Application of Variational Iterations Method for Studying Physically and Geometrically Nonlinear Kirchhoff Nanoplates: A Mathematical Justification

Abstract

We have proposed a development of the variational iteration method (VIM), or extended Kantorovich method, by studying physically nonlinear (FN) or geometrically nonlinear (GN) Kirchhoff nanoplates as an example. The modified couple stress theory was used for modeling size-dependent factors of the Kirchhoff nanoplates. Nested one into the other iteration procedures of the Birger method of variable elasticity parameters, of the variational iteration method (VIM), and of the Newton–Raphson method for physically nonlinear (FN) Kirchhoff nanoplates were constructed. The solution of problems for geometrically nonlinear (GN) Kirchhoff nanoplates was carried out on the basis of the variational iteration method and the Newton–Raphson method. The validity of the results was ensured by the coincidence of the results obtained via several methods of reducing partial differential equations to ordinary differential equations and via the finite difference method. The computational effectiveness of the proposed iterative procedure was demonstrated in terms of both accuracy and performance. A comparison of the results obtained showed that the variational iteration method (VIM) is the most efficient and fastest of all the methods considered both for problems with physical nonlinearity and for geometrically nonlinear problems.

Keywords

• modified couple stress theory
• geometric nonlinearity
• physically nonlinearity
• method of variable elasticity parameters
• variational iteration method
• elastic–plastic bending nanoplates