![Specimen-wise fatigue test results and defect configuration of the experiments by [27]. The linkage of Kitagawa-Takahashi and Frost diagrams is demonstrated for defect size of 450 µm by drawing an additional vertical axis with increasing $K_t$. Keeping the defect size constant but making the defect sharper, as shown in the defect configuration, cracks initiate easier and fatigue limit drops until becoming constant when the cracks become non-propagating. El-Haddad [2] and Siebel-Stieler [31] models were used to describe the fatigue limit from crack non-propagation and initiation, respectively. A scaled prediction by Murakami-Endo model [3] is plotted for comparison.](https://www.researchgate.net/profile/Joona-Vaara-2/publication/390504740/figure/fig6/AS:11431281370261105@1744353090518/Specimen-wise-fatigue-test-results-and-defect-configuration-of-the-experiments-by-27_Q320.jpg)
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Contents lists available at ScienceDirect
International Journal of Fatigue
journal homepage: www.elsevier.com/locate/ijfatigue
Probabilistic description of the cyclic R-curve based on microstructural
barriers
Joona Vaara a,e,∗, Kimmo Kärkkäinen d, Miikka Väntänen b, Jukka Kemppainen c,
Bernd Schönbauer f, Suraj More f, Mari Åman d, Tero Frondelius a,d,e
aWärtsilä, Järvikatu 2-4, 65100 Vaasa, Finland
bGlobal Boiler Works Oy, Lumijoentie 8, 90400 Oulu, Finland
cApplied and Computational Mathematics, University of Oulu, Pentti Kaiteran katu 1, 90014, Finland
dMaterials and Mechanical Engineering, University of Oulu, Pentti Kaiteran katu 1, 90014, Finland
eFaculty of Built Environment, Tampere University, Korkeakoulunkatu 7, 33720, Finland
fInstitute of Physics and Materials Science, University of Natural Resources and Life Sciences (BOKU), Peter-Jordan-Str. 82, 1190 Vienna, Austria
A R T I C L E I N F O
Keywords:
Cyclic R-curve
Fatigue behavior
Probabilistic model
Microstructure
Crack initiation
A B S T R A C T
A model for the probabilistic cyclic R-curve has been derived. The model is based on the commonly used
hypothesis of consecutive microstructural barrier fronts defining the erratic behavior of microstructurally short
cracks and the transition to physically short cracks with declining importance of the microstructural features.
The model can describe the linkage between the traditional cyclic R-curve analyses and the El-Haddad type
Kitagawa-Takahashi diagrams with the asymptotic fatigue limit at small defect sizes. The model fit against
the experimental non-propagating crack lengths perfectly matches the observed and predicted fatigue limit for
several defect types and sizes. The presented framework can be used to analyze any geometry, loading history,
or defect configuration, including defect interaction problems.
1. Introduction
The physical understanding of fatigue has been revolutionized many
times throughout history. Elber’s finding of crack closure [1], models
for physically short crack behavior [2], and non-propagating cracks
emanating from defects defining the fatigue limit [3] truly caused
paradigm shifts in the fatigue research community. The works of Mi-
nakawa, Nakamura, McEvily [4,5] and Tanaka [6] offered further
physical insight by experimentally quantifying the development of the
crack closure or threshold stress intensity factor 𝛥𝐾th, as a function of
the crack length — also known as the cyclic resistance curve (R-curve).
The study of crack closure has been active for several decades from
theoretical, experimental, and simulation perspectives [7–11]. Gradu-
ally, we could see these efforts paying off by providing prospective
physical explanations for several phenomena common in the field of
fatigue: mean stress sensitivity [11,12], physically short crack behav-
ior [13], environmental effects [7,14] and notch fatigue [6,15,16], to
name a few. However, widespread adaptation has yet to be realized,
and phenomenological models are still commonly used.
Complete understanding can still be argued to be lacking. From the
authors’ perspective, the single biggest issue of cyclic R-curve analysis
∗Corresponding author at: Faculty of Built Environment, Tampere University, Korkeakoulunkatu 7, 33720, Finland.
E-mail address: joona.vaara@wartsila.com (J. Vaara).
currently is the transferability of the experimental results from through-
plate cracks to radially growing 3D cracks and defect-initiated fatigue
analysis. As a recent example in the study of Pourheidar et al. [17],
the measured cyclic R-curve resulted in conservative estimates of the
fatigue limit compared to the experiments with artificial defects. This
was the case when the experiments produced shorter non-propagating
cracks (NPCs), but the method showed better correspondence with
those with longer non-propagating cracks. This behavior could be
explained by Miller’s hypothesis [18] of consecutive microstructural
barriers and the diminishing of their role with crack length — explain-
ing the differences between microstructurally short and physically short
cracks. While the hypothesis is widely used [19,20], it has remained
on a qualitative level. Other more recent findings strengthen the hy-
pothesis: Chapetti et al. [20] have emphasized that without defects, the
strongest microstructural barriers, such as grain boundaries, commonly
arrest the microstructurally short cracks. Additionally, multi-phase ma-
terials often exhibit crack initiation in the softer phase and arrest at
or close to the harder phase boundary [21–24]. Crack arrest at the
phase boundary is difficult to explain purely based on crack closure
development and a contribution from microstructural barriers is likely
required. A model capturing the essence of Miller’s hypothesis would
https://doi.org/10.1016/j.ijfatigue.2025.108953
Received 17 January 2025; Received in revised form 10 March 2025; Accepted 22 March 2025
International Journal of Fatigue 198 (2025) 108953
Available online 2 April 2025
0142-1123/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ).
J. Vaara et al.
Nomenclature
cdf Cumulative distribution function
EPFM Elasto-Plastic Fracture Mechanics
FEM Finite Element Method
KT Kitagawa–Takahashi
LEFM Linear Elastic Fracture Mechanics
mb Microstructural barrier
MFM Micromechanical Fracture Mechanics
NPC Non-propagating crack
pdf Probability density function
PICC Plasticity-induced crack closure
RICC Roughness-induced crack closure
𝛽Stress gradient factor
𝜒Relative stress gradient
𝛥
𝐾th Baseline cyclic resistance curve
𝛥CTD Cyclic crack tip displacement
𝛥𝐽 Cyclic J-integral
𝛥𝐾eff Effective stress intensity factor range
𝛥𝐾th,ef f Intrinsic threshold stress intensity factor
range
𝛿𝐾th,min,𝑃 𝑃th quantile of the extra resistance from
microstructural barriers
𝛥𝐾th,0Initial resistance
𝛿𝐾th,𝑖 Extra resistance of a microstructural barrier
𝛥𝐾th,𝑗 Additional resistance due to crack closure
development
𝛥𝐾th Threshold stress intensity factor range
d𝑎Infinitesimal crack advance
d𝑀Number of microstructural barriers at the
crack front within the infinitesimal advance
d𝑠
d𝑁Number of microstructural barrier fronts
within the infinitesimal crack advance d𝑎
d𝑠Infinitesimal distance along crack front
HV Vickers hardness
𝜇th , 𝜎th Scale and shape parameters of the extra
resistance from microstructural barrier
𝜌Notch root radius
𝜎aStress amplitude
𝜎ci Crack initiation limit
𝜎w,0Fatigue limit of plain specimens
𝜎wFatigue limit
area Murakami–Endo square-root area parame-
ter as a defect/crack size indicator
inherently carry information on variance scaling. In contrast, the efforts
to capture the cyclic R-curve’s stochastic characteristics have been mere
regression analyses [17,25]. Formulating such a model and examining
its properties is the key objective of the present study.
The role of crack initiation is worth highlighting. As noted before,
the immediate microstructure around defects is clearly a source of
variance in fatigue tests, not only from the micromechanical fracture
mechanics perspective but also from the crack initiation perspective.
Professor Murakami, in his book, describes one of the experiments
where the fatigue test specimen contained multiple artificial defects;
some defects initiated no cracks, whereas some had a non-propagating
crack on one side, and some had a non-propagating crack on both
sides [26, Chapter 4, p. 43]. Similar behavior was observed in the
experiments of More et al. [27]. The high variance of crack initiation
also hindered the analysis of defect interaction studies in [28,29].
𝑎, 𝛥𝑎 Crack extension
𝑏𝑗Saturation rate of 𝑗th component of crack
closure development
𝑑Average distance between microstructural
barriers along crack front, characteristic
microstructural length
𝑑2Average distance between microstructural
barrier fronts in the crack growth direction,
characteristic microstructural length
𝐹𝑋(⋅)Cumulative distribution function of random
variable 𝑋
𝑓𝑋(⋅)Probability density function of random
variable 𝑋
𝐾𝑡Elastic stress concentration factor
𝑃Probability or 𝑃th quantile
𝑃(A) Probability of event A
𝑅Stress ratio
𝑠Length of crack front
𝑠𝑔Material constant, microstructural length
R-curve Resistance curve
According to the authors’ experience with QT-steels, the fatigue limit
defined by small non-metallic inclusions can be up to 15% higher com-
pared to the Murakami–Endo model prediction, possibly explained by
increased crack initiation resistance. Another study worth mentioning
in this regard, conducted by Spriestersbach et al. [30], verified by
serial grinding of the fatigued specimen that larger, but less critical,
non-metallic inclusions could be found. Again, this suggests that the
microstructure and possibly crack initiation play a role in what was
ultimately observed. To shed some light on this issue, More et al.
[27] systematically studied multiple artificial defects per specimen
and monitored the crack initiation and non-propagating crack lengths.
They concluded that the modified Siebel-Stieler model [31] could
describe the observed crack initiation behavior for multiple defect types
and sizes. Further support for the Siebel-Stieler type crack initiation
behavior can also be found in the earlier experimental study by Schön-
bauer and Mayer [32] and in the recent non-local prediction approach
by Merot et al. [33]. The latter non-local formulation can also quantify
the inevitable statistical size effect related to the increased probability
of finding weaker microstructure in the larger highly stressed volume.
Thus, crack initiation as a whole needs to be understood to distinguish
its role in the fatigue process, observations, and predictions alike. In
this paper, the data presented by More et al. [27] is studied primarily
from the crack non-propagation perspective, but the learnings of crack
initiation need to be included and possibly expanded for the sake of
completeness.
All the efforts in quantifying cyclic R-curves and crack initia-
tion limit often culminate in visualizing the results in the Kitagawa–
Takahashi (KT) diagram, defining the relation between the size of
the defect or crack and fatigue limit. This diagram, together with
defect size distributions, can be further refined for the assessment of
the failure probability of machine components or used in assessing
the severity of observations from non-destructive testing in quality
control. Zerbst and Madia [19] and Maierhofer et al. [15], among
others, have highlighted that the cyclic R-curve co-exists with, and
ultimately defines, the KT-diagram and transition from the physically
short crack regime to the long crack regime. However, the KT-diagrams
from traditional cyclic R-curve analyses are not capable of describing
the asymptotically constant fatigue strength with decreasing defect size
but rather produce a KT-diagram asymptotic to the intrinsic threshold
value, which is unconservative. In practice, an ad hoc fix to this has
been to hard-limit the KT-diagram by the plain fatigue strength [20,34].
International Journal of Fatigue 198 (2025) 108953
2
J. Vaara et al.
On a broader scale, the need for such fixes might compromise the
physical foundation and generalization of a model. In the long run,
this raises issues regarding the transferability of the results to real
component fatigue analysis, which is of major importance in this paper.
The aim of the present paper is to study and discuss this aspect further.
As seen above, when the physical data-generating process has multi-
ple sources of scatter, it is only possible to make thorough conclusions
by dramatically increasing the sample sizes in the experimental cam-
paigns. This paper proposes a different approach to this problem: a
model for the stochastic data-generating process is required to utilize
all available data. In this approach, the observed scatter contains
information on the underlying physical phenomena, which can all be
deduced to provide unique properties to the data-generating process.
Understanding the sources of scatter is key to deriving a physically
founded model for the probabilistic R-curve. In this paper, a model
based on Miller’s hypothesis is proposed for the probabilistic resistance
curve, answering the questions: ‘Given an arbitrary 𝛥𝐾(𝛥𝑎) path, what
is the distribution of the non-propagating crack lengths and probability
of observing them?’. The re-analysis of the systematic measurements
performed by More et al. [27] plays a key role in demonstrating the
model. It will be shown that fatigue from drilled artificial defects,
with two cracks initiating at the free surface, is inevitably a two-crack
interaction problem with certain consequences. Although the results
are promising, it must be acknowledged that this is the first step in
this direction, and much more verification and refinement is required.
The model can unify the phenomena seen for microstructurally short,
physically short, and long cracks. While existing solutions focus more
on the explicit description of crack growth rate [35], the focus of the
proposed approach will be on the analysis of defect-initiated fatigue
limit and capturing the stochastic characteristics of the cyclic R-curve.
The paper is structured as follows: first, we try to form an overview
of the relevant physical mechanisms, their relation to modeling choices,
and resulting phenomena. Then, a probabilistic model for the cyclic
R-curve is derived. The experimental results by More et al. [27] are
revisited before a re-analysis. Next, the parameter estimation problem
is considered. Since the paper is heavy on theoretical considerations,
much of the discussion must take place in the modeling section to
justify the modeling choices. Still, the Discussion section is dedicated
to the results and reflections on the model’s properties.
2. Model
Before deriving the model, a deep dive is required into state-of-
the-art knowledge and how the physical phenomena are reflected in
the modeling choices. Microstructural barriers and heterogeneity as
well as their effects on the development of crack closure are seen as
fundamental pieces in explaining the scatter of non-propagating crack
lengths.
2.1. Non-propagating crack lengths
The model is explained below before the mathematical derivation
is presented. Consider an arbitrary crack with length 𝑎 and crack front
length 𝑠(𝑎). The average distance between microstructural barriers, or
the characteristic microstructural length, is 𝑑. Each microstructural
barrier has an i.i.d. capability to resist crack penetration with a finite
support. These microstructural barriers can describe virtually any-
thing in the microstructure: high-angle grain boundaries, precipitates,
carbides, triple points, or increased roughness-induced crack closure
(RICC) resulting from asymmetric yield [36] when the crack faces a
new grain, to name a few. All of these can be lumped into this single
mechanism, providing identical behavior. On top of this, a baseline
crack closure is assumed to develop as a function of crack extension.
It has been argued that no or little crack closure is developed for
a classical crack initiating from persistent slip bands in shear mode
(stage I) before turning perpendicular to the first principal stress (stage
II) [20,37,38]. In contrast, the simulation studies of plasticity-induced
crack closure (PICC) in [39] showed increased closure development
for an inclined crack. However, for cracks initiated from defects large
enough, it is commonly observed that the stage I crack growth phase
is either very short or skipped altogether [40].
The criterion for crack arrest is that the strength of each mi-
crostructural barrier is higher than the crack driving force, which
corresponds to the minimum strength criterion. Chapetti et al. [20]
advocate for the maximum microstructural strength criterion. With a
maximum strength defining the crack arrest, however, one would need
to assess the distances between such strong barriers and how the crack
penetration between them would affect the crack driving force. The
microstructurally resolved crack growth simulation models, such as the
one presented in [41], can produce rather irregular crack shapes. For
the sake of simplicity, the minimum criterion, promoting regular crack
shape, and ease of analysis was chosen here. With finite support for the
single microstructural barrier’s strength, either model should provide
a similar statistical scaling of the variance with increased crack front
length. The longer the crack front, the more microstructural barriers
the crack front sees, and the less variance there is for either mini-
mum or maximum strength. A schematic drawing of a crack growth
scheme with grains as the microstructural barriers is shown in Fig.
1. The effective crack driving force is initially high, but then crack
closure develops, and a window of crack non-propagation is clearly
visualized. To conclude, this concept is commonly used to explain
the transition from the microstructurally short to the physically short
crack regime [18,20,42]. However, no models exist to describe this
mathematically, which will be tackled next.
Given a crack of length 𝑎, facing 𝑀 microstructural barriers (mb)
at the crack front, the probability of crack arresting is assumed to be
equal to the probability that all microstructural barriers have higher
strength than the crack driving force
𝑃(crack arrest at 𝑎∣𝑀mb)=
𝑀
𝑖=1
𝑃𝛥𝐾(𝑎, 𝑠𝑖)< 𝛥𝐾t h(𝑎, 𝑠𝑖),(1)
where 𝑠𝑖 is the position of the 𝑖th microstructural barrier along the
crack front and 𝛥𝐾th is the corresponding resistance. This corresponds
to the minimum strength defining the crack arrest. The rationale is
that as long as the crack penetrates any microstructural barriers, it is
assumed that the growth will eventually lead to the crack penetrating
the neighboring microstructural barriers, causing a chain reaction until
all microstructural barriers at the crack front are penetrated. This
model will promote a regular shape for the crack, also commonly seen
in experiments.
For a through-plate crack, the crack extension as the driver for crack
closure is a widely used concept, but for a radially growing 3D crack, it
has not been discussed until recently [43]. Consider a radial coordinate
system describing the state of closure at each point on the crack front.
It should be emphasized that the resistance grows only when fractured
material is left behind the crack front. In the cases where the crack can
be assumed to grow rapidly into a shape with almost constant 𝛥𝐾eff
along the crack front, these details can be largely neglected. That is
because the constant 𝛥𝐾eff is then representative of the whole crack
front. For the sake of completeness, the derivation is given for a non-
constant 𝛥𝐾eff along the crack front, as this transient phase might be
interesting to analyze as well.
It is assumed that a baseline crack closure development or cyclic R-
curve exists that can be satisfactorily described by the sum of decaying
exponential functions as in [4,13]
𝛥
𝐾th (𝑎) = 𝛥𝐾th,0+
𝐻
𝑗=1
𝛥𝐾th,𝑗 1 − exp −𝑏𝑗𝑎,(2)
where 𝛥𝐾th,0 is the initial resistance, which for a sharp initial closure-
free crack is equal to the intrinsic threshold stress intensity range
𝛥𝐾th,ef f [15]. Recently, it has been suggested that this applies only
International Journal of Fatigue 198 (2025) 108953
3
J. Vaara et al.
Fig. 1. A schematic drawing of a crack growing from a defect showcasing the model assumptions. The assumption is that the minimum strength along the crack front defines
the crack arrest. The grains represent microstructural barriers with the capability to resist crack growth. The crack is assumed to grow through each row of grains at a time.
Evaluation of local crack driving force and resistance considers the crack to have grown to that point. The limited window for crack non-propagation is a result of crack closure
development.
for positive stress ratios, and it should be corrected with a factor
(1 − 𝑅) for negative stress ratios [12]. The terms 𝛥𝐾th,𝑗 and 𝑏𝑗, 𝑗∈
[1,2,…, 𝐻] denote the additional resistance development and satura-
tion rate, respectively. 𝐻 is the number of terms chosen to describe
the R-curve. Maierhofer et al. [15] have often found two terms to be
sufficient: first, to describe the fast saturation of PICC, and second,
to describe the slower saturation of RICC and oxide-induced crack
closure (OICC). Kärkkäinen et al. [43] noted, based on the simulation
of PICC, that contribution from RICC is most likely required to reach an
agreement with the Murakami-Endo fatigue limit. The form of baseline
closure is interchangeable with any other, and the formulation does not
depend on using exactly this form.
As we are dealing with microstructurally short cracks, strictly speak-
ing, the linear elastic fracture mechanics (LEFM) quantities, such as
stress intensity factor range, should not be used. Instead, elasto-plastic
or micromechanical fracture mechanics (EPFM, MFM) quantities for
the crack driving force, such as cyclic J-integral 𝛥𝐽 or cyclic crack
tip displacement 𝛥CTD, are recommended [35,42]. For the sake of
simplicity, we use the 𝛥𝐾 notation in the derivation, and the reader
should interpret it as the crack driving force. This can also be consid-
ered a choice of coordinate system or mapping of the results; as long
as a bijective relation exists between the true and used quantities, the
results are only presented in different coordinate systems. The largest
discrepancies can be found with very short cracks near the defect.
The EPFM analyses on the crack emanating from surface defects [43]
showed support for interpreting the crack driving force as Murakami
and Endo [44] suggest, i.e., treating the defect as a sharp crack. The
presented probabilistic formulation is invariant with the quantities used
for the crack driving force.
To continue briefly on the microstructurally short fracture mechan-
ics, the question is often also where to put the uncertain elements: to
the crack driving force (e.g., microstructural factors) or the resistance
term (e.g., closure development or microstructural barriers) yielding
crack arrest. The crack facing a microstructural barrier as the arrest
criterion helps mitigate this issue, as it allows for the crack to temporar-
ily grow with a crack driving force lower than the intrinsic threshold.
Effectively, this can be seen to account for the higher crack driving
forces in the microstructurally short crack regime. The conventional ex-
perimental setup for cyclic R-curve avoids these problems by generating
a closure-free crack by compressive pre-cracking. This rather lengthy
but necessary discussion is intended to clarify the authors’ views and
justify the choices made in this paper. The careful reader will notice
that simplicity is often favored rather than over-complicating matters.
Each microstructural barrier is assumed to have intrinsic resistance
against crack penetration. On top of the baseline cyclic R-curve 𝛥
𝐾th ,
this additional resistance should have finite support stemming from the
International Journal of Fatigue 198 (2025) 108953
4
J. Vaara et al.
physical interpretation discussed above. With the minimum strength
criterion defining crack arrest, provided by Eq. (1), only finite support
for the minimum strength is required. Thus, we consider the extra resis-
tance 𝛿𝐾th,𝑖 of each microstructural barrier to be an i.i.d. lognormally
distributed random variable
𝛿𝐾th,𝑖 ∼ ln 𝜇th , 𝜎th ,(3)
where 𝜇th, 𝜎t h are the scale and shape parameters, respectively. The
total resistance of the 𝑖th microstructural barrier can then be expressed
as
𝛥𝐾th,𝑖 (𝑎) = 𝛥
𝐾th (𝑎) + 𝛿𝐾th,𝑖 .(4)
The spatial distribution of microstructural barriers is assumed to
be uniform. Consequently, the number of microstructural barriers d𝑀
faced along infinitesimal advancement of the crack front d𝑠 follows
Poisson distribution with intensity 1∕𝑑,
d𝑀∼ Poi d𝑠
𝑑.(5)
Employing an abbreviation for the instantaneous arrest probability
against one microstructural barrier 𝑝(𝑎, 𝑠)𝑃𝛥𝐾 (𝑎, 𝑠)< 𝛥𝐾th,𝑖 (𝑎, 𝑠),
the probability of crack arrested within the incremental distance along
the crack front is
𝑃(crack arrest at 𝑎, 𝑠 ∣ d𝑀) = 𝑝(𝑎,𝑠)d𝑀.(6)
Marginalization over the number of microstructural barriers d𝑀 yields
𝑃(crack arrest at 𝑎, 𝑠 or no microstructural barriers ∣ d𝑠) = exp −(1 − 𝑝(𝑎, 𝑠)) d𝑠
𝑑.
(7)
Note that full marginalization contains the event ‘no microstructural
barriers at 𝑠 within d𝑠’. This event will be handled later, as its probability
is very high for infinitesimal distances. The probability of crack arrest
of the whole crack front, or having no microstructural barriers, is then
𝑃(crack arrest at 𝑎or no microstructural barriers)
=
𝑠
𝑡=0
𝑃(crack arrest at 𝑎, 𝑡 or no microstructural barriers ∣ d𝑡, mb),
(8)
which is a product integral of Type II, a geometric integral, and can be
written as
𝑃(crack ar rest at 𝑎or no microstructural barriers) = exp −1
𝑑𝑠
0
(1 − 𝑝(𝑎, 𝑡))d𝑡.
(9)
The probability of a crack facing no microstructural barriers is
exp (−𝑠(𝑎)∕𝑑) where 𝑠(𝑎) is the total length of the crack front. Ac-
cording to the assumptions made here, this event cannot cause crack
arrest, so removing it yields the probability for crack arrest given a
microstructural barrier at the crack front
𝑃(crack ar rest at 𝑎∣ mb 1) =
exp −1
𝑑𝑠
0(1 − 𝑝(𝑎, 𝑡))d𝑡− exp −𝑠(𝑎)
𝑑
1 − exp −𝑠(𝑎)
𝑑.
(10)
The microstructural barriers are not perfectly aligned in periodic
rows, and the first microstructural barrier is located randomly about
the defect, especially for a machined surface or artificial defect. Fur-
thermore, the effective crack driving force along the crack front can
have an arbitrary history as a function of the crack length. The length
of the crack front, in the case of a 3D crack growing in a fixed shape,
also increases linearly with crack growth. Given an infinitesimal crack
growth d𝑎, the number of faced microstructural barrier fronts can be
expressed again using the Poisson distribution
d𝑁∼ Poi d𝑎
𝑑2,(11)
where 𝑑2 is the average distance between successive microstructural
barrier fronts. The simplest choice is 𝑑2=𝑑 for the microstructural
barriers representing grain boundaries. One can then write the proba-
bility of crack arrest due to at least one of the microstructural barrier
fronts using order statistics
𝑃(crack front arrest at 𝑎∣ d𝑁)= 1 − 1 − 𝑃(crack arrest at 𝑎∣ mb 1)d𝑁.
(12)
Marginalization over d𝑁 yields
𝑃(crack front arrest at 𝑎∣ d𝑎)= 1 − exp −d𝑎
𝑑2
𝑃(crack arrest at 𝑎∣ mb 1).
(13)
The probability for the crack not to be arrested before 𝑎 can be written
as a product integral
𝑃(no crack arrest before 𝑎)=
𝑎
𝑧=0 1 − 𝑃(crack front arrest at 𝑧∣ d𝑧).
(14)
Again the product integral can be simplified, and the cumulative dis-
tribution function for non-propagating crack lengths can be expressed
as the integral
𝐹𝑎npc (𝑎) = 1 −exp −1
𝑑2𝑎
0
𝑃(crack front arrest at 𝑧∣ mb 1)𝑑𝑧.(15)
Finally, the probability density function for non-propagating crack
lengths can be derived as
𝑓𝑎npc (𝑎) = 1
𝑑2
𝑃(crack front arrest at 𝑎∣ mb 1)1 − 𝐹𝑎npc(𝑎).(16)
The survival function evaluated at sufficiently long crack lengths
gives the probability of not observing non-propagating cracks and, thus,
failure. A sufficiently long crack exceeds the window for cracks to
become non-propagating. Extensive Monte Carlo simulations verified
the analytical model. Note that with constant instantaneous arrest
probability 𝑝 in Eq. (15), the distribution of non-propagating cracks
reduces to an exponential distribution with an expected value of 𝑑2∕𝑝.
This is a useful feature, as shown later in Section 3.2 when considering
the parameter estimation scheme.
To help the reader understand the problem, a close analogy to the
problem scheme can be made with a real-life scenario: Consider a car
traveling with an arbitrary speeding profile. The driver meets an officer on
average every 𝑑 kilometers, and the probability of getting a ticket depends
on their instantaneous speed and the mood or tolerance of the officer. The
equivalent question would be: ‘How far does the driver get before receiving
the first speeding ticket?’
2.2. Properties, assumptions, and limitations
The model captures certain observed phenomena and predicts a
different scaling between through-plate and 3D cracks. More specifi-
cally, according to the model, short 3D cracks should exhibit a higher
resistance than through-plate cracks, and the differences should dis-
appear when the crack front is long enough. This behavior could be
seen in the experimental study by Pourheidar et al. [17]. The model
predicts that the likelihood of crack arrest increases the longer the crack
propagates with a low 𝛥𝐾eff . This means that the model predicts a
loading path dependency. In the analysis of fatigue of artificial defects,
in Section 2.3, it is also assumed that the cracks on both sides of the
defect grow at a similar pace unless arrested. The assumption of radial
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scaling of the cyclic R-curve should be appropriate if the near-field
closure mainly determines the realized closure. Support for this was
found in the study by Kärkkäinen et al. [43].
The assumption that the crack arrest mechanism controlled by
microstructural barrier fronts could fully determine the cyclic R-curve’s
stochastic behavior might be overly simplified. This aspect will be
discussed further in the Discussion Section 5.
One key assumption in the model is that the cyclic R-curve does
not depend on the initial defect or the loading – a common assumption
in the notch fatigue analyses employing the cyclic R-curve concept [6].
The loading independency is obviously only valid relatively close to the
threshold. Although Kärkkäinen et al. [45] show how to incorporate
cyclic R-curve for the analysis of underloads – a procedure that should
be applicable also for overloads – it still needs further verification. The
model will be used to describe the very diverse data of More et al. [27],
revisited in Section 3.1, and the hypotheses will certainly be put to the
test.
While the focus in the present paper has been on uniaxial fatigue,
it does not necessarily mean the framework could not be applied
to multiaxial fatigue. Let us first consider torsional fatigue; if the
differences between the defected material’s torsional and axial fatigue
limits can be described by the differences of 𝛥𝐾 alone [46], it would
imply a similar closure development. If a significant mode II/III crack
growth is observed before branching to mode I crack growth [47], the
crack closure development might be delayed. This is similar to the
stage I cracks without defects in axial loading and could be linked to
the increased critical defect sizes observed in torsional fatigue [26].
These effects can be considered with a proper description of the crack
opening 𝛥𝐾 development and possibly shifting the baseline resistance
curve to larger crack lengths. Thus, for in-phase multiaxial fatigue, the
model could be applied, whereas for out-of-phase loading, it might be
more compromised. A better understanding of the relative roles of each
loading for the development of effective crack closure is needed.
What about arrest in between the microstructural barrier fronts?
This is a very interesting question. However, there are plenty of ex-
perimental observations indicating that the microstructural barriers
cause only temporary retardation of the crack growth rate, and once
penetrated through, the crack growth rate respectively increases [41,
48–50]. Furthermore, a crack loaded near the threshold can appear
to be non-propagating at the surface for a certain period of loading
cycles but eventually continue growth [51]. These observations provide
further support for the chosen crack arrest criterion.
The use of inexact crack driving force should also be repeated for
possible limitations for short cracks, discussed in Section 2.1. This is,
however, not a limitation of the model itself, as the derivation can
be done using any quantities defining the instantaneous crack arrest
probability in Eq. (1). Besides, the stochastic microstructural barriers
as the arrest criterion effectively produce similar behavior as the higher
crack driving forces in the micromechanical fracture mechanics.
2.3. Drilled defects are initially a two-crack interaction problem
It is best to start by showing the most eye-opening evidence from
our in-situ microscope tests to open the discussion on artificial defects.
In Fig. 2, the crack growth history of a single specimen is shown. It
can be seen that the crack on the left side initiates (around 105 cycles)
and grows to a substantial length before the crack on the right side
even initiates after 3 × 107 cycles. Both cracks end up non-propagating
roughly after another 2 × 107 cycles as a constant 𝛥𝐾 ≈ 8 MPam
test was applied after both cracks had initiated. Now, it is clear that
the growth of the first crack enabled the initiation of the second
crack. The values in the images are directly taken from the image-
recognition algorithm – a careful inspection reveals that there was no
crack propagation between Figs. 2(c) and 2(d).
A further consideration about using the present model in the typical
scenario, where the cracks on the sides of the artificial defect (𝑎1,𝑎2)
are of different sizes, gives reason to believe that it should be treated
as a crack interaction problem. Before the cracks coalesce:
•The cyclic R-curve of each crack develops independently of the
other: 𝛥𝐾th(𝑎1) and 𝛥𝐾th (𝑎2)
•The crack driving force of each crack sees the contribution from
the other: 𝛥𝐾1(𝑎1, 𝑎2) and 𝛥𝐾2(𝑎2, 𝑎1).
Performing brief FEM analyses confirmed the usefulness of the
Murakami–Endo hypothesis for 3D cracks [44]: regardless of a one-
sided crack or two cracks of similar total area, the 𝛥𝐾 would be
sufficiently similar. Thus, it was decided to simplify the problem with
the assumption 𝛥𝐾1(𝑎1, 𝑎2) = 𝛥𝐾2(𝑎2, 𝑎1)𝛥𝐾(𝑎1, 𝑎2).
𝛥𝐾(𝑎1, 𝑎2) = 0.65𝛥𝜎 𝜋10−6ar ea0+ area1(𝑎1) + area2(𝑎2)(17)
For the sake of simplicity, we estimate the individual cracks to grow
in a quarter-circular shape. As noted earlier, if the crack grows in a
shape where 𝛥𝐾eff is approximately constant along the crack front,
the problem reduces to a 1D problem; single 𝛥𝐾 and 𝛥𝐾t h values
sufficiently describe the crack. The framework comes with a couple of
interesting properties:
•A crack can temporarily arrest, but if the other crack contin-
ues to grow, the arrested crack will eventually resume growth,
overcoming the strength of the barrier that initially arrested it.
In other words, a total arrest requires both cracks to become
non-propagating. The temporary arrest and resumption of crack
propagation can occur multiple times.
•A consequence of the previous bullet point is that having observed
only the end state of a non-propagating crack pair (𝑎1, 𝑎2) comes
with an infinite number of paths leading to the state.
•Consequently, the likelihood becomes intractable – unless there
is an in-situ crack path measurement. Simulation-based methods,
such as Approximate Bayes Computation, are required to capture
model uncertainty.
•As each crack has its own cyclic R-curve, the path of least resis-
tance for the cracks is retaining a relatively similar size. That is,
for all 𝛥𝐾(𝑎1, 𝑎2) isocurves, the lowest expected 𝛥𝐾th is reached
when 𝑎1=𝑎2.
•Tests employing constant 𝛥𝐾 lose the interaction property. This
has the direct consequence that the interaction effect at the
fatigue limit is less pronounced for large defects and more em-
phasized for small defects as the slope of 𝛥𝐾 development tends
to reduce with increasing defect size.
When the cracks grow together, the crack behaves as a single with
a continuous crack front. This framework should work to analyze
virtually all crack interaction problems. All required is the explicit
relation of the crack driving force between the cracks being studied.
3. Re-analysis of the experimental results by More et al.
The experiments used to analyze the model were presented in detail
in [27], but as they are tightly integrated into the demonstration of the
model, the data is briefly revisited.
3.1. Experimental results revisited
The studied material was 42CrMo4 QT-steel, with its mechanical
properties in Table 1.
An extensive ultrasonic fatigue testing campaign in room temper-
ature was conducted with the specimens containing varying numbers
and types of artificial defects. A total of 12 different defect types of
different sizes and shapes were tested with a stress ratio 𝑅= −1,
focusing on quantifying their respective fatigue limits with a runout
limit of 109 cycles. The series comprised a total of 204 defects in 69
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Fig. 2. Crack evolution around an artificial defect of area = 93 μm. The first crack initiated before the number of cycles 1.861 × 105.
Table 1
Mechanical properties of the studied 42CrMo4 QT-steel.
Tensile strength [MPa] Yield strength [MPa] Elongation at fracture [%] HV
807 660 24 270
specimens. Surfaces of the specimens were electropolished to reduce
potential residual stresses. The smooth specimen gauge section diam-
eters varied from 3.2 mm to 5.2 mm – larger specimens were used with
the larger artificial defects. Each defect was checked for cracks, and the
lengths were measured after the test with a light-optical microscope.
The outcome of the fatigue tests and defect configurations are shown
in Fig. 3, categorizing the specimen-wise results into three categories:
failed, runout with non-propagating cracks (in any of the defects),
and runout without cracks. The defects were manufactured by a high-
precision drill with varying diameters. The number of adjacent holes
would define the total width, which would also be the depth of the
deepest drilling. The depths of the adjacent drillings would follow
the apex angle of the drill, as illustrated in the Figure. Only 6 defect
types with notch root radius less than or equal to 50 μm exhibited non-
propagating cracks, which are examined further in this study. These
comprised a total of 175 defects in 49 specimens. Further details on
the experiments are presented elsewhere [27].
3.1.1. Crack initiation
The classical interpretation is that the crack initiation limit is
the limit for micromechanical elastic shakedown [42,52]. Indeed, the
branching point for observing non-propagating cracks in the early
works of Isibasi was approximated to be the first occurrence of cyclic
plasticity at the notch root [53]. Later on, Nisitani and Endo [54]
considered the crack initiation limit for both shallow and deep notches
to be determined by the relation between the local stress and relative
stress gradient 𝜒. Recently, the Siebel–Stieler model [31] has success-
fully been used to describe crack initiation behavior for multiple defect
sizes and shapes [27,32]:
𝜎ci =𝜎w,0
𝐾𝑡1 + 𝜒𝑠𝑔≈𝜎w,0
𝐾𝑡
1 + 𝛽𝑠𝑔
𝜌
,(18)
where 𝜎w,0 is the fatigue limit of plain specimens, commonly approxi-
mated as 𝜎w,0≈ 1.6𝐻𝑉 with 𝐻𝑉 being the Vicker’s hardness. 𝐾𝑡 is the
elastic stress concentration factor, 𝛽= 2 + 1∕𝐾𝑡 is the stress gradient
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Fig. 3. Specimen-wise fatigue test results and defect configuration of the experiments by [27]. The linkage of Kitagawa–Takahashi and Frost diagrams is demonstrated for defect
size of 450 μm by drawing an additional vertical axis with increasing 𝐾t. Keeping the defect size constant but making the defect sharper, as shown in the defect configuration,
cracks initiate easier and fatigue limit drops until becoming constant when the cracks become non-propagating. El-Haddad [2] and Siebel–Stieler [31] models were used to describe
the fatigue limit from crack non-propagation and initiation, respectively. A scaled prediction by Murakami–Endo model [3] is plotted for comparison.
factor as approximated by Schijve [55], 𝜌 is the notch root radius and
𝑠𝑔 a material constant with units of length. Schönbauer and Mayer [32]
estimated 𝑠𝑔 as the critical defect size.
The crack initiation is effectively eliminated from the equation
in the cyclic R-curve tests by producing a razor-sharp initial notch.
The closure-free pre-crack is then grown under compressive loading
from the notch [56]. In these conditions, the intrinsic threshold stress
intensity factor range 𝛥𝐾t h,eff defines the initial resistance. However, as
pointed out in the Introduction, crack initiation plays a role in cracks
emanating from defects [26–29,32]. With a higher stress ratio 𝑅 and
significant reduction of crack closure, it is also clear that the crack
initiation’s role is increased. With a high enough 𝑅, crack initiation will
define the fatigue limit in real applications. Furthermore, More et al.
[27] and Schönbauer and Mayer [32] showed that large enough defects
behave as blunt notches, for which the fatigue limit is defined solely by
the crack initiation limit.
A statistical analysis of the crack initiation limit revealed that a
normally-distributed factor, with a mean and standard deviation of
1.01 and 0.12, respectively, applied to the Siebel–Stieler prediction (18)
would capture the observed variance of the crack initiation behavior of
the defects for the studied 42CrMo4-QT steel in [27]. Thus, this model
is adopted in the present study to describe the probability of crack
initiation. The microstructure around the defect and potential residual
stresses are the most obvious sources of scatter for crack initiation.
The general crack initiation behavior for sharp defects is similar
to the one for notch fatigue beyond the branching point, described
by Frost and Dugdale [51]: first, there are no cracks, but increasing the
load results in cracks initiating but becoming non-propagating. Finally,
the fatigue limit is met at the loading, where the cracks no longer
arrest. Based on the experiments of More et al. [27], the link between
Kitagawa–Takahashi and Frost diagrams can be retrieved, as shown in
Fig. 3. In the Figure, the Frost diagram is generated for an exemplary
initial defect size of 450 μm of different shapes with varying 𝐾t –
determined by FEM in [27]. Moving vertically in the stress amplitude
axis, when the crack initiation limit of a defect is intersected, one
retrieves a point in the Frost diagram. The critical stress concentration
factor corresponds to the point where fatigue limit defined by crack
non-propagation is intersected – which was sufficiently described by
the El-Haddad model in [27]. For this defect size, the critical stress
concentration factor was 𝐾t,cr ≈ 4, meaning that the fatigue limit
of all but the 5-hole defect configuration would be defined by crack
initiation rather than crack non-propagation. The experimental data
with approximately similar sizes also agree with this sentiment. It can
be seen that the branching point defined by 𝐾t,cr is not a material
constant but depends on the size of the notch or defect.
3.1.2. Non-propagating cracks
The crack initiation, growth, and arrest as a function of crack length
are plotted individually for each defect type in Fig. 4. The plot is
arranged so that one continuous line is plotted per failed specimen
for the defect causing the failure. In these specimens, the judgment of
cracks not initiating from non-critical defects was considered reliable,
but no assessment of non-propagation could be made. In the case
of two non-propagating cracks, the 𝛥𝐾(𝑎1, 𝑎2) development is plotted
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Fig. 4. Non-propagating cracks, denoted with ‘X’, in the experimental data of More et al. [27] for each defect type. Development of stress intensity factor range 𝛥𝐾 as a function of
crack extension 𝑎 from the defects is estimated by Eq. (17) and drawn with a line. The horizontal histograms indicate the number of crack initiation sites with no crack initiation.
See Fig. 3 for the defect configurations and interpretation of the cracks. Note that the scales change between the defect types for improved visibility.
according to Eq. (17) assuming growth at a similar pace until the
smaller crack arrested. After that, only the larger grew. One can see that
the larger the initial defect, typically the longer the non-propagating
cracks. The threshold stress intensity factor range corresponding to
the analyzed fatigue limit by More et al. [27] is plotted as a dashed
horizontal line.
The pair-wise non-propagating crack lengths are plotted in Fig. 5.
No initiation is shown as zero crack length. This result provides support
for the behavior that once both cracks have initiated, the preferred path
is along the 𝑎1=𝑎2 line, as discussed in Section 2.3.
Additionally, three decreasing 𝛥𝐾 tests with through-plate crack
specimens, with 3 mm thickness, were also tested. The 𝛥𝐾 decreasing
paths until crack arrest are plotted in Fig. 8. The resulting crack length
and 𝛥𝐾th pairs were estimated to be (957 μm, 9.29 MPam), (1595 μm,
10.81 MPam) and (3631μm, 9.85 MPam).
This dataset is diverse and, thus, very interesting for the model to
tackle.
3.2. Parameter evaluation
The parameters of the model are:
•Characteristic microstructural lengths 𝑑 and 𝑑2, controlling and
scaling the microstructural variance with crack length.
•Single microstructural barrier strength distribution characterized
by 𝜇th and 𝜎th .
•Baseline R-curve parameterization 𝛥
𝐾.
As discussed in Section 2.3, the investigated drilled artificial defects
have to be treated as a two-crack interaction problem, and the likeli-
hood of two cracks under constant stress amplitude (i.e. 𝛥𝐾 increasing)
becomes intractable. However, if the data consists of single cracks, or
if enough of no crack initiation on the other side of the artificial defect
were observed, the likelihoods derived in Section 2.1 could be used to
identify the model parameters and uncertainty. However, after looking
more closely at the variance scaling, one could find a more simple
procedure to fit the parameters. Given 𝑀 > 1 i.i.d. microstructural
barriers and their respective resistance 𝛿𝐾th,𝑖 along the crack front 𝑠
defined in Eqs. (5) and (3), respectively, the 𝑃th quantiles are given as
𝛿𝐾th,min,𝑃 (𝑎) = 𝐹−1
𝛿𝐾th,𝑖 −𝑑
𝑠(𝑎)ln 1 − 𝑃1 − exp −𝑠(𝑎)
𝑑.(19)
The limit value for 𝑠→0 is simply 𝐹−1
𝛿𝐾th,𝑖 (𝑃). This is a powerful
expression for parameter fitting: plotting the data in (𝑎, 𝛥𝐾) coordinates
reveals the baseline closure 𝛥
𝐾th of Eq. (2) defined by the envelope
of the longest non-propagating crack lengths. Additionally, for a 3D
crack with 𝑠(𝑎=0)=0, the initial variance as 𝑎→0 is defined by the
strength of a single microstructural barrier (Eq. (3)). Lastly, the scaling
of the variance is defined by 𝑑 according to Eq. (19). This parameter
identification scheme is plotted against the experimental data in Fig. 6.
The upper band in Fig. 6 corresponds to the instantaneous arrest
probability of 5%, given a microstructural barrier front, as in Eq. (1).
The microstructural length 𝑑= 20 μm corresponds best with this
material’s prior austenite grain size. In this framework, the microstruc-
tural length is tightly connected to the strength of the microstructural
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Fig. 5. Non-propagating crack lengths (𝑎1, 𝑎2) of [27] initiated from different defect types, denoted with different markers and the same colors as in Fig. 4. 𝛥𝐾 contours by Eq. (17)
are drawn with solid gray lines. See Fig. 3 for the defect configurations and interpretation of the cracks (𝑎1,𝑎2).
Fig. 6. Measured non-propagating crack lengths 𝑎 and corresponding estimated 𝛥𝐾 in the model parameter identification scheme. The envelope of non-propagating cracks is fit
between the two curves drawn as solid black lines. The upper curve defines the minimum strength of microstructural barriers along the crack front, corresponding to 95% quantile
in Eq. (19). A crack growing along this line has a 5% probability of arresting every time it faces a microstructural barrier front. The lower curve corresponds to the baseline crack
closure 𝛥
𝐾th , defined in Eq. (2). If a crack grows below this line, probability of arrest is 100% once it faces a microstructural barrier front. The horizontal distribution is the initial
resistance given a single microstructural barrier. The used parameters are 𝑑= 20 μm, 𝛥𝐾th,0= 4.0 MPam, 𝛥𝐾th,1= 4.5 MPam, ln 𝑏1= −4.2, 𝛥𝐾th,2= 2.0 MPam, ln𝑏2= −6.9,
E[𝛿𝐾th,𝑖 ] = 1.3 MPam, Std[𝛿𝐾th,𝑖 ] = 1.2 MPam. See Fig. 3 for the defect configurations and interpretation of the cracks (𝑎1,𝑎2).
barrier: a longer average distance between the microstructural bar-
riers means that fewer of the microstructural interfaces act as real
microstructural barriers. In other words, choosing a high value for 𝑑
has a censoring effect on some of the weaker microstructural barriers.
On the other hand, if a lower value for 𝑑 was used, then the distribu-
tion defining the strength of microstructural barriers should put more
weight on lower values.
The El-Haddad model fit of More et al. [27] using long crack
threshold value 8.7 MPam matches the observed fatigue limit from
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non-propagating cracks. However, this value is over 2 MPam lower
than the largest measured long crack threshold value. Here, the faster
saturating resistance was fit as 𝛥𝐾th,0+𝛥𝐾t h,1= 8.5 MPam, which
is very close to the long crack threshold value used in the El-Haddad
fit. A slower saturating term was added in the baseline cyclic R-curve
to reach the long crack threshold values. Adding the slower saturating
second term to the baseline R-curve does not interfere with the previous
fit shown in Fig. 6. The fit for long crack threshold values is shown in
Results Section 4 (Fig. 8) to simultaneously demonstrate the model’s
capability to predict the variance in 𝛥𝐾 decreasing tests.
What remains to be fitted is the average distance between the
microstructural barrier fronts 𝑑2. This parameter controls the number
of arrest checks the crack encounters per unit length and will be
vital in defining the share of non-propagating cracks given constant
stress amplitude 𝜎𝑎 or 𝛥𝐾 increasing loading paths. Larger values
should promote a larger variance in the non-propagating crack lengths
and fewer non-propagating cracks. Given that this material’s average
distance between microstructural barriers 𝑑 coincides fairly well with
the prior austenite grain size, 𝑑2=𝑑 is probably the most physically
founded choice.
However, 𝑑2 is also related to the asymptotic fatigue limit of plain
specimens in the KT-diagram. Given an initial defect, with a size
approaching zero area →0, the probability of failure is controlled
almost exclusively by the event of not facing a microstructural barrier
front within the window of non-propagation. In other words, the crack
has to initiate in favorable microstructural surroundings, either a large
enough grain or with only minor microstructural barriers providing
little resistance to the crack propagation. Setting the stress amplitude
equal to the plain specimen fatigue limit 𝜎𝑎=𝜎𝑤,0≈ 1.6𝐻𝑉 and
finding the crack length where 𝛥𝐾 coincides with the baseline closure
development 𝛥
𝐾th yields a reasonable estimate for 𝑑2. With the material
studied here, this procedure would give 𝑑2≈ 23 μm, which is relatively
close to the previously identified 𝑑= 20 μm and further justifies the
assumption that 𝑑2=𝑑. The parameter 𝑑2 defined in this way has
similarities to the intrinsic crack length of the El-Haddad model, and
the faster the cyclic R-curve develops, the closer these two become.
Lastly, the distribution of non-propagating cracks defined in Eq. (16)
with a constant instantaneous crack arrest probability 𝑝 follows an
exponential distribution with an expected value of 𝑑2∕𝑝. Thus, the dis-
tribution of non-propagating cracks, especially those emanating from
the smallest defects, could be used to get a lower support for parameter
𝑑2. The mean length of non-propagating cracks from the 50μm drilled
defects was 22 μm. The histogram of non-propagating crack lengths and
the exponential distribution are shown in Fig. 7. As most specimens
were loaded using different amplitudes and the interaction of cracks
is neglected in this lumped view, this should be considered a rough
estimation. The histogram of non-propagating cracks has a mode, indi-
cating an initial ramp-up of the instantaneous crack arrest probability
(1) following approximate stabilization. Once stabilized, the remain-
ing probability mass follows a truncated exponential distribution. For
the sake of demonstration, the probability density function (Eq. (16))
corresponding to an exemplary instantaneous arrest probability is also
shown. Now, the role and identification procedure for every parameter
is clearly defined.
4. Results
A loading path dependency is predicted with the presented model.
In Fig. 8, the parameter fit and predictive capabilities of the model
(Eq. (15)) for 𝛥𝐾 decreasing tests are shown. The lowest 𝛥𝐾th value and
highest variance are predicted for the steepest decreasing loading path.
The observed threshold values are estimated at the crack propagation
rate d𝑎∕d𝑁= 10−11 m/cycle. Even though the model can predict
some variance for these test results, the non-monotonic behavior of the
observed 𝛥𝐾th values makes it difficult for the model to fully explain
these observations. This is a sign that another source of variance is
likely required, which will be touched on in the Discussion Section 5.
One of the most interesting properties of the cyclic R-curve is its
capability to predict the KT-diagram, shown in Fig. 9. The prediction
curves correspond to 5%, 50%, and 95% quantiles, respectively. The
prediction is an excellent match with the observations. As the defect
sizes increase, the sizes of non-propagating cracks also increase, and
accordingly, the model predicts a reduced variance for the fatigue limit
of these defects. For smaller defects, the steeper 𝛥𝐾 development and
the resulting smaller window of non-propagation also promote higher
variance. The role of crack initiation for each defect type and size can
be compared to the simulated curve without crack initiation resistance,
corresponding to the cyclic R-curve. Generally, the sharper the defect,
the longer it follows the fatigue limit without contribution from crack
initiation. For this experimental data consisting only of the defects
with non-propagating cracks at the fatigue limit, no large effects from
crack initiation can be seen. For this material, the Murakami–Endo
model underestimates the fatigue limit for the smallest defects and
overestimates it for the largest ones.
In Fig. 10, a sample simulation prediction for each specimen of More
et al. [27] is shown, including crack initiation. The resemblance to the
experimental result shown in Fig. 4 is clear. However, it must be noted
that this was a deliberately chosen sample, and while sampling, the
model was seen to predict rather a high variance in the experimen-
tal setup. Given the number of non-initiations in the defects in the
experiments, even with load levels where failures are observed, this
seems plausible. A smaller variance could be observed if the initiations
were pre-defined in the simulation to match the experimental result.
This approach is not problem-free either. The example shown in Fig. 2
demonstrates that one possible crack growth path would be neglected,
where only one crack initiates first and grows until the second crack
initiates.
Another aspect that should be noted is the design of experiments.
The experiments aimed to get results from three categories:
•Runout specimen with no cracks initiated from any of the defects,
•Runout specimen with non-propagating cracks, and
•Failed specimen.
The typical number of specimens available for testing a specific defect
type was five. All but the 1×100 μm hole defects had one to two failing
specimens — so it is built in that once the experimentalist witnesses a
result from one of the categories, they will explore with vastly different
stress amplitudes. This kind of sequential behavior should be mimicked
so that the simulations would produce a more consistent and represen-
tative output. Another thing to note in the experimental results is the
lack of non-propagating cracks longer than the drilling depth, i.e., close
to coalescing into a continuous crack front. The coalescence was not
modeled, which might explain some of the longest non-propagating
cracks in the simulation.
Fig. 11 shows the predicted evolution of non-propagating crack
sizes as a function of loading level for two different initial defect
sizes, area = 93 μm and area = 185 μm, respectively. The com-
mon phenomenon of longer non-propagating cracks with larger initial
defects is captured. However, an interesting detail can be seen with
increasing loading level. The mean size of non-propagating cracks first
increases but then decreases. First, barely any failures are observed,
and increasing the load here results in longer non-propagating cracks,
as 𝛥𝐾 intersects the cyclic R-curve later. Then, the probability of failure
increases with an increase in the stress amplitude, and the 𝛥𝐾 increases
more steeply with crack length. The window for cracks becoming non-
propagating (low 𝛥𝐾eff values) is shorter and occurs earlier. The peak
length is observed earlier for cases with two cracks initiated as opposed
to a single crack because the slope of 𝛥𝐾 is less steep for a single crack.
Another interesting aspect is the evolution of the histogram shape
of two non-propagating crack lengths. With a low probability of failure,
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J. Vaara et al.
Fig. 7. Histogram of non-propagating crack lengths of the 50 μm holes, an exponential distribution with expected value of 20 μm and demonstration of Eq. (16) with given
instantaneous arrest probability. The exponential distribution was shifted by 15 μm in the 𝑥-axis, corresponding to the approximate stabilization of the instantaneous crack arrest
probability.
Fig. 8. Parameter identification scheme and predictive demonstration for the long crack threshold tests. The used parameters are 𝛥𝐾th,0= 4.0 MPam, 𝛥𝐾th,1= 4.5 MPam,
ln 𝑏1= −4.2, 𝛥𝐾th,2= 2.0 MPam, ln 𝑏2= −6.9.
Fig. 9. Predicted Kitagawa–Takahashi diagram with 90% confidence region. The 𝑥-axis corresponds to the initial defect size. The binary non-propagating crack findings in the
experimental data are offset by 5% in the 𝑥-axis for better visibility.
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J. Vaara et al.
Fig. 10. Non-propagating cracks, denoted with ‘X’, as predicted for each defect type. A deliberately picked sample of simulation prediction of the same specimen-wise configuration
as in experiments. Compare to the experimental result shown in Fig. 4. Development of stress intensity factor range 𝛥𝐾 as a function of crack extension 𝑎 from the defects is
estimated by Eq. (17) and drawn with a line. The horizontal histograms indicate the number of crack initiation sites with no crack initiation. See Fig. 3 for the defect configurations
and interpretation of the cracks. Note that the scales change between the defect types for improved visibility.
the non-propagating cracks are more likely to be of similar length,
i.e., highly correlated. Meanwhile, with a higher probability of failure,
the positive correlation is lost. This is explained by the fact that
as the window for non-propagation becomes shorter, the population
having two non-propagating cracks is predominantly described by a
sufficiently strong microstructural barrier front and the simplest crack
growth mode where both cracks arrest only once, and no resumption
of growth occurs. In fact, the simulation suggests that an increased
number of growth resumptions generally results in longer cracks cen-
tered around the correlation line. With the highest loading amplitude
shown in Fig. 11(b), the boundaries of the histogram roughly follow the
constant 𝛥𝐾 contours shown in Fig. 5. Increasing the loading further
results in fewer non-propagating cracks that are generally shorter,
caused by the occasional strong microstructural barrier front.
To support the parameter identification scheme and inspect the
model’s capabilities in the small defect regime, another Kitagawa–
Takahashi diagram is shown in Fig. 12. Unlike the traditional determin-
istic cyclic R-curve analysis, the probabilistic version presented here
can extend predictions below the intrinsic threshold value. The pre-
dicted fatigue limit captures similar asymptotic behavior as commonly
seen in experiments. This behavior is a consequence of the limited
window of the microstructurally short crack susceptible to arrest by
a microstructural barrier front. When it comes to the high predicted
variance, in reality, the material typically has tens or hundreds of these
small defects, and the competing risks scheme [57,58] decreases the
upper quantiles. On the other hand, the lower quantiles can be raised
by the introduction of a micromechanically sound crack initiation
limit. When the defect becomes sub-grain-sized, the model’s prediction
considers practically only the crack. The baseline R-curve (Eq. (2)), fit
against the longest non-propagating cracks in Fig. 6, can be seen to
match well with the lower bound of fatigue limit for large enough ini-
tial defects. Therefore, the described parameter identification scheme
can be seen as quite successful.
5. Discussion
A probabilistic model for predicting non-propagating cracks has
been presented. The model is based on the simple assumption that the
crack front faces successive microstructural barriers that can arrest it
— with minimal additional parameters. The hypothesis is not new, as
it was originally described by Miller [18], and is still being used to
explain various observations [19,20]. However, the probabilistic model
capturing the essence of the hypothesis is novel. The model extends the
cyclic R-curve and defect-based fatigue frameworks, enabling the model
to be used for notch fatigue, among others. The probabilistic model
allows one to analyze data previously considered too scattered and gain
insight into the physical process generating the experimental data. Fit-
ting the cyclic R-curve to the radially growing non-propagating cracks
initiated at artificial defects has not been done before either. Miller [18]
was of the opinion that a major barrier of a certain size would always
exist, and characterizing the behavior of these very small cracks would
not be of importance in fracture mechanics-based component fatigue
assessment. We beg to differ for two reasons: first, without characteriz-
ing this behavior, a lot of the available experimental observations are
not truly utilized, and second, understanding the probabilities of failure
in the physical interpretation of notch and defect-initiated fatigue aids
the common goal of making robust machine component designs that
should last 20 or 30 years, multiple overhauls and possible corrosion.
A schematic presentation of the model’s capabilities in different loading
scenarios is shown in Fig. 13.
Further discussions will be held on the properties of the model,
utilizing the model for crack interaction problems, the role of crack
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Fig. 11. Predicted evolution of the distribution of non-propagating crack sizes for two initial defects (a) 1 × 100μm hole and (b) 2 × 100 μm hole for three different loading levels
around their respective fatigue limit. Brighter regions in the 2D-histogram are predicted to be more probable. The horizontal bar plot indicates the share of results in each category.
Marginal distributions of the non-propagating crack lengths for defects initiating single and two cracks are shown as horizontal 1D-histograms. The fatigue limit 𝜎𝑤 corresponds
to the reference El-Haddad fit shown in Fig. 3.
Fig. 12. Predicted Kitagawa-Takahashi diagram extended to microstructurally small defect regime. The 𝑥-axis corresponds to the initial defect size. The solid lines correspond to
5% and 95% quantiles, and the dashed lines to 50%, where applicable.
initiation, variance in the cyclic R-curves, and finally, the parameters
of the model.
5.1. Properties of the model
The model answers the important questions: ‘Given an arbitrary crack
driving force path, what is the distribution of non-propagating crack lengths,
and the probability of observing them?’. Previous regression analyses cap-
turing the uncertainty of the cyclic R-curve could only rely on quantiles,
i.e., the probability that the cyclic R-curve crosses the nominal loading
path (𝛥𝑎, 𝛥𝐾). As an extreme example, a crack growing at a fixed
quantile of the cyclic R-curve would either become non-propagating
instantaneously with the probability tied to the quantile or be allowed
to grow infinitely. In contrast, the present model states that the longer
(in spatial dimensions) the crack spends in the proximity of the cyclic
R-curve, the more probable it is to arrest. This yields an interesting
loading path dependency on the model predictions. From the successive
microstructural barriers point of view, it is also physically founded.
To supplement the previous analyses, one could use the quantiles
of regression analysis in the current framework as the instantaneous
arrest probability in (1). Furthermore, the probabilistic model extends
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J. Vaara et al.
the prediction range to the microstructurally short crack regime and
successfully describes the linkage between the Kitagawa–Takahashi
diagram from cyclic R-curve analysis and the El-Haddad type curve.
Initiation of closure-free microstructurally short stage I cracks has been
used as a way to describe the asymptotic fatigue limit for decreasing
defect sizes [38]. The mechanism defining the asymptotic fatigue limit
in the present model is simply the probability of crack initiating in a
favorable enough microstructural surrounding not to become arrested
— which in no way contradicts the previously mentioned mechanism
but rather complements it. A delayed crack closure development can
be achieved by decreasing the initial threshold value in the baseline
closure (2) in the present model. In principle, this should be only
applied to small initial defects where stage I initiation is expected
— no stage I initiation was observed for the defects studied here.
However, it remains unclear if both mechanisms are needed or if the
correct behavior can be effectively captured by only one or the other.
Quantifying the crack initiation resistance for microstructurally small
defects in combination with weak grains would complete the picture
and provide keys to assessing statistical size effects.
5.2. Crack interaction
The presented methodology can aid in progressing the analysis of
general crack interaction problems. In fact, it was realized that the
crack growth from artificially drilled holes should be treated as a
two-crack interaction problem. From the physical consideration, cracks
growing on both sides of the drilling inevitably develop crack closure
depending on the crack length, and the interaction of crack driving
force is obvious. One crack can temporarily become non-propagating,
but if the other continues to grow, the other will eventually resume
growth. This raises the question of how big an error would be made if
this was neglected, which can be counter-intuitive. First, let us bring
in theoretical considerations. Both cracks growing equivalently long
should be the path of least resistance for crack closure, meaning any
deviation from this would result in more likely crack arrest and a
higher fatigue limit. As the crack front is split in two, each half is more
prone to arrest, facing a strong microstructural barrier. After the first
crack arrests, the other grows with a milder 𝛥𝐾 development and is
more prone to arrest as well. However, the opposite was observed: the
solution with interaction yielded a lower fatigue limit.
Let us delve deeper into the reasoning. In the proximity of the
fatigue limit of small defects, the crack driving force scales rapidly
with crack advance, and the window for crack non-propagation is
surpassed faster. Therefore, at least one of the two cracks is more likely
to grow large enough not to become non-propagating. Furthermore,
as the initial defects get smaller, their weight in the crack driving
force decreases relative to the crack growth. Eventually, the crack
driving force becomes insensitive to the initial defect size. Thus, the
asymptote in the KT-diagram is described by the stress level required
to grow the crack beyond the non-propagation window before the
first major microstructural barrier front is met, roughly within the
scale of average microstructural barrier spacing. In fact, the effect of
interacting cracks can be relatively well estimated by analyzing two
i.i.d. cracks by Eq. (15) and assessing the probability that at least one
will not become non-propagating. However, this approximation is not
as precise for larger initial defects and non-propagating cracks with
more complicated crack growth dynamics.
5.3. Crack initiation
The crack initiation limit can increase the inferred fatigue limit by
preventing some of the failures that would have occurred at lower
loading levels. The Siebel–Stieler model predicts that each defect type
with a finite rounding radius would depart from the fatigue limit
defined by the cyclic R-curve with increasing size, and crack initiation
would define the fatigue limit. This was verified by More et al. [27],
Fig. 13. Schematic representation of the model’s predictive capabilities in three
different crack growth scenarios: 𝛥𝐾 increasing test with an initial defect (red line), 𝛥𝐾
decreasing test for long cracks (brown line) and asymptotic fatigue limit with negligible
initial defect (green line). The distributions of non-propagating cracks are drafted for
each scenario using the same color.
and extensive support for such behavior was presented already in the
experimental study by Schönbauer and Mayer [32]: a one-hole drilling
large enough would behave like a blunt notch — no non-propagating
cracks could be observed at the fatigue limit. As an additional small
defect was drilled next to the large drilling, barely changing the
area,
the fatigue limit dropped drastically. The non-propagating cracks and
cyclic R-curve again defined it instead of crack initiation. Interestingly,
and as a side note, the Murakami–Endo model predictions closely
follow the observed fatigue limit even in this blunt notch regime. For
the practical application of the model, it is probably wise to assume that
crack initiation always occurs due to a non-metallic inclusion, corro-
sion, or machining scratch to remain conservative. This puts increasing
emphasis on the study of the cyclic R-curve methodology and damage-
tolerant design. Nevertheless, it is important to distinguish that this
mechanism might mask some of the results for the analysis and design
of experiments. This brings us to the next discussion point: the scatter
in results.
5.4. Physics behind the variance of cyclic R-curve
The present model predicts that a crack with a short crack front
can become non-propagating even with a relatively high crack driving
force. This is a result of the assumption that the minimum crack growth
resistance from the microstructural barriers defines the crack arrest.
As the length of the crack front increases, the minimum resistance
eventually stabilizes to the cyclic R-curve measured with a through-
plate crack. This yields an important property that might explain the
discrepancy in experimental results between short cracks initiated from
3D defects and 2D through-plate specimens [17]. The crack front of
a through-plate crack sees a large and roughly constant number of
microstructural barriers as it grows. In contrast, the crack front of a
radially growing crack sees a linearly increasing number — the small
variance in experimental results of Maierhofer et al. [15] with short
through-plate cracks and a relatively high variance in the radially grow-
ing cracks in the results of More et al. [27] support this to some degree.
According to the model, the previously discussed crack interaction as-
pect also decreases the variance compared to a continuous crack front,
which is emphasized with small cracks. A similar phenomenon would
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J. Vaara et al.
occur even if a continuous crack front were effectively discretized
into nearly independent segments controlled by their respective strong
microstructural barriers. It has to be noted that researchers have,
through time, been clever at eliminating the sources of scatter in the
experiments. Research topics with naturally big scatter have yet to be
seen as fruitful. Our approach is slightly different: we believe that the
scatter contains information about the underlying nature of physics and
a model such as this is required to seize it. Such models also help create
new hypotheses for the experimental studies.
The loading path dependency predicted by the present model pro-
vides another source of scatter in certain measurements. The variance
of 𝛥𝐾 decreasing tests of More et al. [27] hinted that another source
of scatter is required.
Asymmetric yielding and roughness-induced crack closure inher-
ently carry a probabilistic element. Besides, the roughness-induced
closure should contain an autocorrelated closure element, i.e., the
previous high closure levels should predict high closure in the near
future, and vice versa. As the individual crack segment is connected to
the neighboring crack segments, extreme behavior should be restricted
via residual stresses that attempt to drive the solution towards a more
continuous mode II displacement field along the crack front. Thus, the
physics and balance of near- and far-field closure should define the
extent of the ‘memory’ the closure influence demonstrates [59]. This
aspect is not considered in the current model and could be considered a
prospective follow-up study. An initial check with the model presented
in [7] does not rule out the possibility of roughness-induced crack
closure explaining the relatively high variance in the long crack cyclic
R-curves exhibited in [17]. Alternatively, oxide-induced crack closure
could provide another explanation for the differences, but it would be
only possible to judge with specimen-wise history. As a naive alterna-
tive, the baseline crack closure could simply be a random variable in
the current model to comply with these observations.
For a more concrete assessment of how much the present model
could explain the discrepancy between through-plate and 3D cracks,
a crack emanating from the smaller (100 μm deep and 500 μm wide)
Electro Discharge Machining (EDM) notch of [17] was analyzed. The
studied material was 25CrMo4-QT steel with slightly lower hardness
than the 42CrMo4-QT studied here (approximately 90% of the ultimate
tensile strength). The 𝑅= −1 baseline cyclic R-curve fit here for
42CrMo4-QT had to be scaled with a factor of 0.8 to match the mean of
25CrMo4-QT – which is significantly more compared to what a typical
hardness scaling of fatigue strength would suggest. In the present
framework, this means that an even larger systematic difference must
exist between the cyclic R-curves determined with through-plate cracks
and defect-initiated cracks. It would follow that the baseline cyclic R-
curve should be lower, and the role of microstructural barriers should
be more emphasized. The relatively wide and shallow rectangular EDM
notch is not a natural shape for the crack, and thus, it is interesting from
the non-homogeneous crack driving force perspective. Here, the EDM
notch and the crack emanating from it was estimated as a semi-elliptical
crack [60]. Treating the notch as a crack initially yields a 45% higher
maximum stress intensity factor compared to the minimum along the
crack front, and the segment with above 90% of maximum 𝛥𝐾 is 187 μm
long. It is clear that this segment acts as the spearhead for crack growth,
and in near-threshold conditions, it would likely determine the crack
arrest. Elber-Paris law with exponent 𝑚= 3 and a radial crack closure
development were assumed to estimate the relative crack growth at
the free surface and at the deepest point. The crack was assumed to
stay in an elliptical shape for the stress intensity factor assessment.
The crack would grow from the initial aspect ratio 0.4 to 0.8 within
approximately 280 μm crack growth from the deepest position. With
this approach, the predicted median fatigue limit would be only 5%
higher compared to that resulting from a deterministic cyclic R-curve
analysis with an average non-propagating crack length of 122 μm. While
the study did not report all non-propagating crack lengths, they ap-
peared to be substantially shorter than what was predicted here for
the 𝑅= −1 case — again indicating an emphasized effect of the
microstructural barriers. However, the stress intensity factors at the
deepest point of a semi-elliptical crack with varying aspect ratios were
noticed to be roughly 15 % to 20 % lower compared to the FEM solution
presented in [17] over a relatively wide range of crack extensions. This
includes the 50 μm to 100 μm range that was discussed to match well
with the reference solutions, but as noted above, this only applies if
the aspect ratio is assumed to remain constant. Thus, the predicted
fatigue limit with deterministic R-curve prediction and varying crack
aspect ratio would be closer to 150 MPa rather than 136.5 MPa. Still,
the observed fatigue limit was roughly 200 MPa, and to explain the
remaining discrepancy and the short non-propagating cracks, the role
of microstructural barriers needs to be increased. Additionally, with
the non-homogeneous crack driving force, the distribution of a single
microstructural barrier’s strength is emphasized. Thus, modifying the
expected value of a single microstructural barrier’s strength from 1.3
MPam to 3.5 MPam led to an increase of predicted median fatigue
limit, matching the observed. This modification would also bring the
cyclic R-curves closer to agreement with the expected difference in
fatigue strength between the two materials. This exercise showed that
•The difference between observed cyclic R-curves between
through-plate and defect-initiated, radially growing cracks is even
larger than anticipated when studying the 42CrMo4-QT steel and
only defect-initiated cracks. In the present framework, it would
correspond with a low baseline crack closure.
•To compensate for the decrease in baseline crack closure, the
role of microstructural barriers needs to be increased. The present
framework is capable of describing the behavior of both types of
specimens.
•The relatively low baseline crack closure provides further support
for the chosen crack arrest criterion in minimum strength along
the crack front.
5.5. On the parameters
In the present formulation, a baseline crack closure is assumed, and
the extra resistance from microstructural barriers has a lower support
of zero. This choice simplifies the parameter identification scheme,
as the baseline crack closure can be assumed to coincide with the
cyclic R-curves measured with through-plate cracks. A schematic plot of
the difference between near-threshold through-plate (microstructurally
short) and defect-initiated (microstructurally small) cracks is shown in
Fig. 14. A microstructurally small crack, loaded at the fatigue limit,
would become non-propagating by a major microstructural barrier
with a 50% probability. Thus, the fatigue limit would be increased
from the one defined by the tangency condition with cyclic R-curve,
corresponding to the microstructurally short crack. Apart from the
perturbations by the microstructural barriers, this offset would result
in a window of approximately constant crack growth rate, according to
the Elber-Paris law. Then again, the characteristic crack growth rate at
the fatigue limit would be descriptive of the strength of microstructural
barriers and could be characterized by fatigue tests with in-situ crack
growth measurement. It was shown that with artificial defect-initiated
non-propagating cracks, the baseline closure curve was estimated as
the envelope curve of the longest non-propagating cracks. However,
reflections against the cyclic R-curves measured in [17] indicated that
the baseline closure should be even lower (see Section 5.4). The data
shows support for the modification of the intrinsic threshold with a
factor of 1 − 𝑅 for negative stress ratios, as proposed by Patriarca
et al. [12]. Given this input, the predicted KT-diagram could closely
reproduce the observed El-Haddad type curve. This supports the idea
of a cyclic R-curve that can be applied universally in fatigue analysis
of different types of specimens or loading histories.
International Journal of Fatigue 198 (2025) 108953
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J. Vaara et al.
Fig. 14. Schematic difference between microstructurally small and short cracks at their respective fatigue limits. The microstructurally small crack faces consecutive microstructural
barrier fronts, manifested as erratic crack growth rates. These barriers can also be seen in the cyclic R-curve as extra obstacles to be penetrated, which requires increased crack
driving force. Similarly, the magnitude of the sudden drops in crack growth rate should contain information on the strength of these microstructural barriers. Microstructurally
short crack has a relatively long crack front that more likely finds a weak spot in the microstructure so that the effect of microstructural barriers is diminished and the resulting
fatigue limit is also smaller.
The presented probabilistic model allows for crack growth even
below intrinsic threshold values. As long as the crack initiates, this fea-
ture effectively represents micromechanical fracture mechanics’ higher
crack driving forces. Nonetheless, the role of micromechanical barriers
is emphasized for the microstructurally short cracks as stressed by
Chapetti et al. [20,61,62]. The model’s results are encouraging, espe-
cially as the model is so simple and only introduces a few additional
parameters. Given that a cyclic R-curve or KT-diagram exists, only
the average microstructural barrier spacings (𝑑, 𝑑2) and the random
variable quantifying the extra resistance 𝛿 𝐾th from a single microstruc-
tural barrier need to be calibrated. These additional parameters can
be inferred from the variance scaling in the cyclic R-curve or the
asymptote of the fatigue limit in the KT-diagram. The assumption
𝑑≈𝑑2 may be physically founded for the cases without significant
anisotropies, further simplifying the parameter identification process.
We are certain that the parameter fit could be optimized further, but
it was not in the scope of the current study. With a minimal amount
of parameters that are clearly defined, it is straightforward to use the
model on a component-level analysis.
6. Conclusions
The conclusions to be drawn are summarized below:
•A stochastic model based on the hypothesis of consecutive mi-
crostructural barriers has been derived as a probabilistic cyclic
R-curve. The non-propagating crack lengths follow an exponen-
tial type distribution with a non-homogeneous rate parameter,
depending on the development of the crack driving force.
•The consecutive microstructural barriers form a relatively sim-
ple model that can effectively describe a plethora of physical
mechanisms. However, the microstructural barriers alone cannot
explain the observed variance of the long crack threshold values.
Roughness-induced crack closure is seen to be most suitable to
describe this missing part.
•The model can predict the fatigue limit via the distribution of
non-propagating cracks and the probability of observing them.
Application of the model to the analysis of defect-initiated fatigue
limit and non-propagating cracks proved highly successful.
•The model extends the concepts of cyclic R-curve and defect-
based fatigue analysis and provides a general framework for
analyzing crack interaction problems. Drilled defects should be
treated as a two-crack interaction problem if the non-propagating
cracks defining the fatigue limit do not coalesce.
•The solution from the probabilistic cyclic R-curve yields the lower
bound of fatigue limit. Crack initiation resistance can only in-
crease the apparent fatigue limit.
•The presented probabilistic cyclic R-curve can describe the tran-
sition through micromechanically short crack to physically short
crack and long crack regimes. The model produces an El-Haddad
type Kitagawa–Takahashi diagram with an asymptotic fatigue
limit. The mechanism is the probability of the crack growing in a
favorable microstructural surrounding without major barriers.
•Qualitative differences in fatigue limits between short through-
plate cracks and defect-initiated radially growing cracks can be
predicted based on the different number of microstructural barri-
ers at the crack front. Cracks growing from small defects would,
on average, exhibit higher resistance to fatigue crack growth com-
pared to through-plate cracks of similar length, where the longer
crack front more probably finds a weak spot in the microstructure.
This discrepancy should be emphasized for materials with strong
International Journal of Fatigue 198 (2025) 108953
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J. Vaara et al.
microstructural barriers. However, more validation is required on
this aspect.
•A loading path dependency of crack non-propagation is deduced.
The longer the crack grows with low 𝛥𝐾eff , the more likely
it is to face a major microstructural barrier and become non-
propagating. This aspect might explain some of the variance in
certain types of experiments.
CRediT authorship contribution statement
Joona Vaara: Writing – review & editing, Writing – original draft,
Visualization, Validation, Software, Methodology, Investigation, For-
mal analysis, Conceptualization. Kimmo Kärkkäinen: Writing – review
& editing, Writing – original draft, Visualization, Methodology, Inves-
tigation, Conceptualization. Miikka Väntänen: Writing – review &
editing, Writing – original draft, Visualization, Methodology, Inves-
tigation, Conceptualization. Jukka Kemppainen: Writing – original
draft, Methodology, Formal analysis, Conceptualization. Bernd Schön-
bauer: Writing – review & editing, Supervision, Resources, Project
administration, Investigation. Suraj More: Writing – review & editing,
Investigation, Data curation. Mari Åman: Writing – review & editing.
Tero Frondelius: Writing – review & editing, Supervision, Resources,
Project administration, Funding acquisition.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
Funded by the European Union (Grant Agreement No. 101058179;
ENGINE). Views and opinions expressed are however those of the
authors only and do not necessarily reflect those of the European Union
or the European Health and Digital Executive Agency. Neither the
European Union nor the granting authority can be held responsible for
them. The corresponding author is grateful for the grant received from
Tekniikan Edistämissäätiö.
Data availability
Data will be made available on request.
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