Abstract
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.
Similar content being viewed by others
References
Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011)
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)
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)
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)
Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids. 59, 2103–2115 (2011)
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)
Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)
Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech. 227, 29–42 (2016)
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)
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)
Grekov, M.A.: A slightly curved crack in an isotropic body. Vestnik Sankt–Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya. (3), 74–80 (2002)
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)
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)
Grekov, M.A., Kostyrko, S.A.: A multilayer film coating with slightly curved boundary. Int. J. Eng. Sci. 89, 61–74 (2015)
Grekov, M.A., Kostyrko, S.A.: Surface effects in an elastic solid with nanosized surface asperities. Int. J. Solids Struct. 96, 153–161 (2016)
Grekov, M.A., Kostyrko, S.A., Vakaeva, A.B.: The model of surface nanorelief within continuum mechanics. AIP Conf. Proc. 1909, 020062 (2017)
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)
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)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975)
Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)
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)
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)
Kostyrko, S.A., Grekov, M.A., Altenbach, H.: A model of nanosized thin film coating with sinusoidal interface. AIP Conf. Proc. 1959, 070017 (2018)
Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Netherlands (1977)
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)
Povstenko, Y.Z.: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41, 1499–1514 (1993)
Romanova, V.A., Balokhonov, R.R.: Numerical analysis of mesoscale surface roughening in a coated plate. Comput. Mater. Sci. 61, 71–75 (2012)
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)
Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A. 453, 853–877 (1997)
Steigmann, D.J., Ogden, R.W.: Elastic surface–substrate interactions. Proc. R. Soc. A. 455, 437–474 (1999)
Tian, L., Rajapakse, R.K.N.D.: Finite element modelling of nanoscale inhomogeneities in an elastic matrix. Comput. Mater. Sci. 41, 44–53 (2007)
Vikulina, YuI, Grekov, M.A., Kostyrko, S.A.: Model of film coating with weakly curved surface. Mech. Solids 45, 778–788 (2010)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work is supported by German Academic Exchange Service and St. Petersburg State University under joint Grant 9.23.819.2017 and Russian Foundation for Basic Research under Grant 18-01-00468.
Rights and permissions
About this article
Cite this article
Kostyrko, S., Grekov, M. & Altenbach, H. Stress concentration analysis of nanosized thin-film coating with rough interface. Continuum Mech. Thermodyn. 31, 1863–1871 (2019). https://doi.org/10.1007/s00161-019-00780-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-019-00780-4