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On the computation of the Hashin-Shtrikman bounds for transversely isotropic two-phase linear elastic fibre-reinforced composites. (English) Zbl 1360.74024
Summary: Although various forms for the Hashin-Shtrikman bounds on the effective elastic properties of inhomogeneous materials have been written down over the last few decades, it is often unclear how to construct and compute such bounds when the material is not of simple type (e.g. isotropic spheres inside an isotropic host phase). Here, we show how to construct, in a straightforward manner, the Hashin-Shtrikman bounds for generally transversely isotropic two-phase particulate composites where the inclusion phase is spheroidal, and its distribution is governed by spheroidal statistics. Note that this case covers a multitude of composites used in applications by taking various limits of the spheroid, including both layered media and long unidirectional composites. Of specific interest in this case is the fact that the corresponding Eshelby and Hill tensors can be derived analytically. That the shape of the inclusions and their distribution can be specified independently is of great utility in composite design. We exhibit the implementation of the computations with several examples.

MSC:
74B05 Classical linear elasticity
74E30 Composite and mixture properties
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[1] Tsai SW (1992) Theory of composite design. Think Composites, Dayton, Ohio
[2] Torrent, D; Sanchez-Dehesa, J, Acoustic cloaking in two dimensions: a feasible approach, New J Phys, 10, 063015, (2008)
[3] Amirkhizi, AV; Tehranian, A; Nemat-Nasser, S, Stress-wave energy management through material anisotropy, Wave Motion, 47, 519-536, (2010) · Zbl 1231.74068
[4] Voigt, W, Ueber die beziehung zwischeden beiden elasticitätsconstanten, Annalen der Physik, 38, 573-587, (1889) · JFM 21.1039.01
[5] Reuss, A, Calculation of the flow limits of mixed crystals on the basis of the plasticity of mono-crystals, Z Angew Math Mech, 9, 49-58, (1929) · JFM 55.1110.02
[6] Eshelby, JD, The determination of the elastic field of an ellipsoidal inclusion, Proc R Soc Lond A, 241, 376-396, (1957) · Zbl 0079.39606
[7] Hashin, Z; Shtrikman, S, On some variational principles in anisotropic and non-homogeneous elasticity, J Mech Phys Solids, 10, 335-342, (1962) · Zbl 0111.41401
[8] Hashin, Z; Shtrikman, S, A variational approach to the theory of the elastic behaviour of multiphase materials, J Mech Phys Solids, 11, 127-140, (1963) · Zbl 0108.36902
[9] Hashin, Z, On the elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry, J Mech Phys Solids, 13, 119-134, (1965)
[10] Hashin, Z, Analysis of properties of fiber composites with anisotropic constituents, J Appl Mech, 46, 543-550, (1979) · Zbl 0436.73013
[11] Walpole, LJ, On bounds for the overall elastic moduli of inhomogeneous systems-I, J Mech Phys Solids, 14, 151-162, (1966) · Zbl 0139.18701
[12] Willis, JR, Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J Mech Phys Solids, 25, 185-202, (1977) · Zbl 0363.73014
[13] Willis, JR, Variational and related methods for the overall properties of composites, Adv Appl Mech, 21, 1-78, (1981) · Zbl 0476.73053
[14] Willis JR (1982) Theory elasticity, of composites mechanics of solids, the Rodney Hill 60th anniversary volume. In: Hopkins HG, Sewell MJ (eds) Pergamon, Oxford, pp 653-686 · Zbl 0139.42601
[15] Willis JR, Bounds on the overall properties of anisotropic composites. In: Mura T, Koya T (eds) Variational methods in mechanics, chap. 21. Oxford University Press, Oxford · Zbl 0759.73064
[16] Ponte Castañeda, P; Willis, JR, The effect of spatial distribution on the effective behaviour of composite materials and cracked media, J Mech Phys Solids, 43, 1919-1951, (1995) · Zbl 0919.73061
[17] Willis, JR; Ericksen, JL (ed.), Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, 247-263, (1986), New York
[18] Hill, R, Elastic properties of reinforced solids: some theoretical principles, J Mech Phys Solids, 11, 357-372, (1963) · Zbl 0114.15804
[19] Talbot, DRS; Willis, JR, Variational principles for inhomogeneous nonlinear media, IMA J Appl Math, 35, 39-54, (1985) · Zbl 0588.73025
[20] Ponte Castañeda, P, The overall constitutive behaviour of nonlinearly elastic composites, Proc R Soc Lond Ser A, 422, 147-171, (1862) · Zbl 0673.73005
[21] Ponte Castañeda, P, A new variational principle and its application to nonlinear heterogeneous systems, SIAM J Appl Math, 52, 1321-1341, (1992) · Zbl 0759.73064
[22] Beran, MJ; Molyneux, J, Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media, Q Appl Math, 24, 107-118, (1966) · Zbl 0139.42601
[23] Milton, GW; Phan-Thien, N, New bounds on the effective moduli of two-component materials, Proc R Soc Lond A, 380, 305-331, (1982) · Zbl 0497.73016
[24] Torquato, S, Random heterogeneous media: microstructure and improved bounds on effective properties, Appl Mech Rev, 44, 37-76, (1991)
[25] Rodrígues-Ramos, R; Guinovart-Díaz, R; Bravo-Castillero, J; Sabina, FJ; Berger, H; Kari, S; Gabbert, U, Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions, Arch Appl Mech, 79, 695-708, (2009) · Zbl 1264.74195
[26] Hill, R, Theory of mechanical properties of fibre-strengthened materials: I. elastic behaviour, J Mech Phys Solids, 12, 199-212, (1964)
[27] Liu, LP, Solutions to the eshelby conjectures, Proc R Soc Lond A, 464, 573-594, (2008) · Zbl 1132.74010
[28] Kang, H; Milton, GW, Solutions to the polya szego conjecture and the weak eshelby conjecture, Arch Ration Mech Anal, 188, 93-116, (2008) · Zbl 1134.74013
[29] Hill, R, The elastic behavior of a crystalline aggregate, Proc Phys Soc Lond Sect A, 65, 349, (1952)
[30] Sevostianov, I; Yilmaz, N; Kushch, V; Levin, VM, Effective elastic properties of matrix composites with transversely isotropic phases, Int J Solids Struct, 42, 455-476, (2005) · Zbl 1143.74319
[31] Kanaun SK, Levin VM (2008) Self-consistent methods for composites. Volume 1-static problems. Springer, Dordrecht · Zbl 1231.74068
[32] Walpole, LJ, Fourth-rank tensors of the thirty two crystal classes: multiplication tables, Proc R Soc Lond A, 391, 149-179, (1984) · Zbl 0521.73005
[33] Backus, G, Long-wave elastic anisotropy produced by horizontal layering, J Geophys Res, 67, 4427-4440, (1962) · Zbl 1369.86005
[34] Withers, PJ, The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium and its relevance to composite materials, Phil Mag A, 59, 759-781, (1989)
[35] Mura T (1991) Micromechanics of defects in solids. Kluwer, Dordrecht
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