contents acronyms and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . iii...

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

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

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

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

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

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

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

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

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-

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

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

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.

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)

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)

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)


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).

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


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


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


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


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].


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


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-


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


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


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.


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.


1.3.3 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)


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


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


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


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.


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〉.


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-


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)


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].


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


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


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


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)


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


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


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]


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


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

contents acronyms and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . iii...

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