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Acta Materialia xxx (2013) xxxxxxModeling fatigue crack growth
resistance of nanocrystalline alloys
Piyas B. Chowdhury a, Huseyin Sehitoglu a,, Richard G. Rateick
b, Hans J. Maier c
aDepartment of Mechanical Science and Engineering, University of
Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL
61801, USAbHoneywell Aerospace, 3520 Westmoor Street, South Bend,
IN 46628, USA
c Institut fur Werkstoffkunde (Materials Science), Leibniz
Universitat Hannover, D-30823 Garbsen, Germany
Received 18 November 2012; received in revised form 16 January
2013; accepted 17 January 2013Abstract
The description of fatigue crack growth in metals has remained
an empirical field. To address the physical processes contributing
tocrack advance a model for fatigue crack growth (FCG) has been
developed utilizing a combined atomisticcontinuum approach. In
par-ticular, the model addresses the important topic of the role of
nanoscale coherent twin boundaries (CTB) on FCG. We make the
centralobservation that FCG is governed by the dislocation glide
resistance and the irreversibility of crack tip displacement, both
influenced bythe presence of CTBs. The energy barriers for
dislocation slip under cyclical conditions are calculated as the
glide dislocation approachesa twin boundary and reacts with the
CTB. The atomistically calculated energy barriers provide input to
a mechanics model for disloca-tions gliding in a forward and
reverse manner. This approach allows the irreversibility of
displacement at the crack tip, defined as thedifference between
forward and reverse flow, to be determined. The simulation results
demonstrate that for both refinement of twin thick-ness and a
decrease in crack tip to twin spacing FCG resistance improves, in
agreement with recent experimental findings reported in
theliterature. 2013 Acta Materialia Inc. Published by Elsevier Ltd.
All rights reserved.
Keywords: Fatigue; Nanocrystalline; Nickel; Damage tolerance;
Coherent twin1. Introduction
Current assessment of materials for damage tolerance isbased on
methodologies that were developed more than40 years ago. These
methodologies are empirical and rulebased, such as the well known
ASME Design Code [1]that treats combined fatigue and creep damage.
Today itremains a challenge to predict material degradation
underfatigue loading conditions utilizing scientific
principles.Compared with unidirectional deformation, fatigue
intro-duces irreversibilities that are characteristic of
cyclicaldeformation. These irreversibilities are a strong
functionof the crystal structure, the alloy composition, and
theinterface interactions. Nanocrystalline materials with
twinboundaries [210] have attracted considerable attentionrecently,
and possess combined strengthening attributes1359-6454/$36.00 2013
Acta Materialia Inc. Published by Elsevier Ltd.
Allhttp://dx.doi.org/10.1016/j.actamat.2013.01.030
Corresponding author.E-mail address: [email protected] (H.
Sehitoglu).
Please cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030with higher
ductility. On the other hand, their fatigue dam-age tolerance
characteristics have received less consider-ation, and the present
paper is geared towards building aframework for the modeling of
fatigue crack growth innanomaterials.
A number of studies have elucidated the strengtheningmechanisms
in nanocrystalline materials under monotonicloading conditions
[24,1115]. Fatigue studies of nano-crystalline metals displaying
higher endurance limits [1620] compared with their coarse grained
counterparts havealso been undertaken. Recent works have also
demonstratedsuperior damage tolerance [5,21] in the presence of
nanoscaletwins, hence the prospect of enhanced overall fatigue
resis-tance with combined monotonic strength holds
considerablepromise. In particular, Singh et al. [5] demonstrated
thatintroducing nanotwins with a gradually diminishing lamel-lar
spacing in ultrafine grained (UFG) Cu substantiallyimproved damage
tolerance metrics such as the thresholdstress intensity range DKth
and, most significantly, therights reserved.
gue crack growth resistance of nanocrystalline alloys. Acta
Mater
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2 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxxnear-threshold crack growth rate da/dN. Moreover, studiesby
Sangid et al. [21] on electro-deposited nanocrystallinenickelcobalt
alloys with a high volume fraction of anneal-ing twins in the
grains further corroborated the existence ofsuperior fatigue crack
growth (FCG) impedance. Whilethese studies point to the
significance of coherent bound-aries on FCG behavior, understanding
the mechanistic ori-gin of such microstructure-driven phenomena
necessitatesa detailed study informed by the underlying
atomistics,capturing the operative cyclical crack tip plasticity at
theappropriate length scale. The current paper has developeda model
for subcritical FCG behavior combining atomisticand continuum
considerations in the presence of twinlamellae of nanoscale
thickness and spacing. The advan-tage of the model is that there
are no adjustable parameters(fitting constants) and crack
propagation occurs due to theirreversibility of plastic flow at
crack tips.
A fatigue crack advances because of the irreversibleglide of
dislocations emitted by the crack-tip, the degreeof which dictates
the net plastic displacement per cycle[2230]. Pippan et al. [2729]
showed that crack tip dis-placement under forward and reverse
loading does not can-cel out because of dislocation annihilation,
resulting infatigue crack advance. We note that microstructural
fac-tors that would influence the degree of glide
irreversibilitymust also alter the FCG rates. Specifically,
microstructuralobstacles, such as coherent twin boundaries (CTBs)
andgrain boundaries (GBs), in the neighborhood of an advanc-ing
crack mean that the slip reversibility is difficult to ascer-tain.
The extent of irreversibility imposed by theseobstacles is a
function of the nature of the slipinterfaceFig. 1. Schematics
representing the focus of the investigation in this paper. In
funder mode III loading) which interact with a nanoscale twin.
Slip-coherent twwork studied the isolated role of twin lamellae
width (t) and the crack to twcoherency of the twin boundaries
allows glissile motion of dislocations on thebehavior are
summarized. In addition to t and d, the glide strength, s0 and the
i
Please cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030interactions. At the
same time, the presence of such inter-faces influences the
resistance to slip propagation so (i.e.the difficulty of plastic
flow advancing past the obstacle,manifested as an elevation of the
unstable fault energycus). cus is the maximum fault energy during
slip establishedfrom the block-like motion of an upper surface
relative to alower one. Inevitably, its extrinsic (modified) level
willchange due to the intersection of slip with interfaces.
Theresulting crack growth rate da/dN is related to the slippaths,
residual dislocations, and conservation of theBurgers vectors as
influenced by the twin width andspacing.
Fig. 1 depicts the forward slip emission from an advanc-ing
fatigue crack and its interaction with a CTB. The nat-ure of the
slipCTB interaction is a function of the type ofincident
dislocation (pure edge, pure screw or mixed).Residual dislocations
with a total Burgers vector br arean outcome of these reactions,
which depend on the inter-face orientation and the resolved shear
stresses of theincoming and outgoing slip systems [31,32].
Variations insuch sliptwin reactions would ultimately modify the
glidepath irreversibility. The fatigue crack growth resistance
isexpected to change with the four factors shown in Fig. 1,the
irreversibility (denoted p), the intrinsic stress so relatedto the
gamma surface (Generalized Stacking Fault Energy),and the twin
thickness t and twin spacing d. If the irrevers-ibility p is 0 no
crack growth can occur. We show that theirreversibility is dictated
by the gamma surface differentialupon forward and reverse flow at
the crack tip.
The prevalence of twins, as in the case of the NiCoalloy seen in
the transmission electron microscopy (TEM)orward load an advancing
fatigue crack emits dislocations (pure screw typein boundary (CTB)
interactions dictate the FCG mechanism. The currentin spacing (d)
on the FCG behavior in a single nanotwinned grain. TheCTB, unlike
incoherent GB. Factors that influence fatigue crack growth
rreversibility, p, under cyclical loading influence fatigue
crack growth rates.
gue crack growth resistance of nanocrystalline alloys. Acta
Mater
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Fig. 2. (a) TEM images of Ni1.62 wt.% Co alloy before the
fatigueexperiment. Notice the prevalence of nanotwins. (b)
Post-fatigue TEMimages of Ni1.62 wt.% Co alloy. A high degree of
sliptwin/GBinteraction is noticeable.
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx 3image
of a pre-FCG experimental specimen (Fig. 2a),improves the FCG
resistance to a considerable degree.Fig. 2b demonstrates an
enhanced degree of dislocationpile-up at twin boundaries and GBs,
indicating slip-medi-ated crack tip plasticity as the primary
deformation mech-anism. Intuitively, the implied improvement in FCG
[5,21]points to a special mechanism(s) involved in cyclical slipCTB
interactions. Hence, one needs to establish the under-lying
governing physics that decide dislocation glide, whichis
susceptible to local stress sources (e.g. GB, CTB, residualsessile
dislocations).
The dislocation gliding mechanism depends on thedislocation core
properties. Glissile motion occurs byPlease cite this article in
press as: Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030alternately
rearranging atomic distortion that proceedsvia successive tearing
and forming of atomic bonds sur-rounding the core structure [33].
The driving shear stressfor such motion scales with the activation
energy barriersfor the translational motion onto a close-packed
slip plane.The energetics of dislocation translation lie in the
relativemotions of the core atoms. Alteration of the
dislocationgliding condition, as influenced by nano-obstacles and
therespective energetics, necessitates a non-continuum model-ing
framework. In that regard, molecular dynamics (MD)allows the
capture and quantification of the physics of slip-ping at the
atomistic length scale. MD simulates the timeevolution of atomic
nuclei (considered as classical Newto-nian particles) by
integrating their equations of motion[34]. The metallic bonding is
modeled through a homoge-neously distributed electron cloud
functional and a pair-wise interaction potential. A semi-empirical
embeddedatom method (EAM) formalism, curve fitted with
experi-mental and/or ab initio material properties, employs
suchmodeling to accurately describe the bonding energy land-scape
[35]. Utilizing MD simulations with an EAM poten-tial Ezaz et al.
[31] quantified the energetics of dislocationglide upon interaction
with twins under monotonic condi-tions. In the literature some
researchers [3638] haveemployed MD-EAMmethods to study massive
cyclical slipemissions leading to nanovoid coalescence as the
crackadvancing mechanism in the presence of GBs. However,the
physics of slip irreversibility accumulation, as theunderlying
incentive for crack tip plasticity, has not yetbeen explored.
In our approach we employ atomistic simulations toreveal the
nature of sliptwin interaction under cyclicalconditions, and the
underlying fault energy barriers. Sucha perspective reveals the
exact role of CTBs as irreversibil-ity-inducing microstructural
elements as well as effectivebarriers to cyclic slip.
Quantification of the cyclical sliptwin reaction energetics allows
the calculation of idealshear stresses for to and fro glide, as
modified by the pres-ence of CTBs and/or residual dislocations. We
incorporatethese atomistically extracted material properties in
fracturemechanics-based formulations to simulate FCG undergo-ing
large scale slip activities. The mechanics simulationsemploy
cyclical irreversibility as the principal driving forceof crack
advancement in the presence of nanotwins. Thecombination of two
different length scale methodologiesis important, in that the
continuum descriptions of FCGutilize input from the governing
atomistic physics. Hencewe obtain an in-depth understanding of FCG
as influencedby nanotwins. Such an insight highlights the role of
somecritical characteristic dimensions associated with
thesenano-obstacles (e.g. twin lamellar thickness and twin tocrack
tip spacing) on the FCG metrics.
2. Methods
To develop a FCG methodology we employed bothatomistic slip-twin
and fracture mechanics-based crack-tipgue crack growth resistance
of nanocrystalline alloys. Acta Mater
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4 P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxxslip
simulations. An open source software LAMMPS(large-scale
atomic/molecular massively parallel simulator)[39] was used to
perform the MD simulations. A semi-infinite discrete dislocation
simulation set-up was thenestablished using input from the MD
results. The combina-tion of these simulations provides a
convenient conduit toexplore some fundamental aspects of the
mechanism ofFCG.
For the MD simulations a nickel single crystal grain
wasconstructed with the crystallographic orientation shown inthe
inset in Fig. 3. This grain contains a coherent twin offinite
thickness. A stress concentrator (atom size void)placed in the
matrix simulates a dislocation source. Thewhole system was
energetically minimized, using the conju-gate gradient (CG)
algorithm [34]. This resulted in an ener-getically stable single
nanotwinned grain. The CGalgorithm iteratively solves atomic
coordinates to reachthe minimum energy of the system within a
predefined con-vergence limit. An acceptance criterion adjusts the
newatomic positions, conjugate to the previous ones that fol-low
the direction of steepest descent on the potential energycurve.
Moreover, enforcement of three-dimensional peri-odic boundary
conditions on the supercell eliminates theeffects of free surface
energy, thereby simulating a systemof bulk material. The supercell
dimensions were configuredsuch that the physical observables (e.g.
temperature, pres-sure, kinetics and potential energy of the
system) convergedto system size independence. In view of the goals
of thepresent work we conducted a number of MD simulationswith
varying source to twin distances as well as twin lamel-lar widths.
Consequently, the supercell size was variedaccordingly to give the
optimally converged dimensionsfor each simulation, avoiding any
artifacts of periodicity.
Cyclical shear was applied to the supercell under straincontrol
conditions. A strain range of emin = 4.46% toemax = 9.22% (i.e. Re
= 0.48) was selected to facilitate slipnucleation from the void
(slip source), and sufficient plasticFig. 3. Cyclical stressstrain
response of a nanotwinned grain with adislocation source (not
shown) in the matrix in the vicinity of the coherenttwin boundary
as obtained by MD simulations. The configuration aboveproduces pure
screw dislocations.
Please cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030flow to provide to
and fro glissile motion across the twin.The MD simulations were run
for a duration of severalhundred picoseconds. Such a timescale is
inherent in MDsimulations, limited by the computational capability.
Ourinvestigation required the calculation of parameters suchas the
local plastic shear strain due to slip, the Burgers vec-tors
thereof, and the energetics of sliptwin reactions.These parameters
are unaffected by the high deformationrates arising from such a
timescale. In order to conductnon-equilibrium MD simulations (i.e.
evolution of the sys-tem under the imposed conditions) we employed
an iso-baricisothermal (NPT) ensemble along with a NoseHoover
thermostat algorithm. Hence, the total number ofatoms N, the
external pressure P, and the temperature T(at 10 K) of the system
were held constant. The dynamicsof deformation proceeded utilizing
the velocity Verlet algo-rithm as the time integrator. Atomistic
snapshots at differ-ent time points were carefully analyzed using
visualmolecular dynamics (VMD) [40] and AtomEye configura-tion
viewer [41]. These visualization tools, combined within-house
MATLAB programs, helped capture the detailsof sliptwin interactions
(e.g. the conservation of Burgersvectors) and calculate fault
energies, glide distance of slip,etc. Volume-averaged virial stress
formulation, neglectingthe kinetic energy contribution [42], was
employed in orderto quantitatively assess the stressstrain response
of thesystem.
One essential part of our investigation was to calculatethe
energetics of complex cyclical sliptwin reactions,necessitating
accurate descriptions of the atomic levelenergy landscape through
EAM formulations [35]. A com-parative study of EAM potentials
available in the literaturedemonstrated that the Foiles and Hoyt
potential [43] pro-vides good agreement between the unstable fault
energycus, the density functional theory (DFT) calculations(254 mJ
m2), and the intrinsic stacking fault energy (cisf)with the
experimental finding (127 mJ m2). The interpla-nar potential energy
profiling incorporating all of theseparameters is termed the
generalized stacking fault energy(GSFE) curve [44], as shown in
Fig. 6. The GSFE repre-sents the energy pathway to create the
lattice distortionof a dislocation along the Burgers vector
direction. A typ-ical GSFE is calculated by sliding one crystalline
half-spaceon top of another on the slip plane along the slip
direction.We utilized the Foiles and Hoyt EAM potential to com-pute
the modified GSFE, as influenced by local stressesduring back and
forth dislocation glide traversing the twin(to be discussed in
detail later). For a more thoroughdescription of the MD simulation
procedures employedin the present work readers are referred to Ezaz
et al. [31].
3. Results
3.1. Molecular dynamics simulations
Fig. 3 shows a typical MD-based cyclical shear stressstrain
response of a nanotwinned grain with a dislocationgue crack growth
resistance of nanocrystalline alloys. Acta Mater
-
Fig. 4. (a) Steady-state cyclical sliptwin interaction for the
forward part of a MD fatigue cycle (for visualization convenience
perfect lattice atoms arerendered invisible and only defect atoms
are shown). The dissociated leading and trailing partials emitted
from the source (not shown) are approaching theclosest CTB. All
vectors (in the matrix and/or twin) are represented in their
respective coordinate frames. (b) Two partials recombining upon
interactingwith a CTB. The emerging leading Shockley partial (pink)
and incident trailing Shockley partial (brown) are shown. The
situation shown depicts themetastable phase of the sliptwin
reactions. More dislocations subsequently nucleate. (c) Nucleation
of multiple dislocations as a result of the incidentdislocationCTB
interaction. (d) A simplified double Thompson tetrahedron depiction
of the dislocation reactions during forward flow at a CTB
depictingconservation of the Burgers vectors. (For interpretation
of the references to colour in this figure legend, the reader is
referred to the web version of thisarticle.)
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx
5source located in the matrix. Cyclical deformation isapplied to
the extent that it facilitates dislocation nucle-ation, and with a
sufficient degree of slip to intersect thetwin, located at a
distance d from the source. The strainrange for subsequent cyclical
loading is set up such thatto and fro dislocation motions occur
across the width tof the twin. Separate MD simulations were carried
out withvarying finite twin lamellar widths and crack to twin
spac-ings to study the influences of these dimensions. With
agradual increase in t or d the external loading needs to
beincreased in order for the slip to reach and traverse theentire
width of the nanotwin. Consequently, a graduallygreater number of
dislocations nucleate with the increasein the applied load. The
multitudes of dislocations undergorelatively more complex forms of
interactions with theCTBs at larger t and/or d. Nevertheless, we
observed a gen-eralized pattern of cyclical sliptwin interaction
with simul-taneous incorporation and transmission of slip for all
casesPlease cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030of varying t and/or
d. Hence the fundamental similaritiesreside in the type of
interaction and the introduction ofirreversible slip activity in
each cycle, irrespective of thenumber of incident dislocations. In
the following sectionthe cyclical sliptwin reaction involving only
one incidentdislocation (pure screw type) is described in detail
for thecase of smaller t and d (requiring a lower applied
load).
As can be seen in Fig. 3, the stressstrain approaches asaturated
response as the cyclical sliptwin interactionmechanism also
achieves a recurrent steady-state. Sincethe MD simulations were
performed on pristine crystals,and at high deformation rates, the
stresses from the MDare high compared with the experimental
stressstrainresponse. However, the dislocation reactions
associatedwith the sliptwin interactions are unaffected, as
verifiedby running simulations at different strain rates.
Therefore,the cyclical stressstrain plots in Fig. 3 are interpreted
onlyto obtain a quantitative estimation of the
quasi-steady-stategue crack growth resistance of nanocrystalline
alloys. Acta Mater
-
Fig. 5. (a) Schematic showing calculation of the slip
irreversibility p. c represents shear strains due to dislocation
glide. (b) Slip irreversibility calculatedfrom MD simulations as a
function of crack tip to twin spacing t with constant d = 80 nm.
(c) Slip irreversibility calculated from MD simulations as
afunction of the crack tip to twin spacing d with constant t = 80
nm.
Fig. 6. Generalized stacking fault energy (GSFE) utilized in our
study forpure nickel, calculated using the EAM potential developed
by Foyles andHoyt [43].
6 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxxresponse of the system under investigation. The next sec-tion
describes a detailed study of the CTBdislocationinteractions to
clarify the reaction type.Please cite this article in press as:
Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.0303.1.1. Cyclical
sliptwin interactions
MD simulations revealed the exact nature of the steady-state
cyclical sliptwin reactions. During forward loading,after elastic
straining, a perfect screw dislocation of Bur-gers vector a
21 1 0 nucleates from the source on the most
favorable slip system with the maximum resolved shearstress.
Immediately after emission the perfect dislocationdissociates into
two Shockley partials (leading and trailing)separated by a ribbon
of stacking fault (Eq. (1)).
a21 1 0Full
! a62 1 1Leading
a61 2 1Trailing
Stacking Fault 1
Under continued application of forward shear loadingthe extended
dislocation (with a leading and a trailingShockley partial) glides
towards the CTB (Fig. 4a). Beingobstructed by the CTB, the two
partials recombine andgenerate new screw dislocations at the site
of incidence(Fig. 4b). The new dislocations similarly dissociate
intoShockley partials. One of the new dislocations is incorpo-rated
into the CTB as twinning partials, and another(extended) is
transmitted inside the twin (Fig. 4c). Eq. (2)summarizes this
reaction. Transmitted partials a
62 1 1T
and a61 1 2T become a6 2 1 1T and a6 1 1 2T, respectively,gue
crack growth resistance of nanocrystalline alloys. Acta Mater
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P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx 7in
the matrix frame. The total Burgers vector is then con-served on
both sides of Eq. (2). Fig. 4d provides a doubleThompsons
tetrahedron depiction of the forward sliptwin reaction.
2
Until the end of the forward loading cycle the twinningpartials
on the CTB carry on gliding in opposite directions,gradually
increasing their separation distance in order tominimize the
elastic strain energy. Because of the glidingof these twinning
partials the twin boundary migratesone atomic layer. The
transmitted partials inside the twincontinue to glide. As the load
is reversed (unloading) thepreviously transmitted leading and
trailing partials (insidethe twin) now reverse their directions of
motion (upon elas-tic relaxation). The returning extended
dislocation interactswith the CTB and undergoes similar
multiplication,thereby creating two new twinning partials on the
CTBand two Shockley partials in the matrix. Eq. (3) describesthe
reverse reaction. The total Burgers vector (upon con-version of
twin dislocations to the matrix frame) on bothsides is
conserved.3Table 1Summary of cyclical steady-state sliptwin
interaction.
Schmid factors (sRSS=s) Sliptwin interaction(s) Inc
Incident CTB Outgoing bs
0.778 1.0 0.778 Transmission, incorporation a2 1
Please cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030With further
unloading the twinning partials continue toglide in opposite
directions, eventually causing twin migra-tion by one atomic layer.
The twin migration process mayinvolve growth or shrinkage of the
twin depending on thedirection of motion of the participant
twinning partials.The matrix Shockley partials continue gliding
towardsthe source until a full dislocation (which also
dissociatesto become extended) of opposite sign nucleates from
thesource, meets with the returning one, and they annihilateeach
other. At the end of unloading another new negativedislocation
(considering the original nucleated slip, at thebeginning of
forward loading, to be of positive type) nucle-ates and glides
towards the twin, repeating the mechanismover subsequent
cycles.
Table 1 summarizes the sliptwin reactions. Here bs, beand br
refer to the screw, edge component, and residual dis-location on
the CTB, respectively. In summary, the reac-tion process involves
transmission of unobstructed slippast the CTB (designated
outgoing), and incorporation ofslip with br in the CTB. The full
dislocations are of purescrew type (which dissociate into partials)
for both the inci-dent and outgoing systems. For all the active
slip systemsthe resolved shear stress sRSS under global applied
stress sis calculated using the formulation:
sRSS rijminj 4In Eq. (4) rij is the remote stress tensor, mi the
slip plane
normal vector, and ni the vector representing the slip
direc-tion. For our case the applied rij is reduced to s13 (s
inFig. 3). As can be seen in Table 1, the magnitudes of theratio of
sRSS to s, defined as the Schmid factor (SF), forthe active slip
systems are fairly high. The maximum SFis operative on the CTB,
which facilitates the incorporationof glissile twinning partials.
The next largest SF acting onthe outgoing slip system inside the
twin assists in the trans-mission of slip past the CTB. The
analyses, as summarizedin Table 1 and Fig. 4, concern forward flow
of slip past theCTB. However, the reverse reaction is modified only
in theform of enhanced resistance (due to the presence of twin-ning
partials in close vicinity) at the CTB. The reverse inci-dence of
slip upon the CTB results in similar interactionproducts from
dislocations at the reaction site. For morecomplex cases involving
a multitude of incident disloca-tions (at larger t and/or d with a
higher applied load) theinteraction type remains fundamentally
identical, undergo-ing simultaneous incorporation and transmission
of slip.The subsequent sections address quantification of the
slipirreversibility and the origin of the discrepancy of forwardvs.
reverse slip resistance at the CTB.ident slip (matrix) Outgoing
slip (twin) Residual slip (CTB)
be bs be br
1 0 0 a2 1 1 0T 0 a2 0 1 1
gue crack growth resistance of nanocrystalline alloys. Acta
Mater
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8 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxx3.1.2. Cyclical slip irreversibilities
Fig. 5a shows a simple example of how irreversible dis-location
glide during cyclical loading can be quantified,considering the
case of a single incident dislocation (atsmaller t or d).
Steady-state cyclical sliptwin interactioninvolves the incidence of
a dissociated full dislocation(screw) on the CTB. The reaction
results in the simulta-neous incorporation and transmission of
extended disloca-tions. The final locations of these dislocations
are atpositions c and e (partials) and d (extended full
disloca-tion, shown as full for simplicity) at the end of
forwardloading. The partials at c and e contribute to migrationof
the twin by one atomic layer. During reverse flow theextended
dislocation at position d is transmitted back intothe matrix, again
leaving new partials at c0 and e0, whichrepulse the partials at c
and e. The returning crack-boundextended dislocation is annihilated
by another incomingdislocation of opposite sign (negative) at
location f. Wecalculated the ratio p between the irreversible
plastic shearstrain cirr and the total plastic shear strain ctotal
generatedby dislocation glide over a cycle using Eq. (5). These c
val-ues represent shear strains due to dislocation glide as
theyappear in Eq. (5). Even though Fig. 5a shows the
cyclicalprocess involving only one incident dislocation, on
gradu-ally increasing t and/or d a number of dislocations
willcontribute to the overall irreversible phenomena in anidentical
manner. The parameter p is then computed asa varying function of t
and d. Fig. 5b and c demonstratesthe results.
p cirrctotal
cce cc0e0 cafcad cdb cce cc0e0 cbf
5
The trend for cyclical crack tip slip irreversibilities p,
ascalculated in Fig. 5b and c, tends to become independentof t and
d at sufficiently large magnitudes. At lower valuesof t and d a
relatively small number of dislocations areemitted from the source
and traverse the entire thicknessof the twin. Consequently, a
shorter spacing between thesource and the twin as well as thinner
twins will expeditethe return of source-bound positive slip, and at
the sametime preclude gliding of negative slip sufficiently
fartheraway from the source. Thus the annihilation process(location
f in Fig. 5a) occurs in very close proximity tothe source (as in
the insets in Fig. 5b and c). As a result,the magnitude of p is low
in the small t and/or d regime,as calculated with Eq. (5). However,
an increase in t or dnecessitates a larger applied load in order
for slip to occurand traverse the twin. This results in a gradually
greaternumber of dislocation emissions. The involvement of
mul-titudes of dislocations leads to an increase in the
pile-upstress during both forward and reverse flow. This resultsin
even greater blockage of returning positive dislocations,thereby
permitting unobstructed negative slip to travelfurther away from
the source. As a result, the annihilationprocess occurs at a
greater distance from the source (asindicated in the insets). Due
to the involvement ofPlease cite this article in press as:
Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030large-scale slip
activity for even higher t or d the annihi-lation point eventually
settles at a stable unvarying posi-tion, corresponding to the
plateau region in Fig. 5b and c.
The underlying origin of glide irreversibilities as influ-enced
by CTBs can be traced back to the discrepancies inthe energy
pathways for slip for forward and reverse trans-mission across
CTBs. Such a phenomenon modifies theresistance of slip penetrating
the CTBs under cyclical con-ditions, as further discussed
below.
3.1.3. Energy barrier and ideal glide strength
The potential energy variationdisplacement relation-ship of a
pair of partial dislocations (an extended full dis-location
separated by a stacking fault) in an otherwiseperfect fcc crystal
is described by the GSFE, as in Fig. 6.During glissile slip motion
sliding of atomic planes occursby overcoming the unstable fault
energy cus (inset inFig. 6). The unmodified GSFE curve points to
the resis-tance of dislocation glide scaling with cus, as imposed
bythe crystal. A Shockley partial dislocation glides on the(1 1 1)
plane with the Burgers vector along the h1 1 2i direc-tion. The
modified GSFE takes the form of an increase incus due to the
presence of the CTB (or any other local stresssources), as shown in
Fig. 7a. In order to compare the rel-ative resistance encountered
during forward and reverseflow the modified cus in the vicinity of
a CTB is calculated(Fig. 7c).
With a view to estimating the resistance stress providedby a CTB
to the approaching slip for back and forth trans-mission we
calculated the modified GSFE in the vicinity ofthe CTB using a
dynamic approach. Considering thedynamic nature of dislocation
glide over time, computingthe variation in the potential energy
difference in somepreselected atoms allows quantification of the
modifiedGSFE (discussed in detail in Appendix A). The cus
valuesderived from these modified energy curves are plotted asa
function of distance normal to the CTB in Fig. 7c. InFig. 7b the
dislocation at A is approaching the CTB butis still unaffected by
the stresses resulting from thematrixtwin interfacial atomic
mismatch. Thus cus at Adenotes the energy barrier that a
dislocation has to over-come when it is gliding freely inside the
crystal. The magni-tude of cus at A matches the peak in Fig. 6,
whichrepresents the energy barrier to unobstructed
gliding,amounting to 254 mJ m2. The local stress generated dueto
atomic mismatch at and around the CTB elevates cusonce the
approaching dislocation is in closer proximity.The maximum energy
barrier that the incident dislocationneeds to overcome is achieved
when the slip interacts withthe CTB, an intermediate step in
formation of the finalreaction products (Fig. 7b and c point B).
The elevatedenergy at point B corresponds to a cus value as high
as340 mJ m2. Therefore the energy path A! B ! C (redcurve)
describes the variation in cus for transmitted disloca-tions during
forward flow. In the course of reverse flow thereturning
dislocation encounters an even greater energybarrier due to the
presence of the dissociated twinninggue crack growth resistance of
nanocrystalline alloys. Acta Mater
-
Fig. 7. (a) Schematic demonstrating the expected increase in
energy barrier (cus) due to the presence of a local stress source.
The dark line depicts theplanar fault energy for dislocation glide
through a perfect crystal, while the green line represents the
enhanced energy encountered in the presence of aCTB. The maximum
slope of the (un)modified GSFE equals the ideal shear stress of the
crystal smax. Ezaz et al. [31] extensively explored the
contributionof local stresses to the fault energetics of sliptwin
interactions. (b) As a pair of Shockley partials approaches the CTB
the energy barrier (cus) is elevated inthe neighborhood of the
twinmatrix interface. cus is maximum at the CTB (position B). At C
the cus is the same as at A. Upon interaction with a CTB apair of
Shockley partials is left on the CTB, while another pair transmits
into the twin. As the transmitted pair glides away from the
interface cus decreasesto the level of a perfect lattice barrier.
During the reverse transmission upon flipping of loading the
returning dislocation encounters enhanced cus due tothe presence of
the Shockley partials on the CTB (at point D). As the returning
extended dislocation is transmitted back into the matrix it
undergoes asimilar multiplication, leaving another pair of twinning
partials on the CTB. Schematic of load cycles in MD simulations
(strain control) showing where A,B, C, D and A0 occur. (c) Change
in unstable energy cus as a pair of partial dislocations approach a
CTB during forward/reverse loading in the MD fatiguecycle. Path A!
B! C provides an energy barrier against forward transmission (red
curve), and C !D ! A0 against reverse transmission. The energy atD
is greater than at B because of the presence of dissociated
Shockley partials on the CTB during reverse loading. (For
interpretation of the references tocolor in this figure legend, the
reader is referred to the web version of this article.)
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx 9
Please cite this article in press as: Chowdhury PB et al.
Modeling fatigue crack growth resistance of nanocrystalline alloys.
Acta Mater(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030
-
Fig. 8. The set-up for dislocation dynamics simulations. A mode
III crackemits a series of screw dislocations that glide away to
interact with ananotwin of finite lamellar width placed at a finite
distance. Positivedislocations assume equilibrium positions xfn and
x
rn during forward and
reverse loading, respectively. Negative dislocations nucleate
during thereverse half cycle, and eventually annihilate returning
positivedislocations.
10 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxxpartials on the CTB. Therefore the reverse energy
barrierfollows the path C! D ! A0 (blue curve). The cus maxi-mum
reaches a magnitude of 452 mJ m2 at point D. Theelevation of the
reverse transmission energy barrier com-pared with the forward
barrier can be attributed to theincrease in local stress around the
CTB due to the residualtwinning partials. The GSFE for slip glide,
as modified inthe above mentioned manner, facilitates calculation
ofthe ideal shear strength of the crystal smax. smax is a func-tion
of cus, and is calculated from the maximum slope ofthe modified or
unmodified GSFE curve (Eq. (6)). Apply-ing corrections for thermal
activation and strain rate tothe plastic flow the ideal critical
glide strength so at roomtemperature and a typical experimental
strain rate can becalculated. Eq. (7) implies that so is also a
function of strainrate _e, temperature (T), and activation volume
(V). Theprocedure for obtaining so is detailed in Appendix B.
smax smaxcus @c@x
max
6
so sosmax; _e; T ; V 7Below we investigate continuum slip
emissions from a
fatigue crack whose glide paths become irreversible uponcyclical
loading. Atomistically computed so values are thenutilized to
characterize the continuum level dislocationglide, and subsequent
fracture mechanics simulations.
3.2. Continuum dislocation simulations
We modeled a pre-existing mode III fatigue crack in thepresence
of a nanotwin. The crack emits a series of screwdislocations. These
dislocations intersect the twin and,eventually, their cyclical
glide paths become irreversiblevia annihilation. The slip glide
resistance (due to latticefriction and penetrating twins)
influences the equilibriumpositions and the total glide path
irreversibility. Cracksadvance by accumulating plastic displacement
at the tip,originating from the irreversibility of cyclical slip.
For agiven crack length (a), twin thickness (t) and twin
positionfrom the crack tip (d) and stress intensity levels (DKIII)
wecan predict the corresponding values of da/dN. da/dN isexpressed
as a function of the equilibrium positions of dis-crete
dislocations at a certain DKIII.
3.2.1. Fracture mechanics calculationsIn the continuum model we
selected a mode III fatigue
crack and the associated emission of pure screw disloca-tions
(Fig. 8). The MD derived glide strengths for a screwdislocation
were utilized. The continuum dislocations canovercome the glide
resistances under the applied externalload, and eventually assume
equilibrium positions. At theequilibrium position a dislocation
emitted from a cracktip experiences three forces: (1) resolved
applied shearstress sApplied; (2) image stress sImage; (3) pile-up
stresssPile-up. With the nucleation of new dislocations the
localstress at the crack tip decreases due to enhanced imageand
pile-up stress. In order to compensate for this decreasePlease cite
this article in press as: Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030the applied load has
to be increased to facilitate furthernucleation. Thus Eq. (8)
summarizes the net shear stresssn acting on the nth
dislocation.
sn sApplied sImage sPile-up 8sn is formulated in Eq. (9).
sn KIIIffiffiffiffiffiffiffiffiffi2pxn
pApplied
lb4pxnImage
lb2pxn
Xin
ffiffiffiffiffiffiffiffiffiffiffixixn
s1
xi xnPile-up
9
Eq. (9) gives sn as a function of KIII, the applied globalstress
intensity factor for mode III loading, l, the shearmodulus, b, the
Burgers vector, and xn, the location ofthe nth dislocation along
its glide path from the source(crack tip). sn ought to be of
sufficiently large magnitudein order for slip to overcome the
unstable energy barrier(cus). Therefore with increasing global
applied loading snneeds to surpass and/or equal so to initiate
glide. ThusEq. (10) provides the conditions for gliding, which is
fur-ther rearranged to formulate Eq. (11).
sn P so 10
KIIIffiffiffiffiffiffiffiffiffi2pxn
p lb4pxn
lb2pxn
Xin
ffiffiffiffiffiffiffiffiffiffiffixixn
s1
xi xn so 0 11
Eq. (11) provides the equilibrium conditions for disloca-tions.
The final equilibrium positions (xi) of all dislocationsduring both
forward and reverse loading (at maximumKIII) were solved from Eq.
(11). These xi values were uti-lized as the input for a FCG rate
formulation (discussedlater in Eq. (15)).
In order to clarify the procedure for the mechanics-based
simulations let us consider a simple case consistingof a very low
applied DKIII such that cyclical crack tip plas-ticity involves
only one discrete dislocation (designated 1 ingue crack growth
resistance of nanocrystalline alloys. Acta Mater
-
Fig. 9. A single dislocation demonstrating the slip trajectory
during da/dN calculations. A mode III crack emits a screw
dislocation (designated 1) duringforward loading which assumes an
equilibrium position at maximum forward load (trajectory shown in
red). During reverse loading (blue trajectory), afterelastic
relaxation the dislocation starts to return, and is annihilated by
a newly nucleated dislocation of opposite sign (designated 2). (For
interpretation ofthe references to color in this figure legend, the
reader is referred to the web version of this article.)
Fig. 10. Irreversibility of crack tip emitted dislocation
activities for threecases of finite nanotwin lamellar spacings.
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx 11Fig.
9) with no obstacle (twin) in the glide path. At a cer-tain time
point in the loading cycle the forward xf1 andreverse xr1
equilibrium positions of the disloction aresolved by setting the
lattice friction stress equal to theapplied resolved shear stress.
In the forward half-cycle asthe applied load is increased the
dislocation continues toglide away until the maximum KIII is
reached. Fig. 9 dem-onstrates the trajectory for the case of a
single dislocationduring forward/reverse loading. In forward
loading a dislo-cation nucleates at a critical KIII value and then
glides awayto assume its final position (red curve). During
reverseloading the dislocation does not immediately start to
returntowards the crack tip because of elastic strain recovery.
Asthe shear stress in the reverse direction exceeds the
latticefriction resistance it starts to glide towards the crack
tipand eventually is annihilated by a newly nucleated
negativedislocation. Continued reverse loading triggers the
nucle-ation of another negative dislocation which repeats
themechanism over another cycle. This simplistic demonstra-tion of
the irreversiblity of a discrete dislocation glide pathover a
fatigue cycle elucidates the fundamental procedureof the
continuum-based simulations.
The introduction of nanoscale twins on the glide path ofslip
modifies the total irreversibility as well as slip obstruc-tion by
the crack. Eq. (12) provides an evaluation of theslip
irreversibility parameter p (previously defined as theration
between cirr and ctotal) as a function of the disloca-tion
positions at equilibrium during forward and reverseflow (denoted by
the superscripts f and r, respectively).
p cirrctotal
Xni1
xri2xfi xri
12
Fig. 10 shows the evolution of p as evaluated with thespecified
values of t at constant d, with a change in theapplied stress
intensity factor range. p increases non-Please cite this article in
press as: Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030linearly (square
root trend) with DKIII, and eventuallyachieves a plateau. The
computed tendency of p highlightsthe functional dependence of
irreversible glide phenomena,and hence the crack tip plasticity, on
the variation in twinthickness t on the nanoscale. This is
consistent with theMD calculations elucidated earlier in Fig. 5b
and c. Inresponse to changes in the twin lamellar width the
equilib-rium positions of dislocations change accordingly.
Thiswould lead to different degrees of irreversibility in the
over-all dislocation glide paths, as implied through Eq. (12).These
results points to a change in FCG rate as a functionof t or d. In
order to further explore the t and/or d depen-dence of FCG a
comparison of da/dN under such condi-tions was evaluated.gue crack
growth resistance of nanocrystalline alloys. Acta Mater
-
Fig. 11. Determination of the critical matrixtwin interface zone
size (2q).
Fig. 12. (b) da/dN vs. DKIII plots demonstrating the influence
of (a) twinwidth and (b) crack to twin distance on FCG and the
threshold properties.
12 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxx3.2.2. FCG simulations
A cyclical crack accumulates plastic displacement on
anincremental basis. If there are n dislocations emitted fromthe
crack tip the plastic displacement at the tip caused byeach
emission contributes to the overall crack extension.Based on the
formalisms introduced earlier the rate ofcrack tip advancement per
cycle can be formulated as:
dadN
Z xfmax0
du 13
dadN
xfmax
2l
Xni1
sfn Dsn 14
In Eq. (13) xfmax is the maximum distance away from thecrack
during forward loading traveled by the farthest dislo-cation, u is
the crack tip displacement as a function of xiand l is the shear
modulus in the slip direction, sfn is theshear stress at the end of
the forward half-cycle (a functionof xfi ), and Dsn is associated
with the distance (xi) traveledby the returning crack bound
dislocations (a function ofxfi xri ). Combination of Eqs. (9) and
(14) leads to theda/dN formulation given in Eq. (15). In this
formalismda/dN is essentially a function of the equilibrium
disloca-tion positions. The solutions for these slip locations
(xi)at the maximum KIII, as obtained from the equilibriumcondition
(Eq. (11)) during forward/reverse flow, provideinput to the da/dN
evaluation. Calculation of da/dN in thismanner inherently
incorporates the atomistically computedideal glide stress as well
as the twin penetration strength.
dadN
xfmaxDKIII2l
ffiffiffiffiffiffi2p
pXni1
1ffiffiffiffixfi
p 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffixfi xri
p !
xfmaxb8p
Xni1
1
xfi 1xfi xri
xfmaxb4p
Xni1
Xji
ffiffiffiffiffiffiffiffiffiffiffixfjxfi
s1
xfj xfi
0@
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixfj
xrjxfi xri
s1
xfj xrj
xfi xri
1A 15
We employed the da/dN formulations from Eq. (15)
toquantitatively investigate the sensitivity of the slip
blockingstrength of nanotwins against FCG. Fig. 11 demonstratesthat
for a twin placed at a large distance d from the cracktip (of the
order of 1000 the Burgers vector) FCGbecomes totally independent of
the resistance of the inter-face to the interacting slip. Thus such
a calculation pro-vides information regarding the critical zone of
influenceof the CTB. As shown in Fig. 11, the critical interface
influ-ence size resides in the plateaux regions for which da/dN
isindependent of the slip-blocking strength at small d values.The
CTB influence zone ranges across a few Burgers vectorvalues,
consistent with the MD findings (Fig. 7c). Determi-nation of the
neutral CTB influence size assists in the calcu-lation of FCG as a
function of t and d.Please cite this article in press as: Chowdhury
PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030Fig. 12a and b shows
how the FCG metrics change withvariations in t and d, as predicted
earlier in the evaluationof cyclical slip irreversibilities in Fig.
10. These plots of da/dN vs. the applied DKIII reveal a t and d
dependence of therate of FCG as well as the threshold behavior. For
smallert and/or d FCG is significantly influenced by the
nanotwingue crack growth resistance of nanocrystalline alloys. Acta
Mater
-
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx
13dimensions. Thinner twins as well as twins placed veryclose to
the crack tip lead to enhanced FCG metrics.4. Discussion
4.1. Cyclical sliptwin interactions and irreversibilities
Earlier works have pointed out the role of the localstress state
and the type of dislocation (screw, edge ormixed) in dictating the
nature of dislocationCTB interac-tions [31,45]. In the present work
we have examined thecyclical sliptwin interaction mechanism for
fatigue cracktip-nucleated dislocations. Table 1 summarizes
theobserved dislocationCTB reactions during forward flow.Our case
consists of pure screw incident dislocations thatglide and
intersect with the CTB. In the literature the inci-dence of a screw
dislocation upon a coherent twin report-edly activates one of two
mechanisms [7,46]: (i)incorporation (absorption) of the incident
dislocation fol-lowed by dissociation into Shockley partials on the
CTB;(ii) direct transmission of the incident dislocation throughthe
twin. We observed that the incident dislocation causesnucleation of
dislocations at the reaction site, as describedin Fig. 4c. As
listed in Table 1, the SFs on the CTB andoutgoing system are 1.0
and 0.778, respectively, which aresufficiently high to facilitate
the reaction observed. Thisreaction consists of simultaneous
incorporation and trans-mission of glissile dislocations on and
across the CTB, asshown in Fig. 4c. As clarified in the recent
literature, block-ing of slip by CTBs (incorporation) promotes to a
consid-erable extent macroscopic ductility, by allowing
dislocationglide along the interface, unlike incoherent GBs [7].
Thisspecial feature of CTBs, along with permitting transmis-sion,
results in an enhancement of the macroscopicmechanical properties.
Sangid et al. [21] ascribed the bal-ance of macroscopic strength
and ductility of nanotwinnedmaterials to the superior FCG metrics.
As per our observa-tions, the types of sliptwin interactions which
promoteboth strength and ductility on the micro-scale are
operativeduring to and fro glissile motions of slip across
nanotwins.
Farkas et al. [36], Nishimura and Miyazaki [37] and Pot-irniche
et al. [38] studied the mechanism of fatigue crackadvancement
through the nucleation and coalescence ofnanovoids formed due to
crack tip slip activity by MD.While these simulation studies
addressed the FCG mecha-nism experimentally observed in especially
the smaller sized(
-
14 P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxxCTB.
The enhanced energy barrier during reverse flowowing to this
residual slip on the CTB would pose a greaterdegree of obstruction
to reversing dislocations. The presentwork quantifies the degree of
glissile irreversibility asrelated to the characteristic
microstructure dimensions, aswell as captures the governing physics
in terms of theunderlying energetics. Even though the present study
islimited to discussions of cases concerning only screw
dislo-cations, similar physics are expected for edge and
mixeddislocations. Pure edge or mixed dislocations leave
residualdislocations on the CTB upon interacting with
coherenttwins, as examined earlier in the literature
[6,7,31,49].Despite the varied nature of sliptwin reactions from
caseto case, the enhanced resistance encountered by reversingslip
and their subsequent annihilation by dislocations ofopposite sign
would occur in a similar generalized patternfor mixed or pure edge
cases. Hence the overall irreversibleglide path pattern would
essentially be identical with simi-lar trends in FCG
characteristics.
4.2. Role of microstructural dimensions on FCG
In the fracture mechanics simulations of a mode III fati-gue
crack emitting screw dislocations the cyclical slip
irre-versibility p is again evaluated as a function of theapplied
stress intensity factor. In Fig. 10 the functionaldependence of p
on DKIII shows a square root trend. Thiscould be attributed to the
involvement of multitudes of dis-locations that glide away to
interact with the nanoscaleobstacle (twin) at larger KIII. The
forces barring glidingof dislocations away from the source
(originating fromincreasing pile-up and image stress) restrict
dislocationmovement to a greater extent at larger KIII.
Mughrabi[50] summarized the estimated cyclical slip
irreversibilities,which were experimentally found to be almost
negligible atlow loading amplitudes (leading to a long fatigue
life), andclose to unity at larger loading amplitudes (resulting in
ashort fatigue life). For diminishing twin lamellae spacingsthe
irreversibility also decreased, as depicted in Fig. 10.The
mechanism lies in the lowered capability of thinnertwins to hold
back dislocations at larger applied KIII,resulting in a low slip
irreversibility.
Another important question regarding the transmissionstress
required for dislocations to penetrate these nanoscaleobstacles is
illustrated in Fig. 11. From the MD simula-tions we can estimate
the critical twinmatrix interfacezone size as normalized by the
Burgers vector within whichthe CTB stress field would effectively
elevate the energybarrier to transmission (Fig. 7c). The effective
range ofthe interface influence zone size is in the range of a few
mul-tiples of the Burgers vector. To determine the role of
thepenetration strength of the nanotwins we looked at theevolution
of da/dN at constant KIII with a varying degreeof CTB influence
zone (i.e. transmission strength of thetwins). The results, shown
in Fig. 11, indicate a neutral pla-teau in da/dN at low 2q/b for
varying twin to crack spac-ings, where q is the distance from the
twinmatrixPlease cite this article in press as: Chowdhury PB et al.
Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030interface. For twins
located far enough from the crackthe rate of FCG is independent of
the influence of disloca-tion transmission stress. We chose a value
of 2q in this neu-tral plateau for subsequent da/dN vs. DKIII
simulations inorder to compare the role of t and/or d on the
FCGproperties.
The continuum simulation framework involving
discretedislocations provides quantitative evidence of a role of
themicrostructural dimensions t and d in FCG. Fig. 12a and
bdemonstrates how the change in any one of these micro-structural
characteristic lengths (with the other being keptconstant) affects
da/dN. As can be seen, both the Parisand threshold regimes are
influenced by such a variationin t or d. For a decrease in either
the width of the nanotwinlamella or the twin to crack tip spacing
da/dN alsodecreases. Crack advancement in the threshold regime
isminuscule by nature, and is characterized by cycle by
cyclediscrete plasticity. Our approach of modeling the crack
tipplasticity accounts for individual dislocation
contributions,thereby faithfully capturing the incremental crack
growthfor both massive and minute slip activities.
The results have important implications in understand-ing mode
II (shear mode) fatigue crack growth, as wellwhere edge
dislocations are emitted from the crack tip.Although at the
continuum level the description of edgeand screw dislocations are
similar, as noted by Pippan,the reactions at the boundaries and
residual dislocationswill differ, necessitating a complete analysis
with corre-sponding MD simulations. This may explain the
funda-mental differences between different threshold levelsobserved
in the literature for mixed mode loading cases.
In summary, the critical microstructural characteristiclengths
(t and d) associated with these obstacles play a pro-nounced role
in FCG simulations. The correlation of FCGmetrics with the change
in these characteristic microstruc-tural dimensions is governed by
variations in the irrevers-ibility and blockage of slip emitted
from the crack tipinteracting with an annealing nanotwin in the
vicinity.FCG progresses via a combination of these two phenom-ena
as influenced by nanotwins. If the CTB permits areduced degree of
cyclical slip irreversibility the cyclicalcrack extension will act
likewise. Therefore, FCG isexpected to occur at varying rates
corresponding to thechanges in these characteristic microstructural
dimensions(t and d). The insight obtained from such observations
fur-ther clarifies the mechanism of cycle by cycle crack
propa-gation. Slip irreversibility increases non-linearly as
thesecharacteristic lengths become greater, eventually
reachingsaturated levels. Similar trends in da/dN with respect
tochanges in spacing of the twin to the crack tip or the
twinlamella thickness are observed in the continuum FCG
sim-ulations. Fig. 12a and b summarizes the variations in da/dN
with these changes in the microstructural
characteristicnanodimensions, thereby mapping the crack
growthregimes. The trends in FCG in these calculations are
con-sistent with earlier experimental findings in the literatureas
reported by Sangid et al. [21] and Singh et al. [5].gue crack
growth resistance of nanocrystalline alloys. Acta Mater
-
Fig. A1. The sensitivity of the calculation of cus (bulk and
CTB) on thelength scale of the chosen tracing atoms. The desired
ranges of l and w(normalized by the theoretical core width of a
screw dislocation 2n [51])for tracing the potential energy
variation due to the oncoming dislocationis highlighted. cus (bulk
and/or CTB) converges with increasing l anddecreasing w.
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx 155.
Conclusions
Utilizing MD and fracture mechanics simulations of dis-crete
dislocation formulations we have studied the mecha-nism of FCG at
the appropriate length scale by quantifyingcyclical slip
irreversibilities. Our goal was to isolate the roleof nanotwins in
the mechanism of FCG, as brought to ourattention by recent
experimental findings. The major con-tributions of this work can be
summarized as follows.
1. The analysis presented in this study underscores the roleof
twin spacing and twin lamellar width on FCG. Wenote that the
influence of these nanodimensions becomesmore prominent when the
twin spacing and twin widthare typically less than 20 nm. The
increase in FCG resis-tance is governed by the modified cyclical
slip irrevers-ibility and dislocation annihilation behavior upon
sliptwin interaction. The FCG metrics, such as da/dN andthe
threshold stress intensity range, are increased to asubstantial
extent on refinement of the nanotwin spacingand thickness. However,
as these characteristic dimen-sions increase their role in FCG
becomes less.
2. The investigation unfolded enhanced energy barriers forslip
to glide across nanoscale CTBs owing to the pres-ence of residual
dislocations during reverse flow undercyclical conditions.
Quantification of the mismatch inenergy barriers to to and fro
glide across twins providesa physical explanation for the
irreversible glissile motionof slip. Such an insight extends our
mechanistic under-standing of previously observed experimental
findingson FCG as influenced by nanoscale twins.
3. Considerable attention has been devoted to
ensuringconvergence of the unstable energy values when choos-ing
fault dimensions of the order of the dislocation core.The
sensitivity of the selected area of atoms to the unsta-ble energy
values was determined for dislocationadvance in the matrix and near
CTBs. In addition, thefriction stress levels were scaled to account
for the strainrate and temperature effects. As a result, the
differentialin friction stress upon forward and reverse loading
wasshown to play the most significant role in FCG.
Acknowledgements
Support for this work was provided primarily by Hon-eywell
Aerospace Corporation. We acknowledge the useof the parallel
computing resource, the Taub cluster, atthe University of
Illinois.
Appendix A
Since dislocation glide occurs via motion of the corethrough
consecutive tearing and forming of atomic bondsaround the core the
sequential rows of atoms on the slipplane ahead of the oncoming
dislocation alternately comewithin the influence of the mobile
core. Therefore, by calcu-Please cite this article in press as:
Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030lating the variation
between the enhanced potential energy(E) of such atoms and the bulk
perfect lattice energy(Eperfect) one would be able to compute the c
surface (usingEq. (A1) after Vitek et al. [44]). We devised a novel
tech-nique of calculating the fault energies during dynamic
slipmotion.
c E EperfectA
A1
Ahead of an oncoming dislocation a group of atoms,designated
tracing atoms (with an area on the slip planeA = wl in Fig. A1),
were carefully selected, where l and ware the distances parallel
and normal to the dislocationline, respectively. The curves in Fig.
A1 substantiate thelength scale independence of modified/unmodified
cus onthe range of the selected tracing area dimensions
(normal-ized by the theoretical core width of a screw
dislocation,2n d where d is the interplanar distance (d 2.03 A
fornickel) between (1 1 1) slip planes [51]).
A large w value spreading beyond the influence of thecore
distortion encompasses lower energy bulk atoms inthe calculation,
therefore the normalized values are suffi-ciently small for
convergence, as shown in Fig. A1. Thevalue of l was also selected
to ensure convergence, as againshown in Fig. A1. The energy of the
tracing atoms was con-firmed to have a cus value consistent with
density functionaltheory. For the case of an extended dislocation,
as the lead-ing Shockley partial approaches the tracing atoms
thevalue of c starts to increase, and achieves a maximum valuecus
when the leading partial passes it. The departing leadingpartial
leaves a stacking fault behind in its wake. At thispoint c assumes
the value cisf. As the trailing partial trans-lates the value of c
in the tracing area again starts togue crack growth resistance of
nanocrystalline alloys. Acta Mater
-
Fig. A2. (a) Schematic demonstrating the atomic configuration of
the fcc structure to illustrate the physics of dislocation
dissociation. (Left) fcc stackingon consecutive (1 1 1) planes.
Silver, green and red atoms represent planes A, B and C,
respectively. (Right) Top projection of the (1 1 1) plane shows
thedissociation of a full dislocation into two partials. Such
dissociation is energetically favored over slipping along b1 due to
the lowered cus in the partialdirections (b2 and b3). (b)
Comparison of the standard (unmodified) GSFE curves for pure nickel
using both molecular statics (MS) and current dynamiccalculation
methods for the case of a full dislocation (b1) dissociating into
two Shockley partials (b2 and b3). (For interpretation of the
references to colourin this figure legend, the reader is referred
to the web version of this article.)
16 P.B. Chowdhury et al. / Acta Materialia xxx (2013)
xxxxxxincrease, eventually decreasing to the perfect lattice
value.Then the value of c returns to zero.
c as a function of the reaction path coordinate computedin this
study (away from the twin boundaries) is based onthe geometry shown
in Fig. A2a and produces the well-known baseline GSFE of the
sliding half-block approach(Fig. A2b).
In order to calculate the modified GSFE due to localstress
sources the tracing atoms are selected at varyingproximities from
the local stress source. Therefore, thestress source would
contribute accordingly to the potentialenergy of the tracing atoms.
Thus one can calculate themodified cus or the whole c displacement
plot as influencedby the stress source in that particular position.
Such a tech-nique is applied to obtain the variation of cus near
the CTBduring forward/reverse flow (Fig. 7c). The maximum
slopecalculated from such a modified whole c displacement (asin
Fig. 7a) provides smax under the influence of local stress,which is
then appropriately scaled to room temperatureand the lower strain
rate, as explained in Appendix B.
Appendix B
Plastic flow (i.e. dislocation glide) as a function of
tem-perature is modeled by an Arrhenius-type equation as fol-lows
[52].
_c _co exp EakT
B1
where Ea is the activation energy barrier, T is the
absolutetemperature (Kelvin), k is the Boltzmann constant, and
_cois a constant associated with the rate of deformation.
Thederivative of Ea with respect to the glide resistant stressso (a
function of temperature and strain rate) providesthe kinetic
signature of plastic deformation, which isPlease cite this article
in press as: Chowdhury PB et al. Modeling fati(2013),
http://dx.doi.org/10.1016/j.actamat.2013.01.030defined as the
activation volume V. V scales with thephysical area swept by the
dislocations. For nanocrystal-line materials or in confined volumes
where there are lim-ited dislocations sources it is of the order of
several b3,while for bulk materials its magnitude can be 1000b3.
Itsmagnitude can be determined experimentally or fromMD
simulations. From the MD simulations the differencein flow stresses
(Ds) for the nucleation of a single disloca-tion loop is calculated
at two different strain rates ( _c2 and_c1) and a constant
temperature, using Eq. (B2) [52]:
V @Ea@s
kT @ ln _c@s
kTDs
ln_c2_c1
B2
We note that Ea is a decreasing function of so. Consid-ering the
linear so dependence of Ea [52] one can write
Ea E V so B3Using a temperature normalization introduced by
Zhu
et al. [53],
so E
V kTV
lnkTNtol_cV
B4
where E is the athermal activation energy barrier, N is
thenumber of nucleation sites, to is the attempt frequency, andl is
the shear modulus. The value of E is established froma knowledge of
the critical stress at 0 K and V. Bothquantities are determined
from the MD simulations con-ducted in this study. Then the above
equation allows thedetermination of so at different strain rates
and tempera-tures. The constants utilized in our work are as
follows:Boltzmann constant k = 1.3806503 1023 m2 kg s2 K1;athermal
activation energy barrier E = 1.3 eV; activationvolume V = 2.25b3;
Burgers vector~b a
61 1 2, where lat-
tice constant a = 3.52 A; number of nucleation sitesN = 100;
attempt frequency to = 3.14 1011 Hz; sheargue crack growth
resistance of nanocrystalline alloys. Acta Mater
-
Table B1The unstable stacking fault energy, the maximum
(critical) stress for slip,and the critical stress so at room
temperature and typical experimentalstrain rates (1104 s1).GSFE
(pure nickel) cus (mJ m
2) smax (GPa) so (GPa)
Intrinsic (unmodified) 254 6 1.8Forward transmission 340 12.6
3.8Reverse transmission 452 21.3 6.4
P.B. Chowdhury et al. / Acta Materialia xxx (2013) xxxxxx
17modulus l = 76 GPa; temperature T = 300 K (room tem-perature);
shear strain rate _c 1 104 s1. Because theMD calculations are
conducted at 10 K and at high strainrates it was deemed necessary
to scale the results for lowerstrain rates and room temperature. We
note that this is stilla topic of current research; the
modifications made aboveand illustrated in Table B1 represent the
current state ofknowledge. Future refinements in this area will
not, how-ever, change the conclusions reached in this work.
The intrinsic (unmodified) cus in Table B1 correspondsto the
unstable stacking fault energy of pure Ni. The mod-ified cus levels
in Table B1 are for a dislocation within thetwin boundary zone, as
explained in the text. The corre-sponding critical stress levels
are encountered during for-ward/reverse transmission, as shown in
Fig. 7c. The smaxvalues are obtained from the slope of the c
displacementcurves from the simulations. The third column is
obtainedfrom Eq. (B4).
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