PSI - Issue 79

Available online at www.sciencedirect.com

ScienceDirect

Procedia Structural Integrity 79 (2026) 524–525

28th International Conference on Fracture and Structural Integrity - 3rd Mediterranean Conference on Fracture and Structural Integrity Preface SabrinaVantadori a , Francesco Iacoviello b *, Giuseppe Andrea Ferro c , Filippo Berto d a Università di Parma, Dipartimento di Ingegneria e Architettura, Italy b Università di Cassino e del Lazio Meridionale, Dipartimento di Ingegneria Civile e Meccanica, Italy c Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Edile e Geotecnica, Italy d "Sapienza" Università di Roma, Dipartimento di Ingegneria Chimica, Materiali, Ambiente, Italy b Department of Mechanical Engineering, © 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 IGF28 - MedFract3 organizers Keywords: Gruppo Italiano frattura; Conference organization; IGF YouTube channel. Preface The Italian Group of Fracture (IGF) is a cultural association devoted to: (i) spreading and promoting works and researches about fracture phenomena, even forming workgroups; (ii) promoting all the activities concerning the development of materials and structure testing standards; (iii) cooperating with foreign associations with the same intents; (iv) organizing meetings, workshops, conferences, debates and courses about fracture phenomena; (v) publishing meetings proceedings, news, journals. Since its foundation in 1982, IGF organized dozens of national and international conferences, workshops and summer schools, both in presence and, more recently, in remote or in a blended version. Since 2007, IGF videorecorded all the events, publishing the video recordings of the presentations in a dedicated YouTube channel (https://www.youtube.com/@IGF_YT), associating a DOI to each video presentation and, also, collected them in ISBN-numbered books available through ESIS-PH (https://www.esis-ph.eu/). The IGF28-MedFract3 was as a joint conference of two different periodical events:

* Corresponding author. Tel.: +3807762993681 E-mail address: iacoviello@unicas.it

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 IGF28 - MedFract3 organizers 10.1016/j.prostr.2026.01.087

Sabrina Vantadori et al. / Procedia Structural Integrity 79 (2026) 524–525

525

- IGF28, 28th International Conference on Fracture and Structural Integrity, that is the biennial event organized by IGF. In its first issues, the event was in Italian and dedicated to the Italian IGF members. In the last years, it was transformed in an international conference with English as main language, but the events were always held in Italy; - MedFract3, 3rd Mediterranean Conference on Fracture and Structural Integrity, is the third edition of an event that is organized every three years. The first edition was held in Athens, in cooperation with the Greek Group of Fracture. The second edition was held in Catania and online. This conference was a resounding success, drawing over 170 participants from 30 different countries. The program featured four distinct invited lectures: - Fretting fatigue: our recent developments on the use of Artificial Neural Networks and on the conduct of complex tests (José Alexander Araújo); - Mean stress effect, stress ratio R in Fatigue Crack Growth Rate description - from Dimensional Analysis to generalization of Paris law (Grzegorz Lesiuk); - The effect of laser induced residual stress on fatigue properties of titanium alloys (Oleg Plekhov); - Fatigue short crack growth prediction of additively manufactured alloy based on ensemble learning (Guian Qian). Five different thematic symposia took place: - Defects in additive manufactured materials and their effect on fracture and fatigue performance. Chairs: S. Vantadori, F. Berto; - Artificial Intelligence and Physics-based Numerical Methods for Fracture and Fatigue Damaging Processes. Chairs: A. Tridello, E. Salvati; - Structural Integrity of Solid-State Welded & Additively Manufactured Metals. Chairs: E. Salvati, S. Bagherifard; - Durability of structural joints: experimental, theoretical and numerical approaches. Chairs: A. Califano, G. Cricrì, V. Giannella, R. Sepe; - Energy Methods for Fatigue Assessment. Chairs: D. Santonocito, P. Foti, G. Risitano, F. Berto. During the conference, IGF Honorary Membership was awarded to José Alexander Araújo, Grzegorz Lesiuk, Oleg Plekhov, and Guian Qian. Additionally, Sabrina Vantadori received the Manson-Coffin IGF Medal , and Giuseppe Andrea Ferro was honored with the IGF Award of Merit . According to the IGF tradition, all the video recordings of the presentations are now available in the IGF YouTube channel and are collected in an ISBN-numbered book available through ESIS-PH. YouTube playlist: https://www.youtube.com/playlist?list=PLT1-2PyZ6QrJH2JaFjeMogheK_CEZFziY Presentations Book: https://www.esis-ph.eu/index.php/eph/catalog/book/268. Looking forward to meeting all of you in the next IGF conferences.

Available online at www.sciencedirect.com

ScienceDirect

Procedia Structural Integrity 79 (2026) 291–297

© 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) Abstract In this study, the micromechanical response of a representative volume element (RVE) under cyclic loading was simulated using the crystal plasticity finite element method (CPFEM) to obtain the local stress-strain response and accumulated plastic strain. Based on the high-fidelity data generated by CPFEM, an incremental neural network (INN) model was constructed. The INN model takes the load ratio and the current accumulated plastic strain as inputs to predict the corresponding accumulated plastic strain increment for a given number of cycles. Compared with traditional fatigue prediction models, this model does not require presetting empirical equations. The results demonstrate that this incremental learning approach can effectively capture the nonlinear evolution of plastic strain with the number of cycles. The developed single-hidden-layer INN model accurately predicts the plastic strain accumulation process in laser powder bed fusion (LPBF) GH4169 (Inconel 718) under cyclic loading and achieves the highest prediction accuracy. 28th International Conference on Fracture and Structural Integrity - 3rd Mediterranean Conference on Fracture and Structural Integrity Accumulated plastic strain prediction of LPBF alloy under fatigue load based on CPFEM and incremental neural network Qinghui Huang 1,2 , Pengbo Wang 1,2 , Lei Bian 3 , Meng Zhao 3 , Pablo Lopez-Crespo 4 , Ivan Sergeichev 5 , Filippo Berto 6 , Wenqi Liu 1,* , Guian Qian 1,* 1 State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing, China 2 School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China 3 Dynamic Machinery Institute of Inner Mongolia, Hohhot, China 4 Department of Civil and Materials Engineering, University of Malaga, Malaga, Spain 5 Skolkovo Institute of Science and Technology, Center for Materials Technologies, Moscow, Russia 6 Department. of Chemical Engineering Materials Environment, Sapienza University of Rome, Rome, Italy

Peer-review under responsibility of IGF28 - MedFract3 organizers Keywords: Neural network, Crystal plasticity, Accumulated plastic strain

* Corresponding author. Tel.: +86 135-5267-7285. E-mail address: liuwenqi@imech.ac.cn, qianguian@imech.ac.cn

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 IGF28 - MedFract3 organizers 10.1016/j.prostr.2025.12.336

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

292

1. Introduction Metal alloys fabricated by laser powder bed fusion (LPBF) technology exhibit complex mechanical responses due to their unique microstructures, such as pores, lack of fusion defects, and anisotropic grain orientation (Herzog et al. (2016)). Particularly under cyclic loading, the accumulation of microscopic plastic strain directly influences the fatigue life and durability of the material (Mellor et al. (2014), Taghizadeh and Zhu (2024)). Therefore, accurately predicting the accumulated plastic strain of LPBF alloys under fatigue loads is crucial for evaluating the reliability of additively manufactured components. The microstructure of LPBF GH4169 (Inconel 718) exhibits significant spatial non-uniformity. During the LPBF process, the coarse-grained and fine-grained regions that tend to form present a spatially ordered bimodal structure (Zhu et al. (2021)). This structure leads to a highly localized plastic strain distribution of the material during cyclic deformation. In recent years, the crystal plastic finite element method (CPFEM) has become a powerful tool for studying the micromechanical behavior of metallic materials (Roters et al. (2010)). By coupling the crystal plasticity constitutive and finite element simulation, CPFEM can reproduce the local stress-strain response and accumulated plastic strain of materials under cyclic loading. For instance, researchers have analyzed the influence of different defect sizes and locations on fatigue indicator parameters (FIPs) through CPFEM simulation, thereby establishing correlations with fatigue life (Hao et al. (2022)). However, the computational cost of CPFEM is high, making it difficult to be directly applied to fatigue analysis and life prediction at the engineering component level. Meanwhile, data-driven methods, especially artificial neural networks (ANNs), have shown great potential in predicting the fatigue behavior of materials. For example, a deep belief network-back propagation (DBN-BP) model has been successfully used to predict the fatigue life of the Ti-6Al-4V fabricated by LPBF in the high-cycle, low-cycle and very-high-cycle fatigue regimes (Le et al. (2021)). However, most existing data-driven models mainly focus on the prediction of macroscopic fatigue life ( N f ), but fail to capture the incremental evolution of plastic strain during cyclic deformation. To address these challenges, this paper innovatively proposes a multiscale framework that combines CPFEM and neural networks (NNs) to accurately predict the accumulated plastic strain of LPBF GH4169 under fatigue loading. This framework fully exploits the advantages of CPFEM in revealing microscopic mechanical responses and the ability of incremental neural networks (INNs) to capture nonlinear temporal evolution, providing a new solution for the accurate fatigue damage assessment and life prediction of in LPBF alloys.

Nomenclature RVE

representative volume element CPFEM crystal plasticity finite element method INN incremental neural network LPBF laser powder bed fusion DBN-BP deep belief network–back propagation RMSE root mean square error MAE mean absolute error

2. Method This paper introduces a research framework that combines multiscale simulation with data-driven methodologies. The core lies in generating high-fidelity data through CPFEM simulation and then training an incremental neural network (INN) to predict the accumulated plastic strain under fatigue loads of LPBF alloys. 2.1 CPFEM theory and simulation Crystal plasticity finite element (CPFE) theory is an advanced numerical simulation method based on material microstructure. It integrates the physical mechanisms of crystal plasticity with the macroscopic finite element

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

293

framework to predict both the micromechanical responses and macroscopic mechanical behavior of polycrystalline metals under complex loading conditions. The core premise is that the macroscopic nonlinear mechanical behavior of materials originates from the collective manifestation of microscopic physical processes within their constituent grains, including crystal orientation, morphological evolution, and dislocation slip (Hutchinson, (1976)). This study focuses on the fabrication of GH4169 (Inconel 718) Ni-based superalloy materials using a commercial EP-M250 LPBF system. To accurately capture the micromechanical behavior of LPBF GH4169, this study developed a CPFEM simulation model based on the representative volume element (RVE) of material. The Voronoi method was employed to construct an equivalent RVE model with an equiaxed structure and random grain orientation. Three distinct strain amplitudes (εₐ) were selected: 0.15%, 0.195% and 0.25%, with a load ratio (R) of -1. According to the experimental data of LPBF GH4169, these model parameters were calibrated, as summarized in Table 1. The accumulated plastic strain was recorded during the first 50 cycles of simulation (Li et al. (2015)). The CPFEM simulation data were subsequently used for training and validation of the neural network model. Table 1. Elasticity and crystal plasticity model parameters of the LPBF GH4169 alloy. C 11 [MPa] C 12 [MPa] C 44 [MPa] 0 γ  n κ [MPa] c 1 , c 2 b Q [MPa] 311145.511 153250.774 78947.3684 0.00025 12 130.5 1,0.5 10 12 2.2 Incremental neural network The INN designed in this paper aims to capture the evolution behavior of plastic strain from incremental value under cyclic load.

Figure 1 Incremental neural network architectures for Δ prediction. The INN adopts the concept of incremental learning to capture the evolution of material response from stepwise data. Instead of directly mapping the total value, the INN predicts the incremental quantity Δ between successive time steps, which effectively embeds the derivative information and augments the dataset (Ma et al. (2021)). The investigated network architectures include: 1) a simple network, 2) a single-hidden-layer network, and 3) a double hidden-layer network, as illustrated in Fig. 1. The model employs strain amplitude ( a ) and the current timestep accumulated plastic strain as input to predict the incremental plastic strain ( ∆ ). The architecture consists of two input layer neurons and one output layer neuron, with all three networks utilizing ReLU activation function. To

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

294

prevent overfitting, dropout regularization (with rates of 0.001 and 0.1) and L2 regularization (with a strength of 0.001) are applied to the hidden layers. The training process incorporates early stopping techniques while visualizing loss values on the validation set. The model is trained to predict ∆ , and the accumulated plastic strain is updated recursively via , +1 = , + ∆ , , starting from an initial value of zero. The forward propagation is defined by Equation (1), and model loss function is quantified by the formula in Equation (2). ( α ) = ∑ ( ℎ ) = 1 (1) Where is the output of the -th input neuron, ( ℎ ) is the weight between the input and hidden layers, and ( ) is the weighted input of the -th hidden neuron (before activation). The ReLU function given by ( )= (0, ) . = 1 ∑ � Δ , − Δ , �� 2 = 1 (2) Where Loss is the mean squared error, is the number of samples, Δ , is the true incremental value, and Δ� , is the predicted incremental value. 3. Results and discussion 3.1 Analysis and prediction of accumulated plastic strain growth Fig. 2 presents the accumulated plastic strain contours for the first fifty cycles under different strain amplitudes ( εₐ ) at R = -1. The red regions indicate areas with higher plastic strain accumulation. Under different strain amplitudes, high-strain hotspots persist at specific grain boundaries, indicating microstructural control over fatigue nucleation sites, which is consistent with CPFEM-based fatigue initiation studies. The figures show that the locations of extreme accumulated plastic strain values are independent of the strain amplitude but are intrinsically related to the grain microstructure itself. Fig. 3 presents the actual accumulated plastic strain evolution curves (from CPFEM simulations) and neural network predictions for LPBF GH4169 under va rying strain amplitudes (0.15%, 0.195% and 0.25%) and - 1 load ratio conditions.

Figure 2 Accumulated plastic strain contour map based on CPFEM.

Figure 3 INN predicted accumulated plastic strain (Pac or ε p,a) curves at different strain amplitudes.

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

295

Figure 4 Schematic diagram of cyclical loading waveform.

The data reveal that as cycle count (t) increases, the accumulated plastic strain of LPBF GH4169 exhibits a nonlinear growth trend, consistent with the microdeformation mechanisms under cyclic loading. During the initial stage, the accumulated plastic strain rises rapidly. This occurs because LPBF GH4169 initially can contains a high density of dislocations, which rapidly move and proliferate under cyclic loading, significantly increasing plastic deformation. Comparative analysis of curves at different strain amplitudes shows that higher amplitudes result in greater accumulated plastic strain at equivalent cycle counts. For instance, at t = 5 (N=50 cycles, as shown in Fig. 4), the 0.25% strain amplitude corresponds to a larger accumulated plastic strain than the 0.15% ampl itude. This occurs because higher strain amplitudes generate greater internal local stresses, which more effectively promote dislocation motion and multiplication, thereby accelerating plastic strain accumulation. The results demonstrate excellent agreement between the neural network- predicted curves and the CPFEM simulation results, particularly at the 0.25% strain amplitude. The INN model effectively captures the gradual deceleration in the rate of plastic strain accumulation as the number of cycles increases. 3.2 Evaluation of model prediction performance Fig. 5 illustrates the trends of training and validation losses over epochs for three neural network models: simple, single-hidden-layer, and double-hidden-layer networks.

Figure 5 Loss curve comparison of different model architectures.

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

296

The simple network shows rapid declines in both training and validation losses during the initial 200 epochs, but stabilizes beyond this threshold with validation losses slightly higher than training losses. This indicates limited capability to capture nonlinear relationships and evidence of underfitting. The single-hidden-layer network demonstrates continuous decreases in both training and validation losses through increased epochs, with validation losses approaching training losses. Notably, when epochs reach 800-1000, validation losses hit their lowest point before stabilizing, suggesting a strong learning capacity to effectively model nonlinear relationships between input parameters and incremental plastic strain, while early stopping techniques successfully mitigated overfitting. The double-hidden-layer network maintains an overall downward trend in training losses, though with fluctuations consistently exceeding those of the single-hidden-layer network. This discrepancy likely stems from its higher complexity and insufficient training sample size to support effective model learning. Fig. 6 and Fig. 7 present the prediction results and performance metrics of three structural neural network models (simple network, single-hidden layer, and double-hidden-layer) on the validation set. In the scatter plot comparing predicted values with actual ones, the single-hidden-layer model achieved a validation set R² value of 0.9797. This indicates great alignment between predictions and actual values, demonstrating the ability to accurately capture the plastic strain accumulation. In contrast, the simple network (R²=0.5449) showed more scattered data points around the diagonal, while the double-hidden layer model (R²=0.8963) exhibited lower consistency in data distribution compared to the single-hidden-layer model. This suggests that the simple network lacks the complexity required to model nonlinear relationships, whereas the double-hidden-layer model may have been affected by overfitting or improper architecture selection.

Figure 6 Predicted results under validation set.

Figure 7 RMSE, R ² and MAE values (under validation set) for different model structures

The bar charts of quantitative indicators such as root mean square error (RMSE), R², and mean absolute error (MAE) further quantify model performance. Regarding RMSE, the single-hidden-layer model (Model 2) achieved the lowest value of 0.0003, significantly lower than the simple network (Model 1: 0.0015) and the double-hidden-layer (Model 3: 0.0007). A lower RMSE indicates smaller deviations between predicted and actual values. For R², the single-hidden layer model reached 0.9797, markedly higher than the simple network (0.5449) and the double-hidden-layer (0.8963). A higher R² value signifies stronger variance interpretation capability and superior predictive performance. In terms of mean absolute error (MAE), the single-hidden-layer model demonstrated the best performance with a value of

Qinghui Huang et al. / Procedia Structural Integrity 79 (2026) 291–297

297

0.0002, followed by the simple network model at 0.0013 and the double-hidden-layer model at 0.0006. MAE measures average absolute error, where lower values indicate more accurate individual predictions. Overall, among these three models, the single-hidden-layer neural network strikes an optimal balance between complexity and generalization capability, making it the preferred choice for predicting accumulated plastic strain under fatigue loading. 4. Conclusions This study established a CPFEM simulation model to capture the plastic strain accumulation process of LPBF GH4169 under cyclic loading. The proposed incremental neural network (INN) model effectively predicts accumulated plastic strain in LPBF GH4169 under fatigue loading. Among three constructed INN models (simple network, single-hidden-layer network, and double-hidden-layer network), the single-hidden-layer network demonstrated best predictive performance. It achieved an R² value of 0.9797 on the validation set, with mean absolute error (MAE) and root mean square error (RMSE) at 0.0002 and 0.0003, respectively. This model accurately captures nonlinear evolution of plastic strain with the number of cycles without requiring predefined empirical equations, thereby overcoming a key limitation of traditional fatigue prediction models. The multiscale framework integrating CPFEM and INN provides a novel solution for the accurate prediction of accumulated plastic strain and fatigue damage assessment in LPBF alloys. In this framework, CPFEM generates high-precision micromechanical data, while the neural networks learn nonlinear evolution patterns of plastic strain from this data, achieving deep integration of multiscale simulation and data-driven methodologies. Acknowledgements This work was funded by the International Partnership Program for Grand Challenges of Chinese Academy of Sciences (025GJHZ2023092GC) and the Strategic Priority Research Program of Chinese Academy of Sciences (XDB0620303). References Herzog D, Seyda V, Wycisk E, Emmelmann C, 2016. Additive manufacturing of metals. Acta Materialia 117, 371–392. Mellor S, Hao L, Zhang D, 2014. Additive manufacturing: A framework for implementation. International Journal of Production Economics 149, 194–201. Taghizadeh M, Zhu Z, 2024. A comprehensive review on metal laser additive manufacturing in space: Modeling and perspectives. Acta Astronautica 222, 403–421. Zhu L, Xue P, Lan Q, et al, 2021. Recent research and development status of laser cladding: A review. Optics & Laser Technology 138, 106915. Roters F, Eisenlohr P, Hantcherli L, Tjahjanto D, Bieler T.R, Raabe D, 2010. Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling. Acta Materialia 58, 1152–1211. Hao X, Liu G, Wang Y, Wu S, Wang Z, 2022. Optimization of investment casting process for K477 superalloy aero-engine turbine nozzle by simulation and experiment. China Foundry 19, 351–358. Le B Q, Tran V H, Bui Q M, Pham V H, 2021. Fatigue life prediction of additively manufactured Ti-6Al-4V alloy using deep belief network back propagation model. Materials Science and Engineering: A 825, 141862. Hutchinson, J., 1976. Bounds and self-consistent-estimates for creep of polycrystalline materials. 348. Li L, Shen L, Proust G, 2015. Fatigue crack initiation life prediction for aluminium alloy 7075 using crystal plasticity finite element simulations. Mechanics of Materials 81, 84–93. Ma X, He X, Tu Z C, 2021. Prediction of fatigue-crack growth with neural network-based increment learning scheme. Engineering Fracture Mechanics 241, 107402.

Available online at www.sciencedirect.com

ScienceDirect

Procedia Structural Integrity 79 (2026) 361–369

© 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 IGF28 - MedFract3 organizers Abstract The statistical evaluation of S-N curves often involves censored data in the form of runouts. While the qualitative e ff ect of runouts on parameter estimation is well understood, their influence on the width and reliability of confidence intervals has not been sys tematically quantified. This study focuses on the bilinear S-N curve model and investigates how varying proportions of runouts in the dataset influence the confidence intervals of key model parameters, including the stress and number of cycles at the knee point, as well as the slope parameters. Using maximum likelihood estimation within a probabilistic framework based on a log-normal distribution, a Monte Carlo simulation is performed in which the proportion of runouts is systematically varied. Confidence inter vals are derived using the likelihood ratio method, which is particularly suitable for an appropriate treatment of runouts. This work provides a quantitative foundation for understanding the impact of runouts on uncertainty in bilinear S-N curve models and o ff ers practical guidance for test planning and model interpretation in fatigue analysis. 28th International Conference on Fracture and Structural Integrity - 3rd Mediterranean Conference on Fracture and Structural Integrity A Coverage Study on the E ff ect of Runouts on Confidence Intervals for Bilinear S-N Curves Felix-Christian Reissner ∗ , Jo¨rg Baumgartner Fraunhofer Institute for Structural Durability and System Reliability LBF, Bartningstraße 47, 64289 Darmstadt, Germany

Keywords: S-N curve; Confidence intervals; Likelihood ratio; Maximum likelihood estimation

1. Introduction

Fatigue design relies on an accurate characterization of the relationship between cyclic load amplitude S a , i and fatigue life N i , commonly represented by S-N curves. These models serve as the foundation for predicting fatigue life, defining design limits, and ensuring structural durability. In engineering practice, S-N curves are typically derived from experimental data containing both failures and runouts, i.e., specimens that have not failed within the test dura tion. The statistical treatment of these censored observations remains a central challenge in fatigue data evaluation. Runouts contain valuable information, as they indicate that the fatigue strength of a specimen exceeds the applied stress level for a given number of cycles. Ignoring them, as done in many conventional evaluations, can lead to bi ased parameter estimates. Modern fatigue analysis therefore incorporates runouts explicitly, for example by maximum

∗ Corresponding author. E-mail address: felix-christian.reissner@lbf.fraunhofer.de (Felix-Christian Reissner).

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 IGF28 - MedFract3 organizers 10.1016/j.prostr.2025.12.346

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

362

likelihood estimation (MLE) frameworks, see Pascual (1999); Meeker et al. (2024); Sto¨rzel and Baumgartner (2021). However, while the influence of runouts on point estimates of S-N parameters is qualitatively understood, their quan titative e ff ect on the accuracy and reliability of confidence intervals has not been systematically assessed. Confidence intervals (CIs) provide a measure of statistical uncertainty and are valuable for evaluating the robustness and reliability of estimated S-N parameters. In particular, they allow engineers to assess the precision of the knee point and slope parameters, which directly influence fatigue design margins. Despite their importance, the computation and interpretation of CIs in fatigue analysis remain inconsistent across standards and guidelines like the FKM Guideline, DIN 50100, ASTM E739-10 or ISO 12107, where either confidence intervals are not provided or nonlinear models are not considered. These limitations motivate a probabilistic framework capable of handling nonlinearity, censored data, and small sam ple sizes typically encountered in fatigue testing. Likelihood-based confidence intervals, and in particular those derived from the profile likelihood ratio, provide a robust and conceptually sound approach to quantify uncertainty under these conditions. They are assumed to provide reliable results under censoring and with small sample sizes Meeker et al. (2017), but systematic validation in the context of S-N curves, especially for bilinear models with runouts, remains scarce. The objective of this work is therefore to quantify the influence of runouts on the coverage behavior of profile likelihood confidence intervals for bilinear S-N models. The bilinear Basquin model is chosen as it represents a practical and widely used formulation. Using a Monte Carlo framework, the coverage probability, i.e., the propor tion of intervals containing the true parameter value, is evaluated for di ff erent proportions of runouts. This allows a quantitative assessment of whether profile-likelihood CIs achieve their nominal confidence levels under realistic test conditions.

2. Background and Theory

The characterization of fatigue behavior is commonly based on S-N data, where S a , i denotes the applied stress amplitude and N i the corresponding number of cycles to failure. In experimental practice, a series of specimens is tested under di ff erent constant load amplitudes, and the resulting pairs ( S a , i , N i ) describe the relation between load and fatigue life. Since the data typically span several orders of magnitude in both stress and lifetime, logarithmic transformations S a , log , i = log 10 ( S a , i ) and N log , i = log 10 ( N i ) are applied, yielding a more tractable representation that reveals approximately linear trends. These trends form the basis of S-N models.

2.1. Bilinear Basquin S-N model

The linear Basquin model Basquin (1910) is one of the best-known S-N models and is widely used. Here, the bilinear Basquin model is employed, as it allows for di ff erent slopes in the long-life and high-cycle regimes, thereby capturing the characteristic change in fatigue behavior near the knee point. Usually, the fatigue life N i is modeled as the dependent variable while the stress amplitude S a , i is the independent variable. This follows the physical relationship between stress amplitude S a , i and fatigue life N i . However, in this work the relationship is reversed. Meeker et al. (2024) show that taking the fatigue life N i as the independent variable simplifies the scatter. Using this approach, the scatter can be modeled as constant over the entire S-N curve, which is a good approximation for most engineering materials. Finally, the bilinear S-N model is given by: S a , log , i =   S a , k , log + 1 k 1 ( N log , i − N k , log ) , if N i ≤ N k , S a , k , log + 1 k 2 ( N log , i − N k , log ) , if N i > N k . (1) Here, N k , log denotes the knee point, S a , k , log the corresponding load amplitude, k 1 the slope before the knee point, and k 2 the slope after the knee point. The scatter of the fatigue data is modeled assuming a normal distribution of the logarithmic load amplitude S a , log , i ,

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

363

which corresponds to a log-normal distribution of the load amplitude S a , i This yields the complete S-N model S a , log , i = g ( N log , i | θ ) where the parameter vector is defined as θ = ( S a , k , log , N k , log , k 1 , k 2 ,σ S , log ) ⊺ .

2.2. Likelihood with failures and runouts

The model parameters θ are estimated by maximum likelihood estimation (MLE). The likelihood function L ( θ ) is constructed based on the probability density function (PDF, see Eq. 2) for failed samples and the survival function (SF, see Eq. 3) for runouts, i.e., right-censored observations. The PDF is

exp  −

2 σ 2  ,

( S a , log , i − µ i ) 2

1 √ 2 πσ 2

2 )

f ( S a , log , i | µ i , σ

(2)

=

where the mean µ i is given by the S-N model µ i = g ( N log , i | θ ) and the variance σ 2 is σ 2 S , log . The SF is

2 )

2 ) ,

SF ( S a , log , i | µ i ,σ

= 1 − F ( S a , log , i | µ i ,σ

(3)

where F ( · ) is the cumulative distribution function of the normal distribution of the logarithmic stress amplitude. With the PDF and SF, the (full) likelihood function for independent observations is

i =  

n 

i = 1 

2 ) 1 − δ i  , where δ

1 , if specimen i is a failure , 0 , if specimen i is a runout .

2 ) δ i · SF ( S

(4)

f ( S a , log , i | µ i ,σ

L ( θ ) =

a , log , i | µ i ,σ

2.3. Profile-likelihood confidence intervals

Beyond point estimation, the likelihood can be used to construct confidence intervals via the profile likelihood . Let the full parameter vector be θ =  λ ψ  ,

where λ is the scalar parameter of interest (e.g., S a , k , log ) and ψ denotes the vector of nuisance parameters. Let L ( ˆ θ ) denote the global maximum likelihood. The profile likelihood is then defined as

ψ 

L ( ˆ θ ) 

L ( λ, ψ )

R ( λ ) = max

(5)

,

i.e., the likelihood of the parameter of interest λ maximized over the nuisance parameters ψ and normalized by the global maximum likelihood L ( ˆ θ ). Therefore, R ( λ ) reaches its maximum value at the maximum likelihood estimate ˆ λ . As shown by (Mavrakakis, 2021, p. 139), the statistic − 2log R ( λ ) converges in distribution to a χ 2 random variable

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

364

with one degree of freedom under standard regularity conditions. Hence, a (1 − α ) confidence set for λ is givenby R ( λ ) ≥ exp − 1 2 χ 2 1 , 1 − α , (6) i.e., the set of λ values where the profile likelihood ratio remains above the cuto ff . Here, α denotes the significance level, representing the probability that the true parameter value lies outside the constructed confidence interval. Graph ically, the confidence limits are defined by the intersections between R ( λ ) and the horizontal cuto ff line, see Fig. 1, where the horizontal cuto ff line is placed at (1 − α ) = 0 . 9.

Fig. 1: Schematic profile likelihood and confidence interval construction.

3. Methodology

3.1. Base model for simulation

The base model for sampling is given by Eq. (1) and the standard deviation σ S , log . The parameters of the base model are given in Table 1. Sampling is performed at evenly spaced logarithmic fatigue-life values. To account for physical plausibility, samples are truncated three standard deviations below S a , log = g ( N log = 4 | θ ). Any truncated sample is resampled uniformly in N between1 · 10 4 and1 · 10 7 until it falls outside the truncation region. To incorporate runouts, the right limit for sampling is extended and samples with fatigue life N > 1 · 10 7 are considered runouts. Load amplitude at the knee point Knee point Slope ( N i ≤ N k ) Slope ( N i > N k ) Standard deviation S a , k N k k 1 k 2 σ S , log 125 1 · 10 6 5 22 0.03 Table 1: Base model parameters used for data generation.

3.2. Design of experiments (DOE)

The aim of this study is to evaluate the actual coverage of profile-likelihood confidence intervals for the given S-N model with and without runouts for a sample size of n = 15. The DOE is summarized in Table 2. For each coverage setting, 1000 simulation runs are conducted, and each profile likelihood is constructed from 100 constrained likelihood evaluations. In this study, profile likelihoods are estimated for the load amplitude at the knee point S a , k , the knee point N k , and the slope k 1 .

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

365

Series Min. number of cycles Max. number of cycles Number of samples Number of runouts N min N max n n ro A 1 · 10 4 1 · 10 7 15 0 B 1 · 10 4 5 . 6 · 10 7 15 3 Table 2: Design of experiments.

3.3. Coverage study procedure

For each parameter, the following steps are performed:

1. Generate S-N data from the base model (Eq. (1)) with the parameter set in Table 1 and standard deviation σ S , log , following the DOE settings in Table 2. 2. Estimate the (constrained) likelihoods across a grid for the parameter of interest λ , maximizing over the nuisance parameters to obtain the profile likelihood. 3. Derive the CI based on Eq. (6). 4. Record whether the true parameter is within the CI (coverage).

4. Results

4.1. Coverage results for the load amplitude at the knee point S a , k

The empirical coverage for the load at the knee point S a , k is shown in Fig. 2. The empirical coverages are comparable across the investigated confidence levels, regardless of the presence of runouts. It should be noted that the presence of runouts suppresses unrealistically low values of S a , k that occasionally occur in Series A. In all cases, the empirical coverage tends to underestimate the nominal confidence level.

Fig. 2: Profile likelihoods and confidence intervals (CI) for the load amplitude at the knee point S a , k .

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

366

The convergence between the empirical histogram of the likelihood-ratio statistic and the reference χ 2 distribution is illustrated in Fig. 3. While the agreement is generally reasonable, some discrepancies remain. The bins fall below the χ 2 curve at low test-statistic values (up to about 0 . 5) and are above at higher values. In addition, the confidence levels based on the χ 2 distribution are shown.

Fig. 3: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the load amplitude at the knee point S a , k .

4.2. Coverage results for the knee point N k

The results for the knee point N k (Fig. 4) are qualitatively similar to those for the load amplitude at the knee point S a , k . The presence of runouts tends to shift the knee point N k toward lower values. As with the load amplitude at the knee point S a , k , empirical coverage underestimates the nominal level across all investigated confidence levels. However, runouts appear to slightly improve empirical coverage for the knee point N k .

Fig. 4: Profile likelihoods and confidence intervals (CI) for the knee point N k .

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

367

Regarding convergence to the χ 2 distribution, the behavior for the knee point N k is similar to that of the load amplitude at the knee point S a , k (Fig. 5), showing slightly better agreement in the presence of runouts. The better agreement is especially noticeable in the first histogram bins corresponding to confidence levels below 0.3. This improvement is also noticeable at higher confidence levels greater than 0.8.

Fig. 5: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the knee point N k .

4.3. Coverage results for the slope k 1

For the slope k 1 (HCF regime, N i ≤ N k ), empirical coverage is consistently below the nominal levels. Unlike the knee-point parameters, the presence of runouts shows little to no e ff ect on the coverage of the slope k 1 . The corresponding likelihood-ratio histograms (Fig. 7) resemble those of the load amplitude at the knee point S a , k and the knee point N k and show the same general under- / overestimation pattern relative to the χ 2 reference.

Fig. 6: Profile likelihoods and confidence intervals (CI) for the slope k 1 .

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

368

Fig. 7: Empirical histogram vs. χ 2 PDF and nominal confidence levels (CL) for the slope k 1 .

4.4. Summary

Figure 8 summarizes the coverage behavior of the load amplitudes at the knee point S a , k , the knee points N k , and the slopes k 1 for confidence levels from 0 . 10 to 0 . 98. All empirical coverages tend to fall below nominal levels. The presence of runouts appears to improve coverage primarily for the knee point N k , while the e ff ect on the load amplitude at the knee point S a , k is minor for nominal confidence levels greater than 0.8, and the e ff ect on the slope k 1 is minor across all investigated confidence levels.

Fig. 8: Summary of empirical coverage vs. nominal confidence level and percent error.

5. Discussion

This study indicates a systematic undercoverage of profile-likelihood confidence intervals in the investigated test setup. Potential contributors include the small sample size and limited identifiability of certain parameters. According to Wilks’ theorem Wilks (1938), convergence is asymptotic, i.e., holds as n → ∞ . Nevertheless, the bias remains moderate for sample size n = 15, typically below 15%, and below 10% for higher confidence levels ( > 0 . 9). Thus, despite the bias, profile-likelihood confidence intervals provide meaningful uncertainty quantification for bilinear S-N models, particularly when censoring due to runouts is appropriately handled in the likelihood. In addition, the bias can likely be reduced by the use of small-sample corrections like Bartlett’s correction Bartlett (1937).

Felix-Christian Reissner et al. / Procedia Structural Integrity 79 (2026) 361–369

369

6. Conclusion

This study systematically quantified the e ff ect of runouts on profile-likelihood confidence intervals in bilinear S-N curve models. Across all investigated parameters, the empirical coverage was consistently below the nominal level, indicating a systematic undercoverage. The bias, however, remained moderate (typically below 15%, and less than 10% for high confidence levels), suggesting that profile-likelihood intervals remain practically useful in fatigue appli cations. Runouts primarily influenced the identifiability of the knee point N k , while their e ff ect on the load amplitude at the knee point S a , k and the slope k 1 was limited. These results highlight the importance of considering censoring ef fects in fatigue test planning and provide practical guidance on the interpretation and reliability of confidence intervals in fatigue analysis. Bartlett, M.S., 1937. Properties of su ffi ciency and statistical tests. Proceedings of the royal society of london. series a-mathematical and physical sciences 160, 268–282. Basquin, O.H., 1910. The exponential law of endurance tests, in: Proceedings of American Society of Testing Materials, pp. 625–630. Mavrakakis, M.C., 2021. Probability and statistical inference. Texts in statistical science, CRC Press, Taylor Francis Group, Boca Raton. Meeker, W.Q., Escobar, L.A., Pascual, F.G., Hong, Y., Liu, P., Falk, W.M., Ananthasayanam, B., 2024. Modern statistical models and methods for estimating fatigue-life and fatigue-strength distributions from experimental data. arXiv:2212.04550 . Meeker, W.Q., Hahn, G.J., Escobar, L.A., 2017. Statistical Intervals A Guide for Practitioners and Researchers. Wiley. Pascual, Francis G.; Meeker, W.Q., 1999. Estimating fatigue curves with the random fatigue-limit model. Technometrics 41, 277–289. doi: 10. 1080/00401706.1999.10485925 . Sto¨rzel, K., Baumgartner, J., 2021. Statistical evaluation of fatigue tests using maximum likelihood. Materials Testing 63, 714–720. doi: 10.1515/ mt-2020-0116 . Wilks, S.S., 1938. The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics 9, 60–62. doi: 10.1214/aoms/1177732360 . References

Available online at www.sciencedirect.com

ScienceDirect

Procedia Structural Integrity 79 (2026) 81–87

28th International Conference on Fracture and Structural Integrity - 3rd Mediterranean Conference on Fracture and Structural Integrity Additive Manufacturing Applied to Maintenance Management

Suzana Lampreia a *, Teresa Morgado b , João Alves c a CINAV – Portuguese Naval Academy, Base Naval de Lisboa, Alfeite-Almada, Portugal; b Instituto Superior de Engenharia de Lisboa, Lisboa, Portugal; c UNIDEMI, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Monte da Caparica, Portugal

Abstract When an organization decides to still use assets with many years of operation, for example ships or equipment’s, some challenges in supply chain of spare parts and systems preservation may become high level of challenge, not only because it isn´t available, but also the access to components if ships are on sea. In the area of spare parts nowadays we have available 3D printers that enhance autonomous capacity and rapid responses to some anomaly on equipment´s. Even if it is resolved for a short-limited time, for a ship on sea, it may mean it can navigate safely to a shore and make a permanent repair restoring ship full performance. This article aims to explore the state of art of 3D metal printers, and the defects after printing spare part. Evaluating the development of technology and implementation on maintenance in industry and in ships, making an analysis and comparative study. By this research it is intended to analyze the appliance of this technology to components of electro-pumps that are implemented on ships.

© 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 IGF28 - MedFract3 organizers Keywords: Addictive Manufacturing; Spare Parts; Ships; Defects

1. Introduction Nowadays, 3D printers for conception of ship models considering a modular construction (Choi et al., 2024), or equipment’s spare parts printing are a reality, also on research centers it is used to modulate and fabricate parts of systems or equipment to build prototypes to show research results. This may be a methodology that will reduce cost

* Corresponding author. Tel.: +351 21 0901 931 E-mail address: suzana.paula.lampreia@marinha.pt

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 IGF28 - MedFract3 organizers 10.1016/j.prostr.2025.12.310

Suzana Lampreia et al. / Procedia Structural Integrity 79 (2026) 81–87

82

of building systems, reduce parts or spare parts fabrication and be more sustainable, because the waste results are reduced relatively to subtractive manufacturing where objects are obtained on demand. (Agawala, 2025)(Coruzzolo et al., 2022) Spare part management on board ships, and especially in old ships, is critical on equipment´s maintenance. In a maritime context, ships often operate in isolated and harsh environments, where logistics on port may be subject to delays, or customs service’s reasons, or just because it is in a mission at sea. This can result in assets downtime, increased costs, and elevated risk to safety and operations. Addictive manufacturing, especially with metal 3D printing, offers on-demand spare part production that can be deployed at sea or in port, or for military ship on a Naval Base workshop. This technology may reduce lead-times and dependence on global logistics networks. A expeditious research on Google Scholar for items such as Maintenance (M), Maintenance on Ships (MS), Addictive Manufacturing (AM), Addictive Manufacturing on Ships (AMS), Addictive Manufacturing for Equipment’s Spare Parts (AMESP), 3D Manufacturing (3D-M), 3D Manufacturing on Ships (3D-MS) and 3D Manufacturing for Equipment’s Spare Parts (3D-MESP) result on a significative number of published articles, Fig. 1.

Fig. 1. Results of research on the present study.

Recent modelling studies in the marine sector demonstrate that integrating addictive manufacturing into spare-part supply chains may reduce delivery time and cost for both high and low value components. Furthermore, addictive manufacturing facilitates decentralized or hybrid supply chain configuration, which can decrease inventory requirements and mitigates supply-chain fragility. (Alzahami, 2025) Nowadays USA Navy implemented aboard warships addictive manufacturing systems to produce functional parts at sea, enhancing maintenance autonomy (Barnabas, 2025). An Australian shipbuilder introduced the largest 3D printer in the US to quickly fabricate mission critical parts, where for example reduced eight months on awaiting time for a fortnight (Jones, 2025). In the present paper will be reviewed the current addictive manufacturing technologies suitable for ship maritime maintenance, particularly metal 3D printing methods, referring the possible occurrence of defects in manufactured spare parts, and presents a SWOT analysis for implementation in maritime electro pumps systems. 2. 3D printers on equipment maintenance In the area of spare parts nowadays we have available 3D printers that enhance autonomous capacity and rapid responses to some anomaly on equipment´s (Sledgers et al, 2023). Even if it is resolved for a short-limited time, for a ship on sea, it may mean it can navigate safely to a shore and make a permanent repair restoring ship full performance. AM may be applied on maritime assets enhancing ships and other maritime infrastructure autonomy and resilience (Kostiti and Nikitakos, 2018).