Weakly-singular, Weak-form Integral Equations for

Cracks in Three-Dimensional Anisotropic Media

Jaroon Rungamornrat1 and Mark E. Mear2

Abstract

Singularity-reduced integral relations are developed for displacement discontinuities in three-dimensional, anisotropic linearly-elastic media. An isolated displacement discontinuity is consideredfirst, and a systematic procedure is followed to develop relations for the displacement and stress.The singularity-reduced relation for the stress is particularly important since it is in a form whichallows a weakly-singular, weak-form traction integral equation to be readily established. The re-sults for a general displacement discontinuity are then specialized to cracks and dislocations; theresults for dislocations appear to be simpler in form than those previously available, and they allowthe connection between earlier developments for dislocations and cracks to be put into perspective.The development employed for an isolated crack is then extended to allow treatment of cracks in afinite domain, and a pair of weakly-singular, weak-form displacement and traction integral equationis established. The integral equations are analogous to those developed for isotropy by Li and Mear[16] and Li et al. [17], but the extension to anisotropy requires an approach quite distinct fromthat previously employed. The displacement and traction integral equations which are establishedcan be combined to obtain a final formulation which is in a symmetric form, and in this way theyserve as the basis for a weakly-singular, symmetric Galerkin boundary element method suitable foranalysis of cracks in anisotropic media.

Keywords: integral equations, boundary element, dislocations, cracks, anisotropy, singularity-reduced, weakly-singular.

1 Introduction

Boundary value problems for cracks in homogeneous, linearly elastic media can be formu-lated conveniently in terms of singular integral equations. The development of these integralequations typically rests upon Somigliana’s identity, which is an expression for the displace-ment at a point in the interior of the body in terms of the displacement and traction actingon the surface of the body. From this identity, a displacement integral equation is readilyobtained but, as is well known, this integral equation is insufficient to properly treat crack

1Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The Universityof Texas at Austin, TX 78712, USA.

2Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin,TX 78712, USA.

1

problems since information about the traction acting on the crack is absent. For this reason,it is necessary to establish a traction integral equation for the crack.

Such a traction integral equation can be directly obtained by using Somigliana’s identityalong with the strain-displacement relations and Hooke’s law. However, the integral equationobtained in this fashion involves a strongly-singular kernel (of order 1/r3) which requiresspecial theoretical and numerical considerations. In particular, in a numerical treatmentof the integral equation it is necessary to satisfy the requirement that the derivative of thedisplacement be continuous (hence ruling out the use of standard Co elements) and to employspecial integration techniques (e.g. [4], [11], [19]).

Toward alleviating the difficulties posed by the strongly-singular kernel, singularity-reduced traction integral equations have been sought through a regularization process in-volving an integration by parts. In the context of three dimensional crack modeling, thefirst such singularity-reduced traction integral equations appears to be the Cauchy-singular(of order 1/r2) relation obtained independently by Bui [6] and Weaver [24]. This integralequation is restricted to mode-I loading of a planar crack in an unbounded domain, but theresult was later generalized by Sladek and Sladek [21] to allow treatment of curved cracksand mixed mode loading. It should be noted that while the kernel appearing in these equa-tions is Cauchy-singular rather than strongly-singular, there remains the requirement thatthe derivative of the crack-face displacement be continuous. To eliminate this requirement, itis necessary to develop a weak-form statement of the traction integral equation in such a waythat an additional integration by parts can be performed to render the kernel weakly-singular(of order 1/r).

The first weakly-singular, weak-form traction integral equation for fracture analysis isthat by Gu and Yew [12]. Their development rests upon Weaver’s [24] Cauchy-singulartraction integral equation and, as such, is restricted to an isolated planar crack subjectedto mode-I loading. A generalization of this work was carried out by Xu and Ortiz [26] whodeveloped a variational boundary integral equation to treat an isolated crack with arbitrarygeometry and mixed-mode loading. To obtain their integral equation, Xu and Ortiz [26]utilized the fact that the crack-face displacement can be represented in terms of a continuousdistribution of dislocation loops. A limitation of this result is that it applies only for a crackin an unbounded domain.

Li and Mear [16] and Li et al. [17] presented a systematic technique for regularizingboth the displacement and traction integral equations associated with three dimensionalisotropic media, and by using this technique they were able to obtain a pair of weakly-singular, weak-form integral equations applicable to cracks in a finite domain. We remarkthat their final formulation is closely related to that of Bonnet [5] for elasticity problemsin the absence of a crack. Li et al. [17] used the pair of weakly-singular displacement andtraction integral equations as the basis for a symmetric Galerkin boundary element method,and they successfully implemented the formulation to allow treatment of general boundaryvalue problems for bodies containing cracks. Their development is, however, restricted toisotropic material behavior.

Modeling anisotropic material behavior within the context of singular integral equations

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is quite challenging due to the complexity of the associated fundamental solutions, andfor this reason work on modeling cracks in three-dimensional anisotropic media is modestin comparison to that for isotropic material behavior. The singularity-reduced tractionintegral developed by Sladek and Sladek [21] applies for general anisotropy, but (in additionto the limitations already mentioned) it explicitly involves the stress fundamental solutionwhich is difficult to compute for anisotropic material behavior. More recently, Sladek et al.[22] established completely regularized integral equations for anisotropic media which areapplicable to cracks in both unbounded and finite domains. There remains the requirementthat the derivative of the crack face displacement be continuous and, further, the integralequations involve both the gradient of the displacement fundamental solution and the stressfundamental solution.

Work toward developing weak-form, weakly-singular traction integral equations for cracksin anisotropic media is particularly limited. Becache et al. [2] utilized a double layer potentialtechnique along with the Fourier transform to obtain a regularized integral equation for threedimensional anisotropic elastodynamics. Their final result is in the frequency domain andis left expressed in terms of the Fourier transform variables associated with the spatialcoordinates. In principle, the result can be specialized to the static case by taking a limit inwhich the frequency tends to zero, but the treatment of this limiting process as well as thesubsequent inversion of the Fourier transform appears to be quite complicated. Recently, Xu[27] utilized Lothe’s [18] expression for the interaction energy between dislocation loops toextend the formulation by Xu and Ortiz [26] to allow treatment of material anisotropy. Theresulting traction integral equation contains a weakly-singular kernel but, its application islimited to a crack in an unbounded domain.

In this paper we develop weakly-singular, weak form integral equations for cracks ingenerally anisotropic finite domains. The integral equations are analogous to those developedfor isotropy by Li and Mear [16] and Li et al. [17], but it should be noted that the extensionto anisotropy is by no means trivial and, in fact, it requires an approach quite distinctfrom that previously employed. The displacement and traction integral equations which areestablished can be combined to obtain a final formulation which is in a symmetric form,and in this way they serve as the basis for the development of a weakly-singular, symmetricGalerkin boundary element method suitable for analysis of cracks in anisotropic media.

We first consider an isolated displacement discontinuity and, by introducing certain spe-cial decompositions for the stress fundamental solution and the strongly singular kernel,systematically develop singularity-reduced integral relations for the displacement and stress.The singularity-reduced relation for the stress is particularly important since it is in a formwhich allows a weakly-singular, weak-form traction integral equation to be readily estab-lished. The results for an isolated crack then form the basis to treat cracks in a finitedomain. Indeed, treatment of a finite domain requires only results obtained for the un-bounded domain, and the final displacement and traction integral equations involve exactlythe same set of weakly-singular kernels.

We remark that while this work is primarily concerned with the modeling of cracks,the singularity-reduced integral relations which are developed for an unbounded domain

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pertain to an arbitrary displacement discontinuity. Included as a special case are dislocationsin anisotropic elastic media, and this case is discussed in detail for the following reasons.Firstly, the results developed here for dislocations (specifically, a line integral relation for thedisplacement field and a line integral relation for interaction energy) appear to be in a formwhich is simpler than those previously available. Secondly, it allows the direct connectionbetween results for dislocations and cracks to be emphasized and, in particular, it allowsearlier (independent) findings for these two types of discontinuities to be put into propercontext.

2 Integral Relations for Discontinuities

Consider a homogeneous, anisotropic, linearly elastic domain containing an isolated displace-ment discontinuity as shown schematically in Figure 1. The surface of the discontinuity iscomprised of an upper and lower surface3 S+

c and S−c , respectively, and these surfaces aregeometrically coincident such that their unit normals (taken to be directed ‘into’ the dis-continuity) satisfy n+

i = −n−i . It is assumed that the domain in which the discontinuity isembedded is free of body force and that remote loading is absent. Further, it is assumedthat, as is typical, the traction acting on the discontinuity is locally self-equilibrated in thatit satisfies t+ = −t− at each point on the geometrically coincident surfaces.

S+c

S−c

x•

Figure 1: Schematic of an isolated discontinuity.

Somigliana’s identity is readily specialized to this case, with the result being an integralrelation giving the displacement u at a point x in the domain in terms of data on the surface

3Here and in what follows, the superscripts + and - are used to indicate that the quantity is associatedwith the ‘upper’ and ‘lower’ surface of the crack, respectively.

4

of the discontinuity. With S ≡ S+c and ni ≡ n+

i , this well known result takes the form

up(x) = −∫

S

Spij(ξ − x) ni(ξ) ∆uj(ξ) dS(ξ) (1)

in which the kernel Spij corresponds to the stress induced by a unit concentrated load acting

in the pth coordinate direction4, and where ∆uj(ξ) = u+j (ξ) − u−j (ξ) denotes the jump in

displacement across the discontinuity (i.e. the relative displacement of the two geometricallycoincident surfaces associated with the displacement discontinuity). Utilizing this form ofSomigliana’s identity, an integral relation for the stress is readily obtained as

σlk(x) =

∫

S

Σlkij (ξ − x) ni(ξ) ∆uj(ξ) dS(ξ) (2)

where

Σlkij ≡ Elkpq

∂Spij

∂ξq

(3)

in which Elkpq are the elastic moduli. These integral relations apply for points x which arenot on the surface of the discontinuity and, owing to the Cauchy-type singular kernel in (1)and the strongly-singular kernel in (2), proper care must be exercised in considering a limitas x approaches this surface. In particular, when (2) is to be utilized to obtain a relation forthe traction on the surface of the discontinuity, it is necessary to properly treat and interpretthe limit in terms of a Hadamaard finite part integral.

We now distinguish two special types of displacement discontinuities: the first is a dislo-cation for which bi ≡ ∆ui is a prescribed constant, and the second is a crack for which thetraction ti ≡ t+i = −t−i is prescribed and the relative displacement ∆ui is to be determined(subject to the condition that it vanishes along the edge of the discontinuity, viz. the crackfront). For the latter case it is important to note that the displacement relation (1) does notcontain information about the traction acting on the discontinuity, hence it is ‘degenerate’in this sense and does not provide a basis for obtaining a useful integral equation for theunknown relative crack-face displacements. Instead, attention must necessarily be directedtoward the stress relation (2) and, specifically, toward its use in obtaining a traction integralequation.

Now, for either of these two types of discontinuities it is of interest to regularize theintegral relations to render them more suitable for numerical analysis. What is soughtis to reduce the ‘strength’ of the singularity associated with the kernels by means of anintegration by parts. Such results are well known in the field of dislocation mechanics (e.g.Hirth and Lothe [14]; Lothe [18]), and more recently there has been significant work directedtowards achieving such regularization for cracks (e.g. Xu and Ortiz [26], Li and Mear [16],

4The use of both superscripts and subscripts in expressing the cartesian components of tensor valuedquantities is simply a matter of notational convenience. See Appendix A for a description of the notationadopted for the fundamental solution.

5

Becache et al. [2], Xu [27]). The purpose of the work presented in this paper is to providea straightforward, complete development of such singularity-reduced integral equations forapplication to displacement discontinuities in anisotropic media.

3 Development of singularity-reduced relations

We first provide an overview of the regularization strategy utilized to obtain singularity-reduced integral relations, after which we develop in detail the kernels which appear in theserelations. The development is carried out in the context of an isolated discontinuity asintroduced above, but we emphasis that the primary objective of this work is to developa full set of weakly-singular, weak-form equations applicable to treatment of cracks in afinite domain, and these results will be established using the results obtained for an isolateddiscontinuity. In this context we remark that whereas a displacement integral equationfollowing from (1) is not essential for treating an isolated crack (when it is the relativedisplacements ∆ui or, more specifically, the stress intensity factors which are sought), itscounterpart for finite domains is relevant for developing strategies to treat general fractureproblems. For this reason, a singularity-reduced displacement integral equation is discussedeven for the case of an isolated discontinuity.

Toward developing the singularity-reduced integral equations, we first establish their ex-istence in terms of certain weakly-singular kernels. Explicit partial differential equationsgoverning these kernels are derived and, after the overall form of the integral equations isestablished, these differential equations are solved by an application of the Radon transform.Results are obtained for general displacement discontinuities, but these results then special-ized for treatment of dislocations and cracks. We begin with the displacement integral equa-tion after which a treatment of the stress is provided leading, finally, to a weakly-singular,weak-form traction integral equation.

3.1 Displacement

As a means to develop a singularity-reduced displacement integral equation, we introducea decomposition for Sp

ij which is analogous to that utilized by Li and Mear [16] in theirtreatment of isotropic media. Specifically, we write

Spij(ξ − x) = F p

ij(ξ − x) + Hpij(ξ − x) (4)

in which

Hpij ≡ δpj

∂

∂ξi

( 1

4πr

)= − 1

4π

(ξi − xi)δpj

r3(5)

where r = ‖ξ − x‖ is the distance between the field point ξ and the source point x. Since

Spij,i = Hp

ij,i = −δpj δ(ξ − x) (6)

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in which δ(ξ − x) is the 3-D Dirac delta function centered at x, it follows that F pij is

divergence free everywhere including the source point x. This fact implies the existence ofthe representation (see Appendix C)

F pij(ξ − x) = εikm

∂Gpmj(ξ − x)

∂ξk

(7)

where the kernel Gpmj is weakly-singular at ξ = x in the sense that it is of order 1/r as

r → 0. Using (4) and (5), and expressing the ‘stress fundamental solution’ Spij in terms of

the ‘displacement fundamental solution’ U ij , equation (7) gives rise to the following system

of partial differential equations to be solved for the kernel Gpmj:

εikm

∂Gpmj(ξ − x)

∂ξk

= Eijab∂Up

a (ξ − x)

∂ξb

− δpj∂

∂ξi

( 1

4πr

)(8)

For isotropic material behavior it is relatively simple to evaluate Gpmj by means of a ‘di-

rect integration’ of these partial differential equations (see Li and Mear [16]), but such anapproach is extremely difficult, if not impossible, for general anisotropy due to the compli-cated nature of the fundamental solution. However, an explicit solution can be obtainedin a straightforward fashion by exploiting the Radon transform, and this solution will bepresented further below.

Now, using (4) and (7) to re-express the stress fundamental solution which appears inthe displacement integral relation (1), and then integrating by parts via Stokes’ theorem, weobtain

up(x) =

∫

S

Gpmj(ξ − x) Dm∆uj(ξ) dS(ξ)−

∮

∂S

Gpmj(ξ − x) ∆uj(ξ) dξm

−∫

S

Hpij(ξ − x) ni(ξ) ∆uj(ξ) dS(ξ) (9)

where

Dm = nlεlsm∂

∂ξs

(10)

is a surface differential operator. We note that the kernel Hpijni (which also arises in the con-

text of integral representations for Laplace’s equation) is weakly-singular hence the integralrelation (9) does in fact involve only weakly-singular kernels. With y ∈ S being a point onthe surface of the discontinuity, the limit x → y is now readily taken with the result

1

2Σup(y) =

∫

S

Gpmj(ξ − y) Dm∆uj(ξ) dS(ξ)−

∮

∂S

Gpmj(ξ − y) ∆uj(ξ) dξm

−∫

S

Hpij(ξ − y) ni(ξ) ∆uj(ξ) dS(ξ) (11)

in which Σup(y) = u+p (y) + u−p (y). While this relation is not of primary interest in our

development, it does provides the additional information required to (separately) determine

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the displacement of the upper and lower surfaces of the discontinuity. Before pursuingthe displacement equation further, we turn our attention to establishing the existence ofsingularity-reduced relations for the stress and traction.

3.2 Stress and traction

The kernel Σlkij (ξ − x) which appears in the stress relation (2) is strongly-singular in the

sense that it is of order 1/r3 as r → 0. Toward establishing a singularity-reduced alternativeto (2), we first note that (see Appendix A)

Σlkij,i(ξ − x) = − ∂

∂ξi

[Eijklδ(ξ − x)

](12)

and

Σlkij,l(ξ − x) = − ∂

∂ξl

[Eijklδ(ξ − x)

](13)

That is, the divergence of Σlkij with respect to either ξi or ξl vanishes everywhere except at the

source point x where it possesses a singularity in terms of the derivative of the Dirac-deltafunction. This observation motivates a decomposition of Σlk

ij (ξ − x) as

Σlkij (ξ − x) = Σlk

ij (ξ − x) + Dlkij (ξ − x) (14)

in which

Dlkij (ξ − x) = −Eijkl δ(ξ − x) (15)

and where Σlkij is divergence free (with respect to ξi and ξl) everywhere including the source

point x. It then follows that there exists a representation in the form (see Appendix C)

Σlkij (ξ − x) = εism

∂

∂ξs

εlrt∂

∂ξr

Ctkmj(ξ − x) (16)

where the kernel Ctkmj is weakly-singular at ξ = x in the sense that it is of order 1/r as r → 0.

Upon combining (14), (15) and (16), and then expressing Σlkij in terms of the displacement

fundamental solution, we obtain the system of differential equations governing Ctkmj as

εism∂

∂ξs

εlrt∂

∂ξr

Ctkmj(ξ − x) = ElkpqEijab

∂2Upa (ξ − x)

∂ξq∂ξb

+ Eijklδ(ξ − x) (17)

We defer solution of these equations until the next section.We now return to (2) and use the results presented above to obtain a singularity-reduced

relation for the stress field induced by the displacement discontinuity. A key observation inthis regard is that, for any point x in the domain (i.e. x /∈ S), the validity of (2) is unalteredby replacing the kernel Σlk

ij with Σlkij ; this follows immediately from the decomposition (14)

8

and the fact that, for x /∈ S, the term involving the integral of the Dirac delta functionvanishes. Combining (2) (with Σlk

ij first replaced by Σlkij ) and the representation (16), inte-

grating by parts via Stokes’ theorem and utilizing the translational property of the kernelCtk

mj to exchange a derivative with respect to ξr for one with respect to −xr, we obtain

σlk(x) = εlrt∂Ωtk

∂xr

(18)

where the quantity

Ωtk(x) = −∮

∂S

Ctkmj(ξ − x) ∆uj(ξ) dξm +

∫

S

Ctkmj(ξ − x) Dm∆uj(ξ) dS(ξ)

(19)

involves only weakly-singular kernels.The relation (18) for the stress at an internal point x is well suited to obtain an integral

equation for the traction acting at a point y ∈ S on the surfaces of discontinuity. Indeed, todo so we simply form nl(y)σlk(x) and take a limit as x → y with the result

tk(y) = DtΩtk(y) (20)

Finally, a weakly-singular weak-form traction integral equation follows by multiplying (20) bya continuous test function ∆uk(y), integrating the result over the surface of the discontinuity,and employing Stokes’ theorem to obtain

∫

S

tk(y) ∆uk(y) dS(y) =

∮

∂S

∆uk(y) Ωtk(y)dyt −∫

S

Dt∆uk(y) Ωtk(y) dS(y)

(21)

3.3 Solution for Gpmj and Ctk

mj

The existence of the kernels Gpmj and Ctk

mj has been established, and it has been shown thatthat these two kernels are governed by the system of differential equations (8) and (17),respectively. In this section we provide a solution for the kernels through an applicationof the Radon transform. (See Appendix B for a summary of certain results concerning theRadon transform which are relevant to our development.) As a starting point, we summarizethe use of the Radon transform to obtain the displacement fundamental solution. Thisprocedure is well known (we note in particular the description given by Bacon et al. [1]),but it is presented here since certain intermediate results will be needed in the developmentto follow.

For a unit point load acting in an unbounded anisotropic solid, Navier’s equation (gov-erning the displacement fundamental solution) takes the form

Eijkl∂2Up

k (ξ − x)

∂ξl∂ξj

= −δip δ(ξ − x) (22)

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Upon an application of the Radon transform we find

zlzjEijkl∂2Up

k (z, α− z · x)

∂2α2= −δip δ(α− z · x) (23)

where f(z, α−z ·x) denotes the transform of a function f(ξ−x) (in the transform domainα, z in which α is a scalar and z is a unit vector). With (z,z)ik ≡ zjEjiklzl, (23) can be

solved for ∂2Upk/∂α2 with the result

∂2Upa (z, α− z · x)

∂α2= −(z,z)−1

ap δ(α− z · x) (24)

where (z, z)−1 denotes the inverse of the tensor (z,z). An application of the inverse Radontransform then leads to

U ij(ξ − x) =

1

8π2r

∮

z·r=0

(z,z)−1ij ds(z) (25)

in which the integral is to be evaluated over a unit circle ‖z‖ = 1 on a plane normal to thevector r = ξ−x as shown schematically in Figure 2. Note that the integrand is well-definedat every point along the contour as a result of the positive definiteness of (z, z) and, clearly,the kernel U i

j is singular only at ξ = x and is of order 1/r as r → 0.

ξ

x

z

Figure 2: Schematic indicating contour of integration for the displacement fundamentalsolution.

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3.3.1 Kernel Gpmj

Taking the Radon transform of (8), we obtain

εikmzk

∂Gpmj(z, α− z · x)

∂α= Eijabzb

∂Upa (z, α− z · x)

∂α− δpjzi

∂

∂α

( 1

4πr

)

(26)

Taking the derivative of (26) with respect to α, noting that (see Appendix B)

∂2

∂α2

( 1

4πr

)= −δ(α− z · x) (27)

and utilizing (24), we obtain

εikmzk

∂2Gpmj(z, α− z · x)

∂α2= Ωijp δ(α− z · x) (28)

in which

Ωijp = δpjzi − Eijdczc(z,z)−1pd (29)

Now, a particular solution of (28) can readily be constructed by expressing Ωijp in termsof the product of εikmzk and a certain function as follows:

Ωijp = δiaΩajp

= δiaδkbzkzbΩajp

= (εikmεabm + δibδka)zkzbΩajp

= εikmεabmzkzbΩajp + zizaΩajp

= εikmzk

[εabmzbΩajp

](30)

Note that the epsilon-delta identity εijkεipq = (δjpδkq − δjqδkp) has been used along with thefact that zaΩajp = 0. With (30) and the fact that εabmzazb = 0, (28) becomes

εikmzk

∂2Gpmj(z, α− z · x)

∂α2= εikmzk

[− εabmzbzc (z,z)−1

pd Eajdc δ(α− z · x)]

(31)

and a particular solution of (31) follows by inspection with the result

∂2Gpmj(z, α− z · x)

∂α2= −εabmzazc (z, z)−1

pd Eajdc δ(α− z · x) (32)

By employing the Radon transform inversion, the kernel Gpmj in physical domain is obtained

as

Gpmj(ξ − x) =

1

8π2r(εabmEajdc)

∮

z·r=0

zbzc (z,z)−1pd ds(z) (33)

As in the case for the displacement fundamental solution (25), the kernel Gpmj is singular

only at ξ = x and is of order 1/r as r → 0.

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3.3.2 Kernel Ctkmj

Upon taking the Radon transform of (17) and utilizing (24), we find that

εismεlrtzszr

∂2Ctkmj(z, α− z · x)

∂α2= Λijkl δ(α− z · x) (34)

where

Λijkl = Eijkl − Eijeo Ednkl zozd (z, z)−1en (35)

Proceeding in a fashion analogous to that used to obtain Gpmj, we seek a particular solution

of (34) by expressing Λijkl in terms of a product between the linear operator εismεlrtzszr anda certain function. With the use of the epsilon-delta identity and the fact that zaΛajkl = 0,we obtain

Λijkl = δiaΛajkl

= δiaδpszszp Λajkl

= (εismεapm + δipδsa)zszpΛajkl

= εismεapmzszp Λajkl + zizaΛajkl

= εismεapmzszp Λajkl (36)

Upon a further manipulation (now ‘on the index l’) we find

Λijkl = εismεapmzszr δprδlbΛajkb

= εismεapmzszr(εlrtεbpt + δlpδrb)Λajkb

= εismεpamεlrtεpbtzszrΛajkb + εismεalmzszbΛajkb

= εismεlrtzszr

[εpamεpbtΛajkb

](37)

in which the epsilon-delta identity has been used yet again along with the fact that zbΛajkb =

0. With (34) and (37), it is evident that a particular solution for ∂2Ctkmj/∂α2 is given by

∂2Ctkmj(z, α− z · x)

∂α2= εpamεpbtΛajkbδ(α− z · x) (38)

and the kernel Ctkmj(ξ − x) is obtained by utilizing the Radon transform inversion formula

with the result

Ctkmj(ξ − x) = − 1

8π2r

∮

z·r=0

εpamεpbtΛajkb ds(z) (39)

By exploiting the identity (z,z)ij (z,z)−1ij = 3, a more concise relation is obtained as

Ctkmj(ξ − x) =

1

8π2rAtkeo

mjdn

∮

z·r=0

zozd (z, z)−1en ds(z) (40)

12

where Atkeomjdn is a constant (which depends only on the moduli) given by

Atkeomjdn = εpamεpbt

(EdnkbEajeo − 1

3EajkbEdneo

)(41)

Note that the kernel Ctkmj is also singular only at ξ = x and is of order 1/r as r → 0.

3.3.3 Summary of the kernels

Let Kikjl be the fourth order tensor defined by

K ikjl ≡

1

8π2r

∮

z·r=0

(z,z)−1ij zkzl ds(z) (42)

Then the displacement fundamental solution U ij , the kernel Gp

mj and the kernel Ctkmj can be

expressed in succinct form in terms of this single tensor as

U ji (ξ − x) = Kik

jk (43)

Gpmj(ξ − x) = εabmEajdc Kpb

dc (44)

Ctkmj(ξ − x) = Atkeo

mjdn Keond (45)

in which Atkeomjdn is given by (41).

We remark that an alternative form for K ikjl in terms of an integral on the real line can

be readily obtained in a fashion analogous to that commonly employed for the displacementfundamental solution. Let z = a cos α + b sin α with α ∈ [0, 2π] and with a, b beingorthonormal vectors which lie in a plane perpendicular to r = (ξ−x). Then it can be shownthat

K ikjl =

1

4π2r

∫ ∞

−∞

Γij(p) Θkl(p)

(1 + p2)|Γ(p)| dp (46)

in which

Γij = (a,a)ij + (a, b)ij + (a, b)jip + (b, b)ij p2 , (47)

ΓijΓjk = |Γ(p)|δik , (48)

Θkl = akal + (akbl + albk)p + bkblp2 , (49)

where p = tan α, |Γ(p)| denotes the determinant of Γ, and (a, b)ij = akEkijlbl. The evalua-tion of this integral (via contour integration) involves, as usual, determining the roots of asextic equation (e.g. [23], [25]), and differs from the standard treatment for the displacementfundamental solution only in a minor way associated with the additional term Θkl and thetwo simple poles at p = ±√−1 which arise from the presence of zkzl in (42).

13

4 Dislocations and cracks

The singularity-reduced integral relations developed above apply to an arbitrary displace-ment discontinuity in an unbounded domain. The special cases of dislocations and cracksare of particular interest, and in this section we discuss these two cases in detail.

4.1 Dislocations

For the special case of a dislocation for which bi ≡ ∆ui is constant, the singularity-reducedintegral equation (9) gives rise to a generalization of Burgers’ equation for anisotropic mediagiven by

up(x) = −∮

∂S

bj Gpmj(ξ − x) dξm − 1

4πbp Ω(x) (50)

where

Ω(x) =4π

3

∫

S

Hpip(ξ − x) ni(ξ) dS(ξ) (51)

is the solid angle to the surface. As shown in Appendix D, the solid angle can be computedin terms of an integral on ∂S as

Ω(x) = 4π

∫

∂S

ωm(ξ, x) dξm (52)

with the ‘generating function’ ωm given by

ωm(ξ,x) =1

3

∫

Γ(x)

εmrt Hprp(ξ − z) dzt (53)

in which Γ(x) is any path, originating at x and extending to infinity, which does not intersectthe discontinuity. For example, when Γ(x) is a straight line we readily obtain

ωm(ξ,x) =1

4π

εmktekpt

(1 + eipi)

1

r(54)

in which p is a unit vector along the path (directed toward x) and ek ≡ (ξk − xk)/r. Apiecewise-linear path is also readily treated, so in this sense it is always possible to choosea path which does not intersect the discontinuity and which is such that a simple closed-form relation for ωm can be obtained. Having determined ωm, we can now evaluate thedisplacement entirely in terms of a line integral as

up(x) = −∫

∂S

bj [Gpmj + δjpωm] dξm (55)

14

We remark that the jump in displacement across the discontinuity is manifested in the factthat the path must be taken to be directed ‘away from’ the discontinuity in such a way thatit does not intersect the surface.

A line integral representation for the stress field is obtained from (18) as

σlk(x) = − εlrt∂

∂xr

∮

∂S

bj Ctkmj(ξ − x)dξm (56)

and a weakly-singular, weak-form traction integral equation follows from (21). A line in-tegral representation for the energy associated with the presence of an array of dislocationloops is also readily obtained. For purposes of discussion, assume now that two dislocationsare present, the first with edge ∂S1 and Burgers’ vector b1

i and the second with edge ∂S2

and Burgers’ vector b2i . Toward developing an expression for the interaction energy Wint

associated with these two dislocations, let σ(1)kl and σ

(2)kl be the stress fields induced by the

first and second dislocation were it present in the absence of the other, so that

Wint =1

2

∫

S1

σ(2)kl n1

k b1l dS +

1

2

∫

S2

σ(1)kl n2

k b2l dS (57)

in which the superscripts on the surfaces of integration and the unit normals serve to indicatewith which dislocation they are associated. Upon use of (56) and an application of Stokes’theorem, it can then be shown that

Wint = −∮

∂S1

∮

∂S2

b1k b2

j Ctkmj(ξ − y) dyt dξm (58)

We remark that the contributions arising from each dislocation in the expression (56) canbe combined to obtain this final simple form due to the fact that, in the integral expressionswhere it occurs, Ctk

mj(ξ − x) can be replaced with Cmjtk (x− ξ) without altering the value of

the integral (see Li and Mear [16] for a discussion of this result); indeed, the specific form ofthe kernel Ctk

mj(x− ξ) to be obtained above satisfies the equality Ctkmj(ξ−x) = Cmj

tk (x− ξ).The self energy for each dislocation has the exact same form as (58) except that the righthand side is to be multiplied by 1/2 and, of course, both of the Burgers’ vectors and lineintegrals which appear are taken to be those associated with the particular dislocation forwhich an expression for the self energy is sought. Note also that, upon a proper interpretationof terms, these expressions for energy follow directly from the weak-form traction integralequation.

4.2 Cracks in an unbounded domain

Consider a crack on which tractions ti ≡ t+i = −t−i are prescribed. What is sought is therelative crack-face displacement ∆ui, and the weakly-singular weak-form traction integralequation provides a basis for a numerical procedure to determine these quantities. Therelative crack-face displacement must vanish along the crack front, and consistent with this

15

we chose the test function ∆ui to also possess this feature. Then the contribution from theline integral terms in (21) vanish and the weak-form traction integral equation simplifies to

∫

S

tk(y) ∆uk(y) dS(y) = −∫

S

∫

S

Ctkmj(ξ − y) [Dt∆uk(y)] [Dm∆uj(ξ)] dS(ξ)dS(y)

(59)

We emphasize that the kernel in this integral equation is weakly-singular (so that the integralsexist in the ordinary sense and the crack displacement data need only be continuous) andthat it represents a symmetric weak-form equation for the unknown crack-face displacements.Once these quantities are determined, other information of interest (i.e. the displacementand stress field and, most importantly, the stress intensity factors) can readily be found.

5 Additional discussion of the kernels

As is clear from the development above, the same kernels necessarily appear in the singularity-reduced integral relations for both dislocations and cracks. In this section we discuss thenon-uniqueness of the kernels and make a connection, where possible, to earlier results ob-tained within the context of modeling dislocations and cracks. Further, we discuss certaingeneral properties of the kernels as well as certain features associated with specific types ofmaterial symmetry.

5.1 Relation to previously obtained kernels

It is evident from the representations (7) and (16) that the kernels Gpmj and Ctk

mj are not

unique (also see Appendix C). Indeed, given a particular pair of kernels Gpmj, C

tkmj, the

quantities Gpmj, C

tkmj for which

Gpmj = Gp

mj +∂Lp

j

∂ξm

, Ctkmj = Ctk

mj +∂M tk

j

∂ξm

+∂Nk

mj

∂ξt

(60)

are also valid kernels. Here Lpj , M tk

j and Nkmj are arbitrary, but for our purposes it suffices to

restrict attention to quantities which are a function of ζ = (ξ−x) and which are homogeneousof degree λ = 0.

Now, for the special case of isotropy, Li and Mear [16] carried out a direct integration of(7) to obtain the particular kernel

Gpmj(ζ) =

1

8π(1− ν)r

[(1− 2ν)εmpj + εajm

ζaζp

r2

](61)

in which ν is Poisson’s ratio. We remark that this kernel is identical to that appearing inBurgers’ equation [7] for the displacement field induced by a dislocation. For anisotropy, weare not aware of a solution for the kernel Gp

mj in the form (33), but a closely related result is

16

given by Leibfried [15] in terms of a Fourier integral. We also note that, when specialized toisotropy, the kernel given by (33) reduces to (61) along with an additional term which canbe expressed in the form ∂Lp

j/∂ξm (see Appendix E).

For isotropy, Li and Mear [16] obtained the particular kernel Ctkmj given by

Ctkmj(ζ) =

µ

4π(1− ν)r

[(1− ν)δktδjm + 2νδkmδjt − δkjδtm − ζkζj

r2δtm

](62)

in which µ is the shear modulus. The first kernel obtained for isotropy appears to be thatof Blin [3] in his analysis of the interaction energy for dislocation loops (also see Hirth andLothe [14]). Blin’s kernel differs from (62) by terms of the form ∂M tk

j /∂ξm and ∂Nkmj/∂ξt

(see Appendix E) and, while the simplicity of (62) may be preferable for numerical analysis,the two kernels are equivalent by (60). We remark that other (equivalent) kernels have beenobtained within the context of integral equation representations for boundary value problemsin isotropic linear elasticity, and we note in particular the work of Nedelec [20] and Bonnet[5].

For anisotropy the only previously available explicit kernel seems to be that developedby Lothe [18] in his analysis of the interaction energy for dislocation loops. Lothe’s kernel isgiven by

Ctkmj(ξ − x) = − 1

8π2r

∮

|z|=1

εumvεwtszvzs

[Eujkw − zczd(z,z)−1

en EujecEdnkw

]dS (63)

and in Appendix E it is demonstrated that the kernel given by (40) (which is significantlysimpler in form) is in fact equivalent to this kernel.

5.2 Properties of the kernels

We now focus attention on the specific kernel Gpmj given by (33) and the specific kernel Ctk

mj

given by (40). Since it will appear in the integral equations to be developed further belowfor a finite domain, we also discuss the displacement fundamental solution. Noting thatthe tensor (z,z) is symmetric and that the integrands in both (33) and (40) are even withrespect to the components of z which appear, we deduce that

Gpmj(ξ − y) = Gp

mj(y − ξ)

Ctkmj(ξ − y) = Ctk

mj(y − ξ) , Ctkmj(ξ − y) = Cmj

tk (ξ − y)

U ji (ξ − y) = U j

i (y − ξ) , U ji (ξ − y) = U i

j(ξ − y)

(64)

Additional properties of interest for the kernels follow by a consideration of elastic materialsymmetry.

For purposes of discussion, assume that the material has a plane of symmetry and intro-duce a local cartesian coordinate system ζ1,ζ2,ζ3 which has its origin at the source pointx and for which ζ1 = 0 defines the plane of material symmetry. Let ξ be a given point andlet ξ∗ be its ‘image’ obtained by a reflection across the plane ζ1 = 0 as shown schematically

17

ζ2, ζ2

ζ1

r

ζ1 x

ξξ∗

r∗

Figure 3: Coordinate system ζ1, ζ2, ζ3 with ζ1 = 0 corresponding to a plane of materialsymmetry.

in Figure 3. With r = (ξ − x) and r∗ = (ξ∗ − x), we have r∗ = r and r∗α = δ∗αβrβ whereGreek indices are used to indicate components with respect to the local coordinate systemand where

δ∗αβ =

−1 , α = β = 1

1 , α = β = 2, 30 , α 6= β

(65)

We now introduce yet other local coordinate system ζ1,ζ2,ζ3 which has its origin at x andsatisfies ζ1 = −ζ1, ζ2 = ζ2 and ζ3 = ζ3. Note that this is the left-handed coordinate systemwhich is obtained through a ‘reflection’ of the ζi system across the plane ζ1 = 0.

Consider the kernel Gpmj and let its components relative to the ζi system be denoted

Gγαβ and its components relative to the ζi system be denoted Gγ

αβ. From the fact thatζ1 = 0 is a plane of material symmetry, we deduce the correspondence

Gγαβ(ξ∗ − x) = Gγ

αβ(ξ − x) (66)

and then upon a coordinate transformation (taking due account of the alternating symbolwhich appears in Gp

mj) we find

Gγαβ(ξ∗ − x) = −δ∗αδδ

∗βηδ

∗γρG

ρδη(ξ

∗ − x)

= −δ∗αδδ∗βηδ

∗γρG

ρδη(ξ − x) (67)

18

Similarly, we obtain

Cγδαβ(ξ∗ − x) = δ∗αηδ

∗βκδ

∗γρδ

∗δλC

ρληκ(ξ − x) (68)

and

Uβα (ξ∗ − x) = δ∗αηδ

∗βκU

κη (ξ − x) (69)

The properties (64) and (67)-(69) play an important role in reducing the computationaleffort involved in evaluating the kernels U j

i , Gpmj, C

tkmj.

6 Cracks in a finite domain

Consider a finite body which contains an embedded or surface breaking crack as shownschematically in Figure 4. The boundary of the domain which the body occupies consistsof an ‘ordinary’ boundary So and the crack surface S+

c ∪ S−c . Consistent with the previousdevelopment for an isolated crack, attention is restricted to cases in which body force isabsent and for which the traction loading on the crack is such that t+i = −t−i . Further, welet Sc ≡ S+

c and, for convenience, we introduce S = So ∪ Sc.

S+c

S+c

S−c

S−c

So

Figure 4: Schematic of a finite body containing cracks.

Now, Somigliana’s identity gives the displacement up at a location x within the domainas

up(x) =

∫

So

Upj (ξ − x) tj(ξ) dS(ξ)−

∫

S

Spij(ξ − x) ni(ξ) vj(ξ) dS(ξ)

(70)

19

where the quantity

vj(ξ) =

uj(ξ) , ξ ∈ So

∆uj(ξ) , ξ ∈ Sc(71)

has been introduced for convenience. From the expression for the displacement, we readilyobtain an integral relation for the stress at points within the domain as

σlk(x) = −∫

So

Sjlk(ξ − x) tj(ξ) dS(ξ) +

∫

S

Σlkij (ξ − x) ni(ξ) vj(ξ) dS(ξ)

(72)

We seek a regularization of these integral expressions and, specifically, we seek a pair ofweakly-singular, weak-form displacement and traction integral equations which form thebasis of a symmetric Galerkin boundary element method analogous to that developed forisotropy by Li et al. [17]. We note that if the boundary value problem under considerationis one in which tractions are prescribed on the entire surface, then a weak-form tractionintegral equation suffices to obtain a symmetric formulation. However, when displacementsare prescribed on a portion of the ordinary boundary a symmetric formulation can onlybe achieved by employing both a weak-form traction integral equation and a weak-formdisplacement integral equation.

Consider first the displacement given by (70). Since the displacement fundamental so-lution is weakly-singular, only the term involving the stress fundamental solution requiresregularization. Such a regularization is readily achieved by employing the decomposition (4)and then integrating by parts over the entire surface via Stokes’ theorem. After carrying outthis process, the limit as x tends to a point on the ordinary boundary can be easily formed,and with y ∈ So being any such point we find

1

2up(y) =

∫

S

Gpmj(ξ − y)Dmvj(ξ)dS(ξ)−

∫

S

ni(ξ)Hpij(ξ − y)vj(ξ)dS(ξ)

+

∫

So

Upj (ξ − y)tj(ξ)dS(ξ) (73)

We remark that a limit in which x tends to a point on the crack surface can also be treated(see (11) for the case of an isolated crack) but, owing to the fact that the crack is subjectedonly to prescribed tractions, such an expression is not required for establishing a symmetricformulation. Upon multiplying (73) by a test function tp and integrating the result over theordinary surface So, we obtain a weakly-singular, weak-form displacement integral as

1

2

∫

So

up(y) tp(y) dS(y) =

∫

So

tp(y)

∫

S

Gpmj(ξ − y)Dmvj(ξ) dS(ξ) dS(y)

−∫

So

tp(y)

∫

S

ni(ξ)Hpij(ξ − y)vj(ξ) dS(ξ) dS(y)

+

∫

So

tp(y)

∫

So

Upi (ξ − y)ti(ξ) dS(ξ) dS(y) (74)

20

Consider next the stress given by (72). To regularize this integral relation, we utilize thedecomposition (4) for the Cauchy-type singular kernel Sj

lk and the decomposition (14) forthe strongly-singular kernel Σlk

ij , and then we carry out an integration by parts vis Stokes’theorem. The resulting singularity-reduced stress relation is then used to form a tractionintegral equation for points y ∈ So ∪ Sc via a limiting process similar to that employed inthe context of an isolated crack. The final form of traction integral equation is given by

c(y)tk(y) = Dt

∫

S

Ctkmj(ξ − x)Dm∆vj(ξ)dS(ξ) + Dt

∫

So

Gjtk(ξ − y)tj(ξ)dS(ξ)

−∫

So

nl(y)Hjlk(ξ − y)tj(ξ)dS(ξ) (75)

where c = 1/2 for y ∈ So whereas c = 1 for y ∈ Sc. Finally, a weak-form traction integralequation is obtained from (75) as

−∫

Sc(y)tk(y)vk(y)dS(y) =

∫

SDtvk(y)

∫

SCtk

mj(ξ − y)Dmvj(ξ)dS(ξ)dS(y)

+∫

SDtvk(y)

∫

So

Gjtk(ξ − y)tj(ξ)dS(ξ)dS(y)

+∫

Svk(y)

∫

So

nl(y)Hjlk(ξ − y)tj(ξ)dS(ξ)dS(y) (76)

where

vk(y) =

uk(y) , y ∈ So

∆uk(y) , y ∈ Sc(77)

in which uk and ∆uk are test functions associated with So and Sc, respectively. The weakly-singular, weak-form displacement integral equation (74) and the weakly-singular, weak-formtraction integral equation (76) represent a generalization (to anisotropic media) of the workby Li and Mear [16] for isotropic media.

7 Conclusion

A systematic procedure has been followed to obtain singularity-reduced integral relationsfor displacement discontinuities in anisotropic, linearly elastic media. An arbitrary isolateddiscontinuity is treated first, and the results are then specialized to dislocations and cracks.For the case of dislocations, the line integral representations obtained for displacement andinteraction energy are in terms of weakly-singular kernels that are simpler in form thanthose previously available. The key result obtained for cracks is a weakly-singular, weak-form traction integral equation which is in terms of a simple kernel well-suited for numericaltreatment.

21

The primary focus of this work has been the development of weakly-singular, weak-formintegral equations for cracks is finite domains, and the development of these integral equa-tions rests upon the results for an isolated discontinuity. Here, both a displacement andtraction integral equation have been obtained, and these can be combined to arrive at asymmetric formulation analogous to that established for isotropic media by Li and Mear[16]. This symmetric formulation constitutes a basis for the development of a symmetricGalerkin boundary element method suitable to treat general boundary value problems forcracks in anisotropic, linearly elastic media. Since the integral equations are only weakly-singular, standard Co elements can be employed in a numerical discretization (as opposedto C1 elements which are required for Cauchy-singular and strongly-singular formulations).Li et al [17]. carried out a complete numerical implementation for isotropic media, and theyfound that highly accurate stress intensity factors can be obtained for complex, mixed-modeproblems. It is expected that a generalization of their technique, by means of the integralequations developed here, will lead to a computational procedure well-suited for analysis ofcracks in generally anisotropic media.

AcknowledgementsThis work was supported by the National Science Foundation through a grant to The Uni-versity of Texas at Austin. The first author (Jaroon Rungamornrat) gratefully acknowledgessupport provided by the Anandamahidol Foundation in the form of an Anandamahidol Schol-arship.

Appendix

A Discussion of U pi , Sp

ij and Σklij

We denote the displacement and stress associated with the unit Kelvin state as Upi , Sp

klsuch that, for a concentrated load Fp acting at a point x, the displacement and stress at apoint ξ are given by ui(ξ − x) = Up

i (ξ − x)Fp and σkl(ξ − x) = Spkl(ξ − x)Fp, respectively.

With partial differentiation with respect to ξi denoted by a comma (i.e. (·),i ≡ ∂(·)/∂ξi),the Kelvin state satisfies

Spkl,k = EklijU

pi,kl = δlp δ(ξ − x) (78)

where δ(ξ − x) is the Dirac-delta function centered at x. For purposes of presentation, werefer to Up

i as the ‘displacement fundamental solution’ and to Spij as the ‘stress fundamental

solution’.Equation (12) is readily established by using (78) and noting the symmetries of the elastic

moduli. Equation (13) follows from

Σlkij,l(ξ − x) = ElkpqS

pij,ql = ElkpqEijmnUp

m,nql

22

= EijmnElkpqUmp,qnl = EijmnS

mlk,ln

= −[Eijklδ(ξ − x)],l (79)

where the reciprocity-relation U ji = U i

j has been employed in addition to (78). We note thatthe observations Σlk

ij,i = 0 for ξ 6= x and Σlkij,l = 0 for ξ 6= x have been utilized in various

earlier investigations (e.g. Becache et al.[2] and Bonnet [5]).

B Radon transform

Here we provide only a brief summary of certain properties of the Radon transform whichare pertinent to our development. See Bacon et al.[1] for a more in depth summary, and see[8, 10, 13] for an extensive development of this integral transform.

The Radon transform involves two independent transform parameters: a unit vector zand a scalar α with −∞ < α < ∞. For a given z and α, the relation z · ξ = α defines aplane in Euclidian space R3, and the Radon transform of a (scalar, vector or tensor valued)function f = f(ξ) is defined in terms of an integral over such planes as

f(z, α) = Rf =

∫

z·ξ=α

f(ξ) dS(ξ) =

∫

R3

f(ξ) δ(α− z · ξ) dV (ξ) (80)

The function f(ξ) is then given in terms of its Radon transform f(z, α) by the inversionformula

f(ξ) = − 1

8π2

∫

||z||=1

∂2f(z, α)

∂α2dS(z) (81)

in which the integral is to be carried out over the surface of the unit sphere ||z|| = 1. Certain

useful properties and results concerning the Radon transform are summarized below.

(1) The Radon transform of a derivative of a function satisfies

R∂f(ξ)

∂ξk

= zk

∂f(z, α)

∂α(82)

(2) The Radon transform of the Dirac-delta function δ(ξ − x) is given by

Rδ(ξ − x) =

∫

R3

δ(ξ − x)δ(α− z · ξ)dV (ξ) = δ(α− z · x) (83)

(3) Recall that

∆( −1

4πr

)= δ(ξ − x) (84)

23

Taking the Radon transform of (84) we have

zizi∂2

∂α2

(1

r

)= −4πRδ(ξ − x) (85)

which, upon noting that zizi = 1 and employing (83), leads to

∂2

∂α2

(1

r

)= −4πδ(α− z · x) (86)

(4) Let f = f(ξ − x) be a function for which

∂2f(z, α− z · x)

∂α2= g(z, α− z · x) δ(α− z · x) (87)

Employing the Radon transform inversion formula we obtain

f(ξ − x) = − 1

8π2

∫

||z||=1

g[z,z · (ξ − x)] δ[z · (ξ − x)] dS(z) (88)

and, since δ(ax) = δ(x)/|a|, it follows that

f(ξ − x) = − 1

8π2r

∫

||z||=1

g[z,z · (ξ − x)] δ(z · e) dS(z) (89)

where r = (ξ − x), r = ||r|| and e = r/r. From the sifting property of the Dirac-deltafunction, equation (89) reduces to (e.g. [9])

f(ξ − x) = − 1

8π2r

∮

z·r=0

g[z,z · (ξ − x)] ds(z) (90)

in which the integral is to be evaluated over the unit circle ||z|| = 1 in a plane for whichz · r = 0.

C Existence and nonuniqueness of Gpmj and Ctk

mj

C.1 Kernel Gpmj

To establish the existence (and nonuniqueness) of the kernel Gpmj, it suffices to show that

the system of linear equations (28) admits an infinite number of solutions. Now, it can bereadily shown that the adjoint of the linear operator L = εikmzk is L∗ = εimkzk and, further,that the null space N of L and the null space N ∗ of L∗ coincide and satisfy

N ∗(εimkzk) = cm = ρzm : ∀ρ ∈ R (91)

24

That the vector on the right hand side of (28) is orthogonal to all elements of N ∗ (i.e. thatthe equations are consistent) follows from

ciΩijpδ(α− z · x) =[δpjzizi − ziEijdczc(z,z)−1

pd

]ρδ(α− z · x)

=[δpj − (z,z)jd(z,z)−1

pd

]ρδ(α− z · x)

=[δpj − δjp

]ρδ(α− z · x)

≡ 0 (92)

and this ensures the existence of ∂2Gpmj/∂α2. Since the dimension of the null space of L is

not zero, the solution for ∂2Gpmj/∂α2 is not unique and, in fact, there is an infinite number of

solutions. From (91) it is evident that, for any particular solution ∂2Gpmj/∂α2, the quantity

∂2Gpmj/∂α2 + zm∂2Lp

j/∂α2 is also a solution. Applying the Radon transform inversion itthen follows that if Gp

mj is a particular kernel, then the quantity Gpmj + ∂Lp

j/∂ξm is alsoa valid kernel for every (suitably well behaved) function Lp

j . If attention is restricted tokernels which are functions of (ξ − x) and which are homogeneous of degree λ = −1, it isthen consistent to restrict attention to functions Lp

j(ξ−x) which are homogeneous of degreeλ = 0.

C.2 Kernel Ctkmj

The existence and nonuniqueness of the kernel Ctkmj is established by showing that (34)

admits an infinite number of solutions. Toward demonstrating this, we first represent thecomponents of the linear operator εismεlrtzszr in matrix form as

[εismεlrtzszr] =

0 −z3Z z2Zz3Z 0 −z1Z

−z2Z z1Z 0

(93)

in which the pth-row and qth-column of the matrix correspond to the indicies i,m, l, tappearing in the linear operator according to p = [3(i − 1) + l] and q = [3(m − 1) + t], thematrix Z is defined by

Z =

0 −z3 z2

z3 0 −z1

−z2 z1 0

(94)

and 0 denotes the 3× 3 matrix for which all entries are zero.Now, since Z is skew-symmetric it follows that the operator εismεlrtzszr is symmetric

and self-adjoint. Further, since the matrix in (93) is singular with rank equal to four, thedimension of the null space of the operator is five. To construct a basis for the null space,we observe that a solution to

εismεlrtzszrcmt = 0 (95)

25

is given by cmt = ηmzt +κtzm in which ηm and κt are arbitrary. It is then easily verified that,by making appropriate choices for ηm and κt, a set of five linearly independent elements c

(i)mt

(with i = 1, 2, 3, 4, 5) can be generated to form a basis for the null space. Indeed, the

particular choices c(i)mt = δimzt for i = 1, 2, 3 and c

(3+i)mt = δitzm for i = 1, 2 serve this

purpose. (Note that the element c(6)mt = δ3tzm can be expressed as a linear combination of

the previous five elements; this is readily shown by expressing the elements c(i)mt in the form

of a 9-vector.) Hence, the null space of εimsεltrzszr is given by

N (εimsεltrzszr) = cmt = ηmzt + κtzm : ∀ ηm, κt ∈ R (96)

A necessary and sufficient condition for ∂2Ctkmj/∂α2 to exist is that the term on the right

hand side of (34) be orthogonal to all elements in the null space of the (self-adjoint) linearoperator εimsεltrzszr, and that this holds can be demonstrated as follows:

cilΛijklδ(α− z · x) = ηi

[zlEijkl − (zlElkpqzq)Eijabzb(z,z)−1

ap

]δ(α− z · x)

+ κl

[ziEijkl − (ziEijabzb)Elkpqzq(z,z)−1

ap

]δ(α− z · x)

= ηi

[zlEijkl − (z, z)kpEijabzb(z, z)−1

ap

]δ(α− z · x)

+ κl

[ziEijkl − (z,z)jaElkpqzq(z, z)−1

ap

]δ(α− z · x)

= ηi

[zlEijkl − zbEijkb

]δ(α− z · x)

+ κl

[ziEijkl − zqElkjq

]δ(α− z · x)

≡ 0 (97)

That ∂2Ctkmj/∂α2 is non-unique is evident from the fact that the dimension of the null

space of εimsεltrzszr is not zero. From (96) it is clear that, for any particular solution∂2Ctk

mj/∂α2, the quantity ∂2Ctkmj/∂α2 + zm∂2M tk

j /∂α2 + zt∂2Nk

mj/∂α2 is also a valid solution.This implies that given any particular kernel Ctk

mj, the quantity Ctkmj +∂M tk

j /∂ξm +∂Nkmj/∂ξt

is also a valid kernel. Here it suffices to restrict attention to functions M tkj (ξ − x) and

Nkmj(ξ − x) which are homogeneous of degree λ = 0.

D Discussion of the solid angle Φ(x)

The quantity Hpip admits a representation in the form

Hpip(ξ − x) = 3εikm

∂ωm(ξ,x)

∂ξk

(98)

where

ωm(ξ,x) =1

3

∫

Γ(x)

εmrt Hprp(ξ − z) dzt (99)

26

in which Γ(x) is any path originating at x and extending to infinity (without passing throughthe field point ξ). To establish this result, we first note that Hp

ip → 0 as r → ∞, that∂Hp

ip/∂ξk = −∂Hpip/∂xk and that ∂Hp

ip/∂ξi = −3δ(ξ − x). Then from (99) along with thedelta-epsilon identity we find

3εikm∂ωm(ξ,x)

∂ξk

=

∫

Γ(x)

∂Hpip(ξ − z)

∂ξt

dzt −∫

Γ(x)

∂Hpkp(ξ − z)

∂ξk

dzi

= −∫

Γ(x)

∂Hpip(ξ − z)

∂zt

dzt

= Hpip(x− ξ) (100)

Substituting (98) into (51) yields

Ω(x) = 4π

∫

S

ni(ξ)εikm∂ωm(ξ,x)

∂ξk

dS(ξ) (101)

which, upon an application of Stokes’ theorem (with x /∈ S and with the path Γ(x) beingsuch that it does not intersect S), gives rise to

Ω(x) = 4π

∫

∂S

ωm(ξ, x) dξm (102)

We remark that a related development (in terms of somewhat different notation) is given byLi and Mear [16].

E Equivalence of kernels and reduction for isotropy

The kernel Gpmj given by (33) and the kernel Ctk

mj given by (40) are valid for general anisotropy.To specialize them to isotropic material behavior we note that for such materials (e.g. [1])

Eklpq = µ(δkpδlq + δkqδlp +

2ν

1− 2νδklδpq

)(103)

(z,z)ij = µ(δij +

1

1− 2νzizj

)(104)

(z,z)−1ij =

1

µ

(δij − 1

2(1− ν)zizj

)(105)

Utilizing these relations we readily reduce (33) to

Gpmj(ζ) = Gp

mj(ζ)(61) + Gpmj(ζ) (106)

(107)

and (40) to

Ctkmj(ζ) = Ctk

mj(ζ)(62) + Ctkmj(ζ) (108)

27

in which Gpmj(61) and Ctk

mj(62) denote the kernels given by (61) and (62), respectively,and where

Gpmj(ζ) =

εjpk

8π

(ζk

r

),m

(109)

Ctkmj(ζ) = − µ

8π

[δtk r,jm + δmj r,kt

]− νµ

4π(1− ν)

[δtj r,km + δmk r,jt

]

+µ

16π(1− ν)

[(ζjζk

r

),mt− (1− 2ν)δjk r,mt

](110)

In these relations, ζ = (ξ − x), r = ||ζ|| and a comma indicates partial differentiation (i.e.(·),m ≡ ∂(·)/∂ξm). Clearly, within the context of the non-uniqueness expressed by (60), theterms Gp

mj and Ctkmj can be discarded without loss. We remark that the kernels Gp

mj(61)

and Ctkmj(62) appear to be in their ‘most reduced form’ in the sense that they do not contain

any additional terms in the form of a gradient of a function which can be discarded by (60).Next we consider Blin’s [3] kernel for isotropic media given by

Ctkmj(ζ) =

µ

4π(1− ν)r

[(1− v)δtkδmj − (1− 2ν)δmkδjt − δjkδmt + εiktεpjm

ζiζp

r2

]

(111)

Upon employing the identity

εumvεwts = δuw(δmtδvs − δmsδvt) + δut(δmsδvw − δmwδvs)

+ δus(δmwδvt − δmtδvw) (112)

it can be shown that this kernel can be re-expressed in the form (108) with the additionalterm Ctk

mj given by

Ctkmj(ζ) =

µ

4π(1− ν)

[(δjkζt

2r− δjtζk

r

),m

+(δjkζm

2r− δmkζj

r

),t

](113)

Clearly the additional term Ctkmj can be discarded without loss.

Finally, we consider Lothe’s kernel for general anisotropy. By employing (112), Lothe’skernel (63) can be re-expresses as

Ctkmj(ζ) = Ctk

mj(ζ)(40) + Ctkmj(ζ) + ¯C

tk

mj(ζ) (114)

where Ctkmj(ζ)(40) denotes the kernel given by (40) and where the additional terms Ctk

mj and

¯Ctk

mj are given by

Ctkmj(ζ) =

Eujku

8π

(ζt

r

),m

=Eujku

8π

(ζm

r

),t

(115)

28

and

¯Ctk

mj(ξ − x) = − 1

8π2r

∮

||z||=1

zmztzczd(z,z)−1en EajecEdnkads (116)

Clearly the term Ctkmj can be discarded without loss (since it is in terms of a gradient with

respect to ξm or ξt). While the term ¯Ctk

mj must itself be expressible as a gradient with respectto ξm or ξt, we have not attempted to directly show this. Rather, we simply verify (by useof the Radon transform) that

εism∂

∂ξs

εlrt

∂ ¯Ctk

mj(ξ − x)

∂ξr

= 0 (117)

hence the term ¯Ctk

mj can also be discarded without loss.

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