Abstract
Composites made of n constituents, or phases, firmly bonded across interfaces, will be considered. Each phase conforms to a constitutive relation of the form
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References
Ben-Amoz, M. 1966. Variational principles in anisotropic and nonhomogeneous elastokinetics. Quart. Appl. Math. XXIV, 82–86.
Beran, M. J. 1965. Use of the variational approach to determine bounds for the effective permittivity of a random composite. Nuovo Cimento 38, 771–782.
Beran, M. J. and McCoy, J. J. 1970. Mean field variations in a statistical sample of heterogeneous linearly elastic solids. Int. J. Solids Struct. 6, 1035–1054.
Budiansky, B. 1965. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227.
Foldy, L. L. 1945. Multiple scattering theory of waves. Phys. Rev. 67, 107–119.
Gerard, P. 1991. Microlocal defect measures. Commun. Partial Dif. Equat. 16, 1761–1794.
Gurtin, M. E. 1964. Variational principles in linear elastodynamics. Arch. Rat. Mech. Anal. 16, 34–50.
Keller, J. B. 1964. Stochastic equations and wave propagation in random media. Proc. Symposia in Applied Mathematics, Vol. XVI, Stochastic Processes in Mathematical Physics and Engineering,pp. 145–170. American Mathematical Society, Providence, RI.
Kinra, V. K. 1984. Acoustical and optical branches of wave propagation in an epoxy matrix containing a random distribution of lead inclusions. Review of Progress in Quantitative Nondestructive Evaluation (D. O. Thompson and D. E. Chimenti, Eds.), pp. 983–992. Plenum, New York.
Kinra, V. K. 1985. Dispersive wave propagation in random particulate composites. Recent Advances in Composites in the United States and Japan, ASTM STP 864 (J. R. Vinson and M. Taya, Eds.), pp. 309–325. ASTM, Philadelphia.
Hashin, Z. and Shtrikman, S. T. 1962. On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342.
Hashin, Z. and Shtrikman, S. T. 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140.
Hill, R. 1965. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222.
John, F. and Nirenberg, L. 1961. On functions of bounded mean oscillation. Communs. Pure Appl. Math. 14, 415–426.
Luciano, R. and Willis, J. R. 2000. Bounds on non-local effective relations for random composites loaded by configuration-dependent body force. J. Mech. Phys Solids 48, 1827–1849.
Maxwell, J. C. 1904 A Treatise on Electricity and Magnetism ( Third edition ). Clarendon Press, Oxford.
Milton, G. W. 1982. Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30, 177–191.
Ponte Castaneda, P. 1991. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39, 45–71.
Ponte Castaneda, P. 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 44, 827–862.
Ponte Castaneda, P. and Tiberio, E. 2000. A second-order homogenization method in finite elasticity and applications to black-filled elastomers. J. Mech. Phys. Solids 48, 1389–1411.
Ponte Castaneda, P. and Willis, J. R. 1999. Variational second-order estimates for nonlinear composites. Proc. R. Soc. Lond. A455, 1799–1811.
Sabina, F. J. and Willis, J. R. 1988. A simple self-consistent analysis of wave propagation in particulate composites. Wave Motion 10, 127–142.
Sabina, F. J. and Willis, J. R. 1993. Self-consistent analysis of waves in a polycrystalline medium. Eur. J. Mechanics A/Solids 12, 265–275.
Sabina, F. J., Smyshlyaev, V. P. and Willis, J. R. 1993a. Self-consistent analysis of waves in a matrix-inclusion composite. I. Aligned spheroidal inclusions. J. Mech. Phys. Solids 41, 1573–1588.
Sabina, F. J., Smyshlyaev, V. P. and Willis, J. R. 1993b. Self-consistent analysis of waves in a matrix-inclusion composite. II. Randomly oriented spheroidal inclusions. J. Mech. Phys. Solids 41, 1589–1598.
Sabina, F. J., Smyshlyaev, V. P. and Willis, J. R. 1993c. Self-consistent analysis of waves in a matrix-inclusion composite. III. A matrix containing penny-shaped cracks. J. Mech. Phys. Solids 41, 1809–1824.
Sayers, C. M. and Smith, R. L. 1983. Ultrasonic velocity and attenuation in an epoxy matrix containing lead inclusions. J. Phy. D: Appl. Phys. 16, 1189–1194.
Schulgasser, K. 1976. Relationship between single-crystal and polycrystal electrical conductivity. J. Appl. Phys. 47, 1880–1886.
Talbot, D. R. S. and Willis, J. R. 1985. Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39–54.
Talbot, D. R. S. and Willis, J. R. 1992. Some simple explicit bounds for the overall behaviour of nonlinear composites. Int. J. Solids Strct. 29, 1981–1987.
Talbot, D. R. S. and Willis, J. R. 1994. Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proc. R. Soc. Lond. A447, 365–384.
Talbot, D. R. S. and Willis, J. R. 1997. Bounds of third order for the overall response of nonlinear composites. J. Mech. Phys. Solids 45, 87–111.
Tartar, L. 1990. H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations. Proc. R. Soc. Edinb. A115, 193–230.
Van Tiel, J. 1984. Convex Analysis. Wiley, New York.
Willis, J. R. 1977. Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185–202.
Willis, J. R. 1980. A polarization approach to the scattering of elastic waves. I. Scattering by a single inclusion. J. Mech. Phys. Solids 28, 287–305.
Willis, J. R. 1981a. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics 21 (C. S. Yih, Ed.), pp. 1–78, Academic Press, New York.
Willis, J. R. 1981b. Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion 3, 1–11.
Willis, J. R. 1982. Elasticity theory of composites. Mechanics of Solids. The Rodney Hill 60th Anniversary Volume (H. G. Hopkins and M. J. Sewell, Eds.), pp. 653–686. Pergamon, Oxford.
Willis, J. R. 1983. The overall elastic response of composite materials. J. Appl. Mech. 50, 1202–1209.
Willis, J. R. 1985 The non-local influence of density variations in a composite. Int. J. Solids Struct. 21, 805–817.
Ying, C. F. and Truell, R. 1956. Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid. J. Appl. Phys. 27, 1086–1097.
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Willis, J.R. (2001). Lectures on Mechanics of Random Media. In: Jeulin, D., Ostoja-Starzewski, M. (eds) Mechanics of Random and Multiscale Microstructures. International Centre for Mechanical Sciences, vol 430. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2780-3_5
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DOI: https://doi.org/10.1007/978-3-7091-2780-3_5
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