9361e6b6-9351-4537-bd7a-f7430ed8b93920241113062541739wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS2224-34291991-874710.37394/232011http://wseas.org/wseas/cms.action?id=4006116202411620241910.37394/232011.2024.19https://wseas.com/journals/mechanics/2024.phpElhassaneBarhdadiDepartment of Mechanical Engineering, Abdelmalak Asaadi University, Sidi Bouafif, 32003 Al-Hoceima, MOROCCOIn this work, the extension of the Milgrom and Shtrikman model to anisotropic composite materials containing n-layered hollow ellipsoidal inclusions, is presented. The effective properties of such materials are determined using the Green function techniques and interfacial operators. Here, the basic unit of the microstructure is a hollow system of contacting concentric ellipsoidal shells, each of which is made of one of the components. Space is packed with such units of different sizes, but the same proportions; the cavity within each such shell system is then packed with similar systems and this continues in an infinite nesting sequence. In the final configuration, the effective properties are inside and outside the basic unit of layered shells (n+1). For n=2 and in the case of isotropic material, it is shown that the effective compressibility covers all ranges of the Hashin-Strikman bounds.111320241113202411311712https://wseas.com/journals/mechanics/2024/a245111-497.pdf10.37394/232011.2024.19.12https://wseas.com/journals/mechanics/2024/a245111-497.pdf10.1016/0020-7225(95)00008-lHerve, E., Zaoui, A., Elastic behaviour of multiply coated fibre reinforced composites, Int. J. Eng. Sci., Vol. 33, No. 10, 1995, pp. 1419–1433, https://doi.org/10.1016/j.ijsolstr.2020.03.013. 10.1115/1.3640682Hashin, Z., The elastic moduli of heterogeneous materials, J. Appl. Mech., Vol. 29, No. 7, 1962, pp.143-150, https://doi.org/10.1115/1.3636446. Hervé, E., Thermal and thermoelastic behaviour of multiply coated inclusionreinforced composites, Int. J. Sol. Stru., Vol. 39, No. 4, 2022, pp. 1041-1058, https://doi.org/10.1016/S0020- 7683(01)00257-8. 10.1115/1.3086336Koutsawa, Y., Cherkaoui, M., Daya, D., Multi-coating inhomogeneities problem for effective viscoelastic properties of particulate composite materials, J. Eng. Mat. Tech., Vol. 131, No. 2, 2009, pp. 1–11, https://doi.org/10.1115/1.3086336. 10.1080/14786431003767033Berbenni, S., Cherakoui, M., Homogenization of multi-coated inclusion-reinforced linear elastic composites with eigenstrains: application to the thermo-elastic behavior, Phil. Maga. & Phil. Mag. L., Vol. 90, No. 22, 2010, pp. 3003-3026, https://doi.org/10.1080/14786431003767033. 10.1016/j.apm.2020.06.005Bonfoh, N, Dizart, F., Sabar, H., New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials, App. Math. Mod., Vol. 87, No. 4, 2020, pp. 584-605, https://doi.org/10.1016/j.apm.2020.06.005. 10.1016/s0065-2156(08)70332-6Walpole, L. J., Elastic behavior of composite materials: theoretical foundations, Ad. App. Mech., Vol. 21, No. 8, 1981, pp. 169-242, https://doi.org/10.1016/S0065- 2156(08)70332-6. 10.1115/1.2712472Barhdadi, E. H., Lipinski, P., Cherkaoui, M., Four phase model: A new formulation to predict the elastic moduli of composites, J. Eng. Mat. Tech., Vol. 129, No. 2, 2007, pp. 313-320, https://doi.org/10.1115/1.2712472. 10.1007/bf01392841Dederichs, P., H., Zeller R., Variational treatment of the elastic constants of disordered materials, Z. Phy. Vol. 259, No. 2, 1973, pp. 103-113, https://doi.org/10.1007/BF01392841. 10.1016/0022-5096(83)90004-2Hill, R., Interfacial operators in the mechanics of composite media, J. Mech. Phy. Sol., Vol. 31, No. 4, 1983, pp. 347-357, https://doi.org/10.1016/0022-5096(83)90004- 2. 10.1063/1.344097Milgrom, M., Shtrikman, S., A layered-shell model of isotropic composites and exact expressions for effective properties, J. App. Ph., Vol. 66, No. 8, 1989, pp. 3429–3436, https://doi.org/10.1063/1.344097. 10.1016/0022-5096(79)90032-2Christensen, R., M., Lo, K., H., Solutions for effective shear properties in three Phase sphere and cylinder models, J. Mech. Phy. Sol., Vol. 27, No. 4, 1979, pp. 315-330, https://doi.org/10.1016/0022-5096(79)90032- 2. 10.1098/rspa.1957.0133Eshelby, J., D., The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. London, Ser. A. Vol. 241, No. 4, 1957, pp. 376-396, https://doi.org/10.1098/rspa.1957.0133. Benveniste, Y., Dvorak, G., J., in The Toshio Mura Anniversary volume: Micromechanics and Inhomogeneity, (eds.), Springer, New York, 1989, pp. 65–81, https://doi.org/10.1007/978-1-4613-8919-4. 10.1115/1.2904286Cherkaoui, M., Sabar, H., Berveiller, M., Micromechanical approach of the coated inclusion problem and applications to composite materials, ASME J. Eng. Mat. Tech., Vol. 116, No. 3, 1994, pp. 274-278, https://doi.org/10.1115/1.2904286. 10.1016/0022-5096(63)90060-7Hashin, Z., Shtrikman, S., A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phy. Sol., Vol. 11, No. 2, 1963, pp. 127-140, https://doi.org/10.1016/0022-5096(63)90060- 7.