International Journal of Fracture

, Volume 177, Issue 2, pp 199–206 | Cite as

On the Compliance Contribution Tensor for a Concave Superspherical Pore

  • Igor Sevostianov
  • Albert Giraud
Letters in Fracture and Micromechanics


This paper focuses on the effect of concavity of pores on elastic properties of porous materials. We consider a pore having a shape of a supersphere of unit radius (x)2p + (y)2p + (z)2p = 1 focusing mostly on the case p ≤ 1. Using FEM analysis of Sevostianov et al (2008), we propose simple approximate formulae for components of the compliance contribution tensor of the supersphere. From these formulae, we identify the microstructural parameter describing its contribution into effective elastic properties. The derivation is illustrated by comparison of the results with the known ones for a spheroidal pore of the aspect ratio γ.


Porous material effective properties non-ellipsoidal pores concave pores supersphere 


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  1. Drach B., Tsukrov I., Gross T., Dietrich S., Weidenmann K., Piat R., Böhlke T. (2011) Numerical modeling of carbon/carbon composites with nanotextured matrix and 3D pores of irregular shapes. Int. J. Sol. Structures 48: 2447–2457CrossRefGoogle Scholar
  2. Eshelby J.D. (1957) The determination of the elastic field on an ellipsoidal inclusion and related problems. Proc. Roy. Soc. L., A 241: 376–392CrossRefGoogle Scholar
  3. Eshelby, J.D. (1961) Elastic inclusions and inhomogeneities. In: Progress in Solid Mechanics V.2 (eds. I. N. Sneddon, R. Hill), Norht-Holland, Amsterdam, 89-140.Google Scholar
  4. Grechka V., Vasconselos I., Kachanov M. (2006) The influence of crack shapes on the effective elasticity of fractured rocks. Geophysics 71: D153–D160Google Scholar
  5. Hill R. (1963) Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11: 357–372CrossRefGoogle Scholar
  6. Huet, C., Navi, P. and Roelfstra, P.E. (1991) A homogenization technique based on Hill’s modification theorem. In: Continuum Models and Discrete Systems, v.2 (ed. G. Maugin 135-143.Google Scholar
  7. Kachanov M., Sevostianov I. (2012) Rice’s internal variables formalism and its implications for the elastic and conductive properties of cracked materials, and for the attempts to relate strength to stiffness. J. Appl. Mech. 79: 031002–1-10Google Scholar
  8. Kachanov M., Tsukrov I., Shafiro B. (1994) Effective moduli of solids with cavities of various shapes. Appl. Mech. Rev. 47: S151–S174CrossRefGoogle Scholar
  9. Mear M.E., Sevostianov I., Kachanov M. (2007) Elastic compliances of non-flat cracks. Int. J. Sol. Structures 44: 6412–6427CrossRefGoogle Scholar
  10. Onaka S. (2001) Averaged Eshelby tensor and elastic strain energyof a superspherical inclusion with uniform eigenstrains. Phil. Mag. Let. 81: 265–272CrossRefGoogle Scholar
  11. Onaka S. (2012) Superspheres: intermediate shapes between spheres and polyhedral. Symmetry 4: 336–343CrossRefGoogle Scholar
  12. Onaka S., Fujii T., Kato M. (2007) Elastic strain energy due to misfit strains in a polyhedralprecipitate composed of low-index planes. Acta Mater. 55: 669–673CrossRefGoogle Scholar
  13. Prokopiev O., Sevostianov I. (2006) On the possibility of approximation of irregular porous microstructure by isolated spheroidal pores. Int. J. Fracture 139: 129–136CrossRefGoogle Scholar
  14. Prokopiev O., Sevostianov I. (2007) Modeling of porous rock: digitization and finite elements versus approximate schemes accounting for pore shapes. Int. J. Fracture 143: 369–375CrossRefGoogle Scholar
  15. Sevostianov, I. and Kachanov, M. (1998) On the relationship between microstructure of the cortical bone and its overall elastic properties, Int. J. Fracture, 92, 1998, pp. L3-L8.Google Scholar
  16. Sevostianov I., Kachanov M. (2002) On the elastic compliances of irregularly shaped cracks. Int. J. Fracture 114: 245–257CrossRefGoogle Scholar
  17. Sevostianov I., Kachanov M. (2007) Relations between compliances of inhomogeneities having the same shape but different elastic constants. Int. J. Eng. Sciences 45: 797–806CrossRefGoogle Scholar
  18. Sevostianov, I., Kováčik, J. and Simančík, F. (2006) Elastic and electric properties of closed-cell aluminum foams. Cross-property connection Materials Science Eng., A-420, 87-99.Google Scholar
  19. Sevostianov I., Kachanov M., Zohdi T. (2008) On computation of the compliance and stiffness contribution tensors of inhomogeneities. Int. J. Sol. Structures 45: 4375–4383CrossRefGoogle Scholar
  20. Tsukrov I., Novak J. (2002) Effective elastic properties of solids with defects of irregular shape. Int. J. Solids Struct. 39: 1539–1555CrossRefGoogle Scholar
  21. Tsukrov I., Novak J. (2004) Effective elastic properties of solids with two-dimensional inclusions of irregular shape. Int. J. Solids Struct. 41, 2004: 6905–6924CrossRefGoogle Scholar
  22. Zimmerman R.W. (1986) Compressibility of two-dimensional cavities of various shapes. J. Appl. Mech. 53: 500–504CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNew Mexico State UniversityLas CrucesUSA
  2. 2.LAEGO-ENSGVandoeuvre-l_es-Nancy CedexFrance

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