Physical nonlinearity in porous functionally graded kirchhoff nano-plates: Modeling and numerical experiment

Abstract

In this paper, mathematical models of porous functionally graded inhomogeneous plates were constructed. The following hypotheses were used as the foundation: the kinematic Kirchhoff model was used; nano effects were considered by the modified couple stress theory; material properties depend on coordinates (x,y,z), and for metals, they also hinge on the stress-strain state at a point in the plate volume, i.e. there is a dependence σi(ei). This dependence can be any analytical function derived from experimentation, which means that it is possible to take into account elastic-plastic deformations according to the deformation theory of plasticity. The ceramic material is resilient. The required variational and differential equations and the boundary conditions were derived from the Lagrange functional. The plate material's porosity was characterised by six types. For numerical solutions, the methods of reduction of the partial differential equation to the ordinary differential equation were applied, as well as iterative methods nested one into another, based on the following techniques: variational iterations method (VIM), Agranovskii-Baglai-Smirnov method (ABSM) and Birger's method of variable parameters of elasticity. For each of these methods, there are theorems proving their convergence. Each problem that describes the stress-strain state of Kirchhoff nano-plates was solved using the following analytical methods: Exact Navier's solution (ENS), Bubnov-Galerkin method (BGM), Levy solution (LS), numerically analytical Kantorovich-Vlasov (KVM) methods, variational iterations method (VIM), Vaindiner method (VaM) and numerical finite difference method (FDM). The convergence of these methods was investigated and the optimal method for calculating such structures, both in terms of accuracy and speed, was determined for the elliptic 2D PDEs of arbitrary area Ω. Numerical results for both elastic nano-plates and elastoplastic nano-plates have been presented.