Two approaches to study the effect of surface stresses in an elastic body with a nearly circular nanodefect

Keywords: nearly circular nanodefect, boundary perturbation method (BPM), finite-element method (FEM), stress concentration, surface stress


Most of the advanced construction and functional materials are elastically nonuniform, moreover, for many of them, the elongated holes and inclusions are typical, which are similar to a cylinder in form. The strength and physicochemical properties of a material, to a great extent, depend on the peculiarities of the strain-stress state of the near-surface and boundary layers of the materials in the heterogeneous systems. The development of the processes of elastic deformation and fracture in these areas, to a large extent, determines the mechanical behavior of a material in general and arouses much interest. The authors study the influence of interfacial stresses on the strain-stress state of elastic bimaterial with smooth waveform interface; consider the 2-D solid mechanics problem of an elastic body with nanoscale boundary surface texture, which appears between the nearly circular inclusion and the matrix. It is expected that a body is situated within a uniform stress field. To solve the problem, the authors used the simplified Gurtin–Murdoch’s surface/interface elasticity model, where the interfacial boundary is the negligibly thin layer exactly bordered on the bulk phases. It is acknowledged that there are no displacement discontinuities on the interfacial boundary, and the stress jump is determined by the effect of surface/interfacial stress according to the generalized Laplace–Young law. Using the boundary perturbation method, the problem solution for each approximation is limited to a singular integrodifferential equation against the unknown surface/interfacial stress. The paper gives the numerical results for the problem to a first approximation. As a result, the authors carry out the comparative analysis of the strain-stress state using the finite-element method and analytical boundary perturbation method.

Author Biographies

Aleksandra B. Vakaeva, St. Petersburg State University, St. Petersburg (Russia)

PhD (Physics and Mathematics), senior lecturer

Gleb M. Shuvalov, St. Petersburg State University, St. Petersburg (Russia)


Sergey A. Kostyrko, St. Petersburg State University, St. Petersburg (Russia)

PhD (Physics and Mathematics), Associate Professor

Olga S. Sedova, St. Petersburg State University, St. Petersburg (Russia)

PhD (Physics and Mathematics), senior lecturer


Duan H.L., Wang J., Karihaloo B.L. Theory of elasticity at the nanoscale. Advances in Applied Mechanics, 2009, vol. 42, pp. 1–68.

Wang J., Huang Z., Duan H., Yu S., Feng X., Wang G., Zhang W., Wang T. Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica, 2011, vol. 24, no. 1, pp. 52–82.

Podstrigach Ya.S., Povstenko Yu.Z. Vvedenie v mekhaniku poverkhnostnykh yavleniy v deformiruemykh tverdykh telakh [Introduction to the mechanics of surface phenomena in deformable solids]. Kiev, Naukova dumka Publ., 1985. 200 p.

Povstenko Ya.Z. Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. Journal of Mechanics and Physics Solids, 1993, vol. 41, no. 9, pp. 1499–1514.

Gibbs J.W. The Scientific Papers of J. Willard Gibbs. London, Longmans-Green, 1906. Vol. 1, 476 p.

Gurtin M.E., Murdoch A.I. Surface stress in solids. International Journal of Solid Structures, 1978, vol. 14, no. 6, pp. 431–440.

Bochkarev A.O., Grekov M.A. Influence of Surface Stresses on the Nanoplate Stiffness and Stability in the Kirsch Problem. Physical Mesomechanics, 2019, vol. 22, no. 3, pp. 209–223.

Grekov M.A., Sergeeva T.S. Interaction of edge dislocation array with biomaterial interface incorporating interface elasticity. International Journal of Engineering Science, 2020, vol. 149, pp. 103233.

Smirnov A.M., Krasnitckii S.A., Gutkin M.Y. Generation of misfit dislocation in a core-shell nanowire near the edge of prismatic core. Acta Materialia, 2020, vol. 186, pp. 494–510.

Miller R.E., Shenoy V.B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 2000, vol. 11, no. 3, pp. 139–147.

Wang W., Zeng Xi., Ding J. Finite element modeling of two-dimentional nanoscale structures with surface effects. World Academy of Science, Engineering and Technology, 2010, vol. 48, no. 12, pp. 426–431.

Tian L., Rajapakse R.K.N.D. Finite element modeling of nanoscale inhomogeneities in an elastic matrix. Computational Materials Science, 2007, vol. 41, no. 5, pp. 568–574.

Vakaeva A.B., Grekov M.A. Investigation of the stress-strain state of an elastic body with almost circular defects. Protsessy upravleniya i ustoychivost, 2014, vol. 1, no. 1, pp. 111–116.

Eremeyev V.A. On effective properties of materials at the nano- and microscales considering surface effects. Acta Mechanica, 2016, vol. 227, no. 1, pp. 29–42.

Eremeyev V.A., Lebedev L.P. Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Continuum Mechanics and Thermodynamics, 2016, vol. 28, pp. 407–422.

Grekov M.A. Fundamental Solution for the Generalized Plane Stress of a Nanoplate. Advanced Structured Materials, 2019, vol. 108, pp. 157–164.

Sedova O.S., Pronina Yu.G. On the choice of equivalent stress for the problem of mechanochemical corrosion of spherical members. Vestnik Sankt-Peterburgskogo universiteta. Seriya 10: Prikladnaya matematika. Informatika. Protsessy upravleniya, 2016, no. 2, pp. 33–44.

Medina H., Hinderliter B. The stress concentration factor for slightly roughened random surfaces: Analytical solution. International Journal of Solid and Structures, 2014, vol. 51, pp. 2012–2018.

Gharahi A., Schiavone P. Effective elastic properties of plane micro polar nanocomposites with flexural effects. International Journal of Mechanical Sciences, 2018, vol. 149, pp. 84–92.

Vakaeva A.B. Stress-strain state of an elastic body with a nearly circular inclusion incorporating interfacial stress. Vektor nauki Tolyattinskogo gosudarstvennogo universiteta, 2017, no. 4, pp. 20–25.

Vakaeva A.B., Grekov M.A. Effect of interfacial stresses in an elastic body with a nanoinclusion. AIP Conference Proceedings, 2018, vol. 1959, p. 070036.

Novozhilov V.V. Teoriya uprugosti [Elasticity theory]. Leningrad, Sudpromgiz Publ., 1958. 374 p.

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, 2004, no. 1, pp. 81–88.

Muskhelishvili N.I. Nekotorye osnovnye zadachi matematicheskoy teorii uprugosti [Some basic problems of the mathematical theory of elasticity]. Moscow, Nauka Publ., 1966. 707 p.

Sharma P., Ganti S., Bhate N. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Applied Physics Letters, 2003, vol. 82, no. 4, pp. 535–537.

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