Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Surface elasticity effect on diffusional growth of surface defects in strained solids

  • 66 Accesses

  • 1 Citations


This paper presents a theoretical approach that allows to predict the nucleation of surface topological defects under the mechanical loading taking into account the thermodynamic and elastic properties of solid surface as well as its geometrical characteristics. Assuming that the surface atomic layers are thermodynamically unstable under the certain conditions, we obtain the evolution equation describing the kinetics of the relief formation in the case of diffusion mass transport activated by the stress field. The rate of growth of surface defects depends on the field of bulk and surface stresses, which vary with the shape and size of the considered defects. To find the stress state, we use the first-order perturbation solution of a 2D boundary value problem formulated in the terms of the constitutive equations of bulk and surface elasticity. The solution of linearized evolution equation gives the critical values of the ridges size and the initial level of stresses, which stabilize surface profile.

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


  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)

  2. 2.

    Asaro, R.J., Tiller, W.A.: Interface morphology development during stress-corrosion cracking: Part I. Via surface diffusion. Metall. Trans. 3, 1789–1796 (1972)

  3. 3.

    Berrehar, J., et al.: Surface patterns on single-crystal films under uniaxial stress: experimental evidence for the Grinfeld instability. Phys. Rev. B 46, 13487–13495 (1992)

  4. 4.

    Colin, J., Grilhe, J., Junqua, N.: Morphological instabilities of a stressed pore channel. Acta Mater. 45, 3835–3841 (1997)

  5. 5.

    Cuenot, S., et al.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)

  6. 6.

    Duan, H.L., Weissmüller, J., Wang, Y.: Instabilities of core–shell heterostructured cylinders due to diffusions and epitaxy: spheroidization and blossom of nanowires. J. Mech. Phys. Solids 56, 1831–1851 (2008)

  7. 7.

    Eremeyev, V.A.: On effective properties of materials at the nano- and microscales considering surface effects. Acta Mech. 227, 29–42 (2016)

  8. 8.

    Eremeev, V.A., Freidin, A.B., Sharipova, L.L.: Nonuniqueness and stability in problems of equilibrium of elastic two-phase bodies. Dokl. Phys. 48, 359–364 (2003)

  9. 9.

    Fartash, A., et al.: Evidence for the supermodulus effect and enhanced hardness in metallic superlattices. Phys. Rev. B 44, 13760–13763 (1991)

  10. 10.

    Freund, L.B.: Evolution of waviness on the surface of a strained elastic solid due to stress-driven diffusion. Int. J. Solids Struct. 28, 911–923 (1995)

  11. 11.

    Gao, H.: Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure. J. Mech. Phys. Solids 42, 741–772 (1994)

  12. 12.

    Grekov, M.A., Kostyrko, S.A.: Surface effects in an elastic solid with nanosized surface asperities. Int. J. Solids Struct. 96, 153–161 (2016)

  13. 13.

    Grekov, M.A., et al.: A periodic set of edge dislocations in an elastic solid with a planar boundary incorporating surface effects. Eng. Fract. Mech. 186, 423–435 (2017)

  14. 14.

    Grinfeld, M.: Instability of the equilibrium of a nonhydrostatically stressed body and a melt. Fluid Dyn. 22, 169–173 (1987)

  15. 15.

    Grinfeld, M.A.: Thermodynamic Methods in the Theory of Heterogeneous Systems. Longman, Harlow Essex (1991)

  16. 16.

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

  17. 17.

    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

  18. 18.

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

  19. 19.

    Kim, J.-H., Vlassak, J.J.: Perturbation analysis of an undulating free surface in a multi-layered structure. Int. J. Solids Struct. 44, 7924–7937 (2007)

  20. 20.

    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. (Eds) VII International Conference on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2017, pp. 1183–1192. CIMNE, Barcelona (2017)

  21. 21.

    Kostyrko, S.A., Grekov, M.A., Altenbach, H.: A model of nanosized thin film coating with sinusoidal interface. AIP Conf. Proc. 1959, 070017 (2018)

  22. 22.

    Kostyrko, S.A., Shuvalov, G.M.: Stability analysis of nanoscale surface patterns in stressed solids. AIP Conf. Proc. 1959, 070016 (2018)

  23. 23.

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

  24. 24.

    Mogilevskaya, S.G., Crouch, S.I., Stolarski, H.K.: Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56, 2298–2327 (2008)

  25. 25.

    Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957)

  26. 26.

    Mullins, W.W., Sekerka, R.F.: Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323–329 (1963)

  27. 27.

    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Dordrecht (1977)

  28. 28.

    Nazarenko, L., Stolarski, H., Altenbach, H.: Thermo-elastic properties of random composites with unidirectional anisotropic short-fibers and interphases. Eur. J. Mech./A Solids 70, 249–266 (2018)

  29. 29.

    Sekerka, R.F.: Morphological stability. J. Cryst. Growth 3, 71–81 (1968)

  30. 30.

    Shenoy, V.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104 (2005)

  31. 31.

    Shuvalov, G.M., Kostyrko, S.A.: Surface self-organization in multilayer film coatings. AIP Conf. Proc. 1909, 020196 (2017)

  32. 32.

    Spencer, B.J., Meiron, D.L.: Nonlinear evolution of the stress-driven morphological instability in a two-dimensional semi-infinite solid. Acta Metall. Mater. 42, 3629–3641 (1994)

  33. 33.

    Srolovitz, D.J.: On the stability of surfaces of stressed solids. Acta Metall. 37, 621–625 (1989)

  34. 34.

    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453, 853–877 (1997)

  35. 35.

    Steigmann, D.J., Ogden, R.W.: Elastic surface–substrate interactions. Proc. R. Soc. A 455, 437–474 (1999)

  36. 36.

    Torii, R.H., Balibar, S.: Helium crystals under stress: the Grinfeld instability. J. Low Temp. Phys. 89, 391–400 (1992)

  37. 37.

    Yang, W.H., Srolovitz, D.J.: Cracklike surface instabilities in stressed solids. Phys. Rev. Lett. 71, 1593–1596 (1993)

  38. 38.

    Yeremeyev, V.A., Freidin, A.B., Sharipova, L.L.: The stability of the equilibrium of two-phase elastic solids. J. Appl. Math. Mech. 71, 61–84 (2007)

  39. 39.

    Wang, X., Schiavone, P.: Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole. Mech. Res. Commun. 52, 57–61 (2013)

Download references

Author information

Correspondence to Sergey Kostyrko.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work is supported by Russian Foundation for Basic Research under Grant 18-01-00468.

Communicated by Victor Eremeyev, Holm Altenbach.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kostyrko, S., Shuvalov, G. Surface elasticity effect on diffusional growth of surface defects in strained solids. Continuum Mech. Thermodyn. 31, 1795–1803 (2019). https://doi.org/10.1007/s00161-019-00756-4

Download citation


  • Surface diffusion
  • Surface stress
  • Evolution equation
  • Boundary perturbation method