On the mechanics of strain localization in plasticity: isotropic and orthotropic, elasto- and rigid-plastic, associated and non-associated models

Abstract

As strain localization is usually prognostics of localized failure in solids and structures, prediction of its occurrence and quantification of its adverse effects are of both theoretical and practical significance. Regarding plastic solids, onset of strain localization was presumed to be coincident with strain bifurcation, and the discontinuous bifurcation analysis was usually adopted to determine the discontinuity orientation though it does not apply to rigid-plastic solids. However, recent studies indicate that strain bifurcation and localization correspond to distinct stages of localized failure and should be dealt with separately. In this work, the mechanics of strain localization is addressed for perfect and softening plasticity in the most general context. Both isotropic and orthotropic, elasto- and rigid-plastic solids with associated and non-associated flow rules are analytically considered and numerically validated, extending our previous work on softening plasticity with associated evolution laws. In addition to Maxwell’s kinematics and continuity of the traction rate for strain bifurcation, a novel necessary condition, i.e., the stress rate objectivity (independent from the discontinuity bandwidth), and the resulting kinematic and static constraints, are derived for the occurrence of strain localization. In particular, the localization angles of the discontinuity band (surface) depend only on the specific stress state and the plastic flow tensor, irrespective from the material elastic constants and from the plastic yield function. Moreover, it is found that a transition stage generally exists in the case of plane strain during which the orientation of plastic flow rotates progressively such that strain localization may occur. Back-to-back numerical predictions of some benchmark problems, involving both perfect and softening plasticity, also justify the analytical results.