Content uploaded by Alessandro Tognan

Author content

All content in this area was uploaded by Alessandro Tognan on Apr 07, 2023

Content may be subject to copyright.

ACCEPTED PREPRINT

Probabilistic Defect-based Modelling of Fatigue Strength for Incomplete

Datasets Assisted by Literature Data

Alessandro Tognana,1,∗, Enrico Salvatia,2

aPolytechnic Department of Engineering and Architecture, University of Udine, Via delle Scienze 206, Udine, 33100, Italy

Abstract

Probabilistic defect-tolerant fatigue design protocols have become the leading paradigms in structural en-

gineering. To eﬀectively deal with this problem, El Haddad’s (EH) curves are generally employed for the

evaluation of the fatigue endurance limit. Herein, the synergic exploitation of Logistic Regression (LR)

and Maximum a Posteriori (MAP) allows for calibrating EH parameters using the sole data from fatigue

characterisation and post-mortem fractography. An extensive literature research provided the ground to in-

troduce, when necessary, prior information for some of the more commonly used metallic alloys. Eventually,

EH curves are retrieved upon a Monte Carlo simulation to support probabilistic engineering practice.

Keywords: Fatigue Endurance Limit, El Haddad, Additive Manufacturing, Logistic Regression, Maximum

a Posteriori Estimation

∗Corresponding author

Email addresses: tognan.alessandro@spes.uniud.it (Alessandro Tognan), enrico.salvati@uniud.it (Enrico Salvati)

1ORCID: 0000-0001-8109-4034

2ORCID: 0000-0002-2883-0538

1

ACCEPTED PREPRINT

1. Introduction

In practical engineering applications, fatigue is widely reported as a detrimental phenomenon respon-

sible for unforeseen yet catastrophic failures [1, 2]. Fatigue failures of metallic components are primarily

triggered at the microstructural level by potentially concomitant material inhomogeneities that act as local

strain/stress concentrators when cyclic loads are applied externally [3]. For many metallic materials, below

a certain applied stress (or strain) amplitude, the propagation or nucleation of cracks from existing defects

is inhibited or at least considerably delayed. Such a threshold is generally known as Stress Intensity Fac-

tor range threshold (∆Kth) for pre-existing cracks or defects, or more in general fatigue endurance limit;

whether an actual non-propagation regime of loaded cracks exists is still a subject of debate.

Unfortunately, as many other material properties or characteristics, a precise evaluation of fatigue

strength in material is highly hampered by a number of uncertainties due to the material inhomogene-

ity at the micro scale level. For instance, the presence of defects in materials is one of the leading source

of scatter when dealing with fatigue experimental data and, regardless of the manufacturing process, any

material unavoidably retains defects. There exists a wide range of defects depending on the considered

length scale and physical origin. Of particular interest are those defects that can be considered as cracks,

e.g., lack-of-fusion or gas pores. A special attention has been recently focused to defects when dealing with

additively manufactured (AMed) materials, nonetheless, defects can also arise from consolidated fabrication

processes, such as casting, welding, etc. [4–9].

Another consequence stemming from manufacturing processes, giving rise to altered or scattered fatigue

performance, is the ubiquitous presence of residual stress (RS) across scales [10–14]. Depending upon its sign

and location, RS can interact with the local state of stress surrounding defects, giving rise to a detrimental

eﬀect when of tensile nature, while compressive RS is generally beneﬁcial as it can induce several stress-

shielding phenomena [3, 15]. Whilst macro scale RS can be eﬀectively accounted for, micro- and nano

scale RS are extremely diﬃcult to evaluate and take into consideration. Such a non-exhaustive list of

inhomogeneities explains why the vast majority of the employed design methodologies rely on deterministic

approaches that simply account for them by arbitrarily decreasing the evaluated fatigue strength of materials

through safety factors [16]. In contrast, more recently, probabilistic approaches have seen an increasing

popularity due to their ability to quantitatively assess the likelihood for a certain failure to occur [17–19].

While the micro scale RS inﬂuence on fatigue strength is far to be understood, the presence of crack-like

defects has been largely studied in the past years. Speciﬁcally, deterministic approaches have been recently

employed for defective materials by using Fracture Mechanics approaches, both for the evaluation of the

fatigue ﬁnite life and the fatigue endurance limit [16, 20–23]. Such a popularity has been promoted lately by

the spread of AM materials. In order to employ this approach, the knowledge of the defect’s characteristic

size is the ﬁrst requirement; generally a length. The approach proposed by Murakami to identify this length

2

ACCEPTED PREPRINT

relies on the deﬁnition of the square root of the projected area (√area) of the defect on the plane normal to

the loading direction, assuming that defects have maximum √area of around 1000 µm and that the crack is

predominantly loaded under mode I [7, 24]. Therefore, Linear Elastic Fracture Mechanics (LEFM) can be

invoked by using √area as the crack characteristic length to deﬁne the Mode I stress intensity factor (SIF)

range ∆Kthat is assumed as the fatigue crack driving force:

∆K=Y∆σqπ√area (1)

where ∆σis the applied stress range, and Yis a factor depending on the distance between the centroid of

the defects and the specimen’s free surface [24].

By exploiting this approach, the inﬂuence of the defect’s size on the evaluation of whether a crack would

propagate from a defect is promptly retrieved through the Kitagawa-Takahashi (KT) diagram provided

that both the fatigue endurance limit for defect-free material ∆σw, and the fatigue crack growth threshold

for long cracks ∆Kth,lc are known [25]. In this instance, the fatigue endurance limit is deﬁned at a ﬁnite

number of reversal to failure – generally 2·106or 107. Therefore, the KT diagram proposes a way to identify

combinations of defect’s dimension and applied load magnitude to estimate the region of propagating cracks

(above the diagram) and the region of non-propagating cracks (beneath the diagram)[26, 27] (see Fig. 1).

Intending to provide a continuous transition across the entire spectrum of crack size to improve the ﬁt

with the experimental data, El Haddad (EH) proposed its renowned equation [28], whose associated curve

serves the same purpose as the KT diagrams. It is worth mentioning that other variants exist, such as that

proposed by Chapetti to account for closure phenomena and short crack growth [29]. According to [30], EH

curves and KT diagrams have been frequently restated in terms of √area when Murakami’s approximations

held. Hence, in those circumstances EH curve was recast as follows:

∆σ= ∆σws√area0

√area0+√area (2)

where √area0represents the EH critical defect’s length, according to LEFM:

√area0=1

π∆Kth,lc

Y∆σw2

(3)

Fig. 1 illustrates an exempliﬁcation of the EH curve (Eq. (2)) and KT diagram. Both are represented as

a function √area and related to a synthetic dataset. In this ﬁgure, “Failed” concisely indicates specimens

characterised by a ﬁnite life and “Run-out” as those specimens lasting more than the predeﬁned fatigue

endurance limit.

Although the last decades have seen several developments focused on the extension of the EH model

to the ﬁnite fatigue life regime in commercial Inconel 718 [20, 21] and its generalisation to account for

both the regimes [26], very little has been done to exploit these models in a probabilistic framework for

3

ACCEPTED PREPRINT

Δσw

Δσ

ΔKth,lc

area

area0

KT

EH

Failed

Run-out

Figure 1: Schematic of El Haddad (EH) curve and Kitagawa-Takahashi (KT) diagram as a function of √area along with a

synthetic dataset. “Failed” and “Run-out” denotes specimens characterised by ﬁnite and inﬁnite fatigue life, respectively.

defect-tolerant design purposes. For instance, a recent study considered a batch of AlSi8Cu3 specimens as

a case study, to conceive and apply a probabilistic approach to determining the EH’s model parameters

[31]. The experimental campaign of this work permitted ∆Kth,lc and ∆σwto be modelled in terms of the

survival probability – referred to as Weibull’s – of the examined specimens. Another probabilistic method for

identifying the EH curve of AM50hp and AZ91hp was developed in [32]. In this instance, a bivariate Weibull

distribution embedding the EH model was used to outline the crack propagation region of the examined

specimens.

As concerns the inputs required to calibrate the outlined models, the knowledge of ∆Kth,lc, ∆σwis

sometimes diﬃcult to be determined experimentally, especially regarding the latter due to the intrinsic

presence of defects in the probed material. Nevertheless, these parameters can be estimated indirectly if a

suﬃcient number of fatigue experimental tests are available at diﬀerent defect sizes and loading magnitudes

– which is one of the key problem addressed in the present manuscript through the exploitation of Machine

Learning (ML) approaches.

In recent years, ML approaches has been employed to forecast the ﬁnite fatigue life of metallic alloys

[33, 34]. Furthermore, the combination of fatigue characterisation and defect descriptors, has found a

good breeding ground at the intersection of ML and ﬁnite fatigue life prediction. Speciﬁcally, location and

morphological traits of defect were fed as inputs into ML predictive models to forecast the number of reversals

to failure [35–42]. Despite this, to the best of the author’s knowledge, ML has never been applied to problems

dealing with fatigue endurance limit. ML can address these outstanding issues and provide a probabilistic

assessment as well. In this respect, datasets similar to that in Fig. 1 are expected to be available either when

experimentally searching for the fatigue endurance limit of a material or in the scientiﬁc literature where

specimens are labelled as either run-out or failed. Since two distinct classes are present, it is thus evident

4

ACCEPTED PREPRINT

that a dichotomous classiﬁcation of the dataset can be naturally introduced. As such, this categorisation

can be automatically tackled through supervised ML classiﬁcation methods, amongst which the authors

recognise the Logistic Regression (LR) as the most suitable, given its probabilistic nature [43].

Although LR was developed in the nineteenth century to model population growth [43], it has become

part of the realm of supervised ML relatively recently [44]. As concerns engineering applications it would

seem that LR has been only exploited, in tandem with digital imagery, to classify spalling eﬀects on concrete

surfaces [45]. Speciﬁcally, LR aims to set out the so-called decision boundary, which is a curve that optimally

separates the members of one class from the others. Generally, the decision boundary contains trainable

parameters identiﬁed through Maximum Likelihood Estimation (MLE). Once these parameters and the

decision boundary are determined, LR is capable of predicting the membership probability for the considered

classes.

The present manuscript develops an LR-based probabilistic framework to estimate the EH curve at

a given probability of failure. Speciﬁcally, the functional form of the EH curve (Eq. (2)) is exploited to

craft an appropriate decision boundary whose trainable parameters are ∆Kth,lc and ∆σw. Since MLE only

provides point estimates of these parameters, Maximum a Posteriori estimation (MAP) is pursued to carry

out the training of ∆Kth,lc and ∆σwwhile providing their resulting probability distribution. A Monte Carlo

approach is followed by sampling the previously evaluated distributions of ∆Kth,lc and ∆σwand computing

the associated EH curves accordingly. Finally, a probabilistic post-processing of the whole set of EH curves is

carried out, thus providing the probabilistic EH curve at a given failure probability. The developed LR-MAP

method is validated by pursuing a common 80/20 random split of the input dataset. This permit generating

the training and test dataset having 80% and 20% the dimension of the input dataset, respectively. LR-MAP

is therefore trained on the sole training set and used to predict data belonging to the test set, i.e. the unseen

data. To enforce the MAP approach when dealing with lack of information from the available experimental

results, this manuscript gathers numerous datasets that include estimates of ∆Kth,lc and ∆σw, or where

the identiﬁcation of the EH curve or KT diagrams was performed, with special attention on those including

complete datasets. A few of these datasets were then considered to develop and assess the eﬀectiveness of

the proposes method. Finally, the advantages, implications and limitations of the proposed ML approach

are widely vetted.

5

ACCEPTED PREPRINT

2. Machine Learning Probabilistic Design Curves Evaluation

2.1. Generic Structure of the Datasets

Before illustrating the theoretical framework of the devised approach it is worth formalising the mathe-

matical structure of the datasets (an example in Fig. 1). In order to reframe the parameter estimation of

the EH curve in terms of ∆Kth,lc and ∆σw, the datasets were conveniently transformed into:

D=n(∆Ki,∆σi), Yi,Ri:Ri∈ {run-out,failed}oi= 1,2, ..., N(4)

where Ndenotes the size of D, ∆Kiwas computed using Eq. (1) given √areai,∆σi. Although Rilabels

the i-th point of the dataset in a descriptive manner, run-out and failed were associated with 0 and 1 for

the sake of the numerical implementation.

In case of datasets showing defects at diﬀerent positions with respect to the sample’s free surface, multiple

values of Yimust be used for the calculation of the SIF range. So, with the purpose of representing all the

samples on a unique diagram, a single Yi,eq is set for the whole dataset while rescaling √areaiaccordingly:

√areai,eq =Y2

i

Y2

i,eq

√areai(5)

which assumes the same SIF for both the equivalent and the original defect. The reader can refer to

Appendix Appendix A for a brief proof of Eq. (5).

2.2. Logistic Regression

According to the classic LR formulation, D(Eq. (4)) is assumed to be generated by a random binary

vector such that (∆K(i),∆σ(i))7→ Ri[46–48]. Therefore, LR is trained as a probabilistic predictor of Ri

which receives the explanatory variables ∆K(i)and ∆σ(i)as inputs. Let p=P[Ri= 1] and, consequently,

1−p=P[Ri= 0] be the probability of occurrence of a failed and run-out specimen, respectively. Complying

with the general framework of LR, pis modelled through the so-called log-odds ratio:

ln p

1−p=H(x,θ) (6)

where xis the vector of the explanatory variables, whereas θis the vector of the trainable parameters

deﬁning the decision boundary H(x,θ). Solving Eq. (6) for pprovides the associate LR equation:

p(H(x,θ)) = 1

1 + e−H(x,θ)(7)

The most widely used LR exploits a linear expression for the decision boundary. Let x= [x1x2] and

θ= [θ1θ2θ3] the vector of the explanatory variables and trainable parameters, respectively. The decision

boundary of the linear LR takes the form H(x,θ) = x1θ1+x2θ2+θ3. In this case, it can be to proven that

H(x,θ) is proportional to the signed distance between xand the line X1θ1+X2θ2+θ3= 0, where X1and

6

ACCEPTED PREPRINT

X2are the independent and dependent variables, respectively. However, a LR model can be also adapted to

deal with an arbitrary decision boundary (other than the linear formulation). Given that the present study

focused on the evaluation of the EH curve for the classiﬁcation of the tested samples, such a signed distance

becomes:

H(x,θ) = χ(x,θ)v

u

u

targmin

t"(t−√area)2+ ∆σ−∆σws√area0

√area0+t!2#(8)

where the explanatory variables are x= [∆K∆σ], the vector of trainable parameters is θ= [∆Kth,lc ∆σw],

and tis a dummy variable that is sought to ﬁnd the minimum distance between xand the EH curve de-

scribed by θ. While the critical defect dimension (√area0) can be evaluated through Eq. (3), and χ(x,θ)

determines the sign of the distance:

χ(x,θ) =

1 if xabove EH curve

−1 if xbeneath EH curve

(9)

It is thus evident that, the higher the H(x,θ), the higher the value of p. By contrast, the lower the

H(x,θ), the less the value of pso that the tested point is likely to be run-out.

2.3. Maximum a Posterior Estimation

As quickly mentioned in Introduction, the parameters of the decision boundary are commonly trained

utilising MLE. This methodology entails building the log-likelihood function upon a given dataset:

log P[x|θ] =

N

X

i=1 Rip(H(xi,θ)) + (1 − Ri) (1 −p(H(xi,θ))) (10)

and then maximising log P[x|θ] to identify the unknown parameters:

ˆ

θ= argmax

θ

[log P[x|θ]] (11)

where ˆ

θis the vector containing the optimal trained parameters.

Although MLE is a probabilistic strategy to train the parameters, it only provides point estimates. Thus,

it cannot ascertain whether potential sources of uncertainty aﬀect the datasets and it does not accept prior

knowledge into the learning process. To overcome this limitation, MAP is used to accomplish the training

task. MAP relies upon Bayes’ theorem:

P[θ|x] = P[x|θ]P[θ]

P[x](12)

where P[θ] and P[θ|x] are the prior and posterior distributions of θ,P[x|θ] is the likelihood function, and

P[x] is the so-called evidence deﬁned by:

P[x] = Zθ

P[x|θ]P[θ]dθ(13)

7

ACCEPTED PREPRINT

which is typically deemed as a normalisation constant. Hereafter, (log-) prior and posterior will be used

without explicitly indicating “distribution” for the sake of brevity. When mentioning the (log-) likelihood,

“function” will no longer be used.

Diﬀerently from MLE, MAP trains the parameters by maximising the log-posterior log P[θ|x]:

ˆ

θ= argmax

θ

[log P[x|θ] + log P[θ]] (14)

where −log P[x] vanishes during the maximisation as it is a constant, and log P[x|θ] is, again, the log-

likelihood (Eq. (10)). The prior P[θ] plays a key role in the formulation of MAP. Speciﬁcally, one can

appropriately tailor P[θ] to encode prior knowledge while training LR predictor by prescribing probability

distribution over the trainable, yet unknown parameters [49]. In this regard, ∆Kth,lc and ∆σwwere initially

hypothesised as independent, hence P[θ] = P[∆Kth,lc]P[∆σw]. According to Eq. (14), the log-prior becomes:

log P[θ] = log P[∆Kth,lc] + log P[∆σw] (15)

Whilst any distribution can be prescribed over θ, Gaussian (N) and Uniform (U) priors were considered

in the present work. The former can be used to introduce strongly informative prior knowledge during the

training phase, and at the same time, it acts as a L2-regulariser for log P[x|θ], thus facilitating the training

process and avoiding possible local maxima of the log-likelihood (see Appendix Appendix B for further

details on L2-regularisation). On the other hand, when none or limited prior knowledge is available, U

priors are preferred. These Upriors do not aﬀect the likelihood, since the corresponding constant term

vanishes while maximising Eq. (14). For this reason, Uis called a non-informative prior. As a special case

for the log-posterior, if Uis imposed on both parameters, MAP reduces to MLE. It should be emphasises

that the distribution of prior knowledge is generally unknown. Nevertheless, a rational approximation that

can be made is the assumption of Gaussian distribution. It is important to state that if a suﬃciently

large dataset is available, this assumption can be readily checked. The user can actually choose the most

appropriate prior to model the distribution of the concerned parameters. In the present work, however,

Gaussian priors were adopted to make the calculation more tractable, while easily introducing the most

common L2-regularisation for the log-likelihood. Furthermore, hypothesising independent distributions for

each element of the prior is typically not restrictive. In fact, this practice is generally advisable rather than

erroneously speculating about the underlying relationship amongst the parameters, thus injecting biased

information into the learning stage.

The maximisation of the log-posterior log P[θ|x] (Eq. (14)) provided the expected value of the parameters,

namely ˆ

θ= [ ˆ

∆Kth,lc ˆ

∆σw] whereby ˆ

√area0was evaluated according to Eq. (3). Correspondingly, the

expected EH curve was determined through Eq. (2):

∆σ=ˆ

∆σwv

u

u

t

ˆ

√area0

ˆ

√area0+√area (16)

8

ACCEPTED PREPRINT

Further operations are required to fully characterise the posterior P[θ|x]. Albeit P[x|ˆ

θ] and P[ˆ

θ] can be

promptly computed by the sole knowledge of ˆ

θ, the integral of P[x] (Eq. (13)) is generally intractable, thus

P[θ|x] does not possess a closed-form expression. As commonly performed in other contexts of ML methods,

Laplace’s approximation for the posterior was invoked [49, 50]. Such an approximation assumes P[θ|x] as

Gaussian and expands it via a second-order Taylor’s expansion centred at ˆ

θ, thus leading to:

P[θ|x]≈1

2π(det H(ˆ

θ)−1)1/2exp −1

2(θ−ˆ

θ)⊤H(ˆ

θ)(θ−ˆ

θ)(17)

where H(ˆ

θ) is the Hessian matrix of −log P[θ|x] evaluated at ˆ

θ. Additionally, Eq. (17) can be succinctly

contracted as follows:

P[θ|x]∼ N(ˆ

θ,H−1) (18)

It is interesting to note that Eq. (17) tacitly endows H(ˆ

θ) with role of covariance matrix, hence H(ˆ

θ) must be

positive (semi-) deﬁnite. The reader can refer to Appendix Appendix C for further details on the derivation

of Laplace’s approximation used herein. Finally, the marginalisation of Eq. (17) allowed the distribution of

each parameter in ˆ

θto be automatically evaluated:

∆Kth,lc ∼ N(ˆ

∆Kth,lc,V[ˆ

∆Kth,lc]) (19)

∆σw∼ N(ˆ

∆σw,V[ˆ

∆σw]) (20)

where V[ˆ

∆Kth,lc] = [H−1]11 and V[ˆ

∆σw] = [H−1]22 correspond to the diagonal terms of H−1(ˆ

θ), which are

in fact the associated variance of each parameter.

2.4. Probabilistic Crack Propagation Region

A Monte Carlo simulation exploited the marginal posterior distributions of the parameters to built a prob-

abilistic fatigue endurance limit curve. This approach commenced with sampling N(ˆ

∆Kth,lc,V[ˆ

∆Kth,lc]) and

N(ˆ

∆σw,V[ˆ

∆σw]) using Sobol’s low-discrepancy sequences via SALib - Sensitivity Analysis Library in Python

[51–53]. The sampling generated MMonte Carlo trials whose j-th element was θ(j)= [∆K(j)

th,lc ∆σ(j)

w]⊤.

Upon computing √area0(j)by Eq. (3), the j-th EH curve turned out to be:

E(j): ∆σ(j)= ∆σ(j)

wv

u

u

t√area0(j)

√area0(j)+√area ∀j= 1,2,...,M(21)

Let E(m)=E[{E(1),E(2) , . . . , E(m)}] be the √area-wise expected value of the history of EH curves up

to the m-th trial. If Mis suﬃciently large, the Central Limit Theorem allows one to readily compute the

√area-wise prediction intervals [54]:

PhE(M)− P(M)≤ E(M+1) ≤ E(M)+P(M)i=β(22)

9

ACCEPTED PREPRINT

where βis conﬁdence level and P(M)is the semi-amplitude of the interval deﬁned as:

P(M)=TβS(M)p1+1/M(23)

In Eq. (23), S(M)=pV[{E(1),E(2) , . . . , E(M)}] is the √area-wise standard deviation of the whole history

of EH curves, and Tβis the 1 −β/2 percentile of Student’s t-distribution with M−1 degrees of freedom.

From a mathematical perspective, Eq. (22) is naturally interpreted as the interval, namely E(M)) ± P(M),

where the (M+ 1)-th EH curve is expected to belong, given the collection of trials E(1),E(2), . . . , E(M), for

a conﬁdence level β. Such a conﬁdence level is of utmost importance during the structural design process

of a mechanical component to deﬁne the acceptable level of risk for a speciﬁc engineering problem.

In order to monitor the convergence of the Monte Carlo simulation, the following indicator was adopted:

ρ(j)=s(E(j)− E(j−1))2

(E(j))2(24)

Essentially, this indicator evaluates the relative residual between two consecutive EH curves E(j)and E(j−1) .

2.5. Computational Algorithm

Algorithm 1 is ﬁnally presented to oﬀer a succinct overview of the computational setting conceived in

the present work.

10

ACCEPTED PREPRINT

Algorithm 1 Computational algorithm for LR in the present study.

— Logistic Regression —

Require: Preparation of Dataset D ▷ Eq. (4)

Transform data points – if necessary – by SIF equivalence ▷Eq. (5)

Require: Functional form of the decision boundary H(x,θ)▷Eq. (8)

— Maximum a Posteriori —

Require: Log-Prior log P[θ] = log P[∆Kth,lc] + log P[∆σw]▷Eq. (15)

Require: Log-likelihood P[x|θ]▷Eq. (10)

Build Log-Posterior log P[θ|x] = log P[x|θ] + log P[θ]

Maximise log-posterior ▷Eq. (14)

Ensure: Optimal parameters ˆ

θ= [ ˆ

∆Kth,lc ˆ

∆σw]

Compute optimal EH curve ▷Eq. (16)

— Probabilistic Crack Propagation Region —

Require: Approximated Posterior Distribution P[θ|x]∼ N(ˆ

θ,H−1)

Require: Marginal Posterior Distributions ∆Kth,lc ∼ N(ˆ

∆Kth,lc,V[ˆ

∆Kth,lc]) ▷Eq. (19)

Require: Marginal Posterior Distributions ∆σw∼ N(ˆ

∆σw,V[ˆ

∆σw]) ▷Eq. (20)

— Monte Carlo Simulation —

for i∈ {1,2,...,M}do

Draw sample from ∆Kth,lc ∼ N(ˆ

∆Kth,lc,V[ˆ

∆Kth,lc]) ▷Eq. (19)

Draw sample from ∆σw∼ N(ˆ

∆σw,V[ˆ

∆σw]) ▷Eq. (20)

Compute & collect the j-th EH curve E(j)▷Eq. (21)

— Check convergence —

if ρ(j)suﬃciently small then ▷Eq. (24)

Terminate simulation

else

Continue simulation

end if

end for

— Prediction Intervals —

Require: Conﬁdence level β

Compute prediction intervals of the collected mEH curves ▷Eq. (22)

11

ACCEPTED PREPRINT

3. Material Datasets

This section attempts summarising experimental datasets available in the current literature devoted to

fatigue tests, on metallic materials containing defects, for the characterisation of fatigue endurance lim-

its. Alongside, this literature survey intends to provide potential practitioners with additional inputs to

both support the application of the presented method and achieve more accurate estimates of ∆Kth,lc and

∆σw. The collected data can refer to the SIF threshold for long cracks ∆Kth,lc and the fatigue limit of a

defect-free specimen ∆σwwhen available. In case of tested materials, for instance, using multiple batches

or by introducing artiﬁcial defects of diﬀerent nature, Tables 1-2 indicate ranges for ∆Kth,lc and ∆σwus-

ing the symbol “−”. In a few cases, multiple references are speciﬁed for a single dataset, meaning that

complementary information for the same material can be retrieved from diﬀerent sources.

The key purpose of this task is the retrieval of relevant information regarding some of most commonly

used metallic materials to enforce the lack of necessary information for a univocal evaluation of the EH

curve given a certain incomplete experimental dataset. In particular, the present Section summarises the

data regarding Fe- and Al-based alloys (Tables 1-2), whereas the reader can refer to Appendix Appendix

D for Ti-, Ni-, and Mg-based alloys (Tables D.5-D.7). The results obtained in such a literature survey are

summarised in Tables 1-D.7, in which all the relevant characteristic conditions are reported alongside, such as

the manufacturing conditions, further treatments, type of loading, and stress ratio R. It is worth remarking

that a comprehensive review on the fatigue strength characterisation for AMed AlSi10Mg and Ti-6Al-4V

can be found in [55]. Alongside, a systematic survey of fatigue properties for many metals, spanning AMed

and cast Fe- and Ti-based alloys both, is presented in [56]. While, an exhaustive literature review on the

fatigue properties of Mg alloys is given in [57].

Amongst the considerable collection of references presented herein, two metallic alloys were selected to

develop and assess the eﬀectiveness of the conceived approach, namely AlSi8Cu3 [31] and AISI316L [58],

both tested at R = −1 through alternating tensile fatigue tests. Regarding the size of the datasets, i.e. the

amount of point data thereof, it is possible to observe that the dataset of the former alloy is about six times

smaller than that of the latter; see Figs. 2(a)-(b). This allowed the proposed method to be tested for two

diﬀerent scenarios, namely small- and large-data regimes. As far as AlSi8Cu3 is concerned, the specimens

were cast and underwent T6 heat treatment and machining. The authors of Ref. [31] stated that the defects

that supposedly initiate cracks were superﬁcial, thus prompting to adopt Y= 0.65 for the calculation of

the SIF range. Nonetheless, upon machining such superﬁcial defects, these turned out to be most akin to

opened cracks, thus substantially changing the scenario. For this reason, the authors of the work decided to

adopt Y= 0.73 for the evaluation of the SIF range [31], and so was adopted in the present paper as well. On

the other hand, the AISI316L samples were built using L-PBF along the vertical direction, then machined

and polished. The authors of Ref. [58] aimed to assess whether the fatigue response had been sensitive

12

ACCEPTED PREPRINT

to defects of diﬀerent nature. To this end, apart from few specimens which did not undergo any further

operations, the remaining ones were alternatively subjected to machining & polishing, or pre-corrosion via

anodic polarisation, or electrical discharge machining to produce hemispherical defects. Additional analyses

revealed that the defects were primarily superﬁcial, whereas only a small fraction were classiﬁed as internal.

In this regard, the SIF range for superﬁcial defects was calculated adopting Y= 0.65 as per semi-circular

cracks, while Y= 0.5 was used for internal defects, in agreement to [4]. Given the negligible number of

internal defects, in the present paper these were converted by setting Yeq = 0.65 and rescaling the respective

√area to √areaeq using Eq. (5).

Fig. 2(a)-(b) shows the datasets of the chosen metallic alloys. The i-th specimen of each dataset were

characterised by √areaiand ∆σi. Diﬀerent markers were used to distinguish failed specimens from those

that run out. Additionally, the markers were coloured according to ∆Kcomputed through Eq. (1), whose

value can be read on the rightmost colour bar.

In agreement with the stages of the LR-MAP approach and the structure of the datasets outlined in

Eq. (4), the algorithm does not necessitate speciﬁc pre-processing of the data. Nonetheless, the user is

committed to consider transforming√areaiwhen defects are characterised by diﬀerent Yi. Additionally, the

approach handles specimens probed at unique fatigue testing conditions, i.e. same mode and stress ratio.

It is worth pointing out that this probabilistic assessment of the fatigue endurance curve is independent

on the distribution of the defect size. In fact, the essential information about the size of the defects observed

in the probed material is encoded into the likelihood function whereby the failure probability is modelled

(see Eq. (7)).

101102103

√area [𝜇m]

102

6 × 101

2 × 102

3 × 102

Δ𝜎[MPa]

Run out

Failed 1.9

2.2

2.6

3.0

3.4

3.8

4.2

4.6

5.0

5.4

5.8

Δ𝐾[MPa √m]

(a)

100101102103

√area [𝜇m]

103

2 × 102

3 × 102

4 × 102

6 × 102

Δ𝜎[MPa]

Run out

Failed 1.3

2.7

4.0

5.4

6.8

8.2

9.5

10.9

12.3

13.6

15.0

Δ𝐾[MPa √m]

(b)

Figure 2: Datasets examined in the present paper for the development of the proposed method. (a) AlSi8Cu3 [31]. (b) AISI316L

[58]. The specimens of both datasets were subjected to alternating tensile fatigue test at R = −1.

13

ACCEPTED PREPRINT

Table 1: Datasets of Fe-based alloys. Additive manufacturing, wrought, and cast are abbreviated as AM, WR, and CS,

respectively. A ∗denotes data that are not available or indicated in the corresponding references.

Material Manufacturing

details

Loading

mode

R ∆Kth,lc ∆σwReference(s)

[MPa√m] [MPa]

AISI304 Machined

Machined,

annealed, polished

Rot. Bending

Rot. Bending

-1

-1

6.44

5.28

293 −297

296 −300

[59]

AISI316L AM (L-PBF)

AM (L-PBF)

Tension

Tension

-1

0.1

9.04

6.40

900

511

[58]

AISI316L AM (L-PBF) Tension 0.1 2.6 −3.4 556 −580 [60, 61]

AISI316L AM (L-PBF) Tension 0.1 9.1 −9.9 ∗[62, 63]

AISI403 WR (Annealed) Tension -1 6.6 780 [64, 65]

X20Cr13 (AISI 420) WR (Annealed) Tension -1 6.6 868 [64–66]

17-4PH (AISI 630) WR Tension -1 6.7 747 −799 [67]

17-4PH (AISI 630) WR (PH H1150) Tension -1 6.7 1126 [68, 64, 66]

17-4 PH (AISI 630) AM (L-PBF)

AM (L-PBF)

AM (L-PBF)

Tension

Tension

Tension

-1

0.1

0.7

7.27

4.06

2.47

2130

∗

∗

[69]

25CrMo4 ∗Tension -1 14.65 694 [27, 70]

Maraging Steel 300 AM

AM

AM

Tension

Tension

Tension

-1 6.26

5.69

5.70

1146

1776

1978

[66]

Mild Steel WR Bending -1 10.4 470 [66, 71]

Mild Steel WR Bending -1 12.4 326 [66, 71]

Mild Steel WR Tension -1 13.0 340 [66, 72]

Nodular Cast Iron CS Tension 0.1 ∗360 [73, 74]

14

ACCEPTED PREPRINT

Table 2: Datasets of Al-based alloys. Additive manufacturing, wrought, and cast are abbreviated as AM, WR, and CS,

respectively. A †in “Loading mode” indicates data obtained through fatigue crack growth test, whereas a ∗denotes data that

are not available or indicated in the corresponding references.

Material Manufacturing

details

Loading

mode

R ∆Kth,lc ∆σwReference(s)

[MPa√m] [MPa]

AlSi10Mg AM (L-PBF, as built)

AM (L-PBF, machined)

Tension

Tension

0.1

0.1

1.04

1.53

155.2

298.2

[75]

AlSi10Mg AM (L-PBF, as-built) Bending 0.1 1.41 174 [76]

AlSi10Mg AM (L-PBF) Tension -1 3.2 −3.6 380 −420 [22]

AlSi10Mg AM Tension -1 ∗ ∗ [77]

AlSi7Mg0.6 AM (L-PBF) Tension -1 0.96 −1.70 ∗[78]

AlSi7Mg0.6 AM Tension 0.1 3.99 −4.72 400 [79]

A356-T6 CS

CS

CS

CS

CS

CS

Tension

Tension

Torsion

Torsion

Tension-Torsion

Tension-Torsion

-1

0

-1

0

-1

0

∗

∗

∗

∗

∗

∗

70 −90

40 −75

50 −87

40 −73

45 −68

35 −45

[80]

A357-T6 (+ Sr) CS

CS

Tension

Tension

-1

0.1

∗

∗

214

94 −138

[81]

AlSi8Cu3 T6 Sr CS Tension -1 1.13 −1.16 84 −100 [82, 83]

AlSi8Cu3 T6 Sr CS - HIP Tension -1 0.69 −1.31 122 −145 [82, 83]

AlSi7Cu0.5Mg T6 Sr CS Tension -1 5.34 −5.36 69.4 −113.7 [84, 83]

AlSi7Cu0.5Mg T6 Na CS Tension -1 5.19 47.8 −71.6 [84, 83]

AlSi7Cu0.5Mg T7 Sr CS Tension -1 4.36 −5.06 75.0 −103.5 [84, 83]

AlSi8Cu3 T6 CS - HIP Tension -1 3.54 212.8 [31]

AS7G06-T6 CS

CS

Tension

Tension

-1

0.1

∗

∗

91

66

[85]

2024 T3 Tension -1 4.8 332 [86]

7075 T6 Tension -1 4.0 336 [86]

7049 ∗ † -3,-2,-1 2 −5∗[87]

6061 Cold-Spray

Rolled

†

†

0.1−0.7

0.1

1.3−2.8

3.8

∗

∗

[88]

15

ACCEPTED PREPRINT

4. Applications of the Method and Discussion

Small dataset case-study (AlSi8Cu3)

The ﬁrst case study presented herein regards an AlSi8Cu3 taken from [31] whose associated dataset is

shown in Figure 2(a).

Figs. 3(a)-(c) summarises the contour plots of the log-likelihood, log-prior and log-posterior. Thanks to

the opportune parametrisation of √area (see Section 2) each contour is a function of ∆Kth,lc and ∆σw.

2 4 6 8 10

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

100

150

200

250

300

350

Δ𝜎𝑤[MPa]

-81.0

-72.9

-64.9

-56.9

-48.9

-40.9

-32.9

-24.8

-16.8

log ℙ[𝐱|𝜃]

(a)

2 4 6 8 10

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

100

150

200

250

300

350

Δ𝜎𝑤[MPa]

-5.6

-4.9

-4.2

-3.5

-2.8

-2.1

-1.4

-0.7

-0.0

log ℙ[𝜃]

(b)

2 4 6 8 10

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

100

150

200

250

300

350

Δ𝜎𝑤[MPa]

MAX

-86.6

-78.0

-69.5

-60.9

-52.4

-43.8

-35.3

-26.8

-18.2

log ℙ[𝜃|𝐱]

(c)

Figure 3: The elements of the MAP regarding the AlSi8Cu3 dataset. (a) Log-likelihood (b) Log-prior (c) Log-posterior.

The log-likelihood log P[x|θ] built upon the considered dataset is displayed in Fig. 3(a). The visual

inspection of log P[x|θ] disclosed the absence of maximum points. As a result, the sole maximisation of the

log-likelihood would not have estimated any of the parameters sought. In particular, the optimiser would

have moved toward ∆σw≃180 MPa and ∆Kth,lc →+∞indeﬁnitely while seeking the maximum.

Although this preliminary assessment seems to hinder the parameter estimation, it concurs with the

conformation of the dataset. A close examination of Fig. 2(a) revealed that failed and run-out specimens

16

ACCEPTED PREPRINT

can be – almost exactly – separated by a horizontal having intercept ∆σw≃180 MPa. Despite matching

the experimental evidence, these results would be unacceptable from an engineering design perspective as

they would consider also unrealistic values of ∆Kth,lc. A speciﬁc choice of priors allowed for circumventing

this issue. Since no information about ∆Kth,lc could be essentially inferred from the dataset, ∆Kth,lc ∼

N(3.3,1.42) MPa√m was prescribed as a prior, whose parameters are the mean and the variance of the

set of ∆Kth,lc obtained by gathering data of similar materials from [82–84]. On the other hand, ∆σwwas

already expected to be around 180 MPa. Therefore, it was suﬃcient to rely upon the experimental evidence

and prescribe a non-informative Uprior for this parameter, hence ∆σw∼ U. The resulting log-prior

log P[θ] = log P[∆Kth,lc] + log P[∆σw] is shown in Fig. 3(b).

Fig. 3(c) shows the log-posterior log P[θ|x] given by the sum of the log-likelihood (Fig. 3(a)) and the

log-prior (Fig. 3(b)). Herein, it is possible to recognise that log P[θ|x] peaked at ˆ

θ= [ ˆ

∆Kth,lc ˆ

∆σw] =

[5.4 187.1], thus providing the expected values of the parameters. Following, these values were exploited

to apply Laplace’s approximation to log P[θ|x] and compute the posterior P[θ|x]. The subsequent marginal-

isation of P[θ|x] provided the distributions of the estimated parameter, i.e. ∆Kth,lc ∼ N(5.4,1.32) MPa√m

and ∆σw∼ N(187.1,20.42) MPa. Fig. 4(a) shows the contour plot of P[θ|x], whereas Fig. 4(b) oﬀers a

graphical representation of the marginal distributions of ∆Kth,lc and ∆σw.

2 4 6 8 10

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

100

150

200

250

300

350

Δ𝜎𝑤[MPa]

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

ℙ[𝜃|𝐱]

(a)

0246810

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

ℙ[Δ𝐾𝑡ℎ,𝑙𝑐]

Δ𝐾𝑡ℎ,𝑙𝑐

Δ𝜎𝑤

100 150 200 250

Δ𝜎𝑤[MPa]

0.000

0.005

0.010

0.015

0.020

ℙ[Δ𝜎𝑤]

(b)

Figure 4: Results of the MAP and marginalisation of the posterior (a) Contour plots of the posterior of AlSi8Cu3 dataset. (b)

Marginal distributions of ∆Kth,lc and ∆σwin black and grey, respectively.

Large dataset case-study (AISI316L)

It is interesting to note that, even in this case, the log-likelihood did not reveal any maximum point, see

Fig 5(a). As a consequence, ∆Kth,lc and ∆σwwould have been, again, unidentiﬁable unless speciﬁc priors

had been deﬁned. Moreover, the datasets shown in Fig. 2(b) did not exhibit any distinctive characteristic,

such as preferred values of the parameters whereby the dataset could have been split – similarly to the former

17

ACCEPTED PREPRINT

case. In order to accomplish the training stage, the following priors ∆Kth,lc ∼ N(6.8,0.282) MPa√m and

∆σw∼ N(864.0,136.82) MPa were prescribed, again, by gathering data of similar materials from [64–69].

The combinations of these priors computed through Eq. (15) is portrayed in Fig. 5(b). Upon injecting these

priors into the learning stage and computing the log-posterior (Fig. 5(c)), the MAP successfully estimated

the expected parameters, ˆ

∆Kth,lc = 7.5 MPa√m and ˆ

∆σw= 1003 MPa. Next, Laplace’s approximation

permitted the posterior to be evaluated, which is displayed in Fig. 6(a). Finally, the marginal posterior

distribution of each parameter was calculated, resulting in ∆Kth,lc ∼ N(7.5,0.32) MPa√m and ∆σw∼

N(1003,1462) MPa. Such marginal distribution are displayed in Fig. 6(b).

5.0 7.5 10.0 12.5 15.0

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

500

750

1000

1250

1500

1750

2000

Δ𝜎𝑤[MPa]

-212.5

-202.6

-192.7

-182.8

-172.9

-163.0

-153.1

-143.2

-133.3

log ℙ[𝐱|𝜃]

(a)

5.0 7.5 10.0 12.5 15.0

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

500

750

1000

1250

1500

1750

2000

Δ𝜎𝑤[MPa]

-580.2

-507.7

-435.2

-362.7

-290.3

-217.8

-145.3

-72.8

-0.3

log ℙ[𝜃]

(b)

5.0 7.5 10.0 12.5 15.0

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

500

750

1000

1250

1500

1750

2000

Δ𝜎𝑤[MPa]

MAX

-770.1

-694.1

-618.2

-542.3

-466.3

-390.4

-314.4

-238.5

-162.6

log ℙ[𝜃|𝐱]

(c)

Figure 5: The elements of the MAP regarding the AISI316L dataset. (a) Log-likelihood (b) Log-prior (c) Log-posterior.

Results and Discussion

Table 3 succinctly gathers the MAP estimates of ∆Kth,lc and ∆σwin terms of 99.7% conﬁdence intervals

and the reference values from the literature. Regarding AlSi8Cu3, the reported intervals satisfactorily match

the data from the literature for both ∆Kth,lc and ∆σwand corroborate the thorough choice of the prior

18

ACCEPTED PREPRINT

5.0 7.5 10.0 12.5 15.0

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

500

750

1000

1250

1500

1750

2000

Δ𝜎𝑤[MPa]

0.000

0.001

0.001

0.002

0.002

0.003

0.004

0.004

ℙ[𝜃|𝐱]

(a)

6.5 7.0 7.5 8.0 8.5

Δ𝐾𝑡ℎ,𝑙𝑐 [MPa√m]

0.00

0.25

0.50

0.75

1.00

1.25

1.50

ℙ[Δ𝐾𝑡ℎ,𝑙𝑐]

Δ𝐾𝑡ℎ,𝑙𝑐

Δ𝜎𝑤

600 800 1000 1200 1400 1600

Δ𝜎𝑤[MPa]

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

ℙ[Δ𝜎𝑤]

(b)

Figure 6: Results of the MAP and marginalisation of the posterior (a) Contour plots of the posterior distribution of AISI316L

dataset. (b) Marginal distributions of ∆Kth,lc and ∆σwin black and grey, respectively.

adopted. Nonetheless, the agreement with the literature was only partially achieved as concerns AISI316L.

Whilst the interval estimates of ∆σwincludes its reference counterpart, the analogous estimate for ∆Kth,lc

revealed the corresponding reference value to be unlikely. This could be interpreted as being a consequence of

the selected prior for ∆Kth,lc. In this regard, the data borrowed from the literature appears tightly centred

around 6.8 MPa √m, thus resulting in a limited standard deviation as high as 0.28 MPa √m. These features

biased the posterior by concentrating its distribution around 7.5 MPa √m with a relatively small standard

deviation of 0.3 MPa √m, thus reﬂecting the traits of the prior. Moreover, the ∆Kth,lc value provided by the

literature is certainly aﬀected by both epistemic and aleatoric errors that undoubtedly aﬀect its reliability.

However, this results and the choice of the priors are not limiting as one can leverage the excellent ﬂexibility

of the conceived methodology. Speciﬁcally, should further data regarding the characterisation of AISI316L

be available, the prior can be readily updated to obtain more and more truthful appraisals of the parameters.

Despite this, the estimates are conservative as compared with the reference values so that they can be deemed

as acceptable given the restrictive data availability.

Table 3: 99.7% probabilistic intervals of the estimated parameters and comparison with the literature.

Material ˆ

∆Kth,lc ±3qV[ˆ

∆Kth,lc]ˆ

∆σw±3qV[ˆ

∆σw] ∆Kth,lc ∆σwRef.

Ref. Ref.

[MPa√m] [MPa] [MPa √m] [MPa]

AlSi8Cu3 5.4±3·1.3 187.1±3·20.4 3.54 212.8 [31]

AISI316L 7.5±3·0.3 1003 ±3·146 9.04 900 [58]

It should be mentioned that the microstructure of the material under examination may aﬀect fatigue

19

ACCEPTED PREPRINT

performance and, consequently, have implications on the evaluated EH parameters. Nevertheless, the results

obtained through this method can be compared to other similar materials having the same microstructural

characteristics, i.e. fabricated employing the same manufacturing conditions. Hence, the retrieved EH

parameters are exclusively valid for the speciﬁc analysed material. In this instance, the sole feature that is

meant to vary is the size of the defect where the fatal fatigue crack originated from. On the other hand,

it is not possible to characterise the long crack fatigue behaviour unless bespoke experimental would be

deployed. In this regard, the intrinsic material defects span (approximately) from 10 µm to 103µm in terms

of the defect representative size. To overcome this limitation the proposed methodology exploits literature

data to estimated rational interval which the EH parameters should belong to.

The obtained marginal posterior distributions of ∆Kth,lc and ∆σwwere utilised to conduct a Monte Carlo

simulation to determine the probabilistic fatigue endurance limit curve, following the procedure laid out in

Section 2.4. The outcomes of the statistical post-processing (see Section 2.4) are graphically summarised

in Fig. 7(a) and Fig. 7(b) for AlSi8Cu3 and AISI316L, respectively. The black solid line indicated in these

ﬁgures is the evaluated EH curve, i.e. the EH curve given by the expected estimated parameters, whereas

the grey solid line is the reference EH curve retrieved from the literature. Additionally, the dot-dashed and

dashed lines are the lower (E(M)−P(M)) and upper (E(M)+P(M)) bound of the prediction interval computed

through Eq.(23). For the sake of convenience, each ﬁgure reports the related experimental dataset. Besides

the mathematical interpretation of the prediction intervals given in Section 2.4, E(M)± P(M)deﬁnes the

probabilistic crack propagation region. In particular, its lower and upper bounds are regarded as the failure

probability thresholds at 2.5% and 97.5% respectively, for the adopted conﬁdence level β= 95% (Eq. (23)).

The evaluated EH curves shown in Fig. 7 concurs well with the corresponding reference ones. This could

be attributed the distinctive marginal posterior of each identiﬁed parameter. In this respect, the reference

values of ∆Kth,lc and ∆σware included within the 99.7% probabilistic interval in Table 3, thus resulting

likely values with respect to the relative marginal posterior. This however is not fully conﬁrmed when

considering ∆Kth,lc of AISI316. Speciﬁcally, its reference value falls outside the probability interval so that

it appears unlikely. Despite this, the evaluated EH curves can be considered as acceptable, since the scope

of this method is estimating the fatigue endurance limit by the sole exploitation of fatigue characterisation

and literature data, rather than seeking a perfect match with the reference curves.

The peculiar characteristics of the marginal posterior are reﬂected on the prediction intervals as well.

During the Monte Carlo simulation, samples are drawn from the marginal posterior distributions of ∆Kth,lc

and ∆σw. Consequently, if the reference values belongs to the 99.7% probabilistic interval, these values are

likely to be drawn from the marginal posteriors. Also, the closer are the reference values to the mode of

the respective marginal posterior the higher the probability of being drawn is. Therefore, the associated

EH curve trials are supposed to increasingly approach the corresponding reference EH curve to a certain

extent. In particular, if the reference values of ∆Kth,lc and ∆σwlays in a suﬃciently narrow neighbourhood

20

ACCEPTED PREPRINT

of the mode, the prediction interval will enclose the reference EH curve with the conﬁdence level β. This

could evidently be seen for AlSi8Cu3 by comparing the values in Table 3 and Fig. 7(a). By contrast, since

the evaluated ∆Kth,lc of AISI316 falls outside the probabilistic interval and far away from the mode of the

posterior, this property does not hold, see Fig. 7(b). Further, Fig. 7(b) also highlight the implication of

the chosen prior. In this respect, the limited scatter of the literature values of ∆Kth,lc drastically narrowed

the prediction band across the long cracks regime. As brieﬂy mentioned earlier this phenomenon can be

mitigated by enriching the prior with data from the literature spanning a wider range of ∆Kth,lc.

Furthermore, a rapid critical inspection of the results shown in Fig. 7 enables one to assess the buoyancy

of the EH curves evaluated at diﬀerent failure probability levels. Particularly, apart from the regions of

the diagram where failed and run-out specimens coexist, each curve reasonably leaves failed and run-out

specimens above and beneath the EH curves, according to their respective level of failure probability. As a

result, each probabilistic EH curve accommodates the natural conformation of the dataset.

In order not to overload the present manuscript, the authors transferred a few comments on the conver-

gence of the Monte Carlo simulation to Appendix Appendix E.

101102103

√area [𝜇m]

102

6 × 101

2 × 102

3 × 102

Δ𝜎[MPa]

2.5%

97.5%

Run-out

Failed Reference EH

Evaluated EH Pred. Band @ 2.5%

Pred. Band @ 97.5%

1.9

2.2

2.6

3.0

3.4

3.8

4.2

4.6

5.0

5.4

5.8

Δ𝐾[MPa √m]

(a)

100101102103

√area [𝜇m]

103

2 × 102

3 × 102

4 × 102

6 × 102

Δ𝜎[MPa]

2.5%

97.5%

Run-out

Failed Reference EH

Evaluated EH Pred. Band @ 2.5%

Pred. Band @ 97.5%

1.3

2.7

4.0

5.4

6.8

8.2

9.5

10.9

12.3

13.6

15.0

Δ𝐾[MPa √m]

(b)

Figure 7: Probabilistic fatigue endurance limit at 2.5% and 97.5% failure probability along with the examined datasets (a)

AlSi8Cu3 (b) AISI316L.

Figure 7 which shows the probabilistic endurance limit curves evaluated using the entire dataset. For

the sake of validating the developed LR-MAP framework, the same curves are retrieved splitting the full

dataset into the training set Dtr, which LR is trained on, and test set Dts, which was held out, such that

Dtr ∪Dts =D,Dtr ∩Dts =∅, and Ntr +Nts =N. Since a typical 80/20 random split was adopted, the

training and test dataset were proportionally partitioned as Ntr = 0.8N, and Nts = 0.2N. Importantly, the

training stage exploited the same priors used earlier.

Table 4 reports the identiﬁed parameters which approaches those in Table 3, thus proving the robustness

21

ACCEPTED PREPRINT

of the LR-MAP approach.

Table 4: 99.7% probabilistic intervals of the estimated parameters and comparison with the literature. This intervals were

retrieved conducting the training stage on the sole Dtr.

Material ˆ

∆Kth,lc ±3qV[ˆ

∆Kth,lc]ˆ

∆σw±3qV[ˆ

∆σw] ∆Kth,lc ∆σwRef.

Ref. Ref.

[MPa√m] [MPa] [MPa √m] [MPa]

AlSi8Cu3 5.2±3·1.3 188.6±3·22.3 3.54 212.8 [31]

AISI316L 7.3±3·0.26 1003 ±3·107 9.04 900 [58]

Figure 8 shows the evaluated EH curves at diﬀerent levels of failure probability obtained by partitioning

the dataset. The points belonging to Dtr are indicated in agreement with the markers of Fig. 7. Whilst,

failed and run-out specimens belonging to Dts are denoted by “plus” and triangle markers. It can be notice

once again that each curve divides failed specimens from run-out specimens, according to their respective

level of failure probability – except for the region were failed and run-out may overlap.

101102103

√area [𝜇m]

102

6 × 101

2 × 102

3 × 102

Δ𝜎[MPa]

2.5%

97.5%

Run-out

Failed

Reference EH

Run-out (Test set)

Failed (Test set)

Evaluated EH

Pred. Band @ 2.5%

Pred. Band @ 97.5%

1.9

2.2

2.6

3.0

3.4

3.8

4.2

4.6

5.0

5.4

5.8

Δ𝐾[MPa √m]

(a)

100101102103

√area [𝜇m]

103

2 × 102

3 × 102

4 × 102

6 × 102

Δ𝜎[MPa]

2.5%

97.5%

Run-out

Failed

Reference EH

Run-out (Test set)

Failed (Test set)

Evaluated EH

Pred. Band @ 2.5%

Pred. Band @ 97.5%

1.3

2.7

4.0

5.4

6.8

8.2

9.5

10.9

12.3

13.6

15.0

Δ𝐾[MPa √m]

(b)

Figure 8: Probabilistic fatigue endurance limit at 2.5% and 97.5% failure probability along with the examined datasets. This

curves were retrieved conducting the training stage on the sole Dtr (a) AlSi8Cu3 (b) AISI316L.

22

ACCEPTED PREPRINT

5. Conclusions

The computational framework proposed herein allows for an accurate probabilistic evaluation of the

fatigue endurance limit in metallic materials containing defects. The underlying method relies on the

elegant combination of LR and MAP which enables for probabilistically evaluating the EH’s parameters

(∆Kth,lc and ∆σw). This methodology seamlessly merges partial fatigue results of a probed material and

the available results taken from the literature that help enforce the physical soundness of the evaluated

parameters. Consequently, a relatively limited number of fatigue tests is suﬃcient to attain a comprehensive

fatigue characterisation and curb the onerousness of its protocol.

The LR-MAP strategy outputs the probability distributions of ∆Kth,lc and ∆σw. Subsequently, these

distribution can be supplied to a Monte Carlo simulation to evaluate the fatigue endurance limit curve

for a speciﬁed level of failure probability, e.g. 2.5%. This outcome can be directly used when designing

engineering structures or components following a probabilistic framework.

Two diﬀerent metallic alloys, i.e. AlSi8Cu3 and AISI316L, were considered to demonstrate the eﬀective-

ness of the LR-MAP appoach. In both examples, the intervention of MAP showed its excellent inclination at

incorporating prior knowledge from the literature, thus leading to univocal estimates of the EH parameters.

Additionally, a 80/20 random split of the dataset enabled for validating the LR-MAP, while demonstrating

its robustness and predictive capabilities.

The LR-MAP method will bring extremely relevant implications in many fatigue design contexts, es-

pecially those dealing with defective materials, such as AMed. Accurate fatigue endurance curves will

increasingly empower the implementation of emerging techniques in numerous advanced engineering appli-

cations.

Acknowledgements

Luca Laurenti (TU Delft) and Andrea Patan`e (Trinity College Dublin) are gratefully acknowledged for

the fruitful discussions and valuable suggestions.

This work has been supported by the project ”CONCERTO – Multiscale modelling/characterisation and

fabrication of nanocomposite ceramics with improved toughness” funded by the MIUR Progetti di Ricerca

di Rilevante Interesse Nazionale (PRIN) Bando 2020 – grant 2020BN5ZW9.

23

ACCEPTED PREPRINT

References

[1] G. S. Campbell and R. Lahey. A survey of serious aircraft accidents involving fatigue fracture. International Journal of

Fatigue, 6(1):25–30, January 1984. ISSN 0142-1123. doi: 10.1016/0142-1123(84)90005-7.

[2] Zahra S. Hosseini, Mohsen Dadfarnia, Brian P. Somerday, Petros Sofronis, and Robert O. Ritchie. On the theoretical

modeling of fatigue crack growth. Journal of the Mechanics and Physics of Solids, 121:341–362, December 2018. ISSN

0022-5096. doi: 10.1016/j.jmps.2018.07.026.

[3] U. Zerbst, M. Madia, C. Klinger, D. Bettge, and Y. Murakami. Defects as a root cause of fatigue failure of metal-

lic components. I: Basic aspects. Engineering Failure Analysis, 97:777–792, March 2019. ISSN 13506307. doi:

10.1016/j.engfailanal.2019.01.055.

[4] Yukitaka Murakami. Metal Fatigue: Eﬀects of Small Defects and Nonmetallic Inclusions. Elsevier, 2019.

[5] Niloofar Sanaei and Ali Fatemi. Defects in additive manufactured metals and their eﬀect on fatigue perfor-

mance: A state-of-the-art review. Progress in Materials Science, 117:100724, April 2021. ISSN 00796425. doi:

10.1016/j.pmatsci.2020.100724.

[6] M. R. Mitchell. Review of the Mechanical Properties of Cast Steels With Emphasis on Fatigue Behavior and the Inﬂuence

of Microdiscontinuities. Journal of Engineering Materials and Technology, 99(4):329–343, October 1977. ISSN 0094-4289.

doi: 10.1115/1.3443549.

[7] Yukitaka Murakami. Material defects as the basis of fatigue design. International Journal of Fatigue, 41:2–10, August

2012. ISSN 01421123. doi: 10.1016/j.ijfatigue.2011.12.001.

[8] Amir Mostafaei, Cang Zhao, Yining He, Seyed Reza Ghiaasiaan, Bo Shi, Shuai Shao, Nima Shamsaei, Ziheng Wu,

Nadia Kouraytem, Tao Sun, Joseph Pauza, Jerard V. Gordon, Bryan Webler, Niranjan D. Parab, Mohammadreza

Asherloo, Qilin Guo, Lianyi Chen, and Anthony D. Rollett. Defects and anomalies in powder bed fusion metal additive

manufacturing. Current Opinion in Solid State and Materials Science, 26(2):100974, April 2022. ISSN 1359-0286. doi:

10.1016/j.cossms.2021.100974.

[9] Hanqing Liu, Jun Song, Haomin Wang, Chuanli Yu, Yaohan Du, Chao He, Qingyuan Wang, and Qiang Chen. Slip-driven

and weld pore assisted fatigue crack nucleation in electron beam welded TC17 titanium alloy joint. International Journal

of Fatigue, 154:106525, January 2022. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2021.106525.

[10] P J Withers. Residual stress and its role in failure. Reports on Progress in Physics, 70(12):2211–2264, December 2007.

ISSN 0034-4885, 1361-6633. doi: 10.1088/0034-4885/70/12/R04.

[11] M.N. James, D.J. Hughes, Z. Chen, H. Lombard, D.G. Hattingh, D. Asquith, J.R. Yates, and P.J. Webster. Residual

stresses and fatigue performance. Engineering Failure Analysis, 14(2):384–395, March 2007. ISSN 13506307. doi:

10.1016/j.engfailanal.2006.02.011.

[12] E. Salvati and A.M. Korsunsky. An analysis of macro- and micro-scale residual stresses of Type I, II and III using

FIB-DIC micro-ring-core milling and crystal plasticity FE modelling. International Journal of Plasticity, 98:123–138,

November 2017. ISSN 07496419. doi: 10.1016/j.ijplas.2017.07.004.

[13] Alessandro Tognan, Lise Sandnes, Giovanni Totis, Marco Sortino, Filippo Berto, Øystein Grong, and Enrico Sal-

vati. Evaluation and Origin of Residual Stress in Hybrid Metal and Extrusion Bonding and Comparison with Fric-

tion Stir Welding. International Journal of Mechanical Sciences, 218:107089, March 2022. ISSN 00207403. doi:

10.1016/j.ijmecsci.2022.107089.

[14] A. Tognan, L. Laurenti, and E. Salvati. Contour Method with Uncertainty Quantiﬁcation: A Robust and Optimised

Framework via Gaussian Process Regression. Experimental Mechanics, April 2022. ISSN 0014-4851, 1741-2765. doi:

10.1007/s11340-022-00842-w.

[15] Enrico Salvati, Hongjia Zhang, Kai Soon Fong, Xu Song, and Alexander M. Korsunsky. Separating plasticity-induced

24

ACCEPTED PREPRINT

closure and residual stress contributions to fatigue crack retardation following an overload. Journal of the Mechanics

and Physics of Solids, 98:222–235, January 2017. ISSN 00225096. doi: 10.1016/j.jmps.2016.10.001.

[16] Yukitaka Murakami, Toshio Takagi, Kentaro Wada, and Hisao Matsunaga. Essential structure of S-N curve: Prediction

of fatigue life and fatigue limit of defective materials and nature of scatter. International Journal of Fatigue, 146:106138,

May 2021. ISSN 01421123. doi: 10.1016/j.ijfatigue.2020.106138.

[17] L. Patriarca, S. Beretta, S. Foletti, A. Riva, and S. Parodi. A probabilistic framework to deﬁne the design stress and

acceptable defects under combined-cycle fatigue conditions. Engineering Fracture Mechanics, 224:106784, February 2020.

ISSN 00137944. doi: 10.1016/j.engfracmech.2019.106784.

[18] D. S. Paolino, G. Chiandussi, and M. Rossetto. A uniﬁed statistical model for S-N fatigue curves: probabilistic deﬁnition.

Fatigue & Fracture of Engineering Materials & Structures, 36(3):187–201, 2013. ISSN 1460-2695. doi: 10.1111/j.1460-

2695.2012.01711.x.

[19] A. Tridello, C. Boursier Niutta, F. Berto, M. M. Tedesco, S. Plano, D. Gabellone, and D. S. Paolino. Design against

fatigue failures: Lower bound P-S-N curves estimation and inﬂuence of runout data. International Journal of Fatigue,

162:106934, September 2022. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2022.106934.

[20] Luke Sheridan. A modiﬁed El-Haddad model for versatile defect tolerant design. International Journal of Fatigue, 145:

106062, April 2021. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.106062.

[21] Luke Sheridan, Joy E. Gockel, and Onome E. Scott-Emuakpor. Stress-defect-life interactions of fatigued addi-

tively manufactured alloy 718. International Journal of Fatigue, 143:106033, February 2021. ISSN 0142-1123. doi:

10.1016/j.ijfatigue.2020.106033.

[22] S. Romano, L. Patriarca, S. Foletti, and S. Beretta. LCF behaviour and a comprehensive life prediction model for

AlSi10Mg obtained by SLM. International Journal of Fatigue, 117:47–62, December 2018. ISSN 01421123. doi:

10.1016/j.ijfatigue.2018.07.030.

[23] Marco Pelegatti, Denis Benasciutti, Francesco De Bona, Alex Lanzutti, Michele Magnan, Jelena Srnec Novak, Enrico

Salvati, Francesco Sordetti, Marco Sortino, Giovanni Totis, and Emanuele Vaglio. On the factors inﬂuencing the elasto-

plastic cyclic response and low cycle fatigue failure of AISI 316L steel produced by laser-powder bed fusion. International

Journal of Fatigue, 165:107224, December 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2022.107224.

[24] Y. Murakami and M. Endo. Eﬀects of defects, inclusions and inhomogeneities on fatigue strength. International Journal

of Fatigue, 16(3):163–182, April 1994. ISSN 0142-1123. doi: 10.1016/0142-1123(94)90001-9.

[25] H. Kitagawa. Applicability of fracture mechanics to very small cracks or the cracks in the early stage. Proc. of 2nd ICM,

Cleveland, 1976, pages 627–631, 1976.

[26] M Ciavarella and F Monno. On the possible generalizations of the Kitagawa–Takahashi diagram and of the El Had-

dad equation to ﬁnite life. International Journal of Fatigue, 28(12):1826–1837, December 2006. ISSN 01421123. doi:

10.1016/j.ijfatigue.2005.12.001.

[27] J. Maierhofer, H. P. G¨anser, and R. Pippan. Modiﬁed Kitagawa–Takahashi diagram accounting for ﬁnite notch depths.

International Journal of Fatigue, 70:503–509, January 2015. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2014.07.007.

[28] M.H. El Haddad, T.H. Topper, and K.N. Smith. Prediction of non propagating cracks. Engineering Fracture Mechanics,

11(3):573–584, January 1979. ISSN 00137944. doi: 10.1016/0013-7944(79)90081-X.

[29] M Chapetti. Fatigue propagation threshold of short cracks under constant amplitude loading. International Journal of

Fatigue, 25(12):1319–1326, December 2003. ISSN 01421123. doi: 10.1016/S0142-1123(03)00065-3.

[30] Uwe Zerbst, Giovanni Bruno, Jean-Yves Buﬃ`ere, Thomas Wegener, Thomas Niendorf, Tao Wu, Xiang Zhang, Nikolai

Kashaev, Giovanni Meneghetti, Nik Hrabe, Mauro Madia, Tiago Werner, Kai Hilgenberg, Martina Koukol´ıkov´a, Radek

Proch´azka, Jan Dˇzugan, Benjamin M¨oller, Stefano Beretta, Alexander Evans, Rainer Wagener, and Kai Schnabel. Dam-

age tolerant design of additively manufactured metallic components subjected to cyclic loading: State of the art and

25

ACCEPTED PREPRINT

challenges. Progress in Materials Science, 121:100786, August 2021. ISSN 0079-6425. doi: 10.1016/j.pmatsci.2021.100786.

[31] R. Aigner, S. Pusterhofer, S. Pomberger, M. Leitner, and M. Stoschka. A probabilistic Kitagawa-Takahashi diagram

for fatigue strength assessment of cast aluminium alloys. Materials Science and Engineering: A, 745:326–334, February

2019. ISSN 0921-5093. doi: 10.1016/j.msea.2018.12.108.

[32] B. K¨ohler, H. Bomas, W. Leis, and L. Kallien. Endurance limit of die-cast magnesium alloys AM50hp and AZ91hp

depending on type and size of internal cavities. International Journal of Fatigue, 44, November 2012. ISSN 01421123.

doi: 10.1016/j.ijfatigue.2012.05.011.

[33] Zhengheng Lian, Minjie Li, and Wencong Lu. Fatigue life prediction of aluminum alloy via knowledge-based machine

learning. International Journal of Fatigue, 157:106716, April 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2021.106716.

[34] E. Maleki, S. Bagherifard, S.M.J. Razavi, M. Bandini, A. du Plessis, F. Berto, and M. Guagliano. On the eﬃciency of

machine learning for fatigue assessment of post-processed additively manufactured AlSi10Mg. International Journal of

Fatigue, 160:106841, July 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2022.106841.

[35] Enrico Salvati, Alessandro Tognan, Luca Laurenti, Marco Pelegatti, and Francesco De Bona. A defect-based physics-

informed machine learning framework for fatigue ﬁnite life prediction in additive manufacturing. Materials & Design,

222:111089, October 2022. ISSN 02641275. doi: 10.1016/j.matdes.2022.111089.

[36] Hongyixi Bao, Shengchuan Wu, Zhengkai Wu, Guozheng Kang, Xin Peng, and Philip J. Withers. A machine-learning fa-

tigue life prediction approach of additively manufactured metals. Engineering Fracture Mechanics, 242:107508, February

2021. ISSN 00137944. doi: 10.1016/j.engfracmech.2020.107508.

[37] Taotao Zhou, Shan Jiang, Te Han, Shun-Peng Zhu, and Yinan Cai. A Physically Consistent Framework for Fatigue

Life Prediction using Probabilistic Physics-Informed Neural Network. International Journal of Fatigue, page 107234,

September 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2022.107234.

[38] Seunghyun Moon, Ruimin Ma, Ross Attardo, Charles Tomonto, Mark Nordin, Paul Wheelock, Michael Glavicic, Maxwell

Layman, Richard Billo, and Tengfei Luo. Impact of surface and pore characteristics on fatigue life of laser powder bed

fusion Ti–6Al–4V alloy described by neural network models. Scientiﬁc Reports, 11(1):20424, December 2021. ISSN

2045-2322. doi: 10.1038/s41598-021-99959-6.

[39] Xin Peng, Shengchuan Wu, Weijian Qian, Jianguang Bao, Yanan Hu, Zhixin Zhan, Guangping Guo, and Philip J.

Withers. The potency of defects on fatigue of additively manufactured metals. International Journal of Mechanical

Sciences, 221:107185, May 2022. ISSN 00207403. doi: 10.1016/j.ijmecsci.2022.107185.

[40] Jun Li, Zhengmao Yang, Guian Qian, and Filippo Berto. Machine learning based very-high-cycle fatigue life prediction

of Ti-6Al-4V alloy fabricated by selective laser melting. International Journal of Fatigue, 158:106764, May 2022. ISSN

01421123. doi: 10.1016/j.ijfatigue.2022.106764.

[41] Lei He, Zhilei Wang, Yuki Ogawa, Hiroyuki Akebono, Atsushi Sugeta, and Yoshiichirou Hayashi. Machine-learning-based

investigation into the eﬀect of defect/inclusion on fatigue behavior in steels. International Journal of Fatigue, 155:106597,

February 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2021.106597.

[42] A Ciampaglia, A. Tridello, D.S. Paolino, and F. Berto. Data driven method for predicting the eﬀect of process parameters

on the fatigue response of additive manufactured alsi10mg parts. International Journal of Fatigue, 170:107500, 2023.

ISSN 0142-1123. doi: https://doi.org/10.1016/j.ijfatigue.2023.107500.

[43] Nicolas Baca¨er. A Short History of Mathematical Population Dynamics. Springer London, London, 2011. ISBN 978-0-

85729-114-1 978-0-85729-115-8. doi: 10.1007/978-0-85729-115-8.

[44] Stephan Dreiseitl and Lucila Ohno-Machado. Logistic regression and artiﬁcial neural network classiﬁcation mod-

els: a methodology review. Journal of Biomedical Informatics, 35(5):352–359, October 2002. ISSN 1532-0464. doi:

10.1016/S1532-0464(03)00034-0.

[45] Nhat-Duc Hoang, Quoc-Lam Nguyen, and Xuan-Linh Tran. Automatic Detection of Concrete Spalling Using Piecewise

26

ACCEPTED PREPRINT

Linear Stochastic Gradient Descent Logistic Regression and Image Texture Analysis. Complexity, 2019:e5910625, July

2019. ISSN 1076-2787. doi: 10.1155/2019/5910625. Publisher: Hindawi.

[46] Alfred DeMaris. A Tutorial in Logistic Regression. Journal of Marriage and Family, 57(4):956–968, 1995. ISSN 0022-

2445. doi: 10.2307/353415. Publisher: [Wiley, National Council on Family Relations].

[47] David G. Kleinbaum and Mitchel Klein. Logistic Regression. Statistics for Biology and Health. Springer New York, New

York, NY, 2010. ISBN 978-1-4419-1741-6 978-1-4419-1742-3. doi: 10.1007/978-1-4419-1742-3.

[48] Joseph M. Hilbe. Practical Guide to Logistic Regression. Chapman and Hall/CRC, New York, March 2016. ISBN

978-0-429-17446-9. doi: 10.1201/b18678.

[49] Kevin P. Murphy. Machine learning: a probabilistic perspective. Adaptive computation and machine learning series.

MIT Press, Cambridge, MA, 2012. ISBN 978-0-262-01802-9.

[50] Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. Adaptive computation

and machine learning. MIT Press, Cambridge, Mass, 2006. ISBN 978-0-262-18253-9. OCLC: ocm61285753.

[51] I. M Sobol’. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Computational

Mathematics and Mathematical Physics, 7(4):86–112, January 1967. ISSN 0041-5553. doi: 10.1016/0041-5553(67)90144-

9.

[52] Jon Herman and Will Usher. SALib: An open-source Python library for Sensitivity Analysis. The Journal of Open

Source Software, 2(9), January 2017. doi: 10.21105/joss.00097. Publisher: The Open Journal.

[53] Takuya Iwanaga, William Usher, and Jonathan Herman. Toward SALib 2.0: Advancing the accessibility and interpretabil-

ity of global sensitivity analyses. Socio-Environmental Systems Modelling, 4:18155, May 2022. doi: 10.18174/sesmo.18155.

[54] William Q. Meeker, Gerald J. Hahn, and Luis A. Escobar. Statistical Intervals: A Guide for Practitioners and Re-

searchers. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA, March 2017. ISBN

978-1-118-59484-1 978-0-471-68717-7. doi: 10.1002/9781118594841.

[55] S. Beretta and S. Romano. A comparison of fatigue strength sensitivity to defects for materials manufactured by

AM or traditional processes. International Journal of Fatigue, 94:178–191, January 2017. ISSN 0142-1123. doi:

10.1016/j.ijfatigue.2016.06.020.

[56] D. Rigon and G. Meneghetti. Engineering estimation of the fatigue limit of wrought and defective additively manufactured

metals for diﬀerent load ratios. International Journal of Fatigue, 154:106530, January 2022. ISSN 01421123. doi:

10.1016/j.ijfatigue.2021.106530.

[57] B. J. Wang, S. D. Wang, D. K. Xu, and E. H. Han. Recent progress in fatigue behavior of Mg alloys in air and aqueous

media: A review. Journal of Materials Science & Technology, 33(10):1075–1086, October 2017. ISSN 1005-0302. doi:

10.1016/j.jmst.2017.07.017.

[58] Pierre Merot, Franck Morel, Etienne Pessard, Linamaria Gallegos Mayorga, Paul Buttin, and Thierry Baﬃe. Fatigue

strength and life assessment of L-PBF 316L stainless steel showing process and corrosion related defects. Engineering

Fracture Mechanics, page 108883, October 2022. ISSN 0013-7944. doi: 10.1016/j.engfracmech.2022.108883.

[59] M. Kuroda and T. J. Marrow. Modelling the eﬀects of surface ﬁnish on fatigue limit in austenitic stainless steels.

Fatigue & Fracture of Engineering Materials & Structures, 31(7):581–598, July 2008. ISSN 8756758X, 14602695. doi:

10.1111/j.1460-2695.2008.01223.x.

[60] Olivier Andreau, Etienne Pessard, Imade Koutiri, Jean-Daniel Penot, Corinne Dupuy, Nicolas Saintier, and Patrice

Peyre. A competition between the contour and hatching zones on the high cycle fatigue behaviour of a 316L stainless

steel: Analyzed using X-ray computed tomography. Materials Science and Engineering: A, 757:146–159, May 2019.

ISSN 0921-5093. doi: 10.1016/j.msea.2019.04.101.

[61] Olivier Andreau, Etienne Pessard, Imade Koutiri, Patrice Peyre, and Nicolas Saintier. Inﬂuence of the position and size of

various deterministic defects on the high cycle fatigue resistance of a 316L steel manufactured by laser powder bed fusion.

27

ACCEPTED PREPRINT

International Journal of Fatigue, 143:105930, February 2021. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.105930.

[62] Jyoti Suryawanshi, K.G. Prashanth, and U. Ramamurty. Mechanical behavior of selective laser melted 316L stainless

steel. Materials Science and Engineering: A, 696:113–121, June 2017. ISSN 09215093. doi: 10.1016/j.msea.2017.04.058.

[63] Xiaoyu Liang, Anis Hor, Camille Robert, Feng Lin, and Franck Morel. Eﬀects of building direction and loading mode on

the high cycle fatigue strength of the laser powder bed fusion 316L. International Journal of Fatigue, 170:107506, May

2023. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2023.107506.

[64] Bernd M. Sch¨onbauer and Herwig Mayer. Eﬀect of small defects on the fatigue strength of martensitic stainless steels.

International Journal of Fatigue, 127:362–375, October 2019. ISSN 01421123. doi: 10.1016/j.ijfatigue.2019.06.021.

[65] Bernd M. Sch¨onbauer and Stefanie E. Stanzl-Tschegg. Inﬂuence of environment on the fatigue crack growth behaviour

of 12% Cr steel. Ultrasonics, 53(8):1399–1405, December 2013. ISSN 0041-624X. doi: 10.1016/j.ultras.2013.02.007.

[66] Daniele Rigon and Giovanni Meneghetti. An engineering estimation of fatigue thresholds from a microstructural size and

Vickers hardness: application to wrought and additively manufactured metals. International Journal of Fatigue, 139:

105796, October 2020. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.105796.

[67] Bernd M. Sch¨onbauer, Keiji Yanase, and Masahiro Endo. The inﬂuence of various types of small defects on the fatigue

limit of precipitation-hardened 17-4PH stainless steel. Theoretical and Applied Fracture Mechanics, 87:35–49, February

2017. ISSN 0167-8442. doi: 10.1016/j.tafmec.2016.10.003.

[68] Bernd M. Sch¨onbauer, Stefanie E. Stanzl-Tschegg, Andrea Perlega, Ronald N. Salzman, Neville F. Rieger, Alan Turn-

bull, Shengqi Zhou, Mikolaj Lukaszewicz, and David Gandy. The inﬂuence of corrosion pits on the fatigue life of

17-4PH steam turbine blade steel. Engineering Fracture Mechanics, 147:158–175, October 2015. ISSN 0013-7944. doi:

10.1016/j.engfracmech.2015.08.011.

[69] Simone Romano, P. D. Nezhadfar, Nima Shamsaei, Mohsen Seiﬁ, and Stefano Beretta. High cycle fatigue behavior and

life prediction for additively manufactured 17-4 PH stainless steel: Eﬀect of sub-surface porosity and surface roughness.

Theoretical and Applied Fracture Mechanics, 106:102477, April 2020. ISSN 0167-8442. doi: 10.1016/j.tafmec.2020.102477.

[70] S. C. Wu, Y. Luo, Z. Shen, L. C. Zhou, W. H. Zhang, and G. Z. Kang. Collaborative crack initiation mechanism of

25CrMo4 alloy steels subjected to foreign object damages. Engineering Fracture Mechanics, 225:106844, February 2020.

ISSN 0013-7944. doi: 10.1016/j.engfracmech.2019.106844.

[71] Yoshikazu Nakai and Keisuke Tanaka. Grain Size Eﬀect on Growth Threshold for Small Surface-Cracks and Long

Through-Cracks Under Cyclic Loading. Proceedings of The Japan Congress on Materials Research, pages 106–112, 1980.

ISSN 0368-3141.

[72] S. Usami and S. Shida. Elastic–Plastic Analysis of the Fatigue Limit for a Material with Small Flaws. Fatigue & Fracture

of Engineering Materials & Structures, 1(4):471–481, 1979. ISSN 1460-2695. doi: 10.1111/j.1460-2695.1979.tb01334.x.

[73] Yves Nadot. Fatigue from Defect: Inﬂuence of Size, Type, Position, Morphology and Loading. International Journal of

Fatigue, 154:106531, January 2022. ISSN 01421123. doi: 10.1016/j.ijfatigue.2021.106531.

[74] A. Nasr, Ch. Bouraoui, R. Fathallah, and Y. Nadot. Probabilistic high cycle fatigue behaviour of nodular cast iron

containing casting defects. Fatigue & Fracture of Engineering Materials & Structures, 32(4):292–309, 2009. ISSN 1460-

2695. doi: 10.1111/j.1460-2695.2009.01330.x.

[75] S. Beretta, L. Patriarca, M. Gargourimotlagh, A. Hardaker, D. Brackett, M. Salimian, J. Gumpinger, and T. Ghidini.

A benchmark activity on the fatigue life assessment of AlSi10Mg components manufactured by L-PBF. Materials &

Design, 218:110713, June 2022. ISSN 0264-1275. doi: 10.1016/j.matdes.2022.110713.

[76] S. Beretta, M. Gargourimotlagh, S. Foletti, A. du Plessis, and M. Riccio. Fatigue strength assessment of “as built”

AlSi10Mg manufactured by SLM with diﬀerent build orientations. International Journal of Fatigue, 139:105737, October

2020. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.105737.

[77] Xiao Niu, Shun-Peng Zhu, Jin-Chao He, Ding Liao, Jos´e A. F. O. Correia, Filippo Berto, and Qingyuan Wang. Defect

28

ACCEPTED PREPRINT

tolerant fatigue assessment of AM materials: Size eﬀect and probabilistic prospects. International Journal of Fatigue,

160:106884, July 2022. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2022.106884.

[78] Timothy D. Piette, Robert J. Warren, Anthony G. Spangenberger, Edward J. Hummelt, and Diana A. Lados. Microstruc-

ture evolution, fatigue crack growth, and ultrasonic fatigue in As-fabricated laser powder bed and conventionally cast

Al–10Si-0.4Mg: A mechanistic understanding and integrated ﬂaw-sensitive fatigue design methods. Materials Science

and Engineering: A, 825:141892, September 2021. ISSN 0921-5093. doi: 10.1016/j.msea.2021.141892.

[79] M. Bonneric, C. Brugger, N. Saintier, A. Castro Moreno, and B. Tranchand. Contribution of the introduction of artiﬁcial

defects by additive manufacturing to the determination of the Kitagawa diagram of Al-Si alloys. Procedia Structural

Integrity, 38:141–148, 2022. ISSN 24523216. doi: 10.1016/j.prostr.2022.03.015.

[80] Mohamed Iben Houria, Yves Nadot, Raouf Fathallah, Matthew Roy, and Daan M. Maijer. Inﬂuence of casting defect

and SDAS on the multiaxial fatigue behaviour of A356-T6 alloy including mean stress eﬀect. International Journal of

Fatigue, 80:90–102, November 2015. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2015.05.012.

[81] Antonio Rotella, Yves Nadot, Micka¨el Piellard, R´emi Augustin, and Michel Fleuriot. Fatigue limit of a cast Al-Si-Mg alloy

(A357-T6) with natural casting shrinkages using ASTM standard X-ray inspection. International Journal of Fatigue,

114:177–188, September 2018. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2018.05.026.

[82] C. Garb, M. Leitner, and F. Gr¨un. Eﬀect of elevated temperature on the fatigue strength of casted AlSi8Cu3 aluminium

alloys. Procedia Structural Integrity, 7:497–504, 2017. ISSN 24523216. doi: 10.1016/j.prostr.2017.11.118.

[83] Christian Garb, Martin Leitner, Bernhard Stauder, Dirk Schnubel, and Florian Gr¨un. Application of modiﬁed Kitagawa-

Takahashi diagram for fatigue strength assessment of cast Al-Si-Cu alloys. International Journal of Fatigue, 111:256–268,

June 2018. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2018.01.030.

[84] C. Garb, M. Leitner, and F. Gr¨un. Application of area-concept to assess fatigue strength of AlSi7Cu0.5Mg

casted components. Engineering Fracture Mechanics, 185:61–71, November 2017. ISSN 0013-7944. doi:

10.1016/j.engfracmech.2017.03.018.

[85] P. Mu, Y. Nadot, C. Nadot-Martin, A. Chabod, I. Serrano-Munoz, and C. Verdu. Inﬂuence of casting defects on the

fatigue behavior of cast aluminum AS7G06-T6. International Journal of Fatigue, 63:97–109, June 2014. ISSN 0142-1123.

doi: 10.1016/j.ijfatigue.2014.01.011.

[86] C. Santus and D. Taylor. Physically short crack propagation in metals during high cycle fatigue. International Journal

of Fatigue, 31(8):1356–1365, August 2009. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2009.03.002.

[87] M. da Fonte, F. Romeiro, M. de Freitas, S.E. Stanzl-Tschegg, E.K. Tschegg, and A.K. Vasud´evan. The eﬀect of mi-

crostructure and environment on fatigue crack growth in 7049 aluminium alloy at negative stress ratios. International

Journal of Fatigue, 25(9):1209–1216, 2003. ISSN 0142-1123. doi: https://doi.org/10.1016/S0142-1123(03)00150-6.

[88] Anastasios G. Gavras, Diana A. Lados, Victor K. Champagne, and Robert J. Warren. Eﬀects of processing on mi-

crostructure evolution and fatigue crack growth mechanisms in cold-spray 6061 aluminum alloy. International Journal

of Fatigue, 110:49–62, 2018. ISSN 0142-1123. doi: https://doi.org/10.1016/j.ijfatigue.2018.01.006.

[89] Romali Biswal, Xiang Zhang, Abdul Khadar Syed, Mustafa Awd, Jialuo Ding, Frank Walther, and Stewart Williams.

Criticality of porosity defects on the fatigue performance of wire + arc additive manufactured titanium alloy. International

Journal of Fatigue, 122:208–217, May 2019. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2019.01.017.

[90] Etienne Pessard, Manon Lavialle, Pascal Laheurte, Paul Didier, and Myriam Brochu. High-cycle fatigue behavior of a

laser powder bed fusion additive manufactured Ti-6Al-4V titanium: Eﬀect of pores and tested volume size. International

Journal of Fatigue, 149:106206, August 2021. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2021.106206.

[91] G. L´eopold, Y. Nadot, T. Billaudeau, and J. Mendez. Inﬂuence of artiﬁcial and casting defects on fatigue strength of

moulded components in Ti-6Al-4V alloy. Fatigue & Fracture of Engineering Materials & Structures, 38(9):1026–1041,

2015. ISSN 1460-2695. doi: 10.1111/ﬀe.12326.

29

ACCEPTED PREPRINT

[92] Hiroshige Masuo, Yuzo Tanaka, Shotaro Morokoshi, Hajime Yagura, Tetsuya Uchida, Yasuhiro Yamamoto, and Yuk-

itaka Murakami. Inﬂuence of defects, surface roughness and HIP on the fatigue strength of Ti-6Al-4V manufactured

by additive manufacturing. International Journal of Fatigue, 117:163–179, December 2018. ISSN 0142-1123. doi:

10.1016/j.ijfatigue.2018.07.020.

[93] J. O Peters and R. O Ritchie. Inﬂuence of foreign-object damage on crack initiation and early crack growth during

high-cycle fatigue of Ti–6Al–4V. Engineering Fracture Mechanics, 67(3):193–207, October 2000. ISSN 0013-7944. doi:

10.1016/S0013-7944(00)00045-X.

[94] M. Benedetti and C. Santus. Building the Kitagawa-Takahashi diagram of ﬂawed materials and components using an

optimized V-notched cylindrical specimen. Engineering Fracture Mechanics, 224:106810, February 2020. ISSN 0013-7944.

doi: 10.1016/j.engfracmech.2019.106810.

[95] Chretien Ga¨elle, Sarrazin-Baudoux Christine, Leost Laurie, and Hervier Zeline. Near-threshold fatigue propagation of

physically short and long cracks in Titanium alloy. Procedia Structural Integrity, 2:950–957, 2016. ISSN 24523216. doi:

10.1016/j.prostr.2016.06.122.

[96] Y. N. Hu, S. C. Wu, Z. K. Wu, X. L. Zhong, S. Ahmed, S. Karabal, X. H. Xiao, H. O. Zhang, and P. J. Withers. A new

approach to correlate the defect population with the fatigue life of selective laser melted Ti-6Al-4V alloy. International

Journal of Fatigue, 136:105584, July 2020. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.105584.

[97] Theo Persenot, Alexis Burr, Guilhem Martin, Jean-Yves Buﬃere, Remy Dendievel, and Eric Maire. Eﬀect of build

orientation on the fatigue properties of as-built Electron Beam Melted Ti-6Al-4V alloy. International Journal of Fatigue,

118:65–76, January 2019. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2018.08.006.

[98] Punit Kumar and Upadrasta Ramamurty. High cycle fatigue in selective laser melted Ti-6Al-4V. Acta Materialia, 194:

305–320, August 2020. ISSN 1359-6454. doi: 10.1016/j.actamat.2020.05.041.

[99] L. Barricelli, L. Patriarca, A. du Plessis, and S. Beretta. Orientation-dependent fatigue assessment of Ti6Al4V manufac-

tured by L-PBF: Size of surface features and shielding eﬀect. International Journal of Fatigue, page 107401, November

2022. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2022.107401.

[100] M. Tarik Hasib, Halsey E. Ostergaard, Qian Liu, Xiaopeng Li, and Jamie J. Kruzic. Tensile and fatigue crack growth

behavior of commercially pure titanium produced by laser powder bed fusion additive manufacturing. Additive Manu-

facturing, 45:102027, 2021. ISSN 2214-8604. doi: https://doi.org/10.1016/j.addma.2021.102027.

[101] Yuwei Zhai, Haize Galarraga, and Diana A. Lados. Microstructure, static properties, and fatigue crack growth mechanisms

in ti-6al-4v fabricated by additive manufacturing: Lens and ebm. Engineering Failure Analysis, 69:3–14, 2016. ISSN

1350-6307. doi: https://doi.org/10.1016/j.engfailanal.2016.05.036. Special issue on the International Conference on

Structural Integrity.

[102] M. Tarik Hasib, Halsey E. Ostergaard, Xiaopeng Li, and Jamie J. Kruzic. Fatigue crack growth behavior of laser powder

bed fusion additive manufactured ti-6al-4v: Roles of post heat treatment and build orientation. International Journal

of Fatigue, 142:105955, 2021. ISSN 0142-1123. doi: https://doi.org/10.1016/j.ijfatigue.2020.105955.

[103] Baohua Nie, Zihua Zhao, Yongzhong Ouyang, Dongchu Chen, Hong Chen, Haibo Sun, and Shu Liu. Eﬀect of Low Cycle

Fatigue Predamage on Very High Cycle Fatigue Behavior of TC21 Titanium Alloy. Materials, 10(12):1384, December

2017. ISSN 1996-1944. doi: 10.3390/ma10121384. Number: 12 Publisher: Multidisciplinary Digital Publishing Institute.

[104] J. R. Poulin, A. Kreitcberg, P. Terriault, and V. Brailovski. Fatigue strength prediction of laser powder bed fusion

processed Inconel 625 specimens with intentionally-seeded porosity: Feasibility study. International Journal of Fatigue,

132:105394, March 2020. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2019.105394.

[105] J.-R. Poulin, V. Brailovski, and P. Terriault. Long fatigue crack propagation behavior of inconel 625 processed by laser

powder bed fusion: Inﬂuence of build orientation and post-processing conditions. International Journal of Fatigue, 116:

634–647, 2018. ISSN 0142-1123. doi: https://doi.org/10.1016/j.ijfatigue.2018.07.008.

30

ACCEPTED PREPRINT

[106] Xiaoan Hu, Zhiyuan Xue, TingTing Ren, Yun Jiang, ChengLi Dong, and Fencheng Liu. On the fatigue crack growth

behaviour of selective laser melting fabricated inconel 625: Eﬀects of build orientation and stress ratio. Fatigue & Fracture

of Engineering Materials & Structures, 43(4):771–787, 2020. doi: https://doi.org/10.1111/ﬀe.13181.

[107] Yoichi Yamashita, Takao Murakami, Rei Mihara, Masami Okada, and Yukitaka Murakami. Defect analysis and fatigue

design basis for Ni-based superalloy 718 manufactured by selective laser melting. International Journal of Fatigue, 117:

485–495, December 2018. ISSN 01421123. doi: 10.1016/j.ijfatigue.2018.08.002.

[108] W. B. Li, J. C. Pang, H. Zhang, S. X. Li, and Z. F. Zhang. The high-cycle fatigue properties of selective laser melted

Inconel 718 at room and elevated temperatures. Materials Science and Engineering: A, 836:142716, March 2022. ISSN

0921-5093. doi: 10.1016/j.msea.2022.142716.

[109] Jayaraj Radhakrishnan, Punit Kumar, Shihao Li, Yakai Zhao, and Upadrasta Ramamurty. Unnotched fatigue of Inconel

718 produced by laser beam-powder bed fusion at 25 and 600°C. Acta Materialia, 225:117565, February 2022. ISSN

1359-6454. doi: 10.1016/j.actamat.2021.117565.

[110] Kun Yang, Qi Huang, Qingyuan Wang, and Qiang Chen. Competing crack initiation behaviors of a laser additively

manufactured nickel-based superalloy in high and very high cycle fatigue regimes. International Journal of Fatigue, 136:

105580, July 2020. ISSN 0142-1123. doi: 10.1016/j.ijfatigue.2020.105580.

[111] Nicola Cersullo, Jon Mardaras, Philippe Emile, Katja Nickel, Vitus Holzinger, and Christian H¨uhne. Eﬀect of Internal

Defects on the Fatigue Behavior of Additive Manufactured Metal Components: A Comparison between Ti6Al4V and

Inconel 718. Materials, 15(19):6882, January 2022. ISSN 1996-1944. doi: 10.3390/ma15196882.

[112] R. Konecn´a, L. Kunz, G. Nicoletto, and A. Baca. Fatigue crack growth behavior of Inconel 718 produced by selective laser

melting. Frattura ed Integrit`a Strutturale, 10(35):31–40, December 2015. ISSN 19718993. doi: 10.3221/IGF-ESIS.35.04.

[113] R. Koneˇcn´a, L. Kunz, G. Nicoletto, and A. Baˇca. Long fatigue crack growth in inconel 718 produced

by selective laser melting. International Journal of Fatigue, 92:499–506, 2016. ISSN 0142-1123. doi:

https://doi.org/10.1016/j.ijfatigue.2016.03.012. Fatigue crack paths 2015.

[114] Xiaolong Li, Wei Li, Muhammad Imran Lashari, Tatsuo Sakai, Ping Wang, Liang Cai, Xiaoming Ding, and Usama Hamid.

Fatigue failure behavior and strength prediction of nickel-based superalloy for turbine blade at elevated temperature.

Engineering Failure Analysis, 136:106191, June 2022. ISSN 1350-6307. doi: 10.1016/j.engfailanal.2022.106191.

[115] M. Papakyriacou, H. Mayer, U. Fuchs, S. E. Stanzl-Tschegg, and R. P. Wei. Inﬂuence of atmospheric moisture on slow

fatigue crack growth at ultrasonic frequency in aluminium and magnesium alloys. Fatigue & Fracture of Engineering

Materials & Structures, 25(8-9):795–804, 2002. ISSN 1460-2695. doi: 10.1046/j.1460-2695.2002.00571.x.

[116] Md.Shahnewaz Bhuiyan and Mutoh Yoshiharu. Two stage S-N curve in corrosion fatigue of extruded magnesium alloy

AZ31. Songklanakarin Journal of Science and Technology, 31, September 2009.

[117] Sotomi Ishihara, Zhenyu Nan, and Takahito Goshima. Eﬀect of microstructure on fatigue behavior of AZ31 mag-

nesium alloy. Materials Science and Engineering: A, 468-470:214–222, November 2007. ISSN 0921-5093. doi:

10.1016/j.msea.2006.09.124.

[118] Sanlong Zheng, Qin Yu, and Yanyao Jiang. An experimental study of fatigue crack propagation in extruded

AZ31B magnesium alloy. International Journal of Fatigue, 47:174–183, February 2013. ISSN 01421123. doi:

10.1016/j.ijfatigue.2012.08.010.

[119] Duke Culbertson and Yanyao Jiang. An experimental study of the orientation eﬀect on fatigue crack propagation

in rolled az31b magnesium alloy. Materials Science and Engineering: A, 676:10–19, 2016. ISSN 0921-5093. doi:

https://doi.org/10.1016/j.msea.2016.08.088.

[120] P Venkateswaran, S Ganesh Sundara Raman, S.D Pathak, Y Miyashita, and Y Mutoh. Fatigue crack growth be-

haviour of a die-cast magnesium alloy az91d. Materials Letters, 58(20):2525–2529, 2004. ISSN 0167-577X. doi:

https://doi.org/10.1016/j.matlet.2004.03.014.

31

ACCEPTED PREPRINT

[121] S. Ishihara, S. Kitagawa, M. R. Qi, H. Shibata, and T. Goshima. Evaluation of distribution of fatigue lives of the extruded

magnesium alloy AZ61. Acta Mechanica, 224(6):1251–1260, June 2013. ISSN 1619-6937. doi: 10.1007/s00707-013-0872-8.

[122] M Bhuiyan, Y Mutoh, T Murai, and S Iwakami. Corrosion fatigue behavior of extruded magnesium alloy AZ61 under

three diﬀerent corrosive environments. International Journal of Fatigue, 30(10-11):1756–1765, October 2008. ISSN

01421123. doi: 10.1016/j.ijfatigue.2008.02.012.

[123] U. Karr, B.M. Sch¨onbauer, and H. Mayer. Near-threshold fatigue crack growth properties of wrought magnesium alloy

AZ61 in ambient air, dry air, and vacuum. Fatigue & Fracture of Engineering Materials & Structures, 41(9):1938–1947,

September 2018. ISSN 8756758X. doi: 10.1111/ﬀe.12832.

[124] Sabrina Alam Khan, Yukio Miyashita, Yoshiharu Mutoh, and Toshikatsu Koike. Eﬀect of anodized layer thickness on

fatigue behavior of magnesium alloy. Materials Science and Engineering: A, 474(1):261–269, February 2008. ISSN

0921-5093. doi: 10.1016/j.msea.2007.04.078.

32

ACCEPTED PREPRINT

Appendix A. SIF Equivalence

Let ∆Kbe the SIF of a given defects characterised by Yand √area, and subjected to ∆σ:

∆K=Y∆σ√area (A.1)

One wish to ﬁnd the equivalent √areaeq for the given defects but imposing Yeq, hence:

∆Keq =Yeq∆σ√areaeq (A.2)

Assuming ∆Keq = ∆Kleads to:

√areaeq =Y2

i

Y2

eq

√area (A.3)

which is Eq. (5).

Appendix B. L2-Regularisation for the Posterior

Let us recall the argument of argmax of Eq. (14), namely the log-posterior without the logarithm of the

evidence (Eq. (13)):

log P[θ|x] = log P[x|θ] + log P[θ] (B.1)

Assuming ∆Klc,th and ∆σwas independent random variables, the prior is:

P[θ] = P[∆Kth,lc]P[∆σw] (B.2)

Hence, Eq. (B.1) becomes:

log P[θ|x] = log P[x|θ] + log P[∆Kth,lc] + log P[∆σw] (B.3)

Let us prescribe over ∆Klc,th and ∆σwtwo Gaussian priors such that:

∆Kth,lc ∼ N(µ∆K, S∆K)⇒P[∆Kth,lc] = 1

p2πS2

∆K

exp "−1

2∆Kth,lc −µ∆K

S∆K2#(B.4)

∆σw∼ N(µ∆σ, S∆σ)⇒P[∆σw] = 1

p2πS2

∆σ

exp "−1

2∆σw−µ∆σ

S∆σ2#(B.5)

where µ∆K,µ∆σ,S∆K, and S∆σare the known mean and the variance of each random variable. Therefore,

Eq. (B.3) turns out to be:

log P[θ|x] = log P[x|θ]−1

2∆Kth,lc −µ∆K

S∆K2

−1

2∆σw−µ∆σ

S∆σ2

+C(B.6)

in which Cis a constant deﬁned by:

C= log 1

p2πS2

∆K

+ log 1

p2πS2

∆σ

(B.7)

33

ACCEPTED PREPRINT

Recasting Eq. (B.6) into Eq. (14) and dropping the constant Cgive:

ˆ

θ= argmax

θ"log P[x|θ]−1

2∆Kth,lc −µ∆K

S∆K2

−1

2∆σw−µ∆σ

S∆σ2#(B.8)

The last equation can be rewritten as follows:

ˆ

θ= argmax

θlog P[x|θ] + λ∆K∥∆Kth,lc −µ∆K∥2

2+λ∆σ∥∆σw−µ∆σ∥2

2(B.9)

which represents the L2-regularised log-likelihood, where λ∆K,λ∆σare two independent regularising weights

λ∆K=−1/2S2

∆Kand λ∆σ=−1/2S2

∆σ, and ∥·∥2denotes the L2-norm. It is worth mentioning that Eq. (B.9)

can be easily customized in case of priors of diﬀerent nature. For instance, if a Uprior is prescribed over

∆σw, Eq. (B.9) transforms into:

ˆ

θ= argmax

θlog P[x|θ] + λ∆K∥∆Kth,lc −µ∆K∥2

2(B.10)

where L2-regularisation acts over the sole random variable ∆Kth,lc . Obviously, the same concept applies

when a Uprior is imposed on ∆Kth,lc. Finally, if Upriors are prescribed over both variables, the log-

likelihood is not regularised, and MAP reduces to the mere MLE.

Appendix C. Laplace’s Approximation for the Posterior

Laplace’s approximation models the posterior as Gaussian and entails a second-order Taylor expansion

of log P[θ|x] at ˆ

θ:

log P[θ|x]≈log P[ˆ

θ|x] + ∇log P[θ|x]θ=ˆ

θ

| {z }

=0

+1

2(θ−ˆ

θ)⊤H(log P[θ|x])θ=ˆ

θ(θ−ˆ

θ) (C.1)

where the ﬁrst-order term, ∇log P[θ|x]|θ=ˆ

θ, is zero as ˆ

θis a maximum point for log P[θ|x]. Additionally,

H(log P[θ|x])|θ=ˆ

θis the Hessian matrix of log P[θ|x] evaluated at ˆ

θ– concisely denoted as H(ˆ

θ) in the

following. Given that H(ˆ

θ) will take the role of covariance matrix of the posterior it has to be positive

(semi-) deﬁnite. However, since ˆ

θis a maximum, H(ˆ

θ) is negative deﬁnite. This can be corrected by

changing the sign of the second-order term:

log P[θ|x]≈log P[ˆ

θ|x]−1

2(θ−ˆ

θ)⊤H(ˆ

θ)(θ−ˆ

θ) (C.2)

where, with abuse of notation, H(ˆ

θ) is now positive deﬁnite. In addition, after this change of sign, H(ˆ

θ)

represents the Hessian matrix of −log P[θ|x] computed at ˆ

θ. The exponentiation of both sides of Eq. (C.2)

gives:

P[θ|x]≈P[ˆ

θ|x] exp −1

2(θ−ˆ

θ)⊤H(ˆ

θ)(θ−ˆ

θ)(C.3)

34

ACCEPTED PREPRINT

In Eq. (C.3), P[ˆ

θ|x] can be conveniently set as the traditional normalisation constant for bivariate Gaussian

distributions:

P[ˆ

θ|x] = 1

2π(det H(ˆ

θ)−1)1/2(C.4)

thus leading to the ﬁnal expression (Eq. (17)):

P[θ|x]≈1

2π(det H(ˆ

θ)−1)1/2exp −1

2(θ−ˆ

θ)⊤H(ˆ

θ)(θ−ˆ

θ)(C.5)

which can be contracted as Eq. (18):

P[θ|x]∼ N(θ|ˆ

θ,H−1) (C.6)

35

ACCEPTED PREPRINT

Appendix D. Additional Material Datasets

This Appendix summarise the data for Ti-, Ni- and Mg-based alloys to facilitate the exploitation of the

proposed method on materials diﬀerent from those analysed in the present manuscript. The data herein are

presented in the same manner as those in Section 3.

Table D.5: Datasets of Ti alloys. Additive manufacturing, wrought, and cast are abbreviated as AM, WR, and CS, respectively.

A†in “Loading mode” indicates data obtained through fatigue crack growth test, whereas a ∗denotes data that are not available

or indicated in the corresponding references.

Material Manufacturing

details

Loading

mode

R ∆Kth,lc ∆σwReference(s)

[MPa√m] [MPa]

Ti-6Al-4V AM (WAAM) Tension 0.1 4.5 540 [89]

Ti-6Al-4V AM (L-PBF) Tension -1 2.33 342 −663 [90]

Ti-6Al-4V AM Tension 0.1 ∗325 [91]

Ti-6Al-4V AM (EBM)

AM (L-PBF)

Tension

Tension

-1

-1

6.82

6.79

1181

1210

[66, 92]

Ti-6Al-4V Mill annealed

bar stock

Tension 0.1 1 −4.3 450 [93]

Ti-6Al-4V AM (L-PBF) Tension -1 7.3 −7.5 753.4 −1086 [94]

Ti-6Al-4V AM (L-PBF) Tension -1 5.6 914 −1086 [86]

Ti-6Al-4V ∗Tension 0.1 3.73 −5.22 160 −200 [95]

Ti-6Al-4V AM (L-PBF) Tension -1 3.9 389 −432 [96]

Ti-6Al-4V AM (EBM) Tension 0.1 3.4 −5.7 ∗[97]

Ti-6Al-4V AM (EBM) Bending -1 5.5 −8.4 680 −1050 [98]

Ti-6Al-4V AM (L-PBF) Tension and

bending

0.1 2 −6.7 195 −221 [99]

Ti-6Al-4V AM (L-PBF) †0.1 1.8−4.1∗[100]

Ti-6Al-4V AM (LENS)

AM (EBM)

†

†

0.1 2.6−4.5

3.2−5

∗

∗

[101]

Ti-6Al-4V AM (L-PBF, as-built)

AM (L-PBF, HIP)

AM (L-PBF, HT 1020)

†

†

†

0.1

0.1

0.1

1.52 −1.82

3.12 −4.53

6.22 −7.29

∗

∗

∗

[102]

TC21 ∗Tension -1 2.8 430 [103]

36

ACCEPTED PREPRINT

Table D.6: Datasets of Ni-based alloys. Additive manufacturing, wrought, and cast are abbreviated as AM, WR, and CS,

respectively. A †in “Loading mode” indicates data obtained through fatigue crack growth test, whereas a ∗denotes data that

are not available or indicated in the corresponding references.

Material Manufacturing

details

Loading

mode

R ∆Kth,lc ∆σwReference(s)

[MPa√m] [MPa]

Inconel 625 AM (L-PBF) Tension 0.1 7.0 590 [104]

Inconel 625 WR

AM (L-PBF, annealed)

AM (L-PBF, HIP)

†

†

†

0.1

0.1

0.1

7.2

7.1−10.6

10.9−11.1

∗

∗

∗

[105]

Inconel 625 AM (L-PBF) †0.1−0.7 5 −10 ∗[106]

Inconel 718 AM (L-PBF)

AM (L-PBF)

Tension

Tension

-1

0.1

6

4

∗

∗

[21, 20]

Inconel 718 AM (L-PBF) Tension -1 13.7 ∗[107]

Inconel 718 AM (L-PBF) Tension -1 12 506 [108]

Inconel 718 AM (L-PBF) Tension -1 1.81 −2.06 250 −325 [109]

Inconel 718 AM (L-PBF) Tension -1 12 946 [110]

Inconel 718 AM (L-PBF) Tension 0.1 ∗ ∗ [111]

Inconel 718 AM (L-PBF) †0.1 1.5 ∗[112]

Inconel 718 AM (L-PBF) †0.1 3.0 ∗[113]

K418 CS

CS

Tension

Tension

-1

0.1

13.85

12.09

∗

∗

[114]

37

ACCEPTED PREPRINT

Table D.7: Datasets of Mg-based alloys. Additive manufacturing, wrought, and cast are abbreviated as AM, WR, and CS,

respectively. A †in “Loading mode” indicates data obtained through fatigue crack growth test, whereas a ∗denotes data that

are not available or indicated in the corresponding references.

Material Manufacturing

details

Loading

mode

R ∆Kth,lc ∆σwReference(s)

[MPa√m] [MPa]

AM50hp Die-CS Conventional Tension 0.1 1.36 −1.62 49 [32]

AZ91hp Die-CS (Conventional)

Die-CS (Vacural)

Tension

Tension

0.1

0.1

∗

1.43 −1.61

44

62

[32]

AS21hp CS †-1 1.30 −2.3∗[115]

AZ31 Extruded Tension -1 0.85 88 [116]

AZ31 Extruded Rot. Bending -1 0.35 ∗[117]

AZ31B Extruded Tension 0.10 −0.75 0.95 −2.43 ∗[118]

AZ31B Rolled

Rolled

Rolled

†

†

†

0.1

0.5

0.75

1.23 −2.00

0.85 −1.54

0.77 −1.07

∗

∗

∗

[119]

AZ91D Die-CS †0.1 4 ∗[120]

AZ61 Extruded Rot. Bending -1 0.39 ∗[121]

AZ61 Extruded Tension -1 0.8 140 −200 [122]

AZ61 WR †-1 1.1−1.9∗[123]

AM60 Anodised Tension 0.1 0.86 72 [124]

AM60hp CS †-1 1.40 −2.4∗[115]

AS21hp CS †-1 1.25 −2.7∗[115]

Appendix E. Convergence of the Monte Carlo Simulation

It is also useful to brieﬂy comment on the behaviour of the convergence indicator ρideﬁned in Eq. (24).

Except for a few initial ﬂuctuations, ρiconverged rapidly toward zero. Although the total number of Monte

Carlo samples was set to M= 12288, Fig. E.9 suggests that 1000 samples would have suﬃced. In fact, after

this number of samples, ρistabilised and exhibited negligible oscillation so that the convergence could have

been considered as achieved.

38

ACCEPTED PREPRINT

(

j

)

Figure E.9: Convergence indicator throughout the Monte Carlo simulation.

39