Focusing on the effect of shape factor on the overall effective properties of heterogeneous materials, the first and the second Eshelby problem related to 3D 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 first Eshelby problem, Eshelby tensor interrelates the resulting strain about inclusion and eigenstrain that would have been experienced inside the inclusion without any external contraire. Calculations of this tensor for superspherical pores-both concave and convex shapes-are performed numerically. Results are given by an integration of derivation of Green''s tensor over volume of the inclusion. Comparisons with the results of Onaka (2001) for convex superspheres show that the performed calculations have an accuracy better than 1%. The current calculations have been done to complete his results.
In the second Eshelby problem, property contribution tensors that characterizes the contribution of an individual inhomogeneity on the overall physical properties have been numerically calculated by using Finite Element Method (FEM).Property contribution tensors of 3D non ellipsoidal inhomogeneities, such as supersphere and superspheroid, have been obtained. Simplified analytical relations have been derived for both compliance contribution tensor and resistivity contribution tensor.
Property contribution tensors have been used to estimate effective elastic properties and effective conductivity of random heterogeneous materials, in the framework of Non-Interaction Approximation, Mori-Tanaka scheme and Maxwell scheme.
Two applications in the field of geomechanics and geophysics have been done. The first application concerns the evaluation of the effective thermal conductivity of oolitic rocks is performed to complete the work of Sevostianov and Giraud (2013) for effective elastic properties. A two step homogenization model has been developed by considering two distinct classes of pores: microporosity (intra oolitic porosity) and meso porosity (inter oolitic porosity). Maxwell homogenization scheme formulated in terms of resistivity contribution tensor has been used for the transition from meso to macroscale. Concave inter oolitic pores of superspherical shape have been taken into account by using resistivity contribution tensor obtained thanks to FEM modelling. Two limiting cases have been considered: ‘dry case'' (air saturated pores) and ‘wet case'' (water liquid saturated pores). Comparisons with experimental data show that variations of effective thermal conductivity with porosity in the most sensitive case of air saturated porosity are correctly reproduced.
Applicability of the replacement relations, initially derived by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It is the second application of newly obtained results on property contribution tensors.
We have considered 3D inhomogeneities of superspherical shape. From the results, it has been seen that these relations are valid only in the convex domain, with an accuracy better than 10%. Replacement relations can not be used in the concave domain for such particular 3D shape.
目录
Chapter 1 Introduction
Chapter 2 Property contribution tensors of 3D non-ellipsoidal inhomogeneity
Chapter 3 Property contribution tensors of superspherical pores
Chapter 4 Property contribution tensors of superspheroidal pores
Chapter 5 Effective thermal conductivity of oolitic rocks using the Maxwell homogenization method
Chapter 6 Accuracy of the replacement relations for materials with non-ellipsoidal inhomogeneities
Chapter 7 Concluding remarks and perspectives
References
Appendix A Analytical S and H tensor of sphere in isotropic case
Appendix B Numerical results of superspherical pore by FEM
Appendix C Numerical integration on the surface of superspheroid
Appendix D Numerical results of superspheroidal pore by FEM
Appendix E Complementary geometrical results related to superspheroidal shapes
Appendix F Complementary geometrical results related to superspherical shapes
