A bi-parametric Wohler curve for high cycle multiaxial fatigue

assessment
L . S U S M E L 1 a n d P. L A Z Z A R I N 2
1

Department of Mechanical Engineering, University of Padova, Padova, Italy, 2Department of Management and Engineering, University of Padova,
Vicenza, Italy
Received in nal form 27 July 2001

A B S T R A C T This paper presents a method for estimating high-cycle fatigue strength under multiaxial

loading conditions. The physical interpretation of the fatigue damage is based on the
theory of cyclic deformation in single crystals. Such a theory is also used to single out
those stress components which can be considered signicant for crack nucleation and
growth in the so-called Stage I regime. Fatigue life estimates are carried out by means
of a modied Wohler curve which can be applied to both smooth and blunt notched
components, subjected to either in-phase or out-of-phase loads. The modied Wohler
curve plots the fatigue strength in terms of the maximum macroscopic shear stress
amplitudes, the reference planewhere such amplitudes have to be evaluatedbeing
thought of as coincident with the fatigue microcrack initiation plane. The position of
the fatigue strength curve also depends on the stress component normal to such a plane
and the phase angle as well. About 450 experimental data taken from the literature are
used to check the accuracy of the method under multiaxial fatigue conditions.
Keywords
crystal.
NOMENCLATURE

cyclic deformation; high cycle fatigue; multiaxial loads; notch; single

E (%)=fatigue strength error index in percentage

kt =inverse slope of the fatigue curves in the modied Wohler diagram
Kf,ax =axial fatigue strength reduction factor
Kf,tors =torsional fatigue strength reduction factor
m=unit vector of the easy glide direction
m*=direction of maximum resolved shear macro-stress
M=generic direction on the D plane
Ms , Ts =Papadopoulos integrals
n=unit vector nornal to the generic D plane
N=number of cycles
Nf =number of cycles to failure
NRef =number of cycles assumed as a reference value
Oxyz=reference frame
Oab=reference frame on the generic plane D
X, Y =parameters depending on the applied loadings
a, b=parameters depending on the material fatigue strength
d=out-of-phase angle
w, h=angles which dene the position of a generic plane D
w*, h*=angles which dene the initiation plane
w, h=angles which dene the plane on which the Ts value is maximum

Correspondence: P. Lazzarin, Department of Management and

Engineering, University of Padova, Stradella S.Nicola 3-36100
Vicenza, Italy.
E-mail: plazzarin@gest.unipd.it

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63

64

L . S U S M E L a n d P. L A Z Z A R I N

D=generic plane
C=accumulated plastic strain
mcpl =microscopic plastic shear strain
mt=microscopic shear stress
mty0 =microscopic initial yield limit of a crystal
r=stress ratio related to the initiation plane
sA =fully reversed axial fatigue limit
2
sH,max =maximum hydrostatic stress
sn,max (w*, h*)=maximum stress normal to the initiation plane
sT =tensile strength
t=shear stress
ta (w*, h*)=shear stress amplitude on the plane of the maximum shear stress
amplitude
tA,Ref =fatigue strength corresponding to NRef cycles
tA =fully reversed torsional fatigue limit
2
tm (w*, h*)=mean shear stress on the initiation plane
tr =resolved shear stress
tr,a =resolved shear stress amplitude
V=easy glide plane

INTRODUCTION

Mechanical components frequently work under multiaxial fatigue loadings and, for this reason, the problem of
the multiaxial fatigue assessment has long been investigated and continues to be investigated by many researchers. The state of the art shows the approaches vary
mainly as a function of the fatigue life and are different
for low-cycle fatigue and high-cycle fatigue. The most
popular low-cycle fatigue life estimation techniques are
based on a strain approach (see, for example, the critical
plane based criteria proposed by Socie and coworkers,14
Brown and Miller5 and Wang and Brown6 as well as the
energy criterion introduced by Ellyin,79 ). These criteria
are sometimes extended to high-cycle fatigue, where the
plastic strain contribution becomes negligible.1012 On
the contrary, all the multiaxial criteria devoted solely to
high-cycle fatigue are based exclusively on stress.
This holds true, for example, for the microscopicapproach-based criteria proposed by Dang Van13 or
Papadopoulos14,15 and for the critical-plane-based criteria due to McDiarmid16,17 Matake18 and Findley.19
Recently, the critical plane approach has been reviewed
and modied by Carpinteri and Spagnoli.20 Their
new criterion correlates the critical plane orientation
with the weighted mean principal stress directions.
Accordingly, the fatigue failure assessment is performed
by considering a non-linear combination of the maximum normal stress and the shear stress amplitude acting
on the critical plane.
The older and universally known, studies on uniaxial

fatigue problems are based on a stress approach and

their inuence on fatigue assessment techniques has been
crucial up to now. This fact results in two main consequences: a large amount of experimental data obtained
under load control is reported in the technical literature,
and the SN Wohler curves continue to be used regularly,
at least in some European countries, by engineers
engaged in fatigue strength problems.
In this paper a multiaxial life estimation method based
on a modied, non-conventional, Wohler curve is presented. The method takes advantage of some recent
ndings of multiaxial fatigue studies and, in particular,
uses the theory of cyclic deformation in single crystals
and the initiation plane concept21,22 to give a physical
interpretation of fatigue damage and to single out those
stress components that are considered to be really signicant to crack initiation and growth during the
so-called Stage I regime, as dened in Millers
papers.23,24
SOME PRELIMINARY DEFINITIONS

Consider a body subjected to a cyclic loading and point

O located on its surface (Fig. 1a). This point is thought
of as the critical point for fatigue strength, so that it is
considered as coincident with the centre of the absolute
reference frame Oxyz (Fig. 1a). The position of a material plane D, having n as normal unit vector, can be
determined by means of the spherical co-ordinates w and
h: the former is the angle between the projection of the
unit vector n on the xy plane and the x-axis, the latter

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65

Fig. 1 Reference frame and denition of

the spherical co-ordinates h and w.

the angle between the normal n and the z-axis22 (see

Fig. 1b).
Considering the generic plane D, in each instant t it
is possible to subdivide the stress state into two components: the normal stress sn (t) and the shear stress t(t).
The amplitude, the mean value and the maximum value
of the stress component normal to D can be expressed
by the following relationships:22
sn,a (w, h)=

G
G

H
H

1
max sn (w, h, t)min sn (w, h, t)
2 tT
tT

sn,m (w, h)=

1
max sn (w, h, t)+min sn (w, h, t)
2 tT
tT

(1)

sn,max (w, h)=sn,a (w, h)+sn,m (w, h)

The denition of the amplitude and the mean value of
the shear stress tangential to a generic plane is much
more complex, mainly because the vector t(t) changes
its magnitude and direction during the cyclic load. In
order to dene ta and tm , we intend here to use the
concept of the minimum circumscribed circle according
to Papadopoulos.22 More precisely, the shear stress
amplitude is equal to the radius of the minimum circle
that circumscribes the curve Y (the curve Y being
described by the tip of the shear stress vector t(t) on the
D plane during the cyclic load), whereas the mean shear
value is dened as the magnitude of the vector that joins
point O with the centre of the minimum circumscribed
circle (Fig. 2).
Finally, it is important to highlight that in the present
paper the stresses will always be referred to the net area.

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

THE MULTIAXIAL FATIGUE DAMAGE

ACCORDING TO THE THEORY OF THE CYCLIC
DEFORMATION IN SINGLE CRYSTALS

Consider the generic body shown in Fig. 3(a). A volume

V containing point O is considered and its dimensions
are dened in such a way that the stress state can be
assumed to be a constant and equal to the value related
to point O. In addition, the material is thought of as
homogeneous and isotropic.
In a crystal the microscopic cyclic stress state creates
persistent slip bands25 and such bands are parallel to an
easy glide plane V [Fig. 3(b)]. After a certain number of
cycles, a microcrack initiates in the crystal because of a
microstress concentration effect due to the presence of
a deep intrusion21,26 [Fig. 3(b)].
The damage phenomenon is mainly inuenced by the
value of a microplastic shear strain acting on the easy
glide plane V and calculated with respect to the easy
glide direction m.21,27 The theory of cyclic deformation
in single crystals makes use of the cumulated plastic
strain C to weigh the fatigue damage of the single crystal.
In particular, C is dened as follows:21
N

C= |mcpl,i m|

(2)

i=1

where mcpl,i is the microscopic plastic shear strain amplitude in the i-th cycle and N is the total number of cycles.
Under the hypothesis of a purely elastic macroscopic
strain, the macroscopic shear stress versus the microscopic plastic shear stress relationship can be expressed

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L . S U S M E L a n d P. L A Z Z A R I N

Fig. 2 ta and tm denitions based on the

minimum inscribed circle concept due to
Papadopoulos.

by the following equation:27,28

tm=h(mcp m)

(3)

where h is a monotonic function. Equations (2) and (3)

show that the cumulated plastic strain value depends
both on the macro-shear stress (calculated along the easy
glide direction m) and the number of cycles to failure
N. Papadopoulos was able to demonstrate27 according
to the elastic shakedown state concept, that the scalar
entity C can be expressed for an innite number of
loading cycles as:
1
C = (ta mmty0 )
2 g

(4)

where g is a dimensional constant positive and mty0 is the

initial yield limit of the crystal.
Equation (4) shows that, when N2, the accumulated plastic strain is proportional only to the amplitude
of the resolved macro-shear stress13,21,22 and no longer
to N.
Note that the proposed considerations are here
expressed under the hypothesis of purely elastic macroscopic deformation and they are related to an easy glide
plane and an easy glide direction of a single crystal. The
situation in a real material is much more complex, mainly
because of the presence of grain boundaries, non-metallic
inclusions, precipitates, defects, and so on. As a consequence, it is not possible to know a priori the easy glide
plane and the direction of the grains which are likely
to break.
Nevertheless, the concepts of easy glide plane/direction can be applied to a real material if two further
simplifying hypotheses are accepted. First, in a polycrystal at room temperature the fatigue cracks occur mainly

in a transcrystalline mode in the persistent slip bands

(PSBs).25 Second, the material can be thought of as
homogeneous and isotropic. Such hypotheses make it
possible to afrm that, from a statistical point of view,
each plane locates an equal number of grains that
have an easy glide plane coincident with the considered
plane [Fig. 4(a),(b)]; moreover, each direction on a plane
locates an equal number of grains that have an easy
glide direction coincident with the analysed direction
[Fig. 4(c)]. These simple remarks, together with Eq. (3)
suggest that the microscopic fatigue crack occurs on the
plane of maximum macroscopic shear stress amplitude
and the maximum fatigue damage is produced along the
direction of maximum macroscopic resolved shear stress
amplitude m*. Since m* direction is implicitly and
unambiguously determined by using the ta denition
proposed by Papadopoulos (Fig. 2), the shear stress
amplitude calculated by means of the minimum circumscribed circle is always proportional to the plastic strain
cumulated by the unfavourably positioned crystals, i.e.
ta is proportional to the fatigue damage of the unfavourably orientated crystals. This last statement shows that
it is not necessary to calculate the fatigue damage in
each direction, as alone by Papadopoulos27 since the ta
denition intrinsically represents the shear stress fatigue
damage calculated along each direction.
Papadopouloss theory was proposed according to the
elastic shakedown state concept. A fatigue limit is reached
if, at the mesoscopic scale, some plastically deforming
grains tend to recover a purely elastic response. In that
case initiation is avoided. In other words, Papadopoulos
uses asymptotic quantities and then deals with innite
fatigue life by comparing the maximum C with the
2
corresponding shakedown value. In order to extend

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Fig. 3 The elementary volume V, the micro-stress state affecting

the single crystal (a) and the persistent slip bands in a single crystal
(b).

Fig. 4 Unfavourably positioned crystals

located by the D1 (a) and D2 (b) planes and
by the m1 and m2 directions on the D plane.

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67

Papadopouloss theory to nite fatigue life, Morel28

proposed a fatigue life prediction method built according
to a microscopic description of the damage accumulation.
The initiation process of a crack is treated as a phenomenon taking place on a scale of the order of a grain or a
few grains. A crack is supposed to initiate as a consequence of the failure of some grains which are less
resistant to plastic deformation. Such grains are assumed
to behave as crystals following three successive phases:
hardening, saturation and softening. In order to accurately describe all these phases, the damage variable
chosen is always the accumulated plastic strain at its
mesoscopic scale. Its estimation requires the location of
the plane subjected to maximum damage. This operation
is achieved by maximizing the measure of the accumulated plastic mesostrain on every material plane of an
elementary volume V.
Morels method, which gives very satisfactory results
in high cycle fatigue28 demonstrated that the fatigue
damage in each phase is always proportional to the
macroscopic resolved shear stress vector acting in an
easy glide direction. It is interesting to note that, by
proceeding on parallel tracks, the combined use of the
single crystal cyclic deformation theory and the minimum circumscribed circle concept makes it possible to
single out the same entity as a fundamental stress component for the multiaxial fatigue damage.
Finally, on the basis of the previous considerations, it
is also possible to state that the plane of maximum
macroscopic shear stress amplitude can be thought of as
coincident with the fatigue microcrack initiation plane.
This statement fully agrees both with Millers Stage I
concept23,24 and the experimental data obtained by Socie
and Bannantine.30 when testing AISI 304, Inconel 718
and SAE 1045 specimens. Such data demonstrated, in
particular, that shear crack nucleation was followed by
crack growth on planes of maximum principal strain,
even if the nal failure was controlled by Mode I.30
Fatigue crack initiation as well as fatigue crack growth

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L . S U S M E L a n d P. L A Z Z A R I N

are conditioned by the stress normal to the initiation

plane. The inuence of this component during the
growth can be explained by transferring Socies fatigue
damage model1,2 to the microscopic scale and by
observing that the normal macro-stresses are transmitted
on the microscopic scale without being altered:27 a
normal traction stress component opens the microcrack
and, consequently, favours growth; on the other hand, a
compression stress component slows down the growth
because of the friction between the faced microcrack
surfaces [Fig. 5(b)]. In the same way, the normal stress
component also inuences the persistent slip band (PSB)
formation process, i.e. the microcrack initiation process,
because a compression component inhibits the PSB
laminar ow, whereas a traction component favours their
ow [Fig. 5(a)].
In conclusion, the theory of cyclic deformation in
single crystals suggests that the fatigue damage in a
polycrystal depends on the maximum shear stress amplitude (determined by the minimum circumscribed circle
concept) and on the stress component normal to the
crack initiation plane. The model presented here should
be used to describe the fatigue damage only during
Stage I (shear crack), according to Millers denition.23,24
Then, from a rigorous point of view, the predicted
fatigue life should be thought of as coincident with the
number of cycles required to exhaust Stage I.
THE NEW METHOD FRAME

Consider a body subjected to a multiaxial cyclic load

[Fig. 1(a)] and dene the plane of maximum shear stress
amplitude by using the minimum circumscribed circle
concept (Fig. 2).
With reference to this plane, it is possible to dene
for the crack initiation plane the stress ratio r as follows:
r=

sn.,max
(w*, h*)
ta

(5)

In Eq. (5) the maximum value of normal stress is used

to take into account the inuence of mean stress on
multiaxial fatigue strength.1,30
It is worth noting that the parameter r makes it
possible to take into account directly the phase angle
inuence on multiaxial fatigue strength, as shown in
Fig. 6. In particular, Fig. 6(b), (c) show that the r ratio
varies as a function of the phase angle d, even when the
applied load amplitudes are kept constant. This holds
true both for axial/torsional loads [Fig. 6(b)] and for a
combined application of an axial load and an internal
pressure [Fig. 6(c)]. Obviously, the presence of out-ofphase loadings results, in general, in variations of r,
shear stress amplitude and initiation plane.
Consider now a modied loglog Wohler diagram
wherethe abscissa is the number of cycles to failure and
the ordinate the shear stress amplitude ta (w*, h*) calculated on the initiation plane [Fig. 7(a)]. Two fatigue
curves are usually available for materials: the fully
reversed axial and the torsional fatigue curves. When
plotted in the modied Wohler diagram, they are represented by two different straight lines: the axial and the
torsional fatigue curves correspond to r=1 and r=0,
respectively [Fig. 7(a)]. In general, such curves have
different values of the inverse slope kt [being kt (r)=
tan a(r)] and are identied by the values tA,Ref (r=1)
and tA,Ref (r=0) at a number of cycles to failure NRef ,
to be assumed as reference value (for example, 2106
cycles in some design codes). If one would use the Von
Mises hypothesis simply to estimate the trend, the ratio
between the torsion and tension reference values is equal
to:
2
tA, (r=0)
2
$1.155
(6)
=
tA, (r=1) 3
2
Then it is natural to think that, in the modied Wohler
diagram, the more r increases, the more the fatigue
curve moves downwards. If experimental data were so

Fig. 5 Application of the Socie fatigue

damage model to interpret both the
persistent slip band laminar ow (a) and the
micro-crack growth (b).

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69

Fig. 6 Inuence of phase angle d on stress components for a cylindrical specimen (a) subjected to a stress state generated by axial force and
internal pressure (b) and by axial force and torsional moment (c, d).

numerous as to make it possible to evaluate the functions

tA,ref (r) and kt (r), a fatigue life prediction for a multiaxial
stress state could be carried out by using the following
expression:

tA,Ref (r)
Nf =
ta (w*, h*)

k(r)

NRef

(7)

where Nf is the number of cycles to failure. Equation

(7) represents a bi-parametric non conventional Wohler
curve given in terms of r and the shear stress component
amplitude. This procedure can also be used to estimate
the fatigue life of notched components, by altering the
fatigue curves by means of the fatigue strength reduction
factor Kf 31 and then involving in the multiaxial fatigue

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

life assessment the nominal net section stresses

[Fig. 7(b)].

FORMALIZATION OF THE METHOD FOR HIGHCYCLE FATIGUE PROBLEMS

The accuracy of the method proposed for multiaxial

fatigue life estimations largely depends on the number
of experimental data used to determine the functions
tA,Ref (r) and kt (r). The precision is obviously expected
to increase with the number of modied Wohler curves
available to perform the calibration of the model.
In most cases, the experimental data of fatigue strength
refer to fully reversed axial and torsional loading con-

70

L . S U S M E L a n d P. L A Z Z A R I N

Fig. 7 Modied Wohler curves (a) and

fatigue curves valid for smooth and notched
components (b).

ditions, so that only two different fatigue limit values

and two different inverse slope values are available to
determine the functions tA,Ref (r) and kt (r). This directly
implies giving tA,Ref (r) and kt (r) only in terms of two
simple linear functions.
By assuming the linearity for tA,Ref (r) and using the
uniaxial and torsional reference shear stresses to calibrate
the model, tA,Ref (r) can be written as follows:
tA,Ref (r)=tA,Ref (r=0)

(8)

+r[tA,Ref (r=1)tA,Ref (r=0)]

The condition to be assured under multiaxial high cycle
fatigue is:
ta (w*, h*)tA,Ref (r)
By introducing Eq. (8) into Eq. (9) one obtains:

(9)

ta (w*, h*)+[tA,Ref (r=0)tA,Ref (r=1)]

sn,max
(w*, h*)
ta

tA,Ref (r=0)

(10)

When reference shear stress values correspond to fatigue

limits, Eq. (10) becomes:

sn,max
sA
(w*, h*)tA
ta (w*, h*)+ tA 2
2
2
2
ta

(11)

It is worth noting that, while the qualitative interpretation of the multiaxial fatigue damage is based on the
theory of cyclic deformation in single crystals, the quantitative evaluation uses a modied Wohler curve in conjunction with the crack initiation plane concept.
In order to be applied, Eq. (11) requires the values of
the fully reversed fatigue limits sA and tA . Such
2
2
values are summarized in Table 1 for all the materials

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71

Table 1 Data related to smooth specimens

Materials

Refs

sA (MPa)
2

tA (MPa)
2

sT (MPa)

Applied loads*

No. data

Carbon steel 0.35% C

Hardened steel 0.51% C
Soft steel 0.1% C
Cast iron 3.87% C
Duraluminium 3.81% Cu
Grey cast iron 3.32% Cu
42CrMo4
34Cr4
30NCD16
CK45
SAE4340
SAE1045
XC18
FGS 800-2
EN24T
25CrMo4
EN25T
St35
0.1% C steel (normalized)
0.4% C steel (normalized)
0.4% C steel (spheroidized)
0.9% C steel (pearlitic)
3% Ni steel
3/3.5% Ni steel
Cr-Va steel
3.5% NiCr steel (normal impact)
3.5% NiCr steel (low impact)
NiCrMo steel (6070 tons)
NiCrMo steel (7580 tons)
NiCr steel
SILAL cast iron
NICROSILAL cast iron
Brass
Hard steel
Soft steel
Carbon steel
Cast iron
CrMo steel
CrMo steel
CrMo steel
CrMo steel
CrMo steel
CrMo steel
NiCrMo steel S81
NiCrMoVa steel
CrMoVa steel DTD551
CrMoVa steel DTD551
CrMoVa steel DTD551
CrMoVa steel DTD551
NiCr steel
NiCr steel
NiCr steel

[42]
[43]
[43]
[43]
[43]
[44]
[45]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[15]
[53, 54]
[55]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[41]
[17]
[17]
[17]
[56]
[57]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]]

215.8
313.9
235.4
96.1
156
143
398
410
660
423
462
211.5
332
294
405
361
476
189
268.6
331.9
274.8
352.0
342.7
352.0
429.1
540.3
509.4
725.6
660.7
810.4
240.8
253.2
83.0
460.0
196.0
261.0
151.0
713.2
688.9
509.0
589.6
593.3
628.3
589.7
660.7
667.8
659.9
706.1
737.7
666.7
653.2
771.9

127.2
196.2
137.3
91.2
100
110
260
256
410
287
286
125.5
186
220
270
228
273
122
151.3
206.9
155.9
240.8
205.3
267.1
257.8
352.0
324.2
484.7
342.7
452.3
219.2
211.5
74.0
275.0
186.0
160.0
92.0
425.3
412.8
306.9
367.6
350.7
366.6
331.9
342.7
398.3
386.5
412.5
447.4
369.7
339.6
452.3

570
694
382
185
443
279
1025
795
1880

PPT
BT
BT
BT
BT
BT
BT
BT
BT
PPT
PPT
PPT
BT
BT
PPPr
PPTPr
PPPr
PPPr
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
PPPr
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT
BT

6
24
15
12
14
15
9
14
10
8
3
5
3
3
11
8
5
8
10
5
5
5
5
5
5
10
5
5
5
5
5
5
2
2
2
3
5
6
6
6
3
6
6
6
6
6
6
3
6
6
6
5

*B, bending; PP, pushpull; T, torsion; Pr, internal/external pressure.

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

621
1530
815
850

430.7
648.8
477.0
847.5
526.4
722.5
751.8
895.3
896.9
1000.3
1242.7
1667.2
230.0
219.2

946.5
946.5
954.0
944.8
944.8
954.0
1103.8
1242.7
1397.1
1397.1
1368.7
1368.7
1398.4
1398.4
1667.2

72

L . S U S M E L a n d P. L A Z Z A R I N

we will consider in the re-analysis of multiaxial fatigue

data. Equation (11) will be applied also to fatigue data
characterized by values of r greater than 1.0.
VALIDATION OF THE METHOD

Equation 11 of the proposed criterion can be rewritten

in the form:
X+aY b

(12)

where parameters X and Y depend on the applied loads

and a and b are, on the contrary, two constants which
depend on the material fatigue properties. The accuracy
of the multiaxial fatigue predictions can be quantied by
using the fatigue strength error index E %:12
E %=

X
1
baY

(13)

if E=0 the fatigue life estimation is exact; if E<0 the

prediction is non-conservative; nally, if E>0 the prediction is conservative.
The degree of accuracy of Eq. (11) has been checked
by using hundreds of fatigue strength data taken from
the literature and already re-organized into a database.32
In particular, the validation of the method involves a
total number of 447 experimental data subdivided into
72 different series. The data refer to cylindrical specimens, both smooth and notched, made of different
materials. Information about the different series of data
is given in Table 1 and in Table 2 for smooth and notched
specimens, respectively. It is worth noting that the
notches are quite blunt, the fatigue strength reduction

factor ranging only between 1.17 and 2.48. It is the

authors opinion that in the presence of severe or sharp
V-shaped notches the method should be re-formulated
on the basis of the notch stress intensity factors (N-SIFs)
of the relevant geometries, by upgrading some procedures already illustrated for plane fatigue problems.3336 Such procedures initially proposed for a
material obeying a linear elastic law, have recently been
extended37 also to cases of small scale plasticity where
plastic N-SIFs substitute elastic N-SIFs. The more
severe the notch is, the easier the identication of the
crack initiation plane should be. It is reasonable to think
that, in the presence of sharp V-shaped notches the
geometry, and no longer the stress ratio r, fully controls
the crack initiation and propagation phenomena. Data
already reported by Ritchie et al.38 related to sharp
V-shaped notches subjected to combined tension and
torsion loads, seem to strongly support this hypothesis.
Figure 8 plots tA as a function of sA , by using the
2
2
data reported in Tables 1 and 2. The diagram shows that
estimating the fully reversed torsion fatigue limit by
using the Von Mises hypothesis is generally a good
approximation which, in most cases, is also in the safe
direction; this judgement holds true for both smooth
and notched specimens. Note that each point of the
diagram represents a series of experimental data. Two
aspects complicated the re-analysis shown in Fig. 8:
(a) sometimes only the mean values were reported in the
original papers and the information about the statistical
dispersion of the experimental data was often not available; (b) some of the published fatigue limits were
obtained by extrapolating run-out data. These consider-

Table 2 Bending/torsion experimental data obtained using notched specimens

Materials

Refs

Specimen shape

sNA (MPa)
2

tNA (MPa)
2

Kf,ax

Kf,tors

No. data

0.4% C steel (normalized)

3% Ni steel
3/3.5% Ni steel
Cr-Va steel
3.5% NiCr steel (normal impact)
3.5% NiCr steel (low impact)
NiCrMo steel (75-80 tons)
CrMo steel
CrMo steel
CrMo steel
CrMo steel
CrMo steel
NiCrMoVa steel
NiCrMoVa steel
CrMoVa steel DTD551
CrMoVa steel DTD551
CrMoVa steel DTD551
CrMoVa steel DTD551
NiCr steel

[41]
[41]
[41]
[41]
[41]
[41]
[41]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]
[58]

Sharp V
Sharp V
Sharp V
Sharp V
Sharp V
Sharp V
Sharp V
Oil hole
Oil hole
Oil hole
Oil hole
Oil hole
Notched
Oil hole
Oil hole
Oil hole
Oil hole
Oil hole
Oil hole

179.1
209.9
302.6
216.1
268.6
247.0
271.7
450.9
223.4
424.7
423.4
423.4
271.7
288.6
300.7
297.8
471.3
448.1
312.4

176.0
151.3
183.7
160.6
236.2
182.2
240.8
295.5
162.1
305.5
300.3
288.4
240.8
211.5
235.4
220.9
354.6
354.9
225.5

1.85
1.63
1.47
1.99
2.01
2.06
2.43
1.53
2.28
1.39
1.40
1.48
2.43
2.29
2.19
2.48
1.57
1.65
2.09

1.18
1.36
1.45
1.61
1.49
1.78
1.42
1.40
1.89
1.20
1.17
1.27
1.42
1.62
1.64
2.03
1.26
1.26
1.51

5
5
5
5
5
5
5
6
6
3
6
6
5
3
6
6
3
6
6

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A B I - PA R A M E T R I C W O H L E R C U R V E

73

tA(MPa)

the other hand the severity of the notches was not so

strong as to overwhelm the role of the shear stress
amplitudes in describing multiaxial fatigue phenomenon.
Finally, it is useful to point out that in the combined
bending/torsion tests (312 in total), parameter r was
greater than 1.0 in 34 cases, its maximum value being
4.0. In the other tests (pushpull, torsion and pressure,
36 data in total), 24 data exhibited a r-value ranging
from 1.1 to 3.2. Equation (11) was seen to work well in
these cases.

sA(MPa)
Fig. 8 Fully reversed torsional fatigue limit versus fully reversed
uniaxial fatigue limit (fatigue limits of the notched specimens being
given in terms of nominal stresses on the net area).

ations also hold true for the multiaxial fatigue data

reported in Figs 9 and 10. Moreover, a rigorous validation of the proposed method should be carried out by
using only fatigue data related either to the fatigue crack
initiation phase or to Stage I fatigue. Unfortunately, the
numbers of cycles of almost all the data reported in the
literature referred to the complete rupture of the specimens and the fatigue limit was extrapolated at a certain
number of cycles. In any case, it is well-known that for
smooth specimens the fatigue crack initiation life is a
large proportion of the total fatigue life and this also
holds substantially true for blunt notched specimens.
However, it is evident that. using total life fatigue data
increases the scatter between predictions and experimental data. Nevertheless, the agreement is very satisfactory,
conrming the validity of the method.
In Fig. 9 the shear stress amplitudes calculated on the
initiation plane ta (w*, h*) are reported versus the estimated multiaxial fatigue limit bar. Figs 9(a), (b) refer
to data obtained with smooth specimens, either in-phase
or out-of-phase. It is evident that there is a good
agreement between the new method and the experimental results: in particular, in the case of in-phase load,
predictions are given within a fatigue strength error
index E of about 20%; in the case of out-of-phase
loads, E ranges between 30%.
Finally, an even more favourable trend can be noticed
in Fig. 9(c), where notched specimen data are taken into
account. The method makes it possible to summarize all
the experimental points in a very restricted error band
(15%<E<15%). It is evident that in the presence of
notches characterized by fatigue strength reductions
factors Kf ranging between 1.17 and 2.48, the accuracy
of theoretical prediction improves. This might be mainly
due to the reduction of the critical volume (the so-called
structural volume) in which fatigue crack initiates; on

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

A COMPARISON WITH PAPADOPOULOS

CRITERION

Some comparative analyses reported in the literature

demonstrated12,39 that in multiaxial high-cycle fatigue
Papadopoulos criterion gives more accurate predictions
than the criteria due to Crossland, Sines, McDiarmid,
Matake and Dang Van.
Papadopoulos criterion is based on the calculation of
the Ts and Ms integrals dened as follows14 (see Fig. 2):

SP
SP P

2p
2
tr,a
(w, h, j) dj

Ts (w, h)=

(14)

j=0

2p

Ms =

w=0

p
2
Ts
(w, h) sin hdhdw

(15)

h=0

where the integrals represent the contribution of many

grains to the damage mechanism. Such integrals were
introduced because the use of the amplitude of shear
stress alone (Dang Vans criterion) did not lead to good
predictions when dealing with out-of-phase loading.
Papadopoulos subdivided the metals into hard or
mild on the basis of the following rule
Mild metals:
tA
2 0.6
sA
2
Hard metals:

0.5

ta
2 0.8
sA
2
In the case of hard materials, by using the Ms integral,
the criterion turns out to be:27
0.6

tA sA /3
2
2
sH,max tA
(16)
2
sA /3
2
On the other hand, in the case of mild metals, a critical
plane-based approach is required: the critical plane,
identied by the angles w and h, is the plane on
which the Ts value is greatest. For these materials

Ms +

L . S U S M E L a n d P. L A Z Z A R I N

ta(w*,h*) (MPa)

In-phase data (smooth specimens)

b-ar (MPa)

ta(w*,h*) (MPa)

Out-of-phase data (smooth specimens)

b-ar (MPa)
In-phase data (notched specimens)

ta(w*,h*) (MPa)

74

b-ar (MPa)

Fig. 9 Experimental shear stress amplitude

ta (w*, h*) versus estimated multiaxial fatigue
limit bar for all in-phase and out-ofphase tests carried out on smooth (a, b) and
notched (c) specimens.

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A B I - PA R A M E T R I C W O H L E R C U R V E

75

Fig. 10 Error frequency distribution

histograms for smooth specimens (a) and
notched specimens (b).

Papadopoulos criterion can be written as follows:14

3p
sA
tA 2 sH,max tA p
(17)
2
2
sA
2
2
The degree of accuracy in the fatigue life predictions of
the method proposed here and Papadopouloss criterion

Ts (w, h)+

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

has been checked by using the error frequency histogram

plotted in Fig. 10(a). As a small number of experimental
data collected from the literature exhibited a ratio
tA
2 0.6
sA
2

76

L . S U S M E L a n d P. L A Z Z A R I N

such data were not considered when applying

Papadopouloss criterion.
In Fig. 10(a) the fatigue strength error index E is
reported in the abscissa and its values have been separated
into 5% amplitude intervals. The ordinate is the error
frequency (%) and this quantity is thought of as the
number of data (expressed in percentage) which have an
error index within each considered DE interval.
Figure 10(a) shows that the performances of the two
criteria are comparable: in fact, about 70% of the
experimental data belong to the interval 10%
<E<10%. The new method has two main advantages
over Papadopoulos criterion:
$

there is a common formulation valid for mild and

hard metals;
for a generic load history, the calculation of the stress
components ta (w*, h*) and sn,max (w*, h*) are easier and
faster than calculations of the Ms and Ts integrals, at
least if one takes advantage of some illuminating
suggestions due to Weber et al.,59

Finally, it is worth noting that in a recent paper, Morel

et al.29 were able to demonstrate that there is a precise
link between Papadopoulos mesoscopic criterion and an
energetic criterion due to Froustey et al.40 in which it
was stated that crack initiation occurs as soon as the
total strain energy density exceeds a critical value. The
paper shows that the accumulated mesoscopic plastic
strain used by Papadopoulos to characterize the endurance limit can be estimated with the global strain energy
density at the macroscopic scale. Moreover, in the presence of in-phase and out-of-phase synchronous sinusoidal constant amplitude loadings, a single analytical
formulation of these criteria can be written either with
stress quantities or with energetic quantities describing
the same physical phenomenon.
A COMPARISON WITH GOUGH AND
McDIARMIDS CRITERIA

A B A BA
2

sx,a
sA
2

B A BA

sA
s
2 1 + x,a
tA
sA
2
2

A B
A B A B A BA

sx,a
1
sA
2

sA
s
2 1 x,a
tA
sA
2
2

sA
2 2
tA
2

1
(19)

The McDiarmid criterion16,17 uses a critical plane

approach, the plane being that on which the shear stress
amplitude achieves its maximum value. McDiarmids
fundamental equation is:
ta (w*, h*)+

tA
2 sn,max (w*, h*)tA
2
2sT

(20)

Note that Eq. (20) respects the formulation provided in

general terms by Eq. (12).
The distribution of the error frequency is shown in
Fig. 10(b), where predictions based on Gough and
McDiarmid criteria are compared with those pertaining
to the present method.
About 98% of the experimental data are assured to
belong to the |E |<10% range if one uses the method
proposed here, whereas by applying Gough and
McDiarmids criteria such a percentage decreases to 20
and 38%, respectively.
Fifteen years ago, when dealing with the effect of
mean stress and stress concentration on fatigue under
combined bending and torsion, McDiarmid had suggested60 for the allowable amplitude of shear stress a
different and more complex expression than Eq. (20). In
such an expression the allowable shear stress depended
on the normal stress amplitude (powered to 1.5) and the
mean stress (powered to 2.0), both evaluated on the
plane of maximum range of shear stress. In 71 tests
considered therein, carried out on specimens made in
the same steel, the fatigue limit was demonstrated to be
predicted within a 8% scatter band, that is with the
same degree of accuracy shown by the present method.
CONCLUSIONS

As far as the authors are aware, the literature reports

two fundamental criteria explicitly proposed to analyse
multiaxial fatigue strength of the notched components:
the Gough criterion41 and the McDiarmid criterion.16,17
Gough demonstrated that the fatigue resistance in the
presence of a stress concentration effect can be well
represented by an ellipse arc obeying the equation:41
txy,a
tA
2

E%=

txa,a
tA
2

sA
2 1
tA
2
(18)

For the ellipse arc, the error index E % turns out to be:

In this paper a new method for multiaxial fatigue life

predictions has been presented. The method is based on
the combined use of the critical plane approach and a
non-conventional bi-parametric Wohler curve. The
curve is given as a function of the shear stress component
calculated on the plane of maximum shear stress amplitude and of the maximum stress normal to this plane:
the orientation of the microcracks is thought of as
dependent on the shear stress whereas, on the other
hand, the micro/macro-crack growth is thought of as
mainly dependent on the normal stress.
The Wohler curve has enabled us to immediately
transfer methodologies already developed and tested in

2002 Blackwell Science Ltd. Fatigue Fract Engng Mater Struct 25, 6378

A B I - PA R A M E T R I C W O H L E R C U R V E

the uniaxial fatigue eld to the multiaxial fatigue eld,

and among these, the statistical re-analysis on the basis
of the log-normal distribution as well as or the use of
the fatigue strength reduction factors Kf for notched
components. Even if in this paper the criterion has been
applied only to the high-cycle fatigue data and fatigue
limits, it can easily be used in the medium fatigue life,
where elasticity continues to play a strongly prevailing
role with respect to plasticity.
To validate the method about 450 data taken from the
literature have been systematically analysed. All data
referred to smooth and notched cylindrical specimens
made with different materials and under different surface
nishing.
Some comparisons have demonstrated that the new
method has the same accuracy and reliability in the
multiaxial fatigue assessment as the Papadopoulos criterion but offers two advantages: rst, the analytical
formulation is the same for hard and mild metals; second,
when a generic multiaxial load history is considered, the
new method results in a reduction of the calculation
time with respect to the time required to determine
Papadoupolos parameters Ms and Ts , at least if one
takes advantage of some skills already proposed by
Weber et al.
Finally, the accuracy of the method has been checked
in the case of components weakened by quite blunt
notches (their fatigue strength reduction factors being
always less than 2.5). The method has been seen to be
more precise than Goughs criterion and a more recent
criterion due to McDiarmid.
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