On the evolution of damage-induced localization in a deformable-director Cosserat continuum

This paper analyzes a small-deformation, size-dependent Cosserat continuum, which in addition to the usual kinematics and balance laws admits a deformable triad of director vectors at each material point. Deformations of these directors are determined by additional director-momentum balance laws. The constitutive equations enrich elastic response with rate-independent damage that reduces the integrity of the effective resistance to distortional deformations of the material in the macro-continuum only, while leaving the stiffnesses associated with the deformable director triad unchanged. A finite element formulation for the resulting field equations is developed. Two benchmark problems (biaxial isochoric loading and a notched plate) are analyzed to assess mesh objectivity and evolution of localization patterns. The results show that, for the examined problems, the Cosserat model with a deformable director triad predicts size-dependent yet mesh-independent shear-band initiation, width, and progression, with band thickness governed by the internal length l rather than by the mesh. In contrast, the classical Cosserat baseline with a rigid director triad retains mesh sensitivity once multiple bands nucleate and evolve. Overall, the benchmark problems examined indicate that the deformable Cosserat model has strong potential to be interpreted as a physically motivated continuum description of localized regions, rather than solely as a numerical regularization device.