Skip to main content
SpringerLink
Log in

Soft Neutral Elastic Inhomogeneities with Membrane-type Interface Conditions

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

A foreign body, called an “inhomogeneity,” when introduced in a host solid disturbs the stress field which is present in it. One can explore the possibility of modifying the contact mechanism between the inhomogeneity and the host body so as to leave the stress field in the host solid undisturbed. If such a procedure succeeds, then the inhomogeneity is called “neutral.” Modification of the contact mechanism between the inhomogeneity and the host solid can be achieved, for example, by a suitably designed thick or thin interphase between them. When the interphase is thin, it can be represented by an “imperfect interface” model. In the present study we consider “soft” inhomogeneities which are more compliant than the host body. A “membrane-type interface” which models a thin and stiff interphase is used in rendering such inhomogeneities neutral. Illustrative examples are constructed for cylindrical neutral inhomogeneities of elliptical cross section under a triaxial loading, and for spheroidal inhomogeneities subjected to an axisymmetric loading.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mansfield, E.H.: Neutral holes in plane sheet—reinforced holes which are equivalent to the uncut sheet. Q. J. Mech. Appl. Math. 6, 370–378 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  2. Richards, R. Jr., Bjorkman, G.S. Jr.: Neutral holes—theory and design. J. Eng. Mech. Div. ASCE 108, 945–960 (1982)

    Google Scholar 

  3. Budiansky, B., Hutchinson, J.W., Evans, A.G.: On neutral holes in tailored sheets. J. Appl. Mech. 60, 1056–1058 (1993)

    MATH  Google Scholar 

  4. Senocak, E., Waas, A.: Neutral cutouts in laminated plates. Mech. Compos. Mater. Struct. 2, 71–89 (1995)

    Google Scholar 

  5. Ru, C.Q.: Interface design of neutral elastic inclusions. Int. J. Solids Struct. 35, 557–572 (1998)

    Article  Google Scholar 

  6. Benveniste, Y., Miloh, T.: Neutral inhomogeneities in conduction phenomena. J. Mech. Phys. Solids 4, 197–208 (1999)

    MathSciNet  Google Scholar 

  7. Milton, G.W., Serkov, S.K.: Neutral coated inclusions in conductivity and anti-plane elasticity. Proc. R. Soc. Lond., A 457, 1973–1997 (2001)

    MATH  ADS  MathSciNet  Google Scholar 

  8. Benveniste, Y., Chen, T.: On the Saint-Venant torsion of composite bars with imperfect interfaces. Proc. R. Soc. Lond., A 457, 231–255 (2001)

    MATH  ADS  MathSciNet  Google Scholar 

  9. Chen, T., Benveniste, Y., Chuang, P.C.: Exact solutions in torsion of composite bars: thickly coated neutral inhomogeneities and composite cylinder assemblages. Proc. R. Soc. Lond., A 458, 1719–1759 (2002)

    MATH  ADS  Google Scholar 

  10. Schiavone, P.: Neutrality of the elliptic inhomogeneity in the case of non-uniform loading. Int. J. Eng. Sci. 41, 2081–2090 (2003)

    Article  MathSciNet  Google Scholar 

  11. Van Vliet, D., Schiavone, P., Mioduchowski, A.: On the design of neutral inhomogeneities in the case of non-uniform loading. Math. Mech. Solids 8, 161–169 (2003)

    Article  MATH  Google Scholar 

  12. Mahboob, M., Schiavone, P.: Designing a neutral inhomogeneity in the case of general non-uniform loading. Appl. Math. Lett. 18, 1312–1318 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Vasudevan, M., Schiavone, P.: New results concerning the identification of neutral inhomogeneities in plane elasticity. Arch. Mech. 58, 45–58 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Vasudevan, M., Schiavone, P.: Design of neutral inhomogeneities in the case of non-uniform loading. Int. J. Eng. Sci. 43, 1081–1091 (2005)

    Article  MathSciNet  Google Scholar 

  15. Bertoldi, K., Bigoni, D., Drugan, W.J.: Structural interfaces in linear elasticity. Part II: Effective properties and neutrality. J. Mech. Phys. Solids 55, 35–63 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Leonhardt, U.: Notes on conformal invisibility devices. New J. Phys. 8, (2006) Art. No.118

  18. Milton, G.W., Nicorovici, N.A.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond., A 462, 3027–3059 (2006)

    ADS  MATH  MathSciNet  Google Scholar 

  19. Milton, G.W., Briane, M., Willis, J.R.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, (2006) Art. No. 248

    Google Scholar 

  20. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic waves. Science 312, 1780–1782 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  21. Bertoldi, K., Bigoni, D., Drugan, W.J.: Structural interfaces in linear elasticity. Part I : Nonlocality and gradient approximations. J. Mech. Phys. Solids 55, 1–34 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150 (1962)

    MATH  MathSciNet  Google Scholar 

  23. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131 (1963)

    Article  ADS  Google Scholar 

  24. Benveniste, Y., Milton, G.W.: New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages. J. Mech. Phys. Solids 51, 1773–1813 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Benveniste, Y.: A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids 54, 708–734 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Benveniste, Y.: Corrigendum to: A general interface model for a three-dimensional curved anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids 54, 708–734 (2006). J. Mech. Phys. Solids, 55, 666–667(2007)

    Google Scholar 

  27. Benveniste, Y., Miloh, T.: Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33, 309–323 (2001)

    Article  Google Scholar 

  28. Gol’denveizer, A.L.: Theory of Elastic Thin Shells. Pergamon, New York (1961)

    Google Scholar 

  29. Chen, T.Y., Chiu, M.S., Weng, C.N.: Derivation of the generalized Young–Laplace equation of curved interfaces in nano-scaled solids. J. Appl. Phys. 100(074308), 1–5 (2006)

    Google Scholar 

  30. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  31. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1975)

    Article  Google Scholar 

  32. Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Q. J. Mech. Appl. Math. 29, 245–275 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  33. Murdoch, A.I.: Some fundamental aspects of surface modelling. J. Elast. 80, 33–35 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Gurtin, M.E., Weissmuller, J., Larché, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)

    ADS  Google Scholar 

  35. Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond., A 453, 853–858 (1997)

    MATH  ADS  MathSciNet  Google Scholar 

  36. Schiavone, P., Ru, C.Q.: Integral equation methods in plane–strain elasticity with boundary reinforcement. Proc. R. Soc. Lond., A 454, 2223–2242 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)

    Article  ADS  Google Scholar 

  38. Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic properties of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)

    Article  ADS  Google Scholar 

  39. Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53, 1574–1596 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P.: Nanoporous materials can be made stiffer than non-porous counterparts by surface modification. Acta Mater. 54, 2983–2990 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, 69978, Israel

    Y. Benveniste & T. Miloh

Authors
  1. Y. Benveniste
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Miloh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Benveniste.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benveniste, Y., Miloh, T. Soft Neutral Elastic Inhomogeneities with Membrane-type Interface Conditions. J Elasticity 88, 87–111 (2007). https://doi.org/10.1007/s10659-007-9115-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-007-9115-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation