Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 1, pp. 233-254, Warsaw 2010

COMPARISON OF SOME SELECTED MULTIAXIAL

FATIGUE FAILURE CRITERIA DEDICATED

FOR SPECTRAL METHOD

Adam Niesłony

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

e-mail: a.nieslony@po.opole.pl

The paper presents the procedure of estimation of fatigue life in the high-
cycle fatigue regime under amultiaxial random loading using the spectral
method.Theprocedureconsistsofapplicationof thepowerspectraldensity
function of the equivalent stress in the fatigue life assessment together
with the known spectral method developed for the uniaxial stress state.
The model proposed by Miles and Dirlik was presented as an example.
Two groups of multiaxial fatigue failure criteria have been distinguished:
the criteria based on the critical plane approach and the criteria using
invariants of the stress state. Two sets of experimental data were used in
order to compare the calculated and experimental fatigue lives. It has been
shown that under the multiaxial random loading the results of fatigue life
calculated according to the consideredmethod arewell correlatedwith the
results of experiments if themultiaxial fatigue failure criterion is properly
selected for mechanical parameters of the tested material.

Keywords: spectralmethod,multiaxial fatigue, high cycle fatigue, random
loading

Notations

ak – coefficients for multiaxial fatigue criteria, k=1, . . . ,6
f – frequency
G(f) – power spectral density function (PSDF)
Gσ(f) – matrix of PSDFs
QM – matrix of coefficients for criterion proposed by Preumont

and Piefort
Qos,Qon – coefficients matrices

l̂k,m̂k, n̂k – for k = 1,2,3, direction cosines of averaged directions of
principal stresses σ1(t)­σ2(t)­σ3(t)



234 A. Niesłony

mσ,mτ – coefficients of inclination of Wöhler’s curves for tension-
compression and torsion, respectively

M+ – expected rate of peaks
Nf – number of cycles to failure
Nσ,Nτ – number of cycles read out from Wöhler’s curves for tension-

compression and torsion, respectively
N(σa) – functionwhich returns number of cycle for given stress ampli-

tude σa from standard fatigue characteristic
N+0 – expected rate of zero crossing with positive slope
p(σa) – probability density function (PDF) of stress amplitudes
I – irregularity factor
rσos,τos – correlation coefficient of histories τos(t) and σon(t)
σeq – equivalent stress
σaf – fatigue strength
σ(t) – vector of components of stress tensor
ξk – k-th moment of PSDF
µ – variance
Γ(·) – gamma function
conj(·) – complex conjugate function

1. Introduction

The known methods of evaluation of fatigue life of machine elements and
structures under random loadings can be divided into two groups. The first
one is the cycle counting approach, based on various cycle counting algori-
thms, which allows obtaining the amplitude histogram of cycles from random
loading (Banvillet et al., 2004; Colombi and Doliński, 2001; Dowling, 1972;
Łagoda et al., 2001, 2005; Macha et al., 2006). The second group uses specific
procedures known form spectral analysis of stochastic processes (Banvillet et
al., 2004; Benasciutti and Tovo, 2006; Łagoda et al., 2005; Macha et al., 2006;
Niesłony andMacha, 2007; Sherratt et al., 2005),where theprobabilitydensity
function (PDF) of stress amplitudes are approximated with several parame-
ters computed from the power spectral density function (PSDF). In these two
groups, final fatigue damage is usually obtained by the damage accumulation
hypothesis, where a standard fatigue characteristic of the material, obtained
under cyclic loading, is applied. Using the linear Palmgren-Miner hypothesis
of damage accumulation, the expected number of cycles to crack initiation Nf



Comparison of some selected multiaxial fatigue... 235

can be derived as follows (Morrow, 1986; Niesłony andMacha, 2007; Sherratt
et al., 2005; Szala, 1998)

Nf =

[ ∞∫

0

p(σa)

N(σa)
dσa

]−1
(1.1)

where p(σa) is the PDF of stress amplitudes and N(σa) represents a function
which returns the number of cycles for the given stress amplitude σa from a
standard fatigue characteristic, i.e. Wöhler curve.

In the spectralmethod, the power spectral density function is usually used
(Dirlik, 1985; Sobczyk and Spencer, 1992) to describe the loading. This func-
tion represents the power distribution versus frequency, i.e. distribution of
the mean square values of particular harmonic components appearing in the
random process and in the frequency bandwidth. Observing the shape of the
PSDF, it is easily to judge how the signal is distributed in the frequency. Ho-
wever, in practice, parameters which would describe the characteristic of the
signal in a quantitative way are required. Usually, the first five moments of
the PSDF are used (Sobczyk and Spencer, 1992)

ξk =

∞∫

0

G(f)fk df for k=0,1, . . . ,4 (1.2)

Using these moments, it is possible to receive very important, from the
point of view of the material fatigue, statistical parameters concerning the
time course of the process, like the variance

µ= ξ0 (1.3)

the expected rate of zero crossing with the positive slope

N+0 =

√
ξ2
ξ0

(1.4)

the expected rate of peaks

M+ =

√
ξ4
ξ2

(1.5)

and the irregularity factor

I =
N+0
M+

(1.6)



236 A. Niesłony

In the case of simple loading states, like uniaxial tension-compression, seve-
ral proposals for determination of the fatigue life using the spectralmethodare
known(NiesłonyandMacha, 2007). In thesemodels, computation isperformed
in the frequency domain and distribution of stress amplitudes is determined
directly from the PSDF. The cycle counting methods, such as the rain flow
algorithm, are not necessary.

An example of such solution was presented by Miles in 1954 (Niesłony
andMacha, 2007). It was elaborated for the Gaussian narrow-band frequency
loading. Using the Palmgren-Miner linear hypothesis of damage accumulation
and Rayleigh probability distribution for approximation of stress amplitudes,
the following equation for the fatigue life can be derived

Nf =
Aσ

(2µσ)
mσ
2 Γ
(
mσ+2
2

) (1.7)

where Aσ = lg(Nf)+mσ lg(σa) is the coefficient ofWöhler’s curves for tension-
compression. The theory presented by Miles was developed by other authors
(Niesłony, 2008). They proposed much more universal solutions giving stress
amplitude distributions similar to those obtained according to the rain flow
algorithm. The proposal formulated by Dirlik in 1985 for the wide-band fre-
quency Gaussian loading belongs to the well-known ones (Dirlik, 1985)

p(σa)=
1√
ξ0

[G1
Q
exp
(−Z
Q

)
+
G2Z

R2
exp
(−Z2
2R2

)
+G3Z exp

(−Z2
2

)]
(1.8)

where

Z =
σa√
µ0

G1 =
2(xm− I2)
1+ I2

G2 =
1− I−G1+G21

1−R
xm =

ξ1
ξ0

√
ξ2
ξ4

G3 =1−G1−G2 R=
I−xm−G21
1− I−G1+G21

Q=
5(I−G3+G2R)

4G1
I =

ξ2√
ξ0ξ4

This formula provides thebest results of approximation of stress amplitude
distributiondeterminedby the rainflowalgorithm(NiesłonyandMacha, 2007;
Sherratt et al., 2005) as comparedwith other approximations, and can beused
for stress histories with narrow- and broad-band frequency spectra. However,
in this case, a complicated form of the PDF, Eq. (1.8) does not allow one to



Comparison of some selected multiaxial fatigue... 237

simplify the equation for fatigue damage, and the integral occurring in Eq.
(1.1) must be solved numerically.

Equation (1.1) is designed for a uniaxial loading and cannot be applied
directly under a multiaxial loading. In this case, the spatial stress state sho-
uld be substituted with the equivalent uniaxial one. To do this, a suitable
fatigue failure criterion shall be used. The criterion allows reducing a triaxial
random stress state to the equivalent uniaxial one in the sense of material
fatigue (Karolczuk andMacha, 2005; Jiang et al., 2007; Niesłony and Sonsino,
2008). Then, the history of the equivalent stress or its suitable probabilistic
characteristics may be applied for calculations of the fatigue life, as under the
uniaxial random loading.

The stress state in the time domain can be defined using the vector of
components of the stress tensor

σ(t)= [σxx(t),σyy(t),σzz(t),σxy(t),σxz(t),σyz(t)] (1.9)

The probabilistic relation between particular components plays an impor-
tant role in material fatigue. Therefore, in the spectral method, these proper-
ties are expressed with a matrix of PSDF Gσ(f) which for the vector, Eq.
(1.9), gives a rectangular 6×6 dimensional matrix

Gσ(f)=





Gxx,xx(f) · · · Gxx,yz(f)
...

...
...

Gyz,xx(f) · · · Gyz,yz(f)




 (1.10)

On the diagonal of the matrix the autospectral density functions and on
the upper and lower triangular part of the matrix the cross-spectral density
functions are placed. Since the relationship Gi,j(f) = conj[Gj,i(f)] for i 6= j
appears, for description of the frequency structure of the random stress state
twenty onePSDFs shouldbeknown.Using the advantage ofmultiaxial fatigue
failure criteria defined in the frequency domain, the PSDF of the equivalent
stress Geq(f) can be computed. Next, the well known uniaxial solutions can
be used, for example the models proposed by Miles, Eq. (1.7) or Dirlik, Eq.
(1.8), where the PSDF of the equivalent stress state should be applied instead
of the uniaxial PSDF (Niesłony and Macha, 2007). The general concept of
using the equivalent uniaxial stress state during fatigue life assessment with
the spectral method under a multiaxial random loading is shown in Fig.1.



238 A. Niesłony

Fig. 1. Flow chart for calculation of fatigue life using spectral method under
multiaxial random loading

2. Multiaxial fatigue failure criteria related to the spectral

method

2.1. Criteria based on the critical plane approach

In 1988 Macha (Macha, 1988, 1996) introduced the multiaxial fatigue fa-
ilure criteria based on the critical plane approach defined in the frequency
domain to the spectral method. He used his criteria defined in the time do-
main, being a linear combination of stress or strain tensor components on the
critical plane (Karolczuk and Macha, 2005). For these criteria, the general
function of material fatigue strength under a multiaxial random loading was
defined as follow

S(t)= {σij(t),P,C} (2.1)

where σij(t) are components of the stress or strain tensor, P – parameters for
determination of critical plane position, and C – parameters characterizing
the material. The surface of limit states determining the fatigue life under a
random complex state of stress is described by the limit value of the fatigue
strength function

max
t
{S(t)}=σaf (2.2)

corresponding to the fatigue strength of the material under the alternating
cycle of uniaxial tension-compression σaf. Expression (2.2) should also be read
as a 100%quantile of the randomvariable S.Dependingon the chosenmethod
for determination of the critical plane position,Macha obtained special forms
of the equivalent stress, for example according to the criterion of maximum
normal stress on the critical plane

σeq(t)=ση(t)= l̂
2
1σxx(t)+ m̂

2
1σyy(t)+ n̂

2
1σzz(t)+2l̂1m̂1σxy(t)

(2.3)
+2l̂1n̂1σxz(t)+2m̂1n̂1σyz(t)



Comparison of some selected multiaxial fatigue... 239

or according to the criterion of maximum shear and normal stresses on the
critical plane

σeq(t)=
1

1+K

{
[l̂21 − l̂23 +K(l̂21 + l̂23)2]σxx(t)+

+[m̂21− m̂23+K(m̂21+ m̂23)2]σyy(t)+
+[n̂21− n̂23+K(n̂21+ n̂23)2]σzz(t)+ (2.4)
+2[l̂1m̂1− l̂3m̂3+K(l̂1+ l̂3)(m̂1+ m̂3)]σxy(t)+
+2[l̂1n̂1− l̂3n̂3+K(l̂1+ l̂3)(n̂1+ n̂3)]σxz(t)+
+2[m̂1n̂1− m̂3n̂3+K(m̂1+ m̂3)(n̂1+ n̂3)]σyz(t)

}

where l̂k, m̂k and n̂k for k = 1,2,3 are suitable directional cosines of avera-
ged directions of principal stresses σ1(t) ­ σ2(t) ­ σ3(t), and K represents
the material constant determined in cyclic fatigue tests. As the criteria are
based on the notion of the critical plane, three methods of determination of
its location are distinguished: weight functions method, variance method and
damage accumulation method. From the spectral point of view, a possibility
is open for including into the calculation algorithm the two latter ones.Macha
reported that the equivalent stress σeq(t) could be understood as the output
signal from a physical system with six inputs for which signals representing
suitable stress tensor components were delivered, Fig.2.

Fig. 2. Interpretation of the damage parameter xeq(t) as an output signal from the
physical systemwith pulse transition functions hk(δ), k=1, . . . ,6 where

signals xk(t), k=1, . . . ,6 were introduced to its input

Thus, theuse of transformation for linear systems, known fromthe spectral
analysis of stochastic processes (Bendat and Piersol, 1980), leads to the final
equation for the equivalent stress history

σeq(t)=
6∑

k=1

akσk(t) (2.5)



240 A. Niesłony

and the PSDF of the equivalent stress

Geq(f)=
6∑

i=1

6∑

j=1

aiajGij(f)=
6∑

i=1

[
a2iGii(f)+2

6∑

j=i+1

aiajGij(f)
]

(2.6)

where σk(t), k=1, . . . ,6, are stress tensor components, Gij(f) – components
of the matrix of PSDF of the stress tensors, and ak – coefficients depending
on the multiaxial fatigue failure criterion, shown in Table 1 for the criteria
(2.3) and (2.4).

The mentioned solution allows one to apply also other linear criteria of
multiaxial fatigue, for example thecriteriaproposedbyŁagodaandOgonowski
(2005). The resulting PSDF of the equivalent stress determined according to
Eq. (2.6) is equivalent to thePSDFdeterminedby theFourier transformof the
equivalent stress σeq(t) obtained in the time domain according to Eq. (2.5).

Table 1. Coefficients ak in σeq(t) expression according to the criterion of
maximumnormal stress,Eq. (2.3) and criterion ofmaximumshear andnormal
stresses, Eq. (2.4), on the critical plane

a Eq. (2.3) Eq. (2.4)

a1 l̂
2
1

l̂21− l̂23 +K(l̂21+ l̂23)2
1+K

a2 m̂
2
1

m̂21− m̂23+K(m̂21+ m̂23)2
1+K

a3 n̂
2
1

n̂21− n̂23+K(n̂21+ n̂23)2
1+K

a4 2l̂1m̂1
2[l̂1m̂1− l̂3m̂3+K(l̂1+ l̂3)(m̂1+ m̂3)]

1+K

a5 2l̂1n̂1
2[l̂1n̂1− l̂3n̂3+K(l̂1+ l̂3)(n̂1+ n̂3)]

1+K

a6 2m̂1n̂1
2[m̂1n̂1− m̂3n̂3+K(m̂1+ m̂3)(n̂1+ n̂3)]

1+K

2.2. Criteria using invariants of the stress tensor

Preumont andPiefort (1994) presented amethod for determination of the
PSDF of the equivalent stress using the Huber-Mises-Hencky hypothesis. To



Comparison of some selected multiaxial fatigue... 241

present the method, let us consider the case of the plane stress state. Under
this condition, the equivalent stress takes the form

σ2eq =σ
2
xx+σ

2
yy −σxxσyy +3σ2xy (2.7)

After defining the vector of stress tensor components σ= [σxx,σyy,σxy]
⊤,

Eq. (2.7) can be written according to the rules of matrix calculus

σ2eq =σ
⊤
QMσ= tr{QM[σσ⊤]} (2.8)

where

QM =





1 −0.5 0
−0.5 1 0
0 0 3




 (2.9)

is thematrix of coefficients for theHuber-Mises-Hencky equivalent stressunder
the plane stress state, σ⊤ is the vector transposed to σ, tr{. . .} is the sum
of components of the main diagonal of the square matrix. On the basis of
Eq. (2.8), the relation for expected values E[·] could be stated

E[σ2eq] =σ
⊤
QMσ= tr{QME[σσ⊤]} (2.10)

Thus, the obtained formula presents themean-square value of the reduced
stress.Themean-square value can be also determined directly fromthematrix
of PSDF of the stress tensor

E[σ2eq] =

∞∫

0

Geq(f) df =

∞∫

0

tr{QMGσ(f)} df (2.11)

where Gσ(f) is the matrix of PSDF (autospectral and cross-spectral) of the
stress vector σ. On the basis of the preceding formulae, Preumont and Pie-
fort postulated a method for determination of PSDF of the equivalent stress
directly from thematrix of PSDF of the stress vector

Geq(f)= tr{QMGσ(f)} (2.12)

This criterion can be related only tomaterials for which the coefficients of
inclination of Wöhler’s curves for tension-compression and torsion are equal

mσ =mτ (2.13)

and the fatigue characteristics of the material satisfy the following equality

σaN =
√
3τaN (2.14)



242 A. Niesłony

where σaN and τaN are amplitudes read out fromWöhler’s curves for a con-
stant number of cycles Nf for tension-compression and torsion, respectively.

Pitoiset (2001) carriedoutmanysimulationsandcompared the fatigue lives
determined in time and frequencydomains.High conformity of the resultswas
reported.This paper also presents an example of calculations of the fatigue life
of a nozzle of the Vulcain motor working in the Ariane V rocket. Let us note
that while determining the reduced PSDF using Eq. (2.12), the interactions
between the stress components σxx and σxy as well as σyy and σxy have not
been taken into account because of the zeros in the coefficient matrix QM,
Eq. (2.9). It is an unfavourable property of this method. The above approach
was applied by Pitoiset et al. (2001) for formulation of the criteria proposed
byMatake andCrossland in the frequency domainwith the sameweakness of
the method.

Bonte et al. (2007) criticized the solution proposed by Preumont and Pie-
fort due to the fact that not all phase shifts between the stress tensor com-
ponents had been taken into account while determining the PSDF of the equ-
ivalent stress. They proposed their own solution and obtained the equivalent
stress including phase displacements. Let us note, however, that their solu-
tion leads to a lower value of the determined equivalent stress which does not
correlate with typical results obtained from fatigue tests. In such cases, the
phase displacement of stress tensor components with constant amplitudes and
frequencies usually influences reduction of the fatigue life, so the determined
equivalent stress shouldbehigher.The limitation expressedbyEqs. (2.13) and
(2.14) is presented in both Preumont and Piefort’s proposals and the solution
developed by Bonte et al. Thus, possibilities of their application are rather
small.

The restriction expressed by Eq. (2.14) seems to be the most important
because awide range ofmaterials does not satisfy it. However, it is possible to
formulate a criterion in the frequency domain without the mentioned restric-
tion, which allows one to obtain correct values under simple loading states,
i.e. tension-compression and torsion (Niesłony, 2008). It is defined directly in
the frequency domain, and its equivalent form in the time domain for random
processes does not exist. It is assumed thatPSDFof shear andnormal stresses
acting on the octahedral plane, Fig.3, strongly influences thematerial fatigue.

The criterion can be written in the following general form

Geq(f)= g[Gτ os(f),Gσon(f),P ] (2.15)

where P is the vector of material constants, Gτ os(f) and Gσos(f) are the
PSDFs of shear and normal stresses on the octahedral plane, respectively.



Comparison of some selected multiaxial fatigue... 243

Fig. 3. Normal and shear stresses acting on the octahedral plane

In order to compute the particular PSDFs, the same procedure proposed by
Pitoised and Premount was used

Gτ os(f)= tr{QosGσ(f)} Gσon(f)= tr{QonGσ(f)} (2.16)

with the suitably defined coefficient matrix

Qos =
1

9






2 −1 −1 0 0 0
−1 2 −1 0 0 0
−1 −1 2 0 0 0
0 0 0 6 0 0
0 0 0 0 6 0
0 0 0 0 0 6






Qon =
1

9






1 1 1 0 0 0
1 1 1 0 0 0
1 1 1 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0






(2.17)
PSDF of the equivalent stress is defined as the sum of these functions

with suitable weights A and B. According to the summation theory for two
one-dimensional stochastic processes, it can be written

Geq(f)=A
2Gτ os(f)+B

2Gσon(f)+2A
2B2Re{Gτ os,σon(f)} (2.18)

where Re{Gτ os,σon(f)} is the real part of cross PSDF of τos(t) and σon(t).
Depending on the material, the ratio of stresses amplitude from the Wöhler
curves for tension-compression and torsion varies, and in a general case de-
pendson the number of cycles. The advanced criteria include that fact in order
to determine the correct equivalent values in simple loading cases, i.e. tension-
compression and torsion. Such solutions can be found in the criteria proposed
by Sonsino, Dang Van (Niesłony and Sonsino, 2008; Karolczuk and Macha,
2005), or Łagoda and Ogonowski (2005). Also in this case, the coefficients A



244 A. Niesłony

and B are especially determined so as to obtain correct results for uniaxial
tension-compression and torsion

A=

√
6

2

σaf
τaf

(2.19)

B=−
√
3rτ os,σos

σaf
τaf
+
1

τaf

√
3r2τ os,σosσ

2
af
+9τ2

af
−3σ2

af

Assuming that the histories τos(t) and σon(t) are correlated, themaximum
value of PSDF of the equivalent stress leads to the following form of the
coefficient B

B=−
√
2A+3 (2.20)

3. Experimental verification

For the verification purposes, experimental data which can be found in the
literature were used. While searching for the suitable data, special attention
was paid to the kind of experimental tests and their complexity. For these
reasons, two sets of experimental results presentedbySimbürger (1975) andby
Niesłony andMacha (2007)were selected. For bothdata sets, the computation
of PSDF of the equivalent stress were done according to four criteria:

C1 – criterion of maximum normal stress on the critical plane, Eq. (2.3),

C2 – criterion of maximum normal and shear stresses on the critical plane,
Eq. (2.4),

C3 – criterion proposed by Preumont and Piefort, Eq. (2.12),

C4 – criterion of shear and normal stresses acting on the octahedral plane,
Eq. (2.18).

3.1. Fatigue data obtained by cyclic bending with torsion

The testing program by Simbürger (1975) comprised plain and notched spe-
cimens made of Ck 45 steel (SAE 1045). The tests were load controlled. A
constant amplitude loading was used with equal frequency of stress tensor
components to perform several tests under bending and/or torsion. The cri-
terion for fatigue failure was the crack initiation defined as crack depth of
0.25mmand it wasmonitored and detected during the tests by the ultrasonic
method. While uniaxial tests for determining only material data, unnotched



Comparison of some selected multiaxial fatigue... 245

plain and hollow specimens were tested, Table 2. In the case of bending, tor-
sion and combined loading, notched specimens, with the notch radius 5mm,
having the theoretical stress concentration factor Ktt = 1.24 under torsion
and Ktb = 1.49 under bending, were tested. Six tests series were carried out
for the notched specimens by bending (S01), torsion (S02), and four combi-
nations of bendingwith torsion (S03-S06). The tests under combined bending
with torsion were carried out for the constant ratio of amplitudes of nominal
stresses τa,xy/σa,xx = 0.575. In this case, the series S03-S06 differed in the
value of phase shift between bending and torsion with values ϕστ =0, 30, 60
and 90 degree. Figure 4 presents geometry of the specimenwith scheme of the
loading.

Fig. 4. Geometry of the specimen with scheme of the loading used by Simbürger

Application of the spectral method presented in the previous section for
fatigue life calculation in the case of constant amplitude loading is incorrect.
They assume that the loading has a random character with normal distribu-
tion, i.e. the Gaussian process, what is not appearing in this case. However,
it is obvious that the components of the vector of the loading are changing
sinusoidally in time and they are characterized by the same frequency. It is
expected that the equivalent stress to be also a constant amplitude coursewith
the frequency equal to the frequency of components of the loading vector. The-
refore, by computing the amplitude of the equivalent stress the variance of this
process can be used

σa,eq =
√
2µeq =

√√√√√2
∞∫

0

Geq(f) df (3.1)

Like during calculations in the time domain, also in the spectral method
the number of cycles to crack initiation is determined directly from the stress-
life characteristic

Ncal =Aσ(σa,eq)
−mσ (3.2)



246 A. Niesłony

It leads to simplification of the calculations, since the estimate of the PDF
of amplitudes is unnecessary. The result is not burdened with uncertainness
of the damage accumulation hypothesis and of approximation of the PDF of
amplitudes. In order to obtain calculated fatigue lives for each considered case
of loading and criterion of multiaxial fatigue, the fatigue limit was not taken
into account during the calculations, Eq. (3.2). The calculation results Ncal
were compared with the test results Nexp in Fig.5.

The quantities on the axes in the figures are presented in a logarithmic or-
der. The dashed linesmark the scatter bandwith coefficient 3which indicates
that the points within the band results from the comparison of two quantities,
x and y, for which the expression 1/3 < y/x < 3 is satisfied. The diagonal
(full line) defines an ideal case forwhich the two comparedquantities are equal
y/x=1.

For thecaseof loadingunderpurebending(S01), the criteriabasedon inva-
riants of the stress state (C3 andC4) are giving the best results in relationship
to the critical plane concept criteria (C1 andC2). Probably, these criteria are
taking the special condition of the stresses arising at the bottom of the notch
into account appropriately. In Fig.5b, only the results of calculations received
according toC2 andC3 criteria are included into the established scatter band.
This is a result of using the relationship σaf/τaf which was applied in these
criteria. In Fig.5c-f, the results of calculations are presented for the combined
loading with phase shifts ϕστ =0, 30, 60 and 90 degrees. It is easy to notice
that the C1 and C2 criteria taking into account the non-proportionality of
the loading are not correct and tend to overestimate the calculated fatigue
life. Under those criteria, worse results of the calculation for greater values of
the phase shifts were received. It can be assumed that the criteria based on
the invariants of the stress state should be used for fatigue life assessment of
machines made of Ck45 steel.

3.2. Fatigue data obtained under random bending with torsion

The data obtained while fatigue tests under combined random bending
with torsion were presented by Niesłony and Macha (2007). Round smooth
specimensmadeof 18G2Asteelwere tested.A typical product, i.e. adrawnbar
was tested. Since it was assumed that a commercially available material could
be subjected to fatigue tests, the purchased material had not been subjected
to any preliminary selection. The main material constants given in Table 2
were obtained from the basic cyclic fatigue tests.

In the tests, histories ofGaussiandistributionsandnarrow frequencybands
were used. The dominating frequency was 20Hz. In Fig.6, geometry of the



Comparison of some selected multiaxial fatigue... 247

Fig. 5. Comparison of the calculated Ncal and experimental Nexp cycles to failure



248 A. Niesłony

Table 2.Parameters ofWöhler’s curves used in the fatigue life calculation

Material
Tension Torsion

σaf [MPa] mσ Nσ τaf [MPa] mτ Nτ

Ck45 390 8.5 200000 275 29 300000

18G2A 204 7.9 1120000 142 12.3 1982900

specimen with scheme of the loading is presented. The tests were performed
for different correlation coefficients rστ between the histories of bending σ(t)
and torsion τ(t) and different variances µσ and µτ. Eight fatigue tests were
performed andmarked with symbols as follows:

• N01 – pure bending,
• N02 – pure torsion,
• N03 – non-proportional bending with torsion, rστ ≈ 0,

√
µσ/µτ ≈ 2

• N04 – non-proportional bending with torsion, rστ ≈ 0,
√
µσ/µτ ≈ 1

• N05 – proportional bending with torsion, rστ =1,
√
µσ/µτ ≈ 2

• N06 – proportional bending with torsion, rστ =1,
√
µσ/µτ ≈ 1

• N07 – non-proportional bending with torsion, rστ ≈ 0.5,
√
µσ/µτ ≈ 1

• N08 – non-proportional bending with torsion, rστ ≈ 0.5,
√
µσ/µτ ≈ 2

Fig. 6. Geometry of the specimen with scheme of the loading presented by Niesłony
andMacha (2007)

In order to determine the fatigue life from PSDF of the equivalent stress,
a suitable spectral formula was derived, assuming that the loading had a nor-
mal probability distribution with the narrow-band frequency spectrum and
damageswas accumulated according to the Serensen-Kogayev hypothesis.The
following final formula was obtained



Comparison of some selected multiaxial fatigue... 249

Tcal =
bSK
M+

[ ∞∫

aSK

p(σa)

N(σa)
dσa

]−1
(3.3)

where

bSK =

√
2µσΓ

(
3
2
,
a2
SK
σ2
af

2µσ

)
e
a
2

SK
σ
2

af

2µσ −aSKσaf
σamax−aSKσaf

(3.4)

is theSerensen-Kogayev coefficient formulated for the spectralmethod, aSK is
the coefficient enabling one to include amplitudes below the fatigue limit and
PDF p(σa) is computed using the Dirlik model, Eq. (1.8). Details according
to the spectral formulation of the coefficient bSK can be found in literature
(Łagoda et al., 2005). The calculation results were compared with the fatigue
test results in Figs.7 and 8.

Fig. 7. Comparison of the calculated Tcal and experimental Texp time to failure for
loading cases N01-N04

The same type of graphwas used for the data set bySimburger.Because of
the random loading, the experimental Texp and calculated Tcal time to failure



250 A. Niesłony

Fig. 8. Comparison of the fatigue life calculated according to the four criteria C1-C4
with the experimental life, loading cases N05-N08

was used in figures instead of the number of cycles. Like previously, only the
criteria C2 and C3 give acceptable results under pure torsion. Worse results
were obtained by application of the criterion C1 because of the big scatter
range.Thecriteriabasedon the invariants of the stress state (C3andC4) tends
to underestimate the computed life what is acceptable from the engineering
point of view. Generally, good results of calculations were obtained owing to
the C2 criterion which can be recommended for the fatigue life assessment of
18G2A steel.

4. Conclusions

The spectral method of fatigue life determination has been successfully ap-
plied under uniaxial and very rarely undermultiaxial random loadings so far.
Some fatigue failure criteria, such as the criterion ofmaximumprincipal stress



Comparison of some selected multiaxial fatigue... 251

or the criterion based on the Huber-Mises-Hencky strength hypothesis, give
satisfactory results only for some constructional materials. From theoretical
considerations it appears that those criteria do not include the influence of
changes of the principal stress directions under a multiaxial loading on the
material fatigue. Therefore, it is necessary to apply these criteria for mate-
rials which are not sensitive to non-proportionality of the loading, i.e. tension-
compressionwith phase shifts or non-correlated random loadings. The criteria
being a linear combination of the stress state components based on the critical
plane approach take into account the non-proportionality of the loading. All
of the presented criteria do not include the loadingmean value, and the range
of their application should bemore verified with experimental data.
The presented multiaxial fatigue failure criteria reduce the three-

dimensional stress state to the equivalent uniaxial one directly in the fre-
quency domain working on PSDFs of stresses. This fact give us possibility to
use the knownuniaxialmodels during calculations, see Fig.1, what is a strong
advantage of this kind of criteria. Because of the calculation rate and good
correlationwith the experimental results (Łagoda et al., 2005), the introduced
method is useful for the fatigue life assessment of real structures under random
or variable amplitude loadings using FEM (Niesłony, 2008).
Development of numerical methods, especially virtual test methods using

FEM joined with multi-body dynamic analysis allows one to state that the
criteria of multiaxial fatigue defined in the frequency domain could be very
interesting in the nearest future.

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ATRBydgoszcz

Porównanie kilku wybranych kryteriów wieloosiowego zniszczenia

zmęczeniowego dedykowanych metodzie spektralnej

Streszczenie

Wpracyprzedstawionometodologięwyznaczania trwałości zmęczeniowejwzakre-
sie dużej liczby cykli przy obciążeniu losowymzwykorzystaniemmetody spektralnej.
Metodologia ta wykorzystuje funkcję gęstości widmowej mocy naprężenia ekwiwa-
lentnego przy wyznaczaniu trwałości zmęczeniowej wraz ze znanymi jednoosiowymi
modelami spektralnymi. Jakoprzykładprzedstawionomodele proponowaneprzezMi-
lesa i Dirlika. Wyróżniono dwie grupy kryteriów wieloosiowego zmęczenia: kryteria
opierające się na pojęciu płaszczyzny krytycznej i kryteria wykorzystujące niezmien-
niki stanu naprężenia.W celu porównania obliczeniowej i eksperymentalnej trwałości



254 A. Niesłony

zmęczeniowej wykorzystano dwa zestawy danych otrzymanych przy kombinacji zgi-
nania ze skręcaniem.Wykazano, że przy wieloosiowymobciążeniu losowym trwałości
obliczeniowe korelują dobrze z otrzymanymi podczas eksperymentu, jeżeli zostanie
zastosowane odpowiednie do właściwości mechanicznych materiału kryterium wielo-
osiowego zniszczenia zmęczeniowego.

Manuscript received May 4, 2009; accepted for print August 24, 2009