Post-buckling Solutions of Hyper-elastic Beam by Canonical Dual Finite Element Method

TL;DR

This paper introduces a canonical dual finite element method for analyzing post-buckling behavior of large deformed beams, enabling the identification of all possible solutions including stable and unstable configurations.

Contribution

The paper develops a novel primal-dual algorithm based on canonical duality theory to solve non-convex post-buckling problems and explicitly formulates a pure complementary energy in finite dimensions.

Findings

Global maximum corresponds to stable buckled state

Unstable states are highly sensitive to element number and loads

Numerical results confirm theoretical predictions and reveal post-bifurcation phenomena

Abstract

Post buckling problem of a large deformed beam is analyzed using canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal-dual algorithm is proposed, which can be used to find all possible solutions of this nonconvex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam. While the local extrema of the pure-complementary energy present unstable deformation states, especially. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.