Random Features of the Fatigue Limit

Extremes 2:2, 165±176 (1999)

# 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Random Features of the Fatigue Limit

THOMAS SVENSSON

Swedish National Testing and Research Institute SE-501 15 BoraÊ s, SwedenE-mail: [email protected]

JACQUES DE MAREÂ

Mathematical Statistics, Chalmers University of Technology, SE-412 96 GoÈteborg, SwedenE-mail: [email protected]

[Received February 3, 1999; Revised September 10, 1999; Accepted September 10, 1999]

Abstract. The classical fatigue limit is often an important characteristic in fatigue design regarding metallic

material. The limit is usually obtained from a staircase test in combination with some assumption about the

statistical distribution of the limit. This distribution can be of a normal, log-normal or of extreme value type and

no particular physical argument gives favor to any speci®c distribution. This leads to a certain ambiguity in the

evaluation of test results which forces the designer to introduce large safety factors.

In order to ®nd a physically based statistical distribution for use in staircase tests to determine the fatigue limit

we present here a random model for the fatigue limit based on the following assumptions;

(i) The square root area model according to Murakami and co-workers is valid,

(ii) the randomness in the fatigue limit is induced by the randomness of the maximum defect size,

(iii) the random maximum defect size has an extreme value distribution of Gumbel type.

This leads to the fatigue limit distribution based on Gumbel (FLG), which is recommended to replace the

normal distribution in the evaluation of staircase fatigue tests in case of hard materials.

It turns out that the skewness of the resulting distribution depends on the coef®cient of variation; with a normal-

like non-skewed distribution at the coef®cient of variation of ®ve percent.

Key words. Fatigue limit, defects, Gumbel distribution

AMS 1991 Subject Classi®cations. 60G70, 62N05

1. Introduction

We intend to investigate the possibility of estimating the fatigue limits of metallic

materials for design purposes. The fatigue limit is de®ned as the highest stress amplitude

for which the material in question has an in®nite life. Since both tests and service

applications last only for ®nite time the de®nition for practical purposes can be interpreted

as an endurance limit, speci®ed for a certain upper limit for the life. This can be expressed

in a formula for the fatigue life, N for a material subjected to an oscillating stress with

amplitude S,

N � f �S� for S4Se

� Ne for S � Se

�;

where f�S� is a functional relationship ( for instance the Basquin equation), Se is the fatigue

limit and Ne is a speci®ed number of cycles which can be related to the application in

question or to a practical limit for reasonable test costs. Often used values of Ne are one,

two or ten million cycles.

The simplest method for the estimation of the fatigue limit is to use empirical formulae

which relate static tests of material strength, such as the ultimate strength, the yield

strength or the hardness to the fatigue limit. One such formula is in suitable units,

Se51:6 HV ;

where HV is the Vickers hardness. The validation of such formulae is questionable,

because the mechanisms that determine different strength characteristics are not always

the same, and sometimes can even be practically independent. This is in particular the case

for hard materials, where defects in the structure have a large in¯uence on the fatigue limit,

but not on static material strength properties.

The most common method for fatigue limit estimations is to make laboratory tests on a

number of specimens and use some statistical method to analyze the results. Another

method is to do a microscopic investigation of cross sections of the material and use an

empirical model to determine the fatigue limit from the distribution of the observed

defects. In this paper we ®rst summarize the advantages and disadvantages of these

methods. It turns out that one important disadvantage with the established laboratory test

method is that the choice of statistical distribution is arbitrary. On the other hand, in the

defect analysis method there are physical arguments for the choice of an extreme value

distribution, but the inspection procedure leads to a huge extrapolation in order to obtain

the fatigue limit. We here show how the distributional knowledge from the second method

can be used to improve the laboratory test evaluation method.

2. Fatigue limit tests

2.1. The staircase method

Awidely used method for the determination of the fatigue limit based on fatigue tests is the

staircase method, also called the up-and-down test strategy [1,2]. It is a sequential test,

where the result from one test determines the stress amplitude for the next. First, the

experimenter must make an initial guess of the fatigue limit S0, choose a stress step size DSand an endurance limit Ne. The ®rst specimen is then subjected to the stress amplitude S0.

If it fails before Ne has been reached, the next specimen is loaded with S1 � S0 ÿ DS,

otherwise with S1 � S0 � DS. This procedure is continued until all specimens have been

tested by choosing the load level for the i-th test to Si � Siÿ1 ÿ DS or Si � Siÿ1 � DS.

In order to analyze the results the fatigue limit Se is regarded as a random variable and

166 SVENSSON AND DE MAREÂ

its statistical properties are estimated. A general statistical tool for such estimations is the

maximum-likelihood method, which chooses that statistical distribution that maximizes

the probability of the obtained experimental result. The method can in this case be

summarized as follows:

1. Assume that the fatigue limit is distributed according to a certain distribution F,

F�s; y� � P�Se � s�, where y is a parameter vector.

2. Let the test levels be s0; s1; . . . ; sm.

3. Find the parameter vector y that maximizes the log-likelihood function l,

l � lnY

si

F�si; y�nsi �1ÿ F�si; y��msi ;

" #

where nsiand msi

are the number of failures and survivors at load level si, respectively.

E.g. in the case when the normal distribution is used the parameter vector y is two-

dimensional, containing the expectation m and the standard deviation s.

The solution of the maximization problem can be obtained numerically and con®dence

intervals for the parameters can be calculated approximately from the second partial

derivative of the log-likelihood function l, based on large sample theory assumptions.

Unfortunately, the assumptions behind the estimation of the con®dence intervals are not

valid for fatigue limit tests in general. Firstly, the sample size is usually small �530� and

the large sample theory cannot be relied on. Secondly, the estimates of the parameter stend to be biased (see for instance [3,4]).

The evaluation of the results from a staircase test can be considerably simpli®ed under

certain circumstances and assumptions. In [1] the maximum-likelihood procedure

described above is approximated resulting in analytical expressions for the estimates.

The procedure is based on two main assumptions: 1. The stress step size DS is less than

twice the standard deviation for the distribution and is kept constant during the whole

staircase test, and 2. the fatigue limit, or a suitable transformation, is normally distributed,

Se*N�m; s�. The resulting simpli®ed procedure does not include any numerical

maximization, but uses only simple analytical expressions. This procedure is widely

used in experimental practice.

Since this simpli®ed procedure is an approximation of the full maximum-likelihood

procedure, the objections to the validity of the result are the same as above, but we must

add more doubts about this simpli®ed version. Namely, the result depends on the initial

choice of the step size, and on the normal distribution assumption.

2.2. Simulations to test the validity of the staircase method

In the reports [3] and [4] the staircase method described above has been tested by

simulations. A certain statistical distribution has been ®xed, a great number of complete

RANDOM FEATURES OF THE FATIGUE LIMIT 167

fatigue tests have been simulated, and the results analyzed. The simulations show that

estimates of medians are satisfactory, but con®dence limits on small percentiles based on

staircase tests of reasonable size (� 30 specimens) are dubious because,

� the estimated standard deviation is biased,

� the test size is too small to justify the large sample theory,

� the underlying statistical distribution is unknown.

The ®rst problem may be accounted for by correction. No analytical solution to this bias

problems have been presented to our knowledge, but approximate correction formulae

should be possible to obtain by simulations. The second problem is mainly a problem of

trade off between accuracy and costs. The third problem is the subject for the present

investigation and it turns out that results from another fatigue limit estimation approach

are useful, namely the defect analysis method.

3. Estimates based on defect content

The methods presented so far only take the global behavior of the material into account

and need no more theory for the nature of fatigue limit than its existence. However,

knowledge about the physical nature of the fatigue limit has increased since the ®rst

fatigue tests were performed and now we have a picture of it that should be useful in

engineering design. For instance Miller claims in [5],

Finally, the paper illustrates why the period of crack initiation should, in most

engineering cases, be regarded as zero; that the important condition for analysis is

whether a crack, irrespective of its size, will or will not propagate; and that the fatigue

lifetime should be equated to crack propagation alone. In this context, the fatigue limit

is not related to the initiation of a crack but to one of the three threshold states which

determine whether or not a crack will continue to grow to failure.

The threshold states depend on micro-structural features of the material and their

functional relationship to the fatigue limit is not well de®ned. Therefore, the direct use of

threshold theories is not possible in engineering design problems. However, the theory

gives rise to methods based on statistical properties of the microstructure, such as the

distribution of defects. A summary of models based on inclusions and defect sizes can be

found in [6].

One of these models is the empirical��areap

parameter model which is based on two

simpli®ed parameters and is therefore suitable for engineering applications. The model is

developed for hard steels, where internal inclusions are crucial for the fatigue limit. It gives

the fatigue limit as a function of the largest defect size and the hardness of the material,

Se � q� ��areamax

p �ÿ1=6; �1�


where areamax is the largest defect size in the volume of the material that is subject to the

maximum stress and q is a function,

q � q�HV ;R�;

where HV is the Vickers hardness of the material, and R is the stress ratio Smin=Smax. The

defect size areamax is de®ned as the area of the projection of the defect on the plane

perpendicular to the applied load direction. The hardness is measured in traditional

material tests. The statistical distribution of the maximum defect size is determined by

area measurements in a number of cross sections of the material. Since the fatigue strength

of the material depends primarily on the weakest link, i.e. the largest defect, statistical

extreme value theory can be applied and only a limited set of distributions need to be

considered. Among these distributions the most popular type for inclusion sizes seems to

be the Gumbel type with the cumulative distribution function,

FV0�x� � P�X � x� � exp ÿ exp ÿ xÿ m

s

� �h i; �2�

where m and s are location and scale parameters, respectively, and V0 is the volume of the

material to which the inclusion distribution applies. The parameters m and s are estimated

from the inclusions observed on a predetermined cross section area. This area corresponds

to a certain volume and the fatigue limit is estimated by extrapolating the distribution to

the interesting material volume.

The assumption of the Gumbel distribution is based on engineering experience. This

distribution is physically unsound in the sense that it allows negative defect sizes.

However, this lack of physical relevance has been found to have no practical importance.

Recent investigations [7] do not violate the Gumbel assumption.

3.1. Extrapolation

The estimated parameters of a distribution function are usually based on a very small

volume. In order to use it for the estimation of the fatigue limit for a certain specimen or

engineering component it must be applied to a volume V corresponding to the amount of

the material that is subjected to the maximum stress. Such an extrapolation procedure

gives an estimate of the largest defect in a specimen, a component, or a number of

components and can be used to predict the fatigue limit distribution by using equation (1)

[8]. This is done under the assumption that the other elements of the formula can be

regarded as deterministic variables or constants.

The fatigue limit estimation based on the distribution of defects does not usually include

any con®dence bounds and no standard procedure for this problem has been published.

Since the inspected volume must be very small compared to the whole interesting volume

a huge extrapolation must be made. Therefore, the uncertainty in the type of distribution

and in the estimated parameters will have a great in¯uence on the con®dence that can be


given to the resulting fatigue limit estimation. However, a number of successful examples

have been demonstrated by Murakami and others where comparisons to fatigue test

estimations show discrepancies less than 10% regarding the mean values. The design

problem of ®nding lower fractiles in the distribution is not as well investigated, but Beretta

and Murakami have recently presented different methods for the estimation of con®dence

bounds [8].

The method is useful only for materials where defects are the dominating source of

fatigue crack nucleation. This is primarily the case for hard materials with the typical

defect size greater than the typical grain size. In softer materials the fatigue limit probably

depends on micro-structural barriers as grain boundaries or pearlite zones and the

application of the��areap

-method in such cases has not been yet investigated.

4. A combined method based on fatigue tests

One of the drawbacks mentioned above that is common to all methods based on fatigue

tests is that the type of the fatigue limit distribution is unknown. The small amount of

information that is used in the test evaluation, failure or not failure, is far from suf®cient to

give much empirical knowledge about the distribution type with reasonable test sizes.

Therefore, the choice of distribution is rather arbitrary which is illustrated by the fact that

the recommendations are for instance normal, log-normal, logistic, extreme value for the

largest observation, or extreme value for the smallest observation [9]. One possibility to

overcome this drawback is to combine the traditional material test method with the defect

contents approach which is shown in the following.

The��areap

-model offers the opportunity to use a distribution family that is partly based

on physical considerations for hard materials and therefore can be expected to be closer to

reality in this case than those arbitrary chosen from traditional statistics. Such a

distribution family can be calculated in the following way:

Assume that the maximum defect size follows the Gumbel distribution (2). Also assume

that the hardness of the material is a deterministic constant. For clarity, write equation (1)

with the random variables as capitals, Se � qAÿ1=6, and calculate the distribution of the

fatigue limit Se,

FSe�s� � P�Se � s� � P�qAÿ1=6 � s� � 1ÿ FA

s

q

� �ÿ6" #

� 1ÿ exp ÿ exp ÿ sÿ6 ÿ mqÿ6

sqÿ6

� �� ; �3�

where we have assumed in accordance with the physics that A, s and q are strictly positive.

Now, let

� � mqÿ6 and t � sqÿ6


and (3) reduces to the fatigue limit distribution function based on Gumbel (FLG),

FSe�s� � 1ÿ exp ÿ exp ÿ sÿ6 ÿ �

t

� �� : �4�

The corresponding density function is obtained by differentiation,

fSe�s� � dFSe

�s�ds

� 6

tsÿ7 exp ÿ exp ÿ sÿ6 ÿ �

t

� �ÿ sÿ6 ÿ �

t

� �:

The fractile sp in this distribution can easily be found by solving the equation,

FSe�sp� � p

giving

sp � � ÿ t ln ln1

1ÿ p

� �� ÿ1=6

: �5�

Equation (5) gives the fractile p of the fatigue limit and is identical with the method

presented in for instance [8]. The advantage of formulating the distribution of the fatigue

limit itself (4) is in the situation when nothing is known about the parameters in the��areamax

p-distribution. In this case the fatigue limit must be estimated from fatigue tests.

But, since we now have a model for the distribution, the parameters � and t can be ®tted to

the experimental results and the estimations of lower fractiles can be improved.

It is interesting to study the principal behavior of the fatigue limit distribution model (4)

and compare this with the recommendations in [9]. The recommendations give a wide

spectrum of distributions, symmetrical such as normal, and skewed in both directions as

the different extreme value distributions. In fact, a study of the FLG distribution (4) shows

that it has a variety of behaviours, depending on the ratio t/�. For ratios in the interval t/�[ �0:05; 0:55� the distribution can be both symmetric or skewed in either direction. For

t=�&0:28 the distribution is essentially symmetric and close to the normal distribution. In

Figures 1±3 we illustrate the different distribution behaviours. The ®gures have been

constructed as follows: three stair case tests (including 500 specimens each to avoid

estimate errors) on FLG distributed fatigue limits with different ratios t/� were simulated.

For each test we made a maximum-likelihood estimation of the distributions based on the

FLG and the normal assumptions respectively. The reason for this approach is to compare

the normal and the FLG distributions ®tted on the same data sets. The coef®cients of

variation (CV), de®ned as the ratio s/m in the resulting normal distribution N�m, s�, are 0.1,

0.05 and 0.01, which correspond to t=� � 0:06; 0:28 and 0.54 respectively.


Figure 1. Probability density functions for a small coef®cient of variation.

Figure 2. Probability density functions for a medium coef®cient of variation.


4.1. Applications

The model has been applied to a number of reported stair case fatigue tests on hard steels.

The results originates from fatigue tests (with 30 specimens for each material) on different

martensitic stainless steels. They contain coef®cients of variations giving distributions

skewed in both directions. In Table 1 some of the results are shown. It can be concluded

that the estimations of the medians are similar for the two evaluation methods, but for the

1% fractiles the differences are more signi®cant. In the case of large coef®cients of

variation the normal distribution tends to give more conservative estimates and in the case

of small coef®cients of variation the opposite is true.

Figure 3. Probability density functions for a large coef®cient of variation.

Table 1. Comparison between different distribution assumptions for real data.

Median (normal) Median (FLG) 1% fractile (normal) 1% fractile (FLG) Coef®cient of variation

554 555 509 507 0.03

726 727 677 670 0.03

727 728 672 662 0.03

594 594 539 536 0.04

606 606 531 537 0.05

584 585 507 508 0.06

714 712 618 619 0.06

763 764 661 676 0.06

616 614 496 513 0.08


5. Validity considerations

5.1. Test simulations

In order to distinguish between the FLG and the normal distribution in a fatigue test it

would be necessary to perform a huge number of tests. A ®rst attempt formulating a test

plan is the following:

1. Do a staircase test with 30 specimens and estimate the parameters for the normal and

FLG distributions, respectively.

2. Find the load level, giving the largest difference in cumulative probability between the

two distributions and test a large number �N� of specimens on this level.

3. Calculate (i) the probability for the obtained result, given that the true distribution is

normal, and (ii) the probability for the obtained result given that the distribution is FLG

and let the ratio between these two probabilities determine which distribution that is

true. Hence when the ratio is at least one we conclude that the distribution is normal and

otherwise FLG.

To ®nd what number of specimens that is needed we simulated tests according to this

plan. In the simulations we chose either the normal or the FLG distribution and checked

the fraction of tests giving a correct result. In Table 2 some of the simulation results are

shown for different true distribution types and for different numbers of tests. The

percentage represents the fraction of correct conclusions about the type of distribution.

Unfortunately, the simulations show that validation by fatigue tests is impossible for a

reasonable number of test specimens.

5.2. Validity of the assumptions

Since validity tests seem to be unrealistic it is necessary to make a close examination of the

assumptions behind the FLG distribution model:

1. The square root area model, i.e. equation (1), is valid. This assumption has been

empirically checked for a great number of materials with the following conclusion [6]:

They concluded from more than 100 experimental data that the prediction error is

Table 2. Percentage of correct conclusions for a fatigue limit distribution

with the coef®cient of variation: 0.075.

N � 200 N � 400 N � 1000

Normal 50% 47% 60%

FLG 72% 73% 75%


mostly less than 10% for notched and cracked specimens having��areap

less than

1000 mm and for HV ranging from 70 to 720.

2. The hardness can be regarded as a deterministic variable in this context, i.e. the

randomness of the maximum defect sizes determines the statistical properties of the

fatigue limit. This assumption may be critical, since hardness measurement results

usually contain a large scatter. However, partly this scatter depends on the

measurement errors, which may dominate compared to the inherent material hardness

variation. This point needs a more careful examination.

3. The distribution of the defects is of Gumbel type. The theory behind the square root

area model is based on the weakest link theory, and it is reasonable to assume that the

distribution of the maximum defect sizes is an extreme value distribution. There exist

three different types of such distributions, namely the type I (Gumbel), the type II

(Frechet) and the type III (Weibull) distribution [10]. We have here used the Gumbel

type, but the other two may be useful. Further investigations of this point are needed.

6. Conclusions

Established methods for investigating the fatigue limit have proven to be useful for the

estimation of the median value of the fatigue limit distribution. This is true both for the

methods based on sequential fatigue tests and for the method based on defect investigations.

Hence, they are useful in applications that intend to compare or improve the quality of

different materials. However, lower fractiles are not satisfactory estimated, which is a

critical drawback in the application of engineering design. In the case of fatigue test methods

one important reason for this drawback is the lack of knowledge about the type of statistical

distribution of the fatigue limit. In the case of defect content methods huge extrapolations

are necessary that give uncertain estimates of the dispersion in the fatigue limit.

An alternative method for the estimation of the fatigue limit distribution has been

presented here that combines the two different approaches and thereby avoid some of the

drawbacks. The method is based on some assumptions, which seem reasonable based on

consideration of the physical background. This physical background should be a strong

argument for chosing the FLG distribution in favor of the traditionally arbitrary choices.

The use of the method is limited to high strength materials, where the fatigue limit is

governed by the maximum inclusion size, but the validity of the model may be extended if

the square root area property could be identi®ed in some other defect type, for instance the

largest grain, the largest notch or the most severe surface roughness.

7. Acknowledgments

This work is supported by the Swedish Institute of Applied Mathematics (ITM), AB

Sandvik Steel, ABB Stal AB, the Stochastic Center at Chalmers University of Technology

and the Swedish National Testing and Research Institute.


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Random Features of the Fatigue Limit