ijf 2007 - norberg olsson - the effect of loaded volume and stress gradient on the fatigue limit

7/31/2019 IJF 2007 - Norberg Olsson - The Effect of Loaded Volume and Stress Gradient on the Fatigue Limit

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The effect of loaded volume and stress gradient on the fatigue limit

S. Norberg a,b, M. Olsson b,*

a Scania CV AB, SE-151 87 So derta lje, Swedenb Solid Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden

Received 11 August 2006; received in revised form 8 November 2006; accepted 22 November 2006Available online 19 January 2007

Abstract

In this paper an investigation of multiaxial stress based criteria and evaluation methods is presented. The criteria are used with thepoint, gradient and volume methods. The purpose is to determine the combination of criteria and methods that is best suited for designagainst the fatigue limit. The evaluation is based on elastic FE-analysis of 15 geometries for which the fatigue limit loads are known. Thepoint method is based on the maximum values of the fatigue stress in each specimen. With the gradient method, the fatigue stress isadjusted with the relative or absolute gradient of the fatigue stress itself. With the volume method, a statistical size effect is considered,by use of a weakest link integral. Thus, the probability of fatigue depends on the fatigue stress distribution. Also, the gradient and vol-ume methods are combined. The results show that the point and gradient methods are not good for prediction of the fatigue limit. It isrecommended to use the volume method in fatigue design. It is accurate enough for prediction of the fatigue limit, straightforward to useand easy to interpret. The choice of method is much more important than the choice of criteria.Ó 2007 Elsevier Ltd. All rights reserved.

Keywords: High cycle fatigue; Weakest link; Statistical size effect; Gradient effect

1. Introduction

1.1. The importance of fatigue assessment

A machine component must meet several requirementsregarding mechanical properties, such as static strength,fatigue strength, and stiffness. It must also be economicalin the sense that the total cost of the whole structure,including both production and use, should be low. Theamount of raw material used and the final volume of the

component are thus important. In a vehicle application itdirectly affects the weight that needs to be transported.Unnecessary weight decreases the possible payload as wellas increases the fuel consumption. Another, often moreimportant, factor to consider is the reliability of the com-ponent or the whole structure. The reliability can be influ-enced by many factors. However, fatigue is usually themost important life-limiting phenomenon.

The designer must balance reliability against total cost.The goal is to achieve an optimum design that incorporatesall the requirements. The industrial need for accurate fati-gue predictions is obvious. Erroneous, or neglected, fatigueassessment during component design can lead to malfunc-tion. The cost that eventually follows from product failurecan be considerable for the producer as well as the con-sumer. Failures can also lead to accidents.

In this paper, the combined influence from fatigueassessment methods and criteria based on elastic FE-anal-

yses of the loaded structures is investigated. The goal of theinvestigation is to identify criteria and methods that aregood for prediction of the fatigue limit loads for a numberof different geometries. In addition, the methods should bequick and easy to use.

1.2. Current methods of fatigue assessment

In [1] different criteria and the fatigue post-processorFemfat were evaluated. It was found that all the evaluatedcriteria were unsatisfactory. It was observed that the

0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2006.11.011

* Corresponding author. Tel.: +46 8 790 75 41; fax: +46 8 411 24 18.E-mail address: [email protected] (M. Olsson).

www.elsevier.com/locate/ijfatigue

Available online at www.sciencedirect.com

International Journal of Fatigue 29 (2007) 2259–2272

International Journalof

Fatigue

mailto:[email protected]

mailto:[email protected]



estimations of the fatigue limit load of the different speci-mens are related to how severe the stress concentration ineach specimen is. This, in conjunction with a difficulty toseparate the criteria, lead to the conclusion that it may bemore important to take the spatial distribution of the stressinto account than to choose among existing criteria. The

investigation presented here attempts to clarify whetherthe large differences in the maximum stresses of the differ-ent specimens, when evaluated at their fatigue limit loads,can be explained with the use of either the statistical sizeeffect or the gradient effect.

Both these effects have been suggested by numerousauthors [2–7]. Both effects are motivated by the fact thatall empirical experience shows that the higher the maxi-mum stress at the fatigue limit load the smaller the volumesubjected to high stresses. A small volume of high stresswith moderate stresses around requires a large stress gradi-ent. This makes it reasonable that the stress gradientshould be related to the allowable value of the maximum

stress. If the distribution of flaws is roughly uniform andthe volume subjected to a high stress is small, the worstflaw will probably be smaller than the worst flaw in a largervolume. A smaller volume can thereby probably tolerate ahigher stress. The weakest link method is derived from suchstatistical reasoning.

1.3. Point method

The point method only requires the time history of thestress tensor to be known at the point. Information aboutthe stress in the surroundings or any gradients or the size

of the specimen is not used. The criteria used in this workare Crossland [8], Dang Van [9], Findley et al. [10], Matake[11], Papadopoulos [12], and Sines [13]. They are evaluatedat all material points as described by Eqs. (1)–(8). In theequations, symbols not explicitly defined should be self explanatory. All criteria are expressed with the use of twoequations. The equation to the left defines a local fatiguestress; the equation to the right compares this with a criticalmaterial value. If this critical value is exceeded at any pointinthe loaded specimen or component, fatigue is predicted tooccur.

The Crossland criterion uses a von Mises stress modifiedwith the maximum hydrostatic stress during the load cycle,

rvM;a þ aCrh;max ¼ bC; bC 6 bcritC : ð1Þ

The subscript ‘‘a’’ means amplitude, ‘‘vM’’ von Mises and‘‘h’’ hydrostatic. All a- and bcrit-parameters are criteria spe-cific material properties and the local value of the b-para-meter is the fatigue stress of each criterion, respectively.The fatigue stress of criterion ‘‘X ’’ is denoted bX , thus,bC is the Crossland fatigue stress, and the a- and bcrit-parameters are named analogously.

The Dang Van criterion is based on the use of a modi-fied stress state with stresses ~rijðt Þ. This is used to computethe maximum shear stress and the hydrostatic tension as

functions of time and they are combined in the criterion,

maxt

ð~smaxðt Þ þ aDVrhðt ÞÞ ¼ bDV; bDV 6 bcritDV: ð2Þ

The modified stress state is defined by using a yield surface.A material that exhibit an infinitely low mixed hardening isassumed. This hardening will lead to the smallest possibleyield surface that contains the time history of the stress ten-

sor. The deviatoric part of the midpoint of this yield sur-face is subtracted from the original stress state. This leadsto a stress state from which all mean shear stresses havebeen removed. Further, the differences between the meansof the principal stresses are removed but the mean hydro-static stress is preserved. Also, all varying parts of the ori-ginal stress state are retained since the stress state ismodified with a constant. In the criterion, the instanta-neous values of the largest shear stress of the modifiedstress state, ~smaxðt Þ ¼ ð~r1ðt Þ À ~r3ðt ÞÞ=2, and the hydrostaticstress are combined and the fatigue measure is defined asthe largest value of this combination during the load cycle.

The Findley criterion is a critical plane criterion and it is

evaluated on the plane on which the expression in theparenthesis in Eq. (3) attains its highest value. The fatiguestress of the criterion equals this maximum value,

maxall planes

ðsa þ aFrn;maxÞ ¼ bF; bF 6 bcritF : ð3Þ

Here, sa is the shear stress amplitude of a plane. The shearstress amplitude is defined by the smallest circumscribedcircle method [14], and rn;max is the largest normal stressthat acts on the plane during the load cycle.

The Matake criterion is a critical plane criterion, like theFindley criterion, but it uses one measure to identify thecritical plane and another to evaluate the loading of it,

maxall planes

ðsaÞ þ aMrÃn;max ¼ bM; bM 6 b

critM : ð4Þ

The normal stress rÃn;max is evaluated on the plane with the

largest shear stress amplitude. It is defined as the largestnormal stress that occurs on that plane during the load cy-cle.

The Papadopolus criterion is a critical plane criterionwhich uses a special type of shear stress,

T a; max þ aPrh; max ¼ bP; bP 6 bcritP ; ð5Þ

T a; max ¼ maxall planes

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

pZ

2p

0

s2aðvÞ dvs : ð6Þ

Here, an effective shear stress amplitude, T a;max is used. It isdefined as an RMS-value of a shear stress amplitude, saðvÞ.On a cutting plane the stress vector, i.e. the traction vector,can be divided into a shear stress vector and a normalstress. The time history of the shear stress vector forms atrajectory on the cutting plane. This trajectory is projectedonto a line in the cutting plane, that line is oriented with theangle v. The shear stress amplitude, saðvÞ, is defined as half the length of this projection, see Fig. 1.

The Sines criterion is similar to the Crossland criterionbut it uses the mean hydrostatic stress instead of the

maximum,

2260 S. Norberg, M. Olsson / International Journal of Fatigue 29 (2007) 2259–2272



rvM;a þ aSrh;mean ¼ bS; bS 6 bcritS : ð7Þ

The mean should here be interpreted as

rh;mean ¼ ðmaxt

rh À mint

rhÞ=2: ð8Þ

1.4. Gradient methods

The maximum local stress at the fatigue limit load isnot the same in a uniformly stressed specimen as in anotched one; the specimen with a stress concentration willtolerate a higher maximum stress. The gradient theory iscommonly used to explain, at least qualitatively, thiseffect. At a stress concentration stresses decrease rapidly,i.e. the gradient is large. This is one motivation for the

gradient theory. Several physical motivations can be con-sidered. It could be that the material actually benefits froma gradient and it can thus withstand a higher maximumstress in the presence of a stress gradient. Or, it is the aver-age stress over a length, area or volume that determineswhether fatigue occurs, or not. If the gradient of a fatiguestress is combined, linearly, with that stress itself, it can beinterpreted to give the value a certain distance away. Thesize of the highly loaded volume is related to the gradient.Also, a strong gradient could lead to non-propagatingcracks.

A criterion usually consists of some shear stress ampli-tude modified with some kind of normal stress. The gradi-ent method causes some further modification, based on thegradient of some stress measure.

In this work, different gradient approaches are investi-gated. Two approaches are based on the criteria describedabove and a third is according to Papadopoulos and Pan-oskaltis [6].

1.4.1. Fatigue stresses adjusted by the absolute and relative

gradients

Both the absolute and relative gradient methods arebased on the point methods discussed earlier. The fatiguestress is modified with the gradient of the fatigue stress

itself. Thus, a gradient adjusted fatigue stress is created.

The adjustments are performed with either the absoluteor the relative gradient.

The absolute gradient method is formulated as

bgrad;abs X ¼ bpoint

X À cabs X grad bpoint

X

À Á : ð9Þ

The subscript ‘‘X ’’ indicates the criterion. The relative gra-

dient modifications are

bgrad;rel X ¼ bpoint

X À crel X

grad bpoint X

À Á bpoint X

: ð10Þ

In both these methods the c-parameter is used to controlhow strong the effect of the gradient adjustment is. Forthe absolute gradient method the c-parameter has a directlength scale interpretation. For the relative gradient meth-od the interpretation of c is not so clear. The c-parametersshould be positive in order to explain results from fatiguetesting, i.e. gradient should reduce the fatigue stress.

1.4.2. The criterion by Papadopoulos and Panoskaltis

In [6] Papadopoulos and Panoskaltis offer a gradientadjusted criterion. It uses the relative gradient of the max-imum hydrostatic tension during a load cycle to modifywhat basically is the Crossland criterion as

bgradPP ¼ ra

vM þ aPPrh; max 1 À cPP

maxt

jgradðrhÞj

rh; max

* + !: ð11Þ

As above, the c-parameter controls the effect of the gradi-ent contribution to the criterion’s fatigue stress. The brack-ets hi in Eq. (11) have the property that they yield zero if is negative and also if rh; max is zero.

1.5. Volume method

The statistical volume effect as described by Weibull[2,3] is used in this work. It is assumed that if the fatigueloading in a sub-volume exceeds the fatigue resistance of the sub-volume, the entire specimen or component will suf-fer fatigue failure. The fatigue stress must be computedfrom a known stress state. In this study, the different fati-gue stresses of the criteria discussed above are used toquantify the fatigue loading. The resistance of the materialis assumed to follow a three parameter Weibull distribu-tion, with parameters r0, ra, and m. Here, r0 is a material

fatigue limit under which the probability of fatigue is zero.The parameters ra and m characterize the scatter and tail of the probability distribution. The fatigue resistance isdescribed for a reference volume, V 0, and it is assumed thatthe fatigue resistances at all points are statistically indepen-dent. This makes it possible to compute the probability of failure for the entire component if the probability of failurefor each sub-volume is known,

P f ¼ 1 ÀY N

n¼1

ð1 À ð p f ÞnÞ: ð12Þ

Here, P f is the total probability of failure and ( p f Þn the

probability of failure for the nth of the N sub-volumes.

Fig. 1. Schematic of the computation of T a;max. The angles u and h definethe normal direction n of a plane. The shear stress amplitude saðvÞ isdefined by projection of the shear stress vector trajectory on a line whosedirection in the plane is given by v.

S. Norberg, M. Olsson / International Journal of Fatigue 29 (2007) 2259–2272 2261



In the continuum limit Eq. (12) in combination, with theWeibull distribution of the fatigue strength, leads to

P f ¼ 1 À eÀR

V

bX Àr0ra

À ÁmdV V 0 : ð13Þ

The statistical method used in this application of the vol-ume method can also be applied to surface methods. Also

combinations of the volume and surface methods havebeen suggested, e.g. [4]. In the present investigation onlythe volume method is used.

2. Experiments

Nishida et al. [15] have performed fatigue tests on spec-imens of the same material but different shapes. The geom-etries of the different specimens are shown in Fig. 2. Thespecimens can be divided into groups. They can first bedescribed by their basic geometry, smooth or with a cir-cumferential stress concentration, either a notch or a step.

In addition, these basic shapes are modified with holesdrilled in the radial direction. The holes go through thespecimens, and have different diameters. All holes arelocated at the root of the notch or step. The specimens weremanufactured from a mild steel, JIS S15C, and electropolished.

The fatigue testing was carried out as rotating four pointbending. The results of the fatigue tests are shown in Table1. It is the nominal bending stress at a circular cross sectionwith diameter ten millimetres that is shown.

3. Computation of criteria

The numerical evaluation of the criteria and methods isperformed with the fatigue post-processor FAST (FatigueAnalysiS Tool) which has been developed to aid in bothcriteria development and fatigue strength assessment of actual components. FAST is described in [1], and only ashort overview is given here.

FAST is built to interact with the Abaqus FEM-code

and consists of two parts. One part is written in Pythonand interacts with Abaqus and the other part is writtenin Matlab and performs the actual computations. Theapplication of FAST is done in several steps. The first stepis to identify which part of the FE-model that should beanalysed. The data needed for the computations are thenextracted from the Abaqus Odb-file, transformed to andsaved in Matlab format. In this work the stresses at theintegration points are used, and also the integration pointvolumes and coordinates. The reason for this is that theAbaqus Odb-file holds stress data for these points.

The computations are done in two steps. The first step

consists of computing commonly used variables in orderto save work by not having to compute them several timesas they are used to compute different criteria. Examples of such variable are the von Mises stress and the largesthydrostatic stress. The second step is the computation of actual criteria which also includes insertion of the fieldvalues of the criteria in the Abaqus Odb-file. This makesit possible to assess the fatigue loading in a viewer. Theform functions of the elements are thus used to inter-and extrapolate the field values throughout the elements.Several of the computations performed require a searchfor a maximum or minimum over one or two variables.An adaptive algorithm that substantially saves on theamount of computations needed was developed for thispurpose, see [1].

The criteria were implemented already with the workdescribed in [1]. The implementations of the gradient andvolume methods are described in the Appendix. Two differ-ent methods for computing the gradient are presented. Theeight parameter version has been used in this work.

This work is solely concerned with specimens tested inrotating bending. The specimens do not have rotationalsymmetry, because of the drilled holes. This necessitatesthat one full revolution is modelled. However, there is nodifference in the resulting stresses caused by whether the

specimen or the load is rotated. This makes is possible to

Fig. 2. Summary of the geometries of the test specimens. The specimensare given names like A0, B5, C10, and D15. The letter refers to thegeometries above. The number refers to the diameter of the hole at the testsection, X in the figure above. The number indicates the diameter of thehole in tenths of millimetres, i.e. B5 has a 0.5 mm hole, and zero meansthat there is no hole. All measurements in the figure are given in

millimetres. For the detailed geometry see Nishida et al. [15].

Table 1The fatigue test results by Nishida et al. [15]

Specimen A0 A10 A15 B0 B5 B10 B15 C0Strength/MPa 190 100 90 120 100 100 95 150Specimen C5 C10 C15 D0 D5 D10 D15Strength/MPa 110 110 105 115 100 90 85

The nominal stress at the smallest cross section, cylindrical with diameter

10 mm, is presented.


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model rotating bending with a rotating bending momentinstead. A rotating bending moment can be described by

Mrotðt Þ ¼ M x sinðxt Þ þ M y cosðxt Þ; ð14Þ

where the two bending moments are perpendicular to eachother and the axial direction of the specimen. Since the

material behaviour is linearly elastic the stress at any point,x, in a specimen can then be described as

rðx; t Þ ¼ rM xðxÞ sinðxt Þ þ rM y

ðxÞ cosðxt Þ: ð15Þ

This allows a simplified FE-analysis where only the stressfields, rM x

and rM y , of the two bending moments Mx and

M y, respectively, are computed. They are combined inFAST to form the stress history of the rotating bendingloading, as in Eq. (15). This means that all codes used tocompute the criteria are adapted to this. This was consid-ered to be the best way to handle the rotation since it al-lows the stress history to be modelled exactly.

4. Evaluation of criteria and methods

4.1. Value of the a-parameter

All point criteria are formulated as a sum of a shearstress amplitude term and some sort of normal stress term.The normal stress term is weighted with an a-parameter.The value of this parameter is used to tune the criteria sothat they give correct predictions for different stress histo-ries. The specimens used for the evaluation have similarstress states. Firstly, this means that the value of thea-parameter is of small significance in this investigation.

Secondly, it would also be very difficult to establish theproper values of the a-parameter; the scatter in the testdata is likely to be more significant than the influence of the a-parameter. Thus, it is chosen to assign values to thea-parameters by other means. The a’s are chosen so thatthe shear stress fatigue limit in fully reversed torsion is80% of the normal stress fatigue limit in push-pull, i.e.sf ¼ 0:80rf . This is a typical relation for mild steels. Theresulting values of the a-parameters are shown in Table2. This determination of the a-parameters also means thatthere is one parameter less to be established during theevaluation.

4.2. Error Index

A measure of the error of the predictions of fatigue isneeded. It is necessary for comparison reasons so that theset of parameters that gives the smallest error can be found.In this work a measure called the Error Index is used. The

Error Index is defined in the same way for the point, gradi-ent, and volume methods. Scale factors, denoted nk for thek th specimen, that relates the predicted and experimentallyfound fatigue limits are used. If the experimental and pre-dicted fatigue limits are equal, then nk ¼ 1. The definitionsof this scale factor for the different methods are discussedbelow for each method.

The Error Index of a combination of one criterion andone method is defined from the values of nk of all specimens,

Error Index ¼X M

i¼1

ðni À 1Þ2; ð16Þ

where M is the number of specimens. In this investigationis M ¼ 15.

4.3. Point method

The spatial distribution of the fatigue stress is not con-sidered when the criteria are evaluated point-wise. Instead,it is the highest value of the fatigue stress, at any point inthe specimen that is used. If the point method can describethe fatigue phenomena correctly with a criterion, the max-imum fatigue stress of that criterion will be the same for allspecimens.

The value of the a-parameter is computed as describedabove. This means that there is no parameter left to fit tothe test data. The results are therefore obtained directlyby computing each criterion. The bcrit-parameter for eachcriterion, in Eqs. (1)–(5) and (7), is chosen as the averageof the maximum fatigue stresses of all specimens for thatcriterion. The fatigue stresses, b X , of the different criteriaare then normalised with the bcrit

X -value of the criterion. Itis the maximum value of the normalized fatigue stress thatis used in the computation of the Error Index.


The gradient methods give local values. This means thatthe evaluation is performed analogously with the evalua-tion of the point method. The difference from the pointmethod is that there is now a parameter that freely canbe assigned a value, the c-parameter. The value of thisparameter is chosen so that the Error Index is minimized.

4.5. Volume method

The volume method is based on a description that givesa density of failure probability in all points of the specimen.This is used to compute the total probability of failure for

the entire specimen. It is not necessarily the point with the

Table 2The values of the a-parameters of the different fatigue criteria

Criterion Value of the a-parameter

Crossland 1.16Dang Van 0.90Findley 0.75Matake 0.60Papadopoulos 0.90Sines –

Note that the Sines fatigue criterion does not have any value assigned tothe a-parameter. The Sines fatigue criterion uses the mean hydrostatictension to modify the shear term. The mean hydrostatic tension is zero forboth the load cases used to assign values to the a-parameters. This makes

it impossible to assign any value.




highest density of failure probability that will show fatiguefailure initiation. Another part with a lesser density but in alarger volume may be more likely to see fatigue initiation.The volume method results in probabilities of failure whichis difficult to compare with the results of the other methods;a probability of failure is unsatisfactory as the result of the

evaluation.If a specimen is analyzed at the experimentally foundfatigue limit load a probability of fatigue will be found.This probability should be 50%; that is the probability of failure for the experimentally found fatigue limit in thisinvestigation. The load applied in the analyses is thereforescaled so that the fatigue probability is 50%. The necessaryload scale factors, nk , are used in the Error Index computa-tion. The parameters of the Weibull distribution are chosenso that the Error Index is minimized.

5. Results

The results of the different criteria are presented withnormalized values of the fatigue stress. The fatigue stressesof each point or gradient criteria are normalized so that theaverage of the maximum values for all specimens is one. Aperfect criterion would have a normalized fatigue stress of one for every specimen. A value higher than one meansthat the criterion predicts fatigue failure at a lower loadthan the actual fatigue limit load, and a value lower thanone means that the criterion consider the actual fatiguelimit load to be safe. In the case of the volume methodthe scale factors of the load needed for a failure probabilityof 50% are used, as discussed above.

5.1. Point method

The point method is evaluated with the largest values of the fatigue stresses for each criterion. The maximum valuesof the normalised fatigue stress for each criterion and spec-imen are shown in Fig. 3. In Table 3 the Error Indices of

the point method are shown, as well as those of the othermethods.The differences between the criteria are small; the differ-

ences between the specimens are large. A direct applicationof the criteria in this way does not give a result that can beused for design purposes. Note that the stress states aresimilar; this means that the observed differences are closelyrelated to the values of the maximum stresses in each spec-imen, which differ substantially for the different specimens.

It should be observed that the errors are systematic.Specimens with both a hole and a notch withstand muchhigher stresses than the smooth one or those with only a

0

0.2

0.4

0.6

0.8

1

1.2

1.4

A0 A10 A15 B0 B5 B10 B15 C0 C5 C10 C15 D0 D5 D10 D15

Crossland Dang Van Findley

Matake Papadopoulos Sines

Specimen

N o r m a l i z

e d f a t i g u e s t r e s s

smooth

with

hole

notched with hole

notchedsmooth

Fig. 3. Results for the normalized fatigue stress of the fatigue criteria for the point method. Note that all fatigue criteria give similar results.

Table 3Error indices of the different methods based on the different criteria

Pointmethod

Relativegradientmethod

Absoutegradientmethod

Volumemethod

Volumeandgradientmethod

Crossland 0.687 0.563 0.244 0.047 0.017Dang Van 0.561 0.487 0.234 0.035 0.021Findley 0.682 0.576 0.238 0.047 0.017Matake 0.682 0.576 0.240 0.047 0.017Papadopoulos 0.559 0.485 0.234 0.035 0.021Sines 0.844 0.737 0.231 0.080 0.014




notch. The ones based on the smooth specimen but withdrilled holes take a middle position.


The results of the relative gradient method are shown in

Fig. 4. Compared with the point method in Fig. 3, the val-ues of the specimens with the smallest holes, B5, C5, andD5 have decreased. The specimens with the medium holes,A10, B10, C10, and D10 show a decrease, but not quite aslarge. The maximum stress is comparable in the specimensthat have holes, but the gradient is dependent on the size of the hole, and does therefore vary much more. When thevalues of the Error Indices in Table 3 are studied, a smallimprovement can be seen when the point and relative gra-dient methods are compared for each criterion.

In Fig. 5 the result of varying the gradient parameter forthe Crossland criterion adjusted with the relative gradientis shown. In Fig. 5a the fatigue stress is shown for all the

different specimens. For a certain value of the gradientparameter the highest value at any integration point inthe FE-model of each specimen, is shown. For small valuesof the gradient parameter the curves are dominated by theintegrations points with high values of the point criteria.These points also exhibit large gradients and they are there-fore superseded by integration points with more moderatevalues of the point criteria but considerably lesser gradientsas the value of the gradient parameter increases the influ-ence of the gradient. In Fig. 5b the normalised values of the fatigue stress adjusted with the relative gradient areshown for the Crossland criterion. The average for all spec-

imens in Fig. 5b is thus one. In addition to the curves of thedifferent specimens the Error Index is shown, the dashed

line. The Error Index curve shows that the value of the rel-ative gradient parameter is of little importance up untilroughly 0.14 MPam and that values beyond that causelarge errors. When the normalized fatigue stresses of thedifferent specimens are studied, it becomes clear that evenat the value of the gradient parameter that gives the best

fit the differences between the specimens are significant.In Fig. 6 the results of the absolute gradient method areshown. The Error Indices are given in Table 3. In Fig. 7results for the absolute gradient method analogous to theseof Fig. 5 are shown. For this case, the Error Index behavesvery differently; there is a clear optimum value, see Fig. 7b.The values of the gradient parameters that give the best fitfor each criterion are given in Table 4.

The gradient criterion by Papadopoulos and Panoskaltiswas also studied. The result is shown in Fig. 8. This crite-rion is similar to the relative gradient method, especiallyfor the stress states used in this evaluation. The differentstress states are basically equal but for the level of stress.

This means that all stresses and fatigue stresses will beroughly proportional. This has the effect that there is nolarge difference between the relative gradient method basedon the Crossland criterion and the Papadopoulos and Pan-oskaltis criterion, respectively. This is confirmed by theircurves, Figs. 4 and 8, and that the Error Index for thePapadopoulos and Panoskaltis criterion is 0.53, which isclose to the 0.56 of the relative gradient criterion basedon the Crossland criterion.

5.3. Volume method

The results of the volume method are shown in Fig. 9and the Error Indices are tabulated in Table 3. The volume

0

0.2

0.4

0.6

0.8

1

1.2

1.4




Specimen N o r m a l i z e d f a t i g u e s t r e s s a d j u

s t e d w i t h t h e r e l a t i v e g r a d i e n t

smooth with hole

notched with hole

notched

smooth

Fig. 4. Results for the normalized fatigue stress adjusted with the relative gradient method. Note that all fatigue criteria give similar results.




method is a significant improvement compared to the pointand gradient methods. It should be noted that the weakestlink method use the Weibull distribution and therefore hasthree parameters available for fitting purposes; the valuesof these parameters that give the best fit for each criterionare shown in Table 5. Nonetheless, the results are muchbetter and the accuracy of the predictions make the volumemethod useable for design purposes. The scale factorsrange from 0.9 to 1.1, with most values clearly in the mid-dle of the interval, i.e. the error of most predictions are less

than 10%.

In Fig. 10 the volume distribution of the fatigue stress byCrossland is shown for all specimens. The figure shows thevolume of the specimen subjected to at least the fatiguestress value indicated on the horizontal axis. The central50 mm of each specimen is used since this is the interestingpart. Fig. 10 shows large variations in both stresses andvolumes subjected to different stresses in the specimens. Itshould be noted that the volume axis of the diagram is log-arithmic. The smooth specimen, A0, is the most evenlyloaded specimen, and D5 show the highest Crossland fati-

gue stress.

Fig. 5. The effect of varying the relative gradient parameter for the Crossland fatigue criterion. (a) Fatigue stress adjusted with the relative gradient for thedifferent specimens. (b) The result of figure (a) is scaled so that the average value is one for all values of the relative gradient parameter. Here the scatter

can be studied. The variation of the Error Index with the relative gradient parameter is also shown.

0

0.2

0.4

0.6

0.8

1

1.2

1.4




Specimen

N o r m a l i z e d f a t i g u e s t r e s s a d j u

s t e d w i t h t h e a b s o l u t e g r a d i e n t

smooth with hole

notched

with hole

notched

smooth

notched

notched

notched with hole

notched with hole

Fig. 6. Results for the normalized fatigue stress adjusted with the absolute gradient method. Note that all fatigue criteria give similar results.




5.4. Combined volume and gradient method

The results with the combination of the gradient andvolume methods are shown in Fig. 11 and the Error Indicesare given in Table 3. It can be observed that this combina-tion gives the nominally best results. This is expected sincethere is an additional parameter used, which will lead tobetter agreement.

However, if the parameters in Table 6 are studied it can

be observed that the sign of the gradient are questionable.The negative sign means that a higher gradient is moresevere than a small one; this is contradictory to how the

Fig. 7. The effect of varying the absolute gradient parameter for the Crossland fatigue criterion. (a) Fatigue stress adjusted with the absolute gradient. (b)

The result of figure (a) is scaled so that the average value is one for all values of the absolute gradient parameter. Here the scatter can be studied. Thevariation of the Error Index with the absolute gradient parameter is also shown.

Table 4Parameter values for the absolute gradient method that minimizes theError Indices

Criterion Absolute gradient parameter (m)

Crossland 0.000225Dang Van 0.000281Findley 0.000240Matake 0.000230Papadopoulos 0.000281Sines 0.000190

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40


Specimen

N o r m a l i z e d g r a d i e n t a d j u

s t e d f a t i g u e s t r e s s

smooth with hole

smoothnotched

notched with hole

Fig. 8. Results for the normalized fatigue stress adjusted with the gradient according to the method by Papadopoulos and Panoskaltis.




gradient method is generally described. A gradient adjustedfatigue stress has been used in the Weibull integral. Theminimization of the Error Index gives the negative signof the gradient parameters.

6. Discussion

In this investigation the fatigue stress is used as the con-nection between the stress cycle and the fatigue process.The process is also influenced by micro structural details,etc. Hence, uncertainty is present and will lead to scatterin fatigue results. The fatigue limit loads for the differentgeometries given in Table 1 are expected to contain suchunknown scatter. It is not possible to know which, if any,value is better than the other. For this reason, all the 15geometries used in this investigation contribute in the sameway to the Error Index. This is different from the procedurein the design situation, where smooth specimens usually areused to find the fatigue limit stress. That value is considered

‘‘correct’’ and it is used for design of other geometries. It isinteresting to note that for the volume criteria, see Fig. 9,the fatigue limit load of the smooth specimen is very wellpredicted. For the fatigue stress, Fig. 3, the relative gradi-ent adjusted fatigue stress, Fig. 4, and the absolute gradientadjusted fatigue stress, Fig. 6, this is not the case.

Besides the scatter of the results, there can also be sys-tematic errors, so that the fatigue stresses based on elasti-cally computed stresses do not represent a measure of thepropensity for fatigue. This could happen if the criterionis poor or if local plasticity at high stress concentrationsis present. From Figs. 3 and 4, the point and relative gra-

dient methods, it is possible to see that there are systematic

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40




Specimen

R e q u i r e d l o a d s c a l e f a

c t o r

smooth with hole

smooth

notched

notched with holenotched

notched

notched with hole

notched with hole

Fig. 9. Required load scale factors for 50% fatigue probability for the volume method.

Table 5Parameters of the Weibull distribution for the volume method that givesthe best fit, minimum Error Index, for each criterion

r0 ðMPaÞ ra ðMPaÞ M

Crossland 243 5.35 3.44Dang Van 141 6.80 4.34

Findley 175 4.00 3.47Matake 141 3.15 3.46Papadopoulos 141 6.91 4.36Sines 160 4.08 3.50

Here, V 0 ¼ 4 cm3.

Fig. 10. Volume distribution of the fatigue stress of the Crossland fatiguecriterion. Each curve represents a specimen loaded at the experimentallyfound fatigue limit load. The Weibull parameter r0, under which thefatigue probability is zero, is also indicated. The volume-axis has a

logarithmic scale.




errors. There is a direct correlation between the type of specimen and the error of the prediction. Since these errorscan not be found for other methods, it seems likely that thefault can be attributed to the used methods. In Fig. 3 it mayalso be seen that the experimentally found fatigue limitload of specimen D5 seems a bit too high.

The purpose of the gradient adjusted fatigue stresses(relative and absolute) is to reduce the modelling error.The gradient adjusted fatigue stresses are non-local correc-tions that can be given different physical interpretations,for example, incorporation of plasticity. The c-parametercontrol how powerful an effect the gradient adjustment willbe. In Fig. 4, the results for the relative gradient adjustedfatigue stresses are shown. The improvement is quite smallcompared to the local fatigue stress. There is still a depen-dence on geometry, similar to that of the point criteria.This is also the case for the results of the gradient criteria,Fig. 8. In Fig. 6, the absolute gradient adjusted fatiguestresses are shown. The results are not quite satisfactory

for design purposes but there is an improvement compared

to the results of the point and relative gradient methods. Itis easier to interpret and understand the meaning of theabsolute gradient adjusted fatigue stress; at a field pointit is the lowest fatigue stress found on a small spherearound the point. In Fig. 12, three contour plots are shownfor specimen B10 and the Crossland criterion; in (a) the

local fatigue stress, in (b) the relative gradient adjusted fati-gue stress and in (c) the absolute gradient adjusted fatiguestress are shown. As can be seen, the relative gradient con-tours are very similar to the local fatigue stress, the adjust-ment is quite small. The absolute gradient adjusted fatiguestress show quite different contours.

An alternative gradient computation with a 20 parame-ter fitting function was also investigated. It did not lead toany significant improvement of the results. Further, it iscomputationally more expensive as well as being more sen-sitive to numerical disturbances. The used FE-meshesresolve the interesting parts of the specimens very welland there are therefore only small errors inherent withassuming that the fatigue stresses are linear within the ele-ments. The possible error with the used eight parameter fit-ting is certainly not on a scale as to affect the conclusions.

The improvement of the gradient method over the pointmethod is small. If the values of point and relative gradientmethods are compared, a small improvement can beobserved when the point and relative gradient methodsare compared for each criterion. The improvement is muchtoo small to make it probable that the relative gradient is aproperty of fundamental significance for the fatigue phe-nomena. Also, the relative gradient results show the samedependence on geometry as the point method. This indi-

cates that the relative gradient method fails to add

0.0

0.2

0.4

0.6

0.8

1.0

1.2



Matake Papadopoulos S ines

Specimen

R e q u i r e d l o a d s c a l e f a c t o r

smooth with hole

smooth notched

notched with hole

notched

notched

notched with hole

notched with hole

Fig. 11. Required load scale factor for 50% fatigue probability for the combined gradient and volume method.

Table 6Parameters that result from minimization of the Error Indices of thecombined volume and gradient method

Criterion r0 ðMPaÞ ra ðMPaÞ m cabs (lm)

Crossland 269 1.59 1.85 À161Dang Van 155 1.75 2.39 À73Findley 194 1.18 1.85 À161

Matake 155 0.94 1.85 À161Papadopoulos 158 1.32 2.18 À88Sines 193 0.88 1.67 À220

Here, V 0 ¼ 4 cm3.




anything important to the point methods. The slight

improvement that does occur can probably be attributedto the addition of a free parameter.

The results in Fig. 9, the statistical volume effect, arevery good. This means that the modelling error is substan-tially reduced, and it is an indication that an essential phys-ical feature of fatigue may be described by the weakest linkreasoning. In Fig. 13, the probability of fatigue failure isshown as a function of the load scale factor, the Crosslandfatigue stress is used. Three geometries are shown, A0, D0and D10, where A0 is the smooth rotating bending speci-men. Specimens D0 and D10 have stress concentrations,with the stress concentration and stress gradient being larg-est in D10. The curves indicate that a strong stress concen-tration will lead to more scatter in the measured fatiguelimit load and quite low load values for which fatigue ispossible, although not very likely. The smooth specimenshows a more ‘‘deterministic’’ behaviour.

The combined gradient and volume method shown inFig. 11 is also good, and there is a clear improvement com-pared to the volume method. However, the gain in accu-racy, comparing Figs. 9 and 11, does not necessarily justify the extra computational effort. If the range of theload scale factor is studied, rather than the Error Index,the improvement is not that clear. The combined methodhas many more specimens that are very close to unity but

the worst ones are almost as bad as the worst ones of the

volume method, they are simply not that many. The gradi-

ent parameter became negative for the combined method.This means that a higher gradient would be more severe

Fig. 12. Contour plots of the fatigue stress of the Crossland fatigue criterion; (a) point method, (b) adjusted with the relative gradient (c) adjusted with theabsolute gradient. Different contour levels are used in these three plots.

Fig. 13. Probability of fatigue failure for three specimens. The type of specimen is indicated with labels and the geometries are shown in Fig. 2.1.Weakest link statistics with the Crossland fatigue stress and a threeparameter Weibull distribution has been used. The parameters are selected

so that the Error Index is minimized.




than a small one; this is contradictory to how the gradienteffect is generally understood. The gradient part of themethod may be considered as a modification of the volumemethod. This means that the differences between the actualdata and the predictions of the volume method should bedecreased by use of the gradient influence. The sign of the gradient parameter should then be related to how thevolume method needs to be corrected. It is not obvious thatthis correction should be in the same ‘‘direction’’ as thegradient modification of the point criteria. Thus, the signof the gradient parameter is not by default the same as inthe gradient criteria.

In Fig. 14, an overview of all the results is shown in onediagram. The height of the bars is the Error Index. Notethat the directions of the base plane have different mean-ings; criterion and method, respectively. What is clear,and most interesting, is that the choice of criterion is notas important as the choice of method. One trend can pos-sibly be seen about the criteria; for some methods the Sinescriterion seems a bit worse than the other, this is mostapparent for the volume method. The importance of taking

the statistical volume effect into account is made clear bythis diagram.

The stress states do not vary much between the criticalparts of the different specimens, where it is close to uniax-ial. The uniaxial stress state is common in critical parts inengineering applications.

7. Conclusions

The results may be summarized in the following points.

Elastic analyses of the fatigue limit loads were per-

formed for 15 different geometries.

Different fatigue criteria evaluated with point, volume,and gradient methods were compared. The deviationfrom accurate predictions was measured with an ErrorIndex that takes all 15 geometries into account.

Point and gradient methods have low predictivecapabilities.

The volume method, as well as the combined volumeand gradient method, has good predictive capabilities.

For design purposes the pure volume method is recom-mended for its combination of being straight-forward touse, giving good results and seeming physically sound.

In this investigation, the choice of fatigue criteria ismuch less important than the choice of method. Thedegree of multiaxiality was very small in the specimensused. The choice of criteria may be of more significancein other cases.

Acknowledgements

This research was made possible through financial sup-port from Scania Truck and The Swedish Research Coun-

cil. This support is gratefully acknowledged. SN alsothanks Scania Truck for the possibility to present this re-search.

Appendix. Computation of volume and gradient methods

Volume

The probability of failure P f for a volume V 0 subjectedto the fatigue stress bX given by the three parameterWeibull distribution is

P f ¼ 1 À eÀ

bXÀr0raÀ Ám

; ðA1Þ

CrosslandDang Van

FindleyMatake

PapadopoulosSines

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

Pointcriteria

Relativegradientcriteria

Absoutegradientcriteria

Volumecriteria

Volumeand gradientcriteria

C r it e r io n

E r r o r I n d e x

M e t h o

d

Fig. 14. Schematic summary of the whole investigation. The Error Index is a measure of the total error in the predictive capabilities for criterion/methodcombinations. Here it is given as a function of both fatigue criterion and method.




where r0, ra and m are the parameters of the Weibull dis-tribution. The value of V 0 is arbitrary but the values of ra,and m will depend on it. In Eq. (A1) P f is set to zero forb X < r0. The parameter r0 can be seen as a threshold stressbelow which fatigue does not occur.

The weakest link principle along with the Weibull distri-

bution, Eq. (A1), gives the result shown in Eq. (13). This isnumerically computed, based on the integration points of the FE-mesh. To each integration point, n, a volume, V n,is associated. To simplify the computations it is assumedthat the fatigue stress is constant in this volume. The dis-cretized version of Eq. (13) is

P f ¼ 1 À eÀP N

n¼1

V nV 0

bn X

Àr0ra

À Ám

: ðA2Þ

Here, N is the number of integration points. The integra-tion point volumes are extracted from the Abaqus Odb-file.

Gradient

To compute the fatigue stresses of the criteria once thegradients are known are straightforward. The difficulty isto compute the gradient.

The stress data as well as all computations this far, i.e.the computation of the point criteria, are located at theFE-integration points. This means that the values of thescalar whose gradient should be computed are known atthe integration points, and only at these points. In orderto compute the gradient, or any derivative, of a functionwhose values are known only at a limited number of points,some approximation of the function must be chosen. Inthis case, the known values were fitted to an analyticalfunction with the least squares method. It is necessary toknow the coordinates of the FE-integration points in orderto fit the function and these coordinates were extractedfrom the Abaqus Odb-file.

In this work only 20-node hexagonal elements with fullintegration are used. This means that each element have 27integration points. The fitting to an analytical function isperformed separately for each element. Two different func-tions are chosen for the fitting. The first fitting functionthat is used is the trilinear function with eight parameters,

f 8ð x; y ; z Þ ¼ A þ Bx þ Cy þ Dz þ Exy þ Fyz þ Gzx þ Hxyz :

ðA3ÞHere, x, y, and z are the Cartesian coordinates of points inthe element. Note that Eq. (A3) will lead to eight unknownsbut 27 equations, one for each integration point. The un-knowns are determined with the least squares method.The second function is a 20 parameter function that coin-cides with the sum of the form functions of the element,

f 20ð x; y ; z Þ ¼ A þ Bx þ Cy þ Dz þ Exy þ Fyz þ Gzx

þ Hxyz þ Jx2 þ Ky 2 þ Lz 2 þ Mx2 y

þ Nxy 2 þ Oy 2 z þ Pyz 2 þ Qz 2 x þ Rzx2

þSx2 yz

þTxy 2 z

þUxyz 2

ðA4

Þ

Also the parameters, A to U , in Eq. (A4), are fitted withthe least squares method. In order to increase the accu-racy of the computed parameters, a Cartesian coordinatesystem with its origin at the centre of the element wasused during the computation of the parameters of eachelement.

When the fitting was performed, the derivatives of x, y,and z were computed and the gradient vector were formedand its length computed as

jrðÞj ¼o

o xðÞ

2

þo

o y ðÞ

2

þo

o z ðÞ

2 !1=2

: ðA5Þ

References

[1] Norberg S, Olsson M. A fast, versatile fatigue post-processor andcriteria evaluation. Int J Fatigue, 0142-1123 2005;27(10–12):1335–41.

[2] Weibull W. A statistical theory of the strength of materials,Handlingar Ingeniorsvetenskapsakademien; 1939. p. 151. ISBN 991-428078-1.

[3] Weibull W. A statistical representaion of fatigue failures in solids,Kungliga Tekniska Hogskolans Handlingar; 1949. p. 27. ISBN 991-269732-4.

[4] Bomas H, Linkewitz T, Mayr P. Application of a weakest-linkconcept to the fatigue limit of the bearing steel SAE 52100 in abainitic condition. Fatigue Fract Eng Mater Struct, 8756-758X1999;22(9):733–41.

[5] Munday EG, Mitchell LD. The maximum-distortion-energy ellipse asa biaxial fatigue criterion in view of gradient effects. Exp Mech, 0014-4851 1989;29(1):12–5.

[6] Papadopoulos IV, Panoskaltsis VP. Invariant formulation of agradient dependent multiaxial high-cycle fatigue criterion. Eng FractMech, 0013-7944 1996;55(4):513–28.

[7] Makkonen M. Notch size effects in the fatigue limit of steel. Int JFatigue, 0142-1123 2003;25(1):17–26.

[8] Crossland B. Effect of large hydrostatic pressures on the torsionalfatigue strength of an alloy steel. In: Proceeding of the internationalconference on fatigue of metals, London & New York; 1956. p. 138– 49. ISBN 99-0981150-2.

[9] Dang Van K. Macro-micro approach in high-cycle multiaxial fatigue.In: McDowell DL, Ellis R, editors. Advances in multiaxial fatigue.ASTM STP 1191. Philadelphia: American Society for Testing andMaterials; 1993. p. 120–30. ISSN 0066-0558.

[10] Findley WN, Coleman JJ, Hanley BC. Theory for combined bendingand torsion fatigue with data for SAE 4340 steel. In: Proceeding of the international conference on fatigue of metals. London & NewYork; 1956. p. 138–149. ISBN 99-0981150-2.

[11] Matake T. An explanation on fatigue limit under combinedstress. Bull Jpn Soc Mech Eng 1977;20(141):257–63. ISSN 0021-3764.

[12] Papadopoulos IV. Long life fatigue under multiaxial loading. Int JFatigue 2001;23(10):839–49.

[13] Sines G. Behaviour of metals under complex static and alternatingstresses. In: Sines G, Waisman JL, editors. Metal fatigue. London:McGraw-Hill; 1959, ISBN 991-053748-6. p. 145–69.

[14] Papadopoulos IV. Critical plane approaches in high-cycle fatigue: onthe definition of the amplitude and mean value of the shear stressacting on the critical plane. Fatigue Fract Eng Mater Struct, 8756-758X 1998;21(3):269–85.

[15] Nishida S, Hattori N, Takaoka Y. Evaluation of strength of double notched specimens. Key Eng Mater, 1013-9826 2003;251– 252:19–24.


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