Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 3, pp. 819-829, Warsaw 2012

50th Anniversary of JTAM

APPLICATION OF SPECTRAL METHOD IN FATIGUE LIFE

ASSESSMENT – DETERMINATION OF CRACK INITIATION

Adam Niesłony, Michał Böhm

Opole University of Technology, Department of Mechanics and Machine Design, Opole, Poland

e-mail: a.nieslony@po.opole.pl; michalboehm@gmail.com

The purpose of this paper is to provide a simple approach for indication of those places
of machine parts, where the fatigue crack initiation begins. This method uses FEM and
the cyclic stress-strain curve for calculating the stress state on specimens. The multiaxial
fatigue failure criterion based on the critical plane approach was used for reduction of the
random spatial stress state into the equivalent uniaxial one. The critical plane position was
established with the use of the variance method, and fatigue life was determined with as-
sumption of linear damage accumulation. Experimental verification of the proposedmethod
was performed with the use of results of fatigue tests for St52.3 structural steel subjected
to biaxial random tension-compression. Cruciform specimens with four holes were tested.
During the tests, the place and time of crack initiation were observed. It was shown, that
the proposedmethod for determination of the places of fatigue crack initiation gave results
well correlated with the data obtained from experiments. The proposed method could be
applied by engineers in order to identify the places and time of fatigue cracks initiation.

Key words: spectral method, multiaxial fatigue, random loading, crack initiation

Notations

ak – coefficients for multiaxial fatigue failure criteria, k=(1, . . . ,6)
E – Young’s modulus of elasticity [MPa]
F1−3,F2−4 – forces applied to specimen arms 1-3 and 2-4, respectively, [kN]
f – frequency [Hz]
L(t) – loading vector [kN]
t – time [s]
Tcal,Texp – calculated and experimental obtained time to crack initiation, respecti-

vely, [s]
GL(f),Gσ(f) – power spectral densitymatrix of loading vector L [kN

2/Hz] and of vector
of strain tensor components [MPa2/Hz], respectively

Q,QH – matrix of transition coefficients [kN
−1] andmatrix of coefficients of gene-

ralized Hooke’s law [MPa], respectively
m – coefficient of inclination ofWöhler’s curve for tension-compression
lη,mη,nη – directional cosines of vector normal to critical plane
M+ – expected rate of peaks
N(∆σ) – function which returns number of cycles for given stress range ∆σ from

standard fatigue characteristic
N0 – number of cycle of knee point of Wöhler’s curve
p(∆σ) – probability density function (PDF) of stress ranges
rF – correlation coefficient according to forces F1−3(t) and F2−4(t)
ε – vector of strain tensor components



820 A. Niesłony,M. Böhm

σeq(t),σaf – equivalent stress and fatigue strength, respectively, [MPa]
∆σtr – stress range according to Neuber’s rule [MPa]
ξk – k-th moment of PSDF
ν – Poisson’s ratio
µ – variance [MPa2]

1. Introduction

From considerations on damaged elements of machines or structures, it appears that material
fatigue in the high-cycle range often appears in form of cracks (Manson, 1965; Stephens and
Fuchs, 2001; Łagoda et al., 2005; Niesłony and Macha, 2007; Schijve, 2009). Such failures are
often observed in geometrically complex elements loaded by many independent random forces.
In such cases, application of traditional methods of fatigue life determination, where the life is
calculated in one point of themaximum fatigue intensity, is difficult because the position of that
point is unknown. Evaluation of fatigue life of such elements is most often conducted by time-
consumingandexpensive experimental tests on thebasis ofwhich fatigue characteristics of entire
elements or structure joints are designated. Received characteristics are highly individual, since
they concern only one geometry and load type, and the problem of fatigue is treated globally.
Therefore, an attempt to solve this problem by an experiment turned out to be insufficient and
economically unjustified.

At the stage of machine element designing, static Finite Element Method (FEM) analyses
are usually performed. As a result, maps of stresses or strains parameters, such as the stresses
reduced according to the Huber-Mises hypothesis, are obtained. The values placed at geometry
of the considered element as colored maps, allow evaluating the correctness of the designed
element under static load. In many cases machine elements are dynamically loaded, so the
designer has to take into account the future service conditions for the fatigue life assessment.
Designers should assess when and where a fatigue crack could initiate. This complex task can
be realised by applying known local methods of fatigue life determination and FEM (Niesłony
and Sonsino, 2006; Niesłony, 2009). As a result of this type of analysis, we obtain a fatigue
damage map similar to the well known stress map, which indicates the area of fatigue crack
initiation for the current loading case. Furthermore, the obtainment of the expected fatigue
life – the necessary parameter for further reliability analysis of the construction is at ease. In
order to use the previous stages of design in an effective way, where we often perform static
FEMcalculations, the fatigue calculations use themodels and results of the static FEManalysis
performing the determination of crack initiation on the basis of postprocessing. In these types
of analysis, the speed of calculations is also very important, becausewhile preparing the fatigue
life maps according to the local methods we need to compute the fatigue life several times,
usually for every node of the finite element mesh. Therefore, the spectral method of fatigue life
assessment is used,which is characterized as beingmore efficient than the cycle countingmethod
(Pitoiset and Preumont, 2000; Łagoda et al., 2005; Niesłony and Sonsino, 2006).

In this paper, the authors present an algorithm which allows one to determine the place of
fatigue crack initiationwith the use of FEM, and its verification based on fatigue tests of special
cruciform specimens being subjected to tension-compression in two perpendicular directions.

2. Tests results used for verifying the fatigue crack initiation

The test results obtained for smooth or notched specimens subjected to combined bendingwith
torsion (Macha andNiesłony, 2011) or tension-compression with torsion (Shamsaei et al., 2011)



Application of spectral method in fatigue life assessment... 821

are usually applied for verification of algorithms of fatigue life assessment under multiaxial lo-
ading. Such tests allow one to load the specimen with any combination of the bendingmoment
or the axial force and the torsional moment. Also, tests of cruciform specimens are performed
in order to verify algorithms of multiaxial fatigue assessment (Zhang and Sakane, 2007). In the
central part of the cruciform specimen, the plane stress state is obtained and suitable configura-
tion of forces applied to the specimen arms, allow one to obtain variable or constant directions
of principal stresses.
Determination of the fatigue crack initiation places in typical specimensused for fatigue tests

seems to be trivial. The crack initiation occurs in one good known place where the maximum
stress concentration can be observed. For smooth specimens, it is the cylindrical surface of
the least diameter, for notched specimens – the neighborhood of the notch bottom (Niesłony,
2010), and for the cruciform specimens – their center (Łagoda et al., 1999). Because of that,
the test results presented in this paper concern special cruciform specimens where the place of
fatigue crack initiation depends on loading configuration. Thus, in typical cruciform specimens
presented inFig. 1a fourholes, 12mmindiameter,weremade.Theywere symmetrically arranged
in relation to the arms along the radius of 26mm, Fig. 1b. The specimens were made of St52.3
steel (sheets 8mm in thickness), the specimen dimensions are presented in Fig. 1c. The chosen
static and cyclic properties of the considered steel are shown in Table 1.

Table 1. Selected material properties of St52.3 structural steel

E [MPa] ν m σaf [MPa] N0

207000 0.28 7.9 205 1.12 ·106

Fig. 1. Typical specimen for fatigue tests at the standMZPK-100 (a) special specimen with holes and
directions of force action (b) dimensions of the specimen (c)

The fatigue tests were performed at the stand MZPK-100 for tests of cruciform specimens
under a biaxial plane stress state.While testing, the specimenswere loaded by a randomhistory
of Gaussian distribution with a wide frequency band. An example of the time course is shown
in Fig. 2. Random histories of forces F1−3(t) and F2−4(t) applied to the specimen arms were



822 A. Niesłony,M. Böhm

obtainedbyfiltration of noise of normaldistribution, following that theywere scaled to a suitable
value of standard deviation for a given sample and arm pairs 1-3 and 2-4. Two series of tests
were carried out. In series K01, the sense of force vector applied in one pair of arms was set
inversely to the second pair F1−3(t)=−F2−4(t), which led to the value of correlation coefficient
rF = −1. In series K02, the senses of applied forces were equal, F1−3(t) = F2−4(t), and the
correlation coefficient rF = 1. During the fatigue tests, the specimen surface near the holes
was observed using optical microscope – time and places of crack occurrences were registered.
Figure 3 shows the initial place and path of the observed fatigue cracks in the specimens during
tests K01 and K02. Table 2 contains time to cracks initiation obtained for particular cracks
during the fatigue tests.

Fig. 2. A fragment of the random loading history registered by the force sensor during fatigue test

Fig. 3. Fatigue crack paths observed during the tests for all tested specimens, seriesK01 (a) andK02 (b)

3. Determination of fatigue crack initiation location

Themain part of the algorithm, showing the location of fatigue crack initiation is presented in
Fig. 4. There are three parts of the algorithm. The first part is responsible for the preparation
for calculations, the next part – for fatigue calculations, and the last one concerns interpretation
of calculation results. The algorithmuses the spectralmethod of fatigue life determination. This
method is very efficient and suitable for the assumed type of loading. In the following subsections
each part of the algorithmwill be presented respectively to the object of analysis – the cruciform
specimen with holes presented in Fig. 1.



Application of spectral method in fatigue life assessment... 823

Table 2.Time to the initiations of fatigue cracks presented in Fig. 3

Spec. rF (series F1−3 F2−4 Tcal Crack Texp
No. of tests) [kN] [kN] [s] No. [s]

1 −1 (K01) 7.645 7.554 1911 1.1 1685
1.2 1803
1.3 2520
1.4 2885
1.5 4320∗

1.6 4320∗

2 −1 (K01) 6.028 5.905 3930 2.1 2711
2.2 2890
2.3 3502
2.4 4461

3 1 (K02) 12.74 12.57 60401 3.1 15210
3.2 15942
3.3 16060
3.4 19650
3.5 20767

4 1 (K02) 13.24 12.97 52517 4.1 15951

5 −1 (K01) 6.115 6.361 4038 5.1 2504
5.2 2845
5.3 3615
5.4 3995
5.5 3995
5.6 4176

∗ – crack detected after the test was stopped

Fig. 4. Algorithm showing places of fatigue crack initiation

3.1. Preparations for fatigue calculations

At this stage, the specimens geometry is modelled and linear-elastic analyses of FEM are
performed. Since the loading of the specimenwas performed by two independent forces F1−3(t)
and F2−4(t), twoFEMmodelshavebeenpreparedand subjected tounitary loads.The symmetry
of the sample was used to load only a quarter of the model. Free mesh option was used in the
Comsol Multiphysics 3.5 FEM program for mesh generation with more detailed mesh near the
holes, see Fig. 5. From the FEM results, the matrix of coefficients of transition Q between the
unit value of global specimen loading L(t)= [F1−3(t),F2−4(t)] and the strain tensor components



824 A. Niesłony,M. Böhm

in the specimen (Niesłony and Sonsino, 2006) can be defined. The strain tensor components are
obtained by the following operation

ε(t)=L(t)Q (3.1)

where

ε(t)= [εxx(t),εyy(t),εzz(t),εxy(t),εxz(t),εyz(t)] (3.2)

Thematrix of transition coefficients is defined as

Q=

[

QxxF1−3 QyyF1−3 QzzF1−3 QxyF1−3 QxzF1−3 QyzF1−3
QxxF2−4 QyyF2−4 QzzF2−4 QxyF2−4 QxzF2−4 QyzF2−4

]

(3.3)

Assuming that the random loadingprocess L(t) is ergodic and stationary, and its probability
distribution can bedescribedby anormal distribution, thepower spectral densitymatrix GL(f)
with the 2×2 size was determined.Material constants describing static and fatigue properties
of St52.3 structural steel were defined according to Table 1.

3.2. Calculation of damage degree

The subdomain of the specimen subjected to fatigue analysis is defined in the following step.
In the considered case, it is the central part of the specimen including holes (see Fig. 5). Because
of the symmetry, also in this case, only the quarter of the analysed area was modeled.

Fig. 5. A part of the specimen for determination of fatigue life maps

The FEM mesh was generated for the considered subdomain. It should be understood that
the FEMmesh is not involved in FEM calculations, but is used only during the interpolation of
results and visualization. This is one of the major advantages of the proposed algorithm, since
it allows the fatigue analysis of selected parts of the structure regardless of the FEM analysis.
Calculations can be performed repeatedly on the basis of interpolated to themesh nodes values
of coefficients Q. The fatigue damagewas determined for each node of themesh; the calculation
steps were the following:

STEP 1 – determination of the power spectral density matrix directly from the matrix of
power spectral density of loading

Gσ(f)= [QQH]
⊤
GL(f)QQH (3.4)

where Q is the matrix of transition coefficients (3.33), QH is the matrix of coefficients of
generalized Hooke’s law written in the form



Application of spectral method in fatigue life assessment... 825

QH =
E

(1+ν)(1−2ν)



















1−ν ν ν 0 0 0
ν 1−ν ν 0 0 0
ν ν 1−ν 0 0 0
0 0 0 1−2ν 0 0
0 0 0 0 1−2ν 0
0 0 0 0 0 1−2ν



















(3.5)

STEP 2 – calculation of power spectral density of the equivalent stress history. During this
stage of calculation, it is needed to choose the right multiaxial fatigue failure criterion. It is
recommended that this criteria should be defined in the frequency domain, which allows the
performance of efficient calculations, without the need to generate stress time courses on the
basis of the PSDmatrix of stress tensor components (3.4). A wider description of those sort of
criteria can be found in many literature positions (Pitoiset and Preumont, 2000; Niesłony and
Macha, 2007; Niesłony, 2010; Cristofori et al., 2011; Susmel andTovo, 2011). In order to perform
calculations, it has been decided to use one of the linearmultiaxial fatigue criteria, which allows
one to determine the PSD function of the equivalent stress course according to the following
formula

Gσeq(f)=aGσ(f)a
T (3.6)

where a is a vector of coefficients resulting from the multiaxial fatigue failure criterion under
the assumed position of the critical plane (Łagoda and Ogonowski, 2005). The criterion of the
maximum normal stress on the critical plane was applied (Grzelak et al., 1991). According to
this criterion, the equivalent stress is expressed as

σeq(t)=ση(t)= l
2
ησxx(t)+m

2
ησyy(t)+n

2
ησzz(t)+2lηMησxy(t)

+2lηnησxz(t)+2mηnησyz(t)
(3.7)

where lη, mη, nη are directional cosines of the vector normal to the critical plane. Basing on
Eq. (3.7), the vector of coefficients from the criterion a can be easily defined

a= [l2η,m
2
η,n
2
η,2lηmη,2lηnη,2mηnη] (3.8)

The criterion is rather simple but it gives good results for stresses occurring at the edges
of plane elements subjected to tension, because in such places there is a stress state similar to
uniaxial tension-compression. The variance method was used for determination of the critical
plane position (Łagoda et al., 2005; Niesłony andMacha, 2007; Susmel, 2010). According to that
method, the critical plane is the plane of themaximum variance of normal stress. The search of
the critical plain position is usually performed through appointing the variance of normal stress
for the closed set of planes i∈ (1,k) and selection of that of the greatest variance value, µmax

µmax =max
i
(aiµa

T
i )=max

i

(

ai

∞
∫

0

Gσ(f) dfa
T
i

)

(3.9)

where µmax is the maximum variance for the considered set of planes, ai – vector of criterion
coefficients for the i-th plane. The set of planes should be suitably wide and it should include
all the possible positions of the critical plane.

STEP 3 – calculation of fatigue life Tcal. Because of the wide loading frequency band, the
Dirlik proposal (Dirlik, 1985; Niesłony, 2010) was chosen for description of the probability di-
stribution of loading cycles stress ranges

p(∆σ)=
1

2
√
ξ0

(K1
K4
e
−Z
K4 +

K2Z

R2
e
−Z
2

2R2 +K3Ze
−Z
2

2

)

(3.10)



826 A. Niesłony,M. Böhm

and

Z =
∆σ

2
√
ξ0

K1 =
2(xm− I

2)

1+ I2
K2 =

1− I−K1+K
2
1

1−R

K3 =1−K1−K2 K4 =
5(I−K3+K2R)

4K1
R=

I−xm−K
2
1

1− I−K1−K
2
1

xm =
ξ1
ξ0

√

ξ2
ξ4

I =
ξ2√
ξ0ξ4

ξk =

∞
∫

0

Gσeq(f)f
k df

The model proposed by Dirlik is often applied and gives good results in the considered
conditions.Moreover, using themodel, ProbabilityDensity Function (PDF) of stress ranges can
be estimated similarly to those obtained using the rainflow cycle counting method. This allows
one to undergo transformation of the obtained PDF according to the Neuber method

p(∆σtr)= g(p(∆σ),E,n
′,K′) (3.11)

where g(·, ·) is the transformation function according to the Neuber rule. The procedure allows
determination of approximate values of the elastic-plastic stresses under cyclic loading. Next,
the fatigue damage accumulation was performed with the use of the linear Palmgren-Miner
hypothesis for the whole stress range

Tcal =

(

M+
∞
∫

0

p(∆σtr)

N(∆σtr)
d∆σtr

)−1

(3.12)

where

M+ =

√

ξ4
ξ2

N(∆σtr)=σ
m
afN0

(∆σtr
2

)−m

It should be noted that only a fewmaterials can be correctly describedwith this hypothesis
(Mroziński, 2011; Szala, 1998). One of them is the St52.3 steel. Other damage summation hypo-
theses are presented and well described for example in (Fatemi and Yang, 1998; Halford, 1997;
Niesłony andMacha, 2007; Read and Shenoi, 1995). The limitations of linear damage accumula-
tion hypotheses are well known to the authors, but it seems that the advantages of this method
which ismainly: extremely easines of using and almost all alternative developedmethods require
at least one or more constants that must be experimentally determined (Manson and Halford,
2006), are relevant to the subject of this paper.

3.3. Visualization and analysis of the results

The fatigue damage map was made for the quarter of the central part of the specimen
(Fig. 5b) including the area of fatigue crack occurrence for both test series. Figure 6 presents
the obtained fatigue damage map using the value of damage logarithm log(1/Tcal). All fatigue
cracks from the specimen presented in Fig. 3 were superimposed on the resulting fatigue map
of the quarter of the FEM modelled sample in order to directly compare the calculations with
experimental results.Thedark areas are characterized by lower fatigue life, and the fatigue crack
initiation canbe expected in thoseplaces.Comparing the obtained fatigue damage contourswith
the cracks, high correlation between the areas of maximumdamage degree and the areas of real
fatigue cracks can be noticed. In the case of test series K01, three such places were found for
each hole, and two such places were found in test series K02 (see Fig. 6). From the calculation



Application of spectral method in fatigue life assessment... 827

results presented in Table 2 it appears that in the case of loading realized under rF = −1,
series K01, the calculated fatigue lifes are a little underrated as compared to the test results.
The calculated fatigue lifes for rF = 1, series K02, are greatly overestimated. It is probably
caused by the occurrence of local plastic strains at the holes edge. Thus, another method of
correction of the stress ranges than the Neuber rule, while fatigue life assessment based on the
stress obtaining from linear-elastic FEM analysis should be considered.

Fig. 6. Fatigue life map for the central part of the specimen for two test series: K01 (a) andK02 (b)

4. Conclusions

Cruciform specimens with holes were subjected to biaxial asynchronous (K01) and synchronous
(K02) tension-compression. Fromanalysis of the test results it appears that the obtained fatigue
life maps (Fig. 6) allow one to predict the areas liable to crack initiation. These areas match
(Fig. 3) with the areas of fatigue cracks initiation determined during the tests. The calculated
fatigue lifes correlate with those obtained from the experiments.

Fatigue calculations were performed independently of linear-elastic analyses used for deter-
mination of the transition coefficients Q. Thus, they could be realized many times for different
loading conditions as post-processing routine. It strongly influences the rate of calculations
because in the considered situation linear-elastic FEManalyses do not need to be repeated.Mo-
reover, application of only a part of geometry for fatigue calculations allows making an efficient
analysis only in one place, where the fatigue crack initiation is expected. In extreme cases, it
may be at only one point (mesh node), but then the fatigue damage maps cannot be drawn.
This fact is important in the fatigue analysis of so-called big FEMmodels (more than 1000 000
degrees of freedom), for which fatigue life calculations would be time-consuming. Selection of
subdomains, where the fatigue life assessment is performed, is a simple solution allowing fatigue
calculations also for big FEMmodels.

The proposed method should be implemented in FEM software and used by engineers for
quick identification of the points of fatigue crack initiation. However, it should be developed for
including the possibility of determination of the crack initiation direction, what is important
for simulation of the crack propagation, and the accuracy of determination of time to crack
initiation should be improved.

Acknowledgement

This paper is realized within the framework of research project No. 2011/01/B/ST8/06850 funded

by National Science Centre in Poland.



828 A. Niesłony,M. Böhm

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Zastosowanie metody spektralnej w wyznaczaniu trwałości zmęczeniowej – określanie

inicjacji pęknięć

Streszczenie

Celemniniejszejpracy jestprzedstawienieprostegopodejściawwyznaczaniutychmiejscwelementach
maszyn, gdzie nastąpi inicjacja pęknięć. Niniejsza metoda wykorzystuje metodę elementów skończonych
(MES) i krzywą cyklicznego odkształcenia w celu obliczenia stanu naprężenia w próbkach. Wieloosiowe
kryterium zmęczenia oparte na założeniach płaszczyzny krytycznej zostało zastosowane w celu redukcji
przestrzennego losowegostanunaprężenianaekwiwalentny jednoosiowy.Położeniepłaszczyznykrytycznej
zostałozdefiniowaneprzypomocymetodywariancji, natomiast trwałośćzmęczeniowazostaławyznaczona
zgodnie z założeniami liniowej hipotezy sumowania uszkodzeń.Weryfikacja eksperymentalna zapropono-
wanejmetody została przeprowadzona na podstawiewynikówbadań zmęczeniowych stali konstrukcyjnej
St52.3 poddanej dwuosiowym losowemu obciążeniu rozciąganiem-ściskaniem. W niniejszych badaniach
zostały zastosowane próbki krzyżowe z czterema otworami. W trakcie badań były rejestrowane miejsca
oraz czas inicjacji pęknięcia. Wykazano, że proponowana metoda wyznaczania miejsc inicjacji pęknięć
dała wyniki pokrywające się z otrzymanymiw trakcie eksperymentu. Proponowanametodamoże zostać
zastosowana przez inżynieróww celu identyfikacji miejsc oraz czasu, w którym nastąpi inicjacja pęknięć
zmęczeniowych.

Manuscript received February 3, 2012; accepted for print March 6, 2012