A case of indispensable localized instability in elastic–plastic solids

The primary instability in homogeneous elastic!plastic bodies with prescribed boundary displacements is studied[ In this event instability is known to arise when Hadamard|s inequality is _rst 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 speci_ed conditions\ or di}use instability is equally possible "which is true for elastic bodies#[ In order to state a rational criterion for distinction between localized and di}use 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 in_nitesimal 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 di}use IMs[ The conclusion is drawn that indispensability of such a localization "treated in the sense of small local! izational volume# is a global\ essentially non!linear e}ect "boundary constraint piecewise linear consti! tutive relation#[