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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. Constr. Build. Mater. 234, 117124. https://doi.org/10.1016/j.conbuildmat.2019.117124
Barthélémy, J.-F., 2026. Echoes: Extended Calculator of HOmogEnization Schemes. https://doi.org/10.5281/zenodo.7348758
Barthélémy, J.-F., 2020. Simplified approach to the derivation of the relationship between Hill polarization tensors of transformed problems and applications. Int. J. Eng. Sci. 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. Int. J. Solids Struct. 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. Transp. 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. Int. J. Eng. Sci. 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. Int. J. Eng. Sci. 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. Int. J. Solids Struct. 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. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 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. Numer. 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. Comput. Mech. 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. Mech. Res. Commun. 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. Mech. Mater. 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. Int. J. Solids Struct. 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. Adv. Appl. Mech. 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. Appl. Mech. Rev. 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 Mag. 36, 367–372.
Love, A.E.H., 1944. Treatise on mathematical theory of elasticity. 4th edition. Treatise Math. Theory Elast. 643.
Masson, R., 2008. New explicit expressions of the Hill polarization tensor for general anisotropic elastic solids. Int. J. Solids Struct. 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. J. Elast. 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/10.1007/978-94-009-3489-4
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 Mater. J. 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. J. Elast. 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. Cem. Concr. Res. 41, 467–476. https://doi.org/10.1016/j.cemconres.2011.01.010
Ponte Castañeda, P., Willis, J.R., 1995. The effect of spatial distribution on the effective behavior of composite materials and cracked media. Journal of the Mechanics and Physics of Solids 43, 1919–1951. https://doi.org/10.1016/0022-5096(95)00058-Q
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 Sér. 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. ACI J. Proc. 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. Geophys. Prospect. 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. Mech. Phys. Creep Shrinkage Durab. Concr. Tribute 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., Giraud, A., 2013. Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape. International Journal of Engineering Science 64, 23–36. https://doi.org/10.1016/j.ijengsci.2012.12.004
Sevostianov, I., Kachanov, M., 2002. On elastic compliances of irregularly shaped cracks. Int. J. Fract. 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. Mech. Mater. 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. Constr. Build. Mater. 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. Int. J. Solids Struct. 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. Cem. Concr. Res. 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. Proc. R. Soc. Lond. Math. Phys. Sci. 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. J. Mech. Phys. 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. Philos. Mag. 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)

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