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An innovative and sustainable methodology for fatigue
characterization and design
Davide Crisafulli
a
, Pietro Foti
b
, Filippo Berto
b
, Giacomo Risitano
a
,
Dario Santonocito
a,*
a
University of Messina, Department of Engineering, Contrada di Dio, 98166 Messina, Italy
b
La Sapienza University of Rome, Department of Chemical, Materials and Environmental Engineering, Via Eudossiana 18, 00184 Rome, Italy
ARTICLE INFO
Keywords:
Fatigue design
Static Thermographic Method
Risitano’s Thermographic Method
Strain Energy Density Method
ABSTRACT
Most failures in engineering components and structures are determined by the fatigue phenom-
enon, known for being lengthy to be experimentally investigated and challenging to be properly
accounted in design. These aspects can represent a barrier for companies working on the edge of
new technologies due to the duty and need to ensure reliability for their products. This study aims
to propose a methodology to couple the economic and sustainability advantages of Thermo-
graphic Methods in investigating the fatigue behaviour of materials with those of local ap-
proaches to provide a simple, yet reliable, fatigue design tool. To pursue this aim we intentionally
conducted in the present paper the minimum required experimental campaign needed by Ther-
mographic Methods to achieve the basic information to calibrate a local approach, the Strain
Energy Density one. The proposed methodology has been satisfactorily validated by applying the
calibrated local approach to foresee the fatigue behaviour of a vast experimental database,
characterized by a variety of notched geometries, retrieved from literature.
1. Introduction
Continuous and outstanding advances in materials such as the developing of high-entropy ceramics [1], alloys with nano-bridged
honeycomb structure [2], titanium produced via near void-free 3D printing [3] or pure titanium with nacre-like surface nanolaminates
[4], as well as manufacturing techniques including innovative rapid liquid printing [5], the combination of spark plasma sintering and
3D printing [6] and other hybrid techniques [7], are nowadays dominating the landscapes of research and industry development
allowing to produce components with shapes and properties able to meet multifunctional purposes among which the demand for an
eco-friendly product life cycle[8–10], e.g. proposing circular economy solution for reducing plastic waste [8], redesign commercial
products [9] or addressing challenges in the biomedical sector [10]. There is, however, another side of the coin. Although the
advancement in the materials and manufacturing capabilities, industries must still ensure the reliability of the produced components
with the same safety degree as before [11]. On this side, the advancement on the design technique struggle to keep the pace with the
others; nevertheless, there is an undeniable need for advanced design approaches coupled with methodologies able to decrease the
time and money demand for the experimental characterization of materials. These two research elds – design approaches and
experimental characterization- set the boundaries to ensure reliability and safety.
* Corresponding author.
E-mail address: dsantonocito@unime.it (D. Santonocito).
Contents lists available at ScienceDirect
Engineering Fracture Mechanics
journal homepage: www.elsevier.com/locate/engfracmech
https://doi.org/10.1016/j.engfracmech.2025.110854
Received 18 October 2024; Received in revised form 26 December 2024; Accepted 20 January 2025
Engineering Fracture Mechanics 315 (2025) 110854
Available online 27 January 2025
0013-7944/© 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
( http://creativecommons.org/licenses/by/4.0/ ).
Nomenclature
c specic heat
c
w
stress range correction factor
dV
p
innitesimal plastic volume
d
ε
p
innitesimal plastic deformation
dQ
p
innitesimal heat generation
E Young’s Modulus
E
c
critical value of the plastic strain energy
E
p
plastic strain energy per unit volume
E
r
residual energy of the material
f frequency
K
f
, K
f
* Fatigue notch factor, Effective fatigue notch factor
K
m
thermoelastic constant
K
t
Theoretical Stress Concentration Factor
m angular coefcient of the regression line
N, N
0
, dN, ΔN Number of cycles
N
f
number of cycles to failure
q intercept of the regression line
Q Energy released as heat
r0centre of the control volume distance perpendicularly to the notch edge
R stress ratio
R
2
coefcient of determination
R
0
control volume radius
R
m
maximum stress
R
p0.2
stress at plastic deformation of 0.2 %
t
int i
i-th intersection time
T
0
initial temperature of the specimen in a static tensile test
TΔ
σ
eq , T
ΔW
stress scatter index, SED scatter index
W1MPa,R0value of SED obtained applying the nominal stress of 1 MPa at a given R0
α
thermal expansion coefcient
βTaylor-Quinney coefcient
ΔT temperature variation
ΔT
j
experimental temperature point
ΔT
1
, ΔT
2
temperature regression subset
ΔT
1,i,
ΔT
2,i
data set for the linear regressions, t ≤t
int i
ΔTj,ipredicted temperature value
ΔT average temperature
ΔT
s
stabilization temperature
ΔW,ΔWNcyclic averaged SED, critical cyclic averaged SED
Δ
σ
N,0fatigue strength at a given number of cycles
ΦEnergy Parameter
ρ
density, llet radius at notch
σ
nom
,
σ
nominal applied stress, average stress
σ
Y
yielding stress
˙
σ
stress rate
σ
a
stress level in a constant amplitude fatigue test
σ
0
,
σ
0 RTM
fatigue limit, fatigue limit estimation by Risitano’s Thermographic Method
σ
lim
limit stress of the Static Thermographic Method
σ
linear
proportional stress value
σ
peak plastic
peak plastic stress at notch tip
Abbreviations
CA constant amplitude
FEA Finite element analysis
IR Infrared
PS probability of survival
RTM Risitano’s Thermographic Method
SED Strain Energy Density
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
2
Engineering components, due to the need to conform with their various functional purposes, bring along numerous sites of
geometrical discontinuity, i.e. notches, which result in stress gradients that limit the possibility to fully exploit the mechanical
properties of the material and expose some locations for triggering the initiations of the fatigue phenomenon [12,13], the cause of most
of the failures in components under service. Acting as the weakest links, the design of such geometrical discontinuity dominates the
entire reliability of a component by governing the fatigue phenomenon [14]. The “weakest-link” concept also denotes a microscopic
model based on the probabilistic framework proposed by Weibull, widely acknowledged for its application in describing the failure
strength of brittle materials. This model evaluates the crack initiation probabilities, which are critical in predicting notch effects under
high-cycle fatigue (HCF) [15] and low-cycle fatigue (LCF) [16] regimes, as well as in scenarios involving multiaxial loading [17].
Consequently, an accurate assessment of the structural integrity of a component passes by a reliable estimation of the notch effect on
the fatigue strength.
Various methodologies can be encountered in literature to deal with this effect [11]. The easiest way to get a rst tentative fatigue
design may be to consider the nominal stress and increase it by a factor equal to the so-called theoretical stress concentration factor, Kt,
i.e. the ratio between the elastic peak stress at the notch tip and the nominal stress value under linear elastic condition. As a matter of
fact, this results in a stress-based point method that aims to account for the notch effect by just considering an equivalent maximum
stress in the component, such as, for example, the Von Mises one, and assuming the same fatigue behaviour exhibited by smooth
laboratory specimens neglecting also some other important effect such as those related to the size of the component under analysis
[18–23] or stress gradient originated from non-uniform stresses that promotes crack initiation and even fatigue fracture [24]. Such an
estimation is usually conservative and would result in oversized components practically nullifying possible advantages achieved on the
manufacturing and material side. The notch effect is indeed affected by several aspects that mitigate the detrimental effects that could
be foreseen according to such a methodology for fatigue design [12].
Experimentally, the notch effect is usually quantied by the so-called fatigue notch factor, Kf, dened as the ratio between the
nominal fatigue strength of a smooth specimen and that of a notched one under the same experimental conditions and the same
number of cycles. However, the dependency of Kf on several parameters going from the material itself to the geometric parameters
describing the notches, makes its experimental investigation unfeasible in practice due to the lengthy and costly testing processes.
Various expressions have been developed over the years by researchers to obtain an estimation of Kf starting from Kt and the various
parameters known to inuence it while accounting on various models [25].
On the other hand, the recently developed manufacturing processes allows for complex geometrical components that limit the
application of traditional notched fatigue design methodologies [11]. Moving away to more advanced approaches, it is worth
mentioning those dealing with notch effect through the local stress–strain condition at the notch tip [26]. However, relying on the
evaluation of point stress and strain values, it follows that the proper application of these methods is strictly dependent on the accuracy
of the strategy employed to obtain such values. Various well-known approaches are available in literature to simplify their evaluation.
For instance, Tao proposed a pseudo-stress correction method for estimating local strains at notches under multiaxial cyclic loading
[27]. Neuber extended the theory of stress concentration to account for shear-strained prismatic bodies governed by arbitrary
nonlinear stress–strain laws [28]. Building upon Neuber’s work, Ye introduced a modied Energy of Strain Energy Density (ESED)
approach to predict the nonlinear stress–strain behaviour at notches [29]. Additionally, Lim applied local stress–strain methodologies
to estimate fatigue crack initiation life in cyclically non-stabilized and non-Masing steels [30].
Nevertheless, a precise estimation can be achieved only through elasto-plastic FEA that, on the other hand, makes their application
computational demanding.
To obtain more user-friendly, yet reliable, fatigue design methodologies, research efforts have been devoted over the years to
develop approaches, named local approaches, based on parameters evaluated as averaged values over the stress or energy eld around
the notch tip. Dealing with the stress-based approaches, it is worth mentioning the Theory of Critical Distances (TCD) [31], that have
been proposed in literature in slightly different versions and in combination with other methods [32–34]. As regard the energy-based
approaches, the averaged Strain Energy Density method (SED) [14,35,36], have been widely validated as a fatigue design method able
to directly account for various effects among which the notch effect one. Over the years, researchers have improved the SED approach,
providing advice and recommendations on its applicability [37], including the case of localized and generalised plasticity [38], and
trying to simplify its application in the case of complex models [39], proposing volume free procedure [40], using a coarse mesh
[41,42], although the method already represented a convenient choice due to the reduced time and computational resources required
thanks to its proven low sensitivity with respect to the model discretization [43].
On experimental mechanics, the adoption of infrared (IR) thermography can severally shorten the required time to assess the
fatigue life of specimens and mechanical components. Indeed, fatigue in materials is a dissipative phenomenon where a large amount
of the mechanical work done on the material is converted into heat and released on the surrounding environment [44,45].
The rst observation of dissipative effect during fatigue tests were performed by Risitano and his School [44,46]. Moving from this
rst observation, some researcher proposed to observe the thermoelastic signal during a fatigue test; especially, the second harmonic of
the temperature signal is related to dissipative effects and can be directly related to the fatigue limit of the material [47–49].
Meneghetti [50] proposed an experimental procedure to directly evaluate the heat energy density per cycle dissipated by a material
undergoing a fatigue test, the Q parameter, allowing the synthetization in a single scatter band of the fatigue life of plain and notched
STM Static Thermographic Method
TCD Theory of critical distances
XRF X-Ray Fluorescence
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
3
specimens. The Thermographic Methods (Static Thermographic Method, STM [51,52], and Risitano’s Thermographic Methods, RTM
[53,54]) are strongly established approaches based on monitoring the temperature trend on the surface of a specimen during a me-
chanical test by using an infrared camera [51–56]. The advantage of these methods is the rapid fatigue characterisation of materials
through uniaxial tensile and fatigue tests [55]. The eld of application ranges from the most common engineering to the most
innovative materials obtained through additive manufacturing [56,57].
This scientic article, leveraging the authors’ expertise in the Thermographic and SED Methods, aims to propose a methodology for
conducting a rapid and sustainable experimental characterization of engineering materials and retrieve the basic information needed
to perform their fatigue assessment of components in the presence of complex geometries.
The stainless steel AISI 304L, a low carbon variant of stainless steel 304, commonly used in a wide range of applications[58–61], i.e.
in petrochemical [58], food [59], mechanical [60] or automotive [61] industries, have been chosen as reference case due to its ductile
behaviour that was expected to result in a marked difference between the values of the theoretical stress concentration factor and the
fatigue notch effect. On some very plastic materials, K
f
differs greatly from K
t
, therefore a design based on S-N curves obtained on
smooth specimens and stress-based approaches leads to an underestimation of the fatigue limit of components with complex geometry
[62]. Such a ductile material, although challenging and controversial for a conventional application of methods such as the SED
approach, originally and rigorously intended for brittle materials, represented, on the other hand, a perfect candidate to prove the
challenges in fatigue design described in the above and pursue the aim of the present study.
2. Methods
In this section the methodologies adopted in the present paper are explained. To account for a rapid fatigue characterization, the
Risitano’s Thermographic Method (RTM) and the Static Thermographic Method (STM) are introduced. They allow to obtain indication
of the fatigue limit of the material with a rapid stress-controlled stepwise fatigue test, adopting the RTM, and the limit stress of rst
damage with a static tensile test, adopting the STM. With RTM it is also possible to retrieve the fatigue curve of the material. On the
other hand, the Strain Energy Density (SED) approach, after being calibrated though the experimental outcomes of the thermographic
methods, provides a useful tool for the design of complex components.
2.1. Risitano’s Thermographic Method
As observed by Risitano [46], the fatigue properties of a material can be estimated by monitoring the surface temperature evolution
of a specimen during a fatigue test with a cyclic loading condition above its fatigue limit [53,54].
The surface temperature trend of a metallic material stressed above its fatigue limit has the typical trend of Fig. 1a. The rst phase is
characterized by a rapid increase in temperature with continuous rearrangement and dislocation motion in the microstructure of the
material. The second phase is characterized by a semi-equilibrium thermal state with a stable increase in temperature (ΔT
st
=cost).
The third phase has a very high increase in temperature, which is associated with the rapid growth of the crack length up to the failure
of the specimen. This phase is characterized by a limited number of cycles compared to phase I and II. The stabilization temperature,
ΔT
st
, is proportional to the applied stress level, and its values are more pronounced when the cyclic stress is higher compared to the
fatigue limit of the material. It is assumed that the fatigue failure occurs when the strain energy of plastic deformations reaches a
constant critical value E
c
, characteristic of each material and equal to the energy to failure per unit volume. Considering E
p
as the
plastic strain energy per unit volume, the cumulative residual lifetime of the specimen, E
r
, after a number of cycles N
0
can be evaluated
as:
(a) (b)
Fig. 1. A) temperature trend during a constant amplitude fatigue test with stress level above the fatigue limit of the material; b) temperature trend
during a stepwise fatigue test. (based on [63]).
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
4
Er=Ec−∫N0
0
EpdN (1)
The plastic strain energy is proportional to the energy released as heat Q in the surrounding environment and, considering that the
stored energy in the specimen is very low compared to the released one, it is possible to state that Q is proportional to the critical strain
energy E
c
of the material. The released heat Q can be easily estimated with a thermal sensor, like infrared cameras, during a fatigue
test.
For a xed testing frequency and stress ratio, it is possible to estimate the Energy Parameter Φ, dened as the integral of the
specimen’s temperature increment versus number of cycles curve. Both the Energy Parameter and the released heat are proportional to
the limit energy of the material (Eq. (2)).
Φ=∫N0
0
ΔT⋅dN∝Ec(2)
By subjecting a single specimen to a stepwise fatigue test and recording the stabilization temperature for each stress increment ΔT
st
i
(Fig. 1b), it is possible to estimate the fatigue limit in the knee region of the stabilization temperature vs applied stress curve. Given
the constancy of the Energy Parameter, it is possible to obtain the S-N curve of the material estimating the number of cycles to failure
with Eq. (3) and adopting just one specimen.
Ni=Φ
ΔTsti
(3)
2.2. Static Thermographic Method
The analysis of the energetic release during a static tensile test can provide useful information also on the fatigue behaviour of the
material. Early studies by Caglioti [64] on the energetic release of metallic specimens showed that when a metallic material is elas-
tically strained, the temperature trend change in a remarkable way as the yielding stress is approaching. The order of magnitude of the
temperature decrement for a metal under adiabatic conditions is of the order of 0.2 K, a value that in practical application has been
always neglected due to the difculty to measure such small temperature variation [64].
Melvin [65,66] correlated the material microstructure with the energetic release of the material. When a tensile stress is applied on
a crystalline structure, like metals, a corresponding elastic dilation is observable. The applied external work, at atomic level, raises the
interatomic potential of the vibrating body in thermal equilibrium and hence increases the mean interatomic separation. To balance
with the deformation, the material, given also the fact that cannot increase the atomic population, cools down. If the applied stress is
removed, the material resumes its previous average interatomic distance without showing effect of the previous deformation; this
holds true up to a load beyond an elastic limit.
The yielding, as observed by Caglioti [64], is associated with a sudden increase in the temperature trend of the specimen,
compensating the thermoelastic cooling of the specimen.
Moving from the previous studies of Geraci [67], Chrysochoos [68] and Plekhov [69], Risitano et al. [70,71] in 2010 analysed the
energy dissipation during a static traction test of a specimen to better identify the end of the thermoelastic phase as the deviation from
the linearity of the temperature versus stress signal.
The Static Thermographic Method [52] moves from the assumption that fatigue failures occur at points where the local stress
condition is amplied by structural micro defects within the material, able to produce plastic deformation in the material’s structure.
Internal defects act as stress raisers compared to the average nominal stress at which the specimen is loaded. If the same average stress
Fig. 2. Temperature trend during a static tensile test with the three different phases of the temperature signal. The transition from Phase I to Phase
II correspond to the rst damage within the material (
σ
lim). (Based on [63]).
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
5
is applied under fatigue test conditions, the specimen will fail.
A static tensile test can be thought as the rst positive half cycle of a tension–compression fatigue test and the temperature trend
can be divided into three phases (Fig. 2). The rst phase has loads low enough to assume that all the crystals are stressed into the elastic
eld and follow the Lord Kelvin’s law (Eq. (4)) (segment OA). In the second phase, the majority of the crystal are elastically stressed,
but a fraction of it is plastically deformed (segment AB). As the stress level increases, in the third phase, all the crystals are plastically
deformed (segment BC). During the static tensile test, a deviation from the linearity is due to the rise of heat generation sources because
of irreversible micro plastic deformations before the macroplastic deformation regime [72], where energy dissipation is present with
irreversible dislocation motion.
The law of temperature variation for an isotropic metallic material under adiabatic test conditions can be written according to
Melvin’s model by neglecting the plastic contribution of the work, leading to Lord Kelvin’s law:
ΔT= −
α
ρ
c⋅T0
σ
nom = − KmT0⋅
σ
nom (4)
Where
α
is the thermal expansion coefcient,
ρ
the density, c the specic heat, K
m
the thermoelastic constant, T
0
the initial
temperature and
σ
nom
the nominal applied stress.
According to this equation, during the perfect thermoelastic phase, the temperature detected on the specimen’s surface exhibits a
linear decrease as the load increases. If a plasticity condition is locally reached near a defect zone of innitesimal volume dV
p
, under an
average stress
σ
(lower than the yielding stress), Eq. (4) is no longer valid and the plastic deformation, d
ε
p
, is the reason of the
innitesimal heat generation dQ
p
that lead to a deviation from the linear trend:
dQp=β
σ
d
ε
p(5)
Where β is the Taylor-Quinney coefcient [73,74], i.e. the percentage of plastic deformation dissipated into heat. The novelty of the
STM consists of a rapid procedure, even faster than other fatigue energy-based methods, to estimate the rst damage of the material.
The identication of the rst local plasticization, followed by heat release during a static traction test, can be dened as the “limit
stress”,
σ
lim
, i.e. the transition stress from Phase I to Phase II. It is the stress at which the released heat modies the linear temperature
trend during a monoaxial static tensile test. The value of the limit stress can be adopted as an estimation of the fatigue limit of the
material.
2.3. Strain Energy Density method
The averaged Strain Energy Density (SED) criterion, in the version proposed in 2001 by Lazzarin and Zambardi, has been widely
used for the prediction of the static and fatigue behaviour of components with sharp V-notches [14]. In the following twenty years of
research, its application has been further extended to consider also different geometries and loading conditions [75]. The method is
Fig. 3. Control volume for: local geometries leading to a stress singularity such as; a) crack and b) sharp V-notch; local geometries leading to a stress
concentration such as: c) U-notch and d) blunt V-notch under mode I loading conditions; e) U-notch and f) blunt V-notch under mixed mode loading
conditions. Figure .
adapted from [82]
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
6
based on the key concept that brittle fracture occurs when the SED value, averaged in a properly dened volume of material in the most
critical area of the analysed component, named control volume, reaches a critical value independent on the local geometry and loading
conditions. Accordingly, the use of the SED method allows to directly consider effects such as the notch effect.
The method, as formalized by Lazzarin and Zambardi [14], is reminiscent of Neuber’s concept of elementary volume and Erdogan
and Sih local mode I dominance[76]; however, it is worth noting that various energy-based criteria can be found in literature to deal
with the fatigue phenomenon [77–81].
Dealing with the fatigue phenomenon, the application of the method relies on the hypothesis that fatigue failure happens within the
linear elastic regime, and it shows a brittle behaviour; in such a condition the parameter to consider is the cyclic averaged SED value
ΔW. The fatigue failure is assumed to occur, at a given number of cycles, when this parameter assumes the critical value, called ΔWN.
The value of averaged SED, in components having stress gradients, is evaluated in a control volume, dened accordingly to the method
theory [75].
The control volume, according to the method theory, has some features: a characteristic length, R
0
[75], assumed to be a material
property; a shape that is dependent on the local geometry (a sector-shaped cylinder with radius R
0
in the case of sharp-notches- see
Fig. 3a) and b) −and a crescent-moon like shape given by two arches with R
0
being the maximum distance between them for blunt
notches- see Fig. 3c) and d); a position that, for the case of blunt notches, depends on the loading conditions resulting in a control
volume having its axis of symmetry passing though the notch curvature centre and the maximum of the rst principal stress eld along
the notch edge and a centre, for its higher arc, located at a distance r0 (function of the geometry through the Eq. (6) [75].
r0=
ρ
•(
π
−2
α
)
(2
π
−2
α
)(6)
The critical cyclic averaged SED for a smooth specimen under uniaxial loading with isotropic linear elastic material behaviour can
be estimated with the following equation [14]:
ΔWN=Δ
σ
2
N,0
2E (7)
Where Δ
σ
N,0 is the fatigue strength at a given number of cycles, N, and at a load ratio R =0 and E is the modulus of elasticity derived
from standard tensile test.
Considering that the method assumes that the cyclic averaged SED value is the one leading to fatigue failures, it also allows for a
direct consideration of the load ratio effect. To properly estimate the value of the cyclic evaluated SED under different load ratios,
taking advantage of the benets introduced by this method, the following equation can be applied to directly evaluate it starting from
the stress range values [83]:
ΔWc =(1−R2)
(1−R)2
Δ
σ
2
2E =cw
Δ
σ
2
2E 0≤R<1 (8)
ΔWc =(1+R2)
(1−R)2
Δ
σ
2
2E =cw
Δ
σ
2
2E −∞<R<0 (9)
Fig. 4. Geometry of the AISI 304L specimen: a) Plain b) V-notch (measures in mm).
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
7
3. Experimental campaign and numerical simulations
3.1. Material properties and specimen’s geometry
Experiments were conducted on AISI 304L specimens of two different geometries showed in Fig. 4. A plain geometry (Fig. 4a) was
adopted to estimate the mechanical properties and the stress–strain curve of the material. A double −edge blunt V-notch geometry was
adopted to estimate the fatigue life of a thick notched detail (Fig. 4b). The opening angle for the notch specimen was xed to 2
α
=135◦,
with a llet radius
ρ
=2 mm. The thickness of the plain and V-notch specimens was chosen equal to 10 mm to obtain a thick notched
detail, and the width is equal to 24 mm; hence, the nominal cross section was equal to 24x10 mm
2
. The choice for an equal nominal
cross section between plain and notched geometries was made to highlight only the inuence of notch effect on the fatigue strength.
Table 1 reports the average chemical composition of AISI 304L stainless steel obtained with the X-Ray Fluorescence (XRF) analysis
with a SPECTRO instrument (AMETEK, Germany).
3.2. Mechanical tests
Quasi-static and fatigue tests were performed with a servo-hydraulic testing machine MTS 810, with a maximum load capacity of
250 kN, at the University of Messina. During the tests the specimen surface temperature was monitored by using the infrared camera
FLIR, model A40, with a thermal sensitivity of 0.08 ◦C at 30 ◦C. Before each test, the surface of the specimen was painted with a thin
layer of matte black paint, to increase its thermal emissivity (up to 0.98) and improve the quality of the emitted thermal signal. The
image acquisition frequency of the infrared camera was set at 5 Hz for quasi-static tensile tests, while an image every 10 cycles was
acquired during fatigue tests. All data results from the tests were processed using Matlab® software (MathWorks).
Quasi-static monotonic tensile tests were conducted adopting stress rates within the range 2–6 MPa/s to ensure adiabatic testing
conditions; in this way, the specimen, due to its very small temperature gradient respect the environment, has no time to exchange
energy. From the tensile tests, the engineering stress, considering the nominal section of the specimen, and temperature trends over
time were evaluated.
Fatigue tests were carried out using a sinusoidal stress wave with a zero-mean value (stress ratio R =-1) and a frequency of 5 Hz.
The specimens were stressed in two different ways: using constant amplitude (CA) stress levels; using a stepwise increase of the stress
level every 10,000 cycles. These two approaches allow to estimate the S-N curve of the material, respectively, in the traditional way
and in a rapid way applying the RTM explained in section 2.1..
3.3. Literature data on AISI 304L
To compare the ndings of this study with other results, the fatigue dataset has been extended with fatigue tests carried out on AISI
304L in various literature articles [84–86]. It is worth underling that the criterion chosen to select the fatigue data in literature relied
solely on the declared nominal material. On the other hand, it must be considered that, although these tests are carried out on the same
nominal material, i.e. AISI 304L, there could still be a variety on the materials themselves; such a variety can indeed be appreciated, as
an example, through the various Young’s Modulus reported in these papers for the same nominal material or, where available, by
hardness and reduction area percentage. It is important to underline that, although fatigue data have been reported also in the low
cycle fatigue (LCF) regime for completeness, the present work limits its analysis, evaluation and conclusions only to fatigue data having
failures above 104 cycles.
Fatigue data, together with the main geometrical features of the specimens retrieved from literature are reported in Fig. 5. This set
of specimens is composed by dogbone-shaped specimens and notched plates (V and U-notches and hole) with a thickness equal to 6
mm, and hourglass shaped specimen with a diameter of 3 mm.
3.4. Numerical simulations
The presence of thick specimens with a quite ductile material required special attention in the estimation of the acting stresses, so it
is worth noting a few considerations.
Performing 2D simulations for a thin plate, the inuence of the thickness can be neglected, and the plane stress hypothesis is valid.
For a thick plate the stress state changes also along the plate thickness and the third component of the stress must be considered, so the
plain strain condition become dominant.
When considering a notched element, it is known that the effect of the stress intensication is proportional to the K
t
factor, dened
as the ratio between the elastic stress at the notch tip and the nominal stress, i.e. the average stress applied on the resistance section.
Under fatigue condition, the mitigation of the effect lead by the stress gradients at notches is quantied by the fatigue notch factor K
f
,
Table 1
Chemical composition of AISI 304L material (in wt. %).
Si Mn Cr Mo Ni Cu Fe
0.53 1.92 18.64 0.29 8.04 0.41 balance
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8
lower than K
t
, and usually evaluated between 10
6
and 10
7
cycles. This factor varies according to the life of the material and is
inuenced by several effects such as the dimension of the notch radius and the reversed yielding effect.
For this reason, the authors performed preliminary 3D simulations campaign not to neglect the inuence of thickness and to
evaluate the fatigue notch factor, K
f
*, in case of fully reversed loading.
The simulations were carried out using Ansys APDL nite element software, adopting a mapped volume mesh with 20-node
hexahedra SOLID186 elements. Only one-eighth of geometry specimen was modelled, taking advantage of its symmetry planes.
The mesh size varies from a minimum of 0.2 mm near the notch (Fig. 6b) to a maximum of 6 mm at the furthest point. The mesh
strategy was inspired by the study by Cicha´
nski [87]. It is well known that austenitic stainless steel presents cyclic softening and phase
transformations due to the arrangement of internal dislocations when subjected to cyclic straining. However, in the present study, due
to the fact that the fatigue test campaign was performed under stress control at R =-1 where, according to the ndings of Nagaishi et al
[88], for notched specimens the local stress ratio is equal to the nominal one, it is reasonable to dene the material behaviour with an
isotropic multilinear plasticity model through the data obtained from a static tensile test (see section 4.1). From each simulation the
average plastic stress was evaluated for the different nominal stress applied.
A second simulation campaign was performed to evaluate the averaged SED near the notch tip adopting a 2D elastic model for the
various notch geometries considered (see Fig. 4b and Fig. 6).Whenever possible, the symmetry of the geometry have been exploited to
reduce the computational time effort.
Fig. 5. A) fatigue dataset considered; b) geometry of the specimens with indication of the markers adopted in the graphs (measures in mm) and
testing frequency.
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A mapped mesh with 8-nodes quad and 6-nodes tria PLANE183 elements was adopted under the hypothesis of plane strain con-
dition. As an example, the adopted mesh with the symmetric boundary condition and the applied tensile load on a quarter of V-notched
specimen is showed in Fig. 6a. The concentration key point technique was applied to achieve a proper mesh near the notch tip, where
the element size reaches 0.025 mm and gradually becomes larger up to the distance of 3 mm. It must be highlighted that, although the
low mesh sensitivity of the method being the SED parameter evaluated directly from the displacement eld [43], the mesh size at the
notch tip had to be ne enough to meet the conditions dened in [40,89]. Indeed, having to carry out a parametric study to calibrate
the method, i.e. determining the value of R0, for the analysed material, the averaged SED had to be rstly evaluated for different values
of the control volume characteristic length. Under the linear elastic regime hypothesis, the averaged SED value has been calculated
applying a traction load in the grip section resulting in a nominal stress at the net cross section of 1 MPa for R0 ranging from 0.1 mm up
to 3 mm with a step of 0.01 mm. Therefore, the mesh size near the notch tip must be equal to at least ¼ of the minimum control volume
radius considered according to [89]. To estimate the SED value at any other stress value, under the linear elastic hypothesis, the
following equation was adopted:
ΔW=W1MPa,R0•Δ
σ
2•cw(10)
Where W1MPa,R0 is the value of SED obtained applying the nominal stress of 1 MPa at a given R0, Δ
σ
is the value of the applied stress
to be referred to, and the term cW depends on the stress ratio adopted (see Eqs. (8) and (9).
Fig. 6b shows a detail of the mesh around the notch tip with the mesh ination caused from the adoption of the concentration key
point technique. For area of the specimen far away from the notch tip, an element dimension of 0.5 mm was used with a free mesh
method.
4. Results and discussion
The experimental test campaign has been performed on AISI 304L. Quasi-static monotonic tensile tests have been carried out
monitoring the temperature evolution of the specimens to nd the rst damage within the material applying the STM. A series of
stepwise fatigue tests have been performed to nd the fatigue limit and the fatigue life of the material with few specimens and in short
amount of time, applying the RTM. Finite element simulations have been performed to investigate the notch effect on thick specimen
and, nally, the SED method has been calibrated through the experimental values acquired in the present paper and validated to
account for the notch effect on the AISI 304L with several plain and notched congurations retrieved from literature. It is worth noting
(a)
(b) (c)
Fig. 6. 2D FEM model with symmetric boundary conditions and tensile load: a) Quarter of the V-notch specimen b) Detail of the mesh near the
notch tip for elasto-plastic simulations; c) Detail of the mesh near the notch tip for SED simulations.
D. Crisafulli et al.
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that, in this phase, the Thermographic Methods allow for a rapid calibration of the control volume radius, instead of making a long test
campaign.
4.1. Mechanical characterization of AISI 304L
A preliminary tensile test campaign has been performed on two plain specimens, to retrieve the mechanical properties of the AISI
304L under study (Table 2). The tensile tests have been performed under displacement control with a velocity of 5 mm/min while an
extensometer with an initial gauge length of L
0
=25 mm has been adopted to estimate the strain of the steel. The yielding stress, Rp0.2,
has been estimated considering a residual plastic deformation of 0.2 %; the ultimate stress, Rm, as the maximum stress achieved by the
specimen and, nally, the Young’s Modulus, E, with a linear regression according to ASTM E111. The stress–strain curve of the second
specimen has been reported in Fig. 7.
4.2. Estimation of the limit stress during static tensile test
Tensile test can be adopted to obtain information regarding the damage state within the material. The adoption of the temperature
as a third parameter, in adjunction to stress and strain, can indicate the transition from a stress state without any plasticity toward a
stress state where some crystals of the material experience irreversible plastic deformation, with a consequent release of heat (Eq. (5)).
It is possible to assess a corresponding macroscopic stress level where this happens, i.e. the limit stress
σ
lim
, and, according to Risitano
[52], if that stress level is applied to the material in a cyclic way, it will experience fatigue failure.
Quasi-static tensile tests have been conducted on thick AISI 304L specimen in the plain and V-notched conguration. The maximum
temperature of a rectangular spot has been measured in the gauge length for the plain geometry and near the notch tip for the V-notch
geometry – see Fig. 8a). The thermal data used is for the region of interest (ROI) with the highest temperature over time, corresponding
to the notch where the crack originates and propagates.
To observe the limit stress, a proper stress rate must be adopted. This is necessary because, if it is too slow, the specimen can
exchange heat with the surrounding environment; while, if it is too fast, the cooling phase of the specimen is very short, and it is
followed by a sudden increase of the temperature starting at the yielding stress of the material. For the three tensile tests on the plain
specimens, stress rates of 2 MPa/s and 4 MPa/s have been adopted.
The raw temperature signal for the plain specimens S-PLAIN-03, referred to the temperature of the specimen at the beginning of the
test, has been reported in Fig. 8b) versus the applied nominal stress and testing time. It is possible to observe a rst cooling phase with
an initial slope, followed by a second cooling phase with a atter slope compared to the rst one.
When the yielding stress of the material is reached, the temperature signal begins to rise, up to the failure of the specimen. To assess
the limit stress, the temperature signal from the beginning of the test up to the yielding stress has been modelled with a bilinear model
composed by n temperature points (see the schematic representation reported in Fig. 8c). Linear regressions of the experimental
temperature points (ΔT
j
) vs. time from the beginning of the test up to the i-th intersection point (ΔT
1,i
data set, t ≤t
int i
) and from the i-
th up to the yielding point (ΔT
2,i
data set, t >t
int i
) have been performed obtaining the predicted j-th temperature (
ΔTj,i) for the i-th
iteration (Eq. (11), with t
int i
the i-th intersection time, m the angular coefcient and q the intercept). An iterative approach (Fig. 9)
evaluates the limit stress (Eq. (12)) by maximizing the coefcient of determination R
2
for the bilinear model (Eq. (13)) of the tem-
perature signal considering the predicted j-th temperature (
ΔTj,i) for the i-th iteration and the average temperature value of the whole
temperature set (ΔT=1
n∑n
j=1ΔTj).
ΔTj,i={ΔT1,i=m1,it+q1,i,t≤tinti
ΔT2,i=m2,it+q2,i,t>tinti(11)
σ
lim =
σ
lim,iwithi :R2=max(R2
i)(12)
R2
i=1−∑n
j=1(ΔTj−
ΔTj,i)2
∑n
j=1(ΔTj−ΔT)2(13)
In Fig. 10, the temperature trends of all the three tested plain specimens have been reported, focusing just on Phase I and II of the
temperature signal. For the S-PLAIN-01 specimen (Fig. 10a), the stress rate of 2 MPa/s leads to a noisier temperature signal compared
to the other two tests performed at 4 MPa/s; however, it is possible to assess the limit stress. The iterative algorithm allows to assess the
limit stress, i.e. the change in the slope of the temperature signal, at a macroscopic stress level equal to 190.5 MPa. The different
Table 2
Mechanical properties of AISI 304L under study.
No. Specimen R
p0.2
[MPa] R
m
[MPa] E [MPa]
S-PL-01 395 612 257,240
S-PL-02 439 622 243,890
Avg. 417 617 250,565
Std.Dev. 31 7 9440
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Fig. 7. Stress-Strain curve of AISI 304L under study (specimen S-PL-02).
Fig. 8. A) measure spot on plain and v-notch specimens. the maximum temperature value has been recorded; b) temperature evolution during a
static tensile test on plain aisi 304l specimen. c) schematic representation of the bilinear mathematical model considered for raw temperature data
tting. the maximization of the r2 leads to the limit stress evaluation.
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adopted stress rates lead to different values of the maximum coefcient of determination, as can be noted in Table 3. The lower the
stress rate, the worst the coefcient of determination and the noisier the temperature signal. The minimum temperature decrement is
within the range of 0.2–0.5 K. For the plain specimens, the average limit stress is equal to
σ
lim Plain
=197.5 ±6.1 MPa.
By monitoring the surface temperature near the notch tip for blunt V-notched specimens, it is possible to assess a limit stress
(Table 4). For this geometry, after some trials, a stress rate of 6 MPa/s has been adopted to highlight the bilinear trend of the tem-
perature signal. The temperature trends for the V-notch specimens have been reported in Fig. 11. Bilinear model has been applied to
the raw temperature data and the coefcient of determination has been estimated iteratively, leading to the best estimation of the limit
stress. For the rst specimen it is difcult to clearly observe the bilinear trend, because the difference in the slopes between phase I and
II is not so pronounced. However, the iterative algorithm allows to estimate the limit stress at
σ
lim
=158.4 MPa. The other two tests
have a more marked bilinear trend. The estimation of the limit stress, in these two cases in equal to 149.4 and 147.5 MPa. The
minimum temperature decrement is within the range of 0.2–0.4 K, with a coefcient of determination higher than 0.82. For the V-
notch specimens the average limit stress is equal to
σ
lim V-notch
=151.8 ±5.8 MPa.
4.3. Fatigue strength estimation with RTM
For the estimation of the fatigue life, two stepwise stress-controlled fatigue tests were performed for each type of geometry. The
temperature variation of the specimen’s surface during the test was monitored with an infrared camera on the area where fatigue
failure happens.
The plain specimens were tested at the frequency of 5 Hz with a fully reversed load (R =-1) starting from a stress level of
σ
a
=100
MPa, increasing the stress step of 10 MPa every 10x10
3
cycles for one test and every 20x10
3
cycles for the other, up to the specimen
failure.
Fig. 12a-b reports the stabilization temperature versus the number of cycles and the applied stress level for the plain AISI 304L
specimens. The stabilization temperatures are evaluated with respect to the initial temperature assumed by the specimen, i.e. at a
number of cycles equal to zero, and they are evaluated for each stress level considering temperature data between 30 % and 90 % of the
temperature signal in the stress level (see zoom in Fig. 12a). The average values of the stabilization temperature with one standard
deviation are reported in the gure.
Comparing the two graphs, the Energy Parameters Φ, evaluated as the subtended area of the temperature curve with respect to the
number of cycles, have very close values; hence, there is a very good match in terms of dissipated energy during the fatigue test of both
specimens. Stress levels from 100 MPa to 160 MPa have shown noisy signal or no relevant increase of the temperature (below 2.5 K).
Signicative increments of the temperature signal can be observed for stress levels of about 170–180 MPa (above 3 K); then tem-
perature continues to increase until the specimen failure. The maximum temperature reached by the specimens is 464.45 K and 455.54
Fig. 9. Block diagram of the computational algorithm illustrating the iterative approach proposed for evaluating the limit stress using STM.
D. Crisafulli et al.
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K, respectively for test 1 and test 2.
To assess the fatigue strength, the stabilization temperatures have been reported vs. the applied stress levels in Fig. 12c-d. As is
possible to observe, when the applied stress increases, the stabilization temperature also increases, as expected; however, at a certain
Fig. 10. Temperature trend vs. applied stress level and time for plain AISI 304L specimens. Bilinear trend tted on raw temperature data. The limit
stress is evaluated as the knee of the two regression lines.
Table 3
Limit stress and maximum coefcient of determination (R
2
) for the quasi-static tensile tests on plain AISI 304L speci-
mens. The R
2
has been obtained by iterative estimation on the bilinear model.
˙
σ
[MPa/s]
σ
lim
[MPa] R
2
max
S-Plain-01 2 190.5 0.6705
S-Plain-02 4 201.5 0.8041
S-Plain-03 4 200.5 0.9143
Avg. 197.5
Std. Dev. 6.1
Table 4
Limit stress and maximum coefcient of determination (R
2
) for the quasi-static tensile tests on V-notch AISI 304L
specimens. The R
2
has been obtained by iterative estimation on the bilinear model.
˙
σ
[MPa/s]
σ
lim
[MPa] R
2
S-V135-01 6 158.4 0.9073
S-V135-02 6 149.5 0.8298
S-V135-03 6 147.5 0.8939
Avg. 151.8
Std. Dev. 5.8
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stress level, there is a knee and the increase in temperature becomes steeper. By tting the data with two regression lines, the rst
below the knee region (ΔT
1
, from 120 to 190 MPa), the latter after the knee region (ΔT
2
, from 200 to 220 MPa), the fatigue limit,
σ
0 RTM
, can be estimated from the intersection of the two regression lines. The results show an average fatigue limit value of
σ
0 RTM Plain
=193.2 ±4.4 MPa for the plain specimens (Fig. 12c-d).
The same testing procedure has been adopted for V-notched specimens at the same load frequency of 5 Hz and stress ratio (R =-1),
increasing the load step of 10 MPa every 10x10
3
cycles.
The temperature behaviour for the notched specimen is very similar to that of the plain specimens. The two stepwise tests for
notched specimens were reported in Fig. 13a-b. The Energy Parameter from the two tests is similar; however, it is one order of
magnitude lower than the plain specimen, even if the cross section is the same. The shorter fatigue life and the lower Energy Parameter
can be addressed to the presence of notch. Fig. 13c-d shows the result of the fatigue limit estimation by RTM. Even in this case there is a
knee region, and it is possible to make two linear regressions below (ΔT
1
, from 120 to 170 MPa) and above (ΔT
2
, from 180 to 210 MPa)
the knee. An average value of
σ
0 RTM Vnotch
=164.8 ±2 MPa turns out to be lower, as expected, due to the presence of the V-notch
which is known to affect the fatigue life of the component.
To validate the fatigue limit estimations by RTM, a series of constant amplitude stress-controlled fatigue tests were performed at
stress levels near the estimated fatigue limit, imposing the run-out conditions at 2x10
6
cycles, i.e. fatigue failure occurs for a number of
cycles after 2x10
6
.
For the plain specimens, run-outs have been reached at stress levels below 180 MPa (N ≥2.14x10
6
), which is in agreement with the
prediction of the STM and RTM (Table 5). For V-notched specimens, run-outs have been reached up to 140 MPa (N ≥2.99x10
6
) and
have shown failure at 160 MPa (N
f
=5.30x10
5
) and 180 MPa (N
f
=2.16x10
5
), conrming the prediction of the STM and RTM. It is to
point out that the required time to obtain information regarding the fatigue limit of AISI 304L is signicantly reduced adopting the
Thermographic Methods; indeed, just few minutes are necessary to retrieve the limit stress (≈5 min) with STM and few hours to
retrieve the fatigue limit with the RTM (≈8h). These values allow engineers to proceed with an early design of the mechanical
component against fatigue failure. On the other hand, the required time per test to obtain information on the fatigue limit throughout
Fig. 11. Temperature trend vs. applied stress level and time for V-notch AISI 304L specimens. Bilinear trend tted on raw temperature data. The
limit stress is evaluated as the knee of the two regression lines.
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constant amplitude fatigue tests and staircase procedure are very long and with a difcult evaluation of the real standard deviation
[90–92], which can lead to an excessive overestimation of the fatigue strength, hence an overestimation of the reliability of the
component [93].
From the stepwise fatigue tests performed on AISI 304L specimens it is possible to predict the fatigue life in terms of applied stress
(
σ
a
) vs. number of cycles exploiting the constancy of the Energy Parameter. Indeed, this parameter is almost constant for plain and V-
notch specimens, once that stress ratio and test frequency are xed. According to Eq. (3), by dividing the Energy Parameter for the i-th
stabilization temperature recorded during the stepwise fatigue test, it is possible to predict the number of cycles to failure for the
specimen. Having known the applied stress level, it is possible to obtain the S-N curve of the material (for plain specimens) or for the
component (notched specimen).
Fig. 14reports the predicted fatigue life for AISI 304L by RTM (blue markers) with the prediction of other researchers (red markers)
and CA fatigue test (full blue markers with arrow). The points predicted by RTM fall near other literature data; in particular, plain
specimens are consistent with the value of strain-controlled tests [94] and smooth specimen [85]. This last dataset shows runout at a
stress level equal to 190 MPa, which is consistent with the ndings of STM and RTM.
On the other hand, V-notch specimens show early failures compared to plain specimens and their trend follows the one of hole
specimen [85].
4.4. Summarize fatigue life with Strain Energy Density
The averaged Strain Energy Density method has been chosen in this study as design approach for the fatigue assessment of notched
components; hence, in the following, we report the comparison between the calibration of the method, for the material AISI 304L, both
starting from the experimental values evaluated through the Thermographic Methods and through a much more larger fatigue
database retrieved from data available in literature. The novelty of the present scientic article is the adoption of the Thermographic
Fig. 12. Stepwise fatigue tests on plain AISI 304L specimen: a) Test 01; b) Test 02. Fatigue limit estimation by RTM on AISI 304L plain specimen: c)
Test 01; d) Test 02.
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Methods to retrieve all the necessary parameters that allow the SED method calibration.
Two different ways have been adopted to calibrate the SED method through the fatigue data presented in this paper and retrieved
from literature. It is important to underline that such calibrations have been done only on fatigue tests presenting failures above 104
cycles; consequently, fatigue data in the low cycle fatigue regime are actually not considered, but have been just reported for com-
pletness.
The rst methodology, shown in Fig. 15a, has been proposed by Milone et al [95], and consists in evaluating the averaged SED
curve vs the number of cycles for different values of the stress ratio R; the calibrated value of the control volume radius R0 is the one
that minimize the scatter index of the fatigue curve in terms of SED, T
ΔW
. The fatigue dataset collected in the present paper resulted in a
control volume radius of R
0
=2.32 mm considering the fatigue data having failure between 10
4
and 2x10
6
cycles. The determined
fatigue scatter band is characterized by a stress-equivalent scatter index of TΔ
σ
eq =1.63 (TΔ
σ
eq =
TΔW
√) considering the fatigue curve
having a probability of survival (P.S.) of 2.3 % and 97.7 % and the fatigue data having failure between 10
4
and 2x10
6
cycles.
The second methodology consists in a procedure similar to the conventional one of the SED method that equates the averaged SED
of two different geometries at 2x10
6
cycles; however, in such a case, the limit stresses derived through the STM method for the plain
Fig. 13. Stepwise fatigue tests on V-notch AISI 304L specimen: a) Test 01; b) Test 02. Fatigue limit estimation by RTM on AISI 304L V-notched
specimen c) Test 01; d) Test 02.
Table 5
Fatigue limit estimation comparison between STM, RTM and constant amplitude fatigue tests.
Plain V-notch Avg Test duration
σ
lim
[MPa] 197.5 ±6.1 151.8 ±5.8 ~5 min
σ
0 RTM
[MPa] 193.2 ±4.4 164.8 ±2.0 ~8h
CA at run-out [MPa] ≤180 ≤140 ~111 h
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and V-notched geometries have been considered to impose the equivalence of the averaged SED value of the two different geometries
considered.
The calibrated value is equal to R0=2.28mm, as shown in Fig. 16a. The employment of these R0 value to summarise the fatigue
data in an averaged SED vs number of cycles data, shown in Fig. 16b, resulted in a fatigue scatter band very similar to the one obtained
through the other procedure; this is due to the fact that the R0 value, determined through the limit stress achieved applying the STM
method, is very close to the one achieved considering the procedure presented by Milone et al [95]. On the other hand, regardless of the
slightly different R0 value, the limited difference between the SED vs number of cycle curve determined according to the different
procedures could be also evident through Fig. 15a in which it can be noticed that the scatter index is not changing abruptly around
R0=2.32mm, the value minimizing the scatter index according to Milone et al [95] methodology.
It must be noted, on the other hand, that the variability in the limit stress determined by the STM method may result in an R0 value
ranging from about 1.9mm to 2.9mm depending on the smooth and V-notch tests considered. However, it is worth noting that, through
the STM method, the R0 calibration is possible with an already high accuracy considering only six static tests, three for each geometry,
while the other methodology available in literature would need at the least two fatigue curves, one for a notched geometry and one for
a smooth geometry, with almost thirty fatigue tests to ensure the statistical validity of the determined fatigue curves (15 specimens for
each condition) and, as a consequence, of the R0 value.
It is worth highlighting that the high value of the control volume radius achieved, compared to other materials [96], can be justied
by the high plasticity at the notch tip and reduced notch effect for AISI 304L, thus, the SED method has been applied in an uncon-
ventional way being the conditions for linear elastic fracture mechanics not accomplished; nevertheless, the achieved results,
considering also the numerosity of the experimental fatigue data considered, lead to satisfying results.
4.5. Notch effect in fatigue life
For ductile steels, the reversed yielding effect and possible presence of plasticity near the notch tip become more predominant at
higher stress and short fatigue life; indeed, it can happen that the ratio between the plain and notched nominal fatigue strengths
assumes value below K
f
, dened as the ratio between the fatigue limit of the plain specimen and the notched one. It is possible to dene
an effective fatigue notch factor K
f
* that varies according to the life of the specimens, due to the plasticity at higher stress levels. It is
dened as the ratio between the peak plastic stress at notch over the nominal stress (
σ
peak plastic
/
σ
nom
).
Numerical 3D FE simulation results of one-eighth of the V-notched specimen under tensile loading conditions have been reported in
Fig. 17. When the applied nominal stress for the V-notch specimen is below
σ
linear
/K
t
(A zone, Fig. 17), with
σ
linear
the proportional
stress value (i.e. last value of stress where the behaviour of the material is linear =280 MPa); no plasticity occurs at the notch tip and
the effective fatigue notch factor is equal to the theoretical stress concentration factor (K
f
*=K
t
, with K
t
=2.49); this behaviour can
happen at low stress levels.
By increasing the applied nominal stress, the plastic stress will also increase considering a strain hardening model, as well as the
plastic volume involved around the notch tip (B zone, Fig. 17.); as a result, the effective fatigue notch factor value gradually decreases
and it is possible to dene a region of local yielding around the notch tip.
Fig. 14. S-N curve for AISI 304L predicted by RTM and comparison with literature data.
D. Crisafulli et al.
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If the applied stress is over the proportional stress value of the material (C zone, Fig. 17.), the extreme phenomenon of full yielding
occurs, and the stress level become uniform over the entire net section of the specimen, equalling the nominal tension (K
f
*=1).
Nagaishi et al. [88] performed fatigue test campaign on circular notched hot-rolled AISI 304 steel and observed a dependency of the
fatigue limit from the stress ratio R; indeed, when R has negative value, the alternating tension–compression promotes the motion of
dislocation [97] resulting in martensitic transformation inside the notch volume leading to a rise in local hardening. The increase of
hardness at the notch tip can suppress fatigue crack initiation.
The presence of high plasticity at the notch tip for AISI 304L can be a justication for the SED high control volume radius obtained;
anyway, SED approach can be applied for the design of notched components.
Fig. 15. A) scatter index vs control volume radius curve with r0 evaluated as the one minimizing the scatter index; b) nominal averaged sed vs.
number of cycles to failure (notice that fatigue data below 104 cycles have been reported only for completeness, but they have been not considered
in SED method calibration although corrected though it in the present gure just for representation purposes).
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5. Conclusions
A rapid methodology has been proposed for the fatigue design of notched details. Indeed, it is not easy for companies to usually
perform a complete fatigue test campaign according to standard due to the lack of material and time. Moreover, for a proper design of
notches, it is necessary to evaluate the energy eld around the notch tip and adopt a design curve. The material under study was an
austenitic AISI 304L steel, where plain and V-notched specimens were retrieved.
Monotonic static tensile tests were performed to obtain the mechanical properties; meanwhile, the evolution of the specimen
temperature was monitored with an infrared camera to apply the Static Thermographic Method (STM). The stress level of transition
between the rst linear cooling phase, in accordance with thermoelastic law, and the second linear cooling phase, having different
slopes, was identied as the limit stress,
σ
lim
, where damage within the material arises. For plain material it was found a limit stress of
σ
lim
=197.5 ±6.1 MPa, while for V-notched specimen it was equal to
σ
lim V-notch
=151.8 ±5.8 MPa.
Fig. 16. A) R0 evaluation considering the limit stress determined through the STM method; b) SED data summary considering the R0 value
determined through the STM method (notice that fatigue data below 104 cycles have been reported only for completeness but they have been not
considered in SED method calibration although corrected though it in the present gure just for representation purposes).
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Engineering Fracture Mechanics 315 (2025) 110854
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To obtain the fatigue limit of the material and compare it with the limit stress from STM, a series of stress-controlled stepwise
fatigue tests were performed applying the Risitano’s Thermographic Method (RTM). A fatigue limit of
σ
0 RTM Plain
=193.2 ±4.4 MPa
was obtained for plain specimens, while
σ
0 RTM Vnotch
=164.8 ±2 MPa for V-notch specimens in well agreement with the estimated of
the STM; thus, both methods provide a rst indication for the design of the mechanical component. With the RTM it was also possible
to obtain, adopting few specimens, the S-N curve of the material.
The design of notched details can be performed adopting the Strain Energy Density (SED) method calibrated based on the values
obtained by the Thermographic Methods; such procedure has been validated by summarizing in a unique SED-based master curve
(ΔW-N) the fatigue data obtained by RTM and other fatigue data from literature.
The novelty of the proposed approach is the estimation of the control volume radius, R
0
, needed to apply the SED method by means
of the STM: a time saving procedure compared to the traditional estimation of this parameter via S-N curve. Even if the value of the
control volume radius is higher compared to other material, this can be addressed to the high ductility of the steel under study and the
presence of a large plastic area around the notch tip.
The proposed rapid methodology allows an easy design of mechanical components adopting few specimens and a short testing time.
CRediT authorship contribution statement
Davide Crisafulli: Writing – original draft, Software, Investigation, Data curation. Pietro Foti: Writing – original draft, Validation,
Methodology, Investigation, Formal analysis. Filippo Berto: Writing – review & editing, Validation, Supervision, Project adminis-
tration, Methodology. Giacomo Risitano: Writing – review & editing, Validation, Supervision, Conceptualization. Dario Santono-
cito: Writing – review & editing, Writing – original draft, Validation, Supervision, Investigation.
Declaration of competing interest
The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to
inuence the work reported in this paper.
Data availability
Data will be made available on request.
References
[1] S. Schweidler, M. Botros, F. Strauss, Q. Wang, Y. Ma, L. Velasco, G. Cadilha Marques, A. Sarkar, C. Kübel, H. Hahn, J. Aghassi-Hagmann, T. Brezesinski, B.
Breitung, High-entropy materials for energy and electronic applications, Nature Reviews Materials 2024 9:4 9 (2024) 266–281. 10.1038/s41578-024-00654-5.
Fig. 17. Change of the fatigue notch factor Kf* during cyclic yielding for V notched AISI 304L specimen.
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
21
[2] P. Kumar, S. Huang, D.H. Cook, K. Chen, U. Ramamurty, X. Tan, R.O. Ritchie, A strong fracture-resistant high-entropy alloy with nano-bridged honeycomb
microstructure intrinsically toughened by 3D-printing, Nature Communications 2024 15:1 15 (2024) 1–9. 10.1038/s41467-024-45178-2.
[3] Z. Qu, Z. Zhang, R. Liu, L. Xu, Y. Zhang, X. Li, Z. Zhao, Q. Duan, S. Wang, S. Li, Y. Ma, X. Shao, R. Yang, J. Eckert, R.O. Ritchie, Z. Zhang, High fatigue resistance
in a titanium alloy via near-void-free 3D printing, Nature 2024 626:8001 626 (2024) 999–1004. 10.1038/s41586-024-07048-1.
[4] X. Zhang, Y. Zhang, Q. Yu, X. Li, X. Wang, J. Wang, C. Jiang, Y. Jia, B. Sun, R. Ritchie, S.-T. Tu, Nacre-like surface nanolaminates induce superior fatigue
resistance in gradient pure titanium, (2024). 10.21203/RS.3.RS-3894539/V1.
[5] Equbal A, Equbal MA, Khan ZA, Badruddin IA. A review on the rapid liquid printing (RLP): future 3D printing technology. Prog Addit Manuf 2024:1–17. https://
doi.org/10.1007/S40964-024-00815-6/FIGURES/4.
[6] Le Cloarec J, Marinel S, Estourn`
es C, Dehmas M, Mani`
ere C. Ti–6Al–4V complex shape production by spark plasma sintering with ceramic–metal sacricial
powder and interface 3D printing. Prog Addit Manuf 2024:1–14. https://doi.org/10.1007/S40964-024-00803-W/FIGURES/10.
[7] Vanerio D, Guagliano M, Bagherifard S. Emerging trends in large format additive manufacturing processes and hybrid techniques. Prog Addit Manuf 2024;2024:
1–28. https://doi.org/10.1007/S40964-024-00771-1.
[8] Tabary SAAB, Fayazfar HR. Circular economy solutions for plastic waste: improving rheological properties of recycled polyethylene terephthalate (r-PET) for
direct 3D printing. Prog Addit Manuf 2024:1–10. https://doi.org/10.1007/S40964-024-00784-W/FIGURES/7.
[9] Chiu MC, Lin YH. Simulation based method considering design for additive manufacturing and supply chain An empirical study of lamp industry. Ind Manag
Data Syst 2016;116:322–48. https://doi.org/10.1108/IMDS-07-2015-0266/FULL/XML.
[10] Peron M, Saporiti N, Shoeibi M, Holmstr¨
om J, Salmi M. Additive manufacturing in the medical sector: from an empirical investigation of challenges and
opportunities toward the design of an ecosystem model. International Journal of Operations and Production Management ahead-of-print 2024. https://doi.org/
10.1108/IJOPM-12-2023-0948/FULL/XML.
[11] Liao D, Zhu SP, Correia JAFO, De Jesus AMP, Berto F. Recent advances on notch effects in metal fatigue: A review. Fatigue Fract Eng Mater Struct 2020;43:
637–59. https://doi.org/10.1111/FFE.13195.
[12] Qylafku G, Azari Z, Gjonaj M, Pluvinage G. On the fatigue failure and life prediction for notched specimens. Mater Sci 1998;34:604–18. https://doi.org/
10.1007/BF02355779/METRICS.
[13] Qu R, Zhang P, Zhang Z. Notch Effect of Materials: Strengthening or Weakening? J Mater Sci Technol 2014;30:599–608. https://doi.org/10.1016/J.
JMST.2014.04.014.
[14] Lazzarin P, Zambardi R. A nite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int J
Fract 2001;112:275–98. https://doi.org/10.1023/A:1013595930617.
[15] Lanning DB, Nicholas T, Palazotto A. HCF notch predictions based on weakest-link failure models. Int J Fatigue 2003;25:835–41. https://doi.org/10.1016/
S0142-1123(03)00156-7.
[16] Liu X, Wang R, Hu D, Mao J. A calibrated weakest-link model for probabilistic assessment of LCF life considering notch size effects. Int J Fatigue 2020;137:
105631. https://doi.org/10.1016/J.IJFATIGUE.2020.105631.
[17] Bomas H, Schleicher M. Application of the weakest-link concept to the endurance limit of notched and multiaxially loaded specimens of carburized steel
16MnCrS5. Fatigue Fract Eng Mater Struct 2005;28:983–95. https://doi.org/10.1111/J.1460-2695.2005.00935.X.
[18] Berto F, Lazzarin P, Wang CH. Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary
thickness. Int J Fract 2004;127:265–82. https://doi.org/10.1023/B:FRAC.0000036846.23180.4d.
[19] Kotousov A, Wang CH. Three-dimensional stress constraint in an elastic plate with a notch. Int J Solids Struct 2002;39:4311–26. https://doi.org/10.1016/
S0020-7683(02)00340-2.
[20] Zhu SP, Ai Y, Liao D, Correia JAFO, De Jesus AMP, Wang Q. Recent advances on size effect in metal fatigue under defects: a review. Int J Fract 2022;234:21–43.
https://doi.org/10.1007/S10704-021-00526-X/FIGURES/12.
[21] Schweiger G, Heckel K. Size effect in randomly loaded specimens. Int J Fatigue 1986;8:231–4. https://doi.org/10.1016/0142-1123(86)90026-5.
[22] Ai Y, Zhu SP, Liao D, Correia JAFO, Souto C, De Jesus AMP, et al. Probabilistic modeling of fatigue life distribution and size effect of components with random
defects. Int J Fatigue 2019;126:165–73. https://doi.org/10.1016/J.IJFATIGUE.2019.05.005.
[23] Makkonen M. Notch size effects in the fatigue limit of steel. Int J Fatigue 2003;25:17–26. https://doi.org/10.1016/S0142-1123(02)00053-1.
[24] Zhu SP, Ye WL, Correia JAFO, Jesus AMP, Wang Q. Stress gradient effect in metal fatigue: Review and solutions. Theor Appl Fract Mech 2022;121:103513.
https://doi.org/10.1016/J.TAFMEC.2022.103513.
[25] Yao W, Xia K, Gu Y. On the fatigue notch factor, Kf. Int J Fatigue 1995;17:245–51. https://doi.org/10.1016/0142-1123(95)93538-D.
[26] Qian GA, Wang MT, Wang L. High cycle fatigue life prediction by local stress-strain method. J Mech Strength 2004;26:275–7.
[27] Tao ZQ, Shang DG, Sun YJ. New pseudo stress correction method for estimating local strains at notch under multiaxial cyclic loading. Int J Fatigue 2017;103:
280–93. https://doi.org/10.1016/J.IJFATIGUE.2017.06.010.
[28] Neuber H. Theory of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary Nonlinear Stress-Strain Law. J Appl Mech 1961;28:544–50.
https://doi.org/10.1115/1.3641780.
[29] Ye D, Matsuoka S, Suzuki N, Maeda Y. Further investigation of Neuber’s rule and the equivalent strain energy density (ESED) method. Int J Fatigue 2004;26:
447–55. https://doi.org/10.1016/J.IJFATIGUE.2003.10.002.
[30] Lim JY, Hong SG, Lee SB. Application of local stress–strain approaches in the prediction of fatigue crack initiation life for cyclically non-stabilized and non-
Masing steel. Int J Fatigue 2005;27:1653–60. https://doi.org/10.1016/J.IJFATIGUE.2005.07.014.
[31] Taylor D. The theory of critical distances. Eng Fract Mech 2008;75:1696–705. https://doi.org/10.1016/J.ENGFRACMECH.2007.04.007.
[32] Carpinteri A, Spagnoli A, Vantadori S. Multiaxial fatigue life estimation in welded joints using the critical plane approach. Int J Fatigue 2009;31:188–96.
https://doi.org/10.1016/J.IJFATIGUE.2008.03.024.
[33] Carpinteri A, Boaretto J, Fortese G, Giordani F, Rodrigues RI, Iturrioz I, et al. Welded joints under multiaxial non-proportional loading. Theor Appl Fract Mech
2018;93:202–10. https://doi.org/10.1016/J.TAFMEC.2017.08.004.
[34] Luo P, Yao W, Li P. A notch critical plane approach of multiaxial fatigue life prediction for metallic notched specimens. Fatigue Fract Eng Mater Struct 2019;42:
854–70. https://doi.org/10.1111/FFE.12956.
[35] Livieri P, Lazzarin P. Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int J Fract
2005;133:247–76. https://doi.org/10.1007/s10704-005-4043-3.
[36] Radaj D, Vormwald M. Advanced methods of fatigue assessment. Springer; 2013.
[37] Fischer C, Fricke W, Rizzo CM. Experiences and recommendations for numerical analyses of notch stress intensity factor and averaged strain energy density. Eng
Fract Mech 2016;165:98–113. https://doi.org/10.1016/j.engfracmech.2016.08.012.
[38] Lazzarin P, Zambardi R. The equivalent strain energy density approach re-formulated and applied to sharp V-shaped notches under localized and generalized
plasticity. Fatigue Fract Eng Mater Struct 2002;25:917–28. https://doi.org/10.1046/j.1460-2695.2002.00543.x.
[39] Meneghetti G, Campagnolo A, Babini V, Riboli M, Spagnoli A. Multiaxial fatigue assessment of welded steel details according to the peak stress method:
Industrial case studies. Int J Fatigue 2019;125:362–80. https://doi.org/10.1016/j.ijfatigue.2019.04.014.
[40] Foti P, Berto F. Some useful expressions and a proof of the validity of the volume free procedure for the SED method application. Eng Fract Mech 2022;274.
https://doi.org/10.1016/j.engfracmech.2022.108818.
[41] Pontesso A, Zappalorto M, Berto F. Energy-based evaluation of the stress concentration factor with highly non-regular coarse meshes: Theoretical formulation
and practical validation. Fatigue Fract Eng Mater Struct 2023;46:1388–400. https://doi.org/10.1111/FFE.13933.
[42] Campagnolo A, Zuin S, Meneghetti G. Averaged strain energy density estimated rapidly from nodal displacements by coarse FE analyses: Cracks under mixed
mode loadings. Fatigue Fract Eng Mater Struct 2020;43:1658–85. https://doi.org/10.1111/ffe.13187.
[43] Lazzarin P, Berto F, Zappalorto M. Rapid calculations of notch stress intensity factors based on averaged strain energy density from coarse meshes: Theoretical
bases and applications. Int J Fatigue 2010;32:1559–67. https://doi.org/10.1016/j.ijfatigue.2010.02.017.
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
22
[44] Curti G, La Rosa G, Orlando M, Risitano A. Analisi tramite infrarosso termico della temperatura limite in prove di fatica. Proceedings XIV Convegno Nazionale
AIAS 1986:211–20.
[45] Iziumova A, Vshivkov A, Prokhorov A, Gachegova E, Davydov D. Heat dissipation and fatigue crack kinetic features of titanium alloy Grade 2 after laser shock
peening. Frattura Ed Integrit`
a Strutturale 2022;16:516–26. https://doi.org/10.3221/IGF-ESIS.62.35.
[46] A. Geraci G. La Rosa A. Risitano L’infrarosso termico nelle applicazioni meccaniche CRES Symposium 1984 Catania Italy 8 9.
[47] Pitarresi G, Patterson EA. A review of the general theory of thermoelastic stress analysis. J Strain Anal Engng Des 2003;38:405–17. https://doi.org/10.1243/
03093240360713469.
[48] Shiozawa D, Inagawa T, Washio T, Sakagami T. Fatigue limit estimation of stainless steels with new dissipated energy data analysis. Procedia Struct Integrity
2016;2:2091–6. https://doi.org/10.1016/J.PROSTR.2016.06.262.
[49] Cappello R, Meneghetti G, Ricotta M, Pitarresi G. On the correlation of temperature harmonic content with energy dissipation in C45 steel samples under fatigue
loading. Mech Mater 2022;168:104271. https://doi.org/10.1016/J.MECHMAT.2022.104271.
[50] Meneghetti G. Analysis of the fatigue strength of a stainless steel based on the energy dissipation. Int J Fatigue 2007;29:81–94. https://doi.org/10.1016/J.
IJFATIGUE.2006.02.043.
[51] A. Risitano, G. Risitano, C. Clienti, Fatigue limit by thermal analysis of specimen surface in mono axial traction test, in: EPJ Web Conf, EDP Sciences, 2010.
10.1051/epjconf/20100638010.
[52] Risitano A, Risitano G. Determining fatigue limits with thermal analysis of static traction tests. Fatigue Fract Eng Mater Struct 2013;36:631–9. https://doi.org/
10.1111/ffe.12030.
[53] La Rosa G, Risitano A. Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components. Int J Fatigue 2000;22:
65–73. www.elsevier.com/locate/ijfatigue.
[54] G. Fargione, A. Geraci, G. La Rosa, A. Risitano, Rapid determination of the fatigue curve by the thermographic method, J. Basic Eng. Trans. ASME, Ser. D 24
(2002) 37. www.elsevier.com/locate/ijfatigue.
[55] Corigliano P, Cucinotta F, Guglielmino E, Risitano G, Santonocito D. Fatigue assessment of a marine structural steel and comparison with Thermographic
Method and Static Thermographic Method. Fatigue Fract Eng Mater Struct 2020;43:734–43. https://doi.org/10.1111/ffe.13158.
[56] Crisafulli D, Fintov´
a S, Santonocito D, D’Andrea D. Microstructural characterization and mechanical behaviour of laser powder Bed Fusion stainless steel 316L.
Theor Appl Fract Mech 2024;131. https://doi.org/10.1016/j.tafmec.2024.104343.
[57] D. Santonocito, Evaluation of fatigue properties of 3D-printed Polyamide-12 by means of energy approach during tensile tests, in: Procedia Structural Integrity,
Elsevier B.V., 2020: pp. 355–363. 10.1016/j.prostr.2020.04.040.
[58] Tawancy HM. Failure Analysis of 304 Stainless Steel Components Used in Petrochemical Industry Applications. Metall Microstruct Anal 2019;8:705–12. https://
doi.org/10.1007/s13632-019-00578-5.
[59] Rossi S, Leso SM, Calovi M. Study of the Corrosion Behavior of Stainless Steel in Food Industry. Materials 2024;17. https://doi.org/10.3390/ma17071617.
[60] Boujelbene M, Kharrat M, Dammak M, Maalej A. Tribological behaviour of the 304L/HDPE contact: application for the knee prostheses. Int J Exp Comput
Biomech 2009;1:214–24. https://doi.org/10.1504/IJECB.2009.0292.
[61] Manwatkar SK, Kuhite KS, Narayana Murty SVS, Ramesh Narayanan P. Metallurgical Analysis of Failed AISI 304L Stainless Steel Tubes Used in Launch Vehicle
Applications. Metallography, Microstructure, and Analysis 4 2015:497–507. https://doi.org/10.1007/s13632-015-0247-8.
[62] Liu M, de Oliveira Miranda AC, Avelar Antunes M, Meggiolaro MA, Pinho de Castro JT. Plastic stress concentration effects in fatigue strength. Int J Fatigue 168
2023:107394. https://doi.org/10.1016/J.IJFATIGUE.2022.107394.
[63] D’Andrea D, Gatto A, Guglielmino E, Risitano G, Santonocito D. A comparison on static and fatigue behaviour between traditional and SLM AISI 316L. J Mech
Sci Technol 2023;37. https://doi.org/10.1007/s12206-022-2103-5.
[64] Giuseppe. Caglioti, A. Ferro Milone, Societ`
a italiana di sica., Mechanical and thermal behaviour of metallic materials : Varenna on Lake Como, Villa
Monastero, 30th June - 10th July 1981, North-Holland Pub. Co, 1982.
[65] A.D. Melvin, A.C. Lucia, G.P. Solomos, G. Volta, D. Emmony, Thermal emission measurements from creep damaged specimens of AISI 316L and Alloy 800H,
Proc. 9th Int. Conf. on Experimental Mechanics Volume 2, Pages 765-773 2 (1990) 765–773.
[66] Melvin AD, Lucia AC, Solomos GP. The thermal response to deformation to fracture of a carbon/epoxy composite laminate. Compos Sci Technol 1993;46:
345–51. https://doi.org/10.1016/0266-3538(93)90180-O.
[67] Geraci A, La Rosa G, Risitano A. Correlation between thermal variation in static test and elastic limit of material using infrared imagery. in: 7th International
Conference on Mechanical Behaviour of Material, The Hague May. 1995.
[68] Chrysochoos A, Louche H. Infrared image processing to analyze the caloric effects accompanying strain localization. Int J Eng Sci 2000;38:1759–88. https://
doi.org/10.1016/S0020-7225(00)00002-1.
[69] Plekhov OA, Naimark OB. Theoretical and experimental study of energy dissipation in the course of strain localization in iron. J Appl Mech Tech Phys 2009;50:
127–36. https://doi.org/10.1007/s10808-009-0018-x.
[70] Clienti C, Fargione G, La Rosa G, Risitano A, Risitano G. A rst approach to the analysis of fatigue parameters by thermal variations in static tests on plastics. Eng
Fract Mech 2010;77:2158–67. https://doi.org/10.1016/j.engfracmech.2010.04.028.
[71] A. Risitano, D. Corallo, G. Risitano, A. Sirugo, Determinazione del limite di fatica mediante prove quasi-statiche, Proceedings XXXIX Convegno Nazionale AIAS
(2010).
[72] Brown N. Observations of microplasticity 1968.
[73] Behrens BA, Chugreev A, Bohne F, Lorenz R. Approach for modelling the Taylor-Quinney coefcient of high strength steels. Procedia Manuf 2019;29:464–71.
https://doi.org/10.1016/j.promfg.2019.02.163.
[74] Rittel D, Zhang LH, Osovski S. The dependence of the Taylor–Quinney coefcient on the dynamic loading mode. J Mech Phys Solids 2017;107:96–114. https://
doi.org/10.1016/j.jmps.2017.06.016.
[75] Berto F, Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches. Mater Sci Engng
R Rep 2014;75:1–48. https://doi.org/10.1016/j.mser.2013.11.001.
[76] Seweryn A. Brittle fracture criterion for structures with sharp notches. Eng Fract Mech 1994;47:673–81. https://doi.org/10.1016/0013-7944(94)90158-9.
[77] Halford GR. The energy required for fatigue(Plastic strain hystersis energy required for fatigue in ferrous and nonferrous metals). J Mater 1966;1:3–18.
[78] W. Macek, R. Branco, J. de Jesus, J.D. Costa, S.P. Zhu, R. Masoudi Nejad, A. Grygu´
c, Strain energy density and entire fracture surface parameters relationship for
LCF life prediction of additively manufactured 18Ni300 steel, 10.1177/10567895241245879 (2024). 10.1177/10567895241245879.
[79] Li DM, Nam WJ, Lee CS. A strain energy-based approach to the low-cycle fatigue damage mechanism in a high-strength spring steel. Metall Mater Trans A Phys
Metall Mater Sci 1998;29:1431–9. https://doi.org/10.1007/S11661-998-0358-9/METRICS.
[80] Lachowicz CT. Calculation of the elastic–plastic strain energy density under cyclic and random loading. Int J Fatigue 2001;23:643–52. https://doi.org/10.1016/
S0142-1123(00)00102-X.
[81] Golos K, Ellyin F. Total Strain Energy Density as a Fatigue Damage Parameter. Advances in Fatigue Science and Technology 1989:849–58. https://doi.org/
10.1007/978-94-009-2277-8_42.
[82] Foti P, Filippi S, Berto F. Fatigue assessment of welded joints by means of the strain energy density method: Numerical predictions and comparison with
eurocode 3. Frattura Ed Integrita Strutturale 2019;13. https://doi.org/10.3221/IGF-ESIS.47.09.
[83] Lazzarin P, Sonsino CM, Zambardi R. A notch stress intensity approach to assess the multiaxial fatigue strength of welded tube-to-ange joints subjected to
combined loadings. Fatigue Fract Eng Mater Struct 2004;27:127–40. https://doi.org/10.1111/J.1460-2695.2004.00733.X.
[84] Meneghetti G, Ricotta M, Atzori B. A synthesis of the push-pull fatigue behaviour of plain and notched stainless steel specimens by using the specic heat loss.
Fatigue Fract Eng Mater Struct 2013;36:1306–22. https://doi.org/10.1111/ffe.12071.
[85] Meneghetti G, Ricotta M. The use of the specic heat loss to analyse the low- and high-cycle fatigue behaviour of plain and notched specimens made of a
stainless steel. Eng Fract Mech 2012;81:2–16. https://doi.org/10.1016/j.engfracmech.2011.06.010.
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
23
[86] Klus´
ak J, Koz´
akov´
a K, Fintov´
a S, Seitl S. Fatigue lifetimes of 1.4306 and 1.4307 stainless steels subjected to ultrasonic loading, in. Procedia Structural Integrity,
Elsevier BV 2022:1369–75. https://doi.org/10.1016/j.prostr.2022.12.174.
[87] A. Cicha´
nski, A. Cicha´
nski, The inuence of mesh morphology on the SCF in 2D FEM analysis of at bars with opposite V-notch under tension, 2016. https://
www.researchgate.net/publication/308612142.
[88] Nagaishi N, Ogawa Y, Okazaki S, Matsunaga H. Fatigue limit of circumferentially notched metastable austenitic stainless steel under positive/negative stress
ratios: Synergistic effects of notch-root mechanistic singularity and phase transformation. Int J Fatigue 2022;165. https://doi.org/10.1016/j.
ijfatigue.2022.107229.
[89] Foti P, Razavi SMJ, Ayatollahi MR, Marsavina L, Berto F. On the application of the volume free strain energy density method to blunt V-notches under mixed
mode condition. Eng Struct 2021;230:111716. https://doi.org/10.1016/j.engstruct.2020.111716.
[90] Wallin KRW. Statistical uncertainty in the fatigue threshold staircase test method 2010. https://doi.org/10.1016/j.ijfatigue.2010.09.013.
[91] R. Pollak, A. Palazotto, T. Nicholas, A simulation-based investigation of the staircase method for fatigue strength testing, (n.d.). 10.1016/j.
mechmat.2005.12.005.
[92] Zhao YX, Zhang Y, He HW. Improved measurement on probabilistic fatigue limits/strengths by test data from staircase test method 2016. https://doi.org/
10.1016/j.ijfatigue.2016.09.010.
[93] Thompson T, Liu J, Hu C. A comparative analysis of step loading and staircase testing for fatigue strength estimation of an engine component. Fatigue Fract Eng
Mater Struct 2023;46:667–81. https://doi.org/10.1111/FFE.13898.
[94] Meneghetti G, Ricotta M, Atzori B. Experimental evaluation of fatigue damage in two-stage loading tests based on the energy dissipation, Proc Inst Mech Eng C J
Mech. Eng Sci 2015;229:1280–91. https://doi.org/10.1177/0954406214559112.
[95] Milone A, Foti P, Filippi S, Landolfo R, Berto F. Evaluation of the inuence of mean stress on the fatigue behavior of notched and smooth medium carbon steel
components through an energetic local approach. Fatigue Fract Eng Mater Struct 2023;46:4315–32. https://doi.org/10.1111/ffe.14139.
[96] Radaj D. State-of-the-art review on the local strain energy density concept and its relation to the J-integral and peak stress method. Fatigue Fract Eng Mater
Struct 2015;38:2–28. https://doi.org/10.1111/ffe.12231.
[97] Mayama T, Sasaki K, Kuroda M. Quantitative evaluations for strain amplitude dependent organization of dislocation structures due to cyclic plasticity in
austenitic stainless steel 316L. Acta Mater 2008;56:2735–43. https://doi.org/10.1016/j.actamat.2008.02.005.
D. Crisafulli et al.
Engineering Fracture Mechanics 315 (2025) 110854
24
