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## Fundamental Eshelby problems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Homogenization schemes**

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

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

**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.

**Organization of the work**

Four parts are involved in this work. In Part 1, background and the theoretical frameworks are briefly overviewed. In Part 2, property contribution tensor such as compliance contribution tensor in elasticity problems and resistivity contribution tensor in thermal conductivity problems will be carefully discussed. Numerical calculations are performed for superspherical and superspheroidal pores with the pursuit of high precision. Three papers related to Eshelby tensor, compliance contribution tensor, resistivity contribution tensor published from 2015 to 2016 in I.J.E.S will be shown entirely (IJES: International Journal of Engineering Science) In Part 3, relative applications in the field of rock mechanics and geophysics will be illustrated. Effective thermal conductivity of oolitic rocks is discussed by using the reformulated Maxwell homogenization approach (paper published in International Journal of Rock Mechanics and Mining Sciences (2015)). The accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities is verified (paper submitted to International Journal of Solids and Structures I.J.S.S. April/2016).

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

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

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

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

**Eshelby tensor of concave superspherical inclusions**

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

This chapter is motived by the work of Onaka (2001) who considered a particular≥3-D non-ellipsoidal superspherical shape of convex curvature (with concavity parameter 1). The novelty of the presented work concerns the extension to concave domain. As an example, concave superspherical shape (with < 1) is useful to model filling materials between spheres as it may be observed in oolithic rocks (oolithes are nearly spheroidal or spherical grains).

After a brief introduction of fundamental physics of the first Eshelby problem in section 1, new computational results for concave superspherical pores will be provided in section 2, to end this chapter, discussions about the related issues and some short remarks will be proposed in section 3.

**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.

**Table of contents :**

**PART 1 INTRODUCTION **

**I General framework**

1.1 Background Context

1.2 Fundamental Eshelby problems

1.3 Homogenization schemes

1.4 Goals of the thesis

1.5 Organization of the work

**PART 2 PROPERTY CONTRIBUTION TENSORS OF MATERIAL CONTAINING 3-D NONELLIPSOIDAL INHOMOGENEITIES **

**II Eshelby tensor of concave superspherical inclusions**

1 INTRODUCTION & BACKGROUND

1.1 The first Eshelby problem for ellipsoidal inclusions

1.2 Isotropic material containing non-ellipsoidal inclusion

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

2.1 Introduction

2.2 Average S-tensor for superspherical inclusion

2.3 Numerical approach

2.4 Discussion

2.5 Conclusions

3 DISCUSSIONS: EFFECT OF THE SHAPE FACTOR ON THE ELASTIC ENERGY

**III Property contribution tensor of superspherical pores**

1 INHOMOGENEITY PROBLEM: BACKGROUND

1.1 The second Eshelby problem for ellipsoidal inhomogeneities

1.2 Homogenization schemes of effective properties

1.3 Isotropic materials containing superspherical inhomogeneities

2 EVALUATION OF THE EFFECTIVE ELASTIC AND CONDUCTIVE PROPERTIES OF A MATERIAL CONTAINING CONCAVE PORE (IJES, 2015B)

2.1 Introduction

2.2 Property contribution tensors for a superspherical inhomogeneity

2.3 Effective properties of a material containing superspherical pores

2.4 Cross-property connections for a material containing superspherical pores

2.5 Concluding remarks

**IV PROPERTY CONTRIBUTION TENSOR OF SUPERSPHEROIDAL PORES **

1 OVERVIEW OF NUMERICAL PROCEDURES

2 COMPLIANCE AND RESISTIVITY CONTRIBUTION TENSORS OF AXISYMMETRIC CONCAVE PORES OF SUPERSPHEROIDAL SHAPE (IJES, 2016)

2.1 Introduction

2.2 Property contribution tensors for a superspheroidal inhomogeneity.

2.2.1 Compliance and resistivity contribution tensors

2.3 Analytical approximations for property contribution tensors of a superspheroidal pore

2.4 Concluding remarks

**PART 3 APPLICATIONS TO THE HETEROGENEOUS ROCK LIKE MATERIALS **

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

1.1 Introduction

1.2 Background results

1.3 Microstructure of a reference porous oolitic limestone

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

1.5 Numerical results

1.6 Conclusion

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

2.1 Introduction.

2.2 Compliance and stiffness contribution tensors and replacement relations for ellipsoidal inhomogeneities.

2.3 Calculation of compliance and stiffness contribution tensors and Hill tensor P for a superspherical inhomogeneity

2.4 Using replacement relations for calculation effective properties of materials.

2.5 Concluding remarks.

**PART 4 CONCLUDING REMARKS AND PERSPECTIVES **

1.1 Concluding remarks for the mains results

1.2 Perspectives

**REFERENCES**