NTN V[Z [S Û[] SN VT`R P]NPW^ · D][\NTN_V[Z [S Û[]_ SN_VT`R P]NPW^ F( F`]RÛ NZQ E( C( EV_PUVR E^qfdrb ò^`h molm^d^qflk fk bkdfkbbofkd j^qbofîp e^p _bbk qeb pr_gb`q lc `lkpfabo,

$Page 1: NTN V[Z [S Û[] SN VT`R P]NPW^ · D][\NTN_V[Z [S Û[]_ SN_VT`R P]NPW^ F( F`]RÛ NZQ E( C( EV_PUVR E^qfdrb ò^`h molm^d^qflk fk bkdfkbbofkd j^qbofîp e^p _bbk qeb pr_gb`q lc `lkpfabo,$
Propagation of short fatigue cracksS. Suresh and R. O. Ritchie

Fatigue crack propagation in engineeringmaterials has been the subject of consider-able research, and extensive review articleshave appeared over the past several years.Most of these investigations focused on thebehaviour of 'long' fatigue cracks, eventhough the characteristics associated withthe extension of small cracks in metals andalloys remain relatively unexplored, despitethe ir unque stionable im portan ce from anengineering standpoint. In this review, themechanics and micromechanisms of thesubcritical growth of short fatigue cracksare examined, and aspects of their propaga-tion behaviour are contrasted with those oflong cracks in terms of fracture mechanics,microstructure, and environment. Cracks aredefined as being short (i) when their lengthis small compared to relevant microstruc-tural di~ensions (a continuum mechanicslimitation), (ii) when their length is smallcompared to the scale of local plasticity (alinear elastic fracture mechanics limitation),or (iii) when they are simply physicallysmall (e.g. ~ O. 5-1 mm). Since all threetypes of short flaw are known to propagatefaster. than (or at least at the same rate as)corresponding long fatigue cracks subjectedto the same nominal driving force, currentdefect tolerant fatigue design procedureswhich utilize long crack data can, in certainapplications, result in overestimates of life-times. The characteristics of the short crackproblem are critically reviewed in the lightof the influences of local plasticity, micro-structure, crack tip environment, growthmechanisms, crack driving force, and thepremature closure of the crack. IMR/137

Professor S. Suresh, BTech, MS, ScD, is in theDivision of Engineering, Brown University,Providence, RI, USA. Professor R. O. Ritchie, MA,PhD, FIM, CEng, is in the Department of MaterialsScience and Mineral Engineering and the LawrenceBerkeley Laboratory, University of California atBerkeley, Cal., USA. Professor Suresh was formerlywith the University of California at Berkeley.

LIST OF SYMBOLSa == crack length

ao == intrinsic crack length, i.e. constantcharacteristic of material ormaterial condition in expressionfor M (equation (24))

~a == increment of crack extension

da/dN == fatigue crack propagation rate

A == constant in cyclic constitutivelaw (Fig. 5)

b == sum of crack length a and blockedslip band zone Wo (equations (18)and (19))

B == thickness of testpiece (Fig. II)

c == depth of edge notch or half lengthof internal notch

C =: half width of surface microcrack

C == experimentally determined scalingconstant (equation (2))

d == proportionality factor dependenton yield strain EO and work harden-ing exponent n (equation (13))

d g == grain size

do == maximum thickness of excessoxide layer

e == nominal difference in crack lengthbetween fatigue crack of length ain unnotched specimen and equiva-lent fatigue crack of length I grow-ing from notch

E == elastic (Young's) modulusE' == effective value of Young's modulus

under different loading conditionsi == function of stress intensity factor

range M and load ratio R(equation (15))

f ij == dimensionless function of polarangle e measured from crack plane(equation (3))

ib ,Iij == universal functions of both polarangle e measured from crackplane and work hardening exponentn (equation (10))

G == strain -energy release rate

J == scalar amplitude .of crack tipstre ss and strain field under non-linear elastic conditions

~J == cyclic component of J

kf == fatigue strength reduction factor

kt == theoretical elastic stress con-centration factor

446 Suresh and Ritchie: Propagation of short fatigue cracks

Meff

RE,R a == strain and stress concentrationfactors

K cl == closure stress intensity factor

K~ == critical value of microscopicstress intensity at tip of slip band

K I == stress intensity factor in mode Iloading

I\Ic == critical stress intensity at failure(plane strain fracture toughness)

Ki¥' == mode II value of microscopicstress intensity at tip of slip band(Fig. 17)

Kl == limiting stress intensity for longcrack emanating from notch

== maximum and minimum stre sssintensities during cycle

== limiting stress intensity for shortcrack emanating from notch

== threshold stress intensity factor

== long crack threshold stressintensity factor

== local mode I and mode II stressintensity factors for non-linearcrack

== nominal stress intensity factorrange (Kmax -Kmin)

== effective stress intensity factorrange (Kmax - Kcl)

== equivalent stress intensity rangefor short crack

Mi == initial stress intensity range

Mth == threshold stress intensity factorrange

DJ( E == pseudo-elastic-plastic strainintensity range

Mo == long crack threshold stressintensity factor range

== length of crack growing from notch

lo == transition size of crack growingfrom notch

m == exponent in Paris power law(equation (2))

m' == exponent in power law for elastic-plastic fatigue crack growth(equation (12))

n, n' == monotonic and cyclic work harden-ing exponents, respectively

J.V == number of cycles

N i == number of cycles necessary toinitiate macrocrack

NP == number of cycles necessary topropagate macrocrack subcriticallyuntil failure

Pel == closure load during fatigue cycle

Pmax, Pmin == maximum and minimum loadsduring fatigue cycle

q == fraction of elastic strain withvalues between O. 5 and 1(equation (22))

Q == dimensionless function of geometry(equation (8))

r == radial distance from crack tip

rmax, r tl == maximum and cyclic sizes ofplastic zone

ry == size of plastic zoner~ == cyclic size of plastic zone at

fatigue limit

R == load or stress ratio (Kmin/Kmax)

R E == ratio of minimum to maximumstrain (Fig. 24)

s == half distance between location ofthickest oxide layer and crack tip

S == compliance of microcrack incompletely opened state

Wo == length of blocked slip band zone

~ We, /). Wp == elastic and plastic components ofof strain -energy density range

x == ratio of mode II to mode I cracktip displacements

a == empirical constant in Peterson kf(equation (32))

(3 == constant in equation (36)

y == non-dimensional fracture surfaceroughness factor

0, at == crack tip opening displacement

0max, ~ a == maximum and range of crack tipopening displacement

o(amax) == crack tip opening displacement atmaximum stress

0(0) == crack tip opening displacement atzero load

/)'0t == crack tip opening displacementrange

E == local strain

Ep == plastic strain

EO == yield strain

~E == local strain range

~Ee, tl Ep == elastic and plastic strain ranges

8 == polar angle measured from crackplane

eo, 81 == angles associated with crackdeflection (Fig. 31)

v == Poisson ratio

p == notch root radius

a == local stress

aoo == nominal stress

acl, amax == closure and maximum stresses

ae == tensile stress corresponding tofatigue limit

Suresh and Ritchie: Propagation of short fatigue cracks 447

or rainflow counting methods); multiaxial stresses;environmental effects; and so forth (for a summary,see Ref. 2). Although based on total life, thisapproach - which is in widespread use, particularlyin the automotive industry - essentially representsdesign against crack initiation, since near thefatigue limit, especially in smooth specimens, mostof the lifetime is spent in the formation of anengineering sized crack. In procedures forpredicting lifetimes, such S-N or low cycle fatigue(LCF) testing might simulate the initiation andearly growth of the fatigue crack within the fullyplastic region of the strain field at some stressconcentrator (see Fig.l).

For safety-critical structures, especiallythose with welded' and riveted components, theapproach is different. There has been a growingawareness that the presence in a material ofdefects below a certain size must be assumed andtaken into account at the design stage. Under suchcircumstances the integrity of a structure willdepend on the lifetime spent in crack propagation,and since the crack initiation stage will be short,the use of conventional S-N total life analyses maylead to dangerous overestimates of life. Suchconsiderations have led to the adoption of theso-called 'defect tolerant' approach in which thefatigue lifetime is asse ssed in terms of the time,or number of cycles, necessary to propagate thelargest undetected crack to failure.3 Here theinitial size of the crack is estimated using non-destructive evaluation or proof tests, whereas thefinal size is defined in terms of· the fracturetoughness K1c, the limit load, or some allowablestrain criterion. This approach, the only one usedfor certain applications in the nuclear and aero-space industries, for example, relies on theintegration of an expression for crack growth,representing a fracture mechanics characterizationof relevant fatigue crack propagation data suitablymodified to account for mean stress effects (e.g.using the Forman equation), variable amplitudeloading (e.g. using the Wheeler or the Willenborgmodel), environmental effects, and so forth, asrequired (for a summary, see Ref. 4). Such expres-sions are generally based on the original Parispower law relationship, i.e. are of the form

Schematic diagram showing various stages offatigue in engineering components and typicallaboratory tests to evaluate fatigue life!

(2)

/Low cycle/ fatigue test

" ..•.•••.~)"'",\ ,,". 't ><'v ........•

Nucleation and-early growth

do/dN == CMm

where C and m are experimentally determinedscaling constants, da/dN is the crack growth

1

air == normal frictional stress fordislocation motion

== local crack tip stresses dependenton distance from crack tip andinclination to crack plane

== threshold stress for no crackgrowth

== yield or flow stre ss

== local stress range

== nominal stress range

== fatigue limit or endurance strength

== threshold stress range for nocrack growth

== shear frictional stress for disloca-tion motion (Fig. 17)

Fatigue fracture s ac count for the vast maj ority ofin-service failures in most engineering structuresand components, either as a result of puremechanical loading or in conjunction with slidingand friction between surfaces (fretting fatigue),rolling contact between surfaces (rolling contactfatigue), aggre ssive environments (environmentallyassisted or corrosion fatigue) or elevated tem-peratures (creep fatigue). Such progressivefracture of materials by incipient growth of flawsunder cyclically varying stresses, termed fatigue,can be categorized into the following discrete yetre la ted phenomena:

(i) initial cyclic damage in the form of cyclichardening or softening

(ii) creation of initial microscopic flaws(microcrack initiation)

(iii) coalescence of these microcracks to form aninitial 'fatal' flaw (microcrack growth)

(iv) subsequent macroscopic propagation of thisflaw (macrocrack growth)

(v) final catastrophic failure or instability.

In engineering terms the first three stages,involving cyclic deformation and microcrackinitiation and growth, are generally classifiedtogether as (macro-) crack initiation, implying theformation of an 'engineering sized' detectablecrack (e.g. of the order of several grain diametersin length). Thus, in such terms, the total fatiguelife N can be defined as the number of cyclesne ce ssary both to initiate a (macro-) crack, N i,and to propagate it sub critically until final failure,Np, Le.

N=Ni+Np . (1)

In fatigue design, where data from laboratorysized specimens are used to predict the lifetimeof more complex components in service (see Fig. 1after Ref.1), this distinction between initiation andpropagation live s can be critical. Conventionalapproache s to fatigue design involve the use ofS-N' curves (stress v. number of cycles), repre-senting the total life re sulting from a given stress(or strain) amplitude, suitably adjusted to take intoaccount effects of mean stre ss (using, for example,Goodman diagrams); effective stress concentrationsat notches (using fatigue strength reduction factorsor local strain analysis); variable amplitude loading(using the Palmgren-Miner cumulative damage law


2 Typical fatigue crack propagation rates(da/dN) for long and short cracks as functionof stress intensity factor range AK

Long crack(LEFM)

log A K----.

Short crockfrom nOlch~

--""

constant-ampl itude loadi ngR = constant

Shari croci

-V\ ,-~\\,,I ~ Long crack threshold ~ Ko

i

(ii) cracks which are of a length comparable to thescale of local plasticity (e.g. small cracksembedded in the plastic zone of a notch or ofa length comparable with their own crack tipplastic zones, typically ~10-2 mm in ultrahighstrength materials and ~0.1-1 mm in lowstrength materials)

(iii) cracks which are simply physically small(e.g.~0.5-1 mm).t

Most investigations to date have focused on thefirst two factors, which represent, respectively, acontinuum mechanics limitation and an LEFMlimitation to current analyses. Here, presumablywith an appropriate micro- or macro-mechanicscharacterization of crack advance, it should bepossible to establish a corre spondence betweendata for the growth rates of long and short cracks.However, physically short flaws, which are 'long'in terms of continuum mechanics and LEFManalyses, have also been shown to propagate morequickly than corresponding long cracks under thesame nominal driving force (see e.g.Refs.11 and12). This may reflect basic differences in thephysical micro-mechanisms associated with theextension of long and short cracks. Thus, physicallyshort cracks represent a limitation in the simili-tude concept of fracture mechanics.

In this article is pre sented a review of theexperimental results that have been obtained andof the way s in which the growth of short fatiguecracks in engineering materials has been inter-preted. The literature on 'conventional' crackinitiation and 'long' crack growth is examined onlybriefly since the se are topics which have beengiven ample coverage in Refs. 4-7 . However, aconcerted effort has been made to review all thecurrently available information on short cracks.Since this is a rapidly expanding field somereferences may have inadvertantly been missed,and the authors. regret any such omissions.

tSuch physically small flaws are sometimesreferred to as 'chemically small' where environ-mental effects are dominant.!l

increment per cycle, and M is the alternatingstre ss intensity factor given by the differencebetween the maximum and minimum stress inten-sities in the fatigue cycle (M == Kmax - Kmin).5In its simplest form equation (2) provides areasonable description of growth rate in the so-called intermediate range of growth rate s, typicallybetween 10-6 and 10-3 mm/cycle. However, itunderestimates propagation rates of higher valuesof M, as final instability is approached (e.g. asKmax -1 K1c), but overestimates propagation ratesat lower values of M( approaching the so-calledlong crack fatigue threshold stress intensity rangeMo,below which long cracks remain dormant orgrow at experimentally undetectable rates.6 Sincethe time for crack initiation is generally taken tobe zero, such defect-tolerant lifetime predictionsare assumed to be inherently conservative andto simulate the macroscopic growth of the fatiguecrack (see Fig. 1). In its most widely used form,that based on linear elastic fracture mechanics(LEFM), the latter approach would imply crackgrowth outside the notch strain field where thecrack tip plastic zone is small compared to thecrack length, i.e. that a long crack propagate sunder nominally elastic conditions. 5, 6

Current practice in the determination of therelevant crack growth 'law' * for a particularmaterial under a particular set of conditions fora given application is to use data from laboratorytests on fatigue crack propagation, characterizedin terms of the linear elastic stress intensity rangeM. However, most of these data have beenobtained from testpieces containing cracks of25 mm or so, whereas many defects encounteredin service are far smaller than this, particularlyin turbine discs and blades, for example. In therelatively few case s where the fatigue behaviour ofsuch short cracks has been studied experimentally(for earlier reviews, see Refs. 7-12), it has beenfound - almost without exception - that, under thesame nominal driving force, the growth rates ofshort cracks are greater than (or at least equal to)the corresponding growth rates of long cracks (seeFig. 2). This implie s a breakdown in the similitudeconcept generally assumed in fracture mechanics,as described in the section 'Similitude concept'below. Furthermore, it suggests that the use ofdata for long cracks in defect tolerant lifetimecalculations for components where the growth ofshort flaw s repre sents a large proportion of thelifetime, can lead to considerable overes~imates.

There are several ways of defining' short'tcracks:

(i) cracks which are of a length comparable to thescale of the microstructure (e.g. of the orderof the grain size)

*Although often termed 'laws', such crack growthrelationships are invariably totally empirical andare derived simply by fitting mathematical equa-tions, of the form of equation (2), to sets ofexperimental data.

t A distinction is sometimes made between smallcracks, which are small in all dimensions, andshort cracks, which are small in all but onedimension (presumably the width), although suchdefinitions are not considered in the pre sentreview.


Situations in which short and long cracks may growdifferently are discussed, from both an engineeringand a mechanistic viewpoint, in terms of (i)appropriate fracture mechanics characterizationand (ii) the physics and mechanisms involved incrack advance. In the former case the short crackproblem is treated principally in terms of elastic-plastic fracture mechanics (EPFM) analyses inwhich the effects of local crack tip plasticity andthe strain fields at ,notches are taken into account,whereas in the latter case difference s in crackgrowth behaviour are examined in terms of therole of crack size and shape, microstructure,environment, and mechanisms for crack closureand extension. First, though, a brief summary isgiven of the fracture mechanics procedures usedto characterize fatigue crack propagation for bothlong and short cracks.

FRACTURE MECHANICS CHARACl'ERIZATIONOF FATIGUE CRACK GROWTH

Linear elastic fracture mechanics

The fracture mechanics approach to correlatingcyclic crack advance begins with characterizingthe local stress and deformation fields at thecrack tip. This is achieved principally throughasymptotic continuum mechanics analyses wherethe functional form of the local singular field isdetermined to within a scalar amplitude factorwhose magnitude is calculated from a completeanalysis of the applied loading and geometry. Forthe linear elastic behaviour of a nominallystationary crack subjected to tensile (mode I)opening, the local crack tip stresses aij can becharacterized in terms of the Kr singularfield:13,14

(J··(r e) = __ 1_ K t·· (Ie) + O(rl/2) +1J ' - (27Tr) 1/2 r.- 1J

lim Uij(r, 8) = \1/2 Kr/ij (8) (3)r~o (27Tr

where Kr is the mode I stress intensity factor, rthe distance ahead of the tip, e the polar anglemeasured from the crack plane, and Iij a dimen-sionless function of e. Similar expressions existfor cracks subjected to pure shear (mode II) andanti -plane strain (mode III). Provided thisasymptotic field can be considered to 'dominate'the vicinity of the crack tip, over a region which islarge compared to the scale of microstructuraldeformation and fracture events involved, thescalar amplitude factor Kr :can be considered as asingle, configuration independent parameter whichuniquely and autonomously characterizes the .localstre ss field ahead of a linear elastic crack and canbe used as a correlator of crack extension.

For cracks subjected to cyclically varying10ads,Kr must be defined at the extremes of thecycle, such that a maximum and a minimum stressintensity (K max and Kmin, respectively) for aparticular crack length can be computed., Accordingto the original analysis by Paris and Erdogan,15the crack growth increment per cycle in fatigue,da/dN, can be described in terms ofa power lawfunction (equation (2)) of the range of Kr, given bythe stress intensity rangeDJ(. It is important to-

note here that such an asymptotic continuummechanics characterization does not require adetailed quantitative knowledge· of the microscopicbehaviour of individual fracture events, and thusthe analysis is independent of the specific micro-mechanism of crack advance.

One of the principal limitations of thisapproach specifically of the adoption of Kr as avalid des~ription of the crack tip field, is that astate of small scale yielding must exist. Fromequation (3) it is apparent that as r-70, stressesbecome infinite at the tip. However, in reality suchstresses are limited by local crack tip yielding,which occurs over a region ahead of the crack tipknown as the plastic zone. Calculations of theextent of this region vary according to the mode ofapplied loading and the geometry of the body,16but a rough estimate of the size ry of the plasticzone ahead of a monotonically loaaed crack isgiven by

r ~ ~ (Kr)2 (4)y 27T ao

where a0 is the yield strength of the material.Provided this extent of local plasticity is smallcompared to the extent of the Kr-field, which isitself small compared to the overall dimensions ofthe body (including the crack length), the plasticzone can be considered as merely a small pertur-bation in the linear elastic field, and the Kr-fieldcan be assumed to dominate the region around thecrack tip. This situation, known as small scaleyielding, occurs only when the size of the plasticzone is at most one-fifteenth of the in -planedimensions of the crack length and the depth ofremaining ligament.

The local yielding ahead of fatigue cracks ismade somewhat more complex by reversed plasti-city. However, following the analysis by Rice17 ofa cyclically stre ssed, elastic-perfectly plasticsolid, plastic superposition of loading and unloadingstress distributions can be used to compute theextent of the plastic zone ahead of a fatigue crack.On loading toK max, a monotonic or maximumplastic zone is formed at the crack tip ofdimension (from equation (4))

r ~~(Kmax)2 (5)max 27T (Jo

However, on unloading from Kmax to Kmin, super-posing an elastic unloading distribution of maximumextent -200 give s rise to re sidual compressivestre sse s of magnitude -(Join a region ahead of thecrack tip. This region is known as the cyclicplastic zone, and its size rDois approximately a·quarter of the size of the monotonic zone:

1 'DJ()2r li. '" 21T (2U

O• (6)

where, strictly speaking, 00 is now the cyclic yieldstrength. Once again, the correlation of Kr withcrack extension by fatigue will be a valid approachprovided small scale yielding conditions apply,namely that rmax be small compared to thein-plane dimensions.

The numerical values of the stress intensityfactors atthe crack tip,Kr, ilK, and so on, cannot

International Metals Reviews, 1984, Vol. 29, No.6


where n is the work hardening exponent, E' theappropriate elastic modulus (== E for plane stressor EI(l - lJ 2) for plane strain), and fij and f{i areuniversal functions of their arguments for planestress and plane strain, respectively. The ampli-tude of this field is the so-called J -integral25 and,analogous to Kr, J uniquely and autonomouslycharacterizes the crack tip field under elastic-plastic conditions, provided there is some degreeof strain hardening. Furthermore, for small scaleyielding J can be directly related to the strainenergy release rate G, and hence to K1, Le.

J == G == KrIE' (linear elastic) (11)

Despite difficulties in establishing the precisemeaning of J as applied to a description of thegrowth of cyclically stressed (non- stationary)cracks, Dowling26 and Dowling and Begley27proposed a power law correlation of fatigue crackgrowth rates under elastic-plastic conditions tothe range of J, i.e.

dal dN a: l:iJ m' (12)

Provided such analysis is fundamentally justified,the use of l:iJ doe s pre sent a fir st approach tocharacterizing the growth of short cracks whichare comparable in size to the extent of local yield-ing. However, as mentioned above, the validity ofthe l:i.J approach is often questioned since itappears to contradict a basic assumption in thedefinition of J - that stress is proportional to thecurrent plastic strain.25 This follows because Jis defined from the deformation theory· of plasticity,which does not allow for the elastic unloading andnon-proportional loading effects which accompany

Value s of lo are generally a small fraction of thenotch root radius p, and for moderate to sharpnotches they generally fall in the range pi 20 topi 4. Dowling21 has further noted that ktaOOvalue sonly 20-30% above ao are sufficient to generate anotch tip plastic zone which engulfs the smallcrack region lo and thus, for small cracks atnotches, LE FM analysis will often be suspect.

Elastic-plastic fracture mechanics

The above example serve s to illustrate one aspectof the short crack problem, where the crack lengthis comparable with the size of the plastic zonearound the notch tip. A similar situation, wheresmall scale yielding conditions may not apply, iswhen the plastic zone at the tip of the fatiguecrack itself is comparable with the crack length,Le. when a"'ry• Since the use of Kr singular fieldsis no longer appropriate in such instance s, alterna-tive asymptotic analyses have been developed todefine the crack tip stress and strain fields in thepresence of more extensive local plasticity (for arecent review, see Ref. 22). Based on the deforma-tion theory of plasticity (Le. non-linear elasticity),the asymptotic form of these local fields for non-linear elastic power hardening solids obeying theconstitutive law a ex: ~B,is given by the Hutchinson-Rice-Rosengren (HRR) singularity as23,24

lim CJij(r, 8) == (E'Jlaijr)n/(n+l) aofij(8,n))r-7o (. (10)

lim Eij(r, 8) = (E'Jlaijr)1/(n+l) ffj(8,n) \r-7o

be determined from the asymptotic analyses, yetcan be computed from the overall geometry andapplied loading conditions; in fact, solutions forKr applicable to a wide range of conditions are nowtabulated in handbooks.18-20 A useful example ofsuch Kr solutions which is particularly relevant tothe short crack problem is that of a crack (oflength l) growing from a notch (of length 2c) -see Fig.3. Modelling the notch as a circular holein an infinite plate under a remotely appliedtensile stre ss dJO

, the limiting analytical Kr solutionfor a short crack emanating from the notch isobtained as

Ks = 1. 12ktaOO(7Tl)1/2, (7)

where kt is the theoretical elastic stress con-centration factor (equal to 3 here) and 1.12 is thefree surface correction factor. However, when thecrack becomes long the limiting stress intensityis obtained by idealizing the geometry so that thenotch becomes part of a long crack of lengtha = c + 1, su ch that

KI == Qaoo

(7Ta)V2 (8)

where Q is a dimensionle ss function of geometry,such as a correction factor to allow for finitewidth. The numerically determined K1 solution forany crack emanating from a notch, shown by thedashed line in Fig. 3, can be seen to be given bythe limiting cases of short and long cracks. Asshown by Dowling,21 the transitional crack sizelo, which can be interpreted as the extent of thelocal notch field, can then be obtained by combiningequations (7) and (8) to give

lo == cl (1.12ktl Q)2 + 1] (9)

3 Linear elastic K1 solutions for crack of lengthl emanating from circular notch of radius cin an infinite plate subjected to remotelyapplied uniaxial tensile stre ss 0-00

; shortcrack (Ks) and long crack (Kl) limitingsolutions and numerical solutions are shown21

ale, Crack Length From Centreline1.0 1.2 1.4 1.6

6.0 f t t t

2::- 8'Vic::Q,):s ~ l ~ ~ ~V) 4.0 - a-(JJ= Remote Tensile StressV)

~if)V)V)

Kp,= c1'{TIa Ks

= 1. 12 kt cffITl~c::0

"Vic::Q,)

E0

2.0~ ". ".

~ "~ /"~ Numerical SolutionI'

00 0.2 0.4 0.6

p,f c, Crack Length Beyond Notch



crack advance.2 8 However, by recognizing thatconstitutive laws for cyclic plasticity (Le. thecyclic stress-strain curve) can be considered interms of stable hysteresis loops, and that suchloops can be mathematically shifted to a commonorigin after each reversal, the criterion of stressbeing proportional to current plastic strain caneffectively be satisfied for cyclic loading. 22,29

An alternative treatment of elastic-plasticfatigue crack growth, which is not subject to therestrictions required by non-linear elasticity, is .to use the concept of crack tip opening displace-ment (CTOD). From equation (10) it is apparentthat the ~ening of the crack faces as r --7 0 variesas r n/(n 1) such that this separation can be usedto define the CTOD (Ot) as the opening where 45°lines emanating back from the crack tip interceptthe crack faces. Thus, for proportional loading

0t = d(Eo,n) J/oB (elastic-plastic) l (13)0t ex: Kl/ooE' (linear elastic) \

where d is a proportionality factor having a valueof '" 0.3-1 depending on the yield strain EO' thework -hardening exponent n, and whether planestress or plane strain is assumed.30 Since 0t,like J, can be taken as a measure of the intensityof the elastic-plastic crack tip fields, it is feasibleto correlate rates of fatigue crack growth to therange of 0t, i.e. the cyclic CTOD ~0t=

da/ dN ex: ~Ot (elastic-plastic) I(14)

da/dN ex: Ml/2ooE' (linear elastic) \

Approaches based on J and 0t are basically equiva-lent for proportional loading, and are of coursevalid under both elastic-plastic and linear elasticconditions. Therefore, they are generally applica-ble to a continuum description of the growth ratebehaviour of cracks that are considered smallbecause their size is comparable with the scale oflocal plasticity. For such short cracks the use ofEPFM rather than LEFM may thus be expected, atleast in part, to normalize difference s in behaviourbetween long and short cracks. However, the shortcrack problem is not simply a breakdown in theapplication of LE FM since the use of elastic-plastic analyses cannot totally normalizedifferences between short and long cracks.Although elastic-plastic analysis is certainlyimportant, differences in the behaviour of long andshort cracks can also be attributed to micro-structural, environmental, and closure effects.Thus the short crack problem relates to a break-down, not simply in LE FM, but in the fra.cturemechanics similitude concept.

Similitude concept

The. application of fracture mechanics to thepropagation of fatigue cracks is based on thepremise that the governing parameter, such as thestress intensity factor Kr or the J -integral, usedto correlate growth rates fully describes thestress and deformation fields in the vicinity of thecrack tip. In addition, it is implicitly assumedthat the concept of similitude (see Fig.4) is valid.This concept implie s that, for two cracks ofdifferent sizes subjected to the same stressintensity (under small scale yielding) in a given

.keOIPre(0) (b)

a short crack, a '" r y; b long crack, a ~ r y

4 Schematic representation of similitude con-cept' which implies that cracks of differinglength a subjected to same nominal drivingforce, e.g. M, have equal plastic zone sizesry ahead of crack and will therefore advanceby equal increments ~a per cycle

material-micro structure-environment system,crack tip plastic zones are equal in size and thestress and strain distributions along the bordersof these zones (ahead of the crack) are identical.Accordingly, equal amounts of crack extension~a are to be expected. For a cyclic load denotedby K max and K min, this dictate s that

~a = da/dN = f(Kmax,Kmin) -= f(M,R) (15)for each cycle, where the stress intensityrange AK = Kmax - Kmin and the load ratioR = Kmin/Kmax. However, this concept of simili-tude cannot be applied when

(i) crack sizes approach the local microstructuraldimensions

(ii) crack sizes are comparable with the extent oflocal plasticity for non-stationary flaw s

(iii) through thickness, out of plane stresses(which are independent of Kr) are different

(iv) crack extension mechanisms are different(v) extensive fatigue crack closure is observed(vi) external environments significantly influence

crack growth,

to name but a few instances.10,12,31,32 Most ofthese mechanisms are specific to the short crackproblem and thus contribute to difference s in thegrowth rate behaviour of long and short cracks atnominally identical driving forces. They arediscussed below in terms of local plasticity. andmicrostructural and environmental factors. Ingeneral, however, their effect on the breakdown ofthe similitude concept for short cracks is that theyinfluence (to varying degree s according to thecrack length, for example) the local driving force(Le. the characterizing parameter Kr or Jeffectively experienced in the region near the tip).It is this 'near tip' parameter that governs crackadvance, not the 'nominal' global value of thisparameter computed by conventional analyses ofexternally applied loads and measurements ofmacroscopic crack lengths. In only a few instanceshave the relationships between the 'near tip' andthe 'nominal' driving forces been quantified,indicating that the methodology for a global fracturemechanics analysis of short crack growth has yetto be developed.

EXPERIMENT AL TECHNIQUES

In the experimental measurement of the propaga-tion and threshold behaviour of small fatiguecracks, many complex problems associated with

International Metals Reviews, 1984, Vol. 29, NO.6

452 Sureslz and Ritclzie: Propagation of short fatigue cracks

reproducibility and scatter can be encountered.This is particularly true since most of theapproaches used so far have involved adaptingprocedures originally designed for measurOing longcracks.

Several experimental techniques are availablefor monitoring the growth rate s of long fatiguecracks (see e.g. Refs. 33-35), including

(i) optical techniques (i.e. with the naked eye ortravelling microscope s, or using high speedcameras)

(ii) methods based on measuring electrical resis-tance or potential using either dc or accurrents

(iii) compliance techniques using mouth opening(clip) or back-face strain gauges

(iv) acoustic emission(v) ultrasonics(vi) eddy current methods,

the first three being the most widely used. Forlong cracks, growth rates above 10-6 mm/ cycleare generally measured by utilizing using suchtechnique s at constant cyclic load levels (i.e.increasing AK with increasing crack length). 34At levels near the threshold, on the other hand,crack growth rates (da/dN;S 10-6 mm/ cycle) andthreshold AKo values for long cracks are normallymeasured using the so-called 'load shedding'(decreasing DK) method (see e.g. Ref. 6). Thisprocedure involves making continual reductionsin AK (starting from an intermediate value) of notmore than 10/'0. At each M level, the cr'ack isthen allowed to propagate over a distance atleast four times the size of the maximum plasticzone generated at the previous higher M level.This load reduction scheme is repeated until thethreshold AKo for no detectable (long) crackadvance is reached. Such procedures,6 typicallyused to monitor the growth and arrest of longfatigue cracks, pose extreme difficulties whenadopted for measuring short cracks. Firstly, onefaces the problem of estimating the depth of ashort crack, emanating from a surface, from itswidth. This requires an empirical calibrationand/or an educated guess. Also, with load sheddingprocedures'used near the threshold, the crack iscontinually growing and so may cease to be 'short'by the time the 'threshold is reached. Moreover,the initiafion stageofa 'major' short crack mayinvolve the linking up of several flaws (e.g.fromcracked'inclusions) at different locations. Whilelow magnification optical techniques are ,not parfi-cularly suit.ed to detecting the presence andmonitoring the growth of short cracks during a'fatigue test, other methods such as the potential,resistance, or compliance teachniques 'may nothave the required resolution and reproducbility to'enable the growth of (part through) short cracksof c'omplex geometry to be characterized. 'Inaddition to the difficulties .as·sociatedwith thelimitations of the crack monitoring Clevice, thegrowth ,of a short crack can 'be a strong functionof local microstructural characterist-icsand'environment (see e.g. Refs. 7-12). This aspect ofshort crack ad~ance can also lead to'poorreproducibility.

Since load shedding proc€dure s are oftentHffi,cult toapply,there is always the question·@f

.International Metals R'eviews., 1984, Vol.>29, NO.6

whether short crack behaviour should be studiedusing an artificial or a natural crack. It is parti-cularly difficult to create an artificial notchwithout damaging the material immediately aheadof it, yet this is essential since it is this region inwhich short crack growth must be studied. Theareas around machined notches contain residualstresses, whereas notches or starter cracksintroduced by electro-discharge machining have alocally melted zone at their tip. Subsequentannealing cannot be guaranteed to remove suchdamage without drastically changing the micro-structure under test. As discussed below, thereare procedure s in which a long crack is grown andmaterial in its wake is then machined off to leavea short crack. Reproducibility is a major problemhere, as is crack shape, since any amount of non-uniformity or 'bowing' in the original long crackwill introduce irregular short cracks. However,one very successful method for rapidly initiatingshort cracks without causing significant damage,but only for aluminium alloys, has been to embrittlethe sample surface before fatigue using small dropsof ink at well separated 10cations.36 Although theprecise mechanism (involving some environmentallyinfluenced fatigue process) is unknown, short crackshave been found to initiate extremely rapidly.

Initiation and growth measurements of short cracks

Optical techniques

Crack monitoring techniques using visual methodshave probably been the most. widely used proceduresto date. For example, using procedures firstadopted by Barsom and McNicol37 for studyingcrack initiation, 'Fine and co-workers3 8,3 9 have·used a metallurgical microscope with a longworking distance for visual and photographicobservations of crack initiation and microcrackgrowth in wide, single edge notch (SEN) samples,of steels and aluminium alloys~ The microscope'was mounted on an x-j:' micrometer base forpositioning and· measuring crack length. A cameraattachment was available for photographicallyrecording the progressive changes in the surfaceof specimens subjected to cyclic loading.

Morris and co-workers40-49u:sed taperedflexural and hourglass shaped fatigue samples tomonitor microcracks in aluminium and titaniumalloys. Cracks were initiated on the surface of thespecimen using various techniques, includingmaking starter notches and marking with felt-tippens. 'Crackpropagation rates were estimated bymonitoring crack size at ,regular interVals bytransferring the specimen to a scanning electronmicroscope (SEM}. The specimens were loaded'in theSEM to estimate the microcrack tip openingdisplacement at the maximum load .. For example,.'an empirical calibration equation was obtained for22J'9T851alum'inium alloy from' several measure-ments 'of 'surface microcracks to estimate the1rdepth a from a,knowledge' of their width ,,2c.The ·expression

a/2e =0.,36'2 + 25.0t(o{.omax)j2'C"-·O.015)

(16)

was'oDtained,47 where o(omax)is theCTOD,at themaximumstre SS,Gmax.


(ii) ranges of stress and plastic strain, obtainedfrom stable cyclic hysteresis loops, wereused to quantify jj.J

(iii) the value of jj.J for such semicircular surfacecrack geometries was given by:

~J ~ 3. 2~ Wea + 5. D6.lVpa (17)

where 6.We and 6.Wp are the elastic and plasticcomponents of the remote strain energydensity ranges (see Fig. 5)

(iv) the scaling constants in equation (17), whichincorporate correction factors for specimengeometry and flaw shape, were derived fromequivalent linear elastic solutions.

Tn the general sense, there are problems withusing either optical or replication techniques inthat they are difficult to apply in hostile environ-ments, such as in corrosive solutions or at elevatedtemperatures. More importantly, they giveinformation only on the surface length of thecrack, and therefore assumptions must be made ifthe internal profile is to be estimated.

Optical techniques have also been used tostudy short cracks at elevated temperatures. Forexample, Sheldon et al.5o designed a microscopewith a long working distance to monitor in situ thegrowth of short and long cracks in nickel-basesuperalloys at temperatures as high as 65DoC.Cracks were initiated at high stress intensitiesat ambient temperature, whereupon the load wassuccessively reduced until the approximate roomtemperature long crack threshold values werereached. Once load shedding was complete, thespecimen was machined and polished to a configura-tion such that a crack only D. D6-D.16mm long wasleft in the test panel. The cross- section of thespecimen was a parallelogram containing the crackat one acute corner. Sheldon et ale claimed thatthe taper allows the location of the crack front tobe defined more accurately, and that the crack inthe taper interacted with only a small amount ofmaterial along its front. This was considered to berepresentative of a naturally occurring smallcrack.

Replication techniques

One of the most widely used techniques for moni-toring the initiation and growth of small flaws hasbeen the replication method. For example,Dowling51,52 used cellulose acetate surfacereplicas for measuring the growth of short cracksduring low cycle fatigue tests on smooth axialspecimens of A533B steel. Tests were interruptedperiodically for replicating with "'D. 1 mm tape,softened with acetone to gain an impre ssion of thespecimen surface. The propagation rate s of suchsurface flaws (of length D. 25-1.75 mm) werecharacterized in terms of EPFM, specificallyinvolving 6.J. Procedures for calculating J fromthe loading portion of the fatigue cycle, summarizedin Fig. 5, involve the following approximations: 51

(i) surface cracks were assumed to have deptha equal to half their surface length 2c

Electrical potential techniques

Electrical potential techniques have also beendeveloped for studying short cracks. Gangloff, forexample, quantified the formation and subcriticalpropagation of small cracks emanating fromartificial surface defects.!!, 53, 54 Flaws wereintroduced along a chord of an hourglass type(low cycle fatigue) specimen (at the minimumdiameter) by either conventional grinding orelectrospark discharge machining. The crackdepth was continuously monitored via dc electricalpotential measurements and an analytical calibra-tion model, 53 and was found to agree reasonablywell with the corre sponding values measuredoptically. The calibration model used was claimedto account for crack shape as well as variations indepth for the elliptical surface flaws utilized in theexperiments. This technique is particularly suitable

~£ ~(j (tl~l/n'Cyclic (J -Eo Curve- -: - + -- 2 2£ 2A

f,'

5 Procedure for estimating ~J from stress-strain hysteresis loops for' growth of small cracks duringlow cycle fatigue of smooth axial specimens51



Remove' ridgeand test

Slot andprecrnckin fatigue

Machinetop

edges

(0)

~~~lSlot and pre crack Grind Test

in fatigue surface

a through thickne ss crack; b thumbnail crack

6 Ways of preparing short cracks58

fatigue crack was found to propagate down theridge as a straight fronted crack, and then into thebulk of the specimen with a semi-elliptical crackfront. Its depth was controlled by stopping thebending after it had spread a certain distanceacross the prepolished shoulders. Finally, theridge was removed to'leave a semi-ellipticalsurface crack in the remaining rectangular section.

Such procedure s have been used to producethrough-thickness and part-through-thicknessshort cracks to measure the thresholds for thearrest of short cracks.59-62 Usami and Shida,59for example, used this teclmique to measure fatiguethresholds in a range of steels for crack sizesbetween 0.1 and 0.3 mm. McCarver and Ritchie60used similar procedures to monitor the thresholdsfor physically short cracks in a wrought nickel-base superalloy, Rene 95. In both these investiga-tions fatigue precracking was performed under afar-field cyclic compre ssive load on bend speci-mens. This results in growth and arrest of thepre crack to a predictable depth which is a functionof the size of the plastic zone as computed fromthe compressive loads. Such compression givesrise to residual tensile stresses which apparentlyact as the driving force for crack growth. *Following precracking and machining away ofmost of the crack, the samples were annealed inan attempt to minimize possible damage in thevicinity of the short crack tip. McCarve randRitchie60 subsequently measured short crackthresholds in such specimens, using a procedureanalogous to defining a fatigue limit in unnotchedspecimens. Samples were cycled at differentinitial stress intensity ranges Mi, and the shortcrack threshold defined in terms of the largestvalue of Mi which did not cause failure (or any

*Recent studies by Sur.esh63 on crack initiationunder compression in a wide range of alloys haverevealed that cracks arrest at a length approxi-mately equal to the cyclic plastic zone size incompression. However, at high cyclic compressiveloads it was found that crack arre st can occur atcritical lengths independent of applied load,apparently asa result of' the development ofclosure behind the crack tip.

Measurements of short crack thresholds

The load shedding procedure commonly u sed toobtain the thre shold M( 0 for no detectable propaga-tion of long cracks is not readily applicable toshort cracks (those of a size smaller than the localplastic zone or the characteristic microstructuraldimension) because of the need to reduce the loadto threshold levels over a considerable distanceof crack growth. However, the procedure can beused for physically short (0.1-0.8 mm) cracks inultrahigh strength materials where both the scaleof local plasticity and the grain size are smallerthan the crack depth, and load shedding procedure scan be sufficiently rapid to enable the near-thre shold region to be approached over smallincrements of crack advance (see e.g. Ref. 57).

Wiltshire and Knott58 used two differentmethods to obtain short 'through -thickne ss' and'thumbnail' crack s in a study of the effe ct of cracklength on fracture toughness. Such proceduresseem suitable for use in evaluating growth andthresholds for short cracks. For through-thicknesscracks in maraging steels a long crack was firstproduced by' fatiguing SEN specimens in bending,as shown in Fig. 6a, with the' sample in the solutionannealed condition. Short through-thickness crackswere then obtained by g-rinding away the uppersurface, and the samples were subsequently agedto develop full strength. Similar procedure s wereused to produce short 'thumbnail' cracks, as shownin Fig.6b. The two top edges of a bend specimenwere machined away to leave a ridge along thecentre,a slot was introduced in the ridge, and thespecimen was fatigue pre cracked in bending. The

for physically short cracks, Le. those 0.5-1 mmlong, and so has been effective in gaining informa-tion on the growth of short fatigue cracks inhigher strength materials. Furthermore, since itis a remote monitoring technique it can be appliedeven in aggressive environments.

Other techniques

Several other techniques have been proposed forthe dete ction and monitoring of small flaw s, andmany, although still in the development stage, showexcellent promise for obtaining measurements inthe micrometre-millimetre range. For example,Nelson and co-workers55 have utilized a methodof monitoring surface acoustic waves to quantifythe depth and crack closure characteristics ofmicroscopic surface fatigue cracks. Using acousticmeasurements of the reflection coefficient ofRayleigh waves incide'nt on the crack, coupled withoptical measurements of surface crack length,these authors were able to claim good accuracyover a range of crack aspect ratios for depthsbetween 50 and 150 /lm (Ref. 55). Several electro-chemical methods have also been proposed for theearly detection of microscopic fatigue damage,notably by Baxter.56 Accuracies of ±10 /lm havebeen reported for the detection of small fatiguecracks in both steels and aluminium alloys throughthe identification of locations where such cracksrupture the surface oxide film. Such rupture sitesare imaged using photoelectron microscopy and,more recently, by measuring the re-anodizationcurrent or\vith the aid of a gel containing iodine. 56



~ 10IEz~~ 8SwC>z<[a: 6>-t:enzw•....4~enenw0::~ 2...J<[

t:z 0104

•

RENE 95Wrought AlloyEnvironment: moist air (30% RH)Frequency: 25 Hz, R=O.I

Initial crock sizes: 0.01-0.20 mm

••

•

flKth=3MNm3/2-.....---.

f06 107

CYCLES TO FAILURE, N

7 Variation of initial stress intensity AKiwith number N of cycles to failure for physically short cracksin wrought Rene 95 (<10 = 1400 MNm-2), showing definition of threshold Mth for 0.01-4>. 2 mm cracks,fatigue tested in moist air at 25 Hz and R = O. 1 (Ref. 60)

0.8

0.7- AI 2219 T851R = -I, CT max ~ 0.9 CTo

0.6 5k cycles, 0.3 Hz

0.5 45% RHE i=73.4I-'-m

~ 0.4ro

0.3

0.2

0.1

0 -0.2 0 0.2 0.4

ala-max

(taken to be the value of a/ amax at the inflectionpoint in the stress-displacement curve), andS = 1:::lJ/ ~a/amax) measured for a > acb where S isa measure of the compliance of the microcrack inthe completely opened state.

Compliance techniques were also used byTanaka and Nakai61,64 for measuring crackclosure levels to examine the growth of shortcracks emanating from notched centre-crackedspecimens of a structural low carbon steel. Here,the. hystere sis loop of load v. CTOD at the centreof the crack was recorded intermittently duringthe test, and the point of crack opening was deter-mined as the inflection point of compliance,

8 Typical microcrack compliance curve for2219 T851 aluminium alloy; 0 is openingacross microcrack, and <1/ CTmaxfraction ofmaximum load applied during fatigue41

evidence of crack growth) in 108 cycles (seeFig. 7).

Measurement of closure of short cracks

The techniques available for routine monitoring ofthe premature closure of long fatigue cracksgenerally cannot be used for short cracks (whosegrowth involves considerable geometrical changes,and whose measurement suffers from irreproduci-bility and scatter) which demand far superiorresolution in measurements. Closure in longfatigue cracks has been measured using opticaltechniques, compliance techniques, methodsinvolving electrical resistance or potential,acoustic emission, and ultrasonics (see e.g. Refs.33 and 34). Most of these techniques do not seemsuitable for use with. short cracks because of theirinsufficient re solution, insensitivity to short crackshape and geometry, and irreproducibility.

Early studies of crack closure for shortcracks were carried out by Morris and Buck,41who used a compliance technique to determine theclosure load for surface microcracks of up to onegrain size produced in triangular specimens ofaluminium alloys subjected to fully reversedloading. Scanning electron microscopy was thenu sed to measure the compliance of surfacemicrocracks. A 'home-made' fixture incorporatedin the electron microscope was utilized to applyknown loads to the samples in situ. High resolu-tion micrographs of the crack were obtained atdifferent load (stress) levels, from which the crackopening 0 was estimated. Figure 8 shows a typicalcompliance curve for a microcrack obtained for2219 T851 aluminium alloy. Morris and Buckdefined the following parameters to characterizecrack closure: 0(0), the crack, opening displacementat zero load, acl/ amax, the closure stress ratio



I,,,/,<""Short surface

/' crocks 0.006 mmto" v 0.5 mm deep

~~J'

J'

,."J' //

/IIII

Long th rough- section

crocks >0.254mm deep

-610

-310

9 Appearance of typical surface microcrackinitiated at ~-phase (Cu2FeAI7) intermetallicin 2219 T851 aluminium alloy fatigue testedin humid air45

10 Difference in propagation rates da/dN offatigue cracks as function of· stre ss intensityfactor range AK for precipitation hardenedaluminium alloys68

magnified by a circuit subtracting the elasticcompliance from the loading curve.

Recently" procedures have been developed tomonitor the behaviour of short fatigue cracks,using stereo-imaging techniques and in situanalyses in the SEM.65,66 Such methods enablethe crack opening (mode I) and sliding- (mode II)displacements to be measured directly as functionsof load for both short and long cracks subjectedto cyclic loads within the electron microscope.

106K,MNm3/2

100

B0

a ()

104 mm mm 0--0 6 2.0 &0() 3 2.0 ~-• 1.5 2.0 0D. 6 0.3 ().0 6 0.5 ()

() .•-5 f)() •

~ 10 ()fJ ••u~ 6'. •u

.••...•.•. dJE fJ ••E

ocP ••... <>8 ••Z"0 () ..""-0 •"0

-6 ()

10 () ..fJ •

()..•~ ()~ 0 •~ 0 ()

~ 0~ 0~ 0

-7 ~10 0

4 10 20 406K,MNm-3/2

11 Difference in propagation rates da/dN of shortand long fatigue cracks as function of stre ssintensity factor range AK for O. 035%C mildsteel of yield strength 00 == 242 MNm-2

Microstructural effects

The first definition of a short fatigue crack (seethe introductory section) refers to cracks whichare of a size comparable to the scale of charac-teristic microstructural features. Figure 9 showsa micrograph of such a typical surface microcrackinitiated at a {3-phase (Cu2FeAI7) intermetallic in2219 T 851 aluminium alloy, taken from work byMorris.45 A number of recent experimentalstudies7-12,37-49,67-76 on the initiation and.growth of cracks in a wide range of materialshave revealed that such short cracks, initiatednear regions of surface roughening caused by theto and fro motion of dislocations or at inclusionsand grain boundaries, propagate at rates which aredifferent from those of equivalent long-cracks whencharacterized in terms of conventional fracturemechanics concepts. For example, it was firstshown by Pearson68 that in precipitation hardenedaluminium alloys, cracks of a size comparableto the average grain diameter grew several timesas quickly as long cracks at nominally identicalalternating stress intensities (Fig. 10). Otherstudies of mild steels69 (Fig. 11), silicon iron7o(Fig. 12) and peak aged 7075 aluminium alloy71(Fig.13), for example, clearly reveal this lack ofcorre spondence between data for long and formicroscopically short cracks. However, theresults obtained by Lankford71, 72 and

RESULTS ON GROWTHOF SHORT FATIGUECRACKS



40

Long Crocks(LEFM)

CRACK LENGTH a

7075 T6 AluminiumAlloygrain sizes dg2 >dg1

-310

C9UAu

E -5E 10z32o'D

Threshold 11K 0

5 10t1K,MNm-3/2

14 Effect of grain size d on growth of micro-structurally short an3long cracks in 7075 T6aluminium alloy <70= 515 MNm-2; micro-cracks grow below long crack thresholdAKo and show growth rate minima approxi-mately where crack length a ro.J dg (Ref. 71)

others39,49,69,70,73,74 have shown that there isgenerally a consistent trend to this variationbetween growth rates for long and short cracks.This trend is depicted clearly in Fig.14 for 7075 T6aluminium alloy, in which growth rate s for long andfor microscopically short (i.e. of a size comparablewith the grain diameter) cracks are plotted as afunction of the linear elastic stre ss intensityrange.71 It is apparent from this figure that thegrowth rates associated with the short cracks areup to two orders of magnitude faster than those ofthe long cracks, and further that such acceleratedshort crack advance occurs at stress intensityranges well below the long crack fatigue thre sholdstress intensity range (MO).71 The initially highergrowth rates of the short cracks can be seen to

'decelerate progressively (and even arrest incertain cases) before merging with the long crackdata at stress intensities close to M0' similar toobservations reported elsewhere by Morris andco-workers,42-46 Kung and Fine, 39 Tanaka andco-workers, 70,75,76 Taylor and Knott,73 andothers. The progressively decreasing growth ratesof the short cracks below the long crack thresholdis intriguing since in terms of conventionalanalyses one would imagine the nominal drivingforce for crack advance (e .g. M) to increase withincreasing crack length (e .g. as given by equations(7) and (8», thereby leading to progressivelyincreasing growth rates. However, the behaviourof microscopically short cracks has beenrationalized in terms of a deceleration of growthas a result of crack closure and through inter-actions with microstructural features - parti-cularly grain boundaries.12,42-4 7,69-76 Forexample, specific observations of grain boundariesimpeding the growth of short cracks have beenreported for aluminium alloys, 42-46,71,72nickel-base superalloys,50 tempered martensitichigh strength steels, 77,78 and lower strength mildsteels. 75,76 Using arfSUments based on micro-plasticity and crack closure effe cts, Morris and

108

20

6

10

Threshold

I

160

I4

I I

40 80·

2a,l1m

2

2 5~K,MNm-3/2

S 20C materia IThrough cracko da/dN vs ~K

R=-ISurface cracka(mm)

A 0.0500.0600.1\

Corner crack00.35~ 1.48

I

20

Arrest

7075 T6 Aluminium Alloy

-310

-6100.5

12 Difference in propagation rates da/dN of shortand long fatigue crack s as function of stre ssintensity factor range AK for 3/oSi iron ofyield strength (70= 431 MNm-2 (Ref. 70)

13 Difference in propagation rate s do / dN of shortand long fatigue cracks as function of stressintensity factor range AI{ for peak agedAI-Zn-Mg alloy (7075 T6) of yield strength<70= 515 lV1Nm-2(Ref. 71)



co-workers44-49 have modelled the process interms of two factors: the cessation of propagationinto a neighbouring grain until a sizeable plasticzone is established, and a retardation in growthrates caused by an elevated crack closure stress.Tanaka and co-workers76, 79, SO similarly con-sidered the impeded growth of microstructurallyshort cracks in terms of the pinning of slip band s,emanating from the crack tip, by the next grainboundary. The results obtained by Lankford71 andshown in Fig. 14 indicate that the minimum crack

growth rate appears to correspond to a cracklength roughly equal to the smallest grain dimen-sion (Le. a '" dg). Furthermore, the depth of thedeceleration 'well' appears· to be determined bythe degree of microplasticity involved in the cracktraversing the boundary. For example, when theorientation between the grain containing the crackand the neighbouring grain was similar, there waslittle deceleration in growth rates at the boundary.Thus a consensus from these studies is that,despite their propagating faster than long cracks,

(0)

0.5 RMaterial 0"0 (MNm2)-1.0

bcu 0 520 C 366e 520 C 194 2

<l~I" 6 Mild steel 289

J:: CD G40 IIb+-' 0 5M41 251 A<l o • SM41 2510.2 • SM-50 373~

• HT 80 726V • 13Cr cost steel 769W Copper

0.1 £ Aluminium 30.4

0.01 0.1 10 100

(b)

a threshold stress AOth,normalized with respect to smooth bar fatigue limit AOe;b threshold stressintensity range Mth, normalized with respect to long crack threshold AI{0

15 Variation of threshold stress and stress intensity with crack length Q, normalized with respect tointrinsic crack length ao = (l/1T)(M 0/A(1e)2(Ref. 76)



short cracks are apparently impeded by thepresence of grain boundaries, which in generalwould be unlikely to affect significantly the localpropagation rates of long cracks. The extent ofreductions in the effective driving force for variousdegrees of such short crack deflections at grainboundaries has recently been calculated bySuresh.81 It is the present authors' opinion thatthe interaction of short cracks with microstructuralfeatures, which leads to the apparent progressivedeceleration of short cracks below the value ofl:J(0 for long cracks, results principally from crackdeflection 81 and associated crack closure 12,61mechanisms. The specific evidence for this isexamined in the 'Discussion' section below.

From such experimental studies, it is readilyapparent that threshold stresses (or stressintensities) associated with long and short cracksare likely to be very different. Conventionalfracture mechanics arguments imply that thethreshold stress intensity range (AKth) for aparticular material should be independent of cracklength (Le . .M{th = Mo,the long crack threshold,is constant). Kitagawa and Takahashi,82 however,first showed that below a critical crack size thethreshold l:J(th for short cracks actually decreasedwith decreasing crac~ length, where the thre sholdstress D.Uthapproached that of the smooth barfatigue limit D.Ue at very short crack lengths (seee.g. Fig. 15). Typical experimental data showingthis decay in the threshold stress intensity rangeAKth at short crack lengths are shown in Fig. 16,taken from the results obtained by Romaniv et al.69

for annealed mild steel, O.45%C austenitic steel,and an AI-Zn-Mg alloy. Several workers have

20

claimed that the critical crack size (below whichM<th is no longer constant with crack size)depends on microstructural and mechanicalfactors.8-1o,12,47,64,69-76,80-83 It has even beensuggested, for example, .that this critical cracklength above which LEFM is applicable is simplyten times the characteristic microstructuraldimension.83 However, from continuum mechanicsarguments, an estimate of this crack size can beobtained in terms of (lj1T)(LlliQ/ D.Ue)2, with bothl:J(0 and D.Ue corrected for a common load ratio.Typical values of this dimension, which effectivelyrepresent the limiting crack size for valid LEFManalysis (see the section 'Fracture mechanicscharacterization of fatigue crack growth' above),vary from 1-10 Mm in ultrahigh strength materials(Le. Uo '" 2000 MNm-2) to 0.1-1 mm in lowstrength materials (Le. Uo '" 200 MNm-2).12

From such experimental results it is generallyconcluded that the threshold condition for nogrowth for long cracks is one of a constant stre ssintensity, Le. DKo, whereas the threshold conditionfor short cracks is one of a constant stress, i.e.the fatigue limit .6Ue or the endurance strength.Such a premise has been shown to be consistentwith the models derived by Tanaka et at. 76 in whichthe threshold for short crack propagation isgoverned by whether the crack tip slip bands areblocked, or can traverse the grain boundary to anadjacent grain. This condition governing whetherthe slip band is blocked, shown in Fig. 1 7 in termsof a critical value of the microscopic stre ssintensity K~ at the tip of the band, yields expres-sions for the fatigue threshold stress Uth andstress intensity Kth:

16 Variation of threshold stress intensity rangeM th with short crack length a in 040. 11aust~itic 0.45CJoC steel, 00 = 550MNm-2,0.035CJoCmild steel, °0 = 242 MNm-2, andAl-Zn-Mg alloy, 00 = 180 MNm-2 (Ref. 69)

66 G40.11 steel6

10N

M"IEz Mild steel 02: 0.c 0-:Z

<J0

5 0

(19)

• (18)

(IX(b)

Kf? 2 *Uth = (1Tb)1/2 + -;; ufr cos-1 (a/ b)

Kth = Uth (1Ta)1/2

= K~(a/b )1/2

+2(al1T)1/2afr cos-1 (alb)

Schematic illustration of a crack tip slip bandblocked by grain boundary and b coplanar slipband emanating from tip of isolated crack76

Crack tip

Grai n boundary

Slip band

Tfr

(0)

Here, for a slip band coplanar to the crack planein the two dimensional model shown in Fig. 17, bis the crack length a plus the width Wo of theblocked slip band zone, and arr is the frictional

17

52o,mm

0.50.22

AI-Mg-Mn 0110o



103 10-1

300MFatigue limits Ultrahigh strength

?~(Je at R::O steel {\J

~ 102 103 IEZ

.c 2:tj .c

<J tj<J

Tempering UTS lif10

Temp.tOe MNm-2

a 100 2238b 300 1737c 470 1683

I d 650 1186 10

10-6 10-5 10-4 10-3 10-2 10-1

Q, in

18 Predicted variation of threshold stress ~Oth at R == 0 with crack length a based on data for 300Multrahigh strength steel (silicon modified AlBl 4340), oil quenched and tempered between 100 and650°C to vary tensile strength (1186 to 2338 NINm-2) (Ref. 6)

stress for dislocation motion in the band. Thelong crack threshold stress intensity Ko thusfollows from equation (19) by taking the limit ofKthwhen wo~a, i.e.

Ko = K~ + 2(2/1T)l!2ofr wOl/2 • (20)whereas the short crack threshold stre ss, i.e. thefatigue limit 0e, follows from equation (18) ata= O,Le.

0e = otr + K~/(1TwO)l/2 • (21)

Since, at the fatigue limit, the slip band can beconsidered to be constrained within a single grain,lDO can be assumed to be approximately half thegrain size dg• The interesting aspect of the blockedslip band model is the prediction (from equations(20) and (21)) that the long crack threshold Koincreases with grain size, whereas the short crackthreshold, or fatigue limit 0e' decreases.76 Suchpredictions are borne out by experiment andhighlight the fact that metallurgical factors whichmay improve resistance to the growth of longcracks (Le. raise Mo) may simultaneously lowerresistance to the initiation and growth of small

cracks (i.e. lower ~oe). 77 This effect is particularlypronounced in ferrous alloys: as shown in Fig.18,increasing the tempering temperature to decreasestrength level in an ultrahigh strength, siliconmodified 4340 steel (300M) lowers the long crackthreshold Mo' yet raises the fatigue limit ~oe(Ref. 6). A second example, shown in Fig. 19, isfrom the work of Usami and Shida,59 who comparedthe threshold behaviour as a function of surfaceroughness (to simulate crack size) of a cast iron(00 = 113 :MNm-2) and a maraging steel (00 = 1906MNm-2). It is quite clear from this work59 thatthe maraging steel, while of substantially higherstrength and having a far superior fatigue re sist-ance at short crack sizes (below 0.1-1 mm), isactually inferior to cast iron from the viewpointof long crack threshold behaviour.

Local plasticity effects

The second definition of a short crack (see theintroductory section) refers to cracks which areof a size comparable to the scale of localplasticity, such as the crack tip plastic zone

o Maraging steel 2.6(eTo ::1906 MNm-2)

• Fe 20 cost iron 5.9(eTo =113MNm-2)

-Calculated valueThreshold forcentre-crocked

plate

10

.cfj<l

10-4 10-3 10-2 10-1 10 102

Surface Roughness, mm19 Comparison of threshold behaviour of cast iron, 0'0 == 113 MNm-2, and maraging steel, (10 =

1906 MNm-2, as function of surface roughness (to simulate crack size)59



generated by the crack itself (near-tip plasticity),or the strain field of a notch or a larger stressconcentration which may encompass the crack inthe vicinity of the notch (notch field plasticity).Each of these two local plasticity effects isexamined below.

given in terms of a crack length a, elastic modulusE, and representative strain range D.Eas

ME = D.E(7Ta)l/2= (q D.Ee+ ~Ep)(7Ta)l/2 (22)

DKeq= QE AE(rra)l/2 (23)

where D-.Eis the sum of the plastic strain rangeD.Epplus part of the elastic strain q AEe,whereO.5<q<1, and Q is the compliance function basedon the equivalent linear elastic Kr solution for theloading geometry in question. Recently, Starkey andSkeUon88 have shown that the D.J and DKeqapproaches are essentially the same up to highplastic strains, and good correlations have beenfound between data for long and short cracks atboth room and elevated temperatures by expressingthe crack growth data in terms of (EM)l/2 orDKeq·

Although analyses of the behaviour of shortcracks in terms of elastic-plastic constitutivelaws often seem necessary, even with the moreappropriate characterization afforded by suchfracture mechanics, it is still often apparent thatshort cracks propagate at somewhat faster rates(Fig. 20). In order to account for this further'breakdown' in continuum mechanics characteriza-tion, EI Haddad and co-workers9,84,89,90 haveproposed an empirical approach based on thenotion of an intrinsic crack length parameter, ao•These authors redefined the stress intensity factorin terms of the physical crack length plus ao, suchthat the stress intensity range which characterizesthe growth of fatigue cracks, independent of cracklength, is given by the equation

M = QAOoo [1T(a + 00)]112 (24)

where Q is the usual geometry factor. 84 The'material-dependent constant' ao was estimatedfrom the limiting conditions of crack length wherethe nominal stress D-.Ooo approaches the fatiguelimit D-.0e when a ~ 0 and where M = Mo, Le.

ao ~ ~ (Mo) 2 (25)1T AOe

The value of the intrinsic crack size ao can beseen to be equivalent to the critical crack sizeabove which Mth becomes constant at the longcrack threshold stress intensity Mo in Figs. 15and 18. In other words, ao is an indication of thesmalle st crack size that can be characterized atthe threshold in terms of LEFM. This intrinsiccrack size approach has been claimed to be aspecial case of Tanaka and co-workers' blockedslip band model76 discussed above, where thefriction stress 0f*rin equation (20) is taken to bezero. Although somewhat physically unrealistic,setting afr = O,Ko = K~, and ao = /'vo in equations(18) and (19) gives76

Ko0th = ---- (26)

[rr(a + Go) ]1/2

K al/2

Kth == (a ~ ao)V2 (27)

which are identical to the expressions derived byEI Haddad et al. 84 In equations (26) and (27),°0 isessentially a fitting parameter that reproducesthe variation of D.0thwith crack length shown in

Q)

163~u

'"EE

Z~10-4 ~

~APPROX 6E

• 0 0.04• 0 0.02• ~0.012.0 0.009+¢ 0.005 10-5

OPEN SYMBOLS as 0.007 in.(0.1778mm)

SCATTERBANO FROM'LARGE SPECIMENS

A 533B STEEL

-310

10-4

-6to

Near-tip plasticity effectsWhile elastic-plastic fracture mechanics analysesseem more suited to the characterization of shortcracks comparable -in size to the extent of self-induced near-tip plasticity, a comparison of theirbehaviour with equivalent long cracks using LEFManalyses (Le. at the same nominal value of M)shows that the short cracks grow much morequickly.8-10,12,47,84 Part of the reason for suchapparently anomalous re suUs lie s not in anyphysical difference ben.veen the behaviour of longand short cracks, but with the inappropriate useof LEFM analyses. This was shown particularlyclearly by Dowling,51 who monitored the growth ofsmall surface cracks in smooth bar specimens ofA533B nuclear pressure vessel steel subjected tofully reversed strain cycling. By analysing thegrowth rates dajdN in terms of D.J, using valuesof J computed from the stress-strain hysteresisloops (described by equation (17) and shown inFig. 5), there was found a closer correspondencebetween the behaviour of long and short cracks(Fig.20). Analogous approaches to the short crackproblem have been suggested in terms of ME, thepseudo-elastic-plastic strain intensity range, 85-8 7or DKeq,the equivalent stress intensity range, 87

10 102b.J, in Ib in-2

20 Variation of fatigue crack growth rates dajdNfor long (a~25 mm) and short (0$0.18 mm)cracks in A533B steel, <10 = 480 MNm-2,

under plastic loading; data analysed in termsof M (Ref. 51)

b.J, mMNm-2

0.0012 0.001 .01


462 Sureslz and Ritchie: Propagation of short fatigue cracks

21 Schematic illustration of crack tip and notchtip plastic strain fields associated with growthof short crack of length l emanating fromnotch of depth c and root radius p

Values of kf approach the theoretical values of ktfor larger notches and in higher strength materials,the degree of agreement being measured in termsof the so- called notch sensitivity index, defined as(kf -l)/(kt -1). Although values. of kt aretabulated in handbooks (see e.g. Ref. 96), thedetermination of Rf generally involves experimentalmeasurements or empirical predictions, such asthe Peterson equation for iron-base \vroughtalloys,97

where p is the notch root radius and a is anempirical constant that depends on the strengthand ductility of the material. Typical values for arange from 0.01 for annealed steels to 0.001 forhighly hardened steels.2 Since such empiricalequations are available only for steels, to avoidalways measuring kf experimentally the so-called'local strain approach', essentially a modificationof the Neuber rule for cyclic loading, has recentlybeen developed (see e.g. Refs. 2, 98, 99). The valuesof kf suggested by Morrow and co-workers98 aredetermined from the relations

(31)

(33)

(34)

(32)

t

XBL 828-10819

(plastic)

(elastic)

f

crock tip plastic zone

plastic zone of notch

notch field boundary

t

unnotched bar endurance limit ~ knotched bar endurance limit -...;:t

t

kf = (k (Jk E)l/2

kf = (~a~EE)l/2/~aoo

kf ~ 1 + (kt - 1)/[1 + (a/p)]

For elastic conditions, equation (29) reduces to

kt = (aEE)l!2/aoo (30)

where a and E are the local stress and strain atthe notch surface, respectively, aoo is the nominalapplied stress, and E is the elastic modulus.

Although elastic stress concentration factorsare sometimes used in fatigue for conservativedesign in the presence of notches, kt is generallyreplaced by kf, the fatigue strength reductionfactor; kf can be considered as the effective stressconcentration under fatigue loading conditionsand is defined for finite life as

Notch field plasticity effectsLocal plasticity also has an important influence onthe initiation and growth of cracks emanating fromnotches. Such cracks are defined as being shortbecause their size is comparable to the extent ofthe strain field of the notch tip plastic zone (seeFig.21).

Stress analyses and failure predictions fornotched components have traditionally involved theuse of theoretical elastic stress concentrationfactors kt or, where plasticity is considered,procedure s such a s those developed by Neube r .95For example, the well-known Neuber rule suggeststhat the elastic stress concentration factor kt,under conditions of plastic deformation, is givenapproximately by the geometric mean of the stressand strain concentration factors, k a and k E'

respectively: 95

kt = (k ak E)V2 (29)

Fig.18. Furthermore, by recomputing both t:J andM( to include 00' El Haddad et al.91 have reanalysedthe short crack data obtained by Dowling51 (seeFig. 20) and claimed a closer correspondencebetween results for long and short cracks. Thisintrinsic or 'fictitious' crack size approach hasalso been used to characterize a short crackemanating from a notch, 64,91 as discussed in thenext section.

Although such an approach is successful inexplaining differences in the growth rate kineticsof long and short cracks obtained from conventionalLE FM analyse s, it is totally empirical, as thephysical· significance of the parameter ao is notunderstood; neither is there any convincingcorrelation between 00 and any characteristicmicrostructural dimension.

A somewhat different approach to rationalizingthe behaviour of long and short cracks, specificallywith respect to the threshold condition, was adoptedby Usami and co-workers.59,92-94 To replace thenotion that the threshold condition is one ofconstant stress for short cracks, but one of aconstant stress intensity range for long cracks,these authors proposed a single criterion that thecyclic plastic zone dimension (r~) at the fatiguelimit is a material constant.59,92-94 They usedthe Dugdale solution to approxima te the size ofthis plastic zone at the fatigue limit (r 5') forR ~ 0, in terms of the yield stress ao and cracksize 0, Le.

r ~F = a [se c (1T~a th/ 4a 0) - 1] (28)

which was shown to reproduce the form of thecurve of ~ath versus crack size shown in Figs. 15,18, and 19. Similar to the model based on anintrinsic crack size, this model can again beconsidered a special case of the blocked slip bandmodel (equations (18)-(21)) by setting Kf? = 0(Ref. 76). By developing similar expre ssions fornegative R ratios, Usami and Shida59 have alsoclaimed to have explained the effects of stressratio and yield strength on the behaviour of shortcracks. However, experimental confirmation of theconstancy of r D.-F at the threshold for long andshort cracks and the validity of the Dugdale solu-tions for the size of the plastic zone in thisinstance has yet to be obtained.



Complete failure

••

• propagation data

o initiation data

Stress concentration factor kt-----.

••

•

no cracks formed

C/)C/)

~(j)

notches, and is in disagreement with the predictionsmade by Dowling21 for this regime (equation (9».

An interesting aspect of crack growth fromnotches is that, after growing a short distance,cracks can arrest completely and become so-callednon -propagating cracks (NPCs), as first demon-strated by Phillips,106 Frost,107 and Frost andDugdale.108 Since then, a number of otherstudies64,89,91,100,103,109-111 have confirmedthat NPCs exist, although the mechanism(s) bywhich a propagating crack becomes an NPC is notunderstood. Stress-strain/life analyses, however,have revealed that NPCs form only at sharpnotche s, above the critical stress concentrationfactor kt (see e.g. Ref s. 100, 109, 111). This isillustrated in Fig. 23, where the long life fatiguestrengths (Le. fatigue limits) are plotted as func-tions of kt. It is apparent from this figure that thethreshold stress for crack initiation, Le. theunnotched fatigue limit AUe divided by kf or kt, isless than the stre ss that would cause completefailure above the critical kt for NPCs.

Fracture mechanics analyses of the plasticityof the notch field suggestthat the condition fornon-propagation involves either the long crackthreshold stress intensity AKo applied to the shortcrack,94 or a threshold strain intensity factorincorporating an intrinsic crack length (ao) term.89As discussed below, the above models fail toexplain physically why NPCs occur since the merepresence of a notch plastic zone does not obviouslygive rise to a driving for ce for crack growth whichpasses through a minimum.

Whether short cracks emanating from notchesarrest or not, their growth, as compared to resultsobtained from conventional LEFM analyses of longcracks, is generally non-unique and significantlyfaster when characterized in terms of t::J( orK max. An example of this behaviour is shown inFig. 24, from the work of Leis and Forte, 110 wheregrowth rates are plotted for cracks ~250 Jlm longpropagating from notches of varying kt, and com-pared to the growth rate s of long crack s in anSAE 1015 mild steel. Hammouda and Miller109have analysed such behaviour in terms of notchplasticity theory, and argue that the total plasticshear displacement, which is taken as the sum ofthe shear displacement arising from (notch) bulkplasticity and from the local crack tip plastic zonefor LEFM controlled growth, determines the growth

coc"Eoc"0.!!!a.a.«

23 Threshold stress for crack initiation, i.e.unnotched fatigue limit ~ae divided by kf orkb as function of Rt (Ref. 111)

• (35)e == 7. 691 l (c / p) 1/2

102103 104 105 106

FatiguC2crack initiation cyclC2s(Ni)

Correlation of fatigue life, based on initiationof engineering sized crack, with so-calledfatigue crack initiation threshold Mjpl/2based on results for HY130 steel, 0"0 == 1000MNm-2, double edge notched specimens of rootradius p between 0.2 and 9. 5 mm (Ref. 37)

22

and that the extent of the notch field is equal to0.13(cp)l/2. This estimate of the size of the notchfield appears to be less accurate for fairly sharp

where AUoo, AU, and AE are the ranges of the

nominal stress, local stress, and local strain,respectively. To compute kf, equation (34), whichis the equation of a rectangular hyperbola (AUAc==constant), must be solved simultaneously with thecyclic constitutive law, which similarly relatescyclic stresses to cyclic strains.

Fracture mechanics analyses, incorporatingboth analytical and numerical procedures (e.g.from equations (7) and (8» have also been utilizedby many authors to treat the problem of notchesin fatigue. In particular, Barsom and McNicol,37Smith and Miller, 100 Dowling,21,101 Kitagawa,102Lukas and Klesnil,103 EI Haddad andco-workers, 91,104,105 and Tanaka and Nakai64have all attempted to characterize the growth ofsmall cracks which are either fully or partiallysubmerged in a notch tip plastic zone. Althoughnot concerned specifically with the growth ofshort cracks, Barsom and McNico137 proposed theconcept of a fatigue crack initiation thresholdM/pV2, which is assumed to be a materialconstant for the initiation of 'engineering sized'cracks from notches of different root radii p (seeFig. 22). With the usual definition of the stressconcentration factor kb however, the concept of aninitiation threshold M/ pl/2 is really the notchedbar fatigue limit AUekt expressed in fracturemechanics terms.7 Smith and Miller,100 on theother hand, assumed that a fatigue crack of lengtha growing in an unnotched specimen is comparableto a fatigue crack of length l growing from a notchwhen both have the same instantaneous velocityunder identical conditions of bulk applied stre SSe

They further suggested that the contribution e(== a - l) to a crack of length l growing from anedge notch of depth c and root radius p can beexpressed as



Kmax, ksi in1/210

SAE1015 Steel

24 Propagation rate da/dN of cracks emanatingfrom notche s as function of maximum stre ssintensity factor Kmax in 0.15/oC mild steel;kt is theoretical elastic stress concentrationfactor, R stress ratio, and HE edge strainratio110

kinetics of such short cracks. If the crack iscompletely submerged in the notch tip plastic zone(see Fig. 21), bulk plasticity conditions dominatethe behaviour. Hammouda and Miller estimatethat the growth rates of the short cracks willdecrease progressively until they arrest or mergewith the long crack LEFM curve, where behaviouris dominated by local plasticity conditions withinthe crack tip plastic zone (see Fig. 25). Severalcontinuum mechanics explanations have beenadvanced for such observations of growth rates ofshort cracks decreasing within the notch tip plasticzone. These explanations have been based on thetotal plastic shear displacement argument (s eeabove),1 09 on ela sti c-plastic analy se s using theJ parameter and the ao concept,89 and on the ideathat the size of the reversed plastic zone is aproperty of the material and is independent ofcrack length.94 Such behaviour, however, isdifficult to comprehend, particularly since thereis a striking similarity between Figs. 25 and 14.In Fig. 14 is shown the progressive decelerationin short crack growth rate s in· the ab sence of anotch, which has been attributed to the impedanceof growth at grain boundaries.71 Thus it must beconcluded that even though the precise mechanismfor the deceleration of short crack growth and theforma tion of NPCs is unclear, factor s such as notchtip plasticity, microplasticity, grain boundaryblocking of slip bands, ce ssation of growth,

Long Crackscrack-tip plasticity(LEFM) control

tz~ou

Threshold) [).Ko~K--.

25 Elastic-plastic and linear elastic characteriza-tion of kinetics of crack growth for shortcrack propagation from notch, as shown inFig. 21 (Ref. 109)

crystallographic reorientation, deflection at grainboundaries, and crack closure may all playasignificant role. In this regard it has been claimedthat, since small cracks are capable of propagatingbelow the long crack threshold Mo value, theymay propagate for some distance until the com-bination of their size and the local stress causesthem to arrest at or below the Mo value.8Although this is a convenient explanation of thebehaviour shown in Figs.14 and 25, the physicalmechanisms that have been suggested for suchbehaviour do not follow from conventional elasticor elastic-plastic notch analyses. However, recentstudies have suggested that the most generalexplanation involves the phenomenon of fatiguecrack closure, and the variation of closure withcrack length.12,61 This is discussed in detailbelow.

Environmental and closure effects

The third definition of a short crack (see theintroductory section), perhaps the most importantfrom a design viewpoint, is the crack which is longcompared to the scale of both the microstructureand the local plasticity, yet simply physicallysmall, Le. typically less than 0.5-1 mm long.Since both continuum mechanics and LEFMcharacterizations of the behaviour of such crackswould be expected to be valid, it is perhapssurprising to find that under certaincircumstances11,48,49,54,60 even physicallyshort cracks grow faster than long cracks undera nominally identical driving force (i.e. at the sameM). This is not at all consistent with the conceptof similitude, which forms the basis of fracturemechanics analyses of subcritical crack growth,and has been attributed primarily to twofactors.32

The first of these is connected with thephenomenon of· crack closure: interference andphysical contact between mating fracture surface sin the wake of the· crack tip can, under positiveloads during the fatigue cycle, lead to effective

Q)

~~.£

10-4 Z~ou

10010Kmax ,MNm-3/2

Stress and edQe straincontrol Re:' R= -1

R£ = - 1 --kt R =-1

o 2.5o 4.4 06 6.2 v

Note: Solid symbols denotecracks of length lessthan 250jlm

10-61

Q)

u~ 10-2

EEzu"0 10-3U



~Keff = Kmox-Kmin

(0) (b)

~ Keff = Kmox -Kcl

(c) (d)

a no closure, and closure induced by: b cyclic plasticity; c corrosion deposits; d rough fracturemorphology

26 Mechanisms of fatigue crack closure; Meff is effective stress intensity range, defined by Kmax - Kcl,where Kcl is stress intensity at which two fracture surfaces come into contac~ (Kcl ~ Kmin)(Ref. 119)

500

• •••• •

•

•

•

• •••

o

0.3

E:1. 0.2

o2 0.1U

100 200 300 400CRACK LENGTH (2o),J.Lm

27 Variation of CTODo at zero load with lengthof small surface cracks in 6Al-2Sn-4Zn~Motitanium alloy (0-0 == 1140 :MNm-2, primarya grain size ~ 4 J1m, fJ grain size ~ 12 Mm),showing reduction in crack closure wi. thdecreasing crack size48

Extensive studies of the behaviour of longcracks, particularly at near-threshold stressintensity levels, have revealed that such closuremechanisms determine to a large extent theeffects on crack growth of load ratio,114-116,123yield strength,116,124,125 grain size,116,120environment, 114-116 ,121 ,122, 125 and variableamplitude cycling, 126-128 and even in the veryexistence of a thre shold for no long crackgrowth.116,121,125,129 However, such microscopicclosure mechanisms are also particularly relevantto the behaviour of short cracks, simply becausetheir action predominate s in the wake of the cracktip. Since small cracks, by definition, have only alimited wake, it is to be expected that the effe ctof crack clo sure will be different for long andshort cracks - that the short crack is likely to beless influenced by closure. Evidence that theextent of crack closure is a function of crack sizehas been reported by James and Morris48 for thegrowth of short cracks in titanium alloys. Bymonitoring the surface CTOD at zero load forcracks from 50 to 500 Jlm long (see Fig. 27), theseauthors concluded that for cracks less thanapproximately 160 }.lm long the extent of crack

closure of the crack. Since the crack cannotpropagate while it remains closed, the net effectof closure is to reduce the nominal stress intensityrange M, computed as Kmax - Kmin from measure-ments of applied load and crack length, to somelower effective value Meff actually experienced atthe crack tip, Le. Meff == Kmax - Kcl, where Kclis the stress intensity at closure (~Kmin).112Several factors can lead to closure, such as theconstraint of surrounding elastic material on theresidual stretch in material elements previouslyplastically strained at the tip (plasticity inducedclosure),112 the presence of corrosion debriswithin the crack (oxide indu ced closure), 113 -116and contact at discrete points between faceted orrough fracture surfaces where significant inelasticmode II crack tip displacements are present(roughness induced closure).116-120 Thesemechanisms of crack closure are illustratedschematically in Fig. 26.

Plasticity induced closure, as first defined byElberl12 from compliance measurements onfatigue cracks in aluminium alloys at high stressintensity ranges, is generally considered to playadominant role under plane stress and is thuspresumed to be of less importance at near-thresholdlevels where mostly plane strain conditions exist.In the latter case, the main contributions to theclosure of fatigue cracks come from oxide depositswithin the crack and from premature contactbetween the fracture surface asperities. Considera-tion of simple geometric models have led Suresh,Ritchie, and co-workersl19-122 to propose thefollowing relationships for such closure:

(Kcl/Kmax)oxide ~(d5 }3/7TSOmaxEo)lJ2 (36)

(Kcl/Kmax)roughness ~ (2yx/(1 + 2yx)]l/2 (37)

where do is the maximum thickness of the excessoxide deposit located a distance 2s from the cracktip, 0max is the maximum CTOD (mode I), x is theratio of mode II to mode I crack tip displacements,EO is the yield strain (ao/E), y is a non-dimensionalroughness factor given by the ratio of the heightof a fracture surface asperity to its width, and {3is a constant of numerical value "'1/32.

International Metals Reviews, 1984, Vol. 29, No. 6


c•..

Z1J

"-

Q)

10-5 ~u

ou10.6

60

20

(b)

LONG ANDSHORT CRACKS

4130 STEEL

R=Q

6K ksi in1/2,20 40

Slashed marks:Region I data

10

10

-610

EEZ -5~IOou

-4Q)IOU~u

"-

10.720 40 60

6K MNm-3/2,29 Fatigue crack propagation rate da/dN as

function of AK for long (a '" 50 mm) andphysically short (a = 0.1--0.8 mm) cracks inAlSI 4130 steel, 00 == 1300 MNm-2, tested inmoist air and in aqueous 3'10NaCI solution 53

similar (see Fig. 29). This phenomenon is not com-pletely understood, but preliminary analyses haveindicated that the effect could be attributable todifference s in the local crack tip environment oflong and short cracks, re sulting mainly from thedifferent electrochemically active surface-to-volume ratios of the cracks and from the influenceof crack length on the solution renewal rate in thecrack tip region.11,54,57

R=Q

3

Notched specimenAaooCMNm-2)

60 0 160.76 0

Cracked specimen-daldN v At<Cl>

U~ 10;)"-EE

10-4Notched soecimen .,&,ooCMNm-2) 1',',60 0 160. ,

10-5 76 0 <SiJCracked specimen ~---daldN v ~eff J1$-daldN v At< ~

0'010-6 ~6

~l~0/

~,0dO,:0,,,(0) ,

10-95 10 20 3 5 10

6K, MNm-3/2 6Keff, MN m-3/2

a da/dN v. nominal stress intensity range AK; b da/dN v. effective stress intensity range Meff, notethat anomalous (sub-threshold) behaviour of short cracks is brought into direct correspondence withconventionally obtained data for long cracks once crack closure is accounted for

28 Variation of crack propagation rate with stress intensity ranges in JIM SM416, O.17%C structuralmild steel, 00 == 194 MNm-2 (Ref. 61)

closure, particularly that induced by roughness,decreased with decreasing crack length. A furtherinfluence of roughness-induced closure was foundby McCarver and Ritchie,6 0 who studied thecry stallographic growth of long and physicallyshort fatigue cracks in Rene 95 nickel-basesuperalloy. Threshold D.Kthvalues for shortcracks (a"'O. 01-0. 20/lm) at low mean stresses(R == 0.1) were found to be 60/0 smaller than forlong cracks (a"'25 mm), yet at high mean stresses(R == O.7) where closure effects were minimal,this difference was not apparent. More recently,Tanaka and Nakai61 have used compliancetechniques to measure the extent of crack closurefor small cracks emanating from notches in a lowcarbon steel and found a marked reduction in'clo sure at short crack lengths. The se author sclaimed that the anomalous (sub-threshold)behaviour of short cracks (as shown in Figs. 14and 25) could be brought into direct correspondencewith conventional long crack data (see the nextsection) by analysing it in terms of Meff (seeFig.28). Thus, at equivalent nominal M( levels,physically short cracks may be expected to pro-pagate faster (and show lower threSholds) thancorresponding long cracks simply because closureproduces a smaller increase in effective stressintensity range at the crack tip in long cracks.Specific mechanisms for this effect are discussedin more detail in the next section.

Chemical and electrochemical effects canalso increase the growth rate of small cracks,and this is particularly relevant to the growth ofphysically short cracks in aggressiveenvironments.11,53,54,57,130 Experiments byGangloff11,54 on high strength AISI 4130 steelstested in NaCI solution revealed the corrosionfatigue crack propagation rates of short cracks(0.1-0.8 mm) to be up to two orders of magnitudefaster than corresponding rates for long cracks(25-60 mm) at the same Mlevel, although thebehaviour in inert atmospheres was essentially



30 Changes in direction of crack advance whencrack tip encounters grain boundaries in7075 T6 aluminium alloy71

threshold at progressively decreasing growthrates (see Fig. 25), and can even arrest to formnon -propagating cracks, 106-110 but such behaviourcan also occur in the absence of a notch and canbe attributed to microstructural factors (see Fig.14).71 Certainly no linear elastic analysis of ashort crack emanating from a notch (see e.g. Refs.21,100) has shown that the crack driving force(e.g. Kr) passes through a minimum as"the crackbegins to extend from the notch, and to the presentauthors' knowledge there is no formal elastic-plastic analysis available which predicts a similarvariation in cr~ck driving force without having toincorporate crack closure effects.12,64,135Although there is no complete continuumme chanic s analy sis, it can be concluded that theanomalous growth rate of short cracks emanatingfrom notches may result in part from the inter-action of the short crack with microstructuralfeatures, and principally from crack closure.

With respect to microstructural features it isgenerally accepted that the pre sence of micrdscopicdiscontinuities, such as grain boundaries, hardsecond phases, or inclusions, plays a somewhatminimal role in influencing the growth of longfatigue crac~s6 (at least for growth rates belowrw10-3 mm/ cycle) because the behaviour isgoverned primarily by average bulk properties.10However, it is clear that this is not the case forsmall cracks whose length is comparable to thesize of microstructural features. For example,for small cracks contained within a single graincyclic slip will be strongly influenced by the 'crystal orientation and the proximity of the grainboundary, re suiting in locally non-linear crackextension.49,71-74 There is now a large body ofevidence showing that the growth of small cracksis impeded by the presence of grain boundaries(see e.g. Fig. 30) by such mechanisms as the block-ing of slip bands76 or containment of the plasticzone49 within the grain, reorientation and re-initiation of the crack as it traverses theboundary, 49,71 and simple cessation of growth atthe boundary. 49 Crack propagation has also beenfound to be halted by harder se cond phase s: induplex ferritic-martensitic steels, small cracks

DISCUSSION

From the above review of expe rimental re sults itis apparent that short fatigue cracks may pres~ntdifficulties in fatigue design simply because theirgrowth behaviour cannot be predicted withcertainty by using the analyses and methodologiesdeveloped for long cracks, for example by makinguse of LEFM results. It is also apparent that theuse of these procedures can lead to overestimatesof the lifetimes of components containing shortcracks because, under the same nominal drivingforce, a short crack invariably propagates morequickly than a corresponding long crack. Thisproblem, of 'lack of similitude', arises for anumber of reasons, including:

(i) inappropriate fracture mechanics charac-terization of the crack driving force for shortcracks subj ected to near-tip and notch fieldplasticity effects

(ii) local microstructural features which do notsubstantially alter the growth of large cracksbut can interact strongly with small cracks 'because they are of comparable size

(iii) lack of similitude associated with crackextension me chanisms

(iv) crack closure effects(v) differences in the local crack tip environments.

Each of these factors is now examined in turn.

Questions on the inappropriate use of LE FM tocharacterize the extension of short cracks havebeen central to the short crack problem (see e.g.Refs. 7-10,12,131,132). In fact, it has often beenclaimed that the short crack problem arises simply:vh.en LEFM analyses become invalid,131 althoughIt IS now clear that this is an oversimplification.Conventional LEFM approache s can be inappro-priate for short cracks, even under nominallyelastic conditions, since the use of the linearelastic singularity to characterize the localstresses on the basis of K1 (Le. equation (3))invariably involves neglecting all terms of orderhigher than r-1/2 (Ref. 133 and 134). However,when a rw ry such higher order terms can have anappreciable effect and therefore should be con-sidered when comparing the behaviour of long andshort cracks.133,134 It is also well recognizedthat one of the main reasons for the breakdown inLEFM analyses for short cracks is the presenceof.exce ssive plasticity over distance s comparablewIth the crack size in the vicinity of the crack tip.This problem has been partly resolved by the useof elastic-plastic fracture mechanics in methodsbased on the J -integral or on the crack tip openingdisplacement, as evidenced by the results obtainedby Dowling51 and shown in Fig.20. It is nowapparent that differences in the behaviour of longand short cracks as revealed by early studies canbe traced to the fact that growth rates were com-pared at equivalent M values and that the use ofthis LEFM parameter did not 'provide an adequatecharacterization of the stress and strain fieldsat the tip of short cracks, where a rw rye However,for short cracks emanating from notclies whereinitial growth occurs within the plastic z~ne of thenotch (see Fig. 21), a continuum mechanics des-cription of the behaviour is less clear. Certainlythere is experimental evidence that such shortcracks can propagate below the long crack

------- ---- GB



were observed to initiate and grow in the softerferrite, only to arrest when they encountered theharder martensite.l36 Many of these effects can beexplained by considering the mechanics of crackdefle ction.

Based on the theoretical analyses by Bilbyet al.137 and Cotterell and Rice,138 recent studiesby Suresh81 have shown that alternatives toprevious interpretations of the way in which acrack tip interacts with a grain boundary can bedeveloped by considering the effect of crackdeflection on the propagation of short fatiguecracks. For a short crack, the low restraint oncyclic slip promotes a predominantly crystal-lographic mode of failure. When a crack tipreaches a grain boundary, it tends to reorientitself in the adjacent grain to advance by thesingle shear mechanism, and can be considerablydeflected by the grain boundary. This phenomenonis illustrated schematically in Fig. 31. The extentof deflection at the grain boundary is a functionof the relative orientations of the most favourableslip systems in the adjoining crystals. For anelastic crack initially inclined at an angle eo tothe mode I growth plane and deflected at the firstgrain boundary through an angle e1 (see Fig.31),approximate estimates of the local stress intensityfactors yield the relation81

K1/Kr = cos2 eo cos3(~ e1)

+ 3 silleo coseo sin(~e1) cos2(~el)(38)

K2/Kr = cos2 eo sin(~e1) cos2(~e1)

- sineo coseo cos(~e1)[1 - 3sin2(~e1)]

(39)

Here, K 1 and K 2 are the near- tip mode I and modeII stress intensity factors, respectively, immediatelyfollowing deflection at the grain boundary, whereasKr is the nominal mode I (far-field) value. For atypical short crack emanating from the surface atan angle of eo ~ 45° and deflected at the grainboundary by e1 ~ 90°, equations (38) and (39) yieldK1 ~ 0.7Kr and K2 ~ O. 35Kr. The effective drivingforce for coplanar growth can then be approximatedas the square root of the sum of the squares ofK1 and K2, such that tJ{eff ~ O.78&(r. Thus con-sideration of crack deflection processes alone canaccount for a significant reduction in driving forceas a crack tip interacts with a grain boundarywhen the way in which a short crack advance s ischaracterized by LEFM. It has been postulated 81that if the extent of defle ction at the grain boundaryis large, the effective cyclic stresses may bereduced to a value smaller than the true thre sholdfor short crack advance (e .g. to the fatigue endur-ance limit) such that complete crack arre st willresult (denoted by curve A in Fig.32). If theeffective cyclic stresses after deflection are abovesuch threshold values, there would be no crackarrest (as denoted by curve B in Fig.32) and onlya temporary deceleration in growth rate. Althoughthe numerical predictions of the deflection modelscan be subject to considerable uncertaintie s whenused to characterize the mechanics of short cracksin metals and alloys, the mechanisms underlyingcrack deflection processes have been shown toprovide a physically meaningful explanation notonly for the role of microstructure in influencingshort crack advance, but also for several fatiguecharacteristics of long cracks under constant81and variable amplitude81,126-128 loading.

In addition to causing a reduction in theeffective driving force, crack deflection mecha-nisms could playa maj or role in enhancingclosure.81,120 For example, the irreversibility of

Pmax

Pmin

Time ,t Time,t

M icrost ruct ura lIyshort cracksiz

"'0'0"UOJo

....J filt + Threshold ~Ko

Log~K--+

32 Variation of fatigue crack growth rate da/dNwith 6.K for both long and microstructurallyshort fatigue cracks; note how growth rate sfor short cracks decrease progressively belowlong crack thre sholdAKo before arresting ormerging with long crack data

( b)

a propagation into first grain; b propagationacross grain boundary

31 Growth and deflection of microstructurallyshort fatigue crack and the re sultant cracktip displacements and closure;, ~o is shortcrack initiation angle and (Jl angle ofdeflection at first grain boundary81


Suresh and Ritchie: Propagation of short fatigue cracks· 469

slip steps and surface oxidation can lead to non-uniform tensile opening and shear displacementsof short cracks.12,81 Given the presence ofserrated fracture surfaces and mode II crack tipdisplacements occurring after deflection, suchnon-uniformities in crack oPening and slidingresult in premature contact between aSPerities,leading to roughness induced closure (see Figs.26and 31). (An ideally elastic crack may not resultin any roughness induced closure, irrespective ofthe extent of deflection.) Experimental measure-ments of crack closure made by Morris andco-workers4 7-4 9 do indeed show that even shortcracks (spanning only a few grain diameters) canclose above the minimum load of the fatiguecycle (see Fig. 27).

A further factor which may contribute todifferences in the behaviour of long and shortcracks is the question of crack shape.10 Even longcracks, running across many grains, are known topossess certain irregularities in their geometry(on a microscopic scale) that result from localinteractions with microstructural features,l 0 yet,at a given 6K, the overall growth behaviour wouldbe expected to be similar. However, on comparinga large crack with a small one this similaritywould seem questionable. Moreover, the earlystages of fatiglie damage often involve the initiationof several small cra.cks, the subsequent growth ofany of which is likely to be strongly influenced bythe presence of the others.37-40

Differences in the behaviour of long and shortcracks may also result from the fact that, at thesame nominal 6K, the crack extension mechanismsmay be radically different. As pointed out bySchijve,10 the restraint of the elastic surroundingon a small crack near a free surface is verydifferent from that experienced at the tip of a longcrack inside the material. For a small, grainsized crack, cyclic slip along the system with thehighest critical resolved shear stress results inmixed mode II and mode I slip band cracking akinto Forsyth's stage I mechanism.139 However, fora long crack, spanning many grains, maintainingsuch slip band cracking in a single direction ineach grain is incompatible with maintaining acoherent crack front. The resulting increasedrestraint on cyclic plasticity will tend to activatefurther slip systems, leading to a non-crystallo-graphic mode of crack advance by alternating orsimultaneous shear, commonly referred to asstriation growth (Forsyth's stage II) .139 At near-threshold levels where the extent of local plasticitycan be small enough to be contained within a singlegrain, even long cracks may propagate via thesingle shear mechanism, the orientation of the slipband cracking changing at each grain boundary andleading to a faceted or zigzag crack path morpho-logy (see Fig.33).117-120

The occurrence of this shear mode of crackextension, with the related development of a facetedfracture surface, has a maj or influence on themagnitude of crack closure effects,116-120 whichmay in turn lead to differences in the behaviourof long and short cracks (see Figs. 26 and 31).

The differences in fatigue characteristicsresulting from crack closure arise from twomain sources. First, since closure results from

the constraint of surrounding elastic material onthe plastic region surrounding the crack, theamount of closure experienced by a small crackat high stress amplitudes in a fully plastic speci-men would be far less than that experienced by alarger crack at lower stress amplitudes in anelastic-plastic or nominally elastic specimen atthe same (nominal) driving force. However, moreimportantly, differences between the effect ofclosure on long and short cracks result from thefact that such closure effects predominate in thewake of the crack tip. Since short cracks - bydefinition - possess a limited wake, it is to beexpected that in general such cracks will besubjected to less closure. Thus, at the samenominal driving force, short cracks may experiencea larger effective value than will the equivalentlong crack. As outlined in the previous section,this can arise in two ways. In the first, withrespect to plasticity induced closure, plasticdeformation in the wake of the crack has to buildup before it can be effective in reducing 6Keff(Ref.112). From analogous studies of the role ofdilatant inelasticity (inelasticity resulting fromphase transformations) on reducing the effectivestress intensity at the crack tip in ceramics,140it has been found that the full effe ct of this closureis felt only when the transformed zone extendsinto the wake of the crack by a distance of fivetimes its forward extent. Although no comparableanalysis has been carried out for plastic deforma-tion in metals, it is to be expected that the role ofthe compressive stresses in the plastic zoneencompassing the wake of the crack would belimited for small cracks of a length comparableto the forward extent of this zone (i.e. for a ("oJ ry).It is believed12,64,133 that this is one of the mainreasons (at least from the per spective of continuummechanics) for non-propagating cracks and alsoexplains why microstructurally short cracks andcracks emanating from notches can propagatebelow the long crack threshold D..K0 (see Figs. 14and 25). Essentially, they can initiate and grow atnominal stress intensities below t::J(0 because ofthe absence of closure effects but, as they increasein length, the build-up of permanent residualplastic strains in their wake means that crackclosure begins to have the effect of progressivelydecreasing the effective AK experienced at thecrack tip, resulting in a progressive reduction incrack growth rate and sometimes complete arre st.

This notion, by which the anomalous behaviourof short cracks below the long crack thre sholdregime and in the strain field of notches (see e.g.Fig. 1) is related primarily to a decrease in crackclosure effects at small crack sizes,12,64,135 hasrecently been substantiated by both numerical135and experimental64 studies. Newman135 hasdemonstrated that by incorporating plasticityinduced closure in finite element computationalmodels of fatigue crack propagation, the progres-sively decreasing growth rates of short cracksemanating from notches could be predicted andshown to be in good agreement with experimentaldata on lower strength steel (see Fig. 34). Tanakaand Nakai64 monitored the growth of similar shortcracks in notched specimens of low strength steelat both R = 0 and 0.4 while simultaneously measur-ing the extent of crack closure. Their data, which



Flow Band

(0)

CrackFlowBands

(b)

c

e

d

f

a, c, e near threshold, ry < dg, stage I, mixed modes I and II; b, d,j higher growth rates, ry > dg,stage II, mode I; c, d, fractographs of nickel-plated 1018 steel; 118 e,f sections of 7075 T6 aluminiumalloy 10

33 Crack opening profiles and morphologies of long fatigue cracks120

show the characteristic decreasing growth ratefor short cracks when plotted in the usual way(in terms of a nominal M(), can be seen to coincidewith the long crack data and to fall on a singlesmooth curve when reanalysed in terms of AKeffincorporating the experimental Kcl measurements(see Fig.28).

A similar situation can arise from the effectof roughne ss induced crack closure promoted byrough, irregular fracture surfaces, particularly ifthe crack extension mechanism involves a strongsingle shear (mixed mode II and mode I)component.118-120 Since a crack of zero lengthcan have no fracture surface and hence cannotundergo roughness induced closure, the develop-ment of such closure is expected to be a strongfunction of crack size,47-49,120,133 as demon-strated by the experimental data obtained by Jamesand Morris48 and shown in Fig. 27. An estimate


of the lower bound to the size of transition crackbelow which roughness induced closure is ineffec-tive (at near-threshold levels) can be appreciatedfrom Figs. 31 and 33 (Ref. 120). A long crackwhich encompasses several grains will, at near-threshold levels, have developed a faceted morpho-logy and, as a result of the incompatibility betweenmating crack surfaces arising from the mode IIcrack tip displacements, will undergo roughnessinduced closure in the manner depicted in Fig. 26.The short crack, however, will not undergo suchclosure while its length remains Ie ss than a graindiameter since it will not have changed directionat a grain boundary, and accordingly will not haveformed a faceted morphology, despite extension viathe same single shear mechanism.

In general, since most of the results that showdifferences between the growth rates of long andshort cracks have been obtained at low stress


Low StrengthCSA Low Strength G40.11 Steel0'0 = 510 MNm-2

R =-1

Centre crack tension(a= 0·35 to 1.2 mm)

Centre crack tension pa =(0.4 to 2 mm)~ P .., Baseline data

o "'''-!h .. Single edge crack tension: .:.. (long crack, a > 2 mm). ..

I •, .& :.

RJ r ·,!J.~ I..

f •f

Prediction

~K,MNm-3/2

34 Comparison of experimental results and numerical predictions of crack propagation rates for smallcracks in centre cracked tensile specimens of eSA G40.11 low strength structural steel 00 =510 1VJNm-2, subjected to high stress levels13 5

intensity ranges, and fatigue-crack propagation oflong cracks in this near-threshold regime isknown to be strongly influenced by crack closureeffects, it seems most likely that the main reasonfor the faster growth of short cracks and the factthat they can propagate below the long crackthreshold Mo is associated with the role ofclosure decreasing with crack size. Resultsobtained by Potter and Yee141 on the behaviour ofshort cracks under variable amplitude loading areconsistent with this notion, since the crack growthtransients (i.e. accelerations and retardations)normally observed following overloads andspectrum loading sequences, which have beenattributed - at least in part - to closure mecha-nisms (see e.g.Ref. 112), were largely absent forshort cracks. In this regard it is useful to com-pare data for short and long cracks at high loadratios, since closure effects are then minimaleven for long cracks. This has been done forcrystallographic near-threshold fatigue in nickel-base alloys,6 0 and the threshold for short cracks,despite being 60~o smaller than the long crackthreshold at R = 0.1, was approximately equal tothe long crack threshold at R = 0.7.

Finally, large differences in the behaviour oflong and short cracks can arise at stress inten-sities well outside the threshold regime, becauseof environmental factors.11, 54,130 As shown inFig. 29, the results obtained by Gangloff11, 54 havedemonstrated that corrosion fatigue crack growthrates of physically short cracks in AISI 4130 steeltested in aqueous NaCI solution can be one to twoorders of magnitude faster than the correspondinggrowth rate s of long cracks at the same M value.This unique environmentally assisted propagation ofshort cracks was attributed to differing local crack

tip environments as a function of crack size v

-Specifically, Gangloff argued that the localconcen-tration of the embrittling species within the crackdepends on the surface-to-volume ratio of thecrack, on the diffusive and convective transport ofthe embrittling medium to the crack tip, and onthe distribution and coverage of active site s forelectrochemical reaction, all processes sensitiveto crack depth, opening displacement, and cracksurface morphology. 54,57 Analogous, yet Ie ssspectacular, environmental crack size effects mayalso arise in gaseous environments or in thepresence of internally charged hydrogen where,for example, the presence of hydrogen may inducean intergranular fracture mode. The rough cracksurfaces that tend to be produced by this failuremechanism would lead to roughness inducedclosure, which again directly influences the longcrack phenomenon of a reduced Meff (see e.g.Ref. 142) .

These differences in the behaviour of fatiguecracks of different size provide clear examples ofhow the fracture mechanics similitude concept canbreak down. The stress intensity, althoughadequately characterizing the mechanical drivingforce for crack extension, can account for neitherthe chemical activity of the crack tip environmentnor the local interaction of the crack with micro-structural features. Since these factors, togetherwith the development of crack closure, are astrong function of crack size, it is actuallyunreasonable to expect the growth behaviour oflong and short cracks to be identical. Thus, in theabsence of the similitude relationship, the analysisand utilization of laboratory fatigue-crack propaga-tion data to predict the performance of in-servicecomponents in which short cracks are present



becomes an extremely complex task; a task whichdemands an immediate and major effort from bothresearchers and practising engineers alike.

CONCLUDlNG REMARKS

The problem of short cracks must be recognizedas one of the most important and challenging topicscurrently faced by researchers in fatigue. Notonly is it a comparatively unexplored areaacademically, but it also raises doubts about theuniversal application to the characterization ofsub-critical flaw growth of fracture mechanics,the misuse of which can result in overestimatesof defect-tolerant lifetimes. It is an area that,repre sents an interface between the fracturemechanics methodologies dealing specifically withthe macroscopic growth of fatigue cracks and theclassical engineering mechanics methodologiesdealing with total life and engineering concepts of(macro-) crack initiation (as depicted in Fig. 1).As discussed in the introductory section, the lastprocess, of macrocrack initiation, is simply thegrowth of short cracks (microcracks). The pro-cess of the initiation of short cracks (microcrackinitiation) has not been treated explicity in thispaper as it has been the subject of severalrecent reviews (see e.g. Refs. 7,143,144). Sufficeit to say that such small cracks tend to initiate atconstituent particle s (i.e. inclusions and inter-metal1ics, as shown in Fig. 9) in commercialmaterials (see e.g. Refs. 39, 41, 145), whereas inpure metals and alloys their initiation is oftenassociated with emerging planar slip bandscalled persistent slip bands (PSBs) (see e.g.Ref. 143, 144, 146, 147). In fcc metals, such smallcracks appear to initiate via a crystallographicstage I mechanism along the PSB, as shown inFig. 35 (Ref. 148), although the specific mechanismsof initiation and their relation to the PSBs varymarkedly from material to material.144

In the pre sent paper an attempt has beenmade to provide a critical overview of recentexperimental studies on the growth of smallfatigue cracks, and specifically to outline themechanical, metallurgical, and environmentalreasons, as to why the behaviour of such cracksshould differ from the behaviour of long cracks.The intention has not been to present a formalanalysis of each of these factors, since in mostcases such an analysis is simply not possible, butrather to give a thorough review of the manyinterdisciplinary factors which may be relevantto the short crack problem. It has been concludedthat differences in the behaviour of short and longcracks are to be expected, and that such differ-ence s can arise from a number of distinctphenomena:

(i) inadequate characterization of the mechanicsof crack tip stress and deformation fields ofshort cracks, including the neglect of higherorder terms for the elastic singularity andthe presence of extensive local crack tipplasticity

(ii) notch tip stress and deformation field effects(for cracks emanating from notches)

(iii) the interaction, including deflection, or shortcracks with microstructural features such asgrain boundaries, inclusion,S, and second phases,of dimensions comparable in size with thecrack length

(iv) differences in crack shape and geometry(v) differences in crack extension mechanism(vi) the effect of crack closure varymg with crack

length(vii) differertces in the local crack tip environments.

Each of these factor s represents a formidablechallenge in fatigue research because of the com-plex nature of both experimental and theoreticalstudies, yet they are of great importance to anunderstanding of the anomalous behaviour of shortflaws. This problem will undoubtedly come to

35 Transmission electron micrograph of stage I small cracks propagating within persistent slip bands,showing ladder -like dislocation substructures in fatigued, polycrystalline, high purity copper. Insetsshow corresponding optical micrograph of cracks and electron diffraction pattern148



18.

17.

20.

19.

P. C. Paris and F .Erdogan: J. Basic Eng.(Trans. ASME D), 1963,85, 528.J. R. Rice: in 'Fracture: an advanced treatise',(ed. H. Liebowitz), Vol. 2,191; 1968, New York,Academic Press.J. R. Rice: in 'Fatigue crack propagation',STP 415, 247; 1967, Philadelphia, Pa,American Society forTesting and Materials.H. Tada, P. C. Paris, and·G.R. Irwin: 'Thestress analysis of cracks handbook'; 1973,Hellertown, Pa, Del Re search Corp.G. C. Sih: 'Handbook of stress intensityfactors'; 1973, Bethlehem, Pa, LehighUniversity .D. P. Rooke and D. J . Cartwright: 'Compendiumof stress intensity factors'; 1975, London,HMSO.N. E. Dowling: Fatigue Eng. Mater. Struct.,1979,2, 129.R.O.Ritchie: J.Eng.Mater. Technol. (Trans.ASME, H), 1983, 105,1.J. vi. Hutchinson: J. M echoPhys. Solids, 1968,16,13.J.R.Rice and G.R.Rosengren: J.Mech.Phys.Solids, 1968, 16, 1.J .R.Rice: J.Appl.Mech. (Trans. ASME),1968, 35, 379.N. E. Dowling: in 'Cracks and fracture', STP601, 19; 1976, Philadelphia, Pa, AmericanSociety for Testing and Materials.N. E. Dowling and J. A. Begley: in 'Mechanicsof crack growth', STP 590, 82; 1976,Philadelphia, Pa, American· Society forTesting and Materials.J .W. Hutchinson and P. C. Paris: in 'E lastic-plastic fracture', STP 668, 37; 1979,Philadelphia, Pa, American Society forTesting and Materials.D. M. Parks: unpublished work, YaleUniversity, 1978.C. F. Shih: J. Mech. Phys. Solids, 1981,29,305.D. Broek and B. N. Leis: in 'Materials,experimentation and design in fatigue', (ed.F.Sherratt and J.B.Sturgeon), 129; 1981,Guildford, Westbury House.R. O. Ritchie and S. Suresh: Mater. Sci. Eng.,1983,57, L27.C.J . Beevers (ed.): 'The measurement ofcrack length and shape during fracture andfatigue'; 1980, Warley, West Midlands,Engineering Materials Advisory ServicesLtd.C. J. Beevers (ed.): 'Advances in cracklength measurement'; 1983, Warley, WestMidlands, Engineering Materials AdvisoryServices Ltd.ASTM Standard E647-83 on 'Test methodfor constant-load -amplitude fatigue crackgrowth rates above 10-8 m/cycle', AnnualBookof ASTM Standards, Section 3, 710;1983, Philadelphia, Pa, American Society forTesting and Materials.W. L. Morris, M.R.'James, and O.Buck: in'Nondestructive evaluation - microstructuralcharacterization and'reliability strategies',(ed. O.Buck and S. M.Wolf), 387; 1981,Warrendale, Pa,Metallurgical Society ofAIME.J. M. Barsom and R. C. McNicol: in 'Fracturetoughness and slow stable cracking', STP


24.

27.

15.

16.

28.

21.

25.

26.

22.23.

29.

32.

30.31.

33.

35.

34.

36.

37.

REFERENCES

assume even greater significance in the futuresince, with improvements in the science andpractice of non-destructive testing, the projectedlifetime of a fatigue flaw in the short crack regimewill become an increasingly larger· proportion ofthe total 1ife .

The work was performed under Grant No. AFOSR-82-0181 from the US Air Force Office of ScientificResearch to the University of California atBerkeley. The authors wish to thank Dr A. H.Rosenstein of AFOSR for his support andencouragement, DrsW. L. Morris and J. Lankfordfor kindly supplying original micrographs, and M.Penton for her help in preparing the manuscript.This review is an expanded version of a paperpresented at the 55th Specialists Meeting of theAGARD Structural Materials Panel in Toronto,Canada, in September 1982.

ACKNOWLEDGMENTS

1. L. F. Coffin: in 'Fatigue and microstructure',(ed. M. Meshii), 1; 1979, Metals Park, Ohio,American Society for Metals.

2. M.R. Mitchell: in 'Fatigue and microstruc-ture', (ed. M.Meshii), 385; 1979, Metals Park,Ohio, American Society for Metals.

3. J.E.Campbell,W.E.Berry, and C.E.Feddersen: 'Damage tolerant designhandbook'; 1972, Columbus, Ohio, Metals andCeramics Information Center, BattelleColumbus Laboratorie s.

4. S. T .Rolfe and J. M.Barsom: 'Fracture andfatigue control in structures: applications offracture mechanics'; 1977, Englewood Cliffs,NJ, Prentice Hall.

5. H.H.Johnson and P.C.Paris: Eng. Fract.Mech., 1968, 1, 3.

6. R. O. Ritchie: Int. Met. Rev., 1979~24, 205.7. M.E. Fine and R.O.Ritchie: in 'Fatigue and

microstructure', (ed. M. Meshii), 245; 1979,Metals Park, Ohio, American Society forMetals.

8. S.J.Hudak: J.Eng.Mater.Technol. (Trans.ASME H), 1981,103, 26.

9. M. H. EI Haddad, T. H. Topper, and B.Mukherjee: J. TesL Eval., 1981, 9, 65.

10. J. Schijve: in 'Fatigue thresholds', (ed. J.Backlundetal.), Vol. 2,881; 1982, Warley,West Midlands, Engineering MaterialsAdvisory Services Ltd.

11. R.P. Gangloff: in 'Advances in crack lengthmeasurement', (ed.C.J.Beevers), 175; 1983,Warley, West Midlands, Engineering MaterialsAdvisory Services Ltd.

12. R. O. Ritchie and S. Suresh: in 'Behaviour ofshort cracks in airframe components',AGARD Conf. Proc. No. 328,1-1; 1983,Neuilly sur Seine, Advisory Group forAerospace Research and Development.

13. M. L. Williams: J.Appl. Mech. (Trans. ASME E),1957, 24, 109.

14. G.R. Irwin: J.Appl.Mech. (Trans. ASME),1957,24.

http://www.maneyonline.com/action/showLinks?crossref=10.1016%2F0013-7944%2868%2990014-3

http://www.maneyonline.com/action/showLinks?crossref=10.1520%2FJTE11646J

http://www.maneyonline.com/action/showLinks?crossref=10.1115%2F1.3224969


559, 183; 1974, Philadelphia, Pa, American 69.Society for Testing and Materials.

38. Y. H.Kim, T .Mura, and M.E. Fine: Metall.Trans., 1978, 9A, 1679.

39. C. Y.Kung and M.E. Fine: Metall. Trans.,1979, lOA, 603. 70.

40. W. L. Morris: Metall. Trans., 1977, BA, 589.41. W.L.Morris and O.Buck: Metall. Trans.,

1977, 8A, 597.42. W. L. Morris: Metall. Trans .., 1978, 9A, 1345.43. R. Chang, W.L.Morris, and O.Buck: Scr. 71.

Metall., 1979, 13, 191.44. W. L. Morris: Metall. Trans., 1979, lOA, 5. 72.45. W. L. Morris: personal communication,

Rockwell Interna tional Science Center, 73.Thousand Oaks, Cal., 1983.

46. W. L. Morris: Metall. Trans., 1980, llA, 1117. 74.47. W.L.Morris, M.R.James, and O.Buck:

Metall. Trans.~ 1981, 12A, 57. 75.48. M.R.James and W.L.Morris: Meta~l. Trans.,

1983, 14A, 153.49. W.L.Morris,M.R.James,and O.Buck: Eng. 76.

Fract. Mech., 1983, 18, 871.50. G. P. Sheldon, T. S. Cook, T. W.Jones, and J. 77.

Lankford: Fatigue Eng. Mater. Struct., 1981,3, 219. 78.

51. N.E. Dowling: in 'Cyclic stress-strain and 79.plastic deformation aspects of fatigue crackgrowth', STP 637, 97; 1977, Philadelphia, Pa,American Society for Testing and Materials.

52. N.E.Dowling: in Proc.ASME 4th Natl Congr. 80.on 'Pressure vessel and piping technology',Portland, Oreg., June 1983, American Society 81.for Mechanical Engineers. 82.

53. R. P. Gangloff: in 'Fatigue crack growthmeasurement and data analysis', (ed. Hudakand Bucci), STP 738,120; 1981, Philadelphia,Pa, American Society for Testing and 83.Materials.

54. R. P.Gangloff: Res Mech.Lett., 1981, 1, 299. 84.55. M. T .Resch, D. V.Nelson, J. C. Shyne, and

G.S.Kino: in Ref.34,p.473.56. W. Baxter: Int. J. Fatigue, 1983, 5, 37. 85.57. R. P. Gangloff: in 'Embrittlement by the

localized crack environment', (ed. R. P. 86.Gangloff); 1984, Warrendale, Pa, Metallurgical 87.Society of AIME, in the press.

58. B. Wiltshire and J . F .Knott: Int. J. Fract., 88.1980,16, R21.

59. S. Usami and S.Shida: Fatigue Eng. Mater. 89.Struct., 1979, 1,471.

60. J. F. McCarver and R. O.Ritchie: Mater. Sci. 90.Eng., 1982, 55,63.

61. K. Tanaka and Y. Nakai: Fatigue Eng. Mater. 91.Struct., 1983,6,315.

62. J .Byrne and T. V. Duggan: in 'Fatigue 92.thresholds', (ed. J. Backlund et al.), Vol. 2,753; 1982, Warley, West Midlands, EngineeringMaterials Advisory Services Ltd.

63. S. Suresh: Eng. Fract. Mech., 1984,20, in 93.the press.

64. K. Tanaka and Y.Nakai: J. Eng. Mater. 94.Technol. (Trans. ASME H), 1984, 106, 192.

65. J. Lankford and D. L.Davidson: Acta Metall.,1983,31,1273.

66. J. Lankford and D. L.Davidson: Fatigue Eng. 95.Mater. Struct., 1983,6,241.

67. F. Guiu, R. Dulniak, and B. C.Edwards: 96.Fatigue Eng. Mater. Struct., 1982,5,311.

68. S. Pearson: Eng. Fract.Mech., 1975,7,235. 97.


O.N.Romaniv, V.N. Siminkovich, and A.N.Tkach: in 'Fatigue thresholds', (ed. J .BYlcklundet al.), Vol. 2, 799; 1982, Warley, WestMidlands, Engineering Materials AdvisoryServices Ltd.K. Tanaka, M.Hojo, and Y.Nakai: in 'Fatiguemechanisms: advances in quantitative mea-surement of fatigue damage', (ed. Lankfordet al.), STP 811,207; 1983, Philadelphia, Pa,American Society for Testing and Materials.J. Lankford: Fatigue Eng. Mater. Struct.,1982, 5, 233.J. Lankford: Fatigue Eng. Mater. Struct.,1983,6,15.D.Taylor and J.F.Knott: Fatigue Eng. Mater.Struct., 1981,4,147.C.W.Brown and M.A. Hicks: Fatigue Eng.Mater. Struct., 1983,6,67.Y.Nakai and K. Tanaka: in 'Proc. 23rdJapan congress on materials research',1980, 106.K. Tanaka, Y.Nakai, and M.Yamashita: Int.J.Fract., 1981, 17, 519.R.O.Ritchie: J.Eng.Mater. Technol. (Trans.ASME H), 1977~99,195.J. Lankford: Eng. Fract.Mech., 1977,9,617.S. Taira, K. Tanaka, and M. Hoshina: in'Fatigue mechanisms', STP 675,135; 1979,Philadelphia, Pa, American Society forTesting and Materials.S. Taira, K. Tanaka, and Y•Nakai: Mech. Res.Commun.,11978, 5,375.S. Suresh: Metall. Trans., 1983, 14A, 2375.H.Kitagawa and S. Takahashi: in 'Proc.2ndinto con£. on mechanical behavior ofmaterials', 627; 1976, Metals Park, Ohio,American Society for Metals.D. Taylor: Fatigue Eng. Mater. Struct., 1982,5,305.M.H.EI Haddad, K.N. Smith, and T. H. Topper:J. Eng. Mater. Technol. (Trans. ASME H),1979,101,42.R. C.Boettner, C. Laird, and A.J .McEvily:Trans. AIME,1965, 233, 379.H.D.Solomon: J.Mater., 1972,7,299.J. R. Haigh and R. P. Skelton: Mater. Sci. Eng.,1978,36,133.M.S. Starkey and R. P. Skelton: Fatigue Eng.Mater. Struct., 1"982,5, 329.M.H.El Haddad, T. H. Topper, and K. N. Smith:Eng. Fract.Mech., 1979,11, 573.M. H.EI Haddad, T. H. Topper, and K.N. Smith:J. Test. Eval.~ 1980,8,301.M.H.EI Haddad, N.E.Dowling, T .H. Topper,and K.N.Smith: Int.J. Fract., 1980,16,15.H.Ohuchida, A.Nishioka, and S. Usami: in'Proc. 3rd into com. on fracture', April 1973,Munich, Verein Deutscher Eisenhiittenlente,Vol. 5, V-443/ A.H. Ohuchida, S. Usami, and A. Nishioka:Bull.Jpn Soc. Mech. Eng., 1975,18,1185.S. Usami: in 'Fatigue thresholds', (ed. J.Backlund et al.), Vol. 1, 205; 1982, Warley,West Midlands, Engineering MaterialsAdvisory Services Ltd.H.Neuber: J. Appl. Mech., (Trans. ASME H),1961, 28, 544.R. E. Peterson: 'Stress concentration factors';1974, New York, Wiley-Inter science .R.E. Peterson: in 'Metal fatigue', (ed. G. Sines


and J. L. Waisman), 293; 1959, New York, 122. S. Suresh and R.O.Ritchie: Scr. Metall.! 1983,McGraw-HilI. 17, 575.

98. T. H. Topper, R. M. Wetzel, and J. Morrow: 123. R. A. Schmidt and P. C. Paris: in 'Progre ssJ. Met., 1969,4,200. in flaw growth and fracture toughness

99. N .E. Dowling, W.R. Brose, andW. K. Wilson: te sting" STP 536, 79; 1973, Philadelphia, Pa,'Fatigue under complex loading - analysis American Society for Testing and Materials.and experiments', 55; 1977, Warrendale, Pa, 124. R.O.Ritchie, S.Suresh, and P.K.Liaw: inSociety of Automotive Engineers. 'Ultrasonic fatigue', (ed.J.M.Wells et al.)~

100. R.A.Smith andK.J.Miller: Int.J.Mech.Sci., 433; 1982, Warrendale, Pa, Metallurgical1977,19,11. Society of AIME.

101. N. E. Dowling: in 'Fracture mechanics', STP 125. S. Suresh and R. O. Ritchie: in 'Concepts of677,247; 1979, Philadelphia, Pa, American fatigue crack growth thresholds', (ed. D. L.Society for Testing and Materials. Davidson and S. Suresh), 227; 1984,

102. H.Kitagawa: in 'Fatigue thresholds' , (ed. J. Warrendale, Pa, Metallurgical Society ofBacklund et al.), Vol. 2, 1051; 1982, Warley, AIME.West Midlands, Engineering Materials 126. S. Suresh: Scr.Metall., 1982,16,995.Advisory Services Ltd. 127. S. Suresh: Eng. Fract. Mech., 1983, 18, 577.

103. P. Lukas and M.Klesnil: Mater. Sci. Eng., 128. S. Suresh and A.K. Vasudevan: in 'Concepts1978,34,61. of fatigue crack growth thresholds', (ed. D. L.

104. M. H. EI Haddad, K. N. Smith, and T. H. Topper: Davidson and S. Suresh); 361; 1984, Warren-in 'Fracture mechanics', STP 677, 274; 1979, dale, Pa, Metallurgical Society of AIME.Philadelphia, Pa, American Society for 129. R.J. Cooke, P.E. Irving, G. S.Booth, and C.J.Testing and Materials. Beevers: Eng. Fracl.Mech., 1975,7,69.

105. T . H. Topper and M. H. EI Haddad: in 130. B. F .Jones: J. Mater. Sci.? 1982, 17,499.'Fatigue thresholds', (ed. J.Backlund et al.), 131. K.J. Miller: Fatigue Eng. Mater. Struct.,Vol. 2,777; 1982, Warley, West Midlands, 1982,5,223.Engineering Materials Advisory Services Ltd. 132. W. T . Chiang and K. J . Miller: Fatigue Eng.

106. C. E . Phillips: in 'Proc. colloq on fatigue' , Mater. Struct., 1982,5,249.1955, Stockholm, IUTUM, 210; 1956, Berlin, 133. A. Talug and K.Reifsnider: in 'Cyclic stress-Springer. strain and plastic deformation aspects of

107. N. E. Frost: J. Mech. Eng. Sci., 1960,2, 109. fatigue crack growth', STP 637, 81; 1977,108. N. E. Frost and D. S. Dugdale: J. Mech. Phys. Philadelphia, Pa, American Society for

Solids, 1957, 5, 182. Testing and Materials.109. M.M.Hammouda and K.J.Miller: in 134. R.J. Allen and J. C. Sinclair: Fatigue Eng.

'Elastic-plastic fracture', STP 668, 703; Mater Struct., 1982,5,343.1979, Philadelphia, Pa, American Society for 135. J . C. Newman, Jr: in 'Behaviour of shortTesting and Materials. cracks in airframe components', AGARD

110. B. N. Leis and T. P. Forte: in 'Fracture Conf. Proc. No. 328,6-1; 1983, Neuilly surmechanics', STP 743, (ed. R. Roberts), 100; Seine, Advisory Group for Aerospace1981, Philadelphia, Pa, American Society for Research and Development.Te sting and Materials. 136. T . Kunio and K. Yamada: in 'Fatigue

111. N.E. Frost, K.J. Marsh, and L. P. Pook: mechanism', STP 675, 342; 1979, Philadelphia,'Metal fatigue '; 1974, Oxford, Clarendon Pa, American Society for Te sting andPress. Materials ..

112. W.Elber: in 'Damage tolerance in aircraft 137. B. A. Bilby, C.E. Cardew, and I. C. Howard:structures', STP 486,230; 1971, Philadelphia, in 'Fracture 1977', (ed. D~M.R. Taplin),Pa, American Society for Testing and Vol. 3,197; New York, Pergamon.Materials. 138. B. Cotterell and J .R.Rice: Int.J. Fract.,

113. K. Endo, K. Komai, and Y. Matsuda: Mem. 1980, 16, 155.Fac. Eng., Kyoto Univ., 1969,31, 25. 139. P. J. E. Forsyth: in 'Crack propagation "114. R.O.Ritchie, S.Suresh, and C.M.Moss: 76; 1962, Cranfield, Bedfordshir~, CranfieldJ. Eng. Mater. Technol. (Trans. ASME H), Press.1980, 102, 293. 140. R.M.McMeeking and A.G.Evans: J.Am.

115. A. T. Stewart: Eng. Fract.Mech., 1980,13, Ceram.Soc., 1982,65,242.463. 141. J.M.PotterandB.G.W.Yee: in 'Behaviour

116. S. Suresh, G. F. Zamiski, and R. O. Ritchie: of short cracks in airframe components',Metall. Trans.? 1981, 12A, 1435. AGARD Conf. Proc.No. 328,4-1; 1983,

117. N.Walker and C.J.Beevers: Fatigue Eng. Neuilly sur Seine, Advisory Group forMater. Struct., 1979, 1, 135. Aerospace Research and Development.

118. K. Minakawa and A.J. McEvily: Scr. Metall.~ 142. K. A.Esaklul, A. G. Wright, and W. W.1981,15,633. Gerberich: Scr. Metall., 1983,17, 1073.

119. R. O.Ritchie and S. Suresh: Metall. Trans., 143. C. Laird: 'Fatigue and microstructure', (ed.1982, 13A, 937. M. Meshii), 149; 1979, Metals Park, Ohio,120. S. Suresh and R. O. Ritchie: Metall. Trans .,. American Society for Metals.1982, 13A, 1627. 144. H. Mughrabi: in 'Strength of metals and

121. S. Suresh, D.M. Parks, and R.O.Ritchie: in alloys', (ed. P. Haasen et aL), Vol. 3, 1615;'Fatigue thresholds', (ed. J. Backlund el at.), 1980, New York, Pergamon.Vol. 1, 391; 1982, Warley, West Midlands, 145. G. Liitjering, H.Daker, and D. Munz: inEngineering Materials Advisory Services 'The microstructure and design of alloys',Ltd. Vol. 1, 427; 1973, London, The Institute of


Documents

NTN V[Z [S Û[] SN VT`R P]NPW^ · D][\NTN_V[Z [S Û[]_ SN_VT`R P]NPW^ F( F`]RÛ NZQ E( C( EV_PUVR E^qfdrb ò^`h molm^d^qflk fk bkdfkbbofkd j^qbofîp e^p _bbk qeb pr_gb`q lc `lkpfabo,