## Abstract

This paper presents a finite element model (FEM) to investigate the effect of prior austenite grain refinement on rolling contact fatigue (RCF). RCF life was determined using continuum damage mechanics (CDM), which simulated material deterioration as a function of cycle. Continuum damage mechanics calculations in this investigation considered the subsurface shear (orthogonal) reversal to be responsible for RCF failure. To establish the CDM critical parameters—resistance stress (σr) and damage rate exponent (m)—torsion stress-life data from open literature of three different grain sizes for the same material was used. It was observed from the torsion S-N (stress-life) data that the resistance stress exhibits a linear relationship with grain diameter. As grain diameter was refined, the resistance stress was found to increase. The damage rate exponent (m) displayed no relation to grain diameter; hence, the average value from the three torsion S-N curves was used in this investigation. In order to assess the effect of grain refinement on RCF life, a series of unique material microstructures were constructed using the Voronoi tessellation process at eight mean grain diameters. Finite element (FE) simulations were devised at three contact pressures, typical of heavily loaded lubricated contacts, and the RCF life was determined for each set of microstructures of a given mean grain diameter. The RCF results at the eight grain diameters indicate that fatigue performance is improved exponentially with finer grain diameter. The observed life improvements from the RCF simulations resulting from grain refinement exhibit good corroboration with existing experimental results found in open literature. A single predictive fatigue life equation was constructed from this investigation’s RCF simulations to evaluate the stochastic RCF performance, given grain diameter and contact pressure, of non-conformal contacts.

## 1 Introduction

In tribo-machines (gas turbines, reciprocating engines, etc.), rolling element bearings (REBs) are frequently used to support significant amount of load while allowing for rotational motion. These precision engineered machine components also help minimize frictional losses. In an industry growing at a compound annual growth rate of 6.85%, with a market size predicted to surpass \$195 billion [1] by 2027, a key factor limiting REB life is rolling contact fatigue (RCF) [2,3]. This type of fatigue, which possesses two predominant failure mechanisms [48], is induced by the rolling motion between the rolling element (e.g., balls, rollers) and the raceway. This rolling motion produces a complex alternating contact stress acting over a small volume [9].

Subsurface originated spalling (SOS) is the prevalent failure mechanism of properly installed REBs operating in a well-maintained environment that is lubricated and clean [6,7,1012]. The fatigue damage accumulation of this failure mechanism is the culmination of recurring localized, alternating contact stresses. Cracks will nucleate subsurface as a result, with the repeated exposure to the RCF stresses promoting crack propagation toward the surface [13]. As a result of the physics of subsurface spalling, detecting cracks before significant amount of material dislodges from the contact is not trivial.

The second of the two predominant RCF failure modes is surface originated pitting (SOP). SOP is present when a fatigue crack initiates on the contacting surface, then propagates into the material along a shallow path. The presence of a surface stress riser (i.e., high surface friction, significant roughness, inadequate lubrication conditions, etc.) manifest surface failures in critical tribological components [1416]. Recently, more surface failures have materialized because of demanding design requirements that drive tribological components to operate in rigorous conditions (i.e., higher loads, speeds, etc.) [1719]. As a result, service life is diminished when modern tribo-machines experience SOP failure [2024].

Regardless of failure mechanism, determining RCF life has been the subject of many investigations since the early 1900’s. Predictive theories by Lundberg and Palmgren [25] and Ioannides and Harris [26] are a few of the most famous and widely implemented theories to predict REB RCF life. The contacting bodies’ stress solution and Weibull probability distribution [27,28] serve as the basis for these predictive theories, which capture the scatter intrinsic to bearing life. A limitation of these predictive theories is their reliance on experimental data to capture the scatter in bearing life. Experimental RCF testing at elevated loads has proven to be taxing [29,30] not the least of which is a product of the advancement in bearing steel technology since the turn of the century. As a result, this poses a particularly challenging problem to the tribological community when it comes time to evaluate new manufacturing techniques or materials in order to determine their efficacy in enhancing fatigue performance.

Since the turn of the twenty-first century, damage mechanics has become a popular tool for researchers studying the rolling contact fatigue problem [3136]. Some of these same researchers [3537] have also found torsion fatigue testing as an effective tool to rapidly ascertain material specific parameters in order to calibrate fatigue damage evolution. This approach carefully follows findings from Littman [7], Lundberg and Palmgren [25] and others who have all indicated that the RCF failure stress is the subsurface shear reversal. Torsion S-N experiments assess the material’s fatigue performance against the RCF failure stress. Sadeghi and coworkers [3544] have been successful in obtaining estimates to fatigue life and spalling profiles that compare well with experimental results by utilizing these torsion-derived material constants within a finite element (FE) framework that also contains continuum damage mechanics (CDM). This framework also implements Voronoi tessellations to model the material microstructure. Throughout Sadeghi and coworker’s investigations, they have demonstrated the ability to capture the stochastic nature of RCF with these tessellations. Miller [45] showed a similar microstructural induced scatter phenomenon. A fundamental aspect of the CDM-FE approach, with torsion-derived material constants, is its independence from Weibull regression parameters produced from expensive and time-consuming fatigue test data.

The estimation of and enhancement to RCF life is well studied and of importance to the REB industry. This is particularly impactful for components in advanced aerospace applications, as these components need to operate for tens of billions of cycles before routine inspection is performed [46]. To address these needs, advancements have been made to manufacturing processes, lubricant formulations, as well as the material and heat treatment processes implemented [19,47,48]. Investigations by Padmanabhan and Wood [49], Wang et al. [50], Santos et al. [51], Nakane et al. [52], Khani Sanij et al. [53], and Kamiya et al. [54] have all shown that grain size (prior austenite) can be refined through repeated cycles of austenization and quenching, forging, or ring rolling [55] followed by tempering. The connection between prior austenite grain size and heat treatment process is important because it is well established [56] that mechanical properties are related to the grain size. The Hall–Petch [57,58] relation, which relates yield stress or hardness to the grain size, is provided in Eq. (1). Note, this relation holds for grains at micron scale, which are typically seen in conventional metals
$σy=σy0+kd$
(1)

The equation indicates that the yield strength (σy) varies with d−0.5, where d is the average grain size. While the Hall–Petch relation has been recognized for over half a century, the relation between grain size and general fatigue properties of a material is not as well established. Early studies by Thompson and Backofen [59] as well as Lukáš and Kunz [60] indicated that fatigue life of a polycrystalline metal under constant stress amplitude is sometimes increased with grain refinement. Later, an investigation by Li et al. [61] demonstrated that the type of constant stress amplitude loading will produce different fatigue responses as grain size is refined. As grain size was refined and the fatigue specimens were subjected to tension–compression (T-C) cyclic loading, fatigue improvement was observed in the low cycle fatigue (LCF); however, little improvement was noticed in the high cycle fatigue (HCF) regime causing the various grain size stress-life (S-N) curves to converge to the same value. The grain size variants of copper specimens were also subjected to cyclic torsion loading, where, a similar trend of life improvement was observed in the LCF regime. Interestingly, the S-N torsion curves did not converge toward a single value in the HCF regime, as was the case in T-C. The S-N curves for torsion remained relatively parallel, indicating that the improvement of the torsion fatigue strength seen in the LCF regime is also present into the HCF regime.

Over the past half century, investigations into the effect of grain size on rolling contact fatigue have been prevalent. Early studies by Santos et al. [51] and Stickels [62] showed the propensity of finer prior austenite grained materials to have enhanced RCF life characteristics. Additional research by Park et al. [63], Ooki [64], Lee et al. [65], Ghodrati et al. [66] and Cao et al. [67] supported the findings of Santos et al. and Stickels by quantifying the RCF life improvement between two grain sizes. Additionally, these researchers indicated that reliability (Weibull slope) remains relatively unchanged as grain size is refined. Although many investigations have explored the effect of grain size on RCF, there has not been a comprehensive computational study quantifying life improvement over the range of high-carbon anti-friction bearing steel grain sizes for REB application. According to ASTM standard A295 [68], high-carbon anti-friction bearing steels, such as AISI 52100, should have a grain structure that is ASTM 8 or finer (i.e., ASTM 8, 9, 10, etc.). ASTM grain size 8 or finer corresponds to a mean grain diameter that is 22.5 microns or finer, per ASTM E112 [69].

Presented in this investigation is finite element model (FEM) used to study the effects of grain refinement on lubricated nonconformal contacts like REBs. The FE model utilizes continuum damage mechanics to assist the prediction RCF life by simulating the deterioration of material and related said deterioration to a quantity of stress cycles. Open literature torsion S-N data of the same material at three grain sizes established the critical parameters (σr, m) used in the CDM calculations. The open literature torsion S-N data detailed that the resistance stress (σr) possesses a negatively correlated linear relationship with grain diameter. Thus, a larger resistance stress is present at finer grains. The other CDM parameter (m) displayed no relation to grain diameter; hence, the mean of the three S-N curves was used in the CDM calculations. Material microstructures were constructed using the Voronoi tessellation process, given mean grain diameter. In this investigation, the effect of mean grain diameter (d) and contact pressure (Ph) were explored to quantify the effects of grain refinement on rolling contact fatigue.

## 2 Modeling Approach

### 2.1 Continuum Damage Mechanics.

Material degradation is characterized in CDM [31] through an internal state variable D. This state variable is defined as
$D=AdA$
(2)
In Eq. (2), the internal state variable D is the ratio between the damaged area Ad of an arbitrary plane in a material (typically represented as voids) and the total (damaged and undamaged) cross-sectional area A of the same plane of material. The damage state variable, D, can take values between 0 and 1, where 0 represents pristine material and 1 fully damaged material. When a material is assumed to be isotropic, the damage state variable reduces from a tensor to a scalar [38,39,70], thereby constructing the following CDM constitutive relationship:
$σij=Cijkl(1−D)εkl$
(3)
In addition to the stress–strain constitutive equation, another function is required to determine the rate at which damage is occurring within the material. This investigation implemented a modified version of Xiao et al.’s [33] HCF damage evolution equation. Shear reversal was considered to be the RCF failure stress [7,25,71] and allowed for the evolution of damage to be expressed as
$(dDdN)shear=[Δτσr(1−D)]m$
(4)
σr and m are the CDM critical parameters and are derived from torsion S-N data. This is accomplished by (1) rearranging Basquin’s [72] law (Eq. (5)) for the number of cycles to failure and (2) integrating the damage evolution rate (Eq. (4)) for the number of cycles
$Δτ2=σf(Nf)a⇒Nf=(2σfΔτ)−1a$
(5)
$∫0NfdN=∫01[σr(1−D)Δτ]mdD⇒Nf=1m+1(σrΔτ)m$
(6)
Equations (5) and (6) are equated and the following relationships are established:
$m=−1a$
(7)
$σr=2σf(1−1a)−a$
(8)

Previous models implementing torsion-derived material constants into the damage rate equation (Eq. (4)) have proven to be successful in predicting RCF life. These models used constant values of σr and m derived from torsion S-N data from either Styri [73] or Shimizu et al. [74]. In this investigation however, this approach has been modified because torsion S-N data from Bomidi et al. [75] indicates that σr is a function of grain diameter.

### 2.2 Fatigue Resistance and Grain Size.

It is well established that various material and mechanical properties can be related to the grain size. Investigations by Lee et al. [65] as well as others have demonstrated that in constant stress amplitude experiments (1) the fatigue improvement is relatively parallel (e.g., fatigue improvement present in LCF regime persists into HCF regime) as grain size is refined and (2) fatigue life is enhanced by grain refinement. Utilizing Basquin’s law (Eq. (5)) in its linearized form facilitates the quantification of these qualitative observations. Taking the logarithm of both sides of Eq. (5) yields the linearized form of Basquin’s law:
$log(Δσ2)=a⋅log(Nf)+log(σf)$
(9)

The first of the aforementioned two qualitative observations suggests fatigue improvement is relatively parallel while grain size is refined. This observation indicates that the slope term (a) in Eq. (9) is impervious to changes in grain diameter. The second observation indicates that the material constant σf is a strong function of grain diameter and increases in magnitude as grain size is refined. It was demonstrated in Eqs. (7) and (8) that the Damage law’s (Eq. (4)) material constants (σr and m) are a function of both σf and a. Therefore, to effectively calibrate the CDM-FE model to investigate the effect of grain size on RCF, torsional fatigue data of various grain sizes for the same material must be used.

For this investigation, AISI 52100 bearing steel torsional fatigue testing conducted by Bomidi et al. [75] was used to implement the high cycle fatigue damage evolution equation (Eq. (4)) as a function of grain size. Bomidi et al. [75] investigated a number of bearing steels, both through and case hardened, and quantified the stress-life (S-N) behavior for each material. For the purposes of this investigation, steels A, C, and D were used since they were all AISI 52100 through hardened bearing steels, but had different mean prior austenite grain sizes. Figure 1 (Bomidi et al. [75]) presents the grain structure of steel C. The grain size measurements performed in Bomidi et al. [75] were in accordance to ASTM E112 [69].

Fig. 1
Fig. 1
Close modal
Torsion S-N data from Bomidi et al. [75] is shown in Fig. 2. Of the three materials used in this investigation (A, C, D), steel A has the finest mean grain diameter, followed by steel C and lastly by steel D. It was observed from the three S-N curves in Fig. 2 that the power law exponential constant, or slope term, (a) does not exhibit a noticeable relation with grain size. Therefore, the average value from the three torsion S-N curves, −0.104, was used. The σf term however, exhibited a negatively correlated, linear relationship with grain diameter and is outlined in Eq. (10):
$σf=2.850−0.029⋅d$
(10)
Fig. 2
Fig. 2
Close modal
Utilizing Eqs. (7) and (8), the grain size dependent CDM critical parameters were established for this investigation
$m=9.588$
(11)
$σr=7.291−0.074⋅d$
(12)
The yield strength (σy) of the three materials, which was determined using the material’s Vickers hardness data in concert with Pavlina and Van Tyne’s [76] linear correlation between yield strength and Vickers hardness. It was observed that the yield strengths of the three materials followed the Hall–Petch relation (∝ d−0.5) with respect to grain diameter. This relation is given by
$σy=0.767d+1.731$
(13)

A summary of the material parameters used in this investigation’s grain size dependent continuum damage mechanics FE model can be found in Table 1.

Table 1

Material damage parameters for AISI 52100 steel evaluated using Torsion SN data [75]

ParameterValue
Half-contact width, b (μm)100
Elastic modulus (GPa)210
Poisson’s ratio0.3
σr (GPa)7.291 − 0.074 · d
m9.588
ParameterValue
Half-contact width, b (μm)100
Elastic modulus (GPa)210
Poisson’s ratio0.3
σr (GPa)7.291 − 0.074 · d
m9.588

### 2.3 Modeling of Material Microstructure.

To model the microstructure of a single-phase polycrystalline material, this investigation adopted the Voronoi tessellations approach. Voronoi tessellations were used because (1) they are quantitatively similar to grain assemblies [77] and (2) provide an approach to capture the effect of microstructural features on fatigue life scatter, fatigue damage initiation, and propagation [7880]. Voronoi cells, used to represent material grains, were constructed as the result of the partition of a Euclidean space into a series of regions. A fundamental requirement of Voronoi tessellations is the enforcement that within a particular region, all points must be closer to said region’s seed point than any other region’s seed point [81]. In this investigation, the edges of the Voronoi cells are recognized to represent the microstructure’s grain boundaries (e.g., weak planes). Thus, the cell edges serve as the location where cracks form (intergranular failure) within the microstructure as a result of damage accumulation.

In this investigation a line contact is studied. A line contact can be approximated as an equivalent contact between two cylinders, which can further be represented as a semi-infinite half-space with an applied Hertzian load. Figure 3 presents a visual representation of the Hertzian load applied on said semi-infinite half-space and provides the general structure of the domains developed for this investigation. Similar to Lorenz et al. [8,43], the domains used in this investigation have Voronoi, coarse triangular, and infinite element regions. The Voronoi cells located within the Voronoi region were discretized in accordance with the centroidal discretization approach using plane strain triangular finite elements. To simulate the stochastic nature of rolling contact fatigue and achieve the objectives of this study, series of unique Voronoi microstructures were constructed at eight mean grain diameters in accordance with ASTM E112 [69]. ASTM A295 [68] outlines that the mean grain diameter of high-carbon anti-friction bearing steel should be ASTM 8 (22.5 μm) or finer [82,83]. Hence, the eight mean grain diameters ranged from 22.5 μm to 5.0 μm at 2.5 μm increments. This range of mean grain diameters corresponds to a range of ASTM grain diameters from ASTM 8 to nearly ASTM 12.5. Figure 4 provides an example of a Voronoi tessellation at each mean grain diameter investigated.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

Figure 3 also identifies the microstructure’s critically stressed region during a rolling pass, the representative volume element (RVE) [39,43]. The RVE is located within Voronoi region and similar to Ref. [37] for computational efficiency purposes, −0.5b to 0.5b in the x direction and 0 to −b in the y direction was the location of interest within the microstructure. Note, b is the Hertzian half-contact width. A coarsening mesh comprised the same type of elements found in the Voronoi region bordered the Voronoi region, with infinite elements surrounding the coarsening mesh.

### 2.4 Fatigue Damage Model.

Rolling contact fatigue loading was simulated by discretely passing a Hertzian pressure distribution and shear surface traction across the surface of a Voronoi microstructure in 5 μm increments similar to Refs. [8,43]. The 5 μm increment offered good resolution of the subsurface stresses, especially at the finest grain diameter of 5 μm, while minimizing computational effort. The condition of a well lubricated contact was simulated through the application of a surface shear stress in addition to the Hertzian contact pressure. The surface shear stress was proportional to the applied contact pressure through a friction coefficient (μ) of 0.05 [84,85]
$q(x)=μ⋅p(x)$
(14)

The RCF life requirements of critical tribological components, like REBs, can range from 106 to 1011 cycles [46,48] depending on application. As a result of operating in the gigacycle fatigue regime, it is inefficient to simulate every fatigue cycle. To address this computational issue, Lemaitre’s [31] jump-in-cycle method was implemented. The jump-in-cycle approach assumes that over a number of cycles (Ni), loading to be piecewise periodic and constant. During that number of cycles, a known value of damage $Dji$ exists at every boundary within the microstructure. The index i indicates which block of cycles or loading pass, whereas j can take values of one to the number of elements or boundaries within the domain. For a given loading block, the FE model is used to determine the range shear of shear stress, $(Δτ)ji$, acting along each boundary. Thus, the damage rate for all elements within the domain for loading pass “i” is

$(dDdN)ji=[(Δτ)jiσr(1−Dji)]m$
(15)
Lemaitre’s jump-in-cycle method assumes a constant damage increment (ΔD) over a block of cycles. In this investigation ΔD was set to 0.2, since other researchers [23,36,37,86] have shown that little improvements to accuracy are obtained, yet significant computational costs are incurred, when ΔD is reduced to 0.1 or 0.05. Therefore, the block of cycles (Ni) can be computed as
$ΔNi=ΔD(dDdN)crit$
(16)
with
$(dDdN)crit=max(dDdN)ji$
(17)
The computed block of cycles is then multiplied by each element’s damage evolution rate to account for the damage accumulated by the RCF process
$Dji+1=Dji+(dDdN)ji⋅ΔNi$
(18)

This process is repeated, new stress fields are determined, and damage calculations are performed. After each block of cycles, every element within the RVE is reviewed, and once an element reaches a critical level of damage [87], a crack is nucleated at the Voronoi boundary via the node release (NR) algorithm. Please refer to Refs. [23,36,37,86] for details of the NR algorithm. The repeated exposure to the RCF stresses will promote crack growth, where eventually, a dominant crack will form. Once a surface crack is present, failure is considered and the sum of each loading block’s number of cycles constitute the fatigue life for the given microstructure. Figure 5 details the progression of crack nucleation and propagation within the CDM-FE RCF simulation. Lastly, in order to capture the scatter intrinsic to RCF, the jump-in-cycle method is performed on 33 unique Voronoi microstructures per mean grain diameter. Weibull analysis on the stochastic fatigue lives characterizes the RCF performance.

Fig. 5
Fig. 5
Close modal

## 3 Results and Discussion

Before the CDM-FE model simulated the fatigue performance across various grain size and contact pressures, an initial RCF study was performed using conditions commonly found in literature. These conditions included a half-contact width of 100 µm, a mean grain diameter of 10 µm (typical of air melt 52100 bearing steel), and a maximum Hertzian pressure of 1.0 GPa. The RCF lives of 33 unique microstructures subjected to the above conditions were determined, Weibull regression was performed, and the results were compared with empirical and computational models from literature. Figure 6 presents the fatigue life dispersion of various life prediction models and the current model. Figure 6 and Table 2 demonstrate that the fatigue prediction of the various life prediction models and current model compare well. Furthermore, Table 2, which details the Weibull parameters (β, η) for each model (i.e., current, Lundberg-Palmgren, etc.), illustrates that the current model’s Weibull slope or scatter (β) corroborates well with experiments [88,89].

Fig. 6
Fig. 6
Close modal
Table 2

Weibull Slope and bearing life for various RCF models (Ph = 1.0 GPa)

SourceWeibull Slope, βBearing life, L10 (cycles)
Current model3.262.70 · 1010
Lundberg and Palmgren [25]1.140.86 · 1010
Raje et al. [35]1.721.06 · 1010
Lorenz et al. [43]3.203.08 · 1010
Harris and Barnsbya [88]0.51 ≤ β ≤ 5.7
Harris and Kotzalasa [89]0.7 ≤ β ≤ 3.5
SourceWeibull Slope, βBearing life, L10 (cycles)
Current model3.262.70 · 1010
Lundberg and Palmgren [25]1.140.86 · 1010
Raje et al. [35]1.721.06 · 1010
Lorenz et al. [43]3.203.08 · 1010
Harris and Barnsbya [88]0.51 ≤ β ≤ 5.7
Harris and Kotzalasa [89]0.7 ≤ β ≤ 3.5
a

Experimental data.

### 3.1 Rolling Contact Fatigue Life Simulations.

Once the efficacy of CDM-FE model was demonstrated, the model was used to study the effect of grain refinement on rolling contact fatigue performance. To achieve this objective, critical simulation parameters, summarized in Table 3, were established. The selection of the eight grain sizes was guided by ASTM A295 [68], whereas the three contact pressures were selected based on typical operating conditions for bearings in engineering applications [48,90].

Table 3

Simulation parameters

ParameterValue
Grain diameter, d (μm)22.5, 20.0, 17.5, 15.0, 12.5, 10.0, 7.5, 5.0
Contact pressure, P (GPa)1.0, 1.5, 2.0
Number of Voronoi domains per mean grain diameter33
ParameterValue
Grain diameter, d (μm)22.5, 20.0, 17.5, 15.0, 12.5, 10.0, 7.5, 5.0
Contact pressure, P (GPa)1.0, 1.5, 2.0
Number of Voronoi domains per mean grain diameter33

Figures 79 present the Weibull distributions, for the variety of grain size and contact pressure studied in this investigation. A summary of the Weibull parameters (β, η) is provided in Table 4. It is clear that for all contact pressures, fatigue life improves as grain size is refined. Equations (15) through (17) offer one explanation to this observation. As discussed in this investigation, the resistance stress (σr) has a linear relation with grain size and increases in magnitude as grain size decreases. As grain size is refined and σr increases, the magnitude of the quotient between shear stress reversal and resistance stress in Eq. (15) decreases. Since subsurface stress, damage exponent (m), and the damage state variable are not a function of grain size, a smaller damage evolution rate is calculated at smaller prior austenite grain sizes. Hence, a smaller critical rate is identified leading to an increase in life prediction (Eq. (16)).

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Table 4

Summary of Weibull parameters

Grain diameter, d (μm)Weibull Slope, βWeibull Scale, η
1.0 GPa1.5 GPa2.0 GPa1.0 GPa1.5 GPa2.0 GPa
22.52.612.342.210.64 · 10100.14 · 1090.09 · 108
20.01.771.751.720.94 · 10100.19 · 1090.13 · 108
17.52.932.722.511.29 · 10100.28 · 1090.16 · 108
15.02.852.742.902.09 · 10100.45 · 1090.28 · 108
12.52.862.762.733.62 · 10100.81 · 1090.53 · 108
10.03.262.902.775.38 · 10101.27 · 1090.84 · 108
7.53.153.293.078.55 · 10102.24 · 1091.48 · 108
5.04.394.204.0812.7 · 10102.96 · 1092.02 · 108
Grain diameter, d (μm)Weibull Slope, βWeibull Scale, η
1.0 GPa1.5 GPa2.0 GPa1.0 GPa1.5 GPa2.0 GPa
22.52.612.342.210.64 · 10100.14 · 1090.09 · 108
20.01.771.751.720.94 · 10100.19 · 1090.13 · 108
17.52.932.722.511.29 · 10100.28 · 1090.16 · 108
15.02.852.742.902.09 · 10100.45 · 1090.28 · 108
12.52.862.762.733.62 · 10100.81 · 1090.53 · 108
10.03.262.902.775.38 · 10101.27 · 1090.84 · 108
7.53.153.293.078.55 · 10102.24 · 1091.48 · 108
5.04.394.204.0812.7 · 10102.96 · 1092.02 · 108

In addition to the improved life predictions due to the grain size dependent resistance stress, the ratio between propagation and initiation life is affected by prior austenite grain refinement. According to Longching et al. [91], the time necessary for the first crack to appear in 52100 bearing steel specimens represented approximately 10% of total life. It is recognized that the assumption by Longching et al. [91] is not a general statement as Beheshti and Khonsari [92] point out, noting different factors (i.e., loading condition, material properties, etc.) may influence this phenomenon. However, Beheshti and Khonsari further mention that the propagation life percentage can sometimes approach 90% total life and is seen as an important phase in the fatigue process [92]. Initiation life in the CDM-FE model is defined as the number of cycles before the first crack appears in the microstructure (Fig. 5). Propagation life on the other hand is the cycle count required for the crack to reach the surface. Note, due to construct of the CDM-FE model, multiple nucleation sites are permitted, and they are allowed to interact with each other throughout the duration of the simulation.

Figure 10 was prepared to illustrate the relation between initiation (solid color region) and propagation life (hatched region). For brevity, this relation is shown for the 10th, 50th, and 90th percentiles of failure for all eight grain diameters at 1.5 GPa Hertzian contact pressure. As illustrated in Fig. 10, the propagation life was the primary contributor to the simulated fatigue lives. Moreover, all cases presented in Fig. 10 exhibiting initiation lives that are greater than the 10% experimentally observed measure. Note, only the 90th percentile of life ratios for the 7.5 µm and 5.0 µm microstructures exhibited a total life to initiation life ratio that approached Longching et al.’s [91] 10% measure. As a result of the unique construction of each series of Voronoi domains per mean grain diameter, some microstructures were more susceptible than others to rapid propagation failure. This was also observed and discussed by Morris et al. [37]. Another contribution to the underprediction of the total life to initiation life ratio lies in the definition of failure within the context of the simulation. In the simulation, RCF life was considered the cycle count when a crack was first detected at the surface, not the point at which a spall appears. As noted by Morris et al. [37], this difference contributes to the underprediction of the total life to initiation life ratio. Nevertheless, in general, the relation between propagation to initiation life increases as grain size is refined. Physically more grain boundaries, which impedes the crack’s propagation, are required to fail before the crack reached the surface. Therefore, resistance is provided to the crack, thereby increasing the fatigue life of the material. This observation is consistent with Refs. [93,94].

Fig. 10
Fig. 10
Close modal

Grain size dependent CDM-FE simulations were run using a single core per simulation on an Intel® Xenon® Processor E5-2660 V3 with 64 GB of RAM and a processor speed of 2.6 GHz. Average run times for the eight mean grain diameters were 1.57 h (22.5 μm), 2.00 h (20 μm), 3.53 h (17.5 μm), 3.70 h (15 μm), 7.03 h (12.5 μm), 23.22 h (10 μm), 60.87 h (7.5 μm), and 139.35 h (5 μm). Note, there was not an appreciable difference in computational cost at different contact pressures; hence, the reported average value listed above represents the average across all three contact pressures at a single mean grain diameter.

### 3.2 Fatigue Life Equation for Grain Refinement.

The simulations from this investigation (Table 4) indicate that prior austenite grain refinement will enhance rolling contact fatigue performance in accordance with findings by Refs. [6267,95]. In order to provide meaningful explanation to RCF improvement produced by grain refinement, a careful examination of the Weibull parameters (β, η) was necessary.

Regression analysis on the Weibull scale or strength parameter (η) reveals a strong exponential relation with grain diameter. This relationship, illustrated in Fig. 11, is of the form:
$η=C1⋅e(C2⋅d)$
(19)
where the coefficients C1 and C2 for each contact pressure are tabulated in Table 5. Coefficient C2 does not follow a distinct trend with contact pressure; however, coefficient C1 exhibits sensitivity to contact pressure. In addition to the distinct exponential relation with grain diameter, the Weibull scale parameter (η) also follows the well-established [3537,66,96,97] power law relation with contact pressure, Fig. 12 and is of the form:
$η=C3⋅(Ph)C4$
(20)
Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
Table 5

Exponential grain diameter relation coefficients

Contact pressure (GPa)Coefficient C1 (millions of cycles)Coefficient C2
1.0307, 576−0.1753
1.57911−0.1849
2.0533−0.1864
Contact pressure (GPa)Coefficient C1 (millions of cycles)Coefficient C2
1.0307, 576−0.1753
1.57911−0.1849
2.0533−0.1864
The Weibull shape or life scatter parameter (β) systematically increases with decreasing grain size. No clear trend with contact pressure was identified, therefore, the following expression for Weibull slope was constructed:
$β=6.615⋅(d)−0.3271$
(21)

Figure 13 illustrates this relation (Eq. (21)) with the values of β that were obtained from the current investigation’s simulations (Table 4). Regardless of contact pressure or grain diameter, the Weibull slopes (β) from the current investigation reside within the experimental range of values recorded by Harris and Barnsby [88] and Harris and Kotzalas [89]. Therefore, the adequacy of this investigation’s grain size-dependent continuum damage mechanics framework to predict fatigue life and scatter is confirmed.

Fig. 13
Fig. 13
Close modal
Once the relations between the Weibull parameters (β, η) and the two study parameters (d, Ph) were established, it was desired to utilize the aforementioned relations to construct a general fatigue life equation capable of predicting fatigue performance and scatter of RCF under elastic fatigue conditions at various grain sizes. It was determined that the fatigue life equation would take the form of a two-parameter Weibull distribution. In order to develop the general predictive fatigue life equation, first, Eqs. (19) and (20) were combined to form a single equation for the Weibull scale parameter (η):
$η=C1⋅e(C2⋅d)⋅C3⋅(Ph)C4$
(22)
It was observed that coefficient C2 did not follow a distinct trend with pressure; hence, the average value of the three data sets is used and recognized as a material constant. This is in accordance with Zaretsky [83]. Similarly, the power law exponential coefficient, C4, did not exhibit correlation with grain diameter; thus, the average value of the eight data sets is used. Setting coefficients C2 and C4 to the average values of their respective sets of data facilitated the merger of coefficients C1 and C3 into a single coefficient C13. Once complete, the functions for β (Eq. (21)) and η (Eq. (22)) were supplied to the aforementioned two-parameter Weibull distribution and rearranged to arrive at the final form of the predictive grain size-dependent fatigue life equation:
$N=η⋅ln(1S)1β=C13⋅e(C2⋅d)⋅(Ph)C4⋅ln(1S)16.612⋅(d)−0.3271$
(23)
$N=312,563e0.1822⋅dPh9.1630⋅ln(1S)(0.1512⋅(d)0.3269)$
(24)

In Eq. (24), the number of cycles to failure (N) is recognized in units of millions of cycles (106), with grain diameter and contact pressure provided in units of micron (μm) and gigapascal (GPa), respectively. Zaretsky [83] formed a similar exponential life expression, but used hardness instead of mean grain diameter. Zaretsky’s exponential slope (C2) or material constant is approximately half the value observed in the current investigation.

Once the predictive life equation was established, it was used to quantify the RCF life for prior austenite grain sizes from three experimental investigations—Park et al. [63], Ooki [64], and Lee et al. [65]. Table 6 provides the respective investigation’s grain diameters, observed life improvement, and the predicted life improvement using the current investigation’s life equation (Eq. (24)). Table 6’s data are also presented graphical via Fig. 14, where, it is observed that current investigation’s life improvement predictions follow the experimentally observed life improvements. For Ooki’s [64] investigation specifically, which refined the mean prior austenite grain size from the typical mean grain diameter of air melt 52100 bearing steel, 10.5 μm, to 4.4 μm, the current investigation’s fatigue life equation predicted RCF improvement within 0.62% of Ooki’s [64] reported improvement.

Fig. 14
Fig. 14
Close modal
Table 6

Summary of RCF life improvement due to grain size refinement with experimental studies

InvestigationGrain diameter, d (μm)Experimental life improvementaPredicted life improvement (Eq. (24))
Park et al. [63]17.902.743.26
12.02
Ooki [64]10.503.623.64
4.40
Lee et al. [65]14.406.385.78
5.90
InvestigationGrain diameter, d (μm)Experimental life improvementaPredicted life improvement (Eq. (24))
Park et al. [63]17.902.743.26
12.02
Ooki [64]10.503.623.64
4.40
Lee et al. [65]14.406.385.78
5.90
a

Life improvement is the ratio of RCF life at the fine grain size to the RCF life at the coarse grain size.

It is imperative to recall that the presented grain size-dependent continuum damage mechanics framework is calibrated via torsion fatigue data and not full-scale bearing fatigue test data. As has been demonstrated by various researchers [29,30], full-scale bearing test results can be time consuming, especially compared with the time needed to evaluate a material under torsional fatigue [3537]. As demonstrated by the life improvement comparisons (Fig. 14), the grain size-dependent CDM-FE model is capable of simulating fatigue life improvements that are similar to existing experimental results all the while capturing the stochastic phenomenon of rolling contact fatigue. Thus, the utility of the grain size-dependent CDM framework, with random Voronoi microstructures and torsion-derived material constants, is the sovereignty from the expensive REB fatigue data’s Weibull parameters to evaluate the effect of grain refinement of RCF performance.

## 4 Summary of Results

In this investigation, a grain size-dependent CDM framework was incorporated into a FEM to investigate the effect of grain refinement on the RCF performance of REBs. Continuum damage mechanics simulated material deterioration as a function of cycle, thereby enabling the calculation of RCF life. Grain size-dependent CDM critical material parameters (σr, m) were established from open literature torsion S-N data. The S-N data used was for the same material of three different grain sizes. The material constant m did not display a relation with grain size, therefore, the average value from the S-N curves was used in this investigation. The resistance stress (σr) demonstrated a negatively correlated linear relation with grain size; thus, a higher resistance stress was observed at smaller mean grain diameters. Material microstructure was modeled using Voronoi tessellations. A series of unique material microstructures were constructed at eight mean grain diameters, then finite element simulations were conducted at three contact pressures. These simulations determined the RCF life for each set of microstructures of a given mean grain diameter and contact pressure. The following results were obtained:

1. RCF life displayed an inclination to improve as grain size is refined, specifically, the Weibull scale parameter (η) exhibited an exponential relation with grain diameter. Furthermore, the observed life improvements derived from the RCF grain refinement simulations exhibited good corroboration with existing experimental results found in open literature.

2. The simulation results enabled the construction of a predictive fatigue life equation that was as a function of the relations between grain size, contact pressure, and the Weibull shape (β) and scale parameters (η). This predictive life equation was then used to evaluate the stochastic RCF performance based on grain diameter and contact pressure of nonconformal Hertzian contacts.

## Acknowledgment

This research was funded by the Air Force Research Laboratory (AFRL), performed at the Mechanical Engineering Tribology Lab (METL), Purdue University, West Lafayette, IN, under U.S. Air Force Contract No. FA8650-14-D-2348 and has received approval for public release (AFRL-2021-1096). The authors would like to acknowledge the support by Mr. Kevin Thompson, AFRL, for guidance and program management oversight. In addition to the support by UES Inc. and the AFRL at WPAFB, the authors would like to express their appreciation to Cummins for their support of this project.

## Conflict of Interest

There are no conflicts of interest.

## Nomenclature

a =

shear reversal power law fit exponent

b =

Hertzian half-contact width (μm)

d =

mean grain diameter (μm)

k =

Hall–Petch constant; strengthening coefficient (Pa · m0.5)

m =

material damage exponent

p =

pressure (Pa)

q =

surface traction (Pa)

A =

sectional area

D =

state variable for damage

E =

Young’s modulus (Pa)

N =

number of rolling contact cycles

S =

survivability or reliability

damaged sectional area

Ci =

predictive life equation coefficients

Cijkl =

stiffness tensor (Pa)

L10 =

bearing life (90% reliability)

Nf =

number of cycles to failure

Ph =

Hertzian pressure (Pa)

$dDdN$ =

damage evolution rate

β =

Weibull scale parameter or slope

Δτ =

shear stress reversal (Pa)

ɛkl =

strain tensor

η =

Weibull characteristic life or shape parameter

μ =

surface friction coefficient

ν =

Poisson’s ratio

σf =

S-N (stress-life) power law material constant (Pa)

σij =

stress tensor (Pa)

σr =

resistance stress (Pa)

σy =

yield stress (Pa)

σy0 =

Hall–Petch constant; frictionless stress or flow stress of a single crystal (Pa)

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