Advertisement

Lectures on Mechanics of Random Media

  • J. R. Willis
Part of the International Centre for Mechanical Sciences book series (CISM, volume 430)

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
(1.1)

Keywords

Variational Principle Random Medium Momentum Density Displacement Boundary Condition Uniform Medium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ben-Amoz, M. 1966. Variational principles in anisotropic and nonhomogeneous elastokinetics. Quart. Appl. Math. XXIV, 82–86.MathSciNetGoogle Scholar
  2. 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.CrossRefGoogle Scholar
  3. 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.CrossRefMATHGoogle Scholar
  4. Budiansky, B. 1965. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227.CrossRefGoogle Scholar
  5. Foldy, L. L. 1945. Multiple scattering theory of waves. Phys. Rev. 67, 107–119.CrossRefMATHMathSciNetGoogle Scholar
  6. Gerard, P. 1991. Microlocal defect measures. Commun. Partial Dif. Equat. 16, 1761–1794.CrossRefMATHMathSciNetGoogle Scholar
  7. Gurtin, M. E. 1964. Variational principles in linear elastodynamics. Arch. Rat. Mech. Anal. 16, 34–50.MATHMathSciNetGoogle Scholar
  8. 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.Google Scholar
  9. 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.Google Scholar
  10. 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.Google Scholar
  11. Hashin, Z. and Shtrikman, S. T. 1962. On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342.CrossRefMathSciNetGoogle Scholar
  12. 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.CrossRefMATHMathSciNetGoogle Scholar
  13. Hill, R. 1965. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222.CrossRefGoogle Scholar
  14. John, F. and Nirenberg, L. 1961. On functions of bounded mean oscillation. Communs. Pure Appl. Math. 14, 415–426.CrossRefMATHMathSciNetGoogle Scholar
  15. 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.CrossRefMATHMathSciNetGoogle Scholar
  16. Maxwell, J. C. 1904 A Treatise on Electricity and Magnetism ( Third edition ). Clarendon Press, Oxford.Google Scholar
  17. Milton, G. W. 1982. Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30, 177–191.CrossRefMATHMathSciNetGoogle Scholar
  18. Ponte Castaneda, P. 1991. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 39, 45–71.CrossRefMATHMathSciNetGoogle Scholar
  19. Ponte Castaneda, P. 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 44, 827–862.CrossRefMATHMathSciNetGoogle Scholar
  20. 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.CrossRefMATHMathSciNetGoogle Scholar
  21. Ponte Castaneda, P. and Willis, J. R. 1999. Variational second-order estimates for nonlinear composites. Proc. R. Soc. Lond. A455, 1799–1811.CrossRefMATHGoogle Scholar
  22. Sabina, F. J. and Willis, J. R. 1988. A simple self-consistent analysis of wave propagation in particulate composites. Wave Motion 10, 127–142.CrossRefMATHGoogle Scholar
  23. 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.MATHGoogle Scholar
  24. 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.CrossRefMATHMathSciNetGoogle Scholar
  25. 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.CrossRefMATHMathSciNetGoogle Scholar
  26. 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.CrossRefMATHMathSciNetGoogle Scholar
  27. 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.CrossRefGoogle Scholar
  28. Schulgasser, K. 1976. Relationship between single-crystal and polycrystal electrical conductivity. J. Appl. Phys. 47, 1880–1886.CrossRefGoogle Scholar
  29. Talbot, D. R. S. and Willis, J. R. 1985. Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39–54.CrossRefMATHMathSciNetGoogle Scholar
  30. 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.CrossRefMATHMathSciNetGoogle Scholar
  31. 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.CrossRefMATHGoogle Scholar
  32. 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.CrossRefMATHMathSciNetGoogle Scholar
  33. 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.CrossRefMATHMathSciNetGoogle Scholar
  34. Van Tiel, J. 1984. Convex Analysis. Wiley, New York.MATHGoogle Scholar
  35. Willis, J. R. 1977. Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185–202.CrossRefMATHGoogle Scholar
  36. 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.CrossRefMATHMathSciNetGoogle Scholar
  37. 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.Google Scholar
  38. Willis, J. R. 1981b. Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion 3, 1–11.CrossRefMATHMathSciNetGoogle Scholar
  39. 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.Google Scholar
  40. Willis, J. R. 1983. The overall elastic response of composite materials. J. Appl. Mech. 50, 1202–1209.CrossRefMATHGoogle Scholar
  41. Willis, J. R. 1985 The non-local influence of density variations in a composite. Int. J. Solids Struct. 21, 805–817.CrossRefMATHGoogle Scholar
  42. 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.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • J. R. Willis
    • 1
  1. 1.University of BathBathUK

Personalised recommendations