Abstract
A system of von Karman equations for a nanoplate has been generalized by introducing effective tangential and flexural stiffnesses and elastic moduli, with regard to surface elasticity and residual surface stresses on the outer surfaces. A modified Kirsch problem was solved for the case of an infinite nanoplate with a circular hole under plane stress in terms of effective elastic moduli. Two forms of local stability loss in this problem and the corresponding critical load for two different elastic characteristics of all plate surfaces were determined numerically and analytically. The dependence of the effective stiffnesses and elastic moduli on the plate thickness, and of the critical load on the hole radius (size effect) was discussed.
Similar content being viewed by others
References
Panin, V.E., Egorushkin, V.E., and Panin, A.V., Physical Mesomechanics of a Deformed Solid as a Multilevel System. I. Physical Fundamentals of the Multilevel Approach, Phys. Mesomech., 2006, vol. 9, no. 3–4, pp. 9–20.
Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., and Wang, T., Surface Stress Effect in Mechanics of Nanostructured Materials, Acta Mech. Solid. Sinica, 2011, vol. 24, pp. 52–82.
Eremeyev, V.A., Altenbach, H., and Morozov, N. F., The Influence of Surface Tension on the Effective Stiffness of Nanosize Plates, Dokl. Phys., 2009, vol. 54, no. 2, pp. 98–100.
Eremeyev, V.A. and Morozov, N.F., The Effective Stiffness of a Nanoporous Rod, Dokl. Phys., 2010, vol. 55, no. 6, pp. 279–282.
Eremeyev, V.A., On Effective Properties of Materials at the Nano- and Microscales Considering Surface Effects, Acta Mech., 2016, vol. 227, no. 1, pp. 29–42.
Goldstein, R.V., Gorodtsov, V.A., and Ustinov, K.B., Effect of Residual Surface Stress and Surface Elasticity on Deformation of Nanometer Spherical Inclusions in an Elastic Matrix, Phys. Mesomech., 2010, vol. 13, no. 5–6, pp. 318–328.
Krivtsov, A.M. and Morozov, N.F., Anomalies in Mechanical Characteristics of Nanometer-Size Objects, Dokl. Phys., 2001, vol. 46, no. 11, pp. 825–827.
Ivanova, E.A., Krivtsov, A.M., and Morozov, N.F., Peculiarities of the Bending-Stiffness Calculation for Nanocrystals, Dokl. Phys., 2002, vol. 47, no. 8, pp. 620–622.
Berinskii, I.E., Krivtsov, A.M., and Kudarova, A.M., Bending Stiffness of a Graphene Sheet, Phys. Mesomech., 2014, vol. 7, no. 4, pp. 356–364.
Miller, R.E. and Shenoy, V.B., Size-Dependent Elastic Properties of Nanosized Structural Elements, Nanotechnology, 2000, vol. 11, pp. 139–147.
Shenoy, V.B., Atomic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces, Phys. Rev. B, 2005, vol. 71, no. 9, pp. 94–104.
Duan, H.L., Wang, J., and Karihaloo, B.L., Theory of Elasticity at the Nanoscale, Adv. Appl. Mech., 2009, no. 42, pp. 1–68.
Gibbs, J.W., The Scientific Papers of J. Willard Gibbs. V. 1, London: Longmans-Green, 1906.
Gurtin, M.E. and Murdoch, A.I., A Continuum Theory of Elastic Material Surfaces, Arch. Ration. Mech. Anal., 1975, vol. 57, no. 4, pp. 291–323.
Gurtin, M.E. and Murdoch, A.I., Surface Stress in Solids, Int. J. Solids Struct., 1978, vol. 14, pp. 431–440.
Lim, C.W. and He, L.H., Size-Dependent Nonlinear Response of Thin Elastic Films with Nano-Scale Thickness, Int. J. Mech. Sci., 2005, vol. 46, no. 11, pp. 1715–1726.
Huang, D.W., Size-Dependent Response of Ultra-Thin Films with Surface Effects, Int. J. Solids Struct., 2008, vol. 45, no. 2, pp. 568–579.
Mogilevskaya, S.G., Crouch, S.L., and Stolarsk, H.K., Multiple Interacting Circular Nano-Inhomogeneities with Surface/Interface Effects, J. Mech. Phys. Solids, 2008, vol. 56, pp. 2298–2327.
Tian, L. and Rajapakse, R.K.N.D., Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity, Trans. ASME. J. Appl. Mech., 2007, vol. 74, no. 5, pp. 568–574.
Tian, L. and Rajapakse, R.K.N.D., Elastic Field of an Isotropic Matrix with Nanoscale Elliptical Inhomogeneity, Int. J. Solids Struct., 2007, vol. 44, pp. 7988–8005.
Grekov, M.A. and Yazovskaya, A.A., The Effect of Surface Elasticity and Residual Surface Stress in an Elastic Body with an Elliptic Nanohole, J. Appl. Math. Mech., 2014, vol. 78, no. 2, pp. 172–180.
Grekov, M.A. and Kostyrko, S.A., Surface Effects in an Elastic Solid with a Nanosized Surface Asperites, Int. J. Solids Struct., 2016, vol. 96, pp. 153–161.
Altenbach, H., Nremeev, V.A., and Morozov, N.F., On Equations of the Linear Theory of Shells with Surface Stresses Taken into Account, Mech. Solids, 2010, vol. 45, no. 3, pp. 331–342.
Ru, C.Q., A Strain-Consistent Plastic Plate Model with Surface Plasticity, Continuum Mech. Thermodyn., 2016, vol. 28, pp. 263–273.
Morozov, N.F., Tovstik, P.P., and Tovstik, T.P., Continuum Model of Multilayer Nanoplate Bending and Oscillation, Fiz. Mezomekh., 2016, vol. 19, no. 6, pp. 27–33.
Bauer, S.M., Kashtanova, S.V., Morozov, N.F., and Semenov, B.N., Stability of a Nanoscale-Thickness Plate Weakened by a Circular Hole, Dokl. Phys., 2014, vol. 59, no. 9, pp. 416–418.
Kirsch, P.G., Die Theorie der Plastizitat und die Bedurfnisse der Festigkeitslehre, Zeitschrift des Vereines deutscher Ingenieure, 1898, vol. 42, pp. 797–807.
Ciarlet, P.G. and Rabier, P., Les Equations de von Karman: Lecture Notes in Mathematics. V. 826, Berlin: SpringerVerlag, 1980.
Papkovich, P.F., Ship Structural Design, V. II, Eningrad: Sudpromgiz, 1941.
Povstenko, Yu.Z., Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids, J. Mech. Phys. Solids, 1993, vol. 41, pp. 1499–1514.
Bochkarev, A.O. and Grekov, M.A., Local Instability of a Plate with a Circular Nanohole under lniaxial Tension, Dokl. Phys., 2014, vol. 59, no. 7, pp. 330–334.
Bochkarev, A.O. and Grekov, M.A., On Symmetrical and Antisymmetrical Buckling of a Plate with Circular Nanohole under Uniaxial Tension, Appl. Math. Sci., 2015, vol. 9, no. 125, pp. 6241–6247.
Bochkarev, A.O. and Grekov, M.A., The Influence of the Surface Stress on the Local Buckling of a Plate with a Circular Nanohole, Proc. Int. Conf. Stabil. Control Proc. in Memory of V.I. Zubov, SCP 2015, 2015, pp. 367–370.
Mihlin, S.G., Variational Methods in Mathematical Physics: Int. Series of Monographs in Pure and Appl. Physics. Vol. 50, Pergamon Press, 1964.
Bochkarev, A.O. and Dal, Y.M., Local Stability of Notched Plastic Plates, Sov. Phys. Dokl., 1989, vol. 308, no. 2, pp. 312–315.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2017, published in Fizicheskaya Mezomekhanika, 2017, Vol. 20, No. 6, pp. 62–76.
Rights and permissions
About this article
Cite this article
Bochkarev, A.O., Grekov, M.A. Influence of Surface Stresses on the Nanoplate Stiffness and Stability in the Kirsch Problem. Phys Mesomech 22, 209–223 (2019). https://doi.org/10.1134/S1029959919030068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1029959919030068