Size effects in dislocation depinning models for plastic yield

0.44 MB·English

by  Zoe Budrikis

Size effects in dislocation depinning models for plastic yield
Zoe Budrikis1 and Stefano Zapperi2,1
1ISI Foundation, Via Alassio 11/c, 10126 Torino, Italy
2CNR-IENI, Via R. Cozzi 53, 20125 Milano, Italy
Typically, theplasticyieldstressof asample isdeterminedfrom astress-strain curvebydefining
a yield strain and reading off the stress required to attain it. However, it is not a priori clear that
yieldstrengthsofmicroscalesamplesmeasuredthiswayshoulddisplaythecorrectfinitesizescaling.
Herewe studyplastic yield as a depinningtransition of a 1+1 dimensional interface, and consider
howfinitesize effectsdependon thechoice ofyield strain, aswell as thepresenceof hardeningand
3 thestrengthofelasticcoupling. Ourresultsindicatethatinsufficientlylargesystems,thechoiceof
1 yield strain is unimportant,but in smaller systems onemust takecare toavoid spurious effects.
0
2 PACSnumbers: 62.20.F-,64.60.an

n
a
I. INTRODUCTION strain increases that ideally are limited only by sample
J
failure.
4
Like other phase transitions, the depinning transition
2 A complete understanding of microscale plasticity is
issubjecttofinitesizeeffects,sothatthedepinningforce
important both for fundamental and technological rea-
] in a system of linear size L has the asymptotic form
sons, and much experimental and theoretical effort has
t
sof breeseunltexthpaetndheadsoenmethrgisedpr—oblaemndinthraetceisntnyoetairnst.uiAtivkeelyy fc(L)=f∞+aL−1/ν (1)
. obvious from our everyday experience of classical bulk
at plasticity —is thatonthe microscaleplastic flow is sub- wheref∞isthedepinningforceinthelimitL→∞andν
m isthefinitesizeexponent,determinedbytheuniversality
ject to sizeeffects, with recentexperiments showingthat
class of the model. A key question that this work aims
- in submicron samples, the yield stress increases with de-
d to address is whether the observed finite size scaling is
creasing sample size [1]. However, the origins of these
n affected by how the depinning transition is measured.
finite size effects are not clear, and many mechanisms
o In particular, we can imitate the experimental proce-
c have been proposed (for reviews, see Refs [2–4]).
dure for measuring yield stress by mapping our simula-
[
In typical experimental studies, the yield stress is de- tionsofinterfacedepinningontoplotsofstressvs plastic

1 termined from measured stress-strain curves by defining strainfromstresscontrolledexperiments. The motionof
v ayieldstrainγY,typicallyafewpercent,andreadingoff a dislocation with Burgers vector b over distance x in a
4 thestressrequiredtoattainsuchastrain. Thisprocedure systemofsizeL2resultsinastrainincreaseofγ =bx/L2,
9
suffers from the fact that measured stress-strain curves andthedrivingforceisproportionaltoanexternalstress.
7
arenot smoothandthe transitionto plastic yield is gen- We can therefore consider a plot of the mean position of
5
. erally preceded by power law distributed avalanches [5]. the interface vs driving force to represent plastic strain
1 Thisraisesthequestionofhowtheyieldstrainshouldbe vs stress,asillustratedinFig.1(a). Inanalogytotheap-
0
selected to ensure that the measured yield stress is not proachtakeninexperimentalanalysis,wedefineatarget
3
1 systematically over- or under-estimated in a way that displacement∆=γL2/btodefinetheonsetofdepinning,
: affects finite size scaling. Furthermore, it is not a pri- and study the effects of varying this target. In what fol-
v ori clear that one can compare experimental results in lows, we will measure lengths in units of b.
i
X whichdifferentthresholdstrainshavebeenused,making We note that this approach to measure the depinning
r it difficult to synthesize results from different studies. threshold raises another issue. It has previously been
a
In this paper, we approach this problem from a theo- pointed out [16, 17, 19] that in a simulation of a system
retical perspective, by studying numericalsimulations of of finite size L M with periodic boundary conditions
×
1+1dimensionaldepinning atzerotemperature. Thisis in both directions, a “natural” choice of M is M Lζ,
thesimplestmodelforplasticyieldinginacrystal: thein- whereζ istheinterfaceroughnessexponentandLζ∼gives
terfacerepresentsadislocation,driventhrougharandom theinterfacewidthatcriticality. Inthisway,oneensures
potentialgeneratedby disorderandotherdislocationsin that for all L, the system contains the same number of
the sample. Depinning has previously been studied in a critical configurations. On the other hand, our target
variety of contexts, using both numerical and analytical scales as L2, so although it is natural for the plasticity
techniques[6–18],anditiswell-knownthatthereexistsa scenario at hand, the system aspect ratios increase with
critical driving force f , above which the interface is de- L. We will see that this affects the distribution of mea-
c
pinned and has finite velocity always, and below which sured depinning forces, as anticipated [16, 17].
the interface will always come to a stop at long times. The depinning models can also be related to strain-
This transition is analogous to plastic yielding, in which controlled experiments, by driving the interface via a
stresses above a finite threshold are able to induce large springofstiffnessk,andadjustingtheexternally-applied
2
stress so that the total strain — the elastic strain of the
spring,partially relievedby the plastic deformationaris-
ing from interface motion — increases smoothly. The
set-up and a typical stress-strain curve are illustrated in
Fig. 1(b). Again in this context, we study the effect of
the choice of yield strain on measured yield stress, and
discusshowthese effectscanbe relatedtostrainharden-
ing in a stress controlled experiment.
The outline of the paper is as follows. In Section II,
we use the quenched Edwards-Wilkinson model for in-
terface depinning as a case study for the role of tar-
get choice on measured properties, considering not only
the finite size scaling of the mean depinning force but
also how the choice of target affects its whole distribu-
tion. The ideas developed in Section II are indepen-
dent of universality class, which we demonstrate in Sec-
tion III via results from the Kardar-Parisi-Zhang(KPZ)
model and a mean field model. Finally, in Section IV
wediscusstwoclosely-relatedextensionstothequenched
Edwards-Wilkinsonmodel, namely strainhardening and
the above-mentioned strain controlled experiments. We
showthatfinitesizeeffectsinthesemodelsaremorecom-
plex than in the other models we study, depending on
both the finite size scaling of the depinning force and
also the hardening amplitude or stiffness of the coupled
spring.
FIG. 1: (a)Typical “stress-strain” (driving force, f, vs mean
II. EFFECTS OF TARGET CHOICE: THE
interface displacement, u¯) curves from simulations of the
QUENCHED EDWARDS-WILKINSON
quenched Edwards-Wilkinson equation, with (circles) and
EQUATION
without (triangles) hardening. The black solid line indicates
the f ∝ u¯ behavior of the model with hardening as u¯ → ∞
We start by considering a simple linear model for the (seeSectionIV).Depinningforcesaredeterminedbyselecting
interplay between elasticity, random pinning and an ex- a target value of u¯ and reading off the driving force required
ternal drive, namely, the quenched Edwards-Wilkinson to attain this target. (b) Data from a hardening simulation
model [6, 7]. This will allow us to discern the effect of recastasastraincontrolledexperimentwithmachinestiffness
the choice of target interface displacement in a straight- k, using the transformations σT = f −ku¯/L2, γT = f/k, as
forwardmanner,givingresultsthatwillaidthe interpre- described in Section IV. The yield stress, σT,y is determined
by selecting a yield strain γ and reading off the stress re-
tation of more complex models that follow. T,y
quired to attain it.
In general, an overdamped elastic interface character-
ized by height function u(x) and subject to a random
potential and a uniform external drive f is described by
overdamped equations of motion is given by ux+1 +ux−1−2ux. We allow only forward
motion of the interface: at each simulation step, a site
1 x may increase u by one unit if the total force acting
x
u˙(x)=f (x) (2a)
Γ tot on it is positive, or remain stationary if the force is zero
=f +f (x,u(x)) η(x,u(x)), (2b) or negative. The random potential is implemented by
el
− drawing a random number for each x,u . In general,
x
where the damping constant Γ is hereafter set to unity. these are from a uniform distribution over [0,1], but in
Theself-interactionoftheinterface,f ,promotesittobe Section III we will also study Gaussian distributed dis-
el
flat, and may be local or long-range, depending on the order,because when allsites are coupledto one another,
model at hand. The self-interaction is in competition the disorder distribution can be important [20]. We use
with the quenched disorderη, which causes the interface periodicboundaryconditionsinthexdirection,andopen
to prefer to passthroughsites with lowpotentialenergy. boundary conditions in the u direction.
InthequenchedEdwards-Wilkinsonmodelthatwestudy In order to generate “stress-strain” curves in a stress-
in this Section, the elastic force at position x on the line controlled simulation, we start with a flat interface with
is given by 2u(x). u = 0 everywhere and f = 0. We increase the drive f
x
∇ ∗
In this work, we study a cellular automaton version adiabatically,by finding the site x with the smallest re-
of this model, by discretizing x and u. The elastic force quired driving force for motion, and increasing f so that
3
ΣHfcL 5´00001....000000015-15004ôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàòôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàòôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôàôôôôôôôôôôôôôôôôôôôòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôòôôôôôôôôôôòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôòôôôôôôôôôôôôôôòôòôôôôôòôôôôòôòôôôôôòôôôôòôòòôòôòôòôòô ô òôôòôôôôô
ôôôôô
1´10-4 ô
ô
10 50 100 1000 10000
L
FIG.3: Standarddeviation in themeasured depinningforce,
σ(f ), plotted against system linear size L. For sufficiently
c
large targets, σ(f )∼L−0.75, as indicated by the solid black
c
line. The target strains used are γ = 0.07 (circles), 0.15
(triangles), 2 (squares). Because the distributions of f are
c
notGaussian (seeFig.4),theerrorbarshavebeenestimated
using themethod in Ref. [21].
dependence becomes apparent: for small γ, f initially
c
grows with L, but for large γ, f (L) decreases monoton-
c
ically.
This result can be understood by considering that
FIG. 2: (a) Mean measured depinning force hf i plotted
c there are two length scales in the direction of interface
againstsystemlinearsizeL,fromsimulationsofthequenched
propagation. The first is the interface width at depin-
Edwards-Wilkinson equation. The target strains used are
ning, which is given by wLζ, with the roughness expo-
γ = 0.01 (circles), 0.07 (triangles), 0.25 (squares) and 2 (in-
verted triangles). For all targets, f approaches the same nentζ =1.25forthequenchedEdwards-Wilkinsonequa-
c
value as L→ ∞, but the behavior for smaller L depends on tion[10,15]andwaconstant. Thesecondlengthscaleis
γ. Error bars (typically smaller than the symbols) represent ourchoice oftarget,whichis γL2. The twolengthscales
thestandarderror. (b)Thesamedata,collapsedusingEq.(1) are equal when
and(3). Theblacksolid lineisproportionaltoL−0.75,which
is the expected finitesize scaling for themodel for large L. L2−ζ = w, (3)
γ
x∗ canmove. This may trigger anavalanchevia changes giving a crossoverlength scale L =(w/γ)1/(2−ζ).
c
in f , and we update sites in parallel until no more can If L < L , the target is shorter than the interface
el c
move. We recordthevalueoff andthemeanpositionof width at criticality, and the simulation is halted during
the interface attained after it has stopped moving, and theearlystagesofinterfacemotion,beforethedepinning
thenincreasethedriveagain. AsmentionedintheIntro- transitioncanoccur. Inthis regime,increasingL —and
duction, the record of driving forces and mean interface thereby the target — should increase the final driving
positions is the “stress-strain”curve,since the motionof force applied before the simulation is stopped. On the
a dislocation by distance x increases the plastic strain other hand, for L > L , we can expect the interface to
c
by x/L2 (we work in units where the Burgers vector b is undergothedepinningtransition,andthestandardfinite
unity). We continue to increasethe drive until the mean sizescalingofEq.(1)shouldhold. AtL ,thereisatran-
c
interface position passes the target γL2. The driving sition from the former behavior to the latter. However,
forcerequiredforthetargettobe attainedisdenotedf , if γ is sufficiently large, the target is always larger than
c
andforsimplicitywewillcallita“depinning”threshold, the interface width, that is, L 0 as γ .
c
→ →∞
even if this is not strictly accurate. Toverifythisinterpretation,wecollapsethedatafrom
Figure 2(a) shows the mean force required for the in- simulationswith differentγ values. FromEq.(3), we see
terfacetoattainthetargetmeanpositionγL2,asafunc- that we must rescale L Lγ−1/(2−ζ). Then from the
→
tion of system linear size L, for a range of γ values be- finite size scaling (1) and the fact that ν =1/(2 ζ) [9],
−
tween 0.01 and 2. In the large-L limit, we find that we rescale (fc f∞) by γ. As seen in Fig. 2(b), the
fc(L) approaches f∞ = 0.424 according to L−1/ν, with datacollapseis−verygood. Furthermore,thatthecorrect
1/ν = 0.75, in agreement with previous studies of finite finite size scaling laws are recoveredfor sufficiently large
size scaling [14, 16]. However, for small L, a distinct γ targets is further evidenced by the measured scaling of
4
æ
(cid:144)HHLL12Ν-ΖÈ 000...121005000ìàìàììàæììàààààæì æìàæìæìàæìæàìææàæììàààààæì
-L(cid:144)f-Γ¥00..005700 æìàææàææàææàæàà
\ àà
Xfc æ
ÈH 0.030 æ
æ
æ
0.020
1 10 100 1000
LΓ1(cid:144)H2-ΖL
FIG. 5: Data collapse for the KPZ model, based on the
same arguments as the quenched Edwards-Wilkinson model
(Eq.1and3),usingfittedparametersζ =0.635,f∞ =0.226,
1/ν =0.55. Thesymbolsrepresentγ =0.1(circles),γ =0.01
(squares) and γ = 0.005 (diamonds). The standard error is
smaller than thesymbol size.
widths. In agreement with Refs. [16, 17], as γ increases,
thedistributionapproachestheGumbellimit,albeitvery
slowly in the tails of the distributions.
FIG.4: (a) Histograms of themeasured depinningforce over III. OTHER SIMPLE DEPINNING MODELS:
105 trials,forL=100,withγ =0.01(circles),1(squares),10 KPZ AND MEAN FIELD
(diamonds) and 100 (triangles). The distributions have been
normalized to have zero mean and unit variance. Error bars
The role of the parameter γ relative to the interface
aresmallerthanthesymbolsize. (b)Cumulativedistribution
width does not depend specifically on the choice of in-
functions,showingconvergencetowardstheGumbeldistribu-
tion(astraightlineonthisdouble-logplot)asγincreases. For teraction fel used in the quenched Edwards-Wilkinson
comparison, in bothplots, theGaussian distribution (dashed equation, and we have verified that models from other
line)andGumbeldistribution(solidthinline)arealsoshown. universality classes are affected by the choice of γ in
the same way. For example, in the KPZ model [22],
where the elastic force f = 2u(x) is supplemented
el
by a nonlinear term proportio∇nal to ( u)2, we mea-
thestandarddeviationoffc(L),asplottedinFig.3. This ∇
has the asymptotic form L−1/ν, as expected from finite sure a roughness exponent of ζ =0.635 (consistent with
other measurements [15, 23]) and a finite size exponent
size scaling.
of 1/ν 0.55. Per Eq. (3), f (L) curves for different
c
We can go beyond the moments of the distribution of γ can b≈e collapsed via the rescaling L γ1/(2−ζ)L,
fc to consider the distribution itself. It has previously (fc f∞) γ−1/(ν(2−ζ))(fc f∞), as show→n in Fig. 5.
been shown [16, 17] that the distribution of f measured − → −
c The quenched Edwards-Wilkinson and KPZ models
in a finite sample of size L M (with open boundary
× both have local interactions, and the interface width
conditions in the M direction) depends on the ratio of scales with length as Lζ. We now turn to a mean-field
M to the interface width wLζ. Heuristically, the system
model [13, 24, 25], in which every site interacts with ev-
can be divided into n=M/(wLζ) subsystems which are
ery other site, and the interface width does not depend
approximatelyindependent, eachwithitsowndepinning
on the length scale it is measured over (ζ =0). We cou-
force. The depinning force in the entire L M system
ple the interface to its center of mass, u¯. The total force
×
is the maximum of these, and if n the measured
→ ∞ on site x is
depinning force is described by extreme value statistics,
in particular, the Gumbel distribution, which has cumu- f =f +J(u¯ u )+η . (4)
lative distribution P(x) = exp( e−(x−µ)/β)), where µ is x − x u,x
−
the modal value of x and the standard deviation of the In Fig. 6 we show the finite size behavior of the de-
distribution is βπ/√6. In Fig. 4, we plot the distribu- pinning force for this model. The finite size exponent
tion of f (rescaledto have zeromean and unit standard takes the mean-field value of ν = 2. For fixed J we can
c
deviation) for L=100, for a range of γ values. The tar- collapsedatafromsimulationswithdifferentγ valuesfol-
gets rangefroma few interfacewidths to 104 interface lowingthe same argumentas forthe quenchedEdwards-
∼
5
(cid:144)14-HX\Lff-c¥000...125000 à ò à òóàá àáòóæàáòóàáàáòóàáòóæàáòóàáòóòóæìòóæàáææìòóàáææàáæìòóæàáìòóìòóìæìàáììæìòæàææìòàææàæìòæìòìòìòæììò wficnitnshrHahgeeAtpeae=orrilssntoeneetpa−sretk,nonhrkadrfaeui¯tanstit/cddoihvLeaniesef2arollhd,oeylrarm,cetrilaffivaodanterieiingctnotnhgtehsniienvefdcaogerimcsrlsiyactpcteiseohlcareanareencesodledqtfudamaiuascinnsieledroptsnse.ctltdnartasaheTctttveiieohnohmeeneemrcexodhdhnotneaevitas.ptrrerprndditolnAaeeahlsnlnlcse.eieifdnnoimangrgetcerxedenferopostfaruemaiccrlncettiie----,,
Γ á ment,onecanimaginethatasubsystemundergoingplas-
0.05 á á tic deformation is coupled to a spring of stiffness k, and
áááááá a total deformation γT is imposed (as illustrated in the
Inset of Fig. 1(b)). In the absence of a plastic subsys-
1 10 102 103 104 105 106
tem, γ would be attained using a stress of σ = kγ ,
ΓL2 per HoToke’s law, but when plastic deformatioTn occurTs,
the spring extension/compression is reduced by u¯/L2 so
that σ is given by
FIG.6: Rescaleddepinningforcevs targetdisplacementγL2, T
inthemeanfieldmodel. Thesolidblacklinesareproportional u¯
to L−1/2. Closed symbols represent data for J = 1, open σT =k(γT − L2). (6)
symbols represent data for J = 0.1, the γ values are 0.005
(squares),0.01(triangles),0.1(circles)and1(diamonds). For Simulations of an interface subject to strain hardening
J =0.1, f∞ ≈0.64, for J =1, f∞ ≤10−3. can be reinterpreted in terms of strain controlled exper-
iments [26, 27], by treating f as kγ and plotting σ vs
T T
γ , as illustrated in Fig. 1(b). We work in units where
T
Wilkinson model, but with the interface width constant. stresses and forces are interchangeable.
As seen in Fig. 6, the data collapse is excellent. Wewillfirstdiscusshowsizeeffectsininterfacedepin-
The coupling strength J affects the finite size scaling ning are affected by strain hardening, and then discuss
in two ways. First, as J increases, the interface becomes size effects in strain controlled experiments.
flatter. Accordingly,asseeninFig.6,forlargercoupling
strengths, the peak in f (L) occurs for smaller targets.
c
ThesecondeffectofJ istoalterthedepinningthreshold, A. Hardening
so that as J increases, f∞ decreases. Heuristically, this
can be understoodby considering that as J increases,so
We have simulated the quenched Edwards-Wilkinson
does the forward force on sites lagging behind the mean
model with an added hardening term. A typical “stress-
interfaceposition,sothattheexternaldrivingforceneeds
strain” curve is shown in Fig. 1(a). For small driving
onlybeabletoovercomethelocalpinningforceonafew
force and interface displacement, the behavior is that of
sites to start a large avalanche.
the quenched Edwards-Wilkinson model without hard-
For completeness, we have also made simulations us-
ening, but as the driving force and displacement are in-
ing Gaussian distributed quenched disorder, with η =
h ii creased, the “stress-strain” curve departs from its zero-
0.5,σ(η ) = 0.2, and find very similar behavior to that
i hardening counterpart. When hardening dominates, if a
observed with uniform-distributed disorder. ∗
force of f has been applied to drive the interface to po-
∗
sition u¯ and it has come to a stop, the force requiredto
start it moving again is
IV. EFFECTS OF HARDENING AND
∗
MACHINE STIFFNESS ∗ u¯
f =f +k . (7)
L2
Finally, we discuss two related scenarios: strain hard-
This is the behavior seen in the “stress-strain” curve,
ening and strain-controlled experiments. In materials
whichapproachesalineofslopek/L2inthelimitu¯ .
subject to hardening, the external stress required to in- →∞
In Fig. 7, we plot typical f vs L curves. With the
duceplasticdeformationgrowswiththeplasticstrain. A c
introduction of hardening,it is no longer possible to col-
simplemodelforthiscanbeincorporatedintothemodel
lapse data for the whole space of parameters γ and k.
for interface depinning, by adding a force term that is
Instead, the observedbehavior depends on kγ. For large
opposedto the interfacemotion,andproportionalto the
kγ, f (L) curves from simulations with the same k can
strain associated with the interface displacement, u¯/L2, c
be collapsed by subtracting kγ from f , but data from
where u¯ is the average of ux over x. The total force on simulations with different k cannot bhecciollapsed onto a
site x is then
single form. Furthermore, if the kγ is small, this proce-
u¯ duredoesnotyielddatacollapse,asexemplifiedinFig.7
fx =fel(x,ux)−η(x,ux)+f −kL2, (5) by the fc(L)−kγ curve for γ =0.01,k=1.
6
0.43
0.425
X\f-kΓc 00..0044..12445512 óõóõóõçóõçóõç ó òóôõæçòóòóôõæçòóòóôõæçòóòóôõæçôõæçôõæçôõæçôõæçòóõæç ó òõæ òæ òõæòææææææ ΣT,y 00..0044..01445512òæ òóõæç òóõæç òóõæç òóõæç òóõæç òóõæçòóõæçòóõæçòóõæç òóõæç òóõæç òæ
óõ ç
0.405 óõ ç ò 0.4òóõç ò ò òá òá òá òáòáòáòá òá òá ò
á
0.410 ææ 5æ0 10ææ0æ 5001000 5000 0.395 á
100 200 500 1000 2000
L
L
FIG. 7: Size dependenceof depinning force for the quenched FIG. 8: Size dependence of the measured yield stress σ
T,y
Edwards-Wilkinsonequationsubjecttohardeningofstrength in strain controlled simulations of the quenched Edwards-
k. ForsufficientlylargekγL2,datafromsimulationswithdif- Wilkinson model, with spring constant k = 10 (open sym-
ferent γ valuescan becollapsed bysubtracting kγ from hfci, bols), and k = 1 (filled symbols). The yield strains used are
togive finitesize scaling of theform described by Eq.(8), as γ =1 (triangles), 0.5 (circles), 0.4 (invertedtriangles) and
T,y
indicatedbytheblacksolid line. Thiscollapsedoesnotwork 0.04 (squares). The black lines indicate the expected finite
for small kγL2, as seen in the data for k =1,γ =0.01 (open size scaling of Eq. (9), with f∞ = 0.424 and 1/ν = 0.75.
circles), which only fall on to the collapsed curve for suffi- When kγ is too small, this scaling breaks down because
T,y
ciently large L. The dashed horizontal line indicates f∞, the the interface never reaches criticality, as seen for the lower
L→∞limitoffc(L)intheabsenceofhardening. Solidsym- two curves(k=10,γT,y =0.04, and k=1,γT,y =0.4).
bols represent k=10 and open symbols represent k =1; the
shapes represent γ = 0.01 (circles), γ = 0.15 (triangles) and
γ =2 (inverted triangles). Error bars represent thestandard
simulations of an interface described by the quenched
error.
Edwards-Wilkinsonequation. For sufficiently large γ ,
T,y
thebehaviordescribedinEq.(1)isrecoveredandσ (L)
T,y
We can interpret this by considering the maximum approaches the same L limit found in stress con-
→ ∞
hardening force applied during simulations, which is ap- trolled simulations, with the same finite size exponent
plied when the interface is at its target: u¯ = γL2. For 1/ν =0.75. However,forsmallerγ ,thispicturebreaks
T,y
small kγ, the hardening force is always negligible, and down.
the size effects are the same as those discussed in previ- We can interpret this as follows. Recall that in stress
ous Sections. When kγ is large, hardening becomes im- controlled simulations of the quenched Edwards Wilkin-
portant. A first approximation to understand its role is sonmodel,thetargetγL2 growsfasterthantheinterface
toimaginethattheinterfaceisdriventothetargetwith- widthwLζ andthereforeasL thecorrectfinitesize
→∞
out hardening, and then the hardening force is turned scalingmustoccur. However,instraincontrolledsimula-
on. Assuming the target is sufficiently large, the force tions,thetargetγ =f /kmaycorrespondtoadriving
T,y y
required to attain it (with zero hardening) is given by force f that is always smaller than the true depinning
y
Eq. (1). Then, from Eq. (7), the force required to con- force (as given in Eq. (1)). In that case, the interface
tinue to move the interface is neverreachescriticalityforanyL,andthestandardfinite
sizescalingdoesnothold. Notethattheminimumstrain
f =f∞+aL−1/ν +kγ, (8) target depends on both the depinning force of the inter-
face and also the spring constant k, as shown in Fig. 8.
where f∞ is the L → ∞ limit of fc(L) when k = 0. If the target is large enough for the interface to reach
This form agrees well with our simulation data, as seen
criticality, then the size effects can be predicted by re-
in Fig. 7. Because of the extra γ-dependent term, it is
interpreting the strain controlled simulation as a stress
not possible to use linear rescalings to eliminate the γ
controlledsimulationwith hardening. The finite size be-
dpeopsseinbdlee.nce of (fc−f∞), and exact data collapse is not havior of f =kγT is given by Eq. (1) and (7), so that
u¯
σT,y =f −kLy2 =f∞+aL−1/ν, (9)
B. Strain controlled experiments
where u¯ is the mean interface displacement when the
y
In strain controlled simulations, the choice of yield total strain is equal to the yield strain. In other words,
strain is more crucial than in stress controlled simula- for sufficiently large yield strains, stress controlled and
tions. In Fig. 8, we plot the size dependence of σ for strain controlled simulations give the same results, and
T,y
several choices of yield strain γ , for strain controlled furthermore, in that case, the measured yield stress in
T,y
7
a strain controlled experiment does not depend on the tal studies of polycrystalline copper samples under ten-
yield strain. sion [28] have revealedthat profiles of the surface height
y(x)(withxalongthesampleaxis)areself-affine,thatis,
W(L) y(x+L) y(x) LH,withH 0.75forim-
≡h| − |i∼ ≈
V. CONCLUSION posedstrains&10%. Thissurfaceprofileisrelatedtothe
variation in the strain, which in a sample of length L is
The question of how to measure yield stress is an im- ∆γ =W(L)/L L−0.25. Imposingtherequirementthat
∼
portant methodological problem in the determination of thetotalstrainshouldbelargerthanitsvariance,amin-
finite size effects in microscale plasticity, which to our imum system size Lc is determined by setting γ = ∆γ.
knowledgehasreceivedlittleattentionpreviously. Inthis In the case of polycrystalline copper, this length scale
paper,we havegainedinsightinto howto avoidspurious turnsouttobesmallenoughtohavenoexperimentalef-
size effects in a simplified model for the onset of plastic fect: usingthe datafromZaiseret al.’s experiments[28],
yield when the yield strain is arbitrarily defined. Reas- Lc 1.6 10−9 m for γtot = 10%. For different mate-
≈ ×
suringly, in all the models we study, for sufficiently large rials and loadingconditions, however,it may be possible
systems the correct finite size scaling is recovered. How- for Lc to be comparable with experimental sample sizes,
ever, for smaller systems, the finite size effects are not leading to intricate size effects.
described by the scaling law (1), and instead depend on
various details of the model under study, a result which
has implications especially for simulation studies of de-
Acknowledgments
pinningandplasticity,whereuntilrecentlyonlyrelatively
small sizes were accessible computationally.
We also observe that the idea that the crossover be- This work is supported by the European Research
tween “small” and “sufficiently large” systems is deter- CouncilthroughtheAdvancedGrant2011SIZEFFECT.
mined by the roughness of a self-affine interface is one WethankMichaelZaiser,ZsoltBertalanandGianfranco
that has broader relevance. For example, experimen- Durin for helpful discussions.
[1] M.D.Uchic,D.M.Dimiduk,J.N.Florando,andW.D. 67, 021602 (2003).
Nix,Science 305, 986 (2004). [16] C. J. Bolech and A. Rosso, Phys. Rev. Lett. 93, 125701
[2] M. Zaiser, Adv.Phys.55, 185 (2006). (2004).
[3] J. R. Greer and J. T. De Hosson, Prog. Mater. Sci. 56, [17] A. Fedorenko, P. Le Doussal, and K. Wiese, Phys. Rev.
654 (2011). E 74, 041110 (2006).
[4] O.Kraft,P.A.Gruber,R.M¨onig,andD.Weygand,Ann. [18] A. A. Middleton, P. Le Doussal, and K. J. Wiese, Phys.
Rev.Mater. Res.40, 293 (2010). Rev. Lett.98, 155701 (2007).
[5] D. M. Dimiduk, C. Woodward, R. LeSar, and M. D. [19] O. Duemmer and W. Krauth, Phys. Rev. E 71, 061601
Uchic,Science 312, 1188 (2006). (2005).
[6] M. V.Feigel’man, Sov.Phys. JETP 58, 1076 (1983). [20] P. M. Duxbury and J. H. Meinke, Phys. Rev. E 64,
[7] R. Bruinsma and G. Aeppli, Phys. Rev. Lett. 52, 1547 036112 (2001).
(1984). [21] D. G. Bonett, Comput. Stat. Data An.50, 775 (2006).
[8] T. Nattermann, S. Stepanow, L.-H. Tang, and [22] M.Kardar,G.Parisi,andY.-C.Zhang,Phys.Rev.Lett.
H.Leschhorn, J. Phys. II France 2, 1483 (1992). 56, 889 (1986).
[9] O.NarayanandD.Fisher,Phys.Rev.B48,7030(1993). [23] Y.-J. Chen, S. Papanikolaou, J. P. Sethna, S. Zapperi,
[10] H.Leschhorn, Physica A 195, 324 (1993). and G. Durin,Phys. Rev.E 84, 061103 (2011).
[11] L. A. N. Amaral, A.-L. Barab´asi, S. V. Buldyrev, S. T. [24] D. S. Fisher, Phys. Rev.Lett. 50, 1486 (1983).
Harrington,S.Havlin,R.Sadr-Lahijany,andH.E.Stan- [25] D. S. Fisher, Phys. Rev.B 31, 1396 (1985).
ley,Phys. Rev.E 51, 4655 (1995). [26] M. Zaiser and P. Moretti, J. Stat. Mech. 2005, P08004
[12] A.RossoandW.Krauth,Phys.Rev.B65,12202(2001). (2005).
[13] J. Vannimenus and B. Derrida, Journal of Statistical [27] M. Zaiser and N. Nikitas, J. Stat. Mech. 2007, P04013
Physics 105, 1 (2001). (2007).
[14] P. Le Doussal, K. Wiese, and P. Chauve, Phys. Rev. B [28] M.Zaiser,F.Grasset,V.Koutsos,andE.Aifantis,Phys.
66, 174201 (2002). Rev. Lett.93, 195507 (2004).
[15] A. Rosso, A. Hartmann, and W. Krauth, Phys. Rev. E