Cumulative Fatigue Damage and Life Prediction Theories a Survey of the State of the Art for...

Int. 1. Fatigue Vol. 20, No. I, pp. 9-34, 1998© 1998 Elsevier Science Ltd. All rights reserved

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Cumulative fatigue damage and lifeprediction theories: a survey of the state ofthe art for homogeneous materials

A. Fatemi* and L. Vangt

*Department of Mechanical, Industrial and Manufacturing Engineering, TheUniversity of Toledo, Toledo, OH 43606, USAtAdvanced Design, Spicer Driveshaft Division, DANA Corporation, Holland, OH43528, USA(Received 21 October 1996; revised 22 March 1997; accepted 15 June 1997)

Fatigue damage increases with applied load cycles in a cumulative manner. Cumulative fatigue damageanalysis plays a key role in life prediction of components and structures subjected to field load histories.Since the introduction of damage accumulation concept by Palmgren about 70 years ago and 'lineardamage rule' by Miner about 50 years ago, the treatment of cumulative fatigue damage has receivedincreasingly more attention. As a result, many damage models have been developed. Even though earlytheories on cumulative fatigue damage have been reviewed by several researchers, no comprehensivereport has appeared recently to review the considerable efforts made since the late 1970s. This articleprovides a comprehensive review of cumulative fatigue damage theories for metals and their alloys,emphasizing the approaches developed between the early 1970s to the early 1990s. These theories aregrouped into six categories: linear damage rules; nonlinear damage curve and two-stage linearizationapproaches; life curve modification methods; approaches based on crack growth concepts; continuumdamage mechanics models; and energy-based theories. © 1998 Elsevier Science Ltd.

(Keywords: cumulative fatigue damage; fatigue damage accumulation; cumulative damage rules; load interaction effects; fatigue life predictions)

INTRODUCTION

Fatigue damage increases with applied cycles in acumulative manner which may lead to fracture. Cumulative fatigue damage is an old, but not yet resolvedproblem, More than seventy years ago, Palmgren1 suggested the concept which is now known as the 'linearrule'. In 1945, Miner2 first expressed this concept ina mathematical form as: D ="i.(n/Nr), where D denotesthe damage, and ni and Nfi are the applied cycles andthe total cycles to failure under ith constant-amplitudeloading level, respectively. Since then, the treatmentof cumulative fatigue damage has received increasinglymore attention. As a result, many related researchpapers are published every year and many differentfatigue damage models have been developed.

Some of the progress on the subject of cumulativefatigue damage has been summarized in several reviewpapers. Newmark3 in a comprehensive early reviewdiscussed several issues relating to cumulative damagein fatigue such as damage cumulation process, damagevs cycle ratio curve, and influence of prestressing oncumulative cycle ratios. Socie and Morrow4 presented a

*Author for correspondence.

9

review of contemporary approaches for fatigue damageanalysis employing smooth specimen material data forpredicting service life of components and structuressubjected to variable loading. The early theories oncumulative fatigue damage have also been reviewedby Kaechele5, Manson6, Leve7, O'Neill8, Schive9,

Laflen and Cook lO and Golos and Ellyin" . However,as pointed out by Manson and Halford12 in 1986, nocomprehensive report has appeared recently to reviewthe considerable effort made since Schive's publication.In addition, no such review has been published sincethe late 1980s.

This review paper provides a comprehensive overview of cumulative fatigue damage theories for metalsand their alloys. Damage models developed before1970s were mainly phenomenological, while those after1970s have gradually become semi-analytical or analytical. Several researchers4

-9 have reviewed the

theories developed before 1970s. These damage rulesare first reviewed in this paper. Then a more detaileddiscussion on the selected approaches developed after1970s is presented. Even though some of the continuumdamage mechanics (COM) models are also mentioned,these approaches are not reviewed in this paper. Animportant application of these models has been in

10 A. Fatemi and L. Yang

damage assessment of inhomogeneous materials. Itshould also be noted that this review paper deals withdamage rules and life prediction aspects of cumulativefatigue damage. Another review paper13 provides acomprehensive overview of cumulative fatigue damagemechanisms and quantifying parameters.

WORK BEFORE 1970s

The phenomenologically-based damage theoriesdeveloped before 1970s were originated from threeearly concepts (discussed below) and attempted toimprove the linear damage rule (LDR). These theoriescan be categorized into five groups: the damage curveapproach (DCA); endurance limit-based approach; SN curve modification approach; two-stage damageapproach; and crack growth-based approach.

Three early conceptsThe history of fatigue damage modeling can be

dated back to 1920s and 1930s. It was Palmgren1 whofirst introduced the concept of linear summation offatigue damage in 1924. French14 first reported thesignificant investigation of the overstress effect onendurance limit in 1933. In 1938, Kommers 15 suggestedusing the change in the endurance limit as a damagemeasure. In 1937, Langer16 first proposed to separatethe fatigue damage process into two stages of crackinitiation and crack propagation. The linear rule wasproposed for each stage. These three early concepts(linear summation, change in endurance limit and twostage damage process) laid the foundation for phenomenological cumulative fatigue damage models.

Linear damage rulesMiner2 first represented the Palmgren linear damage

concept in mathematical form as the LDR presented by:

D = 'ir; = 'in/Nf; (l)

In the LDR, the measure of damage is simply thecycle ratio with basic assumptions of constant workabsorption per cycle, and characteristic amount of workabsorbed at failure. The energy accumulation, therefore,leads to a linear summation of cycle ratio or damage.Failure is deemed to occur when 'iri = 1, where ri isthe cycle ratio corresponding to the ith load level, orri = (nINf );. Damage vs cycle ratio plot (the damagecurve or D-r curve as it is usually called) for this ruleis simply a diagonal straight line, independent of loading levels. In a S-N diagram, the residual life curvescorresponding to different life fractions are essentiallyparallel to the original S-N curve at failure. The maindeficiencies with LDR are its load-level independence,load-sequence independence and lack of load-interaction accountability. In 1949, Machlin17 proposed ametallurgically based cumulative damage theory, whichis basically another form of LDR. In 1950s, Coffinand co-workers 18,19 expressed the LDR in terms ofplastic strain range, which is related to fatigue lifethrough the Coffin-Manson relation. In a later study,Topper and Biggs20 used the strain-based LDR tocorrelate their experimental results. A review on theapplications of the LDR to strain-controlled fatiguedamage analysis was given by Miller21 in 1970. However, due to the inherent deficiencies of the LDR, nomatter which version is used, life prediction based on

this rule is often unsatisfactory. Experimental evidenceunder completely reversed loading condition often iridicates that 'ir; > 1 for a low-to-high (L-H) loadingsequence, and 'ir; < 1 for a high-to-low (H-L) loading sequence.

Marco-Starkey theoryTo remedy the deficiencies associated with the LDR,

Richart and Newmark22 introduced the concept of damage curve (or D-r diagram) in 1948 and speculatedthat the D-r curves ought to be different at differentstress-levels. Upon this concept and the results of loadsequence experiments, Marco and Starkey23 proposedthe first nonlinear load-dependent damage theory in1954, represented by a power relationship, D = I.rji,where Xi is a variable quantity related to the ith loadinglevel. The D-r plots representing this relationship areshown in Figure 1. In this figure, a diagonal straightline represents the Miner rule, which is a special caseof the above equation with Xi = 1. As illustrated byFigure 1, life calculations based on Marco-Starkeytheory would result in 'ir; > 1 for L-H load sequence,and in 'iri < 1 for H-L load sequence.

Damage theories based on endurance limit reductionOn the other hand, the concept of change in endur

ance limit due to prestress exerted an important influence on subsequent cumulative fatigue damageresearch. Kommers24 and Bennett25 further investigatedthe effect of fatigue prestressing on endurance properties using a two-level step loading method. Theirexperimental results suggested that the reduction inendurance strength could be used as a damage measure,but they did not correlate this damage parameter tothe life fraction. This kind of correlation was firstdeduced by Henry26 in 1955 and later by Gatts27,28,and Bluhm29. All of these damage models based onendurance limit reduction are nonlinear and able to

FOR OPERATION AT CTt

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FOLLOWED BY OPERATION AT CT,

,~.LN' .• I

, • l,Z

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CYCLE

Figure 1 Schematic representation of damage vs cycle ratio for theMarco-Starkey theory23

Cumulative fatigue damage and life prediction theories 11

(b) H-L load sequence

Figure 2 Schematic representation of fatigue behavior by therotation method and by the Miner rule for (a) L-H, and (b) H-Lload sequences35

(a) L-H load sequence

Nt N,

Fatigue life, cycles

ACTUAL "2

ACTUAL nz

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N1 Nt

Fatigue life, cycles

I

I"2 BYMINER,

I

I

0"0::::l.... CT,.-.-c..8~

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Damage theories based on crack growth conceptAnother approach in cumulative fatigue damage

analysis is the crack growth concept. On the basis ofthe mechanism of progressive unbounding of atoms asa result of reversed slip induced by stress cycling,Shanlelo introduced a damage theory by definingcrack length as a damage measure in 1952. It wassuggested that the crack growth rate varies with the

0"0

::::l.....-.-c..8~

CJ:ICJ:I0........CJ:I

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In either stage, the LDR is then applied. Manson3?

reverted to Grover's work and proposed the doublelinear damage rule (DLDR) in 1966. This damagemodel and its applications were further examined anddiscussed in Ref. 38. In the original version of DLDR,the two stages were separated by equations of:N1 = Nf - PNf,6 and Nll = PNf·6, where P is a coefficient of the second stage fatigue life. A graphicalrepresentation of DLDR applied to a H-L two-levelstep load sequence is shown in Figure 3. Recently,Bilir39 carried out an experimental investigation withtwo-level cycling on notched 1100 Al specimens. Areasonable agreement between predictions by theDLDR and the experimental data was obtained.

Early theories accounting for load interaction effects

These theories include Corten-Dolon modepo andFreudenthal-Heller31 ,32 approach. Both theories arebased on the modification of the S-N diagram, whichis simply a clockwise rotation of the original S-N linearound a reference point on the line. In the CortenDolon model, a point corresponding to the highestlevel in the load history is selected as the referencepoint, while in the Freudenthal-Heller approach, thisreference is chosen at the stress level corresponding toa fatigue life of 103_104 cycles. Later, Spitzer andCorten33 attempted to further improve the CortenDolon approach. They suggested to obtain the slopeof the modified S-N line from the average result of afew repeated two-step block tests. With rotating bending specimens of SAE 4130 steel, Manson et al.34

,35,

also examined the approach based on the S-N linerotation and convergence concept. They suggested thata point corresponding to a fatigue life between 102

and 103 cycles on the original S-N line can be selectedas the convergence point. Their approach also providesa method for predicting the reduction in endurancelimit due to precycling damage, and is therefore ableto account not only for the load interaction effect, butalso for small cycle damage. Figure 2 shows a schematic representation for two-level L-H and H-L stressing. In these figures, the Miner rule is represented bythe solid lines which are parallel to the original S-Ncurves. It can be seen that the LDR and the S-N linerotation approaches differ in their abilities to accountfor the load interaction effects.

1. damage due to crack initiation, N1 = aN6 and2. damage due to crack propagation, Nll = (l - a)Nf ,

where a is a life fraction factor for the initiationstage.

Two-Stage linear damage theories

The two-stage linear damage approach improves onthe LDR shortcomings, while still retains its simplicityin form. Following Langer's conceptl6

, Grover36 considered cycle ratios for two separate stages in thefatigue damage process of constant amplitude stressing:

account for the load sequence effect. Some of thesemodels can also be used for predicting the instantaneous endurance limit of a material, if the loadinghistory is known. None of these models, however, takeinto account load interaction effects.


1.0

·1

..... "1-+--"2 - TO FAILUREa1a2

STR ESS 0 I-I-tH+lI+H+f+++f~""'-',- TIME

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A,,,,,,,,,,,," ,,-DOUBLE LINEAR DAMAGE RULE,,~, ,, ,

,," ,, , ".(

No I 6"2)",,~ N,: I' N',2 ,,-LINEAR DAMAGE

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N

Z"Nc:o~

~ No 2

a: ~w~

1u>-uC)zz~::!EW 6N 2a: Nr ,2

APPLIED CYCLE RATIO, "1/N',l

Figure 3 Illustration of the double linear damage rule for H-L two-level load cycling38

applied stress level in either a linear or an exponentialmanner. Valluri41 ,4Z presented a crack growth damagemodel in a differential form in 1961. The quantitativedevelopment of the theory is based on concepts derivedfrom dislocation theory and a synthesis of the macroscopic elasto-plastic fracture theory. The equation formulated is in a form similar to that expressed by linearelastic fracture mechanics (LEFM): da/dN = Cf(a)a,where a is the crack length, C is a constant and j(a)is a function which depends on the material and loadingconfiguration. Another damage theory using crackgrowth concept was formulated by Scharton and Crandall43 in 1966. Its mathematical expression is representedby: da/dN = am + 'j(ai), where m is a material constant.

DAMAGE CURVE APPROACH, REFINEDDOUBLE LINEAR DAMAGE RULE ANDDOUBLE DAMAGE CURVE APPROACH

The DCA, refined DLDR, and double damage curveapproach (DDCA) were developed by Manson, Halford, and their associates 1Z

,44 and have many commonfeatures.

Damage curve approachThis approach was developed to refine the original

DLDR through a reliable physical basis. It is recognized that the major manifestation of damage is crackgrowth which involves many complicated processessuch as dislocation agglomeration, subcell formation,multiple micro-crack formation and the independentgrowth of these cracks until they link and form adominant crack. Based on this phenomenological recognition, Manson and Halford44 empirically formulatedthe 'effective crack growth' model that accounts forthe effects of these processes, but without a specificidentification. This model is represented by:

(2)

where ao, a and ar are initial (r = 0), instantaneous,and final (r = 1) crack lengths, respectively; and q is

a function of N in the form q = BNf3 (B and f3 aretwo material constants). Damage is then defined as theratio of instantaneous to final crack length, D = alar.In most cases, ao = 0, and the damage function of theDCA simply becomes:

D = r4 (3)

Obviously, this form is similar to the Marco-Starkeytheoryz3. Through a series of two-level tests, the constant f3 can be determined from the slope of theregression line of the experimental data: that is,log[log(l - rz)llog rd vs 10g(N/Nz). A value of f3 =0.4 was determined in Ref. 44. Furthermore, if areference level, Nn is selected, the other constant, B,can then be expressed as N;f3. Therefore, the exponentq in Equation (3) can be written as q = (N/Nr )f3, whichis load level dependent.

Refined double linear damage ruleThe original DLDR can be refined by linearization

of damage curves defined by DCA model. In therefined DLDR, the knee points in a damage vs cycleratio (D-r) plot, which divide the damage process intotwo phases, are determined by:

Dknee =A(N/NY" and r knee = 1 - (l - A)(N/N)OI

(4)

where .A and a are two constants determined fromregression analysis of the experimental data. Theempirical values of these two constants were found tobe A = 0.35 and a = 0.25 for high strength steels 1Z

,44.

Shi et al.45, have recently used a similar approach to

define the knee points. They proposed a knee pointcoordinate formula based on the two-stage damage rule.

Double-Damage curve approachThis approach is developed by adding a linear term

to the DCA equation with some mathematical manipulation and can be presented as:

D = [(pr)k + (l - pk)~q]llk (5)


(7a)r1 - 'V

D = Ie

1 - 'Yec

HYBRID THEORY

Bui-Quoc and colleagues presented their work dealingwith cumulative fatigue damage under stress-controlledsl and strain-controlled conditionss2 in 1971. Thetheory for stress-controlled fatigue was first developedfrom the hybridization of four prior damage modelsby Henry26, Gatts27, Shanley4° and Valluri41 . It waslater adopted to strain-controlled cycling fatigue. Boththeories were then combined into a 'unified theory's3.Noting the interaction effect under cyclic loadinginvolving several stress levelss4, Bui-Quoc and coworkers.sl ,ss-s8 modified their damage models toaccount for this sequence effect. These damage modelshad already been extended to include high temperaturefatigueS9, creep6ll-63 and creep-fatigue damage64-7l conditions. They were further modified to take into accountnot only the effects of mean stress/strain72

, but alsothe effects of temperature and strain rates73 on fatiguedamage accumulation.

Stress-Controlled version

The main hypotheses in the development of thisdamage theory is that cracks growing in a materialsubjected to cyclic loading lead to a continuousreduction in fatigue strength and endurance limit. Forconvenience, all the parameters in this model wereexpressed by dimensionless ratios with respect to theoriginal endurance limit, aeo' These include the instantaneous endurance limit ratio, 'Ye = ajaeo' the appliedstress ratio, l' = a/aeo, and the critical endurance limitratio, 'Yec = aeJaeo' which corresponds to failure. Adifferential equation for strength evaluation rate wasobtained by combining three fundamental damagetheories:

1. Shanley's power rule of crack growth rate in termsof the maximum cyclic stress;

2. Valluri ,s relation between crack growth and cyclicstress range; and

3. Gatts' damage function described by the secondpower of the stresses in excess of the instantaneousvalue of the endurance limit.

An integration of this differential equation with somemathematical manipulations results in the damage function for the stress-controlled condition as:

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(6)

-- DOUBL.£ DAMAGE CURVE APPROACH--- DOUBL.£ W£AR DAMAGE RUL.£--- DAMAGE CURVE APPROACH

IS\NGL.£ TERM EOUATlON)

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1. 0,--------------------. B an improved representation of data by DDCA overLDR and DCA for 316 stainless steel tested underhigh-to-low loading. The DDCA has also been appliedto two other materials used in the turbo pump bladeof the main engines of the space shuttle49.

The three aforementioned models possess similarcharacteristics. They are all load-level dependent, butdo not account for the load interaction effect andsmall-amplitude cycle damage. With some modificationin procedure, the mean stress equation by Heidmannso

can be incorporated into these damage models. Thedetails of this incorporation can be found in Ref. 12.

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Dknee A(N/NY'p---

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As can be seen from Figure 4, the DDCA representsa continuous damage curve which conforms to theDLDR line in the early portion of the Phase I regime,but blends into the DCA curve which is also close tothe DLDR in Phase II. To evaluate the effectivenessof the developed DDCA, Manson and Halford and coworkers46-48 conducted cumulative damage experimentson both 316 stainless steel and Haynes Alloy 188. Acomparison of the experimental results with the DDCApredictions indicate good agreements. Figure 5 shows

where k is a mathematical exponent to give a close fitto the double linear damage line, and p is a constantmeasured from the slope of the first damage accumulation line in DLDR:

Figure 4 Comparison of the DDCA with DLDR and DCA l2

Figure 5 Improved representation of H-L load interaction tests of316 stainless steel using DDCA as compared with LDR and DCA l2

where 'Yu = aJaeo, and m is a material constant. Thecharacteristic of this equation is shown in Figure 6 as


1.0,.....--------------_

o.a

0."'

0.2

oo 0.2

y • 1./

0."' 0 .•

CYCLE RATIO

Yy • 2.0

0.8 to

The hybrid theory

Henry theory

Miner theory

Figure 6 Characteristic of the hybrid damage function 209 and comparison with the Miner rule and the Henry theory

(7b)

the load parameter for the first level, AI' For the secondload level, however, the load parameter, Az, is replacedby an imaginary strain, A2 ', which is, therefore, called'fictitious load'. To determine the fictitious value, A2 ',

a parameter Y used in regression analysis is proposed:

where LlA = A2 - AI' The value of the materialconstant a is in the range of 0-1. It can be experimentally determined from two-step fatigue tests, or empirically estimated by taking a = 0.5 as a reasonableapproximation58. This approach can be extended tomulti-step loading by defining the interaction parameter Vk' (k = 2, 3'00 .,i) between any two successivestrain levels k - I and k in the same form as Equation(10), but with LlA =Ak - Ak _ I' Under the assumptionthat a multi-step fatigue process accumulates interaction

(8)

(10)

(ILlAI)B2

Y = 1 + BI -~ ,-If3LlA*

where B I, B2 and B3 are constants to be determinedexperimentally; LlA is the difference between strainlevels: LlA = A2 - AI; Y and LlA* are sequencerelated parameters defined as follows for the L-Hincreasing step:

Ar* - A2Y = and LlA* = A* - A (9a)Ar* - A2 ' r I

where Ar* = Af7; and for the H-L decreasing step:

A2 - IY = and LlA* = A - (9b)A2 ' - I I

In the cycle ratio modification approach, the damagefunction in Equation (7)a, (7)b is modified by introducing an exponent, v, to the cycle ratio, rV

• Therefore, vis called a load-interaction parameter. For two-stepcycling, v is related to another parameter, a, by theempirical equation:

Strain-Controlled versionThe conversion of the stress-controlled theory to

strain-controlled version was made simply by replacingthe stress parameters Yx in Equation (7)a, with thecorresponding strain parameters, Ax, which are definedas: Ax = 1 + In (E)Eeo). The symbol 'x' stands fordifferent subscripts. Therefore, the strain-controlled version of hybrid theory can be mathematicallypresented as:

D _ In(EjEeo) _ r-In(EeJEeo) - A - (A/Ar)m

r + (1 - r) A-I

compared with the Henry theory and the LDR. Forlarge y, it is clear that the difference between the twomodels becomes appreciable.

where A = 1 + In(E/Eeo) and Ar = 1 + In(ErfEeo ), inwhich Ee, Eeo and Eec are instantaneous, initial andcritical strain endurance limit, E is the appliedmaximum cyclic strain, and Er is fracture ductility orthe true strain at fracture. The D-r plot of Equation(7)b is similar to Figure 6 described by Equation (7)a.

Both Equations (7)a and (7)b give a nonlinear, loadlevel dependent damage assessment. They also accountfor the effect of reduction in strain endurance limitresulting from prior strain cycling. These modelsimprove life predictions compared to the LDR, butdeviations from experimental results are still found74,mainly due to the load interaction effects which arenot accounted for by this model.

Modified version to account for load interactioneffects

To account for load interaction effects, Bui-Quocdeveloped two approaches to improve the model. Oneis the fictitious load approach51.55,57 and another is thecycle ratio modification approach55,56. The fictitiousload approach was developed only for two-step loadcycling. In this approach, there is no modification of


(13)

effect as well as damage, the interaction parameterappropriate for the ith load level becomes:

Vi = 1 X V2' X V3' X .,. v/ x ... X Vi _ I' X V;' (k = 2, 3, ... ,i)

(11)

Iterative calculations from i = 2 to i = i (i 2:: 2)following a similar procedure presented for two-stepcycling would provide a prediction of the remaininglife fraction for the ith level loading.

THEORIES USING THE CRACK GROWTHCONCEPT

The crack growth concepts developed in 1950s and1960s have enjoyed wide acceptance since cracks aredirectly related to damage, and since modem technology has provided sophisticated tools and techniqueswhich enable measurement of very small cracks in theorder of 1 JLm. Several macro fatigue crack growthmodels based on LEFM concepts were developed inthe early 1970s to account for load interaction effects inthe crack propagation phase (stage II) of the cumulativefatigue damage process. These models attempt toexplain macrocrack growth retardation resulting fromoverloads under variable amplitude loading conditions.After the early 1970s, several new fatigue damagetheories have been developed based on the microcrackgrowth concept. Though some are still phenomenological, most of these newer models better explain thephysics of the damage than those developed before1970s.

Macro fatigue crack growth modelsA popular macro fatigue crack growth retardation

model is the Wheeler modeF5. This model assumesthe crack growth rate to be related to the interactionof crack-tip plastic zones under residual compressivestresses created by overloads. This model modifies theconstant amplitude growth rate equation, da/dN =A(~Kt, by an empirical retardation factor, C:

da/dN = CJA(~Kt] where: Ci = (r p/rmaxY' (12)

Here rpi is the plastic zone size associated with the ithloading cycle, r max is the distance from the currentcrack tip to the largest prior elastic-plastic zone createdby the overload, and p is an empirical shaping parameter depending on material properties and load spectrum. A similar retardation model based on crack tipplasticity is the Willenborg modeF6. This model usesan effective stress intensity factor at the crack tip,(~Keff);' to reduce the applied crack tip stress intensityfactor, ~K;, due to the increased crack tip residualcompressive stress induced by the overloads. Thereduction in the applied ~K is a function of theinstantaneous plastic zone size at the ith load cycleand of the maximum plastic zone size caused bythe overload. Unlike the Wheeler model however, theWillenborg model does not require an empirical shaping parameter.

Based on his experimental observations, Elber77.78suggested that a fatigue crack can close at a remotelyapplied tensile stress due to a zone of compressiveresidual stresses left in the crack tip wake. This resultsin a reduced driving force for fatigue crack growth.The crack tip stress intensity factor driving the crack

is then an effective stress intensity factor based on theeffective stress range, ~Seff = Smax - Sop, where Sopis the crack tip opening stress. Other crack closuremodels have also been developed which include thoseby Newman79,8o, Dill et af.8 I ,82, Fuhring and Seeger83

and de Koning84. The difficulty in using crack closuremodels is in determining the opening stress, Sop. Newman's modeF9 predicts the crack opening stress by aniterative solution procedure for a cycle-by-cycle closurecalculation using detailed finite element programs. Inaddition to the plasticity induced crack closure, otherforms of fatigue crack closure can arise from corrosion(oxide-induced closure), fracture surface roughness(roughness-induced closure), and other microstructuraland environmental factors as categorized by Ritchieand Suresh85-88.

Statistical macrocrack growth models have also beenproposed89,90 in which crack growth rate is related toan effective stress intensity factor range based onprobability-density curve characteristics of the loadspectrum. The effective stress intensity factor rangedescribed in terms of the root-mean-square value ofstress intensity factor range, ~KrmS' proposed by Barsom90 is given by:

~Krms =J(*1~K7 )/nwhere ~Ki is the stress intensity factor in the ith cyclefor a load sequence consisting of n cycles. Thesemodels are empirical and do not account for loadsequence effects such as crack growth rate retardation.

Double exponential lawFor the accumulation of fatigue damage in crack

initiation and stage I growth, Miller and Zachariah91

introduced an exponential relation between the cracklength and elapsed life for each phase. The approachis thus termed double exponential law. In this modeldamage is normalized as: D = a/af, where a and afare instantaneous and final crack lengths, respectively.Later, Ibrahim and Miller92 significantly modified thismodel. Based on the growth mechanism of very smallcracks, crack propagation behavior in stage I was thenmathematically described in a manner similar to thatexpressed by LEFM for stage II growth as:

dadN = <fJ(LlYp)"a (14)

where <P and ex are material constants, and ~Yp is theplastic shear strain range. From this equation, a linearrelationship between the initial cycle ratio, rb and thefinal cycle ratio, r 2 , in two level cycling can be foundfor r l in excess of the initiation boundary rI, I =NI , /Nf, I' To determine the phase boundary betweeninitiation and stage I propagation, data from a seriesof two level strain-controlled tests are then collectedand plotted in the rl - r2 frame. An example of thistype of plot and its comparison with the linear rule isshown in Figure 7. In a further study by Miller andIbrahim93, N; and a; data were correlated with thecorresponding values of plastic shear strain range, ~YP'

through a power function, The phase boundary in theD-r frame is then also defined through ~Yp ' Thedamage equation for stage I propagation can, therefore,be described as:

Figure 8 A summary of the accumulation of fatigue damage atvarious load levels based on the double exponential rule proposedby Ibrahim and Miller93

Short crack theory

Miller and co-workers94-

101 investigated the behaviorof very short cracks and proposed that crack initiationoccurs immediately in metal fatigue, and that thefatigue lifetime is composed entirely of crack propagation from an initial defect size, aQ • The early twophases were renamed as microstructurally short crack(MSC) growth and physically small crack (PSC)growth, rather than as initiation and stage I propagation. Both MSCs and PSCs are elasto-plastic fracturemechanics (EPFM) type cracks. The growth behaviorof MSC cracks is, however, significantly influenced bythe microstructure in addition to the loading condition.The phase boundary between MSCs and PSCs, and thatbetween EPFM and LEFM cracks are schematicallyrepresented in Figure 9. This is a modification ofKitagawa-Takahashi !la-a diagram by Miller97

.98 .

Based on experimental observation and data analysis,crack growth models for MSCs and PSCs were established and mathematically described as96- 98,IOO:

dadN =A(!ly)"(d - a) for MSCs: aQ ::5 a ::5 at (16a)

dadN =B(!ly)f3a - C for PSCs: at ::5 a ::5 af (16b)

where A, B, (X and f3 are constants obtained by fittingof the experimental data; !ly is the shear strain range,at is the crack length corresponding to phase transitionfrom MSC growth to PSC propagation, d representsthe barrier size, and C is the crack growth rate at thethreshold condition. Equation (16)b was also derivedin Ref. 102 for high strain torsional fatigue damageaccumulation. The mathematical forms of Equation(16)a, (16)b seem convenient for application to theanalysis of fatigue damage accumulation. However, thephysics and validity of the short crack theory stillneeds further experimental evidence.

(17)

accumulation in the initiation phase is not yet established, but only schematically indicated by dashedcurves. Difficulties of modeling this damage phase canhardly be overcome, unless the damage mechanismsof this regime are well understood.

Ma-Laird model

Ma and Laird103 found that in the short crack regime,similar to the MSC region defined by Miller, crackpopulation, P, is linearly related to the applied strainamplitude and used life, and can therefore act as adamage indicator. Based on this concept, Ma andLaird proposed a new approach to summing cumulativedamage and predicting fatigue life, which is formulated as:

where (!ly/2)limit is the fatigue limit strain, K = C/Perit

(C is a constant in the strain-life equation), Perit is thecritical crack population at which failure is deemed,and (Xi is the loading history factor corresponding tothe ith load level. Based on the experimental findingsin Ref. 104, Ma and Laird defined (Xi as the ratio of

o

(15)

1.0

A. Fatemi and L. Yang

0.8

1.0

Stage IPropagation

0.60.40.2

fraction of life spent at the

low strain level, r I

- - - _ Jnitiation"-

......

" '\.,

o

-I10

EE

0

-210

J:..a.

"0~

·3u 100'-u

°0-4

100

16

Q1.0

.,J:.... N0 ~..c:.,

"Cl.... >"II- c:0- '-

0 ..lit

C J:0 CI..

J:U0..~

Figure 7 Schematic representation of the cumulative damage curvebased on the modified Ibrahim-Miller model in a L-H two-levelstep test93

I .Or-------.--........--....------.--..,

D = !!.- = (~)(I -r)/(I - rt)

af af

As summarized in Figure 8 for damage lines at variousstrain range levels, the above equation represents abundle of line segments radiated from point (1.0, 1.0)and terminated at the phase boundary defined by (NlNf ,

alaf)' However, the predictive model for damage


CRACKS

CracksPhysically Small

tit II I i

NON 1- PR~PAGATINGHicrostruc turally

Short Cracksd1 dz

VlVl....IXlV!

LOG. CRACK LENGTH

Figure 9 A modified Kitagawa-Takahashi l1u-Q diagram showing boundaries between MSCs and PSCs, and between EPFM cracks andLEFM cracks""

(19b)

1.0 t---------f----------i

crack initi ati0n 3tage craek pro pageti0n3tage.. ..

1.0

(20a)

0.5Cycle retio, r

aa

Figure 10 Schematic representation of damage functions proposedby Vasek and Polak

....cc

.......ccII

Q

£a i ncrea:si ng

Dc 2 ID = Dc + - [em( r ~ ) - I] for propagation: 1/2:s r:S 1

m

(20b)

where Dc = ajar, and m = kNrl2. This is essentiallya linear-exponential model. Figure 10 schematicallyrepresents this approach in the D-r frame.

MORE RECENT THEORIES BASED ON LIFECURVE MODIFICAnONS

The life curve modification approaches introducedbefore the 1970s possess attractive features of relativesimplicity in form, and effectiveness in implementation.Since 1970s several other damage rules have beendeveloped based on life curve modifications. These

D =2Dcr for initiation: D :::; r:::; 1/2

and

a i = (~'Yp) /(~'Yp) (a i :::; 1)2 i 2 max. in pre - loading history

(18)

The model represented by Equation (17) has the abilityto account for the load interaction effects. However, itshould be pointed out that this model predicts a longerlife for H-L strain sequence where a < 1, than forL-H strain sequence where a is always equal to 1.This is in contradiction with the common experimentalobservations in completely reversed loading.

where Vi is the crack growth rate independent ofapplied cycles, k is a coefficient, and ao, ac and ar arethe initial, critical, and final crack lengths, respectively.The critical crack length, ac , defines the transitionfrom initiation phase to propagation phase. Also, themagnitudes of Vi' k and ac are load level dependent.In their experiments, Vasek and Polak found the valuesof these three quantities to increase with increasing theloading level, and ac was reached approximately athalf-life (r = nlNr = 1/2) under constant amplitudecycling. Subsequently, integrating Equation (19)a,(19)b, damage evolution functions can be explicitlyexpressed as:

Vasek-Polak approachBased on their experimental observations and

interpretation, Vasek and Poiak105 identified two damage regimes. In the crack initiation regime, a constantcrack growth rate was proposed, described by:

dadN = Vi for ao :::; a :::; ac (19a)

and in the crack propagation regime, the dependenceof da/dN on the crack length was approximated by alinear relation:

dadN = Vi + k(a - ac ) for ac :::; a :::; ar

the currently applied strain amplitude to the maximumstrain amplitude in the pre-loading history includingthe current cycle:


Figure 11 Schematic representation of the modified S-N curveaccording to the Leipholz approach 113

these bounds are also applicable to nonlinear lifecurves. Bounds are narrowed by the inclusion ofadditional information from the fatigue damage processof crack initiation and propagation. This is simply areplacement of LDR with DLDR. The initiation lifefractions were determined from the empirical relationgiven in Ref. 91. Further improvement110, ofthis modelwas made by considering all parameters to be functionsof the random variables N 1, Nz and Ne• For theseparameters in the bounds, if extreme values associatedwith any desired number of standard deviations areused, statistically optimized bounds will result and datascatter can be bracketed. The extent to which thebounds can bracket the data scatter depends on thechoice of the number of standard deviations. Based onthe mathematical analogy between the fatigue and creepcumulative damage problems, the bound theory wasmodified to predict creep residual time in a two-stageexposure to stresses at and az at a fixed elevatedtemperature. The theory was further extended to creepfatigue interactions. A full presentation of the derivation can be found in Ref. Ill.

Leipholz's approachIn agreement with Freudenthal's and Heller's opinion

that the errors in life predictions based on LDR aredue not to its linear summation but to the assumptionof damage-rate independence of loading levels, LeipholzltZ-114 resumed the concept of replacing the original S-N curve with a modified curve, S-N', whichaccounts for load interaction effect. Leipholz' s modelis represented as:

N'2, = lri,(f3/N/) (22)

where N'2, is the total accumulated life, and f3i andN/ are the frequency of cycles (n/N'2,) and the modifiedlife with loading level ai' respectively. Figure 11describes the typical manner in which the modified SN curve converges to the original curve at a highloading level, and deviates from it at low loadinglevels. The S-N' curve is determined from multi-levelrepeated block tests along with Equation (22). Detailsof the method for obtaining the modified S-N curveare referred to Refs 112,113,115. The experimentalverifications of this modified life theory were given inRefs 113 and 115. Results show that this model canprovide accurate predictions of fatigue lives under

N· O NI,

virg in $- N curve

modified $ - Ncurve

s·I

s

models are load-level dependent and can account forthe load sequence effects.

Subramanyan's knee point approachA knee point-based approach was introduced by

Subramanyan106 based on observations of experimentalresults. In his study, a set of isodamage lines wereintroduced which were postulated to converge to theendurance knee point of the S-N curve. The damageis then defined as the ratio of the slope of an isodamageline to that of the original S-N curve. This implies anassumption that the endurance limit of a materialremains constant at all stages of the damage process.Mathematical expression for any isodamage line caneasily be obtained from this postulation, provided theoriginal S-N curve is linearized and has a knee point.For a loading sequence including i (i 2:: 1) steps, amathematical form for the residual cycle ratio at theith level can be found:

r; = 1 - k- I + [r; _ z + ... + (rz + rj'1)"2 ... ]"i-2}"i- I (21)

where (lk = 10g(Nk + /Ne)/log(N,jNe) for k = 1, 2, ... , i- 1. However, it should be noted that this approachis not valid at stress levels near the fatigue limit ofthe material. There are two reasons for this limitation.One reason is the singularity at the knee point sinceall the isodamage lines pass through this point. Thesecond reason is the nonlinearity that a log-log S-Nplot usually exhibits in the vicinity of the fatigue limit.

Hashin-Rotem modelHashin and Rotem 107 presented a discussion of the

S-N line convergence and rotation approaches in theframework they have devised for cumulative damageanalysis. Two types of convergence were speculated.In the first model, all damage lines pass through theintersection of the original S-N line with the S-axis(called static ultimate). This approach avoids determination of the convergence point as in the earlier S-Ncurve modification models3D--35. In the second model,the convergence point is at the endurance limit. Essentially, this is Subramanyan's concept, which has alreadybeen discussed. Based on the proposed approaches,Hashin and Rotem lO7 performed analytical calculationson two-stage, three-stage, periodic two-stage and amplitude continuously changed cyclic loadings. Experimentswere carried out by Hashin and Laird108 with twostage cycling and the data were used to test the effectiveness of this predictive model characterized by theendurance point convergence. Predicted results werefound to be in good agreement with test data, as wellas with those predicted from the double exponentialdamage rule.

Ben-Amoz's bound theoryFatigue damage is a statistical phenomenon in nature

and test data are inevitably scattered. Based on thisargument, Ben-Amoz lO9 introduced a concept of bandson residual fatigue life instead of seeking a definiteform for a damage rule. This theory states that aresidual life line obtained from the rotation of theoriginal life line would fall in the upper and lowerbounds. For the first approximation, Ben-Amoz represented the two bounds by Miner LDR (a paralleltranslation of S-N line) and Subramanyan's theory (arotation around the endurance limit). He proved that

19

3.0

l .. • ('lb)

2.5 ~o

2.51.5 2.0tot (%l

Loop Tnus. ~ • I (i)

(b) non-Masing type deformation

800 600

t. Slnin

(a) Masing-type deformation

1000 100lUT • (MPa)

Figure 12 Materials exhibiting hysteresis loops with (a) Masingtype deformation, and (b) nonMasing type deformation '26

Regardless of the type of energy model, the conceptused in damage modeling is the same. Both energymodels are essentially similar to Subramanyan's convergence approach106. A power function analogous to~n S-N ~elation :-vas .employed to describe the energylIfe relatIOn, WhICh IS a straight line in a log frame.As illustrated in Figure 13 for a two-level load test,isodamage lines intersect the extension of the originalenergy-life line at the point (Ne*, awe*), rather than at(Ne, aWe) which is the original endurance limit. Thepo~nt (~e*,.awe*) is, therefore, called the 'apparent'fatIgue lImIt. There are several methods to determinethe coordinates Ne* and aWe*. One method is basedon the predictive equation of change in endurance limitsuch as Bui-Quoc's hypothesis55

. Another method l31 isbased on the use of the relation between the thresholdstress intensity factor, aKlh , and the apparent fatiguestress limit in conjunction with the cyclic stress-strainequation. In later modifications, Ellyin et aZy,126,127

(23)1 (aiT )2aw=- --+iT2E 2 m

where iTm is the mean stress.

Models proposed by Ellyin and co-workersKujawski and Ellyin 124 developed a preliminary

damage model by using plastic strain energy densityas a parameter. Theoretically, plastic strain energyabsorbed in a complete cycle can be obtained byintegrating the area included in a hysteresis loop. It is,therefore, also referred to as the hysteresis energy anddenoted by awp

• Another alternative is the mastercurve technique. It has been found l25 that there aretwo types of materials, Masing type and nonMasingtype, as shown in Figure 12. For a Masing material,the master curve can directly be constructed from thecyclic stress-strain curve. For a nonMasing material,however, this is not straight-forward. Ellyin and coworkers ll .

126-129 employed the Jhansale-Topper tech

nique 130 to construct the master curve. Once the mastercurve is constructed, the calculation of awp can be formulated.

It was later found that some inefficiencies wereassociated with the plastic strain energy approach. For~xample, the effect of mean stress cannot be directlyIncorporated in the determination of the hysteresisenergy. Also, for the low strain high-cycle fatigue, theplastic strain energy density is very small. In somecases, though the macroscopic (bulk) response of thematerial is elastic or quasi-elastic, microscopic (local)plastic deformation may still exist in the material dueto the nonuniformity of local strain distribution and/ordue to the strain concentration by high prestraining. Toovercome these shortcomings, Golos and Ellyin11 ,126,127

modified the plastic strain energy-based model by usingtotal strain energy density, awl. The total strain energydensity combines both plastic (awP) and elastic (awe)portions. The elastic portion is thought to be associatedwith the tensile mode and can facilitate crack growth.The calculation of awe is obtained from:

ENERGY BASED DAMAGE THEORIES

Since the report of connection between hysteresisenergy and fatigue behavior by Inglis l16

, many studieshave been carried out on energy methods. Severalfailure criteria based on strain energy were establishedby Morrowll

? and Halford l18 in the 1960s. However,cumulative damage theories based on strain energywere mainly developed in the last two decades. Someenergy-based damage parameters have been proposedsuch as those by Zuchowski 119 and Budiansky andO'Connell 120. It has been realised that an energy-baseddamage parameter can unify the damage caused bydifferent types of loading such as thermal cycling,creep, and fatigue. In conjunction with Glinka's rule l2l

,

it is possible to analyze the damage accumulationof notched specimen or components with the energyapproach. Energy-based damage models can alsoinclude mean stress and multiaxial loads since multiaxial fatigue parameters based on strain energy havebeen developed I22,123.

Cumulative fatigue damage and life prediction theories

repeated block loading. This predictive theory is alsoexpanded to stochastic loading histories 113,115.


AWI

4W2

.AWI

Life curve

Damaqe curve

tI- - - - - - - - - - -,- - -

: I

- - - - - - -'- - - - J.. - - - ~ _..2 -'~------iI I 'I "I I I I

- - -,- 1 ~ ...J - --

I I I 1 ' , '"

n2

n N NeI •

log Nf

Figure 13 Damage line through 'apparent' fatigue limit defined by point (Ne*, ~We*)124

(26)

(27)

(29)

(28)

D = <l>l/l(n' + Ol )(' + 13)] = r'/( n' + Ol)

where a and b are two constants. The incremental rateof plastic strain energy was then derived as:

dW I - n* (dE )' + n*-=4 -- r/3K* -p

dN 1 + n* 2

and the energy accumulation is defined by introducinga parameter called the fraction of plastic strain energy,<I> = W/Wf = r' + /3, Finally, the fatigue damage functionwas constructed as:

where a = (dmiE;4?b-y~, and n' is the cyclic strainhardening exponent. The model represented by Equation (29) is a nonlinear, load-dependent damageaccumulation model. It accounts for the load interactioneffect and the change in strain hardening through thestress response. This damage model is specially suitablefor materials which exhibit cyclic hardening. Experimental verification of this model can be found inRef. 135.

In addition to the energy approaches reviewed above,several other energy-based models also exist. In theearly 1970s, Bui-Quoc '37 conducted an experimental

Model developed by Niu et al.By examining constant strain amplitude test data,

Niu et al. 135,136 found the cyclic strain hardening coef

ficient to change during the cycling process, while thecyclic strain hardening exponent had a negligiblechange. Therefore, a new cyclic stress-strain relationwas proposed as:

dO" (dEp)n* /3-=K* - r2 2

where K* and n* are cyclic strain hardening coefficientand exponent determined near failure (r = n/Nf = 1),and 13 is the cyclic hardening rate. The expression for13 was given as:

fixed the point (Ne*, dWe*) at the intersection of theoriginal energy-life curve extension with a critical damage curve which delineates the boundary betweenfatigue initiation and propagation phases. This criticalcurve can be experimentally determined, similar tothe determination of the Frence curve132. Once theconvergence point is determined, damage lines corresponding to different degrees of damage will convergeto or radiate from it. The energy-based damage modelsproposed by Ellyin and co-workers possess featuressimilar to Subramanyan's model. However, the energyapproaches have no singularity problem at endurancelimit, since the convergence point is selected at the'apparent' fatigue limit below the original knee point.

Leis theory

Leis'33 proposed an energy-based nonlinear historydependent damage model which links the damage parameter to fatigue life in a manner similar to the SmithWatson-Topper parameter I 34:

40"'D = Ef (2Nf?b1+ 40"/ E/ (2Nf )b1+ c, (24)

where 0"/ and E/ are the fatigue strength and ductilitycoefficients, respectively. However, the exponents b,and c, are analogous to but different from the fatiguestrength exponent, b, and fatigue ductility exponent, c.In this model, b , and c, are two variables related to theinstantaneous strain-hardening exponent, nl' through:

- 1 - n,c, = 1 5 and b , = 1 5 (25)+ n, + n,

which are asserted by an analogy to the Morrow'scorrelations 117. From experimental observations, Leisspeculated that the parameter n l can be characterizedas a function of the accumulation of plastic strain,~dEp. The model represented by Equation (24) istherefore an analytical formulation in terms of thedeformation history. Clearly, properly defining thefunction nl = nl(IdEp) is crucial in the application ofthis damage model.


investigation on fatigue damage with five-step loadingtests. Both step-up (increasing loads) and step-down(decreasing loads) experiments were conducted on twosteels. The number of strain levels in multiple-steptests did not have an influence on the cyclic strainhardening coefficient, and the strain ratio had littleeffect on the cyclic stress-strain curves and a negligibleeffect on the total plastic energy at fracture, Wf' Basedon these experimental observations, a model for calculating the value of Wf accumulated during a fatiguedamage process was proposed, given by:

2K'Mn'+fW:

f=!.n~w. = !'rN

fl . - C(n' + I) (30)

I I n' + 1 I I

where ~Wi is the hysteresis energy for the ith loadinglevel, and M and c are material constants in therelation: ~E;Vf =M. However, since it is found l17,118that the total plastic energy at failure is not constantfor most materials, application of this model to thecumulative fatigue damage problems is questionable.

From the viewpoint of crack growth, Radhakrishnan 138, 139 postulated that the crack growth rate is proportional to the plastic strain energy density which islinearly accumulated to failure. For a m-level loadvariation, an expression for predicting the remaininglife fraction at the last load step was, therefore, formulated as:

(31)

where Wfi and W fm are total plastic strain energy atfailure for the ith and the last (mth) levels underconstant amplitude cycling, respectively. This formulation implies that failure occurs when the accumulatedplastic energy reaches the value of W fm in the laststage. This implication excludes the influence of loadinteraction effect on either Wfi or W fm .

A concept similar to Radhakrishnan's was also proposed by Kliman 140 and applied to repeated blocktests with harmonic loading cycles. A linear energyaccumulation was applied. The hysteresis energy ofeach loading block was calculated as the sum of themultiplication of frequency, nbi' by the correspondingplastic strain energy density, ~w;, as Wb = !.~winbi'

Apparently, this hypothesis does not consider the loadsequence effects. In reality, however, it has been experimentally shown141 that the value of Wb changes withthe spectrum sequence pattern in a block. Disregardingloading sequence effects, Kliman defined the damagefraction per block as:

Wb 1Db = - = - !.~W.nb (32)W

fRW

fRI I

where WfR is the total energy at fracture for a givensequence. Failure is deemed to occur when D = DJJf= 1. Based on this damage accumulation model forblock loading, one can calculate the accumulatedenergy following a successive procedure, cycle bycycle and block by block.

CONTINUUM OAMAGE MECHANICSAPPROACHES

Continuum damage mechanics is a relatively new subject in engineering mechanics and deals with the mech-

anical behavior of a deteriorating medium at the continuum scale. This approach is developed based on theoriginal concepts of Kachanov l42 and Rabotnov l43 intreating creep damage problems. The general conceptsand fundamental aspects of this subject can be foundin Refs 144-151. Hult146,147 and Chabochel52 arguedthe importance of COM in damage analysis. The success of COM application in modeling the creep damageprocess has encouraged many researchers to extendthis approach to ductile plastic damage, creep-fatigueinteraction, brittle fracture and fatigue damage. Inaddition to metallic materials, COM can also be appliedto compositesl 53 and concrete materials l54. Krajcinovic l55 and Chaboche l56,157 have reviewed the mainfeatures of the COM approach and its applications.

For the one-dimensional case, Chaboche158 postulated that fatigue damage evolution per cycle can begeneralized by a function of the load condition anddamage state. Tests conducted under completelyreversed strain-controlled conditions provided supportive information. By measuring the changes in tensileload-carrying capacity and using the effective stressconcept, he formulated a nonlinear damage evolutionequation as I58,159:

D = 1 - [1 - r llo - a)p/O + f3) (33)

where f3 is a material constant and a is a function ofthe stress state. This damage model is highly nonlinearin damage evolution and is able to account for themean stress effect. It is, therefore, called a nonlinearcontinuous-damage (NLCO) modeP60. This model hasthree main advantages. First, it allows for the growthof damage below the initial fatigue limit, when thematerial is subjected to prior cycling above the fatiguelimit l60. Second, the model is able to take into accountthe influence of initial hardening effect by introducinga new internal variable which keeps memory of thelargest plastic strain range in the prior loading historyI60.161. Third, mean stress effect is directly incorporated in the model. However, since a scalar damagevariable is employed and the model is written in itsuniaxial form involving the maximum and meanstresses, difficulties will inevitably be present when themodel is extended to multiaxial loading conditions.The main features, advantages and some deficienciesof the NLCO model are summarized in Ref. 160.

Based on the COM concept, many other forms offatigue damage equations have been developed afterChaboche's work l58. Such models include those proposed by Lemaitre and Chabochel45,162, Lemaitre andPlumtreel63, Wang l64

, Wang and LOU l65 and Li et al. 166

Basically, all these COM-based approaches are verysimilar to Chaboche NLCO model in both form andnature. The main differences lie in the number and thecharacteristics of the parameters used in the model, inthe requirements for additional experiments, and intheir applicability.

Socie and co-workers I67,168 applied the LemaitrePlumtree model to the fatigue damage analysis of castiron to account for the influence of defects on fatiguelife. They reported improved life predictions as compared to the Miner rule '67. Plumtree and O'Connor l69

attempted to analyze damage accumulation and fatiguecrack propagation in 6066-T6 aluminum using a modified Lemaitre-Plumtree Model. Hua and Socie l70 alsoevaluated the Chaboche and the Lemaitre-Plumtree


(34)

Figure 14 Illustration of internal stress and effective stress on astress-plastic strain hysteresis looplHO

(J : - a , curv.

a - £ p curve

F

,/ a/

/I

, p0

B

A

A model based on internal and effective stresses

The concept of internal and effective stresses wasgenerated from the discovery that the average dislocation velocity and thus the plastic strain rate is proportionally related to the effective (resolved) stressacting on a dislocation I 76-179. This effective stress, aconcept different from the effective stress defined inCDM approach, is equivalent to the difference betweenthe applied and internal (back) stresses. Ikai and coworkersl8°-183 introduced this concept for the analysisof cumulative fatigue damage. As illustrated in Figure14, the highest level of an elastic range (HG) is definedas the internal stress, ai' and the difference (HC), aa- ai' gives the effective stress, aeff (where aa is theapplied stress amplitude). Based on the 'stress-dip'technique I 84,1 85, a i and aeff for a given applied stresslevel, aa' can experimentally be determined.

Based on their experimental observations under bothconstant and variable amplitude stress cycling, Ikai etal. concluded that internal stress (as a result of elasticinteraction of dislocations) is representative of thefatigue resistance of a material and that the effectivestress above a critical value is responsible for thefatigue deformation, and hence fatigue damage in thematerial. It was speculated that a material under cycling

level through a modification factor which keeps thememory of loading histories. Therefore, Equation (36)represents a load dependent damage model with linearevolution, nonlinear accumulation, and accounts forload interaction effect. Kramer believed the modelcould also be extended to corrosion-fatigue damageanalysis, since corrosive attack promotes the surfacelayer stress. Applications of the model represented byEquation (36) to two and three step load level fatiguetests were reported by Kramer with a 2014-T6 aluminum l73 and by Jeelani et al. with a titanium 6AL-4Valloy'74 and a AISI 4130 steeJl75.

Kramer's suiface layer stress model

Recognizing that information from the surface of afatigued material usually plays an important role indamage analysis, Kramer172 introduced the concept ofsurface layer stress to characterize fatigue damage. Itwas postulated that during fatigue cycling, the specimen surface layer hardens due to a higher dislocationdensity than the interior. Consequently, to attain agiven plastic strain, more stress must be imposed thanwould otherwise be required if the hardened layer werenot present. Kramer defined this additional stress asthe surface layer stress, as. Under constant amplitudecycling, this stress was found to linearly increase withapplied cycles, n, as: as = Sn, where the proportionalitycoefficient S is the increase rate of as and is loadamplitude dependent. This coefficient can be describedas: S =Kaf;, where K and p are material constants. Asthe fatigue process continues, the surface layer stresswould reach a critical value, as*, when a fatal crackforms. This critical stress is found to be independentof the stress amplitude and can be expressed as:as* =SNf • A stress-life equation was, therefore,derived:

2a*U a = ~ (2Nf ) - lip

OTHER DAMAGE THEORIES

models in their investigations of biaxial fatigue. Theyfound the Chaboche model to be better than the Lemaitre-Plumtree model for fatigue damage assessment.

The CDM models aforementioned were mainlydeveloped for uniaxial fatigue loading. Some difficulties arise when these models are extended to multiaxialloading160. To overcome these difficulties, Chow andWei 171 have recently attempted a generalized threedimensional isotropic CDM model by introducing adamage effect tensor. However, due to the complexityof nonproportional multiaxial fatigue problems, a threedimensional anisotropic CDM model does not yet exist.Though the framework was already proposed byChaboche in Ref. 148, great efforts are still needed toobtain an appropriate generalized prediction model forcumulative fatigue damage.

In this equation, 2a.//K is equal to the fatiguestrength coefficient a; and -lip to the fatigue strengthexponent, b. Considering accumulation of the surfacelayer stress directly to quantify the damage process,Kramer defined the damage rule simply as:

D = '}.(as/as*) (35)

Failure is deemed to occur when D = 1. Since a si islinearly developed with applied cycles as indicated bya si = Sn;, Equation (35) is actually another version ofthe LDR. Based on experimental observation of thesurface layer stress evolution under two-level cycling,Kramer modified the previous model to:

(36)

According to this equation, damage evolves linearly ata load level, but sums up nonlinearly from level to


exponent a is a material constant. For an accurateprediction of fatigue life, quantities D , and a must bedetermined by experiments. This damage modelaccounts for the interactive damage resulting fromeither tensile and/or compressive overloads. It alsotakes into account the damage from post-overloadsmall-amplitude cycles below the constant amplitudefatigue limit, which is normally ignored in other damage models. The results in Refs 191,192 indicate thisdamage model to be promising in applications.

A plastic strain evolution modelIn order to represent the relationship between dam

age evolution and changes in mechanical properties,Azari et al. 194 postulated a general form of damagefunction, that is, D = j(Y, X), where Y denotes thedamage and X is the material's property. It followsthat the proper selection of property and evolution ofits change enable an accurate evaluation of fatiguedamage and prediction of fatigue life. In their study,total strain range, dE, was controlled and plastic strainrange, dEp , was then chosen as an evaluation property.Therefore, damage accumulation was expressed as:

D ="i.(dEp - dEpo ) ~ IIC (39)dEpf - dEpo ;

where C is a constant, and dEpo, dEp and dEpf areinitial, present and final values of the plastic strainrange, respectively, dEp is usually a function of theapplied strain, dE, and elapsed cycles, n. For a givendE of constant amplitude, damage can be plotted withrespect to n, which results in a damage evolutioncurve. The area under the curve, A, can be calculatedby integration. Experiments 194 showed that the quantityA divided by the value of N f is a constant (about0,55). Therefore, Azari et al. '94 proposed a criterionfor fatigue life prediction under complex loadingexpressed as:

"i.(A;fNfi) =Constant (40)

The form of this model is simple and its applicationonly requires information from constant amplitude tests.Moreover, in the constant amplitude tests, only theplastic strain response needs to be monitored. Thisdamage model can be applied to multi-level complexloading situations. However, this model cannot accountfor load interaction effects and small cycle damage.

Additional approachesIn addition to the aforementioned models, there still

exist other approaches for cumulative fatigue damageanalysis. In order to cover as many models as possiblewhile avoiding being lengthy, this subsection reviewsthe remaining approaches, but only very briefly.

Fong '95 postulated that fatigue damage rate, as afirst approximation can be linearly related with damageitself, With the initial and final conditions (D = 0 atr = 0 and D = 1 at r = 1 at failure), the integrationof the damage rate equation gives a damage functionin the form of:

reacts in two contradictory manners: suffering damageas a consequence of the effective stress; and beingstrengthened in proportion to the internal stress toendure further stressing. The internal stress was foundto increase through dislocation reaction or strain agingof the material. This mechanism can probably be usedto explain many phenomena in cumulative fatigue damage processes such as the coaxing effect2z,z5, fatiguelimit and its load history dependency, and fatiguefailure caused by small cycles. In addition, the internalstress concept may provide physical interpretations forcyclic creep and cyclic stress relaxation179,186,187, barriers for growth of MSCs97-99,188,189, load interactioneffects, cyclic hardening or softening and possibly otherphenomena. However, this method can not be appliedto damage assessment for fatigue crack propagationprocess, where internal stress measurement becomesmeaningless.

On the basis of the behavior of internal stress andeffective stress, Ikai and co-workers I8o,183 proposed anew approach to fatigue damage accumulation. Thismodel is simply an effective stress version of theMiner rule. In the so called ES-Miner rule, a aefY

log Nf plot is used as the baseline in damage calculation, instead of the conventional S-N diagram, andthe Miner linear summation is then applied,

An overload damage modelBoth tensile and compressive overloads are fre

quently encountered in real loading spectra. Brose etal. 190 conducted a systematic study on the effects ofoverloads on the fatigue behavior. They performedcompletely reversed strain-controlled fatigue tests onsmall smooth specimens with either 10 completelyreversed initial overstrain cycles, or one fully reversedperiodic overstrain cycle at intervals of 105 cycles.Recently, Topper and co-workers191 ,192 carried outintensive studies on this subject by subjecting smallsmooth specimens to uniaxial stress histories consistingof repeated blocks, where each block contained oneunderload or overload and a fixed number of smallfully reversed cycles. A damage model which accountsfor the overload effect was then established to predictthe experimental results. Their work has been reviewedby DuQuesnayl93. In this model, damage summationwas expressed by:

D = "iDol + "iDss + "iDint with D = 1 at failure (37)

where Dol is the damage due to the overloads, Dss isthe damage due to the smaller amplitude cycles in theabsence of overloads (at a steady-state condition), andDint is the additional 'interactive' damage due to thesmaller cycles succeeding the overload. For a periodicoverload history, the interactive damage per block isdetermined through:

(for compressive overload)

(for tensile overload)

(38)

ekr; - ID ="i. -k---l for ki=O

e -(41)

where D , is the interactive damage due to the firstsmaller cycle (N = 1) after the overload, and the

The validity of this model was experimentally evaluated and compared with three other models in Ref. 170


(43)

where Ncat is the calculated life for the present loadingcondition, and e is a correction factor ( =Nexp'INcal').For improved life estimation with the application ofthis relative method, the considered load spectra should

spacing. Then a linear summation of the normalizedPSB density enables life prediction. In another study,Inoue et al.203 developed a multiaxial micro-damageapproach. The damage accumulation was presumed tobe a function of the applied cycles and the intensityof slip b~nds, 'l'(t:;>*, occurring in all grains withcommon N (unit vector normal to the slip plane).Normalized by the intensity at the termination of crackinitiation life where the maximum value of 'l'(t:;)* isreached, a scalar damage variable was then defined by:

~ 'l'(t:;>* (n)-ck/mD(N, n) =-- --- (44)

'l'~ * N(N)max I

where m is the hardening exponent, and c and k arethe exponents in relations: /lE = g(N;)< and /la =h(/lE)k, respectively. In multiaxial fatigue, /la and /lEare replaced by the corresponding equivalent quantities.This model assumes that crack initiation occurs at theneighborhoog of the grain with slip plane N, when thevalue of D(N, n) reaches unity.

Different from others, Abuelfoutouh and Halford204

introduced 'resistance to flow (X),' instead of 'damage(D)', as a scalar state variable. The rate of evolutionof X was defined by a state variable constitutiverelation as:

where b, c and d are material constants; i 2 is thesecond invariant of the deviatory stress tensor; and thesign is positive for a hardening material and negativefor a softening or damaged material. Integration ofEquation (45) gives the evolution rule of X with respectto the applied cycles or life fraction r, which determines a resistance-life (X-r) curve somewhat similarto an upside down damage curve. For complex fatigueblock loading, a concept of local iso-resistance curvewas introduced into the X-r diagram. This enables alevel-by-level successive fatigue life prediction. However, since these distribution curves were just conceptually described in Ref. 204, this approach cannot beapplied unless these curves can practically be determined.

A correction method, called the relative Miner rule,is suggested to be a possible solution for improvingfatigue life prediction accuracy for variable amplitudeor service loading spectra20S-207. The basic assumptionof this method is that for sufficiently similar loadinghistories, the deviations from the Miner linear damagesummation have the same trend and similar relativemagnitudes. Under this hypothesis it is possible fromthe experimental results of a reference loading spectrum, to predict the fatigue life, N', of other loadinghistories with sufficiently similar loading patterns tothe reference pattern. If the life ratio of experimentalto calculated values of the reference loading spectrumis Nexp'INca/, then the corrected prediction of lifebecomes:

(45)

(46)

dXdN = ± ii + hexp(CX + d)

N 'N'=N II~=N lie

cal N' calcal

k 2n. (/la)lIdD-~_l_~

b - ; = 1 (2Nf ); /lah

where d is an interactive exponent derived from baseline data (d = bl[b + c + l] in which band carefatigue strength and ductility exponents), and /lah isthe highest range of stress response in the sequence.For complex loading histories, an event is consideredto be a cycle identified based on the rainflow countingtechniqueI99.2oo. The model suggests that the factor(/laJ/lah)lId modifies the slope of the stress-life curvein a way similar to that in Corten-Dolon Theory30. Asto the i-integral damage model, it characterizes crackgrowth rate in terms of elastic-plastic work requiredto open and extend the crack.

Recently, Pasic201 attempted to combine the fracturemechanics-based damage model with the CDMapproach. In his 'unified' approach, damage in stresscontrolled fatigue is defined as the product of thelongest or 'equivalent' surface crack length and theaccumulated strain range, both in exponential form.Employing fracture mechanics, a relation between cycleratio and evolution of normalized crack length wasfirst derived. The normalized crack length was thenrelated to normalized damage through the effectivestress concept in CDM. Therefore, the damage evolution rule is implied by these two relations. Thismodel, however, lacks experimental verification.

Rather than using macro damage parameters, someinvestigators focused their efforts to search for appropriate micro-variables which best describe fatigueinitiation damage mechanisms and process. Cordero etal.202 formulated cumulative fatigue damage for persistent slip band (PSB) type materials through PSB density, which is defined as the ratio of PSB length to its

Its nonlinearity matches the experimental data fairlywell.

To analyze complex strain histories, Landgrafl96

derived a damage equation based on the linear summation hypothesis from the strain-life equation. Damage per reversal was defined as reciprocal reversals tofailure. Combined with Morrow's mean stress modification 197, the damage rate equation was derived as:

Dlreversal =_1_ = [a/ - am (~_~ + am)]II(h - c)

2Nf E/E /lEe a/(42)

where a/, E/, band c are the low cycle fatigueproperties. The ratio of plastic to elastic strain rangeprovides the experimental input. /lEp and /lEe can bedetermined from a block of steady-state stress-strainresponses. Using /lEri/lEe as a damage parameter, thismethod entails a reversal-by-reversal damage analysisof a complex history.

Also considering variable amplitude histories representative of service load situations, Kurath et al. 198

examined both the effect of selected sub-cyclesequences on fatigue damage, as well as the applicability of plastic work model and i-integral for damagesummation. Plastic work was employed to account forinteraction effects. By introducing an interactive factorinto each summation term in linear damage hypothesis,the damage for a block sequence with k levels wasdefined as:


have basic loading parameters identical or similar tothose in the reference spectrum such asmaximum/minimum amplitudes, mean loads and eventfrequency distribution.

SUMMARY

More than 50 fatigue damage models have been proposed since the Palmgren damage accumulation concept and the Miner LDR were introduced. Most ofthese models are summarized in Tables 1-8. The physical basis, damage expression and the main characteristics of each model are listed in these tables. Theabbreviations used for describing the physical basisand characteristics of each model are defined at thebottom of the tables. In general, damage theoriesdeveloped before 1970s are mainly phenomenological,while those after 1970s are semi-analytical because, tosome extent, they involve damage mechanism(s).

As a whole, six major categories in cumulativefatigue damage modeling exist:

1. linear damage evolution and linear summation;2. nonlinear damage curve and two-stage lineariz

ation approaches;3. life curve modifications to account for load interac-

tions;4. approaches based on crack growth concept;5. models based on CDM; and6. energy-based methods.

No clear boundaries exist among some of theseapproaches. LDRs can not account for load sequenceand interaction effects due to their linear nature. Thefirst nonlinear load-dependent damage theory represented by the power relationship, D = Ir/ i , was proposed by Marco and Starkey in 1954. In two-stagelinearization approaches, the damage process is dividedinto two stages of crack initiation and crack propagation and the LDR is then applied to each stage. Lifecurve modification approaches are based on modifyingthe material S-N curve, are load-level dependent, andcan account for the load sequence effects. Approachesbased on the crack growth concept including macrocrack growth retardation models have enjoyed a widedegree of acceptance since crack growth can directlybe related to the physics of the damage process. CDMapproaches are relatively new approaches, modelingthe material damage process at the continuum scale.These approaches were originally developed to modelcreep damage and later extended to include the fatiguedamage process. Cumulative damage theories based onenergy have mainly been developed since the late1970s and have the potential to unify the damagecaused by different types of loads such as thermalcycling, creep and fatigue.

Though many damage models have been developed,unfortunately, none of them enjoys universal acceptance. Each damage model can only account for oneor several phenomenological factors, such as loaddependence, multiple damage stages, nonlinear damageevolution, load sequence and interaction effects, overload effects, small amplitude cycles below fatigue limitand mean stress. Due to the complexity of the problem,none of the existing predictive models can encompassall of these factors. The applicability of each modelvaries from case to case. Consequently, the Palmgren-

Miner LDR is still dominantly used in design, in spiteof its major shortcomings. Also, the most commonmethod for cumulative damage assessment using LEFMhas been based on integration of a Paris-type crackgrowth rate equation, with modifications to account forload ratio and interaction effects. More efforts in thestudy of cumulative fatigue damage are needed in orderto provide design engineers with a general and reliablefatigue damage analysis and life prediction model.

ACKNOWLEDGEMENTS

Financial support for this project was provided by theEdison Industrial Systems Center and DANA Corporation.

REFERENCES

Palmgren, A., Die Lebensdauer von Kugellagern. Veifahrenstechinik, Berlin, 1924, 68, 339-341.

2 Miner, M. A., Cumulative damage in fatigue. Journal of AppliedMechanics, 1945, 67, AI59-AI64.

3 Newmark, N. M., A review of cumulative damage in fatigue.In Fatigue and Fracture of Metals, ed. W. M. Murray. TheTechnology Press of the MIT-Wiley, New York, NY, 1952, pp.197-228.

4 Socie, D. F. and Morrow, J. D., Review of contemporaryapproaches to fatigue damage analysis. Fracture control reportNo. 24. College of Engineering, University of Illinois, Urbana,IL, December 1976; [also in Risk and Failure Analysis forImproved Peifonnance and Reliability, ed. J. J. Burke and V.Weiss. Plenum, New York, 1976, pp. 141-194.]

5 Kaechele, L., Review and analysis of cumulative-fatigue-damagetheories. RM-3650-PR. The Rand Corporation, Santa Monica,1963.

6 Manson, S. S., Interpretive report on cumulative fatigue damagein the low-cycle range. Welding Journal Research, 1964,43(Supplement), 344s-352s.

7 Leve, H. L., Cumulative damage theories. In Metal Fatigue:Theory and Design, ed. A. F. Madayag. Wiley, New York, NY,1969, pp. 170--203.

8 O'Neill, M. J., A review of some cumulative damage theories.Structures and Materials Report No. 326, Aeronautical ResearchLaboratories, Melbourne, Australia, 1970.

9 Schive, J., The accumulation of fatigue damage in aircraftmaterials and structures. AGARD-AG-157. Advisory Group forAerospace Research and Development, Paris, 1972.

10 Laflen, 1. H. and Cook, T. S., Equivalent damage-a criticalassessment. National Aeronautics and Space Administration Contract Report, NASA CR-167874. NASA, 1982.

11 Golos, K. and Ellyin, F., Generalization of cumulative damagecriterion to multilevel cyclic loading. Theoretical and AppliedFracture Mechanics, 1987, 7, 169-176.

12 Manson, S. S. and Halford, G. R., Re-examination of cumulativefatigue damage analysis-an engineering perspective. Engineering Fracture Mechanics, 1986, 25(5/6), 539-57 I.

13 Yang, L. and Fatemi, A., Cumulative fatigue damage mechanisms and quantifying parameters: a literature review. Journalof Testing and Evaluation, in press.

14 French, H. J., Fatigue and hardening of steels. Transactions,American Society of Steel Treating, 1933, 21, 899-946.

15 Kommers, J. B., The effect of overstressing and understressingin fatigue. Proceedings, American Society for Testing andMaterials, 1938, 38(Part II), 249-268.

16 Langer, B. F., Fatigue failure from stress cycles of varyingamplitude. ASME Journal of Applied Mechanics, 1937, 59,AI60-AI62.

17 Lim, L. C., Tay, Y. K. and Fong, H. S., Fatigue damage andcrack nucleation mechanisms at intermediate strain amplitudes.Acta Metallurgica et Materialia, 1990, 38(4), 595-{i01.

18 Coffin, L. F., Design aspects of high-temperature fatigue withparticular reference to thermal stresses. Transactions of theASME, 1956, 78, 527-532.

19 Baldwin, E. E., Sokol, G. J. and Coffin, L. E, Cyclic strainfatigue studies on AISI 347 stainless steel. Proceedings, American Society for Testing and Materials, 1957, 57, 567-586.


Table 1 Summary of cumulative fatigue damage theories: work before the 1970s

Model Model developer Year Physical basis" Expression Characteristicsh Ref.

Miner LDR Miner 1945 Constant energy D = "in/Nf; = "ir; LDE, nLLD, nLlA, 2absorption per nSC, many Applcycle (CON) (popular), S

Machlin theory Machlin 1949 Constant PI2 LDE, nLLD, nLlA, 208(metallurgic LDR) dislocation

D = "in;JRg; dtnSC, nAppI, C

generation percycle (CON) 0

Strain version of Coffin 1956 Directly converted "in(~E )'Ia LDE, nLLD, nLlA, 18D= I pILDR from stress version e'la nSC, some Appl,

G

Marco-Starkey Marco and 1954 Conceptual (CON) D="ir;,;,x; > I nLDE, LLD, nLlA, 23theory Starkey nAppi, S

Henry theory Henry 1955 Endurance limit D = ~eo - (Te = "i r; nLDE, LLD, nLlA, 22change (PHE) (Teo I + (I - r;)fA; SC, few Appl, G

Gatts theory Gatts 1961 Endurance limit

_Y. '+ 'y, (I - JnLDE, LLD, nLlA, 27change (PHE)

u, [SC, few Appl, C

Ye = - = Y" 1-aco

Yu - I Y" - IBluhm's hypothesis Bluhm 1961 Endurance limit d" = IY("JI - IYe(" - III nLDE, LLD, nLlA, 29

change (CON) SC, nAppi, C

Corten-Dolon Corten and Dolon 1956 Number of damage D = "im;p;nb; nLDE, LLD, LlA, 30model nuclei (CON) SC, few Appl, G

Frudenthal-Heller Frudenthal and 1959 Fictitious life(nw)

nLDE, LLD, LlA, 31theory Heller curve, probabilistic D = "i ~; , where w; is an interaction factor nSC, some Appl,

analysis (sANA) G

Grover's two-stage Grover 1960 Crack initiation n Two-stage LDE, 36damage theory and crack "i a;~; = I for initiation stage LLD, nLlA, nSC,

propagation, two- m· few Appl, Sstage linear "i (I _ ~;)N; = I for propagation stage

evolution (CON)

Double linear Manson et al. 1966 Crack initiation "i 2= "i n; 00 = I for phase ITwo-stage LDE, 37

damage rule and crack N,.; N f - PNf ' LLD, nLlA, SC,(DLDR) 1967 propagation, two- m· m· some Appl, S 38

stage linear "i i- = "i PN:L6 = I for phase IIevolution (EMP) II,i f

Shanley theory Shanley 1952 Crack growth,D = "i( (j<J ) I - r;

nLDE, LLD, nLlA, 40crack length as nSC, few Appl, Gdamage measure

Clc

(PHE)

Valluri theory Valluri 1961 Crack growth and da nLDE, LLD, some 41dislocation, fracture dN = Cf«(T)a Appl, Gmechanics (sANA)

Scharton-Crandall Scharton and 1966 Crack growth da nLDE, LLD, some 43theory Crandall fracture mechanics -=am+'f«(T.. ) Appl, GdN '1

(sANA)

"CON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalhLDE, linear damage evolution; LLD, load level dependent; LlA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'

20

21

22

23

Topper, T. H. and Biggs, W. D., The cyclic straining of mildsteel. Applied Materials Research, 1966, 202-209.Miller, K. J., An experimental linear cumulative-damage law.Journal of Strain Analysis, 1970,5(3), 177-184.Richart, F. E. and Newmark, N. M., A hypothesis for thedetermination of cumulative damage in fatigue. Proceedings,American Society for Testing and Materials, 1948, 48, 767-800.Marco, S. M. and Starkey, W. L., A concept of fatigue damage.Transactions of the ASME, 1954, 76, 627-632.

24

25

26

27

Kommers, J. B., The effect of overstress in fatigue on theendurance life of steel. Proceedings, American Society for Testing and Materials, 1945, 45, 532-541.Bennett, J. A., A study of the damaging effect of fatiguestressing on X4130 steel. Proceedings, American Society forTesting and Materials, 1946, 46, 693-714.Henry, D. L., A theory of fatigue damage accumulation in steel.Transactions of the ASME, 1955, 77, 913-918.Gatts, R R, Application of a cumulative damage concept tofatigue. ASME Journal of Basic Engineering, 1961,83,529-540.


Table 2 Summary of cumulative fatigue damage theories: DCA, refined DLDR and DDCA

Model Model Year Physical Expression Characteristicsb Ref.developer basisa

Damage Manson 1981 Effective D = '2.rq with q = (N/N,)i3 and f3 =0.4 nLDE, LLD, nLlA, 44,12curve and microcrack nSC, some Appl,approach Halford growth (PHE) G(DCM

Refined Manson 1981 Based on D, ='2.(nr/N,); N] =N - NIl B =0.65 Two-stage LDE, 44, 12DLDRc and DCA and

Dn ='2.(nnINn); Nn =BN(N/N)u 0' =0.25LLD, nSC, nLlA,

Halford linearization many Appl, S(EMP)

Double Manson 1986 Based on D = '2.[(pr;)k + (I - Pk)rkq ] Ilk, A =0.35, k =5, nLDE, LLD, nLlA, 12damage curve and both DCA nSC, some Appl,approach Halford and refined P =A(N/N;)U/[I - B(N/N;)U], B =0.65, 0' =0.25, f3 =0.4 C(DDCA)C DLDR

(EMP)

aCON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LlA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'cThe constants were obtained based on experiments with Maraging 300CVM steel, SAE 4130 steel, and Ti-6AI-4V alloy

Table 3 Summary of cumulative fatigue damage theories: hybrid theories

Model

Stressversion"

Model Yeardeveloper

Bui-Quoc et 1971al.

Physical basisa

Hybridization,endurance limitchange (sANA)

Expression

1 - Ye ~ r;D=--=.:,

1 - Yec (. (1 _ .) Y; - (Yhu)m)r, + r, 1

Y; -

m=8

Characteristicsb

nLDE, LLD, nLlA,SC, some Appl, C

Ref.

209

Strainversionc

Fictitiousloadmodificationc

Cycle-ratiomodification"

Bui-Quoc etal.

Bui-Quoc

Bui-Quoc

1971

1981

1982

Transplanted from 1 - Ae r;the stress version D =~ = '2. A. _ (A/A)

(sANA) ec. + (1 _ .)' 'f mr, r, A. - 1,m = 8

Endurance limit r;change, to account D = I -------=-------,. I d" A/ - (A/ /Af)mJ or oa mteractIon ri + (l - r;) \ .' _ 1

effects (sANA) /\,

m=8

Endurance limit r"change, to account D = I A. - (A/A)for load interaction r•. + (l - rJ' ,t m

effects (sANA) A; - I

m=8

nLDE, LLD, nLlA,SC, some Appl, C

nLDE, LLD, LlA,SC, few Appl, C

nLDE, LLD, LlA,SC, some Appl, C

52

51,55

55, 56

aCON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LlA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'cThe constant m = 8 was determined from experiments with A-201 and A-517 steels

28

29

30

31

Gatts, R. R., Cumulative fatigue damage with random loading.ASME Journal of Basic Engineering, 1962, 84, 403-409.Bluhm, J. 1., A note on fatigue damage. Materials Researchand Standards, 1962.Corten, H. T. and Dolon, T. J., Cumulative fatigue damage. InProceedings of the International Conference on Fatigue of Metals. Institution of Mechanical Engineering and American Societyof Mechanical Engineers, 1956, pp. 235-246.Freudenthal, A. M. and Heller, R. A., On stress interaction infatigue and a cumulative damage rule. Journal of the AerospaceSciences, 1959, 26(7), 431-442.

32

33

34

Freudenthal, A. M., Physical and statistical aspects of cumulativedamage. In Colloquium on Fatigue, Stockholm, May 1955.Springer-Verlag, Berlin, 1956, pp. 53--62.Spitzer, R. and Corten, H. T., Effect of loading sequence oncumulative fatigue damage of7071-T6 aluminum alloy. Proceedings, American Society for Testing and Materials, 1961, 61,719-731.Manson, S. S., Nachigall, A. J. and Freche, J. c., A proposednew relation for cumulative fatigue damage in bending. Proceedings, American Society for Testing and Materials, 1961, 61,679-703.


Table 4 Summary of cumulative fatigue damage theories: recent theories based on crack growth

Model Model Year Physical basis' Expression Characteristicsb Ref.developer

Double Miller and 1977 Two-stage crack (r l + r2 - I) nLDE, LLD, 91exponential Zachariah growth (PHE) Nt. 1 =Nt, 1 r

2nLlA, nSC, a

law few Appl, C(1 st version)

Double Ibrahim 1980 Two-stage crack ( I ) (at I) (aD) nLDE, LLD, 92exponential and Miller 1981 growth (PHE & r2 = (1 - r t)~ In -' lin ---: nLlA, nSC, 93law EMP) t, t at at some Appl, C(2nd version) D =~ =( ~r-r)/(I - r,)

af af

Short crack Miller 1982 MSC, PSC, da Clear physical 95-99theory E-P fracture dN =A(dy)"(d - a) for MSCs: aD oS a oS at basis, it is

mechanics difficult to(sANA) da determine the

dN =B(dy)f3a - C for PSCs: at oS a oS atmicro-

parametersinvolved

Ma-Laird Ma and 1989 Crack D ="i.(P/Pcd,) =K"i.n,[(dy/2);u; - (dy/2)tlmit] LlA, few Appl, 104theory Laird population not universal, G

(PHE)

Vasek-Polak Vasek and 1991 Microcrack D =2Dcr for initiation: D oS r oS 1/2 Two-stage 105model Polak kinetics

Dc .nLDE, LLD,

equivalent crack D =Dc + - [e m (2r - I) - I] for propagatIOn: 1/2 oS r oS I nLlA, nSC, fewlength (PHE & m Appl, C

sANA)

"CON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LlA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix on' stands for 'not' or 'non'

Table 5 Summary of cumulative fatigue damage theories: models based on modifying life-curve

Model Modeldeveloper

Year Physical basis' Expression Characteristicsb Ref.

Subramanyan Subramanyan 1976 Convergence to the r; = I - {r; - t + [r; - 2 + ... + (r2 + ri)",. .. ]"; _ 2J "ha; _ I LDE, LlA, nSC, 106model knee-point (CON) some Appl, G

Hashin- Hashin and 1978 Two types of Formulation based on static strength point, and LDE, LlA, nSC, 107Rotem theory Rotem convergence formulation based on endurance limit point some Appl, G

(CON)

Bound theory Ben-Amoz 1990 Upper and lower Bounds formed by Miner rule and Subramanyan LDE, LlA, nSC, 109-11 Ibounds of model; bounds formed by DLDR and Subramanyan few Appl, C

convergence lines model; and statistical bounds(sANA)

Leipholz's Leipholz 1985 Experimental Modified life curve is obtained from repeated multi- LDE, LlA, SC, 112, I 13approach determination level block tests a few Appl, G,

(EXP) E

'CON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LlA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix on' stands for 'not' or 'non'

35

36

37

Manson, S. S., Nachigall, A. J., Ensign, C. R. and Freche, J.C., Further investigation of a relation for cumulative fatiguedamage in bending. ASME Journal of Engineering for Industry,1965, 87, 25-35.Grover, H. J" An observation concerning the cycle ratio incumulative damage. In Symposium on Fatigue of Aircraft Structures, ASTM STP 274. American Society for Testing andMaterials, Philadelphia, PA, 1960, pp. 120-124.Manson, S. S., Interfaces between fatigue, creep, and fracture.International Journal of Fracture Mechanics, 1966, 2, 328-363.

38

39

40

Manson, S. S., Freche, J. C. and Ensign, S. R., Application ofa double linear damage rule to cumulative fatigue, In FatigueCrack Propagation, ASTM STP 415. American Society forTesting and Materials, Philadelphia, PA, 1967, pp. 384--412.Bilir, O. G" Experimental investigation of fatigue damageaccumulation in 1100 Al alloy. International Journal of Fatigue,1991, 13(1), 3-6.Shanley, F. R., A theory of fatigue based on unbonding duringreversed slip, Report P-350. The Rand Corporation, Santa Monica, 1952.


Table 6 Summary of cumulative fatigue damage theories: energy-based damage theories

Model Model Year Physical basisa Expression Characteristicsb Ref.developer

Plastic strain Kujuwski and 1984 Convergence, In the plastic strain energy vs life diagram, isodamage LDE, LIA, SC, 124energy Ellyin plastic strain curves converge to the apparent fatigue limit, rather few Appl, G, E(hysteresis energy, (CON) than to the original limitenergy)

Total strain Golos and 1987 Convergence, In the total strain energy vs life diagram, isodamage LDE, LIA, SC, 11,126,127energy Ellyin total strain curves converge to the apparent fatigue limit, rather some Appl, G,

energy, (sANA) than to the original limit E

Bui-Quoc model Bui-Quoc 1973 Constant total k 2K'M", + IkriNf~ - C(n' + I) Energy version

137plastic energy at W, = njLlWj =~-I-

failure (CON) of LDR, fewAppl, G

Radhakrishnan Radhakrishnan 1978 Crack growth "'il Wfj Another energy 138approach rate is related to r", = I W rj version of LDR,

plastic energy i= I fm few Appl, G(CON)

Kliman Theory Kliman 1984 Block spectrum, Wb I Also energy 140similar basis to Db =- =- kLlWjnbj version of LDR,

the aboveWfR WfR

a few Appl, G(CON)

Niu theory Niu et ai, 1987 Strain hardening D =<p 1/[(n' + a)(l + 13)] = !,r}/(n' + a) nLDE, LLD, 135, 136and plastic LIA, SC, some

strain energy Appl, C(sANA)

Leis model Leis 1988 Related to two 4u' nLDE, LLD, 133exponents in D =_f (2Nj )2b] + 4ur E/(2Nf )h] + c, LIA, SC, nAppi,E

strain-life Gequation (CON)

aCON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LIA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'

41

42

43

44

45

46

47

48

49

50

Valluri, S. R, A unified engineering theory of high stress levelfatigue. Aerospace Engineering, 1961, 20, 18-19.Valluri, S. R, A theory of cumulative damage in fatigue. ReportNo. ARL 182, Aeronautical Research Laboratory, Office ofAerospace Research, United States Air Force, 1961.Scharton, T. D. and Crandall, S. H., Fatigue failure undercomplex stress histories, ASME Journal of Basic Engineering,1966, 88(1), 247-251.Manson, S. S. and Halford, G. R, Practical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damage. International Journal ofFracture, 1981, 17(2), 169-192.Shi, Z., Wang, D, and Xu, H., Two-stage fatigue damagecumulative rule. International Journal of Fatigue, 1992, 14(4),395-398.Manson, S. S. and Halford, G. R, Complexities of high-temperature metal fatigue: some steps toward understanding. IsraelJournal of Technology, 1983, 21, 29-53.Bizon, P. T., Thoma, D. J. and Halford, G. R, Interaction ofhigh cycle and low cycle fatigue of Haynes 188 at 1400 F. InStructure Integrity and Durability of Reusable Space PropulsionSystems, NASA CP-2381. NASA Lewis Research Center, Cleveland,OH, 1985, pp. 129-138.Halford, G. R and Manson, S. S., Reexamination of cumulativefatigue damage laws. In Structure Integrity and Durability ofReusable Space Propulsion Systems, NASA CP-2381. NASA,1985, pp. 139-145.McGaw, M. A., Cumulative fatigue damage models. In LewisStructures Technology, Vol. 3, Structural Integrity Fatigue andFracture Wind Turbines HOST, Proceedings of an Expositionand Symposium of Structures Technology, NASA LewisResearch Center, Cleveland, OH, 1988, pp. 201-211.Heidmann, K. R, Technology for predicting the fatigue life of

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gray cast iron. PhD dissertation, Case Western Reserve University, Cleveland, OH, 1985.Bui-Quoc, T., An interaction effect consideration in cumulativedamage on a mild steel under torsion loading. Proceedings ofthe 5th International Conference on Fracture, Pergamon Press1981, 5, 2625-2633.Bui-Quoc, T., Dubuc, J., Bazergui, A. and Biron, A., Cumulativefatigue damage under strain controlled conditions. Journal ofMaterials, 1971,6(3),718-737.Dubuc, J., Bui-Quoc, T., Bazergui, A. and Biron, A., Unifiedtheory of cumulative damage in metal fatigue. W.R. C. Bulletin,1971, 162, 1-20.Bui-Quoc, T., Damage cumulatif en fatigue. In Fatigue desMateriaux et des Structures, Maloine, Paris, 1980, pp. 313-342(in French).Bui-Quoc, T., Cumulative damage with interaction effect due tofatigue under torsion loading. Experimental Mechanics, 1982,22, 180-187.Bui-Quoc, T., A simplified model for cumulative fatigue damagewith interaction effects. In Proceedings of the 1982 Joint Conference on Experimental Mechanics, Society for ExperimentalStress Analysis, Brookfield Center, CT, 1982, pp. 144-149.Biron, A. and Bui-Quoc, T., Cumulative damage concepts withinteraction effect consideration for fatigue or creep; a perspective. In Transactions of the 6th International Conference onStructural Mechanical Reaction Technology, Paris, France, 1981,L9/1.1-7.Bernard-Connolly, M., Bui-Quoc, T. and Biron, A., Multilevelstrain controlled fatigue on a type 304 stainless steel. ASMEJournal of Engineering Materials and Technology, 1983, 105,188-194.Bui-Quoc, T., High-temperature fatigue-life estimation: extension


Table 7 Summary of cumulative fatigue damage theories: continuum damage mechanics approaches

Model Model Year Physical basisa Expression Characteristicsb Ref.developer

Chaboche model Chaboche 1974 All these D = I - [I - r'/(l- alj'/!] +(3) All are 158models are in similar

Lemaitre- Lemaitre and 1979 based on the D = I - (I - r) 'II] + p) characteristics 163Plumtree model Plumtree effective stress such as nLDE,

1990 concept in D ="kr;'l(l - a) LLD, potential 145,162Lemaitre- Lemaitre and CDM. The to account forChaboche model Chaboche 1992 differences are D = I - (I - r)]/I] +n(Rl] SC and LIA, 164

the number of increasing Appl,Wang model Wang 1990 variables D =Dc - (Dc - Do)(I - r)'-(3 Convenience 165

assumed in the depends on theWang-Lou Wang and damage rate parameters usedmodel Lou equation and

boundaryconditions

Li-Qian-Li Li et al. 1989 Besides above, A Similar 166model a dislocation D = I - (fa)p characteristics to

variable is above, nAppI, Cinvolved (ANA)

Three- Chow and 1991 CDM approach A damage effective tensor was introduced and a Similar 171dimensional Wei in three- generalized three-dimensional isotropic CDM model characteristics,CDM model dimensional was proposed based on effective stress concept nAppI, very C

space (ANA)

aCON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analyticalbLDE, linear damage evolution; LLD, load level dependent; LIA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'

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of a unified theory. Experimental Mechanics, 1975, 15(6),219-225.Bui-Quoc, T., An engineering approach for cumulative damagein metals under creep loading. ASME Journal of EngineeringMaterials and Technology, 1979, 101, 337-343.Bui-Quoc, T. and Biron, A., Cumulative damage with interactioneffect due to creep on a Cr-Mo-V steel. In International Conference on Engineering Aspects of Creep, Vol. I. MechanicalEngineering, London, UK, 1980, pp. 121-125.Bui-Quoc, T. and Biron, A., Interaction effect consideration incumulative damage in metals due to creep. Journal of Mechanical Engineering Science, 1981, 23(6), 281-288.Bui-Quoc, T., Evaluation of creep-rupture data using a newapproach. In Transactions of the 6th International Conferenceon Structural Mechanical Reaction Technology, Paris, France,1981, L7/1.1-8.Bui-Quoc, T., Recent developments of damage concepts appliedto creep fatigue combination. In The 3rd International Seminaron Inelastic Analysis and Life Prediction in High TemperatureEnvironment, Paris, 1981, Paper B3.1.Bui-Quoc, T., Recent developments of continuous damageapproaches for the analysis of material behavior under creepfatigue loading. ASME PVP-59. ASME, 1982, pp. 221-226.Bui-Quoc, T. and Biron, A., A phenomenological approach forthe analysis of combined fatigue and creep. Nuclear EngineeringDesigns, 1982, 71(1), 89-102.Bui-Quoc, T. and Gomuc, R. A damage approach for analyzingthe combining effect under creep-fatigue loading. In Advances inLife Prediction Methods, Proceedings of the ASME InternationalConference, ed. D. A. Woodford and J. R. Whitehead. TheAmerican Society of Mechanical Engineers, New York, NY,1983, pp. 105-113.Gomuc, R. and Bui-Quoc, T., An analysis of the fatigue/creepbehavior of 304 stainless steel using a continuous damageapproach. ASME Journal of Pressure Vessel Technology, 1986,108, 280-288.Gomuc, R., Bui-Quoc, T. and Biron, A., Evaluation of thecreep-fatigue behavior of 2.25 Cr-I Mo steel by a continuousdamage approach. Res Mechanica, 1987, 21(2), 135-154.Bui-Quoc, T., Gomuc, R., Biron, A., Nguyen, H. L. and Masounave, 1.. Elevated temperature fatigue---{;reep behavior of nickelbase superalloy IN 625. In Low Cycle Fatigue, ed. H. D.

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Solomon, G. R. Halford, L. R. Kalsand and B. N. Leis, ASTMSTP 942. American Society for Testing and Materials, Philadelphia, PA, 1988, pp. 470-486.Gomuc, R., Bui-Quoc, T., Biron, A. and Bernard, M., Analysisof type 316 stainless steel behavior under fatigue, creep andcombined fatigue---{;reep loading. ASME Journal of PressureVessel Technology, 1990, 112(3), 240-250.Bui-Quoc, T., Choquet, 1. A. and Biron, A., Cumulative fatiguedamage on large steel specimens under axial programmed loading with nonzero mean stress. ASME Journal of EngineeringMaterials and Technology, 1976, 98, 249-255.Zhang, A., Bui-Quoc, T. and Gomuc, R., A procedure for lowcycle fatigue life prediction for various temperatures and strainrates. ASME Journal of Engineering Materials and Technology,1990, 112, 422-428.Magnin, T. and Ramade, C, The effect of surface microcrackson the cumulative damage during cyclic deformation. In Strengthof Metals and Alloys (ICSMA 8), Proceedings of the 8th International Conference on the Strength of Metals and Alloys,Pergamon Press, Oxford, 1989, pp. 731-735.Wheeler, O. E., Spectrum loading and crack growth. ASMEJournal of Basic Engineering, 1972, D94(1), 181-186.Willenborg, J., Engle, R. M. and Wood, H. A., A crack growthretardation model using an effective stress concept, AFFDL TM71-I-FBR, 1971.Elber, W., Fatigue crack closure under cyclic tension. Engineering Fracture Mechanics, 1970, 2, 37-45.Elber, W., The significance of fatigue crack closure. In DamageTolerance in Aircraft Structures, ASTM STP 486. AmericanSociety for Testing and Materials, Philadelphia, PA, 1971, pp.230-242.Newman, J. C. Jr, A crack closure model for predicting fatiguecrack growth under aircraft spectrum loading. In Methods andModels for Predicting Fatigue Crack Growth under RandomLoading, ASTM STP 748. American Society for Testing andMaterials, Philadelphia, PA, 1981, pp. 53-84.Newman, J. C Jr, Prediction of fatigue crack growth undervariable amplitude and spectrum loading using a closure model.In Design of Fatigue and Fracture Resistant Structures, ASTMSTP 761. American Society for Testing and Materials, Philadelphia, PA, 1982, pp. 255-277.Dill, H. D. and Saff, C. R., Spectrum crack growth prediction


Table 8 SUinmary of cumulative fatigue damage theories: other approaches

Model Model developer Year Physical basisa Expression Characteristicsb Ref.

Surface layer Kramer 1974 Surface layer D = L(erJer,*) LDE, LIA, SC, 172,173stress approach stress change some Appl, G

(PHE)

ES-Miner rule Ikai et al. 1989 Internal stress Applied stress can be resolved into internal and LDE, LIA, SC, 180-183and effective effective stresses. The internal stress is few Appl, C, Estress evolutions representative of the fatigue resistance of a material(sANA) while the effective stress is responsible for the

fatigue damage

Overload Topper et al. 1990 Crack opening D = LDo' + LD" + LD;nt LLD, LIA, SC, 191,192damage model and closure some Appl, G,

(PHE) E

Plastic strain Azari et al. 1984 Plastic strain L( .iEp - .iEpo ) nLDE, LLD, 194evolution model evolution and D= nLIA, nSC, few

accumulation.iEpf - .iEpn ;- l/e

Appl, G(PHE)

Fong theory Fong 1982 Assuming a D = L(ekr; - I)/(ek - 1) for ki=O nLDE, LLD, 195linear damage few Appl, Grate (CON)

Landgraf model Landgraf 1973 Strain version1 [er,.' - erm (.iEp erm ) ]

LDE, nLIA, 196of LDR Dlreversal = - = - +~ nSC, errninvolving mean

2Nf E,.'E .iEe ert l/(b - c)involved, few

stress (CON) Appl, G

Plastic work Kurath et al. 1984 Plastic work, ±2n (.ier)LDE, LIA, nSC, 198

based damage LDR, load D- -~, -' few Appl, Gmodel interaction b - ,~ ,(2Nr), .ierh lid

(CON)

Unified Pasic 1992 CDM and This is an approach combining fracture mechanics Only conceptual 201approach fracture with damage mechanics nAppI, C

mechanics(ANA)

PSB version of Cordero etal. 1988 Persistent slip L(n/Nr;) = DS = I + DiD I LDE, nLLD, 202LDR band density few Appl, C

(PHE)

Micro-damage Inoue et al. 1987 Involving PSBD(N, n) = ['¥*(N)/,¥*(N)rnax](nIN,) - d1m

nLDE, LLD, 203mechanics parameter few Appl, Cmodel (ANA)

A model based Abuelfoutouh 1989 Change in dX/dN = ± J] + bexP(CX + d) nLDE, LLD, 204on resistance-to- and Halford resistance-to- LIA, SC, fewflow flow (ANA) Appl, C

A correction Buch et al. 1982 Using correction N" = Ncai'(Ncxp'INcal') = Neate Relying on 205,206approach factor C experiment,

some Appl

aCON, conceptual; PHE, phenomenological; EXP, experimental; EMP, empirical; ANA, analytical; sANA, semi analytical"LDE, linear damage evolution; LLD, load level dependent; LIA, load interaction accountable; SC, small amplitude cycle damage accountable;Appl, application(s); S, simple; G, general; C, complicated; the suffix 'n' stands for 'not' or 'non'

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methods based on crack surface displacement and contact analyses. In Fatigue Crack Growth under Spectrum Loads, ASTMSTP 595. American Society for Testing and Materials, Philadelphia, PA, 1976, pp. 306-319.Dill, H. D., Saff, C. R. and Potter, J. M., Effect of FighterAttack Spectrum on Crack Growth, ASTM STP 714. AmericanSociety for Testing and Materials, Philadelphia, PA, 1980, pp.205-217.Fuhring, H. and Seeger, T., Dugdale crack closure analysis offatigue cracks under constant amplitude loading. EngineeringFracture Mechanics, 1979, 11, 99-122.de Koning, A. U., A simple crack closure model for predictionof fatigue crack growth rates under variable amplitude loading.In Fracture Mechanics: Thirteenth Conference, ASTM STP 743.

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American Society for Testing and Materials, Philadelphia, PA,1981, pp. 63-85.Ritchie, R. 0., Suresh, S. and Moses, C. M., Near thresholdfatigue crack growth in 21/4 Cr-I Mo pressure vessel steel inair and hydrogen. ASME Journal of Engineering Materials andTechnology, 1980, 102, 293-299.Ritchie, R. O. and Suresh, S., Some considerations on fatiguecrack closure at near-threshold stress intensities due to fracturesurface morphology. Metallurgical Transactions, 1981, 13A,937-940.Suresh, S. and Ritchie, R. 0., A geometrical model for fatiguecrack closure induced by fracture surface morphology. Metallurgical Transactions, 1982, 13A, 1627-1631.Suresh, S. and Ritchie, R. 0., Near-threshold fatigue crack


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