The Composite Eshelby Tensors and their applications to homogenization Article

First Online: 22 November 2007 Received: 25 January 2007 Accepted: 14 August 2007 DOI :
10.1007/s00707-007-0504-2

Cite this article as: Sauer, R.A., Wang, G. & Li, S. Acta Mech (2008) 197: 63. doi:10.1007/s00707-007-0504-2
Summary In recent studies, the exact solutions of the Eshelby tensors for a spherical inclusion in a finite, spherical domain have been obtained for both the Dirichlet- and Neumann boundary value problems, and they have been further applied to the homogenization of composite materials [15], [16]. The present work is an extension to a more general boundary condition, which allows for the continuity of both the displacement and traction field across the interface between RVE (representative volume element) and surrounding composite. A new class of Eshelby tensors is obtained, which depend explicitly on the material properties of the composite, and are therefore termed “the Composite Eshelby Tensors”. These include the Dirichlet- and the Neumann-Eshelby tensors as special cases. We apply the new Eshelby tensors to the homogenization of composite materials, and it is shown that several classical homogenization methods can be unified under a novel method termed the “Dual Eigenstrain Method”. We further propose a modified Hashin-Shtrikman variational principle, and show that the corresponding modified Hashin-Shtrikman bounds, like the Composite Eshelby Tensors, depend explicitly on the composite properties.

References Chiu Y. P. (1977). On the stress field due to initial strains in cuboid surrounded by an infinite elastic space.

J. Appl. Mech. 44: 587–590

MATH Christensen R. M. and Lo K. H. (1979). Solutions for the effective shear properties in three phase sphere and cylinder models.

J. Mech. Phys. Solids 27: 315–330

MATH CrossRef Eshelby J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems.

Proc. Roy. Soc. A 241: 376–396

MATH CrossRef MathSciNet Eshelby J. D. (1959). The elastic field outside an ellipsoidal inclusion.

Proc. Roy. Soc. A 252: 561–569

MATH CrossRef MathSciNet Eshelby, J. D.: Elastic inclusions and inhomogeneities. In: Progress in solid mechanics, Vol. 2. pp. 89–104 (Snedden, N. I., Hill, R., eds.). North-Holland 1961

Hashin Z. and Shtrikman S. (1962a). On some variational principles in anisotropic and nonhomogeneous elasticity.

J. Mech. Phys. Solids 10: 335–342

CrossRef MathSciNet Hashin Z. and Shtrikman S. (1962b). A variational approach to the theory of the elastic behavior of polycrystals.

J. Mech. Phys. Solids 10: 343–352

CrossRef MathSciNet Hashin Z. (1991). The spherical inclusion with imperfect interface.

J. Appl. Mech. 58: 444–449

CrossRef Hashin Z. (2002). The interphase/imperfect interface in elasticity with application to coated fiber composites.

J. Mech. Phys. Solids 50: 2509–2537

MATH CrossRef MathSciNet Hill, R.: New derivations of some elastic extremum principles. In: Progress in applied mechanics – The Prager anniversary volume, pp. 99–106. New York: Macmillan 1963

Hill R. (1965a). Continuum micro-mechanics of elastoplastic polycrystals.

J. Mech. Phys. Solids 13: 89–101

MATH CrossRef Hill R. (1965b). Theory of mechanical properties of fibre-strengthened materials III. Self-consistent model.

J. Mech. Phys. Solids 13: 189–198

CrossRef Jiang B. and Weng G. J. (2004). A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials.

J. Mech. Phys. Solids 52: 1125–1149

MATH CrossRef Li S., Sauer R. and Wang G. (2005). A circular inclusion in a finite domain. I. The Dirichlet-Eshelby problem.

Acta Mech. 179: 67–90

MATH CrossRef Li S., Sauer R. A. and Wang G. (2007a). The Eshelby tensors in a finite spherical domain: I. Theoretical formulations.

J. Appl. Mech. 74: 770–783

CrossRef MathSciNet Li S., Wang G. and Sauer R. A. (2007b). The Eshelby tensors in a finite spherical domain: II. Applications in homogenization.

J. Appl. Mech. 74: 784–797

CrossRef MathSciNet Löhnert, S.: Computational homogenization of microheterogeneous materials at finite strains including damage. Dissertation, Universität Hannover 2004

Luo H. A. and Weng G. J. (1987). On Eshelby’s inclusion problem in a three-phase spherically concentric solid and a modification of Mori-Tanaka’s method.

Mech. Mater. 6: 347–361

CrossRef Luo H. A. and Weng G. J. (1989). On Eshelby’s S-tensor in a three-phase cylindrical concentric solid and the elastic moduli of fibre-reinforced composites.

Mech. Mater. 8: 77–88

CrossRef Marur P. R. (2005). Effective elastic moduli of syntactic foams.

Mater. Lett. 59: 1954–1957

CrossRef Mura T. and Kinoshita N. (1978). The polynomial eigenstrain problem or an anisotropic ellipsoidal inclusion.

Phys. Status Solidi A 48: 447–450

CrossRef Mura T. (1987). Micromechanics of defects in solids, 2nd ed. Martinus Nijhoff, Boston

Nemat-Nasser S. and Hori M. (1999). Micromechanics: overall properties of heterogeneous materials, 2nd ed. Elsevier, Amsterdam

Qiu Y. P. and Weng G. J. (1991). Elastic moduli of thickly coated particle and fiber-reinforced composites.

J. Appl. Mech. 58: 388–398

MATH CrossRef Rodin G. J. (1996). Eshelby's inclusion problem for polygons and polyhedra.

J. Mech. Phys. Solids 44: 1977–1995

CrossRef Saidi F., Bernabé Y. and Reuschlé T. (2005). Uniaxial compression of synthetic, poorly consolidated granular rock with a bimodal grain-size distribution.

Rock Mech. Rock Engng. 38: 129–144

CrossRef Sharma P. and Ganti S. (2004). Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies.

J. Appl. Mech. 71: 663–671

MATH CrossRef Somigliana, C.: Sopra l’equilibrio di un corpo elastico isotropo, pp. 17–19. Il Nuovo Ciemento 1886

Wang G., Li S. and Sauer R. (2005). A circular inclusion in a finite domain. II. The Neumann–Eshelby problem.

Acta Mech. 179: 91–110

CrossRef Willis, J. R.: Variational and related methods for the overall properties of composites. In: Advances in applied mechanics (Yih, C.-S., ed.), Vol. 21, pp. 1–78. New York: Academic Press 1981

© Springer-Verlag Wien 2007

Authors and Affiliations 1. Department of Civil and Environmental Engineering University of California Berkeley USA