$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} % \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\trans}[1]{{{#1}{}^\intercal}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \newcommand{\mat}{\mathsf} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\volt}[1]{{#1}^{-1\circ}} \newcommand{\dcirc}{\overset{\circ}{:}} \newcommand{\jump}[1]{\mathopen{[\![}\,#1\,\mathclose{]\!]}} $$

References

Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. National Bureau of Standards - Applied Mathematics Series - 55, Washington D.C.

Achour, M., Bignonnet, F., Barthélémy, J.-F., Rozière, E., Amiri, O., 2020. Multi-scale modeling of the chloride diffusivity and the elasticity of Portland cement paste. Construction and Building Materials 234, 117124. https://doi.org/10.1016/j.conbuildmat.2019.117124

Barthélémy, J.-F., 2022. Echoes: Extended Calculator of HOmogEnization Schemes. https://doi.org/10.5281/zenodo.19679057

Barthélémy, J.-F., 2020. Simplified approach to the derivation of the relationship between Hill polarization tensors of transformed problems and applications. International Journal of Engineering Science 154, 103326. https://doi.org/10.1016/j.ijengsci.2020.103326

Barthélémy, J.-F., 2009a. Compliance and Hill polarization tensor of a crack in an anisotropic matrix. International Journal of Solids and Structures 46, 4064–4072. https://doi.org/10.1016/j.ijsolstr.2009.08.003

Barthélémy, J.-F., 2009b. Effective Permeability of Media with a Dense Network of Long and Micro Fractures. Transport in Porous Media 76, 153–178. https://doi.org/10.1007/s11242-008-9241-9

Barthélémy, J.-F., Giraud, A., Lavergne, F., Sanahuja, J., 2016. The Eshelby inclusion problem in ageing linear viscoelasticity. Int. J. Solids Struct. 97–98, 530–542. https://doi.org/10.1016/j.ijsolstr.2016.06.035

Barthélémy, J.-F., Giraud, A., Sanahuja, J., Sevostianov, I., 2019. Effective properties of ageing linear viscoelastic media with spheroidal inhomogeneities. International Journal of Engineering Science 144, 103104. https://doi.org/10.1016/j.ijengsci.2019.05.015

Barthélémy, J.-F., Sevostianov, I., Giraud, A., 2021. Micromechanical modeling of a cracked elliptically orthotropic medium. International Journal of Engineering Science 161, 103454. https://doi.org/10.1016/j.ijengsci.2021.103454

Bornert, M., Bretheau, T., Gilormini, P., 2001. Homogénéisation en mécanique des matériaux. Hermes science.

Budiansky, B., O’Connell, R.J., 1976. Elastic moduli of a cracked solid. International Journal of Solids and Structures 12, 81–97. https://doi.org/10.1016/0020-7683(76)90044-5

Dormieux, L., Kondo, D., Ulm, F.-J., 2006. Microporomechanics. John Wiley & Sons, Chichester, West Sussex, England ; Hoboken, NJ. https://doi.org/10.1002/0470032006

Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 241, 376–396. https://doi.org/10.1098/rspa.1957.0133

Espelid, T.O., Genz, A., 1994. DECUHR: An algorithm for automatic integration of singular functions over a hyperrectangular region. Numerical Algorithms 8, 201–220. https://doi.org/10.1007/BF02142691

Gavazzi, A.C., Lagoudas, D.C., 1990. On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Computational Mechanics 7, 13–19. https://doi.org/10.1007/BF00370053

Ghahremani, F., 1977. Numerical evaluation of the stresses and strains in ellipsoidal inclusions in an anisotropic elastic material. Mechanics Research Communications 4, 89–91. https://doi.org/10.1016/0093-6413(77)90018-0

Giraud, A., Sevostianov, I., Kushch, V.I., Cosenza, P., Prêt, D., Barthélémy, J.-F., Trofimov, A., 2019. Effective electrical conductivity of transversely isotropic rocks with arbitrarily oriented ellipsoidal inclusions. Mechanics of Materials 133, 174–192. https://doi.org/10.1016/j.mechmat.2019.03.011

Hervé, E., Zaoui, A., 1993. N-layered inclusion-based micromechanical modelling. Int. J. Engng. Sci. 31, 1–10. https://doi.org/10.1016/0020-7225(93)90059-4

Hervé-Luanco, E., 2014. Elastic behavior of composites containing multi-layer coated particles with imperfect interface bonding conditions and application to size effects and mismatch in these composites. International Journal of Solids and Structures 51, 2865–2877. https://doi.org/10.1016/j.ijsolstr.2014.04.008

Hill, R., 1965. Continuum micro-mechanics of elastioplastic polycrystals. J. Mech. Phys. Solids 13, 89–101.

Hinsen, K., 2025. Khinsen/mittag-leffler.

Kachanov, M., 1993. Elastic Solids with Many Cracks and Related Problems. Advances in Applied Mechanics 30, 259–445. https://doi.org/10.1016/S0065-2156(08)70176-5

Kachanov, M., 1992. Effective Elastic Properties of Cracked Solids: Critical Review of Some Basic Concepts. Applied Mechanics Reviews 45, 304–335. https://doi.org/10.1115/1.3119761

Kachanov, M., Sevostianov, I., 2018. Micromechanics of Materials, with Applications, Solid Mechanics and Its Applications. Springer International Publishing, Cham. https://doi.org/10.1007/978-3-319-76204-3

Kellogg, O.D., 1929. Potential theory. Berlin : Springer-Verlag.

Kreher, W., 1990. Residual stresses and stored elastic energy of composites and polycristals. J. Mech. Phys. Solids 38, 115–128.

Laws, N., 1985. A short note on penny-shaped cracks in transversely isotropic materials. Mech. Mater. 4, 209–212.

Laws, N., 1977. A note on interaction energies associated with cracks in anisotropic solids. Philos. Magazine 36, 367–372.

Love, A.E.H., 1944. Treatise on mathematical theory of elasticity. 4th edition. A Treatise on the Mathematical Theory of Elasticity 643.

Masson, R., 2008. New explicit expressions of the Hill polarization tensor for general anisotropic elastic solids. International Journal of Solids and Structures 45, 757–769. https://doi.org/10.1016/j.ijsolstr.2007.08.035

Milton, G.W., 2002. The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511613357

Morin, L., Gilormini, P., Derrien, K., 2020. Generalized Euclidean Distances for Elasticity Tensors. Journal of Elasticity 138, 221–232. https://doi.org/10.1007/s10659-019-09741-z

Mura, T., 1987. Micromechanics of Defects in Solids, Second Edition. Kluwer Academic. https://doi.org/10.1002/zamm.19890690204

Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Properties of Heterogeneous Materials 2nd Edition, North Holland. ed. North Holland, Amsterdam, The Netherlands.

Papadakis, V.G., Vayenas, C.G., Fardis, M.N., 1991. Physical and Chemical Characteristics Affecting the Durability of Concrete. ACI Materials Journal 88, 186–196. https://doi.org/10.14359/1993

Parnell, W.J., 2016. The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics. Journal of Elasticity 125, 231–294. https://doi.org/10.1007/s10659-016-9573-6

Pichler, B., Hellmich, C., 2011. Upscaling quasi-brittle strength of cement paste and mortar: A multi-scale engineering mechanics model. Cement and Concrete Research 41, 467–476. https://doi.org/10.1016/j.cemconres.2011.01.010

Pouya, A., 2000. Une transformation du problème d’élasticité linéaire en vue d’application au problème de l’inclusion et aux fonctions de Green. C. R. Acad. Sci., Série IIb 328, 437–443.

Pouya, A., Zaoui, A., 2006. A transformation of elastic boundary value problems with application to anisotropic behavior. Int. J. Solids Struct. 43, 4937–4956. https://doi.org/10.1016/j.ijsolstr.2005.06.046

Powers, T.C., Brownyard, T.L., 1946. Studies of the Physical Properties of Hardened Portland Cement Paste. JP 43. https://doi.org/10.14359/15302

Saevik, P.N., Jakobsen, M., Lien, M., Berre, I., 2014. Anisotropic effective conductivity in fractured rocks by explicit effective medium methods. Geophysical Prospecting 62, 1297–1314. https://doi.org/10.1111/1365-2478.12173

Sanahuja, Julien, 2013. Effective behaviour of ageing linear viscoelastic composites: Homogenization approach. Int. J. Solids Struct. 50, 2846–2856. https://doi.org/10.1016/j.ijsolstr.2013.04.023

Sanahuja, J., 2013. Efficient homogenization of ageing creep of random media: Application to solidifying cementitious materials. Mechanics and Physics of Creep, Shrinkage and Durability of Concrete. A Tribute to Zdeněk P. Bažant. CONCREEP-9 201–210. https://doi.org/doi:10.1061/9780784413111.023

Sanahuja, J., Dormieux, L., Chanvillard, G., 2007. Modelling elasticity of a hydrating cement paste. Cem. Concr. Res. 37, 1427–1439.

Sevostianov, I., Kachanov, M., 2002. On elastic compliances of irregularly shaped cracks. International Journal of Fracture 114, 245–257. https://doi.org/10.1023/A:1015534127172

Sevostianov, I., Levin, V., Radi, E., 2015. Effective properties of linear viscoelastic microcracked materials: Application of Maxwell homogenization scheme. Mechanics of Materials 84, 28–43.

Somé, S.C., Barthélémy, J.-F., Mouillet, V., Hammoum, F., Liu, G., 2022. Effect of thermo-oxidative ageing on the rheological properties of bituminous binders and mixes: Experimental study and multi-scale modeling. Construction and Building Materials 344, 128260. https://doi.org/10.1016/j.conbuildmat.2022.128260

Suquet, P., 1997. Effective properties of nonlinear composites, in: Suquet, P. (Ed.), Contin. Micromechanics. SpringerWienNewYork, Udine, Italy, pp. 197–264.

Suvorov, A.P., Dvorak, G.J., 2002. Rate form of the Eshelby and Hill tensors. International Journal of Solids and Structures 39, 5659–5678. https://doi.org/10.1016/S0020-7683(02)00369-4

Tennis, P.D., Jennings, H.M., 2000. A model for two types of calcium silicate hydrate in the microstructure of Portland cement pastes. Cement and Concrete Research 30, 855–863. https://doi.org/10.1016/S0008-8846(00)00257-X

Torquato, S., 2002. Random Heterogeneous Materials, Interdisciplinary Applied Mathematics. Springer New York, New York, NY. https://doi.org/10.1007/978-1-4757-6355-3

Walpole, L.J., 1984. Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 391, 149–179. https://doi.org/10.1098/rspa.1984.0008

Willis, J.R., 1977. Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids 25, 185–202. https://doi.org/10.1016/0022-5096(77)90022-9

Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59, 759–781. https://doi.org/10.1080/01418618908209819

Zaoui, A., 2002. Continuum Micromechanics: Survey. J. Eng. Mech. 128, 808–816. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:8(808)

\(\,\)