Strain localization for constitutive models combining plasticity with elastic degradation

The paper examines the localization properties of dissipative material processes when plasticity and elastic degradation are simultaneously active. In that case the tangent operator comprises two rank-one updates of the elastic tensor in the spirit of multisurface elastoplasticity. The spectral properties of the tangent operator determine failure conditions in the small in terms of loss of uniqueness and discontinuous bifurcation of the incremental response. These properties are investigated with the eigenproblem fro the sum of n rank-one m x m matrices and for n rank-one updates of the m x m identitiy matrix, when the tangent tensor is written in matrix form. Explicit expressions are presented for the localization condition of two dissipative material processes. Conclusions are drawn with respect to the localization indicator for von Mises elastoplasticity combined with scalar damage. It is shown that the interaction of plasticity and damage destabilizes the localization tensor, whereby localization is possible when both mechanisms are hardening.