Determining the Burgers vectors and elastic strain.pdf

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Determining the Burgers vectors and elastic strain energies ofinterface dislocation arrays using anisotropic elasticity theory

A.J. Vattre a,b,, M.J. Demkowicz b

a CEA, DAM, DIF, F-91297 Arpajon, Franceb MIT, Department of Materials Science and Engineering, Cambridge, MA 02139, USA

Received 2 May 2013; received in revised form 7 May 2013; accepted 8 May 2013Available online 18 June 2013

Abstract

A formalism for describing interface dislocation arrays linking the FrankBilby equation and anisotropic elasticity theory under thecondition of vanishing far-field stresses is developed. The present approach enables the determination of a unique reference state forinterface misfit dislocations, within which the Burgers vectors of individual dislocations are defined and allows for the unequal partition-ing of elastic fields between neighboring crystals. The elastic strain energies of interface dislocation arrays are computed using solutionsfor short-range elastic fields. Examples of applications to simple interfaces are given, namely symmetric tilt and twist grain boundaries, aswell as a pure misfit heterophase interface. 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Semicoherent interfaces; Linear elasticity theory; Arrays of dislocations; Interface energy

1. Introduction

Far from being featureless dividing surfaces betweenneighboring crystals, interfaces in polycrystalline solidshave internal structures of their own. These structuresdepend on interface crystallographic character (misorienta-tion and interface plane orientation) and affect the physicaland chemical properties of interfaces, such as interfaceenergy [1], resistivity [2], diffusivity and permeability [3],mechanical properties [4], point defect sink efficiencies [5],and mobilities [6]. To better understand and control the

properties of interfaces, it is desirable to be able to predicttheir internal structures. This paper presents a method forpredicting a specific interface structural feature: the Bur-gers vectors of intrinsic dislocations in semicoherent grainboundaries and heterophase interfaces. This informationis then used to compute interface elastic strain energies.

One way of studying interface structure is throughatomistic simulations, which explicitly account for all theatoms that make up an interface. However, this approachis not always practical or efficient: it can be veryresource-intensive because it requires a separate simulationfor each individual interface. Thus, it does not lend itself torapidly scanning over many different interfaces, for exam-ple if one were searching for trends in interface structuresor for tailored interfaces with a specific structure. Low-cost, analytical techniques for predicting interface structurewould be preferable in such situations.

One widely used analytical approach applies to semico-herent interfaces and describes interface structures in termsof intrinsic dislocations using the closely related FrankBil-by [79] and O-lattice [911] techniques. Both proceduresrequire the selection of a reference state, within which theBurgers vectors of individual interface dislocations aredefined. Because this choice does not affect the calculatedspacing and line directions of interface dislocations (seeSection 2.3), it has sometimes been viewed as if it were arbi-trary. In practice, one of the adjacent crystals [1214] or a

1359-6454/$36.00 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.actamat.2013.05.006

Corresponding author at: CEA, DAM, DIF, F-91297 Arpajon,France. Tel.: +33 169265379.

E-mail address: [email protected] (A.J. Vattre).

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Acta Materialia 61 (2013) 51725187
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median lattice [15] have often been used as the referencestate.

However, the choice of reference state does influence thevalues of far-field stresses, strains and rotations associatedwith interface dislocations. These, in turn, are usually sub-ject to constraints, namely that the far-field stresses be zero

and that the far-field rotations be consistent with a pre-scribed misorientation. Thus, the choice of reference stateis in fact not arbitrary. As discussed by Hirth and co-work-ers [1618], the importance of selecting proper referencestates has often been overlooked, in part because thebest-known applications of interface dislocation modelsare to interfaces of relatively high symmetry, such as sym-metric tilt or twist grain boundaries, for which correct ref-erence states are easy to guess. Furthermore, manyanalyses assume uniform isotropic elasticity, which leadsto equal partitioning of interface dislocation elastic fieldsbetween the neighboring crystals. In general, however,interfaces need not have high symmetry and the neighbor-

ing crystals may have unlike, anisotropic elastic constants.The correct selection of reference states in such generalcases is far more challenging.

The purpose of the present work is to formulate anapproach for determining reference states (and thereforealso Burgers vectors) that give rise to predictions of inter-face dislocation structure whose far-field elastic fields areconsistent with specified far-field stresses and constraintson the crystallographic character of semicoherent inter-faces. Our method accounts for several factors that, tothe best of our knowledge, have not been addressed inother studies, namely: differences in elastic constants

between crystals neighboring an interface, their elasticanisotropy, and unequal partitioning of elastic fieldsbetween them. We use our results to compute the elasticstrain energies of several simple example interfaces, namelysymmetric tilt and twist grain boundaries, as well as puremisfit heterophase interfaces. Applications of our methodto more complex interface types will be presented in a fol-low-on study.

In Section 2, we define certain terms used in the presentwork, describe our approach to modeling interface disloca-tions, and state the constraints on far-field elastic fieldsimposed by interface crystallography. Section 3 introducesour strategy for determining the Burgers vectors of inter-face dislocations using anisotropic elasticity theory. Sec-tion 4 presents complete solutions for elastic fields ofinterface dislocations. Section 5 gives applications to sev-eral examples. Section 6 provides a summary of our mainresults and concluding remarks.

2. Problem definition

In what follows in Section 2, we define the notion ofintroducing Volterra dislocations into a reference statefor constrained interfaces consistent with the FrankBilby

equation that are free of far-field stresses.

2.1. Planar interfaces in linear elastic bicrystals

In our analysis, we consider planar interfaces formed byjoining two semi-infinite linear elastic crystals. We assumethat the crystallography of the interface has been specifiedcompletely. For a grain boundary, this requires five param-

eters: three to describe the relative misorientation betweenneighboring crystals, and two to describe the orientation ofthe grain boundary plane [9]. For a heterophase interface,the number of crystallographic degrees of freedom maybe higher. For example, an interface between two face-cen-tered cubic (fcc) crystals such as Al and Ni would requirethe lattice parameters of the two neighboring metals tobe given in addition to the five parameters needed for agrain boundary. Interfaces between materials with differingcrystal structures may require further parameters.

To describe completely the crystallography of a hetero-phase interface between elements A and B, we adopt thenotion of a reference state for the interface. In the refer-

ence state, the interface is coherent, i.e. the two separatecrystals that meet at the interface are rotated and strained[9,19] such that they are in perfect registry with each otheracross the interface plane after bonding. Thus, the refer-ence state has the interface structure of a single perfectcrystal.

Starting from the reference state, materials A and B aremapped separately into new configurations that yield aninterface with the required crystallographic character andzero far-field stresses, as shown in Fig. 1. Following Hirth,Pond and co-workers [18], we refer to the state of the inter-face after this mapping as the natural state. For a grain

boundary, the maps applied to materials A and B areproper rotations, while for a pure misfit interface they arepure strains. To account for both cases as well as for het-erophase interfaces between misoriented crystals, wedescribe the maps as uniform displacement gradients AFand BF. In the reference state, the neighboring crystalsmight not be stress free, but the interface is coherent. Inthe natural state, the interface is not coherent, but theneighboring crystals are both free of far-field stresses.

This framework is sufficiently general to describe thecrystallography of many commonly studied heterophaseinterfaces, e.g. ones formed by fcc and body-centered cubic(bcc) metals [1,20], but not all. For example, mapping froma common reference state to an interface between a cubicand hexagonal close-packed crystal cannot be accom-plished by a displacement gradient alone and requires aninternal shuffle rearrangement as well [21]. In the presentwork, we restrict ourselves to materials that may bemapped to a common reference state using displacementgradients alone.

The crystallographic considerations described above donot require a single, unique reference state. On the con-trary, an infinite number of new reference states may begenerated from an original one by applying to it any uni-form displacement gradient RF. If the original reference

state may be mapped to the natural state with AF and

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BF, then the new reference state may be mapped to thesame natural state using AFRF

1 and BFRF1. However,

a consistent description of the elastic fields of a discrete dis-location network in an interface of specified crystallogra-phy and free of far-field stresses does require a singlespecific reference state.

2.2. Volterra dislocations in the reference state

The atomic structures of real interfaces are not likethose generated by the linear mappings from a referencestate. Instead, for any given interface crystallography, theatomic structure may undergo a variety of local relaxationsor reconstructions that lower its energy. In many low-mis-orientation grain boundaries and low-misfit heterophaseinterfaces, these changes lead to formation of regions ofcoherency (which generally have low energies) separatedby networks of dislocations. Many such interface disloca-

tion networks have been imaged using transmission elec-tron microscopy [22].There are two common ways of describing interface dis-

locations. In one, they are viewed not as conventional Vol-terra dislocations, but rather as special kinds of interfacedefects with short-range elastic fields that are formed whenthe interface atomic structure in the natural state relaxes[23,24]. The superimposed elastic fields of all such defectsresiding within an interface decay away to zero at longrange and therefore do not alter the far-field stress stateor the crystallography of the natural interface state.

Another descriptionthe one we adopt hereviewsinterface dislocations as genuine Volterra dislocations withresultant elastic stress and strain fields that need not decayto zero at long range. For example, the structure of somepure misfit heterophase interfaces may be described as anarray of equally spaced edge dislocations residing on thesame glide plane [25]. It may be shown that such an arrayof Volterra dislocations has a non-zero far-field stress [26].Certain symmetric tilt grain boundaries may be describedas arrays of edge dislocations lying directly one above theother on separate glide planes. Such Volterra dislocationarrays have zero far-field strains (hence, also zero stresses[26]), but non-zero far-field rotations [27,28]. In general,arrays of Volterra dislocations may have non-zero far-field

strains, rotations, or both.

In the work described here, we model interface disloca-tions as Volterra dislocations that have been introducedinto the reference state, as shown in Fig. 1. We require thatthe far-field stresses due to these dislocations Ar

1dis and Br

1dis

are equal and opposite to the coherency stresses Arc and

Brc in the reference state, respectively, leading to the

removal of all far-field stresses in the natural state:

Arc Ar1dis 0 and; Brc Br1dis 0: 1Although free of long-range stresses, interface dislocationnetworks in the natural state have non-zero short-rangeelastic fields as a result of the superposition of the non-uni-form stress fields of the Volterra dislocation networks andthe uniform coherency stresses in the reference state. Addi-tionally, the far-field rotations due to the Volterra disloca-tions are required to conform to the given interfacecrystallographic character. These requirements restrict thechoice of reference states to a single specific one.

We treat the notion of introducing Volterra dislocationsinto the reference state primarily as a hypothetical opera-tion. However, this operation may be a physically mean-ingful analog of processes occurring at some realinterfaces. For example, the transformation of certaincoherent heterophase interfaces into ones that are notcoherent, but free of far-field stresses, occurs by the depo-sition on the interface of Volterra dislocations that glidethrough the neighboring crystalline layers [25,2931]. Sim-ilarly, subgrain boundaries are thought to assemble fromglide dislocations formed during plastic deformation ofpolycrystals [32].

2.3. Crystallographic constraints on interface dislocations

A variety of shapes of interface dislocation networkshave been observed [22], but here we will limit ourselvesto ones that may be represented by j6 2 arrays of paralleldislocations with Burgers vectors bi, line directions ni, andinter-dislocation spacings di. Following previous investiga-tors [79], we relate these quantities to the density of admis-sible Volterra dislocations in the reference state andinterface crystallography as:

B

Xj

i1

n nidi

p bi AF1

BF

1

p T p;

2

Fig. 1. Mapping from a coherent reference state to the natural state using displacement gradients AF and BF. Volterra dislocations introduced into thereference state remove coherency stresses and may change the misorientation of the neighboring crystals.

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(x1,x2,x3), as shown in Fig. 2b. An interface containingonly one array of straight parallel dislocations is a specialcase of this more general geometrical description. The unitvector normal to the interface is nkx2, with the interfacelocated at x2 = 0: x2 > 0 for material A, and x2 < 0 formaterial B. The dislocation line direction n1 is parallel topo2 and n2 p

o1

, as illustrated in previous studies [9,13,14].A representative interface unit cell of the dislocation

pattern is illustrated in Fig. 2b. Translations of the unit cellby the basis vectors po1 and p

o2 tessellate the interface plane.

It is also convenient to identify a non-orthogonal (oblique)

frame with basis vectors x01;x2;x03 , where x01 po1 n2 andx03kx3 po2

n1. The oriented angle between n2 and n1 isdenoted by /, so that x01 x1 csc / and x03 x3 x1 ctg /.Thus, any position vector in this non-orthogonal framemay be expressed as rx01 po1 x03 po2 .

Due to the periodicity of the interface dislocation struc-ture, it is useful to seek a complete set of wavevectors ksuch that the elastic fields in the interface may be analyzedusing plane waves ei2pkr. The set of all k is convenientlywritten as k np1 mp2 with respect to the reciprocalvectors p1 and p

2 , defined by the orthogonality conditions

pa pob dab, where dab is the Kronecker delta and n, m are

integers.The complete elastic distortion field Dtot is the superpo-sition of the uniform coherency and the Volterra disloca-tion distortions, Dc and Ddis, as discussed in Section 2.2.Following Bonnet [39,40], outside of dislocation cores, Dtotmay be expressed as the biperiodic Fourier series:

Dtotx Dc Ddisx Dc Xk0

ei2pkr Dkx2; 5

where i ffiffiffiffiffiffiffi1p and the sum spans over all non-zero wave-vectors k. The Fourier amplitudes of the complete distor-

tion waves Dk(x2) are required to converge (not necessary

to zero) in the far-field, i.e. x2 ! 1. The componentsk1 and k3 of the wavevector k satisfy:

k r k1 x1 k3 x3 n csc /

po1j j m ctg /

po2j j

x1 mpo2j j

x3: 6

The complete displacement field utot may be found by inte-grating Eq. (5):

utotx

u0

Dc x|fflfflfflfflfflffl{zfflfflfflfflfflffl}

affine part Xk0 ei2pkr uk

x2

uaffx udisx; 7

where u0 is an arbitrary constant displacement. The fieldutot may be decomposed into an affine part uaff correspond-ing to Dc and a biperiodic Fourier series representation ofdisplacement fields udis generated by the Volterradislocations.

The Fourier amplitudes in Eqs. (5) and (7) are deter-mined from linear elasticity in the absence of body forcesand subject to boundary conditions associated with inter-face dislocations. The complete displacement gradientsDtot(x) = grad utot (x) in crystals A and B must fulfill thepartial differential equations of mechanical equilibrium:

divC : grad utotx 0; 8where : denotes the double inner product and C is a fourth-order anisotropic elasticity tensor.

4.2. Complete field solutions

Substituting the displacement field Eq. (7) into Eq. (8),the second-order differential equation applied to both

half-spaces is obtained as follows:

(a) (b)

Fig. 2. (a) Schematic illustration of a planar interface dislocation network formed by bonding materials A and B. (b) The geometry of an interfacecontaining two sets of dislocations described by O-lattice vectors po1 and p

o2 . Open circles represent O-lattice points and filled circles illustrate atoms with

nearly matching positions in materials A and B.



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w1W1 ukx2 w2 W2 Wt2 @ukx2

@ x2 W3 @

2ukx2@ x22

0: 9with w1 = 4p2 and w2 = i2p. Here, t denotes the matrixtranspose and W1, W2 and W3 are 3 3 real matrices re-lated to the wavevectors (i.e. interface geometry) and thestiffness constants (i.e. elasticity) indexed in Voigt notation:

W2 k1c16 k3c56 k1c12 k3c25 k1c14 k3c45k1c66 k3c46 k1c26 k3c24 k1c46 k3c44k1c56

k3c36 k1c25

k3c23 k1c45

k3c34

2

64

3

75W3 Wt3

c66 c26 c46

c22 c24

sym c44

264375: 10

As demonstrated in Appendix A, the complete displace-ment field (7) may be written as expressed in Eq. (A.5), i.e.

utotx u0 Dc x 1i2p

Xk0

ei2pkrX3a1

kaei2ppax2aa

faei2ppax2aa; 11where the eigenvalues pa and eigenvectors aa are calculated

by solving the sextic Eq. (A.3) and the homogeneous linearsystem of Eqs. (A.2), respectively. The asterisk indicatescomplex conjugates of solutions with positive imaginaryparts, i.e. pa3 pa and aa3 aa, indexed by a = 1, 2, 3.The complete elastic strains and stresses are also deducedfrom Eq. (11) by:

Etotx fDtotxg 12

grad utotx grad uttotx

rtotx C : Etotx;12

respectively. Eq. (12a) gives the straindisplacement rela-

tionship, where {Dtot(x)} denotes the symmetric compo-nent of the distortion field, given by Eq. (A.6). Eq. (12b)is the generalized Hookes law for small strains that deter-mines the stress field, as expressed in Eq. (A.8). The generalsolutions of elastic fields of Eqs. (11) and (12) are expressedas linear combinations of the eigenfunctions given by Eq.(A.1), and include ka and fa as complex unknown quanti-ties that are to be determined by the boundary conditions.

The following two sections describe the boundary condi-tions associated with equilibrium interface dislocations:conditions 1 and 2 deal with the far-field elastic fields (Sec-tion 4.3), while conditions 3 and 4 are focused on specificrequirements at the interface (Section 4.4).

4.3. Far-field boundary conditions

Condition 1: convergence of elastic fields

In accordance with Saint Venants principle, the conver-gence of the Fourier amplitudes uk(x2) when x2 ! 1leads to the requirement that Af

a = 0 and Bka = 0. This

condition applies to infinite bicrystals and would not beappropriate for bicrystals terminated with free surfaces.

Condition 2: absence of strains in the far-fields

The elimination of the coherency strains Ec by the far-field strains of the interface Volterra dislocations E1dis is

taken into account by requiring the total elastic strain fieldEtot to decay to zero when x2 ! 1, i.e.lim

x2!1Etotx E1tot Ec E1dis 0; 13

where Ec = {Dc} and E1dis = {D

1dis} is the far-field strain

produced by the interface dislocations. Eq. (13) is equiva-lent to Eqs. (1) expressed using strains rather than stresses.As shown in Appendix B, the far-field distortions, calcu-lated individually for each set of dislocations, i= 1 and 2,and then superposed, are given by Eq. (B.10) as:

D1dis sgnx2 Re X2

i

1

d1i X3

a

1

kai Gai fai Gai: 14

Here, Afa1 Afa2 0 and Bka1 Bka2 0 for the reasons de-

scribed in boundary condition 1. Superimposed bars areused to indicate quantities related to the far-field boundaryconditions, as mentioned in Appendix B. Re stands for thereal part of a complex quantity. In contrast, the complexconstants Ak

ai and B

fai are determined by solving the systemof Eqs. (B.17) with the aid of the complex tensors Ga1 andGa2 given by Eqs. (B.8) and (B.11), respectively.

4.4. Interface boundary conditions

Condition 3: disregistry due to Volterra dislocations

Disregistry is the discontinuity of displacements acrossan interface [26], expressed in terms of the relative displace-ments between neighboring atomic planes. Each disloca-tion produces a stepwise change in disregistry at its corewith magnitude equal to its Burgers vector. The disregistryat x2 = 0 of a network of two sets of dislocations may berepresented by the staircase functions:

Dutotx1;x3 Autotx1;x3 Butotx1;x3

b1 csc / x1po

1j j$ % b2$

x3 ctg / x1po

2j j %; 15

W1 Wt1 k21c11 2k1k3c15 k23c55 k21c16 k1k3c14 c56 k23c45 k21c15 k1k3c13 c55 k23c35

k21c66 2k1k3c46 k23c44 k21c56 k1k3c36 c45 k23c34sym k21c55 2k1k3c35 k23c33

264

375

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as illustrated in Fig. 3, where only one set has been dis-played for clarity. According to Eq. (7), the complete dis-placement discontinuity Dutot at the interface is expressed

as:Dutotx1;x3 Duaffx1;x3 Dudisx1;x3: 16

The left-hand side of Eq. (16) gives the relative displace-ment field Duaff at the interface generated by the uniformmacroscopic distortions ADc and BDc in the affine form:

Duaffx1;x3 Du0 sADc BDcxtx20; 17

where Du0 12 b1 b2 is chosen, without loss of gener-ality. As shown in Fig. 3, Eq. (17) may be interpreted as acontinuous distribution of (fictitious) Volterra dislocationswith infinitesimal Burgers vectors and spacing [8,41].

The right-hand side of Eq. (16) is the displacement dis-continuity Dudis produced by equilibrium interface disloca-tions in the natural state, shown as D in Fig. 1. Accordingto Eq. (7) and (11), the quantity Dudis is given by:

Dudisx1;x3 1i2p

Xk0

ei2pkrX3a1

Aka

Aaa BfaBaa; 18

which may be represented by sawtooth functions[39,42,43], as illustrated in Fig. 3. Using the Fourier sineseries analysis and superposing the sawtooth-shaped func-tions associated with the two sets of dislocations, Eq. (18)

can be expressed as:

D udisx1;x3 X1n1

b1np

sin 2pncsc / x1

po1j j

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{set 1

X1m1

b2mp

sin 2pmx3 ctg / x1

po2j j|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}set 2

: 19

Thus, the boundary condition (19) for equilibrium inter-face dislocations, combined with Eq. (18), leads to a set

of six linear equations:

R1 :

ReX3a1

Aka

Aaa BfaBaa #

ImX3a1

Aka

Aaa BfaBaa 0;

8>>>>>>>:

20

where Im stands for the imaginary part of a complex quan-tity and # is given by:

# b1

nif m 0 nP 1

b2m

if n 0 mP 10 if nm 0 n;mP 1:

8>: 21

Condition 4: no net tractions along the interface

The solution must satisfy the boundary condition:

Artotx1; 0;x3n Brtotx1; 0;x3n; 22where r

tot(x

1, 0, x

3) is reduced to the short-range stress field

produced by the interface equilibrium dislocations whenEqs. (1) are satisfied. In that case, following Eq. (A.8) thetractions at the interface may be written as:

rtotx1; 0;x3n sgnx2Xk0

ei2pkrX3a1

kaha faha; 23

where the subsidiary complex vectors ha are related to thevectors aa by:

ha Wt2 pa W3

aa pa1W1 pa W2aa; 24with hak Hak2, as in Eq. (A.10). Boundary condition (22)together with Eq. (23) leads to the additional system ofsix linear equations:

R2 :

ReX3a1

Aka

Aha BfaBha 0

ImX3a1

Aka

Aha BfaBha 0:

8>>>>>>>:

25

The two latter conditions 3 and 4 may be rewritten in aeigenvalue problem for equilibrium interface dislocationarrays. Indeed, the elastic fields of these dislocations inan anisotropic bicrystal free of far-field strains are givenin terms of the 12 eigenvalues Eval and 12 corresponding

eigenvectors Evec with a = 1, 2, 3, i.e.

Eval fRe Apa; Im Apa; Re Bpa; Im BpagEvec fAaa; Baa; Aha; Bhag:

26

All these quantities are determined by solving a six-dimen-sional eigenvalue problem that may be recast with the aidof Eqs. (24) into the form:

Naa

ha

! pa a

a

ha

!27

where the real nonsymmetric 6 6 matrices N depend onthe wavevectors and the stiffness constants for crystals A

and B through the W matrices given by Eqs. (10), i.e.

Fig. 3. The disregistry Dutot due to interface Volterra dislocations is astaircase function. It may be decomposed into an affine part Duaffgenerated by a uniform distortion (represented by a continuous distribu-tion of fictitious infinitesimal dislocations) and a sawtooth function Dudisassociated with the equilibrium interface dislocations in the natural state.



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N W13 W

t

2 W13

W1 W2W13 Wt2 W2 W13

" #: 28

Finally, the linear systems R1 and R2 are solved numericallyto determine the 12 real constants Ecst, i.e.

Ecst

fRe Ak

a; Im Aka; Re Bf

a; Im Bfa

g;

29

completing the solutions of the elastic fields.

4.5. Interface elastic strain energy

Using the divergence theorem, the elastic strain energyof equilibrium interface dislocation arrays may beexpressed as a surface integral over a unit cell A of theinterface dislocation network:

Ee 12 A

ZZA

rtotx1; 0;x3n Dudisx1;x3 dS; 30

where rtot(x1, 0, x3)n is the traction vector produced at the

interface of interest. Stress fields at dislocation cores di-verge, so regions near the cores must be excluded fromthe integral in Eq. (30). Following standard practice [26],we limit the domain of integration to parts of the interfaceunit cell that are not within a predetermined cutoff distancer0 of the dislocation cores.

5. Example applications

Here, we apply the model described in the forgoing sec-tions to simple example interfaces: symmetric tilt and twistgrain boundaries as well as a pure misfit heterophase inter-

face. The materials properties used in these examples arelisted in Table 1. Interfaces with both misorientationsand misfits will be treated in a separate study.

5.1. Symmetric tilt grain boundary

Pure tilt boundaries that contain one set of dislocationshave been discussed extensively [9,14]. To illustrate andvalidate the present method, we discuss a symmetrical tiltboundary with [001] tilt axis and tilt angle h = 2. The cal-culations are carried out for Cu, which has a moderatelyhigh anisotropy ratio, ACu = 2c44/(c11 c12) = 3.21. Theboundary consists of one set of straight parallel disloca-

tions with Burgers vector content B (see Eq. (2)) expressedas:

B n nd

p

b R1 R1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

T

p

2 sin h=2 p x: 31Here, we have used a median lattice as the obvious refer-ence state: the mapping matrices F have been replaced by

rotation matrices R, with R+ representing a rotation ofthe upper crystal by angle h+ = h/2 about the tilt axisand R

the rotation h

= h/2 of the adjacent lower crys-

tal. Eq. (31) is known as Franks formula [15,45], whichgives the density of interface dislocations needed to createthe tilt boundary. Selecting b = aCu[010]kn, Eq. (31) showsthat n = [001] and d= 10.3567 nm.

We confirmed that the far-field stresses vanish for thischoice of reference lattice, as expected, and that the onlynon-zero stresses are short-ranged. Fig. 4 plots interfacestresses as a function ofx1 and x2 (the stresses are invariantalong the dislocation line direction, x3). The red contour

illustrates where the stresses fall to zero when jx2jP 710 nm (depending on the stress components), showing thattheir range is comparable to the dislocation spacing. Thefar-field rotations may be calculated from the antisymmet-ric part of the far-field distortions, i.e. X1 gD1disf. Theysatisfy X1 X1 T and yield a net non-vanishing rota-tion about the tilt axis, as excepted [46,47]:

- -1 -1 0

0

0:03490

0B@

1CA x1 b

d: 32

The disregistry Du2 tot and the displacement discontinuity

Du2 dis associated with the Volterra and equilibrium tiltboundary dislocations are plotted in Fig. 5a. They are ingood quantitative agreement with the applied boundaryconditions, represented by staircase and sawtooth curves.

The average elastic energy per unit interface area Ee isdetermined for several values of the core cutoff parameterr0. Following Eq. (30), Ee may be written as:

Eer0 12 d

Zdr0r0

r22 totx1; 0; 0Du2 disx1; 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}W

dx1: 33

The variation of stress component r22 tot at x2 = 0 with x1

is plotted as a black line in Fig. 5b. The core region isshaded in gray. Local contributions to the interface elasticenergy W(values of the integrand in Eq. (33)) are plotted inred. The average elastic energy per unit interface area willdepend on the choice of r0. For example, Ee = 142.8mJ m2 with r0 = b/2 and Ee = 167.8 mJ m

2 with r0 = b/3, where b is the magnitude ofb. We attempt to determinean appropriate r0 value by comparing the interface elasticenergies computed with our method to experimentally mea-sured energies of small-angle [00 1] tilt boundaries [48],plotted as solid triangles in Fig. 6. Our calculations usingr0 = b/2 are in good agreement with the experiments up

to $5

, while r0 = b/3 fits better in the range of $512.

Table 1Material properties for copper, niobium, iron, aluminum and nickel. Thevalues of lattice parameters a for all materials are those listed by Gray [44]and elastic components c11, c12 and c44 by Hirth and Lothe [26].

Properties Materials

Symbol Unit Cu Nb Fe Al Ni

a A 3.615 3.301 2.866 4.050 3.524

c11 GPa 168.4 246.0 242.0 108.2 246.5c12 GPa 121.4 134.0 146.5 61.3 147.3

c44 GPa 75.4 28.7 112.0 28.5 124.7

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The classical energy per unit area given by Read andShockley [27], ERS (h) = 1450 h(3 lnh) mJ m2, is alsoshown in Fig. 6. It compares well with our calculations forr0 = b/3.

5.2. Twist grain boundary

As shown in Fig. 7a, small-angle (01 0) twist grainboundaries contain two sets of dislocations, so their dislo-

cation content B can be expressed as:

(a)

(b)

(c)

Fig. 4. Contour plots of stress components (a) r11 tot, (b) r12 tot and (c)r22 tot, for the 2 symmetric tilt boundary described in the text. Thenegative values (compression) are plotted in light gray, and the positivevalues (extension) in dark gray. The stresses decay away over distancescomparable to the interface dislocation spacing. In red, the stress fieldvalues are equal to zero. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

(a)

(b)

Fig. 5. (a) Disregistries Du2 tot (staircase function) and Du2 dis (sawtoothfunction) computed using 100 harmonics for the 2 symmetric tiltboundary described in the text. (b) Stress distribution r22 tot and localelastic energy density Ee at the grain boundary.

Fig. 6. Interface elastic energies Ee computed using two different corecutoff parameters r0 for a [001] tilt grain boundary in Cu as a function ofthe tilt angle h. The gray line shows the ReadShockley solution.

Experimental values are shown with solid triangles [48].



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B n n1d1

p

b1 n n2d2

p

b2

R1 R1

p: 34We consider twist boundaries of angle h = 2 in Cu, wherethe rotation axis is perpendicular to the boundary,x = x2 = [010]. As in the case of the tilt boundary, theobvious reference state for twist boundaries is the medianlattice suggested by Frank [49]. In this state, the totalrotation across the boundary is equally partitioned be-tween the two grains. However, to illustrate the importanceof selecting the correct reference state, we will also considerother possible reference states. A common choice is to useof the adjacent crystal grains as the reference state. There isa continuum of other possible reference states betweenthese two extremes, so we introduce the angle hc = jhto define the rotation of the reference state from the casewhere the upper crystal above the boundary has been cho-sen as the reference lattice. Here, j is a dimensionlessparameter that varies from 0 to 1. Equipartitioning of rota-tions between the adjacent crystals (i.e. the medianlattice) occurs when j = 1/2.

Section 2.3 demonstrated that interface dislocation

geometry is independent of reference state. In this example,

the twist boundary contains an orthogonal grid of disloca-tions with line directions n1 1=

ffiffiffi2

p 1 0 1 andn2 1=

ffiffiffi2

p 1 0 1. The spacings between successive paralleldislocations are d1 = d2 = d= 7.3233 nm. Because of thepure twist misorientations, the coherency stress fields arezero for all possible reference states. Fig. 7b plots the

dependence of non-vanishing far-field stress componentson j. If a reference state with j = 0 is chosen, then theinterface dislocations deviate by 1 from pure screw charac-ter and possess non-zero far-field stress componentsr111 r133 and r111 r133 . This demonstrates thatj = 0 does not represent the correct reference state sinceEqs. 1 (and Eqs. (13) via Eq. (12b)) are not satisfied. Fur-thermore, the far-field rotation with j = 0 does not equal2, but discrepancies of the order of 0.001 between therotation vector component and the prescribed misorienta-tion are found. As j increases, the far-field stresses decreaseand eventually reach zero at j = 1/2, as expected. Theinterface dislocations have perfect screw characters for this

reference state, where non-zero far-field stresses are againobtained when j is increased beyond j = 1/2.

Taking j = 1/2, we calculate the elastic strain energy perunit area Ee for the twist grain boundary using theexpression:

Eer0 12 A

ZZdr0r0

W1 W2 W12 dx1 dx3; 35

with A po1 po2 the area of the interface unit cell. Eq.

(35) is decomposed into self-energy densities W(1) andW(2) for each set of parallel dislocations and the W(12) be-tween the two sets. These energies are obtained from the

separate elasticity solutions for each set of dislocations:

W1 W2 r23 tot 1x1; 0; 0 Du3 dis 1x1; 0 r12 tot 20; 0;x3Du1 dis 20;x3

W12 r23 tot 1x1; 0; 0Du1 dis 20;x3 r12 tot 20; 0;x3Du3 dis 1x1; 0:

36

The local self-and interaction energies are shown in Fig. 8aand b, respectively. The integral of the interaction energyW(12) over area A is zero for any value r0, in agreement

(a)

(b)

Fig. 7. (a) Small-angle twist grain boundary on a (010) plane containingtwo sets of orthogonal dislocations. (b) Dependence of far-field stresses onj for the 2 twist boundary described in the text.

(a) (b)

Fig. 8. Local (a) self-{W(1) + W(2)} and (b) interaction W(12) elasticenergies arising from two sets of orthogonal screw dislocations in a 2

twist boundary on a (010) plane in Cu.

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with the classical dislocation theory result that orthogonalscrew dislocations do not exert any forces on each other[26]. The total elastic energy is plotted in Fig. 9 as a func-tion of the twist angle up to 12 for three core cutoff param-eters: r0 = b1/2, r0 = b1/3 and r0 = b1/4.

5.3. Comparison of tilt and twist grain boundary energies

In this section, we use our model to compare the elasticenergies of small-angle tilt and twist boundaries with iden-tical misorientation angles about the same [001] rotationaxis. We are particularly interested in determining whether

anisotropy influences which of the two has lower energy fora given misorientation. We carry out our calculations on ahypothetical ironniobium alloy, FevNb1v. This is a con-venient choice because Fe and Nb have similar c11 and c12values, as shown in Table 1. Their c44 values, however, dif-fer markedly and therefore their anisotropy ratios do aswell. The elastic constants and lattice parameters of thehypothetical FevNb1v alloy are found by linear interpola-tion between those of Fe and Nb.

Fig. 10a shows the elastic energies of 2 tilt and twistboundaries computed as a function of 0 6 v 6 1 with r0= bFeNb/4, r0 = bFeNb/3, and r0 = bFeNb/2, such thatbFeNb = vbFe + (1

v)bNb. Tilt boundary energies vary

roughly logarithmically with v while twist energies increaselinearly with the increasing v. Fig. 10b compares theseenergies over 0 6 v 6 0.2. The tilt boundary energy ishigher than that of the twist boundary for pure Nb andfor all v 6 0.09. By contrast, the twist boundary energy ishigher than the tilt boundary energy for Fe and forv > 0.09. These findings demonstrate that the relative ener-gies of tilt and twist boundaries may be quite sensitive toanisotropy.

These calculations were performed using several differ-ent dislocation core cutoffs for edge and screw dislocations.In general, one may expect dislocations with screw charac-

ter to have larger cores [26]. Inspection of Fig. 10 demon-

strates that the choice of core cutoff may affect the valueof v at which the crossover in boundary energies occurs,but does not alter the qualitative conclusion that the rela-tive energies of tilt and twist boundaries depend onanisotropy.

5.4. Pure misfit interface

Lastly, we illustrate our model on an AlNi heterophaseinterface. The terminal planes of both adjacent crystals arefcc (010) planes. The [100] and [001] directions of bothcrystals are parallel in the interface plane. Thus, the inter-face is in the cube-on-cube orientation and contains twosets of parallel dislocations. Following Eq. (2), the Burgersvector content B is written as:

B n n1d1

p

b1 n n2d2

p

b2

AlS1rAl NiS1rNi|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}T

p: 37

Fig. 9. Elastic energies per unit area Ee as a function of the rotation angleh of twist grain boundaries along (010) planes in Cu for three core cutoffparameters r0.

(a)

(b)

Fig. 10. Elastic energies per unit area Ee of 2 tilt and twist grainboundaries in an ironniobium alloy, FevNb1v, as a function of v.Calculations are performed for three core cutoff parameters r0 and plottedin the v ranges of (a) 0 6 v 6 1 and (b) 0 6 v 6 0.2 for clarity.



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The reference state for this interface is a crystal orientedidentically to the Al and Ni in their natural state, butstrained such that its lattice constant in the interfaceplane is ac, with aNi 6 ac 6 aAl. Only strains within theinterface are necessary to ensure coherency: normal strainsare not required. Thus, the matrix T in Eq. (37) is

composed of two equibiaxial stretch matrices (norotations), AlS1 = AlEc + I and NiS

1 = NiEc + I, where Irepresents the identity matrix. These mapping matrices de-pend on the ratios of lattice parameters between Al and Niin their natural and reference states, rAl = aAl/acP 1 andrNi = aNi/ac 6 1. The matrix T in Eq. (37) may also berewritten as the difference between the coherency strainsprescribed in Al and Ni:

AlEc NiEc T: 38Following the procedure described in Section 3, we ini-

tially choose Ni as the reference lattice, so that rAl = aAl/aNi and rNi = 1, and identify b1

aNi=ffiffiffi2

p

101

and

b2 aNi=ffiffiffi2p 101. Then, using Eq. (3), we find an interfacethat consists of an orthogonal grid of edge dislocationswith n1 1=

ffiffiffi2

p 1 0 1 and n2 1=ffiffiffi

2p 1 0 1, and the corre-

sponding dislocation spacings d1 = d2 = 1.902 nm. Usingthis choice of reference state, we find that the far-fieldstrains produced by the interface dislocations are:

AlE1dis

0:10133 0 0

0 0 0

0 0 0:10133

264

375

NiE1dis

0:03243 0 0

0 0 00 0 0:03243

264 375;39

such that the matrices in Eqs. (39) satisfy:

AlE1dis NiE1dis T: 40

Combining Eqs. (38) and (40), we find:

AlEc AlE1dis NiEc|ffl{zffl}0

NiE1dis () AlE1tot NiE1tot 0:03243 0 00 0 0

0 0 0:03243

264

375 0;41

with NiEc = 0 here, because Ni has been chosen as the ref-erence lattice. However, according to Eq. (41b), condition

2, given by Eq. (13) is not satisfied since the total far-fieldstrains in each individual material do not decay to zerowhen x2 ! 1. This demonstrates that the initial choiceof reference state is not correct.

To find the correct reference state, we introduce a vari-able d, with 0 6 d 6 1, that interpolates ac between aAl andaNi:

ac daAl 1 daNi: 42It is shown that the far-field strains in Al and Ni are equalfor all d, so that Eq. (41a) is always satisfied, i.e.

AlE1tot NiE1tot with NiEc = 0 ifd = 0 and AlEc = 0 if d = 1.

However, only one unique reference state (corresponding

to a unique value ofd) gives vanishing far-field strains inthe bicrystal in its natural state by satisfying Eq. (13) aswell. The pure misfit interface example serves to show thatEq. (41a) is a necessary, but not sufficient, condition fordetermining the reference state.

The total far-field strain component AlE111 tot in Al is plot-

ted in Fig. 11 as a function ofd and is identical to the compo-nent AlE133 tot, according to the interface symmetry (all other

strain components are zero). Because Eq. (41a) is verified forall d, the same components in Ni give the same plot as inFig. 11. The far-field strains vary linearly with d and becomezero when d = 0.21787, so that ac = 0.36386 nm. This valueofac is the unique coherent reference state for whichthe puremisfit AlNi interface of interest is consistent with theFrankBilby equation. It is closer to aNi than to aAl becauseNi is the stiffer of these two materials and so carries a lowercoherency strain in the reference state. The far-field rotationsare zero for all values ofd, as excepted.

To demonstrate the errors that come about from ignoring

the unequal partitioning of elastic fields and to validate ourcalculation, we recompute ac under the assumptionthat bothsides of the interface have the same stiffness (equal to that ofAl or Ni), but different natural lattice parameters (aAl andaNi, as the original calculation). The ac value we calculatefor this case is in very good agreement with the well-knownapproximate result a 2aAl aNi=aAl aNi 0:37687 nm[7,19,50], corresponding to d = 0.46521. This value, how-ever, is far from the correct lattice parameter of the referencestate when the differing stiffnesses of Al and Ni are taken intoaccount, as illustrated by cross symbols in Fig. 11. It is alsoshown that a deviates from our prediction and is not consis-

tent with the FrankBilby equation when the heterogeneousdistortions of bicrystals are explicitly described atequilibrium.

To investigate the effect of the relative stiffness of theneighboring materials on the reference state, we artificially

Fig. 11. Dependence of the total far-field strain component AlE111 tot in Al

on d for a AlNi heterophase interface. The red dotted line gives theunique reference state, for which the far-field decay to zero and thecoherent parameter ac is defined. The lattice parametera 2aAl aNi=aAl aNi, which is a good approximation for an interfacebetween crystals of different lattice parameters but identical elastic

constants [7,19,50], is marked by a gray cross symbol.

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vary the value of Alc11 for Al, assigning to it a value ofAlc11 = s Alc11, where Alc11 is the elastic constant forvirtual Al, and 0 6 s 6 30. Fig. 12 shows that the latticeparameter of the reference state ac increases with s, i.e. asthe fictitious Al become stiffer, and approaches the latticeparameter of real Al asymptotically. This is to be expectedbecause, as the stiffness of a material increases, the coherencystresses it would carry in the reference state become prohib-itively large, so its coherency strains must decrease. Whens = 0 the coherent parameter is naturally definedby ac $ aNi.To measure the discrepancy of choosing a as the latticeparameter defined in the reference state, Fig. 12 shows thata corresponds to consider a fictitious Al material with

Alc11 assigned to $ 8 Alc11 = 2000 GPa.

6. Concluding remarks and outlook

We have developed a formalism for determining the ref-erence states of semicoherent interfaces that give rise to dis-locations whose far-field elastic fields meet the condition ofvanishing far-field strains and prescribed misorientations.Interface dislocations are viewed as Volterra dislocationsthat have been inserted into this unique reference state,subject to the above-stated constraints. The complete elas-tic fields of these dislocations are calculated using heteroge-

neous anisotropic linear elasticity and interface dislocationconfigurations consistent with the FrankBilby equation.The present model resolves the ambiguity arising fromthe infinite number of reference states available when theFrankBilby equation is analyzed based on geometryalone, i.e. without consideration of the elastic fields.

We have shown several example calculations that illus-trate the importance of accounting for the reference state.An incorrect reference state leads to non-zero far-fieldstresses, incorrect far-field rotations, or both. We have alsoshown that it is important to account for the anisotropy ofelastic constants in the materials joined at the interface.

For example, the degree of anisotropy determines whether,

for a given misorientation angle, pure tilt or pure twistgrain boundaries have lower energies. Finally, the differ-ence in elastic constants between the adjoining crystals isalso important and leads to unequal partitioning of elasticfields. The correct reference state is generally closer to thestiffer of these two crystals.

The method presented has numerous additional applica-tions besides the ones illustrated here. For example, werestricted our attention here to interfaces with unique dislo-cation spacings and line directions. However, in some cases[18], multiple dislocation geometries are consistent with theFrankBilby equation and additional criteria are needed toselect the correct one. The elastic strain energies computedby the method described here may be used as just such a cri-terion: the most likely dislocation geometry is the one withleast strain energy. Another application arises in the analysisof dislocations at heterophase interface in atomistic simula-tions [1]. The correct reference state is needed to characterizethe disregistry associated withmisfit dislocations in such sim-

ulations. Applications of our method to the analysis of fccbcc interfaces will be presented in a follow-on study.

Extensions of the formalism presented here to new prob-lems are also possible. For example, bilayers of finite thick-ness terminated with free surfaces or layered superlatticeswith differing layer thicknesses would give rise to differentsolutions for elastic fields than the ones presented here. Elas-tic interaction energies between interface dislocation arraysandother defects, such as point defects, extrinsic dislocationsor cracks, may also be studied. Finally, more realistic modelsof dislocation cores, such as ones based on the PeierlsNab-arro model [5154], may be incorporated.

Acknowledgements

We thank the following individuals for fruitful discus-sions: R. Bonnet, W. Cai, J.P. Hirth, R.G. Hoagland, X.-Y. Liu, R.C. Pond, J. Wang and A. Rollett. This researchwas funded by the US Department of Energy, Office of Sci-ence, Office of Basic Energy Sciences under Award No.2008LANL1026 through the Center for Materials at Irra-diation and Mechanical Extremes, an Energy Frontier Re-search Center at Los Alamos National Laboratory. M.J.D.acknowledges support from the National Science Founda-tion under Grant No. 1150862 for the work described inSections 2 and 5.

Appendix A. Complete elastic field solutions

The complete solutions for elastic fields associated witha network consisting of two sets of dislocations is treatedby applying the sextic formalism developed by Stroh [35].It may be used to find elastic fields of straight dislocationsin anisotropic linear elastic solids. For non-zero wave vec-tors k, the standard solutions satisfying Eq. (9) can be writ-ten in the form [55]:

ukx2

ei2ppx2 a;

A:1

Fig. 12. Dependence of the coherent lattice parameter of a AlNiheterophase interface as the stiffness constant Alc11 of Al is artificiallyvaried, such that Alc11 = s Alc11 (see text).



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where p and a become the unknowns of the boundary valueproblem. Introducing Eq. (A.1) into Eq. (9), the vector a isfound to satisfy the homogeneous linear system:

W1 pW2 Wt2 p2 W3

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Pa 0; A:2

which is an eigenvalue problem with p as the unknown[35,56]. A non-trivial solution can be found only if:

det P 0: A:3This leads to a sextic equation for p. Due to the positivedefiniteness of elastic strain energy, the solutions of Eq.(A.3) have six imaginary roots, which occur in complexconjugates [34]. It is convenient to arrange the three firstsolutions pa to have positive imaginary parts, indexed bysuperscripts a = 1, 2, 3. The remaining three solutions havenegative imaginary parts, so that pa3 pa, where * indi-cates complex conjugation. The corresponding vectors aa

are also complex conjugates with aa

3

aa

, so that the gen-eral solution may be rewritten as a linear combination ofthe three eigenfunctions:

ukx2 1i2p

X3a1

kaei2ppax2 aa faei2ppax2 aa: A:4

Here, ka and fa are complex scaling parameters that dependon boundary conditions. Substituting Eq. (A.4) into Eq.(7), the complete elastic displacement field may be writtenin terms of a biperiodic Fourier series expansion, i.e.

utotx u0 Dcx 1i2pXk0 e

i2pkr

X3

a1

kaei2ppax2aa

faei2ppax2aa: A:5The elastic distortion derived from Eq. (A.5) is therefore:

Dtotx Dc Xk0

ei2pkrX3a1

kaei2ppax2 Ga faei2ppax2 Ga;

A:6where the complex matrices Ga Gakl are defined by:

Gakl aakk1 dl1 k3 dl3 pa dl2; A:7for the same wave vector k. Moreover, the associated com-

plete stress field is obtained by using Hookes law:

rtotx rc Xk0

ei2pkrX3a1

kaei2ppax2 Ha faei2ppax2 Ha;

A:8where the coherency stress field rc is related to the coher-ency strain by:

rc C : Ec C : fDcg; A:9and the matrices Ha Hakl are defined by:Hakl

k1 cklj1

k3 cklj3

pa cklj2

aaj :

A:10

Appendix B. Far-field solutions for interface dislocations

The far-field elastic fields for dislocation arrays areobtained from the complete expressions derived inAppendix A by determining separately the contributionof each set of dislocations when x2 ! 1 and thensuperposing the individual solutions. The total displace-ment field for two sets of parallel dislocations may berewritten as:

udisx1;x2;x3 udis 1x1;x2 udis 2x1;x2;x3; B:1where subscripts dis1 and dis2 denote the individual sets ofdislocations 1 and 2. Each individual displacement fieldmay be determined by requiring po1

! 1 or po2 ! 1in Eq. (6), giving:

udis 1x1;x2 X1n1

ei2pknx1 unx2

udis 2x1;x2;x3 X1m1

ei2pkmx1k3x3 umx2; B:2

where kn, km and k3 are the three wavelengths defined inEq. (6):

kn n csc /po1j j

; km m ctg /po2j j

and; k3 mpo2j j

: B:3

Without loss of generality, we restrict the calculation to set1. From Appendix A, the displacements un(x2) in Eq.(B.2a) are non-trivial solutions of the second-order differ-ential Eq. (9), written here as:

0 4p2 k2n C1unx2 i2pkn C2 Ct2 @unx2

@x2

C3 @2unx2@x22

; B:4

with C1 = cj1k1, C2 = cj1k2 and C3 = cj2k2. Similar to expres-sion (A.4), the most general solution of Eq. (B.4) is the lin-ear combination of eigenfunctions:

unx2 1i2pn

X3a1

ka1ei2pknp

a1x2 aa1 fa1ei2pknp

a1x2aa1; B:5

where the six complex eigenvalues pa1 and the correspond-ing eigenvectors aa1 are determined by solving the two fol-lowing equations:

det P1 0P1 a

a1 0

with; P1 C1 p1 C2 Ct2 p21C3: B:6

Here, the subscript dis in Eqs. (B.5) and (B.6) is omitted forclarity. Unlike ka and fa in Eq. (A.4), the complex con-stants ka1 and

fa1 in Eq. (B.5) have been divided by n 0[57]. This operation is convenient when evaluating thesum over n of Eq. (B.2a). The elastic distortion field is thenobtained by differentiation of Eqs. (B.2a) and (B.5):

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Ddis 1x1;x2 d11X1n1

ei2pknx1X3a1

ka1ei2pknp

a1x2 Ga1

fa1ei2pknpa1x2 Ga1: B:7

Here, d1 po1

cos / and Ga1 Ga1kl are 3 3 complexmatrices defined by:

Ga1kl aa1k dl1 pa1dl2

: B:8The sum over n in Eq. (B.7) may be calculated analytically,so the far-field distortion D1dis 1 is:

D1dis 1 sgnx2 d11X3a1

ka1Ga1 fa1Ga1: B:9

Similar results are found for the elastic distortion producedby set 2. The total far-field distortion is obtained by addingthe contributions of both dislocation sets:

D1dis

D1dis 1

D1dis 2

sgnx2 ReX2i1

d1iX3a1

kai Gai fai Gai; B:10

where the matrices Ga2 Ga2kl are defined by:Ga2kl aa2k cos /dl1 sin /dl3 pa2dl2

: B:11

The complex eigenvalues pa2 and eigenvectors aa2 are deter-

mined by solving Eqs. (B.6) with the associated matrixP2 defined by:

P2 C12 p1C22 p21C32; B:12where:

C12 cos2 / cj1k1 sin2 / cj3k3 12

sin2/ fcj1k3gC22 cos / fcj1k2g sin / fcj2k3gC32 C3 cj2k2:

B:13

According to Eq. (B.10), the overall long-range stress fieldproduced by set 1 + 2 in both half-spaces is:

r1dis r1dis 1 r1dis 2

sgnx2

X2

i1d1i Re

X3

a1kai H

ai fai Hai; B:14

where Ha1 Ha1kl and Ha2 Ha2kl are defined by:Ha1kl cklj1 pa1 cklj2

aa1k

Ha2kl cos / cklj1 sin / cklj3 pa2 cklj2

aa2k:B:15

Finally, to obtain the complete far-field solutions of thedistortions by Eq. (B.10) and the stresses by Eq. (B.14),the 24 remaining unknown complex constants, i.e.

Ecst Re Akai ; Im Akai ; Re Bfai ; Im Bfai

; B:16for a 2 {1,2,3} and i2 {1, 2}, are found by solving the twofollowing linear systems of 12 equations for each wave vec-

tor k 0:

R11

2 :

ReXk0

X3a1

Akai Aa

ai Bfai Baai bi

ImXk0

X3a1

Akai Aa

ai Bfai Baai 0

ReXk0

X3a1

Akai Ah

ai Bfai Bhai 0

ImXk0

X3a1

Akai Ah

ai Bfai Bhai 0;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

B:17

according to the conditions 3 and 4 for specific require-ments at the interface, as discussed in Section 4.4. The vec-tors ha are related to Ha by

ha1k Ha1k2 and; ha2k Ha2k2: B:18

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