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Modeling of rolling contact fatigue for hard machined components with process-induced residual stress
International Journal of Fatigue 26 (2004) 605–613 www.elsevier.com/locate/ijfatigue
Modeling of rolling contact fatigue for hard machined components with process-induced residual stress Y.B. Guo a,, Mark E. Barkey b a
b
Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA Aerospace Engineering and Mechanics Department, The University of Alabama, Tuscaloosa, AL 35487, USA Received 9 May 2003; received in revised form 15 September 2003; accepted 15 October 2003
Abstract This work has investigated the effect of process-induced residual stress on bearing rolling contact fatigue (RCF). A 2-D finite element simulation model of bearing rolling contact has been developed to incorporate the process-induced residual stress profile as initial conditions. The applied load was modeled using the interaction between the roller and the bearing inner race instead of moving elastic-based Hertzian pressure and tangential traction across the surface. Residual stress profiles of an AISI 52100 steel inner race machined by hard turning and grinding were used to evaluate fatigue damage using the critical plane approach. Three cases using the simulation model were assessed: (a) measured residual stress by hard turning; (b) measured residual stress by grinding; and (c) free of residual stress. In addition, the sensitivity of fatigue damage to normal load, tangential traction, and magnitude of surface residual stress was also investigated. The results from simulation post-processing showed that these distinct residual stress profiles only affect near-surface fatigue damage rather than locations deeper in the subsurface. It was concluded, then, that residual stress affects only near-surface initiated RCF rather than subsurface initiated RCF, which has been demonstrated with some prior experimental data. Furthermore, the applied normal load has a significant effect on fatigue damage, while the effects of magnitudes of friction and surface residual stress are small. Lastly, multiaxial fatigue damage parameters used in conjunction with a critical plane approach can characterize the relative fatigue damage under the influence of process-induced residual stress profiles. # 2003 Elsevier Ltd. All rights reserved. Keywords: Rolling contact fatigue; Finite element analysis; Residual stress; Critical plane; Hard turning; Grinding
1. Introduction Hard turning [1], i.e. turning hardened steels, is a competitive alternative finishing process for making a wide range of precision mechanical components, such as bearing races, gears, cams, shafts, axles, tools, dies, and other components. Compared with grinding, hard turning has the potential to reduce capital investment by some 40%, increase production rate by approximately 30%, and reduce production time by 25 to 30% [2], while maintaining equivalent surface finish characteristics of the components. The most significant difference in surface integrity between hard turning and Corresponding author. Tel.: +1-205-348-2615; fax: +1-205-3486419. E-mail address: [email protected] (Y.B. Guo).
0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.009
grinding is that hard turning may induce a relatively deep ‘‘surface’’ compressive residual stress [3–7] while achieving an equivalent or better surface finish, form, and size accuracy [8]. Matsumoto et al. [9] showed that the subsurface compressive residual stresses increased tensile fatigue strength. Smith [10] showed that hard turning could provide tensile fatigue life equal to or better than grinding. However, none of this research investigated the case of rolling contact fatigue (RCF), which is the major application of hard-machined components. An early experimental study [11] showed that compressive residual stress induced by grinding would improve RCF life. Additionally, deep compressive residual stress was found to be more beneficial to bearing fatigue life than shallower stress of greater magnitude [4,11]. This significant effect of residual stress
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pattern on RCF life has been also observed in an experimental study [12]. Although the experimental data are essential to estimate the effect of residual stress on RCF, these studies generally fail to give an insight into the fatigue damage mechanism, which is determined by interactions between residual stress, service load, and surface material properties in rolling contact. However, the determination of the fatigue damage mechanism is vital to predict RCF life. Since rolling contact involves complex interactions between residual stress, service load, and surface material properties, finite element analysis (FEA) is often used as a means of modeling wheel/rail rolling contact [13–17] . It has been used to show [16] that surface tensile residual stress results in an increase of crack extension rate, while a compressive tensile residual stress suppresses crack extension rate in wheel/rail contact. However, the process-induced residual stress was generally not incorporated in these studies. Further, loading conditions in a precision bearing are significantly different from that in wheel/rail contact. The effect of a pre-existing residual stress profile on rolling contact stress and strains was investigated in a previous study [18], but its effect on fatigue damage is not clear at present. Therefore, the objective of the present investigation is to examine how the process-induced residual stress profile influences fatigue life to crack initiation during multiaxial fatigue loading conditions. From the deformation point of view, the stress state in the vicinity of rolling contact is triaxial, displaying compressive hydrostatic stresses whose magnitudes are usually several times the yield strength of the material [18,19]. Moreover, the loading is non-proportional. From the fatigue damage point of view, the planes of maximum stress and the maximum strain are not synonymous for non-proportional loading. In addition, the stress and strain maxima in a loading cycle do not coincide, unlike the uniaxial loading case for stable materials. There have been a number of multiaxial fatigue life models developed for non-proportional loading, such as the effective stress range approach [20], critical plane approaches [21,22], the SWT approach [23], and the plastic strain energy density approach [24]. Although no particular models have been universally accepted for multiaxial fatigue, critical plane approaches appear to be the most capable. The major advantage of critical plane approaches is that they relate predicted fatigue life to experimentally observed cracking behavior. Critical plane models are based on a physical interpretation of the fatigue process. They are based on the observation that cracks form and grow on a critical plane, which is defined as the plane experiencing maximum damage. They can account for mean stress effects, nonproportional hardening, and the strain path dependency of fatigue life, through the
incorporation of a stress term. In addition, they are also relatively easy to apply for variable amplitude loading analyses. The shear-based parameter proposed by Fatemi and Socie [21], and Fatemi and Kurath [25] were used in this study to investigate the effect of residual stress on RCF.
2. Finite element modeling and analysis The simulation analysis has three parts: FEA modeling approach, residual stress and material properties, and post-processing of simulation results. Each part is detailed in the following sections. 2.1. Finite element model The finite element mesh, Fig. 1, of rolling contact for a roller bearing consists of a smooth roller and a smooth deformable machined inner race, which usually has shortest life. The assembly is represented as a 2-D plane strain model since the width of the inner ring is much larger than the thickness of interest. The rigid roller has a radius R and is modeled as an analytical rigid surface. The smooth machined inner race has a length of 5 mm and a thickness of 0.5–1.1 mm, which is at least 10 times the depth of residual stress distributions. The inner race is about 25 a long in the rolling direction to ensure steady state rolling conditions, where a is the half-contact width between the roller and the inner race. The element size varies from 1.5 to 10 lm in the machined layer so that different residual stress distributions (usually less than 100 lm) can be modeled properly. To accurately model surface deformation, a high density mesh of 6000–12,800 four-node bilinear elements was used depending on a specific residual stress profile. Additionally, locally adaptive meshing techniques were applied to the element layers just below the machined surface to accommodate potential plastic strain of surface material. The inner race considered in this study is free of microstructure changes such as the process-induced white layer. In practice, the phase-transformed surface layer is usually polished away in this type of assembly.
Fig. 1. Finite element mesh of bearing rolling contact. The mesh dimensions are not in proportional in order to show surface elements.
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The roller is free to rotate about its axis, while the roller center moves at the speed V determined as V ¼ ðR dÞ x
ð1Þ
where x is angular rotation speed of the roller and d the radial displacement on the outer surface of the roller. The left and right edges of the inner race were constrained in X direction, while the semi-infinite elements on the bottom were used as quiet boundaries. The roller moves forward and backward a certain distance in one cycle. The advantage of this type of simulation is that steady-state rolling can be achieved in a small number of rolling cycles [26,27]. Further, the test data [28] have shown that the full residual stress is developed in the shortest number of revolutions. Additional rolling cycles do not greatly affect the magnitude and depth of residual stress. Hence, the simulation results after six cycles could represent the stabilized values. Simulation convergence was ensured by using the appropriate number of elements and time increment. Room temperature and process-induced residual stress profiles by hard turning and grinding were used as initial conditions for the 12 simulations performed in this study. All the simulations were implemented using ABAQUS [29] running on a desktop computer. 2.2. Applied loading As reviewed in the Introduction, the applied load may not be accurately modeled by applying the normal pressure based on the elastic-based Hertzian contact theory. Therefore, a new modeling approach [18] has been used in this study to bring into the inner race contact with the roller by applying a radial displacement d, Fig. 1, on the outer surface of the roller. The roller rotates forward and backward on the top surface of the inner race. Frictional behavior is modeled by Coulomb’s law using the friction coefficient in practical operating conditions for a cylindrical roller bearing [30]. Compared with moving an assumed Hertzian pressure and tangential traction across the surface undertaken in the literature [14,27,31], this proposed method has several advantages: (1) controlling applied load, either below or above the shakedown limit, by directly adjusting the magnitude of d; (2) modeling contact load more accurately by avoiding the estimates of elastic-based Hertzian pressure and contact area; (3) the rolling condition (pure rolling or rolling plus sliding) will be automatically determined by the applied load; and (4) enabling input of roller properties to more accurately model interaction between the roller and the inner ring. This approach is suitable for both low and high cycle rolling contact.
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2.3. Material properties The input material property model may have a profound influence on rolling contact fatigue damage. A number of studies [31–36] have reported that different plastic models produced quite different surface deformation and residual stresses. For example, the elasticperfectly plastic model produces much larger surface deformation and residual stress, as compared to the linear kinematic hardening plasticity model [33,35], while the isotropic hardening plasticity model and the linear kinematic hardening plasticity model make little difference [35]. The inner race material is AISI 52100 bearing steel (quenched and tempered to 62 HRC), and its elastic and plastic material properties are listed in Table 1 [37]. Since the material only experiences small plastic strain and low temperature for practical rolling conditions, the range of input data is thought to be suitable and is justified by the following simulation results and discussions. Linear kinematic hardening was used in this study since an experimental study [32] has shown that this material has cyclic hardening behavior in rolling contact. However, cyclic hardening is generally a transient behavior, which occurs only during the early hundreds of cycles in roller bearings. Therefore, the cyclic hardening effect will be assumed to be small for steady-state rolling contact. In addition, the strain hardened layer of an inner race induced by hard turning and grinding is usually removed by a superfinishing process such as polishing in production, therefore, the effect of the strained hardened layer on rolling contact fatigue would be very small. 2.4. Fatigue parameter Based on the observations of mean stress influence on crack initiation and early growth, Socie and coworkers [21,38] modified a multiaxial fatigue parameter Table 1 Material properties of AISI 52100 steel (62 HRC) [37] Density Thermal expansion Young’s modulus Poisson’s ratio Flow stress
7800 kg/m3 v v (11.5E6/ C, 25 C) v v (12.6E6/ C , 204 C) v (201 GPa, 22 C) v (178 GPa, 200 C) v (0.277, 20 C) v (0.269, 200 C) v (1600 MPa, 0, 22 C) v (1867 MPa, 1.73E3, 22 C) v (1672 MPa, 0, 200 C) v (2230 MPa, 9.5E3, 200 C) v (2483 MPa, 26.1E3, 200 C)
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developed earlier by Brown and Miller [39] and others. FP ¼
Dc De rn þ þ 2 2 E
ð2Þ
where Dc=2 is the shear strain amplitude on the critical plane, De=2 is the normal strain amplitude perpendicular to the critical plane, rn is the mean normal stress perpendicular to this plane, E is Young’s modulus, and FP is the fatigue parameter, respectively. The critical plane was defined as the material plane that experiences the maximum shear strain range. In subsequent work, Fatemi and Socie [21], Fatemi and Kurath [25] developed another shear-based parameter with the form, FP ¼
Dc rn;max ð1 þ K Þ 2 ry
ð3Þ
where Dc=2 has the same meaning as that in Eq. (2), rn;max is the maximum normal stress on the critical plane, ry is the yield strength of the material, and K is a material constant. A methodology, Fig. 2, has been developed for the linkage between surface integrity and component life analysis and prediction. The fatigue analysis is conducted as a post-processing step based on the FEA results. In general, the critical plane upon which to evaluate the fatigue damage parameter is not known before the analysis. By surveying a comprehensive set of candiv date planes (in increments of 10 ) for each element below the surface, the critical plane, which experiences maximum shear strain amplitude, Dcmax , was determined in a loading cycle. The corresponding angle for each critical plane will be the fatigue crack angle. Two orthogonal planes were discovered that experienced Dcmax , one of these planes experienced high tensile maximum stress and is expected to govern fatigue failure. The basis of the critical plane approach is that the fatigue damage accumulation on the ‘‘critical plane’’ is highest among all possible planes. When the critical plane is determined, the parameters on the right side of Eqs. (2) and (3) and fatigue parameter for each element will be computed. FP sensitivity to the nonzero constant K was studied by using 0.5 and 1, respectively,
although it is usually assumed to be unity. A FP profile below the component surface can be plotted, then the component with a larger magnitude of peak FP will have a shorter life.
3. Effect of process-induced residual stress on fatigue parameter (FP) The effect of residual stress on fatigue parameter was studied in three cases: residual stresses by turning and grinding in Fig. 3 [4] and residual stress-free. The three simulations were conducted under the same conditions, Table 2. It should be point out that the residual stress patterns by hard turning and grinding are typical when sharp cutting tools are used, and therefore represent a ‘‘class’’ of residual stress patterns in production. All of the simulated results were obtained from steady-state rolling conditions. Fig. 4 shows that FP distributions with peak FP value at 156 lm below the machined surface using model I. The location of peak FP value, i.e. maximum fatigue damage, is much larger than the depth (100 lm) of residual stress distributions in Fig. 3. It demonstrates that the distinct residual stress profiles may not affect subsurface FP distributions or peak FP value and its location, except those near the surface within 20 lm. The effect of residual stresses on near-surface fatigue damage is also shown using model II in Fig. 5, in which the location of maximum FP ratio at 40 lm coincides with that of peak compressive residual stress of the turned sample. The FP ratio of the ground sample to the turned sample is less than unity on the surface, which means surface damage is also shown using model II in Fig. 5, in which the location of maximum FP ratio at 40 lm coincides with that of peak compressive residual stress of the turned sample. The FP ratio of the ground sample to the turned sample is less than unity on the surface, which means surface fatigue damage of the ground sample is less than that of the turned sample. This may be attributed to the fact that the magnitude of compressive residual stress on the ground surface is larger than that of the turned sample. The FP ratio is
Fig. 2. A strategy for the linkage between surface integrity and RCF life analysis.
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Fig. 4.
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Fatigue parameter using model I.
Fig. 3. Measured process-induced residual stresses of the inner ring [4].
larger than unity below the machined surface, which explains the experimental observations [4] that the hard-turned surface has longer rolling contact fatigue life than that of the ground surface. Compared with the difference of near surface FP parameter by model I, Fig. 5 shows that model II gives a much larger depth of the affected FP. It can also be seen that FP ratio increases with the increase of K value. Although both model I and model II capture the effect of residual stress on FP, Fig. 6 shows that model II, with either K ¼ 0:5 or K ¼ 1, gives slightly larger FP values that those by model I. Figs. 7 and 8 are the distributions of shear strain and shear stress below the surface, which shows that the locations of peak shear strain and peak shear stress do not coincide due to the nature of the non-proportional loading in rolling contact. Furthermore, the locations of peak FP, Fig. 6, by model I and II are much smaller than that of peak shear strain, Fig. 7, but are close to the location of peak shear stress in Fig. 8. The FP distributions and ratios demonstrate that distinct residual stress patterns affect neither the subsurface FP magnitude nor its peak location, while the difference in near-surface FP is clear. Therefore, it is reasonable to infer that the initial residual stress would only affect near-surface initiated RCF rather than subsurface initiated RCF for materials susceptible to
fatigue crack initiation by shear-strain or shear-stress type mechanisms. Fig. 9 represents the predicted crack behavior near the machined surface. It is clear that residual stress affects near-surface crack angle, though the crack angle might not be accurate due to the limitation of critical plane method. 4. Parametric studies The frictional and normal load between the roller and the inner race and the magnitude of residual stress profile are important input parameters for the RCF analysis. Therefore, parametric studies have been conducted to investigate RCF sensitivity to these parameters. 4.1. Effect of friction coefficient FP sensitivity to friction coefficient was studied using the value of 0.001 (u1), 0.002 (u2), and 0.004 (u3), while keeping other parameters in Table 2 constant.
Table 2 Simulation conditions of rolling contact Roller radius (mm) No. of cycle Distance per pass (mm) Rotation speed (rpm) Friction coefficient Displacement d (lm)
9.38 6 4 8000 0.0011 10 Fig. 5.
Fatigue parameter ratio using model II.
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Fig. 6.
Fatigue parameter comparison using models I and II. Fig. 8.
Fig. 10 shows that the increase of friction, though by four times, hardly affects the location of subsurface peak FP and FP distributions. However, subsurface peak FP magnitude decreases with the increase of friction coefficient using both models. But, it has a negligible effect on value of the near-surface FP for the friction coefficients considered in this study. Model II with K ¼ 0:5 gives almost same results as those with K ¼ 1. By checking the peak magnitude of subsurface shear stress, the applied load is still below the shakedown limit. It implies that fatigue life would not be reduced appreciably under the friction conditions, which is consistent with the experimental observation. For large values of friction coefficient, the tangential surface tractions for varying friction were reported in another study [40]. 4.2. Effect of normal load The effect of normal load on FP was evaluated by increasing the displacement d of 10 lm (L1), Table 2, to 12.5 lm (L2) and 15 lm (L3), respectively. Other
Fig. 7.
Shear strain distributions below the surface.
Shear stress distributions below the surface.
simulation conditions are the same as those in Table 2. Fig. 11 shows that normal load significantly increases subsurface FP magnitude and slightly shifts the location of peak FP deeper into the machined layer. The effect of normal load on near-surface FP is much smaller than that of subsurface FP. However, FP distribution patterns are almost unaffected by the applied load, although FP magnitude increases significantly. Similar results are also given using model II with K values of 0.5 and 1. When the surface deformation is 12.5 lm, the maximum subsurface shear stress (897 MPa) is close to the shakedown limit (924 MPa), and it (969 MPa) is over the shakedown limit at 15 lm. It indicates that increasing normal load would significantly reduce the component life, while it has little effect on the location of subsurface fatigue crack, which agrees with the related experimental data. 4.3. Effect of surface residual stress magnitude It has been shown that depth of compressive residual stress profiles, Fig. 3, affects near-surface FP. The effect of surface residual stress magnitude on FP was further demonstrated by inputting two measured residual stress profiles, Fig. 12 [11]. The most significant difference between the two profiles is that sample A has a much larger magnitude of surface compressive residual stress than that of sample B, but the depth of compressive residual stress below surface is almost the same for the two profiles. Sample A was prepared by grinding followed by mechanical polishing, sample B was ground similarly to sample A, after which 7–12 lm of metal was removed by electropolishing, producing a much lower magnitude of surface compressive residual stress, while the depth of compressive residual stress is equivalent to sample A. The simulation conditions are listed in Table 3.
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Fig. 9.
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Schematic of the effect of residual stress on crack morphology.
The two distinct residual stress profiles have slight effects on subsurface FP magnitudes, Fig. 13, in which larger surface compressive residual stress produces smaller FP magnitudes. However, model I yields almost the same near-surface damage, while model II shows that near-surface damage of case B is larger than that of case A. It is therefore inferred that sample A has less subsurface fatigue damage and thus has longer RCF life, which is consistent with the general belief and experimental data.
5. Discussion In this work, simulation convergence was ensured by increasing the number of elements and decreasing time increment to obtain stable results. The meshes are fine enough to input process-induced residual stress profiles accurately. Rolling contact reaches steady state after a few passes, which is consistent with the experimental observations [28]. The FP calculation is based on stresses and strains obtained from the Gauss points. Considering the small element size of near-surface region and very low friction in bearings, the values of the Gauss points nearest the contact surface would be considered as the stresses and strains on the surface. The
Fig. 10. The effect of friction coefficient on FP using models I and II (k¼ 1).
negligible effect of tangential load on FP may be contributed to the small values of friction coefficient considered. Although the two models employed yield slightly different FP in the near-surface region, the general trend is similar. This study focused on the effect of residual stress on FP and fatigue life implications. To further predict RCF life using the calculated FP, more material fatigue properties need to be determined, which is beyond the scope of this work. The simulation results demonstrate that initial residual stress affects only near-surface RCF, where fatigue initiation is caused by small scale contact stress perturbations generated by residual stresses and surface material properties in the machined layer. Residual stress hardly influences subsurface initiated RCF, where fatigue crack initiation is caused by shear-strain or shear-stress generated by macroscopic contact and usually occurs in the subsurface region.
6. Conclusions Both models I and II can characterize the effect of residual stress on rolling contact fatigue damage and near-surface fatigue crack angle. The value of constant
Fig. 11. The effect of normal load on FP using models I and II (k¼ 1).
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Distinct residual stress profiles only affect near-surface FP distribution and magnitude rather than those in the subsurface, which indicates that residual stress affects only near-surface initiated RCF, rather than subsurface initiated RCF. Tangential load has a negligible effect on FP using the friction coefficient considered. The normal load will significantly increase FP magnitude and shifts its location of peak magnitude deeper into the surface. Additionally, a larger magnitude of surface compressive residual stress tends to reduce fatigue damage compared with a smaller magnitude upon the depth of compressive residual stress is equivalent. In summary, FP can be a parameter to characterize relative fatigue damage under the influence of processinduced residual stress profiles. Fig. 12. Measured process-induced residual stresses in grinding [11].
Acknowledgements Table 3 Simulation conditions of rolling contact Roller radius (mm) No. of cycle Distance per pass (mm) Rotation speed (rpm) Friction coefficient Displacement d (lm)
This work was supported by DoT/CAVT at The University of Alabama, and is gratefully acknowledged. 6.35 6 4 5000 0.0011 6
K in the fatigue damage parameter (Eq. (3)) has a negligible effect on FP distributions and magnitudes. The location of peak FP value is much larger than the depth of residual stress distributions below surface, and much smaller than that of peak shear strain but close to the location of peak shear stress in rolling contact. The predicted fatigue damage agrees with the experimental observations.
Fig. 13. The effect of surface residual stress on FP using models I and II (k¼ 1).
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Modeling of rolling contact fatigue for hard machined components with process-induced residual stress Y.B. Guo a,, Mark E. Barkey b a
b
Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA Aerospace Engineering and Mechanics Department, The University of Alabama, Tuscaloosa, AL 35487, USA Received 9 May 2003; received in revised form 15 September 2003; accepted 15 October 2003
Abstract This work has investigated the effect of process-induced residual stress on bearing rolling contact fatigue (RCF). A 2-D finite element simulation model of bearing rolling contact has been developed to incorporate the process-induced residual stress profile as initial conditions. The applied load was modeled using the interaction between the roller and the bearing inner race instead of moving elastic-based Hertzian pressure and tangential traction across the surface. Residual stress profiles of an AISI 52100 steel inner race machined by hard turning and grinding were used to evaluate fatigue damage using the critical plane approach. Three cases using the simulation model were assessed: (a) measured residual stress by hard turning; (b) measured residual stress by grinding; and (c) free of residual stress. In addition, the sensitivity of fatigue damage to normal load, tangential traction, and magnitude of surface residual stress was also investigated. The results from simulation post-processing showed that these distinct residual stress profiles only affect near-surface fatigue damage rather than locations deeper in the subsurface. It was concluded, then, that residual stress affects only near-surface initiated RCF rather than subsurface initiated RCF, which has been demonstrated with some prior experimental data. Furthermore, the applied normal load has a significant effect on fatigue damage, while the effects of magnitudes of friction and surface residual stress are small. Lastly, multiaxial fatigue damage parameters used in conjunction with a critical plane approach can characterize the relative fatigue damage under the influence of process-induced residual stress profiles. # 2003 Elsevier Ltd. All rights reserved. Keywords: Rolling contact fatigue; Finite element analysis; Residual stress; Critical plane; Hard turning; Grinding
1. Introduction Hard turning [1], i.e. turning hardened steels, is a competitive alternative finishing process for making a wide range of precision mechanical components, such as bearing races, gears, cams, shafts, axles, tools, dies, and other components. Compared with grinding, hard turning has the potential to reduce capital investment by some 40%, increase production rate by approximately 30%, and reduce production time by 25 to 30% [2], while maintaining equivalent surface finish characteristics of the components. The most significant difference in surface integrity between hard turning and Corresponding author. Tel.: +1-205-348-2615; fax: +1-205-3486419. E-mail address: [email protected] (Y.B. Guo).
0142-1123/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.10.009
grinding is that hard turning may induce a relatively deep ‘‘surface’’ compressive residual stress [3–7] while achieving an equivalent or better surface finish, form, and size accuracy [8]. Matsumoto et al. [9] showed that the subsurface compressive residual stresses increased tensile fatigue strength. Smith [10] showed that hard turning could provide tensile fatigue life equal to or better than grinding. However, none of this research investigated the case of rolling contact fatigue (RCF), which is the major application of hard-machined components. An early experimental study [11] showed that compressive residual stress induced by grinding would improve RCF life. Additionally, deep compressive residual stress was found to be more beneficial to bearing fatigue life than shallower stress of greater magnitude [4,11]. This significant effect of residual stress
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pattern on RCF life has been also observed in an experimental study [12]. Although the experimental data are essential to estimate the effect of residual stress on RCF, these studies generally fail to give an insight into the fatigue damage mechanism, which is determined by interactions between residual stress, service load, and surface material properties in rolling contact. However, the determination of the fatigue damage mechanism is vital to predict RCF life. Since rolling contact involves complex interactions between residual stress, service load, and surface material properties, finite element analysis (FEA) is often used as a means of modeling wheel/rail rolling contact [13–17] . It has been used to show [16] that surface tensile residual stress results in an increase of crack extension rate, while a compressive tensile residual stress suppresses crack extension rate in wheel/rail contact. However, the process-induced residual stress was generally not incorporated in these studies. Further, loading conditions in a precision bearing are significantly different from that in wheel/rail contact. The effect of a pre-existing residual stress profile on rolling contact stress and strains was investigated in a previous study [18], but its effect on fatigue damage is not clear at present. Therefore, the objective of the present investigation is to examine how the process-induced residual stress profile influences fatigue life to crack initiation during multiaxial fatigue loading conditions. From the deformation point of view, the stress state in the vicinity of rolling contact is triaxial, displaying compressive hydrostatic stresses whose magnitudes are usually several times the yield strength of the material [18,19]. Moreover, the loading is non-proportional. From the fatigue damage point of view, the planes of maximum stress and the maximum strain are not synonymous for non-proportional loading. In addition, the stress and strain maxima in a loading cycle do not coincide, unlike the uniaxial loading case for stable materials. There have been a number of multiaxial fatigue life models developed for non-proportional loading, such as the effective stress range approach [20], critical plane approaches [21,22], the SWT approach [23], and the plastic strain energy density approach [24]. Although no particular models have been universally accepted for multiaxial fatigue, critical plane approaches appear to be the most capable. The major advantage of critical plane approaches is that they relate predicted fatigue life to experimentally observed cracking behavior. Critical plane models are based on a physical interpretation of the fatigue process. They are based on the observation that cracks form and grow on a critical plane, which is defined as the plane experiencing maximum damage. They can account for mean stress effects, nonproportional hardening, and the strain path dependency of fatigue life, through the
incorporation of a stress term. In addition, they are also relatively easy to apply for variable amplitude loading analyses. The shear-based parameter proposed by Fatemi and Socie [21], and Fatemi and Kurath [25] were used in this study to investigate the effect of residual stress on RCF.
2. Finite element modeling and analysis The simulation analysis has three parts: FEA modeling approach, residual stress and material properties, and post-processing of simulation results. Each part is detailed in the following sections. 2.1. Finite element model The finite element mesh, Fig. 1, of rolling contact for a roller bearing consists of a smooth roller and a smooth deformable machined inner race, which usually has shortest life. The assembly is represented as a 2-D plane strain model since the width of the inner ring is much larger than the thickness of interest. The rigid roller has a radius R and is modeled as an analytical rigid surface. The smooth machined inner race has a length of 5 mm and a thickness of 0.5–1.1 mm, which is at least 10 times the depth of residual stress distributions. The inner race is about 25 a long in the rolling direction to ensure steady state rolling conditions, where a is the half-contact width between the roller and the inner race. The element size varies from 1.5 to 10 lm in the machined layer so that different residual stress distributions (usually less than 100 lm) can be modeled properly. To accurately model surface deformation, a high density mesh of 6000–12,800 four-node bilinear elements was used depending on a specific residual stress profile. Additionally, locally adaptive meshing techniques were applied to the element layers just below the machined surface to accommodate potential plastic strain of surface material. The inner race considered in this study is free of microstructure changes such as the process-induced white layer. In practice, the phase-transformed surface layer is usually polished away in this type of assembly.
Fig. 1. Finite element mesh of bearing rolling contact. The mesh dimensions are not in proportional in order to show surface elements.
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The roller is free to rotate about its axis, while the roller center moves at the speed V determined as V ¼ ðR dÞ x
ð1Þ
where x is angular rotation speed of the roller and d the radial displacement on the outer surface of the roller. The left and right edges of the inner race were constrained in X direction, while the semi-infinite elements on the bottom were used as quiet boundaries. The roller moves forward and backward a certain distance in one cycle. The advantage of this type of simulation is that steady-state rolling can be achieved in a small number of rolling cycles [26,27]. Further, the test data [28] have shown that the full residual stress is developed in the shortest number of revolutions. Additional rolling cycles do not greatly affect the magnitude and depth of residual stress. Hence, the simulation results after six cycles could represent the stabilized values. Simulation convergence was ensured by using the appropriate number of elements and time increment. Room temperature and process-induced residual stress profiles by hard turning and grinding were used as initial conditions for the 12 simulations performed in this study. All the simulations were implemented using ABAQUS [29] running on a desktop computer. 2.2. Applied loading As reviewed in the Introduction, the applied load may not be accurately modeled by applying the normal pressure based on the elastic-based Hertzian contact theory. Therefore, a new modeling approach [18] has been used in this study to bring into the inner race contact with the roller by applying a radial displacement d, Fig. 1, on the outer surface of the roller. The roller rotates forward and backward on the top surface of the inner race. Frictional behavior is modeled by Coulomb’s law using the friction coefficient in practical operating conditions for a cylindrical roller bearing [30]. Compared with moving an assumed Hertzian pressure and tangential traction across the surface undertaken in the literature [14,27,31], this proposed method has several advantages: (1) controlling applied load, either below or above the shakedown limit, by directly adjusting the magnitude of d; (2) modeling contact load more accurately by avoiding the estimates of elastic-based Hertzian pressure and contact area; (3) the rolling condition (pure rolling or rolling plus sliding) will be automatically determined by the applied load; and (4) enabling input of roller properties to more accurately model interaction between the roller and the inner ring. This approach is suitable for both low and high cycle rolling contact.
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2.3. Material properties The input material property model may have a profound influence on rolling contact fatigue damage. A number of studies [31–36] have reported that different plastic models produced quite different surface deformation and residual stresses. For example, the elasticperfectly plastic model produces much larger surface deformation and residual stress, as compared to the linear kinematic hardening plasticity model [33,35], while the isotropic hardening plasticity model and the linear kinematic hardening plasticity model make little difference [35]. The inner race material is AISI 52100 bearing steel (quenched and tempered to 62 HRC), and its elastic and plastic material properties are listed in Table 1 [37]. Since the material only experiences small plastic strain and low temperature for practical rolling conditions, the range of input data is thought to be suitable and is justified by the following simulation results and discussions. Linear kinematic hardening was used in this study since an experimental study [32] has shown that this material has cyclic hardening behavior in rolling contact. However, cyclic hardening is generally a transient behavior, which occurs only during the early hundreds of cycles in roller bearings. Therefore, the cyclic hardening effect will be assumed to be small for steady-state rolling contact. In addition, the strain hardened layer of an inner race induced by hard turning and grinding is usually removed by a superfinishing process such as polishing in production, therefore, the effect of the strained hardened layer on rolling contact fatigue would be very small. 2.4. Fatigue parameter Based on the observations of mean stress influence on crack initiation and early growth, Socie and coworkers [21,38] modified a multiaxial fatigue parameter Table 1 Material properties of AISI 52100 steel (62 HRC) [37] Density Thermal expansion Young’s modulus Poisson’s ratio Flow stress
7800 kg/m3 v v (11.5E6/ C, 25 C) v v (12.6E6/ C , 204 C) v (201 GPa, 22 C) v (178 GPa, 200 C) v (0.277, 20 C) v (0.269, 200 C) v (1600 MPa, 0, 22 C) v (1867 MPa, 1.73E3, 22 C) v (1672 MPa, 0, 200 C) v (2230 MPa, 9.5E3, 200 C) v (2483 MPa, 26.1E3, 200 C)
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developed earlier by Brown and Miller [39] and others. FP ¼
Dc De rn þ þ 2 2 E
ð2Þ
where Dc=2 is the shear strain amplitude on the critical plane, De=2 is the normal strain amplitude perpendicular to the critical plane, rn is the mean normal stress perpendicular to this plane, E is Young’s modulus, and FP is the fatigue parameter, respectively. The critical plane was defined as the material plane that experiences the maximum shear strain range. In subsequent work, Fatemi and Socie [21], Fatemi and Kurath [25] developed another shear-based parameter with the form, FP ¼
Dc rn;max ð1 þ K Þ 2 ry
ð3Þ
where Dc=2 has the same meaning as that in Eq. (2), rn;max is the maximum normal stress on the critical plane, ry is the yield strength of the material, and K is a material constant. A methodology, Fig. 2, has been developed for the linkage between surface integrity and component life analysis and prediction. The fatigue analysis is conducted as a post-processing step based on the FEA results. In general, the critical plane upon which to evaluate the fatigue damage parameter is not known before the analysis. By surveying a comprehensive set of candiv date planes (in increments of 10 ) for each element below the surface, the critical plane, which experiences maximum shear strain amplitude, Dcmax , was determined in a loading cycle. The corresponding angle for each critical plane will be the fatigue crack angle. Two orthogonal planes were discovered that experienced Dcmax , one of these planes experienced high tensile maximum stress and is expected to govern fatigue failure. The basis of the critical plane approach is that the fatigue damage accumulation on the ‘‘critical plane’’ is highest among all possible planes. When the critical plane is determined, the parameters on the right side of Eqs. (2) and (3) and fatigue parameter for each element will be computed. FP sensitivity to the nonzero constant K was studied by using 0.5 and 1, respectively,
although it is usually assumed to be unity. A FP profile below the component surface can be plotted, then the component with a larger magnitude of peak FP will have a shorter life.
3. Effect of process-induced residual stress on fatigue parameter (FP) The effect of residual stress on fatigue parameter was studied in three cases: residual stresses by turning and grinding in Fig. 3 [4] and residual stress-free. The three simulations were conducted under the same conditions, Table 2. It should be point out that the residual stress patterns by hard turning and grinding are typical when sharp cutting tools are used, and therefore represent a ‘‘class’’ of residual stress patterns in production. All of the simulated results were obtained from steady-state rolling conditions. Fig. 4 shows that FP distributions with peak FP value at 156 lm below the machined surface using model I. The location of peak FP value, i.e. maximum fatigue damage, is much larger than the depth (100 lm) of residual stress distributions in Fig. 3. It demonstrates that the distinct residual stress profiles may not affect subsurface FP distributions or peak FP value and its location, except those near the surface within 20 lm. The effect of residual stresses on near-surface fatigue damage is also shown using model II in Fig. 5, in which the location of maximum FP ratio at 40 lm coincides with that of peak compressive residual stress of the turned sample. The FP ratio of the ground sample to the turned sample is less than unity on the surface, which means surface damage is also shown using model II in Fig. 5, in which the location of maximum FP ratio at 40 lm coincides with that of peak compressive residual stress of the turned sample. The FP ratio of the ground sample to the turned sample is less than unity on the surface, which means surface fatigue damage of the ground sample is less than that of the turned sample. This may be attributed to the fact that the magnitude of compressive residual stress on the ground surface is larger than that of the turned sample. The FP ratio is
Fig. 2. A strategy for the linkage between surface integrity and RCF life analysis.
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Fig. 4.
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Fatigue parameter using model I.
Fig. 3. Measured process-induced residual stresses of the inner ring [4].
larger than unity below the machined surface, which explains the experimental observations [4] that the hard-turned surface has longer rolling contact fatigue life than that of the ground surface. Compared with the difference of near surface FP parameter by model I, Fig. 5 shows that model II gives a much larger depth of the affected FP. It can also be seen that FP ratio increases with the increase of K value. Although both model I and model II capture the effect of residual stress on FP, Fig. 6 shows that model II, with either K ¼ 0:5 or K ¼ 1, gives slightly larger FP values that those by model I. Figs. 7 and 8 are the distributions of shear strain and shear stress below the surface, which shows that the locations of peak shear strain and peak shear stress do not coincide due to the nature of the non-proportional loading in rolling contact. Furthermore, the locations of peak FP, Fig. 6, by model I and II are much smaller than that of peak shear strain, Fig. 7, but are close to the location of peak shear stress in Fig. 8. The FP distributions and ratios demonstrate that distinct residual stress patterns affect neither the subsurface FP magnitude nor its peak location, while the difference in near-surface FP is clear. Therefore, it is reasonable to infer that the initial residual stress would only affect near-surface initiated RCF rather than subsurface initiated RCF for materials susceptible to
fatigue crack initiation by shear-strain or shear-stress type mechanisms. Fig. 9 represents the predicted crack behavior near the machined surface. It is clear that residual stress affects near-surface crack angle, though the crack angle might not be accurate due to the limitation of critical plane method. 4. Parametric studies The frictional and normal load between the roller and the inner race and the magnitude of residual stress profile are important input parameters for the RCF analysis. Therefore, parametric studies have been conducted to investigate RCF sensitivity to these parameters. 4.1. Effect of friction coefficient FP sensitivity to friction coefficient was studied using the value of 0.001 (u1), 0.002 (u2), and 0.004 (u3), while keeping other parameters in Table 2 constant.
Table 2 Simulation conditions of rolling contact Roller radius (mm) No. of cycle Distance per pass (mm) Rotation speed (rpm) Friction coefficient Displacement d (lm)
9.38 6 4 8000 0.0011 10 Fig. 5.
Fatigue parameter ratio using model II.
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Fig. 6.
Fatigue parameter comparison using models I and II. Fig. 8.
Fig. 10 shows that the increase of friction, though by four times, hardly affects the location of subsurface peak FP and FP distributions. However, subsurface peak FP magnitude decreases with the increase of friction coefficient using both models. But, it has a negligible effect on value of the near-surface FP for the friction coefficients considered in this study. Model II with K ¼ 0:5 gives almost same results as those with K ¼ 1. By checking the peak magnitude of subsurface shear stress, the applied load is still below the shakedown limit. It implies that fatigue life would not be reduced appreciably under the friction conditions, which is consistent with the experimental observation. For large values of friction coefficient, the tangential surface tractions for varying friction were reported in another study [40]. 4.2. Effect of normal load The effect of normal load on FP was evaluated by increasing the displacement d of 10 lm (L1), Table 2, to 12.5 lm (L2) and 15 lm (L3), respectively. Other
Fig. 7.
Shear strain distributions below the surface.
Shear stress distributions below the surface.
simulation conditions are the same as those in Table 2. Fig. 11 shows that normal load significantly increases subsurface FP magnitude and slightly shifts the location of peak FP deeper into the machined layer. The effect of normal load on near-surface FP is much smaller than that of subsurface FP. However, FP distribution patterns are almost unaffected by the applied load, although FP magnitude increases significantly. Similar results are also given using model II with K values of 0.5 and 1. When the surface deformation is 12.5 lm, the maximum subsurface shear stress (897 MPa) is close to the shakedown limit (924 MPa), and it (969 MPa) is over the shakedown limit at 15 lm. It indicates that increasing normal load would significantly reduce the component life, while it has little effect on the location of subsurface fatigue crack, which agrees with the related experimental data. 4.3. Effect of surface residual stress magnitude It has been shown that depth of compressive residual stress profiles, Fig. 3, affects near-surface FP. The effect of surface residual stress magnitude on FP was further demonstrated by inputting two measured residual stress profiles, Fig. 12 [11]. The most significant difference between the two profiles is that sample A has a much larger magnitude of surface compressive residual stress than that of sample B, but the depth of compressive residual stress below surface is almost the same for the two profiles. Sample A was prepared by grinding followed by mechanical polishing, sample B was ground similarly to sample A, after which 7–12 lm of metal was removed by electropolishing, producing a much lower magnitude of surface compressive residual stress, while the depth of compressive residual stress is equivalent to sample A. The simulation conditions are listed in Table 3.
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Fig. 9.
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Schematic of the effect of residual stress on crack morphology.
The two distinct residual stress profiles have slight effects on subsurface FP magnitudes, Fig. 13, in which larger surface compressive residual stress produces smaller FP magnitudes. However, model I yields almost the same near-surface damage, while model II shows that near-surface damage of case B is larger than that of case A. It is therefore inferred that sample A has less subsurface fatigue damage and thus has longer RCF life, which is consistent with the general belief and experimental data.
5. Discussion In this work, simulation convergence was ensured by increasing the number of elements and decreasing time increment to obtain stable results. The meshes are fine enough to input process-induced residual stress profiles accurately. Rolling contact reaches steady state after a few passes, which is consistent with the experimental observations [28]. The FP calculation is based on stresses and strains obtained from the Gauss points. Considering the small element size of near-surface region and very low friction in bearings, the values of the Gauss points nearest the contact surface would be considered as the stresses and strains on the surface. The
Fig. 10. The effect of friction coefficient on FP using models I and II (k¼ 1).
negligible effect of tangential load on FP may be contributed to the small values of friction coefficient considered. Although the two models employed yield slightly different FP in the near-surface region, the general trend is similar. This study focused on the effect of residual stress on FP and fatigue life implications. To further predict RCF life using the calculated FP, more material fatigue properties need to be determined, which is beyond the scope of this work. The simulation results demonstrate that initial residual stress affects only near-surface RCF, where fatigue initiation is caused by small scale contact stress perturbations generated by residual stresses and surface material properties in the machined layer. Residual stress hardly influences subsurface initiated RCF, where fatigue crack initiation is caused by shear-strain or shear-stress generated by macroscopic contact and usually occurs in the subsurface region.
6. Conclusions Both models I and II can characterize the effect of residual stress on rolling contact fatigue damage and near-surface fatigue crack angle. The value of constant
Fig. 11. The effect of normal load on FP using models I and II (k¼ 1).
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Distinct residual stress profiles only affect near-surface FP distribution and magnitude rather than those in the subsurface, which indicates that residual stress affects only near-surface initiated RCF, rather than subsurface initiated RCF. Tangential load has a negligible effect on FP using the friction coefficient considered. The normal load will significantly increase FP magnitude and shifts its location of peak magnitude deeper into the surface. Additionally, a larger magnitude of surface compressive residual stress tends to reduce fatigue damage compared with a smaller magnitude upon the depth of compressive residual stress is equivalent. In summary, FP can be a parameter to characterize relative fatigue damage under the influence of processinduced residual stress profiles. Fig. 12. Measured process-induced residual stresses in grinding [11].
Acknowledgements Table 3 Simulation conditions of rolling contact Roller radius (mm) No. of cycle Distance per pass (mm) Rotation speed (rpm) Friction coefficient Displacement d (lm)
This work was supported by DoT/CAVT at The University of Alabama, and is gratefully acknowledged. 6.35 6 4 5000 0.0011 6
K in the fatigue damage parameter (Eq. (3)) has a negligible effect on FP distributions and magnitudes. The location of peak FP value is much larger than the depth of residual stress distributions below surface, and much smaller than that of peak shear strain but close to the location of peak shear stress in rolling contact. The predicted fatigue damage agrees with the experimental observations.
Fig. 13. The effect of surface residual stress on FP using models I and II (k¼ 1).
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