Aus Wissenschaft und Forschung 32 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 Fatigue life prediction of rolling bearings based on damage accumulation considering residual stresses Jae-il Hwang, Gerhard Poll* Dieser Beitrag wurde im Rahmen der 64. Tribologie-Fachtagung 2023 der Gesellschaft für Tribologie (GfT) eingereicht. Die Bestimmung der Ermüdungslebensdauer von Wälzlagern wird empfohlen, unter Verwendung der in ISO 281 definierten Standardberechnungsmodelle durchgeführt zu werden. Diese Modelle sind zuverlässig für rotierende Anwendungen und werden kontinuierlich verbessert. Jedoch, wenn es um Wälzlager geht, die komplexen Belastungsbedingungen ausgesetzt sind, einschließlich oszillierender Bewegungen, wie sie bei Rotorblattlagern von Windturbinen auftreten, fehlt bisher ein validiertes Berechnungsmodell. Im Bereich der Strukturmechanik basiert die Beurteilung der Ermüdungslebensdauer von Stahlbauteilen auf Wöhler Kurven. Variable Betriebslasten werden als Lastkollektiv betrachtet, und die Lastzyklen bis zum Versagen N unter gegebenen Betriebsbedingungen werden unter Anwendung der linearen Schadensakkumulation ermittelt. Diese Arbeit stellt ein neues Modell vor, das die lineare Schädigungsregel mit etablierten Lagertheorien integriert. Alle Wälzkontakte in einem internen Spannungszyklus werden als internes Lastkollektiv betrachtet. Die Bewertung des internen Kollektivs erfolgt anhand von Wöhler Kurven für Torsionsbelastungen. Die Kurzfassung Auswahl der Wöhler Kurve basiert auf dem Spannungskriterium τ o gemäß der Lundberg-Palmgren Theorie. Diese Arbeit umfasst eine schrittweise Anleitung zur Anwendung des neuen Modells zur Berechnung der Ermüdungslebensdauer, unter Verwendung des Zylinderrollenlagers NU 1006 als Referenz-Wälzlager. Die Ergebnisse dieser Studie zeigen eine positive Übereinstimmung zwischen der ermittelten Ermüdungslebensdauer mit dem neuen Modell und jenen, die gemäß ISO 281 ermittelt wurden. Darüber hinaus wird eine neue Methode eingeführt, um Eigenspannungen analytisch zu berücksichtigen, sofern Messwerte verfügbar sind. Basierend auf dieser Untersuchung kann bestätigt werden, dass das neue Modell das Potenzial hat, zuverlässige Ergebnisse zu liefern, ohne die oft in herkömmlichen Berechnungsmodellen erforderlichen Korrekturfaktoren zu benötigen. Schlüsselwörter Lagerermüdungslebensdauer, Wälzkontaktermüdung, oszillierendes Lager, Eigenspannung, Schadensakkumulation, Simple Link Concept The determination of the fatigue life of rolling bearings is recommended to be conducted using the standard calculation models defined in ISO 281. These models yield reliable results for rotating applications. The standard models are continuously refined to align with advancements in bearing production processes and material improvements. However, when dealing with rolling bearings exposed to complex load conditions including oscillatory movements, as seen in rotor blade bearings in wind turbines, a validated calculation model has yet to be established. In the domain of structural Abstract mechanics, fatigue life evaluation for steel components is based on S-N curves. Variable operational loads are treated as load collective, and the load cycle to failure N under given operational conditions is determined through the application of the linear damage rule. This paper introduces a novel model that integrates the linear damage rule with established conventional bearing theories. Within one internal stress cycle, all rolling contacts are regarded as an internal load collective. To evaluate the internal load collective, the stress state at * Dipl.-Ing. Jae-il Hwang and Prof.-Ing. Gerhard Poll Gottfried Wilhelm Leibniz University Hannover Institute of Machine Design and Tribology (IMKT) An der Universitaet 1, 30823 Garbsen TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 32 Introduction - state of the art The foundational mathematical formulation for predicting bearing fatigue life in the relation to maximum orthogonal shear stress τ o,max was introduced by Lundberg and Palmgren in 1947 [1], with subsequent development in 1952 [2]. Drawing from their full-scale fatigue test results, they proposed an additional relationship as follows: (1) where C r is the basic dynamic load rating, P eq is the equivalent bearing load, and p is the bearing life exponent. The Weibull exponent e, representing the slope of the Weibull plot with a 10 % probability of failure, the stress exponent c, signifying the slope of the S-N curve (also referred to as the Wöhler curve), and the depth exponent h, associated with the position z o of the maximum orthogonal shear stress, collectively contribute to the derivation of the bearing life exponent p. This model has been adopted by the International Organization for Standardization (ISO) and has developed into the established standard calculation methodology known as ISO 281. In addition, the German standard calculation method DIN 26 281 has been incorporated into ISO 281. In 1985, Ioannides and Harris [3] proposed the utilization of the fatigue limit stress by modifying the original formulation of Lundberg and Palmgren. Subsequently, the life adjustment factor a ISO was introduced into ISO 281, encompassing the influences of elastic-hydrostatic lubrication (EHL) and the fatigue limit stress related to bearing steel material properties. This led to the establishment of the current state of ISO 281. Unlike the bearing technology domain, fatigue life prediction for steel components in the field of structural mechanics often employs a linear damage accumulation model, originally proposed by Palmgren [4] and further developed by Miner [5]. In the context of high-cycle fatigue (HCF) in the field of structural mechanics, it is postulated that the commencement of early cracks is primarily linked to shear stresses, followed by progression = due to normal stresses that cause crack opening. Consequently, when assessing crack initiation in ductile metallic materials, the stress state is typically analyzed using the maximum normal stress criterion, the maximum shear stress criterion (Tresca theory [6]) or the maximum distortion energy criterion (Von Mises theory [7]), based on S-N curves. First concept of the S-N curve was introduced by Wöhler [8], featuring a single logarithmic scale graph depicting load cycles to failure contingent on external load amplitudes. Subsequently, Basquin [9] proposed plotting the S-N curve on a double logarithmic scale, allowing the Wöhler test results to be represented as a simplified straight-line approximation, referred to as the Basquin-line. The Basquin-line is expressed as: (Eq. 2) where σ a is the local stress amplitude, m is the slope of the Basquin-line, and N D is the load cycle related to the endurance limit stress σ u of the respective material. Below σ u , the material is believed to undergo infinite load cycles without fatigue failure. If the material is subjected to various load amplitudes, damage risk caused by different local stress amplitudes are accumulated linearly with respect to energy absorption at the loaded volume element: (Eq. 3) where n i is the number of the i-th given load cycle for the corresponding load amplitude σ a,i . Accordingly, the ratio of the load cycle n i to the load cycle to failure N i is considered as the partial damage risk of the loaded volume element. According to the Palmgren-Miner linear damage rule, the yielding is assumed to occur, when the sum of all partial damage risks reaches a certain value regarding material properties: (Eq. 4) where k pm is the yielding limit value that was originally suggested as 1 for the aluminium alloy by Miner [5]. The maximum number of load cycles to failure N p is = + + + = + + + = = Aus Wissenschaft und Forschung 33 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 each rolling contact is assessed based on the S-N curve determined in alternating torsional loads. The core rationale behind the selection of an S-N curve lies in the stress criterion τ o based on Lundberg-Palmgren theory, which is incorporated in the new model. This paper includes a comprehensive step-by-step procedure for applying the new model, utilizing the cylindrical roller bearing NU 1006 as a reference. The results of this study indicate a favorable agreement between the fatigue life obtained using the new model and those determined according to ISO 281. Furthermore, a new approach is introduced to analytically account for residual stresses when measured values are available. In this study, by applying this new approach, the results were observed to be in close agreement with the test results. Based on this investigation, it can be confirmed that the new model has the potential to provide reliable results without necessitating the bearing life exponent as well as the correction factors often required in conventional calculation models. Keywords Bearing Fatigue Life, Rolling Contact Fatigue, Oscillating Bearing, Residual Stress, Damage Accumulation, Simple Link Concept TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 33 multaneously, the rolling element rotates about the bearing axis with ω re , and also rotates about its own axis with ω b . This arrangement is depicted in Figure 1. The velocities of the inner and outer contact are respectively expressed as follows: (Eq. 6) (Eq. 7) where r re is the radius of the rolling element and r p is the pitch radius. Consequently, the following relationship between the rolling element and the inner and outer contacts is derived: (Eq. 8) In the case of a stationary outer ring, the rolling element velocity is equivalent to half of the inner ring velocity. Based on this relationship, the inner ring is assumed to rotate with a stationary outer ring from the initial position, which is illustrated in Figure 2.A. At this time, the first rolling contact occurs in the initial position. It should be noted that in this paper, “rolling contact” refers to the condition where the small stressed volume (SSV) is physically in contact with both the rolling element and the outer ring during rotation, with load transfer occurring only through this contact condition. Following the perspective of the Palmgren-Miner linear damage rule, the evaluation of energy resulting from diverse loads is exclusively feasible through accumulation within a single volume element. Thus, all discretized volume elements of the inner ring should be evaluated individually. Nonetheless, since the SSV on the inner ring is theoretically most loaded with the maximum load occurred in the initial position than other volume elements during rotating operation, using the linear damage accumulation rule may be appropriate by considering only the SSV. As the inner ring continues its rotation, a second rolling contact occurs at the critical angle, depicted in Figure 2.B. The critical angle θ crit is determined as: (Eq. 9) with (Eq. 10) where Z denotes the number of rolling elements. The plus sign is indicated for the inner ring and minus sign for the outer ring. Following one complete revolution of the inner ring, the position of the rolling element is displaced by a certain distance from the initial position of the SSV, as illustrated in Figure 2.C. This distance changes each time the inner ring rotates, and it returns to the initial position after a certain number of revolutions. In the reference bearing, the SSV was observed to = = = = + = ( + ) = ( ± ) = Aus Wissenschaft und Forschung 34 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 determined using the ratio of the collective load cycles to the sum of damage risks: (Eq. 5) where p denotes the probability of failure in this study. This approach has continuously evolved and has found widespread application in various metal fatigue analyses. This study introduces a new perspective on bearing fatigue life prediction and presents innovative solutions, which may be applied for any operating conditions regardless of bearing types. Kinematic of rolling bearings - introduction of internal stress cycle Within the proposed model in this paper, all stress states of the loaded volume at each rolling contact are evaluated, employing the S-N curve as described by Basquin to quantify partial damage risks. In this context, a comprehensive understanding of the kinematic of rolling bearings is necessary to ascertain the accurate contact condition. In this section, the cylindrical roller bearing NU 1006 with an outer diameter of 55 mm is employed as a case study to exemplify the process of determining the rolling contacts that occur during rotating operation. General assumptions are: • Neglecting dynamic effects caused by inertial and centrifugal forces • Constant contact angle α for the inner and outer rings • Occurrence of pure rolling without sliding between the bearing components • Neglecting deflections at the rolling contact Consider the inner ring of the reference bearing affixed to a rigid shaft rotating at an angular velocity of ω ic , and the outer ring rotating at an angular velocity of ω oc . Si- = = IC OC IC OC p IC re OC re Bearing axis Figure 1: Kinematic of a rotating bearing TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 34 cyclically return to its initial position. This periodic phenomenon is referred to as internal bearing behavior that lead to the internal stress cycle in this study. Figure 3 shows the distance variation as a function of the number of rolling contacts for the reference bearing. Each bar length denotes the distance from the initial position of the SSV to the respective rolling contact. In the reference bearing, it is clearly observed that the SSV returns to the initial position after every 10 revolutions of the inner ring (n cyc = 10 revolutions), which corresponds to 97 rolling contacts (n i = 97 rolling contacts). Moreover, this phenomenon was also observed to appear in other rotating bearings such as in an angular contact ball bearing 7208 with an outer diameter of 80 mm, as well as in a four-point rotor blade bearing with an outer diameter of approximately 2.4 meters, regardless external forces. New model based on damage accumulation - procedure for use In this section, the new calculation model is introduced for bearing fatigue life prediction. In the new model, the internal load collective is determined by evaluating all rolling contacts n i within the internal stress cycle n cyc Aus Wissenschaft und Forschung 35 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 First one revolution One internal stress cycle 97 50 i Rolling contact number for one single internal stress cycle cyc Figure 3: Periodical occurrence of the Internal stress cycle in the reference bearing NU 1006 due to the internal bearing dynamic behavior: n i = 97 contacts and n cyc = 10 revolutions; from [10] stationary A) Initial position rotating Small Stressed Volume 1 𝜃 crit B) First rolling contact 𝜔 oc = 0 ∆𝜑 C) Displacement of the contact position after one complete revolution 𝜔 ic Figure 2: Change in rolling contact positions associated with the small stress volume on the inner ring during rotating operation; A) initial position of the SSV with the maximum contact pressure, B) First rolling contact occurs after the critical angle, C) Occurrence of displacement of the contact position from the initial position of the small stressed volume to the next rolling contact after one complete revolution TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 35 correspond to the probability of failure for the calculated fatigue life. A detailed procedure for using the new model can be found in [10]. In this paper, supplementary clarifications are provided additionally. For a simplified approach to utilizing the new model for fatigue life prediction of the reference bearing, it is assumed that the bearing is not tilted during rotating operation, and the maximum contact pressure in every rolling contacts occur in the middle of the raceways. Step 0: Pre-condition The internal stress cycle must be found to determine the internal load collective. To proceed with the further calculations, values for n i and n cyc are required. Step 1: Determination of the load distribution in statically loaded bearings In the new model, the determination of the load distribution Q is essential. In this study, the calculation method recommended by DIN 26 281 is employed. For ball bearings, the change in contact angle due to external loads can be accounted for at this step. Subsequently, rolling element forces q i for all rolling contacts n i along the circumference of the respective ring within n cyc are established. The internal load collective is established using q i values obtained in this step. Step 2: Determination of the contact pressures For each rolling element force q i within the internal stress cycle, the contact pressure is determined based on the Hertzian theory [11] recommended within the new model. In order to enhance computational efficiency when dealing with numerous rolling contacts within n cyc , the simplified method proposed by Brewe and Hamrock [12] can be useful. Aus Wissenschaft und Forschung 36 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 due to the internal bearing dynamic behavior. The effective rolling element loads q i are depending on the load distribution Q, characterized by the load distribution factor ϵ. The loaded volume at the rolling contact is evaluated using orthogonal shear stress τ o considering the fatigue limit shear stress τ u , mainly based on the S-N curve. As per the Palmgren-Miner linear damage rule, every rolling contact in the internal load collective is regarded as a partial damage that contributes to the overall cumulative damage risk, denoted as D. Indeed, each rolling contact within the internal stress cycle is individually considered, taking into account the actual load occurrences in the SSV. Therefore, the new model incorporates the Lundberg-Palmgren theory associated with the stress criterion τ o , and the Ioannides-Harris theory related to τ u , into the framework of the damage accumulation theory. The evaluation of the stress distribution is conducted in a layered manner along the circumference of the respective ring with a new defined critical volume in Figure 4.B. The concept of the critical volume is introduced in alignment with the Lundberg-Palmgren original formulation, as shown in Figure 4.A. In fact, in rolling contact, stress values exceeding τ u appear below z o in the loaded volume. To address this, the new model introduces a redefined critical volume that is twice the size of the volume suggested in the Palmgren-Miner theory. In this model, fatigue is assumed to occur when D = 1. Hence, the premise is established that among the layers analyzed from the contact surface to the bearing core within the newly defined critical volume (as illustrated in Figure 4.B), fatigue occurs in the layer where the sum of damage risks D first attains the specified critical threshold. The probability of failure indicated by the S-N curve is assumed to 2 b A) Critical volume suggested by Lundberg and Palmgren 𝑧 o 𝑧 u raceway rolling direction 1 z-layer number 𝑗 … Orthogonal shear stress profile for each z-layer (case of a pure external radial force) … Evaluate up to j -th z-layer 𝑉 HP ~𝑎𝑧 u (2π𝑟 r ) with 𝑧 u = 2𝑧 o Stressed volume B) Critical volume proposed in the new model (depth 𝑧 u related to the fatigue limit stress 𝜏 u ) +𝜏 o −𝜏 o 0 𝑛 i 𝜏 da,1 … +𝜏 o −𝜏 o 0 𝑛 i … +𝜏 o −𝜏 o 0 𝑛 i … Figure 4: Definition of critical volume; A) critical volume proposed by Lundberg and Palmgren from [3], B) Suggestion of new critical volume V HP and the layer-wise evaluation of the stressed volume in the new model TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 36 Parameter Wert 10.1 10 8 load cycles 370 ~ 380 MPa Step 3: Determination of the stress distributions In this step, it is recommended to compute the stress state under a line contact according to Johnson’s approach [13], and under an elliptical contact according to Sackfield-Hills approach [14]. The contact pressure and stress distribution can also be determined using FEM. In both scenarios, employing interpolation technique can be advantageous. Step 4: Determination of partial damages using the S-N curve An S-N curve is employed to evaluate the stress state induced by the rolling element force q i with respect to the use of the Palmgren-Miner linear damage rule. In the new model, it is recommended to use the S-N curve of bearing steel 100Cr6 (AISI 52100) determined under alternating torsional load, as used in [10]. The primary rationale for choosing this specific S-N curve lies in the selection of the orthogonal shear stress as the stress criterion regarding the stress state in the rolling contact. Regarding the Basquin formula described in (Eq. 2), the S-N curve is characterized by three distinct parameters; m, τ u , and N D , the values of which are found in Table 1. where n s always corresponds to 1 indicating that this rolling contact has occurred once within the internal stress cycle. Therefore, the expression 1/ n i signifies the partial damage risk at one rolling contact as a proportion of all rolling contacts n i . It should be noted that the calculation of the partial damages in this step is only related to one single internal stress cycle. Step 5: Determination of the fatigue life of the rolling bearing All partial damages determined in one z-layer are accumulated linearly according to the Palmgren-Miner linear damage rule using (Eq. 3). Since the partial damages occurring only in one internal stress cycle are considered in the previous step, the repetition of the internal stress cycle is mathematically expressed as follows to obtain the sum of damage risk during entire operation: (Eq. 13) where j denotes the number of z-layers, and p signifies the probability of failure. As a result, the minimum value of the load cycles related to the repetition of the internal stress cycle along the z-layers will be represent the fatigue life of the respective rolling bearing. Micro cracks are assumed to be initiated at the depth of this layer due to varied orthogonal shear stress amplitudes during rotating operation. For varying loads with oscillating movements, define the total simulation duration as an internal stress cycle. Evaluate all volume elements on the raceway of both inner and outer rings to find the maximum loaded volume element, accounting for potential differences due to oscillation movements. In this way, damage risks at all volume elements of both rings can be determined, including the highest damage risk. Introduction of simple link concept - consideration of residual stresses In this study, the simple link concept is employed to incorporate residual stress analytically, building upon the general explanation presented in [10]. Further details and additional clarifications are provided in this paper. It should be noted that this approach is applied solely when the measured residual stress is available. According to the work of Voskamp [15], a stable phase emerges after a certain number of bearing revolutions (> 10 3 ~ 10 4 ). This perspective is also consistent within the realm of structural mechanics. For the purpose of the simple link concept, the measurement of the residual stress after this range may be sufficient. In contrast to other methods like Dang Van criterion [16], this approach assumes that the residual stress state σ rs does not precisely reflect the hydrostatic state, as the residual stress component in the z-direction is nearly zero. In general, the measured = , = Aus Wissenschaft und Forschung 37 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 Parameter Wert 10.1 10 8 load cycles 370 ~ 380 MPa Table 1: Parameter values for the chosen S-N curve The orthogonal stress flow at one rolling contact experiences maxima and minima values for each layer, as seen in Figure 4.B. Consequently, the SSV is presumed to be subject to loading due to the difference τ da = τ o,max - τ o,min ; between the maximum and minimum values of orthogonal shear stresses that appears differently for each z-layer. One circumferential volume layer resulting from rotations of the SSV is loaded by every rolling contact n i . For one rolling contact, N i is expressed by modifying (Eq. 2) as: (Eq. 11) Given that the S-N curve is described in terms of stress amplitude, a limit stress value multiplied by a factor of 2 is employed with respect to τ da . This calculation must be performed for all rolling contacts n i within n cyc , and then for all z-layers within the new defined critical volume. Each rolling contact will contribute to occur one partial damage value d i . Consequently, the partial damage can be determined by following relationship: (Eq. 12) = , = TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 37 (Eq. 17) Here, σ tot,3 is presumed to be identical to the total stress tensor in the x-direction since the contact width 2b is presumed to be equal in length regarding line contact (see Figure 4.A). As a result, the equivalent stress state including the residual stress is obtained using the determined principal stresses: (Eq. 18) By applying this equivalent stress state to the simple linked relationship of (Eq. 14), the orthogonal shear stress state is obtained, which may include the residual stress state. Comparison of the results - validation of the new model The fatigue test results of the reference bearing NU 1006 were provided by FVA project 866 I, involving over 200 bearings tested at IMKT. Additionally, the fatigue life of the reference bearing was calculated using Bearinx, a tool capable of providing the fatigue life based on ISO 281. In Figure 5, B 10 from the tests including L 10r and L 10mr from Bearinx are compared with the calculated results obtained using the new model. , = , = , Aus Wissenschaft und Forschung 38 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 residual stress state σ rs can be combined with the initial stress state σ ini resulting from the load that corresponds to the rolling element force within the rolling bearing to obtain total stress state σ tot . Analytical integrating residual stresses is challenging, especially in the bearing technology due to the stress criterion τ o not affecting the stress matrix σ tot . The simple link concept proposes a practical solution using the ratio of maximum Hertzian contact pressure to maximum orthogonal shear stress as well as to maximum equivalent stress. In the case of the reference bearing NU 1006 with a line contact, the maximum Hertzian contact pressure corresponds to 0.25 times the maximum orthogonal shear stress (p h,max = 0.25 τ o,max ) and corresponds to 0.57 times the maximum equivalent stress according to the Von Mises theory (p h,max = 0.57·σ eq,max ). Consequently, both relationships are linked to derive a new relationship: (Eq. 14) According to the Mohr’s circle, three principle stresses can be determined from the total stress state σ tot : = . = . . , = , , , , + , + , = , , , , + , (Eq. 15) (Eq. 16) , , + , , + , , Figure 5: Comparison of the results with 10 % probability of failure for the reference bearing; B 10 from the test results, N 10 obtained using the new model, N 10,rs obtained using the new model with the additional incorporation of the simple link concept, L 10 determined according to ISO 281; from [10] TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 38 This study confirms that the calculation methods outlined in ISO 281 provides acceptable results for predicting fatigue life in rotating bearings. The new model, utilizing a τ u -value of 380 MPa, closely aligns with L 10mr . With the utilization of the new model, the influence of fatigue limit stress on fatigue life can be observed clearly. It’s noteworthy that the test result curve B 10 doesn’t exhibit a linear trend. To investigate the influence of residual stresses on the fatigue life, the simple link concept was additionally applied with the measured values for two load levels (p h,max,ir of 3.8 GPa and 2.5 GPa). Interestingly, by additional applying the simple link concept, the results agree closely with B 10 , when using the limit stress value of 370 MPa. These results indicate that higher loads lead to increase residual stress levels, which lead to increasing the fatigue life, especially in the HCF regime (10.5 times longer for the high load and 1.6 times longer for the low load). It should be noted that the new model does not currently account for the influences of EHL and does not yet incorporate the influence of the size of loaded volume on fatigue life and probability of failure with respect to the stress criterion. Based on these research findings, it can be expected that the critical volume size of the reference bearing NU 1006 closely corresponds to the volume size loaded by external forces on the specimen employed to establish the S-N curve. Acknowledgments The results presented in this paper were obtained within the project of Design of Highly Loaded Slewing Bearings (HBDV) with the grant number of 01183488/ 1. The authors would like to thank the German Federal Ministry for Economic Affairs and Climate Action (BMWK) for the financial and organizational support of this project. The authors also are deeply grateful to the German Federation of Industrial Cooperative Research Associations (AiF) for providing the experimental results obtained within the project of FVA 866 I - Einfluss kurzfristiger Überlasten auf die Lebensdauer von Wälzlagern (grant number of 20733 N). Reference [1] Lundberg, G. and Palmgren, A. Dynamic Capacity of Rolling Bearings; Acta Polytechnic Mechanical Engineering Series, 1(3), Generalstabens Litografiska Anstalts Förlag, Stockholm, Sweden, 1947. [2] Lundberg, G. and Palmgren, A. Dynamic Capacity of Roller Bearings, Handlingar Proc., No. 210, The Royal Swedish Academy of Engineering Sciences, Stockholm, Sweden, 1952 [3] Ioannides, E. and Harris, T.A. A new fatigue life model for rolling bearings, Journal of Tribology, 107, pp. 367- 378, 1985. [4] Palmgren, A. Die Lebensdauer von Kugellagern (Life length of roller bearings or durability of ball bearings), Z. des. Vereines Deutsch. Ingenieure (ZVDI) 14, pp. 339- 341, 1924 [5] Miner, M. A. Cumulative damage in fatigue, J. Appl. Mech. 12, A159-A164, 1945 [6] Tresca, H. E. Sur l’ecoulement des corps solides soumis a de fortes pressions. rue de Seine-Saint-Germain, 10, près l’Institut: Imprimerie de Gauthier-Villars, successeur de Mallet-Bachelier, 1864 [7] Mises, R. v. Mechanik der festen Körper im plastischdeformablen Zustand, Nachrichten Ges. Wiss. Göttingen, Mathematisch-Physikalische Kl., pp. 582-592, 1913 [8] Wöhler, A. Über die Versuche zur Ermittlung der Festigkeit von Achsen, welche in den Werkstätten der Niederschlesisch-Märkischen Eisenbahn zu Frankfurt a. d. O. angestellt sind, Z. für Bauwes. 13, pp. 233-258, 1863 [9] Basquin, O. H. The exponential law of endurance tests, Proceedings-American Soc. Test. Mater. 10, pp. 625-630, 1910 [10] Hwang, J. and Poll, G. A new approach for the prediction of fatigue life in rolling bearings based on damage accumulation theory considering residual stresses, Frontiers in Manufacturing Technology, 2, 2022 [11] Hertz, H. Über die Berührung fester elastische Körper und über die Härte. Verhandlungen des Vereins zur Beförderung des Gewerbefleisses, Leipzig, 1882 [12] Brewe, D. E. and Hamrock, B. J. Simplified solution for elliptical-contact deformation between two elastic solids, J. Lubr. Technol., 99, pp. 485-487, 1977 [13] Johnson, K. L. Contact mechanics, 95, Cambridge University Press, 1985 [14] Sackfield, A. and Hills, D. Some useful results in the classical hertz contact problem, J. Strain Analysis Eng. Des. 18, pp. 101-105, 1983 [15] Voskamp, A. P. Microstructural changes during rolling contact fatigue: Metal fatigue in the subsurface region of deep groove ball bearing inner rings, Ph.D. thesis, Netherlands, Delft University of Technology, 1997 [16] Dang Van K. Sur la resistance la fatigue des metaux, Science Technique Armement, 47 (3), 1973. Aus Wissenschaft und Forschung 39 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0021 TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 39