A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

Abstract

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; 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 slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional 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 more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.