Dislocation patterning in a 2D continuum theory of dislocations

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by  Istvan Groma

Dislocation patterning in a 2D continuum theory of dislocations
Istv´an Groma∗ and P´eter Dus´an Isp´anovity
Department of Materials Physics, Eo¨tv¨os University Budapest, H-1517 Budapest POB 32, Hungary
Michael Zaiser
Institute for Materials Simulation, Department of Materials Science,
Friedrich-Alexander University Erlangen-Nu¨rnberg (FAU), Dr.-Mack-Str. 77, 90762 Fu¨rth, Germany
Understanding the spontaneous emergence of dislocation patterns during plastic deformation is
a long standing challenge in dislocation theory. During the past decades several phenomenological
continuum models of dislocation patterning were proposed, but few of them (if any) are derived
6 from microscopic considerations through systematic and controlled averaging procedures. In this
1 paperwepresenta2Dcontinuumtheorythatisobtainedbysystematicaveragingoftheequationsof
0
motionofdiscretedislocations. Itisshownthatintheevolutionequationsofthedislocationdensities
2
diffusionliketermsneglectedinearlierconsiderationsplayacrucialroleinthelengthscaleselection

b ofthedislocationdensityfluctuations. Itisalsoshownthattheformulatedcontinuumtheorycanbe
e derived from an averaged energy functional using the framework of phase field theories. However,
F in order to account for the flow stress one has in that case to introduce a nontrivial dislocation
mobility function, which proves to be crucial for the instability leading to patterning.
7

PACSnumbers: 64.70.Pf,61.20.Lc,81.05.Kf,61.72.Bb
]
i
c
s
- I. INTRODUCTION tic length scale of the dislocation patterns is an order of
l magnitude larger than the dislocation spacing5. So, to
r
t be able to detect patterning in a DDD simulation one
m Shortly after the first images of dislocations were seen
has to work with systems containing a large amount of
in TEM it was realized that the dislocation distribution
t. inadeformedcrystallinematerialispracticallyneverho- dislocation lines: to see 3-5 pattern wavelengths in each
a
spatial direction, the system size should be 30-50 times
m mogeneous. Depending on the slip geometry, the mode √
larger than 1/ ρ. Especially in 3D this is a rather hard
of loading and the temperature, rather different pat-
d- tern morphologies (e.g. cell1, labyrinth2, vein3 or wall4 task. Although irregular clusters or veins are regularly
observed in simulations16–18 clear evidence of the emer-
n structures) emerge. There are, however, two important
o feature common to all these patterns: It is almost al- gence of a characteristic length scale has not been pub-
c lished so far.
ways observed that the characteristic wavelength Λ of
[
the patterns is proportional to the dislocation spacing, During the past decade continuum theories of disloca-
√
2 Λ ∝ 1/ ρ where ρ is the total dislocation density, and tions derived by rigorous homogenization of the evolu-
v inversely proportional to the stress at which the pat- tion equations of individual dislocations have been pro-
1 terns have formed. These relationships are commonly posedin2Dsingleslip19–23bythepresentauthorsandby
3
referred to as “law of similitude” (for a general overview Mesarovic et.al.24. Later these models were extended to
8
see Ref.5). multiple slip by Limkumnerd and Van der Giessen25. In
7
0 Sincetheearly1960sseveraltheoreticalandnumerical order to obtain closed sets of evolution equations for the
. attempts have been made to model dislocation pattern dislocation densities, assumptions about the correlation
1
formation. Thefirstmodelswerebasedonanalogieswith properties of dislocation systems have to be made, but
0
6 other physical problems like spinodal decomposition6, thereareessentialdifferenceswithearlierphenomenolog-
1 patterning in chemical reaction-diffusion systems7,8, in- ical models: Not only can these assumptions be shown
: ternal energy minimization9, or noise-induced phase to be consistent with the fundamental scaling proper-
v
i transitions10,11. Since, however, it is difficult to see how ties of dislocation systems20,26, but also the numerical
X these models are related to the rather specific proper- parameters entering the theories can be deduced from
r tiesofdislocations(likelongrangescalefreeinteractions, DDD simulations in a systematic manner without fitting
a
motion on well defined slip planes, or different types of them in an ad-hoc manner to desired results. As a con-
short-range effects, etc.) they have to be considered as sequence, the models are predictive and can be directly
attemptstoreproducesomephenomenologicalaspectsof validated by comparing their results to the outcomes of
thepatternsbasedonheuristicanalogies,ratherthande- DDD simulations20,21,23,27.
riving them from the physics of dislocation systems. Since 2D models are not able to account for several
To identify the key ingredients responsible for the effects playing an important role in the evolution of the
emergence of inhomogeneous dislocation patterms, dis- dislocationnetwork, mostimportantlydislocationmulti-
cretedislocationdynamics(DDD)simulationisapromis- plication and junction formation, several 3D continuum
ingpossibility12–15. Themajordifficulty,however,isthat theories have been proposed. For example Acharya28
according to experimental observations the characteris- and later on Chen et.al.29 proposed models in which the
2
coarse grained dislocation density (Nye’s) tensor plays closed set of equations an assumption similar to the “lo-
a central role. This tensor provides information about cal density approximation” often used for many-electron
the distribution of ’geometrically necessary’ dislocations systems is used. After this, it is shown that the same
with excess Burgers vector. However, incipient dislo- evolution equations can be derived in a complementary
cation patterns are often associated with modulations manner,usingtheformalismofphasefieldtheories,from
in the total density of dislocations rather than modula- a functional which expresses the energy of the disloca-
tions of the Burgers vector content. Hence it is doubtful tion system as a functional of the dislocation densities.
whether models which concentrate on the transport of In the last part, by linear stability analysis of the trivial
excess Burgers vector only can capture patterning. solution of the field equations the mechanisms for char-
Applying statistical approaches, El-Azab30 and Sed- acteristic length scale selection in dislocation patterning
lacek et.al.31,32 suggested methods to handle curved dis- are discussed.
locations, but the evolution equations obtained are valid
only for quite specific situations. Considerable progress
towards a generic statistical theory of dislocation mo- II. DENSITY BASED REPRESENTATION OF A
tion in 3D has been made by Hochrainer et.al.33–35 by DISLOCATION SYSTEM: LINKING MICRO- TO
MESOSCALE
deriving a theory of dislocation density transport which
applies to systems of three-dimensionally curved dislo-
cations and can represent the evolution of generic dislo- Let us consider a system of N parallel edge disloca-
cation systems comprising not only ’geometrically neces- tions with line vectors (cid:126)l = (0,0,1) and Burgers vectors
sary’ but also ’statistically stored’ dislocations with zero (cid:126)b = ±(b,0,0). The force in the slip plane acting on a
±
net Burgers vector. Depending on the desired accuracy, dislocation is bτ where τ is the sum of the shear stresses
the approach of Hochrainer allows to systematically de- generated by the other dislocations, and the stress τ
ext
rive density-based theories of increasing complexity. Re- arising from external boundary displacements or trac-
cently this work was complemented by the derivation of tions. It is commonly assumed that the velocity of a dis-
matching energy functionals based upon averaging the location is proportional to the shear stress acting on the
elastic energy functionals of the corresponding discrete dislocation (over-damped dynamics)19,21. So, the equa-
dislocation systems36. In parallel, it was demonstrated tionofthemotionoftheithdislocationpositionedinthe
howsuchenergyfunctionalscanbeusedtoderiveclosed- xy plane at point (cid:126)r is
i
formdislocationdynamicsequationswhichareconsistent
 
not only with thermodynamics, but also with the con- N
straints imposed by the ways in which dislocations move ddxti =M0bτ((cid:126)ri)=M0bi (cid:88) sjτind((cid:126)ri−(cid:126)rj)+τext
in 3D37. j=1,j(cid:54)=i
Concerning dislocation patterning, the general struc- (1)
tureofcontinuumtheoriesthatisrequiredforpredicting where M0 is the dislocation mobility, si = bi/b = ±1
dislocation patterns that are compatible with the “prin- is the sign of the ith dislocation (in the following often
cipleofsimilitude”hasbeenrecentlydiscussedbyZaiser labeled ’+’ for s = 1 and ’−’ for s = −1), τext is the
andSandfeld26. Itwasarguedthatnootherlengthscales external stress, and
√
except the dislocation spacing (1/ ρ) should appear in
bµ x(x2−y2)
such theories - in other words, such theories ought to be τ ((cid:126)r)= (2)
scale free. In Ref.26 the authors also discussed a possible ind 2π(1−ν) (x2+y2)2
extension of the 2D continuum theory of Groma et.al21
istheshearstressgeneratedbyadislocationinaninfinite
leading to the instability of the homogeneous dislocation
medium. Here µ is the shear modulus and ν is Poisson’s
density in a deforming crystal (details are discussed be-
ratio.
low).
After ensemble averaging as explained in detail
Some remarkable steps towards modeling pattern for- elsewhere19,21,22 one arrives at the following evolution
mation have been made by Kratochvil et.al.38 and re- equations:
cently in a 3D mean field theory by Xia and El-Azab39.
Inordertoobtainpatterns,however,inbothmodelsspe-
cific microscopic dislocation mechanisms (sweeping nar- ∂ ρ ((cid:126)r,t)+M b∂ [ρ ((cid:126)r,t)τ (3)
t + 0 x + ext
row dipoles by moving curved dislocations, or cross slip, +(cid:82) {ρ ((cid:126)r,(cid:126)r(cid:48),t)−ρ ((cid:126)r,(cid:126)r(cid:48),t)} τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48)(cid:3)=0,
++ +− ind
respectively) had to be invoked. In the present paper we
adopt a more minimalistic approach where we consider
no other mechanisms apart from the elastic interaction ∂ ρ ((cid:126)r,t)−M b∂ [ρ ((cid:126)r,t)τ (4)
t − 0 x − ext
of dislocation lines. We analyze in detail the proper- −(cid:82) {ρ ((cid:126)r,(cid:126)r(cid:48),t)−ρ ((cid:126)r,(cid:126)r(cid:48),t)} τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48)(cid:3)=0
−− −+ ind
ties of a 2D single slip continuum theory of dislocations
that is a generalization of the theory we have proposed where ρ ((cid:126)r) and ρ ((cid:126)r,(cid:126)r(cid:48)) with s,s(cid:48) ∈ {+,−} are the
s s,s(cid:48)
earlier19–23. In the first part, the general structure of ensemble averaged one and two-particle dislocation den-
the dislocation field equations is outlined. To obtain a sityfunctionscorrespondingtothesignsindicatedbythe
3
subscripts. It should be mentioned that Eqs. (4,3) are depend on dislocation-dislocation correlations. Finally
exact, i.e. no assumptions have to be made to derive let us introduce the quantities
them, but certainly they do not represent a closed set of
τ +τ
equations. In order to arrive at a closed set of equations τ = + −, (13)
one has to make some closure approximation to express v 2
thetermsdependingonthetwoparticledensityfunctions τ = τ+−τ−. (14)
as functionals of the one particle densities (or one has to a 2
go to higher order densities.) The rest of this section is
With these quantities Eqs. (8,9) read
aboutsuggestingaclosureapproximationconsistentwith
discrete dislocation simulation results.
∂ ρ ((cid:126)r,t)+M b∂ {ρ [τ +τ +τ +τ ]}=0, (15)
For the further considerations it is useful to introduce t + 0 x + ext sc v a
the pair-correlation functions dss(cid:48)((cid:126)r1,(cid:126)r2) defined by the ∂tρ−((cid:126)r,t)−M0b∂x{ρ−[τext+τsc+τv−τa]}=0.(16)
relation
In explicit form the stresses τ and τ are given by
v a
ρ ((cid:126)r ,(cid:126)r ,t)=ρ ((cid:126)r )ρ ((cid:126)r )(1+d ((cid:126)r ,(cid:126)r )) (5)
ss(cid:48) 1 2 s 1 s(cid:48) 2 ss(cid:48) 1 2
τ ((cid:126)r)= (cid:82) [ρ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))+κ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))]
AccordingtoDDDsimulationsthepair-correlationfunc- v a s
∗τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48), (17)
tionsdefinedabovedecaytozerowithinafewdislocation ind
spacings20. As a result of this, if the total dislocation τ ((cid:126)r)= (cid:82) [ρ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))+κ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))]
a p a(cid:48)
density ρ = ρ+ +ρ− varies slowly enough in space, we ∗τind((cid:126)r−(cid:126)r(cid:48))d2r(cid:48), (18)
canassumethatthecorrelationfunctionsdependexplic-
itlyonlyontherelativecoordinate(cid:126)r1−(cid:126)r2,seeRefs.21,22. with
The direct (cid:126)r (or (cid:126)r ) dependence appears only through
1 2
the local dislocation density, i.e., d = 1(d +d +d +d ), (19)
s 2 ++ −− +− −+
dss(cid:48)((cid:126)r1,(cid:126)r2)=dss(cid:48)((cid:126)r1−(cid:126)r2,ρ((cid:126)r1)). (6) dp = 12(d+++d−−−d+−−d−+), (20)
d = 1(d −d −d +d ), (21)
(Since dss(cid:48) is short ranged in (cid:126)r1 −(cid:126)r2, it does not make a 2 ++ −− +− −+
anydifferenceifintheaboveexpressionρ((cid:126)r )isreplaced d = 1(d −d +d −d ). (22)
1 a(cid:48) 2 ++ −− +− −+
by ρ((cid:126)r ).)
2
In the case of a weakly polarized dislocation arrange- We note some symmetry properties of the pair corre-
ment where ρ −ρ (cid:28)ρ, the only relevant length scale lation functions: (i) the functions d and d must be
+ − ++ −−
istheaveragedislocationspacing. So, fordimensionality invariant under a swap of the two dislocations and thus
reasons the ρ dependence of d has to be of the form represent even functions of (cid:126)r; (ii) for dislocations with
ss(cid:48)
different signs one gets from the definition of correlation
(cid:112)
dss(cid:48)((cid:126)r1,(cid:126)r2)=dss(cid:48)(((cid:126)r1−(cid:126)r2) ρ((cid:126)r1)). (7) functions that d+−((cid:126)r)=d−+(−(cid:126)r). Hence ds((cid:126)r) and dp((cid:126)r)
are even functions, while the difference d −d ap-
By substituting Eq. (5) into Eqs. (3,4) and introduc- +− −+
pearing in d and d is an odd function.
ingtheGNDdislocationdensityκ=ρ −ρ onearrives a a(cid:48)
+ −
For the further considerations it is useful to introduce
at
the notations
∂ ρ ((cid:126)r,t)+M b∂ {ρ [τ +τ +τ ]}=0 (8)
t + 0 x + ext sc + (cid:90)
τ ((cid:126)r)=− ρ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48) (23)
f a ind
∂ ρ ((cid:126)r,t)−M b∂ {ρ [τ +τ +τ ]}=0 (9)
t − 0 x − ext sc −
referred to “friction stress” hereafter,
where
(cid:90) (cid:90)
τsc((cid:126)r)= τind((cid:126)r−(cid:126)r(cid:48))κ((cid:126)r(cid:48))d2r(cid:48), (10) τb((cid:126)r)= κ((cid:126)r(cid:48))ds((cid:126)r−(cid:126)r(cid:48))τind((cid:126)r−(cid:126)r(cid:48))d2r(cid:48) (24)
commonly called the “self consistent” or “mean field”
commonly called “back stress”,
stress, is a non-local functional of the GND density,
whereas the stresses (cid:90)
(cid:90) τd((cid:126)r)= ρ((cid:126)r(cid:48))dp((cid:126)r−(cid:126)r(cid:48))τind((cid:126)r−(cid:126)r(cid:48))d2r(cid:48) (25)
τ ((cid:126)r)= [ρ ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))
+ + ++
called “diffusion stress”, and
−ρ ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))]τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48), (11)
− +− ind
(cid:90)
and τ(cid:48)((cid:126)r)= κ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48). (26)
f a(cid:48) ind
(cid:90)
τ ((cid:126)r)=− [ρ ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))
− − −−
Since d and d are even functions in Eqs. (23,26)
++ −−
−ρ ((cid:126)r(cid:48))d ((cid:126)r−(cid:126)r(cid:48))]τ ((cid:126)r−(cid:126)r(cid:48))d2r(cid:48) (12) for nearly homogeneous systems the contribution of the
+ −+ ind
4
(cid:90)
differenced++−d−− toτf((cid:126)r)andτf(cid:48)((cid:126)r)canbeneglected τb((cid:126)r) = −∂xκ((cid:126)r) x˜ds((cid:126)r˜)τind((cid:126)r˜)d2r˜
resulting in
D
= −Gb ∂ κ((cid:126)r), (35)
τ(cid:48)((cid:126)r)= κ((cid:126)r)τ ((cid:126)r). (27) ρ x
f ρ((cid:126)r) f
√ √
where (cid:126)r˜ = ρ(cid:126)r, x˜ = ρx, and G = µ . With the
2π(1−ν)
From Eqs. (17,18,27) one gets same notations we find
(cid:90)
τv =−τf +τb (28) τd((cid:126)r) = −∂xρ((cid:126)r) x˜dp((cid:126)r˜)τind((cid:126)r˜)d2r˜
and A
= −Gb ∂ ρ((cid:126)r). (36)
κ ρ x
τ = τ +τ (29)
a ρ f d With Eqs. (28,35,29,36) the evolution equations (32,32)
read
After substituting Eqs. (28,18) into Eqs. (15,16) one
(cid:26) (cid:27)
concludes κ
∂ ρ=−M b∂ κτ −GbD ∂ κ−GbA∂ ρ ,
t 0 x mf ρ x x
∂ ρ ((cid:126)r,t)=
t + (37)
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
−M b∂ ρ τ +τ − 1− κ τ +τ (30) (cid:26) (cid:20) (cid:18) κ2(cid:19) (cid:21)
0 x + mf b ρ f d ∂tκ=−M0b∂x ρ τmf − 1− ρ2 τf
(cid:27)
κ
−GbD∂ κ−GbA ∂ ρ . (38)
∂ ρ ((cid:126)r,t)= x ρ x
t −
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27)
κ
It is important to point out that in general the corre-
+M b∂ ρ τ +τ − 1+ τ −τ (31)
0 x − mf b ρ f d lation functions are stress dependent. As a consequence,
the parameters C, D, and A introduced above can de-
with τmf =τext+τsc. pend on the long-range stress τmf, which is in general
Byaddingandsubtractingtheaboveequationsweob- a non-local functional of the excess dislocation density
tain κ. More precisely, from dimensionality reasons it follows
that parameters may depend on the dimensionless pa-
√
∂tρ((cid:126)r,t)= rameter τmf/(µb ρ). From the symmetry properties of
−M b∂ [κτ +κτ +ρτ ] (32) the correlation function one can easily see that C is an
0 x mf b d
odd, D, and A are even functions of τ . As a conse-
∂ κ((cid:126)r,t)= mf
t
quence, at τ = 0, C vanishes, while D, and A have
(cid:20) κ2 (cid:21) mf
−M b∂ ρτ +ρτ −ρτ + τ +κτ . (33) finite values and so they can be approximated up to sec-
0 x mf b f ρ f d ond order in τ by constants.
mf
To establish the stress dependence of the parameter C
Since according to DDD simulations19,21 and theoret- we note that due to the relation (see Ref.21)
ical arguments22,36, the correlation functions decay to
√
zero faster than algebraically on scales |x−x(cid:48)|(cid:29)1/ ρ, 1
∂ κ((cid:126)r,t)=− ∂ γ˙((cid:126)r,t), (39)
in the above expressions for τ and τ the densities ρ((cid:126)r(cid:48)) t b x
v a
andκ((cid:126)r(cid:48))canbeapproximatedbytheirTaylorexpansion
an explicit expression for the plastic shear rate in a ho-
around the point (cid:126)r. Assuming that the spatial deriva-
mogeneous system is given by
tives of the densities are small on the scale of the mean
√ √
dislocation spacing, ∂xκ/κ (cid:28) ρ,∂xρ/ρ (cid:28) ρ, we can (cid:20) (cid:18) κ2(cid:19) (cid:21)
retain only the lowest-order nonvanishing terms21. Since γ˙ =ρbM0 τmf − 1− ρ2 τf . (40)
τ (x,y) = −τ (−x,y) and τ (x,y) = τ (x,−y),
ind ind ind ind √
from the symmetry properties of the correlation func- where τ = µbC ρ. If we consider a system without
f
tions mentioned above one concludes that up to second excess dislocations, such a system exhibits a finite flow
order stress due to formation of dislocation dipoles or multi-
poles. For stresses below the flow stress, the strain rate
(cid:90)
τ ((cid:126)r) = −(cid:112)ρ((cid:126)r) d ((cid:126)r˜)τ ((cid:126)r˜)d2r˜ is zero. It must therefore be
f a ind
√
− ∂xρx3ρ/(2(cid:126)r)(cid:90) da((cid:126)r˜)x˜2τind((cid:126)r˜)d2r˜ C =(cid:26)αα,µτbm√fρ, ||ττmmff||<≥ααµµbb√ρρ (41)
(cid:18) (cid:19)
(cid:112) η Here the former case corresponds to stresses below the
= −µbC ρ((cid:126)r) 1+ ∂ ρ((cid:126)r) , (34)
ρ2 xx flow stress, and the latter case to stresses above the
5
flow stress. In a system where excess dislocations are introduced via ad-hoc assumptions, we assume that the
present, the excess dislocations cannot be pinned by number of dislocations is conserved, i.e., we consider the
dipole/multipole formation but their effective mobility limit f(ρ ,ρ ) = 0 and focus on the ρ dependence of
+ − ±
is strongly reduced. Only in the limit κ=ρ the effective the velocities v .
±
mobility of the dislocations reaches the value M of the We adopt the standard formalism of phase field the-
0
free dislocation. ories of conserved quantities. Assuming proportionality
In conclusion we note that apart from the actual form between fluxes and driving forces we have
of τ the evolution equations were derived earlier21,22.
d (cid:26) (cid:20) (cid:21)(cid:27)
1+ζ δP 1−ζ δP
FTihneelpaonssdibVlealidmepnoarirtea4n0c,4e1o.fWτdehaalssobneoetnertehcaetn,tlfyorraaisreadthbeyr v+ = −M0 ∂x 2 δρ+ − 2 δρ− (43)
special dislocation configuration where dislocations are (cid:26) (cid:20)1+ζ δP 1−ζ δP (cid:21)(cid:27)
v = −M ∂ − (44)
artificially placed on periodically arranged slip planes, a − 0 x 2 δρ 2 δρ
− +
diffusion like term proportional to ∂ ρ has been derived
x
by Dogge et. al.42. However, in this type of analysis an where P[ρ ,ρ ] is the phase field functional and ζ is a
+ −
artificiallengthscaleparameterisintroducedintermsof parameter. FromEqs. (42,43,44)theevolutionequations
the slip plane spacing and, as demonstrated elsewhere43, for the dislocation densities derive as
the results depend crucially on the difficult-to-justify as- (cid:26) (cid:20) (cid:21)(cid:27)
δP δP
sumptionofastrictlyperiodicarrangementofactiveslip ∂ ρ −∂ ρ M ∂ +ζ∂ = 0, (45)
t + x + 0 xδκ x δρ
planes.
(cid:26) (cid:20) (cid:21)(cid:27)
Since the parameters D and A are directly related to δP δP
∂ ρ +∂ ρ M ∂ −ζ∂ = 0. (46)
the correlation function, it should be possible to deter- t − x − 0 xδκ x δρ
mine their actual values from correlation functions ob-
Accordingly we find
tained by DDD simulations. For reasons that will be
discussed elsewhere, this is however difficult unless ana- (cid:26) (cid:27)
δP δP
lyticalapproximationsforthesefunctionsareknown,and ∂tρ = ∂x κM0∂xδκ +ζρM0∂x δρ , (47)
indirect methods are more reliable. Thus, for nontrivial
(cid:26) (cid:27)
systems like a slip channel under load21, the dislocation δP δP
∂ κ = ∂ ρM ∂ +ζκM ∂ . (48)
configurationaroundhardinclusions,27, andtheinduced t x 0 xδκ 0 x δρ
excessdislocationssurroundinganygivendislocationina
dislocationsystem36,44,theparameterD hasbeendeter- Inpreviousderivations22thetermsproportionaltoδP/δρ
were not considered (ζ = 0). As discussed below, these
mined by direct comparison of DDD simulation results
terms are closely related to τ introduced above. Con-
and solutions of the continuum equations. D is found d
cerning the actual form of P[ρ ,ρ ] it is useful to split
to be in the range of 0.25 to 0.8, while A is found to be + −
about 1.25 by Valdenaire41. it into two parts, the “mean field” or “self consistent”
part P and the “correlation” part P which are defined
sc c
below.
Inordertoobtaintheequationforthemeanfieldstress
III. VARIATIONAL APPROACH
τ from a variational principle for P we represent the
mf sc
associated elastic energy using the Airy stress function
Wehaveshowninearlierworkthattheevolutionequa-
formalism. By taking
tions for the two densities of positive and negative dislo-
cations as studied above can be cast into the framework (cid:90) (cid:20) 1−ν (cid:21)
of phase field theories22,23,44,45. The terms proportional Psc[χ,ρ+,ρ−]= − 4µ ((cid:52)χ)2+bχ∂yκ d2r (49)
to τ , however, were not included into the earlier consid-
a
erations. We now demonstrate that these terms can be the minimum condition
equally obtained from an appropriate energy functional
δP
using the phase field formalism. sc =0 (50)
δχ
For a system of straight parallel edge dislocations
with Burgers vectors parallel to the x axis the evolu- leads to the equation
tion equations of dislocation densities ρ and ρ have
+ −
the form22,42 1−ν (cid:52)2χ=b∂ κ (51)
2µ y
∂ ρ +∂ [ρ v ]=±f(ρ ,ρ ) (42)
t ± x ± ± + −
where χ is the Airy stress function from which the shear
in which we consider only dislocation glide, climb is ne- stress derives via τ = ∂ ∂ χ. The general solution of
mf x y
glected. Herev±istheglidevelocityofpositiveornega- Eq.(51)isτ givenbyEq.(10)plustheexternalstress.
mf
tivesigneddislocations,andf(ρ ,ρ )isatermaccount- Substituting Eq. (49) into (44) and (43) one gets
+ −
ing for dislocation multiplication or annihilation. Since
multiplication terms cannot be derived for 2D systems ∂tρ+((cid:126)r,t)+M0b∂x(ρ+τmf)=0, (52)
(straight dislocations cannot multiply) but need to be ∂ ρ ((cid:126)r,t)−M b∂ (ρ τ )=0. (53)
t − 0 x − mf
6
P merelyrecoversthemeanfieldpartofthedislocation were M(x) is a nontrivial mobility function defined as
sc
velocities v and v but not the terms which are related
+ −
to dislocation-dislocation correlations. It thus needs to  κ2x if |x|<x
becomplementedbya’correlation’partofthephasefield  ρ2 (cid:104) (cid:16) (cid:17)(cid:105) 0
foufnacstiimonialal.raWveeruasgeinagfostrrmatweghyicahscuasnedbeabdeorvievefodrbtyhemderainvs- M(x)=M0sgn(x) |x|−x0 1− κρ22 if |x|(cid:61)x0
ing forces, but applied to the elastic energy functional of (60)
√
the discrete dislocation system36. For the present dislo- with x0 = αµb2 ρ (see fig.1). It is easy to see that this
cationsystemtheresulting‘correlation’partofthephase
field functional is given by
M
(cid:90) (cid:20) (cid:18) ρ (cid:19) Gb2Dκ2(cid:21) M
P = Gb2Aρln + d2r. (54) 0
corr ρ 2 ρ
0
Thisexpressionistantamounttousingalocaldensityap- κ2x 450
proximation for the correlation part (the functional con- ρ2
tainsonlyonthelocalvaluesofthedislocationdensities,
not any gradients or non-local expressions). x x
0
Weconsiderweaklypolarizeddislocationarrangements
where κ/ρ (cid:28) 1 and ∂ ρ/ρ (cid:28) ρ1/2. By neglecting terms Figure 1: The M(x) mobility function
x
of higher than first order in κ/ρ and ∂ ρ/ρ3/2, we find
x
that mobility function recovers Eq. (38) within the general
framework of a phase field theory.
∂ ρ = As concluding remarks of this discussion it should be
t +
(cid:20) (cid:18) D A (cid:19)(cid:21) noted that:
−∂ ρ M b τ −Gb ∂ κ−Gbζ ∂ ρ ,(55)
x + 0 mf ρ x ρ x
• Since thermal energies are typically 4-5 orders of
∂tρ− = magnitude lower than the elastic energies associ-
(cid:20) (cid:18) D A (cid:19)(cid:21) ated with the presence of dislocations36, entropic
+∂ ρ M b τ −Gb ∂ κ+Gbζ ∂ ρ .(56)
x − 0 mf ρ x ρ x contributionstothedislocationfreeenergyareneg-
ligibleuptothemeltingpoint. Therequirementof
From the above equations the evolution equations for κ thermodynamic consistency of any theory in this
and ρ read case reduces to the trivial requirement that the
elasticenergymustdecreaseandcanneverincrease
(cid:26) (cid:27)
κ during system evolution (the latter would imply a
∂ ρ=−M b∂ κτ −GbD ∂ κ−GbA∂ ρ , (57)
t 0 x mf ρ x x transfer of energy from the heat bath to the elastic
energy of the crystal). The comparison of the evo-
lution equations obtained by direct averaging and
(cid:26) (cid:27)
κ the phase field formalism indicates that the evolu-
∂ κ=−M b∂ ρτ −GbD∂ κ−GbA ∂ ρ . (58)
t 0 x mf x ρ x tion equations of the dislocation densities can nev-
ertheless be cast into the phase field framework.
With ζ = 1, apart from the term containing the “fric-
tion” stress τ , Eqs. (57, 58) are equivalent to Eqs. (37, • The irrelevance of thermal fluctuations makes it
f
38). So,withtheappropriateformofthecorrelationterm mandatory to introduce a nontrivial on/off type
in the phase field functional (a form which derives from mobility function. The reason is that dislocations,
ensemble averaging the energy functional of the discrete as they move through the crystal, not only experi-
dislocation system), by applying the standard formalism encetheaverageenergyexpressedbythefunctional
ofphasefieldtheorieswerecovertheevolutionequations P, but also energy fluctuations on scales compara-
ofthedislocationdensitiesderivedbyensembleaveraging ble to the dislocation spacing. The magnitude of
√
the equations of motion of individual dislocations. How- these fluctuations scales like αGb ρ, as discussed
ever, the flow stress which plays a crucial role in plastic e.g. by Zaiser and Moretti46. Since thermal fluc-
deformation of any material can not be directly derived tuations of sufficient magnitude are not available,
within the traditional framework of phase field theories. the work required to overcome these fluctuations
To resolve this issue we modify Eq. (48) to allow for a and to enable sustained dislocation motion must
non-linear dependency on the driving force δP/δκ. The be provided by the local stress. This is reflected
modified equation is given by by the mobility functions M(x) which introduce a
contribution akin to dry friction into the dynamics
(cid:26) (cid:18) (cid:19) (cid:27)
δP δP - for the derivation of a similar friction-like stress
∂ κ = ∂ ρM ∂ +κM ∂ (59)
t x xδκ 0 x δρ contribution in 3D (see36).
7
• The forms of the phase field functional and the V. PATTERN FORMATION
mobility functions suggested here represent only
the simplest possible approximation, which is cor- In the following we discuss under what conditions the
rect for weakly polarized and weakly inhomoge- evolution equations derived above can lead to instability
neous dislocation arrangements only. For some resultingindislocationpatternformation. Onecaneasily
specific problems like dislocation distribution next see that the trivial homogeneous solution ρ = ρ , κ = 0
0
toboundaries,orstronglyinhomogeneoussystems, and τ = τ satisfies Eqs. (37,38,51), where ρ and τ
mf 0 0 0
onemayhavetoconsideradditionalterms(seee.g. are constants representing the initial dislocation density
Ref.23). and the external shear stress, respectively. The stability
of the trivial solution can be analyzed by applying the
standard method of linear stability analysis. One can
IV. TIME VARIATION OF THE PHASE FIELD
easily see that nontrivial behavior can happen only in
FUNCTIONAL √
the flowing regime i.e. if |τ | > αµb ρ , so we consider
0 0
only this case.
The time derivative of the phase field functional is By adding small perturbations to the dislocation den-
dP δP δP δP sities and the Airy stress function in the form
= ∂ ρ+ ∂ κ+ ∂ χ (61)
dt δρ t δκ t δχ t
ρ((cid:126)r,t) = ρ+δρ((cid:126)r,t)
but due to the condition (50) the third term vanishes. κ((cid:126)r,t) = δκ((cid:126)r,t) (68)
For simplicity in the following the phase field functional
χ((cid:126)r,t) = τ xy+δχ((cid:126)r,t)
0
isalwaysevaluatedatχ=χ definedbyδP/δχ| =
min χmin
0. Hence and keeping only the leading terms in the perturbations,
dP (cid:18)δP δP (cid:19)(cid:12)(cid:12) equations (37,38,51) become
dt = δρ∂tρ+ δκ∂tκ (cid:12)(cid:12) . (62)
χ=χmin ∂tδρ = M0∂x[GbA∂xδρ−τ0δκ] (69)
By substituting Eqs. (47,59) into Eq. (62) we obtain
after partial integration ∂ δκ = −M Θ ∂ [ρ ∂ ∂ δχ−GbD∂ δκ] (70)
t 0 f x 0 x y x
√
ddPt = −(cid:26)∂xδδPρ(cid:27)(cid:26)κM0∂xδδPκ +ρM0∂xδδPρ(cid:27) −M0Θf(cid:20)τ∗−αµb 2ρ0(cid:21)∂xδρ.
(cid:26) (cid:27)(cid:26) (cid:20) (cid:21) (cid:27)
δP δP δP
− ∂ ρM ∂ +κM ∂ (.63)
xδκ xδκ 0 x δρ (cid:52)2δχ=4πGb∂ δκ (71)
y
√ √
If |∂xδP/δκ|<αµb2 ρ In these expressions, τ∗ = τ0 −αµb ρ0, and the step
function Θ = Θ(τ∗) is zero if the applied stress is be-
dP (cid:18)∂ δP (cid:19)(cid:18)ρ, κ (cid:19)(cid:18)∂ δP (cid:19) f
=−M xδρ xδρ . (64) low the flow stress in the homogeneous reference state,
dt 0 ∂xδδPκ κ, κ2/ρ ∂xδδPκ and 1 otherwise. To obtain the above equations it was
taken into account that the first-order variation of the
Since M is positive and the matrix
0
flow stress is given by
(cid:18) (cid:19)
ρ, κ √
κ, κ2/ρ (65) αµb ρ0δρ
δτ = . (72)
f 2 ρ
0
is positive definite it follows that dP/dt ≤ 0. In the
√
flowing regime (|∂ δP/δκ|>bαµb2 ρ) we find that The solution of Eqs. (71,69,70) can be found in the
x
form
dP (cid:18)∂ δP (cid:19)(cid:18)ρ, κ(cid:19)(cid:18)∂ δP (cid:19)
dt =−M0 ∂xδδPρ κ, 0 ∂xδδPρ (66)  δρ   δρ0  (cid:18)λ √ (cid:19)
(cid:18) δP(cid:19) xδκ(cid:18) δP(cid:19) xδκ δδχκ =δδχκ0 exp t0t+i ρ0(cid:126)k(cid:126)r (73)
− M ρ ∂ sgn ∂ × 0
0 xδκ xδκ
(cid:20)(cid:12)(cid:12)(cid:12)(cid:12)∂xδδPκ(cid:12)(cid:12)(cid:12)(cid:12)−αµb2√ρ(cid:18)1− κρ22(cid:19)(cid:21) (67) wabeoreve(cid:126)kfiosramdiimnteonsEioqnsl.ess(7q1u,6a9n,t7it0y).iAnfttehresuflbowstiintugtirneggitmhee
(Θ =1) we get
f
ThisagainensuresthatdP/dt≤0. So,wefoundinboth
casesthatthephasefieldfunctionalcannotincreasedur- (cid:18) λ+Akx2, i(γ˙(cid:48)+2α(cid:48))kx (cid:19)(cid:18) δρ (cid:19)=0 (74)
ing the evolution of the system. Since our phase field i(γ˙(cid:48)−α(cid:48))k , λ+Dk2+T((cid:126)k) δκ
x x
functional is tantamount to the averaged elastic energy
functional, this ensures thermodynamic consistency of where the notations t =b2Gρ /B, T((cid:126)k)=4πk2k2/|(cid:126)k|4,
√ 0 0 x y
our theory. γ˙(cid:48) = τ /(Gb ρ ), and α(cid:48) = π(1−ν)α were introduced.
∗ 0
8
Note that in the above equations each of the parameters
are dimensionless and γ˙(cid:48) is proportional to the average
0.4
shear rate γ˙ =M b2ρ τ∗.
0 0
Eq. (74) has nontrivial solutions if 0.2
(λ+Ak2)(λ+Dk2+T((cid:126)k))+k2β =0 (75) λ+ 0
x x x
with β =(γ˙(cid:48)+2α(cid:48))(γ˙(cid:48)−α(cid:48)). This leads to -0.2
(A+D)k2+T((cid:126)k) -0.4
λ =− x
± 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
(cid:113)[(D+A)k2+T((cid:126)k)]2−4k2[β+A(Dk2+T((cid:126)k))] kx
x x x
± .
2
Figure 3: The λ (k ,0) function at A = 1, D = 1 and β =
(76) + x
−1.
It follows that the condition for the existence of growing
perturbations (λ>0) is
For negative β, the λ (k ,k ) function has two equal
+ x y
maxima along the x axis located at
[β+AT((cid:126)k)+ADk2]<0. (77)
x
(cid:113)
T((cid:126)k)cannotbenegativeanditvanishesif(cid:126)k isparallelto k2 =−2β−1+ 1+ (A4−ADD)2
either the x or to the y axis. Thus, β <0 is a necessary x (A−D)2
and sufficient condition for instability. This condition
It should be stressed that according to Eq. (73) the ac-
requires that (i) the system is in the flowing phase and
(ii) γ˙(cid:48) must be smaller than α(cid:48). In this case there ex- t√ual wave vector of the fastest growing perturbation is
ists a region in the(cid:126)k space in which perturbations grow. ρ0(cid:126)kmax. So, in agreement with the principle of simili-
tude observed experimentally the characteristic pattern
Perturbations with wave vectors outside this region de- √
wavelengthscaleswiththedislocationspacing1/ ρ . It
cay in time (see figs. 2,3). This results in a length scale 0
is important to note at this point that the diffusion-like
τ termintroducedhereplaysacrucialroleincharacter-
d
istic wavelength selection. At A = 0 (corresponding to
τ = 0, see Eq. (36)) perturbations of all wave vectors
d
0.4 would grow and there would be no mode of maximum
1 growth rate. In accordance with this, by analyzing the
0.2
stability of the homogeneous solution of the 3D contin-
0
0.5 uum theory of dislocations proposed by Hochrainer et.
-0.2 al33–35, Sandfeld and Zaiser concluded47 that the mean
field and the flow stresses generate instability but they
y 0 -0.4
k
do not result in length scale selection.
-0.6
Within the general framework introduced in the sec-
-0.5
-0.8 ond section there is another way leading to dislocation
-1 pattern formation which can operate even if A = 0, but
-1
whichrequirestheconsiderationofhigher-ordergradients
-1.2
inthedislocationdensities. Untilnowwehaveneglected
-1 -0.5 0 0.5 1
the term proportional to ∂ ρ/ρ3/2 in the expression for
xx
kx the flow stress in Eq. (34). Without going into the de-
tails of the derivation one can find that with this term,
but with A = 0, the evolution equations (69,70) get the
Figure 2: The λ+(kx,ky) function at A = 1, D = 1 and form
β = −1. The function is positive within the region marked
by the contour line λ (k ,k )=0.
+ x y
∂ δρ=−M b∂ [τ δκ] (79)
t 0 x 0
selection corresponding to the fastest growing periodic
perturbation(cid:126)k defined by the condition
max
(cid:12) ∂ δκ=−M bΘ ∂ [ρ ∂ ∂ δχ−GbD∂ δκ] (80)
dλ+((cid:126)k)(cid:12)(cid:12) =0 (78) t (cid:20) 0 f x √0ρx y Gbη x (cid:21)
d(cid:126)k (cid:12)(cid:12)(cid:126)kmax −M0bΘf τ∗−αµb 20∂xδρ− √ρ0∂xxxδρ ,
9
whereη isaconstantandforsimplicitythetermsrelated Irrespective of the pattern selection mechanism and in
toτ areneglected. Aftersubstitutingthesolutiongiven line with previous work47, we find that there are two re-
a
byEq. (73)intoEqs. (79,80)intheflowingregime(Θ = quirements for patterning: First, the system must be in
f
1) we get theplasticallydeformingphase,second,therateofshear
must not be too high (γ˙(cid:48) ≤α(cid:48)). This condition indicates
(cid:18) λ, i(γ˙(cid:48)+2α(cid:48))kx (cid:19)(cid:18) δρ (cid:19)=0 that patterning as studied here can not be understood
ik [(γ˙(cid:48)−α(cid:48))+ηk2], λ+Dk2+T((cid:126)k) δκ as a energy minimization process, despite the fact that
x x x
the dynamics which we investigate minimizes an energy
(81)
functional. This seemingly paradox statement becomes
Eq. (81) has nontrivial solutions if clearer if we consider the limit α→0 where the mobility
functions become trivial. In this limit, the critical strain
λ(λ+Dk2+T((cid:126)k))+k2[(γ˙(cid:48)+2α(cid:48))(γ˙(cid:48)−α(cid:48)+ηk2)]=0 rate where patterning vanishes goes to zero. Thus, the
x x x
patterning is an effect of the non-trivial mobility func-
(82)
tion which introduces a strongly non-linear, dry-friction
like behavior into the system. This aspect of the prob-
leading to
lem, whichcontradictsthelowenergyparadigmandem-
Dk2+T((cid:126)k) phasizes the dynamic nature of the patterning process,
λ1,2 =− x 2 (83) clearly should be further studied by extending the anal-
ysis into the non-linear regime.
(cid:113)
[Dk2+T((cid:126)k)]2−4k2[(γ˙(cid:48)+2α(cid:48))(γ˙(cid:48)−α(cid:48)+ηk2)] In the limit of low strain rates, γ˙ (cid:28)ρbM τ , the se-
x x x 0 ext
±
2 lectedpatternwavelengthbecomesindependentonstrain
(84) rate. The predictions in this regime agree well with ex-
perimental observations: With A = 1.2541, D = 0.2544,
The condition to growing perturbation is α = 0.2, and ν = 0.3, we find a preferred wave-vector
√
|k | =≈ 0.42 ρ corresponding to a wavelength of about
x
(γ˙(cid:48)−α(cid:48))+ηk2 <0 (85) 15 dislocation spacings, in good agreement with typi-
x
cal observations. The preferred patterns corresponds to
Provided that η > 0 and γ˙(cid:48) < α(cid:48) there is again a region dislocation walls perpendicular to the active slip plane,
in the(cid:126)k space in which perturbations grow, and one can again in agreement with observations and discrete simu-
again find the wavelength corresponding to the fastest lations. This agreement does not mean that the present,
growing perturbation. It should be noted that in this very simple considerations alone provide a complete the-
casethelengthscaleselectioniscausedbyasecondorder ory of dislocation patterning – in particular the essen-
effectinthesensethattheterm∂ ρ/ρ3/2 isobtainedby tial aspect of dislocation multiplication, and hence work
xx
the second order Taylor expansion in Eq. (17) while τ hardening, ismissing. However, itindicatesthatwemay
d
given by Eq. (36) corresponds to a first order one. capturesomeoftheessentialfeaturesoftherealprocess.
VI. CONCLUSIONS
Acknowledgments
In summary the general framework explained in detail
in section II and III is able to account for the emergence The authors are grateful to Prof. Alponse Finel for
of growing fluctuations in dislocation density leading to drawingtheirattentiontothefactthatthe“asymmetric”
pattern formation in single slip. The primary source of stresscomponentneglectedinearlierconsiderationsmay
√
instabilityisthe ρtypeofdependenceoftheflowstress, play a significant role. Financial supports of the Hun-
but alone it cannot lead to length scale selection. As garian Scientific Research Fund (OTKA) under contract
it is shown above there are two alternative ways (the numbers K-105335 and PD-105256 and of the European
diffusion-like term associated with the stress τ , or the Commission under grant agreement No. CIG-321842 are
d
∂ typetermintheflowstress)leadingtocharacteristic alsoacknowledged. PDIissupportedbytheJ´anosBolyai
xx
length scale of the dislocation patten. Scholarship of the Hungarian Academy of Sciences.
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