Faligue Flacl. Епgпg Mater. Struct.Yol. 16, No. l0, рр. 1041-1059, 1993Printed in Great ВriЫп. All riфts reserved

875б-75Ех/93 $6.00 + 0.00Соругiфt @ !993 Fatigue & Fracture of

Engineering Маtегiаlý & Structureý Ltd

А RANDOMIZED ENDURANCE LIMIT IN FATIGUEDAMAGE ACCUMULATION MODELS

V. А. КошочStrelochnikov ЗЗ|21З, Ekateгinburg, Russia

AЫract-The use of гапdоm раrаmеtеrý fог describing S-N curves геqчiгеs ýpecial caution. А РесчliаrFatigue Curves @FC) method based on а lineaT damage accumulation rule is proposed for modelling thefatigue damage accumulation process. The fatigue properties of а соmропепt аге ргеsепtеd in the РFСfоrm Ьу randomizing its епdчrапсе limit which facilitates а ýimple technique to predict fatigue life andresidual life distributions. Possible usages of the models and their restrictions аrе illustrated fоr someсопrmоп lbad histories of а component. Раrtiсчlаг attention is paid to the stochastic геlаtiопs betweenгапdоm variables having in some саýеý imрrореr distributions. Тhе рrоЬlеm of ехрегimепtаl testing ofthe models is also discussed.

NОМЕNСLЛТURЕ

SуmЬоЬ

D: equivalent пчmьг of cyclesд' : mathematical expectation

Jft):probability density function of епdчrапсе limit;@): probability density function of fatigue lifeg(y) : computed density function of fatigue life

й : load str€ýý historyrz:slope ofan S-N счrче

,пR, ýR: diýtгibution раrаmеtегs of епdчгапсе limit,??N, JN : distribution раrаmеtеrs of fatigue life

.lГ: fatigue lifeNо: пчmЬr of cycles п,hеп an S-N счrче has ап inflexion point

Jй*: residual fatigue lifeШ(ý): gдi*" life at а conýtant amplitude лS

пl: пlшlьr of cycles of а ceTtain amplitude ý frоm history }p(S) : probability density function of rапdоm amplitudes

R: stгess ratio,s: conýtant amplitude of cyclic loading

sri.:lower limit of integrationýп = епdчrапсе limit at а stresý rаtiо Rýi: lоwег bound of епdчrапсе limit гапgе

с, f, у : gamma-distribution рагаmеtеrsс, с: Weibull distгibution раrаmеtеrý

Г( )=gamma fчпсtiопд:cumulative damage

лч: unit damageдь: block damageо: standard deйation

Nоtаtiопs

РДg : peculiar fatigue curves(*), (") = Models and ехргimепtаl estimations

1и1

1042 V. А. Корпlоч

INTRODUCTION

Stochastic models of fatigue damage accumulation, used to predict fatigue life distributions оf аcomponent чпdеr various loading histoгies, have been investigated Ьу Мiпеr's rule Ьу mапуrеsеаrсhеs e.g. [1-3]. Опе simple concept to introduce probability characters into the descriptionof component fatigue properties is the randomization of some раrаmеtеrs of S-N счrчеý arrd theсаrryiпg out of fчrthеr calculations with random variables Ьу means of pгobability thеоriеs [4--5],This simple and very promising idea, which does not change the ехtегiоr fоrrп ofl а darnageaccumulation гчlе фагtiсчlаrlу а linear one) essentially widens the utilization rsпg€ of deteгnrinisticmodels of fatigue life prediction. Fчrthеrmоrе these models do not greatly iпсrеаsе the size of theinput data, at least at the stage of prediction. Ноwечег, not to violate Ёhе iпtеrпаl hаrrлопу ofdamage accumulation models and to achieve the results that do not contгadict experinrents andсопrmоп sense, spcial cautions аrе rеqчiгеd when гапdоmйiпg the S-N счrче раrаrпеtеrý.

Тhгее possible models аrе studied fоr randomizing the епdчгапсе limit of епdчrапсе (ý_N)curveý, which аrе described Ьу роwеr functions. Тhе obtained stochastic models of fatigue damageaccumulation based on Ресчliаr Fatigue Счrчеs аге of ап ехtrеmеlу simple fоrm and can Ье easilydeгived frоm deteгministic models of liпеаr damage accumulation.

МЛIN DETERMINISTIC RЕLЛТIОNS

The main deterministic relations coupling S-N curves, Мiпеr's damage accumulation rчlе, andloading histoгies with the values of fatigue life and the residual life of а component аrе listed brieflyin Table 1. Some explanations аrе set forth below.

Fatigue епdurапсе (ý-.lй) сиruеs

When calculating the fatigue life of а structure оr соmропепt S-N счrчеs mау Ье considered tohave а йdе distribution:

nr(s): {ЛЬts*/s)', ý:;l (l)

where ý denotes а constant amplitude of cyclic stress, лý* denotes an endurance limit, R is а stressratio, lYo is the пчmЬег of cycles whеrе the S-N счrvе exhibits an inflexion point, and rи is а роwеrfасtоr of the curves. In а double logarithmic plot Eq (1) takes the fогm:

los jч(лs) : ftog ЛЬ + m(log,S* - log лý), ý > ý*--О-'\-/ [о , ý(Sп (2)

Sometimes а semi-logarithmic plot is also used.Ву rергеsепtiпg S-N curves in such а mаппеr it is assessed that if the stгess amplitude is

less than the епdчrапсе limit SR, then fаilчrе does not оссчr, even after ап infinite пчmЬеr ofcycles.

Equations (1) and (2) describ quantile оr mеап values of fatigue lives obtained frоm standardfatigue tests. Непсе, these equations аrе assigned median values of fatigue lives.

Liпеаr dшlage ассwпulаtiоп rule

Uпdеr rеаl service conditions а constant amplitude load history seldom оссчrý. When computingthe fatigue life of а component subjected to complex load histories thеrеfоrе some rule of fatiguedamage accumulation should Ье used. In рагtiсчlаr, the linear damage accumulation rчlе (Miner's

Fatigue damage accumulation models

ТаЬlе l

rulе) exhibits а wide sресtrчm of behaviour and basic statements may Ье formulated in thefollowing way: i) the unit damage, р", is rеlаtеd to every load cycle having an amplitude S, suchthat

д,:l/N(S) (3)

whеrе N(ý) is а quantile оr mеап value of fatigue life determined Ьу the S-N счrче; ii) damagesummation is саrriеd out without any dependence to the оrdеr of application of the load cyclesthеrеЬу allowing us to grочр identical cycles in tеrms of amplitude and to present cumulativedamage in the fоrm of an algebraic sum оr integral:

p:I u,/N(ý)

whеrе п, is the пчmЬеr of cycles with an amplitude ý; iii) the failure of а specimen is а result ofа load history, й, when the cumulative damage длl reaches unity:

p{h\ : I (4)

The most typical load histoгies and formulae fоr computing the unit damage and cumulativedamage аrе listed in the second and third column of Table 1 which contains the basic formulaeused to estimate fatigue lives in а deterministic way.

FЕмs l6/l{K

lйз

History of Loading UnitDamage

Ассчmч l atedDamag,e Li fB ReB idца 1

Li fe

Regular

1

* N(S)

ЁJ =пi] =q

п

N(S)

1N*-р

Nгgв

N(S)-n

Di fferent

j-r_п.

,. _\ L

l,-- N(5 )Li=1

N r,еб

(1-л)N(ý ),l

stoehast i с

aо

г f (S)dSн =l-" J N(9)

о

р =пн1

ý=-F,

Nrеg

N-n

Repoated

.l

гr n... -\Ij-- /

-

-д-N(S).ll=1

=пьFь1

N. я-ьU ,ь

Nгбg

N. -пьь

V. А. Корг:лоч

Fatigue life

The fatigue life under constant loading is defined as the iпчеrsе variable to the unit damage pu,which has the sense ofa раrt ofthe used rеsочrсе. In case ofa паrrоw band stochastic load рrосеss,а unit damage is introduced Йth the integral fоrmulа (the second column, Table 1) Ьу virtue ofwhich the fatigue life can Ье calculated now in terms of rеduсеd load cycles with the rапdоmamplitude лS distributed Ьу а сеrtаiп law p(S).

When а block history is applied to а specimen the unit damage mау Ье defined as а damage,ро, accumulated fоr the block, and the fatigue life is calculated Ьу the пчmЬеr of the load blocks(the forth line, Table 1).

The residual fatigue life is defined as а ргоdчсt of the damage, (1-р), yet not used Ьу the totalpredicted life ,lй оr as the difference between the fatigue life .Л/ and the пчmЬеr of prestressed cycles,п, (the fоrth column, Table l).

Моrе detailed explanations of the deterministic method of computing fatigue lives based onMiner's and similar rчlеs can Ье found in mапу rеfеrепсеs to the fatigue ргоЬlеm.

СUМIJLЛТIУЕ DАМЛGЕ MODELS

Peculiar fatigue curaes

As pointed out above the most simple technique fоr fоrmiпg probabilistic relations, whichdescribe the stochastic fatigue properties of structural components, is the randomization ofthe епduгапсе limit Sp. In this рареr, thгее possible models of rапdоm епdчrапсе limit usageаrе considered, involving minimum information. The restrictions of the models аrе alsodiscussed.

Suppose that at least the median S-N счгче, equations (l) and (2), estimations of themathematical expectations, Д,SR, and standard deviation, d5", 8г€ known.

Considering ý* as а random чаriаЬlе in equations (1) and (2), distributed Ьу some probabilisticlaw, we can describe the stochastic fatigue рrореrtiеs of а set of identical specimens Ьу а familyof гапdоm S-N счrчеs whеrе ечеrу sample of ,S* соrrеsропds with its own S-N счrче. This statisticalassumption reflects а hypothetical idea that every specimen posseýses its own ресчliаг S-Nсчrче, called Peculiar Fatigue Curves (РFС), to indicate their principal diffеrепсе from the S-Nсчrчеs. So each РFС describes some S-N счrче and consequently the fatigue рrореrtiеs of асеrtаiп specimen that, hоwечеr, сап not Ье obtained Ьу tests since ечеrу specimen can only failonce, At the same time the hypothesis gives rich possibilities for developing various modelsincluding those considered hеге. The idea of РFС secms to have Ьсеп рrороsеd in [б]. Thedistгibutions of rапdоm variables to describe s-N счrчеs must геflесt the statistical laws of аspecimens fatigue behaviour. Fог that fеаsоп, the main requirement of stochastic cumulativedamage models аге fог all the rапdоm variables descгibing S-N счrчеs to Ье in statisticalсоrrеlаtiоп.

The lоg-поrmаl and Weibull distгibutions аrе most рорчlаr in treating experimental fatigue livesчпdеr constant-amplitude loading. These distributions аге also чеry convenient fоr the modelsоffеrеd hеrе fоr describing the scatter of fatigue lives, since, if only fог equations (1) and (2), thelog-normal and Weibull distributions fоrm families of one type Йth respect to the роwеrdependence between the rапdоm variables: у:ах-. Thus ifthe distгibutions аrе taken advantageof, fоr the раrаmеtеr ý*, they will lead to the same type of distributions for the fatigue lives N.Using оthеr distributions of S* fоr the fatigue счrче given Ьу equations (1) and (2), without duepreCautions, may cause еrrопеочs гesults.

Fatigue damage accumulation models lM5

Log -поrmаl distributioп

Let the distribution of S* Ье

whеrе the раrаmеtеrs ,??R

folloйng mаппеr:

л(х):#+*о(_*л,_ ), I)0 (5)

and s* аrе ехрrеssеd Ьу ЕлSа and the standard deviation о"" in the

s" : {loge log[1 * (o"*/ES*)2]}'/2

,r?R = log ES* - log[1 * (о"*/Д,S*)']/2

whеrе

log Д,S* : Zlк * s'*/2 log е and о"*/дSр : (lQGпЛов")z _ l)ttZ

Now сопsidеr thrее possible cases of гandomization of fatigue счrчеs Ьу the раrаmеtеr Sp withthe chosen distribution (5), although analogous results сап also Ье dегiчеd rvith the Weibulldistribution,

Model lAssume that РFС аrе defined Ьу the equation:

nr(S):шo(S*/S)-, s>0 (6)

whеrе,S* is а rапdоm variable with the density functionfr(x) of equation (5).

If the rапgе of the distribution is taken frоm 0 to оо then РFС йll fоrm а set of parallel linesin а double logarithmic plot as shown in Fig. 1а. Then the density functionfr(y) of fatigue livesN(S) fоr all possible values of the load amplitude S is of ап ordinary log normal type (5) withthe раrаmеtеrs:

ýц : lиýа and rпп : log Ng * m(m* - log S) (7)

and the mеап value Д log /r(,S) :lа?ц, &s а function of ý coincides with the initial счrче (2) andextends below the inflexion point \ with the same slope.

In this case the раrаmеtеr ,S* has lost the sense of ап епdчrапсе limit and the inflexion of РFСdoes not оссчr.

Model 2In the folloйng alteгnative, an attempt is made to eliminate the previous contradiction of а lost

епdчrапсе limit of the РFС. Suppose, hоwечеr, the раrаmtеr S* has а certain lоwеr Ьочпd Si.:SK > ,Si. Тhеп the distribution of ,S* mау Ье defined as log-normal with thrее раrаmеtеrs includingthe lоwег Ьоuпd лSi; i.e.

f,(x)=ir#=m*o(-ц"!-+-),"'' (s)

whеrе и* and sa have оthеr meanings than in equation (5). In this case, the essence of the modelis that every РFС has the unique endurance limit Si.

Let РFс have the fоrm:

N(S):лЬ(S*/S)', S>^ýi (9)

о(L

ФDэЁcLЕ

al,v,Фl-

aл

о 4ооaL

Ф зоотэ#Е zoog

а)aьоlE

а

о 4(Юсо зоооэЕР zoots

ФоФLа

1о' 1обNчmЬеr of Cycles

1осto Fоilчrе

1оо

Number Fоilчrе105 10Gof Cycles to

f.(y)l

f.,(x)

' l\aUrl|L)el (Jl rrYCleý [(, г аl]Urе

Fig. 1. Ресчliаг Fatigue Curves, (а) Model l: lоg-поrmаl distгibutions оГ ап епduгапсе limit and fatigue

fifЙ see equation (о.lь) Model 2: lоg-погmаl distribution of an endurance limit йth а lоwег bound anddistribution ofthe Гatigue tife; see equations (5) and (l0) гespectively. (с) Model 3: lоg-погmаl distribution

of ап епdчrапсе limit апd imргореr distribution of the fatigue life.

10' 1d 1о0 1о7NчmЬеr of Cycles to Гоilчrе

l046

Fatigue damage accumulation models lM7

with the random раrаmеtеr S* having the distribution given Ьу equation (8). Since the log-normal

distribution with the threshold does not generate а set of distributions of the same type with rеsресt

to the роwеr function transform, the corresponding life distribution of N takes а sufficiently

соmрlех form:

|og eytl^- t у-(;--тZS-N

/ > ЛЬ(Si./лS)'"_о( ),

Si/S)]- иш][и log[y - Nfr(y): (2т )l/25* 1у t/, _ nra/, (S;/s)]

(10)

where и* and ýN аrе determined Ьу equation (7).

The set of РFС constructed in such а mаппеr аrе bounded frоm the left with the сurvе No(Si/S)'(Fig. 1Ь). The РFС in the form of equations (s){lO) rеmоче the еrrоr of the previous variant, but

ihe-serious ргоЬlеm of selecting the раrаmеtеrs лSf., Рlцо аtrd s" arises. Additionally, fitting the

mathematical expectation Е log N(S), defined Ьу Model 2, with the initial experimental mеап сurvе

described Ьу equation (2), is somewhat vague. Fчrthеrmоrе the same situation has оссurrеd again

and the раrаmеtеr ,s* has lost the sense of ап епdчrапсе limit and is considered as unique fоr allсurчеS.

Model3The previous models rеflесt in some degree the possible representation of the РFс fоrm obtained

ьу randomizing the раrаmеtеr s*, but the elimination of the endurance limit in Model l and the

existence of а unique епdчrапсе limit sfr fоr all РFс in Model 2 cannot Ье considered as а

sufficiently acceptable method of constructing РFс since they do not rеflесt the rеаl паturе of the

епdчrапсе limit with its range of scatter. Тhеrеfоrе every РFс of the fоrm given Ьу Eq. (1) should

have its оwп endurance limit, i.e. it is assumed that equation (5) rеаllу defines the scatter of the

endurance limit and that ечеrу РFС reaching the value Nо has а break point, as shown in Fig. lc.So РFС аrе assumed to Ье set exactly Ьу equations (1) or (2) with the lоg-поrmаl distribution

1(-т) of the раrаmеtеГ ,S*, which now may Ье called an endurance limit defining every single

рFс.Given а set of РFС, an imрrореr distribution,fr(;r), of fatigue lives N оссurs due to the рrеsепсе

of the РFс break point, since аftеr achieving the value Nо, an infinite пчmьеr of cycles to failure

оссчrs fоr all congtant amplitudes S. This mау Ье acceptable since, when loading at 1аrgе

amplitudes, the base пчmЬеr of cycles isn't achieved in practice, and in consequence, the

distributions truncation йll not Ье чеrу noticeable. When loading with small amplitudes,

unfinished tests often take place, giving а rеаsоп to suppose the real existence of endurance limits

for some specimens.Thus, in virtue of equations (l)-(2) between the endurance limit Sд and the life N fоr ечеrу value

of s the imрrореr distributionfr(y) сап Ье expressed Ьу means of Eqn (5) in the following mаппеr:

foj):f,(y), уе(О, Nо) (1 l)

whеrе the раrаmеtеrs ,?4N and s" аrе defined similarly to equation (7).

The mathematical expectation of life given Ьу the imрrореr distribution, equation (l1), takes an

infinite value. Ноwечег, given а sеt r4(лS) : {п(S) < tro}, the transformed mathematical expectation

Е[пr*(лs):Д[ff(S)I{,4(s)}], (valid only fоr this set) takes а finite value. The deviation of the

mathematical expectation, determined Ьу Model 3, frоm the experimental s-N сurvе and the

probability of truncating fatigue lives, p{l (s)}, as а function of s fоr given values of Ло, m апd

ь tog s* аrе shown in Fig. 3. It is seen that the difference between the Model estimation and the

ехерrimепtаl curve, Eq. (2), can Ье acceptable.

V. А. Kopttov

Fatigue cumulatiue dаmаgе

The damage accumulated чпdеr some load history fоr all thrее Models considered above isdetermined Ьу the deterministic fоrmчlае, see Table 1, since the damage summation Ьу Miner'srчlе originates frоm some РFС. It-is апоthеr matter that we don't know what РFС is involved indamage accumulation. But this is not obligatory because the existence of the hypothesis is enoughfor tгeating РFС рrореrlу as rапdоm objects defined Ьу the раrаmеtеr ,S*. As fаr as finding thevalued of cumulative damage, оr mоrе precisely, its distributions, these should Ье included in alllife prediction problems. Сопсrеtе examples of treating РFС in fatigue damage acccumulation willЬе rероrtеd belorv.

Log-Nurnber of Cycles to Fоilчrе

Strеss Amplitude МРо

Fig. 2. Соmраrisоп of the average endurance curves fоr Model 3. (а) ехрегimепtаl епdчrапсе lines andthe dесгеаsiпg счrче due to ап impropr life distribution. ф) probability of truncating fatigue lives.

о(L

о2DэЁо.Е

а,оФýtа

эодо|_&

Fatigue damage accumulation models lM9

LIFE PREDICTION

The рrоЬlеm of finding the distribution of lives has аlrеаdу been described above to some extent

when deteгmining the distributions f,(y), i: l, 3,4. The life distгibution of а specimen subjected

to а block оr step-wise load can Ье found in the same way following with the rules of distribution

раrаmеtеr computing of the functionally connected rапdоm variables-Fоr instance consider the чеrу interesting and instructive case of паrrоw band loading when the

integral fоrm of estimating the unit damage аге to Ье taken advantage of (see Table l).

Life predictioп of а соmропепt subjected to stochastic loading

In case of а stationary ergodic loading рrосеss applied to а specimen with а density function,

/(,S), of cyclic amplitude ý the unit damage, р, is determined Ьу linear Miner's rulе aS

(l2)

where N(S) rерrеSепts some РFС fоr ечегу single value of S" determined Ьу fatigue рrореrtiеs ofа specimen, and ý,. is а lоwеr integration limit being different fог each proposed Model. In Modell integrating is done frоm ýi- :0; in Model 2 frоm the deterministic епdurапсе limit,,S,,. : S;;in Model 3 frоm the гапdоm endurance limit ý;," : ,SK. In truth, the necessity of а согrесtunderstanding, fог example putting the integration limits in а ргореr оrdеr in equation (l2), as well

as the соrrесt treatment of rапdоm variables have led the аuthог to the рrоЬlеm of modelling the

specimens' fatigue properties йth РFС and hence the арреагапсе of these three Models, and оthеrprevious attempts [6].

The life distribution of а specimen, in ассоrdапсе with Table 1, is determined as an inverse value

of the unit damage being averaged Ьу amplitudes S. Тhеrеfоrе, the fоrmчlа frоm rеfеrепсеs [4,5]

fоr characteristic life, calculated using equation (l), may Ье accepted assuming the deterministic

сhаrасtеr of the епdurапсе limit and the s-N счrчеs.дssumе now, аý in [5], that the amplitude distribution of а load process is described Ьу а highly

flexible gamma-distribution with thrее раrаmеtегs:

Г* /(s)asР :

Jo,_ лr(S)

p(s) : #Lrs}-

l exp(-fS"). (1 3)

ву selecting the раrаmеtеrs с, Р, 7 аррrорriаtеlу, various distributions can Ье obtained such as

eiponential, seminormal, weibull, Raleigh, and others. The characteristic life expressed in the

пumьеr of reduced load cycles to failure, ц йth the distribution equation (13) takes the fоrm

}У: Ш9Sfi Г(у/а).В'l" (14)Г((у + и)с-', PSfi-)

rrhеrе Г(*) and Г(*,*) denote complete and incomplete gamma-functions, Equation (14), in this

case, defines а stochastic relation between the random variables Sa and n under stochastic loading

of the observed type. As in equation (12), in ассогdапсе with three variants of роssiЫеrandomizaiton of ý*, the раrаmеtеr ý,. standing in the denominator of equation (14) also takes

the thrее forms described above.Unfortunately, Ьу гandomizing ýр, due to the rаthеr complex nature of equation (14), analytical

expressions of the life distribution hаче not Ьееп obtained. То оvеrсоmе this difficulty the

computational рrосеdчrе fоr determining distribution functions may Ье proposed for the thrее

Models of РFС,

1050 V. А. KopTqov

Ехаmрlе 1

As the first example of fatigue life computation Ьу this аррrоасh, consider the loading of pillarsof а travelling gantry сгапе, obtained via stгаiп measurements, with long stationary data beingргосеssеd Ьу the rainflow method and load amplitudes approximated Ьу the Weibull distributionwith shape and fоrm parameters, с : 5'7,6 апd а :2.2:

p(S) : (с/с) (S/c)"-'exp[-(,S/c)f (15)

The welded joints wеге studied using strain gages. The structure was made of steel with апаvеrаgе strength limit 380МРа and an епdчrапсе limit ES_, -78МРа taking the stressconcentration fасtоr into account. The standard deviation о"_, is taken as equal to 15 МРа.The standard S-N счrче has the рагаmеtеrs; No:2 х 106 and m:'7.5; asymmetric cycle fасtоr,ф:0.2. In finding the рагаmеtеrs of equation (15) asymmetric cylces wеrе transformed intosymmetric ones.

In this саýе, the characteristic value of life, equation (14), takes the fоrm:

пr- No(S*/c)'(16)Г(l + m|а, (ý,./с),)

The plots of equation (1б), with changing ý,.: S_, (Model 3) fоr different values of m, areshown in Fig. 3 Ьу solid lines and Ьу the dashed lines fоr constant value ýi-:0 (Model 1). It iSseen that fоr sufrciently lаrgе values of the епdчгапсе limit the evaluation will Ье stored since thedenominator in equation (16) will not increase the fatigue life values N. Plots fоr Model 2 аrе notshown becaues they look like the dashed lines in Fig. 3, but disposed а little higher than thosepresented.

а=2.2

1

Епdчrопсе Limit МРо

Fig. 3 . Plots of the characteristic function of fatigue lives Гоr diffeгent values of the епdчrапсе curve slope;equation (16).

1\ох

Е:,,6.Ёо.р1r,оъо,lFо!-Ф

_оЕэz,

-'-a-_С-'-

m-Еm-7rn=Om=5

Fatigue damage accumulation models l05l

Fоr Models 1 and 2, the life distribution of N is determined Ьу means of fоrmчlае (5) and (10)respectively, but the constants аrе to Ье placed in the соrrесt mаппеr.

Fоr ,Model 3, the computational method of finding the distribution of iy' is used with the knowndistribution of the епdчrапсе limit equation (5), and the given function equation (16) connectingit йth the fatigue lives. The characteristic functions of fatigue life аrе shown in Fig. 4(а) Ьу thesolid lines for Model 1 and 3. The роwеr approximation of equation (16) fоr Model 3, withchanging ý,., is shown in this Figure also. Fig. 4(Ь) illustrates the difference between the densityfunctionfr(y), log-normal, and g(y), having no analytical ехрrеssiоп, that have Ьееп evaluated

fixed limit

сhопgiпg limit

роwеr oppr

Епdurопсе Limit МРо

fixed limit

сhопgiпg |irnit

роyuог opproximotaon

1о 1d_r.lо-,i to 10, 10 ý 1о { 10 10 1оNumber of Cycles to Fоilчrе

Fig. 4. Plots of (а) characteristic and (Ь) probability density Гunctions оГ fatigue lives calculated Ьу theModels.

ьохФ

доrLоlDФ(,ооьФ-оЕэz

аIоххосФэсг0)i-

tJ-

1052 V. А. Kopxov

Ьу Model 1 and 3 (solid lines); the density function fоr the роwеr approximation Ьу Model 3 is

shown also Ьу the dashed line.It should ЬЪ pointed out for Model 3 that restriction of а possible rапgе of .л[ frоm the right

end Ьу Л/о is fullfilled necessarily, and has been taken into account, though there is по йsuаldiffегепсе between the imрrорег distribution and the ргорег опе, since the density function ofamplitudes p(s) is disposed sufficiently high Ьу the ,s-axis. If the density function of ý is moved

down the л9-ахь (i.e. the amplitudes s аrе less than in the present case), the density function ofN will Ь moved to the гight on the N-axis, the restгiction of the distribution g(y) becoming mоrе

арраrепt and its diffегепсе frоm/,(у) mоrе impressive.

RЕSIDUЛL LIFE PREDICTION

The finding of distribution functions fоr residual life is based also on the global assumption ofthe рrеsепсе огргс fоr every specimen and оп а posterioriinformation that Ьу а predicted moment

in tbe, чпdеr а фчеп load history, а studied specimen has not failed. Then, if the preceding load

history is known precisely, the set of curves having not been realized can Ье separated frоm all

рFс ipace[7] and as а consequence а posteriori distribution function сап Ье evaluated.

In case oi rur.o* band оr constant amplitude loading the distribution function of residual life

is obtained very easily Ьу analytical distribution functions, f,(Y), i:1, 3, оr Ьу computed ones,

g(y): fоr this, the constant и1, which means the пчmЬеr of геаl оr геduсеd cycles, realized Ьу

ihi predicted moment, аrе to Ье subtracted frоm the random чагiаьlе ildetermined Ьу one of these

distributions and then the lack of failure Ьу the time пчmьеr п, is to Ье taken into account.

Two stress lmel loadiпg

Consider in mоrе detail the case of two StrеSs level loading. At first а specimen is рrеstrеssеdfоr а given пчmьеr of cycles n, with an amplitude s' and after that it is tested at а stress ofamplitude s2 during the residual пчmьеr of cycles to failure, N*. . In а deterministic case of residual

life prediction, iй*, is defined Ьу the equation (see Table 1)

i{*. : Лr(1 _ дr): rйr(l _ пrlN), (l7)

where дr, is cumulative damage fоr cycles п1. Remembering now that Nl and N2 аrе rаПdОm

variables, defined mоrеочеr Ьу the Same rапdоm чаriаЬlе ,Sp, and also that equation (l7) is to Ье

fulfilled almost ечеrywhеrе, we can make а residual life prediction чпdеr а two StrеSs load history.

The equality of the following events is easily established Ьу the ргоьаьiltу (the set of the events

when the equation is not fulfilled is а set of probability that mеаsчrеs zero)

P{N(sl) > пr}: P{S* > S(n,)}, whеге ,S(n,):,Sr(иr/No)'/' (18)

as rrell as the rеlаtiоп fоr the value of residual life

N*:N(&)-Ь, where D:иr(ýr/Sz)' (l9)

using equations (l8) and (l9) the life distribution is obtained, expressed Ьу means of the

distгibution of the random чаriаЬlе N(Sz)

P{lr*, <у}: P{rr(sr) < D +у/N(&) > Ь}:

Р{Ь <N(й) < Ь + y\lP(b) and Р(Ь): P{.lT (Sr) > Ь}

(20)

Fatigue damage accumulation models 1053

Fоrmulа (20) and the рrоЬlеms connected with it have been сопsidегеd in detail in [7], thеrеfоrеonly Fig. 5 is given to illustrate the proposed rule fоr evaluating the residual life density functionf'(y).It should Ье pointed out only that fоr multiple amplitude stress fatigue tests Ь takes the fогm:

j_lд - I и,(ý/S;).u - ,:,

In case of non-stationary loading, а mоrе sophisticated рrосеdurе fоr evaluating the residual lifedistribution is rеquirеd. Fоr the рrеsепt, the rеqчiгеd сhаrасtегs have been obtained without usingап explicit fоrm of the геlаtiопs'fоr determining cumulative damage distributions. Моrе detaileddiscussion of this рrоЬlеm mау Ье found in [8].

TESTING ОF MODELS

In оrdеr to evaluate the validity and adequacy of the proposed Models, special statistical trialsand tests аге to Ье conducted uпdеr various load histories, having residual life samples of Eq. (20),particularly чпdег two stress level too.

The lack of in-house fatigue tests rеqчirеd the аuthоr to use rеfеrепсе sources whеrе uпfоrtч-nately, qualitative statistical tгials with sufficiently representative sampling of residual lives wегеnot found. Ноwечеr, it was possible to estimate the Model's рrореrtiеs frоm а few statistical tests,using fatigue data frоm two рареrs.

Ехаmрlе 2In rеfеrепсе [9] а пчmЬеr of tests fог determining the residual lives of plate specimens of А-7

steel subjected to а two-stress history of axial tension, Йth а prestгess level ý, and test stress S,was considered, The input information rеqЙrеd fоr constructing the Modcls in this case was themean S-N curve hайпg the fоrm, in semi-logarithmic cooгdinates,

log Enr(S) : log iйо * и(S* _ .S) (21)

Ь NчmЬеr of Cycles to Fоilчrе

Fig. 5. Fогmiпg а posteriori life density Гчпсtiоп fгоm с рrюrj опе.

х;lJ1лсоох=_оо-орtL

1054 V. д. Коршоч

where /,, : 1.045 х lO-a is known and the раrаmеtеr ,S* and ЛIо аrе not defined, but these mау Ье

оrdеrеd at оur discretion. The small amount of test data conducted when determining the s-N сurvе

was the rеаsоп fоr estimating the standard deviation, ,iц, fоr the whole group of lives obtained fоrthe different constant amplitudes, which can Ье taken to Ье equal to 0.1-{).2. The lоwеr bound ofS* in [9] had not been found.

Assign now РFС Model 1 in the fоrm

N(S): лIо19-(sп-л (22)

whеrе ,ý* has the поrmаl distribution, N(S) has the log-normal one and the constant iйо is defined

accounting fоr the difference between the logarithm of the mеап value and the mean logarithm ofthe value Ъr л ь ассоrdапсе with equation (5). Thus if one sets in а rечеrsе way N0: 107 and

Sп:0.14, fоr instance, then the аYеrаgе value, й*:169З4, is found using equation (2l) as well

ai the estimation of the standard deviation jк: l3з9. so all the parameters of the distribution ofS" and N(,S) have been defined.

The testing of adequacy for describing the sсаttеr of fatigue lives with а residual life distribution,

equation (20), using the lоg-погmаl distribution has no Sense Since the largest пчmЬеr of repeated

tests under'fixed values of S, and & did not exceed fочr specimens, and thеrе is no possibility

thеrеfоrе to say something significant about the statistical inference. However, the basic relationto determine residual life is still given, and the linear properties of damage accumulation and the

inherence of fatigue properties can Ье taken advantage of in the following way.

Ечегу РFС of а given specimen gепеrаtеs the whole set of cumulative damage lines defined Ьу

the load amplitudes, and two of them аrе shown in Fig. 6. These lines аrе of an hypothetical

сhаrасtеr, of course, and in fact what lines аrе in accordance with the specimen we do not know.

дt the same time in virtue of the initial assumptions stated fоr these Models, it is known that а

сеrtаiп value of the раrаmеtеr S* is set fоr ечеrу specimen, Ьу which the values of N(,S1), iй(,S2)

and N.. аrе determined uniquely and in consequence the lines of cumulative damage.

Fоr the given Model of РFС and load history, D is determined as

6 _ 10й(sl -s2) (2з)

which rерrеsепts the пumЬеr of cycles at the stress S, being equivalent to the cumulative damage

when loading йth s1. Figure б illustrates the рrореrtу of а раrаllеl tгansition frоm опе cumulative

damage line to апоthеr. Furthеr, knoйng the sample lives of all tested specimens which combine

the фчеп пчmЬеr of prestress cycles with the value of residual life N*, the sample life values N(^Sr)

can Ье rеstоrеd. These valueý mау then Ье соmрагеd йth the initial sample lives at this stress

amplitude S, since

N(,Sr):D+N*,. (24)

is valid.The mеап value and Standard deviation: Л*, sfi, and ;V, .i", frоm Model 1 (*) and tests (^) аrе

given in Table 2 where the second column contains the пчmЬеr of specimens used in constructing

ih. Mod.l'. estimations, and the last column lists the mеап values of fatigue lives rv obtained frоm

the s-N счrче. It is seen that the estimation of mean values as well as standard deviations аrе

different, though these аrе not significantly in contradiction with the test data. The failure to

estimate the fatigue рrореrtiеs mау well Ье explained Ьу the small values of the sample sizes.

conduct now the additional estimation of the Model's quality using all 32 values of residual lives

obtained under different load histories. Substituting equation (24) into equation (22) апd reversing

equation (22) the statistical relation between S* and N(,Sr) can Ье found. Then using all 32 tests

Fatigue damage accumulation models l055

ь N...NчmЬеr of Cycles

Fig. 6. Cumulative dаmаgе r*.firJ.уТ:ý,rllfr'З:ъrfi;n;1ý line to апоthег Ьу the equivalence rule.

values of N* in equation (24) the same пumЬеr of sample values of S* results. Thus, hаЙпg foundthe value of lи*, соmраrе them with the statistical characteristics of the sample obtained Ьу means

of Model 1 as described above. As а геsчlt, the mеап value of the sample, mý:17026, and the

standard deviation, sfi : 1293. If апоthеr value of s" is used the same close averages and mоrеclose values of the standard deviation can Ье obtained. It is seen on а поrmаl probability рареr(Fig. 7а), the поrmаl distribution does not fit well to the sample of ,S*.

Consider опе mоrе ruп of tests uпdеr two stress loading with other type of specimens made

of Д-7 steel, which is also given in [9]. The S-N curve has the same fоrm, equation (21), andthe раrаmеtеr estimations аrе respectively: и:1.18 х 10-а, ДЬ:106, Jп:0.17, Йl:45l97,JK: 1439.

Similar Table 3 includes the Model 1 values and experimental valucs of the statisticalcharacteristics of fatigue lives. The пчmЬеr of specimens subjected to ап identical load in these tests

is less than in the previous case that is why only the estimations of mеап values of N(Sr) аrе

Table 2, Statistical сhаrасtеrý of геstоrеd life values from Model 1 апd frоm ехреrimепtаl data

pýrпчmЬеr

of эре-сimепg

Mode1 1 Tests ý-N сurче

NiЁ

'/{ 5*, н

3200034500з7000

L4в1_0

2L9ozL15650010з016

7э989б5820393э4

z6446o

68850

41з93

зо77

265с00146000в0000

Фо|оЕоо

1000 psi :6.895 МРа.

l056 V. А. Коршоч

Епdчrопсе Limit МРо

Епdчrопсе Limit МРоFig. 7. Nоrmаl probability plots of the епdчгапсе limits restored Ьу the Models. Data frоm two Mmples

taken frоm геfегепсе [9].

given. As above, the distribution of ,S" was estimated with the mеап and standard deviation:mi:36442, sfi:1403. The поrmаl probability plot is shown in Fig. 7Ь.

Ехаmрlе 3In rеfеrепсе [l0] (used fоr testing the adequacy of Models) а two-stress loading of specimens was

conducted and mоrе detailed and representative data аrе cited. The specimens of 75S-T aluminumalloy wеrе subjected to cyclic amplitudes Sr :40000 psi and й : 35000 psi. Соrгеsропdiпgачеrаgеý and sИпdаrd deviations of fatigue lives fог these stгess amplitudes чпdеr constant loadingarc Йr: 320000, frz: 16'17000, J, : 255000 and .i, = 846000 cycles.

99

t95Фо!*Ф(L5оФ.2.роэЕэ5о

1

о.1I

99

Ёsбооýво(L

Ф5о

3rоэЕ5эо

Fatigue damage accumulation models

ТаЫе 3. Mean value estimations Ьу the Models and teýts

pSiпчmЬеr

of sре-cimeng

МоdеI L S-N сцrче

н F

4Е0005эо0056000

L44

5

7э372e15650064оао

1оооооо45о00052000

1000 psi : 6.895 МРа.

The average S-N счrче is well approximated Ьу the line in double logarithmic coordinates withа slope factor m:l2,1.

We поw trу to test the Models Ьу treating equation (20) directly: the distгibution ofгesidual life чпdеr these loads is to Ье fitted with the log-normal distгibution truncated frоm theleft.

The most representative tests аrе fоr 65% and 85% of the average fatigue life и, of ,S, amplitude.In ассоrdапсе with Eq (l9) fоr such а load history, the value of D becomes 1.03 х 106 and1.35 х ,106 respectively. Histograms of гesidual lives, the curves of initial and tгчпсаtеd Modelsdensity functions аrе shown in Fig. 8. Chi-square goodness of fit testing fоr Model 1 as well asfоr Model 3 gives а significance level less than 0.01, though fоr Model 3 truncated both frоm theleft and гight chisquare values аrе а little lаrgеr. The usage of Коlmоgоrоч-Smirпоч tests hаs alsonot given mоrе reliable results.

DISCUSSION

The ргороsеd stochastic superstructure placed очеr the deteгministic Мiпег's rule possesses аmаjоr positive feature namely that damage accumulation, fatigue input data, and the method oflife prediction аrе interconnected elements fог а closed reliability model fоr estimating componentfatigue life. Simplicity of the Model's stochastic strчсtчrеs without exact input data аге alsoattractive, giЙng the possibility of their usage in mапу common practical cases of component lifeprediction. Nаtчrаl interdependence of the life and the residual life prediction рrоЬlеm is easilytraced йth no additional input being rеqчiгеd.

Some discrepancy in the Model's estimations may Ье explained as follows.

(i) The рrосеss of damage accumulation can not Ье completely deterministic Ьу explaining thelife time scatter from а dispersal of inherent mechanical рrорегtiеs of specimens.

(ii) Few data wеrе used.(iii) The initial distribution of lives аrе not quite log-normal. This was the соmmоп used

assumption made Ьу аuthог. As а rчlе, when checking samples Ьу statistical tests,affirmative results аге seldom obtained Ьу а given distribution.

Ноwечег а significant confirmation of the рrеsепсе of а specimen peculiar inherent рrореrtiеsis illustrated in Fig. 7.

. The ачthоr hopes that due to а lack of simple and quality stochastic models for cumulativedamage, this аррrоасh сап Ье Ьгочght to acceptable engineering level.

V, А. Koprqov

NчmЬеr of Cycles to Fоilчrе (xtO6)

NчmЬеr of Cycles to Fоilчrе. (xt06)

Fig. 8. Histogгams of residual life, initial lоg-поrmаl distribution, and predicted truncated Models'distгibution: (а) а рrеstгеss of б5%, ф) а рrеstгеss of 85%. Data fгоm [10].

CONCL[JDING RЕМЛRКS

In the рrеsепt рареr, which is related to stochastic models of cumulative damage, thrее possibleModels of randomizing the епdчrапсе limit of S-N счrчеs аге characterized. Constructing РесuliаrFatigue Счrчеs (РFС) Ьу the аррrоасh рrеsепtеd enables one to find а natural stochastic spreadfоr the deterministic life prediction models and to check the validity of the proposed Models Ьу

с)сФэсгФt_

L_

о

-9эЕэо

(,сФэсгФlE

L_

о

_9эЕэо

t.эЬхtо'2

Fatigue damage accumulation models 1059

means of statistical tests. The aim of this investigation is to рrеsепt the соrrесt manipulation ofrапdоm variables in relation to the fatigue life prediction рrоЬlеm.

REFERENCES

l. Z. W. ВirпЬачm and S. С. Saunders (1968) А probabilistic iпtегрrеtаfiоп of Miner's rulе. SIДМ J. Appl.Math. 16(3), 637452.

2. S. С. Sачпdегs (1970) А probabilistic interpretation of Мiпеr's rulе. II. SIДМ J. Дррl. Math. l9(l),251-265.

3. Т. Shimokawa and S. Tanaka (1980) А statistical consideration of Мiпег's Rule. Int. J. Fatigue 2(5),l65-170.

4. V. V. Bolotin (1984) Lifu predictioп of mасhiпеs апd structures (in Russian). Maschinostroenije, Moscow.5. А. S. Gusev (1989) Fatigue streпgth апd suruioability of structures uпdеr rапdоm loadiпg (in Russian).

Maschinostroenije, Moscow.6. V. L. Raiher (l982) Some сопsеqчепсеý fгоm the twо-раrаmеtеrs model of duгаЬilitу scatter (in Russian).

Uсhепуе zapipki СДНI |3(l), 130*l33.7. V. А. Kopnov (1993) Residual life, liпеаr fatigue damage accumulation and optimal stopping. Reliability

Епgiпееriпg апd System Safety 4aQ), З19-З25.8. V. А. Kopnov and S. А. Timashev (1990) Optimal stopping ргоЬlеm of invisible fatigue cumulative

procoss (in Russian). Problemy mаshiпоstrоепiуа i nadegnosti mаshiп, No. l, 65-70.9. F. Е. Kchart, Jг. and N. М. Nеwmаrk (1948) Ап hypothesis fоr the determination of cumulative damage

in fatigue. Proc, ASTM 48,767-198.l0. Т. J. Dolan апd Н. F. Вrоwп (1952) Effect of рriог repeated prestressing on the fatigue life of 75S-T

aluminum. Proc. дSТМ 52,73з*'140.