Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 349-361, Warsaw 2007

FATIGUE LIFE ESTIMATION OF NOTCHED SPECIMENS

UNDER BENDING AND TORSION WITH STRAIN ENERGY

DENSITY PARAMETER

Tadeusz Łagoda

Paweł Ogonowski

Faculty of Mechanical Engineering, Opole University of Technology

e-mail: t.lagoda@po.opole.pl

This paper presents an energy damage parameter based on the critical
plane under a complex loading state and stress concentration.Two crite-
ria including combinations of the normal and shear strain energy density
parameters have been proposed. The plane in which the normal or she-
ar strain energy density parameter reaches its maximum is assumed to
be the critical plane. The predictions of the proposed model have been
comparedwith constant amplitude fatigue tests under combinedpropor-
tional bending and torsion carried out on ring-notched specimens made
of 42CrMo4V steel.

Key words: multiaxial fatigue, notch, energy approach

Notations

C – coefficient determining stress in circumferential direction de-
pending on its concentration

E – Young’s modulus
i,j,k – unit vectors in Cartesian coordinate system
Kt – theoretical stress concentration factor

l̂η,m̂η, n̂η – direction cosines of unit vector η

l̂s,m̂s, n̂s – direction cosines of unit vector s
Nf – number of cycles to failure
Rp0.2 – yield stress
Rm – ultimate tensile strength
sgn[i,j] – two-argument logical function of signs of i, j variables
t – time
W – strain energy density parameter



350 T. Łagoda, P. Ogonowski

x,y,z – spatial coordinates of specimen
β,κ – coefficients determining criterion form obtained from unia-

xial fatigue test
ε – strain
ν – Poisson’s ratio
γ – engineering shear strain
σ – normal stress
τ – shear stress

Subscripts
a – amplitude s – shear
b – bending t – torsion
eq – equivalent η – normal

1. Introduction

Problems connected with notches are among the most important and widely
tested ones in thematerial fatigue. In such cases, stresses can bemuch greater
than calculated nominal ones and they cause decrease of the fatigue strength.

Problems connectedwith notch occur in the case of section discontinuities.
They cause local stress concentrations which can bemuch greater than calcu-
lated ones. Their magnitudes depend on notch geometry, material properties,
loadingpath, etc.Thenotch occurrence always leads to fatigue strength.Thus,
definition of an efficientmethod for fatigue life determination under stress con-
centrationbelongs to themost importantandwidely testedproblemsof fatigue
of materials.

The first stage of fatigue life determination under stress concentration sho-
uld include the definition of local stresses and elastic-plastic strains in the
notch root. Neuber’s method (Neuber, 1961) or the strain energy densityme-
thod (ESED) (Molski andGlinka, 1981) are themost widely applied for their
determination under cyclic loading. Both methods were generalized to a mul-
tiaxial loading state (Moftakar et al., 1995). The two methods differ only in
the way of determination of the plastic strain energy density. Let us note
thatNeuber’smethod gives overestimated values of strains and stresses in the
notch root, whereas the values obtained with ESED are lowered. The results
obtained according to the model proposed in Łagoda and Macha (1998) are
included between the results obtained according to the models proposed by
Molski and Glinka (1981), Neuber (1961). In the literature, we can also find
other energy models formulated by Inoue et al. (1996) and Ye et al. (2004)



Fatigue life estimation on notched specimens... 351

as well as the empirical model proposed by Grubisic and Sonsino (1982). De-
termination of local stresses and strains in the notch root requires application
of Hencky’s constitutive equations and complex numerical calculations. Un-
der a non-proportional loading, the calculations become even more complex
because the influence of the loading path must be taken into account and an
incremental notation of the chosenmethod should be used.Moreover, in order
to determine relations between stresses and plastic strains, it is necessary to
apply a model of plasticity (for example, the model formulated by Mróz or
Chu). Thus, pseudoelastic strains and stresses are often applied in fatigue cal-
culations. In Susmel (2004), a criterion based on the assumption of the elastic
stress state is proposed; the elastic stress state occurs in the notch neighbo-
urhood and it is determined with the finite element method. The fatigue life
has been determined fromnormal and shear stresses in the plane ofmaximum
normal stresses. Elastic stresses are also used in the criterion proposedbyGru-
bisic and Simburger (1976), where themaximum shear stress was assumed as
the equivalent stress amplitude.
In previous papers by Kardas et al. (2004), Łagoda and Macha (2000),

Sonsino andŁagoda (2004), the strain energy density parameter in the critical
planewasapplied for fatigue life estimation of smooth specimens. In thispaper,
the same parameter is used for notched components.
The aim of this paper is to formulate an energy based criterion formulated

according to stress and strain histories of multiaxial fatigue including stress
concentration. The predictions of the proposed model have been compared
with constant amplitude fatigue tests under combined proportional bending
and torsion carried out on ring-notched specimens made of 42CrMo4V steel
(Pötter, 2000).

2. Energy damage parameter under complex loading state

2.1. The strain energy density parameter in the uniaxial loading

In the uniaxial loading, the strain energy density parameter SEDP (Łago-
da, 2001) is defined as

Waxx(t)=
1

2
σaxx(t)εaxx(t)sgn[σaxx(t),εaxx(t)] (2.1)

where

sgn[i,j] =
sgn[i]+ sgn[j]

2
(2.2)

For a constant amplitude loading, we have the amplitude of SEDP

Waxx =
1

2
σaxxεaxx (2.3)



352 T. Łagoda, P. Ogonowski

and for an elastic material

Waxx =
σ2axx
2E
(1−νC) (2.4)

where 0¬C ¬ ν.

An interpretation of this coefficient, C, is shown in Fig.1. Its approximate
value can be determined from

C =
1.84ν

Kt
(Kt−1)

1−ν (2.5)

Fig. 1. The coefficient C depending on the stress concentration factor

The function given by Eq. (2.5) was based on calculations with the finite
elementmethod for different notch geometries, Kt =Kta for the axial loading
and Kt =Ktb for bending.

2.2. The strain energy density parameter in a multiaxial loading

For the proposed damage parameter under stress concentration, the as-
sumptions are the same as those for smooth specimens (Kardas et al., 2004;
Łagoda and Macha, 2000; łagoda et al., 1999; Łagoda and Ogonowski, 2005;
Sonsino and Łagoda, 2004, i.e.

• Fatigue cracking is caused by the part of strain energy density corre-
sponding to the work of normal stress ση(t) on the normal strain εη(t),
i.e. Wη(t) and the work of shear stress on shear strain εηs(t) in the
direction s in the plane with the normal η, i.e. Wηs(t);

• The direction s on the critical plane is themean direction, in which the
shear strain energy density reaches its maximum, Wηsmax(t);



Fatigue life estimation on notched specimens... 353

• In the limit state, the material effort is determined by the maximum
linear combination of the strain energy density parameters Wη(t) and
Wηs(t), which satisfies the following equation under a randommultiaxial
loading

Weq(t)=βWηs(t)+κWη(t) (2.6)

where

Wη(t)=
1

2
ση(t)εη(t)sgn[ση(t),εη(t)]

Wηs(t)=
1

2
τηs(t)εηs(t)sgn[τηs(t),εηs(t)] (2.7)

εηs(t)=
1

2
γηs(t)

The constants β and κ and the chosen criterion defining the critical
plane position lead to selection of one from the two special forms of
criterion (2.6).

2.2.1. The criterion of the strain energy density parameter on the plane determined
by normal strain energy density

The equivalent strain energy density parameter is a linear combination of
normal and shear strain energy densities. Participation of particular energies
in the damage process depends on the coefficients β and κ. The critical plane
is determined by the normal strain energy density parameter. It is assumed
that the critical planewith the normal η is determined by the normal loading.
The position of the vector s is defined by one of the directions determined by
the given shear loading, where the scalar product

η ·s=0 (2.8)

as it is presented in Fig.2

η= l̂ηi+ m̂ηj+ n̂ηk s= l̂si+ m̂sj+ n̂sk (2.9)

The following equation for the equivalent strain energy density parameter
results from (2.6)

Weq(t)=βWηs(t)+Wη(t) (2.10)

Considering stress and strain states for pure torsion and tension-
compression or bending under a constant-amplitude loading, it is possible to
determine relationships between the coefficients β and κ.

For bending on the plane of maximum tension, we obtain the same values
as in thecase of torsion, so it is notpossible todetermine the coefficient β in an



354 T. Łagoda, P. Ogonowski

Fig. 2. Interpretation of the normal stress ση(t), normal strain εη(t), shear
stress τηs(t) and shear strain εηs(t) acting in the s direction, on the plane with the

normal η

analytical way. This coefficient can be selected for a given material from non-
proportional tests. It could be done in amplitude-constant amplitude fatigue
tests with a phase shift equal to π/2 between bending and torsion.

Criterion (2.10) is proposed for cast irons and welded steel joints (Łagoda
and Ogonowski, 2005).

2.2.2. The criterion of the strain energy density parameter on the plane determined
by shear strain energy density

The equivalent strain energy density is a linear combination of normal and
shear strain energydensities.Participation ofparticular energies in thedamage
process is dependent on the coefficients β and κ as shown in Section 2.1. In
such a case, the critical plane is determined by the shear strain energy density
parameter. It is assumed that Q=Waf and that the critical plane with the
normal η and tangent s is determined as themean position of one of the two
planes, where themaximum shear strain energy density occurs. As previously,
the equivalent strain energy density parameter is determined from Eq. (2.6).

Considering the stress and strain states for pure torsion and tension-
compression or bendingundera constant-amplitude loading,we can determine
equations relating the coefficients β with κ.

Thus, the β parameter is

β= k2
1−υC

1+υ
(2.11)



Fatigue life estimation on notched specimens... 355

and κ parametr is

κ=
[4−k2(1−C2)](1−υC)

(1−υ)(1+C)2
(2.12)

where

k(Nf)=
σaxx(Nf)

τaxy(Nf)
(2.13)

Having found the coefficients β and κ, Eq. (2.7)1 takes the following form

Weq(t)= k
21−υC

1+υ
Wηs(t)+

[4−k2(1−C2)](1−υC)

(1−υ)(1+C)2
Wη(t) (2.14)

Let us note that for C = 0 equation (2.14) is equivalent to the parameter
used in the proposed criterion for smooth specimens, see Kardas et al. (2004),
ŁagodaandMacha (2000), ŁagodaandOgonowski (2005), SonsinoandŁagoda
(2004).

Criterion (2.14) is proposed for steel and aluminiumalloy aswell aswelded
alloys (Łagoda and Küppers, 2006; Łagoda andOgonowski, 2005).

3. Verification of proposed parameter

Specimensmadeof 42CrMo4Vsteelwere testedbyPötter (2000).Notched spe-
cimenswere subjected to tests as shown inFig.3.Calculationswere performed
for bending, torsion and combined bending with torsion. Static properties of
the steel tested are given in Table 1, and its chemical composition – in Ta-
ble 2. The calculated stress concentration factors are Ktb = 2.0 for bending
and Ktt =1.5 for torsion.

Fig. 3. Geometry of tested specimens



356 T. Łagoda, P. Ogonowski

Table 1. Static properties of 42CrMo4V steel (Neuber, 1961)

Rp0.2 [MPa] Rm [MPa] RA E [GPa] ν

743 920 69 210 0.3

Table 2.Chemical composition of 42CrMo4V steel in % (Neuber, 1961)

C Mn Si P S Cr Mo Fe

0.38-0.45 0.5-0.8 < 0.4 < 0.035 < 0.03 0.9-1.2 0.15-0.30 remaining

Thecriterionofmaximumparameter on theplaneofmaximumshear strain
energy density was used for verification. In the tests, C = ν for Ktb ­ 2.0was
assumed, so Eq. (2.14) was reduced to the following form

Weq(t)= k
2(1−υ)Wηs(t)+

4−k2(1−υ)2

1+υ
Wη(t) (3.1)

For a constant amplitude loading, criterion (3.1) reduces to

Waeq = k
2(1−υ)Waηs+

4−k2(1−υ)2

1+υ
Waη (3.2)

In the proposed parameter, the criterion form depends on the coefficient
k(Nf). In Kardas et al. (2004), the coefficient k was determined at the level
of fatigue limit for bending. Pötter (2000) used the stress criterion where the
ratio of shear to normal stresses was determined for 105 cycles. However,
the considered material is characterized by a large difference of slopes on the
Wöhler curve for bending mσ = 5.7 and for torsion mτ = 11.9 (Fig.4),
so assuming the stress ratio for the determined number of cycles does not
give good results (Fig.5 and Fig.6). In Fig.5, the coefficient k is: k(Nf) =
k(105 cycles) and in Fig.6 – k(Nf) = k(σaf/τaf) = k(2.5 ·10

5 cycles). Much
better results could be obtained if the coefficient k is a number of cycles
dependent (Fig.7) according to relationship (2.13). Thus, k and the criterion
form take different values for successive numbers of cycles. In Table 3, values
of constant k as well as β and κ for three numbers of cycles are presented.

Table 3.Values of the constant k and coefficients β and κ depending on
the number of cycles

Nf σa τa
k β κ

[cycles] [MPa] [MPa]

104 687 355 6.67 4.67 0.56

105 406 292 3.43 2.40 1.78

2.5 ·105 330 271 2.65 1.85 2.08



Fatigue life estimation on notched specimens... 357

Fig. 4.Wöhler’s curves for bending and torsion in the stress notation

Fig. 5.Wöhler’s curves in the energy approach based on calculations against the
graph for pure bending (k for the fatigue limit in bending)

It appears from Fig.5 and Fig.6 that most results for torsion are located
outside the scatter bandwith a factor of 2, characteristic for pure bending. In
the case of k for 105 cycles, also under the combined bending with torsion,
some points are located outside the scatter band. If the coefficient k is deter-
minedby an iterativemethodversus the number of cycles, all of the results are
included in the scatter bandwith a factor of 2, as in the case of pure bending.



358 T. Łagoda, P. Ogonowski

Fig. 6.Wöhler’s curve in the energy approach based on calculations against the
graph for pure bending (k for 105 cycles)

Fig. 7.Wöhler’s curve in the energy approach based on calculations against the
graph for pure bending (k determined versus the number of cycles)

4. Conclusions

• In this paper, an energy damage parameter is proposed for a combined
loading by bending with torsion when stress concentration occurs.

• Two forms of the parameter are proposed, which assume that the plane
of maximumnormal or shear strain energy density is the critical plane.



Fatigue life estimation on notched specimens... 359

• The proposed parameter was investigated for 42CrMo4V steel, and the
best results were obtained if the coefficient k depended on the num-
ber of cycles for the criterion of the strain energy density parameter
in the critical plane. The critical plane is one of the planes where the
shear strain energy density parameter reaches its maximum (maximum
damage).

• Further verification of the proposed model should be done for other
materials, other stress concentration factors and other loadings.

Acknowledgement

Theauthorswould like to thank the staffofTUClausthal, especiallyProf.H.Zen-

ner for hospitality, discussion and giving us the test results.

References

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Ocena trwałości zmęczeniowej próbek z karbem poddanych zginaniu ze

skręceniem z wykorzystaniem parametru gęstości energii odkształceń

Streszczenie

Wpracy przedstawiono energetyczny parametr uszkodzenia oparty na płaszczyź-
nie krytycznej w warunkach złożonego stanu obciążenia i spiętrzenia naprężeń. Za-
proponowano dwa kryteria uwzględniające kombinacje parametrów gęstości energii



Fatigue life estimation on notched specimens... 361

odkształcenia normalnego i postaciowego. Za płaszczyznę krytyczną przyjęto płasz-
czyznę,wktórej parametr gęstości energii odkształcenianormalnego lubpostaciowego
osiąga wartośćmaksymalną.Wyniki obliczeń uzyskane za pomocą zaproponowanego
modelu porównano z wynikami stałoamplitudowych badań zmęczeniowych próbek ze
stali 42CrMo4V z karbem obrączkowym wykonanych w warunkach kombinacji pro-
porcjonalnego zginania ze skręcaniem.

Manuscript received June 1, 2006; accepted for print January 17, 2007