A discrete mechanics approach to dislocation dynamics in ...personal.us.es/mpariza/img/pdf_revistas_indexadas/11.pdfJournal of the Mechanics and Physics of Solids 55 (2007) 615–647

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Journal of the Mechanics and Physics of Solids

55 (2007) 615–647

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A discrete mechanics approach to dislocationdynamics in BCC crystals

A. Ramasubramaniama, M.P. Arizab, M. Ortiza,�

aDivision of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USAbEscuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092-Sevilla, Spain

Received 18 March 2006; received in revised form 10 August 2006; accepted 12 August 2006

Abstract

A discrete mechanics approach to modeling the dynamics of dislocations in BCC single crystals is

presented. Ideas are borrowed from discrete differential calculus and algebraic topology and suitably

adapted to crystal lattices. In particular, the extension of a crystal lattice to a CW complex allows for

convenient manipulation of forms and fields defined over the crystal. Dislocations are treated within

the theory as energy-minimizing structures that lead to locally lattice-invariant but globally

incompatible eigendeformations. The discrete nature of the theory eliminates the need for

regularization of the core singularity and inherently allows for dislocation reactions and complicated

topological transitions. The quantization of slip to integer multiples of the Burgers’ vector leads to a

large integer optimization problem. A novel approach to solving this NP-hard problem based on

considerations of metastability is proposed. A numerical example that applies the method to study

the emanation of dislocation loops from a point source of dilatation in a large BCC crystal is

presented. The structure and energetics of BCC screw dislocation cores, as obtained via the present

formulation, are also considered and shown to be in good agreement with available atomistic studies.

The method thus provides a realistic avenue for mesoscale simulations of dislocation based crystal

plasticity with fully atomistic resolution.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Dislocation dynamics; Crystal plasticity; Discrete mechanics; Computational mechanics; Multiscale

modeling

see front matter r 2006 Elsevier Ltd. All rights reserved.

.jmps.2006.08.005

nding author.

dresses: [email protected] (A. Ramasubramaniam), [email protected] (M.P. Ariza),

altech.edu (M. Ortiz).

www.elsevier.com/locate/jmps

dx.doi.org/10.1016/j.jmps.2006.08.005

mailto:[email protected]

mailto:[email protected]

ARTICLE IN PRESSA. Ramasubramaniam et al. / J. Mech. Phys. Solids 55 (2007) 615–647616

1. Introduction

This article is concerned with the applications of discrete theories of crystal elasticity todislocation based crystal plasticity. In a recent publication, Ariza and Ortiz (2005) appliedtools from algebraic topology to formulate a discrete mechanics of crystal lattices. Thethrust of that work was primarily the development of analytical tools for lattice models.The present work focuses on the application of those tools to dislocations in body-centeredcubic (BCC) crystals.Discrete dislocation modeling is a widely used computational methodology for assessing

the mechanical behavior of crystals. Most, if not all, of these methods are based on thelinear elastic theory of dislocations. In spite of this simplicity, much insight hasnevertheless been gained into the mechanisms of deformation and hardening in crystals.Discrete dislocation models have thus provided a useful link between crystal defects andthe macroscopic response of crystals. Additionally, discrete dislocation models are able toaddress a regime that is inaccessible to both atomistic calculations, owing tocomputational limitations, as well as to continuum plasticity theory which operates atlarger length-scales. There are a variety of strategies that are adopted in constructing thesemodels which may roughly be grouped into line-tracking based methods (e.g. Kubin andCanova, 1992; Zbib et al., 2002; Ghoniem et al., 2000; Madec et al., 2002) and non-tracking methods such as the phasefield model (Wang et al., 2001, 2004; Koslowski et al.,2002; Koslowski and Ortiz, 2004; Garroni and Muller, 2003, 2004) and the level-setmethod (Xiang et al., 2003, 2004). Line-tracking methods require the discretization ofdislocation lines by some means and subsequent expensive computations of theinteractions of these discrete segments. There are additional requirements of special rulesto account for topological transitions arising from dislocation reactions, annihilation, etc.Moreover, the necessity of having to regularize the dislocation core to avoid singularitiesinherent in the linear elastic theory still persists in these methods. The phasefield approachdoes away with expensive line-tracking and tracks an order parameter instead, dislocationsemerging as part of the post-processing. However, phenomenological gradient termswhich do not exist in the usual dislocation theory require to be added to the energy to keepdislocation cores from spreading. The level-set method does not require any suchunconventional energy terms and can handle dislocation climb and cross-slip elegantly.The issue of smearing out the core singularity still persists in this method though. It alsoremains to be seen if this method can be used to simulate large ensembles of dislocationsrather than unit processes. A common drawback of prevalent dislocation dynamics modelsarises from the use of isotropic elasticity in most cases. Furthermore, there is no notion of adislocation core-structure nor is there a natural separation of slip-planes. The issue ofdislocation nucleation is typically handled in an ad hoc manner and mobility laws requireto be prescribed a priori.The theory presented in the work of Ariza and Ortiz and extended here, attempts

in part to rectify several of the aforementioned limitations of dislocation dynamicsmodels. The very first step of acknowledging the discreteness of the lattice itself endowsthe theory with several desirable features. The model inherits atomistic resolutionnaturally which automatically eliminates divergences associated with core-cutoffs. Theelasticity of the lattice can be accounted for by using empirical potentials which is amarked improvement upon the use of isotropic elasticity theory. Each dislocation line nowinherits a true core-structure which is dependent upon the interatomic potential.

ARTICLE IN PRESSA. Ramasubramaniam et al. / J. Mech. Phys. Solids 55 (2007) 615–647 617

Topological transitions are automatically accounted for without the need for anyexplicit rules of interaction. The computational advantages that derive from the discreteapproach are also noteworthy. The use of algebraic topology greatly facilitatesbookkeeping in numerical calculations through the judicious use of appropriate operators.Additionally, the formalism can be easily translated to the Fourier domain which allowsfor the construction of fast and efficient computational schemes for large systems.Simulations at the mesoscale with fully atomistic resolution thus become a distinctpossibility in our approach.

The theoretical aspects of constructing a discrete mechanics on crystal lattices havebeen discussed at length in Ariza and Ortiz (2005). More in-depth discussions ofalgebraic topology may be found in the books by Munkres (1984) or Hatcher (2002).We direct the interested reader to those works and review only the essential elementsof the theory and its specialization to BCC crystals in Section 2. In Section 3, wediscuss the application of the theory to harmonic BCC lattices. The emphasis is onusing Mura’s theory of eigendis-tortions (Mura, 1987) to compute the stored energyof dislocation ensembles within the discrete setting. The procedure is entirely general andcan handle the complete set of slip-systems with ease. Additionally, we present someelegant geometrical interpretations of dislocation loops in crystals that emergeautomatically from the theory. We also discuss the incorporation of empirical potentialsin the discrete approach and obtain force-constants for the particular case of theFinnis–Sinclair potential (Finnis and Sinclair, 1984) which we use subsequently in ourcomputations. The outcome of the eigendistortion approach combined with theFinnis–Sinclair potentials is to provide a quadratic g-surface for our crystal. To verifythe consequences of this simplifying assumption we conduct numerical tests in Section 4 ona variety of BCC metals. In particular, we determine the core-energy and structure of12h1 1 1i screw dislocations and show these to be in good agreement with atomisticcalculations.

As mentioned previously, the discrete approach combined with the Fourier transformformalism allows for efficient large-scale calculations of slip distributions in crystals.Since crystallography constrains slip to occur in quanta of Burgers vectors, a naiveformulation of the problem of finding energy-minimizing slip distributions in a crystalessentially leads to an integer optimization problem. Such problems are known to be NPcomplete (e.g. Nemhauser and Wolsey, 1988) and are entirely non-trivial to solve. We,however, use a different approach that circumvents this hard (and extremely large)optimization problem by developing an iterative search algorithm that seeks low-energymetastable equilibrium states of the crystal. The approach is thus in keeping with the spiritof the low-energy dislocation structure (LEDS) hypothesis that is commonly believed to bethe underlying principle of plastic deformation in crystals (Kuhlmann-Wilsdorf, 1999). Thedetails of our algorithm and its application to dislocation dynamics are discussed inSection 5. In particular, we show that the algorithm reduces the problem of findingcomplex dislocation distributions to local equilibration calculations at a set of out-of-equilibrium bonds which in turn is a tiny fraction of the total number of bonds inthe crystal. The computational advantage gained is thus significant. Topologicaltransitions are automatically accounted for by this method with no additional effortwhatsoever. We demonstrate the efficacy of the approach with an example of dislocationemission from a point source of dilatation in a crystal which is a first order approximationto a pressurized void.


2. The lattice complex of BCC crystals

The formulation of a mechanics of defective lattices requires consideration of fields thatare defined on atoms, atomic bonds and other elements of the lattice, and relationsamongst them. In order to facilitate the manipulation of these fields and the expression ofthe laws of mechanics it becomes essential to adopt efficient ‘bookkeeping’ procedures. Thefield of algebraic topology has greatly perfected such bookkeeping procedures and thus weshall borrow from that field basic tools of the trade such as the differential andcodifferential operators.The BCC Bravais lattice is generated by the basis fð�a=2; a=2; a=2Þ; ða=2;�a=2; a=2Þ;ða=2; a=2;�a=2Þg. Following Ariza and Ortiz, we regard the crystal lattice as a chaincomplex, i.e., as a set of interconnected vertices, or 0-cells; elementary segments, or 1-cells;elementary areas, or 2-cells; and elementary volumes, or 3-cells. These cells are shown inFigs. 1–3 and are chosen so as to contain the 1

2h1 1 1i slip directions and f1 1 0g slip-planes.

For reference, the primary slip-systems of a BCC lattice are tabulated in Table 1 and onemay ascertain directly that the elementary sets indeed account for all of these systems. Werecall that a complex Bravais lattice may be regarded as a collection of N simple Bravaissublattices. Thus, an element of a 3D complex Bravais lattice can be indexed by means ofthree integers l � ðl1; l2; l3Þ 2 Z3 and a label a 2 f1; . . . ;Ng that designates which of the N

simple Bravais sublattices the element belongs to. It is evident from Figs. 1–3 that thevertex set can be indexed as a simple Bravais lattice; the elementary segment set as acomplex Bravais lattice comprising seven simple sublattices; the elementary area set as acomplex Bravais lattice comprising 12 simple sublattices; and the elementary volume set asa complex Bravais lattice comprising 12 sublattices. The particular indexing scheme used inthis paper is shown in Figs. 1–3, where �1 ¼ ð1; 0; 0Þ, �2 ¼ ð0; 1; 0Þ, �3 ¼ ð0; 0; 1Þ,�4 ¼ ð1; 1; 1Þ, �5 ¼ ð0; 1; 1Þ, �6 ¼ ð1; 0; 1Þ and �7 ¼ ð1; 1; 0Þ. We shall denote by Np thenumber of sublattices of cells of dimension p. Thus, N0 ¼ 1, N1 ¼ 7, N2 ¼ 12 and

l

l+∈4

l-∈5

l-∈4

l+∈1

l+∈6

l+∈2

l+∈3

l+∈7

l+∈1

l+∈6

l+∈5

l+∈2

l-∈3

l+∈7

l

(l,1)

(l, 6)(l-∈ 5,

5)

(l-∈ 1

,1) (l-∈ 4

,4)

(l, 7

)(l-∈

7,7)

(l, 5)

(l, 2)

(l-∈2 ,2)

(l, 4

)

(l, 3)

(l-∈3 ,3)

(l-∈6, 6)

(a) (b)

Fig. 1. Lattice complex representation of the body-centered cubic lattice and complex lattice indexing scheme:

(a) vertex set, (b) elementary segment set.

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32

4 16

7

5

8

101112 9

(a) (b)

(c)

Fig. 2. Lattice complex representation of the body-centered cubic lattice and complex lattice indexing scheme.

Elementary area set.

A. Ramasubramaniam et al. / J. Mech. Phys. Solids 55 (2007) 615–647 619

N3 ¼ 12. We shall also use the notation ep � ðl ; aÞ, l 2 Z3, a 2 f1; . . . ;Npg, to designate acell of dimension p, and the symbol Ep to designate the set of all cells of dimension p.

The state of the crystal will be described in terms of real-valued functions over Ep, orp-forms. We shall denote by Op the set of p-forms over the lattice. In particular, we shalldenote by ep the p-form that takes the value 1 at ep and 0 elsewhere. Then, a general p formadmits the representation

o ¼X

ep2Ep

f ðepÞep, (1)

where f : Ep ! R and f ðepÞ may be regarded as the value of o at ep. We note that theforms ep generate all other forms, in the sense of (1). The dual Op of Op is the set of p-fields

ARTICLE IN PRESS

1

2

33

44

5566

(a) (b)

(c) (d)

(e) (f)

Fig. 3. Lattice complex representation of the body-centered cubic lattice and complex lattice indexing scheme.

Elementary volume set.

A. Ramasubramaniam et al. / J. Mech. Phys. Solids 55 (2007) 615–647620

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Table 1

BCC slip-systems in Schmid and Boas’ nomenclature

Slip-system A2 A3 A6 B2 B4 B5ffiffiffi3p

s ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1�ffiffiffi2p

m ð0 1 1Þ ð1 0 1Þ ð1 1 0Þ ð0 1 1Þ ð1 0 1Þ ð1 1 0Þ

Slip-system C1 C3 C5 D1 D4 D6ffiffiffi3p

s ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1� ½1 1 1�ffiffiffi2p

m ð0 1 1Þ ð1 0 1Þ ð1 1 0Þ ð0 1 1Þ ð1 0 1Þ ð1 1 0Þ

The unit normal to the slip plane is denoted by m and the unit vector along the Burgers vector by s. Note that the

vectors are expressed in Cartesian coordinates and not in the crystal basis.


over the lattice. By a mild abuse of notation, we shall denote by ep the field that takes avalue of 1 on the p-cell ep and vanishes elsewhere. With this convention, fields have therepresentation

L ¼X

ep2Ep

gðepÞep, (2)

where g : Ep ! R and gðepÞ may be regarded as the value of L at ep. The duality pairingmay then variously be expressed in the equivalent forms

LðoÞ � hL;oi ¼X

ep2Ep

f ðepÞgðepÞ. (3)

The formulation of the equations governing the mechanics of crystals is greatlyfacilitated by the introduction of discrete differential and codifferential operators on formsand fields, respectively. By linearity, it suffices to define the differential and codifferential1

of the generators ep and ep, respectively, with

de0ðlÞ ¼ � e1ðl; 1Þ � e1ðl; 2Þ � e1ðl; 3Þ � e1ðl; 4Þ

� e1ðl; 5Þ � e1ðl; 6Þ � e1ðl; 7Þ þ e1ðl � �1; 1Þ

þ e1ðl � �2; 2Þ þ e1ðl � �3; 3Þ þ e1ðl � �4; 4Þ

þ e1ðl � �5; 5Þ þ e1ðl � �6; 6Þ þ e1ðl � �7; 7Þ, ð4aÞ

de1ðl; 1Þ ¼ � e2ðl; 2Þ þ e2ðl þ �1; 4Þ � e2ðl þ �1; 5Þ

þ e2ðl; 7Þ � e2ðl; 10Þ þ e2ðl þ �1; 12Þ, ð4bÞ

de1ðl; 5Þ ¼ �e2ðl þ �2; 1Þ � e2ðl � �1; 2Þ þ e2ðl þ �3; 3Þ þ e2ðl þ �4; 4Þ, (4c)

de2ðl; 1Þ ¼ �e3ðl � �2; 1Þ þ e3ðl � �7; 5Þ, (4d)

de1ðl; 1Þ ¼ e0ðl þ �1Þ � e0ðlÞ, (4e)

1Throughout this article, the symbol ‘‘d’’ will always denote the codifferential operator—any departure from

this convention will be made explicit.


de1ðl; 5Þ ¼ e0ðl þ �5Þ � e0ðlÞ, (4f)

de2ðl; 1Þ ¼ �e1ðl � �2; 5Þ þ e1ðl � �2; 2Þ þ e1ðl; 3Þ, (4g)

de3ðl; 1Þ ¼ �e2ðl þ �2; 1Þ þ e2ðl þ �5; 7Þ þ e2ðl; 8Þ � e2ðl þ �4; 4Þ (4h)

and all symmetry-related identities (cf. Ariza and Ortiz, 2005). By linearity, the differentialof a p-form Op 3 o ¼

Pf ðepÞe

p is the ðpþ 1Þ-form

do ¼X

ep2Ep

f ðepÞdep (5)

and the codifferential of a p-field Op 3 L ¼P

gðepÞep is the ðp� 1Þ-field

dL ¼X

ep2Ep

gðepÞdep. (6)

It is readily verified from these definitions that

d2 ¼ 0, ð7aÞ

d2 ¼ 0, ð7bÞ

hdL;oi ¼ hL;doi. ð7cÞ

We shall also find it useful to express lattice sums in integral notation. Thus, let A be asubset of Ep and let a 2 Op. Then, if a ¼

Pf ðepÞep we shall writeZ

A

a ¼Xep2A

f ðepÞ. (8)

A precis of properties of discrete differential operators and integrals may be found in Arizaand Ortiz (2005).The discrete Fourier transform provides an additional means of exploiting the

translation invariance of lattice complexes. Thus, the discrete Fourier transform of thep-form o 2 Op is (cf. Appendix A)

oðh; aÞ ¼Xl2Zn

oðl; aÞe�ih�l (9)

and its inverse is

oðl; aÞ ¼1

ð2pÞn

Z½�p;p�n

oðh; aÞeih�l dny. (10)

The Fourier and inverse Fourier transform of fields may be defined likewise: thecorresponding DFT representation of the differential is

cdoðh; aÞ ¼XN

b¼1

QabðhÞoðh;bÞ, (11)

where the matrices

QT1 ¼ ðe

iy1 � 1; eiy2 � 1; eiy3 � 1; eiy4 � 1; eiy5 � 1; eiy6 � 1; eiy7 � 1Þ, (12a)


QT2 ¼

0 �1 0 e�iy1 �e�iy1 0 1 0 0 �1 0 e�iy1

e�iy2 0 �1 0 0 �e�iy2 0 1 0 �e�iy2 0 1

1 0 �e�iy3 0 �1 0 e�iy3 0 �1 0 e�iy3 0

0 1 0 �e�iy4 0 e�iy4 0 �1 1 0 �e�iy4 0

�e�iy2 �eiy1 e�iy3 e�iy4 0 0 0 0 0 0 0 0

0 0 0 0 e�iy1 �e�iy4 �e�iy3 eiy2 0 0 0 0

0 0 0 0 0 0 0 0 �eiy3 e�iy2 e�iy4 �e�iy1

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA,

ð12bÞ

QT3 ¼

�eiy2 0 0 0 eiy7 0

0 0 1 0 �1 0

0 �eiy3 eiy6 0 0 0

�eiy4 eiy4 0 0 0 0

0 0 �eiy1 0 0 eiy7

0 0 �eiy4 eiy4 0 0

eiy5 0 0 �eiy3 0 0

1 0 0 0 0 �1

0 1 0 �1 0 0

0 eiy5 0 0 0 �eiy2

0 0 0 0 �eiy4 eiy4

0 0 0 eiy6 �eiy1 0

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

ð12cÞ

define the exterior derivatives of forms in their Fourier representation. It follows directlyfrom (7c) that the codifferential has the Fourier representation

cdLðh; aÞ ¼XN

b¼1

PabðhÞLðh;bÞ, (13)

where

P ¼ Qy, (14)

where Qy ¼ ðQ�ÞT is the Hermitian transpose of Q.

3. The stored energy of discrete dislocations

In the present work, we focus primarily on developing a discrete theory of harmonic

crystals. This is an inevitable fallout of the desire to use tools such as the Fourier transformfor analytical purposes and for large-scale computations. Before delving into the details ofthe discrete theory, we recall first some well-known notions from the continuumstandpoint to provide a point of reference. Within the harmonic approach, the energyof a crystal is a convex function of the displacement field. The underlying crystalline


structure, however, allows for displacements that leave the lattice-invariant. The totalenergy of a crystal is thus a non-convex function of the displacements when crystal-lographic slip is allowed. While this non-convexity is what allows for the emergence oflattice defects such as dislocations in the first place, it would seemingly stymie our efforts touse the harmonic approximation. This deficiency can be remedied by recourse to the theoryof eigendeformations (Mura, 1987; Ortiz and Phillips, 1999). To this end, recall that theplastic distortion in a slipped crystal may be written as

bij ¼XN

a¼1

gasai maj , (15)

the sum in a running over all the available slip-systems in the crystal. The plastic distortionis, however, not arbitrary and is constrained by crystallography. Thus the plastic distortionis built from lattice preserving deformations such as crystallographic slip and is referred toas an eigendeformation. The elastic energy of the crystal is a functional of the displacementfield and the eigendeformation and is given by

E½u; b� ¼Z

V

1

2Cijklðui;j � bijÞðuk;l � bklÞdV , (16)

which is now quadratic in the elastic distortion field beij ¼ ui;j � bij and piecewise quadratic

in ui;j where Cijkl are the usual elastic moduli. This approach will be employed in ananalogous manner to accommodate crystallographic slip within the discrete formulation.

3.1. Eigendeformations in discrete lattices

The energy of a harmonic crystal admits the representations (Ariza and Ortiz, 2005)

EðuÞ ¼

ZE1

ZE1

1

2Bikðe1; e

01Þduiðe1Þdukðe

01Þ �

1

2hB du;dui (17)

and

EðuÞ ¼

ZE0

ZE0

1

2Aikðe0; e

00Þuiðe0Þukðe

00Þ �

1

2hAu; ui, (18)

where u : E0! Rn is the displacement field, and A : E20 ! symRn�n and B : E2

1! symRn�n

collect the harmonic force-constants of the crystal (E0 and E1 being the vertex set and bondset of the crystal, respectively). By translation invariance we have the representations

B du ¼ W � du, (19a)

Au ¼ U � u, (19b)

where W and U are the force-constant fields of the lattice and � denotes the convolutionoperation (cf. Appendix A). Since Eqs. (19a) and (19b) are in convolution form, an applicationof Parseval’s identity and the convolution theorem yields the alternative representations

EðuÞ ¼1

ð2pÞn

Z½�p;p�n

1

2hWðhÞcduðhÞ;cdu

�

ðhÞidny, (20a)

EðuÞ ¼1

ð2pÞn

Z½�p;p�n

1

2hUðhÞuðhÞ; u�ðhÞidny, (20b)


where we write

hWðhÞcduðhÞ;cdu�

ðhÞi �XN

a¼1

XN

b¼1

Cik

h

ab

! cduiðh; aÞddu�kðh;bÞ, (21a)

hUðhÞuðhÞ; u�ðhÞi � FikðhÞuiðhÞu�kðhÞ (21b)

for shorthand. The preceding representations show that the force-constant fields are related as

U ¼ QT1 WQ�1, (22)

Q1 being the matrix of the Fourier representation of the differential of a 1-form introducedpreviously in Eq. (12).

Consider a homogeneous lattice-invariant deformation of the form

F ¼ I þxd

b�m, (23)

where m is the unit normal to the slip plane, b is the Burgers vector, d is the interplanardistance and x 2 Z is the magnitude of the slip. The resulting eigendeformation 1-formb 2 O1 is

b ¼ ðF � IÞdxðe1Þ ¼ ðdxðe1Þ �mÞxd

b. (24)

Note that the vector quantity dxðe1Þ may be thought of physically as an oriented bond (anelement of the 1-cell set illustrated previously); the preceding expression for theeigendeformation thus measures how the bond changes in length and orientation as aresult of the applied deformation gradient. Also note that the product dxðe1Þ �m is zerowhen the corresponding 1-cell lies in the slip plane and is equal to the interplanar distanced otherwise. Now allow for crystallographic slip on M crystallographic systems defined byBurgers vectors and normals ðbs;msÞ, s ¼ 1; . . . ;M, respectively. The resulting eigende-formation 1-form may immediately be written down from the preceding expression as

b ¼XMs¼1

Xe12E1ðm

sÞ

xsðe1Þb

se1, (25)

where xs : E1ðmsÞ ! Z is the integer-valued slip field corresponding to slip-system s and

E1ðmsÞ ¼ fe1 2 E1; s:t: dxðe1Þ �m

sa0g. We shall assume that exact, or compatible,eigendeformations of this general type cost no energy. In the spirit of the eigendeformationtheory, the elastic energy may be assumed to be of the form

Eðu; nÞ ¼1

2hBðdu� bÞ;du� bi (26)

which replaces Eq. (17) in the presence of crystallographic slip. Clearly, if b ¼ dv, i.e., if theeigendeformations are compatible, then the energy-minimizing displacements are u ¼ vand E ¼ 0. However, because slip is crystallographically constrained, b must necessarily beof the form in Eq. (25) and, therefore, is not compatible in general. By virtue of this lack ofcompatibility, a general distribution of slip induces residual stresses in the lattice and anon-vanishing elastic energy, or stored energy.

An elegant geometrical interpretation of eigendeformations can be given with the aid ofa discrete version of the Nye dislocation density tensor (Nye, 1953; Ariza and Ortiz, 2005),


namely,

a ¼ db. (27)

This simple relation may be regarded as the discrete version of Kroner’s formula (Kroner,1958). From Eq. (27) and by recalling the relations in Eq. (7c), it follows immediately that

da ¼ 0 (28)

which generalizes the conservation of Burgers vector identity. Clearly, if the eigendeforma-tions are compatible, i.e., if b ¼ dv, then a ¼ d2v ¼ 0. The dislocation density a hencemeasures the degree of incompatibility of the eigendeformations. Note that a is a 2-form,a 2 O2, and, therefore, it is defined on the 2-cells of the crystal. Also note that in view ofEq. (25), the dislocation density can be written as

a ¼XMs¼1

Xe12E1ðm

sÞ

xsðe1Þb

s de1 �XMs¼1

as, (29)

i.e., as the sum of dislocation densities from individual slip-systems. Each as is orientedalong the Burgers vector bs and is in turn obtained by the superposition of elementarydislocation loops de1 with multiplicity xs. The unit dislocation loop or ‘‘loopon’’ (Ortiz andPhillips, 1999) thus consists of a ring of oriented 2-cells incident on a 1-cell that supports aslip of unit magnitude. The loopons for a 2D square lattice and for the BCC lattice areillustrated in Fig. 4. In particular, in the 2D case the loopon appears as a dislocationdipole.If the distribution of eigendeformations is known, and in the absence of additional

constraints, the energy of the lattice can be readily minimized with respect to thedisplacement field. Suppose that the crystal is acted upon by a distribution of forcesf : E0! Rn. The total potential energy of the lattice is then

F ðu; nÞ ¼ Eðu; nÞ � hf ; ui. (30)

Minimization of F ðu; nÞ with respect to u yields the equilibrium equation

Au ¼ f þ dBb, (31)

where dBb may be regarded as a distribution of eigenforces corresponding to theeigendeformations b. The equilibrium displacements are2

u ¼ A�1ðf þ dBbÞ � u0 þ A�1dBb, (32)

where u0 ¼ A�1f is the displacement field induced by the applied forces in the absence ofeigendeformations. Conditions under which the minimum problem just described is well-posed and delivers a unique energy-minimizing displacement field have been given in Arizaand Ortiz (2005). The corresponding minimum potential energy is

F ðbÞ ¼ 12hBb; bi � 1

2hA�1ðf þ dBbÞ; f þ dBbi

¼ 12hBb; bi � 1

2hA�1dBb; dBbi � hA�1dBb; f i � 1

2hA�1f ; f i

¼ 12hBb; bi � 1

2hA�1dBb; dBbi � hBb;du0i �

12hAu0; u0i. ð33Þ

2In deriving this relation we use Eq. (26) and the identity EðuÞ ¼ 12hB du;dui ¼ 1

2hAu; ui (cf. Eqs. (17) and (18)).

ARTICLE IN PRESS

(a)

(b)

Fig. 4. Elementary dislocation loops or ‘‘loopons’’ in (a) a square lattice and (b) a BCC lattice. For the square

lattice, the loopons appear as dislocation dipoles with dislocation segments normal to the plane of the figure. In

the BCC case, loopons are constituted of rings of 2-cells incident on 1-cells. Such elementary dislocations may be

constructed by assigning unit eigendeformation to a bond e1 and systematically applying the differential operator

(Eq. (4b)) to the 1-form bðe1Þ.


The first two terms

EðbÞ ¼ 12hBb; bi � 1

2hA�1dBb; dBbi (34)

in Eq. (33) give the self-energy of the distribution of lattice defects represented by theeigendeformation field b, or stored energy; the third term in Eq. (33) is the interactionenergy between the lattice defects and the applied forces; and the fourth term in Eq. (33) isthe elastic energy of the applied forces.

Recall that the crystal under consideration possesses M slip-systems and itseigendeformations admit the representation Eq. (25) in terms of an integer-valued slipfield n ¼ fxs; s ¼ 1; . . . ;Mg. Then, the stored energy Eq. (34) can be written in the form

EðnÞ ¼1

2hHn; ni, (35)


where the operator H is defined by the identity

hHn; ni ¼ hBb;bi � hA�1dBb; dBbi (36)

in which b and n are related through Eq. (25). By translation invariance we must have

Hn ¼ Y � n (37)

for some discrete hardening-moduli field Y, and Eq. (35) admits the Fourier representation

EðnÞ ¼1

ð2pÞn

Z½�p;p�n

1

2hYðhÞnðhÞ; n

�ðyÞidh, (38)

where we write

hYðhÞnðyÞ; n�ðhÞi �

XMr¼1

XMs¼1

XN

a¼1

XN

b¼1

Urs

h

ab

� �x

rðh; aÞx

s�ðh;bÞ (39)

for shorthand. The individual components, Urs, of the matrix Y ðjl�l0ja b Þmay be interpreted as

the interaction energy between two loopons with Burgers vectors br and bs generated by2-cells incident on 1-cells e1ðl; aÞ and e1ðl

0;bÞ, respectively. Also note that Y is completelyknown from the force-constants W. If the crystal is additionally acted upon by a force fieldf , the resulting potential energy is (cf. Eq. (33))

F ðnÞ ¼ EðnÞ � hs; ni � 12hA�1f ; f i, (40)

where the forcing field t follows from the identity

hs; ni ¼ hf ;A�1dBbi (41)

in which b and n are again related through Eq. (25). We note that sðe1Þ is the energeticforce conjugate to nðe1Þ and, therefore, may be regarded as a collection of M discretePeach–Koehler forces, or resolved shear-stresses, acting on the 1-cell e1. According to thisinterpretation, Hn represents the resolved shear-stress field resulting from a slipdistribution n, and, therefore, H may be regarded as an atomic-level hardening matrix.Finally, we require to find the slip distribution n that minimizes the potential energy F

ðnÞ. This is an integer optimization problem and is known to be NP complete—solving it isthus an entirely non-trivial task. We will return to this issue subsequently in Section 5where we illustrate that minimization procedure with a concrete example. For now, wecontinue with the issue of computing the energy of slipped crystal and consider next theincorporation of interatomic potentials within our framework.

3.2. Interatomic potentials and the harmonic approximation

The use of empirical or semi-empirical potentials for modeling defects in solids iscommonplace. There are several variants of such potentials, e.g. pairwise potentials such asthe classical Lennard–Jones potential; pairwise functionals such as the embedded-atommethod (EAM) for FCC metals (Daw, 1990), the modified embedded-atom method(MEAM) for BCC metals (Yuan et al., 2003), the Finnis–Sinclair potential for transitionmetals (Finnis and Sinclair, 1984); Stillinger–Weber potentials for covalent crystals(Stillinger and Weber, 1985); among several others. In order to apply analytical tools suchas the Fourier transform, it is necessary to adapt these potentials to harmonic crystals—this essentially involves obtaining the corresponding force-constants from a given


interatomic potential. We specialize the discussion to the EAM family below noting thatthe procedure is otherwise completely general.

The energy of a crystal may be written within the EAM as

E ¼1

2

Xi;j

V ijðrijÞ þX

i

F iðriÞ, (42a)

ri ¼X

j

FijðrijÞ, (42b)

where i; j; . . . label the atoms, rij ¼ jxi � xjj are the interatomic distances, xi are the atomicpositions, V is the pair potential, r is the electronic density, and F is the embedding energy.For nearest-neighbor interactions, the corresponding expressions of the energy usingdiscrete lattice calculus are

E ¼1

2

ZE0

ZE0

V ðjx� x0jÞ þ

ZE0

F ðrðe0ÞÞ, (43a)

rðe0Þ ¼Z

E0

Fðjx� x0jÞ (43b)

which may equivalently be written as

E ¼

ZE1

V ðjdxðe1ÞjÞ þ

ZE0

F ðrðe0ÞÞ, (44a)

rðe0Þ ¼Z

E1\Stðe0Þ

Fðjdxðe1ÞjÞ, (44b)

where Stðe0Þ is used to denote the star of vertex e0 (e.g. Munkres, 1984). The first andsecond linearizations of this energy are3

dE ¼

ZE1

DV ðjdxðe1ÞjÞdxiðe1Þ

jdxðe1Þjdbiðe1Þ

þ

ZE0

DF ðrðe0ÞÞZ

E1\Stðe0Þ

DFðjdxðe1ÞjÞdxiðe1Þ

jdxðe1ÞjdbiðeiÞ

( ), ð45aÞ

d2E ¼Z

E1

D2V ðjdxðe1ÞjÞdxiðe1Þ dxjðe1Þ

jdxðe1Þj2

�þDV ðjdxðe1ÞjÞ

1

jdxðe1Þjdij �

dxiðe1Þdxjðe1Þ

jdxðe1Þj2

� ��dbiðe1Þdbjðe1Þ

þ

ZE0

D2F ðrðe0ÞÞZ

E1\Stðe0Þ


jdxðe1Þjdbiðe1Þ

!

3Here, ‘‘d’’ denotes the variation and is not to be confused with the codifferential operator while dij is the usual

Kronecker delta.


�

ZE1\Stðe0Þ

DFðjdxðe1ÞjÞdxjðe1Þ

jdxðe1Þjdbjðe1Þ

!

þ

ZE0

D2F ðrðe00ÞÞZ

E1\Stðe0Þ


jdxðe1Þjdbiðe1Þ

!

�

ZE1\Stðe0Þ

DFðjdxðe01ÞjÞdxjðe

01Þ

jdxðe01Þjdbjðe

01Þ

!

þ

ZE0

½DF ðrðe0ÞÞ þDF ðrðe00ÞÞ�Z

E1\Stðe0Þ

D2Fðjdxðe1ÞjÞdxiðe1Þdxjðe1Þ

jdxðe1Þj2

(

þDFðjdxðe1ÞjÞ1

jdxðe1Þjdij �

dxiðe1Þdxjðe1Þ

jdxðe1Þj2

� ��dbiðe1Þdbjðe1Þ, ð45bÞ

where Df ð�Þ denotes the derivative of f with respect to its argument. The correspondingforce-constants are

Bðe1; e1Þ ¼ PI þQdxðe1Þ � dxðe1Þ

jdxðe1Þj2

, (46a)

Bðe1; e01Þ ¼ R

dxðe1Þ � dxðe01Þ

jdxðe1Þjjdxðe01Þjif fe1; e

01g have a common vertex, (46b)

Bðe1; e01Þ ¼ 0 otherwise, (46c)

where, using the notation r � jdxðe1Þj and r0 � jdxðe01Þj,

P ¼1

r½DV ðrÞ þDFðrÞfDF ðrðe0ÞÞ þDF ðrðe00ÞÞg�, (47a)

Q ¼ D2V ðrÞ �DV ðrÞ1

rþ DF ðrðe0ÞÞ þDF ðrðe00ÞÞ� �

D2FðrÞ �DFðrÞ1

r

þ fD2F ðrðe0ÞÞ þD2F ðrðe00ÞÞgðDFðrÞÞ2, ð47bÞ

R ¼ D2F ðrÞDFðrÞDFðr0Þ. (47c)

Specific expressions for V , F and F and their parametrization for specific materials can befound in the specialized literature.Although it has become clear over the past few years that an accurate description of

interatomic forces in BCC transition metals necessitates the inclusion of angular terms(Carlsson, 1991) not included in either the Finnis–Sinclair or EAM type potentials, we usethem in our present analysis for the sake of simplicity. The force-constants for the specialcase of the Finnis–Sinclair potential (Finnis and Sinclair, 1984) have the following terms

V ðrÞ ¼ ðr� cÞ2ðc0 þ c1rþ c2r2Þ, (48a)

FðrÞ ¼ ðr� dÞ2, (48b)

F ðrÞ ¼ �Affiffiffirp

, (48c)

where c0, c1, c2, c, d and A are fitting parameters that can be found using experimental data(see Tables 2 and 3).

ARTICLE IN PRESS

Table 2

Fitting parameters for the Finnis–Sinclair potential [adapted from Finnis and Sinclair (1984)]

d ðAÞ A ðeVÞ c ðAÞ c0 c1 c2

Tungsten 4.400224 1.896373 3.25 47.1346499 �33.7665655 6.2541999

Tantalum 4.076980 2.591061 4.20 1.2157373 0.0271471 �0.1217350

Vanadium 3.692767 2.010637 3.80 �0.8816318 1.4907756 �0.3976370

Molybdenum 4.114825 1.887117 3.25 43.4475218 �31.9332978 6.0804249

Table 3

Lattice parameter a and elastic constants (Mbar)

a ðAÞ c11 c12 c44 c011 c012 c044

Tungsten 3.1652 5.224 2.044 1.606 5.217 2.041 1.604

Tantalum 3.3058 2.660 1.612 0.824 2.657 1.610 0.823

Vanadium 3.0399 2.279 1.187 0.426 2.276 1.185 0.425

Molybdenum 3.1472 4.647 1.615 1.089 4.641 1.613 1.088

The unprimed and primed values represent the elastic constants obtained from the original Finnis–Sinclair

potential and after linearization, respectively.


4. The BCC screw dislocation: core-energy and structure

The core-structure of dislocations, in particular of those in screw orientations, is widelybelieved to be the controlling factor for plastic deformation at low temperatures in BCCmetals. This phenomenon is attributable to the reduced mobility of screw dislocationsowing to their non-planar core-structure and was first described by Hirsch (1960) in termsof material symmetry. Several subsequent studies have built upon this observation andused atomistic models to study in detail the structure and energetics of the screwdislocation core (e.g. Vitek, 1974; Xu and Moriarty, 1996; Yang et al., 2001). Our interesthere is to use the discrete mechanics model to compute the core-structure and energy of[1 1 1] screw dislocations in a variety of BCC metals, viz. tungsten, tantalum, vanadiumand molybdenum, and compare them with existing estimates from the atomistic literature.This exercise provides the most basic validation test for the discrete model before going onto study large dislocation ensembles.

Within linear elasticity theory, the formation energy per unit length of an infinite screwdislocation in a cubic crystal may be expressed in the form

Ef ¼ Efcore þ A ln

R

a0

� �, (49)

where the first term, Efcore, represents the ‘‘core-energy’’ stored in a cylinder of radius a0

about the dislocation line and the second term represents the elastic energy (Foreman,1955) stored in a hollow cylinder of outer and inner radii, R and a0, respectively. Thecoefficient A is a function of the elastic moduli C11, C12 and C44 of the crystal, the directioncosines of the dislocation line with respect to the cubic axes and the Burgers vector (Head,1964; Stroh, 1958). It should be noted that the core-energy is not a physical quantity in the


sense that it depends upon the chosen cutoff radius a0. The formation energy per unitlength of an infinite screw dislocation is plotted in Fig. 5 as a function of lnðR=a0Þ forvarious BCC metals using the discrete approach. The core-energy and the elastic prefactorA are then the y-intercept and slope, respectively, of the linear fit to the data.As a check, the coefficient A is computed directly using the elastic moduli from the

Finnis–Sinclair potential and from the discrete theory (slopes of the linear fits in Fig. 5)and tabulated in Table 4—as expected, these values are in close agreement.A more detailed comparison of the results from the discrete theory with existing values

from the atomistic literature is carried out next. In particular, we compare our results withthe calculations of Yang et al. (2001) and Xu and Moriarty (1996) for tantalum andmolybdenum, respectively, which use multi-ion interatomic potentials derived from themodel generalized pseudopotential theory (MGPT). As mentioned previously, the core-energy is sensitive to the selected core-cutoff and, for purposes of comparison, we adoptthe choice of the above authors and select our core-cutoffs as a0 ¼ 1:75b for tantalum and

2.2

2.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.4

1.2

1.6

1.8

Ef /b

(eV

/A)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

In (R/a0)

Tantalum

Molybdenum

Tungsten

Vanadium

Fig. 5. Energy of a screw dislocation on the A6 slip-system as a function of lnðR=a0Þ as obtained with the discrete

formulation.

Table 4

Comparison of the coefficient A ðeV=AÞ calculated from Finnis and Sinclair’s elastic moduli and from the discrete

theory

AF2S Adiscrete

Tungsten 0.597 0.567

Tantalum 0.244 0.269

Vanadium 0.161 0.147

Molybdenum 0.453 0.466


a0 ¼ 2b for molybdenum. The results obtained from the discrete theory are compared withthe aforementioned studies in Figs. 6 and 7. For tantalum, we estimate a value of0:221 eV=A for the core-energy and 0:269 eV=A for the elastic prefactor which is inexcellent agreement with the values of 0:21 and 0:26 eV=A, respectively, as determined byYang et al. (2001). Similarly, for molybdenum we estimate a value of 0:828 eV=A for thecore-energy and 0:466 eV=A for the elastic prefactor which, once again, is in goodagreement with the respective estimates of 0.888 and 0:5186 eV=A of Xu and Moriarty(1996).

Finally we consider the issue of the structure of the dislocation core as obtained from thediscrete theory. Several detailed descriptions of the structure of BCC screw dislocationcores can be found in the atomistic literature (Vitek, 1974; Xu and Moriarty, 1996;Duesbery and Vitek, 1998; Ismail-Beigi and Arias, 2000; Segall et al., 2001; Yang et al.,2001; Li et al., 2004). The common underlying conclusion that emerges from these studiesis that the screw dislocation core is essentially non-planar and shows approximate three-fold symmetry about the [1 1 1] axis. We recall that it is this non-planar core-structure thatis believed to greatly reduce the mobility of BCC screw dislocations. A common techniqueadopted in atomistic simulations of dislocation core-structure is to use a quadrupolararrangement of dislocations with periodic boundary conditions enforced on the simulationcell. Periodicity eliminates free surface effects while the quadrupolar arrangement ensuresthat there are no grain boundaries and hence no misfit at the cell boundary (Bigger et al.,1992). We adopt an identical approach and simulate a periodic cell of molybdenum with ascrew dislocation quadrupole on the A6 slip-system. The individual dislocations aredefined by applying appropriate eigendeformations on the relevant 1-cells. Thedisplacements of the atoms resulting from this distribution of eigendeformations maythen be readily computed via Eq. (32). The resulting core-structure can be visualized inseveral ways—we show in Fig. 8 the core-structure of one of the dislocations using the

1.0

0.8

0.6

0.4

0.2

0.0

Ef /b

(eV

/A)

-1 0 1 2 3

In (R/1.75b)

Ef/b=0.21+0.26 In (R/1.75b) [Yang et al.]

present work

Ef/b=0.221+0.269 In (R/1.75b)

Fig. 6. [1 1 1] screw dislocation formation energy in Ta as a function of lnðR=1:75bÞ as obtained with the discrete

formulation. The results of Yang et al. (2001) are shown for comparison.

ARTICLE IN PRESS

Ef /b

(eV

/A)

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4-1 0 1 2

In (R/2b)

Ef/b=0.88+0.5186 In (R/2b) [Xu & Moriarty]present workEf/b=0.828+0.466 In (R/2b)

Fig. 7. [1 1 1] screw dislocation formation energy in Mo as a function of lnðR=2bÞ as obtained with the discrete

formulation. The results of Xu and Moriarty (1996) are shown for comparison.

[110]

[112]−

Fig. 8. Differential displacement map of screw dislocation in Mo using the discrete approach. The dashed lines

are a guide to visualizing the three-fold symmetry.


conventional differential displacement (DD) map (Vitek et al., 1970). It may be seen thatthe discrete approach indeed leads to a core-structure that shows the expected three-foldrotational point-group symmetry about the [1 1 1] axis. The structure also compares well


with the first-principles calculations of Ismail-Beigi and Arias (2000) for the, so-called,‘‘easy core’’ structure in molybdenum. Our discrete approach is thus seen to produce bothqualitatively reasonable core-structures and quantitatively correct core-energies.

In concluding this section, we make a few comments regarding the determination ofcore-energy and core-structure using the discrete approach. The dislocation core, asobtained from the present approach, is always expected to be relatively compact owing tothe harmonic nature of the crystal. Thus, the fact that the core-structure obtained herecompares well with the calculations of Ismail-Beigi and Arias (2000) is indeed fortuitous.In fact, our dislocation cores do not, and cannot, show the more extended three-foldstructure obtained by Xu and Moriarty (1996) and Li et al. (2004), among others—whetherthe core is in reality quite compact or more dissociated is an entirely different debate whichwe are not equipped to address with the present method. Nevertheless, the key point tonote is that the seemingly simple discrete model indeed leads to qualitatively reasonablecore-structures and quantitatively accurate core-energies. This is in itself a significantimprovement over standard continuum dislocation models which are entirely obivious tothe underlying atomistic nature of the core.

5. Dislocation dynamics

The problem of understanding slip activity in plastically worked metals is of long-standing interest in physical metallurgy. In particular, the formation of dislocationnetworks in crystals undergoing low temperature non-creep dislocation plasticity is ofconsiderable interest and is currently understood in terms of the so-called LEDS principle(e.g. Kuhlmann-Wilsdorf (1999)). Computational methods to model the dynamics ofdislocation ensembles typically exist as older line-tracking based dislocation dynamicspioneered by Kubin and coworkers (Kubin and Canova, 1992; Madec et al., 2002) or morerecent ones such as level-set methods (Xiang et al., 2003, 2004) and phase-field methods(Wang et al., 2001, 2004). For a general overview on the myriad dislocation dynamicsapproaches currently being employed, we refer the reader to the review by Bulatov (2002).While each of these approaches has its individual strengths and limitations, an importantdrawback is the reliance on continuum elasticity theory to model material response.Atomistic information is thus lost in this method of coarse-graining. In contrast, thediscrete method presented here allows for calculations at the mesoscale with fully atomic-level resolution. At the same time, the discrete Fourier transform formalism allows forfaster and larger computations at much longer time-scales than would be possible withconventional atomistic methods. Complex topological transitions are also easily handledwithin the method as there is no explicit tracking involved. In the following, we proceed toformulate the numerical method and highlight its advantageous features.

5.1. Equilibrium dislocation structures

We have focused thus far only on the energetics of dislocations in crystals. Dislocationsin an ensemble experience a variety of inelastic interactions with each other, with thelattice, with impurity atoms, second-phase particles, forest-dislocations and so on. Theseprocesses are dissipative and irreversible in general and hence need to be accounted forthrough kinetic laws. Following the work of Ortiz and Stainier (1999) and Ortiz andRepetto (1999), these irreversible interactions may be built into the variational framework


through recourse to time discretization via an incremental potential

W ½nnþ1; nn� ¼ F ½nnþ1� � F ½nn� þ Dtfnnþ1 � nn

Dt

� �, (50)

where nn and nnþ1 represent the slip distributions at times tn and tn þ Dt, respectively. Tacitin this formulation is the assumption that the dissipative forces derive from the kineticpotential f. The updated slip distribution may be obtained from the minimum principle

infnnþ12Z

MW ½nnþ1; nn�, (51)

which we note is an integer optimization problem and is known to be NP complete (see,e.g. Nemhauser and Wolsey (1988) for an overview of the subject). The difficulty in solvinglarge integer optimization problems4 is considerable and it is highly unlikely thattraditional integer optimization methods that work well in lower dimensions will provefeasible in our case. Furthermore, while ‘‘last-resort’’ optimization methods such assimulated annealing can certainly be used, the computational effort required is expected tobe beyond most available resources. It is thus clear that effecting the integer minimizationin anything but a small cluster of atoms is, for all practical purposes, an intractable taskand we are required to seek other alternatives to simulate reasonable system sizes.One approach that proves more tractable is to seek metastable equilibrium configura-

tions of the crystal. Note first, that perturbing nnþ1! nnþ1 þ dnnþ1 in Eq. (50) andcomputing the first variation of W gives us the driving force that is work-conjugate to theslip distribution as

Pnþ1 ¼ Y � nnþ1 � snþ1 þqf

q_n. (52)

For the crystal to be in a state of metastable equilibrium, this driving force computed ateach 1-cell must not cause any further dislocation nucleation. Put differently, if we define acritical shear, tcðe1; rÞ, required to nucleate an elementary dislocation loop on slip-system r

it follows that the inequalities

�tcðe1; rÞpPnþ1ðe1; rÞptcðe1; rÞ (53)

must be satisfied on every slip-system at every 1-cell in the crystal. We shall refer to Eq. (53)as the ‘‘metastability condition’’ henceforth. It is straightforward to show using Eq. (50)that the critical shears are given by tcðe1; rÞ ¼ 1

2hHde1;r; de1;ri where de1;r is a delta-function

slip field defined as

de01;r0 ðe1; rÞ ¼

1 if e1 ¼ e01 and r ¼ r0;

0 otherwise:

�(54)

It is important to note that the critical shears are in no way prescribed a priori—they arecomputed from the discrete hardening-moduli as described above (and are thus materialspecific). Our remaining task now is to actually find metastable slip distributions inthe crystal.

4Recall that a crystal with N 1-cells and M slip-systems has M �N independent slip-field variables

xs : E1ðmsÞ ! Z, s ¼ 1; . . . ;M. Even for a small estimate of N106 lattice sites, it is clear that the dimensionality

of the space over which the integer optimization problem must be solved is extremely large.


5.1.1. Procedure for computing metastable equilibria

At first glance, it would appear that in seeking to satisfy the metastability conditions, wehave merely replaced the original integer optimization problem with another largecombinatorial problem. One obvious advantage though is that we are no longer searchingfor unique solutions and we merely need a way of finding admissible solutions, i.e., thosethat satisfy the metastability conditions. The only additional requirement of such anadmissible solution is that it satisfies the LEDS hypothesis which postulates that theamong the LEDS that are in equilibrium with the applied tractions and are accessible tothe dislocation ensemble, the one actually formed most nearly minimizes the total energy(Kuhlmann-Wilsdorf, 1999). Our purpose is thus served if we begin an iterative search forinteger slip distributions in the vicinity of the absolute minimizer of the potential energy.We restrict attention for the present to the case where there are no dissipative effectspresent, i.e., the kinetic potential is set to zero which is tantamount to minimizing thepotential energy albeit over the field of integers. The absolute minimizer nuðlÞ 2 R

M isreadily computed via its Fourier transform as nuðhÞ ¼ Y

�1ðhÞsðhÞ. We now proceed from

the absolute minimum nuðlÞ 2 RM to states of metastable equilibrium nðlÞ 2 ZM via thefollowing algorithm:

(1)
The unconstrained solution nu is projected to the nearest integers to obtain n 2 ZM .While nu, by definition, satisfies the metastability conditions, there is no guarantee that n
does so and this must be ascertained bond-wise. We expect intuitively that this integerprojection would render a metastable solution for most of the slipped regions except at afew sites, most notably, at the intersections of dislocation lines and at dislocation cores.

(2)
The entire lattice is swept over to determine the out-of-equilibrium bonds. Ourexpectation that this set is located at dislocation junctions and cores is indeed borneout in practice (see e.g., Fig. 10 and the relevant discussion). We are thus faced with theproblem of equilibrating a small set of bonds which is but a tiny fraction of the totalnumber of bonds in the crystal. Of course, this fraction is expected to increase withincreasing dislocation activity in the crystal but this does not render the algorithminefficient in our experience.
(3)
From the set of out-of-equilibrium bonds, choose the bond with the maximummagnitude of overstress Pðe1; rÞ and perform a ‘‘spin-flip’’ xðe1; rÞ ! xðe1; rÞ þ de1;r ifPðe1; rÞ4tcðe1; rÞ or xðe1; rÞ ! xðe1; rÞ � de1;r if Pðe1; rÞo� tcðe1; rÞ. This procedure ineffect adds or removes an elementary dislocation loop at the bond under consideration.
(4)
Altering the value of xðe1; rÞ at a single bond alters the overstress Pðe01; r0Þ throughout
the lattice. Accordingly, recalculate Pðe01; r0Þ for the entire lattice. At first, this appears

to be a large computation involving the use of the fast Fourier transform (FFT).However, note that this may be reduced to a relatively simple process by computingand tabulating once and for all, at the very outset, a set of lattice Greens functionsGðe1; r; sÞ for a unit slip field. Thus, the effect of altering xðe1; rÞ at lattice site l by de1;r

leads to a new resolved shear-stress distribution at site l0,

Pðl0; a; sÞnew ¼ Pðl0; a; sÞold þ Gðl0 � l; a; s; rÞde1;r. (55)

(5)
Return to step (2) and iterate until convergence. The final iterate is that nðlÞ 2 ZM
which satisfies the metastability conditions everywhere. We find in practice that it isindeed possible to equilibrate all bonds in the crystal by this method.


Our algorithm for finding a metastable slip distribution in a crystal essentially reduces tofew equilibration calculations at a small fraction of lattice sites. This is a remarkablereduction in the overall dimensionality of the problem. We also note that the algorithm isembarrassingly parallel and allows for efficient simulations of large crystals with fullyatomistic resolution. Finally, we also note that the same algorithm can be appliedincrementally, with minor modifications, when irreversible mechanisms require to beaccounted for. The incremental procedure combined with the metastable equilibriumsearch is expected to naturally allow for path-dependence and hysteresis.

5.2. Application to a point of dilatation

The foregoing solution procedure is now applied to the case of a point of dilatation(Fig. 9), which serves as a first approximation to a void subjected to internal pressure.Assuming, for convenience, that the applied loading is collinear with the body-diagonals,the distribution of forces may be represented as

f ¼ P½sAfe0ðl þ �1Þ � e0ðl � �1Þg þ sDfe0ðl þ �2Þ � e0ðl � �2Þg

þ sCfe0ðl þ �3Þ � e0ðl � �3Þg þ sBfe0ðl þ �4Þ � e0ðl � �4Þg�, ð56Þ

where sA ¼ ½1; 1; 1�=ffiffiffi3p

, sB ¼ ½1; 1; 1�=ffiffiffi3p

, sC ¼ ½1; 1; 1�=ffiffiffi3p

, sD ¼ ½1; 1; 1�=ffiffiffi3p

. Wefurther assume periodic boundary conditions on the crystal to allow for the use of thediscrete Fourier transform. The unit cell is now a set Y Zn such that the translatesfY þ LiAi;L 2 Zng define a partition of Z3 for some translation vectors Ai 2 Zn. The voidis taken to be at the center of the crystal, l ¼ 0; it follows that the Fourier transform of theforce distribution is

f ðhÞ ¼Xl2Y

f ðlÞe�iy�l ¼ �2iP½sA sinðy1Þ þ sD sinðy2Þ þ sC sinðy3Þ þ sB sinðy4Þ�, (57)

where yi ¼ h � �i and the wave-vector h is now restricted to the Brillouin zone Z defined asthe intersection of the lattice spanned by the reciprocal basis and ½�p; p�n. Following the

P

l

l-ε7

l+ε1

l-ε1

l-ε2

l-ε3

l+ε4

l-ε5

l-ε6

l+ε7

l+ε2

l+ε3

l+ε6

l+ε5

xy

z

P

P

PP

P

P

P

Fig. 9. Point of dilatation in a BCC lattice. The forces are taken to be collinear with the body-diagonals here.


discussion in Section 3, the potential energy of the crystal given by Eq. (40) can beconveniently computed via the Fourier transform5 as

F ½n� ¼ 12hðW� W

yQ1U

�1Qy1WÞb; b

�i � hðU

�1Qy1WÞ

y f ; b�i. (58)

The eigendeformation b is in turn related to the slip n through

b ¼XN

s¼1

½dxðe1Þ �ms�

dxsðe1Þb

se1, (59)

d being the distance between consecutive slip-planes, which finally allows us to determinethe potential energy as

F ½n� ¼ 12hYn; n

�i � hs; n

�i, (60)

where

Y ¼ TTðW� WyQ1U

�1Qy1WÞT, (61a)

s ¼ TTðU�1

Qy1WÞy f , (61b)

T ¼½dxðe1Þ �m

s�

dbs. (61c)

It follows immediately that the slip distribution n ¼ Y�1

s is a minimizer of the potentialenergy, provided the force distribution has zero mean, i.e., f ð0Þ ¼ 0, and is unique up torigid translations (Ariza and Ortiz, 2005). It is also evident that such a nðhÞ 2 CM will yieldan absolute minimizer nuðlÞ 2 RM , and will not provide integer minimizers in general.

The equilibration procedure is demonstrated on a sample cell of 1 million vanadiumatoms. The magnitude of the point forces applied at the corners of the cube is taken to beP ¼ 3 eV=A. The slip distribution obtained by projecting the unconstrained minimizer tothe nearest integers is indicated in Fig. 10(a) by the red atoms. Note that the slipdistribution is supported in reality on the 1-cells or the bonds of the lattice. However,illustrating all the slipped 1-cells or, for that matter, the 2-cells on which non-zerodislocation densities a are supported is excessively detailed and hinders clarity invisualization. We hence show only those 0-cells or slipped atoms to which the 1-cells thatsupport non-zero xðei; rÞ are connected. The indexing strategy of the elementary sets allowsfor each 0-cell to be uniquely associated with seven 1-cells. We also recall that the actualdislocation consists of elementary loops that are supported on 2-cells incident on theslipped 1-cells. The green atoms indicate 0-cells that are connected to out-of-equilibrium1-cells and may be noted to lie predominantly along dislocation intersections and cores asconjectured previously. Note that the set of green atoms consists of both red atoms, whichare already slipped, and unslipped atoms where slip is incipient. The final result of thisequilibration process is the slip distribution shown in Fig. 10(b). The information gleanedfrom the slip distribution can also be processed to visualize the actual slip distributions onvarious slip-planes. As an example, we illustrate in Fig. 11 slices of the crystal along theð1 1 0Þ and ð1 1 0Þ directions showing iso-contours of the A6 and B5 slip-systems, which canbe identified by a minor abuse of the discrete terminology as dislocation lines. A single

5For notational ease, the argument h will be dropped now onward, being evident from the context.

ARTICLE IN PRESS

Fig. 10. Simulation cell containing 1 million vanadium atoms. (a) The red spheres indicate slipped atoms as

computed from the integer projection of the absolute minimizer. The green spheres indicate atoms where the

metastability conditions are not satisfied. (b) Slipped atoms after equilibration.


ARTICLE IN PRESS

Fig. 11. Iso-contours of slip on (1 1 0) and ð1 1 0Þ planes corresponding to systems A6 and B5, respectively. The

planes labeled ‘‘M.P.’’ (mid-plane) divide the crystal in half. The scale along the z-axis is highly exaggerated—the

planes are in reality consecutive. Slip is mostly confined to only a couple of planes ahead of and behind the mid-

planes of the crystal owing to the geometry of dilatation source. The (1 1 0) planes show more spread out

dislocation loops unlike the loops on the ð1 1 0Þ planes which are much more confined and highly elongated.


plane from each set is also shown in Fig. 12 for a clearer view of the individualdislocations. Finally, we also illustrate the slip distribution obtained for increasing loads inFig. 13. As expected the size of the dislocation ensemble increases with increasing load and,additionally, more slip-systems are activated to accommodate higher loads. While the slipdistributions obtained in this example are relatively uncomplicated, it is worth remarking

ARTICLE IN PRESS

0 20 40 60

-60

0

20

40

60

3210

-1-2-3

ξA6

0 50

0

20

40

60

210

-1-2-3-4-5

ξB5

-40

-20

-40

-60

-20

-60 -40 -20

-50

(a)

(b)

Fig. 12. Enlarged view of the individual planes labeled ‘‘M.P.’’ in Fig. 11 showing a more detailed view.


that the method naturally allows for far more complex distributions and topologicaltransitions without additional effort. More realistic models that account for the energeticsof dislocation junctions can also be readily incorporated into the method.

ARTICLE IN PRESS

Fig. 13. Slip distributions obtained for loads (a) P ¼ 2:5 eV=A, (b) P ¼ 3 eV=A and (c) P ¼ 3:5 eV=A. The

relatively smaller change in the slip distribution going from (b) to (c) as opposed to (a) to (b) is due to confinement

effects from the image cells.


In concluding this section, we make a few comments on the actual implementation. Thecomputations were performed on COMPAQ Alpha-server 617MHz nodes running MPI.The FFTs are performed using a parallel implementation of FFTW (Frigo and Johnson,2005). The computational effort mostly involves the calculation of the absolute minimizerand the equilibration procedure—the former decouples mode-wise in Fourier space whilethe latter is essentially a local procedure. The computation thus scales well with the numberof processors. The simulation cells illustrated in this article contain 1 million atoms—this ismostly for ease of visualization and interpretation of results. Our methodology can,however, handle significantly larger cells without any difficulty and we have been able tosimulate cell sizes of approximately 60 million atoms typically using 70–80 processors.6

6The interested reader is referred to http://aero.caltech.edu/ashwin for 3D visualizations and animations from

these simulations.

http://aero.caltech.edu/ashwin

http://aero.caltech.edu/ashwin


6. Summary and conclusions

We have presented a discrete modeling approach for dislocation based plasticity in BCCsingle crystals. The method uses elementary notions from algebraic topology incombination with the eigendeformation theory to construct a mechanics of slippedharmonic lattices. Linearized interatomic potentials are incorporated in a straightforwardmanner within the theory thus allowing for realistic dislocations with atomistic cores. Thepredicted core-structure and core-energies are found to compare favorably with resultsfrom more sophisticated computational approaches. The harmonic approximation, at thesame time, allows for the use of the FFT and thereby enables consideration of largecomputational cells. In this work, we have presented large-scale examples of thecomputation of nearly energy-minimizing slip-distributions emanating from a pointsource of dilatation. The technique relies on formulating suitable metastability conditionsand searching for integer-valued metastable equilibria in the vicinity of a non-integerglobal minimum. The overall dimensionality of the problem undergoes a considerablereduction with equilibration calculations being limited to dislocation cores and junctions.It is also worth noting that topological transitions emerge as a byproduct with noadditional effort. All in all, the method combines the realism of discrete lattice models withthe convenience of continuum models and presents a useful tool for seamless multiscalemodeling of dislocation plasticity in crystalline solids.There are several avenues for refinement in the present theory. The harmonic treatment

leads to a quadratic g-surface for the crystal which is an immense simplification. Efforts arepresently underway to relax this constraint and incorporate more general g-surfaces whichwould allow for partial dislocations, stacking faults, etc. An immediate application wouldbe to FCC metals where core-spreading is known to be of importance. The treatment ofLEDS is limited to static (and reversible) cases for the present. It is possible to incorporateirreversibility arising, for example, from Peierls stresses or phonon-drag as pointed out inSection 5.1. Such extensions are currently under investigation. Finally, our methodpresents a very feasible approach for computations of strain hardening and hysteresis inrealistic single crystals with fully atomistic resolution.

Acknowledgment

We gratefully acknowledge the support of the Department of Energy through Caltech’sASC Center for the Simulation of the Dynamic Response of Materials.

Appendix A. The discrete Fourier transform

A.1. Definition and fundamental properties

The discrete Fourier transform of f :Zn ! R is

f ðhÞ ¼Xl2Zn

f ðlÞe�ih�l . (A.1)


In addition we have

f ðlÞ ¼1

ð2pÞn

Z½�p;p�n

f ðhÞe�ih�l dny (A.2)

which is the inversion formula for the discrete Fourier transform. It follows from thisexpression that f ð�lÞ is the Fourier-series coefficient of f ðhÞ. We have the identitydf � g ¼ f g (A.3)

which is often referred to as the convolution theorem. Suppose in addition that f ; g aresquare-summable. ThenX

l2Zn

f ðlÞg�ðlÞ ¼1

ð2pÞn

Z½�p;p�n

f ðhÞg�ðhÞdny (A.4)

which is the Parserval identity for the discrete Fourier transform.

A.2. Periodic functions

The extension of the Fourier transform formalism to periodic functions is of particularinterest in applications. Consider a set Y Zn, the unit cell, such that the translatesfY þ LiAi;L 2 Zng, for some translation vectors Ai 2 Rn, i ¼ 1; . . . ; n, define a partition ofZn. Let Ai be the corresponding dual basis, Bi ¼ 2pAi the reciprocal basis. A latticefunction f : Zn ! R is Y -periodic if f ðlÞ ¼ f ðl þ LiAiÞ, for all L 2 Zn. Proceeding formally,the discrete Fourier transform of a Y -periodic lattice function can be written in the form

f ðtÞ ¼XL2Zn

Xl2Y

f ðlÞe�it�ðlþLjAj Þ ¼1

jY j

Xl2Y

f ðlÞeit�l

( )jY j

XL2Zn

eit�ðLjAj Þ

( ), ðA:5Þ

where jY j is the number of points in Y . But

jY jXL2Zn

e�it�ðLjAj Þ ¼ ð2pÞn

XH2Zn

dðt �HiBiÞ (A.6)

and, hence,

f ðtÞ ¼ð2pÞn

jY j

Xy2Z

f ðhÞdðt � hÞ, (A.7)

where

f ðhÞ ¼Xl2Y

f ðlÞe�ih�l (A.8)

and Z is the intersection of the lattice spanned by Bi and ½�p; p�n. In addition, the inverseFourier transform specializes to

f ðlÞ ¼1

jY j

Xy2Z

f ðhÞeiðhÞ�l . (A.9)

For periodic functions, Parseval’s identity takes the formXl2Y

f ðlÞg�ðlÞ ¼1

jY j

Xy2Z

f ðhÞg�ðhÞ. (A.10)


Likewise, let f and g be complex-valued lattice functions, the latter periodic. Insertingrepresentation (A.7) into the convolution theorem gives

ðdf � gÞðtÞ ¼ f ðtÞð2pÞn

jY j

Xy2z

gðhÞdðt � hÞ

!¼ð2pÞn

jY j

Xy2z

f ðhÞgðhÞdðt � hÞ, (A.11)

whence it follows that

ðdf � gÞðhÞ ¼ f ðhÞgðhÞ. (A.12)

Finally, the average of a periodic function follows as

hf i ¼1

jY j

Xl2Y

f ðlÞ ¼1

jY jf 0. (A.13)

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