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Predicting fatigue resistance of nano-twinned materials:Part II – Effective threshold stress intensity factor range
Piyas B. Chowdhury
a
, Huseyin Sehitoglu
a,
⇑
, Richard G. Rateick
b
a
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801, USA
b
Honeywell Aerospace, 3520 Westmoor St., South Bend, IN 46628, USA
a r t i c l e i n f o
Article history:
Available online xxxx
Keywords:
Slip irreversibilityAnnealing twinShort crack growthMicrostructureFatigue threshold
a b s t r a c t
The determination of
D
K
th,eff
for fatigue crack growth has been a challenging task. A model is forwardedto assess this parameter in presence of nano-scale twins for Cu, Ni andAl. The model utilizes the conceptofirreversibilityofcrackemitteddislocationglide.Incrementalcrackextensionisformulatedonthebasisof force balance of continuumdislocations during cyclic ﬂow. Peierls stresses for free and twin boundaryrestricted glide, computed in Part I, constitute an essential ingredient in devising crack growth thresholdfounded on relative positions of dislocations. Predicted
D
K
th,eff
for relatively short cracks is shown to besubstantiallyenhancedwithareﬁnementintwinlamellarthicknessandtwinspacingasindicatedexper-imentally in recent literature. The theoretical values of
D
K
th,eff
in Ni, Cu and Al are in good agreementwith experimental literature for longer cracks as well. Saturation effects are observed in
D
K
th,eff
levelswith respect to gradual increase in twin nano-dimensions as well as crack length. Mechanistic srcinof these observations is discussed.
2014 Elsevier Ltd. All rights reserved.
1. Background
Threshold stress intensity range,
D
K
th
is the most widely usedparameter to assess fatigue crack growth resistance [1]. Neverthe-less, experimental determination of this quantity has remained arather challenging task to-date. Theoretically, there are a signiﬁ-cant number of studies in the literature to predict this metric forcoarse-grained materials [2–10]. These works have establishedtheimportanceof different mesoscalefactors underlying thephys-ical process of crack growth onset such as lattice frictional stress,slip irreversibility and microstructural obstacles (e.g. grain or twinboundaries). The present study is geared toward developing amodel for evaluating effective threshold stress intensity factorrange,
D
K
th,eff
for nano-materials with annealing twins. Consider-ation of nano-twinned materials in the current undertaking is of special interest in view of recent literature [11,12] reporting supe-rior damage resistance therein. Existence of enhanced monotonicmechanical properties of twinned nano-materials has been con-ﬁrmed in previous literature [13,14]. The underlying microme-chanics related to the observed macroscopic properties has beenaddressed in earlier literature [15–17]. In Part I of this work, wehave studied the micromechanics of cyclic plasticity associatedwithanano-scaletwininanon-continuumframeworkwithaviewto isolating its role on an advancing fatigue crack. In particular, wehavecomputedlatticefrictionalstressinﬂuencedbyatwinbound-ary during cyclic ﬂow which serves as one crucial constituent inmodeling fatigue crack propagation in the present work. Threefcc materials are considered for study, namely, Ni, Cu and Al. InPart II, we predict nano-scale twin-inﬂuenced
D
K
th,eff
levels onthe basis of cyclic irreversibility of discrete dislocations emanatingfrom a fatigue crack.It has been proposed that there exist two different
D
K
th
levels – (a) a microstructural threshold (for short cracks) and (b)a mechanical threshold (for longer cracks) [18]. Fig. 1 illustrates
thedistinctionbetweentheadvancementofashortcrackatmicro-structure level (characterized by ﬂuctuating rate) and stable prop-agation of a microstructure-insensitive longer crack (in Parisregime). Therelativesizeof amicroscopiccrackandtheassociatedextent of near-tip slip activities are small compared to the hostgrain dimension (typically several microns) [19]. It progresses byshear mechanism along the critical slip system with maximumresolved shear stress [20] known as the so-called stage I growthmechanism. At this stage, the advancing micro-crack may be com-pletely arrested at a grain level obstacle such as a grain or a twinboundary [7,18,21–26]. Once a small crack is arrested at such aninterface, the minimum applied
D
K
level required to advance thecrack in the adjacent grain is considered the corresponding micro-
http://dx.doi.org/10.1016/j.ijfatigue.2014.06.0060142-1123/
2014 Elsevier Ltd. All rights reserved.
⇑
Corresponding author. Tel.: +1 217 333 4112; fax: +1 217 244 6534.
E-mail address:
huseyin@illinois.edu (H. Sehitoglu).International Journal of Fatigue xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
International Journal of Fatigue
journal homepage: www.elsevier.com/locate/ijfatigue
Pleasecitethisarticleinpressas:ChowdhuryPBetal. Predictingfatigueresistanceofnano-twinnedmaterials:PartII–Effectivethresholdstressintensityfactor range. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.06.006
structural threshold [18]. Upon overcoming the microstructuralthreshold, the short crack undergoes periodic retardation atvarious micro-barriers to a gradually subsiding effect [27].Therefore, this threshold can be deemed as an important metricrepresenting intrinsic material resistance to cracking as uniquelydecided by material microstructure. The behavior of a micro-crackcan, therefore, be understood only by considering grain level near-tip microplasticity [28]. Nevertheless, experimental characteriza-tion of micro-crack propagation remains a rather challenging taskto-date due to limitations of measurement technique at thatlengthscale.Asashortcrackbecomesprogressivelylonger,theroleof microstructural barriers diminishes until the crack grows sufﬁ-cientlylongtoreachastablepropagationstage(theso-calledstageII growth or Paris growth regime) [29]. This growth period is char-acterizedbytherelativesizeofcrackbeingsigniﬁcantlylargerthanindividual grains as well as associated plastic zone (accumulatedover cycles) which encompasses multitudes of grains. The stageII crack progresses subcritically in a stable manner independentof material microstructure until the catastrophic failure. At thisstage, the crack can be physically discerned by conventionaldetection technique. The microstructure-independent growth of amacroscopically long crack is governed only by applied stressand crack length which can be conveniently described by linearelastic fracture mechanics (LEFM) [30,31]. From a LEFM perspec-tive, the propagation threshold for a macroscopically long crack(i.e. mechanical threshold) is deﬁned as the limiting
D
K
levelbelow which no discernible crack advance is detected with theaid of current experimental methods. Therefore, the early stage Igrowth period for a shorter crack (prior to it becoming noticeablylong) is considered as the so-called initiation period for a moredetectable longer crack. Evidently, the LEFM-modeled crackgrowth characterization is dependent on the precision of experi-mental measurement technique to-date. Nevertheless, theseexperiments, based on load shedding methods, are subjected tohigh costs and test matrixes prone to produce data affected byhistory. In that regard, it is worthwhile to develop a theoreticalframeworkcapableofpredicting
D
K
th
atcriticalmicroscopicstageswhich may provide invaluable insight for studying inherentmaterial impedance to cracking. From a theoretical standpoint, anumberofworkspreviouslyundertakenhaveexaminedthefactorsgoverning the
D
K
th
discussed as follows.Upon carefully reviewing earlier threshold models, three dis-tinct categories exist on the basis of slip characteristics as summa-rized in Table 1 – (i) slip emission, (ii) slip blockage, and (iii) slipirreversibility models. In the slip emission models [3,6,9,10,32],the primary input for threshold prediction consists of the criticalshear stress,
s
, for dislocation nucleation, crack-tip to dislocationspacing,
x
, and the relative orientation of the active slip systemwith respect to the crack path,
h
(as shown in the schematic of Table 1). On the other hand, there are a number of experimentalobservations afﬁrming the role of obstacles such as grain bound-aries (GB) on short crack deceleration and/or arrest. GBs provideimpedance to the crack-emitted slips, and hence necessitategreater applied shear stress,
s
, for further slip emission due toshielding effects. Relative ease of slip transfer across GBs (i.e. com-plete blockage versus full transmission) strongly depends on thelocal stress state at the reaction site and the geometry of incidentdislocations and GBs [16]. Distinct outcomes of slip-GB reactionswould signiﬁcantly affect the threshold condition. Hence, slipblockage models [4,8,33] considered the difﬁculties in the forwardplastic ﬂow (an enhanced
s
), crack length,
a
o
, and the proximity of thecrack-tiptotheGB,
d
, tobethedecidingfactorsfortheonsetof crack propagation. In these studies, the threshold condition wasexpressed in terms of the critical
D
K
level below which a crack isincapable of transmitting strain to a neighboring grain. Moreover,upon cyclic loading, GBs inﬂict irreversibility of slip glide whichleads to accumulation of permanent strain at crack-tip. Therefore,slip irreversibility models [5,34–38] considered the non-zero dif-ferential between the forward and reverse plasticity. The degreeof irreversible glide of dislocation dictates the net permanentcrack-tip extension i.e. completely reversible cyclic ﬂow wouldcausenoeffectivecrackgrowth.Theinterplaywithmicrostructural
Fig. 1.
Aschematicdemonstratingthecrackgrowthregimesdescribedbyconventionallinearelasticfracturemechanics(LEFM)andmicroplasticity.LEFMapproachiswidelyused to characterize the experimentally detectable stable propagation (for long cracks). The mechanical
D
K
th
is considered as the limiting
D
K
level below which nodiscernible crack growth could be detected by conventional technique. On the other hand, a microscopically advancing crack may be arrested at a grain level obstacle (e.g. atwin boundary) during its early stage of propagation. The microstructural threshold could be deemed as the minimum required
D
K
level to advance the crack past such anobstacle. Upon overcoming microstructural
D
K
th
, the micro-crack progresses with intermittent and subsiding deceleration at various obstacles, eventually to grow longenough to be discerned by LEFM based measurement methods.2
P.B. Chowdhury et al./International Journal of Fatigue xxx (2014) xxx–xxx
Pleasecitethisarticleinpress as:ChowdhuryPBetal. Predictingfatigueresistanceofnano-twinnedmaterials:PartII–Effectivethresholdstressintensityfactor range. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.06.006
obstacles(e.g.GBs)wouldmodifytheextentofslipirreversibilities,thereby affecting the fatigue crack growth onset. Discrete slipmodel by Pippan et al. [5,35–37], in particular, demonstrated thatthe dynamic annihilation of dislocations of opposite signs leads tothe overall irreversible glide paths. Hence, the interfacial obstaclesare found to serve as both slip blockage and slip irreversibility-inducing microstructural elements.In light of the aforementioned discussion, we note that thephysics of microstructure-dependent crack advance involves thecombined effects of grain crystallography, lattice resistance todiscrete slip glide, and irreversibilities of near-tip and interfacialplasticity. Evidently, the governing metallurgical variables spannon-continuum to continuum considerations. Hence, in Part I, wehave studied the discrete lattice effects on dislocation glide asinﬂuenced by a nano-scale twin. Currently, in Part II of the study,atomistically computed lattice frictional stress from Part I is uti-lizedtopredict
D
K
th,eff
basedontheirreversibilityofcrack-emitteddiscrete dislocations as imposed by nano-scale twins.
2. Proposed fatigue crack growth model
In order to estimate twin-induced
D
K
th
levels, we conduct frac-ture mechanics based crack growth simulations. Fig. 2 illustratestheconﬁgurationusedforthispurpose. Asemi-inﬁnitesinglecrys-tal with a corner crack is constructed as shown. Twins of ﬁnitelamellar thickness,
t
, and inter-twin spacing,
d
are placed aheadof the advancing crack. Cyclic farﬁeld loading is applied on thesystem. A series of elastically interacting dislocations emanatefrom the crack-tip. The forward and reverse dislocation motionsare shown in red
1
and blue.
Inthisapproach,plasticityonlyaheadofthecrack-tipisconsid-ered i.e. excluding any wake plasticity. Hence, the
D
K
th
computedin this manner would essentially be effective threshold stressintensity factor range,
D
K
th,eff
. Fig. 3 shows the types of the dislo-cation associated with the respective mode of loading. Mode Iand II cracks emit pure edge dislocations while screw dislocationsemanate from a mode III crack. These dislocations assume equilib-riumpositions at the end of forward and reverse loading, balancedby the lattice resistance stress (for free as well as twin-restrictedglide). Upon cyclic loading, there is a net irreversibility of disloca-tion glide trajectories that leads to crack-tip plastic displacementper cycle.
2.1. Cyclic slip irreversibility causing permanent crack extension
Fig. 4 schematically demonstrates the mechanism of the irre-versibility of crack-emitted dislocations (for the sake of simplicity,no twin is shown). Upon reversing the load, two scenarios arepossible leading to irreversible glide. The crack at the peak of theforwardloading(denotedA) hasatotal of
n
numberof discretedislocations ahead at their equilibrium positions, denoted by
x
i
(from the crack-tip) corresponding to the
i
-th dislocation(
i
=1,2,3,
. . .
,
n
). The crack-tip displacement is a function of net
Table 1
Summary of notable slip-based
D
K
th
models in the literature.
Models Investigators Threshold SchematicSlip emission [3,6,9,10,32]
D
K
th
f
(
s
,
x
,
h
)
s
=criticalstress to nucleate slip
x
=crack-tipto dislocation distance
h
=angle between slip and crack pathSlip blockage [4,8,33]
D
K
th
f
(
s
,
a
o
,
d
)
a
o
=initialcrack length
d
=crack-tip to GBdistanceSlip irreversibility [5,34–38]
D
K
th
f
(
D
s
)
D
s
=forwardand reverseﬂow stress differential
1
For interpretation of color in Fig. 2, the reader is referred to the web version of this article.
P.B. Chowdhury et al./International Journal of Fatigue xxx (2014) xxx–xxx
3
Pleasecitethisarticleinpressas:ChowdhuryPBetal. Predictingfatigueresistanceofnano-twinnedmaterials:PartII–Effectivethresholdstressintensityfactor range. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.06.006
irreversibleBurgersvectorsoveracycle.Apermanentdisplacementcorresponding to one Burgers vector (1b), upon complete reversalof load, is necessary to sustain the crack growth at its minimumrate. Below this level, the crack would cease advancing. Suchscenario could arise by two possibilities at the end of the reverseloading (designated B). Firstly, (
n
1) number of dislocations of opposite sign (labeled ‘negative’ in the follow-up discussion) canemanate and annihilate the (
n
1) crack-bound returning positiveslip. It is noted that the nucleation of a negative dislocation alongthe same slip system from the crack nulliﬁes the step (equal to1b) created due to the prior positive dislocation emission. There-fore, emission of
n
number of positive dislocations in forward loadand subsequently (
n
1) negative slip during reverse load wouldresult in a net cyclic displacement of 1b at the cracktip. Secondscenario involves no negative slip nucleation whatsoever andreturningof(
n
1)positiveslipbacktothecrack-tip.Thissituationalso leaves one dislocation trajectory irreversible (i.e. leading topermanent tip extension corresponding to 1b). The former case ismore likely to occur for no obstacle or obstacles placed at a largedistance away from the crack-tip, allowing for negative slip nucle-ation during reverse shear. On the other hand, the latter situationwouldemergeforthecaseofanobstacleplacedintheclosevicinityof the tip, with stronger shielding effects to forward ﬂow (for bothpositive and negative slip). In the following section, we establishthe force balance of dislocations.
2.2. Force equilibrium during dislocation glide
At any stage of loading, the shear stress components acting onthe
i
-th dislocation are: (a) applied shear stress,
s
applied
(whichdrives the dislocation glide from or toward the crack dependingon forward or reverse load), (b) image stress,
s
image
(whichalways drives the dislocation toward the crack) and (c) pile-upstress,
s
pile-up
(which acts against the glide). Therefore, thenet shear stress
s
i
acting on the
i
-th dislocation could bewritten as follows (where,
i
=1,2,3,
. . .
,
n
;
n
is the total numberof dislocation).
Fig. 2.
A semi-inﬁnite body with a corner crack is set up to conduct fracturemechanics simulations. Ahead of the crack, one (or two) twin of ﬁnite thickness
t
isplaced.Whenfarﬁeldload(modeI,IIorIII)isappliedontheconﬁguration,thecrackemits elastically interacting discrete (mathematical) dislocation. These dislocationsassume equilibrium positions, upon overcoming the necessary glide strength (bothpristine lattice frictional resistance as well as twin boundary-induced). A formu-lation for crack growth rate (
da
/
dN
) could be derived founded on the differences inforward and reverse dislocation positions (as an outcome of irreversibility due toannihilation).
Fig. 3.
Different loading modes (I, II and III) and associated dislocation types areillustrated. Mode I and II cracks emit pure edge dislocation while mode III loadinggenerates pure screw dislocation.
Fig. 4.
In order for the crack to commence advancement, a minimum crack growthrate of one Burgers (1b) vector per cycle is required. If
n
number of dislocationsnucleate and assume positions ahead of the crack tip, two distinct possiblescenarios may lead to a cyclic tip plasticity equivalent to one Burgers vector. Onepossibilityisthe nucleationof (
n
1)numberof negativedislocationsthat meet upwith the returning dislocations and simultaneously nullifying the positive dis-placements created at the cracktip, leaving only 1b (corresponding to obstacleplaced far away). Another outcome is the return of (
n
1) previously nucleatedpositive dislocations to the crack (no annihilation) with 1b of remaining tipdisplacement (corresponding obstacle placed close to the crack). In the currentmodel,
D
K
th,eff
for the crack advance is deﬁned as the
D
K
level corresponding a
da
/
dN
of 1b, upon reaching an obstacle.4
P.B. Chowdhury et al./International Journal of Fatigue xxx (2014) xxx–xxx
Pleasecitethisarticleinpress as:ChowdhuryPBetal. Predictingfatigueresistanceofnano-twinnedmaterials:PartII–Effectivethresholdstressintensityfactor range. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.06.006
s
i
¼
s
applied
s
image
s
pile-up
ð
1
Þ
The
i
-th dislocation starts gliding upon overcoming the lattice fric-tional stress,
s
P
which leads to the following equilibrium conditionas in Eq. (2).
s
i
P
s
P
ð
2
Þ
The values of
s
P
for free and twin boundary-restricted motion(during forward or reverse glide) have been computed in Table 3in Part I. The terms,
s
applied
,
s
image
and
s
pile-up
can be expressed asa function of dislocation positions,
x
i
(where,
i
=1,2,3,
. . .
,
n
).Hence, we have Eq. (3) representing the force balance betweenthe net acting shear stress and the lattice resistance for
i
-thdislocation (
i
=1, 2, 3,
. . .
,
n
) in a shorthand form.
K
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
p
x
i
p
|ﬄﬄﬄ{zﬄﬄﬄ}
applied
A
2
x
i
|{z}
image
A
X
j
¼
n j
–
i
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
j
x
i
s
1
x
j
x
i
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
pile-up
s
P
¼
0
;
ð
i
¼
1
;
2
;
3
;
...
n
Þ ð
3
Þ
where
K
is stressintensityfactor,
l
shearmodulus,
b
Burgersvectormagnitude, and
A
is
l
b
2
p
ð
1
m
Þ
and
l
b
2
p
for edge and screw dislocationrespectively(
t
isPoisson’sratio).Eq.(3)representsasetof
n
numberof equilibriumexpressions for
n
dislocations. Atanyinstantof load-ing, all these equilibrium equations are satisﬁed simultaneously,andsolvedfor
x
i
foralldislocations.Atomisticallycomputed
s
P
levelsdeterminethepositionofeachdislocationunderfreeortwinbound-ary-restricted glide condition. For instance, when a certain disloca-tion, upon gliding away from the crack-tip, reaches the twinboundary (located at a particular distance,
d
), the increasedlatticeresistance
s
P
forforwardﬂowpasttheboundary(fromTable3inPartI)isassignedinEq.(3)forthatparticulardislocation.Similarly,during reverse ﬂow corresponding Peierls stress at the interface isutilized.Net irreversible glide path is decided by the ﬁnal positions atthe end of forward and reverse loading (to be deﬁned as
x
f i
and
x
r i
in the following section). Irreversibility of dislocation glide patheventually causes net plastic displacement at the crack-tip. Hence,inthefollowingsection, wederivethecycliccrackgrowthrateasafunction of the ﬁnal dislocation positions.
2.3. Crack growth rate formulation
Fatiguecrackgrowthrate,
da
/
dN
canbecomputedbythediffer-ences in forward and reverse crack-tip displacement (
u
f
and
u
r
respectively) along the crack propagation path given by Eq. (4)[34].
dadN
¼ð
u
f
u
r
Þ
cos
h
ð
4
Þ
where
h
is the angle between the slip plane and crack growth path.Inthecurrentmodel,ithasbeenconsidered70
formodeIandzerofor mode II and III. The displacements during each half cycle (for-ward,
u
f
and reverse,
u
r
) are given by Eq. (5).
du
¼
12
l
Z
s
dx
ð
5
Þ
In view of the discreteness of dislocations, the integral isreplaced by discrete summation, leading to the following expres-sion of
u
f
and
u
r
(for
h
=0
) as follows.
u
f
¼
x
f
1
2
l
X
i
¼
ni
¼
1
s
f i
ð
6
Þ
u
r
¼
x
f
1
2
l
X
i
¼
ni
¼
1
D
s
r i
ð
7
Þ
Delta (
D
) in Eq. (7) indicates that distance parameters during thereverse loading (i.e. crack displacement and dislocations positions,
x
i
) are computed considering the crack-tip at srcin. From Eqs. (4),(6), and (7), we obtain the formulation for
da
/
dN
as follows.
dadN
¼
x
f
1
2
l
X
i
¼
ni
¼
1
ð
s
f i
D
s
r i
Þ ð
8
Þ
where
x
f
1
is the ﬁnal location of the ﬁrst emitted dislocation (i.e.
i
=1)whichrepresentstheextentofforwardﬂowactivitiesi.e.plas-tic zone size. Rewriting Eq. (8) in terms of full expressions of indi-vidual shear stress terms (i.e.
s
applied
,
s
image
and
s
pile-up
), weobtain the ﬁnal expression for the
da
/
dN
as in Eq. (9). (for
h
=0
).
dadN
¼
x
f
1
D
K
2
l
ﬃﬃﬃﬃﬃﬃﬃ
2
p
p
X
i
¼
ni
¼
1
1
ﬃﬃﬃﬃ ﬃ
x
f i
q
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
f i
x
r i
q 0B@1CA
A x
f
1
4
l
X
i
¼
ni
¼
1
1
x
f i
1
x
f i
x
r i
!
A x
f
1
2
l
X
i
¼
ni
¼
1
X
j
¼
n j
–
i
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
f j
x
f i
!v uut
1
x
f j
x
f i
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
f j
x
r j
x
f i
x
r i
!v uut
1
ð
x
f j
x
r j
Þð
x
f i
x
r i
Þ
0@1A
ð
9
Þ
Eq.(9)canbeusedtoevaluate
da
/
dN
levelsasafunctionofﬁnaldis-location positions at the end of forward and reverse loading (
x
f i
and
x
r i
respectively). It provides a closed form expression for
da
/
dN
as aconsequence of differences in forward and reverse dislocation posi-tions. The crack advances during the simulations and dislocationevolution occurs ahead of the crack-tip. Upon advancing for sub-stantial number of growth cycles, the remaining dislocation arraysfromprecedingcyclesleadtosaturationeffectsintermsofthecracksize (as discussed in the following section).
2.4. Crack growth simulation in presence of nano-twin
As an example, Fig. 5a illustrates the calculation of twin-inﬂu-enced
D
K
th,eff
levels from
da
/
dN
versus
D
K
II
plots for the case of amode II crack (edge dislocation) for Ni. In the inset of Fig. 5a, acrack with initial length
a
o
of 25
l
m advances under a constantapplied farﬁeld stress range
D
r
xy
under a load ratio
R
equal to
1 i.e. under completely reversed shearing. A twin of preselectedlamellar thickness of 35nm is placed en route to the crack propa-gation. The left hand side insets depict dislocation arrangements(at
K
max
and
K
min
) for the advancing crack sufﬁciently far fromthe nearest coherent twin boundary (CTB). At
K
max
, a number of dislocations are emitted and piled-up against the CTB. Some dislo-cations have overcome the CTB resistance for forward ﬂow andpositioned themselves inside the twin at an angle to incident slippath. This angle is conﬁgured corresponding to the one observedin molecular dynamics simulations during slip transmission pasta CTB (for pure edge case) as reported in Part I. As follows fromFig. 5a inset, the farthest transmitted dislocation situated on theother side of the CTB is still far fromthe right hand side CTB. Uponreversal, the dislocations closest to the crack-tip commence theircrack-bound glide. Some of reverse-gliding dislocations return tocrack until dislocations of opposite sign emanate and annihilatethe rest of the crack-bound slip until reaching
K
min
. The inset con-ﬁguration at
K
min
illustrates the dislocation positions at the end of the cycle.The crack continues to grow until it starts decelerating uponapproaching the nearest coherent twin boundary (CTB) on the left.At this interface, the crack growthrate
da
/
dN
reaches its minimumvalue. At the beginning of the simulation, the farﬁeld
D
r
xy
level isappropriately adjustedsoas to reacha
da
/
dN
of one Burgersvector(1b) in the ﬁrst encounter with the CTB. The dislocation conﬁgura-tions (at
K
max
and
K
min
) for the crack in very close proximity to theCTB are illustrated in the right hand side inset of Fig. 5a. At
K
max
,
P.B. Chowdhury et al./International Journal of Fatigue xxx (2014) xxx–xxx
5
Pleasecitethisarticleinpressas:ChowdhuryPBetal. Predictingfatigueresistanceofnano-twinnedmaterials:PartII–Effectivethresholdstressintensityfactor range. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.06.006

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