We investigate structural compatibility issues in finite element discretisations of Cosserat micropolar elasticity at finite strains. In these models, independent rotational degrees of freedom are described by a micro-rotation field $\overline{\boldsymbol{R}} \in SO(3)$. Its interaction with the deformation gradient $\boldsymbol{F}$ through the Cosserat strain $\overline{\boldsymbol{R}}^T\boldsymbol{F} - \boldsymbol{I}$ is frequently associated with shear- and membrane-locking in reduced-dimensional formulations. However, the latter is in fact a more general structure-preservation problem inherited from the three-dimensional model. Namely,
while standard Lagrange discretisations of $\boldsymbol{F}_h = \nabla \boldsymbol{\varphi}_h$ can exactly represent constant stretches, the discrete rotations $\overline{\boldsymbol{R}}_h$, which lie in a nonlinear Lie-group without a canonical polynomial interpolation, generally do not coincide with $\mathrm{polar},\boldsymbol{F}_h$. This inconsistency hinders the reconstruction of identity strains and induces locking effects.
To overcome these difficulties, we propose a geometric structure-preserving interpolation method that enforces compatibility between the discrete kinematic fields.
We present the formulation, examine appropriate discrete pairings and their ability to preserve objectivity, and provide numerical examples demonstrating the potential of the method to reduce locking and improve physical fidelity in Cosserat finite element models.
Schek L., Lewintan P., Müller W., Muench I., Zilian A., Bordas S., Neff P., *Sky A., 2026, \textbf{A structure-preserving discretisation of SO(3)-rotation fields for finite Cosserat micropolar elasticity}, \textit{arXiv}, arXiv:2602.15147.