Rank-one updates and failure indicators in multi-dissipative materials

Multi-dissipative constitutive descriptions for irreversible material degradation result in tangent operators with multiple rank-one updates of the elastic tensors: multisurface elastoplasticity, elastoplasticity combined with elastic degradation and multicrack models. The spectral properties of these tangent operators determine failure conditions at the material level in terms of loss of uniqueness and discontinuous bifurcation of the incremental response. These properties can be investigated by studying the eigensolution for the sum of n rank-one mxm matrices and for n rank-one updates of the mxm identity matrix, when the tangent tensors are given in matrix form. Explicit derivations for such eigenproblem are presented. On the light of these new results derivations of faliure analysis for single-dissipative materials are re-analyzed and for double-dissipative mechanisms given explicitly. Conclusions are drawn with respect to the failure indicators for elastoplasticity combined with damage. In particular, it is shown that the interaction of dissipation processes destabilizes the tangent operators.