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.