II Eshelby Tensor of Concave Superspherical Inclusions

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Thèse de DOCTORAT Préparée à L’Université de Lorraine - Laboratoire Georessources –UMR7359

Présentée par Fengjuan CHEN Pour obtenir le grade de Docteur en SCIENCES DE L’UNIVERSITE DE LORRAINE Spécialité Mécanique - Génie Civil

Modélisation Micromécanique de Milieux Poreux Hétérogènes et Applications aux Roches Oolithiques

Ph-D Thesis Micromechanical Modelling of Heterogeneous Porous Materials with Application to Oolitic Rocks

Soutenance publiquement le 24 Octobre 2016 Jury composé de

M.KACHANOV Professeur Tufts University, USA Rapporteur V.KUSHCH Professeur Ukraine National Academy of Science, UA Rapporteur Y.GUEGUEN Professeur Ecole Nationale Supérieure, Paris, France Examinateur I.SEVOSTIANOV Professeur New Mexico State University, USA Examinateur A.GIRAUD Professeur ENSG, Nancy, France Directeur D.GRGIC Maître de conférences ENSG, Nancy, France Co-directeur

For my father, for my mother,

"It's not that I'm so smart, it's just that I stay with problems longer "

Albert Einstein

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Acknowledgement

Considering the challenge, I would not accomplish my Ph.D. without supports from many individuals. My appreciation goes out to them, and I would like to acknowledge some of them here.

I would like to express my most sincere gratitude to my supervisors Prof. Albert Giraud and Dr. Dragan Grgic for their full supports throughout my doctoral study at multiscale hydrogeomechanical group of Georessources laboratory of university of Lorraine. Their constructive scientific suggestions, friendly encouragements and particularly their great patience for a young researcher, supported me to enjoy my research and life in my Ph.D.

I would like to express my appreciation to the committee of my Ph.D. defense, most of them have traveled to France to attend my final defense. Their insightful suggestions and valuable remarks enriched my knowledge and encouraged me for a further exploration.

My gratitude goes to Prof. Mark Kachanov and Prof. Volodymyr Kushch for their constructive report and indications. I’m grateful for the presence of Prof. Yves Guguén as the president of the defense committee.

I would like to thank Prof. Igor Sevostianov. It is my honor for his presence, as well as his kindness supports and helps. As a mentor of my research carrier and life, he has warmly welcomed me several times during my Ph.D. to his team at New Mexico State University. Without his innovative ideas and his programmatic proposals, our works would have no opportunity for publication; also my dissertation would lose a part of knowledge contribution.

I would like also to thank every member of Georessources laboratory for their kindness.

Finally, this dissertation is dedicated to my parents for their permanent love, support and encouragement.

Abstract

Focusing on the effect of shape factor on the overall effective properties of heterogeneous materials, the 1st and the 2nd Eshelby problem related to 3-D non-ellipsoidal inhomogeneities with a specific application to oolitic rocks have been discussed in the current work. Particular attention is focused on concaves shapes such as supersphere and superspheroid. For rocks, they may represent pores or solid mineral materials embbeded in the surrounding rock matrix.

In the 1st Eshelby problem, Eshelby tensor interrelates the resulting strain about inclusion and eigenstrain that would have been experienced inside the inclusion without any external contraire. Calculations of this tensor for superspherical pores– both concave and convex shapes – are performed numerically. Results are given by an integration of derivation of Green’s tensor over volume of the inclusion. Comparisons with the results of Onaka (2001) for convex superspheres show that the performed calculations have an accuracy better than 1%. The current calculations have been done to complete his results.

In the 2nd Eshelby problem, property contribution tensors that characterizes the contribution of an individual inhomogeneity on the overall physical properties have been numerically calculated by using Finite Element Method (FEM). Property contribution tensors of 3D non ellipsoidal inhomogeneities, such as supersphere and superspheroid, have been obtained. Simplified analytical relations have been derived for both compliance contribution tensor and resistivity contribution tensor.

Property contribution tensors have been used to estimate effective elastic properties and effective conductivity of random heterogeneous materials, in the framework of Non-Interaction Approximation, Mori-Tanaka scheme and Maxwell scheme.

Two applications in the field of geomechanics and geophysics have been done. The first application concerns the evaluation of the effective thermal conductivity of oolitic rocks is performed to complete the work of Sevostianov and Giraud (2013) for effective elastic properties. A two step homogenization model has been developed by considering two distinct classes of pores: microporosity (intra oolitic porosity) and meso porosity (inter oolitic porosity). Maxwell homogenization scheme formulated in terms of resistivity contribution tensor has been used for the transition from meso to macroscale. Concave inter oolitic pores of superspherical shape have been taken into account by using resistivity contribution tensor obtained thanks to FEM modelling. Two limiting cases have been considered: ‘dry case’ (air saturated pores) and ‘wet case’ (water liquid saturated pores). Comparisons with experimental data show that

VII variations of effective thermal conductivity with porosity in the most sensitive case of air saturated porosity are correctly reproduced.

Applicability of the replacement relations, initially derived by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It is the second application of newly obtained results on property contribution tensors.

We have considered 3D inhomogeneities of superspherical shape. From the results, it has been seen that these relations are valid only in the convex domain, with an accuracy better than 10%. Replacement relations can not be used in the concave domain for such particular 3D shape.

Keywords: Homogenization, heterogeneous material, inhomogeneity, concave, supersphere, superspheroid, effective elasticity, effective thermal conductivity, cross-property connection.

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Contents

ACKNOWLEDGEMENT ...... V

ABSTRACT ...... VII

LIST OF FIGURES...... XIII

LIST OF TABLES ...... XIX

LIST OF NOTATIONS ...... XXI

RESUME DE LA THESE...... XXV

0.1 Contexte ...... xxvii 0.2 Objectifs de la thèse ...... xxix 0.3 Problèmes d’Eshelby ...... xxx 0.4 Applications aux roches hétérogènes ...... xxxvi 0.5 Conclusions et perspectives ...... xxxviii

PART 1 INTRODUCTION ...... 1

I General framework ...... 3 1.1 Background Context ...... 3 1.2 Fundamental Eshelby problems ...... 9 1.3 Homogenization schemes ...... 11 1.4 Goals of the thesis ...... 11 1.5 Organization of the work ...... 12

PART 2 PROPERTY CONTRIBUTION TENSORS OF MATERIAL CONTAINING 3-D NON- ELLIPSOIDAL INHOMOGENEITIES ...... 13

II Eshelby tensor of concave superspherical inclusions ...... 17

1 INTRODUCTION & BACKGROUND ...... 19 1.1 The first Eshelby problem for ellipsoidal inclusions ...... 19 1.2 Isotropic material containing non-ellipsoidal inclusion ...... 20

2 NUMERICAL EVALUATION OF THE ESHELBY TENSOR FOR A CONCAVE INCLUSION (IJES, 2015A) ...... 23 2.1 Introduction ...... 23

2.2 Average S-tensor for superspherical inclusion ...... 25 2.3 Numerical approach ...... 27 2.4 Discussion ...... 31 2.5 Conclusions ...... 33

3 DISCUSSIONS: EFFECT OF THE SHAPE FACTOR ON THE ELASTIC ENERGY ...... 35 III Property contribution tensor of superspherical pores ...... 37

1 INHOMOGENEITY PROBLEM: BACKGROUND ...... 39 1.1 The second Eshelby problem for ellipsoidal inhomogeneities ...... 39 1.2 Homogenization schemes of effective properties ...... 40 1.3 Isotropic materials containing superspherical inhomogeneities ...... 43

2 EVALUATION OF THE EFFECTIVE ELASTIC AND CONDUCTIVE PROPERTIES OF A MATERIAL CONTAINING CONCAVE

PORE (IJES, 2015B) ...... 45 2.1 Introduction ...... 46 2.2 Property contribution tensors for a superspherical inhomogeneity ...... 47 2.3 Effective properties of a material containing superspherical pores ...... 55 2.4 Cross-property connections for a material containing superspherical pores ...... 58 2.5 Concluding remarks ...... 59

IV PROPERTY CONTRIBUTION TENSOR OF SUPERSPHEROIDAL PORES ...... 63

1 OVERVIEW OF NUMERICAL PROCEDURES ...... 65

2 COMPLIANCE AND RESISTIVITY CONTRIBUTION TENSORS OF AXISYMMETRIC CONCAVE PORES OF

SUPERSPHEROIDAL SHAPE...... 67

(IJES, 2016) ...... 67 2.1 Introduction ...... 68 2.2 Property contribution tensors for a superspheroidal inhomogeneity...... 72 2.2.1 Compliance and resistivity contribution tensors ...... 72 2.3 Analytical approximations for property contribution tensors of a superspheroidal pore ...... 76 2.4 Concluding remarks ...... 82

PART 3 APPLICATIONS TO THE HETEROGENEOUS ROCK LIKE MATERIALS ...... 83

V Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method (IJRMMS, 2015)...... 87

1.1 Introduction ...... 89 1.2 Background results ...... 91 1.3 Microstructure of a reference porous oolitic limestone ...... 93 1.4 A two-scale porosity model for effective thermal conductivity of isotropic porous oolitic rocks ...... 96 1.5 Numerical results ...... 101 1.6 Conclusion ...... 106 VI Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities (IJSS, 2016) ...... 107 2.1 Introduction...... 109 2.2 Compliance and stiffness contribution tensors and replacement relations for ellipsoidal inhomogeneities...... 114 2.3 Calculation of compliance and stiffness contribution tensors and Hill tensor P for a superspherical inhomogeneity ...... 116 2.4 Using replacement relations for calculation effective properties of materials...... 119 2.5 Concluding remarks...... 125

PART 4 CONCLUDING REMARKS AND PERSPECTIVES ...... 127

1.1 Concluding remarks for the mains results ...... 129 1.2 Perspectives ...... 132

REFERENCES ...... 135

APPENDIX ...... 145

Appendix A: analytical S and H tensor of sphere in isotropic case ...... 145 Appendix B: numerical results of superspherical pore by FEM ...... 147 Appendix C: numerical integration on the surface of superspheroid ...... 149 Appendix D: Numerical results of superspheroidal pore by FEM ...... 153 Appendix E: complementary geometrical results related to superspheroidal shapes ..... 157 Appendix F: complementary geometrical results related to superspherical shapes ...... 161

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List of figures

FIG.0.1: OBSERVATIONS AU MEB SUR LE CALCAIRE OOLITIQUE DE LAVOUX (LA PREMIERE IMAGE EN HAUT A GAUCHE CORRESPOND A UNE SECTION POLIE) (GRGIC 2011)...... XXVIII

FIG.0.2: UN MACRO-PORE CONCAVE ILLUSTRE CETTE IMAGE MEB DU CALCAIRE OOLITIQUE DE LAVOUX ...... XXVIII

FIG.0.3: (A).COMPARAISON DES RESULTATS NUMERIQUES ACTUEL DES PORES SUPERSPHERIQUES AVEC CEUX D’ONAKA (2001) POUR LA FORME CONVEXE (B). DEPENDANCE DES TRACES DU TENSEUR DE HILL SUR LE COEFFICIENT DE FORME (CHEN ET AL. 2015A)...... XXXI

FIG.0.4: MODELE NUMERIQUE D’UN MATERIAU SOLIDE CONTENANT UNE INHOMOGENEITE ISOLEE DU TYPE SUPERSPHERE (1/8 DE L’INHOMOGENEITE) (A) ET DU TYPE SUPERSPHEROÏDE (INHOMOGENEITE ENTIERE) (B) ET LEURS CHAMPS DE DEPLACEMENTS...... XXXIII

FIG.0.5: TENSEUR DE SOUPLESSE EVALUEPAR LES DEUX METHODES DE NORMALISATION (A) SUPERSPHERE (B) SUPERSPHEROÏDE (CHEN ET AL.2015 B) ...... XXXIV

FIG.0.6: TENSEUR DE RESISTIVITE EVALUE PAR LES DEUX METHODES DE NORMALISATION (A) SUPERSPHERE (B) SUPERSPHEROÏDE (SEVOSTIANOV ET AL.2016) ...... XXXV

FIG.0.7: POUR CERTAINES FRACTIONS VOLUMIQUES ϕ , DEPENDANCE DU MODULE D’ELASTICITE ISOSTATIQUE ET DU MODULE DE CISAILLEMENT SUR LES PROPRIETES ELASTIQUES DES CALCITES CONTENANT DES PORES VIDES PAR DES SCHEMAS D’HOMOGENEISATIONS DIFFERENTS...... XXXVII

FIG.1.1: METHOD OF MULTIPLE SCALES (MICRO » MINI » MACRO) (GUÉGUEN AND PALCIAUSKAS 1992) ...... 4

FIG.1.2: REAL HETEROGENEOUS MEDIUM (LEFT) AND EFFECTIVE HOMOGENEOUS MEDIUM (RIGHT). (GUÉGUEN AND PALCIAUSKAS 1992) ...... 4

FIG.1.3: ILLUSTRATION OF ELLIPSOIDAL AND NON-ELLIPSOIDAL SHAPES ...... 6

FIG.1.4: LOW MAGNIFICATION SEM IMAGES OF IRON ORE (IRONSTONE) SHOWING THE OOLITIC TEXTURE. CORTEXES ARE MADE BY POROUS AND CONCENTRIC LAYERS OF GOETHITE. (A) CRACK SURFACE. (B) POLISHED THIN SECTION. (GRGIC 2001)...... 7

XIII

FIG.1.5: HIGH MAGNIFICATION SEM IMAGES OF OOLITIC IRON CORE SHOWING CONCAVE INTER- OOLITIC CARBONATE CRYSTALS...... 7

FIG.1.6: HIGH MAGNIFICATION SEM IMAGE OF OOLITIC IRON ORE SHOWING A CONCAVE INTER- OOLITIC CRYSTAL OF SIDERITE. (GRGIC ET AL. 2013) ...... 8

FIG.1.7: LOW AND HIGH MAGNIFICATION SEM OBSERVATIONS ON OOLITIC LAVOUX LIMESTONE (THE FIRST IMAGE CORRESPONDS TO A POLISHED THIN SECTION). (GRGIC 2011) ...... 8

FIG.1.8: SEM IMAGE OF OOLITIC LAVOUX LIMESTONE SHOWING A CONCAVE MACRO-PORE...... 9

FIG.2.1: VOLUME OF THE SUPERSPHERE INDEPENDENT ON THE CONCAVITY PARAMETER P (SUPERSPHERE WITH P=0.35 IS SHOWN IN THE RIGHT) ...... 26

FIG.2.2: SHAPE OF CONVEX SUPERSPHERES AT SEVERAL VALUES OF P (IMAGES PRODUCED BY GOCAD) ...... 27

FIG.2.3: ONE-EIGHT OF SPHERE OF UNIT RADIUS PRODUCED BY GOCAD AND THE MESH USED ...... 28

FIG.2.4: FUNCTION F (P), DEFINED BY (2.8), CALCULATED WITH DIFFERENT MESH DENSITY, COMPARED TO RESULTS OF ONAKA (2001)...... 28

FIG.2.5: COMPARISON OF THE CALCULATED E WITH MESH DENSITY 80 NODES PER EDGE WITH S1212 THE RESULTS OF ONAKA (2001) ...... 29

FIG.2.6: CONCAVE HYPERSPHERES (IMAGES PRODUCED BY GOCAD)...... 30

FIG.2.7: NON-ZERO COMPONENTS OF THE AVERAGE ESHELBY TENSOR FOR DIFFERENT VALUES OF P ...... 30

FIG.2.8: TRACES OF THE HILL TENSOR FOR DIFFERENT VALUES OF P ...... 32

FIG.2.9: NORMALIZED STRAIN ENERGY FOR EIGENSTRAIN OF TYPE A AND B FOR CONCAVE SUPERSPHERICAL PORES, COMPARED WITH THE ONE OF ONAKA (2001) FOR CONVEX RANGE...... 36

FIG.3.1: (A) REMOTELY LOADING AND THE BOUNDARY CONDITIONS FOR ELASTICITY PROBLEMS (B) HEAT FLUX VECTORS AND THE GRADIENT OF TEMPERATURE FOR THERMAL CONDUCTIVITY PROBLEMS...... 44

FIG.3.2: DIFFERENT SUPERSPHERICAL SHAPE AT DIFFERENT VALUES OF P. INSET: DEPENDENCE OF THE SUPERSPHERE VOLUME ON P ...... 49

FIG.3.3: REFERENCE MESH FOR CONCAVE SUPERSPHERE (P=0.35) (1/8 F THE MESHED SUPERSPHERE ON THE LEFT, DETAILS AT THE TOP RIGHT CORNER AND THREE-NODES TRIANGULAR FINITE ELEMENTS ON THE SURFACE AT THE BOTTOM RIGHT CORNER) ...... 50

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FIG.3.4: NON-ZERO COMPONENT OF COMPLIANCE CONTRIBUTION TENSOR FOR DIFFERENT VALUES OF P (FROM 0.2 TO 5) ...... 53

FIG.3.5: LINEAR APPROXIMATION OF THE NON-ZERO COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR NORMALIZED ACCORDING TO (2.32). R-SQUARED VALUES OF THE APPROXIMATIONS ARE INDICATED ...... 53

FIG.3.6: DEPENDENCE OF THE RESISTIVITY CONTRIBUTION TENSOR ON P (A) AND ITS LINEAR APPROXIMATION (B)...... 54

FIG.4.1: BOUNDARY CONDITIONS FOR MODELLING OF MATERIALS CONTAINING CONCAVE SUPERSPHEROIDAL PORES ...... 66

FIG.4.2: TEMPERATURE GRADIENT AND CONVECTION OF HEAT FLUX FOR THERMAL CONDUCTIVITY PROBLEMS ...... 66

FIG.4.3: PORES OF CONCAVE SHAPES TYPICAL IN ROCK MECHANICS: (A) SANDSTONE (FROM EMMANUEL ET AL. 2010); (B) HARZBURGITE (FROM WARK ET AL.2003) ...... 68

FIG.4.4: AXISYMMETRIC SUPERSPHEROID (“SUPEREGG”) FOR DIFFERENT VALUES OF P...... 71

FIG.4.5: (A) TOTAL SURFACE AREAS OF A SUPERSPHEROID OF CONCAVITY P AND OBLATE SPHEROID OF ASPECT RATIO γ NORMALIZED TO THE SURFACE AREA OF THE SUPERSCRIBED UNIT SPHERE: (B) NORMALIZED VOLUMES OF THE SUPERSPHEROID AND SPHEROID ...... 71

FIG.4.6: 3D REFERENCE FINITE ELEMENT MESH FOR CONCAVE SUPEREGG (P=0.3) ...... 74

FIG.4.7: NON-ZERO COMPONENT OF COMPLIANCE CONTRIBUTION TENSOR (A) AND RESISTIVITY CONTRIBUTION TENSOR (B) AS FUNCTIONS OF THE CONCAVITY PARAMETER P ...... 75

FIG.4.8: LINEAR APPROXIMATION OF THE NON-ZERO COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR (A) AND RESISTIVITY CONTRIBUTION TENSOR (B) NORMALIZED ACCORDING TO (4.14) AND (4.15) ...... 79

FIG.4.9: NON-ZERO COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR OF OBLATE SPHEROIDAL PORE USING NORMALIZATION (4.14) ...... 79

FIG.4.10: COMPARISON OF APPROXIMATION (4.23) WITH NUMERICAL VALUES OF NON-ZERO COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR FOR A SUPERSPHERICAL PORE ...... 80

FIG.4.11: (A) DEPENDENCE OF THE RESISTIVITY CONTRIBUTION TENSOR FOR A SPHEROIDAL PORE NORMALIZED ACCORDING TO (4.15). (B) COMPARISON OF THE APPROXIMATION (4.24) WITH NUMERICAL RESULTS ...... 81

FIG.5.1: SCANNING ELECTRON MICROSCOPY (SEM) IMAGES OF LAVOUX LIMESTONE (SEE GIRAUD AND SEVOSTIANOV 2013)...... 94

FIG.5.2: FIRST HOMOGENIZATION STEP: MICROPORES INSIDE OOLITE CORE ARE HOMOGENIZED BY USING SELF-CONSISTENT METHOD (2D REPRESENTATION OF A 3D MICROSTRUCTURE) ...... 97

FIG.5.3: SECOND HOMOGENIZATION STEP: TRANSITION FROM MESOSCALE TO MACROSCALE WITH MAXWELL HOMOGENIZATION METHOD, CASE OF SIMPLIFIED MODEL (ELLIPSOIDAL APPROXIMATION FOR ALL THE PORES)...... 100

FIG.5.4: MESOPOROSITY: ELLIPSOIDAL PORES (B1) AND CONCAVE PORES (B2) (GIRAUD AND SEVOSTIANOV 2013) ...... 100

FIG.5.5: PORES SATURATED WITH AIR: EXPERIMENTAL THERMAL CONDUCTIVITY AS A FUNCTION OF POROSITY, FOR PURE CALCITIC LIMESTONES (ROBERTSON 1988) ...... 101

FIG.5.6: PORES SATURATED WITH WATER LIQUID: EXPERIMENTAL THERMAL CONDUCTIVITY AS A FUNCTION OF POROSITY FOR PURE CALCITIC LIMESTONES (ROBERTSON 1988) ...... 101

FIG.5.7: PORES SATURATED WITH AIR: EFFECTIVE THERMAL AS A FUNCTION OF MESOPOROSITY WITH

B2=RANDOM DISTRIBUTION CRACKS (OBLATE γ = 0.05 ) ...... 103

FIG.5.8: PORES SATURATED WITH AIR: EFFECTIVE THERMAL CONDUCTIVITY AS A FUNCTION OF

MESOPOROSITY WITH B2=RANDOM DISTRIBUTION OF NEEDLES (PROLATE γ = 20 )...... 103

FIG.5.9: PORES SATURATED WITH LIQUID WATER: EFFECTIVE THERMAL CONDUCTIVITY AS A

FUNCTION OF MESOPOROSITY WITH B2=RANDOM DISTRIBUTION OF CRACKS (OBLATE γ = .050 ) .... 103

FIG.5.10: PORES SATURATED WITH LIQUID WATER: EFFECTIVE THERMAL CONDUCTIVITY AS A

FUNCTION OF MESOPOROSITY WITH B2 RANDOM DISTRIBUTION OF NEEDLES (PROLATE γ = 20 ) .... 103

FIG.5.11: PORES SATURATED WITH AIR: EFFECTIVE THERMAL CONDUCTIVITY AS A FUNCTION OF

MESOPOROSITY WITH B2=RANDOM DISTRIBUTION OF SUPERSPHERE ( p = 0.35 ) ...... 105

FIG.5.12: PORES SATURATED WITH LIQUID WATER: EFFECTIVE THERMAL CONDUCTIVITY AS A

FUNCTION OF MESOPOROSITY WITH B2 DISTRIBUTION OF SUPERSPHERES ( p = 0.35 ) ...... 105

FIG.5.13: PORES SATURATED WITH AIR: RESISTIVITY CONTRIBUTION TENSOR OF SPHEROID AS A

SP−air FUNCTION OF PORE ASPECT RATIOγ , COMPARISON WITH SUPERSPHERE ( , H b2 = 0.529 ) ...... 105

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FIG.5.14: PORES SATURATED WITH LIQUID WATER: RESISTIVITY CONTRIBUTION TENSOR OF SPHEROID

AS A FUNCTION OF PORE ASPECT RATIO γ ,COMPARISON WITH SUPERSPHERE ( p = 0.35 ,

SP−lq H b2 = 0.448 ) ...... 105

FIG.6.1: SUPERSPHERICAL INHOMOGENEITY FOR DIFFERENT VALUES OF P ...... 113

FIG.6.2: DEPENDENCES OF F (P) DEFINED BY EQUATION (6.20) (A) AND COMPONENT P1212 OF HILL’S TENSOR GIVEN BY (6.19) (B) ON THE CONCAVITY PARAMETER P ...... 117

FIG.6.3: COMPARISON OF THE COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR CALCULATED BY EQUATION (6.22B) WITH ONES OBTAINED BY CHEN ET AL. (2015B) USING FEM ... 118

FIG.6.4: DEPENDENCE OF THE BULK MODULI OF CALCITE WITH CRUDE OIL FILLED PORES ON THE BULK MODULUS OF DRY CALCITE AT SEVERAL VOLUME FRACTIONS OF PORES CALCULATED BY DIFFERENT METHODS ...... 123

FIG.6.5: DEPENDENCE OF THE BULK AND SHEAR MODULI OF CALCITE WITH CLAY FILLED PORES ON THE ELASTIC PROPERTIES OF DRY CALCITE AT SEVERAL VOLUME FRACTIONS OF PORES CALCULATED BY DIFFERENT METHODS...... 124

FIG.6.6: DEPENDENCE OF THE BULK MODULI OF CALCITE WITH CRUDE OIL FILLED PORES ON THE VOLUME FRACTIONS OF PORES AT SEVERAL VALUES OF THE BULK MODULUS OF DRY CALCITE CALCULATED BY DIFFERENT METHODS ...... 124

FIG.6.7: DEPENDENCE OF THE BULK AND SHEAR MODULI OF CALCITE WITH CLAY FILLED PORES ON THE VOLUME FRACTIONS OF PORES AT SEVERAL VALUES OF THE ELASTIC MODULI OF DRY CALCITE CALCULATED BY DIFFERENT METHODS ...... 125

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List of tables

TABLE B1. CALCULATED COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR AND THE RESISTIVITY CONTRIBUTION TENSOR ...... 147

TABLE B2. NUMBER OF ELEMENTS AND NODES OF T3 MESH ...... 148

TABLE D1. CALCULATED COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR ...... 153

TABLE D2. NUMBER OF ELEMENTS AND NODES OF THE MESH B (MORE REFINED ON SUPERSPHEROID SURFACE) ...... 154

TABLE D3. RELATIVE ERRORS ON COMPONENTS OF COMPLIANCE TENSOR BETWEEN MESH A AND MESH B ...... 155

TABLE D4. COMPONENTS OF RESISTIVITY CONTRIBUTION TENSOR OF SUPERSPHEROID ...... 156

TABLE E1: EXACT VALUES OF DIMENSION LESS VOLUME AND SURFACE AREA OF A SUPERSPHEROID (A =1, ) FOR CONCAVITY PARAMETER ) ...... 157

TABLE F1. COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR FOR A SUPERSPHERICAL PORE:

H ijkl′ CALCULATED BY (3.67) FROM COMPONENTS OF HILL TENSOR AND H ijkl CALCULATED BY FEM 161

TABLE F2. MATERIAL PROPERTIES USED FOR CALCULATION OF THE EFFECTIVE PROPERTIES...... 162

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List of notations

σ Stress vector

ε Strain vector

A,B,C Shape factor p Concavity parameter

γ Aspect ratio a,b,c semi-principal axes of length

* Eigenstrain inside inclusion εij

Ω Domain occupied by inclusion or inhomogeneity

Ω∂ Surface of inclusion or inhomogeneity

S E Eshelby tensor

... Average over volume

Position vector of a point on the surface x = ( x1 ,x2 ,x3 )

Component of resulting strain around inclusion εij

G Green’s function

δ kl Kronecker delta symbol

V* Volume of inclusion or inhomogeneity

V Volume of matrix including inhomogeneity

V0 Volume of the unit sphere

Γ Euler gamma function

ν 0 Poisson’s ratio of matrix

XXI

P ,Q Hill’s tensor

µ , λ Lamé parameters

Stiffness tensor of matrix C0

C1 Stiffness tensor of inhomogeneity

Compliance tensor of matrix S0

S1 Compliance tensor of inhomogeneity

H Compliance contribution tensor

N Stiffness contribution tensor

∆ε Extra strain due to inhomogeneity

σ ∞ Remotely applied stress

∇T Temperature gradient q Heat flux vector

R Resistivity contribution tensor n Inward unit normal vector to the boundary

ex , e y , ez Basis unit normal vectors u Displacement vector

Si Surface area of i-th element

S * The total surface area of the superspheroid

L∗ Certain characteristic length of inhomogeneity

J Fourth-rank unit tensor

G0 Shear modulus of matrix

K0 Bulk modulus of matrix

k0 , λ0 Thermal conductivity of matrix

k1 , λ1 Thermal conductivity of inhomogeneity

XXII

η Geometric parameter of concavity parameter

Function of Poisson’s ratio α k , αG , H K , H G

ϕ , f Porosity

N el Number of finite elements on boundary surface

nG Number of Gauss integration points per elements

i i i Weights and coordinates of Gauss integration points wG ,ξG ,ηG

el j )( hop Function of u0 and n0

L , L1 , L2 Distance from the center of inhomogeneity

g Shape factor in terms of function of the spheroid’s aspect ratio

P, Λ, I Upper case bold symbols refer to second order tensors a,ε,σ Lower case bold symbols refer to first order tensors

I conductivity of the oolite at the mesoscale λ0

λa conductivity of the intra-oolitic porous phase

λ0 conductivity of micritic solid grains

Pi Pressure inside the inhomogeneities produced by external load

τ i Deviatoric stresses inside the inhomogeneities

γ i Deviatoric strains inside the inhomogeneities ei Pressure inside the inhomogeneities produced by the

corresponding volumetric strains

XXIII

XXIV

Résumé de la thèse

XXV

XXVI

0.1 Contexte

Ce travail se place dans le cadre de l’homogénéisation des milieux hétérogènes aléatoires, naturels ou synthétiques.

Les méthodes de changement d’échelle ou d’homogénéisation sont utilisées pour définir des propriétés effectives, propriétés élastiques, conductivité, à une échelle dite "macroscopique" où le matériau hétérogène peut être considéré comme un matériau homogène équivalent. Ces méthodes permettent de remonter au niveau des propriétés effectives les informations microstructurales : propriétés physiques des constituants, effets de forme, taille, et de distribution spatiale etc. Elles seront utilisées dans ce travail, dans le cadre des approches dites par tenseurs d’Eshelby basées sur la solution du problème fondamental d’une inclusion, ou hétérogénéité isolée en milieu infini.

Les applications envisagées concernent principalement les matériaux hétérogènes de type roches poreuses. Les matériaux poreux peuvent d’ailleurs être vus comme un cas particulier de matériaux hétérogènes dont certains composants sont constitués de fluides (air et/ou eau liquide, hydrocarbure etc.).

Les hétérogénéités, ou inhomogénéités (les deux termes sont synonymes dans ce qui suit), représentent aussi bien des composants solides (inclusions minérales solides) que fluides (pores). A titre d’exemple dans le cas des argilites, les hétérogénéités minérales solides peuvent représenter des cristaux de calcite, quartz, etc. noyées dans une matrice argileuse.

On s’intéressera en particulier aux roches carbonatées poreuses, et plus précisément aux calcaires oolithiques, largement étudiées dans le cadre de l'étude de faisabilité du stockage géologique de gaz tels que le dioxyde de carbone CO2 (Sterpenich et al. 2009, Makhloufi et al. 2013, Grgic 2011). Le calcaire oolithique de Lavoux (Nguyen et al. 2011, Giraud et al. 2012) a été choisi en tant que roche de référence pour les applications des modèles micromécaniques analytiques et numériques.

Le calcaire oolithique est principalement composé de calcite et sa microstructure est caractérisée par un ensemble d’oolithes cimentées par des cristaux de calcite syntaxiale (calcite spathique d’une taille de plusieurs centaines de micromètres). L’observation au MEB (microscopie électronique à balayage) de cette roche a montré (1) deux types de pores : micro à l’intérieur des oolites et macro (avec une taille maximale de 100 µm) entre les oolithes (Fig.0.2). Certains macro-pores ont une forme concave. (2) Les oolites sont constituées de

XXVII couches concentriques de micro-calcite (avec une taille moyenne d'environ 3 µm) et leur diamètre moyen est d'environ 400 µm (Fig. 0.1 ) (Grgic 2011).

Fig.0.1: Observations au MEB sur le calcaire oolitique de Lavoux (la première image en haut à gauche correspond à une section polie) (Grgic 2011).

Fig.0.2: Un macro-pore concave illustré cette image MEB du calcaire oolitique de Lavoux

La plupart des méthodes d’homogénéisation des milieux aléatoires utilisées dans le cadre des approches d’Eshelby sont basées sur la solution fondamentale de l’hétérogénéité ou inhomogénéité ellipsoïdale isolée en milieu infini, obtenue par Eshelby (1957, 1961). Par

XXVIII conséquence les inhomogénéités sont généralement supposées ellipsoïdales dans ce type d’approche.

Des extensions aux hétérogénéités isolées de formes non ellipsoïdales ont été étudiées ces dernières décennies, par exemple, les polygones et polyèdres (Rodin 1996), des pores de formes irrégulières de matériaux composites carbone-carbone (Drach et al. 2011), des fissures planes (Fabrikant 1989, Sevostianov et Kachanov 2002), des fissures planes recoupées (Grechka, Vasconselos et Kachanov 2006), des fissures non planes (Mear, Sevostianov et Kachanov 2007) et des fissures traversant les pores (Kachanov et Sevostianov 2012).

L’exemple des roches oolithiques montre l’insuffisance d’une représentation ellipsoidale, et par conséquence convexe, des hétérogénéités.

La représentation ellipsoidale, 3D, est riche dans le sens où inclut de manière exacte ou approchée des géométries aussi différentes que les sphères, sphéroides, les ‘aiguilles’, les cylindres de sections elliptique ou circulaire, les fissures circulaires (et le cas limite ‘penny shaped crack’) ou planes. Une limitation de cette représentation est la convexité de la courbure de sa surface. Les observations de microstructure de roches oolithiques mettent notamment en évidence des formes concaves pour les matériaux de remplissage inter oolithique, que ce soit les pores inter oolithiques ou des remplissages de type calcite syntaxiale.

0.2 Objectifs de la thèse

L'objectif de ce travail est d'étudier les effets de formes d’hétérogénéités tridimensionnelles non ellipsoïdales sur les propriétés effectives de matériaux hétérogènes. Plus précisément nous cherchons à caractériser les paramètres géométriques dont l’influence sur les propriétés effectives est majeure : coefficients de forme (ou rapport d’aspect), concavité ou convexité, etc. De nombreuses études ont été réalisées dans le cas bidimensionnel mais très peu dans le cas tridimensionnel qui nous intéresse. Cela est dû au fait que des méthodes de résolution numérique (éléments finis notamment) doivent être utilisées dans le cas tridimensionnel pour résoudre le problème de l’inhomogénéité isolée en milieu infini, dans le cas des problèmes élastiques ou de diffusion linéaire stationnaire, dès que la forme de l’inhomogénéité est non ellipsoïdale (et concave en particulier).

Une attention particulière est portée dans ce travail aux formes concaves. Comme discutée par Giraud et Sevostianov (2013), la forme concave de pore et de cristaux inter-oolithiques du calcaire oolithique de Lavoux (France) est décrite de manière approchée par l’équation de la

XXIX

2 p 2 p 2 p supersphère x1 + x2 + x3 ≤ 1 avec p le coefficient de forme (Onaka 2001, 2012, Zohdi et Wriggers 2008). Elle est un cas particulier de la forme générale de type superellipsoïde. La supersphère et les tenseurs de contributions associés respectent une symétrie cubique. On ne considèrera que des distributions isotropes d’inhomogénéités supersphériques, ce qui correspond à un matériau effectif (macroscopique) isotrope. L’anisotropie effective n’est pas étudiée dans ce travail.

Une autre forme, également convexe ou concave, le supersphéroïde, décrite par l’équation

p 2 2 2 p ()x1 + x2 + x3 ≤ 1. Cette forme respecte une symétrie de révolution, et les tenseurs de contribution associés respectent la symétrie isotrope transverse (ceci dans le cas où la matrice environnante est isotrope). De manière similaire au cas de la supersphère, on ne considèrera dans ce travail que des distributions isotropes d’inhomogénéités supersphéroidales ce qui, combiné à une isotropie des constituants, conduit à un matériau effectif isotrope.

Les applications envisagées concernent les géomatériaux de type roches hétérogènes. Les tenseurs de contribution élastique sont déterminés numériquement. Sur cette base, les relations simplifiées sont fournies pour l'utilisation de ces résultats dans le cadre des méthodes de type EMA (Effective Medium Approximation) telles que les approximations NIA (Non Interation Approximation), Maxwell, Mori-Tanaka.

0.3 Problèmes d’Eshelby

0.3.1 Le premier problème d’Eshelby pour des inclusions 3-D non ellipsoïdales

J.D. Eshelby a déterminé la solution élastique autour et à l'intérieur d'une inclusion ellipsoïdale isolée dans une matrice élastique isotrope infinie (Eshelby 1957). Dans ce cas précis, à la fois l'inclusion et la matrice sont constituées du même matériau. Ce problème fondamental est connu comme le premier problème d’Eshelby ou problème de l'inclusion. Eshelby a démontré un résultat d'une importance majeure en micromécanique : l'uniformité des champs de contraintes et de déformations autour et à l'intérieur de l'inclusion ellipsoïdale soumise à un chargement uniforme aux limites infinies.

Dans le cas des formes non ellipsoïdales et en particulier pour les formes non ellipsoïdales concaves tridimensionnelles étudiées dans cette thèse, cette uniformité n'est plus vraie. Ces nouveaux résultats sont fournis par des approches numériques dans la gamme concave et

XXX convexe dans le domaine élastique (Chen et al. 2015 a), cependant ils ne sont pas liés aux propriétés effectives.

La fonction de Green exacte (solution de Kelvin en élasticité linéaire isotrope) est intégrée numériquement à la surface de l'inclusion supersphérique afin d’obtenir le tenseur d’Eshelby moyen (Rodin 1996, Mura 1987, Maekenscoff 1998, Onaka 2001). Du fait de la symétrie cubique de la supersphère et des relations importantes proposées par Milgrom and Shtrikman

(1992), une seule composante non nulle doit être évaluée.

0.26 Convex

Results of Onaka 0.25 (2001) for p > 1

0.24 Our calculations

(a) p 0.23 0.1 1.0 10 100

0.045 Traces 0.040

0.035

0.030

0.025 (b) 0.020 p 0.2 1.00 2.00 3.00 4.00 5.00 0

Fig.0.3: (a).Comparaison des résultats numériques actuel des pores supersphériques avec ceux d’Onaka (2001) pour la forme convexe (b). Dépendance des traces du tenseur de Hill sur le coefficient de forme (Chen et al. 2015a).

XXXI

Une comparaison est faite, à titre de validation de la méthode d’intégration numérique développée, avec les résultats d’Onaka (2001) pour la supersphère de forme convexe (ligne en pointillés sur le graphique de la figure 0.3a), et le taux d'erreur est inférieur à 1%. Les calculs actuels sont effectués pour compléter les résultats d’Onaka (2001).

Le tenseur de Hill est lié au tenseur d’Eshelby par une relation linéaire, des nouveaux résultats concernant les traces du tenseur de Hill montrent qu’ils sont fortement dépendants du coefficient de forme (Fig. 0.3b), surtout dans le domaine de concave.

0.3.2 Le deuxième problème d’Eshelby pour des hétérogénéités 3-D non ellipsoïdales

Le deuxième problème d’Eshelby ou le problème de l'inhomogénéité est beaucoup plus important pour les applications aux matériaux hétérogènes. On considère dans ce cas une matrice élastique infinie, contenant une hétérogénéité constituée par un matériau élastique ayant des propriétés différentes, soumise à une charge uniforme (contrainte ou déformation) aux limites infinies.

On peut souligner que les deux problèmes d’Eshelby sont reliés par de simples relations linéaires uniquement dans le cas particulier des inclusions ou des hétérogénéités ellipsoïdales. Cela est dû à la propriété d'uniformité des champs de contrainte et de déformation qui ne vaut que pour les formes ellipsoïdales (c’est la conjecture d’Eshelby, voir parmi de nombreuses références Ammari et al. 2010). En raison de l'équivalence mathématique entre les deux problèmes d'Eshelby pour l'hétérogénéité de forme ellipsoïdale, différents tenseurs caractérisant les interactions élastiques peuvent être utilisés, car ils sont liés les uns aux autres par des relations linéaires : le tenseur d'Eshelby, le tenseur de Hill, le tenseur de rigidité et le tenseur de souplesse (Hill 1965, Walpole 1969, Kunin and Sosnina 1971, Sevostianov and Kachanov 1999, 2002, 2007).

Cependant, l’équivalence n'est plus vraie dans le cas des formes non ellipsoïdales. Il faut alors définir soigneusement les tenseurs propres caractérisant les propriétés élastiques entre l’hétérogénéité et la matrice environnante. Le deuxième problème d’Eshelby est au coeur des travaux de cette thèse.

La méthode des éléments finis est utilisée pour résoudre le problème de l’inhomogénéité du type supersphère ou supersphéroïde, isolé dans une matrice infinie, et obtenir le tenseur de souplesse (Hill 1963, Kachanov et al. 1994). Les résultats sont donnés par l’intégration du produit vectoriel sur le volume de l’hétérogénéité. Pour le problème d’élasticité, il s’agit du

XXXII vecteur de déplacement et du vecteur normal. Les champs de déplacement pour la supersphère et le supersphéroïde sont illustrés sur la figure 0.4.

(a) : supersphère

(b) : supersphéroïd

Fig.0.4: Modèle numérique d’un matériau solide contenant une inhomogénéité isolée du type supersphère (1/8 de l’inhomogénéité) (a) et du type supersphéroïde (inhomogénéité entière) (b) et leurs champs de déplacements.

Deux méthodes de normalisation sont souvent appliquées. Dans le cas où on s’intéresse à l’effet du volume donné sur le coefficient de forme, le tenseur H est formulé en termes de ratio

V* /V où V* est le volume d’hétérogénéité et V est le volume total de la matrice y compris l’hétérogénéité. Autrement, si on s’intéresse à l’effet du coefficient de forme de l’hétérogénéité

XXXIII

3 sur le tenseur, H tenseur est formulé en termes de L* /V où L* est généralement considéré comme la distance la plus longue entre deux indices de l’hétérogénéité.

La première méthode de normalisation n’est pas adaptée au cas limite où le volume de l’inhomogénéité V* tend vers zéro, ce qui inclut en partiulier les fissures.

Les résultats pour la supersphère et le supersphéroïde sont illustrés sur la figure 0.5 (Chen et al. 2015 b).

(a) : supersphère ~ H ijkl H ijkl 8.0 10.0 H1111 ~ 5p −1 ~ S H1111 = H1111 8.0 4 6.0 R2 = 0.998587 H1212

H 6.0 ~ 5p −1 ~ S 1122 H1212 = H1212 4.0 4 R2 = 0.998564 4.0

~ 5p −1 ~ − H −= H S 2.0 1122 4 1122 2.0 2 R = 0.998183 0.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 0.2 0.4 0.6 0.8 1.0 p p

(b) : supersphéroïd

30 ~ 3.0 E0H3333 0HE 3333 0HE 1122 ~ HE HE E0H1313 20 0 1313 0 1133 2.0 HE ~ 0 1111 E0H1111 ~ 10 1.0 E0H1122 ~ E0H1133 0 0

-10 -1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 p p Fig.0.5: Tenseur de souplesse évaluépar les deux méthodes de normalisation (a) supersphère (b) supersphéroïde (Chen et al.2015 b)

XXXIV

Les problèmes linéaires de conductivité/résistivité thermique sont étudiés en utilisant la même méthodologie (Sevostianov et Kachanov 2002). L’intégration est donnée par le produit vectoriel du gradient de température et du vecteur normal. De même, le tenseur de résistivité est également évalué par deux méthodes de normalisation (Sevostianov et al. 2016), les résultats sont illustrés sur la figure 0.6 (Chen et al. 2015 b). L’analyse des invariants du tenseur de contribution à la souplesse H ("compliance contribution tenseur") de la supersphère et du supersphéroïde est également présentée.

(a) : supersphère 3.0 2.0

~ 5p −1 ~S R11 −= R11 2.5 1.5 4 R 2 = 0.9917

2.0 1.0

1.5 0.5

1.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 0.2 0.4 0.6 0.8 1.0 p p

(b) : supersphéroïd

20 2.0 ~ Rk 330 Rk 330 ~ Rk 110 Rk 110 15 1.5

10 1.0

5 0.5

0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 p p Fig.0.6: Tenseur de résistivité évalué par les deux méthodes de normalisation (a) supersphère (b) supersphéroïde (Sevostianov et al.2016)

XXXV

0.4 Applications aux roches hétérogènes

Les résultats théoriques et numériques concernant les tenseurs de résistivité et compliance sont +appliqués aux propriétés effectives de roches hétérogènes, via les méthodes d’homogénéisation des milieux aléatoires. Ces applications envisagées dans ce travail concernent les propriétés effectives des roches hétérogènes, d’une part la conductivité thermique et, d’autre part les propriétés élastiques et poroélastiques.

La conductivité thermique effective des roches oolithiques est discutée en utilisant le schéma de Maxwell récemment reformulé par I. Sevostianov et coauteurs (Sevostianov et Giraud 2013, Sevostianov 2014).

Ces résultats ont fait l’objet d’une publication, Giraud et al. (2015). Le modèle proposé initialement par Giraud et Sevostianov (2013) pour des hétérogénéités ayant une forme concave, par exemple la forme supersphérique, est appliqué dans ce papier. Le schéma de Maxwell est formulé en terme de tenseur de résistivité. Par rapport aux travaux antérieurs sur les matériaux étudiés, les calcaires oolithiques, la nouveauté est la prise en compte de la forme concave et 3D de certains pores inter oolithiques. Les tendances obtenues sont cohérentes avec les données bibliographiques pour la conductivité thermique de ce type de matériau, dans deux cas particuliers : le cas sec (les pores sont saturés d’air) et le cas humide (les pores sont saturés à l’eau liquide). Un point à noter est que le modèle morphologique présenté dans Giraud et Sevostianov (2013) pour l’estimation des propriétés effectives élastiques a été utilisé pour la conductivité thermique.

La seconde application présentée concerne l’extension des relations dites de substitution, établies par Sevostianov et Kachanov (2007), Les applications dans le cas d’hétérogénéités ellipsoïdales isotropes et/ou anisotropes au cas non ellipsoidal. Ces relations concernent des hétérogénéités de même forme mais de paramètres élastiques différents (par exemple pores saturés ou secs et matrice solide). Appliquées aux tenseurs de souplesse, ces relations permettent de définir dans un cadre général anisotrope les relations de compatibilité micro- macro dites de Biot-Gassmann, très importantes en poroélasticité linéaire. La nouveauté est la prise en compte d’hétérogénéités 3D de forme non ellipsoidales et les résultats ont fait l’objet d’un article récemment soumis à IJSS (soumis en mai 2016, en cours d’expertise).

XXXVI

Keff K0 1.0

0.8 φ = 10.0

0.6 φ = 25.0

0.4 φ = 50.0 (a) 0.2 p 0.0 0.2 0.4 0.6 0.8 1.0 K K dry 0

Geff G0 1.0

φ = 10.0 0.8

φ = 25.0 0.6

φ = 50.0 0.4

(b) 0.2 p 0.0 0.2 0.4 0.6 0.8 1.0 G G dry 0

Gassmann1951 CizShapiro2007 NIA MoriTanaka1973

Maxwell1873

Fig.0.7: Pour certaines fractions volumiques ϕ , dépendance du module d’élasticité isostatique et du module de cisaillement sur les propriétés élastiques des calcites contenant des pores vides par des schémas d’homogénéisations différents.

La comparaison des résultats estimés par ces relations avec ceux qui sont fournis par la méthode aux éléments finis montre que ces relations de substitution ne sont utilisables, avec une erreur relative inférieure à 10 %, dans le cas non ellipsoidal et pour des formes de type supersphère, que pour des hétérogénéités de forme convexe. Dans le cas contraire, en particulier dans le domaine concave, l’erreur relative est importante et ces relations ne peuvent être utilisées.

XXXVII

Les applications de ses résultats concernent les propriétés effectives élastiques et poroélastiques de roches réservoirs contenant des hétérogénéités remplies par des argiles ou par des fluides (gaz, eau, huiles, etc.). A titre d’exemple, le module d’incompressibilité et le module de cisaillement estimés par ces relations sont illustrés sur la figure 0.7.

0.5 Conclusions et perspectives

Dans le cadre de l’étude des propriétés effectives des matériaux hétérogènes, le travail présenté concerne des hétérogénéités tridimensionnelles, non ellipsoïdales de formes concave et/ou convexe, telles que supersphère et supersphéroïde.

Les tenseurs de contribution d’une hétérogénéité isolée, contribution à la souplesse d’une part (problème élastique) et à la résistivité d’autre part (problème de diffusion linéaire stationnaire), sont calculés numériquement par éléments finis. Les nouveaux résultats obtenus ont permis de proposer des relations analytiques approchées et utiles dans le cadre des méthodes d’homogénéisation dites EMA (Effective Medium Approximation) : en citera en particulier les schémas NIA (Non Interaction Approximation), Maxwell, Mori-Tanaka.

Concernant les applications aux roches hétérogènes, les propriétés effectives thermiques des roches hétérogènes telles que les calcaires oolitiques ont été evaluées pour compléter l’analyse de Giraud et Sevostianov (2013) dans les problèmes d’élasticité. En outre, les relations de substitution initialement proposées par Sevostianov and Kachanov (2007) ont été élargies aux hétérogénéités non ellipsoïdales .

Un résultat important est que, dans le cas de formes de type supersphère, ces relations simplifiées ne peuvent être utilisées que dans le cas convexe (le seuil limite de validité considéré est de 10 % d’écart relatif).

Parmi les perspectives, on citera l’extension des résultats présentés aux roches isotropes transverses telles que les argilites, marnes ou schistes.

La prise en compte d’hétérogénéités 3D de forme concave telles que celles étudiées dans ce travail, mais dans le cas de matériaux hétérogènes à constituants anisotropes d’une part et à anisotropie effective d’autre part sera une perspective importante pour les roche

XXXVIII

1 Part

Part 1 Introduction

ChaPter 1

I General framework

1.1 Background Context

Many natural substances and man-made materials are heterogeneous, their physical properties depend on volume fraction, distribution, orientation and individual properties of their constituted elements. Oolitic rocks (e.g., limestone, ironstone) are typical examples of such materials.

Heterogeneity of materials can be considered differently depending on the scales of measurement: micro-, mini- and macroscale. At microscale, all components, such as the skeletal portion which is often solid and pores which are generally filled with a liquid or gas of porous materials, can be observed. At miniscale or called as mesoscale, a representative volume element (RVE) that constitutes a sufficiently large, statistically representative pattern of the microstructure should be considered, it is the smallest volume over which a measurement could be acquired that will yield a value representative of the whole. At macroscale, a solution for the posed problem may be found. Method of multiple scales raises a possibility to evaluate macroscopic behaviors of heterogeneous materials via their microstructures (Fig. 1.1). This method can be applied to a medium that are heterogeneous in small scale, but homogeneous in big scale.

In other words, in the context of effective problems, effective properties of such material are shown by an integration of physical parameters over the volume of RVE, the latter parameters could be, for example, loading rate, displacement, temperature, etc. This procedure is called homogenization where the real heterogeneous materials is considered to be equivalent to a homogeneous material with certain effective properties (Fig. 1.2). Homogeneity (statistic) is therefore a necessary condition to apply this homogenization approach.

Ch.1: General framework

Porous carbonate rocks such as oolitic limestones are extensively investigated in the framework of feasibility study of geological storage of residual gases such as carbon dioxide CO2 (Sterpenich et al. 2009, Makhloufi et al. 2013). The Lavoux limestone (Nguyen et al. 2011, Giraud et al. 2012) will be chosen as a reference rock for applications of analytical and numerical micromechanical models.

100 µm 1 mm 1 cm

Fig.1.1: Method of multiple scales (micro » mini » macro) (Guéguen and Palciauskas 1992)

Fig.1.2: Real heterogeneous medium (left) and effective homogeneous medium (right). (Guéguen and Palciauskas 1992)

Effect of shape paramter on the overall properties of materials containing 3-D non-ellipsoidal inhomogeneities will be mainly discussed in this work.

Staring from the simplest case of ellipsoid, its general description with semi-principal axes of length a, b, c is:

x 2 x 2 x 2 1 + 2 + 3 = 1 (1.1) a 2 b2 c 2

Ch.1: General framework

In numerous literatures, inhomogeneities contained in solid materials are typically assumed to be ellipsoids of the same identical aspect ratio, this unrealistic assumption is widely responsible for insufficient linkage between methods of micromechanics and materials science applications. A special case of ellipsoid is described by the following equation with a = b , γ = c/a , namely spheroid:

x 2 x 2 x 2 1 + 2 + 3 = 1 (1.2) a 2 a 2 (aγ )2 It is oblate with aspect ratio γ < 1, prolate withγ > 1 and spherical with γ = 1 , analytical solutions of effective problems related to these shapes are well known (Mura 1987, Nemat Nasser and Hori 1993, Kachanov et al. 2003). In limiting cases, the shape becomes strongly oblate (γ << 1) including perfectly rigid disk and crack-like pore (flat cracks, no-flat cracks, intersected and branched cracks). Otherwise, it becomes strongly prolate (γ >> 1) including perfectly rigid cylindrical fiber and cylindrical pore.

Only limited number of numerical results and approximated estimates are known for 3-D non-ellipsoidal inhomogeneities since the end of 20th century. Several shapes such as polygons and polyhedral (Rodin 1996), pores of irregular shapes of carbon-carbon composite materials (Drach et al. 2011), planar cracks (Fabrikant 1989, Sevostianov and Kachanov 2002), intersecting planar cracks (Grechka, Vasconselos and Kachanov 2006), non-planar cracks (Mear, Sevostianov and Kachanov 2007) and cracks growing from pores (Kachanov and Sevostianov 2012) have been discussed.

Particular attention is paid to concave shapes, the only one that has been discussed in the litteratures is supersphere (Onaka 2001, 2012, Zohdi and Wriggers 2008, Eq.9.2) and it is a particular case of superellipsoid:

x 2 p + x 2 p + x 2 p = 1 1 2 3 (1.3) where p is called concavity parameter. While, another concave shape, namely superspheroid, which respects axisymmetric symmetry describing by:

2 2 p 2 p ()x1 + x2 + x3 = 1 (1.4) The diversity of shape according to equation (1.1-1.4) is shown in Figure 1.3.

Ch.1: General framework

Ellipsoid Non-ellipsoid

x2 y 2 z2 + + = 1 a = 1,b = 60 ,. c = .40 a 2 b2 c2

Special case :

2 ()x2 + y 2  z  +   = 1 a = 1 2   a  γ ⋅ a 

Superspheroid Supersphere

p=0.35 p=0.35

p 2 p 2 p 2 p 2 p γ = 50 (. oblate ) 2 2 ()x1 + x2 + x3 = 1 x1 + x2 + x3 = 1 γ = 2 ( prolate ) γ = 1 ( sphere )

Analytical known Unknown

Fig.1.3: Illustration of ellipsoidal and non-ellipsoidal shapes

Grgic (2001) showed, from SEM (Scanning Electron Microscopy) observations on iron ore (i.e., oolitic ironstone from Lorraine, France), that some macro-pores and inter-oolitic crystallized minerals of carbonate composition (calcite, siderite) have concave shapes (Fig. 1.4). Other SEM observations done in GeoRessources Laboratory show high magnifications images of these concave carbonate crystals (Fig. 1.5). Calcite and siderite fill the pores between oolitic prolate spheroidal grains (oolites). Figure 1.6 shows a SEM image of a very characteristic concave inter-oolitic crystal of siderite between 3 oolites of prolate spheroid shape (Grgic et al. 2013).

Figure 1.7 represents some SEM images of an oolitic limestone (from Lavoux, France). This rock is mainly composed of calcite and its microstructure is characterized by an assembly of (almost spherical) oolites cemented by large syntaxial crystals of calcite (sparry calcite of several hundred µm of size). There are two kinds of pores: micro inside oolites and macro between oolites. Oolites are made up of concentric layers of micro-calcite (with an average size around 3 µm) and their mean diameter is about 400 µm. Generally speaking, the diameter l of

Ch.1: General framework concentric layers are in the range of 100 µm < l < 1000 µm. Figure 1.8 shows a concave macro- pore (with a maximum size of 100 µm) between oolitic grains.

Fig.1.4: Low magnification SEM images of iron ore (ironstone) showing the oolitic texture. Cortexes are made by porous and concentric layers of goethite. (a) Crack surface. (b) Polished thin section. (Grgic 2001).

Fig.1.5: High magnification SEM images of oolitic iron core showing concave inter-oolitic carbonate crystals.

Ch.1: General framework

Fig.1.6: High magnification SEM image of oolitic iron ore showing a concave inter-oolitic crystal of siderite. (Grgic et al. 2013)

Fig.1.7: Low and high magnification SEM observations on oolitic Lavoux limestone (the first image corresponds to a polished thin section). (Grgic 2011)

Ch.1: General framework

Fig.1.8: SEM image of oolitic Lavoux limestone showing a concave macro-pore.

1.2 Fundamental Eshelby problems

In the 1950s, J. D. Eshelby determined the elastic field about and within an isolated ellipsoidal inclusion embedded in an infinite isotropic elastic matrix (see Eshelby 1957). It is worth to emphasize that both inclusion and matrix are constituted by the same material.

This fundamental problem is known as the First Eshelby problem or the Inclusion problem. Eshelby has obtained result of major importance in micromechanics: uniformity of strain and stress fields inside and around ellipsoidal inclusion submitted to remotely uniform loading.

The Second Eshelby problem or the Inhomogeneity problem is much more important for practical applications concerning heterogeneous materials. One considers in this case an infinite elastic matrix, containing an ellipsoidal inhomogeneity constituted by an elastic material with different properties, submitted to uniform remote loading (uniform stress or uniform strain imposed on infinite boundary).

It may be emphasized that the two problems are equivalent only in the particular case of ellipsoidal inclusions or inhomogeneities as it is related to the property of uniformity of stress and strain fields which holds only for ellipsoidal shapes (this is known as Eshelby’s conjecture, see among many others Ammari et al. 2010).

Ch.1: General framework

Concerning rock materials, inhomogeneities may represent pores (elastic material with zero elastic moduli) or solid mineral inhomogeneities (for example inclusions of calcite, quartz etc. in argillaceous rocks) embedded in a surrounding rock matrix.

Due to the mathematical equivalence between the two Eshelby problems for inhomogeneities of ellipsoidal shapes, different tensor characterizing elastic interactions may be used as they are related to each other by linear relations: Eshelby tensor, Hill tensor, stiffness contribution tensor of compliance contribution tensor (Hill 1965, Walpole 1969, Kunin and Sosnina 1971, Sevostianov and Kachanov 1999, 2002, 2007).

However, this is not the case for non-ellipsoidal shapes and in particular for the 3-D non- ellipsoidal shapes investigated in this work. One has to carefully define the proper tensors characterizing elastic between inhomogeneity and the surrounding matrix and the second Eshelby problem will be the focus of the work.

New results will be provided for the first Eshelby problem of superspherical inclusion mainly to complete previous work of Onaka (Onaka 2001) in the convex range but the results (averaged Eshelby tensor), will be not used in homogenization scheme as it would be irrelevant.

For the second Eshelby problem, the proper tensors are the stiffness contribution tensor or the compliance contribution tensors defined by Sevostianov and Kachanov (2011, 2013).

To the author’s knowledge analytical results are only available in the case of ellipsoidal inhomogeneities embedded in an isotropic matrix, or aligned ellipsoidal inhomogeneities aligned in the direction of a transversely isotropic matrix (Withers 1989, Sevostianov et al. 2005, Levin and Markov 2005)

Due to this, semi analytical and numerical method will be used to solve the Eshelby problems related to non-ellipsoidal inclusions and inhomogeneities:

The exact elastic Green function will be numerically integrated on the surface of the superspherical inclusion for the first Eshelby problem to obtain the averaged Eshelby tensor (Rodin 1996, Mura 1987, Maekenscoff 1998, Onaka 2001)

The finite element method will be used to solve the problem of the isolated superspherical and/or superspheroidal inhomogeneity and to obtain the compliance contribution tensor (Hill 1963, Kachanov et al. 1994)

The linear conductivity/resistivity problems will be investigated by using the same methodology (Sevostianov and Kachanov 2002)

Ch.1: General framework

1.3 Homogenization schemes

By neglecting interaction effects, Non-Interaction Approximation (NIA) constitutes the simplest approach for effective problems, taking advantage of its simplicity, the main focus could be addressed to the shape effect of shape factor. Its accuracy holds for low concentration of inhomogeneity. Importance of NIA approach is shown in two aspects: (1) it constitutes the basic building block for other approximation schemes that account interactions by placing non- interacting inhomogeneities into certain "effective environment" (effective matrix or effective field). (2) the explicit cross-property interrelating changes in the elastic and conductive properties due to inhomogeneities that are established in NIA remains efficient at substantial concentrations (since interactions produce similar effects on each of the two properties).

Mori-Tanaka scheme and Maxwell scheme are largely investigated in the case where the interaction affects could be accounted at substantial concentration. Their accuracies are generally appreciated, in particular, Maxwell scheme is considered to be the most accurate homogenization scheme. Interactions are taken into account by the following manners: (1) In Mori-Tanaka scheme, the isolated inhomogeneity is assumed to be subjected to an effective stress field. (2) In Maxwell scheme, the far field generated due to the inhomogeneities is equal to the far field produced by a fictitious domain of certain shape that possesses unknown effective properties.

1.4 Goals of the thesis

The main focus of this work is to investigate the effect of the concavity parameter of 3-D non-ellipsoidal inhomogeneities on the overall effective properties in the framework of Eshelby tensor approach and micromechanics of random heterogeneous media. To the author’s acknowledge this is novel in 3D context as many results are available in 2D case. Property contribution tensors will be determined numerically by using appropriate numerical method such as finite element. On this basis, approximate simplified relations will be provided for use of these results in NIA, Maxwell, Mori-Tanaka approximations.

As previously mentioned, 3D inhomogeneites (solid or fluid filled pores) of concave shapes may be observed in rock materials such as oolitic rocks. Applications of newly calculated property contribution tensors to effective properties of such heterogeneous materials, via relevant homogenization schemes, will be presented. It completes previous known results, for such materials based on ellipsoidal approximation for all inhomogenities shapes.

Ch.1: General framework

1.5 Organization of the work

Four parts are involved in this work. In Part 1, background and the theoretical frameworks are briefly overviewed. In Part 2, property contribution tensor such as compliance contribution tensor in elasticity problems and resistivity contribution tensor in thermal conductivity problems will be carefully discussed. Numerical calculations are performed for superspherical and superspheroidal pores with the pursuit of high precision. Three papers related to Eshelby tensor, compliance contribution tensor, resistivity contribution tensor published from 2015 to 2016 in I.J.E.S will be shown entirely (IJES: International Journal of Engineering Science)

In Part 3, relative applications in the field of rock mechanics and geophysics will be illustrated. Effective thermal conductivity of oolitic rocks is discussed by using the reformulated Maxwell homogenization approach (paper published in International Journal of Rock Mechanics and Mining Sciences (2015)). The accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities is verified (paper submitted to International Journal of Solids and Structures I.J.S.S. April/2016).

To conclude, in Part 4, some essential results that have been done in this work will be reminded and a brief perspective of the future work will be mentioned.

2 Part Part 2 Property contribution tensors of material containing 3-D non-ellipsoidal inhomogeneities

Preface

"Eshelby problems" and their explicit results of uniformity of strain and stress fields around and inside region Ω occupied by an ellipsoidal inclusion or inhomogeneity submitted to a remotely uniform loading at infinity lead to remarkable revolutions in solid mechanics and in micromechanics of the 20th century.

Both inclusion- and inhomogeneity problem are involved in "Eshelby problems. Solution of inclusion problem such as the resulting strain around Ω is interrelated to an natural existing eigenstrain by a fourth-rank Eshelby tensor S E or Hill’s tensor P , whereas solution of inhomogeneity problem are given by property contribution tensor, which turns into by compliance contribution tensor H in elasticity problems or by resistivity contribution tensor R in thermal conductivity problems. If Ω is ellipsoidal, both problems are mathematically equivalent and their solutions are interrelated by linear equations.

Nevertheless, one has to carefully distinguish the differences between the two Eshelby problems for non-ellipsoidal inhomogeneities, in particular for 3-D cases. In the context of effective problems, only property contribution tensors characterizing contribution of individual inhomogeneity on the overall effective properties are involved in homogenization schemes.

ChaPter 2

II Eshelby tensor of concave superspherical inclusions

For determining the elastic fields of homogeneous materials containing non-ellipsoidal regions of the same elastic properties, the first Eshelby problem has been largely investigated. It could be traced back to its original proposition of Eshelby (1957, 1961) who considered only ellipsoidal shapes for isotropic medium.

This chapter is motived by the work of Onaka (2001) who considered a particular 3-D non- ellipsoidal superspherical shape of convex curvature (with concavity parameter 1). The novelty of the presented work concerns the extension to concave domain. As an� ≥example, concave superspherical shape (with < 1) is useful to model filling materials between spheres as it may be observed in oolithic rocks (oolithes are nearly spheroidal or spherical grains). � After a brief introduction of fundamental physics of the first Eshelby problem in section 1, new computational results for concave superspherical pores will be provided in section 2, to end this chapter, discussions about the related issues and some short remarks will be proposed in section 3.

SeCtion 1

1 Introduction & background

1.1 The first Eshelby problem for ellipsoidal inclusions

For a two-phase heterogeneous material which is composed by an elastically homogeneous matrix and a region Ω possessing the same elastic properties withthe surrounding materials, an elastic field is generated around Ω due to an eigenstrain that would have been naturally experienced inside Ω without any external constraint. The said problem named the first Eshelby problem is generally known as "inclusion problem" or "eigenstrain problem ". Region Ω is called inclusion.

For determining the elastic strain and stress around the inclusion, J.D.Eshelby firstly considered the said problem associated with an ellipsoidal inclusion with a uniform eigenstrain (Eshelby 1957, 1961). At the boundary of the inclusion, a strain ε( x) is generated against the direction of eigenstrain ε* for keeping the initial shape of the boundary Ω∂ .Submitting Hooke’s law into equilibrium equation gives an explicit expression of ε( x) :

∂ ∂G ()x − x′ ε x = C 0 ε ∗ mj Vd ′ (2.1) mp () ijkl kl ∫ ∂x p Ω ∂xi′

0 where Cijkl represent stiffness tensor or elastic tensor, notation "0" and 1"" are used to denote the region Ω and the surrounding matrix in the whole work and x ∉ Ω , x′ ∈ Ω . One has to notice that Green tensor G( x − x′) is the key element for solving the problem.

Mura (1987, eq.11.1 -11.14) has derived that the strain field inside an ellipsoidal inclusion is a function independent of position vector x , implying equation (2.1), the strain field is uniform inside the ellipsoidal inclusion. This result was established by Eshelby (1957).

Ch.2-1: Introduction & background

An alternative form of the above expression is given in terms of displacement:

∂G ()x − x′ u x = C 0 ε ∗ mj Vd ′ (2.2) m () ijkl kl ∫ Ω ∂xi′

This reflects the physically presentation of Green tensor: the displacement at position x ∉ Ω generated by body force which is implied at x′ ∈ Ω .

For isotropic material, explicit form of Green function G( x − x′) is expressed in the following form (Mura 1987):

1  xx  G ()x − x′ = ()3 − 4ν δ + ji (2.3) ij  ij 2  16πµ()1 −ν x  x 

′ 2 2 2 Where xi = xi − xi , x = ()()x1 − x1′ + x2 − x2′ + ()x3 − x3′ and δij = 1 if i = j otherwise

δij = 0 .

In the framework of the first Eshelby problem, the resulting strain is interrelated to the eigenstrain by the following relation:

E ∗ ε ij ()()x = Sijkl x ε kl (2.4)

Where the fourth-rank tensor S E is called Eshelby tensor. Combination of equation (2.1) and (2.4) gives explicitly the exact form of Eshelby tensor of ellipsoidal inclusion, by following the divergence theorem, one obtains:

∂ ∂G ()x − x′ ∂ S E ( x) = C 0 mj Vd ′ = C 0 G x − x′ dS (2.5) mpkl ijkl ∫ ijkl ∫ mj () ∂x p Ω ∂xi′ ∂x p Ω∂

1.2 Isotropic material containing non-ellipsoidal inclusion

Ellipsoidal assumption could only be adopted in limited cases and non-ellipsoidal shapes are widely investigated in a large number of case. The current work is motived by the issues related to 3-D non-ellipsoidal inclusions, a particular attention is paid to concave superspherical pores.

One has to notice that the uniformity of strain and stress is uniquely relevant for ellipsoidal shape, while for non-ellipsoidal inclusions, non-constancy appears. The "average value"

Ch.2-1: Introduction & background method is efficient when dealing with such problems. For non-ellipsoidal inclusions, the

1  ∂u ∂u  average strain is expressed as follows by implying its definitionε =  i + j  : ij 2  ∂x ∂x   j i 

1 ∆ε = − ()u ( x)n + u ( x)n dS (2.6) ij ∫Ω i j j i 2V* with unit normal vector n directed inwards the inclusion,  denotes the average value of a physical parameter over volume. Additionally, the following expression can be given by averaging equation (2.4) under Hill’s (1963) condition:

∆ε = S E : ε* = S E : ε* (2.7) Equating (2.6) and (2.7) yields the explicit form of average Eshelby tensor. For uniform eigenstrain, one obtains:

E 1 Sijkl = − ()ui ( x)n j + u j ( x)ni dS (2.8) * ∫Ω 2Vε kl

SeCtion 2

2 Numerical evaluation of the Eshelby tensor for a concave inclusion (IJES, 2015a)

F.CHEN, A.Giraud, I.Sevostianov, D.Grgic, Numerical evaluation of the Eshelby tensor for a concave superspherical inclusion, International Journal of Engineering Science, 93 (2015) 51- 58

Abstract

We calculate Eshelby tensor for inclusions of non-ellipsoidal shape. We focus on the

2 p 2 p 2 p superspherical shape described by equation x1 + x2 + x3 ≤ 1. It is convex when p > 5.0 and concave when p < 5.0 . We propose a numerical approach to perform integration on the surface of the superspherical inclusion necessary to compute the average Eshelby tensor. Validation of the method is done by comparison of the results with analytical solutions for a spherical inclusion ( p = 1) and with numerical results of Onaka (2001) ( p > 1).

2.1 Introduction

We discuss the first Eshelby problem for an inclusion (eigenstrain) of a concave shape, focusing on the case of a superspherical inclusion. Note that the term "Eshelby problem" actually covers two problems (known as the first and the second problems):

I. Inclusion problem. It is also called eigenstrain or transformation problem. Region Ω in an

* infinite linear elastic solid, called inclusion, undergoes prescribed eigenstrain εij which is not

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion necessarily uniform in Ω (although is was assumed uniform in the original work of Eshelby 1957). The eigenstrain is small strain that would have existed in Ω without the constraint of

* the surrounding. The presence of εij may be due to inelastic deformation, thermal expansion, phase transformation, etc. It is usually known and treated as a prescribed quantity. Region Ω is then deformed elastically into the initial shape and inserted back.

II. Inhomogeneity problem. Region Ω in the infinite linear elastic solid called inhomogeneity has elastic properties that are different from the ones of the surrounding solid. The latter is subjected to remotely applied loading. If Ω has the ellipsoidal shape, analysis simplifies considerably.

In both problems, of interest are the resulting stresses and strains (inside and outside Ω and stress concentration on the boundary). Actually, both Eshelby problem can be modeled by certain distribution of fictitious body force. Problem I – the Inclusion problem – is generally simpler: its solution is given in terms of an integral, over the region Ω , of derivatives of Green’s tensor, whereas Problem II – the Inhomogeneity problem – leads to an integral equation. If the domain Ω is ellipsoidal, the two problems are mathematically equivalent (otherwise they are not) and their solution is given in closed form, in terms of elliptic functions (that reduce to elementary functions if Ω has the spheroidal shape).

Eshelby (1957) considered the case of the isotropic material and uniform eigenstrain. Later contributions extended his results to anisotropic materials (Kinoshita and Mura 1971, Mura 1987), non-uniform loading (Asaro and Barnett 1975), and non-linear behavior of an inhomogeneity (Lyssiak and Sevostianov 1997). Eshelby problems involving non-ellipsoidal inclusions and inhomogeneities are more complex than the ones for the ellipsoidal shape: stresses and strains produced by uniform applied loads are not constant inside Ω , and there is no connection between the inclusion and the inhomogeneity problems that would be exact. Generally, calculation of the Eshelby tensor S E reduce to Taylor series approximations or to numerical evaluation of the harmonic and bi-harmonic potentials φ and ϕ and their derivatives.

Non-constancy of strains inside a non-ellipsoidal domain Ω gives rise to the concept of their average values over Ω . The idea was formulated by Rodin (1996) in the context of solving of the first Eshelby problem for polygons and polyhedral. Later the similar approach was used by Onaka (2001) and Onaka et al. (2002) who considered convex supersphere ( p > 1) and toroidal

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion inclusions respectively. To the best of our knowledge, no results have been obtained for Eshelby problem for inclusions of concave shape. This case is the focus of our paper.

2.2 Average S-tensor for superspherical inclusion

E * Eshelby tensor Sijkl relates eigenstrain ε ij to the resulting strain ε ij inside the inclusion:

E * ε ij = Sijklε ij (2. 9)

For non-ellipsoidal inclusions, this tensor is non-uniform and, following Rodin (1996) we

E E consider its average. Rodin has mentioned that the traces Siijj and Sijij (summation over , ji )

E are independent of the inclusion shape. Eshelby tensor Sijkl reflects the geometric symmetry of the inclusion’s shape

An important connection between components of Eshelby tensor has been derived by

E Milgrom and Shtrikman (1992). They showed that invariants of average Eshelby tensor Sijij

E and Siijj are independent of the inhomogeneity shape and therefore can be calculated from ones for sphere:

E 1 +ν 0 E Siijj = ,Sijij = 3 (2.10) 1 −ν 0 In the present work, we focus on a superspherical inclusion of unit radius described by:

2 p 2 p 2 p x1 + x2 + x3 = 1 (2.11)

Onaka (2012) reported that this equation was probably first proposed by Lamé. Parameter p < 5.0 controls the extent of concavity and will be called concavity factor. When p = 1, the superspherical inclusion becomes a sphere. It becomes convex when p > 5.0 and concave when p < 5.0 (the case of interest here). The volume changes with p according to Onaka (2001) (Fig.2.1):

2 []Γ()21 p 3 V = (2.12) * 3p2 Γ()23 p

∞ − xt −1 where Γ( x) = te dt , V* varies strongly. ∫0

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

E For a superspherical inclusion, Eshelby tensor Sijkl exhibits the cubic symmetry (Onaka

2001):

E E E S1111 = S2222 = S3333 E E E (2.13) S1212 = S2323 = S3131

E E E E E E S1122 = S2233 = S3311 = S2211 = S3322 = S1133 In this case, Milgrom and Shtrikman relations (Eq. (2.10)) take the form:

E E S1111 + 2 S1212 = 1 (2.14) E E 1 +ν 0 3 S1111 + 6 S1122 = 1 −ν 0 where ν 0 is Poisson’s ratio of the matrix. So that there is only one independent component of Eshelby tensor in this case. In addition,

1 f ( p) E (2.15) S1212 = + 3 8π(1 −ν 0 ) with

1   xx   21 ′ (2.16) f ( p) =   3 1 Sdn n2dS V ∫ΩΩ ∫ x  *   1  

Eq. (2.16) expresses the dependence of the average Eshelby tensor on parameter p.

V* ( p) 10

10-1

10-3

10 -5 0.1 1.0 10 100 p

Fig.2.1: Volume of the supersphere independent on the concavity parameter p (supersphere with p=0.35 is shown in the right)

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

2.3 Numerical approach

2.3.1 Average S-tensor for convex supersphere

In the x1 − x2 − x3 Cartesian coordinate system, a convex superspherical inclusion of unit radius can be described by the following equation:

2 p 2 p 2 p x1 + x2 + x3 ≤ 1 ( p ≥ 0.5 ) (2.17) All shapes between an octahedron and a cube can be described as superspheres with different shape factors p (Fig. 2.2). Due to geometric symmetry, only 1 8 of the real shape

( x1 > 0, x2 > 0, x3 > 0) has to be considered for calculation of Eshelby tensor.

p=0.5 p=1

p=3 p=10

Fig.2.2: Shape of convex superspheres at several values of p (Images produced by gOcad)

In order to validate the proposed method, a sphere of unit radius is used as a checkpoint. 1 8 of a sphere of unit radius produced by gOcad in shown in Figure 2.3. 15291 nodes and 30,103 triangles are generated in the model.

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

E E E E For a spherical inclusion, the exact values S1111 = S1111 = 0.523177 , S1122 = S1122 =

E E 0.047935 , S1212 = S1212 = 0.238411 (see appendix A for details). Comparison of the calculations with these values gives the relative computational error between 0.13% and 0.66%, with accuracy of order of 10−3 .

Fig.2.3: One-eight of sphere of unit radius produced by gOcad and the mesh used

Further comparison has been done with the results of Onaka (2001) for convex superspheres (dashed line in Figs. 2.4 and 2.5), the accuracy is better than 1% in all cases.

f ( p)

-1.3 Convex shape

-1.4 Results of Onaka (2001) for p > 1

-1.5 Mesh density 30 nodes per edge

-1.6 Mesh density 50 nodes per edge -1.7 Mesh density 80 nodes per edge -1.8 0.1 1 10 100 p

Fig.2.4: Function f (p), defined by (2.8), calculated with different mesh density, compared to results of Onaka (2001).

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

E S1212 0.26 Convex shape

Results of Onaka (2001) for p > 1

0.25

Our calculations 0.24

0.23 0.1 1 10 100 p

E Fig.2.5: Comparison of the calculated S1212 with mesh density 80 nodes per edge with the results of Onaka (2001)

2.3.2 Average S-tensor for a concave inclusion problem ( p < 0.5 )

Figure 2.6 illustrates the change of the concave supersphere from octahedron to three mutually perpendicular needle-type branches.

When the shape parameter becomes small, for example p = 0.1 , the cross product and gradient method for calculating surface area and normal vector are no longer applicable because of unacceptable numerical error. Hence we consider only the shape parameter changes from p = 0.35 to p = 0.2 .

E Figure 2.4 and 2.5 illustrate dependence of f ( p) defined by Eq. (2.42) and S1212 on the concavity parameter p . It is seen that generally dependence of the Eshelby tensor on p is not

E very strong (as p varies from 0.1 to 100, S1212 for example is in the interval between 0.23 and 0.26), Figure 2.7 illustrates the dependence of the non-zero components of Eshelby tensor for different values of p .

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

p=0.5 p=0.35

p=0.2 p=0.1

Fig.2.6: Concave hyperspheres (images produced by gOcad).

0.6

0.5 E S1212 0.4 E S1111 0.3

E S1122 0.2

0.1

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig.2.7: Non-zero components of the average Eshelby tensor for different values of p

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

2.4 Discussion

* In the first Eshelby problem with εij being generally non-uniform in Ω , the far-field

* asymptotics is independent of the shape of Ω and of the distribution of ε kl within Ω . It is determined by the product:

* * ε kl ( x′)dx′ = ε kl V* ∫Ω (2.18) Indeed, by using Green’s function and divergence theorem, the displacement both inside Ω and outside Ω takes the form:

∂Gmj ( x − x′) * um( x) = Cijkl ε kl ( x′) Vd ′ ∫Ω ∂x′ i (2.19) implying

∂ ∂Gmj ( x − x′) * ε mn( x) = Cijkl ε kl ( x′) Vd ′ ∂x ∫Ω ∂x′ n i (2.20) In the far-field asymptotics, the distance ( x − x′) between the point of observation and points of Ω is much larger than linear dimensions of Ω . Then

0 ∂Gmj ( x − x′) * ∂Gmj ( x − x ) * ε kl ( x′) Vd ′ ≈ − ε kl ( x′) Vd ′ ∫Ω ∂x′ ∂x ∫Ω i i (2.21) where x0 is an arbitrary point inside Ω . Since

* * ε kl ( x′)dx′ = ε kl V* ∫Ω the far displacement field is determined by the average eigenstrain in Ω , but not by the shape of Ω . The shape-independence allows one to derive the general expression for the far-field of an inclusion of any shape, by considering for example, the spherical inclusion (Sevostianov and Kachanov 2011):

1 V* * * * ε ij ( x) = 3 {(1 − 2ν 0 )(2ε ij − ε kkδ ij ) + 3(1 − 2ν 0 )ε kkα α ji 8π(1 −ν 0 ) r (2.22) * * * * + 6ν 0 (ε ilα α jl + ε jlα α li ) + 3ε klα αlk δ ij −15ε klα α α αlkji } This equation constitutes general expression for far-field asymptotic of an inclusion of any

* * shape. In particular, the far strain field due to uniform hydrostatic eigenstrain ε kl = ε δ kl in volume Ω is expressed by:

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

(2.23) that coincides with the strain field of the center of expansion of intensity

(2.24)

0.045 Traces 0.040

0.035

0.030

0.025

0.020 p 0.20 1.00 2.00 3.00 4.00 5.00

Fig.2.8: Traces of the Hill tensor for different values of p

As mentioned in the Introduction, the traces and (summation over indices ) are independent of the inclusion shape. Note that the traces and of the Hill tensor that is related to Eshelby tensor :

(2.25) is shape-dependent (Fig. 2.8). It can be explained on example of isotropic material, for which the stiffness tensor is:

and

(2.26)

and, therefore traces and are not proportional to correspondent traces of Eshelby tensor.

Ch.2-2: Numerical evaluation of the Eshelby tensor for a concave inclusion

2.5 Conclusions

We calculated Eshelby tensor of a superspherical inclusion focusing on concave shapes. We developed a numerical approach to perform integration on the surface of the inclusion. We

E E showed in particular that traces Siijj and Sijij are shape independent while the same traces

of Hill’s tensor Piijj and Pijij show mild dependence on the inclusion shape. We have to note that Eshelby tensor is related to the first Eshelby problem only and it can be used to evaluate the elastic field inside and around the inclusion due to the presence of eigenstrain. Eshelby tensor cannot used to calculate effective properties of heterogeneous materials (as is sometimes erroneously stated in literature)

Acknowledgments

The author appreciate the supports of gOcad research team of Lorraine University, and also gratefully acknowledge the support from the FP7 Project TAMER IRSES-GA-2013-610547.

SeCtion 3

3 Discussions: effect of the shape factor on the elastic energy

Elastic strain energy Es of a material containing Ω with a uniform eigenstrain is derived by Mura (1987):

1 * 1 * Es −= εij σ ij dx −= V*εij σ ij (2.27) 2 ∫Ω 2

For a superspherical pore:

0 E * * σ ij = Cijkl ()Sklmn ε mn − ε kl (2.28)

Submitting (2.28) into (2.27) yields the elastic strain energy of a material containing a superspherical pore:

1 * 0 E * * E = − V*ε []C ()S ε − ε (2.29) S 2 ij ijkl klmn mn kl

To evaluate the effect of the shape factor on the strain energy of such material, we consider normal strain energy

E E = s (2.30) normal * 2 V* ()ε

* * * * by assuming two purely deviatoric eigenstrains, denoted by A ( ε12 = ε 21 = ε A and other εij = 0 )

ε * and B (ε * = ε * = − B ,ε * = ε * and other ε * = 0 ) (Onaka 2001). 11 22 2 33 B ij

Explicit expression of the normal strain energy can be obtained by submitting equation (2.29) into (2.30). For eigenstrain of type A and B, we have:

A A ES E Enormal = * 2 = 2µ ()1 − 2 S1212 (2.31) V* (ε A )

Ch.2-3: Discussions & concluding remarks

E B  9 1 − 2ν  E B = S = 2µ  S E − 0  (2.32) normal * 2  1212  V* (ε B )  4 (14 −ν 0 ) 

Newly obtained results on Eshelby tensor of a superspherical pore can be used to calculate the normal strain energy, the dependencies of the strain energy on shape factor – concavity describing derivation of the shape from a sphere - are illustrated in Figure 2.9. One observes that, when shear by eigenstrain occurs on parallel to flat surface of inclusion (type A), dependency of the strain energy on the shape factor monotonically increases for an inclusion of concave shape and decreases for an inclusion of convex shape; when shear by eigenstrain occurs along plains inclined to flat surface of inclusion (type B), diametrically opposed results are obtained. For a spherical inclusion, the strain energy is independent from eigenstrain.

1.15

1.10 A sphere Enormal /Enormal

1.05 Type B B sphere Enormal /Enormal 1.00

Type A

0.95 Results of Onaka (2001) for p > 1 0.90 0.2 0.5 1.0 2.0 3.0 4.0 5.0 p

Fig.2.9: Dependency of the normal strain energy on shape factor for a material containing a superspherical pore undergoes eigenstrain of type A and B

ChaPter 3

III Property contribution tensor of superspherical pores

In the framework of the second Eshelby problem, the determination of the property contribution tensors of 3-D non-ellipsoidal inhomogeneity is the central interest in the issues related to effective problems. Supersphere, as stated in the previous chapter, will be considered as the shape of randomly oriented inhomogeneities contained in an elastic extended solid.

Introduction of inhomogeneities into homogeneous materials produces extra strain in elasticity problems and extra temperature gradient in thermal conductivity problems, property contribution tensor turns respectively into compliance contribution tensor H and resistivity contribution tensor R .

This chapter is motived by the work of Sevostianov and Giraud (2012) who have derived approximatively analytical expression for the components of compliance contribution tensor of elasticity problems. Nevertheless, the accuracy of the latter need to be improved since they are built on some partly unstable numerical results of Sevostianov, Kachanov and Zohdi (2008). Moreover, another goal of this chapter is to extend the current calculations to conductivity problems.

Semi-analytical approaches are carried out for evaluating property contribution tensors for both elasticity and conductivity problems and they are illustrated in section 2. As usual, this chapter will be started by the background introduction and be ended by discussions and concluding remarks.

SeCtion 1

1 Inhomogeneity problem: background

1.1 The second Eshelby problem for ellipsoidal inhomogeneities

Similar to the first Eshelby problem excluding the two-phase composite is submitted to a remotely applied uniform loading at infinity, and region Ω namelynhomogeneity i possesses a contractive elastic properties with the solid matrix. This problems is called the second Eshelby problem or "inhomogeneity problem".

Elastic deformations occurs on the boundary of inhomogeneity due to the presence of the said loading. Extra strain or stress due to the appearance of inhomogeneity is related to the said deformation by certain equation of integral over V* . The influences caused by the introduction of the inhomogeneities on the overall physical properties are the main concerns of the current work.

Thus, the second Eshelby problem is highly important, in particular for applications related to effective properties of heterogeneous rocks materials (limestone oolitic, calcite with pores filled by clay, crude oil, etc…). Elastic tensors such as compliance contribution tensor H or stiffness contribution tensor N - its dual version - are used in this problem to characterizing the contribution of the individual inhomogeneity on the effective properties.

For ellipsoidal inhomogeneities, ascribe to the mathematical equivalence of the two Eshelby problems, characterizing elastic tensors associated with the two problems are interconnected one with another by linear relations due to the uniformity of strain and stress inside Ω, H and N are interrelated by the following relations (Sevostianov and Kachanov 2007):

H = −S0 : N : S0 or equivalently, N = −C0 : H : C0 (3.1)

Where S0 is the compliance tensor of the matrix. While the stiffness contribution tensor is related to the Hill’s tensor by a linear relation:

Ch.3-1: Inhomogeneity problem: background

−1 −1 N = [](C1 − C0 ) + P (3.2)

The same concern occurs for conductivity problems, the extra temperature gradient is distributed over V* due to the introduction of heat flux, characterizing conductive tensors such as resistivity contribution tensor R and conductivity contribution tensor K - its dual version - is interconnected by the following linear relations:

2 −1 C −1 2 −1 C −1 K = −k0 R = []()k1 − k0 + P , or equivalently, R = −r0 K = []()r1 − r0 + Q (3.3)

Where P c and Q c represent the first and the second Hill’s tensor in conductivity problem. k and r denote second-rank conductivity and resistivity tensors.

These characterizing tensors are generally qualified as property contribution tensor. Kachanov and Sevostianov (2013) mentioned in their work that the summation of property contribution tensors of all inhomogeneities contained in materials gives the microstructural parameter and the effective properties are function of such parameter. Thus, the prior importance of the identification of H and R appear. N and K tensor could be obtained by using expressions (3.1) and (3.3).

For elasticity problems, two different normalization methods are provided with different focuses. As discussed by Sevostianov et al. (2008), if the posed-question is the evaluation of the changes of a given volume V* on shape factor, H tensor should be scaled by V* /V . where

V* represents volume of inhomogeneity and V is the total volume of matrix including the

inhomogeneity. The effect of volume V* is emphasized at strong concavity. Meanwhile, another normalization method which gives generally linear approximations is as well largely used, it raises the possibility to evaluate the changes of H tensor as the inhomogeneity shape

3 is changed in certain specific way, H tensor is stated by L* /V where L* is taken as the longest distance between two indices of the inhomogeneity. For conductivity problems, R tensor could be evaluated by the same principle.

1.2 Homogenization schemes of effective properties

The problems related to effective properties have been discussed since a long time in the context of several physical properties. Sevostianov et al. (2005) have discussed the general effective elastic properties for transversely-isotropic materials containing 3-D non ellipsoidal

Ch.3-1: Inhomogeneity problem: background inhomogeneities of diverse shapes such as strongly oblate spheroidal, strongly prolate spheroidal and spherical inhomogeneity by using single inhomogeneity approach. Wang et al. (2006) have developed a unifying equations for five fundamental effective thermal conductivity structural models by using some rules based on structure volume fractions for heterogeneous materials, Giraud et al. (2012) have discussed effective poroelastic coefficient of isotropic limestone oolitic rocks (Lavoux) in the framework of Hashin Composite Sphere Assemblage (CSA, Hashin 1962) and self-consistent models. Observing from SEM images (Grgic 2011) of the said rock shows that the pores between oolitic prolate spheroidal grains are concaves, their shape can be approximately described by supersphere (Giraud and Sevostianov 2013).

For the aim of evaluating effective properties of homogeneous materials containing inhomogeneities of diverse shapes, in particular for the studied 3-D non-ellipsoidal shapes, several homogenization schemas are provided, they could be divided into two categories:

At limited concentration of smallness, Non-interaction Approximations (NIA) is largely used due to its simplicity. Sevostianov and Kachanov (2013) have discussed effective properties of heterogeneous materials and have pointed out that the effect of shape factor could be focusely studied by using the said method, other approximations that can account the interactions of inhomogeneities could be expressed on the base of the NIA method.

Two different ways for accounting the interactions are mentioned in the paper of Sevostianov et al. (2005): effective matrix method or effective field method. Some largely used approximations such as Mori-Tanaka scheme, Maxwell scheme are established by using the latter method. The interactions are taken into account by placing the "isolated" inhomogeneities of Non-Interaction Approximation into some matrix with certain effective properties.

Classical bounding techniques for determining the upper bound and lower bound for effective constants are reviewed briefly in the framework of NIA, Mori-Tanaka and Maxwell scheme.

For a general two-phase heterogeneous materials containing matrix and inhomogeneity with bulk modulus K or shear modulus G and volume fraction ϕ (ϕ0 + ϕ1 = 1), the effective bulk and shear modulus given by Voigt (1889) and Reuss (1929) bounds are expressed as follows:

K K K R = 10 ≤ K ≤ ϕ K + ϕ K = K V eff ϕ K + ϕ K eff 00 11 eff 01 10 (3.4) R G G10 V Geff = ≤ Geff ≤ ϕ G00 + ϕ G11 = Geff ϕ G01 + ϕ G10

Ch.3-1: Inhomogeneity problem: background

These bounds are rather elementary and they show only the extremes limits of effective values. While narrower Hashin-Shtrikman bounds proposed by Hashin-Shtrikman (1963) is expressed as:

ϕ ϕ k + 0 ≤ k ≤ k + 1 (3.5) 1 ϕ 1 eff 0 ϕ 1 1 + 0 + 3k1 k0 − k1 3k0 k1 − k0

With k0 ≥ k1 . In the contrary case, the inequalities should be reversed.

With Non-Interaction Approximtion, summation of compliance- or stiffness contributions tensors of inhomogeneities yields the effective bulk and shear modulus for elastic properties of an isotopic matrix with randomly oriented inhomogeneities of identical shape:

K0 G0 Keff = , Geff = (3.6) 1 +ϕ 1 A 1 + ϕ1B

For conductivity problems, similar expression is given for effective conductivity modulus:

k k = 0 (3.7) eff 1 + ϕ C where A , B and C are shape factors.

With Mori-Tanaka scheme, the effective field acting on each inhomogeneity is equal to the average over the matrix, one obtains the effective bulk and shear modulus for elastic properties:

K G K = 0 , G = 0 (3.8) eff ϕ eff ϕ 1 + 1 A 1 + 1 B 1 − ϕ1 1 − ϕ1

and the effective conductivity modulus for conductivity problems:

k k = 0 (3.9) eff ϕ 1 + 1 C 1 − ϕ1

When it comes to the matter of Maxwell scheme which is probably the most accurate one compared with all the other approximation methods, the effective bulk and shear modulus for material containing randomly oriented spheroidal pores of identical aspect ratios are illustrated:

K 1 − pBϕ G 1 − pCϕ eff = K , eff = G (3.10) K0 1 + pB[]1 − ϕK G0 1 + pC[]1 − ϕG

Ch.3-1: Inhomogeneity problem: background

Where ϕ K ,ϕG are functions of physical parameter of composite materials, such as Poisson ratio of the matrix (see Sevostianov and Kachanov 2013).

1.3 Isotropic materials containing superspherical inhomogeneities

Unlike ellipsoidal inhomogeneities, the solutions for non-ellipsoidal inhomogeneities should be performed semi-analytically. Focusing on concave shapes, the property contribution tensors are carefully evaluated for superspherical pores, for both elasticity and conductivity problems.

The superspherical pore is modelled by a cavity of unite radius with surfaces described by

2 p 2 p 2 p geometric equation x1 + x2 + x3 = 1, the solid matrix is modelled by a block with a side length of 10 times bigger than the said pore. The most possibly refined mesh is imposed on the surface of pore, and gradually ticking meshes are exerted through the whole volume of the matrix from the interface matrix-inhomogeneity to its infinite boundary.

For elasticity problems, 1/8 pore need to be modelled and the physical parameters applied in the modelling for evaluating compliance contribution tensor of concave superspheroidal pores are stated as follows:

ν 0 = 0.3 ⇔ Poisson’s ratio of the matrix

2 E0 = 1 N /m ⇔ Young’s modulus of the matrix

σ ∞ = 0.001 N /m2 ⇔ Uniform loading at infinity of the matrix

Remotely applied loadings are submitted along − e3 (yellow surface), zero-displacement boundary conditions (green, blue and red surfaces) are implied on the plans perpendicular to x, y and z axes, as shown in figure 3.1 (a). Cubic symmetry of supersphere leads to an overall isotropy, one obtains:

H1111 = H 2222 = H 3333

H1212 = H 2121 = H1313 = H 3131 = H 2323 = H 3232 H = H = H = H = H = H 1122 2211 1133 3311 2233 3322 (3.11)

only three non-zero components H1111 ,H1122 and H1212 should be evaluated.

Ch.3.1: Inhomogeneity problem: background

Thermal conductivity problems Elasticity problems

Tupper

σ ∞

Displacement Convection of Displacement x=0 Heat flux = 0 y=0

y Displacement Tbuttom Inclusion z=0 x 1/8 shape 1/4 shape

(a) (b)

Fig.3.1: (a) Remotely loading and the boundary conditions for elasticity problems (b) Heat flux vectors and the gradient of temperature for thermal conductivity problems.

Similarly, for conductivity problems, heat flux vector (blue surfaces in figure 3.1(b)) are defined by:

q = 0 W /m2 (3.12) and the temperatures (red surfaces) imposed on the two infinity surface are:

T = 20 °C and T = − 20 °C (3.13) upper buttom

Due to the gradient of temperature ∇T , 1/4 shape need to be modelled and only one non- zero component of the second-rank resistivity contribution tensor should be calculated since:

R11 = R22 = R33 (3.14)

SeCtion 2

2 Evaluation of the effective elastic and conductive properties of a material containing concave pore (IJES, 2015b)

F.CHEN, I.Sevostianov, A.Giraud, D.Grgic, Evaluation of the effective elastic and conductive properties of a material containing concave pores, International Journal of Engineering Science, 97 (2015) 60-68

Abstract

We calculate effective properties of a porous materials with non-ellipsoidal concave pores.

2 p 2 p 2 p The pore shape is described by equation of a supersphere x1 + x2 + x3 = 1 that convex when p > 5.0 and concave when p < 5.0 . Compliance and resistivity contribution tensors for a superspherical pore are calculated using finite element method and approximated by analytical expressions for p <1 . These tensors are used to evaluate effective elastic and conductive properties of a material with superspherical pores via non-interaction approximation, Mori- Tanaka scheme and Maxwell scheme. We show that the geometrical parameters entering expressions for the elastic moduli and conductivity are the same and establish cross-property connection for such materials. These connections coincide with ones for a material with spherical pores.

Keywords: concave pore, supersphere, effective properties, homogenization, Maxwell scheme, Mori-Tanaka scheme, cross-property connections.

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore

2.1 Introduction

In this paper we develop a semi-analytical approach to evaluate effective elastic properties and thermal conductivity of a material containing pores of non-ellipsoidal shape focusing on concave superspherical pores. The key quantity in the problem of the effective elastic and conductive properties of a heterogeneous material are property contribution tensors that gives the extra strain or temperature gradient produced by introduction of the inhomogeneity into the material subjected to uniform stress field or heat flux. Alternatively, one can use the dual stiffness contribution tensor or conductivity contribution tensor (Sevostianov and Kachanov 2007).

Although, various materials science applications require quantitative characterization of inhomogeneities of irregular shape, most of the existing results are based on Eshelby’s (1957, 1961) solution for an ellipsoidal inhomogeneity. While for 2-D non-elliptical inhomogeneities many analytical and numerical results have been obtained (Zimmerman 1986, Kachanov et al. 1994, Tsukrov and Novak 2002, 2004), only a limited number of numerical results and approximate estimates are available for more complex 3-D shapes. Compliance contribution tensors for several examples of pores of irregular shape typical for carbon-carbon composites have been calculated by Drach et al. (2011) using FEM. The authors showed that pores of irregular shapes can be sometimes approximated by ellipsoids (in agreement with earlier experimental works of Prokopiev and Sevostianov (2006, 2007)). It is difficult, however, to make any conclusions from the results of Drach et al. (2011) (Tables 1 and 2 in their paper) regarding effect of any particular irregularity factor. In the narrower context of irregularly shaped cracks, certain results were obtained by Sevostianov and Kachanov (2002), Grechka et al. (2006), Mear et al. (2007), Kachanov and Sevostianov (2012).

The only analytical model that can account for concave shape of the pores has been developed by Sevostianov and Giraud (2012) and applied by Giraud and Sevostianov (2013) to calculation of the overall elastic properties of oolitic rock. This approach, however is based on computational results of Sevostianov et al. (2008) where stability of the calculations is rather poor and, as we discuss below, the accuracy of the calculations is insufficient.

The property contribution tensor is used as the basic building block to evaluate effective elastic and conductive properties of a material containing concave superspherical pores. For

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore this goal we use (1) Mori-Tanaka scheme (Mori and Tanaka 1973) in the form given by Benveniste (1986) and (2) Maxwell scheme (Maxwell 1873) in the form proposed by Sevostianov and Giraud (2013). These schemes, in particular, are known to have good agreement with experimental data on elastic and conductive properties of porous materials.

Remark. Some results have been obtained in the context of calculation of Eshelby tensor for a non-ellipsoidal inclusion. We mention results of Rodin (1996) for polyhedral shapes, Onaka (2001) for a concave supersphere; Onaka et al. (2002) for a toroidal inclusion and Chen et al. (2015a) for a convex supersphere. We have to point out however, that Eshelby tensor for non- ellipsoidal inhomogeneities has nothing in common with the problem of effective properties of heterogeneous material. Unfortunately, several publications already appeared where authors erroneously try to calculate effective properties of materials with non-ellipsoidal inhomogeneities using Eshelby tensor (see, for example Hashemi et al. (2009) where results of Onaka (2001) are used to calculate effective properties of a composite with cuboidal inhomogeneities). This mistake is a consequence of the misleading overestimation of the role of Eshelby tensor in micromechanics.

2.2 Property contribution tensors for a superspherical inhomogeneity

2.2.1 Compliance and resistivity contribution tensors

These tensors have been first introduced in the context of pores and cracks by Horii and Nemat-Nasser (1983) (see also detailed discussion in the book of Nemat-Nasser and Hori 1993). Components of this tensor were calculated for 2-D pores of various shape and 3-D ellipsoidal pores by Kachanov et al. (1994). For general case of elastic inhomogeneities, these tensors were introduced and calculated for ellipsoidal shapes by Sevostianov and Kachanov (1999, 2002). In this sub-section we briefly describe the physical meaning of the compliance contribution tensor and discuss how it may be calculated for a superspherical pore

We consider a homogeneous isotropic elastic material (matrix), with the compliance tensor

S0 containing an inhomogeneity, of volumeV* , of a different material with the compliance

tensor S1 . The compliance contribution tensor of the inhomogeneity is a fourth-rank tensor H that gives the extra strain (per reference volume V ) due to its presence:

V V ∆ ε = * H : σ ∞ , or, in components, ∆ε = * H σ ∞ (3.15) V ij V ijkl kl

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore

∞ where σ kl are remotely applied stresses that are assumed to be uniform within V in the absence of the inhomogeneity. For an ellipsoidal inhomogeneity, its compliance contribution tensor is expressed in terms of tensor Q - one of two Hill’s tensors (Hill 1965, Walpole 1969) as

−1 −1 H = []()S1 − S0 + Q , (3.16) Sevostianov and Kachanov (2011) showed that the far-field asymptotes of the elastic fields generated by an inhomogeneity determines its contribution to the effective elastic properties and vice versa. The latter result, in particular, allows formulation of the Maxwell homogenization scheme in its terms (Sevostianov and Giraud 2013).

For a pore, the additional strain due to its presence is calculated as an integral over its boundary Ω∂

− 1 ∆εij = ()nu ji + u nij dS (3.17) 2V ∫ Ω∂ where u and n denote displacements on the pore boundary and a unit normal to it (directed inwards the pore). The representation (3.17) directly follows from application of the divergence theorem to a solid containing a pore (see, for example, Kachanov et al. 1994).

The resistivity contribution tensor has been introduced by Sevostianov and Kachanov (2002) in the context of the cross-property connection between elastic and conductive properties of heterogeneous materials. We assume that the background material of volume V with the isotropic thermal conductivity k0 contains an isolated inhomogeneity of volume V* with the isotropic thermal conductivityk1 . Limiting cases k1 = 0 and k1 = ∞ correspond to an insulating and a superconducting inhomogeneities. Assuming linear relation between temperature gradient ∇T and the heat flux vector q per volume (Fourier law) for both the constituents, the change in ∇T required to maintain the same heat flux if the inhomogeneity is introduced as (Sevostianov and Kachanov 2002):

V ∆()∇T = * R ⋅ q (3.18) V where the symmetric second-rank tensor R is called the resistivity contribution tensor of an inhomogeneity.

For an insulating inhomogeneity (like pore, for example), expression similar to (3.17) can be written. The additional temperature gradient due to the presence of such inhomogeneity can be represented as an integral

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore

 ∂T  − 1 ∆  = TnidS  ∂x  V ∫ Ω∂  i  (3.19) where T and n are temperature and inward unit normal to the boundary

2.2.2 Property contribution tensors for a superspherical pore

We now focus on the case when surface Ω∂ is a supersphere of unit radius, i.e. is described by the following equation:

2 p 2 p 2 p x1 + x2 + x3 = 1 (3.20) Parameter p may be called concavity factor or a parameter of deviation from spherical shape. When p = 1, the supersphere becomes the ordinary sphere. The sphere is transformed into a cube as p → ∞ . For p > 0.5, its shape is convex, for p < 0.5 it is concave. Examples of this shape are shown in Figure 3.2 for different values of p .

As p decreases, the volume of the supersphere

3   1  2  3  V∗ = 2Γ  3p Γ  (3.21)   2 p   2 p 

∞ (where Γ()x = ∫ te xt −− 1dt is gamma function) varies dramatically (inset in Figure 3.2). 0 3   1  2  3  V∗ = 2Γ  3p Γ    2 p   2 p 

10-1 p=0.1 p=0.2 p=0.35

10-3

p 10-5 p=10 p=1 p=0.5 0.0 0.2 0.4 0.6 0.8 1.0 Fig.3.2: Different superspherical shape at different values of p. Inset: dependence of the supersphere volume on p

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore To evaluate integrals (3.17) and (3.19), 3D finite element modellings was performed by using NX Simulation software developed by Siemens. Linear isoparametric Lagrange finite elements has been used: 4-nodes tetrahedrons has been used for volume discretization and the surface of the supersphere was discretized by 3-nodes triangles (T3) which corresponds to borders of volume finite elements belonging to the supersphere boundary. The reference mesh is presented in Figure 3.3. Due to the symmetries of the problem, only 1/8 of the supersphere can be considered, and symmetry conditions have been imposed (see appendix B for details, table B2). This mesh is composed of

259728 nodes

639436 4-nodes tetrahedron finite elements

22496 3-nodes triangular finite elements on the supersphere surface

Fig.3.3: Reference mesh for concave supersphere (p=0.35) (1/8 f the meshed supersphere on the left, details at the top right corner and three-nodes triangular finite elements on the surface at the bottom right corner)

For the conduction problem and the determination of resistivity contribution tensor, the 1/4 of the supersphere needs to be considered as a remote condition of uniform and uniaxial thermal flux vector is imposed on the infinite boundary. In this case, only two symmetry planes with

respective unite normal vectors ex and e y have been considered.

Equation (3.17) can be discretized as:

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore − 1 − 1 N ∆εij = ()nu ji + u nij dS = ()nu ji + u nij Si (3.22) ∫ Ω∂ ∑ 2V 2V i=1

Where N is the number of elements on boundary surface, ui and ni are projections of displacement vector u and unite normal vector n located at the centroid point of each triangle element on i direction, respectively and the summation is over all the elements at the supersphere boundary.

Centroid point is the product of trisection of the coordinates of the three nodes. For example, a triangle composed by node A, B and C:

 x A + x B + xC x0 = 1 1 1  1 3  A B C  0 x2 + x2 + x2 x2 = (3.23)  3 x A + x B + xC  0 3 3 3 x3 =  3

The same interpolation is used to calculate displacement at the centroid point as a function of nodal displacements obtained by FEM

 u + u + u u = iA iB iC  i 3   u jA + u jB + u jC u j = (3.24)  3 u + u + u  kA kB kC uk =  3 The normal vector to an implicitly defined surface is proportional to the gradient of the function f so that

 ∂f ∂f ∂f  N = (N ,N ,N ) =  , ,  (3.25) i j k   ∂x1 0 ∂x2 0 ∂x3 0  =xx 11 =xx 22 =xx 33  Normal vector of the linear triangular finite element is calculated at the centroid point which is a natural candidature to represent triangle ABC. Unit normal vector is then

N n = ()n ,n ,n = (3.26) i j k N.N

Surface area Si of the linear triangular finite element may be easily obtained as the cross product

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore 1 S = AC × AB (3.27) i 2 and relation (3.22) may be calculated by summing all surface element contributions. The same procedure has been used to evaluate expression (3.19)

Figure 3.4 shows the dependence of the compliance contribution tensor components on the concavity parameter p varying from 0.2 to 5 (see appendix B for details, table B1, exact solutions of H tensor for sphere of unit radius is found in appendix A for details). Note, that as discussed by Sevostianov et al. (2008) and Sevostianov and Giraud (2012), to increase the accuracy of the calculations, extra strain due to the cavity ∆ ε in (3.15) may be normalized to

3 L∗ where L∗ is certain characteristic length of the pore (for example, the maximal distance between its far points):

L3 ~ ∆ ε = ∗ H : σ ∞ (3.28) V ~ Certainly, tensors H and H entering relations (3.17) and (3.19) are different since they represent different normalizations of the pore contribution to the effective compliance. ~ Sevostianov and Giraud (2012) shown that tensor H for a concave superspherical pore ( p < 0.5) can be approximated by linear functions of the concavity factor p . Unfortunately, their results are based on low accuracy calculations of Sevostianov et al. (2008). Our results, ~ allows more accurate representation of tensor H . They are illustrated in Figure 3.5. Note that effect of pore becomes negligible at p = 0.2 . Due to that, one can write for the components of ~ H :

~ 5p − 1 ~ H = H S (3.29) ijkl 4 ijkl ~ Connecting now expressions for Hijkl and Hijkl corresponding to the two normalizations (3.15) 3 and (3.29) and taking into account that L3 = V (where V is the volume of the unit sphere), ∗ 4π 0 0 we get

()53 p − 1  V0  ~ S H ijkl =  Hijkl (3.30) 16π V∗ ()p  For a spherical pore ( p = 1) (see Kachanov et al. 2003)

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore

~ S 4π S  HG  1  H K  1  H = H = 2π   J- II  +  II  (3.31) 3 2G0  3  3K0  3  1 where J = ()δ δ + δ δ and II = δ δ ; the two terms in the brackets give the ijkl 2 ik jl il jk ijkl ij kl deviatoric and hydrostatic parts of the extra strain due to the spherical pore; and coefficients

10()1 −ν 0 1 −ν 0 HG = , H K = . (3.32) 7 − 5ν 0 1 − 2ν 0

8.0

6.0 H1111

4.0

H1212

2.0

H1122 0 . 0 p 0.0 1.0 2.0 3.0 4.0 5.0

Fig.3.4: Non-zero component of compliance contribution tensor for different values of p (from 0.2 to 5)

10.0 ~ 5p −1 ~ H = H S 1111 4 1111 8.0 R2 = 0.998587

6.0 ~ 5p −1 ~ H = H S 1212 4 1212 R2 = 0.998564 4.0

2.0 ~ 5p −1 ~ − H −= H S 1122 4 1122 p 0.0 R2 = 0.998183 0.2 0.4 0.6 0.8 1.0

Fig.3.5: Linear Approximation of the non-zero components of the compliance contribution tensor normalized according to (2.32). R-squared values of the approximations are indicated

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore Then, the following simple approximate relations are obtained for the compliance contribution tensor of a superspherical pore:

()53 p − 1  V0  HG  1  H K  1  H =    J- II  +  II  (3.33) 8 V∗ ()p 2G0  3  3K0  3  Formula (3.33) identify the microstructural parameter for approximate characterization of the effect of the superspherical pore on elastic properties as

 3  ()5p − 1 p2Γ  ()53 p − 1  V0  3π  2 p  η =   = 3 . (3.34) 8 V∗ ()p  4   1  Γ    2 p  Note that this parameter differs from one introduced by Sevostianov and Giraud (2012) because of the insufficient accuracy of their estimates. 3.0 (a)

2.5

2.0

R11 1.5

1.0 p 0.0 1.0 2.0 3.0 4.0 5.0

2.0 (b)

1.5 ~ 5p − 1 ~S R11 = − R11 1.0 4 R2 = 0.9917

0.5

0.0 p 0.2 0.4 0.6 0.8 1.0

Fig.3.6: Dependence of the resistivity contribution tensor on p (a) and its linear approximation (b).

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore

Figure 3.6(a), shows dependence of the resistivity contribution R11 ( = R22 = R33 ) on the concavity parameter p (see appendix B for details, table B1). Introducing normalization of

3 ∆()∇T to L∗ we rewrite (3.16) as

()L 3 ~ ∆()∇T = * R ⋅ q (3.35) V ~ and observe that components of R show linear dependence on the concavity parameter (Fig 3.6b) for p ≤ 1 . Moreover, similarly to the compliance contribution tensor, resistivity contribution tensor vanishes at p = 0.2 .

Due to that, we can write

~ 5p − 1 ~ R = R S (3.36) ij 4 ij

~S where Rij = 2πδij k0 is the resistivity contribution tensor of a spherical pore with normalization (3.35). Finally, it leads to the following expression for the resistivity contribution tensor of the superspherical pore:

()53 p − 1  V0  k0Rij =  δij = ηδij (3.37) 8 V∗ ()p 

Remark. Comparison of the expressions (3.32) and (3.37) indicates that the microstructural parameter describing contribution of the superspherical pore into elastic properties is the same for resistivity. In particular it leads to explicit cross-property connections (Sevostianov and Kachanov 2002, 2008). Moreover, the cross-property connection has to be of the same form as for spherical pores.

2.3 Effective properties of a material containing superspherical pores

In this section we discuss overall elastic and conductive properties of a material containing multiple superspherical pores. Strictly speaking, interactions should be incorporated into the property contribution tensors, since they affect compliance and resistivity contributions of individual pores. Such parameters would depend on mutual positions of pores in the way that is relevant for the interaction mechanics. The effective property would then be a linear function of such a parameter (as implied by summation of the individual property contributions). As discussed by Sevostianov and Kachanov (2013), such an approach – incorporating interactions

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore into the microstructural parameter – may not be practical, although it may, possibly, be realized in some cases. Note that most widely used homogenization schemes can be reformulated in terms property contribution tensors derived for non-interacting inhomogeneities. Namely this approach is used below.

2.3.1 Non-interaction approximation

This approximation is reasonably accurate at low concentration of inhomogeneities ("dilute limit"). If interaction between the inhomogeneities is neglected, each inhomogeneity can be assumed to be subjected to the same remotely applied field. Contributions of the inhomogeneities into the change in the temperature gradient and overall strain be treated separately. Then, the total temperature gradient can be written as

1 1  ()()n n  T q V R q ()∇ i = i + ∑ * ij  j (3.38) k0 V  n  and total strain

0 1  ()()n n  ε = S σ + V H σ ij ijkl kl ∑ * ijkl  kl (3.39) V  n  For isotropic mixture of superspherical pores it yields the following expressions for effective conductivity k and effective bulk and shear moduli, K and G (subscript NIA indicates non- interaction approximation):

k K G k = 0 ,K = 0 ,G = 0 NIA 1 + ϕη NIA 1 + ϕηH NIA 1 + ϕηH K G (3.40)

where ϕ is the overall porosity, shape factor η is defined by (3.34) and parameters H K and

HG are given by (3.32). Formulas (3.40) highlight the fundamental importance of the property contribution tensors: it is them that have to be summed up, in the context of the effective material properties. The sums properly reflect compliance contributions of individual inhomogeneities.

Remark: Its dual version – dilute scheme – yields linear expressions of equation (3.40), effective conductivity k and effective bulk and shear moduli, K and G (subscript DIL indicates dilute scheme) are expressed as following:

k = k (1 − ϕη),K = K (1 − ϕηH ),G = G (1 − ϕηH ) DIL 0 DIL 0 K DIL 0 G (3.41)

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore By considering the circular cracks and its crack density parameter, computational simulations show that dilute scheme narrows drastically the range of the crack density parameter where Non-Interaction Approxiamtion remains adequate.

2.3.2 Mori-Tanaka scheme

It belongs to the class of effective field homogenization schemes where each inhomogeneity, treated as a single one, is placed into the unaltered matrix material; interactions are accounted for by assuming that the inhomogeneity is subjected to the field that differs from the remotely applied one. The basic idea of the method has roots in works of Mossotti (Feynman et al. (1964), chapter 11). The Mori-Tanaka scheme (Mori and Tanaka 1973) as interpreted by Benveniste (1986) is based on the assumption that the effective field acting on each inhomogeneity is equal to the average over the matrix. Then the macroscopic properties may be calculated from the non-interaction approximation with appropriate change of the remotely applied field. For a porous material of porosity ϕ , this average is related to remotely applied field by multiplying the latter by the factor 1 ()1 − ϕ so that the effective elastic and conductive properties are obtained from the NIA result by replacement ϕ → ϕ ()1 − ϕ correctly predicting that effective conductivity and shear and Young’s moduli approach zero as ϕ → 1:

k K G k = 0 ,K = 0 ,G = 0 MT ϕ MT ϕ MT ϕ 1 + η 1 + ηH K 1 + ηHG 1 − ϕ 1 − ϕ 1 − ϕ (3.42)

(superscript MT indicates that the effective properties are calculated in the framework of Mori- Tanaka scheme)

2.3.3 Maxwell homogenization scheme

It is probably the oldest one. Originally proposed by Maxwell (1873) for electrical conductivity of a material containing randomly located spherical inhomogeneities, it was further developed in a number of works. In the context of elastic properties, McCartney and Kelly (2008) and McCartney (2010) formulated it for a material containing either random or aligned ellipsoidal inhomogeneities of identical aspect ratios. We are using the interpretation proposed by Sevostianov and Giraud (2013), where Maxwell scheme is formulated in terms of property contribution tensors.

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore The basic idea of the method is that the far field produced by the considered set of inhomogeneities is equated to the far field produced by a fictitious domain of certain shape that possesses the (yet unknown) effective properties. Note that the choice of the mentioned shape is non-trivial. Sevostianov (2014) developed an approach that allows one to find this shape. For an isotropic mixture of inhomogeneities in isotropic matrix the shape is spherical. In this case (Sevostianov and Giraud 2013, Sevostianov and Kachanov 2013)

3 − 2ϕη 1 − ϕη H KαK 1 − ϕη HGαG kM = k0 ,KM = K0 ,GM = G0 (3.43) 3 + ϕη 1 + ϕη H K ()1 − αK 1 + ϕη HG ()1 − αG

where

2 1 − 2ν 1 7 − 5ν α = 0 ,α = 0 K 3 1 −ν G 15 1 −ν 0 0 (3.44) and superscript M indicates Maxwell homogenization scheme.

2.4 Cross-property connections for a material containing superspherical pores

We now discuss the possibility to extend explicit connections between overall elastic and conductive properties to the materials containing superspherical inhomogeneities. Cross- property connections have been developed by Sevostianov and Kachanov (2002, 2008) for non- interacting spheroidal inhomogeneities. Note that experimental data on diverse heterogeneous materials show that the connections continue to hold at finite, or even large, concentrations of inhomogeneities that may have non-spheroidal, “irregular” shapes (see, for example, Sevostianov et al. 2006). It may be explained by the hypothesis that irregularity factors affect the elastic and the conductive properties in a similar way, so that the connection between the two is not affected much. As follows from expressions (3.44) it is indeed so. In the framework of non-interaction approximation, one can write

K0 − K NIA k0 − kNIA 1 −ν 0 k0 − kNIA = H K = K NIA kNIA 1 − 2ν 0 kNIA G − G k − k 10()1 −ν k − k 0 NIA = H 0 NIA = 0 0 NIA G G k 7 − 5ν k NIA NIA 0 NIA (3.45)

Cross-property connection (3.45) is identical to one given by Sevostianov and Kachanov (2008) for spherical pores. It supports the hypothesis formulated above. The same form of the

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore cross-property connection will take place if the effective properties are calculated using Mori- Tanaka effective field scheme:

K − K 1 −ν k − k G − G 10()1 −ν k − k 0 MTS = 0 0 MTS , 0 MTS = 0 0 MTS K 1 − 2ν k G 7 − 5ν k MTS 0 MTS MTS 0 MTS (3.46)

Cross-property connections in the framework of Maxwell homogenization scheme can be written in a similar way if we observe first that H αKK = H αGG = 32 and, therefore

1 − 2ϕη 1 − 2ϕη K = K ,G = G M 0 3 + ϕηβ M 0 3 + ϕηβ K G (3.47)

1 +ν 0 8 − 10ν 0 where β K = and βG = . Expressing now ηϕ from the first equations (3.45) as 1 −ν 0 ()15 −ν 0

3()k − k ϕη = 0 M 2k + k 0 M (3.48) and substituting this relation into (3.47), we obtain cross-property connections in the framework of Maxwell homogenization scheme:

K − K ()()22 k + k + 3 k − k ()β + 2 0 M = 0 M 0 M K K 7k − 4k M M 0 (3.49) G − G (22 k + k )+ 3 (k − k )(β + 2 ) 0 M = 0 M 0 M G GM 7kM − 4k0

Expressions (3.49) obviously coincide with ones for spherical pores.

2.5 Concluding remarks

In the present work, we discussed effective elastic and conductive properties of a material containing pores of superspherical shape which boundary is described by equation

2 p 2 p 2 p x1 + x2 + x3 = 1. We showed that, for p < 1, contributions of an individual pore into both elastic and electric properties is described by the same geometric parameter  3  ()5p − 1 p2Γ  3π  2 p  η = 3 . 4   1  Γ    2 p 

Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore Effective elastic and conductive properties are then expressed in terms of this parameter in the frameworks of non-interaction approximation, Mori-Tanaka effective field method and Maxwell homogenization scheme. Since the microstructural parameter is the same for elastic and conductive properties, it is possible to write explicit cross-property connections that coincide with ones for a material containing multiple spherical pores. It support the hypothesis that irregularity factors affect the two types of properties in a similar way.

Acknowledgement

ChaPter 4

IV Property contribution tensor of superspheroidal pores

p 2 p  x2 + x2  x Superspheroidal’s surface is described by implicit equation  1 2  + 3 = 1 . This  a  a γ shape is generally convex or concave depending on p parameter. In the convex range, it is also known as "superegg" by referring to Danish poet and scientist Piet Hein (1905-1996) who 4 defined for this the following values p = 1.25,γ = . Geometrically speaking, as a special case 3 of superellipsoid, this shape is (in the convex range) a product of revolution obtained by rotating a curve that lies between the ellipse and the rectangle along an axe.

This shape respects symmetry of revolution and related contribution tensors respect transverse isotropy (surrounding matrix material is supposed isotropic). As previously stated, we only consider in this work isotropic effective material related to random (isotropic) distribution of superspheroidal inhomogeneities. This shape completes the superspherical shape which respect cubic symmetry.

After a brief overview of numerical procedures, new results evaluated by two different normalization methods are provided in section 2.

SeCtion 1

1 Overview of numerical procedures

A new concave shape possessing the symmetry of revolution is introduced and named as "superspheroid" in the current work. After the discussions about supersphere which is considered as the first concave shape, superspheroidal shape shows its particular interest in the two-phase modelling in transversely isotropic effective media (case of aligned distribution) or isotropic effective media (case of a random distribution). The effective elastic properties and thermal conductivity of isotropic material containing such pores are performed semi- analytically.

Singal inhomogneity of superspheroidal shape with axe of ration e3 is treated in FEM procedures, the composite is considered to be isotropic in x1 − x2 plan and to be transversely isotropic in x1 − x3 and x2 − x3 plan.

In elasticity problems, different uniform loadings such as σ 33 , σ11 ,σ 23 and σ12 should be applied at the infinity of the matrix separately, an example of σ 33 is illustrated in figure 4.1(a) with zero-displacement boundary conditions (green, blue and red surfaces) implied on the plans perpendicular to x, y and z axes, as illustrated in figure 4.1 (b).

Five non-zero components of compliance contribution tensor such as H 3333 ,H1122 ,H1313 ,

H1133 and H1313 need to be evaluated by using the following relations:

H1111 = H 2222 ,H1122 = H 2211 ,H1313 = H 2323 ,H1133 = H 2233 ,H1313 = H 2323 (4.1)

Remark: Each uniform loading allows only to calculate one or two non-zero components:

For details, σ 33 gives H1133 and H 3333 , σ11 gives H1111 a nd H1122 , σ 23 gives H 2323 .

Ch. 4-1: Overview of numerical procedures

Displacement_y=0

σ ∞

Displacement_x=0

∞ σ y x Displacement_z=0 (a) (b)

Fig.4.1: Boundary conditions for modelling of materials containing concave superspheroidal pores

In conductivity problems, two non-zero components R11 and R33 need to be calculated by taking advantage of the following equivalence:

R11 = R22 (4.2)

Temperatures (yellow surfaces) are implied on the same surfaces with the elasticity problems and heat flux (blue surfaces) are assumed to be zero. An example of the applied temperature gradient ∇T along e3 giving R33 is shown in figure 4.2. For details, ∇T along e1 gives R11 .

Tupper

Convection of Heat flux = 0

Tbuttom y x

Fig.4.2: Temperature gradient and convection of heat flux for thermal conductivity problems

SeCtion 2

2 Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheroidal shape

(IJES, 2016)

I.Sevostianov, F.CHEN, A.Giraud, D.Grgic, Compliance and resistivity contribution tensors of axisymmetric concave pores, International Journal of Engineering Science, 101 (2016) 14- 28

Abstract

We focus on the effect of non-ellipsoidal axisymmetric concave pore on overall properties of porous materials. This effect is described by compliance and resistivity contribution tensors.

2 2 p 2 p The pore shape is described by equation ( x1 + x2 ) + x3 = 1 that is convex when p > 0.5 and concave when p < 0.5. The limiting case p → 0 corresponds to a combination of a circular crack of unit radius and a needle of unit half-length normal to the crack, p→∞ describes a circular cylinder and p =1 - a unit sphere. Compliance and resistivity contribution tensors for a superspheroidal pore are calculated using finite element method and approximated by analytical expressions for p <1. These results allow evaluation of the effective elastic and conductive properties of a material with concave pores using various homogenization methods.

Keywords: Axisymmetric shape, superspheroid, superegg, compliance contribution tensor, resistivity contribution tensor, concave pore.

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

2.1 Introduction

In this paper, we discuss microstructures that contain pores of concave shapes that are frequent in both naturally occurring and man-made materials. In particular, concave pores are typical for microstructures of geomaterials (Wark et al. 2003, Emmanuel et al. 2010). Figure 4.3 illustrates backscattered electron images of sandstone and harzburgite with pores of concave shape. In analyses of the effective properties of materials with irregular microstructure, much attention has been paid to interactions between inhomogeneities. Their shapes – the factor at least as important as interactions – received much less attention; the inhomogeneities are typically assumed to be ellipsoids of identical aspect ratios. This unrealistic assumption is largely responsible for insufficient linkage between methods of micromechanics and materials science applications. In this paper, we discuss the property contribution tensors that give the extra strain or temperature gradient produced by introduction of the pore into the material subjected to uniform stress field or heat flux. Namely these tensors serve as the basic building block to evaluate effective elastic and conductive properties of heterogeneous materials (Kachanov and Sevostianov 2005).

Fig.4.3: Pores of concave shapes typical in rock mechanics: (a) sandstone (from Emmanuel et al. 2010); (b) harzburgite (from Wark et al.2003)

Compliance contribution tensor have been first introduced in the context of ellipsoidal pores and cracks by Horii and Nemat-Nasser (1983). Components of this tensor were calculated for 2-D pores of various shape and 3-D ellipsoidal pores by Kachanov et al. (1994). For general case of elastic inhomogeneities, these tensors were introduced and calculated for ellipsoidal shapes by Sevostianov and Kachanov (1999). Kushch and Sevostianov (2015) established the link between compliance contribution tensor and dipole moments. The resistivity contribution tensor R has been introduced by Sevostianov and Kachanov (2002a) in the context of the cross-

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape property connection between elastic and conductive properties of heterogeneous materials. Kushch and Sevostianov (2014) developed explicit relations between resistivity contribution tensor and dipole moments.

Most of the results on property contribution tensors are obtained for ellipsoidal inhomogeneities in terms of Eshelby tensor (Eshelby 1957, 1961, Mura 1987). We have to point out, however, that, for non-ellipsoidal inhomogeneities, this link is lost. Eshelby tensor is irrelevant for the problem of effective properties of heterogeneous material with non-ellipsoidal inhomogeneities, and compliance contribution tensor, therefore, has to be calculated independently.

While for 2-D non-elliptical inhomogeneities many analytical and numerical results have been obtained (Zimmerman 1986; Kachanov et al. 1994; Tsukrov and Novak 2002, 2004), only a limited number of numerical results and approximate estimates are available for non- ellipsoidal 3-D shapes. Compliance contribution tensors for several examples of pores of irregular shape typical for carbon-carbon composites have been calculated by Drach et al. (2011) using FEM. The authors give the values of the components of compliance contribution tensors for several specific shapes (Tables 1 and 2 in their paper), but did not discuss effect of any particular irregularity factor. In the narrower context of irregularly shaped cracks, certain results were obtained for compliance contribution tensors by Fabrikant (1989), Sevostianov and Kachanov (2002a) (planar cracks), Grechka et al. (2006) (intersecting planar cracks), Mear et al. (2007) (non-planar cracks), and Kachanov and Sevostianov (2012) (cracks growing from pores).

The only concave shape that has been discussed in literature in the context of effective material properties is a supersphere – shape described by equation x2 p + y 2 p + z2 p = 1 that is convex when p > 0.5 and concave when p < 0.5. This equation is a particular case of superellipsoidal equation (Zohdi and Wriggers (2008), eq. 9.2). It has been introduced by Lamé (1818) where concave - convex cases have been discussed (see Guitart 2009, Barbin and Guitart 2013), Effect of a superspherical pore on overall elastic properties was first analyzed by Sevostianov et al. (2008) using FEM. Their results have been used by Sevostianov and Giraud (2012) to derive approximate analytical expressions for the components of the compliance contribution tensor. This derivation have recently been corrected and extended to resistivity contribution tensor by Chen et al. (2015b). The authors showed that, for, contributions of an

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

2 p 2 p 2 p individual superspherical pore x1 + x2 + x3 = 1 into both elastic and conductive properties are described by the same geometric parameter

 3  (5p − 1) p2Γ  3π  2 p  η = 3 (4.3) 4   1  Γ    2 p  and expressed effective elastic and conductive properties in terms of this parameter using various homogenization schemes. Since the microstructural parameter is the same for elastic and conductive properties, the authors also obtained explicit cross-property connections between the said properties that coincide with ones for a material containing multiple spherical pores.

In the text to follow, we study another shape -axisymmetric one described by

2 p ( x2 + x2 ) p x 1 2 + 3 = 1 (4.4) a 2 p a 2 pγ 2 p This shape may be called a “superegg” following Danish mathematician, poet, and designer Piet Hein who used concavity parameter p = 1.25, and aspect ratio γ = 4/3 (see Hein, Matematiksider.dk). In this paper we prefer dissociate the generic shape, superspheroid, to this particular superegg case, and superspheroid name will be adopted. We restrict ourselves by the case a = γ = 1 and focus the analysis on the concavity parameter p. The concavity range corresponds to the condition 2 p < 1, and the convex range to 1 < 2 p (Figure 4.4). The limiting case p → 0 corresponds to a circular crack of unit radius crossed by a needle along the axis of revolution. The case p = 1 corresponds to a unit sphere. In the limit p→∞, the superspheroid tends to a circular cylinder of radius a =1 and length L = 2a = 2.

The volume V* of the superspheroid (4.4) is

3 V* = 4πa γg( p) (4.5)

Where

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

1 + 2 p   1  Γ Γ  1 2 p p g( p) =     (4.6) 3  3  Γ   2 p  and Γ denotes Euler gamma function.

p =1 p = .50

p = .40 p = .30 p = .20

Fig.4.4: Axisymmetric superspheroid (“superegg”) for different values of p.

1.0 1.0 S S sphere V Vsphere 0.9 0.8

0.8 0.6

0.7 0.4

0.6 0.2

(a) (b) 0.5 γ , p 0.0 γ , p 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig.4.5: (a) Total surface areas of a superspheroid of concavity p and oblate spheroid of aspect ratio γ normalized to the surface area of the superscribed unit sphere: (b) normalized volumes of the superspheroid and spheroid

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

The total surface area of the superspheroid is

S * = 4π a 2 γ j γ ,( p) (4.7) with

u=1 1 j γ ,( p) = (1 − u2 p ) 21 p )( 1 + u( p−24 )(1 − u2 p ) 1/( p−2) du (4.8) ∫u=0 γ 2 Exact expression of integral (4.8) may be obtained for particular values of p (see in appendix C analytical results of total surface area and volume, for p = 0; 1/4; 1/3; 1/2; 1; ∞).The dependences of the surface area and volume of the superspheroid on the concavity parameter p are illustrated in Figure 4.5. Corresponding quantities for a spheroid (ellipsoid of revolution) are presented for comparison.

2.2 Property contribution tensors for a superspheroidal inhomogeneity.

2.2.1 Compliance and resistivity contribution tensors

We first discuss the compliance contribution tensor and consider a homogeneous elastic

0 material (matrix), with the compliance tensor S containing an inhomogeneity, of volumeV* , of a different material with the compliance tensor S1 . The compliance contribution tensor of the inhomogeneity is a fourth-rank tensor H that gives the extra strain (per reference volume V) due to its presence:

V* ∞ V* ∞ ∆ ε = H : σ ∆εij = Hijklσ kl V or, in components, V (4.9)

∞ where σ kl are remotely applied stresses that are assumed to be uniform within V in the absence of the inhomogeneity. Sevostianov and Kachanov (2011) showed that the far-field asymptotes of the elastic fields generated by an inhomogeneity determines its contribution to the effective elastic properties and vice versa. For a pore, the additional strain due to its presence is calculated as an integral over its boundary ∂Ω

− 1 ∆εij = ()nu ji + nu ij dS (4.10) 2V ∫ Ω∂ where u and n denote displacements on the pore boundary and a unit normal to it (directed inwards the pore). The representation (4.11) directly follows from application of the divergence theorem to a solid containing a pore (see, for example, Kachanov et al. 1994).

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

Considering resistivity contribution, we assume that the background material of volume V with the thermal conductivity k0 contains an isolated inhomogeneity of volume V1 with the

thermal conductivity k1 . Assuming linear relation between temperature gradient ∇T and the heat flux vector q per volume (Fourier law) for both the constituents, the change in ∇T required to maintain the same heat flux if the inhomogeneity is introduced as (Sevostianov and Kachanov 2002a):

V ∆()∇T = * R ⋅ q (4.11) V where the symmetric second-rank tensor R is called the resistivity contribution tensor of an inhomogeneity. For an insulating inhomogeneity (like pore, for example), expression similar to (4.11) can be written. The additional temperature gradient due to the presence of such inhomogeneity can be represented as an integral

 ∂T  − 1 ∆  = TnidS (4.12)   ∫ Ω∂  ∂xi  V where T and n are temperature and inward unit normal to the boundary.

2.2.2 Numerical procedure to determine compliance and resistivity contribution tensors of a superspheroidal pore.

Equation (4.10) can be discretized as:

Neln G − 1 − 1  i el j )( i i  ∆εop = ()nu po + u nop dS =  wGhop ()ξG ,ηG  (4.13) ∫ Ω∂ ∑ ∑ 2V 2V j=1 i = 1 

Where N el the number of finite elements on boundary surface, nG is the number of Gauss

i i i integration points per element, wG ,ξG ,ηG denote weights, and coordinates of Gauss integration

el j )( points in the reference plane. hop denotes a function of u0 and n0 , components of displacement vector u and unit inward normal vector n, calculated by interpolation at every Gauss integration point of each surface triangular element on o, p directions, respectively and the summation is over all the elements at the superspheroid boundary (see appendix C).

Due to the axial symmetry of the superspheroidal pore, its compliance contribution tensor H is transversely isotropic, and characterized by 5 independent components (hereafter, e3 is along the symmetry axis). To evaluate integral (4.13), 3D finite element modelling have been performed with NX Simulation software developed by Siemens. Quadratic isoparametric

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

Lagrange finite elements have been used: 10 -nodes tetrahedrons for volume discretization and 6-nodes triangles (T6), which correspond to borders of volume finite elements belonging to the boundary, for discretization of the surface of the superspheroid.

3D finite element meshes are presented in Figure 4.6. Transition zones allowing size progression are considered from the boundary of superspheroid to the infinite boundary which are imposed on the face of cubes of length 2 L = 40 (distance L = 20 to the center of superspheroid of unit radius). Transition zones correspond to cubes with faces (on the Cartesian coordinate axis) at distances L1 = 2 , L2 = 5 from the center. A high density mesh is considered

∞ ∞ in the first cube of length 2L1 = 4 surrounding the superspheroid. Three loading casesσ 33 , σ11

∞ and σ 23 are necessary to determine the 5 components H1111, H1122 , H1133, H 3333, H 2323 .

p=0.30

Inside view

0 1 2 5 10

0 1 1.5 On superegg Entire mesh

106134 nodes 1177256 nodes

53066 quadratic 890505 quadratic elements elements

Fig.4.6: 3D reference Finite Element mesh for concave superegg (p=0.3)

Numerical integration of (4.10) has been performed by using standard Gauss-Legendre quadrature formula with quadratic surface (3D modelling) finite elements. Components of unit vector n have been calculated on Gauss integration points by using exact relations detailed in appendix E. The total surface area has been numerically calculated by using Gauss-Legendre quadrature formula on the surface mesh of the superspheroid (see appendix C for details). It has

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape been compared to values obtained by numerical integration of relation (4.7) in the general case, and to exact values obtained for particular values of concavity parameter p = 0.25, p = 0.33, p = 0.5, p = 1. It allows one to check the accuracy of the surface meshing and this is necessary for low values of concavity parameter p.

30 20

E0H3333 E0H1122 0Rk 33

E0H1313 E0H1133 20 15 k0R11

E0H1111

10 10

(b) 5 0

(a) p p -10 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Fig.4.7: Non-zero component of compliance contribution tensor (a) and resistivity contribution tensor (b) as functions of the concavity parameter p

Quadratic isoparametric 3D, 10 nodes Lagrange isoparametric tetrahedrons have been used. 18 values of concavity parameter have been considered, regularly distributed between p = 0.2 and p =1 (except of p = 0.33 which has been added for comparison with analytical total surface area), and for each value of p, two finite element meshes have been considered and compared: mesh (a) with a relative coarser discretization of the surface of superspheroid and mesh (b) with a refined discretization of the surface (see Table D2 for the detail of number of nodes and elements of mesh b). A test of geometrical precision of the finite element mesh have been performed by comparing the total surface area numerically calculated by using Gauss Legendre quadrature on the basis of the surface mesh of the superspheroid to exact values given in Table E1 for p = 0.25; 0.33; 0.5; 1.

Numerical results for Hijkl components are given in Table D1 and Figure 4.7(a).

Relative errors on components Hijkl obtained by comparing mesh (a) and mesh (b) are given in Table D3 All relative error are lower than 1 %. Due to the sharp shape of superspheroid in

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape the low range of concavity parameter ( p ≤ 0.3), a very refined finite element mesh is necessary on the outer surface of the pore and its vicinity. Complementary finite element modelling have been performed for the concave range p < 0.5 with the highest refinement inside the cube

surrounding the superspheroid ( 2L1 = 4 ).

The same 3D finite element meshes have been used for the heat conduction problem: the complete superspheroid has been modelled with quadratic isoparametric Lagrange finite elements. Two distinct uniform remote flux loadings, with non-zero components q1 , q3 are necessary to respectively determine the two distinct components R11 , R33 of the transversely isotropic resistivity contribution tensor.

The calculations have been performed with mesh (b). 3D Finite Element Modelling have been completed by 2D-axisymmetrical modelling, for the loading case q3 which respects symmetry of revolution, to obtain complementary results for R33 component. Isoparametric

Lagrange quadratic finite elements have been used with two dimensional 6-node triangular finite elements in the host matrix and 3-node quadratic Lagrange finite element on the superspheroid boundary. 2D-axisymmetric analysis allowed substantial accuracy increase in the low range of concavity parameter ( 0.2 ≤ p ≤ 0.35 ). Convergence of results with refinement of 2D-axis meshes has been obtained in all the range 0.2 ≤ p ≤ 1, with relative error

lower than 1 %. Numerical results for R11 , R33 components of contribution resistivity tensor are given in Table D4 and Figure 4.7 (b).

2.3 Analytical approximations for property contribution tensors of a superspheroidal pore

Note, that as discussed by Sevostianov et al. (2008) and Sevostianov and Giraud (2012), to increase the accuracy of the calculations, extra strain due to the cavity ∆ε in (4.9) may be

3 alternatively normalized to L* where L* is certain characteristic length of the pore (for example, the maximal distance between its far points):

L3 ~ ∆ ε = * H : σ ∞ (4.14) V ~ Certainly, tensors H and H entering relations (4.9) and (4.14) are different since they represent different normalizations of the pore contribution to the effective compliance.

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

~ Sevostianov and Giraud (2012) and Chen et al. (2015b) shown that tensor H for a concave superspherical pore ( p < 0.5) can be approximated by linear functions of the concavity factor p . Similar observation has been done by Chen et al. (2015b) in the context of resistivity

3 contribution tensor: normalization of ∆(∇T ) to L* yields

L3 ~ ∆()∇T = * R ⋅ q (4.15) V ~ and components of R show linear dependence on the concavity parameter for p ≤ 1.

Let us check this approach for the superspheroidal pore described by (4.4). Taking into account that L* = 1 (superspheroid of unit radius), we get the following connection between ~ ~ components of tensors H and H , and R and R : ~ ~ Hijkl = V* ( p)Hijkl , Rij = V* ( p)Rij (4.16) ~ ~ so that components Hijkl and Rij can be calculated from the data in Tables D1 and D4 using (4.5) and (4.6). Figure 4.8 illustrates the accuracy of linear approximation for these component. It is ~ ~ ~ seen that while H 3333 , H1313 and R33 allow reasonably accurate linear approximation, ~ ~ components H1111 and R11 show noticeable deviation from the linear behavior. Moreover, as seen from Table D1 and Figure 4.7, H1111 , R11 , and H1122 components (i.e being normalized according to 4.9) are almost constant on the entire range of variation of p.

To find analytical approximation for compliance and resistivity tensors, we first consider a spheroidal pore of aspect ratio γ with the equation of the surface

x2 x2 x2 1 + 2 + 3 = 1 (4.17) a2 a2 γ a 22 Components of the compliance contribution tensor for such a pore are given by (Sevostianov et al. 2006):

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

κ 0( f0 − f1 ) 1 H1111 = 2 + ; 2(4κ 0 − 1)G0 []2κ 0( f0 − f1 ) − (4κ 0 − 1) f0 G0 []1 − 2 f0 + κ 0( f0 − f1 )

4κ 0 − 1 − 6κ f00 + 2 f0 − 2κ f10 H3333 = 2 ; 4(4κ 0 − 1)µ0 []2κ 0( f0 − f1 ) − (4κ 0 − 1) f0 κ ( f − f ) 1 H = 0 0 1 − ; 1122 2(4κ − 1)G []2κ ( f − f ) − (4κ − 1) f 2 G []1 − 2 f + κ ( f − f ) 0 0 0 0 1 0 0 0 0 0 0 1 (4.18) f0 − 2κ 0( f0 + f1 ) H1133 = 2 ; 4(4κ 0 − 1)µ0 []2κ 0( f0 − f1 ) − (4κ 0 − 1) f0

H1111 + H1122 1 H1212 = = ; 2 G0 []1 − 2 f0 + κ 0( f0 − f1 ) 4 H1313 = ; 4µ0 []f0 + 4κ f10 where

2 2 1 3K0 + µ0 γ (1 − g) γ 2 κ 0 = = , f0 = 2 , f1 = 2 [](2γ + 1)g − 3 (4.19) (12 −υ0 ) 3K0 + 4µ0 2(γ − 1) 4(γ − 1) and shape factor g is expressed in terms of the spheroid’s aspect ratio γ as follows

 1 1 − γ 2  arctan , oblate ()γ < 1  γ 1 − γ 2 γ g(γ ) =  (4.20) 1 γ + 1 − γ 2  In , prolate ()γ > 1  2 2 2γ γ − 1 γ − γ − 1

~ For oblate spheroid, components Hijkl , shown in Figure 4.9, are obtained from (4.18) via multiplication by the aspect ratio γ . Comparing the curves in Figs 4.9 and 4.8(a), one can see ~ ~ ~ that H 3333 , H1313 and H1133 for a superspheroid (4.4) plotted as functions of the concavity parameter p almost coincide with corresponding curves for spheroid (4.17) plotted as functions of the aspect ratio γ . The only difference is that the curves for a spheroid have to be slightly shifted to the right. Interestingly, the best fit is given by shifting α equal to the difference between p and γ at which volumes of the superspheroid (4.4) and spheroid (4.17) vanishes (see Fig 4.5b). In other words, when

p − α γ = (4.21) 1 − α

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

~ ~ E0H3333 E0H1122 3.0 2.0 ~ ~ E H E0H1313 0 1133 ~ k0R33 ~ E H 0 1111 ~ k0R11 2.0 1.5

1.0 1.0

0.0 0.5

(a) (b) p -1.0 p 0.0

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

Fig.4.8: Linear approximation of the non-zero components of the compliance contribution tensor (a) and resistivity contribution tensor (b) normalized according to (4.14) and (4.15)

3.0 γ E0H3333

γ E0H1313 2.0

γ E0H1111

1.0 γ E0H1122

γ E0H1133 0.0

γ -1.0 0.0 0.2 0.4 0.6 0.8 1.0

Fig.4.9: Non-zero components of the compliance contribution tensor of oblate spheroidal pore using normalization (4.14)

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

3.0 3.0 (a) p −α (b) γ = 1−α 2.0 2.0

1.0 1.0

0.0 0.0

-1.0 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 γ p, p ~ E H ~ 0 3333 E0H1133 E0H1111

~ E0H1313 E0H1122

Fig.4.10: Comparison of approximation (4.23) with numerical values of non-zero components of the compliance contribution tensor for a superspherical pore

In our calculations, we used α = 0.15 – at this value, ratio of the volume of the superspheroid to the volume of superscribed sphere is smaller than 0.01. We cannot say the same for ~ ~ components H1111 , H1122 , however. On the other side, components H1111 and H1122 (i.e. components of the compliance contribution tensor normalized by 4.9) are almost independent of p and coincide with the corresponding components for a sphere. An important observation is that components of the resistivity contribution tensor for a superspheroid behave in the same ~ way: R11 is almost independent of the concavity parameter p , while R33 can be accurately ~ approximated by R33 for a spheroid of aspect ratio (4.21). Figure 4.10 and 4.11 provide an illustration.

Taking into account that expression of the compliance and resistivity contribution tensors for a spherical pore are, (see Kachanov et al. 2003, Sevostianov et al. 2006)

(13 −υ ) H = 0 []10(1 + υ )J − (1 + 5υ )II (74 − 5υ )(1 + υ )G 0 0 0 0 0 (4.22) 3 Rij = δij 2k0

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

1 (where J = (δ δ + δ δ ) and II = δ δ are fourth rank unit tensors), the following ijkl 2 jlik jkil ijkl ij kl approximate representation for components of compliance and resistivity contribution tensors for a superspheroid (4.4) can be written:

(13 −υ0 )(9 + 5υ0 ) − (13 −υ0 )(1 + 5υ0 ) H1111 = ; H1122 = 4(7 − 5υ0 )(1 + υ0 )µ0 4(7 − 5υ0 )(1 + υ0 )µ0

15(1 −υ0 ) H1212 = 4(7 − 5υ0 )µ0

γ ( p) 1 −υ0 (1 + υ0 ) − (1 + 2υ0 ) f0 − f1 H 3333 = 2 (4.23) V* ( p) 4µ0(1 + υ0 ) ( f0 − f1 ) − (1 + υ0 ) f0

γ ( p) 1 −υ0 −υ f00 − f1 H1133 = 2 V* ( p) 4µ0(1 + υ0 () f0 − f1 ) − (1 + υ0 ) f0

γ ( p) 1 −υ0 1 H1313 = V* ( p) µ0 (1 −υ0 ) f0 + 2 f1 and 3 γ ( p) 1 R = R = ; R = 11 22 2k 33 V ( p) 2 f 0 * 0 (4.24)

2.0 2.0 (a) (b)

1.5 1.5

p −α γ = 1.0 1.0 1−α

~ γ Rk Rk 0.5 0 33 0.5 0 33 Rk γ 0Rk 11 0 11 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 γ γ p,

Fig.4.11: (a) Dependence of the resistivity contribution tensor for a spheroidal pore normalized according to (4.15). (b) Comparison of the approximation (4.24) with numerical results

where functions f0 and f1 have to be calculated by (4.19) for γ (p) given by (4.21); volume of the superspheroid V∗(p) is given by expressions (4.5) and (4.6).

Ch.4-2: Compliance and resistivity contribution tensors of axisymmetric concave pores of superspheoidal shape

2.4 Concluding remarks

In the present work, we discussed compliance and resistivity contribution tensors of a

2 2 p 2 p concave axisymmetric pore which boundary is described by equation( x1 + x2 ) + x3 = 1 , and related contribution tensors are transversely isotropic. The present work completes recent paper Chen et al. (2015b) devoted to a concave superspherical pore (which leads to cubic contribution tensors).

We showed that, for p <1, contributions of an individual pore into elastic and electric can be approximated by expressions (4.23). Note also, that for the studied range of variation of the concavity parameter 0.2 < p < 1, these formulas may be further simplified as follows:

(13 −υ0 )(9 + 5υ0 ) − (13 −υ0 )(1 + 5υ0 ) H1111 ≈ ; H1122 ≈ 4(7 − 5υ0 )(1 + υ0 )µ0 4(7 − 5υ0 )(1 + υ0 )µ0 15(1 −υ ) H ≈ 0 1212 4(7 − 5υ )µ 0 0 γ ( p) (13 −υ0 )(9 + 5υ0 ) 1 − γ ( p) (12 −υ )10 H 3333 ≈ + V* ( p) 4(7 − 5υ0 )(1 + υ0 )µ0 V* ( p) πµ0

γ ( p) (13 −υ0 )(1 + 5υ0 ) H1133 ≈ − V* ( p) 4(7 − 5υ0 )(1 + υ0 )µ0

γ ( p) 15(1 −υ0 ) 1 − γ ( p) 1 −υ0 H1313 ≈ + (4.25) V* ( p) 4(7 − 5υ0 )µ0 V* ( p) π(2 −υ0 )µ0 3 (3π − 4 γ () p) + 4 R11 = R22 ≈ ; R33 = (4.26) 2k0 2πV* ( p) These formulas serve as the basic building block for calculation of the overall elastic and conductive properties of a material containing multiple concave pores.

Acknowledgement

Financial support from the FP7 Project TAMER IRSES-GA-2013-610547 and New Mexico Space Grant Consortium contained in the NASA Cooperative Agreement NNX13AB19A to New Mexico State University are gratefully acknowledged. The authors are also indebted to René Guitart for providing detailed information son Lamé’s contributions in mathematical physics.

Part 3

Part 3 Applications to the heterogeneous rock like materials

Preface

In wide-ranging engineering applications, quantitative characterization of inhomogeneities of 3-D non-ellipsoidal shapes has received particularly close attention in recent years. Predictions of the effective properties of heterogeneous materials show its particular importance in their integrations into the wide engineering utilizations.

In this part, effective thermal conductivity of oolitic rocks by using reformulated Maxwell homogenization method will be discussed in Chapter 1. Sevostianov and Giraud (2013) have evaluated effective elastic properties of such rocks containing superspherical pores, this work is done to complete their results by approximating superspherical pores by the best fit ellipsoid. The new estimations show a better accuracy due to newly developed numerical method.

Chapter 2 is considered as an extension of replacement relations that was originally derived explicitly by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, the applicability of the replacement relations for isotropic materials containing non-ellipsoidal inhomogeneities will be checked in the current chapter, this may lead to a straightforward solution of effective problems where these relations are well-funded. These relations might be applied to anisotropic materials, their relevancy for non-ellipsoidal inhomogeneities need to be verified. In practical utilization, replacement relations can be used in numerous applications in geomechanics and geophysics

ChaPter 5

V Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method (IJRMMS, 2015)

A.Giraud, I.Sevostianov, F.CHEN, D.Grgic, Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method, International Journal of Rock Mechanics & Mining Sciences 80(2015) 379-387

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

Abstract

The present work focuses on effective thermal conductivity of oolitic limestones, characterized by an assemblage of porous grains (oolites), mesopores and solid grains. Two distinct scales of pores, micropores or intra oolitic pores and mesopores or inter oolitic pores are taken into account. At the first step, micropores are homogenized inside the oolites by using self-consistent homogenization scheme. The second homogenization step describing transition from the mesoscale to the macroscale, is performed by using a recent reformulation of the Maxwell homogenization scheme (Sevostianov and Giraud 2013). At the mesoscale, porous oolitic inclusions are quasi spherical whereas two families of mesopores are considered according to analysis of photomicrographs: (1) randomly oriented oblate spheroidal pores and (2) concave pores. The proposed model is compared to a simplified one when all the pores are of ellipsoidal shape. The relevancy of the ellipsoidal approximation is then evaluated. In particular, the influence of the shape of the mesopores on the overall thermal conductivity is discussed

Keywords: effective thermal conductivity, Maxwell homogenization method, oolitic limestone, double porosity, concave pore

1.1 Introduction

The present work focuses on the determination of overall thermal conductivity of oolitic limestone that is modelled as a heterogeneous material composed of an assemblage of quasi- spherical porous grains (oolites, o), mesopores (b) and solid matrix (sparitic cement c) (Giraud et al. 2012, Grgic 2011, Nguyen et al. 2011, Ghabelzloo et al. 2009). For this goal, we use recently reformulated Maxwell's homogenization method for elastic composites (Sevostianov and Giraud 2013) which was successfully applied to the estimate of effective elastic constants of oolitic. This method has also been explored in Levin et al. (2012), Sevostianov (2014), Sevostianov et al. (2015) for mechanical properties and in Kushch et al. (2014) and Sevostianov et al. (2014) for conductivity. It must be emphasized that Maxwell's homogenization model has been presented in Zimmerman (1989) for the prediction of the thermal conductivity of fluid- saturated rocks. In particular, it has been shown in Zimmerman (1989) that this model allows us to predict overall thermal conductivity in a wide range of rock-microstructure type of sedimentary or crystalline rocks, by considering randomly distributed spheroidal pores.

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

One specific point related to the microstructure of heterogeneous oolitic porous rocks is the multiscale structure of pore space. The complex realistic pore structure can be simplified and the total porosity can be decomposed into two scale separated classes of pores: intra-oolitic pores or micropores, at the micro-scale, and inter-oolitic pores or mesopores at the mesoscale. The microstructural model presented in Giraud and Sevostianov (2013) is adopted in this paper. It is applicable if there is a separation of scale between micropores and macropores. The novelty of the model consists in the account for concave pores modelled by superspherical ones. Reformulation of Maxwell's homogenization method in terms of the resistivity contribution tensors allows accounting for non-ellipsoidal shape of such pores using the numerically evaluated contribution tensors (Chen et al. 2015b) when no analytical solution can be obtained. This model is then compared to a simple one in which the concave pores are replaced by the best ellipsoidal approximates.

Extensive study of the effective properties of carbonate rocks taking into account a multiscale description, with applications to elasticity, electrical conductivity, thermal conductivity and permeability in the context of a cross property analysis have been presented by Markov and coauthors Kazatchenko et al. (2006), Kazatchenko et al. (2007), Markov et al. (2009), Markov et al. (2014) and Aquino-López et al. (2015). In these works, two distinct classes of pores are considered: primary small scale pores and secondary mesoscopic pores. Small scale pores are similar to intra-oolitic pores of oolitic rocks, and secondary pores similar to inter-oolitic pores. A very complete description of the secondary (large scale) porosity is presented in Kazatchenko et al. (2006) by introducing four types of pores: vugs (quasi-spherical inclusions), quasi vugs (oblate ellipsoids), channels (prolate ellipsoids), and cracks (strongly oblate inclusions). The presence of an interphase coating oolites, an Interfacial Transition Zone (ITZ), is not taken into account in this paper but it is certainly important in porous oolitic rocks similar to the cement based material (see for example Lutz and Zimmerman (2005), for a study of the influence of Interfacial Transition Zone on effective conductivity).

Among many papers related to characterization of thermal conductivity of porous heterogeneous rocks, Alishaev et al. (2012) present an experimental characterization including a large sensitive study of physical parameters, in a wide temperature range. An extensive literature review on thermal conductivity measurements in carbonate rocks may be found in Thomas et al. (1973). Importantly, overall isotropy of thermal conductivity is observed in most cases. The authors pointed out that anisotropic single crystals, including calcite, show directional differences in thermal conductivity, but randomly oriented polycrystalline

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method aggregates produce an overall isotropic thermal effective conductivity. This result is consistent with the developed approach in the present paper. Random distribution of calcite phase mineral is accounted through averaging of the calcite mineral thermal conductivity (see Suquet et al. 2001) and by using equivalent isotropic conductivity. A detailed review of thermal conductivity data sets for geomaterials made of natural soil particles, crushed rock particles and sedimentary rock including analysis of influence of particle shape, grains pore-size distributions, fluid saturated (air and liquid water) is given in Côté and Konrad (2009). It may be noticed that Markov et al. (2015) have recently studied thermal conductivity of gas saturated porous materials taking into account methods of statistical physics and rarefied gas dynamics. They have shown that in the slip flow regime corresponding to low Knudsen number, it is necessary to use the method of rarefied gas dynamics to correctly predict the effective thermal conductivity. These phenomena may be of major importance in nanoporous materials and porous rocks such as gas shales. In the present study we focused on the micro- and meso- porosities of oolitic limestones, respective pore size correspond to the standard hydrodynamic regime and these phenomena have been neglected.

1.2 Background results

Hereafter we define some notations and recall some results which are needed later. Upper case bold symbols P, Λ, I refer to second order tensors, lower case bold symbols, as a,ε,σ refer to first order tensors. I represents the second-order, identity tensor ( δij denotes

Kronecker delta symbol, δij = 1ifi = j , δij = 0 other-wise). For the sake of simplicity only the polarization tensors related to spheroidal inclusions are recalled. The general case of the 3D ellipsoid could be also considered in what follows. Oblate spheroid ( 0 < γ < 1) and prolate spheroïd (1 < γ ), with symmetry axis 3 are described by the equation

x2 x2 x2 x ∈ Ω ⇔ 1 + 2 + 3 ≤ ,01 ≤ γ ≤ 1 a 2 a 2 a γ 22 (5.1)

a1 = a2 = a,a3 = aγ Hill Polarization tensor of a spheroidal inclusion in an infinite isotropic elastic medium (λ denotes the conductivity of the infinite isotropic medium surrounding the spherical inclusion) is a transversely isotropic second order tensor

Q γ )( 1 − 2Q γ )( P = ΙT + Ι N λ λ (5.2)

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

Ι N = e3 ⊗ e3, ΙT = e1 ⊗ e1 + e2 ⊗ e2 , I = I N + IT (5.3) with (Torquato 2002, Carslaw and Jaeger 1959, among many others)

 1  1  γ  v   1 + 1 − arctan  ,if γ < 1      2  u  v  γ   1  ,if γ = 1 Q(γ ) =  3 (5.4) 1  1  γ  γ + u   1 + 1 − In  ,if γ > 1     2  u  2 u  γ − u   2 2  v = 1 − γ ,u = γ − 1 Hill Polarization tensor of a spherical inclusion (γ=1 and Q=1/3)

1 P = Ι (5.5) 3λ Q tensor (Sevostianov and Giraud 2013, Eq. 2.9) for Maxwell homogenization scheme

Q = Λ.(I − P.Λ) (5.6) In the case of a sphere (using 5.5 and Λ = λΙ )

2 Q = λΙ (5.7) 3 Conductivity contribution tensor (Sevostianov and Mishuris 2014)

−1 −1 Ni = []P + ( Λi − Λ) (5.8) Resistivity contribution tensor (Sevostianov and Mishuris 2014)

H = − −1 ⋅ ⋅ −1 i Λ Νi Λ (5.9) Dilute concentration tensor

−1 Ai = []I + P ⋅( Λi − Λ) (5.10) In this paper, the reformulation of the Maxwell homogenization scheme in terms of compliance contribution tensor (5.9) will be used. For a spherical inclusion i(λi ) embedded in an infinite isotropic matrix

N = N I,H = H I, A = A I i i i i i i (5.11) 3λ A = i 2λ + λ i (5.12) and

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

λ − λ H = i A ,N −= (λ − λ )A (5.13) i λ2 i i ii

In the case of an isotropic ellipsoidal inclusion with conductivity tensor Λi = λi I surrounded by an infinite isotropic matrix with conductivity tensor Λ = λI (5.1) relation (5.5) gives

 1 1    Ai = λ IT + I N  (5.14)  λ 1( − Q(γ )) + λiQ γ )( 2λQ γ )( + λi 1( − 2Q(γ ))  Orientational average (case of a random orientation distribution)

1 a = Tr )( Ia (5.15) 3 One obtains spherical (isotropic) second order tensors

Ai = Ai I,Hi = Hi I (5.16) with

λ  2 1    g λ,( λi γ ), =  +  (5.17) 3  λ 1( − Q(γ )) + λiQ γ )( 2λQ γ )( + λi 1( − 2Q(γ )) 

Ai = g(λ,λi γ ), (5.18) One verifies relations

λ − λ H = i A (5.19) i λ2 i 1.3 Microstructure of a reference porous oolitic limestone

One presents in this section some microstructural observations of a reference porous oolitic rock chosen for this work, an oolitic limestone from Lavoux (West of France, Sterpenich et al. 2009) whose micro-structure has been detailed Giraud and Sevostianov (2013). One may also refer to Nguyen et al. (2011) and Giraud et al. (2012), for details. Oolites are quasi-spherical grains constituted by concentric porous layers. The oolite microstructure are constituted by an assemblage of calcite grains (micrite, solid grains with diameter range 1–5 μm) and micropores. More precisely oolites contain quasi-spherical grains composed of concentric layers, diameter range 100 μm to 1 mm, the layers are composed of an assemblage of micropores and micrite grains. Micrite or microcrystalline calcite is composed of spherical grains constituting solid oolitic phase, diameter range 1–5 μm. Sparitic calcite cement or sparite, spar calcite, diameter range 20–100 μm, it corresponds to the solid phase at the mesoscale (Fig. 5.1). Solid phase of Lavoux limestone, i.e. micritic grains inside oolites and sparitic cement between oolites, is a

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method quasi-mono-mineral material constituted of pure calcite (solid volume fraction of calcite,

s fcalcite = Ωcalcite Ωs ≈ 0.98 ).

1 mm 100 µm

30 µm

Fig.5.1: Scanning Electron Microscopy (SEM) images of Lavoux limestone (see Giraud and Sevostianov 2013).

The total pore volume fraction of Lavoux limestone varies from 0.15 to 0.30, it is decomposed into two classes of pores: inter-oolitic pores (mesopores) and intra-oolitic pores (micropores) of approximately equal partial porosities Giraud et al. (2012). As in Giraud and Sevostianov (2013), a sensitive study is performed in this paper on a relatively wide range of porosity. Mesoscopic pores are divided into two types of pores: oblate spheroidal pores of aspect ratio γ = 0.2 (index b1 in what follows) and ellipsoidal pores (index b2) replacing superspherical concave pores taken into account in Giraud and Sevostianov (2013).

Two cases are considered, flat pores similar to cracks are represented by oblate spheroidal pores of aspect ratio γ = 0.05, and elongated pores cracks are represented by prolate spheroidal pores of aspect ratio γ = 20. The partial porosities of the two kinds of pores are assumed equal. Flat and elongated pores may respectively correspond to particular cases of secondary pores of carbonate rocks described in Kazatchenko et al. (2006): channels (prolate ellipsoids, as elongated pores), and cracks (flat oblate ellipsoids).

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

At the macroscopic level, experimental characterizations of the Lavoux limestone show that the overall elastic behavior is not far from isotropy. It results from an isotropic or random distribution of constituents: oolites, sparitic cement and mesopores. Thermal conductivity needs to be measured on this reference material (and experimental study is currently performed and results will be presented in a next paper), and it is assumed that the overall conductivity is also isotropic. See Thomas et al. (1973) for an extensive review on thermal conductivity of carbonate rocks which confirms reason-ably overall isotropy in most cases. The thermal

−1 −1 conductivity of the pure calcite mineral will be assumed equal to λc = 3.3Wm K (according to Vasseur et al. (1995) and Guéguen et al. (1997)). The numerical values of thermal

−1 −1 conductivities for liquid and air are respectively λl = 0.5984Wm K and

−1 −1 λg = 0.0255Wm K (according to Clauser and Huenges 1995).

1.3.1 Volume fractions and constituents

As in Giraud et al. (2012), Giraud and Sevostianov (2013) three different scales may be identified. First the smallest scale, referred as microscopic scale, corresponds to the intra- granular or intra-oolitic level. Second the intermediate scale, referred as mesoscopic scale, corresponds to the scale of oolite grains, syntaxial calcite grains (referred as sparitic calcite cement, and inter-oolitic pores. Third the largest scale, referred as macroscopic scale, corresponds to a large representative volume elements compared to the oolite size, inter-oolitic pores and syntaxial calcite grains. At the mesoscale, one considers a four phase composite material composed of poroelastic oolites (o) (constituted by solid grains and intra-oolitic pores, or micropores), inter-oolitic mesopores divided into two families b1 and b2, and sparitic cement (or syntaxial calcite) constituted by pure calcite grains (index c).

A two scale porosity is considered as two population of voids may be identified. Intra-oolitic voids of spherical or ellipsoidal shape, with an average diameter of 0.1 μm, referred as micropores in what follows, with index a. Inter-oolitic voids of ellipsoidal shape, with an average diameter approximately of 10 μm, referred as mesopores b, divided into two families of equal volume fraction, indexes b1 and b2. The total volumes occupied by the phases write

Ω = Ω0 + Ωb + Ωc (5.20) with

Ωb = Ωb1 + Ωb2 (5.21)

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

One has

Ω = Ω0 + Ωb1 + Ωb2 + Ωc (5.22) with corresponding volume fractions

Ω Ω Ω Ω f = 0 , f = b1 , f = b2 , f = c 0 Ω b1 Ω b2 Ω c Ω (5.23)

f0 + fb1 + fb2 + fc = 1 One defines the total volume of intra-oolitic pores

S Ω0 = Ωa + Ω0 (5.24) and the porosity of the oolite phase at the mesoscopic scale

Ωa fa = (5.25) Ω0 The total pore volume can be expressed as

Ω = Ω + Ω + Ω p a b1 b2 (5.26) and the total porosity

Ω Ω Ω f = a + b1 + b2 = ff + f + f (5.27) p Ω Ω Ω a 0 b1 b2

Representative data for the reference material studied in this paper are close to f p = 0.26 ,

f0 = 0.74 , a ff 0 = 0.14 , fa = 0.19 , fb = 0.12 (Giraud et al. 2012) but a sensitivity study will be performed on the volume fractions of the different phases.

In sensitive study, as in Giraud and Sevostianov (2013), we will consider fb1 = fb2 = fb /2 .

1.4 A two-scale porosity model for effective thermal conductivity of isotropic porous oolitic rocks

1.4.1 First step: homogenization of micropores and solid grains inside oolites

The first step represents the transition from the microscopic scale to the mesoscopic scale. Oolite pores are homogenized and the result of the first step is the porous oolite (Fig.5.2).

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

Step 1 : Homogenization of micropores in oolite

Step 1 : Microscopic level to mesoscopic level Oolitic

Porous oolitic o Oolite pores Solid oolite

Fig.5.2: First homogenization step: micropores inside oolite core are homogenized by using self-consistent method (2D representation of a 3D microstructure)

As previously indicated, the granular and random micro-structure of oolites conduces to choose the self-consistent approximation, originally due to Bruggeman (1935) and, for the step

I I. By respectively denoting λ0 ,λa ,λ0 conductivity of the oolite at the mesoscale, conductivity of the intra-oolitic porous phase, conductivity of micritic solid grains ( λ0 = λc ), the well-known self-consistent approximations for a two phase material with spherical particles is the positive root of the quadratic equation (see [Torquato 2002, formula (18.13), p. 463])

f λaa (1 − fa )λ0 1 I + I − = 0 (5.28) λa + 2λ0 λa + 2λ0 3 It’s well known solution writes (see [Torquato 2002, formula (18.14), p. 463], with

d = 3, f2 = 1 − f1 )

1 2 h(λ1,λ2 , f1 ) = ()α12 + α12 + 8λ λ21 (5.29) 4

α12 = λ1(3 f1 − 1) + λ2(2 − 3 f1 ) (5.30) and then self-consistent approximation writes

I λ0 = h(λa ,λ0 , fa ) (5.31)

It may be noticed that, in the tested range of microporosity fa < 0.2 , as we consider spherical particles, numerical differences between Self-Consistent approximation, and other approximations such as Maxwell or Mori Tanaka would not be very significant. In the present

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method model, the impact on overall thermal conductivity is limited when compared to the second homogenization step which corresponds to the mesoscopic scale. This comment is restricted to the tested range of microporosity and it would be irrelevant for a wider range.

1.4.2 Second step: transition from the mesoscopic scale to the macroscopic scale with ellipsoidal pores

At the mesoscale one considers a four phase heterogeneous medium, which is composed of porous oolites, nearly spherical and randomly distributed, it is the main phase ( f0 ≈ 0.74 in the reference case), pure solid calcite referred as sparitic cement (index c), mesopores modelled as two distinct families of ellipsoids randomly distributed in orientation. Reformulation of Maxwell homogenization scheme recently by Sevostianov and coauthors in the context of elasticity and conductivity problems (Sevostianov and Giraud 2013, Giraud and Sevostianov 2013, Sevostianov 2014, Sevostianov and Mishuris 2014) is then used for the transition from the mesoscopic scale to the macroscopic scale. More precisely reformulation of Maxwell homogenization scheme in terms of resistivity contribution tensor provides (see Giraud and Sevostianov 2013 relation 5.10 using compliance contribution tensor) the simple scalar relation established under assumption of macro-isotropy.

−1 eff  1  −1 2   λ =  + f H + f H + f H − λ  − 1 (5.32) MX  ()b1 b1 b2 b2 0 0 c    λc  3  

with H b1 , H b2 and H 0 respectively given by formula (5.19) – (5.13)

λc − λb λc − λb H b1 = 2 g(λc ,λb ,γ b1 ),H b2 = 2 g(λc ,λb ,γ b2 ) (5.33) λc λc 3(λ − λI ) c 0 (5.34) H 0 = I λc(2λc + λ0 ) From relation (5.32) one may easily deduce an explicit formula for the effective thermal conductivity:

eff 3 − 2λc ( fb1H b1 + fb2 H b2 + f0 H 0 ) λMX = λc (5.35) 3 + λc ( fb1H b1 + fb2 H b2 + f0 H 0 )

It should be emphasized that a more complex description of the mesoscale porosity could be easily introduced in the reformulation of Maxwell homogenization scheme in terms of resistivity contribution. As an example, the fourth type of secondary pores distinguished in

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

Kazatchenko et al. (2006) for carbonate rocks could be introduced as randomly oriented spheroids are considered and relation (5.32) would be replaced by

−1 −1 eff  1  −1 2   λ =  + f H + f H + f H + f H + Hf − λ  (5.36) MX  ()b1 b1 b2 b2 b3 b3 b4 b4 00 c    λc  3  

with fbi and H bi respectively denoting volume fraction and resistivity contribution tensor of the mesoscale pore bi. As oolites are not perfectly spherical, a random distribution of the best ellipsoidal approximation of oolites (oblate spheroids with aspect ratio close toγ = 0.7 ) could be also used and relation (5.32) would be replaced by

−1 −1 eff  1  −1 2   λ =  + f H + f H + Hf − λ  (5.37) MX  ()b1 b1 b2 b2 00 c    λc  3  

with H 0 given by a relation similar to (5.33). In further works we will also take into account more realistic shapes such as concave shapes (Chen et al. 2015b). As previously indicated, ellipsoidal shape is adopted for simplicity and in this case the best ellipsoidal approximation of more complex shapes needs to be used (Fig.5.3).

It may be noticed that relations (5.32)–(5.36)–(5.32) are very similar to relation (4.9) of paper Sevostianov and Giraud (2013), and relations (5.13)–(5.17) of paper Giraud and Sevostianov (2013), which have been obtained in elasticity.

1.4.3 Second step: transition from the mesoscopic scale to the macroscopic scale with concave pores

As in Giraud and Sevostianov (2013), concave pores (b2) are approximated by super-spheres

2 p 2 p 2 p x1 + x2 + x3 = 1 (5.38) of concavity factor p = 0.35 (Fig.5.4).

For pores of the shape of supersphere of concavity factor p, approximate expressions for components of the resistivity contribution tensor have recently be obtained by Chen et al. (2015b) (see Chen et al. 2015a, for the elasticity problem) thanks to a numerical method (Finite Element Method). As this tensor is spherical (hydrostatic) one have

H b2= H b2 (5.39)

Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

Numerical values of the resistivity contribution tensors are given in the next section, as they depend on the thermal conductivity of the surrounding matrix (λc ) and the inclusion (liquid water and air will be considered).

Step 2 : Maxwell Homogenization scheme Meso pores (b2) (randomly oriented) Homogenized Sparitic (c) Meso pores (b1)

Sparitic cement

Fig.5.3: Second homogenization step: transition from mesoscale to macroscale with Maxwell homogenization method, case of simplified model (ellipsoidal approximation for all the pores).

Fig 5.4: Mesoporosity: ellipsoidal pores (b1) and concave pores (b2) (Giraud and Sevostianov 2013)

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1.5 Numerical results

1.5.1 Experimental data

We present in this section a comparison of the developed micro–macro model with experimental results for relatively pure calcitic limestones presented in Robertson (1988). Oolitic limestones studied in this paper belong to this type of limestone. Intra-oolite solid grains and inter-oolite sparitic cement are composed of calcite. Extensive bibliographical review and data relative to thermal conductivity of limestones may be found in Alishaev et al. (2012), Popov et al. (2003) , and in Zimmerman (1989) for porous rocks more generally.

Air Saturated – Experimental data Liquid Saturated – Experimental data eff eff λMX 3.0 λMX 3.0 2.5 2.5

2.0 2.0 1.5 1.5 1.0 1.0

0.5 0.5 fb fb 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig.5.5: Pores saturated with air: experimental Fig.5.6: Pores saturated with water liquid: thermal conductivity as a function of porosity, for experimental thermal conductivity as a function of pure calcitic limestones (Robertson 1988) porosity for pure calcitic limestones (Robertson 1988) Experimental results are presented in Figs. 5.5 and 5.6 and they cover a wide range of porosity (upper porosity is close to 0.7 which corresponds to highly porous rocks). In this paper, we restrict the study to the range 0 < fb < 0.3which is relatively large for oolitic limestones. Upper value of thermal conductivity is given by the conductivity of the calcitic solid phase and it corresponds to the case of zero porosity. As expected, as conductivities of air and liquid water are lower than that of solid, effective thermal conductivity decreases with increasing values of porosity. In both cases, pore space respectively fully saturated by air or liquid water, effective thermal conductivity – porosity curve may be accurately fitted by a linear function of porosity (see Figs.5.5 and 5.6). As experimental data collected in bibliographical review are not restricted to the detailed investigated microstructure type (oolitic limestone rocks with two- scale porosity), the aim of the comparison is not to very accurately fit the data but mainly to

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Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method recover qualitative results, in particular effects of porosity and saturating fluid on overall thermal conductivity.

1.5.2 Simplified model based on ellipsoidal approximation for all pore families

The sensitive study is similar to the one presented in Giraud and Sevostianov (2013) . For the simplified model based on ellipsoidal approximation, mesopore family b2 is modelled by flat oblate pores close to cracks ( γ = 0.05 ) or elongated prolate pores (γ = 20 ) instead of concave pores of superspherical shape.

−1 −1 Two cases are considered: porosity fully saturated with liquid water ( λ = 0.5984Wm K ),

−1 −1 wet case, and porosity fully saturated with air ( λa = 0.0255Wm K ), dry case.

Figure 5.7–5.10 illustrate dependence of the effective thermal conductivity on the mesoporosity (assuming that b1 and b2 pore families have the same volume fractions) at different levels of the volume fraction of oolites ( f0 = 0 04 05 06 .,.,.,. 7 ).

As it has been observed in elasticity Giraud and Sevostianov (2013), distribution of the volume concentration between spar calcite and porous oolitic grains plays minor role as compare to volume fraction of pores.

As expected, to the higher conductivity contrast between solid calcite mineral and air

λ g ≈ 129.4 (5.40) λ compared to the corresponding ratio between solid calcite and liquid water

λ c ≈ 5.5 (5.41) λ the higher impact of the porosity on the overall thermal conductivity is observed in the dry case. Due to this higher contrast, the case of the air saturated porosity is the most interesting for a micro–macro characterization because it is more sensitive to pore shape than the case of water liquid saturated (air saturated porosity is similar to an insulating phase). Comparisons between model and experiments show that flat oblate spheroidal pores (γ = 0.05 ) overestimate the effect of porosity on the overall in the case of air saturated pores. This overestimate is less pronounced in the case of liquid saturated pores. In both cases, liquid and air, prediction of model are more accurate with prolate spheroidal elongated pores (γ = 20 ) than crack similar oblate pores.

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Air Saturated – Air Saturated – eff eff λMX λMX 3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

1.0 1.0

0.5 0.5

fb fb 0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Fig.5.7: Pores saturated with air: effective thermal Fig.5.8: Pores saturated with air: effective thermal as a function of mesoporosity with b2=random conductivity as a function of Mesoporosity with distribution cracks (oblate ) b2=random distribution of needles (prolate )

Water liquid Saturated – Water liquid Saturated – eff λMX eff

3.0 λMX 3.0

2.5 2.5

2.0 2.0

1.5 1.5

fb fb 1.0 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Fig.5.9: Pores saturated with liquid water: effective Fig.5.10: Pores saturated with liquid water: effective thermal conductivity as a function of mesoporosity thermal conductivity as a function of mesoporosity with b2=random distribution of cracks (oblate with b2 random distribution of needles (prolate ) )

Remark: In Figure 5.7-5.10, -blue: f0 = 0.4, yellow f0 = 0.5, green, f0 = 0.6, red f0 = 0.7, experimental data taken from Robertson (1988) (for interpretation of the reference to color in this figure caption, the reader is referred to the web version of this paper)

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Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method

1.5.3. Model including concave pores family

One only replaces ellipsoidal pore family b2 by superspheres of concavity factor p = 0.35 and the same sensitivity study is per-formed. Preceding values of resistivity contribution tensor (formula 5.33) needs to be replaced by the following numerical values (see Chen et al. 2015b, for details on calculation of resistivity contribution tensors):

sup ersph H b2 (λc ,λ ) = 0.447518 (5.42)

sup ersph H b2 (λc ,λg ) = 0.528849 (5.43) and relation (5.35) can be used. Figs. 5.11 and 5.12 illustrate dependence of the effective thermal conductivity on the mesoporosity at different levels of the volume fraction of oolites. Comparison with experimental data shows that the model accurately predicts effect of porosity and effect of saturating fluid. Similar to the model with prolate elongated pores, variations of effective thermal conductivity with porosity in the most sensitive case of air saturated porosity are correctly reproduced. Microstructural observations (see Fig. 5.4) show that concave pore shapes are more relevant than elongated pore shapes to describe the mesopores filling the space between spherical oolites. Experimental data correspond to purely calcitic limestones, and it is not restricted to oolitic lime-stones. A more accurate validation of the model by comparison with experimental data specific to oolitic limestone will be necessary. Experimental study of effective thermal conductivity of Lavoux oolitic limestone is in progress and experimental results will be presented in a next paper. On the basis of the preceding results, a best ellipsoidal approximation can be proposed for this material by minimizing the difference between the numerical values of the resistivity contribution tensor respectively obtained for the supersphere (5.43)– (5.42) and for the spheroids (5.33), for the two cases (air saturated and water liquid saturated, Figs. 5.13 and 5.14)

Air−min λb2 ≈ 0.31 (5.44)

Lq−min λb2 ≈ 0.11 (5.45)

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Air Saturated – Supersphere Water liquid saturated – Supersphere eff eff λMX λMX 3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

1.0 1.0

0.5 0.5

fb fb 0.0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Fig.5.11: Pores saturated with air: effective thermal Fig.5.12: Pores saturated with liquid water: effective conductivity as a function of mesoporosity with thermal conductivity as a function of mesoporosity b2=random distribution of supersphere ( p = 0.35 ) with b2 distribution of superspheres ( p = 0.35 )

Remark: In Figure 5.11-5.12, -blue: f0 = 0.4, yellow f0 = 0.5, green, f0 = 0.6, red f0 = 0.7, experimental data taken from Robertson (1988) (For interpretation of the reference to color in this figure caption, the reader is referred to the web version of this paper)

Air Saturated Liquid (water) H b2 H b2

1.4 0.50

1.2

1.0 0.45

0.8

0.40

0.6

γ γ

0.05 0.10 0.50 1.0 5 10 0.05 0.10 0.50 5 10

1 Fig.5.13: Pores saturated with air: resistivity Fig.5.14: Pores saturated with liquid water: resistivity contribution tensor of spheroid as a function of pore contribution tensor of spheroid as a function of pore aspect ratio , comparison with supersphere aspect ratio ,comparison with supersphere ( , ) ( , )

Resistivity contribution tensor of a superspherical inclusion de-pends on the shape parameter (concavity parameter p) and on the contrast between respective thermal conductivities of

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Ch.5: Applications – Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method inclusion and matrix. This contrast is very different in the two extreme cases investigated (see 5.40 and 5.41). As a consequence, the best ellipsoidal approximation depends on the conductivities of constituents, and it yields different aspect ratios in the two cases.

1.6 Conclusion

The main factor affecting elastic properties of oolitic limestone is the pore space geometry. As identified in Giraud and Sevostianov (2013) some mesopores are concave and may be approximated by superspherical shape (Chen et al. 2015a, Chen et al. 2015b). A simplified model is presented by using ellipsoidal approximation for all the pores. This model could be used as a first approach for estimating overall thermal conductivity, its advantage being the simplicity of the corresponding homogenization model. Numerical results confirm the potential importance of the shape of mesopores for the conductivity problem (Giraud and Sevostianov 2013). In the two particular cases presented, air saturated and liquid water saturated pores, the comparison between resistivity contribution tensors of spheroids allows us to define in each case the best ellipsoidal approximation for a given superspherical shape.

Comparison between multi-scale model based on Maxwell homogenization method and experimental data shows that it correctly predicts effects of porosity and saturating fluid on overall thermal conductivity, when taking into account concave pore of superspherical shape. These results are interesting in relation to cross-property analysis between elastic coefficients and thermal conductivity. It confirms previous results obtained with a similar upscaling model for the same material, for the prediction of elastic coefficients Giraud and Sevostianov (2013). Experimental characterization of the thermal conductivity of such oolitic limestones, at different saturation ratios is also in progress and comparison with numerical results needs to be performed.

Acknowledgement

Financial support from the from TAMER European project (Trans-Atlantic Micromechanics Evolving Research Materials containing inhomogeneities of diverse physical properties, shapes and orientations), FP7 Project TAMER IRSES-GA-2013-610547, New Mexico Space Grant Consortium contained in the NASA Co-operative Agreement NNX13AB19A to New Mexico State University, and NSF grant DMR-1229558 are gratefully acknowledged.

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ChaPter 6

VI Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities (IJSS, 2016)

F.CHEN, I.Sevostianov, A.Giraud, D.Grgic, Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities, International Journal of Solids and Structures. 000 (2016) 1-8.

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

Abstract

In this paper, we focus on replacement relation that links the property contribution tensors of inhomogeneities having the same shape but different elastic properties. We check the possibility to apply the relations, originally derived for ellipsoidal inhomogeneities (Sevostianov and Kachanov 2007) to ones of non-ellipsoidal shape. We discuss inhomogeneities of superspherical shape, described by equation x 2 p + y 2 p + z 2 p ≤ 1 and show that the replacement relations can be used in the rank of convex shapes ( p > .50 ), while for concave shapes the error is significant. In practical applications, it means that for materials with convex inhomogeneities results obtained for effective elastic constants of a porous material can be used to approximately evaluate effective properties of a composite of the same morphology.

Keywords: Replacement relation, effective properties, supersphere

2.1 Introduction.

We focus on the problem of predicting the change in overall elastic properties of a material upon the change in pore-fill material properties. The problem in its present form was first addressed by Sevostianov and Kachanov (2007) who derived exact explicit relations that link the contributions of inhomogeneities having the same shape but different elastic constants, to the overall elastic properties. The relations are exact for ellipsoids and may be used as approximations for certain non-ellipsoidal shapes – the authors shown that relations can be used with satisfactory accuracy for a cube and for various 2-D shapes. This approach also constitutes an approximate connection between the first Eshelby’s problem (the eigenstrain problem) and the second one (the inhomogeneity problem), for non-ellipsoidal shapes.

Replacement relations play an important role in geomechanics in the context of the effect of saturation on seismic properties of rock. This problem was first addressed Gassmann (1951) who proposed the following relation expressing bulk and shear moduli K and G of fully saturated rock in terms of the elastic properties of dry rock:

dry 2 K ()1 − K K dry dry 0 eff 0 ; G = G (6.1) K eff = K eff + dry eff eff 1 − φ − K eff K0 + φ K0 K1 where subscripts “0”, “1” and “eff” note elastic constants of the matrix material, material filling the pores and effective properties of a composite (material with saturated pores), respectively;

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

dry φ is volume fraction of the inhomogeneities (porosity for the material with unfilled pores); K eff

dry and Geff are bulk and shear moduli of the porous material of the same morphology.

Gassmann’s equation is widely used in geophysics to predict effect of different fluid saturations on the wave speed. However, as mentioned by Han and Batzle (2004) porosity, density, and velocity are not independent and values must be kept consistent and constrained. They proposed to use Voigt-Reuss bounds and critical porosity limits constrain to get upper and lower bounds of the fluid-saturation effect on bulk modulus and compared the results with experimental observations. Further development of Gassmann’s equation was done by Ciz and Shapiro (2007) who obtained the following replacement relations for a particular case of an isotropic two-phase composite with isotropic constituents:

φK dry ()K − K + K ()K − K dry K ≈ K eff 0 1 1 0 eff eff 0 φ()K − K ()K − K dry + φK dry ()K − K + K ()K − K dry 0 1 0 eff eff 0 1 1 0 eff (6.2) dry dry φGeff ()G0 − G1 + G1 ()G0 − Geff Geff ≈ G0 dry dry dry φ()G0 − G1 ()G0 − Geff + φGeff ()G0 − G1 + G1 ()G0 − Geff where subscripts “0”, “1” and “eff” note elastic constants of the matrix material, material filling the pores and effective properties of a composite, respectively; φ is volume fraction of the

dry dry inhomogeneities (porosity for the material with unfilled pores); K eff and Geff are bulk and shear moduli of the porous material of the same morphology. The accuracy of the expressions (6.2) is unknown, as mentioned by Saxena and Mavko (2014a), who derived replacement relations (they use term “substitution relations”) under assumption that the strains and stresses inside the inhomogeneities are uniform (overall properties and properties of the constituents are isotropic).

For an isotropic rocks containing respectively inhomogeneities of the same shape, but different elastic constants, denoted by “A” and “B”, they got the following expression for the effective bulk modulus:

(K − K )(K − K ) ' 0 B 0 A 1)( 2)( (6.3) (K A − KB )α1 + (GA − GB )α2 = φ 1)( 2)( (Keff − Keff ) (K0 − Keff )(K0 − Keff ) where

ee P P K K K K  1 τ Aτ B  γ Aγ B α = BA = BA ; and α ' = A B α = A B  ij ij  = 2 ij ij 1 e e P P 2 G G 2 G G  2 P P  ee BA BA A B A B  BA  BA

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

Subscripts “ 1)( ” and “(2) ” referred to the composite containing “A” and “B”, respectively.

i Overbar stays for the average over the volume, Pi and e ( i ≡ A or B ) are the pressure inside the inhomogeneities produced by external load and the corresponding volumetric strains; τ i and γ i in equation (6.3) are deviatoric stresses and strains inside the inhomogeneities. The expression for the effective shear modulus can be written in the same manner:

(G − G )(G − G ) ( ) ( ) ' 0 B 0 A ( 1)( 2 )( ) (6.4) GA − GB β1 + K A − K B β 2 = φ 1)( 2 )( Geff − Geff (G0 − Geff )(G0 − Geff ) with

A B A B A B 1 τ ij τ ij 1 γ ij γ ij G G G G  P P  1 ee β = = and β ' = A B β = A B  ij ij  = BA 1 2 τ Aτ B 2 γ A γ B 2 K K 2 K K  τ Aτ B  4 γ A γ B 12 12 12 12 A B A B  12 12  12 12 The assumption used in the derivation is equivalent to the statement that the inhomogeneities are ellipsoids subjected to the uniform external field (Eshelby 1957, Lubarda and Markenscoff 1998, Rodin 1996).

Importance of the replacement relations for geophysical applications has been discussed in recent papers of Saxena and Mavko (2014b, 2015). In the general area of the calculation of effective properties of heterogeneous materials, if the accuracy of the replacement relations is sufficient, we see two main applications:

A: Effective properties of a composite can be calculated from the effective properties of a porous material having the same morphology.

B: The effective properties of a heterogeneous material can be evaluated using Eshelby tensors of the individual inhomogeneities.

Both of these applications are of crucial importance.

Calculation of effective properties of composites from the effective properties of porous materials gives us a possibility to reduce the amount of calculations of the property contribution tensors of inhomogeneity. Note that while for 2-D non-elliptical inhomogeneities many analytical and numerical results have been obtained (Zimmerman 1986, Kachanov et al. 1994; Tsukrov and Novak 2002, 2004), only a limited number of numerical results and approximate estimates are available for non-ellipsoidal 3-D shapes. Several examples of pores of irregular shape typical for carbon-carbon composites have been analyzed in the context of their contribution into overall compliances by Drach et al. (2011) using FEM. In the narrower context

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities of compliance of irregularly shaped cracks, certain results were obtained by Fabrikant (1989), Sevostianov and Kachanov (2002a) (planar cracks), Grechka et al. (2006) (intersecting planar cracks), Mear et al. (2007) (non-planar cracks), and Kachanov and Sevostianov (2012) (cracks growing from pores and cracks with partial contact between the faces). Concave inhomogeneity shapes have been analyzed by Sevostianov et al. (2008), Sevostianov and Giraud (2012), Chen et al. (2015b) and Sevostianov et al. (2016). Argatov and Sevostianov (2011) found the contribution of a thin rigid toroidal inhomogeneity into overall stiffness. The replacement relations allows to use these results for materials containing inhomogeneities of arbitrary properties.

Using Eshelby tensor allows one to reduce solving integral equations to evaluation of integrals (Mura 1987). For non-ellipsoidal shapes several results have been obtained by Rodin (1996) in the context of polygons and polyhedral. Onaka (2001) and Onaka et al. (2002) obtained expressions for the components of Eshelby tensor for a convex supersphere and toroidal inclusion respectively. Chen et al. (2015a) provided results for Eshelby tensor for a concave and convex supersphere. Generally, the Eshelby tensor for non-ellipsoidal inclusions is irrelevant for the problem of effective properties of heterogeneous material. In some papers, however, authors erroneously try to calculate effective properties of materials with non- ellipsoidal inhomogeneities using Eshelby tensor (see, for example Hashemi et al. (2009) where results of Onaka (2001) are used to calculate effective properties of a composite with cuboidal inhomogeneities). Replacement relations, when their accuracy is acceptable, allow using of this approach as an approximation.

Remark. Irrelevance of Eshelby tensor for calculation of the effective properties of a material containing non-ellipsoidal inhomogeneities immediately follows from Eshelby (1957) paper (see also book of Mura, 1987, where the results are better structured) and the paper of Lubarda, and Markenscoff (1998). Indeed, to calculate effective elastic properties using one of the standard micromechanical schemes (one particle approximations) one has to know the average strain field inside a single representative inhomogeneity embedded in the infinite space (see, for example, review of Markov, 2000, where details of derivation are given for several methods of homogenization). This problem is reduced to solving an integral equation for the strain tensor. If (and only if!) the strain field is uniform inside the inhomogeneity, it can be taken out of the integral and the equation is reduced to an algebraic one with the coefficients expressed in terms of the Eshelby tensor components. Such a reduction is possible in the case of ellipsoidal inhomogeneity only (Lubarda, and Markenscoff, 1998). Other shapes require

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities solving the integral equation (see, for example, recent result of Radi and Sevostianov (2016), on a material containing toroidal inhomogeneities) and Eshelby tensor cannot be used for evaluation of the elastic fields inside a single inhomogeneity and, therefore, is irrelevant for calculation of the effective properties.

In the text to follow, we provide numerical analysis of the applicability of the replacement relations to the case of the inhomogeneities of irregular shape on example of a supersphere of unit radius. This shape is described by the following equation:

2 p 2 p 2 p x1 + x2 + x3 = 1 (6.5) We call parameter p a concavity factor or a parameter of deviation from spherical shape. For p > 0.5 , its shape is convex, for p < 0.5 it is concave. When p = 1, the supersphere becomes the ordinary sphere. The sphere is transformed into a cube as p → ∞ . Examples of this shape are shown in figure 6.1 for different values of p .

p=5 p=2.5 p=1.5 p=1

p=0.5 p=0.4 p=0.3 p=0.2

Fig.6.1: Superspherical inhomogeneity for different values of p

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

2.2 Compliance and stiffness contribution tensors and replacement relations for ellipsoidal inhomogeneities.

Property contribution tensors are used in the context of homogenization problems to describe contribution of a single inhomogeneity into the property of interest. In the context of the effective elastic properties, one can use compliance contribution tensor of an inhomogeneity H that gives the extra strain produced by introduction of the inhomogeneity into the otherwise uniform stress field. Alternatively, one can use the dual stiffness contribution tensor N that gives the extra stress due to inhomogeneity when it is placed into the otherwise uniform strain field. Compliance contribution tensors have been first introduced in the context of pores and cracks by Horii and Nemat-Nasser (1983) (see also detailed discussion in the book of Nemat- Nasser and Hori 1993). Components of this tensor were calculated for 2-D pores of various shape and 3-D ellipsoidal pores in isotropic material by Kachanov et al. (1994). For general case of elastic inhomogeneities, these tensors were introduced and calculated (for ellipsoidal shapes) by Sevostianov and Kachanov (1999, 2002). Sevostianov et al. (2005) calculated components of this tensor for a spheroidal inhomogeneity embedded in a transversely-isotropic material.

We first consider a homogeneous elastic material (matrix), with the compliance and

∗ stiffness tensors S0 and C0 . It contains an inhomogeneity, of volumeV , of a different material with the compliance and stiffness tensors S1 and C1 . The contribution of the inhomogeneity to the overall strain, per representative volume V (the extra strain, as compared to the homogeneous matrix) is given by the fourth-rank tensor – the compliance contribution tensor of the inhomogeneity – defined by

V * ∆ ε = H : σ ∞ (6.6) V where σ ∞ is the “remotely applied” stress field, that, in absence of the inhomogeneity, would have been uniform within its site (“homogeneous boundary conditions”, Hashin 1983); a colon denotes contraction over two indices. Similarly, the stiffness contribution tensor N , dual to H , can be introduced:

V * ∆ σ = N : ε ∞ (6.7) V where ε ∞ is the “remotely applied” strain. 114

Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

For an ellipsoidal inhomogeneity, tensors H - and N -tensors are given by:

−1 −1 −1 −1 H = [](S1 − S0 ) + Q , N = [](C1 − C0 ) + P (6.8)

Hill’s tensor Pmnrs is calculated in terms of the gradient of the second rank Green’s tensor

Gij ()x − x′ as

∂ ∂G ()x − x′ P ()x ≡ mj dx′ (6.9) mpij ∂x ∫ ∂x′ p V1 i where parentheses at subscripts denote symmetrization with respect to m ↔ p and i ↔ j ; Hill’s tensor has the same symmetry as the tensor of elastic constants:

Pijkl = Pjikl = Pijlk = Pklij (6.10) Second Hill’s tensor Q is related to as follows (Walpole 1966):

0 0 Qijkl = Cijmn ()J mnkl − PmnrsCrskl (6.11)

Hereafter, J ijkl = (δ δljik + δilδ kj /) 2 and the inverse of symmetric (with respect to i ↔ j and −1 −1 −1 k ↔ l ) fourth-rank tensor X ijkl is defined by X ijmn X mnkl = X ijmn X mnkl = J ijkl .

From the expressions (6.8), one can observe that P and Q depend only on the inhomogeneity shape (and elastic properties of the surrounding material), but not on its elastic constants. Thus, these equations written for two inhomogeneities “A” and “B” having the same shape but different elastic constants (and placed in the same matrix) will contain the same Q and P . Excluding P and Q from them yields the following replacement relations (Sevostianov and Kachanov 2007):

−1 −1 −1 −1 H A − H B = ()()SA − S0 − SB − S0 (6.12)

−1 −1 −1 −1 N A − N B = ()()C A − C0 − CB − C0 (6.13) In particular, if material “B” is either rigid or represents a pore, the above relations take the form:

−1 −1 −1 −1 −1 −1 N A − N rigid = ()C A − C0 ; H A − H pore = ()SA − S0 (6.14) If the inhomogeneity A is a pore and B is perfectly rigid these relations simplify further:

−1 −1 −1 −1 N pore − N rigid −= S0 ; H pore − Hrigid = C0 (6.15)

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

For a non-ellipsoidal shape, tensors P and Q are not constant inside the inhomogeneity,

P = P ()x , Q = Q ()x and relations (6.8) do not take place. However, in some cases, these expressions can be used approximately with tensors P = P()x , Q = Q()x being replaced by their volume averages P and Q . It also leads to the approximation of the replacement relations (6.12, 6.13) since they follow from the representation (6.8). In the next section we check the accuracy of such approach for an inhomogeneity of a superspherical shape (6.5).

2.3 Calculation of compliance and stiffness contribution tensors and Hill tensor P for a superspherical inhomogeneity

To calculate P for a superspherical shape, we use its connection with Eshelby’s tensor S E :

E 0 Pijkl ( x) = Sijmn( x)Smnkl (6.16) and observation of Milgrom and Shtrikman (1992) on independency of the traces of Eshelby

E E tensor Siijj and Sijij on the inhomogeneity shape and can be calculated from the ones for a sphere:

E 1 +ν 0 E Siijj = ,Sijij = 3 (6.17) 1 −ν 0

Accounting for cubic symmetry of the superspherical inhomogeneity (Fig. 6.2), it leads to the following relations for isotropic matrix with shear modulus G0 and Poisson’s ratioν 0 :

1 −ν 0 2ν 0 1 P1111 + P1122 + 2P1212 = 1 − 2ν 0 1 − 2ν 0 2G0 (6.18) 1 1 − 2ν 0 P1111 + 2P1122 = 6G0 1 −ν 0

So that average tensor Pijkl has only two independent components. That can be calculated as (Onaka 2001)

1 1 f ( p)  P1212 =  +  (6.19) G0 3 8π(1 −ν 0 ) with

1   xx '   f ( p) = 21 n dS n dS (6.20) *   3 1  2  V ∫ΩΩ ∫ x  

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21 where x k = xk − xk′,x = ( xx kk ) ,k = ,21 , x = ( x1 ,x2 ,x3 ) is position vector of a point on the surface, n1,n2 are the projections of the outwards unit vector normal to the surface of

* supersphere on xx 21 plane and V is the volume of the inhomogeneity. For the supersphere

∞ − xt −1 (6.5), V* is expressed in terms of Gamma function Γ x)( = te dt as ∫0

8m2 []Γ()m 3 1 V = , m = (6.21) * 3 Γ()3m 2 p Due to the geometric symmetry, only 1 8 of the real shape ( x > 0, y > 0,z > 0) need to be

considered for calculations. Figure 6.2 shows dependences of f ( p) and P1212 on the concavity parameter p by numerical integration of (6.19).

Combining expressions (6.8) and (6.11), we can calculate now components of the approximate compliance contribution tensor H′ for the superspherical inhomogeneity as

1 0 −1 0 0 −1 Hijkl′ = [](Sijkl − Sijkl ) + Cijmn ()J mnkl − PmnrsCrskl (6.22a)

1 0 −1 For a pore, (Sijkl − Sijkl ) vanishes and expression (6.22a) can be rewritten as:

0 0 −1 Hijkl′ = []Cijmn ()J mnkl − PmnrsCrskl (6.22b)

f ( p) P1212 -1.3 0.34 Convex shapes Convex shapes -1.4 0.33 -1.5 0.32 -1.6 0.31 -1.7 (a) (b) -1.8 p 0.30 p 0.1 1.0 10 100 0.1 1.0 10 100 Onaka (2001) for p > 1 Chen et al (2015)

Fig.6.2: Dependences of f (p) defined by equation (6.20) (a) and component P1212 of Hill’s tensor given by (6.19) (b) on the concavity parameter p

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

Note that tensor H′ is approximate since relation (6.8) is valid for ellipsoidal shapes only. The accuracy of the approximation can be evaluated comparing (6.22) with the numerical solution.

Components of tensor Hijkl has been directly calculated by Chen et al. (2015b) by FEM with

Siemens NX 1.0. Figure 6.3 illustrates the comparison of the components of tensors Hijkl′ and

Hijkl in dependence on the concavity parameter p (for reader’s convenience, the results are also given in Table F1 of annex F).

H1111 − H1122 8.0 2.5 Convex shapes (a) Convex shapes (b) 6.0 2.0

1.5 4.0 1.0

2.0 0.5 p p 0.0 0.0 0.1 1.0 10.0 0.1 1.0 10.0

H1212 6.0 Convex shapes (c) 5.0 FEM calculations

4.0 Evaluation via 3.0 replacement relations

2.0 p 1.0 0.1 1.0 10.0

Fig.6.3: Comparison of the components of the compliance contribution tensor calculated by equation (6.22b) with ones obtained by Chen et al. (2015b) using FEM

We observe that the accuracy is better than 10% for all components of Hijkl for convex shapes ( p > 0 5). . For concave shapes, however, the errors are significant indicating that expressions (6.22) produce insufficient accuracy. Therefore, the replacement relations (6.12), derived from them cannot be used and components of the compliance and stiffness contribution

118

Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities tensors for concave inhomogeneities need to be calculated numerically for each specific set of inhomogeneity’s elastic constants.

2.4 Using replacement relations for calculation effective properties of materials.

Below we use the replacement relations to express effective elastic properties of a material containing multiple superspherical inhomogeneities in terms of the effective properties of a porous material having the same morphology. Note that most widely used homogenization schemes can be reformulated in terms of property contribution tensors derived for non- interacting inhomogeneities (Sevostianov and Kachanov 2013). Below, we calculate effective properties in the framework of the non-interaction approximation and using Mori-Tanaka and Maxwell homogenization schemes.

2.4.1 Non-interaction approximation.

This approximation is reasonably accurate at low concentration of inhomogeneities ("dilute limit"). If interaction between the inhomogeneities is neglected, each inhomogeneity can be assumed to be subjected to the same remotely applied field and the effective properties can be expressed as the summation of the contributions of the matrix and the individual inhomogeneities so that the total over volume V strain is given by

0 1  ()()n n  ε S σ V H σ ij = ijkl kl + ∑ * ijkl  kl (6.23) V  n 

For material containing inhomogeneities “A”, effective compliance tensor can be expressed as:

NIA Seff = S0 + φ H A (6.24)

In particular, for a material containing dry pores of the same shapes, its effective compliance is

NIA Sdry = S0 + φ H pore (6.25)

These two expressions can be interrelated using replacement relation (6.22) as:

NIA −1 −1 −1 Seff = S0 + φ[]()SA − S0 + φ()Sdry − S0 (6.26)

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

For isotropic mixture of the inhomogeneities, (6.26) yields the following expressions for

NIA NIA effective bulk and shear moduli, Keff and Geff (subscript NIA indicates non-interaction approximation)

φ K NIAK (K − K ) + K K (K − K NIA ) K NIA = dry 0 0 A A 0 0 dry eff φ K (K − K ) + K (K − K NIA ) 0 0 A A 0 dry (6.27) NIA NIA NIA φ Gdry G0(G0 − GA ) + GAG0(G0 − Gdry ) Geff = NIA φ G0(G0 − GA ) + GA(G0 − Gdry )

NIA NIA in terms of K dry and Gdry of the dry material given by

NIA K0 NIA G0 Kdry = ,Gdry = , (6.28) 1 + φηH K 1 + φηHG where ϕ is the overall porosity,

1 −ν 0 10(1 −ν 0 ) H k = ,HG = (6.29) 1 − 2ν 0 7 − 5ν 0 and η is the shape factor that for the superspherical shape has the following form (Chen et al. 2015b)

()53 p − 1  V0  η =   (6.30) 8 V∗ ()p 

(V∗ ()p is the volume of the supersphere given by (6.21) and V0 = 4π 3 is the volume of the ordinary sphere of the unit radius).

Expressions (6.27) and (6.28) involve volume fraction of the inhomogeneities. They may be rewritten in the form independent of volume fraction, but involving shape factor η instead:

(K − K ) + ηH K K NIA = K NIAK 0 A K A eff dry 0 K (K − K ) + ηH K K NIA 0 0 A K A dry (6.31)

NIA NIA (G0 − GA ) + ηHGGA Geff = Gdry G0 NIA G0(G0 − GA ) + ηHGGAGdry

In most applications, however, the shape factor is not a perfectly determinable parameter and form (6.27) looks like a more appropriate one. Non-interaction approximation serves as the basic building block for Mori-Tanaka and Maxwell scheme as indicated by Sevostianov and Kachanov (2013) and Sevostianov and Giraud (2013), respectively.

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2.4.2 Mori-Tanaka scheme

It belongs to the class of effective field homogenization schemes where each inhomogeneity, treated as a single one, is placed into the unaltered matrix material; interactions are accounted for by assuming that the inhomogeneity is subjected to the field that differs from the remotely applied one. The basic idea of the method has roots in works of Mossotti (Feynman et al. 1964, chapter 11). The Mori-Tanaka scheme (Mori and Tanaka 1973) as interpreted by Benveniste (1986) is based on the assumption that the effective field acting on each inhomogeneity is equal to the average over the matrix. Then the macroscopic properties may be calculated from the non-interaction approximation with appropriate change of the remotely applied field. Mori- Tanaka scheme is probably the most widely used homogenization scheme due to its simplicity and accurate predictions.

For material containing inhomogeneities “A”, effective compliance tensor can be expressed in terms of H A as (6.24):

MT −1 −1 Seff = S0 + φ H A : []φ ()SA − S0 : H A + ()1 − φ J (6.32)

−1 In particular, for dry material ()Spore − S0 → 0 and

φ S MT = S + H (6.33) dry 0 ()1 − φ pore

MT MT Using replacement relation (6.12) we can now express Seff in terms of Sdry as

MT −1 MT −1 −1 Seff = S0 + φ [](SA − S0 ) + φ (Sdry − S0 ) (6.34)

In the case of isotropic microstructure,

K (K − K MT ) + φ K MT (K − K ) K MT = K A 0 dry dry 0 A eff 0 K (K − K MT ) + φ K (K − K ) A 0 dry 0 0 A (6.35) MT MT MT GA(G0 − Gdry ) + φ Gdry (G0 − GA ) Geff = G0 MT GA(G0 − Gdry ) + φ G0(G0 − GA ) where

K G K MT = 0 ,G MT = 0 (6.36) dry φ dry φ 1 + ηH 1 + ηH ()1 − φ K ()1 − φ G

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

and η , H K , and H G are given by (6.30) and (6.29). Expressions (6.35) can also be re-written in alternative form where volume fraction does not appear:

K ()K − (1 −ηH )K MT + K MT (K − K ) K MT = K A 0 K dry dry 0 A eff 0 K ()K − (1 −ηH )K MT + K (K − K ) A 0 K dry 0 0 A (6.37) MT MT MT GA ()G0 − (1 −ηHG )Gdry + Gdry (G0 − GA ) Geff = G0 MT GA ()G0 − (1 −ηHG )Gdry + G0(G0 − GA )

2.4.3 Maxwell scheme

It is proposed by Maxwell (1873) for electrical conductivity of a material containing randomly located spherical inhomogeneities is probably the oldest homogenization scheme. We use its interpretation proposed by Sevostianov and Giraud (2013), where Maxwell scheme is formulated in terms of property contribution tensors. The basic idea of the method is that the far field produced by the considered set of inhomogeneities is equated to the far field produced by a fictitious domain of certain shape that possesses the (yet unknown) effective properties. It yields the following expression for the effective compliance tensor:

M −1 Seff = S0 + φ H A : []J − φ QΩ : H A (6.38)

where QΩ is fourth-rank tensor (Hill’s tensor) that depends on the shape of the domain Ω . For a porous material

M −1 Seff = S0 + φ H pore : []J − φ QΩ : H pore (6.39) Replacement relation (6.12) yields then:

M −1 M −1 −1 Seff = S0 + φ [](SA − S0 ) + φ (Sdry − S0 ) (6.40)

In particular, for isotropic materials:

M M M K0 [ K A(K0 − Kdry ) + φ Kdry (K0 − K A )] Keff = M K A(K0 − Kdry ) + φ K0(K0 − K A ) M M (6.41) M G0 [GA(G0 − Gdry ) + φ Gdry (G0 − GA )] Geff = M GA(G0 − Gdry ) + φ G0(G0 − GA ) with effective bulk and shear moduli for dry material given by:

M 1 − ϕηH αkk M 1 − ϕηH αGG Kdry = K0 ,Gdry = G0 (6.42) 1 + ϕηH k (1 − αk ) 1 + ϕηHG (1 − αG ) where

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

2 1 − 2ν 0 1 7 − 5ν 0 α k = ,αG = (6.43) 3 1 −ν 0 15 1 −ν 0

Alternative, volume fraction independent form of the equations (6.41) is

M M M K dry (K 0 − K A ) + K A ()η H k (K dry (1 − α k ) + K0α k ) K eff = K 0 M K0 (K0 − K A ) + K A ()η H k (K dry (1 − α k ) + K0α k )

M M (6.44) M Gdry (G0 − GA ) + GA ()η H G (Gdry (1 − αG ) + G0αG ) Geff = G0 M G0 (G0 − GA ) + GA ()η H G (Gdry (1 − αG ) + G0αG )

To illustrate the replacement relations, we consider calcite matrix with pores filled with either crude oil (liquid) or clay (solid). The properties of the matrix material (calcite) and two possible fillers are given in Table F2 of Annex F. Figure 6.4 and 6.5 illustrate dependence of the isotropic elastic moduli of a porous material with pores filled by oil and clay (respectively) on corresponding elastic moduli of the dry material. An interesting observation is that equations (6.1) (classical Gassmann’s equation), (6.2) (equations of Ciz and Shapiro) and replacement relations given by three different micromechanical homogenization schemes (6.27, 6.35, and 6.41) completely coincide. Note, that, in the latter three equations, shape of the inhomogeneities does not enter.

Keff / K0 1.0

0.8 φ = 10.0 Gassmann1951 CizShapiro2007 φ = 25.0 NIA 0.6 Mori-Tanaka1973 Maxwell1873 0.4 φ = 50.0

0.2

0.0 0.2 0.4 Kdry / K0.60 0.8 1.0

Fig.6.4: Dependence of the bulk moduli of calcite with crude oil filled pores on the bulk modulus of dry calcite at several volume fractions of pores calculated by different methods

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

K K G G eff 0 eff 0

1.0 1.0

φ = 10.0 φ = 10.0 0.8 0.8

φ = 25.0 0.6 φ = 25.0 0.6

φ = 50.0 0.4 φ = 50.0 0.4 (a) (b) 0.2 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 K K G G dry 0 dry 0

Gassmann1951 CizShapiro2007 NIA MoriTanaka1973

Maxwell1873

Fig.6.5: Dependence of the bulk and shear moduli of calcite with clay filled pores on the elastic properties of dry calcite at several volume fractions of pores calculated by different methods

Figure 6.6 and 6.7 illustrate dependence of the elastic moduli of the materials with filled pores on the volume fraction of the inhomogeneities at several values of the elastic moduli of the dry material.

Keff /K0

1.0

Kdry K0 = 8.0 0.9 Gassmann1951 0.8 CizShapiro2007

Kdry K0 = 6.0 0.7 NIA Mori-Tanaka1973 0.6 Kdry K0 = 4.0 Maxwell1873 0.5

0.4 0.0 0.2 0.4 0.6 0.8 1.0 ϕ

Fig.6.6: Dependence of the bulk moduli of calcite with crude oil filled pores on the volume fractions of pores at several values of the bulk modulus of dry calcite calculated by different methods

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

G G Keff / K0 eff 0 1.0 1.0 (a) (b) K K = 8.0 G G = 8.0 0.9 dry 0 0.9 dry 0

0.8 0.8 G G = 6.0 Kdry K0 = 6.0 dry 0 0.7 0.7 0.6 0.6 G G = 4.0 0.5 K K = 4.0 0.5 dry 0 dry 0 0.4 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

ϕ ϕ

Gassmann1951 CizShapiro2007 NIA MoriTanaka1973

Maxwell1873

Fig.6.7: Dependence of the bulk and shear moduli of calcite with clay filled pores on the volume fractions of pores at several values of the elastic moduli of dry calcite calculated by different methods

2.5 Concluding remarks.

We verified applicability of the replacement relations derived by Sevostianov and Kachanov (2007) for a material containing ellipsoidal inhomogeneities. These relations link the property contribution tensors of inhomogeneities having the same shape but different elastic properties. For non-ellipsoidal shapes, the replacement relations can be used as an approximation. We assessed the accuracy of this approximation focusing our attention on inhomogeneities of

2 p 2 p 2 p superspherical shape (described by equation x1 + x2 + x3 ≤ 1). It is shown that for convex shapes (concavity parameter p > 0.5 ) the accuracy of the representation of the property contribution tensor in terms of the Hill’s tensor is better than 10% while for concave shapes the error produced by such a representation is significant. Therefore, the replacement relations derived from this representation can be used in the case of the convex inhomogeneities only. Note, that the replacement relations can be written in the framework of any homogenization scheme. We showed, how appropriate relations can be obtained in the frameworks of non- interaction approximation and two most widely used homogenization methods – Mori-Tanaka scheme and Maxwell scheme. As seen, from the structure of formulas (6.26), (6.34), and (6.40), different homogenization schemes produce the same replacement relations (assuming that the properties of materials with dry and filled pores are calculated with the same method, of course).

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Ch.6: Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities

It is related to the fact that the effective properties in the mentioned schemes are expressed in terms of the compliance contribution tensor for a single inhomogeneity (for a pore, this tensor is just an inverse of the second Hill’s tensor Q ).

We illustrated the replacement relations on a simple example of an isotropic microstructure and showed that the results completely coincide with the prediction of the classical Gassman’s formula and with the formula proposed by Ciz and Shapiro (2007). The advantage of the Sevostianov and Kachanov replacement relations, as compared to ones of Gassman and Ciz and Shapiro is the possibility to use them for anisotropic materials (formulas (6.26), (6.34), and (6.40)).

Acknowledgements

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Part 4

Part 4 Concluding Remarks and Perspectives

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Concluding remarks and perspectives

128

Concluding remarks and perspectives

1.1 Concluding remarks for the mains results

The presented work is focused on the determination of property contribution tensors of three dimensional non-ellipsoidal inhomogeneities of concave or complex shape (Part II). Application to these tensor to effective properties of heterogeneous rock like materials (Part III) are presented.

Limestone oolitic porous rock, such as Lavoux limestone, is chosen as a reference rock due to its large utilizations, serves as rock reservoir, etc.

From SEM images of limestone oolitic (Grgic 2011), concave pores are observed between prolate spheroidal oolite grains. The shape can be approximately described by a supersphere (Giraud and Sevostianov 2013).

The elastic field (strain/stress, displacement and strain energy) for such materials associated with non-ellipsoidal shapes are determined numerically in the current work. Particular attention is paid to the issues related to the inhomogeneities having concave superspherical or superspheroidal shape.

1.1.1 Results for the first Eshelby problem

In the framework of the first Eshelby problem, the concept of fourth-rank Eshelby tensor is proposed to interrelate the elastic field around to the known eigenstrain inside inclusion. The knowledge of Eshelby tensor leads us to the determination of the elastic strain and stress field.

New results of Eshelby tensor and strain energies for superspherical pores in concave range are provided in chapter 2. The accuracy of the current calculations is better than 1% and it has been verified by the comparison with the ones of Onaka (2001) for convex superspheres. Concluding from the newly calculated results, general depencency of the Eshelby tensor on shape factor is not significant, which means, the elastic field changes gently when the shape of inclusions changes widely.

2 p 2 p 2 p Supersphere is described by equation x1 + x2 + x3 ≤ 1 and its shape varies from a three-perpendicular cracks to a cube-like morphology for 0.2 ≤ p ≤ 5 , the corresponding numerical analysis are performed carefully starting from p = 0.2 and end to p = 5 with an incremental of ∆ p = 0.5 . The upper stop mark is chosen as p = 5 since the volume of

129

Concluding remarks and perspectives

inclusion starting change not significantly for a large increase in p and the numerical prediction of Eshelby tensor remains almost constant. Moreover, the isotropic projections of Eshelby tensor are showed to be independent and the ones of Hill’s tensor are strongly dependent.

It is worth to notice that, for 3-D non-ellipsoidal inhomogeneities, Eshelby tensor is not relevant to effective properties, only the property contribution is involved. Thus, the main attention in this work is the second Eshelby problem. The particular attention is paid to the issues related to the inhomogeneities having concave superspherical or superspheroidal shape.

1.1.2 Results for the second Eshelby problem

Property contribution tensor, which turns into compliance contribution tensor H for elasticity problems and resistivity contribution tensor R for conductivity problems, are the key elements to identify in the framework of the second Eshelby problem. They are later used in Non-Interaction Approximation, Mori-Tanaka scheme and Maxwell scheme.

By implying FEM semi-analytical approach with refined linear mesh (T3 on surface of inhomogeneity), H and R are performed by two normalization methods. The first one reflects the influence of a given volume on a shape factor when the latter changes from 0.2 to 5, this effect is emphasized at strong concavity, while the second one allows generally a linear approximation where the changes of property contribution tensor when the shape of inhomogeneity changes in a specific way need to be analyzed.

The lower value of concavity parameter has been chosen as p = 0.2 for the following reasons: (1) serious computational difficulties are experienced for the reconstruction of highly concave surface (2) volume of inhomogeneity decreases very strongly around p = 0.2

For cubic superspherical pores, interrelating the two normalization methods gives an important geometric parameter that is used to describe the contribution of an individual pore  3  ()5p − 1 p2Γ  3π  2 p  into both elastic and electric properties: η = 3 for p < 1. One has to notice 4   1  Γ    2 p  that it is different from the one of Sevostianov and Giraud (2012) and it shows a better accuracy. Due to this, effective elastic and conductive properties for material containing superspherical

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Concluding remarks and perspectives

pores could be rewritten in terms of the newly obtained geometric parameter in the framework of Non-Interaction Approximation, Mori-Tanaka scheme and Maxwell scheme.

Both elastic and conductive properties of materials having the same microstructures can be expressed by the same geometric parameter, thus the cross-property connections which interrelate the changes of elastic and conductive properties could be written explicitly. The expressions of the said connections for superspherical pores and identical to the ones of spherical pores, this confirm that the irregularity factors affect the elastic and conductive properties in a similar way.

For axisymmetric superspheroidal inhomogeneity with a rotation axis of e3 which boundary

2 2 p 2 p is described by ()x1 + x2 + x3 ≤ 1, conclusions could be made from the newly obtained ~ ~ ~ numerical results. On one side, the three non-zeros components such as H 3333 , H1133 and H1313 (normalized by the second normalization method) show their linear behaviors on shape ~ ~ parameter p , but not for the others H1111 and H1122 . On the other side, H1111 and H1122 (normalized by the first normalization method) show their shape independency on shape parameter p and coincide with the corresponding components for sphere. Interestingly, the ~ same conclusions could be made for resistivity contribution tensor. R 33 behaves linearly on

shape parameter p and R 11 is independent of p . Connecting the two normalization methods gives linear approximations for all components of compliance and resistivity contribution tensor. Particularly, these representations could be further simplified for concave superspheroidal pores in the range of 0.2 < p < 1. The latter serves as the basis for calculating effective properties of materials containing multiple concave pores.

1.1.3 Applications to heterogeneous rock like materials

Two applications about porous rocks are investigated in the current work.

The first application concerns the evaluation of the effective thermal conductivity of oolitic rocks is performed to complete the work of Sevostianov and Giraud (2013) for effective elastic properties. A two step homogenization model has been developed by considering two distinct classes of pores: microporosity (intra oolitic porosity) and meso porosity (inter oolitic porosity). Maxwell homogenization scheme formulated in terms of resistivity contribution tensor has been used for the transition from meso to macroscale. Concave inter oolitic pores of superspherical

131

Concluding remarks and perspectives

shape have been taken into account by using resistivity contribution tensor obtained thanks to FEM modelling. Two limiting cases have been considered: ‘dry case’ (air saturated pores) and ‘wet case’ (water liquid saturated pores). Comparisons with experimental data show that variations of effective thermal conductivity with porosity in the most sensitive case of air saturated porosity are correctly reproduced.

Applicability of the replacement relations, initially derived by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It it the second application of newly obtained results on property contribution tensors.

1.2 Perspectives

Perspectives related to Part 1

Moreover, with an ambition to more generally analyze the influences of shape factors on the overall properties, the changes of property contribution tensors on aspect ratio γ by fixing concavity parameter p need to be evaluated in the framework of the second Eshelby problem.

Limiting case of perfectly rigid 3D concave inhomogeneities of non ellipsoidal shape could be also interesting to study, to complete analysis of property contribution tensors related to pores.

Contrast of elastic and conductive properties between inhomogeneity and surrounding matrix could be also another important parameter to study. It could complete existing studies related to ellipsoidal inhomogeneities and generalize to the case of non ellipsoidal 3D convex/concave inhomogeneities.

The current work is restricted to analysis of isotropic effective media. Extension to anisotropy could be also an important issue. The reference non ellipsoidal shapes introduced, superspheres and superspheroids, could be useful to study anisotropic effective materials containing such convex and/or concave inhomogeneities. In the simplest case of orientation distributions respecting the same symmetries and same symmetry axis than inhomogeneities, cubic and transverse isotropic effective materials could be studied.

132

Concluding remarks and perspectives

Extension to anisotropic properties of constituent could be also an important issue.

As an example, for the first Eshelby problem, generalization to transverse isotropy would imply to consider exact Green function defined by Pan and Chou (1976) instead of the Kelvin solution of the isotropic case. For the second Eshelby problem, FE modellings need to be performed considering anisotropic constituents and it may be straithforward by using standards FE computer codes.

Perspectives related to Part 2

Numerical results presented for thermal conductivity of oolitic limestones need to be compared with experimental data obtained on the considered reference limestone (Lavoux limestone). It would be convenient to measure the thermal conductivity in the two limiting cases, dry case and wet case, to compare the experimental and predicted effects of porosity on the effective tharmal conductivity

133

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144

Appendix

Appendix A: analytical S and H tensor of sphere in isotropic case

In elastic problem, the exact solution of Eshelby tensor and compliance contribution tensor for spherical inclusions of unit radius in isotropic material is illustrated:

E 7 − 5ν 0 S1111 = 15(1 −ν 0 )

E 1 − 5ν 0 S1122 = − (A.1) 15(1 −ν 0 )

E 4 − 5ν 0 S1212 = 15(1 −ν 0 ) and

s s 3 (1 −ν 0 )(9 + 5ν 0 ) E0 H1111 = E0 H 3333 = 2 7 − 5ν 0

s s 3 (1 −ν 0 )(1 + 5ν 0 ) E0 H1122 = E0 H1133 = − (A.2) 2 7 − 5ν 0 2 s 15 1 −ν 0 E0 H 2323 = 2 7 − 5ν 0

where ν 0 is Poisson’s ratio of matrix.

145

Appendix

146

Appendix

Appendix B: numerical results of superspherical pore by FEM

Table B1. Calculated components of the compliance contribution tensor and the resistivity contribution tensor

p H1111 H1122 H1133 R11 V*(p) 0.20 -2.252067 7.769440 5.047077 2.846482 0.020923 0.25 -1.130373 4.816686 2.981758 2.116599 0.088891 0.30 -0.845580 3.382321 2.093818 1.760289 0.227063 0.35 -0.672997 2.810947 1.740110 1.668871 0.435782 0.40 -0.604294 2.523916 1.562424 1.642101 0.701513 0.45 -0.584079 2.426625 1.502196 1.593402 1.006303 0.50 -0.560089 2.303054 1.425700 1.571938 1.333360 0.55 -0.539260 2.221076 1.374952 1.545565 1.669101 0.60 -0.524058 2.166984 1.341466 1.529875 2.003596 0.65 -0.501285 2.077241 1.285911 1.522164 2.330014 0.70 -0.485789 1.999407 1.237728 1.513326 2.643944 0.75 -0.477704 1.972500 1.221072 1.508698 2.942742 0.80 -0.472486 1.958481 1.212393 1.505054 3.225153 0.85 -0.472140 1.961806 1.214451 1.502997 3.490662 0.90 -0.470814 1.965580 1.216788 1.501279 3.739395 0.95 -0.470107 1.971549 1.220483 1.501193 3.971861 1.00 -0.475593 1.998992 1.237471 1.511107 4.188791 1.50 -0.443077 1.949215 1.206657 1.528976 5.696528 2.00 -0.426956 1.943592 1.203176 1.54037 6.481977 2.50 -0.419892 1.948496 1.206212 1.551237 6.930352 3.00 -0.411795 1.954874 1.210160 1.558827 7.207667 3.50 -0.406305 1.954060 1.209656 1.565641 7.390316 4.00 -0.403603 1.976868 1.223775 1.571108 7.516700 4.50 -0.400512 1.973865 1.221916 1.573139 7.607653 5.00 -0.398529 1.983064 1.238864 1.573172 7.675234

147

Appendix

Table B2. Number of elements and nodes of T3 mesh

p Total number of node on the surface Total number of element on the surface of of supersphere supersphere

0.20 7576 31920

0.25 8464 37896

0.30 20072 82624

0.35 16168 66752

0.40 12240 50576

0.45 19448 79552

0.50 18152 72984

0.55 18024 76672

0.60 19248 86128

0.65 21400 92496

0.70 22600 101048

0.75 24864 111896

0.80 20600 94216

0.85 24688 114552

0.90 22296 104864

0.95 20264 95544

1.00 22496 108560

1.50 22616 114064

2.00 19472 96592

2.50 18392 94704

3.00 31504 160432

3.50 34536 179968

4.00 31248 161976

4.50 32672 159520

5.00 35088 178024

148

Appendix

Appendix C: numerical integration on the surface of superspheroid

The surface of superspheroid is discretized with quadratic isoparametric 6 -node Lagrange triangular finite elements (T6, or TRIA6) and standard Gauss-Legendre quadrature rule (see among others Zienkiewicz and Taylor 2000) is used to numerically calculate the surface integral

i 1 1 ε = ()nu lk + nu kl dS (C.1) kl Ωi ∫ Ω∂ i 2

i )( Curvilinear coordinates (ξ,η) , and node coordinates xk are related by shape functions N ξ η),( i :

6 ( , ) ( , ) i )( xk ξ η = ∑ Ni ξ η xk (C.2) i=1 Standard notations are used, ξ represents curvilinear coordinate in the reference plane

whereas uk represents k component of displacement vector. For the 6 -nodes triangular

element, Ni denotes the global numbering of the node in the 3D finite element mesh, 1 ≤ i ≤ 6 denotes local numbering in the reference plane ξ η),( . Nodes numbered

( (N1(ξ = 0,η = 0),N 2(ξ = 1,η = 0),N3(ξ = 0,η = 1)) correspond to vertices of triangular 1 1 1 1 elements, and nodes (N (ξ = ,η = 0),N (ξ = ,η = ),N (ξ = 0,η = )) correspond to 4 2 5 2 2 6 2 mid-edge nodes. First partial derivative of spatial coordinates write ( k = 1, 2, 3).

∂x (ξ,η) 6 ∂N (ξ,η) ∂x (ξ,η) 6 ∂N (ξ,η) k i i )( k i i )( (C.3) = ∑ xk , = ∑ xk ∂ξ i=1 ∂ξ ∂η i=1 ∂η Shape functions and derivatives may be found in standard finite element textbooks (see among others Zienkiewicz and Taylor (2000)). One defines metric tensor gij related to

curvilinear coordinates(τ1 = ξ,τ 2 = η) , ,( ji = 21 ), .

∂xk ∂xk ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 gij = = + + (C.4) ∂τ i ∂τ j ∂τ i ∂τ j ∂τ i ∂τ j ∂τ i ∂τ j and

∂x ∂x ∂x ∂x ∂x ∂x g = k k , g = k k , g = k k (c.5) 11 ∂ξ ∂ξ 22 ∂η ∂η 12 ∂ξ ∂η The area of the triangle may be calculated thanks to the following surface integral in the reference plane

149

Appendix

1 1−ξ I = dξ g g − g 2 dη (C.6) ∫ξ =0 ∫η=0 11 22 12

Taking into account interpolation formula (C.3), gij components write

2 3  6 ∂N ξ,( η)  g i x i )( 11 = ∑ ∑ k  k=1  i =1 ∂ξ 

2 3  6 ∂N ξ,( η)  g i x i )( 22 = ∑ ∑ k  k=1  i =1 ∂η  3  6 ∂N ξ,( η)  6 ∂N ξ,( η)  g i x i )( i x i )( 12 = ∑ ∑ k ∑ k  (C.7) k=1  i =1 ∂ξ  i=1 ∂η  Returning to the problem of calculation of integral (C.1), the contribution of the quadratic triangular element T6 takes the form

1 1−ξ I el = dξ f (ξ,η) g g − g 2 dη (C.8) op ∫ξ =0 ∫η=0 op 11 22 12

where the function f op (ξ,η) represents

1 f ξ,( η) = (u ξ,( η)n ξ,( η) + u ξ,( η)n ξ,( η)) (C.9) op 2 0 p p 0

Displacement components u0 and up are calculated by using the same interpolation formula

than the coordinates xi

6 ( , ) ( , ) i )( uk ξ η = ∑ Ni ξ η uk (C.10) i=1

i )( uk represents the component k of nodal vector displacement, at node i (i = 1 ,...6). np(ξ,η) is a known function of the spatial coordinates (see appendix E). Finally, integral (C.8) may be simply expressed as

1 1−ξ el Iop = dξ hop ξ η),( dη (C.11) ∫ξ =0 ∫η=0 With

2 hop(ξ,η) = fop(ξ,η) g11g22 − g12 (C.12)

Numerical integration of (C.11) is performed by using standard Gauss Legendre quadrature rules

nG el i ( i , i ) Iop = ∑ wGhop ξG ηG (C.13) i=1

150

Appendix

i i i where nG denotes the number of Gauss integration points in a T 6 finite element, ξG ,ηG , wG respectively denote the local coordinates of Gauss integration points and corresponding weight.

1 3 1 2 2 3 4 4 We consider nG = 4 , with ξG = ξG = ηG = ηG = 51 , ξG = ηG = 53 , ξG = ηG = 31 ,

1 2 3 4 wG = wG = wG = 25 (24 * 4) and wG = − 27 (24 * 4) (Zienkiewicz and Taylor 2000). Finally, the integral on the whole superegg surface is obtained by summing the contribution of all the finite elements. By denoting N el the total number of T 6 finite elements on the superspheroidal boundary, the surface integral writes

Nel nG  i el j )( i i  ( , ) Iop = ∑ ∑ wGhop ξG ηG  (C.14) j=1 i = 1 

151

Appendix

152

Appendix

Appendix D: Numerical results of superspheroidal pore by FEM

Table D1. Calculated components of the compliance contribution tensor

p H1111 H1122 H1133 H3333 H1313

0.20 1.887796 -0.477459 -0.964299 27.731870 8.024717

0.25 1.808015 -0.419517 -0.842265 12.354200 3.654240

0.30 1.819960 -0.420280 -0.783590 7.440550 2.558693

0.33 1.840120 -0.421878 -0.739390 5.895420 1.901430

0.35 1.854860 -0.424255 -0.720632 5.217026 1.742868

0.40 1.894290 -0.434329 -0.681435 4.065400 1.506599

0.45 1.918003 -0.443068 -0.642262 3.352013 1.388400

0.50 1.937640 -0.451064 -0.611918 2.916446 1.323735

0.55 1.952870 -0.459580 -0.587291 2.639410 1.289338

0.60 1.963770 -0.466431 -0.568153 2.456980 1.269780

0.65 1.973077 -0.472413 -0.553477 2.332502 1.269780

0.70 1.979451 -0.477634 -0.541160 2.241809 1.258690

0.75 1.984812 -0.482264 -0.531049 2.174690 1.252894

0.80 1.989424 -0.486274 -0.522567 2.122560 1.251279

0.85 1.993319 -0.489852 -0.515269 2.081938 1.250837

0.90 1.996558 -0.492969 -0.508869 2.049180 1.250650

0.95 1.999258 -0.496250 -0.501454 2.027340 1.250050

1.00 2.001203 -0.498053 -0.498057 2.001212 1.249900

153

Appendix

Table D2. Number of elements and nodes of the mesh b (more refined on superspheroid surface)

Total number of Total number of Total number of Total number of

p node of the nodes on the finite element on finite element on entire mesh surface of the entire model surface of superspheroid superspheroid

0.20 1080757 76882 814773 38440

0.25 1160349 101834 877469 50916

0.30 1177256 106134 890505 53066

0.33 1264321 118634 957219 59316

0.35 1178130 111834 891698 55916

0.40 1272154 118286 963115 59142

0.45 1295544 124278 981481 62438

0.50 1234343 113666 934203 56832

0.55 1159415 108834 877349 54416

0.60 1203043 112366 910622 56182

0.65 1294027 123234 980283 61616

0.70 1249954 118362 951796 54180

0.75 1162313 115218 878187 47608

0.80 1229546 113126 930723 56562

0.85 1296402 120558 981749 60278

0.90 1254071 112170 949123 56084

0.95 1277832 118978 967682 59488

1.00 1267595 117058 959745 58528

154

Appendix

Table D3. Relative errors on components of compliance tensor between mesh a and mesh b

p H1111 H1122 H1133 H3333 H1313

0.20 0.21% 0.46% 0.17% 0.41% 0.42%

0.25 0.07% 0.18% 1.32% 0.43% 0.33%

0.30 0.33% 0.42% 2.50% 0.10% 0.83%

0.33 0.02% 0.06% 1.47% 0.44% 0.31%

0.35 0.01% 0.04% 0.11% 0.40% 0.27%

0.40 0.00% 0.01% 0.13% 0.40% 0.16%

0.45 0.00% 0.00% 0.11% 0.29% 0.07%

0.50 0.01% 0.00% 0.11% 0.21% 0.06%

0.55 0.01% 0.01% 0.07% 0.16% 0.02%

0.60 0.01% 0.01% 0.05% 0.08% 0.02%

0.65 0.02% 0.02% 0.04% 0.05% 0.91%

0.70 0.01% 0.01% 0.02% 0.03% 0.18%

0.75 0.01% 0.01% 0.04% 0.04% 0.23%

0.80 0.01% 0.01% 0.02% 0.02% 0.22%

0.85 0.01% 0.01% 0.02% 0.02% 0.11%

0.90 0.01% 0.01% 0.01% 0.01% 0.06%

0.95 0.27% 0.03% 0.02% 0.02% 0.02%

1.00 0.00% 0.00% 0.00% 0.00% 0.04%

155

Appendix

Table D4. Components of resistivity contribution tensor of superspheroid

p R11 R33

0.20 2.02468100 15.1971865

0.25 1.71555700 6.631911

0.30 1.64000000 3.939221

0.35 1.54856700 2.823389

0.40 1.52899800 2.289616

0.45 1.51192200 1.999021

0.50 1.50630000 1.832213

0.55 1.50554600 1.728305

0.60 1.50589500 1.662567

0.65 1.50754400 1.616467

0.70 1.50912600 1.587973

0.75 1.50990200 1.559419

0.80 1.50992300 1.539122

0.85 1.50890400 1.523779

0.90 1.50704300 1.509388

0.95 1.50725300 1.505122

1.00 1..50124400 1. .496256

156

Appendix

Appendix E: complementary geometrical results related to superspheroidal shapes

E.1 Total surface area of a superspheroid: exact results

Some analytical results may be obtained for particular values of concavity parameter p.

The two limiting cases may be recovered

1 lim j γ = )1( p)( = (E.1) p→0 2

lim pg )( = 0 (E.2) p→0 3 lim j γ = )1( p)( = (E.3) p ∞→ 2 3 lim 3g p)( = (E.4) p ∞→ 2

Table E1: Exact values of dimension less volume and surface area of a superspheroid (a =1, ) for concavity parameter )

p 3g( p) j p)( j ( p) sphere γ = )1( γ = )1( 3g (S) g( p) 0 0 1 ∞ .0 353 = .0 500 2

1 1 .0 554 .16 631 .0 413 = .0 100 4 10 1 8 3 .7 875 .0 465 ≈ .0 229 = .0 600 3 35 5 1 1 2 .4 243 .0 595 = .0 500 ≈ .0 707 2 2 2

1 1 1 3 1

∞ 3 3 3 .1 837 = .1 500 2 2

157

Appendix

E.2 Unit vector to the boundary of superspheroid: derivation from implicit surface equation

The normal vector to an implicitly defined surface

2 p ( x2 + x2 ) p x f ( x , x , x ) = 1 2 + 3 − 1 = 0 (E.5) 1 2 3 a2 p a2 pγ 2 p is the gradient vector on the surface

∂ f ∂ f ∂ f N = grad f = e1 + e2 + e3 (E.6) ∂x1 ∂x2 ∂x3 Relations (E.5-E.6) gives (in what follows one considers outward normal vector)

∂ f 2 p 2 2 p−1)( ∂ f 2 p 2 2 p−1)( N1 = = 2 p x1( x1 + x2 ) , N 2 = = 2 p x2( x1 + x2 ) (E.7) ∂x1 a ∂x2 a

∂ f 2 p 1 ( p−12 ) N3 = = 2 p 2 p x3 sign( x3 ) (E.8) ∂x3 a γ As the surface of the superspheroid is a surface of revolution, cylindrical coordinate frame (r,

ϕ , x3 with 0 ≤ r, 0 ≤ ϕ ≤ 2π ) allows to express the results in compact form. One replaces x1

and x2 by their expression as functions of radius r and polar angle ϕ

2 2  x2  r = x1 + x2 ,ϕ = arctan  (E.9)  x1 

x1 = rcosϕ, x2 = rsinϕ (E.10) The normal vector N writes

2 p 1 ( p−12 ) N = ()r( p−12 )(cosϕe + sinϕe + x sign(z )e ) (E.11) a 2 p 1 2 γ 2 p 3 3 3

Cylindrical coordinate’s vector basis (e r ,eφ ,e 3 ) writes

e r = cosϕe1 + sinϕe 2 , eϕ = − sinϕe1 + cosϕe 2 (E.12) and

2 p  1 ( p−12 )  N = r( p−12 ) e + x sign( x )e  (E.13) 2 p  r 2 p 3 3 3  a  γ 

158

Appendix

One recovers the known result: the normal vector to a surface of revolution belongs to a meridian plane (φ = constant plane), its component Nφ is equal to zero. A normal unit vector may be easily deduced from relation (E.13).

N n = (E.14) N ⋅ N and

1 ( p−12 ) 1 ( p−12 ) n = ()r e r + 2 p x3 sign( x3 )e 3 ) ( p−24 ) 1 ( p−24 ) γ r + 4 p x3 γ (E.15)

159

Appendix

160

Appendix

Appendix F: complementary geometrical results related to superspherical shapes

Table F1. Components of the compliance contribution tensor for a superspherical pore: H ijkl′

calculated by (3.67) from components of Hill tensor and H ijkl calculated by FEM

p H′ H′ H′ 1111 H1111 1122 H1122 1212 H1212

0.20 1.84152 7.769440 -0.395762 -2.252067 1.33843 5.047077

0.25 1.89651 4.816686 -0.423257 -1.130373 1.30152 2.981758

0.30 1.95315 3.382321 -0.451574 -0.845580 1.26801 2.093818

0.35 1.99962 2.810947 -0.474807 -0.672997 1.24339 1.740110

0.40 2.03362 2.523916 -0.491809 -0.604294 1.22679 1.502424

0.45 2.05558 2.426625 -0.502788 -0.584079 1.21664 1.402196

0.50 2.07061 2.303054 -0.510306 -0.560089 1.20994 1.355700

0.55 2.07033 2.221076 -0.510164 -0.539260 1.21006 1.314952

0.60 2.0684 2.166984 -0.509198 -0.524058 1.21091 1.281466

0.65 2.0626 2.077241 -0.506302 -0.501285 1.21348 1.258911

0.70 2.05373 1.999407 -0.501863 -0.485789 1.21748 1.257728

0.75 2.04494 1.972500 -0.497469 -0.477704 1.2215 1.251072

0.80 2.03624 1.958481 -0.493120 -0.472486 1.22555 1.248393

0.85 2.02763 1.961806 -0.488815 -0.472140 1.22963 1.244451

0.90 2.01911 1.965580 -0.484554 -0.470814 1.23373 1.242788

0.95 2.01067 1.971549 -0.480337 -0.470107 1.23786 1.243955

1.00 2.00187 1.998992 -0.475934 -0.475593 1.24225 1.247471

1.50 1.95584 1.949215 -0.455220 -0.443077 1.26953 1.246657

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Appendix

2.00 1.92186 1.943592 -0.435933 -0.426956 1.28600 1.243176

2.50 1.91435 1.948496 -0.432173 -0.419892 1.29482 1.242212

3.00 1.89714 1.954872 -0.423567 -0.411793 1.30113 1.248160

3.50 1.88985 1.954060 -0.419925 -0.406305 1.30575 1.250656

4.00 1.88623 1.976868 -0.418117 -0.403603 1.30943 1.258775

4.50 1.87905 1.973865 -0.414525 -0.400512 1.31168 1.258916

5.00 1.87847 1.964475 -0.414300 -0.395101 1.31398 1.258864

Table F2. Material properties used for calculation of the effective properties.

Material Bulk modulus(GPa) Shear modulus(Gpa)

Calcite (Matrix)(Sone and Zobak 2013) 70.2 29

Crude oil(liquid) (Jessup 1930) 2.35 --

Clay(solid)(Sone and Zobak 2013) 12 6

162

Summary

Title: Micromechanical Modelling of Heterogenous Porous Materials with Application to Oolitic Rocks

Keywords: Homogenization, heterogeneous material, inhomogeneity, concave, supersphere, superspheroid, effective elasticity, effective thermal conductivity, cross-property connection.

Summary:

163

Summary

Applicability of the replacement relations, initially derived by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It it the second application of newly obtained results on property contribution tensors.

164

Résumé

Titre : Modélisation Micromécanique de Milieux Poreux Hétérogènes et Applications aux Roches Oolithiques

Mots clés : Homogénéisation, matériaux hétérogènes, concave, supersphère, supersphéroïde, propriétés élastiques effectives, conductivité effective, roches oolithiques.

Résumé :

La problématique suivie dans ce travail est la détermination des propriétés effectives, élastiques et conductivité, de matériaux poreux hétérogènes tels que des roches, les roches oolithiques en particulier, en relation avec leur microstructure.

Le cadre théorique adopté est celui de l’homogénéisation des milieux hétérogènes aléatoires et on suit les approches par tenseurs d’Eshelby. Ces approches sont basées sur la résolution des problèmes d’Eshelby : le problème de l’inclusion (premier problème) et le problème de l’inhomogénéité (second problème) isolées dans un milieu infini. La solution de ces problèmes de référence est analytique, en élasticité linéaire isotrope et en diffusion linéaire stationnaire, dans le cas d’inhomogénéités 2D ou 3D de type ellipsoïde. Elle conduit à la définition de tenseurs caractérisant les interactions entre l’inclusion/inhomogénéité et le milieu environnant. On utilise dans ce travail les tenseurs de contribution relatifs à une inhomogénéité isolée, définis par Kachanov et Sevostianov 2013, contributions à la souplesse (élasticité) et à la résistivité (conductivité). Ces tenseurs au cœur des méthodes d’homogénéisation de type EMA (Effective Medium Approximation), et en particulier les schémas NIA (Non Interaction Approximation), Mori Tanaka et Maxwell

Ce travail est centré sur la caractérisation des paramètres géométriques microstructuraux dont l’influence sur les propriétés effectives est majeure. On étudie en particulier les effets de forme des inhomogénéités, la nouveauté est l’aspect 3D.

Les observations microstructurales de roches oolithiques, dont le calcaire de référence de Lavoux, mettent en évidence des hétérogénéités de forme 3D et concave. En particulier les matériaux de remplissage inter-oolithes, pores ou calcite syntaxiale. Ces formes peuvent être observées sur d’autres matériaux hétérogènes et ont été peu étudiées dans le cadre

165

Résumé micromécanique. Cela nécessite de considérer des formes non ellipsoïdales et de résoudre numériquement les problèmes d’Eshelby.

Le cœur de ce travail est consacré à la détermination des tenseurs de contribution d’inhomogénéités 3D convexes ou concaves de type supersphère (à symétrie cubique) ou supersphéroïde (à symétrie de révolution).

Le premier problème d’Eshelby a été résolu, dans le cas de la supersphère, par intégration numérique de la fonction de Green exacte (solution de Kelvin dans le cas isotrope) sur la surface de l’inclusion. Des modélisations 3D aux éléments finis ont permis de résoudre le second problème d’Eshelby et d’obtenir les tenseurs de contribution à la souplesse et à la résistivité pour les superphère et supersphéroïde. Sur la base des résultats numériques, des relations analytiques simplifiées ont été proposées pour les tenseurs de contribution sous forme de fonctions des paramètres élastiques des constituants et du paramètre adimensionnel p caractérisant la concavité. Un résultat important, dans le cas de la superphère et dans le domaine concave, est l’identification d’un même paramètre géométrique pour les tenseurs de contribution à la souplesse et à la résistivité.

Les résultats numériques et théoriques obtenus sont appliqués à deux problèmes : l’estimation de la conductivité thermique effective de roches calcaires oolithiques d’une part et l’étude de l’extension des relations dites de substitution définies par Kachanov et Sevostianov 2007 au cas non ellipsoïdal d’autre part. Pour le premier problème, un modèle micromécanique de type Maxwell, à deux échelles a permis de retrouver les résultats expérimentaux disponibles dans la littérature, en en particulier l’influence de la porosité sur la conductivité thermique effective dans les cas sec et humide.

Dans le cas du second problème, les résultats obtenus ont permis de montrer que la validité de relations de substitution est restreinte, dans le cas non ellipsoïdal et en considérant une forme d’inhomogénéité de type supersphère, au domaine convexe seulement.

166