Abstract
A two-dimensional model of an elastic body at nanoscale is considered as a half-plane under the action of a periodic load at the boundary. An additional surface stress, and constitutive equations of the Gurtin–Murdoch surface linear elasticity are assumed. Using Goursat–Kolosov complex potentials and Muskhelisvili technique, the solution of the boundary value problem in the case of an arbitrary load is reduced to a hypersingular integral equation in an unknown surface stress. For the case of a periodic load, the solution of this equation is found in the form of Fourier series. The influence of the surface stress on the stresses at the boundary of the half-plane under the tangential and normal periodic loading is analyzed. In particular, it is found out the size effect which becomes apparent in the dependence of the stresses on a length of the load period of the order 10 nm. Moreover, the tangential stresses appear under the action of the normal loads.
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References
Altenbach, H., Eremeev, V.A., Morozov, N.F.: On equations of the linear theory of shells with surface stresses taken into account. Mech. Solids 45, 331–342 (2010)
Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)
Goldstein, R.V., Gorodtsov, V.A., Ustinov, K.V.: Effect of residual stress and surface elasticity on deformation of nanometer spherical inclusions in an elastic matrix. Phys. Mesomech. 13, 318–328 (2010)
Grekov, M.A.: A singular plane problem in the theory of elasticity. St. Petersburg University, St. Petersburg (2001) (in Russ.)
Grekov, M.A., Kostyrko, C.A.: Instability of a flat surface of a film coating due to surface diffusion. Vestnik St. Petersburg University, Ser. 10(1):46–54 (2007) (in Russ.)
Grekov, M.A., Morozov, N.F.: Surface effects and problems of nanomechanics. J. Ningbo Univ. 25, 60–63 (2012)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57, 146–147 (1975)
Linkov, A.M.: Boundary Integral Equations in Elastisity Theory. Kluwer, Dordrecht (2002)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1963)
Podstrigach, Y.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids. Naukova Dumka, Kiev (1985) (in Russ.)
Tian, L., Rajapakse, R.K.N.D.: Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. Trans. ASME J. Appl. Mech. 74, 568–574 (2007)
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Grekov, M.A., Vikulina, Y.I. (2013). Effect of a Type of Loading on Stresses at a Planar Boundary of a Nanomaterial. In: Altenbach, H., Morozov, N. (eds) Surface Effects in Solid Mechanics. Advanced Structured Materials, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35783-1_6
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DOI: https://doi.org/10.1007/978-3-642-35783-1_6
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