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Mechanics Research Communications 46 (2012) 62– 70

Contents lists available at SciVerse ScienceDirect

Mechanics Research Communications

jo ur nal homep age : www.elsev ier .com/ locate /mechrescom

Kröner’s formula for dislocation loops revisited

Nicolas Van GoethemUniversidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, CMAF, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

a r t i c l e i n f o

Article history:Received 11 February 2012Received in revised form 17 August 2012Available online xxx

Keywords:Dislocation loopsDislocation density tensorContortion tensorStrain incompatibilitySingle crystals

a b s t r a c t

In this communication, our aim is to respond to open questions which arose from a previous publication(Van Goethem, N., 2011. Strain incompatibility in single crystals: Kröner’s formula revisited. Journal ofElasticity 103 (1), 95–111) where a new Kröner’s formula was proved for a set of skew dislocation anddisclination lines: (i) Does the new formula hold for a dislocation loop? (ii) Which new terms appear dueto the line curvature? In this work we validate by complete calculation of distributional type a generalKröner’s formula for two classical examples of dislocation loops.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Several dislocation theories coexist in the literature and in general each emphasizes a particular aspect of the physics of dislocations anddisclinations. For a physical approach let us refer to Kröner (1980), Dascalu and Maugin (1994), Hehl and Obukhov (2007), Lazar (2007), andLazar and Hehl (2010). In engineering practice, empirical models are for instance used in the context of single crystal growing from the melt(Müller and Friedrich, 2004; Müller et al., 2004). These models are in general rather crude extensions of models available for polycrystals,whereas the physics of single crystals which we here consider is radically different for one main reason. Since the dislocations satisfy aconservation law and since there are no internal boundaries, dislocation may form curves which are comparable with the characteristiclength of the crystal. So, separation of scales can hardly be done, and one is forced to analyze the properties of dislocations and/or disclinationcurves, even if a macroscopic thermomechanical model is adopted.

As soon as the mesoscale is considered1 (thus also at the macroscale, cf. Van Goethem, 2012), one crucial governing equation holdingfor both static and dynamic descriptions of dislocations, is what we called in previous contributions (Van Goethem and Dupret, 2012a; VanGoethem, 2010, 2011b) the “Kröner’s formula”. This formula relates the linear elastic strain incompatibility to dislocation and disclinationdensity tensors. This means that as soon as these densities are known, the strain E� must satisfy a geometrical constraint, which canroughly be stated as follows: the Ricci (i.e., curvature) tensor associated to the elastic metric g = I − 2E� is directly related to the contortioncurl, being the contortion �� an alternative tensorial expression of the dislocation density. For a discussion on the non-Riemannian natureof the dislocated crystal, see Kröner (1980) and the recent contributions (Maugin, 2003; Hehl and Obukhov, 2007; Kleinert, 2010; VanGoethem, 2010; Lazar and Hehl, 2010). Let us emphasize that in, e.g., Kröner (1980) and Kleinert (1989) the mesoscopic “Kröner’s formula”,namely inc E� = −�� × ∇ , follows in a straightforward manner from an “elastic–plastic” displacement gradient (or distortion) decompositionpostulate, which itself requires the selection of a particular reference configuration.2 Therefore, it is not admissible in our approach whichavoids any such arbitrary reference body prescription.

In Van Goethem (2011b) a new “Kröner’s formula” was proven in the absence of disclinations and under precise field assumptionsfor a finite set of skew isolated (i.e., with no accumulation sets) rectilinear defects. It turned out that the formula we proved was not theformula classically reported in the literature (Kröner, 1980; Kleinert, 1989), since an additional term generated by the edge segments of adislocation curve happens to be directly related to the scalar curvature of g, or, equivalently, to the trace of ��.

E-mail address: vangoeth@ptmat.fc.ul.pt1 In our approach, mesoscopic fields are identified with a �-superscript, which is removed as their macroscopic counterparts are considered.2 Elastic and plastic decomposition of the strain can classically be considered. However, no rigorous such decomposition holds for the distortion since in the absence of a

well-defined privileged reference configuration, there is no constitutive law which would define the elastic and plastic parts of the rotation tensor.

0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechrescom.2012.08.009

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N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70 63

In this communication, our aim is to respond to open questions which arose from Van Goethem (2011b):

(1) does the new formula hold for a dislocation loop?(2) which new terms appear due to the line curvature?

Therefore, the main part of the present work is devoted to verify by complete calculation of distributional type that a general Kröner’sformula (i.e., an extension of that proven in Van Goethem (2011b), which we here state as Conjecture 1) holds true for two classicalexamples of dislocation loops.

Let us emphasize that Kröner’s formulae are crucial since they relate the mechanical to the defect internal variables in any completethermodynamic model of dislocations. Whereas it could seem artificial to consider sets of rectilinear dislocation lines, as in Van Goethemand Dupret (2012a,b), and Van Goethem (2011b), this is no more the case for dislocation loops, which are the most common type ofdislocations observed in single crystal growing from the melt. So, the topic of this paper has a potential scientific impact for engineer andtechnology-oriented applications (Müller et al., 2004).

2. Preliminary results at the mesoscale: the basis of the distributional approach

The basis of the distributional approach can be found with more detail in the two references (Van Goethem and Dupret, 2012a,b), wherethe defect lines were assumed parallel to the z-axis, with a resulting elastic strain independent of z (in fact, those lines are the edge andscrew dislocations and the wedge disclination). Hence they could be treated as a set of points in the plane. These two introductory workspaved the way for the first application of the theory to 3D dislocations and disclinations in an elastic medium (Van Goethem, 2011b), wherethe lines were not assumed parallel anymore and where in addition to the 3 above-mentioned families of defects, we included the twistdisclination. The main results of these three papers are first recalled.

Let us emphasize that in the present work, disclinations, which is a rarer kind of defects in single crystals growing from the melt, arenot considered.

Notations 1. For a second-order tensor E, we introduce the left (resp. right) curl operator ∇× (resp. ×∇), i.e., (∇ × E)ij = �ikl∂kElj and3

(E ×∇)ij = �lkj∂kEil (otherwise written, (E × ∇)T = − ∇ × ET), where ET denotes the transpose of E.The incompatibility tensor associated to the symmetric second-order tensor E writes as

inc E := −∇ × E × ∇ = ∇ × (∇ × E)T ,

i.e., written componentwise, (inc E)ij = �ikm�jln∂k∂lEmn.

The assumed open and connected domain is denoted by ˝, the dislocation(s) are indicated by L ⊂ ˝, and ˝L stands for \ L.In the sequel, we say that a symmetric tensor Emn is compatible on U ⊂ if �kpm�lqn∂p∂qEmn vanishes in U. Moreover, as soon as

E ∈ L1loc

(˝�, R3×3), the incompatibility of E, inc E is a distribution (Schwartz, 1957), that is, a linear and continuous form on the space of test

functions C∞c (˝).

Assumption 1 (Planar loop). Let L ⊂ be a torsion-free dislocation loop with is assumed homoeomorphic to the circle4 and has a Lipschitzcontinuous tangent vector.5

Assumption 2 (3D elastic strain). The linear strain E�mn is a given symmetric Ls

loc(˝)-tensor compatible on ˝�

L, with 1 ≤ s < 2. In otherwords, the incompatibility tensor, as defined by the distribution ��

kl:= �kpm�lqn∂p∂qE�

mn, vanishes everywhere on ˝�L.

Definition 1 (Dislocation densities).

Dislocation density: ˛� := �ıL ⊗ B� (˛�ij := �iB

�j ıL) (2.1)

Mesoscopic contortion: �� := ˛� − I

2tr ˛� (��

ij := ˛�ij − 1

2ıij˛

�kk), (2.2)

where ıL denotes the 1-dimensional Hausdorff measure concentrated on L, and � the unit tangent vector to L.

The following classical theorem is easily proven from the relation ∇ · ˛� = 0 (see, e.g. Kleinert, 1989).

Theorem 1 (Conservation laws). Isolated dislocations are either closed or end at the boundary of ˝.

It has been proven in Van Goethem and Dupret (2012a,b) that at the mesoscale and for a set parallel rectilinear defects, strain incom-patibility satisfies the following theorem.

Theorem 2 (Incompatibility of Volterra dislocations). For a set of isolated parallel dislocations L, incompatibility is the following first-ordersymmetric tensor distribution,

Kröner’s formula: � = inc E� = −�� × ∇ = ∇ × (��)T . (2.3)

This result correspond to the Kröner’s formula as reported in, e.g., Kröner (1980) and Kleinert (1989).Moreover, it turns out that this formula is no longer true without a correction term if the lines are not parallel. It has been proven in

Van Goethem (2011b) that at the mesoscale and for a set of skew rectilinear defects, strain incompatibility satisfies the following theorem.

3 This notation is preferred to the other as found in Van Goethem (2011a,b) with the opposite sign convention for the ×∇ operator.4 That is, is represented by a continuous stretching and bending of the circle.5 Therefore its curvature exists almost everywhere and is bounded.

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64 N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70

Fig. 1. Left: pure edge dislocation loop. Right: the planar loop dislocation. The out-of-plane normal � is pointing downwards.

Theorem 3 (Incompatibility of a set of skew defect lines). Under Assumptions 2 and an assumption on the concentration properties of theFrank tensor (cf. Van Goethem, 2011b), strain incompatibility for a set of isolated skew (rectilinear) defect lines L is the following first-ordersymmetric tensor distribution:

��mn = incmn(E�) := �mkp�nlq∂k∂lE�

pq =∑L∈

[�Lm�L

k��Lkn]m↔n, (2.4)

where the Einstein tensor reads

��Lkn = ��L

kn − ıkn

2��L

pp, (2.5)

the Kröner’s tensor is defined by

��Lkn = �kij∂i�

�Lnj , (2.6)

with the defect contortion �� as given by Definition 1, and with symbol Am↔n meaning that sum of tensor Amn and its transposed Anm is taken.Moreover, �L indicates the tangent vector of L (here understood as uniformly extended in a neighbourhood of L6).

Remark 1 (Link with Kröner’s formula). In 2D elasticity (i.e., when the strain is independent of z) and for a single dislocation L along thez-axis, Kröner’s formula, �� =− �� × ∇ (cf. Kröner, 1980; Kleinert, 1989) is easily recovered from the general formula (2.4) (cf. Van Goethemand Dupret, 2012a; Van Goethem, 2011b).

3. The corrected Kröner’s formulae for dislocation loops

Let be the inward normal to the loop L and introduce the normal vector � : = × � in such a way that the right-handed orthonormalbasis {�, , �} is defined and satisfy the usual Frenet’s formulae on L: in particular we recall that ∂s� = � and ∂s = − ��, with s the curvilinearabcissa of L and � the curvature of L (Fig. 1).

The aim of this paper is to generalize Eq. (2.4) for a loop. However, it is beyond the scope of this communication to give a complete proofof this formula. Let us nevertheless gather some hints towards the general formula by scrutating the proof of Theorem 3 of Van Goethem(2011b). First of all, let us remark that a loop is a limit of countable skew segments, whereby Theorem 3 appears as a first step.

As a first adaptation, if the dislocation is a curve it must be closed by Theorem 1 in such a way that all quantities which appear at thesegment end-points mutually cancel. The second modification comes from identity �n∂i˛

�kj

= �k∂i˛�nj

(cf. Eq. (4.10) in Van Goethem, 2011b)which does not hold anymore if the line is curved, unless replaced by

�n∂i˛�kj = �k∂i˛

�nj − ��i�nkl�lB

�j ıL. (3.1)

Briefly stated, the line curvature comes into play (at least for the – non conservative7 edge dislocation loop) and the formula for a loop isinferred as follows.

Conjecture 1 (Incompatibility of planar loops). For a planar loop L, incompatibility is the following first-order symmetric tensor distribution

��mn = incmn(E�) = [�m�k��

kn + �m�n12

(B�k�k)�ıL]m↔n (3.2)

where ��kn

= ��kn

− ıkn2 ��

pp and ��kn

= �kij∂i��nj

are defined by (2.5) and (2.6), and with � the tangent vector of L (here understood, as �, as multipliedby a unit cut-off function around L).

Formula (3.2) will be verified by complete hand calculation in Section 4 on two examples of dislocation loops, namely the conservativeplanar loop (i.e., with vanishing out-of-plane Burgers vector component), and the pure edge dislocation loop (Fig. 1). In a first step, let usdetermine the differences between this formula and the classical Kröner’s formula.

6 That is, multiplied by a unit cut-off function around L, where the exact expression of this cut off function (which must be smooth, with compact support, and take unitvalues in a neibourhood of L), and in particular its width has no importance since it is ultimately multiplied by a concentrated distribution.

7 Meaning that the out-of-plane Burgers vector component does not vanish.

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N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70 65

3.1. Case A. The pure edge dislocation loop

Let us consider a loop dislocation L lying on the e1 − e2 plane, with Burgers vector B�3 with e3 the normal vector to the plane.

Contortion of the pure edge dislocation loop: one has ˛�ij

= B�j�iıL, ˛�

kk= 0 and hence ��

ij= ˛�

ij.

Difference between Kröner’s and corrected Kröner’s formula: it follows that ��ij

= B�z

(�ipz∂p(�jıL)

)and ��

iz= 0. Moreover since �˛� �� = ˛

we compute

��˛ˇ = B�

z (�˛� �ˇ∂� ıL + ˛ˇ�ıL)

and ���� = B�

z

(−�∂�ıL + �ıL

). It results that ��

˛ˇ= ��

˛ˇ− 1/2ı˛ˇ��

�� = B�z (�˛� �ˇ∂� ıL +˛ˇ�ıL) + 1/2ı˛ˇB�

z (�∂�ıL − �ıL).Then, the corrected Kröner’s formula, as given by Eq. (3.2) of Conjecture 1 writes as

��˛ˇ = [�˛����

�ˇ]˛↔ˇ + �˛�ˇB�z �ıL = −�˛�ˇB�

z �∂�ıL. (3.3)

3.2. Case B. The conservative planar loop: (a) ˛ and zz-components

Contortion of the conservative planar loop: let us consider a loop dislocation L lying on the e1 − e2 plane with Burgers vector B�i

= B�� (the

antiplanar component B�z = 0 – here Greek indices take their values in {1, 2}). We have ˛�

ij= B�

� �iı�jıL, ˛�kk

= B�� �� ıL and hence

��ij = ˛�

ij − 12

ıij˛�kk = B�

� �iı�jıL − 12

ıijB�� �� ıL. (3.4)

Difference between Kröner’s and corrected Kröner’s formula: from (3.4), the planar components of Kröner’s formula read ��˛ˇ

= �˛pq∂p��ˇq

=−�˛� ∂z��

ˇ�(since ��

ˇz= 0) and hence

��˛ˇ = B�

� ∂zıL(

−�˛� �ˇ + 12

�� �˛ˇ

)with its trace (recall that ˛ = �˛� �� ) ��

�� = �B��∂zıL. It results that

��˛ˇ = ��

˛ˇ − 12

����ı˛ˇ = B�

� ∂zıL(

−�˛� �ˇ + 12

�� �˛ˇ − 12

ı˛ˇ�B��

).

The corrected Kröner’s formula, as given by Eq. (3.2) of Conjecture 1 thus reads

��˛ˇ =

(�˛�ˇ(B�

��) − (�˛ˇ + �ˇ˛)B�

���

2

)∂zıL. (3.5)

Moreover ��zz = 0 since �z = 0 while ��

zz = �� ∂� ��z

= 0 since ��z

= 0.

3.3. Case B. the conservative planar loop: (b) ˛z-components

Since ��zz = −1/2B�

���ıL, we compute from (3.4) the antiplane components of Kröner’s formula, viz., ���z = ��ˇ∂ˇ

(−1/2B�

���ıL)

=��ˇ

(−1/2B�

���∂ˇıL − 1/2B���ˇ��ıL

)while ��

�� = 0. It results that ��˛z = ��

˛z . Incompatibility as given by Eq. (3.2) of Conjecture 1 reads(recall that �� ��ˇ = − ˇ)

��˛z = �˛

B����

2ˇ∂ˇıL. (3.6)

4. Validation of the corrected Kröner’s formulae

We must verify that

��mn = incmn

(E�

):= �mpi�nqj∂p∂qE�

ij (4.1)

is a concentrated distribution on L and coincides with Eqs. (3.3), (3.5), and (3.6).In the spirit of our distributional approach, Eq. (3.5) can be verified by hand computations as soon as an expression E� of the strain

associated to the conservative planar dislocation is known. Referring to Kleinert (1989), the strain expression for any curve L reads

E�ij(x) = B�

r

8��uvr[�j�l�vil]i↔j

∮L

��(x)∂u

(1

‖x − x′‖)

dL(x′) + B�r

8�(1 − )�klr

∮L

��(x′)∂i∂j∂l‖x − x′‖dL(x′) (4.2)

where ∂ refers to the derivation with respect to x. Of course, the strain as expressed by (4.2) is compatible outside L. However, as aL1

loc(˝)-function, it is a concentrated distribution on L as the following distributional expression of incompatibility will show:

〈��mn, ϕ〉 = 〈E�

ij, �mpi�nqj∂p∂qϕ〉 = − B�r

8��uvr�mpi�nqj[�j�l�vil]i↔j〈

∮L

��(x′)(

1‖x − x′‖

)dL(x′), ∂p∂u∂qϕ〉

+⟨

B�r

8�(1 − )�mpi�nqj�klr

∮L

��(x′)∂l‖x − x′‖dL(x′), ∂p∂q∂i∂jϕ

⟩, (4.3)

where the last term identically vanishes by the smoothness of ϕ.

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66 N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70

Let us denote ‖x− x′ ‖ by R, ��(x′)dL(x′) by dx′� , −�(x′)dL(x′) by dC ′

� , and recall that in 3D,

�1R

= ∂2i

1R

= 4�ıx′ (x) = �′ 1R

= ∂′2i

1R

= 4�ıx(x′), (4.4)

where the ′ symbol in �′, or in general in ∂′, refers to a derivation with respect to x′ (and of course, ∂i indicates the derivative with respectto xi, with xi the ith Cartesian coordinates). We recall that ıx′ (x) is the shifted Dirac delta satisfying 〈ıx′ , ϕ〉 = ϕ(x′) for every test functionϕ. Moreover, ϕ will always denote a 3D test function with compact support in ˝�.

Local basis components are also used, by which we mean the following. Consider a smooth planar loop L whose interior surfaceis star-shaped w.r.t. to one interior point. Let x ∈ ˝L and xL be its closest projection on L. So, the tangent vector �(xL) at xL satisfies�(xL) · (x − xL) = 0. Moreover, let us introduce its two orthonormal unit vector �(xL) and (xL) such that {�, �, } form a direct orthonormallocal basis at xL (i.e., with pointing inwards the loop). The planar components of � and will in the sequel be denoted by �˛ and ˇ with�ˇ˛�˛ = ˇ and ˛, ∈ {1, 2} (or any other dumb Greek index except � and which are kept to designate the tangent and normal vectors).

We adopt the notation r := (xi − xLi)i(xL) and z := (xi − xLi)�i(xL). In particular, the loop L lies in the plane {z = 0}. Moreover, ∂r : = � ∂�

and ∂s : = �� ∂� with the arc parameter denoted by s such that ds = � ′idx′

i where � ′i= �i(x′). So, in the local basis at xL, one has the radius R

satisfying R2(x, xL) = ‖x − xL‖2 = z2 + r2 is independent of s. In particular the following expression for the Laplacian holds in the polar base{, �, �} with polar coordinates8 {r, s, z} and for a scalar function f:

�f = ∂2r f + 1

r∂rf + 1

r2∂sf + ∂2

z f. (4.5)

4.1. Case A. Validation the pure edge dislocation loop

For the planar pure edge dislocation loop, r = 3(“=z”), and hence

〈��mn, ϕ〉 = −B�

z

8�� ��mpi�nqj[�j�l��il]i↔j〈

∮L

dx′�

R, ∂p∂q∂ ϕ〉. (4.6)

Observe from the identity (with the convention �ab : = �zab)

� ��mpi�nqj[�j�l��il]i↔j = ı�q(ıpn� m − ımn� p) + ı�p(ıqm� n − ımn� q) + � pımqın� + � qım�ınp + ıpq(�n ım� + �m ın�)

that the first two terms (i.e., with ı�q and ı�p) provide a vanishing contribution because the curve is a loop, and that the second two (i.e.,with � p and � q) also provide a vanishing contribution by the smoothness of the test-function, in such a way that (4.6) simply rewrites as

〈��mn, ϕ〉 = −B�

z

8�(�n ım� + �m ın�)〈

∮L

�1R

dx′�, ∂ ϕ〉.

On the one hand, observe that ��zn and �mz identically vanish, and on the other that also the terms ��

mnn and m��mn do vanish since by

(4.4),

�m ın�〈∮L

�1R

dx′�, n∂ ϕ〉 = 4��m

∮L

′�∂′

ϕdx′� = 0

while, in the local basis (recall that ∂s = � ∂ ),

�n ım�〈n

∮L

�1R

dx′�, ∂ ϕ〉 = −〈� ∂

∮L

1R

dx′m, �ϕ〉 = 0.

So, the only nonzero component is ��mn�m�n with (recall that �n�n = − and � ′

kdx′

k = ds′)

〈��mn�m�n, ϕ〉 = B�

z

∮L∂′

ϕds′,

in such a way that

��˛ˇ = −B�

z �˛�ˇ ∂ ıL, (4.7)

where we recall that ��mz = 0. Therefore, by (4.7), the announced Eq. (3.3) is recovered.

4.2. Case B. Validation of the conservative planar loop: (a) ˛ and zz-components

For the planar dislocation loop, consider (4.3) with r = . Moreover we shall consider the planar components of incompatibility, i.e., takem = and n = ˇ. Let us decompose the computations in subterms:

��˛ˇ =

take u=z in (4.3)︷︸︸︷��{1}

˛ˇ+

take u=� in(4.3)︷︸︸︷��{2}

˛ˇ=

take u=z in (4.3)︷ ︸︸ ︷��{11}

˛ˇ+ ��{12}

˛ˇ+

take u=� in (4.3)︷ ︸︸ ︷��{21}

˛ˇ+ ��{22}

mn

and compute each term separately.

8 As the loop is homeomorphic to the circle and is assumed to enclose a star-shaped surface, s can be considered as an angular parameter.

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N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70 67

For u = z (we have consequently set v = � and r = ),

〈��{1}mn , ϕ〉 =

B�

8�� ��mpi�nqj[�j�l��il]i↔j〈

∮L

dx′�

R, ∂p∂q∂zϕ〉.

From the identity

� ��mpi�nqj[�j�l��il]i↔j = ı�q(ıpn� m − ımn� p) + ı�p(ıqm� n − ımn� q) + � pımqın� + � qım�ınp + ıpq(�n ım� + �m ın�), (4.8)

it can be observed that the first two terms provide a vanishing contribution because the curve is a loop. Observe first that ��zz identically

vanishes by (4.8).The last one provides the {11}-contribution, namely

B�

8�(�n ım� + �m ın�)〈

∮L

�1R

dx′�, ∂zϕ〉 =

B�

2(�n ım� + �m ın�)

∮L∂′

zϕ(x′)dx′�,

and hence, in the local basis, ��{11}mz = 0 and from ˛ = �˛ � ,

��{11}˛ˇ

= �˛�ˇ(B� )∂zıL − (�˛ˇ + �ˇ˛)

B� �

2∂zıL. (4.9)

Moreover, the two central terms term of (4.8), collectively labelled by {12}, yield

〈��{12}mn , ϕ〉 = −

B�

8�

[〈∮L� p∂′

p1R

dx′n, ∂m∂zϕ〉

]m↔n

. (4.10)

Observe on the one hand that its trace

〈��{12}kk

, ϕ〉 = −B�

4�〈∮L∂′

k1R

dx′k, � p∂p∂zϕ〉

identically vanishes, since the curve is a loop, while the first diagonal entry,

〈��{12}mn �m�n, ϕ〉 =

B�

8�

[〈�n∂s

∮L

1R

dx′n, � p∂p∂zϕ〉

]m↔n

,

vanishes (because ∂s of the integral vanishes), whereby all diagonal terms vanish since dx′z = 0.

Moreover the off-diagonal component of (4.10) writes in the local basis as (recall that ∂r = � ∂� )

〈��{12}mn m�n, ϕ〉 = −

B�

8�� p〈�n∂p

∮L

1R

dx′n, ∂r∂zϕ〉

showing only one non-vanishing contribution for ∂p = p∂r. Hence, recalling (4.5) with f = R,

〈��{12}mn m�n, ϕ〉 =

B�

8�〈� �n

∮L∂2

r1R

dx′n, ∂zϕ〉 =

B�

8�〈� �n

∮L

(�

1R

+(

1R3

− ∂2z

1R

)− 1

R2∂2

s1R

)dx′

n, ∂zϕ〉

=B�

8�〈� �n

∮L

�1R

dx′n, ∂zϕ〉 =

B�

8�〈�

∮L

�1R

dx′n, �n∂zϕ〉

yielding by (4.4),

��{12}˛ˇ

= −B�

�

2

(�˛ˇ + �ˇ˛

)∂zıL. (4.11)

For u = � , while r = , m = and n = (and where we have consequently set v = z), one obtains from the identity (where the last termvanishes since ı�z = 0)

[�j�l�vil]i↔j = ıjvı�i + ıivı�j − 2ıijı�v,

that

〈��{2}˛ˇ

, ϕ〉 =B�

8��� �˛��ˇ�〈

∮L∂�

1R

dx′� + ∂�

1R

dx′�, ∂� ∂zϕ〉,

and hence, by the identity �˛��ˇ� = ı˛ˇı�� − ı˛�ı�ˇ, the definition dC ′� = −�dL(x′) (i.e., dx′

ˇ = �ˇ�dC ′�) with �ˇ = − �ˇ�� and since L is a

loop,

〈��{2}˛ˇ

, ϕ〉 = −B�

8�〈∮L��

(��ˇ∂′

˛1R

+ ��˛∂′ˇ

1R

)dC ′

�, ∂� ∂zϕ〉.

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68 N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70

Noticing that since dC ′z = 0, i must be equal to z, one rewrites

〈��{2}˛ˇ

, ϕ〉 = − B�q

8�〈∮L�jpq

(�ikn∂′

m1R

+ �ikm∂′n

1R

)dC ′

k, ∂p∂iϕ〉ım˛ınˇıjz (4.12)

which by the identity (since k must be equal to �) �jpq�ikn = ıji(ıpkıqn − ıpnıqk) = ıpkıqn − ıpnıqk and by the Divergence theorem yields

〈��{2}˛ˇ

, ϕ〉 = − B�q

8�〈∮L∂′

k∂′p

1R

dS′,(

(ıpkıqˇ − ıpˇıqk)∂˛ + (ıpkıq˛ − ıp˛ıqk)∂ˇ

)∂zϕ〉

= −B�q

8�〈∫

BL�′ 1

RdS′,

(ıqˇ∂˛ + �q˛∂ˇ

)∂zϕ〉 + B�

q

4�〈∮L∂′

˛∂′ˇ

1R

dS′, ∂q∂zϕ〉. (4.13)

Let us introduce �A the characteristic function of the set A and j the inwards normal of its boundary ∂A. Recall the relation∫A

ıx(x′)dS(x′) = �A with ∂j�A = −jı∂A. (4.14)

Then, (4.13) rewrites as

〈��{2}˛ˇ

, ϕ〉 =B�

2〈(ı ˇ∂˛ + ı ˛∂ˇ)�BL (x), ∂zϕ〉 +

B�

4�〈∫

BL∂′

˛∂′ˇ

1R

dS′, ∂ ∂zϕ〉

= −B�

2〈(ı ˇ˛ + ı ˛ˇ)ıL, ∂zϕ〉 +

B�

4�〈∫

BL∂′

˛∂′ˇ

1R

dS′, ∂ ∂zϕ〉 (4.15)

=B�

2〈(ı ˇ˛ + ı ˛ˇ)∂zıL, ϕ〉 −

B�

4�〈∂˛

∮L

1R

dC ′ˇ, ∂ ∂zϕ〉, (4.16)

whose first term as labeled by {21} rewrites as

��{21}˛ˇ

= (�ˇ˛ + �˛ˇ)B�

�

2∂zıL + ˛ˇ

(B�

)∂zıL. (4.17)

Concerning the second term of (4.15) (or (4.16)), labeled by {22}, it can be observed that it is symmetric with, as expressed in the localbasis, vanishing entries along �. Moreover, its trace reads

〈��{22}˛˛ , ϕ〉 = B�

〈�BL , ∂ ∂zϕ〉 = −B� 〈 ∂zıL, ϕ〉,

and equals the only non vanishing diagonal entry in the local basis, i.e., the ˛ˇ-entry, and hence

��{22}˛ˇ

= −˛ˇ

(B�

)∂zıL. (4.18)

Hence, summing (4.9), (4.11), (4.17) and (4.18) entails that

��˛ˇ = �˛�ˇ

(B�

� �

)∂zıL − (�˛ˇ + �ˇ˛)

B�� ��

2∂zıL, (4.19)

which is recognized as Eq. (3.5), achieving this second verification.

4.3. Case B. Validation of the conservative planar loop: (b) ˛z-components

For the planar dislocation loop, r = , it results that for m = 3 = z, n = ˇ, one has p = �, i = and hence

〈��zˇ, ϕ〉 =

−B�

8��uv ��˛�ˇqj[ıjvı�˛ + ı˛vı�j − 2ı˛jı�v]〈

∮L

dx′�

R, ∂�∂q∂uϕ〉. (4.20)

For u = 3 = z, v = � and since j /= 3 (otherwise all terms inside the bracket would vanish), q must be equal to 3 (i.e., q = z). Let us analyzeseparately each of the 3 terms of (4.20) arising from the 3 terms inside the bracket of (4.20):

(i)B�

8���˛(�� �ˇ�)〈

∮L

dx′

R, ∂�∂2

z ϕ〉 =B�

ˇ

8�〈∮L

dC ′p

R, ∂p∂2

z ϕ〉 =B�

ˇ

2〈∫

BLıx(x′)dS′, ∂2

z ϕ〉

(ii)B�

8�(�� ���)�ˇ�〈

∮L

dx′�

R, ∂�∂2

z ϕ〉 =B�

8�〈∮L

dC ′ˇ

R, ∂ ∂2

z ϕ〉

(iii)−B�

4��� (��˛�ˇ˛)〈

∮L

dx′�

R, ∂�∂2

z ϕ〉 =−B�

4�〈∮L

dC ′

R, ∂ˇ∂2

z ϕ〉,

whereby the sum of these three terms (i.e., (i) + (ii) + (iii)) reads

〈��{1}zˇ

, ϕ〉 =B�

ˇ

2〈�BL , ∂2

z ϕ〉 −B�

8�〈∫

BL∂′

∂′ˇ

1R

dS′, ∂2z ϕ〉. (4.21)

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N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70 69

Observe that the first term of the RHS of (4.21) vanishes by (4.14), and hence

〈��{1}zˇ

, ϕ〉 = −B�

8�〈∫

BL∂′

∂′ˇ

1R

dS′, ∂2z ϕ〉. (4.22)

Moreover, 〈∫

BL∂′

∂′ˇ

1R dS′, ∂2

z ϕ〉 = 〈∫

BL∂′

ˇ1R dS′, ∂ ∂2

z ϕ〉 = −〈∂

∫BL

∂′ˇ

1R dS′, ∂2

z ϕ〉 is symmetric with, as expressed in the local basis, vanishing

entries along � (because ∂s of the integral vanishes). Moreover, its trace vanishes by (4.14), 〈4��BL , ∂2z ϕ〉 = 0, implying that the whole

term (4.22) vanishes.For u = �, v = 3 = z, one is left with (we have chosen q = � , since j = 3)

〈��{2}zˇ

, ϕ〉 =B�

8�(�� ���)�ˇ� 〈

∮L

1R

dx′�, ∂�∂�∂� ϕ〉 =

B�

8�(�� ���)�ˇ� 〈

∮L∂′

�1R

dx′�, ∂�∂� ϕ〉

which from �� ��� = ı��ı � − ı��ı � (and � ′ dx = ds′) rewrites as

〈��{2}zˇ

, ϕ〉 = −B�

8��ˇ� 〈∂�

∮L

1R

dx′ , �ϕ〉,

whose projection along ,

〈��{2}zˇ

ˇ, ϕ〉 =−B�

8�〈�� ∂�

∮L

1R

dx′ , �ϕ〉,

vanishes. Moreover, its projection along reads (recall that �ˇ� �ˇ = − � )

〈��{2}zˇ

�ˇ, ϕ〉 =B�

8�〈�ˇ�ˇ�

∮L

�1R

dx′ , ∂� ϕ〉 =

B�

2〈∮L

ıx(x′)dx′ , −� ∂� ϕ〉

yielding

��{2}zˇ

= �ˇ

� B�

2� ∂� ıL. (4.23)

By summing (4.22) and (4.23) and since ��˛z = ��

z˛, we recover the announced Eq. (3.6).

5. Concluding remarks

In the first part of this communication, that is Section 3, a general Kröner’s formula, stated as Conjecture 1 and holding for a dislocationloop, has been inferred, directly from a previously established result as found in (Van Goethem, 2011b). This formula basically relates theelastic strain incompatibility inc E� to a projection on the dislocation line L of the semi-deviatoric part of the (right-) curl of the contortiontensor, viz. ��×∇ , where we recall that �� is related to the dislocation density ˛� by the (same semi-deviatoric) relation �� = ˛� − I/2tr ˛�.Moreover, in this new formula, a curvature-dependent term appears which however vanishes if the dislocation has no climb component,i.e., is conservative.

In a second part, that is Section 4, the explicit differences between this new formula and the classically reported Kröner’s formula hasbeen exhibited for two types of dislocation loops, namely the pure edge dislocation loop (with an out-of-plane Burgers vector, therebysometimes referred to as non conservative), and the planar loop with planar Burgers vector. In particular, it happens that the new termsappearing in Conjecture 1 are mandatory as soon as the dislocation exhibits a planar Burgers vector.

Finally, the last part of this communication consisted in a direct validation of the new formula as based on the theory of distributions.This rather technical part is the core of the present contribution.

We believe that this new formula can have an impact for crystal growth practice, since in the presence of dislocations, scale separationcan hardly be done in any realistic thermodynamic model accounting for dislocation creation and/or movement (cf., e.g., Zbib et al., 2005)outside equilibrium. Therefore, even at the macroscale, dislocation loops may appear while interacting with any other defect types (suchas point defects, see Van Goethem et al., 2008).

In this paper, we have established a formula which explicitly relates the elastic strain incompatibility to the dislocation density. Thisgeometric relation still holds if motion of dislocations is considered. At the macroscale a dynamic model with incompatibility as driving“force” has been proposed in Van Goethem (2011a, 2012). The new mesoscopic formula will then play a role but a crucial step prior toreach crystal growth models at the macroscale (cf. Zaiser, 2004, for a state-of-the art) is to homogenize the mesoscopic results establishedso far. This will be considered in a forthcoming work in the spirit of Van Goethem (2012).

Another link with the macroscale is related to the dislocation motion on a prescribed set of glide planes. The following mathematicalissue appears: how does the mesoscopic model conduct us to these privileged, macroscopic, directions of motion? Specifically, at themesoscale, many fields are likely to be of bounded deformation (in the sense of Temam and Strang (1980)) and hence concentrated effectsappear on the set of jump points of the displacement: do they correspond to the glide planes? This is the subject of a work in preparation(Van Goethem, in preparation). Again, since the new formula proved in this paper holds for 3D loops (in the sense that the strain of theseloops depends on 3 space variables), one open question is to show that the distributional approach allows us to rigorously give a meaningto the mesoscopic and macroscopic fields, and to infer from this multiscale approach the required fields of interest to reach a completemacroscopic model, which is still missing for crystal growth practitioners. Let us emphasize that the homogenization of the mesoscopicformula is a hard task which remains to be done.

Author's personal copy

70 N. Van Goethem / Mechanics Research Communications 46 (2012) 62– 70

Acknowledgements

The author has been supported by Fundac ão para a Ciência e a Tecnologia (Ciência 2007, PEst OE/MAT/UI0209/2011 and FCT Project:PTDC/EME-PME/108751/2008).

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