Skip to main content
SpringerLink
Log in

An elastic plate bending equation of second-order accuracy

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A study is carried out of a thin plate of constant thickness made of linearly elastic material which is transversally isotropic and heterogeneous in the thickness direction. Asymptotic expansions in powers of the relative plate thickness are constructed, and the bending equation of second-order accuracy (the SA model) is delivered. The results of the SA model are compared with the Kirchhoff–Love classical model and with the Timoshenko–Reissner (TR) model, as well as with the exact solution. To this end, some problems for a functionally gradient plate bending, and for a multi-layer plate bending and free vibration are solved and analysed. The range of plate heterogeneity, for which the error of the approximate models is small, is established. The TR model and the SA model are proved to yields results close to each other and the exact results for a very broad range of heterogeneity. That is why the generalized TR model for one-layered homogeneous transversely isotropic plate is proposed. Parameters of this model are chosen so that the results are close to the exact results and the results by the SA model. For the Navier boundary conditions, the analytical solution of 3D problems for a rectangular heterogeneous plate is constructed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kirchhoff, G.: Vorlesungen über mathematische Physik. Mechanik, Leipzig (1876). [in German]

    MATH  Google Scholar 

  2. Love, A.E.H.: A Treatise on the Mathematical Theory Elasticity. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

  3. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)

  4. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. Trans. ASME J. Appl. Mech. 12, 69–77 (1945)

    MathSciNet  MATH  Google Scholar 

  5. Chernykh, K.F., Rodionova, V.A., Titaev, B.F.: Applied Theory of Anisotropic Plates and Shells. St.Petersburg University Press, Saint Petersburg (1996). [in Russian]

    Google Scholar 

  6. Goldenweizer, A.L.: Theory of Elastic Thin Shells. Pergamon Press, Oxford (1961)

    Google Scholar 

  7. Tovstik, P.E., Tovstik, T.P.: A thin-plate bending equation of second-order accuracy. Dokl. Phys. 59(8), 389–392 (2014)

    Article  Google Scholar 

  8. Tovstik, P.E.: On the asymptotic character of approximate models of beams, plates and shells, Vestnik St.Petersburg Univ. Mathematics. Allerton Press, NewYork 3, 49–54 (2007)

  9. Vetyukov, Y., Kuzin, A., Krommer, M.: Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates. Int. J. Solids Struct. 48, 12–23 (2011)

    Article  MATH  Google Scholar 

  10. Eremeev, V.A., Zubov, L.M.: Mechanics of Elastic Shells. Nauka, Moscow (2008). [in Russian]

    Google Scholar 

  11. Altenbach, H., Mikhasev, G.I. (eds.): Shell and Membrane Theories in Mechanics and Biology. Springer, Berlin (2014)

    Google Scholar 

  12. Tovstik, P.E., Tovstik, T.P.: Two-dimensional model of anisotropic of shells. Shell structures: theory and Applications. Proceedings of the 10th SSTA 2013 Conference, 3, 153–156 (2014)

  13. Andrianov, I.V., Danishevskyy, V.V., Weichert, D.: Boundary layers in fibrous composite materials. Acta Mech. 216, 3–15 (2011)

    Article  MATH  Google Scholar 

  14. Tovstik, P.E., Tovstik, T.P.: On the 2D models of plates and shells including shear. ZAMM. 87(2), 160–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tovstik, P.E., Tovstik, T.P.: Two-dimensional linear model of elastic shell accounting for general anisotropy of material. Acta Mech. 225(3), 647–661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tovstik, P.E.: Two-dimensional models of plates made of an anisotropic material. Dokl. Phy. 54(4), 205–209 (2009)

    Article  MathSciNet  Google Scholar 

  17. Tovstik, P.E., Tovstik, T.P.: Two-dimensional model of plate made of anisotropic inhomogeneous material. Proceedings of the ICNAAM-2014. AIP Conf. Proc. 1648, art. no.300011 (2015)

  18. Birman, V.: Plate structures. Solid mechanics and its applications, vol. 178. Springer, Netherlands (2011)

  19. Grossi, R.O.: Boundary value problems for anisotropic plates with internal line hinges. Acta Mech. 223(1), 125–144 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Reddy, J.N.: A refined nonlinear theory of plates with transvere shear deformation. Int. J. Solids and Struct. 20, 881–896 (1894)

    Article  MATH  Google Scholar 

  21. Bauer, S., Voronkova, E.: Nonclassical theories of bending analysis of orthotropic circular plate. Shell structures theory and application Proceedings of the 10th SSTA 2013 Conference V. 3, 57–60 (2014)

  22. Reddy, J., Wang, C.: An overview of the relationship between the classical and shear deformation plate theories. Compos. Sci. Technol. 60, 2327–2335 (2000)

    Article  Google Scholar 

  23. Barretta, R.: Analogies between Kirchhoff plates and Saint-Venant beams under flexure. Acta Mech. 225(7), 2075–2083 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Batista, M.: An exact theory of the bending of transversely inextensible elastic plates. Acta Mech. 226(9), 2899–2924 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, G.Y., Gao, X.-L., Wang, J.Z.: A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects. Acta Mech. 226(12), 4073–4085 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Donnell, L.H.: Beams, Plates and Shells. McGraw-Hill, NewYork (1976)

    MATH  Google Scholar 

  27. Timoshenko, S.P.: Strength of Materials. Van Vistrand, New York (1956)

    MATH  Google Scholar 

  28. Ambartsumjan, S.A.: Theory of anisotropic shells. Progress in Materias Science. Ser. II. Stanford, Conn.: Technomic (1970)

  29. Jawad, M.H.: Theory and Design of Plate and Shell Structures. Springer, Berlin (1994)

    Book  Google Scholar 

  30. Altenbach, H.: Theories of laminated and sandwich plates. Overv. Mech. Compos. Mater. 34, 333–349 (1998)

    Google Scholar 

  31. Eremeev, V.A., Ivanova, E.A., Altenbach, H., Morozov, N.F.: On effective stiffness of a three-layered plate with a core filled with a capillary fluid. Shell structures theory and application Proceedings of the 10th SSTA 2013 Conference. Vol. 3, 85–88 (2014)

  32. Morozov, N.F., Tovstik, P.E.: Bending of two-layer beam with non-rigid contact between layers. Appl. Math. and Mech. 75, 77–84 (2011)

    Article  MATH  Google Scholar 

  33. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells. CRC, Boca Raton (2004)

    MATH  Google Scholar 

  34. Aghalovyan, L.A.: On the classes of problems for deformable one-layer and multilayer thin bodies solvable by the asymptotic method. Mech. Compos. Mater. 47(1), 59–72 (2011)

    Article  Google Scholar 

  35. Asnafi, A., Abedi, M.: A complete analogical study on the dynamic stability analysis of isotropic functionally graded plates subjected to lateral stochastic loads. Acta Mech. 226(7), 2347–2363 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tovstik, P.E., Tovstik, T.P.: Free vibrations of anisotropic beam. Vestnik St.Peterdurg Univ. Ser. 1,1(59), 4, 599–608 (2014)

  37. Goldenweizer, A.L., Lidsky, V.B., Tovstik, P.E.: Free Vibrations of Thin Elastic Shells. Nauka, Moscow (1979). [in Russian]

    Google Scholar 

  38. Tovstik, P.E.: Vibrations and stability of a prestressed plate on an elastic foundation. Appl. Math. Mech. 73(1), 77–87 (2009)

  39. Grossi, R.O., Raffo, J.: Natural vibrations of anisotropic plates with several internal line hinges. Acta Mech. 224(11), 2677–2697 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sarangi, S.K., Ray, M.C.: Active damping of geometrically nonlinear vibrations of laminated composite plates using vertically reinforced 1–3 piezoelectric composites. Acta Mech. 222(3–4), 363–380 (2011)

    Article  MATH  Google Scholar 

  41. Morozov, N.F., Tovstik, P.E.: On chessboard buckling modes in compressed materials. Acta Mech. 223, 1769–1776 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kienzler, R., Schneider, P.: Comparison of various linear plate theories in the light of a consistent second order approximation. Shell structures theory and application. Proceedings of the 10th SSTA 2013 Conference V. 3, 109–112 (2014)

  43. Tovstik, P.E., Tovstik, T.P.: Generalized Timoshenko-Reissner models for beams and plates, strongly heterogeneous in the thickness direction. ZAMM (2016). doi:10.1002/zamm.201600052

  44. Morozov N.F., Tovstik, P.E., Tovstik, T.P.: Generalized Timoshenko–Reissner model for a multilayer plate. Mech. Solids 51, 527–537 (2016)

  45. Morozov, N.F., Tovstik, P.E., Tovstik, T.P.: The Timoshenko-Reissner generalized model of a plate highly nonuniform in thickness. Dokl. Phys. 61(8), 394–398 (2016)

    Article  Google Scholar 

  46. Morozov N.F., Tovstik P.E., Tovstik T.P.: Continuum model of multilowered nano-plate. Dokl. Phys. 61(11), 567–570 (2016)

  47. Tovstik, P.E., Tovstik, T.M.: Bending stiffness of a multilayered plate // ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering. 3, Pages 3423–3435 (2016)

  48. Berdichevsky, V.L.: An asymptotic theory of sandwich plates. Int. J. of Eng. Sci. (2009). doi:10.1016/j.ij.engsci.2009.09.001

Download references

Acknowledgements

The work is supported by Russian Foundation for Basic Research, Grants 16-01-00580-a and 16-51-52025 MNT-a.

Author information

Authors and Affiliations

  1. Institute of Problems of Mechanical Engineering, St. Petersburg, Russia

    Petr Tovstik & Tatiana Tovstik

Authors
  1. Petr Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatiana Tovstik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tovstik, P., Tovstik, T. An elastic plate bending equation of second-order accuracy. Acta Mech 228, 3403–3419 (2017). https://doi.org/10.1007/s00707-017-1880-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1880-x

Search

Navigation