A framework for geometrically nonlinear continuum damage mechanics

The objective of this work is the derivation of a framework for geometrically nonlinear continuum damage mechanics which allows for the description of damage by second order tensors. To this end, so called fictitious undamaged or rather microscopic configurations are considered which are related to the macroscopic configurations by linear tangent maps which allow for the interpretation as damage deformation gradients. The corresponding Finger tensors defined in the macroscopic configuration are then understood as damaged metrics for measuring the free energy for given strains. Thereby, the underlying motivation is provided by the hypothesis of strain energy equivalence between microscopic and macroscopic configurations. Based on the standard dissipative material approach the constitutive framework is completed by stresses, damage stresses, a damage condition and the associated evolution laws for the damage metric and the internal hardening variable. The relationship of the present damage metric based theory to the classical understanding of damage as an area reduction is highlighted. Finally, a simple prototype model problem is presented which allows for an efficient and accurate algorithmic treatment. Numerical examples demonstrate the applicability of the proposed framework