BIFURCATIONS IN ELASTIC PLASTIC MATERIALS

When strain-softening, elastic-plastic materials are loaded into the plastic regime they often exhibit deformations that are localized into small regions at some point in the loading process. This intense localized deformation limits the formability of materials and will often quickly lead to failure with continued loading. Localized deformation is often associated with satisfaction of the classical discontinuous bifurcation criterion. Here we propose that the loss of strong ellipticity criterion should be used in place of the classical discontinuous bifurcation criterion as a necessary condition for localization. The application of the strong ellipticity criterion implies that a bifurcation mode associated with loss of positive definiteness of the symmetric part of the acoustic tensor must be identified rather than a mode associated with the first zero eigenvalue of the acoustic tensor itself. The eigensystem for the symmetric part of the tangent modulus tensor is obtained for several different plasticity models. This eigensystem provides information about deformation modes associated with both diffuse and discontinuous bifurcations. Material properties, boundary conditions and body geometry are all shown to affect the diffuse and localized deformation modes that are generated. Numerous experimental observations of necking and localization in metal specimens subject to various boundary conditions are explained with the proposed approach.