A case of indispensable localized instability in elastic–plastic solids

Abstract

The primary instability in homogeneous elastic-plastic bodies with prescribed boundary displacements is studied. In this event instability is known to arise when Hadamards inequality is first violated, this violation being the very condition for the localized instability to be possible in principle. The question is posed whether localized instability is the only possible type of instability under specified conditions, or diffuse instability is equally possible (which is true for elastic bodies) .In order to state a rational criterion for distinction between localized and diffuse instability modes (IMs) (which are treated in full generality as mutually complementary notions without any a priori prescriptions regarding the mode of deformation) , it is proposed to characterize IMs by means of some quantitative measure of localization named the localizational volume. The latter evaluates the volume of that part of a body, where relatively great incremental strains are concentrated (this property of proposed measure is proved) .The main result established is that in the problem under consideration any primary IM is characterized by infinitesimal value of localizational volume, i.e. all the primary IMs appear to be localized in such a volumetric sense, which means at least the absence of diffuse IMs.The conclusion is drawn that indispensability of such a localization (treated in the sense of small localizational volume) is a global, essentially non-linear effect (boundary constraint+piecewise−linear constitutive relation) .