A parametric approach to irregular fatigue protection is presented. The method proposed consists of two parts: empirical determination of certain characteristics of a material by means of a relatively small number of well-defined standard tests, and arithmetical application of the results obtained to arbitrary loading histories. The following groups of parameters are thus taken into account: (1) the variations of the mean stress, (2) the interaction of these variations and the superposed oscillating stresses, (3) the spectrum of the oscillating-stress amplitudes, and (4) the sequence of the oscillating-stress amplitudes. It is pointed out that only experimental verification can throw sufficient light upon possibilities and limitations of this (or any other) prediction method

Erismann, T. H.

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A PARAMETRIC APPROACH TO IRREGULAR
FATIGUE PREDIC TION
By T. H. Erismann
Federal Laboratories for Testing Materials and Research
Dubendorf, Switzerland
SUMMARY
The method proposed consists of two parts: empirical determination of certain
characteristics of a material by means of a relatively small number of well-defined
standard tests, and arithmetical application of the results obtained to arbitrary loading
histories. The following groups of parameters are thus taken into account: the varia-
tions of the mean stress, the interaction of these variations and the superposed oscil-
lating stresses, the spectrum of the oscillating-stress amplitudes, and the sequence of
the oscillating-stress amplitudes. It is pointed out that only experimental verification
can throw sufficient light upon possibilities and limitations of this (or any other) pre-
diction method.
FUNDAMENTALS OF PARAMETRIC APPROACH
The fundamental procedural scheme of the method evolved in this paper consists
of the following phases:
(1) Determination of a number of characteristics of the material by means of
standard tests
(2) Prediction of the fatigue life of the material on the basis of these characteris-
tics and an analysis of the expected loading history.
The problem thus raised can, in principle, always be solved since for a specified
accuracy there will always be a finite number of tests from which the necessary data for
a satisfactory processing of the second phase can be drawn. There is, consequently, a
problem of interpolation which can be solved with a finite number of base points. The
question remains how to attain the goal economically. In view of the present state of
development of servo-hydraulic testing equipment and digital computers, neither irregu-
lar stress sequences in the standard tests nor extensive algorithms in analysis and
evaluation are prohibitive.
The progress of our knowledge of fatigue strength has in the last few decades
become an alarming "parameter explosion." Thus, in order to avoid undue complica-
tions, no mention will be made of parameter groups connected with notch effect or
environmental influences and nothing but the loading history will be considered.
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Consequently,the following effects will be taken into account:
(1) Meanstress effect (indexM), influence of the chronological curve of the mean
stress
(2) Interaction effect (indexJ), influence of systematically occurring correlations
betweenthe effects of meanand oscillating stresses
(3) Spectral effect (index P), effect of the statistical distribution of the oscillating-
stress variations
(4) Sequenceeffect (indexQ), effect of the sequencein which the individual stress
variations follow oneanother.
The parameters linked to these effects will be called in this paper "M.I.S.S. parameters"
from their initial letters.
Owingto the inherent complexity of the problem, a certain number of simplifying
assumptionsmust be made. In particular it is assumedthat
(1) Miner's rule is applicable with sufficient accuracy to sufficiently narrow sec-
tions of a stress spectrum
(2) The influence of the mean-stress effect canbe described with sufficient accuracy
by the statistical amplitude distribution of the variations occurring in the mean stress
(3) Every increment of a loading history producesan incremental interaction effect
approximately proportional to its meanstress level andto its "linear-damage increment"
(according to Miner's rule)
(4) The effect of the sequenceextendsmainly to the "coarse sequence"(that is, to
changesof the spectrum over longer periods of the loading history), whereas the "fine
sequence" (that is, the individual sequence of the single-stress cycles) is of minor impor-
tance in practice, provided the process under consideration can be described with suffi-
cient accuracy by stochastic characteristics
(5) The effects of the M.I.S.S. parameters allow with sufficient accuracy a linear
(in one exceptional case, a quadratic} interpolation when the logarithm of the "Miner sum"
V V ni
= --... (1)
_, _ N ii
(which according to Miner's rule should always be equal to 1) is used as the interpolation
value. In equation (1), n i is the number of cycles applied at stress level whereas Nj
is the total number of cycles to cause failure at stress level.
It is not possible here to justify these assumptions. A justification of these assump-
tions is given in reference 1.
430
OUTLINE OF THE M.I.S.S. METHOD
The methodoutlined here starts from relation (1); it being clearly understood,how-
ever, that a sum not equal to 1 is permissible. In accordancewith the assumptions
made,the four M.I.S.S.parameters PM' PJ, Pp, and PQ are definedand
= f(PM,Pj,Pp, PQ) . . . (2)
is postulated. This postulation means that the spectrum, as in Miner's rule, is still the
most important group of parameters, but not the only decisive one.
Thus, the first phase of the method results in the performance of an unequivocally
defined series of standard tests with well-defined M.I.S.S. parameters and the determina-
tion of the resulting standard Miner sums _ which are to be considered as charac-
:.J S
teristics of the material.
The second phase, that is, the application to an expected loading history with a
given nominal fatigue life, is divided into the following partial phases:
(1) Analysis of the loading history according to equation (1). The result is the
Woehler-Miner
sum _ W"
(2) Analysis of the loading history according to well-defined formulae for deter-
mining their M.I.S.S. parameters.
(3) Application of the M.I.S.S. parameters as a means of interpolation in the results
field of the standard Miner sums.
(4) Formation of the quotient
The result is the Effective Miner sum _E"
_E/_W which indicates the chances of survival of
_" ]k-"
the
specimen. For _._E/2_JW > 1, survival of the loading history is to be expected.
The first and fourth phases are carried out according to the known algorithms of
Miner's rule. The interpolation in the third phase must mainly be carried out linearly
for log_. Figure 1 shows theinterpolationfor PM and Pj, with Pp and PQ
held constant. The heights of the column at its four edges represent the results of four
standard tests. The result of the interpolation is the height (heavy dotted line) for PM
and Pj. Only the standard tests and the stress-history analysis need tobe considered
in more detail.
431
MATHEMATICAL TREATMENT OF PARAMETER GROUPS
It would be impossible to give a complete review of the mathematical deductions
applied to obtain appropriate equations for the calculation of the M.I.S.S. parameters.
It must be referred, therefore, to more detailed works on the subject. (See refs. 1 and 2.)
The following remarks are made in order to give a general idea of the logical structure
of the method.
All the equations used in this connection are based solely on the simplifying
assumptions made. For instance, the second assumption means only that the mean
stress effect for given values of the other parameters is determined by the "spectrum of
the mean stress" so that Miner's rule may be considered as applicable to the mean
Thus, an extremely simple definition of the mean-stress parameter is obtained;stress.
that is,
PM _ m= (3)• • •
-J Na_w
where m(_ is the number of mean-stress variations of the value (_, and N a is the
cycle number pertaining to _ according to the (_-N curve. The division by the
Woehler-Miner sum _" is used for normalizing purposes.
_._W
The spectrum of a loading history is represented in principle by a parabolic
approximation so that theoretically, three parameters are needed for the mean value, the
average slope, and the curvature of the parabola. The first of these parameters, how-
ever, is insignificant since in the chosen representation, the mean value of the parabola
is given. It is a particular feature of the method that the ordinal numbers i of certain
well-defined stresses ai are used as values of the independent variable of the approxi-
mation so that the respective numbers n i are expressed by
ni . . .
- c 0+ c 1 -i+ c 2 .i 2 (4)Ni .....W
where Co, Cl, and c 2 are the spectral parameters Pp.
In a more or less analogous way, the parameters for interaction and sequence, pj
and PQ, are deduced from the assumptions made. Both are connected with variations
of the spectral parameter Cl; thus, predominance of high- or low-stress amplitudes
under particular conditions is indicated. Although Pj establishes a correlation with
the mean stress (_m (taking into account such phenomena as the increased dynamic
forces due to increased payload of a vehicle), PQ accounts for variations undergone in
432
the course of the loading history as a whole (as encounteredowing to more frequent over-
load whenan aircraft is transformed from passengerto cargo transport); thus, it serves
as a safeguardagainst unpleasantsurprises, becausemanymaterials have a shorter
fatigue life whenfirst subjectedto low- andthen to high-stress amplitudes.
The influence of parameter variations uponloading histories is illustrated in fig-
ures 2, 3, and 4. In figure 2, the oscillating stresses are not present for negativemean
stresses becauseof the choice of constantsin the equations. In figure 3 the stress
amplitudes are higher in section II than section I; thus, a higher value of c1 is indi-
cated. In figure 4, the variation of c2 results in predominanceeither of extreme (high
and low) or of medium amplitudes. It will be observed that in figure 3, section II shows
higher stress amplitudes than section I; thus, a higher value of c1 is indicated. Vari-
ation of c2 results in predominanceeither of extreme (high andlow) or of medium
amplitudes (fig. 4).
STANDARD TESTS AND PRACTICAL APPLICATION
Since linear interpolation of log ) has been accepted for all parameters
(excluding Cl) , variations through all possible combinations other than the meaningless
ones (PM = 0i; PJ + 0) would give 36 base points which can be found from 36 tests
(plus repetitions). The cost is considerable but is certainly justified for important
materials.
Experience will show whether all 36 standard tests are really required. The num-
ber depends on how far the multidimensional functions of figure 1 can be approximated by
plane surfaces. If, for example, it should be found that, apart from Pp, the function
surface can be considered as sufficiently plane, then only the inclinations of this plane
would have to be known; that is, for every additional parameter only one single base point
would have to be determined and nine tests would suffice. Probably the truth lies between
these two extremes.
An actual "cooking recipe" for the details of the method is found in reference 1.
There the individual steps are not only commented on but are also compiled in tabular
form so that the economic establishment of suitable computer programs is possible.
POSSIBILITIES AND LIMITATIONS
By the method described, it is possible to determine the probable fatigue life of a
given material under a given loading history without having to carry out a large number
of loading tests. As a matter of fact, the standard Miner sums as characteristics of a
material, together with the _-N curve, should contain enough data to predict the fatigue
433
life of a material with sufficient accuracy even in irregular stress sequences. The main
point obviously is ' What is meantby "sufficient accuracy"?' Of course, the M.I.S.S.
methodis more expensivethan the determination of a number of a-N curves, althoughnot
impossibly so. In return, it refines the results obtainedfrom Miner's rule (whichare
derived only from a-N curves). Only systematic tests with different materials cangive
a definite answer. Suchtests shouldprove worthwhile, for even Miner's rule in its
present form is better than it is usually believed to be. Thus, it shouldusually be possi-
ble to supplementit by introducing additional parameters (meanstress, interaction, and
sequence)so that the result can satisfy the demandsof practice.
It must be stressed in this connectionthat the testing equipmentavailable shouldbe
improved in the senseof cheaperandfaster executionof large numbers of irregular
fatigue tests. Only sucha developmentwill guaranteein reasonable time the acquisition
of the information necessaryfor anefficient verification of the methodsproposedhere
or elsewhere.
REFERENCES
1. Erismann, T.H.: Ein Verfahren zur Absch_'tzungder Lebensdauervon Materialien
bei unregelm_'ssigenBelastungsfolgen. SchweizerArch. angew.Wiss. Tech. 36,
1970,pp. 57, 103.
2. Erismann, T.H.: Fatigue-Life Prediction Under Irregular Stress Conditions. J.
Strain Analysis, vol. 5, 1970,p. 207.
BIBLIOGRAPHY
1. Miner, M.A.: Cumulative Damagein Fatigue. Trans. ASME,Ser. A, J. Appl. Mech.,
vol. 67, 1945,p. 159.
2. Palmgren, A.: Die LebensdauervonKugellagern. ZVDI, Bd. 68, 1924,p. 339.
434
tog Z
Figure I.- Interpolation for PM and Pj.
:o_ _:o
Figure 2.- Characteristic stress curves for various values of PM and Pj.
435
PQ=O _-
PQ#o m
Figure 3.- Characteristic loading histories for PQ = 0 and PQ _ 0 and PM = PJ = O.
C2<0
C2>0
Figure 4.- Characteristic stress curves for various values ol the spectrum curvature parameter c 2.
436


A parametric approach to irregular fatigue prediction

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