Loss of uniqueness and bifurcation vs instability: some remarks

Abstract

ABSTRACT The present work is an attempt to clarify the different notions of bifurcation and loss of stability. Although these notions have been already discussed by many authors, we believe that they still deserve some discussion. In fact, many different “definitions” of stability exist. This is not a problem for conservative systems, for which it turns out that all these definitions do coincide. Nor is this a problem if the system incorporates some viscous dissipation. However, for other cases, e.g. non conservative systems incorporating dry friction dissipation, which are of relevance in geomechanics, this is very important, because different definitions may yield different results. We recall first the definition of Lyapunov stability, which is related to the dynamic study of the influence of initial conditions on the solution of the mechanical equations. In this framework, we have a look at the linearization of the equations and its consequences. Then we go back to Hill's work and try to give the basis of the so-called “Hill stability criterion”. Finally, we “define” an other stability criterion which has been criticized by Hill in some early paper. Through the study of a very simple mechanical system, we exhibit some differences between all these notions, showing that a system can be stable according to one criterion but unstable according to other criteria. It is necessary to mention which definition of stability is used. Any existing stability criterion has to be thought of as an heuristic method. On the contrary, bifurcation or loss of uniqueness is a more clear concept. Shear band localization, controlability, and invertibility can be actually seen as particular cases of bifurcation. However, bifurcation criteria have to be considered carefully, since they almost always incorporate linearization, which in most cases is not strictly justified. Some examples are given of general results which have been obtained without the use of any linearization procedure.