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International Journal of Fatigue 30 (2008) 1678–1689
International Journalof Fatigue
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Analysis of surface crack growth under rolling contact fatigue
D. Canadinc *, H. Sehitoglu, K. Verzal
University of Illinois at Urbana-Champaign, Department of Mechanical Science and Engineering, 1206 W. Green Street, Urbana, IL 61801, USA Received 7 May 2007; received in revised form 25 October 2007; accepted 4 November 2007 Available online 22 January 2008
Abstract Understanding the fatigue crack growth phenomenon in railheads requires a study of driving forces, such as the crack tip opening and sliding displacements, under repeated rolling contact. Finite element simulations, allowing elastic–plastic deformation, and mixed-mode crack growth laws were utilized to demonstrate that the fatigue crack growth rates display a minimum after a ?nite amount of crack advance. These results have implications in designing strategies for optimum grinding or wear rates to limit fatigue crack growth, and thereby prolong rail life. During the simulations, the crack was allowed to advance, permitting residual deformations and stresses to be retained from cycle to cycle. The opening and closure of crack surfaces, under forward and reverse slip and stick conditions were monitored. Normal pressures of 1500 MPa and 2000 MPa, along with shear traction ratios in the range of ?0.4 to 0.4 were investigated for a varying crack size of 3–15 mm. An interesting ?nding was that the crack tip opening displacements decreased while the crack tip sliding displacements increased with increasing crack length. ? 2007 Elsevier Ltd. All rights reserved.
Keywords: Surface crack; Rolling contact; Fatigue; Crack tip opening; Crack tip sliding; Crack growth rate
1. Background and motivation Extending the rail life has signi?cant economic bene?ts to the railroad industry. Currently, the concerns surrounding the rail are on the maintenance costs and increasing the margin of safety. Catastrophic rail failure can occur due to initiation of surface cracks at the railhead followed by their advance into the bulk of the rail. At a ?rst glance, very low wear rates would be bene?cial in rails, however, there are some advantages in controlled wear to curtail the fatigue cracks advancing in the railhead. We note that both wear and rolling contact fatigue (RCF) mechanisms operate simultaneously. As a result, an optimum wear rate (or grinding rate) under a given operating condition could extend the rail life [1] and mitigate catastrophic failure. The wear rates have been determined from laboratory experiments in the past and can be used to predict the ?eld
* Corresponding author. Present address: Koc University, Department of Mechanical Engineering, Istanbul, Turkey. Tel.: +90 212 338 1891; fax: +90 212 338 1548. E-mail address: dcanadinc@ku.edu.tr (D. Canadinc).
data, however, despite a signi?cant body of work, a clear understanding of fatigue crack growth behavior under contact has not been established. Admittedly, this is a rather complex problem involving opening and sliding of crack surfaces, high closure forces under compression, and residual stress ?elds. Therefore, the present work focuses on simulation of all these factors that a?ect crack growth behavior. As stated earlier, there have been numerous models proposed to predict the wear rates as a function of contact pressure, surface conditions and material hardness [2]. Although empirical, these models are su?cient to deliver the wear rate under various loading conditions. On the other hand, the prediction of fatigue crack growth rates under these conditions is rather complex and has not been explored fully. Previous attempts [3,4] have been mostly focused on either fatigue crack initiation or determination of crack driving forces under elastic contact. The excessive plastic deformation in rolling contact limits the use of linear-elastic fracture mechanics (LEFM) [5–15]. Generally, the fatigue cracks that initiate on free surfaces undergo both sliding and opening mode displace-
0142-1123/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.11.002
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ments (Stage I), and eventually turn to the plane of maximum tensile stress (Stage II) [11]. In RCF, however; cracks remain predominantly in Stage I due to the compressive stresses normal to the crack faces [12]. An added factor is the presence of external shear tractions that can produce a tensile stress, a Mode I crack opening, in addition to the Mode II sliding displacements. Our previous work on crack growth in Stage I (shear cracks) forms the basis for our simulations in this study [8]. We extend the work to determination of crack tip opening (DUI) and crack tip sliding displacements (DUII) under repeated rolling contact. There have been a number of investigations that focused on determining stress intensity levels for cracks subjected to RCF [13–15]. The DKI (stress intensity factor range (SIF)) was shown to approach zero beyond a ‘critical’ crack length [13] which depends on the loading conditions and orientation of the crack [14]. In a recent study [15] focusing on the surface cracks under RCF, linear elastic ?nite element (FE) simulations were utilized to calculate the SIF ranges (DKI and DKII) and a decrease in the SIF range occurred due to increasing friction between the crack faces upon crack advance. However, the previous studies performed to predict crack growth rates and fatigue life under RCF loading [16,17] were con?ned to elastic deformations. At the same time, the high wheel loads are generating deformation in the rail heads well beyond the elastic regime. It is known that the residual plastic deformation due to crack growth, and therefore the accumulated residual stresses [8] in?uence the crack growth behavior drastically. To our knowledge, this is the ?rst detailed investigation on the crack driving forces in half-plane contact accounting for elastic–plastic deformation history. We make a distinction between ‘‘static’’ and ‘‘propagating’’ cracks and show that allowing the crack ‘‘propagation’’ during the simulations produces a better description of the driving forces. There are recent works focusing on predicting the rail life under rolling contact [1,17,18] recognizing the competition between the RCF and wear [17]. The elimination of surface crack by wear was forwarded as a strategy to improve rail life. But a minimum in the crack growth rate was predicted to be of the order of few microns, a rather small distance. Also, the fatigue crack growth simulations were not explicitly considered to ascertain the precise length which coincides with the minimum fatigue crack growth rate. This minimum (which we identify as several millimeters) depends non-trivially on the applied pressure, the friction between the crack surfaces, and the relative Mode I and Mode II components, which are all analyzed in the present study. As a result of the systematic analyses presented herein we postulate the following crack growth history. Initially small cracks propagate under RCF loading, however until a critical crack length is reached, the crack propagation rate decreases as the crack tip moves away from the contact stress ?eld. At this critical crack length, the crack growth rate is at a minimum (Fig. 1). Beyond this minimum point,
Fig. 1. Schematic illustrating the life cycle of a crack; and the competition between crack growth rate and wear rate.
however; crack propagation rate increases as the crack advances possibly because of the decrease in compressive closure forces. In the remainder of this paper, these di?erences and the basis for the proposed crack growth transients for a propagating crack are explained in detail in the light of extensive numerical simulations. 2. Approach A detailed numerical investigation was carried out in which over 150 RCF simulations were conducted. Each simulation required 2 h of computational time on a 1.3 GHz processor and was performed using an IBM pSeries 690 supercomputer. The FE mesh (using ABAQUS) utilized over 25,000 elements and a surface crack inclined 45° to the half-plane (Fig. 2). A 45° inclination was selected because this orientation represents the plane of maximum shear stress range throughout the rolling contact (RC) cycle when a tangential force is applied [3]. The mesh is re?ned in the region where the crack faces are in contact, and at the crack tip. A typical element in the re?ned region (with ?ne mesh) is 0.5 mm wide and 0.2 mm tall with four nodes. However, smaller elements 0.25 mm wide and 0.1 mm tall were utilized along the crack faces and around the crack, such that 50 elements were placed along the crack length on each face of a 5 mm crack, and the plastic zone encompasses 10–20 elements around the crack tip (Fig. 2). Since the contact area width is small in comparison to the curvature of the rail, the plane strain condition is utilized in the simulations. Both faces of the crack were de?ned as contact surfaces in the ABAQUS input ?le following the standard procedure, yet no special elements were utilized. Friction was also allowed in order to account for slip and stick between the faces of the crack. The coordinate system for the model, along with a schematic representation of the normal and shear loading, is shown in Fig. 2 where ‘e’ is the distance between the load center and crack mouth, ‘a’ is the Hertzian half contact width, and Q and P are the tangential force and normal pressure distribution on the contact region. In the simulations, the load translates from the left to the right hand side of the mesh.
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Fig. 2. Coordinate systems and load de?nition associated with crack behavior simulations, and the ?nite element mesh model with a close-up view of the crack-tip area. For p0 = 1500 MPa, a = 13.9 mm; and for p0 = 2000 MPa, a = 18.6 mm.
The material studied in the FE simulations is a pearlitic rail steel. The material behavior is represented by kinematic hardening, and shear stress–shear strain data for the pearlitic steel under pure torsion of tubular specimens (obtained experimentally) is included in Fig. 3. The fracture strain in shear is approximately 1.35. To model the semi-in?nite elastic halfspace, in?nite elements were implemented at the boundaries to ensure that a ?nite size mesh would provide reliable results [19–21]. The results from the FE model correlate well with a semi-analytical model for stresses developed by Jiang and Sehitoglu [4]. A schematic showing crack displacements and the coordinate systems used in this study is given in Fig. 4. The dots on the crack faces represent nodal elements in the FE model that initially touch each other in the unloaded state [8]. In order to simulate rolling contact fatigue loading the Hertzian pressure is traversed across the half-plane, and
Fig. 3. Shear stress–shear strain data for pearlitic rail steel from a pure torsion test to fracture.
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Fig. 4. Schematic illustrating the UI and the UII displacements on the crack face.
the displacements are monitored every cycle. Under repeated rolling contact the crack opens and the crack faces slide relative to one another. The amount of opening is described by UI and the amount of relative sliding is denoted as UII. These variables are de?ned as U I ? U I;UPPER ? U I;LOWER ?1? U II ? U II;UPPER ? U II;LOWER where UPPER and LOWER refer to the crack ?anks. Throughout the loading cycle, UI and UII vary, and the ranges of UI and UII are of utmost interest for determining crack driving force in fatigue. The total opening and sliding range throughout the loading cycle is de?ned as DU I ? U I?MAX ? U I?MIN ?2? DU II ? U II?MAX ? U II?MIN The UI?MIN equals zero since the crack tip is closed at some point throughout the loading cycle in every simulation, and DUI and UI?MAX become equivalent. The maximum value of UII, referred to as UII?MAX, is obtained throughout the loading cycle, and similarly, UII?MIN denotes the minimum value of UII throughout the rolling RC cycle. By de?nition, DUI P 0 and DUII P 0. The ranges of relative crack opening and sliding are of great importance as these values govern fatigue crack growth rates. Depending on the loading conditions, the DUI and the DUII can take on a variety of di?erent values. The following de?nitions are implemented to describe the state of the crack [22]: 1. If UI > 0 at every point up to the crack tip (totally unzipped crack) the crack is said to be ‘‘fully open’’ (Fig. 5). Alternatively, a crack is ‘‘closed’’ if UI = 0 at every node along the crack face. If UI is greater than zero at some nodes, and zero at others, the crack is described as ‘‘partially open’’. 2. If UII is non-zero at any point along the crack face, the crack is said to have ‘‘partially slipped’’. However, if UII is non-zero at every point up to the crack tip, the crack
Fig. 5. Schematic illustrating open and closed cracks throughout the loading cycle for loading corresponding to p0 = 1500 MPa, Q/P = 0.2, l = 0.1, and a crack of length 4.95 mm for a static crack.
has ‘‘fully slipped’’. The crack is referred to as ‘‘sticking’’ if UII does not change (DUII = 0) during the loading cycle (Fig. 6). 3. Forward slip will be de?ned when UII increases, and reverse slip is implied by a decrease in the UII (Fig. 6). Figs. 5 and 6 display the concepts associated with crack opening and closure along with forward and reverse slip. Fig. 5 shows the opening behavior for the crack tip of a 4.95 mm long crack subjected to a load with p0 = 1500 MPa and Q/P = 0.2. A positive Q/P ratio implies that the tangential loading occurs in the same direction as rolling, as seen in Fig. 2. The crack is closed, or UI = 0, for Q/P = 0.2 until the load has passed over the crack mouth and is located at e/a % 1.25. The crack opening continues to increase until e/a % 1.8. The crack tip remains open even after the load has translated past the crack due to the residual
Fig. 6. Schematic illustrating forward and reverse slip throughout the loading cycle for loading corresponding to p0 = 1500 MPa, Q/P = 0, l = 0.1, and a crack of length 4.95 mm for a static crack.
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stresses and plastic deformation built up around the crack tip during the loading cycle. The range of crack tip opening displacement throughout the loading cycle, DUI, is indicated on Fig. 5. Fig. 6 displays the crack sliding behavior under pure rolling conditions (Q/P = 0). The crack has fully slipped throughout the entire loading cycle. As the load approaches the crack mouth and e/a < ?1.0, reverse slip occurs, and the magnitude continues to increase as the load approaches the crack. As the load reaches e/a % ?1.0, forward slip begins to occur. As the load passes over the crack mouth, sticking occurs as UII remains constant. Both reverse and forward slip take place again as e/a > 0.5 and the load travels away from the crack. As demonstrated by the pure rolling case, multiple regions of reverse and forward slip can occur during a single RC loading cycle. The de?nition of DUII for the entire loading pass is given schematically in Fig. 6. 3. Simulation results Following the methodology outlined in the previous section, FE simulations were carried out systematically. First, the crack opening and sliding displacements of stationary cracks were calculated for varying Hertzian pressures and Q/P ratios. The analyses were then extended to propagating cracks, and the crack growth rates, along with the crack opening and sliding displacements, were calculated for different load-Q/P ratio combinations. All FE simulations allowed for elastic–plastic deformation. In the remainder of this very section, the details of the calculations and the corresponding results are presented. 3.1. Opening and sliding behavior under di?erent shear tractions While the magnitude of crack tip opening and sliding vary with the applied load, general observations about crack behavior can be made depending on the direction of the tangential loading. The variation of UI and UII at the crack tip can be seen in Figs. 7 and 8 throughout a loading cycle for Q/P = ?0.2, Q/P = 0, and Q/P = 0.2. The crack opening behavior for a positive tangential load (Q/P = 0.2) was discussed in detail in the previous section, and it is compared to the opening behavior for di?erent tangential loading in Fig. 7. The crack opening behavior for negative tangential loading (Q/P = ?0.2) is much di?erent than for either pure rolling or positive tangential loading. One interesting feature when Q/P = ?0.2 is that the maximum crack tip opening occurs at e/ a % ?1.5, and the crack tip and crack faces are open before the loading reaches the crack mouth. Under a ?nite tangential force, residual UI persists after the load has translated past the crack. The crack opening displacements under pure rolling (Q/P = 0) with p0 = 1500 MPa are much different than those under ?nite tangential force since the crack tip never becomes open. For Q/P = 0, the crack faces
Fig. 7. Variation of the crack tip opening displacement throughout the loading cycle for di?erent values of Q/P. The simulations use a Hertzian peak stress of p0 = 1500 MPa and a static crack of length 4.95 mm with a coe?cient of friction between the crack faces of 0.1.
Fig. 8. Variation of the crack tip sliding displacement throughout the loading cycle for di?erent values of Q/P. The simulations use a Hertzian peak stress of p0 = 1500 MPa and a static crack of length 4.95 mm with a coe?cient of friction between the crack faces of 0.1.
away from the crack tip open during the loading cycle; however, the crack tip remains closed throughout the entire loading cycle as seen in Fig. 7. In other words, the crack becomes partially open under pure rolling. The crack sliding behavior under pure rolling conditions was outlined in detail in the previous section, and is compared to ?nite Q/P ratios in Fig. 8. Under a positive tangential load, reverse crack slip occurs almost immediately as UII becomes negative. When the load approaches e/a % ?1.0, forward slip takes place. As the load continues to pass over the crack mouth, the positive crack slip persists and UII continues to increase until e/a % 1.0. The negative tangential force (Q/P = ?0.2) alters the crack sliding behavior drastically. The negative tangential load causes UII P 0 until e/a % 0.5, and the UII?MIN occurs after the
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load has passed over the crack (e/a > 0). For ?nite tangential loading, residual UII persists and can be seen in Fig. 8. For the resulting RC passes, these residual displacements represent the initial values for further deformation. Their presence a?ects the DUII in subsequent cycles and produces a distinction between ‘propagating’ and ‘static’ cracks. 3.2. Static cracks We ?rst focus on the behavior of static cracks under contact loading. A static crack, also referred to as a ‘‘saw’’ crack, refers to a crack that is present without any residual plastic deformation from previous loading cycles. The plastic deformation history of the surrounding medium during the crack growth process has been ignored to reduce computational time and to preliminarily investigate the impact of di?erent variables on crack tip displacements. First, a crack of 4.95 mm in length was considered, with a coe?cient of friction l = 0.1 between the crack faces. Two di?erent normal loads were applied with peak contact stresses of p0 = 1500 MPa and p0 = 2000 MPa, and the Q/P ratio was systematically varied. The results for the maximum crack opening displacement can be found in Fig. 9, and the range of crack tip sliding, DUII, is presented as a function of the Q/P ratio in Fig. 10. The crack opening and sliding displacements both increase with increasing Q/P magnitude and higher normal loads. The ranges of crack tip opening and sliding are o?-centered toward negative Q/P values when p0 = 2000 MPa. The loading condition of p0 = 2000 MPa produces signi?cantly more plastic deformation than p0 = 1500 MPa. Therefore, the residual stresses, and deformation at the crack tip that occur during the loading cycle lead to the o?set of the DUI and the DUII values for p0 = 2000 MPa. The corresponding details of the plastic deformation are documented by Verzal [23].
Fig. 10. Crack tip sliding displacements versus Q/P for a Hertzian contact with peak loads of p0 = 1500 MPa and p0 = 2000 MPa. The data is for a static crack of length 4.95 mm and a friction coe?cient of 0.1 between the crack faces.
Next, the combination of a normal load with a maximum Hertzian pressure of p0 = 1500 MPa and a Q/P ratio of 0.2 was considered. Both the crack length and friction coe?cients between the crack faces were varied. Even though the material on the crack faces remains the same, lubrication and surface roughness can change the coe?cient of friction quite signi?cantly, which can subsequently in?uence the behavior at the crack tip. During reverse slip when e/a < 0, the normal force tends to reduce the contact pressure between the crack faces and minimizes the role of the friction coe?cient. During forward slip when e/a > 0, the normal force increases the contact force between the crack faces and maximizes the e?ect of friction. When the friction coe?cient between the crack faces is low (l 6 0.2), the crack length and UII?MAX increase proportionally (Fig. 11). Nevertheless, as the friction coe?cient between the crack faces becomes larger (l = 0.35),
Fig. 9. Crack tip opening displacements versus Q/P ratio for Hertzian contact loads with peak pressures of p0 = 1500 MPa and p0 = 2000 MPa. The data is for a static crack of length 4.95 mm and a friction coe?cient of 0.1 between the crack faces.
Fig. 11. UII?MAX versus crack length plotted for di?erent coe?cients of friction. A Hertzian load with p0 = 1500 MPa and Q/P = 0.2 was applied to static cracks.
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UII?MAX decreases with increasing crack length. Longer cracks with high friction coe?cients between the crack faces shield the crack tip from the applied stress ?eld because the total frictional force along the crack face resisting slip is greater. Frictional shielding of the crack tip can be illustrated by de?ning the crack sliding as a function of the distance from the crack tip (Fig. 12). The coe?cient of friction between the crack faces has a signi?cant in?uence on the UII?MAX pro?le along the crack face. When l = 0.3, the crack tip is almost completely shielded from sliding and the slip pro?le becomes concave upward. Therefore, the UII?MAX behavior as a function of crack length is the opposite for low coe?cients of friction compared to high coe?cients of friction. When analyzing the e?ect of crack length on DUII (Fig. 13), it can be clearly observed that the range of crack tip sliding increases as if it obeys a power law for low coef?cients of friction, while remaining fairly constant for higher coe?cients of friction. For high coe?cients of friction, UII?MAXdecreases with increasing crack length while reverse slip increases since reverse slip is less dependent on the friction coe?cient. Consequently, DUII remains fairly constant. In contrast, UII?MAX and UII?MIN both increase with increasing crack length for low friction coef?cients between the crack faces, so DUII also increases with increasing crack length. Overall, as the coe?cient of friction between the crack faces increases, the crack tip sliding displacements decrease. Intuitively, the friction coe?cient should not a?ect the DUI since the crack faces are not in contact when the crack is open. However, a higher friction coe?cient shields the crack tip which alters the pro?le of plastic zone at the crack tip. Consequently, the DUI decreases as the friction between the crack faces increases (Fig. 14). Most interestingly, the crack opening displacements also decrease as the crack length increases. The applied stress in the x-direction (rx) is the main driver of crack opening, and rx is
Fig. 13. DUII versus crack length plotted for di?erent coe?cients of friction. A Hertzian load with p0 = 1500 MPa and Q/P = 0.2 was applied to static cracks.
Fig. 14. DUI versus static crack length plotted for di?erent coe?cients of friction. A Hertzian load with p0 = 1500 MPa and Q/P = 0.2 was applied.
expressed as a function of depth into the material (Fig. 15). It is shown for a variety of Q/P ratios that rx is two to three times larger at a depth of 3 mm than it is at 8 mm. Since rx decays as the depth increases, the DUI decreases as the crack length increases. 3.3. Propagating cracks All of the simulations in the previous section were concerned with static cracks, or cracks with no prior plastic deformation history. This simpli?cation is not realistic because a residual stress ?eld will accumulate due to the repeated RC loading, as well as around the crack tip, as the crack advances. In this section, the explanations regarding the analysis of propagating cracks are presented. In the corresponding FE simulations, a crack of l = 2.8 mm initial length was subjected to RCF until it grew to a length of l = 8.0 mm. For some cases of particular interest, a modi?ed mesh was used that allowed the crack to grow up to
Fig. 12. UII?MAX as a function of the distance from the crack tip for a 5 mm static surface crack for di?erent coe?cients of friction. A Hertzian load with p0 = 1500 MPa and a Q/P ratio of 0.2 was applied.
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Fig. 15. Depth versus rx for di?erent values of tangential loading under a Hertzian normal load with p0 = 1500 MPa. The intermediate data lines correspond to Q/P = 0.3 and Q/P = 0.2.
Fig. 17. Comparison of crack tip sliding displacements for propagating and static cracks subjected to a load of p0 = 2000 MPa and Q/P = 0.2.
15.0 mm. The ‘propagating crack’ results were compared to the ‘static crack’ results to analyze the in?uence of the residual stress ?elds and residual displacements on the crack tip behavior. A detailed study was performed to determine the role of the point of release of the crack during the cycle, or the loading state at which the crack propagates. Within this analysis, the crack was advanced at UI?MAX, UII?MAX, and UII?MIN. DUI and DUII were found to be independent from the point of release, and in the subsequent simulations the cracks were extended at the point of maximum positive slip, UII?MAX [23]. As a result of the static crack analysis, it was found that UII?MAX increases with crack length for low values of friction (l 6 0.2) and decreases with crack length for high values of friction, such as l = 0.35 (Fig. 11). However, for propagating cracks; UII?MAX decreases when subjected to a load with p0 = 2000 MPa and Q/P = 0.2, with a friction coe?cient of 0.2 between the crack faces (Fig. 16). A com-
parison of the range of crack tip sliding, DUII, between the static and propagating cracks for friction coe?cients of 0.1 and 0.2 is presented in Fig. 17 for the same loading conditions. The range of crack tip sliding is smaller for propagating cracks than for static cracks. 3.4. Crack growth rates The simplest model for predicting mixed mode crack growth rates is an additive model based on the type of loading experienced by the crack [11]. For a plane strain RC surface crack, Mode I and Mode II loading occurs, and the crack growth rate can be de?ned as da ? C 3 ?DK I ?m3 ? C 4 ?DK II ?m4 dN ?3?
where C3, C4, m3, and m4 are empirically determined constants. If the crack tip displacements are used [24], the crack growth rate can be de?ned as da m m ? C 1 ?DU I ? 1 ? C 2 ?DU II ? 2 dN ?4?
where C1, C2, m1, and m2 are empirical constants (Fig. 18); however di?erent than those present in Eq. (3). The determination of the constants applicable to RCF crack propagation requires a signi?cant amount of experimental e?ort, which is beyond the scope of this very paper. Conse-
Fig. 16. UII?MAXand UII?MIN versus crack length for propagating and static cracks with a friction coe?cient of 0.2 when subjected to a load of p0 = 2000 MPa and Q/P = 0.2.
Fig. 18. Schematic de?ning the empirical constants of Eqs. (4) and (5).
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quently, the crack growth rates will be normalized in this study in the following manner da C2 m m =C 1 ? ?DU I ? 1 ? ?DU II ? 2 ?5? dN C1 If the crack growth equation is applied to the RC behavior for propagating cracks, the crack growth rate as a function of crack length can be derived for a surface crack under RC loading. In order to investigate typical crack growth rates, ?rst a propagating crack subjected to a normal load of p0 = 1500 MPa with Q/P = 0.2 and a friction coe?cient of 0.2 between the crack faces was considered. For this simulation, a modi?ed mesh was implemented allowing crack tip displacements to be calculated from 3 mm to 15 mm. The crack growth rate as a function of crack length was calculated for three di?erent values of the constant C2/ C1, with m1 = m2 = 2 (Fig. 19). When investigating the crack tip displacements from the FE model, the DUI was found to decrease with increasing crack length while the DUII increased. Additionally, Eq. (5) contains a component of crack growth due to Mode I loading and another component due to Mode II loading. Consequently, the crack growth rate due to Mode I loading decreases as the crack length increases, whereas the crack propagation rate due to Mode II deformation increases with crack length. Since one component of the growth rate is increasing while the other is decreasing with increasing crack length, a minimum exists on the total crack growth curve. As C2/C1 decreases, the crack growth caused by DUII decreases relative to DUI, and the minimum point in the crack growth rate function shifts toward longer crack lengths. A minimum does not exist between 3 mm and 12 mm when C2/ C1 = 0.1 because Mode II crack growth is already dominating at 3 mm. For this case, the minimum is expected at a crack length shorter than 3 mm.
Fig. 20 displays a comparison of the crack growth rate for Hertzian contact loads with peak stresses of p0 = 1500 MPa and p0 = 2000 MPa. For these simulations, the Q/P ratio was 0.2 and a friction coe?cient was varied. The constants implemented for the crack growth model were C2/C1 = 0.01 and m1 = m2 = 2. The most obvious feature of this graph is the di?erence in crack growth rate magnitudes between the two loading scenarios. The crack growth rate for p0 = 2000 MPa is almost an order of magnitude larger than the propagation rate for p0 = 1500 MPa. An interesting feature is the location of the minimum. The minimum for p0 = 2000 MPa occurs at a crack length of approximately 4 mm where the p0 = 1500 MPa load reaches a minimum around 7–8 mm. When p0 = 2000 MPa, the DUII displacement increases with crack length. However, the DUI displacements decrease very gradually and the minimum of the function occurs at a shorter crack length. When p0 = 1500 MPa, DUI displacements typically decrease signi?cantly when the crack length is longer than 5 mm while the DUII displacements increase gradually. Therefore, the minimum of the overall growth rate function is reached after the large decrease in opening displacements. If the opening displacements for p0 = 2000 MPa decrease signi?cantly for cracks longer than the ones considered in this study, it is possible that another minimum in the overall growth rate function could exist at crack lengths longer than 10.0 mm. The crack growth rates can also be analyzed depending on the coe?cient of friction (l) between the crack faces. As expected, the crack growth rates decrease as l increases because the higher frictional force reduces DUI and DUII. When p0 = 1500 MPa, it can be seen that the trends in the crack growth rate change when l = 0.35. When l = 0.1 or l = 0.2, the DUII displacements make a signi?cant contribution to the crack growth rate. When l = 0.35, the DUI displacements dominate the crack growth rate for crack lengths between 3 mm and 10 mm due to the drastic reduction in sliding displacement. There-
Fig. 19. Normalized crack growth rate for di?erent ratios of C2/C1 when m1 = m2 = 2. The propagating crack simulations were conducted with a peak Hertzian load of p0 = 1500 MPa, Q/P = 0.2, and a friction coe?cient of 0.2 between the crack faces.
Fig. 20. Normalized crack growth rate for di?erent values of the Hertzian peak contact stress. For the propagating crack simulations, Q/P = 0.2, m1 = m2 = 2, and C2/C1 = 0.01.
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fore, the crack growth rate for l = 0.35 decreases gradually as the crack tip opening displacements decrease. It is expected that a minimum would occur in the crack growth function for l = 0.35 as the crack becomes long enough, such that the DUI displacements approach 0. 4. Discussion and ?nal remarks 4.1. Crack propagation Propagating cracks generate an evolving plastic zone around the crack tip and residual displacements along the crack faces as the crack advances. The residual displacements in the wake of the crack tip lead to smaller crack tip displacements when comparing a propagating crack to a static crack under identical loading [8,25]. The role of the plastic zone generated by a fatigue (propagating) crack was previously investigated, and a comparison of crack opening displacements from a fatigue crack to those from a stationary (static) crack was presented [25]. Accordingly, under uniaxial loading, the crack tip opening displacement was found to be lower than the static displacements at the crack tip and along the crack faces, due to residual displacements. The current study revealed that, for propagating cracks released at UII?MAX, a residual compressive stress exists at the crack tip due to the residual displacements. Consequently, the friction between the crack faces increases and the relative sliding decreases. For instance, for p0 = 2000 MPa and Q/P = 0.2 with a friction coe?cient of 0.2 between the crack faces, the magnitude of UII?MAX, the maximum relative sliding between the faces of a propagating crack, is much smaller as compared with that in the case of a static crack (Fig. 16). Additionally, the increased friction stemming from the compressive residual stress ?eld brings about a decrease in both DUI (Fig. 14) and DUII (Fig. 17). These observations stand in good agreement with the ?ndings of our previous work [8,25]. We note that, in the present investigation, a minimum in the da/dN curve exists due to a competition between Mode I and Mode II displacements as the crack grows. Previous works [1,18] attributed the origin of such a minimum to the decay in contact stress ?eld and increased bending stresses. On the contrary, the present work postulates that the minimum depends entirely on the relative contributions from DUI and DUII due to the applied contact stress, which changes with increasing crack length. For instance, for the case of p0 = 1500 MPa, DUI displacements were observed to decrease signi?cantly (Fig. 14) when the crack length is longer than 5 mm, while the DUII displacements increased gradually (Fig. 13). As a result, the minimum of the overall crack growth rate function was attained after the large decrease in opening displacements (Figs. 19 and 20). Since bending stresses are not considered in the current simulations, the interplay between DUI and DUII is su?cient to explain the minimum.
4.2. Competition between wear and crack propagation The current results demonstrated that, although both crack propagation due to RCF and wear lead to rail failure, an optimum combination of these two mechanisms may yield a prolonged rail life. Speci?cally, a crack propagation rate that is greater than the wear rate leads to rail failure due to crack propagation. However, if the wear rate is greater, then the crack will be worn away before reaching a critical length, yet wear will be the mechanism responsible for rail deterioration, and ultimately adverse changes in the rail contour. This would cause severe contact at rail corners and a?ect the performance of the rail vehicle. Nevertheless, an optimum combination of wear and crack growth rates could be of signi?cant bene?t in curtailing rail failure. Namely, wear at the crack mouth could limit the crack length to avoid rapid crack growth. These cracks could eventually lead to branching and catastrophic failure. The crack growth rate itself depends on the crack opening and sliding displacements, and thus the crack length and the minimum point of a crack growth rate function are of crucial importance in optimizing the competition between the rates of crack propagation and wear. To demonstrate this argument, wear experiments were carried out at the University of She?eld to experimentally determine the wear rates for various Hertzian contact peak pressures [26]. Disc-on-disc tests were conducted to simulate the RC between rail and wheel, and the disc materials were chosen accordingly. Wear rates were determined by monitoring the weight loss and dimensional changes in each disc. Some of the results from these wear tests are compared to the crack growth rates obtained from the current simulations, further emphasizing the importance of the knowledge of the minimum crack growth rate and the corresponding crack length (Figs. 21 and 22). For instance, for a propagating crack that advanced to 8 mm in length under a Hertzian contact load with a peak pressure of p0 = 2000 MPa, for a friction coe?cient of 0.1 between the crack faces, the normalized wear rate (3 · 10?7 mm/
Fig. 21. Comparison of normalized wear rates (dashed lines), and normalized crack growth rates of Fig. 19.
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propagating crack. For a given friction coe?cient between the crack faces, the minimum crack growth rate can assume di?erent values, which is dictated by p0 and C2/ C1 ratio (Figs. 19–22). 4.3. On the simulations It was earlier demonstrated that the crack mouth will open but the crack tip will remain closed for a surface crack in an elastic half-space under pure rolling Hertzian contact loading, for any given peak pressure [14]. Nontheless, our analyses pointed out that the validity of this statement strongly depends on the peak pressures; the crack tip remains closed under pure rolling Hertzian contact loading for p0 = 1500 MPa, whereas signi?cant plastic deformation and crack tip opening displacements are present for the case of p0 = 2000 MPa (Fig. 9). We note that the disagreement between the results of Fujimoto et al. [14] and the current ?ndings stems mainly from the di?erent material models and crack advance procedure. Speci?cally, both static and propagating cracks of lengths in the 2–5 mm range were investigated, where the propagating cracks were allowed to grow to 8 mm, and 15 mm in speci?c cases, which are much larger than the microstructural features that are known to a?ect the crack growth, such as pearlite spacing and pearlite colony size. Furthermore, this is the regime where ‘‘continuum’’ analysis provides reliable results. Such crack length necessitates the consideration of EPFM – as done in the current paper – since the crack is short, and the plastic zone size to crack length ratio is large, where the crack growth is elastic–plastic in nature. Two recent works utilizing the concept of EPFM in the analysis of crack growth [18,27] considered extremely short cracks, where the crack lengths varied between 0.05 and 0.4 mm. In our case, we would not ?nd a minimum in this range. We note that the crack lengths utilized in the current study are an order of magnitude greater. Accordingly, most surface cracks of critical length in the railheads fall into the range of 2–5 mm, and usually possess an angle of about 45° with the contact surface for the material considered. 5. Conclusions Based on the work presented herein, we draw the following conclusions: 1. Normalized crack growth rates as a function of crack length display a minimum because DUI decreases with crack length while DUII increases with crack length. The previous works have predicted the existence of a minimum in the da/dN function with respect to crack length, and attributed the minimum to decaying contact stress ?elds and increasing bending stress ?elds. In the present study, we demonstrated that higher normal loads lead to higher crack propagation rates, while higher friction coe?cients between the crack surfaces decrease the crack growth rate.
Fig. 22. Comparison of normalized wear rates (dashed lines), and normalized crack growth rates of Fig. 20.
cycle, assuming C1 = 1) is considerably lower than the normalized crack propagation rate (6 · 10?5 mm/cycle, assuming C1 = 1) (Fig. 22), implying that the rail failure will take place due to rapid growth of initially small surface cracks. We note that the presentation of the crack growth rate (da/dN) data (Figs. 19–22) in a normalized fashion (with respect to C1) is not coincidental. The reason is that the empirical constants C1 and C2 are not available, and further experimental e?ort is needed in order to establish these values for the RCF case. However, this is beyond the scope of the current work. Moreover, normalization allows the assumption of any values for the C1 and C2, and therefore enables the observation of trends in general for various possible scenarios. Previously, the life cycle of a crack was described as a four-stage phenomenon [1,18]: Crack initiation by lowcycle fatigue (LCF), ratcheting damage and early crack growth are followed by lengthening of the crack, leading to higher stress concentrations, and therefore higher crack propagation rates. Beyond a critical crack length, the tip of the crack moves away from the contact stress ?eld, resulting in a reduction of the crack growth rate. However, once the crack is su?ciently long, bending stresses lead to rail failure [1,18]. Nevertheless, the results of the current study depict an alternative scenario: Initially small cracks propagate due to repeated RC loading, yet the crack propagation rate decreases as the crack tip moves away from the contact stress ?eld (it should be noted that the current work does not consider crack initiation by LCF). Once a critical crack length is reached, the crack growth rate assumes its minimum (Fig. 1). Beyond this minimum point, however; crack propagation rate increases concomitant with the increasing crack length. In fact, the current simulation results support that DUI decreases with increasing crack length for all coe?cients of friction (Fig. 14) whereas an increase in DUII is evident as the crack length increases (Fig. 13). Since the crack growth rate (da/dN) is a function of both DUI and DUII (Eq. (4)), crack growth rate history similar to the one illustrated in Fig. 1 is expected for a
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2. Propagating cracks generate a plastic deformation history around the crack faces and the crack tip. Consequently, the magnitudes of DUI and DUII are lower for a propagating crack when compared to a static crack. The results also point toward di?erent crack tip sliding trends with increasing crack length between static and propagating cracks. 3. The friction coe?cient between the crack faces can drastically alter the sliding displacements experienced by the crack tip. As the friction coe?cient approaches 0.35, the frictional forces shield the crack tip from the stress ?eld and the UII?MAX decreases with increasing crack length. Consequently, the range of crack tip sliding remains nearly constant with increasing crack length when the friction coe?cient is 0.35 while sliding displacements increase with increasing crack length for lower friction coe?cients. 4. The magnitude and direction of tangential loading can signi?cantly alter the opening and sliding behavior at the crack tip of a 45° surface crack during a rolling contact (RC) cycle. Unlike elastic analysis, the results indicate that crack tip opening displacements can occur under pure rolling (Q/P = 0) contact conditions if the applied load causes plastic deformation. 5. From a design point of view, the knowledge of crack growth rate as a function of crack length is of crucial importance. An optimum combination of crack propagation and wear rates is expected to yield a prolonged rail life.
Acknowledgements This study was funded by Transportation Technology Center, Inc. (TTCI), a subdivision of Association of American Railroads (AAR). The numerical simulations were carried out on an IBM pSeries 690 supercomputer at the National Center for Supercomputing Applications (NCSA), University of Illinois at Urbana-Champaign. The wear test results used in this study were obtained at the University of She?eld, United Kingdom. References
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