On application of classical Eshelby approach to calculating effective elastic moduli of dispersed composites

Conference paper

Abstract

The problem of finding effective elastic moduli of media with spheroid inclusions in case of small concentration of these inclusions is addressed. A number of particular solutions, both known and new, were obtained as limit transitions and asymptotical expansion of the general solution, based on Eshelby’s approach. A special attention was paid to determining the ranges of applicability of the obtained asymptotical solutions. It was shown that for spheroid inclusions the areas of applicability of the asymptotic solutions are determined by two parameters: the ratio of elastic moduli of the inclusion and the matrix and aspect ratio of the inclusions.

Keywords

Effective properties Eshelby’s tensor Inclusion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chow TS (1977). Elastic moduli of filled polymers: The effects of particle shape. J Appl Phys 48: 4072–4075 CrossRefGoogle Scholar
  2. Eshelby JD (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc Lon A 241: 376–396 MATHCrossRefMathSciNetGoogle Scholar
  3. Eshelby JD (1961). Elastic inclusions and inhomogeneities. In: Sneddon, IN and Hill, R (eds) Progress in Solid Mechanics, vol 2, pp 89–140. North-Holland, Amsterdam Google Scholar
  4. Hashin Z (1988). The differential scheme and its application to cracked materials. J Mech Phys Solids 36: 719–733 MATHCrossRefMathSciNetGoogle Scholar
  5. Kovalenko YF and Salganik RL (1977). Treshinovatye neodnorodnosti i ih vlianie na effektivnye mehanicheskie harakteristiki (Crack-like Inhomogeneities and their Influence on Effective Mechanical Characteristics). Izv AN SSSR Mech Tv Tela No 5: 76–86 Google Scholar
  6. Krivoglas MA and Cherevko AS (1959). Ob uprugih modulah tverdoi smesi (On elastic moduli of solid mixture). Fizika metallov i metalovedenie 8(2): 161–164 Google Scholar
  7. Kroner E (1958). Berechnung der Elastischen Konstanten des Vielkristalls aus den Konstanten der Einkristalls. Z Phys 151: 504–518 CrossRefGoogle Scholar
  8. Mura T (1982). Micromechanics of defects in solids. Martinus Nijhoff Publishers, The Hague-Boston Google Scholar
  9. Odegard GM, Gates TS, Wise KE, Park C, Siochi EJ (2002) Constitutive modeling of nanotube-reinforced polymer composites. NASA/CR-2002-211760 ICASE Report No/2002-27Google Scholar
  10. Roscoe RA (1973). Isotropic composites with elastic and viscoelastic phases: General bounds for the moduli and solutions for special geometries. Rheol Acta 12: 404–411 CrossRefGoogle Scholar
  11. Salganik RL (1973). Mehanika tel s bolshim chislom treshin (Mechanics of Bodies with Many Cracks). Izv AN SSSR Mech Tv Tela No 4: 149–158 Google Scholar
  12. Salganik RL (1974). Protsessy perenosa v telah s bolshim chislom treshin (Transition Processes in Bodies with Many Cracks). Inzhenerno-fizicheskii zhurnal Tom 27(6): 1069–1075 Google Scholar
  13. Tucker CLIII and Liang E (1999). Stiffness predictions for unidirectional short-fiber composites: review and evaluation. Composites Sci Technol 59: 655–671 CrossRefGoogle Scholar
  14. Ustinov KB (2003a). Ob opredelenii effektivnyh uprugih harakteristik dvuhfaznyh sred. Sluchai izolirovannyh neodnorodnostei v forme ellipsoidov vrashenia (On determining the effective characteristics of elastic two phases media. The case of spheroid inclusions). Uspehi Mehaniki 2: 126–168 Google Scholar
  15. Ustinov KB (2003b) Ob opredelenii effektivnyh uprugih harakteristik dvuhfaznyh sred. Sluchai izolirovannyh uploshennyh neodnorodnostei i oblasti primenimosti asimptotik. (On determining effective elastic characteristics of two phases media Flat inclusions and ranges of applicability of asymptotics). Preprint IPM RAS N735, 24 ppGoogle Scholar
  16. Walpole LJ (1969). On the overall elastic moduli of composite materials. J Mech Phys Solids 17: 235–251 MATHCrossRefGoogle Scholar
  17. Walsh JB (1965). The effect of cracks on the compressibility of rocks. J Geophys Res 70(2): 381–389 MATHCrossRefGoogle Scholar
  18. Wu TT (1966). The effect of inclusion shape on the elastic moduli of a two-phase material. Int J Solids Struct 2: 1–8 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institute for Problems in MechanicsMoscowRussian Federation

Personalised recommendations