nanoscale planar field projections of atomic decohesion

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Philosophical Magazine,Vol. ??, No. ?, Month?? 2006, 1–31

Nanoscale planar field projections of atomic decohesion and slip5 in crystalline solids. Part I. A crack-tip cohesive zone

S. T. CHOI and K.-S. KIM*

Division of Engineering, Brown University, Providence, RI 02912, USA

(Received 22 22 22; in final form 22 22 22)

The field projection method of Hong and Kim (2004) to identify the crack-tip10 cohesive zone constitutive relations in an isotropic elastic solid is extended to a

nanoscale planar field projection of a cohesive crack tip on an interface betweentwo anisotropic solids. This formulation is applicable to the elastic field of acohesive crack tip on an interface or in a homogeneous material for anycombination of anisotropies. This method is based on a new orthogonal

15 eigenfunction expansion of the elastic field around an interfacial cohesivecrack, as well as on the effective use of interaction J integrals. The nanoscaleplanar field projection is applied to characterizing a crack-tip cohesive zonenaturally arising in the fields of atomistics. The atomistic fields analysed areobtained from molecular statics simulations of decohesion in a gold single crystal

20 along a ½�1�12� direction in a (111) plane, for which the interatomic interactions aredescribed by an embedded atom method potential. The field projection providescohesive traction, interface separation, and the surface-stress gradient causedby the gradual variation of surface formation within the cohesive zone. Therefore,the cohesive traction and surface energy gradient can be measured as functions

25 of the cohesive zone displacements. The introduction of an atomistic hybridreference configuration for the deformation analysis has made it possible tocomplete the field projection and to evaluate the energy release rate of decohesionwith high precision. The results of the hybrid analyses of the atomistics andcontinuum show that there is a nanoscale mechanism of decohesion lattice

30 trapping or hardening caused by the characteristics of non-local atomisticdeformations near the crack tip. These characteristics are represented by surfacerelaxation and the development of surface stresses in the cohesive zone.

1. Introduction

The energetics of nanoscale fracture, adhesion and slip processes in crystalline solids35 are typically characterized by non-local and nonlinear deformation of the process

zone surrounded by a non-uniform field of long-range interaction in a low symmetry,i.e. an anisotropic system. Historically, such nanoscale processes have often beenanalysed in the view of cohesive zone models in the context of field theory [1–8].As an illustration, figure 1a shows the atomic positions near a crack tip on a �27

40 grain boundary in gold simulated by the embedded atom method (EAM) [9]. Detailsof such simulations will be discussed later in the atomistics section. Such atomic

*Corresponding author. Email: [email protected]

Philosophical Magazine

ISSN 1478–6435 print/ISSN 1478–6443 online � 2006 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/14786430601110372

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decohesion and/or slip processes in nanoscale are to be analysed from the perspectiveof a continuum cohesive zone model as depicted in figure 1b. In the cohesive zonemodel, the surrounding elastic fields in equilibrium with the state of cohesive zoneseparation represent the non-uniform deformation characteristics of a long-rangeinteraction with external loading agencies, while the cohesive zone constitutiverelations epitomise the non-local and nonlinear deformation characteristics involvedin such interatomic separation processes. A major energy flux from the externalloading to the fracture process zone, upon virtual crack growth, is transmittedthrough the non-uniform deformation field of a long range interaction. Therefore, itis desirable to characterise the deformation field based on the energetic interactionintegrals, e.g. interaction J integral [10], between the atomistic deformation fields(figure 1a) and the elastic fields of a cohesive zone model (figure 1b). Recently, Hongand Kim [11] found that, due to the crack geometry of the near-field boundary, theelastic fields of cohesive zone models can be decomposed into sets of eigenfunctionfields, mutually orthogonal to each other for interaction J integrals. Theeigenfunction fields form bases for the interaction integrals to characterise atomicdecohesion and/or slip processes. Such a characterizing scheme of decohesion and/orslip processes through interactions with the eigenfunction fields of a cohesive zonemodel is called a planar field projection method (FPM). A formulation of the planarFPM for a crack-tip cohesive zone will be developed in this paper (Part I), whileanother formulation for a dislocation-core cohesive zone will be presented in a sequelpaper (Part II). Once the atomic decohesion process is characterized by the planarFPM, the field-theoretical aspects of the cohesive zone model will be particularlyuseful in studying the energetic stability of fracture, adhesion, and slip processes.

Figure 1. Deformation field near a crack with a cohesive zone size of 2c on an interfacebetween two anisotropic solids: (a) an atomistic deformation field of decohesion along the �27grain boundary simulated using the EAM; and (b) a mathematical field model of decohesioninteraction with atomistic deformation field. The black and white circles in (a) represent atomsin two alternating ð�110Þ atomic planes.

2 S. T. Choi and K.-S. Kim


Whereas, in crystalline solids, the far field lattice deformation surrounding afracture process zone can be analysed by elastic field theories, the discrete andquantum mechanical nature of atomic debonding in the process zone has beeneffectively modelled with atomistic simulations [12–18]. In atomistic simulations,fracture-process modelling requires simulation techniques that can handle compu-tationally large numbers of atoms with sufficient accuracy. For such reasons, theatomic resolution aspects of fracture processes have been historically modelled byatomic statics, i.e. atomistics, or molecular dynamics simulations with various semi-empirical interatomic potentials, employing limited ab initio calculations to enhancethe accuracy of the simulations [19]. The interatomic potentials used in the fracturestudies include multi-body potentials in general [20–22] and EAM potentials formetals [16, 18, 23]. One of the major difficulties in the atomistic simulation ofnanoscale fracture processes is creating a reduced representation of the decohesionprocesses for various modes of separation from the simulation results. Such difficultystems from the fact that the energetics of atomistic separation processes in thenanoscale is highly non-local and nonlinear. Such non-local and nonlinearbehaviours are treated as cohesive zone constitutive relations in a cohesive zonemodel [5, 6, 8]. The energetic analysis of fracture processes with a cohesive zonemodel is particularly useful for solving technologically important problems ofcontrolling the various nanoscale mechanisms of the toughening or embrittlementof solids, including nanoscale solid solution toughening, hydrogen embrittlement,and effects of grain-boundary solute segregation [24].

In conventional cohesive zone models, the cohesive zone is typically character-ized by the separation of two cohesive surfaces and associated tractions, assumingthat the tractions are in direct equilibrium with the stress state in the surroundingelastic body. In such cohesive zone models, tractions between the surrounding elasticbody and cohesive zone surfaces are continuous across the cohesive zone. However,in nanoscale atomic decohesion processes, the surface stresses in the cohesivesurfaces must be included in the cohesive zone model, and the tractions between theelastic body and cohesive surfaces become discontinuous across the cohesive zone.Thus, the cohesive zone constitutive relations include not only the separations ofcohesive surfaces and the associated work-conjugate tractions, but also the cohesive-zone centre line (CCL) displacements and associated work-conjugate surface stresses.The importance of the surface stresses in surface energetics has been well noted bySchiotz and Carlsson [18] and Wu and Wang [25], but it has not been well reflected incohesive zone models. Furthermore, in conventional cohesive zone models, attentionhas been focused mainly on finding the surrounding elastic field for a known state ofthe cohesive zone, i.e. separation or traction distributions used to solve boundaryvalue problems. However, identifying the cohesive zone constitutive relations fromthe state of the surrounding far field is an inverse problem. A solution procedure forsuch inverse problems is called the field projection method (FPM). Recently, Hongand Kim [11] have expressed a general form of the elastic fields of a crack tip with acohesive zone in a homogeneous isotropic material in terms of an eigenfunctionexpansion of complex functions in the Muskhelishvili formalism [26]. They alsoshowed an effective use of the eigenfunction expansion in an energeticallymeaningful FPM, providing an inversion method based on interaction J integrals[10]. However, a class of eigenfunctions that corresponds to the surface stresses in the

Atomic decohesion and slip in crystalline solids. Part I 3


cohesive zone was not included in their formulation. This class of eigenfunctions isimportant because surface stresses play significant roles in nanoscale fractureprocesses. The FPM with a complete set of eigenfunctions is considered particularlyuseful for a multiscale analysis of fracture processes with atomistic simulations andfor the experimental measurement of nanoscale cohesive zone properties, includingsurface stresses.

Another important aspect of nanoscale fracture processes in crystalline materialsis that deformation develops in a low symmetry system. In other words, the elasticproperties are anisotropic and the separation or slip processes are concentrated in alimited set of interatomic planes. Therefore, the elastic fields surrounding a cohesivezone must be analysed with anisotropic elasticity. A brief background on themathematical descriptions of anisotropic elastic crack tip fields follows. Eshelbyet al. [27], Stroh [28], and Lekhnitskii [29] developed a linear theory of anisotropicelasticity for a generalized two-dimensional deformation, known as the Strohformalism. Using the Stroh formalism, Suo [30] expressed the asymptotic elastic fieldof a sharp crack on the interface in an anisotropic bimaterial whose near-tip stressfield has oscillatory characteristics in general. Suo [30] decomposed the asymptoticcrack-tip stress fields in accordance with the eigenvector direction of the interfacialcrack tip characteristic equation for mathematical convenience. Subsequently,Qu and Li [31] obtained another eigenfunction expansion of an interfacial crack,of which stress intensity factors degenerate to conventional stress intensity factors asthe bimaterial elastic properties reduce to those of homogeneous anisotropicmaterials. In doing so, Qu and Li [31] introduced a new matrix function that plays animportant role in representing the oscillatory characteristics of an interfacial cracktip. Later, Beom and Atluri [32] and Ting [33] independently identified expressions ofthe generalized Dundurs parameters of dissimilar anisotropic materials, whichdegenerate to conventional Dundurs parameters [34] as the elastic anisotropy isreduced to isotropy. Beom and Atluri [35] later considered the general Hilbert arcproblems to obtain a complete eigenfunction expansion of an interfacial crack, whichincludes the admissible regular fields, supplementing the non-regular field of Qu andLi [31], at the crack tip.

In this paper, a complete set of eigenfunctions for an interfacial cohesive cracktip field in an anisotropic bimaterial is provided and the corresponding FPM is alsopresented. The complete set includes a subset of eigenfunctions corresponding to theelastic fields caused by surface stresses in the cohesive zone. The complete set ofeigenfunctions is orthogonalized for interaction J integrals between the conjugateeigenfunctions within the set. The mathematical structure of the formulation isgeneral and is also applicable to cohesive crack tip fields in homogeneous materials,regardless whether the material is anisotropic or isotropic. Interaction J integralsbetween atomistic deformation fields at the nanoscale and the eigenfunction fields ofthe mathematical model build up the nanoscale planar FPM for characterizing cracktip decohesion and/or slip processes. While the method is applicable to eitherexperimentally measured or atomistically simulated deformation fields, the method istested to characterize the atomic decohesion processes in gold with deformationfields from EAM simulations. The test results show that the FPM measures thebounds of the surface energy, as well as the anisotropic surface stress of a solidsurface, as a function of cohesive zone displacements. In addition, it is found that



the field-projection-nominal (FPN) peak strength of the atomic decohesion issubstantially lower, e.g. approximately 4GPa, than the conventionally estimatedrigid-separation-nominal (RSN) strength, e.g. approximately 15GPa, for gold (111)separation. Furthermore, this hybrid method of decohesion analysis revealed thatthere is a nanoscale mechanism of decohesion lattice trapping, caused by surfacestresses, prior to the dislocation emission or crack growth from a sharp crack tip.

The structure of this paper is as follows. In section 2, the configurational balancein a nanoscale crack-tip cohesive zone model is discussed in terms of conservationintegrals, including the surface and interface energies and stresses. In section 3, theeigenfunction expansion is carried out to completely describe the elastic fields nearan interfacial crack-tip cohesive zone. Then, a J-orthogonal representation of theeigenfunction expansion is made in section 4 to develop the nanoscale FPM insection 5. Subsequently, in section 6, the FPM is applied to the deformation fieldsnear a crack tip on a (111) plane with a prospective crack growth direction of ½�1�12�,generated by EAM simulations. In section 7, the results of the analysis carried out insection 6 are discussed. Finally, in section 8, some conclusions are provided.In addition, three appendices, A, B, and C, are attached to provide necessaryinformation on the Stroh matrices, a complex-function representation of theanisotropic elastic field of a bimaterial interface crack tip, and the orthogonalizationof the interaction J integral and orthogonal polynomials.

Throughout this paper, a lower case bold Latin character represents a three-component column matrix and an upper case bold Latin character represents a threeby three matrix. All italic bold characters indicate tensor quantities, whereas { }applied to a tensor stands for a matrix composed of the Cartesian components of thetensor associated with the Cartesian coordinates x1, x2 and x3, for which the basevectors are represented by ej, j¼ 1, 2 and 3, respectively. The symbol I stands for theidentity matrix. It is remarked that a summation convention is not employed inthis article. Subscript indices in h i refer to upper material with 1 and lower materialwith 2. A prime over a function represents the derivative with respect to theassociated argument. A tilde over a matrix symbol indicates that the matrix isdiagonal; a tilde over a variable designates a normalized variable; a hat over afunction symbol represents an auxiliary function; and bars over any charactersdenote complex conjugates.

2. Configurational force balance in a nanoscale crack-tip cohesive zone model

A schematic of a nanoscale crack-tip cohesive zone model is shown in figure 2,whereas the coordinates to describe the cohesive zone model are depicted infigure 1b. The cohesive zone model is composed of two elements: one for a bodyregion (figure 2a) and the other for a surface region (figure 2b). For the body region,the conventional elastic field description is applicable. Tractions at the boundarybetween the surface element and the body element along the cohesive zone,�c� x� c, are denoted as tþ for the upper part and t� for the lower part of thecohesive zone. The tractions are related to the stress fields at the boundary of theupper body region h1i, rþ, as tþ¼ rþ � e2, and at the boundary of the lower body



region h2i, r�, as t�¼ r� � e2. Since the two elements are regarded in reversible static

equilibrium in this model, the proper conservation integrals can be applied for the

configurational force balances. A relevant conservation integral for this geometry is

the J integral [36], defined as:

J�½S � ¼

Z�

e1 � nð Þ ’ ½symðruÞ� � e1 � ruð Þ � r � n� �

ds, ð1Þ

where S symbolizes an elastic field collectively representing the elastic displacement

field, u, and the elastic stress field, r; the vector n denotes the outward unit normal of

the integral path, �; and the strain energy density, ’, is a function of the symmetric

part of the displacement gradient, sym(ru), which is the linear strain. The J integral

vanishes for a closed contour that encloses a regular region.In cohesive zone models, the J integral along a contour, �, that starts from the

lower crack face and ends at the upper crack face of the body region, has an energy

balance relationship defined as

J� ¼ �h1i þ �h2i � �h12i� �

, ð2Þ

where � represents the surface energy density of materials h1i or h2i or the

interface h12i.When the contour �0 only encloses the cohesive zone boundary in the body

region, the J integral can be expressed as

J�0¼

Z c

�c

tþ �@uþ

@x1� t� �

@u�

@x1

� �dx1

¼

Z c

�c

tþ � t��

�1

2

@uþ

@x1þ@u�

@x1

� �� þ

1

2tþ þ t��

�@uþ

@x1�@u�

@x1

� � dx1

¼

Z c

�c

s �@ua

@x1þ ta �

@d

@x1

� �dx1, ð3Þ

Σ0<1> g0<1>Σ0<12>g0<12>

g0<2>Σ0<2>

<1>

<2> t −

−t+

− t−

t+

D Γ

−cc

(a)

(b)

Figure 2. Schematics of the configurational force balance in a nanoscale crack-tip cohesivezone model: (a) a diagram of the bulk region, illustrating the domain of the interactionJ integrals encompassing a cohesive crack tip; and (b) a diagram of the surface-interface regionshowing the configurational force balance in the region.



where s is the traction jump, tþ� t�; d is the displacement jump, uþ� u�; ta is theaverage traction, (tþþ t�)/2; and ua is the average displacement, (uþþ u�)/2.The average displacement is the cohesive-zone centre line (CCL) displacement.In conventional cohesive zone models, s vanishes; however, in nanoscale cohesivezone models, the average surface stress, Ra(¼(�þþ'�)/2), along the cohesive zonecan balance the traction jump as sþr �Ra

¼ 0, neglecting the second order effect ofCCL curvature. For a one-dimensional cohesive zone as shown in figure 2b, the netforce balance of the surface element becomesZ c

�c

�1dx1 ¼ �0h1i þ�0h2i ��0h12i, ð4Þ

where �0h�i for �¼ 1, 2 or 12 means �11h�i on a free surface for surface stresses or ona fully bonded interface for an interface stress. On a free surface, �¼ �0þ'0: "

s witha surface strain of "s [25].

When the elastic field is linear, two independent equilibrium fields, S[r, u] andS½r, u�, can be superposed and the interaction J integral, J int

� ½S, S � [10], between Sand S along the path � is defined as

J int� ½S, S � ¼ J�½Sþ S � � J�½S � � J�½S �

¼

Z�

e1 � nð Þ r : ruð Þ � e1 � ruð Þ � r � n� e1 � ruð Þ � r � n½ �ds: ð5Þ

The J�[S ] integral, as well as the interaction integral, J int� ½S, S �, are path independent

among homologous paths that enclose a cohesive zone. Since the interactionJ integral provides a way of interrogating an unknown field with a known set offields, a well characterized set of elastic eigenfunction fields of a crack-tip cohesivezone is developed in the following sections; these functions can be used for suchinterrogations.

3. Eigenfunction expansion of interfacial cohesive crack-tip elastic fields

A displacement field, u, and the corresponding stress field, r, of an anisotropic elasticmedium in static equilibrium without a body force in a two-dimensional (x1, x2)plane can be expressed in a compact form as [27, 28]

fug ¼ 2Re AfðzÞ½ �, ð6Þ

e1 � rf g

e2 � rf g

¼ 2Re

�B~�f 0ðzÞBf 0ðzÞ

, ð7Þ

where fðzÞ �P3

j¼1 fjðzjÞfejg is a column matrix representation of the three analyticfunctions of the respective complex variables zj¼ x1þ pj x2; the Stroh eigenvalue, pj,and the Stroh matrices, A and B, are defined in appendix A. The matrix ~� is adiagonal matrix with the components pj �jk, where �jk is the Kronecker delta. Theargument z represents zj for the corresponding j’th row of the function matrix f(z).



An anisotropic bimaterial with its interface on the x1 axis has a traction-free,

semi-infinite interfacial crack along x15�c and a cohesive zone on �c� x1� c, as

shown in figure 1. The elastic fields around the cohesive zone can be described by the

superposition of the non-singular eigenfunctions of sharp-crack-tip elastic fields with

respect to x1¼�c and x1¼ c, as Hong and Kim [11] did for a cohesive crack tip field

in a homogeneous isotropic solid. In addition, the elastic field caused by the surface

stress along the cohesive zone faces must be superposed for nanoscale cohesive crack

tip fields; the surface stress appears to be a traction jump between the two faces of

the cohesive zone. Based on the eigenfunction expression, equation (B1) in

appendix B, of a sharp-crack-tip elastic field, the column matrix function f 0(z) of

the bimaterial cohesive crack tip field can be described as

f 0h�iðzÞ ¼1

2B�1h�i I� ð�1Þ

�ib½ � Y ðzþ cÞ�i"� �

gðzÞffiffiffiffiffiffiffiffiffiffiffizþ cp

� Y ðz� cÞ�i"� �

hðzÞffiffiffiffiffiffiffiffiffiffiffiz� cp� �

þi


�a½ � qðzÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 � c2p

þ rðzÞh i

, ð8Þ

for �¼ 1 or 2 referring to materials 1 or 2, respectively. The four analytic matrix

functions g(z), h(z), q(z), and r(z) can be expressed as a series,P1

n¼0 GnðzÞgn,P1n¼0 HnðzÞhn,

P1n¼0 QnðzÞqn, and

P1n¼0 RnðzÞrn, for which Gn(z), Hn(z), Qn(z), and

Rn(z) are real matrix base functions, for example, GnðzÞ ¼ �GnðzÞ, and gn, hn, qn,

and rn are column matrices of real coefficients. The generalized Dundurs

parameter real matrices a and b, the bimaterial oscillatory index ", and the

bimaterial real function matrix Y[�(z)] for an arbitrary function �(z) are defined

in appendix B for anisotropic bimaterials. It is worth noting that the oscillatory

fields near the two ends of the cohesive zone resulting from the eigenfunction

expansion given in equation (3) do not give physically unacceptable phenomena

because it does not include a term of diverging stress in the eigenfunction

expansion.The decomposition of the function f 0h�i(z) into g(z), h(z), q(z), and r(z) in

equation (8) is analogous to that made by Hong and Kim [11] for a cohesive

crack tip field in a homogeneous isotropic solid. The function g(z) is responsible

for the traction distribution at the cohesive zone, whereas the function h(z) is

responsible for the separation variations along the cohesive zone. To take into

account the elastic field of the surface energy gradient along the cohesive zone,

which was ignored by Hong and Kim [11], the q(z) function is introduced here.

The term q(z) is important for nanoscale fracture problems; however, it is

negligible for large-scale fracture problems in general. Including a regular field

near the cohesive zone represented by the function r(z), for which the

corresponding traction along x1 axis vanishes, the holomorphic function f 0h�i(z)

in equation (8) constructs a complete set of eigenfunction expansions near

a cohesive crack tip on an interface between two anisotropic solids. In the

following section, the eigenfunctions are orthogonalized to the orthogonal base

functions,GonðzÞ, Ho

nðzÞ, QonðzÞ, and Ro

nðzÞ with respect to the interaction

J integral [10]. The eigenfunction expansion based on such base functions is

called the J-orthogonal representation.



4. J-Orthogonal representation of the cohesive zone eigenfunction expansion

In this section, a set of well characterized eigenfunction fields for a cohesive zone isdeveloped and these eigenfunction fields can also be conveniently used to interrogateother elastic fields with interaction J integrals. In particular, the interaction will bebetween configurational work-conjugate sets of eigenfunctions. When the interactionJ integral, equation (5), is applied to the contour �0 along the faces of the cohesivezone, the traction and displacement gradient along the cohesive zone are required toevaluate the interaction J integral. The traction along the cohesive zone, equation (7),is expressed explicitly with the eigenfunction expansion in equation (8) as

e2 � rðx1Þ� �

h� i¼ Y ðcþ x1Þ

�i"� �

gðx1Þffiffiffiffiffiffiffiffiffiffiffiffifficþ x1p

þ ð�1Þ� I� ð�1Þ��½ �qðx1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � x21

q, ð9Þ

for �¼ 1 or 2 referring to materials 1 or 2, respectively; while the displacementgradient along the cohesive zone is given by

e1 � ruðx1Þ� �

h�i¼ Im M�1h�i I� ð�1Þ

�ib½ �

n oY ðcþ x1Þ

�i"� �

gðx1Þffiffiffiffiffiffiffiffiffiffiffiffifficþ x1p

þ ð�1Þ�L�1h�i I� ð�1Þ�ib½ �Y ðc� x1Þ

�i"e�"� �

hðx1Þffiffiffiffiffiffiffiffiffiffiffiffiffic� x1p

� ð�1Þ�Wh�i I� ð�1Þ��½ �qðx1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � x21

qþ L�1h�i I� ð�1Þ

��½ �rðx1Þ,ð10Þ

for �¼ 1 or 2 referring to materials 1 or 2, respectively. Definitions of theelastic property matrices M, L, and W used in equation (10) are given in appendicesA and B.

Then, by inserting equations (9)–(10) into equation (5), the expression of theinteraction J integral is reduced to

J int�o½S, S � ¼ �

Z c

�c

e2 �rðx1Þ� �T

e1 � ruðx1Þ� �

þ e2 � rðx1Þ� �T

e1 � ruðx1Þ� �h ih i

dx1

¼

Z c

�c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2�x21

qgTðx1ÞU

�1Y �cðx1Þ½ �hðx1Þþ gTðx1ÞU

�1Y �cðx1Þ½ �hðx1Þn o

dx1

þ 4

Z c

�c


qqTðx1Þ Lh1i þLh2i

� ��1rðx1Þþ q

Tðx1Þ Lh1i þLh2i

� ��1rðx1Þ

h idx1,

ð11Þ

where the double bracket � � �½ �½ � indicates a jump across the cohesive zone; the elasticproperty matrix,U�1, is defined as U�1 ¼ ðL�1h1i þ L�1h2i ÞðIþ b

2Þ; and the function�c(x1)¼ [(c�x1)/(cþ x1)]

�i" cosh �". In deriving the second equality of equation (11),various identity relationships of equation (B5) in appendix B were used, and onlyfour terms survived among the sixteen interaction pairs between {g(z), h(z), q(z), r(z)}and {gðzÞ, hðzÞ, qðzÞ, rðzÞ}. The result shows the distinct configurational-work-conjugate pairs of {g(z), h(z)} and {q(z), r(z)}. Equation (11) can be furtherreduced by constructing orthogonal base functions of g(z), h(z), q(z), and r(z) asGo

nð ~z Þ ¼ V�T ~Pnð ~z ÞVT, Ho

nð ~z Þ ¼�V�T ~Pnð ~z Þ �V

T, Qonð ~z Þ ¼ Unð ~z ÞI, and Ro

nð ~z Þ ¼ Unð ~z ÞI,where ~z denotes z/c; V denotes a Hermitian matrix defined in appendix C; ~Pnð ~z Þdenotes the diagonal matrix functions of normalized Jacobi polynomials with thecomponents �jkP

ðaj, �ajÞn ð ~z Þ=

ðaj, �ajÞn ; UnðzÞ ¼ P ð1=2, 1=2Þn ðzÞ= ð1=2, 1=2Þn denotes the



Chebyshev polynomials of the second kind; and ða, bÞn denotes the normalizationfactors of the Jacobi polynomial P ða, bÞn ðzÞ. The explicit expression of ða, bÞn and thedetails of the orthogonalization with Jacobi polynomials for interaction J integralsare provided in appendix C.

With the orthogonal eigenfunctions, the stress function f 0h�i(z) is thenexpressed as

f 0h�iðzÞ ¼XNn¼0

Gonh�ið ~z Þgn þHo

nh�ið ~z Þhn þQonh�ið ~z Þqn þ Ro

nh�ið ~z Þrn

n o, ð12Þ

where gn, hn, qn, and rn are column matrices of real coefficients, and

Gonh�ið ~z Þ ¼

1

2

ffiffiffiffiffiffiffiffiffiffiffi~zþ 1p

B�1h�i I� ð�1Þ�ib½ �Y ð ~zþ 1Þ�i"

� �Go

nð ~z Þ,

Honh�ið ~z Þ ¼ �

1

2

ffiffiffiffiffiffiffiffiffiffiffi~z� 1p

B�1h�i I� ð�1Þ�ib½ �Y ð ~z� 1Þ�i"

� �Y�1 cosh�"½ �Ho

nð ~z Þ,

Qonh�ið ~z Þ ¼

i

2

ffiffiffiffiffiffiffiffiffiffiffiffiffi~z2 � 1p

B�1h�i I� ð�1Þ�a½ �Qo

nð ~z Þ,

Ronh�ið ~z Þ ¼

i


�a½ �Ronð ~z Þ: ð13Þ

Expressing the auxiliary field, as well as the actual field of interest, in the sameform of eigenfunction expansion, equation (12), and substituting the eigenfunctionsinto equation (11), the interaction integral, J int

�o½S, S �, is obtained in terms of the

coefficients of the eigenfunctions:

J int�o½S, S � ¼

�c

2

XNn¼0

gTnU�1hnþ g

TnU�1hnþ 4qTn Lh1i þLh2i

� ��1rnþ 4q

Tn Lh1i þLh2i� ��1

rn

h i:

ð14Þ

When the auxiliary field is chosen to be the actual field, the interaction integralbecomes twice the value of the J integral. As shown in equation (14),the eigenfunction fields selectively interact with other fields in the interactionJ integral. Therefore, the interaction J integrals with the eigenfunctions can be usedas filters for the fields to be interrogated. In addition, the interaction J integral ispath independent so that the integral can be evaluated at some distance from thecohesive zone. In other words, the cohesive zone characteristics can be assessed usingfar-field data. The method of assessing local properties with far-field data with anFPM is discussed in the following section.

5. Field projection method for a J-equivalent crack-tip cohesive zone

In the previous section it was shown that, with the base eigenfunctions, an infiniteseries of four real column matrices, gn, hn, qn, and rn, completely determine the fieldsurrounding the cohesive zone in the cohesive zone model. In this section, formulasto determine the four real column matrices with interaction J integrals are provided.



The four real column matrices, gn, hn, qn, and rn, are responsible for cohesive zone

tractions, separations, surface stresses, and cohesive-zone centre line (CCL)

displacements, respectively. If an auxiliary field of SðkÞhn

for which hn ¼ fekg

(k¼ 1, 2, 3) and gn ¼ qn ¼ rn ¼ 0, i.e. f 0h�iðzÞ ¼ Honh�ið ~z Þfekg, is used in equation

(14), the interaction J integral is simply expressed as J int�o½S, S

ðkÞhn� ¼ ð�c=2ÞgTnU

�1fekg.

Subsequently, this relationship can be easily inverted to become

gn ¼2

�cU J int

� ½S, Sð1Þhn�, J int

� ½S, Sð2Þhn�, J int

� ½S, Sð3Þhn�

h iT, n ¼ 1, 2, . . . ,N: ð15aÞ

Similarly, with an auxiliary field, S ðkÞgn , of f0h�iðzÞ ¼ Go

nh�ið ~z Þfekg for k¼ 1, 2, 3,

hn ¼2

�cU J int

� ½S, Sð1Þgn�, J int

� ½S, Sð2Þgn�, J int

� ½S, Sð3Þgn�

h iT, n ¼ 1, 2, . . . ,N ð15bÞ

is obtained; with an auxiliary field, S ðkÞrn , of f0h�iðzÞ ¼ Ro


qn ¼1

2�cLh1i þ Lh2i� �

J int� ½S, S

ð1Þrn�, J int

� ½S, Sð2Þrn�, J int

� ½S, Sð3Þrn�

h iT, n ¼ 1, 2, . . . ,N;

ð15cÞ

and finally with an auxiliary field, S ðkÞqn , of f0h�iðzÞ ¼ Qo


rn ¼1

2�cLh1i þ Lh2i� �

J int� ½S, S

ð1Þqn�, J int

� ½S, Sð2Þqn�, J int

� ½S, Sð3Þqn�

h iT, n ¼ 1, 2, . . . ,N:

ð15dÞ

As shown in equations (15), for an arbitrary elastic field of a cohesive zone model,

the field near the cohesive zone determined by gn, hn, qn, and rn can be assessed by

interaction J integrals along a contour � at a distance away from the cohesive zone.

This is an attractive feature of the FPM for experimental measurements or atomistic

simulations of the cohesive zone characteristics. As long as the elastic-field data in a

region that encloses the cohesive zone at some distance is available, cohesive zone

characteristics can be evaluated. When the FPM is applied to an elastic field that is

linearly elastic all the way close to a planar cohesive zone, as shown in figure 1b or

figure 2, the assessment can achieve an arbitrary accuracy with a large number, N, in

equations (15). However, when the method is applied to an elastic field that embraces

not only a planar cohesive zone but also a non-local and/or nonlinear bulk

deformation region near the cohesive zone, the results of the FPM with a finite

number N must be interpreted as J-equivalent cohesive zone behaviour. In other

words, the FPM is regarded as a mechanical test in which the region enclosed by the

elastic field is energetically stimulated by the J-interaction integral operations with

appropriate eigenfunction fields, and the gn, hn, qn, and rn values determined by the

interaction J integrals are considered as responses to such mechanical stimuli.

The responses of the J-evaluated gn are interpreted as cohesive zone tractions of a

J-equivalent cohesive zone, while those of the J-evaluated hn are understood as

cohesive zone separation gradients of the J-equivalent cohesive zone. Similarly, the

responses of the J-evaluated qn are taken as the surface stress gradient of

the J-equivalent cohesive zone, and the responses of the J-evaluated rn are taken

as the CCL displacement gradient of the J-equivalent cohesive zone.



In the following section, the FPM is utilized to probe the J-equivalent cohesive

zone characteristics of atomic decohesion in a ½�1�12� direction on a (111) plane of a

gold single crystal, with crack tip deformation fields from EAM simulations. In the

analysis, it is found that N¼ 3 with an optimum cohesive zone size is enough to

describe the J-equivalent cohesive zone characteristics, since the self-equilibriumfields of the high spatial frequency traction distribution at the cohesive zone

rapidly decay away from the cohesive zone. Even though the field

projection scheme described so far is formulated for general anisotropic bimaterials,

degenerate cases in which one or both of the constituent materials are isotropic can

be dealt with in the same framework, as discussed by Choi et al. [37], for a sharp

crack tip field. Also, it is worth mentioning that for special combinations ofanisotropic bimaterials, e.g. symmetric tilt grains, of which the bimaterial matrix,

M�1h1i þ

�M�1

h2i , is a real symmetric matrix, the oscillatory stress fields disappear and

many of the mathematical expressions involved in the field projection can be reduced

to very simple forms with the relations b¼ 0 ("¼ 0), Y[�(z)]¼ I, V �V�T¼ L�1h1i þ L�1h2i ,

and ~PnðzÞ ¼ IUnðzÞ.

6. EAM simulation and field projection of atomic decohesion in gold

6.1. Atomistic deformation analysis with EAM simulations

In a cohesive zone framework, the constitutive relations of the cohesive zone areregarded as phenomenological material characteristics that can be evaluated by the

FPM described in the previous section. The FPM is applicable to the decohesion

processes in homogeneous elastic solids, as well as to those of interfaces in

anisotropic elastic bimaterials. The application of the method is also valid for any

elastic deformation fields of cohesive crack-tips obtained either by experimental

measurements or by computational simulations. In this section, the FPM is applied

to analysing nanoscale decohesion processes and their associated deformation fieldsin a fcc gold crystal, obtained by atomistic simulations. The EAM potential of Foiles

et al. [9] is employed for the atomistic simulations since the potential is known to be

quite reliable for atomistic simulations of atoms with nearly-filled d-orbital electronic

configurations such as gold. Foiles et al. [9] provided the potential tuned to fit

various empirical material parameters including the lattice constant of gold,

a¼ 4.08 A, the three elastic constants of C11¼ 183.0GPa, C12¼ 159.0GPa, and

C44¼ 45.0GPa, the sublimation energy of 3.93 eV, and the vacancy formation energyof 1.03 eV. They also reported that the potential gives a (111) surface energy of

0.79 Jm�2 and a surface relaxation of 0.1 A contraction in the first layer of a (111)

free surface.In this study, the crack geometry of a (111) plane decohesion is considered.

A semi-infinite crack is on a (111) plane with its crack tip lying along a ½�110� direction

and its prospective propagation direction in ½�1�12�. A Cartesian coordinate system is

assigned with the x 01 axis along ½�1�12�, the x 02 axis along [111], and the x 03 axis along½�110� for the deformation analysis.



In the atomistic field of an embedded atom method potential, the deformation isnon-local and it can be induced by nearby density gradients; for example, the surfacerelaxation is evident near a free surface in the atomistics. Therefore, a localdeformation associated with a stress field must be delineated from such atomisticnon-local deformations near a free surface [9]. Here, ‘‘local’’ constitutive relationsignifies that the stress is only a function of the stress-inducing strain at a local point.However, it is quite difficult to find the stress-inducing deformation from the finalequilibrium positions of the atoms for a crack tip geometry, because the stress-freereference configuration is not well defined near the crack tip for the non-localatomistics. In the following, a construction of an approximate stress-free referenceconfiguration is introduced. The approximate reference configuration is close to astress-free state away from the crack tip. Since the field projection method utilizesthe far-field information, it is sufficient to use the approximate referenceconfiguration, in extracting the crack tip cohesive zone constitutive relations withthe field projection method. Derivation of the stress-inducing deformation fromthe approximate stress-free reference configuration involves four atomisticconfigurations.

Figure 3 shows the four configurations of the atomistic states. The black andwhite atoms in figure 3 represent atoms in different ð�110Þ atomic planes, e.g. the Aand B atomic planes, respectively, which are considered to be the planes of thegeneralized two-dimensional deformation. The top half of figure 3a shows a perfectlattice configuration and the bottom bright region depicts a continuum representa-tion of the perfect lattice. Figure 3b exhibits an atomistic equilibrium state of adeformed configuration. The top half of figure 3c demonstrates an atomisticconfiguration relaxed near a free surface along the x 01 axis and the bottom darkregion illustrates a continuum representation of the relaxed configuration. Figure 3dshows a hybrid configuration of the perfect lattice configuration (a) and the relaxedfree-surface configuration (c).

The deformed configuration in equilibrium, shown in figure 3b, was generated byrelaxing the atomic positions from an imposed initial configuration, while the atomicpositions at an outer boundary layer were held fixed. The initial configurationcorresponds to a linear elastic crack-tip displacement field of a mixed mode withKI ¼ 0:38MPa

ffiffiffiffimp

and KII ¼ �0:10MPaffiffiffiffimp

. The initial condition was imposed on98,496 gold atoms in the region of three dimensions: 54

ffiffiffi6p

a� 76ffiffiffi3p

a�ffiffiffi2p

a. Then,a conjugate gradient method of minimizing the total energy of the system was used torelax the atomic positions in the bulk region, while the atoms at the outer boundarylayer, with a thickness of 2a, were held fixed and periodic boundary conditions wereapplied in a ½�110� direction. The total energy of the system at each stage of therelaxation was evaluated using the EAM potential of Foiles et al. [9]. The modemixity of the initial condition was chosen to suppress the localized deformationalong the [112] direction on a ð11�1Þ plane, which could have emitted a dislocationduring the relaxation if the mode mixity had not been applied. In the followingsection, the decohesion process zone ahead of the crack tip in figure 3b is analysedwith the FPM developed in the previous sections.

In the following deformation analysis, the atomic positions on only one atomicplane, e.g. plane A in figure 3, are used for the planar field projection analysis. Here,the j th atomic position in a perfect lattice (figure 3a) is denoted as x0j

½P�, in the cracked



configuration (figure 3b) as x0j½C�, in a free surface relaxed configuration (figure 3c) as

x0j½F�, and in a hybrid reference configuration (figure 3d) as x0j

½R�. The hybrid reference

position is defined as

x0j½R� �

jj

�x0j½F� þ 1�

jj

�

� �x0j½P�, ð16Þ

where the angle is measured from the prospective crack propagation direction to

the line segment between x0j½P� and the crack tip. We also define the deformation

gradient from dx 0½P� to dx 0½C� as F[C], from dx 0½P� to dx 0½R� as F[R], and from dx 0½R� to dx 0½C�as F[H]. Every deformation gradient of a configuration relative to a perfect lattice

configuration, such as F[C] or F[R], can be defined in the Voronoi cell [38] of the jth

atom in the perfect lattice configuration as

F j �1

A0

XNk¼1

x0jðkÞ � x0 j

2

!� nk

" #�Lk

0, ð17Þ

(a) Perfect-lattice configuration (b) Cracked configuration

(c) Free-surface configuration (d) Angularly-hybridized configuration

Figure 3. The kinematics of an atomistic deformation near a crack tip, expressed withcombinations of the deformation gradients F[C] and F[R]. Three nominal stress-free referenceconfigurations are shown for (a) bulk deformation, (c) deformation near a free surface, and(d) deformation near a crack tip, whereas an equilibrium configuration is shown in (b) for anatomistic deformation field near a crack tip opening along a ½�1�12� direction in a (111) plane ingold. The lower halves of (a), (c), and (d) depict continuum representations.



where nk and �Lk0 are the outward unit normal vector and the length of the k’th side

of the Voronoi cell, respectively; x0j(k) is the position of the nearest atom in the nk

direction; and A0 is the area of the Voronoi cell. Equation (17) is for a two-dimensional case with N¼ 4. Then, the deformation gradient F j

½H� of the crackedconfiguration (b) with respect to the hybrid reference configuration (d) is given byF½H� ¼ F½C�F

�1½R� . Defining the deformation gradient at a surface atomic site, the

relative displacement at the free-surface-side face of the Voronoi cell, with respect tothe displacement at the atomic site, is chosen to be equal to the negative of thedisplacement at the opposite face of the Voronoi cell. When the hybrid referenceconfiguration is used to evaluate the deformation gradient, the deformation gradientnear the free surface is scarcely sensitive to the choice of the relative displacement.

When an atomistic deformation field is analysed with a local elasticity theory,we must consider two major effects of the high atomic density gradients near a freesurface. One is the surface stress effect and the other is the surface relaxation effect.The surface stress effect is taken into consideration in the configurational forcebalance, as shown in figure 2. The surface relaxation effect is dealt with by the properchoice of an effective reference configuration for assessing the deformation thatinduces bulk stress. The hybrid reference configuration, x0j

½R� of figure 3d, is used inthis paper as the effective reference configuration. Since the FPM uses an elastic fieldat a distance away from the crack tip, the hybrid reference configuration is asatisfactory choice. Figure 4a shows a vector plot of the displacement gradient in thex 02 direction, (F12[P], F22[P]� 1), with respect to the perfect lattice referenceconfiguration. The plot exhibits a very large displacement gradient of the surfacerelaxation, which cannot be used for evaluating the bulk stress induced by elasticdeformation. Figure 4b shows (F12[H], F22[H]� 1) with respect to the hybrid referenceconfiguration. The figure shows the elimination of the artefact of the displacementgradients caused by the surface relaxation. Therefore, the hybrid deformationgradient was used for the linear strain tensor, e ¼ ðF½H� þ FT

½H�Þ=2� I, to evaluate thebulk stress, r¼C : e, at a region of small strain, where the interaction J integrals ofthe FPM were carried out. For the integration, the deformation gradient was linearlyinterpolated between the atomic positions. For numerical accuracy the interactionintegrals, J int

�o½S, S � in equation (5) were evaluated with domain integrals [39]. The

width of the integration domain, D in figure 2, was set to be 20 A, while therectangular inner boundary of the domain surrounded the cohesive zone at adistance d, having a dimension of 2d by 2(cþ d) with d¼ 10 A.

6.2. Field projection of the atomistic deformation field

The interaction domain integrals, J intD ½S, S �, between the atomistic deformation

field, S[r, u], e.g. figure 3b, obtained from the EAM analysis and the auxiliaryeigenfunction fields S r, u½ � developed in section 5 are calculated for the FPM toevaluate the cohesive zone characteristics. The cohesive zone characteristics aredescribed by gn, hn, qn, and rn (n¼ 0, 1, 2, . . . ,N) of the equations (15) and thecohesive zone end locations. In the following analysis, n up to 2 is used so thatthe cohesive zone characteristics are effectively described with 38 parameters:three components of the twelve coefficient vectors and the two end positions of



the cohesive zone. The two end positions of the cohesive zone were chosen asxa1 ¼ �14 A and xb1 ¼ 109 A (c¼ 61.5 A) by trial and error so that at the left endposition, xa1, and the cohesive zone normal separation, �2, obtained from the FPMis as close as possible to that of the applied linear elastic K-field, and at the rightend position, xb1, the cohesive zone normal traction, t2, has the same value as that

(a)

(b)

x 2′ (Å

)x 2′ (

Å)

x1′ (Å)

x1′ (Å)

Figure 4. Vector plots of the displacement gradient in x0

2 direction, ð@u1=@x0

2, @u2=@x0

2Þ,at each atomic site for the atomistic deformation of figure 3b. Displacements are measuredwith respect to (a) a perfect configuration and (b) the hybrid configuration of figure 3d.



of the applied K-stress field. The location of the left end position, xa1 ¼ �14 A,

estimated by the FPM is approximately one lattice distance, 4 A, from the

location, x 01 ffi �10:0 A, where the distance between the nearest atoms in the

upper and lower surfaces is approximately equal to the cut off radius of

interatomic interaction in gold, 5.55 A. The difference is believed to be caused by

the surface stress employed in the cohesive zone model and by the choice of the

criterion to match the normal separation, �2, obtained from the FPM to that of

the applied linear elastic K-field.Figure 5a and b, respectively, shows the cohesive zone tractions and

separations determined by the FPM for the atomistic deformation field simulated

with the displacement boundary conditions of the applied linear elastic K-field.

The cohesive zone shear traction, t1, shown in figure 5a deviates from that of the

applied linear elastic K-field near the right end of the cohesive zone. In this

crystallographic orientation of a face-centred cubic crystal, the opening and

shearing modes are coupled together from the linear elastic point of view.

In other words, the tangential separation is related not only to the shear traction,

but also to the normal traction and similarly for the normal separation; thus, the

opening and the shearing modes should be considered together. The mixed mode

dependence is approximately 3% of the primary mode dependence in this orien-

tation. If the mixed mode dependence is ignored, the two modes can be separated

and the cohesive zone shear traction, t1, at the right end position, xb1, can be

matched to that of the applied K-stress field by adjusting xb1 ¼ 160 A(c¼ 87.0 A),

as shown with the dash-dot curve in figure 5a. However, since the cohesive zone

shear tractions for the two different cohesive zone sizes give similar results for the

surface stress and surface energy distributions in the cohesive zone, the two end

positions of the shear cohesive zone are set to be the same as those of the normal

cohesive zone. The shear separations shown in figure 5b for the two different

cohesive zone sizes are very close to each other. The normal separation, shown as

a dark solid curve in figure 5, exhibits a zone of negative separation between

x 01 ffi 18 A and x 01 ffi 62 A. This is a sub-region of the negative normal separation

modulus in the cohesive zone, which may reflect the tendency of phase

transformation such as twinning or unstable shear localization in a certain slip

plane near the crack tip.Figure 6a and b show the surface stress gradients and centre line

displacements of the cohesive zone, respectively. A careful observation of the

equilibrium atomic positions near the crack tip reveals that the interatomic crack

plane is not flat: it is curved. Therefore, the surface stress and the curvature can

sustain the normal traction jump, i.e. the normal surface stress gradient. T-stress

is believed to cause significant variations of the curvature to change the normal

surface stress gradient. The relationships between the cohesive zone tractions and

separations are plotted in figure 6c. The maximum normal traction in the

cohesive zone is 4.06GPa when �2¼ 0.29 A. The cohesive zone surface stress is

plotted against the normal separation along the cohesive zone in figure 6d. The

surface stress is a monotonically increasing function of the normal separations, �2,except for the small �2. The surface stress of the (111) free surface in the [112]

orientation is 2.06Nm�1.



−20 0 20 40 60 80 100 120 140 160 180−4

−2

0

2

4

−15

0

15

30

45

60

75

90

105

120 t2 (c = 61.5 Å) t1 (c = 61.5 Å) t1 (c = 87.0 Å) σ22 (K field) σ12 (K field)

Tra

ctio

ns (

GPa

)

x1′ (Å)

−20 0 20 40 60 80 100 120 140 160 180

x1′ (Å)

x 2′ (Å

)x 2′ (

Å)

(a)

(b)

−4

−2

0

2

4

6

−15

0

15

30

45

60

75

90

105

120

135

δ2 (c = 61.5 Å) δ1 (c = 61.5 Å) δ1 (c = 87.0 Å) δ2 (K field) δ1 (K field)

Sepa

ratio

ns (

Å)

Figure 5. (a) Cohesive zone traction distributions of the near-crack-tip field. The endpositions of the cohesive zone are chosen as xa1 ¼ �14 A and xb1 ¼ 109 A (c¼ 61.5 A),for which traction t2 matches well with the applied K-field. Traction t1 matches betterwith the applied K-field for an independent choice, for t1 only, of xa1 ¼ �14 A andxb1 ¼ 160 A (c¼ 87.0 A) (dash-dot line). The atomic positions near the crack tip simulatedby the EAM, the same as those in figure 3b, are also shown for comparison. (b) Cohesive zoneseparations for xa1 ¼ �14 A and xb1 ¼ 109 A (c¼ 61.5 A). Another measure (dash-dot line) ofthe shear separation is also plotted for the choice of xa1 ¼ �14 A and xb1 ¼ 160 A (c¼ 87.0 A).The atomic positions near the crack tip simulated by the EAM, the same as those in figure 3b,are also shown for comparison.



7. Discussions

Before the results of the field projection for atomistic decohesion in gold arediscussed, three major approximations employed in this analysis are explainedbriefly. One is for deformation non-locality, another for deformation nonlinearity,and the other for the numerical integration employed in the FPM. Firstly, thedeformation field analysed by the EAM is inherently non-local. The non-locality isprojected onto a planar region while the material behaviour at the atomistic scale inthe remainder of the bulk region is treated with a local theory of linear elasticity.Secondly, there is some nonlinear deformation outside the cohesive zone. Thenonlinearity is also projected onto the cohesive zone in this field projection scheme.Finally, since the deformation gradients are linearly interpolated between atomicpositions, the lattice constant of gold, a¼ 4.08 A, acts as a fundamental length scaleand the integration becomes inaccurate along a contour close to the cohesive zonefor a small c/a. As Hong and Kim [11] noted, the interaction integral turns out to besubtracting a large number from another large number to obtain a small value of the

−20 0 20 40 60 80 100 120−0.6

−0.4

−0.2

0.0

0.2

0.4

τ1

τ2Surf

ace-

stre

ss g

radi

ents

(G

Pa)

(a)

x1′ (Å)

−1.0

−0.5

0.0

0.5

1.0

u1a

u2aC

CL

dis

plac

emen

ts (

Å)

(b)

x1′ (Å)

−20 0 20 40 60 80 100 120

−2 −1 0 1 2 3 4 5 6−2

0

2

4

6

t2-δ2

t1-δ1

Tra

ctio

ns (

GPa

)

Separations (Å)

a

b cde

fa

b

c

d

ef

(c)

−1 0 1 2 3 4 5−0.5

0.0

0.5

1.0

1.5

2.0

2.5

Surf

ace

stre

sses

(Jm

−2)

δ2 (Å)

(d)

Figure 6. (a) Surface stress gradients along the cohesive zone. (b) Cohesive-zone centre line(CCL) displacements at the cohesive zone. (c) Cohesive zone traction-separation relations.(d) Cohesive zone surface stresses.



integral for a large c/a and a large number of series terms, n. Therefore,the integration is performed at an optimum distance from the cohesive zone witha finite n. The truncation of the series of eigenfunctions for the field projectionimplies a low spatial frequency representation of the distribution of the cohesive zonecharacteristics. It is also worth noting that the linear strain tensor is used in theinteraction integral to guarantee path independence, and thus, the domain ofthe integral avoids a region of large rotation.

The results of the field projection presented in the previous section show variouscohesive zone characteristics that can be interpreted in a continuum point of view foratomistic decohesion in a discrete lattice system of a gold crystal. Comparisonsbetween the characteristics in the view of a continuum and those derived somewhatdirectly from the atomic positions are discussed here. In figure 7a, the field projectednormal and shear separations shown in figure 5a are compared with the atomic siteseparations between adjacent atomic planes across the cohesive zone. The atomic siteseparations stand for the difference between the displacement at an atomic site onone of the two separating atomic planes and the displacement at the conjugatecounter point on the opposite atomic plane of separation. The conjugate counterpoint is a Lagrangian point where a vector normal to the separating atomic plane atan atomic site intersects with the opposite separating atomic plane in the referenceconfiguration. The atomic site normal separations, shown as solid circles in figure 7a,display a sharp transition at the crack tip, while the atomic site shear separations,shown as open circles, demonstrate a smooth transition. The distribution of theatomic site shear separations is far more spread in the cohesive zone than that of thenormal separations because the interatomic plane of decohesion is an easy-glide slipplane. On the other hand, the field projected normal separation oscillates smoothlyalong the atomic site separation distribution. The field projected shear separationclosely follows the distribution of the atomic site shear separation. The fieldprojected separation distributions resemble low frequency approximations of theatomic site separations in the cohesive zone. It is also worth noting that the atomicsite separations are evaluated with the displacements on two separating atomicplanes, while the continuum separations are determined by the displacementsextrapolated to the mid-plane of the two separating atomic planes.

Once the separations in a cohesive zone are known, the tractions in the cohesivezone have traditionally been estimated with rigid separation potentials, e.g. Roseet al. [40], or with generalized stacking fault energy, e.g. Vitek [41]. For a combinednormal and shear separation of two atomic planes, two simple processes ofseparation can be considered. One is the rigid separation for which the relativeatomic positions are held rigid, respectively, in each volume of the two separatingparts; the other is the affine separation for which all atomic inter-planes parallel tothe decohesion plane of interest experience the same separations. If the interatomicpotential were local, the energy per unit area required for a rigid separation, i.e. therigid separation potential, would be identical to that for a corresponding affineseparation. In figure 7b, the rigid separation potential corresponding to every atomicsite separation in the cohesive zone is plotted with a dark solid circle, while the affinepotential at every matching site is mapped out with an open circle. The twoseparation processes are substantially different with respect to each other indicatingthat the interatomic potential is non-local, particularly for large deformations.



The non-locality in the EAM potential occurs for two major reasons. One is that anatom interacts with atoms beyond the nearest neighbours, with a cut off radius of5.55 A in gold, and the interatomic potential is anharmonic; the other is that theembedding term in the EAM potential is a nonlinear function of electron density and

−2

0

2

4

6(a)

(b)

Sepa

ratio

ns (

Å)

δ2 obtained from FPM

δ1 obtained from FPM

δ2 directly from atomic positions

δ1 directly from atomic positions

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

Pote

ntia

ls a

nd s

urfa

ce s

tres

ss (

Jm−2

)

Decohesion potential obtained from FPM Contribution from surface stress Normal separation potential of FPM Shear separation potential of FPM Rigid separation potential of EAM Affine separation potential of EAM 2Σ (x1

′ ); Σ (x1′ ) = surface stress

−20 0 20 40 60 80 100 120

x1′ (Å)

−20 0 20 40 60 80 100 120

x1′ (Å)

Figure 7. (a) Cohesive zone separations. The solid lines represent the separation profilesobtained by the FPM along the crack plane. The open and solid circles indicate shear andnormal atomic-site separations, respectively, between adjacent atomic planes across thecohesive zone. (b) Field-projected separation potentials and surface stress are plotted with therigid and affine separation potentials of EAM, shown as solid and open circles, respectively.



the background electron density is a superposition of the atomic electron densities ofatoms hosting the embedding. The latter reason is solely responsible for the

difference between the complete separation energy of the rigid and affineseparations, while both reasons are accountable for the non-local elastic behaviourof the solid when the strain gradient is large. This fact signifies that the fracture

energy of nanoscale decohesion depends on the history of the deformations adjacentto the decohesion process zone, even without a dislocation motion. In fracturemechanics terms, the critical stress intensity factors for a nanoscale decohesionprocess depend on various non-singular stress terms, including the T-stress; the

fracture toughness depends on external loading. In addition, the non-local effects,together with the discreteness of the lattice, determine the lattice trapping toughnessvariation of the crack tip at the nanoscale.

Figure 7b also shows the field-projected decohesion potential �(x0 ), with a dark

solid curve, as a function of the position x 01 along the cohesive zone:

� x 01� �¼

Z x 01

c

ta2d�2dx1þ ta1

d�1dx1

� �dx1 þ

Z x 01

c

�2dua2dx1þ �1

dua1dx1

� �dx1: ð18Þ

The decohesion potential is composed of the separation potentials expressed asthe first integral in equation (18) and also contributions from the surface stressarticulated in the second integral. The separation potential can be decomposed into anormal separation potential (the thin dashed line in figure 7b) and a shear separation

potential (the thin dash-dot curve in figure 7b). The contribution from the surfacestress is so small that it cannot be seen in the plot of figure 7b. The completedecohesion potential at the left end of the cohesive zone is 1.407 Jm�2, which is

approximately 2% smaller than the J integral value around the cohesive zone. Thisdifference is believed to be a truncation error of the eigenfunction series in the FPM.The potential shows that the major decohesion process occurs in the range of�14 A < x 01 < 18 A, and a region of negative potential appears at approximately

18 A< x 01 < 65 A. Twice the surface stress, 2�, distributed in the cohesive zone isplotted in the thick dashed line in figure 7b. The surface stress mostly develops in therange of �14 A < x 01 < 50 A. These region sizes represent the effects of the atomistic

deformation non-locality reflected on the cohesive zone model through a low-spatial-frequency field projection. The fully developed surface stress in the ½�1�12�direction on the (111) face is estimated to be 1.05Nm�1 in this analysis.

Regarding the distribution of potentials in the cohesive zone, the sizes of thecohesive zone and the sub-zone of the negative potential are surprisingly large.

Although the cut off spatial frequency involved in the field projection was low for thegiven cohesive zone size, the cohesive zone size was an optimization parameter forthe field projections shown in figure 7b. Therefore, the large optimum size of the

cohesive zone, 12.3 nm, and the existence of a relatively large sub-zone of negativepotential appear to indicate the tendency for unstable phase transformations, such astwining or dislocation emission, near the crack tip. Indeed, localized twinning was

observed ahead of the crack tip when the magnitude of the applied KII was increased,keeping the sign of the mode mixity. When the sign of the mode mixity was reversed,a dislocation was emitted on a ð11�1Þ plane in a [112] direction, as shown in figure 3b.The tendency of the non-local elastic field to drive such unstable deformations near



the crack tip is believed to be projected on the cohesive zone as a zone of negativedecohesion potential or negative modulus.

The energy release rate of the crack was also evaluated by the J integral andinteraction J integrals, based on various deformation gradients that were defineddifferently for the atomistic deformation field. The inner contours of the domainintegral were squares made of jx 01j ¼ c0 and jx 02j ¼ c 0 with a constant c0 ranging from5 to 200 A. The results are shown in figure 8. Regarding the field without an appliedT-stress shown in figure 8(a), the energy release rates, as evaluated by the J integralwith the hybrid deformation gradient F[H], are shown with open circles and they arepath independent for a c0 larger than approximately 60 A. Even in the range 5 A� c0 � 60 A, the J integral values are nearly path independent. The J values areapproximately 1.5–2% lower than the applied nominal J value, 1.475 Jm�2, of thedisplacement field at the outer boundary layer of the modelling region. Thedifferences are believed to be mainly due to the relaxation from the initial field to anequilibrium field within a finite modelling volume. The results of the evaluation ofthe J integrals with the deformation gradient, F[C], relative to a perfect latticeconfiguration are plotted with solid circles and underestimate the energy release rateby approximately 10%. The energy release rate was also evaluated by interactionJ integrals with both the hybrid deformation gradient, F[H], and the deformationgradient, F[C], relative to a perfect lattice configuration; the former is plotted withopen squares and the latter with solid squares. For the auxiliary field of theinteraction J integrals, the singular K-field of a crack tip in an anisotropic elasticsolid with the same orientation was employed. Then, the complex stress intensity

20 40 60 80 100 120 140 160 180 2001.20

1.25

1.30

1.35

1.40

1.45

1.50

Ene

rgy

rele

ase

rate

(Jm

−2)

Average inner distance of integration domain (Å)

Complete decohesionpotential = 1.407 Jm−2

Potential without surface stresscontribution = 1.404 Jm−2

Figure 8. Energy release rates evaluated by various conservation integrals with differentintegral domains. The thick solid line is the applied nominal energy release rate, 1.475 Jm�2.The evaluation of J integrals with F[H] is shown by open circles; J integrals with F[C] by solidcircles; interaction J integrals with F[H] by ‘þ’ marks; and interaction J integrals with F[C] by‘�’ marks.



factor K was converted to the energy release rate. The energy release rates evaluatedby the interaction J integrals were almost identical to the J values computed with thehybrid deformation gradient. The interaction J integral evaluations were insensitiveto the choice of reference configurations between the perfect and hybridconfigurations. The energy release rates estimated with the low spatial frequencyfield projection are also outlined in figure 8. The energy release rate assessed by thecomplete decohesion energy is the same as that assessed by the complete separationenergy; the contributions from the surface stress are negligible for this field. In orderto see the path dependence of the field projection, one-eighth of the perimeter size ofthe rectangular contour was used as the average contour size in this plot. The energyrelease rates estimated by the field projection were independent of the contour size.

As shown in figure 8, the low spatial frequency field projection with an n up to 2in equation (15) imparts an approximately 2% truncation error in the energy releaserate estimation. If the Griffith criterion [42] were satisfied for an equilibrium crack-tip field in the nanoscale, the energy release rates evaluated by the J integral would beidentical to 2�0 for a field without T-stress and 2�0þ 2�0"T for a field with T-stress.On the contrary, the near-crack-tip non-local deformation field allows variation ofthe atomistic configuration to induce an apparent lattice trapping barrier for crackgrowth. Considering that the critical energy release rate,g, for a crack growth rate, _l,should satisfy the dissipation condition, ðg� 2�Þ _l 0, there is an admissible rangeof surface energy at grec � 2� � gadv, where grec and gadv denote the critical energyrelease rates for crack recession (or healing) and advancement, respectively.

In order to see the lattice trapping effect, the applied mode I stress intensityfactor was varied in the range of 0:375MPa

ffiffiffiffimp� KI � 0:385MPa

ffiffiffiffimp

, while themode II stress intensity factor was maintained at KII ¼ �0:10 MPa

ffiffiffiffimp

. Neithercrack propagation nor healing was observed in the range of0:376 MPa

ffiffiffiffimp� KI � 0:384 MPa

ffiffiffiffimp

. However, when KI ¼ 0:375 MPaffiffiffiffimp

orKI ¼ 0:385 MPa

ffiffiffiffimp

was applied, a dislocation was emitted from the crack tip onthe ð11�1Þ plane along the [112] direction. It is an interesting result that a crack doesnot advance or heal. Therefore, it can be known that Grec� 1.38 Jm�2 and1.44 Jm�2�Gadv. Considering that the 2�0 of the (111) free surface is approximately1.58 Jm�1 for the EAM potential [9], the lattice trapping magnitude of the energyrelease rate for crack healing is more than 14% of 2�0. Although a full study of thecritical conditions of lattice trapping and dislocation emission is beyond the scope ofthis paper, it has been demonstrated that a proper choice of hybrid referenceconfiguration makes it possible to evaluate the J integral values accurately and thus,to study such critical crack tip conditions. While conventional J integral evaluationswith a proper hybrid configuration can provide accurate energy release rates,the FPM offers a means of studying the cohesive zone characteristics, in particularthe sizes of the non-local and nonlinear deformation zones and definitions oftractions, at the atomistic length scale in the cohesive zone.

Figure 9 shows the states of the nominal normal separations and tractions in thecohesive zone measured by the rigid separation potential (RSP) method and by theFPM. In figure 9a, the solid circles indicate the states determined by the RSP methodat the atomic sites in the cohesive zone. The states of the RSP of the normalseparations with various constant shear separations are also illustrated with variouscurves. The nominal normal peak stress is approximately 14–15GPa; although it can



be reduced by a certain amount with a superimposed shear separation, it is largerthan 10GPa for shear separations less than 80% of the unstable stacking shearseparation. However, the field projected states of the normal separations are quitedifferent from those of the RSP and are plotted with a thick line in the figure. TheFPM gives a normal peak stress of only 4GPa. Figure 9b exhibits the atomicpositions near the decohesion process zone. The fifth atom is at the RSP state of12GPa nominal normal peak stress; however, the FPM measures the peak stress of4GPa at the tenth atom in the numbering sequence. The RSP measurements indicatethat the interatomic forces are transmitted through the atomic bonds of the secondatom and beyond; however, the FPM shows that the forces are transmitted throughthose of the first atom and beyond. In the RSP measurements, only one isolatedatom, number 5, experiences a nominal normal stress above 4GPa; the nominaltraction at such an isolated site cannot be assessed reliably by the RSP. The FPMprovides an estimation of the traction from extrapolations of the adjacentinteratomic states. The field projection shows that the cohesive zone tractions aremore broadly spread than the distribution measured by the RSP method.

−2 −1 0 1 2 3 4 5−4

−2

0

2

4

6

8

10

12

14

16t2-δ2 from FPM

δ1/δ1p = 0.0

δ1/δ1p = −0.2

δ1/δ1p = −0.4

δ1/δ1p = −0.6

δ1/δ1p = −0.8

From atomistics

Tra

ctio

n, t 2

(G

Pa)

Separation, δ2 (Å)

12

3

4

5

6

1′2′3′4′5′6′

(a)

(b)

Figure 9. (a) Comparison between two nominal states of cohesive zone separation measuredby the RSP method (solid circles) and the FPM (open circles) at cohesive zone atomic sites.(b) Atomistic deformation configuration near the crack tip and the cohesive zone atomic sites.



8. Conclusions

A general formulation of a planar elastic field projection has been derived foridentifying the cohesive zone constitutive relations from the static or steady statedynamic elastic fields of a cohesive crack tip on an interface between two anisotropicsolids. The formulation is also applicable to the elastic field of a cohesive crack tip ina homogeneous solid whether it is isotropic or anisotropic. A new framework ofcohesive zone models, including the surface stress effects, has been introduced todevelop the nanoscale FPM. A new eigenfunction expansion of the elastic fieldsurrounding a crack-tip cohesive zone made it possible to develop the FPM. Theeigenfunctions were orthogonalized for interaction J integrals based on the Jacobipolynomials. The field projection using the interaction J integrals providesJ-equivalent cohesive tractions, interface displacements and separations, as well asthe surface energy gradients caused by the gradual variation of the surface formationwithin the cohesive zone. In particular, the FPM can define tractions and stresseswithin and near a cohesive zone more objectively than the conventional rigidseparation potential method. In addition, the field projection measurement canidentify the energetic states of decohesion in the cohesive zone and it provides ameans to compare the experimental measurements and computational simulations ofatomistic decohesion or slip processes on the same grounds.

The efficacy of the FPM has been demonstrated with nanoscale deformationfields of EAM atomistic simulations for crystal decohesion along a ½�1�12� direction ina gold (111) plane. It has been noticed that two major free surface effects, i.e. surfacerelaxation and surface stress, play significant roles in the atomistic elastic fieldanalysis near a crack tip. The surface stress has been correctly incorporated into therelationship of the configurational force balance in the cohesive zone model and ahybrid reference configuration has been introduced to deal with the surfacerelaxation effect. These two new treatments have made it possible to study latticetrapping in crack growth and the validity of the Griffith criterion with conservationintegrals. It has been found that the lattice trapping potential is larger than 14% of2�0 for the healing process of a gold (111) plane in a [112] direction. The low spatialfrequency field projections have revealed that the decohesion potential along thecohesive zone is quite different from the prediction based on the rigid separationpotential. The peak normal traction, approximately 4GPa, in the cohesive zone hasbeen found to be much lower than the 15GPa expected from the rigid separationpotential. This result clearly shows that the rigid separation potential overestimatesthe peak stress of separation in the crack tip region of highly non-local deformation.It has been also noticed that the critical energy release rates for various crack tipprocesses, such as crack growth and dislocation emission, depend heavily on themodes, such as KII/KI and the T-stress, of external loading on the crack tip at thenanoscale. The field projection also showed that a sub-zone of negative potentialexists within the cohesive zone, reflecting the tendency of some unstabledeformations near the crack tip and that the effective cohesive zone size variesdepending on the mode of external loading. Half of the effective zone size, which ispresumably close to the radius of the non-local deformation zone, was measured tobe approximately 40 A in an applied singular field. The FPM, together with thevarious analysis techniques developed in this paper, will be useful in studying



decohesion, dislocation emission, single asperity friction, and grain boundary sliding

and separations in crystalline solids in the nanoscale.

Acknowledgements

This work was supported by the MRSEC Program of the National Science

Foundation under Award Number DMR-0079964 and the Post-Doctoral

Fellowship Program of Korea Science & Engineering Foundation (KOSEF).

The authors are grateful to Dr. Pranav Shrotriya for his help with the code

Dynano87 for the atomistic simulations and Dr. Foiles for making the code available

to the public on ftp://146.246.250.1.

Appendix A. Stroh matrices A, B, and M�1

Among the expressions of the displacement and stress fields of an anisotropic elastic

medium presented in equations (1) and (2) of the main text, the terms entering the

expressions of the boundary conditions along the entire x1 axis are

e1 � ruf g ¼ 2Re Af 0ðzÞ½ �, ðA1aÞ

e2 � rf g ¼ 2Re Bf 0ðzÞ½ �, ðA1bÞ

where e1 and e2 represent the base vector of the (x1, x2) Cartesian coordinate. The

symbol A represents a complex matrix that satisfies the Stroh [28] characteristic

equations

fTag þ Tb þ TTb

� �~�þ Tcf g~�

2h i

AT¼ 0 ðA2Þ

with Ta¼ e1 �C � e1, Tb¼ e1 �C � e2, Tc¼ e2 �C � e2, C denotes the elastic moduli tensor

of the solid, and ~� is the diagonal Stroh eigenvalue matrix with the components of

pj�jk. The matrix B is given by

B ¼ TTb

� �Aþ fTcgA~�: ðA3Þ

Stroh [28] showed that A and B are non-singular for three distinct complex pairs of pjand the matrix

M�1 � iAB�1 ðA4Þ

is a positive definite Hermitian matrix. Here, i ¼ffiffiffiffiffiffiffi�1p

and ( )�1 stands for the inverse

of the matrix. The explicit expressions of A, B, and M�1 in terms of elastic constants

are given by Suo [30] (Suo used the symbols L and B in his paper for B and M�1 in

this paper, respectively) and Ting [43].



Appendix B. A complex function representation of the anisotropic elastic field of a

bimaterial interface crack tip

The eigenfunction expansion around a sharp interfacial crack tip lying on the

negative x1 axis in an anisotropic bimaterial is given by Beom and Atluri [32] as

f 0h�iðzÞ ¼ B�1h�i

hI� ð�1Þ�ib

iY z�i"� � kðzÞ

2ffiffiffiffiffiffiffiffi2�zp þ i

hI� ð�1Þ�a

imðzÞ

� �, ðB1Þ

for �¼ 1 or 2 referring to materials 1 or 2, respectively, and where k(z) and m(z) are

real analytic functions that satisfy kðzÞ ¼ �kðzÞ and mðzÞ ¼ �mðzÞ, respectively.

The generalized Dundurs parameter matrices a and b for dissimilar anisotropic

materials are defined as follows [32, 33]:

a ¼ Lh1i � Lh2i� �

Lh1i þ Lh2i� ��1

, b ¼ L�1h1i þ L�1h2i

� ��1Wh1i �Wh2i� �

, ðB2Þ

where L�1¼Re(M�1) is a symmetric real matrix and W¼�Im(M�1) an antisym-

metric real matrix and, thus, a and b are also real matrices. The bimaterial parameter

" that appears in the argument of the function Y[z�i"] is an oscillatory index which is

related to b by

" ¼1

2�ln

1�

1þ

� �, ¼ �

1

2tr b2� � 1

2

: ðB3Þ

The real matrix function Y[�(z)] is explicitly defined in terms of b by Qu and

Li [31] as

Y �ðzÞ½ � � Iþi

2�ðzÞ � ��ðzÞ½ �bþ

1

21�

1

2�ðzÞ þ ��ðzÞ½ �

� �b2, ðB4Þ

where �(z) is an arbitrary function of z. The matrix function Y[�(z)] plays an

important role in representing the oscillatory fields near the crack tip. Certain

properties of the matrix function Y[�(z)] are used for various derivations in this

paper, and they are summarized as follows [31, 35]:

Y �1ðzÞ½ �Y �2ðzÞ½ � ¼ Y �1ðzÞ�2ðzÞ½ �, ðB5aÞ

Y �ð �zÞ½ � ¼ Y �ðzÞ½ �, ðB5bÞ

UYT �ð �zÞ½ �U�1 ¼ Y ��ðzÞ½ �, ðB5cÞ

Y ��ðx1ei�Þ

� �Iþ ibð Þ

�1¼ Y xi"1 cosh�"

� �, ðB5dÞ

Iþ ibð ÞY �ðx1ei�Þ

� �¼ I� ibð ÞY �ðx1e�i�Þ½ �, ðB5eÞ

where � denotes an arbitrary function while � indicates the function �(z)¼ z�i".



Appendix C. Orthogonalization of the interaction J integral and orthogonal

polynomials

In order to orthogonalize the interaction J integral of equation (8), the matrix

function Y[�(z)] for �(z)¼ z�i" should be diagonalized. From the definition of Y[�(z)]in equation (B4) and the property of a Hermitian matrix, as prepared by Beom and

Atluri [35], the matrix function Y[�(z)] can be diagonalized as:

Y �ðzÞ½ � ¼ �V�T ~K �ðzÞ½ � �V

Tor U�1Y �ðzÞ½ � ¼ V ~K �ðzÞ½ � �V

T, ðC1Þ

in which ~K �ðzÞ½ � ¼ diag �ðzÞ, ��ðzÞ, 1� �

and V� (v1, v2, v3), where v1, v2ð¼ v1Þ, and

v3ð¼ �v3Þ are eigenvectors corresponding to the eigenpairs (�", v1), (", v2), and (0, v3)

of ð �M�1h1i þM�1

h2i ÞðM�1h1i þ

�M�1

h2i Þ�1v ¼ e2�"v. Therefore, by using equation (C1), the

interaction integral in equation (8) is rewritten as:

J int�"½S, S � ¼

Z c

�c


qgT0 ðx1Þ

~K �cðx1Þ½ �h0ðx1Þþ gT0 ðx1Þ

~K �cðx1Þ½ �h0ðx1Þn o

dx1

þ 4

Z c

�c


qqTðx1Þ Lh1i þLh2i

� ��1rðx1Þþ q

Tðx1Þ Lh1i þLh2i

� ��1rðx1Þ

h idx1

¼ c

Z 1

�1

gT0 ðcsÞ ~xðsÞh0ðcsÞþ gT0 ðcsÞ ~xðsÞh0ðcsÞ

h ids

þ 4c

Z 1

�1

ffiffiffiffiffiffiffiffiffiffiffiffi1� s2p

qTðcsÞ Lh1i þLh2i� ��1

rðcsÞþ qTðcsÞ Lh1i þLh2i

� ��1rðcsÞ

h ids,

ðC2Þ

where g0(x1)¼VTg(x1), �c(x1)¼ [(c� x1)/(cþx1)]�i", h0ðx1Þ ¼ VTY cosh�"½ �hðx1Þ,

and s¼ x1/c. The weight function ~xðsÞ is a diagonal matrix with the components

of !j(s)�jk and

xjðsÞ ¼ ð1� sÞaj ð1þ sÞ �aj j ¼ 1, 2, 3ð Þ, ðC3Þ

in which the exponents are written in terms of " as a1¼ 1/2� i", a2¼ 1/2þ i", anda3¼ 1/2. Then, the analytic functions g0(cs) and h0(cs) can be expanded in terms of

Jacobi polynomials, denoted as P ða, bÞn ðsÞ, which are orthogonal at the interval [�1,1]

with a weight function of u(s) as

Z 1

�1

Pða, bÞm ðsÞ

ða, bÞm

Pða, bÞn ðsÞ

ða, bÞn

xðsÞds ¼�

2�mn, ðC4Þ

where

ða,bÞn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2aþbþ2�ðnþ aþ 1Þ�ðnþ bþ 1Þ=½�ð2nþ aþ bþ 1Þ�ðnþ 1Þ�ðnþ aþ bþ 1Þ�

p



and �(z) is a gamma function. The first two terms and the recurrence relations of the

Jacobi polynomials are given by

Pða, bÞ0 ðsÞ ¼ 1,

Pða,bÞ1 ðsÞ ¼

a� b

2þ 1þ

aþ b

2

� �s,

2 nþ 1ð Þ nþ aþ bþ 1ð Þ

2nþ aþ bþ 1ð Þ 2nþ aþ bþ 2ð ÞPða,bÞnþ1 ðsÞ

¼a2 � b2� �

2nþ aþ bð Þ 2nþ aþ bþ 2ð Þþ s

Pða,bÞn ðsÞ

�2 nþ að Þ nþ bð Þ

2nþ aþ bð Þ 2nþ aþ bþ 1ð ÞPða, bÞn�1 ðsÞ: ðC5Þ

On the other hand, q(cs) and r(cs) may be expanded in terms of the Chebyshev

polynomials of the second kind, denoted by Un(s), which are a special case of theJacobi polynomials P ða,bÞn ðsÞ with a¼ b¼ 1/2.

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Corrections

1. Page 1, (Footnote of S. T. Choi): Present address: Micro Device & Systems Lab, SAIT, P.O. Box 111,

Suwon 440-600, Republic of Korea.

2. Page 1, 9th line: “Hong and Kim (2004)” should read “Hong and Kim (2003)”.

3. Page 4, 14th line: Remove “generalized” ; Change “… known as the Stroh …” to “…e.g. the Stroh

(displacement) …”

4. Page 7, 11th line: “ 0 0 : sγ γ= + Σ ε ” should read “ 0 0 : sγ γ= + Σ ε ” (bold).

5. Page 7, 12th line: “ sε ” should read “ sε ” (bold).

6. Page 7, 3rd line from the bottom: “ ” should read “ ” (bold). λ λ

7. Page 9, in equation (10): “{ }1 1( ) ...u x µ⋅∇e ” should read “{ }1 1( ) ...u xµ

⋅∇e ”.

8. Page 9, in equation (10): α ’s should be non-italic and bold.

9. Page 9, 4th line from the bottom: “ , ( ) ( )o T Tn z z−=G V VnP ( ) ( )o T T

n z z−=H V VnP

n z z−= VPn

” should read

“ G V , ( ) ( )o T T ( ) ( )o T Tn z z−= VPnH V ” (non-italic).

10. Page 9, the last line: “ ( , ) ( , )( )j j j ja a a ajk n nzδ θP ; (1 2,1 2) (1 2,1 2)( ) ( )n n nU z z θ= P ” should read

“ ( , ) ( , )( )j j j ja a a ajk n nP zδ θ ; (1 2,1 2) (1 2,1 2)( ) ( )n n nU z P z θ= ” (non bold).

11. Page 10, 2nd line: “ ” should read “ ” ” (non bold). ( , ) ( )a bn zP ( , ) ( )a b

nP z

12. Page 10, equations (12) and (13): equations (12) and (13) should read, respectively,

{0

( ) ( ) ( ) ( ) ( )N

o o o on n n n n n n

n

z z z zµ µ µ µ µ< > < > < > < > < >=

′ = + + +∑f }nzg h q rG H Q R , (12)

[ ]

1

1 1

2 1

1

1( ) 1 ( 1) ( 1) ( ),2

1( ) 1 ( 1) ( 1) cosh ( ),2

( ) 1 ( 1) ( ),2

( ) ( 1)2

o in n

o i on n

o on n

on

z z i z z

z z i z

iz z z

iz

µ εµ µ

µ εµ µ

µµ µ

µµ µ

πε

− −< > < >

− − −< > < >

−< > < >

−< > < >

⎡ ⎤ ⎡ ⎤= + − − +⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= − − − − −⎣ ⎦ ⎣ ⎦

⎡ ⎤= − − −⎣ ⎦

⎡ ⎤= − −⎣

B I β Y G

B I β Y Y H

B I α Q

B I α

G

H

Q

R ( ).on z⎦R

o

z (13)

13. Page 11, 4th line: “ { }ˆ ( ) ( )onz zµ µ< > < >′ =f H e k ” should read “ { }ˆ ( ) ( )o

n kz zµ µ< > < >′ =f eH ”.

14. Page 11, 8th line: “ { }ˆ ( ) ( )onz zµ µ< > < >′ =f G e k ” should read “ { }ˆ ( ) ( )o

n kz zµ µ< > < >′ =f eG ”.

15. Page 11, 10th line: “ { }ˆ ( ) ( )onz zµ µ< > < >′ =f R e k ” should read “ { }ˆ ( ) ( )o

n kz zµ µ< > < >′ =f eR ”.

16. Page 11, 12th line: “ { }ˆ ( ) ( )onz zµ µ< > < >′ =f Q e k ” should read “ { }ˆ ( ) ( )o

n kz zµ µ< > < >′ =f eQ ”.

17. Page 12, 16th line: “ ” should read “ ” (non-italic). ( ) ( )n nz U z= IP ( ) ( )n nz U z=P I

18. Page 15, 22nd line: “ ” should read “12[P] 22[P]( , −F F 1) 1)12[P] 22[P]( , F F − ” (non-bold).

19. Page 15, 25th line: “ ” should read “12[H] 22[H]( , −F F 1) 1)12[H] 22[H]( , F F − ” (non-bold).

20. Page 15, 4th line from the bottom: “of the equations (15)” should “of the equation (15)”.

21. Page 17, 9th line: “Figure 5a and b, respectively, shows” should read “Figure 5a and b, respectively,

show”.

22. Page 17, 29th line: “in figure 5” should read “in figure 5b”.

23. Page 24, 18th line from the bottom: “ 2rec 1.38 JmG −≤ ” should read “ ”. 2

rec 1.38 Jm−≤G

24. Page 24, 17th line from the bottom: “ ” should read “ ”. 2adv1.44 Jm G− ≤ 2

adv1.44 Jm− ≤ G

25. Page 27, in equations (A2) and (A3): λ should be bold.

26. Page 27, 10th line from the bottom: “λ ” should read “λ ” (bold).

27. Page 29, in equations (C1) and (C2): Λ should be non-italic.

28. Page 29, 8th line: should be non-italic and “Λ 2 1( )=v v ” should read “ 2 1( )=v v ”.

29. Page 29, 11th line from the bottom: “ [ ]0 1 1( ) cosh ( )Tx xπε=h V Y h ” should read

“ [ ]0 1 1( ) cosh ( )Tx xπε=h V Y h ” (bar on TV to indicate complex conjugate).

30. We provide the high quality artwork in the attachment for Figures 1, 3, 4 and 9(b) as requested in the

query.

31. We have corrected the legend of Figure 7(a) and provide the corrected figure as an attachment.

2

nanoscale planar field projections of atomic decohesion

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