Discrete Crystal Elasticity and Discrete Dislocations in Crystals

Arch. Rational Mech. Anal. 178 (2005) 149–226

Digital Object Identifier (DOI) 10.1007/s00205-005-0391-4

**Discrete** **Crystal** **Elasticity** **and** **Discrete**

**Dislocations** **in** **Crystal**s

M. P. Ariza & M. Ortiz

Communicated by the Editors

Abstract

This article is concerned with the development of a discrete theory of crystal

elasticity **and** dislocations **in** crystals. The theory is founded upon suitable adaptations

to crystal lattices of elements of algebraic topology **and** differential calculus

such as cha**in** complexes **and** homology groups, differential forms **and** operators,

**and** a theory of **in**tegration of forms. In particular, we def**in**e the lattice complex

of a number of commonly encountered lattices, **in**clud**in**g body-centered cubic **and**

face-centered cubic lattices. We show that material frame **in**difference naturally

leads to discrete notions of stress **and** stra**in** **in** lattices. Lattice defects such as dislocations

are **in**troduced by means of locally lattice-**in**variant (but globally **in**compatible)

eigendeformations. The geometrical framework affords discrete analogs of

fundamental objects **and** relations of the theory of l**in**ear elastic dislocations, such

as the dislocation density tensor, the equation of conservation of Burgers vector,

Kröner’s relation **and** Mura’s formula for the stored energy. We additionally supply

conditions for the existence of equilibrium displacement fields; we show that

l**in**ear elasticity is recovered as the Ɣ-limit of harmonic lattice statics as the lattice

parameter becomes vanish**in**gly small; we compute the Ɣ-limit of dilute dislocation

distributions of dislocations; **and** we show that the theory of cont**in**uously distributed

l**in**ear elastic dislocations is recovered as the Ɣ-limit of the stored energy as

the lattice parameter **and** Burgers vectors become vanish**in**gly small.

Contents

1. Introduction .....................................150

2. **Crystal** lattices ....................................155

2.1. Elementary properties of crystal lattices ...................156

2.2. The cha**in** complex of a crystal lattice ....................158

2.3. Examples of lattice complexes ........................165

150 M. P. Ariza & M. Ortiz

3. Calculus on lattices .................................178

3.1. The differential complex of a lattice .....................178

3.2. Integration on lattices .............................182

3.3. The Gauss l**in**k**in**g number ..........................183

3.4. Examples of l**in**k**in**g relations .........................184

4. Harmonic lattices ..................................185

4.1. The mechanics of lattices ...........................185

4.2. Equilibrium problem .............................191

4.3. Cont**in**uum limit ................................194

4.4. Simple illustrative examples .........................197

5. Eigendeformation theory of crystallographic slip ..................199

5.1. The energy of a plastically deformed crystal .................199

5.2. Examples of dislocation systems .......................201

5.3. The stored energy of a fixed distribution of dislocations ...........202

5.4. Simple illustrative examples of stored energies ................207

5.5. Cont**in**uum limit of the stored energy .....................209

Appendix A. The discrete Fourier transform ......................219

Appendix A.1. Def**in**ition **and** fundamental properties ...............219

Appendix A.2. Wave-number representation ....................220

Appendix A.3. Periodic functions ..........................221

Appendix A.4. Sampl**in**g **and** **in**terpolation .....................222

1. Introduction

The work presented **in** this article is concerned with the development of a discrete

theory of crystal elasticity **and** dislocations **in** crystals. The classical treatment

of harmonic crystal lattices **and** their defects, e.g., with**in** the context of lattice statics

(cf., e.g., [4, 45]) **and** eigendeformations (cf., e.g., [36]), becomes unwieldy when

applied to general boundary value problems **and** conceals the essential mathematical

structure of the theory, often impair**in**g **in**sight **and** h**in**der**in**g analysis. Instead,

here we endeavor to develop a theory founded upon suitable adaptations to crystal

lattices of elements of algebraic topology **and** differential calculus such as cha**in**

complexes **and** homology groups, differential forms **and** operators, **and** a theory

of **in**tegration of forms. These elements, **and** their connection to l**in**ear elasticity

**and** topological defects, are often taken for granted **in** R n but need to be carefully

def**in**ed **and** formalized for general crystal lattices. The result**in**g discrete geometrical

framework enables the formulation of a mechanics of lattices **in** a manner

that parallels closely – **and** extends **in** important respects – the classical cont**in**uum

theories.

While the theory of l**in**ear elastic dislocations **in** crystals is well known (e.g.,

[19, 3, 36]), it may st**and** a brief review, especially as a po**in**t of reference for the

discrete theory. The classical geometrical theory of l**in**ear elastic dislocations (e. g.,

[47, 23, 30, 48, 36, 20–22]) starts by posit**in**g the existence of an elastic distortion

field β e with the def**in****in**g property that

∫

E(β e 1

) =

R 3 2 c ij klβij e βe kldx (1)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 151

Fig. 1. Schematic representation of a perfect Volterra dislocation **in** a l**in**ear elastic solid.

The plane S of unit normal m is the slip plane; D is the dislocation loop. The displacement

field jumps discont**in**uously across the slip area (shown as the shaded region of the slip plane

bounded by D) by one Burgers vector b. The presence of the dislocation may be detected

by l**in**k**in**g its elastic deformation field aga**in**st a Burgers circuit C.

is the stra**in** energy of the crystal. Here, c ij kl = c jikl = c ij lk = c klij denote the

elastic moduli of the crystal. A Volterra dislocation of Burgers vector b supported

on a closed curve D is a field β e such that

∫

βij e dx j =−L**in**k(C, D)b i (2)

for all closed curves C (cf., Fig. 1). Here

C

L**in**k(C 1 ,C 2 ) = 1

8π

∫C 1

∫

C 2

(dx 1 × dx 2 ) · (x 1 − x 2 )

|x 1 − x 2 | 3 (3)

is the Gauss l**in**k**in**g number of two loops C 1 **and** C 2 **in** R 3 (e.g., [21]). The test

circuits C used **in** (2) to detect the presence of a dislocation are known as Burgers

circuits. Consider now an element of oriented area A with boundary C. Then, the

total Burgers vector of all dislocations cross**in**g A is

∫

b i (A) =− βij e dx j . (4)

By Stoke’s theorem we have

∫

b i (A) =

where ndS is the element of oriented area **and**

A

C

α il n l dS, (5)

α il =−β e ij,k e kj l (6)

is Nye’s dislocation density tensor [38]. The relation (6) between the elastic distortion

field **and** the dislocation density field was presented by Kröner [26]. It follows

immediately from this relation that

α il,l = 0, (7)

i.e., the divergence of the dislocation density field is zero. This means, **in** particular,

that the l**in**k between the dislocation l**in**es **and** any closed surface, i.e., the number

152 M. P. Ariza & M. Ortiz

of dislocation signed cross**in**gs through the surface, is zero. This property of dislocation

l**in**es is sometimes referred to as the conservation of Burgers vector property.

Evidently, α measures the failure of the elastic distortion β e to be a gradient **and**

completely describes the distribution of dislocations **in** the crystal. In particular, if

α = 0 it follows that

β e ij = u i,j , (8)

i.e., β e is the gradient of a displacement field u, **and** (1) reduces to the stra**in** energy

of a l**in**ear elastic solid.

At room temperature, dislocations **in** metals are predom**in**antly the result of

processes of crystallographic slip that build upon energetically **and** k**in**etically

favorable lattice-preserv**in**g shears. By its crystallographic nature, slip results **in**

dislocations that lie on well-characterized crystal planes **and** has Burgers vectors

co**in**cident with well-characterized crystal directions. Each possible slip plane **and**

Burgers vector pair def**in**es a slip system. The set of slip systems of a crystal is

closely related to its crystal structure **and** often is known experimentally [19]. In a

cont**in**uum sett**in**g, crystallographic slip may be described by identify**in**g the elastic

distortion β e with the Lebesgue-regular part of the distributional gradient of the

displacement field u (as **in** the Lebesgue decomposition theorem), **and** requir**in**g

the rema**in****in**g s**in**gular part to be supported on a collection S of crystallographic

slip planes. The extent of crystallographic slip on a plane **in** S is described by the

displacement jump [[u]] across the plane. This displacement jump is required to

be lattice preserv**in**g **and**, hence, an **in**teger l**in**ear comb**in**ation of Burgers vectors

conta**in**ed with**in** the slip plane. Thus, the unknown fields of the theory are the

elastic distortion β e , which equals the displacement gradient ∇u **in** R 3 \S, **and**

the displacement jump [[u]] on S. The elastic distortion β e follows directly from

the m**in**imization of the stra**in** energy (1) over R 3 \S with respect to the displacement

field u, subject to the prescribed [[u]] across S. This problem belongs to a classical

class of problems **in** l**in**ear elasticity know as ‘cut surfaces’ [36]. By contrast,

the evolution of the displacement jump [[u]] on S is governed by k**in**etics, **in**clud**in**g

lattice friction, cross slip, **and** short-range dislocation-dislocation **in**teractions

result**in**g **in** jogs, junctions **and** other reaction products.

There are several essential shortcom**in**gs of the theory of l**in**ear-elastic dislocations

just outl**in**ed which are a direct consequence of the cont**in**uum character of the

theory. Firstly, l**in**ear-elastic Volterra dislocations as described above have **in**f**in**ite

stra**in** energy. The conventional ‘fix’for this unphysical situation is to exclude **in** the

computation of the stra**in** energy a small tube of material, or core, around the dislocation

l**in**e of radius a 0 , the ‘cut-off radius’ (cf., e.g., [19, 3]). The result**in**g stra**in**

energies then diverge logarithmically **in** a 0 . The **in**troduction of a core is **in**tended

to account for the discreteness of the crystal lattice **and** its relaxation **in** the vic**in**ity

of the dislocation l**in**e. The cut-off radius is an ad-hoc parameter extraneous to

l**in**ear elasticity which must be determ**in**ed by fitt**in**g to experiment. In addition, the

core cut-off radius is a poor representation of the structure of dislocation cores **in**

general.

A superior regularization of the theory of l**in**ear-elastic dislocations may be

accomplished by relax**in**g the requirement that the displacement jump [[u]] across

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 153

the s**in**gular set S be an **in**teger l**in**ear comb**in**ation of **in**-plane Burgers vectors,

**and** **in**stead **in**troduc**in**g a Peierls potential φ [40] such that the total energy of the

crystal is

∫

E(u) =

R 3 \S

∫

1

2 c ij klu i,j u k,l dx + φ([[u]])dS. (9)

S

The Peierls potential on a slip plane is periodic **in** shear **and** exhibits the translation

**in**variance of the lattice (cf., e.g., [39]). The displacement field then follows by the

m**in**imization of E(u) under the action of applied loads **and** subject to k**in**etic or

topological constra**in**ts such as lattice friction, dislocation-dislocation **and** dislocation-obstacle

**in**teractions [24, 14, 25]. While the Peierls framework represents a

vast improvement over the cut-off radius regularization, it falls short **in** a number

of important aspects. Thus, s**in**ce the crystal is regarded as a cont**in**uum, there is no

natural notion of spac**in**g between slip planes. Of course, the potential slip planes

may be placed at the appropriate crystallographic **in**terplanar distance, but then the

assumption of **in**terplanar l**in**ear elasticity becomes questionable.

A natural approach to the mechanics of crystal defects which is free of the

pathologies just described is to acknowledge the discrete nature of crystal lattices

from the outset. However, this requires foundational developments **in** two ma**in**

areas: the formulation of a discrete differential calculus on crystal lattices; **and** the

application of the said discrete differential calculus to the formulation of a discrete

mechanics of lattices, **in**clud**in**g lattice defects. Thus, the geometrical theory of

crystal defects outl**in**ed above relies heavily on the homotopy of closed Burgers

circuits **in** R 3 as a device for characteriz**in**g l**in**e defects. Homotopy also underlies

fundamental rules govern**in**g dislocation reactions, such as Frank’s rule (e.g.,

[19]) **and** topological transitions such as dislocation junctions [48]. In addition, the

geometrical theory of crystal defects is tacitly founded on the homology of R 3 .It

is well known that the s**in**gular homology of R n is H p (R n ) = Z if p = 0, **and**

H p (R n ) = 0ifp ≧ 1 (e.g., [34]). The identity H 0 (R n ) = Z follows directly from

the path-connectedness of R n , i.e., the fact that every pair of po**in**ts **in** R n can be

jo**in**ed by a cont**in**uous path. The identities H p (R n ) = 0, p ≧ 1 imply **in** particular

that every closed loop **in** R n is the boundary of a surface **and** is contractible to a

po**in**t, **and** that every closed surface is the boundary of a volume **and** is likewise contractible

to a po**in**t. These properties of R 3 , **and** related topological **in**variants such

as the Gauss l**in**k**in**g number, underlie the def**in**ition (2) of a Volterra dislocation.

F**in**ally, the familiar differential operators of R 3 , such as div, grad **and** curl, **and** the

**in**tegration of forms of various orders, **in**clud**in**g Stoke’s theorem, arise frequently

**in** relations such as (1), (2), (5), (6), (7) **and** (8).

In lay**in**g the groundwork for a discrete theory, these elements of algebraic

topology **and** differential calculus need to be extended to crystal lattices. At the

most fundamental level, the notions of po**in**t, curve, surface **and** volume **and** their

boundary **and** coboundary connections need to be carefully redef**in**ed. These objects

supply the natural doma**in**s of def**in**ition of mechanical fields **and** forms, as well as

their natural doma**in**s of **in**tegration. **Crystal** lattices can be endowed with this requisite

structure by regard**in**g them as complexes (not necessarily simplicial). This

154 M. P. Ariza & M. Ortiz

notion of crystal lattice is greatly exp**and**ed with respect to traditional representations

of lattices [4, 44], **and** proves powerful as a foundation for a mechanics of

lattices. The construction of lattice complexes is not entirely trivial, as the complex

must be translation **in**variant, possess the symmetries of the crystal, **and** have the

homology of the imbedd**in**g space, e.g., R n . In addition, the lattice complex must

conta**in** all the operative slip systems of the crystal. In Section 2 we def**in**e the lattice

complex of a number of commonly encountered lattices, **in**clud**in**g body-centered

cubic **and** face-centered cubic lattices.

In Section 3, we **in**troduce the differential complex of a lattice, **in**clud**in**g fields

**and** forms of all orders, the differential **and** codifferential operators, the wedge

product, the Hodge-∗ operator, **in**tegrals of forms of all orders, a discrete Stoke’s

theorem, a discrete Helmholtz-Hodge decomposition, **and** a discrete l**in**k**in**g number.

These tools enable the formulation of a discrete mechanics of crystal lattices

**and** lattice defects that is formally **in**dist**in**guishable from the cont**in**uum theory.

In this approach, all specificity regard**in**g the cont**in**uum or discrete nature of the

system – **and** the structure of the crystal – is subsumed **in** the def**in**ition of the

lattice complex **and** differential structures, **and** the physical govern**in**g equations

are identical for all systems. In this sense, the present work may be regarded as part

of an ongo**in**g effort to develop mechanical **and** physical theories of discrete objects

us**in**g discrete differential calculus (cf., e.g., [18, 28] **and** the extensive literature

review there**in**).

In Section 4, we proceed to develop a mechanics of lattices **and** lattice defects.

A key step **in** that regard is the realization that the essential structure of the theory is

dictated by material-frame **in**difference, i.e., by the **in**variance of the energy of the

crystal lattice under translations **and** rotations. Indeed, we show that material-frame

**in**difference results **in** a natural discrete version of the Cauchy-Green deformation

tensor field, **and** upon l**in**earization, of the small-stra**in** tensor field. The correspond**in**g

def**in**itions of stress, traction **and** equilibrium, follow from duality **and**

stationarity. For harmonic lattices, the theory precisely identifies the form of the

stress-stra**in** relations. It bears emphasis that these relations have a special structure

due to material-frame **in**difference, **and** that the force-displacement relations of

the crystal lattice **in**herit that structure. Conversely, arbitrary force constants violate

material-frame **in**difference **in** general, with the unphysical consequence that lattice

rotations may **in**duce stra**in** energy. The restrictions placed on the force constants

of a crystal lattice by material-frame **in**difference do not appear to have been fully

appreciated **in** the lattice statics literature (for notable exceptions, cf., [43, 45]).

In all these developments, the discrete differential calculus framework proves

of great value. For **in**stance, with**in** that framework the various mechanical fields

f**in**d a clear def**in**ition as functions over the lattice complex: the displacement field

is a 0-form; the deformation field is a 1-form; the stra**in**-tensor, metric-tensor **and**

stress fields are scalar-valued functions over pairs of 1-cells; **and** so on. It is also

**in**terest**in**g to note the notion of a tensor field which emerges from the theory. Thus,

for **in**stance, the stra**in** **and** stress tensors, which provide examples of second-order

tensors, are real-valued functions of pairs of cells. Also, the notion of metric that

emerges naturally from the theory is noteworthy: the natural way to def**in**e the metric

properties of the lattice is to provide the **in**ner products of the differential of the

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 155

position vector for every pair of 1-cells. The availability of an efficient theory of

**in**tegration of forms, **in**clud**in**g a Stoke’s theorem, is also **in**valuable **in** deriv**in**g the

equations of equilibrium of the crystal **and** establish**in**g their structure.

In Section 5, we **in**troduce the notion of discrete dislocation by means of the

st**and**ard eigendeformation device (cf., [36, 39] for overviews of eigendeformations

**in** the context of dislocation mechanics). Uniform eigendeformations represent

lattice-**in**variant that cost no energy. Non-uniform eigendeformations are not

gradients of a displacement field **in** general **and**, therefore, **in**troduce dislocations

**in**to the lattice. The structure of the result**in**g dislocations is particularly reveal**in**g.

Thus, it follows that the set of all discrete dislocation loops can be identified with

the group B 2 of 2-coboundaries of the lattice; **and** the set of all Burgers circuits can

be identified with the group Z 1 of 1-cycles of the lattice. The Burgers-circuit test (2)

simply reflects the duality between these two groups. In particular, the coboundaries

of all 1-cells of the lattice, which constitute a generator of B 2 , may be regarded

as elementary dislocation loops. All rema**in****in**g dislocations loops are generated by

the elementary loops **in** the sense of cha**in**s. Likewise, the boundaries of all 2-cells

of the lattice constitute a generator of Z 1 **and** may be regarded as elementary – or

the smallest possible – Burgers circuits. The theory also provides discrete versions

of fundamental relations such as Kröner’s formula (6), the conservation of Burgers

vector relation (7), **and** others. The geometrical **and** physical **in**sight **in**to the structure

of discrete dislocations afforded by the geometrical foundations of the theory

can hardly be overemphasized.

While the primary focus of this paper is the formulation of a discrete mechanics

of lattices, we nevertheless delve briefly **in**to some questions of analysis. Thus, for

**in**stance, we give conditions for the existence of equilibrium displacement fields

(Propositions 2 **and** 3) **and** discuss the equilibrium problem for slip **and** dislocation

fields. We also show that l**in**ear elasticity is recovered as the Ɣ-limit of the harmonic

lattice statics problem, as the lattice parameter becomes vanish**in**gly small

(Proposition 4). We also **in**vestigate two cont**in**uum limits of the stored energy of

a dislocation ensemble. The first limit is atta**in**ed by the application of a scal**in**g

transformation which exp**and**s – **and** rarefies – the dislocation ensemble, while

leav**in**g the lattice parameter unchanged (Proposition 5). The second limit of the

stored energy is obta**in**ed by scal**in**g down the lattice size **and** the Burgers vector

sizes simultaneously, **and** leads to the classical theory of cont**in**uously distributed

dislocations (Proposition 6). In conclusion we note that the **in**vestigation of the

cont**in**uum limits just described fits with**in** current efforts to underst**and** cont**in**uum

models as the limits of discrete systems (cf., [7] **and** the references there**in**).

2. **Crystal** lattices

The development of a general theory of the mechanics of lattices **and** lattice

defects is greatly facilitated by a judicious use of elements of algebraic topology **and**

differential calculus. These elements greatly streaml**in**e the notation **and**, perhaps

more importantly, shed considerable **in**sight **in**to the general structure of the theory,

especially **in** the presence of lattice defects. We beg**in** by regard**in**g crystal lattices

156 M. P. Ariza & M. Ortiz

as cha**in** complexes, a view that greatly exp**and**s previous mathematical models of

lattices. In particular, the lattice complex encodes all **in**formation regard**in**g po**in**ts,

curves, surfaces, volumes of the lattice **and** their adjacency relations. These algebraic

elements provide the basis for the subsequent def**in**ition of a discrete lattice

calculus. For **in**stance, the boundary **and** coboundary operators **in**duce differential

**and** codifferential operators on forms **and** fields, **and** the cup product of cocha**in**s

**in**duces a wedge product of forms. Provided that it is appropriately def**in**ed, the

lattice complex also encodes the geometry of the slip systems **and** discrete dislocations

of the correspond**in**g crystal class. We develop the complexes of commonly

encountered lattices, such as the face-centered cubic lattice **and** the body-centered

cubic lattice, **and** provide **in** tabular form all the fundamental operations required

**in** applications.

2.1. Elementary properties of crystal lattices

2.1.1. Notation. We denote the group of **in**tegers by Z, the real numbers by R **and**

the complex numbers by C. In **in**dicial expressions **in**volv**in**g vectors **and** tensors

we adopt E**in**ste**in**’s summation convention unless explicitly stated.

The most elementary view of a crystal lattice is as a discrete subgroup [44]

L ={x(l) = l i a i ,l∈ Z n } (10)

of R n , where {a 1 ,...,a n } are l**in**early **in**dependent vectors def**in****in**g a lattice basis.

The dual basis {a 1 ,...,a n } is characterized by the property

a i · a j = δ i j , (11)

where the dot denotes scalar product **in** R n . Follow**in**g common notation, we shall

denote by the volume of the unit cell of the lattice, or atomic volume. The

dual **and** reciprocal lattices are the lattices spanned by the dual basis **and** by

{2πa 1 ,...,2πa n }, respectively. We recall that the first Brillou**in** zone B is the

Voronoi cell of the reciprocal lattice at the orig**in**. The volume of the dual unit cell

is 1/, whereas the volume |B| of the first Brillou**in** zone, **and** of the reciprocal

unit cell, is (2π) n /.

A complex lattice may be def**in**ed as a collection of **in**terpenetrat**in**g simple

lattices hav**in**g the same basis. Thus, the po**in**t set of a complex lattice is of the

form:

L ={x(l,α) = l i a i + r α ,l∈ Z n ,α= 1,...,N}, (12)

where α labels each constituent simple sublattice, **and** r α is the relative translation

vector of the α sublattice. Certa**in** crystal lattices **and** certa**in** sets of objects that

arise naturally **in** the mechanics of crystals, such as the set of atomic bonds, possess

the translational symmetry of a complex lattice. Even when a crystal lattice can be

described as a simple lattice, it may be convenient to describe it as a complex lattice

for **in**dex**in**g purposes. For **in**stance, a body-centered cubic lattice can variously be

described as a simple lattice or as a complex lattice consist**in**g of two superposed

simple-cubic lattices.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 157

We recall that a symmetry of the lattice is an orthogonal transformation Q ∈

O(n) that maps the lattice **in**to itself (cf., e.g., [12]). The set S of all symmetry

transformations is a subgroup of O(n) known as the symmetry group of the lattice.

In three dimensions there are 32 possible symmetry groups correspond**in**g to each

of the 32 crystallographic po**in**t groups [6]. If the **in**version transformation belongs

to S, then the lattice is said to be centrosymmetric. A simple lattice is necessarily

centrosymmetric. We also recall that two lattices L 1 **and** L 2 def**in**e the same Bravais

lattice if there exists a l**in**ear isomorphism φ : R n → R n such that L 2 = φ(L 1 )

**and** S(L 2 ) = φS(L 1 )φ −1 .Forn = 2 there are exactly 5 Bravais lattices, whereas

for n = 3 there are 14 Bravais lattices [44].

A crystallographic plane is conventionally def**in**ed by its Miller **in**dices m ∈ Z n ,

def**in**ed such that the po**in**ts {a 1 /m 1 ,...,a n /m n } are the **in**tercepts of the plane with

the coord**in**ate axes. A family of parallel crystallographic planes may be def**in**ed

implicitly by the equation

x · k ∈ Z (13)

for some dual vector k, where the symbol · signifies the duality pair**in**g **in** R n .Itis

readily shown that the components of the vector k **in** the dual basis a i co**in**cide with

the Miller **in**dices. The distance between consecutive parallel planes is important

**in** underst**and****in**g processes of crystallographic slip. A trite calculation gives the

distance between the two planes as

d −1 =|k|, (14)

where |·|denotes the st**and**ard norm **in** R n .

The choice of lattice basis is clearly not unique.Any n-tuple (a

1 ′ ,...,a′ n ) related

l**in**early to the orig**in**al basis as a

i ′ = µ j

i

a j also def**in**es a lattice basis of the same

orientation provided that µ ∈ SP(Z n ) [12], i.e., µ j

i

∈ Z **and** det(µ) =±1. A

far-reach**in**g consequence of this fact is that any aff**in**e mapp**in**g F on R n of the

form

F = µ j

i

a j ⊗ a i (15)

leaves the lattice **in**variant. A particularly important class of lattice-**in**variant deformations

of relevance to processes of crystallographic slip is

F = I + (s j a j ) ⊗ (m i a i ) (16)

with s, m ∈ Z n **and** s i m i = 0. These simple-shear deformations represent uniform

crystallographic slip through a Burgers vector b = s j a j on the plane of normal

m i a i . Energetic considerations, such as the exam**in**ation of core energies of l**in**ear-elastic

dislocations, as well as k**in**etic considerations, such as those based on

l**in**ear-elastic estimates of lattice friction, suggest that crystallographic slip occurs

preferentially on closed-packed planes, i.e., planes for which the **in**terplanar distance

d, equation (14), is maximized. A tabulation of commonly observed slip

planes may be found **in** the treatise of Hirth &Lothe [19].

158 M. P. Ariza & M. Ortiz

2.2. The cha**in** complex of a crystal lattice

2.2.1. Notation. Throughout this section we follow the notation **and** nomenclature

of [34], which may be consulted for further details **and** background. We denote the

**in**terior of a subset A of a topological space by IntA, its boundary by BdA, **and** its

closure by Ā. We also denote by B n the unit n-ball, i.e., the set of all po**in**ts x of R n

such that |x| ≦ 1, where |·|denotes the st**and**ard norm **in** R n . The unit sphere S n−1

is the set of po**in**ts for which |x| =1. If f : G → H is a group homomorphism,

we denote by kerf the kernel of f , i.e., the subgroup f −1 ({0}) of G, **and** by imf

the image of f , i.e., the subgroup f (G) of H . Given two abelian groups G **and** H

we denote by Hom(G, H ) the abelian group of all homomorphisms of G **in**to H .

In the preced**in**g elementary view, a crystal lattice is regarded merely as a po**in**t

set, each site be**in**g occupied by an atom. A more complete view of a crystal lattice

**in**cludes objects such as atomic bonds, or pairs of atoms, elementary areas **and** elementary

volumes. We shall refer to these higher-dimensionality objects as p-cells,

where p is the dimension of the cell. p-cells supply the natural support for def**in****in**g

functions aris**in**g **in** the mechanics of lattices, such as displacement **and** deformation

fields **and** dislocation densities. They also supply the natural framework for

def**in****in**g discrete differential operators **and** **in**tegrals. Naturally, we require the collection

of all p-cells to be translation **and** symmetry **in**variant. There are a number

of additional restrictions on p-cells that give mathematical expression to the **in**tuitive

notion of a perfect, or defect-free, crystal lattice. In this section, we proceed to

develop this extended view of crystals.

Recall that a space e is called a cell of dimension p if it is homeomorphic with

B p , i.e., if there exists a cont**in**uous bijective function f : B p → e such that its

**in**verse is also cont**in**uous. It is called an open cell if it is homeomorphic to Int B p .

We shall use the notation e p **and** the term p-cell when we wish to make explicit

the dimension of the cell. We recall that a regular CW complex is a space X **and**

a collection of disjo**in**t open cells whose union is X such that: (i) X is Hausdorff;

(ii) for each open cell e there exists a homeomorphism f : B p → X that maps

IntB p onto e **and** carries BdB p **in**to a f**in**ite union of open cells, each of dimension

less than p; (iii) a set A is closed **in** X if A ∩ē is closed **in** ē for each cell e.

X is the underly**in**g space of the complex. By a slight abuse of notation we shall

use the symbol X to refer both to the complex **and** to the underly**in**g space. In all

subsequent developments the underly**in**g space is always R n **and** hence condition

(i) is automatically satisfied. The cells conta**in**ed **in** another cell e are the faces of e.

The faces of e different from e itself are called the proper faces of e. The notation

e ′ ≼ e or e ≽ e ′ signifies that e ′ is a face of e, whereas the notation e ′ ≺ e or e ≻ e ′

signifies that e ′ is a proper face of e. The subspace X p of X that is the union of the

open cells of X of dimension at most p is a subcomplex of X called the p-skeleton

of X. In particular, X 0 is the set of 0-cells or vertex set. The dimension of a complex

is the largest dimension of a cell of X.

We def**in**e an n-dimensional lattice complex to be a CW complex such that:

(A1) The underly**in**g space is all of R n .

(A2) The vertex set def**in**es an n-dimensional lattice.

(A3) The cell set is translation **and** symmetry **in**variant.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 159

For n ≦ 3, we shall refer to the 1-cells of a lattice complex as elementary segments,

to the 2-cells as elementary areas, **and** to the 3-cells as elementary volumes. The

requirement that these elements form a CW complex gives mathematical expression

to the **in**tuitive notions that p-cells should be non-**in**tersect**in**g, **and** their boundaries

the union of cells of lower dimension. Implied **in** axiom A1 is the requirement that

the union of all cells **in** the complex be R n . Translation **and** symmetry **in**variance

require that if a cell is **in** X, so must be all translates of the cell by translation vectors

of the lattice, **and** all symmetry-related cells. The symmetry **and** CW complex

requirements just stated place fundamental restrictions on the choice of elementary

segment, elementary area **and** elementary volume sets.

There are additional restrictions on a lattice complex that are of a homological

nature **and** that give mathematical expression to the **in**tuitive notion of a perfect,

defect-free lattice. As noted **in** the **in**troduction, the geometrical theory of l**in**earelastic

po**in**t **and** l**in**e defects is (tacitly or explicitly) built around the homology of

R 3 , specifically on the fact that every closed loop **in** R 3 is the boundary of a surface

**and** is contractible to a po**in**t, **and** that every closed surface is the boundary of a

volume **and** is likewise contractible to a po**in**t. Non-trivial homology groups are

then associated with distributions of po**in**t **and** l**in**e defects **in** the cont**in**uum (e g.,

[47, 23, 30, 48, 36, 20–22]).

In order to carry these tools over to the discrete sett**in**g, we proceed to formulate

a homology of crystal lattices. We beg**in** by def**in****in**g the cha**in** complex of the

lattice. By convention, vertices have one orientation only. Elementary segments,

or 1-cells, are oriented by order**in**g their vertices. We shall use the symbol [v, v ′ ]

to denote the elementary segment def**in**ed by the vertices {v, v ′ } together with the

orientation determ**in**ed by the order**in**g (v, v ′ ). Two adjacent oriented elementary

segments of the form [v, v ′ ] **and** [v ′ ,v ′′ ] are said to be oriented consistently. An

elementary area, or 2-cell, is oriented by assign**in**g consistent orientations to each

of its elementary segments. An elementary volume, or 3-cell, has two possible

orientations, **in**ward **and** outward. Thus, a 2-cell is consistently oriented with an

outwardly (or **in**wardly) oriented 3-cell if the boundary of the 2-cell is oriented

clockwise (respectively, counter-clockwise) when seen from **in**side the 3-cell. We

shall denote by [v 0 ,...,v p ] the p-simplex def**in**ed by the vertices {v 0 ,...,v p }

together with the orientation determ**in**ed by the order**in**g (v 0 ,...,v p ) (cf., e.g.,

[34], § 39, for a general def**in**ition of the orientation of a cell).

We recall that a p-cha**in** **in** a CW complex X is a function c from the set of

oriented p-cells of X to the **in**tegers such that: (i) c(e) =−c(e ′ ) if e **and** e ′ are

opposite orientations of the same cell; **and** (ii) c(e) = 0 for all but f**in**itely-many

oriented p-cells e. Note that p-cha**in**s are added by add**in**g their values. We shall

use the notation c p when we wish to make explicit the dimension of the cha**in**. The

result**in**g group of oriented p-cha**in**s is denoted C p .Inann-dimensional complex,

C p is the trivial group if pn.Ife is an oriented cell, the elementary

cha**in** correspond**in**g to e is def**in**ed by: (i) c(e) = 1; (ii) c(e ′ ) =−c(e) if e ′ is the

opposite orientation of e; **and** (iii) c = 0 otherwise. By an abuse of notation we

shall use the same symbol to denote a cell **and** the correspond**in**g elementary cha**in**.

It is straightforward to show (cf., e.g., [34], Lemma 5.1) that C p is free-abelian, i.e.,

it is abelian **and** it has a basis. Indeed, once all cells are oriented, each p-cha**in** c p

160 M. P. Ariza & M. Ortiz

can be written uniquely as a f**in**ite l**in**ear comb**in**ation c p = ∑ n i e i of elementary

p-cha**in**s e i with **in**teger coefficients n i . Thus, the cha**in** c assigns the **in**teger value

n i to e i , −n i to −e i **and** 0 to all oriented p-cells not appear**in**g **in** the sum.

Any function f from the oriented p-cells of X to an abelian group G extends

uniquely to a homomorphism C p → G provided that f(−e p ) =−f(e p ) for all

oriented p-cells e p . The boundary operator is a homomorphism ∂ p : C p → C p−1

that can be def**in**ed **in** this manner by sett**in**g

∂ p e p =

∑

±e p−1 , (17)

e p−1 ≺e p

where the sign is chosen as + (or −) if the orientation of e p−1 co**in**cides with

(respectively, is the opposite of) that **in**duced by e p . For **in**stance, if e =[v, v ′ ]

is an oriented elementary segment, then ∂ 1 e = v ′ − v. It is readily verified that

∂ p−1 ◦∂ p = 0. Indeed, ∂ p−1 ◦∂ p e p = ∑ e p−1 ≺e p

∑e p−2 ≺e p−1

±e p−2 , **and** the terms

**in** these two sums cancel **in** pairs. Then, ker∂ p is called the group of p-cycles **and**

denoted Z p , while im∂ p+1 is called the group of p-boundaries **and** denoted B p .

Also H p = Z p /B p is the pth homology group of the lattice complex. Henceforth

we shall omit the dimensional subscript from ∂ p whenever it can be deduced from

the context.

We shall say that a lattice complex X is perfect, or defect-free, if its homology

is identical to the homology of the underly**in**g space. For **in**stance, a lattice **in** R n

is perfect if H p = Z for p = 0 for H p = 0, if p ≧ 1, i.e., if the homology of

the complex is identical to the s**in**gular homology of R n (cf., e.g., [34], § 29, for

a general discussion of s**in**gular homology). The identity H 0 = Z follows simply

by ensur**in**g that X is path connected (cf., e.g., [34], Theorem 7.2), i.e., that every

pair of po**in**ts **in** X is jo**in**ed by a cont**in**uous path of elementary segments. This

requires all lattice sites to be connected by elementary segments, without ‘hang**in**g’

sites. The identities H p = 0, p ≧ 1, imply **in** particular that every 1-cycle **in** X

is the boundary of a 2-cha**in**, **and** that every 2-cycle is the boundary of a 3-cha**in**.

In the applications of **in**terest here, it is important that a lattice complex possess

a reference configuration that is defect-free. In order to ensure this property we

append the follow**in**g axiom:

(A4) The lattice complex is perfect.

This requirement places fundamental restrictions on the choice of cell set. In applications,

it also provides a useful screen**in**g test of the admissibility of a lattice

complex, especially **in** three dimensions.

We note that, for a given lattice, the choice of lattice complex is not uniquely

determ**in**ed by axioms (A1)–(A4) **in** general. Consider for **in**stance an n-dimensional

simple lattice hav**in**g the trivial symmetry group {E,I}, where E **and** I are

the identity **and** **in**version transformations, respectively. Then, for any lattice basis

we can construct a complex **in** which the n-cells are the parallelepiped def**in**ed by

the basis vectors **and** its translates, **and** the rema**in****in**g cells are chosen so that the

complex is homeomorphic to the simple-cubic complex described **in** Section 2.3.4.

Clearly, each lattice basis determ**in**es a lattice complex, **and** the choice of complex

is not unique. In applications, the lattice complex must be carefully constructed

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 161

so that it conta**in**s all the crystallographic elements of **in**terest, **in**clud**in**g the full

complement of slip planes, Burgers vectors, dislocation loops, Burgers circuits,

**in**terstitial sites, **and** others. These additional requirements tend to reduce the latitude–or

elim**in**ate it completely–**in** the choice of lattice complex.

We additionally def**in**e the group C p of p-cocha**in**s as the group Hom(C p , Z) of

all homomorphisms of C p **in**to Z.Ifc p is a p-cocha**in** **and** c p is a p-cha**in**, we denote

by 〈c p ,c p 〉 the value of c p on c p . The coboundary operator δ p : C p → C p+1 is

then def**in**ed by the identity

〈δ p c p ,c p+1 〉=〈c p ,∂ p+1 c p+1 〉. (18)

From the identity ∂ p−1 ◦ ∂ p = 0, it follows immediately that δ p+1 ◦ δ p = 0. Let

e ∗ be the elementary cocha**in** whose value is 1 on a cell e **and** 0 on all other cells.

The mapp**in**g ( ∗ ) extends to an isomorphism from C p to the subgroup of C p of

cocha**in**s of f**in**ite support, whose **in**verse will also be denoted by ( ∗ ). We shall also

simply write e p = (e p ) ∗ **and** e p = (e p ) ∗ **and** leave the operator ∗ implied. With

this notation, an arbitrary cocha**in** can be expressed as the formal (possibly **in**f**in**ite)

sum c p = ∑ n i ei ∗, where n i ∈ Z, **and** we have the identity δ p c p = ∑ n i δ p ei ∗.

Thus, the coboundary operator can be def**in**ed by specify**in**g δ p e p on each oriented

p-cell e p , namely,

δ p e p =

∑

±e p+1 , (19)

e p+1 ≻e p

where the sign is chosen as + (or −) if the orientation of e p co**in**cides with (respectively,

is the opposite of) that **in**duced by e p+1 . Then, kerδ p is called the group

of cocycles **and** denoted Z p , while imδ p−1 is called the group of coboundaries

**and** denoted B p . Also H p = Z p /B p is the pth cohomology group of the lattice

complex. As before, we shall henceforth omit the dimensional subscript from δ p

whenever it can be deduced from the context.

As subsequent developments will demonstrate (cf. Sections 5.1 **and** 5.2), the

cycle group Z 1 of a crystal lattice is the group of all Burgers circuits. The generators

of Z 1 may be **in**terpreted as elementary Burgers circuits. The coboundary group

B 2 of a crystal lattice is the group of all dislocation loops. This makes precise the

sense **in** which Burgers circuits **and** dislocation loops are dual to each other. The

generators of B 2 may be **in**terpreted as a set of elementary dislocation loops.

In addition, as we shall see **in** Section 3, the boundary **and** coboundary operators

just def**in**ed **in**duce a discrete lattice analog of the de Rham differential complex **in**

R n . In particular, the cup product of cocha**in**s provides the basis for the **in**troduction

of a wedge product of discrete forms. Recall that **in** a simplicial complex X, the

cup product of c p ∈ C p **and** c q ∈ C q is def**in**ed by the identity

〈c p ∪ c q , [v 0 ,...,v p+q ]〉=〈c p , [v 0 ,...,v p ]〉 〈c q , [v p ,...,v p+q ]〉 (20)

if [v 0 ,...,v p+q ] is a simplex such that v 0 < ···

162 M. P. Ariza & M. Ortiz

(cf., e.g., [34]). It is clear from its def**in**ition that the cup product depends on the

choice of order**in**g of the vertices. For crystal lattices, the natural partial order**in**g

of the vertices is obta**in**ed by **in**troduc**in**g a translation-**in**variant orientation of the

1-cells such that {e 1 (l, 1), ..., e 1 (l, N 1 )} have e 0 (l) as their first vertex, **and** then

lett**in**g v 1

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 163

Hence, L p (e p−1 ,e p ) is also the value at e p of the cocha**in** obta**in**ed by apply**in**g the

coboundary operator to the elementary cocha**in** e p−1 .As **in** the case of the boundary

**and** coboundary operators, **in** the sequel we shall omit the dimensional label from

the matrices L p whenever it can be deduced from the context.

In the special case of lattice complexes, the functions L can be given more

explicit **in**dexed representations. Start by giv**in**g the cells of the lattice complex

translation-**in**variant orientations. Introduce an equivalence relation T **in** C p that

identifies two cha**in**s when one can be obta**in**ed from the other by a translation.

Denote by [c p ] the equivalence class of c p **and** by C p /T the quotient set of equivalence

classes. By translation **in**variance, the equivalence class [e p ] of a p-cell can

be **in**dexed as a Bravais lattice, **and** the lattices of all p-cell classes jo**in**tly def**in**e

a complex Bravais lattice. Thus, a p-cell **in** an n-dimensional lattice complex can

be **in**dexed as e p (l, α), where l ∈ Z n , α ∈{1,...,N p }, **and** N p is the number of

simple p-cell sublattices. Us**in**g this **in**dex**in**g, lattice cha**in**s **and** cocha**in**s have the

representation

c p = ∑ ∑

N p

c p (l, α)e p (l, α),

l∈Z n α=1

c p = ∑ ∑

N p

c p (l, α)e p (l, α),

l∈Z n α=1

(28a)

(28b)

where we write c p (l, α) ≡ c p

(

ep (l, α) ) **and** c p (l, α) ≡〈c p ,e p (l, α)〉, respectively.

An application of the boundary operator to (28a) yields

∂c p = ∑ ∑

N p

c p (l, α)∂e p (l, α). (29)

l∈Z n α=1

In addition, by translation **in**variance,

∑

β=1

∂e p (l, α) = ∑ N p−1

L αβ (l − l ′ )e p−1 (l ′ ,β), (30)

l ′ ∈Z n

where the function L αβ (·) takes **in**teger values **and** has f**in**ite support. Insert**in**g (30)

**in**to (29) yields

(∂c p )(l ′ ,β)= ∑ ∑

N p

L αβ (l − l ′ )c p (l, α). (31)

l∈Z n α=1

By duality, the coboundary operator follows likewise as

∑

β=1

(δc p−1 )(l, α) = ∑ N p−1

L αβ (l − l ′ )c p−1 (l ′ ,β). (32)

l ′ ∈Z n

It therefore follows that the matrix representation of the boundary **and** coboundary

operators is completely determ**in**ed by the coefficients L αβ (l) **in** (30).

164 M. P. Ariza & M. Ortiz

The discrete Fourier transform provides an additional means of exploit**in**g the

translation **in**variance of lattice complexes. Thus, the discrete Fourier transform of

the p-cha**in** c p is (cf., Appendix A)

ĉ p (θ, α) = ∑

l∈Z n c p (l, α)e −iθ·l , (33)

**and** its **in**verse is

c p (l, α) = 1 ∫

(2π) n ĉ p (θ, α)e iθ·l dθ. (34)

[−π,π] n

S**in**ce cha**in**s take nonzero values over f**in**itely many cells, the sum (33) also conta**in**s

f**in**itely many terms **and** ĉ p (·,α) ∈ C ∞ ([−π, π] n ). As noted **in** Appendix A,

c p (−l,α) is the Fourier-series coefficient of ĉ p (θ, α), **and** hence the discrete Fourier

transform def**in**es an isomorphism φ p between C p **and** the group Ĉ p of functions

from [−π, π] n to C N p

that have f**in**itely-many nonzero **in**teger Fourier-series

coefficients. Let Ĉ p = Hom(Ĉ p , Z). We def**in**e the discrete Fourier transform of

co-cha**in**s φ p : C p → Ĉ p as the dual isomorphism of φp

−1 , i.e., by the relation

φ p (c p ) = c p ◦ φ −1 , or, equivalently, by the identity

p

Parseval’s identity then yields the representation of φ p :

ĉ p (θ, α) = ∑

**and** the representation of its **in**verse:

c p (l, α) = 1 ∫

(2π) n

〈c p ,c p 〉=〈ĉ p , ĉ p 〉, (35)

l∈Z n c p (l, α)e −iθ·l , (36)

[−π,π] n ĉ p (θ, α)e iθ·l dθ. (37)

Thus, c p (−l,α) is the Fourier-series coefficient of ĉ p (θ, α), **and** hence Ĉ p can

be identified with the group of functions from [−π, π] n to C N p

that have (possibly

**in**f**in**itely-many nonzero) **in**teger Fourier-series coefficients. We def**in**e the

boundary operator ˆ∂ p : Ĉ p → Ĉ p−1 so that φ p def**in**es a cha**in** map between the

cha**in** complexes C ={C p ,∂ p } **and** C ˆ ={Ĉ p , ˆ∂ p }, i.e., by enforc**in**g the identity:

ˆ∂ p ◦ φ p = φ p−1 ◦ ∂ p for all p ≧ 1 (cf., e.g., [34], § 13, for a general for the

def**in**ition of a cha**in** map). The group Hˆ

p = ker ˆ∂ p /im ˆ∂ p+1 is the pth homology

group of the cha**in** complex C. ˆ S**in**ce φ p is the dual of φ p , it is a cocha**in** map, **and**

the coboundary operator follows equivalently as the dual of ˆ∂ p or from the identity

ˆδ p ◦ φ p = φ p+1 ◦ δ p . The group Hˆ

p = ker ˆδ p /im ˆδ p−1 is the pth cohomology

group of the cha**in** complex C. ˆ

S**in**ce φ is an isomorphism, it def**in**es a cha**in** equivalence **and** **in**duces a homology

isomorphism between C **and** C. ˆ It therefore follows that Cˆ

can be used **in**terchangeably

**in** place of C **in** order to compute the homology of X. This is significant

**in** practice, s**in**ce the operators ˆ∂ **and** ˆδ are po**in**twise algebraic operators. Thus,

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 165

apply**in**g the discrete Fourier transform to both sides of (31) **and** **in**vok**in**g the convolution

theorem (cf., Appendix A) we obta**in**

ˆ∂ĉ p (θ, β) =

∑

N p

P βα (θ)ĉ p (θ, α), (38)

α=1

where we write

By duality we immediately have

P βα (θ) = ∑

ˆδĉ p−1 (θ, α) =

l∈Z n L αβ (l)e iθ·l . (39)

N ∑p−1

Q αβ (θ)ĉ p−1 (θ, β), (40)

β=1

where we write

Q αβ (θ) = Pβα ∗ (θ). (41)

Thus, the matrices P p (θ) represent the boundary operators ˆ∂ p , whereas their hermitian

transpose Q p (θ) represent the coboundary operators ˆδ p−1 . In particular, the

verification that the lattice complex is perfect reduces to the **in**vestigation of the

kernels **and** images of the matrices P p , namely, for fixed θ ̸= 0 **and** p ≧ 1, we must

have kerP p = imP p+1 . Then it follows that B p = Z p **and** H p = 0, as required.

2.3. Examples of lattice complexes

In this section we collect the cha**in** complex representations of: the monatomic

cha**in**, the square lattice, the hexagonal lattice, the simple cubic (SC) lattice, the

face-centered cubic (FCC) lattice **and** the body-centered cubic (BCC) lattice. The

monatomic cha**in**, the square lattice, the hexagonal lattice, **and** the SC lattice are

discussed ma**in**ly for purposes of illustration. The square, SC **and** FCC lattices furnish

examples of lattice complexes which are not simplicial. α-Polonium has the

SC structure, which is otherwise rare. Metals which crystallize **in** the FCC class

at room temperature **in**clude: Alum**in**um, copper, gold, lead, nickel, plat**in**um, rhodium,

silver **and** thorium; whereas metals which crystallize **in** the BCC class at room

temperature **in**clude: Chromium, iron, lithium, molybdenum, niobium, potassium,

rubidium, sodium, tantalum, tungsten **and** vanadium. We recall that the symmetry

group of cubic crystals is the octahedral group O, which is of order 24 **and** is generated

by the transformations {E, C 2 (6), C 3 , C3 2(8), C 4, C4 3(6), C2 4

(3)}, where E

is the identity **and** C 2 , C 3 **and** C 4 are the abelian groups of rotations about 2, 3 **and**

4-fold axes [16].

The homology of these lattices is most readily analyzed with the aid of the

matrix representations P p (θ) **and** Q p (θ) of the operators ˆ∂ p **and** ˆδ p−1 , respectively.

In all of the examples collected below, the follow**in**g properties are readily

verified, e.g., with the aid of the s**in**gular-value decomposition of P p :

166 M. P. Ariza & M. Ortiz

(i) P p P p+1 = 0, 1 ≦ p

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 167

2.3.2. The square lattice. The square Bravais lattice is generated by the basis

{(a, 0), (0,a)} **and** its symmetry group is C 4 , the abelian group of two-dimensional

rotations by π/2. The lattice complex representation of the square lattice

is shown **in** Fig. 3. The square lattice furnishes an example of a lattice complex

that is not simplicial. The connectedness, **and** translation **and** symmetry **in**variance

requirements are satisfied by choos**in**g the elementary areas to be squares, as shown

**in** Fig. 3. It is evident from this figure that the vertex **and** elementary area sets can

be **in**dexed as two-dimensional Bravais lattices. By contrast, there are two types of

elementary segments, namely, horizontal segments **and** vertical segments. Hence,

the segment set has the structure of a complex Bravais lattice compris**in**g two simple

sublattices. We choose the **in**dex**in**g scheme shown **in** Fig. 3, where ε 1 = (1, 0)

**and** ε 2 = (0, 1). The boundary **and** coboundary operators are determ**in**ed by the

rules

∂e 1 (l, 1) = e 0 (l + ε 1 ) − e 0 (l),

∂e 2 (l) = e 1 (l, 1) + e 1 (l + ε 1 , 2) − e 1 (l + ε 2 , 1) − e 1 (l, 2),

δe 0 (l) =−e 1 (l, 1) − e 1 (l, 2) + e 1 (l − ε 1 , 1) + e 1 (l − ε 2 , 2),

δe 1 (l, 1) = e 2 (l) − e 2 (l − ε 2 ),

(46a)

(46b)

(46c)

(46d)

**and** all the symmetry-related identities. The correspond**in**g DFT representation

(39) is

Q 1 = ( e iθ 1

− 1, e iθ 2

− 1 ) ,

(47a)

( ) 1 − e

iθ 2

Q 2 =

e iθ . (47b)

1

− 1

A cup product can be def**in**ed by multiply**in**g cells **in** the 1-skeleton, as **in** the

monatomic cha**in**, **and** append**in**g the follow**in**g non-zero products:

e 1 (l, 1) ∪ e 1 (l + ε 1 , 2) = e 2 (l),

e 1 (l, 2) ∪ e 1 (l + ε 2 , 1) =−e 2 (l),

e 0 (l) ∪ e 2 (l) = e 2 (l),

e 2 (l) ∪ e 0 (l + ε 1 + ε 2 ) = e 2 (l).

(48a)

(48b)

(48c)

(48d)

(l+ 2 ,1) l+ 1 + 2

2

(l,2)l+

l (l+ 1 ,2)

l+ 1

l

(l,1)

Fig. 3. Lattice complex representation of the square lattice **and** **in**dex**in**g scheme.

168 M. P. Ariza & M. Ortiz

It is straightforward to verify that this product is associative **and** satisfies the coboundary

formula (21). It is also **in**terest**in**g to note that every 2-cell is generated

**in** two different ways by the multiplication of two pairs of 1-cells. In addition there

are exactly two vertices which act on a 2-cell, one on the left **and** the other on the

right.

2.3.3. The hexagonal lattice. The hexagonal lattice is generated by the basis

{(a, 0), (a/2, √ 3a/2)} **and** its symmetry group is C 6 , the abelian group of twodimensional

rotations by π/3. The lattice complex representation of the hexagonal

lattice is shown **in** Fig. 4. It is evident from this figure that there are three types

of elementary segments **and** two types of elementary areas. Hence, the elementary

segment **and** area sets have the structure of complex lattices compris**in**g three

**and** two simple sublattices, respectively. We choose the **in**dex**in**g scheme shown **in**

Fig. 4, where ε 1 = (1, 0), ε 2 = (0, 1) **and** ε 3 = ε 2 − ε 1 = (−1, 1). The boundary

**and** coboundary operators are determ**in**ed by the rules

∂e 1 (l, 1) = e 0 (l + ε 1 ) − e 0 (l),

(49a)

∂e 2 (l, 1) = e 1 (l, 1) + e 1 (l + ε 1 , 3) − e 1 (l, 2),

(49b)

δe 0 (l) =−e 1 (l, 1) − e 1 (l, 2) − e 1 (l, 3)

+e 1 (l − ε 1 , 1) + e 1 (l − ε 2 , 2) + e 1 (l − ε 3 , 3), (49c)

δe 1 (l, 1) = e 2 (l, 1) − e 2 (l − ε 3 , 2),

(49d)

**and** all symmetry-related identities. The correspond**in**g DFT representation (39) is

Q 1 = (e iθ 1

− 1, e iθ 2

− 1, e iθ 3

− 1),

⎛

1 −e iθ ⎞

3

Q 2 = ⎝ −1 1 ⎠ ,

e iθ 1

−1

(50a)

(50b)

where θ 3 = θ 2 − θ 1 . A cup product can be def**in**ed by adopt**in**g the partial order**in**g

of the vertices implied by the orientation of the 1-cells. Cells **in** the 1-skeleton are

Fig. 4. Lattice complex representation of a hexagonal lattice **and** **in**dex**in**g scheme.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 169

l+ 3

l+ 1+ 3 l+ 1+ 2+ 3

l+ 2+ 3

l

l+ 2

l+ 1 l+ 1+ 2

(a)

(l+1,3)

(l+ 3,1)

(l,3)

(l,1)

(l+ 1+ 3,2)

l

(l+ 1,2)

(b)

(l+ 3,2)

(l,2)

(l+ 2+ 3,1)

(l+1+2,3)

(l+2,3)

(l+ 2,1)

( l + 3,3)

( l ,2)

( l + 1,1)

( l ,1)

( l + 2,2)

l

l

( l ,3)

(c)

(d)

Fig. 5. Lattice complex representation of the simple-cubic lattice **and** **in**dex**in**g scheme. (a)

Vertex set. (b) Elementary segment set. (c) Elementary area set. (d) Elementary volume set.

multiplied as **in** the monatomic cha**in**. The rema**in****in**g non-zero products are

e 0 (l) ∪ e 2 (l, 1) = e 2 (l, 1),

e 2 (l, 1) ∪ e 0 (l + ε 2 ) = e 2 (l, 1),

e 0 (l) ∪ e 2 (l, 2) =−e 2 (l, 2),

e 2 (l, 2) ∪ e 0 (l + ε 2 ) =−e 2 (l, 2),

e 1 (l, 1) ∪ e 1 (l + ε 1 , 3) = e 2 (l, 1),

e 1 (l, 3) ∪ e 1 (l + ε 3 , 1) =−e 2 (l, 2).

(51a)

(51b)

(51c)

(51d)

(51e)

(51f)

It is **in**terest**in**g to note that there is exactly one pair of 1-cells which generates each

1-cell by multiplication. Likewise, there is one vertex which operates on each 1-cell

from the left, **and** another which operates from the right.

2.3.4. The simple cubic lattice. The simple-cubic (SC) Bravais lattice is generated

by the basis {(a, 0, 0), (0,a,0), (0, 0,a)}. The lattice complex representation

170 M. P. Ariza & M. Ortiz

of the SC lattice is shown **in** Fig. 5. The cubic lattice furnishes an example of a

three-dimensional lattice complex that is CW but not simplicial. The connectedness,

**and** translation **and** symmetry **in**variance requirements are satisfied by choos**in**g the

elementary volumes to be cubes, as shown **in** Fig. 5c. It is evident from this figure

that the vertex **and** elementary volume sets can be **in**dexed as three-dimensional

Bravais lattices. We also note that there are three types of elementary segments,

along each of the cube directions, **and** three types of elementary areas. Hence, the

segment **and** area sets have the structure of complex lattices compris**in**g of three

simple sublattices each. We choose the **in**dex**in**g scheme shown **in** Fig. 5, where

ε 1 = (1, 0, 0), ε 2 = (0, 1, 0), **and** ε 3 = (0, 0, 1). The boundary **and** coboundary

operators are determ**in**ed by the rules

∂e 1 (l, 1) = e 0 (l + ε 1 ) − e 0 (l),

(52a)

∂e 2 (l, 1) = e 1 (l, 2) + e 1 (l + ε 2 , 3) − e 1 (l + ε 3 , 2) − e 1 (l, 3), (52b)

∂e 3 (l) = e 2 (l + ε 1 , 1) − e 2 (l, 1) + e 2 (l + ε 2 , 2) − e 2 (l, 2)

+e 2 (l + ε 3 , 3) − e 2 (l, 3), (52c)

δe 0 (l) =−e 1 (l, 1) − e 1 (l, 2) − e 1 (l, 3) + e 1 (l − ε 1 , 1)

+e 1 (l − ε 2 , 2) + e 1 (l − ε 3 , 3), (52d)

δe 1 (l, 1) =−e 2 (l, 2) + e 2 (l − ε 3 , 2) + e 2 (l, 3) − e 2 (l − ε 2 , 3), (52e)

δe 2 (l, 1) =−e 3 (l) + e 3 (l − ε 1 ),

(52f)

**and** all the symmetry-related identities. The correspond**in**g DFT representation (39)

is

Q 1 = ( e iθ 1

− 1, e iθ 2

− 1, e iθ 3

− 1 ) ,

⎛

0, e iθ 3

− 1, 1 − e iθ ⎞

2

Q 2 = ⎝ 1 − e iθ 3, 0, e iθ 1

− 1 ⎠ ,

e iθ 2

− 1, 1 − e iθ 1, 0

⎛

e iθ ⎞

1

− 1

Q 3 = ⎝ e iθ 2

− 1 ⎠ .

e iθ 3

− 1

(53a)

(53b)

(53c)

A cup product can be def**in**ed by multiply**in**g cells **in** the 2-skeleton, as **in** the square

lattice, **and** append**in**g the follow**in**g non-zero products:

e 2 (l, 1) ∪ e 1 (l + ε 2 + ε 3 , 1) = e 3 (l),

e 1 (l, 1) ∪ e 2 (l + ε 1 , 1) = e 3 (l),

e 2 (l, 2) ∪ e 1 (l + ε 1 + ε 3 , 2) = e 3 (l),

e 1 (l, 2) ∪ e 2 (l + ε 2 , 2) = e 3 (l),

e 0 (l) ∪ e 3 (l) = e 3 (l),

e 3 (l) ∪ e 0 (l + ε 1 + ε 2 + ε 3 ) = e 3 (l).

(54a)

(54b)

(54c)

(54d)

(54e)

(54f)

It is straightforward to verify that this product is associative **and** satisfies the

coboundary formula (21). As **in** the case of the square lattice, it is also **in**terest**in**g

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 171

l- 6

l- 5

l+ 1

l- 3 l- 4

l+ 5

l+ 2

l+ 4

l+ 3

l

l- 2

l- 1

l+ 6

(a)

(l- 6,6)

(l,2)

(l- 1,1)

(l- 3,3)

l

(b)

(l,1)

(l,4) (l- 4,4)

(l,5)

(l,3)

(l- 5,5)

(l- 2,2)

(l,6)

Fig. 6. Lattice complex representation of the face-centered cubic lattice **and** **in**dex**in**g

scheme. (a) Vextex set. (b) Elementary segment set.

to note that every 3-cell is generated **in** four different ways by the multiplication of

certa**in** pairs of 1 **and** 2-cells. In addition there are exactly two vertices which act

on a 3-cell, one on the left **and** the other on the right.

2.3.5. The face-centered cubic lattice. The face-centered cubic (FCC) Bravais

lattice is generated by the basis {(0,a/2,a/2), (a/2, 0,a/2), (a/2,a/2, 0)}. A

lattice complex representation of the FCC lattice is shown **in** Figs. 6, 7 **and** 8. The

complex is chosen so as to conta**in** the commonly observed

2 1 〈110〉 slip directions

**and** {111} slip planes. It is evident from the figures that the vertex set can be **in**dexed

as a simple lattice; the elementary segment set as a complex lattice compris**in**g six

simple sublattices; the elementary area set as a complex lattice compris**in**g eight

simple sublattices; **and** the elementary volume set as a complex lattice compris**in**g

of three sublattices. We choose the **in**dex**in**g scheme, shown **in** Figs. 6, 7 **and** 8,

where ε 1 = (1, 0, 0), ε 2 = (0, 1, 0), ε 3 = (0, 0, 1), ε 4 = ε 2 − ε 1 = (−1, 1, 0),

ε 5 = ε 3 − ε 1 = (−1, 0, 1) **and** ε 6 = ε 3 − ε 2 = (0, −1, 1). The boundary **and**

coboundary operators are determ**in**ed by the rules

∂e 1 (l, 1) = e 0 (l + ε 1 ) − e 0 (l),

(55a)

∂e 2 (l, 1) = e 1 (l, 5) − e 1 (l + ε 6 , 4) − e 1 (l, 6),

(55b)

∂e 3 (l, 1) = e 2 (l + ε 2 , 1) + e 2 (l + ε 2 , 4) + e 2 (l + ε 2 , 6) + e 2 (l + ε 2 , 8)

−e 2 (l + ε 5 , 5)− e 2 (l + ε 5 , 7)− e 2 (l + ε 5 , 2)−e 2 (l + ε 5 , 3), (55c)

∂e 3 (l, 2) = e 2 (l + ε 6 , 3) − e 2 (l + ε 3 , 6) + e 2 (l + ε 5 , 7) − e 2 (l, 1), (55d)

δe 0 (l) =−e 1 (l, 1) − e 1 (l, 2) − e 1 (l, 3) − e 1 (l, 4)

−e 1 (l, 5) − e 1 (l, 6), +e 1 (l − ε 1 , 1)+e 1 (l − ε 2 , 2)

+e 1 (l − ε 3 , 3)+e 1 (l − ε 4 , 4)+ e 1 (l − ε 5 , 5)+e 1 (l − ε 6 , 6), (55e)

172 M. P. Ariza & M. Ortiz

2

3

1

4

(a)

(b)

7

5

6

8

(c)

(d)

Fig. 7. Lattice complex representation of the face-centered cubic lattice **and** **in**dex**in**g

scheme. Elementary area set.

δe 1 (l, 1) = e 2 (l, 7) − e 2 (l + ε 1 , 8) − e 2 (l, 5) + e 2 (l + ε 1 , 6),

δe 2 (l, 1) = e 3 (l − ε 2 , 1) − e 3 (l, 2),

(55f)

(55g)

**and** all symmetry-related identities. The correspond**in**g DFT representation (39) is

Q 1 = (e iθ 1

− 1, e iθ 2

− 1, e iθ 3

− 1, e iθ 4

− 1, e iθ 5

− 1, e iθ 6

− 1), (56a)

⎛

0 0 0 0 −1 e −iθ 1

1 −e −iθ ⎞

1

0 0 −1 e −iθ 2

1 −e −iθ 2

0 0

Q 2 =

0 0 e −iθ 6

−e −iθ 2

0 0 −e −iθ 5 e −iθ 1

⎜ −e iθ 6

e −iθ 5 0 0 −e iθ 1

e −iθ 2

0 0

, (56b)

⎟

⎝ 1 −e −iθ 5 0 0 0 0 e −iθ 5 −1 ⎠

−1 e −iθ 6

−e −iθ 6

1 0 0 0 0

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 173

1

2

(a)

(b)

3

(c)

Fig. 8. Lattice complex representation of the face-centered cubic lattice **and** **in**dex**in**g

scheme. Elementary volume set.

⎛

e iθ ⎞

2

−1 0

−e iθ 5 0 e iθ 3

−e iθ 5 e iθ 6

0

Q 3 =

e iθ 2

0 −e iθ 2

−e iθ 5 0 1

. (56c)

⎜ e iθ 2

−e iθ 3

0

⎟

⎝ −e iθ 5 e iθ 5 0 ⎠

e iθ 2

0 −e iθ 1

A cup product can be def**in**ed by multiply**in**g cells **in** the 2-skeleton, as **in** the

hexagonal lattice, **and** append**in**g the follow**in**g non-zero products:

174 M. P. Ariza & M. Ortiz

e 2 (l + ε 2 , 6) ∪ e 1 (l + ε 2 , 5) = e 3 (l, 1),

e 2 (l + ε 2 , 4) ∪ e 1 (l + ε 3 , 4) = e 3 (l, 1),

e 2 (l + ε 5 , 2) ∪ e 1 (l + ε 5 , 2) = e 3 (l, 1),

e 2 (l + ε 5 , 7) ∪ e 1 (l + ε 3 , 4) = e 3 (l, 1),

e 2 (l, 1) ∪ e 1 (l + ε 5 , 1) = e 3 (l, 2),

e 0 (l) ∪ e 3 (l, 1) = e 3 (l, 1),

e 3 (l, 1) ∪ e 0 (l + ε 3 + ε 4 ) = e 3 (l, 1),

e 0 (l) ∪ e 3 (l, 2) = e 3 (l, 2),

e 3 (l, 2) ∪ e 0 (l + ε 3 ) = e 3 (l, 2),

(57a)

(57b)

(57c)

(57d)

(57e)

(57f)

(57g)

(57h)

(57i)

**and** others related to these by symmetry. A straightforward calculation shows

that this product is associative **and** satisfies the coboundary formula (21). The

non-simplicial cell is obta**in**ed by the multiplication of four pairs of 1 **and** 2-cells,

whereas the two simplicial cells are generated by one pair each. All 3-cells are acted

upon by exactly two vertices, one act**in**g on the left **and** another on the right.

2.3.6. The body-centered cubic lattice. The body-centered cubic (BCC) Bravais

lattice is generated by the basis {(−a/2,a/2,a/2), (a/2, −a/2,a/2),

(a/2,a/2, −a/2)}. A lattice complex representation of the BCC lattice is shown

**in** Figs. 9, 10 **and** 11. The complex is chosen so as to conta**in** the

2 1 〈111〉 slip directions

**and** {110} slip planes. Complexes conta**in****in**g other commonly observed slip

planes, such as {112}, may be constructed likewise. It is evident from the figures

that the vertex set can be **in**dexed as a simple lattice; the elementary segment set as

a complex lattice compris**in**g of seven simple sublattices; the elementary area set

l+ 7

l- 3

l- 7

l+ 1

l+ 2 l+ 4

l- 5

l- 6 l

l+ 6

l+ 5

l- 4 l- 2

l- 1 l+ 3

(l-6,6)

(l-1,1)

(l,5)

(l-3,3)

(l-4,4)

(l,7)

(l,2) (l-2,2)

(l,4)

l

(l-7,7)

(l,3)

(l,1)

(l-5,5)

(l,6)

(a)

(b)

Fig. 9. Lattice complex representation of the body-centered cubic lattice **and** complex lattice

**in**dex**in**g scheme. (a) Vextex set. (b) Elementary segment set.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 175

3

2

8

7

4 1

5

6

(a)

(b)

11

12 9

10

(c)

Fig. 10. Lattice complex representation of the body-centered cubic lattice **and** complex

lattice **in**dex**in**g scheme. Elementary area set.

as a complex lattice compris**in**g of twelve simple sublattices; **and** the elementary

volume set as a complex lattice compris**in**g of twelve sublattices. We choose the

**in**dex**in**g scheme shown **in** Figs. 9, 10 **and** 11, where ε 1 = (1, 0, 0), ε 2 = (0, 1, 0),

ε 3 = (0, 0, 1), ε 4 = (1, 1, 1), ε 5 = (0, 1, 1), ε 6 = (1, 0, 1) **and** ε 7 = (1, 1, 0). The

boundary **and** coboundary operators are determ**in**ed by the rules

∂e 1 (l, 1) = e 0 (l + ε 1 ) − e 0 (l),

(58a)

∂e 1 (l, 5) = e 0 (l + ε 5 ) − e 0 (l),

(58b)

∂e 2 (l, 1) =−e 1 (l − ε 2 , 5) + e 1 (l − ε 2 , 2) + e 1 (l, 3),

(58c)

∂e 3 (l, 1) =−e 2 (l + ε 2 , 1) + e 2 (l + ε 5 , 7) + e 2 (l, 8) − e 2 (l + ε 4 , 4), (58d)

176 M. P. Ariza & M. Ortiz

1

2

(a)

(b)

3

4

(c)

(d)

5

6

(e)

(f)

Fig. 11. Lattice complex representation of the body-centered cubic lattice **and** complex

lattice **in**dex**in**g scheme. Elementary volume set.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 177

δe 0 (l) =−e 1 (l, 1) − e 1 (l, 2) − e 1 (l, 3) − e 1 (l, 4)

−e 1 (l, 5) − e 1 (l, 6) − e 1 (l, 7) + e 1 (l − ε 1 , 1)

+e 1 (l − ε 2 , 2) + e 1 (l − ε 3 , 3) + e 1 (l − ε 4 , 4)

+e 1 (l − ε 5 , 5) + e 1 (l − ε 6 , 6) + e 1 (l − ε 7 , 7), (58e)

δe 1 (l, 1) =−e 2 (l, 2) + e 2 (l + ε 1 , 4) − e 2 (l + ε 1 , 5)

+e 2 (l, 7) − e 2 (l, 10) + e 2 (l + ε 1 , 12), (58f)

δe 1 (l, 5) =−e 2 (l + ε 2 , 1) − e 2 (l − ε 1 , 2) + e 2 (l + ε 3 , 3)

+e 2 (l + ε 4 , 4), (58g)

δe 2 (l, 1) =−e 3 (l − ε 2 , 1) + e 3 (l − ε 7 , 5),

(58h)

**and** all symmetry-related identities. The correspond**in**g DFT representation (39) is

Q 1 = (e iθ 1

− 1, e iθ 2

− 1, e iθ 3

− 1, e iθ 4

− 1, e iθ 5

− 1, e iθ 6

− 1, e iθ 7

− 1), (59a)

⎛

Q 2 =

⎜

⎝

0 −1 0 e −iθ 1 −e −iθ 1 0 1 0 0 −1 0 e −iθ 1

e −iθ 2 0 −1 0 0 −e −iθ 2 0 1 0 −e −iθ 2 0 1

1 0 −e −iθ 3 0 −1 0 e −iθ 3 0 −1 0 e −iθ 3 0

0 1 0 −e −iθ 4 0 e −iθ 4 0 −1 1 0 −e −iθ 4 0

−e −iθ 2 −e iθ 1 e −iθ 3 e −iθ 4 0 0 0 0 0 0 0 0

0 0 0 0 e −iθ 1 −e −iθ 4 −e −iθ 3 e iθ 2 0 0 0 0

0 0 0 0 0 0 0 0 −e iθ 3 e −iθ 2 e −iθ 4 −e −iθ 1

⎞

,

⎟

⎠

(59b)

⎛

Q 3 =

⎜

⎝

−e iθ 2

0 0 0 e iθ 7

0

0 0 1 0 −1 0

0 −e iθ 3

e iθ 6

0 0 0

−e iθ 4

e iθ 4

0 0 0 0

0 0 −e iθ 1

0 0 e iθ 7

0 0 −e iθ 4

e iθ 4

0 0

e iθ 5 0 0 −e iθ 3

0 0

1 0 0 0 0 −1

0 1 0 −1 0 0

0 e iθ 5 0 0 0 −e iθ 2

0 0 0 0 −e iθ 4

e iθ 4

0 0 0 e iθ 6

−e iθ 1

0

⎞

. (59c)

⎟

⎠

A cup product is **in**duced by adopt**in**g the partial order**in**g of the vertices implied by

the orientation of 1-cells. Cells **in** the 2-skeleton are multiplied as **in** the hexagonal

complex. The rema**in****in**g non-zero products are

e 0 (l) ∪ e 3 (l, 1) = e 3 (l, 1),

(60a)

e 3 (l, 1) ∪ e 0 (l + ε 4 ) = e 3 (l, 1),

(60b)

e 2 (l + ε 2 , 1) ∪ e 1 (l + ε 5 , 1) =−e 3 (l, 1),

(60c)

e 1 (l, 2) ∪ e 2 (l + ε 5 , 7) = e 3 (l, 1),

(60d)

**and** those obta**in**ed from these by symmetry. Aga**in** it is **in**terest**in**g to note the

uniqueness of the representations of the 3-cells as products of their faces.

178 M. P. Ariza & M. Ortiz

3. Calculus on lattices

The objective of this section is to develop a lattice analog of the de Rham complex

**in** R n (cf., e.g., [5]). In particular, we wish to extend to lattices the notions of

forms **and** differential operators, as well as the theory of **in**tegration of forms **and** currents.

This extension results **in** natural def**in**itions of boundaries **and** generalizations

of Stoke’s theorem, the Hodge decomposition theorem, the Gauss l**in**k**in**g number

**and** other classical results **and** objects. As we shall see, the result**in**g framework

is ideally suited for the development of a mechanics of lattices, with **and** without

defects. In particular, the discrete version of the Gauss l**in**k**in**g number which we

develop **in** Section 3.3 provides a natural measure of the degree of entanglement

of a dislocation ensemble. This entanglement contributes to the work-harden**in**g of

crystals. In addition, the discrete analog of discrete dislocations loops **and** Burgers

circuits will be **in**troduced **in** Section 5.1. As **in** the cont**in**uum sett**in**g, the discrete

l**in**k**in**g number also provides a natural device for relat**in**g those classes **and** types

of objects.

3.1. The differential complex of a lattice

Consider a lattice complex X **and** its cha**in** complex {C p ,∂}. Let E p be the set

of p-cells of the lattice. A real-valued exterior differential form of order p, ora

p-form, is a cocha**in** with real coefficients. Thus, p-forms have the representation

ω = ∑

e p ∈E p

f(e p )e p , (61)

where f : E p → R. Also p-forms with coefficients **in** an arbitrary abelian group

G may be def**in**ed likewise. The case G = R n will arise frequently **in** subsequent

applications. In this context, we shall refer to the elementary p-cocha**in**s e p as elementary

p-forms, **and** we shall denote by p the set of p-forms over the lattice.

The value of the p-form (61) on a general p-cha**in** c p is

〈ω,c p 〉= ∑

e p ∈E p

f(e p )〈e p ,c p 〉. (62)

The exterior derivative of a p-form is the (p + 1)-form

dω = ∑

e p ∈E p

f(e p )δe p . (63)

S**in**ce δ 2 = 0, we have immediately d 2 = 0. It also follows immediately from the

properties of the coboundary operator that

〈dω,c p+1 〉=〈ω,∂c p+1 〉 (64)

for all p-forms ω ∈ p . Let now Y be a subcomplex of X. Let α, β ∈ p **and**

suppose that α |Y = β |Y . Then,

〈α, ∂e p+1 〉=〈β,∂e p+1 〉 (65)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 179

**and**

〈dα,e p+1 〉=〈dβ, e p+1 〉 (66)

for all (p + 1)-cells e p+1 of Y . Hence, dα |Y = dβ |Y , which shows that d is a local

operator. All the above def**in**itions **and** operations rely solely on the abelian-group

structure of R under addition **and**, therefore, extend immediately to forms with

coefficients **in** an arbitrary abelian group G. When such an extension is necessary,

we shall denote by p (X, G) the set of p-forms over X with coefficients **in** G.

The wedge-product of a real-valued p-form α = ∑ fe p **and** a real-valued

q-form β = ∑ ge q may be def**in**ed as the real-valued (p + q)-form

α ∧ β = ∑ ∑

f(e p )g(e q )e p ∪ e q . (67)

e q ∈E q

e p ∈E p

The wedge product confers ∗ =⊕ n p=0 p a graded algebra structure, **and** it **in**herits

the essential properties of the cup product, namely, bil**in**earity, associativity, **and**

the coboundary formula

d(α ∧ β) = (dα) ∧ β + (−1) p α ∧ (dβ) (68)

for all α ∈ p **and** β ∈ q .

The graded algebra ∗ together with the operator d may be regarded as a discrete

de Rham complex on the lattice X. A form α ∈ p is closed if dα = 0, **and**

exact if there is a form β ∈ p−1 such that α = dβ. The kernel of d are the closed

forms **and** the image of d are the exact forms. The pth de Rham cohomology of X

is the vector space H p = (kerd ∩ p )/(imd ∩ p ). The cohomology of X is the

direct sum H ∗ =⊕ p∈Z H p .InR n , Po**in**caré’s lemma states that H ∗ = R if n = 0

**and** H ∗ = 0 otherwise. We will require the cohomology of a perfect n-dimensional

lattice to be identical to that of R n .

Considerations of duality lead naturally to the def**in**ition of p-vector fields **and**

the codifferential operator. A real-valued vector field of order p,orap-vector field,

is a cha**in** with real coefficients **and** arbitrary (not necessarily f**in**ite) support. Thus,

p-vector fields have the representation

u = ∑

f(e p )e p , (69)

e p ∈E p

where f : E p → R. We shall denote by p the space of all p-vector fields over

X. Ifu = ∑ fe p is a p-vector field **and** α = ∑ ge p is a p-form we write

〈α, u〉 = ∑

f(e p )g(e p ). (70)

e p ∈E p

We def**in**e the codifferential operator δ : p → p−1 by the identity

〈α, δu〉 =〈dα,u〉 (71)

180 M. P. Ariza & M. Ortiz

for all α ∈ p−1 . It follows from (18) that the codifferential operator of a p-vector

field u = ∑ fe p admits the representation

δu = ∑

f(e p )∂e p . (72)

e p ∈E p

The preced**in**g def**in**itions **and** operations reta**in** there mean**in**g for p-forms **and** p-

vector fields with coefficients **in** an **in**ner-product space H . In applications the case

H = R n , endowed with the st**and**ard **in**ner product, will arise frequently. However,

**in** this simple case we may equivalently perform all differential operations **and**

duality operations component by component.

The exterior derivative of a p-form α **in**herits the matrix representation

dα(e p+1 ) = ∑

L(e p ,e p+1 )α(e p ) (73)

e p ∈E p

directly from (63) **and** the matrix representation (27) of the coboundary operator.

In addition, the complex Bravais lattice representation (32) **and** the discrete Fourier

representation (40) of the coboundary operator carry over mutatis mut**and**is to

the exterior derivative. Likewise, the codifferential operator of a p-vector field u

**in**herits the matrix representation

δu(e p−1 ) = ∑

L(e p−1 ,e p )u(e p ) (74)

e p ∈E p

directly from (72) **and** the matrix representation (26). Here aga**in**, the complex

Bravais lattice representation (31) **and** the discrete Fourier representation (38) of

the boundary operator carry over mutatis mut**and**is to the codifferential.

Given forms α = ∑ fe p , β = ∑ ge p **and** fields u = ∑ ae p , v = ∑ be p

tak**in**g values **in** a Hilbert space H , the correspond**in**g L 2 -products are

〈α, β〉 = ∑

〈f(e p ), g(e p )〉,

(75a)

e p ∈E p

〈u, v〉 = ∑

e p ∈E p

〈a(e p ), b(e p )〉.

(75b)

These products enable the **in**troduction of the flat **and** sharp operators, ♭ : p → p

**and** ♯ : p → p , respectively, through the usual identities (cf., [2])

〈α, u ♭ 〉=〈α, u〉,

(76a)

〈α ♯ ,u〉=〈α, u〉. (76b)

Clearly, α ♯ = ∑ fe p **and** u ♭ = ∑ ae p , **and** thus the flat **and** sharp operators simply

extend to forms **and** fields with the mapp**in**gs ∗:C p → C p **and** ∗:C p → C p

def**in**ed **in** Section 2.2. Us**in**g the sharp **and** flat operators it is possible to extend the

differential **and** codifferential operators to fields **and** forms, respectively, by sett**in**g:

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 181

δα = (δ(α ♯ )) ♭ **and** du = (d(u ♭ )) ♯ . It is also possible to def**in**e a discrete Hodge

∗-operator, i.e., **and** isomorphism ∗: p → n−p , through the identity

〈α, β〉 =〈α ∧∗β,µ〉 =〈β ∧∗α, µ〉, (77)

where µ = ∑ e n is the unit n-cha**in**, or volume form, tak**in**g the value 1 on

all n-cells. By l**in**earity, **in** suffices to specify the ∗-operator for all elementary

p-cocha**in**s e p . It can be readily verified that ∗e p is the unique elementary (n − p)-

cocha**in** e n−p such that e p ∪ e n−p = e n for some elementary n-cocha**in** e n . Indeed,

〈α ∧∗e p ,µ〉=α(e p ) =〈α, e p 〉 (78)

**and** (77) is satisfied identically. Thus, the Hodge ∗-operator is related to the cap

product (cf. Section 2.2) through the identity

∗α = ∑

f(e p )(e p ∩ µ) ∗ , (79)

e p ∈E p

where α = ∑ fe p .

As **in** the conventional Hodge-de Rham theory we can def**in**e the discrete

Laplace-de Rham operator : p → p by the formula = dδ + δd. A form α

such that α = 0 is said to be harmonic. We shall denote by K p the vector space

of harmonic p-forms. Then we have the follow**in**g discrete analog of the classical

Hodge decomposition theorem.

Proposition 1. The follow**in**g equation holds:

where the decomposition is L 2 -orthogonal.

p = d p−1 ⊕ δ p+1 ⊕ K p , (80)

Proof. (cf., [2], Theorem 7.5.3). Let α ∈ p . Then α = 0 if **and** only if dα = 0

**and** δα = 0. Indeed, if dα = 0 **and** δα = 0 then α = dδα + δdα = 0. Conversely,

let α = 0. Then 〈α, α〉 =〈dδα + δdα, α〉 =〈dδα, α〉 +〈δdα, α〉 =

〈dα,dα〉+〈δα, δα〉 =0, whence it follows that dα = 0 **and** δα = 0. Suppose now

that ω = dα + δβ + γ , with ω ∈ p , α ∈ p−1 , β ∈ p+1 **and** γ ∈ K p . Then

〈dα,δβ〉 =〈d 2 α, β〉 =〈α, δ 2 β〉=0. In addition, 〈dα,γ〉 =〈α, δγ 〉=0, **and**

〈δβ, γ 〉=〈β,dγ〉 =0. Thus, the spaces d p−1 , δ p+1 **and** K p are orthogonal.

Let now C p be the orthogonal complement of d p−1 ⊕ δ p+1 . Clearly, K p ⊂ C p .

Conversely, let γ ∈ C p . Then 〈dα,γ〉=〈α, δγ 〉=0 for all α ∈ p−1 , **and** δγ = 0.

In addition, 〈δβ, γ 〉=〈β,dγ〉 =0 for all β ∈ p+1 , **and** dγ = 0. Hence, γ = 0,

**and** C p ⊂ K p . ⊓⊔

The Hodge decomposition theorem supplies the follow**in**g direct l**in**k between

harmonic functions **and** cohomology.

Corollary 1. K p **and** H p are isomorphic.

182 M. P. Ariza & M. Ortiz

Proof. (cf., [2], Corollary 7.5.4). Let γ ∈ K p . Then γ = 0, dγ = 0 **and**

γ ∈ kerd p . Hence, K p can be mapped to kerd p by **in**clusion, **and** then to H p by

projection, i.e., by assign**in**g to γ its equivalence class [γ ]. We wish to show that

the mapp**in**g K p → H p thus def**in**ed is an isomorphism. Suppose [γ ]=0. This

requires that there is a β ∈ p−1 such that γ = dβ, i.e., that γ be exact. But,

s**in**ce γ is harmonic we additionally have δγ = 0. Therefore, 〈γ,γ〉=〈γ,dβ〉=

〈δγ, β〉 =0. Hence γ = 0 **and** the map γ →[γ ] is one-to-one. Suppose now that

[ω] ∈H p . By the Hodge theorem we can write ω = dα + δβ + γ , where γ ∈ K p .

But 0 = dω = dδβ. Hence 〈δβ, δβ〉 =〈β,dδβ〉 =0 **and** δβ = 0. This leaves

ω = dα + γ **and** [ω] =[γ ], **and** thus the map γ →[γ ] is onto. ⊓⊔

For perfect lattices, the Hodge decomposition theorem reduces to the follow**in**g

form.

Corollary 2. Let X be a perfect lattice complex. Let ω be a p-form over X. Then,

where

ω = dα + δβ, (81)

α = −1 δω,

β = −1 dω.

(82a)

(82b)

Proof. For a perfect lattice, K p = H p = 0 **and** the Hodge decomposition reduces

to (81). In addition, we have the identities: d −1 δω = −1 dδω **and** δ −1 dω =

−1 δdω. The first of these identities is verified as follows: d −1 δω = (dδ +

δd)d −1 δω = dδd −1 δω = d(δd+dδ) −1 δω = d −1 δω = dδω. The second

identity is verified as follows: δ −1 dω = (δd + dδ)δ −1 dω = δdδ −1 dω =

δ(dδ + δd) −1 dω = δ −1 dω = δdω. By virtue of the preced**in**g identities we

have dα + δβ = d −1 δω + δ −1 dω = −1 dδω + −1 δdω = −1 (dδ + δd)ω

= −1 ω = ω. ⊓⊔

3.2. Integration on lattices

The characteristic cha**in** of a subset A of E p is

χ A = ∑

e p . (83)

e p ∈A

The **in**tegral of a p-form α over A is def**in**ed as

∫

α ≡〈α, χ A 〉. (84)

A

We may equivalently regard 〈α, χ A 〉 as the action of A on the form α, **and** A as a

current, i.e., a l**in**ear functional on forms (cf., e.g., [31]). If α = ∑ f(e p )e p then

∫

α = ∑

f(e p ). (85)

A

e p ∈A

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 183

Suppose now that A ⊂ E p+1 **and** ω ∈ p . Then

∫

dω =〈dω,χ A 〉=〈ω,∂χ A 〉. (86)

A

We def**in**e the **in**tegral of ω over the boundary of A as

∫

ω =〈ω,∂χ A 〉. (87)

∂A

Then we have

∫ ∫

dω =

A

∂A

ω, (88)

which may be regarded as a discrete version of Stoke’s theorem. As an example of

an application, the **in**tegration-by-parts formula

∫ ∫

∫

α ∧ β = (dα) ∧ β + (−1) p α ∧ (dβ), (89)

∂A

A

where α ∈ p , β ∈ q , **and** A ⊂ E p+q+1 , follows directly by **in**tegrat**in**g the

coboundary formula (68) over A **and** apply**in**g Stoke’s theorem to the left-h**and**

side of the result**in**g equation.

A

3.3. The Gauss l**in**k**in**g number

Throughout this section we restrict our attention to three-dimensional lattices,

i.e., n = 3. The l**in**k**in**g number of two 1-forms ω,ω ′ ∈ 1 is def**in**ed to be

∫

L**in**k(ω, ω ′ ) = ω ∧ dω ′ . (90)

E 3

From the **in**tegration by parts formula (89) it follows that

∫

L**in**k(ω, ω ′ ) = dω ∧ ω ′ . (91)

E 3

Clearly, L**in**k(ω, ω ′ ) = 0 if either ω or ω ′ is exact. In particular, let A ∈ E 0 **and** let

χ A be its characteristic cha**in**, regarded as a 0-form. Then dχ A is a

1-form which represents the closed oriented boundary ∂A of A. It then follows

that L**in**k(dχ A ,ω ′ ) = 0, i.e., the l**in**k**in**g number of a closed oriented surface with

any other 1-form is necessarily zero. Suppose **in**stead that ω = χ B ∧ dχ A for some

B ∈ E 0 **in**tersect**in**g A. Then, ω represents a subset C of ∂A determ**in**ed by its

**in**tersection with B. In addition, it follows from an application of the coboundary

formula (68) that dω = dχ B ∧ dχ A = dχ A ∧ dχ B , which represents the loop D

obta**in**ed by **in**tersect**in**g the boundaries ∂A **and** ∂B. Likewise, let ω ′ = χ B ′ ∧ dχ A ′

for some dist**in**ct but **in**tersect**in**g sets A ′ ,B ′ ∈ E 0 , **and** let the oriented area C ′

184 M. P. Ariza & M. Ortiz

**and** oriented loop D ′ be def**in**ed as before. Now suppose that D **and** C ′ **in**tersect

transversally, i.e., dω ∧ ω ′ is supported on isolated 3-cells. Then,

L**in**k(D, D ′ ) = ∑ D∩C ′ ±1, (92)

where the sum extends over the support of dω∧ ω ′ **and** the sign is positive at **in**tersections

where the orientations of D **and** C ′ are consistent, **and** negative otherwise.

It is readily shown (e.g., [5], §17) that L**in**k(D, D ′ ) is **in**dependent of the choices

made **in** the preced**in**g def**in**ition (cf., [5] also for the connection between the l**in**k**in**g

number **and** the Hopf **in**variant).

If the lattice is perfect, then ω may be written **in** the form (81). Insert**in**g this

representation **in**to (90) we obta**in**

∫

∫

L**in**k(ω, ω ′ ) = dα ∧ dω ′ + δβ ∧ dω ′ . (93)

E 3 E 3

But the first term on the right-h**and** side vanishes by virtue of (89) **and** we have

∫

L**in**k(ω, ω ′ ) = δβ ∧ dω ′ . (94)

E 3

Furthermore, **in**sert**in**g (82b) **in**to this expression we obta**in**

∫

L**in**k(ω, ω ′ ) = δ −1 dω ∧ dω ′ . (95)

E 3

This identity generalizes the well-known **in**tegral formula (3) for the Gauss l**in**k**in**g

number of two loops C 1 **and** C 2 **in** R 3 (e.g., [21]).

3.4. Examples of l**in**k**in**g relations

By way of illustration, we may exam**in**e the l**in**k**in**g number for the cases of the

simple cubic, BCC **and** FCC lattices described **in** Sections 2.3.4, 2.3.6 **and** 2.3.5.

By the bil**in**earity of the l**in**k**in**g number, it suffices to provide rules for the l**in**k**in**g

of pairs of elementary 1-cells.

3.4.1. The simple cubic lattice. In the simple cubic lattice the l**in**ks that can be

formed with a 1-cell, such as e 1 (l, 2), are:

L**in**k ( e 1 (l + ε 2 + ε 3 , 1), e 1 (l, 2) ) = 1,

L**in**k ( e 1 (l + ε 1 + ε 2 , 3), e 1 (l, 2) ) = 1,

L**in**k ( e 1 (l + ε 2 , 3), e 1 (l, 2) ) = 1,

L**in**k ( e 1 (l + ε 2 , 1), e 1 (l, 2) ) = 1.

(96a)

(96b)

(96c)

(96d)

All other l**in**k**in**g relations may be deduced by symmetry.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 185

3.4.2. The body-centered cubic lattice. In the BCC lattice the 1-cells **in** the cube

direction, such as e 1 (l, 5), e 1 (l, 6) **and** e 1 (l, 7), do not form any l**in**ks. The l**in**ks

that can be formed with a diagonal 1-cell, such as e 1 (l, 1), are:

L**in**k ( e 1 (l + ε 6 , 2), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l + ε 7 , 3), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l + ε 1 , 2), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l + ε 1 , 3), e 1 (l, 1) ) = 1,

(97a)

(97b)

(97c)

(97d)

with all other l**in**k**in**g relations follow**in**g by symmetry.

3.4.3. The face-centered cubic lattice. In the FCC lattice all 1-cells are symmetry

related. It therefore suffices to enumerate the l**in**ks that can be formed with one

1-cell, with all other l**in**k**in**g relations follow**in**g by symmetry. Tak**in**g e 1 (l, 1) as

the representative 1-cell we obta**in**

L**in**k ( e 1 (l − ε 4 , 4), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l + ε 2 , 6), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l + ε 1 , 4), e 1 (l, 1) ) = 1,

L**in**k ( e 1 (l − ε 5 , 6), e 1 (l, 1) ) = 1,

(98a)

(98b)

(98c)

(98d)

with all other l**in**k**in**g relations follow**in**g by symmetry.

4. Harmonic lattices

In this section we apply the mach**in**ery developed **in** the preced**in**g sections

to the formulation of a mechanics of lattices. An important observation is that the

essential structure of that mechanics follows from material-frame **in**difference, possibly

comb**in**ed with additional assumptions of locality. In particular, **in**variance of

the energy with respect to rigid-body rotations **and** translations determ**in**es the form

of the deformation **and** stress fields, both for f**in**ite **and** l**in**earized k**in**ematics. The

equilibrium equations then follow from stationarity, **and** can be given a compact

expression us**in**g discrete differential operators. In addition, we briefly delve **in**to

basic questions of analysis perta**in****in**g to the equilibrium problem, **and** **in**vestigate

the cont**in**uum limit of the energy function of harmonic lattices **in** the sense of

Ɣ-convergence.

4.1. The mechanics of lattices

Consider a crystal lattice described by its correspond**in**g lattice complex X.A

deformation of a crystal is a mapp**in**g y : E 0 → R n . Thus, y(e 0 ) is the position of

vertex e 0 ∈ E 0 **in** the deformed configuration of the crystal. We shall assume that

the energy of a deformed crystal can be written as a function of the form E(y),

i.e., as a function E : 0 (X, R n ) → R. For **in**stance E(y) may be approximated

186 M. P. Ariza & M. Ortiz

by means of a suitable empirical potential. Examples of commonly used empirical

or semi-empirical potentials **in**clude: pairwise potentials such as Lennard-Jones

[1]; the embedded-atom method (EAM) for FCC metals [11]; the modified embedded-atom

method (MEAM) for BCC metals [50]; the F**in**nis-S**in**clair potential

for transition metals [13]; MGPT pseudo-potentials [32]; Still**in**ger-Weber for

covalent crystals [46]; bond-order potentials [41]; **and** others.

Energy functions can often be simplified by recourse to suitable assumptions

regard**in**g the nature of the atomic **in**teractions. Atoms **in** a crystal can be grouped

by shells, accord**in**g to their distance to a reference atom. Then, nearest-neighbor

**in**teractions refer to **in**teractions extend**in**g to the first shell of atoms; next-to-nearest

neighbor **in**teractions consist of **in**teractions extend**in**g to the second shell of atoms;

**and** so on. Some empirical potentials restrict the range of **in**teractions between atoms

to a few atom shells. For **in**stance, the simplest implementations of the EAM account

for nearest-neighbor **in**teractions only. Another common set of restrictions underly**in**g

empirical potentials concerns the extent of multi-body **in**teractions. Thus, the

energy function E(y) is said to consist of two-body **in**teractions if it can be written

as a sum over energy functions depend**in**g on the positions of pairs of atoms; it

is said to consist of three-body **in**teractions if E(y) can be written as a sum over

functions depend**in**g of the positions of three atoms; **and** so on. For **in**stance, the

Lennard-Jones potential accounts for pairwise **in**teractions only.

The energy function E(y) must be material-frame **in**different, i.e., **in**variant

under superposed rigid-body motions. Thus we must have

E(Ry + c) = E(y) (99)

for all R ∈ SO(n) **and** c ∈ R n . Let y 1 ∼ y 2 if **and** only if y 2 = y 1 + c for some

c ∈ R n . Then ∼ is an equivalence relation, **and** translation **in**variance requires that E

be expressible as a function of the quotient space 0 (X, R n )/ ∼. But 0 (X, R n )/ ∼

may be identified with the range of 0 (X, R n ) by d. Thus, if y 1 ∼ y 2 then clearly

dy 1 = dy 2 . Conversely, if dy 1 = dy 2 , then y 1 **and** y 2 differ by a constant. Hence

it follows that the energy function is expressible as a function of the deformation

dy : E 1 → R n . We shall denote E(dy) the result**in**g energy function. Rotation

**in**variance further requires that

E(Rdy) = E(dy) (100)

for all R ∈ SO(n), i.e., the function E(dy) is isotropic. From Cauchy’s representation

formula for isotropic functions of an arbitrary number of vectors (cf., e.g., [49])

it follows that the energy must be expressible as a function E(C) of the discrete

right Cauchy-Green deformation field C : E 1 × E 1 → R def**in**ed as

C(e 1 ,e ′ 1 ) = dy(e 1) · dy(e ′ 1 ) (101)

for every pair of 1-cells e 1 **and** e ′ 1

. Note that no assumption of locality is implied

**in** the representation E(C), which **in** pr**in**ciple may depend on the value of C for

all pairs of 1-cells.

The stable configurations of **in**terest are the m**in**imizers of E(y), possibly over

a f**in**ite doma**in** subject to appropriate boundary conditions **and** under the action of

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 187

applied loads. However, the energy l**and**scape def**in**ed by E(y) is typically of great

complexity, with numerous energy wells, e.g., correspond**in**g to lattice-**in**variant

deformations. In order to make progress we shall resort to two ma**in** approximations,

namely, the harmonic approximation **and** the eigendeformation formalism.

A harmonic crystal is a crystal whose energy is quadratic **in** the displacement

field u(e 0 ) = y(e 0 ) − x(e 0 ). We may obta**in** the harmonic approximation of the

energy by exp**and****in**g the function E(x + u) **in** the Taylor series of u. The result is

a quadratic function of the form

∫

1

E(u) =

∫E 1

∫E 1

∫E 1 E 1

2 C(e 1,e 1 ′ ,e′′ 1 ,e′′′ 1 )ε(e 1,e 1 ′ )ε(e′′ 1 ,e′′′ 1 )≡ 1 〈Cε,ε〉, (102)

2

where ε : E 1 × E 1 → R, def**in**ed as

ε(e 1 ,e 1 ′ ) = 1 [

du(e1 ) · dx(e 1 ′ 2

) + dx(e 1) · du(e 1 ′ )] = ε(e 1 ′ ,e 1), (103)

is the discrete stra**in** field. We shall also denote by ε(u) the operator that maps

a displacement to its correspond**in**g stra**in** field. Then, the energy (102) can be

rewritten as

E(u) = 1 〈Cε(u), ε(u)〉. (104)

2

The coefficients C : E1 4 → R are harmonic moduli which measure the strength of

the **in**teractions between 1-cells of the lattice. Ow**in**g to the symmetry of the stra**in**

field, **in** (102) we may suppose

C(e 1 ,e 1 ′ ,e′′ 1 ,e′′′ 1 ) = C(e′ 1 ,e 1,e 1 ′′ ,e′′′ 1 ) = C(e 1,e 1 ′ ,e′′′ 1 ,e′′ 1 ) (105)

without loss of generality. We verify that the discrete stra**in** field is **in**deed **in**variant

under superposed rigid-body displacements, i.e., under transformations of the

form u → u + c + ωx, c ∈ R n , ω ∈ so(3), **and** therefore the energy (102) is

**in**f**in**itesimally material-frame **in**different. Insert**in**g (103) **in**to (102) we obta**in** the

representation

where

E(u) =

∫E 1

∫

1

E 1

2 du(e 1) · B(e 1 ,e 1 ′ )du(e′ 1 ) ≡ 1 〈Bdu,du〉, (106)

2

∫

B(e 1 ,e 1 ′ ) = C(e 1 ,e 1 ′′ ,e′ 1 ,e′′′ 1 )dx(e′′ 1 ) ⊗ dx(e′′′ 1 ). (107)

E 1

In addition, us**in**g the representation (73) of the differential we obta**in**

where

E(u) =

∫E 0

∫

E 1

∫

A(e 0 ,e ′ 0 ) = ∫

1

E 0

2 u(e 0) · A(e 0 ,e 0 ′ )u(e′ 0 ) ≡ 1 〈Au, u〉, (108)

2

E 1

∫

L(e 0 ,e 1 )L(e 0 ′ ,e′ 1 )B(e 1,e 1 ′ ). (109)

E 1

188 M. P. Ariza & M. Ortiz

The function A : E 2 0 → symRn×n collects the harmonic force constants of the

crystal. All moduli aris**in**g **in** the preced**in**g representations of the energy must be

translation **and** symmetry **in**variant. In addition, we have the symmetry relations

C(e 1 ,e 1 ′ ,e′′ 1 ,e′′′ 1 ) = C(e′′ 1 ,e′′′ 1 ,e 1,e 1 ′ ), (110a)

B(e 1 ,e 1 ′ ) = BT (e 1 ′ ,e 1),

(110b)

A(e 0 ,e 0 ′ ) = AT (e 0 ′ ,e 0).

(110c)

F**in**ally, by translation **in**variance we have the representations

Bdu = ∗ du,

Au = ∗ u,

(111a)

(111b)

where **and** are the force-constant fields of the lattice. S**in**ce (111a) **and** (111b)

are **in** convolution form, an application of Parseval’s identity **and** the convolution

theorem yields the additional representations:

E(u) = 1

(2π) n ∫

[−π,π] n 1

2 〈 ˆ(θ)û(θ), û ∗ (θ)〉 dθ, (112a)

E(u) = 1 ∫

1

(2π) n [−π,π] n 2 〈 ˆ(θ)P ∗ (θ)û(θ), P (θ)û ∗ (θ)〉 dθ, (112b)

which will be extensively used **in** subsequent analyses.

Tak**in**g variations of (102) we obta**in**

where

Ė =

∫E 1

∫

σ(e 1 ,e ′ 1 ) = ∫

or, **in** **in**variant notation,

σ(e 1 ,e 1 ′ )˙ε(e 1,e 1 ′ ) ≡〈σ, ˙ε〉, (113)

E 1

E 1

∫

E 1

C(e 1 ,e ′ 1 ,e′′ 1 ,e′′′ 1 )ε(e′′ 1 ,e′′′ 1 ) (114)

σ = Cε (115)

is the discrete stress field. This relation may be regarded as a discrete version of

Hooke’s law. We note that the stress field is symmetric, i.e.,

σ(e 1 ,e ′ 1 ) = σ(e′ 1 ,e 1). (116)

Insert**in**g (103) **in** (113) we further obta**in**

∫

Ė = τ(e 1 ) · d ˙u(e 1 ) ≡〈τ,d ˙u〉,

E 1

(117)

where

τ(e 1 ) =

∫

σ(e 1 ,e 1 ′ )dx(e′ 1 )

E 1 (118)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 189

or, **in** **in**variant notation,

τ = σdx (119)

is an **in**ternal force field. The equilibrium equation is, therefore,

δτ = 0. (120)

The equilibrium conditions across the boundary of a doma**in** can be ascerta**in**ed

likewise. Let A ⊂ E 1 . Then (117) can be split as

∫

∫

Ė = τ(e 1 ) · d ˙u(e 1 ) + τ(e 1 ) · d ˙u(e 1 )

A

E 1 −A

∫

∫

= χ A τ(e 1 ) · d ˙u(e 1 ) + χ E1 −Aτ(e 1 ) · d ˙u(e 1 ), (121)

E 1 E 1

**and** stationarity requires that

δ(χ A τ)(e 0 ) + δ(χ E1 −Aτ)(e 0 ) = 0 (122)

which, for vertices e 0 on the boundary ∂A, may be regarded as a discrete traction-equilibrium

equation. An equation of equilibrium **in** terms of the displacement

field can be obta**in**ed as follows. From (120) we have that, for any test field

v ∈ 0 (X, R n ),

0 =〈δτ, v〉 =〈τ,dv〉=〈σdx,dv〉=〈Cε(u), ε(v)〉 =〈Au, v〉 (123)

**and**, hence, the equations of equilibrium take the form

δτ = Au = 0, (124)

which can also be derived by tak**in**g variations of (104) directly.

Further restrictions on the force constants, which facilitate their identification,

are obta**in**ed from macroscopic properties of the lattice such as the elastic moduli.

In order to make this connection, consider a stress field σ : E 1 × E 1 → R over the

lattice **and** let τ : E 1 → R n be the correspond**in**g **in**ternal force field. Suppose **in**

addition that an **in**cremental deformation ˙u : E 0 → R n is applied to the stressed

lattice. Let Y be a subcomplex of X consist**in**g of n-cells **and** their faces, henceforth

referred to as a sub-body, **and** let |Y | be its volume. The tractions on the boundary

of Y are given by δ(χ E1 (Y )τ), equation (122), **and** their deformation power is

∫

Ė(Y) = δ(χ E1 (Y )τ)(e 0 ) ·˙u(e 0 )

E

∫ 0

= (χ E1 (Y )τ)(e 1 ) · d ˙u(e 1 )

E

∫ 1

= τ(e 1 ) · d ˙u(e 1 )

E 1 (Y )

∫ ∫

= σ(e 1 ,e 1 ′ )dx(e′ 1 ) · d ˙u(e 1)

E 1 (Y ) E

∫ ∫ 1

= σ(e 1 ,e 1 ′ )˙ε(e 1,e 1 ′ ). (125)

E 1 (Y ) E 1

190 M. P. Ariza & M. Ortiz

Suppose that the **in**cremental deformation is of the form d ˙u = ˙¯εdx, where ˙¯ε ∈

symR n×n represents a uniform macroscopic stra**in** rate. For this particular case we

have

{∫

}

Ė(Y) =

σ(e 1 ,e 1

∫E ′ )dx(e 1) ⊗ dx(e 1 ′ ) · ˙¯ε (126)

1

E 1 (Y )

**and**, hence, the average macroscopic stress over Y is

¯σ(Y) = 1 ∫ ∫

σ(e 1 ,e 1 ′ |Y |

)dx(e 1) ⊗ dx(e 1 ′ )

E 1 (Y ) E 1

= 1 ∫

τ(e 1 ) ⊗ dx(e 1 ), (127)

|Y |

E 1 (Y )

so that Ė(Y) =|Y |¯σ(Y)· ˙¯ε. Assume now that the state of stress is itself the result

of a uniform macroscopic stra**in** ¯ε ∈ symR n×n , i.e.,

{∫ ∫

}

σ(e 1 ,e 1 ′ ) = C(e 1 ,e 1 ′ ,e′′ 1 ,e′′′ 1 )dx(e′′ 1 ) ⊗ dx(e′′′ 1 ) ¯ε, (128)

E 1 E 1

which follows from (114) by sett**in**g du =¯εdx. S**in**ce, **in** this case, the state of

stress **and** deformation of the lattice is uniform, it follows that, for sufficiently large

Y , ¯σ(Y)converges to

where

¯σ ij = c ij kl ¯ε kl , (129)

c ij kl ∼ 1 ∫

∫

C(e 1 ,e 1

|Y | E 1 (Y )

∫E 1

∫E ′ ,e′′ 1 ,e′′′ 1 )

1 E 1

dx i (e 1 )dx j (e 1 ′ )dx k(e 1 ′′ )dx l(e 1 ′′′ )

∼ 1

|Y |

∫

E 1 (Y )

∫

E 1

B ik (e 1 ,e ′′

1 )dx j (e 1 )dx l (e ′′

1 ) (130)

are the elastic moduli of the lattice. We note that the requisite major **and** m**in**or

symmetries c ij kl = c klij = c jikl = c ij lk of the elastic moduli follow directly from

(105).

An alternative expression for the elastic moduli **in** terms of force constants may

be obta**in**ed as follows: **in**sert representation (73) of dx **in**to (130) to obta**in**

c ij kl ∼ 1 ∫ ∫

L(e 0 ,e 1 )L(e 0

|Y |

∫E 0

∫E ′ ,e′ 1 )B ik(e 1 ,e 1 ′ )x j (e 0 )x l (e 0 ′ ). (131)

0 E 1 (Y ) E 1

Us**in**g (109) we have, asymptotically for large Y ,

c ij kl ∼ 1 ∫ ∫

A ik (e 0 ,e 0 ′ |Y |

)x j (e 0 )x 1 (e 0 ′ ) (132)

E 0 (y) E 0

or, equivalently,

c ij kl ∼− 1 ∫ ∫

A ik (e 0 ,e 0 ′ 2|Y |

)(x j (e 0 ) − x j (e 0 ′ ))(x l(e 0 ) − x l (e 0 ′ )) (133)

E 0 (Y ) E 0

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 191

For a simple lattice, by translation **in**variance **and** representation (111b) we have,

explicitly,

c ij kl = −1 ∑

ik (l)x j (l)x l (l), (134)

2

l∈Z n

where the elastic moduli are now expressed simply **in** terms of the force constants

of the lattice.

It is **in**terest**in**g to note that the m**in**or symmetries of c ij kl are not evident **in**

(134). Indeed, those m**in**or symmetries do not hold for arbitrary choices of the

force-constant field , but only for force-constant fields of the form (109) dictated

by material-frame **in**difference. The restrictions imposed by material-frame

**in**difference on the force constants have been addressed **in** [43, 45], but have not

always been recognized **in** the literature, where the force constants are often mistakenly

regarded as fundamental **and** subject to no restrictions other than material

symmetry.

4.2. Equilibrium problem

Suppose now that a crystal lattice is acted upon by a distribution of forces

f ∈ 0 . We wish to determ**in**e the equilibrium configurations of the crystal, i. e.,

the solutions of the equilibrium equation

Au = f (135)

under appropriate boundary **and** forc**in**g conditions. Problems aris**in**g **in** applications

frequently fall **in**to the follow**in**g categories: (i) f has bounded support **and**

u decays to 0 at **in**f**in**ity; (ii) f **and** u are periodic; (iii) f **and** u are periodic **in**

certa**in** directions, **and** f has bounded support **and** u decays to 0 at **in**f**in**ity **in** the

rema**in****in**g directions. Examples of problems of the first type **in**clude the deformation

of a crystal under the action of an applied po**in**t load or **in** the presence of

a dilatation po**in**t. Taylor dislocation lattices fall **in** the second class of problems,

straight dislocations **in** the third.

The equilibrium equation (135) may be analyzed – **and** its solutions characterized

– by means of the fundamental solution formalism (cf., e.g., [42]). Recall that,

by translation **in**variance, the equilibrium equation can be expressed **in** convolution

form as

∗ u = f, (136)

where is the force-constant field of the lattice. The follow**in**g proposition covers

a broad class of problems often encountered **in** applications.

Proposition 2. Suppose that

(i) has f**in**ite support;

(ii) ˆ −1 ∈ L 1 ([−π, π] n );

(iii) f has f**in**ite support.

192 M. P. Ariza & M. Ortiz

Let

G(l) = 1

(2π) n ∫

[−π,π] n ˆ −1 (θ)e iθ·l dθ. (137)

Then,

is a solution of the equilibrium equation (136).

u = G ∗ f (138)

Note that assumptions (i) **and** (iii) state that **and** f are zero everywhere except

on a f**in**ite set of vertices, whereas assumption (ii) implies that ˆ is **in**vertible almost

everywhere **in** [−π, π] n **and** its **in**verse is **in**tegrable. The function G is the fundamental

solution, or Green’s function, of the lattice. Condition (ii) of the proposition

is easily verified directly for specific force-constant models. Clearly, solutions need

not be unique **in** general, e.g., they may be determ**in**ed up to translations **and** **in**f**in**itesimal

rotations.

Proof. We verify that, s**in**ce ˆ −1 (θ)e iθ·l ∈ L 1 ([−π, π] n ), G(l) **in** (137) is def**in**ed

for all l ∈ Z n . In addition,

( ∗ G)(l) = ∑

(l − l ′ )G(l ′ )

l ′ ∈Z n

= ∑

l ′ ∈Z n (l − l ′ )

{ 1

(2π) n ∫

[−π,π] n

}

ˆ −1 (θ)e iθ·l dθ . (139)

S**in**ce the sum over l extends over a f**in**ite number of terms, we have

( ∗ G)(l) = 1 ∫

{ }

∑

ˆ −1

(2π) n (θ) (l − l ′ )e iθ·l dθ

[−π,π] n

l ′ ∈Z n

= 1 ∫

(2π) n Ie iθ·l dθ

[−π,π] n

= Iδ(l), (140)

where

δ(l) =

{

1 ifl = 0,

0 otherwise,

(141)

**and** I is the identity **in** R n×n . Also, we note that (138) is well def**in**ed s**in**ce by

assumption f has f**in**ite support. Then

( ∗ u)(l) = ∑ { }

∑

(l − l ′ ) G(l ′ − l ′′ )f (l ′′ ) . (142)

l ′ ∈Z n l ′′ ∈Z n

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 193

S**in**ce both sums are f**in**ite, we have

( ∗ u)(l) = ∑ { }

∑

(l − l ′ )G(l ′ − l ′′ ) f(l ′′ )

l ′′ ∈Z n l ′ ∈Z n

= ∑

δ(l − l ′′ )f (l ′′ )

l ′′ ∈Z n

= f(l), (143)

which shows that u is **in**deed a solution of (136). ⊓⊔

The solution of the periodic problem is elementary. Suppose that the periodic

unit cell of the problem is **in** the form of a set Y ⊂ Z n , **and** that the translates

{Y + L i A i ,L∈ Z n } def**in**e a partition of Z 3 for some translation vectors

A i ∈ Z n , i = 1,...,n. Let A i be the correspond**in**g dual basis, **and** B i = 2πA i

the reciprocal basis. Us**in**g the convolution theorem for periodic lattice functions,

equation (A.293), the equilibrium equations (136) reduce to the f**in**ite problem

ˆ(θ)û(θ) = f(θ), ˆ θ ∈ Z, (144)

where Z is the **in**tersection of the lattice spanned by B i **and** [−π, π] n , **and** f(θ) ˆ

is given by (A.289). Suppos**in**g that ˆ(θ) is **in**vertible for all θ ̸= 0, the sole

rema**in****in**g difficulty **in** solv**in**g (144) is that ˆ(θ) necessarily vanishes at the orig**in**.

Thus, the solvability of problem (144) necessitates fˆ

0 = 0. In view of (A.294), this

condition simply amounts to the requirement that the average force 〈f 〉 be zero,

i.e., the applied forces f(l)must be **in** static equilibrium. The solution û(θ) is then

**in**determ**in**ate at θ = 0, i.e., the lattice displacement field u(l) is determ**in**ed up

to a rigid translation. This situation can also be understood from the st**and**po**in**t of

the Fredholm alternative. Thus, consider the operator (T û)(θ) = ˆ(θ)û(θ). The

kernel N (T ) is the space of rigid translations. For problem (144) to have solutions,

fˆ

must be orthogonal to N (T ), i.e., fˆ

0 must be zero or, equivalently, f(l) must

have zero average. If this condition is satisfied, the solution is determ**in**ed modulo

N (T ), i.e., up to a rigid translation. If û(θ) is a solution of (144), the correspond**in**g

lattice displacement field u(l) follows by an application of the **in**verse discrete

Fourier transform formula (A.290).

Conditions for existence **and** uniqueness of the solutions of the equilibrium

problem follow from st**and**ard theory (e.g., [10]). To this end, recall that the space

X = H 1 (R n )/R n , def**in**ed as the space of equivalence classes with respect to

the relation: u ∼ v ⇔ u − v is a constant, is a Banach space under the norm

|| ˙u|| X = ||∇u|| L 2, where u ∈˙u, ˙u ∈ X (cf., e.g., [8], proposition 3.40). For present

purposes, it is advantageous to express the equilibrium problem **in** variational form.

To this end, let f ∈ X ∗ **and** let

{

E(u) −〈f, u〉 if supp(û) ∈[−π, π] n ,

F (u) =

(145)

+∞ otherwise,

be the potential energy of the crystal, where the energy E(u) is def**in**ed as **in** (112a).

Then, we wish to f**in**d the m**in**imum

m X (F ) = **in**f F (u) (146)

u∈X

194 M. P. Ariza & M. Ortiz

**and** the set of m**in**imizers

M X (F ) ={u ∈ X s. t. F (u) = m X (F )}. (147)

The follow**in**g proposition collects st**and**ard conditions guarantee**in**g the existence

**and** uniqueness of solutions of problem (146).

Proposition 3. Suppose that ˆ is measurable **and** that there exists a constant C>0

such that

〈 ˆ(θ)ζ, ζ ∗ 〉 ≧ C|θ| 2 |ζ | 2 , ∀θ ∈[−π, π] n ,ζ∈ C n . (148)

Then F has a unique m**in**imum **in** X.

Proof. Under the assumptions, F is sequentially coercive **in** X. In fact, the sets

{F ≦ t} are bounded **in** the reflexive space X **and** therefore relatively compact **in** its

weak topology. In addition, F is cont**in**uous **in** the strong topology of X **and** convex,

**and** therefore sequentially lower semicont**in**uous **in** the weak topology of X. Hence,

F has a m**in**imum **in** X. The uniqueness of the m**in**imizer follows directly from the

strict convexity of F . Thus, let u, v ∈ M X (F ) so that F (u) = F(v) = m X (F ). If

u ̸= v we have

( u

F

2 + v )

< 1 2 2 F (u) + 1 2 F(v) = m X(F ), (149)

which is impossible. Hence u = v.

⊓⊔

The results cited **in** this proof are st**and**ard (e.g., [10], Examples 1.14, 1.22 **and**

1.23, Propositions 1.18 **and** 1.20, Theorem 1.15).

4.3. Cont**in**uum limit

In this section we study the cont**in**uum limit of the equilibrium problem (135),

i.e., the behavior of the lattice as the lattice parameter is allowed to become vanish**in**gly

small, or, equivalently, **in** the long wavelength limit. It is widely appreciated

that harmonic lattices behave like elastic cont**in**ua **in** that limit. A common way

of underst**and****in**g this limit**in**g behavior is to refer to the asymptotic form of the

dynamical matrix, or, equivalently, of the phonon dispersion relation as the wave

number k → 0. Here, **in**stead, we seek to underst**and** the cont**in**uum limit **in** the

sense of the Ɣ-convergence of the energy functional (104). As it is well known,

Ɣ-convergence of the energy functional also guarantees convergence of the energy

m**in**imizers under rather general conditions (cf., e.g., [10], Corollary 7.24). For

simplicity, throughout this section we restrict our attention to simple lattices.

We beg**in** by adopt**in**g the wave number representation of the Fourier transform

(cf. Section A.2), which is more natural **in** the cont**in**uum sett**in**g. In this representation

the energy (112a) takes the form

E(u) = 1 1

(2π)

∫B

n 2 〈D(k)û(k), û∗ (k)〉 dk, (150)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 195

where

D(k) =

1 ˆ(k) 2

= 1 ∑

(l)e −ik·x(l) (151)

l∈Z n

is the dynamical matrix of the lattice. For a simple lattice **and** assum**in**g sufficient

differentiability, the properties

D T (k) = D ∗ (k),

D(−k) = D T (k),

D(0) = 0,

∂ k D(0) = 0

(152a)

(152b)

(152c)

(152d)

follow simply from reciprocity, translation **in**variance **and** the centrosymmetry of

the lattice. In particular, property (152a) implies that D(k) is hermitian. The behavior

of D(k) for small k is of particular importance **in** the cont**in**uum limit. From

(151), (152c), (152d) **and** (134) we obta**in**

lim

ε→0 ε−2 D ik (εk) = lim ε −2 1 ∑

(l)e −iεk·x(l) =c ij kl k j k l ≡ (D 0 ) ik (k), (153)

ε→0

l∈Z n

where D 0 is the dynamical matrix of the l**in**ear elastic solid of moduli c ij kl .

Consider now the sequence of functions E ε : H 1 (R n ) → ¯R def**in**ed as:

{ 1

E ε (u) =

(2π)

∫B/ε 1 n 2 〈ε−2 D(εk)û(k), û ∗ (k)〉 dk if supp(û) ∈ B/ε,

(154)

+∞ otherwise,

obta**in**ed by scal**in**g down the lattice size by a factor of ε. We wish to ascerta**in** the

limit**in**g behavior of the sequence E ε as ε → 0. We surmise that the likely limit is

the l**in**ear elastic energy

∫

E 0 (u) =

R n 1

2 c ij klu i,j u k,l dx = 1

(2π) n ∫

R n 1

2 c ij klk j k l û i û ∗ k

dk, (155)

where c ij kl are the elastic moduli of the material **and** a comma denotes partial differentiation.

Simple conditions under which this limit is **in**deed realized are provided

by the follow**in**g proposition.

Proposition 4. Suppose that:

(i) for every ζ ∈ C n the function 〈D(·)ζ, ζ ∗ 〉 is measurable on B;

(ii) there is a constant C such that

for a. e. k ∈ B **and** for every ζ ∈ C n ;

0 ≦ 〈D(k)ζ, ζ ∗ 〉 ≦ C|k| 2 |ζ | 2 (156)

196 M. P. Ariza & M. Ortiz

(iii) for every ζ ∈ C n , the functions ε −2 〈D(εk)ζ, ζ ∗ 〉 converge for a. e. k to

〈D 0 (k)ζ, ζ ∗ 〉=c ij kl k j k l ζ i ζk ∗ (157)

Then,

Ɣ − lim E ε (u) = E 0 (u) (158)

ε→0

**in** the weak topology of H 1 (R n ).

Proof. Let u ∈ H 1 (R n ), **and** let u ε the sequence of H 1 (R n )-functions converg**in**g

to u obta**in**ed by restrict**in**g û to B/ε (cf. Section A.4.1). Then,

E ε (u ε ) = 1 1

(2π)

∫B/ε

n 2 〈ε−2 D(εk)û(k), û ∗ (k)〉 dk. (159)

By assumptions (ii)–(iii) **and** dom**in**ated convergence, it follows that

∫

∫

〈ε −2 D(εk)û(k), û ∗ (k)〉dk = 〈D 0 (k)û(k), û ∗ (k)〉dk (160)

lim

ε→0

B/ε

**and**, hence,

lim E ε(u ε ) = E 0 (u). (161)

ε→0

Let now u ε ⇀u**in** H 1 (R n ). We need to show that

E 0 (u) ≦ lim **in**f E ε(u ε ). (162)

ε→0

Suppose that lim **in**f ε→0 E ε (u ε )

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 197

If D(k) is sufficiently smooth at the orig**in** then, by (152c) **and** (152d),

lim

ε→0 ε−2 D ik (εk) = 1 ∂ 2 D ik

(0)k j k l . (167)

2 ∂k j ∂k l

Compar**in**g this expression with (153) we f**in**d the identity

c ij kl + c ilkj = ∂2 D ik

∂k j ∂k l

(0), (168)

which provides an alternative means, equivalent to (134), of comput**in**g the elastic

moduli of the lattice. Identity (168) shows that, as expected, the elastic moduli fully

describe the behavior of the lattice **in** the cont**in**uum limit.

If the first of **in**equalities (156) can be strengthened to

C|k| 2 |ζ | 2 ≦ 〈D(k)ζ, ζ ∗ 〉 (169)

for some constant C>0, for all ε>0, a. e. k ∈ B **and** for every ζ ∈ C n , then

the sequence of functions E ε (u) is equicoercive **in** the space X = H 1 (R n )/R n

def**in**ed **in** Section 4.2. If, **in** addition, the conditions of Theorem 3 are satisfied,

then the unique m**in**imizer of E ε (u) converges weakly **in** X to the unique m**in**imizer

of E 0 (u) (cf., e.g., [10] for the connection between equicoercivity **and** convergence

of m**in**imizers).

4.4. Simple illustrative examples

The consideration of complex **in**teratomic potentials describ**in**g specific materials

is beyond the scope of this paper. Therefore, we shall conf**in**e our attention

to two simple examples, namely, the square **and** simple cubic lattices with nearest

neighbor **in**teractions, ma**in**ly for purposes of illustration **and** for develop**in**g **in**sight

**and** **in**tuition.

4.4.1. The square lattice. As a first simple example we consider the case of a

square lattice undergo**in**g anti-plane shear. The lattice complex is as described **in**

Section 2.3.2. In the nearest-neighbor approximation, the s**in**gle non-zero harmonic

force constant is C(e 1 ,e 1 ) = µ/a, where µ is the shear modulus of the crystal.

The rema**in****in**g nonzero force constants (111a) **and** (111b) are

( ) ( )

0 0

= = µa, (170)

1, 1 2, 2

**and**

(1, 0) = (0, 1) = (−1, 0) = (0, −1) =−µa, (171a)

(0, 0) =−(1, 0) − (0, 1) − (−1, 0) − (0, −1) = 4µa. (171b)

198 M. P. Ariza & M. Ortiz

Here ( l

α,β)

designates the force constant that couples du(0,α) to du(l, β), **and**

(l) designates the force constant that couples u(0) **and** u(l). A straightforward

computation yields

(

ˆ(θ) = 4µa s**in** 2 θ 1

2 + θ )

s**in**2 2

, (172)

2

whence the dynamical matrix follows as

D(k) = 4µ (

a 2 s**in** 2 k 1a

2 + k )

2a

s**in**2 , (173)

2

which **in** this simple example reduces to a scalar. We verify that (1/2)∂ k1 ∂ k1 D(0) =

µ **and** (1/2)∂ k2 ∂ k2 D(0) = µ, as required.

4.4.2. The simple cubic lattice. Consider now the simple cubic lattice complex

described **in** Section 2.3.4. In the nearest-neighbor approximation, the non-zero

harmonic force constants are: C(e 1 ,e 1 ,e 1 ,e 1 ) ≡ F ; C(e 1 ,e 1 ,e

1 ′ ,e′ 1

) ≡ G; **and**

C(e 1 ,e

1 ′ ,e 1,e

1 ′ ) ≡ H . Here e 1 **and** e

1 ′ denote 1-cells hav**in**g a common vertex. Thus,

F is the strength of the pairwise component of the energy, G measures the strength

of the coupl**in**g between the elongations of adjacent 1-cells, or Poisson effect, **and**

H measures the strength of the bond-angle **in**teractions. The correspond**in**g forceconstant

fields are

⎛

⎞

( ) F 0 0

0

= a 2 ⎝ 0 4H 0 ⎠ ,

(174a)

1, 1

0 0 4H

( ) 0

1, 2

⎛ ⎞

0 G 0

= a 2 ⎝ H 00⎠ ,

0 0 0

(174b)

**and**

⎛

⎞

( 1, 0, 0 ) −F 0 0

= a 2 ⎝ 0 −4H 0 ⎠ ,

(175a)

0 0 −4H

⎛

⎞

( 1, 1, 0 ) 0 −G − H 0

= a 2 ⎝ −H − G 0 0⎠ ,

(175b)

0 0 0

⎛

⎞

( 0, 0, 0 ) 2F + 16H 0 0

= a 2 ⎝ 0 2F + 16H 0 ⎠ , (175c)

0 0 2F + 16H

whence all the rema**in****in**g force constants follow by symmetry. Lengthy but straightforward

calculations yield

ˆ 11 (θ) = a

[4F 2 s**in** 2 θ (

1

2 + 16H s**in** 2 θ 2

2 + θ )]

s**in**2 3

,

2

(176a)

ˆ 12 (θ) = 4a 2 (G + H)s**in** θ 1 s**in** θ 2 ,

(176b)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 199

whence the dynamical matrix follows as

D 11 (k) = 1 [

4F s**in** 2 k (

1a

a 2 + 16H s**in** 2 k 2a

2 + k )]

3a

s**in**2 , (177a)

2

D 12 (k) = 4 a (G + H)s**in**(k 1a)s**in**(k 2 a)

(177b)

with all rema**in****in**g components follow**in**g by symmetry. F**in**ally, the elastic moduli

follow from (168) as

c 11 = aF,

c 12 = 4aG,

c 44 = 4aH.

(178a)

(178b)

(178c)

Thus, for this simple model all the forces constants can be identified directly from

the elastic moduli **and** the lattice parameter.

5. Eigendeformation theory of crystallographic slip

The total energy of a crystal is **in**variant under lattice-**in**variant deformations

(cf. Section 2.1) **and**, therefore, it is a nonconvex function of the atomic displacements.

This lack of convexity **in** turn permits the emergence of lattice defects such

as dislocations. A seem**in**g deficiency of the harmonic approximation described

earlier – which would appear to disqualify it as a suitable framework for the study

of dislocations – is that the energy (108) is a convex function of the displacements

**and**, **in** particular, not **in**variant under lattice-**in**variant deformations. However, this

limitation can be overcome by recourse to the theory of eigendeformations (cf.,

e.g., [36]), which we proceed to formulate **in** this section.

5.1. The energy of a plastically deformed crystal

Consider a uniform lattice-**in**variant deformation of the form (16), correspond**in**g

to a homogeneous crystallographic slip on planes of normal m i a i through a

translation vector s j a j . This uniform deformation has the matrix representation

F = I + ξ b ⊗ m, (179)

d

where m is the unit normal to the slip plane, b is the Burgers vector, d is the distance

between consecutive slip planes, **and** ξ ∈ Z is the slip amplitude **in** quanta of

Burgers vector. The correspond**in**g l**in**earized eigendeformation 1-form β ∈ 1 is

β(e 1 ) ≈ (F − I)dx(e 1 ) = (dx(e 1 ) · m) ξb. (180)

We note that, by a suitable choice of lattice complex, 1-cells either lie on a slip

plane, **in** which case dx(e 1 ) · m = 0, or jo**in** two consecutive slip planes, **in** which

200 M. P. Ariza & M. Ortiz

case dx(e 1 ) · m = d. Let E 1 (m) be the set of 1-cells of the second type, i.e.,

E 1 (m) ={e 1 ∈ E 1 , s. t. dx(e 1 ) · m = d}. Then we have the representation

β =

∑

ξbe 1 . (181)

e 1 ∈E 1 (m)

The general eigendeformation 1-form result**in**g from the activation of the slip system

def**in**ed by b **and** m is obta**in**ed by localiz**in**g (181) 1-cell by 1-cell, with the

result

β =

∑

ξ(e 1 )b e 1 , (182)

e 1 ∈E 1 (m)

where now ξ is an **in**teger-valued function on E 1 (m). Suppose now that crystallographic

slip can take place on N crystallographic systems def**in**ed by Burgers

vectors **and** normals (b s ,m s ), s = 1,...,N, respectively. The result**in**g eigendeformation

1-form is

β =

N∑

∑

s=1 e 1 ∈E 1 (m s )

ξ s (e 1 )b s e 1 , (183)

where ξ s : E 1 (m s ) → Z is the slip field correspond**in**g to slip system s. We shall

assume that exact, or compatible, eigendeformations of this general type cost no

energy. A simple device for build**in**g this feature **in**to the theory is to assume an

elastic energy of the form

E(u, ξ) = 1 〈B(du − β), du − β〉, (184)

2

which replaces representation (106) **in** the presence of eigendeformations. Clearly,

if β = dv, i.e., if the eigendeformations are exact, or compatible, then the energym**in**imiz**in**g

displacements are u = v **and** E = 0. However, because slip is crystallographically

constra**in**ed, β must necessarily be of the form (183) **and**, therefore,

is not compatible **in** general. By virtue of this lack of compatibility, a general distribution

of slip **in**duces residual stresses **in** the lattice **and** a nonvanish**in**g elastic

energy, or stored energy.

A compell**in**g geometrical **in**terpretation of eigendeformations can be given with

the aid of a discrete version of the Nye dislocation density tensor [38], namely,

α = dβ. (185)

This simple relation may be regarded as the discrete version of Kröner’s formula

(6) (see [26]). From (185) it follows immediately that

dα = 0, (186)

which generalizes the conservation of the Burgers vector identity (7). Note that α

is a 2-form, i.e., α ∈ 2 . Clearly, if the eigendeformations are compatible, i.e., if

β = dv, then α = d 2 v = 0. Thus, the dislocation density 2-form α measures the

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 201

degree of **in**compatibility of the eigendeformations. Furthermore, it follows from

the def**in**ition (90) of the l**in**k**in**g number that

∫

ω ∧ α = L**in**k(ω, β). (187)

E 3

Thus, suppose that α is supported on a collection of closed loops, **and** ω is a subset

of dχ A , where χ A is the characteristic cha**in** of a subset A ∈ E 0 . Then by the **in**tersection

**in**terpretation of the l**in**k**in**g number it follows that (187) counts the number

of signed **in**tersections of the dislocation loops with ω. In particular, if ω = dχ A ,

then L**in**k(ω, β) = 0, which shows that the number of dislocation loops enter**in**g a

closed surface must necessarily equal the number of dislocation loops exit**in**g the

surface.

Further **in**sight **in**to the geometry of dislocations may be derived by **in**sert**in**g

representation (183) **in**to (185), which yields

α =

N∑

∑

s=1 e 1 ∈E 1 (m s )

ξ s (e 1 )b s δe 1 ≡

N∑

α s . (188)

Thus, each slip system s contributes a certa**in** dislocation density α s to α. Furthermore,

the dislocation density α s has a constant direction b s **and** consists of the

superposition of elementary dislocation loops δe 1 ∈ B 2 with multiplicities ξ s (e 1 ).

Each elementary loop δe 1 is an elementary 2-coboundary. Equivalently, s**in**ce the

lattice complex is assumed to be perfect, the elementary dislocation loops may be

regarded as 2-cocycles surround**in**g the 1-cells of the lattice, hence their designation

as loops.

s=1

5.2. Examples of dislocation systems

A compilation of experimentally observed slip systems metallic crystals may

be found **in** [19]. Thus, for **in**stance, FCC crystals are commonly found to deform

plastically through the activation of 12 slip systems consist**in**g of {111} slip planes

**and** [110] slip directions. Likewise, BCC crystals exhibit activity on 24 ma**in** slip

systems consist**in**g of {110} **and** {112} slip planes **and** [111] slip directions. Table 1

collects the slip directions **and** plane normals for the square, hexagonal, simple

cubic, FCC **and** BCC crystals for ease of reference.

In perfect lattices, the sets {∂e 2 (l, α), l ∈ Z n ,α = 1,...,N 2 } **and** {δe 1 (l, α),

l ∈ Z n ,α = 1,...,N 1 } generate Z 1 **and** B 2 , respectively. These 1-cycles **and**

2-coboundaries may be regarded as sets of elementary Burgers circuits **and** dislocation

loops, respectively. All Burgers circuits **and** dislocation distributions can

be obta**in**ed by tak**in**g **in**teger comb**in**ations of the elementary circuits **and** loops,

respectively. In two-dimensional lattices, the elementary loops take the form of

dipoles. The geometry of the elementary circuits **and** dipoles of the square **and**

hexagonal lattices is shown **in** Fig. 12 by way of illustration. In three-dimensional

lattices, the elementary dislocation loops consist of r**in**gs of oriented 2-cells **in**cident

on 1-cells. In particular, it is somewhat mislead**in**g – **and** best avoided – to

represent elementary dislocation loops as closed curves **in** space. The geometry of

202 M. P. Ariza & M. Ortiz

Table 1. Slip-system sets **in** Schmid **and** Boas’ nomenclature. The vector m is the unit normal

to the slip plane, **and** s is the unit vector **in** the direction of the Burgers vector of the

system. All vectors are expressed **in** cartesian coord**in**ates.

Slip System A3 B3

Square s [001] [001]

m (100) (010)

Slip System A3 B3 C3

Hexagonal s [001] [001] [001]

2m (0¯20) ( √ 3¯10) ( √ 310)

Slip System A2 A3 B1 B3 C1 C2

SC s [010] [001] [100] [001] [100] [010]

m (100) (100) (010) (010) (001) (001)

FCC

Slip

√

System A2 A3 A6 B2 B4 B5

√ 2s [01¯1] [101] [¯1¯10] [0¯11] [10¯1] [¯110]

3m (¯111) (¯111) (¯111) (111) (111) (111)

Slip

√

System C1 C3 C5 D1 D4 D6

√ 2s [011] [¯10¯1] [1¯10] [0¯1¯1] [¯101] [110]

3m (¯1¯11) (¯1¯11) (¯1¯11) (1¯11) (1¯11) (1¯11)

BCC

Slip

√

System A2 A3 A6 B2 B4 B5

√ 3 s [¯111] [¯111] [¯111] [111] [111] [111]

2 m (0¯11) (101) (110) (0¯11) (¯101) (¯110)

Slip

√

System C1 C3 C5 D1 D4 D6

√ 3 s [11¯1] [11¯1] [11¯1] [1¯11] [1¯11] [1¯11]

2 m (011) (101) (¯110) (011) (¯101) (110)

the elementary Burgers circuits **and** dislocation loops of the cubic, FCC **and** BCC

lattices is shown **in** Fig. 13.

5.3. The stored energy of a fixed distribution of dislocations

If the distribution of eigendeformations is known, **and** **in** the absence of additional

constra**in**ts, the energy of the lattice can be readily m**in**imized with respect

to the displacement field. Suppose that the crystal is acted upon by a distribution

of forces f ∈ 0 (X, R n ). The total potential energy of the lattice is then

F (u, ξ) = E(u, ξ) −〈f, u〉. (189)

M**in**imization of F (u, ξ) with respect to u yields the equilibrium equation

Au = f + δBβ, (190)

where δBβ may be regarded as a distribution of eigenforces correspond**in**g to the

eigendeformations β. The equilibrium displacements, are, therefore,

u = A −1 (f + δBβ) ≡ u 0 + A −1 δBβ, (191)

where u 0 = A −1 f is the displacement field **in**duced by the applied forces **in** the

absence of eigendeformations **and** we write, formally,

A −1 f ≡ G ∗ f (192)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 203

(a)

(b)

(c) (d) (e)

Fig. 12. Elementary dislocation dipoles (generators of the group B 2 ) for the square (a **and** b)

**and** hexagonal lattices (c, d **and** e). The dipoles consist of a dislocation segment po**in**t**in**g **in**to

the plane (shown as ⊗ ), **and** a dislocation segment po**in**t**in**g away from the plane (shown as

⊙ ). The correspond**in**g Burgers circuits (generators of the group Z1 ) are shown as oriented

circles. The 1-cell to which the dipoles are attached is shown as an oriented 1-cell.

(cf. equation (138)). Conditions under which the m**in**imum problem just described

is well-posed **and** delivers a unique energy-m**in**imiz**in**g displacement field have

been given **in** Section 4.2. The correspond**in**g m**in**imum potential energy is

F(β) = 1 2 〈Bβ, β〉−1 2 〈A−1 (f + δBβ), f + δBβ〉

The first two terms

= 1 2 〈Bβ, β〉−1 2 〈A−1 δBβ, δBβ〉−〈A −1 δBβ, f 〉− 1 2 〈A−1 f, f 〉

= 1 2 〈Bβ, β〉−1 2 〈A−1 δBβ, δBβ〉−〈Bβ, du 0 〉− 1 2 〈Au 0,u 0 〉. (193)

E(β) = 1 2 〈Bβ, β〉−1 2 〈A−1 δBβ, δBβ〉 (194)

204 M. P. Ariza & M. Ortiz

(a)

(b)

(c)

(d)

Fig. 13. Elementary dislocation loops (generators of the group B 2 ) for: (a) the simple cubic

lattice; (b) the face-centered cubic lattice; **and** (c) **and** (d) the body-centered cubic lattice. The

rema**in****in**g elementary circuits **and** loops may be obta**in**ed by symmetry from those shown

**in** the figure. The elementary loops consist of r**in**gs of oriented 2-cells **in**cident on a 1-cell.

The oriented 2-cells are the elementary Burgers circuits (generators of the group Z 1 ). The

1-cell to which the elementary dislocation loops are attached is shown as an oriented 1-cell.

**in** (193) give the self-energy of the distribution of lattice defects represented by the

eigendeformation field β, or stored energy; the third term **in** (193) is the **in**teraction

energy between the lattice defects **and** the applied forces; **and** the fourth term **in**

(193) is the elastic energy of the applied forces.

In the theory of cont**in**uously distributed elastic dislocations, a result of Mura

[35] shows that the energy E(β) can **in** fact be expressed directly as a function E(α)

of the dislocation density field α, **and** is **in**dependent of the choice of slip distribution

β used to **in**duce α. In particular, two distributions of slip which differ by an

exact form **and**, hence, represent the same dislocation density, have the same stored

energy. In l**in**ear elasticity, this situation also arises **in** the theory of cut surfaces,

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 205

where an application of Stoke’s theorem shows that the energy is **in**dependent of

the choice of cut [3]. The correspond**in**g discrete analog may be derived as follows.

By virtue of the Hodge-Helmholtz decomposition (81) for perfect lattices we have

the respresentation

β = dv + δ −1 α, (195)

where v = −1 δβ, (cf., equation (82a)). Insert**in**g this representation **in** (194) **and**

redef**in****in**g u−v as u, an altogether identical derivation to that lead**in**g to (193) now

gives

E(α) = 1 2 〈Bδ−1 α, δ −1 α〉− 1 2 〈A−1 δBδ −1 α, δBδ −1 α〉 (196)

for the stored energy of the crystal, **and**

F(α) = E(α) −〈Bδ −1 α, du 0 〉− 1 2 〈Au 0,u 0 〉 (197)

for the potential energy. We thus verify that, as **in** the case of l**in**ear elastic dislocations,

the stored energy correspond**in**g to a distribution of slip β can be expressed

directly **in** terms of the dislocation density α = dβ.

Suppose that the crystal under consideration possesses N slip systems **and** its

eigendeformations admit the representation (183) **in** terms of an **in**teger-valued slip

field ξ ≡{ξ s ,s = 1,...,N}. Because of the **in**teger-valuedness condition, each

component ξ s correspond**in**g to a particular slip system s def**in**es a 1-cocha**in** **and**,

consequently, ξ belongs to the group C1 N . Additionally we recall that values of ξ

s

on 1-cells e 1 such that dx(e 1 ) is conta**in**ed **in** the correspond**in**g slip plane, i.e.,

such that dx(e 1 ) · m s = 0, contribute noth**in**g to the eigendeformation field β **and**,

therefore, may be set to zero for def**in**iteness. This normalization will be left implied

– but is tacitly **in** force – throughout all subsequent developments. Then, the stored

energy (194) can be written **in** the form

where the operator H is def**in**ed by the identity

E(ξ) = 1 〈Hξ,ξ〉, (198)

2

〈Hξ,ξ〉= 1 2 〈Bβ, β〉−1 2 〈A−1 δBβ, δBβ〉, (199)

**in** which β **and** ξ are related through (183). By translation **in**variance we must have

Hξ = ϒ ∗ ξ (200)

for some discrete harden**in**g-moduli field ϒ.Ifξ ∈ l 2 , then (198) admits the Fourier

representation

E(ξ) = 1 ∫

1

(2π) n [−π,π] n 2 〈 ˆϒ(θ)ˆξ(θ), ˆξ ∗ (θ)〉 dθ. (201)

206 M. P. Ariza & M. Ortiz

If, **in** addition, the crystal is acted upon by a force field f , the result**in**g potential

energy is (cf. equation (193))

F(ξ) = E(ξ) −〈τ,ξ〉− 1 2 〈A−1 f, f 〉, (202)

where the forc**in**g field τ follows from the identity

〈τ,ξ〉=〈f, A −1 δBβ〉, (203)

**in** which β **and** ξ are aga**in** related through (183). We note that τ(e 1 ) is the energetic

force conjugate to ξ(e 1 ) **and**, therefore, may be regarded as a collection of N discrete

Peach-Koehler forces, or resolved shear stresses, act**in**g on the 1-cell e 1 . Accord**in**g

to this **in**terpretation, Hξ represents the resolved shear-stress field result**in**g from

a slip distribution ξ, **and** therefore H can be regarded as an atomic-level harden**in**g

matrix (cf., e.g., [9] for an account of the classical notion of a harden**in**g matrix **in**

crystal plasticity).

The preced**in**g framework may be taken as a basis for formulat**in**g abstract versions

of a number of classical problems **in** dislocation mechanics. For **in**stance, a

central problem **in** physical metallurgy **and** crystal plasticity is the characterization

of the dislocation structures that arise as a result of the plastic work**in**g of ductile

crystals (cf., e.g., [17, 27]). Insight **in**to those structures can sometimes be derived

from knowledge of energy-m**in**imiz**in**g dislocation structures. The correspond**in**g

variational problem concerns the m**in**imization of (197) subject to the constra**in**t

(186) of conservation of Burgers vector **and** the additional constra**in**t

||α|| 1 ≡

N∑

s=1

∑

e 2 ∈E 2

|dξ s (e 2 )|=M (204)

that fixes the total mass of dislocation **in** the crystal. The result**in**g problem lacks

lower-semicont**in**uity **and** its m**in**imizers may be expected to exhibit f**in**e oscillations.

Numerical studies of energy-m**in**imiz**in**g dislocation structures do **in**deed

exhibit such patterns [37, 29].

Another longst**and****in**g problem **in** physical metallurgy concerns the determ**in**ation

of the patterns of slip activity which occur **in** plastically worked metals

**and** attendant macroscopic properties such as yield stresses **and** harden**in**g rates

(cf., e.g., [33]). On a first approach to the problem, it might seem tempt**in**g to

attempt the m**in**imization of the potential energy (202) as a means of characteriz**in**g

the slip fields likely to occur under the action of applied loads. However, this

approach fails due to the lack of coerciveness of the function F(ξ). We demonstrate

this lack of coerciveness by means of a counterexample. Consider the case

of a square lattice undergo**in**g anti-plane shear, cf. Section 5.4.1. Suppose that the

lattice deforms under the action of a po**in**t load f > 0 applied to the orig**in**. For

h ∈ Z, h ≧ 1, consider a sequence of displacements such that u h (l) = hb at l = 0,

**and** u h (l) = 0 elsewhere. In addition, let (ξ h ) A3 (l, 1) =−h, (ξ h ) B3 (l, 2) =−h,

(ξ h ) A3 (l − ε 1 , 1) = h, (ξ h ) B3 (l − ε 2 , 2) = hb, **and** ξ h = 0 otherwise. Then, it can

be readily verified that du h = β h **and** α h = 0, **and** thus the stra**in**-energy of the

crystal is zero. Therefore, F(ξ h ) =−hf b which tends to −∞ as h →+∞. Thus,

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 207

F(ξ) lacks coerciveness **and** the correspond**in**g m**in**imum problem fails to deliver

solutions **in** general.

This degeneracy can be remedied, **and** the mathematical problem rendered well

posed, by a careful account**in**g of physical sources of resistance, or obstacles, to

the motion of dislocations (cf., e.g., [19]). For **in**stance, when two dislocation l**in**es

cross each other, reactions take place that result **in** jogs, junctions **and** other reaction

products. This source of work harden**in**g is referred to as forest harden**in**g.A

simple limit**in**g case is obta**in**ed by assum**in**g that an **in**f**in**ite energy is required for

dislocations to cross each other. This effectively rules outs cross**in**gs, a constra**in**t

that can be expressed **in** terms of the l**in**k**in**g number (90) **in** the form

L**in**k ( ξ r (e 1 ), ξ s (e ′ 1 )) = 0, ∀e 1 ,e ′ 1 ∈ E 1,r,s= 1,...,N, r ̸= s, (205)

This constra**in**t **in**troduces a topological obstruction that restricts the configurations

accessible to the dislocation ensemble. A rigorous analysis of the **in**teraction

between dislocations **and** obstacles has recently been performed by Garroni **and**

Müller [14, 15].

5.4. Simple illustrative examples of stored energies

In this section we present two simple examples of stored energy correspond**in**g

to the simple cubic **and** square lattices. While these examples are of limited relevance

to actual materials, they nevertheless serve the purpose of illustrat**in**g the

structure of stored energies.

5.4.1. The square lattice. A particularly simple example concerns a square lattice

undergo**in**g anti-plane shear. The nearest-neighbor force constants **and** dynamical

matrix for this case have been collected **in** (170), (171a), (171b) **and** (173). The

relation between the eigendeformations **and** the slip field is

β ( e 1 (l, 1) ) (

= bξ A3 e1 (l, 1) ) ,

β ( e 1 (l, 2) ) (

= bξ B3 e1 (l, 2) ) ,

(206a)

(206b)

where the nomenclature for the slip systems can be referred to **in** Table 1. In addition,

we conventionally set ξ A3

(

e1 (l, 2) ) = 0 **and** ξ B3

(

e1 (l, 1) ) = 0. From (184)

the energy of the crystal per unit length **in** the anti-plane shear direction follows as

E(u, ξ) = 1

(2π) 2 ∫ π

−π

∫ π

M**in**imization with respect to u gives

−π

µ {

|(e

iθ 1

− 1)û(θ) − bˆξ A3 (θ)| 2

2

+|(e iθ 2

− 1)û(θ) − bˆξ B3 (θ)| 2} dθ 1 dθ 2 . (207)

û(θ) = i b 2

s**in** θ 1

2 ˆη A3 (θ) + s**in** θ 2

2 ˆη B3 (θ)

s**in** 2 θ 1

2

+ s**in** 2 θ 2

2

, (208)

208 M. P. Ariza & M. Ortiz

**and** the correspond**in**g stored energy per unit length is computed to be

E(ξ) = 1

(2π) 2 ∫ π

−π

∫ π

−π

µb 2 | s**in** θ 2

2 ˆη A3 (θ) − s**in** θ 1

2 ˆη B3 (θ)| 2

2 s**in** 2 θ 1

2

+ s**in** 2 θ dθ 1 dθ 2 , (209)

2

2

where we write b 2 ≡|b| 2 **and**

ˆη A3 (θ) = ˆξ A3 (θ)e iθ1/2 ,

ˆη B3 (θ) = ˆξ B3 (θ)e iθ 2/2

(210a)

(210b)

may be regarded as translates of the fields ξ A3 **and** ξ B3 to the center of the correspond**in**g

1-cells. Alternatively, us**in**g (47b) **and** (185), the energy (209) may be

recast **in** the form

E(α) = 1

(2π) 2 ∫ π

−π

∫ π

−π

µb 2

2

|ˆα(θ)| 2

s**in** 2 θ 1

2

+ s**in** 2 θ 2

2

dθ 1 dθ 2 , (211)

which is a special case of (196) **and** exemplifies the general result that the stored

energy of a crystal can be written **in** terms of the dislocation density α. In the special

case **in** which slip is conf**in**ed to a s**in**gle plane, e.g., the plane {e 1 (l, 2) s. t. l 2 = 0},

the energies (209) **and** (211) reduce to

E(ξ) = 1 ∫ π

µb 2

2π −π 2

∣

∣s**in** θ 1

2

∣ ∣∣

√1 + s**in** 2 θ 1

2

|ˆξ B3 (θ 1 )| 2 dθ 1 (212)

**and**

E(α) = 1 ∫ π

µb 2 |ˆα(θ 1 )| 2

dθ 1 , (213)

2π −π 2

√s**in** 2 θ 1

2

+ s**in** 4 θ 1

2

respectively.

5.4.2. The simple cubic lattice. The relation between eigendeformations **and** the

slip field **in** a simple-cubic lattice is

β ( e 1 (l, 1) ) (

= b A2 ξ A2 e1 (l, 1) ) (

+ b A3 ξ A3 e1 (l, 1) ) ,

β ( e 1 (l, 2) ) (

= b B1 ξ B1 e1 (l, 2) ) (

+ b B3 ξ B3 e1 (l, 2) ) ,

β ( e 1 (l, 3) ) (

= b C1 ξ C1 e1 (l, 3) ) (

+ b C2 ξ C2 e1 (l, 3) ) ,

(214a)

(214b)

(214c)

where the designation of the slip systems is as **in** Table 1. In addition, the components

of ξ not appear**in**g **in** the preced**in**g relations are conventionally set to

zero. Suppose, for simplicity, that activity takes place **in** the s**in**gle system C1 **and**

that c 12 = 0 **and** c 11 = 2c 44 . The elastic moduli thus obta**in**ed are isotropic, **and**

Poisson’s ratio is zero. Assume, **in** addition, that ˆξ C1 is **in**dependent of θ 1 , correspond**in**g

to a distribution of straight screw dislocations. Then, us**in**g moduli (174a)

**and** (174b) the energy per unit length of dislocation is computed to be

E(ξ) = 1 ∫ π ∫ π

(2π) 2 2ab 2 s**in** 2 θ 2

H

2

s**in** 2 θ 2

2

+ s**in** 2 θ |ˆξ C1 | 2 dθ 2 dθ 3 . (215)

3

2

−π

−π

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 209

F**in**ally, if slip is conf**in**ed to the s**in**gle plane {e 1 (l, 3) s. t. l 3 = 0}, the energy per

unit length reduces to

∣

E(ξ) = 1 ∫ ∣

π ∣s**in** θ 1 ∣∣

2ab 2 2

H

|ˆξ C1 |

(2π) −π

√1 2 dθ 2 . (216)

+ s**in** 2 θ 1

2

We verify that (212) is recovered from this expression by mak**in**g the identification

4aH = c 44 = µ.

5.5. Cont**in**uum limit of the stored energy

We conclude this article by return**in**g to the connection between the discrete **and**

cont**in**uum theories **in**vestigated **in** Section 4.3. Here aga**in**, the aim of the analysis is

to identify limit**in**g situations **in** which the mechanics of the crystal can effectively

be described by means of l**in**ear elasticity, thus achiev**in**g considerable simplification

with respect to the full discrete theory. However, the cont**in**uum limit of the

stored energy is somewhat subtle due to the logarithmic divergence of the energy

of l**in**ear elastic dislocations. This difficulty notwithst**and****in**g, a well-def**in**ed cont**in**uum

limit is atta**in**ed by lett**in**g the dislocations become well-separated, i.e., **in**

the dilute limit. Another useful limit of the discrete theory is atta**in**ed by lett**in**g both

the lattice size **and** the Burgers vector become vanish**in**gly small. As shown subsequently,

this limit co**in**cides with the classical theory of cont**in**uously-distributed

dislocations (cf., e.g., [36]).

5.5.1. Well-separated or dilute dislocations. Recall that the stored energy has

the representation (201) **in** terms of the harden**in**g moduli ˆϒ(θ) **and** the complex

Bravais lattice representation ˆξ(θ,α) of the discrete slip field. Recall that the support

of the slip field ξ s is E1 s ={e 1 ∈ E 1 , s. t. dx(e 1 ) · m s = d s }. With a view

to facilitat**in**g the passage to the cont**in**uum, we rewrite the stored energy (201) **in**

wavenumber form **and** require the Fourier transforms ˆξ s of the slip functions to be

supported on the Brillou**in** zone B of the lattice. The result**in**g form of the stored

energy is

E(ξ) = 1 1

(2π)

∫B

n 2 〈G(k)ˆξ(k), ˆξ ∗ (k)〉 dk, (217)

where

G rs (k) = 1

∑

α∈I r ∑

β∈I s

ˆϒ rs

αβ (k)eik·(r α−r β )

(218)

**and** is the volume of the unit cell of the lattice. In this expression, the labels r

**and** s refer to a pair of slip systems, whereas the labels α **and** β range over the

sublattices of E1 r **and** Es 1 , respectively, **and** r α is the shift of sublattice α.

In order to atta**in** the limit of **in**terest we proceed to scale the slip fields **in** such

a way as to **in**creas**in**gly separate the dislocations. Consider a slip system s **and** let

(c1 s,...,cs n ) be a Bravais basis for the crystal such that (cs 1 ,...,cs n−1

) spans the

210 M. P. Ariza & M. Ortiz

slip planes of the system, **and** let (cs 1,...,cn s ) be the correspond**in**g dual basis. Let

ξ s be the slip function for system s **and** let h ∈ Z, h ≧ 1. The appropriate scaled

slip function is

where ξ is extended by periodicity outside B **and**

ˆξ s h (k) = ws h (k)ˆξ s (hk), (219)

n−1

wh s (k) = ∏

j=1

1 − e −ihk·cs j

1 − e −ik·cs j

≡ qs (hk)

q s (k)

(220)

is a slip-plane w**in**dow function. The effect of this scal**in**g transformation may be

ascerta**in**ed by apply**in**g it to the function

{

ξ s (l ′ 1 ifl ′ = l,

) =

(221)

0 otherwise.

The Fourier transform of this function is ˆξ s (k) = e −ik·x(l) , **and**, hence, **in** this

case ˆξ s h (k) = ws h (k)e−ik·x(hl) . Thus, by l**in**earity, the scal**in**g transformation maps

the value ξ s (l) at x(l) of a general slip function ξ s onto the area {x(l + ν), ν i =

1,...,h, i = 1,...,n− 1} of the correspond**in**g slip plane, which **in** turn has the

effect of exp**and****in**g the dislocation l**in**es with**in** their planes. In addition, the scal**in**g

transformation separates the planes conta**in****in**g the dislocations. We additionally

def**in**e the cont**in**uum w**in**dow functions

n−1

w0 s (k) = ∏

j=1

1 − e −ik·cs j

ik · c s j

≡ qs (k)

q0 s (222)

(k),

**and** **in**troduce the diagonal matrices W h = diag{wh 1,...,wN h } **and** W 0 =

diag{w0 1,...,wN 0

} for notational convenience. The properties

**and**

|h 1−n w s h (h−1 η)| ≦ 1 (223)

lim

h→∞ |h1−n wh s (h−1 η)| =w 0 (η) (224)

are noted for subsequent reference.

It should be carefully noted that **in** the scal**in**g just described, the scaled dislocation

ensemble consists of long straight segments of directions def**in**ed by the

vectors (c1 s,...,cs n−1

). Evidently, the correspond**in**g limit**in**g energy depends on

the choice of these vectors. However, the dislocation segments of many crystals

classes exhibit preferred directions. For **in**stance, dislocations **in** BCC crystals are

known to consist predom**in**antly of screw **and** edge segments. In these cases, the

vectors (c1 s,...,cs n−1

) may conveniently be taken to co**in**cide with the preferred

orientations of the dislocation segments.

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 211

Next we wish to elucidate the behavior of the stored energy of the crystal under

the scal**in**g transformation just def**in**ed, i.e., the behavior of the sequence of functions

E h (ξ) = E(ξ h )

= 1 1

(2π)

∫B

n 2 〈G(k)W h(k)ˆξ(hk), Wh ∗ (k)ˆξ ∗ (hk)〉 dk. (225)

Alternatively, effect**in**g a change of variables to η = hk we obta**in**

E h (ξ) =

h−n 1

(2π)

∫hB

n 2 〈G(h−1 η)W h (h −1 η)ˆξ(η),Wh ∗ (h−1 η)ˆξ ∗ (η)〉 dη

= 1 1

(2π)

∫B

n 2 〈G h(k)ˆξ(k), ˆξ ∗ (k)〉 dk, (226)

where

G h (k) = h −n

∑

ν∈Z n ∩[−h,h] n (W h GW ∗ h )( h −1 (k + 2πν i a i ) ) . (227)

The limit of E h (ξ) as h →∞now characterizes the stored energy of a crystal

conta**in****in**g well-separated, or dilute, dislocations.

The follow**in**g lemmas set the stage for the determ**in**ation of the dilute limit of

the stored energy. We beg**in** by ascerta**in****in**g the form

of G(k) **in** the long wavelength limit.

Lemma 1. Suppose that:

(i) there is a constant C>0 such that

G 0 (k) = lim

ε→0

G(εk) (228)

C|k| 2 |ζ | 2 ≦ 〈D(k)ζ, ζ ∗ 〉 (229)

for a. e. k ∈ B **and** for every ζ ∈ C n ;

(ii) for every ζ ∈ C n , the functions ε −2 〈D(εk)ζ, ζ ∗ 〉 converge for a. e. k to

Then,

〈D 0 (k)ζ, ζ ∗ 〉=c ij kl k j k l ζ i ζ ∗ k . (230)

for a. e. k.

G rs

0 (k) = [ c ij kl − (D0 −1 ) ] bi

r b

pm(k)c pqij c mnkl k q k n

d r mr k

s

j

d s ms l . (231)

212 M. P. Ariza & M. Ortiz

Proof. Fix k throughout. From (183) **and** (184) we have

〈G(k)ζ, ζ ∗ 〉=m**in**

c∈C n〈E(k)[Q(k)c − η(k)], [Q(k)c − η(k)]∗ 〉 (232)

for all ζ ∈ C N , where Q(k) is the matrix representation (41) of d 0 ,

**and**

η(k) =−

N∑

r=1

ζ r [m r · i∂ k Q(k)] br

d r (233)

E(k) = 1 2 ˆ(k). (234)

By (109) we have the identity

Scal**in**g k **in** (232) we obta**in**

〈D(k)c, c ∗ 〉=〈E(k)Q(k)c, [Q(k)c] ∗ 〉. (235)

〈G(εk)ζ, ζ ∗ 〉=m**in**

c∈C n〈E(εk)[Q(εk)c − η(εk)], [Q(εk)c − η(εk)]∗ 〉

= m**in**

c∈C n〈E(εk)[Q(εk)ε−1 c−η(εk)], [Q(εk)ε −1 c−η(εk)] ∗ 〉. (236)

By assumption (229) the quadratic functions 〈ε −2 D(εk)c, c ∗ 〉 are convex **and** equicoercive.

Therefore, we have

〈G 0 (k)ζ, ζ ∗ 〉=lim 〈G(εk)ζ, ζ ∗ 〉

ε→0

= m**in** lim

c∈C n ε→0 〈E(εk)[Q(εk)ε−1 c−η(εk)],[Q(εk)ε −1 c−η(εk)] ∗ 〉. (237)

But,

lim

ε→0 ε−1 Q(εk) = k · ∂ k Q(0),

(238a)

lim η(εk) =− ∑

N ζ r [m r · i∂ k Q(0)] br

ε→0 d r , (238b)

r=1

〈E(0)a · ∂ k Q(0)c, [a · ∂ k Q(0)c] ∗ 〉=c ij kl c i a j ck ∗ a∗ l ,

(238c)

where **in** arriv**in**g at the last identity we have made use of (168) **and** (235). Hence,

(

)(

∗

N∑

N∑

〈G 0 (k)ζ, ζ ∗ 〉=m**in** c

c∈C n ij kl ic i k j − ζ r si r mr j ic k k l − ζ s sk l) s ms . (239)

r=1

A direct calculation of the m**in**imizer f**in**ally gives (231).

We note that G 0 (k) is determ**in**ed by the properties of the limit**in**g l**in**ear elastic

cont**in**uum, namely, by the elasticity of the crystal through the elastic moduli c ij kl

**and** the elastic dynamical matrix (153). The follow**in**g lemma reveals the essential

structure of the cont**in**uum kernel G 0 (k).

s=1

⊓⊔

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 213

Lemma 2. Suppose that the elastic moduli c ij kl are positive def**in**ite **and** f**in**ite.

Then, there is a constant C>0 such that

〈G 0 (k)ζ, ζ ∗ 〉 ≦ CN

N∑

r=1

Proof. By (239) **and** convexity we have

〈G 0 (k)ζ, ζ ∗ 1

〉 ≦ m**in**

c∈C n N

1

≦ m**in**

c∈C n N

= CN

N∑

r=1

(1 − (k · mr ) 2

|k| 2 )

|ζ r | 2 . (240)

N∑

) (ick

c ij kl

(ic i k j − Nζ r m r i sr j k l − Nζ r m r k sr l

r=1

N∑

C|ic ⊗ k − Nζ r m r ⊗ s r | 2 (241)

r=1

(1 − (k · mr ) 2

|k| 2 )

|ζ r | 2 . ⊓⊔

F**in**ally, **in** order to **in**vestigate the Ɣ-convergence of the energy E h (ξ) we need

to ascerta**in** the precise manner **in** which it scales with h. Beg**in** by not**in**g that

h 2−n G h (k) =

∑

ν∈Z n ∩[−h,h] n (

(h 1−n W h )G(h 1−n W ∗ h ))( h −1 (k + 2πν i a i ) ) . (242)

Thus, **in** view of (224) **and** def**in**ition (228) we expect

h 2−n G h (k) ∼

∑

ν∈Z n ∩[−h,h] n (W 0 G 0 W ∗ 0 )(k + 2πν ia i ) (243)

for large h. The behavior of the right-h**and** side of this asymptotic equation is

established by the follow**in**g lemma.

Lemma 3. Let n ≧ 2, Z ∋ h>1. Suppose that the elastic moduli C ij kl are positive

def**in**ite **and** f**in**ite. Then, there is a constant C>0 such that

1

log h

for all ζ ∈ C N .

∑

ν∈Z n ∩[−h,h] n 〈(W 0 G 0 W ∗ 0 )(k + 2πν ia i )ζ, ζ ∗ 〉 ≦ C|ζ | 2 (244)

Proof. Fix k **and** ζ ∈ C N . By lemma 2 there is a constant C>0 such that

〈G 0 (k)W 0 (k)ζ, W ∗ 0 (k)ζ ∗ 〉 ≦ C

N∑

r=1

(1 − (k · mr ) 2

|k| 2 )

|w r 0 (k)|2 |ζ r | 2 . (245)

) ∗

214 M. P. Ariza & M. Ortiz

Let kp r = k − (k · mr )m r be the projection of k onto the slip planes of system r **and**

let θi r = k · ci r. Then,

(1 − (k · mr ) 2 )

|k| 2 |w0 r (k)|2 = |kr p |2

|k| 2 |wr 0 (k)|2

∑ n−1

i=1

≦ C

(θ i r ∑ )2

ni=1

(θi r)2 |wr 0 (k)|2

But

h∑

h∑

ν i =−h ν n =−h

= 2

∣ s**in** θ i

r 2 ∣

≦ 2

∣ s**in** θ i

r 2 ∣

≦ 2

∣ s**in** θ i

r 2 ∣

= 2

∣ s**in** θ i

r 2 ∣

4 s**in** 2 1 2 (θ r i

+ 2πν i )

∑ nk=1

(θ r k + 2πν k) 2

h∑

h∑

ν i =−h ν n =−h

h∑

h∑

ν i =−h ν n =−h

h∑

∞∑

ν i =−h ν n =−∞

ν i =−h

By virtue of the bound

⎛

∑n−1

⎜

n−1 ∏

≦ C ⎝

i=1

j=1

j̸=i

2 ∣ ∣ s**in**

1

2

(θ r i

+ 2πν i ) ∣ ∣

∑ nk=1

(θ r k + 2πν k) 2

⎞

s**in** 2 2 1 θ j

r ⎟

( ) 2 12

θj

r

2 ∣ ∣ s**in**

1

2

(θ r i

+ 2πν i ) ∣ ∣

(θ r i

+ 2πν i ) 2 + (θ r n + 2πν n) 2

2 ∣ ∣ s**in**

1

2

(θ r i

+ 2πν i ) ∣ ∣

⎠ 4 s**in**2 2 1 θ i

r ∑ nk=1

(θk r . (246)

)2

(θi r + 2πν i ) 2 + (θn r + 2πν n) 2

h∑ 1 ∣ s**in**

1

2

(θi r + 2πν i ) ∣ ∣ s**in**h(θ

r

i

+ 2πν i ) ∣ |θi r + 2πν i | cosh(θi r + 2πν i ) − cos θn

r . (247)

s**in** η 2 s**in**h η ≦ C ≈ 2.23743 (248)

cosh η − cos ξ

**in** the range η ∈[0, ∞), ξ ∈[−π, π], we further have

h∑

h∑

ν i =−h ν n =−h

4 s**in** 2 1 2 (θ r i

+ 2πν i )

∑ nk=1

(θ r k + 2πν k) 2

≦ 2C

∣ s**in** θ i

r h∑

1

2 ∣ |θ r ν i =−h i

+ 2πν i |

≦ 2C

∣ s**in** θ i

r ( 1

2 ∣ |θi r| + 1 )

π log h

≦ C

(1 + 1 )

π log h . (249)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 215

In addition we have

h∑

ν j =−h

Comb**in****in**g the preced**in**g bounds we obta**in**

∑

ν∈Z n ∩[−h,h] n 〈(W 0 G 0 W ∗ 0 )(k + 2πν ia i )ζ, ζ ∗ 〉 ≦ C

4 s**in** 2 1 2 (θ r j + 2πν j )

(θ r j + 2πν j ) 2 ≦ 1. (250)

(1 + 1 π log h )

|ζ | 2 , (251)

which proves the assertion.

⊓⊔

The preced**in**g lemma suggests the behavior E h (ξ) ∼ h n−2 log h for large h.

Therefore, we **in**troduce the scaled sequence of functions

where

F h (ξ) = h2−n

log h E h(ξ) = 1 1

(2π)

∫B

n 2 〈K h(k)ˆξ(k), ˆξ ∗ (k)〉 dk, (252)

K h (k) = h2−n

log h G h(k). (253)

The behavior of the sequence F h (ξ) as h →∞is characterized by the follow**in**g

proposition.

Proposition 5. Let n ≧ 2, Z ∋ h>1. Suppose that:

(i) for every ζ ∈ C N the function 〈K h (·)ζ, ζ ∗ 〉 is measurable on B;

(ii) there is a constant C such that

0 ≦ 〈K h (k)ζ, ζ ∗ 〉 ≦ C|ζ | 2 (254)

for a. e. k ∈ B **and** for every ζ ∈ C N ;

(iii) for every ζ ∈ C N , the function 〈K h (k)ζ, ζ ∗ 〉 converges for a. e. k to 〈K 0 (k)ζ,

ζ ∗ 〉 as h →∞.

Let

Then,

F 0 (ξ) = 1 1

(2π)

∫B

n 2 〈K 0(k)ˆξ(k), ˆξ ∗ (k)〉 dk. (255)

**in** X ={ξ ∈ C N 1 s. t. ˆξ ∈[L 2 (B)] N }.

Ɣ − lim

h→∞ F h = F 0 (256)

216 M. P. Ariza & M. Ortiz

Proof. Let ξ ∈ X. Then, from assumptions (i)–(iii) **and** dom**in**ated convergence,

it follows that F h (ξ) → F 0 (ξ). Hence, F h converges to F 0 po**in**twise. Consider

now the space Y ={ξ : E 1 → R N s. t. ˆξ ∈[L 2 (B)] N } obta**in**ed by lift**in**g the

**in**teger-valuedness constra**in**t **and** allow**in**g the slip fields to take real values. By

assumption (ii) it follows that the functions F h are convex **and** equibounded, hence

equi-lower semicont**in**uous, over Y . S**in**ce X is a closed subspace of Y it follows that

the sequence F h is likewise equi-lower semicont**in**uous over X. The proposition

then follows from a st**and**ard result regard**in**g the equivalence of po**in**twise convergence

of equi-lower semicont**in**uous sequences **and** Ɣ-convergence (see [10],

Proposition 5.9). ⊓⊔

The conditions of the theorem may be verified simply for specific force constant

models. The h n−2 log h scal**in**g of the stored energy E h (ξ) is consistent with the

known solutions from l**in**ear elasticity for po**in**t dislocations **in** two dimensions **and**

for dislocation loops **in** three dimensions (cf., e.g., [19]). The factor h n−2 accounts

for the **in**crease **in** dislocation length, or dislocation stretch**in**g, with **in**creas**in**g h.

The factor log h corresponds to the familiar logarithmic divergence of the dislocation

energy with the size of the doma**in** occupied by the dislocations. It bears

emphasis that the limit**in**g energy (255) **in**volves an **in**tegral over the Brillou**in** zone

B **and** is def**in**ed for discrete slip fields. In this particular sense, the limit just derived

reta**in**s the discreteness of the lattice on the scale of the dislocation cores **and**, as

a consequence, rema**in**s free of the logarithmic core-divergence of l**in**ear elastic

dislocation mechanics.

5.5.2. Cont**in**uously-distributed dislocations. Lemma 2 establishes the structure

of the cont**in**uum kernel G 0 (k) **and**, by extension, sets the natural functional

framework for the analysis of cont**in**uously distributed dislocations. Thus, the bound

(240) suggests that the natural norm for a cont**in**uum slip field η : R n → R N is

||η|| X =

[ N

∑

s=1

∫ (

1

(2π) n 1 − (k · ms ) 2 )

1/2

|k| 2 |ˆη s (k)| dk] 2 . (257)

Correspond**in**gly, we may def**in**e the space X of cont**in**uum slip fields as the space

of distributions result**in**g from the completion of C0

∞ under the norm (257). S**in**ce

0 ≦ 1 − (k · ms ) 2

|k| 2 ≦ 1, (258)

it follows that

||η|| X ≦ ||η|| L 2 (259)

**and** L 2 (R n , R N ) is a subspace of X. Moreover, it follows from (259) that sequences

converg**in**g strongly **in** L 2 also convergence strongly **in** X.

In addition to L 2 -fields, the space X conta**in**s discont**in**uous slip fields such as

η s = f(xp s )δ(xs n ), (260)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 217

where (x1 s,...,xs n ) is a local system of cartesian coord**in**ates for system s, with

(x1 s,...,xs n−1 ) spann**in**g the **in**-plane directions **and** xs n measur**in**g distance **in** the

direction normal to the slip planes; δ is the Dirac measure; **and**, for simplicity, we

assume simple slip **and** set η r = 0, r ̸= s. Indeed, **in** this case,

||η|| 2 X = 1

(2π) n ∫ |k

s

p | 2

|k s | 2 | ˆ f(k s p )|2 dk s

=

∫

1

(2π) n−1

|k s p || ˆ f(k s p )|2 dk s p =||f ||2 1/2 . (261)

Thus, the slip field (260) is **in** X provided that f ∈ H 1/2 (R n−1 ).

For ε>0, consider now the sequence of stored energy functions E ε : X → ¯R:

⎧ ∫

1 12

(2π)

⎪⎨

n 〈G(εk)ˆξ(k), ˆξ ∗ (k)〉 dk if supp(ˆξ s ) ⊂ ε −1 B,

**and** ξ s (εx(l)) ∈ εZ,

E ε (ξ) =

l ∈ Z

⎪⎩

n (262)

,s= 1,...,N,

+∞ otherwise,

obta**in**ed by scal**in**g down both the crystal lattice **and** the Burgers vectors by a factor

of ε simultaneously. In addition, consider the follow**in**g c**and**idate limit**in**g energy

E 0 : X → ¯R:

E 0 (η) = 1

(2π) n ∫ 1

2 〈G 0(k) ˆη(k), ˆη ∗ (k)〉 dk. (263)

The behavior of the sequence E ε (ξ) as ε → 0 is characterized by the follow**in**g

proposition.

Proposition 6. Suppose that:

(i) for every ζ ∈ C N the function 〈G(·)ζ, ζ ∗ 〉 is measurable on B;

(ii) there is a constant C such that

0 ≦ 〈G(k)ζ, ζ ∗ 〉 ≦ C

N∑

r=1

(1 − (k · mr ) 2

|k| 2 )

|ζ r | 2 (264)

for a. e. k ∈ B **and** for every ζ ∈ C N ;

(iii) for every ζ ∈ C N , the functions 〈G(εk)ζ, ζ ∗ 〉 converge for a. e. k to 〈G 0 (k)ζ, ζ ∗ 〉

as ε → 0.

Then,

**in** the weak topology of X.

Ɣ − lim

ε→0

E ε = E 0 (265)

218 M. P. Ariza & M. Ortiz

Proof. Let η ∈ X. Let u i ∈ C0

∞ be such that u i → η **in** X. Approximate u i by

simple functions tak**in**g values **in** εZ **and** converg**in**g strongly to u i **in** X. F**in**ally,

sample the simple functions po**in**twise on a lattice of size ε (cf. Section A.4.2).

Let ξ ε be a diagonal sequence extracted from the preced**in**g sequences. Then ξ ε

satisfies the **in**teger-valuedness constra**in**t **in** (262) **and** converges strongly to η **in**

X. By assumption (ii) we have

N

〈G(εk)ˆξ ε (k), ˆξ ε ∗ (k)〉 ≦ C ∑

(1 − (k · mr ) 2 )

|k| 2 |ˆξ ε r (k)|2 . (266)

r=1

By assumption (iii) **and** the dom**in**ated convergence theorem it follows that

∫

∫

〈G(εk)ˆξ ε (k), ˆξ ε ∗ (k)〉dk = 〈G 0 (k) ˆη(k), ˆη ∗ (k)〉dk (267)

**and**, hence,

lim

ε→0

lim E ε(ˆξ ε ) = E 0 (η). (268)

ε→0

Let now η ε ⇀η**in** X. We need to show that

E 0 (η) ≦ lim **in**f E ε(η ε ). (269)

ε→0

Suppose that lim **in**f ε→0 E ε (η ε )

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 219

The energy (263) follows directly by m**in**imiz**in**g out the displacement field

**in** cont**in**uum plasticity with the aid of the Fourier transform. However, the preced**in**g

discussion does add to that formal result by provid**in**g an underst**and****in**g

of the precise manner **in** which the cont**in**uously distributed dislocation limit is

approached, **and** an underst**and****in**g of the natural functional sett**in**g **in** which to

couch that limit. Thus, for **in**stance, it is **in**terest**in**g to note that the energy (263) is

well-def**in**ed for discont**in**uous slip of the form (260), provided that the slip distribution

f ∈ H 1/2 (R n−1 ) on the slip plane. By contrast, the limit**in**g theory does not

support Volterra dislocations, s**in**ce a piecewise constant function f ∈ H 1/2 (R n−1 )

is necessarily constant [14].

Appendix A. The discrete Fourier transform

Appendix A.1. Def**in**ition **and** fundamental properties

Beg**in** by **in**troduc**in**g the usual l p spaces of lattice functions. Let f be a complex-valued

lattice function, i.e., a function f : Z n → C. Its l p -norm is

||f || p =

{ ∑

l∈Z n |f(l)| p } 1/p

.

(A.274)

Let l p be the correspond**in**g Banach spaces. In particular, l 1 can be turned **in**to a

Banach algebra with **in**volution by tak**in**g the convolution

(f ∗ g)(l) = ∑

f(l− l ′ )g(l ′ )

(A.275)

l ′ ∈Z n

as the multiplication operation **and** complex conjugation as the **in**volution (cf., e.g.,

[42]). The discrete Fourier transform of f ∈ l 1 is

f(θ)= ˆ

∑

f(l)e −iθ·l .

(A.276)

l∈Z n

In addition we have

f(l)= 1

(2π) n ∫

[−π,π] n

ˆ f(θ)e iθ·l dθ,

(A.277)

which is the **in**version formula for the discrete Fourier transform. It follows from

this expression that f(−l) is the Fourier-series coefficient of f(θ). ˆ The discrete

Fourier transform characterizes all complex homomorphisms of the Banach algebra

l 1 . In particular we have the identity

̂f ∗ g = ˆ f ĝ,

(A.278)

220 M. P. Ariza & M. Ortiz

which is often referred to as the convolution theorem. Suppose **in** addition that

f, g ∈ l 2 . Then, from the identity

∑

( ∫

)

1

(2π) n f(θ)e ˆ iθ·l dθ g ∗ (l)

l∈Z n [−π,π] n

= 1 ∫

( )

∑

ˆ

(2π) n f(θ) g ∗ (l)e −iθ·l dθ (A.279)

[−π,π] n l∈Z n

we obta**in**

∑

f(l)g ∗ (l) = 1 ∫

(2π) n f(θ)ĝ ˆ ∗ (θ)dθ, (A.280)

l∈Z n [−π,π] n

which is the Parserval identity for the discrete Fourier transform. If, **in** particular,

g = f , Parseval’s identity reduces to

∑

|f(l)| 2 = 1 ∫

(2π) n | f(θ)| ˆ 2 dθ,

(A.281)

l∈Z n [−π,π] n

which shows that the DFT is an l 2 -isometry. S**in**ce l ∞ ⊂ ̂l 1 is dense **in** l 2 , it follows

that the DFT has a unique cont**in**uous extension to an isometric isomorphism of l 2

onto L 2 ([−π, π] n ).

Appendix A.2. Wave-number representation

The ord**in**ary Fourier transform of cont**in**uum fields is most often expressed **in**

terms of wave-numbers. In applications concerned with the pass**in**g to the cont**in**uum,

it is therefore natural to adopt a wave-number vector representation of the

discrete Fourier transform. In particular, this representation ensures that the discrete

**and** ord**in**ary Fourier transforms of physical fields of the same k**in**d have match**in**g

units. We beg**in** by effect**in**g a change of variables **in** all discrete Fourier transforms

f(θ)from ˆ angles θ to wave-number vector k ≡ θ i a i . In addition, we multiply f(θ) ˆ

by the atomic volume. The result**in**g function f(k) ˆ is supported on the Brillou**in**

zone B **in** dual space, **and** is given directly by

f(k)= ˆ ∑

f(l)e −ik·x(l) ,

(A.282)

l∈Z n

where x(l) = l i a i are the coord**in**ates of the vertices of the lattice. We note that the

identity k · x(l) = θ · l holds, so that (A.282) is **in**deed equivalent to (A.276) except

for the scal**in**g factor of . The **in**verse mapp**in**g is given by

f(l)= 1 ˆ

(2π)

∫B

n f(k)e ik·x(l) dk.

(A.283)

In this representation the convolution (A.275) becomes

(f ∗ g)(l) = ∑

f(l− l ′ )g(l ′ )

(A.284)

l ′ ∈Z n

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 221

**and** the convolution theorem rema**in**s of the form (A.278). In addition, the Parseval

identity (A.280) now is

∑

f(l)g ∗ (l) = 1

(2π) n f(k)ĝ ˆ

l∈Z

∫B

∗ (k)dk,

n

which establishes an isometric isomorphism of l 2 onto L 2 (B).

(A.285)

Appendix A.3. Periodic functions

The extension of the Fourier transform formalism to periodic functions is of

particular **in**terest **in** applications. Consider a set Y ⊂ Z n , the unit cell, such that the

translates {Y +L i A i ,L∈ Z n }, for some translation vectors A i ∈ Z n , i = 1,...,n,

def**in**e a partition of Z n . Let A i be the correspond**in**g dual basis, **and** B i = 2πA i the

reciprocal basis.A lattice function f : Z n → R is Y -periodic if f(l)= f(l+L i A i ),

for all L ∈ Z n . Proceed**in**g formally, the discrete Fourier transform of a Y -periodic

lattice function can be written **in** the form

f(t) ˆ = ∑ ∑

f(l)e −it·(l+Lj A j )

L∈Z n

= 1

|Y |

l∈Y

{ }{

∑

f(l)e it·l |Y | ∑ }

e it·(Lj A j )

, (A.286)

l∈Y

L∈Z n

where |Y | is the number of po**in**ts **in** Y . But

|Y | ∑ e it·(Lj A j ) = (2π) ∑ n

L∈Z n

**and**, hence,

where

f(t)= ˆ

(2π)n

|Y |

H ∈Z n δ(t − H i B i )

∑

f(θ)δ(t ˆ − θ),

θ∈Z

f(θ)= ˆ

∑ f(l)e −iθ·l

l∈Y

(A.287)

(A.288)

(A.289)

**and** Z is the **in**tersection of the lattice spanned by B i **and** [−π, π] n . In addition,

the **in**verse Fourier transform specializes to

f(l)= 1 ∑

f(θ)e ˆ iθ·l .

(A.290)

|Y |

θ∈Z

For periodic functions, Parseval’s identity takes the form

∑

f(l)g ∗ (l) = 1 ∑

f(θ)ĝ ˆ ∗ (θ).

|Y |

l∈Y

θ∈Z

(A.291)

222 M. P. Ariza & M. Ortiz

Likewise, let f **and** g be complex-valued lattice functions, where the latter is periodic.

Insert**in**g representation (A.288) **in**to the convolution theorem gives

(

)

(f ̂∗ g)(t) = f(t) ˆ

(2π) n ∑

ĝ(θ)δ(t − θ)

|Y |

whence it follows that

= (2π)n

|Y |

θ∈Z

∑

f(θ)ĝ(θ)δ(t ˆ − θ),

θ∈Z

̂ (f ∗ g)(θ) = ˆ f(θ)ĝ(θ).

(A.292)

(A.293)

F**in**ally, the average of a periodic function follows as

〈f 〉= 1

|Y |

∑

l∈Y

f(l)= 1

|Y | ˆ f 0 .

(A.294)

Appendix A.4. Sampl**in**g **and** **in**terpolation

In pass**in**g to the cont**in**uum the question arises of how to sample a function over

R n on a lattice **and**, conversely, how to extend a lattice function to R n . The former

operation may be regarded as sampl**in**g a cont**in**uum function, whereas the latter

operation may be regarded as **in**terpolat**in**g a lattice function. Evidently, functions

can be sampled **and** **in**terpolated **in** a variety of ways. In this section we collect the

particular schemes that arise **in** applications.

Appendix A.4.1. Filter**in**g. A simple device for sampl**in**g a function f ∈ H s is

to restrict its ord**in**ary Fourier transform f(k)to ˆ the Brillou**in** zone B of the lattice,

i.e., to set

ŝ B f(k)= ˆ f(k), k ∈ B. (A.295)

The correspond**in**g lattice function then follows from an application of the **in**verse

discrete Fourier transform (A.283), with the result

s B f(l)= 1 ˆ

(2π)

∫B

n f(k)e ik·x(l) dk

(A.296)

for l ∈ Z n . Conversely, for every lattice function f(l)we can def**in**e an **in**terpolant

over R n of the form

i B f(x)= 1 ˆ

(2π)

∫B

n f(k)e ik·x dk.

(A.297)

A projection P B : H s → H s can now be def**in**ed as P B = i B ◦ s B , i.e., by apply**in**g

the sampl**in**g **and** **in**terpolation operations **in** succession. Thus, for every f ∈ H s

we have

̂P B f(k)=

{

f(k) if k ∈ B,

0 otherwise.

(A.298)

**Discrete** **Crystal** **Elasticity** **and** **Discrete** **Dislocations** **in** **Crystal**s 223

Clearly, the projection is orthogonal. Suppose now that the lattice is scaled down

**in** size by a factor ε → 0. The Brillou**in** zones of the scaled lattices are B ε = B/ε.

Let f ∈ H s **and** f ε = P Bε f . Then it follows trivially that fˆ

ε → fˆ

po**in**twise. In

addition,

|| fˆ

ε − f ˆ|| 2 s = 1 ∫

(2π) n | f(k)| ˆ 2 dµ s (k), (A.299)

R n \B ε

where dµ s (k) = (1 +|k| 2 ) s dk, which, by dom**in**ated convergence, shows that

f ε → f **in** H s .

AppendixA.4.2. Po**in**twise sampl**in**g. An alternative scheme for sampl**in**g a function

on a lattice is to sample the function po**in**twise, i.e.,

s B f(l)= f(x(l))

(A.300)

for l ∈ Z n , provided that the po**in**twise value of f at x(l) is def**in**ed. This method

of sampl**in**g has the property of preserv**in**g the range of the function, i. e., if f takes

values **in** a subset A ⊂ R, then so does s B f . In terms of Fourier transforms we

have

ŝ B f(k) = ∑

f(x(l))e −ik·x(l)

l∈Z n

= ∑ { ∫

}

1

(2π) n f(k ˆ ′ )e ik′·x(l) dk ′ e −ik·x(l)

l∈Z n

= 1

(2π) n ∫ { ∑

l∈Z n e i(k−k′ )·x(l)

}

f(k ˆ ′ )dk ′

= 1

(2π) n ∫ { (2π) n ∑ m∈Z n δ(k − k ′ − 2πm i a i )

}

ˆ f(k)dk.(A.301)

Hence, the discrete Fourier transform of the sampled lattice function is given by

the Poisson summation formula

ŝ B f(k)= ∑ m∈Z n

ˆ f(k+ 2πm i a i ), k ∈ B. (A.302)

Conversely, for every lattice function f(l)we def**in**e an **in**terpolant over R n of the

form (A.297). As **in** the preced**in**g section, a projection P B : L 2 → L 2 can now be

def**in**ed as P B = i B ◦s B , i.e., by apply**in**g the sampl**in**g **and** **in**terpolation operations

**in** succession. Thus, if f ∈ L 2 **and** f B = P B f we have

̂P B f(k)=

{ ∑

m∈Z n ˆ f(k+ 2πm i a i ) if k ∈ B,

0 otherwise.

(A.303)

It is verified that

224 M. P. Ariza & M. Ortiz

( )

P B f(x(l)) = 1 ∑

(2π)

∫B

n f(k+ ˆ 2πm i a i ) e ik·x(l) dk

m∈Z n

= 1

(2π) n ∫

ˆ f(k)e ik·x(l) dk = f (x(l)),

(A.304)

which shows that the projection preserves the value of the function on the lattice

**and**, hence, is orthogonal. Suppose now that the lattice is scaled down **in** size by a

factor ε → 0. The Brillou**in** zones of the scaled lattices are B ε = B ε . Let f ∈ L 2

**and** let f ε = P Bε f . Then, we have

ˆ f ε (k) =

{ ∑

m∈Z n ˆ f(k+ ε −1 2πm i a i ) if k ∈ B ε ,

0 otherwise.

(A.305)

We verify that

|| fˆ

ε − f ˆ|| 2 0

= 1 ∫

(2π) n | fˆ

ε (k) − f(k)| ˆ 2 dk + 1 ∫

B ε

(2π) n | f(k)| ˆ 2 dk

R n \B ε

= 1 ∫ ∣ ∣∣

∑

(2π) n f(k+ ˆ ε −1 2πm i a i ) ∣ 2 dk + 1 ∫

B ε

(2π) n | f(k)| ˆ 2 dk

m∈Z n \{0}

R n \B ε

≦ 1 ∫

∑

(2π) n | f(k+ ˆ ε −1 2πm i a i )| 2 dk + 1 ∫

B ε

(2π) n | f(k)| ˆ 2 dk

m∈Z n \{0}

R n \B ε

∫

1

= 2

(2π) n | f(k)| ˆ 2 dk.

(A.306)

R n \B ε

Hence, by dom**in**ated convergence, || ˆ f ε − ˆ f || 0 → 0asε → 0.

Acknowledgements. The authors gratefully acknowledge the support of the Department of

Energy through Caltech’s ASCI ASAP Center for the Simulation of the Dynamic Response

of Materials. MO wishes to gratefully acknowledge support received from NSF through

an ITR grant on Multiscale Model**in**g **and** Simulation **and** Caltech’s Center for Integrative

Multiscale Model**in**g **and** Simulation. We are greatly **in**debted to Sergio Conti,Adriana

Garroni **and** Stefan Müller for technical guidance **and** for many helpful discussions **and**

suggestions. This work was partially carried dur**in**g MO’s stay at the Max Planck Institute

for Mathematics **in** the Sciences of Leipzig, Germany, under the auspices of the Humboldt

Foundation. MO gratefully acknowledges the f**in**ancial support provided by the Foundation

**and** the hospitality extended by the Institute.

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Division of Eng**in**eer**in**g **and** Applied Science

California Institute of Technology

Pasadena, CA 91125, USA

e-mail: ortiz@aero.caltech.edu

**and**

Escuela Técnica Superior de Ingenieros Industriales

Universidad de Seville

e-mail: mpariza@us.es

(Accepted January 10, 2005)

Published onl**in**e September 16, 2005 – © Spr**in**ger-Verlag (2005)