A group-theoretic approach to the bifurcation analysis of elastic frameworks with symmetry

TL;DR

This paper introduces a group-theoretic method for analyzing bifurcations in elastic frameworks with symmetry, using a linearized formulation of Cosserat rod assemblies to predict symmetry-breaking bifurcations.

Contribution

It develops a linear-space formulation for elastic frameworks with symmetry, enabling natural application of group-theoretic bifurcation analysis and handling multiple bifurcations with symmetry reduction techniques.

Findings

Successfully analyzed a tripod framework for bifurcations.

Extended methodology to a hexagonal space frame.

Identified simple and double bifurcation points in examples.

Abstract

We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic frameworks is not straightforward. We consider frames comprising assemblages of Cosserat rods, and the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a linear space. The field equations comprise the assembly of all rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the symmetry-group action, and the implementation of group-theoretic methods is now natural within the linear-space context. All potential generic, symmetry-breaking bifurcations are predicted apriori. We then employ an open-source path-following code, which can detect and compute simple, onedimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a hexagonal space frame via the same approach. The tripod and the hexagonal dome both exhibit simple and double bifurcation points.