some aspects of the numerical solution of equilibrium problems in finite elasticity

Description

Analytical and computational aspects of solution paths for nonlinear equations are examined, with emphasis on problems in which there are many parameters. The solution to problems of this type is described by an equilibrium hypersurface and methods are presented which allow for the determination of the various features of this surface. These include methods for following numerically any curve on the primary surface, and for determining on such a curve all the singular points (both limit and bifurcation points). Further methods are then presented which allow branching onto secondary paths (subsets of secondary surfaces) from bifurcation points in order to trace out these paths and so determine the bifurcation behaviour of the problem considered. To complete the analysis of the equilibrium surface methods are developed to trace the loci of singular po in ts. The locus of a bifurcation point determines the intersection of the primary and secondary equilibrium surfaces while the loci of limit points allow for the determination of regions of stable and unstable behaviour on the equilibrium surface. These methods are applicable to any system of nonlinear equations but the particular application here is to systems of equations obtained from the finite element approximation of boundary-value problems in elasticity. Attention is restricted to plane boundary-value problems involving incompressible hyperelastic materials. The strain-energy function used to characterise these materials is based on a symmetric function of the principal stretches All of the above ideas are investigated nu me rica lly for the pro bl em of a pressurised rubber cylinder. subjected to axial extension. This problem contains two identifiable loading parameters and exhibits complex limit and bifurcation behaviour, which is studied in some de tail.