Fatigue Fracc. Engng Muter. Struct. Vol. 18, No. 3, pp. 397-41 1, 1995 8756-758X/95 $6.00 + 0.

00
Printed in Great Britain. All rights reserved Copyright 0 1995 Fatigue & Fracture of
Engineering Materials & Structures Ltd

HIGH CYCLE FATIGUE AND A FINITE ELEMENT ANALYSIS

P. BALLARD', K. DANGVAN', A. DEPERROIS~and Y. V. PAPADOPOULOS~
Laboratoire de Mecanique des Solides, Ecole Polytechnique, 91 128 Palaiseau a d e x , France
'Hispano-Suiza, rue du Capitaine Guynemer, B.P. 60,92270 Bois-Colombes, France
'IISI, TP 210,21020 Ispra, Italy

Abstract-We aim to deveIop a systematic method of designing structures, by finite element methods, for
high cycle fatigue under periodic constant load systems. After having defined a precise terminology, we
quickly list those multiaxial fatigue criteria which can be found in the literature. Some criteria, derived
from a microscopic approach (Dang Van's, Papadopoulos' and Deperrois' criteria) are extensively
presented. The criteria which can be reasonably retained for numerical analyses of structures are
underlined and compared to one anoth :r. As a conclusion, we describe a high cycle fatigue CAD system
which can be derived from this analysis.

INTRODUCTION
The fundamental problem we deal with concerns whether or not a mechanical component (of
given geometry and known metallic material) submitted to a periodic constant load system will
break after a finite number of cycles? This problem has been investigated since the beginning of
the nineteenth century and many important fatigue test campaigns have been carried out in order
to derive useful fatigue criteria. The first of these were naturally uniaxial and led to some basic
experimental observations:
Fatigue is a local phenomenon and the fatigue behaviour at a point in a specimen is governed
by the history of the macroscopic stress at that point.
In uniaxial experiments, a fatigue limit strength can be distinguished, i.e. there is a maximum
value of the local stress amplitude under which no fatigue crack can be observed whatever
the applied number of cycles.
This fatigue limit strength depends of course on the material but also on the type of applied
load (bending, twisting, etc.).
The macroscopic local behaviour remains elastic when the local stress is approximately at
the fatigue limit strength, except sometimes at stress concentrations. But, even in those cases
where macroscopic plastic deformation can locally be observed, elastic shakedown is reached
after a little number of cycles and during the following cycles the macroscopic behaviour
is elastic.
In uniaxial tests, the fatigue limit strength (i.e. the maximum admissible value of the local
stress amplitude) depends on a superimposed static loading, that is the mean stress value.
In uniaxial tests, the influence of a superimposed static torque on the fatigue limit strength is
negligible.

*This paper was one of several presentations read at Euromech 297 Conference held in Corsica on 1-4 September 1992,
chaired by Professor K. Dang Van, address above.

391
398 P. BALLARDet al.

These results led to the formation of uniaxial empirical fatigue criteria such as Goodman’s line,
Gerber’s parabola, Marin’s ellipse, Haigh’s diagram, etc. Unfortunately, these criteria may only be
applied in the case of local uniaxial stress. This condition rarely occurs especially in complex
structures. In order to deal with the fatigue behaviour of complex structures, a fatigue criterion
must be multiaxial. Such a criterion is nothing but a local condition which must be fulfilled by
the history of the stress tensor at the considered point.
We can now be more precise. We have seen that in high cycle fatigue, the material remains
macroscopically elastic. We have restricted ourselves to periodic constant load systems. Therefore,
the macroscopic stress & - at a given point 5 is a periodic function of time t:
- t + T )= gx,
3 T V t g(5, - t)
and the general form of a fatigue criterion (in case of a periodic constant load system) must be:
f{gcx,t)/t E LO, 771 G 0 (2)
The meaning of such a criterion is that if the condition is satisfied at each point 5 of the structure,
no fatigue crack will be observed whatever the number of applied cycles. Otherwise, a fatigue
crack is likely to appear at each point of the structure where the condition is violated. Such a
criterion must be applicable to any stress periodic constant function of time and must be of course
intrinsic, i.e. independent on the spatial coordinate system in which stresses are expressed.
In the following sections, we list and discuss high cycle fatigue multiaxial criteria which can be
found in the literature. We present a microscopical approach, detail Dang Van’s, Papadopoulos’
and Deperrois’ criteria and compare them. We also show how this approach allow to develop a
systematical method of designing structures submitted to periodic constant load systems (i.e. a
high cycle fatigue CAD system).

DEFINITIONS AND NOTATIONS

We now define carefully some quantities which appear frequently in high cycle fatigue studies.
It is easy to check that any quantity introduced is intrinsic and can be evaluated whatever periodic
constant function of time the stress follows. We consider an arbitrary point _x in the structure and,
in order to reduce notations, the argument _x is omitted.
Classification of periodic loading paths
Since the behaviour of the material can be assumed elastic and the load system is periodic and
of constant amplitude, then the stress at a given point of the structure is periodic and can be
expressed by Eq. (1) in the form:

3T V t &(t
- + T )= &-( t ) (3)
Here, g ( t ) is moving along a closed curve 4 (see Fig. 1) in stress space. This curve is the local
loading path and the set of values taken by g(t ) during a period is named the stress cycle.
If the loading path is a line segment in stress space, it will be called a linear loading path (see
Fig. 2). The function:

is a classical example of a linear loading path. It is clear that even in the case of a linear loading
path, the eigen-directions of the stress tensor may vary with time.
 High cycle fatigue and a finite element analysis 399

(b)
Fig. 1. Periodic loading path in stress space. Fig. 2. Linear loading paths. (a) simple and (b) complex.

A linear loading path will be said to be radial or proportional if it is supported by a line

emanating from the origin, and alternate if it is a line segment centered on the origin.
Mathematical structure of the stress space
It is important to report on the mathematical structure of the stress space because this will
provide us with a useful geometrical interpretation of some fatigue criteria. The stress space is
obviously a six dimensional vector space and we may construct a scalar product for it:

where : is the diadic product of tensors. The stress space is therefore an euclidean vector space
and the scalar product classically induces a metric on the space.
The deviatoric part -S of the stress tensor - is defined by:
_s=g-9’Id
- - - (6)
where - denotes the unit tensor of the second order and 9 the hydrostatic pressure, i.e.:

A stress tensor is said to be spheric (or isotropic) if it is proportional to Id. It is said to be the
deviatoric if its trace is zero. The set of deviatoric tensors is a five dimen3onal linear manifold
(hyperplane) in stress space. The set of spheric tensors is a one dimensional linear manifold (line).
Moreover, it is easy to check that the spheric line is perpendicular to the deviatoric hyperplane.
The direct sum of these subspaces is the whole space and so, a stress tensor is the sum of its
spheric part and its deviatoric part. This breakdown plays a particularly important role in fatigue.
When g(t)is describing the curve 4, then s ( t ) is describing a curve 4’ which is the orthogonal
projectron of 4 on the deviatoric hyperplane.

Some dejnitions
For a computational approach, it is necessary to define precisely some quantities that are
functions of the stress cycle. Certain of these quantities are classical in fatigue, some others less
so, but, all of them are intrinsic. We will see afterwards that local hydrostatic pressure, tension
and shear-stress play a role in high cycle fatigue behaviour. We define here some intrinsic measures
of these quantities during a local load cycle.
400 P.BALLARDet al.

Hydrostatic pressure
The instantaneous hydrostatic pressure is:
Y(t ) = f tr g(
- t)
The maximum hydrostatic pressure, the mean hydrostatic pressure and the hydrostatic pressure
amplitude are:
Ym,, = max Y(t ) (9)
t

Y m = l? L TY(t)dt
(10)

Ya = max Y(t) - min P(t ) (11)

t t

Tension
Let us now consider a material plane P with normal _n. The instantaneous stress vector gn(t)
-
relative to P is:
gn(t) = s ( t ) * ~ (12)
The orthogonal projection t) of the stress tensor on the normal to the plane P of the stress
is called the instantaneous tension vector relative to P and is defined by:
Nf!?,t) = (_n&l(t))_n (13)
The instantaneous tension relative to P is:
N(_n,t) = 3 .-g t)._n ( 14)
and the maximum tension, mean tension and the tension amplitude relative to P are:

Shear-stress
The tangential part C(_n,t) to the plane P of the vector stress is named the instantaneous
shear-stress vector relative to P. It is the perpendicular projection of Znon the plane P (see Fig. 3):
C(_n,t ) = gw_n- (_n&)._nh
The shear-stress 11 C(_n,t)II relative to P is:
llC(_n,011 = Jllg(t)_nllz
-(_n.p.d2 ( 19)
During a period of loading, the extremity of the shear-stress vector C(_n,t) relative to a plane P is
moving along a closed curve $ of the plane P (see Fig. 3). We define the shear-stress amplitude
Ca(_n)on this plane as the half-diameter of the curve $. The diameter of a plane closed curve is
the length of the longest line segment which can be drawn inside the curve. Thus, we have:
 High cycle fatigue and a finite element analysis 401

Fig. 3. Evolution of the shear-stress C@,t ) on a material plane P during a periodic constant loading.

which can be expressed, from Eq. (18), as:

C E * C ~ ( ~ ~ ) - ~ ( ~ ~ )(21)
I*~I~
Another way to evaluate the shear during a loading period is to use the deviatoric part _s(t)of
the stress tensor. We have seen previously that during a loadiing period, z ( t ) describes a dosed
curve 4’ in the deviatoric hyperplane. Let us recall that the stress <pace is an euclidean
vector space (by Eq. ( 5 ) ) and the diameter of a closed curve in an euclidean vector space is, by
definition, the length of the longest line segment which can be drawn inside the curve. Hence the
diameter of the curve 4‘ is:

as the half-diameter of the curve 4’

We define the octahedral shear-stress amplitude denoted by Fa
multiplied by the factor 1/&:

All these quantities may be evaluated whatever the local load cycle and their expressions may
be simplified in the case of a linear local load cycle.

TRADITIONAL MULTIAXIAL FATIGUE CRITERIA

Historical summary
From a historical point of view, it seems that the first fatigue tests were conducted in 1829 by
Albert in Germany. Later on, some important test campaigns were carried out in the last half of
the nineteenth century and in the first half of the twentieth century (let us cite Wohler, Stanton,
Gough, Haigh, Gould, Locati, etc. .. .) for various kinds of loadings. At the end of each campaign,
a fatigue criterion was published. The first ones were only applicable to uniaxial loadings.
Afterwards, some multiaxial tests were performed (for example in-phase twisting-bending) in order
to generalize the criterion to multiaxial cases. As an illustration, we can cite the work of Gough
and Pollard [ 11, and Nishihara and Kawamoto [2]. Since 1950, researchers have tried to deduce
a general form of the multiaxial criteria on the basis of physical observations. These criteria are
all based on the following physical observation. A fatigue crack, once created, propagates first
along a shear plane. Forsyth [3] named this the propagation first step. Having reached a certain
length, the crack is deflected and propagates in a plane perpendicular to the maximum tensile
402 P. BALLARDet al.

principal stress direction. This is Forsyth’s propagation second step. Since, at its beginning, the
crack propagates along a shear plane, many authors assumed that fatigue crack initiation was
governed by the shear-stress. Since it is admitted that the tension normal to the shear plane also
influences fatigue behaviour, it was natural to formulate a fatigue criterion using the shear stress
and the tension relative to the same plane. Referring to this idea, we can cite Stulen-Cummings’
criterion [4], and those of Findley [S], and McDiarmid [6]. Some other authors have used the
Von Mises measure or octahedral shear stress instead of the shear stress itself. As examples we
can cite Sines’ criterion [7], and those of Crossland [8], and Kakuno-Kawada [9].
Conclusions on traditional fatigue criteria
There are numerous fatigue criteria in the literature but there is not yet a universally accepted
approach. A complete survey of high-cycle fatigue criteria can be found in [lo]. In view of the
computation of complex structures, we can only consider those criteria which apply whatever the
loading path in the stress space. If they are compared on the basis of:
0 agreement with experimental results in multiaxial fatigue
0 numerical computation duration
0 number of material constants to identify
it seems that the more efficient one, in view of numerical computation of structures, is Crossland’s
criterion. Using notations introduced previously, Crossland’s criterion can be expressed as:
Z, + a%,,ax< b (24)
where a and b are two material constants which can be identified from two different fatigue limit
strength measurements, for example in alternate bending and alternate twisting.

A MICROSCOPIC APPROACH

A microscopic approach was initiated in 1973 by Dang Van [ll]. The basic principle of that
work was to derive a multiaxial fatigue criterion from a model of what happens at the scale of
a grain.
The basic assumption
During a fatigue test which involves a large number of cycles, the stress at a macroscopic scale
remains elastic. However, it is well known that at the microscopic scale, a metal is not isotropic
nor homogeneous and instead is constituted from crystals of random orientation. This induces
local fluctuations of the macroscopic stress and defines the microscopic stress. Such an analysis
enables us to give a physical interpretation to the fatigue phenomena. Indeed, we can now easily
imagine that the local microscopic stress can locally exceed the yield strength in certain unfavorably
oriented grains, whereas the macroscopic stress remains elastic. If the cyclic plastic response of the
grain to the loading is not elastic shakedown, some micro-cracks will appear. These micro-cracks
then join to form a crack of detectable size. This is Forsyth’s propagation first step. The crack
is then macroscopic and propagates according to fracture mechanics rules. This is Forsyth’s
propagation second step. This phase is ended by the rupture of the specimen.
It is of course possible to initiate a macroscopic crack which will not propagate and then, no
fatigue rupture will occur. However, the loading threshold of initiation and the loading threshold
of propagation are sufficiently close to each other, at least in many materials, that they may be
assimilated. Thus, the fatigue threshold can be assumed to be the crack initiation threshold.
 High cycle fatigue and a finite element analysis 403

From this analysis, we can formulate the fundamental assumption of the microscopic approach
to fatigue as: “fatigue rupture will not occur if and only if the response of the grains most
unfavorably oriented and subjected to the microscopic fluctuations of the periodic loading, is
elastic shakedown”. The following criteria are all based on this hypothesis. We will now present
the theoretical background of this analysis.
Micro and macro scale relationships
The mechanical state of a continuous medium is described by macroscopical stress and strain.
These quantities are defined at each point _x of the solid. More precisely, we consider an elementary
volume V ( 5 )assumed to be isotropic and homogeneous, in which the stress g(x)and the strain
&) are uniform. This is the macroscopic scale. In reality, in the case of metds, there are in the
Glementary volume V(x)many crystals of different shapes, orientations and internal states. This
leads us to define a microscopic scale which can account for the fluctuations at the scale of a grain
in VQ). The mechanical state of the grain is defined by the stress g ( y ; _x) and the strain ~- (- y_x);
where y is the grain position in a local coordinate system associated?; V ( x )(see Fig. 4).
In hygh cycle fatigue experiments, the macroscopic behaviour of a specimen remains elastic.
From this, we can conclude that the number of grains which undergo plasticity is very low.
Therefore, we can also admit that the strain in a grain is determined by the strain of the matrix
surrounding it:
- - +p
- =_E*g
& - =_Ee- -
+_Ep (25)
In this equation, we have omitted the _x and y parameters to reduce the notations. As far as there
is no detectable plastification at the macroscopic scale, we can neglect EP,
-
g- = E”
- =_Ee- +_E*- (26)
We can also make the assumption that the grain has the same modulus as the surrounding matrix:
ie the macroscopic and the microscopic elastic modulii tensors are equal.
L= 1 (27)
The preceding hypothesis is known as the Lin-Taylor model which implies,
g- = 1 :_E”- + 2p:P- (28)

E- tYi

t
Fig. 4. Microscopic and macroscopic scales for a polycrystalline metal.
404 et al.
P. BALLARD

since _EP
-
is deviatoric. This can also be written as,
-0 = &
- - 2@- (29)
We quantify the microscopic residual stress -p as,
-

-p- = -2pf - (30)

so that:

-0=g+p
- - -- (31)
As a consequence of Eq. (30), parameter -p is deviatoric, and the microscopic and macroscopic
hydrostatic pressures are equal,
9 = +tr &‘=$
- tr -_a (32)
From these equations, it is clear that the microscopic stress calculation is equivalent to the
determination of the microscopic plastic strain -5” in the crystal.
Dung Vun’s criterion
From the previous analysis, we can say that no fatigue rupture will occur at a point in the
structure if the microscopic response at that point is elastic shakedown. This is the case if dt)
- can
be written as:
-g ( t ) = &-( t ) + p--* (33)
where -p* is constant in time and -~ ( tsatisfies
) the crystal plasticity criterion:
-
Vtf(=a(t))G 0 (34)
We need now to postulate a crystal plasticity criterion. According to Schmidt’s law, it would be
natural to write it as
vt v_nlid?, t )I1 b (35)
where llg(_n,t ) 1 is the shear-stress (see Eq. (19)) associated with _o(t).However, experimental results
show a dependence of fatigue strength on hydrostatic pressure and so Dang Van postulated in
1973 [11] that,
V t V _n llg(_n,t )II + UP< b (36)
where 9= 5 tr _o
- = 3 tr &
- is the hydrostatic pressure. Dang Van’s criterion is then:

The value _n* at which the maximum is reached clearly defines the plane in which the fatigue crack
will begin to propagate when the criterion is not satisfied.
To apply this criterion, we must be able to express _c(_n, t) as a function of g(t).In other terms,
this criterion must be completed by an evaluation for -p* and so Dang Van sGggested that,
g(_n,t ) = C(_n,t ) +z*(n) (38)
where z*(_n)= p*_n is the residual shear-stress on each plane of normal _n. As long as the microscopic
plasticity criteson on that plane is a circle centered on the origin, it is natural to choose _z*(_n) as
the vector Mo where M is the centre of the smallest circle in which the curve $(_n) is included
(see Fig. 5).
 High cycle fatigue and a finite element analysis 405

Macroscopic path
/ Microscopic path
~~ ~~ ~ ~~~ ~

Fig. 5. Residual shear-stressconstruction.

The residual shear-stress is then independently constructed for each plane, and this is not
compatible in general with the existence of a tensor -p* such that,
-
vrj I*(!) = -p.*_n
-
but, it is justifiable if we consider that different crystals are involved when _n varies.
Since ths criterion can also be expressed as:

we name
z ( t ) = max lIc(9, t ) II (40)
n
and represent the loading path in the (9(t ) , z( t)) plane (see Fig. 6 ) and this is Dang Van’s diagram
of the loading path. If the loading curve remains beneath Dang Van’s line no fatigue crack will
appear, otherwise rupture is presumed.
The issue here is that the double maximisation leads to a considerable computation time and
so, with the aim to reduce the computation time, Papadopoufos proposed his criterion.

Papadopoulos ’ criterion
Criterionformulation
Papadopoulos’ idea [lo] was to use Von Mises’ criterion instead of Schmidt’s law for the grain
plasticity criterion. It is expressed by:

Fig. 6. Examples of Dang Van’s diagram in case of (a) no rupture and (b) rupture.
406 P.BALLARDet al.

where 3 is the microscopic stress deviator, k the shear strength and _a and b are kinematical and
isotro$c strain-hardening parameters. From the micro-macro scale relkionships, Eq. (41)becomes:

where _S is the macroscopic stress deviator and p is the microscopic residual stress tensor. From
Melan’i adaptation theorem [121, the response’of the grain will be elastic shakedown provided
that there exist a tensor <*
- and a number k* not “too great” such that Eq. (43)is fulfilled:
V t $ ( S ( t ) - -z * ) : (-S ( t ) - z-* ) <(k*)2 (43)
If the response of the grain is adapted, it is possible to calculate *; and k*. Papadopoulos showed
in [lo] that they can be well approximated by the following algorithm, that is <* - is the value of
_z- such that:

and k* is given by:

These quantities have a geometrical interpretation, namely that ?* and f i k * are respectively the
centre and the radius of the smallest hypersphere in which t6e closed curve 4’ (described by
the macroscopic stress deviator S(t))is included.
To satisfy the adaptation theorem, we must still express that k* is not too great. Papadopoulos
postulated that
k* + upmx < b
which is his multiaxial fatigue criterion.
Numerical implementation algorithm
Papadopoulos’ criterion could be implemented using a numerical solution to the min-max
problem, Eq.(44). This would lead to too great a computation time in the case of a complex
structure analysis. This computation duration may be reduced by using some results of geometry
in point spaces. Numerically, the problem is to find the smallest hypersphere (in a 5D space) in
which a set of points is included. It is well known that this hypersphere exists and is unique. It
can be quickly determined using the following algorithm.
Determination of the smallest hypersphere in which a set of points is included
We give an algorithm to determine the solution in the 2D case. Let us consider a finite set of
points {Mi/l< i < n } in a plane. We want to determine the smallest circle in which are all the
points. We examine all pairs of points {Mi, Mj} and check if all points are in the circle of diameter
MiMj. We stop as soon as this condition is realized. If it is realized for no pairs, we examine all
the triplets {Mi,Mj, Mk}and check if all points are included in the circle which is circumscript to
the triangle {Mi,Mj, Mk}. It can be proved, using Karatheodory’s theorem [ 131, that this condition
will be realized at least once. This algorithm, which can be easily generalized to the 5D case,
allows a very quick numerical computation of Papadopoulos’ criterion.
Dang Van’s criterion second formulation
On the basis of the previous paragraph results, Dang Van et al. slightly modified the original
Dang Van’s criterion in 1987 [ 14,161 in order to make its computation quicker.
 High cycle fatigue and a finite element analysis 407

If _s(t) is the microscopic stress deviator, and if the microscopic response has reached elastic
shakedown, then we have:
+
-_s( t) = -#( t) p*
-- (47)
Following Papadopoulos' work, it is natural to assume that - p * is the center of the smallest
hypersphere of the deviatoric hyperplane in which the macro'scopic deviatoric loading path
is included. As - p * can be easily determined by mean of the previous algorithm, then g- ( t ) is
known in function=&'(t).
- At instant t, the maximum shear-stress z(t) is given by:

where a,(t) and aIII(t)are respectively the greatest and the smallest eigenvalue of a(t). As -p* is
supposed to have been determined and g(t ) to be known, the instantaneous maximum microsc'opic
stress can easily be computed. Dang Van's criterion can then be expressed naturally as:
max [z( t) + aP(t ) ] < b (49)
t

This criterion is very close to the original Dang Van's criterion but is much easier and faster to
compute. This approach has been applied with success in many practical cases, see for example
Refs [ 14- 161.
Comparison of Papadopoulos' criterion with experimental results
Papadopoulos has systematically compared his predictions to the experimental results available
in the multiaxial fatigue literature. For each experiment, he has calculated k* and -flux + b for
the critical loading. If these values are equal, the criterion prediction is perfect. This analysis is
represented in Fig. 7 for linear loading paths and for general periodic loading paths. The dashed
line represents the predictions from the criterion and C is a normalisation constant.
We see that Papadopoulos' predictions are quite satisfactory for linear loading paths but not so
precise for other loading paths and particularly for multiaxial out-of-phase loadings. To improve
this particular aspect, Deperrois [ 171 proposed a new criterion in 1991.

Fig. 7. Experimental results and Papadopoulos' predictions in the cases of (a) linear loading paths and
(b) some general periodic loading paths.
408 P. BALLARDet al.

Deperrois’ criterion
The main disadvantage of Papadopoulos’ criterion (which is also common with Crossland’s and
Dang Van’s criteria) is that it does not make any difference between linear and complex loading
paths (see Fig. 8), whereas a difference is demonstrated in experiments.
In deviatoric stress space, we know that the extremity of the vector _S( t ) describes a closed curve
4’. We have seen that Papadopoulos’ measure for fatigue induced damage was the radius Gk* of
the smallest sphere in which 4’ is included. To make a difference between linear and complex
loadings, Deperrois’ proposal is as follows: we consider the longest vector Dl of the deviatoric
stress space (or subspace) which can be drawn inside 4’. Let Pl be the hyperplane of the deviatoric
stress space which is orthogonal to 4’ and 4; the orthogonal projection of 4’ on Pl. Then D2 is
the longest vector of PI which can be drawn inside 4;. Also Pz is the hyperplane of Pl which
is orthogonal to Q2,etc. .., Thus, following this algorithm, we can define Q1,D2, Q3,D4 and D5.
Deperrois then defines:

and his criterion is expressed by:

+
A(@) a%ax <b (51)
We can make two remarks:
1. In most of the usual cases which are encountered, D3, D4 and D5 are null vectors, and only
Dl and D2 need to be calculated.
2. As we shall see later, in the case of linear loadings, Deperrois’, Papadopoulos’ and Crossland’s
criteria are strictly equivalent.

1.0 I

I
0.0 .

w 0.6 .

4f Heidenrich

0.2 - piqx +ENSAMBordcaux

A Issler
OMcDilvmid
XDeitmann

0.0 *’ .
4 I

Fig. 8. Complex and linear loading

paths in deviatoric stress space.
 High cycle fatigue and a finite element analysis 409

In the case of complex loading paths, Deperrois compared his predictions with experimental results
as did Papadopoulos. Results are displayed in Fig. 9. We see that a satisfying result occurs.

DISCUSSION
According to us, in view of the numerical designing of a complex structure submitted to a
constant periodic loading, the multiaxial high cycle fatigue criteria to be retained are Crossland’s,
Dang Van’s, Papadopoulos’ and Deperrois’ criteria.
Comparative analysis of these criteria
Prediction realism
It can easily be proved by geometrical considerations that in the case of linear loading paths,
Crossland’s, Papadopoulos’ and Deperrois’ criteria are strictly identical. In what follows, we say
that a criterion is more conservative than another when its predictions are more pessimistic
whatever the loading path. It can be proved (the proof is very heavy but elementary)that Deperrois’
criterion is more conservative than Papadopoulos’ which is more conservative than Crossland‘s.
Dang Van’s criterion is more difficult to compare with the others. However, Papadopoulos has
shown [lo] that in case of a proportional loading path, Papadopoulos’ criterion is more conserva-
tive than Dang Van’s. As a conclusion, we can say that, at this time, the high cycle fatigue criterion
for general periodic loading paths which presents the best accordance with experimental results
seems to be Deperrois’ criterion.
Computation time
As our final aim is numerical implementation, we must consider computation time. The original
Dang Van’s criterion is expensive to compute and can hardly be applied to the computation of
complex structures. However, Dang Van’s criterion reformulation allows the fatigue computation
of complex structure. Crossland’s, Papadopoulos’ and Deperrois’ criteria are very quick to compute.
Identijication of material constants
All the criteria require the identification of two material constants a and b. These two constants
may be calculated for example on the basis of an experimental determination of the fatigue limit
strength in alternate bending f and in alternate twisting t. For Dang Van’s criterion, we then have:

and, for Crossland‘s, Papadopoulos’, Deperrois’ criteria:

-
t-flJ3
a= , and b = t (53)
f 13
A high cycle fatigue CAD system
Here we illustrate how we derive a systematical finite element method to analyse the high cycle
fatigue behaviour of complex structures submitted to periodic constant load systems, e.g., we wish
to determine whether or not a mechanical component with a given geometry, made with a given
material and submitted to a given periodic loading will undergo an infinite number of cycles
without breaking. In most cases, it is possible to answer this problem by mean of finite element
computations. It is of course clear that, since the answer is via numerical calculations, it remains
approximate.
410 P.BALLARDeb at.

Let us consider a complex structure submitted to a periodic constant load system (denoted
formally by F ( t), 0 ,< t < T) and we aim to determine whether or not this structure will undergo
an infinite number of cycles without initiating a high cycle fatigue crack. We need two fundamental
tools: a classical elastic static finite element software and a routine in which our selected “best”
multiaxial fatigue criteria are implemented. We have developed such a routine under the name
SOLSTICE and Crossland’s, Dang Van’s, Papadopoulos’ criteria are implemented in this routine.
It is then possible to approach the solution in three steps:

1. We construct a geometrical finite elements discretization of the structure and we discretize

the load system in n instants. We get n load systems F,, F,, . ..,F, corresponding to the
n instants t,, t,, .. ., t,.
2. We perform n elastic finite element computations corresponding to the n load systems. As a
result, we get the stress tensor at each node of the structure for each of the n instants of
the cycle.
3. Using SOLSTICE, we apply a fatigue criterion (for example Deperrois’) at each node of the
structure. As a result, we get at each node of the structure a non-dimensional number whose
significance is as follows. If the number is negative then no fatigue crack will appear whatever
the number of cycles applied. If the number is positive a fatigue crack will initiate at a finite
number of cycles.

It is then easy, using a graphic postprocessor, to draw a fatigue criterion isovalue picture. Such a
picture shows the most fatigue-sensitive parts of the structure.

REFERENCES
1. H. J. Gough, H. V. Pollard and W. J. Clenshaw (1951) Some experiments on the resistance of metals to
fatigue under combined stress. Memo 2522, Aeronautical Research Council. HMSO, London.
2. T. Nishihara and M. Kawamoto (1941) The strength of metals under combined alternating bending and
torsion. Memoirs of the College of Engineering, Kyoto Imperial University, 10(6), 177-201.
3. P. J. E. Forsyth (1961) A two stage process of fatigue crack growth. In: Proc. Fatigue Crack Propagation
Symposium, Cranfield. pp. 76-94.
4. F. B. Stulen and H. N. Cummings (1954) A failure criterion for multiaxial fatigue stresses. Proc. ASTM,
54, 822-835.
5. W. N. Findley (1957) Fatigue of metals under combined stresses. Trans. ASME, 79, 1337-1348.
6. D. L. McDiarmid (1991) A general criterion for high-cycle multiaxial fatigue failure. Fatigue Fract. Engng
Mater. Struct., 14(4), 429-453.
7. G. Sines (1959) Behavior of metals under complex static and alternating stresses. In: Metal Fatigue
(Edited by G. Sines and J. L. Waisman), McGraw Hill, New York. pp. 145-169.
8. B. Crossland (1956) Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy
steel. In: Proc. lnt. Conf. Fatigue of Metals, Institution of Mechanical Engineers, London. pp. 138-149.
9. H. Kakuno and Y. Kawada (1979) A new criterion of fatigue strength of a round bar subjected to
combined static and repeated bending and torsion. Fatigue Engng Mater. Struct., 2, 229-236.
10. Y. V. Papadopoulos (1987) Fatigue polycyclique des mBtaux: une nouvelle approche. PhD thesis, h o l e
Nationale des Ponts & Chausskes, Paris.
11. K. Dang Van (1973) Sur la resistance A la fatigue des mktaux. Sciences et Techniques de 1’Armement.
3eme fascicule, pp. 647-722.
12. J. Mandel, J. Zarka and B. Halphen (1977) Adaptation d‘une structure Blastoplastique A Ccrouissage
cidmatique. Mech. Res. Comm., 4( 5), 309-314.
13. S. R. Lay (1982) Convex Sets and their Applications. John Wiley and Sons, New York.
14. K. Dang Van, Y. V. Papadopoulos, B. Griveau and 0. Message (1987) Sur le calcul des structures
soumises A la fatigue multiaxiale. In: Materiaux et Structures, HERMES Bditeur, Paris.
 High cycle fatigue and a finite element analysis 411

15. K. Dang Van and Y. V. Papadopoulos (1987) Multiaxial fatigue failure criterion: a new approach. In:
Proc. Third Int. Con$ Fatigue and Fatigue Thresholds, Fatigue 87, University of Virginia, Charlottesville,
Virginia, USA.
16. K. Dang Van, B. Griveau and 0. Message (1989) On a new multiaxial fatigue limit criterion: theory and
application. In: Biaxial and Multiaxial Fatigue, EGF 3 (Edited by M. W. Brown and K. J. Miller),
Mechanical Engineering Publications, London. pp. 479-496.
17. A. Deperrois (1991) Sur le calcul de limites d’endurance des aciers. PhD thesis, Ecole Polytechnique, Paris.