Local bifurcation analysis of circular von-K\'arm\'an plate with Kirchhoff rod boundary

TL;DR

This paper performs a detailed local bifurcation analysis of a circular von-Kármán plate with Kirchhoff rod boundary conditions, combining symmetry reduction, theoretical bifurcation curve derivation, and numerical validation to understand buckling and post-buckling behavior.

Contribution

It introduces a symmetry-based reduction approach and Lyapunov Schmidt analysis for bifurcation in a nonlinear elastic plate-rod system, validated by finite element simulations.

Findings

Critical points are confirmed as bifurcation points.

Bifurcation nature changes under tension.

Theoretical bifurcation curves match numerical results near critical points.

Abstract

Symmetry based reduction is applied to the buckling of a circular von-Karman plate with Kirchhoff rod boundary, where a mismatch between the edge length and the perimeter of plate is treated as the bifurcation parameter. A nonlinear operator formulation describes the equilibrium of the elastic rod plate system. The critical points, as potential bifurcation points, are stated using the linearized operator. The symmetry of null space for each critical point is identified as a subgroup of the complete symmetry group of nonlinear problem, the equivariance associated with the nonlinear operator is used in this process. Sufficient evidence is provided for each critical point to be a bifurcation point for the symmetry reduced problem and post buckling analysis is carried out using Lyapunov Schmidt reduction. Bifurcation curves are obtained till quadratic order in bifurcation parameter away from each critical value. Theoretical results for bifurcation curves are validated against the numerical simulation based on a symmetry reduced finite element method for some illustrative examples of critical points. A numerical study is carried out for the dependence of the coefficient of quadratic term in the bifurcation parameter when structural parameters are varied in a neighborhood of four fixed sets of structural parameters. Numerical results based on a symmetry reduced finite element analysis confirm that the nonlinear solution agrees with the local theoretical behavior close to a critical point but deviates further away from it. Using these tools, two main conclusions are reached. First it is observed that the critical points of the linearized problem are indeed bifurcation points. Second, an alteration in the nature of bifurcation is observed during the parameter sweep study when the plate is in tension.