PSI - Issue 68

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European Conference on Fracture 2024 Preface Željko Božić a, *, Robert Basan b , Goran Vukelić c , Siegfried Schmauder d , Leslie Banks Sills e , Aleksandar Sedmak f , Francesco Iacoviello g a University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lučića 5, 10000 Zagreb, Croatia b University of Rijeka, Faculty of Engineering, Vukovarska 58, HR-51000 Rijeka, Croatia c University of Rijeka, Faculty of Maritime Studies, Marine Engineering Dept., Studentska 2, HR-51000 Rijeka, Croatia d University of Stuttgart, Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), Pfaffenwaldring 32, Stuttgart, Germany e Tel Aviv University, School of Mechanical Engineering, Tel Aviv 6997801, Israel European Conference on Fracture 2024 Preface Željko Božić a, *, Robert Basan b , Goran Vukelić c , Siegfried Schmauder d , Leslie Banks Sills e , Aleksandar Sedmak f , Francesco Iacoviello g a University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana Lučića 5, 10000 Zagreb, Croatia b University of Rijeka, Faculty of Engineering, Vukovarska 58, HR-51000 Rijeka, Croatia c University of Rijeka, Faculty of Maritime Studies, Marine Engineering Dept., Studentska 2, HR-51000 Rijeka, Croatia d University of Stuttgart, Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), Pfaffenwaldring 32, Stuttgart, Germany e Tel Aviv University, School of Mechanical Engineering, Tel Aviv 6997801, Israel f Facuty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia g Università di Cassino e del Lazio Meridionale, via G. DI Biasio 43, 03043, Cassino (FR), Italy The 24th European Conference on Fracture, ECF24, was held on-site in Zagreb, Croatia, with possible online participation, from August 26 – 30, 2024. The ECF24 conference was organized by the European Structural Integrity Society (ESIS) and the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. The ECF24 conference was a forum for discussing current and future trends in experimental, analytical and numerical fracture mechanics, fatigue, structural integrity assessment, failure analysis, and other important topics in the field. The participants of the ECF24 conference had the opportunity to exchange ideas with leading researchers across a wide range of disciplines and get acquainted with the state-of-the-art in theory and applications of fatigue and fracture mechanics. ECF24 managed to bring together 650 scientists and engineers from 54 countries from around the world. Nine Plenary Lectures were presented at the ECF24 conference by distinguished scientists and researchers. In total 680 Extended abstracts were accepted. 90 parallel sessions were held, with 535 Oral presentations, including 50 Online presentations. In Poster Sessions 73 posters were presented. 17 thematic symposia were organized within the The 24th European Conference on Fracture, ECF24, was held on-site in Zagreb, Croatia, with possible online participation, from August 26 – 30, 2024. The ECF24 conference was organized by the European Structural Integrity Society (ESIS) and the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb. The ECF24 conference was a forum for discussing current and future trends in experimental, analytical and numerical fracture mechanics, fatigue, structural integrity assessment, failure analysis, and other important topics in the field. The participants of the ECF24 conference had the opportunity to exchange ideas with leading researchers across a wide range of disciplines and get acquainted with the state-of-the-art in theory and applications of fatigue and fracture mechanics. ECF24 managed to bring together 650 scientists and engineers from 54 countries from around the world. Nine Plenary Lectures were presented at the ECF24 conference by distinguished scientists and researchers. In total 680 Extended abstracts were accepted. 90 parallel sessions were held, with 535 Oral presentations, including 50 Online presentations. In Poster Sessions 73 posters were presented. 17 thematic symposia were organized within the © 2025 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 ECF24 organizers f Facuty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia g Università di Cassino e del Lazio Meridionale, via G. DI Biasio 43, 03043, Cassino (FR), Italy

* Corresponding author. Tel.: +385 1 6168 536; fax: +385 1 6156 940. E-mail address: zeljko.bozic@fsb.unizg.hr

2452-3216 © 2025 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 ECF24 organizers 10.1016/j.prostr.2025.06.014 2452-3216 © 2025 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 ECF24 organizers 2452-3216 © 2025 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 ECF24 organizers * Corresponding author. Tel.: +385 1 6168 536; fax: +385 1 6156 940. E-mail address: zeljko.bozic@fsb.unizg.hr

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conference, 14 of which were organized by ESIS Technical Committees. 195 full papers are selected after a peer review process to be published in this volume, following the tradition of previous ESIS conferences. The ESIS Summer School “Fatigue and Fracture Modelling and Analysis” was organized prior to the conference on August 24 and 25, 2024. The Summer School provided scientists, researchers and engineers from academia and industry with the opportunity to hear from the renowned speakers about fundamental aspects of fracture mechanics, multiaxial fatigue of metallic alloys, advanced approaches such as multiscale materials modelling and local approaches for fatigue design. We would like to take this opportunity to thank all participants of the ECF24 conference for their contributions and the many high-quality papers. Many thanks to the organizers of the Thematic Symposia who helped to structure ECF24 as an efficient and smooth-running event. We would like to thank ESIS ExCo members, all colleagues and everyone who was involved in the organization of the ECF24 conference for their great effort, support and commitment.

Guest Editors of the Procedia Structural Integrity ECF24 Conference Proceedings: Željko Božić, University of Zagreb, Croatia Robert Basan, University of Rijeka, Croatia Goran Vukelić, University of Rijeka, Croatia Siegfried Schmauder, University of Stuttgart, Germany Leslie Banks-Sills, Tel Aviv University, Israel Aleksandar Sedmak, University of Belgrade, Serbia Francesco Iacoviello, Università di Cassino e del Lazio Meridionale – DICeM, Italy

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European Conference on Fracture 2024 Acceleration and deceleration of unzipping supershear fracture propagating along a straight perforation line consisting of small scale cracks Koji Uenishi a,b, *, Kaichi Akimoto b , Masanao Sekine b a Department of Advanced Energy, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8561 Chiba, Japan b Department of Aeronautics and Astronautics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, 113-8656 Tokyo, Japan Abstract We experimentally investigate two-dimensional dynamic fracture propagation along a straight perforation line containing multiple small-scale cracks. If there exists only one single perforation line in a brittle polycarbonate specimen, dynamic fracture due to application of external quasi-static loading seems propagating unidirectionally in an “unzipping” way along the perforation line, simply linking the edges of the cracks without diverting and branching. However, a closer look into the photographs experimentally obtained by a high-speed video camera shows that fracture propagation is not always unidirectional and its speed also fluctuates back-and-forth between subsonic and supershear levels, i.e. the speed can be lower or higher than the relevant shear wave speed of the specimen during propagation. Generally, the propagation speed becomes maximum just before the fracture leaves the edge of a preexisting small-scale crack and it is low just before the fracture approaches the edge of the next crack in the perforation line. Furthermore, the fluctuation of fracture propagation speed at supershear levels may generate multiple Mach (shock) wavefronts with different Mach angles. © 2025 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 ECF24 organizers Keywords: Unzipping fracture; Supershear fracture; Multiple Mach wavefronts European Conference on Fracture 2024 Acceleration and deceleration of unzipping supershear fracture propagating along a straight perforation line consisting of small scale cracks Koji Uenishi a,b, *, Kaichi Akimoto b , Masanao Sekine b a Department of Advanced Energy, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8561 Chiba, Japan b Department of Aeronautics and Astronautics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, 113-8656 Tokyo, Japan Abstract We experimentally investigate two-dimensional dynamic fracture propagation along a straight perforation line containing multiple small-scale cracks. If there exists only one single perforation line in a brittle polycarbonate specimen, dynamic fracture due to application of external quasi-static loading seems propagating unidirectionally in an “unzipping” way along the perforation line, simply linking the edges of the cracks without diverting and branching. However, a closer look into the photographs experimentally obtained by a high-speed video camera shows that fracture propagation is not always unidirectional and its speed also fluctuates back-and-forth between subsonic and supershear levels, i.e. the speed can be lower or higher than the relevant shear wave speed of the specimen during propagation. Generally, the propagation speed becomes maximum just before the fracture leaves the edge of a preexisting small-scale crack and it is low just before the fracture approaches the edge of the next crack in the perforation line. Furthermore, the fluctuation of fracture propagation speed at supershear levels may generate multiple Mach (shock) wavefronts with different Mach angles. © 2025 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 ECF24 organizers Keywords: Unzipping fracture; Supershear fracture; Multiple Mach wavefronts © 2025 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 ECF24 organizers

* Corresponding author. Tel.: +81-4-7136-3824; fax: +81-4-7136-3824. E-mail address: uenishi@k.u-tokyo.ac.jp * Corresponding author. Tel.: +81-4-7136-3824; fax: +81-4-7136-3824. E-mail address: uenishi@k.u-tokyo.ac.jp

2452-3216 © 2025 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 ECF24 organizers 2452-3216 © 2025 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 ECF24 organizers

2452-3216 © 2025 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 ECF24 organizers 10.1016/j.prostr.2025.06.096

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1. Introduction We have been investigating the multiscale fracture behavior of brittle materials (Gomez et al., 2020), in particular, local fracture behavior in a global crack system that may be related to, for example, a cluster of earthquakes and earthquake swarms induced by multiple local fractures in a global geological fault system. So far, through the laboratory two-dimensional fracture experiments utilizing the technique of dynamic photoelasticity in conjunction with high-speed cinematography, we have found that the local fracture behavior in polycarbonate specimens with sets of preexisting small-scale parallel cracks strongly depends on the external (quasi-) static / impact loading conditions as well as on the initial inclination angle and distribution pattern of the sets of parallel cracks. For instance, if a number of perforation lines consisting of small-scale cracks are preset in a brittle polycarbonate specimen by a digitally controlled laser cutter, fractures can easily jump to distant places and propagate back-and-forth inside the specimen and they do not always propagate along one of the perforation lines and break the specimen in an “unzipping” fashion (Uenishi and Nagasawa, 2023; Uenishi et al., 2024). However, the mechanical details behind this rather unexpected behavior of (un)fracture along, across or beyond perforation lines have not been thoroughly understood yet. Therefore, here, we study the fundamental characteristics of “unzipping” fracture related to perforation lines containing small-scale cracks. 2. Fracture propagating along a straight perforation line consisting of small-scale cracks In the study on fracture behavior of perforated materials (e.g. Tokiyoshi et al., 2001; Dastjerdi et al. 2011; Ma et al., 2015; Carlsson and Isaksson, 2019; Lin et al., 2023; Peng et al., 2023; Wang et al., 2023), instead of straight cracks, circular holes are usually aligned in a single or two straight line(s). Here, in order to comprehend basic two dimensional fracture behavior of multiple, truly straight small-scale cracks aligned in a single line, polycarbonate specimens with two geometrically different configurations are prepared by a digitally controlled laser cutter as shown in Figs. 1 and 2. In the relatively “dense” first experiment, in a transparent birefringent brittle specimen (140 mm ´ 45 mm ´ 2 mm), cracks of length 3 mm are placed at a spatial distance of 1 mm while in the relatively “coarse” second experiment, cracks of length 2 mm at a spatial distance of 8 mm are preset. Both specimens are subjected to tension at a constant strain rate of 1.67 ´ 10 - 1 /s using a tensile testing machine, and dynamic behavior and isochromatic fringe patterns inside the specimens are visualized photoelastically with a high-speed video camera at a frame rate of 400,000 frames per second (fps) (Fig. 1) or 200,000 fps (Fig. 2). The mass density, shear modulus and Poisson’s ratio of the polycarbonate are 1,200 kg/m 3 , 820 MPa and 0.37, and with these material properties, the longitudinal (P) and shear (S) wave speeds are calculated to be some 1,820 m/s and 830 m/s. Figure 1(a) shows the dynamic development of fracture along the perforation line with relatively densely distributed small-scale cracks indicated in Fig. 1(b). If there exists only one single perforation line consisting of densely distributed small-scale cracks in a specimen, as easily expected, dynamic fracture can propagate in an “unzipping” way along the perforation line, unidirectionally from right to left linking the edges of the cracks without diverting the direction of propagation and branching. However, the experimentally obtained photographs in Fig. 1(a) clearly show that the speed of fracture propagation fluctuates, repeatedly every 7.5 µ s, between two levels, a high supershear level (at a level above the relevant shear wave speed of the specimen; e.g. at time 12.5, 20, 27.5 µ s) and a low subsonic level (e.g. at 17.5, 25 µ s). The supershear fracture propagation can be simply identified by the Mach (shock) wavefronts indicated in red in Fig. 1(a) just before the fracture leaves the edge of a preexisting small-scale crack, and its speed is calculated from the experimental photographs to be about 1,200 m/s. That is, the Mach number with respect to the S wave speed is M S = 1,200 m/s / 830 m/s = 1.45 (Mach number with respect to the P wave speed M P = 1,200 m/s / 1,820 m/s = 0.66). On the other hand, the subsonic fracture speed just before the fracture approaches the edge of the next crack in the perforation line is evaluated to be some 200 m/s, with the Mach number M S = 0.24 ( M P = 0.11). Thus, during propagation, the Mach number M S fluctuates approximately between 0.24 and 1.45. However, the apparent average speed of fracture propagation is evaluated to be (3 + 1) mm / 7.5 µ s = 530 m/s, which is in a subsonic range and well below the shear wave speed, with the Mach number M S = 0.64 ( M P = 0.29), and the supershear fracture may not be recognized if only global fracture behavior is observed at larger time intervals. In Fig. 2, when the perforation line contain coarsely distributed small-scale cracks, the speed of fracture propagation seems to fluctuate even within supershear levels, and as seen in the central section at time 40 and 45 µ s,

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Fig. 1. (a) Dynamic fracture development along a straight perforation line with relatively densely distributed small-scale cracks in a polycarbonate specimen shown in (b) [unit: mm]. The specimen is under quasi-static tension at a constant strain rate of 1.67 ´ 10 - 1 /s, and the dynamic fracture behavior is visualized using a high-speed video camera at a frame rate of 400,000 frames per second (fps). As can be seen from the isochromatic fringe patterns (contours of the maximum in-plane shear stress), the propagation speed of fracture fluctuates between low (wavefronts indicated in blue) and supershear (above the shear wave speed of the specimen with Mach (shock) wavefronts in red) levels. this fluctuation at supershear levels can generate multiple Mach wavefronts with different Mach angles (indicated in red). The fracture behavior seems more complex than the previous case and the fracture propagates not only from right to left but also from left to right starting at the right edges of preexisting small-scale cracks, and therefore, the physical background behind the multiple Mach wavefronts generated by the dynamic fracture is still unclear. Together with the preexisting small-scale crack, if merged, subsonic fracture from right to left and that from left to right may seem to have an apparent speed exceeding the shear wave speed of the specimen. Further investigation including more thorough experimental observations is needed. 3. Conclusions We have shown two typical examples of fracture propagation along a single straight perforation line with small scale cracks in a two-dimensional polycarbonate specimen under quasi-static tension. If the preexisting small-scale cracks are densely distributed, the dynamic fracture does propagate unidirectionally in an “unzipping” way along the

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Fig. 2. (a) Same as in Fig. 1, but now the specimen has a straight perforation line with relatively coarsely distributed small-scale cracks illustrated in (b) [unit: mm]. Again, quasi-static tension at a constant strain rate of 1.67 ´ 10 - 1 /s is applied to the specimen, and the fracture behavior is observed with a high-speed video camera at a frame rate of 200,000 fps. The fluctuation of fracture propagation speed at supershear levels seems to generate multiple Mach wavefronts with different Mach angles indicated in red. perforation line, but its propagation speed is not constant. The speed rather fluctuates between subsonic and supershear levels, with the maximum just before the fracture leaves the edge of a preexisting small-scale crack and low values just before the fracture approaches the edge of the next crack. It is possible that the (pre-)existence of the small-scale cracks can accelerate the fracture propagation speed to a higher level. If the small-scale cracks along the perforation line is less densely distributed, the speed of fracture propagation may fluctuate at supershear levels, generating multiple Mach wavefronts around the tip of the propagating fracture, but its mechanical details are still open.

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Acknowledgements The research has been financially supported by the Japan Society for the Promotion of Science (JSPS) through the “KAKENHI: Grant-in-Aid for Scientific Research (C)” Program under grant number 23K04021. References Carlsson, J., Isaksson, P., 2019. Crack Dynamics and Crack Tip Shielding in a Material Containing Pores Analysed by a Phase Field Method. Engineering Fracture Mechanics 206, 526–540. Dastjerdi, M. H., Rübesam, M., Rüter, D., Himmel, J., Kanoun, O., 2011. Non Destructive Testing for Cracks in Perforated Sheet Metals. IEEE 8th International Multi-Conference on Systems, Signals & Devices, 5 pages. Gomez, Q., Uenishi, K., Ionescu, I. R., 2020. Quasi-Static versus Dynamic Stability Associated with Local Damage Models. Engineering Failure Analysis 111, 104476. Lin, R., Dong, M., Lan, S., Lou, M., 2023. Numerical Simulation of Liquid Water Transport in Perforated Cracks of Microporous Layer. Energy 262, 125372. Ma, H., Pang, X., Zeng, J., Wang, Q., Wen, B., 2015. Effects of Gear Crack Propagation Paths on Vibration Responses of the Perforated Gear System. Mechanical Systems and Signal Processing 62-63, 113–128. Peng, X., Chen, Z., Bobaru, F., 2023. Accurate Predictions of Dynamic Fracture in Perforated Plates. International Journal of Fracture 244, 61– 84. Tokiyoshi, T., Kawashima, F., Igari, T., Kino, H., 2001. Crack Propagation Life Prediction of a Perforated Plate under Thermal Fatigue. International Journal of Vessels and Piping 78, 837 – 845. Uenishi, K., Fujimoto, M., Akimoto, K., 2024. On the Diversity of Fracture Behavior in a Brittle Solid with Sets of Preexisting Small-Scale Cracks. Procedia Structural Integrity 61, 108–114. Uenishi, K., Nagasawa, K. 2023. Brittle Fracture Development through Sets of Preexisting Small-Scale Cracks under Quasi-Static and Dynamic Impact Loading. Mechanics of Advanced Materials and Structures 30, 3710 – 3720. Wang, C., Hu, M., Cheng, L., Cheng, B., Ji, X., Ren, Y., Wang, S., Li, J., 2023. Radial Impact Fracture Characteristics and Crack Initiation Criterion of Concentric Perforated Granite after High Temperature-Water Cycle. Engineering Fracture Mechanics 286, 109288.

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European Conference on Fracture 2024 A cyclic resistance curve approach to estimating fatigue limit in the presence of over- and underloads Kimmo Ka¨rkka¨inen a, ∗ , Joona Vaara c,b , Miikka Va¨nta¨nen d ,MariÅman a , Tero Frondelius c,a,b a Materials and Mechanical Engineering, Pentti Kaiteran katu 1, 90014 University of Oulu, Finland b Faculty of Built Environment, Tampere University, Korkeakoulunkatu 7, 33720, Finland c R & D and Engineering, Wa¨rtsila¨, P.O.Box 244, 65101, Vaasa, Finland d Global Boiler Works Oy, Lumijoentie 8, 90400 Oulu, Finland Abstract Tensile overloads (OL) and compressive underloads (UL) can have a significant e ff ect on fatigue crack propagation and fatigue limit. This article provides a simple method for quantitatively estimating the fatigue limit in the presence of sporadic over- and underloads via a cyclic resistance curve (R-curve) analysis. Finite element modeling results show that both loading spikes can ef fectively reset the R-curve by removing crack closure. It is shown that recurrent application of either loading spike can significantly reduce the e ff ective fatigue limit. The saturation value of the e ff ective fatigue limit is governed by the intrinsic threshold ∆ K th , e ff and the rate of saturation depends on the loading spike type. © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of ECF24 organizers. Keywords: Variable amplitude loading; Crack closure; Fracture mechanics; Fatigue strength; Numerical modeling European Conference on Fracture 2024 A cyclic resistance curve approach to estimating fatigue limit in the presence of over- and underloads Kimmo Ka¨rkka¨inen a, ∗ , Joona Vaara c,b , Miikka Va¨nta¨nen d ,MariÅman a , Tero Frondelius c,a,b a Materials and Mechanical Engineering, Pentti Kaiteran katu 1, 90014 University of Oulu, Finland b Faculty of Built Environment, Tampere University, Korkeakoulunkatu 7, 33720, Finland c R & D and Engineering, Wa¨rtsila¨, P.O.Box 244, 65101, Vaasa, Finland d Global Boiler Works Oy, Lumijoentie 8, 90400 Oulu, Finland Abstract Tensile overloads (OL) and compressive underloads (UL) can have a significant e ff ect on fatigue crack propagation and fatigue limit. This article provides a simple method for quantitatively estimating the fatigue limit in the presence of sporadic over- and underloads via a cyclic resistance curve (R-curve) analysis. Finite element modeling results show that both loading spikes can ef fectively reset the R-curve by removing crack closure. It is shown that recurrent application of either loading spike can significantly reduce the e ff ective fatigue limit. The saturation value of the e ff ective fatigue limit is governed by the intrinsic threshold ∆ K th , e ff and the rate of saturation depends on the loading spike type. © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of ECF24 organizers. Keywords: Variable amplitude loading; Crack closure; Fracture mechanics; Fatigue strength; Numerical modeling © 2025 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 ECF24 organizers Predicting the fatigue limit of a mechanical component in service is crucial for safe operation. Components see not only constant amplitude loading, but often encounter various loading spikes. Encountering unexpected loading spikes increases the uncertainty of a component’s residual fatigue strength, leading to costs related to increased inspection frequency or component failure. The e ff orts to study the e ff ect of over- and underloads have not been directed to near-threshold loading conditions or fatigue limit assessment to the necessary degree. It is the authors’ view that the current state of literature does not provide a definitive, physically based method to account for over- and underloads in fatigue limit estimation. The present study takes some of the necessary steps towards rectifying this issue. It has been known for very long that tensile overloads and compressive underloads can have a distinct influence on fatigue crack propagation (Dieter et al., 1954; Wheeler, 1972; Topper and Yu, 1985; Mlikota et al., 2017; Chen et al., Predicting the fatigue limit of a mechanical component in service is crucial for safe operation. Components see not only constant amplitude loading, but often encounter various loading spikes. Encountering unexpected loading spikes increases the uncertainty of a component’s residual fatigue strength, leading to costs related to increased inspection frequency or component failure. The e ff orts to study the e ff ect of over- and underloads have not been directed to near-threshold loading conditions or fatigue limit assessment to the necessary degree. It is the authors’ view that the current state of literature does not provide a definitive, physically based method to account for over- and underloads in fatigue limit estimation. The present study takes some of the necessary steps towards rectifying this issue. It has been known for very long that tensile overloads and compressive underloads can have a distinct influence on fatigue crack propagation (Dieter et al., 1954; Wheeler, 1972; Topper and Yu, 1985; Mlikota et al., 2017; Chen et al., 1. Introduction 1. Introduction

2452-3216 © 2025 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 ECF24 organizers 10.1016/j.prostr.2025.06.110 ∗ Corresponding author E-mail address: kimmo.karkkainen@oulu.fi (Kimmo Ka¨rkka¨inen). 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of ECF24 organizers. ∗ Corresponding author E-mail address: kimmo.karkkainen@oulu.fi (Kimmo Ka¨rkka¨inen). 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of ECF24 organizers.

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2020; Liang et al., 2022). It is commonly reported that crack growth rate is reduced after overloads (Wheeler, 1972; Topper and Yu, 1985; Chen et al., 2020) and increased after underloads (Topper and Yu, 1985; Liang et al., 2022), making a single instance of the former beneficial, and one of the latter harmful in terms of fatigue life. However, the same may not apply to recurrent loading spikes, or their influence on fatigue limit defined by short crack arrest (Dieter et al., 1954; Fleck, 1985; Pompetzki et al., 1990; Murakami and Endo, 1994). This can be understood to be a combined result of two mechanisms. Firstly, the direct damaging e ff ect of the large load cycles becomes important if frequently applied. Secondly, both over- and underloads can facilitate crack propagation even with base load amplitudes below the constant amplitude fatigue limit, as shown by Pompetzki et al. (1990). In other words, applying either loading spike may resume the propagation of non-propagating cracks. An explanation can be found in the loss of crack closure, caused by crack tip blunting following an overload, or crack flank compression following an underload (Maierhofer et al., 2018; Ka¨rkka¨inen et al., 2024a,b). While the inadequacy of a linear damage summation (Miner, 1945) in describing variable amplitude fatigue (Watson and Topper, 1972; Fleck, 1985; Murakami and Endo, 2023) likely stems from fatigue as a whole being an inherently non-linear crack growth problem, the second mechanism described above has received little recognition and is expected to contribute to the deviation. The current article focuses on this mechanism, assuming a finite number of infrequent loading spikes and thereby neglecting the ipso facto damaging e ff ect of the large load cycles, whose unrestricted application would ultimately render the fatigue limit assessment in terms of base loading amplitude meaningless. The cyclic resistance curve (R-curve) depicts the development of crack growth resistance as a function of crack length (Tanaka and Akiniwa, 1988; Zerbst and Madia, 2015; Maierhofer et al., 2018). A lower bound for the fatigue limit, σ w , is defined by a tangency condition between the nominal driving force, ∆ K , and R-curve, ∆ K th . The R curve analysis assumes crack initiation, i.e., crack initiation resistance cannot increase the fatigue limit within the framework, which is why the term lower bound is used. Short crack arrest is, nevertheless, often the fatigue limit defining mechanism as crack-like defects are an exceedingly common source of failure in real components (Murakami and Endo, 1994). Thus, a fatigue limit R-curve analysis can be regarded as conservative and reasonable, and therefore adopted in the current study. This article directly builds upon earlier important findings. In specific, Maierhofer et al. (2018) schematically show the e ff ect of an underload on the R-curve, where crack closure is momentarily weakened, and the R-curve is partially reset. Ka¨rkka¨inen et al. (2024b) provide simulation results supporting this phenomenon and establish an analytical framework serving as a basis for the present study. Here the e ff ect of both over- and underloads on fatigue limit is quantitatively assessed. The article is structured as follows. First, details on the finite element modeling used to obtain the plasticity-induced crack closure response to loading spikes are briefly given. Second, the analysis framework and assumptions are described. Finally, results from the analysis are presented, along with relevant discussion, followed by the conclusions of the study. In this section, the numerical crack propagation model and plasticity-induced crack closure results are presented. The modeling framework is largely similar to those used in earlier studies (Ka¨rkka¨inen et al., 2023, 2024a,b), which is why only necessary details are briefly described here. The model geometry corresponds to a 2D plate with dimensions 10 mm x 10 mm, with a sharp initial crack 0.5 mm in total length in the middle. Plane stress constraint condition is chosen. Quarter symmetry is used with the appropriate symmetry boundary conditions imposed on the left and bottom sides (see Fig. 1(a)). The latter boundary condition is modified during the analysis to allow for crack propagation. A hard frictionless contact is defined between the crack surface and an analytical rigid surface corresponding to the other crack flank. Finite element mesh carries the same principle as in previous studies (Ka¨rkka¨inen et al., 2024b); a fine structured mesh consisting of fully integrated linear quadrilateral plane stress elements is used near the crack path, and the rest of the model is freely meshed with the same element type. A minimum element size of 2.5 µ m is chosen in accordance with the literature recommendations, being at least 1 / 10 of the theoretical forward plastic zone size (Oplt et al., 2019; Rice, 1967). A continuum material model with linear kinematic hardening is defined, corresponding to a medium strength steel. Material parameters are Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0 . 3, hardening ratio H / E = 0 . 05, yield strength σ y = 500 MPa, and Vickers hardness HV = 200kgf / mm 2 . An example of the material model behavior in a 2. Modeling

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Fig. 1. Finite element model details. (a) Geometry. (b) Loading pattern. (c) Material model.

uniaxial stress state is presented in Fig. 1(c). Loading consists of cyclic fully reversed ( R = − 1) constant amplitude base loading with a single over- or underload, applied as uniform stress to the top surface of the plate. Stress amplitude of the base loading corresponds to the constant amplitude fatigue limit determined by R-curve analysis, σ a = σ w , CA = 147 MPa (see Section 3 and Fig. 4). Two di ff erent over- and underload magnitudes are considered, normalized by yield strength, σ OL = [0 . 5 , 0 . 75] σ y = [250 , 375] MPa and σ UL = [ − 0 . 75 , − 1 . 0] σ y = [ − 375 , − 500]MPa. Crack propagation is realized by releasing the current crack tip node at maximum load every other load cycle (see Fig. 1(b)). One additional step of maximum load is given after node release. The result, opening level σ op /σ max , is obtained from the final loading step of each pattern. The final length of the crack is such that the initial crack length is doubled, i.e., total crack extension ∆ a = a init = 250 µ m, corresponding to 100 crack growth increments of one element size. A single over- or underload is applied at ∆ a = 100 µ m, i.e., at ∆ a / a init = 0 . 4. The level of plasticity-induced crack closure is measured with the opening level, σ op /σ max , defined by the first node contact criterion; the crack is considered open when the displacement of the first node behind the crack tip is positive. The opening level development during crack propagation is presented in Fig 2. Overloads cause a delayed, transient increase of plasticity-induced crack closure, whereas underloads simply reset the crack closure development. Importantly, either loading spike is able to momentarily remove crack closure.

Fig. 2. Opening level ( σ op /σ max ) development as a function of normalized crack extension ( ∆ a / a init ). A single (a) overload or (b) underload is applied at ∆ a / a init = 0 . 4.

3. Analysis

This section describes and expands on the analysis method originally proposed for underloads by Ka¨rkka¨inen et al. (2024b), in an aim to provide a tool for estimating the fatigue limit in the presence of sporadic loading spikes. The

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e ff ect of over- and underloads on the fatigue limit can be analyzed in a simple and e ff ective manner using the cyclic resistance curve, or R-curve (Tanaka and Akiniwa, 1988; Maierhofer et al., 2018). It is commonly assumed that the development of crack closure corresponds to the R-curve, allowing for the transients in crack closure to be linked to the changes in the R-curve. The modeling results presented in the preceding section show that both over- and underloads can cause a resetting of crack closure development, but overloads are also able to significantly alter the subsequent closure levels. Thus, a resetting of the R-curve can be assumed to follow either loading spike, but the shape of the new R-curve is altered with overloads. The R-curves are modified by adjusting the ∆ K th , lc and a 0 parameters (see Eq. (1)). Fig. 3 presents the R-curves based on the simulated crack closure response, which are used in the present analysis. Notice that for the fatigue limit R-curve analysis, only the initial part of the R-curve is important.

Fig. 3. (a) Simulated crack closure response to an over- or underloads. (b) Corresponding R-curves. The closure response after an underload corresponds to an unaltered R-curve. With overloads, R-curves are modified by adjusting the ∆ K th , lc and a 0 parameters (see Eq. (1)).

Another issue is determining when during crack propagation loading spikes would likely occur. Present analysis assumes sporadic or relatively rare loading spikes in otherwise constant amplitude base loading below the constant amplitude fatigue limit. In this case, the majority of the service time is spent in the state where the crack has arrested at the intersection point of the nominal driving force and the R-curve. A sporadic over- or underload is most likely to occur at this stage, and consequently assumed so in this analysis. As the loading spikes are assumed to be infrequent and finite, their inherent damaging e ff ect can be neglected in this analysis. The proposed method is schematically illustrated for the underload case in Fig. 4 and encapsulated in the following assumptions, whose justifications were discussed above. Overloads follow the same principle; the only di ff erence is that an altered R-curve (Fig. 3(b)) is used from the first overload onwards. • R-curve is reset by a loading spike. • R-curve shape is altered by an overload but not by an underload. • A loading spike occurs at the point marking crack arrest, i.e., the intersection point of the ∆ K -curve and R-curve. Based on these considerations, the e ff ective fatigue limit, σ w , e ff , reduced by single or numerous over- or underloads, can be computed as a function of the number of over- or underloads, n . A condition for crack arrest in terms of R-curve analysis is the equality of ∆ K and cyclic R-curve ∆ K th (Zerbst and Madia, 2015; Maierhofer et al., 2018), ∆ K =∆ K th ⇔    2 σ Y √ π ( a init +∆ a ) =∆ K th , lc  ∆ a + a ∗ − a s ∆ a + a ∗ + a 0 − a s a ∗ = a 0 ( ∆ K th , e ff / ∆ K th , lc ) 2 1 − ( ∆ K th , e ff / ∆ K th , lc ) 2 , a 0 = 1 π  ∆ K th , lc Y ∆ σ th , 0  2 , (1) where σ is the nominal stress amplitude, Y shape factor, a init initial crack length, ∆ a crack advance, ∆ K th , lc long crack threshold stress intensity factor range, ∆ K th , e ff intrinsic threshold stress intensity factor range, and ∆ σ th , 0 material threshold stress range. A new solution-dependent variable, a s , is introduced, which shifts the R-curve by a length equal to ∆ a of the previous arrested crack. At the e ff ective fatigue limit, ∆ K is a tangent of the final R-curve. As Eq.

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Fig. 4. Schematic illustration of the cyclic resistance curve analysis (Ka¨rkka¨inen et al., 2024b). Over- and underloads can be thought to reset the R-curve. A loading history containing three sporadic underloads is considered in this instance. Overloads follow the same principle; the only di ff erence is that an altered R-curve (Fig. 3(b)) is used from the first overload onwards.

(1) transforms into a second order polynomial equation, the tangent condition takes the form

b 2 2 − 4 b 1 b 3 = 0 , where   

b 1 = 4 πσ 2 2 b 2 = ( a 0 + a ∗ + a init − a s ) b 1 − ∆ K 2 th , lc b 3 = ( a 0 + a ∗ − a s ) a init b 1 − ( a ∗ − a s ) ∆ K 2 th , lc . w , e ff Y

(2)

As the problem of finding σ w , e ff ( n ) is recursive in nature, no closed form solution exists. However, the numerical implementation is rather simple. The constant amplitude fatigue limit used for the base loading, σ w , CA = 147MPa, is obtained when the number of loading spikes is zero. The parameter values used are Y = 1, a init = 250 µ m, ∆ K th , lc = 15MPa √ m, ∆ K th , e ff = 2 . 5MPa √ m, and ∆ σ th , 0 = 2 × 1 . 6 HV = 640MPa.

4. Results and discussion

The analysis results concerning the e ff ect of over- and underloads on fatigue limit are provided next, along with relevant discussion. Fig. 5 presents the reduction of σ w , e ff in terms of the ratio to the constant amplitude fatigue limit, σ w , CA , as a function of the number of loading spikes, n . The reduction of the e ff ective fatigue limit is substantial in all cases, and is primarily a consequence of crack growth. It can be seen that σ w , e ff /σ w , CA saturates with a large enough number of underloads in the loading history. Eq. (3) provides the limit value of σ w , e ff , governed by the intrinsic threshold ∆ K th , e ff and initial crack size a init ; once the stress amplitude is lowered enough for the initial ∆ K to equal ∆ K th , e ff , no crack propagation occurs. It is worth noting that Eq. (3) corresponds to the intrinsic threshold line in the Kitagawa–Takahashi-diagram.

∆ K th , e ff 2 Y √ π a init

lim n →∞

σ w , e ff ( n ) =

(3)

A considerable di ff erence in the rate of saturation of σ w , e ff exists depending on the loading spike, or R-curve shape. The significant delay of the R-curve saturation corresponding to the large overload (Fig. 3(b)) translates into an increased sensitivity of σ w , e ff to these loading spikes; fatigue limit is reduced 30 % after only one large overload. This result is qualitatively in alignment with the experimental data from Pompetzki et al. (1990), who considered relatively large over- and underloads (near yield strength) and found overloads to be more e ff ective at lowering the fatigue limit. For most other loading spikes, a pure resetting of the R-curve (unaltered R-curve shape), corresponding to the underload case herein, can be assumed for a simple and conservative analysis. However, the question of how large of

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Fig. 5. The reduction of fatigue limit ( σ w , e ff /σ w , CA ) as a function of the number of over- or underloads ( n ). Corresponding analytical limit value from Eq. (3) is drawn with dashed horizontal lines for validation. The range of n between zero and one is shown on a linear scale.

a loading spike is required to fully reset the R-curve is worthy of discussion. An explicit answer proves di ffi cult, as the problem depends on a multitude of factors, such as base loading stress ratio, crack length, and Bauschinger e ff ect, to name a few. As seen in Fig. 2, all loading spike magnitudes considered here were able to momentarily remove crack closure, even in fully reversed loading. For overloads, the authors’ simulation experience suggests that roughly a 30 % increase in maximum load may give rise to strong enough crack tip blunting to remove crack closure. On the other hand, a 20 % increase has been reported to produce a visible beach-mark on the crack surface (Vosikovsky and Rivard, 1981). As a crack can carry compression but not tension, the R-curve is generally less sensitive to underloads, but the underload response is highly influenced by mean stress, Bauschinger e ff ect, and notch plasticity (Antunes et al., 2019; Ka¨rkka¨inen et al., 2024b). It is worth noting that the present analysis considers only plasticity-induced crack closure, whereas the R-curve consists of the total closure influence, including also roughness and oxide-induced crack closure, for example. How ever, plasticity-induced crack closure is the fastest to saturate (Maierhofer et al., 2018), and thus dominant in the initial part of the R-curve most relevant for the current analysis. Additionally, in plane stress conditions assumed here, the role of plasticity-induced crack closure is elevated (McClung et al., 1991; Pippan et al., 2002; Ka¨rkka¨inen et al., 2024b). Also, the momentary removal of closure following either loading spike applies not only to plasticity-induced crack closure, but to all types of contact between the crack surfaces. As the present analysis assumes relatively rare loading spikes, the nominal load increase is not considered. With frequent overloads, the direct damaging e ff ect of the overload cycles may result in finite life, as also reported by Pompetzki et al. (1990), making the fatigue limit analysis in terms of the base loading amplitude meaningless. The mechanism lowering fatigue limit will nonetheless coexist with the direct e ff ect of recurrent overloads. The present analysis provides a promising tool for estimating the residual fatigue limit a ff ected by a finite number of sporadic loading spikes, although more work is needed for the experimental validation of the current analysis, as relevant results especially for underloads are not readily available in literature.

5. Conclusions

The present article demonstrated a simple method for estimating the fatigue limit in the presence of sporadic over and underloads. Conclusions of the study are as follows.

• Recurrent over- and underloads are detrimental to fatigue limit; even a decrease to one third of the constant amplitude fatigue limit is possible with numerous over- and / or underloads. • Both over- and underloads can remove existing closure and reset the R-curve, enabling existing non-propagating cracks to resume growth. Large overloads may delay R-curve saturation and significantly promote the decline of the e ff ective fatigue limit. • E ff ective fatigue limit saturates with enough over- / underloads. The limit value is governed by intrinsic threshold stress intensity factor range and initial crack size.

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• Present analysis assumes rare overloads. Damaging e ff ect of the actual overload cycles exists in addition to the demonstrated fatigue limit reducing mechanism, which can ultimately result in finite life.

Acknowledgements

Funded by the European Union (Grant Agreement No. 101058179; ENGINE). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Health and Digital Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. The authors wish to acknowledge CSC – IT Center for Science, Finland, for computational resources. The corresponding author wishes to thank Tauno To¨nning Foundation for a personal research grant.

References

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