Properties of Discontinuous Bifurcations Solutions in Elasto-Plasticity

Explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic-plastic materials are presented. It then follows that the classical value of the critical hardening modulus derived by Rice (1976. Proc. 14th IUTAM Congr., Delft, The Netherlands, pp. 207–220, North Holland, Amsterdam.) is, in fact, the only possible one. Furthermore, from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface. Explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non-associated volumetric flow rules while the devialoric portion is associated. The considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening. The result obtained by Rudnicki and Rice (1975, J. Mech. Phys. Solids 23, 371–394) for a Drucker-Pruger material appear as a special case. Other criteria that are investigated arc those of Coulomb and Rankine.