Stress concentration analysis of nanosized thin-film coating with rough interface

  • Sergey KostyrkoEmail author
  • Mikhail Grekov
  • Holm Altenbach
Original Article


The boundary perturbation method combined with the superposition principle is used to calculate the stress concentration along the arbitrary curved interface of an isotropic thin film coherently bonded to a substrate. In the case of plane strain conditions, the boundary value problem is formulated for a four-phase system involving two-dimensional constitutive equations for bulk materials and one-dimensional equations of Gurtin–Murdoch model for surface and interface. Static boundary conditions are formulated in the form of generalized Young–Laplace equations. Kinematic boundary conditions describe the continuous of displacements across the surface and interphase regions. Using Goursat–Kolosov complex potentials, the system of boundary equations is reduced to a system of the integral equations via first-order boundary perturbation method. Finally, the solution of boundary value problem is obtained in terms of Fourier series. The numerical analysis is then carried out using the practically important properties of ultra-thin-film materials.


Stress concentration Interface roughness Thin film Boundary perturbation method 


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  1. 1.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the existence of solution in the linear elasticity with surface stresses. Z. Angew. Math. Mech. 90, 231–240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bashkankova, E.A., Vakaeva, A.B., Grekov, M.A.: Perturbation method in the problem on a nearly circular hole in an elastic plane. Mech. Solids 50, 198–207 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids. 59, 2103–2115 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cuenot, S., Fretigny, C., Demoustier-Champagne, S., Nysten, B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)CrossRefGoogle Scholar
  8. 8.
    Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech. 227, 29–42 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eremeyev, V.A., Lebedev, L.P.: Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity. Cont. Mech. Therm. 28, 407–422 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gao, X., Huang, Z., Fang, D.: Curvature-dependent interfacial energy and its effects on the elastic properties of nanomaterials. Int. J. Solids Struct. 113–114, 100–107 (2017)CrossRefGoogle Scholar
  11. 11.
    Grekov, M.A.: A slightly curved crack in an isotropic body. Vestnik Sankt–Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya. (3), 74–80 (2002)Google Scholar
  12. 12.
    Grekov, M.A.: The perturbation approach for a two-component composite with a slightly curved interface. Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya. (1), 81–88 (2004)Google Scholar
  13. 13.
    Grekov, M.A., Kostyrko, S.A.: A film coating on a rough surface of an elastic body. J. Appl. Math. Mech. 77, 79–90 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grekov, M.A., Kostyrko, S.A.: A multilayer film coating with slightly curved boundary. Int. J. Eng. Sci. 89, 61–74 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grekov, M.A., Kostyrko, S.A.: Surface effects in an elastic solid with nanosized surface asperities. Int. J. Solids Struct. 96, 153–161 (2016)CrossRefGoogle Scholar
  16. 16.
    Grekov, M.A., Kostyrko, S.A., Vakaeva, A.B.: The model of surface nanorelief within continuum mechanics. AIP Conf. Proc. 1909, 020062 (2017)CrossRefGoogle Scholar
  17. 17.
    Grekov, M.A., Sergeeva, T.S., Pronina, Y.G., Sedova, O.S.: A periodic set of edge dislocations in an elastic solid with a planar boundary incorporating surface effects. Eng. Fract. Mech. 186, 423–435 (2017)CrossRefGoogle Scholar
  18. 18.
    Grekov, M.A., Vakaeva, A.B.: Effect of nanosized asperities at the surface of a nanohole. Proc. VII Europ. Congr. Comput. Meth. Appl. Sci. Eng. 4(1), 7875–7885 (2016)CrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kim, H.-K., et al.: Suppression of interface roughness between \(\text{ BaTiO }_3\) film and substrate by \(\text{ Si }_3\text{ N }_4\) buffer layer regarding aerosol deposition process. J. Alloys Compd. 653, 69–76 (2015)CrossRefGoogle Scholar
  22. 22.
    Kostyrko, S.A., Altenbach, H., Grekov, M.A.: Stress concentration in ultra-thin film coating with undulated surface profile. In: Papadrakasis, M., Oñate, E., Schrefler, B.: VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2017, pp. 1183–1192. CIMNE, Barcelona (2017)Google Scholar
  23. 23.
    Kostyrko, S.A., Grekov, M.A., Altenbach, H.: A model of nanosized thin film coating with sinusoidal interface. AIP Conf. Proc. 1959, 070017 (2018)CrossRefGoogle Scholar
  24. 24.
    Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)ADSCrossRefGoogle Scholar
  25. 25.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Netherlands (1977)CrossRefGoogle Scholar
  26. 26.
    Nazarenko, L., Stolarski, H., Altenbach, H.: Effective properties of short-fiber composites with Gurtin–Murdoch model of interphase. Int. J. Solids Struct. 97–98, 75–78 (2016)CrossRefGoogle Scholar
  27. 27.
    Povstenko, Y.Z.: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41, 1499–1514 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Romanova, V.A., Balokhonov, R.R.: Numerical analysis of mesoscale surface roughening in a coated plate. Comput. Mater. Sci. 61, 71–75 (2012)CrossRefGoogle Scholar
  29. 29.
    Ru, C.Q.: Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. Sci. China Phys.Mech. Astron. 53, 536–544 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A. 453, 853–877 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Steigmann, D.J., Ogden, R.W.: Elastic surface–substrate interactions. Proc. R. Soc. A. 455, 437–474 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tian, L., Rajapakse, R.K.N.D.: Finite element modelling of nanoscale inhomogeneities in an elastic matrix. Comput. Mater. Sci. 41, 44–53 (2007)CrossRefGoogle Scholar
  33. 33.
    Vikulina, YuI, Grekov, M.A., Kostyrko, S.A.: Model of film coating with weakly curved surface. Mech. Solids 45, 778–788 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    Zemlyanova, A.Y., Mogilevskaya, S.G.: Circular inhomogeneity with Steigmann–Ogden interface: local fields, neutrality, and Maxwell’s type approximation formula. Int. J. Solids Struct. 135, 85–98 (2018)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Otto von Guericke UniversityMagdeburgGermany

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