Review Article A Review on Fatigue Life Prediction Methods for...

Review ArticleA Review on Fatigue Life Prediction Methods for Metals

E Santecchia1 A M S Hamouda1 F Musharavati1 E Zalnezhad2 M Cabibbo3

M El Mehtedi3 and S Spigarelli3

1Mechanical and Industrial Engineering Department College of Engineering Qatar University Doha 2713 Qatar2Department of Mechanical Engineering Hanyang University 222 Wangsimni-ro Seongdong-gu Seoul 133-791 Republic of Korea3Dipartimento di Ingegneria Industriale e Scienze Matematiche (DIISM) Universita Politecnica delle Marche 60131 Ancona Italy

Correspondence should be addressed to E Zalnezhad erfan zalnezhadyahoocom

Received 19 June 2016 Accepted 17 August 2016

Academic Editor Philip Eisenlohr

Copyright copy 2016 E Santecchia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Metallic materials are extensively used in engineering structures and fatigue failure is one of the most common failure modes ofmetal structures Fatigue phenomena occur when a material is subjected to fluctuating stresses and strains which lead to failuredue to damage accumulation Different methods including the Palmgren-Miner linear damage rule- (LDR-) based multiaxialand variable amplitude loading stochastic-based energy-based and continuum damage mechanics methods forecast fatiguelife This paper reviews fatigue life prediction techniques for metallic materials An ideal fatigue life prediction model shouldinclude the main features of those already established methods and its implementation in simulation systems could help engineersand scientists in different applications In conclusion LDR-based multiaxial and variable amplitude loading stochastic-basedcontinuum damage mechanics and energy-based methods are easy realistic microstructure dependent well timed and damageconnected respectively for the ideal prediction model

1 Introduction

Avoiding or rather delaying the failure of any componentsubjected to cyclic loadings is a crucial issue that must beaddressed during preliminary design In order to have a fullpicture of the situation further attention must be given alsoto processing parameters given the strong influence thatthey have on the microstructure of the cast materials andtherefore on their properties

Fatigue damage is among themajor issues in engineeringbecause it increases with the number of applied loading cyclesin a cumulative manner and can lead to fracture and failureof the considered part Therefore the prediction of fatiguelife has an outstanding importance that must be consideredduring the design step of a mechanical component [1]

The fatigue life prediction methods can be divided intotwo main groups according to the particular approach usedThe first group is made up of models based on the predictionof crack nucleation using a combination of damage evolutionrule and criteria based on stressstrain of components The

key point of this approach is the lack of dependence fromloading and specimen geometry being the fatigue life deter-mined only by a stressstrain criterion [2]

The approach of the second group is based instead oncontinuumdamagemechanics (CDM) inwhich fatigue life ispredicted computing a damage parameter cycle by cycle [3]

Generally the life prediction of elements subjected tofatigue is based on the ldquosafe-liferdquo approach [4] coupledwith the rules of linear cumulative damage (Palmgren [5]and Miner [2]) Indeed the so-called Palmgren-Miner lineardamage rule (LDR) is widely applied owing to its intrinsicsimplicity but it also has some major drawbacks that need tobe considered [6] Moreover some metallic materials exhibithighly nonlinear fatigue damage evolution which is loaddependent and is totally neglected by the linear damage rule[7]Themajor assumption of theMiner rule is to consider thefatigue limit as a material constant while a number of studiesshowed its load amplitude-sequence dependence [8ndash10]

Various other theories and models have been developedin order to predict the fatigue life of loaded structures [11ndash24]

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 9573524 26 pageshttpdxdoiorg10115520169573524

2 Advances in Materials Science and Engineering

Among all the available techniques periodic in situ mea-surements have been proposed in order to calculate themacrocrack initiation probability [25]

The limitations of fracture mechanics motivated thedevelopment of local approaches based on continuum dam-age mechanics (CDM) for micromechanics models [26] Theadvantages of CDM lie in the effects that the presence ofmicrostructural defects (voids discontinuities and inhomo-geneities) has on key quantities that can be observed andmeasured at the macroscopic level (ie Poissonrsquos ratio andstiffness) From a life prediction point of view CDM is partic-ularly useful in order to model the accumulation of damagein amaterial prior to the formation of a detectable defect (ega crack) [27]TheCDM approach has been further developedby Lemaitre [28 29] Later on the thermodynamics ofirreversible process provided the necessary scientific basisto justify CDM as a theory [30] and in the framework ofinternal variable theory of thermodynamics ChandrakanthandPandey [31] developed an isotropic ductile plastic damagemodel De Jesus et al [32] formulated a fatigue modelinvolving a CDM approach based on an explicit definitionof fatigue damage while Xiao et al [33] predicted high-cyclefatigue life implementing a thermodynamics-based CDMmodel

Bhattacharya and Ellingwood [26] predicted the crackinitiation life for strain-controlled fatigue loading using athermodynamics-based CDM model where the equations ofdamage growthwere expressed in terms of theHelmholtz freeenergy

Based on the characteristics of the fatigue damage somenonlinear damage cumulative theories continuum damagemechanics approaches and energy-based damage methodshave been proposed and developed [11ndash14 21ndash23 34 35]

Given the strict connection between the hysteresis energyand the fatigue behavior of materials expressed firstly byInglis [36] energy methods were developed for fatigue lifeprediction using strain energy (plastic energy elastic energyor the summation of both) as the key damage parameteraccounting for load sequence and cumulative damage [18 37ndash45]

Lately using statistical methods Makkonen [19] pro-posed a new way to build design curves in order to studythe crack initiation and to get a fatigue life estimation for anymaterial

A very interesting fatigue life prediction approach basedon fracture mechanics methods has been proposed by Ghi-dini and Dalle Donne [46] In this work they demonstratedthat using widespread aerospace fracture mechanics-basedpackages it is possible to get a good prediction on the fatiguelife of pristine precorroded base and friction stir weldedspecimens even under variable amplitude loads and residualstresses conditions [46]

In the present review paper various prediction methodsdeveloped so far are discussed Particular emphasis will begiven to the prediction of the crack initiation and growthstages having a key role in the overall fatigue life predic-tion The theories of damage accumulation and continuumdamagemechanics are explained and the predictionmethodsbased on these two approaches are discussed in detail

2 Prediction Methods

According to Makkonen [19] the total fatigue life of acomponent can be divided into three phases (i) crackinitiation (ii) stable crack growth and (iii) unstable crackgrowth Crack initiation accounts for approximately 40ndash90of the total fatigue life being the phase with the longesttime duration [19] Crack initiation may stop at barriers (eggrain boundaries) for a long time sometimes the cracks stopcompletely at this level and they never reach the critical sizeleading to the stable growth

A power law formulated by Paris and Erdogan [47] iscommonly used to model the stable fatigue crack growth

119889119886

119889119873= 119862 sdot Δ119870

119898

(1)

and the fatigue life 119873 is obtained from the following integra-tion

119873 = int

119886119891

119886119894

119889119886

119862 sdot (Δ119870)119898 (2)

where Δ119870 is the stress intensity factor range while 119862 and 119898

are material-related constants The integration limits 119886119894and

119886119891correspond to the initial and final fatigue crack lengthsAccording to the elastic-plastic fracture mechanics

(EPFM) [48ndash50] the crack propagation theory can beexpressed as

119889119886

119889119873= 119862

1015840

Δ1198691198981015840

(3)

where Δ119869 is the 119869 integral range corresponding to (1) while1198621015840 and 119898

1015840 are constantsA generalization of the Paris law has been recently

proposed by Pugno et al [51] where an instantaneous crackpropagation rate is obtained by an interpolating procedurewhich works on the integrated form of the crack propagationlaw (in terms of 119878-119873 curve) in the imposed condition ofconsistency with Wohlerrsquos law [52] for uncracked material[51 53]

21 Prediction of Fatigue Crack Initiation Fatigue cracks havebeen a matter of research for a long time [54] Hachim etal [55] addressed the maintenance planning issue for a steelS355 structure predicting the number of priming cycles ofa fatigue crack The probabilistic analysis of failure showedthat the priming stage or rather the crack initiation stageaccounts formore than 90of piece lifeMoreover the resultsshowed that the propagation phase could be neglected whena large number of testing cycles are performed [55]

Tanaka and Mura [56] were pioneers concerning thestudy of fatigue crack initiation in ductile materials usingthe concept of slip plastic flow The crack starts to formwhen the surface energy and the stored energy (given by thedislocations accumulations) become equal thus turning thedislocation dipoles layers into a free surface [56 57]

In an additional paper [58] the same authors modeledthe fatigue crack initiation by classifying cracks at first as

Advances in Materials Science and Engineering 3

(i) crack initiation from inclusions (type A) (ii) inclusioncracking by impinging slip bands (type B) and (iii) slip bandcrack emanating from an uncracked inclusion (type C) Thetype A crack initiation from a completely debonded inclusionwas treated like crack initiation from a void (notch) Theinitiation of type B cracks at the matrix-particle interface isdue to the impingement of slip bands on the particles butonly on those having a smaller size compared to the slip bandwidth This effect inhibits the dislocations movements Thefatigue crack is generated when the dislocation dipoles get toa level of self-strain energy corresponding to a critical valueFor crack initiation along a slip band the dislocation dipoleaccumulation can be described as follows

(Δ120591 minus 2119896)11987312

1= [

8120583119882119904

120587119889]

12

(4)

where119882119904is the specific fracture energy per unit area along the

slip band 119896 is the friction stress of dislocation 120583 is the shearmodulus Δ120591 is the shear stress range and 119889 is the grain size[58] Type C was approximated by the problem of dislocationpile-up under the stress distribution in a homogeneousinfinite plane Type A mechanism was reported in highstrength steels while the other two were observed in highstrength aluminum alloys The quantitative relations derivedbyTanaka andMura [58] correlated the properties ofmatricesand inclusions as well as the size of the latter with the fatiguestrength decrease at a given crack initiation life and with thereduction of the crack initiation life at a given constant rangeof the applied stress [58]

Dang-Van [59] also considered the local plastic flow asessential for the crack initiation and he attempted to give anew approach in order to quantify the fatigue crack initiation[59]

Mura and Nakasone [60] expanded Dang-Vanrsquos workto calculate the Gibbs free energy change for fatigue cracknucleation from piled up dislocation dipoles

Assuming that only a fraction of all the dislocations inthe slip band contributes to the crack initiation Chan [61]proposed a further evolution of this theory

Considering the criterion of minimum strain energyaccumulation within slip bands Venkataraman et al [62ndash64] generalized the dislocation dipole model and developeda stress-initiation life relation predicting a grain-size depen-dence which was in contrast with the Tanaka and Muratheory [56 58]

(Δ120591 minus 2119896)120572

119894= 037 (

120583119889

119890ℎ)(

120574119904

120583119889)

12

(5)

where 120574119904is the surface-energy term and 119890 is the slip-

irreversibility factor (0 lt 119890 lt 1) This highlighted the needto incorporate key parameters like crack and microstructuralsizes to getmore accuratemicrostructure-based fatigue crackinitiation models [61]

Other microstructure-based fatigue crack growthmodelswere developed and verified by Chan and coworkers [61 65ndash67]

Notch crack Short cracks

Fatiguelimit

Microstructurallyshort cracks

Physically smallcracks

EPFMtype

cracks

log crack length

d1 d2 d3

log

stres

s ran

geΔ

Zero

Crack speed

LEFMtype

cracks

Nonpropagating cracks

Figure 1 A modified Kitagawa-Takahashi Δ120590-119886 diagram showingboundaries betweenMSCs and PSCs and between EPFM cracks andLEFM cracks [69 71]

Concerning the metal fatigue after the investigation ofvery-short cracks behavior Miller and coworkers proposedthe immediate crack initiation model [68ndash71] The early twophases of the crack follow the elastoplastic fracturemechanics(EPFM) and were renamed as (i) microstructurally shortcrack (MSC) growth and (ii) physically small crack (PSC)growth Figure 1 shows the modified Kitagawa-Takahashidiagram highlighting the phase boundaries between MSCand PSC [69 71]

The crack dimension has been identified as a crucialfactor by a number of authors because short fatigue cracks(having a small length compared to the scale of local plasticityor to the key microstructural dimension or simply smallerthan 1-2mm) in metals grow at faster rate and lower nominalstress compared to large cracks [72 73]

211 Acoustic Second Harmonic Generation Kulkarni etal [25] proposed a probabilistic method to predict themacrocrack initiation due to fatigue damage Using acousticnonlinearity the damage prior to macrocrack initiation wasquantified and the data collected were then used to performa probabilistic analysis The probabilistic fatigue damageanalysis results from the combination of a suitable damageevolution equation and a procedure to calculate the proba-bility of a macrocrack initiation the Monte Carlo methodin this particular case Indeed when transmitting a singlefrequency wave through the specimen the distortion givenby the material nonlinearity generates second higher levelharmonics having amplitudes increasing with the materialnonlinearity As a result both the accumulated damageand material nonlinearity can be characterized by the ratio1198602119860

1 where 119860

2is the amplitude of the second harmonic

and 1198601is the amplitude of the fundamental one The ratio

is expected to increase with the progress of the damageaccumulation It is important to point out that this 119860

2119860

1

acoustic nonlinearity characterization [74] differs from theapproach given by Morris et al [75]

In the work of Ogi et al [74] two different signals weretransmitted separately into a specimen one at resonancefrequency 119891

119903and the other at half of this frequency (119891

1199032)


Macrocrack initiation5

4

3

2

1

00 1

A2A

1(10minus3)

NNf

Figure 2 Typical evolution of the ratio 1198602(119891

1199032)119860

1(119891

119903) for 025

mass C steel 119873119891= 56 000 [74]

The transmission of a signal at 119891119903frequency generates a mea-

sured amplitude 1198601(119891

119903) and while the signal is transmitted

at frequency 1198911199032 the amplitude 119860

2(119891

1199032) was received The

measurement of both signals ensures the higher accuracy ofthis method [74] Figure 2 shows that the 119860

2(119891

1199032)119860

1(119891

119903)

ratio increases nearly monotonically and at the point of themacrocrack initiation a distinct peak can be observed Thisresult suggests that the state of damage in a specimen duringfatigue tests can be tracked by measuring the ratio 119860

2119860

1

According to the model of Ogi et al [74] Kulkarni et al[25] showed that the scalar damage function can be written as119863(119873) designating the damage state in a sample at a particularfatigue cycle The value119863 = 0 corresponds to the no-damagesituation while 119863 = 1 denotes the appearance of the firstmacrocrackThe damage evolution with the number of cyclesis given by the following equation

119889119863

119889119873=

1

119873119888

(Δ1205902 minus 119903

119888(120590)

Δ1205902)

119898

1

(1 minus 119863)119899 (6)

When Δ1205902 is higher than the endurance limit (119903119888(120590))

otherwise the rate 119889119863119889119873 equals zero

212 Probability of Crack Initiation on Defects Melanderand Larsson [76] used the Poisson statistics to calculate theprobability119875

119909of a fatigue life smaller than119909 cycles Excluding

the probability of fatigue crack nucleation at inclusions 119875119909

can be written as

119875119909= 1 minus exp (minus120582

119909) (7)

where 120582119909is the number of inclusions per unit volume

Therefore (7) shows the probability to find at least oneinclusion in a unity volume that would lead to fatigue life nothigher than 119909 cycles

In order to calculate the probability of fatigue failure119875 deBussac and Lautridou [77] used a similar approach In theirmodel given a defect of size 119863 located in a volume adjacentto the surface the probability of fatigue crack initiation was

assumed to be equal to that of encountering a discontinuitywith the same dimension

119875 = 1 minus exp (minus119873120592119863) (8)

with 119873120592as the number of defects per unit volume having

diameter 119863 As in the model developed by Melander andLarsson [76] also in this case an equal crack initiation powerfor different defects having the same size is assumed [77 78]

In order to account for the fact that the fatigue crackinitiation can occur at the surface and inside a material deBussac [79] defined the probability to find discontinuities ofa given size at the surface or at the subsurfaceGiven a numberof load cycles 119873

0 the survival probability 119875 is determined

as the product of the survival probabilities of surface andsubsurface failures

119875 = [1 minus 119901119904(119863

119904)] [1 minus 119901

120592(119863

120592)] (9)

where 119863119904and 119863

120592are the diameters of the discontinuities

in surface and subsurface leading to 1198730loads life 119901

119904(119863

119904)

and 119901120592(119863

120592) are the probabilities of finding a defect larger

than 119863119904and 119863

120592at surface and subsurface respectively It

must be stressed that this model does not rely on the type ofdiscontinuity but only on its size [79]

Manonukul and Dunne [80] studied the fatigue crackinitiation in polycrystalline metals addressing the peculiar-ities of high-cycle fatigue (HCF) and low-cycle fatigue (LCF)The proposed approach for the prediction of fatigue cracksinitiation is based on the critical accumulated slip propertyof a material the key idea is that when the critical slip isachieved within the microstructure crack initiation shouldhave occurredThe authors developed a finite-element modelfor the nickel-based alloy C263 where using crystal plasticitya representative region of the material (containing about 60grains) was modeled taking into account only two materialsproperties (i) grain morphology and (ii) crystallographicorientation

The influence on the fatigue life due to the initial con-ditions of the specimens was studied deeply by Makkonen[81 82] who addressed in particular the size of the specimensand the notch size effects both in the case of steel as testingmaterial

The probability of crack initiation and propagation froman inclusion depends on its size and shape as well as on thespecimen size because it is easier to find a large inclusion ina big component rather than in a small specimen [77 79 83ndash85]

22 Fatigue Crack Growth Modeling The growth of a crackis the major manifestation of damage and is a complexphenomenon involving several processes such as (i) disloca-tion agglomeration (ii) subcell formation and (iii) multiplemicrocracks formation (independently growing up to theirconnection) and subsequent dominant crack formation [39]

The dimensions of cracks are crucial for modeling theirgrowth An engineering analysis is possible considering therelationships between the crack growth rates associated withthe stress intensity factor accounting for the stress conditions


at the crack tip Of particular interest is the behavior andmodeling of small and short cracks [4 86] in order todetermine the conditions leading to cracks growth up to alevel at which the linear elastic fracture mechanics (LEFM)theory becomes relevant Fatigue cracks can be classified asshort if one of their dimensions is large compared to themicrostructure while the small cracks have all dimensionssimilar or smaller than those of the largest microstructuralfeature [87]

Tanaka and Matsuoka studied the crack growth in anumber of steels and determined a proper best-fit relation forroom temperature growth conditions [88 89]

221 Deterministic Crack Growth Models While the Paris-Erdogan [48] model is valid only in the macrocrack rangea deterministic fatigue crack model can be obtained startingfrom the short crack growth model presented by Newman Jr[93] Considering119873 cycles and a medium crack length 120583 thecrack growth rate can be calculated as follows

119889120583

119889119873= exp [119898 ln (Δ119870eff) + 119887] 120583 (119873

0) = 120583

0gt 0 (10)

whereΔ119870eff is the linear elastic effective stress intensity factorrange and119898 and 119887 are the slope and the intercept of the linearinterpolation of the (log scale)Δ119870eff minus119889120583119889119873 respectively Inorder to determine the crack growth rate Spencer Jr et al [9495] used the cubic polynomial fit in ln (Δ119870eff ) Therefore thecrack growth rate equation can be written in the continuous-time setting as follows [96 97]

119889120583

119889119905=

(120597Φ120597119878) sdot (119889119878119889119905)

(1 minus 120597Φ) 120597120583 120583

0(1199050) = 120583

0gt 0 (11)

Manson and Halford [98] introduced an effective crackgrowth model accounting for the processes taking placemeanwhile using

119886 = 1198860+ (119886

119891minus 119886

0) 119903

119902

(12)

where 1198860 119886 and 119886

119891are initial (119903 = 0) instantaneous and final

(119903 = 1) cracks lengths respectively while 119902 is a function of119873in the form 119902 = 119861119873

120573 (B and 120573 are materialrsquos constants)A fracture mechanics-based analysis addressing bridges

and other steel structures details has been made by Righini-otis and Chryssanthopoulos [99] accounting for the accept-ability of flaws in fusion welded structures [100]

222 Stochastic Crack Growth Models The growth of afatigue crack can be also modeled by nonlinear stochasticequations satisfying the Ito conditions [94ndash101] Given that119864[119888(120596 119905

0)] = 120583

0and cov[119888(120596 119905

0)] = 119875

0 the stochastic

differential equation for the crack growth process 119888(120596 119905) canbe written according to the deterministic damage dynamicsas

119889119888 (120596 119905)

119889119905= exp[119911 (120596 119905) minus

1205902

119911(119905)

2] sdot

119889120583

119889119905forall119905 ge 119905

0 (13)

If 120596 and 119905 represent the point and time of the sam-ple in the stochastic process the auxiliary process 119911(120596 119905)

is assumed to be a stationary Gauss-Markov one havingvariance 120590

2

119911(119905) which implies the rational condition of a

lognormal-distributed crack growth [96]In order to clarify the influence of the fracture peculiar-

ities on the failure probability of a fatigue loaded structureMaljaars et al [102] used the linear elastic fracture mechanics(LEFM) theory to develop a probabilistic model

Ishikawa et al [103] proposed the Tsurui-Ishikawamodelwhile Yazdani and Albrecht [104] investigated the applicationof probabilistic LEFM to the prediction of the inspectioninterval of cover plates in highway bridges As for the weldedstructures a comprehensive overview of probabilistic fatigueassessment models can be found in the paper of Lukic andCremona [105] In this study the effect of almost all relevantrandom variables on the failure probability is treated [105]

The key feature of the work of Maljaars et al [102] withrespect to other LEFM-based fatigue assessment studies [87103ndash105] is that it accounts for the fact that at any moment intime a large stress cycle causing fracture can occurThereforethe probability of failure in case of fatigue loaded structurescan be calculated combining all the failure probabilities for alltime intervals

Considering the stress ranges as randomly distributedthe expectation of 119889119886119889119873 can be written as a function of theexpectation of the stress range Δ120590 as follows (see (14))

119864(119889119886

119889119873)

= 1198601119864 [Δ120590

1198981]Δ120590trΔ120590th

sdot ([119861nom119861

]

119875

119862load119862glob119862scf119862sif119884119886radic120587119886)

1198981

sdot sdot sdot

+ 1198602119864 [Δ120590

1198982]infin

Δ120590tr

sdot ([119861nom119861

]

119901


1198982

(14)

119864 [Δ120590119898

]1199042

1199041= int

1199042

1199041

119904119898

119891Δ120590

(119904) 119889119904 (15)

where 119891Δ120590

(119904) is the probability density function of the stressranges Δ120590 and 119904 represents the stress range steps The 119862-factors are the uncertainties of the fluctuating load mode In(14)119861nom represents the plate thickness (considering aweldedplate) used in the calculation of the stress while 119875 is thethickness exponent The probabilistic LEFM model appliedon fatigue loaded structures typical of civil enegineeringshowed that modeling the uncertainity factors is the keyduring the assessment of the failure probability which is quiteindependent of the particular failure criterion The partialfactors introduced to meet the reliability requirements ofcivil engineering structures and derived for various values ofthe reliability index (120573) appeared to be insensitive to otherparameters such as load spectrum and geometry [102]


23 Stochastic Methods for Fatigue Life Prediction Ting andLawrence [106] showed that for an Al-Si alloy there was aremarkable difference between the size distributions of thecasting pores and of those that initiated dominant fatiguecracks

In order to consider also the size of the defects Todinov[78] classified them into categories according to their size andthen divided them into groups according to the probabilityof fatigue crack initiation Other than type and size anadditional separation was done when needed based on theshape of the defects Considering 119872 minus 1 groups (where 119872

is the index reserved for the matrix) having a 119901119894average

probability of fatigue crack initiation each it is possible tocalculate the probability 119875

119894that at least one fatigue crack

initiated in the 119894th group as

119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)

where (1 minus 119901119894)119873119894 is the probability that none of the grouprsquos

defects initiated a fatigue crack while 119873119894represents the

number of discontinuities in the group The fatigue crack issupposed to be generated on the 119895th defect of the 119894th group(119895 = 1119873

119894)Theprobability119875

119894is not affected by the presence of

other groups of defects because they do not affect the fatiguestress range

Figure 3 shows that given a circle having unit area andcorresponding to the matrix the area of the overlappingdomains located in this circle is numerically equal to the totalprobabilities 119875

119894 The 119894 index domain overlaps with greater

index domains thus resulting in the following relationsbetween the average fatigue lives of the groups 119871

119894le 119871

119894+1le

sdot sdot sdot le 119871119872

[78] 119891119894is the frequency of failure or rather

the probability that in the 119894th group of defects a dominantfatigue crack initiates The shortest fatigue life is given bythe cracks generated in the first group (all dominant) andtherefore119891

1= 119875

1 Recurrent equations can be used to express

the dependence between the probabilities and the fatiguefailure frequencies

1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875

2) sdot sdot sdot (1 minus 119875

119872minus1)

(17)

This can be reduced to

119875119894=

119891119894

1 minus sum119894minus1

119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)

Considering a new distribution 1198731015840

119894of defects in the groups

the probabilities 1198751015840

119894are calculated according to (16) while

failure frequencies are calculated as follows

1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1

Figure 3 Probabilities that at least one fatigue crack had beennucleated in the ith group of defects (PM is the probability for nucle-ation in the matrix) Fatigue failure frequencies f i (the probabilityfor initiation of a dominant fatigue crack in the ith group) arenumerically equal to the area of the nonoverlapped regions (adaptedfrom [78])

With the new failure frequencies being 1198911015840

119894(119894 = 1119872) and the

average fatigue lives being 119871119894 119894 = 1119872 the expected fatigue

life 119871 can be determined as follows [78]

119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)

The Monte-Carlo simulation showed that in the case of castaluminum alloys the fatigue life depends more on the typeand size of the defects where the fatigue crack arises than onother parameters concerning the fatigue crack initiation andpropagation In general it can be stated that the probability offatigue crack initiation from discontinuities and the variationof their sizes produce a large scatter in the fatigue life [78]

A nonlinear stochastic model for the real-time computa-tions of the fatigue crack dynamics has been developed byRayand Tangirala [96]

Ray [107] presented a stochastic approach to model thefatigue crack damage of metallic materials The probabilitydistribution function was generated in a close form withoutsolving the differential equations this allowed building algo-rithms for real-time fatigue life predictions [107]

Many static failure criteria such as Shokrieh and LessardTsay-Hill Tsai-Wu and Hashin can be converted into fatiguefailure criterion by replacing the static strength with fatiguestrength in the failure criterion [108ndash111]


1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2

For operation at 1 followed byoperation at 3


Cycle ratio niNi

Dam

age

D

Figure 4 A schematic representation for the Marco-Starkey theory[90]

24 Cumulative Damage Models for Fatigue Lifetime Calcula-tion Themost popular cumulative damagemodel for fatiguelife prediction is based on the Palmer intuition [5] of a linearaccumulation Miner [2] was the first researcher who wrotethe mathematical form of this theory The Palmgren-Minerrule also known as linear damage accumulation rule (LDR)stated that at the failure the value of the fatigue damage 119863

reaches the unity [2]

119863 = sum119899119894

119873119894

= 1 (21)

The LDR theory is widely used owing to its intrinsic simplic-ity but it leans on some basic assumptions that strongly affectits accuracy such as (i) the characteristic amount of workabsorbed at the failure and (ii) the constant work absorbedper cycle [39] From the load sequence point of view the lackof consideration leads to experimental results that are lowerthan those obtained by applying the Miner rule under thesame loading conditions for high-to-low load sequence andhigher results for the opposite sequence

In order to overcome the LDR shortcomings firstlyRichart and Newmark [112] proposed the damage curvecorrelating the damage and the cycle ratio (119863 minus 119899

119894119873

119894

diagram) Based on this curve andwith the purpose of furtherimproving the LDR theory accuracy the first nonlinear loaddependent damage accumulation theory was suggested byMarco and Starkey [90]

119863 = sum(119899119894

119873119894

)

119862119894

(22)

The Palmgren-Miner rule is a particular case of the Marco-Starkey theory where 119862

119894= 1 as reported in Figure 4

The effect of the load sequence is highlighted in Figure 4since for low-to-high loads sum(119899

119894119873

119894) gt 1 while for the high-

to-low sequence the summation is lower than the unity [90]Other interesting theories accounting for load interaction

effects can be found in literature [113ndash122]The two-stage linear damage theory [123 124] was for-

mulated in order to account for two types of damage due to(i) crack initiation (119873

119894= 120572119873

119891 where 120572 is the life fraction

factor for the initiation stage) and (ii) crack propagation(119873

119894= (1 minus 120572)119873

119891) under constant amplitude stressing [123

124] This led to the double linear damage theory (DLDR)developed by Manson [125] Manson et al further developedthe DLDR providing its refined form and moving to thedamage curve approach (DCA) and the double damage curveapproach (DDCA) [98 126 127] These theories lean onthe fundamental basis that the crack growth is the majormanifestation of the damage [39] and were successfullyapplied on steels and space shuttles components (turbo pumpblades and engines) [128 129]

In order to account for the sequence effects theoriesinvolving stress-controlled and strain-controlled [130ndash135]cumulative fatigue damage were combined under the so-called hybrid theories of Bui-Quoc and coworkers [136 137]Further improvements have been made by the same authorsin order to account for the sequence effect when the cyclicloading includes different stress levels [138ndash141] temperature[142 143] and creep [144ndash150]

Starting from the Palmgren-Miner rule Zhu et al [92]developed a new accumulation damage model in order toaccount for the load sequence and to investigate how thestresses below the fatigue limit affect the damage inductionThe specimens were subjected to two-stress as well as multi-level tests and the authors established a fuzzy set method topredict the life and to analyze the evolution of the damage

Hashin and Rotem [151] Ben-Amoz [152] Subramanyan[153] and Leipholz [154] proposed in the past a varietyof life curve modification theories also various nonlinearcumulative damage fatigue life prediction methods can befound in literature [155ndash158]

Recently Sun et al [159] developed a cumulative damagemodel for fatigue regimes such as high-cycle and very-high-cycle regimes including in the calculation some keyparameters such as (i) tensile strength of materials (ii) sizesof fine grain area (FGA) and (iii) sizes of inclusions Fatiguetests on GCr15 bearing steel showed a good agreement withthis model [159]

241 Fatigue Life Prediction at Variable Amplitude LoadingUsually the fatigue life prediction is carried out combiningthe material properties obtained by constant amplitude lab-oratory tests and the damage accumulation hypothesis byMiner andPalmgren [2 5] Popular drawbacks of thismethodare the lack of validity of the Palmgren-Miner rule accountingfor sequential effects residual stresses and threshold effectsbut the biggest one is that for loading cycles having amplitudebelow the fatigue limit the resulting fatigue life according tothe LDR is endless (119873 = infin) [160] This is not acceptable


especially when variable amplitude loadings are applied [6161]

To overcome these issues a variety of approaches hasbeen proposed and can be found in literature [6 162ndash168]Among these a stress-based approach was introduced by theBasquin relation [162 163]

120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)

where 1205901015840119891119864 and 119887 are the fatigue strength coefficient Youngrsquos

modulus and the fatigue strength exponent respectivelyThevalues of these three terms should be determined experi-mentally A modification of the Basquin relation has beenproposed by Gassner [164] in order to predict the failures ofmaterials and components in service

A strain-based approach was developed by means of theCoffin-Manson relation

120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840

119891correspond to plastic strain amplitude

fatigue ductility exponent and fatigue ductility respectively[51 169] The combination of (24) and (25) leads to a widelyused total strain life expression

120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)

which has been implemented in fatigue life calculationsoftware

A different approachwas proposed by Zhu et al [92] whoextended the Miner rule to different load sequences with theaid of fuzzy sets

A detailed description of the effects of variable amplitudeloading on fatigue crack growth is reported in the papers ofSkorupa [170 171]

Schutz and Heuler [167] presented a relative Miner rulewhich is built using constant amplitude tests to estimate theparameter 120573 afterwards spectrum reference tests are used toestimate the parameter 120572 According to Minerrsquos equation [2](see (21)) for every reference spectrum test the failure occursat

119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)

where 119873119891is the number of cycles to failure and 119873pred is the

predicted life according to Palmgren-MinerWhenmore thanone reference test is conducted the geometric mean value isusedThe failure is predicted at a damage sum of119863lowast (not 1 asin the Palmgren-Miner equation) and the number of cyclesto failure becomes

119873lowast

119894= 119863

lowast

sdot 119873119894 (27)

A stochastic life prediction based on thePalmgren-Miner rulehas been developed by Liu andMahadevan [172]Theirmodelinvolves a nonliner fatigue damage accumulation rule and

accounts for the fatigue limit dependence on loading keepingthe calculations as simple as possible Considering an appliedrandom multiblock loading the fatigue limit is given by

sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)

where 120596119894is the loading cycle distribution and119860

119894is a material

parameter related to the level of stressJarfall [173] and Olsson [165] proposed methods where

the parameter 120572 needs to be estimated from laboratory testswhile the exponent 120573 is assumed as known

In the model suggested by Johannesson et al [174] loadspectra considered during laboratory reference tests werescaled to different levels The authors defined 119878eq as theequivalent load amplitude for each individual spectrum as

119878eq = 120573radicsum

119896

V119896119878120573

119896 (29)

where 120573 is the Basquin equation (119873 = 120572119878minus120573) exponent

Considering the load amplitude 119878119896 its relative frequency of

occurrence in the spectrum is expressed by V119896 The features

of the load spectra such as shape and scaling are chosenin order to give different equivalent load amplitudes Themodified Basquin equation is then used to estimate thematerial parameters 120572 and 120573

119873 = 120572119878minus120573

eq (30)

This estimation can be done combining the Maximum-Likelihood method [175] with the numerical optimizationConsidering a new load spectrum V

119896

119896 119896 = 1 2 the

fatigue life prediction results from the following calculations

= minus120573

eq where eq = radicsum

119896

V119896120573

119896 (31)

Using continuum damage mechanics Cheng and Plumtree[13] developed a nonlinear damage accumulation modelbased on ductility exhaustion Considering that in general thefatigue failure occurs when the damage119863 equals or exceeds acritical damage value119863

119862 the damage criterion can be written

as

119863 ge 119863119862fatigue failure (32)

In the case of multilevel tests the cumulative damage iscalculated considering that 119899

119894is the cycle having a level stress

amplitude of Δ120590119886

119894(119894 = 1 2 3 ) while 119873

119894is the number of

cycles to failure Therefore the fatigue lives will be 1198731 119873

2

and1198733at the respective stress amplitudesΔ120590

119886

1Δ120590

119886

2 andΔ120590

119886

3

The damage for a single level test can be written as [13]

1198631= 119863

119862

1 minus [1 minus (1198991

1198731

)

1(1+120595)

]

1(1+1205731)

(33)


where 120595 is the ductility and 120573 is a material constant Thecumulative damage for a two-stage loading process can betherefore written as

1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)

In order to account for variable amplitude loading under thenominal fatigue limit Svensson [15] proposed an extension ofthe Palmgren-Miner rule considering that the fatigue limit ofa material decreases as the damage increases owing to dam-age accumulation due to crack growth The effects on fatiguebehavior and cyclic deformation due to the load sequencewere investigated for stainless steel 304L and aluminum alloy7075-T6 by Colin and Fatemi [176] applying strain- andload-controlled tests under variable amplitude loading Theinvestigation under different load sequences showed that forboth materials the L-H sequence led to a bigger sum (longerlife) compared to the H-L sequence

A fewmodels explaining the macrocrack growth retarda-tion effect under variable amplitude loading can be found inliterature [177ndash183] also combined with crack closure effects[179]

Recently starting from the model proposed by Kwofieand Rahbar [184 185] using the fatigue driving stress param-eter (function of applied cyclic stress number of loadingcycles and number of cycles to failure) Zuo et al [186]suggested a new nonlinear model for fatigue life predictionunder variable ampitude loading conditions particularlysuitable for multilevel load spectra This model is based on aproper modification of the Palmgren-Minerrsquos linear damageaccumulation rule and the complete failure (damage) of acomponent takes place when

119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)

120573119894is the expended life fraction at the loading stress 119878

119894 and

119873119891119894is the failure life of 119878

119894[184] Compared with the Marco-

Starkey [90] LDR modification the present model is lesscomputationally expensive and is easier to use comparedto other nonlinear models In order to account for randomloading conditions Aıd et al [160] developed an algorithmbased on the 119878-119873 curve further modifed by Benkabouche etal [187]

242 Fatigue Life Prediction underMultiaxial Loading Condi-tions Generally speaking a load on an engineering compo-nent in service can be applied on different axis contemporary(multiaxial) instead of only one (uniaxial) Moreover theloads applied on different planes can be proportional (inphase) or nonproportional (out of phase) Other typicalconditions that can vary are changes in the principal axes

or a rotation of these with respect to time a deflection inthe crack path an overloading induced retardation effecta nonproportional straining effect a multiaxial stressstrainstate and many more [188ndash190] These factors make themultiaxial fatigue life prediction very complicated A numberof theories and life predictions addressing this issue have beenproposed in the past [15 191ndash215]

For the multiaxial fatigue life prediction critical planeapproaches linked to the fatigue damage of the material canbe found in literature these approaches are based eitheron the maximum shear failure plane or on the maximumprincipal stress (or strain) failure plane [190] The criticalplane is defined as the plane with maximum fatigue damageand can be used to predict proportional or nonproportionalloading conditions [216] The prediction methods can bebased on the maximum principal plane or on the maximumshear plane failure mode and also on energy approachesThe models can be classified into three big groups as (i)stress-based models involving the Findley et al [217] andMcDiarmid [218] parameters (ii) strain-based models (ieBrown and Miller [195]) and (iii) stress- and strain-basedmodels involving the Goudarzi et al [194] parameter forshear failure and the Smith-Watson-Topper (SWT) parameterfor tensile failure [216 219] Stress-based damage modelsare useful under high-cycle fatigue regimes since plasticdeformation is almost negligible while strain-based criteriaare applied under low-cycle fatigue regimes but can be alsoconsidered for the high-cycle fatigue

The application of these methodologies on the fatigue lifeprediction of Inconel 718 has been recently made by Filippiniet al [220]

Among all the possible multiaxial fatigue criteria thosebelonging to Sines and Waisman [221] and Crossland [222]resulted in being very easy to apply and have been extensivelyused for engineering design [223]

In the case of steels by combining the Roessle-Fatemimethod with the Fatemi-Socie parameter it is possible toestimate the fatigue limit for loadings being in or out of phaseand the hardness is used as the only materialrsquos parameter[216 224]

In particular Carpinteri and Spagnoli [206] proposed apredictionmethod for hardmetals based on the critical planedetermination a nonlinear combination of the shear stressamplitude and the maximum normal stress acting on thecritical plane led to the fatigue life prediction According to(37) the multiaxial stress state can be transformed into anequivalent uniaxial stress

120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)

where 119862119886is the shear stress amplitude119873max is the maximum

normal stress and 120590119886119891

120591119886119891

is the endurance limit ratio [206]Further modifications proposed for this model are availablein literature [225ndash227]

Papadopoulos [228] also proposed a critical plane modelin order to predict multiaxial high-cycle fatigue life using astress approach [218 229 230]


For high-cycle fatigue Liu and Mahadevan [208] devel-oped a criterion based on the critical plane approach butthis model can be even used for the prediction of fatigue lifeunder different loading conditions such as (i) in phase (ii)out of phase and (iii) constant amplitude The same authors[231] proposed also a unified characteristic plane approachfor isotropic and anisotropic materials where the crackinginformation is not required

Bannantine and Socie [191] stated that there is a particularplane which undergoes the maximum damage and this iswhere the fatigue damage events occur All the strains areprojected on the considered plane and the normal strain(shear strain) is then cycle-counted using the rainflow-counting algorithm [232]The strain to be counted is selectedaccording to the predominant cracking mode for the consid-eredmaterialThe fatigue damage connected with every cyclecan be calculated with the tensile mode (see (38)) or the shearmode (see (39))

Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)

where for each cycle Δ120576 is the strain range and 120590max is themaximum normal stress The maximum shear strain rangeis given by Δ120574 while the maximum normal stress on themaximum shear plane is expressed by 120590

119899max Other constantsand symbols are explained deeply in the paper of Goudarzi etal [194] where a modification of Brown and Millerrsquos criticalplane approach [195] was suggested It is worth mentioningthat among the conclusions reached by Fatemi and Socie[195] it is stated that since by varying the combinations ofloading and materials different cracking modes are obtaineda theory based on fixed parameters would not be able to pre-dict all themultiaxial fatigue situationsTherefore this modelcan be applied only to combinations of loading and materialsresulting in a shear failureThe Bannantine and Sociemethod[191] is unable to account for the cracks branching or forthe consequent involvement of the multistage cracks growthprocess Therefore this model should be considered onlywhen the loading history is composed of repeated blocksof applied loads characterized by short length because thecracks would grow essentially in one direction only [189]

Given that cycling plastic deformation leads to fatiguefailure Wang and Brown [189 192 233 234] identified thata multiaxial loading sequence can be assimilated into cyclesand therefore the fatigue endurance prediction for a generalmultiaxial random loading depends on three independentvariables (i) damage accumulation (ii) cycle counting and(iii) damage evaluation for each cycle Owing to the keyassumption of the fatigue crack growth being controlled bythe maximum shear strain this method has been developedonly for medium cycle fatigue (MCF) and low cycle fatigue

(LCF) [189 233] Application of theWang and Brown criteriashowed that allowing changes of the critical plane at everyreversal a loading history composed of long blocks is suitablefor fatigue life prediction [192]

It is worth mentioning that a load path alteration (torsionbefore tension or tension before torsion etc) in multiaxialfatigue strongly affects the fatigue life [235ndash237]

A situation of constant amplitude multiaxial loading(proportional and nonproportional) was used to develop anew fatigue life prediction method by Papadopoulos [228]

Aid et al [238] applied an already developed DamageStress Model [160 239ndash241] for uniaxial loadings to study amultiaxial situation combining it with thematerialrsquos strengthcurve (119878-119873 curve) and the equivalent uniaxial stress

More recently Ince andGlinka [242] used the generalizedstrain energy (GSE) and the generalized strain amplitude(GSA) as fatigue damage parameters for multiaxial fatiguelife predictions Considering the GSA the multiaxial fatiguedamage parameter can be expressed as follows

Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)

where the components of the shear (120591 120574) and normal (120590 120576)strain energies can be spotted Application of this model toIncoloy 901 super alloy 7075-T561 aluminum alloy 1045HRC55 steel and ASTM A723 steel gave good agreement withexperimental results [242]

A comparison of various prediction methods for mul-tiaxial variable amplitude loading conditions under high-cycle fatigue has been recently performed by Wang et al[243] Results showed that for aluminumalloy 7075-T651 themaxium shear stess together with themain auxiliary channelscounting is suitable for multiaxial fatigue life predictions

25 Energy-Based Theories for Fatigue Life Prediction Asmentioned before in this paragraph linear damage accu-mulation methods are widely used owing to their clar-ity These methods are characterized by three fundamentalassumptions (i) at the beginning of each loading cycle thematerial acts as it is in the virgin state (ii) the damageaccumulation rate is constant over each loading cycle and(iii) the cycles are in ascending order of magnitude despitethe real order of occurrence [18] These assumptions allowpredicting the fatigue failure for high cycles (HCF) in asufficiently appropriate manner but the same cannot be saidfor the low-cycle fatigue (LCF) where the dominant failuremechanism is identified as the macroscopic strain

In order to describe and predict the damage process andtherefore the fatigue life under these two regimes a unifiedtheory based on the total strain energy density was presentedby Ellyin and coworkers [12 244 245]

The total strain energy per cycle can be calculated as thesum of the plastic (Δ119882

119901) and elastic (Δ119882119890) strain energies

Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)

Loop traces 2 = f(2)

(a) Masing-type deformation

1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)

(b) Non-Masing type deformation

Figure 5Materials exhibiting hysteresis loops with (a)Masing-type deformation and (b) non-Masing type deformation (Adapted from [39])

The plastic portion of strain is the one causing the damagewhile the elastic portion associated with the tensile stressfacilitates the crack growth [12] This theory can be appliedto Masing and non-Masing materials [246] (Figure 5) andin both cases a master curve can be drawn (translatingloop along its linear response portion for the non-Masingmaterials)

Therefore the cyclic plastic strain can be written asfollows

Non-Masing behavior is

Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)

Ideal Masing behavior is

Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)

where 119899lowast and 119899

1015840 are the cyclic strain hardening exponentsof the mater curve and of the idealized Masing materialrespectively Further improvements to this theory have beenmade in order to account for the crack initiation andpropagation stages [245ndash247]

Concerning the low-cycle fatigue (LCF) Paris and Erdo-gan [47] proposed an energy failure criterion based on theenergy expenditure during fatigue crack growth

Interesting energy-based theories were proposed also byXiaode et al [248] after performing fatigue tests underconstant strain amplitude and finding out that a new cyclicstress-strain relation could have been suggested given thatthe cyclic strain hardening coefficient varies during the testsas introduced also by Leis [249]

The theory of Radhakrishnan [250 251] postulating theproportionality between plastic strain energy density andcrack growth rate is worth mentioning The prediction of thelife that remains at them-load variation can be expressed as

119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898

are the total plastic strain energy atfailure for the 119894th and the 119898th levels respectively undercycles at constant amplitude In the case of harmonic loadingcycles Kliman [252] proposed a similar concept for fatiguelife prediction

Based on the energy principle and on a cumulativedamage parameter Kreiser et al [18] developed a nonlineardamage accumulation model (NLDA) which is particularlysuitable for materials and structures subjected to loadingsof high amplitude applied for low cycles (substantial plasticstrain) This model starting from the Ellyin-Golos approach[12] accounts for the load history sequence which reflectsin the progressive damage accumulation Considering theLCF Coffin-Manson relation [253] the plastic strain rangewas considered as an appropriate damage parameter formetals showing stable hysteresis while in the case of unstablehysteresis also the stress rangemust be included [18 254 255]The cumulative damage function (120593) depends on a materialparameter (119901

119898) and on the cumulative damage parameter

(120595) and it can be defined using the ratio between the plasticenergy density (ΔWp) and the positive (tensile) elastic strainenergy density (Δ119882

+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)


With the universal slopesmethodManson [256 257] derivedthe following equation for the fatigue life prediction

Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)

where 120576119891is the fracture ductility and 120590

119861is the tensile strength

As can be seen in (39) the values of the slopes have beenuniversalized by Manson and are equal to minus06 for the elasticpart and minus012 for the plastic part

Considering a wide range of materials steels aluminumand titanium alloysMuralidharan andManson [258] deriveda modified universal slopes method in order to estimatethe fatigue features only from the tensile tests data obtainedat temperatures in the subcreep range This method gives ahigher accuracy than the original one and is described by thefollowing equation

Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)

Obtaining the elastic slope from tensile strength Mitchell[259] proposed a modified elastic strain life curve A furthermodification of this method was proposed by Lee et al [260]in order to estimate the life of some Ni-Based superalloysexposed to high temperatures

Baumel and Seeger [261] established a uniform materialslaw that uses only the tensile strength as input data In thecase of some steels (low-alloyed or unalloyed) the equationcan be expressed as follows

Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)

For titanium and aluminum alloys the equation is modifiedas follows

Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)

The application of these techniques to the prediction ofthe fatigue life of gray cast iron under conditions of hightemperatures has been recently conducted by K-O Lee andS B Lee [262]

1

1

1

p1

p2 p3

V

Figure 6 Three groups of defects characterized by individualprobabilities 119901

1= 02 119901

2= 03 and 119901

3= 04 of fatigue crack

initiation used in the Monte Carlo simulations The region withvolume 119881 is characterized by a fatigue stress range Δ120590 [91]

Under multiaxial loading conditions the fatigue life canbe also estimated using energy criteria based on elastic energy[40 263 264] plastic energy [40 265ndash267] or a combinationof these two [219 268ndash276]

26 Probability Distribution of Fatigue Life Controlled byDefects Todinov [91] proposed a method to determine themost deleterious group of defects affecting the fatigue lifeFor a given material the effect of discontinuities on thecumulative fatigue distribution can be calculated [91] Asalready proposed in a previous work by the same author [78]the defects have been divided into groups according to theirtype and size each group being characterized by a differentaverage fatigue life The individual probability 119901

119894depends on

the fatigue stress rangeΔ120590 but not on other groups of defectsConsidering a volume region 119881 (Figure 6) subjected to

fatigue loading and having a fatigue stress range Δ120590 theprobability that at least one fatigue crack will start in thisvolume can be obtained by subtracting from the unity theno-fatigue crack initiation (in 119881) probability Therefore theprobability equation 119875

0

(119903)can be written as follows

1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)

where 120583 represents the defectsrsquo density reducing the fatiguelife to a number of cycles lower than 119909The absence of fatiguecrack initiation at Δ120590 can be defined by

1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)


which can be simplified as follows

1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)

Equation (53) shows how to obtain the fatigue crack initiationprobability (1198750) at a fatigue stress range Δ120590 This probability1198750 equals the probability of at least a single fatigue crack

initiation in a space volume 119881 and is given by

119866 (Δ120590) = 1 minus 119890minus119881sum119872119894=1 120583119894119901119894(Δ120590) (54)

The equivalence in (54) leads to a fatigue life smaller than orequal to 119909 cycles as reported in

119865 (119883 le 119909) = 1 minus 119890minus119881sum119909 120583119909119901119909(Δ120590) (55)

This calculation is extended only to the defectsrsquo groupsinducing a fatigue life equal to or smaller than 119909 cycles Thesame groups are considered for the expected fatigue life 119871

determination obtained by calculating the weighted averageof the fatigue lives 119871

119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)

where 119891119894(119894 = 0119872) are the fatigue failure frequencies

The most influential group of defects for the fatigue life isdetermined by calculating the expected fatigue life rise whenthe 119894th group of defects is removed

Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)

Therefore the largest value of Δ119871119898

corresponding to theremoval of the119898-indexed group of defects identifies themostdeleterious ones

Su [277] studied the connection between the microstruc-ture and the fatigue properties using a probabilistic approachThe key assumption in this case is that the fatigue lifedecreasesmonotonically with the size of the localmicrostruc-tural features which are responsible for the generation ofthe fatigue crack Addressing the cast aluminum topicSu [277] obtained that micropores are the crucial featureaffecting the fatigue life It must be stressed out that thelocal microstructural feature having the highest impact onthe fatigue properties depends on the particularmaterial (egfor cast aluminum alloy 319 [278] fatigue cracks are used toinitiate frommicropores [278ndash281]) Caton et al [281] found acorrelation between the critical pore size and the fatigue lifethe latter being monotonically decreasing with the porosity

sizeTherefore the growth of a fatigue crack can be calculatedas follows

119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)

where 119886 is the crack length whose initial value is equal to thepore diameter of the discontinuity that generates the crackIntegrating (58) the relationship between the initial pore sizeand the fatigue life can be written as follows

119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)

where 1198630is the crack length while 119862 119904 and 119905 are material

parameters 120590119886 120590

119910 and 120576max are the alternating stress magni-

tude the yield stress and the maximum strain respectivelyIt can be useful to invert (58) in order to express the criticalpore dimension as a function of the fatigue life

1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)

27 ContinuumDamageMechanics (CDM)Models for FatigueLife Prediction Starting from the concept that the accumu-lation of damage due to environmental conditions andorservice loading is a random phenomenon Bhattacharayaand Ellingwood [27 282] developed a continuum damagemechanics (CDM) basedmodel for the fatigue life predictionIn particular starting from fundamental thermodynamicconditions they proposed a stochastic ductile damage growthmodel [27] The damage accumulation equations resultingfrom other CDM-based approaches suffer usually from alack of continuity with the first principles of mechanics andthermodynamics [283] owing to their start from either adissipation potential function or a kinetic equation of dam-age growth Battacharaya and Ellingwood [27] consideredinstead the dissipative nature of damage accumulation andaccounted for the thermodynamics laws that rule it [284]If a system in diathermal contact with a heat reservoir isconsidered the rate of energy dissipation can be calculatedfrom the first two laws of thermodynamics and can bewrittenas follows according to the deterministic formulation

Γ equiv minus119864+ minus

120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)

where119870119864is the kinetic energy and119882 is the work done on the

system The Helmholtz free energy Ψ(120579 120576 119863) is a function ofthe damage variable 119863 the symmetric strain tensor (120576) andthe temperature (120579)

The system is at the near equilibrium state and is subjectedto continuous and rapid transitions of its microstates causingrandom fluctuations of state variables around their meanvalues Therefore the stochastic approach [27] is the most


suitable to describe the free energy variation and as per (61)the first one can be written as follows

120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)

where 119861(119905) represents the free energy random fluctuation 1199050

is the initial equilibrium state and 119905 is an arbitrary instant oftime (119905 gt 119905

0) For a body that undergoes an isotropic damage

due to uniaxial loading the following can be assumed

120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin

is the far-field stress acting normal to the surfaceand 120595

119863is the partial derivative of the free energy per unit

volume (120595) with respect to 119863 The quantity 119904119887 which is

equal to (1205972

119861)(120597120576120597119881) represents the random fluctuationimposed on the stress field existing within the deformablebody Assuming that 119904

119887is described by the Langevin equation

it is possible to write what follows

119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888

2are positive constants while 120585(120576) corresponds

to the Gaussian white noise indexed with the strain Inthis particular case three key assumptions are necessary(i) 119904

119887should be a zero-mean process characterized by

equiprobable positive and negative values (ii) compared tothe macroscopic rate of damage change the fluctuation rateis extremely rapid and (iii) the 119904

119887mean-square fluctuation

should be time or strain independent [27] Consideringthe scale of timestrain typical of structural mechanics thefluctuations are extremely rapid and therefore the damagegrowth stochastic differential equation can be written asfollows [285]

119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)

where 119882(120576) is the standard Wiener process This formu-lation allows considering the presence of negative damageincrements over a limited time interval at the microscaleeven though the calculated increment of damage should benonnegative in absence of repair

According to Battacharaya and Ellingwood [27] in pres-ence of a uniaxialmonotonic loading the free energy per unitvolume can be calculated as follows

120595 = int120590119889120576 minus 120574 (66)

In (66) 120574 is the energy associated with the defects formation(per unit volume) due to damage evolution With the aid ofthe Ramberg-Osgood monotonic stress-strain relations thetotal strain can be estimated as follows

120576 =

119864+ (

119870)

119872

(67)

where the first term is the elastic strain (120576119890) and the second

is the plastic strain (120576119901) while 119872 and 119870 are the hardening

exponent andmodulus respectivelyThe second term of (66)can be instead estimated as

120574 =3

4120590119891119863 (68)

where120590119891is the failure strain assuming that (i) discontinuities

are microspheres not interacting with each other and havingdifferent sizes (ii) stress amplifications can be neglected and(iii) there is a linear relation between force and displacementat the microscale Equation (65) can be rewritten as follows

119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)

where 119860 and 119861 are coefficients depending on 120576119901 120590

119891 119872

119870 and 1205760 If the materials properties Ω = 120576

0 120590

119891 119870119872

are considered deterministic and 1198630

= 119863(1205760) is the initial

damage that can be either deterministic or Gaussian theresult is that damage is a Gaussian process as defined by thedamage variable [27]

Concerning nonlinear models Dattoma et al [286]proposed theory applied on a uniaxial model based oncontinuum damage mechanics [287] In the formulation ofthis nonlinear model [286] the authors started from thenonlinear load dependent damage rule firstly formulated byMarco and Starkey [90] as follows

119863 =

119899

sum

119894=1

119903119909119894

119894 (70)

where 119909119894is a coefficient depending on the 119894th load The

experimental results showed a good agreement with the datacalculated with this method although for each load it isnecessary to recalculate the 119909

119894coefficients

The mechanical deterioration connected to fatigue andcreep through the continuum damage theory was introducedby Kachanov [288] and Rabotnov [289] Later Chaboche andLemaitre [290 291] formulated a nonlinear damage evolutionequation so that the load parameters and the damage variable119863 result in being nondissociable

120575119863 = 119891 (119863 120590) 120575119899 (71)

where 119899 is the number of cycles at a given stress amplitudeand 120590 is the stress amplitude [291 292]

This fatigue damage can be better defined as follows

120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)

where 1205731198720 and 119887 depend on the material while 120572 depends

on the loading 120590max and 120590med are the maximum and themean stress of the cycle respectively The stress amplitude 120590

119886

is calculated as 120590119886

= 120590max minus 120590med This approach has been


adopted by various authors [13 14 33 292] with integrationsandmodifications All these CDMmodels were developed forthe uniaxial case but also thermomechanical models basedon mechanics of the continuous medium can be found inliterature [11 293] According to Dattoma et al [286] thenumber of cycles to failure (119873

119891) for a given load can be

written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)

which is a good approximation of the linear relation betweenlog 119878 and log119873 [33] The main result highlighted by thisformulation is that damage is an irreversible degradationprocess which increases monotonically with the appliedcycles (see (74)) Moreover the higher the load the larger thefatigue damage (see (74))

120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)

This model was employed for the fatigue life calculationof a railway axle built with 30NiCrMoV12 running onto aEuropean line for about 3000 km taking care of its load his-tory Final experimental tests were conducted with high-lowlow-high and random sequence using cylindrical specimensResults showed that this model has a good capacity to predictthe final rupture when complex load histories are considered[286 287]

A lot of fatigue life prediction theories based on nonlinearcontinuum damage mechanics can be found in literature [1314 39 294ndash298] addressing various situations such as fatiguecombined to creep uniaxial fatigue and ductile failure

28 Other Approaches Addressing the particular case ofaluminum alloys Chaussumier and coworkers developeda multicrack model for fatigue life prediction using thecoalescence and long and short crack growth laws [299]Based on this work further studies on the prediction offatigue life of aluminum alloys have been conducted by thesame authors [300 301] The approach used in the paperof Suraratchai et al [300] is worth mentioning where theeffect of the machined surface roughness on aluminumalloys fatigue life was addressed Considering an industrialframe the particular purpose of this work was the fatiguelife prediction of components when changing machiningparameters and processes in order to avoid tests that couldbe expensive and time-consuming Accounting for othermethods present in literature [302ndash309] that usually considerthe surface roughness as a notch effect in terms of stressthe theory developed by Suraratchai et al [299] modeledhow the geometric surface condition affects the fatigueproperties of structures In this theory [299] the surfaceroughness is considered responsible for the generation of alocal stress concentration controlling the possible surfacecrack propagation or nonpropagation

Stre

ss

Damaging

Strengthening

and damaging

UselessLow-amplitude loads

Fatigue limit

Fatigue life

U

L

S-N curve

Figure 7 Load region and its strengthening and damaging effect(adapted from [92])

Thermomechanical fatigue (TMF) mainly related tosingle crystal superalloys operating at high temperatures wasstudied by Staroselsky and Cassenti [310] and the combina-tion of creep fatigue and plasticity was also addressed [311]The low-cycle fatigue-creep (LCF-C) prediction has beenstudied by several authors [70 312ndash317] for steels and super-alloys applications Strain range partitioning- (SRP-) basedfatigue life predictions [318ndash320] or frequency modifiedfatigue life (FMFL) [320 321] can be also found in this field

Zhu et al [92] used the fuzzy set method to predict thefatigue life of specimen under loadings slightly lower thanthe fatigue limit accounting for strengthening and damagingeffects as well as for the load sequence and load interactionA schematic representation of this theory is reported inFigure 7

Other particular fatigue life prediction models can befound in literature based on significant variations of physicaland microstructural properties [322ndash327] on thermody-namic entropy [328 329] and on entropy index of stressinteraction and crack severity index of effective stress [330]

Even though a look at the frequency domain for thefatigue life estimation has been given in the past [331ndash334]recently new criteria have been developed by several authorshighlighting the rise of a brand new category of fatigue lifeestimation criteria that will surely get into the engineeringspotlights [335ndash342]

3 Summary

Starting from the introduction of the linear damage rule ahuge number of fatigue life predictionmodels have been pro-posed yet none of these can be universally accepted Authorsall over the world put efforts into modifying and extendingthe already existing theories in order to account for all thevariables playing key roles during cyclic applications of load-ings The complexity of the fatigue problem makes this topicactual and interesting theories using new approaches arisecontinuously The range of application of each model variesfrom case to case and depending on the particular applica-tion and to the reliability factors that must be considered


LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading

Ideal prediction model

Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d

Microstructuredependent

Figure 8 Features of an ideal fatigue life prediction model

researchers can opt for multivariable models to be computedor approaches easier to handle leading to a ldquosafe-liferdquo modelAs reported in Figure 8 an ideal fatigue life prediction modelshould include the main features of those already establishedand its implementation in simulation systems could helpengineers and scientists in a number of applications

Competing InterestsThe authors declare that they have no competing interests

AcknowledgmentsThis research was made possible by a NPRP award NPRP5ndash423ndash2ndash167 from the Qatar National Research Fund (amember of the Qatar Foundation)

References

[1] R I Stephens A Fatemi R Stephens and H O Fuchs MetalFatigue in Engineering JohnWiley amp Sons New York NY USA2000

[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 pp A159ndashA164 1945

[3] G Ayoub M Naıt-Abdelaziz F Zaıri and J M GloaguenldquoMultiaxial fatigue life prediction of rubber-like materialsusing the continuum damage mechanics approachrdquo ProcediaEngineering vol 2 no 1 pp 985ndash993 2010

[4] S Suresh Fatigue of Materials Cambridge University PressCambridge UK 1991

[5] A Palmgren ldquoDie lebensdauer von kugellagernrdquo Zeitschrift desVereins Deutscher Ingenieure vol 68 pp 339ndash341 1924

[6] J Schijve Fatigue of Structures and Materials Springer Scienceamp Business Media 2001

[7] H F S G Pereira A M P de Jesus A A Fernandes and AS Ribeiro ldquoAnalysis of fatigue damage under block loading in alow carbon steelrdquo Strain vol 44 no 6 pp 429ndash439 2008

[8] L Xi and Z Songlin ldquoStrengthening of transmission gear underlow-amplitude loadsrdquoMaterials Science and Engineering A vol488 no 1-2 pp 55ndash63 2008

[9] X Lu and S L Zheng ldquoStrengthening and damaging under low-amplitude loads below the fatigue limitrdquo International Journal ofFatigue vol 31 no 2 pp 341ndash345 2009


[10] L Xi andZ Songlin ldquoChanges inmechanical properties of vehi-cle components after strengthening under low-amplitude loadsbelow the fatigue limitrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 32 no 10 pp 847ndash855 2009

[11] S Oller O Salomon and E Onate ldquoA continuum mechanicsmodel for mechanical fatigue analysisrdquo Computational Materi-als Science vol 32 no 2 pp 175ndash195 2005

[12] E Zalnezhad A A D Sarhan and M Hamdi ldquoInvestigatingthe fretting fatigue life of thin film titanium nitride coatedaerospace Al7075-T6 alloyrdquo Materials Science and EngineeringA vol 559 pp 436ndash446 2013

[13] G X Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998

[14] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999

[15] T Svensson ldquoCumulative fatigue damage taking the thresholdinto accountrdquo Fatigue amp Fracture of Engineering Materials ampStructures vol 25 no 8-9 pp 871ndash875 2002

[16] M Jono ldquoFatigue damage and crack growth under variableamplitude loading with reference to the counting methods ofstress-strain rangesrdquo International Journal of Fatigue vol 27 no8 pp 1006ndash1015 2005

[17] E Zalnezhad A A D M Sarhan and M Hamdi ldquoPredictionof TiN coating adhesion strength on aerospace AL7075-T6 alloyusing fuzzy rule based systemrdquo International Journal of PrecisionEngineering and Manufacturing vol 13 no 8 pp 1453ndash14592012

[18] D Kreiser S X Jia J J Han and M Dhanasekar ldquoA nonlineardamage accumulation model for shakedown failurerdquo Interna-tional Journal of Fatigue vol 29 no 8 pp 1523ndash1530 2007

[19] M Makkonen ldquoPredicting the total fatigue life in metalsrdquoInternational Journal of Fatigue vol 31 no 7 pp 1163ndash1175 2009

[20] G H Majzoobi R Hojjati M Nematian E Zalnejad A RAhmadkhani and E Hanifepoor ldquoA new device for frettingfatigue testingrdquo Transactions of the Indian Institute of Metalsvol 63 no 2-3 pp 493ndash497 2010

[21] S-P Zhu and H-Z Huang ldquoA generalized frequencyseparation-strain energy damage function model for lowcycle fatigue-creep life predictionrdquo Fatigue and Fracture ofEngineering Materials and Structures vol 33 no 4 pp 227ndash2372010

[22] P Grammenoudis D Reckwerth and C Tsakmakis ldquoContin-uum damage models based on energy equivalence part Imdashisotropic material responserdquo International Journal of DamageMechanics vol 18 no 1 pp 31ndash63 2009

[23] P Grammenoudis D Reckwerth and C Tsakmakis ldquoContin-uum damage models based on energy equivalence part IImdashanisotropic material responserdquo International Journal of DamageMechanics vol 18 no 1 pp 65ndash91 2009

[24] E Rejovitzky and E Altus ldquoOn single damage variable modelsfor fatiguerdquo International Journal of Damage Mechanics vol 22no 2 pp 268ndash284 2013

[25] S S Kulkarni L Sun B Moran S Krishnaswamy and J DAchenbach ldquoA probabilistic method to predict fatigue crackinitiationrdquo International Journal of Fracture vol 137 no 1 pp9ndash17 2006

[26] B Bhattacharya and B Ellingwood ldquoA new CDM-basedapproach to structural deteriorationrdquo International Journal ofSolids and Structures vol 36 no 12 pp 1757ndash1779 1999

[27] B Bhattacharya and B Ellingwood ldquoA CDM analysis ofstochastic ductile damage growth and reliabilityrdquo ProbabilisticEngineering Mechanics vol 14 no 1-2 pp 45ndash54 1999

[28] J Lemaitre A Course on Damage Mechanics Springer NewYork NY USA 1992

[29] J Lemaitre ldquoEvaluation of dissipation and damage in metalsunder dynamic loadingrdquo in Proceedings of the InternationalConference on theMechanical Behavior ofMaterials I (ICMI rsquo71)Kyoto Japan August 1971

[30] D Krajcinovic and G U Fonseka ldquoThe continuous damagetheory of brittle materialsrdquo Journal of Applied Mechanics vol48 pp 809ndash824 1981

[31] S Chandrakanth and P C Pandey ldquoAn isotropic damage modelfor ductile materialrdquo Engineering Fracture Mechanics vol 50no 4 pp 457ndash465 1995

[32] A M P De Jesus A S Ribeiro and A A Fernandes ldquoFiniteelementmodeling of fatigue damage using a continuumdamagemechanics approachrdquo Journal of Pressure Vessel Technology vol127 no 2 pp 157ndash164 2005

[33] Y-C Xiao S Li and Z Gao ldquoA continuum damage mechanicsmodel for high cycle fatiguerdquo International Journal of Fatiguevol 20 no 7 pp 503ndash508 1998

[34] O E Scott-Emuakpor H Shen T George and C Cross ldquoAnenergy-based uniaxial fatigue life prediction method for com-monly used gas turbine enginematerialsrdquo Journal of Engineeringfor Gas Turbines and Power vol 130 no 6 Article ID 0625042008

[35] W Cui ldquoA state-of-the-art review on fatigue life predictionmethods for metal structuresrdquo Journal of Marine Science andTechnology vol 7 no 1 pp 43ndash56 2002

[36] N P Inglis ldquoHysteresis and fatigue of Wohler rotating can-tilever specimenrdquoMetallurgist vol 3 pp 23ndash27 1927

[37] R Zuchowski ldquoSpecific strain work as both failure criterion andmaterial damage measurerdquo Res Mechanica vol 27 no 4 pp309ndash322 1989

[38] B Budiansky and R J OrsquoConnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[39] E Zalnezhad A A D M Sarhan and M Hamdi ldquoSurfacehardness prediction of CrN thin film coating on AL7075-T6alloy using fuzzy logic systemrdquo International Journal of PrecisionEngineering andManufacturing vol 14 no 3 pp 467ndash473 2013

[40] E Macha and C M Sonsino ldquoEnergy criteria of multiaxialfatigue failurerdquo Fatigue and Fracture of Engineering Materialsand Structures vol 22 no 12 pp 1053ndash1070 1999

[41] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoin Proceedings of the ASTM International Fraction Dampingand Cyclic Plasticity 1965

[42] E Zalnezhad A A D Sarhan and M Hamdi ldquoA fuzzy logicbasedmodel to predict surface hardness of thin filmTiN coatingon aerospace AL7075-T6 alloyrdquo The International Journal ofAdvancedManufacturing Technology vol 68 no 1 pp 415ndash4232013

[43] Y-N Fan H-J Shi and K Tokuda ldquoA generalized hysteresisenergy method for fatigue and creep-fatigue life prediction of316L(N)rdquo Materials Science and Engineering A vol 625 pp205ndash212 2015

[44] T Letcher M-H H Shen O E Scott-Emuakpor T Georgeand C Cross ldquoAn energy-based critical fatigue life predictionmethod for AL6061-T6rdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 35 no 9 pp 861ndash870 2012


[45] O Scott-Emuakpor T George C Cross and M-H H ShenldquoMulti-axial fatigue-life prediction via a strain-energy methodrdquoAIAA Journal vol 48 no 1 pp 63ndash72 2010

[46] T Ghidini and C Dalle Donne ldquoFatigue life predictions usingfracture mechanics methodsrdquo Engineering Fracture Mechanicsvol 76 no 1 pp 134ndash148 2009

[47] P Paris and F Erdogan ldquoA critical analysis of crack propagationlawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash5331963

[48] R P Skelton T Vilhelmsen and G A Webster ldquoEnergycriteria and cumulative damage during fatigue crack growthrdquoInternational Journal of Fatigue vol 20 no 9 pp 641ndash649 1998

[49] G SWang ldquoAnEPFManalysis of crack initiation stable growthand instabilityrdquo Engineering Fracture Mechanics vol 50 no 2pp 261ndash282 1995

[50] J R Rice ldquoA path independent integral and the approximateanalysis of strain concentration by notches and cracksrdquo Journalof Applied Mechanics vol 35 no 2 pp 379ndash386 1968

[51] N Pugno M Ciavarella P Cornetti and A Carpinteri ldquoAgeneralized Parisrsquo law for fatigue crack growthrdquo Journal of theMechanics and Physics of Solids vol 54 no 7 pp 1333ndash13492006

[52] A Wohler ldquoTests to determine the forces acting on railwaywagon axles and the capacity of resistance of the axlesrdquoZBauwvol 10 pp 583ndash616 1860

[53] N Pugno P Cornetti and A Carpinteri ldquoNew unified lawsin fatigue from the Wohlerrsquos to the Parisrsquo regimerdquo EngineeringFracture Mechanics vol 74 no 4 pp 595ndash601 2007

[54] N F Mott ldquoA theory of the origin of fatigue cracksrdquo ActaMetallurgica vol 6 no 3 pp 195ndash197 1958

[55] A Hachim H Farid M El Ghorba K El Had A Akefand M Chergui ldquoPrediction and evolution of the fatiguecrack initiation in S355 Steel by the probabilistic methodrdquoInternational Journal of Engineering Science vol 1 pp 16ndash212012

[56] K Tanaka and T Mura ldquoA dislocation model for fatigue crackinitiationrdquo Journal of Applied Mechanics vol 48 no 1 pp 97ndash103 1981

[57] R Kapoor V S H Rao R S Mishra J A Baumann andG Grant ldquoProbabilistic fatigue life prediction model for alloyswith defects applied to A206rdquoActaMaterialia vol 59 no 9 pp3447ndash3462 2011

[58] K Tanaka and T Mura ldquoA theory of fatigue crack initiation atinclusionsrdquo Metallurgical Transactions A vol 13 no 1 pp 117ndash123 1982

[59] K Dang-Van ldquoMacro-micro approach in high-cycle multiaxialfatiguerdquo in Advances in Multiaxial Fatigue ASTM 1993

[60] T Mura and Y Nakasone ldquoTheory of fatigue crack initiation insolidsrdquo Journal of Applied Mechanics vol 57 no 1 pp 1ndash6 1990

[61] K S Chan ldquoA microstructure-based fatigue-crack-initiationmodelrdquoMetallurgical and Materials Transactions A vol 34 no1 pp 43ndash58 2003

[62] G Venkataraman Y-W Chung Y Nakasone and T MuraldquoFree energy formulation of fatigue crack initiation alongpersistent slip bands calculation of S-N curves and crackdepthsrdquo Acta Metallurgica et Materialia vol 38 no 1 pp 31ndash40 1990

[63] G Venkataraman Y W Chung and T Mura ldquoApplication ofminimum energy formalism in a multiple slip band model forfatiguemdashI Calculation of slip band spacingsrdquoActaMetallurgicaet Materialia vol 39 no 11 pp 2621ndash2629 1991

[64] G Venkataraman Y W Chung and T Mura ldquoApplication ofminimum energy formalism in a multiple slip band model forfatiguemdashII Crack nucleation and derivation of a generalisedCoffin-Manson lawrdquoActaMetallurgica et Materialia vol 39 no11 pp 2631ndash2638 1991

[65] K S Chan ldquoScaling laws for fatigue crackrdquo MetallurgicalTransactions A vol 24 no 11 pp 2473ndash2486 1993

[66] K S Chan and T Y Torng ldquoA probabilistic treatment ofmicrostructural effects on fatigue crack growth of large cracksrdquoJournal of Engineering Materials and Technology vol 118 no 3pp 379ndash386 1996

[67] K S Chan and M P Enright ldquoProbabilistic micromechanicalmodeling of fatigue-life variability in an 120572 + 120573 Ti alloyrdquoMetallurgical and Materials Transactions A vol 36 no 10 pp2621ndash2631 2005

[68] K J Miller ldquoThe short crack problemrdquo Fatigue amp Fracture ofEngineering Materials amp Structures vol 5 no 3 pp 223ndash2321982

[69] K J Miller ldquoThe behavior of short fatigue cracks and theirinitiation part IImdasha general summaryrdquo Fatigue amp Fracture ofEngineering Materials amp Structures vol 10 no 2 pp 93ndash1131987

[70] KJ Miller ldquoMetal fatigue a new perspectiverdquo in Topics inFracture and Fatigue Springer Berlin Germany 1992

[71] K J Miller ldquoThe behavior of short fatigue cracks and their ini-tiationsrdquo inMechanicanical Behavior of Materials-V PergamonPress 1987

[72] S Suresh and R O Ritchie ldquoPropagation of short fatiguecracksrdquo InternationalMetals Reviews vol 29 pp 445ndash476 1984

[73] M Amiri and M Modarres ldquoShort fatigue crack initiation andgrowth modeling in aluminum 7075-T6rdquo Journal of MechanicalEngineering Science vol 229 no 7 pp 1206ndash1214 2015

[74] H Ogi M Hirao and S Aoki ldquoNoncontact monitoring ofsurface-wave nonlinearity for predicting the remaining life offatigued steelsrdquo Journal of Applied Physics vol 90 no 1 pp 438ndash442 2001

[75] W L Morris O Buck and R V Inman ldquoAcoustic harmonicgeneration due to fatigue damage in highminusstrength aluminumrdquoJournal of Applied Physics vol 50 no 11 pp 6737ndash6741 1979

[76] A Melander and M Larsson ldquoThe effect of stress amplitudeon the cause of fatigue crack initiation in a spring steelrdquoInternational Journal of Fatigue vol 15 no 2 pp 119ndash131 1993

[77] A de Bussac and J C Lautridou ldquoA probabilistic model forprediction of LCF surface crack initiation in PM alloysrdquo Fatigueamp Fracture of Engineering Materials amp Structures vol 16 no 8pp 861ndash874 1993

[78] M T Todinov ldquoA probabilistic method for predicting fatiguelife controlled by defectsrdquoMaterials Science and Engineering Avol 255 no 1-2 pp 117ndash123 1998

[79] A de Bussac ldquoPrediction of the competition between surfaceand internal fatigue crack initiation in PM alloysrdquo Fatigue ampFracture of Engineering Materials amp Structures vol 17 no 11 pp1319ndash1325 1994

[80] A Manonukul and F P E Dunne ldquoHigh- and low-cyclefatigue crack initiation using polycrystal plasticityrdquo Proceedingsof the Royal Society of London A Mathematical Physical andEngineering Sciences vol 460 no 2047 pp 1881ndash1903 2004

[81] MMakkonen ldquoStatistical size effect in the fatigue limit of steelrdquoInternational Journal of Fatigue vol 23 no 5 pp 395ndash402 2001

[82] M Makkonen ldquoNotch size effects in the fatigue limit of steelrdquoInternational Journal of Fatigue vol 25 no 1 pp 17ndash26 2002


[83] A Pineau ldquoSuperalloy discs durability and damage tolerance inrelation to inclusionsrdquo inHigh Temperature Materials for PowerEngineering Part II Kluwer Academic New York NY USA1990

[84] S Deyber F Alexandre J Vaissaud and A Pineau ldquoProbabilis-tic life ofDA718 for aircraft engine disksrdquo in Superalloys 718 625706 and Derivatives E A Loria Ed pp 97ndash110 The MineralsMetals amp Materials Society 2005

[85] A Pineau and S D Antolovich ldquoHigh temperature fatigueof nickel-base superalloysmdasha review with special emphasison deformation modes and oxidationrdquo Engineering FailureAnalysis vol 16 no 8 pp 2668ndash2697 2009

[86] D Taylor ldquoGeometrical effects in fatigue a unifying theoreticalmodelrdquo International Journal of Fatigue vol 21 no 5 pp 413ndash420 2000

[87] M K Chryssanthopoulos and T D Righiniotis ldquoFatiguereliability of welded steel structuresrdquo Journal of ConstructionalSteel Research vol 62 no 11 pp 1199ndash1209 2006

[88] K Tanaka and S Matsuoka ldquoA tentative explanation for twoparameters C andm in Paris equation of fatigue crack growthrdquoInternational Journal of Fracture vol 13 no 5 pp 563ndash583 1977

[89] K Tanaka ldquoA correlation of ΔK th-value with the exponentm in the equation of fatigue crack growth for various steelsrdquoInternational Journal of Fracture vol 15 pp 57ndash68 1979

[90] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransactions of the ASME vol 76 pp 627ndash632 1954

[91] M T Todinov ldquoProbability distribution of fatigue life controlledby defectsrdquoComputers and Structures vol 79 no 3 pp 313ndash3182001

[92] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011

[93] J C Newman Jr ldquoA crack-closure model for predicting fatiguecrack growth under aircraft loadingrdquo in Method and Modelsfor Predicting Fatigue Crack Growth under Random LoadingASTM 1983

[94] B F Spencer Jr and J Tang ldquoMarkov process model for fatiguecrack growthrdquo Journal of Engineering Mechanics vol 114 no 12pp 2134ndash2157 1989

[95] B F Spencer Jr J Tang and M E Artley ldquoStochastic approachto modeling fatigue crack growthrdquo AIAA journal vol 27 no 11pp 1628ndash1635 1989

[96] A Ray and S Tangirala ldquoStochastic modeling of fatigue crackdynamics for on-line failure prognosticsrdquo IEEE Transactions onControl Systems Technology vol 4 no 4 pp 443ndash451 1996

[97] V V Bolotin Prediction of Service Life for Machines andStructures ASME Press 1989

[98] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981

[99] T D Righiniotis and M K Chryssanthopoulos ldquoProbabilisticfatigue analysis under constant amplitude loadingrdquo Journal ofConstructional Steel Research vol 59 no 7 pp 867ndash886 2003

[100] BSI ldquoGuide on methods for assessing the acceptability of flawsin fusion welded structuresrdquo BS 7910 2000

[101] A H Jazwinski Stochastic Processes and Filtering TheoryCourier Corporation 2007

[102] J Maljaars H M G M Steenbergen and A C W MVrouwenvelder ldquoProbabilistic model for fatigue crack growthand fracture of welded joints in civil engineering structuresrdquoInternational Journal of Fatigue vol 38 pp 108ndash117 2012

[103] H Ishikawa A Tsurui H Tanaka and H Ishikawa ldquoReliabilityassessment of structures based upon probabilistic fracturemechanicsrdquoProbabilistic EngineeringMechanics vol 8 no 1 pp43ndash56 1993

[104] N Yazdani and P Albrecht ldquoProbabilistic fracture mechanicsapplication to highway bridgesrdquo Engineering Fracture Mechan-ics vol 37 no 5 pp 969ndash985 1990

[105] M Lukic and C Cremona ldquoProbabilistic assessment of weldedjoints versus fatigue and fracturerdquo Journal of Structural Engi-neering vol 127 no 2 pp 211ndash218 2001

[106] J C Ting and F V Lawrence ldquoModeling the long-life fatiguebehavior of a cast aluminum alloyrdquo Fatigue amp Fracture ofEngineering Materials and Structures vol 16 no 6 pp 631ndash6471993

[107] A Ray ldquoStochastic modeling of fatigue crack damage for riskanalysis and remaining life predictionrdquo Journal of DynamicSystems Measurement and Control ASME vol 121 no 3 pp386ndash393 1999

[108] M M Shokrieh and L B Lessard ldquoProgressive fatigue damagemodeling of composite materials part I modelingrdquo Journal ofComposite Materials vol 34 no 13 pp 1056ndash1080 2000

[109] XDiao L B Lessard andMM Shokrieh ldquoStatisticalmodel formultiaxial fatigue behavior of unidirectional pliesrdquo CompositesScience and Technology vol 59 no 13 pp 2025ndash2035 1999

[110] S Baradaran W J Basirun E Zalnezhad M Hamdi A A DSarhan and Y Alias ldquoFabrication and deformation behaviourof multilayer Al

2O3TiTiO

2nanotube arraysrdquo Journal of the

Mechanical Behavior of Biomedical Materials vol 20 pp 272ndash282 2013

[111] Z Hashin ldquoFailure criteria for unidirectional fiber compositesrdquoJournal of Applied Mechanics vol 47 pp 324ndash334 1980

[112] F E Richart and N M Newmark ldquoA hypothesis for thedetermination of cumulative damage in fatiguerdquo Proceedings ofthe American Society for Testing Materials vol 48 pp 767ndash8001948

[113] H T Corten and T J Dolan ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of MetalsInstitution ofMechanical Engineering and American Society ofMechanical Engineers 1956

[114] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and a cumulative damage rulerdquo Journal of the AerospaceSciences vol 26 no 7 pp 431ndash442 1959

[115] A A D Sarhan E Zalnezhad andMHamdi ldquoThe influence ofhigher surface hardness on fretting fatigue life of hard anodizedaerospace AL7075-T6 alloyrdquo Materials Science and EngineeringA vol 1 pp 23ndash40 2012

[116] R Spitzer and H T Corten ldquoEffect of loading sequenceon cumulative fatigue damage of 7071-T6 aluminum alloyrdquoProceedings ASTM vol 61 pp 679ndash703 1961

[117] R Badaliance ldquoFatigue life prediction metals and compositesrdquoin Fracture Mechanics Methodology Martinus Nijhoff 1984

[118] S S Manson A J Nachtigall C R Ensign and J C FrecheldquoFurther investigation of a relation for cumulative fatiguedamage in bendingrdquo Journal of Engineering for Industry vol 87no 1 pp 25ndash35 1965

[119] Z Marciniak D Rozumek and E Macha ldquoFatigue livesof 18G2A and 10HNAP steels under variable amplitude and


random non-proportional bending with torsion loadingrdquo Inter-national Journal of Fatigue vol 30 no 5 pp 800ndash813 2008

[120] E Zalnezhad A A D Sarhan and M Hamdi ldquoA fuzzy logicbased model to predict surface hardness of thin film TiNcoating on aerospace AL7075-T6 alloyrdquo International Journal ofAdvancedManufacturing Technology vol 68 no 1 pp 415ndash4232013

[121] H Gao H-Z Huang Z Lv F-J Zuo and H-K Wang ldquoAnimproved Corten-Dolanrsquos model based on damage and stressstate effectsrdquo Journal of Mechanical Science and Technology vol29 no 8 pp 3215ndash3223 2015

[122] S-P ZhuH-ZHuang Y Liu L-P He andQ Liao ldquoA practicalmethod for determining the Corten-Dolan exponent and itsapplication to fatigue life predictionrdquo International Journal ofTurbo and Jet Engines vol 29 no 2 pp 79ndash87 2012

[123] T Nicholas High Cycle Fatigue A Mechanics of MaterialsPerspective Elsevier 2006

[124] H J Grover ldquoAn observation concerning the cycle ratio incumulative damagerdquo in Proceedings of the ASTM Symposium onFatigue Aircraft Structures January 1960

[125] S S Manson ldquoInterfaces between fatigue creep and fracturerdquoInternational Journal of FractureMechanics vol 2 no 1 pp 328ndash363 1966

[126] S S Manson and G R Halford ldquoRe-examination of cumu-lative fatigue damage analysismdashan engineering perspectiverdquoEngineering Fracture Mechanics vol 25 no 5-6 pp 539ndash5711986

[127] S S Zhao-Feng W De-Jun and X Hao ldquoTwo-stage fatiguedamage cumulative rulerdquo International Journal of Fatigue vol14 no 6 pp 395ndash398 1992

[128] S S Manson and G R Halford ldquoComplexities of high-temperature metal fatigue some steps toward understandingrdquoIsrael Journal of Technology vol 21 pp 29ndash53 1983

[129] M A McGaw Cumulative Fatigue Damage Models NASALewis Structures Technology 1988

[130] D L Henry ldquoA theory of fatigue damage accumulation in steelrdquoASME Transactions vol 77 pp 913ndash918 1955

[131] R R Gatts ldquoApplication of a cumulative damage concept tofatiguerdquo Journal of Basic Engineering T vol 83 pp 529ndash5401961

[132] F R Shanley ldquoA theory of fatigue based on unbonding duringreversed sliprdquo Report P-350 The Rand Corporation 1952

[133] S R Valluri ldquoA unified engineering theory of high stress levelfatiguerdquo Tech Rep ARL-181 1961

[134] T Bui-Quoc ldquoAn interaction effect consideration in cumulativedamage on a mild steel under torsion loadingrdquo in Proceedingsof the 5th International Conference on Fracture vol 5 pp 2625ndash2633 Cannes France 1982

[135] T Bui-Quoc J Dubuc A Bazergui and A Biron ldquoCumulativefatigue damage under strain controlled conditionsrdquo Journal ofMaterials vol 3 pp 718ndash737 1971

[136] J Dubuc T Bui-Quoc A Barzegui and A Biron ldquoUnifiedtheory of cumulative damage in metal fatiguerdquoWRC Bulletinvol 162 pp 1ndash20 1971

[137] T Bui-Quoc J A Choquet and A Biron ldquoCumulative fatiguedamage on large steel specimens under axial programmedloading with nonzero mean stressrdquo Journal of EngineeringMaterials and Technology ASME vol 98 no 3 pp 249ndash2551976

[138] T Bui-Quoc ldquoCumulative damage with interaction effect dueto fatigue under torsion loadingmdashan analysis of the interaction

effect on the basis of a continuous-damage concept is madeusing published test results obtained from two-step fatigue testsat room temperaturerdquo Experimental Mechanics vol 22 no 5pp 180ndash187 1982

[139] T Bui-Quoc ldquoA simplified model for cumulative damagewith interaction effectsrdquo in Proceedings of the Joint Conferenceon Experimental Mechanics Society for Experimental StressAnalysis May 1982

[140] A Biron and T Bui-Quoc ldquoCumulative damage concepts withinteraction effect consideration for fatigue or creep a perspec-tiverdquo in Proceedings of the Transactions of the 6th InternationalConference on StructuralMechanical Reaction Technology ParisFrance 1981

[141] M Bernard-Connolly T Bui-Quoc and A Biron ldquoMultilevelstrain controlled fatigue on a type 304 stainless steelrdquo Journal ofEngineering Materials and Technology ASME vol 105 no 3 pp188ndash194 1983

[142] T Bui-Quoc ldquoHigh-temperature fatigue-life estimation exten-sion of a unified theoryrdquo Experimental Mechanics vol 15 no 6pp 219ndash225 1975

[143] A Zhang T Bui-Quoc and R Gomuc ldquoA procedure forlow cycle fatigue life prediction for various temperatures andstrain ratesrdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 112 no 4 pp 422ndash428 1990

[144] T Bui-Qoc ldquoAn engineering approach for cumulative damageinmetals under creep loadingrdquo Journal of EngineeringMaterialsand Technology vol 101 no 4 pp 337ndash343 1979

[145] T Bui-Quoc and A Biron ldquoCumulative damage with interac-tion effect due to creep on aCr-Mo-V steelrdquo in Proceedings of theInternational Conference on Engineering Aspects of Creep 1980

[146] T Bui-Quoc and A Biron ldquoInteraction effect considerationin cumulative damage in metals due to creeprdquo Journal ofMechanical Engineering Science vol 23 no 6 pp 281ndash288 1981

[147] T Bui-Quoc ldquoRecent developments of continuous damageapproaches for the analysis of material behavior under creep-fatigue loadingrdquo Tech Rep CONF-820601 Section of AppliedMechanics 1982

[148] E Zalnezhad S Baradaran A R Bushroa and A A D SarhanldquoMechanical property enhancement of Ti-6Al-4V bymultilayerthin solid film TiTiO

2nanotubular array coating for biomedi-

cal applicationrdquoMetallurgical andMaterials Transactions A vol45 no 2 pp 785ndash797 2014

[149] R Gomuc T Bui-Quoc and A Biron ldquoEvaluation of the creepfatigue behavior of 225 Cr-1 Mo steel by a continuous damageapproachrdquo Res Mechanica vol 21 pp 135ndash154 1987

[150] R Gomuc T Bui-Quoc A Biron and M Bernard ldquoAnalysisof type 316 stainless steel behavior under fatigue creep andcombined fatigue-creep loadingrdquo Journal of Pressure VesselTechnology vol 112 no 3 pp 240ndash250 1990

[151] Z Hashin and A Rotem ldquoA cumulative damage theory offatigue failurerdquo Materials Science and Engineering vol 34 no2 pp 147ndash160 1978

[152] M Ben-Amoz ldquoA cumulative damage theory for fatigue lifepredictionrdquo Engineering Fracture Mechanics vol 37 no 2 pp341ndash347 1990

[153] S Subramanyan ldquoA cumulative damage rule based on the kneepoint of the S-N curverdquo Journal of Engineering Materials andTechnology vol 98 no 4 pp 316ndash321 1976

[154] H H E Leipholz ldquoOn themodified S-N curve for metal fatigueprediction and its experimental verificationrdquo Engineering Frac-ture Mechanics vol 23 no 3 pp 495ndash505 1986


[155] H Gao H-Z Huang S-P Zhu Y-F Li and R Yuan ldquoAmodified nonlinear damage accumulationmodel for fatigue lifeprediction considering load interaction effectsrdquo The ScientificWorld Journal vol 2014 Article ID 164378 7 pages 2014

[156] P J Huffman and S P Beckman ldquoA non-linear damageaccumulation fatigue model for predicting strain life at variableamplitude loadings based on constant amplitude fatigue datardquoInternational Journal of Fatigue vol 48 pp 165ndash169 2013

[157] R Yuan H Li H-Z Huang S-P Zhu and H Gao ldquoAnonlinear fatigue damage accumulation model consideringstrength degradation and its applications to fatigue reliabilityanalysisrdquo International Journal of Damage Mechanics vol 24no 5 pp 646ndash662 2015

[158] Z Lv H-Z Huang S-P Zhu H Gao and F Zuo ldquoA modifiednonlinear fatigue damage accumulation modelrdquo InternationalJournal of Damage Mechanics vol 24 no 2 pp 168ndash181 2014

[159] C Sun J Xie A Zhao Z Lei and Y Hong ldquoA cumulativedamage model for fatigue life estimation of high-strength steelsin high-cycle and very-high-cycle fatigue regimesrdquo Fatigue ampFracture of Engineering Materials amp Structures vol 35 no 7 pp638ndash647 2012

[160] A Aıd A Amrouche B B Bouiadjra M Benguediab andG Mesmacque ldquoFatigue life prediction under variable loadingbased on a new damage modelrdquo Materials and Design vol 32no 1 pp 183ndash191 2011

[161] C Bathias ldquoThere is no infinite fatigue life inmetallicmaterialsrdquoFatigueampFracture of EngineeringMaterialsamp Structures vol 22no 7 pp 559ndash565 1999

[162] O H Basquin ldquoThe exponential law of endurance testsrdquoProceedings of the American Society for Testing and Materialsvol 10 pp 625ndash630 1910

[163] F Kun H A Carmona J S Andrade Jr and H J HerrmannldquoUniversality behind basquinrsquos law of fatiguerdquo Physical ReviewLetters vol 100 no 9 Article ID 094301 4 pages 2008

[164] E Gassner ldquoBetriebsfestigkeit eine Bemessungsgrundlagefur Konstruktionsteile mit statistisch wechselnden Betriebs-beanspruchungenrdquo Konstruktion vol 6 pp 97ndash104 1954

[165] K E Olsson ldquoFatigue reliability predictionrdquo ScandinavianJournal of Metallurgy vol 18 no 4 pp 176ndash180 1989

[166] W Schutz ldquoThe significance of service load data for fatigue lifeanalysisrdquo in Fatigue Design ESIS16 MEP 1993

[167] W Schutz and P Heuler ldquoMinerrsquos rule revisitedrdquo in Proceedingsof the AGARD-SMP Meeting Industrinlagen-Betriebsgesell-shaft GmbH Ottobrunn Germany 1993

[168] P C McKeighan and N Ranganathan Fatigue Testing andAnalysis Under Variable Amplitude Loading Conditions ASTMInternational West Conshohocken Pa USA 1995

[169] D Rial H Kebir EWintrebert and J-M Roelandt ldquoMultiaxialfatigue analysis of a metal flexible piperdquo Materials and Designvol 54 pp 796ndash804 2014

[170] M Skorupa ldquoLoad interaction effects during fatigue crackgrowth under variable amplitude loadingmdasha literature reviewPart I empirical trendsrdquo Fatigue amp Fracture of EngineeringMaterials amp Structures vol 21 no 8 pp 987ndash1006 1998

[171] M Skorupa ldquoLoad interaction effects during fatigue crackgrowth under variable amplitude loadingmdasha literature reviewPart II qualitative interpretationrdquo Fatigue amp Fracture of Engi-neeringMaterials amp Structures vol 22 no 10 pp 905ndash926 1999

[172] Y Liu and S Mahadevan ldquoStochastic fatigue damage modelingunder variable amplitude loadingrdquo International Journal ofFatigue vol 29 no 6 pp 1149ndash1161 2007

[173] L Jarfall ldquoPrediction of fatigue life and crack growth usingspectrum test datardquo SAAB-SCANIA Report TKH R-3379 1985

[174] P Johannesson T Svensson and J de Mare ldquoFatigue lifeprediction based on variable amplitude testsmdashmethodologyrdquoInternational Journal of Fatigue vol 27 no 8 pp 954ndash965 2005

[175] J W Harris and H Stocker ldquoMaximum likelihood methodrdquo inHandbook ofMathematics and Computational Science SpringerNew York NY USA 1998

[176] J Colin and A Fatemi ldquoVariable amplitude cyclic deformationand fatigue behaviour of stainless steel 304L including stepperiodic and random loadingsrdquo Fatigue and Fracture of Engi-neering Materials and Structures vol 33 no 4 pp 205ndash2202010

[177] S Kumai and Y Higo ldquoEffects of delamination on fatigue crackgrowth retardation after single tensile overloads in 8090 Al-Lialloysrdquo Materials Science and Engineering A vol 221 no 1-2pp 154ndash162 1996

[178] V Zitounis and P E Irving ldquoFatigue crack acceleration effectsduring tensile underloads in 7010 and 8090 aluminium alloysrdquoInternational Journal of Fatigue vol 29 no 1 pp 108ndash118 2007

[179] O E Wheeler ldquoSpectrum loading and crack growthrdquo Journal ofBasic Engineering Transactions of the ASME vol 94 no 1 pp181ndash186 1972

[180] J Willenborg R M Engle and H A Wood ldquoA crack growthretardation model using an effective stress conceptrdquo Tech RepAFFDL TM-71-1-FBR 1971

[181] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970

[182] R O Ritchie S Suresh and C M Moses ldquoNear-thresholdfatigue crack growth in 2 14 Cr-1Mo pressure vessel steel in airand hydrogenrdquo Journal of EngineeringMaterials andTechnologyvol 102 no 3 pp 293ndash299 1980

[183] R O Ritchie and S Suresh ldquoSome considerations on fatiguecrack closure at near-threshold stress intensities due to fracturesurface morphologyrdquo Metallurgical Transactions A vol 13 no5 pp 937ndash940 1982

[184] S Kwofie and N Rahbar ldquoA fatigue driving stress approach todamage and life prediction under variable amplitude loadingrdquoInternational Journal of Damage Mechanics vol 22 no 3 pp393ndash404 2013

[185] S Kwofie and N Rahbar ldquoAn equivalent driving force modelfor crack growth prediction under different stress ratiosrdquoInternational Journal of Fatigue vol 33 no 9 pp 1199ndash12042011

[186] F-J Zuo H-Z Huang S-P Zhu Z Lv andH Gao ldquoFatigue lifeprediction under variable amplitude loading using a non-lineardamage accumulation modelrdquo International Journal of DamageMechanics vol 24 no 5 pp 767ndash784 2015

[187] S Benkabouche H Guechichi A Amrouche andM Benkhet-tab ldquoAmodified nonlinear fatigue damage accumulationmodelunder multiaxial variable amplitude loadingrdquo InternationalJournal of Mechanical Sciences vol 100 pp 180ndash194 2015

[188] D L McDowell and R Ellis Advances in Multiaxial FatigueASTM International West Conshohocken Pa USA 1993

[189] C H Wang and M W Brown ldquoLife prediction techniques forvariable amplitude multiaxial fatiguemdashpart 1 theoriesrdquo Journalof Engineering Materials and Technology vol 118 no 3 pp 367ndash370 1996

[190] A Fatemi andN Shamsaei ldquoMultiaxial fatigue an overview andsome approximation models for life estimationrdquo InternationalJournal of Fatigue vol 33 no 8 pp 948ndash958 2011


[191] J A Bannantine and D F Socie ldquoA variable amplitudemultiaxial fatigue life prediction methodrdquo in Proceedings ofthe 3rd International Conference on BiaxialMultiaxial FatigueStuttgart Germany April 1989

[192] C H Wang and M W Brown ldquoA study of the deforma-tion behaviour under multiaxial loadingrdquo European Journal ofMechanics AmdashSolids vol 13 pp 175ndash188 1994

[193] D Socie ldquoA summary and interpretation of the society ofautomotive engineers biaxial testing programrdquo in MultiaxialFatigue Analysis and Experiments Society of Automotive Engi-neers Inc 1989

[194] S Goudarzi K Khojier H Savaloni and E Zalnezhad ldquoOnthe dependence of mechanical and tribological properties ofsputtered chromium nitride thin films on deposition powerrdquoAdvanced Materials Research vol 829 pp 352ndash356 2014

[195] M W Brown and K J Miller ldquoA theory for fatigue under mul-tiaxial stress-strain conditionsrdquo Proceedings of the Institution ofMechanical Engineers vol 187 pp 745ndash755 1973

[196] E Krempl ldquoThe influence of state stress on low-cycle fatigue ofstructural materialsrdquo ASTM STP 1974

[197] Y S Garud ldquoMultiaxial fatigue a survey of the state of the artrdquoJournal of Testing and Evaluation vol 9 no 3 pp 165ndash178 1981

[198] E Zalnezhad A A D Sarhan and M Hamdi ldquoFretting fatiguelife evaluation of multilayer Cr-CrN-coated Al7075-T6 withhigher adhesion strengthmdashfuzzy logic approachrdquo InternationalJournal of AdvancedManufacturing Technology vol 69 no 5ndash8pp 1153ndash1164 2013

[199] M De Freitas B Li and J L T Santos ldquoA numerical approachfor high-cycle fatigue life prediction with multiaxial loadingrdquoin Multiaxial Fatigue and Deformation Testing and PredictionASTM 2000

[200] J A Bannantine and D F Socie Multiaxial Fatigue LifeEstimation Techniques Advances in Fatigue Life PredictionTechniques ASTM 1992

[201] M Mitchell and L Landgraf Advances in Fatigue LifetimePredictive Techniques ASTM 1992

[202] S Kalluri and J P Bonacuse Multiaxial Fatigue and Defor-mation Testing and Prediction ASTM International WestConshohocken Pa USA 2000

[203] C A Goncalves J A Araujo and E N Mamiya ldquoMultiaxialfatigue a stress based criterion for hard metalsrdquo InternationalJournal of Fatigue vol 27 no 2 pp 177ndash187 2005

[204] E Zalnezhad A A D Sarhan and M Hamdi ldquoInvestigatingthe effects of hard anodizing parameters on surface hardnessof hard anodized aerospace AL7075-T6 alloy using fuzzy logicapproach for fretting fatigue applicationrdquo International Journalof AdvancedManufacturing Technology vol 68 no 1ndash4 pp 453ndash464 2013

[205] I V Papadopoulos P Davoli C Gorla M Filippini andA Bernasconi ldquoA comparative study of multiaxial high-cyclefatigue criteria for metalsrdquo International Journal of Fatigue vol19 no 3 pp 219ndash235 1997

[206] A Carpinteri and A Spagnoli ldquoMultiaxial high-cycle fatiguecriterion for hard metalsrdquo International Journal of Fatigue vol23 no 2 pp 135ndash145 2001

[207] G Q Sun D Wang and D G Shang ldquoTime-dependent multi-axial fatigue and life prediction for nickel-based GH4169 alloyrdquoFatigue and Fracture of Engineering Materials and Structuresvol 36 no 10 pp 1039ndash1050 2013

[208] Y Liu and S Mahadevan ldquoMultiaxial high-cycle fatigue cri-terion and life prediction for metalsrdquo International Journal ofFatigue vol 27 no 7 pp 790ndash800 2005

[209] H J Gough and H V Pollard ldquoThe strength of metals undercombined alternating stressrdquo Proceedings of the Institution ofMechanical Engineers vol 131 pp 3ndash18 1935

[210] H J Gough H V Pollard and W J Clenshaw Some Experi-ments on the Resistance of Metals to Fatigue Under CombinedStresses vol 2522 ofAeronautical ResearchCouncil Reports 1951

[211] H Kakuno and Y Kawada ldquoA new criterion of fatigue strengthof a round bar subjected to combined static and repeated bend-ing and torsionrdquo Fatigue amp Fracture of Engineering Materialsand Structures vol 2 no 2 pp 229ndash236 1979

[212] E Zalnezhad A A D Sarhan and P Jahanshahi ldquoA new fret-ting fatigue testing machine design utilizing rotating-bendingprinciple approachrdquo The International Journal of AdvancedManufacturing Technology vol 70 no 9ndash12 pp 2211ndash2219 2014

[213] Y-Y Wang and W-X Yao ldquoEvaluation and comparison ofseveral multiaxial fatigue criteriardquo International Journal ofFatigue vol 26 no 1 pp 17ndash25 2004

[214] A Fatemi and P Kurath ldquoMultiaxial fatigue life predictionsunder the influence of mean-stressesrdquo Journal of EngineeringMaterials and Technology vol 110 no 4 pp 380ndash388 1988

[215] W F Findley ldquoModified theory of fatigue under combinedstressrdquo Proceedings of the Society for Experimental Stress Anal-ysis vol 14 pp 35ndash46 1956

[216] N Shamsaei and A Fatemi ldquoEffect of hardness on multiaxialfatigue behaviour and some simple approximations for steelsrdquoFatigueampFracture of EngineeringMaterials amp Structures vol 32no 8 pp 631ndash646 2009

[217] W N Findley J J Coleman and B C Hanley ldquoTheory frocombined bending and torsion fatigue with data for SAE 4340steelrdquo in Proceedings of the International Conference on Fatigueof Metals Institute of Mechanical Engineers London UKSeptember 1956

[218] D LMcDiarmid ldquoA shear stress based critical plane criterion ofmultiaxial fatigue failure for design and life predictionrdquo Fatigueamp Fracture of Engineering Materials amp Structures vol 17 pp1475ndash1485 1994

[219] R N Smith P P Watson and T H Topper ldquoA stress-strainparameter for the fatigue of metalsrdquo Journal of MaterialsChemistry vol 5 pp 767ndash778 1970

[220] M Filippini S Foletti and G Pasquero ldquoAssessment of mul-tiaxial fatigue life prediction methodologies for Inconel 718rdquoProcedia Engineering vol 2 no 1 pp 2347ndash2356 2010

[221] G Sines and J L Waisman Metal Fatigue McGraw-Hill NewYork NY USA 1959

[222] B Crossland ldquoEffect of large hydrostatic pressures on thetorsional fatigue strength of an alloy steelrdquo in Proceedings ofthe International Conference on Fatigue of Metals London UK1956

[223] A Carpinteri M De Freitas and A Spagnoli BiaxialMultiax-ial Fatigue and Fracture Elsevier 2003

[224] M L Roessle and A Fatemi ldquoStrain-controlled fatigue prop-erties of steels and some simple approximationsrdquo InternationalJournal of Fatigue vol 22 no 6 pp 495ndash511 2000

[225] A Carpinteri A Spagnoli and S Vantadori ldquoMultiaxialfatigue life estimation in welded joints using the critical planeapproachrdquo International Journal of Fatigue vol 31 no 1 pp 188ndash196 2009

[226] A Carpinteri A Spagnoli and S Vantadori ldquoMultiaxial fatigueassessment using a simplified critical plane-based criterionrdquoInternational Journal of Fatigue vol 33 no 8 pp 969ndash976 2011


[227] A Carpinteri C Ronchei A Spagnoli and S Vantadori ldquoOnthe use of the Prismatic Hull method in a critical plane-basedmultiaxial fatigue criterionrdquo International Journal of Fatiguevol 68 pp 159ndash167 2014

[228] I V Papadopoulos ldquoLong life fatigue undermultiaxial loadingrdquoInternational Journal of Fatigue vol 23 no 10 pp 839ndash8492001

[229] J L Robert ldquoFatigue life prediction under periodical or randommultiaxial stress statesrdquo in Automation in Fatigue and FractureTesting and Analysis ASTM 1994

[230] F Morel ldquoCritical plane approach for life prediction of highcycle fatigue under multiaxial variable amplitude loadingrdquoInternational Journal of Fatigue vol 22 no 2 pp 101ndash119 2000

[231] Y Liu and S Mahadevan ldquoA unified multiaxial fatigue damagemodel for isotropic and anisotropic materialsrdquo InternationalJournal of Fatigue vol 29 no 2 pp 347ndash359 2007

[232] Y MurakamiThe RainflowMethod in Fatigue The Tatsuo EndoMemorial Volume Butterworth-Heinemann 1992

[233] C H Wang and M W Brown ldquoA path-independent parameterfor fatigue under proportional and non-proportional loadingrdquoFatigueamp Fracture of EngineeringMaterials amp Structures vol 16no 12 pp 1285ndash1298 1993

[234] D Skibicki Phenomena and Computational Models of Non-Proportional Fatigue of Materials Springer 2014

[235] K J Miller ldquoMaterials science perspective of metal fatigueresistancerdquo Materials Science and Technology vol 9 no 6 pp453ndash462 1993

[236] D F Socie and G B Marquis Multiaxial Fatigue Society ofAutomotive Engineers 2000

[237] N Shamsaei M Gladskyi K Panasovskyi S Shukaev and AFatemi ldquoMultiaxial fatigue of titanium including step loadingand load path alteration and sequence effectsrdquo InternationalJournal of Fatigue vol 32 no 11 pp 1862ndash1874 2010

[238] A Aid M Bendouba L Aminallah A Amrouche N Bensed-diq and M Benguediab ldquoAn equivalent stress process forfatigue life estimation undermultiaxial loadings based on a newnon linear damage modelrdquo Materials Science and EngineeringA vol 538 pp 20ndash27 2012

[239] G Mesmacque S Garcia A Amrouche and C Rubio-Gonzalez ldquoSequential law in multiaxial fatigue a new damageindicatorrdquo International Journal of Fatigue vol 27 no 4 pp 461ndash467 2005

[240] S Garcia A Amrouche G Mesmacque X Decoopman andC Rubio ldquoFatigue damage accumulation of cold expanded holein aluminum alloys subjected to block loadingrdquo InternationalJournal of Fatigue vol 27 no 10ndash12 pp 1347ndash1353 2005

[241] A Aid J Chalet A Amrouche and G Mesmacque ldquoA newdamage indicator from blocks loading to random loadingrdquoin Proceedings of the Fatigue Design Conference Paris FranceNovember 2005

[242] A Ince andG Glinka ldquoA generalized fatigue damage parameterfor multiaxial fatigue life prediction under proportional andnon-proportional loadingsrdquo International Journal of Fatiguevol 62 pp 34ndash41 2014

[243] C Wang D-G Shang X-W Wang H Chen and J-Z LiuldquoComparison ofHCF life predictionmethods based on differentcritical planes under multiaxial variable amplitude loadingrdquoFatigue amp Fracture of Engineering Materials amp Structures vol38 no 4 pp 392ndash401 2015

[244] F Ellyin andDKujawski ldquoAn energy-based fatigue failure crite-rionrdquo inMicrostructure and Mechanical Behaviour of MaterialsEAMS 1986

[245] K Golos and F Ellyin ldquoGeneralization of cumulative damagecriterion to multilevel cyclic loadingrdquo Theoretical and AppliedFracture Mechanics vol 7 no 3 pp 169ndash176 1987

[246] G Masing ldquoEigenspannungen und verfestigung beim messing(Self stretching and hardening for brass)rdquo in Proceedings ofthe 2nd International Congress for Applied Mechanics ZurichSwitzerland 1926

[247] K Golos and F Ellyin ldquoTotal strain energy density as afatigue damage parameterrdquo in Advances in Fatigue Science andTechnology vol 159 of NATO ASI Series pp 849ndash858 SpringerBerlin Germany 1989

[248] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987

[249] B N Leis ldquoA nonlinear history-dependent damage model forlow cycle fatiguerdquo in Low Cycle Fatigue ASTM 1988

[250] V M Radhakrishnan ldquoCumulative damage in low-cyclefatiguerdquo Experimental Mechanics vol 18 no 8 pp 292ndash2961978

[251] V M Radhakrishnan ldquoAnalysis of low cycle fatigue based onhysteresis energyrdquo Fatigue amp Fracture of Engineering Materialsamp Structures vol 3 no 1 pp 75ndash84 1980

[252] V Kliman ldquoFatigue life prediction for a material under pro-grammable loading using the cyclic stress-strain propertiesrdquoMaterials Science and Engineering vol 68 no 1 pp 1ndash10 1984

[253] J H Huang Y Si L G Zheng and X H Dong ldquoA dislocationmodel of low-cycle fatigue damage and derivation of the Coffin-Manson equationrdquoMaterials Letters vol 15 no 3 pp 212ndash2161992

[254] J Polak K Obrtlik andM Hajek ldquoCyclic plasticity in type 316laustenitic stainless steelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 17 no 7 pp 773ndash782 1994

[255] J S Santner and M E Fine ldquoThe hysteretic plastic work asa failure criterion in a Coffin-Manson type relationrdquo ScriptaMetallurgica vol 11 no 2 pp 159ndash162 1977

[256] S S Manson ldquoFatigue a complex subjectmdashsome simpleapproximationsrdquo Experimental Mechanics vol 5 pp 193ndash2261965

[257] S SManson andG R Halford Fatigue andDurability ofMetalsat High Temperatures ASM InternationalWest ConshohockenPa USA 2009

[258] UMuralidharan and S SManson ldquoAmodified universal slopesequation for estimation of fatigue characteristics of metalsrdquoJournal of Engineering Materials and Technology vol 110 no 1pp 55ndash58 1988

[259] M R Mitchell Fatigue and Microstructures American SocietyFor Metals 1979

[260] K-O Lee K-H Bae and S-B Lee ldquoComparison of predictionmethods for low-cycle fatigue life of HIP superalloys at elevatedtemperatures for turbopump reliabilityrdquo Materials Science andEngineering A vol 519 no 1-2 pp 112ndash120 2009

[261] A Baumel and T Seeger Materials Data for Cyclic LoadingElsevier 1990

[262] K-O Lee and S B Lee ldquoA comparison of methods for predict-ing the fatigue life of gray cast iron at elevated temperaturesrdquoFatigue amp Fracture of Engineering Materials amp Structures vol39 no 4 pp 439ndash452 2016

[263] G Sines ldquoFailure ofmaterials under combined repeated stresseswith superimposed static stressesrdquo Tech Rep NAAC-TN-34951955


[264] H Majors D D Mills and C W McGregor ldquoFatigue undercombined pulsating stressesrdquo Journal of Applied Mechanics vol16 pp 269ndash276 1949

[265] D Lefebvre K W Neale and F Ellyin ldquoA criterion for low-cycle fatigue failure under biaxial states of stressrdquo Journal ofEngineering Materials and Technology vol 103 no 1 pp 1ndash61981

[266] Y S Garud ldquoA new approach to the evaluation of fatigueundermultiaxial loadingsrdquo Journal of EngineeringMaterials andTechnology vol 103 no 2 pp 118ndash125 1981

[267] K Kanazawa K J Miller andMW Brown ldquoLow-cycle fatigueunder out-of-phase loading conditionsrdquo Journal of EngineeringMaterials and Technology vol 99 no 3 pp 222ndash228 1977

[268] F Ellyin ldquoA criterion for fatigue under multiaxial states ofstressrdquo Mechanics Research Communications vol 1 no 4 pp219ndash224 1974

[269] G Glinka G Shen and A Plumtree ldquoMultiaxial fatigue strainenergy density parameter related to the critical fracture planerdquoFatigueamp Fracture of EngineeringMaterials amp Structures vol 18no 1 pp 37ndash46 1995

[270] E Zalnezhad A A D Sarhan and M Hamdi ldquoAdhesionstrength predicting of CrCrN coated Al7075 using fuzzy logicsystem for fretting fatigue life enhancementrdquo in Proceedings oftheWorld Congress on Engineering and Computer Science vol 1pp 2ndash8 2013

[271] D F Socie ldquoMultiaxial fatigue damage modelsrdquo Journal ofEngineering Materials and Technology vol 109 no 4 pp 293ndash298 1987

[272] C C Chu F A Conle and J J Bonnen Multiaxial Stress-strain Modeling and Fatigue Life Prediction of SAE Axle ShaftsAdvances in Multiaxial Fatigue ASTM 1993

[273] T Lagoda and E Macha ldquoGeneralization of energy-basedmultiaxial fatigue criteria to random loadingrdquo in MultiaxialFatigue and Deformation Testing and Prediction ASTM 2000

[274] E Zalnezhad and A A D Sarhan ldquoMultilayer thin film CrNcoating on aerospace AL7075-T6 alloy for surface integrityenhancementrdquo International Journal of Advanced Manufactur-ing Technology vol 72 no 9 pp 1491ndash1502 2014

[275] T Łagoda ldquoEnergy models for fatigue life estimation underuniaxial random loading Part I the model elaborationrdquo Inter-national Journal of Fatigue vol 23 no 6 pp 467ndash480 2001

[276] T Łagoda ldquoEnergy models for fatigue life estimation underuniaxial random loading Part II verification of the modelrdquoInternational Journal of Fatigue vol 23 no 6 pp 481ndash489 2001

[277] X Su ldquoToward an understanding of local variability of fatiguestrength with microstructuresrdquo International Journal of Fatiguevol 30 no 6 pp 1007ndash1015 2008

[278] J M Boileau and J E Allison ldquoThe effect of solidificationtime and heat treatment on the fatigue properties of a cast 319aluminum alloyrdquo Metallurgical and Materials Transactions APhysical Metallurgy and Materials Science vol 34 no 9 pp1807ndash1820 2003

[279] E Zalnezhad A A D Sarhan and M Hamdi ldquoOptimizing thePVD TiN thin film coatingrsquos parameters on aerospace AL7075-T6 alloy for higher coating hardness and adhesion with bettertribological properties of the coating surfacerdquoThe InternationalJournal of Advanced Manufacturing Technology vol 64 no 1pp 281ndash290 2013

[280] B Skallerud T Iveland and G Harkegard ldquoFatigue life assess-ment of aluminum alloys with casting defectsrdquo EngineeringFracture Mechanics vol 44 no 6 pp 857ndash874 1993

[281] M J Caton J W Jones J E Allison J C J Newman and R SPiascik Fatigue Crack GrowthThresholds Endurance Limits andDesign ASTM 2000

[282] B Bhattacharya and B Ellingwood ldquoContinuum damagemechanics analysis of fatigue crack initiationrdquo InternationalJournal of Fatigue vol 20 no 9 pp 631ndash639 1998

[283] B Bhattacharya A damage mechanics based approach to struc-tural deterioration and reliability [PhD thesis] Johns HopkinsUniversity 1997

[284] N R Hansen and H L Schreyer ldquoA thermodynamicallyconsistent framework for theories of elastoplasticity coupledwith damagerdquo International Journal of Solids and Structures vol31 no 3 pp 359ndash389 1994

[285] C W Gardiner Handbook of Stochastic Methods for PhysicsChemistry and the Natural Sciences Springer New York NYUSA 1985

[286] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006

[287] S Giancane R Nobile F W Panella and V Dattoma ldquoFatiguelife prediction of notched components based on a new nonlin-ear Continuum Damage Mechanics modelrdquo Procedia Engineer-ing vol 2 no 1 pp 1317ndash1325 2010

[288] LM Kachanov Introduction to ContinuumDamageMechanicsMartinus Nijhoff 1986

[289] Y N Rabotnov Creep Problems Under in Structural MembersJohn Wiley amp Sons New York NY USA 1969

[290] J-L Chaboche ldquoContinuous damage mechanicsmdasha tool todescribe phenomena before crack initiationrdquo Nuclear Engineer-ing and Design vol 64 no 2 pp 233ndash247 1981

[291] J Lamaitre and J L Chaboche Mechanics of Solid MaterialsCambridge University Press Cambridge UK 1990

[292] O Salomon SOller andEOnate ldquoFatigue analysis ofmaterialsand structures using continuum damage modelrdquo InternationalJournal of Forming Processes vol 5 pp 493ndash503 2002

[293] V Velay G Bernhart D Delagnes and L Penazzi ldquoA con-tinuum damage model applied to high-temperature fatiguelifetime prediction of a martensitic tool steelrdquo Fatigue andFracture of Engineering Materials and Structures vol 28 no 11pp 1009ndash1023 2005

[294] R Yuan H Li H-Z Huang S-P Zhu and Y-F Li ldquoA new non-linear continuum damage mechanics model for the fatigue lifeprediction under variable loadingrdquoMechanika vol 19 no 5 pp506ndash511 2013

[295] E Santecchia A M S Hamouda F Musharavati E ZalnezhadM Cabibbo and S Spigarelli ldquoWear resistance investigation oftitanium nitride-based coatingsrdquo Ceramics International vol 9pp 10349ndash10379 2015

[296] J J Ping M Guang S Yi and X S Bo ldquoA continuum damagemechanics model for creep fatigue life assessment of a steamturbine rotorrdquo International Journal of Pressure Vessels andPiping vol 78 pp 59ndash64 2001

[297] J JianPing M Guang S Yi and X SongBo ldquoAn effectivecontinuum damage mechanics model for creep-fatigue lifeassessment of a steam turbine rotorrdquo International Journal ofPressure Vessels and Piping vol 80 no 6 pp 389ndash396 2003

[298] R Brighenti and A Carpinteri ldquoA notch multiaxial-fatigueapproach based on damage mechanicsrdquo International Journal ofFatigue vol 39 pp 122ndash133 2012


[299] M Chaussumier M Shahzad C Mabru R Chieragatti andF Rezaı-Aria ldquoA fatigue multi-site cracks model using coales-cence short and long crack growth laws for anodized alu-minum alloysrdquo Procedia Engineering vol 2 pp 995ndash1004 2010

[300] M Suraratchai J Limido C Mabru and R ChieragattildquoModelling the influence of machined surface roughness on thefatigue life of aluminium alloyrdquo International Journal of Fatiguevol 30 no 12 pp 2119ndash2126 2008

[301] M Chaussumier C Mabru M Shahzad R Chieragatti and FRezaı-Aria ldquoA predictive fatigue life model for anodized 7050aluminium alloyrdquo International Journal of Fatigue vol 48 pp205ndash213 2013

[302] R E Peterson Stress Concentration Factors John Wiley andSons 1974

[303] E Mohseni E Zalnezhad A A D Sarhan and A R BushroaldquoA study on surface modification of Al7075-T6 alloy againstfretting fatigue phenomenonrdquo Advances in Materials Scienceand Engineering vol 2014 Article ID 474723 17 pages 2014

[304] D Arola and C L Williams ldquoEstimating the fatigue stress con-centration factor of machined surfacesrdquo International Journal ofFatigue vol 24 no 9 pp 923ndash930 2002

[305] S K As B Skallerud B W Tveiten and B Holme ldquoFatiguelife prediction of machined components using finite elementanalysis of surface topographyrdquo International Journal of Fatiguevol 27 no 10ndash12 pp 1590ndash1596 2005

[306] D M Jafarlou E Zalnezhad A S Hamouda G Faraji N A BMardi and M A H Mohamed ldquoEvaluation of the mechanicalproperties of AA 6063 processed by severe plastic deformationrdquoMetallurgical and Materials Transactions A vol 46 no 5 pp2172ndash2184 2015

[307] M H El Haddad N E Dowling T H Topper and K N SmithldquoJ-integral applications for short fatigue cracks at notchesrdquoInternational Journal of Fracture vol 16 no 1 pp 15ndash30 1980

[308] D Taylor and O M Clancy ldquoThe fatigue performance ofmachined surfacesrdquo Fatigue amp Fracture of EngineeringMaterialsamp Structures vol 14 pp 329ndash336 1991

[309] S Andrews and H Sehitoglu ldquoA computer model for fatiguecrack growth from rough surfacesrdquo International Journal ofFatigue vol 22 no 7 pp 619ndash630 2000

[310] A Staroselsky and B N Cassenti ldquoCreep plasticity and fatigueof single crystal superalloyrdquo International Journal of Solids andStructures vol 48 no 13 pp 2060ndash2075 2011

[311] D Rubesa Lifetime Prediction and Constitutive Modeling forCreep-Fatigue Interaction Gebruder Borntraeger 1996

[312] J S Zhang High Temperature Deformation and Fracture ofMaterials Woodhead Publishing 2010

[313] Y-C Chiou and M-C Yip ldquoAn energy-based damage parame-ter for the life prediction of aisi 304 stainless steel subjected tomean strainrdquo Journal of the Chinese Institute of Engineers vol29 no 3 pp 507ndash517 2006

[314] Z Fan X Chen L Chen and J Jiang ldquoFatigue-creep behaviorof 125Cr05Mo steel at high temperature and its life predictionrdquoInternational Journal of Fatigue vol 29 no 6 pp 1174ndash11832007

[315] W M Payten D W Dean and K U Snowden ldquoA strainenergy density method for the prediction of creep-fatiguedamage in high temperature componentsrdquo Materials Scienceand Engineering A vol 527 no 7-8 pp 1920ndash1925 2010

[316] S-P Zhu H-Z Huang H Q Li R Sun and M J ZuoldquoA new ductility exhaustion model for high temperature lowcycle fatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011

[317] T Gocmez A Awarke and S Pischinger ldquoA new low cyclefatigue criterion for isothermal and out-of-phase thermome-chanical loadingrdquo International Journal of Fatigue vol 32 no4 pp 769ndash779 2010

[318] S Baradaran E Zalnezhad W J Basirun et al ldquoStatisticaloptimization and fretting fatigue study of ZrZrO

2nanotubular

array coating on Tindash6Alndash4Vrdquo Surface and Coatings Technologyvol 258 pp 979ndash990 2014

[319] J F Saltsman and G R Halford ldquoAn update on the total strainversion of SRPrdquo in Low Cycle Fatigue ASTM 1987

[320] J P Pedron and A Pineau ldquoThe effect of microstructure andenvironment on the crack growth behaviour of Inconel 718 alloyat 650 ∘Cunder fatigue creep and combined loadingrdquoMaterialsScience and Engineering vol 56 no 2 pp 143ndash156 1982

[321] L F Coffin Fatigue Annual Reviews 1972[322] M Bao and D G Shang ldquoNew prediction method of fatigue

life under random loadingrdquo Journal of Beijing PolytechnicUniversity vol 27 pp 434ndash436 2001

[323] T Łagoda E Macha and A Niesłony ldquoFatigue life calculationby means of the cycle counting and spectral methods undermultiaxial random loadingrdquo Fatigue amp Fracture of EngineeringMaterials amp Structures vol 28 no 4 pp 409ndash420 2005

[324] D G Pavlou ldquoA phenomenological fatigue damage accumu-lation rule based on hardness increasing for the 2024-T42aluminumrdquoEngineering Structures vol 24 no 11 pp 1363ndash13682002

[325] L Y Xie ldquoDefinition of the effective stress for fatigue damagerdquoChinese Journal of Applied Mechanics vol 9 pp 32ndash36 1992

[326] L Wang Z Wang and J Zhao ldquoLife prediction by ferrite-pearlite microstructural simulation of short fatigue cracks athigh temperaturerdquo International Journal of Fatigue vol 80 pp349ndash356 2015

[327] S R Yeratapally M G Glavicic M Hardy and M D SangidldquoMicrostructure based fatigue life prediction framework forpolycrystalline nickel-base superalloys with emphasis on therole played by twin boundaries in crack initiationrdquo Acta Mate-rialia vol 107 pp 152ndash167 2016

[328] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety of London A Mathematical Physical and EngineeringSciences vol 466 no 2114 pp 423ndash438 2010

[329] E Macha T Łagoda A Niesłony and D Kardas ldquoFatiguelife under variable-amplitude loading according to the cycle-counting and spectral methodsrdquo Materials Science vol 42 no3 pp 416ndash425 2006

[330] J Kim J Yi J Kim G Zi and J S Kong ldquoFatigue life predictionmethodology using entropy index of stress interaction andcrack severity index of effective stressrdquo International Journal ofDamage Mechanics vol 22 no 3 pp 375ndash392 2013

[331] L D Lutes and C E Larsen ldquoImproved spectral method forvariable amplitude fatigue predictionrdquo Journal of StructuralEngineering vol 116 no 4 pp 1149ndash1164 1990

[332] G Petrucci and B Zuccarello ldquoFatigue life prediction underwide band random loadingrdquo International Journal of Fatiguevol 27 pp 1183ndash1195 2004

[333] C Braccesi F Cianetti G Lori and D Pioli ldquoA frequencymethod for fatigue life estimation of mechanical componentsunder bimodal random stress processrdquo Structural Durabilityand Health Monitoring vol 1 no 4 pp 277ndash290 2005

[334] C Braccesi F Cianetti G Lori and D Pioli ldquoThe frequencydomain approach in virtual fatigue estimation of non-linear


systems the problem of non-Gaussian states of stressrdquo Interna-tional Journal of Fatigue vol 31 no 4 pp 766ndash775 2009

[335] A Carpinteri A Spagnoli C Ronchei D Scorza and S Van-tadori ldquoCritical plane criterion for fatigue life calculation timeand frequency domain formulationsrdquo Procedia Engineering vol101 pp 518ndash523 2015

[336] C Braccesi F Cianetti and L Tomassini ldquoAn innovative modalapproach for frequency domain stress recovery and fatiguedamage evaluationrdquo International Journal of Fatigue vol 91 part2 pp 382ndash396 2016

[337] A Niesłony and M Bohm ldquoFrequency-domain fatigue lifeestimation with mean stress correctionrdquo International Journalof Fatigue vol 91 pp 373ndash381 2016

[338] K Walat ldquoSelection of the best method for assessment of thefatigue life of aluminium alloys based on the root of the scattermean-square valuerdquo Acta Mechanica et Automatica vol 6 no1 pp 86ndash88 2012

[339] C Braccesi F Cianetti G Lori and D Pioli ldquoEvaluation ofmechanical component fatigue behavior under random loadsindirect frequency domain methodrdquo International Journal ofFatigue vol 61 pp 141ndash150 2014

[340] C Braccesi F Cianetti and L Tomassini ldquoRandom fatigue Anew frequency domain criterion for the damage evaluation ofmechanical componentsrdquo International Journal of Fatigue vol70 pp 417ndash427 2015

[341] C E Larsen and T Irvine ldquoA review of spectral methods forvariable amplitude fatigue prediction and new resultsrdquo ProcediaEngineering vol 101 pp 243ndash250 2015

[342] C Braccesi F Cianetti and L Tomassini ldquoValidation of a newmethod for frequency domain dynamic simulation and damageevaluation of mechanical components modelled with modalapproachrdquo Procedia Engineering vol 101 pp 493ndash500 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


Among all the available techniques periodic in situ mea-surements have been proposed in order to calculate themacrocrack initiation probability [25]

The limitations of fracture mechanics motivated thedevelopment of local approaches based on continuum dam-age mechanics (CDM) for micromechanics models [26] Theadvantages of CDM lie in the effects that the presence ofmicrostructural defects (voids discontinuities and inhomo-geneities) has on key quantities that can be observed andmeasured at the macroscopic level (ie Poissonrsquos ratio andstiffness) From a life prediction point of view CDM is partic-ularly useful in order to model the accumulation of damagein amaterial prior to the formation of a detectable defect (ega crack) [27]TheCDM approach has been further developedby Lemaitre [28 29] Later on the thermodynamics ofirreversible process provided the necessary scientific basisto justify CDM as a theory [30] and in the framework ofinternal variable theory of thermodynamics ChandrakanthandPandey [31] developed an isotropic ductile plastic damagemodel De Jesus et al [32] formulated a fatigue modelinvolving a CDM approach based on an explicit definitionof fatigue damage while Xiao et al [33] predicted high-cyclefatigue life implementing a thermodynamics-based CDMmodel

Bhattacharya and Ellingwood [26] predicted the crackinitiation life for strain-controlled fatigue loading using athermodynamics-based CDM model where the equations ofdamage growthwere expressed in terms of theHelmholtz freeenergy

Based on the characteristics of the fatigue damage somenonlinear damage cumulative theories continuum damagemechanics approaches and energy-based damage methodshave been proposed and developed [11ndash14 21ndash23 34 35]

Given the strict connection between the hysteresis energyand the fatigue behavior of materials expressed firstly byInglis [36] energy methods were developed for fatigue lifeprediction using strain energy (plastic energy elastic energyor the summation of both) as the key damage parameteraccounting for load sequence and cumulative damage [18 37ndash45]

Lately using statistical methods Makkonen [19] pro-posed a new way to build design curves in order to studythe crack initiation and to get a fatigue life estimation for anymaterial

A very interesting fatigue life prediction approach basedon fracture mechanics methods has been proposed by Ghi-dini and Dalle Donne [46] In this work they demonstratedthat using widespread aerospace fracture mechanics-basedpackages it is possible to get a good prediction on the fatiguelife of pristine precorroded base and friction stir weldedspecimens even under variable amplitude loads and residualstresses conditions [46]

In the present review paper various prediction methodsdeveloped so far are discussed Particular emphasis will begiven to the prediction of the crack initiation and growthstages having a key role in the overall fatigue life predic-tion The theories of damage accumulation and continuumdamagemechanics are explained and the predictionmethodsbased on these two approaches are discussed in detail

2 Prediction Methods

According to Makkonen [19] the total fatigue life of acomponent can be divided into three phases (i) crackinitiation (ii) stable crack growth and (iii) unstable crackgrowth Crack initiation accounts for approximately 40ndash90of the total fatigue life being the phase with the longesttime duration [19] Crack initiation may stop at barriers (eggrain boundaries) for a long time sometimes the cracks stopcompletely at this level and they never reach the critical sizeleading to the stable growth

A power law formulated by Paris and Erdogan [47] iscommonly used to model the stable fatigue crack growth

119889119886

119889119873= 119862 sdot Δ119870

119898

(1)

and the fatigue life 119873 is obtained from the following integra-tion

119873 = int

119886119891

119886119894

119889119886

119862 sdot (Δ119870)119898 (2)

where Δ119870 is the stress intensity factor range while 119862 and 119898

are material-related constants The integration limits 119886119894and

119886119891correspond to the initial and final fatigue crack lengthsAccording to the elastic-plastic fracture mechanics

(EPFM) [48ndash50] the crack propagation theory can beexpressed as

119889119886

119889119873= 119862

1015840

Δ1198691198981015840

(3)

where Δ119869 is the 119869 integral range corresponding to (1) while1198621015840 and 119898

1015840 are constantsA generalization of the Paris law has been recently

proposed by Pugno et al [51] where an instantaneous crackpropagation rate is obtained by an interpolating procedurewhich works on the integrated form of the crack propagationlaw (in terms of 119878-119873 curve) in the imposed condition ofconsistency with Wohlerrsquos law [52] for uncracked material[51 53]

21 Prediction of Fatigue Crack Initiation Fatigue cracks havebeen a matter of research for a long time [54] Hachim etal [55] addressed the maintenance planning issue for a steelS355 structure predicting the number of priming cycles ofa fatigue crack The probabilistic analysis of failure showedthat the priming stage or rather the crack initiation stageaccounts formore than 90of piece lifeMoreover the resultsshowed that the propagation phase could be neglected whena large number of testing cycles are performed [55]

Tanaka and Mura [56] were pioneers concerning thestudy of fatigue crack initiation in ductile materials usingthe concept of slip plastic flow The crack starts to formwhen the surface energy and the stored energy (given by thedislocations accumulations) become equal thus turning thedislocation dipoles layers into a free surface [56 57]

In an additional paper [58] the same authors modeledthe fatigue crack initiation by classifying cracks at first as



(Δ120591 minus 2119896)11987312

1= [

8120583119882119904

120587119889]

12

(4)







(Δ120591 minus 2119896)120572

119894= 037 (

120583119889

119890ℎ)(

120574119904

120583119889)

12

(5)





Fatiguelimit



EPFMtype

cracks

log crack length

d1 d2 d3

log

stres

s ran

geΔ

Zero

Crack speed

LEFMtype

cracks






1 where 119860




2119860

1




1199032)



4

3

2

1

00 1

A2A

1(10minus3)

NNf


1199032)119860

1(119891

119903) for 025

mass C steel 119873119891= 56 000 [74]





2(119891



2(119891

1199032)119860

1(119891

119903)


2119860

1


119889119863

119889119873=

1

119873119888

(Δ1205902 minus 119903

119888(120590)

Δ1205902)

119898

1

(1 minus 119863)119899 (6)






can be written as

119875119909= 1 minus exp (minus120582

119909) (7)





119875 = 1 minus exp (minus119873120592119863) (8)






119875 = [1 minus 119901119904(119863

119904)] [1 minus 119901

120592(119863

120592)] (9)

where 119863119904and 119863



119904(119863

119904)

and 119901120592(119863


than 119863119904and 119863












119889120583

119889119873= exp [119898 ln (Δ119870eff) + 119887] 120583 (119873

0) = 120583

0gt 0 (10)


119889120583

119889119905=

(120597Φ120597119878) sdot (119889119878119889119905)

(1 minus 120597Φ) 120597120583 120583

0(1199050) = 120583

0gt 0 (11)


119886 = 1198860+ (119886

119891minus 119886

0) 119903

119902

(12)

where 1198860 119886 and 119886






0)] = 120583

0and cov[119888(120596 119905

0)] = 119875

0 the stochastic


119889119888 (120596 119905)

119889119905= exp[119911 (120596 119905) minus

1205902

119911(119905)

2] sdot

119889120583

119889119905forall119905 ge 119905

0 (13)



2







119864(119889119886

119889119873)

= 1198601119864 [Δ120590

1198981]Δ120590trΔ120590th

sdot ([119861nom119861

]

119875


1198981

sdot sdot sdot

+ 1198602119864 [Δ120590

1198982]infin

Δ120590tr

sdot ([119861nom119861

]

119901


1198982

(14)

119864 [Δ120590119898

]1199042

1199041= int

1199042

1199041

119904119898

119891Δ120590

(119904) 119889119904 (15)

where 119891Δ120590









119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)










119894le 119871

119894+1le




1= 119875



1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875


119872minus1)

(17)


119875119894=

119891119894


119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)






1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1



119894(119894 = 1119872) and the



119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)






1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





(Δ120591 minus 2119896)11987312

1= [

8120583119882119904

120587119889]

12

(4)







(Δ120591 minus 2119896)120572

119894= 037 (

120583119889

119890ℎ)(

120574119904

120583119889)

12

(5)





Fatiguelimit



EPFMtype

cracks

log crack length

d1 d2 d3

log

stres

s ran

geΔ

Zero

Crack speed

LEFMtype

cracks






1 where 119860




2119860

1




1199032)



4

3

2

1

00 1

A2A

1(10minus3)

NNf


1199032)119860

1(119891

119903) for 025

mass C steel 119873119891= 56 000 [74]





2(119891



2(119891

1199032)119860

1(119891

119903)


2119860

1


119889119863

119889119873=

1

119873119888

(Δ1205902 minus 119903

119888(120590)

Δ1205902)

119898

1

(1 minus 119863)119899 (6)






can be written as

119875119909= 1 minus exp (minus120582

119909) (7)





119875 = 1 minus exp (minus119873120592119863) (8)






119875 = [1 minus 119901119904(119863

119904)] [1 minus 119901

120592(119863

120592)] (9)

where 119863119904and 119863



119904(119863

119904)

and 119901120592(119863


than 119863119904and 119863












119889120583

119889119873= exp [119898 ln (Δ119870eff) + 119887] 120583 (119873

0) = 120583

0gt 0 (10)


119889120583

119889119905=

(120597Φ120597119878) sdot (119889119878119889119905)

(1 minus 120597Φ) 120597120583 120583

0(1199050) = 120583

0gt 0 (11)


119886 = 1198860+ (119886

119891minus 119886

0) 119903

119902

(12)

where 1198860 119886 and 119886






0)] = 120583

0and cov[119888(120596 119905

0)] = 119875

0 the stochastic


119889119888 (120596 119905)

119889119905= exp[119911 (120596 119905) minus

1205902

119911(119905)

2] sdot

119889120583

119889119905forall119905 ge 119905

0 (13)



2







119864(119889119886

119889119873)

= 1198601119864 [Δ120590

1198981]Δ120590trΔ120590th

sdot ([119861nom119861

]

119875


1198981

sdot sdot sdot

+ 1198602119864 [Δ120590

1198982]infin

Δ120590tr

sdot ([119861nom119861

]

119901


1198982

(14)

119864 [Δ120590119898

]1199042

1199041= int

1199042

1199041

119904119898

119891Δ120590

(119904) 119889119904 (15)

where 119891Δ120590









119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)










119894le 119871

119894+1le




1= 119875



1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875


119872minus1)

(17)


119875119894=

119891119894


119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)






1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1



119894(119894 = 1119872) and the



119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)






1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





4

3

2

1

00 1

A2A

1(10minus3)

NNf


1199032)119860

1(119891

119903) for 025

mass C steel 119873119891= 56 000 [74]





2(119891



2(119891

1199032)119860

1(119891

119903)


2119860

1


119889119863

119889119873=

1

119873119888

(Δ1205902 minus 119903

119888(120590)

Δ1205902)

119898

1

(1 minus 119863)119899 (6)






can be written as

119875119909= 1 minus exp (minus120582

119909) (7)





119875 = 1 minus exp (minus119873120592119863) (8)






119875 = [1 minus 119901119904(119863

119904)] [1 minus 119901

120592(119863

120592)] (9)

where 119863119904and 119863



119904(119863

119904)

and 119901120592(119863


than 119863119904and 119863












119889120583

119889119873= exp [119898 ln (Δ119870eff) + 119887] 120583 (119873

0) = 120583

0gt 0 (10)


119889120583

119889119905=

(120597Φ120597119878) sdot (119889119878119889119905)

(1 minus 120597Φ) 120597120583 120583

0(1199050) = 120583

0gt 0 (11)


119886 = 1198860+ (119886

119891minus 119886

0) 119903

119902

(12)

where 1198860 119886 and 119886






0)] = 120583

0and cov[119888(120596 119905

0)] = 119875

0 the stochastic


119889119888 (120596 119905)

119889119905= exp[119911 (120596 119905) minus

1205902

119911(119905)

2] sdot

119889120583

119889119905forall119905 ge 119905

0 (13)



2







119864(119889119886

119889119873)

= 1198601119864 [Δ120590

1198981]Δ120590trΔ120590th

sdot ([119861nom119861

]

119875


1198981

sdot sdot sdot

+ 1198602119864 [Δ120590

1198982]infin

Δ120590tr

sdot ([119861nom119861

]

119901


1198982

(14)

119864 [Δ120590119898

]1199042

1199041= int

1199042

1199041

119904119898

119891Δ120590

(119904) 119889119904 (15)

where 119891Δ120590









119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)










119894le 119871

119894+1le




1= 119875



1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875


119872minus1)

(17)


119875119894=

119891119894


119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)






1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1



119894(119894 = 1119872) and the



119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)






1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119889120583

119889119873= exp [119898 ln (Δ119870eff) + 119887] 120583 (119873

0) = 120583

0gt 0 (10)


119889120583

119889119905=

(120597Φ120597119878) sdot (119889119878119889119905)

(1 minus 120597Φ) 120597120583 120583

0(1199050) = 120583

0gt 0 (11)


119886 = 1198860+ (119886

119891minus 119886

0) 119903

119902

(12)

where 1198860 119886 and 119886






0)] = 120583

0and cov[119888(120596 119905

0)] = 119875

0 the stochastic


119889119888 (120596 119905)

119889119905= exp[119911 (120596 119905) minus

1205902

119911(119905)

2] sdot

119889120583

119889119905forall119905 ge 119905

0 (13)



2







119864(119889119886

119889119873)

= 1198601119864 [Δ120590

1198981]Δ120590trΔ120590th

sdot ([119861nom119861

]

119875


1198981

sdot sdot sdot

+ 1198602119864 [Δ120590

1198982]infin

Δ120590tr

sdot ([119861nom119861

]

119901


1198982

(14)

119864 [Δ120590119898

]1199042

1199041= int

1199042

1199041

119904119898

119891Δ120590

(119904) 119889119904 (15)

where 119891Δ120590









119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)










119894le 119871

119894+1le




1= 119875



1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875


119872minus1)

(17)


119875119894=

119891119894


119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)






1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1



119894(119894 = 1119872) and the



119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)






1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










119875119894= 1 minus (1 minus 119901

119894)119873119894

(16)










119894le 119871

119894+1le




1= 119875



1198911= 119875

1

1198912= 119875

2(1 minus 119875

1)

1198913= 119875

3(1 minus 119875

1) (1 minus 119875

2)

119891119872

= 119875119872

(1 minus 1198751) (1 minus 119875


119872minus1)

(17)


119875119894=

119891119894


119895minus1119891119895

=119891119894

sum119872

119895=119894119891119895

119894 = 1119872 (18)






1198911015840

119894= 119875

1015840

119894(1 minus

119894minus1

sum

119895=1

1198911015840

119895) 119894 = 2119872 (19)

f1 = P1fi Pi

f2 P2

fMPM = 1



119894(119894 = 1119872) and the



119871 =

119872

sum

119894=1

1198911015840

119894119871119894

= 1198751015840

11198711+ 119875

1015840

2(1 minus 119875

1015840

1) 119871

2+ sdot sdot sdot

+ 1198751015840

119872(1 minus 119875

1015840

1) (1 minus 119875

1015840


1015840

119872minus1) 119871

119872

(20)






1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




1 gt 2 gt 3

sumniNi

= (AB + CD) lt 1

i = 1 2

i = 1 2

sumniNi

= (AB + ED) gt 1

00 05

1

1

A E B C D

3

1

2



Cycle ratio niNi

Dam

age

D




119863 = sum119899119894

119873119894

= 1 (21)



119894119873

119894


119863 = sum(119899119894

119873119894

)

119862119894

(22)




119894119873





119894= 120572119873



119894= (1 minus 120572)119873











120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






120590119886=

(119864 sdot Δ120576)

2= 120590

1015840

119891(2119873

119891)119887

(23)




120576119901

119886=

(Δ120576119901

)

2= 120576

1015840

119891(2119873

119891)119888

(24)

with 120576119901

119886 119888 and 120576

1015840



120576119886= 120576

119901

119886+ 120576

119890

119886= 120576

1015840

119891(2119873

119891)119888

+ (1205901015840

119891

119864) (2119873

119891)119887

(25)





119863lowast

=119873119891

sum

119894

V119894119878120573

119894=

119873119891

119873pred (26)



119873lowast

119894= 119863

lowast

sdot 119873119894 (27)



sum119899119894

119873119894

sum1

119860119894120596

119894+ 1 minus 119860

119894

(28)


119894is a material





119896

V119896119878120573

119896 (29)





119873 = 120572119878minus120573

eq (30)


119896

119896 119896 = 1 2 the


= minus120573


119896

V119896120573

119896 (31)



as





119894(119894 = 1 2 3 ) while 119873



2


119886

1Δ120590

119886

2 andΔ120590

119886

3


1198631= 119863

119862


1198731

)

1(1+120595)

]

1(1+1205731)

(33)



1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





1198991

1198731

+ (1198992

1198732

)

(1minus120593)[(1+1205732)(1+1205731)]

= 1 (34)




119863 =

119899

sum

119894=1

120573119894

ln119873119891119894

ln1198731198911

= 1 (35)

where

120573119894=

119899119894

119873119891119894

(36)


119894 and











120590eq = radic1198732

max + (120590119886119891

120591119886119891

)

2

1198622

119886 (37)



120591119886119891






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






Δ120576

2120590max =

(1205901015840

119891)2

119864(2119873

119891)2119887

+ 1205901015840

1198911205761015840

119891(2119873

119891)119887+119888

(38)

Δ1006704120574

2equiv

Δ120574

2(1 +

120590119899max

120590119910

)

=(120591

lowast

119891)2

119866(2119873

119891)119887

+ 120574lowast

119891(2119873

119891)119888

(39)









Δ120576lowast

gen

2= (

120591max1205911015840119891

Δ120574119890

2+

Δ120574119901

2+

120590119899max

1205901015840119891

Δ120576119890

119899

2+

Δ120576119901

119899

2)

max

= 119891 (119873119891)

(40)







Δ119882119905

= Δ119882119901

+ Δ119882119890

(41)


stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




stre

ss

strain

Cyclic C2 = f(2)



1000

800

600

400

200

0

800

600

400

200

0

0

Δ ()

0

Δ

(MPa

)

Master curve

302520151005

30252015

4000

2002 0

00015151005

Δlowast ()

Δlowast

(MPa

)






Δ119882119901

=1 minus 119899

lowast

1 + 119899lowast(Δ120590 minus 120575120590

0) Δ120576

119901

+ 1205751205900Δ120576

119901

(42)


Δ119882119901

=1 minus 119899

1015840

1 + 1198991015840Δ120590Δ120576

119901

(43)






119903119898

= 1 minus

119898minus1

sum

119894=1

119882119891119894

119882119891119898

119903119894 (44)

where 119882119891119894

and 119882119891119898





+

119890)

120593 = 119891 (120595 119901119898) =

1

log 10 (Δ119882119901Δ119882+

119890) (45)



Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Δ120576 = Δ120576119890+ Δ120576

119901= 35

120590119861

119864119873

minus012

119891+ 120576

06

119891119873

minus06

119891 (46)





Δ120576 = 117 (120590119861

119864)

0832

119873minus009

119891

+ 002661205760155

119891(120590119861

119864)

minus053

119873minus056

119891

(47)



Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 150120590119861

119864(2119873

119891)minus0087

+ 059120595 (119873119891)minus058

(48)

where

120595 = 1 when 120590119861

119864le 0003

120595 = 1375 minus 1250120590119861

119864when 120590

119861

119864gt 0003

(49)


Δ120576

2=

Δ120576119890

2+

Δ120576119901

2

= 167120590119861

119864(2119873

119891)minus0095

+ 035 (2119873119891)minus069

(50)


1

1

1

p1

p2 p3

V


1= 02 119901

2= 03 and 119901





119894depends on



0


1198750

(119903)=

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(51)


1198750

=

infin

sum

119903=0

1198750

(119903)=

infin

sum

119903=0

(120583119881)119903

119890minus120583119881

119903[1 minus 119901 (Δ120590)]

119903

(52)



1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





1198750

= 119890minus120583119881

infin

sum

119903=0

[120583119881 (1 minus 119901 (Δ120590))]119903

119903= 119890

minus120583119881

119890120583119881[1minus119901(Δ120590)]

= 119890minus120583119881sdot119901(Δ120590)

(53)








119894

119871 = 11989101198710+

119872

sum

119894=1

119891119894119871119894 (56)



Δ119871119894= minus119891

119894119871119894+

119866119894

1 minus 119866119894

(11989101198710+

119872

sum

119895=119894+1

119891119895119871119895) (57)





119889119886

119889119873= 119862[(120576max

120590119886

120590119910

)

119904

119886]

119905

(58)


119873 = 119870(119886minus119905+1

119891minus 119863

minus119905+1

0)

where 119870 =1

1 minus 119905119862minus1

(120576max120590119886

120590119910

)

minus119904119905

(59)


parameters 120590119886 120590



1198630= [119886

minus119905+1

119891minus (1 minus 119905) 119862(120576max

120590119886

120590119910

)

119904119905

119873]

1(1minus119905)

(60)



120597Ψ

120597120576sdot 120576 minus

120597Ψ

120597119863 ge 0 (61)






120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





120575Ψ (119905) = 120575int

119905

1199050

( minus 119864) 119889119905 minus 120575int

119905

1199050

Γ 119889119905 + 120575119861 (119905) asymp 0 (62)





120590infin

+ 120595119863

119889119863

119889120576+ 119904

119887= 0 (63)

where 120590infin




equal to (1205972




119889119904119887

119889120576= minus119888

1119904119887+ radic119888

2120585 (120576) (64)

where 1198881and 119888







119889119863 (120576) = minus120590infin

120595119863

119889120576 minusradic119888

21198881

120595119863

119889119882 (120576) (65)



120595 = int120590119889120576 minus 120574 (66)


120576 =

119864+ (

119870)

119872

(67)




120574 =3

4120590119891119863 (68)



119889119863 (120576119901) = 119860 (120576

119901) (1 minus 119863 (120576

119901)) 119889120576

119901

+ 119861 (120576119901) 119889119882(120576

119901)

(69)


119891 119872


0 120590

119891 119870119872


= 119863(1205760) is the initial



119863 =

119899

sum

119894=1

119903119909119894

119894 (70)



119894coefficients


120575119863 = 119891 (119863 120590) 120575119899 (71)



120575119863 = [1 minus (1 minus 119863)120573+1

]120572(120590max 120590med)

sdot [120590119886

1198720(1 minus 119887120590med) (1 minus 119863)

]

120573

120575119899

(72)



119886






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






written as follows

119873119891

=1

1 minus 120572

1

1 + 120573[

120590119886

1198720

]

minus120573

(73)


120575119863

120575119899gt 0 (74)

1205752

119863

120575119899120575120590gt 0 (75)




Stre

ss

Damaging

Strengthening

and damaging


Fatigue limit

Fatigue life

U

L

S-N curve






3 Summary



LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




LDR-based

Continuum damage

mechanics

Energy-based

Stochastic-based

Multiaxial and

variable amplitude

loading


Easy methodRea

listic

meth

od

Well timed

Dam

age

conn

ecte

d






References


















































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials











































































































2O3TiTiO





























































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




















































































































































































































2nanotubular



































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Review Article A Review on Fatigue Life Prediction Methods for...

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