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An integral equation formulation of two- and three-dimensional nanoscale inhomogeneities

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Abstract

An integral equation formulation of two- and three-dimensional infinite isotropic medium with nanoscale inhomogeneities is presented in this paper. The Gurtin–Murdoch interface constitutive relation is used to model the continuity conditions along the internal interfaces between the matrix and inhomogeneities. The Poisson’s ratios of both the matrix and inhomogeneities are assumed to be the same. The proposed integral formulation only contains the unknown interface displacements and their derivatives. In order to solve the nanoscale inhomogeneities, the displacement integral equation is used when the source points are acting on the interfaces between the matrix and inhomogeneities. Thus, the resulting system of equations can be formulated so that the interface displacements can be obtained. Furthermore, the stresses at points being in the matrix and nanoscale inhomogeneities can be calculated using the stress integral equation formulation. Numerical results from the present method are in good agreement with those from the conventional sub-domain boundary element method and the analytical method.

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References

  1. Brebbia CA, Dominguez J (1992) Boundary elements—an introduction course. Computational Mechanics Publications, New York

    Google Scholar 

  2. Dong CY, Pan E (2011) Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects. Eng Anal Bound Elem 35: 996–1002

    Article  MathSciNet  Google Scholar 

  3. Dong CY, Lo SH, Cheung YK (2003) Interaction between coated inclusions and cracks in an infinite isotropic elastic medium. Eng Anal Bound Elem 27: 871–884

    Article  MATH  Google Scholar 

  4. Duan HL, Wang J, Huang ZP, Karihaloo BL (2005) Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J Mech Phys Solids 53: 1574–1596

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng YS, Yao RS, Zhang LD (2004) Preparation and optical properties of SnO2/SiO2 nanocomposite. Chinese Phys Lett 21: 1374

    Article  Google Scholar 

  6. Gao W, Yu SW, Huang GY (2006) Finite element characterization of the size-dependent mechanical behavior in nanosystems. Nanotechnology 17: 1118–1122

    Article  Google Scholar 

  7. He LH, Li ZR (2006) Impact of surface stress on stress concentration. Int J Solids Struct 43: 6208–6219

    Article  MATH  Google Scholar 

  8. Jammes M, Mogilevskaya SG, Crouch SL (2009) Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes. Eng Anal Bound Elem 33: 233–248

    Article  MathSciNet  Google Scholar 

  9. Javili A, Steinmann P (2010) A finite element framework for continua with boundary energies. Part II: The three-dimensional case. Comput Methods Appl Mech Eng 199: 755–765

    Article  MathSciNet  MATH  Google Scholar 

  10. Leite LGS, Coda HB, Venturini WS (2003) Two-dimensional solids reinforced by thin bars using the boundary element method. Eng Anal Bound Elem 27: 193–201

    Article  MATH  Google Scholar 

  11. Li S, Sauer R, Wang G (2007) The Eshelby tensors in a finite spherical domain—Part I: Theoretical formulations. J Appl Mech 74: 770–784

    Article  MathSciNet  Google Scholar 

  12. Li S, Wang G, Sauer R (2007) The Eshelby tensors in a finite spherical domain—Part II: applications to homogenization. J Appl Mech 74: 784–798

    Article  MathSciNet  Google Scholar 

  13. Lim CW, Li ZR, He LH (2006) Size dependent, non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress. Int J Solids Struct 43: 5055–5065

    Article  MATH  Google Scholar 

  14. Link S, El-Sayed MA (2003) Optical properties and ultrafast dynamics of metallic nanocrystals. Ann Rev Phys Chem 54: 331–366

    Article  Google Scholar 

  15. Luo J, Wang X (2009) On the anti-plane shear of an elliptic nano inhomogeneity. Eur J Mech A/Solids 28: 926–934

    Article  MATH  Google Scholar 

  16. Mogilevskaya SG, Crouch SL, Stolarski HK (2008) Multiple interacting circular nano-inhomogeneities with surface/interface effects. J Mech Phys Solids 56: 2298–2327

    Article  MathSciNet  MATH  Google Scholar 

  17. Mogilevskaya SG, Crouch SL, Ballarini R, Stolarski HK (2009) Interaction between a crack and a circular inhomogeneity with interface stiffness and tension. Int J Fract 159: 191–207

    Article  Google Scholar 

  18. Ou ZY, Wang GF, Wang TJ (2008) Effect of residual surface tension on the stress concentration around a nanosized spheroidal cavity. Int J Eng Sci 46: 475–485

    Article  Google Scholar 

  19. Ricci A, Ricciardi C (2010) A new finite element approach for studying the effect of surface stress on microstructures. Sens Actuators A 159: 141–148

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Sharma P, Wheeler LT (2007) Size-dependent elastic state of ellipsoidal nanoinclusions incorporating surface/interface tension. ASME J Appl Mech 74: 447–454

    Article  MathSciNet  MATH  Google Scholar 

  22. She H, Wang B (2009) Finite element analysis of conical, dome and truncated InAs quantum dots with consideration of surface effects. Semicond Sci Technol 24: 025002

    Article  Google Scholar 

  23. Tian L, Rajapakse RKND (2007) Analytical solution for size-dependent elastic field of a nano-scale circular inhomogeneity. ASME J Appl Mech 74: 568–574

    Article  MATH  Google Scholar 

  24. Tian L, Rajapakse RKND (2007) Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int J Solids Struct 44: 7988–8005

    Article  MATH  Google Scholar 

  25. Tian L, Rajapakse RKND (2007) Finite element modelling of nanoscale inhomogeneities in an elastic matrix. Comput Mater Sci 41: 44–53

    Article  Google Scholar 

  26. Veprek S (2003) Superhard and functional nanocomposites formed by self-organization in comparison with hardening of coatings by energetic iron bombardment during their deposition. Rev Adv Mater Sci 5: 6–16

    Google Scholar 

  27. Vollath D, Szabo DV (2004) Synthesis and properties of nanocomposites. Adv Eng Mater 6: 117–127

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    C. Y. Dong

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  1. C. Y. Dong
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Correspondence to C. Y. Dong.

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Dong, C.Y. An integral equation formulation of two- and three-dimensional nanoscale inhomogeneities. Comput Mech 49, 309–318 (2012). https://doi.org/10.1007/s00466-011-0640-3

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  • DOI: https://doi.org/10.1007/s00466-011-0640-3

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