J Strain Analysis Fatigue crack growth ‘‘overload …croft/papers/183-Mech-Syn-OL...transient fatigue crack growth rate retardation produced by a single overload cycle known as

Original Article

J Strain Analysis47(2) 83–94� IMechE 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0309324711435197sdj.sagepub.com

Fatigue crack growth ‘‘overload effect’’:mechanistic insights from in-situsynchrotron measurements

Mark Croft1,2, Najeh Jisrawi1,3, Alexander Ignatov1, Ronald L Holtz4 andZhong Zhong2

AbstractSynchrotron-based, high-energy X-ray diffraction measurements are used to study the local strain fields underlying thetransient fatigue crack growth rate retardation produced by a single overload cycle known as the overload effect.Specifically, 4140 steel compact tension specimens fatigued for varying levels of crack growth after an overload cyclehave been studied with in-situ diffraction under varying external loads. The load responses of the strain at the overload-position, versus at the crack tip, are focused upon in detail. The large compressive residual strain at the overload-pointis observed to remain essentially unchanged even after the overload-point is left in the wake of the propagating cracktip. The differential strain-load response at the crack-tip/overload position before and immediately after the overload isseen to be unchanged. Once the overload point is behind the crack tip, a highly nonlinear behavior is observed in whichthe load response of the strain field transfers from the overload -point to the crack tip when the load exceeds a criticalvalue. The results are discussed in terms of plasticity-induced crack face contact at the overload point as an importantlocal mechanism contributing to the ‘‘overload effect’’ in this specific system.

KeywordsFatigue, strain, X-ray, synchrotron, overload

Date received: 22 August 2011; accepted: 25 November 2011

Introduction

Most models for predicting fatigue crack growth arebased upon the local crack tip strain/stress fields, whichheretofore have been largely inaccessible experimen-tally.1 This situation has changed dramatically over thepast several years with the evolution of high-energy,synchrotron-based X-ray diffraction microscopic strainmapping techniques.2–15 One problem that these newtechniques can shed new light upon is the ‘‘overloadeffect’’, in which a single tensile overload, in an other-wise constant amplitude fatigue experiment, causes apronounced transient retardation in the rate of crackgrowth (Figure 1(a)).14–21 In real-world variable ampli-tude fatigue loading, such overload (OL) events havebeen shown to play important roles in the lengtheningof the fatigue life of structural components. The scien-tific debate over the detailed mechanics underlying theoverload effect has been discussed at length in the liter-ature.14–21 Two opposing interpretations have arisen inthis debate regarding the origin of the overload effect.One view argues that it is entirely due to residual

stresses in front of the crack tip;19 while the other, morecustomary, view is that it is due to plasticity-inducedcrack closure behind the crack tip.22–26

A recent tour de force, in-situ, synchrotron-diffrac-tion/mechanics-modeling article by Steuwer et al.14

studied the overload effect on thick, fine-grained Al-alloy specimens. This study14 did not, however, producedecisive results on the issue of the importance of behindversus in-front-of crack effects in the overload effect.Croft et al.5–7 have reported extensive synchrotron-based energy dispersive X-ray diffraction (EDXRD)

1Department of Physics, Rutgers University, USA2National Synchrotron Light Source, Brookhaven National Laboratory,

USA3Department of Applied Physics, University of Sharjah, United Arab

Emirates4Naval Research Laboratory, Washington DC, USA

Corresponding author:

Mark Croft, Department of Physics, Rutgers University, Piscataway, NJ

08854, USA.

Email: [email protected]

measurements on a series of 4140 steel fatigue speci-mens spanning behavior from prior to, to well after, anoverload, including as a function of in-situ loading.This work reported the details of the strain distribution,its load response and evolution with crack growth sub-sequent to an OL, and also quantitatively addressedsome modeling predictions.26,27 This work5–7 did not,however, directly address the controversy over the ori-gin of the overload effect.

The recent work of Belnoue et al.15 is very importantto this question. They have performed detailed two-dimensional (2D) strain mapping measurements oncompact-tension geometry Ti–6Al–4V specimens fati-gued beyond an overload to points where the crackgrowth rate is respectively maximally retarded and fullyrecovered (to the pre-overload rate). They confirmed15

the existence of a compressive strain behind the cracktip in specimens fatigued beyond an overload as well as

the disappearance of this compression under in-situloading. Moreover they used a multivariable leastsquares fitting analysis to integrate the full 2D strainfields into a linear elastic fracture mechanics model.Belnoue et al.15 invoked plasticity induced crack clo-sure as the origin of the behind-crack-tip-compressionand its load response.

Lee et al.16,17 have performed neutron scatteringstrain field measurements on a 6.35mm thick, CT-geo-metry, HASTELLOY C-2000 alloy specimen with in-situ fatigue crack growth spanning an overload cycle.These studies used 2 to 4mm3 gauge volumes for profil-ing the residual strains in the crack plane; and for deter-mining the load response at 13 load levels at positionsin-front-of, at and behind the crack tip. Similar toCroft et al.6,7 they observed a load-induced transfer ofcompressive strain from the ‘‘blunting region’’ (the OLposition) to the crack tip. They interpreted this in termsof gradual crack opening under load.17

Colombo et al.28 and Christopher et al.29,30 have per-formed careful, systematic experiments on fatigue crackphotoelasticity on thin, transparent birefringent poly-carbonate materials. They have measured pre-overloadand post-overload (fatigued beyond the OL point) spe-cimens. Their analysis indicates the key role of crack-face interactions behind the crack tip in contributing tocrack growth retardation following overloads, in addi-tion to residual stresses generated ahead of the cracktip. Moreover, they have applied detailed analyticaland numerical modeling of their results for a variety ofrelated phenomena.

Steuwer et al.14 have noted that much has beenlearned from such local strain field studies,5–7,14–17 theyalso aptly noted that significant amounts of synchro-tron beam time would be required to decisively eluci-date the fundamental micromechanical origins of theoverload effect. The general scarcity of high energy X-ray beam time availability demands that a comprehen-sive picture of the existing synchrotron results, and thefull weight of the inferences they motivate, be madevery clear. This paper thus focuses on expanding theresults and discussion of a nonlinear ‘‘load transfereffect’’ previously described in references.6,7

Specifically, it will be emphasized that these results arehighly consistent with a microscopic mechanism for theOL effect in which crack face contact at the OL posi-tion, when it is behind the crack tip, is important in atleast this specific set of experiments.

Experimental

EDXRD measurements

The EDXRD measurements discussed here were per-formed at the Brookhaven National Synchrotron LightSource (NSLS) on the superconducting wiggler beamline X17-B1. The experimental set-up has beendescribed previously.3–7 Basically, ‘‘white beam’’ inci-dent radiation is scattered at a fixed angle 2u (2u=12�

Figure 1. (a) The crack growth rate da/dN plotted versuscrack length, a, for a 4140 steel CT specimen, fatigued underconstant amplitude conditions with an overload cycle. The cracklength is referenced with respect to the crack length at overloadaOL = 25.4 mm and da/dN is normalized to the pre-overloadcrack growth rate of 2.05 (10)24 mm/cycle. The nominalpositions along the growth rate curve where in-situ loadingstudies on similarly prepared samples were carried out, arelabeled: f, for constant amplitude fatigued specimen; f + ol, forthe f specimen subjected in-situ to an OL (overload) cycle; fo,for a specimen prepared with a terminal OL cycle; max-ret, for aspecimen exhibiting maximum crack growth rate retardation(i.e. fatigued beyond the OL cycle the critical distance,Dac~0.4 mm); and 50%-ret for a specimen fatigued toapproximately 50% recovery of the pre-OL growth rate. (b) Aschematic of the sequence of fatigued specimens and the in-situload levels used in this study The load sequences points (circles)for each of the specimens are indicated. The max-ret and 50%-retpoints occur respectively at ~6(10)3 and ~2(10)4 cycles, N,beyond the overload point.

84 Journal of Strain Analysis 47(2)

in this work). The incident/ diffracted beams are highlycollimated, thereby defining the small gauge volume.2

The energies, E, of the scattered Bragg peaks are givenby E=hc/[2dhkl sin(u)] where dhkl is the inter-atomicplane with Miller indices (hkl), c is the speed of light,and h is Plank’s constant.

In this work the positional variation of the intera-tomic spacing (d) of the most intense (321) Bragg linehas been used to determine the strain e=[(d–d0)/d0]= [(E0–E)/E0]. The reference d0 (and E0) representthe stress-free lattice spacing (Bragg line energy) andwere fixed by the Bragg line position far from thecrack.5–7 Similar strain results were also obtained bythe Rutgers group by fitting a set of 7–10 Bragg lines.2,5

The use of multiple Bragg lines for strain determinationhas been used on a number of studies.2,8,14 It is impor-tant to note that since these X-ray strain measurementsprobe inter-atomic distances they explicitly render onlythe elastic component of the strain. The relative strainresolution in these EDXRD measurements is in the1024–1025 range.2,4,8,14

In this paper the coordinate system used (see Figure2) defines the x-direction as the direction of the crackgrowth, the y-direction as along the tensile stress direc-tion (with y=0 set at the crack plane) and the z direc-tion as through the 4mm thickness of the specimen(with z=0 being the specimen center). In the series ofspecimens studied x=0 has been set at the terminationof the constant amplitude fatigue crack length orequivalently the position where the overload wasapplied.

The x–y cross-section of the diffraction/gauge vol-ume was a square of 50–60mm on a side. The y–zgauge volume (GV) cross-section was an elongated reg-ular parallelogram (50–60mm) along y and with the zextent being ;300mm (see Croft et al.2,5 for further dis-cussion of the GV). The scattering angle of 2u=12�allowed the localization of the GV around the center

plane of the specimen. Indeed, detailed strain mappingof the crack plane shows that the GV sampled regionswell away from edge effects7 and in a locally flat regionof the crack front. For future synchrotron experimentsit is useful to emphasize that the scattering angle of2u=12� in the present work is particularly large forEDXRD measurements, leading to a better localizationof the GV in the beam (z) direction. This is importantbecause any uniform slope/curvature or local jagged-ness in the crack front can lead to a spatial averagingand degradation of the crack tip strain features. In sys-tems where the spatial extent of the OL compressionregion (see below) is small, such averaging over crackfront excursions out of the GV could render itunobservable.

The scattering vector, in this case, was inclined atonly 6� with respect to the y-direction, yielding, to anexcellent approximation, the eyy strain component.Despite the tri-axial character of the stresses in thesesystems it will be assumed, for interpretation purposeshere, that the behavior of the strains eyy and Deyy canbe used as indicators of the behavior of the stresses syy

and Dsyy. Indeed excellent agreement in many aspectsof the results with the multi strain component andderived stress variations of Steuwer et al.14 providesstrong motivation for the utility of this assumption.

Strain measurements of the CT specimens under in-situ loading, were performed using a fixture constructedto open the crack mouths by using a jack screw.7 Theload at the screw pivot point was measured by a digitaltransducer techniques load monitor thereby allowingthe calculation of tensile load, F, at the pins of the CTspecimen (see Figure 2).

Materials and preparation

The specimens in this study were all 4mm thick, 4140alloy steel in the normalized condition, compact tension(CT) specimens as described in the references.5,6 Cracklength measurements were made via the dc potentialdrop method.5,6 The fatigue preparations of the varioussamples and crack growth rate curve are summarizedschematically in Figures 1(a) and 1(b). The crackgrowth rate, da/dN (where N is the number of fatiguecycles) curve is shown in Figure 1(a), and is normalizedto the pre-overload crack growth rate. This da/dNcurve shows the typical overload retardation effect.18,19

This curve was used to estimate the crack lengths towhich the cracks in subsequent specimens should begrown in order to explore the strain fields correspond-ing to key points on the da/dN curve. The points atwhich specimens were investigated are indicated on thegrowth rate curve. A schematic of the specimens andin-situ loading experiments is shown in Figure 2 andwill now be discussed in detail.

All specimens were first prepared with a fatiguecrack length of 25.4mm under constant amplitude fati-gue conditions with: Kmax=39.6MPam1/2; andKmin=R Kmax with R=0.1. At the 25.4mm crack

Figure 2. A schematic of the compact tension (CT) specimensand notations used in this work. For clarity the schematic cracksfor the max-ret and 50%-ret specimens have been displacedvertically from the f and fo cracks at the notch. The coordinatedirections are also shown (see text for discussion).

Croft et al. 85

length, the load, applied to the pins of the CT specimen,corresponding to this Kmax was F=Fmax=3.8kN.The constant amplitude fatigued specimen (f) had nofurther fatiguing. All of the other specimens were sub-jected to an overload with FOL=2Fmax. That is, asingle peak with KOL=79.2MPa m1/2. The fatigue-overloaded (fo) sample fatiguing was terminated imme-diately after the overload. Two samples were subjectedto additional constant amplitude fatigue after the over-load, with the initial pre-overload K=Kmax, to propa-gate the crack specific distances beyond the overloadpoint. The sample designated max-ret specimen wasfatigued to the critical distance Dac;0.4mm beyond theOL crack length corresponding to the point of maxi-mum crack growth retardation. Similarly, the 50%-retspecimen was fatigued ;1.5mm crack extensionbeyond the OL point, reaching approximately 50%recovery of the pre-OL crack growth rate.

Results

Before and immediately after OL response

Figure 3(a) shows the eyy strain profiles, along x (crackgrowth) direction, for the constant amplitude fatigued,f-specimen, at the external loads of F=0 andF=Fmax. The classic enhancement of the responseapproaching the crack tip (at x=0) is clear.1 Since theload-induced stress/strain change at the crack tip is cru-cially important to the crack growth process, the strainresponse, Deyy(F)= eyy(F)2 eyy(F=0), at the crack tipis plotted in Figure 3(b). Although there are only twopoints in this ‘‘before OL’’ plot it serves as a referencefor subsequent discussion of other load-series results.

Also shown in Figure 3(a) is the eyy profile for thesame f-specimen overloaded to F=FOL=2Fmax

(labeled ‘‘during OL’’ in the figure). Note this fatigue-sequence/load point is labeled as ‘‘f+ ol’’ in the Figure1 schematics. The plasticity induced saturation of theresponse near the crack tip can clearly be seen by thesmall increase in the eyy value at the tip betweenF=Fmax and FOL as compared to the large eyy changebetween F=0 and Fmax. This point is underscored inFigure 3(b), where the ‘‘during OL’’ Deyy(F) loadresponse curve follows the before OL curve up to Fmax,and then breaks sharply to the low Deyy(FOL) value.

Post-OL response

Figure 4(a) shows the eyy load response for the fo speci-men (see Figure 1 schematics). Here the main portionof the figure shows the x-direction profiles (longitudinalto the crack) in the y=0 crack plane. The F/Fmax=0,residual strain, eyy profile is dominated by the large,sharp compressive feature centered on the crack tip(x=0). This feature is associated with the reverse plas-ticity generated by the misfit elongated region near thetip generated in the overload cycle.1,14 A series of eyyprofiles for varying in-situ loads are also shown inFigure 4(a). The load response at the crack tip can beseen to be large, although the maximum strain at Fmax

is substantially reduced below the pre-overload value(see Figure 3(a)).

Figure 4(b) shows the eyy profiles along the y-direction (traversing the crack plane at the tip position)for a series of external loads. All of these strain profilesexhibit a sharp peak at the crack tip (y=0) positionand a long-range fall off in the elastic strain above andbelow the crack plane. Here the peak strain is domi-nated by the residual, compressive strain for F/Fmax \ 1/2 and by the applied tensile stress for largerloads.

Figure 3. (a) eyy strain profiles along the x (crack growth) direction, and in the y = 0 crack plane, at the external loads of F=0 andF = Fmax for the f-specimen before the overload. Also shown is the eyy(F = FOL = 2 Fmax) strain profile on the same specimen (labeledf + ol in the previous schematic) loaded to the overload (labeled ‘‘during OL’’ in this figure). Note that the x = 0 is defined as theposition of the overload in this and subsequent figures. (b) A plot of the load-induced change in the strain Deyy(F) = eyy(F)- eyy(F = 0)in the vicinity of the crack tip for the before, and during OL cases. For these two curves the eyy(F = 0) used was the before OL value.For comparison the tip, Deyy(F) curve for the after overload case is also shown. For the after OL case the eyy(F = 0) used was thatafter the OL (see next figure and the related discussion in the text). (OL = overload.).

86 Journal of Strain Analysis 47(2)

The change in the tip response, Deyy(F) = eyy(F)-eyy(F=0), for this fo specimen is plotted in Figure 3(b)(see ‘‘after OL’’ curve). Despite the very large residualcompressive strain, eyy(F=0), it should be noted thatthe overall strain response after the overload, is essen-tially the same as before the overload. This is notentirely surprising since the plastic deformation has dis-placed the starting point deep into the compressive sideof the stress–strain curve and the linear elastic responsewould be expected to span approximately twice theyield stress. Moreover, the fact that the crack growthrate immediately after the overload, is unchanged isalso remarkably well correlated with the unchangedDeyy(F) curve (as discussed below).

Maximum retardation point

Figure 5 shows the eyy load response for the max-retspecimen (see Figure 1 schematics). The load responseof this max-ret specimen manifests the onset of anonlinear, critical load, Fc-type behavior in which thelow-load response occurs at the OL-position and istransferred to the crack tip in the high load regime.6,7

This load response transfer can be seen by comparingthe eyy profiles in Figure 5(a), which evolve from a neg-ative peak at the OL-position, for F=0, to a positivepeak at the crack tip, for F=Fmax. This critical load,Fc-type behavior will be better resolved and clearer inthe 50%-ret specimen discussed subsequently.

In Figure 5(c) the load response curve, Deyy(F), atthe OL point (x=0) shows: a linear rise with load untilF=0.5Fmax; a decrease in slope between F=0.5Fmax

and F=0.75Fmax; and a clear saturation with nochange between F=0.75Fmax and F=Fmax. Thusthere is an abrupt cut off in the response at the OLposition at the ‘‘critical load’’ Fc;0.75.

The Deyy(F) response at the tip position also shows alinear response between F=0 and F=0.5Fmax; how-ever, this response is much less than that at the OLposition. Thus the initial strain response is dominantlyat the OL position. An increase in the slope of the tipresponse curve occurs between F=0.5Fmax andF=0.75Fmax, with a still greater increase betweenF=0.75Fmax and F=Fmax. Indeed the slope of thetip response between F=0.75Fmax and F=Fmax isessentially the same as that of the OL response at lowload. Thus the strain response enhancement at thecrack tip mirrors the cut off in the response at the OLposition.

The quality of the data of the load series data inFigure 5(a) is sufficiently good to empirically define aload and position dependent eyy response function (orsusceptibility), essentially a dimensionless complianceparameter normalized to the peak load at overload,x(x,f), where f=F/Fmax via

x(x, f)=Deyy(x, f)

Dfð1Þ

Using an interpolation fit to all of the curves in Figure5(a) allows the construction of such an experimentalresponse-function covering the intervals indicated inFigure 5(a); 1 [0 \ F \ 0.25Fmax], 2 [0.25Fmax \ F\ 0.5Fmax], 3 [0.5 Fmax \ F \ 0.75Fmax] and 4[0.75Fmax \ F \ Fmax]. The resulting response func-tion, x, curves are shown in Figure 5(b). Evaluation ofthese x(x,f) at the OL position (x=0) and the tip posi-tion x;0.4 recovers the slope of the Deyy(F) curves inFigure 3(b). In general however such response functioncurves contain much greater information on the spatialextent of the strain response to load all along the crackplane.

Figure 4. (a) eyy strain profiles along the x direction, and in the y = 0 crack plane, for a series of external loads between F = 0 andF = Fmax for the fo specimen after the overload. (b) eyy strain profiles along the y direction (transverse to the crack plane at the cracktip, x = 0, position) for a series of external loads between F=0 and F = Fmax for the same fo specimen. Note that the strain changesDeyy(F) = eyy(F) –eyy(F = 0) in the vicinity of the crack tip derived from these data appear as the ‘‘after OL’’ curve in the inset of theprevious figure. (OL = overload).

Croft et al. 87

The x(x,f) curves for the intervals 1–4 are shown inFigure 5(b). Over intervals 1 and 2 the response func-tion is sharply peaked at the OL-position, while beinglow and constant in the region of the crack tip. A smallshift in the x-coordinate of the x OL-peak, toward thetip, between interval 1 and 2 is also noticeable. Overinterval 3, the x becomes distinctly bimodal with adegraded peak near the OL position and a new peak

arising near the crack tip position. This dramaticallyindicates the onset of the OL-to-tip load response trans-fer process. Interestingly (as addressed in the discussionsection) the x-position of the OL peak is shifted notice-ably toward the tip in interval 3. Over interval 4 the x

at the OL position has been totally suppressed and arobust peak has emerged at the crack tip position. Theresponse function curves provide a very tangible illus-tration of the transfer of strain response from the over-load position to the crack tip position above the criticalload.

In general a theoretically or experimentally (withsmaller F intervals) defined response function (or sus-ceptibility), x, could be used to predict the strain changebetween any two loads f1 and f2 via

eyy x, f2ð Þ � eyy x, f1ð Þ=ðf2f1

x x, fð Þdf ð2Þ

Approximately, x will be very small at the crack tipand large at the overload position for loads less thanthe critical load. Conversely, x will be very small at theoverload and larger at the crack tip position for loadsgreater than the critical load. Knowledge of such aresponse function would be useful in modeling the fulldetails of varying fatigue loading scenarios.

50% recovery

In the 50%-ret specimen the OL and crack tip strainfeatures are most widely separated and the critical loadFc-type behavior is most clearly observed. Figure 6shows the eyy load response for the 50%-ret specimen(see Figure 1 schematics). The eyy profiles at differentloads shown in Figure 6 clearly show the large negativepeak at the OL position disappear continuously andcompletely between F=0 and F=Fmax 3/8.Concomitantly, strain fields near the crack tip manifestlittle change in the same low load region. However, forF . Fc=Fmax 3/8 the tensile peak near the crack tipgrows rapidly with increasing load.

In the inset of Figure 6 the load response curves,Deyy(F), at the OL and crack tip positions, are shownalong with the post-overload curve for reference.Interestingly for F \ Fc the 50%-ret specimen Deyy(F)response at the OL position is almost identical to thatof the post-OL. Referring back to the inset of Figure 3,one can say more generally that there is essentiallyequality between Deyy(F) responses of: the pre-OL (atthe tip position); the post-OL (at the tip/OL position);and the 50%-ret response for F \ Fc at the OLposition.

For F . Fc the 50%-ret response clearly saturatesat a load independent value. For F \ Fc (see Figure 4)the 50%-ret load response at the crack tip is very small.For F . Fc, on the other hand, the slope of theresponse curve becomes large and comparable to thatof the post-OL and 50%-ret (F \ Fc) curves.

Figure 5. (a) eyy strain profiles along the x (crack growth)direction, and in the y = 0 crack plane, for the specimenexhibiting maximum crack growth retardation (max-ret), forvarying external loads between F = 0 and F = Fmax. Note thelabels 1–4 in the figure indicates the steps between whichdifference/susceptibility curves (shown in the bottom of thisfigure) were calculated. (b) The eyy strain susceptibility profiles,x(x,f) , where f = F/Fmax, along the x (crack growth) direction forthe external load intervals 1-4 indicated in Figure 5(a). Seeequation (1) for the definition of x(x,f). (c) Plots of the load-induced change in the strain Deyy(F) = eyy(F)2eyy(F = 0) in thevicinities of the crack tip and of the OL positions. The post-OL(overload) curve from Figure 3(b) is included for reference.

88 Journal of Strain Analysis 47(2)

Fc variation with crack length

In Figure 7 the critical load Fc (normalized to Fmax) atwhich the load response transfers from the OL positionto the crack tip position is plotted versus the crackextension beyond the overload position. For F \ Fc

the response at the crack tip is greatly suppressed lead-ing to the suppression of the fatigue damage and crack

growth driving force in this portion of the load cycle.The normalized crack growth rate da/dN (norm),reproduced in Figure 7 is understandably anti-correlated with the Fc variation.

As demonstrated earlier only when F . Fc does thecrack tip strain respond and thus only when F . Fc

should the cyclic damage (crack growth driving force)be operative. Consequently the normalized differentialforce DF/Fmax=12 Fc/Fmax should provide a measureof the effective cyclic applied force which is active incrack propagation. Despite there only being threex-points and despite the modest number of loads usedto determine Fc, the plot of DF/Fmax (in Figure 7) iswell correlated with the crack growth rate curve.

Figure 6. A comparison of the in-crack-plane (y = 0) eyy- strain profiles (along the x-direction) for the 50%-ret. specimen with sevenexternal loads between F = 0 and Fmax. Inset: plots of the load induced change in the strain Deyy(F) = eyy(F)2eyy(F = 0) in the vicinitiesof the crack tip and of the OL-position. Note the fo specimen, ‘‘after OL’’Deyy(F) curve has been included for reference.(OL = overload).

Figure 7. A series of quantities determined in this work areplotted versus the tip extension beyond the OL (overload)position. To illustrate the correlation of these quantities withcrack growth the normalized crack growth rate da/dN (norm) isreproduced here (left scale) from Figure 1(a). The critical loadsFc (normalized to Fmax) is plotted with the right scale. Theeffective portion differential cyclic load DF/Fmax = 1–Fc/Fmax overwhich the load actively contributes to crack propagation isplotted on the left scale. Finally the normalized strain differenceDeyy(norm) = [eyy(F = Fmax,x)– eyy(F = 0,x)]/ Deyy(before OL) isplotted on the left scale. Here the normalization factorDeyy(before OL) = [eyy(F = Fmax,x = 0)– eyy(F = 0,x = 0)] is thestrain difference before the OL.

Figure 8. An exaggerated-for-illustration schematic of the OL(overload) material ligament deformation in and after the OLcycle. The (a)–(d) indicates a schematic series of situationsdescribed in the text.

Croft et al. 89

The total cyclic variation in the strain Deyy at thecrack tip can provide a microscopic measure of thecyclic crack growth driving force. The normalizedstrain difference (at the crack tip) Deyy(norm) =[eyy(F=Fmax,x )2 eyy(F=0,x)]/ Deyy(before OL) isplotted on the left scale in Figure 7. Here the normaliza-tion factor Deyy(before OL) = [eyy(F=Fmax,x=0)2eyy(F=0,x=0)] is the crack tip strain difference beforethe OL. Again the correlation between Deyy(norm) andthe crack growth rate curve is good. These resultsunderscore the importance of the magnitude of theeffective differential variation of the stress at the cracktip in driving crack growth.

Discussion

Perhaps the most striking of the above results is thehighly nonlinear elastic response at the OL-positionwhen it is behind the crack tip. Specifically, the strainresponse at the OL-position is elastic up until a criticalload Fc, above which it saturates at a load independentvalue. That this OL-position effect is intimately associ-ated with the OL crack growth retardation is empiri-cally clear from the simultaneous crossover fromstrong-suppression to sharp-enhancement of theresponse at the crack tip at precisely the same Fc.

Although much more sophisticated modeling iscalled for in the future, these results motivate an imme-diate and straightforward interpretation. This interpre-tation is based upon the fact that two surfaces inphysical contact, and under compression, exert a nor-mal force, Ns, upon one another that will exhibit linearelastic response as long as there is surface contact. Thisnormal force will naturally transmit a compressive elas-tic strain field in the subsurface material. When contactbetween the surfaces is lost (i.e. Ns!0) the elasticresponse ceases and saturates at a constant value. Thediscussion below is motivated by the microscopic straindata and not by any pre-existing models. Nevertheless,it is emphasized that the simplified interpretation pre-sented here has been proposed as part of more sophisti-cated modeling in the past.20–25

The first interpretive step involves the strong plasticdeformation of the ligament of the material directlyin front of the crack tip by the OL (see schematic inFigure 8). The ligament of material immediately infront of the crack tip before the OL has a characteristicsize, Figure 8(a). Upon application of the FOL this liga-ment undergoes strong elasto-plastic elongation underthe tensile overload, Figure 8(b). At the peak of theoverload (F=FOL) strong plastic deformation rendersthis ligament permanently elongated with respect to itspre-overload length. Referring to Figure 3 the loss ofthe response in the elastic-eyy (as measured by diffrac-tion) between F=Fmax and F=FOL clearly reflectsthis plasticity.

Unloading the specimen to F=0, i.e. from Figure8(b) to 8(c), yields an elastic shrinkage of this ligament.

The transverse ligament Poisson-bulging (shown inFigure 8(c) and 8(d)) ignores the asymmetry at thecrack tip; however, this effect is assumed here to be sec-ond order for simplified illustrative purposes.Moreover, as the OL position is left behind by the pro-pagating crack tip this asymmetry becomes smaller.Upon unloading to F=0 this tensile-induced plasticelongation is retained, so that the ligament is now toolong to ‘‘fit’’ into the previous volume it occupied. Thusthe OL-ligament feels a compressive misfit elastic force(Fel in figure) from the surrounding material and exertsa Newtonian elastic reaction compressive stress field onits surroundings.

The existence of this post-overload elastic stress fieldat F=0 is clear from the sharp compressive peaks ineyy in Figure 4. In the literature this OL-feature is oftenreferred as the process or reverse plastic zone.14 Theresponse of this OL-feature under tensile loading imme-diately after the OL, is linear elastic with the same slopeas the crack tip response before the OL as seen in theinset of Figure 3.

After the fatigue crack has grown entirely throughthe now fractured, OL-ligament, the misfit ligamentcreated by the OL is still elongated with respect to thesurrounding material. Therefore the crack face ends ofthe elongated OL-ligament, albeit fractured into twoparts, exert similar compressive stresses in the interac-tion with the surrounding medium, as illustrated (d) inthe Figure 8 schematic. Figures 5 and 6 illustrate thiseffect by the persistence of a essentially unchanged OL-feature (at F=0) behind the crack tip in the max-retand 50%-ret specimens.

Applying a varying external load F to the fractured,elongated, OL-ligament, Figure 9 shows a schematic ofthe load response of the fractured-OL-ligament behindthe crack tip as in (d) of Figure 8. For F=0 (far left inFigure 9) the OL ligament feels a compression due tothe normal, crack face contact force (Ns) and the elasticforce, Fel ,of the surrounding medium (as discussedabove). Here Ns=Fel is required for equilibrium. Aslong as contact between the two fractured ligamentpieces persists the normal force remains finite. The nor-mal force communicates the compressive stress betweenthe two separate OL-ligament portions as effectively asif they were still attached. As the external tensile forceis increased however, the compression is relaxed, andFel (and Ns) are reduced (see center schematic in theFigure 9). At a critical external, Fc, when the com-pressed OL ligament is fully relaxed, Ns (along withFel) first becomes 0. For all external loads F . Fc,Ns=Fel=0 since the OL ligament pieces are no lon-ger in contact as indicated in the schematic on the farright in Figure 9.

The reaction force to Fel creates a long range com-pressive stress field in the surrounding medium. Thework done by the varying tensile load goes into decreas-ing the elastic stress of the surrounding medium, thenormal crack face contact force, and the elastic exten-sion of the OL ligament. This work is diverted from

90 Journal of Strain Analysis 47(2)

creating stress and doing fatigue damage at the cracktip. Once F . Fc subsequent tensile loading is free toinduce a tensile tip stress, stip (as indicated in the farright schematic in Figure 9). There was of course someresidual and low load stress at the tip below Fc also.

The clearest example of this behavior in the data isin the 50%-ret specimen results, where a dramaticallyclear Fc-type behavior is seen in Figure 6. For F \ Fc,while the crack faces are in contact, a linear elasticstrain response at the OL position, precisely equal tothe pre- and post-OL tip response, is seen.Concomitantly very little response at the crack tip posi-tion is seen. For F . Fc, when the crack faces are notin contact: a saturated strain response with no loadcoupled change is seen at the OL position; and thestrain response at the crack tip position is large with anelastic slope essentially the same as the pre- and post-OL tip response.

In the max-ret specimen case the OL- and crack-tip- positions are somewhat less resolved. Nevertheless,

a similar Fc-type behavior (see Figure 5) is seen. Indeed,strong peaks in the strain load response susceptibilityfunction illustrate the OL to tip load-response transferin detail (see Figure 5(b)).

The modest shift of the OL position response func-tion (x) peak toward the crack tip with increasing loadwas noted in the discussion of the max ret results inFigure 5(b). Such a shift would be consistent with theexpected opening of the crack faces moving from thespecimen notch toward the crack tip with increasingtensile load, F. Thus for an OL-ligament of finite width,the loss of crack face contact would be expected tomove toward the crack tip with increasing load in thevicinity of Fc, as observed. This expectation should beverified in more detailed studies.

Figure 10 shows a contour plot of the residual(F=0) eyy-strain in the plane spanning the regionabove and below the y=0, crack plane for the max-retspecimen. Two features illustrated by this contour plotshould be noted: the massive (in magnitude and spatialextent) compressive strain field at the OL-point nowbehind the crack tip; and the small island of tensilestrain at the crack tip position. The massive OL-pointcompression, behind the crack tip, is consistent withthe crack face contact at the plastically deformed OLfeature position as discussed earlier.

It has been argued19 that plasticity associated withcrack tip processes always opens the crack rather thanclosing it. It would follow therefore that the crack facecontact proposed here would be expected to contributea tensile residual stress/strain at the crack tip. Thus theisland of tensile strain at the crack tip, in Figure 10, isalso consistent with the presence of crack face contactat the OL-point behind the crack tip. As seen abovehowever, the diversion of the strain response to the OLposition appears to underlay the crack growth retarda-tion for F \ Fc. Thus while the small residual tensilestrain feature at the crack tip appears to underscore theconsistency of the crack face contact at the OL-point,the very low response to load of this feature, forF \ Fc, implies its lack of relevance for driving crackgrowth in this regime.

Thus the residual and highly nonlinear load inducedstrain responses are all qualitatively consistent with thesimple schematic/interpretation illustrated in Figures 8and 9. Indeed, in the authors’ opinion, it is difficult toimage another self-consistent, alternative explanationthat can account for all of the observations. In the past,other authors have proposed sophisticated and detailedmodels involving such plasticity-induced crack facecontact. They did so moreover with in the absence ofsuch persuasive microscopic data as presented here.Elber21,22 proposed OL-plasticity-induced crack facecontact and McClung25 reviewed the concept. Fleckand coworkers23,24 presented extremely similar argu-ments based upon OL plasticity-induced humps alongthe crack face coming into and losing contact behindthe crack tip. Interestingly, the recent OL-effect finiteelement analysis of Pommier et al.,20 invoked node

Figure 10. A contour map of the residual eyy- strain in the x–yplane for the max-ret specimen. Note the large compressivestrain field centered on the OL (overload)-point and the smallisland of tensile strain centered on the crack tip position. Alsonote the longer-range tensile eyy- strain that appears beyondx~1.5 mm, consistent with the results presented in Croft et al.6

Figure 9. A schematic illustration of an interpretation for theload-induced transfer of strain response from the OL(overload)-position to the tip. The fractured OL-ligament behindthe crack tip is pictured. The applied external load increasesfrom left to right. The normal surface force ,Ns, and the elasticforce of the surrounding medium, Fel, are discussed in the text.

Croft et al. 91

points released behind the crack tip which were mod-eled by nonlinear springs with high stiffness in com-pression and null in tension. This is of course preciselythe elastic behavior at the OL-position used in theinterpretation presented here. There have been severalinteresting works by Kujawski31 and Stoychev andKujaswki.32 taking a fresh look at overload retardationphenomena contrasting the customary approach ofanalyzing these effects in terms of a single ‘‘effective’’stress intensity parameter, DKeff; versus a two-parameter approach requiring both a cyclic, DK, andmaximum, Kmax, driving forces. Their arguments areparticularly clear that a single-parameter approach isnecessarily incomplete in describing overload effects. Infact, Colombo et al.28 and Christopher et al.29,30 in thework cited in the Introduction, likewise have suggestedclear argument that several stress intensity factors areneeded to correctly describe overload effects, althoughtheir approach is quite different from that of Kujawski.It is beyond the scope of the current paper to addressall of the subtle issues surrounding various two-parameter models versus more customary approaches.The present authors’ results do show that, on the localcrack tip level, both the maximum elastic strain andstrain range are affected by the overload. On the otherhand, the correlation of the Deyy dependence with thecrack growth rate in Figure 7 is suggestive that Deyy,not the peak strain, is the dominant driver. A muchhigher density of local strain data, as a function ofcrack growth beyond the OL, is clearly required toresolve the relative importance of peak strain versusstrain range in the two-parameter models.

An issue that remains unresolved by the authors’ cur-rent work is the effect of stress components sxx, on thephenomenon described above. In preliminary studies bythe authors’ group, the measured exx strain is dominantlycompressive at the OL point behind the crack tip. Asmaller Poisson effect tensile anomaly in the exx compo-nent is superimposed on the compressive effect. Thissuperposition is, however, quite dependent on the distancefrom the crack plane. This would be expected due to thediffering boundary conditions on portions of the OL liga-ment protruding above versus below the crack face. Dueto the large amount of synchrotron beam time requiredfor proper systematic studies, the present authors’ resultsare not sufficient to expand on this subject at present.Future scattering experiments should include equallydetailed mapping studies of the exx strain component par-allel to the crack propagation direction. Also needed areexperimental studies accessing the third strain component,ezz, high-resolution studies in this direction will be moredifficult due to the elongation of the gauge volume andthe thickness of the material in the z-direction.

Summary

Detailed strain field profiling, under in-situ loading, ofa sequence of specimens with an overload embedded in

their fatigue history have been presented. An interpreta-tion has been proposed whereby the key strain field fea-tures and nonlinear responses to external loading canbe heuristically understood via a simple interpretation.The interpretation involves the following.

(a) The creation of an OL-induced plastically dilatedligament at the crack tip.

(b) The misfit of this dilated OL-ligament in the speci-men at zero external load leading to a robust com-pressive elastic strain field.

(c) The persistence of the OL-misfit and its compres-sive strain field even after the fractured OL-liga-ment is left well behind in the wake of thepropagating crack tip. The presence of a normalforce between the fractured, but contacting, sur-faces supports the compression (but not tension).

(d) The linear elastic relaxation of the OL-compres-sion (behind the crack tip) by external tensile load-ing up until a critical load, Fc, while the fracturedportions of the OL ligament are in contact. Herethe crack tip load response is suppressed by thediversion of the work done to the OL ligament forF \ Fc.

(e) The termination of the OL ligament responsewhen the OL ligaments loose contact and the con-comitant transfer of load-induced elastic responseto the tip position for F . Fc.

In view of the many model variations of crack faceinteractions and the debate surrounding them, theauthors wish to circumscribe the interpretation here tothe specific materials and conditions studied.Nevertheless the data presented appears to heavily sup-port the importance of the role of crack face contactbehind the crack tip in the overload crack growth retar-dation effect in this particular system. Future in-situsynchrotron experimental work on a single specimenfatigued and overloaded in-situ at a higher density offatigue crack lengths beyond the OL are most definitelycalled for to verify and elucidate the effects reportedhere. Future modeling of other systems or conditionsshould be based upon similar microscopic experimentalstrain field mapping/profiling studies.

More important than the particular microscopicinterpretation proposed here is the fact that detailed,local strain field measurements motivated the bottomup interpretation. The expanded availability of shortlength scale synchrotron based strain profiling offersthe opportunity for a paradigm change in understand-ing the mechanical properties of materials. In somecases the local strain fields can be used to check theassumptions in existing theoretical modeling at thelocal level. Indeed, as emphasized above, the OL effectinterpretation invoked here was not new but has beenproposed by investigators with deeper insight, but lessdirect/local data, in the past. In cases like the OL-effectthe local strain field mapping can provide a detailedpicture so that appropriate underlying mechanisms and

92 Journal of Strain Analysis 47(2)

quantitative model modifications can be supported, orchallenged, from the bottom up.

Funding

This work was supported by Office of Naval Research(ONR) under contract no. N00014-04-1-0194 andDURIP ONR N00014-02-1-0772. Utilization of theNSLS was supported by US Department of Energycontract DE-AC02-76CH00016.

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Appendix

Notation

CT compact tension (as in fatiguesample geometry)

da/dN crack length (a) growth rate(where N is the number of fatiguecycles)

Fel the elastic force of thesurrounding medium on the OLligament

DF/Fmax 12Fc/Fmax

the normalized differentialapplied force

Fmax the maximum applied cyclicfatigue load (at the CT specimenmounting pins)

Fmin=R Fmax

with R=0.1the maximum applied cyclicfatigue load (at the mountingpins)

FOL=2 Fmax the overload applied load (atmounting pins)

f, fo, max-ret, and50%-ret specimens

the constant amplitude fatigued,fatigued plus overloaded, themaximum crack growthretardation, and 50% recovery offatigue crack growth ratespecimens respectively

Kmax maximum stress intensity factor(in the fatigue cycle)

Kmin=R Kmax

with R=0.1minimum stress intensity factor(in the fatigue cycle)

Ns the crack face contact forceOL overloadeyy strain component along the y-

direction (perpendicular to thecrack propagation direction x)

Deyy strain the change in eyy betweentwo loads

Deyy(F) = eyy(F)2eyy(F=0)

strain the change in eyy betweenzero and F loads

eyy(F,x)] the strain as a function ofexternal load, F, and position, x,in the crack plane

Deyy(norm)=[eyy(F=Fmax,x)2eyy(F=0,x)]/Deyy(before OL):

the change in strain normalized tothe pre-OL value

x(x,f), where f=F/Fmax

the spatial and load dependentstrain response function (strainsusceptibility)

94 Journal of Strain Analysis 47(2)