the motion of deformable ellipsoids in power-law viscous materials: formulation and numerical...

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Journal of Structural Geology 50 (2013) 22e34

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Journal of Structural Geology

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The motion of deformable ellipsoids in power-law viscous materials: Formulationand numerical implementation of a micromechanical approach applicable to flowpartitioning and heterogeneous deformation in Earth’s lithosphere

Dazhi Jiang*

Department of Earth Sciences, The University of Western Ontario, 1151 Richmond Street, London, Ontario N6A 5B7, Canada

a r t i c l e i n f o

Article history:Received 1 December 2011Received in revised form22 June 2012Accepted 27 June 2012Available online 11 July 2012

Keywords:Eshelby theoryFlow partitioningViscous flowNumerical modelingHeterogeneous deformationHigh-strain zones

* Tel.:þ 1 519 661 3192; fax:þ 1 519 661 3198.E-mail address: [email protected].

0191-8141/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.jsg.2012.06.011

a b s t r a c t

Earth’s lithosphere is heterogeneous in rheology on a wide range of observation scales. When subjectedto a tectonic deformation, the incurred flow field can vary significantly from one rheologically distinctelement to another and the flow field in an individual element is generally different from the bulkaveraged flow field. Kinematic and mechanical models for high-strain zones provide the relationsbetween prescribed tectonic boundary conditions and the resulting bulk flow field. They do not deter-mine how structures and fabrics observed on local and small scales form. To bridge the scale gapbetween the bulk flow field and minor structures, Eshelby’s formalism extended for general power-lawviscous materials is shown to be a powerful means. This paper first gives a complete presentation ofEshelby’s formalism, from the classic elastic inclusion problem, to Newtonian viscous materials, and tothe most general case of a power-law viscous inhomogeneity embedded in a general power-law viscousmedium. The formulation is then implemented numerically. The implications and potential applicationsof the approach are discussed. It is concluded that the general Eshelby formalism together with the self-consistent method is a powerful and physically sound means to tackle large plastic deformation of Earth’slithosphere.

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1. Introduction

Rocks are composed of rheologically heterogeneous elementsover a wide scale range of observations. When subjected toa tectonic deformation, the incurred flow field in the heteroge-neous rock mass is more complex than it would be in a rheologi-cally homogeneous material. In the latter, the flow field variessmoothly in space and usually has a simple relationship with theimposed boundary conditions. In the former, the flow field is morelumpy e varying considerably from one element to another and, toa less degree, within individual elements. As a result, the flow fieldinside a rheologically distinct element does not in general beara simple relationship with the far-field tectonic boundary condi-tions. This presents a major problem to geologists who usedeformation structures and fabrics on outcrop and smaller scales(‘minor structures’ hereafter) to infer deformation boundaryconditions on the tectonic scale much larger than the structuresthemselves (e.g., Simpson and Schmid, 1983; Hanmer andPasschier, 1991).

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To simplify the flow field in a heterogeneous body, the normalcontinuum mechanics approach is to replace the field quantities ata point by their averaged values over a suitable RepresentativeVolume Element (RVE) centered at that point (cf. Batchelor, 2000,pp. 4e5; Ranalli, 1995, pp. 5e6; Lister and Williams, 1983; Li andWang, 2008, pp. 78e80). This “smoothed-out” and simpler flowfield is called the bulk or macroscopic flow field which captures thevariation of the flow on the scale greater than the RVE but has noinformation for smaller (less than RVE) scale complexities in theflow. All current models for high-strain zones, kinematic ormechanical, are ones for the bulk flow field arising froma prescribed boundary condition (e.g., Ramsay and Graham, 1970;Ramberg, 1975; Ramsay, 1980; Sanderson and Marchini, 1984;Simpson and De Paor, 1993; Fossen and Tikoff, 1993; Robin andCruden, 1994; Dutton, 1997; Jiang and Williams, 1998; Lin et al.,1998; Passchier, 1998; Jiang, 2007c). Minor structures insidenatural high-strain zones however are small features compared tothe RVE and generally owe their formation to flow field variationson scales below the RVE. Models for the bulk scale flow field cannotbe used directly to interpret minor structures.

To understand minor structures we must establish the rela-tionship between the bulk flow field, which can be related to the

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D. Jiang / Journal of Structural Geology 50 (2013) 22e34 23

boundary conditions, and flow fields inside individual, rheologi-cally distinct, elements which control the development ofobserved minor structures. Lister and Williams (1983) describedthe deviation of the local flow field from the bulk one as “flowfield partitioning”. We need a physically sound means to governflow partitioning to bridge the scale gap problem highlightedabove.

From a physical point of view, flow field partitioning occurswhere there are mechanical interactions between a rheologicallydistinct element and the surrounding medium. Eshelby (1957,1959) pioneered a means to deal with such interactions for anisolated elastic region in an infinite elastic field. His approach hassince been extended to general viscous materials, linear or power-law, isotropic or anisotropic, and has led to the development ofa new and active discipline of continuum mechanics calledmicromechanics (Mura, 1987; Nemat-Nasser and Hori, 1999; Quand Cherkaoui, 2006; Hutchinson, 1976; Lebensohn and Tomé,1993). As will be shown in this paper, Eshelby’s inhomogeneityformalism for general power-law viscous materials and the self-consistent method (Kröner, 1961; Hill, 1965, 1966; Hutchinson,1976; Mura, 1987; Molinari et al., 1987; Lebensohn and Tomé,1993) provide a powerful means for tackling flow partitioning innatural deformations.

In this contribution, I first give a complete presentation ofEshelby’s formalism, from the classic elastic inclusion problem, toNewtonian viscous materials, and to the most general case ofa power-law viscous inhomogeneity embedded in a power-lawviscous medium. I then implement the formulation numericallyfor practical use. Mathcad worksheets are provided in the onlinesupplementary data for interactive numerical simulation andresult visualization. To make the paper less dependent on thematerials science literature so that the reader can follow the flow ofideas more easily, I have reproduced some published equations insome detail. Finally I discuss the implications and potential appli-cations of the approach in geology. A full account of the self-consistent formulation and its implementation will be presentedin a separate contribution.

2. Nomenclature

Throughout this paper, I will use a combination of Cartesiantensor notation and a standard tensor/matrix notation. Tensorsare represented by bold-face letters. Fourth order tensors will berepresented by uppercase bold-face letters. As far as possible,second order tensors will be represented by lowercase bold-faceletters. A few exceptions are made for some tensors to beconsistent with commonly used notations in earlier papers.Scalars and scalar components of tensors are represented by italicletters. The summation convention is assumed unless declaredotherwise whereby a repeated index represents summation overthe value of 1, 2, 3 for the index. The double contracted product oftwo tensors is denoted by ‘:’. Thus the ijkl-components of A:B,where both A and B are fourth order tensors, are AijmnBmnkl, theij-components of A:b, where b is a second order tensor, areAijmnbmn, and a:b is a scalar, aijbij. The product between 2 s ordertensors or between two matrices is denoted as ab, withij-components being aikbkj.

The following three fourth order unit tensors are used in thispaper:

Jijkl ¼ dikdjl; JSijkl ¼12

�Jijkl þ Jjikl

�; JAijkl ¼

12

�Jijkl � Jjikl

�where dij¼ 1 for i¼ j and 0 for is j. The three tensors J, JS, and JA arereferred to, respectively, as the fourth order unit tensor, the fourth

order symmetric unit tensor, and the fourth order anti-symmetricunit tensor (Li and Wang, 2008, p, 8).

a1, a2, a3, ai the semi-axes of an ellipsoid (first, second, third,general)

A strain rate partitioning tensor (fourth order)b1, b2, b3, b4, b5, b6, bl base set of 6 orthonormal symmetric

second-order tensors, (l ¼ 1, 2, . 6)C, Cinh, CM elastic moduli tensor (general, of the inhomogeneity, of

the matrix medium)C 6 � 6 matrix of the elastic modulidt time step for numerical computationDaxis diagonal matrix of the strain rates for the three semi-axes

of an ellipsoide, eC, e*, eM, einh, ~e elastic strain tensor (general, constrained, eigen-,

in the far-field matrix medium, in aninhomogeneity, difference between theinhomogeneity and the far-fieldmatrixmedium)

E(q,k) elliptic integral of the second kind3, 3

inh, 3E

3M, 3

0, ~3viscous strain rate tensor (general, of aninhomogeneity, of an ellipsoid, of the matrixmedium, back-extrapolated term in tangentlinearization, difference between the ellipsoidand the far-field matrix medium)

F(q,k) elliptic integral of the first kindH;H

_Hill’s constraint tensor, its inverse (also known as theinteraction tensor)

I second order unit tensorJ, JS, JA 4th order unit tensor (general, symmetric, anti-

symmetric)J the J-integrals in calculation of Eshelby tensorsL velocity gradient tensor of matrix flow (second order)M, MM, Minh, M(tan), M(sec) 4th order viscous compliances tensor

(general, of the matrix medium, of theinhomogeneity, tangent, secant)

Q matrix defined by the orientation of an ellipsoidQ angular velocity tensor of an ellipsoidq1, f1, q2 spherical angles defining the orientation of an ellipsoidn, nM stress exponent (general, of the matrix material)r, reff, r0 viscosity ratio between ellipsoid and matrix medium

(Newtonian, effective where one or both the ellipsoid andthe matrix medium are power law, effective at the matrixmedium strain rate state

s, sinh, sM, ~s Cauchy stress tensor (general, in the inhomogeneity,in the matrix medium, difference between theinhomogeneity and the matrix medium)

S, P symmetric Eshelby tensor, anti-symmetric Eshelby tensorT, TS, TA 4th order Green interaction tensor, symmetric Green

interaction tensor, anti-symmetric Green interactiontensor

winh, wE, wM, ~w vorticity in an inhomogeneity, in an ellipsoid, inthe far-field matrix medium, vorticity differencebetween the ellipsoid and the far-field matrixmedium

uinh, uM elastic rotation tensor in the inhomogeneity, elasticrotation tensor in the remote matrix medium

u incremental angle of rotationu normalized incremental angular rotation tensor

3. The interaction between an elastic inclusion/inhomogeneity and the embedding infinite elastic medium:the classic Eshelby formalism

Eshelby (1957, 1959) considers the elastic field of an infiniteuniform elastic body caused by a “region” he called “inclusion”

D. Jiang / Journal of Structural Geology 50 (2013) 22e3424

inside the body undergoing a “change of shape and size” (Fig. 1a)which amounts to a stress-free strain field, if the inclusionwere cutout from and therefore unconstrained by the elastic matrixmedium (Fig. 1b and c), that Eshelby (1957, 1959) called the“transformation strain” (eT in Fig. 1c). Later, Mura (1987) referred totransformation strains generally as eigenstrains (e*). Since theinclusion is constrained by the surrounding matrix medium, itinteracts with the matrix medium and reaches a constrainedequilibrium stress and strain state. Eshelby (1957, p. 384, hisEq. (3.5)) proved that if the inclusion is ellipsoidal, then the stressand strain fields inside the inclusion in the constrained state areuniform. The strain, eC, and rotation, uC, in the ellipsoidal inclusionare related to the eigenstrain by two fourth order Eshelby tensors:

eC ¼ S : e* (1)

and

uC ¼ P : e* (2)

where S and P are respectively the symmetric and anti-symmetricEshelby tensors.

C

TeCe Cω+

C T:e S eC T:ω Π e

a

c

Fig. 1. The classic Eshelby inclusion problem. (a) In a homogeneous solid free of stress and sthe inclusion region having undergone transformation were cut out, the remaining solid bodstrain field (eT) that changes U to UT. The inclusion in the transformed state is stress-free butamounts to “force” the transformed inclusion UT to fit back into the void in (b) and perfectinside the inclusion as well as in the surrounding medium. As long as the inclusion is ella uniform field of strain, eC, and rotation, uC, which are related to the transformation strainby the general concept of eigenstrain. See text for details.

Eshelby called an ellipsoidal region, linearly elastic but havingdifferent elastic moduli from the embedding matrix medium, anellipsoidal inhomogeneity. To solve for the stressestrain field in anisolated ellipsoidal inhomogeneity embedded in an infinite elasticbody subjected to a far-field uniform elastic strain, Eshelby (1957)had the brilliant idea that an ellipsoidal inhomogeneity canalways be replaced, uniquely, by an “equivalent inclusion”with theright eigenstrain field so that the stress state inside the ellipsoidalregion and in the surrounding matrix medium is the same as whenthe inhomogeneity is present.

Such an “equivalent inclusion” is found by setting

einh ¼ eC þ eM (3)

and

Cinh : einh ¼ CM :�eC þ eM � e*

�(4)

where e is the second order strain tensor and C the fourth orderelastic moduli tensor. The sub- or super-script “inh” and “M” standfor inhomogeneity and matrix medium respectively. On the right-

b

d

C

train has a region U, called inclusion, to undergo “transformation” (shape change). (b) Ify would remain stress- and strain-free. (c) The “transformation” amounts to a uniformwith transformation strain. (d) Because the inclusion is constrained in the solid, whichly “glue” the interface, the misfit induces stress and deformation (strain and rotation)ipsoidal, the equilibrated shape and orientation of the inclusion UC are described byfield eT by two Eshelby tensors (c). Since Mura (1987), transformation strain is replaced


hand side of Eq. (4), the eigenstrain e* is subtracted from the totalstrain in the inclusion to calculate the stress because the eigen-strain state corresponds to the stress-free state of the inclusion.

Combining Eqs. (1), (3) and (4) yields:

einh ¼hJS � Sþ S : C�1

M : Cinhi�1

: eM (5)

Combining Eqs. (2)e(4) yields:

uinh ¼ uM þP : S�1 :�einh � eM

�(6)

Eqs. (5) and (6) explicitly relate the strain field in the far-fieldmatrix medium to the strain and rotation in the inhomogeneity.In other words, Eqs. (5) and (6) specify how the far-field elasticdeformation is partitioned into the deformation field inside theinhomogeneity. We wish to develop similar relations for naturaldeformation of rocks to address the interaction of an individualelement with the surrounding rock masses.

Applying the general Hook’s law (s ¼ C:e, s being the Cauchystress tensor), using Eqs. (1) and (3) to get rid of e* and eC in Eq. (4)leads to:

~s ¼ �CM :�S�1 � JS

�: ~e ¼ �H : ~e; H ¼ CM :

�S�1 � JS

�(7a)

~e ¼ ��S�1 � JS

��1: MM : ~s ¼ �H

_: ~s; H

_ ¼�S�1 � JS

��1: MM

(7b)

where the tilde stands for the difference in the quantity betweeninside the inhomogeneity and in the matrix medium (e.g.,~s ¼ sinh �sM). MM is the matrix medium compliances tensor(inverse of CM),H is calledHill’s constraint tensor (Qu and Cherkaoui,2006, pp. 90, 315), and H

_, the inverse of H, is known as the inter-

action tensor (Lebensohn and Tomé, 1993).Similarly, using Eqs. (2) and (3) to get rid of e* and eC in Eq. (4)

leads to:

~u ¼ P : S�1 : ~e (8)

Eqs. (7) and (8) summarize Eshelby’s solutions to the elastic inho-mogeneity problem. In the following, wewill extend this formalismto progressively more general materials. Since an “inhomogeneity”can always be replaced by an “equivalent inclusion” and since onlyellipsoidal inclusions are considered in this paper, no distinctionwill be made between “ellipsoidal inhomogeneity” and “ellipsoidalinclusion” hereafter. They will be simply referred to as “ellipsoids”.

4. Partitioning of flow between a viscous inhomogeneousellipsoid and the embedding viscous material

4.1. General linearly viscous materials

Theory of linear elasticity and theory of Newtonian fluids areformally equivalent (e.g., Spencer, 1980, pp. 110e118). The principleof superposition (Fung, 1965, pp. 3) applies to both linear elasticsolids and Newtonian fluids. Because of this, the above Eshelby’ssolutions (Eqs. (7) and (8)) for linear elastic materials can be applieddirectly to Newtonian viscousmaterials. Bilby et al. (1975) and Bilbyand Kolbuszewski (1977) have given equivalent equations forisotropic and incompressible Newtonian materials by letting thePoisson’s ratio term in the original Eshelby equations tend to 0.5.

The common assumption that viscous materials are incom-pressible implies that their constitutive equations relate only the

deviatoric part of the stress tensor to the strain rate tensor. Oneconsequence of this is that the viscous compliances tensor (seebelow) is non-invertible. To apply the equivalent Eshelbyformalism developed for elastic materials (Eqs. (7) and (8)), enmasse, to Newtonian materials requires the existence of theviscous moduli tensor which is the inverse of the viscous compli-ances tensor. There are two different but equivalent ways to dealwith this issue. One is to regard incompressibility as an additionalkinematic condition and the viscous compliances tensor asrelating only the deviatoric stresses (5 independent components)to the viscous strain rates (5 independent components). In thiscase, it is necessary to define a different Eshelby tensor for viscousmaterials, Sv, in replacement of S for compressible materials.Components of Sv are the same as S except for the Sviijj (nosummation) terms which must always satisfy Svii11 þ Svii22 þ Svii33 ¼0 to ensure incompressibility (see Lebensohn et al., 1998). Asecond way is to simply regard viscous materials as slightlycompressible (with a very small, but non-zero, compressibility) inall the calculations. The bulkmodulus is set to approaching infinityto achieve incompressibility. In this case, the constitutive equationstill relates the Cauchy stresses (6 independent components) tothe viscous strain rates (6 components because of the material’scompressibility). The second approach is used in this paper as itallows direct adoption of the elastic Eshelby formalism for appli-cation in viscous materials.

Because of full equivalence between linear elasticity and New-tonian viscosity, Eqs. (7) and (8) can be rewritten for Newtonianmaterials as:

~3¼ �H_

: ~s (9a)

~w ¼ P : S�1 : ~3 (9b)

H_ ¼

�S�1 � JS

��1: MM (9c)

where 3,s, andw are respectively the viscous strain rate, stress, andvorticity in replacement of elastic strain, stress, and rotation in Eqs.(7) and (8);MM in Eq. (9c) is now the viscous compliances tensor forthe matrix material.

Similar to Eqs. (5) and (6), we can derive an explicit form forgeneral Newtonian viscous materials from Eq. (9) by using thegeneral constitutive equation 3¼ M:s:

3E ¼

hJS � Sþ S : MM : M�1

E

i�1: 3

M (10a)

wE ¼ wM þP : S�1 :�

3E � 3

M�

(10b)

where ME is the viscous compliances tensor for the ellipsoid.Where both the matrix medium and the ellipsoid are isotropic,

the MM : M�1E term in Eq. (10a) is reduced to the scalar viscosity

ratio, r, of the inhomogeneity to the matrix medium and Eq. (10a) issimplified to:

3E ¼

hJS þ ðr � 1ÞS

i�1: 3

M (11)

4.2. General power-law viscous materials

The rheology of rocks in the ductile lithosphere is commonlyconsidered to be power law viscous (Kohlstedt et al., 1995). Theelegance of Eshelby’s equivalent-inclusion approach arises from the


principle of superposition (Fung,1965, p. 3) which does not apply torheologically nonlinear materials. To take full benefit of the Eshelbyformalism, the rheological equation must be first linearized. Theidea of linearization is based on the fact that at any given stress andstrain rate state the actual nonlinear rheology may be approxi-mated by a linear one in the vicinity of that state. One can regarda progressive finite deformation as being achieved throughsuccessive infinitesimal steps, in each of which a linear approxi-mate rheology is used and the Eshelby formalism is valid. Forpower-law viscous materials, the most commonly used lineariza-tion is the tangent linearization based on the first-order Taylorexpansion of the original constitutive relation (Hutchinson, 1976;Molinari et al., 1987; Lebensohn and Tomé, 1993).

For a power-law viscous material, the constitutive equation canbe written in a pseudo-linear form (e.g., Hutchinson, 1976, p. 104,his Eq. (2.5)):

3ðsÞ ¼ MðsecÞðsÞ : s (12)

where M(sec)(s) is referred to as the secant viscous compliancestensor which depends on the current state of deviatoric stress (orstrain rate). For an isotropic incompressible power-law material, itis reduced to a scalar function of the deviatoric stresses (or strainrates), and its inverse is the well-known effective viscosity(e.g., Ranalli, 1995, p. 78).

A first-order Taylor expansion of Eq. (12) in the vicinity ofa stressestrain rate state is:

3ðsÞ ¼ 3ðs0Þþv 3

vs

��s¼s0

: ðs�s0Þ ¼ nMðsecÞðs0Þ :sþð1�nÞ 3ðs0Þ

¼ MðtanÞðs0Þ :sþ 30

(13a)where

MðtanÞ ¼ v 3

vs

��s¼s0

¼ nMðsecÞ (13b)

and 30 ¼ (1 � n) 3(s0) are respectively the tangent compliances

tensor and the back extrapolated term (Hutchinson, 1976; Molinariet al., 1987; Lebensohn and Tomé, 1993); n is the power-law stressexponent for the material.

In the vicinity of the current stressestrain rate state, the linearconstitutive relationship (Eq. (13a)) can be used to replace Eq. (12)so that the Eshelby’s formalism applies. In this case, the interactiontensor (Eq. (9c)) becomes:

H_ ¼

�S�1 � JS

��1: MðtanÞ

M ¼ nM�S�1 � JS

��1: MðsecÞ

M (14)

where nM is the power-law stress exponent of the matrix material.An explicit relationship between the strain rate tensor in the

ellipsoid and that in the far-field matrix medium for the generalcase of a power-law ellipsoid in a power-law viscous matrix can beobtained by inserting Eq. (14) into Eq. (9a) and rearranging:

3E ¼

hJS � Sþ nMS : MðsecÞ

M : MðsecÞ�1E

i�1:hJS þ ðnM � 1ÞS

i: 3

M

(15)

The vorticity relationship can also be obtained by using Eq. (9b)together with the above relation.

Eq. (15) is simplified to the following form if both the matrixmedium and the ellipsoid are incompressible isotropic power-lawmaterials:

3E ¼

hJS þ

�nMreff � 1

�Si�1

:hJS þ ðnM � 1ÞS

i: 3

M (16)

because in such a case the term MðsecÞM : MðsecÞ�1

E in Eq. (15) isreduced to the scalar effective viscosity ratio reff between theellipsoid and the matrix medium.

Eq. (16) coincides with Eq. (11) for Newtonianmaterials (nM¼ 1)and reff is replaced by the constant r. Where the ellipsoid is ofpower-law but the matrix medium is Newtonian, Eq. (16) becomes:

3E ¼

hJS þ

�reff � 1

�Si�1

: 3M (17)

which differs from Eq. (11) only in the viscosity ratio term.Mancktelow (2011) considered a power-law elliptical inclusion

embedded in a 2D Newtonian matrix, a case to which Eq. (17)applies. He suggested to the effect that Eq. (17) also applieswhere thematrixmaterial is also of power-law. This amounts to usethe secant compliances in the interaction equation, leading toH_ ¼ ðS�1 � JSÞ�1 : MðsecÞ

M , rather than using the linearized tangentcompliances, leading to Eq. (14). This will generally overestimatethe strain rate in the ellipsoid if it is stronger than the matrix andunderestimate the strain rate in the ellipsoid if it is weaker (Zhong,2012). Although Eq. (17) appears to have incorporated the effectiveviscosities in the viscosity ratio term, it does not incorporate thetangent linearization of the constitutive equation for the matrixmedium to ensure that the Eshelby formalism can be applied. Eq.(16) should be used where the matrix medium is of power-law.

5. Algorithm for numerical implementation

The relationships between the flow field in an ellipsoid and thatin the far-fieldmatrixmedium for themost general case of a power-law viscous ellipsoid in a power-law viscous medium can besummarized as:

3E ¼ A : 3

M (18a)

wE ¼ wM þP : S�1 :�A � JS

�3M (18b)

where A ¼hJS � Sþ nM S : MðsecÞ

M : MðsecÞ�1E

i�1: ½JS þ ðnM � 1ÞS� is

here called the strain rate partitioning tensor. Eq. (18a) relates thestrain rates in the ellipsoid to the far-field strain rates and Eq. (18b)the vorticity in the ellipsoid to the far field flow.

To apply Eq. (18) for the progressive deformation of an ellipsoid,an algorithm must be developed to so that all terms in the equa-tions can be calculated and the state of the ellipsoid can be tracked.

5.1. Eshelby tensors in isotropic viscous materials

In isotropic and incompressible materials, the Eshelby tensorsare rather simple. The non-zero components of S and P are, fromexpressions of Eshelby (1957):

Siiii ¼34p

a2i Jii; Siijj ¼34p

a2j Jij; Sijij ¼38p

�a2i þ a2j

�Jij

Pijij ¼ Pijji ¼Jj � Ji

8p; Pjiij ¼ Pjiji ¼ �Pijij

(19)

where ai (i ¼ 1,2,3) are the ellipsoid’s semi-axis lengths and theJ-terms can be defined in terms of 2 elliptic functions:

Fðq; kÞ ¼Zq0

dw�1� k2sin2 w

�12

; Eðq; kÞ ¼Zq0

�1� k2sin2w

�12dw

(20)

where q ¼ arcsinðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ða3=a1Þ2

qÞ; k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða21 � a22Þ=ða21 � a23Þ

q.


Specifically, the J-terms for four different situations are asfollows:

For triaxial ellipsoids (a1 > a2 > a3):

J1 ¼ 4pa1a2a3�a21 � a22

��a21 � a23

�1=2 ½Fðq; kÞ � Eðq; kÞ�

J3 ¼ 4pa1a2a3�a22 � a23

��a21 � a23

�1=2"a2�a21 � a23

�1=2a1a3

� Eðq; kÞ#;

J2 ¼ 4p� J1 � J3 (21)

and the remaining J-terms follow from the following formulae1:

Jij ¼Jj � Ji

3�a2i � a2j

�; Jii ¼4p3a2i

� Jij � Jik; ðisjskÞ (22)

For oblate spheroids (a1 ¼ a2 > a3):

J1 ¼ J2 ¼ 2pa21a3�a21 � a23

�3=2"cos�1

�a3a1

� a3a1

1� a23

a21

!1=2#;

J3 ¼ 4p� 2J1

J13 ¼ J23 ¼ J1 � J3

3�a23 � a21

�; J11 ¼ J22 ¼ p

a21� 34J13;

J12 ¼ J11

3; J33 ¼ 4p

3a23� 2J13 (23)

For prolate spheroids (a1 > a2 ¼ a3):

J2 ¼ J3 ¼ 2pa1a23�a21 � a23

�3=2"a1a3

a21a23

� 1

!1=2

�cosh�1�a1a3

#;

J1 ¼ 4p� 2J2

J12 ¼ J13 ¼ J2 � J1

3�a21 � a22

�; J11 ¼ 4p3a21

� 2J12;

J22 ¼ J33 ¼ p

a22� 34J13; J23 ¼ J22

3(24)

For spheres (a1 ¼ a2 ¼ a3):

J1 ¼ J2 ¼ J3 ¼ 4p3; J11 ¼ J22 ¼ J33 ¼ 4p

5a21; J12 ¼ J13

¼ J23 ¼ J11

3(25)

5.2. Eshelby tensors in anisotropic viscous materials

For ellipsoids in a general anisotropic viscous material,Lebensohn et al. (1998, 2004) gave the following procedure to

1 Note there is a factor of 3 difference between Eshelby (1957) Jij terms and thedefinition in Mura (1987) and Qu and Cherkaoui (2006). Eshelby’s definition is usedhere as in Jiang (2007b, 2012).

calculate the Eshelby tensors. The procedure stems from using theGreen functionsmethod to solve the set of differential equations forthe Eshelby inhomogeneity problem.

First construct a 3 � 3 matrix function from the viscouscompliances tensor M of the matrix medium after the constitutiveequation is linearized:

AikðxÞ ¼ M�1imknxmxn (26)

where x is a unit vector defined by spherical angles:

x ¼0@ sin q cos f

sin q sin fcos q

1A (27)

For a power-law matrix material, M should be the linearizedcompliances such as the tangent compliances.

Second, calculate the fourth order Green interaction tensor Tdefined by:

Tijkl ¼a1a2a34p

Z2p0

Zp0

xjxlA�1ik ðxÞh

ða1x1Þ2þða2x2Þ2þða3x3Þ2i3=2sin q dqdf

(28)

Third, calculate the symmetric, TS, and anti-symmetric, TA,Green tensors:

TS ¼ JS : T : JS; TA ¼ JA : T : JS (29)

from which the Eshelby tensors are finally:

S ¼ TS : M�1; P ¼ TA : M�1 (30)

AsM�1 is a symmetric fourth order tensor, its contractionwith JS

reproduces itself. Inserting Eq. (29) into Eq. (30), we arrive at thefollowing two expressions for calculating Eshelby tensors directlyfrom T:

S ¼ JS : T : M�1; P ¼ JA : T : M�1 (31)

Numerical integration of Eq. (28) is by the GausseLabattomethod. The integration sphere (0 � 4 � p, 0 � q � 2p) is dividedinto 8 equal quadrants, 4 in one hemisphere, and 10 integrationnodes are used in each quadrant.

5.3. Inversion of fourth order tensors

Many fourth order tensors and their inverse tensors have beenused in the formulation so far. All fourth order tensors in this paper,except for P, T, and TA, are symmetric with respect to interchangeof the leading pair of indices and also of terminal pair(i.e., Bijkl ¼ Bjikl ¼ Bijlk). Only inversion of such symmetric fourthorder tensors is required in this paper. In this sense, the inverse ofa fourth order symmetric tensor B is a unique fourth order tensorB�1, also symmetric, satisfying:

B�1 : B ¼ B : B�1 ¼ JS (32)

The method in many textbooks for inverting a general fourthorder symmetric tensor (e.g., Nye, 1957; Mura, 1987, pp. 502e504;Nemat-Nasser and Hori, 1999, pp. 638e641) is very tedious. I useamuch simpler method described by Lebensohn et al. (1998) whichinvolves introducing a base set of 6 orthonormal symmetricsecond-order tensors, bl (l ¼ 1, 2, . 6), defined as:

b1 ¼ 1ffiffiffi6

p

0BB@

�1 0 0

0 �1 0

0 0 2

1CCA ; b2 ¼ 1ffiffiffi

2p

0BB@

�1 0 0

0 1 0

0 0 0


2p

0BB@

0 0 0

0 0 1

0 1 0

1CCA

b4 ¼ 1ffiffiffi2

p

0BB@

0 0 1

0 0 0

1 0 0


2p

0BB@

0 1 0

1 0 0

0 0 0


3p

0BB@

1 0 0

0 1 0

0 0 1

1CCA


With this base set, a symmetric fourth order tensor like C is firsttransformed to a 6 � 6 matrix C by:

Clh ¼ Cijklblijb

hkl; ðl;h ¼ 1;2;.6Þ (33)

Matrix C can be readily inverted. The inversion of C can then beobtained by transforming the 6 � 6 matrix, C�1

lh , back to a fourthorder tensor using the base set of orthonormal symmetric tensors:

C�1ijkl ¼ C�1

l blijbhkl (34)

5.4. Iterative procedure for power-law materials

In a power-law material, the viscous compliances tensor is nota material constant but depends on the current state of deviatoricstresses or the strain rates. This makes it difficult to assign an initialeffective viscosity ratio (Eq. (16)) between a power law ellipsoid toits matrix medium because the actual strain rates, 3

E, in the ellip-soid is unknown a priori. An iterative procedure is used in calcu-lating 3

E, starting with an initial viscosity ratio defined ata reference strain rate for both the matrix medium and the ellip-soid. Calculated 3

E is used as input to calculate the new viscosityratio for the next step of calculation of 3

E. This iterative procedurecontinues until the current viscosity ratio coincides with the laststep ratio within a specified tolerance. Specifically, the effectiveviscosities of both the matrix medium and the ellipsoid can beexpressed in terms of a reference state effective viscosity(Schmalholz et al., 2008; Mancktelow, 2011):

hMeff ¼

3MII30

!1�nMnM

hM0 ; hEeff ¼

3EII30

!1�nEnE

hE0

where 3II is the second invariant of the strain rate tensor, 30 thesecond invariant of the reference strain rate tensor, h0 the effectiveviscosity at the reference strain rate state, and superscripts ‘E’ and‘M’ stand for the ellipsoid and the matrix medium respectively.Since the strain rates tensor for the matrix medium is given ina simulation, it can be used as the reference state, and an iterativeequation for the effective viscosity ratio is:�hEeff

�iþ1

hM0¼

3EII iþ1

3EII i

!1�nEnE

�hEeff

�i

hM0or riþ1

eff ¼

3EII iþ1

3EII i

!1�nEnE

rieff

The iterative procedure terminates when��riþ1eff � rieff

��<error.

5.5. Strain and rotation of the ellipsoid

The shape and rotation of the ellipsoid are governed by thefollowing evolution equations (Jiang, 2012):

da ¼ Daxisa; Daxisij ¼

0; if isjE and a ¼ @ a1

a2A (35)
dt 3ij; if i ¼ j
0a3

1

dQdt

¼ �QQ (36)

where Q is the matrix defined by the orientation of the ellipsoid(three spherical angles, Jiang, 2007a,b, 2012) and Q is the angularvelocity tensor of the ellipsoid’s axes which is defined by Jiang(2007b, 2012):

Q ¼ wE �wE0(37)

where wE0 is the vorticity in the ellipsoid measured relative to theframe tracking the ellipsoid’s own semi-axes. The components ofwE0 expressed in the ellipsoid’s own axes are (Goddard and Miller,1967; Bilby and Kolbuszewski, 1977; Jiang, 2012):

wE0

ij ¼

8>><>>:

a2i þ a2ja2i � a2j

3Eij; aisaj

wEij; ai ¼ aj

(38)

Mancktelow (2011) regarded wE0 as the internal vorticity in thesense of Astarita (1979) and Means et al. (1980) for the flow in theellipsoid. As the internal vorticity is measured relative to theinstantaneous stretching axes of the strain rate tensor (Astarita,1979; Jiang, 1999, 2010) which for the ellipsoid is the principalaxes of 3

E. The latter generally have a spin component (angularvelocity) relative to the ellipsoid’s semi-axes. Therefore wE0 gener-ally cannot be the internal vorticity for the flow field inside theellipsoid. To obtain the internal vorticity, wE0 must be partitionedusing the method of Jiang (1999, 2010).

The following approximation (e.g., Basar and Weichert, 2000,p. 31) is used here to solve Eqs. (35) and (36) instead of thefourth-order RungeeKutta method by Jiang (2007b, 2012):

aiðt þ dtÞzaiðtÞ$exp�Daxisii dt

�ðno sum over iÞ (39)

and

Q ðtþdtÞzexpð�QdtÞ$Q ðtÞ ¼ ðIþsinu$uþð1�cosuÞ$u2Þ$Q ðtÞ(40)

where I is the second order unit tensor, dt is the step length ofcalculation, u is half the Euclidean norm of the skew symmetrictensor Qdt, and u is defined as:

u ¼ �Qdtu

(41)

The approximations of Eqs. (39) and (40) are computationally

end

start

end

start

end

startba c

Fig. 2. Comparison of the 4th order RungeeKutta approximation used in Jiang (2007a,b, 2012) and the Rodrigues approximation used in this paper for a rigid prolate ellipsoid in

a simple shearing flow. The simple shearing flow is dextral horizontal shearing on an eastewest vertical zone with the velocity gradient tensor being L ¼ 0 1 00 0 00 0 0

!. The shape of

the ellipsoid is 5:1:1. The initial orientation of the long axis is 45� due north. (a) Rotation path of the object from the RungeeKutta approximation. dt ¼ 0.05, 1000 steps of total runand plots are every 10 steps of calculations. (b) Rotation path from the Rodrigues approximation for the same simulation dt, steps and interval between outputs. (c) Rotation pathwith dt reduced to 0.025, total steps 2000 and the interval between outputs 20 steps of calculation. The accuracy in (a) is excellent, the path follows nearly perfectly Jeffery (1922)orbit even when the shear strain reaches 50. In (b) the path drifts only slightly from the perfect Jeffery orbit. The accuracy is further improved in (c) when dt is reduced by half. TheRodrigues approximation is acceptable.


more efficient than the RungeeKutta method. Fig. 2 compares theaccuracy of the approximation here with the RungeeKutta methodfor rotation. The current method greatly improves the computationspeed for the same level of accuracy.

The above algorithm has been implemented in MathCad whichis a user friendly, spreadsheet-like mathematics applicationallowing full interactive calculation and result visualization. Thegeneral principle of the implementation is described in Jiang(2007b, 2012). A set of worksheets necessary for simulating themotion of a single ellipsoid or the motion of a group of non-interacting ellipsoids are provided as online supplementary dataassociated with this paper. An instruction on using the programsis given in Appendix A.

6. Applications and discussion

6.1. Applications of the general Eshelby formalism

The formulation and implementation aforementioned can beused to investigate the motion of an individual deformable elementof general power-law rheology embedded in a general power-lawmaterial. It can be used to track the state of a system of non-interacting deformable elements embedded in a homogeneouspower-law matrix medium. Fig. 3 presents three examples ofsimulation for the history of a single ellipsoid in a given flow. Fig. 4presents one example of simulation for the evolution of a system of300 deformable ellipsoids in a transpressional flow field.

Natural examples where the motion history of a singledeformable element becomes important are the behavior ofdeformable porphyroclasts in mylonites and the partitioned flowfield in a rheologically distinct rock unit, such as a plutonic body,enclosed in an orogen-scale tectonic zone subjected to plate-scaleboundary conditions. For instance the behavior of a porphyro-clastic mica fish in an ultra-mylonite matrix may be studied byregarding the clast as anisotropic power-law viscous inhomoge-neity embedded in an isotropic power-law viscous matrix. Thepartitioned flow field in a rheologically distinct rock body becomesimportant when the body is enclosed in a large deformation zoneand minor structures are developed inside the body. In this case,the rock body can be regarded as an Eshelby inhomogeneity(discussed below) so that Eq. (18) can be used.

The partitioning equations (Eq. (18)) are fully based on micro-mechanics in contrast to some kinematically prescribed

partitioning studies (e.g., Lister and Williams, 1983; Jiang, 1994a,b;Jones et al., 2005; Hudleston, 1999) which considered only straincompatibility and ignored the principle of stress equilibrium(Fletcher and Pollard, 1999; Jiang and Williams, 1999). The kine-matic approach itself cannot go very far because there are aninfinite number of kinematically permissible paths to deforma volume of rock from one configuration to another (e.g., Passchieret al., 2005) and investigating some ad hoc prescribed paths mayhelp to fit some specific field data but does little in advancing ourunderstanding generally. The general Eshelby formalism (Eq. (18))has the potential to address all scenarios of flow partitioning aslong as the tangent linearization of the rheology is valid andtreating individual rock elements as ellipsoidal inhomogeneitiesare reasonable (discussed below).

Where the deformable elements are small compared to the scaleof observation and the concentration of the elements is low, theinteraction among the elements can be neglected. Natural situa-tions of this include matrix-supported conglomerates andporphyroblast-bearing rocks where the clast content is low. Theformulation in this paper can be directly used to track the shapefabric development defined by the elements.

The approach can also be extended to deforming systemscomprising multi-scale elements where large deforming elementscontain fabric-defining elements of much smaller scales. In thiscase, the instantaneous flow field inside the large enclosingelement, determined from the Eshelby formalism, forms theboundary condition dictating the motion of fabric-definingelements enclosed in the large element. Such treatment has thepotential to address multi-order, multi-scale fabric developmentthat is common in Earth’s lithosphere.

6.2. Discussion

As elegant as it is, the general Eshelby formalism (Eq. (18)) isbased on the assumption of an isolated ellipsoidal object embeddedin a rheologically homogeneous medium. Geological elements,from mineral grains in a rock to large rock bodies like plutons ina tectonic belt, are generally irregular 3D bodies embedded ina heterogeneous rock mass. At a first glance, the Eshelby formalismwould appear to have little relevance to natural deformation ofrocks in Earth’s lithosphere. Is it reasonable to regard geologicalelements as ellipsoidal in shape? How is the interaction amongdifferent elements addressed?

b

2

3

4

5

6

7

0 100 200 300 400 500 600 700 800 900 1000

1

2

3

steps

r

g

0 0.5 1 1.50

0.5

1

1.5

2

ln(a2/a3)

ln(a

1/a2

)

c

ln(a2/a3)

ln(a

1/a2

)

a

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

a1

a2

a3

ln(a2/a3)

ln(a

1/a2

)

e

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

f

a1 a2

a3a1

a2a3

d

Fig. 3. Results from three numerical experiments using Single-Clast. The matrix flow field is the same as for Fig. 2. In all three numerical experiments, the initial orientations of theellipsoids are all the same (axis 1: 45�e330� , axis 2: 15� , 225� , and axis 3: 41�, 122�) but the shapes are different. The matrix medium and the ellipsoids are power-law viscous. Thematrix medium stress exponent is Nm ¼ 3 and the ellipsoids’ stress exponent is Nc ¼ 4. The initial effective viscosity ratios are all 2 at the matrix field strain rate. Experiment 1 (a andb) is for an initially prolate object with axial ratios 5:3:3. Experiment 2 (c,d) is for an initially triaxial ellipsoid with axial ratios 5:3:1, and Experiment 3 (e and f) is for an initiallyoblate object with axial ratios 5:5:3. (a) and (b) are respectively the shape and principal axes orientations for Experiment 1. (c) and (d) are for Experiment 2. And e and (f) are forExperiment 3. The green square dots are in the upper hemisphere. (g) The actual ratios of the effective viscosity of the respective ellipsoid to the matrix effective viscosity for thethree experiments (numbers on the curves correspond to the Experiment). Although the assigned initial viscosity ratios of the ellipsoid to the matrix medium are all 2, the actualviscosity ratios vary with ellipsoid initial shapes and with time as the ellipsoids deform and the strain rates inside them vary. dt ¼ 0.0125, total steps of run 1000 (therefore shearstrain is 12.5), and 10 steps of calculations between outputs for all three experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)


These two questions cannot be answered in a simple quantita-tive sense because a closed form mathematic expression forpotential errors arising from irregular shapes and inhomogeneousmatrix rheology is not available. However, it is instructive to reviewsome achievements in materials science and geology where thesimplified treatment has been made that individual elements aretaken as ellipsoids and the entire heterogeneous material asa homogeneous medium. Budiansky and Mangasarian (1960) werethe first to treat individual grains in a polycrystalline material asellipsoidal inhomogeneities embedded in the polycrystalline

medium which is taken to be mechanically homogeneous (Qu andCherkaoui, 2006). Their treatment has been the basis for the wideapplication of the Eshelby formalism in micromechanics to obtainthe macroscopic effective mechanical properties of a material fromthe properties of its constituent elements (Mura, 1987, pp.421e433; Nemat-Nasser and Hori, 1999; Part I). The generalsuccess of this approach is strong evidence that treating mineralgrains as ellipsoidal Eshelby inhomogeneities is physically sound. Inmany theoretical considerations and application, dislocations,stacking faults, cracks, weakened zones, and other discontinuities

a b c

d e f

h

ln( )a /a32

)(nla1

/a2

0 1 2 3 4 50

1

2

3g

ln( )a /a32

)(nla1

/a2

0 1 2 3 4 50

1

2

3

Fig. 4. An example of applying Multi-Clast for a system of 300 deformable ellipsoids. The ellipsoids were initially uniform randomly oriented in 3D. The initial shapes are: long andintermediate axes vary uniformly between 10 and 1, and the short axes are fixed at 1. The initial effective viscosity ratios of the ellipsoids to the matrix medium are between 1 and 5.Both the matrix medium and the ellipsoids are power law viscous and have the same stress exponent of 3. The flow field is an eastewest vertical transpression zone with dextral

sense of shear. The velocity gradient is L ¼ 0 1 00 �0:364 00 0 0:364

!corresponding to a transpression convergence angle of 20� . dt ¼ 0.0125. (a), (b), and (c) are respectively the

orientations of the long, intermediate, and short axes after 120 steps of computation. (d), (e), and (f) the same set of orientations after 240 steps of computation. (g): The shapes ofall 300 ellipsoids plotted on a logarithm Flinn diagram after 120 steps of computation. (h) The shapes of all ellipsoids after 240 steps of computation.


have been treated as Eshelby “ellipsoids” (e.g., Mura, 1987, pp.15e20, 240e379; Rudnicki, 1977; Healy et al., 2006; Exner andDabrowski, 2010). In geology literature, rigid or deformable clastshave commonly been treated as ellipsoidal objects (e.g., Ramsay,1967, pp. 209e221; Dunnet, 1969; Gay, 1968; Ghosh and Ram-berg, 1976; Je�zek et al., 2002). Many physical experiments(e.g., Ferguson, 1979; Arbaret et al., 2001; Ghosh and Ramberg,1976) and theoretical considerations (Willis, 1977) find that therotational behavior of non-ellipsoidal objects is very close to that oftheir best-fit ellipsoids. It should also be pointed out that the shapeof an ellipsoid can be made to vary widely to fit natural objects bychanging the relative length of its three semi-axes from rod-like

(a1 / N, a2 ¼ a3 / 0), to spherical (a1 z a2 z a3), to plane-like(a1 z a2 [ a3 / 0), and many other shapes.

Kröner (1961) and Hill (1965, 1966) proposed the following self-consistent method to address the interactions among inhomoge-neities. Imagine a composite material schematically shown in Fig. 5comprising a groundmass phase and dispersed inhomogeneousphases. Where the concentration of inhomogeneous phases is solow that the inhomogeneities are practically non-interacting, theEshelby formalism (Eq. (18)) applies perfectly and the property ofthe groundmass phase is the property of the matrix mediumsurrounding each and every inhomogeneity. As the volume fractionof the inhomogeneous phase increases, the effect of interactions

0 0.2 0.4 0.6 0.8

2

4

6

8

10

Voigt

Ruess

self-consistent

0

ciFig. 6. HEM viscosity as a function of the concentration of inhomogeneous phases. Theinhomogeneous phases are made of 500 Newtonian grains uniform randomly orientedin 3D space with their long axes varying uniformly between 1 and 10, intermediateaxes between 1 and 10, and the short axes all fixed at 1. The groundmass phase hasrelative viscosity of 1 and the inhomogeneous elements have relative viscositiesbetween 1 and 25, uniform distribution. The viscosity of the HEM can be calculated byan iterative procedure using Eq. (42). Upper bound Voigt average (blue line) and lowerbound Ruess average (green curve) are also shown. (For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version of this article.)

c3

ci

c2

c10

3

i

2

1c0

Fig. 5. A composite material comprising a groundmass phase (white, with viscosity h0and volume concentration c0) and inhomogeneous phases (with their respectivevolume concentrations cr and viscosities hr). Where the inhomogeneous phases aredilute (c0 close to 1), each inhomogeneity can be regarded as an isolated Eshelbyinhomogeneity. When c0 decreases, the presence of inhomogeneous phases influencesthe effective property of the medium surrounding each inhomogeneity. The propertyof the whole ensemble of the material is represented by a “homogeneous effectivemedium” (HEM) whose mechanical property is determined by a self-consistentmethod. See text for more details.


among inhomogeneities may be addressed by considering thecontribution of the inhomogeneities to the mechanical property ofthe matrix medium. The matrix medium surrounding any indi-vidual inhomogeneity is not the groundmass phase alone but isa “homogeneous effective medium” (HEM) which behaves effec-tively as the groundmass and all inhomogeneities combined.Although the properties of the groundmass phase and the inho-mogeneous phases are known, the property of HEM is unknown,a priori, but can be determined self-consistently. Mura (1987, pp.430e433) gave the relationship for the viscosity of the HEM, h, ifall inhomogeneities are isotropic and randomly oriented in space:

h ¼ h0 þXmr¼1

crðhr � h0Þhhþ 2Sr1212ðhr � hÞ (42)

where cr, hr, and Sr1212 (r ¼ 1 to m, m being the total number ofinhomogeneous phases) are respectively the concentration,viscosity, and Eshelby tensor component of the rth inhomogeneousphase; c0 and h0 are the concentration and viscosity of thegroundmass phase. Eq. (42) applies to the situation where thegroundmass phase disappears like in a polycrystalline materialwhen c0 ¼ 0 and h0 is taken as 0 as well. Fig. 6 presents an exampleof using Eq. (42) for a composite material made of 500 grainsuniform randomly oriented in 3D space in a groundmass phase.

With the self-consistent method, Hutchinson (1976) developeda viscoplastic self-consistent (VPSC) model for large plastic defor-mation of polycrystalline materials and related the secant formula-tion to the tangent formulation (Qu and Cherkaoui, 2006, p. 314).Sincehiswork, theVPSCmodelhasbeenadoptedbymanyauthors forstudying large plastic deformation of polycrystalline materials.Molinarietal. (1987)applied this to texture simulation inpolycrystals.

The self-consistent method of Eq. (42) is a first step as it assumesthat the orientations of the inhomogeneities are uniform randomlydistributed in space. Lebensohn and Tomé (1993) developed a fullyself-consistent approach for texture simulation in which the HEMproperty is updated self-consistently every step of computation toaddress the buildup of HEM anisotropy. The VPSC formalism ofLebensohn and Tomé (1993) has been used quite successfully bymany scientists to simulate texture development in metals andcrustal and mantle rocks (e.g., Barber et al., 1994; Tommasi et al.,2009; Wenk et al., 1989, 2011; Wenk and Christie, 1991).

The success of the VPSC approach is strong evidence that thegeneral Eshelby formalism together with the self-consistentmethod forms a powerful and physically sound means to tacklelarge plastic deformation of heterogeneous materials. It is only bya simple analogy that we apply the approach toward naturaldeformation of rocks in Earth’s crust and mantle. Analogous toindividual crystal grains in a polycrystalline material, at any givenscale of observation we may regard each rheologically distinct rockelement as an Eshelby inhomogeneity embedded in a ‘poly-element continuum’ which is the deforming rock mass in the RVEcentered on the element. Central to the Eshelby formalism and self-consistent approach is the assumption that the intra-elementdeformation is homogeneous. This is clearly an approximationand the flow field inside an element from the Eshelby solutionsshould be regarded as the average field in that element. Where theintra-element heterogeneity of deformation becomes significant,the element itself should be considered as an aggregate made ofsub-elements or a more sophisticated approach, like the clusterscheme of Tomé and Canova (1998), may be necessary.

The formulation in this paper covers the general Eshelbyformalism; a complete formulation for the self-consistent approachconsidering the mechanical anisotropy development in the HEM isout of the scope of this paper and will be presented in a separatecontribution.

7. Conclusions

A complete formulation is presented of the Eshelby’s formalismfor general viscous power-law materials. The formulation isimplemented numerically. The approach allows investigation of themotion of individual deformable objects of general viscousrheology in a deforming viscous material. It also enables tracking ofshape fabric development defined by a population of non-interacting deformable viscous objects. The relationship between


the flow field inside an ellipsoidal object and the surroundingbackground flow field can be used to investigate the partitioning offlow into rheologically distinct elements.

To take full benefit of the Eshelby formalism, grains in a poly-crystalline material have been treated as Eshelby inhomogeneitiesembedded in a homogeneous effective medium in materialsscience. The simplified approach has been tested by numerousapplications inmaterials science and is the basis inmicromechanicsfor obtaining the macroscopic properties of heterogeneous mate-rials from properties of their constituent parts. By analogy, at anygiven scale of observationwemay regard rheologically distinct rockelements as Eshelby inhomogeneities embedded in a homogeneouseffectivemediumwhich is the deforming rockmass. This treatmentallows the elegant Eshelby formalism to be applicable but ignoresany intra-element heterogeneity in deformation. Eshelby solutionsrepresent the average deformation in an element where the intra-element deformation is heterogeneous.

To obtain the rheological property of the HEM, the self-consistent method is used. A simple self-consistent method ispresented which assumes that the orientations of the constituentelements are randomly oriented. A fully self-consistent formulationthat considers the evolution of the HEM property will be presentedin a separate contribution.

The general Eshelby formalism together with the self-consistentmethod for determining the evolution of HEM property formsa self-contained and powerful means to tackle heterogeneousprogressive deformations in Earth’s lithosphere.

Acknowledgments

I thank Elena Druguet, Paul Bons, and David Iacopini for criticaleditorial and review comments. Parts of this paper were presentedat the GSA Penrose Conference on “Deformation Localization inRocks: New Advances” held between 27 June and 2 July 2011 in Capde Creus, Spain. I thank the conveners and the organizingcommittee for making the conference a great experience. Discus-sion with participants at the conference was helpful. Support fromCanada’s NSERC and China NSF (40828001) is acknowledged.

Appendix A. Instruction on the use of the MathCad Programs

For more details of the general implementation, the user shouldrefer to Jiang (2007a,b,c, 2012) in addition to this paper. Themacroscopic flow field is given by a velocity gradient tensor, L,which is provided as an input in the form of a 3 � 3 matrix. Theorientation of an ellipsoid is defined by three spherical angles (q1,f1, q2) and its shape by three semi-axis lengths (a1, a2, a3), generallyin the order of (a1 � a2 � a3) for easy recognition. It is possible forthe axes to change their relative lengths in a progressive defor-mation and the ordering of the three axes not in the suggestedorder is acceptable. The initial ratio of effective viscosity of theellipsoid to that of the matrix medium, r0, defined at the strain ratestate of the matrix medium must also be given as input. Thereforethe complete initial state of an ellipsoid, x0, is defined by a 1 � 7column matrix, the transpose of ð q1 f1 q2 a1 a2 a3 r0 Þ.The finite deformation is achieved by successive increments andthe final deformation state is given by STEPS of computations.

There are 8 modular worksheets in the supplementary data.To model the motion of a single ellipsoid, launch Single-

Clast.xmcd. Ensure that reference links to all the modules arecurrent and active. Provide the following input parameters:

L: the bulk flow field velocity gradient tensorx0: the initial state of the ellipsoidNm: the stress exponents for the matrix medium

Nc: the stress exponent for the ellipsoiddt: time step for computationSTEPS: the total steps of computationmm: the number of computation steps between output sets,must be a factor of STEPS

The choice of dt for a simulation must ensure that each step ofthe computation represents an infinitesimal deformation. Thismeans kLijkdt � 1 must be satisfied, where k:k stands for anappropriate norm, such as the Euclidean norm, of a tensor. For ourapproximation (Eqs. (39) and (40)), numerical experiments showthat if kLijkedt < 0:025, the local error is so small that furtherreducing leads to no significant difference in the result after manythousand steps of computation, equivalent to a shear strain of >20for the case of a progressive simple shear.

After input parameters are assigned, evaluate the worksheet.When the computation is completed, the rotation paths for thethree semi-axes are presented separately on an equal-areaprojection. Upper and lower hemispheres projections are repre-sented by different symbols.

To model the motion of a group of ellipsoids, launch Multi-Clast.xmcd. Ensure that the reference link to all the modules iscurrent and active. Provide the required input parameters whichare:

L: the bulk flow field velocity gradient tensorN: the initial state of all ellipsoids. Because each ellipsoidrequires 8 parametersð q1 f1 q2 a1 a2 a3 r0 Nc Þ to define its initial state, fora system of n ellipsoids,N is a n � 8 matrix.Nm: the stress exponents for the matrixdt: time step for computationSTEPS: the total steps of computation

Evaluate the worksheet. When the computation is completed,the shapes of all the ellipsoids are plotted in a Flinn plot and theorientations of the ellipsoids are plotted in three equal-area plotsrespectively for a1-, a2-, and a3-axes.

Appendix B. Supplementary material

Supplementary material associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.jsg.2012.06.011.

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