Necas Center for Mathematical Modeling

Simulation of interacting dislocationsglide in a channel of a persistent slip

band

J. Kristan and J. Kratochvıl and V. Minarik and M. Benes

Preprint no. 2008-021

Research Team 3Faculty of Nuclear Science of the Czech Technical University

Trojanova 13, 120 00 Praha 2http://ncmm.karlin.mff.cuni.cz/

Simulation of interacting dislocations glide

in a channel of a persistent slip band

JOSEF KRISTAN∗†, JAN KRATOCHVIL‡, VOJTECH MINARIK§, and MICHAL BENES§

† Faculty of Mathematics and Physics, Charles University, Mathematical Institute,

Sokolovska 83, 186 57 Prague, Czech Republic

‡ Faculty of Civil Engineering, Czech Technical University in Prague, Department of Physics,

Thakurova 7, 166 29 Prague, Czech Republic

§ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague,

Department of Mathematics, Trojanova 13, 120 00 Prague, Czech Republic

Abstract

The aim of the paper is to present and analyze in details a simulation methodbased on a parametric description of interacting glide dislocations. The dislocationsare modelled as planar flexible curves employing the flowing finite volume methodand the method of lines. The evolving positions and shapes of the dislocations aregoverned by the equation of motion where each infinitesimal dislocation segment issubjected to a line tension, interactions with segments of other dislocations and thestress field imposed by loading conditions. As an example the proposed method isemployed to evaluate a bow-out of dislocations from the walls in a channel of a per-sistent slip band (PSB) and their subsequent interactions. Two different situationsare considered: (i) so called “stress-control” where the stress in the channel inducedby boundary conditions is assumed to be uniform, (ii) “strain-control” where thesum of elastic and plastic strain is uniform.

Keywords : dislocation bowing, dislocation interaction, line-tension, curvaturedriven equation

1 Introduction

Discrete dislocation dynamics (DDD) has now become attractive and helpful tool for in-vestigation and modelling of elementary processes of plastic deformation of crystalline

∗Corresponding author. Email: kristan1@email.cz

1

materials. In early versions, the collective behavior of dislocations was explored by directnumerical simulations considering infinitely long parallel straight dislocations. However,there was a need for extension of the DDD methodology to the more physical, yet, consid-erably more complex conditions of three-dimensional DDD computer simulations of plas-tic deformation; applications of these models spanned a number of important mesoscaleplasticity problems; see e.g. [1–6].

The methods developed for three-dimensional dislocation dynamics can be categorizedinto two groups, according to the line discretization scheme and representation of a generalcurved dislocation segment. The first method is based on an edge-screw or edge-mixed-screw discretization of the dislocation lines; e.g. [1–4]. The basic idea of the approach isthat the discrete segments move on a discrete network similar to the natural crystal latticebut orders of magnitude larger. The second category of methods simulates dislocationsas flexible lines discretized either by curved segments [5, 6] or by line segments [7–9].

In recent papers [7, 8] the method of flexible lines discretized by line segments hasbeen applied in the simulation of a dislocation moving in a field of dislocation loops. In[9] the simulation method has been employed to estimate a so-call saturation stress in aPSB–channel formed in fatigued metals. There two unlike originally screw dislocationsin parallel slip planes were pushed by applied stress in the opposite directions till theyformed a dipole. As the stress was increased the dislocations became more curved, untilthey escaped one another. The stress needed to accomplish that was interpreted as thesaturation stress. The simulations [9, 10] were used as an alternative to the analyticalstudies of this problem by Brown [11, 12] and Mughrabi & Pschenitzka [13]. For moredetails the reader is referred to [9, 10].

In the present paper the model and the numerical method of solutions, which is basedon DDD simulations modelling interacting dislocations as flexible lines, are describedin details. As an example, we follow a generation of glide dislocations by the processof bowing-out from walls of a PSB–channel and their mutual encounter. The evolvingpositions and shapes of the glide dislocations are governed by the equation of motion whereeach dislocation segment is subjected to a line tension, to interactions with segments ofother dislocations and the walls, and to the stress field imposed by loading conditions.However, instead of solving the full PSB boundary value problem (BVP)(a more detaileddiscussion is given in [9, 10]), two simplified loading situations are considered; either stressin the channel is assumed to be uniform (“stress-control”) or a sum of elastic and plasticstrain is kept uniform (“strain-control”).

2 Mathematical Model

Governing equation. Inspired by papers [17–19], a gliding dislocation curve Γ(t) attime t is described by a C1–smooth (not necessarily closed) one dimensional non-selfintersecting curve in a plane R

2. It can be parameterized by a smooth vector function~X(u, t) : S × I → R

2, where u is a parameter from a fixed interval S = [U1, U2], U2 > U1,

2

U1 U2

u

xO Xx(u, t)

~Xu(u, t)

ϕ(u, t)Γ(t)

z

Xz(u, t)

Figure 1: The mapping Γ(t) = Image( ~X(·, t)) of the interval [U1, U2] into the glide planexOz.

and I = [0, T ) is a time interval; i.e. the dislocation curve is given as

Γ(t) = Image( ~X(·, t)) := { ~X(u, t); u ∈ S}

and for which the local length g = |∂u~X| > 0 (| · | denotes the Euclidean norm in R

3 and∂ξF = ∂F/∂ξ). A motion of gliding dislocation then corresponds to evolution of familyof planar curves {Γ(t)}t≥0 from initial configuration Γ0 = Γ(0) at t = 0. The mappingis shown schematically in figure 1. Since at low temperature plastic deformation thedislocations move along the crystallographic planes, we assume the set { ~X(u, t); (u, t) ∈S × I} is a subset of the corresponding glide plane in the Cartesian coordinate systemidentified (for simplicity) with the xz–plane.

The tangent and normal vectors to the dislocation line in the glide plane are denoted∂u

~X and ∂u~X⊥, respectively; the outward normal vector ∂u

~X⊥ is defined in such a waythat det(∂u

~X, ∂u~X⊥) = 1 (det(~x, ~y) denotes the determinant of the 2 × 2 matrix with

column vectors ~x and ~y). A unit arc–length parameterization of a curve Γ is denoted by

s and it satisfies |∂s~X(s, t)| = 1 for any s and t. Furthermore, the arc–length parame-

terization is related to the original parameterization u via the equality ds = g du. Theinterval of values of the arc–length parameter depends on the curve Γ; more precisely,s ∈ [0, LΓ(t)], where LΓ(t) is the total length of the curve Γ at time t. Then ~t = ∂s

~X

and ~n = ∂s~X⊥ represent unit tangent and normal vectors in the glide plane, respectively:

~t = ∂s~X = ∂u

~Xdu/ds = ∂u~X/|∂u

~X| and ~n = ∂s~X⊥ = ∂u

~X⊥/|∂u~X|.

The glide of a planar dislocation segment is governed by a linear viscous law assumedin the form of the standard mean curvature flow equation [17, 21]

Bv = Tκ + F , (1)

3

where the material constant B is the drag coefficient, v = v(s, t) is the magnitude of thenormal velocity ~v of the evolving segment of the curve, T = T (s, t) is the line tension, andthe local curvature κ = κ(s, t) of the curve in the direction ~n at s is defined by Frenet’s

formula: ∂2s~X = κ∂s

~X⊥. The magnitude of the driving force F = F (s, t) in eq. (1) can

be written as F = bτeff ; b is the magnitude of the Burgers vector ~b and τeff represents thelocal resolved shear stress acting on the dislocation segment which is specified in the nextsubsection.

The scalar eq. (1), using v = ∂t~X · ∂s

~X⊥, can be written in the form of an intrinsicdiffusion equation,

B∂t~X = T∂2

s~X + F∂s

~X⊥ . (2)

The vector eq. (2) is of the type ∂t~X = β~n, which is a special case of general geometric

equation ∂t~X = β~n + α~t; β and α represent suitable smooth functions. Neither for closed

curves nor open curves with pinned end points the presence of a tangential velocity α inthe position vector equation has impact on the shape of the evolving curve. Therefore,a natural setting α ≡ 0 has been chosen for analytical as well as numerical treatment.However, as it was shown in [20], for general curvature driven motions a suitable tangentialvelocity may significantly stabilize numerical computations. It prevents the Lagrangianalgorithm from its main drawbacks, the merging of numerical grid points and it also allowsfor large time steps without loosing stability.

Commenting the line tension force approximation Tκ employed in eqs. (1) and (2)let us note that the curved dislocation feels its own elastic field of as a straighteningself-force. The self-interaction would require computation of the stress caused by onepiece of dislocation at the location of another, so that self-stress effects with logarithmicsingularities would occur, see [22]. However, as was shown by Schmid and Kirschner [23],in the limit of mild curvature the self-stress is proportional to the line tension, which isthe approximation introduced by de Wit and Koehler [24]. A drawback of the presentmodel is that the line tension approximation could negatively influence the shape of thedislocation close to the wall where the curvature is the highest.In the present computations we accept an orientation dependent line tension T = T (ϕ),see [24, 25],

T (ϕ) = E +∂2E

∂ϕ2= Eedge(1 − 2ν + 3 cos2(ϕ)), Eedge =

µb2

4π(1 − ν)log

R

r0

Here, E is the elastic energy per unit length of a long straight dislocation and Eedge theenergy of dislocation of edge character; µ is the shear modulus, b is the magnitude ofthe Burgers vector, ν the Poisson’s ratio and ϕ(s, t) the angle between the tangent tothe dislocation segment and the Burgers vector. R has the dimensions of the crystal ifthere is only one dislocation in the crystal and r0 ≈ b is the radius of the dislocation core.Especially, Tscrew = Eedge(1 + ν) and Tedge = Eedge(1 − 2ν) are the line tensions for thescrew and edge dislocation, respectively.

4

Resolved shear stress. Four contributions to the resolved shear stress are considered:

τeff = τdisl + τwall + τapp + τ0 .

τdisl is the resolved shear stress exerted by other gliding dislocations, τwall represents thewall interaction (treated as the elastic field of rigid edge dipoles) and τapp approximatesthe stress in the channel determined by the applied boundary conditions (applied stress).In the present considerations an influence of a friction stress and debris left by shuttlingdislocations is incorporated in the term τ0.Shear stress exerted by a dislocation. The resolved shear stress in the channel at ~r = (x, y)caused by the elastic field of a curved dislocation Γ can be expressed as (omitting thedependence on time t)

τdisl(~r) =

∫

Γ

τd(~r, s)ds . (3)

Here, τd(~r, s) is the resolved shear stress at a point ~r exerted by a dislocation segment dsof Γ; τd will be specified in the next section. The integral is taken along the curve Γ attime t. In our model, the general shape of a dislocation line with no symmetry constraintsis considered.Wall interaction. The long range stress caused by the different deformability of thewalls and the channel is incorporated into the stress τapp determined by the boundaryconditions. However, there are also short range interactions among the dipolar loopsclustered in the walls and parts of dislocations close to the walls. Following Brown [12],these short range interactions are incorporated in the model as elastic fields of fixed rigidedge dipoles located at the surface of the walls. Segments of gliding dislocations depositedat the walls are trapped in the elastic potential valleys produced by the dipoles parallel tothe walls. A sufficiently strong stress can bow-out a dislocation segment from the wall intothe channel as modeled in this paper. Each dipole is formed by the two edge dislocationsin a stable equilibrium configuration. For simplicity, the centers of dipoles are placed inthe slip planes of dislocations (the effects of the dipolar structure of the walls have beenexplored in more detail in [16]. The height of individual dipoles, hdip, controls the distanceof the elastic valleys from the walls as well as their strength. The shear stress field ofthe dipole consists of the stress field of edge dislocations. The shear stress produced at apoint ~r = (x, y) by an infinitely long straight edge dislocation, located at the origin alongthe z–axis, is

τedge(x, y) = ±µb

2π(1 − ν)

x(x2 − y2)

(x2 + y2)2. (4)

The sign “+” or “−” depends on the orientation of the dislocation line representing thecorresponding edge dislocation. Here, µ is the shear modulus, b the magnitude of theBurgers vector and ν the Poisson ratio.Applied stress. As already mentioned in the Introduction, instead of solving the fullBVP of the stress distribution in the channel, two simplified limit cases are considered:(i) the “stress controlled regime” in which the applied stress τapp in the channel is keptuniform (the same assumption was employed in the original composite model proposed

5

by Mughrabi [27, 28]), and (ii) the “strain controlled regime” in which the total strainεtot remains uniform. We argue that the reality lie between these two limits.

In the stress-control, the elastic strain γe, coupled to the stress by Hooke’s law, γe =τapp/µ, remains uniform. Therefore, it cannot adjust to the generally nonuniform plasticstrain produced by the dislocation glide (the compatibility of total strain in the channelis violated). Such artificial rigidity causes the stress level to be higher than in reality.In the numerical simulations the applied stress is the same at each point of the glidingdislocation and in the present paper we explore a special case τapp = const.

In the strain-control, the total shear strain as a sum of an elastic part γe and a plasticpart γp, εtot = τapp/µ + γp, is assumed to be uniform in the channel. It is not requiredthat the stress in the channel satisfies the stress equilibrium; only the equilibrium ofthe forces exerted on the dislocation lines are guaranteed by the equation of motion (1).Accordingly, the resulting applied stress is smaller than in reality.

To estimate the plastic strain represented by a slip γp the considered dislocations aretaken as representatives of glide dislocations in the channel. The rate of change in plasticslip (at a fixed position in the space) is given by Orowan equation ∂γp/∂t = bv; b isthe magnitude of the Burgers vector, density of glide dislocations and v their averagevelocity. In agreement with the observation [28] that the density of glide dislocations inPSB–channels practically does not change during the deformation process, is taken asa material constant. Under these simplifying assumptions the infinitesimal plastic straindγp carried by a dislocation segment of length dl at a point s of gliding dislocation Γ andat time t is

dγp(s, t) = bv(s, t) dtdl = b dS(s, t) ,

and consequently plastic strain carried by dislocation segment dl during the time interval[t0, t],

γp(s, t) = b

∫ t

t0

dS(s, t) = bS(s, t0, t) .

The integral is taken along the time interval [t0, t]. Initially γp(s, t0) = 0. S(s, t0, t)denotes the area slipped by the dislocation segment dl at the point s in the time interval[t0, t]. The applied stress τapp exerted on a dislocation segment of length dl at the points and time t can be then explicitly expressed as

τapp(s, t)dl = µ[εtot(t)dl − bS(s, t0, t)] . (5)

Recall that the observe length dl of fixed dislocation segment at s can change in time,i.e. during the evolution of a dislocation curve. In the numerical simulations we explorea special case εtot(t) = εt, where ε is a constant.

3 Numerical method

For the numerical simulations we employed a semi-implicit scheme and the discretizationbased on the flowing finite volume approach in space [20] and the method of lines in

6

~Xi−1

~Xi

~Xi+1

~Xi−

1

2

~Xi+

1

2

Γ(t)

~X0

~XM

di

di+1

Figure 2: Discretization of the curve.

time [21]. By discretization in space the governing equations are reduced to an ODEsystem which is solved by the standard Runge–Kutta method of the fourth order withfixed time step.

Discretization. In the numerical scheme a smooth dislocation curve Γ(t) is represented

by a M -sided moving polygon P(t) =⋃M

i=1[~Xi−1, ~Xi], i.e. the curve is approximated by

M linear segments [ ~Xi−1, ~Xi], i = 1, . . . ,M . M + 1 is a constant number of points onthe curve. In the arc–length parameterization s the points are denoted by the subindexi, i = 0, . . . ,M :

~Xi = ~Xi(t) = ~X(si, t) ; 0 = s0 < s1 < . . . < sM = LΓ(t) ,

where LΓ(t) is the total length of the curve Γ at time t; see figure 2.

The linear segments [ ~Xi−1, ~Xi], i = 1, . . . ,M , are called the flowing finite volumes. We

construct also dual volumes Vi = [ ~Xi−1/2, ~Xi+1/2] ≈ [ ~Xi−1/2, ~Xi] ∪ [ ~Xi, ~Xi+1/2] for i =

1, . . . ,M − 1. For an arbitrary j, ~Xj+1/2 denotes the midpoint of a line segment ~Xj~Xj+1,

i.e. ~Xj+1/2 = 12( ~Xj + ~Xj+1) (j = 0, . . . ,M − 1). Note that ~X(i−1)+1/2 ≡ ~Xi−1/2.

System of equations. The equation of motion for the point ~Xi is derived from eq. (2).As the interior and the end points of the polygon behave differently, they are treatedseparately.Interior points of the polygon. Integrating evolution eq. (2) at time t over a dual volumeVi, i = 1, . . . ,M − 1, we get

B

∫

Vi

∂t~Xds =

∫

Vi

Ti∂2s~Xds + b

∫

Vi

τi∂s~X⊥ds ,

where the discrete quantities Ti and τi are constant over the dual volume Vi at the corre-sponding point ~Xi; Ti is an approximation of a line tension T , τi is an approximation ofτeff . Using the Newton-Leibniz formula and constant approximation of the quantities, we

7

have at any time t:

Bdi + di+1

2

d ~Xi

dt= Ti[∂s

~X]i+ 1

2

i− 1

2

+ bτi[ ~X⊥]

i+ 1

2

i− 1

2

.

di = | ~Xi− ~Xi−1| is the distance of the two neighboring points ~Xi and ~Xi−1 of the polygon.

By taking central differences (of the first order) in space we obtain (∂s~X)i+1/2 = ( ~Xi+1 −

~Xi)/di+1 for i = 0, . . . ,M − 1. In result, the equation of motion for the corresponding

point ~Xi in the dual volume Vi can be written in the form, i = 1, . . . ,M − 1,

Bd ~Xi

dt=

2Ti

di + di+1

(

~Xi+1 − ~Xi

di+1

−~Xi − ~Xi−1

di

)

+2bτi

di + di+1

~X⊥i+1 −

~X⊥i−1

2. (6)

In eq. (6) we employ an orientation dependent line tension Ti = Eedge (1 − 2ν + 3 cos2(ϕi)),with

cos(ϕi) = (∂s~X)i ·

~b

b≈

~Xi+1 − ~Xi−1

di + di+1

·~b

b.

The symbol “·” stands for the scalar product of vectors.End points of the polygon. Next, we derive equations of motion for the end points of thepolygon: ~X0 and ~XM . For that, degenerate dual volumes V+

0 = [ ~X0, ~X1/2] for ~X0 and

V−M = [ ~XM−1/2, ~XM ] for ~XM are constructed. Integrating the evolution eq. (2) in dual

volumes V+0 and V−

M , we have

Bd0

2

d ~X0

dt= T0[∂s

~X]1

2

0 + bτ0[ ~X⊥]

1

2

0 ; BdM

2

d ~XM

dt= TM [∂s

~X]MM−1/2 + bτM [ ~X⊥]MM−1/2 .

As above, using central differences in space we get (∂s~X)0 = ( ~X1− ~X0)/d1 and (∂s

~X)M =

( ~XM − ~XM−1)/dM . Note, that (∂s~X)0 = (∂s

~X)1/2 and (∂s~X)M = (∂s

~X)M−1/2. Therefore,the first terms on the right-hand sides of the last equations vanish. This is in agreementwith the fact that for the straight segments [ ~X0, ~X1/2] and [ ~XM−1/2, ~XM ] the curvature κ

is zero. Finally, we exploit the definition of the points ~Xj+1/2 for j = 0,M − 1 and thecorresponding equations of motion can be written in the form

Bd ~X0

dt=

2F0

d1 + d1

( ~X⊥1 − ~X⊥

0 ) ; Bd ~XM

dt=

2FM

dM + dM

( ~X⊥M − ~X⊥

M−1) . (7)

The eqs. (6) and (7) are supplemented with initial and boundary conditions. An exampleof such conditions is given in the next section.

Resolved shear stress. In the numerical implementation of the model the componentsof the resolved shear stress, i.e. the resolved shear stress exerted by other gliding dislo-cations τdisl, the wall interaction τwall and the applied stress τapp, have to be adjusted tothe dislocation representation as a moving polygon.

8

Shear stress exerted by a dislocation. In τdisl given by eq. (3) we employ for evaluation ofτd(~r, s) de Wit’s formula [29] for the stress field of finite straight dislocation segment AB;

τd(~r, s) ≈ τ (AB)xy = τxy(~r − ~XB) − τxy(~r − ~XA) , (8)

where ~XA and ~XB are the end points of a straight segment of the polygon at s. UsingDevincre’s formula (see equation (25) in [31]), derived directly from de Wit’s formula,the stress components τij generated at ~r by the semi-infinite straight dislocation of the

Burgers vector ~b, with the end point at ~r0, and of a line direction ~ξ, are given, within theframe of isotropic elasticity by

τij(~r − ~r0) =µ

4π

1

R(R + L)

{

(~b × ~P )iξj + (~b × ~P )jξi −1

1 − ν

(

(~b × ~ξ)iPj + (~b × ~ξ)jPi

)

−(~b × ~η) · ~ξ

1 − ν

[

δij + ξiξj + (ηiξj + ηjξi + Lξiξj)R + L

R2+ ηiηj

2R + L

R2(R + L)

]}

.

(9)

The geometrical meaning of the symbols can be seen in figure (3). In eq. (9): i, j ∈ {1, 2, 3}and the following symbols are used: R marks the magnitude of the positional vector~R = ~r − ~r0; η the magnitude of the vector ~η = ~R − L~ξ (which is perpendicular to the

straight dislocation); L = ~R · ~ξ is the the projection of vector ~R to the dislocation line;

Pi and Pj are the components of the vector ~P = ~R−R~ξ; ξi and ξj the components of the

vector ~ξ; finally, ηi and ηj are the components of the vector ~η. δij stands for Kroneckerdelta. The symbols “·” and “×” mean the scalar and cross product of vectors, respectively.

In the approximation of τd(~r, s) introduced by eq. (8) the length of the straight seg-ments of a polygon can be, in principle, as small as requires the accuracy of the numericalmethod. However, de Wit’s formula (9) is correct only for a semi-infinite dislocation ina homogeneous infinite crystal. The formula should be corrected by image forces at thewalls, which are not incorporated in the present version of the model.

In the investigation of mutual interactions of gliding dislocations the stress exertedby other dislocation can be computed as a sum of the stress contributions produced bystraight finite segments of its polygon. Therefore, we can use formulae (8) and (9). Let

the polygons P(t) =⋃M

i=1[~Xi−1, ~Xi] and P

′

(t) =⋃M

i=1[~X

′

i−1, ~X′

i ] be approximations of the

dislocations Γ and Γ′

, respectively. Then, the shear stress acting at the point ~Xi of thepolygon P is a sum of the stress contributions by all the straight segments ~X

′

j~X

′

j+1 of the

polygon P′

(to simplify the notation, time dependence is omitted):

τxy( ~Xi) =

∫

Γ′

τd( ~Xi, s′)ds′ ≈

M−1∑

j=0

τ( ~X

′

j~X′

j+1)

xy =M−1∑

j=0

(

τxy( ~Xi − ~X′

j+1) − τxy( ~Xi − ~X′

j))

.

The shear stress τxy( ~Xi) is constant through a dual volume Vi. In the above evaluatedstress the orientation of the dislocation curves plays an important role which has to be

9

O

~r

~ξ~η

~b

~R

L~ξ

~r0

Figure 3: Vectors appearing in the definition of the stress tensor generated by semi–infinite straight dislocation. The unit vector ~ξ is parallel to the dislocation line, ~R is thevectorial distance between a point ~r0 of the dislocation line and an arbitrary point ~r atwhich the stress produced by the semi–infinite straight dislocation is calculated. ~η is thecomponent of ~R perpendicular to ~ξ.

taken into account: in the case of the same character of dislocations, the correspondinginteraction shear stress is repulsive, in the latter case it is attractive. To obtain the largestpossible interaction stress, the distance of the corresponding slip planes of the curves isequal to the critical separation distance for which the screw dislocations of opposite signnearly annihilate mutually by cross slip (although such process is excluded in the presentmodel). The orientation of Γ and Γ

′

is specified in the next section.Wall interaction. As was mentioned in section 2, paragraph Wall interaction, each dis-location wall is created by dislocation dipoles. We assume that each moving dislocationin the vicinity of the wall feels strongly the elastic stress field only by the nearest dislo-cation dipole forming that wall. The elastic shear stress field produced by the dipole ata point ~r = (x, y) in the glide plane of a mobile dislocation can be computed by usingeq. (4). Let (x1, y1) and (x2, y2) be the relative coordinates between the point ~r and thepositions of straight dislocations forming a suitable dipole, respectively. By using eq. (4),the expression for the elastic shear stress field at ~r is

τwall(x, y) = ±µb

2π(1 − ν)

(

x1(x21 − y2

1)

(x21 + y2

1)2

−x2(x

22 − y2

2)

(x22 + y2

2)2

)

.

The sign, either “+” or “−”, depends on the type of dipole. Here, µ is the shear modulus,b the magnitude of the Burgers vector and ν the Poisson ratio.

In the following considerations we will distinguish between different time levels. Forthis let introduce the notation: Pj is a polygonal approximation of Γ(t) at time t = tj,where the time node tj =

∑j−1k=1 ∆tk is the j-th time (t0 = 0) with the time step ∆tj.

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Γ(tj)

Γ(tj+1)

~Xji+1

~Xj

i+ 1

2

~Xj

i− 1

2

~Xji−1

~Xji

~Xj+1

i−1~X

j+1

i− 1

2

~Xj+1

i

~Xj+1

i+ 1

2

~Xj+1

i+1

~nj+1

i

Figure 4: Slipped area by dislocation segment in corresponding dual volume.

At t0 = 0 the initial P0 is an approximation of Γ0. The polygon Pj =⋃M

i=1[~Xj

i−1,~Xj

i ] is

made by the discrete plane points ~Xji , where i = 1, . . . ,M , denotes a space discretization

and j = 0, 1, . . ., denotes a discrete time stepping.Applied stress. In the stress-control regime, the applied stress τapp is uniform at eachpoint of dislocation line. It is a prescribed function of time and it approximates the stressinduced by the loading conditions. In the numerical simulations we explore the case ofconstant applied stress, i.e. the constant approximation of applied stress τapp in dual

volume Vji at corresponding point ~Xj

i is

(τ ji )app = τapp = const.

In the total strain-control regime, for evaluation of applied stress τapp in eq. (5), onehas to specify an area ∆Sj+1

i slipped in time interval (tj, tj+1) by a dual volume Vji in the

corresponding point ~Xi, i = 1, . . . ,M − 1. For simplicity, the area ∆Sj+1i is replaced by

the area of parallelogram with the vertices { ~Xji−1/2,

~Xji+1/2,

~Xj+1i−1/2,

~Xj+1i+1/2}, i.e. ∆Sj+1

i ≈

|~nj+1i ×~t j+1

i |, see figure (4), where we labeled the “normal” vector ~nj+1i = ~Xj+1

i − ~Xji for

i = 1, . . . ,M − 1, and the unit “tangent” vector

~t j+1i ≈

1

2

(

~Xj+1i+1 − ~Xj+1

i−1

dj+1i+1 + dj+1

i

+~Xj

i+1 −~Xj

i−1

dji+1 + dj

i

)

for i = 1, . . . ,M − 1 .

d ji = | ~Xj

i −~Xj

i−1| is the distance of the two neighboring points ~Xji and ~Xj

i−1 of a polygon

at time tj. For i = 0,M we set up ∆Sji = 0 for all j in the case of evolution a bowing

dislocations pinned in two fixed end points.The total area S j

i slipped during evolution from initial time t0 = 0 to current time tj is

11

Height of wall dipoles hdip = 5 nmWidth of channel dc = 1200 nmWidth of dislocation walls dw = 150 nmSpacing between slip planes h = hc = 42 nmMagnitude of the Burgers vector b = 0.256 nmShear modulus µ = 42.1 GPaPoisson ratio ν = 0.33Density of glide dislocations ≈ 1013 m−2

Drag coefficient B = 1.0 · 10−5 Pa sEnergy of edge dislocation Eedge ≈ 2.3 nJ m−1

Friction stress τ0 = 5 MPa

Table 1: Parameters used in the simulations.

S ji =

∑

i ∆S ji and a constant approximation of the applied stress τapp in dual volume Vj

i

at the corresponding point ~Xji is

(τ ji )app = µ[εtj − bS j

i ] ,

where ε is a prescribed constant.

4 Results and Discussion

The proposed simulation method might give a new insight into elementary processes ofdislocation interactions in a PSB–channel. In [9] we studied the process of passing of twoglide dislocations. These simulations provided satisfactory estimates of the stress neededfor passing of the two dislocations stretched across the width of the channel; their endpoints were fixed at the walls. However, such initial conditions are rather artificial. Herewe present the simulations that take into account a process of dislocations bowing out ofthe walls. The configurations of the dislocations during their motion are controlled bythe initial conditions, by the geometrical constraints of the channel and the walls, andby the stress field in the channel. If not specified otherwise, the material parameterssummarized in table 1 are used. The height of the wall dipoles hdip = 5 nm is identifiedwith the average height of dipolar loops which are the main building blocks of the walls[26]. The values of the parameters dc, dw, b, µ and ν correspond to cyclically deformedcopper at room temperature [13, 27]. In [27, 28] Mughrabi presented the value of thedensity of screw dislocations in the PSB–channel ≈ 1013 m−2. The energy of edgedislocations Eedge ≈ 2.3 nJ m−1 is chosen in such a way that the line tensions of edge andscrew dislocations, Tedge ≈ 0.8 nN and Tscrew ≈ 3.0 nN, are comparable to the experimentalvalues [27, 28].General geometry of the channel and walls. In figures 5 and 6 we can see a draft of the sim-ulation space. In figure 5 a cut through the channel and the walls in a perpendicular direc-tion is shown. There are displayed the positions of the two (initially edge) dislocations and

12

hc

dwdc dw

x

y

O

~b

Γ′

Γ

hdip

hdip

dipole

wall wallchannel

xz − plane

Figure 5: A model of PSB - a perpendicular cut through the channel.

the positions of four infinitely-long dipoles. The coordinates of the midpoints of the rigiddipoles in the xy–plane are (−600 nm, 0 nm), (+600 nm, 0 nm), (−600 nm, hc = 42 nm) and(+600 nm, hc = 42 nm), respectively. The characteristic lengths of the geometry are shownas well. In figure 6 a 3D-geometry is shown. There is a schematic plot of dislocation curvesΓ and Γ′ which bow-out to the channel from the walls. The dislocation Γ glide in thexz–plane, Γ′ in the plane parallel to the xz–plane and the distance of these two planesis hc = 42 nm. Each dislocation is during the evolution pinned at the end points, inparticular in the xyz–system, Γ at D1(560 nm, 0 nm, 0 nm), D2(560 nm, 0 nm, 700 nm) andΓ′ at D′

1(−560 nm, 42 nm,−500 nm), D′2(−560 nm, 42 nm, 0 nm), see figure 6.

Simulations. As the first step, the crystal is not exposed to the loading conditions, i.e.τapp = 0 MPa in the stress-control and εtot = 0 in the strain-control regime. Let theinitial configuration of Γ and Γ′ be identified with the linear segments D1D2 and D′

1D′2

shown as he dotted lines in fig. 6. The vectors ~t and ~t′ determine the initial orientationsof the dislocation curves. These vectors are chosen in such a way that the dislocationdipoles representing the walls attract the dislocation curves inwards to the correspondingwalls. Such positions of the trapped dislocations will be starting positions for the bow-out simulations. A starting position of Γ, the time level t = 0 s, is shown in fig. 7. Ina large part of its length, the dislocation segment is straight and parallel to the walldipoles. The midpoint of the curve Γ in the starting position is located in the xyz-spaceat (593 nm, 0 nm, 350 nm) and the midpoint of Γ′ is at (−593 nm, 42 nm, 250 nm).

When the dislocations Γ and Γ′ are exposed to a sufficiently strong loading conditions,they bow-out and expand into the channel. We use τapp = 40 MPa in the stress-controland ε = 9.3 · 10−4 s−1 in the strain-control. Fig. 7 shows in detail sequential positionsof the curve Γ bowing out from the wall in the stress-control; the sequential positions ofboth Γ and Γ′ can be seen in fig. 8. In addition, the trajectories of some selected points ofΓ are shown as dotted lines, e.g. p126 denotes the midpoint of Γ. For the strain-control

13

hc

dc

D1

D2

~t

D′

1

D′

2

~t′

Γ′

Γ

Figure 6: A model of PSB in three-dimensional space.

the sequential positions of both Γ and Γ′ can be seen in figs. 9 (a) and 9 (b).As long as the dislocations are far from each other, they almost do not feel the elastic

field of the other dislocation. On the other hand, if the dislocations are close enoughthe interaction shear stress can be very strong and influence the dislocations’ shapes andtheir “contact”. The sequential positions of the dislocations shown in figs. 8 and 9 for thestress-control and the strain-control, respectively, demonstrate rather big difference in thedislocations’ shapes and the extend of their “contact”. These upper and lower estimatesof the bow-out and passing processes emphasize the need to evaluate correctly the stressfield in the channel by solving a BVP. Moreover, the strain-control simulation in fig. 9 (a)and (b) differ in the starting positions. The simulation in fig. 9 (a) started at the samepositions as in the stress-control, while in the simulation in fig. 9 (b) the straight segmentsD1D2 and D′

1D′2 were used as the starting positions. The differences seen in fig. 9 (a) and

(b) indicate a sensitivity of the bow-out to details of modeling of a close range interactionbetween a dislocation and a wall.

5 Discussion

As was mentioned in section 2, a tangential velocity of the points of the dislocation lines ina curvature driven motion has no impact on the shape of the evolving curves. Therefore, anintroduction of a tangential velocity can provide a convenient redistribution of the gridpoints Xi of the polygons approximating the dislocations. Two methods are currentlybeing tested. The first version of the redistribution keeps the grid points equidistant.The method prevents the main drawback of the present simulation that in faster movingparts of the curve the length of the linear segments of the polygon increases faster than inthe rest of the curve. Another possibility is to design the redistribution which increasesthe density of the grid points in the parts of a high curvature. Both methods improve the

14

accuracy of the numerical method, may reduce a computational time and allow larger timesteps without loss of stability. The simulations based on these methods will be publishedin a subsequent paper.

From the computer simulation point of view a higher accuracy and efficiency of thenumerical method would be useless without a simultaneous improvement of the modelitself. At least two improvements are needed: (i) instead of the “stress-control” and“strain-control” estimates of the upper and lower limits of the stress field in a PSB–channel, formulate and solve the corresponding BVP; (ii) develop a more realistic modelof the close interaction between the dislocations and the walls of PSB.

The BVP means to consider a PSB sandwiched between two elastic half-spaces whichapproximate the parts of the crystal filled with a vein structure. Far from the inter-faces between the PSB and the vein structure, the half-spaces can be thought of as beingexposed to loading conditions of the specimen. The PSB has a composite structure ofwalls and channels. The plastic properties of the relatively rigid walls consisting mainlyof edge dipolar loops are controlled by polarization of the loops, absorption of dislo-cations deposited at the walls, and the loop and dislocation annihilation. Much softerplastic properties of the channels are governed by dislocation glide. The stress field in thechannel is strongly influenced by the dislocation motion. For a computer simulation thespecification, formulation, and solution of the PSB–BVP presents a challenging task. Thecomputation method requires a combination of the discrete dislocation dynamics and thefinite element method. The homogenization method for a discrete-continuum simulationof dislocation dynamics [33] is a successful attempt to design such combination.

In the present paper the local forces exerted on the glide dislocations by the PSB wallswere approximated by the elastic stress fields of the fixed rigid edge dipoles. However,details of the stress field at the walls may influence the bow-out process. Therefore, itwould be more realistic to model the walls as clusters of the edge dipolar loops. The wallsof such consistency were introduced in [16] for the study of the passing stress and thedeposition of the dislocations at the walls in the stress-control regime.

6 Conclusions

The interacting dislocations are modeled in the arc-length parameterization as movingflexible planar curves. The glide of a dislocation segment is governed by a linear viscouslaw assumed in the form of the standard mean curvature flow equation where each in-finitesimal dislocation segment is subjected to the line tension, the interactions with othermoving dislocations, and the stress field imposed by the loading conditions.

The presented DDD simulation method is suitable for study of elementary microscopicmechanisms of plastic deformation. We employed an semi-implicit scheme and the dis-cretization based on flowing finite volume approach in space and the method of lines intime. By discretization in space the governing equations are reduced to a ODE systemsolved by standard Runge–Kutta method of the fourth order with fixed time step.

We employed the proposed simulation method in modelling of a generation of glide

15

dislocations by the process of bowing out from the walls of a PSB–channel. The shortrange interactions among the dislocations and the walls are incorporated in the modelas the elastic stress fields of the edge dipoles located at the surface of the walls. Twodifferent physical situations are explored: (i) the stress-control and (ii) the strain-controlregime. In the first case the stress in the channel induced by the boundary conditions isassumed to be uniform, in the later case a sum of the elastic and plastic strain is uniform.These two situations are rough substitute for a solution of PSB boundary value problem.

Acknowledgements

We are grateful to Prof. H. Mughrabi and Doc. R. Sedlacek for the inspiring discussions,valuable comments and critical reading of the manuscript. The first and the third authorswere partly supported by Necas Center for Mathematical Modelling, project LC06052 ofthe Ministry of Education, Youth and Sports of the Czech Republic. The research of thesecond author has been supported by grants VZ-MSMT 6840770021. The fourth authorwas partly supported by the project No. MSM 6840770010 ”Applied Mathematics inTechnical and Physical Sciences” of the Ministry of Education, Youth and Sports of theCzech Republic.

200 400 600 800 1000

0

200

400

600

7

65

43

2

p 200

p 175

p 150

p 126

p 100

p 75

z-a

xis

(n

m)

x-axis (nm)

p 50

stress-control

time level t:

(1) 0 s

(2) 0,11 s

(3) 0.17 s

(4) 0.21 s

(5) 0.25 s

(6) 0.31 s

(7) 0.74 s

1

Figure 7: upper figure: A 3-D plot of the end position of the curve Γ and the dipole atthe corresponding wall; bottom figure: A detail of positions of bowing out curve Γ fromthe wall in the stress-control regime.

16

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1969 (2001)

18

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200

-800

-600

-400

-200

0

200

400

600

800

4

4

1

12

23

3

p 200

p 175

p 150

p 126

p 100

p 75

z-a

xis

(n

m)

x-axis (nm)

p 50

stress-control

time level t:

(1) 0.17 s

(2) 0.74 s

(3) 1.42 s

(4) 1.76 s

(a)

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200

-800

-600

-400

-200

0

200

400

600

800

1

p 200

p 175p 150

p 126

p 100

p 75

z-a

xis

(n

m)

x-axis (nm)

p 50

stress-control

time level t:

(1) 2.26 s

1

(b)

Figure 8: A sequence of positions of the curves in the stress-control.

19

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200

-800

-600

-400

-200

0

200

400

600

800

3

6

3

strain-control

time level t:

(1) 0 s

(2) 1.73 s

(3) 1.78 s

(4) 1.95 s

(5) 2.65 s

(6) 5.09 s

(7) 5.56 s

z-a

xis

(n

m)

x-axis (nm)

54

7

54

76

(a)

-1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200

-800

-600

-400

-200

0

200

400

600

800

5 4 32

z-a

xis

(n

m)

x-axis (nm)

strain control

time level t:

(1) 2.77 s

(2) 4.73 s

(3) 6.05 s

(4) 7.33 s

(5) 9.27 s

1

54

32

1

(b)

Figure 9: A sequence of positions of the curves in the strain-control.

20