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Nonlinear elasticity of composite materials

Landau coefficients in dispersions of spherical and cylindrical inclusions
  • S. Giordano
  • P. L. Palla
  • L. Colombo
Mesoscopic and Nanoscale Systems

Abstract

We investigate the elastic properties of model composites, consisting in a dispersion of nonlinear (spherical or cylindrical) inhomogeneities into a linear solid matrix. Both phases are considered isotropic. Under the simplifying hypotheses of small deformation for the material body and of small volume fraction of the embedded phase, we develop a homogenization procedure based on the Eshelby theory, aimed at describing nonlinear features. We obtain the bulk and shear moduli and Landau coefficients of the overall material in terms of the elastic behavior of the constituents and of their volume fractions. The mixing laws for the nonlinear properties describe a complex scenario where possible strong amplifications of the nonlinearities may arise in some given conditions.

PACS

62.25.-g Mechanical properties of nanoscale systems 62.23.Pq Composites 81.40.Jj Elasticity and anelasticity, stress-strain relations 

References

  1. S. Torquato, Phys. Rev. Lett. 79, 681 (1997)Google Scholar
  2. Z. Hashin, S. Shtrikman, J. Appl. Phys. 33, 3125 (1962)Google Scholar
  3. Z. Hashin, S. Shtrikman, J. Mech. Phys. Solids 10, 335 (1962)Google Scholar
  4. W.F. Brown, J. Chem. Phys. 23, 1514 (1955)Google Scholar
  5. S. Torquato, J. Mech. Phys. Solids 45, 1421 (1997)Google Scholar
  6. S. Torquato, J. Mech. Phys. Solids 46, 1411 (1998)Google Scholar
  7. L.K.H. Van Beek, Progress in Dielectric (Heywood, London, 1967), Vol. 7, p. 71Google Scholar
  8. S. Giordano, W. Rocchia, J. Appl. Phys. 98, 104101 (2005)Google Scholar
  9. L.J. Walpole, Adv. Appl. Mech. 11, 169 (1981)Google Scholar
  10. R.M. Christensen, Mechanics of composite materials (Dover Publication Inc., New York, 2005)Google Scholar
  11. Z. Hashin, J. Appl. Mech. 50, 481 (1983)Google Scholar
  12. N. Norris, Mech. Mat. 4, 1 (1985)Google Scholar
  13. R. McLaughlin, Int. J. Eng. Sci. 15, 237 (1977)Google Scholar
  14. S. Giordano, Eur. J. Mech. A/Solids 22, 885 (2003)Google Scholar
  15. M. Kachanov, I. Sevostianov, Int. J. Solids Struct. 42, 309 (2005)Google Scholar
  16. Heterogeneous Media: Micromechanics Modeling Methods and Simulations edited by K.Z. Markov, L. Preziozi (Birkhauser, Boston, 2000)Google Scholar
  17. M. Kachanov, Appl. Mech. Rev. 45, 305 (1992)Google Scholar
  18. S. Giordano, L. Colombo, Phys. Rev. Lett. 98, 055503 (2007)Google Scholar
  19. S. Giordano, L. Colombo, Eng. Frac. Mech. 74, 1983 (2007)Google Scholar
  20. P. Gilormini, F. Montheillet, J. Mech. Phys. Solids 34, 97 (1986)Google Scholar
  21. P.P. Castañeda, J. Mech. Phys. Solids 39, 45 (1991)Google Scholar
  22. P. Suquet, J. Mech. Phys. Solids 41, 981 (1993)Google Scholar
  23. L.V. Gibiansky, S. Torquato, J. Appl. Phys. 84, 301 (1998)Google Scholar
  24. L.V. Gibiansky, S. Torquato, J. Appl. Phys. 84, 5969 (1998)Google Scholar
  25. T. Mura, Micromechanics of defects in solids (Kluwer Academic Publishers, Dordrecht, 1987)Google Scholar
  26. R.J. Atkin, N. Fox, An introduction to the theory of elasticity (Dover Publication Inc., New York, 1980)Google Scholar
  27. V.V. Novozhilov, Foundations of the nonlinear theory of elasticity (Dover Publication Inc., New York, 1999)Google Scholar
  28. T.K. Ballabh, M. Paul, T.R. Middya, A.N. Basu, Phys. Rev. B 45, 2761 (1992)Google Scholar
  29. A.E.H. Love, A treatise on the mathematical theory of elasticity (Dover Publication Inc., New York, 2002)Google Scholar
  30. L.D. Landau, E.M. Lifschitz, Theory of Elasticity, Course of Theoretical Physics, 3rd edn. (Butterworth Heinemann, Oxford, 1986), Vol. 7Google Scholar
  31. S. Giordano, L. Colombo, P. Palla, Europhys. Lett. 83, 66003 (2008)Google Scholar
  32. J.D. Eshelby, Proc. R. Soc. London A 241, 376 (1957)Google Scholar
  33. J.D. Eshelby, Proc. R. Soc. London A 252, 561 (1959)Google Scholar
  34. I. Sevostianov, A. Vakulenko, Int. J. Fracture 107, L9 (2000)Google Scholar
  35. I.Y. Tsvelodub, J. Appl. Mech. Tech. Phys. 41, 734 (2000)Google Scholar
  36. I.Y. Tsvelodub, J. Appl. Mech. Tech. Phys. 45, 69 (2004)Google Scholar
  37. S. Catheline, J.-L. Gennisson, M. Fink, J. Acoust. Soc. Am. 114, 3087 (2003)Google Scholar
  38. S. Catheline, J.-L. Gennisson, M. Tanter, M. Fink, Phys. Rev. Lett. 91, 164301 (2003)Google Scholar
  39. V. Holy, G. Springholz, M. Pinczolits, G. Bauer, Phys. Rev. Lett. 83, 356 (1999)Google Scholar
  40. M. Schmidbauer, S. Seydmohamadi, D. Grigoriev, Z.M. Wang, Y.I. Mazur, P. Schäfer, M. Hanke, R. Köhler, G.J. Salamo, Phys. Rev. Lett. 96, 066108 (2006)Google Scholar
  41. A. Mattoni, L. Colombo, F. Cleri, Phys. Rev. B 70, 094108 (2004)Google Scholar
  42. R.M. Christensen, Proc. R. Soc. London, Ser. A 440, 461 (1993)Google Scholar
  43. R.W. Zimmerman, Mechanics of Materials 12, 17 (1991)Google Scholar
  44. R.W. Zimmerman, Appl. Mech. Rev. 47, S38 (1994)Google Scholar
  45. J. Li, C. Papadopoulos, J.M. Xu, M. Moskovits, Appl. Phys. Lett. 75, 367 (1999)Google Scholar
  46. Z. Wang, P. Ciselli, T. Peijs, Nanotechnology 18, 455709 (2007)Google Scholar
  47. S. Wang, Z. Liang, T. Liu, B. Wang, C. Zhang, Nanotechnology, 17, 1551 (2006)Google Scholar
  48. S. Iwamoto, H. Yano, A.N. Nakagaito, M. Nogi, Appl. Phys. A 81, 1109 (2005)Google Scholar
  49. M. Nogi, K. Abe, K. Handa, F. Nakatsubo, S. Ifuku, H. Yano, Appl. Phys. Lett. 89, 233123 (2006)Google Scholar
  50. M.F. Thorpe, P.N. Sen, J. Acoust. Soc. Am. 77, 1674 (1985)Google Scholar
  51. R. Hill, J. Mech. Phys. Solids 12, 199 (1964)Google Scholar
  52. R. Hill, J. Mech. Phys. Solids 12, 213 (1964)Google Scholar
  53. R. Hill, J. Mech. Phys. Solids 13, 189 (1965)Google Scholar
  54. K.A. Snyder, E.J. Garboczi, J. Appl. Phys. 72, 5948 (1992)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Sardinian Laboratory for Computational Materials Science (SLACS, INFM-CNR)SardiniaItaly
  2. 2.Department of PhysicsUniversity of Cagliari, Cittadella UniversitariaMonserrato (Ca)Italy

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