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R. Sunder 1

Unraveling the Science of Variable AmplitudeFatigue

ABSTRACT: Conventional methods to estimate variable-amplitude fatiguelife revolve either around cumulative damage analysis using the local stress-strain approach, or, around one of the crack growth load interaction models.Despite advances in modeling the mechanics of fatigue, none of these meth-ods can faithfully reproduce the near-threshold variable amplitude fatigueresponse that determines the durability of machines and structures primarilybecause they fail to model the science behind the residual stress effect. Re-sidual stress effects have a strong bearing on metal fatigue and owe theirinuence to the moderation of crack-tip surface chemistry and surfacephysics. This demands the treatment of threshold stress intensity as a vari-able, sensitive to load history. The correct estimation of crack closure is alsocrucial to determining the variable amplitude fatigue response and demandsassessment of the cyclic plastic zone stress-strain response.

KEYWORDS: fatigue crack growth, variable-amplitude loading, crackclosure, residual stress

Introduction

Many complex phenomena of engineering signicance including heat transfer,

stress/strain distribution in materials and built-up structures, their dynamicresponse, and even uid ow have been understood to a point where analytical

Manuscript received May 2, 2011; accepted for publication November 1, 2011; publishedonline December 2011.1 BiSS Research, 41A 1A Cross, AECS 2nd Stage, Bangalore 560094, India, e-mail:[email protected] at the 11th ASTM/ESIS Symposium on Fatigue and Fracture Mechanics, Ana-heim, CA, USA, May 17-20, 2011. Submitted for publication in ASTM STP.

Cite as: Sunder, R., “Unraveling the Science of Variable Amplitude Fatigue,” J. ASTM Intl. , Vol. 9, No. 1. doi:10.1520/JAI103940.

Copyright VC 2012 by ASTM International, 100 Barr Harbor Drive, PO Box C700, WestConshohocken, PA 19428-2959.

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Reprinted from JAI , Vol. 9, No. 1doi:10.1520/JAI103940

Available online at www.astm.org/JAI

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and numerical modeling, practically from rst principles, can simulate theactual process with amazing consistency. In stark contrast, the science of metal

fatigue has remained largely empirical even after 150 years of intense study. In-credible improvements have been effected in the safety and useful life of suchheavily stressed transportation vehicles such as aircraft and automobiles.These were made possible to a large extent by advances in analytical techniquesrelated to stress-strain distribution in materials and structures under bothstatic and dynamic conditions, and in the area of materials engineering. Thequality of computer-aided design through solid modeling and nite elementanalysis permits even less experienced engineers to ensure a uniform distribu-tion of stresses and avoid localized stress concentration, so that adequate

safety factors can be provided without substantially increasing weight. Finally,fracture mechanics combined with improvements in non-destructive evalua-tion (NDE) allows “on-condition maintenance,” whereby structures andmachines can be periodically inspected and repaired or retired only if neces-sary—“if NDE does not reveal a defect, the structure must be good till the nextinspection.”

A brief review of progress in understanding metal fatigue is made below inan attempt to explain its enigmatic nature. This is followed by a description of two major operative mechanisms that control variable-amplitude fatigue, crack closure, and residual stress. The implications of the synergy of the two inde-pendent phenomena are discussed. The paper concludes with a description of new avenues for research that follow from the discovery of the science behindthe residual stress effect and improved crack closure measurement.

Metal Fatigue—A Chronological Brief

Crucial Early Observation— Railway engineers in the early 19th centurywere shocked to discover that wagon axles made from high quality ductilesteel could inexplicably break like glass, even though operating stress levels

were far less than the tested static strength of these superior quality steels.Thus, the same material would show a “brous” (ductile) fracture when it failsstatically and a “crystalline” (brittle) one when it fails under very long termrepeated loading of low magnitude [1]. This gave birth to the speculation(‘theory’ at the time), that cyclic loading can induce metallurgical transforma-tions even at ambient temperature, forcing local brittle failure along crystallo-graphic planes. Steam from the locomotive owing past axles was cited as onepossibility [2]. The present study proposes, in part, to show that while suchconclusions may seem delusive, the factual signicance of the “crystalline”appearance of high cycle fatigue fractures appears to have been overlooked fortoo long.

Significance of Cyclic Loading— Wohler’s experiments in the mid-nineteenth century opened up metal fatigue to engineering applications [3].He established the concept of the S-N curve that relates fatigue life to the am-plitude of cyclic loading. By performing tests at higher stress amplitudes,Wohler showed that fatigue fractures could retain the “brous” appearance

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associated with static fracture. 2 He also established the idea of a fatigue limitand its relationship with mean stress. In so doing, Wohler put in place the idea

of fatigue being sensitive to both the amplitude and mean level of cyclic load-ing and also the machinery of empirical correlation that continues to serve asthe foundation of fatigue analyses. The signicance of Wohler’s work must be judged against the background of prevailing speculative interpretations of thetime along with the backdrop of the Industrial Revolution. Scientic advanceof the discipline came much later through its association with cyclic slip, assummarised in Fig. 1. This perception served as virtual blinders, clouding formore than a century, a pertinent but inconvenient question: if fatigue is indeeddriven by cyclic slip, why is fatigue life and particularly, fatigue limit, so sensi-

tive to mean stress?3

The link between cyclic plastic strain, reversed slip, anddislocation dynamics appeared to hold much more promise given the nebu-lous nature of the mean stress effect. Additionally, with the subsequent discov-ery of crack closure (to which we will return), the mean stress effect alsoappears to have been treated as effectively ‘closed.’

Cumulative Damage and Service Load Environment— Service loading typi-cally involves a mix of cycles of varying magnitude and asymmetry, with thelargest load occurring extremely rarely in actual usage, if at all. 4 Merely ensur-ing that stresses due to the largest expected load do not exceed the fatigue limitis an impractically safe design proposition except, perhaps, in civil structures.The Miner Rule 5 introduced in the early 20th century attempts to resolve thisproblem by suggesting that the remaining life in a given variable-amplitudeload history undergoes a continuous cycle-by-cycle fractional decrementexpressed as the inverse of total fatigue life after each load cycle [13]. Thus, forany given arbitrary load sequence, failure is associated with the sum of

2) In commenting on Wohler’s collection of laboratory fatigue fractures displayed at theParis Exhibition in 1867, Anon. prophetically observed “M. Wohler’s modest exhibitionmay have been overlooked by ninety nine out of a hundred professional visitors to the Ex-hibition, yet we believe ourselves justied in saying that his scientic and patient experi-ments will be referred to long after the majority of those things which have drawn ashower of medals and ribbons upon themselves at present will be dismissed and for-gotten” [4]. Indeed, in terms of value, Wohler’s lifetime effort appears formidable evengiven today’s experimental resources. Just consolidating the results of his fatigue experi-ments under a vast variety of conditions involving axial, shear, and torsional loadingwould constitute a meaningful research effort.3) Particularly considering that cyclic slip is mean stress insensitive! From the publishedliterature, only Manson’s expression of hope that “a meaningful rationale for the mean-stress effect would be a noteworthy achievement over the coming 25 years” [5] appears tosuggest awareness of the enigma surrounding an important but unresolved phenomenon.4) Examples are the occasional potholes for automobiles and turbulent weather for air-craft. Careless driving over deep potholes and a ight straight into a storm may serve asextreme design considerations.5) Though it is known this way, actually, the rule was proposed some 20 years earlier byPalmgren in Europe.

22 JAI STP 1546 ON FATIGUE AND FRACTURE MECHANICS

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FIG. 1— A brief on metal fatigue. (a) Typical fatigue test results obtained in Wohler’stime [7] shown as tables of max applied stress (fully reversed in tension and compression by rotation-bending) versus cycles to failure. (b) Test results of Wohler and Baush-inger for different steels showing that the fatigue limit is mean stress sensitive [8].Many decades later, these came to be better known as the Goodman diagram [9]. (c) Anew understanding of fatigue emerged with the association of yield with dislocationmovement. Mott’s analog between slip and the ease of moving a fold in a carpet and [10,11] helps explain the formation of persistent slip bands (PSBs) (d) [12]. This, inturn, readily explains why fatigue life is controlled by the plastic strain range (e). (f)Cycles A, B, and C, being identical in magnitude, will cause the same extent of reversed slip or cyclic plastic strain. They ought to result in the same fatigue life, but do not, as shown by Wohler and Bauschinger in (b). This has been an enduring enigma surround-ing metal fatigue.

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cumulative fractional damage from successive load cycles attaining unity. Theidea of cumulative damage is purely notional, carries no scientic rationale,

and is not associated with any entity that could be monitored in real time.Nevertheless, it held out the promise of practical application in designing fordesired nite life, such as the warranty period for non-safety critical engineeredproducts. Any such optimism was soon dashed by Gassner’s experiments undermulti-step programmed block loading [14]. He established that the actual dam-age sum at failure can uctuate wildly, depending on the mix of programmedloads, i.e., that fatigue damage is not linearly cumulative. In the tumultuousyears preceding WWII, Gassner proceeded to develop empirical proceduresinvolving testing under a simulated service environment, in order to obtain fa-

tigue life curves valid for a given material, component, joint, or even structuralassembly, subject to the statistical equivalent of a given service load history.Thus, while Gassner’s effort did nally come up with an engineering solution, itdid so without casting any light on why metal fatigue is so sensitive to loadsequence. Continued emphasis on laboratory testing under a simulated serviceenvironment underscores the signicance of load sequence sensitivity. In themeantime, some four decades after Gassner experiments, the rst analytical ba-sis to account for it emerged in the form of the local stress-strain (LSS)approach.

Local Stress-Strain Approach

Figure 2 summarises the LSS approach that is based on the principle that notch fatigue response will be the same as smooth specimen fatigue response to the simulated notch root stress-strain response. Due to the hysteretic 6 nature of thenotch root inelastic stress-strain response, local tensile yield during an overloadwill cause a downward shift in the local stress response to subsequent elasticloading. Assuming that fatigue is a localized phenomenon, it would follow thataccounting for sequence sensitivity of metal fatigue hinges on the capability tosimulate the notch root inelastic response and then translate that response intolocal stress-strain cycles, identiable for the purpose of a cumulative fatiguedamage estimate after correcting for sequence sensitive local mean stress. TheLSS approach is built around several important advances in applied mechanics.Neuber came up with a simple equation that relates remote elastic loading tolocal inelastic stress-strain at a notch root subject to shear [15]. This wasassumed to be extendable to the axial stress-strain response. A simultaneous so-lution of Neuber’s equation with the Ramberg-Osgood equation [16] yields thelocal inelastic stress-strain response to a given applied load. In the late 1960s,

6) Deviation from linear response due to yield imposes hysteresis upon load reversal. As aconsequence, local stress and strain at any point of time need not be uniquely related toapplied load. They will become sensitive to load history and also to the direction of theload change. Quite simply, hysteresis induces either reduced local stress at the cost of increased local strain, or vice versa.


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FIG. 2— Fatigue damage caused by the two sequences shown in (a) would appear simi-lar, gauging from the smooth specimen elastic response in (b). However, if the two sequences are applied on a notch root seeing the local inelastic response as in (c), thelocal mean stress in cycles B and E will be dissimilar. Thus, if Miner’s Rule appeared to apply to (b), it needs to be adapted to (c) by accounting for load sequence sensitivity of the notch root mean stress. (d) and (e) Local Stress Strain (LSS) approach serves as the foundation of contemporary industrial fatigue design. It incorporates (d) Neuber conversion based on the Masing model of material stress-strain memory [17,18], (e) Rain- ow cycle counting to determine closed fatigue cycles, (f) damage estimates using strain-life data and Miner’s Rule. In practice, case (b) also exhibits load sequence sensi-tivity, rendering the LSS approach questionable.

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Wetzel [17] employed the emerging power of digital computers to combine alinearized Masing model representing material memory effect in stress-strain

response 7 with the Neuber equation into a numerical model, capable of realisti-cally simulating the notch root cyclic inelastic response to an arbitrary appliedload sequence. This made it possible, for the rst time, to visualize the effect of load history in inducing changes to notch root residual stress and therebyaccount for its effect on fatigue damage [18]. Around the same time, Endo [19]came up with the Rainow cycle counting technique to identify closed fatiguecycles from an arbitrary random sequence of peaks and valleys, which is typicalof the service load environment. 8

The early 1970s nally saw the emergence of a numerical apparatus built

around the Neuber conversion, the Masing model, Rainow, and cumulativedamage estimates to calculate notch fatigue life. A timely addition to fatiguetechnology in the 1960s were computer controlled servo-hydraulic testingmachines. They permitted the determination of cyclic stress-strain characteris-tics for use in modeling the material response. They also permitted testingunder both total strain and plastic strain control, so as to obtain strain-life dataunder highly controlled conditions.

The LSS apparatus was amenable to variations in terms of equations to cal-culate damage and correct it for sequence-sensitive mean stress. It was alsoopen to sophistication in terms of accounting for strain hardening and soften-ing, stress relaxation, and creep-fatigue interaction. 9 Continuous advancementin computing power combined with its integration with nite element analysesnow permit the digital simulation of the cyclic stress-strain response at hotspots in a structure for design optimization and durability assurance. Such soft-ware packages form the backbone of contemporary industrial fatigue design.Even so, fatigue critical components are released into the market only after rsttesting their durability and structural integrity in the laboratory under simu-lated service conditions.

The continued need for component-level testing may not merely be a mea-

sure of insurance against the unexpected, but an acknowledgment of the

7) The stress-strain curve of a material can be divided into a number of linear segments.Metals have this amazing property to remember exactly “how much” they have deformedalong each linear segment and, therefore, how much more they can afford to deformalong the same segment. Thus, having exhausted one, their response will move on alongthe next segment and so on. By simulating this response, one can digitally simulate atension-compression stress-strain response in a manner that will be remarkably similarto that of real materials.8) The salient feature of Rainow is its physical consistency. Rainow counted cycles willalways correspond to fully closed stress-strain hysteresis loops required to estimate cu-mulative fatigue damage. Previous cycle counting techniques did not carry a physicalbasis.9) This opened the opportunity for the research community to come up with fairly diverseways of computing damage through a variety of corrections employed to suit observedempirical results, while essentially using the same technique to compute inputs in theform of local stress and strain.


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unknown with regard to variable amplitude fatigue. This possibility is under-scored by a serious shortcoming of the LSS approach, as illustrated in Fig. 3.

For all its sophistication, even the most modern machinery of notch fatigue sim-ulation cannot explain sequence sensitivity under a fully elastic notch rootresponse. Designers strive to ensure that local stresses never exceed yield. Thiseffectively implies that if machines and structures respond in real life the waythey do in simulation, there will be no local inelasticity. 10 Experience showshowever, that while the notch root stress-strain response in real life may remainelastic and therefore, sequence insensitive, sequence effects, in fact, becomemore signicant with reducing overall stress level. This serious anomaly appearsto have remained largely unnoticed in the shadow of the elegance of numerical

simulation.Limitations of the LSS approach should not come as a surprise. In scienticterms, advances over what Wohler had originally conceived some 150 years ear-lier were restricted to the newfound ability to accurately determine the localstress strain response at fatigue critical locations. Note that local stress andstrain amplitude is load sequence independent .11 Their estimation does notactually require the elaborate cycle-by-cycle numerical simulation provided bystate-of-the-art software. The only reason for resorting to cycle-by-cycle simula-tion is to determine sequence sensitive local mean stress. If, indeed, this sensi-tivity disappears under a fully elastic response, there must be other reasons formetal fatigue being load sequence sensitive. The LSS approach elegantly han-dles the mechanics of the notch root response, however. it fails to address thescience behind how such mechanics induce fatigue damage and, particularly,why such damage may be sensitive to mean stress. Viewing fatigue as largely aprocess of crack growth opens the possibility of resolving this problem ( Fig. 4).

The impressive analytical machinery upon which the LSS approach is basedmay indeed provide an accurate picture of the sequence-sensitive notch rootcyclic inelastic stress strain and cycle-by-cycle variation in residual stress underservice loading. However, fatigue crack growth consumes the bulk of total fa-

tigue life and unlike a notch root, the crack tip will, by denition, always see aninelastic cyclic response. Thus, once a crack appears, sequence effects will notonly continue to prevail under the elastic notch root response, but may evenbecome dominant, given the nature of near-threshold crack growth sensitivityto overloads. Obviously, one cannot hope to harmonize variable amplitude fa-tigue test results obtained using the LSS and fracture mechanics approaches asshown in Figs. 4( c) and 4( d).

10) Note that cyclic inelasticity demands the exceedance of twice the yield stress, render-ing it even more improbable in durable designs. However, even such designs often ulti-mately fail in fatigue, suggesting that in real-life cracks can form and grow even in theevent of totally elastic notch root response.11) Local stress and strain amplitude are uniquely related to applied stress amplitude bythe Neuber and Ramberg-Osgood equations, stress concentration factor, Young’s modu-lus, the strain hardening exponent, and cyclic strength coefcient. Applied mean stressand mean strain do not gure in the relationship.

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FIG. 3— (a) Computed fatigue life versus local elastic design stress using the LSS approach for an airframe structural Al-alloy under typical ghter aircraft (FALSTAFF [20]) and transport aircraft (TWIST [21]) load spectra [22]. The shaded area is the esti-mated potential variation due to load sequence rearrangement. Note that curves for both spectra merge into a single line below twice the yield stress (800 MPa), when cyclic slip turns negligible. (b) Schematic notch root response for symmetric load spectrum, and (c) response for asymmetric spectra such as FALSTAFF and TWIST. Even assum-

ing twice the yield strain at the highest load, only symmetric spectra such as rotating parts seeing fully reversed loading are likely to experience cyclic inelastic conditions.Others, as in (c) will not see cyclic inelasticity and, according to the LSS approach, should not exhibit sequence sensitivity. However,in practice they do, and do so to a sig-nicant extent, undermining the credibility of the LSS approach. Sequence effects obviously have to do with the nature of fatigue crack growth. Crack tip response will alwaysbe sequence sensitive because the crack tip will always see a cyclic inelastic response.


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FIG. 4— (a) Fatigue as a crack growth process. Advances in non-destructive inspectiontechnology are likely to increase demands on the ability to model the growth of smaller cracks at lower growth rates. (b) Fractograph of natural crack formation and growth

under 3-step programmed loading in an Al-alloy out of an inclusion seen at bottom left. Each band corresponds to 2000 cycles and is indicative of the reproducibility of the fa-tigue crack growth process even at small crack size and low growth rates [25]. (a) and(b) Are suggestive of fatigue as a crack growth process, sensitive to crack tip cyclic response, rather than of cumulative damage at the notch root. (c) and (d) Range and damage exceedance (RDE) curves computed for Al-alloy L73/2014-T6 under FALSTAFF and TWIST load spectra [26]. 1—Rainow counted cycle range; 2—damage contribu-tion calculated using the LSS approach at 800 MPa (see Fig. 3( a )), and contribution to fatigue crack extension for a small crack [3] and long crack [4]. Note that in FAL-STAFF, just 10% of the cycles (the largest) contribute in excess of 90% of the damage.This explains why the MiniFALSTAFF and FALSTAFF spectra yield similar results. Onthe contrary, in the case of the TWIST spectrum, the LSS and fracture mechanics approach provide contradictory results, with the former wrongly indicating that just some 2% of the cycles contribute all the damage, while in actual experience, the smaller cycles control damage. As shown by curves 3 and 4, when small cycles determine crack growth, load interaction effects gain in importance. This underscores the signicance of the near-threshold behaviour and its potential load sequence sensitivity.

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Modern fatigue critical structures including most airframes are periodicallyinspected for cracks. If no cracks are observed, the structure is released for fur-

ther service until the next scheduled inspection. This implies indenite usage,provided cracks, if detected, are immediately repaired, or the part is replaced.The cost of repair will eventually determine “retirement for cause” [23]. Thecost of inspection, along with its periodicity, will determine the overall econom-ics of operation. In this scheme, the enforced periodicity of inspection is deter-mined by the quality and reliability of non-destructive inspection (NDI), whichneeds to be matched by the ability to correctly estimate the residual life of thestructure with such a crack. Obviously, neither the actual initial defect size(assuming it is smaller than NDI-detectable size) nor the ability to correctly

model very early growth carry value in a condition monitoring scheme.From the overall standpoint of durability assessment, understanding fa-tigue crack growth response below NDI-detectable crack size becomes valuablein the event there is a demand for an extended period of service before rst inspec-tion . It assumes even more importance when the component is not subject toinspection. Additionally, it certainly offers the promise of just doing away alto-gether with the obsolete concept of cumulative fatigue damage. The potentialfor doing so is supported by the highly reproducible growth bands in Fig. 4( b)even at incredibly small crack sizes.

As a rule, the quality of life estimate is inversely proportional to life [24].Assuming the bulk of that life is exhausted by crack growth, the study of nearthreshold variable-amplitude crack growth becomes extremely important.Indeed, the potential for the advancement and application of fracture mechan-ics in structural design over the last four decades has largely overshadowedopportunities presented by the LSS approach.

Fracture Mechanics Approach

With the birth of linear elastic fracture mechanics, the stress intensity factor K

became available, that serves several important purposes. Here, K is, in effect, asimilarity criterion, to which both residual strength and fatigue crack kineticscan be related (see Fig. 5). Paris showed that the fatigue crack growth rate da/dN correlates with the cyclic stress intensity range D K [27]. This was a turningpoint in the advancement of fatigue research. In contrast to a notional parame-ter called cumulative damage, a quantiable parameter in the form of crack sizewas now available to characterize damage. Further, K permits the unication of experimental data for a given material, irrespective of cracked body geometry,crack size, shape, and applied load level. In effect, K is to a cracked body whatstress is to a smooth uniform section specimen. Using K, experimental crack growth data obtained on simple laboratory coupons could be readily extrapo-lated to structural components of engineering interest.

Crack Growth Load Interaction Models

The 1960s saw much progress in unraveling the mystery behind the loadsequence effect researched forty years earlier by Gassner that had debunked the


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FIG. 5— Stress intensity factor K as a similarity criterion for fatigue crack growth. (a)Stress intensity for crack subject to uniform remote stress [1] increases with crack size which is the inverse of the case of rivet (point) load [2]. Correspondingly, the growth rate, da/dN will also vary differently with crack size. Yet, as shown in (b), da/dN for thetwo cases will fall into a single scatter band when plotted against the stress intensity range [28]. Experience shows, however, that the relationship (b) combined with K arenot sufcient similarity criteria for engineering applications. Consider the schematic of the loads in (c) on a transport aircraft at A—take-off and climb, B—cruise, and C— descent and landing (load level on a transport liner gradually drops due to mass reduc-tion from fuel consumption). Crack growth curves will vary as shown in (d), depending on the mere rearrangement of loads [29]. Cycles covering a few thousand ights and re- arranged to form a Hi-Lo programmed sequence will yield a crack growth life about four times greater than if applied as is. This is attributed to load interaction mecha-nisms including crack closure, residual stress, and crack front incompatibility.

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Palmgren-Miner Rule. The advent of precision servo-hydraulics based test sys-tems allowed systematic experiments on variable amplitude fatigue crack

growth. These permitted the study of crack growth rate transients after over-loads and underloads superposed on baseline constant amplitude loading.Experiments came up with the astonishing nding that applying a tensile over-load , in fact, ends up retarding further crack growth even if the crack would havesubstantially incremented during the overload. It was also found that compres-sive overloads (inappropriately called “underloads”) could, in effect, erase theretarding effect of a previous overload. These observations revealed that under variable amplitude loading, the order in which different loads are applied inu-ences the rate of crack advance in a manner that could not be readily explained

by considerations of solid mechanics (see Fig. 5( c) and 5( d)). Clearly, the mate-rial at the crack tip appeared to “remember” what previously transpired in amanner that affected its subsequent fatigue resistance. The search was on forload interaction mechanisms that may be responsible for sequence effects.

Wheeler [30] and Willenborg [31] came up with empirical models on theconsideration that the tensile monotonic plastic zone ahead of the crack tip willact as a wedge squeezed by the elastic matrix to create a zone of compressive re-sidual stresses at the crack tip (see Fig. 6). If an overload is applied, this plasticzone will increase in size as a square function of the overload ratio, leading to asubstantial increase in the near-tip compressive stress. To account for thiseffect, Wheeler introduced a transient retardation factor as a power function of the ratio of remaining crack extension in the overload plastic zone to the size of this zone with constants empirically selected to approximate experimentalobservations. Willenborg interpreted the same effect in terms of a reduced“effective” stress ratio due to increased compressive residual stress, also with atransient function to t real observations. This model relies on Walker’s equa-tion correcting the growth rate for the stress ratio [32].

If, in the 1970s, the LSS approach was already incorporated into commer-cially available industrial software for fatigue design, the Wheeler and Willenborg

models were also brought into the market for the safe-life and fail-safe design of aircraft structures and later, into the nuclear, piping, energy, railroad, automo-tive, and other industries. Forty yearslater, software built around these modelscontinues to dominate industrial fatigue design. Even so, safety critical designsare invariably tested in the laboratory under simulated service conditions.

Fatigue Crack Closure

Just when it seemed that the Wheeler and Willenborg models appeared to holdpromise in application, if not in scientic conviction, Elber’s [33] discovery of crack closure (Fig. 6( e)) nally developed a mechanism that actually makes scientic sense and can be analytically modeled using fracture mechanics con-cepts. Newman [34], de Koning [35], and others came up with numericalmodels of how the plastically stretched wake behind the crack tip effectivelycloses even under tensile load. This was a milestone in the analytical simulationof the mean stress effect in metal fatigue. What is more, the new approach wasable to simulate, with reasonable conviction, the consequences of tensile and


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FIG. 6— General scheme of load interaction models in current use. The action of a tensile overload (a) is described in (b)-(d). A is the monotonic plastic zone from baselineloading and B, the cyclic plastic zone. C is the overload plastic zone and D, the cyclic plastic zone due to overload, that vanishes upon the next tensile cycle. E is the crack wake zone squeezed into bearing by the surrounding stretched material from the plastic zone. (b) Indicates the crack tip picture upon the application of tensile overload. (c)Shows the picture when the crack is almost through the overload plastic zone, and (d)indicates crack tip growing through overload stretched wake. (e) Crack tip response toload sequence 1-5, shown in the inset. Laser interferometry [36] estimates over 0.15 mm gauge length after deducting the elastic response. The loop shape unambiguously underscores the portion of load cycle when the crack was open. Also note that closure is cycle sequence insensitive (2,4 and 1,5 indicate similar closure level). This is proof that closure is insensitive to the cyclic plastic zone response (to crack-tip residual stress). According to both the Wheeler and Willenborg models, compressive stresses in

the overload plastic zone will retard crack growth until the baseline monotonic plastic zone begins to exit the overload plastic zone, as in (c). Using Elber’s closure model, re-tarded growth will persist for some distance beyond the overload plastic zone (d). Nei-ther the Wheeler/Willenborg nor the closure models can explain the possible differencesin crack extension between cycles 2,4 and 1,5. In fact, the rst two actually model closure, even if they may profess to model the residual stress effect!

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compressive overloads. 12 Clinching evidence appeared by way of the ability of theclosure model to explain accelerated crack growth after a step-wise increase in

load and the ner aspect of delayed retardation after an overload. There was noway for the Wheeler and Willenborg models to explain such behaviour. Closureconsiderations make it obvious and simple. When a tensile overload is applied, ittakes some crack growth for the overload induced wake with extra stretch to takeeffect. Therefore, retardation is not immediate. In fact the crack may even mo-mentarily accelerate because the the overload itself opens up the crack, causing areduction in closure stress. However, when a compressive overload is applied theconsequent reduction in closure stress is immediate. Crack closure based modelswere thus able to simulate, through mechanics based computations, many seem-

ingly complex load sequence effects that had hitherto appeared inexplicable. Incrack closure, a scientic explanation at long last seemed available for the effectof both mean stress (stress ratio) and residual stress. All other load interactionmechanisms appeared either insignicant, were perhaps manifested through clo-sure, or, an outright gment of imagination. Or so it seemed.

The 1970s and 1980s saw the publication of over a thousand papers related tocrack closure. The bandwagon soon became an overcrowded train, with individualcoaches representing the variety of sources of crack closure. As it were, Elber’s dis-covery was “merely” of plasticity induced closure. To this were added oxide-inducedclosure, roughness-induced closure, and asperity-induced closure. It was then sug-gested that closure is but one shielding mechanism for a fatigue crack, with the fur-ther division of shielding into extrinsic and intrinsic. Therefore, closure was nowbracketed with crack tip shielding mechanisms such as uncracked bres in the crack wake, or, higher stiffness bres ahead of it. As a consequence, if everything seemedsimple and straightforward as illustrated by Elber’s early work, a much more com-plex and confusing picture seemed to emerge from subsequent research.

The cause of closure has not been helped by an unfortunate aspect of its mea-surement. Unlike parameters that can be directly measured, such as dimensions orweight, or at least by an easy to dene and strictly reproducible process such as mod-

ulus of elasticity, yield stress, or ultimate stress, crack closure measurement carries aheavy measure of interpretation. An annexure to ASTM E647 with a recommendedpractice for closure measurement is a good example of a technique that deliversmeasurements of little practical value. Remote measurement of crack opening dis-placement representing contact response integrated way beyond intervals actuallyaffecting closure carries only a remote chance of correlation with an actual value. 13

12) The Wheeler and Willenborg models could not account for the effect of compressiveoverloads.13) Closure induces a certain wedge opening stress intensity to compensate for the appliedstress falling below closure stress. The contribution to the stress intensity of a point force inthe crack wake will be inversely proportional to its distance from the crack-tip. Assuming com-pressive yield stress upon wake contact, the depth of relevance to closure is of the order of amonotonic plastic zone size. Displacement measurements made remote from this zone of inuence cannot be expected to sense the crack tip response with the desired sensitivity.Indeed, there are no published data showing credible closure measurements under variableamplitude loading.


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The issue is further complicated by difculties in mechanism isolation to eliminateambiguity in the interpretation of the results. For example, would it be fair to attrib-

ute retarded crack growth to roughness induced closure, when the very occurrenceof roughness may have also reduced the intensity of the crack tip stress eld by aragged crack front and possible multiple plane separation?

The technique in Fig. 6( e) involving near-tip laser indentation interferome-try 14 [36] and fractography using the ‘Closure Block’ 15 [37] are exceptions thatdeliver reproducible and scientically defendable results. Unfortunately, theseare not amenable to easy implementation in routine engineering laboratorymeasurements.

Load Sequence Sensitivity of Individual Crack Extension MechanismsWe now proceed to analyse different stages of fatigue crack growth associatedmechanisms and how they may be affected by the variable-amplitude environ-ment. Measurable fatigue crack growth rates range from less than atomic spac-ing, right up to 1 mm/cycle, a potential variation of at least eight orders of magnitude ( Fig. 4( a)). There are not many phenomena of engineering relevance,with such a wide swing in kinetics. Crack extension itself occurs in an environ-ment of several competing mechanisms, with individual mechanisms dominat-ing selected intervals of growth rate. Add to this the different ways in which

ambient conditions can affect individual mechanisms. It would, therefore,come as a surprise if any single crack extension mechanism can describe theprocess. Even more surprising would be a single load interaction model comingup with consistent estimates of variable-amplitude crack growth rates. Forclarity, we broadly divide crack kinetics into three distinct ranges of the crack growth rate and proceed to examine how the dominant crack extension mecha-nism in each range responds to variable-amplitude loading. Before doing so, wedene a basic assumption that is required to distinguish fracture mechanicsbased analysis from cumulative damage concepts.

History Effect on Crack Extension— Consider crack extension in identicalcycles A, B, and C shown in Fig. 7( a) with different loading histories. Case (a)involves constant amplitude loading. Cases (b) and (c) involve prior cycling atincreased loading amplitude, causing greater near-tip cyclic slip. Based on cu-mulative fatigue damage considerations, one should expect crack extension C toexceed B and for both to exceed A due to greater “prior damage,” However,there appears to be absolutely no empirical evidence to suggest such a possibil-ity! 16 Fracture mechanics based models of variable-amplitude fatigue, in fact,

14) With a working gage length of the same order as the plastic zone size, this technique issensitive to the inelastic stress-strain response within the cyclic plastic zone as seen inFig. 6( e).15) The technique proceeds on the premise that given constant K max , there is no other ex-planation for equal striation spacing under varying K min other than equal D K eff .16) Not necessarily because such a possibility does not exist, but rather, because of the lim-itations in experimental techniques to address the question in quantiable terms.

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FIG. 7— (a) According to the cumulative damage concept, identical load cycles A, B, and C may extend the crack differently because of the different load history preceding each of them. In contrast, all crack growth models ignore the possibility of damage tomaterial ahead of the crack tip. This understanding is central to analytical modeling of load history effects. (b) The three growth rate regimes and their associated fractures for

an Al-alloy. Crack extension in a cycle under variable amplitude loading may fall into any of these three regimes, depending on its magnitude. (c) During the rising half cycle shown in the inset, the crack will rst extend by brittle micro-fracture (BMF) over a -nite number of atomic layers embrittled by instantaneous surface diffusion (ii), andthen switch to shear extension (iii), suggesting striation formation by the mode change(iv) [39]. Any further increase in load beyond 2 may induce a disproportionately higher quasi-static crack extension. This explains why striations marking individual cycles are seen only over a very narrow range of growth rate.


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simply ignore it. They assume that the crack extension in the next cycle is drivenonly by the magnitude of that cycle. The prevailing understanding of crack

growth load interaction effects is also based exclusively on variables that controlcrack kinetics in the next load cycle. It ignores any prior “slip-reversal damage”to the crack tip. In the absence of compelling arguments to the contrary, weshall ignore any prior damage and its effect in considering dominant crack extension mechanisms and how they respond to variable amplitude loading. Indoing so, we make an important assumption that the fatigue crack can extendunder each load cycle. 17

Dominant Crack Extension Mechanisms— At the commencement of the rising

half of a new load cycle, the dominant crack extension mechanism is still anunknown. Crack extension will commence by a yet to be dened mechanism oncethe load excursion exceeds a certain threshold value. It will soon transform to stria-tion mode as the stress intensity falls into the Paris regime and then proceeds toextend through local quasi-static fracture in the event K approaches critical values(see Fig. 7( b)). Each of these three stages occupies a nite but overlapping intervalof crack growth rates, with the rst transition occurring around 10 4 mm/cycleand the second one depending largely on the stress ratio, around 10 2 mm/cycle.With the increasing stress ratio, this last transition will progressively move intolower growth rates because of the onset of quasi-static fracture leading to a short-ened Paris interval. Note that the different stages in crack growth are associatedwith the change in growth rate over several orders of magnitude. Higher ordergrowth rates will necessarily be associated with a mix of mechanisms 18 (seeFig. 7( c)), though the last mechanism to switch-in would emerge as the dominantone by virtue of its disproportionately large contribution to crack extension.

The above rationale suggests that in variable amplitude fatigue, a variety of crack extension mechanisms will continuously leave an imprint on the fracturesurface and their mix will depend on the load spectrum. A corresponding mix of load interaction mechanisms may also continuously prevail. We now proceed to

consider in greater detail, individual crack extension mechanisms and howeach one may be sequence sensitive. In doing so, less signicant load interactionmechanisms such as crack-tip blunting/resharpening, history-induced phasetransformations, and other such effects whose inuence cannot be deemed deci-sive or quantiable are ignored.

High-End Growth Rates

The crack tip will see critical conditions associated with catastrophic fracturewhen K approaches K c associated with static fracture. Such local failure is

17) Crack growth rates less than atomic spacing are readily explained by the possibility of local crack extension occurring at different points on the crack front at different times[38].18) After all, the crack tip at the commencement of rising load half-cycle, “does not yetknow” the extent to which it will be loaded. It will switch sequentially to the “mechanismof least resistance to crack extension” corresponding to the instantaneous load increment.

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attributed to quasi-static rupture of the material directly ahead of the crack tip.If the material is inherently brittle, it will simply cleave locally. If it is ductile, as

is the case with most aerospace structural materials, at least two simultaneousmechanisms are likely. Stable crack growth by shear can be either Mode II orMode III. This typically occurs at the specimen edge, where plane stress condi-tions promote shear ligament formation and gradually spread inward, becauseligament formation demands crack extension. 19 A little deeper, and particularlygiven a straight crack front, plane strain conditions associated with constraintcan prevail, leading to the buildup of hydrostatic tension 20 that can result instatic rupture by microvoid coalescence (essentially, an analog of cavitation inliquids), seen on the fracture surface as clusters of microscopic cavities, irrefu-

table evidence that local failure was instantaneous. Note that because condi-tions of constraint develop at some distance from the crack tip, crack jump ortunneling by microvoid coalescence will invariably be accompanied by a shearof the interim ligament at the very tip of the crack that remained under planestress. A third mechanism is typical of Al-alloys and the proliferation in them of secondary particulates that are natural barriers to slip. As a consequence, if sizeable slip is involved that covers a distance exceeding their average spacing, astrain localization will result, leading to a shear fracture along interconnectingplanes between particulates. This leads to the appearance on the fracture sur-face of a disproportionately high density of particulate voids, that should not beconfused with microvoid coalescence associated with static fracture as was thecase in. An example of a mix of the two appears in Fig. 7( b) (also, see Fig. 10( b)).Being a highly localized phenomenon, such ruptures may occur momentarilyand only at one or a few points ahead of the crack front. This, in macroscopicterms, will show up as increasingly accelerated fatigue cracking as K max undercyclic loading approaches K c .21 One may expect that as the ratio K max /K capproaches unity, the crack growth rate will approach innity (static fracture).

19) As a rule of thumb, the crack needs to extend over an interval of at least half the speci-men thickness in order for the front to completely rotate to shear mode. Quite simply,front rotation also demands extension.20) Liquids follow Pascal’s Law. Applying pressure at any point will result in all ends of theconstraining container seeing that pressure. This is what drives uid power technology.Solids are different from liquids in their resistance to sliding (shear or slip), which isinnitely higher than viscosity in liquids. Therefore, when a smooth solid specimen ispulled, it will readily transversely contract, as seen on a rubber band. However, if forsome reason such a contraction is inhibited by external or internal conditions (con-straint), a hydrostatic response will result, whereby tension will be experienced in alldirections. An example of hydrostatic tension in the response of secondary particulates isforthcoming. A stress gradient serves as a natural constraint and can result in a near-hydrostatic local response.21) In the presence of a substantial quasi-static crack extension, one can hear audible pop-ins. Much lower levels of such an extension can be picked up by acoustic emission, whichoften serves as a tool for on-line structural diagnostics. This is used in industry to “hear”defects growing in a structure and to locate them by triangulation, much like GPS posi-tioning systems.


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Forman et al. introduced such a correction into the crack growth rate equationwhich otherwise only carried two material constants to be determined by statis-

tical analysis of laboratory data. The correction kicks in only at higher values of D K, or at a very high stress ratio, where K max gets closer to K c even at lower D K.

Critical conditions associated with local quasi-static crack extension requirehigh stress and strain levels. Since these will be tied to the top end of the localstress-strain hysteresis loop, they may be immune to hysteretic effects andtherefore insensitive to load history. Also at these levels, crack closure has prac-tically no role to play because the process is driven by the maximum drivingforce, rather than its range. There is, however, some possibility of effects attrib-utable to strain hardening or softening that may affect local fracture resistance

and will be stress history sensitive. Importantly, the crack-tip stress-strainresponse will be extremely sensitive to local constraint. This will vary across thethickness and will also be determined by instantaneous crack front orientationas well as shape, that is, in effect, determined by the cumulative preceding crack extension. Of all the load history related parameters, this one appears worthy of analytical consideration at a high growth rate. To do so, one may treat K c as acrack front related parameter varying between a low of K 1c associated withplane strain and a high of K c , associated with plane stress and therein introducea history sensitive component into the Forman equation to account forsequence sensitivity of high end growth rates.

In summary, the effect on high end growth rates of the loading history maybe accounted for by correcting K and K c for crack front shape and orientation.Parameters such as crack closure and residual stress will have little bearing onhigh-end growth rates.

Intermediate (Paris Regime) Range Growth Rates

The Paris Regime is characterized by a log linear relationship between D K andda/dN over a range of growth rates covering the interval 10 4–10 2 mm/cycle

with nonlinearity at the high end coming in due to the quasi-static componentand with the lower end overlapping with near-threshold fatigue response. Theinterval is dominated by cyclic slip driven crack-tip extension, according to a va-riety of schemes proposed in the literature [41]. A reasonably straight crack front is conducive to transgranular slip along preferred planes and one canreadily accept the possibility within individual grains of highly reproducibleextent of stretch and compression in successive load cycles that leave behindstriation bands with near digital precision. 22

There are different ways in which a crack can extend over a load cycle in a pre-dominantly slip dominated mode. The rst is by deformation (as opposed to

22) Reference [42] describes an experiment that involved “punching” onto the fracture sur-face of fatigue striations representing binary code of text strings in much the same way asinformation is stored on digital media. This would not be possible without precisely re-producible cycle-by-cycle fatigue crack extension at the microscopic scale and serves as acompelling argument in favour of fractography as a dependable tool not only in failureanalysis, but also for the quantitative validation of crack growth models.

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fracture), whereby the shear stretch produced in the rising half cycle cannot befully reversed upon unloading, resulting in a fold, as indicated by the well-known

Laird model [43]. From this, follows the unexpected conclusion that the crack extends during unloading. The second possibility is that the crack extends by shearfracture [44], whose extent is determined by rising load excursion exceeding a cer-tain threshold level over which microscopic stable crack extension occurs, but notunstable (even if localized) fracture. Reversed deformation during unloading willessentially prepare a sharp crack for extension in the next cycle. 23 The third possi-bility is a combination of the two, leading to a somewhat greater crack extensionconsidering that the crack will continue to grow during unloading as well. All threepossibilities are supported by observations of extremely well dened striations that

mark the fatigue fracture surface, though the textbook understanding is of fatiguecrack extension by deformation (slip), not shear fracture.Assuming that the crack-tip response is controlled exclusively by the cyclic

stress-strain curve and the extent of change in stress intensity, crack extension inthis range should be insensitive to the applied stress ratio and to near-tip meanstress (i.e., residual stress). Mean stress insensitivity is the very essence of a processdriven by slip alone. It follows that any sensitivity of intermediate range crack growth rates to the stress ratio and to the load history may be attributed largely, if not solely, to crack closure. An inevitable conclusion then would be that if theWheeler and Willenborg models indeed correctly simulate intermediate range vari-able amplitude behaviour, they may be merely appearing to do so by the happycoincidence of fudged closure response. Indeed, if fatigue crack growth is predomi-nantly slip driven, the only plausible explanation for the stress ratio and load historyeffects is fatigue crack closure controlling the effective range of the stress intensity.

All three possible ways of crack extension by slip previously listed carry cer-tain implications that go beyond insensitivity to residual stress, stress ratio, andstress history. They imply cycle-by-cycle striation formation. They also implyrelative immunity of the Paris Regime to the environmental effect (assumingslip is environment independent) and to cycling frequency (assuming rate-

insensitivity of slip over the practical range of frequency). Sensitivity to environ-ment and frequency increases at lower growth rates associated with thresholdsand at much higher rates associated with sustained load cracking, creep, etc.

There are two curious features of intermediate range crack growth whosesignicance appears to have remained largely unnoticed over the ve decades of study by high resolution electron fractography. One is the surprisingly narrowband of growth rates (usually within one or two orders ofmagnitude of varia-tion) over which discernible striations are observed. 24 The other is the surpris-ing absence of striations in vacuum.

23) A blunt crack tip offers multiple parallel slip planes that will contribute to cumulativestretch by dissipating total strain. A sharp crack restricts the number of shear planes andthereby encourages shear fracture by focusing shear strain into fewer slip planes.24) The resolution of electron fractography is adequate to resolve a crack extension lessthan 10 6 mm/cycle, but one seldom sees striations at growth rate less than 10 4 mm/cycle.


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Near Threshold Fatigue Crack Response

Indeed, why are striations not discernible in vacuum? And why are we usuallyunable to see striations in atmospheric fractures at growth rates below10 4 mm/cycle, even if electron microscopes can resolve features one hundredtimes smaller? The controversial brittle micro-fracture (BMF) model of near-threshold crack growth 25 appears to provide the answer [45] (Fig. 8). The fa-tigue crack tip represents an extreme stress concentrator. Associated with suchstress concentration is an extreme stress gradient that in turn induces condi-tions of severe near-tip constraint because the surrounding lightly stressed ma-terial does not permit local necking. This, in turn, induces conditions of hydrostatic loading: application of tensile load normal to the crack plane causesincreasing tensile stresses in the transverse direction as well. Stress in the thirddirection along the major crack axis will be somewhat relaxed, at least at thetip, because the free crack tip surface is free to move inward into the material.However, the “diaphragm” stresses stretching the crack tip surface in two direc-tions will increase the inter-atomic distance along the loading axis while notallowing transverse spacing to reduce. Such conditions are conducive to theactivation of surface physics (diffusion of active species into surface layers) andsurface chemistry (chemical reaction with active species), leading to acceleratedtransgranular 26 fatigue crack extension [46].

In a careful study on the near-threshold fatigue fracture mode of anAl-alloy, Gangloff et al. observe that crack extension occurred along crystallo-graphic slip planes [47]. This by itself need not imply that crack extensionoccurred by slip unless it can also be shown that the fracture plane was orientedappropriately with respect to the loading direction 27 as is the case with ductileresponse and striation formation in the Paris Regime. Once a surface layer hasbeen embrittled, it may not matter whether Mode I (tensile rupture) or II (slip)is involved. If Mode II was indeed involved, it would lead to the formation of shear lips and progressive rotation of the fatigue fracture plane by 45 . How-ever, atmospheric fatigue fracture surfaces in the near-threshold regime do nottend to develop shear lips. They remain at and normal to the loading plane.

Compelling evidence in support of the previous rationale comes by way of fatigue fractures obtained in salt water, air, and vacuum [48,49] under identicalloading conditions. The authors attributed the delayed transition to shear modein salt water and air out of early Mode I cracking to the adverse effect of envi-ronment on resistance to Mode I. A crack tip stress state is determined by shear-lip formation, which, in turn, is driven by dominant macroscopic mode of crack

25) Against the general perception of metal fatigue being associated with cyclic slip (defor-mation), the BMF model suggests that near-threshold fatigue crack extension occurs byfracture.26) This is not to be confused with the mechanism of stress corrosion cracking associatedwith the intergranular short circuit diffusion of active species that essentially leads tocrack extension by grain separation.27) Just as delamination in composites can occur either by Mode I or Mode II.

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FIG. 8— The science behind the residual stress effect in metal fatigue crack growth. (a)When an argon bubble is inserted under ruthenium monolayers, the stretched topinstantaneously attracts active species, while the compressed region at the root of theblister repels them [46]. (b) According to the BMF theory, the same holds true at the fa-tigue crack tip [45]: the active species is moisture at room temperature that is repelled from the crack tip at minimum load, 1. During the rising half-cycle, moisture molecules are attracted by the rising stresses at the crack tip. They react with metal to form metal oxide and hydroxide to release hydrogen that diffuses into the substrate to embrittle and fracture the affected surface layers under rising stress. (c) The surface physics and chemistry described in (b) will be affected by the crack tip stress history as shown by the schematic repeat action of load sequence 1-7. (d) If closure is reduced or absent (Lo-

Sop), cycles 2-3 and 5-6 will see hysteretic crack-tip stress-strain response. Higher stress causes more BMF at 2-3 than at 5-6. (e) However, if the crack is partially closed during2-3 and 5-6, both cycles will see similar reduced local stress and therefore, equally re-tarded crack extension. (b)–(e) Underscores the signicance of the cyclic plastic zone response in controlling atmospheric sub-critical fatigue crack growth. Closure andWheeler/Willenborg models are incapable of explaining cycle-by-cycle hysteretic loadinteraction effects in fatigue crack growth.


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extension, rather than by applied D K 28 or even the growth rate. Thus, in high vacuum, shear lips will form earlier than in salt water or even air, even if vac-

uum growth rates will be much lower, given similar loading conditions. Pippanet al. have observed that the fatigue crack stays sharp in air and turns blunt in vacuum 29 [50] but failed to draw conclusions on how this may reect on thecrack extension mode. Embrittled surface layers will also exhibit reduced elon-gation. As a consequence, the crack may extend by BMF before the potentialonset of slip on neighboring planes that promote blunting, or, on the sameplane, but deeper into the substrate.

In room temperature atmospheric fatigue, BMF appears to be primarilypromoted by surface diffusion of hydrogen released by the reaction of moisture

with the crack tip surface resulting in oxide and hydroxide formation. Oxidationappears to be an unlikely factor in BMF, a conclusion prompted by the retardednear-threshold fatigue crack growth in dry oxygen observed by Bowles [51]. Intests on an Al-alloy, Bowles also observed that when the environment isswitched from laboratory air to dry oxygen, striations gradually disappear, leav-ing a surface akin to that obtained in vacuum. 30 This observation also points tothe potential role of BMF in striation formation. The BMF controls the near-threshold fatigue response that extends up to a growth rate of between 10 5 and10 4 mm/cycle, suggesting that surface physics and chemistry do affect tens,but perhaps not hundreds or thousands of atomic layers at the crack tip. Per-haps crack extension by the BMF (mode I) over such a distance in the course of the rising load half cycle, when followed by subsequent crack extension eitherby shear in Mode II, or, by folding of shear stretched crack tip surface, or, by acombination of the two leaves that distinct wavy pattern one associates withwell-dened striations. Striation formation may thus require two distinctly dif-ferent crack extension mechanisms to operate sequentially (as shown by theschematic in Fig. 7( c)). If only one of them operates as in the case below theParis Regime (only BMF and no slip) or in high vacuum (only slip and no BMF),no discernible contrasting topographical feature may result to mark the pro-

gress of the crack front.Just as room temperature near threshold fatigue is closely linked withcycle-by-cycle crack extension by the BMF of crack-tip surface layers embrittledby surface physics and chemistry, a similar process may control elevated

28) The ratio of plastic zone size to thickness is often treated as a reection of the stressstate. Implicit in this assumption is a at and straight crack front. In reality, a curved(tongue shaped) crack front or one that is tilted will both promote plane stress due toliga-ment response.29) Interestingly, having obtained lucid evidence about the cause (sensitivity of crack-tipdeformation to environment), the authors seem to have failed to draw the logical conclu-sion about its effect (sensitivity of the crack extension mode to the environment)!30) Their ‘gradual’ rather than immediate disappearance also raises the intriguing ques-tion of hydrogen consumption. Does hydrogen get consumed by embrittlement, or does itescape upon BMF to affect the next layer? Partial consumption can explain the momen-tary persistence of BMF into vacuum. It may also explain sustained accelerated internalcracking as in gigacycle fatigue.

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temperature transgranular fatigue crack growth. The latter is accelerated by theenhanced oxidation of crack tip layers that can considerably exceed the depth of

moisture related surface diffusion by hydrogen. In both cases, crack extensionis transgranular and involves cycle-by-cycle crack tip surface activity that accel-erates crack extension by comparison to vacuum fatigue response. For this rea-son, in both cases, the threshold stress intensity will be much less than in high vacuum. It thus emerges, that, if near-threshold behaviour is sensitive to diffu-sion kinetics, threshold stress intensity ought to be controlled by the cyclic plas-tic zone response!

Threshold Stress Intensity

While near-threshold fatigue crack growth behaviour has long been connectedwith crack-tip surface physics and with surface chemistry [52, 53], there seemsto have been a general failure to appreciate the connection between the kineticsof surface activity and near tip hydrostatic stress, and the sensitivity of the latterto the stress history and to stress ratio. This may be attributed to the prevailingstereotype of the crack tip essentially seeing an elastic, ideally plastic cyclicresponse, implying a local stress ratio of R ¼ 1 (zero mean stress), irrespectiveof the applied stress ratio and stress history. Such an assumption may haveassisted crack-tip elasto-plastic stress-strain analyses and may also have been

appropriate for the Paris Regime with its sizeable cycle plastic zones. However,it appears to have clouded the signicance of the hysteretic stress-strainresponse within the cyclic plastic zone in moderating near-threshold diffusionactivity at room temperature and chemical reactivity at elevated temperatures.On the contrary, the signicance of residual stress is well known and appreci-ated in stress corrosion cracking, as is practiced in assessing heat affected zonesin welding and crack growth in an aggressive environment. There is, however,an extremely important distinction between stress corrosion cracking andatmospheric near-threshold fatigue crack growth. Near threshold fatigue crack growth is affected by near-tip stress zones that are hydrostatic and microscopicin comparison to those considered in stress corrosion cracking.

When crack-tip cyclic slip recedes below a certain threshold, crack growthwill practically cease in vacuum. This vacuum threshold stress intensity range isabout three times greater than effective D K th in air. Crack growth in air at D K less than vacuum D K th cannot obviously be explained on considerations of slip.The difference between the two when raised to the power of four and above,covers a growth rate variation in air, exceeding two orders of magnitude. The in-terim interval thus covers a vital segment of the sub-Paris regime atmosphericfatigue response that may well account for the bulk of total fatigue life. 31 Fa-tigue kinetics over this interval will obviously be determined by the ability of

31) In observing fatigue fractures, one may be inclined to associate the bulk of the fatigueprocess with the largest observable area of the fatigue fracture. However, the bulk of fa-tigue life may, in fact, have been consumed in early crack growth. While assessing fatiguefractures, it may be important not to ignore that, albeit small, region covering the crack initiation area.


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hydrostatic stresses within the cyclic plastic zone to moderate crack-tip surfacechemistry and surface physics. These stresses are the sum of the crack-tip mean

stress associated with the current stress ratio, history sensitive, residual stressand their hysteretic variation while the crack remains open. 32

The consequences of such a “crack-tip cyclic diffusion pump” may be var-ied. An arrested crack tip will progressively lose its resistance under the persis-tent onslaught of diffusing active species. Hydrogen trapped in the rising loadhalf-cycle will not be released upon unloading. Oxidation at an elevated temper-ature accelerated during the rising load half-cycle will not be reversed uponunloading. This implies that the crack front will be inclined to straighten itself even if the crack does not uniformly extend in successive cycles. Over each cycle

that the tip does not give way, surface layers are likely to see a little moreembrittlement. At the same time, interstitial diffusion is a self-retarding processbecause diffused layers represent barriers to newer and deeper diffusion. This iswhy the effect in question is unlikely to signicantly inuence growth rates inexcess of 10 4 mm/cycle. Another measure of the effect can emerge from a com-parison of Paris Regime growth rates in air and high vacuum. Their differenceis substantially less than under near-threshold conditions. Thus, while a fatiguecrack in air can grow at 10 5 mm/cycle, it may just remain arrested under high vacuum given the same loading conditions.

Significance of Cyclic Plastic Zone Response in Variable Amplitude Fatigue

Near-tip residual stress is highly sensitive to load history and can vary substan-tially on a cycle-by-cycle basis. Figure 8( c)–8( e) schematically illustrates thecycle-by-cycle near-tip stress history for a fully open and for a partially closedfatigue crack. Consider the repeated action of cycles 1-7. Identical embeddedcycles 2-3 and 5-6 will see the consequences of the hysteretic crack tip responsewithin the cyclic plastic zone. In both cases, near tip stress will be well belowthat under constant amplitude loading because of the compressive stress intro-duced by tensile overload 4. As a consequence, cycles 2-3 and 5-6 will see re-tarded crack extension. However, the retardation in 5-6 will be greater becauselower local stress reduces the diffusion activity with the crack growth tendingtowards vacuum response. Importantly, the difference between 2-3 and 5-6being hysteretic can be seen on a cycle-by-cycle basis. This is possible only if thecrack is fully open during the embedded cycles as in Fig. 8( d) and with closurestress well below the minimum stress in the two cycles. The hysteretic variationwill cease in the event of partial closure as shown in Fig. 8( e). Both cycles willbe equally retarded in this case, being rendered cycle-sequence insensitive dueto partial closure. Cycle-sequence sensitivity is a term alien to conventionalmodeling based on the Wheeler, Willenborg, or Elber models. 33 The signicance

32) An important consequence of this possibility is that a partially closed crack will not seecycle-sequence sensitivity, a feature to be addressed further in the text.33) Curiously, interpretation of notch root fatigue response universally proceeds on this very understanding, and has remained unquestioned, even in the absence of any scienticrationale for the notch root mean stress effect!

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of the cyclic plastic zone and the possibility of cycle-sequence sensitivitybecomes obvious only when viewed from the perspective of the BMF model.

Figure 9 describes the extension of this new understanding to re-interpretthe simple case of tensile and compressive overloads and how their action maybe modeled. Consider the case of identical cycles AB, DE, and GH in Fig. 9( a)and 9( b). We assume the stress ratio to be sufciently high in order to precludethe possibility of crack closure. Near-tip cyclic stress strain response for thesecycles appears as Fig.9( c). We see that the baseline cycle AB would be associatedwith a certain near-tip stress r A. If the stress ratio of this cycle had been highersuch that A and C were equal, the local stress would have risen to r C . In theevent of tensile overload as in Fig. 9( a), the following cycle ED sees a deep drop

in the near-tip stress to r E . However, if a compressive overload followed the ten-sile overload as in Fig. 9( b), the following cycle GH will see an increased localmean stress due to the preceding yield in compression at F. The new meanstress will still be lower than in AB. Assuming a unique relationship between thenear-tip mean stress and threshold stress intensity, the three identical loadcycles in question will follow different near-threshold da/dN curves as, indicatedin Fig. 9( d). Empirical determination of these modied da/dN curves of relevance to variable amplitude fatigue requires specially designed experimentsinvolving the controlled variation of near-tip mean stress. Note that, given theimpact of compressive overloads leading to increased tensile near-tip stresses,there is no reason why the left extreme of these curves cannot tend towardszero. Note also, that the right extreme for the da/dN curve is a high vacuumresponse that effectively limits the extent to which compressive residual stresscan retard the fatigue process.

Figure 9( e) and 9( f ) assist in understanding transients associated with near-tip residual stress response after an overload. Here, A and C are the monotonicbaseline and overload plastic zones, respectively; D and B are the associatedcyclic plastic zones, respectively. Figure 9( f ) shows the near-tip response at theboundary of zone D. Since this point will see a fully elastic response, one may

assume that as the crack tip approaches this point, any hysteretic effects seen inFig. 9( c) will disappear. This means that beyond this point, from a residualstress perspective, crack growth will be identical for the two cases in Fig. 9( a)and 9( b). The memory about the compressive overload stands is erased fromthis point. It also appears possible that, after some crack extension and wellbefore the boundary of C, near-tip stresses will be restored to the levels associ-ated with the baseline conditions and one should, therefore, not see the extentof the retardation zone assumed by the Wheeler and Willenborg models.

In summary, quite independent of crack closure, the residual stress effect ismanifested through the response of near-tip elements within the cyclic plasticzone to stress history. Their response determines instantaneous D K th , suggest-ing that the near-threshold da/dN versus D K curve is not a material constantand needs denition on a cycle-by-cycle basis as a function of near-tip mean ormaximum stress. As a consequence, the hysteretic near-tip stress variation indu-ces cycle-sequence sensitivity in near-threshold crack extension, provided thecrack is fully open during the given cycle. In the event of partial crack closure,cycle-sequence sensitivity is not possible because the minimum crack-tip stress


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FIG. 9— The new perspective of how tensile and compressive overloads distort the fa-tigue process. (a) Tensile overload; and (b) compressive overload following a tensile overload. (c) Crack tip stress-strain response showing the effect of overloads on localmean stress (crack-tip residual stress). Tensile overload pushes local stress into compression (ED), but if a compressive overload follows, local stress will rise (GH), thoughnot to the baseline value (AB). (d) Near threshold crack growth rates can swing dramat-ically depending on crack tip stress. (e) Overload cyclic plastic zone is small by comparison to the tensile overload plastic zone. Therefore, any sequence sensitive hysteretic effect will disappear on its boundary, as seen in (f). This implies that beyond this point,it will not matter whether a compressive overload followed the tensile one. However, due to the combined action of closure and residual stress, most of the load-interaction effect, bordering on crack retardation and possible momentary arrest, would have been exhausted within the cyclic plastic zone. Conventional modeling techniques cannot reproduce these effects because they ignore the cyclic plastic zone response and its effect on threshold.

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in the cycle is practically tied to the lowest possible crack-tip stress (see Fig.8( e)).

Upon application of a tensile overload, the impact of the associated residualcompressive stress is immediate. This combines with the delayed developmentof closure awaiting wake build up. As a consequence, retardation will be imme-diate in the event of the near-threshold response and delayed in the event of theParis Regime response or in high vacuum. 34 Closure related retardation due tooverload will vanish only after the crack tip has extended well outside the over-load plastic zone of the crack tip (see Fig. 6( d)). In contrast, the hysteretic na-ture of residual stress effects will disappear at the boundary of the overloadcyclic plastic zone and the retarding effect of residual stress will altogether dis-

appear well before the crack tip exits the overload plastic zone as the near tipstresses approach baseline values. This point has no connection with the pointwhere crack closure reaches its maximum. Thus, the combined action of crack tip residual stress and closure will be limited in the crack extension interval.However, over this small interval, retardation is likely to border on crack arrest.The closure model accounts for only part of what happens except in the partialcase of Paris Regime growth rates. Additionally, the Willenborg and Wheelermodels altogether ignore the cyclic plastic zone response and treat the transientprocess as a continuously changing one over the entire monotonic plastic zone.

The ramications of the deviation from reality of all existing approaches tocrack growth estimates under variable amplitude loading can be judged fromtwo important practical considerations of computation. First, the baselinecyclic plastic zone where hysteretic effects dominate will be well under 10% of the overload monotonic plastic zone size. 35 Second, computed residual fatiguelife, being an integral of the growth rate function, will accumulate errors incomputed transient growth rates. This suggests the questionability of obtainingreasonable crack growth estimates using available models. The suggestion mayappear preposterous when viewed against the operating framework of techni-ques currently in use to handle variable amplitude fatigue. An examination of

the empirical evidence and denition of the emerging perspective is, therefore,pertinent.

The Experimental Evidence

A series of experiments were performed to verify each of the conclusions thatfollow from interpreting variable-amplitude fatigue using the BMF model com-bined with closure. To avoid speculation, each experiment was speciallydesigned to deliver irrefutable fractographic evidence [54–63]. These highlight

34) Published fractographic data showing delayed retardation are restricted to the ParisRegime—they show striations.35) Plastic zone size ratio is given as the square of the ratio of overload stress intensity tohalf the baseline effective stress intensity range because cyclic plastic zone size is deter-mined by twice the yield stress required for reverse yield.


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quantiable effects for which there appears to be no alternate interpretation. 36

The experiments imply the important criterion of falsiability.

It was a chance discovery that initiated this research in the late 1990s. Earlyexperiments on Al-alloy specimens using closure-free high stress ratio cyclingwere performed in search of a correlation between the applied stress and shortcrack response [54] (the so-called short crack effect). The experiments were per-formed under programmed loading with three steps of identical amplitude but varying mean stress. Figure 4( b) shows a typical fatigue fracture from theseexperiments. Clearly visible are sets of three bands of crack extension associatedwith each set of three steps of loading. Note that each band is caused by a fewthousand load cycles and is not to be confused with striations from individual

cycles seen in the Paris Regime.A noticeable difference in the crack growth rate is observed between individual steps at small crack size in Fig. 4( b). This difference gradually tapers outto uniform crack growth rate as the crack grows much larger. The embeddedcycles were placed on the rising half of the major cycle. 37 The authors correlatedmeasured crack growth rates with the maximum local notch root stress in individual steps and came up with an empirical equation to account for the shortcrack effect as a function of local maximum stress [54]. At the time, thisapproach was considered consistent with the prevailing notions of the so-called“short crack effect,” where parameters such as local stress were consideredessential to explain what the stress intensity range could not. We did not con-sider the possibility that the steps with lower mean stress may experience thebenecial effect of preceding stressing at a higher level. We believed that havingensured the crack was fully open by keeping stress ratios high, no load interac-tion effects were possible. Sometime after the publication of this work, achance 38 discovery was made of equally spaced concentric circular bandswithin voids on the fatigue fracture surface left behind by secondary particu-lates [55].

Several conclusions crucial to unraveling the nature of metal fatigue

emerged from the detailed study of fatigue voids. While it has long been knownthat fatigue cracks form at the notch root, the new evidence conrms the possi-bility that early fatigue kinetics are the consequence of several competing mech-anisms operating at different locations. At a high applied stress level promotingintense reverse slip, the notch root surface is likely to develop several crack

36) A few early experiments involved the analysis of striation patterns. The rest involvedestimates of spacing between marker bands employed to unambiguously characterize mi-croscopic crack extension over thousands of near-threshold load cycles that cannot, intheir individual capacity, produce discernible growth marks. This technique permitsquantitative estimates of crack extension without a limitation on the minimum growthrate. The pictures reproduced in this paper reach down to 10 8 mm/cycle.37) Had they been placed on the falling half, the retardation effect would have been muchmore dramatic given the hysteretic response. At the time, the authors were not aware of the phenomenon involved.38) In routine electron microscopy particulate voids are usually ignored as dark, feature-less cavities.

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origins almost simultaneously [61]. Plane stress conditions at the surface com-bined with assistance from the environment 39 appear to dominate. With a

decreasing stress level, the number of such sites will diminish, with a generaltendency towards eventual sub-surface initiation. 40 One may speculate that con-straint in the interior will promote local defect growth by microscopic failurethrough modes other than planar slip, which prefers plane stress conditions.

In Al-alloys, innumerable secondary particulates lying beneath the notchroot appear to bear evidence to the consequences of cyclic hydrostatic stressesoperating in the constrained region beneath the notch. These induce the grad-ual separation by interfacial fatigue cracking of the secondary particulate fromthe matrix. Cyclic hydrostatic loading action is apparent from the simultaneous

onset and identical growth rate of typically six (even more in the case of theirregular shape of the particulate) penny shaped interface cracks covering all sixsides of the particulate (see Fig. 10). The smallest crack size seen is of the orderof 0.125 l m, which may represent the smallest reproducible and traceable fa-tigue crack observed in research practice. The bands also indicate an incrediblylow growth rate down to 10 8 mm/cycle. The generally uniform spacing of theconcentric bands is of practical signicance, suggesting that the interfacialcrack growth rate appeared to be insensitive to change in the mean stress inindividual steps of the programmed load sequence employed. This was in con-trast to the major short crack at the same proximity to the notch root! Surely,the effect that caused growth rates to be different between steps in the majorshort crack as seen in Fig. 4( b) ought to have also have inuenced the interfacialcrack growth! However, they apparently did not, after all.

There was, however, an important difference between the conditions underwhich the two cracks grew. Unlike the major crack originating from the surfaceand continuously exposed to the environment, interfacial cracks around sec-ondary particulates grow in ideal vacuum. This is conrmed by simultaneouscracking around the particulate that could not have progressed without cyclichydrostatic tensile stresses, and these in turn will disappear once the particulate

is exposed and constraint disappears. There was obviously something linkednot with the macro-mechanics of the notch response, but rather, with themicro-mechanism of crack extension that seemed to determine fatigue resist-ance. A possibility has now emerged that vacuum inhibits the root cause for themean (residual) stress effect in metal fatigue. It was also possible that in air, itwas not the applied mean stress itself, but the sequence of its change (load his-tory) that was responsible. Perhaps, indeed, vacuum does disable residual(mean) stress related effects?

Reference [56] describes an experiment dedicated to conclusively isolatethe role of environment in near-threshold fatigue by falsication. The experi-ment involved testing to failure under the same three-step programmed loading,but alternating between air and vacuum every given number of blocks. The

39) In Al-alloys, interfacial environmental attack causes early pitting through the separa-tion of secondary particulates on the notch surface. Each pit is a potential initial defect.40) Gigacycle fatigue is almost always associated with internal crack formation.


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FIG. 10— Fatigue voids and microvoids [55]. (a) Proof that individual voids seen on Al- alloy fatigue fractures were formed by fatigue-separation of secondary particulates fromthe matrix and not due to high K max quasi-static failure as claimed in [40]. Evidence of interfacial cracking under three-step programmed loading (inset). Clear, equally spacedbands marked by marker loads between steps indicate that the change in the mean stress level did not have any effect on the crack extension due to the 2000 cycles in each step. The schematic shows cyclic hydrostatic forces responsible for the cracking. (b) Rare picture of the secondary particulate that remained bonded to the fatigue fracture.The area immediately around the particulate is evidently formed by fatigue. The surrounding area is marked by clusters of microvoids that coalesced to cause quasi-static crack extension. Microvoids are form

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