Case study of multiaxial criteria for rolling contact fatigue of bearing steels

A rolling contact fatigue (RCF) differs from the classic fatigue in a stress-state. Nowadays, a prediction of the RCF is still not on the sufficient level. A lot of researchers tried to apply different multiaxial fatigue criteria (MFC) to the RCF, respectively they modified some or even proposed new one. Our paper focuses on assessment of bearing life estimation based on mentioned methods with experimental validation in laboratory. Comparison and summarization of different methods used in MFC life estimation is presented, with inclusion of fatigue material properties, hardness and probability. Mainly bearing steels are used for evaluation and region of high-cycle and giga-cycle fatigue.


Introduction
The rolling bearing is exposed to many types of failure during operation.These types are well described in the publication of Harris [1] who presents analytic methods for predict these failures.This study exclusively deals with the failure named Rolling Contact Fatigue (RCF) which leads to pitting (Fig. 1).It is not yet too wellknown to what extent the classic fatigue and the RCF are related.The difference is that the crack initiates on the surface of part within classic fatigue, whereas the crack initiates in subsurface layer in the RCF (macropitting) (Fig. 3).It is caused by cyclic contact loading which leads to a specific distribution of stress state and maximal shear stress beneath the surface (Fig. 2).Publications [2][3][4][5] describe an analogy between the classic and the rolling contact fatigue, on the basis of which Multiaxial Fatigue Criteria (MFC) can be used for the RCF, even though they were originally derived for the classic fatigue.3. Crack due to rolling contact fatigue [6] Nowadays, it is well-known that due to stress raisers on the surface of bearing raceways, such as scratches, surfaces inclusions or other inhomogeneity, RCF cracks can initiate [7][8][9].In this case, the failure is called micropitting.Micropitting is typical for modern high strength steels or bearing steels.
RCF can occur, for example, at the contact of a train wheel with rails or at gears.RCF at rolling bearings is specific because of the fact that the chrome bearing steel shows such resistance to fatigue that cracks occur in the gigacycle (ultra-high) region of fatigue.
Commonly used methodology for the assessment of service-life of rolling bearings is based on ISO 281 or more recently on the ISO 16 281 standard.They have certain limitations which proposed methodology eliminates.The main benefit is the opportunity of using the Finite Element Method (FEM), which is how the stress state in rolling bearing is analyzed.Using MFC and this stress state is possible to evaluate the servicelife.There are some studies of appropriateness of MFC for this problem [10][11][12] but it has not been established yet which is the best.Therefore, a verification of MFC was proposed.

Multiaxial fatigue criteria
Multiaxial fatigue criteria can be divided into several groups.This study primarily deals with the criteria for high-cycle fatigue.They divide into stress-based (or stress invariant based), based on critical plane approach and energy-based (or integral approach).According to the research of appropriateness, some of them are described below.

Dang Van
In literature, for simulations of life ended by RCF, we can frequently encounter the Dang Van criterion [16], proposed in form (1) [10][11][13][14][15].  max ()is the instantaneous maximal shear stress,  ℎ () is the instantaneous hydrostatic stress,   is the fatigue limit in reversal torque and   is the fatigue limit of fully reversed tension-compression or bending of rotation.Even though this criterion is used very often, it is inappropriate (too benevolent) [14,17,18], because it does not consider the significant influence of negative hydrostatic stress.

Dang Van with locus by Desimone et. al
As mentioned above, the Dang Van criterion does not provide appropriate results for life, so Desimone et.al [14] proposed an approach for its refinement.A failure locus for negative  ℎ in (1) modifies   into a constant value (2).
Fig. 4. Conservative locus for the Dang Van criterion applied to rolling/sliding contact [11] This modification creates the failure locus illustrated in Fig. 4. Recent papers confirmed this work with experiments even for bearing steels [11,19].

Liu and Mahadevan
The Liu and Mahadevan criterion [20,21] provides a good match of life results with an experiment even for bearing steel.[11] provides a proof.The Liu and Mahadevan criterion is based on two characteristic planes -the critical plane, which is a plane of maximal amplitude of shear stress, and the plane of fatigue failure, where the macrocrack occurs.An angle difference between these two planes  can be calculated using (3) and (4).
Using ( 5) and ( 6), the necessary parameters can be calculated and finally, using (7), the criterion is set up.   is the amplitude of equivalent stress,  , is the amplitude of normal stress,  , is the mean value of normal stress,  , is the amplitude of maximal shear stress, and  ,  is the amplitude of hydrostatic stress. ) For the evaluation of a safety fatigue factor or even for the evaluation of the number of cycles up to a failure, a comparison with fatigue limit or with the S-N curve can be used (8).
For the Crossland criterion,   (9), which is the centre of the smallest hypersphere circumscribing the load path in the deviatoric space (more in [26]), and stress deviator at the generic instant (), must be evaluated.Now, it is possible to calculate (10) the amplitude of the square root of the second invariant of the deviatoric component of the stress tensor √ 2, .Then, the final form can be stated as (11). √

Papadopoulos
This criterion [27], based on mesoscopic scale, is proposed as the best agreement with out-of-phase bending and torsion experimental results for hard metals, where in (3) equation ( 12) applies for .The criterion is estimated as (13).
0.557 ≤  ≤ 0.8 In a simplified form, it is possible to say that √〈  2 〉 is "the volumetric root mean square of the amplitude of the resolved shear stress on all possible planes" [10].

Roessle-Fatemi
One of the aims is the inclusion of the surface hardness to the calculation of rolling bearing durability.Poeppelman [28] used the Roessle-Fatemi criterion [29] for the fatigue evaluation of high hardness steels with appropriate results.In this study, this criterion is verified even for bearing steel.
The criterion is in fundamentals different from the others mentioned.It is based on the Coffin-Manson curve (strain-life approach).This means that it presumes plastic deformations.The criterion uses the Brinell hardness  to calculate the fatigue strength coefficient (14) and the fatigue ductility coefficient (15).
′ is in  and  is the modulus of elasticity, also in .Fatigue strength exponent  was determined to be −0.09and the fatigue ductility exponent  to be 0.56.
The final criterion involves only the material hardness and the modulus of elasticity (16).
[30] describes an inclusion of multiaxiality in this criterion.They use von-Mises, Tresca and maximum principal stress criteria.

Proposed methodology of numerical solution
The first part is the evaluation of stress state in the subsurface layer of raceway.This can be achieved using the Hertz theory at simple tasks.At more complex tasks, using ISO 16 281 standard or using finite element method (FEM) is needed.The application of MFC follows.Criteria intended for high-cycle fatigue are proposed in form given by equation (17).  is the reduced stress characteristic for each criterion, and  is the limit value, e.g.fatigue limit.However, in this form they provide only the information of the fatigue safety factor.This study suggests using methodology proposed by Liu and Mahadevan [20] to extend the inequation (17) by the dependence on the number of cycles up to a failure (18), which allows a prediction of service-life.
(  ) ≤ (  ) (18) For example, the Dang Van criterion (mentioned in (1)) can be converted into being the dependent on number of cycles up to a failure using (19) and (20) as follows. σ → τ W (  ) =  −1 (20)  −1 replaces   . −1 is a designation of shear stress in which the failure occurs for given number of cycles according to the fatigue test.Therefore, it is important to know such a part of S-N curve that describes the region of fatigue in which the failure of rolling bearings is expected.
A generalized form of the algorithm of numerical calculation of life may look as follows: It means that at the beginning we define a variable  > 0, then we choose ε as a step that we use for finding Nf as a trial and error method.If the inequation (18) does not apply, the algorithm is ended and the final value Nf is found.The use of logarithm coordinates is preferable.
It is important to affect the difference between the number of revolutions, which is stated by basic durability  10 , and the number of cycles to which the bearing is exposed during this durability.A description of the calculation of the number of cycles is in [31].It depends on the size and number of rolling elements.After several mathematical adjustments, it is possible to calculate it as (21).The parameters are described in Fig. 5,  is the number of rolling elements (or balls).

Application of the multiaxial fatigue criteria using material characteristics based on hardness
This methodology is purely empirical, but it provides good results (see the verification below).S-N curve is  14) and (15).The necessary coefficients describing the S-N curve can be calculated as (22).The final form of the S-N curve is presented in (23).Variable  is number of cycles and it is in logarithm form.

Verification
Based on the described principle, some criteria where tested.For the verification, the experiment AXMAT II was chosen.FEM model of integral parts of the AXMAT test rig was created.The experiment and the numerical analysis are described in this section.

Experiment
The experimental RCF apparatus employed in this study is a flat washer-type RCF test-rig with acoustic emission and vibration monitoring systems.This special test-rig, shown in Fig. 6, was designed for life tests of thrust bearings (smaller size) and an evaluation of the rolling contact fatigue resistance of the material [32].It consists of a mechanical loading lever, an electrical motor, a specimen holder, a catch driver, a supporting frame and a monitoring system.

Fig. 6. Test rig AXMAT II [32]
The speed of the electrical motor can be adjusted by a frequency converter to the required level.This allows standard RCF tests to be performed, including tests at low speeds.The upper ring of the test bearing is clamped in the holder and the lower ring is fastened in the catch driver.In the case of testing material specimens, the specimen was fastened in the holder in place of the upper bearing ring (the geometry of the specimen is shown in Fig. 7).The holder is stationary, and the catch driver is driven by a shaft attached to an electrical motor.For a standard RCF test, the speed was set at 2150 min -1 .The holder is equipped with a polyamide safety element to offload in case of specimen overheating.The Hertzian contact stress for rolling contact tests can be set using the combination of weights in the range from 2000 to 6000 MPa.

S-N curve is not determined due to brittleness of material
In this study, the specimen was loaded by force which should cause the contact stress of 5000 MPa according to Hertzian calculation.This load causes plastic deformations, but the number of cycles up to failure is still in the high-cycle region.The reason of higher value of the load is that the experiment was very time consuming (100 -140 h per one specimen).Material characteristics of the tested material are in Table 1.This bearing steel was hardened and it is brittle (for brittle materials is in Eq. ( 3)  ≥ 1 [11].).Therefore, the performance of fatigue tests was problematic, as shown in Fig. 8. Fatigue limit was determined (also in Table 1).

Numerical analysis
The finite-element model of integral parts of the AXMAT test rig is created.These parts were housing washer, balls and the specimen.Thrust loading of this assembly was cyclic symmetric, therefore, only this symmetrical part of assembly was modelled (see Fig. 9).The material of the specimen was ideal elastic-plastic with bilinear behaviour.The elastic modulus  = 210 and the Poisson's ratio  = 0.3 were used.For contact solution, augmented Lagrangian formulation was used.
Due to the plasticity, deformation shows a dent on the raceway surface of the specimen (Fig. 10).Therefore, contact pressure did not reach the stated value 5000, but 4179  (Fig. 11).
In the next part of the durability solution, the stress state of the specimen in the form of principal stresses was exported from the strain-stress analysis to the fatigue analysis.In -axis only data under the contact point were used (Fig. 12).The stress results are shown in Fig. 13.Due to the plastic strain, peak-to-peak amplitude of maximal shear stress was not variable from a certain point beneath the surface.history.After each load of the analysed point, complete unload follows.Therefore, load history appears as shown in Fig. 14 and the amplitude and the mean stresses can be easily calculated.These data provided input for the criteria.The final results of the number of cycles up to failure was recalculated to a number of rotations using equation (21).Here, angle  was equal to /2, thus, the number of rotations were always equal to number of cycles divided by half of number of balls.Then, number of rotations is recalculated on time in hours.
From fatigue limit of the material of the specimen was established: • No failure -criterion Dang Van, Crossland, Papadopoulos and Dang Van with modified locus • Failure -criterion Liu Mahadevan However, this study proposed an approach to life estimation in combination with setting the S-N curve from hardness and following calculation using MFC.The Gaussian distribution for hardness input is included.Therefore, results are given with their probability of failure.It is chosen probability 0.5, 0.9, 0.95 and 0.99.Results from this methodology are shown in Fig. 15.

Discussion
Measuring the S-N fatigue curve material characteristic of the 100Cr6MnSi6-4 bearing steel is problematic.Therefore, MFC cannot be used for life estimation, only for the failure prediction.Only the Liu and Mahadevan criterion predicts the failure in this case.MFC in combination with hardness properties can be effectively used for life estimation.From these results it is obvious that the Liu Mahadevan criterion is the most appropriate from all tested criteria.
Yet, there are several influences which were not included in the analysis.The most critical is that torsional fatigue tests were not performed.Also, the AXMAT experiment was not properly set up for time reasons (100 -140 h per specimen).Contact pressure exceeded the appropriate limit and calculations could be influenced by plastic strains.A better way is set the loading with no following plastic deformation, but time can be significantly increased.Then, influences of macro-and micro-slippages and influences of roughness of surfaces still are not included in the analysis.
Due to the suggested verification of this study, more levels of loading were not tested.Also, more different types of material and different hardness of specimen surfaces should be tested in the future.

Conclusion
The methodology of the evaluation and the verification of the durability of rolling bearing has been proposed.Durability was evaluated as quantification of servicelife, which finishes together with the pitting occurrence.For the verification, the use of test-rig named AXMAT II was proposed.The 100Cr6MnSi6-4 hardened bearing steel was tested.Weibull curve was drawn from these tests.
The procedure of durability calculation was described.Stress state or strain state respectively in area beneath the surface of raceways was input for calculation.This state together with material characteristics gained from fatigue tests input to multiaxial fatigue criteria.They were used in calculation and results were compared with the Weibull curve.The Liu and Mahadevan criterion seems to be the most appropriate for this application at this moment.But with the increasing number of tests, other criteria can be more appropriate for the rolling contact fatigue of rolling bearing.

Fig. 8 .
Fig. 8. Fatigue test of hardened bearing steel 100Cr6MnSi6-4.S-N curve is not determined due to brittleness of material

Fig. 15 .
Fig. 15.Comparison of experimental results of 100Cr6MnSi6-4 bearing steel and MFC (using hardness as input) results

Table 1 .
Material characteristics (means) of applied material of hardened bearing steel 100Cr6MnSi6-4