Theory of thin elastic plate: history and current state of the problem

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Abstract

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

About the authors

V. V. Vasil'ev

Central Research Institute of Special Machine Building

Author for correspondence.
Email: vvvas@dol.ru
Russian Federation, Hotkovo

References

  1. Vasil'ev V.V. Mechanics of structures made of composite materials. M.: Mashinostroenie, 1988. 270 pp. [in Russian]
  2. Todhunter L., Pearson K. A history of the theory of elasticity and the strength of materials. N.-Y.: Dover, 1960. Pt. 1. 762 p. Pt. 2. 546 p.
  3. Kirhgof G. Mechanics. M: Publ. of the USSR Academy of Sciences, 1962. 402 pp. [in Russian]
  4. Thomson U., Tait P. Treatise on Natural Philosophy. M., Izhevsk: SIC “Regular and Chaotic Dynamics” Izhevsk Institute of Computer Research, 2011. Part 2. 560 p. [in Russian]
  5. Vasil'ev V.V. On Kirchhoff and Thomson-Tait transformations in classical plate theory // RAS bulletin. Sol. Mech. 2012. № 5. P. 98–107. [in Russian]
  6. Carrera E., Elishakoff I., Petrolo M. Who needs refined structural theories? // Compos. Struct. 2021. V. 264. P. 1–16. https://doi.org/10.1016/j.compstruct.2021.113671
  7. Vasil'ev V.V. On the theory of thin plates // RAS bulletin. Sol. Mech. 1992. № 3. P. 26–47. [in Russian]
  8. Vasil'ev V.V. Classical plate theory – history and modern analysis // RAS bulletin. Sol. Mech. 1998. № 3. P. 46–58. [in Russian]
  9. Vasiliev V.V. Modern conceptions of plate theory // Compos. Struct. 2000. V. 48. № 1–3. P. 39–48. https://doi.org/10.1016/S0263-8223(99)00071-9
  10. Hencky H. Uber die Berucksichtigung der Schubverzerrung in ebenen Platten // Ing.-Archiv. 1947. V. 16. P. 72–76. https://doi.org/10.1007/BF00534518
  11. Bolle L. Contribution au problem lineaire de flexion d’une plaque elastique // Bull. Tech. Suisse Romander. 1947. V. 11. P. 32.
  12. Vasil'ev V.V., Lur'e S.A. On the problem of constructing non-classical theories of plates // RAS bulletin. Sol. Mech. 1990. № 2. P. 158–167. [in Russian]
  13. Vasiliev V.V., Lurie S.A. On refined theories of beams, plates and shells // J. Compos. Mater. 1992. V. 26. № 4. P. 546–557. https://doi.org/10.1177/002199839202600405
  14. Zhilin P.A. On classical plate theory and the Kelvin-Tait transformation // RAS bulletin. Sol. Mech. 1995. № 4. P. 134–140. [in Russian]
  15. Sheremet'ev M.P., Peleh B.L., Dyachina O.P. Investigation of the influence of shear deformation on the bending of a square plate by a concentrated force // Applied Mechanics. 1968. Vol. 4. Iss. 4. P. 1–7. [in Russian]
  16. Timoshenko S.P., Voinovskii-Kriger S. Plates and shells. M.: Fizmatgiz, 1963. 635 pp. [in Russian]
  17. Alfutov N.A. On some paradoxes of the theory of thin plates // RAS bulletin. Sol. Mech. 1992. № 3. P. 65–72. [in Russian]
  18. Vasil'ev V.V. Torsion of a square isotropic plate by angular forces and distributed moments // RAS bulletin. Sol. Mech. 2017. № 2. P. 20–31. [in Russian]
  19. Timoshenko S.P. History of the strength of materials science. M.: URSS, 2001. 536 c.
  20. Nadai A. Die elastischen Platten. Berlin.: Verlag von Julins Springer, 1925. 125 p.
  21. Methods of static tests of reinforced plastics. Reference manual ed. by Y.M. Tarnopolskiy. Riga: Zinatne, 1972. 227 pp. [in Russian]
  22. Jamielita G. On the winding paths of the theory of plates // Mechanika Teoretyczna i Stosowana. J. Theor. Appl. Mech. 1993. V. 31. № 2. P. 312–327.
  23. Timoshenko S.P. On the correction for shear of the differential equation for transverse vibrations of prismatic bars // Phil. Mag. 1921. V. 41. № 245. P. 744–746. https://doi.org/10.1080/14786442108636264
  24. Elishakoff I. Handbook on Timoshenko-Ehrenfest beam and Uflyand-Mindlin plate theories. World scientific publ. Co. 2020. 769 p.
  25. Uflyand Ya.S. Wave propagation in transverse vibrations of rods and plates // J. App. Math. Mech. 1948. Vol. 12. Iss. 8. P. 287–300. [in Russian]
  26. Mindlin R.D. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates //J. Appl. Mech. 1951. V. 18(1). P. 31–38. https://doi.org/10.1115/1.4010217
  27. Reissner E. On the theory of bending of elastic plates // J. Math. Phys. 1944. V. 23. № 4. P. 184–191.
  28. Reissner E. The effect of transverse shear deformation on bending of elastic plates // Trans. ASME. 1945. V. 15. P. A69-A77.
  29. Voloh K.Ju. On the classical theory of plates // J. App. Math. Mech. 1994. Vol. 58. Iss. 6. P. 156–165. [in Russian]
  30. Darevskii V.M. On static boundary conditions in the classical theory of plates and shells // RAS bulletin. Sol. Mech. 1995. № 4. P. 129–132. [in Russian]
  31. Gol'denveizer A.L. On approximate methods of calculation of thin elastic shells and plates // RAS bulletin. Sol. Mech. 1997. № 3. P. 134–149. [in Russian]
  32. Gol'denveizer A.L. Algorithms for asymptotic construction of linear two-dimensional thin shell theory and Saint-Venant's principle // J. App. Math. Mech. 1994. Vol. 58. Iss. 6. P. 96–108. [in Russian]
  33. Zhilin P.A. On the Poisson and Kirchhoff plate theories in terms of modern plate theory // RAS bulletin. Sol. Mech. 1992. № 3. P. 48–64. [in Russian]
  34. Gol'denveizer A.L. On the bending theory of Reissner plates // Proceedings of the USSR Academy of Sciences. Dept. of Technical Sciences. 1958. № 4. P. 99–109. [in Russian]
  35. Vasil'ev V.V. On the asymptotic justification method of the theory of plates // RAS bulletin. Sol. Mech. 1997. № 3. P. 150–155. [in Russian]
  36. Gol'denveizer A.L. Note on the article by V.V. Vasiliev “On the asymptotic method of justification of the theory of plates” // RAS bulletin. Sol. Mech. 1997. № 4. P. 150–158. [in Russian]
  37. Slivker V.I. Structural mechanics: variation bases. Textbook. M: Publ. of the Association of Civil Engineering Universities, 2005. 736 p. [in Russian]

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