Int. 1. Fatigue Vol. 20, No. I, pp. 9-34, 1998 1998 Elsevier Science Ltd. All rights reservedPrinted in Great Britain0142-1123/98/$19.00+.00PU: 80142-1123(97)00081-9Cumulative fatigue damage and lifeprediction theories: a survey of the state ofthe art for homogeneous materialsA. 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 interac-tion effects; fatigue life predictions)INTRODUCTIONFatigue damage increases with applied cycles in acumulative manner which may lead to fracture. Cumu-lative fatigue damage is an old, but not yet resolvedproblem, More than seventy years ago, Palmgren1 sug-gested the concept which is now known as the 'linearrule'. In 1945, Miner2first 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. Newmark3in 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 Morrow4presented a*Author for correspondence.9review 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 CooklOand 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 over-view 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 ana-lytical. Several researchers4- 9have reviewed thetheories 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 in10 A. Fatemi and L. Yangdamage assessment of inhomogeneous materials. Itshould also be noted that this review paper deals withdamage rules and life prediction aspects of cumulativefatigue damage. Another review paper13provides acomprehensive overview of cumulative fatigue damagemechanisms and quantifying parameters.WORK BEFORE 1970sThe 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; S-N curve modification approach; two-stage damageapproach; and crack growth-based approach.Three early conceptsThe history of fatigue damage modeling can bedated back to 1920s and 1930s. It was Palmgren1whofirst introduced the concept of linear summation offatigue damage in 1924. French14first reported thesignificant investigation of the overstress effect onendurance limit in 1933. In 1938, Kommers15suggestedusing the change in the endurance limit as a damagemeasure. In 1937, Langer16first 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 two-stage damage process) laid the foundation for phenom-enological cumulative fatigue damage models.Linear damage rulesMiner2first represented the Palmgren linear damageconcept 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 load-ing 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-interac-tion accountability. In 1949, Machlin17proposed ametallurgically based cumulative damage theory, whichis basically another form of LDR. In 1950s, Coffinand co-workers18,19 expressed the LDR in terms ofplastic strain range, which is related to fatigue lifethrough the Coffin-Manson relation. In a later study,Topper and Biggs20used the strain-based LDR tocorrelate their experimental results. A review on theapplications of the LDR to strain-controlled fatiguedamage analysis was given by Miller21in 1970. How-ever, due to the inherent deficiencies of the LDR, nomatter which version is used, life prediction based onthis rule is often unsatisfactory. Experimental evidenceunder completely reversed loading condition often iridi-cates that 'ir; > 1 for a low-to-high (L-H) loadingsequence, and 'ir; < 1 for a high-to-low (H-L) load-ing sequence.Marco-Starkey theoryTo remedy the deficiencies associated with the LDR,Richart and Newmark22introduced the concept of dam-age 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 influ-ence on subsequent cumulative fatigue damageresearch. Kommers24and Bennett25further investigatedthe effect of fatigue prestressing on endurance proper-ties 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 toFOR OPERATION AT CTtFOLLOWED IY OPERATION AT CT3"" nL If (AI + CD I ( Ii I Z IoFOR OPERATION AT CT3FOLLOWED BY OPERATION AT CT, LN' . I, l,Z A E C 0CYCLEFigure 1 Schematic representation of damage vs cycle ratio for theMarco-Starkey theory23Cumulative fatigue damage and life prediction theories 11(b) H-L load sequenceFigure 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 sequenceNt N,Fatigue life, cyclesACTUAL "2ACTUAL nz"2 BY MINERN1 NtFatigue life, cyclesII"2 BYMINER,II0"0::::l....CT,.-.-c..8~CJ:ICJ:I0.......CJ:I"00.-.-c..c..