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Sapienza Università di Roma
Dottorato di Ricerca in Fisica
Scuola di dottorato “Vito Volterra”
Fluctuations in DeformingMaterials
Thesis submitted to obtain the degree of
“Dottore di Ricerca” - Doctor Philosophiæ
PhD in Physics - XXI cycle - October 2008
by
Fabio Leoni
Program Coordinator Thesis Advisors
Prof. Enzo Marinari Dr. Stefano Zapperi
Prof. Luciano Pietronero
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Contents
Acronyms and abbreviations . . . . . . . . . . . . . . . . . . .
. . . . . . . iii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . v
Introduction 1
1 Crystal plasticity 7
1.1 Continuum macroscopic description of crystal plasticity . .
. . . . . . 7
1.2 Plasticity at the micro scale: dislocations and fluctuation
phenomena 11
1.3 Theoretical approaches . . . . . . . . . . . . . . . . . . .
. . . . . . . 16
1.3.1 Statistical and stochastic approaches . . . . . . . . . .
. . . . 17
1.3.2 Discrete dislocation dynamics . . . . . . . . . . . . . .
. . . . 19
1.3.3 Stochastic continuum models . . . . . . . . . . . . . . .
. . . 23
1.3.4 Phase-field models . . . . . . . . . . . . . . . . . . . .
. . . . 24
2 Grain boundary diffusion in a crystal 27
2.1 Dislocation assemblies . . . . . . . . . . . . . . . . . . .
. . . . . . . 27
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
. . . . . 30
2.2.1 Dislocation model for grain boundary dynamics . . . . . .
. . 30
2.2.2 Flat grain boundary . . . . . . . . . . . . . . . . . . .
. . . . 33
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ii CONTENTS
2.2.3 Flexible grain boundary . . . . . . . . . . . . . . . . .
. . . . 36
2.2.4 Continuum theory . . . . . . . . . . . . . . . . . . . . .
. . . 39
3 Slip line formation at the initial stage of plastic
deformation 43
3.1 Slip lines . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
3.1.1 Experimental investigation of slip lines . . . . . . . . .
. . . . 43
3.1.2 The yielding transition . . . . . . . . . . . . . . . . .
. . . . . 46
3.2 Dislocation pile-up model for slip lines . . . . . . . . . .
. . . . . . . 46
3.2.1 Dislocations pile-up . . . . . . . . . . . . . . . . . . .
. . . . . 46
3.2.2 The double-ended pile-up model . . . . . . . . . . . . . .
. . . 47
3.2.3 Absence of phase transition without pinning centers . . .
. . . 51
3.2.4 Slip line formation as a non-equilibrium critical
phenomenon . 54
4 Instabilities in plastic flow: the Portevin-Le Chatelier
effect 61
4.1 Instabilities in plastic flow . . . . . . . . . . . . . . .
. . . . . . . . . 61
4.1.1 The Lüders band . . . . . . . . . . . . . . . . . . . . .
. . . . 62
4.1.2 The Portevin-Le Chatelier effect . . . . . . . . . . . . .
. . . . 64
4.1.3 Experimental investigations of PLC features . . . . . . .
. . . 67
4.2 Mesoscopic approach to modelling the PLC effect . . . . . .
. . . . . 69
4.2.1 Mesoscopic approaches . . . . . . . . . . . . . . . . . .
. . . . 69
4.2.2 The Ananthakrishna model . . . . . . . . . . . . . . . . .
. . 72
4.3 Discrete dislocation dynamics approach to the PLC effect . .
. . . . . 75
4.3.1 A single dislocation in a cloud of pinning centers . . . .
. . . . 75
4.3.2 Array of dislocations in presence of mobile pinning
centers . . 78
5 Stick-Slip dynamics in sheared granular media 83
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CONTENTS iii
5.1 Granular media and laboratory experiments . . . . . . . . .
. . . . . 83
5.1.1 Granular media . . . . . . . . . . . . . . . . . . . . . .
. . . . 83
5.1.2 Scaling and laboratory experiments . . . . . . . . . . . .
. . . 85
5.1.3 Granular medium sheared in a Couette geometry . . . . . .
. 87
5.2 Analogy between sheared granular media and dislocations
dynamics 92
5.2.1 Crackling noise . . . . . . . . . . . . . . . . . . . . .
. . . . . 92
5.2.2 Stick-slip behavior, time scales and SOC in SGM and PLC .
. 93
5.3 Friction in solids . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 95
5.3.1 Historical introduction . . . . . . . . . . . . . . . . .
. . . . . 95
5.3.2 Static friction at the micro-scale . . . . . . . . . . . .
. . . . . 96
5.3.3 Geometric age: the state variable . . . . . . . . . . . .
. . . . 99
5.3.4 Velocity jump experiments: the rate variable . . . . . . .
. . . 102
5.3.5 Rate and state formulation: the Rice-Ruina friction law .
. . . 103
5.4 Friction in granular media: a rate and state formulation . .
. . . . . . 105
5.4.1 A phenomenological approach to asymmetry and hysteresis .
. 105
5.4.2 A rate and state approach . . . . . . . . . . . . . . . .
. . . . 108
Conclusions 113
Appendix 117
A Linearized fluctuation theory 117
B 2d Ornstein-Uhlenbek process 121
Bibliography 127
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Acronyms and abbreviations
2d two-dimensional space3d three-dimensional spaceABBM
Alessandro, Beatrice, Bertotti and Montorsi modelASEP asymmetric
simple exclusion processCA cellular automataCN crackling noiseDD
dislocation dynamicsDSA dynamical strain agingGB grain boundaryGM
granular mediaGW Greenwood-Williamson modelHAGB high angle grain
boundaryLAGB low angle grain boundaryLYP lower yield pointMCIs
multi contact interfacesMD molecular dynamicsPLC Portevin-Le
Chatelier effectPK Peach-KoehlerPN Peierls-NabarroRR Rice-Ruina
lawSEM scanning electron micrographSOC self organized
criticalitySoS solid on solidSGM sheared granular mediaSRS strain
rate sensitivityTEM transmission electron microscopyUYP upper yield
point
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Nomenclature
This list contains notations which are used repeatedly.
b Burgers vectorσ stress tensorσl local stressG shear modulusΓ,
µ, χ damping coefficientsǫ strainǫel elastic strainǫpl plastic
strainǫL Lüders strainǫ̇ strain rateΘ hardening coefficientθ
position coordinateφ state variableξp pinning correlation lengthC
Hooke’s tensorC(...) displacement correlatorS strain rate
sensitivityD diffusion coefficientτ time constantτd delay timeNp
number of pinning centersfp pinning forceρm mobile dislocation
densityρim immobile dislocation densityρc Cottrell’s dislocation
densityr position vectorxcm center of mass of the grain
boundary
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vi Nomenclature
W 2 mean-square width of the grain boundaryA area of the grain
boundary〈·〉 average over configurationsρ densityη Gaussian noisek
spring constantV velocityf0 pinning force constanttw whiting timeρ̄
total dislocation densityL system sizeN number of dislocationsFN N
dislocations distribution functionNc cumulative frequencyJ
dislocation fluxψ scaled stressP plasticity indexH(·) Heaviside
step functionα density exponentβ velocity exponentν correlation
function exponentz dynamical exponentθp velocity relaxation
exponentζp density relaxation exponentγp profile exponentφp
velocity profile size exponentψp density profile size exponentfσ
non-dimensional correlation functionξσ stress fluctuation
correlation lengthǫc correlation strainτE acoustic energy bursts
exponentkǫ strain burst exponentt∗ nucleation timeXP position of
the pinning center Pvs stationary velocityρs stationary densityσint
interaction stress between dislocationsσimg stress due to image
dislocationsσeff effective stressΣr real contact surface
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Introduction
Since the advent of the statistical physics, we gained better
and new understanding
of phenomena described by a set of many interconnected degrees
of freedom. With
the improvement of experimental techniques, it has become
possible to investigate
materials at smaller and smaller scales and then more and more
phenomena should
be described in statistical terms.
In this dissertation, we try to gain a better understanding of
fluctuation phe-
nomena in deforming materials, investigating some aspects of
crystal plasticity and
friction between solids and granular materials, employing a
statistical physics ap-
proach. In both these two area of physics, a continuum
macroscopic model (such as
thermodynamics) has been employed to describe many phenomena. In
the contin-
uum models of plasticity, crystalline solids are considered as
homogeneous continua
which, in the absence of so-called plastic instabilities, deform
under homogeneous
loads in a smooth and spatially homogeneous manner because
fluctuations are sup-
posed to average out above a scale assumed to be small in
comparison with the
dimensions of the deforming body.
When a material is deformed plastically, dislocation line length
increases due
multiplication processes. Initially, the mutual interaction is
not significant as the
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2 Introduction
total dislocation density is not significant. In the course of
plastic deformation,
dislocation density increases. Thus, the spatial arrangement of
dislocations evolves
continuously. When dislocation density is relatively small, the
plastic flow properties
of the bulk material may simply reflect the behavior of isolated
mobile dislocations.
In contrast, as dislocations multiply, both local and long
ranged interactions of
dislocations become important. Thus, the contribution of
dislocation interactions to
the flow stress increases in course of plastic flow. Eventually,
collective behavior sets-
in, characterized by the emergence of dislocation-rich and
dislocation-poor regions.
Sometimes, rather well defined spatial patterns can be observed
beyond a certain
critical stress, strain, or dislocation density. Even when we
see ‘ordinary’ and smooth
behaviour of materials during deformation, at the micron scale
the system undergoes
abrupt changes, defects proliferate randomly and rearrange. The
system deforms
in an intermittent manner: what appears as a regular process on
large scales of
common observation, is characterised by apparently irregular
bursts and intricate
spatiotemporal signatures on microscopic scales. Intermittent
dislocation motion is
a generic feature of plasticity, not only in micron-scale
samples, as shown recently by
acoustic emission measurements and by earlier reports from slip
line kinematography
in macroscopic samples.
As a consequence, recent theoretical investigations have focused
on models that
describe the evolution of the dislocation densities in time and
space during plastic
flow. These approaches do not assume, a priori, that a certain
structure is formed
with characteristic geometrical dimensions, but attempt to
obtain these patterns.
We shall discuss different approaches to study deforming
materials, with particular
attention to models which describe plastic flow in terms of the
motion of discrete
lattice dislocations. The general problem of three-dimensional
interacting disloca-
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Introduction 3
tions in a medium is a formidable theoretical task. Some
problems for example
arise from the necessity to describe a system of flexible lines
with non-conserved
line length. Moreover, in a finite body the displacements and
stresses around a
dislocation depend on the external surface. Often the boundary
conditions can be
satisfied by placing an ‘image dislocation’ outside the crystal
in such a manner that
its stress field cancels that of the real dislocation at the
surface. But only for specific
geometries the results are simple. Linear assemblies, however,
such as pileups and
low-angle grain boundaries (LAGB) are often encountered
experimentally during re-
laxation processes and provide a nice exemplification for the
problem of dislocation
motions.
Granular physics is still a mixture of many different concepts,
modeling tools,
and phenomenological theories. A unified description exists for
a dilute system of
rapid grains with the kinetic theory of dissipative gases.
However the situation
that represents the vast majority of experimentally and
practically relevant cases is
that for which densities approaches close packing limit. In this
case no universal
description exists to date. Here we describe the shear response
of a granular medium
in a Couette geometry. These types of dense granular systems are
characterized by
stress chains. The key point is that in this type of granular
systems, forces are
inhomogeneous and intermittent if the system is deformed.
What characterises plastic and granular media that we have
briefly introduced,
as well as many more systems, is the role of randomness.
Geological faulting occurs
on randomly disordered substrates. Rupture is governed by random
forces exerted
by defects. Topological defects in deforming crystals are
subject to stress fields gen-
erated within their random arrangements. As the reader may
notice, microstructural
randomness, i.e. disorder, plays a key role. It does not simply
perturb the response
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4 Introduction
of the system to an external load. It induces a novel class of
phenomena, including
intermittency, avalanches, spatial heterogeneity and
roughening.
The origin of the noise, or randomness, is attributed to the
significantly fast time
scales of a large number of degrees of freedom. However, the
strength of the noise
itself may remain a parameter or it must be determined by other
considerations.
There could be several microscopic dislocation (or granular)
mechanisms that can
contribute to the noise and it may not be easy to model the
statistical properties
of the noise starting from microscopic mechanisms. An ubiquitous
type of noise
in nature arise from Brownian motion or random walk of
microscopic components.
For example, it has been employed successfully to explain the
Barkhausen crack-
ling noise. A generalization of Brownian motion is the bounded
random walk, or
Ornstein-Uhlenbeck process, that we have employed to schematize
the properties of
noise arising in dislocation dynamics problems as well as in
solid-granular friction.
The present manuscript is organised in the following manner.
Chapter I: here we study the diffusion of a LAGB in a
crystalline potential, clari-
fying the role of its internal degrees of freedom by means of
numerical simulations,
and comparing these results with those of the molecular dynamics
(MD) method.
Chapter II: as second dislocation assemblies system, here we
consider the problem
of slip line formation, employing the double-ended pile-up model
in which a new
way to schematize dislocation sources in two dimensions is
proposed. The model
explains experimental observations in fcc alloys of a transition
from homogeneous to
inhomogeneous slip, with increasing impurity concentration,
confirming the hypoth-
esis of critical behaviour and corroborating the conceptual
framework of a yielding
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Introduction 5
transition.
Chapter III: it is devoted to the study of instability in
plastic flow with partic-
ular attention to the Portevin-Le Catelier effect. Also in this
case we follow the
discrete dislocation dynamics approach, trying to explain the
phenomena as due to
the emergence of a collective motion of many interacting
dislocations in a landscape
of mobile impurities.
Chapter IV: finally, we analyse the shear response of a granular
media. To cope with
this problem means study what type of friction law is able to
reproduce dynamical
features, especially in the stick-slip regime. We follows in
first place a phenomeno-
logical approach to try to reproduce memory and asymmetry
effects. Then we use a
modified version of the rate and state friction law employed to
study solid on solid
friction.
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Chapter 1Crystal plasticity
1.1 Continuum macroscopic description of crystal
plasticity
In the elastic theory of solid bodies, topological defects are
disregarded and in the
continuum description fluctuations are supposed to average out
above the scale of a
representative volume element which is assumed to be small in
comparison with the
dimensions of the deforming body. The continuum description of
crystal plasticity
follows this scheme. Plastic deformation is usually defined as a
deformation that
persists after the driving force applied to the body is removed.
The geometrical
measure of deformations in a body is described by the strain
tensor ǫ. In a small
strain formulation the total strain tensor is considered as the
sum of an elastic and
a plastic part: ǫ = ǫel + ǫpl. In a linearly elastic material
the elastic strain is related
to the stress σ via ǫel = C−1σ where C is Hooke’s tensor of
elastic moduli. If the
stress is removed, the plastic strain ǫpl remains. The plastic
strain tensor ǫpl can be
expressed in terms of the shear strains on the different slip
systems as
ǫpl =∑
i
ǫiM i (1.1)
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8 Crystal plasticity
Figure 1.1: Different types of rate-independent plasticity: (a)
‘ideally plastic’ be-haviour; (b) plasticity with strain
hardening.
where i distinguishes the different slip systems, ǫi are the
respective scalar shear
strains, and M i is the projection tensor. If one excludes, as
in the present discus-
sion, transformation-induced plasticity (where deformation is
due to a stress-driven
transition between crystalline phases with different lattice
structures) any occurring
atomic rearrangements must preserve the crystal lattice
structure. This implies that
deformation must occur by shear of adjacent lattice planes. The
component of the
shear stress resolved in the direction i of slip is called
resolved shear stress σi. It is
the inner product of the projection tensor M i and the stress
tensor σ: σi = M i ·σ.
In crystal plasticity one has to provide constitutive relations
between stress and
strain and their time derivatives. These relations express the
fact that deforma-
tion or flow of the continuum proceeds against internal forces
or, conversely, that
stresses acting on the continuum produce deformations. The most
general isotropic,
homogeneous stress-strain relation is of the form
F (σ, σ̇, σ̈, ..., ǫ, ǫ̇, ǫ̈, ...) = 0. (1.2)
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1.1 Continuum macroscopic description of crystal plasticity
9
Derivatives of higher than second order in the previous relation
are of little practi-
cal importance, as their use requires the specification of a
larger number of initial
conditions than are usually available on the basis of physical
considerations. The
relations in Eq.1.2 can be reduced to scalar relations, so in
the following we adopt
a scalar formulation considering shear deformation on a single
slip system. The
simplest elastic-plastic constitutive relation, called ideal
plasticity, is rate indepen-
dent (F (σ, ǫ) = 0). In this case (Fig.1.1a), the plastic strain
increases indefinitely
when the stress reaches a critical value σy (yield stress). This
means that a material
cannot support any shear stress above the yield stress: at yield
the solid behaves as
an ideal fluid. Upon unloading it immediately becomes again an
elastic solid, while
the permanent shear deformation remains. A straightforward
generalization of this
relation is to assume, in the plastic regime, the stress σ as an
increasing function of
strain (strain hardening). In the stress-strain graph (Fig.1.1b)
the onset of plasticity
in this case corresponds to a change in slope, called the strain
hardening coefficient
Θ, or the tangent modulus.
When the relaxation processes in the continuum are due to the
interaction be-
tween plastic and viscous behavior, the resulting stress-strain
relations are referred
to as viscoplastic. In the viscoplastic regime, solids exhibit
permanent deformations
but continue to undergo creep flow as a function of time under
the influence of the
applied stress. Since the viscous flow is assumed to be
incompressible, the stress-
strain relation involves only the stress, strain and their
derivatives (F (σ, σ̇, ǫ, ǫ̇) = 0).
Linear viscoplasticity assumes that the shear strain rate ǫ̇ is
zero up to the yield
stress and then increases linearly with stress (see
Fig.1.2a),
ǫ̇ =1
µ
(|σ| − σy)sign(σ) |σ| > σy,
0 else ,(1.3)
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10 Crystal plasticity
Figure 1.2: Linear viscoplasticity: (a) strain rate vs. stress;
(b) stressstrain curvesfor driving at different rates ǫ̇1 >
ǫ̇2.
where the sign function indicates that the direction of the
shear depends on the sign
of the driving stress, and µ is a viscoplastic rate coefficient.
Linear viscoplasticity can
be generalized to account for strain hardening, by assuming in
Eq.1.3 that the critical
stress is a function of plastic strain, σy = σy(ǫ) (see
Fig.1.2b). Other generalizations
may include nonlinear stress-strain relationships, and
additional variables to take
into account the effect of temperature and material
microstructure.
Usually a strain hardening viscoplastic material responds to the
homogeneous
driving stress in a spatially homogeneous manner, but under
certain conditions,
small perturbations in the flow may become undamped. In this
case plastic insta-
bility appear. Considering the linear viscoplastic flow equation
ǫ̇ = 1/µ(σ − σy(ǫ)),
perturbations become undamped for Θ < 0 (strain softening) or
for µ < 0 (strain
rate softening). Plastic instability give rise to phenomena as
non linear oscillations,
travelling waves, patterning and chaos (see Chap.4).
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1.2 Plasticity at the micro scale: dislocations and
fluctuationphenomena 11
1.2 Plasticity at the micro scale: dislocations and
fluctuation phenomena
In the continuum description of crystal plasticity, fluctuations
are supposed aver-
age out above a characteristic scale (see Sec.1.1). Experimental
observations that
plastic deformation of metals proceeds by the formation of slip
bands or slip pack-
ets (see Chap.3) of width of the order of micron, required a new
interpretation.
Understanding plasticity at the microscopic level, for which
fluctuations become
important, require the introduction of the concept of
dislocations. Thanks to the
advent of x-ray diffraction, it was found that the intensity of
x-ray beams reflected
from crystals was about 20 times grater than that expected for a
beam reflected from
a perfect crystal. In the last case the intensity would be low
because of the long ab-
sorption path provided by multiple internal reflections. To
account for these results,
it was assumed that real crystals consisted of small, roughly
equiaxed crystallites,
in the range of micron or less, slightly misoriented with
respect to one another,
with the boundaries between them consisting of amorphous
material. These mis-
orientations are described by means of topological line defects
called dislocations.
The elementary mechanism of crystallographic slip is not the
rigid sliding of lat-
tice planes above each other, but the expansion of a slipped
area. Then, in other
words, the boundary of a slipped area is the dislocation. A
dislocation is charac-
terized globally by the slip plane and by the dislocation
displacement vector, or the
so called Burgers vector b, of the corresponding slip system,
and locally by a unit
tangent vector t to the line direction. Depending on the angle
between t and b, the
dislocations are distinguished in edge (t⊥b), screw (t ‖ b) and
mixed. The edge
dislocation can be realized imagining to introduce or remove an
extra half plane of
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12 Crystal plasticity
Figure 1.3: Dislocations in solids, characterized by direction t
(see the red line) andBurgers b vector (red vector). The Burgers
vector can be obtained as the differencebetween a closed loop (blue
line) in an undeformed section of crystal and a similarloop around
a dislocation. For edge dislocations t⊥b, while for screw ones t ‖
b.
atoms through the perfect crystal (Fig.1.3). The screw
dislocation can be obtained
imagining to cut a crystal along a plane and slipping one half
across the other by
a lattice vector (Fig.1.3). The plain specified by the vectors b
and t is said glide
plain (or line for a screw dislocation) because the dislocation
can move easiest on
it respect to other plains. Usually dislocation climb is
disadvantaged. In general a
dislocation is of mixed type, having a character between pure
edge and pure screw.
Its Burgers vector can be resolved into the screw component bs =
(b · t)t and the
edge component be = t x (b x t). In the continuum theory, the
local Burgers
vector is given by the line integral, taken in a right-handed
sense relative to t, of
the elastic displacement u around the dislocation: b =∮
(∂u/∂l)dl. A shear stress
σ acting in the dislocation slip system creates a so called
Peach-Koehler force per
unit length on the dislocation. The component of this force in
the dislocation glide
-
1.2 Plasticity at the micro scale: dislocations and
fluctuationphenomena 13
Figure 1.4: Representation of the analogy between caterpillar
and dislocation mo-tion.
plane is given by
F PK = σb[t x n], (1.4)
where n is the unit vector normal to the glide plane. This force
moves the dislocation
such as to shear the material in the direction imposed by the
slip geometry and the
sign of the stress σ. The dislocation line can move through the
crystal in the direction
of the applied stress by breaking only one line of atomic bonds
at a time (see the
schematic picture in Fig.1.4 for edge dislocations). This
requires far less energy than
displacing many atoms.
The dislocation itself is a source of stress, since the lattice
around it is distorted.
The stress field induced by a dislocation in the host material
is of long-range nature.
In the case of curved dislocations, the long-range nature of the
self-interaction stress
acts to keep the dislocation straight. Non local elastic
properties arise naturally from
long range dislocation interactions and influence dramatically
statics and dynamics
of these systems in the presence of disorder, as we will see in
the following chapters.
Due to the discreteness of dislocations, plastic flow on the
microscopic scale of
individual dislocations is necessarily inhomogeneous in space
and often, due to the
presence of localized obstacles, intermittent in time. In the
continuum formulation
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14 Crystal plasticity
Figure 1.5: Mechanical behavior at room temperature for pure Ni
microsampleshaving a 〈134〉 orientation [2]. (A) Stress-strain
curves for microsamples ranging insize from 40 to 5 µm in diameter,
as well as the stress-strain curve for a bulk singlecrystal having
approximate dimensions 2.6x2.6x7.4 mm. (B) A scanning
electronmicrograph (SEM) image of a 20 µm diameter microsample
tested to ∼ 4% strain.
of plasticity (Sec.1.1), microscopic deformation localization
and intermittency are
irrelevant as soon as multiple defects are involved, since the
incoherent superposition
of individual defect motions would result in a smooth and
approximately homoge-
neous flow. In plastically deformed microcrystals, internal
dislocations avalanches
lead to jumps in the stress-strain curve (strain bursts),
whereas in macroscopic sam-
ples plasticity appears as a smooth process [1]. In Ref.[2]
measurements of plastic
yielding for single crystals of micrometer size dimensions are
reported (see Fig.1.5).
By means of three-dimensional simulations of the dynamics of
interacting disloca-
tions, in Ref.[1] the dependence of dislocations avalanches on
microcrystal size are
determined in agreement with the results obtained in [2].
Another tool for monitoring dislocation activity during plastic
deformation is
the acoustic emission recordings. As dislocations move, they
lose energy interacting
with phonons and electrons in the crystal. In many situations,
the motion proceeds
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1.2 Plasticity at the micro scale: dislocations and
fluctuationphenomena 15
Figure 1.6: Statistical properties of the acoustic energy bursts
recorded in ice singlecrystals under constant stress [3]. The main
figure shows the power law distributionof energy bursts for the
different loading steps. The fit yields an exponent τE =1.60± 0.05.
σe is the resolved shear stress acting across the glide plane (σe =
0.030MPa - 0.086 MPa). Inset, a typical recorded acoustic
signal.
in an over-damped manner, due to large drag force. In the case
in which dislocations
are sufficiently rapid, the energy loss is not only dissipated
by heat generation, but
part of the energy is emitted in the form of travelling acoustic
waves which can be
detected as an acoustic signal. In general, acoustic emission
during a deformation
avalanche may be produced by temporally correlated dislocation
motions at different
locations which, owing to long-range interactions between
dislocations, may form
spatial patterns. In Ref.[3] acoustic emission measurements on
stressed ice single
crystals indicate that dislocations move in a scale-free
intermittent fashion. Fig.1.6
shows the distribution of energy bursts and a typical recorded
acoustic signal in the
inset. These results are confirmed by numerical simulations of a
model of interacting
dislocations in Ref.[3].
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16 Crystal plasticity
1.3 Theoretical approaches
Modelling plastic deformation phenomena taking into account
inhomogeneity at the
dislocation level offers fundamental advantages compared to
continuum mechanics
approaches. First of all, in this manner, when is possible to
perform a coarse grain-
ing (for example as in the case of the Orowan relation), one can
relate macroscopic
aspects of plasticity to microscopic properties to establish,
for instance, a physi-
cal basis of the empirical viscoplastic flow rules used in
continuum mechanics. On
the other hand, dislocation dynamical approaches allow to
account for the intrinsic
length scales, such as the grain size, the mesh length of a
dislocation network or
the cross-slip height, which is necessary to understand the
formation of spatial dis-
location structures, as persistent slip bands [4], and plastic
instabilities, as Lüders
band and the PLC effect [5]. The different theoretical
approaches to the investiga-
tion of plasticity differ in the way the deformation state is
represented, however the
different approaches fit themselves into a consistent picture of
plastic yielding as a
non-equilibrium phase transition. In the following we shall
discuss models which de-
scribe plastic flow in terms of the motion of discrete lattice
dislocations (subsection
1.3.2); phenomenological models which account for the influence
of the underlying
dislocation dynamics in terms of fluctuations in the local
evolution stress-strain re-
lationship (subsection 1.3.3); phase-field models which describe
plasticity in terms
of shear strain on crystallographic slip systems, but resolve
the strain on a scale
where individual dislocations can be identified as localized
gradients of the strain
field (subsection 1.3.4). The resulting description of
heterogeneity and avalanches
phenomena in plastic flow of each of these models appear to be
mutually consistent.
-
1.3 Theoretical approaches 17
1.3.1 Statistical and stochastic approaches
In principle one can think, following a statistical mechanics
point of view, to per-
form an averaging procedure to eventually connect the
microscopic properties to
the macroscopic properties of plastic deformations. In this
sense, an appropriate
quantity for averaging is the distribution function of N
dislocations. While the gen-
eral approach is known in the context of condensed matter
physics, in the case of
plasticity two issues complicate the attempts to set up the
equivalent statistical de-
scription. The first major complication arises from the fact
that dislocations are line
defects described by the associated Burgers vector and line
element, even though
this can be dealt with in principle. The second difficulty
arises from the fact that
plasticity is inherently a highly dissipative process wherein a
large part of the work
done on the system is lost in the form of heat. But in general
one needs to deal with
thermal as well athermal dislocation activated processes.
Stochastic methods are
well suited to handle athermal fluctuations. The origin of the
noise is attributed to
the significantly fast time scales of a large number of degrees
of freedom. However,
the strength of the noise itself may remain a parameter or it
must be determined
by other considerations. There could be several microscopic
dislocation mechanisms
that can contribute to the noise and it may not be easy to model
the statistical
properties of the noise starting from microscopic
mechanisms.
A useful stochastic method is the Langevin approach in which the
variable under
question is assumed to evolve in a stochastic manner. In
particular, in the follow-
ing we introduce overdamped Langevin equations to describe
dislocation positions
(discrete dislocation dynamics), and to describe the local shear
strain (stochastic
continuum models). One prerequisite for using a overdamped
Langevin approach
-
18 Crystal plasticity
is that the order parameter variable(s) should have a
substantially slow evolution
compared to the random noise. Let us now examine the
experimental evidence in
support of the separation of time scales in plastic flow to see
if it is meaningful to
take a Langevin approach to dislocation dynamics. Early studies
on f.c.c. metals
have shown that slip in these materials occurs in a localized
manner (with a typical
active slip volume of ∼ 10−10m3 and time duration up to the
order of seconds) with
several dislocations moving coherently in the form of avalanches
[4]. The local shear
strain rate was found to be nearly six orders larger than the
applied strain rate.
Similarly, the duration between the occurrence of the slip line
bundles exceeds six
orders of magnitude. Thus, there appears to be a clear
separation of time scales
between the duration of occurrence of the slip lines and the
time scale associated
with the applied strain rate. This suggests that there are
internal processes that
occur at considerably faster time scale than the macroscopic
measurable quantities
such as the mean shear strain rate, shear stress, etc.
From the stochastic point of view, one feature of dislocation
motion that is im-
portant is the intermittent jerky motion of dislocations during
which they undergo
repeated pinning and unpinning, the latter occurring when the
threshold for un-
pinning is reached. Once unpinned, dislocation segments move in
bursts only to
be arrested at subsequent locations. This actually implies that
dislocations are at
rest most of the time. One should expect anomalous fluctuations
when dislocations
are on the verge of the threshold [6], and usually at any given
time there will be a
certain proportion of dislocation configurations which will be
near the threshold of
unpinning. In other words, the entire process of pinning and
unpinning is a typical
unstable system and such systems are subject to anomalous
fluctuations [7, 8, 9].
Thus, the jerky nature of dislocation motion is suggestive of a
high degree of fluc-
-
1.3 Theoretical approaches 19
tuations in the velocities of dislocations. Indeed, numerical
simulations show that
there is a large dispersion in the distribution of dislocation
velocities. There is also
evidence from the numerical integration of the equations of
motion wherein disloca-
tion interaction is mediated by long-range stress fields. This
shows high degree of
fluctuations in the internal stress [10].
1.3.2 Discrete dislocation dynamics
The dislocation dynamics model describes the motion of
interacting lattice dislo-
cations by means of deterministic equations in which randomness,
heterogeneity
and stochastic behavior are introduced through the probabilistic
choice of initial
conditions and noisy sources (due to moving impurities in the
case of systems mod-
elling the PLC effect, as discussed in Chap.4). In two
dimensional (2d) dislocation
dynamics, dislocations are idealized as straight parallel lines,
while in the three di-
mensional (3d) case, they are considered as flexible and
reactive lines. This means
that each part interact with the rest of the line. Then the
elastic stress field due to
a dislocation line in 3d can be obtained by considering the line
as being composed
of elementary segments of infinitesimal length interacting each
other [11].
In 2d, dislocations are envisaged as charged point particles
moving in a plane
driven by the external stress and subject to mutual interactions
through long range
stress fields. Usually, dislocation motion is assumed to occur
in an over-damped
manner. If the glide of a dislocation is governed by electron
and phonon drag forces,
this lead to a linear relationship between the Peach-Koehler
force and the dislocation
velocity in the glide plane. Assuming that N dislocations
parallel to the z direction
intersect the xy plane at ri, the equation of motion for a
dislocation in the xy plane
-
20 Crystal plasticity
is [12]
µ∂tri = sib
[
σliext +∑
j 6=i
sjσliljint (ri − rj)
]
, (1.5)
where the magnitude b of the Burgers vector is assumed to be the
same for each
dislocation, and si = ±1 refers to its sign. µ is the
dislocation drag coefficient that
is temperature-dependent, li denotes the slip plane, σliext is
the external shear stress
resolved along the li slip plane and σijint is the shear stress
on the dislocation at ri
in the slip plane li by a dislocation at rj in the slip plane
lj. In the case of an
edge dislocation with Burgers vector along the x direction for
example, considering
rj = 0, σint is given by
σint =Gb
2π(1 − ν)xi(x
2i − y2i )
(x2i − y2i )2, (1.6)
where G is the shear modulus and ν the Poisson’s ratio. The
local shear strain rate
created by the dislocation system in the position r in this case
is
ǫ̇(r) = b∑
j
sj∂txjδ(r − rj). (1.7)
Integrating the previous equation is possible to obtain the
phenomenological Orowan
relation
ǫ̇ = bρm〈v〉, (1.8)
where ρm is the mobile dislocation density and 〈v〉 is the
average velocity of the
system of dislocations.
Usually, under the effect of external forces, dislocations are
driven through a
disordered landscape and rearrange into complex assemblies. In
plastic deformation
processes the role of elasticity and disorder are crucial.
Random disorder, in the
-
1.3 Theoretical approaches 21
form of impurities, fluctuations of defect densities and other
spatial heterogeneities,
is responsible for a wealth of phenomena including surface
roughening, non-linear
dynamic response, stick-slip behavior and temporal
intermittency. Under the ef-
fect of an external force, these systems exhibit a complex
behavior arising from the
competition between elasticity and disorder. Disorder tends to
perturb the system,
which reacts by opposing elastic restoring forces. This complex
small scale dynamics
determines the macroscopic behavior of irreversibly deforming
materials. In some
cases, modification of the equation of motion Eq.1.5 to account
for impurities can
be done easily. A simplification of Eq.1.5 occurs if
dislocations can only glide on the
slip plane of the respective crystallographic slip systems,
neglecting the possibility
of climb in others planes. In this case the specification of the
glide plane (li, lj)
can be disregarded. Dislocation climb permits to move outside
the sliding plane.
Climb occurs by point defect emission or annihilation that
enable dislocations to
move circumventing otherwise insurmountable obstacles [5]. Then
the driving force
for dislocation climb is usually the movement of vacancies
through a crystal lattice
because these defects predominates in the diffusion of mass.
Defects that influence
in particular the dislocation dynamics are solute atoms,
particle inclusions or dislo-
cations in other slip systems. The interaction between
dislocations and these type of
defects is studied in the framework of depinning transitions.
This depinning transi-
tion of individual dislocations gliding on their slip plane has
been investigated in the
past in order to explain solid solution hardening, that is, the
increase of the yield
stress value when solute atoms are present in a crystal. In the
discrete formulation
of dislocation motion (Eq.1.5), the effect of the pinning force
is introduced adding
an actracttive term∑
p fp(ri − rp). The detailed shape of the individual pinning
force is inessential for most purposes, provided it is of short
range nature.
-
22 Crystal plasticity
3d dislocation dynamics consider the evolution of a system of
flexible and reactive
lines. Problems arising from the necessity to describe a system
of flexible lines are the
non conserved line length, the need to maintain the connectivity
of these lines, and
the fact that two intersecting dislocations of different systems
may react to create
segments of a third slip system. In general one has to rely on
numerical simulations
for investigating the dynamic evolution of such systems under
externally applied
loads. Both 2d and 3d dislocations dynamics have to cope with
the fact that the
dislocations are endowed with long range (1/r) stress field
which cannot easily be
truncated, and the simulations therefore tend to be
computationally expensive.
Simulations assuming a linear relationship between stress and
dislocation veloc-
ity, as in Eq.1.5, are very common both for 2d and 3d
dislocation systems. However,
it is important to note that experimental results on dislocation
velocities show a
much more complicated picture. At low stresses a strongly
non-linear (exponential)
stress dependence is observed. For the dynamics of dislocation
systems this may
be important, since at not too high applied stresses the motion
of dislocations is
strongly intermittent. They spend most of the time in low stress
configurations,
which therefore control the mean velocity, and experience high
stresses only during
intermittent jumps between such configurations. An easy and
computationally effi-
cient [13] expedient for mimicking the effect of a strongly
non-linear stress-velocity
relationship is to use extremal dynamics or automaton models
where the dislocation
moves (on the time scale considered) instantaneously to the next
stable position
whenever the stress exceeds a given threshold, and remains
stationary otherwise.
-
1.3 Theoretical approaches 23
1.3.3 Stochastic continuum models
An alternative approach to modelling fluctuations and collective
phenomena in plas-
tic deformation consists of continuum models in which
microstructural heterogeneity
and randomness are included in a phenomenological way. Stresses
associated with
individual dislocations are not resolved, but enter the
formulation implicitly. Stress
and strain are considered as mesoscopic fields. In the bulk of
materials, heterogene-
ity and randomness are resolved on the mesoscopic scale, but
they are homogeneous
on the macroscopic scale. Adopting a viscoplastic constitutive
relation of the type
discussed in Sec.1.1, the evolution of the local shear strain ǫ
is given by
µ∂tǫ =
(|σ + σi| − σy(ǫ,∇, r))sign(σ + σi), |σ + σi| > σy(ǫ,∇,
r),
0 else ,(1.9)
with σ the externally applied resolved shear stress, σi the
internal shear stress due to
heterogeneities, and σy(ǫ,∇, r) the local yield stress.
Randomness and heterogeneity
at the micro scale are taken into account phenomenologically
allowing for a stochastic
dependence of the yield stress on space and time. In fact in
plastically deforming
crystals, the local yield stress reflects the dynamics of
interacting dislocations on
scales below the spatial resolution of the mesoscopic scale
[14]. Internal length scales
of the material can be taken into account allowing the yield
stress to depend on the
second gradient of strain [14]. These considerations led to the
following relation
σy(ǫ,∇, r) = −C∂2ǫ
∂x2+ δσ(r, ǫ), (1.10)
where δσ(r, ǫ) is the fluctuating part of the stress associated
to dislocation dynam-
ics resolved below the mesoscopic scale. For a two dimensional
system of straight
parallel dislocations of density ρ, it has the approximate
correlation properties
〈δσ〉 = 0, 〈δσ(r, ǫ)δσ(r + r′, ǫ+ ǫ′)〉 = 〈δσ2〉fσ(r′/ξσ, ǫ′/ǫc),
(1.11)
-
24 Crystal plasticity
where 〈δσ2〉 ≃ G2b2ρ, ξσ ≃ 1/√ρ is the correlation length of the
fluctuating stress
field created by dislocations, ǫc ≃ b√ρ the correlation strain,
and fσ is a non dimen-
sional correlation function. Hardening may be introduced
allowing the dislocation
density, through the amplitude 〈δσ2〉 of local stress
fluctuations, to increase with
local strain.
In Chap.2 both the discrete dislocation dynamics and the
continuum approach
are employed to study the diffusion of a grain boundary in a
Peierls-Nabarro (PN)
potential, that reflect the periodic structure of the atomic
lattice.
1.3.4 Phase-field models
These models are similar to continuum approaches (Sec. 1.1),
however, the strains
are resolved on a microscopic scale on which individual
dislocations appear as local-
ized gradients in the shear strain fields on the respective slip
systems. Lattice strains
are described on quasi-atomic scale, for which both the elastic
and plastic strain are
described by the relative displacement between two slipping
lattice planes divide by
their separation. The evolution of the local shear strains is
derived from an elastic
energy functional E[ǫ] which includes not only long range
elasticity but also the
Peierls-Nabarro energy of the crystal. Koslowski et al. [15]
proposed a phase-field
model in the case of deformation on a single slip system. This
model envisages the
two dimensional distribution of slip on a single slip plane z =
0 in a three dimen-
sional continuum. The numerical simulation of such models is
computationally even
more expensive than 3d dislocation dynamics simulations.
The plastic deformation field ǫ(x, y) is represented as an
integer-valued field with
time-discrete dynamics. The deformation field at time i + 1 is
obtained from field
-
1.3 Theoretical approaches 25
at time i by minimizing the incremental work function
W [ǫ(x, y, i+ 1)|ǫ(x, y, i)] = E[ǫ(x, y, i+ 1)] − E[ǫ(x, y,
i)]
+∫
f(x, y) |ǫ(x, y, i+ 1) − ǫ(x, y, i)| dxdy.(1.12)
The function f(x, y) denotes the energy that is dissipated in
changing the strain
at the point (x, y) by a unit amount. It is described as a local
obstacle strength,
obtained as a sum over randomly distributed point obstacles.
By increasing the external stress by small amounts and carrying
out a sequence
of minimization steps of W , Koslowski et al. [15] found that
dynamics of this model
exhibits strain bursts with scale invariant size distribution of
strain increments with
the relative exponent kǫ ≃ 1.8. While this result compares
reasonably well with
experimental observations, in the model dislocation sources do
not exist and have
to be created ex nihil, as was pointed out in [16]. In spite of
this, the model has
several interesting features. In particular, performing a
spatial and temporal coarse
graining of the discrete-time dynamics, one obtain
∂ǫ(x, y)
∂t=δE ′(ǫ)
δǫ+ f(x, y, ǫ), (1.13)
that is the equation of motion of a two dimensional manifold
with long range elastic-
ity moving through a random medium because the expression of the
coarse-grained
energy functional E ′ is such that the interaction kernel scales
in proportion with the
wavevector modulus q (see [16]). This model is in the
universality class of mean field
depinning, as the model studied by Moretti et al. [17] that can
be treated within
the framework of Koslowski [16].
It remains a task for future work to generalize the model to
more than one slip
plane and investigate the possibility of self-pinning of the
dislocation system in the
-
26 Crystal plasticity
absence of quenched disorder, which characterizes the discrete
dislocation dynamics
simulations.
-
Chapter 2Grain boundary diffusion in a crystal
2.1 Dislocation assemblies
The behavior of isolated dislocations within a disordered
landscape has been widely
investigated in the past [18, 19]. While the behavior of an
isolated dislocation pushed
through a random distribution of obstacles is at present quite
well understood, the
results do not necessarily carry over to the more realistic case
of collective disloca-
tion motion. Dislocations interact via long range stresses and
rearrange according
to a collective behavior, which may lead to intriguing jamming
and avalanche-like
phenomena even in the absence of immobile obstacles [7].
Developing an analytic
approach for the solution of the depinning problem of a random
distribution of dis-
locations is at present considered a hard task. If one
disregards core effect, which
play a role only at small distances, the interaction between two
dislocations comes
down to the interaction between one and the stress field
generated by the other, or
the Peach-Koehler (PK) force. In several manners dislocations
can be assembled
together in a stable configuration. But only some of these
corresponds to systems
of physical interest. It is not infrequent, however, to observe
simple and regular
dislocation arrangements in deforming metals. The two more
frequently disloca-
-
28 Grain boundary diffusion in a crystal
tion assemblies found in materials are the grain boundary (GB),
discussed in this
chapter, and pile up, discussed in chapters 3 and 4 in two
particular cases.
A GB is the interface where two single crystals of different
orientation join in
such a manner that the material is continuous across the
boundary. Polycrystalline
materials are composed of a number of small single crystal
grains bonded together
by grain boundaries. In general, a GB can be curved, but in
thermal equilibrium it
is planar in order to minimize the boundary area and hence the
boundary energy.
The GB is not crystalline but consist of a thin amorphous layer
between the grains.
Understanding grain boundaries, and in general interface
kinetics in materials
is an important theoretical and practical problem, since this
process influences the
microstructure, such as the grain size, the texture, and the
interface type. From the
theoretical point of view, the study of processes that involve
surface and interface
properties gained significant interest in non-equilibrium
statistical mechanics [20,
21]. In particular, dislocations [16, 18] and grain boundaries
[17, 22] provide a
concrete example of driven elastic manifolds in random media
[8]. Other example
of this general problem are domain walls in ferromagnets [23,
24], flux line in type
II superconductors [25, 26], contact lines [27, 28] and crack
fronts [29, 30]. From
the point of view of applications, understanding GB kinetics has
a great importance
for polycrystalline materials, since the resulting grain
microstructure determines
material properties such as strength, hardness, resistance to
corrosion, conductivity
etc. [31]. The problem is particularly important for
nanocrystalline plasticity where
external stress induces a rearrangement of the grains by grain
boundary motion [32].
Hence the ambitious goal of these studies is to be able to
control the microstructural
properties of polycrystals.
Several approaches have been employed in the literature to study
GB kinet-
-
2.1 Dislocation assemblies 29
ics. Ref. [33] employs molecular dynamics (MD) simulations with
appropriate inter
atomic interactions to study the diffusion of grain boundaries
at the atomic scale
[33]. The method allows to quantify the mobility of grain
boundaries and to compare
the results with experiments [33]. While MD simulations provide
a very accurate
description of the dynamics, the method suffers from numerical
limitations and it
is difficult to reach the asymptotic regime. An alternative
method is provided by
the Langevin approach in which the GB is assumed to evolve
stochastically in an
external potential [34]. The dynamics of the underlying
crystalline medium enters
in the problem only through the noise term (due to lattice
vibrations) and the pe-
riodic potential (Peierls-Nabarro). Hence, the equations of
motion of the atoms
or molecules are not directly relevant. Indeed there is
experimental evidence in
supporting of separation of time scales in plastic flow [5] and
it is thus possible to
integrate out the fast degrees of freedom (atomic vibrations)
and consider only the
slow ones (dislocations position).
The two fundamental types of grain boundaries are the so called
tilt and twist
GB. In the general case a boundary is of mixed character,
containing both tilt and
twist components. A GB can be obtained imagining to take two
coincident lattice,
then rotate one of these around an axis: if the axis is parallel
to the boundary plane
we have a tilt GB, while if the axis is perpendicular to the
boundary plane we have
a twist GB. Moreover, depending on the magnitude of
misorientation between two
grains, we have low angle grain boundary (LAGB) and high angle
grain boundary
(HAGB).
Here we study the evolution of a LAGB in a crystalline material
by the Langevin
approach. The LAGB is treated as an array of interacting
dislocations performing
a thermally activated motion in a periodic (Peierls-Nabarro)
potential without any
-
30 Grain boundary diffusion in a crystal
external shear stress, which would instead be relevant for
nanocrystalline plasticity
[32]. Models similar to that considered here have been employed
in the past to
study the conductivity of superionic conductors [35, 36], the
relaxational dynamics
of rotators [37] and Josephson tunneling junctions [34]. Notice
that the crucial role
played by long-range stresses is often disregarded in analyzing
GB deformation. On
the other hand, it has been shown in Ref.[17] that a surface
tension approximation
for the GB stiffness is inappropriate and one has to consider
explicitly non-local
interactions. The present model incorporates this effect in the
equations of motion.
We simulate the set of Langevin equations numerically to
describe the GB kinet-
ics and its fluctuations [38]. The results are in good agreement
with MD simulations
[33] and allow to clarify the origin of the short time
deviations from the diffusive
behavior observed in Ref.[33]. In addition, a linearized version
of the model can be
treated analytically and the asymptotic results are found to be
in good agreement
with the simulations.
2.2 Diffusion of a grain boundary in a Peierls-
Nabarro potential
2.2.1 Dislocation model for grain boundary dynamics
To study the LAGB dynamics we consider a phenomenological
mesoscopic approach
[38]. It is schematized as an array of straight dislocations
that interact with each
other through long-range stress fields and with the crystalline
Peierls-Nabarro (PN)
potential. The GB is composed by N dislocations where
configurations are re-
peated ad infinitum because of periodic boundary conditions
along the y direction.
Each dislocation has Burgers vector of modulus b parallel to the
x axis and the
distance between two adjacent dislocations along the y direction
is fixed to be a
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
31
Figure 2.1: A regularly spaced low angle grain boundary where
the dislocations’Burgers vector is parallel to the x axis. The
ideal configuration is plotted withstraight dashed lines in the
plane yz, whereas the solid lines represent their possibleglide
deformations within the slip plane xz.
(see Fig.2.1). Each straight dislocation interacts with the
lattice and with others
dislocations through long-range stress fields. The effect of the
lattice over each n-th
dislocation can be decomposed as the sum of three
contributions:
• FPN(xn) = −A Gb2πr0 sin(2πxnb
), the PN force where A is the area of the GB,
G is the shear modulus and r0 the inter-atomic distance;
• -µẋn(t), the average effect of the lattice fluctuations where
µ is the viscosity
coefficient;
• µηn(t), the impulsive effect of the lattice fluctuations
assumed to be Gaussian
for the central limit theorem and uncorrelated in space and
time: 〈ηn(t)〉 = 0,
〈ηn(t)ηm(t′)〉 = Dδnmδ(t− t′) where D is the diffusion
coefficient of the GB.
-
32 Grain boundary diffusion in a crystal
The long-range stress field exercised by all the other
dislocations over the n-th,
the Peach-Koehler force F n,NPK (x,y), is computed considering
the image dislocations
method to comply with periodic boundary conditions along the y
direction. Making
use of calculations in [11, 39] one can find the following
expression
F n,NPK (x,y) = − Gb2π
N2a2(1 − ν)∑N
m=1(xn − xm)·
·{cosh[2π(xn − xm)/Na] cos[2π(yn − ym)/Na] − 1}{cosh[2π(xn −
xm)/Na] − cos[2π(yn − ym)/Na]}2,
(2.1)
where ν is the Poisson’s ratio, yn = n · a and ym = m · a.
Finally the over-damped
Langevin equation [34] for the GB reads
µẋn(t) = FPN(xn) + Fn,NPK (x,y) + µηn(t), (2.2)
for n = 1, ..., N , or rather
ẋn(t) = −A Gb2πr0µ sin(2πxnb
) − Gb2π
N2a2(1 − ν)µ∑N
m=1(xn − xm)·
·{cosh[2π(xn − xm)/Na] cos[2π(yn − ym)/Na] − 1}{cosh[2π(xn −
xm)/Na] − cos[2π(yn − ym)/Na]}2+ ηn(t).
(2.3)
To indicate the amplitude of the FPN and FPK forces we introduce
respectively the
parameters APN = AGb/2πr0µ and APK = Gb2π/a2(1 − ν)µ.
The key quantities that we consider in order to characterize the
dynamics of the
GB are:
• the mean-square displacement of the center of mass, ∆x2cm(t) =
〈x2cm(t)〉 −
〈xcm(t)〉2 = 〈xn(t)2〉 − 〈xn(t)〉2 where xcm(t) = xn = 1/N
∑Nn=1 xn(t);
• the mean-square width W 2(t) = 〈x2n〉 − 〈(xn)2〉.
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
33
In the following, we first analyze the case of a flat GB for
which a comparison
with MD simulations approach [33] is made. Next we consider the
full flexible
description of the GB. Finally, we discuss a linearized version
of the model that can
be treated analytically.
2.2.2 Flat grain boundary
For many applications a good approximation is to consider a flat
GB with a single
degree of freedom, for which F n,NPK (x,y) = 0 and xn(t) =
xcm(t) for n = 1, ..., N . In
other words, the flat GB is described by the following
equation
ẋcm(t) = −AGb
2πr0µsin(
2πxcmb
) + η(t), (2.4)
where the correlation properties of the thermal fluctuations
are: 〈η(t)〉 = 0 and
〈η(t)η(t′)〉 = Dδ(t − t′). This type of equation, also known as
the Kramers equa-
tion with periodic potential, has been extensively studied in
the literature [34]. In
particular, the mean-square displacement is known to display a
combination of os-
cillatory and diffusive behavior [34, 40]. Different dynamical
regimes are found as
the potential strength or the friction varies [40]. In fact, we
show next that this
simple model allows to understand the short-time deviations from
diffusive behavior
observed in MD simulations [33].
Integrating Eq. 2.4 with the initial condition xcm(0) = 0 by
means of computer
simulations, we have compared the mean-square displacement
∆x2cm(t) to the one
obtained from MD simulation in Ref. [33]. The parameters
employed are: A =
2746.4123 Å2, G = 25.5 GPa, b = r0 = 4 Å, while the fitted
ones are D and µ to
the values of 0.0155 Å2/ps and 0.24967 · 10−6 m−2Js
respectively.
In Fig. 2.2 the comparison between the mean-square displacement
∆x2cm(t) ob-
tained integrating Eq. 2.4 and the one obtained from MD
simulation is displayed to-
-
34 Grain boundary diffusion in a crystal
gether with the mean-square displacement of the renormalized
free Brownian motion
(described by the equation: ẋ = η(t) with 〈η(t)〉 = 0 and
〈η(t)η(t′)〉 = DRδ(t− t′)).
The agreement between the two simulations is extremely good. For
higher times
(t > 80ps), the mean-square displacement tends to the
renormalized Brownian mo-
tion. Hence taking explicitly into account the sinusoidal
Peierls-Nabarro force in the
Langevin equation allows to describe the mean-square
displacement for early times
of the dynamics.
0 20 40 60 80t [ps]
0
0.5
1
∆xcm
2 [Å
2 ]
Molecular Dynamic simulations, Ref. [18]renormalized Brownian
motion Langevin approach
Figure 2.2: Mean-square displacement ∆x2cm(t) of the flat grain
boundary compari-son between molecular dynamic simulation [33]
(green line) and Langevin approachsimulation (dashed blue line).
The parameters employed are: A = 2746.4123 Å2,G = 25.5 GPa, µ =
0.24967 ·10−6 m−2Js, b = r0 = 4 Å, D = 0.0155 Å2/ps. Bothtype of
simulations predicts a linear dependence in time of ∆x2cm(t) for
long timesrepresented in the figure by the renormalized free
Brownian motion (straight blackline).
One can deduce in a simplified intuitive way the temporal
evolution of the mean-
square displacement starting by the transition probability
density P for small times
(small ∆t) [34]
P (x, t+ ∆t|x′, t) = 12√πD∆t
e−[x−x′−FPN (x)∆t]2
4D∆t . (2.5)
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
35
Next we consider the transition probability P01(x0 + ∆x1, t0 +
∆t|x0, t0) to run
from the point x0 at time t0 to the point x1 = x0 + ∆x1 at time
t1 = t0 + ∆t and
P12(x0 +∆x1 +∆x2, t0 +2∆t|x0 +∆x1, t0 +∆t) to run from x1 at t1
to x2 = x1 +∆x2at t2 = t1 + ∆t
P01 =1
2√πD∆t
e−[∆x1−FPN (x0)∆t]2
4D∆t
P12 =1
2√πD∆t
e−[∆x2−FPN (x0+∆x1)∆t]2
4D∆t .
(2.6)
For a free Brownian motion (FPN(x) = 0) the condition P01 = P12
implies ∆x1 =
∆x2 and stochastic displacements are space independent. If we
impose this condition
in presence of a periodic force FPN(x) = −dUPN(x)/dx, we
obtain
P01 = P12 ⇒ ∆x1 − FPN(x0)∆t = ∆x2 − FPN(x0 + ∆x1)∆t ⇒
⇒ ∆x2 = ∆x1[
1 +dFPN (x)
dx
∣
∣
∣
x0∆t
]
,(2.7)
and then
∆x2 ≷ ∆x1 ifdFPNdx
∣
∣
∣
∣
x0
≷ 0 or ratherd2UPNdx2
∣
∣
∣
∣
x0
≶ 0. (2.8)
This result implies that, with the initial condition xcm(0) = 0,
if the potential
UPN(x) is convex (concave) the mean-square displacement curve is
concave (con-
vex). In the case of the PN potential, we find indeed that the
mean-square dis-
placement curve should display upper and lower deviations from
the straight line,
corresponding to a renormalized free Brownian motion, depending
in d2UPN(x)/dx2.
These deviations decrease in time so that for large times the
curve should approach
a straight line [34].
-
36 Grain boundary diffusion in a crystal
1 10 100 1000 10000 1e+05 1e+06t
0.01
0.1
1
10
100
1000
W2
/ b2
APK
=0.0896 APN
=0
APK
=0.0896 APN
=0.1
APK
=0.0896 APN
=0.2
APK
=0.0896 APN
=0.4
APK
=0.0896 APN
=0.5
APK
=0.0896 APN
=0.6
APK
=0.14 APN
=0
APK
=0.14 APN
=0.1
APK
=0.14 APN
=0.2
APK
=0.14 APN
=0.4
Figure 2.3: Mean-square width of the grain boundary, W 2(t), in
the case of b = 2πand N = 32 for two typical situations. If the
noise is high respect to APK and toAPN , the GB exfoliate
(continuum line), while if the noise is small respect to APK orto
APN , the GB reaches a stationary state (dashed line) for which
W
2(t) saturatesafter a certain time.
2.2.3 Flexible grain boundary
A more general description of the GB considers its internal
deformation and the
dynamics is described by Eq. 2.3. The dynamical behavior of the
GB depends on the
amplitude of the three terms in the right-hand side of Eq. 2.3.
The parameters that
characterize the behavior of the GB are a, b, APK , APN and D.
Varying the values
of these parameters, in the long-time limit the GB can either
exfoliate (when the
noise, D, is high enough with respect to APK and to APN) or
reach a stationary state
(when the noise, D, is small when compared to APK or to APN).
The asymptotic
behavior can be read off from the width W 2(t) that keeps on
increasing when the
GB exfoliates and saturates when the GB remains stable. In Fig.
2.3 the comparison
between these two typical situations is displayed in the case of
N = 32, a = 3π,
b = 2π, APN = 0, 0.1, 0.2, 0.4, D = 0.25 and APK = 0.14 for the
case in which the
GB remains stable, while APK = 0.0896 for the case in which the
GB exfoliates. It
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
37
0.01 0.1A
PK
100
101
102
103
104
105
W2
/ b2
t = 106
t = 105
t = 104
(a)0.1 1
D
100
102
104
106
W2
/ b2
t = 106
t = 105
t = 104
(b)
Figure 2.4: Mean-square width W 2 in terms of Burgers vector b =
2π in Log-Logscale. (a) graphic of W 2 dependence on the parameter
APK keeping fixed APN = 0and D = 0.25 for three different times.
(b) graphic of W 2 dependence on theparameter D keeping fixed APN =
0 and APK = 0.14 for three different times.
is possible to observe from Fig. 2.6 that when the curvature of
W 2(t) changes sign
the GB becomes unstable. To better visualize this crossover, we
report in Fig. 2.4a
and Fig. 2.4b the value of the width W 2 varying APK and D
respectively, for three
different times. In Fig. 2.4a is possible to observe the
increase of W 2 in time for
small values of APK , while for APK > 0.104, W2 saturates
towards its stationary
values. In Fig. 2.4b is possible to observe a similar behavior:
W 2 increases in time
for large values of D, while for D < 0.275, W 2 saturates
towards its stationary
values.
In what follows, we analyze the dynamical behavior of the stable
GB for a = 3π,
b = 2π, APN = 0, 0.4, APK = 0.14 and D = 0.25. In Fig. 2.5 the
average position
of the GB center of mass ∆x2cm(t) is displayed with and without
the PN force in
Log-Log scale for N = 32, 64, 128, 256, 512.
For long times in both cases we have a linear behavior ∆x2cm(t)
∼ t, but for
short times, in the presence of the PN force, there is a clear
deviation from linearity.
This result confirms the conclusion made in the previous
section, that the PN force
-
38 Grain boundary diffusion in a crystal
1 10 100 1000 10000 1e+05t
10-5
10-4
10-3
10-2
10-1
100
101
102
∆xcm
2 / b
2
N = 32N = 64N = 128N = 256N = 512
(a)1 10 100 1000 10000 1e+05
t10
-5
10-4
10-3
10-2
10-1
100
∆xcm
2 / b
2
N = 32N = 64N = 128N = 256N = 512
(b)
Figure 2.5: Mean-square displacement of the center of mass of
the grain boundary,∆x2cm(t) , for Langevin approach simulation
without the PN force (a) and with thePN force (b). ∆x2cm(t) for b =
2π and N = 32, 64, 128, 256, 512 is displayed inLog-Log scale.
1 10 100 1000 10000 1e+05t
0.01
0.1
1
W2
/ b2
N = 32N = 64N = 128N = 256N = 512
(a)1 10 100 1000 10000 1e+05
t
0.01
0.1
1
W2
/ b2
N = 32N = 64N = 128N = 256N = 512
(b)
Figure 2.6: Mean-square width of the grain boundary, W 2(t), for
Langevin approachsimulation without the PN force (a) and with the
PN force (b). W 2(t) for b = 2πand N = 32, 64, 128, 256, 512 is
displayed in Log-Log scale.
is the cause for the deviation from linearity of ∆x2cm(t) for
short times observed
in Ref. [33]. Next we characterize the morphology of the GB
through the width
W 2(t). In Fig. 2.6 W 2(t) for N = 32, 64, 128, 256, 512 is
displayed with and without
the PN force in Log-Log scale. In the absence of the PN force
(Fig. 2.6a) the time
dependence of W 2(t) is qualitatively similar to the same case
but with linearized
PK force discussed in the next section, while in the presence of
the PN force, for
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
39
APN = 0.4 (Fig. 2.6b), W2(t) exhibit a plateau for intermediate
times.
2.2.4 Continuum theory
It is possible to develop an analytic expression in the
continuum limit (a → 0,
N → ∞ and L = Na = const.) for short or long times for W 2(t) in
absence of the
PN force linearizing the PK force. The equation of motion for
FPN = 0 and FPK
linearized is
ẋn(t) = −Gb2
2π(1 − ν)µ
N∑
m=1
xn − xm(yn − ym)2
+ ηn(t). (2.9)
To obtain the short time behavior is sufficient to rewrite Eq.
2.9 as a generalized
Ornstein-Uhlenbeck process [34]
ẋn(t) =N
∑
m=1
gnmxm + ηn(t). (2.10)
The general solution of Eq. 2.10 is
xn(t) =
∫ ∞
0
N∑
m=1
Gnm(t′)ηm(t− t′)dt′, (2.11)
with {Gnm(t)} = Ĝ(t) = eĝt = 1 + ĝt + ĝ2t2/2 + ... (where 1
= {δij}). From the
definition of W 2(t), results
W 2(t) =D
N
N∑
n,m=1
∫ t
0
G2nm(t′)dt′ − D
N2
N∑
n,m,l=1
∫ t
0
Gnm(t′)Glm(t
′)dt′. (2.12)
Replacing the Taylor expansion of the Ĝ matrix in Eq. 2.12 one
obtains for short
times (t ≪ 1/‖ĝ‖) that W 2(t) = (1 − 1/N)Dt + o(t) and in the
continuum limit
W 2(t) = Dt+o(t). To obtain the long times behavior of W 2(t),
we rewrite Eq. 2.9 in
Fourier space [17, 41]. Employing the decomposition xm =
1/L∑
k exp(−ikam)xk,
we obtain
ẋk = −Gb2
2π(1 − ν)µa2L
∞∑
m=−∞
eikam∞
∑
n=−∞
∑
k′(e−ik′am − e−ik′an)xk′
(m− n)2 + ηk. (2.13)
-
40 Grain boundary diffusion in a crystal
The first term in the right-hand side of Eq. 2.13 can be
rewritten as
− Gb2
2π(1 − ν)µa2L∑
k′
xk′∞
∑
m=−∞
ei(k−k′)am(
∞∑
d=−∞
1 − eik′add2
), (2.14)
where d = m− n. Using the following results∞
∑
d=1
1
d2=π2
6,
∞∑
d=1
cos(cd)
d2=π2
6− π|c|
2+c2
4, (2.15)
we obtain
∞∑
d=−∞
1 − eik′add2
= 2∞
∑
d=1
1 − cos(k′ad)d2
= π|k′|a− k′2a2
2, (2.16)
so that Eq. 2.13 becomes
ẋk = −Gb2
2π(1 − ν)µa2 (π|k| −k2a
2)xk + ηk. (2.17)
In the long time limit (large x, small k) the k2 term can be
neglected. Finally
in the continuum limit we replace yn, ym by the continuum
variables y, y′ and
〈η(y, t)η(y′, t′)〉 = aDδ(y − y′)δ(t− t′).
Thus Eq.2.13 becomes
ẋk = −Gb2
2(1 − ν)µa2 |k|xk + ηk = −APK2π
|k|xk + ηk. (2.18)
This type of equation is frequently encountered in problems of
interface motion in
presence of fluctuations (see Appendix A). Eq. 2.18 can be
solved exactly and W 2(t)
is given by [21, 42]
W 2(t) =aD
APK[ln(
L
a) + ln(1 − e−2APKt/L)]. (2.19)
To reproduce the continuum limit by simulations of Eq. 2.9 we
would need a very
large GB, with N ≫ 512. Thus a comparison between Eq. 2.19 and
the simulations
-
2.2 Diffusion of a grain boundary in a Peierls-Nabarro potential
41
1 10 100 1000 10000 1e+05 1e+06t
0.01
0.1
1
W2
/ b2
simulation for N=32 computation in long time
approximationcomputation in small time approximation
Figure 2.7: Comparison between the mean-square width W 2(t) for
linearized FPKand FPN = 0 in the case of b = 2π and N = 32 computed
by numerical simulationand the continuum theoretical prediction in
the short and long times limit. The blackline represents the
simulated data, the green line the long times theoretical
predictionwith fitted parameters and the blue line the short times
theoretical prediction.
16 32 64 128 256 512 1024N
0.3
0.4
0.5
0.6
W2 s
/ b2
FIT: y = 10.425 + 1.7962 ln(x)A
PN = 0.4
APN
= 0
Figure 2.8: Size dependence for the saturation value of the
mean-square width W 2s inLog scale for the abscissa (x-axis) with b
= 2π. As can be seen in figure, in the casein which the PN force is
absent (APN = 0) the relation is logarithmic (W
2s ∼ logN),
while in presence of the PN force (APN 6= 0) a slight deviation
from the logarithmicdependence is observed.
for small N is possible only by introducing some effective
parameters in Eq. 2.19.
In Fig. 2.7 W 2(t) computed by the simulations with N = 32 (for
which the better
statistics is available) is compared with the fitted theoretical
prediction for short and
long times. Eq. 2.19 also predicts that the saturated value of
width (W 2s ) exhibits
-
42 Grain boundary diffusion in a crystal
a logarithmic dependence on the GB length L = N/a: W 2s ∼ logN .
In the case in
which FPK is not linearized this result is also confirmed by
numerical simulations,
showing that W 2s increases logarithmically with N when FPN = 0
(see Fig. 2.8). In
presence of a periodic potential (FPN > 0), however, we
observe a deviation from
the logarithmic growth at large N . This suggests that the
Peierls-Nabarro potential
may set a limit to the GB roughness.
In this chapter we have investigated the diffusion of a
regularly spaced low-
angle grain boundary in a crystalline material. A typical
computational method to
describe the dynamics of the grain boundary is to perform
deterministic molecular
dynamics simulations with appropriate inter atomic interactions
[33]. Here we have
employed the over-damped Langevin approach to obtain a long time
description of
the dynamics, but in particular to perform a comparison with
molecular dynamics
simulations for a specific material [33]. The first results is
the interpretation of the
early times behavior of the mean-square displacement ∆x2cm(t).
The deviation for
early times of ∆x2cm(t) by the case of the renormalized Brownian
motion, that holds
for long times, can be interpreted as the effect on the
dislocations of the periodicity
of the lattice giving rise to the Peierls-Nabarro potential,
irrespective of internal
degrees of freedom of GB. Secondly the description of the
dynamic (∆x2cm(t)) and
the morphology (W 2(t)) of the grain boundary by means of
over-damped Langevin
equations is in qualitatively good agreement with its behavior
in real materials, so
this approach can be considered an useful tool for these
studies.
-
Chapter 3Slip line formation at the initial stage of
plastic deformation
In the present chapter we discuss the growth of a slip line in a
plastically deforming
crystal and study the relevant quantities by numerical
simulation of a double-ended
pile-up model with a dislocation source at one end, and an
absorbing wall at the
other end. By means of this model we find that in presence of
defects, the pile-up
undergoes a second order non-equilibrium phase transition as a
function of stress,
which can be characterized by finite size scaling. Open boundary
conditions are
responsible for non-homogeneous density and velocity profiles,
changing the uni-
versality class from the one expected in the case of the
corresponding homogeneous
depinning transition. In the end, our findings allow to
reinterpret earlier experiments
on slip line kinematography.
3.1 Slip lines
3.1.1 Experimental investigation of slip lines
An enormous body of observations of slip lines by light and
electron microscopy,
and of dislocations structure by X-ray topography [43, 44], etch
pitting [45] and in
-
44 CHAPTER3. SLIP LINE FORMATION
particular by transmission electron microscopy (TEM) [46] has
been accumulated
by now. X-ray methods are restricted to very perfect crystals
and to very low
dislocations velocities owing to long exposure times; TEM
methods are restricted
to moderate velocities and very small regions of the exceedingly
thin specimen; and
etch-pitting methods work only at the surface of the rather
perfect crystals. The
kinematography of slip lines by light microscopy provides a
rather useful compromise
between high resolution in time (& 0.2ms) and in slip step
height (& 100 Å), and
a wide field of view (≃ 0.3 mm). In addition, the observations
can be performed
at all stages of deformation during non stationary and
stationary regimes. Thus it
reveals the correlation of slip over wide distances at a
reasonable local resolution
and, as a non destructive technique, can be easily applied for
dynamics studies.
By their nature slip line observations are restricted to the
surface of the crystal.
Therefore, their relevance to obtain information on the process
in the bulk, which
clearly govern the plastic behavior of a bulk material, may be
questioned [4], and has
to be proved in every special case. Examples are known with
different dislocation
and slip structures near the surface and in the bulk, while
others do not show
differences, at least in stage I deformation [4].
Slip lines at the microscopic scale can be seen as steps
produced when dislocations
emerge on the crystal surface during plastic deformation. A nice
picture of a tensile
tested wire can be seen in Fig.3.1. These steps result in a
complex terraces structure
observed on the surface of materials. Early studies [47] on the
fine structure of the
slip lines suggest that the slip height morphology could be
self-affine.
As discussed in Chap.1.3, intermittent dislocation motion is a
generic feature of
plasticity, not only in micron-scale samples (for example in
acoustic emission mea-
surements [48, 3]) and by earlier reports from slip line
kinematography in macro-
-
3.1 Slip lines 45
Figure 3.1: Localized deformation of a tensile tested bamboo
structured aluminumwire of 25µm diameter.
Figure 3.2: Dislocation groups and multipole arrangements in
Cu-30% Zn observedin TEM after deformation into stage I at room
temperature [4]
-
46 CHAPTER3. SLIP LINE FORMATION
scopic samples ([49, 50, 51]). A nice illustration of
dislocation pile-up in a macro-
scopic sample observed in TEM can be seen in Fig.3.2,
3.1.2 The yielding transition
The experimental observations on dislocations pile-up, some of
which reported in
the previous subsection, lead to the idea that plastic yielding
is a non-equilibrium
critical point [16], similar to the jamming transition observed
in soft and glassy
materials or the depinning transition for disordered elastic
manifolds [8].
The yielding transition has been investigated in various
dislocation models, from
the dynamics of an individual flexible dislocation interacting
with quenched random
impurities such as solute atoms [19, 18], to the dynamics of
several rigid dislocations
moving on single slip systems [7, 14]. While these models
provided a good under-
standing of the yielding transition in simplified conditions,
less is known about the
role of dislocation nucleation and multiplication for the
critical behavior. In very
clean single crystals, with a very small initial dislocation
density, dislocations are
most likely nucleated from sources present at the surface of the
sample, where it
easy to find defects (steps, scratches) acting as stress
concentrators. In this case,
the onset of plasticity corresponds to the creation and
propagation of slip bands
through the entire cross section of the crystal.
3.2 Dislocation pile-up model for slip lines
3.2.1 Dislocations pile-up
A slip band can be envisaged as a queue of dislocations, a
pile-up, pushed through
a series of obstacles (solute atoms or immobile dislocations
from other glide planes).
Experimentally slip lines can terminate or propagate depending
on the value of the
-
3.2 Dislocation pile-up model for slip lines 47
shear stress, temperature, crystal structure and types of
defects [4]. A transition
from homogeneous to inhomogeneous slip, with increasing impurity
concentration,
is observed experimentally in fcc alloys [52, 53]. Here, we
investigate the dynamics
of a double-ended pile-up in presence of defects, with a source
of dislocations at one
end and an absorbing wall at the other (see Ref. [54]). In the
model, the stress
dependence of the stationary dislocation density, velocity and
strain rate obey finite
size scaling. This is indicative of a second order
non-equilibrium phase transition.
The scaling exponents, however, are different from those found
in the corresponding
homogeneous system, where nucleation is not considered and the
dislocation density
is constant [17].
A possible way to understand the necessity of the introduction
of pinning centers
(quenched disorder) in order to have a phase transition in the
double-ended pile-up
model, is to apply an Imry-Ma-like argument [55] as performed in
Ref. [17] in the
case of a LAGB and a pile-up with periodic boundary
conditions.
3.2.2 The double-ended pile-up model
We consider the pile-up as a group of identical edge or screw
straight dislocations
parallel to the z axis that can move in the positive x direction
in the plane y = 0
when the net force acting on them is positive. This corresponds
to an effective one-
dimensional model in which the dislocations are generated from a
source in the left
side of a line of length L (see Fig.3.3).
They interacts with each other and with disordered stress
landscape provided by
solute atoms, or other defects (quenched disorder) and disappear
when they reach
the right side of the line. The dislocations have coordinates
xi, with i = 1, ...N ,
where N = N(t) depends on time t. The dislocation at i = 1 is
immobile and
-
48 CHAPTER3. SLIP LINE FORMATION
Figure 3.3: At the top, we show a three dimensional sketch of a
slip plane line containing a pile-upof dislocations emitted by a
source placed on the left surface and absorbed at the right hand
sideof the sample. Our simplified model is shown in the bottom part
of the figure: dislocations moveon a line of length L and interact
with point impurities. The source is modeled as an
immobiledislocation with varying Burgers vector placed at x1. In
order to take into account the effect ofopen boundary conditions,
interaction stresses include the contribution from an infinite
series ofimage dislocations.
represents the source, as we discuss below. The dislocations for
i = 2, ...N are
mobile and have constant Burgers vector bi = b. The Burgers
vector is directed
along x for edge dislocations and along z for screw ones.
To describe the dynamics of mobile dislocations we use an
over-damped equation,
so that the velocity of dislocations depends linearly on the
resolved shear stress
exerted on it. This approximation is valid provided that the
dislocation velocities
are small in comparison to the stress wave speeds and provided
the time of pile-up
formation is large compared with the characteristic time
constant of the material, as
verified in typical situations [56]. The equation of motion for
the mobile dislocations
-
3.2 Dislocation pile-up model for slip lines 49
is given by
χdxidt
= bi(σ +∑N
j=1
(j 6= i)
σinti,j + σimgi ) +
∑
P
f(xi −XP ), (3.1)
where χ is an effective viscosity and σ is the external stress.
The interaction stress
σinti,j between dislocations i and j is computed taking into
account the image stresses
of dislocation j due to the open boundary conditions, while
σimgi is due to the
interaction between dislocation i and its own images. A compact
expression of the
interaction and image stresses can be obtained by performing the
sum over the