Abstract
The accurate calibration of semi-empirical fatigue models against experimental evidence is a critical step for achieving reliable predictions. Amongst many semi-empirical fatigue models, El Haddad’s (EH) curve is widely exploited to characterise the fatigue endurance limit of defect-laden and cracked metallic alloys. A few deterministic computational models exist in this respect, however, they lack a robust probabilistic perspective and their implementation code is not publicly accessible. The authors of the present work have recently exploited Maximum a Posteriori (MAP) to robustly and probabilistically estimate EH’s curves, even in case of data scarcity or incomplete datasets, combining experimental evidence and prior knowledge taken from literature. Whilst the implementation scheme was published, the associated code was not made available. Hereby, the authors present B-FADE, a flexible open-source Python package, aimed at releasing the implementation of the MAP approach with improvements, as well as several pre- and post-processing utilities to facilitate its deployment. The package is conferred with a sufficient level of abstraction, thus turning out to be easily extensible to future implementation of other relevant fatigue models.
Keywords: El Haddad, Defects, Fatigue endurance limit, Maximum a posteriori, Python
Subject terms: Mechanical engineering, Software, Structural materials
Introduction
El Haddad’s (EH) curve is a semi-empirical model based on Linear Elastic Fracture Mechanics (LEFM) that allows practitioners to estimate the fatigue endurance limit of metals as a function of material’s cracks or defects characteristic size, for a predefined runout threshold in terms of the number of cycles (
), and applied stress ratio (R)1,2. Without loss of generality, the author will refer to defective materials, however, the same reasoning can straightforwardly be applied to other solids containing cracks. To do so, the application of EH’s curve necessitates defects to be regarded as cracks, endowed with a characteristic crack-length and having such a dimension much smaller than the whole probed material’s cross section. As comprehensively reviewed in3, Murakami’s rationale can be followed for several metallic alloys manufactured through various techniques, for instance casting and additive manufacturing. In this respect, defects are related to the renowned 
, i.e. the square root of their surface projection onto the plane orthogonal to the direction of the applied stress range 
4,5. The EH formulation of the fatigue endurance limit assumes the stress intensity factor (SIF) range as the fatigue crack driving force. Adopting 
for the crack-like defect length, the SIF is defined as:
 |
1 |
where Y is given by the distance of defect’s centroid to the free-surface of the fatigued sample6. EH’s curve is mathematically expressed as:
 |
2 |
in which 
is the fatigue endurance limit of the defect-free specimen, and 
is EH’s critical length:
 |
3 |
where, in turn, depends upon the SIF range threshold for long cracks 
. From a phenomenological perspective, EH’s curve partitions the 
plane into the regions of propagating and non-propagating cracks. Hence, prospective runout and failed samples with a given 
shall predominantly be collocated beneath and above the curve, respectively, as schematically portrayed in Fig. 1. Under this traditional phenomenological behaviour of EH’s curve, runouts are regarded as samples that are expected to survive at least 
cycles, whereas failed samples are those meant to fail prior to 
.
Fig. 1.
Pictorial representation of EH’s curve along with a fictitious dataset over a log-log plot. Runout and failed samples are primarily located above and beneath the curve. Runouts as those sample whose fatigue life exceeds the fatigue runout threshold (
). Failed samples are those specimens that collapse prior to 
.
The robust estimation of 
and 
is vital when employing EH’s curve in the context of defective or cracked metallic materials. Nonetheless, estimating 
is rather prohibitive since producing ideally defect-free metallic parts without altering the intrinsic fatigue properties of the materials is hardly attainable. In contrast, determining 
is feasible through established fatigue crack growth (FCG) tests, albeit not always affordable. When 
cannot experimentally be determined, a viable solution would be to infer EH’s parameters (
and 
) by exploiting classic uniaxial constant amplitude fatigue tests at different magnitude of amplitude, thus exploiting post-mortem fractography to evaluate the characteristic dimension (
) of the defect responsible for fatigue crack nucleation and failure (killer defect). Conversely, if 
is independently retrievable through dedicated experiment, such information can act as prior knowledge, thus being extremely beneficial to achieve even more precise estimation of EH’s parameters. It is important to highlight that, fatigue data of defective materials incorporates relevant epistemic and aleatoric uncertainty, thus leading to scattered fatigue response7–13. Hence, 
and 
urge to be probabilistically determined to account for these uncertainties.
EH’s curve has undergone several advancements, specifically regarding its generalisation to the finite fatigue life regime. Accordingly, several deterministic identification methods for EH’s curves have been emerged. For instance, Ciavarella exploited fatigue data in conjunction with fatigue crack growth (FCG) models to identify EH’s curves at targeted fatigue lives14. Besides, Sheridan used Paris’15, Donahue’s model16, and NASGRO equation17,18 to address the same issue in regards to the commercial Inconel 71819,20. It is worth mentioning Chapetti’s work that aimed to extend EH’s to the short-cracks regime while identifying EH’s parameters considering crack closure effects as well21,22. This approach has recently found application in the context of additive manufacturing23,24. Being practical and readily applicable, the EH curve has been exploited for in numerous scenarios3. Despite this, it should be noted that the EH curve characterises the fatigue endurance limit for a single fatigue load condition. Therefore, the EH curve does not allow for capturing the effects of other factors, e.g. R and load frequency, which are known to impact both the endurance limit and lifetime25,26 – the latter, however, may have an effect only under particular conditions27.
The literature reports a few methods to address the identification of EH’s curves from a probabilistic standpoint. Aigner et al. inferred both 
and 
through a Weibull’s distribution to model the survival probability of the analysed batch of AlSi8Cu3 samples28. Eventually, the parameters of the distribution were identified by combining FCG and fatigue tests, and post-mortem fractography data. Another strategy sharing some of the aspects with the former approach was devised in regards to AM50hp and AZ91hp Mg alloys29. More recently, the authors of the present paper exploited Maximum a Posteriori Estimation (MAP) and Machine Learning (ML) to probabilistically determine 
and 
. The key concept of the approach was morphing the behaviour of EH’s curve (Fig. 1) into a dichotomous classification problem via Logistic Regression (LR)30. The proposed computational algorithm was designed to process only uniaxial constant amplitude fatigue data retrieved at several levels of 
, and the corresponding size, 
, of the “killer” defect that triggered fatigue failure. The MAP approach effectively dealt with incomplete datasets and small-data regimes, thanks to the possibility of introducing relevant information from the literature by shaping appropriate prior (distributions). Furthermore, statistical post-processing allowed for the retrieval of EH’s curve at different failure probabilities31. Given the relatively young age of the MAP approach, its potential has only been exploited in32, to the best of the authors’ knowledge.
Notwithstanding many specific approaches to identifying EH’s curve, the literature seems to lack handy publicly available software packages to assist practitioners with this crucial issue. To bridge this gap, the present work pursues the ambitious objective of providing the research community with a novel open-source Python package, namely B-FADE, which revises and encodes the MAP-based approach formerly conceived and presented in31. An illustrative example is presented with regard to EH’s curve, thereby showing its workflow to prospective users. Additionally, the work aims to propose a flexible and extensible code intending to subsequently address MAP for numerous recurrent fatigue models besides EH’s curve. Herein, the distinctive features, and limitations of B-FADE are critically vetted.
Mathematical framework of B-FADE
General setting
Given a target curve 
described by a set of parameters gathered in the vector 
, B-FADE is designed to solve MAP, i.e. it maximises the posterior (distribution) 
of 
after observing the training dataset 
. To facilitate the calculations the logarithm is formerly applied to 
, whose monotonicity lead to the following equivalent formulation of MAP:
 |
4 |
In fact, B-FADE solves for 
since it uses minimize from scipy.optimise33. In the last relationship, 
is given by Bayes’ theorem:
 |
5 |
hence:
 |
6 |
in which 
, 
, are respectively the log-likelihood (function) and the log-prior, whereas 
i.e. logarithm of the so-called evidence, is neglected in the implementation since it is a constant as vanishes while maximising Eq. (4). As concerns the term 
, namely the log-likelihood, B-FADE implements Bernoulli likelihood since it is currently devoted to classification problems:
 |
7 |
where 
is the ground-truth label, and 
is predicted label via LR:
 |
8 |
where 
is the signed distance between the training point 
to the target curve having unknown parameters 
. In regards to the prior 
, B-FADE initially prescribes independent uniform probability distributions over 
. Nonetheless, the user can place over 
any pertinent distribution based on their prior knowledge subsequently.
After estimating 
via Eq. (4) and the corresponding curve 
, B-FADE carries out additional operations to approximate the full posterior. The variational approximation of 
is computed34, i.e. the posterior is represented as a multivariate Gaussian distribution whose mean is 
:
 |
9 |
where 
is the inverse Hessian matrix of 
, which act as a covariance matrix.
Upon approximating the posterior, B-FADE can compute the prediction interval for the identified curve 
:
 |
10 |
B-FADE does so by sampling 
curves from Eq. (9) in a Monte Carlo fashion. The sampling can be performed either by considering the full Hessian matrix, or by neglecting its anti-diagonal entries. The former option, however, allows for taking into account the potential correlation that might arise from 
. In any case, Eq. (10) shows the mean, 
, and semi-amplitude of the interval, 
, of the 
curve trials:
 |
11 |
where 
is the prescribed confidence level, 
is the standard deviation of the curve trials. Lastly, 
is 
percentile of Student’s t-distribution.
Besides the prediction interval, which were only originally envisaged in31, B-FADE also offers the computation of the predictive posterior (distribution), which can be useful to predict the output class y unseen data 
. The predictive posterior is mathematically stated as:
 |
12 |
which can be post-processed in terms of the expected value 
(mean) and the uncertainty 
(standard deviation) to probabilistically characterise the predictions. Still, the integration is well-known to be analytically unfeasible, so B-FADE implements the following approximation:
 |
13 |
where the samples 
are drawn from 
.
El Haddad curves
If one wishes to identify EH’s curves, they would consider Eqs. (2–3) and define 
as the vector of trainable parameters. It should be noted that the present approach is designed to addresses the identification of the 
and 
for a given stress ratio R, in agreement with the traditional formulation of the EH curve. Therefore, under a given R the user would collate the following training dataset containing 
samples after fatigue tests and fractography:
 |
14 |
which contains the outcomes of the fatigue tests at 
with reference to a prescribed fatigue endurance limit 
, i.e. 
(runout) and 
(failed), and the fractographic characterisation of the killer defect’s size, namely 
. The vector of the input features shall be denoted as 
hereinafter.
In case of defects characterised by multiple 
, the SIF equivalence is used to rescale 
accordingly, thus enabling defects to be represented over a unique diagram:
 |
15 |
where 
is a predefined reference value.
Upon possibly prescribing appropriate prior 
, the dataset is processed via Eq. (5) to perform the MAP estimation of the parameters sought. It is worth commenting how 
is computed (Eq. 7) with reference to the application of the LR, see Eq. (8). Particularly regarding EH’s curve, B-FADE computes the signed distance across two stages. The first, entails finding the minimum-distance point 
along the target EH’s curve over log-log plane, thus solving:
 |
16 |
The second, involves calculating the signed distance between 
and 
using the common Euclidean norm:
 |
17 |
where 
gives the sign according to the relative position of 
to the target EH’s curve:
 |
18 |
Although the outlined approach slightly differs from that developed in Ref.31 regarding the computation of the signed distance, the authors believe that the revised framework closely reflect the functional form of EH’s, which is traditionally represented on the log-log plane. This approach is deemed to deliver more accurate and reliable estimations while leading to sufficient gradient magnitude for 
to ensure an appropriate transition between the considered classes.
As mentioned earlier, one can compute both the prediction interval of the estimated EH’s curve and the predictive posterior according to the associated variational approximation of the posterior after running MAP. In agreement with31, the curves enclosing the prediction interval represent the fatigue endurance limit curve at selected failure probabilities. Since the prediction interval is retrieved via Monte Carlo simulations, it offers a frequentist evaluation of the propagating crack region. As shall be clearer later with reference to the illustrative example, the prediction interval, besides its practical interpretation, it also provides a measure of data scatter. Precisely, the more the data is scattered the wider the interval tends to be. Furthermore, the more outliers, i.e. runout surrounding failed and vice versa, the more the interval widens. For this reason such interval shall be referred to as scatter band in the following. On the other hand, Equation Eq. (13) enables one to estimate the propagating crack region in a Bayesian manner, thereby rigorously aligning with the developed MAP methodology. However, it is important to note that the application of one method does not exclude the use of the other. Specifically, users can compute the EH curves at different failure probabilities through the frequentist approach. Alongside this, practitioners can precisely compute the class of unseen data, including the uncertainty over the predicted outcome, in a Bayesian fashion. Since the approaches offer complementary insights, it was decided to implement both in B-FADE.
Overview of B-FADE
The application programming interface (API) resort to a set of specific modules, as depicted in Fig. 2.
Fig. 2.
Architecture of B-FADE. Modules are framed inside rounded-corner boxes. CamelCase monospaced words are classes. Lowercase monospaced words represents class methods. Lines terminating with triangle markers indicate subclasses. Please note that the figure reports only the core methods. Ancillary methods, such as those for inspection purposes, are not shown for conciseness. util.py conceptually lists the contents of the module. Please refer to the source code for the contents not shown herein.
The module abstract.py contains three classes endowing B-FADE with sufficient abstractness, thus making it particularly suited for further extensions. The class AbstractCurve provides a generic representation for describing the curves of fatigue semi-empirical models, whereby the subclass ElHaddadCurve is instantiated, whose concrete method equation implements EH’s curve (Eqs. 2–3)). Alongside, AbstractBayes incorporates the methods to carry out MAP for the concrete representation of fatigue models. ElHaddadBayes is subclassed from AbstractBayes while implementing the concrete method predictor. Such a method encodes the computation of the LR and signed distance as shown in Eq. (8–16), respectively.
Regarding data preparation, B-FADE conveys two main classes in dataset.py. The former is SyntheticDataset, which incorporates a method to generate datasets for testing purposes. The latter is Dataset, which is a generic class that provides a common interface to structure the acquisition of datasets. In this instance, ElHaddadDataset is subclassed from the latter, and it implements methods to partition and pre-process the set of experimental observations, such as the SIF equivalence of Eq. (15).
The module statistics.py contains the class MonteCarlo, which is devoted to sampling the posterior (Eq. 12) and computing the scatter band (Eq. 10–11). Alongside this class, Distribution is designed to wrap scipy distributions33, thereby facilitating the prescription of priors.
As concerns the inspection of input data, intermediate stages of MAP, and results, the module viewers.py provides the users with three viewers to probe the methods of the classes above – similarly to a model-view-controller design pattern35. In particular, BayesViewer and LaplacePosteriorViewer are subclassed from AbstractMAPViewer. The former is dedicated to inspecting the contours of the elements of Bayes’ theorem (Eq. 5), i.e. log-prior, -likelihood, and -posterior, whereas the latter allows displaying the approximated posterior (Eq. 13) and its marginal distributions. Further, PreProViewer class generates a canvas which permits the user to display the input data, the scatter band, and the predictive posterior over the plane where the identified curve is expected to lay in.
Lastly, util.py collects several utilities to support the execution the previous modules – their explanation is deferred to the documentation for brevity.
Workflow for El Haddad curves and discussion
To showcase the workflow of B-FADE, the literature dataset of a Ti6Al4V manufactured via both Selective Laser Melting (SLM) and Electron Beam Melting (EBM) taken from Ref.36,37 was considered as a case study. The material underwent constant stress amplitude fatigue tests at 
, and the fatigue endurance limit was referred to 
cycles.
Data acquisition and pre-processing
B-FADE primarily reads data from comma-separated files and spreadsheets. Irrespective of the format, the files must report dedicated columns listing 
, 
, 
for each i-th killer defect (or prospective killer defects for runout samples). An additional column must be reserved to a binary label 
indicating whether each sample is either runout (
) or failed (
). In this case, all defects are regarded as surface and referred to 
; therefore, an equivalent 
must be calculated beforehand according to Eq. (15), if necessary.
After acquiring and preparing the dataset, it is possible to obtain a preliminary overview of the experimental observations using PreProViewer from viewers.py. Analogously to what is shown in36, it was decided to collate the samples of both manufacturing processes into a single dataset, as portrayed in Fig. 3.
Fig. 3.
Dataset of the Ti6Al4V manufactured via both SLM and EBM from36,37. Runout and failed specimens are represented as circle and cross markers. Whilst the rightmost colour bar shows the SIF range 
associated with each sample.
Maximum a posteriori
Let us assume 
priors are prescribed over 
, i.e. in the absence of prior knowledge. Thus, MAP (Eq. 4) reduces to Maximum Likelihood Estimation (MLE). Before proceeding, it is worth visualising 
invoking BayesViewer in viewers.py, see Fig. 4a). Therein, it is possible to observe that the log-likelihood possesses a maximum surrounded by the contour line holding the highest value. The execution of MAP, in fact MLE in this case, succeeded in providing the expected values: 
and 
and the associated EH’s curve 
The results are reported in Fig. 8 and transferred to Appendix “MLE estimate of El Haddad curve” in order not to overload the main content of the present manuscript. Therein, the displayed identified curve approaches that in Ref.36, and it can be regarded as plausible given the limited amount of data. Still, little mismatch can be noticed with particular regard to 
, and 
may overestimate the fatigue endurance limit of the considered material. Moreover, MLE is rather limiting as it can only compute the expected values of the parameters compared to MAP, which can calculate the probabilistic intervals where the parameters are meant to belong.
Fig. 4.
Contour plots of the MAP elements. (a) Log-likelihood. (b) Log-prior. (c) Log-posterior. The marker in this panel indicates the position of the maximum, i.e. 
, and 
.
Fig. 8.
MLE Estimate of EH’s curve for the considered dataset and comparison with the curve of Ref.36.
For these reasons, prior information from other appraisals in the literature was exploited38,39. Accordingly, the following priors were placed over the parameters sought, namely 
. The resulting log-prior 
is shown in Fig. 4b, whereas the 
is given Fig. 4c, which clearly indicates the existence of a maximum point surrounded by the highest contour line. Therefore, MAP was executed and converged to the anticipated maximum located at 
and 
. The curve associated with these values shall be referred to as 
. Following MAP, B-FADE computed the variational approximation of the posterior in agreement with Eq. (9), i.e. 
. The contour of its joint distribution is shown in Fig. 5a, whereas its marginal distributions are given in Fig. 5b,c. Quantitively, these marginal distribution turned out to be 
and 
.
Fig. 5.
Variational approximation of the posterior. (a) Contour plot of the joint distribution of the posterior. (b) Marginal distribution of 
. (c) Marginal distribution of 
.
Frequentist characterisation of the propagating crack region
Since significant correlation owing to 
can be noticed in Fig. 5a, the joint posterior is sampled 
times. A Monte Carlo simulation was run, thus retrieving the scatter band of 
for a confidence level 
Accordingly, the retrieved lower and upper bounds of the scatter band, i.e. 
and 
correspond to fatigue endurance the limit at 0.5% and 99.5% failure probability level. This yields to a natural interpretation of the obtained bounds: the experimental points located surrounding the lower and upper bounds shall be characterised by 0.5% and 99.5% failure probability, respectively. For instance, an unseen datum close to the upper bound (99.5% curve) is expect to fail, i.e. it holds label 1, with 99.5% probability – analogously for the 0.5% curve. It is worth remarking that the scatter band was retrieved in a Monte Carlo fashion, so it offers a frequentist characterisation of the propagating crack region.
The bounds of the scatter band are shown along with 
in Fig. 6. As concerns 
, the expected curve well captures the global pattern of 
and it evidently aims to optimally separate the failed specimens from runouts. Although there exists only partial agreement between the identified curve and that of Ref36 (represented in light blue and orange, respectively), it is crucial to emphasise that the objective of B-FADE is not to achieve a perfect match with the reference curve. Instead, the goal is to infer the curve based on available data and potentially prior information from the literature. Hence, discrepancies in 
results may arise if the reference curve is determined using alternative methodologies. It is worth mentioning that prior experimental appraisals of 
, when available, would permit the construction of a statistical distribution to craft a prior, thus achieving a considerably more accurate estimate of the sought parameters. Nevertheless, 
is more than satisfactory for the intended purpose.
Fig. 6.
Frequentist characterisation of the propagating crack region via in terms of the scatter band along with the estimated (optimal) EH’s curve.
Owing to Bayesian Inference, the width of the prediction bands relates to the distribution of the data: the higher the point density, the narrower the prediction bands are. The opposite phenomenon is observed towards the edges of the plot where experimental evidence becomes progressively scarce. Additionally, the information available where the training points are concentrated makes the expected curve closely align with the observed data pattern while limiting the parameters’ variability and uncertainty. Bayesian Inference also ensures that more confident predictions and even narrower bands can be obtained if additional data or prior information are provided. Furthermore, the scatter band also encloses the “reference” curve from Ref36, thereby deeming it admissible for the current dataset. As mentioned earlier, the scatter band is influenced by the dispersion of the data. Since the data points are appropriately arranged and few outliers are present, the amplitude of the scatter band is significantly limited. Furthermore, given the configuration of the dataset, a sharp transition is modelled between failures and runouts. This is also reflected in a sudden change between the region of non-propagating and propagating cracks.
Bayesian characterisation of the propagating crack region
Combining ElHaddadBayes and PreProViewer, one can compute and display the predictive posterior, namely Eq. (13), thus predicting the failure probability of prospective experimental samples 
i.e. 
Specifically, ElHaddadBayes probed the predictive posterior 50 times over a regular grid in the KT diagram, i.e. 
having a resolution of 
points. This resulted in 125000 evaluation points where 
was predicted. The grid was selected to sufficiently cover the crack size and fatigue load spectrum. In agreement with Section General Setting, the predicted values were post-processed to calculate the mean 
and the uncertainty 
at each evaluation point. The contour of the mean and uncertainty are shown in Fig. 7a,b, respectively, along with the scatter band independently determined via Eqs. (10–11). In particular, these contours offers the Bayesian representation of the propagating crack region, where the contour lines represent iso-failure probability curves. This approach allows one to readily characterise the failure probability 
(Eq. 12) for unseen data 
. In this case, 
is expressed in terms of the expected value 
(mean, Fig. 7a), which takes values in [0, 1] (
and 
), and the related uncertainty 
(standard deviation, Fig. 7b). Therefore, practitioners can straightforwardly express the predictions as 
.
Fig. 7.
Bayesian characterisation of the propagating crack region via contour plots of the predictive posterior. (a) Expected value (mean), 
. (b) Uncertainty (standard deviation), 
. Each panel portrays the scatter band independently computed by Eqs. (10–11).
Intriguingly, the contour’s gradient depletes within the scatter band. This foreshadows a reasonable equivalence between the methods, despite their different fundamental nature, i.e. frequentist (Monte Carlo) and fully Bayesian, respectively. The obtained results also corroborate the interpretation of the scatter band. As can be seen in Fig. 7a, the contour of mean approaches the anticipated level of failure probability in correspondence of the lower and upper bound of the scatter band derived from the adopted 
.
Prospective impact
The release of B-FADE constitutes a substantial advancement for the fatigue & fracture mechanics community. Specifically, this software package intends to address the need for available open-source codes to estimate the parameters of semi-empirical fatigue models. B-FADE implements an intuitive interface for the MAP approach of Ref.31, making it particularly suitable for being routinely employed to design structural engineering components against fatigue failure. Although current version of the package only targets the estimation of EH’s curves, its level of abstractness does not prevent further developments to tackle fatigue models other than EH’s. It should be acknowledged, however, that B-FADE tackles the identification of the EH’s curve according to its traditional formulation, i.e. under a single fatigue test condition. Therefore, one needs to identify separate EH curves for each load conditions and use them under different scenarios.
A major advantage of B-FADE over deterministic algorithms is the potential to provide probability distributions of the parameters sought, e.g. 
and 
for the specific case of EH’s curves. Consequently, this allows for computing the failure probability of samples rather than forecasting point-valued predictions, thus readily ascertaining the robustness of the predictions. Furthermore, as shown in the illustrative example, B-FADE effectively handles small datasets, thus limiting the commitment to extensive experimental data collection and reducing the related onerousness.
The strength that distinguishes B-FADE from other existing methods stems from injecting prior knowledge from the literature or experimental evidence during the parameters’ identification. In this respect, practitioners are conferred with the ability to shape any prior for incorporating information deemed relevant to the examined dataset.
Conclusions
The present manuscript illustrated the open-source Python package B-FADE and described its API, which improves and wraps the MAP approach, previously conceived by the authors in31. Although the current version of B-FADE is specifically implemented and validated for the Bayesian identification algorithm to determine EH’s parameters, i.e. 
and 
, under a specific fatigue test condition, further developments are planned to extend the code to other fatigue models. As concerns EH’s curves, however, besides offering the frequentist Monte Carlo estimation of the propagating crack region, B-FADE also provides a full Bayesian computation thereof – the two methods yield consistency. Additionally, the package implements data pre-processing and results post-processing utilities to assist users throughout the computational stages.
The illustrative case study on the Bayesian identification of EH’s curve for the considered Ti6Al4V literature dataset, demonstrated B-FADE ’s capability to deal with small-data regimes. Furthermore, B-FADE showed its potential to inject prior knowledge from the literature, thereby ensuring physically consistent results. Lastly, the Bayesian evaluation of the propagating crack region allows one to easily forecast the failure probability of prospective unseen data.
In conclusion, B-FADE shall represent a practical resource to accurately estimate EH’s curve, thus holding the promise of being routinely used for the design against fatigue failure of cracked or defective metallic materials, and generalised to other semi-empirical fatigue models.
Acknowledgements
This work has been supported by the following projects “NextGenerationEU – PNRR M4.C2.1.1 – PRIN 2022 – Codice 20229BM9EL – CUP G53D23001140006” funded by the European Union.
Appendix: MLE estimate of El Haddad curve
See Fig. 8.
Author contributions
A.T. and E.S. conceptualised and developed the methodology. A.T. carried out the implementation of software. A.T. wrote the original draft. E.S. Acquired funding resources. All authors reviewed the manuscript.
Data availability
The dataset considered herein was taken from the literature; see the referenced papers for additional information.
Code availability
The source code of B-FADE along with tutorial examples is publicly available at https://github.com/aletgn/b-fade, whereas its documentation is published at https://b-fade.readthedocs.io/en/latest/index.html. The package is available at https://pypi.org/project/b-fade/, so one can install it via, for instance, pip install –user b-fade+.
Declarations
Competing interests
The authors declare no competing interests.
Consent for publication
All the authors consent to the content of the manuscript.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The dataset considered herein was taken from the literature; see the referenced papers for additional information.
The source code of B-FADE along with tutorial examples is publicly available at https://github.com/aletgn/b-fade, whereas its documentation is published at https://b-fade.readthedocs.io/en/latest/index.html. The package is available at https://pypi.org/project/b-fade/, so one can install it via, for instance, pip install –user b-fade+.