PSI - Issue 46

Volume 46 - 2023

5th International Conference on Structural Intergrity and Durability

Željko Boži ć , Siegfried Schmauder, Katarina Monkova, George Pantazopoulos, Sergio Baragetti, Francesco Iacoviello

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5th International Conference on Structural Integrity and Durability A method to reduce computation time in finite element simulations of deep rolling 5th International Conference on Structural Integrity and Durability

J. Bialowas a *, M. Pletz b , S. Gapp a , J. Maierhofer a a Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben, Austria b Designing Plastics and Composite Materials, Department of Polymer Engineering and Science, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700 Leoben, Austria J. Bialowas a *, M. Pletz b , S. Gapp a , J. Maierhofer a a Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben, Austria b Designing Plastics and Composite Materials, Department of Polymer Engineering and Science, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700 Leoben, Austria

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers Abstract Wheelset axles are railway components which have to meet the highest safety requirements. Compressive residual stresses at the surface increase the endurance limit of wheelset axles which must withstand rotating bending. Deep rolling is a surface treatment method to induce such favorable stresses, but their prediction is still challenging both with experimental and computational methods. The computation time for such a finite element analysis is high. Explicit computations with the use of mass scaling and the application of a coarser mesh are commonly used possibilities to reduce the computation time. In this work, a solution for an implicit model of the deep rolling process will be presented which uses a combination of coupled boundary conditions and so called “shadow” elements. With this setup it is possible to reduce the computation time by a factor of 25 compared to previous models with similar accuracy and resolution. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers. Keywords: deep rolling; wheelset axle; work hardening; coupled boundary conditions; finite element modeling 1. Introduction Deep rolling is a surface treatment process with a wide range of applications. The aircraft, automotive and railroad industries are researching the effects of deep rolling on their components such as turbine blades by Bäcker et al. Abstract Wheelset axles are railway components which have to meet the highest safety requirements. Compressive residual stresses at the surface increase the endurance limit of wheelset axles which must withstand rotating bending. Deep rolling is a surface treatment method to induce such favorable stresses, but their prediction is still challenging both with experimental and computational methods. The computation time for such a finite element analysis is high. Explicit computations with the use of mass scaling and the application of a coarser mesh are commonly used possibilities to reduce the computation time. In this work, a solution for an implicit model of the deep rolling process will be presented which uses a combination of coupled boundary conditions and so called “shadow” elements. With this setup it is possible to reduce the computation time by a factor of 25 compared to previous models with similar accuracy and resolution. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers. Keywords: deep rolling; wheelset axle; work hardening; coupled boundary conditions; finite element modeling 1. Introduction Deep rolling is a surface treatment process with a wide range of applications. The aircraft, automotive and railroad industries are researching the effects of deep rolling on their components such as turbine blades by Bäcker et al.

* Corresponding author. Tel.: +43-3842-45922-549 E-mail address: jakob.bialowas@mcl.at * Corresponding author. Tel.: +43-3842-45922-549 E-mail address: jakob.bialowas@mcl.at

2452-3216 © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers. 2452-3216 © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers.

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers 10.1016/j.prostr.2023.06.009

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(2010), crankshafts by Choi and Pan (2009) and wheelset axles by Zerbst et al. (2013). The latter will be used as an example of application in this article. Wheelset axles are among the most safety-critical components of the rolling stock. Therefore, manufacturers and operators seek for possibilities to increase the reliability and duration of inspection intervals of these components. A promising approach to extend inspection intervals is to use mechanical surface treatments like the deep rolling process, see Nalla et al. (2003), Regazzi et al. (2020). Fig. 1 shows a schematic of the deep rolling process for cylindrical geometries, e.g. wheelset axles. Thereby, a machine, similar to a lathe, presses a work roller with a distinct force onto the surface of an axle. The axle rotates around its axis while the work roller moves in axial direction creating a spiral processing path on the surface of the axle. The distance in axial direction of the tool covered in one turn of the axle is called feed, depicted with the variable f in Fig. 1. The contact pressure introduces plastic strains in the surface and near-surface area of the component. This plastic deformation remains and induces residual stresses. The remaining residual stresses in axial direction vary in depth starting with compressive stresses in the vicinity of the surface, change to tensile at a certain depth where they reach a maximum, and finally fall towards zero with further increasing depth (cf. Fig. 3 (a) for a residual stress profile of a deep rolled wheelset axle). Regazzi et al. (2017) describes the stress in axial direction (σ zz ) as the key stress component of wheelset axles because the predominant load of axles is rotating bending. Despite the fact that this process is already frequently employed, the knowledge about actual advantages of this manufacturing step and detailed information on the residual stress state are limited. Maierhofer et al. (2014) suggested an analytical approach towards optimization of the process, where a process model was used for the calculation of residual stresses based on the Hertzian theory. This model gives a rough estimate of the depth of resulting compressive stresses. The sole information about the magnitude of stress on the surface and the penetration depth are not always sufficient for further calculations. The process parameters as well as the geometry of the components and used tools have a significant influence on the strain and stress fields. For this reason, a method is needed that provides additional insight into the strain and stress distribution. Non-destructive experimental approaches as X-ray diffraction can provide meaningful results for residual stresses for the surface and near surface area but deteriorate in accuracy with measuring depth. Destructive testing as for example the hole drilling method exhibit similar problems, especially when the plastic deformations are close to the yield point, as pointed out by Schajer and Whitehead (2018). Although the measuring depth is higher (up to 2 mm), it is not sufficient for the measurement of residual stresses of deep rolled axles. In order to find the ideal settings of the deep rolling parameters computational methods can overcome some of the difficulties of experimental methods. Multiple authors already performed simulations of the deep rolling process with the finite element method. There are three central features of such simulations: (i) the material model, (ii) the choice of the simplified model geometry and the corresponding boundary conditions, and (iii) the evaluated region of interest: (i) The material model must account for the cyclic nature of this process. The work roller not only rolls over the surface in tangential direction but also in axial direction, rolling over a distinct point on the surface multiple times.

Fig. 1 Simplified representation of the deep rolling process with the process-determining parameters: load ( L ), feed ( f ), edge radius of the work roller ( R WR ), work roller diameter ( D WR ) and axle diameter ( D axle )

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These multiple rollovers cause a complex stress state and plastic deformations at the surface. Perenda et al. (2015) used for these requirements a combined isotropic-kinematic hardening material law that is calibrated to a small number of cycles. (ii) The most common simplification of the axle geometry for deep rolling simulations is a cylinder sector. Both the aperture angle and the length of the sector must not only be chosen appropriately to the process parameters feed and load but also to the geometries of the axle and the work roller. However, the boundary conditions at the remaining free surfaces in tangential direction of these cutouts represent one of the difficulties of modelling the deep rolling process. Previous works try to replace the remaining axle of the cutout cylinder sector with various boundary conditions, see Perenda et al. (2016) and Majzoobi et al. (2016). (iii) So far, these boundary conditions have resulted in models that are divided into two domains, one near the center and one close to the boundaries, see for example Klocke et al. (2009). Most modeling approaches evaluate the results in this central domain called region of interest. The remaining geometry called auxiliary region serves the sole purpose of minimizing the boundary effects on the region of interest. Recently Meyer et al. (2021) proposed the use of periodic boundary conditions on cyclic wheel rail simulations. This method minimizes the auxiliary region and provides evenly distributed results. Moreover, it significantly reduces the number of degrees of freedom and thus reduces the computation time. The following chapter gives a guideline to implement the cyclic material behaviour and a simplified geometry in an implicit finite element model using such coupled boundary conditions and so-called shadow elements with the commercial software Abaqus. 2. Method In the first step of modeling the deep rolling process, the geometry of the axle is reduced to a representative cylinder sector as depicted in Fig. 2. The second simplification is a change in the kinematics of the process, because the computation is more efficient when elements move as little as possible. The work roller is modeled as an analytical rigid fully defined by D WR and R WR , see Fig. 1. Therefore, the axle remains stationary and the roller performs the relative movement exclusively. The kinematics are modeled with connector elements based on Balland et al. (2013). This setup allows for both a force and a displacement-controlled process. Furthermore, this can impose every motion of the roller from gliding to rolling, with or without considering friction. The choice of a cylinder sector as a simplified geometry inevitably leads to the work roller having to be lifted off and reset at the end of each rollover. This is accomplished by dividing each rollover into five steps: preload, load, rolling, unload and return, as indicated in Fig. 2 (a). The purpose of the preload step is to establish contact between the work roller and the surface of the axle. Since this surface changes after each rollover, the exact position of it is unknown before the analysis. The setting of the contact controls is adjusted in this step to allow a force-controlled movement of the roller towards the surface (Abaqus). In the following steps, this setting is reset to the automatic setting and is only activated again in the preload step of the following rollover. Although the simplifications transform the continuous process of deep rolling into a discontinuous one, the aim is to implement the continuous characteristics of this process in the model. In previous models the rolling starts and ends with sufficient distance to the boundaries in tangential direction generating inhomogeneous distributions of results, especially close to these boundaries. These numerical artifacts, created by boundary effects, extend over large areas of the model and allow results to be extracted only in a small central region. A step towards a more homogeneous distribution of results would be to roll over the entire surface up to and including the boundaries. Coupling these boundaries with each other instead of applying an ordinary bearing can further reduce boundary effects. This creates a periodic cell that behaves as if several work rollers were simultaneously deep rolling the axle. Therefore, the displacement degrees of freedom of the nodes on the tangential faces of the cylinder sector are coupled (cf. coupled boundary conditions in Fig. 2). In order to remain independent of the aperture angle of the cylinder sector, the coupling requires local coordinate systems that are deflected from each other by the aperture angle. Thus, the behavior of the nodes � � � and � � � � in Fig. 2 (b) coincides in the respective coordinate system. The faces of the cylinder sector in axial direction are supported similar to a locating and non-locating bearing arrangement: One face is fixed and the other face is free to deform in axial direction.

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Following the surface of the cylinder sector there are additional elements, so-called shadow elements (cf. Fig. 2). This additional surface has a thickness of one element in the radial direction. The purpose of these elements is to extend the rolled over surface so that the work roller can roll beyond the edge of the actual cylinder sector (cf. exit point on surface in Fig. 2 (b)). Rolling over edges usually causes problems in finite element simulation, such as buckling or unrealistic stress concentrations. For this purpose, the surface nodes of these shadow elements are coupled with nodes on the surface of the cylinder sector, as shown schematically with � � � and � � � � in Fig. 2 (b). Due to the coupling, the elements of the cylinder sector experience the same deformation as the actually rolled over shadow elements. Only, the coupling of both, the boundary nodes and the surface nodes enables the work roller to move over the edges. The work roller rolls until it reaches the end of the coupled rolling path, which is also the beginning of the following true rolling path, see Fig. 2 (b). A minimum elastic modulus is assigned to the shadow elements to avoid numerical instabilities but at the same time to keep the influence on the cylinder sector as small as possible. The features described so far allow the work roller to exit and entry onto the cylinder sector simultaneously, but neglect the axial portion of the work roller’s motion per revolution, which is called feed. The work roller moves along a spiral processing path, causing an axial displacement of the exit and entry points on the cylinder sector. In order to ensure that the coupled surface nodes are in one row of the mesh while respecting the feed, the cylinder sector is

(a)

(b)

Fig. 2 Schematic of the implemented simulation model: (a) model kinematics including coupled boundary conditions, shadow elements and simulation steps for one rollover divided into preload, load, rolling, unload and return, (b) details of node coupling at the boundaries and the surface, true and coupled rolling path, and the distortion angle of the cylinder sector θ .

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distorted by the angle θ , as indicated in Fig. 2 (b). This distortion axially offsets the exit and entry point by exactly that portion of the feed that results from deep rolling of the remaining axle. This measure allows a continuous change of the feed, which can be selected independently of the mesh and component size. A Python script controls the creation of the entire model with all geometry and process parameters. An implemented search algorithm allows the coupling of individual nodes, which change their position depending on the selected geometry and feed. The python script couples the process parameter feed f to the geometry parameter distortion angle of the cylinder sector θ , resulting in a different geometry with an altered feed. This setup allows creating models with different parameter combinations and performing parameter studies for the deep rolling of cylindrical components, not only wheelset axles. 3. Results and Discussion Fig. 3 (a) shows a stress contour plot of a first attempt to model the deep rolling process with the possibility to roll over the edges of a cylinder sector with a total aperture angle of 45°. This model is built up from two parts, a plastic region of interest in the center surrounded by an elastic part. The roller touches down on the elastic part and rolls over the entire plastic region before lifting off again on the elastic part on the other side. This model is considered as a reference for the updated model described in this article. Fig. 3 (b) shows the stress distribution of the model described in chapter 2 with an aperture angle of 6°. The distribution of the residual stresses in tangential direction is more homogeneous compared to the reference model with a 45° aperture angle in Fig. 3 (a). Furthermore, the number of elements is significantly smaller (200,000 for the reference model and 80,000 for the updated model), although the mesh size in tangential and radial direction could be decreased by approx. 50% per side. Thus, the mesh of the updated model is finer and can better reproduce the high stress and strain gradients. The biggest difference between the two models is the computation time. It is reduced from about 24 hours per rollover for the 45° aperture angle reference model to less than one hour for the updated model with 6° aperture angle using an Intel Xeon CPU with 64 GB memory for both computations. To obtain a steady state after deep rolling, it is necessary to reach a certain number of rollovers in the simulation. The evaluation path in Fig. 3 is located at half the length of the cylinder sector, which is also half of the rolled-over length in the axial direction (i.e. the number of rollovers times the feed rate). The stress state in this path is already influenced before the work roller comes into direct contact with this path and continues to change after the work roller rolls over the zone behind the path. Considering the material model and possible values of the process parameters for

(a)

(b)

Fig. 3 Contour plots of the residual σ zz stresses, the position of the evaluation path and the modeled movement of the work roller: (a) Reference model with 45° aperture angle of the cylinder sector and (b) the corresponding updated model with 6° aperture angle with an additional detail view.

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(a)

(b)

Fig. 4 (a) Residual σ zz stress and (b) equivalent plastic strain ε p,eq distributions of the updated model with aperture angles 6° and 12° and the reference model with an aperture angle of 45°. The two curves for the updated models nearly coincide.

wheelset axles, at least 20 rollovers are required to reach a steady state in the region of interest, which leads to a total computation time of less than 20 hours. Fig. 4 shows the σ zz stress and the equivalent plastic strain ε p, eq distributions evaluated along the evaluation path of Fig. 3. The results labeled 45° correspond to the reference model in Fig. 3 (a) and those labeled 6° and 12° to the updated model in Fig. 3 (b). The good agreement between the results of the 6° and 12° model shows that a reduction to a 6° aperture angle is acceptable, since they were calculated with the same element size. The differing results between 6° and 45° in Fig. 4 (a) can be explained by the choice of a finer mesh in tangential and axial direction, since smaller elements can better resolve the high stress gradients towards the surface. The differing results in Fig. 4 (b) can be explained with the same argument, although the difference for the equivalent plastic strain ε p, eq is even more pronounced. Moreover, this indicates that the element size of the reference model does not lead to a complete convergence of the plastic deformations in the deep rolled zone. The simulation methodology with periodic boundary conditions is equivalent to as if several work rollers were deep rolling the axle simultaneously. This means that in the model with an aperture angle of 6°, there are 60 rollers distributed around the entire circumference one behind the other, each of which deep rolling an angle of 6°. By further reducing the opening angle, the stress fields of the individual work rollers could influence each other and thus falsify the results. Furthermore, it must be ensured that the size of the contact patch corresponds to a fraction of the surface of the cylinder sector in order to exclude the overlapping of the contact by the shadow elements. 4. Conclusion and Outlook A general method for finite element simulations of the deep rolling process has been developed. By combining the ideas of Meyer et al. (2021) with previous models of the deep rolling process, it is possible to reduce the computation time of such simulations by a factor of 25, generating results with an even higher accuracy. The proposed method not only predicts more detailed residual stress fields for a certain set of parameters significantly faster but also allows studying a wider range of these parameters. In future simulations, the uncoupled zone (cf. Fig. 2 (b)) between the coupled elements can be used to investigate the effects of additional features like cracks, notches, or other defects. Also preceding manufacturing steps such as heat treatment should be taken into account in order to consider the influence of already existing residual stresses on the deep rolling process.

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Acknowledgements The authors gratefully acknowledge the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (IC-MPPE)” (Project No 859480). This program is supported by the Austrian Federal Ministries for Climate Action, Environment, Energy, Mobility, Innovation and Technology (BMK) and for Digital and Economic Affairs (BMDW), represented by the Austrian research funding association (FFG), and the federal states of Styria, Upper Austria and Tyrol. References Abaqus. Dassault Systèmes. Bäcker V., Klocke F., Wegner H., Timmer A., Grzhibovskis R., Rjasanow S. 2010. Analysis of the deep rolling process on turbine blades using the FEM/BEM-coupling. IOP Conf. Ser.: Mater. Sci. Eng. 10, 12134. Balland P., Tabourot L., Degre F., Moreau V. 2013. An investigation of the mechanics of roller burnishing through finite element simulation and experiments. International Journal of Machine Tools and Manufacture 65, 29–36. Choi K., Pan J. 2009. Simulations of stress distributions in crankshaft sections under fillet rolling and bending fatigue tests. International Journal of Fatigue 31, 544–557. Klocke F., Bäcker V., Timmer A., Wegner H. 2009. Innovative FE-analysis of the roller burnishing process for different geometries. Maierhofer J., Gänser H.-P., Pippan R. 2014. Prozessmodell zum Einbringen von Eigenspannungen durch Festwalzen. Mat.-wiss. u. Werkstofftech 45, 982–989. Majzoobi G.H., Zare Jouneghani F., Khademi E. 2016. Experimental and numerical studies on the effect of deep rolling on bending fretting fatigue resistance of Al7075. Int J Adv Manuf Technol 82, 2137–2148. Meyer K.A., Skrypnyk R., Pletz M. 2021. Efficient 3d finite element modeling of cyclic elasto-plastic rolling contact. Tribology International 161, 107053. Nalla R., Altenberger I., Noster U., Liu G., Scholtes B., Ritchie R. 2003. On the influence of mechanical surface treatments—deep rolling and laser shock peening—on the fatigue behavior of Ti–6Al–4V at ambient and elevated temperatures. Materials Science and Engineering: A 355, 216–230. Perenda J., Trajkovski J., Žerovnik A., Prebil I. 2015. Residual stresses after deep rolling of a torsion bar made from high strength steel. Journal of Materials Processing Technology 218, 89–98. Perenda J., Trajkovski J., Žerovnik A., Prebil I. 2016. Modeling and experimental validation of the surface residual stresses induced by deep rolling and presetting of a torsion bar. Int J Mater Form 9, 435–448. Regazzi D., Cantini S., Cervello S., Foletti S. 2017. Optimization of the cold-rolling process to enhance service life of railway axles. Procedia Structural Integrity 7, 399–406. Regazzi D., Cantini S., Cervello S., Foletti S., Pourheidar A., Beretta S. 2020. Improving fatigue resistance of railway axles by cold rolling: Process optimisation and new experimental evidences. International Journal of Fatigue 137, 105603. Schajer G, Whitehead P 2018. Hole-Drilling Method for Measuring Residual Stresses. Morgan & Claypool Publishers, San Rafael: 188 pp. Zerbst U., Beretta S., Köhler G., Lawton A., Vormwald M., Beier H., Klinger C., Černý I., Rudlin J., Heckel T., Klingbeil D. 2013. Safe life and damage tolerance aspects of railway axles – A review. Engineering Fracture Mechanics 98, 214–271.

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers Abstract Determination of the expected fatigue life of components operating under rolling-sliding contact loading (gears, bearings, wheels) is quite challenging due to the complex stresses and strains in the material which are constantly changing during the loading cycles. Additional challenge is a lack of the exact properties of the material that components are made of, which is especially pronounced in cases of heat-treated components in which values of the material properties can vary significantly within the component. Based on the results of fatigue testing of involute spur gears made of differently heat-treated steel 42CrMo4 steel (normalized, quenched&tempered) reported in literature, fatigue life analyses of gear teeth flanks were performed using the previously proposed multiaxial fatigue life calculation model based on Fatemi-Socie critical plane based crack initiation criterion. Since the actual cyclic and fatigue material parameters were unavailable, they were estimated using the estimation method developed specifically for 42CrMo4 steel using monotonic properties of the materials which were reported in corresponding studies. Comparisons of experimental and calculated fatigue lives i.e. load carrying capacities show very good agreement thus confirming the applicability of estimation of the advanced material parameters in fatigue and failure analyses of the actual components subjected to rolling sliding contact loading. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers Keywords : fatigue life ; gears ; rolling-sliding contact ; critical plane ; multiaxial fatigue ; fatigue parameters estimation 5th International Conference on Structural Integrity and Durability Application of material fatigue parameters estimation in analysis of rolling-sliding contact fatigue of gears Robert Basan a, *, Tea Marohnić a , Željko Božić b a University of Rijeka, Faculty of Engineering, Vukovarska 58, HR-51000 Rijeka, Croatia b University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lu č i ć a 5, HR-10000 Zagreb, Croatia

* Corresponding author. Tel.: +385 51 651 530 ; fax: +385 51 651 416. E-mail address: robert.basan@riteh.hr

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Nomenclature b

fatigue strength exponent (-) fatigue ductility exponent (-) Young’s modulus (Nmm -2 ) shear modulus (Nmm -2 ) Brinell hardness (HB) Vickers hardness (HV)

c

E G

HB HV

material parameter in the Fatemi-Socie crack initiation criterion (-)

k

yield stress (Nmm

-2 )

R e

 f '  n  f '

fatigue ductility coefficient (-)

maximum normal stress acting on critical plane (Nmm -2 )

max

fatigue strength coefficient (Nmm -2 ) maximum shear strain amplitude (-) number of load reversals (-)

 (  max /2)/2

2 N f

1. Introduction Determination of load-carrying capacity or durability of engineering elements, components and structures is, with the exception of the simplest ones, quite complex task. Intricate geometries, complex loading, variable operating conditions as well as numerous influences need to be taken into account. This is particularly true for highly loaded components exposed to rolling-sliding contact loading such as gears, rollers or bearings because combination of geometry, loading and operating conditions result in a complex multiaxial states of stresses and strains in the material (Aberšek (2004), Dowling (1993)). Simplified representation of the problem which already reveals part of the complexity is shown in Figure 1 where using very simplified terms such as shear loading/stress and shear strength, different scenarios are presented and corresponding type of fatigue damage that occurs at the gear teeth flanks are shown (Hyde (1996)).

Surface damage (pitting)

Deep subsurface damage (case crushing)

Subsurface damage (flake pitting, spalling)

No damage

Stress

Distance from surface

Shear fatigue strength

Shear stress

Fig. 1. Simplified representation of shear fatigue strength and shear stresses at gear teeth flanks and related typical fatigue damage types/forms

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Numerous approaches and calculation models and methodologies have been proposed in the literature for dealing with the task of the determination of the load-carrying capacity and durability i.e. fatigue lives of gears and other components operating in rolling-sliding contact loading regime. These range from early, empirical data-based works and simplified calculation models by Pederson and Rice (1961), Sandberg (1981), to those using Von Mises stress as a relevant parameter as in Glodež et al. (1996). Stress-based approach ws used in Zwirlein and Wieland (1983) and strain-based models were used by Šraml et al. (2003) and Šraml and Flašker (2007). More recent approaches for modeling and determining fatigue lives involve critical plane-based fatigue crack initiation criterion originally proposed by Fatemi and Socie (1988). Most recent approaches implement multi-scale i.e. micro-structurally-based fatigue crack initiation modelling as done by Mlikota et al. (2021) and critical resolved shear stress as proposed by Mlikota and Schmauder (2020). Regardless of the actual problem and/or selected approach, one of the greater challenges which is very often encountered and that must be overcome in calculation and analysis of both monotonically and cyclically loaded components is a lack of exact properties of the material that they are made of, which is especially pronounced in cases of heat-treated components in which values of material properties can vary significantly within the component. Experimentally determined materials properties and parameters or detailed materials response are very costly in terms of time and financial means, and testing equipment is usually unavailable. Thus, using existing data and information available from published references or other sources of data are often resorted to. However, another solution can also be considered - using various estimation methods with which needed and missing material data can be estimated based on partially available information on material state, condition and monotonic properties. An example of this is a study on large-diameter slewing bearings of onshore wind turbines made from induction hardened and tempered steel 42CrMo4 done by Friederici et al. (2021) where extensive experimental work was done in order to extract material specimens and then test them in order to characterize the material and where estimation methods could perhaps have been implemented in conjuction with simple non-destructive testing methods which were done. In a failure analysis of a ruptured compressor pressure vessel, Vukelic et al. (2021) use well-known possibility to estimate the ultimate strength of the pressure vessel steel using measured values of Vickers hardness. For potential further studies, estimation of strain-life fatigue parameters can be performed in the similar manner. For the purpose of static and fatigue analyses of coil spring, Pastorcic et al. (2021) also estimate ultimate strength from Vickers hardness while fatigue parameters needed are taken from the available published reference. Papadopoulou et al. (2019) use the fatigue parameters needed for strain-life fatigue analysis from the software for numerical analysis. In this case, possibility of estimating them from monotonic properties might have been explored since the samples of failed roll steel pins were available and hardness measurement might have been measured (and possibly static tensile testing as well). 2. Fatigue life calculation model and materials parameters estimation methods Aim of the present study is to evaluate applicability of estimation methods for analysis of load-carrying capacity and durability of gear teeth flanks. For this, multiaxial fatigue life calculation model comprising mathematical model of rolling-sliding line contact combined with a multiaxial fatigue life calculation model based on Fatemi-Socie (FS) critical plane crack initiation criterion, previously proposed by Basan and Marohnić (2019) is used. Besides determination of number of load reversals to crack initiation and fatigue failure, model enables estimation of most likely locations of cracks initiation and based on critical planes orientation also the damage type they are likely to form. In its original form, FS criterion uses shear fatigue parameters but using equivalent deformation (von Mises) criterion, it can be written in the form based on axial fatigue strain-life parameters which are estimated in most of proposed estimation methods:



  

  



max

  

   



max n



f 

2





 3 2

 c

(1)

b

1

2

k

N

N



f   

f

f

2

R G

3

e

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Significant number of estimation methods enabling determination of cyclic stress-strain and strain-life fatigue parameters based on monotonic and other material have been proposed in the literature such as Manson (1965), Bäumel and Seeger (1990) or Roessle and Fatemi (2000). New ones are continually and intensively being developed - especially the ones based on artificial neural networks and other machine-learning based approaches and models since they enable inclusion of more input variables i.e. monotonic properties and modelling of much more complex relations among monotonic and cyclic/fatigue parameters than is possible using conventional approaches. However, for the practical applications and purposes, models using less input parameters are often more advantageous since number of monotonic properties and other material information based on which estimation is to be performed are not always available nor easily obtainable. In this paper, approach to estimation previously proposed in Basan et al. (2010) and related expressions (2–5) and developed in Basan et al. (2015), Basan and Marohnić (2019) for estimation of Basquin-Coffin-Manson fatigue parameters of low-alloy steel 42CrMo4 are used. Estimation expressions use Brinell hardness as the principal input variable and specific feature is that individual fatigue parameters are not treated independently.

HB HB          12,59 12981 4 2

   

  f

5,919

b

(2)

10



E

1

   

2

  

HB

HB

(3)

6,3

b







31,056 26781

2

   

   

HB

HB

   4

6,403

c





1951,7 132520

(4)

  10 

f

1

   

2 HB HB 356 328103

  

(5)

1,955

c

 

 

3. Analysis and results For the evaluation of the applicability of fatigue parameters estimation method(s) for determination of fatigue lives of gear teeth flanks subjected to repeating rolling-sliding contact loading, the results of an extensive experimental study of numerous gear pairs by Niemann and Bötsch (1966) were used. In the study, number of gear pairs for which main gearing parameters are given in Table 1., made of differently quenched and tempered low-alloy steel 42CrMo4 were tested. Information on material condition of each gear pair i.e. hardness after the heat treatment as well as operating conditions are given in Table 2. Further details are available in the original reference.

Table 1. Geometrical parameters of analyzed gear pairs. Parameter

Pinion

Wheel

normal module, m n

3 mm

normal pressure angle,  n

20 

23

38

number of teeth, z 1,2

0

0

profile shift coefficients, x 1,2

facewidth, b 1,2

20 mm

20 mm

1,65

transverse contact ratio,  

center distance, a

91,5 mm

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Table 2. Gears’ material condition and hardness, loading and estimated coefficient of friction in mesh. Parameter Gear pair 1 Gear pair 2 Gear pair 3 Gear pair 4 material and condition 42CrMo4, Q&T 42CrMo4, Q&T 42CrMo4, Q&T 42CrMo4, Q&T yield stress, R e (Nmm -2 ) 540 627 872 961 Brinell hardness, HB (HB) 215 240 310 325 total normal force, F bt (N) 2696 3204 4527 5061 coefficient of friction, estimated 0,04

Values of Vickers hardness given in the study were recalculated to Brinell hardness using the expression (6):

(6)

5000 2407,5 5752500   

HB

HV

Values of individual strain-life fatigue parameters determined for the material of every analyzed gear pair using expressions (2) and (3) are given in Table 3.

Table 3. Values of Basquin-Coffin-Manson fatigue parameters estimated from Brinell hardness. Parameter Gear pair 1 Gear pair 2 Gear pair 3

Gear pair 4

material and condition Brinell hardness, HB (HB)

42CrMo4, Q&T 42CrMo4, Q&T 42CrMo4, Q&T 42CrMo4, Q&T

215

240

310

325

fatigue strength coefficient,  f '/ E (-) fatigue strength exponent, b (-) fatigue ductility coefficient,  f ' (-) fatigue ductility exponent, c (-)

0,00491 -0,087 0,9605 -0,670

0,00537 -0,0842

0,00667 -0,0788 1,0157

0,0069 -0,078 0,9707 -0,733

1,031

-0,6866

-0,726

Experimental and fatigue lives calculated using estimated fatigue parameters were compared through ratios of fatigue lives of individual gear pairs and maximum fatigue life of particular gear pair (gear pair 3 in experimental study and gear pair 4 in calculations). From diagram in Figure 2 it can be seen that increase in hardness of gear teeth flanks’ material results in increased experimental fatigue lives of tested gear pairs. Calculated fatigue lives follow this change very closely with minor deviation notable only between materials with small differences in hardness (310 HB and 325 HB).

2 N f,exp /2 N f,exp_max 2 N f,calc /2 N f,calc_max

Brinell hardness of gear pairs teeth flanks, HB (HB)

Fig. 2. Comparison of relative experimental and calculated fatigue lives (2 N f /2 N f, max ) of tested gear pairs’ teeth flanks.

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4. Conclusions Expressions for estimation of required fatigue data have been developed on material datasets completely independent from the material data available in the experimental study of analyzed gear pairs. Considering that gears’ materials were quite similar and that individual gear pairs were loaded differently, it can be concluded that results of the present studies show quite good agreement between experimental and calculated durability of analized gear pairs. With validity of multiaxial fatigue life calculation model based on Fatemi-Socie (FS) critical plane crack initiation criterion already confirmed in previous analyses and studies, it seems that selected estimation method and related expressions can be successfully applied for estimation of advanced material parameters in fatigue and failure analyses of actual components subjected to rolling-sliding contact loading. Further analyses and validations will be performed with extension to numerical simulations and studies of surface-hardened components whose material and related properties and parameters vary with the distance from the surface. Acknowledgements This work has been supported in part by Croatian Science Foundation under the project IP-2020-02-5764 and by the University of Rijeka under the project number uniri-tehnic-18-116. References Aberšek, B., Flašker, J., 2004. How gears break. Witpress. Basan, R., Marohnić, M., 2019. Multiaxial fatigue life calculation model for components in rolling - sliding line contact with application to gears. Fatigue Fract Eng Mater Struct, 1–16. Basan, R., Marohnić, T., Franulović, M., 2015. Estimation of Fatigue Parameters of 42CrMo4 Steels. Proceedings of the 36th International Conference on Mechanics of Materials. Darmstadt, Germany. Basan, R., Rubeša, D., Franulović, M., Križan, B., 2010. A novel approach to the estimation of strain life fatigue parameters, Procedia Engineering 2, 41–426. Bäumel, A., Seeger, T., 1990. Materials data for cyclic loading – Supplement 1. Elsevier, Amsterdam, Netherlands. Dowling, N.E., 1993. Mechanical behavior of materials. Prentice-Hall International, New Jersey, United States. Fatemi, A., Socie, D.F., 1988. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract Eng Mater Struct 11, 149–165. Friederici, V., Schumacher, J., Clausen, B., 2021. Influence of local differences in microstructure and hardness on the fatigue behaviour of a slewing bearing steel. Procedia Structural Integrity 31, 8–14. Glodež, S., Flašker, J., Ren, Z., Pehan, S., 1996. Optimisation of hardened layer thickness on gears. Conference Proceedings of 4th Symposium Design 96. Opatija, Croatia. Glodež, S., Ren, Z., Flašker, J., 1999. Surface fatigue of gear teeth flanks. Comput Struct 73, 475-483. Hyde, R.S., 1996. Contact fatigue of hardened steel, in ASM Handbook Vol. 19, “Fatigue and Fracture”. ASM International, 1996. Manson, S.S., 1965. Fatigue: A complex subject – some simple approximations. Exp Mech SESA 5(7), 193–226. Mlikota, M., Schmauder, S., 2020. A Newly Discovered Relation between the Critical Resolved Shear Stress and the Fatigue Endurance Limit for Metallic Materials. Metals 10, 803. Mlikota, M., Schmauder, S., Dogahe, K., Božić, Ž., 2021. Influence of local residual stresses on fatigue crack initiation. Procedia Structural Integrity 31, 3–7. Niemann, G., Bötsch, H., 1966. Neue Versuchsergebnisse zur Zahnflanken-Tragfähigkeit von Stirnrädern aus Vergütungsstahl. Konstruktion 12, 481-491. Pederson, R., Rice, S.L., 1961. Case crushing of carburized and hardened gears. Transactions of SAE. Roessle, M.L., Fatemi, A., 2000. Strain–controlled fatigue properties of steels and some simple approximations. Int J Fatigue 22, 495–511. Sandberg, E., 1981. A calculation method for subsurface fatigue. Conference Proceedings of International Symposium on Gearing & Power Transmissions. Tokyo, Japan. Šraml, M., Flašker, J., 2007. Computational approach to contact fatigue damage initiation analysis of gear teeth flanks. Int J Adv Manuf Tech 31, 1066-1075. Šraml, M., Flašker, J., Potrč, I., 2003. Numerical procedure for predicting the rolling contact fatigue crack initiation. Int. J. Fatigue 25, 585-595. Vukelic, G., Vizentin, G., Bozic, Z., Rukavina, L., 2021. Failure analysis of a ruptured compressor pressure vessel. Procedia Structural Integrity 31, 28–32. Zwirlein, O., Wieland, W.P., 1983. Case depth for induction hardened slewing bearing rings. International Off-Highway Meeting & Exposition. Milwaukee: Society of Automotive Engineers, Inc.

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers Numerical investigations by means of elastic–plastic finite element analyses were carried out to estimate the local stress conditions acting in the blade root. The local stresses and strains were post-processed using the critical plane approach and advanced fatigue damage parameters. Finally, the estimated numerical lifetimes were compared with the experimental results of the component tests on original-sized end stage blades. © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers Keywords: component test; end stage blades; multiaxial fatigue; finite elemente analysis; fatigue assessment; lifetime evaluation 1. Introduction The increased use of regenerative energies results in modified requirements for fossil power plants regarding extended partial load efficiency, accelerated load changes, and the climate policy-based desire for an increase in 5th International Conference on Structural Integrity and Durability Component tests and numerical investigations to determine the lifetime and failure behavior of end stage blades L. Frank a, *, S. Weihe a a Materials Testing Institue (MPA) University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany Abstract The paper focuses on experimental and numerical fatigue assessment procedures to evaluate the influence of multiaxial stress state caused by high centrifugal forces superimposed with bending loads due to blade vibrations on the lifetime of end stage blades from steam turbines. The experimental investigations on original-sized end stage blades were carried out on a test rig especially developed for high forces and multicomponent force application based on detailed numerical simulations by the MPA University of Stuttgart. During the fractographic post-examinations of the tested blades using magnetic particle inspection, light and scanning electron microscopes, two competing damage mechanisms were identified, which occurred at different locations.

* Corresponding author. Tel.: +49 (0) 711-685-63954 E-mail address: lukas.frank@mpa.uni-stuttgart.de

2452-3216 © 2021 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSID 2021 Organizers 10.1016/j.prostr.2023.06.002