On the Determination of Bifurcation and Limit Points

Abstract

Necessary and sufficient conditions for the occurrence of a bifurcation in the equilibrium path of a discrete structural system are established as a consequence of the degeneracy of the solution of the rate problem at a critical point. Such result is based on the properties of the elastic-plastic rate problem formulated as a linear complementarity problem (LCP) in terms of plastic multipliers (the moduli of the plastic strain rate vectors) as basic unknowns. The conditions here given allow to distinguish, both theoretically and practically, among bounded bifurcations, unbounded bifurcations, limit points, and unloading points. All of the needed quantities depend either on the starting situation or on the actual known term increment; there is no need to compute eigenvalues or eigenvectors of stiffness matrices. The results obtained can be seen as a refinement, for the discrete elastic-plastic problem, of the uniqueness theory given by Hill. The refinement allows covering the case of vector-valued yield functions and clearly distinguishing, in operative terms, between different types of critical/limit points.