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..