On failure indicators in multi-dissipative materials

Multi-dissipative constitutive descriptions of irreversible material degradation result in tangent operators that are made up of multiple rank-one updates of the elasticity tensors: multisurface elastoplasticity, plastic yielding combined with elastic degradation and multicrack models are representative examples. 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 failure properties are analysed herein by studying the eigensolution of the sum of n rank-one (mxm) matrices, and the eigensolution of n tank-one updates of the (mxm) identity matrix, whereby the tangent material tensors are written in matrix form. Analytical eigensolutions are presented and interpreted mechanically in terms of continuous and discontinuous failure indicators. In the light of these spectral properties, the failure indicators of single-dissipative materials are revisited and explicit results are presented For double-dissipative models in the form of plasticity combined with elastic-damage. In particular, it is shown that the activation and interaction of two dissipation processes may destabilize the tangent operators beyond the state resulting from a single active mechanism.