Elastic-plastic deformation of nanoplates. The method of~variational iterations (extended Kantorovich method)

A. D. Tebyakin ; A. V. Krysko ; M. V. Zhigalov ; V. A. Krysko

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 4, pp. 494-505.

Résumé

In this paper, a mathematical model is constructed based on the deformation theory of plasticity for studying the stress-strain state of Kirchhoff nanoplates (nanoeffects are taken into account according to the modified moment theory of elasticity). An economical and correct iterative method for calculating the stress-strain state of nanoplates has been developed  — the method of variational iterations (the extended Kantorovich method). The method of variational iterations (the extended Kantorovich method) has the advantage over the Bubnov – Galerkin or Ritz method in that it does not require specifying a system of approximating functions satisfying boundary conditions, because the method of variational iterations builds a system of approximating functions at each iteration, which follows from solving an ordinary differential equation after applying the Kantorovich procedure. The correctness of the method is ensured by the convergence theorems of the method of variable elasticity parameters by I. I. Vorovich, Yu. P. Krasovsky and the convergence theorems of the method of variational iterations by V. A. Krysko, V. F. Kirichenko. In addition, the reliability of the solutions for elastic Kirchhoff nanoplates obtained using the variational iteration method is ensured by comparison with the exact Navier solution and solutions using Bubnov – Galerkin methods in higher approximations, finite differences and finite elements. The developed method and the methodology for calculating elastic-plastic deformation of Kirchhoff nanoplates, which is based on this method, are effective in terms of machine time costs compared with the methods of Bubnov – Galerkin in higher approximations, finite differences, Kantorovich – Vlasov, Weindiner and especially finite elements. The influence of the nano coefficient, the types of dependences of strain intensity (stress intensity on the elastic-plastic behavior of the nanoplates) has been studied.

Bibliographie

[1] Kiener D., Motz C., Schobert T., Jenko M., Dehm G., “Determination of mechanical properties of copper at the micron scale”, Advanced Engineering Materials, 8:11 (2006), 1119–1125 | DOI

[2] Huber N., Nix W., Gao H., “Identification of elastic-plastic material parameters from pyramidal indentation of thin films”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 458 (2002), 1593–1620 | DOI

[3] Ristinmaa M., Vecchi M., “Use of couple-stress theory in elasto-plasticity”, Computer Methods in Applied Mechanics and Engineering, 136:3–4 (1996), 205–224 | DOI | MR

[4] Darvishvand A., Zajkani A., “Strain gradient micromechanical modeling of substrate- supported crystalline microplates subjected to permanent in-plane and out-of-plane tractions”, Mechanics Based Design of Structures and Machines, 2021, no. 7, 969–985 | DOI

[5] Darvishvand A., Zajkani A., “Comparative modeling of power hardening micro-scale metallic plates based on lower and higher-order strain gradient plasticity theories”, Metals and Materials International, 27:6 (2021), 1392–1402 | DOI

[6] Tun M. A., “Plastic buckling of moderately thick annular plates”, International Journal of Structural Stability and Dynamics, 5:3 (2005), 337–357 | DOI

[7] Lanzoni L., Radi E., Nobili A., “Ultimate carrying capacity of elastic-plastic plates on a Pasternak foundation”, Journal of Applied Mechanics, 81:5 (2014), 051013, 9 pp. | DOI

[8] Schunck T. E., “Zur Knickfestigkeit schwach gekrümmter zylindrischer Schalen”, Ingenieur-Archiv, 4 (1933), 394–414 (in German) | DOI

[9] Zhukov E. E., “Variational method of successive approximations to the calculation of thin rectangular plates”, Calculation of Thin-Walled Spatial Structures, ed. A. P. Rzhanitsyn, Stroyizdat, M., 1964, 27–35 (in Russian)

[10] Kerr A. D., “An extended Kantorovich method for solution of eigenvalue problem”, International Journal of Solids and Structures, 5:6 (1969), 559–572 | DOI

[11] Yuan S., Jin Y., “Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method”, Composite Structures, 66:6 (1998), 861–867 | DOI

[12] Shufrin I., Rabinovitch O., Eisenberger M., “Buckling of symmetrically laminated rectangular plates with general boundary conditions — a semi analytical approach”, Composite Structures, 82:4 (2008), 521–537 | DOI

[13] Eisenberger M., Shufrin I., “The extended Kantorovich method for vibration analysis of plates”, Analysis and Design of Plated Structures, v. 1, Stability, eds. N. E. Shanmugam, Chieng Ming Wang, Woodhead Publishing, Cambridge, 2007, 192–218 | DOI

[14] Awrejcewicz J., Krysko-jr. V. A., Kalutsky L. A., Zhigalov M. V., Krysko V. A., “Review of the methods of transition from partial to ordinary differential equations: from macro- to nano-structural dynamics”, Archives of Computational Methods in Engineering, 28 (2021), 4781–4813 | DOI | MR

[15] Kirichenko V. F., Krysko V. A., “Method of variational iterations in the theory of plates and its substantiation”, Applied Mechanics, 17:4 (1981), 71–76 (in Russian) | MR

[16] Krysko A. V., Awrejcewicz J., Zhigalov M. V., Krysko V. A., “On the contact interaction between two rectangular plates”, Nonlinear Dynamics, 85 (2016), 2729–2748 | DOI

[17] Awrejcewicz J., Krysko V. A., Zhigalov M. V., Krysko A. V., “Contact interaction of two rectangular plates made from different materials with an account of physical non-linearity”, Nonlinear Dynamics, 91 (2018), 1191–1211 | DOI

[18] Awrejcewicz J., Krysko-jr V. A., Kalutsky L. A., Krysko V. A., “Computing static behavior of flexible rectangular von Karman plates in fast and reliable way”, International Journal of Non-Linear Mechanics, 146 (2022), 104162 | DOI

[19] Awrejcewicz J., Krysko A. V., Smirnov A., Kalutsky L. A., Zhigalov M. V., Krysko V. A., “Mathematical modeling and methods of analysis of generalized functionally gradient porous nanobeams and nanoplates subjected to temperature field”, Meccanica, 57 (2022), 1591–1616 | DOI | MR

[20] Yang F., Chong A. C. M., Lam D. C. C., Tong P., “Couple stress based strain gradient theory for elasticity”, International Journal of Solids and Structures, 39:10 (2002), 2731–2743 | DOI

[21] Vorovich I. I., Krasovsky Yu. P., “On the method of elastic solutions”, Doklady Mathematics, 126:4 (1959), 740–743 (in Russian)

[22] Prandtl L., “Ein Gedankenmodell zur kinetischen Theorie der festen Körper”, ZAMM: Zeitschrift Für Angewandte Mathematik Und Mechanik, 8:2 (1928), 85–106 (in German) | DOI

[23] Ohashi Y., Murakami S., “The elasto-plastic bending of a clamped thin circular plate”, Applied Mechanics, v. 1, ed. H. Görtler, Springer, Berlin–Heidelberg, 1966, 212–223 | DOI

[24] Temperature coefficient of linear expansion of steel, Thermalinfo (in Russian) (accessed 3 August 2022)

[25] Birger I. A., “Some general methods for solving problems in the theory of plasticity”, Applied Mathematics and Mechanics, 15:6 (1951), 765–770 (in Russian) | MR

[26] Kantorovich L. V., Krylov V. I., Approximate Methods of Higher Analysis, Fizmatgiz, M., 1962, 708 pp. (in Russian) | MR

[27] Vlasov V. V., General Shell Theory, Gostekhizdat, M., 1949, 784 pp. (in Russian) | MR

[28] Weindiner A. I., “On a new form of Fourier series and the choice of the best Fourier polynomials”, Computational Mathematics and Mathematical Physics, 7:1 (1967), 240–251 | DOI | DOI | MR