Localization conditions and diffused necking for damage plastic solids

We start by generalizing the one dimensional Considère condition to a maximum power condition for arbitrary three dimensional loadings. In particular, proportional loadings with arbitrary confining pressures and Lode angles are considered. It is shown that the maximum power localization criterion for J2 material does not agree with experimental observations. The constitutive relationship of damage plastic solids is then introduced to the maximum power condition for the diffused necking. We show that the onset of localization is a spontaneous consequence of the evolution of plastic damage as the weakening rate increases. Emphasis is given to the extensively studied sheet metal forming. We show that the governing factor of the localization condition for damage plastic solids is not the damage itself, but the resulting effect of the rate of the weakening from the plasticity induced damage, which is a function of the stress states on the loading path. Effects of the pressure sensitivity, the Lode angle dependence, the damage accumulation and weakening are explored through parametric studies. Examples are given for several metallic alloys that exhibit different shapes of the forming limit curve (FLC). The predicted strain components at the onset of localization agree well with the experimental results.