Theory of material instability in incrementally nonlinear plasticity

Material instability in time-independent elastic-plastic solids is studied as a phenomenon strictly related to qualitative properties of an incremental constitutive law. A broad class of incrementally nonlinear material models is considered which encompasses classical elastoplasticity with a single smooth yield surface, as well as multi-mode plasticity with many internal mechanisms of inelastic deformation which give the yield-surface vertex effect. Three types of instability are investigated: with respect to internal microstructural rearrangements, for deviations from uniform deformation under boundary displacement control, and under flexible constraints corresponding to deformation-sensitive loading. It is shown that the respective instability criteria are different and, moreover, dependent on whether the instability concerns a single equilibrium state or a process of quasi-static deformation. Instability of equilibrium is of dynamic type, while instability of a process going through stable equilibrium states is related to a continuous spectrum of quasi-static bifurcation points along the deformation path. Basic concepts are outlined by the example of a one-dimensional discretized tensile bar and extended to incipient localization of deformation in a three-dimensional continuum. Under symmetry restrictions imposed on an incrementally nonlinear constitutive law, a unified approach to material instabilities of various types is presented which is based on the single energy criterion. By specifying the incremental energy consumption to second-order terms and determining the circumstances in which it fails to be minimized along a fundamental deformation path, the onset of material instability of a selected type can be estimated.