Review Article
A Review on Fatigue Life Prediction Methods for Metals
E. Santecchia,
1
A. M. S. Hamouda,
1
F. Musharavati,
1
E. Zalnezhad,
2
M. Cabibbo,
3
M. El Mehtedi,
3
and S. Spigarelli
3
1
Mechanical and Industrial Engineering Department, College of Engineering, Qatar University, Doha 2713, Qatar
2
Department of Mechanical Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea
3
Dipartimento di Ingegneria Industriale e Scienze Matematiche (DIISM), Universit
`
a Politecnica delle Mar che, 60131 Ancona, Italy
Correspondence should be addressed to E. Zalnezhad; erfan
zalnezhad@yahoo.com
Received June ; Accepted August
Academic Editor: Philip Eisenlohr
Copyright © E. Santecchia et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Metallic materials are extensively used in engineering structures and fatigue failure is one of the most common failure modes of
metal structures. Fatigue p henomena occur when a material is subjected to uctuating str esses and strains, which lead to failure
due to damage accumulation. Dierent methods, including the Palmgren-Miner linear damage rule- (LDR-) based, multiaxial
and variable amplitude loading, stochastic-based, energy-based, and continuum damage mechanics methods, forecast fatigue
life. is paper reviews fatigue life prediction techniques for metallic mater ials. An ideal fatigue life prediction model should
include the main features of those already established methods, and its implementation in simulation systems could help engineers
and scientists in dierent applications. In conclusion, LDR-based, multiaxial and variable amplitude loading, stochastic-based,
continuum damage mechanics, and energy-based methods are easy, realistic, microstructure dependent, well timed, and damage
connected, respectively, for the ideal prediction model.
1. Introduction
Avoiding or rather delaying the failure of any component
subjected to cyclic loadings is a crucial issue that must be
addressed during preliminary design. In order to have a full
picture of the situation, further attention must be given also
to processing parameters, given the strong inuence that
they have on the microstructure of the cast materials and,
therefore, on their properties.
Fatigue damage is among the major issues in engineering,
because it incre ases with the number of applied loading cycles
in a cumulative manner, and can lead to fracture and failure
of the considered part. erefore, the prediction of fatigue
life has an outstanding importance that must be considered
during the design step of a mechanical component [].
e fatigue life prediction methods can be divided into
two main groups, according to the p articular approach used.
erstgroupismadeupofmodelsbasedontheprediction
of crack nucleation, using a combination of damage evolution
rule and criteria based on stress/strain of components. e
key point of this approach is the lack of dependence from
loading and specimen geometry, being the fatigue life deter-
mined only by a stress/strain criterion [].
eapproachofthesecondgroupisbasedinsteadon
continuum damage mechanics (CDM), in which fatigue life is
predicted computing a damage parameter cycle by cycle [].
Generally, the life prediction of elements subjected to
fatigue is based on the “safe-life” approach [], coupled
with the rules of linear cumulative damage (Palmgren []
and Miner []). Indeed, the so-called Palmgren-Miner linear
damage r ule (LDR) is widely applied owing to its intrinsic
simplicity, but it also has some major drawbacks that need to
be considered []. Moreover, some metallic materials exhibit
highly nonlinear fatigue damage evolution, which is load
dependent and is totally neglected by the linear damage rule
[]. e major assumption of the Miner rule is to consider the
fatigue limit as a material constant, while a number of studies
showed its load amplitude-sequence dependence [–].
Various other theories and models have been developed
in order to predict the fatigue life of loaded struc tures [–].
Hindawi Publishing Corporation
Advances in Materials Science and Engineering
Volume 2016, Article ID 9573524, 26 pages
http://dx.doi.org/10.1155/2016/9573524
Advances in Materials Science and Engineering
Among all the available techniques, periodic in situ mea-
surements have been proposed, in order to calculate the
macrocrack initiation probability [].
e limitations of fracture mechanics motivated the
development of local approaches based on continuum dam-
age mechanics (CDM) for micromechanics models []. e
advantages of CDM lie in the eects that the presence of
microstructural defects (voids, discontinuities, and inhomo-
geneities) has on key quantities that can be observed and
measured at the macroscopic level (i.e., Poisson’s ratio and
stiness). From a life prediction p oint of view, CDM is partic-
ularly useful in order to model the accumulation of damage
in a material prior to the formation of a detectable defect (e.g.,
acrack)[].eCDMapproachhasbeenfurtherdeveloped
by Lemaitre [, ]. Later on, the thermodynamics of
irreversible process provided the necessary scientic basis
to justify CDM as a theory [] and, in the framework of
internal variable theory of thermodynamics, Chandrakanth
and Pandey [] developed an isotropic ductile plastic damage
model. De Jesus et al. [] formulated a fa tigue model
involving a CDM approach based on an explicit denition
of fatigue damage, while Xiao et al. [] predicted high-cycle
fatigue life implementing a thermodynamics-based CDM
model.
Bhattacharya and Ellingwood [] predicted the crack
initiation life for strain-controlled fatigue loading, using a
thermodynamics-based CDM model where the equations of
damage growth were expressed in terms of the Helmholtz free
energy.
Basedonthecharacteristicsofthefatiguedamage,some
nonlinear damage cumulative theories, continuum damage
mechanics approaches, and energy-based damage methods
have been proposed and developed [–, –, , ].
Given the strict connection between the hysteresis energy
and the fatigue behavior of materials, expressed rstly by
Inglis [], energy methods were developed for fatigue life
prediction using strain energy (plastic energy, elastic energy,
or the summation of both) as the key damage parameter,
accounting for load sequence and cumulative damage [, –
].
Lately, using statistical methods, Makkonen [] pro-
posed a new way to build design c urves, in order to study
the crack initiation and to get a fatigue life estimation for any
material.
A very interesting fatigue life prediction approach based
onfracturemechanicsmethodshasbeenproposedbyGhi-
dini and Dalle Donne []. In this work they demonstrated
that, using widespread aerospace fracture mechanics-based
packages, it is possible to get a good prediction on the fatigue
life of pristine, precorroded base, and friction stir welded
specimens, even under variable amplitude loads and residual
stresses conditions [].
In the present review paper , various prediction methods
developed so far are discussed. Particular emphasis will be
given to the prediction of the crack initiation and growth
stages, having a key role in the overall fatigue life predic-
tion. e theories of damage accumulation and continuum
damage mechanics are explained and the prediction methods
based on t hese two approaches are discussed in detail.
2. Prediction Methods
According to Makkonen [], the total fatigue life of a
component can be divided into three phases: (i) crack
initiation, (ii) stable crack growth, and (iii) unstable crack
growth. Crack initiation accounts for approximately –%
of the total fatigue life, being the phase with the longest
time duration []. Crack initiation may stop at barriers (e.g.,
grain boundaries) for a long time; sometimes the cracks stop
completely at this level and they never reach the critical size
leading to the stable growth.
A power law formulated by Paris and Erdogan [] is
commonly used to model the stable fatigue crack growth:
=⋅
𝑚
,
()
and the fatigue life is obtained from the following integra-
tion:
=
𝑎
𝑓
𝑎
𝑖
⋅
(
)
𝑚
,
()
where is the stress intensity factor range, while and
are material-related constants. e integration limits
𝑖
and
𝑓
correspond to the initial and nal fatigue crack lengths.
According to the elastic-plastic fracture mechanics
(EPFM) [–], t he crack propagation theory can be
expressed as
=
𝑚
,
()
where is the integral range corresponding to (), while
and
are constants.
A generalization of the Paris law has been recently
proposed by Pugno et al. [], where an instantaneous crack
propagation rate is obtained by an interpolating procedure,
which works on the integrated form of the crack propagation
law (in terms of - curve), in the imposed condition of
consistency with W
¨
ohler’s law [] for uncracked material
[, ].
2.1. Prediction of Fatigue Crack In itiation. Fatigue cracks have
been a matter of research for a long time []. Hachim et
al. [] addressed the maintenance planning issue for a steel
S structure, predicting the number of priming cycles of
a fatigue crack. e probabilistic analysis of failure showed
that the priming stage, or rather the crack initiation stage,
accounts for more than % of piece life. Moreover, the results
showedthatthepropagationphasecouldbeneglectedwhen
a large number of testing cycles are performed [].
Tanaka and Mura [] were pioneers concerning the
study of fatigue crack initiation in ductile materials, using
the concept of slip plastic ow. e crack starts to form
when the surface energy and the stored energy (given by the
dislocations accumulations) become equal, thus turning the
dislocation dipoles layers into a free surface [, ].
In an additional paper [], the same authors modeled
the fatigue crack initiation by classifying cracks at rst as
Advances in Materials Science and Engineering
(i) crack initiation from inclusions (type A), (ii) inclusion
cracking by impinging slip bands (typ e B), and (iii) slip band
crack emanating from an uncracked inclusion (type C). e
type A crack initiation from a completely debonded inclusion
was treated like crack initiation from a void (notch). e
initiation of typ e B cracks at the matrix-particle interface is
due to the impingement of slip bands on the particles, but
only on those having a smaller size compared to the slip band
width. is eect inhibits the dislocations movements. e
fatigue crack is generated when the dislocation dipoles get to
a level of self-strain energy corresponding to a critical value.
For crack initiation along a slip band, the dislocation dipole
accumulation can be described as follows:
(
−2
)
1/2
1
=
8
𝑠
1/2
,
()
where
𝑠
is the specic fracture energy per unit area along the
slip band, is the friction stress of dislocation, is the shear
modulus, is the shear stress range, and is the grain size
[]. Type C was approximated by the problem of dislocation
pile-up under the stress distribution in a homogeneous,
innite plane. Type A mechanism was reported in high
strength steels, while the other two were observed in high
strength aluminum alloys. e quantitative relations derived
by Tanaka and Mura [] correlated t he properties of matrices
and inclusions, as well as the size of the latter, with the fatigue
strength decrease at a given crack initiation life and with the
reduction of the crack initiation life at a given constant range
of the applied stress [].
Dang-Van [] also considered the local plastic ow as
essential for the crack initiation, and he attempted to give a
new approach in order to quantify the fatigue crack initiation
[].
Mura and Nakasone [] expanded Dang-Van’s work
tocalculatetheGibbsfreeenergychangeforfatiguecrack
nucleation from piled up dislocation dipoles.
Assuming that only a f raction of all the dislocations in
theslipbandcontributestothecrackinitiation,Chan[]
proposed a further evolution of this theory.
Considering the criterion of minimum strain energy
accumulation within slip bands, Venkataraman et al. [–
] generalized the dislocation dipole model and developed
a stress-initiation life relation predicting a grain-size depen-
dence, which was in contrast with the Tanaka and Mura
theory [, ]:
(
−2
)
𝛼
𝑖
=0.37
𝑠
1/2
,
()
where
𝑠
is the surface-energy term and is the slip-
irreversibility factor (0<<1). is highlighted the need
to incorporate key parameters like crack and microstructural
sizes, to get more accurate microstructure-based fatigue crack
initiation models [].
Other microstructure-based fatigue crack growth models
were developed and veried by Chan and coworkers [, –
].
Notch crack
Short cracks
Fatigue
limit
Microstructurally
short cracks
Physically small
cracks
EPFM
type
cracks
log crack length
d
1
d
2
d
3
log stress range Δ
Zero
Crack speed
LEFM
type
cracks
Nonpropagating cracks
F : A modied Kitagawa-Takahashi -diagram, showing
boundaries between MSCs and PSCs and between EPFM cracks and
LEFM cracks [, ].
Concerning the metal fatigue, aer the investigation of
very-short cracks behavior, Miller and coworkers proposed
the immediate crack initiation model [–]. e early two
phases of the crack follow the elastoplastic fracture mechanics
(EPFM) and were renamed as (i) microstructurally short
crack (MSC) growth and (ii) physically small crack (PSC)
growth. Figure shows the modied Kitagawa-Takahashi
diagram, highlighting the phase boundaries between MSC
and PSC [, ].
e crack dimension has been identied as a crucial
factor by a number of a uthors, because short fatigue cracks
(havingasmalllengthcomparedtothescaleoflocalplasticity,
or to the key microstructural dimension, or simply smaller
than - mm) in metals grow at faster rate and lower nominal
stress com pared to large cracks [, ].
2.1.1. A coustic Second H a rmonic Genera tion. Kulkarni et
al.[]proposedaprobabilisticmethodtopredictthe
macrocrack initiation due to fatigue damage. Using acoustic
nonlinearity, the damage prior to macrocrack initiation was
quantied, and the data collected were then used to perform
a probabilistic analysis. e probabilistic fatigue damage
analysis results from the combination of a suitable damage
evolution equation and a procedure to calculate the proba-
bility of a macrocrack initiation, the Monte Carlo method
in this particular case. Indeed, when transmitting a single
frequency wave through the specimen, the distortion given
by the material nonlinearity generates second higher level
harmonics, having amplitudes increasing with the material
nonlinearity. As a result, both the accumulated damage
and material nonlinearity can be characterized by the ratio
2
/
1
,where
2
is the amplitude of the second harmonic
and
1
is the amplitude of the fundamental one. e ratio
is expected to increase with the progress of the damage
accumulation. It is important to point out that this
2
/
1
acoustic nonlinearity characterization [] diers from the
approach given by Morris et al. [].
In the work of Ogi et al. [] two dierent signals were
transmitted separately into a specimen, one a t resonance
frequency
𝑟
andtheotherathalfofthisfrequency(
𝑟
/2).
Advances in Materials Science and Engineering
Macrocrack initiation
5
4
3
2
1
0
01
A
2
/A
1
(10
−3
)
N/N
f
F : T ypical evolution of the ratio
2
(
𝑟
/2)/
1
(
𝑟
)for .
mass% C steel,
𝑓
=56,000[].
e transmission of a signal at
𝑟
frequency generates a mea-
sured amplitude
1
(
𝑟
), and while the signal is transmitted
at frequency
𝑟
/2,theamplitude
2
(
𝑟
/2)was received. e
measurement of both signals ensures the higher accuracy of
this method []. Figure shows that the
2
(
𝑟
/2)/
1
(
𝑟
)
ratio increases nearly monotonically, and at the point of the
macrocrack initiation a distinct peak can be observed. is
result suggests that the state of damage in a specimen during
fatigue tests can be tracked by measuring the ratio
2
/
1
.
According to the model of Ogi et al. [], Kulkarni et al.
[] showed that the scalar damage function can be written as
(),designatingthedamagestateinasampleataparticular
fatigue cycle. e value =0corresponds to the no-damage
situation, while =1denotes the appearance of the rst
macrocrack.edamageevolutionwiththenumberofcycles
is given by the following equation:
=
1
𝑐
/2−
𝑐
(
)
/2
𝑚
1
(
1−
)
𝑛
. ()
When /2 is higher than the endurance limit (
𝑐
(
)),
otherwise the rate /equals zero.
2.1.2. Probability of Crack Initiation on Defects. Melander
and Larsson [ ] used the Poisson statistics to calculate the
probability
𝑥
of a fatigue life smaller t han cycles. Excluding
the probability of fatigue crack nucleation at inclusions,
𝑥
canbewrittenas
𝑥
=1−exp −
𝑥
,
()
where
𝑥
is the number of inclusions per unit volume.
erefore, () shows the probability to nd at least one
inclusion in a unity volume that would lead to fatigue life not
higher than cycles.
In order to calculate the probability of fatigue failure ,de
Bussac and Lautridou [] used a similar approach. In their
model, given a defect of size located in a volume adjacent
to the surface, the probability of fatigue crack initiation was
assumedtobeequaltothatofencounteringadiscontinuity
with the same dimension:
=1−exp −
𝜐
,
()
with
𝜐
as the number of defects per unit volume having
diameter . As in the model developed by Melander and
Larsson [], also in this case an equal crack initiation power
for dierent defects having the same size is assumed [, ].
In order to account for the fact that the fatigue crack
initiation can occur at the surface and inside a materi al, de
Bussac [] dened the probability to nd discontinuities of
a given size at the surface or at the subsurface. Given a number
of load cycles
0
,thesurvivalprobabilityis determined
as the product of the survival probabilities of surface and
subsurface failures:
=1−
𝑠
𝑠
1−
𝜐
𝜐
,
()
where
𝑠
and
𝜐
are the diameters of the discontinuities
in surface and subsurface leading to
0
loads life.
𝑠
(
𝑠
)
and
𝜐
(
𝜐
) are the probabilit ies of nding a defect larger
than
𝑠
and
𝜐
at surface and subsurface, respectively. It
must be stressed that this model does not rely on the type o f
discontinuity but only on its size [].
Manonukul and Dunne [] studied the fatigue crack
initiation in polycrystalline metals addressing the peculiar-
ities of high-cycle fatigue (HCF) and low-cycle fatigue (LCF).
e proposed approach for the prediction of fatigue cracks
initiation is based on the critical accumulated slip property
of a material; the key idea is that when the critical slip is
achieved within the microstructure, crack initiation should
have occurred. e authors develo ped a nite-element model
for the nickel-based alloy C where, using crystal plasticity,
a representative region of the material (containing about
grains) was modeled, taking into account only two materials
properties: (i) grain morphology and (ii) crystallographic
orientation.
e inuence on the fatigue life due to the initial c on-
ditions of the specimens was studied deeply by Makkonen
[, ], who addressed in particular the size of the specimens
and the notch size eects, both in the case of steel as testing
material.
e probability of crack initiation and propagation from
an inclusion depends on its size and shape as well as on the
specimen size, because it is easier to nd a large inclusion in
a big component rather than in a small specimen [, , –
].
2.2. Fatigue Crack Gr owth Modeling. e growth of a crack
is the major manifestation of damage and is a complex
phenomenon involving several processes such as (i) disloca-
tion agglomeration, (ii) subcell formation, and (iii) multiple
microcracks formation (independently growing up to their
connection) and subsequent dominant crack formation [].
e dimensions of cracks are crucial for modeling their
growth. An engineering analysis is possible considering the
relationships between the crack growth rates associated with
the stress intensity factor, accounting for the stress conditions
Advances in Materials Science and Engineering
at the crack tip. Of particular interest is the behavior and
modeling of small and short cracks [, ], in order to
determine the conditions leading to cracks growth up to a
level at which t he linear elastic fracture mechanics (LEFM)
theory becomes relevant. Fatigue cracks can be classied as
short if one of their dimensions is large compared to the
microstructure, while the small cracks have all dimensions
similarorsmallerthanthoseofthelargestmicrostructural
feature [].
TanakaandMatsuokastudiedthecrackgrowthina
number of steels and determined a proper best-t relation for
room temperature growth conditions [, ].
2.2.1. Deterministic Crack Growth Models. While the P aris-
Erdogan [] model is valid only in the macrocrack range,
a deterministic fatigue crack model can be obtained starting
from the short crack growth model presented by Newman Jr.
[]. Considering cycles and a medium crack length ,the
crack growth rate can be calculated as follows:
=exp ln
e
+;
0
=
0
>0,
()
where
e
is the linear elastic eective stress intensity factor
range and and aretheslopeandtheinterceptofthelinear
interpolation of the (log scale)
e
−/,respectively.In
order to determine the crack growth rate, Spencer Jr. et al. [,
]usedthecubicpolynomialtinln(
e
). erefore, the
crack growth rate equation can be written in the continuous-
time setting as follows [, ]:
=
(
Φ/
)
⋅
(
/
)
(
1−Φ
)
/
;
0
0
=
0
>0.
()
Manson and Halford [] introduced an eective crack
growth model a ccounting for the processes taking place
meanwhile, using
=
0
+
𝑓
−
0
𝑞
, ()
where
0
, ,and
𝑓
are initial (=0), instantaneous, and nal
(=1) cracks lengths, respectively, while is a function of
in the form =
𝛽
(B and are ma terial’s constants).
A fracture mechanics-based analysis addressing bridges
and other steel structures details has been made by Righini-
otis and Chryssanthopoulos [], accounting for the accept-
abilityofawsinfusionweldedstructures[].
2.2.2. Stochastic Crack Growth Models. e growth of a
fatigue crack can be also modeled by nonlinear stochastic
equations satisfying the It
ˆ
o conditions [–]. Given that
[(,
0
)] =
0
and cov[(,
0
)] =
0
, the stochastic
dierential equation for the crack g rowth process (,)can
be written according to the deterministic damage dynamics
as
(
,
)
=exp
(
,
)
−
2
𝑧
(
)
2
⋅
∀≥
0
.
()
If and represent the point and time of the sam-
ple in the stochastic process, the auxiliary process (,)
is assumed to be a stationary Gauss-M a rkov one having
variance
2
𝑧
(), which implies the rational condition of a
lognormal-distributed crack growth [].
In order to clarify the inuence of the fracture peculiar-
ities on the failure probability of a fatigue loaded structure,
Maljaars et al. [] used the linear elastic f racture mechanics
(LEFM) theory to develop a probabilistic model.
Ishikawa et al. [] proposed the Tsurui-Ishikawa model,
while Yazdani and Albrecht [] investigated the application
of probabilistic LEFM to the prediction of the inspection
interval of cover plates in highway bridges. As for the welded
structures, a comprehensive overview of probabilistic fatigue
assessment models can be found in the paper of Luki
´
cand
Cremona []. In this study, the eect of almost all relevant
random variables on the failure probability is treated [].
e key feature of the work of Maljaars et al. [] with
respect to other LEFM-based fatigue assessment studies [,
–] is that it accounts for the fact that, at any moment in
time,alargestresscyclecausingfracturecanoccur.erefore,
the probability of failure in case of fat igue loaded structures
can be calculated combining all the failure probabilities for all
time intervals.
Considering the stress ranges as randomly distributed,
the expectation of /can be written as a function of the
expectat i on of the stress range , as follows (s ee ()):
=
1
𝑚
1
Δ𝜎
tr
Δ𝜎
th
⋅
nom
𝑃
load
glob
scf
sif
𝑎
𝑚
1
⋅⋅⋅
+
2
𝑚
2
∞
Δ𝜎
tr
⋅
nom
𝑝
load
glob
scf
sif
𝑎
𝑚
2
,
()
𝑚
𝑠
2
𝑠
1
=
𝑠
2
𝑠
1
𝑚
Δ𝜎
(
)
,
()
where
Δ𝜎
()is the probability density function of the stress
ranges and represents the stress range steps. e -
factors are the uncertainties of the uctuating load mode. In
()
nom
represents the plate t hickness (considering a welded
plate) used in the calculation of the stress, while is the
thickness exponent. e probabilistic LEFM model applied
on fatigue loaded structures typical of civil enegineering
showed that modeling the uncertainity factors is the key
during the assessment of the failure probability, which is quite
independent of the particular failure criterion. e partial
factors introduced to meet the reliability requirements of
civil engineering struc tures and derived for various values of
the reliability index ()appeared to be insensitive to other
parameters such as load spectrum and geometry [].