On Nonlinear Dynamic Theory of Thin Plates with Surface Stresses

  • Holm Altenbach
  • Victor A. EremeyevEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 114)


We discuss the modelling of dynamics of thin plates considering surface stresses according to Gurtin–Murdoch surface elasticity. Taking into account the surface mass density we derive the two-dimensional (2D) equations of motion. For the reduction of the three-dimensional (3D) motion equations to the 2D ones we use the trough-the-thickness integration procedure. As a result, the 2D dynamic parameters of the plate depend not only on the density distribution in the bulk but also on the surface mass density.


Thin plate Surface stresses Equations of motion Gurtin-Murdoch approach 



The second author acknowledges financial support from the Russian Science Foundation under the grant “Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base” (No 15-19-10008-P).


  1. 1.
    Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)ADSCrossRefGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Altenbach, H., Eremeyev, V.A.: On the elastic plates and shells with residual surface stresses. Procedia IUTAM 21, 25–32 (2017)CrossRefGoogle Scholar
  4. 4.
    Altenbach, H., Eremeyev, V.A.: Thin-walled structural elements: classification, classical and advanced theories, new applications. In: Altenbach, H., Eremeyev, V.A. (eds.) Shell-like Structures: Advanced Theories and Applications, CISM International Centre for Mechanical Sciences, vol. 572, pp. 1–62. Springer International Publishing, Berlin (2017)CrossRefGoogle Scholar
  5. 5.
    Altenbach, H., Eremeyev, V.A.: Bending of a three-layered plate with surface stresses. In: Altenbach, H., Carrera, E., Kulikov, G. (eds.) Analysis and Modelling of Advanced Structures and Smart Systems. Advanced Structured Materials, vol. 81, pp. 1–10. Springer Nature, Singapore (2018)CrossRefGoogle Scholar
  6. 6.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Linear theory of shells taking into account surface stresses. Dokl. Phys. 54(12), 531 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    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(3), 331–342 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    Altenbach, H., Belyaev, A., Eremeyev, V.A., Krivtsov, A., Porubov, A.V. (eds.): Dynamical Processes in Generalized Continua and Structures. Advanced Structured Materials, vol. 103. Springer, Cham (2019)Google Scholar
  9. 9.
    Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych. Wydawnictwo IPPT PAN, Warszawa, Nieliniowa teoria i metoda elementów skończonych (in Polish) (2004)Google Scholar
  11. 11.
    Duan, H.L, Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Advances in Applied Mechanic, vol. 42, pp. 1–68. Elsevier, Amsterdam (2008)Google Scholar
  12. 12.
    Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech. 227(1), 29–42 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Eremeyev, V.A., Altenbach, H.: Basics of mechanics of micropolar shells. In: Altenbach, H., Eremeyev, V.A. (eds.) Shell-like Structures: Advanced Theories and Applications, CISM International Centre for Mechanical Sciences, vol. 572, pp. 63–111. Springer, Berlin (2017)CrossRefGoogle Scholar
  14. 14.
    Eremeyev, V.A., Altenbach, H., Morozov, N.F.: The influence of surface tension on the effective stiffness of nanosize plates. Dokl. Phys. 54(2), 98–100 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Eremeyev, V.A., Rosi, G., Naili, S.: Surface/interfacial anti-plane waves in solids with surface energy. Mech. Res. Commun. 74, 8–13 (2016)CrossRefGoogle Scholar
  16. 16.
    Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018)CrossRefGoogle Scholar
  17. 17.
    Eremeyev, V.A., Rosi, G., Naili, S.: Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses. Math. Mech. Solids (2018). Scholar
  18. 18.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57(4), 291–323 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)CrossRefGoogle Scholar
  20. 20.
    Han, Z., Mogilevskaya, S.G., Schillinger, D.: Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann-Ogden interfaces. Int. J. Solids Struct. 147, 166–182 (2018)CrossRefGoogle Scholar
  21. 21.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65(1), 010802 (2013)CrossRefGoogle Scholar
  23. 23.
    Laplace, P.S.: Sur l’action capillaire. supplément à la théorie de l’action capillaire. In: Traité de mécanique céleste, vol. 4. Supplement 1, Livre X, Gauthier–Villars et fils, Paris, pp. 771–777 (1805)Google Scholar
  24. 24.
    Laplace, P.S.: À la théorie de l’action capillaire. supplément à la théorie de l’action capillaire. In: Traité de mécanique céleste, vol. 4. Supplement 2, Livre X, Gauthier–Villars et fils, Paris, pp. 909–945 (1806)Google Scholar
  25. 25.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)CrossRefGoogle Scholar
  26. 26.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  27. 27.
    Longley, W.R., van Name, R.G. (eds.): The Collected Works of J. Willard Gibbs, PHD., LL.D., vol I Thermodynamics. Longmans, New York (1928)Google Scholar
  28. 28.
    Lurie, A.I.: Nonlinear Theory of Elasticity (translated by K.A. Lurie). North-Holland Series in Applied Mathematics and Mechanics, vol. 36. North-Holland, Amsterdam (1990)Google Scholar
  29. 29.
    Lurie, A.I.: Theory of Elasticity (translated by A.K. Belyaev). Foundations of Engineering Mechanics. Springer, Berlin-Heidelberg (2005)Google Scholar
  30. 30.
    Naumenko, K., Altenbach, H.: Modelling of Creep for Structural Analysis. Foundations of Engineering Mechanics. Springer, Berlin (2007)CrossRefGoogle Scholar
  31. 31.
    Nazarenko, L., Stolarski, H., Altenbach, H.: Effective properties of short-fiber composites with Gurtin-Murdoch model of interphase. Int. J. Solids Struct. 97, 75–88 (2016)CrossRefGoogle Scholar
  32. 32.
    Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Eng. Sci. 49(10), 1112–1124 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Poisson, S.D.: Nouvelle théorie de l’action capillaire. Bachelier Père et Fils, Paris (1831)CrossRefGoogle Scholar
  34. 34.
    Ru, C.Q.: A strain-consistent elastic plate model with surface elasticity. Contin. Mech. Thermodyn. 28(1–2), 263–273 (2016)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. A 455(1982), 437–474 (1999)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    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 Mech. Solida Sinea 24, 52–82 (2011)CrossRefGoogle Scholar
  38. 38.
    Young, T.: An essay on the cohesion of fluids. Philos. Trans. R. Soc. Lond. 95, 65–87 (1805)ADSCrossRefGoogle Scholar
  39. 39.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für MechanikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland
  3. 3.Southern Federal UniversityRostov on DonRussia
  4. 4.Southern Scientific Center of RASciRostov on DonRussia

Personalised recommendations