 4000 Limit of proportionality R0.01 MPa 500 20 Limit of elasticity R0.05 MPa 600 20 Proof stress R0.2 MPa 730 25 Tensile strength Rm MPa 1190 40 Reduction of area at rupture Z – 0.42 0.03 Ultimation at rupture A – 0.14 0.02 In the course of rolling of a wheel over a rail with a crack, one can observe a very complicated distribution of the stress intensity factors along the crack front. Generally, the crack grows under nonproportionalmixedmodes I, II, III of load cycles. It is impossible to design an experiment inwhich the crack front would be loaded in such away.Most of experimentswithmixedmode loadings are performed as proportional I+II, I+III or (very rarely) II+III load cycles. Thenext problemrefers to thematerial the rails aremade of.At present, some experimental results for nonproportional mixed mode I+II cycles for British rail steel BS11 are available (Bold et al., 1991; Brown et al., 1996;Wong et al., 1996). However, some experiments made by the Author proved that the rate of crack growth of Polish rail steel 900A could be 2 to 3 times lower than the same parameter of BS11 steel. 3.2. Distribution of the wheel-rail contact stress The wear process of wheels and rails changes their geometry and shape of the contact zone. The crack growth changes also the rail geometry. Figure 2 presents the difference between the shape of a rail above the crack and the nominal shape of a UIC60 rail. The lines bear numbers indicating the vertical difference in millimetres. It canbe seen that, for the examined rail, themaximal value of thedifferen- ce between the nominal and measured shapes of the rail reach about 2.2mm for regions away from the crack, and 2.6mm above the crack. This depression of the rail surface is caused by wear observed between the rail and the wheel. In the zone above the crack, wear of crack faces and water penetration into the increasing crack cannot be neglected either. These processes, which lead to a nonzero crack thickness, were presented by Pyrzanowski (2006). Application of experimental results to numerical models... 735 Fig. 2. Difference between the rail shape under a crack and the nominal shape of a UIC60-type rail Due to changes in the rail height, the shape of the rolling strip is also modified. It is shown in Fig.3, where measured shapes of rolling strips are indicated as dark areas against the background of a fair rail. Fig. 3. Measured shape of rolling strips: (a) part of a rail about 800mm long with four cracks, (b) magnified part of the rail with one crack indicated in Fig.3a One rolling strip is divided into two or three parts above the crack (see Fig.3). The first one is close to the gauge corner, approximately in the same 736 P. Pyrzanowski place as without the crack, and the second one – is near the field corner of the rail. Consequently, one area of contact pressure is divided into two or three elements. Figure 4 shows shape of the rolling strip assumed in numerical calculations and the contact stress distribution calculated for a worn rail and wheel for one position of the wheel. Fig. 4. Distribution of contact stresses: (a) contourmap against the background of rolling strips and crack front, (b) perspective view The presented distribution of contact stresses cannot be calculated assu- ming Hertz’s theory. 3.3. Shape of the crack The investigations of the crack shape should address the three following issues: shapeof the crack inmacro-scale, shapeof the crack faces inmicro-scale (often called roughness) and crack thickness. 3.3.1. Shape of the crack in macro-scale A very important geometrical problem related to a crack appearing in the contact zone is its shape. Inmost of analyses, one assumes a very simple shape of the crack. For 3D models, it is usually a semi-elliptical shallow-angle part of a plane, open towards the upper side of the rail in the rail-wheel contact zone. Thesemodels were used byKaneta andMurakami (1991), Bogdański et al. (1998), Bogdański andBrown (2002). However, experimental investigations of rolling contact fatigue cracks have proved that their shapes are muchmore complicated. The investigationsmade by theAuthor (Pyrzanowski andMruk, 2000) allow one to propose a standard shape of the ”squat” type crack as Application of experimental results to numerical models... 737 shown (for a large-size crack) inFig.5. The contour lines illustrate the vertical distance from the highest point of the crack face and are drownevery 0.25mm. Grey areas showing real rolling strips depict the wear process. The black bold line, close to the gauge corner of the crack (in the bottom part of the crack in Fig.5), indicates the crack mouth. Fig. 5. Standard shape of a ”squat” type crack The measured crack shape was used in numerical models (Pyrzanowski, 2005a,b,c). Similar profiles of the crack, used in 2Dmodels, are usually very simple and do not correspond to real shapes. These profiles aremodelled as straight lines propagating at a shallow angle near the surface (Bower, 1988). A bent crack consists of a few line segments (Dubourg andVillechaise, 1992), a kinked crack with single or double branches (Bogdański, 2002) or a straight line parallel to the rolling surface (Komvopoulos and Cho, 1997). 2D models of rails with cracks can be useful in verification of assumptions or calculations of cases which cannot be solved using 3D models, e.g. with the fluid flow of liquids existing in the cracks.However, the investigations donebyPyrzanowski (2004) show that for 2Dmodels the results very strongly depend on the crack shape, and the real shape cannot be neglected. 3.3.2. Shape of the crack faces in micro-scale The shape of the crack faces in micro-scale depends on fracture parame- ters during crack growth and processes that take place in the crack. As shown by Kobayashi et al. (1997), it is possible to deduct the load spectrum para- meters from fatigue failure surfaces using 3D high-resolution elevation maps. 738 P. Pyrzanowski For cracks appearing in the contact zone, such investigations have not been done yet. Pyrzanowski andMruk (2000) analysed surfaces of the ”squat” type crack using a S3P Penthometer for measuring the distribution of roughness along the base lines 4mm long situated in various locations on the crack face. Figure 6 presents an exemplary course of crack roughness and histograms of periods of roughness for the some base. Fig. 6. Crack face roughness: (a) distribution along the base, (b) histogram of roughness The crack face roughness determined experimentally could be used in nu- merical calculations (Olzak and Stupnicki, 1999a). In practice, however, it is very complicated. Olzak and Stupnicki (2001) presented a discussion of two models of crack face interactions, namely, with crack faces covered with layers of a worn out material, the properties of which were determined basing on results of experi- ments, andwith crack faces covered withmicro-asperities, which forced crack dilatation accompanying tangential displacements. All the above-mentionedmodels used in numerical simulations allow us to draw a conclusion that amore realistic representation of interactions between the crack faces is strongly needed. 3.3.3. Thickness of the crack The process of thickness formation and growth in a ”squat” type crack is stimulated by combined effects of very high contact stresses between a wheel and a rail head, wear process andwater penetration into the increasing crack. These processes were accurately described by Pyrzanowski (2006). A simple methodwas used formeasuring the crack thickness, and the results are shown Application of experimental results to numerical models... 739 in Fig.7. The calculated volume of the crack was equal to 115±30mm3 for a crack 77mm long. Fig. 7. Distribution of crack thickness δ The knowledge of the crack thickness distribution is very important, espe- cially for cracks appearing within the contact zone, when the crack grows under multimodal conditions with compression stresses normal to the crack faces, or in situationswhenother effects, e.g. fluidfilling,may exist. Numerical calculations (Pyrzanowski, 2004) show that values of stress intensity factors for models with a non zero crack thickness may be a few times higher than those obtained for models with the crack thickness neglected. 3.4. Crack faces interaction Since the problem of crack face roughness is very complicated in view of both measurement and application, it seems that a better way may be the measuring of interactions between crack faces. Special samples (small cubes) were produced of slices cut out from railheads in which ”squat” type cracks were detected. A cube was selected because the investigated crack should be relatively plain andperpendicular to the front surface of the sample.Then, the selected cube was welded by using of a laser beam to the rest of the sample, which then was mounted to the clamping grips. The sample was subjected to normal Pn and tangential Pt forces (see Fig.8a). The components of the di- splacement vectors of the sample surface points were registered using theGHI approach with a holographic plate mounted near the sample support point. Thismethod is very convenient, since only one hologram is necessary to obtain the data needed for determination of all three components of the displacement vector, ensuring additionally that the displacements of the sample due tomo- vements of the loading frame are eliminated. Both the registration procedure 740 P. Pyrzanowski of interferograms and the way of their reconstruction were presented in detail by Tu et al. (1997) and Szpakowska et al. (1998). Fig. 8. Crack faces interactions: (a) sample with loading scheme, (b) diagram of the interaction factor f versus crack length The distribution of the interaction factor f versus the crack length was calculated by means of the loading scheme and analysis procedure described by Pyrzanowski and Stupnicki (2001a,b). The presence of micro-slip between the crack faces causes that the total magnitude of tangential stress tensor ΣSxy is limited for the current total normal stress component ΣSyy, hence the value of the interaction coefficient at a given point of the crack face can be determined as f = ΣSxy ΣSyy (3.1) The course of the factor f for the investigated squat-type of crack along its length is presented in Fig.8b. Its value is relatively small along the greater part of the crack length, and reaching a high value at the crack tip. The interaction coefficient f can be used as Coulomb’s friction coefficient, and its value for the internal part of the ”squat” crack can be assumed as 0.35. 3.5. Liquid penetration into the crack Avery important problemarising in railhead cracks consists in penetration of cracksbywater (Bogdański et al., 1996;Bower, 1988;KanetaandMurakami, 1991). The stress intensity factors for cracks filled with water could be much greater than for empty cracks (Pyrzanowski, 2005c). The fact that water does appear in ”squat” cracks was proved experimentally. Figure 9 shows traces of the liquid flowing out of the crack (Fig.9a) and small grains on the crack face (Fig.9b). These grains are detached from the surface and can be removed by water outflow from the crack. The chemical constitution with a large content Application of experimental results to numerical models... 741 of oxygen shows that the grains consist of iron oxide, which is the product of iron corrosion in the presence of water. Fig. 9. Evidences of water presence in the crack: (a) traces of the liquid flowed out from the crack, (b) small grains of iron oxide 4. Experimental verification of calculations The next, very important, role of experiments consists in the verification of models and calculations. These investigations should be done in situ, on real objects, loaded in a real manner. For the analysed case, it must be a real ”squat” type of crack in a railhead, loaded by an engine. Such investigations can provide, first of all, information about the crack growth rate. Observation of the crack shape inmacro-scale (Fig.10a) shows visible lines along which the crack front is arrested during summer season. Ultrasound measurements of the crack front position show that the crack grows in the co- planar direction only in the winter season. The knowledge of the load history, i.e. the number of train axles going over the crack (about 27000 engine axes and 423000 carriage axles) and the distance between the arrested crack front lines (7mm) allows us to calculate the average rate of the crack growth. The value of this parameter is 250 to 400nm/cycle in the direction coincident with the rolling direction of trains. If the surface of the crack is badly damaged, it becomes difficult to read the crack growth rate. However, for some parts of the crack, it is still possible (see Fig.10b). The growth rate for the same direction is 250 to 350nm/cycle. 742 P. Pyrzanowski Fig. 10. Experimental verification of the crack growth rate 5. Application of measured parameters to numerical calculations Themeasured parameters can be used in numerical calculations. In the most complex model, the real 3D geometry and the existence of water in the crack should be included. Such a model has not been solved yet. In this paper, two models will be briefly discussed: 2D model with 1D liquid flow and 3D one regarding changes of the geometry due to wear. 5.1. 2D model with 1D liquid flow The advanced model of interaction between a liquid flowing in the crack and crack faces was developed byOlzak andPiechna (2003) andPyrzanowski andOlzak (2004). Pyrzanowski (2005c) described the samemodel of the liquid and crack faces interaction in connectionwith the real geometry of the ”squat” crack. In that model, the following assumptions were made: • A plane specimen was loaded by a rolling cylinder of diameter 900mm. The maximum contact stresses were 450MPa which correspond to ave- rage stresses between the rail and wheel on a rolling strip. • Speed of the rolling cylinder was 30m/s. • Geometry of the crack corresponded to the measured crack geometry of the rail cross-section under the rolling strip. • Mechanical thickness of the crack corresponded to themeasured one (see Fig.11). • Residual hydraulics thickness 30µm in the vicinity of the crack mouth dropped to 5µmat the crack front (see Fig.11). Application of experimental results to numerical models... 743 • The atmospheric pressure was 0.1MPa at the crack mouth. • The friction coefficient was equal to zero. • The coefficient of absolute viscosity of the liquid was 0.001Ns/m2. Fig. 11. Crack thickness: (a) initial mechanical, (b) residual hydraulic, (c) initial total hydraulic Figure 12 shows some results obtained by means of the presented model. The maximal value of the liquid pressure was 265MPa (at the crack front, for the cylinder axis 7mm before crack front – see Fig.12a) and the maximal value of mechanical thickness of the crack was 111µm (12mm from the crack front for the cylinder axis 30mm before the crack front – see Fig.12b). Fig. 12. Distribution of (a) liquid pressure, (b) mechanical thickness versus position along the crack length l and position of the cylinder axis x Figure 13 depicts the influence of the liquid on the distribution of stress intensity factors during rolling and results for the crack with and without the liquid. The resulting distributions of stress intensity factors allow one to calculate the crack growth rate. For the crack without the liquid, this value was about 744 P. Pyrzanowski Fig. 13. Distribution of the stress intensity factors KI and KII during rolling 7nm/cycle and for the crack with the liquid – about 1200nm/cycle, assuming that Brown’s equations are valid. The value 1200nm/cycle is 2 to 3 times greater than that observed. The difference between Polish and British steel (described in Section 3.2) or insufficient precision of the assumptions can be the reason for these differences. 5.2. 3D model regarding changes of geometry due to wear process If the presence of a liquid in the crack is neglected, it becomes possible to solve the problem of the 3D model of a rail with the crack and a wheel. Such a model, in which changes in the geometry caused by wear processes were also introduced, was developed byPyrzanowski (2000a,c). In this model, real geometry of the rail-wheel contact zone and rolling strips (see Section 3.2 and Fig.4), shape of the crack in macro-scale (Section 3.3.1 and Fig.5) and distribution of the crack thickness (Section 3.3.3 and Fig.7) were taken into account. The finite element model of the rail is shown in Fig.14. The whole model comprised 83040 elements and 183830 nodes. The crack length was 77 millimetres. The assumed friction coefficient between the rail and wheel was 0.15, and between the crack faces 0.3. The contact load between the rail andwheel was 100kN, which corresponded to load caused by the engine. The wheel was rolled over the rail over a distance of 110mm, from the position x=−43mm to x= 67mm (the crack range extended from x=−27mm to x=50mm – see the crack coordinate system in Fig.14b). Application of experimental results to numerical models... 745 Fig. 14. A 3D FEMmodel of a rail with a crack: (a) mesh of the whole rail (b) magnifiedmesh over the crack. Arrows indicate the rolling direction of the wheel As a result, one could calculate the distributions of stress intensity factors versus the position on the crack front s andposition of the rail axis in relation to x. Then, the distribution of the crack growth rate along the crack frontwas calculated (Fig.15). The presenteddiagram shows that the co-planar direction of growth is possible only for parts of the crack front in the vicinity of the crackmouth, see grey lines in Fig.15. For the remaining part of the crack, the calculations showadominant tendency of branchgrowing,whose value attains maximum in the front part of the crack just under the rolling strip (black lines in Fig.15). This corresponds to crack growth in summer,when the crack front is arrested (see Section 4 and Fig.10a). Fig. 15. Distribution of the crack growth rate along the crack front 746 P. Pyrzanowski 6. Conclusions The presented results of experimental investigations and numerical calcula- tions show that at present there exist experimentalmethods for determination of parameters which are necessary for numerical models of very complicated problems related to cracks existing in the contact zone. Especially, the follo- wing parameters should be taken into account: • Real geometry of bodies in contact, including the effects of wear. • Geometry of the crack in macro-scale. • Geometry of crack faces in micro-scale or crack faces interaction. • Distribution of the crack thickness. • Influence of other parameters like presence of a liquid, etc. The computing power of contemporary computersmakes it possible to develop very complex 3D models of bodies with cracks. The results of such calculations can be verified using experimental investi- gations in situ. References 1. Bochenek A., 1999, Badanie przyczyn i sposoby zapobiegania pękaniu szyn UIC obrabianych cieplnie, Mat. Konf. 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Huty Katowice – Produkcja i eksploatacja szyn kolejowych, Rogoźnik, 32-48 Wykorzystanie wyników badań eksperymentalnych w modelach numerycznych pęknięć zmęczeniowych rozwijających się w strefie kontaktu tocznego Streszczenie Wpracy opisanomodele numerycznepęknięć zmęczeniowych typu ”squat” rozwi- jających sięw strefie kontaktu tocznegokoło-szynawgłówkach szyn kolejowych.Przy tworzeniu tych modeli wykorzystano wyniki badań eksperymentalnych. Opisano na- stępujące eksperymenty: badania terenowe prędkości rozwoju pęknięcia oraz kształtu szczeliny w makro- i mikroskali oraz grubości szczeliny, a także pomiar oddziały- wania pomiędzy brzegami pęknięcia. Uwzględnienie w obliczeniach wyników badań eksperymentalnych pozwala bardziej wiarygodnie prognozować rozwój pęknięć i czas dopuszczalnej eksploatacji szyn. Manuscript received April 4, 2006; accepted for print May 24, 2006