Geodesic
Postbuckling of a uniformly compressed simply supported plate with free in-plane translating edges
Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 1, pp. 143-154.

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The postbuckling of a Kirchhoff isotropic simply supported plate is considered in detail. The in-plane displacements on the edges of the plate are not constrained. The solution is obtained using the principle of the total potential energy stationarity. The expression for energy is written in the three versions: in terms of the Biot strain, the Cauchy–Green strain, and the strain corresponding to Föppl–von Kármán plate theory. Some approximate solution is constructed by the classical Ritz method. The basis functions are taken in the form of Legendre polynomials and their linear combinations. The obtained equilibrium path is rather similar to the classical equilibrium path of compressed shells. We show the failure of Föppl–von Kármán theory under large deflections. Using the Biot strain and the Cauchy–Green strain leads to the discrepancy between the results of at most 5%. We demonstrate the high accuracy and convergence of the approximate solution.
Keywords: postbuckling, plate, limit point, Biot strain, Cauchy–Green strain. 
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V. B. Myntiuk. Postbuckling of a uniformly compressed simply supported plate with free in-plane translating edges. Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 1, pp. 143-154. http://geodesic.mathdoc.fr/item/SJIM_2020_23_1_a11/

[1] Föppl A., Vorlesungen über Technische Mechanik, v. 5, B. G. Teubner, Leipzig, 1907 | Zbl

[2] T. Von Kármán, “Festigkeitsproblem im Maschinenbau”, Encyk. D. Math. Wiss, 4 (1910), 311–385

[3] P. G. Ciarlet, Plates and Junctions in Elastic Multi-Structures, Springer-Verl, Paris–Masson–Berlin, 1990 | MR | Zbl

[4] C. Y. Chia, Nonlinear Analysis of Plates, Mcgraw-Hill, N.Y., 1980

[5] K. M. Liew, J. Wang, M. J. Tan, S. Rajendran, “Postbuckling analysis of laminated composite plates”, Comput. Methods Appl. Mech. Eng., 195 (2006), 551–570 | DOI | Zbl

[6] O. O. Ochoa, J. N. Reddy, Finite Element Analysis of Composite Laminates, Kluwer Acad. Publ., Dordrecht, 1992

[7] L. C. Shiau, T. Y. Wu, “Application of the finite element method to postbuckling analysis of laminated plates”, AIAA J., 33:12 (1995), 2379–2385 | DOI | Zbl

[8] T. Le-Manh, J. Lee, “Postbuckling of laminated composite plates using NURBS-based isogeometric analysis”, Compos. Struct., 109:1 (2014), 286–293 | DOI

[9] S. A. Khalilov, V. B. Mintyuk, “Postbuckling analysis of flexible elastic frames”, J. Appl. Indust. Math., 12:1 (2018), 28–39 | DOI | MR

[10] W. Z. Chien, “The intrinsic theory of thin shells and plates. I. General theory”, Quart. Appl. Math., 1:4 (1944), 297–327 | DOI | MR

[11] V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, N.Y., 1953 | MR | Zbl

[12] Kh. M. Mushtari, K. Z. Galimov, Nonlinear Theory of Elastic Shells, Tatknigoizdat, Kazan, 1957 (in Russian)

[13] W. T. Koiter, “A consistent first approximation in the general theory of thin elastic shells”, Proc. 1 IUTAM Symposium on Shell Theory (Delft, 1959), Amsterdam, 1959, 12–33 | MR

[14] Sanders J. L. Jr., “Nonlinear theories for thin shells”, Quart. Appl. Math., 21:1 (1961), 21–36 | DOI | MR

[15] V. V. Novozhilov, K. F. Chernykh, “On the “true” measures of stresses and deformations in nonlinear mechanics of a deformable body”, Izv. Vyssh. Uchebn. Zaved. Mekh. Tverd. Tela, 1987, no. 4, 191–197

[16] H. Bufler, “The Biot stresses in nonlinear elasticity and the associated generalized variational principles”, Ing. Arch., 55 (1985), 450–462 | DOI | Zbl

[17] A. H. Nayfeh, P. F. Pai, Linear and Nonlinear Structural Mechanics, Wiley, N.Y., 2004 | MR | Zbl

[18] N. A. Alumáe, “On the Presentation of Fundamental Correlations of the Nonlinear Theory of Shells”, Prikl. Mat. Mekh., 20:1 (1956), 136–139 | MR

[19] S. N. Atluri, “Alternate stress and conjugate strain measures and mixed variational formulations involving rigid rotations for computational analyses of finitely deformed solids with application to plates and shells”, Comput. Structures, 18:1 (1983), 93–116 | DOI | MR

[20] W. Pietraszkiewicz, “Geometrically nonlinear theories of thin elastic shells”, Adv. Mech., 12:1 (1989), 51–130 | MR

[21] V. B. Myntiuk, “Biot stress and strain in thin-plate theory for large deformations”, J. Appl. Indust. Math., 12:3 (2018), 501–509 | DOI | MR | Zbl

[22] P. F. Pai, A. H. Nayfeh, “A new method for the modeling of geometric nonlinearities in structures”, Comput. Structures, 53:4 (1994), 877–895 | DOI | Zbl