Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure --- Equations in Differential Form

TL;DR

This paper applies the homotopy analysis method (HAM) to find convergent series solutions for the large deflection of a circular thin plate under uniform pressure, demonstrating HAM's advantages over traditional perturbation methods in nonlinear cases.

Contribution

The paper introduces the homotopy analysis method (HAM) as a new analytic approach for solving highly nonlinear Von Kármán plate equations under arbitrary boundary conditions.

Findings

HAM provides convergent solutions even for high deflections ($w(0)/h>20$).

HAM outperforms traditional perturbation methods in nonlinear regimes.

All perturbation methods are special cases of HAM.

Abstract

The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von K{\'a}rm{\'a}n \cite{Karman}. {This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM).} Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von K{\'a}rm{\'a}n's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problems