Computational Approach to Improve Bearings by Residual Stresses Based on Their Required Bearing Fatigue Life

Roller bearings are typically made of 100Cr6 bearing steel (German standard corresponding to AISI52100). The investigated bearings of type NU206 have cylindrical rolling elements (rollers), which are held by a polyamide cage, Fig. 1 shows the bearing’s components. In roller bearings, the contact between raceway and rolling elements is exposed to cyclic load due to the rotational motion. Alternating stresses are induced in varying depths beneath the surface of the contacting partners. The depth of the highest load induced stresses can be calculated according to the Hertzian contact theory including the approaches of Karas [1] and Föppl [2]. The normal loads onto the surface cause changes of the microstructure and may trigger crack formation beneath the surface of the raceways in the depth of highest load induced stresses.

According to Voskamp, bearing fatigue life for roller bearings can be separated into three phases [3]:

•  The first phase, the so-called shakedown-phase. Within the first 1000 revolutions plastic micro deformation occurs on the raceway. These deformations lead to work-hardening of the material and a resulting decrease of the retained austenite in the martensitic matrix and a moderate change in compressive residual stresses. The influence depends on the level of the load.

• The second phase, the steady state phase, features no plastic deformations and no changes of the microstructure. The duration of the stationary phase depends on the applied stresses. For loads above the so-called ‘Shakedown Limit’ the third phase starts immediately, whereas for very low loads the bearing remains in the second phase forever and no failure due to fatigue takes place.

•  The third phase, the instability phase. The inner boundary layers, conditioned in the shakedown phase, lose the ability to carry the load elastically. A local change of the microstructure takes place. Due to the phase transformation and the complex residual stress state, the material is modified. The microplastic deformations transform into macroplastic deformations. With persistent cyclic load on the inner boundary layer, cracks grow to the surface and finally pittings occur on the raceway. The depth of these pittings is equal to the depth of the highest load induced stresses. In practical applications, this failure type (Fig. 2) on the surface is one of the main failure modes of bearings (under ideal lubrication conditions and avoidance of lubricant contamination).

To increase the duration of the steady state phase, residual stresses can be induced to the inner boundary zone in the depth of the highest load induced stresses. Thus, the shakedown phase can be forestalled. To achieve pre-induced residual stresses, bearing surfaces can be reinforced with a short period of overload in the shakedown-phase, as well as in the manufacturing process. To show this, Hacke subjected NU206 bearings to a two stage test [4]. In a first step, the tests were run at a higher load for 1.5 million revolutions. The test was then continued at normal load. The tests were performed under full film lubrication with a specific film thickness of l ≥ 2. It could be shown that the bearing fatigue life increased drastically. The manufacturing process can be used to induce residual stresses not only to the near surface zone but to the subsurface region, too. The mechanical surface modification process deep rolling is one effective possibility to induce the compressive residual stresses. Within the process, a hard ball is pressed on a surface of a rotating part. A hydraulic pressure of up to pw = 600 bar can be used for the process. Neubauer et al. investigated the influence of manufacturing processes and induced residual stresses on bearing fatigue life. A computational model was set up and compared to bearing fatigue life tests [5, 6]. For the application of bearings, it is of interest to define a desired bearing fatigue life and calculate the demanded surface properties. Thus, the manufacturing process can be selected to fulfil these specifications. The aim of this paper is to introduce an inverse computational model for this application. Therefore, the calculation of bearings fatigue life will be described shortly. As a second step, the inverse computational model will be presented. From the results of the calculation, a manufacturing process will be chosen to manufacture roller bearing inner rings. At the end, the resulting bearings fatigue life will be determined experimentally

The influence of the residual stresses on the bearing fatigue life can be understood by taking a deeper look on the computational approaches. Roller bearings are highly stressed due to the cyclic contact load. To calculate the bearing fatigue life, the theoretical approaches are based on the weakest link concept by Weibull [7]. Weibull developed the following equation:

S: Probability of survival

n(s): Material dependent scale parameter

V: Damage risk volume [mm³]

For homogeneous materials the following correlation has to be chosen [7]:

with su, so and m as experimentally detected constants, s is the applied stress.

Lundberg and Palmgren developed a model for the calculation of bearing fatigue life, based on Weibull [8]. They expected a failure underneath the bearings surface for high orthogonal stresses and for imperfections of the material. In contrast to Weibull, it is expected that cracks due to fatigue occur on a weak point below the surface, e.g. an inclusion. Such cracks may extend further to the surface and finally lead to pittings. The probability of survival can be calculated as a function of the number of revolutions, the maximal orthogonal stresses, the stressed volume and the depth of the highest load induced stress.

N: Number of load cycles

to: Maximal orthogonal shear stress [MPa]

zo: Depth of the max. orthogonal shear stress [mm]

c: Exponent for stress criterion

h: Factor in the bearing life equation

e: Exponent in the bearing life equation (standardized Weibull slope)

In the 1980s Ioannides and Harris introduced an advanced model for the calculation of bearing fatigue life based on Lundberg and Palmgren [9]. The key point for the model is a statistical connection between the probability of surviving and a stress dependent fatigue criterion si or ti , respectively, for a volume element of the bearings surface. The probability of surviving for a volume element of a bearing is described by the following equation:

A_i is an independent random variable and H(x) a heaviside function with

The resulting stress field is used as input for the fatigue life model of Ioannides et al. [10]. The equation is:

τ_i: Fatigue stress criterion

τ_u: Shear stress fatigue limit (Pa)

z’: Stress weighted depth (depth from the surface) (m)

h: Factor in the bearing life equation

In order to account for the depth of the maximum shear stress the weighted depth z’ is introduced:

The model of Ioannides et al. [10] includes a stress fatigue limit tu and a fatigue stress criterion. For the term ti as fatigue stress criterion the criterion of Dang-Van et al. [11] is selected:

τ_Omax: Max. shear stress (Pa)

k_hyd: Weighting factor for hydrostatic stresses

p′ _hyd : Hydrostatic stress including compressive stresses

The Dang-Van criterion includes the maximum orthogonal shear stress and the local hydrostatic pressure, p’_hyd, corrected for residual and hoop stress. With increased compressive residual stresses the hydrostatic pressure is increased, thus the critical fatigue stress is reduced. The critical fatigue stress is highest in the zone of the highest load induced stresses beneath the surface. Concludingly, by applying residual stresses in this depth, the computational approach predicts that the bearing fatigue life is influenced positively.

In order to assess the influence of the magnitude and depth of residual stresses on bearing fatigue life, Neubauer previously developed a computational method [5, 12]. A finite element model of a bearing inner ring was set up to compute the resulting three-dimensional stress field from the superposition of load stresses and residual stresses. Figure 3 depicts the flow chart of the computational model. The input variables are the external load and the residual stress depth profile. The external load is transformed into a Hertzian pressure distribution. The residual stress depth profiles are defined as initial stresses in the model. One result of the finite element (FE) model are the orthogonal shear stresses beneath the surface. The stresses are required for the forward calculation model based on Ioannides et al. [10], which is computed in MathWorks® Matlab. The method allows determining the resulting bearing fatigue life based on the initial residual stress state of the bearings surface.

The inverse model is based on an iterative reverse calculation of the bearing fatigue life model of Ioannides et al. [10]. It is set up to determine the required residual stress state for reaching a specified bearing fatigue life. The inputs into the computational model are the stresses on the surface, which are simulated based on the load and geometry of the contact between roller and bearing surface. The load and basic rating life L10 (bearing life that 90% of a sufficiently large group of identical bearings reach or exceed before fatigue appears). Output is the required residual stress state in the bearing surface to achieve a bearing fatigue life as required for a given application. The flow chart of the inverse model is shown in Figure 4.

The calculation is performed in Matlab, the flow chart is shown in Figure 5. For the model, the Dang-Van criterion as well as the approach of Ioannides and Harris are used. Due to the approach including a heaviside function the intrinsic stresses cannot be calculated directly, see eqn (9).

In a first step of the calculation, the intrinsic stresses are generated for one point randomly based on previous results of the forward calculation method by Neubauer. The early calculations proved a dependency of the residual stress formed in higher depth and achievable bearing fatigue life. Distribution functions of the residual stress depth profiles originating from the bearings surface are calculated based on the Matlab curve fitting tool.

To calculate the Dang-Van criterion the input based on the shear stress, hydrostatic pressure and residual stresses are used. As the approach of Ioannides and Harris considers that only stresses above the stress fatigue limit result in fatigue, the Dang-Van criterion is used for calculations. The model of Ioannides and Harris can be transformed by introducing two terms.

For the generation of the residual stresses both equations are calculated and balanced. The iterations have to be continued, until both terms converge. After finishing the iteration the resulting residual stress state is calculated and plotted. Figure 6 shows calculated residual stress states for pre-defined bearing fatigue life under a contact pressure of 3,000 MPa. The graphs illustrate the trend that for increasing bearing fatigue life the residual stresses have to be increased and induced into higher depths. For increased bearing rating life the value and depth of the compressive residual stresses have to be adjusted. Figure 7 depicts the calculated residual stress state for a contact pressure of 2,300 MPa and a pre-defined bearing rating life L10 = 1,750 h. Due to the lower contact pressure the depth of highest load induced stress is in lower depths compared to the graphs in Figure 6. The compressive residual stresses should be highest in a depth of approx. 150 µm.

To proof the presented model, machining tests were conducted. For the tests roller bearing inner rings type NU206 were machined by hard turning and deep rolling on a high precision lathe Hembrug Microturn 100. Within the experiments, the influence of process parameters on the residual stress depth profile was analysed. The residual stresses were measured with an X-ray diffractometer Seiffert XRD 3000 P using CrKa radiation. The experimental procedure and results are described in detail by Denkena et al. [13, 14].

The deep rolling process influences the residual stress state significantly. Figure 8 summarizes the experimental results. An increasing rolling pressure pw leads to higher amplitude and larger depth of the compressive stresses. The size of the rolling tool, dk, affects the resulting depth of the residual stress profile. The effect of the overlap factor u can be compared with the effect of the shake down phase in bearings. An increased overlap factor results in an increased number of rolling processes, which leads to higher compressive residual stresses. From the results, a regression model is generated to calculate the resulting subsurface properties by the processes hard turning and deep rolling. To achieve the residual stress state depicted in Figure 7 the process parameters summarized in Table 1 were chosen. The achieved residual stress state is shown in Figure 9.

To verify that with the chosen residual stress state resulting from the computational model the requested bearing fatigue life is achieved, experiments were conducted. Bearings without and with beneficial residual stress states were taken for the tests. Standard NU206 bearing inner rings, featuring no significant residual stresses in higher depth, were compared to bearings with a beneficial residual stress state induced by the manufacturing process. By increasing the pressure for the deep rolling process, the magnitude of the residual stresses can be enhanced whereas by increasing the diameter of the rolling tool the stress field is extended into greater depths. Modified bearing inner rings and standard parts for the rollers and outer rings were subjected to the tests.

The evaluation of the experimental bearing endurance time was performed at six identical 4-bearing endurance test benches. Each of the test rigs allows for running four equal roller bearings of type NU206 C3 at the same time. Thus, higher statistic coverage can be achieved faster in the experiments. All bearings are exposed to the same radial load in the test rig. The applied load is measured with a load cell during testing. The tests were performed under full film lubrication with a specific film thickness of l ≥ 2. The bearings are supplied with tempered oil (at 60°C). A synthetic oil was chosen with a kinematic viscosity of h40 = 68 mm2/s. Bearing failures due to pitting are detected with a condition monitoring system. The radial load was chosen with C/P = 4 equal to a maximum Hertzian contact pressure of pmax = 2,300 MPa in the bearing. The rotational speed was 4,050 min−1. The tests were performed according to the sudden death test principle. When the fatigue life of one of the bearings is reached and a pitting has formed, the other three bearings are suspended. Figure 10 shows a schematic of the test rig.

After failure, the bearings were disassembled and analysed. For the statistical evaluation, the maximum-likelihood method is applied. Thus, the bearings without failure are accounted for as “suspended”. The standard honed bearings reached an L10 of 274 h, and an L50 of about 7,000 h. In comparison, for bearings with pre-induced residual stresses on the inner rings, an L10 of 1,924 h and an L50 of 8,446 h could be observed. The experimentally validated bearing life is close to the calculated L10 of 1,750 h (Figure 7).

For these bearings, the residual stress state remained nearly constant over the test duration. After an ultra-long running time of 378 Mio revolutions, the residual stresses start to decrease in the zone of the highest load induced stresses. Thus, in accordance with Voskamp [3], the breakdown phase is forestalled and the duration of the steady state phase could be increased.

Applying residual stresses on bearing surfaces offers the possibility to increase the bearing fatigue life drastically. To equip these bearings with the desired surface properties, a calculation procedure was established. The heaviside function of the bearing fatigue life model by Ioannides and Harris requires an iterative approach to determine the optimum residual stresses based on the Dang-Van criterion. The calculated residual stress state was compared to measurements showing a good agreement. The inverse computational model allows for choosing the proper manufacturing processes particularly deep rolling processes delivering the desired stress state. Thus, the manufacturing effort can be adjusted according to the desired bearing properties. The bearings can be equipped with improved properties and machines designed regarding improved resource efficiency. Especially, for applications with complicated and expensive replacement of failed bearings (such as windmills and aerospace), such approaches are valuable.