Skip to main content
SpringerLink
Log in

Two-dimensional linear models of multilayered anisotropic plates

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

Abstract

A two-dimensional model describing the multilayered anisotropic plate deformations is proposed. The plate is assumed to consist of some orthotropic layers with arbitrary orientation of axes relative to the plate frame. The studied multilayered plate is replaced by the equivalent plate composed of a monoclinic material with piecewise elastic modules. An asymptotic solution is constructed for long-wave deformations. This problem was solved earlier in the first approximation; however, the obtained solution is not applicable for the case in which the stiffness of layers differs essentially from each other. The second asymptotic approximation is constructed in the present paper. It takes into account the effects of transversal shear and the normal fibers extension. Some special cases resulting in simple equations are studied in detail. The asymptotic solution error is estimated by comparison with the exact three-dimensional solutions for some test examples.

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 uber Mathematische Physik. Mechanik, Leipzig (1876). (in German)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Donnell, L.H.: Beams, Plates and Shells. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  5. Novozhilov, V.: Theory of Thin Shells. Wolters-Noordhoff, Groningen (1970)

    Google Scholar 

  6. Goldenweizer, A.L.: Theory of Elastic Thin Shells. Nauka, Moscow (1976). (in Russian)

    Google Scholar 

  7. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 4(242). Ser. 6. (1921)

  8. 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 

  9. Vekua, I.N.: On one method of calculating prismatic shells. Trudy Tbilis. Mat. Inst. 21, 191–259 (1955). (in Russian)

    Google Scholar 

  10. Chernykh, K.F., Rodionova, V.A., Titaev, B.F.: Applied Theory of Anisotropic Plates and Shells. St. Petersburg Univ. Press, St. Petersburg (1996). (in Russian)

    Google Scholar 

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

    Google Scholar 

  12. Agolovyan, L.A.: Asymptotic Theory of Anisotropic Plates and Shells, p. 414. Nauka, Moscow (1997). (in Russian)

    Google Scholar 

  13. 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 

  14. Vetyukov, Y., Kuzin, A., Krommer, M.: Asymptotic splitting of the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates. Int. J. Solids Struct. 40, 12–23 (2011)

    Article  MATH  Google Scholar 

  15. Schnieder, P., Kienzler, R.: An algorithm for the automatisation of pseudo reductions of PDE systems arising from the uniform-approximation technique. In: Altenbach, H., Eremeyev, V.A. (eds.) Shell-like structures. Non-classical theories and applications, pp. 377–390. Springer, Berlin (2011)

    Chapter  Google Scholar 

  16. Schneider, P., Kienzler, R.: A Reissner-type plate theory for monoclinic material derived by extending the uniform-approximation technique by orthogonal tensor decompositions of nth-order gradients. Meccanica 52, 2143–2167 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Tovstik, P.E.: On the asymptotic character of the approximate models of beams, plates and shells. Vestn. St. Petersb. Univ. Math. 40(3), 188–192 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhilin, P.A.: On the Poisson and Kirchhoff plate theory from the point of view of the modern plate theory. Izv. RAS Mech. Solids 3, 48–64 (1992). (in Russian)

    Google Scholar 

  20. Andrianov, I.V., Danishevs’kyy, V.V., Weichert, D.: Boundary layers in fibrous composite materials. Acta Mech. 216, 3–15 (2011)

    Article  MATH  Google Scholar 

  21. Tovstik, P.E., Tovstik, T.P.: Two-dimensional models of shells made of an anisotropic material. Acta Mech. 225(3), 647–661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tovstik, P.E., Tovstik, T.P.: On the two-dimensional models of plates and shells including the transversal shear. ZAMM 87(2), 160–171 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Morozov, N.F., Tovstik, P.E., Tovstik, T.P.: Generalized Timoshenko–Reissner model for a multi-layer plate. Mech. Solids 51(5), 527–537 (2016)

    Article  Google Scholar 

  24. Tovstik, P.E., Tovstik, T.P.: Generalized Timoshenko–Reissner models for beams and plates, strongly heterogeneous in the thickness direction. ZAMM 97(3), 296–308 (2017)

    Article  MathSciNet  Google Scholar 

  25. Tovstik, P.E., Tovstik, T.P.: An elastic plate bending equation of second-order accuracy. Acta Mech. 228(10), 3403–3419 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Morozov, N.F., Tovstik, P.E., Tovstik, T.P.: Generalized Timoshenko–Reissner model for a multilayer plate. Mech. Solids 51(5), 527–537 (2016)

    Article  Google Scholar 

  27. Morozov, N.F., Tovstik, P.E., Tovstik, T.P.: A continuum model of a multi-layer nanosheet. Dokl. Phys. 61(11), 567–570 (2016)

    Article  Google Scholar 

  28. Morozov, N.F., Tovstik, P.E., Tovstik, T.P.: Free vibrations of a transversely isotropic plate with application to a multilayer nano-plate. In: Altenbach, H., Goldstein, R., Murashkin, E. (eds.) Mechanics for Materials and Technologies. Advanced Structured Materials, vol 46. Springer, Cham (2017)

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

    Article  Google Scholar 

  30. Tovstik, P.E., Tovstik, T.P.: Equations of equilibrium for a strongly heterogeneous shallow shell. Dokl. Phys. 62(11), 522–526 (2017)

    Article  MATH  Google Scholar 

  31. Grigolyuk, E.I., Kulikov, G.M.: Multilayer Reinforced Shells: Calculation of Pneumatic Tires. Mashinostroenie, Moscow (1988). (in Russian)

    Google Scholar 

  32. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells, p. 831. CRC Press, Boca Raton (2004)

    Book  MATH  Google Scholar 

  33. 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 

  34. Reddy, J.N.: A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20, 881–896 (1994)

    Article  MATH  Google Scholar 

  35. Birman, V.: Plate structures. In: Solid Mechanics and Its Applications, vol. 178 (2011). https://doi.org/10.1007/978-94-007-1715-2

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

    Google Scholar 

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

    Article  Google Scholar 

  38. Altenbach, H.: Theories for laminated and sandwich plates. A review. Mech. Compos. Mater. 34, 243–252 (1998)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. 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 

  41. Mikhasev, G., Botogova, M.: Effect of edge shears and diaphragms on buckling of thin laminated medium-length cylindrical shells with low effective shear modulus under external pressure. Acta Mech. 228, 2119–2140 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tovstik, P.E., Tovstik, T.P.: Two-dimensional model of a plate made of an anisotropic inhomogeneous material. Mech. Solids 52(2), 144–154 (2017)

    Article  MATH  Google Scholar 

  43. Tovstik, P.E., Tovstik, T.P., Naumova, N.V.: Long-wave oscillations and waves in anisotropic beams. Vestn. St. Petersb. Univ. Math. 50(2), 198–207 (2017)

    Article  MATH  Google Scholar 

  44. Morozov, N.F., Belyaev, A.K., Tovstik, P.E., Tovstik, T.P.: Two-dimensional equations of second order accuracy for a multilayered plate with orthotropic layers. Dokl. Phys. 63(11), 471–475 (2018)

    Article  Google Scholar 

  45. Parshina, L.V., Ryabov, V.M., Yartsev, B.A.: Energy dissipation during vibrations of non-uniform composite structures. 1. Formulation of the problem. 2. Method of solution. 3. Numerical experiment. Vestn. St. Petersb. Univ. Math. vol. 51, No. 2, No. 3, No. 4 (2018)

Download references

Acknowledgements

The study is supported by the Russian Foundation of Basic Researches, Grants 19.01.00208-a and 16.51.52025-a.

Author information

Authors and Affiliations

  1. St. Perersburg State University, Universitetskaya nab. 7-9, St. Petersburg, Russia, 199034

    A. K. Belyaev, N. F. Morozov & P. E. Tovstik

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg V.O., Bolshoj pr., 61, St. Petersburg, Russia, 199178

    A. K. Belyaev, N. F. Morozov, P. E. Tovstik & T. P. Tovstik

Authors
  1. A. K. Belyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. F. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. E. Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. T. P. Tovstik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. P. Tovstik.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belyaev, A.K., Morozov, N.F., Tovstik, P.E. et al. Two-dimensional linear models of multilayered anisotropic plates. Acta Mech 230, 2891–2904 (2019). https://doi.org/10.1007/s00707-019-02405-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-019-02405-y

Search

Navigation