Prof Giraud Albert

Dr Dragan

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

Considering the challenge, I would not accomplish my Ph.D. without supports from many individuals.

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 [START_REF] Sterpenich | Experimental ageing of oolitic limestones under CO2 storage conditions. Petrographical and chemical evidence[END_REF][START_REF] Makhloufi | Impact of sedimentology and diagenesis on the petrophysical properties of a tight oolitic carbonate reservoir. The case of the Oolithe Blanche Formation (Bathonian[END_REF][START_REF] Grgic | Influence of CO2 on the long-term chemomechanical behavior of an oolitic limestone[END_REF]. Le calcaire oolithique de Lavoux [START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF] inhomogénéité ellipsoïdale isolée en milieu infini, obtenue par [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | Elastic inclusons and inhomogeneities[END_REF]. Par 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 [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF], des pores de formes irrégulières de matériaux composites carbone-carbone [START_REF] Drach | Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes[END_REF], des fissures planes [START_REF] Fabrikant | Applications of Potential Theory in Mechanics[END_REF], Sevostianov et Kachanov 2002), des fissures planes recoupées [START_REF] Grechka | The influence of crack shapes on the effective elasticity of fractured rocks[END_REF], des fissures non planes [START_REF] Mear | Elastic compliances of non-flat cracks[END_REF] et des fissures traversant les pores [START_REF] Kachanov | Rice's internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness[END_REF]. 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

XXX supersphère 1 2 3 2 2 2 1 ≤ + + p p p x x x
avec p le coefficient de forme [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF][START_REF] Onaka | Superspheres: Intermediate Shapes between Spheres and Polyhedra[END_REF], 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 ( )

1 2 3 2 2 2 1 ≤ + + p p x x x
. 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.

Problèmes d'Eshelby

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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]. 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 XXXI convexe dans le domaine élastique [START_REF] Chen | Numerical evaluation of the Eshelby tensor for a concave superspherical inclusion[END_REF], 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 [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF][START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF][START_REF] Maekenscoff | Inclusions with constant eigenstress[END_REF][START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF]. Du fait de la symétrie cubique de la supersphère et des relations importantes proposées par [START_REF] Milgrom | The energy of inclusions in linear media: Exact shape independent relations[END_REF], une seule composante non nulle doit être évaluée.

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

Le deuxième problème d' [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF][START_REF] Walpole | On the overall elastic moduli of composite materials[END_REF][START_REF] Kunin | Ellipsoidal inhomogeneity in the elastic medium[END_REF][START_REF] Sevostianov | Compliance tensor of ellipsoidal inclusion[END_REF][START_REF] Torquato | Random Heterogenous Materials[END_REF], 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 [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF][START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]

XXXV

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 [START_REF] Sevostianov | Compliance and resistivity contribution tensors of axisymmetric concave pores[END_REF], 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. Ces résultats ont fait l'objet d'une publication, [START_REF] Chen | Numerical evaluation of the Eshelby tensor for a concave superspherical inclusion[END_REF]. Le modèle proposé initialement par Giraud et Sevostianov (2013) 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.

Conclusions et perspectives

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.

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 [START_REF] Sterpenich | Experimental ageing of oolitic limestones under CO2 storage conditions. Petrographical and chemical evidence[END_REF][START_REF] Makhloufi | Impact of sedimentology and diagenesis on the petrophysical properties of a tight oolitic carbonate reservoir. The case of the Oolithe Blanche Formation (Bathonian[END_REF]. The Lavoux limestone [START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF], Giraud et al. 2012) will be chosen as a reference rock for applications of analytical and numerical micromechanical models. 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:

1 2 2 3 2 2 2 2 2 1 = + + c x b x a x (1.1) 100 µm 1 mm 1 cm
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 , analytical solutions of effective problems related to these shapes are well known [START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF][START_REF] Nemat-Nasser | Micromechanics: Overall Properties of Heterogeneous Solids[END_REF][START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF]. In limiting cases, the shape becomes strongly oblate ( 1 << γ

) including perfectly rigid disk and crack-like pore (flat cracks, no-flat cracks, intersected and branched cracks). Otherwise, it becomes strongly prolate ( 1 >> γ

) 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 20 th century. Several shapes such as polygons and polyhedral [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF], pores of irregular shapes of carbon-carbon composite materials [START_REF] Drach | Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes[END_REF], planar cracks (Fabrikant 1989, Sevostianov andKachanov 2002), intersecting planar cracks [START_REF] Grechka | The influence of crack shapes on the effective elasticity of fractured rocks[END_REF], non-planar cracks [START_REF] Mear | Elastic compliances of non-flat cracks[END_REF] and cracks growing from pores [START_REF] Kachanov | Rice's internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness[END_REF]) have been discussed. Particular attention is paid to concave shapes, the only one that has been discussed in the litteratures is supersphere [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF][START_REF] Onaka | Superspheres: Intermediate Shapes between Spheres and Polyhedra[END_REF], Zohdi and Wriggers 2008, Eq.9.2) and it is a particular case of superellipsoid:

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

1 2 3 2 2 2 1 = + + p p x x x (1.4)
The diversity of shape according to equation (1.1-1.4) is shown in Figure 1.3. [START_REF] Grgic | Modélisation du comportement à court et à long terme des roches de la formation ferrifère lorraine[END_REF] 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 [START_REF] Grgic | Impact of chemical weathering on micro/macromechanical properties of oolithic iron ore[END_REF].

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 concentric layers are in the range of 100 µm < l < 1000 µm. Figure 1.8 shows a concave macropore (with a maximum size of 100 µm) between oolitic grains.

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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF]. 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 [START_REF] Ammari | Progress on the strong Eshelby's conjecture and extremal structures for the elastic moment tensor[END_REF].

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 [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF][START_REF] Walpole | On the overall elastic moduli of composite materials[END_REF][START_REF] Kunin | Ellipsoidal inhomogeneity in the elastic medium[END_REF][START_REF] Sevostianov | Compliance tensor of ellipsoidal inclusion[END_REF][START_REF] Torquato | Random Heterogenous Materials[END_REF], 2007).

However, this is not the case for non-ellipsoidal shapes and in particular for the 3-D nonellipsoidal 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 [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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 [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]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[START_REF] Levin | Elastic properties of inhomogeneous transversely isotropic rocks[END_REF] 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 [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF][START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF][START_REF] Maekenscoff | Inclusions with constant eigenstress[END_REF][START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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 [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF][START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF] The linear conductivity/resistivity problems will be investigated by using the same methodology (Sevostianov and Kachanov 2002)

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 noninteracting 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.

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. 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.

Organization of the work

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

Part 2

Preface "Eshelby problems" and 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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | Elastic inclusons and inhomogeneities[END_REF] who considered only ellipsoidal shapes for isotropic medium.

This chapter is motived by the work of [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] who considered a particular 3-D nonellipsoidal 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.

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 with the 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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | Elastic inclusons and inhomogeneities[END_REF]. At the boundary of the inclusion, a strain ) ( x ε 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 )

( x ε : ( ) ( ) ∫ Ω * ′ ′ ∂ ′ - ∂ ∂ ∂ = V d x G x C i mj p kl ijkl mp x x x ε ε 0 (2.1)
where 0 ijkl C 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

Ω ∉ x , Ω ∈ ′ x . One has to notice that Green tensor ) ( x x G ′ -
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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF].

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

( ) ( ) ∫ Ω * ′ ′ ∂ ′ - ∂ = V d x G C u i mj kl ijkl m x x x ε 0 (2.2)
This reflects the physically presentation of Green tensor: the displacement at position Ω ∉ x generated by body force which is implied at

Ω ∈ ′ x .
For isotropic material, explicit form of Green function

) ( x x G ′ -
is expressed in the following form [START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF]:

( ) ( ) ( )       + - - = ′ - 2 4 3 1 16 1 x x x x G j i ij ij δ ν ν πµ x x (2.3)
Where

i i i x x x ′ - = , ( ) ( ) ( ) 2 3 3 2 2 2 2 1 1 x x x x x x x ′ - + ′ - + ′ - = and j i if ij = = 1 δ otherwise 0 = ij δ .
In the framework of the first Eshelby problem, the resulting strain is interrelated to the eigenstrain by the following relation:

( ) ( ) * = kl E ijkl ij S ε ε x x (2.4)
Where the fourth-rank tensor E S 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:

( ) ( ) ∫ ∫ Ω ∂ Ω ′ - ∂ ∂ = ′ ′ ∂ ′ - ∂ ∂ ∂ = dS G x C V d x G x C S mj p ijkl i mj p ijkl E mpkl x x x x x 0 0 ) ( (2.5)

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" method is efficient when dealing with such problems. For non-ellipsoidal inclusions, the average strain is expressed as follows by implying its definition

        ∂ ∂ + ∂ ∂ = i j j i ij x u x u 2 1 ε : ( ) ∫ Ω + - = ∆ dS n x u n x u V i j j i ij ) ( ) ( * 2 1 ε (2.6)
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 [START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF] condition:

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

( ) ∫ Ω + - = dS n x u n x u V S i j j i kl E ijkl ) ( ) ( * ε 2 1 (2.8)

Abstract

We calculate Eshelby tensor for inclusions of non-ellipsoidal shape. We focus on the superspherical shape described by equation 1

2 3 2 2 2 1 ≤ + + p p p x x x
. It is convex when 5 . 0 > p and concave when 5 . 0 < p . 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 ( 1 = p ) and with numerical results of [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF]

( 1 > p ).

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 * ij ε which is not 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 * ij ε 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). [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF] considered the case of the isotropic material and uniform eigenstrain. Later contributions extended his results to anisotropic materials [START_REF] Kinoshita | Elastic fields of inclusions in anisotropic media[END_REF]Mura 1971, Mura 1987), non-uniform loading [START_REF] Asaro | The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion[END_REF], and non-linear behavior of an inhomogeneity [START_REF] Lyssiak | Zur Frage der polynomialen Konservativit at (On the problem of polynomial conservation)[END_REF]. 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

E S 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 [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] and [START_REF] Onaka | Elasticstates of doughnut-like inclusions with uniform eigenstrains treated by averaged Eshelby tensors[END_REF] who considered convex supersphere ( 1 > p ) and toroidal 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.

Average S-tensor for superspherical inclusion

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

* ij E ijkl ij S ε ε = (2. 9)
For non-ellipsoidal inclusions, this tensor is non-uniform and, following [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF] we consider its average. Rodin has mentioned that the traces E iijj S and E ijij S (summation over j i, )

are independent of the inclusion shape. Eshelby tensor E ijkl S reflects the geometric symmetry of the inclusion's shape An important connection between components of Eshelby tensor has been derived by [START_REF] Milgrom | The energy of inclusions in linear media: Exact shape independent relations[END_REF]. They showed that invariants of average Eshelby tensor

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

3 1 1 0 0 = - + = E ijij E iijj S S , ν ν (2.10)
In the present work, we focus on a superspherical inclusion of unit radius described by: (

) [ ] ( ) p p p V 2 3 2 1 3 2 3 2 * Γ Γ =
(2.12)

where dt t e x x t 1 0 - ∞ - ∫ = Γ ) ( , *
V varies strongly.

For a superspherical inclusion, Eshelby tensor E ijkl S exhibits the cubic symmetry [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF]:

E E E E E E E E E E E E S S
+ = + = + E E E E S S S S
(2.14) where 0 ν is Poisson's ratio of the matrix. So that there is only one independent component of Eshelby tensor in this case. In addition,

) ( ) ( 0 1212 1 8 3 1 ν π - + = p f S E (2.15) with               ′ = ∫ ∫ Ω Ω dS n S d n x x x V p f 2 1 3 1 2 1 1 * ) (
(2.16) Eq. (2.16) expresses the dependence of the average Eshelby tensor on parameter p.

Numerical approach 2.3.1 Average S-tensor for convex supersphere

In the

3 2 1 x x x
--Cartesian coordinate system, a convex superspherical inclusion of unit radius can be described by the following equation:

1 2 3 2 2 2 1 ≤ + + p p p x x x ( 5 0. ≥ p ) (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 8 1 of the real shape

( 0 0 0 3 2 1 > > > x x x , ,
) has to be considered for calculation of Eshelby tensor. When the shape parameter becomes small, for example 1 0. = p , 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

35 0. = p to 2 0. = p .

Discussion

In the first Eshelby problem with * ij ε being generally non-uniform in Ω , the far-field asymptotics is independent of the shape of Ω and of the distribution of * kl ε within Ω . It is determined by the product:

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

V d x G C u kl i mj ijkl m ′ ′ ′ ∂ ′ - ∂ = ∫ Ω ) ( ) ( ) ( * x x x x ε (2.19) implying V d x G x C kl i mj n ijkl mn ′ ′ ′ ∂ ′ - ∂ ∂ ∂ = ∫ Ω ) ( ) ( ) ( * x x x x ε ε (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

V d x G V d x G kl i mj kl i mj ′ ′ ∂ - ∂ - ≈ ′ ′ ′ ∂ ′ - ∂ ∫ ∫ Ω Ω ) ( ) ( ) ( ) ( * * x x x x x x ε ε 0 (2.21) where 0 x is an arbitrary point inside Ω . Since * * * ) ( V d kl kl ε ε = ′ ′ ∫ Ω x x
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 [START_REF] Sevostianov | Elastic fields generated by inhomogeneities: far-field asymptotics, its shape dependence and relation to the effective elastic properties[END_REF]:

{ } l k j i kl ij l k kl l i jl j l il j i kk ij kk ij ij r V α α α α ε δ α α ε α α ε α α ε ν α α ε ν δ ε ε ν ν π ε * * * * * * * * ) ( ) ( ) )( ( ) ( ) ( 15 3 6 2 1 3 2 2 1 1 8 1 0 0 0 3 0 - + + + - + - - - = x (2.22)
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

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 showed in particular that traces E iijj S and E ijij S are shape independent while the same traces of Hill's tensor iijj P and ijij P 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)

SeCtion 3

3 Discussions: effect of the shape factor on the elastic energy Elastic strain energy s E of a material containing Ω with a uniform eigenstrain is derived by [START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF]:

ij ij ij ij s V dx E σ ε σ ε * * * 2 1 2 1 - = - = ∫ Ω (2.27)
For a superspherical pore:

( ) * * kl mn E klmn ijkl ij S C ε ε σ - = 0
(2.28) Submitting (2.28) into (2.27) yields the elastic strain energy of a material containing a superspherical pore:

( ) [ ] * * * * kl mn E klmn ijkl ij S S C V E ε ε ε - - = 0 2 1 (2.29)
To evaluate the effect of the shape factor on the strain energy of such material, we consider normal strain energy

( ) 2 * * ε V E E s normal =
(2.30) by assuming two purely deviatoric eigenstrains, denoted by A (

* * * A ε ε ε = = 21 12 and other 0 = * ij ε ) and B ( * * * * * , B B ε ε ε ε ε = - = = 33 22 11 2 and other 0 = * ij ε
) [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF].

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:

( )

E A A S A normal S V E E 1212 2 2 1 2 - = = µ ε ) ( * * (2.31)         - - - = = ) ( ) ( * * 0 0 1212 2 1 4 2 1 4 9 2 ν ν µ ε E B B S B normal S V E E (2.32)
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. 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 [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] 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 [START_REF] Sevostianov | On computation of the compliance and stiffness contribution tensors of inhomogeneities[END_REF].

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. Where 0 S is the compliance tensor of the matrix. While the stiffness contribution tensor is related to the Hill's tensor by a linear relation:

[ ] 1 1 0 1 - -+ - = P C C N ) ( (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:

( ) [ ] 1 1 1 2 0 0 - - + - = - = C k P k k R K
, or equivalently,

( ) [ ] 1 1 1 2 0 0 - - + - = - = C r Q r r K R (3.3)
Where c P and c Q 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. [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF]

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 inhomogeneities of diverse shapes such as strongly oblate spheroidal, strongly prolate spheroidal and spherical inhomogeneity by using single inhomogeneity approach. [START_REF] Wang | A new approach to modelling the effective thermal conductivity of heterogeneous materials[END_REF] 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, [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF] have discussed effective poroelastic coefficient of isotropic limestone oolitic rocks (Lavoux) in the framework of Hashin Composite Sphere Assemblage (CSA, [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF]) and self-consistent models. Observing from SEM images [START_REF] Grgic | Influence of CO2 on the long-term chemomechanical behavior of an oolitic limestone[END_REF] of the said rock shows that the pores between oolitic prolate spheroidal grains are concaves, their shape can be approximately described by supersphere [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF].

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. [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF] 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 (

1 1 0 = + ϕ ϕ
), the effective bulk and shear modulus given by Voigt (1889) and [START_REF] Reuss | Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle[END_REF] bounds are expressed as follows:

V eff eff R eff V eff eff R eff G G G G G G G G G K K K K K K K K K = + ≤ ≤ + = = + ≤ ≤ + = 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ (3.4)
These bounds are rather elementary and they show only the extremes limits of effective values.

While narrower Hashin-Shtrikman bounds proposed by [START_REF] Hashin | A variational approach to the theory of the elastic behavior of multiphase materials[END_REF] is expressed as:

0 1 0 0 1 0 1 0 1 1 0 1 1 3 1 3 k k k k k k k k k eff - + + ≤ ≤ - + + ϕ ϕ ϕ ϕ (3.5) With 1 0 k k ≥ .
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:

A K K eff 1 0 1 ϕ + = , B G G eff 1 0 1 ϕ + = (3.6)
For conductivity problems, similar expression is given for effective conductivity modulus:

C k k eff ϕ + = 1 0 (3.7)
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:

A K K eff 1 1 0 1 1 ϕ ϕ - + = , B G G eff 1 1 0 1 1 ϕ ϕ - + = (3.8)
and the effective conductivity modulus for conductivity problems:

C k k eff 1 1 0 1 1 ϕ ϕ - + = (3.9)
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 K eff pB pB K K ϕ ϕ - + - = 1 1 1 0 , [ ] G G eff pC pC G G ϕ ϕ - + - = 1 1 1 0 (3.10)
Where G K ϕ ϕ , are functions of physical parameter of composite materials, such as Poisson ratio of the matrix (see [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF].

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 geometric equation

1 2 3 2 2 2 1 = + + p p p x x x
, 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: . 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.

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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF].

Although, various materials science applications require quantitative characterization of inhomogeneities of irregular shape, most of the existing results are based on [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | Elastic inclusons and inhomogeneities[END_REF] solution for an ellipsoidal inhomogeneity. While for 2-D non-elliptical inhomogeneities many analytical and numerical results have been obtained (Zimmerman 1986[START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF][START_REF] Tsukrov | Effective elastic properties f solids with defects of irregular shape[END_REF], 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 [START_REF] Drach | Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes[END_REF] using FEM. The authors showed that pores of irregular shapes can be sometimes approximated by ellipsoids (in agreement with earlier experimental works of [START_REF] Prokopiev | On the possibility of approximation of irregular porous microstructure by isolated spheroidal pores[END_REF]Sevostianov (2006, 2007)). It is difficult, however, to make any conclusions from the results of [START_REF] Drach | Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes[END_REF] (Tables 1 and2 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), [START_REF] Grechka | The influence of crack shapes on the effective elasticity of fractured rocks[END_REF], [START_REF] Mear | Elastic compliances of non-flat cracks[END_REF], [START_REF] Kachanov | Rice's internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness[END_REF].

The only analytical model that can account for concave shape of the pores has been developed

by [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] and applied by [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF] 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 this goal we use (1) Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] in the form given by [START_REF] Benveniste | On the Mori-Tanaka method for cracked solids[END_REF] and (2) Maxwell scheme [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] in the form proposed by [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF]. 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 [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF] for polyhedral shapes, Onaka (2001) for a concave supersphere; [START_REF] Onaka | Elasticstates of doughnut-like inclusions with uniform eigenstrains treated by averaged Eshelby tensors[END_REF] for a toroidal inclusion and Chen et al.

(2015a) for a convex supersphere. We have to point out however, that Eshelby tensor for nonellipsoidal 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 [START_REF] Hashemi | Composites with superspherical inhomogeneities[END_REF] where results of [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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.

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 [START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]. For general case of elastic inhomogeneities, these tensors were introduced and calculated for ellipsoidal shapes by [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]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 0 S containing an inhomogeneity, of volume * V , of a different material with the compliance tensor 1

S . 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:

∞ = ∆ σ H ε : V V * , or, in components, ∞ = ∆ kl ijkl ij H V V σ ε * (3.15)
where ∞ kl σ 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 [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF][START_REF] Walpole | On the overall elastic moduli of composite materials[END_REF] as

( ) [ ] 1 1 - - + - = Q S S H 0 1 , (3.16)
Sevostianov and [START_REF] Sevostianov | Elastic fields generated by inhomogeneities: far-field asymptotics, its shape dependence and relation to the effective elastic properties[END_REF] 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 [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF].

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

n u n u V i j j i ij ∫ Ω ∂ + - = ∆ 2 1 ε (3.17)
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, [START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF].

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 0 k contains an isolated inhomogeneity of volume * V with the isotropic thermal conductivity 1 k . Limiting cases

0 1 = k and ∞ = 1 k
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):

( ) q R ⋅ = ∇ ∆ V V T * (3.18)
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

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

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: As p decreases, the volume of the supersphere 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 x e and y e have been considered.

      Γ             Γ = * p p p V 2 3 3 2 1 2 2 3 (3.
Equation (3.17) can be discretized as:

( ) ( ) ∑ ∫ = Ω ∂ + - = + - = ∆ N i i i j j i i j j i ij S n u n u V dS n u n u V 1 2 1 2 1 ε (3.22)
Where N is the number of elements on boundary surface, i u and i n 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:

         + + = + + = + + = 3 3 3 3 3 3 0 3 2 2 2 0 2 1 1 1 0 1 C B A C B A C B A x x x x x x x x x x x x (3.23)
The same interpolation is used to calculate displacement at the centroid point as a function The normal vector to an implicitly defined surface is proportional to the gradient of the function f so that

        ∂ ∂ ∂ ∂ ∂ ∂ = = = = = 0 3 3 0 2 2 0 1 1 3 2 1 x x x x x x k j i x f x f x f N N N N , , ) , , (
(3.25)

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

( ) N N N n n n n k j i . , , = = (3.
∞ * = ∆ σ H ε : Ṽ L 3 (3.28)
Certainly, tensors H and H ~ entering relations (3.17 (where 0 V is the volume of the unit sphere), we get

( ) ( ) S ijkl ijkl H p V V p H       - = * 0 16 1 5 3 π (3.30)
For a spherical pore ( 1 = p ) (see [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF]) Then, the following simple approximate relations are obtained for the compliance contribution tensor of a superspherical pore:

            +       = = II II J 3 1 3 3 1 2 2 3 4 0 0 K H - G H K G π π S S
( ) ( )             +             - = * II II J 3 1 3 3 1 2 8 1 5 3 0 0 0 K H - G H p V V p K G

H

(3.33) Formula (3.33) identify the microstructural parameter for approximate characterization of the effect of the superspherical pore on elastic properties as

( ) ( ) ( ) 3 2 0 2 1 2 3 1 5 4 3 8 1 5 3             Γ       Γ - =       - = * p p p p p V V p π η .
(3.34)

Note that this parameter differs from one introduced by [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] because of the insufficient accuracy of their estimates. ) on the concavity parameter p (see appendix B for details, table B1). Introducing normalization of ( ) Due to that, we can write

T ∇ ∆ to 3 * L we rewrite (3.16) as ( ) ( ) q R ⋅ = ∇ ∆ * V L T 3 (3.
S ij ij R p R 4 1 5 - = (3.36) where 0 2 k R ij S ij πδ = ~
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:

( ) ( ) ij ij ij p V V p R k ηδ δ =       - = * 0 0 8 1 5 3 (3.37)
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 [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]Kachanov 2002, 2008). Moreover, the cross-property connection has to be of the same form as for spherical pores.

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 [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF], such an approach -incorporating interactions 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.

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

( ) ( ) ( ) j n n ij n i i q R V V q k T       + = ∇ ∑ * 1 1 0 (3.38)
and total strain

( ) ( ) kl n n ijkl n kl ijkl ij H V V S σ σ ε       + = ∑ * 1 0 (3.39)
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 noninteraction approximation):

G NIA K NIA NIA H G G H K K k k ϕη ϕη ϕη + = + = + = 1 1 1 0 0 0 , , (3.40)
where ϕ is the overall porosity, shape factor η is defined by (3.34) and parameters K H and

) ( ), ( ), ( G DIL K DIL DIL H G G H K K k k ϕη ϕη ϕη - = - = - = 1 1 1 0 0 0 (3.41)
Ch.3-2: Evaluation of the effective elastic and conductive properties of a material containing concave pore 57 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.

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 [START_REF] Feynman | The Feynman lectures on physics: 2[END_REF], chapter 11). The Mori-Tanaka scheme [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] as interpreted by [START_REF] Benveniste | On the Mori-Tanaka method for cracked solids[END_REF] 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 ( )

ϕ - 1 1
so that the effective elastic and conductive properties are obtained from the NIA result by replacement

( ) ϕ ϕ ϕ - →
1 correctly predicting that effective conductivity and shear and Young's moduli approach zero as 1 → ϕ

: G MT K MT MT H G G H K K k k η ϕ ϕ η ϕ ϕ η ϕ ϕ - + = - + = - + = 1 1 1 1 1 1 0 0 0 , ,
(3.42) (superscript MT indicates that the effective properties are calculated in the framework of Mori-Tanaka scheme)

Maxwell homogenization scheme

It is probably the oldest one. Originally proposed by [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] 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, [START_REF] Mccartney | Maxwell's far-field methodology applied to the prediction of properties of multi-phase isotropic particuate composites[END_REF] and [START_REF] Mccartney | Maxwell's far-field methodology predicting elastic properties of multiphase composites reinforced with aligned transversely isotropic spheroids[END_REF] formulated it for a material containing either random or aligned ellipsoidal inhomogeneities of identical aspect ratios. We are using the interpretation proposed by [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF], 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. Note that the choice of the mentioned shape is non-trivial. [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF] 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 [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]Giraud 2013, Sevostianov and[START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF])

( ) ( ) G G G G M K K K K M M H H G G H H K K k k α η ϕ α η ϕ α η ϕ α η ϕ ϕη η ϕ - + - = - + - = + - = 1 1 1 1 1 1 3 2 3 0 0 0 , , ( 3

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. Crossproperty connections have been developed by [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]Kachanov (2002, 2008) for noninteracting 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, [START_REF] Sevostianov | Elastic and electric properties of closed-cell aluminum foams. Cross-property connection[END_REF]). 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 ( )

NIA NIA NIA NIA G NIA NIA NIA NIA NIA NIA K NIA NIA k k k k k k H G G G k k k k k k H K K K - - - = - = - - - - = - = - 0 0 0 0 0 0 0 0 0 0 5 7 1 10 2 1 1 ν ν ν ν (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 cross-property connection will take place if the effective properties are calculated using Mori-Tanaka effective field scheme:

( )

MTS MTS MTS MTS MTS MTS MTS MTS k k k G G G k k k K K K - - - = - - - - = -
( ) ( )( ) ( ) ( )( ) 0 0 0 0 0 0 0 0 4 7 2 3 2 2 4 7 2 3 2 2 k k k k k k G G G k k k k k k K K K M G M M M M M K M M M M - + - + + = - - + - + + = - β β (3.49)
Expressions (3.49) obviously coincide with ones for spherical pores.

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

1 2 3 2 2 2 1 = + + p p p x x x
. We showed that, for 1 < p , contributions of an individual pore into both elastic and electric properties is described by the same geometric parameter ( )

3 2 2 1 2 3 1 5 4 3             Γ       Γ - = p p p p π η .
ChaPter 4

IV Property contribution tensor of superspheroidal pores

Superspheroidal's surface is described by implicit equation 1

2 3 2 2 2 1 = +       + p p a x a x x γ
. This 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 (1905Hein ( -1996) ) who defined for this the following values 3

4 25 1 = = γ , .

p

. Geometrically speaking, as a special case 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

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 semianalytically.

Singal inhomogneity of superspheroidal shape with axe of ration 3 e is treated in FEM procedures, the composite is considered to be isotropic in

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 [START_REF] Wark | Reassessment of pore shapes in microstructurally equilibrated rocks, with implications for permeability of the upper mantle[END_REF][START_REF] Emmanuel | Interfacial energy effects and the evolution of pre size distributions during quartz precipitation in sandstone[END_REF]). 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 [START_REF] Kachanov | On quantitative characterization of microstructures an effective properties[END_REF]. 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 [START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]. For general case of elastic inhomogeneities, these tensors were introduced and calculated for ellipsoidal shapes by [START_REF] Sevostianov | Compliance tensor of ellipsoidal inclusion[END_REF]. [START_REF] Kushch | Dipole moments, property contribution tensors and effective stiffness of anisotropic particulate composites[END_REF] 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-property connection between elastic and conductive properties of heterogeneous materials. [START_REF] Kushch | Effective conductivity of composite with imperfect contact between elliptic fibers and matrix: Maxwell's homogenization scheme[END_REF] 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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Eshelby | Elastic inclusons and inhomogeneities[END_REF][START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF]. 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;[START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]Tsukrov andNovak 2002, 2004), only a limited number of numerical results and approximate estimates are available for nonellipsoidal 3-D shapes. Compliance contribution tensors for several examples of pores of irregular shape typical for carbon-carbon composites have been calculated by [START_REF] Drach | Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes[END_REF] using FEM. The authors give the values of the components of compliance contribution tensors for several specific shapes (Tables 1 and2 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 [START_REF] Fabrikant | Applications of Potential Theory in Mechanics[END_REF] The only concave shape that has been discussed in literature in the context of effective material properties is a supersphere -shape described by equation 1

2 2 2 = + + p p p z y x
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

            Γ       Γ - = p p p p ) ( π η (4.3)
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 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

) ( * p g a V γ π 3 4 = (4.5) Where       Γ       Γ       + Γ = p p p p p g 2 3 1 2 2 1 3 1 ) ( (4.6)
and Γ denotes Euler gamma function. The total surface area of the superspheroid is

) , ( * p j a S γ γ π 2 4 = (4.7) with du u u u p j p p p p u u p ) / ( ) ( ) ( ) ( ) ( ) , ( 2 1 2 2 4 2 2 1 1 0 2 1 1 1 1 - - = = - + - = ∫ γ γ (4.8)
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.

Property contribution tensors for a superspheroidal inhomogeneity.

Compliance and resistivity contribution tensors

We first discuss the compliance contribution tensor and consider a homogeneous elastic material (matrix), with the compliance tensor 0 S containing an inhomogeneity, of volume * V , of a different material with the compliance tensor 1 S . 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:

∞ = ∆ σ H ε : V V * or, in components, ∞ = ∆ kl ijkl ij H V V σ ε * (4.9)
where ∞ kl σ are remotely applied stresses that are assumed to be uniform within V in the absence of the inhomogeneity. [START_REF] Sevostianov | Elastic fields generated by inhomogeneities: far-field asymptotics, its shape dependence and relation to the effective elastic properties[END_REF] 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 (

)

dS n u n u V i j j i ij ∫ Ω ∂ + - = ∆ 2 1 ε (4.10)
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, [START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF].

Considering resistivity contribution, we assume that the background material of volume V with the thermal conductivity 0 k contains an isolated inhomogeneity of volume V1 with the thermal conductivity 1 k . 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):

( )

q R ⋅ = ∇ ∆ V V T * (4.11)
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

dS Tn V x T i i ∫ Ω ∂ - =         ∂ ∂ ∆ 1 (4.12)
where T and n are temperature and inward unit normal to the boundary.

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

Equation (4.10) can be discretized as:

( ) ( ) ∑ ∑ ∫ = = Ω ∂       - = + - = ∆ el G N j n i i G i G j el op i G o p p o op h w V dS n u n u V 1 1 2 1 2 1 η ξ ε , ) ( (4.13)
Where el N the number of finite elements on boundary surface, G n is the number of Gauss integration points per element,

i G i G i G w η ξ , ,
denote weights, and coordinates of Gauss integration points in the reference plane. 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 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 ijkl H components are given in Table D1 and Figure 4.7(a).

Relative errors on components ijkl H 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 ).

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 1 q , 3 q are necessary to respectively determine the two distinct components 11 R , 33 R 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 3 q which respects symmetry of revolution, to obtain complementary results for 33 R 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 ( 35 0 2 0 . .

≤ ≤ p

). Convergence of results with refinement of 2D-axis meshes has been obtained in all the range 1 2 0 ≤ ≤ p .

, with relative error lower than 1 %. Numerical results for 11 R , 33 R components of contribution resistivity tensor are given in Table D4 and Figure 4.7 (b).

Analytical approximations for property contribution tensors of a superspheroidal pore

Note, that as discussed by Sevostianov et al. (2008) and [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF], to increase the accuracy of the calculations, extra strain due to the cavity ε ∆ in (4.9) may be alternatively normalized to 3 * L where * L is certain characteristic length of the pore (for example, the maximal distance between its far points):

∞ = ∆ σ H ε : * V L 3 (4.14)
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. [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] and [START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF] shown that tensor H ~for a concave superspherical pore ( 5 0.

< p

) can be approximated by linear functions of the concavity factor p . Similar observation has been done by [START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF] in the context of resistivity contribution tensor: normalization of

) ( T ∇ ∆ to 3 * L yields ( ) q R ⋅ = ∇ ∆ * V L T 3 (4.
ijkl ijkl H p V H ) ( ~* = , ij ij R p V R ) ( ~* = (4.16) so that components ijkl H ~and ij
R ~can be calculated from the data in Tables D1 andD4 H 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

1 2 2 2 3 2 2 2 2 2 1 = + + a x a x a x γ (4.17)
Components of the compliance contribution tensor for such a pore are given by [START_REF] Sevostianov | Elastic and electric properties of closed-cell aluminum foams. Cross-property connection[END_REF]

): [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ; ; ) ( ; ) ( ) ( ) ( ) ( ; ) ( ) ( ) ( ) ( ) ( ; ) ( ) ( ) ( ; ) ( ) ( ) ( ) ( ) ( 1 0 0 0 1313 1 0 0 0 0 1122 1111 1212 2 0 0 1 0 0 0 0 1 0 0 0 1133 1 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 1122 2 0 0 1 0 0 0 0 1 0 0 0 0 0 3333 1 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 1111 4 4 4 2 1 1 2 1 4 2 1 4 4 2 2 1 1 1 4 2 1 4 2 1 4 2 1 4 4 2 2 6 1 4 2 1 1 1 4 2 1 4 2 f f H f f f G H H H f f f f f f H f f f G f f f G f f H f f f f f f H f f f G f f f G f f H κ µ κ κ κ µ κ κ κ κ κ κ κ κ κ µ κ κ κ κ κ κ κ κ κ + = - + - = + = - - - - + - = - + - - - - - - - = - - - - - + - - = - + - + - - - - - = (4.18) where [ ] 3 1 2 1 4 1 2 1 4 3 3 1 2 1 2 2 2 1 2 2 0 0 0 0 0 0 0 - + - = - - = + + = - = g f g f K K ) ( ) ( , ) ( ) ( , ) ( γ γ γ γ γ µ µ υ κ (4.19)
and shape factor g is expressed in terms of the spheroid's aspect ratio γ as follows ( ) 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 Taking into account that expression of the compliance and resistivity contribution tensors for a spherical pore are, (see [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF], Sevostianov et al. 2006) [ ]

( )        > - - - + - < - - = 1 1 1 1 2 1 1 1 1 1 2 2 2 2 2 γ γ γ γ γ γ γ γ γ γ γ γ γ
ij ij k R G δ υ υ υ υ υ 0 0 0 0 0 0 0 2 3 5 1 1 10 1 5 7 4 1 3 = + - + + - - = II J H ) ( ) ( ) )( ( ) (
f f p V p H f f f f f p V p H f f f f f p V p H H H H + - - = + - - - - + - = + - - - + - + + - = - - = + - + - - = + - + - = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) )( ( ; ) )( (

Concluding remarks

In the present work, we discussed compliance and resistivity contribution tensors of a concave axisymmetric pore which boundary is described by equation 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: These formulas serve as the basic building block for calculation of the overall elastic and conductive properties of a material containing multiple concave pores.

µ υ υ υ υ γ πµ υ γ µ υ υ υ υ γ µ υ υ µ υ υ υ υ µ υ υ υ υ ) )( ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) )( ( ) ( ) ( ) ( ) ( ) )( ( ) )( ( ; ) )( ( ) )( ( * * * + - + - - ≈ - - + + - + - ≈ - - ≈ + - + - - ≈ + - + - ≈ p V p H p V p p V p H H H H 0 0 0 0 0 0 1313 2 1 1 5 7 4 1 15 µ υ π υ γ µ υ υ γ ) ( ) ( ) ( ) ( ) ( ) ( ) ( * * - - - + - - ≈ p V p p V p H (4.25) ) ( ) ( ) ( ; * p V p R k R R π γ π

Part 3 Applications to the heterogeneous rock like materials

Part 3

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. [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF] 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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF] 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

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 quasispherical porous grains (oolites, o), mesopores (b) and solid matrix (sparitic cement c) [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF][START_REF] Grgic | Influence of CO2 on the long-term chemomechanical behavior of an oolitic limestone[END_REF][START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF], Ghabelzloo et al. 2009). For this goal, we use recently reformulated Maxwell's homogenization method for elastic composites [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF] which was successfully applied to the estimate of effective elastic constants of oolitic. This method has also been explored in [START_REF] Levin | Generalized Maxwell's scheme for homogenization of poroelastic composites[END_REF], [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF], [START_REF] Sevostianov | Effective properties of linear viscoelastic microcracked materials: Application of Maxwell homogenization scheme[END_REF] for mechanical properties and in [START_REF] Kushch | Effective conductivity of composite with imperfect contact between elliptic fibers and matrix: Maxwell's homogenization scheme[END_REF][START_REF] Sevostianov | Effective thermal conductivity of a composite with thermos-ssensitive constituents and related problems[END_REF] 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 fluidsaturated 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.

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 [START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF]) 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 [START_REF] Kazatchenko | Simulation of acoustic velocities, electrical and thermal conductivities using united pore-structure model of double-porosity carbonate rocks[END_REF], [START_REF] Kazatchenko | Joint inversion of conventional well logs for evaluation of double-porosity carbonate formations[END_REF], [START_REF] Markov | Conductivity of carbonate formations with microfracture systems[END_REF][START_REF] Markov | Determination of electrical conductivity of double-porosity formations by using generalized differential effective medium approximation[END_REF] and [START_REF] Aquino-López | Modeling and inversion of elastic wave velocities and electrical conductivity in clastic formations with structural and dispersed shales[END_REF]. 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 [START_REF] Kazatchenko | Simulation of acoustic velocities, electrical and thermal conductivities using united pore-structure model of double-porosity carbonate rocks[END_REF] 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 [START_REF] Lutz | Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite[END_REF], 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, [START_REF] Alishaev | Effective thermal conductivity of fluid-saturated rocks:Experiment and modeling[END_REF] 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 [START_REF] Thomas | Thermal conductivity of carbonate rocks[END_REF]. 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 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 [START_REF] Suquet | Rappels de calcul tensoriel et d'élasticité[END_REF] 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 [START_REF] Côté | Assessment of structure effects on the thermal conductivity of two-phase porous geomaterials[END_REF]. It may be noticed that [START_REF] Markov | Effective thermal conductivity of inhomogeneous medium containing gas-filled inclusions[END_REF] 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 mesoporosities of oolitic limestones, respective pore size correspond to the standard hydrodynamic regime and these phenomena have been neglected.

Background results

Hereafter we define some notations and recall some results which are needed later. Upper ), with symmetry axis 3 are described by the equation

γ γ γ a a a a a a x a x a x = = = ≤ ≤ ≤ + + ⇔ Ω ∈ 3 2 1 2 2 2 3 2 2 2 2 2 1 1 0 1 , , x
(5.1)

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

N T Q Q Ι Ι P λ γ λ γ ) ( ) ( 2 1 - + =
(5.2) (5.3) with (Torquato 2002, Carslaw and[START_REF] Carslaw | Conduction of Heat in Solids[END_REF], among many others)

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

Ι P λ 3 1 =
(5.5)

Q tensor (Sevostianov and Giraud 2013, Eq. 2.9) for Maxwell homogenization scheme

) ( P.Λ I Λ. Q - =
(5.6)

In the case of a sphere (using 5.5 and

Ι Λ λ = ) Ι Q λ 3 2 =
(5.7)

Conductivity contribution tensor [START_REF] Sevostianov | Effective thermal conductivity of a composite with thermos-ssensitive constituents and related problems[END_REF])

[ ] 1 1 - - - + = ) ( Λ Λ P N i i (5.8)
Resistivity contribution tensor [START_REF] Sevostianov | Effective thermal conductivity of a composite with thermos-ssensitive constituents and related problems[END_REF])

1 1 - - ⋅ ⋅ - = Λ Ν Λ H i i (5.9) Dilute concentration tensor [ ] 1 - - ⋅ + = ) ( Λ Λ P I A i i (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 i λ embedded in an infinite isotropic matrix

I A I H I N i i i i i i A H N = = = , ,
(5.11)

i i A λ λ λ + = 2 3
(5.12)

and i i i i i i A N A H ) ( , λ λ λ λ λ - - = - = 2 (5.13)
In the case of an isotropic ellipsoidal inclusion with conductivity tensor

I Λ i i λ = surrounded
by an infinite isotropic matrix with conductivity tensor (5.15)

I Λ λ = (5.1) relation (5.5) gives         - + + + - = N i T i i Q Q Q Q I I A )) ( 2 1 ( ) ( 2 1 ) ( )) ( 1 ( 1 γ λ γ λ γ λ γ λ λ (5.
One obtains spherical (isotropic) second order tensors

I H I A i i i i H A = = , (5.16) with         - + + + - = )) ( 2 1 ( ) ( 2 1 ) ( )) ( 1 ( 2 3 ) , , ( γ λ γ λ γ λ γ λ λ γ λ λ Q Q Q Q g i i i (5.17) ) , , ( γ λ λ i i g A =
(5.18) One verifies relations

i i i A H 2 λ λ λ - =
(5.19)

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, [START_REF] Sterpenich | Experimental ageing of oolitic limestones under CO2 storage conditions. Petrographical and chemical evidence[END_REF]) whose micro-structure has been detailed [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF]. One may also refer to [START_REF] Nguyen | A composite sphere assemblage model for porous oolitic rocks[END_REF] and [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF], 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 quasi-mono-mineral material constituted of pure calcite (solid volume fraction of calcite, 98 0.

≈ Ω Ω = s calcite s calcite f
). 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 [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF]. As in [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF], 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 [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF].

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 [START_REF] Kazatchenko | Simulation of acoustic velocities, electrical and thermal conductivities using united pore-structure model of double-porosity carbonate rocks[END_REF]: channels (prolate ellipsoids, as elongated pores), and cracks (flat oblate ellipsoids).

mm 100 µm

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 [START_REF] Thomas | Thermal conductivity of carbonate rocks[END_REF] for an extensive review on thermal conductivity of carbonate rocks which confirms reason-ably overall isotropy in most cases. The thermal conductivity of the pure calcite mineral will be assumed equal to [START_REF] Vasseur | Thermal conductivity estimation in sedimentary bassins[END_REF] and [START_REF] Guéguen | Microstructures, percolation thresholds, and rock physical properties[END_REF]). The numerical values of thermal conductivities for liquid and air are respectively

1 1 3 3 - - = K Wm c . λ (according to

Volume fractions and constituents

As in [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF], [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF] three different scales may be identified. First the smallest scale, referred as microscopic scale, corresponds to the intragranular 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

c b Ω + Ω + Ω = Ω 0 (5.20) with 2 1 b b b Ω + Ω = Ω (5.21) One has c b b Ω + Ω + Ω + Ω = Ω 2 1 0 (5.22) with corresponding volume fractions 1 2 1 0 2 2 1 1 0 0 = + + + Ω Ω = Ω Ω = Ω Ω = Ω Ω = c b b c c b b b b f f f f f f f f , , ,
(5.23) One defines the total volume of intra-oolitic pores

S a 0 0 Ω + Ω = Ω (5.24)
and the porosity of the oolite phase at the mesoscopic scale

0 Ω Ω = a a f
(5.25)

The total pore volume can be expressed as

2 1 b b a p Ω + Ω + Ω = Ω (5.26)
and the total porosity

2 1 0 2 1 b b a b b a p f f f f f + + = Ω Ω + Ω Ω + Ω Ω =
(5.27)

Representative data for the reference material studied in this paper are close to 26 0. [START_REF] Giraud | Effective poroelastic coefficients of isotropic oolitic rocks with micro and meso porosities[END_REF]) but a sensitivity study will be performed on the volume fractions of the different phases.

= p f , 74 0 0 . = f , 14 0 0 . = f f a , 19 0. = a f , 12 0. = b f
In sensitive study, as in [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF], we will consider 2

2 1 / b b b f f f = = .

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). ), 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]) 0 3 1 2 1 2 0 0 0 = - + - + + I a a I a a a f f λ λ λ λ λ λ ) (
(5.28) It's well known solution writes (see [Torquato 2002, formula (18.14), p. 463], with

1 2 1 3 f f d - = = , ) ( ) 2 1 2 12 12 1 2 1 8 4 1 λ λ α α λ λ + + = ) , , ( f h (5.29) ) ( ) ( 1 2 1 1 12 3 2 1 3 f f - + - = λ λ α
(5.30) and then self-consistent approximation writes Step 1 : Microscopic level to mesoscopic level

Step 1 : Homogenization of micropores in oolite 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.

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 ( 74 ( )

1 3 2 1 1 1 0 0 2 2 1 1 -               - + + + = - - c b b b b c eff MX H f H f H f λ λ λ (5.32) with 1 b H , 2 b
H and 0 H respectively given by formula (5.19) -(5.13)

) , , ( ), , , ( 2 2 2 1 2 1 b b c c b c b b b c c b c b g H g H γ λ λ λ λ λ γ λ λ λ λ λ - = - =
(5.33)

) ( ) ( I c c I c H 0 0 0 2 3 λ λ λ λ λ + - =
(5.34)

From relation (5.32) one may easily deduce an explicit formula for the effective thermal conductivity:

) ( 3 ) ( 2 3 0 0 2 2 1 1 0 0 2 2 1 1 H f H f H f H f H f H f b b b b c b b b b c c eff MX + + + + + - = λ λ λ λ (5.35)
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 [START_REF] Kazatchenko | Simulation of acoustic velocities, electrical and thermal conductivities using united pore-structure model of double-porosity carbonate rocks[END_REF] for carbonate rocks could be introduced as randomly oriented spheroids are considered and relation (5.32) would be replaced by ( )

1 1 1 0 0 4 4 3 3 2 2 1 1 3 2 1 - - -               - + + + + + = c b b b b b b b b c eff MX H f H f H f H f H f λ λ λ (5.36)
with bi f and bi H 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 to 7 0.

= γ

) could be also used and relation (5.32) would be replaced by ( )

1 1 1 0 0 2 2 1 1 3 2 1 - - -               - + + + = c b b b b c eff MX H f H f H f λ λ λ
(5.37) with 0 H given by a relation similar to (5.33). In further works we will also take into account more realistic shapes such as concave shapes [START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF]). 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 [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF], and relations (5.13)-(5.17) of paper [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF], which have been obtained in elasticity.

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

As in [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF], concave pores (b2) are approximated by super-spheres Step 2 : Maxwell Homogenization scheme

Numerical results

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 [START_REF] Alishaev | Effective thermal conductivity of fluid-saturated rocks:Experiment and modeling[END_REF], [START_REF] Popov | Interrelations between thermal conductivity and other physical properties of rocks : experimental data[END_REF] , and in Zimmerman (1989) for porous rocks more generally.

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 3 0 0 . < < b f which 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 twoscale porosity), the aim of the comparison is not to very accurately fit the data but mainly to

Simplified model based on ellipsoidal approximation for all pore families

The sensitive study is similar to the one presented in [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF] . For

= f

).

As it has been observed in elasticity [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF], 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 (5.41) 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 ( 05 0.

= γ

) 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 ( 20 = γ

) than crack similar oblate pores. 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.

Conclusion

The main factor affecting elastic properties of oolitic limestone is the pore space geometry.

As identified in [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF] some mesopores are concave and may be approximated by superspherical shape (Chen et al. 2015a[START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF]. 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 [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF]). 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 [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF].

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.

φ is volume fraction of the inhomogeneities (porosity for the material with unfilled pores); Gassmann's equation is widely used in geophysics to predict effect of different fluid saturations on the wave speed. However, as mentioned by [START_REF] Han | Gassmann's equation and fluid-saturation effects on seismic velocities[END_REF] 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 [START_REF] Ciz | Generalization of Gassmann equations for porous media saturated with a solid material[END_REF] who obtained the following replacement relations for a particular case of an isotropic two-phase composite with isotropic constituents:

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) (
G G G G G G G G G G G G G G G G G G K K K K K K K K K K K K K K K K K K - + - + - - - + - ≈ - + - + - - - + - ≈ 0 1 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 φ φ φ φ φ φ (6.2)
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 inhomogeneities (porosity for the material with unfilled pores); dry eff K and dry eff G 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:

) ( ) )( ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ' 2 1 2 0 1 0 0 0 2 1 eff eff eff eff A B B A B A K K K K K K K K K K G G K K - - - - - = - + - φ α α (6.
B A B ij A ij B A B ij A ij B A B A B A B A e e P P G G K K G G K K γ γ τ τ α α 2 2 1 2 2 =         = = '
of compliance of irregularly shaped cracks, certain results were obtained by [START_REF] Fabrikant | Applications of Potential Theory in Mechanics[END_REF], Sevostianov and Kachanov (2002a) (planar cracks), [START_REF] Grechka | The influence of crack shapes on the effective elasticity of fractured rocks[END_REF] (intersecting planar cracks), [START_REF] Mear | Elastic compliances of non-flat cracks[END_REF] (non-planar cracks), and [START_REF] Kachanov | Rice's internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness[END_REF] (cracks growing from pores and cracks with partial contact between the faces). Concave inhomogeneity shapes have been analyzed by Sevostianov et al. (2008), [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]Giraud (2012), Chen et al. (2015b) and [START_REF] Sevostianov | Compliance and resistivity contribution tensors of axisymmetric concave pores[END_REF]. [START_REF] Argatov | Rigid toroidal inhomogeneity in an elastic medium[END_REF] 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 [START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF]. For non-ellipsoidal shapes several results have been obtained by [START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF] in the context of polygons and polyhedral. [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] and [START_REF] Onaka | Elasticstates of doughnut-like inclusions with uniform eigenstrains treated by averaged Eshelby tensors[END_REF] 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 nonellipsoidal inhomogeneities using Eshelby tensor (see, for example [START_REF] Hashemi | Composites with superspherical inhomogeneities[END_REF] where results of [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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 [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF] paper (see also book of [START_REF] Mura | Micromechanics of Defects in Solids, Second, Revised Edition[END_REF], where the results are better structured) and the paper of [START_REF] Lubarda | On the Absence of Eshelby Property for Ellipsoidal Inclusions[END_REF]. 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 [START_REF] Markov | Elementary micromechanics of heterogeneous media[END_REF], 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 [START_REF] Lubarda | On the Absence of Eshelby Property for Ellipsoidal Inclusions[END_REF]. Other shapes require solving the integral equation (see, for example, recent result of [START_REF] Radi | Toroidal insulating inhomogeneity in an infinite space and related problems[END_REF], 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: [START_REF] Nemat-Nasser | Micromechanics: Overall Properties of Heterogeneous Solids[END_REF]. Components of this tensor were calculated for 2-D pores of various shape and 3-D ellipsoidal pores in isotropic material by [START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]. For general case of elastic inhomogeneities, these tensors were introduced and calculated (for ellipsoidal shapes) by [START_REF] Sevostianov | On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites[END_REF]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

∞ = ∆ σ H ε : V V * (6.6)
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:

∞ = ∆ ε N σ : V V * (6.7)
where ∞ ε is the "remotely applied" strain.

For an ellipsoidal inhomogeneity, tensors H -and N -tensors are given by:

[ ] 1 1 0 1 - -+ - = Q S S H ) ( , [ ] 1 1 0 1 - -+ - = P C C N ) ( (6.8)
Hill's tensor mnrs P is calculated in terms of the gradient of the second rank Green's tensor ( )

x x ′ - ij G as ( ) ( ) ∫ ′ ′ ∂ ′ - ∂ ∂ ∂ ≡ 1 V i mj p mpij d x G x P x x x x
(6.9)

where parentheses at subscripts denote symmetrization with respect to p m ↔ and j i ↔ ;

Hill's tensor has the same symmetry as the tensor of elastic constants: ( )

0 0 rskl mnrs mnkl ijmn ijkl C P J C Q - = (6.11) Hereafter, 2 / ) ( kj il lj ik ijkl J δ δ δ δ + =
and the inverse of symmetric (with respect to

j i ↔ and l k ↔ ) fourth-rank tensor 1 - ijkl X is defined by ijkl mnkl ijmn mnkl ijmn J X X X X = = - - 1 1 .
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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF]:

( ) ( ) 1 0 1 0 1 1 - - - - - - - = - S S S S H H B A B A (6.12) ( ) ( ) 1 0 1 0 1 1 - - - - - - - = - C C C C N N B A B A (6.13)
In particular, if material "B" is either rigid or represents a pore, the above relations take the form:

( ) 1 0 1 1 - - - - = - C C N N A rigid A ; ( ) 1 0 1 1 - - - - = - S S H H A pore A (6.14)
If the inhomogeneity A is a pore and B is perfectly rigid these relations simplify further:

0 1 1 S N N - = -- - rigid pore ; 0 1 1 C H H = -- - rigid pore (6.15)
For a non-ellipsoidal shape, tensors P and Q are not constant inside the inhomogeneity, ( )

x P P = , ( ) x Q Q =
and relations (6.8) do not take place. However, in some cases, these expressions can be used approximately with tensors ( )

x P P = , ( ) x Q Q =
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).

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 and E ijij S on the inhomogeneity shape and can be calculated from the ones for a sphere: (6.18) So that average tensor ijkl P has only two independent components. That can be calculated as [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF]) V is the volume of the inhomogeneity. For the supersphere (6.5), * V is expressed in terms of Gamma function

      - + = ) ( ) ( 0 0 1212 1 8 3 1 1 ν π p f G P (6.19) with             = ∫ ∫ Ω Ω dS n dS n x x x V p f 2 1 3 2 1 1 ' * ) ( (6.20) where 2 1 2 1 , , ) ( , = = ′ - = k x x x x x x k k k k k , ) , , (
dt t e x x t 1 0 - ∞ - ∫ = Γ ) ( as ( ) [ ] ( ) m m m V 3 3 8 3 2 Γ Γ = * , p m 2 1 = (6.21)
Due to the geometric symmetry, only 8 1 of the real shape

( ) [ ] 1 0 0 1 0 1 - - - + - = ′ rskl mnrs mnkl ijmn ijkl ijkl ijkl C P J C S S H ) ( (6.22a) For a pore, 1 0 1 - - ) ( ijkl ijkl S S
vanishes and expression (6.22a) can be rewritten as: We observe that the accuracy is better than 10% for all components of ijkl H for convex shapes ( ) . 5 0 > p . 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 tensors for concave inhomogeneities need to be calculated numerically for each specific set of inhomogeneity's elastic constants.

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 noninteracting inhomogeneities [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF]. Below, we calculate effective properties in the framework of the non-interaction approximation and using Mori-Tanaka and Maxwell homogenization schemes.

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

( ) ( ) kl n n ijkl n kl ijkl ij H V V S σ σ ε       + = ∑ * 1 0 (6.23)
For material containing inhomogeneities "A", effective compliance tensor can be expressed as:

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( NIA dry A A NIA dry A A NIA dry NIA eff NIA dry A A NIA dry A A NIA dry NIA eff G G G G G G G G G G G G G G G K K K K K K K K K K K K K K K - + - - + - = - + - - + - = 0
ν ν ν ν - - = - - = ) ( , G k H H (6.29)
and η is the shape factor that for the superspherical shape has the following form (Chen et

al. 2015b) ( ) ( )       - = * p V V p 0 8 1 5 3 η (6.30) ( ( ) p V *
is the volume of the supersphere given by (6.21) and 3 4

0 π = V
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:

NIA dry A G A A G A NIA dry NIA eff NIA dry A K A A K A NIA dry NIA eff G G H G G G G H G G G G G K K H K K K K H K K K K K η η η η + - + - = + - + - = ) ( ) ( ) ( ) ( 0 0 0 0 0 0 0 0 (6.31)
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 [START_REF] Sevostianov | Non-interaction approximation in the problem of effective properties[END_REF] and [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF], respectively.

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 [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] as interpreted by [START_REF] Benveniste | On the Mori-Tanaka method for cracked solids[END_REF] 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 A H as (6.24): In the case of isotropic microstructure,

( ) ( ) [ ] 1 1 1 - - - + - + = J H S S H S S 0 0 φ φ φ A A A
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( A MT dry A A MT dry MT dry A MT eff A MT dry A A MT dry MT dry A MT eff G G G G G G G G G G G G G G K K K K K K K K K K K K K K - + - - + - = - + - - + - = 0 0 0 0 0 0 0 0 0 0 0 0 φ φ φ φ (6.35) where ( ) ( ) G MT dry K MT dry H G G H K K η φ φ η φ φ - + = - + = 1 1 1 1 0 0 , ( 6
( ) ( ) ( ) (
)

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( A MT dry G A A MT dry MT dry G A MT eff A MT dry K A A MT dry MT dry K A MT eff G G G G H G G G G G G H G G G G K K K K H K K K K K K H K K K K - + - - - + - - = - + - - - + - - = 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 η η η η (6.37)

Maxwell scheme

It is proposed by [START_REF] Maxwell | A Treatise on Electricity and Magnetism[END_REF] for electrical conductivity of a material containing randomly located spherical inhomogeneities is probably the oldest homogenization scheme. We use its interpretation proposed by [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF], 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:

[ ] 1 - Ω - + = A A M eff J H Q H S S 0 : : φ φ (6.38)
where Ω Q is fourth-rank tensor (Hill's tensor) that depends on the shape of the domain Ω . For Replacement relation (6.12) yields then:

[ ] 1 1 0 1 0 0 - - - - + - + = ) ( ) ( S S S S S S M dry A M eff φ φ (6.40)
In particular, for isotropic materials:

) ( ) ( )] ( ) ( [ ) ( ) ( )] ( ) ( [ A M dry A A M dry M dry A M eff A M dry A A M dry M dry A M eff G G G G G G G G G G G G G G K K K K K K K K K K K K K K - + - - + - = - + - - + - = 0 0 0 0 0 0 0 0 0 0 0 0 φ φ φ φ (6.41)
with effective bulk and shear moduli for dry material given by: Alternative, volume fraction independent form of the equations (6.41) is

) ( , ) ( G G G G M dry k k k k M dry H H G G H H K K α ϕη α ϕη α ϕη α ϕη - + - = - + - = 1 1 1 1 1 1 0 0 (6.
( ) ( ) ( ) (
)

) ) ( ( ) ( ) ) ( ( ) ( ) ) ( ( ) ( ) ) ( ( ) ( G G M dry G A A G G M dry G A A M dry M eff k k M dry k A A k k M dry k A A M dry M eff G G H G G G G G G H G G G G G G K K H K K K K K K H K K K K K K α α η α α η α α η α α η 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 + - + - + - + - = + - + - + - + - = (6.44)
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.

Concluding remarks.

We verified applicability of the replacement relations derived by [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF] 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 superspherical shape (described by equation

1 2 3 2 2 2 1 ≤ + + p p p x x x
). It is shown that for convex shapes (concavity parameter 5 0.

> p

) 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 noninteraction 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). 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 [START_REF] Ciz | Generalization of Gassmann equations for porous media saturated with a solid material[END_REF]. 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)).

Part 4 Concluding Remarks and Perspectives

Part 4

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 [START_REF] Grgic | Influence of CO2 on the long-term chemomechanical behavior of an oolitic limestone[END_REF], concave pores are observed between prolate spheroidal oolite grains. The shape can be approximately described by a supersphere [START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF].

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.

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 [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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.

Supersphere is described by equation 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.

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 2 0. = p 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 2 0. = p 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 into both elastic and electric properties:

( ) 3 2 2 1 2 3 1 5 4 3             Γ       Γ - = p p p p π η for 1 < p
. One has to notice that it is different from the one of [START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] and it shows a better accuracy.

Due to this, effective elastic and conductive properties for material containing superspherical 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 3 e which boundary is described by ( ) (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. 33 R ~ behaves linearly on shape parameter p and 11 R 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 1 2 0 < < p .

1
. The latter serves as the basis for calculating effective properties of materials containing multiple concave pores.

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 [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF] 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 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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF] for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It it 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.

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.

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

E.2 Unit vector to the boundary of superspheroid: derivation from implicit surface equation

The normal vector to an implicitly defined surface 0 1 As the surface of the superspheroid is a surface of revolution, cylindrical coordinate frame (r, ϕ , 3

x with 0 ≤ r, 0 ≤ ϕ ≤ 2π ) allows to express the results in compact form. One replaces 1 x and 2

x by their expression as functions of radius r and polar angle ϕ

1. 1

 1 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

g

  Shape factor in terms of function of the spheroid's aspect ratio 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.

  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 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

 0 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

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

Fig. 0

 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).

Fig. 0

 0 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.

  Fig.0.5: Tenseur de souplesse évaluépar les deux méthodes de normalisation (a) supersphère (b) supersphéroïde (Chen et al.2015 b)

Fig. 0 . 6 :

 06 Fig.0.6: Tenseur de résistivité évalué par les deux méthodes de normalisation (a) supersphère (b) supersphéroïde(Sevostianov et al.2016)

Fig. 0

 0 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.

Fig. 1

 1 Fig.1.1: Method of multiple scales (micro » mini » macro)[START_REF] Guéguen | Introduction à la physique des roches, Herann, éditeurs des sciences et des arts[END_REF]

Fig. 1

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

Fig. 1

 1 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

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

Fig. 1

 1 Fig.1.6: High magnification SEM image of oolitic iron ore showing a concave inter-oolitic crystal of siderite.([START_REF] Grgic | Impact of chemical weathering on micro/macromechanical properties of oolithic iron ore[END_REF]

  reported that this equation was probably first proposed by Lamé. of interest here). The volume changes with p according to[START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] (Fig.2.1):

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

Fig. 2

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

  A for details). Comparison of the calculations with these values gives the relative computational error between 0.13% and 0

Fig. 2

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

Fig. 2

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

Figure 2

 2 Fig.2.5: Comparison of the calculated ES 1212 with mesh density 80 nodes per edge with the results ofOnaka (2001)

Figure

  Figure 2.4 and 2.5 illustrate dependence of ) ( p f defined by Eq. (2.42) and

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

Fig. 2

 2 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

  second Eshelby problem for ellipsoidal inhomogeneitiesSimilar to the first Eshelby problem excluding the two-phase composite is submitted to a remotely applied uniform loading at infinity, and region Ω namely inhomogeneity 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[START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF]):

σ

  Uniform loading at infinity of the matrixRemotely applied loadings are submitted along 3 e -(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 figure3.1 (a). Cubic symmetry of supersphere leads to an overall isotropy, one obtains:

Fig. 3 2

 3 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.

  p may be called concavity factor or a parameter of deviation from spherical shape.When 1 = p, the supersphere becomes the ordinary sphere. The sphere is transformed into concave. Examples of this shape are shown in Figure3.2 for different values of p .

  ) varies dramatically (inset in Figure3.2).

Fig. 3

 3 Fig.3.2: Different superspherical shape at different values of p. Inset: dependence of the supersphere volume on p

  ) and (3.19) are different since they represent different normalizations of the pore contribution to the effective compliance.[START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] shown that tensor H ~ for approximated by linear functions of the concavity factor p . Unfortunately, their results are based on low accuracy calculations ofSevostianov et al. (2008). Our results, allows more accurate representation of tensor H ~. They are illustrated in Figure3.5. Note that effect of pore becomes negligible at 2 0. = p . Due to that, one can write for the components of H

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

Fig. 3

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

GH

  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:

  indicates Maxwell homogenization scheme.

  connections in the framework of Maxwell homogenization scheme can be written in a similar way if we observe first that 3 relation into (3.47), we obtain cross-property connections in the framework of Maxwell homogenization scheme:

Fig. 4

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

Fig. 4

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

Fig. 4

 4 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)

  , 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).

( 1818 )

 1818 where concave -convex cases have been discussed (seeGuitart 2009, Barbin and[START_REF] Barbin | Mathematical physics in the style of Cabriel Lamé and the Treatise of Emile Mathieu[END_REF], Effect of a superspherical pore on overall elastic properties was first analyzed bySevostianov et al. (2008) using FEM. Their results have been used by[START_REF] Sevostianov | On the compliance contribution tensor for a concave superspherical pore[END_REF] 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[START_REF] Chen | Evaluation of the effective elastic and conductive properties of materials containing concave pores[END_REF]. The authors showed that, for, contributions of an

Fig. 4

 4 Fig.4.4: Axisymmetric superspheroid ("superegg") for different values of p.

Fig. 4

 4 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

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

  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.

Fig. 4

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

  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

  15) and components of R ~ show linear dependence on the concavity parameter for 1 ≤ p .Let us check this approach for the superspheroidal pore described by (4unit radius), we get the following connection between components of tensors H and H ~, and R and R ~:

FigR~ can be accurately approximated by 33 R~

 33 Fig 4.5b). In other words, when

  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

  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).

  order tensors. I represents the second-order, identity tensor ( ). 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 (

Fig. 5

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

  [START_REF] Clauser | Thermal conductivity of rocks and minerals[END_REF].

Fig. 5

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

  may be noticed that, in the tested range of microporosity 2 0. < a f, 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

  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.

  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) (seeChen et al. 2015a, for the elasticity problem) thanks to a numerical method (Finite Element Method). As this tensor is spherical (hydrostatic) the resistivity contribution tensors are given in the next section, as they depend on the thermal conductivity of the surrounding matrix ( c λ ) and the inclusion (liquid water and air will be considered).

Fig. 5

 5 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 :

 4 Fig 5.4: Mesoporosity: ellipsoidal pores (b1) and concave pores (b2)[START_REF] Sevostianov | Micromechanical modeling of the effective elastic properties of oolitic limestone[END_REF]

Fig. 5

 5 Fig.5.5: Pores saturated with air: experimental thermal conductivity as a function of porosity, for pure calcitic limestones(Robertson 1988)

  the simplified model based on ellipsoidal approximation, mesopore family b2 is modelled by flat oblate pores close to cracks

Figure 5

 5 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 ( 7 0 6 0 5 0 4 0 0

Remark:

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

G

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

  p a concavity factor or a parameter of deviation from spherical shape. 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 .

Fig

  Fig.6.1: Superspherical inhomogeneity for different values of p

  contains an inhomogeneity, of volume * V , of a different material with the compliance and stiffness tensors 1 S and 1 C . 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

  s tensor Q is related to as follows[START_REF] Walpole | On bounds for the overall elastic moduli of inhomogeneous systems -I[END_REF]):

  [START_REF] Milgrom | The energy of inclusions in linear media: Exact shape independent relations[END_REF] on independency of the traces of Eshelby tensor E iijj

  symmetry of the superspherical inhomogeneity (Fig.6.2), it leads to the following relations for isotropic matrix with shear modulus 0 G and Poisson's ratio 0

P

  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

H

  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

  of the inhomogeneities, (6.26) yields the following expressions for effective bulk and shear moduli,

  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

.

  The upper stop mark is chosen as 5 = p since the volume of 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.

FEν

  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 is Poisson's ratio of matrix.

  derivatives may be found in standard finite element textbooks (see among othersZienkiewicz and Taylor (2000)). One defines metric tensor ij g area of the triangle may be calculated thanks to the following surface integral in the reference plane problem of calculation of integral (C.1), the contribution of the quadratic triangular element T6 takes the form component k of nodal vector displacement, at node i ( function of the spatial coordinates (see appendix E). Finally, integral (C.8the whole superegg surface is obtained by summing the contribution of all the finite elements. By denoting el N the total number of 6 T finite elements on the superspheroidal boundary, the surface integral writes

TABLE F1

 F1

. COMPONENTS OF THE COMPLIANCE CONTRIBUTION TENSOR FOR A SUPERSPHERICAL PORE: ijkl H ′ CALCULATED BY (3.67) FROM COMPONENTS OF HILL TENSOR AND ijkl H CALCULATED BY FEM 161

TABLE F2

 F2

					List of notations
	σ			Stress vector
	ε			Strain vector
	B A , ,	C	Shape factor
	p			Concavity parameter
	γ				Aspect ratio
	b a , ,	c	semi-principal axes of length
	ε	* ij			Eigenstrain inside inclusion
	Ω			Domain occupied by inclusion or inhomogeneity
	Ω ∂			Surface of inclusion or inhomogeneity
					Eshelby tensor
	. . .			Average over volume
			(	,	,	)
					XX XXI

. MATERIAL PROPERTIES USED FOR CALCULATION OF THE EFFECTIVE PROPERTIES. .......... 162 E S

Part 1 Introduction Part 1 ChaPter 1 I General framework 1.1 Background Context

	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
	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.
	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

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

[START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF]

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).

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

	SeCtion 2
	2

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

  )

	dry		dry
	eff		eff
	eff	dry	dry	dry
		eff	eff	eff
	dry		dry
	eff		eff
	eff	dry	dry	dry
		eff	eff	eff

  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 ijkl H 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 ijkl H ′ and
	1212

ijkl

H in dependence on the concavity parameter p (for reader's convenience, the results are also given in Table

F1

of annex F). 1111 H 1122 H -

  .36) and η , K H , and G H 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:

  , conclusions could be made from the newly obtained numerical results. On one side, the three non-zeros components such as 3333H by the second normalization method) show their linear behaviors on shape parameter p , but not for the others 1111H ~and 1122 H ~.On the other side, 1111 H and 1122 H

	2 1 x	+	x	2 2	p	+	x	3	2	p	≤
											~,	1133 ~ and 1313 H H

~

(normalized

Table B2 . Number of elements and nodes of T3 mesh

 B2

	p	Total number of node on the surface	Total number of element on the surface of
			of supersphere		supersphere
	0.20		7576		31920
	0.20 0.25	-2.252067	7.769440 8464	5.047077	2.846482 37896	0.020923
	0.25 0.30	-1.130373	4.816686 20072	2.981758	2.116599 82624	0.088891
	0.30 0.35	-0.845580	3.382321 16168	2.093818	1.760289 66752	0.227063
	0.35 0.40	-0.672997	2.810947 12240	1.740110	1.668871 50576	0.435782
	0.40 0.45	-0.604294	2.523916 19448	1.562424	1.642101 79552	0.701513
	0.45	-0.584079	2.426625	1.502196	1.593402	1.006303
	0.50 0.50	-0.560089	2.303054 18152	1.425700	1.571938 72984	1.333360
	0.55 0.55	-0.539260	2.221076 18024	1.374952	1.545565 76672	1.669101
	0.60 0.60	-0.524058	2.166984 19248	1.341466	1.529875 86128	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

Table D2 . Number of elements and nodes of the mesh b (more refined on superspheroid surface)

 D2

		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

Table D3 . Relative errors on components of compliance tensor between mesh a and mesh b

 D3

	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%

Table F1 . Components of the compliance contribution tensor for a superspherical pore: ijkl H ′ calculated by (3.67) from components of Hill tensor and ijkl H calculated by FEM p

 F1

	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	1111 H ′ 1.88623	1111 1.976868 H	1122 H ′ -0.418117	1122 -0.403603 H	1212 H ′ 1.30943	1212 1.258775 H
	0.20 4.50	1.84152 1.87905	7.769440 1.973865	-0.395762 -0.414525	-2.252067 -0.400512	1.33843 1.31168	5.047077 1.258916
	0.25 5.00	1.89651 1.87847	4.816686 1.964475	-0.423257 -0.414300	-1.130373 -0.395101	1.30152 1.31398	2.981758 1.258864
	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

Table F2 . Material properties used for calculation of the effective properties.

 F2 Cela nécessite de considérer des formes non ellipsoïdales et de résoudre numériquement les problèmes d'Eshelby.Le coeur 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.

	micromécanique.
	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

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

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.

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. 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. 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. Acknowledgements 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.

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.

XII XIII

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.

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.

The pore shape is described by equation

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.

ChaPter 5

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

A.

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 [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF]. 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

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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF]) to ones of non-ellipsoidal shape. We discuss inhomogeneities of superspherical shape, described by equation 1

and show that the replacement relations can be used in the rank of convex shapes ( 5 0. p >

), 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

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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF] 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 [START_REF] Gassmann | Über die elastizität porpöser medien: Vierteljahrsschrift der Naturforschenden Gesellscaft in Zurich[END_REF] 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:

( )

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;

Subscripts " ) (1 " and " )

( 2 " referred to the composite containing "A" and "B", respectively.

Overbar stays for the average over the volume, i P and i e ( A i ≡ or B ) are the pressure inside the inhomogeneities produced by external load and the corresponding volumetric strains; i τ and i γ 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:

The assumption used in the derivation is equivalent to the statement that the inhomogeneities are ellipsoids subjected to the uniform external field [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF][START_REF] Lubarda | On the Absence of Eshelby Property for Ellipsoidal Inclusions[END_REF][START_REF] Rodin | Eshelby's inclusion problem for polygons and polyhedral[END_REF].

Importance of the replacement relations for geophysical applications has been discussed in recent papers of Saxena andMavko (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[START_REF] Kachanov | Effective moduli of solids with cavities of various shapes[END_REF]Tsukrov andNovak 2002, 2004)

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:

Appendix B: numerical results of superspherical pore by FEM

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

Curvilinear coordinates

) , ( η ξ , and node coordinates

x are related by shape functions ) , ( η ξ i N :

correspond to vertices of triangular elements, and nodes

)) , ( ), , ( ), , ( (

Appendix D: Numerical results of superspheroidal pore by FEM

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

Table E1: Exact values of dimension less volume and surface area of a superspheroid (a =1,

) for concavity parameter ) One recovers the known result: the normal vector to a surface of revolution belongs to a meridian plane ( = φ constant plane), its component φ N is equal to zero. A normal unit vector may be easily deduced from relation (E.13).

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

Summary:

Focusing on the effect of shape factor on the overall effective properties of heterogeneous materials, the 1 st and the 2 nd 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 1 st 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 [START_REF] Onaka | Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains[END_REF] 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 2 nd 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.

Summary 164

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 [START_REF] Sevostianov | Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape[END_REF] 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 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 [START_REF] Sevostianov | Relations between compliances of inhomogeneities having the same shape but different elastic constants[END_REF] for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It it 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.